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Q: How to say correctly? The question is mostly for native speakers. What is best:
"Prices for Linux app" or "Linux app prices"
and why?
A:
"Prices for Linux app" or "Linux app prices"
In both your examples, "prices" is plural, "app" is singular.
"Prices for Linux app" is problematic, firstly because it suggests there are multiple prices (plural) for a single app. And if, for some reason, that is what you meant - multiple prices for one product - it would be missing an article before "Linux app".
"Linux app prices" is fine. It could be an idiomatic way of referring to the prices (plural) of any Linux app, or a range of Linux apps. It doesn't matter that 'app' appears singular.
It could be compared to saying "Greek vacation prices", which could refer to a range of prices of different Greek vacations. But you couldn't say "The prices of Greek Vacation" - you need the plural 'vacations'.
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"redpajama_set_name": "RedPajamaStackExchange"
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Harman Lines is a military installation of the British Army, which is part of Warminster Garrison on the Salisbury Plain. Harman Lines is on Sack Road in Warminster, Wiltshire, United Kingdom.
History
In 1965, Warminster saw a large expansion, and the 1966 Defence White Paper expanded its role. The nearby Land Warfare Centre gained a 'Tank Demonstration Squadron', later 'Armoured', which was a detached armoured unit from its regiment. In 1975, the custom-built Harman Lines was opened on Sack Hill.
As part of the 1998 Strategic Defence Review, a new Joint Nuclear, Biological and Chemical Regiment was formed based at RAF Honington and saw the 1st Royal Tank Regiment dispersed. As part of this change, A Squadron moved to Harman Lines as part of the new unit under the Combined Arms Training Centre. The site remained the home of the Armoured Demonstration Squadron until 2014, when it became the permanent home of, Falcon Squadron, RTR, part of the Defence CBRN Wing.
Current garrison
The current garrison includes:
Falcon Area Surveillance and Reconnaissance Squadron, Royal Tank Regiment (28 Engineer Regiment)
Close Support Troop, Royal Engineers (detached from 28 Engineer Regiment, supporting Falcon Squadron)
Land Warfare Centre Battlegroup Light Aid Detachment, Royal Electrical and Mechanical Engineers
See also
Warminster Garrison
Waterloo Lines
Battlesbury Barracks
Footnotes
References
Installations of the British Army
Warminster
Barracks in England
Buildings and structures in Wiltshire
Military history of Wiltshire
Military installations of the United Kingdom
Wiltshire
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_Para mis más queridos, Rosie, Bryana, y Cid. Y por supuesto, para Toñita._
CONTENTS
Foreword, by Paul Salopek
Preface
Introduction, by Chantal Martineau
Spirit
Terra
Alchemy
Fiesta
Galería
FOREWORD
_Paul Salopek_
_Agave tequilana_ "Weber Azul" is a delicate and waxy blue, like certain hues of woodsmoke, or like the blued shadows that pool in the rincons of the Sierra Madre Occidental, the wild mountain range that knuckles down the western edge of the Mexican state of Jalisco.
I grew up in Jalisco, the birthplace of tequila, the famed spirit these blue agave plants so generously yield. Blue agaves lined the dirt streets of my _colonia._ We schoolboys carved our names and much worse into their fleshy spikes of leaves. We deployed their tall flower stalks—some towered so high they poked rain out of the passing gray monsoon clouds—as spears. Once we were old enough, and sometimes not old enough, we sipped the fermented juices of the blue agave, a universal rite of passage in the Mexican altiplano.
Unlike its brother and sister liquors (whiskeys, gins, vodkas, chachas, ryes, bourbons, scotch, etc.), the raw material of tequila is neither seeds nor roots. No, it is the plant's very core, its heart. The good _jimador_ walks his glaucous blue fields—in the gentle foothills, let's say, above old Zapopan—appraising his crop before the harvest. He looks for many things: The cactuslike agaves must be between four and eighteen years old to give up their best juice. The moisture in the soil and season of the year are also important. The dry weeks just before the first rains are best, as they concentrate the sugars in the agave's swollen _corazón._
The good harvester is always listening to this enigmatic heart. He knows that within it beats the wingflick of the bat that pollinated his plants. It holds the pulse of the yellow sun, the long, lingering cores of summer afternoons. It drums with the raindrops that in Mexico fall straight and silver and hard. He takes the agave's heart in his rough hands and squeezes its essence into a small glass. He offers it up to you.
I regret to report that I do not drink tequila often anymore. My body has seen too many wars. But I remember the color of those plantations—their mesmerizing cool hues in the rippling heat of the subtropics. The blues of childhood.
PREFACE
Tequila, like Mexico, is _mestizaje_ —a coalescence. When pulque, the fermented nectar of Mexico's indigenous world, embraced the Spaniards' copper _alambiques,_ or stills, tequila was born.
Mexico's iconic drink is deeply rooted in a past that is both complex and immense. As early as the sixteenth century, the national drink of Mexico was known as _vino de mezcal,_ from the Spanish word _vino,_ for wine, and the Nahuatl word _mezcal,_ for agave.
The _mezcal_ of the Nahuatl culture played an enormous role in the lives of Mesoamericans. Not only was agave critical for sustenance; it could also be used in the making of shelter, clothing, and tools. Mayahuel, a Venus-like divinity that personifies the maguey plant, became the symbol of fertility for the Aztecs.
The town of Tequila, or Tecuillan, Nahuatl for a "place of work and cutting," is where land, agave, and people came together to produce the iconic spirit of Mexico. It is here, and in other towns in Jalisco, that I set out to explore the contemporary world of tequila. My search led me to the holy trinity of tequila makers—Cuervo, Herradura, and Sauza—which began in 1758 with Cuervo's mass distillation of blue agave sugars. I also sought out artisanal _tequileras_ committed to the traditional craft of tequila-making, from harvest to bottle.
In this landscape of blue agave, I discovered traditions of culture and religion—ancient and modern, indigenous and foreign. All were a reminder of my own complex Spanish and Native American mestizo heritage. Childhood memories resurfaced, decades after I stepped across the Rio Grande into the United States as an immigrant child of seven.
The photographs from this journey reflect the mystical space where the weight of history and the bounty of the earth blend into a spirit called tequila. It is the elixir that remains the guardian of Mexico's landscape, tradition, and identity—indeed, the ancient lord of fire with a savage smile.
INTRODUCTION
_Chantal Martineau_
A silvery blue plant. A crystal clear liquid. How one becomes the other isn't a miracle. It involves science, craft, and a good deal of hard work. The iconic liquid export, distilled from the native blue agave plant, has been made in Mexico for at least four hundred years—some believe centuries longer. The spirit and its raw material have been intricately tied to Mexico's history, culture, and mythology since the time of the Aztecs.
Most of us don't think of tequila as a cultural product. Maybe the idea that a bottle on the back bar at the local watering hole might be culturally and historically significant doesn't occur to us. Or perhaps it's the memory of that first (or worst) experience with the spirit. Shots with salt and lime, or frozen margaritas—these were once the standard introduction to tequila.
These days tequila is often sipped neat, like fine scotch or cognac, or mixed into artisanal cocktails that have nothing to do with a margarita, much less a premixed one. The spirit comes bottled in exquisite crystal decanters, and a number of brands are backed by celebrities or billionaire entrepreneurs. It's not just that the spirit has changed—although it has, as much better incarnations are more readily available than decades ago. Public perception of tequila is also shifting. No longer is it deemed party fuel, something to be slammed during spring break by hooting coeds. It can be a spirit with sophistication and intrigue. And yes, it is a cultural symbol for Mexico, one with a rich and complicated history. Just how it evolved from a rustic regional specialty to a luxury good is a long and winding tale.
It's a story that begins with a native plant, an extraordinary plant, called agave. Agave are large perennial succulents that look like aloe and are often mistaken for cactuses. Taxonomically speaking, they are more closely related to asparagus. More than two hundred varieties exist, and almost all are indigenous to Mexico. Blue Weber ( _agave, tequilana_ ) is the variety used to make tequila. Many others, like espadín, the genetic ancestor of blue Weber, are used to make mezcal. Henequén, a variety that thrives in the southern part of the country, was widely used in textile manufacturing in the early part of the twentieth century (until DuPont invented nylon in the 1930s, devastating the henequén industry).
Take a drive through tequila country and you'll see waves upon waves of steely blue from the road. These are agave fields. Researchers believe that agave has been consumed in Mexico for at least eleven thousand years, making it one of the earliest cultivated crops in the Americas alongside squash and maize. Agave was used for everything from food and shelter to cultural production. Early Mexicans were known to weave its dried leaves into clothing, bedding, and roofs for their homes. The fine needle at the tip of each spiny leaf was used to sew fabric and for drawing, and in the bloodletting ceremonies the Aztecs were so fond of. This deep reliance and connection to the agave is echoed in its prominent place in Aztec mythology.
Mayahuel is the deity most often associated with agave. She is often depicted as having four hundred breasts to feed the Centzon Totochtin—the four hundred rabbit gods of intoxication. Numerous versions of the legend exist, but most begin with Mayahuel's grandmother, the evil goddess Tzitzimitl, who lived in the heavens and devoured light. Mayahuel lived with her grandmother, sleeping all day and night, in a perfect state of wellbeing. One day the god Quetzalcoatl decides to fly up to the heavens to destroy the evil goddess. Instead he finds Mayahuel. She awakens, and they fall in love at first sight. Quetzalcoatl takes her back to earth so they can be together, but they are forced to hide from Tzitzimitl, who is livid at her granddaughter's disappearance. The couple hides in different places around the earth until, finally, they decide to disguise themselves as a tree. Tzitzimitl discovers them and, in a fury, rends the tree in half, killing Mayahuel. Quetzalcoatl flies into a rage and destroys Tzitzimitl. He returns to his lover's remains and buries them. The first agave sprouts from the dirt that covers her bones: a gift from the gods. Quetzalcoatl drinks its sweet sap and rejoices.
Thousands of years before tequila was made, Mesoamerican tribes brewed pulque, a sort of beer, from the agave's fermented sap, called aguamiel (honey water). This tart, milky, mildly effervescent drink was taken during sacred rituals as a way to commune with the gods. The Aztecs deemed it sacrosanct and prohibited common consumption. Pulque was for the exclusive elect—holy men—to drink during religious ceremonies. Public drunkenness was punishable by death, but pulque was prescribed to the sick, the elderly, and pregnant women. Spanish conquistadors disapproved of pulque and called for a ban on the drink, but it didn't last long. The Spanish Crown soon came up with a better solution: to tax pulque. By the mid-eighteenth century, the tax collected on pulque was one of Mexico's most important sources of government revenue.
It's been said that the Spanish brought distillation to Mexico, as they did to the other parts of the New World they conquered. But Mexican researchers have collected evidence suggesting that indigenous people were distilling long before the Spanish invaded. Early artifacts have been found near the Colima volcano, in the state of Jalisco, not too far from the heart of tequila country. The artifacts included vessels that may have been used to distill agave and figurines depicting people drinking from tiny cups, much smaller than those used for pulque. Could it have been mezcal? Scientists on both sides of the border are studying these artifacts, said to date as far back as 1500 BC.
Whoever brought distillation to Mexico, it was early Mexicans who applied it to agave. The technology spread from pueblo to pueblo until communities across the region were distilling the agave varieties that were indigenous to the area. The spirit was named mezcal, from the Nahuatl _mexcalli,_ a combination of the words for agave or maguey _(metl_ ) and oven-cooked ( _ixcalli_ ). What we know as tequila was once one of Mexico's _many mezcales,_ known as _mezcal de Tequila_ or, more accurately, _vino de mezcal de la región de Tequila._
Tequila, like Champagne or Cognac, is not just the name of a product but also refers to a place. The small town in the state of Jalisco is about an hour's drive from Guadalajara, Mexico's second-largest city. The mezcal from Tequila was widely known to be of high quality and, given its proximity to Guadalajara, had a ready-made clientele. By the late nineteenth century, the wealthiest and most famous tequila-producing families in town—the Cuervos and the Sauzas, who still produce some of the biggest brands today—decided to try taking their spirit across the border. Mezcal de Tequila was entered into spirits competitions in the United States and was soon being sold under the moniker Mexican whiskey or Mexican brandy. Americans liked this Mexican whiskey, and its popularity north of the border boosted its popularity back home. Soon mezcal de Tequila became known simply as tequila.
The process for making tequila has evolved since the earliest days of the Cuervos and the Sauzas. But the role of the _jimadores_ —the men tasked with harvesting agave for tequila production—has remained the same, passed from generation to generation. It's still common to find families working the fields together—fathers, sons, brothers, uncles, cousins. The jimador is a symbol of Mexican fortitude and resilience. The harvest, or _jima,_ is backbreaking work. Each plant must be dug out of the ground at the root—no small task at up to two hundred pounds each. The jimador uses his _coa,_ a rounded hoe, to slice off the agave's large, heavy leaves. When he's done, nothing is left but the heart, known as the _piña_ because it looks like a pineapple. This part of the plant gets trucked to the distillery to be cooked, crushed, fermented, and distilled into tequila.
In wine production, connoisseurs talk about terroir. It's a term from the world of French wine that refers to the way the soil, climate, elevation, and topography of a place are married to influence the flavor. Tequila, too, can express terroir. In the Tequila valley, the soil is volcanic and ashy gray, and the piñas grow earthy and spicy. In the highlands, where the soil is brick red and iron-rich, the agave ripens fruity and sweeter. The higher altitude, six thousand feet and more above sea level, translates into drier, sunnier days and cooler nights, and local tequila producers believe the fluctuating temperatures result in more complex flavors. Valley producers, for their part, lay claim to the oldest agave-growing lands. At up to four thousand feet in elevation, the area is misleadingly referred to as the lowlands.
Once the piñas arrive at the distillery, they must be cooked, as the Aztecs' word for mescal _—mexcalli,_ or oven-cooked agave—implies. This can be done in several ways. Traditionally, the halved or quartered piñas are loaded into a brick oven, where they are steamed over several days until they are brown and caramelized. More modern facilities use an autoclave, a sort of pressure-cooker that takes less time and can impart more floral and citrus notes to the tequila. High-volume brands tend to use a diffuser, a machine the size of a train car that processes the agave using hot water.
After cooking, the agave is milled. Few distilleries are still equipped with a tahona, a traditional stone mill that consists of a two-ton wheel carved out of volcanic rock affixed to the center of a round stone pit. The wheel is turned around and around, either pulled by a mule or a mechanized tractor, and gradually crushes the cooked agave on the pit's stone floor. The juices from the crushed agave are collected into fermentation tanks. Most distilleries today use a mechanical shredder, a contraption borrowed from the rum industry for use in grinding sugar cane.
During the fermentation stage, the agave mash sits in wooden or steel vats, bubbling and frothing as it slowly converts its sugar to alcohol. A number of distilleries rely on ambient yeasts—already present in the distillery, floating in the air, and on the very skins of the plants—while others inoculate their mash with synthetic or lab-raised yeasts. It usually takes a few days for the yeasts to devour the sugars, leaving alcohol in their wake. The process also produces carbon dioxide, which escapes into the air.
Distillation is the final step in making tequila. Most distilleries use alembic copper pot stills to bring the liquid to a boil and trap the alcoholic vapors. The distiller's job is to separate the volatile first part of the distillate, known as the heads, and the methanol-heavy second part, known as the tails, from the best part of the distillate, the heart. After two distillation runs, the resulting tequila can be bottled as is, clear and un-aged. This is known as _plata_ (silver) or _blanco_ (white) tequila and is most often enjoyed mixed into cocktails, although purists prefer to sip it neat. Alternatively, the liquid can be put into oak barrels—usually former American whiskey or bourbon barrels—for aging. A _reposado_ (rested) tequila is aged from two months to just under a year, at least a year is required for _añejo_ , and a minimum of three years is required for _extra_ _añejo._ Reposado tequila, with its gently caramelized notes, is often used in cocktails. Añejo and extra añejo tequilas tend to be served on their own, similar to a good after-dinner whiskey or brandy.
Over the years tequila's popularity has ebbed and flowed. In Mexico, a number of tequila producers recall their parents or grandparents drinking it only on the sly. In polite company, scotch or cognac was served. In the United States, tequila has experienced a few surges in popularity, such as during Prohibition. Just as Canadian whiskey flooded the border to the north, tequila was smuggled into southern states, to the delight of thirsty Americans. Later, the crooner Bing Crosby became involved in the business of importing tequila after traveling to Mexico and falling in love with the Herradura brand. In 1958 the hit song "Tequila" (and its unforgettable refrain) by the Champs brought the spirit renewed attention. Another song, Jimmy Buffett's "Margaritaville," hit the charts in 1977. By the 1990s the United States was in the grips of a full-blown tequila boom, but it would be some years before Americans discovered "good" tequila. At the time, there was no such thing as a premium label. The shots and margaritas flowed.
You might say we are currently experiencing a second boom. In the last decade or so we've seen an influx of luxury tequilas made from 100 percent agave. Most people don't realize that their first—or worst—tequila probably fell into the category made with just 51 percent agave distillate and up to 49 percent unnamed "other sugars" (typically derived from corn or sugar cane). For decades 100 percent agave tequila was hard to come by. What's more, consumers weren't educated enough about tequila to know that they might prefer it. The spirit came to the United States and spread around the world with no shortage of myths about psychotropic side effects and worms at the bottom of the bottle. There was little information about its true origins.
The rise of 100 percent agave tequila should be considered a boon. After all, it's a purer, more traditional take on tequila, harkening back to the days when all tequila was 100 percent agave. It wasn't until the mid-twentieth century, as demand started to swell, that producers grew concerned over agave shortages. To keep up with demand over the next two decades, they incrementally reduced the amount of agave required in tequila. In 1974 tequila earned its appellation of origin, a status that, like Bordeaux wine and Comté cheese, identifies it as a distinct product associated with a specific geographic region. It was a crucial step in the fight against counterfeiting and adulteration of the spirit. Legally, to be called tequila, the spirit must be made according to strict regulations and within a designated region comprising the entire state of Jalisco and municipalities in four other states: Guanajuato, Michoacán, Nayarit, and Tamaulipas. By the time the appellation of origin was achieved, however, the amount of agave required in a bottle of tequila had been cut to about half. The rest of the ingredients could come from anywhere.
It's baffling to think that a bottle of tequila, Mexico's iconic liquid export, could be half, well, not tequila. But as demand for tequila grew over the years, shortages and their proposed solutions became inevitable. Blue agave is no easy plant to farm, and the market for it is volatile; the price rises and falls like that of oil or gold. The seemingly unbreakable cycle of gluts and shortages can largely be attributed to the nature of the plant. Think of the usual variables farmers contend with—extreme weather conditions, pests, disease—and multiply these by seven years, which is, on average, how long it takes agave to reach maturity. (Other varieties of agave can take ten, fifteen, twenty, even thirty years to ripen.) Consider grapes; they ripen over a season and grow back each year. But once an agave plant is harvested, it dies. Another must be planted in its place.
It's all part of the romance of agave. A nocturnal plant, it comes alive at dusk. In the wild, an agave grows a tall stalk, a _quiote_ , from its heart. This is its sexual organ. At the top of the stalk sprout tiny flowers, which are pollinated by all sorts of nocturnal creatures. Moths and bats are drawn to the agave's flowers, which open to release their heady perfume as the sun is setting. These night fliers feed on the flowers and scatter their seeds. For the purpose of tequila production, the quiote is cut, conserving all the plant's energy for growing a big, sweet heart. The plant is propagated asexually, using cuttings, the way you might with a house plant. Agaves grow shoots, called _hijuelos,_ which can be cut and replanted to flourish on their own. Some researchers have warned that multiple generations of this type of cloning threatens the plant's genetic robustness. Farming as a monoculture already reduces biodiversity within the plant population. Sustainable growing practices are a top concern for many tequila producers, and several studies into the plant's genetics have been initiated.
For the time being, tequila is experiencing a true golden age. The world's love affair with the spirit has grown the category about 150 percent since 2002, and today the industry benefits some seventy thousand families in Mexico. The premium sector has ballooned some 700 percent over the same period. Now is a good time to drink good tequila, to be sure. When shopping for a bottle, keep an eye out for a few things. First, look for the phrase "hecho en Mexico" (made in Mexico). It's the front line in the battle against counterfeiting. Second, look for the mention of 100 percent agave. This helps ensure that you're drinking a high-quality spirit made with an eye to tradition. Finally check the NOM ( _norma oficial de México_ ) number, located on every bottle of bona fide tequila. This will tell you what distillery the spirit was made in. Only about 150 distilleries exist within the boundaries of the tequila region, but there are at least 700 brands. In other words, most distilleries make multiple labels. By the same token, many brands that are exported are not available domestically and vice versa. A few gems are available both to Mexicans and abroad and are well liked on both sides of the border.
Thankfully, shots with salt and lime are no longer the most accepted way to drink tequila. In enlightened circles, they're downright frowned upon. It's certainly not a traditional way of enjoying the spirit. In Mexico it is sipped and often paired with _sangrita,_ or "little blood"—a nonalcoholic drink, usually tomato or fruit juice based, with a kick. Sangrita's acidity is said to complement the acidity in the tequila. They make for a great aperitif, opening up the appetite for a meal. As for margaritas, they shouldn't be avoided, by any means, but they aren't the only cocktails being made with tequila. There are Manhattan variations made with reposado and contemporary twists on the Old Fashioned that use añejo. But a classic, refreshing margarita, one made with fresh-squeezed lime juice and maybe a dash of pure agave nectar, is nothing to sneer at. Made with great tequila, it can be a great drink.
For a true purist, a glass of good blanco, served neat, is ideal. Its aromas transport you to the rolling green foothills of Jalisco, blanketed in blue. Here the jimador harvests the imposing plants and the distiller turns them into liquid silver, and it becomes obvious that tequila is not only a cultural product but also an agricultural one—the essence of a swath of land and the people who work it, distilled.
SPIRIT
At sunrise, light came calling. I set out from Atotonilco el Alto, in Jalisco, en route to the most Mexican of towns—Arandas. Fields of blue agave flanked the road as if standing guard over the spirit that once fueled the Mexican Revolution. The morning was crisp, and the sun hid behind a mountainside.
A day or two earlier I had studied the landscape, and I knew I could make some compelling images if the light was good.
The rows of agave were symmetrical, as they had been for the past six to eight years and would remain until the harvester, the _jimador_ , came calling. The jimador must be a master of his craft; the sharp daggers of the agave are merciless and unforgiving. They demand respect.
As my eyes scanned the hills, I saw movement. (Movement will always stop a hunter as well as a photographer, likely because reportage photography is half hunt and half art.) A speck of red at the top of the hill stirred my interest. It was a courageous horse named Carablanca, an Iberian import that has toiled the land for centuries.
Uphill and down, row after row, this workhorse carved the earth. When plow hit stone, Carablanca stalled at the anchored rock. The horse caught its breath, regained its footing, and continued along the millennial path that is too well known to man and beast.
Since childhood I have loved horses, and Carablanca was no exception. I could sense the horse's strength as he pulled the weight of the plow, the exhaustion in the white stare of his eyes.
Echoes of the horse's neighing triggered a challenge for me fueled by pure adrenalin. My climb to the top of the hill was strenuous, but the majestic view made it worthwhile. I could now see that there were two horses, not one. They were being driven by a father-and-son team—a master and his young, unwilling apprentice. How often the sweat of one's brow is inherited and not chosen.
I've been a photographer many years, and I am still overwhelmed by a feeling of great respect for my subjects. Sometimes they welcome me into their world with blind trust, and other times they turn me away.
I began to take photographs amid this beautiful backdrop. I scrambled to keep up with the horses and their humans, to frame, compose, catch my breath—and defend myself from the painful stabs of the agave. I struggled to keep up, but finally my shoes were destroyed, hopelessly damaged by a single morning trekking with people who were born into a life so distant from mine. I watched the horse strain each time the plow hit stone, yet both man and beast continued the task of the hour, of the day, of a lifetime.
Later in the day, my aching back forced me to plunk down onto the floor. The cool tile provided relief from the heat of the afternoon. I glanced to my left and saw Rafael, a semiretired jimador. He sat in silence. His face reminded me of a long-lost cousin on my father's side. The feeling of family was immediate. As I sat up, I noticed that the jimador had a stack of _hijuelos_ , agave pups.
I couldn't resist asking if I could take his portrait. With a pronounced shyness, he agreed. I looked around and quickly found a space with lovely reflective soft light and an adobe wall as a backdrop. I asked Rafael to follow me with his hijuelos.
As I framed the image, I was struck by his stoic pose, the honest confession of a hard life. Cousin or not, Rafael felt like a brother, or like that childhood friend who gets lost in time and years later reappears as an old man. For that brief moment we were family.
EL ASCENSO THE ASCENT
_Tequila, Jalisco_
EL SEÑOR DEL FUEGO LORD OF FIRE
_Tequila, Jalisco_
AGAVE AZUL BLUE AGAVE
_Arandas, Jalisco_
HIGHLANDS OF JALISCO
_Atotonilco el Alto, Jalisco_
CORAZÓN DE AGAVE AGAVE HEART
_Tequila, Jalisco_
LOS CAMPOS DE AMATITÁN FIELDS OF AMATITÁN
_Amatitán, Jalisco_
EL JIMADOR CON HIJUELOS HARVESTER WITH AGAVE SPROUTS
_El Arenal, Jalisco_
LA LABRANZA THE TILLING
_Atotonilco el Alto, Jalisco_
ANTE EL ALTAR BEFORE THE ALTAR
_Amatitán, Jalisco_
TERRA
The jimadores are resilient men traversing spiny fields of blue, thrusting their coa tools into magnificent agave to slice off a crown of daggers. They reveal the sacred heart of Mayahuel, the agave queen of the Aztecs. They are the centurions of tequila and guardians of a ritual born of earth, sky, and fire—a constant harvest, day in and day out.
By the lunch hour, the jimadores had laid thin steaks on a comal—an iron griddle—and were savoring their tacos with the pleasure that comes from eating in the company of Mother Nature.
Hours earlier, six agave _piñas_ , or cores, had been sacrificed and arranged to make a circular fire. As the tentacles from the fire began to reach for the skyline, I thought about how much the jimadores were enjoying their hard-earned meal.
When the sun finally dipped below the horizon, I wondered who, from the land of Jalisco, was the first to build this agave fire? Was this ovenlike innovation the accidental origin of mezcal and tequila, or did the lightning described in indigenous folklore strike the ancient wild agave of the Aztecs and release its sweetness into the world?
The piña of the blue agave takes eight to ten years to reach its harvesting prime.
A jimador might explain that the male agave develops a _cogollo_ , a waxy accumulation of dwarfed leaves that must be removed before the roasting process. Agave syrup can become bitter if the cogollo is not removed. The female agave core will eventually send up a stalk known as a _quiote_ that blooms when it reaches maturity, after ten to twelve years. Once in bloom, cross-pollination occurs with assistance of bats, potentially giving life to new agaves. Growers normally do not allow cross-pollination to occur, relying instead on the _hijuelos_ , pup sprouts that emerge from the root system. The hijuelos are harvested and replanted to continue the cycle.
TIERRA, ESPINA, Y AGAVE TERRA, THORN, AND AGAVE
_Arandas, Jalisco_
EL CABALLITO AGAVERO AGAVE COLT
_El Arenal, Jalisco_
TEQUILANA AZUL BLUE TEQUILANA
_Amatitán, Jalisco_
EL CAMPO AZUL BLUE FIELD
_Arandas, Jalisco_
AGAVE AZUL AL A TARDECER BLUE AGAVE AT SUNSET
_Tequila, Jalisco_
LOS JIMADORES AL AMANECER HARVESTERS AT SUNRISE
_Tequila, Jalisco_
LAS SOMBRAS EMPEDRADAS COBBLESTONED SHADOWS
_Arandas, Jalisco_
LA TRILOGÍA AGAVERA AGAVE TRILOGY
_On the road to Chome, Jalisco_
EL JIMADOR THE HARVESTER
_La Barca, Jalisco_
EL AFILADOR THE GRINDER
_Amatitán, Jalisco_
LA DERROTA EN EL CAMPO THE DEFEAT IN THE FIELD
_La Barca, Jalisco_
EL GUARDIÁN DE LOS AGAVES GUARDIAN OF THE AGAVES
_Arandas, Jalisco_
EN REPOSO AT REST
_La Barca, Jalisco_
EL ARBOLITO AGAVERO LITTLE TREE OF THE AGA VES
_Atotonilco el Alto, Jalisco_
AGAVE AZUL BLUE AGAVE
_La Barca, Jalisco_
LOS CAMINANTES THE WAYFARERS
_La Barca, Jalisco_
EL ATARDECER DUSK
_La Barca, Jalisco_
FUEGO CON FUEGO FIRE WITH FIRE
_La Barca, Jalisco_
ALCHEMY
Tucked in the corner of modern Mexican history is La Cristiada, the Cristero Rebellion, a three-year struggle in the late 1920s primarily aimed at the separation of church and state.
The president at the time, Plutarco Elías Calles, wanted to eliminate the Catholic Church's influence in the country, born out of the anticlerical articles of the Mexican Constitution of 1917. Given Mexicans' popular allegiance to Catholicism, a massive armed uprising ensued.
During those challenging years many Cristero soldiers took refuge in the cavernous tequila distilleries, where tunnels hid the persecuted and allowed their escape to safer ground. In some cases these dark tunnels now serve as perfect aging bodegas for that blessed elixir called tequila.
It is said that tequila fueled the Mexican Revolution of 1910—a symbolic protest by indigenous foot soldiers and revolutionaries against the cognac and brandy of the elite. It wasn't until the 1950s, at the peak of the Golden Age of Cinema, that Mexicans truly became enamored with their national drink, watching their movie idols project machismo and tequila on the big screen.
Tequila Sauza was the first distillery to export tequila to the United States in 1873, when three barrels passed through El Paso del Norte, present-day El Paso, Texas. Today the United States purchases more tequila than Mexico, and the spirit is so popular around the world that 80 percent of the product is exported.
Tequila has undeniably become Mexico's great ambassador to the world.
The craft of making tequila requires many tools, among them the tahona. The Arabs named it, the Spaniards brought it to Mexico, and today the most demanding of tequileros won't make tequila without it. The tahona is composed of an enormous grindstone wheel, usually made of volcanic rock and set in a pit, that mashes the succulent pulp of cooked agave piñas.
Traditionally driven by a horse or mule, most tahonas are now pulled by a tractor or other motorized device. Unlike mechanized mashing processes, the massive weight of the stone crushes every fiber of the agave heart, extracting tastes and aromas rooted deep in the rough strands.
LA PUERTA THE DOOR
_Tequila, Jalisco_
EL OBRERO THE WORKER
_Tequila, Jalisco_
LOS HACHEROS THE AX-MEN
_Tequila, Jalisco_
SEMILLAS DE AGAVE AZUL BLUE AGAVE SEEDS
_El Arenal, Jalisco_
EL MEXICANO THE MEXICAN
_Amatitán, Jalisco_
AGAVE EN ALTO RAISED AGAVE
_Amatitán, Jalisco_
LA CARGA THE LOAD
_Arandas, Jalisco_
EL PICADOR THE PICKER
_Atotonilco el Alto, Jalisco_
COA Y AGAVE COA TOOL AND AGAVE
_Amatitán, Jalisco_
LA ESCALERA THE LADDER
_Arandas, Jalisco_
EL MURO DE AGA VE AGAVE WALL
_Arandas, Jalisco_
HORNO A LA ESPERA WAITING OVEN
_Arandas, Jalisco_
HORNO CERRADO CLOSED OVEN
_Amatitán, Jalisco_
EL GUÍA THE GUIDE
_Amatitán, Jalisco_
EL CARGADOR DE AGAVE THE AGAVE LOADER
_Amatitán, Jalisco_
LA ODA AL HORNO ODE TO THE OVEN
_Amatitán, Jalisco_
EL LEGADO THE LEGACY
_Amatitán, Jalisco_
LA PUERTA AL PASADO DOOR TO THE PAST
_Amatitán, Jalisco_
EL ALAMBIQUE DE COBRE COPPER STILL
_Amatitán, Jalisco_
EL SIGLO XVIII EIGHTEENTH CENTURY
_Tequila, Jalisco_
HOMBRE CON MAZO MAN WITH MALLET
_El Arenal, Jalisco_
LA CUCHILLA DE LUZ DAGGER OF LIGHT
_El Arenal, Jalisco_
AGAVE DULCE SWEET AGAVE
_Arandas, Jalisco_
LAS HUELLAS DE TAHONA TRACES OF TAHONA
_Arandas, Jalisco_
LA TAHONA DESEQUILIBRADA UNBALANCED TAHONA
_Arandas, Jalisco_
LA MOLIENDA THE GRINDING
_Tequila, Jalisco_
LA TERRAZA SUN TERRACE
_Tequila, Jalisco_
EL AGUAMIEL AGAVE JUICE
_Tequila, Jalisco_
LA FERMENTACIÓN FERMENTATION
_Arandas, Jalsico_
EL TEQUILERO THE WORKER
_Arandas, falsico_
LOS BARRILES BARRELS
_Amatitán, Jalisco_
BAJO LA LUZ IN THE LIGHT
_El Arenal, Jalisco_
EN RESERVA IN RESERVE
_Atotonilco el Alto, Jalisco_
EL ALTAR DE TEQUILA ALTAR FOR TEQUILA
_El Arenal, Jalisco_
RESERVA DE LA FAMILIA FAMILY RESERVE
_Tequila, Jalisco_
EL BURRO DONKEY
_Amatitán, Jalisco_
FIESTA
It was easy to see that Francisco had lived a few years—fifty-nine, to be exact. He was a no-nonsense weekend charro who doubled up during the week as the watchman and gatekeeper for a distillery in Arenal.
His face wore the scars of time with raw elegance. His body spoke tales of horsemanship, of drinking and afternoon charreadas.
The morning I saw him, Francisco wore a fine starched charro shirt with two horseshoes stitched in gold, one to the right of his heart and the other to the left. There was no doubt that between those two horseshoes raced his pride and his Mexico. A true charro at the doorstep of tequila.
I arrived in Atotonilco el Alto, in the highlands of Jalisco, home of some of the best tequila distilleries, where the iron-rich soil, elevation, and climate favor the blue agave plants. Standing guard to the agave fields is the pitahayo cactus, which eventually morphs into a spiny tree bearing the exotic pitahaya fruit. Most of Mexico's beauty is entrusted to ancient and adolescent landscapes where sunlight, blue agaves, and the pitahayo claim their territorial majesty.
Atotonilco el Alto refuses to surrender its rich culture and tradition to modernity. Everywhere I saw indigenous cornucopias of corn, tamales, champurrado, tacos, and roasted pumpkin seeds, of corner stores with treasures of chile-infused candy and pan dulce. Strings of _papel picado_ , paper banners, fluttered under the spell of colors radiating from a fiery collision between the indigo tint of dusk and the deep orange of a mercury streetlamp.
On a cool December evening a raucous crowd gathered to commemorate the apparition of Mexico's beloved Virgin of Guadalupe. A nervous group of young and old surrounded _el torito_ , a traditional wooden makeshift bull armed with a small arsenal of fireworks and explosive chasers. When the torito is hoisted atop a man's shoulders, it takes on a life of its own and begins chasing anything that moves—like a brave bull would instinctively do.
With one violent swing of the bull's head, the fireworks rocketed into the waiting crowd. Ghostlike revelers danced in and out of the smoke, possessed by wild exhilaration. In this moment euphoria triumphed over every man, woman, and child.
In Amatitán, I walked to the town's quaint cemetery near the entrance to Hacienda Herradura in the hopes of finding the tomb of a famed charro or tequilero. Instead I found fire—a burning bush in the middle of a cemetery. A mirage of tequila on fire, that same fire of conquest and revolution, the conquest that gave birth to the character of Mexico and tequila itself.
CHARRO, CABALLO, Y TEQUILA CHARRO, HORSE, AND TEQUILA
_Arandas, Jalisco_
PASO A PASO STEP BY STEP
_Amatitán, Jalisco_
LAS TEQUILAS DE TEQUILA TEQUILA'S TEQUILA
_Tequila, Jalisco_
FRANCISCO CON TEQUILA FRANCISCO WITH TEQUILA
_El Arenal, Jalisco_
LA SILLA CHAIR
_El Arenal, Jalisco_
LOS COLORES DE FIESTA FIESTA COLORS
_Atotonilco el Alto, Jalisco_
EL ANUNCIO THE ANNOUNCEMENT
_Tequila, Jalisco_
PERRO EN EXALTACIÓN DOG IN EXALTATION
_Tequila, Jalisco_
LA EUFORIA EUPHORIA
_Tequila, Jalisco_
EL CIELO THE SKY
_Tequila, Jalisco_
DESCANSADO ENTRE LOS AGAVES RESTING AMONG THE AGAVES
_Amatitán, Jalisco_
LA VIDA ARDIENTE ARDENT LIFE
_Amatitán, Jalisco_
¡VIVA MÉXICO!
_Arandas, Jalisco_
GALERÍA
With camera in hand, pleading for moments that can be relived in perpetuity, I have met plenty.
This photographic suite is primarily born of film, of transgressions in time and the consequence of light. The glamour of a digital camera is abandoned, and art must be defined in the space of a twelve-exposure roll of film.
The intellect leans toward making photographs worthy of transcending time. These twelve plates reflect that incessant noble pursuit called fine art.
LA NOPALERA PRICKLY PEAR TREE
_Arandas, Jalisco_
AL AMANECER AT SUNRISE
_Tequila, Jalisco_
EL NIDO THE NEST
_La Barca, Jalisco_
EL SILENCIOSO THE SILENT ONE
_Arandas, Jalisco_
EL SACRIFICIO DE MAYAHUEL MAYAHUEL'S SACRIFICE
_Amatitán, Jalisco_
HEMBRA Y MACHO FEMALE AND MALE
_Arandas, Jalisco_
EN CASA DEL AGAVE PLACE OF THE AGAVE
_El Arenal, Jalisco_
EL CENTURIÓN THE CENTURION
_La Barca, Jalisco_
¡AGAVE, LEVÁNTATE Y CAMINA! AGAVE, GET UP AND WALK!
_El Arenal, Jalisco_
LA FORTALEZA FORTITUDE
_El Arenal, Jalisco_
LA TAHONA THE GRINDSTONE
_Arandas, Jalisco_
EL AGAVE EN REPOSO AGAVE AT REST
_El Arenal, Jalisco_
Published Trinity University Press
San Antonio, Texas 78212
Copyright © 2017 by Joel Salcido
All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher.
Book design by Kristina Kachele Design, llc
Cover: Tequila typographic design by Rob Story
ISBN 978-1-59534-824-1 ebook
Trinity University Press strives to produce its books using methods and materials in an environmentally sensitive manner. We favor working with manufacturers that practice sustainable management of all natural resources, produce paper using recycled stock, and manage forests with the best possible practices for people, biodiversity, and sustainability. The press is a member of the Green Press Initiative, a nonprofit program dedicated to supporting publishers in their efforts to reduce their impacts on endangered forests, climate change, and forest-dependent communities.
The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI 39.48–1992.
CIP data on file at the Library of Congress
21 20 19 18 17 5 4 3 2 1
JOEL SALCIDO grew in Mexico and the United States. As a staff photographer for the _El Paso Times_ he documented the Tarahumara Indians and covered the 1985 earthquake in Mexico. He has also traveled extensively in Latin America for _USA Today._ His photographs appear in the collections of the Museum of Fine Arts Houston, the El Paso Museum of Art, the University of Texas Harry Ransom Humanities Center, and the Wittliff Collections at Texas State University. Additional acquisitions have been by the Federal Reserve Bank, the University of Texas at San Antonio, and the University of International Business and Economics in Beijing. The photograph _Atotonilco el Alto_ was recently added to Mexico's National Art Heritage Series. Salcido lives in Austin.
Table of Contents
1. Cover
2. Title Page
3. Dedication
4. Contents
5. Foreword
6. Preface
7. Introduction
8. Spirit
9. Terra
10. Alchemy
11. Fiesta
12. Galeria
13. Copyright
14. About the Author
# Guide
1. Cover
2. Contents
3. Title Page
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{
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Editors' delegation calls on Information Minister
by Fast Kashmir
Zulfkar assures adequate advertisement support to small, medium newspapers
Srinagar, May 31: The Minister for Information and Food, Civil Supplies and Consumer Affairs (FCS&CA), Chowdhary Zulfkar Ali today said the government would continue to provide necessary financial support to the newspapers by way of advertisements to ensure sustainability of especially small and medium publications.
The Minister said this during his interaction with the office bearers and members of the Joint Forum of Newspaper Editors (JFNE), who called on him here today.
The delegation comprised Rasheed Rahi, Mohammad Iqbal, Rashid Rahil, Farooq Ahmad Wani, Muzaffar Ahmad Khan, Zahoor Gulzar, Feroz Ali, Arshad Hussain, Zubair Ahmad Wani, Mukhtar Ahmad Wani, Farooq Ahmad, Javid Sofi and others.
The Minister said the Department of Information and Public Relations (DIPR) has been asked to ensure maximum transparency in advertisement distribution and subsequent release of funds strictly in tune with the guidelines laid down in the Print Media Advertisement Policy-2016. He said the Government has already notified the criteria for categorization of the newspapers for advertisement distribution and fixation of the revised rate structure for each category of the newspapers in tune with their circulation, reach and quality of content.
The Minister said that the government wants to offer as much support as possible on every count to the newspaper sector in the state to ensure its sustainability and professional growth. He said not only the local newspapers, but the national newspapers are also being provided financial support by the state government and out of Rs 32 crore payments made to the newspapers last fiscal, Rs 2.53 crore were paid to the national newspapers. He said the government is also providing regular financial support to wire services and the news agencies including PTI and UNI, and some local news agencies by way of monthly subscription charges.
Zulfkar Ali said the Department has already taken various measures to ensure utmost transparency in distribution of advertisements and in a significant move in this regard, for the first time ever, the details of the payments made to individual publications have been uploaded on the official website of the DIPR. He said it will be a regular feature in the future with the details of advertisements issued by DIPR to the newspapers and the payments made thereof being uploaded on the official website of the Department on quarterly basis.
He said the Department is also in the process of formulating a separate Advertisement Policy for the Electronic and Digital Media. He said the revised Accreditation Rules are being notified shortly.
Calling upon the media to play a responsive and responsible role for the larger good of the society, the Minister said it is imperative for the media play a constructive role in highlighting the real issues confronting the people of the State on developmental, educational and socio-economic fronts. He also stressed the need for distancing reporting from advertisements.
ATM robberies on rise in Valley
Fast Kashmir
|
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Borussia Dortmund was the first German club to win a European trophy, defeating Liverpool 2-1 in 1966 in the Cup Winners, Cup. Long a rival in Westphalia to Schalke 04, the Borussen have been German champions only six times. Three of those titles, and a European Champions Cup, have come since 1995. It is the only Germany club to be publicly listed, but has struggled on and off the field in recent years.
Lothar Emmerich scored 115 goals in 183 games for Dortmund between 1963 and 1969 but is best remembered for his pass that set up the West German goal that forced the famous 1966 World Cup final into extra time.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,508
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Kultura zgrčenih kostura je kultura brončanog doba, koja predstavlja prvu razvojnu fazu brončanog doba u srednjoj Europi (druga je kultura grobnih tumula, a treća kultura žarnih polja).
Hrvatska enciklopedija (LZMK) ovako piše o njoj: "U razvoju srednjoeuropskih kulturnih područja izdvajaju se tri bitne faze obilježene načinom pokapanja mrtvaca: u najstarijoj fazi mrtvaci se pokapaju u zgrčenu položaju, u srednjoj fazi pod zemljanim ili kamenim tumulima, dok se u kasnoj spaljuju, a pepeo se u pravilu polaže u žare (kultura polja sa žarama). U svakom razdoblju brončanoga doba izdvojilo se više lokalnih kulturnih skupina (lužička, únětička, vatinska)".
Izvori
Arheološke kulture
Brončano doba
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,683
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ACCEPTED
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
null
#### Original name
Phoma petiolorum Desm., 1847
### Remarks
null
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,990
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THIS IS A BORZOI BOOK
PUBLISHED BY ALFRED A. KNOPF
Copyright © 2012 by Verlyn Klinkenborg
All rights reserved. Published in the United States by Alfred A. Knopf, a division of Random House, Inc., New York, and in Canada by Random House of Canada Limited, Toronto.
www.aaknopf.com
Knopf, Borzoi Books, and the colophon are registered trademarks of
Random House, Inc.
Grateful acknowledgment is made to Farrar, Straus and Giroux, LLC, for permission to reprint excerpts from the following: "John Wayne:
A Love Song" from _Slouching Towards Bethlehem_ by Joan Didion.
Copyright © 1966, 1968, renewed 1996 by Joan Didion.
Reprinted by permission of Farrar, Straus and Giroux, LLC.
Library of Congress Cataloging-in-Publication Data
Klinkenborg, Verlyn.
Several short sentences about writing / Verlyn Klinkenborg. — 1st ed.
p. cm.
"This is a Borzoi Book."
eISBN: 978-0-307-95849-5
1. Authorship. 2. Rhetoric. I. Title.
PN151.K47 2012
808—dc23 2011050745
Jacket design by Jason Booher
v3.1
To John, Jill, and Jake
The subject is _there_ only by the grace of the author's language.
—Joyce Carol Oates
# PROLOGUE
This is a book of first steps. Their meaning will change as your experience changes. This book contains the bones of many arguments and observations—a vertebra here, a mandible there—but the whole skeleton is what you make of it. You'll find as much about thought and perception here as you will about language. There are no rules, only experiments.
The premise of this book is that most of the received wisdom about how writing works is not only wrong but harmful. This is not an assumption. It's a conclusion.
Like most received wisdom, what people think they know about writing works in subtle, subterranean ways. For some reason, we seem to believe most strongly in the stuff that gets into our heads without our knowing or remembering how it got there. What we think we know about writing sounds plausible. It confirms our generally false ideas about creativity and genius. But none of this means it's true.
What I've learned about writing I've learned by trial and error, which is how most writers have learned. I had to overcome my academic training, which taught me to write in a way that was useless to me (and almost everyone else). Unlearning what I learned in college—teaching myself to write well—is the basis of what I know. So is a lifetime of reading and a love of language. The rest comes from years of writing and teaching writing. The ideas and suggestions in this book have been tested again and again, by me, by my students, and by writers who have figured these things out for themselves.
A couple of cautions before you begin. This book isn't meant to replace the received wisdom. "Received" means untested, untried, repeated out of habit. Everything in this book is meant to be tested all over again, by you. You decide what works for you. This is perhaps the most important thing I have to say. There's no gospel here, no orthodoxy, no dogma. Part of the struggle in learning to write is learning to ignore what isn't useful to you and pay attention to what is. If that means arguing with me as you read this book, so be it.
This is a book full of starting points. Perhaps they'll help you find enough clarity in your own mind and your own writing to discover what it means to write. I don't mean "write the way I do" or "write the way they do." I mean "write the way you do."
Here's a starting point. You may have no idea what way you write. I hope this book will help you find out.
_Note: I use the word "piece" a lot. It means whatever you're writing, whatever the genre, whatever the length_.
# Contents
_Cover_
_Title Page_
_Copyright_
_Dedication_
_Epigraph_
Prologue
First Page
_Acknowledgments_
_A Note About the Author_
_Other Books by This Author_
#
Here, in short, is what I want to tell you.
Know what each sentence says,
What it doesn't say,
And what it implies.
Of these, the hardest is knowing what each sentence actually says.
At first, it will help to make short sentences,
Short enough to feel the variations in length.
Leave space between them for the things that words can't really say.
Pay attention to rhythm, first and last.
Imagine it this way:
One by one, each sentence takes the stage.
It says the very thing it comes into existence to say.
Then it leaves the stage.
It doesn't help the next one up or the previous one down.
It doesn't wave to its friends in the audience
Or pause to be acknowledged or applauded.
It doesn't talk about what it's saying.
It simply says its piece and leaves the stage.
This isn't the whole art of writing well.
It isn't even most of it.
But it's a place to begin, and to begin from again and again.
Short sentences aren't hard to make.
The difficulty is forcing yourself to keep them short.
There are innumerable ways to write badly.
The usual way is making sentences that don't say what you think they do.
Which can the reader possibly believe? Your sentences or you?
The only link between you and the reader is the sentence you're making.
There's no sign of your intention apart from the sentences themselves,
And every sentence has its own motives, its own commitments,
Quite apart from yours.
It adheres to a set of rules—grammar, syntax, the history and customs of the language, a world of echoes and allusions and social cues—that pay no heed to your intentions,
If you don't heed those rules.
It's hard to pay attention to what your words are actually saying.
As opposed to what you mean to say or what you think they're saying.
Knowing what you're trying to say is always important.
But knowing what you've actually said is crucial.
It's easier to tell what you're saying in a short sentence.
You've been taught to believe that short sentences are childish,
Merely a first step toward writing longer sentences.
You'd like to think your education has carried you well past short sentences.
But you've been delivered into a wilderness of false assumptions and bad habits,
A desert of jargon and weak constructions, a land of linguistic barbarism,
A place where it's nearly impossible to write with clarity or directness,
Without clichés or meaningless phrases.
True, you can sound quite grown-up, quite authoritative, in the manner of college professors and journalists and experts in every field.
(You may _be_ a college professor, a journalist, or an expert in some field.)
How well do they write?
How much do you enjoy reading them?
You'll make long sentences again, but they'll be short sentences at heart.
Sentences listening for the silence around them.
Listening for their own pulse.
Here's an experiment:
Pay attention to all the noise in your head as you go about writing.
Much of it is what you already know about writing, which includes:
The voices of former teachers, usually uttering rules.
Rules like, "Don't begin sentences with 'and.' "
(It's okay. You can begin sentences with "and.")
The things everybody knows or assumes about writers and how they work,
Whether they're true or not.
The things you feel you must or mustn't do, without really knowing why.
The things that make you wonder, "Am I allowed to...?"
(Yes, you're allowed to. Not forever and always, but until you decide for yourself what works and what doesn't.)
Write these things down—the contents of the noise in your head as you write.
You can't revise or discard what you don't consciously recognize.
These assumptions and prohibitions and obligations are the imprint of your education and the culture you live in.
Distrust them.
What you don't know about writing is also a form of knowledge, though much harder to grasp.
Try to discern the shape of what you don't know and why you don't know it,
Whenever you get a glimpse of your ignorance.
Don't fear it or be embarrassed by it.
Acknowledge it.
What you don't know and why you don't know it are information too.
Let's make a simple list from the preceding lines:
1. What you've been taught.
2. What you assume is true because you've heard it repeated by others.
3. What you feel, no matter how subtle.
4. What you don't know.
5. What you learn from your own experience.
These are the ways we know nearly everything about the world around us.
Keep them in mind, especially when you begin to think about _what_ to write and _how_ to write about it.
Let's think about what you already know.
In your head, you'll probably find two models for writing.
One is the familiar model taught in high school and college—a matter of outlines and drafts and transitions and topic sentences and argument.
The other model is its antithesis—the way poets and novelists are often thought to write.
Words used to describe this second model include "genius," "inspiration," "flow," and "natural," sometimes even "organic."
Both models are useless.
I should qualify that sentence.
Both models are completely useless.
Loosely linked to these models are two assumptions:
1. Many people assume there's a correlation between sentence length and the sophistication or complexity of an idea or thought—even intelligence generally.
There isn't.
2. Many people assume there's a correlation between the reader's experience while reading and the writer's experience while writing—her state of mind, her ease or difficulty in putting words together.
There isn't.
You can say smart, interesting, complicated things using short sentences.
How long is a good idea?
Does it become less good if it's expressed in two sentences instead of one?
Learn to distrust words like "genius," "inspiration," "flow," "natural," and "organic" when you think about your work.
(Don't use them when you talk about it either.)
They have nothing to do with writing
And everything to do with venerating writers.
Why short sentences?
They'll sound strange for a while until you can hear what they're capable of.
But they carry you back to a prose you can control,
To a stage in your education when your diction—your vocabulary—was under control too.
Short sentences make it easier to examine the properties of the sentence.
(Learn to diagram sentences. It's easy.)
They help eliminate transitions.
They make ambiguity less likely and easier to detect.
There's nothing wrong with well-made, strongly constructed, purposeful long sentences.
But long sentences often tend to collapse or break down or become opaque or trip over their awkwardness.
They're pasted together with false syntax
And rely on words like "with" and "as" to lengthen the sentence.
They're short on verbs, weak in syntactic vigor,
Full of floating, unattached phrases, often out of position.
And worse—the end of the sentence commonly forgets its beginning,
As if the sentence were a long, weary road to the wrong place.
Writing short sentences restores clarity, the directness of subject and verb.
It forces you to discard the strong elements of long sentences,
Like relative pronouns and subordinate clauses,
And the weak ones as well:
Prepositional chains, passive constructions, and dependent phrases.
Writing short sentences will help you write strong, balanced sentences of any length.
Strong, lengthy sentences are really just strong, short sentences joined in various ways.
You don't have to write short sentences forever.
Only until you find a compelling reason for a long sentence
That's as clear and direct as a short sentence.
You'll be tempted to say, "But short sentences sound so choppy."
Only a string of choppy sentences sounds choppy.
Think about variation and rhythm,
The rhythm created by two or three sentences working together,
Rhythm as sound and echo but also rhythm as placement.
Learn to use the position of a sentence, the position of a word—
First? last?—as an intensifier, an accent in itself.
Can a short sentence sound like a harbinger? An adumbration?
Can it sound like a reprise or a coda?
Listen.
How short is short?
That depends on the length of the sentences you're used to writing.
One way to keep sentences short is to keep the space between them as empty as possible.
I don't mean the space between the period at the end of one sentence and the first word of the next.
I mean the space between the period and the _subject_ of the next sentence.
That space often gets filled with unnecessary words.
Most sentences need no preamble—or postlude.
It's perfectly possible to make wretched short sentences.
But it's hard to go on making them for long because they sound so wretched
And because it's easy to fix them.
Making them longer is _not_ the way to fix them.
To make short sentences, you need to remove every unnecessary word.
Your idea of _necessary_ will change as your experience changes.
The fact that you've included a word in the sentence you're making
Says nothing about its necessity.
See which words the sentence can live without,
No matter how inconspicuous they are.
Every word is optional until it proves to be essential,
Something you can only determine by removing words one by one
And seeing what's lost or gained.
Listen for the sentence that's revealed as you remove one word after another.
You'll hear the improvement when you find it.
Try, for instance, removing the word "the."
See when the sentence can do without it and when it can't.
Without extraneous words or phrases or clauses, there will be room for implication.
The longer the sentence, the less it's able to imply,
And writing by implication should be one of your goals.
Implication is almost nonexistent in the prose that surrounds you,
The prose of law, science, business, journalism, and most academic fields.
It was nonexistent in the way you were taught to write.
That means you don't know how to use one of a writer's most important tools:
The ability to suggest more than the words seem to allow,
The ability to speak to the reader in silence.
Why are we talking about sentences?
Why not talk about the work as a whole, about shape, form, genre, the book, the feature story, the profile, even the paragraph?
The answer is simple.
Your job as a writer is making sentences.
Most of your time will be spent making sentences in your head.
In your head.
Did no one ever tell you this?
_That_ is the writer's life.
Never imagine you've left the level of the sentence behind.
Most of the sentences you make will need to be killed.
The rest will need to be fixed.
This will be true for a long time.
The hard part now is deciding which to kill and which to fix and how to fix them.
This will get much, much easier, but the decision making will never end.
A writer's real work is the endless winnowing of sentences,
The relentless exploration of possibilities,
The effort, over and over again, to see in what you started out to say
The possibility of saying something you didn't know you could.
Shape, form, structure, genre, the whole—these have a way of clarifying themselves when sentences become clear.
Once you can actually see your thoughts and perceptions,
It's surprising how easy it is to arrange them or discover their arrangement.
This always comes as a revelation.
What we're working on precedes genre.
For our purposes, genre is meaningless.
It's a method of shelving books and awarding prizes.
Every form of writing turns the world into language.
Fiction and nonfiction resemble each other far more closely than they do any actual event.
Their techniques are essentially the same, apart from sheer invention.
This is not to disparage accuracy, sound research, and impartiality.
Those are wonderful tools for novelists.
I'm interested in the genre of the sentence,
The genre that's always overlooked.
Many writers seem to believe we live in a universe of well-defined literary forms:
The memoir, the profile, the feature, the first novel, the book proposal,
A list of predetermined, prescriptive linguistic shapes
Heaped on a wagon and headed to market.
Writers worry about these shapes and their dictates
Long before they're able to make sentences worth reading.
They aspire to be nature writers,
Forgetting that nature, as a subject, is only as valid as your writing makes it.
They feel the formal burden of the memoir pressing upon them,
Though there's no such thing.
They believe that writing prose is as formulaic as writing a screenplay,
As ruled as a sonnet.
They believe the genre they've chosen
Determines the way they should write,
Complete with a road map, if only they could find it.
But genres are merely outlines by another name.
Better to be discovering what's worth discovering,
Noticing what you notice,
And putting it into sentences that, from the very beginning,
Open the reader's trust and curiosity,
Creating a willingness in the reader to see what you've discovered,
No matter what genre you call it.
Or, better yet, make the reader forget about genre completely.
If you make strong, supple sentences,
Improvise, understand and exploit your mistakes,
Keep yourself open to the possibilities each sentence creates,
Keep yourself open to thought itself,
And read like a writer,
You can write in any form.
You already possess some important assets.
You know how to talk.
How to read.
And, presumably, how to listen.
You've grown up in language.
You have the evidence of your senses.
The upwelling of your emotions.
The persistent flow of thoughts through your mind.
The habit of talking to yourself or staging conversations in your head.
Imagination and memory.
With luck, you were read aloud to as a child.
So you know how sentences sound when read aloud
And how stories are shaped and a great deal about rhythm,
Almost as much as you did when you were ten years old.
You may even have the capacity of knowing what interests you—
Or, better yet, knowing how to detect what interests you.
You're also two people, writer and reader.
This is a tremendous asset.
You can only become a better writer by becoming a better reader.
You have far more experience as a reader than you do as a writer.
You've read millions of words arranged by other writers.
How many sentences have you made so far?
But you've been taught to read in a way that tells you almost nothing about how to write
Or what's really to be found in the books you read.
You were taught that reading is extraction.
You learned to gather something called meaning from what you read,
As if the words themselves were merely smoke signals
Blowing away in the breeze, leaving a trace of cognition in the brain.
You've been taught, too, that writing is the business of depositing _meaning_ to be extracted later,
That a sentence is the transcription of a thought, the husk of an idea,
Valuable only for what it transmits or contains, not for what it is.
You've been taught to overlook the character of the prose in front of you in order to get at its _meaning_.
You overlook the shape of the sentence itself for the _meaning_ it contains,
Which means that while you were reading,
All those millions of words passed by
Without teaching you how to make sentences.
We take for granted, as a premise barely worth examining, that changing the words in a sentence—even the order of words—must have an effect on its meaning.
And yet we think and read and write as if the fit between language and meaning were approximate,
As though many different sentences were capable of meaning the same thing.
Our conventional idea of meaning is something like,
"what can be restated."
It means a summary.
It means "in other words."
You know how to theorize and summarize,
How to identify ideologies in the texts you read.
You do very well on the reading comprehension portion of the test.
But no one said a word about following a trail of common sense
Through the underbrush of the sentences themselves.
No one showed you the affinities at work among those thickets of ink
Or explained that the whole life of the language
Lies in the solidity of the sentence and cannot be extracted.
Writing well and reading well mean paying attention to _all_ the subtleties embodied in a sentence
In its exact form and no other.
How many subtleties?
What kinds?
That depends on how perceptive you become.
No two sentences are the same unless they're exactly the same, word for word.
(And, in a lifetime of writing, it's unlikely you'll ever write the same sentence twice.)
Any variation in wording changes the nuances that emanate from the sentence.
Discovering those nuances, and using them, are parts of the writer's job.
We'll discover a few shortly.
But first, what if _meaning_ isn't the sole purpose of the sentence?
What if it's only the chief attribute among many, a tool, among others, that helps the writer shape or revise the sentence?
What if the virtue, the value, of the sentence is the sentence itself and not its extractable meaning?
What if you wrote as though sentences can't be summarized?
What if you value every one of a sentence's attributes and not merely its _meaning_?
Strangely enough, this is how you read when you were a child.
Children read repetitively and with incredible exactitude.
They demand the very sentence—word for word—and no other.
The _meaning_ of the sentence is never a substitute for the sentence itself,
Not to a six-year-old.
This is still an excellent way to read.
The purpose of a sentence is to say what it has to say but also to be itself,
Not merely a substrate for the extraction of _meaning_.
The words in a sentence have a degree of specificity or concreteness.
They have complex histories.
They derive from dense contexts—literature, culture, the worlds of work.
They've been shaped by centuries of writing,
Centuries of utterance by living human beings.
They resonate with the ghosts of all their earlier forms.
The sentence itself has a rhythm.
It has velocity.
It uses metaphor and simile
Or hyperbole or metonymy or alliteration or internal rhyme or one of hundreds of other rhetorical devices.
It helps define the dramatic gesture that you—the writer—are making in the piece.
It stirs or gratifies the reader's expectations, on many levels.
It identifies the reader.
It gives the reader pause.
It names the world, using the actual names the world already contains.
Perhaps it renames the world.
And this is only the beginning.
You're the curator of all these qualities in the sentences you make,
Which lie there almost unnoticed
If you're interested only in extracting or depositing _meaning_.
Your task isn't merely to write with these qualities in mind.
It's to read with them in mind too,
You're learning to write, gathering materials to write
From everything you read as well as everything you write
And from everything you notice in the world around you.
Learning to write begins anywhere, at any time in life.
There's another trouble with _meaning_.
We've been taught to believe it comes near the end.
As if the job of all those sentences were to ferry us along to the place where meaning is enacted—to "the point,"
Just before the conclusion,
Which restates "the point."
This is especially true in the school model of writing.
Remember the papers you wrote?
Trying to save that one good idea till the very end?
Hoping to create the illusion that it followed logically from the previous paragraphs?
You were stalling until you had ten pages.
Much of what's taught under the name of expository writing could be called "The Anxiety of Sequence."
Its premise is this:
To get where you're going, you have to begin in just the right place
And take the proper path,
Which depends on knowing where you plan to conclude.
This is like not knowing where to begin a journey
Until you decide where you want it to end.
Begin in the wrong place, make the wrong turn,
And there's no getting where you want to go.
Why not begin where you already are?
Is there only the one way to get where you're going?
You were taught in school that each sentence
Rests on all the others like a single card in a house of cards,
A carefully constructed house of logic,
Fragile and easily dislodged.
That's one reason school papers often begin with several false starts.
The piece proceeds after the third introductory paragraph
And usually has two conclusions.
You were taught so much about outlining and transitions and the appearance of logic.
Perhaps you face the difficulties you do
Because you were taught so much about outlining
And transitions and the appearance of logic.
You were given a model of writing in which the sentences,
Inextricably overlapping, seamlessly transitioning,
Point forward toward the conclusion that justifies their existence.
In school you learned to write as if the reader
Were in constant danger of getting lost,
A problem you were taught to solve not by writing clearly
But by shackling your sentences and paragraphs together.
Think about transitions.
Remember how it goes?
Late in the paragraph you prepare for the transition to the next paragraph—
The great leap over the void, across that yawning indentation.
You were taught the art of the flying trapeze,
But not how to write.
Why were you taught to dwell on transitions?
It was assumed that you can't write clearly
And that even if you could write clearly,
The reader needs a handrail through your prose.
What does that say about the reader?
That the reader is essentially passive and in need of constant herding.
Are _you_ that kind of reader?
Do you tumble, uncomprehending, through the gaps between paragraphs?
Do you trip over ellipses?
Do you require constant supervision while walking down corridors of prose?
Do you lose the writer's train of thought unless you're reminded of it constantly?
No sentence can afford to be merely transitional.
If you've written clearly—
And you know what you've said and implied
As surely as you know what you haven't said—
The reader will never get lost reading your prose
Or have trouble following you _without_ transitions.
A reader is likelier to get lost cutting his way through
The jungle of transitions than crossing the gap of a well-made ellipsis.
And what about topic sentences?
Their only purpose is to announce the subject of the paragraph you're about to read,
As if you'd never figure it out otherwise.
In journalism, the equivalent of the topic sentence is the notorious "nut graf,"
A paragraph that tells you the content of the article you're about to read,
As if you couldn't proceed without a précis.
The obsession with transition negates a basic truth about writing,
A magical truth.
You can get anywhere from anywhere,
Always and almost instantly.
The gap between sentences is sometimes a pause for breath
And sometimes an echoing void.
And if you can get anywhere from anywhere,
You can start anywhere
And end anywhere.
There is no single necessary order.
Here's another basic truth.
Prose isn't validated by a terminal _meaning_.
If you love to read—as surely you must—you love being wherever you find yourself in the book you're reading,
Happy to be in the presence of every sentence as it passes by,
Not biding your time until the _meaning_ comes along.
Writing isn't a conveyer belt bearing the reader to "the point" at the end of the piece, where the _meaning_ will be revealed.
Good writing is significant everywhere,
Delightful everywhere.
The transitions you use should exist for the love of transition,
To employ and honor our abiding affection
For the _turn_ that so often takes place in our reading,
The _turn_ when the story changes or redirects itself.
They recall the moment, as children, when we came upon the phrase
"And then one day."
You know exactly how those four words feel.
You know exactly what they do.
When you get lost in your writing, remember them.
Don't use them: think about the possibilities they contain.
The ability to gather and redirect,
To rise above the level of the prose and look around,
As if you were standing in a crow's nest
Looking out over a sea of words,
Detecting a shift in the wind,
A change in the current,
A new impetus in your expectations,
And pointing it out to the reader.
_That's_ a transition.
You knew how it worked and felt
When you were barely old enough to read.
The transitions you were taught in school are merely
The nervous stitching together of sentences and paragraphs,
A cross-hatching of self-reference.
The syntactic result of all that connecting and transitioning—
Linking sentences and paragraphs to each other—
Is the very stuff you cut away in order to write
Short sentences
And make yourself intelligible again.
The extra space you feel between short sentences is mostly
The missing apparatus of transition and connection.
The anxiety about transition isn't caused only by
The prospect of a reader losing her way between paragraphs.
It's also caused by the period at the end of a sentence,
As if the period marks a boundary of comprehension.
This is one of the ways long sentences happen.
Most overcrowded sentences can be broken apart easily.
They became overcrowded because the words and phrases and thoughts they contain
Somehow seemed to belong together
In the shelter found to the left of the period,
The writer huddling words and phrases together into a single long sentence.
Related ideas coexisting side by side in two or three short sentences
Doesn't seem to be good enough:
They must live together in the same ramshackle sentence.
A crowded sentence betrays the writer's worry that the reader won't follow the prose
If parted by a period.
It also betrays the writer's lassitude,
The lazy shuffling of words together into a single sentence
Instead of deciding what really matters
And finding the verbal energy to construct separate sentences.
A single crowded sentence means giving up all the possible relations
Among shorter sentences—the friction, the tension,
The static electricity that builds up between them.
A single crowded sentence has only itself to relate to,
Only an enervated communion among its parts.
What else were you learning?
You were learning to dodge the "I" in your prose—
And yet to sound coy when indulging in it,
As though you were writing in front of a mirror.
In writing nonfiction, were you ever asked to _be_ the narrator,
To speak directly to the reader,
To decide what dramatic gesture you were making and act upon it?
Were you asked to write in order to be heard, to be listened to?
Asked to write a piece that mattered to you?
Was there ever a satisfactory answer to the question,
"Why am I telling you this?"
Besides "It's due on Monday"?
You were taught the perfect insincerity of the writing exercise,
Asked to write pieces in which you didn't and couldn't believe.
You learned a strange ventriloquism,
Saying things you were implicitly being asked to say,
Knowing that no one was really listening.
You were being taught to write as part of a transaction that had
Almost nothing to do with real communication,
Learning to treat the making of sentences as busywork,
A groping for words, an act of drudgery,
A way of dressing up your meaning or your argument with almost no attention to the character of the words or sentences you were using,
Unless you were trying to imitate
The stiff and impersonal manner of "formal" prose.
You were also learning to distrust the reader and yourself.
Do you remember feeling, when you were writing a paper for school,
That your vocabulary was steadily shrinking?
By the end, the same few words seemed to be buzzing
Around and around in your head, like flies weary of feeding.
That's a symptom of boredom.
You were bored from the start and for good reason.
You were repeatedly asked to persuade or demonstrate or argue,
To reiterate or prove or recite or exemplify,
To go through the motions of writing.
You were almost never asked to notice or observe, witness or testify.
You were being taught to manage the evidence gathered from other authorities
Instead of cultivating your own—
To simulate logic
But not to write so clearly that
What you were saying seemed self-evident.
You were also learning to divorce your experience as a reader
From your inexperience as a writer.
When what you needed most was to trust your experience as a reader.
In school, we're taught—or we absorb the idea—that writing
Flows out of the creative writer like lava down the slope of a volcano.
An uninterruptible stream.
And yet we study the work itself as if its molten fire had hardened into rock.
But the work isn't an eruption from the author's brain.
It doesn't merely flow.
And it remains more dynamic, as written—on the page—than we let ourselves imagine.
We forget something fundamental as we read:
Every sentence could have been otherwise but isn't.
We can't see all the decisions that led to the final shape of the sentence.
But we can see the residue of those decisions.
If you look at the manuscripts of writers—
Handwritten drafts preserved in museums and libraries—
You can often see the changes they made scribbled between the lines.
What you can't see are the changes they made in their heads before those sentences were ever inscribed.
If you could look through the spaces between the sentences,
Through the door into the writing room, into that writer's head,
You'd see that every word was different once
And that the writer was contemplating
An incalculable number of differences,
Feeling her way among the alternatives that presented themselves,
Until settling upon words that were finally written down,
Then revised over and over again—
Before they were printed, published, reprinted in anthologies,
And treated as though they'd been carved in stone.
It was all change until the very last second.
Every work of literature is the result of thousands and thousands of decisions.
Intricate, minute decisions—this word or that, here or where, now or later, again and again.
It's the living tissue of a writer's choices,
Not the fossil record of an ancient, inspired race.
Interrogate those choices.
Imagine the reason behind each sentence.
Why is it shaped just this way and not some other way?
Why that choice of words?
Why that phrasing?
Why that rhythm?
The purpose of these questions isn't to construct a theory,
A hypothesis about how or why the writer writes.
The purpose is to help you notice the shape of what lies before you.
The answers to these questions may be nothing more than
Noticing the effect of asking them.
Imagine reading Jane Austen or James Baldwin and wondering,
Why is the sentence this way and not another way?
That sounds like a trivial or unanswerable question.
Until you imagine revising the sentence, giving it a different rhythm,
Substituting a different word, a different structure.
Revise a sentence by Austen or Baldwin?
Why not? It's an experiment.
Try it, and you begin to glimpse the inherent necessity binding the writer's choices together.
You begin to see the invisible tensions that arc from line to line,
Paragraph to paragraph, page to page.
These aren't constructions of logic or meaning.
They're echoes and responses, moments of candor and their aftereffects,
Feats of resilience and attention, sound and impulsion.
"Why is this sentence this way?"
Finds its answer in
"Why is that sentence that way?"
Which sounds circular, until you begin to understand how
Each variation shapes and affirms others,
Creating the restraint of good prose,
A balance of forces and internal tensions
That make the minutest effects discernible.
This isn't a description of the writer's genius or inspiration or intention.
It describes the way every sentence influences every other sentence.
It describes the writer's alertness to her sentences.
The way her sentences listen to one another.
Prose is the residue, the consequence, of the writer's choices,
Choices about the shape of each sentence
And how each sentence shapes the others.
That's how we need to read, as writers—
Paying attention to the decisions embedded in each sentence,
Decisions visible in the structure of the sentence itself.
What you write—what you send out into the world to be read—
Is the residue of the choices and decisions _you_ make.
Choices and decisions _you_ are responsible for.
And what are the choices?
That's like asking, what are the nuances?
It depends on how perceptive you become.
The central fact of your education is this:
You've been taught to believe that what you discover by thinking,
By examining your own thoughts and perceptions,
Is unimportant and unauthorized.
As a result, you fear thinking,
And you don't believe your thoughts are interesting,
Because you haven't learned to be interested in them.
There's another possibility:
You may be interested in your thoughts,
But they don't have much to do with anything you've ever been asked to write.
The same is true of what you notice.
You don't even notice what you notice,
Because nothing in your education has taught you that what you notice is important.
And if you do notice something that interests you,
It doesn't have much to do with anything you've ever been asked to write.
But everything you notice is important.
Let me say that a different way:
If you notice something, it's because it's important.
But what you notice depends on what you allow yourself to notice,
And that depends on what you feel authorized, permitted to notice
In a world where we're trained to disregard our perceptions.
Who's going to give you the authority to feel that what _you_ notice is important?
It will have to be you.
The authority you feel has a great deal to do with how you write, and what you write,
With your ability to pay attention to the shape and meaning of your own thoughts
And the value of your own perceptions.
Being a writer is an act of perpetual self-authorization.
No matter who you are.
Only you can authorize yourself.
You do that by writing well, by constant discovery.
No one else can authorize you.
No one.
This doesn't happen overnight.
It's as gradual as the improvement in your writing.
Start by learning to recognize what interests you.
Most people have been taught that what they notice doesn't matter,
So they never learn how to notice,
Not even what interests them.
Or they assume that the world has been completely pre-noticed,
Already sifted and sorted and categorized
By everyone else, by people with real authority.
And so they write about pre-authorized subjects in pre-authorized language.
Why do I say this?
When students are free to write anything they want,
What they write first are pieces they hope look like something they saw published somewhere
About subjects they believe are pre-authorized
Because someone has already written about them
In pieces they hoped looked like something they saw published somewhere.
A first piece of that kind is a tacit way of taking shelter under the authority of someone else's perceptions.
It's also a way of saying, "I know you're not really interested in what I think or notice."
But that's the very thing the reader _is_ interested in
If your sentences allow him to be.
Is it possible to practice noticing?
I think so.
But I also think it requires a suspension of yearning
And a pause in the desire to be pouring something out of yourself.
Noticing is about letting yourself out into the world,
Rather than siphoning the world into you
In order to transmute it into words.
Practicing noticing will also help you learn more about patience
And the nature of your mind.
Noticing means thinking with all your senses.
It's also an exercise in not writing.
So what is noticing?
A pinpoint of awareness,
The detail that stands out amid all the details.
It's catching your sleeve on the thorn of the thing you notice
And paying attention as you free yourself.
It requires no gear, no special tools, no apparatus.
You practice noticing as part of your ordinary life.
What do you notice? Whatever you notice.
Behavior, thought, overheard words, light, resemblance,
Emotion, totality, particularity,
Whatever you find in the habitat of your perceptions,
Anything, no matter how minute,
Whether you're working or reading or taking the subway.
The pattern is particular to you,
An element in what gets construed as "style."
What you notice has no meaning.
Be sure to assign it none.
It doesn't represent or symbolize
Or belong to some world theory or allegory of perception.
Don't put words to it.
And don't collect it. Let it slip away.
Be patient for the next thing you notice.
There's always an urge among writers
To turn fleeting observations and momentary glimpses
Into metaphors and "material" as quickly as possible,
As if every perception ended in a trope,
As if the writer were a dynamo
Turning the world into words.
The goal is the opposite:
To get your words, your phrases,
As close as you can to the solidity,
The materiality of the world you're noticing.
Rushing to notice never works,
Nor does trying to notice.
Attention requires a cunning passivity.
Let yourself wonder why this thing, this instant, this suddenness, caught your attention.
What you're noticing isn't only what struck you.
It's also how your mind, your attention, gets from place to place,
From the steady current of your thoughts to their sudden interruption.
Notice what you notice and let it go.
You can also make sentences the way you go about practicing noticing.
Catching a phrase in your head,
Exploring the possibilities it occasions,
Then releasing it,
Making nothing more than a vanishing sentence,
Which you do not transfix in some collection of sentences
Or etherize in a jar.
You'll never run out of noticings,
And there are more than enough sentences to let a few go.
The urge to write is so strong.
Aspiring writers want so badly to be pouring something out of themselves.
You need a place where you can practice noticing and making sentences—
Observations of genuine clarity,
Sentences of vigor, invention, and self-perception.
That place would be your mind.
What you get in return for this gathering and releasing
Is habit, ease, trust, and a sense of abundance that sustains your writing.
And your mind never relinquishes what really matters.
As you practice noticing, notice how thickly particled
With names the world around you is.
This will gradually become part of your noticing,
Looking not for words to make us see the way you saw—
But for the names of what you've noticed,
Names that announce the whatness of the world
To a single species.
It's hard to grasp at first the density, the specificity
With which the world has been named.
This is a planet of overlapping lexicons,
Generation after generation, trade after trade,
Expedition after expedition sent out to bring home
Name upon name, terms of identity in endless degrees of intricacy,
And all at hand, if you look for them.
Don't neglect such a rich linguistic inheritance.
It's your business to know the names of things,
To recover them if necessary and use them.
This isn't merely a matter of expanding your vocabulary.
It's a matter of understanding that everything you see and know
About your presence in this moment of perception
Is overlaid by a parallel habitat of language,
Names that lie tacit until you summon them.
And yet you've been taught to make sentences
In which inert verbs act abstractly upon faceless nouns,
To write on a theoretical basis, which deprives the world of its content,
And to use passive constructions, which absolve everyone of responsibility.
What's a metaphor in the prose you were taught to write?
A stage prop, a paraphrase, a clarification, at best,
Nearly always cumbersome, bordering on cliché,
Almost always timid, rarely serious, usually self-conscious,
And too often stretched out over three or four sentences
In order to create an extended metaphor,
Which is a cruel analogical death.
A true metaphor is a swift and violent twisting of language,
A renaming of the already named.
It's meant to expire in a sudden flash of light
And to reveal—in that burst of illumination—
A correspondence that must be literally accurate.
Any give in the metaphor, any indeterminacy,
And it becomes a cloud of smoke, not a flash of light.
Like any rhetorical device, the less you use it, the more effective it is.
Try making prose with a poetic seriousness about its tools—
Rhythm, twists of language, the capacity to show the reader
What lies beyond expression,
But with the gaits of prose and a plainness in reserve
That poetry rarely possesses, an exalted plainness.
One of the hardest things about learning to read well is learning to believe that every sentence has been consciously, purposely shaped by the writer.
This is only credible in the presence of excellent writing.
You may notice, as you write, that sentences often volunteer a shape of their own
And supply their own words as if they anticipated your thinking.
Those sentences are nearly always unacceptable,
Dull and unvarying, yielding only a small number of possible structures
And only the most predictable phrases, the inevitable clichés.
A cliché is dead matter.
It causes gangrene in the prose around it, and sooner or later it eats your brain.
You can't fix a cliché by using it ironically.
You can't make it less gangrenous by appearing to
"quote" it or invert it or joke about it.
A cliché isn't just a familiar, overused saying.
It's the debris of someone else's thinking,
Any group of words that seem to cluster together "naturally"
And enlist in your sentence.
The only thing to do with a cliché is send it to the sports page
Or the speechwriters, where it will live forever.
Volunteer sentences occur because you're not considering the actual sentence you're making.
You're looking past it toward your meaning somewhere down the road,
Or toward the intent of the whole piece.
Somehow that seems more important than the sentence you're actually making,
Though your meaning and the intent of the whole piece
Depend entirely on the sentence you're making.
In fact, you're distracted from the sentence by your intention
And by wondering how soon you'll be done.
You're distracted from the only thing of any value to the reader.
Volunteer sentences are the relics of your education
And the desire to emulate the grown-up, workaday prose that surrounds you,
Which is made overwhelmingly of sentences that are banal and structurally thoughtless.
A volunteer sentence is almost always a perfunctory sentence.
That can change.
But only after years of questioning the shapes of sentences you read,
And every sentence you write.
Don't let the word "years" alarm you.
Think of it as months and months and months and months.
You may think a volunteer sentence is an inspired one
Simply because it volunteers.
This is one reason to abandon the idea of inspiration.
All the idea of inspiration will do
Is stop you from revising a volunteer sentence.
Only revision will tell you whether a sentence that offers itself is worth keeping.
The writer's job isn't accepting sentences.
The job is making them, word by word.
Volunteer sentences,
Volunteer subjects,
Volunteer structures.
Avoid them all.
Most aspiring writers write too soon.
They think writing is a transitive act instead of an intransitive one.
Everything they know about writing—all those images of writers writing—
Hastens them to the desk,
Where they sit perched over the keyboard or pen in hand,
Caught in an anticipatory gesture,
Eyes intent on the possibilities of the screen,
Poised at the brink of thought, but not actually thinking,
As though by leaning forward a sentence will tip out of their heads
And onto the page.
But writing isn't performed upon a device or in a state of anticipation.
Consider the bad habit of typing, preliminarily, two or three words—
A natural start to the sentence, you think—
And then waiting for the rest of the sentence to reveal itself.
But after two or three words the sentence is already foredoomed,
Its structure predetermined.
Two or three words, and you've already reduced the remaining choices
To a small, depressing handful.
It's shocking to realize how quickly you become wedded to those two or three words,
How hard it is to abandon them for an alternative.
Sometimes, going over your work, you discover
That you can't remember how every sentence came to take its shape.
You come across vestiges of unconsciousness in your prose,
Amnesiac stretches where sentences seem to have written themselves.
This is not a good thing.
When the work is really complete, the writer knows how each sentence got that way,
What choices were made.
You become not only a living concordance of your work, able to say where almost any word appears.
You also carry within you the memory of all the decisions you made while shaping your prose,
Decisions invisible to the reader except in the residue of your prose.
It sounds impossible to know so much about what you've written.
And yet it's inevitable.
Something you don't even have to think about
If you've been thinking about your sentences.
This brings us back to the difficulty of knowing what your sentences actually say.
The problem most writers face isn't writing.
It's consciousness.
Attention.
Noticing.
That includes noticing language.
The fundamental act of revision is literally becoming conscious of the sentence,
Seeing it for what it is, word for word, as a shape, and in relation to all the other sentences in the piece.
This is surprisingly hard to do at first
Because our reading habits are impatient and extractive.
And because we've been blinded to the actuality of prose—
Its physical substance—
By the pursuit of _meaning_.
The very nature of reading encourages us to believe we're looking _through_ the prose to worlds on the other side of the ink.
The familiarity with which we know our own lives is sometimes disabling.
Writing is a special instance of that.
In responding to your own prose, you're responding in some sense to yourself,
And no matter how hard you look, you're almost invisible to yourself,
Camouflaged by familiarity.
One basic strategy for revision is becoming a stranger to what you've written.
Try reading your work aloud.
The ear is much smarter than the eye,
If only because it's also slower
And because the eye can't see rhythm or hear unwanted repetition.
But how should you read aloud?
There's self-awareness even in this,
A tendency to overdramatize or become self-conscious,
To read as though the words weren't yours,
Mechanically, without listening,
As though you were somehow hiding from their sound
Or merely fulfilling a rote obligation.
Try reading the words on the page as though they were meant to be spoken plainly
To a listener who is both you and not you—
An imaginary listener seated not too far away.
That way your attention isn't only on the words you're reading.
It's on the transmission of those words.
As you read aloud, catch the rhythm of the sentences without overemphasizing it.
Read so the listener can hear the shape of the syntax.
_You_ be the listener, not another person.
You'll be stopping often.
Reading aloud forces you to choose _how_ you'll read aloud,
What character you'll play, what version of yourself you'll present,
What dramatic gesture you're making as you read.
The act of writing requires exactly the same thing,
Though we pretend it doesn't—
Another good reason to read your work aloud.
If you don't know what I mean by rhythm,
Imagine a singer's phrasing of the lyric in a song.
In prose, it's subtler, the beat and the music quieter.
Try reading aloud some of everything you read, no matter what it is,
A couple of paragraphs from the newspaper or a textbook or a novel or a poem.
Especially a poem.
This is how you begin to understand rhythm and its absence.
It will also improve your ability to read aloud,
Which will help you discern the underlying textures of your prose.
How well you read aloud reveals how well you understand the syntax of a sentence.
Do you remember, in school, going around the room,
Each student in turn reading a paragraph out loud?
Remember how well some students read and others, how badly?
It was a difference in comprehension,
Not of the sentence's _meaning_ ,
But of its texture, pace, structure, actuality.
Don't read straight through without stopping.
Read until your ear detects a problem.
Stop there.
How will you know there's a problem?
Something will sound funny.
You'll feel a subtle disturbance, a nameless, barely discernible tremor inside you.
You won't say, "Aha! That pronoun has the wrong antecedent!"
(Though soon you will.)
You'll simply feel that something's wrong, without knowing what.
(This also happens when you're reading silently, but less emphatically.)
Pay attention now:
No matter how much you know or learn about syntax, grammar, and rhetoric,
This small internal quaver, this inner disturbance,
Is the most useful evidence you'll ever get.
Someday, you'll be able to articulate what causes it.
But for now, what's important is to notice it.
Noticing is always the goal.
Actually, the goal right now is noticing that you're noticing.
One day merely noticing will be enough.
You already experience these faint stirrings in the presence of sentences,
But you didn't know they mattered.
This turns out to be true of many things you notice.
No one taught you to disregard these inner sensations.
No one taught you to be aware of them either.
No one even acknowledged that they exist.
You thought they weren't significant—
Mainly because they were occurring within _you_.
And what do _you_ know (you're always tempted to ask)?
You know a lot, especially in a preconscious kind of way.
Notice those stirrings now, and keep noticing.
Never stop.
They're a sign of your skill and experience as a reader,
And they're immensely useful to you as a writer always.
Writing requires a high degree of inner alertness,
Especially when things are going wrong.
Soon you'll know exactly how to find the things that are going wrong
As well as the things that are going right.
But until then—and even long after—you'll find it easier to detect a problem by the disturbance it causes inside you.
This means paying attention not only to your writing but to your emotions.
I don't mean large-scale emotions—sad, mournful, depressed, suicidal, elated.
I mean a pale and nameless unease,
As if a poorly constructed sentence could make you slightly homesick.
The faint vertigo caused by an ambiguity you can't quite detect.
The malaise given off by an awkwardness in the syntax.
You won't be able to name the feeling a syntactical problem causes.
It doesn't have a name.
What matters is what it points to.
Find out what's causing it and fix it
Even if you're not sure how.
Here's another way to make your prose look less familiar.
Turn every sentence into its own paragraph.
(Hit Return after every period. If writing by hand, begin each new sentence at the left margin.)
What happens?
A sudden, graphic display of the length of your sentences
And, better yet, their relative length—how it varies, or doesn't vary, from one to the next.
Variation is the life of prose, in length and in structure.
Having all your sentences in a column, one above the other, makes them easier to examine.
Suddenly you see similarities in shape.
You notice, for instance, how your sentences cling to each other
Instead of accepting their separateness.
And you can begin to ask questions—simple ones—that will help you understand how to revise
And make better sentences.
How many sentences begin with the subject?
How many begin with an opening phrase _before_ the subject?
Or with a word like "When" or "Since" or "While" or "Because"?
How many begin with "There" or "It"?
What kinds of nouns do you see?
Abstractions? Generalizations?
Multisyllabic Latinate nouns ending in "-ion"?
Or are they the solid names of actual things?
Is the subject of the sentence an actor capable of performing the action of the verb?
Can you adjust the sentence so it is?
Or does the subject of the sentence hide the action of entities that are able to act—humans, for instance?
How close is the subject to its verb?
Are they separated by an inserted phrase?
What does that do to the velocity of the sentence?
How many of the verbs are variants of "to be"—"is," "are," "were," "was," and so on?
Are the verbs active, energetic?
Or do they merely connect or arrange or present or relate?
Are the constructions passive?
How often does the word "as" appear, and in which of its many senses?
Are you using "with" as a preposition or as a false conjunction, a false relative pronoun?
Are there inadvertent repetitions—words repeated unintentionally?
Is every phrase in its proper place, every word?
Is everything next to what it should be next to?
Anything outright ungrammatical?
Words used improperly?
Do verbs that require direct objects (transitive verbs) lack them?
If there's a modifying phrase at the start of the sentence, does it modify the subject of the sentence? (It must.)
Can the sentence be broken in two or three?
Do these questions sound overly technical to you?
They're basic.
But they raise another mistaken assumption about writing.
Many people assume there's an inherent conflict between creativity and a critical, analytic awareness of the medium you work in.
They assume that the creative artist works unconsciously And that knowing too much about matters like grammar and syntax diminishes or blunts creativity.
This is nonsense.
You don't need to be an expert in grammar and syntax to write well.
But you do need to know the difference between transitive and intransitive verbs.
Between active and passive constructions.
The relation between a pronoun and its antecedent.
All the parts of speech.
The different verb tenses.
The nature of participles and their role as modifiers.
The subtleties of prepositions—the hardest part of speech even for native speakers of English.
You need a toolbox of rhetorical devices, like irony, hyperbole,
And the various kinds of analogy.
You need an ever-growing vocabulary—and with it the awareness that most words carry several meanings.
You need to look up even familiar words every time you have a doubt
And especially when you don't have a doubt.
That is, very often.
That is, every time you write.
The history of a word is part of its meaning,
Sometimes even the better part of its meaning.
You're responsible for the nuances of the words you use.
How else can you use exactly the right shade of meaning?
How else can the reader trust you?
You can't disclaim this responsibility.
Those nuances are embedded in etymology.
A good example: the word _autopsia_ ,
Which I came across while working on a book of my own.
I thought—by inference from the context and by analogy with "autopsy"—
That _autopsia_ meant a collection of stuffed or dissected animals,
The sort of collection a natural historian might accumulate.
But if I had thought more carefully and considered the word's roots,
I would have realized that it means
A collection of objects one has "seen for oneself."
I discovered that when I looked up the familiar word "autopsy,"
Which means, etymologically, to see for oneself.
You'll need to look up nearly every word you use for longer than you think.
It's the only way to be sure of meaning
And etymology
And pronunciation, which has a bearing on rhythm.
Think of your vocabulary as your _autopsia—_
Words you've actually seen for yourself.
If you don't know the language of grammar and syntax, try this.
Begin with the parts of speech.
Copy or print out a couple of pages by an author whose work you like.
(For example, the opening of John McPhee's _Coming into the Country_.)
Gather some colored pens or pencils.
Choose one color and circle all the nouns.
Pause to consider them.
Then choose a different color and circle all the verbs.
Pause again.
Ditto the articles, adverbs, adjectives, prepositions, conjunctions, and interjections.
Anything left over?
There shouldn't be.
This will clarify the parts of speech, and it will help you see how the author uses them.
If a word puzzles you, look it up.
A good dictionary will tell you what part of speech it is.
Don't just imagine doing this someday.
Do it. It's interesting.
Now try a slightly harder version of this experiment on a separate copy.
Circle the direct objects.
The indirect objects.
The participles.
The relative pronouns.
The metaphors and similes and analogies.
Any word that seems to be used in a way that distorts its meaning.
Any particularly rhythmic phrases or sentences.
Any spot where you sense a change in direction or time or voice.
Any phrase that interests you.
Any word that stops you.
Anything you _notice_ , whether you think it matters or not.
It matters because you noticed it.
Do any of the words surprise you or call attention to themselves?
I'm not asking whether you know what they mean.
You've already looked up every word you don't know.
Haven't you?
It's easier to answer a comparative question
Than a question that depends on an implicit standard of judgment.
"Is this sentence longer than that one?" is easier to answer than
"Is that word poetic?"
You don't need much experience to tell whether one sentence is longer than another.
But you do to say whether a word is poetic in usage.
That experience is easy to come by.
It's called "reading poetry."
Turn to the poets.
Learn from them.
At first, ask comparative questions.
They'll help you understand how writing works.
Examine the quantities that appear in prose—
The things that can be measured or counted:
Rhythms, patterns of repetition, length of words and sentences,
Length of paragraphs, the breadth of an ellipsis.
Examine the distance between sentences.
Some lie close together, making small steps forward.
Others seem to stand well apart from each other,
Advancing the piece discontinuously.
These kinds of questions will help you understand
The character of what you're reading and how it was made.
We take for granted that what we love in our reading is
Perception, wisdom, poetry, wit, irony.
Yet it's surprising how often what we love is really found in the material structure, the concrete details, the rhythm of the sentences.
Ask yourself questions about the words you find—especially less ordinary or less familiar words.
(But query the familiar ones too.)
Where do they come from?
What line of work are they in?
Who's likely to use them?
And in what context?
This will remind you that every word carries a social freight.
Now perform the same experiment with an author whose work has a different feel.
(Try Joan Didion's essay "Some Dreamers of the Golden Dream" in _Slouching Towards Bethlehem_.)
Then try it again with a page from a very different context—
A business article or a best seller or a critical essay in an academic journal.
What do you find?
Different patterns of usage, different lexicons,
Which create, in turn, different textures and rhythms in the sentences themselves.
Try the same experiment on some pages of your own prose.
What do you notice?
Make some comparative lists:
How does the menagerie of Didion's nouns compare with the menagerie of McPhee's nouns or the nouns in a critical essay or a business article—or something you're writing?
Ask the same question about verbs and sentence structures.
Ask yourself too how present the writer feels to the reader.
How strong is your sense of the speaker or narrator?
How is that sense created, and where do you detect it?
Imagine it this way:
Every piece is an ecosystem of words and structures and rhythms.
How rich and diverse is the ecosystem in each of these pieces?
From which do you derive the most pleasure?
And why?
Be patient with yourself and the things you discover.
This isn't a test.
Every reader will notice different things.
You won't know the significance of everything you notice.
Don't let that deter you.
Don't try to give a _meaning_ to the things you notice.
Just observe them.
Again, the effect of these discoveries may be nothing more than
Noticing the effect of making these discoveries.
You'll become much more adept at seeing how your own sentences are shaped and where they succeed and go awry.
Thinking in terms of grammar and syntax is also a good way to make your sentences seem less familiar.
Suddenly you're looking at their bones and muscles,
The way they're joined and the kinetics of their movement.
But notice.
The point of learning the fundamental language of grammar and syntax
Isn't correctness or obeying the rules.
It's keeping the rules from obtruding themselves upon the reader
Because you've ignored them.
The reader is just like you,
Full of subtle, distracting feelings when things are going wrong in a sentence.
Every reader is always two readers.
One reads with a deep, intuitive feel for the way language works
And yet with overwhelming literalness.
This reader (no matter what he consciously knows about grammar or syntax) is troubled by mistakes, misspellings,
And especially the syntactical miscues that cause ambiguity.
This reader will always stumble over your errors.
If a sentence offers an ambiguous path—two ways of being read—this reader will always take the wrong one.
The other reader—literate, curious, adaptable, intelligent, open-minded—
Will follow you anywhere you want to go
As long as your prose is clear.
(More on this excellent person soon.)
Every reader is both of these readers in one.
Write for both together.
Here's another reason for learning the basics of grammar and syntax:
Syntactic and grammatical accuracy is the precondition for being sure
Your sentences say what you think they say.
It's no trouble to learn these things.
Knowing them doesn't detract from your creativity or spontaneity.
What you're learning isn't a code of ethics or the rules of proper behavior in decent society.
It isn't etiquette or table manners.
You aren't being handcuffed or detained or taught to curtsy
Or forced to wear a borrowed tie at a fancy restaurant
Or compromising your standards.
You're merely learning what is:
The names of the kinds of words, their relation to each other, and their functions.
Like a painter's knowledge of color and the laws of perspective,
A jazz musician's knowledge of chord structures and his instrument.
Your job as a writer is making sentences.
Your other jobs include fixing sentences, killing sentences, and arranging sentences.
If this is the case—making, fixing, killing, arranging—how can your writing possibly flow?
It can't.
Flow is something the reader experiences, not the writer.
A writer may write painstakingly,
Assembling the work slowly, like a mosaic,
Fitting and refitting sentences and paragraphs over the years.
And yet to the reader the writing may seem to flow.
The reader's experience of your prose has nothing to do with how hard or easy it was for you to make.
You're not writing for a reader in the mirror whose psychological state reflects your own.
You have only your own working world to consider.
The reader reads in another world entirely.
So why not give up the idea of "flow" and accept the basic truth about writing?
It's hard work, and it's been hard work for everyone all along.
There's a good reason to believe this, apart from the fact that it's true.
If you think that writing—the act of composition—should flow, and it doesn't, what are you likely to feel?
Obstructed, defeated, inadequate, blocked, perhaps even stupid.
The idea of writer's block, in its ordinary sense,
Exists largely because of the notion that writing should flow.
But if you accept that writing is hard work,
And that's what it feels like while you're writing,
Then everything is just as it should be.
Your labor isn't a sign of defeat.
It's a sign of engagement.
The difference is all in your mind, but _what_ a difference.
The difficulty of writing isn't a sign of failure.
It's simply the nature of the work itself.
For the writer, the word "flow" is a trap.
So is any word that suggests that writing is a spontaneous emission.
Writing doesn't flow, unless you're plagiarizing or collecting clichés or enlisting volunteer sentences.
You'll experience certain kinds of suddenness as you work:
The illusion that time is passing quickly,
An episode of unusual mental clarity,
An almost unnoticed transition from one mood to another.
The piece you're working on may take a jump forward,
And you notice the jump instead of the hours and days of thinking that enabled it.
Everything may flow when you're setting thoughts down on paper.
But that's jotting, not writing.
"Flow" means effusion, a spontaneous outpouring of sentences.
But what it really, secretly means is easy writing.
The more you know about making sentences, the easier it is to fix them,
To get out of trouble, to find the really good sentences—
The better sentences—hiding beneath the skin of your thinking.
What matters isn't how fluidly the sentences are emitted.
Only how good they are.
It's easy to believe in "flow" if you can't feel the difference between a dead sentence and a living one Or see the ambiguities you're accidentally creating.
In other words, "flow" is often a synonym for ignorance and laziness.
It's also a sign of haste, the urge to be done.
Why do I say all this?
Because so many writers worry that their writing isn't flowing.
They worry that they can't live up to expectations—to a cultural illusion—
And they get in trouble with themselves because of it.
Don't underestimate how hard it is to discard a cliché like "flow," even in your own assumptions.
Like so much else about writing well,
Getting rid of useless, even harmful, ideas is hard work.
What lurks behind "flow"?
Above all, the idea of naturalness.
"Natural" is a word that invites suspicion.
It should always present itself in quotation marks,
A sign that its meaning is slippery.
Humans can justify almost anything by calling it natural.
Naturalness is the pervasive myth—the one to root out of your head.
There's nothing natural about writing except the tendency to assume that it's natural,
Thanks to a false analogy with talking.
The connection between talking and writing is nearly as complex as the connection between reading and writing.
You probably don't remember learning to talk as a child.
You probably do remember learning to shape letters and spell words.
Talking is natural.
Writing is not.
Most children can say words before they're two and speak in sentences before they're three.
They can sing the alphabet song almost as soon as they can sing.
But they can't write the alphabet until they can hold an instrument of writing.
It may seem strange that the manual dexterity needed to hold a pencil—or use a keyboard—comes later than the lingual and mental dexterity needed to speak.
But it does.
In writing, there's always a separateness,
The sense of manipulating a tool for producing words at arm's length,
Out there at the ends of your fingers,
Unlike speaking, which arises invisibly from within, like thought and breath.
In writing, there's a psychological separateness too,
The sense of watching yourself think and thinking about it as you do,
A self-consciousness that interrupts the movement of your thoughts
If you experience it while talking.
Humans have a language instinct
But not necessarily a writing instinct.
The difference between talking and writing
Is the difference between breathing and singing well.
It takes years of work to write well,
And only part of that is learning to type.
"Natural," like flow, is also an effect in the reader's mind.
It doesn't describe the act of writing.
It describes the effect of writing.
And like "flow," "natural" is one of the words behind writer's block.
So let's suppose there's no such thing as writer's block.
There's loss of confidence
And forgetting to think
And failing to prepare
And not reading enough
And giving up on patience
And hastening to write
And fearing your audience
And never really trying to understand how sentences work.
Above all, there's never learning to trust yourself
Or your capacity to learn or think or perceive.
People will continue to believe that writing is natural.
This harms only writers who believe it themselves.
And yet good prose often sounds spoken,
As if the writer—or the reader reading aloud—were _saying_ the sentences.
(This isn't the same as sounding colloquial.)
But the arc of education—and the arc of emulation—are usually
Away from spokenness and toward the unspeakable,
Toward longer, more convoluted sentences
Using more elaborate syntax and more jargon-like diction.
There's nothing natural about making sentences that sound spoken,
No matter how natural they sound.
What are their characteristics?
They're fairly short.
They're rhythmic, often with the rhythms of actual speech.
The diction is simple—very few multisyllabic words.
So is the construction—almost no suspended phrases or dependent clauses.
This simplicity makes the rhythm more perceptible.
There's also an acute awareness of the listener's attention and understanding,
A sense of contextual alertness, and a vivid sense of the unspoken.
These are all qualities worth building into your prose.
They must be created, discovered, revealed, constructed.
They don't appear "naturally."
It's always worth asking yourself if you can imagine _saying_ a sentence
And adjusting it until you can.
Just as it's always useful to ask yourself, "What _exactly_ am I trying to say?"
The answer to that question is often the sentence you need to write down.
When your prose begins to stiffen and your thoughts get stuffy,
It's sometimes worth reworking the piece you're writing as if it were
A letter or a long e-mail to a friend,
Someone who knows you well but hasn't seen you in a while.
What happens?
The prose relaxes, the sentences grow more informal.
You remember to use contractions,
Even the words grow shorter.
Suddenly things are clearer and simpler and more direct, as if they were being spoken.
But something else happens too.
There's suddenly a wider variety of tone, an emotional latitude,
A sense that the reader will be able to fill in the gaps,
Even the possibility of humor.
Why the difference?
It isn't the change in genre.
It's the change in the reader.
You're writing to someone who knows you, who understands your allusions,
Your patterns of speech, who's quick and empathetic
In reading your thoughts and feelings, whether they're spoken or unspoken.
What makes this reader valuable is a sense of connection and kinship,
An intuitive grasp of what you say and don't say.
You can make any piece feel like an informal letter
By using the generic characteristics of an informal letter.
But it's far easier to get that feel
By writing to the reader you imagine reading it.
The reader you construct in your imagination
Changes the way you write almost without your noticing it.
Behind "flow" there's something else,
Even something ecstatic—
The priority of thoughts over sentences.
Thoughts leaping ahead, words barely keeping up,
A hectic chase.
Or the other way around,
Sentences spinning out of each other, one after the next,
Phrase eliciting phrase, words—if not sentences—rushing ahead of thought.
It feels like inspiration.
We've all had these moments.
They're enticing.
The mistake is overvaluing them.
You have an effusion one day.
It spawns a piece.
As the piece evolves, you try to protect those original, effusive sentences.
Only to realize, at last, that what you're writing won't come together until they've been removed or revised.
What were you trying to protect?
The memory of the excitement you felt when those words "came to you."
(Where did they "come" from?)
You were protecting the memory of the excitement of really concentrating,
Of paying close attention to your thoughts and, perhaps, your sentences,
The excitement of feeling the galvanic link between language and thought.
That excitement matters, and the memory of it is worth preserving,
Even if those sentences aren't.
Concentration, attention, excitement, will be part of your working state.
Daily.
Flow, inspiration—the spontaneous emission of sentences—will not.
That distinction is worth keeping in mind.
Write consciously, deliberately.
Learn how to get out of trouble.
Learn how to free yourself when you're stuck.
Learn how to know what you're doing when you're making sentences.
The workings of your unconscious mind,
The current of your subterranean thoughts and intuitions,
The flickerings of insight and instinct—
These will always surface, if you write clearly enough to let them.
But they're only some of the tools of your daily work,
Which is making sentences.
The most damaging and obstructive cluster of ideas you face as a writer are nearly all related to the idea of "flow."
Like "genius."
And "sincerity."
And "inspiration."
Distrust these words.
They stand for cherished myths, but myths nonetheless.
"Inspiration" is what gets you to the keyboard,
And that's where it leaves you.
Inspiration is about the swift transitions of thought,
Sudden realizations,
Almost all of them carefully prepared for by continuous thinking.
Inspiration has nothing to do with the sustained effort of making prose.
You'll have many serendipitous moments while writing.
You'll learn to expect them.
But "inspiration," as it's commonly used, is just another word for "flow."
Think of all the requirements writers imagine for themselves:
A cabin in the woods
A plain wooden table
Absolute silence
A favorite pen
A favorite ink
A favorite blank book
A favorite typewriter
A favorite laptop
A favorite writing program
A large advance
A yellow pad
A wastebasket
A shotgun
The early light of morning
The moon at night
A rainy afternoon
A thunderstorm with high winds
The first snow of winter
A cup of coffee in just the right cup
A beer
A mug of green tea
A bourbon
Solitude
Sooner or later the need for any one of these will prevent you from writing.
Anything you think you need in order to write—
Or be "inspired" to write or "get in the mood" to write—
Becomes a prohibition when it's lacking.
Learn to write anywhere, at any time, in any conditions,
With anything, starting from nowhere.
All you really need is your head, the one indispensable requirement.
If you consciously shape your writing,
Know its ins and outs, understand its subtleties—
Then you know exactly what you're doing
And are therefore manipulating the reader.
Which, of course, you are.
We hate the thought of being manipulated,
And yet reading means surrendering to the manipulations of the author's prose.
This is an experience we love, love so much, in fact,
That we hope to be able to manipulate readers ourselves someday.
Readers usually choose not to think of it that way.
They prefer not to think of it at all.
But you should.
One of the few sad parts about writing is that it's almost impossible to surrender to the manipulation of your own prose.
(It's just as well.)
What prevents it is the memory of all the choices you've made
Hovering around every sentence you've written.
In writing, it's impossible to express sincerity sincerely.
That is, just by being sincere.
You really mean what you mean to say.
You feel an intense sincerity burning inside you.
And yet your sentences feel choked or formulaic.
Writing can't convey sincerity—or any other emotion or mood in the writer—simply because you feel it.
We believe so strongly in sincerity and naturalness of expression in writing that we're almost unable to see how false this belief is.
If you want the reader to feel your sincerity, your sentences have to enact sincerity—verbally, syntactically, even rhythmically.
They have to reveal the signs of sincerity—a modesty and directness—
Just as you do when you're talking sincerely.
If you speak sincerely with someone
But in a voice and manner that suggest you're being ironic,
Who would believe you're sincere?
Sincerity is a dramatic role for you and your sentences.
That makes it sound insincere.
But the apparently insincere manipulation of language is the tool that persuades us of your sincerity.
There is no simple, sincere, "natural" space or role for you to occupy in your writing.
Writing is always a gesture requiring your dramatic presence, no matter how subtle—
A presence made up of rhetorical choices:
Choices about who you are in relation to your subject and your reader,
Choices about your presence in the piece, about diction, structure, and the rigor or casualness with which your sentences are constructed or linked.
The emotional power the reader feels
Depends on how clearly you know what your words are doing.
That clarity isn't natural.
It's artificial, the result of hard work.
_You_ be the narrator.
Let _us_ be the readers.
You'll discover that being the narrator is not the same as being yourself.
It's a role, and a dramatic one.
Absorb it and inhabit it.
You're always building a habitation in your prose,
A place from which you speak to the reader.
You're never merely, sincerely yourself.
The question then becomes, who are you?
That's a question every piece needs to answer.
Novelists, short-story writers, and poets understand the gesture in their writing.
They know they occupy a dramatic role and a rhetorical space.
They're rarely afraid to _be_ the narrator or actor, to perform the act of telling a story,
Even if they're telling it under the guise of their own name.
The sense of who you are, what role you choose to play,
What gesture you make toward the reader—
These things are far more important than ideas of
"style" or "voice."
Asked to write short sentences, writers often say,
"But what about my style? What will that do to my voice?"
As if they're sure they have one.
"Style" and "voice" are passive constructs,
Markers of individuality, bow ties of self.
They have more to do with what the writer makes of himself
Than how the reader experiences his prose.
The idea of "voice" at least implies a notion of dramatic presence,
A sense of the writer's gesture.
But what's a writer's "style"?
Style is an expression of the interest you take in the making of every sentence.
It emerges, almost without intent, from your engagement with each sentence.
It's the discoveries you make in the making of the prose itself.
We assume that style is self-expression.
It can be, but only in this sense:
It's the fusion of your command of language and your commitment to your own intent,
Even as your intent shifts under the weight and opportunity
Of the discoveries you make as you work,
Discoveries that are linguistic, conceptual, structural, imaginative.
This doesn't sound like a useful or conventional definition of style Or much like self-expression.
But it does clarify an important thought:
"Style" shouldn't linger in your awareness.
You don't need to think about style.
It's as likely to appear in the character of your thinking,
The shape of your ideas, your sense of humor or irony,
As it is in any "stylistic" markers in the prose itself.
But this will only be true if your prose is clear enough to reveal the character of your thinking, the shape of your ideas, and your sense of humor or irony.
Where ambiguity rules, there is no "style"—or anything else worth having.
Pursue clarity instead.
In the pursuit of clarity, style reveals itself.
Your clarity will differ from anyone else's without your intending to make it differ.
Years later, looking back over your collected works,
You can contemplate your style at leisure.
But for now you have more important things to think about.
Like revision.
All writing is revision.
That's not what you learned in school.
In school you learned to write a draft and then revise.
But imagine this:
You begin to compose a sentence in your head.
You don't write it down.
You let the sentence play through your mind again.
(It's only six words long.)
You replace one or two of the words.
You adjust the rhythm by changing the verb.
You discard the metaphor.
You decide you like the sentence.
You write it down.
Is this composition?
Or revision?
It's both.
Composing a sentence always involves revision
Unless you write down the words of a sentence exactly as they pop into your head.
And why would you do that?
You look at the sentence you've written down.
You choose a simpler noun and a stronger verb.
Is this revision merely because the sentence was already written down?
Or is it composition too?
It makes no difference.
As it's taught in school, revision means little more than
Correction after the fact
(Or possibly proofreading, a completely forgotten but invaluable skill).
This becomes clear when students ask if they can revise a piece.
They mean can they fix what already exists,
Adjust a sentence here or there,
Move a couple of paragraphs.
They never mean
"Let me reimagine the piece completely,
Beginning with my approach to the subject
And keeping only the handful of sentences that actually worked."
Writers at every level of skill experience the tyranny of what exists.
It can be overwhelming—the inertia of the paragraphs and pages you've already composed, the sentences you've already written,
No matter how rough they are.
Whether you love what you've written or not,
Those sentences have the virtue of already existing,
Which makes them better than sentences that don't exist.
Or so it seems.
And yet because they're rough and provisional,
They form an overlapping grid with unsuspected gaps,
A network that seems to defy revision.
Fixing one sentence almost always means fixing another and then another
As though revision were an infinitely recursive act.
There's almost never enough time for revision, if revision comes after the fact
And if it's really revision.
So let's change things.
Try this instead:
Revise at the point of composition.
Compose at the point of revision.
Accept no provisional sentences.
Make no drafts
And no draft sentences.
Bring the sentence you're working on as close to its final state as you can
Before you write it down and after.
Do the same for the next sentence
And right on through to the end.
Think of composition and revision as the same thing,
Different versions of thinking,
Philosophically indistinguishable.
The usual premise is that composition brings something new to the page
And revision fixes it.
This is a useless distinction, and it creates a false sense of priority—
A belief that the writer's real work is making newness out of nothing,
As if creativity only takes place where the ink stops and the blank page begins,
Where the cursor sits blinking.
As if newness couldn't originate between sentences or within a sentence.
As if revision were essentially secondary and uncreative.
Revision (or composition) just as often means
Writing from the middle—from the many middles—and not the end.
The end—where the page goes blank—has no priority.
You're not reading, picking up where you left off.
You're writing.
You left off everywhere at once.
You may have left off in the middle of a paragraph many pages ago,
And everything since has been a detour.
You may find the path you're looking for only by taking a detour.
As you begin rereading a piece you're working on,
Don't hurry to resume work at the end.
Treat every sentence you read as if it were still under revision.
Composing and revising at the same time won't be easy at first.
You'll make sentences that seem finished and then find flaws in them.
Finding flaws is how you learn to make better sentences.
Enjoy it.
You can't prevent yourself from repeating a mistake you haven't noticed.
You'll have to read your work many, many times to find all the problems embedded in it.
Even experienced writers have to do this.
Some flaws do a wonderful job of hiding.
So, you'll be revising each sentence as you compose it.
Composing each sentence as you revise it.
And you'll read and reread every sentence you make many dozens of times,
Sifting out problems as they materialize in front of you.
You'll be looking for flaws.
But also for opportunities—and for missed opportunities:
Things you might have said, ideas you might have developed,
Connections you might have made.
Revision isn't only the act of composition.
Revision is thinking applied to language,
An opening and reopening of discovery,
A search for the sentence that says the thing you had no idea you could say
Hidden inside the sentence you're making.
Revision is the writer's reading,
The habit of noticing choices,
Noticing that every sentence might be otherwise but isn't.
Language writhes with urgency to be saying something.
Your job is to understand and control that urgency.
At first, what you mean to say will emerge
By setting aside the things you don't mean to say
As well as trying to say the thing itself.
Learning to do this will take some time.
You'll feel as though you've bogged down,
As though you'll never find your way to the end.
But you'll also find yourself making discoveries you never could have predicted,
Finding thoughts you never knew existed because they didn't exist
Until you were exploring sentences for their implicit possibilities.
With practice, this will become a more efficient and more creative way to write,
A way of discovering what you didn't know you could say,
Which also means learning something important about yourself.
It's also a vastly more interesting way to write.
With practice, it will become more efficient in every sense,
Faster, more accurate, and far more direct than the way you were taught to write.
You'll learn to trust it implicitly
And yourself as well.
Where do sentences come from?
How do they reveal themselves in your thinking?
We like to think we move from thought to expression,
With no more fuss than a handshake.
Sometimes you know just what you want to say,
And you find the words to say exactly that.
But just as often what you want to say emerges as the sentence takes shape.
The thought isn't primary or absolute.
The thought is only a hint.
Language offers guidance and resistance both.
The sentence _becomes_ the thought by bringing it fully into being.
We assume that thought shapes the sentence.
But thought and sentence are always a collaboration,
The sum of what can be said and what you're trying to say.
In writing you love, the sentences are endlessly various.
How do you find that variety in your own prose?
One way is by looking for sameness, uniformity, and working against it.
You wouldn't repeat the same words over and over again,
So why repeat so many sentence structures?
Better to look for that variety as you're thinking and writing,
Giving as much attention to the shape of the sentence
As you do to what you're trying to say.
Sometimes a rhythm insinuates itself.
You find yourself listening for echoes, opportunities.
Sometimes you find yourself watching the traces of words,
Phrases, memories, flitting through your mind.
Each of these can engender a sentence, offer a shape.
Be responsive to the variations that present themselves as you think.
Soon, you'll grasp that sentences originate and take their endless variety
From within you, from your reading,
Your tactile memory for rhythms,
Your sense of the playfulness at the heart of the language,
Your perception of the world.
You'll learn to let a single word, a simple rhythm,
A thing you've noticed,
Generate a sentence you didn't expect.
This requires a change in your mental habits.
And a reconsideration of how you work.
In the outline and draft model of writing, thinking is largely done up front.
Outlining means organizing the sequence of your meanings, not your sentences.
It derogates the making of sentences.
It ignores the suddenness of thought,
The surprises to be found in the making of sentences.
It knows nothing of the thoughtfulness you'll discover as you work.
It prevents discovery within the act of writing.
It says, planning is one thing, writing another,
And discovery has nothing to do with it.
It overemphasizes logic and chronology
Because they offer apparently "natural" structures.
It preserves the cohesiveness of your research
And leaves you with a heap of provisional sentences,
Which are supposed to sketch the thoughts you've already outlined.
It fails to realize that writing comes from writing.
You're more likely to find the right path—
The interesting path through your subject and thoughts—
In a sentence-by-sentence search than in an outline.
The standard model wastes the contemplative space of writing.
Can you think all the good thoughts in advance?
Outlining has at least as much to do with rescuing the writer from himself
As it does with planning the shape of the piece.
It's meant to free you from thinking as you write.
It provides a catwalk across the open spaces in your mind
To keep you from falling into rumination as you write.
You'll never know what you think until you escape your outline.
The purpose of an outline is also to conserve your material, to distribute it evenly so that _meaning_ discloses itself near the end.
Here's a better approach.
Squander your material.
Don't ration it, saving the best for last.
You don't know what the best is.
Or the last.
Use it up.
There's plenty more where that came from.
You won't make new discoveries until you need them.
What writers fear most is running out of material.
The sound of a writer's fears is the sound of nothing—
No typing, no clicking, no scratching of pens.
But you can only run out of material
If you haven't been thinking or noticing.
Try this:
No outline.
Research, reading, noticing, interviewing, traveling, paying attention, note taking—all the work you do to understand the subject, whatever it is, whatever kind of piece you're writing.
Reread your notes, and take notes on them.
And again.
Take notes on your thoughts.
Most of all, take notes on what _interests_ you.
Be certain you've marked out what _interests_ you.
Don't make an outline from your notes.
Don't turn your notes into a road map for the sentences to come.
Reread your notes.
No matter how long or short they are.
Then think.
And think again.
Learn to be patient in the presence of your thoughts.
Learn to be equally patient in the presence of a new sentence or a phrase you like.
Let yourself pause and work on that sentence.
In your head.
Don't write it down.
Be patient.
Pay attention to everything you're thinking.
Notice your thoughts,
See if you can feel your awareness illuminating them.
If you're paying attention, you'll notice that some of your thoughts interest you and some don't.
How can you tell?
You'll stop and rethink the thought,
Pause in its presence.
Let the thoughts that interest you distract you.
Ask yourself about them.
Why do they interest you?
What were you thinking about before they appeared?
Then come back to the main sequence,
Unless you've discovered a better main sequence
By following a thought you're interested in.
Don't try to distinguish between thinking and making sentences.
Pretend they're the same thing.
Don't rush your thinking.
Don't rush to make sentences.
See what happens when you try to put words to a thought that interests you.
See what words the thought itself is presenting and try making a sentence out of them,
A sentence like the ones we've been talking about, with rhythm and clarity and balance.
_Not_ a volunteer sentence.
See if the thought you're interested in becomes sharper and clearer by making a sentence from it.
It may become more obscure.
What does that tell you?
Don't panic, keep working at it.
If you make a sentence while thinking,
It doesn't mean you have to make more sentences immediately.
You can go back to thinking and see what the business of making a sentence stirred up in you.
It may have dislodged other thoughts, other connections.
No one will teach you how to wait while you think or what to wait for while you're thinking.
You'll have to teach yourself.
Above all, you'll have to teach yourself to be patient.
Trying this once or twice won't do.
It's a skill, not an instinct.
You may have to try it in small increments,
And you may have to cling to a partial outline for a while.
That's okay,
As long as you're prepared to abandon it.
It's a map of the places you may end up not going.
Practice,
And you'll learn to trust the agility and capacity of your thinking.
You'll learn that you don't have to set aside inviolate chunks of time to think.
You'll find yourself working—thinking, making sentences—in the brief intervals of your ordinary life,
In increments no longer than a few seconds.
How long does a thought take?
Or a sentence?
Your thinking will help you discover what interests you in the subject you've chosen
No matter how indirectly or elliptically or obliquely connected it is.
Any thread, any perception, any link, any phrase, any intuition.
You may discover an orderly way to go about this,
Or you may move through your thoughts in ways you can't predict.
It makes no difference.
Resist the temptation to start organizing and structuring your thoughts too soon,
Boxing them in, forcing them into genre-sized containers.
Postpone the search for order, for the single line through the piece.
Let your thoughts overlap and collide and see what they dislodge.
How do you begin to write?
Look for a sentence that interests you.
A sentence that might begin the piece.
Don't look too hard.
Just try out some sentences.
Lots of them.
See how they sound.
Do any of them sound first?
Discard them readily, easily, with no sense of loss,
Then try out some more.
This is important.
Get used to discarding sentences.
You're holding an audition.
Many sentences will try out.
One gets the part.
You'll recognize it less from the character of the sentence itself than from the promise it contains—promise for the sentences to come.
This will get easier with practice.
Don't be alarmed if it takes a day or two of trying out sentences
Before you find the promising one.
It may only be promising enough to lead you to the real first sentence.
Be casual about this.
Look for a sentence that interests you,
A sentence whose possibilities you like because of the potential you see in its wake.
I don't mean a "fantastic first sentence" or one that sounds "introductory."
I don't mean a sentence that sounds first because it sounds like other first sentences you've read.
I don't mean the kind of first sentence teachers sometimes talk about—the one that _grabs_ the reader.
The reader doesn't need grabbing.
She needs to feel your interest in the sentence you've chosen to make.
Nothing more.
What makes the first sentence interesting?
Its exact shape and what it says
And the possibility it creates for another sentence.
A beginning needs no éclat, no cleverness, no tricks,
No coyly hidden awareness of where the piece will take us.
The opening sentence is only creating an opening for the next sentence.
But there's also nothing incidental about that first sentence.
You—your role as a writer, the role you construct, your presence to the reader—you and your first sentence begin together.
You want to _begin_ the piece, not _introduce_ it, which is the difference between a first sentence already moving at speed and a first sentence that wants to generalize while clearing its throat.
The beginning is one sentence long.
It leads to the next sentence and is largely indistinguishable from other sentences leading to the next sentence.
So many writers stumble by making the first sentence try to do too much
And end up making every sentence try to do the same.
Out of all the possibilities created by the first sentence,
Make a second sentence, full of more possibilities, even disconnected ones.
See if you can write the sentence that _arises_ from the first sentence,
Not the sentence that _follows_ from it,
Even if that means the second sentence lies at some distance from the first.
The second sentence you write may turn out
Not to be the second sentence after all.
It may be the ninth.
The sentence isn't burdened by the question, where will it go?
The piece is now two sentences long.
Not two sentences plus the missing pages that haunt you.
The next step is to make the piece three sentences long.
Don't worry about trajectory or sequence.
Don't look further ahead than two or three sentences.
And don't _plan_ those sentences.
Write them in your head instead.
Resist the temptation to rush ahead to see where they're pointing.
What matters isn't where they're pointing
But what interests you in the sentence you're making,
Which you may have to discover as you make it.
Don't steer the sentences where you want them to go.
See if you can follow them there.
They may be going in many different directions at first.
Don't get trapped by the thought of writing sequentially.
If you uncover a sentence that seems to belong to an earlier passage,
Go back to that passage and work there.
You have no idea what you're going to say
Until you discover what you want to say
As you make the sentences that say it.
Every sentence is optional until it proves otherwise.
Writing is the work of discovery.
_Imagine_ sentences instead of writing them.
Keep them imaginary until you're happy with them.
An imaginary sentence somehow feels less _bound_ than one you've written down.
Making sentences soon ceases to be a separate act
And becomes part of the process of thinking.
Aren't you already thinking in sentences?
You'll discover that the act of making sentences in your head—
Composing and revising at the same time,
Making them sharper and more accurate—
Tends to uncover thoughts you didn't know you had,
Allowing you to say things you didn't know you knew how to say
In sentences stronger than you knew you could make.
Soon you find yourself _expecting_ to say things you didn't know you knew how to say.
You'll get into the habit of surprising yourself.
The reader will feel the freshness of the discovery in the prose
Because the writer almost always reveals the excitement of making a discovery
In the rhythm and the vividness of the sentences themselves.
But you distrust your memory.
You're afraid you'll forget the sentences you're imagining.
Why would you?
They're important.
Write them down, just in case.
And then go back to thinking—imagining sentences and their possibilities,
Feeling your way into each new opportunity.
But imagine the sentence in its entirety—and the next one too—before writing anything down.
Sit back from the keyboard or notepad.
Sit back, and continue to think.
That's where the work gets done.
You may glimpse where the piece might go.
You may even see your way through to the end.
If you do, you'll feel a fresh anxiety about forgetting,
As if the forest were closing in on the path you see before you can reach
The warm, welcoming cottage where writing is over.
But thought isn't as fleeting as you think, nor does it come completely unbidden.
If the thought was worth having, you'll rediscover it or find a better one.
The fear of forgetting and the rush to be done are closely related.
You'll learn to trust your memory as you work,
Though it isn't even a matter of trusting your memory.
You'll realize that thinking and remembering are almost indistinguishable.
You're not only imagining sentences you want to write down.
You're also reexploring your subject, sifting your research
And all the elements that make up your subject
Even as you're imagining sentences.
Soon the distinction between thinking about your subject and
Thinking about sentences vanishes.
You'll have stopped making sentences in quarantine,
In the special ward set aside for sentence making once the outline is finished,
The way you were taught in school.
Instead, writing becomes intrinsic to the act of thinking,
Completely intertwined with it.
You're also learning to trust the ability to work in your head
And learning how your mind works,
Which is something you may not have noticed before.
We're always hastening to be done writing,
But we're also hastening to get out of the presence of our thoughts.
Everything about thinking makes us nervous.
We don't believe there's much of value to be found there.
We don't know when we'll come to the end of our thoughts,
But we think it may be soon.
Why?
Your mind is silent yet filled with voices and uncertainty.
The uncertainty you feel is one of the places sentences will come from,
And experience will make your uncertainty more certain.
Stop fearing what you'll find as you think.
Give yourself over to this experiment.
Your intentions will diverge from themselves.
Your starting point may lead to places you didn't imagine,
Places that ask you to reconsider your starting point.
You may feel yourself clinging to your original intention.
Why?
Because it came first?
Why not follow the crosscurrents of your thinking
And see where they lead?
I don't mean follow them blindly.
Allow your thinking to adjust your intentions in the light of your discoveries.
This may mean relinquishing your original intention
If you find a better one as you write.
The piece you're writing is simply the one that happens to get written.
If you'd begun another way, made a different turn, even started in a different mood,
A different piece would have come into being.
The writer's world is full of parallel universes.
You discover, word by word, the one you discover.
Ten minutes later—another hour of thought—and you would have found your way into a different universe.
The piece is permeable to the world around it.
It's responsive to time itself, to the very hour of its creation.
This is an immensely freeing thing to understand.
It liberates you from the anxiety of sequence,
The fear that there's only one way through your subject,
Only one useful approach.
Learn to accept the discontinuity between yourself and what you write,
The discontinuity between your will, your intention, your plan
And the discoveries you make as you work.
Abandon the idea of predetermination,
The shaping force of your intention,
Until you've given it up for good.
Bring your intentions, by all means, but accept that the language we use
Is a language of accidentals, always skewing away from the course we set.
This is something not to mourn but to revel in—
Not only for the friction and sideslip inherent in the language
But for freeing us from the narrowness of our preconceptions.
Imagine this:
The piece you're writing is about what you find in the piece you're writing.
Nothing else.
No matter how factual, how nonfictional, how purposeful a piece it is.
Sooner or later, you'll become more interested in what you're able to say on the page and less interested in your intentions.
You'll rely less on the priority of your intentions and more on the immediacy of writing.
It may sound as if I'm describing a formless sort of writing.
Not at all.
Form is discovery too.
It's perfectly possible to write this way even when constricted by
A narrow subject, a small space, and a tight deadline.
How do you decide what works?
What do you do when your sentences seem to waver in quality and value before your eyes?
You read what you've written, and it looks good.
You read it again, and it looks bad.
You read it a third time, and now you can't tell.
Your emerging skill as a reader will help.
You'll read your sentences against the backdrop of all the rest of your reading.
You'll get better at examining your own choices—the ones you've already made
And the ones you see waiting to be made as you reread what you've written.
Before long you'll notice possibilities you would have been blind to once upon a time.
You'll see that some of your sentences are still conjectural.
You'll stop seeing only the narrow procession of the sentences you've made
And start noticing the thoughts and implications surrounding them.
You'll become strangely aware of what you've chosen not to say
And how that affects the sound of your sentences.
Writers too often respond to whole chunks of what they've written,
Whether it's a paragraph or the entire piece.
You read it and think, "This is terrible," and throw the whole thing away with a sinking heart.
You read it and think, "This is terrific," with a smile and set it aside, done.
Both responses end your engagement with what you've written.
There's a better way.
Start by fixing the sentences that need fixing.
(There will surely be some. If you can't find any, look harder, or begin rereading this book.)
Explore the possibilities that open up.
Continue making small, incremental changes
At the sentence level wherever you see problems,
With no priority given to the beginning or end of the piece.
Listen for rhythm.
Keep reading and rereading what you've written.
Anything that strikes you—anything that causes a subtle, inward sensation of discomfort, an inner alarm, no matter how faint—stop there and figure out what's going on.
It may have to do not with the sentence itself
But with its relation to some other sentence.
There's no rush.
It's surprising where these incremental changes lead,
How they solidify what seems to be unstable,
How they open up directions you hadn't glimpsed before.
You may find that the most important section of the piece—a section you haven't written yet—emerges from the gap created when you break a long sentence in two.
It's true that the simplest revision is deletion.
But there's often a fine sentence lurking within a bad sentence,
A better sentence hiding under a good sentence.
Work word by word until you discover it.
Don't try to fix an existing sentence with minimal effort,
Without reimagining it.
You can almost never fix a sentence—
Or find the better sentence within it—
By using only the words it already contains.
If they were the right words already, the sentence probably wouldn't need fixing.
And yet writers sit staring at a flawed sentence as if it were a Rubik's Cube,
Trying to shift the same words round and round until they find the solution.
Take note of this point: it will save you a lot of frustration.
This applies to paragraphs too.
You may not be able to fix the paragraph using only the sentences it already contains.
How soon will you be getting good?
Why not ask how soon you'll be getting clear?
Look for improvement wherever you find it,
And build on every improvement.
But don't look for too much improvement all at once.
Finding a flaw is an improvement.
So is discarding an unnecessary word or using a stronger verb.
Writing even one clear, balanced, rhythmic sentence is an accomplishment.
It prepares the way for more good sentences.
It teaches you how you respond, inwardly, to a successful sentence of your own making.
If you write a good sentence, how will you know it's good?
You may _know_ it's good, feel certain about it.
But you're likelier to sense an inward difference,
A subtle feeling telling you this sentence isn't the same as the others.
Even beginning writers notice this.
Learning that feeling is important.
It's a guide and an incentive to making more good sentences.
This doesn't happen in a void.
It happens against the backdrop of your constant reading,
Your unending exposure to superb sentences.
And then one day
You'll write a sentence that says more than its words alone can say.
You'll know that it says what you mean without having said it,
And you'll know that the reader knows it too.
This will sound impossible until you've done it once.
Then you'll see how possible it is, and how inviting.
It lets the reader complete the thought.
It sets an echo in motion.
This is writing by implication.
Don't let the success of a sentence or a paragraph or a piece deter you.
Some writers freeze, fearing the next one won't be as good.
Some writers polish a single paragraph until it glows,
Fearing that the next paragraph will ruin it somehow.
Accept it: you'll surely fail again and just as surely succeed.
There's nothing linear or steady in your growth as a writer.
And the moment you find yourself getting good at one thing,
It's time to push on into unsafe terrain.
To do this work requires a balance between your commitment to the sentences you're making
And the knowledge that each of them could be otherwise.
Some should be otherwise; some shouldn't.
Make that simple distinction again and again
And you'll get good at making that simple distinction,
Which is the foundation of writing.
You'll learn to live somewhere between certainty and flux.
You'll learn to remember that your sentences don't acquire their final inertia
Until _you_ release them.
There's nothing permanent in the state of being written down.
Your sentences, written down, are in the condition of waiting to be examined.
You commit yourself to each sentence as you make it,
And to each sentence as you fix it,
Retaining the capacity to change everything and
Always remembering to work from the small-scale—
The scale of the sentence—upward.
Rejoicing and despair aren't very good tools for revising.
Curiosity, patience, and the ability to improvise are.
So is the ability to remain open to the work and let it remain open to you.
Don't confuse order with linearity.
You'll find more than enough order in the thoughts and sentences that interest you.
By order I mean merely connections—
Some close, some oblique, some elliptical—
Order of any kind you choose to create, any way you choose to move.
Don't give in to the memory of your school writing,
The claustrophobic feeling that there's only one right order of arguing, proving, demonstrating,
The assumption that logic persuades the reader
Instead of the clarity of what you're saying.
There's little actual logic in good writing.
There's a current of thoughts and ideas and observations.
Some may be linked by evidence.
One point may substantiate or corroborate another.
But what passes for logic or argument is usually little more than a succession of ideas
Connected mostly by proximity and analogy.
Writing doesn't prove anything,
And it only rarely persuades.
It does something much better.
It attests.
It witnesses.
It shares your interest in what you've noticed.
It reports on the nature of your attention.
It suggests the possibilities of the world around you.
The evidence of the world as it presents itself to you.
Proof is for mathematicians.
Logic is for philosophers.
We have testimony.
The logic of writing, as you learned it in school,
Turns out to mean little more than an obsession with transition
And the scattering of rhetorical tics—overused, nearly meaningless words and phrases.
In fact.
Indeed.
On the one hand.
On the other hand.
Therefore.
Moreover.
However.
In one respect.
Of course.
Whereas.
Thus.
These are logical indicators. Emphasizers. Intensifiers.
They insist upon logic whether it exists or not.
They often come first in the sentence,
Trying to steer the reader's understanding from the front,
As if the reader were incapable of following a logical shift in the middle of the sentence,
As if the sentence had been written in the order the writer thought of the words,
Without any reconsideration.
These words take the reader's head between their hands and force her to look where they want her to.
Imagine how obnoxious that is,
That persistent effort to predetermine and overgovern the reader's response.
These phrases also obscure the content of your sentences.
If a piece is truly assured in its order, no matter how connected or oblique,
It needs no logical indicators.
It will be obvious when one sentence negates or affirms another.
These words betray the writer's anxiety,
The false belief that proof is necessary and possible,
That persuasion is just a "thus" away.
They also try to bolster the apparent authority of your piece
By echoing the apparent authority of other people
Who can't write and who distrust their own thinking.
A simple experiment:
Try removing "but" wherever you can,
And see if the sense of negation or contradiction—
The feel of a reversal taking place—isn't still present.
"But" is always preferable to "however,"
Except in the rare cases where "however" is preferable to "but,"
Which has everything to do with rhythm, formality, and context.
And yes, you may begin a sentence with "but."
Another example of linearity: chronology.
Chronology will always offer itself as the "natural" means of telling a story or recounting an event.
But there's nothing "natural" about moving chronologically in writing.
It's a rhetorical choice among many choices, and usually a dull one at that.
It feels like a privileged choice only by analogy with the sequence of our own lives.
Chronology in our lives is "natural" in a limited sense.
We live on time's arrow,
And our days and nights follow the clock.
But there's no such thing as thought's arrow
Or mood's arrow
Or memory's arrow.
Consider your interior life—what you feel and think and the ways you remember.
How much of it is chronological in order?
Brief segments may be imbued with the orderliness of time.
But in their relation to each other the elements of our internal lives are more likely to be associative, even dissociative,
Linked in ways that have nothing to do with the clock or the day by day of life itself.
Writing is often an appeal not to the order of our chronological lives But to the order of our internal lives,
Which is nonchronological and, in fact, unorderly.
Resist chronology.
It will always try to impose itself.
Break the flow of time once it begins.
Better yet, resist it from the start.
If there's a pleasure in seeing time revealed as we read,
There's an equal pleasure in seeing it suspended, violated,
And broken as only writing can do.
Use the simple past tense—
Avoiding the layering of several pasts—
And give the reader clear temporal clues when needed.
It takes a skillful writer to make the ordinary motion of time engaging.
Narrative is harder to write than almost anything else.
Novels contain far less chronological narrative than you think.
Take a page from almost any novelist.
Look carefully at each sentence.
How many propel the story forward in time?
And how many are devoted to enriching our sense of place and character?
Our lives are full of endings.
The sun goes down every day.
We ask for the check.
Eventually it comes.
How broad a hint does it take to make a reader who lives on a planet full of endings
Feel the end of your piece approaching?
You've already written the ending you need.
You didn't see it, because you were looking for something more dramatic.
The reader saw the end coming from miles away.
When I say resist chronology, I also mean resist the chronology of observation.
Why report on events in the order you observed them?
Why stick to the sequence in which things happened
Unless there's a good reason for it?
I also mean the chronology of the evidence you gather,
The way it tends to clump together in your notes and pieces,
Lumps of this and lumps of that as if every word
Adhered stickily to a cohort of equally sticky words:
The transcript of the interview, the quotation from an article,
The sequence of your impressions as you arrive on the scene.
Your job isn't to arrange chunks of evidence,
Chunks of the world in the order you gather them.
Your job is to _atomize_ everything you touch,
To dissect your evidence into its details and particulars and Resist the inherent jargon of your subject,
Breaking apart every clod of words you come across.
Your job is to undo the adhesiveness of the evidence you've gathered,
Its tendency to clump into indissoluble units.
Dissolve them.
Pay attention only to what interests you in it.
Break the complexity of what you've learned into the very small pieces of a mosaic
Shaped not by the clumping of evidence but by your conscious decisions as a writer.
Use the one detail you need as you need it.
Beware of the way it sticks to other details.
Why reproduce the whole scene when only one moment matters?
Use only the quotation you need where you think it belongs,
And only the very bit that matters.
Use only the words _you_ choose.
Writing is a way of ordering perception, but it's just as often a reordering of perception in a form peculiar to the writer's discovery.
Telling takes the order _you_ want it to, which may have nothing to do with the order that seems "natural," the order that volunteers itself.
The order of what you're writing is determined by your interest in the material And the sense you make of it and by your presence to the reader.
You're not just filling space now.
Here's an experiment:
Copy or print out a couple of pages from a nonfiction work you admire, something not purely memoir.
(For example, the opening of Truman Capote's _In Cold Blood_.)
Underline each fact or assertion, every detail of landscape or character or time or causation.
Remember that everything you're reading—
The very scene in your mind that emerges from the page—
Is a construct of assembled facts.
Then ask yourself, how does the author know these things?
See if you can imagine, or guess, the source of his evidence for each detail,
Each quotation, the particles of every description.
Make a list of those sources.
I don't mean merely the books and articles the author may have read.
I mean people interviewed, repeated visits to the site
On days when the weather differed in different seasons.
I mean police logs and newspapers, hearsay and rumor,
County records and tax rolls, photo albums and gravestones,
Anything the author might have touched upon to make these pages.
It adds up to quite a collection,
From which the author has taken, bit by bit, only the elements he wants.
The evidence has been atomized.
Each minute detail has been removed from the immediate neighborhood of
Its original context—where it was first found or noticed or transcribed—
And given a new neighborhood in the web of the prose itself,
Where newly autonomous facts surround it,
Each of them relocated too.
This new neighborhood is governed by many forces, including rhythm.
Above all, it's governed by the writer's needs.
Not by chronology or logic or spatial sequence
Or any other organization that seems, at first glance, to be "natural."
There will be moments of chronology,
Moments of analysis and reflection,
Moments of visual movement,
Like a tracking shot in the movies.
There will be moments of stationary depth,
Like a landscape drawing by Rembrandt.
But the order of the piece is not determined by any single one of them.
It can have many orders, all flowing into one,
Which is the reader's experience.
You were taught in school to repose on the authority of the evidence you gathered,
The resplendent figures you quoted.
You remember those papers, filled with great lumps of quotation.
Your sentences piloted around them like a ship among icebergs.
But what if you were to muster your own authority?
I don't mean making up facts and quotations.
I mean, what if the reader trusted your prose,
Listened with interest to what you're saying
For the sake of what you're saying,
Instead of noting the complacency, the deference, even the ceremony
With which you bow to the authorities you cite?
What if the reader believed, somehow, in you?
Listened for your voice, not the voices of others?
Watched for your perceptions?
What if the reader felt your authority
And thought about quoting _you_?
In our world—the writing world—
Authority always rests in the hands of the reader,
Who can simply close the book and choose another.
The most fashionable novels and the greatest poems
Cannot force you to read themselves.
Authority always belongs to the reader.
A reader who's opened a book to its first page is in a tender predicament,
Whether she's standing in the aisle of a bookstore or sitting at home.
All the authority belongs to her—the authority to close the book.
And yet she's willing—yearning—to surrender her authority to the author
And keep reading.
Readers exercise their authority almost unconsciously
In their search for the authority that belongs to the author.
As a reader, you know the feeling of looking up after eighty pages and wondering how you got there,
The sense of immersion, of entering a shared but private space.
All the authority a writer ever possesses is the authority the reader grants him.
Yet the reader grants it in response to her sense of the writer's authority.
Authority arises only from clarity of language and clarity of perception.
Authority is how the reader's trust is engaged.
"Authority" is another word for the implicit bond between writer and reader,
The desire to keep reading.
The desire to follow the writer wherever she goes.
The question isn't, can the reader follow you?
That's a matter of grammar and syntax.
The question is, will the reader follow you?
You've been told again and again that you have to seduce the reader,
Sell the story in the very first paragraph.
(Nonsense, but it explains a lot of bad writing.)
The reader isn't looking for the tease of a single paragraph,
Or numbingly clever prose, or sentences full of self-exhibition.
The reader is in love with continuity, with extent, with duration,
Above all with presence—the feeling that each sentence isn't merely a static construct but _inhabited_ by the writer.
Examine yourself while reading and see if that isn't true.
Rhythm is a vital source of the writer's authority.
If the sentences were shaped any other way, the rhythm would be completely different.
Rhythm comes to the reader as a precursor of many things.
It anticipates the intelligibility of the sentence.
It grounds the tongue and the mind.
It creates balance and propulsion.
It's deeply assuring and worth getting right.
Most of all:
Authority arises from the way you write,
Not from the subject you write about.
No subject is so good that it can redeem indifferent writing.
But good writing can make almost any subject interesting.
That's the point of my epigraph from Joyce Carol Oates:
"The subject is _there_ only by the grace of the author's language."
Your grace, your authority, doesn't borrow the subject's validity:
It creates it.
The subject can never justify your prose or redeem its failures.
When it comes to writing, the intensity of the writer's feelings and
The power of the subject mean almost nothing.
We only glimpse that power and intensity
In the power and intensity of the prose.
Yet somehow we believe that subject is everything.
We believe the writer _is_ her story
And that her authority somehow depends on what's happened in her life,
That her authority is authenticity.
People clamor to tell their stories in words.
This doesn't make them writers,
Nor does it make their stories matter.
If _you_ are your story, where do you get another?
If you understand how to build silence and patience and clarity into your prose,
How to construct sentences that are limber and rhythmic and precise
And filled with perception,
You can write about anything, even yourself.
You may feel uncomfortable with the word "authority."
Perhaps it sounds dominant, overbearing, "authoritarian." You may need to work on the problem of self-deprecation,
Self-distrust,
Especially when it comes to noticing the world around you
And what you're able to say about it.
You may be used to denying your perceptions and dismissing your awareness.
You may be caught in a constant state of demurral
Or have the habit of belittling yourself.
Watch for the chronic language of self-disparagement,
The moments when you say, "My problem is..."
Or "It doesn't matter what I think."
If you say these kinds of things, you probably say them out of habit, almost unconsciously.
This is a product of your education too, at home and at school.
Pay attention to it.
Recognize how harmful it is.
Its message—subliminal and overt—is that your perceptions are worthless.
Do everything you can to subvert this habit.
The most subversive thing you can do is to write clearly and directly,
Asserting the facts as you understand them,
Your perceptions as you've gathered them.
You'll ground your own authority in the language itself
As your sentences become better and better.
You may need to write for yourself for a while,
And listen only to the language.
That's okay.
The first person who needs to be persuaded of your authority
Is you.
Don't make it impossible to persuade yourself.
Part of the trouble may be this:
You're afraid your ideas aren't good enough,
Your sentences not clever or original enough.
But what if your ideas are coherent and thoughtful?
What if your perceptions are accurate and true?
Your sentences clear and direct?
What if allowing us to see what's accurate and true is among the best work writing can do?
Saying the obvious thing briefly and clearly and
Observing the critical detail are hard enough.
It's surprising how often ideas that seem obvious to you
Are in no way apparent to the reader.
And vice versa.
What seems like common sense to you may come as a revelation to the reader.
The only sure test of your ideas is whether they interest you
And arouse your own expectations—
The capacity for surprise that you discover as you work.
One purpose of writing—its central purpose—is to offer your testimony
About the character of existence at this moment.
It will be part of your job to say how things are,
To attest to life as it is.
This will feel strange at first.
You'll wonder whether you're allowed to say things that sound
Not merely observant but true,
And not only true in carefully framed, limited circumstances,
But true for all of us and, perhaps, for all time.
Who asked you to say how things are?
Where do you get the authority to do any of this?
The answer is yours to find.
Some people think that discipline is imposed from without,
Regular hours, strict containment, rigorous exclusion.
Some people think discipline is revealed from within,
Enlightenment, purity, solidity of intent.
Discipline is nothing more than interest and expectation, a looking forward.
It's never hard to work when you're interested in what you're working on.
But what if you hate what you're working on?
It helps to examine the content of your loathing.
What is it you hate?
The movement of your ideas?
The nature of your prose?
The obligations and prohibitions you still secretly honor?
The rules and fears you cling to?
Does it feel as though every word you set down is part of an inescapable trap?
As though you're following a logic or order that's not your own?
Building a maze with nothing but dead ends?
Writing in a language you would never say?
It's surprising how often the trouble with a piece of writing
Has nothing to do with the writing itself.
The trouble is anything that keeps you from looking with undiverted attention at what you're thinking and trying to say,
At how you're trying to say it and what the sentence is revealing.
Anything that keeps you from watching the foreground of your mind.
True discipline is remembering and recovering—inventing if necessary—what interests _you_.
If it doesn't interest _you_ , how could it possibly interest anyone else?
The problem may not be the sentences at all.
It may be the expectations that seem to emerge as you write,
The different audiences you're trying to please,
The criticisms you imagine,
The conventions you're obeying without actually choosing them,
The constraints of genre,
Not to mention an endless army of volunteer sentences.
Are you writing on a truly blank screen or piece of paper?
Or are you writing on a palimpsest of rules and regulations,
Things you think you must do, methods you must conform to?
Make yourself aware of the forces getting in the way of your writing.
You may be creating syntactical and logical patterns that cast themselves forward
Into future sentences and end up constricting you.
Parallelisms and contrasts, for instance.
They seem to offer structure and guidance, but they're tying your hands.
Notice how instinctively you grasp any pattern, any parallel,
Any connection that promises to help you define how
The next few sentences, the next few paragraphs, will lay themselves out.
It's as though you can't help wanting the piece to move faster or seem easier to write.
Resist that instinct.
You'll recognize the feeling when that happens, the sense of being trapped,
Coerced into writing a sentence of a predetermined shape.
One of the most powerful feelings a writer experiences while working
Is a sense of obligation, of _having_ to make a sentence or a paragraph
This way or that way, being obliged to write _that_ sentence or _that_ paragraph.
It's a terrible feeling and always a sign of trouble.
Question that obligation. See if you can think your way around it.
Avoiding what you feel you _must_ write is as much a part of writing
As discovering what you didn't know you could write.
Every sentence is entitled to structural freedom.
Yet part of the writer's economy is sometimes finding
The simplest, most direct route, making a simpler, plainer sentence,
Accepting that in the variety of sentences you make,
Some will do their work most effectively if they do it in a straightforward manner.
Plain sentences are as purposeful and efficient as the sentences that seem to resonate.
The way to recover your interest, your discipline, often begins on a small scale,
By fixing sentences, working on syntax,
Looking for problems in the sentences themselves.
As you work—sentence by sentence, thought by thought,
Making this phrase better, that verb stronger—your mood will lift too.
The work will rescue you instead of you rescuing it.
Now it's time to talk about the _other_ reader.
Perhaps you've been wondering about her all along.
Not the reader with a genius for taking you literally,
Who always makes the wrong turn in ambiguous sentences
And stumbles over syntactical blunders,
But the other one who lives beside her in the very same brain:
Literate, curious, adaptable, intelligent, and open-minded.
Let's begin by presupposing she exists,
Which is more than your education presupposes.
Nearly everything you've been taught about writing
Assumes that the reader is plodding at best,
Always distracted and needing a surfeit of superficial cleverness
To keep his head pointed toward the text.
You'll find that assumption all around you.
We remove the unfamiliar words for him
So he'll never have the chance to learn them.
We over-reason for him, filling our prose with approximations of logic,
So he'll feel he's had a good think.
The ordinary reader—the ordinary audience—is a barren conceit.
It guarantees a shared mediocrity.
Don't preconceive the reader's limitations.
They'll become your own.
To write well, it isn't enough for you to read differently.
Imagine the reader reading differently too,
Alive to the movement of language
And the qualities of writing that depend
On an unspoken understanding between writer and reader:
Wit, irony, inference, and implication.
Imagine a reader you can trust.
This sounds like a simple imperative.
But the difference between writing for the reader implicit in your education
And writing for one you trust is the difference between writing clumsily,
Using all the grappling hooks of transition and false logic,
And writing well, able to move briskly and freely,
Going anywhere from anywhere almost instantly.
All your life you've been reading books that trusted you,
Trusted your intelligence, your keenness,
Your ability to feel an invisible wink,
To follow any trail,
Even while you were learning in school not to trust the reader.
The books that trusted you most may be the ones you love best.
And what happens if you trust the reader?
All the devices of distrust fall away,
The pretense of logic, the obsession with transition,
The creeping, incremental movement of sentences,
Sentences stepping on each other's heels.
With them go all the devices
Meant to overawe the reader, that aping of authority
Which even young writers learn so soon and so well—
A prose about hierarchy and its demarcations
Rather than the authority of clarity and directness.
Why would you try to overawe a reader you can trust?
The reader you can trust is a reader predisposed to trust you.
In that reciprocity lies the joy of writing and reading.
You can't trust the reader without also trusting yourself.
The trustworthy reader is alert to the way your sentences
Create promises and contracts.
These implicit promises are a descant running through your sentences.
If you write ambiguous sentences, you create a state of uncontrolled implication,
And among those implications are commitments to the reader that can't be fulfilled
Because the writer isn't aware of them.
But the reader feels them being made and broken, again and again.
Your sentences come one by one onto the stage
And leave it one by one, without assisting each other.
But they listen intently to one another,
With special attention to the promises they've made.
In the syntax and rhythm of the sentences,
In the pace of thought, the intensity of movement,
The crescendo and decrescendo,
The trustworthy reader learns the writer's habitude and how to move with it.
You converse, in a sense, with the voice on the other side of the ink.
That kind of reading is the pleasure of being summoned out of ourselves by the grace,
The ferocity, the skill of the writing before us.
How else to explain our love of even difficult writers?
Their agility evokes our agility.
We move at their speed, elliptically, obliquely,
However they move.
Imagine a cellist playing one of Bach's solo suites.
Does he consider his audience?
(Did Bach, for that matter?)
Does he play the suite differently to audiences
Of different incomes and educations and social backgrounds?
No. The work selects its audience.
You'll be tempted to ask, "Who is the reader?"
The better question is always, "Who am I to the reader?"
And also, "How many versions of 'I' are present in this piece?"
Who said there had to be only one?
Sooner or later, you'll also wonder,
"What can I expect the reader to know?"
It's a perplexing question for writers,
A way of asking, "How much of my world overlaps with the reader's?"
Hidden behind that philosophical question is a more practical one:
"How much do I have to explain?"
It helps to remember that your prose is going to be read Against two different backdrops:
What the reader knows about reading and what the reader knows about life.
It's surprising how many writers forget the life part.
Trusting the reader is a way of controlling
The temptation to over-narrate, over-describe, over-interpret, over-signify.
It lets the reader share the burden of comprehension.
This is part of the constant negotiation between writers and readers.
A good reader will follow a good writer wherever she goes,
And the good writer will do all she can to help.
That's why learning to read your own work as a _reader_ ,
Not as its writer, is so helpful.
Learn to trust yourself as the reader.
You'll never know another more thoroughly.
Instead of writing for an imaginary audience of readers, however large or small,
Try writing for the reader in yourself,
A stand-in for the reader you trust,
Who's always at hand and always consistent.
Like being the narrator, this is a kind of role-playing—
Impersonating the literal-minded reader and the trusting reader at the same time.
It means trying to come to your work
Without the immense foreknowledge of having written it.
And it means imagining the reader's experience
As he gathers what he knows about your piece
Only from what each sentence reveals, one after the other.
This would be impossible if you hadn't spent so much time
As a reader yourself making your way through other writers' works,
Orienting yourself in unfamiliar worlds, sentence by sentence,
Learning characters and deciphering plots, word by word,
Absorbing arguments, and tracing the meditative currents of essays,
Undaunted by the newness of the next thing you read.
Being your own reader doesn't mean you're writing only for yourself.
It isn't solipsism or egocentricity.
It's one of the writer's important economies,
A faith in the kinship between you and the reader who isn't you,
The assurance that what interests you will interest the reader If your sentences warrant it.
This you will have to take on trust—that you and the reader
Are more alike than you like to think.
Otherwise how would any of this be possible?
Besides being your own reader, you're also your own editor.
Your only editor.
Your writing is your responsibility, first and last, in every detail.
No one will fix it or clean it up.
It's your job to be clear, precise, intelligent, resourceful, poetic, and wise
In prose of staggering clarity, all of it perfectly proofread.
That's why you became a writer, isn't it?
You're not responsible for your readers' ignorance,
And they're not responsible for your erudition.
Know what you want to know, learn what you want to learn,
Use what you want to use,
Without worrying whether you're wandering out of the reader's depth.
At the same time show a tender care
For the reader's attention, his knowledge of place and time,
His sense of his whereabouts in the pages before him.
Pause now and then to make sure he's with you.
Bring him up to the crow's nest to get a feel for the current and where you're headed.
This sounds contradictory, I know.
But then, so much of writing is.
When will you be done?
This isn't about getting to the end of the writing day and out of your head at last.
It's about knowing when a piece is finished.
This question is a variant of "How will I know when to stop revising?"
A question that rings with a certain fear,
As though you might spawn a thousand mutations of every sentence
With no means of natural selection.
"Done" isn't absolute or arbitrary.
Nor is it really about learning your limits as a writer.
It's a compromise.
This is another of the writer's economies,
Knowing how far to push a piece and when to let it go.
There's sometimes a relenting when you stop at last.
More time, more money, more research, more intelligence,
Might have made a difference.
But there's sometimes a certainty too,
The knowledge that this particular parallel universe is now complete.
"Done enough" sounds too callow to describe the compromise,
So call it "perfection enough,"
As perfect as possible under the circumstances.
There's no objective measure of "done."
It's an assurance within yourself,
A response to the work that's as much feeling as judgment,
A feeling derived from your rich experience of the completeness
Of all the books you've read in your life.
At first, "done" will come too soon.
You'll think you're finished only to find, again,
How well some sentences hide ambiguity.
Or "done" may seem too remote, an unimaginable state of perfection
Achieved after infinite revision.
But in a certain sense, the completeness of your piece
Will have been inherent in it all along.
You come upon it with one final revision, one final fix.
It's likely to take you by surprise.
You're urging forward every word, every phrase,
Every sentence, every rhythm, until they find their balance, their coherence.
You may think you know, at the start, what you want that coherence to feel like.
But it won't.
So let's not talk about "done."
It's premature, something you'll discover for yourself when the time comes.
You'll need to know two things:
You won't write a final sentence and then "The End,"
And the distance to completion will change with the changes in the way you write.
The better question now is the more fearful one:
"How will I know when to stop revising?"
You may not be able to tell yet whether your revisions are really improvements.
So revise toward brevity—remove words instead of adding them.
Toward directness—language that isn't evasive or periphrastic.
Toward simplicity—in construction and word choice.
Toward clarity—a constant lookout for ambiguity.
Toward rhythm—where it's lacking.
Toward literalness—as an antidote to obscurity.
Toward implication—the silent utterance of your sentences.
Toward variation—always.
Toward silence—leave some.
Toward the name of the world—yours to discover.
Toward presence—the quiet authority of your prose.
And when things are really working,
That's when it's time to break what already works,
And keep breaking it
Until you find what's next.
I began this book by writing,
Know what each sentence says,
What it doesn't say,
And what it implies.
The way to keep going?
Never stop reading.
Say more than you thought you knew how to say
In sentences better than you ever imagined
For the reader who reads between the lines.
_Some Prose and Some Questions_
Here are some passages to experiment with as you read this book.
But let's set aside a few responses:
Whether you like a passage or not.
Whether you like the author or not.
Whether or not you think the author likes you.
Whether you like what a passage is saying.
Those aren't useful layers of response for these experiments.
Let's also set aside the question of meaning, significance.
Don't be concerned with what the author is "trying to say."
Each of these passages—with one or two possible exceptions—is bound to other passages in the work it was taken from and is therefore incomplete.
Watch yourself carefully.
You may be trying to discover things that can later be converted into meaning.
You may also find yourself trying to describe the style of the prose or its ideological or theoretical content.
Don't.
The most valuable thoughts may be the ones that begin, "I don't know if this is important but..." or "This will sound like nitpicking..."
All you're doing is noticing what you notice.
Try to resist deciding whether what you notice is important or not.
Of course it is, even if you can't say precisely what it is you've noticed.
Begin by reading these passages aloud.
I've already listed some of the questions you might ask about these passages (on this page).
You will think of many others.
MY BANDANNA IS ROLLED on the diagonal and retains water fairly well. I keep it knotted around my head, and now and again dip it into the river. The water is forty-six degrees. Against the temples, it is refrigerant and relieving. This has done away with the headaches that the sun caused in days before. The Arctic sun—penetrating, intense—seems not so much to shine as to strike. Even the trickles of water that run down my T-shirt feel good. Meanwhile, the river—the clearest, purest water I have ever seen flowing over rocks—breaks the light into flashes and sends them upward into the eyes. The headaches have reminded me of the kind that are sometimes caused by altitude, but, for all the fact that we have come down through mountains, we have not been higher than a few hundred feet above the level of the sea. Drifting now—a canoe, two kayaks—and thanking God it is not my turn in either of the kayaks, I lift my fish rod from the tines of a caribou rack (lashed there in mid-canoe to the duffel) and send a line flying toward a wall of bedrock by the edge of the stream. A grayling comes up and, after some hesitation, takes the lure and runs with it for a time. I disengage the lure and let the grayling go, being mindful not to wipe my hands on my shirt. Several days in use, the shirt is approaching filthy, but here among grizzly bears I would prefer to stink of humanity than of fish.
JOHN MCPHEE, _Coming into the Country_
THE DISTANCE FROM NEW ORLEANS to Alexandria is about 190 miles. The first 90 miles, from New Orleans to Baton Rouge, are on a throughway, a straight, fast road on the east side of the Mississippi, far enough back from the bank to avoid meanders, and high enough over the marshes to obviate bridges. There is nothing worth a long look. The bayous parallel the road on either side like stagnant, weed-strangled ditches, but their life is discreetly subsurface—snapping turtles, garfish, water moccasins and alligators. The mammals are water rats and muskrats and nutria, a third kind of rat. The nutria, particularly ferocious, is expropriating the other rats. Bird life, on the day we drove through, was a patrol of turkey buzzards looking down for rat cadavers. There pressed down on the landscape a smell like water that householders have inadvertently left flowers in while they went off for a summer holiday. It was an ideal setting for talk about politics.
A. J. LIEBLING, _The Earl of Louisiana_
THE TIDE WAS OUT. So far as the eye could see there stretched the matted bents of the mudflats: a soft monotony blended of grey and green and blue and purple. It had a quilted look, for the thousands of rivulets, which cast a network over it, followed the same course day in, day out, and had worn down the mud into channels between the hummocks some feet deep. To the small creatures which lived here this must have been a most fantastic landscape. At the bottom of these deep channels the tiny streams, only a few inches wide, had their established, deeply graven waterfalls, their rapids which tested to the utmost the gallantry of straws, and lakes with bays and beaches; and on the islands grass roots found purchase on the mud by gripping it and one another so that they grew into cushions of jungle, one plant rising on another like minute vegetable pagodas. The scene was incised and overstuffed with profligate ingenuity; and it was odd to think of all this elaboration being wiped out twice in every twenty-four hours, the rivulets losing their identities in the rough inundation of the tide, the springing grasses, so obstinate in their intention of making dry land out of mud, becoming the bottom of the sea. There was the same spendthrift and impermanent fabrication going on at ground level as there was over our heads, where great clouds, momentarily like castles, temples, mountains, and giant birds, were blown by the cleansing winter wind to the edges of the sky, here not clipped away by hills or streets and astonishingly far apart. There could not have been a more generous scene, nor one which was less suited to receive the remains of Mr. Setty, who from infancy had been so deeply involved in calculation, and so unhappily, who had tried keeping figures outside his head and got sent to prison for it, and had kept them inside his head and got killed for it.
REBECCA WEST, "Mr. Setty and Mr. Hume"
TO MY AUNT MAE—Mary Elizabeth Davenport Morrow (1881–1964), whose diary when I saw it after her death turned out to be a list of places, with dates, she and Uncle Buzzie (Julius Allen Morrow, 1885–1970) had visited over the years, never driving over thirty miles an hour, places like Toccoa Falls, Georgia, and Antreville, South Carolina, as well as random sentences athwart the page, two of which face down indifference, "My father was a horse doctor, but not a common horse doctor" and "Nobody has ever loved me as much as I have loved them"—and a Mrs. Cora Shiflett, a neighbor on East Franklin Street, Anderson, South Carolina, I owe my love of reading.
GUY DAVENPORT, "On Reading"
EVEN WHEN YOU WATCH the process of coal-extraction you probably only watch it for a short time, and it is not until you begin making a few calculations that you realise what a stupendous task the "fillers" are performing. Normally each man has to clear a space four or five yards wide. The cutter has undermined the coal to the depth of five feet, so that if the seam of coal is three or four feet high, each man has to cut out, break up and load on to the belt something between seven and twelve cubic yards of coal. This is to say, taking a cubic yard as weighing twenty-seven hundredweight, that each man is shifting coal at a speed approaching two tons an hour. I have just enough experience of pick and shovel work to be able to grasp what this means. When I am digging trenches in my garden, if I shift two tons of earth during the afternoon, I feel that I have earned my tea. But earth is tractable stuff compared with coal, and I don't have to work kneeling down, a thousand feet underground, in suffocating heat and swallowing coal dust with every breath I take; nor do I have to walk a mile bent double before I begin. The miner's job would be as much beyond my power as it would be to perform on the flying trapeze or to win the Grand National. I am not a manual labourer and please God I never shall be one, but there are some kinds of manual work that I could do if I had to. At a pitch I could be a tolerable road-sweeper or an inefficient gardener or even a tenth-rate farm hand. But by no conceivable amount of effort or training could I become a coalminer; the work would kill me in a few weeks.
GEORGE ORWELL, _The Road to Wigan Pier_
WHAT IS THE CHINESE WAR LIKE? Well, at least it isn't like wars in history books. You know, those lucid tidy maps of battles one used to study, the flanks like neat little cubes, the pincer movements working with mathematical precision, the reinforcements never failing to arrive. War isn't like that. War is bombing an already disused arsenal, missing it and killing a few old women. War is lying in a stable with a gangrenous leg. War is drinking hot water in a barn and worrying about one's wife. War is a handful of lost and terrified men in the mountains, shooting at something moving in the undergrowth. War is waiting for days with nothing to do, shouting down a dead telephone, going without sleep and sex and a wash. War is untidy, inefficient, obscene, and largely a matter of chance.
W. H. AUDEN, 1939
THE WINTER FIRES OF NEW YORK burn everywhere like the ghats in Benares. On the valueless land north of the ship canal some children, dressed like aviators, are burning a Christmas tree. An ashcan is blazing on the banks of the river. Rubbish fires glow in the backyards of Harlem. Farther south, where a slum is being cleared, there is a large conflagration of old lathes. Another rubbish barrel and another Christmas tree are burning on Ninety-sixth Street. On the curb at Eighty-third Street an old wicker table is being consumed with fire. In a vacant lot in the fifties some children are burning a mattress. South of the United Nations there is a big fire of cardboard cartons behind a grocery store. Many fires burn in the gutters and backyards of the slums; there are bonfires of wooden crates in front of the fish market and on Battery Park, untended, an iron basket, full of waste, lights the gloom as all these other fires do on a winter dusk when the dark begins to fall before the lights go on.
JOHN CHEEVER, _Journals_
IN THE SUMMER OF 1943 I was eight, and my father and mother and small brother and I were at Peterson Field in Colorado Springs. A hot wind blew through that summer, blew until it seemed that before August broke, all the dust in Kansas would be in Colorado, would have drifted over the tar-paper barracks and the temporary strip and stopped only when it hit Pikes Peak. There was not much to do, a summer like that: there was the day they brought in the first B-29, an event to remember but scarcely a vacation program. There was an Officers' Club, but no swimming pool; all the Officers' Club had of interest was artificial blue rain behind the bar. The rain interested me a good deal, but I could not spend the summer watching it, and so we went, my brother and I, to the movies.
We went three and four afternoons a week, sat on folding chairs in the darkened Quonset hut which served as a theater, and it was there, that summer of 1943 while the hot wind blew outside, that I first saw John Wayne. Saw the walk, heard the voice. Heard him tell the girl in a picture called _War of the Wildcats_ that he would build her a house, "at the bend in the river where the cottonwoods grow."
As it happened I did not grow up to be the kind of woman who is the heroine in a Western, and although the men I have known have had many virtues and have taken me to live in many places I have come to love, they have never been John Wayne, and they have never taken me to that bend in the river where the cottonwoods grow. Deep in that part of my heart where the artificial rain forever falls, that is still the line I wait to hear.
JOAN DIDION, "John Wayne: A Love Song,"
in _Slouching Towards Bethlehem_
I GREW UP IN THE MIDWEST and despised horses. The ones I rode struck me as stupid and untrustworthy. I went to Wyoming when I was young, and the ones there were worse. On a cold morning, two out of three would buck you down. They were, I felt, an ugly necessity for where a truck wouldn't go.
I've been kicked, stepped on, and bitten. Bitten I liked least. My most trustworthy saddle horse leaned over once while I was cinching him up and clamped on my upper leg, turning the thigh into what looked like a Central American sunset. I threw him down on the ground, half-hitched his feet together, and put a tarp over him. I let him up two hours later: he thought I was the greatest man in the world, one he wouldn't think of biting. Horses only remember the end of the story.
TOM MCGUANE, "Roping, from A to B,"
in _An Outside Chance_
THIS MORNING, an invasion of tiny black ants. One by one they appear, out of nowhere—that's their charm too!—moving single file across the white Parsons table where I am sitting, trying without much success to write a poem. A poem of only three or four lines is what I want, something short, tight, mean, I want it to hurt like a white-hot wire up the nostrils, small and compact and turned in upon itself with the density of a hunk of rock from Jupiter...
But here come the ants: harbingers, you might say, of spring. One by one they appear on the dazzling white table and one by one I kill them with a forefinger, my deft right forefinger, mashing each against the surface of the table and dropping it into a wastebasket at my side. Idle labor, mesmerizing, effortless, and I'm curious as to how long I can do it, sit here in the brilliant March sunshine killing ants with my right forefinger, how long I, and the ants, can keep it up.
After a while I realize that I can do it a long time. And that I've written my poem.
JOYCE CAROL OATES, "Against Nature"
I CARE NOT TO BE CARRIED with the tide, that smoothly bears human life to eternity; and reluct at the inevitable course of destiny. I am in love with this green earth; the face of town and country; the unspeakable rural solitudes, and the sweet security of streets. I would set up my tabernacle here. I am content to stand still at the age to which I am arrived; I, and my friends: to be no younger, no richer, no handsomer. I do not want to be weaned by age; or drop, like mellow fruit, as they say, into the grave. Any alteration, on this earth of mine, in diet or in lodging, puzzles and discomposes me. My household-gods plant a terrible fixed foot, and are not rooted up without blood. They do not willingly seek Lavinian shores. A new state of being staggers me.
Sun, and sky, and breeze, and solitary walks, and summer holidays, and the greenness of fields, and the delicious juices of meats and fishes, and society, and the cheerful glass, and candle-light, and fire-side conversations, and innocent vanities, and jests, and _irony itself—_ do these things go out with life?
CHARLES LAMB, "New Year's Eve," 1821
Consider each sentence on its own, disconnected from the whole.
(That's inevitably one of the ways you consider the sentences _you_ make.)
Pay attention to the ordinary way you might say something.
Think of that as the backdrop for the questions you ask these sentences.
See anything peculiar?
A phrasing you didn't expect?
A rhythm more pronounced than its surroundings?
A word in a position that sounds odd?
Note in each passage how variant in structure the sentences are—no two quite the same in shape.
And how, when they're invariant, you can feel the reason why. (See Auden.)
How closely—or how loosely—are the sentences in a passage bound to each other?
Can you feel a gap between them—something indiscernible going unsaid?
Are there sentences that explain or fulfill the sentence that precedes them?
That build a rhythm with other sentences?
Sentences that misdirect or decoy the reader?
A sentence that's working harder than the others—doing more to draw or turn the reader's attention?
Let yourself ask the question why.
Why is the author choosing this word, writing that sentence that way?
Don't expect to find _an_ answer.
Expect to find some possibilities.
Here are some examples.
Why does John McPhee use the word "refrigerant"?
Why not simply "cooling"?
And why "the temples" and not "my temples"?
And why "days before," as if this were legend, or "level of the sea" instead of "sea level"?
Notice the word "there" in the parenthetical phrase near the end of the passage.
Can you feel how it orients us? How it situates McPhee?
Take a look at the verbs (and the sentence structures) in the passage by A. J. Liebling.
For the most part, they're pretty plain: "is," "are," "was."
So where does the life of this passage come from?
Repetition does one kind of work here, another, vastly different kind in the passage by Auden.
Consider the economy of using the verb "to obviate."
Can you see that Liebling has left the vase out of the penultimate sentence?
And how much it would complicate things to include it?
Slow down a great deal as you read the passage by Rebecca West.
If necessary, take a pen or pencil and indicate the breaks between phrases or clauses where she hasn't already used a mark of punctuation.
Then read it aloud again.
In this passage, it would be useful to sort through the nouns and see what kinds there are, what categories they fall into.
It would also be useful to break this into a series of short sentences—as short as possible ("The tide was out")—and then watch how West recombines them into the texture, the rhythm, of her prose.
Look for the modifiers in this passage—individual words but also, more important, entire phrases.
Pay attention to the way they extend and structure West's sentences.
In Guy Davenport, too, it's worth seeing if you can find the separate assertions that underlie this extraordinary single sentence and turn them into their own sentences:
_Over the years, she and Uncle Buzzie had visited places like Toccoa Falls, Georgia, and Antreville, South Carolina. They never drove faster than thirty miles an hour_.
Then look at the words that are left behind.
This is a sentence built by suspension—a sense of equivalence—not subordination. How does that come about?
And _why_ would Davenport write a sentence like this?
In George Orwell, what's the difference between the sentence he gives us, "When I am digging trenches in my garden, if I shift two tons of earth during the afternoon, I feel that I have earned my tea," and _If I shift two tons of earth during the afternoon when I am digging trenches in my garden, I feel that I have earned my tea_?
There's nothing elaborate or uncommon about the words that appear in any of these sentences.
But notice how patient he is as he builds this passage, step-by-step, rising from calculation to astonishment to a kind of self-abnegation.
And yet can you feel the lack of thus-ness—the way these sentences stand side by side, each one on its own feet?
How tempting it would have been to supply this paragraph with logical indicators to bind the sentences together.
This passage by W. H. Auden was written as a radio talk for the BBC—to be read aloud, or rather spoken into a microphone.
And yet apart from "Well, at least" and "You know"—which suggest an intimate, colloquial connection with his audience—there's nothing here that doesn't sound written to be read silently.
The sentences that begin "War is..." look repetitive, but it's worth noting how un-repetitive they really are, structurally and rhythmically.
Listen, too, for the restraint in this passage, the things Auden refrains from doing.
For instance: he avoids the temptation to build a crescendo into the sentences beginning "War is..."
He also doesn't let these sentences become more parallel than they need to be.
Though everything is just as he wants it in this paragraph, it doesn't feel overdetermined.
He's paying more attention to what the reader hears than he is to the possibility of building more pattern into his prose.
The passage from John Cheever's _Journals_ —what kinds of journals are these?
Clearly not the kind we often mean: daily jottings, notes on life and living, an informal archive of emotions and events.
At play here is perception—the gathering in his mind of all these fires—but also the play of sentence making: How many ways are there to say something is burning?
This passage has no larger purpose than to exist, to work out, for a moment, the possibilities of some sentences.
And yet we feel its inadvertent testimony—the vision of a city lit up by small conflagrations, a city where children instinctively gather near the flames.
Those children dressed like aviators persist.
Look carefully at the rhythm of the last sentence, how it keeps wanting to settle into a familiar, steady beat and how Cheever keeps it from doing so.
And if you lose your sense of what rhythm is, simply remember the line Joan Didion quotes in this passage: "at the bend in the river where the cottonwoods grow."
But look for the counter-rhythms, the passages that are more abrupt, less sinuous.
Where do those staccato pulses come from?
This is also a passage to help you remember how brief—and how extensive—rhythm can be.
The first paragraph is neutral, almost plain, except when Didion turns to the hot wind and the dust—an extended rhythm.
And also the very end: "and so we went, my brother and I, to the movies"—a brief rhythmic pulse to close the paragraph and open what follows.
Remove the word "forever" from the last sentence.
The first time you read the sentence that way, it seems to stumble over itself.
But can you read it without "forever" and still find the rhythm of its opening clause?
Now reinsert "forever."
This is a test you need to be performing on your own sentences.
In the passage from Tom McGuane, think about the sentence "Bitten I liked least."
A "correct" way to say this might be: _I liked being bitten least_.
But there's a moment of discovery in the gap between "I've been kicked, stepped on, and bitten. Bitten I liked least"—a chance to intensify the compression, the terseness, you can feel all through this passage.
Also a chance, in that bitten-off sentence, to intensify the reader's sense of the writer's character.
The essay containing this passage by Joyce Carol Oates also contains the epigraph I borrow from her.
The ellipsis at the end of the first paragraph is hers.
I'll let you find the questions and discover the experiments to perform here.
We've been taught to create a kind of vocal uniformity in our prose—one voice, one tone, a very narrow band of ourselves visible to the reader.
Charles Lamb defies all this, gloriously.
There is something luminescent and completely various in the versions of himself he presents in this essay—as though each emotion revealed a different Lamb.
Don't worry about "Lavinian."
But do pause to admire his use of the verb "reluct"—the root of the only form we use, "reluctance."
And just as Lamb is the most various of all these authors, in the character he presents the reader, this is also the most various passage rhythmically and in the shape of its sentences.
How hard now to say, simply, "I am in love with this green earth."
_Some Practical Problems_
Reading these sentences—and my commentary on them—you'll be tempted to side with the writer, to think, "I know what he means" or "I can see what she's saying." But that's because it feels so normal to try to deduce the meaning of the sentence instead of observing what its words actually say. We're so trained to read for meaning—to look through the sentence to what we think is the author's intention—that in our search for it we're prepared to disregard the literal significance of the prose itself.
You may also be tempted to say, "Maybe the writer wants it that way." But you can only judge intentionality in context. If all the sentences in a piece are clear and sharp, then perhaps—perhaps!—we can say that a slightly aberrant sentence is intentional, if there seems to be a reason for it. But if many of the sentences in a piece are unclear, ambiguous, or weak, we have to assume that intention is irrelevant—indiscernible at best. We have to assume the writer lacks control.
These sentences were written by excellent college students who went on to be very good writers. They—the sentences, that is—are no worse and no better than the sentences you'll come across on any day, in any medium, anywhere.
She didn't trust him with his accent or his gentle demeanor.
Note how "with" distorts the sentence. It could read, _She didn't trust him_. Or _She didn't trust his accent or his gentle demeanor_. And note too that "trust him with" is an active, meaningful locution in English, for example, _She didn't want to trust him with her new car_. You can feel the pressure of that locution in this sentence, even though it doesn't belong there.
I despise the feeling of something falling apart in your body.
Note the shift from the first to the second person.
This makes no sense unless the author is despising you.
The second person requires some attention, or it easily goes awry.
Erica wobbled uncertainly as she tried to sit down on the stool next to me.
Can one wobble _certainly_? "Uncertainly" is implicit in "wobble" and "tried." An example of the kind of redundancy that adverbs often create.
The buffet of diseases, cancers, viruses, and overall deteriorations our present world has to offer is impressive and wary.
Several problems, beginning with an unworkable metaphor: the "wary buffet." The adjectives at the end of the sentence must modify the subject. The author of this sentence has completely lost track of the beginning by the time he reaches the end.
By the time I was 11 the milieu of doubts, questions, and skepticism had culminated into a daunting tangle of despondent confusion.
The trouble? Words used incorrectly: "milieu" and "culminated." By "milieu," he means something like "mix" or "combination." No such thing as "culminating into"—"culminating in" is possible. This would be a tolerable sentence if it said, _By the time I was eleven, my doubts, questions, and skepticism had turned into despondent confusion_. Note the metaphorical feeling of "tangle"—it wants to be more literal, and plural, than "despondent confusion" allows. A tangle of feelings, yes. A tangle of confusion, no.
Her hair, dyed black, is neatly quaffed.
A delightful sentence if the author means that her hair is easily imbibed. The word is "coiffed." This problem is solved by using the dictionary.
Grimy rinds of snow still squat along the northern walls of buildings.
"Grimy rinds of snow" is good. But look what the word "squat" does. It animates the already metaphorical "rinds." In regular life, rinds don't squat.
This is a room where mornings are had in loneliness, and evenings are had in relief.
A nice attempt flawed by a very weak verb—"are had." Slight additional confusion because, as a locution, "to be had" has a very different meaning, as in "We were had." And why the weak verb? The effort to heighten the parallel between mornings and evenings. But could the sentence be as powerful without a parallel structure? Certainly. Shanghaied by a syntactical choice.
His gaze was fixed to the ground with an occasional glance at the horizon for a brief inspection of the distance to the next knoll.
_He gazed at the ground, but sometimes he glanced at the horizon_. Note how "was fixed" throws everything into disarray. There's a kind of verbal poverty in this sentence. Why only one verb? And a flawed one at that? Note how "with an occasional glance" and "for a brief inspection" are trying to do the work of verbs. But they can't. "To glance" and "to inspect" have been turned into nouns, which depletes their energy. Note too that "gaze" is the origin of much of this trouble. The verb again gets turned into a noun, and a noun that's incapable of action. "He," as the subject, offers better verb choices than "his gaze."
The lifejacket dug into my armpits as heavy boots pulled my legs down.
"As" is nearly always trouble. What's it doing here?
Insisting on simultaneity? "And" would work as well and more simply. Note the feeling that these could be someone else's—anyone else's—boots. And note too the feeling that the legs are going down without the rest of this writer's body.
On occasion Etta James may faintly be heard singing in the kitchen, a creek throughout the floorboards or whining from the attic ceiling.
Etta James seems to be whining from the attic ceiling and also to have become, somehow, a stream flowing through the house. Chaos. Instead of "On occasion" try "sometimes." "Creek" does not equal "creak." From the attic—not from the attic ceiling, which is the ceiling _over_ the attic, not over the room below the attic. This attempt to list some of the sounds in a house goes completely awry—and why? Because one verb—"may faintly be heard"—is trying to govern "creak" and "whining." And why? Because the sentence is fundamentally passive. _You can hear Etta James singing faintly in the kitchen. You can also hear creaking in the floorboards and a whining from the attic_. And what if we revise further and remove the verbs of perception? _Sometimes Etta James sings faintly from the kitchen. The floorboards creak, and the ceiling whines_.
So I have found myself inside this intensely feminine world in much the same way as the crossing-dressing men in _Some Like It Hot_ when they infiltrate an all-woman jazz band.
A typo makes it sound as though the author means these men dress up as street crossings. What's the trouble here? The men in _Some Like It Hot_ (and the actors who play them) have names, but the author doesn't know them, or won't use them if he does. He isn't sure whether he can count on the reader's knowing the movie. If he was sure, he wouldn't have added "when they infiltrate an all-woman jazz band." Also, "much like the cross-dressing men" is better than "in much the same way as the cross-dressing men" simply because it uses fewer words. Even better is "much like Tony Curtis and Jack Lemmon."
In the last row, I sat between my mother and father, the latter of whom was to die three months later.
"In the last row" sounds oddly placed, though that depends on context. Note that "the latter of whom" over-specifies—and over-formalizes—a much simpler construction: _I sat between my mother and father, who was to die three months later_. Notice that there's no confusion about who will be dying, even without "the latter of whom." That phrase and the forward-looking past tense verb—"was to die"—do a very good job of draining any emotion out of this sentence. Also, there's an odd sense of intentionality in "was to die." What's wrong with this version: _I sat between my mother and father, who died three months later_? It's ambiguous. Do they both die? Its ambiguity can be resolved easily enough: _I sat between my mother and my father, who died three months later_.
Throughout the year, the chubby pigeons would perch on the south-facing roof of her home.
This sentence implies that svelte pigeons perched elsewhere. Removing "the" from "the chubby pigeons" helps. As does turning "would perch" into "perched." The habitual nature of their perching—which is what "would perch" is meant to suggest—has already been established by "throughout the year."
Since I was little, Wallace has been around.
Notice how different this version of the sentence sounds: _Wallace has been around since I was little_. What's the difference? "Since." In the original, it has the ambiguous overtone of "because," which vanishes in the revision.
He shines from stem to stern due to his three chunky gold rings and spiffy dress shoes.
Perhaps, if he's wearing chunky gold rings on his stem and spiffy dress shoes on his stern. But I doubt he is. Betrayed by a cliché. "Due to" does the work a verb should be doing.
A lot of the older campers still look to me as someone they can confide in.
A syntactical trap. The sentence starts out reading as though it might say something like, "A lot of the older campers still look to me like children." Saying "still regard me" would help. But what if the sentence simply said, _A lot of the older campers still confide in me_?
There is an old man who lives there.
How did this sentence escape fixing? Because it became invisible, that's how. _An old man lives there_.
When the Germans sensed the end of the war, they stopped actively murdering the newborn babies in the camp.
This has the unfortunate effect of implying that "passively murdering" is possible.
All of them are in some kind of ill-fitting jacket.
Of course it's an ill-fitting jacket. All of them are in it. Plural persons in a singular jacket.
The family photos on my mother's side are scarce.
Look at what the word "the" does. It posits the existence of family photos which are then scarce, as if they'd run away from home. Remove "the" and the sentence makes much more sense.
Diane's death must have shocked the family into the realization that in the end death would come for them all.
"Realizing that death," not "the realization that in the end death." Replace the noun phrase with a verb form.
Children in this age group are generally developing social awareness skills where they become aware of the role of the self in relation to others.
"Where" is trying vainly to serve as a relative pronoun, which it's not. All it really succeeds in doing is holding a redundancy apart. _Children in this age-group are generally just becoming aware of the role of the self in relation to others_.
Denizens of America's northwest corner are not exceptionally fond of products derived from the flesh of swine, of course; however they, like any Americans, were keen to be the beneficiaries of the excessive political patronage known as "pork barrel legislation."
This is an attempt at humor by being orotund. Is it really true that people in Washington and Oregon don't like bacon? After the semicolon, the pork becomes metaphorical. This sentence is studded with the attributes of prose as it is usually taught today. It's periphrastic, illogical while insisting on its logic, and awkwardly metaphorical, and the author is buried somewhere under the rubble.
I studied abroad the fall of my junior year in France.
Betrayed by two words that like to stick together: "study abroad." _I studied in France the fall of my junior year_. The abroadness will be implicit, unless this sentence was written in France.
I am not a typical daughter of Seattle, though I do love the Mariners with a passion that can cause the uninitiated ballpark companion to flee Safeco Field for fear of losing an eardrum.
The last half of the sentence can be paraphrased thus: _I scream so loud it hurts my friend's ear_. Nothing is gained by loosing this avalanche of words.
As I dive in, the water gives me its usual jolt in the stomach, but I welcome the shock and hang for a moment in the closest approximation of flight I know.
Note how time runs backward and then stops in this sentence. First the diver feels the jolt of the water but welcomes the shock even in mid-dive, before he has hit the water.
The two families line up like competing rugby teams. The bride takes the place of the rugby ball standing between the two groups in full wedding regalia.
A metaphor that begins to work and then fails completely. The bride as rugby ball is deeply unfortunate.
Whatever ghosts were left in Pompeii had long disappeared, leaving behind twisted plaster casts and British families on holiday.
There's a nice idea behind this sentence, but "leaving behind" must modify "ghosts," which makes them sound as though they'd littered the site with British families on holiday.
Exempt from army duty until the close of the war because of a lame leg it was noticed at the shooting range that he was a good shot and he was told that someone like him could be useful at the front.
Automatic revision: make this two sentences, probably three. Note that "it" was exempt from army duty. And "it" has a lame leg. The difficulty? The passive voice—"it was noticed... that" and "he was told that"—governs the sentence. Any sentence containing "it was noticed that" qualifies for instant demolition and reconstruction. Everything about this sentence—"army duty," "the close of the war," "someone like him could be useful"—is fuzzy.
One at a time I pull my feet up to my knees and wipe my legs dry.
Written by a contortionist. With telescoping legs, it sounds like.
In front of you a grassy meadow dusted with colorful flora extends.
Translated from the German. This sentence is trying to be expressive, descriptive. But "dusted"? "Colorful flora" hides the names of all the wildflowers in that meadow—names that could be looked up and used. This sentence is trying hard not to say, _There's a grassy meadow_. But why? At least that doesn't sound like a peculiar inversion of ordinary syntax.
Throughout the trip we had tried, despite our cameras and the other tourists, to pretend that we were true Parisians. We ordered in broken sentences and struck up awkward conversations with anyone willing to put up with us.
Note that the second sentence is implicitly going to tell us what true Parisians are like. Not like we thought.
About halfway through, I saw to my utmost horror that I could not apply a particular symbol to what I was typing.
Why be hyperbolic here? Hang on to your utmost horror. You may need it for something genuinely horrifying. "Apply a particular symbol to what I was typing"? I think the writer means he didn't know the keystroke for the symbol he wanted to use. Writers often try to be humorous by being hyperbolic. They never succeed.
There are at least eight places to eat crappy food within eyesight from where you stand.
"From where you stand" is unnecessary. Just plain "nearby" would be a welcome substitute for "within eyesight from where you stand."
Those walking on solid ground might lose their hats in a gust of wind, but the manic depressive stands on a seasonal tightrope.
Notice that "but" is being asked to do what it can't possibly do: yoke together these two clauses. We have no idea what contradiction might exist here, if only because the first clause sounds literally plausible while the second is metaphorically confusing. "A seasonal tightrope"?
Occasional cars flash past us.
"Occasional" modifies "cars." That's the problem. How can a car be occasional? This use turns an adverb into an adjective. "Occasional," like "random" and "typical" and even "stereotypical," is often—or even occasionally—used in an almost meaningless way. Don't make time or frequency an attribute of the vehicle. Let the time or frequency indicator stand on its own. _Cars flash past us now and then_.
In a steep inner gorge, a sandstone amphitheatre has formed with a ribbon waterfall cascading from its hundred-foot precipice.
What's the problem? "With." It's trying to be both conjunction and relative pronoun when all it can be is a plain preposition. It also obscures the motion of the ribbon waterfall which "cascades" from a (not "its") hundred-foot precipice.
Displayed in glass cases and lit up like jewelry, tuna is cherished in Japan.
If the tuna comes first, the sentence won't sound so strange. _Tuna is cherished in Japan, even displayed in glass cases and lit up like jewelry_.
An array of noble mountains surround the valley. Robust with pines, spruces, and aspens, they are colorfully vibrant in the warm months.
Can you feel the emptiness of the modifiers here—"noble," "robust," "colorfully vibrant"? And note that summarizing word, "array." _Mountains surround the valley_.
My dad and I are similar in that we both hate "stuff."
"Are similar in that we" equals "both." _My dad and I both hate "stuff."_ Note that you can even do without "both."
He hunched his shoulders, placed one arm on his left leg, and slid into the passenger seat before reaching across his body for his seatbelt.
Can you actually visualize this action? No. Descriptions of physical action require incredible care because we read them with our bodies as well as our brains.
With those closest to me, I don't think twice about taking an unsolicited bite, whereas with newer acquaintances, I worry my probing fork will injure a tenuous relationship.
This writer is having trouble managing her stilts. _I don't worry about stealing a bite from friends. But I do if it's a new acquaintance_. "Whereas," "unsolicited," "tenuous relationship"—all trouble, as is the fact that the literal fork implicit in the first half of the sentence suddenly performs a metaphorical action in the second half.
We hold a mythological view of soldiers. We see them fly off filled with ideology and return wearied and homesick.
Note how the first sentence leads us to expect an explanation in the second sentence—an explanation of what the "mythological view" is. But it isn't forthcoming. And note too the ambiguity in the second sentence. It sounds very much as though we see the soldiers off and then return home ourselves feeling wearied and homesick. Filled with ideology?
Lovers on blankets support the theory that Sevillanos are more public with their affection than lovers anywhere else in the world.
But lovers without blankets disagree. The writer needs to return to the fundamental question: What am I trying to say? Losing "theory" would be a good place to start.
The small tube of sunblock weighs little in my right hand.
But it's much heavier when I hold it in my left. This sentence seems to be trying to say, _I'm holding a small tube of sunblock in my right hand_.
The watching of Super Bowl commercials has truly become a sacred tradition in this country.
"Watching," not "the watching of"—verb form versus noun phrase. This sentence dies by overemphasis. Get rid of "truly" and "sacred." In other words, let the sentence relax and trust that the reader will take your point. You don't have to be so insistent.
Even with my drapes pulled back, there's not much visibility from where I sit.
The possessive pronoun goes awry. This writer seems to be wearing curtains instead of eyeglasses. "The drapes." And how about _I can't see much from where I sit_? "There's not much visibility" could mean there's a heavy ground fog in the room. That's how the word "visibility" is used outside the peculiar world of this sentence.
The architecture was gray and beautiful and old and stretched out in all directions.
No, the buildings—but never the architecture.
The woman is twenty-eight, with the leathery sun-worn skin of a retired couple from Florida.
What a strange woman this is, to have the skin of a retired Florida couple in her possession. How could her sun-worn skin (nice phrase) resemble that of a couple? Why is she being compared to two people? It must have been some couple to have only one skin.
My mother had started planning this pilgrimage since the day I was born.
"Since"? And why the compound past? _My mother started planning this pilgrimage the day I was born_. Or _My mother has been planning this pilgrimage since the day I was_ _born_. The continuity of "since" requires the continuity of "has been planning."
She looked straight at me as she pulled the steering wheel around to make a graceful turn as she shook her head and said: "What a waste; I just don't get it."
"As" runs amok. Breaking this into at least two sentences would help eliminate "as."
The Lincoln-Marti School resides in Little Havana, Miami.
"Resides"? The author is clearly trying not to use "is." But why? It's simple and economical and doesn't make the reader feel as though the Lincoln-Marti School had retired to Florida.
Bread, rice, and bananas constituted my diet.
Ugly in so many ways. Why not choose a subject that is capable of eating? _I ate mostly bread, rice, and bananas_. "Constituted" is the kind of verb—abstract, dull, essentially passive, academic—that should immediately send you hunting for a stronger, more active one.
I had never seen the word "hubris" and allowed my familiarity with the similarly concluded word "debris" to guide my pronunciation.
Can you hear the writer's distrust of the reader (or of herself), as though we might not notice that "hubris" and "debris" both end in "ris"? _I'd never seen the word "hubris," so I pronounced it like "debris."_
I recall listening to a ten-minute soliloquy concerning the tomatoes in the refrigerator, which then moved from item to item on the shelves and kitchen table.
Can you feel the tomatoes moving about in this sentence? "Which" wants to point to "soliloquy" but can't.
The small houses are considered "quaint," with their well-manicured lawns and expensive landscaping of topiary, hedges, well-pruned fruit trees, cobbled footpaths lined with decorative pathlights.
Look what "with" does. It allows the writer to jumble together a pile of nouns and adjectives without going to the bother of constructing a sentence using verbs. Notice how "with" replaces syntactical possibilities that would make this a much stronger sentence.
If there is one landmark here, it is not the 335,024-square-foot anchor store so big it takes up two buildings.
After the opening phrase, we expect to learn what the one landmark is, not what it is not. A promise to the reader is not fulfilled.
While hospitals charge hundreds of thousands of rupees for a prosthetic leg, Jaipur Foot charges two thousand.
The first word is doing the steering, as if readers won't grasp the internal logic of the sentence unless we get a good long glimpse of it coming. _Hospitals charge hundreds of thousands of rupees for a prosthetic leg. Jaipur Foot charges two thousand_. The contrast is perfectly clear even without pointing to it.
Then the whole cat rises and stretches, arching its back and driving each leg into the ground until it quivers.
Pronoun problems. The antecedent of "it" is clearly meant to be "leg," and yet technically it is "ground"—the noun closest to the pronoun. This cat makes the ground quiver.
After seven hours of zooming, dropping, soaring and twisting through the air, the day had come to its natural end.
The day loves to zoom, drop, soar, and twist through the air. The opening phrase _must_ modify the subject of the sentence. But it doesn't.
Los Angeles is the largest city in California with a population of over 4,997,340 spanning 498 square miles.
What does this sentence actually say? That of all the cities in California with a population of over 4,997,340 spanning 498 square miles, Los Angeles is the largest, somehow. Again, the word "with" is the culprit. A comma after California would help, but the sentence would still be weak syntactically. "With" is not remotely strong enough to sustain a sentence like this.
Every morning, just before 9 a.m., artists begin to funnel into the building, as one by one, with deliberate steps, they get off the Paratransit bus.
The artists somehow funnel into the building while at the same time getting off the Paratransit bus. The problem? The overall sequence of the sentence, certainly. How about getting them off the bus first and then having them funnel into the building? But the problem is also "as," which insists on simultaneity. Note that reality is simply imperceptible in this sentence.
A world away from his son and his granddaughters, after over a year of suffering of Alzheimer's and diabetes, he died alone in a hospital in Taiwan.
A simple, effective revision—put the right phrase in the right place. _After months of suffering from Alzheimer's and diabetes, he died alone in a hospital in Taiwan, a world away from his son and granddaughters_.
Her clothes were nondescript, a white t-shirt and jean shorts.
And yet the writer can describe them. How about _She wore a white T-shirt and jean shorts_? The nature of her clothing—its plainness, its simplicity, even its nondescriptness—will be apparent to the reader.
Under the bridges connecting the canals' sides, there are shacks built from scrap metal and wood.
As if the bridges had a function other than connecting the sides of the canals. _Under the bridges are shacks built from scrap metal and wood_.
What we do share in common, though, is our voice.
"Share" implies "in common." And note how the force of the word "though" is already implicit in the very structure of the first four words—"what we do share." _What we do share is our voice_.
As you enter town, you're guided down Main Street by two Quaker burial grounds, one on either side of the road.
These Quaker burial grounds sound rather like traffic cops. The only way to fix this sentence is to back out of it entirely. Instead of writing from the perspective of a procession into town, describe the town and then let someone (if necessary) proceed into it.
In the distance, elephants and buffalo lumber across ancestral stomping grounds.
We say, "These are my old stomping grounds," without implying that we actually stomped there. And yet somehow, in this sentence, it's hard to avoid the picture of elephants and buffalo stomping on their ancestral grounds. What the writer is trying to say has been distorted by the unbidden presence of a cliché.
There are constantly trucks flowing in and out of National Meats.
The natural subject of this sentence is "trucks." _Trucks flow in and out of National Meats_. But trucks are bad at flowing. Perhaps "A stream of trucks flows..." But perhaps "flow" is a bigger problem than it's worth.
It was on a heated summer day when my partner Heather and I were assigned to patrol Riverside County.
"Heated"? This makes the day sound as though it had been warmed artificially or was perhaps angry with someone. And note how cumbersome the overall structure of the sentence is—largely because it begins with "It was on a..." _On a hot summer day, my partner Heather and I were assigned to patrol Riverside County_. This is less emphatic than the writer's version, but then the writer's version is also clumsy, and we can't tell what's really being emphasized or set up.
Following the fence we built the day before, we had come across a mud wallow, the calling card of wild pigs and a sore in ranchers' sides.
The pasts are confused. "Had built" and "came." Asking a wallow to be a pig calling card and a rancher's sore at the same time is asking too much.
By 1556, biographer Giorgio Vasari had written that _The Last Supper_ was ruined.
This sounds like an assertion about Vasari rather than an assertion about _The Last Supper_. What the sentence is trying to say is this: as early as 1556, some people thought _The Last Supper_ was already ruined. This sentence is telling us something different. It could be saying, for instance, that at last, by 1556, the notoriously slow-writing Giorgio Vasari was able to write that _The Last Supper_ was ruined. Also, note how inadequate that defining epithet—"biographer"—is. This is a journalistic habit, and a bad one.
We knew our miserable one mile per hour trekking rate was decreasing exponentially as we pressed on.
This sentence sounds neatly scientific, doesn't it? And yet it indulges in false specificity. If their rate of progress really did decrease exponentially, they'd never get there. Note how the action of walking—which should elicit a verb—vanishes from this sentence, thanks to "our miserable one mile per hour trekking rate."
I leaned against the parapet as the wind blasted me and looked out over the sea.
The wind likes to look out over the sea whenever it can. Who doesn't? Note how useless "as" is here.
She's wearing tapered, elastic-waist jeans that hang 2 inches above her ankles, and an oversized white sweatshirt with a blobby pink heart in the center.
How low is she wearing her jeans? We have to be able to picture how she actually wears them. At the very least, the sentence shouldn't actively prevent us from picturing it, the way this one does.
Kaneisha desperately wanted me to buy new clothes because mine were so bad that they embarrass everyone in their proximity.
Notice how the author suddenly seems to have vanished from her clothing. _Kaneisha desperately wanted me to buy new clothes because mine were so embarrassing_.
It is especially nice to sit there in the evening, when the sun has just set or is in the process of setting.
Or is only a few minutes away from beginning the process of setting or perhaps even beginning the process of just having finished setting. Simplify: when the sun is setting. That's enough.
My hometown gets less annual rainfall than Atlanta or Boston, but somehow we've acquired the gray reputation. Fine by me—our bad rap keeps too many Californians from moving north.
Note how the end of the second sentence completely contradicts the writer's meaning. It says, literally, that she wishes more Californians would move north. Get rid of "too many" and the sentence begins to say something sensible.
But perhaps living in Nevada would enable me an appreciation of beauty in scarcity.
"An appreciation of"? A noun phrase. No energy. Static. "To appreciate"? A verb. Better, but not perfect. Never substitute noun phrases for verbs. "Enable me"? How about this: _But perhaps living in Nevada would teach me to see the beauty in scarcity_?
The waves are loud as they crash against the beach.
Remove "as." Assume that "crash" contains the quality of loudness. _The waves crash against the beach_.
It is a comforting smell, evoking Proustian memories of sledding and snow and knitted mittens.
Yes, we now associate all sensory memory with Marcel Proust. We just didn't know he spent so much time sledding in his knitted mittens.
The world is calm, quiet, indifferent, moving at its own pace unfettered by the frenzy of human activity.
The writer has no idea what "fetter" means, much less "unfettered."
Capitola is outside the picture window.
It would be alarming if Capitola was _inside_ the picture window. Even the slightest effort will produce a more useful, descriptive sentence.
The week before my flight home departed, we decided to explore drugs.
How about "the week before I flew home"? That yields a verb instead of a noun and removes the unnecessary "departed." And the phrase "explore drugs"? Strictly a billboard cliché. No one would ever say, "Let's explore drugs," except ironically. No trace of irony here.
The view from the small balcony was of other apartment buildings.
What tells you that this sentence needs revising? How about the appalling "was of"? Can you feel how this sentence was written? Beginning with "the view" seemed to make sense. But note how summarizing that word is and how it excludes the presence of anyone capable of doing the viewing.
I have never found words, never heard or read any, that would have alleviated the aching and emptiness I always felt following death.
This sentence was written by someone who has died many times. Also "would have" is unnecessary. _I have never found words that ease the aching and emptiness I feel_. Make the dying happen in a separate sentence.
My cousin's baby gurgled in the middle of the carpeted living room, and mourners moved around her, tall legs gathering near a cheese plate.
This sentence is trying to adopt the baby's perspective—not a bad idea. But don't those tall legs sound disembodied? They may have gathered near a cheese plate, but one wonders how they managed to eat.
But when the Plains Indians hunted bison, they used every bit of the kill, from its meat to its dung and its hooves to its bone marrow; whites were notoriously wasteful.
Two sentences to begin with. Notice the awkwardness of the "from... to" structure, which implies a spectrum of possibilities. It's nearly always awkward. Instead: _they used every bit of the kill—meat, dung, hooves, bone marrow, and so on_.
And despite Crosby's distinguishable crooning, _White Christmas_ 's Danny Kaye always got the laughs.
What does "distinguishable" mean here? It means that when Bing Crosby croons, we can tell he's crooning.
The author probably means "distinctive." And note what this sentence doesn't say: Bing Crosby and Danny Kaye are both in _White Christmas_. This sentence makes it sound as though only Kaye is.
She exasperatedly cleans it up with her napkin and gets him more milk.
It takes only a fleeting thought to realize that "exasperatedly" is grotesque, a destroyer of rhythm. The writer will need to convey exasperation some other way.
I had to travel to Spain to see smoking on widespread display.
Note how inactive and unpeopled this is. Smoking is on widespread display in Spain, but who's doing the smoking?
The white steeple sits on the side of a hill across from the Long Ridge Tavern.
And yet one wonders what happened to the church beneath it. Steeples don't sit, except perhaps on the churches below them. They rise.
While one can lament the decrease in imagination that the television show's imagery perpetuates or the loss of emphasis placed on people's distinct intonation, such criticisms discredit the television version by comparison to an incorrigible other, not by its own merit.
One has to be _taught_ to write like this. And then one has to be taught _not_ to write like this. Barely intelligible.
Melissa later told me that a random man offered her $800 to spank him.
Whatever you think of this man, he was certainly not random. He was the very man who was going to offer Melissa $800 to spank him. "Random" has an actual meaning, and this is not it.
The air was hot and damp under the awning of branches and leaves that hung over us.
A common mistake—making explicit what's already implicit. _The air was hot and damp under the awning of branches and leaves_.
The 61–59 contest will be remembered by Duke's imposing size, Butler's resiliency, and the game's final play.
This sentence says that Duke's size, Butler's resilience, and the game's final play will remember the 61–59 contest. The problem? "By" instead of "for."
In the last ten years or so, the revival of the Los Angeles River has emerged as a major policy priority, as activists have successfully convinced public officials that revitalizing the LA River will help them fix the city's worst troubles.
What is the word "as" doing here? Merely joining two sentences and making a single awkward one. Remove "as" and you have two passable sentences.
Cacti are supposed to be the pinnacle of survival.
There's a thought here, though what it is is nearly impossible to say. "The pinnacle of survival" makes no sense. And note that "are supposed to be" could imply the author's disappointment with cacti.
My mom and I drove to where the flames were.
Or perhaps _drove toward the fire_?
#
_Acknowledgments_
Thanks to Dan Frank and, as always, to Flip Brophy. I'm grateful for support from the Ucross Foundation and the John Simon Guggenheim Memorial Foundation. I've been aided and helpfully challenged by colleagues in several English departments, including those at Fordham University, where I first started teaching creative writing, Harvard University, and, especially, Pomona College.
My greatest debt is to my students over the past thirty years. They've taught me far more than I've ever managed to teach them, and they've made teaching writing one of the great joys of my life.
#
A NOTE ABOUT THE AUTHOR
Verlyn Klinkenborg is a member of the editorial board of _The New York Times_ , to which he also contributes meditations about his small farm. Those pieces were collected in _The Rural Life_. His other books include _Making Hay_ , _The Last Fine Time_ , and _Timothy; or, Notes of an Abject Reptile_. He lives in Austerlitz, New York.
# ALSO BY VERLYN KLINKENBORG
_Timothy; or, Notes of an Abject Reptile_
_The Rural Life_
_The Last Fine Time_
_Making Hay_
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Questions for Written Answer
Questions for Written Answer for 2 December 2020
Tabled on 1 December 2020 and due for answer by 15 December 2020.
Lord Alton of Liverpool to ask Her Majesty's Government which contracts they awarded to PestFix for the provision of personal protection equipment; what was the total value of such contracts; and what value of equipment supplied under such contracts was designated as (1) unfit for use in the NHS, (2) fit for clinical use only, (3) suitable for supply or sale to departments other t… Item text truncated - open link to view full item text
Lord Alton of Liverpool to ask Her Majesty's Government which contracts they awarded to Ayanda Capital for the provision of personal protection equipment; what was the total value of such contracts; and what value of equipment supplied under such contracts were designated as (1) unfit for use in the NHS, (2) fit for clinical use only, (3) suitable for supply or sale to departments… Item text truncated - open link to view full item text
Lord Alton of Liverpool to ask Her Majesty's Government which contracts they awarded to PPE Medpro for the provision of personal protection equipment; what was the total value of such contracts; and what value of equipment supplied under such contracts were designated as (1) unfit for use in the NHS, (2) fit for clinical use only, (3) suitable for supply or sale to departments oth… Item text truncated - open link to view full item text
Lord Alton of Liverpool to ask Her Majesty's Government which contracts they awarded to Purple Surgical for the provision of personal protection equipment; what was the total value of such contracts; and what value of equipment supplied under such contracts were designated as (1) unfit for use in the NHS, (2) fit for clinical use only, (3) suitable for supply or sale to department… Item text truncated - open link to view full item text
Lord Alton of Liverpool to ask Her Majesty's Government what assessment they have made of reports (1) that Muslim and Christian minorities are being persecuted in India, and (2) that such persecution has increased during the COVID-19 pandemic following accusations that those minorities had spread COVID-19; and what discussions they have had with the government of India about (a) t… Item text truncated - open link to view full item text
Earl Attlee to ask Her Majesty's Government what formal quality assurance assessment, or similar, is required to be undertaken before an organisation can be accredited (1) to perform, and (2) to analyse, COVID-19 tests; and what is the minimum standard those organisations must meet before accreditation. HL10945
Lord Boateng to ask Her Majesty's Government what assessment they have made of the case for responding to COVID-19-related sovereign debt issues in Africa (1) by allocating special drawing rights by the International Monetary Fund to the worst affected countries, and (2) extending the G20's debt service suspension initiative to permit the participation of multilateral development… Item text truncated - open link to view full item text
Lord Boateng to ask Her Majesty's Government what plans they have (1) to support the creation of a permanent mechanism under the UN for the systematic comprehensive and permanent restructuring of sovereign debt, and (2) to set up a programme of continuing technical assistance to improve (a) debt transparency, and (b) the debt management offices, of those countries worst affected b… Item text truncated - open link to view full item text
Lord Browne of Belmont to ask Her Majesty's Government how many former military bases they are currently using for other purposes. HL10948
Lord Browne of Belmont to ask Her Majesty's Government what steps they intend to take to support people with diabetes who return to work in their normal workplace during the COVID-19 pandemic. HL10949
Lord Browne of Belmont to ask Her Majesty's Government what discussions they have had with the Northern Ireland Executive and the government of the Republic of Ireland about addressing cross-border crime during the Christmas period. HL10950
Lord Browne of Belmont to ask Her Majesty's Government how many successful prosecutions have resulted from the activities of Driver and Vehicle Licensing Agency enforcement officers in (1) 2019, and (2) 2020 to date. HL10951
Lord Browne of Belmont to ask Her Majesty's Government what (1) schemes, and (2) initiatives, they plan to introduce (a) to strengthen the Union, and (b) foster more links between England, Scotland, Wales and Northern Ireland. HL10952
Baroness Coussins to ask Her Majesty's Government, further to the statement in the Spending Review 2020, published on 25 November, that the Department for Education settlement "provides funding to prepare for a UK-wide domestic alternative to Erasmus+, in the event that the UK no longer participates in Erasmus+, to fund outward global education mobilities", whether such funding wo… Item text truncated - open link to view full item text
Baroness Coussins to ask Her Majesty's Government what assessment they have made, if any, of the domestic scheme to replace Erasmus+ set up by Switzerland; and if no such assessment has been made, what plans they have to assess the effectiveness of the Swiss replacement scheme before committing to a UK domestic alternative. HL10954
Baroness Coussins to ask Her Majesty's Government whether, in addition to university student placements, any domestic alternative to Erasmus+ would cover (1) school exchanges for pupils and teachers, (2) technical education and workplace training, (3) work shadowing and youth work, (4) sports, (5) older people, (6) disadvantaged areas, (7) arts projects, and (8) people with disabi… Item text truncated - open link to view full item text
Lord Dodds of Duncairn to ask Her Majesty's Government what plans they have to consult Northern Ireland businesses about legislation on anti-avoidance measures for qualifying Northern Ireland goods being sent to the rest of the UK. HL10956
Lord Hay of Ballyore to ask Her Majesty's Government how many people with attention deficit hyperactivity disorder they have supported in 2020. HL10957
Lord Hay of Ballyore to ask Her Majesty's Government what assessment they have made of the effect of the lockdown restrictions in place to address the COVID-19 pandemic on the airline industry over the Christmas period. HL10958
Lord Hay of Ballyore to ask Her Majesty's Government what assessment they have made of the effect of the lockdown restrictions in place to address the COVID-19 pandemic on the small business sector over the Christmas period. HL10959
Lord Hay of Ballyore to ask Her Majesty's Government what steps they are taking to encourage business travel (1) to help the airline industry, (2) to increase tourism, and (3) to support airport workers back into full-time employment. HL10960
Lord Hay of Ballyore to ask Her Majesty's Government what resources they have made available to support Belfast International Airport during the COVID-19 Pandemic. HL10961
Baroness Hayman of Ullock to ask Her Majesty's Government what progress they have made in ensuring that care homes are granted insurance indemnity under the same terms as provided for the NHS under the Coronavirus Act 2020. HL10962
Baroness Hayman of Ullock to ask Her Majesty's Government what financial support they intend to provide to care homes to enable such homes to offer twice weekly tests to two visitors for each care home resident. HL10963
Baroness Hayman of Ullock to ask Her Majesty's Government what assessment they have made of Methodist Homes' care home visitor policy. HL10964
Baroness Hayman of Ullock to ask Her Majesty's Government what steps they are taking to offer COVID-19 vaccinations to family carers of care home residents. HL10965
Baroness Hayman of Ullock to ask Her Majesty's Government when they estimate that family members and friends of care home residents will be provided with twice weekly testing to enable them to visit; and whether they still plan to implement that policy before 25 December. HL10966
Baroness Hayter of Kentish Town to ask Her Majesty's Government, further to the findings of the report by the Alcohol Health Alliance UK Small change: alcohol at pocket money prices, published in November, what steps they are taking to limit the sale of cheap, strong alcohol. HL10967
Baroness Hayter of Kentish Town to ask Her Majesty's Government what steps they are taking to ensure the new alcohol duty system (1) protects public health, and (2) helps to reduce alcohol harm. HL10968
Baroness Hayter of Kentish Town to ask Her Majesty's Government what assessment they have made of the health impacts of freezing alcohol duty in the 2020 Budget. HL10969
Lord Hussain to ask Her Majesty's Government what discussions they have had with the other permanent members of the UN Security Council about (1) the four letters written by the UN rapporteurs on torture, arbitrary detentions, extra-judiciary and custodial killings in Indian-administered Jammu and Kashmir to the government of India, and (2) the absence of any response to date by t… Item text truncated - open link to view full item text
Lord Lamont of Lerwick to ask Her Majesty's Government when they expect to reply to the letter sent by Lord Lamont of Lerwick to the Chancellor of the Exchequer on 1 September. HL10971
Lord Lancaster of Kimbolton to ask Her Majesty's Government, further to the Written Answer by Baroness Sugg on 18 November (HL10238), what estimate they have made of the total cost of the infrastructure work required on Ascension Island. HL10972
Lord Lancaster of Kimbolton to ask Her Majesty's Government, further to the Written Answer by Baroness Sugg on 18 November (HL10238), what application has been made to the Conflict Stability and Security Fund for infrastructure work on Ascension Island for the 2021/22 financial year onwards. HL10973
Lord Lancaster of Kimbolton to ask Her Majesty's Government, further to the Written Answer by Baroness Sugg on 18 November (HL10238), how much revenue has been raised by the government of Ascension Island through taxes in each of the last five years; and what proportion of that revenue has been allocated to infrastructure improvement. HL10974
Lord Lancaster of Kimbolton to ask Her Majesty's Government, further to the Written Answer by Baroness Williams of Trafford on 18 November (HL9994), how many (1) complaints related to non-compliance of section 54 of the Modern Slavery Act 2015 they have received, and (2) subsequent court injunctions the Home Secretary has applied for. HL10975
Lord Lancaster of Kimbolton to ask Her Majesty's Government, further to the Written Answer by Baroness Williams of Trafford on 18 November (HL9994), when they plan to implement the changes to strengthen and future proof transparency of the Modern Slavery Act 2015 highlighted in their response to the Independent Review of the Modern Slavery Act, published on 22 September. HL10976
Lord Lucas to ask Her Majesty's Government, further to the High Court judgment in R (on the application of) Quincy Bell and A -v- Tavistock and Portman NHS Trust and others [2020] EWHC 3274, issued on 1 December, what plans they have to (1) prevent, or (2) discourage, GenderGP.com from marketing puberty blocking drugs to anyone under the age of 16; and whether they intend to issue… Item text truncated - open link to view full item text
Baroness McIntosh of Pickering to ask Her Majesty's Government what steps they are taking (1) to prevent the sale of counterfeit drugs to developing countries, and (2) to ensure that only patented drugs are sold to those countries. [I] HL10978 [I]
Baroness Parminter to ask Her Majesty's Government how many sites of special scientific interest are on the defence estate; how much land those sites cover; and how many such sites are in a favourable condition. HL10979
Lord Pendry to ask Her Majesty's Government, further to the debate on 26 November (HL Deb, cols 403–23), whether they have made any assessment of the need to provide portable toilets for use by heavy goods vehicle drivers using the port of Dover after 1 January 2021; and if so, how many such toilets they estimate will be required. HL10980
Lord Pendry to ask Her Majesty's Government, further to the debate on 26 November (HL Deb, cols 403–23), whether they have any plans to procure portable toilets for use by heavy goods vehicle drivers using the port of Dover after 1 January 2021; and if so, what steps they are taking to ensure that Ministers in any such procurement process declare (1) personal interests, and (2) in… Item text truncated - open link to view full item text
Lord Porter of Spalding to ask Her Majesty's Government whether councils will be given discretion over the payment of the £500 self-isolation Test and Trace support grant so that councils can pay those claimants who do not submit their claim within two weeks, providing a legitimate reason is given for the delay in the application. HL10982 [I]
Baroness Ritchie of Downpatrick to ask Her Majesty's Government what assessment they have made of the effect of uncertainty over the UK's relationship with the EU after the transition period for the UK's departure from the EU on the economy. HL10983
Baroness Ritchie of Downpatrick to ask Her Majesty's Government what assessment they have made of the effect of the Spending Review 2020 on the UK's economic recovery from the COVID-19 pandemic. HL10984
Baroness Ritchie of Downpatrick to ask Her Majesty's Government what assessment they have made of the impact of the withdrawal of the Coronavirus Job Retention Bonus Scheme on business planning. HL10985
Baroness Ritchie of Downpatrick to ask Her Majesty's Government what plans they have to publish any modelling from September 2020 on the economic effect of (1) introducing the COVID-19 'circuit-breaker' recommended by the Scientific Advisory Group for Emergencies, and (2) the introduction of a lockdown to address the COVID-19 pandemic in November 2020 for England. HL10986
Baroness Ritchie of Downpatrick to ask Her Majesty's Government what representations they have received from organisations representing people who are ineligible for COVID-19 financial support schemes. HL10987
Baroness Ritchie of Downpatrick to ask Her Majesty's Government what assessment they have made of the comparative effect of the COVID-19 financial support package on regional economies. HL10988
The Lord Bishop of St Albans to ask Her Majesty's Government what plans they have to provide funding to charitable hospice care providers who have lost community funding during the COVID-19 pandemic. HL10989
The Lord Bishop of St Albans to ask Her Majesty's Government what plans they have to increase the funding that clinical commissioning groups provide to the charitable hospice sector. HL10990
The Lord Bishop of St Albans to ask Her Majesty's Government what plans they have to provide funding to Rennie Grove Hospice Care to mitigate the impact of the COVID-19 pandemic. HL10991
The Earl of Sandwich to ask Her Majesty's Government what assessment they have made of the report by Save the Children Defenceless: The impact of the Israeli military detention system on Palestinian children, published on 29 October; and what response they have made to that report. HL10992
Lord Scriven to ask Her Majesty's Government what assessment they have made of reports that Sheffield City Council has instructed care homes in Sheffield not to use COVID-19 lateral flow tests for admitting visitors due to concerns that such tests lack accuracy. HL10993
Lord Smith of Finsbury to ask Her Majesty's Government what plans they have to allow the temporary reduced rate of VAT of five per cent for certain attractions to apply where a concert is being streamed as live, but for technical reasons was required to be pre-recorded. HL10994 [I]
Lord Taylor of Warwick to ask Her Majesty's Government what plans they have to allow first year university students from the EU, who have not yet moved to the UK, to qualify for EU pre–settled status. HL10995
Lord Taylor of Warwick to ask Her Majesty's Government what plans they have to increase basic digital skills education programmes for adults. HL10996
Lord Taylor of Warwick to ask Her Majesty's Government what steps they are taking to provide up to date training for Job Centre staff on the skills that employers are currently seeking. HL10997
Lord Taylor of Warwick to ask Her Majesty's Government what steps they are taking to remove barriers to higher education for Gypsy, Roma and Traveller pupils. HL10998
Baroness Tonge to ask Her Majesty's Government what representations they have made to the government of Israel about ensuring that any Israel Defence Force soldier who is found guilty of (1) killing, or (2) injuring, any Palestinian is sentenced appropriately. HL10999
Baroness Tonge to ask Her Majesty's Government what assessment they have made of reports that Palestinians living in Sheikh Jarrah are being evicted by the government of Israel; and what steps they are taking in response to any such assessment. HL11000
Lord Truscott to ask Her Majesty's Government what assessment they have made of the BBC Panorama investigation into (1) shared ownership schemes, and (2) abuses of the leasehold system. HL11001
Lord Truscott to ask Her Majesty's Government what plans they have to reform the leasehold system of ownership during 2021. HL11002
Lord Truscott to ask Her Majesty's Government whether they have sufficient quantities of effective alternative vaccines for the entire UK population following the recent AstraZeneca test results. HL11003
Lord Watson of Invergowrie to ask Her Majesty's Government, further to the Written Answer by Baroness Berridge on 10 November (HL9605), how many tutors under the National Tutor Programme were supporting schools (1) at the start of November, and (2) at the start of December. HL11004
Lord Watson of Invergowrie to ask Her Majesty's Government, further to the Written Answer by Lord Bethell on 24 November (HL10313), when they estimate that the revised project timeline for the National Institute of Health and Care Excellence quality standard on foetal alcohol spectrum disorder will be published. HL11005
Lord West of Spithead to ask Her Majesty's Government whether overseas territories which are eligible for Official Development Assistance will be able to apply for large-scale project funding that is equivalent to that provided through Darwin Plus. HL11006
Lord West of Spithead to ask Her Majesty's Government, further to the statement on the Integrated Review of Foreign, Defence, Security and Development policy by the Prime Minister on 19 November (HC Deb, col 495), when they plan to order the three fleet solid support ships. HL11007
Lord West of Spithead to ask Her Majesty's Government, further to their commitment in The Conservative and Unionist Party Manifesto 2019, published on 24 November 2019, to invest "£500 million in youth services for young people", what plans they have to make this investment; and why this investment did not appear in the Spending Review 2020, published on 25 November. HL11008
Lord West of Spithead to ask Her Majesty's Government what assessment they have made of (1) the distinction between commercial and military offshore unexploded ordnance disposal by deflagration, and (2) the effectiveness of deflagration in the offshore disposal of unexploded ordnance. HL11009
Lord West of Spithead to ask Her Majesty's Government what assessment they have made of (1) possible sites in the United Kingdom for the dismantling of ships and the disposal of any hazardous materials that they might contain, and (2) the potential global demand for, and value of, such operations. HL11010
Legend for business item text
† Item that is new or altered.
[I] Indicates that the member concerned has a relevant registered interest.
* Item is an oral question.
Numbers starting HL are unique identifiers for written questions, Bills or papers.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,426
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Q: Using Monotouch with iSQL SDK Is it possible to use the iSQL SDK (www.mobilefoo.com) with MonoTouch considering that the iSQL SDK is a client library meant for use with XCODE and objective-c code?
A: Yes - not a complex task but a bit time-consuming. Although you should only have to do it once (until they extend their libraries.)
Instructions on how to bind to Objective C libraries and types are here: http://wiki.monotouch.net/HowTo/Interop/Consuming_3rd_Party_ObjC_Libraries
It might well be worth contacting mobilefoo about this too as I'd imagine they'd be interested and might even help.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,145
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E-text prepared by Steven desJardins and the Project Gutenberg Online
Distributed Proofreading Team (http://www.pgdp.net)
Transcriber's note:
This novel was originally serialized in four installments in
_All-Story Weekly_ magazine from December 30, 1916, to January
20, 1917. The original breaks in the serial have been retained,
but summaries of previous events preceding the second, third,
and fourth installments have been moved to the end of this
e-book. The Table of Contents which follows the introduction
was created for this electronic edition.
NUALA O'MALLEY
by
H. BEDFORD-JONES
ALL-STORY WEEKLY
VOL. LXVI NUMBER 2
SATURDAY, DECEMBER 30, 1916
NUALA O'MALLEY
by
H. Bedford-Jones
Author of "Malay Gold," "The Ghost Hill," "John Solomon, Supercargo,"
etc.
This is a stirring, entrancing story of Erin when Cromwell was
campaigning, and when the fighting heritage that is every Irishman's
found vent through sword and ax and fire. You meet Brian Buidh, Brian of
the Yellow Hair, more thrilling than even your favorite movie hero; and
as for Nuala herself--well, just wait till you meet her!--THE EDITOR.
TABLE OF CONTENTS
Part I
I. THE BLACK WOMAN. 177
II. THE BEGINNING OF THE STORM. 179
III. THE DARK MASTER. 182
IV. BRIAN LEANS ON HIS SWORD. 186
V. YELLOW BRIAN RIDES SOUTH. 191
VI. BRIAN TAKES CAPTIVES. 196
VII. THE BIRD DAUGHTER. 201
Part II
VIII. HOW BRIAN WAS NETTED. 419
IX. THE NAILING OF BRIAN. 424
X. IN BERTRAGH CASTLE. 429
XI. THE BAITING OF CATHBARR. 434
XII. HOW THE DARK MASTER WAS RUINED. 438
Part III
XIII. BRIAN RIDES TO VENGEANCE. 659
XIV. HOW THE STORM FARED NORTH. 664
XV. WHAT HAPPENED AT THE TARN. 670
XVI. BRIAN GETS HIS SWORD AGAIN. 674
XVII. BRIAN GOES A CRUISING. 679
Part IV
XVIII. BRIAN YIELDS BERTRAGH. 137
XIX. BRIAN MEETS THE BLACK WOMAN. 142
XX. THE STORM BURSTS. 147
XXI. CATHBARR YIELDS UP HIS AX. 151
XXII. THE STORM OF MEN COMES TO REST. 155
CHAPTER I.
THE BLACK WOMAN.
The horseman reined in as his jaded steed scrambled up the shelving
bank, and for a space sat there motionless, for which the horse gave
mute thanks. The moon was struggling to heave through fleecy clouds, as
it was hard on midnight; in the half obscurity the rider gazed around
suspiciously.
There was nothing in sight to cause any man fear. Behind him rippled the
Dee, and all around was desolation. Ardee itself lay a good two miles in
the rear, burned and laid waste six weeks before, and ten miles to the
south lay Drogheda. Indeed, as the horseman gazed about, he caught
sight of a faint glare on the horizon that drew a bitter word from his
lips.
Dismounting with some difficulty, owing to his cloak and Spanish hat, he
examined a long, raking gash in his horse's flank; then flung off hat
and cloak and calmly proceeded to bind up his own naked shoulder
beneath.
His was a strange figure, indeed, now that he stood revealed. He wore no
clothing save breeches and high riding-boots; an enormous sword without
a sheath was girt about his waist, and the caked blood on his shoulder
and cheek made his fair skin stand out with startling contrast.
About his shoulders fell long hair of ruddy yellow, while his face was
young and yet very bitter, tortured by both physical and mental
anguish, as it seemed. He bound up the deep slash in his shoulder with a
strip of cloth torn from his cloak, felt his wealed cheek tenderly, then
flung the cloak about him again and drew down his broad-brimmed hat as
he turned to his weary horse.
"Well, my friend," and his voice sounded whimsical for all its rich
tone, "you've had a change of masters to-day, eh? I'd like to spare you,
but man's life is first, though Heaven knows it's worth little in
Ireland this day!" With that he reeled and caught at the saddle for
support, put down his head, and sobbed unrestrainedly.
"Oh, my God!" he groaned at length, straightening himself to shake a
clenched and blood-splashed fist at the sky. "Where were You this day?
God! God! The blood of men on Thine altars--"
"Faith, you must be new come to Ireland, then!"
At the shrill, mocking voice the man whirled about and his huge blade
was out like a flash. But only a cackling laugh answered him, as down
from the bank above slipped a perfect hag of a creature, and he drew
back in alarm. At that instant the moon flooded out; his sudden motion
had flung off his wide hat, and he stood staring at the wrinkled
creature whose scanty garments and thin-shredded gray locks were pierced
by a pair of weird brown eyes.
Then he quivered indeed, and even the poor horse took a step backward,
for the old woman had flung up her arms with a shrill cry as she gazed
on the yellow-haired young man.
"The O'Neill!" The words seemed to burst from her involuntarily. She
craned forward, her hands twisting at her ragged shawl, and a flood of
Gaelic poured from her lips as she stared at the awe-struck man.
"Are you, then, the earl, come back from the dead? Ghost of Tyr-owen,
why stand you here idle in the gap of Ulster, where once Cuculain fought
against the host of Meave? Do you also stand here to fight as he
fought--"
"Peace, mad-woman!" exclaimed the young man, stooping after his hat.
"Peace, and be off out of my way, for I have far to ride."
The Gaelic words came roughly and brokenly from him, but the old hag
took no heed. Instead, she advanced swiftly and laid her hand on his
arm, still gazing into his face with a great wonder on her wrinkled
features.
"Who are you?" she whispered. "Tell the Black Woman your name, if you
are no ghost! For even as you stand now, once did these eyes see the
great earl himself."
"I am from Drogheda," answered the man, something very like fear stamped
on his powerful and bitter-touched young face. "My name is Brian Buidh,
and I ride to join Owen Ruadh--"
"Liar!" The old woman spat forth the word with a cackle of laughter.
"Oh, you cannot fool the Black Woman, Yellow Brian! Listen--Brian your
name is, and Yellow Brian your name shall be indeed, since this is your
will. Owen Ruadh O'Neill lies at the O'Reilly stead at Lough Oughter,
but you shall never ride to war behind him, Brian Buidh! No--the Black
Woman tells you, and the Black Woman knows. Instead, you shall ride into
the west, and there shall be a storm of men--a storm of men behind you
and before you--"
"For the love of Heaven, have done!" cried Yellow Brian, shrinking
before her, and yet with anger in his face. "Are you crazed, woman?
Drogheda has fallen; O'Neill must join with the royalists, and never
shall I ride into the west. Be off, for I have no money."
He turned to mount, but again she stopped him. It seemed to him that
there was strange power in that withered hand which rested so lightly on
his arm.
"The Black Woman needs no money, Yellow Brian," she cackled merrily.
"You shall meet me once again, on a black day for you; and when you meet
with Cathbarr of the Ax you shall remember me, Brian Buidh; and when
you ride into the west and meet with the Bird Daughter you shall
remember me.
"So go, Yellow Brian, upon whose heart is stamped the red hand of the
O'Neills! _Beannacht leath!_"
"_Beannacht leath_," repeated the man thickly.
There was a rustle of bushes, and he was alone, wiping the cold sweat
from his face.
"Woman or fiend!" he muttered hoarsely. "How did she know that last?
Yes, she was crazed, no doubt. I suppose that I do look like the
earl--since he was my grandfather!"
And with a bitter laugh he climbed into the saddle and pushed his horse
up the bank. The bushes closed behind him, the night closed over him,
but it was long ere the weird words of the old hag who called herself
the Black Woman were closed from his mind.
For, after all, Yellow Brian was of right not alone an O'Neill, but The
O'Neill.
CHAPTER II.
THE BEGINNING OF THE STORM.
The people of every nation--that is, the tillers of the soil, the people
who form the backbone of their race--are in continual expectancy of a
Man and a Day. Theirs is always the, perhaps, dumb hope, but still the
hope, that in their future lie these two things, a Man and a Day.
Sometimes the Man has come and the Day has failed; sometimes the Day has
come and there has been no Man to use it; but now all Ireland had swept
up in a wild roar, knowing that the Man and the Day had come together.
And so, in truth, they had. Owen, the Ruadh, or red, O'Neill, had fought
a desperate struggle against the royalists. Little by little he had
cemented his own people together, his personal qualities and his
splendid generalship had overborne all else, and the victory of Benburb
had crowned the whole. Then Owen Ruadh was stricken down with sickness,
Cromwell landed and stormed Drogheda, and Yellow Brian had fought clear
and fled away to the kinsman he had never seen.
Now, standing on the castle ramparts overlooking Lough Oughter, Yellow
Brian stared moodily out at the lake. His identity had been revealed to
none, and the name of Brian Buidh had little meaning to any in Ireland.
Years since he who was The O'Neill, the same whom the English called
Earl of Tyr-owen, had fled with his family from the land. His eldest son
John had settled at the Spanish court.
John was a spineless man, unworthy son of a great father, content to
idle away his life in ease and quiet. And it was in the court of Spain
that Brian O'Neill had been born, with only an old Irishwoman to nurse
him and teach him the tongue and tidings of Ireland which his father
cared nothing for.
Yellow Brian had written out these things, sending the letter to the
sick general who lay within the castle. His terrible news of Drogheda
had created consternation, but already O'Neill's forces had been sent to
join the royalists against the common foe. All Ireland was distraught by
war. Royalist, patriot, and Parliament man fought each against the
other, and the only man who could have faced Cromwell lay sick unto
death.
The Day was passing, the Man was passing, and shadow lay upon all the
land.
A man came up and touched Yellow Brian's arm, with word that Owen Ruadh
would see him at once. Brian nodded, following. He was well garbed now,
and a steel jack glittered from beneath his dark-red cloak as he strode
along. Upon his strong-set face brooded bitterness, but his eyes were
young for all their cold blue, and his ruddy hair shone like spun gold
in the sunlight; while his firm mouth and chin, his erect figure, and
his massive shoulders gained him more than one look of appreciation
from the clustered O'Reillys.
He followed the attendant to a large room, whose huge mantel was carven
with the red hand and supporting lions of the clan Reilly, and passed
over to the bed beside the window. He had requested to see O'Neill
alone, and the attendant withdrew silently. Brian approached the bed,
and stood looking down at the man who was passing from Ireland.
Sharp and bright were the eyes as ever, but the red beard was grayed and
the face was waxen; a spark of color came to it, as Owen Ruadh stretched
forth a hand to take that of his visitor.
"Brian O'Neill!" he exclaimed, in a voice singularly like that of Brian
himself. "Welcome, kinsman! But why the silence you enjoined in your
letter?"
"My name is Yellow Brian," answered the younger man somberly. "I have
none other, general. You know the gist of my story, and here is the
rest. I broke with my father, for he would hear nothing of my coming to
Ireland. So I cast off his name and left him to his cursed idleness,
reaching Drogheda barely in time to take part in the siege. I managed to
cut through, as you know, and meant to take service with you--"
He paused, for words did not come easily to him, as with all his race. A
low groan broke from the crippled warrior.
"Too late, kinsman, too late! Cromwell is come, and I will never sit a
horse again--ah, no protests, lad! How old are you?"
"Twenty-three."
"By my faith, you look thirty! Lad, my heart is sore for you. I am
wasted and broken. I have no money, and Cromwell will shatter all before
him; I can do naught save give you advice."
"I want naught," broke in Brian quickly, a little glint as of ice in his
blue eyes. "Not for that did I cast off my name and come to--"
"Tut, tut, lad!" O'Neill reproved him gently. "I understand, so say no
more of that matter. You are Brian Buidh, but to me you are my kinsman,
the rightful head of my house. You can do two things, Yellow
Brian--either follow my advice, or go down to ruin with all Ireland. Now
say, which shall it be?"
Brian gazed at him with thoughtful face. What was the meaning of this
dark speech? As he looked into the keen, death-smitten eyes of the man
who might have saved Ireland, he smiled a little.
"I see naught but ruin, Owen Ruadh," he replied slowly. "I care little
for my life, having no ties left on this earth--"
"Oh, nonsense!" broke in the other impatiently. "You are young, lad--the
bitterness will soon pass, trust me. Now see, here is my advice, such
advice as I would give no other man alive. I am dying, Yellow Brian.
Well, I know that Cromwell will break down all I have built up, and I
can see no brightness for my country. But for you I can see much. You
are young, powerful, the last of the old race; you look strangely like
the old earl, Brian!"
The younger man started. For the first time in many days he remembered
that crazed hag he had met by the Dee water the night of Drogheda.
"Now, harken well. I tell you that our house lies in the dust, Brian;
there is no hope for it or for any O'Neill. But for Yellow Brian there
is hope. You must carve out a holding for yourself, for you are a ruler
of men by your face, lad. Go into Galway, and there, where Cromwell's
men will have hardest fighting of all, gather a force and make head. I
have heard strange tales of a man who has done this very thing--they say
he has seized on a castle somewhere near Bertraghboy Bay, in Galway,
and-- But I am getting weak, Brian lad. Hearken well--Ireland is lost;
carve out now for your own hand, for the Red Hand of the old house, lad!
And take this for my sake."
Almost whispering the last words, Owen Ruadh took from his finger a
signet graven deeply with the Red Hand of Tyr-owen. Brian accepted it
gravely, kissed the hand that gave it, and with tears choking his
throat, left the chamber of the man who was passing from Ireland.
He had been there a brief fifteen minutes, yet it seemed that an age had
passed. Both he and the sick man had said much in few words, for they
were both men who spared speech and did much. But Brian had received a
great wrench.
As he had said, he had cast off his father, for the grandfather's blood
ran riot within him, and had kindled to burning rage against the
sluggard who had made his name a thing of reproach in all lands. With
the overstrong bitterness of youth he had meant to die sword in hand,
fighting for Ireland. The few burning words of Owen Ruadh had stripped
all this false heroism from him, however, and had sent a flame of sanity
into his brain.
Brian returned slowly to the round tower, and stood looking out over the
waters, for the castle was built on an island in the lake a mile from
shore. It was nearing sunset, and snow was in the air--the first snow,
for this was the end of September.
"Ruin--the storm of men!" He repeated unconsciously the words of the hag
who had stopped him by the Dee water. "What shall I do? Which is the
part of a man, after all; to fall for Ireland or to hew out new lands
and found a new house in the west? By my hilt! That old hag told me
truly after all!"
At that thought he stood silent, his eyes troubled. What was this fate
which seemed to drive him into the west, instead of leading him to the
flame of swords as he had so long hoped and dreamed? Death meant little
to him; honor meant much. All his life he had lived in Spain, yet it had
been a double life. He had ridden and hunted and learned arms with the
young nobles of the court, but he had talked and sorrowed and dreamed
with the old Irishwoman who had nursed him.
After all, it is often the dreams of the youth which determine the
career of the man, he reflected.
Which path should he take? As he stood there struggling with himself,
his hand went unconsciously to his long, powerful jaw; it was a gesture
habitual with him when in deep thought--which he seldom was, however.
Now the youth in him spoke for death, now the sanity which had flashed
into his brain from that of the sick man spoke for the life of deeds and
renown which lay in the west.
An incident might turn him either way--and the incident came in the
shape of a very tall old man who wore the Irish garb of belted,
long-sleeved tunic and woolen hose, with iron-soled shoes. The old man's
face was cunning, but his eyes were bright and keen and deep gray; his
gray hair hung low to conceal his lopped ears, and there hung about him
an indescribable air of shrewdness faced with apparent openness of
heart.
Brian glanced at him, remembered that he had heard him called Turlough
Wolf, and looked away carelessly, absorbed in his own thought. But the
old man halted abruptly with an exclamation:
"_Corp na diaoul!_ Where got you that face and that gesture, Drogheda
man?"
Brian looked at him, frowning.
"What mean you, Turlough Wolf?"
The other stared, his thin jaw fallen.
"Why--why," he stammered, "I thought it had been The O'Neill come to
life again! When I was a boy I have seen the earl hold his hand to his
chin--often, often! And--and you look like him, Brian Buidh---"
"Nonsense!" Brian forced a laugh, but as he folded his arms again the
glitter of O'Neill's ring on his finger caught the sharp gray eyes.
Turlough Wolf started.
"Listen!" he said, coming forward insinuatingly. "Yellow Brian, no man
knows who you are, nor do I ask. But Turlough Wolf knows a man when he
sees one, a chieftain among men. I owe no man service; but if you will
need a swift brain, a cunning hand, and an eye that can read the hearts
of men, I will serve you."
Brian looked down into the shrewd face in wonder, then waved an
impatient hand.
"No use, Turlough Wolf. I have no money to pay for service, and to-night
I must ride out to seek I know not what--nay, whether I ride west or
east or south, I know not!"
He turned abruptly, wishing to close the matter, but the old man laid a
restraining hand on his shoulder.
"I seek no money, Yellow Brian. I seek only a master such as yourself; a
man who is a master among men, and whom I can set higher still if he
will heed my counsels. I am old, you are young; I know all parts of the
land by heart, from the Mayo shore to Youghal, and I am skilled at many
things. Take my service and you will not regret it."
Brian hesitated. After all, he considered, the thing came close to being
uncanny. The Black Woman by Dee water; Owen Ruadh himself, and now this
Ulysseslike Turlough Wolf--whither was fate driving him? Was he really
to meet such persons as the Bird Daughter and Cathbarr of the Ax, or
were they only the figment of a crazed old woman's brain?
So he hesitated, gazing down into those clear gray eyes. And as he
looked it seemed to him that he found strange things in them, strange
urgings that touched the chords of his soul. After all, adventure lay in
the west, and he was young!
"Good!" he said, gravely extending his hand. "To-night we ride to the
west, you and I. Come; let us see O'Reilly about horses."
And this was the beginning of the storm of men that came upon the west.
CHAPTER III.
THE DARK MASTER.
"There are two things, Yellow Brian, for you to mind. First, you must
have men at your back who know you for their master; second, you must
stand alone, giving and receiving aid from no man or party in the
land."
Brian nodded and stored away the words in his heart, for in their three
weeks of wandering he had learned that Turlough Wolf was better aid than
many men. It was his doing that, when they had chanced on a party of
ravagers beyond Carrick, Yellow Brian had been led into strife with
their leader. The upshot of that matter was that there was a dead rover;
Yellow Brian had a dozen horsemen behind him and money in his purse, and
of the dozen none but feared utterly this silent man who fought like a
fiend.
To the dozen had been added others--four Scotch plunderers strayed from
Hamilton's horse and half a dozen Breffnians from Ormond's army, who had
been driven out of Munster by the rising of the Parliament men there.
They were a sadly mixed score, of all races and creeds, but were
fighting ruffians to a man, and were bound together by Brian's solemn
pledge that he himself would slay any who quarreled. The result was
peace.
So now, with a good score of men behind him, Yellow Brian had ridden
down into Galway, was past Lough Corrib and Iar Connaught, and was hard
upon Connemara.
There was a thin snow upon the hills, and the bleak wind presaged more;
but the score of men sang lustily as they rode. Two days before they had
come upon a dozen strayed Royalist plunderers, and had gained great
store of food and drink--particularly drink. So all were well content
for the time being.
"Turlough," asked Brian suddenly, as they rode side by side, "did you
ever hear of one called the Black Woman?"
The Wolf crossed himself and grimaced.
"That I have, Yellow Brian, but dimly. They say she deals in magic and
sorcery, and no good comes of meeting with her. But stop--there are
horsemen on the road! Scatter the men, and quickly; let us two bide
here."
There was cunning in the advice, for the two had come to a bend in the
road and the men were a hundred yards behind them. Brian drew rein at
sight of a score of men a scant quarter-mile away and riding up the hill
toward them. He knew that they must also have been seen, but his men
would still be out of sight, so he turned with a quick word:
"Off into the rocks, men! If I raise my sword, come and strike. Off!"
As he spoke he bared that same huge cut-or-thrust brand he had borne
from Drogheda and set the point on his boot. Instantly the men scattered
on either side the road, where black rocks thrust up from the snow, and
within two minutes they and their horses had disappeared.
The riders below came steadily forward in a clump, and Brian saw old
Turlough staring with bulging eyes. Then the Wolf half caught at his
bridle, as if minded to fly, and his hands were trembling.
"What ails you, man?" smiled Brian. "Are they magicians and sorcerers,
then?"
"No, _fareer gair_--worse luck!" blurted out the other. "Look at the
little man who rides first, Yellow Brian!"
Brian squinted against the snow-glare, and saw that the leader of the
approaching party seemed indeed to be a little man with hunched
shoulders and head that glinted steel.
"A hunchback!" he exclaimed. "Well, who is he?"
"The Dark Master--O'Donnell More himself! It is in my mind that this is
a black day, Brian Buidh. O'Donnell More is the master of all men at
craft, and the match of most men at weapons. Beware of him, master,
beware! I had thought that he was still under siege at Bertragh Castle,
else I had never taken this road."
"Nonsense!" laughed out Brian joyously, drinking in the clear afternoon
air. "So much the more honor if we slay him, Turlough Wolf! Let him
match me at weapons, or you at wits, if he can!"
Turlough muttered something and drew back behind Brian's steed with
pallid face. Yellow Brian, however, having a sure trust in his own right
arm and his hidden men, scanned the approaching O'Donnell curiously,
seeking what had inspired such unwonted fear in the old gray Wolf.
He could find nothing ominous in that hunched figure, save its mail-coat
and steel helm. Yet the face was peculiar. Over a drooping mustache of
black flared forth two intense black eyes. Brian noted this, and the
thin, curved nose and prominent chin, and laughed again.
"Who is this Dark Master, Turlough?"
The other shivered slightly. "He is an O'Donnell from the north, come
here some ten years since--he seized on Bertragh even as we intend
seizing on a stead, and has since done evil things in the land. Now
hush, for they say the wind bears him idle talk."
Brian's thin lips curved a trifle scornfully, but he kept silence,
watching the approaching men. At fifty yards' distance they halted.
Their leader eyed the motionless pair for a moment and then slowly rode
on alone, waving back his followers. And Yellow Brian made a strange
figure, with his ruddy hair streaming from beneath his steel cap and the
bright, naked sword rising up from toe to head beside him.
"Well?" O'Donnell More's voice was deep and harsh, though Brian
afterward found that it could be changed to suit its owner's mood. "Who
are you thus disputing my passage?"
"I am Brian Buidh," came Brian's curt reply. "As for dispute, that is as
you will."
"Yellow Brian?" The black brows shot up in surprise. "A strange name.
Whence come you, and seeking what?"
"I seek men, O'Donnell More." Brian swiftly determined that this was a
man who might give him aid, a man after his own heart. "Whence I come is
my affair. Give me men, and I will repay with gold."
"What need have you of men, Yellow Brian," came the sardonic answer,
"when your own lie hidden among the rocks?"
Now indeed Brian started, whereat the other smiled grimly.
"How knew you that?"
"If you recognized me from afar, you had not stayed to meet me unless
you had men," stated O'Donnell shrewdly enough.
"True," said Brian, and laughed out. "Well said, O'Donnell. I have a
score, and want another score. I will match mine against yours, or make
a pact, as you desire."
The Dark Master sat fingering his sword-hilt and considered. With the
black brows down and the black eyes fixed on him, Brian suddenly began
to like the man less.
"I will give you service," returned O'Donnell at last.
Brian smiled. "Men serve me, not I them."
At this curt answer O'Donnell looked black, then fell into thought, his
shoulders hunched up and his head drawn in like the head of a turtle.
Brian wished now that he had struck first and talked afterward.
Finally the Dark Master looked up with a slow smile.
"Welcome to you, Brian of the hard eyes and hollow cheeks," he said.
"_Slaintahut!_ I will not give you men, but I will give you the loan of
men if you will do me one of two favors. Ten miles to the south of here
there is an old tower on a cliff, and in the tower dwells a man with
certain companions who sets me at naught. On an island out near Golam
Head is a castle where a woman rules, who has also set me at naught. Go,
reduce either of these twain, and I will lend you twoscore men for three
months."
Brian sat his great horse and looked at the Dark Master. He would have
sought advice from Turlough Wolf, save that he did not like to turn his
back on those burning eyes. After all, the pact was not a bad one.
"These enemies of yours--who are they, and what force have they?"
The Dark Master chuckled, and his head shot out from between his
shoulders.
"The man is called Cathbarr of the Ax, and he is a hard man to fight,
for he has ten men like himself, axmen all. The woman cannot fight, but
she has a swift mind, many men, and her name is Nuala O'Malley, of the
O'Malleys of Erris."
"I had sooner fight a man than a woman," returned Brian slowly. "Also,
this Cathbarr of the Ax has fewer men. I will do you this favor,
O'Donnell Dubh."
He gave no sign of the wonder that had shot into his mind at the name of
Cathbarr, except that his blue eyes seemed changed suddenly to cold ice.
The Dark Master saw the change, and his smile withered. Brian, watching
him, reflected that this malformed freebooter could be venomous-looking
at times.
"I have passed my word," O'Donnell the Black made curt answer. "Fetch
either of the twain to Bertragh, dead or alive, and you have the loan of
twoscore men for three months, free. Is it a pact?"
"It is a pact," answered Brian, and at that the other galloped back to
his men.
Brian swung his sword and flung it high into the air; before it had
flashed down to nestle in his palm again, his men were scrambling into
the road. He sheathed the sword, smiling a little, and turned to
Turlough.
"Well? To your mind or not, Wolf?"
"My father saw the Brown Geraldine at Dublin," responded that worthy,
scratching the gray beard which had begun to sprout. "They broke his
bones with the back of an ax and swung him out in a cage until he died,
and after. He made pacts too easily."
"Well?" asked Brian again, but a dull flush crossed his cheeks.
"I gave you my rede," said Turlough sullenly. "I said to stand alone,
receiving aid from neither man nor faction. Now there is mischief to be
repaired."
"Then my sword shall repair it," said Brian, and ordered the men to
swing in after him. "Guide us to this tower of Cathbarr's, for my honor
is in my own keeping."
They swung about and headed to the south and the sea.
The hill-paths, which Turlough Wolf seemed to know perfectly, were
cruelly hard on the horses; none were as yet trodden down, for the snow
was fresh, and all the west coast lay desolate. The plague had stricken
Galway and Mayo heavily that year, smiting the mountains with death.
Some few parties of Roundhead horse had come through, because they
feared God and Ireton more than the plague, and some Royalists had fled
up from the south for much the same reason.
In any case, Yellow Brian found all the land desolate, and liked it. The
more wasted the land, he reflected, the more chance for that sword of
his to find swinging-room. As he had ridden, news had come from the
east--news of the Wexford killing and the curse that was come upon the
land. Owen Ruadh O'Neill was not yet dead, but Brian knew that he had
prophesied truly. Ireland's day was gloaming fast.
Despite the dismal tone of Turlough Wolf, Brian told himself that he had
done a good day's work. O'Donnell Dubh would keep his word beyond any
question. As for the man he was to slay, the only part of it which
troubled Brian was the prediction of the Black Woman at the Dee water.
She had known him, and had prophesied O'Neill's death, and had spoken of
the west and this Cathbarr of the Ax. After all, however, she might have
shot a chance shaft which had gone true. Brian had no faith in magic.
All that afternoon he rode on, Turlough Wolf ahead of him, the men
behind. They feared and hated the old Wolf as much as they feared and
loved Brian.
Progress was slow, owing to the bad paths, the snow, and sundry changes
of direction, so that when night fell they had covered but eight miles
of the ten. Turlough suggested that they push on and finish their
business at a stroke, but Brian curtly refused. So the men made camp in
lee of a cliff and proceeded to feast away the last of their provisions
and wine, in confidence that on the morrow they would have more, or else
would need none.
Brian and Turlough built a fire apart, and after their repast Brian
broke silence with a request for information about Cathbarr. It was his
first speech since the parting with the Dark Master.
"I never heard of him," responded Turlough. "No doubt he is some outlaw
who has become a thorn in the Dark Master's flesh. With the woman it is
different."
"Tell me of her," said Brian, gazing into the fire.
"She is an O'Malley, and, like all the clan, makes much of ships and
seamen and little of horses and riders. When the Dark Master came, ten
years ago, he slew her father and mother by treachery, and would have
slain her but that her men carried her off. She was a child then. Now
she is a woman, very bitter against O'Donnell Dubh, and is allied with
the Parliament so that her ships may have the run of the seas, it is
said. O'Donnell takes sides with no faction, but caters to all. He lays
nets and snares, and men fall into them, and he laughs."
"Why is Nuala O'Malley called the Bird Daughter?" asked Brian quietly.
At this question old Turlough rose on his elbow, and in his wide, gray
eyes was set mingled fear and wonder.
"_M'anam an diaoul!_" he spat out. "Who are you to know this thing?"
"Answer my question," returned Brian, hiding his own surprise.
"Seven years ago, master, I was at Sligo Bay with O'Dowda when Hamilton
cut us to pieces. Nuala O'Malley had brought us some powder--she was but
a slip of a girl then. In the evening I was down at the ship when I saw
her come from below, a hooded pigeon in her hands. She whispered in the
bird's ear, set off the hood, and the bird flew into the night. I named
her Bird Daughter, but no other man knew the name."
"Then a woman did," chuckled Brian dryly. "It was but a carrier pigeon,
Turlough; I have seen them used in Spain. Now listen to me."
With that he told him of the Black Woman and his weird meeting at Dee
water. Old Turlough listened in no little amazement, for he was full of
superstitious fancies, but Brian said nothing of his own name. The
uncanny prophecies, however, which now seemed on the road to fulfilment
were enough to give any man pause.
When he had finished, a very subdued Turlough Wolf stated that the Black
Woman was an old hag who wandered all over the land, that some called
her crazy and others thought her inspired, and that his own belief was
that she was a banshee, no less.
At this Brian saw the thing in a more rational light. The old woman knew
of this nook in the west, and, attracted to him by his resemblance to
the long-dead earl, she had endeavored to steer him thither. After all,
it was quite simple.
Of course, old Turlough swore that he had never breathed his name of
Bird Daughter to a living soul, and that it was but a name he had used
in his own mind for the slim girl who had fetched powder from the south.
Brian chuckled, guessing that Turlough was not the only one who had seen
carrier pigeons used, and who had ascribed the thing to higher powers.
The incident served the purpose of establishing a firmer intimacy
between Brian and the old man, however, and convinced Turlough that his
master was destined to fly high. Nor through all the storm of men that
befell after did Turlough again breathe reproof as he had dared that
day.
"I begin to see that your advice was good, Turlough Wolf," said Brian
the next morning, as he rode shivering from camp. "As to making my men
know me for their master, that troubles me little; but I think it will
be a hard matter to avoid making pacts, and to stand alone."
"Lean on your sword," grunted old Turlough. "To my notion, such
friendship as that huge blade of yours can give is better than good.
Order men ahead."
Brian nodded and sent two of the men ahead as scouts, with the Wolf
himself. For the better part of an hour they made slow headway among the
rocks, and then emerged suddenly on the <DW72> leading down to the cliffs
and sea. Turlough pointed to the left.
"There lies the tower, if I mistake not."
Drawing rein, Brian saw at once why he had been sent on this errand.
Cathbarr's tower was an old ruin at the end of a long and narrow
headland--indeed, at high tide most of the headland would be covered,
for it was low and yet beyond shot of the cliffs. Except from the water,
it was almost impregnable; cannon might have reached it from shore, but
two axmen could have held the narrow way against an army.
Brian laughed softly and ordered the men to remain where they were.
"What are you going to do, master?" queried old Turlough anxiously.
"I am going to lean on my sword, as you advised me," chuckled Brian, and
rode on alone.
CHAPTER IV.
BRIAN LEANS ON HIS SWORD.
As he had foreseen, Brian was allowed to ride across the narrow neck of
land where his men would have had to battle for progress. It was from no
mere bravado that he had gone forward alone to the tower, but because
men were worth saving, and he believed that his own sword was a match
for any ax. If this ruffian Cathbarr was a freebooting outlaw, he would
be willing enough to stake his ten men on his prowess, and Yellow Brian
was very anxious to have those ten axmen behind him.
At the top of the tower men watched and steel glistened, and as Brian
rode up to the low gateway, it was flung open and a man strode out.
This man hardly came up to Brian's conception of an outlaw, except as to
stature.
He was a good six feet four, reflected Brian as he drew rein and waited,
and was built in proportion--or, rather, out of proportion. His
shoulders and chest seemed tremendous, and a long mail-shirt reached to
his knees; his hair was short-clipped and brown, and beneath his curly
brown beard Brian made out a massive face, wide-set brown eyes, and an
air not so much ruffianly as of cheerful good-humor.
Brian had no need to ask his name, however, for in one hand he carried a
weapon such as had seldom seen the light since powder had come to
Ireland. It was an ax, some five feet from haft to helve; double-bladed,
each blade eight inches long, curved back slightly, and two inches thick
by twice as much wide. The edges, which came down sharply from the
thickness, were not overkeen, and were not meant to be so. When the
thing struck, that was the end of what stood before it.
"_Cead mile failte!_" cried Cathbarr of the Ax in a deep, rumbling
voice, his white teeth flashing through his beard in a smile. "A hundred
thousand welcomes to you, swordsman! Are you come to capture my lordly
castle?"
"No; your men," laughed Brian, liking this huge, merry giant on the
instant. "I am come from O'Donnell Dubh to reduce you and fetch you to
him."
The smile froze on the giant's face.
"I am sorry for that, yellow one! I like your face and your thews, and
to find that you serve the black traitor of Bertragh is an ill thing."
"I serve no man," answered Brian easily. "I need men. If I conquer you,
O'Donnell lends me twoscore men for three months; also, by conquering
you I win your men to me, which makes fifty. With my seventy men, I
shall fall to work."
"By my faith, a ready reckoner!" and Cathbarr grinned again. "Get down
and fight."
Brian swung out of the saddle and led his horse to one side. They were
not so badly matched, he reflected. Cathbarr's head was bared, while he
had steel cap and jack; but for some reason he felt hesitant at thought
of killing this merry giant.
"Not so bad," he said, baring his five-foot blade and holding it up
against the huge ax. "Not so bad, eh?"
Cathbarr burst into a laugh.
"It will grieve me to crush your skull, dear man," he rumbled. "What a
pair we would make, matched against that Dark Master! But enough.
Ready?"
Brian nodded slightly, and the long ax flashed up.
Now, Brian O'Neill had served a stiff apprenticeship at weapons, and had
faced many men whose eyes boded him death, but here, for the first time
in all his life, he felt the self-confidence stricken out of him.
As Cathbarr heaved up his ax, he became a different man. All the good
cheer fled out of his face; his curly brown beard seemed to stand out
about his head like snakes, and the massiveness of his body was
reflected in the battle-fury of his face. He needed no blows to rouse
him into madness; but with the ax swinging like a reed about him, he
came rushing at Brian, a giant come to earth from of old time. His men
on the tower set up a wild yell of encouragement.
Brian leaped swiftly aside and, thinking to end the fight at a blow,
brought down his sword against the descending ax-haft. Sparks flew--the
haft was bound with iron; Brian only saved himself from falling by a
miracle.
Then began a strange battle of feet against brawn, for Cathbarr rushed
and rushed again, but ever Brian slipped away from the falling ax, nor
was he able to strike back. The play of that ax was a marvel to behold;
it was shield and weapon in one, and it seemed no heavier than a thing
of wood as it whirled. Twice Brian got in his point against the
mail-coat without effect, and twice the ax brushed his shoulder, so that
he gave over thrusting. He knew that he was fighting for his life
indeed.
An instant later he discovered that fact anew as a glancing touch of the
ax drove off his steel cap and sent him staggering back a dozen paces,
reeling and clutching at the air. To his amazement Cathbarr did not
follow him, but stood waiting for him to recover; he had not looked for
such courtesy on the west coast.
He sprang back into his defense, desperate now. Again the ax whirled,
seeming a part of the giant himself, and Brian knew that he was lost if
he waited for it. So, instead of waiting, he leaped under the blow,
dropped his sword, and drove up his fist into the bearded chin, now
flecked with foam.
It was a cruel blow. Cathbarr grunted, his head rocked back, and he
swayed on his feet. Before he could recover, Brian had set his thigh
against him, caught his arm, and sent him whirling to the ground, ax and
all. Then he picked up his sword and stood leaning on it, panting.
Cathbarr sat up and gazed around blankly, until his gaze fell on the
waiting figure. Brian looked at him, smiling slightly, and the eyes of
the two men met and clinched. As if he had been a child caught doing
wrong, the giant grinned and wiped the foam from his beard.
"Was that fair fighting, yellow man?" he asked.
"No," laughed Brian. "It was unfair, Cathbarr; but I think my fists can
best your ax yet."
Slowly the giant got to his feet. To Brian's surprise he left his ax
where it lay and came forward with extended hand.
"Had you claimed that blow as fair," he rumbled, "I would have slain
you. Now I love you, yellow man. Let us make a pact together. What is
your name?"
They struck hands, and Brian felt a great thrill of admiration for this
man whose terrible strength enclosed the simple heart of a child. But
he shook his head.
"I make no pacts, Cathbarr. My name is Brian Buidh. I made pact with the
Dark Master, and now I am sorry for it; yet it must be held to, for I
see no way out of it. But wait--I have a cunning man whose wit may help
us here."
He turned and flung up his sword in the air. His men rode down to the
narrow causeway, while from the tower came shouts warning Cathbarr
against treachery. But the giant only grinned again, and Brian shouted
to Turlough Wolf to come on alone.
Old Turlough obeyed in no little wonder. When he came up Brian told him
what had chanced--that out of enmity had arisen friendship.
"But," he concluded, trouble in his heart, "you must find me a way out,
Turlough. I have passed my word to O'Donness to reduce Cathbarr; to do
that I must slay him, or he me. I see little honor either way."
"Few men find honor in their dealings with the Dark Master," grumbled
Turlough, looking from Cathbarr to Brian. "Yet, if you want a way out,
it is an easy matter. Cathbarr of the Ax, give service to my master.
Thus, Brian Buidh, you shall reduce Cathbarr; yet the Dark Master said
naught of giving up this man to him."
"Good!" cried Brian, eagerness in his blue eyes, and swung on the giant.
"Will you give me your service, friend, and follow me? There shall be a
storm of men--" He paused abruptly as the words fell from his lips, but
he had said enough.
"I give you service, Yellow Brian," rumbled Cathbarr, taking his hand
again, and his strong, white teeth flashed through his beard. "I will
follow you, and my men, and there shall be firm friendship between us.
Is it good?"
"It is good!" exclaimed Brian, his heart singing. But Turlough laughed
harshly.
"So you have again broken my rede, Brian Buidh, for this man knows you
not as his master, but names you his friend. I bade you take, not give."
"It was your own advice," retorted Brian, laughing.
"Aye, since you asked it, I found the way out. But you have not
conquered him."
"He conquered me by not telling a lie," said Cathbarr simply. "I serve
him."
Turlough eyed them keenly, heard how the fight had gone, and then
suddenly comprehended what manner of man this huge, bearded fellow was.
His face cleared, and without a word he clasped Cathbarr's hand, and
asked Brian for orders.
"How far from here is Bertragh Castle?" questioned Brian.
"It overlooks Bertraghboy Bay," answered the giant. "Bide here till
noon, while my men bring in their horses from the hills, and with the
night we can arrive there."
To this Brian assented, well pleased that Cathbarr had horses. Turlough
went back to bring up his men, and Brian entered the tower that served
Cathbarr for castle. It was a small place, but strong; the ten men who
took his hand and gave him service were cut after the pattern of their
master--huge fellows all, O'Flahertys from the mountains who had
followed Cathbarr down to loot the coast, with no ill success.
It was a strange tale that he heard, while he and his men ate and drank
with their new comrades. For some months Cathbarr had maintained himself
here, raiding O'Donnell's lands chiefly and making his ax feared through
all the coast. In fact, the giant had attempted his own errand--to set
himself up in power; but he had gone about it like a child.
The Dark Master had come against him with a hundred men, and after
losing a score and more at the causeway, had tried to starve him out. At
that Cathbarr had calmly stolen away by boat, raided O'Donnell's
choicest farms overnight, and was back with his plunder before the Dark
Master guessed his absence. After this O'Donnell had kept watch and ward
upon his lands, with better results; Cathbarr occupied himself with
raiding against the scattered parties of plunderers in the hills, and
had won some booty.
Brian discovered many things during the hour or two he waited for the
horses to be fetched in. Chief of these was that he had set himself a
difficult nut to crack. The Dark Master held a strong castle, with rich
farms around it, and could summon at need some three hundred men to his
standard. In short, Brian found that O'Donnell held the very position he
himself wanted to hold--and was like to keep it.
"Of course," he thought soberly, reflecting on his future course, "if I
come off clear to-night I can ride with my seventy men to a better
place. And yet--I don't know! What better place than this? It will be no
long time before hoofs are in the land, for Royalist and Roundhead and
Ulsterman will be storming through the hills; Galway will be the last to
give in to Cromwell, of a certainty. When the hurricane falls, I want a
roof to shelter me--and whom could I turn out better than this
O'Donnell?"
Cathbarr's tower was too small to serve him as a fortalice, for it was
barely large enough to shelter the eleven axmen. Suddenly an idea
flashed across Brian's mind. Why not a union with this O'Malley woman
against the Dark Master?
Upon the thought, he rose and went out to the ice-rimmed shore below the
tower, where he paced up and down, considering the matter. After all, it
would do no harm, and there were great possibilities in it. He returned
to the tower at sound of shouts and clattering hoofs, and took Turlough
aside.
"Turlough Wolf, in your advice you spoke against making pacts with men,
but you said nothing of women. It is my purpose to send you to this
O'Malley castle, to propose a pact with Nuala O'Malley against the Dark
Master. You can tell her that I have a hundred horsemen behind me--for
I will have them. Will you do this, bearing her word back to me?"
Turlough plucked moodily at his ragged beard.
"I see no harm in such a pact, master," he replied thoughtfully. "As to
reaching the Bird Daughter, that is another matter. I think that I can
do it, however. When shall I start, and where shall I find you again?"
Brian reflected a moment.
"Start now, Turlough. Cathbarr and I will have no need of advice this
night, for we shall either fight our way clear, or else the Dark Master
will keep to his word. When you return, you will find me here; if I am
not here, I will leave a man here to give you word of me."
"I am to say that you have a hundred horsemen behind you?" Turlough's
sharp eyes swept to Brian's half-questioningly.
"Say a hundred and a half," laughed out Brian, "and trust your silver
tongue for the rest, old Wolf! Never fear, I will have the men. But mind
this, Turlough. I will make no other pact with her than this, against
the Dark Master. It may be that when I have driven him forth I may fly
after other game."
"Men have sought to drive the Dark Master forth," quoth Turlough, "and
their heads have rotted above his gate. Take heed lest there be an empty
spike there this night, Yellow Brian!"
But Brian only laughed shortly, and bade the old man affectionate
farewell, for he knew that Turlough loved him. And when Turlough had
ridden somberly away, Brian felt a strange sense of desertion, of loss,
that was no whit inspired by Turlough's gloomy last words. He shook it
off, however, at gripping hands again with Cathbarr. The axmen had
gathered most of their loot and buried what was of value, for Brian had
determined to return here from Bertragh and make use of the tower until
he had heard from Turlough's errand.
So now, at the head of thirty men, he rode across the narrow causeway
with Cathbarr of the Ax at his side for friend and guide. The giant did
not yet quite comprehend exactly what plan had flashed across the brain
of old Turlough, so as they rode Brian made the thing clearer to him.
When the simple and straightforward Cathbarr grasped the matter, he
smote his horse's neck with a bellow of laughter.
"Ho! So you bring me before the Dark Master ax in hand, reduced to
_your_ service instead of his, my men added to yours--oh, it is a jest,
brother, a jest! I think that O'Donnell will slay us both on the spot!"
"Not if your axmen are true," retorted Brian.
Cathbarr laughed again. "They fear me and they love me, brother," he
cried, gazing back at the file of horsemen. "Your own men fear you and
love you also. Therefore we are men alike."
Brian began to love the man for his utter simplicity, save where there
was killing in hand. Cathbarr seemed in reality to have the heart of a
child, impulsive and passionate to an extreme, and there was always a
certain rugged power in his bearing which bespoke him a true Flaherty of
the mountains. His men were like himself in this respect, and after they
had fraternized with Brian's men they began to feel the same unbounded
surety in Yellow Brian as Cathbarr expressed. Their axes were the usual
splay-bladed affairs that their grandfathers had used under Red Hugh at
the Yellow Ford, nor indeed in all his life had Brian ever seen another
ax like to that of Cathbarr's.
They rode through the afternoon while a light snow fell and a keen east
wind cut down from the peaks of the Twelve Pins, until the shaggy horses
slithered along with tails tucked tight beneath them. But there was good
cheer in the company, for the news had spread of how Yellow Brian would
have seventy men behind him that night. When the darkness began to fall,
Bertragh Castle came in sight far below--a gray crag jutting up from
the plain, scarped and embattled, the sea behind it and the watch-fires
of men twinkling from its keep. All about lay farms and steads, and the
lowing of byred cattle rose on the evening air when the snow ceased.
"Be careful not to drink or eat in that hall," warned Cathbarr blackly.
"Ill comes of it to all who accept hospitality there."
Brian nodded and rode on in silence, for there were parties of horsemen
and pikemen down below and the blare of horns shrilled up. Evidently the
riders on the hills had been seen from afar.
As they reached the lower ground Brian was aware of a band of men riding
to meet them, and halted. Through the dusk came a score of armed
horsemen, and their leader inquired their business, shouting from a safe
distance. Brian returned the shout.
"I am Yellow Brian, and I seek O'Donnell Dubh according to a pact made
with him yesterday. I have reduced Cathbarr of the Ax, and am come in
peace."
"You are expected," called the other, riding up with his men. "The Dark
Master is waiting for you."
And Brian rode on to Bertragh, not without some forebodings.
CHAPTER V.
YELLOW BRIAN RIDES SOUTH.
Outside the castle gates, where cressets flared over the snow, an old
seneschal appeared and ordered Brian to leave his men outside. To this
the men made some objection, but Brian laughed softly.
"Bide where you are," he said. "You shall not be slain unless I am slain
inside."
The O'Donnells watched him and Cathbarr with no little wonder, and the
two men made a fine pair as they marched across the creaking drawbridge.
Though Cathbarr topped Brian by half a head, there was no doubt as to
which was the nobler man; the giant gazed around him with amazed eyes,
but Brian held his head high and strode in with a smile flickering on
his lips. But his blue eyes were very sharp that night.
He saw the crowded men in the courtyard, many of them armed with
muskets, their matches burning, and noted also that the Dark Master
possessed some half-dozen bastards--immense, nine-foot pieces mounted on
huge carriages, with their eight-pound balls piled beside them. In those
days it was no small thing to own such cannon in the west of Ireland,
and Brian eyed them approvingly as he passed through the courtyard. He
was beginning to count them as his own.
Cathbarr had told him that the Dark Master had brought many O'Donnells
down from the north to settle the farms and lands beyond the castle, but
Brian saw that these were not all. The garrison was a riffraff of all
the armies that had wasted Ireland, and they were fighting men fit for
their work.
Brian entered the hall, with Cathbarr muttering oaths a pace behind him.
The hall was high, lit with cressets, and beside a huge fireplace sat
the Dark Master in a carved chair of black wood, an old harper sitting
opposite. Behind Brian and Cathbarr flocked in men until the hall was
well filled.
Brian found the penetrating eyes fixed on him as he advanced, but in
them was no surprise or fear, and O'Donnell calmly stroked his drooping
mustache as he watched. Cathbarr still followed behind, bearing that
great ax of his, and Brian stopped a few paces from the hearth as the
Dark Master spoke.
"Welcome to Bertragh, Yellow Brian. I had not looked for you so soon."
"No." Brian's voice rang out richly in the stillness. "But I am here,
O'Donnell Dubh, to claim my two-score men. I have reduced Cathbarr of
the Ax."
For the first time the hunched O'Donnell seemed to notice Cathbarr. His
black eyes flickered curiously to the giant, then he smiled sourly.
"If he is reduced, why does he not kneel, Brian of the hard eyes?"
"Kneel," ordered Brian.
Cathbarr flushed and his beard began to stand out, but he obeyed. There
was no great love in his face as he knelt, holding to his ax, and gazed
at O'Donnell.
"Throw your ax into the fire," said the Dark Master, his voice smooth as
silk.
"Do not," exclaimed Brian, and his eyes grew bitterly cold as they
clinched with those of the Dark Master. Over the latter's pallid face
crept a slow red fire, and his head drew back between his shoulders. Men
held their breaths.
"O'Donnell," went on Brian slowly, "I have fulfilled my pact. I have
reduced Cathbarr of the Ax--but he serves me and not you. Since I have
conquered him as you bade, I call on you to carry out the pact and lend
me two-score men for three months, scat-free."
If Brian had wanted any testimony as to O'Donnell's iron hand, he had
it. His words, with all they implied, would have drawn a howl of rage
from the retainers of any other chief in the land, but the men behind
and around him only grew more silent.
As for the Dark Master, the red hue died slowly from his face, though
his head remained drawn in, and still his eyes held those of Brian. When
he spoke, it was as if he were musing aloud.
"So, Brian of the hard eyes, you have some courage, eh? _Duar na
Criosd!_ Little did I ever think that a man would come to me and borrow
my own men that he might make war upon me! Is this your thought, Yellow
Brian?"
"You have sharp ears, Dark Master," said Brian dryly, and a chuckle
passed through the crowd. "In time I might take this castle, it is true.
Just now I have other things in mind, however, and I shall not fall upon
you until there has passed gage of battle between us."
"Thanks for so much," smiled the other slowly, though the red crept up
to his cheek-bones faintly. Brian seemed perfectly at his ease, as
indeed he was. "And what if I fell upon you first?"
"I am liker to offer battle than accept it, O'Donnell."
"Now, that is a good answer," said the Dark Master, while a whisper
floated around the hall. "I would be glad to have you at my back, Yellow
Brian, for men who ride behind me are like to win much."
Brian laughed a little.
"Some day I may be at your back, O'Donnell Dubh, and in that day I may
win all that you have, from life to goods."
To his blank amazement, O'Donnell only threw out his head and chuckled;
but it was an evil chuckle, and there was venom gleaming in his black
eyes.
"I think that it were best for me to slay you here, Brian of the hard
eyes, to slay you and this Cathbarr of the Ax. It seems to my mind that
it is anything but good to turn you loose upon the land, for I hear a
storm of hoofs in the air, and dead men are riding on the wind, and
there is a whisper--"
He paused, drew his cloak about him, and gazed down at his foot. That
pause was more dreadful than speech, for the crowded men moved not a
finger, so that Brian all but thought that he and the Dark Master were
alone. Then his face blanched a trifle. For, whether it were some
uncanny play of mind or very truth, it seemed to him that from the wide
fireplace there did indeed come a faint ring of hoofs and clash of
steel; the long cressets over them suddenly flickered smokingly, though
no draft crossed their faces.
Then indeed Brian knew that his fate hung upon the Dark Master's
thoughts, and he drew himself up a little straighter, and his blue eyes
glinted colder than any ice as his hand closed upon his sword-hilt. But
at the slight motion O'Donnell looked up keenly.
"You have ridden hard, Brian. Pause and sup with me--"
"I did not come to eat or drink," said Brian sternly. "Also, I am weary
of this talking. Now fulfil your pact, Dark Master, or be shamed before
all your men."
"Are you for Royalist or Parliament?" asked O'Donnell, as if he had not
heard.
"I am for Brian Buidh."
"Take two-score men and begone," and the other rose. To his surprise,
Brian found that, despite the hunched back, O'Donnell was as tall as
himself. The black eyes flamed out at him for an instant. "I will keep
my honor, though I regret it later, Yellow Brian. Go, with your men.
When next we meet your head shall grin over my gates."
"Thanks for so much," retorted Brian mockingly, though he drew a swift
breath of relief. "My head serves me too well to render it easily. _Slan
leat_, O'Donnell!"
"_Slan leat_," repeated the Dark Master and turned his back, gazing down
at the fire.
Brian turned and strode down the hall, Cathbarr at his heels. When they
reached the courtyard he found men saddling in haste, and an officer
saluted him gravely.
"Two-score men are at your orders, Yellow Brian."
"Let them follow me," said Brian curtly. "And who quarrels with my men,
dies."
To that there was no dispute. The drawbridge clanked down once more,
Brian and Cathbarr mounted and rode out to where the thirty waited
grimly, and after them came the forty men from the garrison. Cathbarr,
who trusted the Dark Master little, set his ten axmen in the van,
followed with Brian, and the sixty followed them into the night.
"I think we came out of that well, brother," said the giant softly.
"Where do we ride?"
"To your tower, for the night. After that, in search of more men."
"Toward Galway or Slyne Head?"
"Wherever there are men."
After that they rode on in silence, while the men behind fraternized
freely. All were of the same stamp, and indeed the two-score already
were as willing to serve Brian as O'Donnell, since they had witnessed
that scene in the castle hall.
Brian wondered dully what the outcome of all this was to be. The strain
of facing O'Donnell and bearding him in his own den had been no light
one, but he knew that Cathbarr had spoken truth in saying that they were
well out of it. The Dark Master, he thought, was a man well worth
fighting. To take his castle was not like turning out a chieftain of
some ancient family, with his clan about him for miles around; O'Donnell
had seized upon the place himself, his men were reavers and outlaws, and
the castle was a strong one.
Then there was the O'Malley alliance. Brian had it in mind to beset the
Dark Master by sea and land at once, for all the O'Malley clan had been
seamen and rovers from time immemorial, while he himself preferred men
and horses at his back. In calmer mood now, he reflected that Turlough
might not return for a week, and there was food and fodder for seventy
men and horses to be obtained.
If he rode toward Galway he would have to plunder the patriots, which
went against the grain. But in lower Galway and Clare things were
different. That winter no army held to winter quarters save that of
Cromwell, and between Limerick and Galway there was a wild rout of men
out of half a dozen armies, the plague had swept off all but the
seafaring folk, and men held only what their swords could guard.
So Brian determined that he would ride toward the south.
He realized well that his men must be drawn together by fighting, that
they must learn a perfect confidence in him, and that they must earn
their sustenance for the time being. Cathbarr already knew of old
Turlough's mission, and of course approved, since in his eyes Brian
could do no wrong. What was more, reflected Brian, he could not make
this alliance empty-handed. He must get men and spare horses, stores and
powder, and some muskets or pistols if possible, for few of his men
carried more than sword or perhaps a sorry pistolet or ancient
bombardule out of date a generation since.
"A storm of men!" he muttered as he gazed at the stars. "A storm of men!
Did that Black Woman speak truly, I wonder? And what dark magic was that
which passed to-night?"
But no answer came to his questions save that the cold stars chilled him
to the bone. Since they had no better place to seek, they returned to
Cathbarr's tower, but it was long past midnight when they reached it,
and the men were nodding in their saddles. As barely a dozen could crowd
into the place, the rest were forced to camp outside in the snow, but
roaring fires and some little food put them in good humor and it was no
hardship to any of them.
"It has been a strange two days for us twain," said Brian as he and
Cathbarr divided a scorched bannock one of the Scots had hastily turned
out over the coals.
"Yes," smiled the giant into his beard, his deep-throated bull's voice
rumbling through their tiny room. "But it is in my mind that there are
stranger days ahead of us, Brian Buidh. A witch-woman once told me that
I would meet my death from water and fire together, brother, in a cause
not mine own."
"You are not bound to my service," replied Brian.
"But I am bound to you, for I like you," answered Cathbarr, and his hand
crushed down on Brian's. That night they slept together beneath the same
blanket, and though after that they spoke few words of love or
friendship, the two men drew ever closer each to the other in all
things.
It had indeed been a strange two days for him, thought Brian as he
roused up the camp late the next morning and set out sentries in the
hills. He had met the Dark Master on the first, and on the second he had
met Cathbarr, then had forced the Dark Master into lending him men
against his will. Now, after a scant three days beyond Lough Corrib, he
had twined his fate with that of other men, had set his heart upon
winning Bertragh Castle, and had won both a stout friend and a stout
enemy.
For he counted O'Donnell as a foe, in which he was not far wrong.
However, there was no time to be wasted, for fodder was exceeding
scanty, and Brian himself had no heart for idleness. As he had resolved
on his course during that return ride the night before, he gathered his
men together and briefly ordered them to be ready to ride at noon, and
to Cathbarr alone he outlined his plan. Then he picked two of the axmen
who knew the country roundabout, and ten from among those O'Donnell had
loaned him, and took them aside and told them of Turlough Wolf, who
would come before long.
"You will bide here," he concluded, "and bid him wait for me. I shall
return this side of ten days. And mind you, if there is feud or
treachery among you so that one man's blood is let, then I will exact a
tenfold vengeance from both men."
The twelve, who were sturdy ruffians and well able to hold the place
against any sudden attack by the Dark Master, looked into the ice-blue
eyes for an instant, and straightway vowed that there would be neither
treachery nor quarreling among them. And Brian guessed shrewdly that he
had inspired some little fear in their hearts.
So that at high noon they rode away to the east, threescore strong, with
Brian and Cathbarr and the remaining eight axmen in the van. Brian did
not spare either man or horse that day, for there was little food left
them; when midnight came they had slipped past Galway and were ready to
ride south, though they all went to rest supperless.
With the morning Brian found that two of the men had slipped off and
were busy plundering a hill-farm a mile away, where an old woman lived
alone. He promptly had them brought before him, and bade them take up
their weapons.
"I am no executioner," he said as he bared his huge sword. "I am a
teacher of lessons, and my lessons must be learned."
When they rode away from that place, leaving the two men buried under
cairns, Brian was well assured that there would be no more ravaging by
his men, though they died of hunger.
However, it proved that there was no great chance of this, for Brian
drove such a storm past Slieve Aughty as had not been heard of in
generations. Of all that chanced in those seven days ere he set his face
to the north again, not much has survived, for there were greater storms
to come afterward, and more talked-of fighting. But certain things were
done which had a sequel.
By the fifth day Brian had swept past Gort toward Lough Graney, and
turned west by Crusheen, which he passed through with a hundred horsemen
at his heels. Two days before he had struck upon fifty Ulstermen who
were working north from Munster, and what were left of them after the
meeting took service with him. From them he learned that O'Neill was
dying or dead, and that the Royalists and Confederacy men were paralyzed
through the south.
They had left Crusheen ten miles behind them on the fifth day, when
Cathbarr laid his hand on Brian's knee and pointed to the left, where a
hill rose against the sky.
"Look there, _boucal_--when the birds fly from the _ceanabhan_, seek for
snakes!"
Brian drew rein. Gazing at the long <DW72>s of moor-grass that rose
across the hill, he saw a sudden flight of blackbirds from over the
crest; they flew toward him, then swerved swiftly and darted to the
right. Brian called up two of his men who knew the country, and asked
them what lay over the hill.
"The Ennis road to Mal Bay," they replied, and he sent them ahead to
scout.
Before he reached the hill-crest they were back with word that an
"army" was on the road, and Brian pushed forward with Cathbarr to see
for himself. Slipping from their horses, they gained the hilltop and
looked over on the winding road beyond. Neither of them spoke, but
Brian's eyes glinted suddenly, for he beheld a train of four wagons
convoyed by some two hundred troopers. He touched Cathbarr and they
returned.
"A party of Ormond's Scottish troopers," he said quietly when they had
rejoined the men. "Cathbarr, take thirty men and work around them. When
you strike, I will lead over the hill and flank them."
The giant nodded, picked his men, and rode away. Brian led his seventy
closer to the rise of ground, and as they waited they could hear the
creaking of wagons and the snap of whips. It was a Royalist convoy, and
since there was no love between the Scots and the Irish of any party,
Brian's men were hungry for the fight.
They got their fill that day.
A rippling shout, a scattering of shots, and Brian spurred forward. The
road wound a hundred yards below, and Cathbarr had already fallen on the
vanguard. The Scots were riding forward to whelm him when Brian's men
drove down with a wild yell and smote the length of their flank.
Brian hewed his way to the side of Cathbarr, and then the sword and ax
flashed side by side. The captain in command of the troopers pistoled
Cathbarr's horse, but the huge ax met his steel cap and Cathbarr was
mounted again. Meanwhile, Brian was engaged with a cornet who had great
skill at fencing, and his huge Spanish blade touched the young officer
lightly until the Scot pulled forth a pistol, and at that Brian smote
with the edge.
The muskets and pistols of the troopers worked sad havoc among Brian's
men at first, but there was no chance to reload, and when the officers
had gone down the Scots lost heart. They would have trusted to no Gaelic
oaths, for men got no quarter in the west, but when Brian shouted at
them in English they listened to him right willingly. A score broke away
and galloped breakneck for the south again, and perhaps fifty had gone
down; the rest gathered about the wagons stared at Brian and Cathbarr in
superstitious awe as the two lowered bloody ax and sword and offered
terms.
"I offer service to you," said Brian. "I am Brian Buidh, and if you will
ride with me you shall find war. Those who wish may return to Ennis."
Now, at the most Brian had some seventy-five men left, and those
clustered at the wagons were over a hundred and a score, with muskets.
But their officers were down, they had received no pay for a year and
more, and they were for the most part Macdonalds of the Isles, who loved
freebooting better than army work. So out of them all only ten men chose
to ride to Ennis again, and Cathbarr shook his head as they departed.
"It seems to me that ill shall come of this," he said, and wiped his ax
clean.
Brian laughed shortly and dismounted. He found that the wagons contained
powder, stores, and muskets; so after placing the wounded in them, he
rode north to Corrofin that day with close to two hundred men at his
back. Staying that night at Corrofin, he hanged ten of the Scots for
plundering, rested his horses for two days, and set his face homeward
with the surety that his men knew him for master.
The storm of men was gathering fast.
CHAPTER VI.
BRIAN TAKES CAPTIVES.
"_Failte abhaile!_ Welcome, Yellow Brian!"
"So you won back before me, eh?" Brian swung down from his horse and
gripped hands with old Turlough Wolf. "Get the men camped, Cathbarr,
then join us."
Turlough's cunning eyes rested on the wagons and weary horsemen, and he
nodded approvingly as Brian told him of what had chanced.
"Said I not that you were a master of men?" he chuckled quietly, as he
turned to follow into Cathbarr's tower. "But it is easier to master men
than women, Brian. I bear you a bitter rede from the Bird Daughter,
master."
"Hard words fare ill on empty stomachs," quoth Brian. "Keep it till I
have eaten."
When Cathbarr had joined them and they had dined well on Royalist stores
and wine, Turlough made report on his mission. It seemed that he had met
with a party of the O'Malleys at the head of Kilkieran Bay at the close
of his first day's ride, and after hearing his errand they had taken him
in their ship out to Gorumna Isle, where stood the hold of Nuala, the
Bird Daughter. And somewhat to his own amazement, Turlough had found
that by this same name she was known along the whole coast.
He reported that it was a strong place, for the castle had been built by
her father; that she had two large ships and five small ones, and that
both ships and castle were defended by all manner of "shot"--meaning
cannon. She had just returned from Kinsale, where she had been aiding
Blake hold Prince Rupert's fleet in the bay. Now Rupert had slipped
away, and after plundering a French ship with wines, she had come home
again.
"She seems a woman of heart," smiled Brian. "What of her looks?"
"I did not see her." Turlough shook his head. "She ordered my message
written out, so she has some clerkly learning. She took an hour to
ponder it, master, then set me ashore with this message.
"'Tell Yellow Brian,' she ordered, 'that I claim tribute from Golam Head
to Slyne. I will make no pact with him until he pay me tribute; and if I
find him on my land I will set him in chains above my water-gate.'"
Brian felt no little dismay at this, for he had counted strongly on
alliance with this Bird Daughter.
However, Turlough proceeded to set forth the reasons for such a message,
as he had conceived them within his shrewd mind. First, it seemed that
the pestilence had visited Gorumna in the absence of its mistress, and
that the Dark Master had caught a score of the O'Malleys who had been
wrecked in Bertraghboy Bay, promptly hanging them all. Between the
plague and the hanging Nuala had a bare fourscore men left within the
castle, and she counted Brian's offer as a ruse on the part of
O'Donnell, for she was strongly afraid of treachery.
"There is more pride than power in that message," commented Cathbarr
easily. "The Dark Master has stripped away all her lands along the
coast, and save for Kilkieran Bay she has little left. Let us fall on
her, brother, and take what _is_ left."
Brian laughed at this naive counsel, looking at Turlough. But the old
Wolf said nothing, brooding over the fire, and Brian reflected within
himself.
He had come into a merciless feud, that he knew well. If he was to enter
upon it he must banish all pity from his heart, which was no easy thing
for him; but Turlough related things he had heard which speedily changed
his mind. There were tales of O'Donnell's ridings through the land, of
men slaughtered and women carried off to people his castle; of
treachery, and worse.
It was also whispered that the Dark Master had made alliance with
certain pirates from the north coast.
However, Brian knew that he must reach some decision regarding his own
men, and that speedily. The three talked long that night, setting aside
the question of the O'Malley alliance for the time being. Brian had some
two hundred men to house and horses to feed; he had good store of
provision and powder, but Cathbarr's little tower was utterly useless to
house the tenth of them all, while the stores would have to be
sheltered. Then O'Donnell might fling his men on them at any moment,
which would mean disaster in their present position.
Cathbarr suggested an attack on Bertragh castle, but Turlough dissented.
"When we strike, we must strike to win," he said shrewdly. "The Dark
Master has more men than we, and the sea is at his back, and they say he
is a warlock to boot."
The giant stared and crossed himself at talk of warlocks, but Brian
laughed out.
"I have a plan," he said, fingering his sword. "O'Donnell watches all
the hill-paths like a hawk, even now in winter. Those wagons are of no
great use to us, and we can store the goods here in the tower for the
present. Get it done to-night, Cathbarr, and get the accouterments from
two of those largest Scots for yourself and me."
Turlough Wolf chuckled suddenly, and Brian knew that the old man had
pierced to something of his plan. But not all.
"Turlough," he went on as the scheme came to him more clearly, "at dawn
ride out with a hundred men to that hill-road where first we met the
Dark Master. Hide the men in the hills, and be ready to ride hard when
the time comes. Cathbarr, before the dawn breaks have the wagons start
out with twenty of the Scots troopers as escort. Bid as many more as can
lie down in the wagons and cover up close with their muskets. Send a man
or two with them to guide to that hill-road of which I spoke. We will
ride after and catch them up shortly after sunrise."
"Good!" roared out the giant, whose brains lay all in his ax. "And the
Dark Master will swoop down to the feast, eh?"
"He will not," returned Brian dryly. "He will send two or threescore men
upon us, and it is my purpose to take as many of these prisoner as may
be."
Cathbarr stared, and Turlough's gray eyes squinted up at Brian.
"How is this, master?" he asked inquiringly. "It is too good a trap to
waste on prisoners--"
"My plan is my plan," said Brian briefly. "I am not making war on
O'Donnell, but I intend to pay tribute to the Bird Daughter, and that
right soon. While we are gone have a score of men remain here and build
huts on the cliffs, Cathbarr."
Turlough fell to staring into the fire, divining the plan at length, and
Cathbarr went out to fulfil his orders. Brian knew well that there was
danger in the scheme, but he determined to deal with one thing at a
time, and thoroughly. Just at present he was intent on forming an
alliance with Nuala O'Malley, for ships and cannon were needful before
he could nip the Dark Master in his hold. It was going to cost the lives
of men, and he made up his mind not to pause for that. If he was to live
and make head it must be by the strong hand alone--the Red Hand of
Tyr-owen; and he looked down at the ring of Owen Ruadh and took it for a
symbol, as his ancestors had taken it.
Before they went to rest Turlough pointed out that if the hills were
watched he and his hundred would be noted, so Brian bade him hit back
toward Lough Corrib and then to come straight down upon the main road.
It might be that he could overcome the Dark Master's men of himself, and
if not, he would hold them until Turlough came up.
With this plan arranged, then, the four wagons set forth under the cold
stars, with thirty Scots lying hidden and twenty riding before and
behind. With the first gleam of dawn Turlough and his hundred cantered
off to the northeast, and an hour later Brian and Cathbarr put on the
buff coats and steel jacks of the troopers, with the wide morions; took
a pair of loaded pistols, and galloped after the slow-moving wagons.
Brian wore his Spanish blade, but Cathbarr had sent his ax ahead with
the troopers.
They caught up with the wagons when the latter were entering upon the
road proper out of the hill-track they had followed. The first snows had
vanished for the most part, leaving bleak, gaunt hills and rugged crags
that twisted with soft fog. The sun struck the fog away, however, and as
Brian rode on he gazed up at the purple mountains on his right, and down
at the purple bog to his left, and caught the gleam of the Bertraghboy
water out beyond. He laughed as he drank in the keen air of morning.
"Best get your edge ready, Cathbarr of the Ax!"
Cathbarr grunted, and slung the heavy hammer-ax at his saddlebow. One of
the guides, who were from the Dark Master's twoscore men, pointed to a
twisted peak on their right, whence an almost invisible spiral of gray
smoke wound up.
"The signal, Yellow Brian," he grinned, cheerfully giving away his
secrets. In fact, all those twoscore men rather hoped that their old
master would be crushed by Brian, for so long as there was booty in
sight they cared not whom they served.
Half an hour later Brian saw ahead of him that same bend of road where
first he and Turlough had met O'Donnell Dubh. But there was no sign of
Turlough, and he cantered ahead to see if the O'Donnell men were below.
As he did so a bullet sang past his ear, and he whirled to see half a
dozen of his men go down beneath a storm of lead from the hillsides; at
the same instant some three-score men came scrambling down from among
the rocks--those same rocks where he had first laid ambush for the Dark
Master.
And riders were coming up on the road below!
He was caught very neatly, and caught by more men than he had looked
for. The remainder of the twenty gathered behind him and Cathbarr, and
the thirty rose among the wagons and for a moment stopped the assault
with their musketry; but before the smoke had cleared away two-score
horsemen came thundering up the road from behind the curve, and struck.
"Albanach! Albanach!"
The wild yells shrilled up, and the Scots troopers knew that they were
fighting without quarter in sight, for the "Albanach," as they were
termed in Gaelic, gave and got little mercy in Ireland. The saddles of
the fallen were filled from the men in the wagons, and leaving the
musketeers to hold off the unmounted men, Brian plunged into the swirl
of fighting horsemen and joined Cathbarr.
The odds were heavy, but the big claymores of the Scots were heavier
still. Side by side, Brian and Cathbarr plunged through the ranks, sword
biting and ax smiting, until they stood almost alone among the
O'Donnells, for their men had been borne back. Then the giant bellowed
and his ax crushed down a man stabbing at Brian's horse; Brian pistoled
one who struck at Cathbarr's back, and pressing their horses head to
tail they faced the circle of men, while behind them roared the battle.
For a moment the O'Donnells held off, recognizing the pair, then one of
them spurred forward with a howl of delight.
"_Dhar mo lamh_, Yellow Brian--your head to our gates!"
Brian thrust unexpectedly, and the man went over his horse's tail as the
ring closed in. So far Cathbarr had forgotten his pistols, but now he
used them, and took a bullet-crease across his neck in return; then the
ax and sword heaved up together, and the ring surged back. A skean went
home in Cathbarr's horse, however, and the giant plunged down, but with
that Brian spurred and went at the O'Donnells with the point of his
blade. This sort of fighting was new to them, and when Brian had spitted
three of them he heard Cathbarr's ax crunch down once more.
They were still cut off from the wagons, but there came a wild drumming
of hoofs, and wilder yells from the men on the hillside. Like a
thunder-burst, Turlough and his hundred broke on the battle. The
O'Donnells were swallowed up, stamped flat; the unmounted men fled among
the rocks, Turlough's men after them, and a dozen horsemen went
streaming down the road.
It was hard to make the maddened Scots take prisoners, but Brian did it,
and when Turlough's men came back he found that they had in all thirty
captives. Some forty of the attackers had fallen and the rest had fled.
Since all his captives expected no less than a quick death, Brian
ordered ten of them bound on spare horses, of which there were plenty.
He himself had lost twenty-three of his Scots, and the remaining score
of captives cheerfully took service under him. Then, picking out one of
them, he gave the man a horse and told him to ride home.
"Tell your master, O'Donnell Dubh," he said, "that his men made this
attack on me, and therefore there is war between us."
The man grinned and departed at a gallop, and word passed through the
men that the Dark Master had found his match at last. As to this,
however, they were fated to change their opinion later.
"Now," said Brian to old Turlough, as between them they bound up a slash
in Cathbarr's thigh, "do you put the wounded in the wagons and begone
home again. Set out sentries against an attack from O'Donnell, and
scatter a score of men out along the roads to watch for other parties.
You might pick up another score of recruits, Turlough Wolf."
Turlough shook his head and tugged at his beard.
"Best take me with you, master, instead of this overgrown ox. You may
need brains in dealing with the Bird Daughter, and he has no more brains
than strew his ax-edge. Also he is wounded."
Brian pondered this, while Cathbarr furtively shook a fist at Turlough.
There was wisdom in the advice, but on the other hand Brian did not like
to leave his precious two hundred men in care of Cathbarr. If the Dark
Master attacked suddenly, as he was like to do, brains would be more
needed than brawn.
On the other hand, he counted on Cathbarr's open face removing the
evident suspicion that the smooth-tongued Turlough had raised in Gorumna
Isle. It had been a mistake, he saw plainly, to send such an emissary on
his mission. Picturing this woman who led her own ships to war, he
limned her in his mind as a large-boned, flat-breasted, wide-hipped
creature--and with good reason. He had seen women fighting at Drogheda
and he had seen them in other places as he rode to the rest, for in
those days many a woman took her slain lord's _skean fada_ and drew
blood for Ireland before she was cut down. And when women rode to battle
there was no mercy asked or given, from Royalist or Confederate or
Parliament man.
Nuala O'Malley was a woman of blood, said Brian to himself, and he would
give her blood for her help.
So he curtly refused Turlough's advice, saw that the ten bridles of his
bound and mounted captives were lined together, and beckoned to
Cathbarr. Before they rode off, however, they doffed their Scot
accouterments and took back their own garments, after which Cathbarr led
the way over the hills to Kilkieran Bay, and Turlough took command of
the force in sullen ill-humor.
The morning was still young, for the attack had taken place a short two
hours after sunrise and had soon been quelled. Beyond a slashed thigh
and a red-creased neck, Cathbarr of the Ax was unhurt, and Brian had
received no scratch. If the ten captives wondered why they were bound
and their comrades freed, they said nothing of it.
Even after seeing what he had of the merciless war in Ireland, Brian had
much ado in making up his mind to hold to the plan he had formed on the
previous evening. These ten ruffians were scoundrels enough, to judge by
looks, and yet they were men; and he had been raised in no such school
of war as this, where surrender meant slaughter without pity. However,
he determined to do what he could for them, and he would have held to
this determination had it not been for what chanced when they rode down
to the little fishing village where Turlough had met the O'Malley men.
They arrived just as the evening was darkling, after a hard day's ride.
As they came within sight of the place, which lay at the head of
Kilkieran water, Brian made out that a small galley was pulled up on
shore, and there were a number of men about the huts. Upon the approach
of the two chiefs with their file of captives there was an instant
scurry of figures; women ran to the huts, and a dozen or more roughly
clad men appeared with pikes and muskets. Brian held up his hand in sign
of peace and rode slowly onward, Cathbarr at his side, to within a dozen
paces of the huts.
"Who are you?" cried out one of the musketeers. "Be off!"
"Bark less, dog," said Brian, scorn in his eye. "We seek Nuala O'Malley.
Take us out to Gorumna Isle in your boat."
"What seek ye with the Bird Daughter?" queried the other suspiciously.
"Her business, not yours."
The seamen gazed at them doubtfully, then a number of other men came
from the huts, well-armed. One of these set up a cry, pointing at the
captives, and a burst of yells answered him from the rest. Next instant
Brian and Cathbarr had their weapons out and were facing an excited
crowd of men.
"Be silent, dogs!" bellowed Cathbarr, and his voice quelled the uproar.
"What means this attack? Would you have the Bird Daughter strip you with
whips, fools?"
The spokesman stood out, his dark face quivering with fury as he
pointed.
"That is as it may be, axman, but first those bound men shall die. One
is the man who slew my brother, nailing him to his own door till he
died; another is he who burned Lame Art's wife and child last
Whit-Sunday--"
"There is he who lopped my husband's hands and nose! Slay him!" shrieked
out a hag as she burst forward. Brian held out his sword and she drew
back, but instantly others had taken up the cry.
"And the devil who hung Blind Ulick!"
"There is he who--"
In that brief moment Brian heard things too horrible for speech. The ten
bound men had grouped together, some pale as death, others laughing
defiantly. But as the crowd surged forward Brian held up his sword, and
they paused to listen; he knew now that there was no more pity in his
heart for these black ruffians of O'Donnell's.
"Let the Bird Daughter render judgment upon them," he shouted. "Friends,
take us to the Bird Daughter and let her do as she will, for I bear
these men to her alone."
At that the crowd fell silent, but their leader gave a rapid order, and
half a dozen men ran down to the strand. Another order, and the maddened
villagers gave back as the seamen closed about Brian and Cathbarr and
their captives.
"Come," said the leader roughly. "You shall go to Gorumna Isle with us,
strange men, but I do not think that you shall ever come back again."
"Nor do I," grinned Cathbarr in the ear of Brian, as they left their
horses to the fishermen, unbound the prisoners from their steeds, and
made their way down to the galley. Brian looked at his friend, and they
both smiled grimly.
CHAPTER VII.
THE BIRD DAUGHTER.
"Now, _there_ is a castle worth the taking, Yellow Brian!" said
Cathbarr.
Brian nodded, his eyes shining in the starlight. After a pull of a long
seven miles down the bay, the galley had rounded into the northern end
of Gorumna Isle, guided by a high beacon set among the stars. As they
drew nearer Brian made out that this beacon was set on the tower of a
high pile of masonry black against the sky, lit here and there by
cressets, and it was plain that the Bird Daughter kept good watch since
they had more than once been hailed in passing the islands.
Once turned into the harbor, Brian found suddenly that they were among
ships, many of them small galleys, but two of good size which bore
riding-lights. Again they responded to hails, and without warning a few
torches blazed out ahead of them. Then it was seen that the castle was
built with its lower part close on the water, and its upper part rising
on the crag. In reality, as he found later, it was two castles in one,
as of necessity it had to be. Were the opposite isles held by an enemy,
and hostile ships in the little harbor, the higher towers running up the
crag could dominate all, and the lower castle could be abandoned without
danger.
Even in the starlight Brian's trained soldier's eye made out something
of this. Then the leader of the seamen came and stood beside them, for
during the two-hours' trip he had talked somewhat with Cathbarr and had
come to look with more respect on Brian himself. That was only natural,
for seamen ever like those men who talk least.
"Strangers," he said with rough courtesy, "a word in your ear. If you
would gain speech with the Lady Nuala, deal not with her as with me.
Send in your names and your business, and you may perchance get to see
her in the morning, or a week hence, as she may choose."
"Thanks," answered Brian. "But my will is not like to hang upon hers."
The seaman shrugged his shoulders, the oars were put in, and they
floated up to where the torches flared. Here there was a landing-place
of hewn stone, with a gate lying open beyond it, and armed men waiting.
One of these, from his bunch of huge keys and air of authority, Brian
knew for the seneschal.
"_M'anam go'n Dhia!_" he growled, peering down into the boat as it
ground on the stone, "what fish have you there?"
"Two salmon and ten herring, Muiertach," laughed one of the men. Brian
and his friend stepped out while the ten prisoners were prodded after
them, and Brian found the seneschal looking him over with some wonder,
hands on hips.
"Well! A giant with a devil's ax, and Cuculain, the Royal Hound, come to
life again! Who are you, yellow man, and who is this axman, and who are
these ten bound men?"
Brian was minded to answer curtly enough, but he looked at the seneschal
and remembered the seaman's kindly warning. Under his eye the laugh
withered suddenly on the seneschal's lips.
"These ten men belong to me, Muiertach. Go, tell the Bird Daughter that
Brian Buidh and Cathbarr of the Ax have come to her, bringing tribute as
she demanded."
Now it was that Cathbarr, who had asked no questions all that day,
perceived for the first time the reason of their fighting and hard
riding, and what the manner of that tribute was. He broke into a great
bellow of laughter so that the rough-clad seamen stared at him in
wonder, but at a word from Brian he quieted instantly.
"In the morning the message shall be delivered, Brian Buidh," returned
burly Muiertach with a glimmer of respect in his voice. "And now render
up your weapons, so that we may treat you as guests--"
"So you sea-rovers are afraid of two men, lest they capture your hold?"
Brian's biting words brought a deep flush to Muiertach's face.
"No weapons do we render," he went on, his voice cold as his eyes. "We
come as guests, seneschal, and our business is not with you. Take these
ten men to your dungeons, take us to guest chambers and give us to eat,
and see that we have speech with the Bird Daughter before to-morrow's
sun is high."
At this Muiertach growled something into his beard, but turned with a
gesture of assent. His men closed around the captives, while Brian and
Cathbarr followed him into the castle, the giant still chuckling to
himself with great rumbles of laughter.
"Let strict watch be kept over these two," said Muiertach in English to
one of the torchmen who accompanied them, thinking he would not be
understood.
"You may yet get a touch of the whip for that order," said Brian in the
same tongue.
Stricken with amazement, Muiertach turned and stared at him, jaw
dropping, while Cathbarr glanced from one to the other in perplexity.
Brian smiled.
"Lead on, and talk less."
With tenfold respect, the seneschal obeyed. Now Brian saw that this
castle was indeed a stronghold, and might easily be defended by fewer
men than it had. The inner walls of the lower castle were well lined
with falcons and falconets, while on the towers above peered out heavier
cannon, which he took for culverins from their length of nose. Crossing
the courtyard, they entered the building itself, and Muiertach led them
through upward-winding corridors, studded with cressets and with here
and there a recessed _prie-dieu_ in the wall.
From the snatches of talk behind the doors they passed, Brian guessed
that this lower castle was occupied by the garrison. In this he was
right, for with torchmen before and behind them they emerged into the
cold night air again and climbed upward, coming to a gate in the wall of
the upper castle. This stood open, but it clanged shut behind them, and
after crossing a steep courtyard they entered a second and broader
corridor.
Muiertach led them up a long flight of stairs, then another, and finally
flung open a heavy door. It was evident that they were lodged in one of
the towers.
"Rest sound and fear not to eat our food," said the seneschal.
"_Beannacht leath!_"
"Blessing on you," responded Brian and Cathbarr together, and entered.
For a wonder, Brian found that the chamber was lighted with candles,
which Cathbarr examined with no little awe. Also, it contained a very
good bed, on which the giant looked with suspicion. The hard stone walls
were hung with tattered tapestries, and before they had settled well
into their chairs two men entered with food and wine of the best.
"Not so bad," smiled Brian as they ate. "How come your wounds, brother?"
"Those scratches? Bah!" And the giant gurgled down half a quart of
Canary at a stretch. "You are not going to sleep on that bed of cloths?"
"That I am," laughed Brian, "and soon, for I am overweary with riding.
Try it, Cathbarr, and you will be glad of it."
"Not I! Since there is no bracken here the floor is good enough for me.
Eh, but this sea-woman will have a thought in her mind over your
message, brother!"
Brian chuckled, but he was too weary with that day's work to talk or
think, and when the remnants of their meal had been removed and their
door shut, he gratefully sought the first bed he had known for weeks.
After some laughing persuasion he prevailed on the suspicious Cathbarr
to blow out the candles, and upon that he fell asleep.
When he wakened it was broad daylight, and Cathbarr was still snoring
with his ax looped about his wrist as usual. Brian, feeling like a new
man, went to the open casement and looked out.
He found himself gazing through a three-foot stone wall, and as he was
doubtless in one of the towers, this argued that the lower walls were
twelve feet thick or more. The lower castle was hid from him, but his
view was toward the upper bay and included the harbor. The two larger
ships, which were small caracks, but large for the west coast in that
day, bore six guns on a side, and Brian saw that they were being
scrubbed and made shipshape. The Bird Daughter must be a woman of some
scrupulousness, he reflected. Beyond the brown sails of two
fishing-boats, and low, storm-boding clouds over the farther hills,
there was nothing more in sight.
As Cathbarr still wore his long mail-shirt, Brian kicked him awake, and
after his first bellowing yawn their door opened and men brought in jars
of water. When the giant's wounds had been dressed, under protest, and
they had broken their fast, the seneschal appeared.
"Chieftains," he said respectfully, "the Lady Nuala has received your
message and will have speech with you this afternoon. Until then she
wishes that you keep your chamber, since she knows not your mind in this
visit."
"That is but fair," assented Brian.
Cathbarr grumbled, but there was no help for it, since they were
virtually prisoners. The day passed slowly, and toward noon storm drew
down on the harbor and snow eddied in their casement. With that, they
fell to polishing their weapons; Brian procured a razor and a
much-needed shave, and Cathbarr furbished up his huge ax until it glowed
like silver.
Finally Muiertach appeared. Brian slung the great sword across his back,
and they followed the seneschal down to the courtyard. Here they were
joined by the captive O'Donnells and the seamen who had brought them to
the castle, and Muiertach led them to the great hall.
The father of this O'Malley woman must have been a man of parts, thought
Brian as he gazed around. The hall was scantily filled with, perhaps,
three-score men ranged along the walls, and at the farther end was a low
dais where a huge log fire roared high. The beams were hung with a few
pennons and ship-ensigns, and on the dais were placed a half-dozen
chairs. Behind one of these stood two women, and in the chair, calmly
facing the hall, sat the Bird Daughter.
Brian caught his breath sharply, and his blue eyes flickered flame as he
saw her. Never in his life had his gaze met such a woman--not in all the
land of Spain or elsewhere in Ireland.
At this time Nuala O'Malley was twenty years old, and ten of those years
had been passed either on shipboard or here in Gorumna Isle. As one
chronicler describes her, "She was not tall, but neither was she small
of stature, and when she stood on a ship's deck there was no tossing
could cause her to stumble. Her hair was not blue, but neither was it
black, and her eyes were very deep and bright, violet in color, and set
wide in her head. Her nose was neither small nor large, her cheeks were
ever red with the wind off the sea, her mouth was finely curved, but
tight-set withal, and she had more chin than women are wont to have. She
was very lissom in body, but her head never drooped."
And that is a most excellent description of the Bird Daughter, in fewer
words than most men might use to-day.
But of all this Brian noted at the moment only that before him sat a
girl-woman whose calm poise and confident power struck out at him like a
vibrant presence. Like himself, she wore a cloak of dark red, but no
steel jack glittered beneath it; there was a torque of ancient gold
about her neck, and her hair was caught up and hidden beneath a small
cap of red.
Brian thought of the woman he had painted in his mind, then laughed
softly. She caught the laugh on his face, and comprehended it, and was
pleased; then as she watched him very calmly, it seemed to Brian that
her sheer beauty was a thing of deception. It must be, for she was
surely a woman of blood. He had known enough of beautiful women, who
played the parts of men, to know that on the far side of their beauty
was neither mercy nor love nor compassion, that their lovers were many
steps to ambition, and that they were venomous. So his smile died away,
and his blue eyes glittered cold and dark, and this the Bird Daughter
saw also.
Now, there was no man on the dais save Muiertach, who mounted the two
steps with his keys jangling. As Brian would have gone after him, two
pikemen stepped forward to intervene. Brian looked into their eyes and
they drew back again. He and Cathbarr mounted to the dais, and he bowed
a low, courtly, Spanish bow, of which the Bird Daughter took no note.
Instead he heard her voice, very low and penetrating, and she was
speaking to the two pikemen.
"Go out into the courtyard," she said, "and give each other five lashes.
This is because you dared insult a guest, and because you drew back
after insulting him. Go!"
The two pikemen, rather pale under their beards, handed over their pikes
to comrades and strode out of the hall. She turned to Brian, speaking
still in Gaelic:
"Welcome, Brian Buidh. You have come to bring me tribute?"
"Yes, Lady Nuala, and the tribute is these ten men of the Dark
Master's."
She looked at Cathbarr; her eyes swept over his ax. Then she looked
again at Brian, and spoke to Muiertach in English.
"Truly, I have seldom seen such a man as this--"
A swift look of warning flashed over the seneschal's face, and Brian
laughed.
"Lady," he said in the same tongue, "he is Cathbarr of the Ax, and he
will be a good man to stand with us against the Dark Master."
She betrayed no surprise, except that a little tinge of red crept to her
temples.
"I did not know you spoke English, Brian Buidh. Still, it was not to
Cathbarr that I referred."
At that it was Brian's turn to redden, and mentally he cursed himself.
There was no evil in this woman's heart, he saw at once. For an instant
he was confused and taken aback. Then she smiled, slowly rose, and
tendered him her hand. Going to one knee, he put her fingers to his
lips.
"Now sit, Yellow Brian," she said, "and let us talk. First, these
captives of yours. Do you in truth bring them as a tribute? How do I
know they are O'Donnell's men?"
"Ask these seamen of yours," laughed Brian, seating himself beside her.
Cathbarr remained standing and leaning on his ax, looking like some
giant of the old times.
She took him at his word, and when she had heard from the seamen certain
tales of what cruelties the ten prisoners had done, her violet eyes
suddenly turned black and an angry pallor drove across her face.
"That is enough," she interrupted curtly. "Take them out and hang them."
The men were led away, and Brian saw that her hands were tightly
clenched, but whether in fury or in fear of herself he could not tell.
Then she turned to him, looking straightly into his face, and on the
instant Brian knew that if this girl-woman bade him go to his death, he
would go, laughing.
"Tell me of yourself, Brian Buidh. Of what family are you? By the ring
on your finger you are an O'Neill; yet I have heard nothing of such a
man as yourself leading that sept. When your messenger came to me, I
read cunning in his face, and took it for a trap set by the Dark Master;
but now that I have seen you and Cathbarr of the Ax, I will take fealty
from you if you wish to serve me."
Brian smiled a little.
"Serve you I would, lady, but not in fealty. I take fealty and do not
give it. My name is indeed Brian Buidh, and as for that ring, it was a
gift from Owen Ruadh."
"Owen Ruadh died two days since," she said softly, watching his face. "I
had word of it this morning."
At that he started, and Cathbarr's eyes widened in fear of magic. Owen
Ruadh had lain on the other side of Ireland, and three months would have
been fast for such news to travel. But Brian nodded sadly.
"Carrier pigeons, eh?" he said in English and paused. He knew not why,
but his loneliness seemed stricken into his heart on a sudden; he who
neither explained nor asked for explanation from any man, felt impelled
to open his life to this girl-woman. He crushed down the impulse, yet
not entirely.
"Perhaps, Lady Nuala, there shall be greater confidence between us in
time, and so I truly desire. But know this much--I am better born than
any man in Ireland--aye, than Clanrickard himself; and I am here in the
west to seek a new name and a new power. It is in my mind to take
O'Donnell's castle from him, lady. I have some two hundred men, of whom
the Dark Master himself lent me twoscore, and in alliance with your
ships we could reduce him."
"How is this, Brian? You say he _lent_ you twoscore men?"
He laughed and explained the fashion of that loan; and when he had
finished a great laugh ran down the hall, and the Bird Daughter herself
was chuckling. Then he waited for her answer, and it was not long in
coming.
"There is some reason in your plan, Brian Buidh, but more reason against
it. The castle that O'Donnell holds was formerly my father's. If you
held it, there would be no peace between us, unless you gave fealty to
me, which I see plainly you will not do. I claim that castle, and shall
always claim it."
"Then it seems that I am held in a cleft stick," smiled Brian easily,
"since I will give fealty to none save the king, or Parliament. You are
allied with the Roundheads, I understand?"
She nodded, watching him gravely.
"Yes. Cromwell is master of the country, and I am not minded to butt my
head against a wall, Brian Buidh. If I am to hold to the little that is
left me, I shall need all my strength."
"And that is not much, lady. Your coasts are plague-smitten, your men
reduced, and Cromwell has not yet won all the country. Galway will be
the last to fall, indeed. But as to Bertragh Castle, why should you not
sell your rights in it to me?"
At his first words a helpless anger flashed into her face, succeeded by
a still more helpless pride.
"No, I will not sell what I have been unable to conquer back, Brian
Buidh. If there were any way out of this difficulty with honor, I would
take it; for I tell you frankly that I would make alliance with you if I
could."
Brian gazed at her, reading her heart, and fighting vainly against the
impulse that rose within him. Twice he tried to speak and could not,
while she watched the conflict in his face and wondered. He wished
vainly that he had Turlough's cunning brain to aid him now.
"Lady," he said at last, biting his lips, "I will do this. I will give
you fealty for the holding of Bertragh Castle, keeping it ever at your
service, but for this alone. When we have taken it, it may be that I
shall render it back after I have won a better for myself; yet, because
I would sit at your side and have equal honor with you, and because we
have need of each other, I will give you the service that I would grant
to no man alive. Is it good?"
For an instant he thought that she was about to break forth in eager
assent, then she sank back in her chair, while breathless silence filled
the hall. She gazed down at the floor, her face flushing deeply, and
finally looked up again, sadly.
"I do not desire pity or compassion, Brian Buidh," she said simply, and
her eyes held tears of helpless anger.
Then Brian saw that she had pierced his mind, for which he was both
sorry and glad. He knew well there were other castles to be had for the
taking, and there was nothing to prevent his riding on past Slyne Head
and winning them--except for his meeting with this girl-woman. Therefore
he lied, and if she knew it, she gave no sign.
"You mistake me, lady," he said earnestly, his blue eyes softening
darkly.
"I propose this only as a stepping-stone to my own ambition. Soon there
will be a sweep of war through the coasts, and I would have a roof over
my head. Is it good?"
She rose and held out her hands to him.
"It is good, Brian Buidh. Give me fealty-oath, for Bertragh Castle
alone."
And he gave it, and his words were drowned in a roar of cheers that
stormed down the hall, for the O'Malleys had heard all that passed.
An hour later Cathbarr of the Ax was despatched in a swift galley to
bear the tidings to Turlough, and bid him make ready for a swift and
sharp campaign.
Through the remainder of that afternoon and evening Brian sat beside the
Bird Daughter, and he found his tongue loosened most astonishingly, for
him. He told her some part of his story, though not his name, while in
turn he learned of her life, and of how her father and mother had been
slain by O'Donnell through blackest treachery.
The more he saw of her, the more clearly he read her heart and the more
he gave her deeper fealty than had passed his lips in the oath of
service. As for her, she had met Blake and others of the Roundhead
captains on her cruises, deadly earnest men all; but in the earnestness
of Brian she found somewhat more besides, though she said nothing of it
then. It was arranged between them that in three days they would meet
before Bertragh Castle, by sea and land, and the Dark Master would be
speedily wiped out.
With the morning Brian set forth to join his men in the largest sailing
galley, for a wild gale was sweeping down from Iar Connaught. But the
O'Malleys were skilled seamen who laughed at wind and waves, and Brian
kissed the hand of the Bird Daughter as he stepped aboard, with never a
thought of the storm of men that was coming down upon them both, and of
the blacker storm which the Dark Master was brewing in his heart.
TO BE CONTINUED NEXT WEEK. Don't forget this magazine is issued weekly,
and that you will get the continuation of this story without waiting a
month.
Nuala O'Malley
by H. Bedford-Jones
Author of "Malay Gold," "The Ghost Hill," "John Solomon, Supercargo,"
etc.
This story began in the All-Story Weekly for December 30.
CHAPTER VIII.
HOW BRIAN WAS NETTED.
The Dark Master sat in his dark hall, brooding.
It was a bad morning, for there was a sweep of wind and black cloud
mingled with snow bearing out of the north; and since the great hall,
with its huge fireplace, was the warmest part of the castle, as many of
the men as could do so had drifted thither, but without making any undue
disturbance over it.
For that matter, they might have passed unseen, since the hall was black
as night save for a single cresset above the fireplace. Here sat the
Dark Master, a little oaken table before him on which his breakfast had
rested, and at his side crouched a long, lean wolfhound that nuzzled him
unheeded. On the other side the table sat the old _seanachie_, who was
blind, and who fingered the strings of his harp with odd twangings and
mutterings, but without coherence, for O'Donnell had bade him keep
silence.
"Go and see what the weather is," commanded the Dark Master. A man rose
and ran outside, while other men came in with wood. Their master
motioned them away, although the fire had sunk down into embers.
"A gale from the north, which is turning to the eastward, with snow,
master."
"Remain outside, and bring me word what changes hap, and of all that you
see or hear. Waste no time about it."
The Dark Master drew his cloak about his humped shoulders, and in the
flickering dim light from overhead his face stood out in all its ghastly
pallor, accentuated by the dead black hair and mustache. But his eyes
were burning strangely, and when they saw it the men drew back, and more
than one sought the outer chill in preference to staying.
Now O'Donnell Dubh stared into the embers and muttered below his breath,
while, as if in response, a little flickering whirlwind of gray ash rose
up and fell back again, so that it blew over the embers and deadened
them. The muscles of the Dark Master's face contracted until his teeth
flashed out in a silent snarl.
"I could have slain, and I did not," he whispered as if to himself. "But
there is still time, and I will not be a fool again!"
The watching men shivered, for it seemed that the wind scurried down the
wide chimney and again blew up the gray ash until the embers glowed
through a white coating. But the wind wrought more than this, for it
brought down from the gray clouds a whispering murmur that drifted
through the hall, and in that murmur were mingled the sounds of beating
hoofs and ringing steel and shrieking men.
"Are watchers posted over the hills and the paths and the Galway roads?"
spoke out the Dark Master as he gazed into the ashes.
"They are watching, master," answered a deep voice from the darkness.
Suddenly the wolfhound raised its head and stared into the ashes also,
as if it saw something there that no man saw, for the bristles lifted on
its neck, and it whined a little. O'Donnell dropped his hand to the thin
muzzle, and the dog was quiet again. But after that the men stared at
the fireplace with frightened eyes.
"There is still time, though one has escaped me," said the Dark Master,
looking up suddenly at his sightless harper, who seemed to fall
atrembling beneath the look. "The one who has escaped matters not, for
his bane comes not at my hands. It is the other whom I shall slay--Brian
Buidh of the hard eyes. Then the Bird Daughter. But it seems to me that
one stands in my path of whom I do not know."
He brooded over the ashes as his head sank between his shoulders like a
turtle's head. Then once again the wind swooped down on the castle, and
whistled down the chimney, and filled the great hall with a thin noise
like the death-rattle of men. The cresset wavered and fell to smoking
overhead.
The Dark Master reached his hand across the table and caught the hand of
the blind harper and spread it out on the oak. A little shudder shook
the old man, and as if against his will he spread out his other hand
likewise, his two hands lying between those of the Dark Master. Then
there fell a terrible and awestruck silence on the hall.
The stillness was perfect, and continued for a long while. Slowly
occurred a weird and strange thing, for, although no blast whimpered
down the chimney, the ashes fell away from the embers, which began to
glow more redly and set out the forms of the Dark Master and the blind
harper in a ruddy light. Suddenly a man pointed to the feet of the Dark
Master, and would have cried out but that another man struck him back.
For the ashes had drifted out from the fireplace, flake after flake, and
were settling about the feet of the Dark Master beneath the table. They
rose slowly into a little gray pile; then one of the men shrieked in
horror at the sight, and the Dark Master threw out his head.
"Slay him," he said quietly and drew in his head once more, staring at
the table.
There was a thudding blow and a groan, then the stillness of death. The
ashes were quiet; the fire glowed ruddily. After a little there came a
soft whirl of soot down the chimney, blackening the embers. The soot
rose and fell, rose and fell, again and again; it was as if an eddying
draft of wind were trying to raise it. Finally it was lifted, but it
only whirled about and about over the embers, like a shape drawn
together by some uncanny force.
The Dark Master raised his head as a clash of steel and the voice of the
watcher came from the outer doorway.
"Master, the blast thickens with black fog!"
"Remain on watch," said O'Donnell, and his head fell.
But through the hall men's hands went out to one another in the
darkness. For storm-driven fog was not a thing that many men had seen
even on the west coast, and when it did happen men said that a warlock
was at work. There was not far to seek for the warlock in this case,
muttered the O'Donnells.
Now the Dark Master looked into the fireplace and that whirling figure
of soot raised itself anew and began its unearthly dance over the
embers. After no long time men saw that the pile of gray ashes under the
table was lifting also, lifting and whirling as though the wind spun it;
but there was no wind.
"There is a man to be blinded," said the Dark Master. "Let him be
blinded with fog and snow, and the men with him, and let the wind come
out of the east and drive him to this place."
Slowly, so slowly that no man could afterward say where there was
beginning or end, the whirling figure of soot dissipated; and little by
little the dancing stream of gray ashes drifted back into the fireplace;
then it also dissipated, seeming to pass up the chimney, so that the
embers glowed red and naked.
"_Seanachie_," said the Dark Master in a terribly piercing voice, "who
is this standing in my way, standing between me and Brian of the hard
eyes?"
The blind harper began to tremble, but again came the clash and the
watcher's voice from the doorway.
"Master, there is snow mingled with the fog, and the wind is shifting to
the eastward."
"Light the beacon and remain on watch," said the Dark Master. But at the
watcher's word new terror seized on the men in the hall.
"_Seanachie_, who stands in my way? Speak!"
The beard of the blind harper quivered and rose as if the wind lifted
it, but men felt no wind through the hall. Then the old man began to
writhe in his chair, and twisted to take his hands from the table, but
he could not, although only he alone held them there. Suddenly his
mouth opened, and a voice that was not his voice made answer:
"Master, two people stand in your way."
"Describe them," said the Dark Master, and those near by saw that sweat
was running down his face, despite the coldness of the hall. After a
moment's silence the old harper spoke again; he had lost his eyes twenty
years since, yet he spoke of seeing.
"Master, I see two people but dimly. One is a man, huge of stature and
standing like Laeg the hero, the friend of the hero Cuculain, leaning
upon an ax--"
"That is Cathbarr of the Ax," broke in the Dark Master. "His bane comes
not at my hands. Who is the other?"
Again the old harper seemed to struggle, and his voice came more
faintly:
"I cannot see, master. I think it is a woman--"
"That is the Bird Daughter," quoth the Dark Master.
"Nay, it is an old woman, but she blinds me--"
And the harper fell silent, writhing, until horror gripped those who
looked on. O'Donnell leaned forward, his head sticking straight out and
his eyes blazing.
"What do you see, _seanachie_? Speak!"
"I see men," and the old harper's voice rose in a great shriek. "A storm
of men and of hoofs, and red snow on the ground, and fire over the snow,
and the man of the ax laughing terribly. And I see other men riding
hard; men with long hair and the flag of England in their midst--and
Cuculain smites them--Cuculain of the yellow hair--the Royal Hound of
Ulster smites them and scatters them--"
"_Liar!_"
With the hoarse word the Dark Master leaned forward and smote the blind
harper with his fist, so that the old man slid from his chair senseless.
Upon that the Dark Master swung around with his teeth bared and his head
drawn in like the head of a snake about to strike.
"Lights!" he roared. "Lights! Bear the _seanachie_ to his chamber, and
send men to ring in the harbor and build beacons on the headlands.
Hasten, you dogs, or I'll strip the flesh from you with whips!"
Under his voice and his flaming eyes the hall sprang into life, while
the men carried out the blind harper and one of their own number who had
been stricken with madness at what he had seen. Then the hall blazed up
with cressets, logs were flung on the fire, and parties of men set out
to build beacons and guard the bay as the Dark Master had given command.
And when word was spread abroad among the others of what had chanced in
the hall that morning, Red Murrough, the Dark Master's lieutenant, swore
a great oath.
"If that Cuculain of whom the _seanachie_ spoke be not the man Brian
Buidh, then may I go down to hell alive!"
And the men, who feared Red Murrough's heavy hand and hated him,
muttered that he would be like to travel that same road whether living
or dead, in which there was some truth.
While these things took place in the hall at Bertragh--and they were
told later to Brian by many who had seen them and heard them, all
telling the same tale--Brian and his sailing galley was making hard
weather of it. Six of the O'Malleys had been sent with him to manage the
galley, for he was no seaman and had placed himself in their hands; and
after rounding into Kilkieran Bay from the castle harbor and reaching
out across the mouth of the bay toward Carna, intending to reach
Cathbarr's tower direct, the blast came down on them, and even the
O'Malleys looked stern.
Sterner yet they looked when Brian cried that Golam Head was veiling in
fog behind them, and with that the wind swerved almost in a moment and
swept down out of the east, bearing fog and snow with it. Nor was this
all, for the shift of wind bore against the seas and swept down
currents and whirlpools out of the bay, and after the snow and black fog
shrieked down upon them, the seamen straightway fell to praying.
"Get up and bail!" shouted Brian, kicking them to their feet, for the
seas were sweeping over the counter. The helmsman groaned and bade him
desist, and almost at the same instant their mast crashed over the bow,
breaking the back of one seaman, and the galley broached to.
With that the O'Malleys ceased praying and fell to work with a will,
getting out the sweeps and bailing. The mingling of snow, shrieking
wind, and black fog had been too much for their superstitious natures,
but made no impression on Brian, for the simple reason that he did not
see why fog and wind should not come together. After he understood their
fears better he shamed them into savage energy by his laughter, and
since the broken-backed man had gone overboard, took his sweep and set
his muscles to work.
They made shift to keep the craft before the wind, but presently Brian
found that half the men's fear sprang from the fact that the fog and
snow blinded them, shutting out the land, and that the shifting wind had
completely bewildered them. When he asked for their compass, their
leader grunted:
"No need have we for a compass on this boat, Brian Buidh, save when
warlocks turn the fog and wind upon us. I warrant that were it not for
the fog, we would be safe in port ere now. As it is, the Virgin alone
knows where we are or whither going."
"This is some of the Dark Master's wizardry," growled out another.
"Before we hung those men of his last night, they said that the winds
would bear word of it to the dark one, _cead mile mollaght_ on him!"
"Add another thousand curses for me," ordered Brian, "but keep to the
bailing, or I'll give you a taste of my foot! And no more talk of
warlocks."
The five men fell silent, and indeed they needed all their breath, for
the struggle was a desperate one. Instead of lessening, the fog only
increased with time, and even Brian began to perceive the marvel in it
as swirl after swirl of darkness swept over them. Yet, since the wind
was from the east, he reasoned, it would naturally blow out the fog from
the bogs and low lands. But this explanation was received in dour
silence by the men, so he said no more.
There was no doubt that Cathbarr had reached home safely, since the
night had been fair enough for the winter season. An hour passed, and
then another, still without a lessening of the eery storm; and the nerve
of the seamen was beginning to give way under the strain, when the
helmsman let out a wild yell:
"A light ahead! A beacon!"
The rowers twisted about with shouts of joy, and Brian perceived a
faint, ruddy light against the sky. Also, the fog began to lessen
somewhat; and upon making out that the beacon undoubtedly came from a
high tower or crag, the shout passed around that they had headed back to
Gorumna with the shifting wind.
This heartened them all greatly, the more so since the gale drove them
straight onward toward the beacon. The fog closed down again, but the
ruddy glare pierced through it; and of a sudden there was no more fog
about them--only a blinding thick snow, which made all things grotesque.
Then two more beacons were made out, lower than the first, and the men
yelled joyously that fires had been lighted on either side the harbor to
guide them in. And so they had been, but otherwise than the men thought.
Half frozen with the cold, they drove on through the snow and spray
until at length they swept in between the guiding fires and scanned the
shores for landing. Then the snow ceased, though the hurricane howled
down behind them with redoubled fury; and as they floated in against a
low, rocky shore, silence of wild consternation fell on them all. For
they had come to Bertragh Castle, and fifty feet away a score of men
were waiting, while others were running down with torches.
Even in that moment of terrible dismay, Brian noted their muskets, and
how the lighted matches flared like fireflies in the wind.
"Trapped!" groaned one of the men, and they would have rowed out again
into the teeth of the storm had not Brian stayed them.
"No use, comrades. They have muskets, and there are cannon up above. Row
in, and if we must die, then let us die like men and not cowards."
Seeing no help for it, the men growled assent, and they drifted slowly
in, all standing ready with drawn swords, while Brian's Spanish blade
flared in the prow. Then in the midst of the gathered men he saw a dark
figure with hunched shoulders, sword in hand. As he turned to the seamen
behind him, there was a glitter in his blue eyes colder than the icy
blast behind them.
"There is the Dark Master, comrades! Let him be first to fall."
They drove up on the shore, and Brian leaped out, with the men behind
him. Still the group above stood silent until the voice of O'Donnell
sheared through the gale. "Fire, and drop Yellow Brian first."
So there was to be no word of quarter! As the thought shot like fire
through Brian's mind, he leaped forward with a shout. A ragged stream of
musketry broke out from the men gathered on the higher rocks, and he
heard the bullets whistle. He paid no heed to the seamen who followed
him, however. His eyes were fixed on the Dark Master's figure, and with
only one thought in his mind he plunged ahead.
More and more muskets spattered out; a bullet splashed against his jack,
and another; something caught his steel cap and tore it away, and a hot
stab shot through his neck. But the group of men was only a dozen paces
from him now, and a wild yell broke from his lips as he saw O'Donnell
step forward to meet him.
Then only did he remember Turlough's speech on the day of that first
meeting with the Dark Master--"The master of all men at craft and the
match of most men at weapons"--and he knew that, despite the hunched
shoulders, this O'Donnell must be no mean fighter. But the next instant
he was gazing into the evil eyes, and their blades had crossed.
Flaming with his anger, Brian forced the attack savagely; then a sharp
thrust against his jack showed him that O'Donnell was armed with a
rapier, and he fell to the point with some caution. With the first
moment of play, he knew that he faced a master of fence; yet almost upon
the thought his blade ripped into the Dark Master's arm.
Involuntarily he drew back, but O'Donnell caught the falling sword in
his left hand and lunged forward viciously. Just as the blades met
again, Brian saw a match go to a musket barely six paces away. He
whirled aside, but too late, for the musket roared out, and a drift of
stars poured into his brain. Then he fell.
Like a flash the Dark Master leaped at the man who had fired and spitted
him through the throat; the others drew back in swift terror, for
O'Donnell was frothing at the mouth, and his face was the face of a
madman. With a bitter laugh he turned and rolled Brian over with his
foot. The five seamen had gone down under the bullets.
"He is only stunned," said Red Murrough. "Shall I finish it?"
"If you want to die with him, yes. Carry him in, and we will nail him up
to the gates to-morrow."
And the clouds fell asunder, and the stars came out, cold and beautiful.
CHAPTER IX.
THE NAILING OF BRIAN.
Brian woke in darkness, with pain tearing at his head and heaviness upon
his hands and feet. When he tried to put his hand to his head, that
heaviness was explained; for he could not, and thick iron struck dull
against stone.
He lay there, and thought leaped into his brain, and he felt very bitter
of spirit, but chiefly for those men who had come with him, and because
he had failed before the Dark Master's hand.
It was cold, bitterly cold, and thin snow lay around him, so that he
knew that he was in some tower or prison that faced to the east. It was
from that direction that the snow had driven, as he had sore cause to
know, and he wondered if the Dark Master had had any hand in that
driving. But this he was not to know for many days.
It was the cold which had awakened him from his unconsciousness, he
guessed. By dint of shifting his position somewhat, he managed to get
his back against a wall, and so got his hands to his head. In such
fashion he made out that his hair was matted and frozen with blood, and
his neck also, where a bullet had plowed through the muscles on the
right side. His head-wound was no more than a jagged tear which had
split half his scalp, but had not hurt the bone, as he found after some
feeling. Then he dropped his hands again, for the chains that bound him
to the wall were very heavy. It must be night, for light would come
where snow had come, and there was no light.
Now, having found that he was not like to die, at least from his wounds,
he set about stretching to lie down again, and found some straw on the
floor. He drew it up with his feet and gathered it about him; it was
dank and smelled vilely, but at the least it gave his frozen body some
warmth, so that he fell asleep after a time.
When he wakened again, it was to find men around him and a narrow strip
of cold sunlight coming through a high slit in the wall of his prison.
From the sound of breakers that seemed to roar from below him, he
conjectured that he was in a sea-facing tower of the castle, in which he
was right.
The men, who were led by Red Murrough, gave him bread and meat and
wine, but they offered no word and would answer no questions. So he ate
and drank, and felt life and strength creeping back into his bones. He
concluded that it must be the day after his arrival.
Now Red Murrough beckoned to the hoary old seneschal, whose red-rimmed
eyes glittered evilly. The old man shook his keys and stooped over
Brian, unlocking the hasp which bound him to the wall-ring. The
oppressive silence of these men struck a chill through Brian, but he
came to his feet readily enough as Murrough jerked his shoulder.
He followed out into a corridor, and the men closed around him, going
with him down-stairs and along other passageways. Brian wondered as to
his fate and what manner of death he was going to die; yet it seemed to
him that death was an impossible and far-off thing where he was
concerned.
He expected no less than death from the Dark Master, but at the same
time it was very hard to believe that he was going to that fate. He was
by no means afraid to die, but he felt that he would like to see the
Bird Daughter once more. Also, he had always thought of fate as coming
to him suddenly and swiftly in battle or foray; and to be deliberately
done to death in cold blood by hanging or otherwise was not as he would
have wished.
"At least," he thought without any great comfort, "Cathbarr and Turlough
will avenge me on the Dark Master--though I had liefer be living when
that was done!"
In one of the larger and lower corridors they came on two men bearing a
body, sewed for burial. Murrough stopped his party and growled out
something.
"It is the _seanachie_," answered one of the bearers. "Since the Dark
Master struck him yester-morn he has not spoken, and he died last
night."
Upon this Red Murrough crossed himself, as did the rest, muttered into
his tangle of red beard, and motioned Brian forward.
This wider passage gave through a doorway upon the great hall. There was
no dais, but the Dark Master was seated before the huge fireplace, his
wolf-hound crouched down at his side. The hall was pierced near the roof
with openings, and lower down with loopholes, so that when the sun shone
outside it was bright enough.
Red Murrough led Brian forward, the clank of the heavy chain-links
echoing hollowly through the place, but O'Donnell Dubh did not look up
until the two men stood a scant four paces from him. Then his head came
out from between his rounded shoulders and his eyes spat fire at Brian.
"A poor ending to proud talk, Brian Buidh!"
Brian tried to smile, but with ill success, for he was chilled to the
bone and there was blood on his face.
"I am not yet dead, O'Donnell."
"You will be soon enough," the Dark Master chuckled, and the hall
thrilled with evil laughter. In the eyes of all Brian had proven himself
the weaker man and therefore deserved his fate. "What of this O'Malley
journey of yours, eh?"
Brian made no answer, save that his strong lips clamped shut, and his
blue eyes narrowed a little. O'Donnell laughed and began to stroke his
wolf-hound.
"I have many messengers and many servants, Yellow Brian, and there is
little my enemies do which is not told me. Even now men are riding hard
and fast to trap Cathbarr of the Ax and your following."
At that Brian laughed, remembering Turlough Wolf and his cunning.
"I think this trapping will prove a hard matter, Dark Master."
"That is as it may be. Now, Brian Buidh, death is hard upon you, and
neither an easy nor a swift one. Before you die there are two things
which I would know from your lips."
Brian looked at him, but without speaking. The Dark Master had thrust
out his head, his hand still lingering on the wolfhound's neck, and his
pallid face, drooping mustache, and high brow were very evil to gaze
upon. Brian, eying that thin-nostriled, cruel nose, and the undershot
jaw of the man, read no mercy there.
"First, who _are_ you, Brian Buidh? Are you an O'Neill, as that ring of
yours would testify, or are you an O'Malley come down from the western
isles?"
At that Brian laughed out harshly. "Ask those servants of which you
boast, Dark Master. Poor they must be if they cannot tell you even the
names of your enemies!"
"Well answered!" grinned the other, and chuckled again to himself as
though the reply had indeed pleased him hugely. "I would that you served
me, Brian of the hard eyes; I suppose that you are some left-hand scion
of the Tyr-owens by some woman overseas, and the O'Neill bastards were
ever as strong in arm as the true sons. Yet you might have made pact
with me, whereas now your head shall sit on my gates, after your bones
are broken and you have been nailed to a door."
"Fools talk over-much of killing, but wise men smite first and talk
after," Brian said contemptuously. He saw that the Dark Master was
somewhat in doubt over slaying him, since if he were indeed an O'Neill
there might be bitter vengeance looked for, or if he belonged to any
other of the great families.
"Quite true," countered the Dark Master mockingly, and with much relish.
"Therein you were a fool, not to slay when first we met, instead of
making pacts. Who will repay me my two-score men, Brian of the hollow
cheeks?"
"The Bird Daughter, perhaps," smiled Brian, "since two days ago she hung
ten of those men I took in my ambuscade."
This stung O'Donnell, and his men with him. One low, deep growl swirled
down the hall, and the Dark Master snarled as his lips bared back from
his teeth. Brian laughed out again, standing very tall and straight, and
his chains clanked a little and stilled the murmur. He saw that
O'Donnell wore his own Spanish blade, and the sight angered him.
"There is another thing I would know," said the Dark Master slowly.
"Tell me this thing, Brian Buidh, and I will turn you out of my gates a
free man."
Brian looked keenly at him and saw that the promise was given in
earnest. He wondered what the thing might be, and was not long in
learning.
"You came hither from Gorumna Castle," went on O'Donnell, fixing him
with his black flaming eyes. "Tell me what force of men is in that
place, Brian of the hard eyes, and for this service you shall be set
free."
"Now I know that you are a fool, O'Donnell Dubh," and Brian's voice rang
out merrily. "I have heard many tales of your wizardry and your servants
and your watchers, but when an unknown man comes to you, his name is
hidden from you; and all your black art cannot so much as tell you the
number of your enemies! Now slay me and have done, for you have wasted
much breath this day, and so have I, and it goes ill in my mind to waste
speech on fools."
"You refuse then?" O'Donnell peered up at him, but Brian set his face
hard and made no reply. With a little sigh the Dark Master leaned back
in his chair and motioned to Red Murrough to come forward.
"Strip him," he said evenly, and at the word a great howl rang out from
all the watching men, like the howl of wolves when they scent blood in
the air.
Murrough in turn signed to two of his men. These came forward and
stripped off what clothes had been left to Brian, so that he stood naked
before them. In that moment he was minded to spring on the Dark Master
and crush him with his chains, but he saw that Red Murrough held a
flint-lock pistolet cocked, and knew it would be useless. Also, if he
had to die, he was minded to do it like a man and not to shame the blood
of Tyr-owen, either by seeking death or by shrinking at its face.
Now there passed a murmur through the hall, and even the Dark Master's
evil features glowed a little; for Brian's body was very fair and slim
and white, yet these judges of men saw that he was like a thing of
steel, and that beneath the satin skin his body was all rippling sinew.
Red Murrough drew out a hasp, brought his chained hands together, and
caught the chain close to his wrists, so that his hands were bound
close.
"Now," said the Dark Master, settling back and stroking his wolfhound as
if he were watching some curious spectacle, "do with him as we did with
Con O'More last Candlemas. But let us work slowly, for there is no
haste, and we must break his will. In the end we will nail him to the
door, and finish by breaking all his bones. It will be very interesting,
eh?"
A fierce howl and clash of steel answered him from the men. At another
sign from Red Murrough, Brian felt himself jerked to the floor suddenly,
and his hands were drawn up over his head. His wrist-chains were
fastened to an iron ring set in the floor, and his ankles to another,
and he stared up at the ceiling-rafters of the hall, watching the motes
drift past overhead in the reaching sunbeams. It all seemed very unreal
to him.
"First that long hair of his," said the Dark Master quietly.
Murrough went to the fire and returned with a blazing stick. Brian's
gold-red hair had flung back from his head, along the floor, and
presently he felt it burning, until his head was scorched and his brain
began to roast and there was the smell of burnt hair rising from him.
Then Murrough's rough hand brushed over his torn scalp, quelling the
fire, but it did not quell the agony that wrenched Brian.
"Paint him," ordered O'Donnell.
Again Murrough went to the fireplace, and returned with a long white-hot
iron which had lain among the embers. This he touched to Brian's right
shoulder, so that the stench of scorched flesh sizzled up in a thin
stream, and followed the iron down across the white breast and thigh,
until it stopped at the knee, and there was a swath of red and blackened
flesh down Brian's body. Yet he had not moved or flinched.
Then Murrough touched the iron to his left shoulder and drew it very
slowly down his left side. One of the watching men went sick with the
smell and went out vomiting. A second swath of red and black rose on the
white flesh, and beneath it all Brian felt his senses swirling. Try as
he would he could not repress one long shudder, at which a wild yell of
delight shrilled up--and then he fainted.
"Take him away," said the Dark Master, smiling a little, as he leaned
forward and saw that Brian had indeed swooned with the pain. "To-morrow
we will paint his back with the whip."
So they loosened him from the iron rings, and four men lifted him and
carried him out. As they passed across the courtyard another came by
with a pail of sea-water, which they flung over him; the salt entered
into his wounds, washing away the blackness from his scalp, and slowly
the life came back to him after he had been chained again in his
tower-room and left alone.
He was sorry for this, because he thought that he had died under the
iron. He found a pitcher of water beside him, and after drinking a
little he spent the rest in washing out the salt from his flesh, though
every motion was terrible in its torture. So great was the pain that
gasping sobs shook him, though he stared up dry-eyed at the stones, and
a great desire for death came upon him.
"Slay me, oh God!" he groaned, shuddering again in his anguish. "Slay
me, for I am helpless and cannot slay myself!"
As if in answer, there came a soft laugh from somewhere overhead, and
the voice of the Dark Master.
"There is no God in Bertragh Castle save O'Donnell, Brian Buidh!"
The blasphemy shocked him into his senses, which had wandered. Now he
knew that from some hidden place the Dark Master was watching him and
listening for his ravings, and upon that Brian sternly caught his lips
together and said no more, though he prayed hard within himself. A cloak
had been laid near-by him, and when he had covered himself somewhat
against the cold, though with great pain in the doing, he lay quiet.
The cold crept into him and for a space he was seized with chills that
sent new thrills of pain through his burned body, for he could not
repress them. After a time he relapsed slowly into numbed
unconsciousness, waking from time to time, and so the hours dragged away
until the night came.
Then men brought him more food and wine and straw, and he managed to
sleep a bit during the darkness, in utmost misery. But after the day had
come, and more wine had stirred his blood redly, Murrough fetched him to
his feet and bade him follow. Brian did it, though walking was agony,
for his pride was stronger even than his torture.
He was halted in the courtyard, found the Dark Master and his men
gathered there, and knew that more torture was to come upon him. After a
single scornful glance the Dark Master ordered him triced up to a post,
which was done. Brian saw a man standing by with a long whip, but gained
a brief respite as the drawbridge was lowered to admit a messenger
mounted on a shaggy hill-pony. O'Donnell bade him make haste with his
errand.
"The word has come, master, that five hundred of Lord Burke's pikemen
are on the road from Galway and will be close by within a day or so."
"And what of Cathbarr of the Ax?" queried the Dark Master. Brian's
heart caught at the words, then his head fell again at the response.
"They have scattered in the mountains, it is said, master."
"Murrough, have men sent to meet these royalists with food and wines,
and if they are bound hither we will entreat them softly and send them
home again empty. Now let us enjoy Brian Buidh a while--though he has
stood up but poorly. It is in my mind that we will nail him up
to-morrow."
With that Brian felt the whip stroking across his naked back. His
muscles corded and heaved up in horrible contraction, but no sound broke
from him; again and again the hide whip licked about him until he felt
the warm blood running down his legs, and then with merciful suddenness
all things went black, and he hung limp against the post.
"Take him back," ordered the Dark Master in disgust. "Why, that boy we
cut up the other side of Clifden had more strength than this fool!"
"His strength went out of him with his hair," grinned Red Murrough, and
they carried Brian to his prison.
The Dark Master had spoken truly, however. Brian's strength lay not so
much in brute muscles, though he had enough of them, as in his nervous
energy; and the slow horror of his burning hair and of that iron which
had twice raked the length of his body had come close to destroying his
whole nervous system. Other men might have endured the same thing and
laughed the next day, but Brian was high-strung and tense, and while his
will was still strong, his physical endurance was shattered.
With the next morning, this fact had become quite evident to the general
disgust of all within Bertragh Castle. The Dark Master himself visited
the cell, and upon finding that Brian was lost in a half stupor and
muttering words in Spanish which no one understood, he angrily ordered
that he be revived and finished with that afternoon.
Red Murrough set about the task with savage determination. By dint of
sea water externally and mingled wine and uisquebagh internally he had
Brian wakened to a semblance of himself before midday. Then food, oil,
and bandages about his wounds, and in another hour Brian was feeling
like a new man.
He was under no misapprehension as to the cause of this kindness, but
cared little. So keenly had he suffered that he was glad to reach the
end, and he walked out behind Red Murrough that afternoon with a ghastly
face, but with firm mouth and firmer stride, though he was very weak and
half-drunk with the liquors he had swallowed.
His fetters were unlocked and he was led to the doorway of the great
hall, with the Dark Master and his men watching eagerly. Red Murrough,
with an evil grin, pressed his back to the door and held up his left arm
against the heavy wood. Brian was half-conscious of another man who bore
a heavy mallet and spikes, and whose breath came foul on his face as he
pressed something cold against the extended left hand.
Then Brian saw the mallet swing back, heard a sickening crunch, and with
a terrible pain shooting to his soul, fell asleep.
CHAPTER X.
IN BERTRAGH CASTLE.
Now, of what befell after that nail had been driven through his hand,
Brian learned afterward; though at the time he was unconscious and
seemed like to remain so. Hardly had he sagged forward limply when two
men came riding up to the gates demanding instant admittance. One of
these was of the Dark Master's band, the other was a certain Colonel
James Vere, of the garrison which held Galway for the king.
O'Donnell, who suddenly found himself with greater things on hand than
the nailing of a prisoner, ordered Brian left where he lay for the
present, and had the drawbridge lowered in all haste. Colonel Vere, who
had late been in rebellion against his gracious majesty, was now joined
with Ormond's men against the common enemy, and was in command of that
force of five hundred pikemen which had been marching to the west.
Knowing this, the Dark Master made ready to set his house in order,
since it was known that Vere's men were only a few hours away. Hardly
had the garrison gone to their posts, leaving Brian in the center of a
little group about the hall doorway, when Colonel Vere rode in and was
received in as stately fashion as possible by the Dark Master. It was
not for nothing that O'Donnell had trimmed his sails to the blast, since
he was on very good terms with all in Galway.
"Welcome," he exclaimed with a low bow as Vere swung down from his
saddle. "Your men received the provision I sent off yesterday?"
"Aye, and thankful we were!" cried the other cheerily, for he was a
red-faced man of forty, a Munsterman and half-English, and loved his
bottle. "Hearing certain news from one of your men I made bold to ride
ahead in all haste, O'Donnell."
"News?" repeated the Dark Master softly. "And of what nature, Colonel
Vere?"
"Why, of one Brian Buidh, or Yellow Brian." At this the Dark Master
began to finger the Spanish blade he had taken from Brian, and for a
second Vere was very near to death, had he known it.
"What of him, Colonel Vere?"
"Why, the rogue had the impudence to come down on a convoy of powder and
stores, last week, going from the Archbishop at Ennis to Malbay, for our
use. Not only this, but a hundred of our rascally Scots deserted to him,
he slipped past us at Galway, and I was in hopes you could give me word
of him when I hit over this way. You're something of a ravager yourself,
sink me if you aren't!" and he dug the Dark Master jovially in the
ribs.
"Yes," murmured O'Donnell thoughtfully, "so they say, Colonel Vere. But
only when Parliament men come past, you understand. So you heard that
this Yellow Brian was here?"
"Aye, and that you were doing him to death," coolly responded Vere, and
his eyes flickered to the white form on the stones. "Zounds! What's
this?"
"Yellow Brian," responded the Dark Master dryly. "What do you want with
him?"
"Eh? Why, I'll take him back to Galway and hang him! I've a dozen of the
Scots he was fool enough to let loose, and when my men come up they'll
identify him readily enough."
"Unless he's dead," chuckled O'Donnell. "Well, if you want him you may
have him and welcome. So now come in and sample some prime sack I took
from the O'Malleys last year."
"With all the honors," responded Vere gallantly, and as they strode past
Brian the Dark Master hastily directed that he be washed and tended and
brought back to his right mind as soon as might be.
This order, and the conversation preceding it, gave Red Murrough some
cause for thought. So it was that when Brian wakened once more in his
cell, as evening was falling, he found the fetters on him indeed, but
Red Murrough had bound up his wounds, dressed his sundered hand-bones,
and was sitting watching him reflectively. It had occurred to the Dark
Master's lieutenant that there might be something made out of this man,
who seemed wanted in several places at once.
Therefore it was that while Brian made an excellent meal for a man
swathed from crown to knees in bandages, Red Murrough poured into his
ear the tale of what had chanced in the courtyard, and why it was that
he was not at this moment nailed to the castle door. Brian collected his
energy with some effort.
"Well, what of it?" he asked weakly.
"Just this, Yellow Brian," and Murrough stroked his matted red beard
easily. "O'Donnell will make a good thing out of handing you over to the
royalists, who mean to hang you in style, it seems. Now, it is in my
mind that it might advantage you somewhat if you were not moved thence
for a few days--indeed, you might even escape, for I think you are not
without friends."
"Eh?" Brian stared up at him wonderingly. "What does it matter to you?"
"Nothing, whether you live or die. But you are in my care, and if I
report that you are in too bad shape to be moved--which you are
not--then this Colonel Vere will camp outside our castle until you are
handed over to him. You will gain a few days in which to get your wits
back, and the rest is in your hands."
"I had not thought you loved me so much," and despite his agony Brian
forced out a bitter laugh.
"Not I! Faith, I had liefer see you nailed--but a service may be paid
for."
"I have no money," Brian closed his eyes wearily.
"No, but you have friends," and Murrough leaned forward. "Promise me a
clerkly writing to the Bird Daughter's men, or to your own men, ordering
that I be paid ten English pounds, and it is done."
"With pleasure," smiled Brian wryly. "Also, if I escape, I will spare
your life one day, Red Murrough."
"Good. Then play your part." And Murrough departed well pleased with his
acumen.
And indeed, the man carried out his bargain more than faithfully. One
visit assured the Dark Master that this broken, burned, cloth-swathed
man was helpless to harm him further, and after that he gave Brian
little thought.
As Murrough had reckoned Brian's swoop on the convoy had given him some
notoriety, and more than once Brian himself remembered Cathbarr's dark
presage after he had let the ten Scots go free to Ennis; Colonel Vere
was anxious to carry him back to Galway for an example to other
freebooters, and he was quite content to bide at Bertragh Castle until
his prisoner could travel.
For that matter the other officers of his command were quite as content
as he himself, since all were men from the south-country who loved good
wines, and the Dark Master had better store of these than the empty
royalist commissariat.
As for the Dark Master, Murrough reported to Brian that he also was well
content. Cromwell was sweeping like an avenging flame from Kilkenny to
Mallow and Ormond was helpless before him; both king's men and Irish
Confederacy men were pouring out of the South in despair, but the two
had finally joined forces and the final stand would take place in the
West. In fact, it seemed that things were dark for Parliament, despite
Cromwell's activity, and the Dark Master was only one of many such who
counted strongly on the rumors that the new king, Charles II, was on his
way to Ireland with aid from France.
And indeed he was at that time; but Charles, then and later, was more
apt at starting a thing than at finishing it.
Red Murrough lost no time in getting his "clerkly writing," luckily for
himself. On the morning after his agreement he brought Brian a quill,
and blood for lack of ink, and sheepskin. Brian wrote the order for ten
pounds, promising to honor it himself if he escaped.
This, however, did not seem likely, and even Murrough frankly stated
that it was impossible. But Brian was tended well, and his perfect
health was a strong asset. His head had been little more than scorched,
and the scalp-wound stayed clean; after the first day there came a
festering in his broken hand, but Murrough washed it out with vinegar
which ate out the wound and cleansed it, after which he bound it firmly
in wooden splints and it promised well.
More than once Brian laughed grimly at the care he was getting, to the
simple end that he should hang over Galway gates as a warning to the
City of the Tribes and to all who entered the ancient Connacian town.
For in that day Galway was a second Venice, and its commerce made rich
plundering for the O'Malley's both of Gorumna and of Erris in the North,
though the war had somewhat dimmed the glory of the fourteen great
merchant families.
Brian wondered often what had become of Cathbarr and his two hundred
men, and Murrough could give him little satisfaction. It was known that
the force had slipped away from Cathbarr's tower and had vanished; Brian
guessed that Turlough had either led them north, or else into the
western mountains where the O'Flahertys held savage rule. However, it
was certain that neither the Dark Master nor the royalists had scattered
them as yet.
So Brian lay in his tower four days and might have lain there four-score
more by dint of Red Murrough's lies, had it not been that on the fourth
evening Colonel Vere managed to stay unexpectedly sober. Being thus
sober, it occurred to him that he had best make sure he had the right
man by the heels. So he ordered his ten Scots troopers in from the camp
outside the walls, and the Dark Master sent for Brian to be identified.
"I'll have you carried down," said Red Murrough on coming for him. "Play
the part, _ma boucal_, and when these royalists get into their cups
again they'll forget all that is in their heads. Here's a cup of wine
before ye go, and another for myself. _Slainte!_"
"_Slainte_," repeated Brian, and went forth to play his part.
When the four men, with Red Murrough at their head, carried him down
into the great hall, Brian found it no little changed. Tables were set
along the walls, each of them being some ten feet in length by two wide,
of massive oak, and in the center was another at which sat O'Donnell,
Colonel Vere, and one or two other officers. Besides these there were a
score more of the royalist officers mingled with the Dark Master's men,
and it seemed that there would be few sober men in that hall by
midnight, from the appearance of things. Only the ten Scots stood calm
and dour before the fireplace.
After that first quick glance around, Brian lay with his head back and
his eyes closed, careful not to excite O'Donnell's suspicion that he was
stronger than he seemed. He was set down in front of the ten Scots, and
there was an eager craning forward of men to look at him, for his name
was better known than himself.
"Zounds!" swore Vere thickly. "The man has a strong and clean-cut face,
O'Donnell! Strike me dead if he does not look like that painting of
O'Neill, the Tyrone Earl, that hangs in the castle at Dublin! Though for
that matter there is little enough of his face to be seen. You must have
borne hardly on him with your cursed tortures."
"I fancy he is an O'Neill bastard," returned the Dark Master lightly.
Brian felt the red creep into his face, but he knew that he was helpless
in his chains, and he lay quiet. "Is he your man, Vere?"
"How the devil should I know?" Vere turned to the troopers and spoke in
English. "Well, boys, is this the fellow we're after? Speak up now!"
"It's no' sae easy tae ken," returned one cautiously. "Yon man has the
look o' Brian Buidh, aye."
"Devil take you!" cried Vere irritably. "Do you mean to say yes or no?
Speak out, one of you!"
"Weel, Colonel," answered another cannily, "Jock here has the right of
it. I wouldna swear tae the pawky carl, but I'd ken the een o' him full
weel. An I had a peep in his een, sir. I'm thinkin' I'd ken their
de'il's look. Eh, lads?"
Since it seemed agreed that they would know Brian better by his hard
blue eyes than by what they could see of his face, the exasperated Vere
commanded that he be made open them if he were unconscious.
"Run your hand down his body, Murrough," ordered the Dark Master
cynically.
Red Murrough leaned over Brian, and the latter opened his eyes without
waiting for the rough command to be obeyed. Instantly the Scots broke
into a chorus of recognition as Brian's gaze fell on them. Vere looked
at him with an admiring laugh.
"Sink me, but the man has eyes! Well, so much the better for the ladies,
eh? Now that this is over, give the lad a rouse and send him back to his
cell."
He waved the Scots to begone, and rose cup in hand. Smiling evilly, the
Dark Master joined him in the toast to Brian, and a yell of delight
broke from the crowd as they caught the jest and joined in. O'Donnell
was just motioning Murrough to have Brian taken away, when there came a
sudden interruption, as a man hastened up the hall. It was one of Vere's
pikemen.
"There is a party of four horsemen just outside our camp, colonel. One
of them bade us get safe-conduct for him from O'Donnell Dubh, upon his
honor."
"Eh?" the Dark Master snarled suddenly. "What was his name, fool?"
"Cathbarr of the Ax, lord."
A thrill shot through Brian, and he tried feebly to sit up. The Dark
Master flashed him a glance. The hall had fallen silent.
"His business?"
"He bears word from one called the Bird Daughter, he said."
While the royalists stared, wondering what all this boded, O'Donnell bit
his lips in thought. Finally he nodded.
"Let the man enter, and tell him that he has my honor for his
safe-conduct."
Vere nodded, and the pikeman departed. Instantly the hall broke into
uproar, but leaving the table, the Dark Master crossed swiftly to
Brian, and bent over him.
"Either swear to keep silence, or I have you gagged."
"I promise," mumbled Brian as if he were very weak. The Dark Master
ordered him carried behind one of the tables close by, and a cloak flung
over him. When it had been done, Brian found that he could see without
being seen, which was the intent of O'Donnell.
Meanwhile the Dark Master was telling Vere and the other officers of
Cathbarr, it seemed, and Vere hastily collected his wine-stricken
senses.
"Nuala O'Malley, eh?" he exclaimed when the Dark Master had finished.
"She is the one who has held Gorumna Castle and would make no treaty
with us, though she has more than once sent us powder, I understand."
"I will talk with you later concerning her," returned O'Donnell. "She is
allied with Parliament, they say, and it might be well for all of us if
ships were sent against her place from Galway, and she were reduced."
Brian saw that things were going badly. The Dark Master seemed to be
playing his cards well, and was doubtless thinking of throwing off the
cloak and openly allying himself with the royalist cause. In this way he
could secure help against Gorumna in the shape of Galway ships and men,
and it was like to go hard with the Bird Daughter in such case.
However, Vere had no power to treat of such things, as Brian well knew.
Also, Nuala had told him herself that her ships had not preyed on the
commerce of Galway's merchants, but only on certain foreign caracks
which free-traded along the coast. Therefore the Galwegians were not apt
to make a troublesome enemy in haste, even if she were proved to be in
alliance with Cromwell.
None the less, the Dark Master was plainly thinking of making an effort
in this direction, and Brian knew that the Bird Daughter was in no shape
to carry things with a high hand in Galway town.
He saw Vere and the Dark Master talking earnestly together across the
table, but could not hear their words--and it was well, indeed, for him
that he could not. As he was to find shortly, O'Donnell's quick brain
had already grasped at what lay behind Cathbarr's coming, or something
of it, and he had formed the devilish scheme on the instant--that scheme
which was to result in many things then undreamed of.
"If I had followed Turlough's rede, there when I first met this devil,"
thought Brian bitterly, "I had slain him upon the road, and that would
have been an end of it. Well, I think that I shall heed Turlough Wolf
next time--if there is a next time."
Brian looked out from his shelter with troubled eyes, for there was
something in the wind of which he had no inkling. He saw Vere break into
a sudden coarse laugh, and a great light of evil triumph shot across
O'Donnell's face. Then the Dark Master gained his feet, gathered his
cloak about his hunched shoulders, and sent Murrough to stand guard over
Brian with a pistol and to shoot if he spoke out.
"Surely he cannot be going back on his word, passed before so many men?"
thought Brian bitterly. "No, that would shame him before all Galway, and
he is proud in his way. But what the devil can be forward?"
To that he obtained no answer. The Dark Master shoved his table back
toward the fireplace, and placed his chair in front of it beside that of
Colonel Vere. It seemed to Brian that the stage was being set for some
grim scene, and a great fear seized on him lest harm was in truth meant
toward Cathbarr.
No doubt the giant had been in communication with the Bird Daughter, and
it had been ascertained that the galley had come to grief at Bertragh
Castle. A sudden thrill of hope darted through Brian. Was it possible
that Cathbarr had led down his men and placed them in readiness to
attack? Yet such a thing would have been madness--to set a scant two
hundred against Vere's pikemen and the Dark Master's force combined!
But Brian knew that Turlough Wolf was at large, and Turlough's brain was
more cunning than most.
If he could only get free, he thought, he might still be able to do
something. He could ride, though it would mean bitter pain, and his
sword-arm was still good--but he had got no farther than this when there
came a tramping of feet, and in the doorway appeared Cathbarr, his
mighty ax in hand, with the O'Donnells around him as jackals surround a
lion.
CHAPTER XI.
THE BAITING OF CATHBARR.
The bearded giant still wore the long mail-shirt that reached to his
knees, and he paused at the doorway with his eyes roving about the hall.
Well did Brian know whom he sought, but it was vain, for Cathbarr could
not see him where he lay.
Then Brian saw that the ax had been changed, and wondered at it. One of
the long, back-curving blades had been rubbed down with files, so that
it was very tapering and thin like an ordinary ax-blade, while the other
was still the blunt, heavy thing it had always been. Brian read the
cunning of Turlough Wolf in that handiwork, and in fact the great ax was
thus rendered tenfold more deadly.
The Dark Master waited quietly until Cathbarr began a slow advance up
the hall, all eyes fixed on him in no little wonder. Then O'Donnell
raised a hand, stopping him.
"Let us have your message, Cathbarr."
The giant halted and dropped the ax-head, leaning on the haft of the
weapon. He took his time about replying, however, and his eyes still
roved about the hall ceaselessly and uneasily. Then of a sudden he gave
over the search, and gazed straight at the Dark Master with a swift
word:
"Have you slain him?"
"Slain who, Cathbarr?" queried O'Donnell, with a thin smile.
"_Duar na Criosd!_" bellowed Cathbarr with sudden fury. "Who but my
friend Brian?"
"Oh!" The Dark Master laughed and eased back in his chair. "No, he's
still alive, Cathbarr? Is your message from the Bird Daughter in his
regard?"
"Yes." Cathbarr fought for self-control, the breast of his mail shirt
rising and falling, his bloodshot eyes beginning to circle about the
place once more in a helpless and angry wonder.
"O'Donnell Dubh," he went on at last, "Nuala O'Malley sends you this
word. Give Brian Buidh over to her, and she will pay you what ransom you
demand."
"What alliance is there between Brian and her?" asked O'Donnell softly.
"Brian has given her service, and I have," Cathbarr flung up his head.
"Our men lie in Gorumna Castle, there are ships coming from Erris and
the isles, and if Brian be slain we shall bear on this hold and give no
quarter. We have four hundred men now, and five ships are coming from
the North."
The Dark Master gazed quietly at the giant, Vere taking no part in the
talk. But Brian, watching also, saw that which brought a mocking smile
to O'Donnell's pallid face. Cathbarr had no fear of any man, and lies
did not come easily to his lips; when he spoke of the force lying in
Gorumna, and of help from Erris, his face gave him away. Brian saw
Turlough behind that tale, but Cathbarr was no man to carry it off with
success.
"Well," laughed the Dark Master, "none the less shall Brian be slain.
Carry back that word to Nuala O'Malley."
Cathbarr's mighty chest heaved like a barrel near to bursting. Brian was
minded to break his promise, but Murrough's pistol was at his head, and
he could but lie quietly and watch. The giant's face flushed somewhat.
"I have not finished," said he. "My business for the Bird Daughter is
done in truth, but now I have to speak a word of my own."
"Let us hear it," returned O'Donnell.
"It is this." Cathbarr drew himself up. "I am more your enemy than is
Brian. Let him go, O'Donnell Dubh, and take me in his place, for I love
him."
A sudden amazed silence fell on every man there, and but for Murrough's
warning hand Brian would have sat up. O'Donnell's jaw fell for an
instant, then his head drew in between his shoulders, he put a hand to
Vere's arm, and whispered something. The royalist nodded, a grin on his
coarse face, and the Dark Master settled back easily. Cathbarr still
stood waiting, the ax held out before him, and a glory in his wide eyes.
"I would sooner hold you than Brian," and O'Donnell spoke softly. "If
you will to take his place and die in his stead, Cathbarr, then loose
that ax of yours."
Brian saw that Cathbarr was lost indeed, for the Dark Master was not
likely to give over his pact with the royalists so easily. Cathbarr
heaved up his ax with a great laugh, like a child; he brought it down on
the stones, but if he had meant to break it the effort was vain. The
huge weapon clanged down and bounded high out of his two hands, so that
men drew back in awe; but the ax whirled twice in the cresset-light,
then fell and slithered over the flagging beneath a table, and no man
touched it.
"Take me," said Cathbarr simply.
"Nay," answered the Dark Master calmly, though his eyes flamed, "kneel
down."
Cathbarr stood breathing heavily for an instant, then slowly obeyed.
Brian saw that his curly beard was beginning to stand out from his face,
but no word came from him as he went to his knees.
"Now," went on the Dark Master, "pray me for Brian's life, mighty one."
The giant struggled with himself, for humiliation came hard to him. Then
his voice fell curiously low, terrible in its self-restraint.
"I pray you for the life of Yellow Brian, O'Donnell."
Brian forced himself up, thinking to cry out a warning before it was too
late; but Murrough's hand closed over his mouth and forced him back
relentlessly.
"Bring ropes," said the Dark Master, and ordered Cathbarr to his feet.
Men hastened out, and returned with a length of rope, binding the
giant's arms behind his back, from elbow to wrist. Then the Dark Master
laughed harshly, but Vere leaned toward him, his face troubled.
"Do not carry this thing farther, O'Donnell," said the royalist
hoarsely. "This man is a fool, but he has a great heart. Let be."
For answer the Dark Master whirled on him with such fury in his snarl
that Vere drew back hastily, and no more words passed between them at
that time. O'Donnell rose and walked down the hall toward Cathbarr, in
his hand a little switch that he used upon that wolfhound of his.
"Now," he said softly, yet his voice pierced hard through the dead
stillness, "in token that your humility in this affair is without guile,
Cathbarr of the Ax, bow your head to me."
The giant obeyed, closing his eyes. The Dark Master lifted his hand and
cut him twice across the head with his switch, while Brian gasped in
amazement and looked for Cathbarr to strike out with his foot. But
although the giant shuddered, he made no move, and the Dark Master
strode back to his seat with a laugh. Then Cathbarr raised his face, and
Brian saw that it was terribly convulsed.
"Do with me as you wish," he said, still in that low voice. "But now let
Brian be freed in my presence."
The Dark Master flung back his head in a laugh, and when the men saw his
jest, a great howl of derision rang up to the rafters. Only Vere's
officers looked on with black faces, for it was plain that this affair
was none of their liking. A look of simple wonder came into Cathbarr's
wide-set eyes.
"Why do you not loose him?" he asked quietly.
"Fetch the man out, Murrough," ordered the Dark Master. "Shoot him if he
speaks."
Now, whether through some shred of mercy--for he knew well that Brian
would cry out--or for some other reason, Murrough leaned down swiftly to
Brian's ear.
"Careful," he whispered as he motioned his men forward. "Play the part,
and mind that this thing is not yet finished."
The warning came in good time, and cooled Brian's raging impulse. He was
lifted from behind the table, his chains clanking, and laid upon it;
Cathbarr gave a great start and bellowed out one furious word:
"Dead!"
"Nay," smiled the Dark Master. "His eyes are open, and he is but weak
with his wounds, Cathbarr. Now say--would you sooner that we cut off
that right hand of his, or blinded him? One of these things I shall do
before I loose him, for I said only that I would take your life for
his."
Brian saw that the Dark Master was only playing with the giant, for well
he knew that Vere wanted to take him back to Galway whole and sound. But
Cathbarr knew nothing of this, and as the whole terrible trickery
flashed over his simple mind he lifted a face that was dark with blood
and passion.
"Do not play with me!" he cried out, his voice deep and angry. "Loose
him!"
Then O'Donnell leaned back in his chair, laughing with his men, and
waved a careless hand toward Vere.
"He is not mine," he grinned. "I have given him to the royalists, for
hanging at Galway. You, however, are now mine to slay."
Whether the Dark Master indeed meant to break his plighted faith, Brian
never knew. Cathbarr took a single step forward, his curly beard
writhing and standing out, and his whole face so terrible to look on
that all laughter was stricken dead in the hall.
"You lied to me!" he cried hoarsely. "You lied to me!"
O'Donnell laughed.
"Aye, Cathbarr. Your master goes back to Galway to be hung--he is out of
my hands, but you are in them. However, since I have passed my word on
your safe-conduct, I think that I may hold to it."
But the giant had not heard him. Throwing back his head, he gave one
deep groan of anguish, and his shoulders began to move very slowly as
his chest heaved up. All the while his eyes were fixed on the Dark
Master, while the whole hall watched him in awe; not even Brian or
O'Donnell himself guessed what that slow movement of Cathbarr's body
boded.
"Best put chains upon him, Murrough," said the Dark Master, his teeth
shining under his drooping mustache.
Vere cried out in sudden wonder.
"'Fore Gad! Look!"
Then indeed the Dark Master looked, and sprang to his feet, and one
great shout of alarm and fear shrilled up from those watching. For as
Cathbarr stood there, the veins had suddenly come out on his face and
neck, and with a dull sound the ropes had broken on his arms, and he was
free.
Murrough rushed forward, and his pistol spat fire. Cathbarr, with his
eyes still on the Dark Master, put out a hand and Murrough went whirling
away with a dull groan. Then the giant rushed.
O'Donnell did not stay for that meeting, but slipped away like a shadow
into his surging men, yelling at them to fire. There were few muskets in
the hall, however, and an instant later Cathbarr had reached the table
where Vere still sat astounded. He brought down a fist on the royalist's
steel cap, and Vere coughed horribly and fell out of his chair with his
skull crushed.
Now a musket roared out, and another. But Cathbarr caught up the oaken
table and faced around on the men who were surging forward at him;
lifting the ten-foot table as though it were paper, he bellowed
something and rushed at them, casting the table in a great heave. It
fell squarely on the front rank, and then indeed fear came upon the
hall. For Cathbarr's foot had struck against his ax, and he rose with it
in his hand.
There was a din of screams and shouts, for half the men were struggling
to get out of the hall and the rest were rushing to get at Cathbarr.
Another musket crashed, and in the smoke Brian saw the giant stagger,
recover, and go bellowing into the crowd.
Brian struggled from the table, groaned with pain, and then stood
watching. He could walk, but his weakness and the chains on his wrists
and ankles hindered him from being of any advantage to Cathbarr, though
he lifted his voice in a shout of encouragement.
Cathbarr heard the shout, and roared out with delight. A musket-ball had
cut across his forehead, and with the blood dripping from his beard he
looked more like a demon than a man. The huge ax flashed in the smoky
light, and before it men groaned and shrieked and gave back; it cleaved
steel and flesh, or smashed helms and heads together, and the Dark
Master had slipped from the place, so that his men had no leader.
Over the roar of fear-mad men, over the storm of shrieks and shouts,
over the dust and smoke, rose the mighty bellow of Cathbarr and the
thudding blows of his ax. The royalist officers were fighting around the
doorway, while O'Donnell's men were trying to make head against the
giant, but he swept through them like a whirlwind, awing them more by
his ferocious aspect and his mad rage than by the half-seen effect of
his terrific strength.
Little by little they eddied out from the door. Men lay all about,
tables were overturned, and through the crowd swirled the terrible ax,
leaving a path of dead in its wake. Brian staggered to the motionless
form of Colonel Vere, and reaching down drew a pistol from the dead
man's belt. His strength was flooding back to him, and in spite of the
agony caused by every movement, he clanked slowly down toward the door.
At sight of his chained and bandage-swathed figure a wild shriek welled
up, and when he laughed and fired into the midst of them all opposition
ceased.
Cathbarr still sought the Dark Master, raging back and forth, smiting
and smiting with never a pause in the flaillike sweep of his long arms.
He saw Brian standing there, and emitted a wild bellow of joy, but never
ceased from his smiting. Out through the door poured a stream of
maddened figures, for blind panic had come on every man there, and
Cathbarr's was not the only weapon that drew blood as the men fought for
exit.
Brian laughed again, for now he knew that he would die in no long time,
but it would not be under the torturers. Cathbarr cleared the hall, sent
the last man flying out with an arm lopped from him, and swung to the
huge doors after kicking two or three bodies from his way. When the beam
had dropped into place and they were alone with the dead and dying, he
turned to Brian and flung out his arms.
"Careful!" exclaimed Brian, seizing his hand. "None of your bear-hugs,
old friend," and he swiftly told of his tortures. Tears ran down the
giant's blood-strewn face as he listened, and with the tenderness of a
woman he picked up Brian and carried him back to a table, setting him on
it.
"First for these chains, brother," he cried, going back for his ax. "We
may yet win out against these devils."
"Small chance," smiled Brian grimly. "I cannot swing a blade, and we
cannot hold this hall for long. Besides, you have some wounds."
Cathbarr roared out a laugh, exuberantly as a boy, and carefully spread
Brian's legs open on the table.
"Hold quiet!" he cautioned, and swung up the ax. Down it flashed, the
thinner blade sheared through the chain an inch from Brian's ankle and
split the oak beneath, and Cathbarr drew back for a second blow.
Four times he struck, and the blows smote off the chains from each wrist
and ankle, although the locked rings still remained. But Brian was free,
and when he gained his feet he found the exercise had somewhat loosened
his muscles, and he picked up a sword.
"We can at least die fighting, Cathbarr," he said, and looked into the
giant's eyes. "And, brother, I thank you."
"Nonsense!" blurted out Cathbarr, wiping the blood from his eyes and
grinning through his beard. "Turlough Wolf has our men hidden around
this royalist camp, and the Bird Daughter has a boat outside the castle.
We cannot get through the royalists, but there is a chance that we can
get to the shore. Besides, she has ships and men coming from her kinsmen
in the North. Now, how shall we get away?"
Brian shook his head. "I can hardly walk, Cathbarr, to say nothing of
swimming or fighting. There is a rear door out of the hall, yonder, but
no use trying it."
"Perchance I have still some strength," grinned Cathbarr, picking up his
ax. "Let us have a look at that rear door, before they come at us with
muskets."
CHAPTER XII.
HOW THE DARK MASTER WAS RUINED.
The fear that had come upon the O'Donnells was so great that not until
pikemen entered the castle from the camp could the Dark Master get men
at the doors of the hall. And this proved the salvation of Brian and
Cathbarr, for when they left the hall by the rear door and slipped
through the corridors, they came out upon the rear or seaward
battlements of the castle.
These they found denuded of men, while from the courtyard and front of
the keep were rising shouts and batterings, whereat Cathbarr chuckled.
"They are all drawn around to the front, brother. Now, how to get down
from here?"
Brian looked around in the starlight, but saw that there was no gate or
other opening in the walls. He began to lose hope again; once the Dark
Master had burst into the great hall he would scatter men over the whole
castle, and their shrift would be short. At this point the walls were
some thirty feet high, and pointing out to the sea stood four of the
bastards, with balls piled beside them.
"Now if we had a rope," he said, "the matter would not be hard. Is that
boat near the shore?"
"Not so far that I cannot make them hear," grinned Cathbarr, opening his
mouth to shout, but Brian stopped him.
"Be careful--do you want to draw down the O'Donnells likewise? Now, cut
the ropes from these cannon, and if we have time we shall yet get down
safe."
Cathbarr rushed off in delight, and began hewing at the recoil-ropes
which bound the bastards and their carriages to their places. Brian
followed him, seizing the ropes and trying to knot the strands hastily
and with no little pain to himself; but now the hope of escape began to
thrill through him, and for the first time since sighting the Dark
Master's stronghold he began to think that he might yet get away.
However, he could do little knotting with one hand, and not until
Cathbarr impatiently took over the task was it finished. At the same
instant a great burst of yells rose over the castle.
"Hasten!" cried Brian, as the other began fastening the line to a
cannon. "I can use one hand--"
"Save your strength," grunted Cathbarr, lifting him after swinging the
loop of his ax around his neck. "Catch me about the neck with your good
arm, and trust me for the rest, brother."
Brian did as he was ordered, since there was no time for lowering him
down. The giant scrambled over the edge, gripping the twisting rope, and
Brian tightened his lips to keep down his groans, for the agony was
cruel to him. He was forced against the body of Cathbarr, and swirl
after swirl of pain went over him at each touch on his burns.
The giant grunted once or twice, for he had many slight wounds also, but
with the rope gripped in hands and feet, he lowered away steadily. At
length they reached the ground, and the scattered rocks along the shore
were but a few yards away.
Cathbarr sent his bull-like voice roaring out at the stars, while Brian
clung weakly to him and searched the waters. He could see nothing, but
suddenly there drifted in a faint shout, and Cathbarr bellowed once
more.
"Swim for it," said Brian, as torches began to move along the walls
above. "If those cannon are not loaded, we're safe."
Cathbarr nodded, and caught up the body of Brian tenderly enough in one
arm, as he splashed out. The icy water shocked Brian's brain awake and
drove the pain out of him momentarily, and before Cathbarr was
waist-deep he heard a hail and saw the dark shape of a galley
approaching.
Muskets flashed out from the walls, and their bullets whistled overhead,
but five minutes later Brian was on the galley, Cathbarr was clambering
over the side, and the light boat was being rowed out again.
Brian thought his senses were slipping away when he found Nuala O'Malley
herself holding his head as he lay in the stern, while men flung cloaks
around him; but warm tears dripped on his face, and she patted his arm
soothingly.
"Lie quiet," she said, but Brian would not, for already his brain was
leaping ahead, and he knew that there was work to be done.
"Tell me," he asked eagerly, "are my men camped around the royalists?
Is help indeed coming to you from the North?"
"Yes," she replied, trying to quiet him. "A pigeon came in from Erris
to-day, with word that two ships with men were on the way to help me.
When I returned from the South and found that the plague had been at
Gorumna, I sent off asking for help, and now it is coming."
"Then send word to Turlough!" cried Brian eagerly. "Tell him to throw my
men on the royalist camp _to-night_ and drive the pikemen into the
castle! Colonel Vere is dead, and there is such confusion that all will
think we have more than two hundred men. If we can leaguer them there
until your ships come, we may win all at a blow!"
Nuala found instantly that there was meat in the plan, and as they were
rowing out to meet one of her caracks, promised to send in the galley
with word to Turlough when they got aboard the larger ship.
This they were no great while in doing. Brian knew nothing of it, for
upon the Bird Daughter's word he had dropped away into a faint once
more. With this Nuala O'Malley was quite content, so that when Brian
wakened he was greatly refreshed and found himself lying bandaged on a
bunk with the sunlight coming through a stern-port beside him, and the
Bird Daughter watching him with food and drink ready.
"Take of this first," she smiled; "then we will talk."
Brian obeyed, being very thirsty and ravenously hungered. He had little
pain except when he tried to move, and so he ate as he lay, propped up
with folded garments, and watched the Bird Daughter. She refused to
speak until he had eaten the meat and cakes she had fetched, but when he
smiled and asked for a razor her grave face rippled with frank laughter,
and her deep violet eyes danced as they looked into his.
"I am sorry I have none," she said mockingly. "So you must wait till we
come to port again. Just at present we are off Slyne Head and bearing
northward."
"What!" Brian stared at her. "Are you in jest?"
It appeared that she was not, for she was sailing north to meet those
ships of her kinsmen, and to hasten them back with her. Meantime
Cathbarr had been sent ashore to meet Turlough and hold the Dark Master
and his royalists in check. Nuala had sent fifty of her men to join
Turlough, left twenty to hold her castle, and had ten with her upon the
carack. It seemed likely that Turlough and Cathbarr could hold the Dark
Master penned up for a few days at least, even with fewer men; if they
could not, said Nuala shortly, they had best sit at spinning-wheels for
the rest of their lives.
"You are a wonderful girl!" said Brian, and fell asleep again.
He remembered little of that voyage, for they met two caracks crowded
with men off Innishark that afternoon, found they were the expected
O'Malleys from the North, and turned back with them at once. Brian
wakened again that same evening, but Nuala refused to let him go on deck
until the following morning, when they sighted Bertraghboy Bay. Then
Brian discarded most of his bandages, dressed, and, with his left arm in
a sling, joined the Bird Daughter on the quarterdeck. He found that his
burns were well on toward healing, for he could walk slowly without
great pain, and had every confidence that he could sit a horse if need
be.
Sailing past Bertragh Castle, the three ships went on up the bay and
cast anchor. It was not hard to see that Turlough and Cathbarr had done
their work well, for in passing the castle they had made out that the
royalist pikemen had been driven inside, and there was some musketry to
be heard at times. No sooner had the anchor-cables roared out, indeed,
than a band of men came riding toward the shore, and Nuala sent off a
boat for them. She had known nothing of Cathbarr's deeds at the castle
until Brian had told her of them, and on seeing that the giant was among
those coming off, she smiled at Brian.
"Now you shall see how a girl can conquer a giant, Yellow Brian!"
Brian laughed and waved a hand to Turlough, who was beside Cathbarr in
the boat. As the men came over the rail, Nuala quietly pushed him aside
and faced the giant, sharply bidding him kneel. Cathbarr had been all
for rushing forward to Brian, and obeyed with an ill grace, when Nuala
quickly leaned forward and kissed him on the brow.
"That is for bravery and faith," she said. "Truly, I would that you
served me!"
Poor Cathbarr grew redder than the Bird Daughter's cloak. He started to
his feet, gazed around sheepishly, found all men laughing at him--and
did the best thing he could have done, which was to go to his knees
again and put Nuala's hand to his lips.
"While my master serves you, I serve you," he blurted out, and this
answer must have pleased Nuala mightily, for she flushed, laughed, and
bade all down into the cabin.
Brian greeted Turlough with no little joy, but beyond assurances that
all went well, gained no knowledge of what had happened. Nuala had sent
for the O'Malley chieftains, and proposed to hold a conference at once.
The O'Malleys arrived from the other ships in a scant five
minutes--dark, silent men who spoke little, but spoke to the point. Art
Bocagh, or the Lame, had had one leg hamstrung in his youth, but Brian
took him for a dangerous man in battle; while his cousin Shaun the
Little was a very short man with tremendous shoulders.
Nuala took her seat at the head of the stern-cabin table, and the
position of affairs was gone over carefully.
It seemed that no sooner had Turlough learned from Cathbarr of what had
taken place in the castle, and that Brian was safe on shipboard, than
he drove his men down pell-mell on the camp, just before dawn. Any other
man would have been exhausted by the events of that night, but Cathbarr
had led them in the assault. The result had been that, with hardly any
resistance, they had slain some four-score of the pikemen, and would
have captured or slain them all had it not been for the Dark Master's
cannon which drove them back.
The better part of the royalist officers had fallen, either then or
under the ax of Cathbarr in the hall of the castle. In fact, after
learning that he had slain some nineteen persons on that occasion,
Cathbarr had taken no few airs upon himself. Vanity was to him as
natural as to a child, and Brian hugely enjoyed watching the giant
strut. However, what remained of Vere's five hundred pikemen were in the
castle, joined to the Dark Master's men; and Turlough's advice was that
since there must be some seven hundred mouths to feed, the safest plan
was to bide close and force the fight to come to them, rather than to
take it to O'Donnell.
"There is reason against that, Turlough Wolf," said Brian quickly. "The
Dark Master has men on the hills, and if news is borne to Galway of what
has happened, we are like to have a larger army on our heels than we can
cope with."
"I have attended to O'Donnell's watchers," said Turlough grimly. "When
Cathbarr bore word of the pact from Gorumna Castle, I sent out horsemen
and we swept the hills bare of men. O'Donnell has no more than are in
the castle, and a score of our own men are on the roads, watching for
any ill."
"How many men have we in all?" spoke up Lame Art O'Malley. "In our ships
there are sixty men we can spare for land battle."
"That gives us three hundred in all," replied Turlough to Nuala's
questioning glance. "If we take a strong position we should sweep most
of O'Donnell's men away at the first charge."
"There you are wrong," said Brian, shaking his head. "Those pikemen are
bad foes for cavalry, and our two hundred horsemen would shatter on them
if they stood firm."
"Not if we choose our ground," said the Bird Daughter, her eyes
flashing. "Nay, _I_ am master here, my friends! Now this is my rede. We
shall not waste men by attacking the castle, unless forced to it by an
army from Galway. Instead, we will wait until the Dark Master is driven
out by hunger; then we will fall on him and destroy him utterly.
"Yellow Brian, you have some knowledge of war, and you shall take this
matter in charge. Cathbarr, do you command fifty horse, with the men
from our ships here, and keep the Dark Master in play. With the
remainder, we shall wait in whatever spot Brian shall choose, and before
many days are sped I think that Bertragh will be mine again."
The Bird Daughter had her way, since none could find much against her
plan; and that afternoon Brian went ashore with her and the O'Malleys,
leaving the three ships at anchor under a small guard. Turlough had made
camp a short mile from the castle, on a little hill among the farms;
both Nuala and the O'Malley men were somewhat surprised at finding the
O'Donnell women and children safe and untouched in their own steads.
"I saw to that," laughed Turlough, slanting his crafty eyes at Brian. "I
had but to threaten them in Brian's name, and the men only were slain."
"I think that you are a hard master," laughed Nuala, but Brian smiled
and pointed to his men, who were pouring out to meet him with shouts of
joy.
"All men do not rule by fear alone, Bird Daughter," he said quietly. She
gave him a quick glance. "I found these men riffraff of the wars, and
while they have no such love for me as Cathbarr here, I think they had
liefer follow me than any other leader."
After that Nuala said little concerning Brian's discipline.
That night Nuala and Brian took up headquarters at one of the larger
farms, and while Cathbarr went before the castle to keep the Dark Master
in check and allow none to leave the place, they called in a number of
those men O'Donnell had loaned to Brian, and questioned them about the
provisioning of the castle.
From these they found that there was good store of all things for the
usual garrison, but with seven hundred men to feed the Dark Master would
be forced out speedily. So with the dawn Brian and Turlough rode forth
to select a battleground, and while Brian was very sore and riding
caused him great pain at first, he soon found himself in better shape.
Turlough picked a hollow in the road a mile farther from the castle,
flanked on either hand by woods and hillsides where men might lie
hidden. Brian found it good, and that afternoon a part of their horsemen
were shifted thither in readiness.
* * * * *
For the next three days there was little done. Twice the Dark Master
attempted sallies with what few horsemen he had left, but on each
occasion Cathbarr's horse smote his men and drove them back. To be sure,
O'Donnell thundered with his bastards, but the guns only burned up good
powder, for Brian would allow no assault made.
By Turlough's advice, however, they brought about the Dark Master's fall
through certain prisoners made in the two sallies.
These captives were led through the depleted central camp, though they
knew nothing of that picked place farther back. Having been allowed to
see what men Brian had here, Turlough slyly drove Cathbarr into parading
his vanity before them; and in all innocence the giant told how he could
put the Dark Master's men to flight single-handed, and of his anxiety
lest the O'Donnells should fear to fight in the open. What was more,
Brian affected to be utterly shattered by his wounds, and with that the
prisoners were sent back with a message offering quarter to all within
the castle save the Dark Master himself.
Early the next morning a horseman came riding fast from Cathbarr with
word that the garrison was stirring. Without delay, Brian donned a
mail-shirt, bound his useless left arm to his side, and mounted. The
Bird Daughter insisted on accompanying him, and stilled his dismayed
protests by asserting her feudal superiority; in the end she had her
way.
Leaving her kinsmen and a hundred more men to dispute O'Donnell's
passage and give back slowly before him with Cathbarr, she and Brian
rode to their men among the trees on the hillsides over the hollow in
the road. Here they had a hundred and fifty men, composed of the Scots
troopers and the pick of the others, and Nuala took one side of the road
while Brian took the other. Then, being well hidden, they waited.
Brian was savagely determined to slay the Dark Master that day, and came
near to doing it. Presently a man galloped up to say that O'Donnell and
six hundred men were on the road, having left the rest to hold the
castle. A little later Cathbarr's retreating force came in sight, and
after them marched O'Donnell. He had deployed his muskets in front and
rear, and rode in the midst of his pikemen, whose banner of England blew
out bravely in the morning wind.
At the edge of the dip in the road Cathbarr led his men in full flight
down the hollow and up the farther rise, where he halted as if to
dispute the Dark Master further. There were barely a dozen mounted men
with O'Donnell, and he made no pursuit, but marched steadily along with
his muskets pecking at Cathbarr's men. When he had come between the
wooded hillsides, however, Cathbarr came charging down the road; the
pikemen settled their pikes three deep to receive him, and with that
Brian led out his men among the trees and swooped down with an ax
swinging in his right hand.
Alive to his danger, the Dark Master tried to receive his charge, but
at that instant Nuala's men burst down on the other flank. Brian headed
his men, and at sight of him a yell of dismay went up from the
O'Donnells. A moment later the pikemen's array was broken and the fight
disintegrated into a wild affray wherein the horsemen had much the
better of it.
Brian tried to cut his way to the Dark Master, but when O'Donnell saw
the pikemen shattered he knew that the day was lost. He gathered his
dozen horsemen and went at Cathbarr viciously; Brian saw the two meet,
saw O'Donnell's blade slip under the ax and Cathbarr go from the saddle,
then the Dark Master had broken through the ring and was riding hard for
the North.
Brian wheeled his horse instantly, found the Bird Daughter at his side,
and with a score of men behind them they rode out of the battle in
pursuit. It proved useless, however, for the Dark Master had the better
horseflesh; after half an hour he was gaining rapidly, and with a bitter
groan Brian drew rein at last.
"No use, Nuala," he said. "I must wait until my strength has come back
to me, for I have done too much and can go no farther."
The girl reined in beside him, and her hand went out to his, and he
found himself gazing deep into her eyes.
"For what you have done, Brian," she said simply, "thanks. Now let us
ride back, for I think there is work before us, and we shall see the
Dark Master soon enough."
"I am not minded to wait his coming," quoth Yellow Brian darkly, and
they returned.
TO BE CONTINUED NEXT WEEK. Don't forget this magazine is issued weekly,
and that you will get the continuation of this story without waiting a
month.
Nuala O'Malley
by H. Bedford-Jones
Author of "Malay Gold," "The Ghost Hill," "John Solomon, Supercargo,"
etc.
This story began in the All-Story Weekly for December 30.
CHAPTER XIII.
BRIAN RIDES TO VENGEANCE.
"Then you are intent on this vengeance, master?" asked Turlough
thoughtfully.
"Yes," answered Brian. "I here take oath that I will never cut hair nor
beard again until I have seen the Dark Master dead."
"You are not like to have a chance at your hair very soon," laughed out
Lame Art O'Malley. "But that is a good oath, Yellow Brian."
"Then I think this is a better plan," spoke up Turlough Wolf. "Give me
ten men, Brian, and I will go to Galway. I will soon get traces of
O'Donnell; and if he goes into the north to get men of his own sept"
(tribe or family), "as I think most likely, I will send back word, and
we can follow him."
"Do it," said Brian, and Turlough was gone that night.
This discussion took place in the hollow, where the fight was soon over
after the flight of the Dark Master. Out of the six hundred who had left
the castle, two hundred had been O'Donnell's men. Half of these remained
and took service with Brian at once. Of the four hundred pikemen, three
hundred had gone down fighting like the stubborn south-country men they
were, and the rest took service with Nuala O'Malley. They were most of
them Kerry men, and well disposed toward ships and piracy.
Brian had lost in all fifty men in that battle, while the Dark Master
had given Cathbarr a goodly thrust through the shoulder, which had let
out most of the giant's vanity and promised to give the huge ax some
time to rest and rust. So, then, Brian found himself heading two hundred
and fifty men of his own, with Nuala's hundred O'Malleys, when they rode
down again to Bertragh Castle.
This had been left in charge of a hundred men under Red Murrough, who
had not been slain, but only wounded by Cathbarr's fist, that night in
the great hall. Having left a party to bring in the wounded in wagons
from the farms, they arrived before the castle shortly after noon.
Cathbarr was left in charge of the camp, and Brian rode up to the gates
with Nuala and her two kinsmen, with a flag of truce.
Murrough and his men were put into consternation by the news Brian gave
them. After much stroking of his matted beard, Murrough proposed to
surrender the castle on condition that he hold his post of lieutenant.
Brian laughed, for he had other views on the subject.
"You sold your master, and you will have no chance to sell me, Murrough.
I will give you the ten pounds I owe you and a good horse. Refuse, and I
slay you when we storm the castle."
The end of that matter was that Murrough assented. An hour later he
opened the gates, his men taking service with the rest under Brian.
Then, having obtained his ten English pounds and a horse, he waved
farewell to his men and rode away; and what became of him after that is
not set forth in the chronicle, so he comes no more into this tale.
Nuala loaded her fifty men into her carack, and sent them home that
night to Gorumna in case of need, proposing to follow later with Lame
Art, Shaun the Little, and her Kerry recruits. The O'Malley cousins
intended going south, since their affair had been so unexpectedly
ended, and picking up a Spanish ship or two before returning home.
"And now, what of your plans?" asked Nuala, as she and Brian sat
together that night before the huge fireplace in the hall, where Brian
had been burned and where Cathbarr had fought so well. "Of course, we
can settle rents later on."
"When there are farms to gather rents from," laughed Brian, stretching
out easily. He lifted his bandaged left hand, gazing at it. "First, I am
minded to rest here and wait for news from Galway. The bones in this
hand of mine are not broken, from what I can make out, and it will soon
knit. As soon as may be, I shall ride after the Dark Master; when I have
paid my debts, I will then be in shape to look for a castle for myself."
"Then you are determined to kill O'Donnell?" and she looked at him
sidewise.
"He has my Spanish blade," said Brian. "It is good Toledo steel, and I
want it back again."
"You have three hundred and fifty men here," she observed. "Can you feed
them?"
"You have food in Gorumna--send me some. When I am well again I shall
ride with most of them, which will lessen the burden. With the spring I
will take lands between here and Slyne Head, for now I am strong enough
to defend what I take."
"I shall also send you some of my pigeons, Brian. They are born and bred
on Gorumna Isle, and if you tie a message to them they will--"
"I know," nodded Brian. "I have seen them used in Spain."
With that she described how she used these pigeons, and Brian saw that
it was not by strength alone that this girl had maintained her position.
She kept men in Galway, Kinvarra, and elsewhere, as far south as the
Shannon and as far north as Erris, with others at Limerick and Tuam and
Castlebar. In this wise she got news of what was passing in Connaught
and Munster before most men had it, and more than one foreign ship had
found her caracks waiting for it through the same means, since she held
a privateer commission given her by Blake to legalize her sea-roving.
Also, she had pigeons which carried return messages, chiefly to her
kinsmen in Erris.
"And what is your goal, Bird Daughter?" Brian turned to her, his blue
eyes clinching on her violet ones. "What will the end of all this wild
life of yours be?"
"I do not know," she answered him, and turned away from his eyes to
stare down into the fire. "In the end I may be forced into marriage,
though I think not, for I have some will of my own in that regard." She
laughed out suddenly and looked up. "Two years ago Stephen Lynch sent me
a fair screed in all the glory of his chevron and three shamrocks and
wolf crest, saying that he was coming in one of his ships to marry me."
"And did he ever come?" smiled Brian.
"Yes; but I took his ship from him and sent him home again by road, tied
to a horse," she rippled out merrily. "Poor Stephen! The Bodkins never
let the Lynches hear the last of it until Stephen fell fighting against
Coote, and there was an end of it and him, too. When are you going to
tell me your name, Brian?"
At the sudden question Brian was tempted, but forbore.
"When I have slain the Dark Master," he laughed.
"Then you are likely to be bearded worse than Cathbarr," she mocked him
gaily. "Unless, indeed, you break that oath you swore this morning."
"Not I," returned Brian shortly. "I am not given to light oaths or light
pacts, Bird Daughter. I think I shall get me a ship and go cruising some
day."
"Come with me," she said, rising, "and you may win food and wine without
begging from your overlord. Well, now for that chamber Cathbarr fixed up
for me. _Beannacht leath!_"
Somewhat to his surprise, the next morning Brian found that Nuala was
extremely businesslike and even curt. Knowing little of women, he tried
to find wherein he had offended; failed utterly, and gave over the
attempt on seeing that Nuala preferred the company of Cathbarr.
Then, remembering that kiss she had given the giant aboard ship, he
concluded that the Bird Daughter was drawn by the physical magnificence
of the man, which gave him a little bitterness. So he merely set his jaw
the harder and said nothing of the thing that lay in his heart to any
one. For that matter, he was not quite sure himself what the thing was;
but he knew that he had never seen a woman such as the Bird Daughter in
all his life, and was not apt to find another.
Turlough having departed on his mission, Brian fell back on Cathbarr to
act as lieutenant; with Nuala herself, the work of getting the castle in
shape proceeded apace. The Bertragh hold was built on a cliff that rose
from the plain on the one hand, and sloped down to the water on the
other; had the Dark Master not fallen into Turlough's trap, he might
have turned out the pikemen to shift for themselves and have held the
castle with his own men for as long as he wished.
Indeed, Brian found that the removal of danger and the taking of the
castle had somewhat puffed up his men, lessening their fear of him. So,
on the second day, he quelled a free fight that rose among them, hanged
ten of the worst, and after this the others became as lambs before him.
Upon exploring the castle, Brian was delighted to find it well equipped
in all things except prisoners. The Dark Master had had little use for
captives, it seemed, and his dungeons were in sad disrepair. However,
there was good store of powder, provisions in moderation, a well within
the castle, and no lack of arms and munitions of war. Brian promptly
took the chamber of O'Donnell for his own use--a large tower-room well
furnished in English style, and having the luxury of a fireplace
besides.
The construction of the building was simple--a large stone structure
with embattled walls, running down close to the sea behind and rising
above the plain in front. Save for the courtyard, the walls were not
separated from the building proper, and there was one high tower, on
which the flagstaff had been shattered since O'Donnell had taken the
place, for he was not given to flags and display. Besides a dozen of the
large bastards, there were five falcons, with plenty of ball.
Therefore, Brian had good reason to be satisfied with his new home. The
only thing that rankled was that he held it not for himself, but for the
Bird Daughter; and he was determined that when he had settled scores
with the Dark Master he would only remain here until he had secured a
hold for himself, free of all service.
But settling with O'Donnell Dubh was the first duty he had. Brian
recalled his torture and the agony of Cathbarr every time he entered the
hall. The iron rings that had been in the floor he had already torn out,
while Nuala had taken for her own the lonely wolfhound, which had been
left behind by the Dark Master. But Brian, who put all his desire for
vengeance in the wish to "get back his Spanish blade," could hardly turn
around without having some phase of his sufferings brought back to him.
The men who had been thrown out along the roads had fetched in word that
the Dark Master had ridden for Galway, so Brian had great hopes that
Turlough would bring back some definite news. If O'Donnell settled in
the city, he was determined to go in at all risks and seek out his enemy
face to face; the O'Malleys were on good terms with the Bodkins, who in
old Galway played _Capulet_ to the _Montague_ of the Lynch family, and
he would be able to command some help in that quarter.
* * * * *
On the fifth day after the castle had been taken, a galley came over
from Gorumna Castle bearing news. Cromwell had failed before Duncannon,
and promised to fail again at Waterford, and hope was rising high among
the royalists, while O'Neill's Ulster army was biding its time in the
north until a new leader was chosen by the Confederacy to make head with
Ormond against the Parliament armies.
Upon this the O'Malley rovers were impatient to revictual at Gorumna and
be off to the south after plunder, so Nuala decided to leave Bertragh
the next morning. That night, after Cathbarr had drunk himself asleep
and the O'Malleys had sought their ships, the Bird Daughter unexpectedly
became very cordial toward Brian once more, and they sat up late before
the fireplace.
Brian did not understand it, but he was quite willing to accept it, and
when the talk turned on personal matters he was careful to ask no
questions concerning Nuala's plans for the future. Instead, he told her
tales of his life at the Spanish court, which interested her vastly,
until in the end she broke forth with a passionate outburst.
"Oh, I wish I were a man!" she cried softly and eagerly, looking into
the red embers. "All my life I have been among men, and yet not of them;
I have had to do with guns and ships and powder, and I think I have not
done so ill, yet I have had dreams of other things--things which I
hardly know myself."
Astonished though he was at her sudden unfolding of herself, Brian
looked at her gravely, his blue eyes very soft as he pierced to her
thought.
"Yes," he said gently, "you are a woman, Bird Daughter--and if you were
a man I think that you might have gain, but others would have great
loss."
"Eh?" She looked straightly at him, unfearing his half-expressed
thought. "I do not seek idle compliments, Yellow Brian, from those who
serve me."
Brian flushed a little.
"It is hard to receive compliments gracefully," he said, and at that she
also , but laughed, her eyes still on his.
"There, give grace to my rude tongue, Brian! Of course you meant it--but
why?"
"Because there is no woman like you, Nuala--so able to weld men into
union, so vibrant with inner power, and yet so womanly withal. It is no
little honor to have known you, to have--"
"I wish you would tell me your name, Yellow Brian!"
There was woman's cunning in the placing of that answer, and it took
Brian all aback. For a moment he was near to blurting out his whole
story; then he took shame for letting a girl's face so run away with
him. None the less, he knew well that it was her heart as well as her
face, and her spirit as well as her heart, that had captured him; yet,
because he had had no dealings with women since leaving Spain some
months before, he told himself that if the Bird Daughter had other women
near by to compare herself with, less attraction might be found in her.
But he did not pause long upon that thought, sweeping his blue eyes to
hers in a smile.
"If you had been a man, Nuala, you had never had fealty from me."
"So--then it _was_ pity?" and swift anger leaped into her face.
"Was it pity that drove Cathbarr to proffer his life for mine?" parried
Brian, his eyes grave. He felt a great impulse to speak out all that was
in him, but crushed it down. Her eyes met his, and held there for a long
moment. Then she spoke very calmly:
"When will you take that cruise with me, Yellow Brian?"
"When I have won my Spanish blade again," he smiled, and after that they
talked no more of intimate things, yet Brian's heart was glad within
him.
With the next morning the Bird Daughter said farewell and went aboard
Lame Art's carack. Sorry was Brian to see her go, for he had come to
count much on her fine backing and inspiring courage, and knew not if he
would ever see her again. As the ships raised anchor, Cathbarr suddenly
let off the bastards with a great roar and raised on the shattered
flag-pole an ensign he had secretly obtained from Shaun the Little. The
ship-cannon barked out in brave answer and hoisted ensigns likewise; but
as Brian looked up at the flag overhead, his despondent mood was not
heartened. The three-masted ship of the O'Malleys flew above him, where
he had much rather flown the red hand of his own house.
"When I have slain the Dark Master," he thought, watching from those
same sea-facing battlements where he and Cathbarr had descended, as the
two caracks leaped off to the south, "and when I have established myself
in some hold, be it never so small, then I shall take back my name again
and let the red hand hold what it has gripped. But not until these
things have been done, for Brian O'Neill will give fealty to none--no,
not even to the Bird Daughter herself."
Thus he thought in his proud bitterness, reckoning not on what the
future was to bring forth. However, he had lost his idea that Nuala
might love Cathbarr, and had great gladness of it.
Now there was work to be done, and Brian soon found himself too busy to
bother his mind with thoughts of bitterness. Cathbarr had done no little
drinking, so that his wound was turning bad, and in no little alarm
Brian banished all liquors from him and tended him carefully. Taking a
lesson from Red Murrough, he washed out the wound with vinegar, and
found that this had its effect.
Since Brian was irked at having to rely on others for his supplies, he
rode to all the outlying farms and sent off the families there under
escort, with sufficient money to keep them and take them to their homes
in the north. Many of them chose to remain, and certain of his men knew
of women-folk they wished to bring hither, so that Brian saw he would
not lack for farmers and settlers. Enough fodder was obtained to keep
his horses for a time; but as this did not satisfy him, he set forth
after four days on a cattle-raid to the northeast, riding past the
Manturks toward Ashford with ninety men.
He was gone on that raid five days; found to his great joy that his
strength had returned to him, and also found a small party of Royalist
horse near Lough Corrib. These had been buying up cattle for the Galway
garrison, and had collected fifty head; but on Brian's approach they did
not stay for dispute, but fled.
So Brian cheerfully sent the fifty head of cattle home with as many men,
and with the others swept around through the mountains. With him were
two of Cathbarr's axmen, and they led him to the hold occupied by
Murrough O'Flaherty of the Kine, where Brian stayed half a day. He
concluded a friendship with the mountaineers, promising them powder in
exchange for cattle, and they promised, in turn, that within three weeks
they would fetch a hundred kine down to Castle Bertragh.
Having thus assured himself of both food and stock for his farms, he
rode home again, to find great news awaiting him.
First, there had come a galley from Gorumna with wine and stores. Nuala
sent word that her men in Galway had informed her the Dark Master was
there, but in no high favor with Lord Burke and the other commanders.
Second, one of Turlough Wolf's men had come in with news which had
caused Cathbarr to have the men in all readiness against Brian's return.
The Dark Master was indeed in Galway town, and had made small head with
his suit for men, having related that Vere and his pikemen were lost.
However, he had been promised some help, provided he could gather any
force of his own and would hold Bertragh for the Royalists. Cromwell had
been driven back at Waterford, but Cork had risen for him, and his men
had entered there.
So the Dark Master was going to the north to get him men in Sligo, as
Turlough had predicted he would do, and his plan was to raise a force,
bring down those Donegal pirates with whom he was in alliance, and set
on Bertragh by sea and land, as Brian himself had aimed at doing.
Turlough said that he was following, but would leave men at Swineford
and Tobercurry with further news of what happed.
"Good!" cried Brian joyfully. "Cathbarr, have a hundred and fifty men
saddled at dawn--what is this?"
Turlough's messenger handed him a paper. It was a safe-conduct issued by
the Confederacy and Royalist leaders in the name of one Stephen Burke,
and where the wily Wolf had gotten it the messenger did not know. But it
might come in useful, since there were few parliament men in Sligo and
Mayo, and Brian tucked it away with a laugh.
"Then to the north at dawn--and O'Donnell shall not escape me this
time!"
CHAPTER XIV.
HOW THE STORM FARED NORTH.
Now, it was no easy matter for a band of horsemen to ride from Galway to
Sligo in that day, unless they were known men and rode for the king or
the Confederacy. Scattered bands of men had come into the west from
Ulster and Leinster, and these had driven out what Parliament men had
landed; through the early years of the war Owen Ruadh's men had swept
all the west country, and now the land was resting, waiting for the
storm that was fated to come upon it when the rest of Ireland had been
crushed under the heel of Ireton. Enniskillen alone, in Fermanagh, held
out for Parliament.
So, while the larger towns were all under Irish authority, the
hill-country was full of seething parties from all armies, most of them
being ravagers and outlaws who would fear to lay hand on so large a
party as Brian's. But little Brian cared for them, and without let or
pause he drove north to Ashford and so into the lowlands.
Knowing that he must return again by the same way, he avoided the larger
towns and pushed hard for Swineford, where he would find word from
Turlough. More than once he met parties of men on the road, but these
were not anxious to question him, and it was not until he was riding
around Claremorris that men began to feel his heavy hand.
With Lough Garra falling behind on the left, and Claremorris at safe
distance on the right, Brian was clattering along on the third morning.
His men carried muskets slung at their saddles, with bandoliers of
cartridges at their waists ready for quick action; and well it was that
they were so prepared. Searching ahead with narrowed eyes, Brian caught
a quick glint of steel on the road, and in no long time he made out a
party of a hundred men riding toward him. Brian got ready both his ax
and his safe-conduct, and rode forward without pause.
Now, he had brought with him most of those Scots troopers he had taken
into service, and as the other party drew near he heard a swift yell of
"Albanach!" that boded no good. But Brian shouted to them and asked who
they were.
"None of your affair!" answered their leader, a huge, dark man. "Who are
you?"
"Stephen Burke from Galway," answered Brian; but before the words left
him he saw a musket flash, and one of his men fell.
Upon that, no more words were wasted. Brian threw up his ax and dug in
his spurs, with his men behind; and when they loosed their muskets they
rode on the hundred with butts swinging. This was a new kind of warfare
in Connaught, and before Brian's ax had struck twice the field was won.
From two prisoners he found that the band was composed of a levy of the
O'Connors out of the Storm Mountains.
"That is not well for our return," said one of his lieutenants. "We will
have the whole country up after this battle, and we have lost ten men."
"Then we shall have the more need of recruits," quoth Brian, and let his
prisoners go free, since they would take no service, but only cursed
him.
However, Brian was not ill pleased, since he found that he was nearly
sped of his wounds, though his left hand gave him some trouble at times.
His pleasure was speedily cured, for when they camped that night on the
hither side of Kiltarnagh there came a rush of men toward dawn, and
before they were beaten off twenty of Brian's men were dead. Five
prisoners were taken, and when two of these had been hung, the other
three confessed that the attack had been made by certain O'Connors from
the southern end of Lough Conn, to whose villages fugitives had come
from the affray of the previous morning.
With that, Brian took counsel with some of his men who knew the country,
and it was their advice that he give up the ride and return home.
"I will not," said Brian shortly. "This war was not of my seeking, but
thirty of my men have been slain. Guide me to these villages, and I will
take blood-fine."
This he did because he needs must. His men did no ravaging, and were in
need of provisions, while he was minded to fill up his ranks. Also, by
taking sharp vengeance, he knew that on his return he was not like to be
molested.
So he turned aside and rode fast for Lough Conn, which he reached the
next evening, and there came a storm of men on all that country. Twice
through the days that followed Brian had to fight hard--once against a
muster of the O'Connors, and once against a large force of ravaging
hillsmen under one Fitzgerald. Him Brian slew with a blow of his ax that
went from shoulder to saddle.
From his men he gained fifty recruits and no small booty, both of money
and horses; and from the O'Connors he took bitter blood-fine for his
slain men in spare horses and provisions.
These doings are set down briefly in the chronicle; but when Brian
turned east again, with Swineford a hard day's ride away, he once more
had a hundred and fifty men at his back, with a good store of all
things, while his name was one that spread fear. He left his men camped
two miles out of Swineford, on the Moy, and rode next morning into the
town with a dozen horsemen only.
In the town was quartered a small force of Maguires from Fermanagh, and
as he rode in Brian was halted by their leader, who gave him the sele of
the day and asked his name. Brian held out his passport, and after
Maguire had fumbled over it and pretended that he could read, he gave it
back with a grin and Brian passed on with another.
The seal of the Confederacy on the safe-conduct was quite enough for any
man in these parts, however.
Brian had not ridden a hundred paces farther before he saw one of
Turlough's men beckoning to him from the door of an inn, so he left his
troopers to drink outside and passed within. Turlough's man joined him
at a table, and there Brian gained news of the most cheering.
Six days before this the Dark Master had arrived at Swineford, with
Turlough an hour behind him. The old Wolf, whose cunning made up for his
lack of courage, had made shift to get two of O'Donnell's dozen men
embroiled with the Maguires. The upshot of that had been a fight,
followed by a delay of two days for investigation; finally the Dark
Master had slipped away, his two men had promptly been hung, and
Turlough had meantime gone ahead to prepare fresh delays at Bellahy and
Tobercurry. He had four men left with him, though he had left Bertragh
with ten.
"Then O'Donnell has four days' start of me," reflected Brian. "If
Turlough can hold him, we will catch him at Sligo at latest."
He left the inn and rode back to his camp, where he had the men on the
road in ten minutes. Tobercurry was only fifteen miles north, and
putting his horses to a gallop, Brian rode hard and fast until that
afternoon he came into the place. He found no garrison, but, instead,
was met by old Turlough himself, with a bandaged head and two wounded
men.
"_Mile failte!_" cried Turlough joyously, running forward to kiss
Brian's hand in wild delight. "You are well come, master! Is all well
down below?"
"All well, old friend," laughed Brian, swinging down to clasp the old
man in his arms. "Where is the Dark Master?"
"Where we shall catch him in a forked stick presently," chuckled
Turlough, wagging his beard. "Get these wild men of yours out of the
town, and come into the inn with me to talk. I have all the Dark
Master's plans, master, and we have only to strike."
Brian ordered his men to camp a mile outside town and to do no
plundering, so they clattered off, to the great relief of the townfolk.
"Now," said Brian, when they two were sitting across a table, "what has
passed that you are bound up? Have you been fighting?"
"Well, after a fashion," grimaced Turlough disgustedly. "I was here
ahead of the Dark Master, and raised the townpeople against him for a
plunderer. When he came up the road was full of men; but the devil slew
two and wounded two of my own men, cut his way through the rest, and as
I fled north my horse flung me and bruised my head. Has the castle
fallen?"
"Yes," laughed Brian, and related what had happed at Bertragh. "Have I
time to bide here and eat?"
Turlough yeasaid this and sent the inn-master bustling for food and
wine. When this was set before them, Turlough Wolf told his tale,
beginning with the statement that two of O'Donnell's men had been
captured when he cut through the townfolk and rode off.
"Where are they?" asked Brian quickly, his eyes narrowing.
"Hanged," chuckled the old man succinctly. "At Galway I could make out
nothing more than the word I sent you by messenger, so I came north
after O'Donnell Dubh, taking very good care that he saw nothing of me."
"I'll warrant that," laughed Brian. "We met your man at Swineford."
"Then no need to tell what passed there. Well, I said that we caught two
of his men here, and I got back into the town just in time to keep the
folk from hanging them to the church steeple."
"Eh?" Brian stared, with his mouth full. "Why, I thought you said--"
"_Dhar mo lamh_, give me time to finish, master!" Turlough hesitated a
little, evidently in some fear. "We took them into the churchyard and
burned them a little, and so got out of them all the Dark Master's
plans. Then the priest shrived them, and I let the townfolk hang them."
Brian looked across the table, his blue eyes like ice and his nostrils
quivering with anger; the old man slanted up his gray eyes and turned
uneasily in his seat, for well he knew what Brian would say to this.
"That was ill done, Turlough Wolf. If you had not served me so well, you
would repent that work. By my faith, I am minded to hang you at their
side!"
Brian meant it, for the torture of men made him furious.
"I am no fool to spare mad dogs," muttered Turlough sullenly. "It was
the Dark Master who lopped these ears of mine eight years gone."
"Tell your tale," said Brian curtly and fell to eating again.
"I found tidings both good and bad, master. From Galway the Dark Master
had sent messengers to his kin in Donegal, bidding them send aid south;
also, he sent to certain pirates north of Sligo Bay. From Sligo to the
Erne all that land is desolate, and has been so these six years, and the
O'Donnells from Lough Swilly have set up a pirate hold near Millhaven.
It was to these that the Dark Master sent also.
"He has appointed a meeting-place in the hills beyond Drumcliff, at a
certain mountain named Clochaun, or the Stone. Now, whether you think
my craft evil or good, master, it is yet gainful to us."
This much Brian was forced to acknowledge, though for many days
afterward he was still angry at Turlough for torturing and hanging those
men. He had no scruples about a downright hanging, but torturing was a
very different matter, and one of which he had tasted himself.
"Well, what is your advice in this?"
"We can do one of two things, master. The one is to ride on to Sligo and
fall on him when he comes south again with his men; the other is to ride
hard after him and catch him, then fall on the Millhaven men, then meet
the O'Donnells who are coming south to join him at the Stone Mountain
with the rest."
"The first plan is more cautious," said Brian thoughtfully; "but to
strike him when he has his men around him would be to repeat what we
have done. I like the other way the better."
"It is both safer and yet more dangerous, master. Safer in that we smite
him and his men separately, and more dangerous because we shall be in
the heart of a wild country, without supplies, and with no aid in case
we are defeated."
"It is more to my mind to talk of winning than losing," grunted Brian.
"I have spare horses and money with which to buy provisions. Also, I
think that I shall stamp flat that pirate nest at Millhaven, and set up
my own banner there."
"Then you have a banner of your own, master?" Turlough squinted up
slyly, for it was the first hint Brian had given him of what lay behind
his nickname.
"Aye!" laughed Brian as the wine warmed him. "And it shall bear the Red
Hand of Tyr-owen, old Wolf; but first to catch the Dark Master. Now let
us go, for we shall ride to the Stone Mountain and see what haps there."
Upon that they rode forth from the town, and all the townfolk bade the
crafty Turlough farewell, and gave him gifts for warning them against
the "plunderers." Turlough looked up at the two bodies swinging in the
wind as they passed the church-tower, and put his tongue in his cheek,
but Brian said no more on the subject.
That night they camped outside the town, and Brian bought all the
provision that the people would sell. This he loaded on the spare
horses, and the next morning they set off for the north.
* * * * *
Now, in that fighting by Lough Conn, Brian had taken a shrewd clip which
had reopened the bullet-tear over his scalp. Added to this, he was not
yet in all of his former strength, and the hard ride to Tobercurry had
set his blood to heating; wherefore it was that before coming to Sligo
Brian was heavy with fever and was shaken with chill. A hard snow was
driving through the night, and Turlough sent most of the men around the
city to wait for them on the other side the Garravogue to avoid danger.
There was no garrison in Sligo, however. The old castle which Red Hugh
O'Donnell had fought over in the old days was ruined; the grand
monastery, built by Brian of Tyr-erril, had been burned by Hamilton's
men, together with the town itself, and Sligo was well-nigh desolate.
Turlough got shelter in a hovel, however; managed to put Brian into a
miserable bed, and gave him a brew to drink. With the morning Brian
found his fever gone, but weakness was on him.
They stayed in Sligo town all that day and the next night, and upon
dawn, Brian insisted on riding north once more, against Turlough's
protests. However, no ill came of it, for Brian was well used to riding,
and the exercise gave him strength, though they made but a short march
that day past the round tower of Drumcliff, halting in the hills.
As Turlough Wolf knew where the Stone Mountain was they had no use for
guides. It lay only another day's march ahead of them, and there was
some danger that their quarry would descry their coming and flee away to
Millhaven.
"This is my rede, master;" said Turlough, "that you and I ride ahead
with a few men to see how things go, and leave our men to follow. The
hills are empty of rovers, for there is naught to plunder; but it were
well to know if the Dark Master has joined with those friends of his."
"That seems good advice," said Brian, and, taking a dozen men, they rode
forward warily, sending out other parties to scout also.
Over them towered the whiteness of the Stone Mountain, for snow lay
thickly on all things. Brian gazed up at the gray-jutted crags, but his
thoughts were not all with the Dark Master. Him he already accounted
slain, and he was thinking of that Millhaven stronghold.
One day his own banner should fly there, he told himself. There must be
a good harbor, else the northern pirates had never settled down to hold
the place; and since all the country roundabout lay bleak and unsettled
of men, the vision came to him of first taking the place, and then
fetching O'Neills from the east and north to settle the lands around.
They would flock to him when his condition was made known, and that
Cromwell's men would shatter the royalists and confederacy Brian saw
clearly, as Owen Ruadh had foretold him.
Already the house of Tyr-owen was scattered and fallen, as the greater
house of Tyr-connall had been before it, for when the last earl had fled
from the land, there had been only the younger branch to hold the sept
together. Owen Ruadh was the final glory of that branch, and now Brian
entertained the vision of transplanting the Red Hand and of making his
rule strong in the west.
But other men had entertained the same vision before him, and it had
remained a vision, and no more; and the high hopes of Brian himself were
fated to be driven upon the rocks of destiny before many days had passed
over.
With the afternoon the little party stood on the lower <DW72>s of the
Stone Mountain itself, and Turlough drew the shape of the place in the
snow with his pike-haft.
"Here are we," he explained, "on the southern <DW72>s. A half-mile ahead
of us is a valley with a small and fast-rushing water, where we shall
make camp this night if the Dark Master be not before us. And if he is
not, then he will be on the northern side, where there are two
well-sheltered valleys with water running, fit for the meeting-place and
camp of men. Here is the easternmost, but, as I remember it, the snow
fills the valley somewhat in winter. The other holds a small lake called
the Dubh Linn, or Black Tarn, and in one of these we shall find the Dark
Master, unless he is here before us."
"Well, let us ride on and see to that," said Brian, and they did so.
However, they found the valley deserted and empty, and picked a place
for camp, sending back a horseman to bring up the force. They could make
out no smoke rising from the mountain, nor dared they light fires until
after dark for fear of alarming O'Donnell; but when the force came up,
Brian sent out scouts to bring in what word might be had.
"Where got you such knowledge of this wilderness?" he asked Turlough
that night when the fires were blazing and the men were warmed and fed.
The old man narrowed his gray eyes and chuckled a little.
"I have been in many armies, master, though I have fought not; and I
have been outlawed twice by the English, in the old days. This was
always a good place to flee to."
Brian laughed and said no more. That night the men rested well, and
Brian himself got sleep which sent strength into him and served him well
in the days to come, for it was long before he was to sleep again, save
as he rode, nodding in the saddle.
Not until nearly dawn did the last of the scouts straggle in. None of
these bore any news, and all agreed that no signs could they find of any
large band of men, nor of any men at all. Turlough heard their reports,
letting Brian sleep, and only when the last man came in were any tidings
brought. This man bore a strip of sheepskin, which, he said, an old
woman had given him to bear to his master.
"A woman!" exclaimed Turlough, scanning the written words on the
sheepskin, but unable to read them. "What is she like? It is a strange
thing if women bide on Slieve Clochaun! Was there any stead near by?"
"None," replied the man, who trembled with something more than cold.
"_M'anam go'n Dhia!_ She was a witch woman, or worse, Turlough Wolf. She
leaped out of the snow in my path, told me to bear that skin to Yellow
Brian, and vanished in a burst of fire. How could she not have been a
devil?"
"Nonsense!" grunted Turlough, though he suddenly laid the strip of skin
down. "You are overwarm with _uisquebagh_, man. What was this woman
like? Was she clad all in black?"
"Faith, I did not stop to see," grinned the man sheepishly.
Turlough stroked his beard, while the men went off to eat and sleep. He
gazed at the strip of skin, and twice stretched out his hand toward it,
with his eye on the fire, but each time drew back. Then he glanced
around craftily, found he was alone, and took from under his cloak a
small, brass crucifix. With this he touched the skin, found that nothing
happened, and rose with a nod. The dawn was just breaking in the east.
"There is no sorcery in it, at least," he muttered; "but I think it
bodes no great good to us. Ho, Brian!"
Brian woke and sprang up. Turlough handed him the strip of skin, saying
no word, and when Brian had held it to the light of the embers, he
looked up suddenly.
"Whence came this?"
"What does it say first?" returned Turlough uneasily.
"News!" cried Brian, his blue eyes aflame with eagerness. "It says that
O'Donnell bides alone by the Black Tarn, and that his horsemen from the
north are camped two miles beyond the mountain, waiting for him, and
that he has made pact with the Millhaven pirates and they have left for
their stronghold. Answer me--whence came this? It is written in good
English writing, man!"
Then Turlough told of what had chanced, and when he had done, Brian
stared into his gray eyes with a great wonder. Twice he tried to speak,
but his lips were dry.
"The Black Woman!" he muttered thickly. "Can it be, Turlough? Who is
she?"
"That was my thought, master," said Turlough. "Who she is none know save
herself; but she deals with no good. This may be a trap; let us ride
south again, and at once, lest evil come upon us."
"South? Not I," laughed Brian, though his face was pale. "To horse,
men!"
And at his ringing shout the camp awoke, and Brian saw his vengeance
drawing near.
CHAPTER XV.
WHAT HAPPENED AT THE TARN.
It had been long, indeed, since Brian had given thought to his meeting
with the Black Woman on the other side of Ireland. In that brief
meeting, the Black Woman had spoken of seeing the old earl, his
grandfather, in his youth. Yet it was forty years since the two earls,
O'Donnell and O'Neill, had fled together from Ireland, and even then
Tyr-owen had been an old man. Unless this Black Woman was close on a
hundred years of age, Brian could not see how she had known Hugh O'Neill
in his youth.
The mere fact that she had recognized him there in the moonlight was
proof of her true speaking, however. Brian could no longer hide from
himself that her words had some strange prophecy in them. She had
foretold his meeting with Cathbarr and with the Bird Daughter, though,
indeed, she might have been attempting only to guide him on the path
which he had afterward followed.
While the men were saddling, Brian called Turlough and told of the hag's
word that she would meet him again "on a black day for him."
"Now, what think you she meant by that, Turlough? Is this the meeting?"
"No, master, for it is no meeting. It may be as you think, and that she
was but trying to lead you into the west; yet, for my part, I call it
sorcery," and the old man crossed himself, for, like better men than
himself, Turlough ascribed all he could not fathom to magic. "It seems
to me that she is some witch who is hanging on your tracks, and that
when--"
"Oh, nonsense!" laughed Brian, flinging the matter from his mind. "At
any rate, she has served me well this time. Now, what rede shall we
follow in this matter, and shall we capture and slay the Dark Master
first, or fall on his men first, or both together?"
"It is ill to sunder a force of men, master," quoth Turlough. "If those
horsemen of O'Donnell's are encamped in a valley two miles to the north,
it is a vale of which I know well. But we must mind this--if O'Donnell
gets safe into Galway again with either these horsemen or those
Millhaven pirates of his clan, he will drive hard against Bertragh."
"The Dark Master shall come no more to Galway," said Brian grimly,
fingering his ax. "Now finish, and quickly."
"I have a plan in my mind, master; but unless we slay the Dark Master,
it is like to fail us. Let us send a hundred of the men around to the
north, for I will tell them how to ride, so that by this night they can
fall upon those men of his and scatter them in the darkness, and drive
them south where we can slay them utterly at our wills. If we drove them
back whence they came, there would be little craft in it, and it is to
my liking to do a thing well or not at all."
"A true word there," nodded Brian, his eyes gleaming. "I think those men
are as good as dead now, Turlough. Speak on."
"With fifty men, master, you and I can reach the valley of the Dubh
Linn. We cannot do it with horses, unless we ride around to the north,
and in that there would be danger of striking on the Dark Master's
scouts. But while our hundred are circling far around, we with fifty can
go over the mountain by valleys and paths I know of, so that by this
evening we will come to the Black Tarn and strike the Dark Master as our
hundred men fall on his camp. That is my--"
"Good!" cried Brian, leaping up eagerly. "Then we--"
"Hold, master!" And Turlough caught his arm, quickly staying him. When
Brian looked down he read a sudden fear in the old man's gray eyes.
"That was my first rede, Yellow Brian, and you would do well to hear my
second also."
"Say it," said Brian, and glanced at the brightening sky.
"My second rede is this. That message might be a trap to ensnare us,
though I have two minds about this Black Woman. But if we fail to slay
the Dark Master at the Black Tarn, we are like to have an ill time."
"Why so?" asked Brian, for he could see no likelihood of that. "I said
that we would slay him."
"Master, do you hold the lives of men in your keeping?" In the gray eyes
leaped a swift horror that amazed Brian. "I tell you that if the Dark
Master escapes from our hand, and his men are driven past our fifty into
the south, he will ride hard before us into Galway. I see evil in that
first rede of mine, Yellow Brian. I see evil in it--"
He broke off, staring past Brian with fixed and unseeing eyes, his face
rigid.
"Turlough, are you mad?" Brian seized the other's shoulder, shaking him
harshly. The old man shivered a little, and sanity came back into his
eyes as they met the icy blue of Brian's. "What daftness is upon you,
man?"
"I know not, master," whimpered old Turlough feebly. "Do as you will."
"Then I will to follow your rede, divide my men as you say, and when we
have slain the Dark Master, we will cut off the last of these O'Donnells
of his, ride to Millhaven and take that hold, and send word to the Bird
Daughter that she may keep Bertragh Castle and send Cathbarr north to
me. Now go, and tell a hundred of the men how to ride around this
mountain; then be ready to guide me over it to the Black Tarn."
"You are a hard man, Yellow Brian," said Turlough, and turned him about
and did as Brian had ordered.
None the less, Brian gave some thought to that second rede of
Turlough's. He saw clearly enough that with the northern horsemen driven
past, scattered though they might be, they could be cut off to a man if
the Dark Master were slain. But if O'Donnell should escape by some trick
of fate, he could gather up his men and drive south.
"If he does that, there will be slaying between Sligo and Galway," swore
Brian quickly. "But I cannot see that he will escape me here. When
another day breaks, I shall have won my Spanish blade again--and then
ho! for the Red Hand of Tyr-owen!"
So Brian laughed and donned his jack and back-piece, while Turlough drew
plans in the snow and showed the leaders of the hundred how to sweep
around without discovery so that they might fall on the northern
horsemen at eve.
Brian had grown into an older and grimmer man since the day he had stood
beside the bed of Owen Ruadh O'Neill, short though the time had been.
Youth was still in his face when he smiled out, but suffering had
deepened his eyes and sunk his cheeks and drawn the skin tighter over
that powerful jaw of his. When he had armed, he stood in thought for a
little, with hand on jaw in his instinctive gesture, and wakened
suddenly to find old Turlough bending the knee before him.
"Now I know of what blood you come, Yellow Brian," said the old man
softly. "I saw Hugh O'Neill, the great earl, standing even as you stand
now, on the morning when we slew the English at the Yellow Ford."
"Man, man!" exclaimed Brian in wonder; "that battle was fought fifty
years ago, and yet you say that you were there?"
"I was the earl's horse-boy, master." And Brian saw tears on the old
man's beard. "I loved him, and I was at the flight of the earls ten
years after, going with Tyr-owen to Italy, and it was these hands laid
him in his grave, master; master, have faith in me--"
Brian put down his hands to those of Turlough, his heart strangely
softened.
"He was my grandfather," he said simply, and Turlough broke down and
wept like a child.
When they left their horses and the camp behind, Brian followed
Turlough, feeling like a new man. He had lightened his heart of a great
load, and he wished that he had talked of these things with Turlough
Wolf long before this. Now he understood why the old man had offered him
service as he stood in that attitude on the battlements of O'Reilly's
castle after leaving Owen Ruadh, and he understood the love that
Turlough bore him, and the silence the old man had kept on the matter,
though it must have ever been deep in his heart to speak out.
No more words passed between them, nor did Brian tell Turlough more of
his story until long after; but of this there was no need. As they
climbed higher on the mountain they could see the hundred horsemen
filing off to the eastward; but soon these were lost sight of as
Turlough led Brian and the fifty through the valleys and deep openings,
which were drifted deep in snow, making progress slow and wearisome.
Indeed, Brian thought afterward that this hard traveling might have been
responsible for what chanced on the other side of the mountain.
On the higher crests and ridges there was little snow, however, and
Turlough seemed to know every inch of the place by heart, though more
than once Brian gave himself up for lost in the maze of smaller peaks
and the twisted paths they followed. Most of the fifty Turlough had
chosen from those hillmen who had joined Brian by Lough Conn, so that
they were not unused to such climbing, and remained with spirits
unshaken by the vast loneliness that surrounded them, and to which other
men might have succumbed somewhat.
Brian himself was no little awed by the desolate grandeur of the Stone
Mountain, but he only wrapped his cloak more closely about him, and
swore that the Dark Master should yield up the Spanish blade before many
more hours.
And so indeed it was done, though not as Brian looked for.
Until long after noon the band wended their way with great toil and pain
over the flanks of the mountain, until Turlough led Brian out to a point
of black rock and motioned toward the valleys below them.
"There to the left," he said, "is the valley of the Black Tarn. Do you
see that smoke, Brian, and that dark spot between the trees and the
lake?"
Brian looked, squinting because of the snow-glare. Leading down from the
side of the mountain itself was a valley--long, and widening gradually
to the plain, where a dark wood swallowed it up. Almost under his feet,
as it were, was a small, round lake deep in the rock, with a small,
frozen-over outlet that was lost in the snow.
But farther down the valley-<DW72>s there were trees, and among them
horses tethered and a fire strewing smoke on the air close beside.
Between this little wood and the tarn itself there stood a low house of
thatch with smoke also rising from it, and from the other fire among the
trees came a sheen of steel caps and jacks, where were men.
But to Brian all these things were very small and hard to make out
distinctly, as if he were looking at some carven mimicry, such as
children are wont to use in play.
"Now come," said Turlough Wolf. "It is no easy task getting there
without being discovered, and the way is long."
Brian found, indeed, that to avoid being seen from below they must
needs take a roundabout way; but when the afternoon was far spent they
had come to a snow-filled hollow among the rocks which Turlough declared
was just over the edge of that valley-<DW72> where stood the low house.
Turlough said that in his day that house had not stood there, and he
knew nothing of it.
Since there could be no talk of lighting a fire, Brian's men huddled
together in the hollow, and ate and drank cheerlessly. Brian was minded
to meet the Dark Master and win his Spanish blade with his own hand, so
he ordered that his men pass on after dark and make ready to fall upon
those men who were camped at the wood, but to hold off until he and
Turlough had smitten the Dark Master in that little thatched house,
where he was most like to be found. Turlough yeasaid this plan, for he
trusted greatly to Brian's strength.
At length they set out under the cold stars, and Brian's men were very
weary, but promised to do all as he had commanded. He and Turlough set
off alone over the hill, and when they had come to the hill-crest after
much toiling through the snow they looked down and found the house a
hundred yards below them.
"Let us go down cautiously," said Turlough, "for I think we can peer
through the thatch and plan our stroke well."
So they struck down openly across the hill-<DW72>, and found that there
was none on guard. The door of the house was fast shut, but Turlough
strode cautiously in the trampled snow around the house, where, at the
side, a spark of firelight glittered through the loose thatch. To this
he led Brian, and Brian stooped down and looked through the cranny,
while Turlough went farther and fared as well.
There was but one room in the hut, and it was well lighted by the fire
that glittered merrily on the hearth. Sitting not far away, but with his
back to Brian, was a man; he sat on a stool, and there seemed to be a
wide earthenware bowl of water or some dark liquid on the floor between
his feet into which he was staring. In his bent-down position his
rounded shoulders stood up stark against the fire, and Brian knew this
was the Dark Master.
His hand went to the pistol in his belt, but since there was no other
man in the hut, he thought it shame to murder O'Donnell as he sat, and
made up his mind to go around to the door and burst in. He saw his own
great sword slung across the Dark Master's back, but even as he stirred
to rise, O'Donnell's voice came to him, low and vibrant, so that he bode
where he was and listened.
"I cannot make out the figures," muttered the Dark Master, still staring
down into the bowl of dark water. "The man has the face of Yellow Brian,
yet he is swart; the woman I sure never saw before. _Corp na diaoul!_
What is the meaning of this? Who stands in my way?"
Brian paused in no little astonishment, and stole a glance aside to see
old Turlough crossing himself fervently. It struck his mind that he had
chanced on some sorcery here, and, remembering the tales he had heard of
the Dark Master's work, he laughed a little and settled down. He was
minded to see what this thing might be; but he made his pistol ready in
case the magic told O'Donnell of his danger.
"It is some great man," came the Dark Master's voice again. "There is
something broidered on his-- By my soul, it is the Red Hand of Tyr-owen!
It is The O'Neill himself--the earl-- Is Yellow Brian of his blood,
then?"
At hearing this Brian crouched closer, in some fear and more wonder. Was
the Dark Master in reality seeing such figures in that water-bowl? Then
the man must be either mad or--or figures were there. Now O'Donnell's
voice rose stronger:
"Which of these twain stands now in my way? It is not Yellow Brian. Ah,
the earl is slipping away, and the woman is smiling. One of his loves,
belike, for he had many; she is fair, wondrous fair! Ah, what's this?"
Brian saw the dark figure crouch lower, as if in astonishment.
"Changing, changing! Is it this woman who stands in my way, then?
Toothless and grinning, crouched low over a stick, rags and tatters and
wisps of gray hair--"
The Dark Master paused in his jerky speech, stiffened as if in wild
amazement at that which he beheld, and a sudden cry broke from him,
sharp and awestruck:
"The Black Woman!"
Then Brian straightened up, feeling Turlough's hand touch his; but for a
space he stood silent while his mind cast out for what the Dark Master's
words meant.
In a flash it came to him. Through some black dealings O'Donnell had in
truth pictured The O'Neill in that bowl, and with him a woman he had
loved and who loved him; and this was no other than she whom Brian had
known as the Black Woman, now become an old hag indeed, with only the
memories of her fair youth and her love behind her. And this was why she
had recognized him and why she had evidently watched over him since that
first meeting, out of the love she had borne the earl, his grandsire, in
days now buried under many bitter years.
The two men looked into each other's eyes, and Brian saw that Turlough's
jaw had dropped loosely, and that fright had stricken the old man almost
out of his senses. With that Brian felt his own fear take wings. He
laughed a little as his grip closed on the haft of his ax, and the cold
star-glint seemed to shine back again from his eyes.
"Bide here if you will," he smiled quietly. "I have my work to do."
And, turning with the word, he strode quickly to the door, just as there
came a great cry from within the place.
CHAPTER XVI.
BRIAN GETS HIS SWORD AGAIN.
Brian pushed the door open, and it gave easily to his fist. Gazing
within he saw the Dark Master standing over the shattered bowl, whose
liquid flowed down toward the hearth and hissed on the embers; plainly,
the Dark Master had seen nothing good in that water, for he had
shattered the bowl with his foot, and his teeth were snarling under his
drooping mustache.
"I am come," said Brian, laughing grimly as he stood in the doorway.
O'Donnell whirled, gripping at his sword.
Now, whether there was magic on the place, as Turlough ever swore, or
whether the opening of the door had made a draft, as Brian thought more
likely, a strange thing happened.
Brian had raised his pistol in his left hand, meaning to kill the Dark
Master without pity in that first moment. Out of the hearth came a great
swirl of ashes and red embers, flying toward the door and closing around
O'Donnell; as Brian pressed the trigger the ashes smote him in a
blinding swirl, and a harsh laugh answered the roar of the pistol.
With a curse Brian cleared his eyes of the light ash and reached with
his ax at the dim figure of the Dark Master, nigh hid with ashes and
powder-smoke. From down the vale came other shots and cries, and he knew
his men had struck on that small camp lying there; but at this O'Donnell
gave him other things to think of.
That was a great fight, for Brian was little used to ax-play and had
much ado to parry the keen thrusts of his own Spanish blade; the roof
was too low to give room for a swing, and when the Dark Master had
lunged him back to the door again, he knew that he had done ill. So with
another bitter curse Brian flung the ax from his hand and ripped out the
long, Irish dagger that hung at his girdle.
For all his wrath he had taken good heed to fling the ax aright, and the
broad flat of it took the Dark Master full in the chest and bore him
back, reeling and shouting for his men. Before he could recover Brian
leaped at him, caught O'Donnell's sword wrist in his left hand, and
aimed a deadly stroke with his _skean_.
The blow went true, but the steel turned aside from the Dark Master's
mail-shirt; O'Donnell caught his wrist in turn, and there the two stood
heaving each at the other for a long minute. Brian's eyes struck cold
and hard into the evil features of the Dark Master; the other's breath
came hot on his cheeks, and so beastlike was the man's face that Brian
half expected those snarling teeth to close snapping at his throat. But
the Dark Master was strong, for all his hunched shoulders.
Then a great flame of vengeance seemed to cleave Brian's soul, and with
a curt laugh he threw out his strength and flung the Dark Master back
bodily so that he fell into the hearth and burst the mud chimney and the
thatched wall behind. Before he could rise again Brian had whipped out
his other pistol and fired; he saw the man's figure writhe aside, then
up through the powder-smoke rose a burning brand that smote him over the
brow heavily. At the same instant the scattered sparks caught the
thatch, and the whole house broke into flame.
Brian's eyes found the dark figure once more and he rushed forward. At
the broken heap of mud from the chimney his feet struck on the sword,
which had fallen from the Dark Master's hand, and he caught it up with a
cry of joy and bore forward.
That brief instant of delay lost him his quarry, however. Brian flung
through the shattered wall, with the whole structure flaming up behind
him; he saw a dark figure on the snow and ran at it, only to find
himself striking at Turlough Wolf, and stayed his hand barely in time.
"Where is he?" he panted hoarsely, looking around with fierce eyes.
Then he caught the Dark Master's figure running across the snow toward
that camp amid the trees, where fighting was still forward and men were
shouting and firing. Brian rushed off, with Turlough staggering after
him; but with a sob of despairing anger he saw the Dark Master flit into
the trees, and heard his voice ringing at his men.
It turned out afterward that Brian's fifty men, weary and chilled, had
made a somewhat heartless assault on the score of horsemen camped in the
trees; therefore, instead of carrying O'Donnell's men off their feet and
cutting them down straightway, they were held off for a little.
The Dark Master knew that he was lost if he stayed long in that place,
however, and when Brian reached the clump of trees he found that he was
too late. With two or three men behind him, O'Donnell had cut through
Brian's men and was galloping away. Brian groaned savagely, leaped at a
mounted man and dragged him from the saddle, and was just springing up
when Turlough caught and stayed him.
"Wait, master!" panted the old man in desperate fear of the surging men
around him, but in more desperate fear for Brian. "This is madness, for
I ordered our fifty horses fetched around--"
"Bide here for them, then!" said Brian, and swung up into the saddle.
One of the Dark Master's men barred his way, and Brian's blade went
through his throat; then he was off after the four figures who by now
were far distant toward the dark forest that swallowed up the valley
ahead.
The cold night air cleared his brain, however, and after a moment he
drew rein with bitterness upon him. Turlough had spoken rightly, for to
ride after those four men with his naked sword alone was in truth
madness. So he came back again to where the last of the hemmed-in
horsemen was being cut out of his saddle, and when his men gathered
about him with a shout, his tongue gave them little joy.
"You are fools," he said harshly, "for the Dark Master has escaped us.
Take these horses, fifteen of you, and ride. Let five men go to bring in
our horses with all speed, and let ten more scatter out in search of our
hundred men. These are not more than two miles distant, and in an hour I
must ride from here. See to it that you return with the men and horses
by then, or shift for yourselves."
"That is too much," spoke out a burly fellow angrily. "We have been
climbing all day, and have----"
Brian said no word, but leaned down from his saddle and his Spanish
blade flickered in the light. The man fell and lay quiet, while the
others drew back in black fear.
"I am master here," said Brian coldly, when a long instant had passed.
"Go."
There was no more muttering among his recruits, either then or later. He
dismounted, saw that the O'Donnells had been slain to the last man, and
joined Turlough at the campfire. Food and drink had been found in the
camp, and a flagon of wine heartened Brian greatly.
"Now give me your rede, Turlough Wolf," he said. "I have failed in this
matter, and it seems that ill shall come of it."
"So I foretold, master, but we may still remedy the ill if we catch
O'Donnell. I think that by now his horsemen are scattered, and this
burning hut will draw our own men thither. Before midnight they will be
here, and we can ride forth. I think that the Dark Master will gather
what men are left him and strike down for Galway."
"Two men may ride the same road," quoth Brian grimly, and set his naked
blade in his belt. He saw that before him lay some fighting and much
hard riding, so inside the next hour he had his men full-fed. Before
this was finished the spare horses and those of his men came in, for
Turlough had ordered them to start at noon and ride around in case of
need.
Brian determined to spare neither men nor horseflesh on that riding, and
when his men were mounted he set out across the night to meet his
hundred, and to hear what had been done at the camp two miles distant.
As the moon was rising he met them; and if he was glad at the meeting,
they were twice glad.
They had found the camp and had lain off it until after dark as
Turlough had bidden them, the more so since there were two-score over a
hundred men there. But at length they had ridden down as if they were
fresh come from the north, and had twice ridden through the camp before
the O'Donnells were well awake, though it had been sharp work. The
result had been that a score of Brian's men had fallen, they had slain a
full half of the O'Donnells, and the rest had been driven and scattered
southward. Brian's men had plundered their camp and were weary, so that
when they heard of what had chanced at the Black Tarn they were somewhat
less than half willing to ride farther.
But Brian speedily persuaded them to that course, and Turlough led them
all to the south on the way to Sligo.
Bitterness and heaviness of heart dwelt deep in Brian that night, and
for some time to come. With the escape of the Dark Master, whether it
had been by magic or craft, all his visions had burst; he must ride away
from the pirate hold at Millhaven, he saw that he would lose many men on
his way south, and yet there lay no choice before him. He had scotched
the snake, and now he must kill it. If the Dark Master reached Galway
town in safety, those O'Donnells from Millhaven would be around by sea
to meet him, and the royalists would lend him men and guns to go against
Bertragh in their cause.
"Is there any likelihood that the Dark Master will miss those scattered
men of his?" he asked Turlough, who rode on his right hand.
"Little, master. There is but the one road south to Sligo at this
season, and it is great wonder indeed that the scattered men did not
fall on us at the Black Tarn in seeking their master. But with only
seventy-five men or so I do not think they will bide our coming."
"Nor do I," and Brian laughed grimly as he thought of that fight with
his enemy.
Certain men had been wounded in those frays, and he left them to follow
after him, so that he turned south with a hundred and a score men at his
back. He did not think that the Dark Master would face him, but since
those men were all O'Donnells who would obey him utterly, he looked to
have some fighting; in which he was not far wrong.
An hour after the day was broken they thundered up to the bridge that
spanned the Garravogue, and ten wild and silent men were holding that
bridge behind an overturned cart for barricade. Brian would waste no men
on a storm, but slew six of the men with musketry and rode over the
other four; even so, those four brought down three of his men before
they were done with.
Brian baited the horses in Sligo, remaining there a scant half-hour.
From the townfolk he learned that the Dark Master was but two hours
ahead of him, and Brian had great hopes of running him to earth that
same day. So he set forth again and they rode hard to Ballsadare, at the
south branch of Sligo Bay, and on to Coolany at the edge of the Storm
Mountains.
At this latter place they found different work, however, for here was a
small garrison of Cavan pikemen who stopped them, lined with their pikes
three deep across the road before the church. Brian was no long time in
learning that the Dark Master had spread word of him as a plunderer and
Parliament man.
"I have no time to waste on you," he said shortly to the leader of the
pikemen. "Here is a safe-conduct, and I am Stephen Burke."
"None the less, you must stay until I have looked into this," said the
other, pulling out his pistol with some determination.
"Stay I will not, but I think you shall," replied Brian, and thrust as
the man fired. The bullet glanced from his jack, but the officer fell
back among his pikes, and Brian spurred after him in great anger. His
Scots troopers were in the van, or what was left of them, and they came
down galloping, and rode over the pikemen leaving a sea of smitten men
in the roadway behind.
Also, ten of Brian's men were left.
By the evening they were back at Tobercurry again, where Turlough had
hung those two men after torturing them. The Dark Master was something
over an hour ahead of them, and he had stayed to fire the church and the
town. Brian's heart was sore for the townfolk, but he could pause no
longer than to bait horses and men, since he looked for hard riding that
night; however, he gave what money and plunder he had to the townfolk
and got a blessing in return, and so rode forth again as the stars
peeped out.
"There are Maguires in Swineford, master," said old Turlough with a
cunning, sidelong look.
"I met them coming north," laughed Brian softly. "They will prove good
men to avoid, so I think that we shall ride around that burg."
Brian thought that he could get through the Maguires, but he intended to
take no chances. However, they had gained to within five miles of
Swineford and had halted to blow the horses, when one of the scouts came
riding back to say that a score of farmers with three carts were
approaching from the town.
Presently they came on them--a black mass swinging down the road, which
was very boggy on either hand. Neither Brian nor Turlough smelt any ill
in this until they were within a hundred paces of the party, when
suddenly the carts were swung across the road and a score of muskets
spat death into Brian's men.
"Back!" shouted Brian, when his men would have charged. "We have no time
and lives to waste on this party--what shall we do, Turlough? The fields
are all bog."
"We cannot well ride around," said Turlough, when they had ridden back a
little, leaving dead men on the road. "But a little way back is a path
that leads out and around Swineford. Put ten men here to keep these
O'Donnells from following us, and we will make a short cut to the Moy
near Kiltanmugh. It was a clever trick, this!"
It was indeed, and it had cost Brian a round score of men, so that he
followed Turlough out into the open land with less than a hundred men
behind him. His fury abated before dawn, when they had splashed across
the Moy and came upon the road once more, but he saw that the O'Donnells
were willing enough to die if the Dark Master might escape, and he
became more cautious.
When the night fell again they were far south of Claremorris, but a
score of horses had foundered and he was forced to leave more men
behind. Until evening Turlough led him at a distance from the main
roads, then they struck into good riding again and save for one detour
to avoid Tuam would have a clear road between themselves and Galway,
which Brian meant to reach before dawn unless his own horse foundered
with the rest.
Of the Dark Master they heard nothing until they were fording the Clare
north of Tuam, when two men gave them word that a scant half-hour before
some two-score horsemen had fled past them toward Tuam.
"Good!" cried Brian. "Now, Turlough, lead us around Tuam, and I think we
shall finish this thing long before the day comes."
Said Turlough sourly, "Every horse down is a man gone, master," but to
that Brian only laughed and set in his spurs.
So now they let gallop through the darkness, trusting more to Turlough's
wits than to their horses' feet; for Brian knew that if his own beasts
were spent, those of the Dark Master were no better unless he were to
get mounts at Tuam. That would be hard, however, for there were no
horses to be had save far in the mountains where the war had not swept
all things away.
No sooner had they reached the road again beyond Tuam than it seemed to
Brian that he heard the faint drum of hoofs ahead of him, and at that he
gave a shout and drove on with such of his men storming behind as might
come. Many of them had gone down, indeed, but now all wakened from their
nodding sleep and kept close, though here and there one dropped out.
Turlough, whose steed had been the best of all save Brian's, kept at his
master's flank.
They were hard on Claregalway when Brian saw his quarry first--a deep
mass of men far ahead on an open stretch of road. Then he knew that the
race was nearly won, and for all that his beast was sobbing under his
thighs, he raced ahead, and laughed out loud when a little band cut off
from the main body of the Dark Master's men. There were fifteen or less
who waited his coming with pistols ready, but Brian rode hardily at
them, their balls whistled overhead or past, and he was on them.
The shock of the meeting came near to unseating him, and sent one of the
foe sprawling, horse and man; Brian cut another to the chin and thrust
the life from a third, and before the first sword had slithered on his
steel-cap his men had swept aside the devoted fifteen, and he was riding
on. O'Donnell had straightened his party for nothing.
Now the Dark Master was riding for his life, and knew it. Some few of
his men fell out with spent beasts, and these Brian's party rode over,
taking and giving but one blow, or none at all. When Claregalway drew up
ahead, cold and gray under the stars, Brian was but two hundred yards
behind with forty men still behind him, while O'Donnell had not half so
many.
As he thundered down to the river Brian had drawn as much ahead of
Turlough and the others as he was behind the Dark Master. He shouted
back to those of his men whose matches were lit to loose off their
muskets, but before the first pan had flashed out he saw the O'Donnells
draw rein and wheel at the bridge-head, while two of their number drove
clattering on into the town.
Now, had Brian chosen to wait for his men things would have fallen out
differently; but this he would not do, for he thought to break through
these as he had done with the others. So he went at them with naked
sword, his heart raging within him and his face set and cold like stone.
He was still fifty paces from the bridge-head when their pistols
spattered out; the men behind dared not fire for fear of hitting him, so
that Brian had all the fight for himself.
He came near to having none, for at that first discharge a pistol-ball
split his jack and lodged in his buff-coat over his heart, while another
came between his arm and his side, drawing blood a little from both;
while a third and worse went into his horse between the fore shoulders.
Brian felt the poor beast falter shudderingly, and pause; then the
O'Donnells shouted greatly and closed about him, thinking to slay him
before his men could come up.
Brian saw a long _skean_ plunge into his horse's neck, and in terrible
anger he smote with the edge, so that a hand and arm hung down from the
dagger, a ghastly thing to see. But the poor steed was dead with that
blow, and Brian had but time to fling himself headlong ere the horse
rolled over.
The leap saved his life, for the O'Donnells were striking fast at him.
Brian rose up between two of them, dragged one down with his left hand
and thrust the other under the arm, and tried to leap up into the
saddle. But as he did so his own men struck, so that the horses were
swept together and pinned Brian's legs between them, and he hung
helpless.
In that instant he saw an ax swinging above him and flung back his head,
but not enough, for the ax fell, and Brian went down under the horses.
Save for three of his men who saw the thing and stood over him, Brian
would have been trampled to death on the spot. These O'Donnells were no
loose fighting-men, and they smote shrewdly against the press of
Brian's greater numbers, while their wild cry rose high over the shrill
of steel. When Brian's men knew that he was down, however, they struck
such blows as they knew not they had in them, and quarter was not asked
or offered in that battle by the bridge.
The fight was not ended until the last O'Donnell went down in a swirl
and clash of steel. Then Turlough, who had kept well out of it according
to his wont, pushed through and fell upon Brian's body. When Brian
opened his eyes his head was still ringing, while his men were bathing
him with water. After an instant he sat up and gazed around.
"The Dark Master--did you catch him?"
"Nay, our thought was all for you, master," answered Turlough.
Brian groaned in great bitterness, but said no word. He knew that his
chance was gone from him for that time, and as he looked around his
heart sank within him. Half of his men had slipped down and lay sleeping
among the dead, and the rest could scarce stay in their saddles for
weariness and lack of sleep. But Turlough sprang up and gazed at the
graying sky with fear in his face.
"Up, master!" he cried fiercely. "We must still ride hard, for the Dark
Master will send out a troop of horse from Galway to catch us, and we
must get past that town before the sun is high!"
So the sleeping were roused in haste, the wounded were put in saddle,
and with their beasts staggering under them, those that were left of
Brian's men closed around him and rode over the bridge through
Claregalway.
CHAPTER XVII.
BRIAN GOES A CRUISING.
Above the head of Bertraghboy Bay there was a swooping curve in the hill
road. It was at this same curve that Brian Buidh had first met the Dark
Master, and it was here he had set that trap which had won him tribute
for the Bird Daughter. When first he had ridden that road Brian had had
a score of lusty men at his back; on the second occasion he had headed a
hundred and four-score; but when he drew rein there a week after that
fight at Claregalway bridge there was with him only old Turlough Wolf,
and their horses were sorry skeletons like themselves.
"We are somewhat worse than when we twain started out together," laughed
Brian bitterly. "Then we had full bellies at the least, but now we have
naught."
"There are men coming, master," said Turlough, hanging weakly to his
saddle. "I think they are our castle watchers."
Very gaunt was Brian that day, and nigh spent with his wounds and hunger
and weariness. During the week that had passed since the Dark Master
slipped away from him, nothing but evil had come upon him.
First they had tried to slip past to the north of the city, and had
reached the Lough Corrib River, and could even faintly hear the bells of
St. Nicholas below, when a half-troop of horse fell upon them. Then in
desperation Brian's men smote for the last time, and put the royalists
to flight; but there Brian lost the most of his men. However, he got
fresh horses, and so fled eastward again when more men were seen
approaching.
What chanced in the six days following is not fully set forth, for Brian
got little glory from it. One by one he lost his men, and at length was
forced north again to the shores of Lough Corrib, with men riding hot
and fast to catch him. With Turlough Wolf alone left to him, he had made
shift to cross the lake in a leaky fisherman's boat, the horses swimming
behind, and so came into the O'Flahertys' country.
There word had also gone forth against him, but because of the pact
between them, Murrough of the Kine sped him in peace through Iar
Connaught, and at length Brian had won home again with joyless heart.
As Turlough said, men were coming, and they were Brian's own men who
watched the roads. From them he got food and wine and two fresh horses,
and with the afternoon they rode down to Bertragh in worse shape than
they had ridden from it. Brian was the less heartened when he saw two of
Nuala O'Malley's ships in the bay, and knew that she must be at the
castle.
Indeed, before they reached the gates the Bird Daughter rode out to meet
them, with Cathbarr striding before her. When the woman saw Brian's face
her violet eyes filled with tears, and when he dismounted and kissed her
hand and would have spoken, she stayed him.
"Nay, we know enough of the story for now, Brian. First rest and eat,
then talk."
Brian guessed straightway that pigeons had come from her men in Galway
telling of those ridings about the city, and that she had come over to
Bertragh in anxiety; and this was the truth indeed.
Turlough Wolf hied him away and slept, but Brian sat about a table in
the hall with Cathbarr and Nuala. He was very worn and weary, but when
he had eaten and drunk he refused to sleep yet a while, and told how
that storm had fared north and what had come of it.
"So I have lost a hundred and fifty hard-won men," he concluded
gloomily. "I would not grudge them if the Dark Master had fallen, but he
is in Galway, and the Millhaven pirates will be down to meet him, and
that means war on Bertragh."
"I will be glad of that," said Cathbarr simply. "I am sound again and
have been sharpening up this ax of mine."
Nuala smiled and put her hand across the table to lay it on Brian's.
"Success would be of little worth, Yellow Brian," she said softly, and
her eyes steadied him, "if it were won without reverses. Few men have
the luck to win always, and a touch of defeat is not an ill thing,
perhaps. When we had this news of you from Galway, a week since, I sent
off a galley to find Blake at the Cove of Cork and seek aid of him. Also
my kinsmen will return to Gorumna before going home to Erris, and we are
not in hard case here. So now get rested, Brian Buidh, and afterward we
will see what may be done. Those Millhaven men have not yet passed
Erris, or I would have word of it by pigeon, so they have doubtless
delayed to plunder in Sligo or Killala."
Brian looked into her eyes, and from that moment he began to put behind
him all thoughts of capturing that Millhaven castle for himself or of
placing himself out of touch with Nuala O'Malley. He went to his chamber
as she bade, and slept that night and the next day and the night after,
waking on the second morning still empty of sleep and seeming more weary
than when he had laid down.
This was but seeming, however, and when he had bathed and eaten he felt
more like himself than for many a day.
Cathbarr had departed at dawn with a wagon-load of powder to trade for
kine with his O'Flaherty kinsmen in the hills, and before Brian had
broken his fast one of the galleys from Gorumna came over with three
pigeons for Nuala. The cage was brought to her as she sat at meat with
Brian in the hall, and she opened the tiny messages with all the
delighted anticipation of a girl.
"This is from that galley I sent to Cork," she exclaimed, laying down
the first. "It merely reports safe arrival and the delivery of my letter
to Blake, who is leaving there before long. Now for the--ah!"
"Good news or bad?" smiled Brian easily, as animation flashed into her
face. She looked up at him with a rippling laugh.
"Both, Brian! This is from Erris, and says that the O'Donnell seamen
have made a landing at Ballycastle under Downpatrick Head, and will
likely put to sea again in a day or two. They will give Erris a wide
berth, never fear, and that means that they will make no pause until
they come to Galway."
The third message was from Galway itself, and said that the Dark Master
was biding the coming of those Millhaven men, and had been promised both
horsemen and shot if they came, so that Bertragh might be taken and held
for Ireland against the Parliament.
"It is not taken yet," laughed Nuala as old Turlough came shuffling up,
and they gave him the sele of the day merrily enough. "You had best keep
these birds, Brian, so that if there is any need you may send me
messages to Gorumna. Now, shall we bide here until the Dark Master comes
against us?"
"I thought you were going to take me cruising with you?" smiled Brian,
but at that Turlough struck in and asked what the messages were. When he
had heard them he stood pulling at his gray beard for a little, then
turned to Brian.
"How is your body, master?"
"Well enough," said Brian, feeling his head. "Save for this beard, which
now I may not cut for a time."
He intended to abide by that oath of his, and so his beard was growing
out and his hair as well, of which latter he was glad.
Since he had ever kept his face clean shaven, however, the beard was not
to his liking. He was quite unaware that it built out his face greatly
and made him grimmer-looking than before, and yet so young were his blue
eyes except when he was in anger that it was not hard for Nuala to
believe that he was only two years older than herself.
None the less, she made great sport of his beard, saying that it curled
at the end like a drake's tail, as indeed it did; and as Brian only
repaid her laughter with the open wonder and admiration that he held for
her, there was great good-comradeship between them.
"There is still one chance for stopping the Dark Master," said Turlough
thoughtfully. "If we cut off those pirate ships on their way south he is
not like to get much help from Galway."
"Oh--and I never thought of it!" cried Nuala, staring at him.
Turlough chuckled. "That was spoken like a woman, mistress! If the rede
seems good we could lay aboard men from here for fighting, and sail out
with those two ships of yours."
Now Brian's heart filled with new hope, and after no long discussion
they decided to adopt the plan. Nuala was of the opinion that a short
cruise would do Brian great good, so they decided to set off that
evening in her two ships, leaving Turlough to keep the castle against
Cathbarr's return.
Had they taken Turlough Wolf with them or had Brian been less
close-mouthed on his return from that cruise, the evil that befell might
have been averted. The old man was cunning and swift at piercing beneath
the craft of other men and turning it back upon themselves; but as
Brian's mind lost its bitterness at his own failure it gained joy at
being with the Bird Daughter, while Nuala had no less friendship and
liking for him, so that neither of them gave much thought to O'Donnell
Dubh who lay in Galway and bided his time after his own fashion.
Once having reached their decision, they hastened it somewhat and sent
men and muskets aboard the two ships at noon. Nuala wished to sail first
to Gorumna Castle and make all safe there, then reach back for Slyne
Head. She proposed that Brian take one carack and she the other, but at
this Brian laughed.
"No, lady--I am no seaman, and I am your guest on this cruise, so I go
with you."
"Well, you shall have good guesting," she answered, flushing a little,
but her eyes not flinching from his, and so they went aboard her ship
together.
Having two hundred men still, Brian had put fifty on each ship in case
they met with those pirates, who were like to give good battle. Also
Turlough had hopes that many of Brian's men would win home from that
riding of his yet, since a large part of them had dropped out by the way
or had been left behind with wounds. And in the end, indeed, fifty or
less did find their way back.
Before night they made Gorumna Castle, and Brian found why they had come
here first. With her Kerry recruits, Nuala had a hundred and eighty men,
so she had set to work to build a tower and small keep on the opposite
island, that Gorumna itself might be more easily defended. Also she had
taken some falconets and two bastards out of a large French ship, and
had set about building a battery outside the castle that would overlook
the harbor.
"That will be better than good when it is done," said Brian approvingly.
"But you had best get it done speedily. When we come back from this
cruise you shall take this hundred men of mine, for I will not need them
until the Dark Master comes, and of that we shall have good warning."
This she was glad of, and she was glad because Brian had found her work
well planned; nor did either of them suspect what grief that loan of a
hundred men was to bring upon Brian.
They paused only to sup at Gorumna, then set forth again, and by dawn
were off Slyne Head with a light breeze behind them. Nuala would take no
chance of missing those Millhaven men, so instead of going north among
the islands she turned her ships and beat off Slyne all that day, seeing
no sail save fishing-craft.
Those were pleasant hours for Brian, for the sea was fair and he had
naught to do but sit with the Bird Daughter. He found himself drawn ever
closer to her, admiring her wit and fairness as he did, and he fancied
that she was by no means unwilling to talk with him and open her mind
as she did to few men. Yet he remembered that he was no more than her
vassal, a landless man in truth.
That night the two caracks separated, standing well off the land and
keeping good watch, but no sign did they catch of the O'Donnell pirates.
Toward morning a stiff wind came upon them from the west, and Brian's
men, being all landsmen, got no great joy out of that cruise.
"This wind is like to hold," said Nuala, laughing as she stood on the
poop with Brian that morning and watched the decks. "I am afraid that we
might as well give over this attempt, Brian. Your men will be in no
shape to fight. What think you?"
"Right," nodded Brian slowly, for he saw that those men of his were
worse than useless with their sickness.
So they turned about and drove before the wind, but before ever they had
got past Slyne Head the men aloft descried a sail to the south that
seemed like a large galley. Nuala signaled the other carack to bear down
with her, and presently they made out that it was a large sailing
galley, which headed straight for them.
"That is none of my ships," exclaimed Nuala, watching. "It seems strange
that she does not flee before us, Brian. She bears no ensign, yet she
must be from these parts, and would naturally have some fear of
pirates."
Brian looked at her rather than the ship, and thought her a fine
picture, with her body swinging a little to the sway of the deck and the
wind blowing her red cloak around her. The galley came straight for them
as if seeking speech, however, and when a falconet was fired from the
carack without charge, she lowered her sail and put out her sweeps,
coming straight for them.
Nuala sped a word to her sailing-master, and the men let down the sails
with shouting and great creaking of ropes. The Bird Daughter stood under
the high poop bulwark, and now she turned to Brian.
"Do you speak with them and find their business, for it seems to me that
all is not as it should be, and they would likely know me too well."
Brian nodded, and when the galley had come under their lee he saw that
she was well laden, and had for crew a dozen rough-looking men. One of
these replied to his hail.
"We are come from Galway, lord, with a gift of stores and wines from
O'Donnell Dubh to certain friends of his whom we came to meet. Are you
those friends, as we think?"
Brian started in surprise, but needed no word from Nuala. He saw that
the Dark Master must have sent this galley out to meet the Millhaven
men, and that the crew had taken the two caracks for those pirate ships.
"We are the O'Donnells from Millhaven," he shouted, and ordered the
seaman to cast down ropes to the galley. Her master, a stout man with
bushy black beard, waved a hand in reply, and after another moment the
two craft ground together. The master of the galley got aboard over the
low waist of the carack, and Brian ordered a dozen of his own
green-faced men down into the smaller ship. At this the galley's master
stared somewhat, but came up to the poop.
"Lord, O'Donnell sends you these stores with a message. I am Con Teague
of Galway."
"Let us have it," ordered Brian, liking the looks of the man not at all.
"He bade us say that he was leaving Galway to-morrow at dawn with a
force of men, and that you should meet him at Bertragh Castle and fall
on that place to take it."
"That is good," laughed Brian. "Now learn that you have found the wrong
ships, my man. We are not the Millhaven pirates, but I am Brian Buidh,
who holds Bertragh; and here is the Lady Nuala, for whom I hold it."
At that Nuala came forward, and Teague looked greatly astonished, as
well he might, and all the Bird Daughter's men fell roaring with
laughter. But he could make no resistance, and stood chapfallen while
Brian talked with Nuala.
"I must back to the Castle," he said, "and see if this news be true. Do
you go on to Gorumna with my men, and I will let loose a pigeon to you.
If the Dark Master is indeed on the way, then come with all the men you
can spare, and it will go hard if we do not best his royalists, and the
pirates later when the latter come."
This was clearly the best plan, so Brian sent Teague down into the
galley and followed him, as the light ship was faster than the caracks.
Replacing half of Teague's men with O'Malleys, he had the ropes cast
off, waved his hand at Nuala, and they drove to the eastward and
Bertragh Castle.
Teague made so much moan over losing his ship that Brian promised it
back to him when they had reached the castle; the stores and wine,
however, he accounted good spoils of war. This put the seaman in better
mood, and by noon the fast galley had covered the twenty miles to
Bertragh, and cast down her anchor in the little bay beyond the castle,
that same bay where Brian had come to grief through O'Donnell's sorcery.
The men crowded down to meet him joyfully, and Brian found that Cathbarr
had come home safe with his beeves and was hungry for fight. No sign had
been heard of the Dark Master along the roads, however, so Brian set
Turlough in charge of getting the stores and wine-casks off the galley,
and fell to work putting the castle in shape for defense.
Since there was no need of loosing a pigeon until word came that the
Dark Master was actually on the way, he sent out men to have a beacon
built on the hills at the bay's head as soon as the enemy was sighted.
What with seeing that the bastards and other shot were cleaned and
loaded, and stationing his hundred men to the best advantage, he found
that the afternoon soon wore away.
"Those are good wines," said Turlough when they sat at meat that
evening, the men eating below in the courtyard around fires. "But I do
not like that ship-master."
So far Brian had said nothing of how the galley had been taken, save
that they had chanced on it at sea and had heard from Teague that the
Dark Master might be on them in another day. As for the O'Malleys, they
kept to themselves and talked not at all, so that neither Turlough nor
Cathbarr had heard the way of that capture.
"Is she unladen?" asked Brian.
"All save a few barrels. That ship-master was so eager to be off,"
grunted old Turlough spitefully, "that I stayed the work and put a guard
on the galley until morning."
"Give the men a cask of the best wine," ordered Brian shortly.
Having taken upon himself the duties of seneschal, Turlough departed
grumbling. While he was gone, Brian's tongue was a little loosened with
wine, so that he told Cathbarr of how he had taken the galley, at which
the giant bellowed with laughter. Presently from the courtyard came
shouting and singing, and Turlough appeared with a beaker of wine.
"The men like it well enough," he said, "yet to me it seems soured.
Taste it, Brian; if it be so, then you have made a poor haul on that
cruise."
Brian sipped the wine, and in truth it seemed to have soured. Cathbarr
made little of that, and would have drunken it except that his clumsy
hand knocked it from the table and emptied it all. But as it happened,
that mischance saved his life.
A little after, Brian pulled out a Spanish pipe he had got that day from
one of the O'Malleys, with some tobacco, and began puffing in great
good-humor, for it was long since he had tasted tobacco. Cathbarr
watched in awe, never having seen this done before, so that Brian and
Turlough had great fun with him. All his life the giant had lived in the
mountains and he knew no more than his ax had taught him; though he had
seen men smoke before, he had ever accounted it sorcery of some kind,
nor could Brian get him to as much as touch the pipe with his finger.
Brian was sorry that the wine had proved sour; the butts were huge ones,
and he had counted on their lasting him and his men all the winter
through. However, he dismissed the matter from his mind and fell to
talking with Turlough and Cathbarr over their arrangements in case of an
attack. In the midst, one of the men who had been watching from the
tower ran in to say that he had caught sight of a beacon on the hills,
which meant that the arch-enemy was on the road.
"Good!" exclaimed Brian, springing up. "Turlough, go fetch me that cage
of pigeons. Cathbarr, see that the men are set on the walls--"
He had got no further than this when there came a strange noise from the
doorway. Turning, he saw a man staggering forward, choking as he came,
and recognized him as one of the Bird Daughter's seamen. The fellow held
a bloody sword in his hand.
"What's this?" cried Brian angrily, noting that there was silence upon
the court-yard. "Has there been wrangling again--"
"Death!" coughed the O'Malley, staring at him with starting, terrible
eyes. "Con Teague--I slew him--too--too late--"
"Man, what is forward?" Brian leaped out and caught the seaman in his
arms, for the fellow's head was rolling on his shoulders.
"Death!" whispered the man again. "They are--all dead--"
His head fell back in death, and the sword fell from his hand with a
clatter. But from Cathbarr, who had gone to the doorway, came one
terrible shout of grief and rage.
"Brian! Our men lie dead--"
"I think the Dark Master has sent us a kindly gift," quoth Turlough
Wolf, as Brian rose with horror in his face and let the seaman's body
fall. "Now I know why that wine was sour, master!"
TO BE CONCLUDED NEXT WEEK. Don't forget this magazine is issued weekly,
and that you will get the conclusion of this story without waiting a
month.
Nuala O'Malley
by H. Bedford-Jones
Author of "Malay Gold," "The Ghost Hill," "John Solomon, Supercargo,"
etc.
This story began in the All-Story Weekly for December 30.
CHAPTER XVIII.
BRIAN YIELDS BERTRAGH.
"I dare not trust birds alone in this strait, Cathbarr. Go to that
galley with the two O'Malleys and hasten to Gorumna. Bid the Bird
Daughter stay and wait further word from me; but take those hundred men
of mine with her galleys, and hasten back. If the beacon on the tower is
burning, I will be here; if not, and if I can make terms, I will meet
you at that tower of yours. Now hasten!"
"But--"
"For God's love go, or my heart will burst!"
Brian sank down on the horse-stone with a groan, and Cathbarr, catching
up his ax, fled through the open gates and was gone into the night.
Brian gazed up after him, and on the hills he saw that dim beacon-fire
heralding the Dark Master.
The six men guarding the galley, two of them being O'Malleys, and three
men who had watched on the tower, were all that remained alive in
Bertragh besides Turlough and Brian. The men had drunk deep of that
poisoned wine; when Con Teague and his men tried to get away after a few
had died, they were slain. But so swift was the poison that only one of
the O'Malleys had lived to reach Brian.
The fires still burned brightly, and before some of them meat was
burning. Sitting in blank despair on a horse-block, Brian saw the dead
bodies of a few less than a hundred men lying there. Turlough Wolf and
his six gave over trying to put life into any of them, and now the old
man came and put his hand on Brian's shoulder.
"Where has Cathbarr of the Ax gone, master?"
Brian told him dully, and Turlough nodded approval, having at length
learned all the story of how that galley had been taken.
"Master, there was deep cunning in this. O'Donnell sent that galley to
you, or, rather, to the Bird Daughter, and he had spies watching. Had
the Gorumna men drunk of that brew, he would have fallen on there; but
here came the galley, and now he comes over the hills. And we are few to
meet him."
"We will be more when the men come in from the hill-roads before him,"
and Brian rose up with heavy heart, forcing himself to the task. "Send
out a man to haste them in and to warn what men there be at the farms.
Also let him send a wagon or two, that these dead may be carried out
before the Dark Master falls on us. Send two men to the tower to build a
beacon, for Cathbarr will not be back before to-morrow night."
Brian went to the stables where the three carrier-pigeons were caged,
and fetched the cage to the great hall. Here he wrote what had happened,
with his plan, in small space, fastened it under the wing of a bird, and
let loose the pigeon from the courtyard.
Stunned though he was by the sudden and terrible blow, Brian had seized
on the only course left him. If he could make shift to hold the castle
at all, he would do so; if not, he must make terms and get off to
Gorumna that he might take vengeance for this dastardly stroke that had
been dealt him.
Nuala had nigh three hundred men in her castle, and he felt that all was
not yet lost, even should he have to yield Bertragh. The Dark Master
would hardly have a large force with him, and he would know nothing of
those hundred men Brian had loaned Nuala; so Brian reckoned that if he
could get away, O'Donnell would think him a broken man who could do no
further against him.
"Well, that's looking too far ahead," thought Brian very wearily.
"Perchance I am broken, indeed, since I have lost two hundred and a half
of men without gain."
An hour later rode in a score of men with wagons, and fell to work
getting the dead out of the castle, though for burying there was no
time. This score, and two more who came in later, were all the men left
to Brian; they reported that the Dark Master would be on them by
daybreak, with two hundred Scots troopers and one horse cannon.
"His friends proved niggardly, then," laughed Brian drearily. "We have
but to hold the place till to-morrow night, friends, and the O'Malleys
will relieve us. Now, one man to watch and the rest of us to rest, for
there is work ahead."
Brian, indeed, got some sleep that night, but it was shot through with
visions of those poisoned men of his, and their twisted faces gibbered
at him, and he thought they shrieked and howled for revenge. When he was
roused at dawn, he found the meaning of those noises, since a great
storm was sweeping down out of the west, and the farther wore the day,
the worse grew the storm.
"Is Heaven itself fighting against us?" he thought bitterly, watching
the sea from the battlements. "Against this blast Nuala cannot reach me,
if she will."
He got little time to brood, however. Before he had broken his fast the
Dark Master's horsemen came in sight--two hundred braw Scots, with
wagons and a cannon following after. It was no large force, but Brian
found afterward that it was the best the Dark Master could get, since
the Galway Irish cared nothing whether the Scots lived or died.
They halted and spread out, half a mile from the castle, and Brian saw
that the men were being quartered on the farms round about. Bitterly he
wished that he had his lost men, for with them he could have sent those
Scots flying home again; but now he was helpless.
With the gates shut and the bastards loaded with bullets to sweep the
approach, Brian sent his twenty men to the battlements and watched, with
Turlough beside him. It was plain that no offensive operations were
under way as yet, and an hour passed quietly; then ten men rode down to
the castle under a white flag, and foremost of them was the Dark Master.
"Now, if I were in your place, master," said Turlough, slanting his eyes
up at Brian in his shrewd way, "I would loose those bastards and sweep
the road bare."
"You are not in my place," said Brian, and the Wolf held his peace.
The Dark Master looked at those bodies piled between the castle and the
shore, and it was easy to see that he was laughing and pointing them out
to the Scots. At that Brian heard his men mutter no little, and he
himself clenched his nails into his palms and cursed bitterly; but he
forbade his men to fire and they durst not disobey him. The party rode
up under the walls, and the Dark Master grinned at Brian standing above.
"You have great drunkards, Yellow Brian," he called mockingly. "Have all
your men drunk themselves to death?"
Brian answered him not, but fingered his hilt; even at that distance the
Dark Master seemed to feel the icy blue eyes upon him, for his leer
vanished.
"Yield to us, Yellow Brian," he continued, shooting up his head from
betwixt his shoulders. "I do not think you have many men in that
castle."
"I have enough to hold you till more come," answered Brian.
"Mayhap, and mayhap not," and O'Donnell laughed again. "Keep a watch to
seaward, Yellow Brian, and when you see four sail turning the headland,
judge if those two caracks of the Bird Daughter's are like to help you."
"If you have no more to say, get you gone," said Brian, feeling the
anger in him rising beyond endurance. The Dark Master looked along the
walls for a moment, then signed to his men, and they rode off through
the driving snow again.
Turlough looked at Brian and Brian at him, and the same thought was in
the minds of both. If those Millhaven men had four ships driving down
before that storm, as seemed probable enough, the Bird Daughter's two
little caracks would never land men under the guns of Bertragh.
About noon the snow fell less thickly, though the storm had risen to
great power, and Brian made out that the Scots were bringing forward
that cannon of theirs. Having some little knowledge of artillery
himself, he drew the charge of bullets from a bastard and put in more
powder, then put the bullets back, a full bag of them. He did the same
with two more of the bastards on that wall, and when the Scots had
halted aimed all three very carefully, and set men by them to fire at
his order. The Scots were turning their cannon about, a score of men
being in their party, and Brian judged that they were eight hundred
paces away--just within range of his bastards.
"The Dark Master lost this hold because he had too many men," he said to
Turlough, "and we shall lose it because we have too few; but we will
make better use of these shot than did he. Fire, men!"
The three men brought down their linstocks and ran for it, having seen
that extra charge of powder set in the cannon. But none of the pieces
burst, though they roared loud enough and leaped at their recoil-ropes
like mad things. When the white smoke shredded down the wind, Brian's
men yelled in great delight, for those Scots and horses about the cannon
were stricken down or fleeing, and the piece had not yet been loaded.
"They will get little joy of that cannon," said Brian grimly, and went
in to meat.
During the rest of the day the cannon stood there silent, dead horses
and men around it; nor was any further attack made. Brian knew well that
having found him prepared, the Dark Master would now attack at night and
hard did Brian pray that the storm might abate from the west, or at
least shift around, so that Nuala's ships could come to his aid.
Instead, the gale only swooped down the wilder, and seemed like to hold
a day or more, as indeed it did. About mid-afternoon Turlough came and
beckoned him silently out to the rear or seaward battlement and pointed
out.
No words passed between the two men, nor were any needed; beating around
the southern headland were four flecks of white that Brian knew for
ships coming from the west with the storm, and he saw that for once the
Dark Master had told the truth.
"I have some skill at war," he said to Turlough that afternoon when they
had seen the four ships weather past them and anchor a mile up the bay;
"and since the Dark Master's troopers are also skilled at that game,
they will fall to work without waste of time or men. We may look to have
the dry moat filled with fascines to-night and our gates blown in with
petards. At the worst, we can hold that tower, where the powder is
stored."
If he had had more men, Brian would have slung the bastards down from
the high walls and set them in the courtyard where they could sweep the
gates when these had been blown in. But they weighed a ton and half
each, and there was no time to build shears to let them down, even had
they had spars and ropes at hand. So Brian set them to cover the
approach, and had the smaller falcons brought down to the courtyard, all
five, where he trained them on the gates and loaded them with bullets
heavily.
"Turlough and I will fire these ourselves," he told his men that evening
as they made supper together, the men looking forward to the night's
work with great joy. "Do the rest of you gather on either hand by the
stables, with spare muskets and pistols."
So this was done as he said. Because of the storm Brian did not light
his beacon after all, but he stocked the tower with food and wine, and
told his men to get there, if they could, when the rest was taken. That
tower had Brian's chamber in the lower part and a ladder in the upper
part, where was great store of powder.
The five falcons were set in front of the hall doorway, where once Brian
had come near to being nailed. Brian loosed another of the pigeons,
telling Nuala how things chanced, and of the four pirate ships, and set
the last bird in the tower in case of need, which proved a lucky thing
for him in the end.
Brian and his men slept after meat, while Turlough Wolf remained
watching. It was wearing well on to midnight when the old man woke them
all, and Brian went to the walls to hear a thud of hoofs and a murmur of
men coming across the wind to him. He sent off men to loose the loaded
guns on the outer walls at random, and then suddenly flung lighted
cressets over the gates.
A wild yell answered this, and bullets from the men who were filling the
dry moat, while others scrambled across it and charged up to the gates
with small powder-kegs and petards ready. This was not done without
scathe, however; Brian's men loosed their muskets, and one by one the
heavy bastards thundered out across the snow, though the result was hard
to see in the darkness.
There came a ragged flash of musketry in reply, and that abandoned
cannon roared out lustily, though its ball passed far overhead. Brian
stood on a demi-bastion that half flanked the gates, and after firing
his pistol into the men below, he leaped down the steps into the
courtyard and joined Turlough behind the falcons.
"One at a time, Turlough. They'll have the gates down in a minute."
While he waited for the storm to fall, Brian saw that two or three of
his men had been hit. He wondered dully that the Dark Master had not
made a general assault, and concluded that he must wish to save men. It
was a long moment that dragged down on him; then a splash of light burst
up, the gates were driven inward and shattered, and with a great roar
there fell a rain of riven beams and stones and dirt.
Sheltering in the hall doorway, Brian and Turlough stayed unmoving
through an instant of black silence. Out of it broke a wild Scots yell,
and in the light of the courtyard cressets a wave of men surged up in
the breach. Brian's linstock fell on a falcon, and the little gun barked
a hail of bullets across the Scots; Turlough's gun followed suit, and
the first lines of men went down in a struggling mass.
The Dark Master was not to be beaten this time, however. Another wave of
Scots swept up, with a mass of men behind them. While some of Brian's
men tried to get the two falcons reloaded, a storm of bullets swept
across the courtyard, and Brian saw Turlough turn and run for it through
the doorway, while two of the men fell over a falcon.
But as the first line of men broke into the courtyard, Brian fired the
remaining three cannon as fast as he could touch linstock to powder. The
bullet-hail tore the front ranks to shreds, but through the darkling
smoke-cloud he saw other men come leaping, and knew that the game was
up.
On the next instant his men had closed around him, muskets were stabbing
the powder-smoke, and Brian fell to work with his Spanish blade.
O'Donnells and Scots together heaved up against them, but Brian's point
weaved out between cutlas and claymore and bit out men's lives until
the mass of men surged back again like the backleash of a wave that
comes against a wall.
Brian heard the Dark Master's voice from somewhere, and with that
muskets spat from the gloom and bullets thudded around him. One slapped
his steel cap away and another nicked his ear, and a third came so close
across his eyes that he felt the hot breath of it; but his men fared in
worse case than that, for they were clutching and reeling and fallen,
and Brian leaped across the last of them into the hall with bullets
driving at his back-piece.
As he ran through the hall he knew that his falcons had punished
O'Donnell's men heavily, and that his twenty men had not fallen without
some payment for their lives. None the less, Bertragh Castle was now
lost to him and to the Bird Daughter; but he thought it likely that he
would yet make a play that might nip O'Donnell in the midst of his
success.
In this Brian was a true O'Neill and the true luck of the Red Hand had
seemed to dog him, for he had lost all his men without suffering a
defeat, and now that he was beaten down, he was planning to strike
heaviest.
He gained the tower well enough, and found Turlough there to receive
him, with food and wine and loaded pistols. They soon had the door of
the lower chamber fast barred and clamped, and Brian flung himself down
on his bed, panting, but unwounded to speak of.
"Now sleep, master," said the old man. "They will search elsewhere, and
finding this door closed will do naught here until the morning."
Brian laughed a little.
"It is not easy to sleep after fighting, Turlough. I think that now I
will send off that last pigeon, so give me that quill yonder."
With great care Brian wrote his message, telling what had passed, and
saying that he hoped to ride free from the castle next morning. In that
case he would be at Cathbarr's tower before evening came, and he told
Nuala to have all her men landed there at once, since she could hope to
do nothing by sea against the pirate ships.
When the writing was bound to the pigeon's wing he loosed the bird
through the seaward casement, and bade Turlough blow out their
flickering oil-light.
After eating and drinking a little, they lay down to sleep. Men came and
pounded at the door, then departed growling; but Turlough had guessed
aright. The Dark Master was plainly speeding the search for Brian
elsewhere, and since there was no sign of life from the powder-tower, he
did not molest this until close to dawn. Then Brian was wakened by a
shock at the door, and he heard the Dark Master's voice outside
directing his men. Still he seemed to have no thought that Brian was
there, but wanted to get at the powder and into his own chamber again.
Brian took up his pistols and went to a loophole opening on the
battlements, while Turlough still crouched on the bed in no little fear.
Finding that the Dark Master stood out of his sight, Brian fired at two
of the men under the door, and they fell; then he raised his voice above
the shouting that came from outside.
"O'Donnell, are you there?"
The uproar died away, and the other's voice came to him.
"So you are trapped at last, Brian Buidh! Now yield and I promise you a
swift hanging."
"Not I," laughed Brian curtly. "There is no lack of powder here,
O'Donnell Dubh, and one of my men holds a pistol ready for it."
At this he glanced at Turlough, who grimaced. But from outside came a
sudden yell of alarm, and Brian saw a few fleeing figures, while
O'Donnell shouted at his men in furious rage. Brian called out to him
again:
"Give me a horse and let me go free with the one man left me, or else I
will blow up both tower and castle, and you will have little gain for my
death."
"Would you trust my word in this?" cried the Dark Master. Brian smiled.
"Yes, as you must trust mine to leave no fuse in the powder when I am
gone."
Then fell silence. Brian hated O'Donnell, as he knew he was hated in
return; and so great was the hatred between them that he felt
instinctively he could trust the Dark Master to send him out free. It
seemed to him that the other would sooner have him go broken and crushed
than do him to death, for that would be a greater revenge. Moreover, the
Dark Master could know nothing of those men at Gorumna and would have
little fear of the Bird Daughter.
And it befell exactly as Brian thought.
"I agree," cried the Dark Master, stepping out in the dawn-light boldly.
"You shall go forth empty as you came, Yellow Brian. What of those
two-score men you owe me?"
"The time is not yet up," returned Brian, beginning to unbar the door,
and he laughed at the mocking voice.
CHAPTER XIX.
BRIAN MEETS THE BLACK WOMAN.
"The storm is over, master, or will be by this night."
"Too late now, Turlough."
Brian and the old man stood in the courtyard, while the Dark Master was
seeing to horses being made ready for them. Drawing his cloak farther
about his hunched shoulders, the latter turned to Brian with a mocking
sneer.
"Now farewell, Brian Buidh, and forget not to repay that loan, if you
can gather enough men together. When you come again, you will find me
here. A merry riding to you. _Beannacht leath!_"
Brian looked at him grimly.
"Your curse would make better company than your blessing, O'Donnell," he
said, and turned to his horse with no more words.
The Scots who were standing around gave vent to a murmur of approval,
and Brian saw the black looks passing between them and the wild
O'Donnells. The Highlanders had done murdering enough in Ireland since
Hamilton brought them over, but they were outspoken men, who had little
love for poisoners; and as Brian settled into the saddle with his huge
sword slung across his back, he caught more than one word of muttered
approval, which the Dark Master was powerless to check.
So Yellow Brian rode out from the castle he had lost, with Turlough Wolf
at his heels, and his heart was very sore. Once across the filled-in
moat and he saw fifty men at work by the shore, loading the dead into
boats to be buried in the bay, for the ground was hard-frozen.
Parties of Scots troopers and the horseless O'Donnells were scattered
over the farmlands and country ahead, but these offered no menace as the
two horsemen rode slowly through them. For all his bitterness, Brian
noted that the four pirate ships had been brought around into the bay
before the castle, into which the Scots had moved, while a great number
of the O'Donnells had landed and were hastily throwing up brush huts on
the height above the shore, evidently intending to camp there for the
present.
That was a dark leave-taking for Brian, since he had lost so many men
and his castle to boot. Yet more than once he looked back on Bertragh,
and when they came to the last rise of ground before the track wound
into the hills and woods, he drew rein and pointed back with a curt
laugh.
"This night I shall return, Turlough, and I think we shall catch the
Dark Master off his guard at last. If we throw part of our men on that
camp at dawn and the rest upon the castle, the tables may yet be
turned."
"A good rede, Brian O'Neill," nodded the old Wolf approvingly. At thus
hearing his name Brian flung Turlough one lightning-swift glance, then
pulled out his Spanish sword and threw it high, and caught it again with
a great shout.
"Tyr-owen! _Slainte!_"
With that he put spurs to his horse and rode on with better heart,
striving to forget his troubles in thinking of the stroke he would deal
that night. If those three pigeons had won clear to Gorumna, he would
find Nuala and her men waiting at Cathbarr's tower, and before the dawn
they would be back again and over the hills.
So they rode onward, and presently came to a stretch of forest, dark
against the snow. Suddenly Turlough drew up with a frightened glance
around.
"Master--what is that wail? If I ever heard a banshee, that is the cry!
Beware of the Little People, master--"
"Nonsense!" exclaimed Brian, drawing rein also and listening. He heard a
faint, sobbing cry come from ahead, and so mournful was it, so charged
with wild grief, that for an instant his heart stood still, and the
color fled from his face.
"It is some woman wailing her dead, Turlough," he said at length,
although doubtfully. "Yet I have never heard a _caoine_ like it; but
onward, and let us see."
"Wait, master!" implored the old man. "Let us cut over the hills and go
by another path--"
"Go, if you are afraid," returned Brian, and spurred forward. The other
hesitated, but followed unwillingly, and a moment later Brian came upon
the cause of that mournful wailing, as the trees closed about them and
the road wound into a hollow.
The dingle was so sheltered by the brooding pines that there was little
snow, except on the track itself, and no wind. Under the spreading
splay-boughs to the right was what seemed to be a heap of rags and
tatters, though the wailing cry ceased as the two riders clattered down,
with Turlough keeping well behind Brian.
The latter drew rein, seeing that the creature under the pine-boughs was
some old crone whose grief seemed more bitter still than his own.
"What is wrong, mother?" he cried cheerily. "Are you from one of the
Bertragh farms?"
The tattered heap moved slightly, and a wrinkled, withered face peered
up at him.
"Nay, I come from farther than that," and to his surprise there was a
mocking note in her voice, though it was weak. "That is a good horse of
yours, _ma boucal_; he must trot sixteen miles to the hour, eh?"
"All of that, mother," returned Brian, wondering if the old crone was
out of her senses. "Was it you whom I heard wailing a moment ago? Where
is your home?"
The old woman broke into a cackle of hideous laughter.
"My home, is it? Once I had a home, Yellow Brian--and it was in
Dungannon, with Tyr-owen and Cormac and Art and the noblest of the
chiefs of Ulster to do me honor! Have you forgotten me, Brian O'Neill,
since we met at the Dee Water?"
Then Brian gave a great cry, and swung down to earth, for now he
recognized the Black Woman. But as he strode toward her she tried to
rise and failed, and forth from the midst of her rags came a quick gush
of red blood. Brian leaped forward and caught her in his arms, pitying
her.
"I knew you," she gasped out weakly, clutching at his shoulder. "I knew
you, son of Tyr-owen! You had yellow hair, but your face was the face I
once loved, the face of the great Hugh--"
She stopped abruptly, and her words were lost in a choking gasp as blood
came from her mouth. Brian swore.
"_Mile Mollaght!_ What has happened here, woman? Are you wounded?"
"Aye, those dogs of O'Donnells," she moaned feebly. Then new strength
came to her, and she peered up with another cackle. "But did I not tell
wisely, son? Have you not found Cathbarr of the Ax and the Bird Daughter
even as I foretold?"
"Yes, yes," returned Brian impatiently. "Where are you wounded, mother?
We can take you--"
"Peace, avic," she cried. "They came on me last night, and my life is
gone. You shall take vengeance for the old _calliagh_, Brian--but first
I must talk. Do you know who I am, avic--or who I was, rather?"
"How should I know that, mother?" answered Brian. "Old Turlough Wolf,
yonder, swears you are some witch--"
"Turlough!" The hag raised herself on his arm, cackling. "So the old
Wolf is still living! Do _you_ know me, Turlough? Do you remember the
sorrowful day of the earl's flight?"
Old Turlough, who had ridden closer, bent over and looked down, fear in
his face. Suddenly he straightened up again with a wild cry.
"Noreen of Breffny! By my hand, it is the earl's love!"
"Aye, the earl's love!" she gasped out, falling back. "I was his love in
truth, Yellow Brian, and he loved me above all the rest, though
another's hand closed his eyes and laid him to earth in Rome. I knew you
would come, Brian--I saw you at Drogheda, though you saw me not, and I
bade you come here into the West, and I have watched over you--"
She coughed horribly, clutching at Brian's arm. He stared down at her in
amazement, for the incredible story seemed true enough. This old hag had
been that Noreen of Breffny of whom he had heard much--the fairest maid
of the North, whom the great earl had loved to the last, though the
church had not blessed their union.
Brian's old Irish nurse had often told him of the "Breffny lily," and it
was bitter and hard to realize that this ancient hag, withered and
shrunk and done to death by the Dark Master's men, had been the fairest
maid in Ulster. She gasped out a little more of her story, and Brian
found that his wild surmises had been true; after seeing him and
recognizing him for one of the earl's house, she had instantly led his
mind to this part of the country, being aware of the strife between
O'Donnell and Nuala O'Malley. It had been a crazed notion enough, and
since then she had kept as near to him as possible in the half-sane idea
that she might help him.
How she had managed to do it ever remained a mystery to Brian, since his
marches had been none of the slowest, but she had done so.
"Where are--your men?" she exclaimed after a little. Brian told her what
had chanced at the castle, and she broke out in a last wild cackling
laugh.
"Tyr-owen's luck!" she cried. "Betrayed and blasted, betrayed and
blasted--but the root of the tree is still strong, Yellow Brian--give me
your blessing, master--give Noreen your blessing before you go to Rome,
Hugh _mo mhuirnin_--"
Brian's face blanched and his hands trembled, for he saw that her
wandering mind took him for his grandsire.
"_Dhia agus mhuire orth_," he murmured, and with a little sob the Black
Woman died.
Silence fell upon the dingle, as Brian gazed down at the woman his
grandfather had loved, and whose love had been no less. Then Turlough
pushed his horse closer, looking down with a shrewd leer.
"Said she not that it would be a black day when you met her again,
master?" he queried with awe in his voice. "I think--"
"Keep silence!" commanded Brian shortly. "Get down from that horse and
dig a grave."
"But the ground is frozen--" began old Turlough in dismay. Brian gave
him one look, and the old man hastily dismounted, crossing himself and
mumbling.
Brian joined him, and they managed to scoop out a shallow grave with
knife and sword, laid the old woman in it, and covered her up again. It
was a sorry burial for the love of the great earl, but it was the best
they could do.
Shaken more than he cared to admit, Brian mounted and rode on in
silence. As he had thought, there was nothing supernatural about this
weird Black Woman, except, perhaps, the manner in which she had
contrived to keep close to him. She had warned him at the Stone
Mountain, and she must have been keeping close to Bertragh ever since,
unseen by any, with her unhinged mind driving her forward relentlessly.
"Poor woman!" he thought darkly, gazing into the hills ahead. "There has
been little luck to any who ever followed an O'Neill or loved an
O'Neill! And now it seems likely that the same ill luck of all my family
is to dog my heels, bringing me up to the heights, only to cast me down
lower than before. Well, I may fall, but it shall not be until I have
dragged down the Dark Master. If I fall not I may yet best the ill-luck
and conquer Millhaven for my own."
With that his mind leaped ahead again as the plan outlined itself to
him. The O'Donnell pirates must have brought their whole force to the
Dark Master's aid, and if he could but cut off that camp of theirs
between the castle and the shore, Nuala O'Malley might bring her two
ships against the weakened four and take them all.
Then, when the castle had fallen, he could sail north to Millhaven,
reduce the stronghold there, and let fly his own banner at last. It was
a good plan, but it hung on many things.
With a short laugh at his own fancies he turned in the saddle as the
voice of Turlough broke into his musings.
"I mind the last time I saw the poor woman back yonder, master. It was
just before the great flight, and I mind now that she was not so
ill-looking even then, though she was well past her youth, and that was
forty years ago. Tyr-connall's bag-pipe men were blowing as we marched
to Lough Swilly, and two earls rode in front when the poor _caillin_
rushed out and flung herself under Tyr-owen's horse--oh, _Mhuire as
truagh, Mhuire as truagh_ for the old days! And when the earl died, her
name was on his lips, and I came home again to find her disappeared. Oh,
what sorrow for the old days! Would that I had died in Rome with the
princes--"
"Stop that wailing," interrupted Brian sternly, for the old man was
lashing himself into a frenzy of grief. "Put spurs to that horse of
yours, Turlough, for we must reach Cathbarr's tower by noon if possible
in order to start the men off over the hills. It'll be a long night's
march, and I've no time to be idling here on the road."
Upon which he dug in his spurs and urged his steed into a gallop, and in
order to keep up, Turlough Wolf had to give over his laments and do
likewise. Brian forced himself to bend all his energies toward carrying
out his final desperate plan, but he silently vowed that the old woman
who had so foully been cut down by the O'Donnells should not die
unavenged.
On they galloped without pause, gained the head of Bertraghboy Bay, and
swung to the east on the last stretch of the trip. The storm which had
arisen so inopportunely was now dying away, and the sun was breaking
through the gray clouds; when they turned out from the main track into
the hill-paths that led to Cathbarr's tower, the rough ground made them
slow their pace. When they were still three miles from the tower,
however, Brian gave a shout.
"Men, Turlough! Cathbarr has sent out men to meet us!"
So, indeed, it proved, and five minutes later a dozen men met them with
yells of delighted welcome. From these overjoyed fellows Brian quickly
learned that Cathbarr was at the tower and that Nuala O'Malley had just
arrived there.
So, leaving them to follow, he and Turlough went on at their best speed,
and twenty minutes later they topped that same long rise from which
Brian had first gazed down on the little promontory where stood
Cathbarr's tower. But now, as he saw what lay beneath, he drew up with a
shout of amazement.
For around the tower and at the base at the neck of land were camped a
goodly force of men, while at anchor near the tower lay--not Nuala's two
ships alone, but also those other two of her kinsmen!
"Those two O'Malleys have returned from the south," exclaimed Turlough
in wild delight. "That means more men and ships, master--we will cut off
those Millhaven pirates to a man!"
Brian sent out a long shout, but his arrival had already been noted. As
he rode down the <DW72>, men poured from the camp and tower, and ahead of
them all came Cathbarr of the Ax, with Nuala and Lame Art and Shaun the
Little behind him.
"Welcome!" bellowed the giant with a huge laugh, pulling Brian from his
horse with a great hug of delight. "Welcome, brother!"
Brian escaped from his grip and bowed over the Bird Daughter's hand. As
he rose, he saw that her face had lost its ruddy hue, and that her eyes
were ringed with darkness. Before he could speak she smiled and gripped
his hand.
"The birds came safe, and we know all. Yesterday arrived these kinsmen
of mine, and their force is joined to our own, Yellow Brian--"
Brian held up his hand, halting her suddenly, and silence fell on the
men who had crowded around. For a moment he gazed into her deep eyes,
then flung up his head and his voice rang clear and stern in the
stillness.
"Lady Nuala," he said quietly, "I promised you that when I slew the Dark
Master I would tell you my name. Before another day has passed I shall
have slain him; and now I tell you and your kinsmen that I renounce all
fealty to you."
At this the Bird Daughter started, staring in amazement, while an abrupt
oath burst from Lame Art. Brian went on calmly.
"This I do because it is not meet that The O'Neill should give fealty to
any, Lady Nuala. I am Brian O'Neill, of right The O'Neill and Earl of
Tyr-owen, though these are empty titles. And this night you and I shall
fall on Bertragh together, Bird Daughter, and when we have won it again
it shall be yours as of old."
And amid a great roar of shouts welling up around him Brian bowed to
Nuala.
"Then, Brian O'Neill," she said, quieting the tumult a little, "am I to
understand that you wish to make pact with me, and to receive no
reward?"
For a moment he gazed openly and frankly into her eyes, and under his
look the red crept into her cheeks again; yet her own eyes did not
flinch.
Brian laughed out.
"Yes, lady! It may be that I shall have a reward to ask of you, but that
may not be until I have won back what I have lost for you."
"And what if the reward be too great?"
"Why, that shall be for you to say!" and Brian laughed again. "Is it
agreed, Bird Daughter?"
For an instant he thought she meant to refuse, as she drew herself up
and met his level eyes; the men around held their breaths, and the
O'Malley chiefs glanced at each other in puzzled wonder. Then her quick
laugh rippled out and she gave him her hand.
"Agreed, Brian--and I hope that you can shave that yellow beard of yours
by to-morrow!"
And the great yell that went up from the men drowned all else in Brian's
ears.
CHAPTER XX.
THE STORM BURSTS.
"Now, the first thing is to see what force of men we have," said Brian,
after the midday meal. They were all gathered in Cathbarr's tower before
a log fire, and were preparing the plan of campaign.
"I have my hundred and eighty men," said Nuala. "When that last pigeon
came from you I set out at once. With the hundred men under Cathbarr, we
have close to three hundred. You can take them all, for my kinsmen here
have enough and to spare to handle my two ships as well as theirs."
"Good!" exclaimed Brian, as the two O'Malleys nodded. "I think that by
striking at dawn we shall find most of the O'Donnells ashore or in the
castle, and if you time your sailing to strike on their four ships at
the same time we may easily take castle, camp, and ships at one blow."
"If all went as men planned we would not need to pray Heaven for aid,"
quoth Shaun the Little sententiously. Brian glanced at him.
"Eh? What do you mean by that?"
"Nothing," returned the wide-shouldered seaman with a shrug. "Except
that there may be more to it than we think, Brian."
"The Dark Master will not suspect your return so suddenly," spoke up
Nuala. "Pay no heed to Shaun, Brian--he was ever a croaker. When think
you we had best start?"
"I am no seaman," laughed Brian. "Get there at dawn, that is all. I will
send on my men at once, then; since we have only two horses, Cathbarr
and I will ride after them later and catch them up. Will you take the
men, Turlough, or bide here out of danger?"
"I think it will be safest with the Lady Nuala," hesitated the old man
craftily.
"Little you know her, then," roared Lame Art, his cousin joining in the
laugh.
So Turlough had decided, however, and he stuck to it. Brian then
described closely how the four pirate ships lay in the bay under
Bertragh, while Shaun went out to arrange the distribution of his men on
Nuala's ships.
The arrangements having been perfected, Brian saw his three hundred men
troop off on their march over the hills, after which he told Nuala at
greater length all that had taken place in the castle since his parting
with her at sea. Bitter and unrestrained were the curses of the
O'Malleys as they heard of how his men had been poisoned, while Nuala's
eyes flamed forth anger.
"There shall be no quarter to these O'Donnells," she cried hotly. "Those
whom we take shall hang, and the Scots with them--"
"Not the Scots," exclaimed Brian quickly. "They are honest men enough,
Nuala, and may serve us well as recruits. If we find them in the castle,
as I think we shall, we may leave them there until we have finished the
Millhaven men; however, it is possible that my men will find the castle
almost unguarded, and so take it at the first blow. However that turns
out, the Dark Master shall not escape us this time."
During the afternoon, when the two O'Malleys were busily getting their
ships in order for the coming fray, Brian sat in the tower with Nuala.
He told her freely of himself, and although neither of them referred to
that reward of which he had spoken at their meeting, Brian knew well
that he would claim it.
He did not conceal from himself that the Black Woman had guided him to
more than conquest by sword. The Bird Daughter was such a woman as he
had dreamed of, but had never found at the Spanish court, and he knew
that whether there was love in her heart or not, his own soul was in her
keeping.
Perhaps he was not the only one who knew this, for as Lame Art rowed out
with his cousin, the latter nodded back at the tower.
"What think you of this ally, Art Bocagh? Could he be truly the Earl's
grandson?"
"I know not," grunted the other. "But I do not care whether he be Brian
Buidh or Brian O'Neill or Brian the devil--he is such a man as I would
fain see sitting in Gorumna Castle, Shaun!"
And Shaun the Little nodded with a grin.
When the sun began its westering, Brian and Cathbarr rode back from the
tower with food and weapons at their saddle-bows, and they paused at the
hill-crest to watch the four ships weigh anchor and up sail, then went
on into the hills. They were to meet their men at that valley where the
Dark Master had been defeated and broken in the first siege, and jogged
along slowly, resting as they rode.
"Brother," said Cathbarr suddenly, fingering the haft of his ax and
looking at Brian, "do you remember my telling you, that night after we
had bearded the Dark Master and got the loan of those two-score men, how
an old witch-woman had predicted my fate?"
"Yes," returned Brian, with a sharp glance. In the giant's face there
was only a simple good-humor, however, mingled with a childlike
confidence in all things. "And I told you that you were not bound to my
service."
"No, but I am bound to your friendship," laughed Cathbarr rumblingly. "I
can well understand how I might die in a cause not mine own, since I am
fighting for you; but I cannot see how death is to come upon me through
water and fire, brother!"
"Nonsense," smiled Brian. "Death is far from your heels, brother, unless
you are seeking it."
"Not I, Brian. I neither seek nor avoid if the time comes. Only I wish
that witch-woman had told me a little more--"
"Keep your mind off it, Cathbarr," said Brian. "In Spain the Moriscoes
say that the fate of man is written on his forehead, and God is just."
"What the devil do I care about that?" bellowed Cathbarr. "I care not
when I die, brother--but I want to strike a blow or two first, and how
can that be done if death comes by water and fire?"
"Well, take heart," laughed Brian, seeing the cause of the other's
anxiety. "You are not like to die from that cause to-night, and I
promise you blows enough and to spare."
Cathbarr grunted and said no more. The last of the storm had fled away,
and the two men rode through a glittering sunset and a clear, cold
evening that promised well for the morrow.
They traveled easily, and it was hard on midnight when a sentry stopped
them half a mile from the hollow where the men were resting. Brian noted
with approval that no fires had been lighted, and he and Cathbarr at
once lay down to get an hour's sleep among the men.
Two hours before daybreak the camp was astir, and Brian gathered his
lieutenants to arrange the attack. Thinking that the Dark Master would
be in the castle, he and Cathbarr took a hundred men for that attack,
ordering the rest to get as close to the camp as might be, but not to
attack until he had struck on the castle, and to cut off the O'Donnells
from their ships. Then, assured that the plan was understood, he and
Cathbarr loaded their pistols and set out with the hundred.
Brian ordered his men to give quarter to all the Scots who would accept
it, if they got inside the castle, and as they marched forward through
the darkness he found to his delight that O'Donnell seemed to have no
sentries out.
"We have caught the black fox this time," muttered Cathbarr, after they
had passed the camp-fires without discovery and the black mass of the
castle loomed up ahead. "They will hardly have repaired those gates by
now, brother."
Brian nodded, and ordered his men to rest, barely a hundred paces from
the castle. Since there was no need of attacking before dawn, in order
to let Nuala come up the bay, he went forward with Cathbarr to look at
the gates.
These, as nearly as he could tell, were still shattered in; there were
fires in the courtyard, and sentries were on the wall, but their watch
was lax and the two below were not discovered. They rejoined the
hundred, and Brian bade Cathbarr follow him through the hall to that
chamber he himself had occupied in the tower, where O'Donnell was most
likely to be found.
"Well, no use of delaying further," he said, when at length the grayness
of dawn began to dull the starlight. Since to light matches would have
meant discovery, he had brought with him those hundred Kerry pikemen
Nuala had recruited after the dark Master's defeat, and he passed on the
word to follow.
The mass of men gained the moat before a challenge rang out from above,
and with that Brian leaped forward at the gates. A musket roared out,
and another, but Brian and Cathbarr were in the courtyard before the
Scots awakened. A startled group barred their way to the hall, then
Brian thrust once, the huge ax crashed down, and they were through.
Other men were sleeping in the hall, but Brian did not stop to battle
here, running through before the half-awakened figures sensed what was
forward. A great din of clashing steel and yells was rising from the
court; then he and Cathbarr gained the seaward battlements and rushed at
the Dark Master's chamber. The door was open--it was empty.
For a moment the two stared at each other in blank dismay. With a yell,
a half-dozen Scots swirled down on them, but Brian threw up his hand.
"The castle is mine," he shouted. "You shall have quarter!"
The Scots halted, and when two or three of the Kerry pikemen dashed up
with news that the rest of the garrison had been cut down or given
quarter, they surrendered.
Brian's first question was as to O'Donnell.
"Either at the camp or aboard one of his kinsmen's ships," returned one
of the prisoners. "They were carousing all last evening."
At the same instant Cathbarr caught Brian's arm and whirled him about.
"Listen, brother!"
So swift had been Brian's attack that the castle had been won in a scant
three minutes. Now, as he listened, there came a ragged roar of
musketry, pierced by yells, and he knew that the camp was attacked.
With that, a sudden fear came on him that he would again be outwitted.
There was a thin mist driving in from the sea which would be dissipated
with the daybreak, and if the Dark Master was on one of the ships he
might get away before Nuala's caracks could arrive. Brian had been so
certain that he would find O'Donnell in the castle that the
disappointment was a bitter one, but he knew that there was no time to
lose.
"Come," he ordered Cathbarr quickly, "get a score of the men and to the
camp. Leave the others here to hold the castle if need be."
As he strode through the courtyard and the sullen groups of Scots
prisoners, he directed the Kerry men to load the bastards on the walls
and give what help might be in destroying the pirate ships. Then, with
Cathbarr and twenty eager men at his back, he set off for the camp at a
run, fearful that he might yet be too late.
The day was brightening fast, and from the camp rose a mighty din of
shouts and steel and musketry. Brian's men had charged after one hasty
volley, but their leader gave a groan of dismay as he saw that instead
of attacking from the seaward side as he had ordered, they were pouring
into the camp from the land side.
O'Donnell must have landed the greater part of his men, for Brian's
force was being held in check, though they had swept in among the brush
huts. Over the tumult Brian heard the piercing voice of the Dark Master,
and with a flame of rage hot in his mind he sped forward and found
himself confronted by a yelling mass of O'Donnells.
Then fell a sterner battle than any Brian had waged. In the lessening
obscurity it was hard to tell friend from foe, since the mist was
swirling in off the water and holding down the powder-smoke. Brian saved
his pistols, and, with Cathbarr at his side, struck into the wild,
shaggy-haired northern men; they were armed with ax and sword and skean,
and Brian soon found himself hard beset despite the pikemen behind.
The Spanish blade licked in and out like a tongue of steel, and Brian's
skill stood him in good stead that morn. Ax and broadsword crashed at
him, and as he wore no armor save a steel cap, he more than once gave
himself up for lost. But ever his thin, five-foot steel drove home to
the mark, and ever Cathbarr's great ax hammered and clove at his side,
so that the fight surged back and forth among the huts, as it was
surging on the other side where was the Dark Master, holding off the
main attack.
Little by little the mist eddied away, however, and the day began to
break. A fresh surge of the wild O'Donnells bore down on Brian's party,
and as they did so a man rose up from among the wounded and stabbed at
Brian with his skean. Brian kicked the arm aside, but slipped in blood
and snow and went down; as a yell shrilled up from the pirates, Cathbarr
leaped forward over him, swinging his ax mightily. With the blunt end he
caught one man full in the face, then drove down his sharp edge and
clove another head to waist. For an instant he was unable to get out his
ax, but Brian thrust up and drove death to a third, then stood on his
feet again.
At the same instant there came a roar from across the camp where his
main body of men were engaged, and Brian thrilled to the sound. As he
afterward found, it was done by Turlough's cunning word; but up over the
din of battle rose the great shout that struck dismay to the pirates and
heartened Brian himself to new efforts.
"Tyr-owen! Tyr-owen!"
With a bellow of "Tyr-owen!" Cathbarr went at the foe, and Brian joined
him with his own battle-cry on his lips for the first time in his life.
The shout swelled louder and louder, and among the huts Brian got a
glimpse of the Dark Master. In vain he tried to break through the
Millhaven men, however; they stood like a wall, dying as they fought,
but giving no ground until the ax and the sword had cloven a way,
although the remnant of the twenty pikemen were fighting like fiends.
Suddenly a yell of dismay went up from the O'Donnell ranks, and they
broke in wild confusion. Leaning on his sword and panting for breath,
Brian looked around and saw what had shattered them so swiftly.
While the stubborn fight had raged, the eastern sky had been streaming
and bursting into flame. Now, sharply outlined against the crimson
water, appeared Nuala's four ships close on those of the pirates. Even
as he looked, Brian saw their cannon spit out white smoke, while from
behind came a deeper thunder as the castle's guns sent their heavy balls
over the pirate ships.
These were anchored a hundred yards from shore, and Brian saw the danger
that betided as the stream of fugitives swept down toward the boats.
Nuala's ships were undermanned, for he had counted on cutting off most
of the pirates in the camp; should the Dark Master get to the ships with
his men, things were like to go hard.
"To the boats!" cried Brian to Cathbarr, and leaping over the dead, the
two joined their men and poured down on the shore.
The Dark Master himself stood by one of the boats, and others were
filling fast with men as they were shoved down. Brian tried to cut his
way to O'Donnell, but before he could do so the Dark Master had leaped
aboard and oars were out. Fully aware of their danger, those of the
pirates who could do so got into their boats and lay off the shore,
while others splashed aboard; Brian led his men down with a rush,
cutting down man after man, splashing out into the swirling water and
hacking at those in the boats, but all in vain. Some half-dozen of the
boats got off, crowded with men, while the remnant of the pirates held
off Brian's force that their master might escape.
Drawing out of the fight, Brian pulled forth his pistols and emptied
them both at the figure of O'Donnell. He saw the Dark Master reel, and
the rower next him plunged forward over the bows, but the next moment
O'Donnell had taken up the oar himself and was at work in mad haste.
Brian groaned and flung away his pistols.
Those aboard the pirate ships had already cut the cables and were
striving to make sail, for there was a light off-shore breeze in their
favor, with an ebbing tide. The O'Malley ships were close on them,
however, and as the cannon crashed out anew the masts of one O'Donnell
ship crashed over. But the Dark Master's boat was alongside another of
the ships, whose sails were streaming up, and now his cannon began to
answer those of Nuala.
But Brian stood in bitterness, unmindful of the wild yells of his men,
for once more the Dark Master had escaped his hand at the last moment.
Shaun the Little had been correct in his "croakings."
CHAPTER XXI.
CATHBARR YIELDS UP HIS AX.
Brian gazed out at the scene before him in dull despair. So close were
the ships that he could clearly make out Nuala's figure, with its
shimmering mail and red cloak, on the poop of the foremost.
Her second carack had fallen behind, a shot having sent its foremast
overside, but the other two ships were driving in. All three were
lowering sail, for the Dark Master's craft were unable to get out of the
bay and were giving over the attempt; his disabled ship was sending
over its men to reinforce him, and Brian saw all his own efforts gone
for nothing.
There came a new burst of cannon, and through the veil of smoke he
perceived that Nuala was laying her carack alongside one of the pirate
ships. But it was not that on which stood the Dark Master; his was the
ship closest to the castle, and Lame Art was bearing down on him, while
Shaun the Little stood for the third, spitting out a final broadside as
he came about and lowered sail.
The crowding men on the shore had fallen silent as they watched the
impending conflict, but now Brian felt Cathbarr touch his arm, and
turned.
"Why so doleful, brother?" grinned the giant; though blood dripped into
his beard from a light slash over the brow, his eyes were as clear and
childlike as ever, and the rage of battle had gone from him. "Let us
join in that fight, you and I?"
"Eh?" Brian started, staring at him. "How may that be?"
"Ho, here is our captain given way to despair!" bellowed Cathbarr, and
his fist smote down on Brian's back. "Wake up, brother! We have three
boats here, and we can still strike a blow or two!"
Now Brian wakened to life indeed. He saw the three boats on the shore,
with dead men hanging over them, and leaped instantly into action.
"Push out those boats--get the oars, there!" he shouted, leaping down to
help shove them out. The men saw his intent, and sprang to work with a
howl of delight.
In no long time the dead were flung out, and the boats pushed down until
they were afloat. Brian leaped into one, Cathbarr into another, and men
piled in after them until the craft were almost awash.
An eddy in the veil of smoke that hung over the bay showed Brian that
Lame Art's ship had grappled with that of O'Donnell, and with renewed
confidence thrilling in him, he shouted to his men to get aboard the
O'Malley ship. The Bertragh cannon had ceased to thunder as the ships
came together, but from the ships balls were hailing, musketry was
crackling, and the water was tearing into spurting jets around the
boats.
Brian's men fell to their oars in sorry fashion enough, but they made up
in energy what they lacked in skill. Driving past Nuala's ship, Brian
saw that she had also grappled and that the battle was raging over her
bulwarks, but sorely tempted to turn aside though he was, he waved his
men on.
They rowed close under the ship to which she was fastened, and as they
sped past the O'Donnells saw them, and gave them a scattering volley.
One or two of Brian's men went down, and a cry broke from him as he saw
a round shot heaved over into his third boat, sinking her; then they
were past, and bearing down on Art Bocagh's ship.
"Tyr-owen for O'Malley!"
Cathbarr's bellow rose over the tumult, and his boat crashed into the
waist of the ship just as Brian leaped up into the mizzen-chains. His
feet gained hold on a triced-up port, and as he looked down he saw a
swell heave up the two boats, then bring them down together with a
splintering smash.
The result was dire confusion. None of the men were seamen, but some of
them gained the side of Brian, others scrambled in through the ports,
and more than one of them fell short and went down. Standing in the
sinking boat with the water swirling about his ankles, Cathbarr caught
up his ax and leaped; a moment later Brian was over the bulwarks with
the giant at his side, and the O'Malleys welcomed them with a yell of
joy.
They were badly needed, indeed. The Dark Master had led his men in
furious onslaught across the waist of the ship, and Art Bocagh was being
beaten back to the poop despite his stubborn resistance. Brian saw that
the Dark Master's men far outnumbered Art's, while from the rigging of
each ship musketeers were sending down bullets into the melee. With a
shout, Brian and Cathbarr led their men on the O'Donnell flank, and the
tide of battle turned.
At the first instant the rush of men bore Brian against the Dark Master,
who was fighting like a demon. Brian caught the snarl on the other's
pallid face, and struck savagely; O'Donnell parried the blow with his
skean and returned it, but Brian warded with his left arm and swept down
his blade. The Dark Master flung himself back, but not far enough, and
Brian saw the point rip open the pallid cheek. Even as he pressed his
advantage, however, another surge of men separated them.
Now Brian gave over every thought save that of reaching his enemy again,
and fell on the O'Donnells with stark madness in his face. A pistol
roared into his stubbly beard and the ball carried off his steel cap,
but he cut down the man and pressed into the midst of the pirates,
cutting and thrusting in terrible rage.
At sight of him men bore back; the icy flame in his eyes took the heart
from those who faced him, and behind rose Cathbarr's wild bellows as the
giant hewed through after Brian. Back went the pirates, and farther
back. Brian found that he had cut his way to Lame Art, and with a yell
the forces joined and swept on the Dark Master's men.
O'Donnell had vanished, and now his men were swept back to the bulwarks
and over to their own deck. Here they made a brief stand; then Cathbarr
leaped over into the midst and his ax crushed down two men at once;
Brian followed him, and for an instant it seemed that they would sweep
all before them.
Just then, however, Lame Art toppled from the bulwarks with a bullet
through him from above, and the Dark Master's disappearance was
explained by a rain of grenades that whirled among the O'Malleys. They
gave back in dismay, Brian and Cathbarr were forced after them, and the
Dark Master himself led his men in a mad stream over the bulwarks once
more.
There was no stopping them now. The death of Art Bocagh had disheartened
his men, and amid flashing steel and spurting fire Brian and Cathbarr
retreated to the quarterdeck. Here they had a brief breathing space
until the pirates came at them anew, and with such fury that three of
them gained a footing to one side. Brian went at them with a shout,
thrust one man through the body, sent a second back with his bare fist,
and as the third man struck down at him a pikeman transfixed the man
before the blow could fall.
The boarders drew back, but as they did so a great heave of the grinding
ships broke the hastily flung grapplings. The ships were borne apart,
and the Dark Master with most of his men remained in the waist of the
O'Malley ship.
This gave a new turn to the conflict. O'Donnell had to master the ship
to win free, and when Brian saw this he gave a great laugh and rejoined
Cathbarr. A quick glance around showed him that Nuala was slowly winning
her grappled decks, while Shaun the Little was hanging off and sending
his cannon crashing into the third pirate ship. The two disabled craft
were slowly drawing together with the tide, which was forcing all eight
into the bay, and were pounding away with their guns as they came.
Now the combat resolved itself into a desperate struggle for possession
of the quarterdeck, which Brian and Cathbarr held. The Dark Master's men
swarmed up at them bravely enough, but the ax and sword flashed up and
down, and time after time the Millhaven men fell back, unable to win a
footing. Twice the Dark Master himself led them, snarling with baffled
rage, but the first time a pikeman thrust him down and the second time
Cathbarr's ax glanced from his helm.
O'Donnell reeled back and was lost to sight for a time.
"That was a poor blow," grunted the giant in disgust. "'Ware, brother!
Stand aside!"
Brian leaped away as the men behind him ran out a falcon and sent its
blast into the crowd below in the waist. A dozen men went down under
that storm of death, but almost at the same moment a grenade burst
behind the falcon, and with that Brian was driven back as a keg of
powder tore out half the quarterdeck in a bursting wall of flame and
smoke.
Barely had the shattering roar died out when Brian's reeling senses
caught a wild yell of dismay from his men.
"Fire! The ship is afire forward!"
Brian saw that the grenades had indeed fired the ship forward, while the
explosion had sent the quarterdeck into a burst of fire also, and the
lowered but unfurled sails were roaring up in flame.
Up poured the O'Malleys, and Brian staggered back to the poop. He had a
vision of the great form of Cathbarr heaving up through the smoke,
blackened and bleeding, but with the ax whirling like a leaf and smiting
down men; then Brian gained the poop, helped the giant up, and with the
few men left they turned to drive down the pirates, who were striving
desperately to win the ship before it was too late.
As he stood with Cathbarr at the narrow break of the poop, beating down
man after man, Brian knew that it was only a question of time now, for
the whole ship was breaking into flame forward. Suddenly he felt a tug
at his buff coat, and looked down to see his belt fall away, sundered at
his side by a bullet. He thought little of it, for he had half a dozen
slight wounds, and turned to smite down at a man who had leaped for the
poop; as his sword sheared through helm and skull, there came another
tug, and Brian felt a bullet scrape along his ribs.
The O'Donnells drew back momentarily, and in the brief pause Brian saw
the figure of the Dark Master by the starboard rail in the waist, aiming
up at him with a pistol, while two men behind him were hastily charging
others. Cathbarr saw the action also, and hastily flung Brian aside, but
too late. A burst of smoke flooded over the waist, and Brian caught the
pistol-flash through it, as the ball ripped his left arm from shoulder
to elbow. Then the pirates were at the poop again, and the waist was
shut out by the flooding smoke as the wind drove it down from forward.
With a scant dozen men behind them, Brian and Cathbarr once more beat
the enemy back; the giant swung his ax less lightly now, and seemed to
be covered with wounds, though most of them were slight. Brian still
eyed the waist for another glimpse of the Dark Master, but the smoke was
thick and he could see nothing. In the lull he flung a wan smile at
Cathbarr, who stood leaning on his ax, his mail-shirt shredded and
bloody.
"Are you getting your fill of battle, brother?"
"Aye," grinned the giant, "and we had best swim for it in another minute
or the ship--look! _M'anam an diaoul!_ Look!"
At his excited yell Brian turned, as a ball whistled between them. There
below, in a boat half full of dead, but with two men at the oars, stood
the Dark Master, just lowering his pistol. He flung the empty weapon up
at Brian with a hoarse yell of anger, and passed from sight beneath the
ship's counter, toward the stern.
Realizing only that his enemy was escaping, Brian whirled and darted for
the poop-cabins. He was dimly conscious of a mass of figures behind,
amid whom stood Cathbarr with the ax heaving up and down, then he was in
the cabins. Jerking open the door to the stern-walk, he saw the Dark
Master's boat directly underneath, hardly six feet from him.
"Tyr-owen!" yelled Brian, and dropping his sword, but holding his skean
firmly, he hurdled the stern-walk railing and leaped.
At that wild shout the Dark Master looked up, but he was too late. Brian
hurtled down, his body striking O'Donnell full in the chest and driving
him over on top of the two rowers, so that all four men sprawled out
over the dead. For an instant the shock drove the breath out of Brian,
then he felt a hand close on his throat, and struck out with his skean.
One of the rowers gurgled and fell back, and Brian rolled over just as
steel sank into his side. Giddy and still breathless, he gained his
knees to find the Dark Master thrusting at him from the stern, while at
his side the other rower was rising. Brian brought up his fist, caught
the man full on the chin, and drove him backward over the gunwale. The
lurch of the boat flung the Dark Master forward, Brian felt a sickening
wrench of pain as the sword pierced his shoulder and tore loose from
O'Donnell's hand, then he had clutched his enemy's throat, and his skean
went home.
Spent though both men were, the sting of the steel woke the Dark Master
to a burst of energy. As the two fell over the thwarts, he twisted above
and bore Brian down and tried to break the grip on his throat, but could
not. For the second time in his life Brian felt that he had a wild
animal in his grasp; the sight of the snarling face, the venomous black
eyes, and the consciousness that his own strength was slowly ebbing, all
roused him to a last great effort.
The smoke-pall had shut out everything but that wolfish face, and as he
writhed up even that seemed to dim and blur before his eyes, so that in
desperate fear he struck out again and again, blindly. The blows fell
harmless enough, for all his strength was going into that right hand of
his; he did not know that his fingers were crushing out the Dark
Master's life, that O'Donnell's face was purple and his hands feebly
beating the air.
Brian knew only that the terrible face was hidden from him by some loss
of vision, some horrible failure of sight due to his weakness. Suddenly
there was a great crash at his side, and he thought that a huge ax with
iron twisted around its haft had fallen from the sky and sheared away
half the gunnel of the boat. He struck out again with his skean, and
felt the blow go home--and with that there came a terrific, blinding
roar. The smoke-veil was rent apart by a sheet of flame, Brian realized
that the burning ship must have blown up, and then a blast of hot wind
drove down against him and smote his senses from him.
CHAPTER XXII.
THE STORM OF MEN COMES TO REST.
"Very well, Turlough. Tell Captain Peyton that I will give him an answer
to his message to-night, then bid my kinsman Shaun entertain him in the
hall, with the other officers. Send some food up here, and I may come
down later."
"And, mistress--you will tell me if--"
"Surely. Now go."
Brian tried to open his eyes, but could not. He tried to move, but could
not; and realized at length that he was lying on a bed, and that a
bandage was on his head and others on his limbs.
Suddenly a hand fell on his cheek, and a thrill shot through him; his
beard had been shaved away, for he could feel the softness of the hand
against his chin. He felt the hand passed over his mouth--and he kissed
it.
There was a startled gasp, then the soft hand returned to his cheek.
"Brian! Are you awake at last?"
"I seem to be," he said, though his voice sounded more like a whisper.
"Is that you, Nuala? Where are we?"
"Yes, it is I," came her voice softly, and something warm splashed on
his cheek. "Oh, Brian! I so feared that--that you were dead!"
The hand moved away, and he moved uneasily, to feel pain through his
body.
"Nay, put back your hand!" he said. He tried to smile. "There, that's
better. Where are we, Nuala? On your ship?"
"No, Brian--at Gorumna. But I forgot. Turlough said you must not talk--"
"Oh, curse Turlough," he cried in irritation. "Gorumna? What has
happened? Where is the Dark Master?"
"Lie still or I must leave you!" she cried sharply, and he obeyed. "The
Dark Master's head is over the gate, Brian. It is two days since the
fight."
"Take that bandage from my eyes, Nuala," he said. After a minute her
hands went to his head, and as he felt the bandage removed, light
dazzled him, and he shut his eyes with a groan. Then he opened them
again, and gradually he made out the figure of Nuala leaning over him,
while a cresset shed light from above.
"Tell me what has happened," said Brian quietly, as he tried again to
move and failed. "Why am I helpless here?"
"Because you are wounded," she replied softly. "Please lie quiet, Brian!
I will tell you all that has chanced."
"Where is Cathbarr! Did we win?"
"Yes, we won; but--but Cathbarr--he must have flung away his ax before
the ship exploded, for we found it sticking in your boat, and--"
Her voice broke, and a pang of bitterness shot through Brian as he
remembered it all now. He groaned.
"And I left him there to die! Oh, coward that I am--coward, and false to
my friend--"
A great sob shook his body, but Nuala's hands fell on his face, and
there was fear in her voice when she answered him.
"No, Brian--don't say that! If any one's fault, it was Shaun's for not
coming sooner to your aid. Cathbarr died as he would have wished, and
indeed as he always thought he would die. But now listen, Brian, for I
have news."
So, leaning over him, she swiftly told him of what had passed. The
O'Donnells had been defeated and slain to the last man; one of their
ships was sunk, and the other three captured, and her men held Bertragh.
As she and Shaun O'Malley lay refitting and gathering their wounded that
same afternoon, a Parliament ship had come in from the south, bearing an
answer to the appeal she had sent to Blake at the Cove of Cork.
He had not only sent her powder and supplies, but had sent her a blank
commission from Cromwell, which would be filled in upon her definite
allegiance to the Commonwealth. The commission guaranteed her possession
of Gorumna and Bertragh and the lands she claimed, and promised that
when the royalists were driven from Galway the grant would be confirmed
by Parliament.
"I am to answer Captain Peyton to-night, Brian," she finished, her eyes
dancing. "And Shaun is going to remain and hold Bertragh for me--"
"What's that?" cried Brian. "Hold Bertragh? Am I then wounded so sore
that I cannot draw sword again?"
"No," and her laugh rippled out. "Turlough says that you will be as well
as ever in a month, Brian. But since you withdrew your fealty to me, I
had to find another servant!"
"I had forgotten that," answered Brian moodily. He stared up at her
face, and as he met her eyes saw the color flow up to her temples.
"You have slain the Dark Master as you promised, Brian," she said
quietly. "And have you forgotten also that you meant to claim a reward
from me for that deed?"
Brian laughed, and his face softened as happiness laid hold upon his
heart.
"I have not forgotten that, Nuala; but now I am not going to ask that
reward in the same way I had intended."
"How do you mean, Brian?" she asked gravely, though her eyes widened a
trifle as if in quick fear.
"This, dear lady," he smiled. "When you answer Captain Peyton, let the
commission be made out in the name of Nuala O'Neill--and take my fealty
for what is left to me of life, Nuala."
He looked up steadily, knowing that all things hung on that instant.
"Well, to tell the truth, Brian," and for a moment she seemed to
hesitate, so that Brian felt a sudden shock, "I--I delayed answering him
in--in that hope!"
And her face came down to his.
[Transcriber's Note: The following synopsis originally appeared at the
beginning of the second installment.]
The scene is laid in Ireland during Cromwell's time, when the whole
country was in arms for or against the various parties. Brian Buidh, or
Brian of the Yellow Hair, himself The O'Neill, comes home from Spain,
where he had been brought up to fight for his country. After a
mysterious warning from the Black Woman, an old hag, he wins forty men
from O'Donnell More, the Black Master, by a trick, and wins the
friendship of Turlough Wolf and Cathbarr of the Ax. His intention is to
gather a storm of men and hold an independent place near Galway. He
forms an alliance with Nuala O'Malley, known as the Bird Daughter
because of her carrier pigeons, for the purpose of recovering her
castle, Bertragh, which O'Donnell had won years before from her parents
by black treachery.
[Transcriber's Note: The following synopsis originally appeared at the
beginning of the third installment.]
The scene is laid in Ireland during Cromwell's time, when the whole
country was in arms for or against the various parties. Brian Buidh, or
Brian of the Yellow Hair, himself The O'Neill, comes home from Spain,
where he had been brought up to fight for his country. After a
mysterious warning from the Black Woman, an old hag, he wins forty men
from O'Donnell More, the Black Master, by a trick, and wins the
friendship of Turlough Wolf and Cathbarr of the Ax. His intention is to
gather a storm of men and hold an independent place near Galway. He
forms an alliance with Nuala O'Malley, known as the Bird Daughter
because of her carrier pigeons, for the purpose of recovering her
castle, Bertragh, which O'Donnell had won years before from her parents
by black treachery.
By warlock arts O'Donnell More brings Brian and a handful of men through
a snowstorm to Bertragh and makes him prisoner. He proceeds to torture
him fiendishly, ending by nailing him to the castle door by one hand.
Just then Colonel James Vere, British officer, arrives, and demands
Brian in order to hang him comfortably in Galway. Red Murrough,
O'Donnell's lieutenant, agrees, for the promise of ten English pounds,
to pretend that Brian is worse off than he is so that he may take longer
to recover. Cathbarr comes in, and offers to take Brian's place if
O'Donnell will release Brian; and when the Black Master makes fun of
him, he goes berserk and cleans out the hall, escaping with Brian to
Nuala. Then they besiege and best O'Donnell, who escapes.
[Transcriber's Note: The following synopsis originally appeared at the
beginning of the fourth and final installment.]
The scene is laid in Ireland during Cromwell's time, when the whole
country was in arms for or against the various parties. Brian Buidh, or
Brian of the Yellow Hair, himself The O'Neill, comes home from Spain,
where he had been brought up to fight for his country. After a
mysterious warning from the Black Woman, an old hag, he wins forty men
from O'Donnell More, the Black Master, by a trick, and wins the
friendship of Turlough Wolf and Cathbarr of the Ax. His intention is to
gather a storm of men and hold an independent place near Galway. He
forms an alliance with Nuala O'Malley, known as the Bird Daughter
because of her carrier pigeons, for the purpose of recovering her
castle, Bertragh, which O'Donnell had won years before from her parents
by black treachery.
By warlock arts O'Donnell More brings Brian and a handful of men through
a snowstorm to Bertragh and makes him prisoner. He proceeds to torture
him fiendishly, ending by nailing him to the castle door by one hand.
Just then Colonel James Vere, British officer, arrives, and demands
Brian in order to hang him comfortably in Galway. Red Murrough,
O'Donnell's lieutenant, agrees, for the promise of ten English pounds,
to pretend that Brian is worse off than he is so that he may take longer
to recover. Cathbarr comes in, and offers to take Brian's place if
O'Donnell will release Brian; and when the Black Master makes fun of
him, he goes berserk and cleans out the hall, escaping with Brian to
Nuala. Then they besiege and best O'Donnell, who escapes.
Brian goes after O'Donnell with a couple of hundred men, having
recovered from his hurts, and all but catches him in a valley, just as
he is working some kind of a divination with a bowl of water. Brian gets
back his Spanish sword, but O'Donnell escapes with some of his men, and
Brian loses all of his in chasing him to keep him from joining with his
pirate friends. Brian and Turlough get back to Bertragh exhausted. He
goes cruising with Nuala, and they meet a small vessel laden with wine
and food for some of O'Donnell's men. Brian goes back with it to
Bertragh, while Nuala goes on to Gorumna Castle, her own home. But the
captured wine proves to be poisoned--it is a trick of the Black
Master's.
* * * * *
Transcriber's note:
The following typographical errors present in the original magazine
publication have been corrected for this electronic edition.
In Chapter V, a missing quotation mark was added after "I am for Brian
Buidh."
In Chapter VI, "Dhar mo lamb" was changed to "Dhar mo lamh".
In Chapter VII, "which were small carracks" was changed to "which were
small caracks". (While "carrack" is the more common English spelling,
the author used "carack" consistently elsewhere in the text.)
In Chapter XI, a missing quotation mark was added after "I would take
your life for his."
In Chapter XII, a missing period was added after "shifted thither in
readiness".
In Chapter XIII, "coming ing in one of his ships to marry me" was
changed to "coming in one of his ships to marry me", and "Beannact
leath!" was changed to "Beannacht leath!".
In Chapter XIV, a missing quotation mark was added after "has joined
with those friends of his".
In Chapter XVI, "those of the Dark Maser were no better" was changed to
"those of the Dark Master were no better".
In Chapter XVIII, a missing quotation mark was added after "They'll have
the gates down in a minute."
In Chapter XIX, "Mhuire as truagh, Muire as truagh" was changed to
"Mhuire as truagh, Mhuire as truagh".
In Chapter XXII, a missing comma was added after "curse Turlough".
No other corrections were made to the original text.
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
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Q: How to Create a JSON Tree from a Tabulated Hierarchy in Python I'm sure there's an elegant way of doing this in Python (or at a push, Javascript), but for the life of me I can't see it...
I have a CSV file of the form:
ID, Name, Description
A, A-name,
A100, A100-name, A100-desc
A110, A110-name, A110-desc
A111, A111-name, A111-desc
A112, A112-name, A112-desc
A113, A113-name, A113-desc
A120, A120-name, A120-desc
A131, A131-name, A131-desc
A200, A200-name, A200-desc
B, B-name,
B100, B100-name, B100-desc
B130, B130-name, B130-desc
B131, B131-name, B131-desc
B140, B140-name, B140-desc
and I want to generate a hierarchical JSON structure so I can visualise the data in theJIT.
var json = {
"id": "aUniqueIdentifier",
"name": "usually a nodes name",
"data": {
"some key": "some value",
"some other key": "some other value"
},
"children": [ *other nodes or empty* ]
};
My plan was to map ID to id, Name to name, Description to data.desc, and organise the hierarchy so that:
*
*Root is parent of A and B
*A is parent of A100 and A200
*A100 is parent of A110 and A120
*A110 is parent of A111, A112 and A113
*B is parent of B100
*B100 is parent of B130 and B140
*B130 is parent of B131
There is also a pathological case in the otherwise regular ordering by ID, where A100 is parent of A131 (the expected A130 is not present).
I was hoping to find an elegant Python solution to this, but it's defeating me at the moment, even ignoring the pathological case...
A: This does it...
import csv
import json
class Node(dict):
def __init__(self, (nid, name, ndescr)):
dict.__init__(self)
self['id'] = nid
self['name'] = name.lstrip() # you have badly formed csv....
self['description'] = ndescr.lstrip()
self['children'] = []
def add_node(self, node):
for child in self['children']:
if child.is_parent(node):
child.add_node(node)
break
else:
self['children'].append(node)
def is_parent(self, node):
if len(self['id']) == 4 and self['id'][-1] == '0':
return node['id'].startswith(self['id'][:-1])
return node['id'].startswith(self['id'])
class RootNode(Node):
def __init__(self):
Node.__init__(self, ('Root', '', ''))
def is_parent(self, node):
return True
def pretty_print(node, i=0):
print '%sID=%s NAME=%s %s' % ('\t' * i, node['id'], node['name'], node['description'])
for child in node['children']:
pretty_print(child, i + 1)
def main():
with open('input.csv') as f:
f.readline() # Skip first line
root = RootNode()
for node in map(Node, csv.reader(f)):
root.add_node(node)
pretty_print(root)
print json.dumps(root)
if __name__ == '__main__':
main()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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|
\section{Introduction}
\label{sec:intro}
\input{texfiles/introduction}
\section{Related work}
\input{texfiles/relatedwork}
\section{Methodology}
\input{texfiles/methodology}
\section{Normalizing Flows}
\label{sec:nf}
\input{texfiles/Normalizing Flows/maths}
\subsection{Architectures}
\label{sec:architectures}
\input{texfiles/Normalizing Flows/architectures}
\subsection{Glow}
\label{sec:glow}
\input{texfiles/Normalizing Flows/glow}
\subsection{Wavelet Flow}
\label{sec:waveletflow}
\input{texfiles/Normalizing Flows/waveletflow}
\section{Dataset}
\input{texfiles/datasets}
\section{Results}
\label{sec:results}
\input{texfiles/results}
\section{Discussion}
\label{sec:majhead}
\input{texfiles/discussion}
\section{Conclusion}
\label{sec:print}
\input{texfiles/conclusion}
\clearpage
\printbibliography
\onecolumn
\section*{Appendix}
\subsection{Mathematical background}
\label{sec:mathback}
A bijection between distributions should adhere to the law of total probability as
\begin{equation}
\int p(x)dx= \int p(y)dy.
\end{equation}
We are only concerned in the absolute change of volume, hence the sign of it is disregarded. We can represent a sample in $Y$ in terms of a sample in $X$. As shown below in the change of variables formula (Eqn.~\ref{change of var}), we can evaluate $p_{\vect{y}}(Y)$ and maintain the total density through any transformation.
\begin{equation}
p(y)=p(x)\left|\frac{dx}{dy}\right|.
\label{change of var}
\end{equation}
This local-linear change approximation is in fact the Jacobian determinant. Given that $y=f(x)$ and thus $x=f^{-1}(y)$ we can write
\begin{equation}
p_Y(y)=p_X(f^{-1}(y))\left|\operatorname{det}\frac{\partial f^{-1}}{\partial y}\right|.
\end{equation}
We denote a sample and its transformation as
\begin{equation}
\begin{aligned}
\mathbf{z}_{i-1} & \sim p_{i-1}\left(\mathbf{z}_{i-1}\right) \\
\mathbf{z}_{i} &=f_{i}\left(\mathbf{z}_{i-1}\right), \mathbf{z}_{i-1}=f_{i}^{-1}\left(\mathbf{z}_{i}\right). \\
\end{aligned}
\end{equation}
Given the chain of the transformations
\begin{equation}
\mathbf{x}=\mathbf{z}_{K}=f_{K} \circ f_{K-1} \circ \cdots f_{1}\left(\mathbf{z}_{0}\right)
\end{equation}
the log-likelihood of the observed data $\vect{x}$ is concisely written as
\begin{equation}
\begin{aligned}
\log p(\mathbf{x})=\log p_{0}\left(\mathbf{z}_{0}\right)-\sum_{i=1}^{K} \log \left|\operatorname{det} \frac{d f_{i}}{d \mathbf{z}_{i-1}}\right|
\end{aligned}
\end{equation}
See appendix \ref{A} for the complete derivation. Notice how the log-likelihood of $p(\vect{x})$ only requires inference on the base distribution from the latent vector $\vect{z_0}$. All other contributions to the log-likelihood are from the absolute value of Jacobian determinant of the bijections. An ideal bijection in NFs should be:
\begin{itemize}
\item invertible in all cases
\item cheap to invert
\item have an easy to compute Jacobian determinant
\item expressive enough to acquire the target distribution.
\end{itemize}
For evaluation, bits per dimension (BPD) is used. BPD is defined as
\begin{equation}
bpd(\vect{x}) = \frac{\operatorname{log} p(\vect{x})}{W \times H \times C \times \operatorname{log}(2)}
\end{equation}
where $H$, $W$ and $C$ stand for height, width and channels of the image.
\subsubsection{Simple flows}
Elementwise bijections are the most simple form of transformations in NFs. Let $h: \mathbb{R} \mapsto \mathbb{R}$ be a bijection of scalar value. Then given vector $\vect{x} \in \mathbb{R}$ we define a flow as
\begin{equation}
\begin{aligned}
g(\vect{x})=(h(x_1), h(x_2),...,h(x_D))^T.
\end{aligned}
\end{equation}
Here the Jacobian determinant is just the product of the absolute values of the derivative of $h$. A logical extension towards element-wise flows are linear mappings as
\begin{equation}
g(\vect{x})=\matr{A}\vect{x}+\vect{b},
\end{equation}
where the observation matrix and bias are $\matr{A} \in \mathbb{R}^{DxD}$ and $\mathbf{b} \in \mathbb{R}^{D}$, respectively. Since the Jacobian determinant is $\det(\matr{A})$, several restrictions can be placed on the observation matrix to make the computation of the Jacobian determinant more efficient, such as forcing it to be diagonal or triangular. Furthermore, \textit{LU decomposition} on the observation matrix has shown to reduce computation complexity to $\mathcal{O}(D)$ and $\mathcal{O}(D^2)$ in the forward and backwards pass, respectively \cite{triang}. Element-wise and linear flows are not expressive enough to capture complex distributions. The output of these flows are always in the same family of distributions as the input. This is because the flows are not able to expand and contract densities. Planar flows and radial flows~\cite{nfvae} are able to do this. The bijection of the planar flow can be written as
\begin{equation}
g(\vect{x})=\vect{x} + \vect{u}h(\vect{w}^T\vect{x}+\vect{b}).
\end{equation}
For radial flows, the expansion and contraction is around a particular point as
\begin{equation}
g(\vect{x})=\vect{x} + \frac{\beta}{\alpha \left|\vect{x}-\vect{x}_0\right|}(\vect{x}-\vect{x}_0).
\end{equation}
Both flows need many consecutive bijections in order to model complex distributions. This can get computationally expensive. Also, the invertibility of the flow holds for only specific conditions. Their practicality has mostly shown in variational inference, as is done in the cited publication.
\subsubsection{Coupling Flows}
\label{coupling}
Dinh \textit{et al.}~\cite{nice} introduced a method for highly expressive transformations referred to as \textit{coupling} flows. Before passing through the bijection, the input $\vect{x} \in \mathbb{R}$ will be split into two subspaces $\vect{x}_A \in \mathbb{R}^d$ and $\vect{x}_B \in \mathbb{R}^{D-d}$. The bijection function is $h(\bullet, \theta(\bullet)): \mathbb{R}^d \mapsto \mathbb{R}^d$. With the \textit{conditioner} $\theta$ being any arbitrary function, we can regard the coupling flow $g:\mathbb{R}^D \mapsto \mathbb{R}^D$ as functions
\begin{equation}
\begin{aligned}
\vect{y}_A &= h(\vect{x}_A,\theta(\vect{x}_B))\\
\vect{y}_B &= \vect{x}_B.
\end{aligned}
\end{equation}
For the invertibility of the total flow we require the conditioner to be invertible as well. The inverse can be stated as
\begin{equation}
\begin{aligned}
\vect{x}_A &= h^{-1}(\vect{y}_A,\theta(\vect{x}_B))\\
\vect{x}_B &= \vect{y}_B.
\end{aligned}
\end{equation}
If we consider the Jacobian
\begin{equation}
\matr{J}_h=\left[\begin{array}{cc}
\frac{\partial\vect{y}_A}{\partial\vect{x}_A} & \frac{\partial \vect{y}_A}{\partial x_{B}} \\
0 & \mathbbm{I}_d
\end{array}\right]
\end{equation}
it can be seen that the determinant equals $\det \matr{J}_h = \frac{\partial \vect{y}_A}{\partial \vect{x}_B}$. The Jacobian is a triangular matrix in the case of a element-wise operating conditioner. See Figure~\ref{fig:coupling} for an illustration of the conventional coupling flow. Most state-of-the-art models are affine coupling-based NFs such as Glow (section \ref{sec:glow}) and variants of it (Section \ref{sec:waveletflow}). Other methods such as Nonlinear squared, Continuous Mixed CDF, Spline, Neural Autoregressive, Sum-of-Square and Real-and-Discrete coupling flows~\cite{nlsf, flow++, neuralimportantcesampling, cubic, neuralsplinef, neuralautoregressive, sos, rad} exist but has not seen equal success.
\begin{figure}[tp]
\begin{minipage}[b]{.48\linewidth}
\centering
\centerline{\includegraphics[width=4.0cm]{images/conventionalcoupling.png}}
\centerline{\textbf{A}}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering
\centerline{\includegraphics[width=4.0cm]{images/multiscalecoupling.png}}
\centerline{\textbf{B}}
\end{minipage}
\caption{(A) Conventional and (B) multi-scale coupling architecture in the normalizing direction}
\label{fig:coupling}
\end{figure}
In regular coupling flows, the conditioner is always dependent on the split vector $\vect{x}_B$. It is possible to have the conditioner be independent of $\vect{x}_B$ to develop something called a \textit{multi-scale} flow~\cite{realnvp} (Figure \ref{fig:coupling}). A \textit{multi-scale} flow introduces dimension to a bijective transformation $g$ in the generative direction. In the normalizing direction, the dimension of the vector that undergoes the bijection $g$ decreases. This is analogous to some degree with filters in convolutional neural networks (CNN) when the coupling vector becomes smaller as we traverse deeper in the network where it enables the network to capture more multi-scale information. This works especially well with natural images of landscapes because these images often contain a general structure over the whole canvas.
\section{Normalizing Flows derivation}
\label{A}
A bijection between distributions should adhere to the law of total probability as
\begin{equation}
\int p(x)dx= \int p(y)dy.
\end{equation}
The absolute change in volume is only relevant. In this fashion we can represent a sample in $Y$ in terms of a sample in $X$ as
\begin{equation}
p(y)=p(x)\left|\frac{dx}{dy}\right|.
\end{equation}
We now observe two variables in this transformation. We have a Uniform(0,1) distribution being linearly transformed in to any arbitrary parallelogram without shift with the matrix.
\begin{equation}
T=\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\end{equation}
Now, the change in absolute volume of the parallelogram is
\begin{equation}
\left|ad-cb\right|=\left|\operatorname{det}(T)\right|.
\end{equation}
Now for a multidimensional non-linear transformation, one approximates this with infinite, infinitesimally parallelograms to track the global change in volume. This local-linear change approximation is in fact the Jacobian determinant. Given that $y=f(x)$ thus $x=f^{-1}(y)$ we can write
\begin{equation}
p_Y(y)=p_X(f^{-1}(y))\left|\operatorname{det}\frac{\partial f^{-1}}{\partial x}\right|.
\end{equation}
Based on a chain of these operations as has been shown in Figure \ref{fig:flow} one can write
\begin{subequations}
\begin{align}
\mathbf{z}_{i-1} & \sim p_{i-1}\left(\mathbf{z}_{i-1}\right) \\
\mathbf{z}_{i} &=f_{i}\left(\mathbf{z}_{i-1}\right), \mathbf{z}_{i-1}=f_{i}^{-1}\left(\mathbf{z}_{i}\right)
\end{align}
\end{subequations}
and thus estimate the probability as
\begin{equation}
\begin{aligned}
p_{i}\left(\mathbf{z}_{i}\right) &=p_{i-1}\left(f_{i}^{-1}\left(\mathbf{z}_{i}\right)\right)\left|\operatorname{det} \frac{d f_{i}^{-1}}{d \mathbf{z}_{i}}\right|.
\end{aligned}
\end{equation}
The inverse function theorem and the property $\operatorname{det}(\matr{M}^{-1})=\operatorname{det}(\matr{M})^{-1}$ for invertible matrices $\matr{M}$ allows us to rewrite the transformation as
\begin{subequations}
\begin{align}
p_{i}\left(\mathbf{z}_{i}\right) &=p_{i-1}\left(f_{i}^{-1}\left(\mathbf{z}_{i}\right)\right)\left|\operatorname{det} \frac{d f_{i}^{-1}}{d \mathbf{z}_{i}}\right| \\
&=p_{i-1}\left(\mathbf{z}_{i-1}\right)\left|\operatorname{det}\left(\frac{d f_{i}}{d \mathbf{z}_{i-1}}\right)^{-1}\right| \\
&=p_{i-1}\left(\mathbf{z}_{i-1}\right)\left|\operatorname{det} \frac{d f_{i}}{d \mathbf{z}_{i-1}}\right|^{-1}
\end{align}
\end{subequations}
and we can state the likelihood of a transformation as
\begin{equation}
\begin{aligned}
\log p_{i}\left(\mathbf{z}_{i}\right) &=\log p_{i-1}\left(\mathbf{z}_{i-1}\right)-\log \left|\operatorname{det} \frac{d f_{i}}{d \mathbf{z}_{i-1}}\right|.
\end{aligned}
\end{equation}
Given the chain of the transformations
\begin{equation}
\mathbf{x}=\mathbf{z}_{K} =f_{K} \circ f_{K-1} \circ \cdots f_{1}\left(\mathbf{z}_{0}\right) \\
\end{equation}
and the log-likelihood of the observed data $\vect{x}$
\begin{subequations}
\begin{align}
\log p(\mathbf{x})=\log p_{K}\left(\mathbf{z}_{K}\right) &=\log p_{K-1}\left(\mathbf{z}_{K-1}\right)-\log \left|\operatorname{det} \frac{d f_{K}}{d \mathbf{z}_{K-1}}\right| \\
&=\log p_{K-2}\left(\mathbf{z}_{K-2}\right)-\log \left|\operatorname{det} \frac{d f_{K-1}}{d \mathbf{z}_{K-2}}\right|-\log \left|\operatorname{det} \frac{d f_{K}}{d \mathbf{z}_{K-1}}\right| \\
&=\ldots \notag
\end{align}
\end{subequations}
or concisely written as
\begin{equation}
\begin{aligned}
\log p(\mathbf{x})=\log p_{0}\left(\mathbf{z}_{0}\right)-\sum_{i=1}^{K} \log \left|\operatorname{det} \frac{d f_{i}}{d \mathbf{z}_{i-1}}\right|
\end{aligned}
\end{equation}
\section{Low level frequency dependent distributions}
\label{B}
\begin{figure}[htp]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=9cm]{images/multinoiselowlvltrain.png}}
\caption{Low level likelihood estimations of the malignant out-of-dataset images with multiplicative Gaussian noise of 0, $0.05^2$ and $0.1^2$ variance from top to bottom.}
\label{fig:ood-fr4}
\end{minipage}
\end{figure}
\begin{figure}[htp]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=9cm]{images/multinoiselowlvl.png}}
\caption{Low level likelihood estimations of the malignant out-of-dataset images with multiplicative Gaussian noise of 0, $0.05^2$ and $0.1^2$ variance from top to bottom.}
\label{fig:ood-fr5}
\end{minipage}
\end{figure}
\begin{figure}[htp]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=9cm]{images/multinoiselowlvlOOD.png}}
\caption{Low level likelihood estimations of the malignant out-of-dataset images with multiplicative Gaussian noise of 0, $0.05^2$ and $0.1^2$ variance from top to bottom.}
\label{fig:ood-fr6}
\end{minipage}
\end{figure}
\section{Samples from GLOW and Wavelet Flow}
\label{C}
\begin{figure}[hbp]
\begin{minipage}[b]{.48\linewidth}
\centering
\centerline{\includegraphics[width=7.0cm]{images/sapmplesisicc.png}}
\centerline{\textbf{A} GLOW}
\end{minipage}
\hfill
\begin{minipage}[b]{0.48\linewidth}
\centering
\centerline{\includegraphics[width=7.0cm]{images/samplewaveletflow.png}}
\centerline{\textbf{B} Wavelet Flow}
\end{minipage}
\caption{Generated images representing benign melanoma}
\label{fig:ood-wvlt2}
\vspace{-0.5cm}
\end{figure}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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#include <aws/greengrass/model/Subscription.h>
#include <aws/core/utils/json/JsonSerializer.h>
#include <utility>
using namespace Aws::Utils::Json;
using namespace Aws::Utils;
namespace Aws
{
namespace Greengrass
{
namespace Model
{
Subscription::Subscription() :
m_idHasBeenSet(false),
m_sourceHasBeenSet(false),
m_subjectHasBeenSet(false),
m_targetHasBeenSet(false)
{
}
Subscription::Subscription(JsonView jsonValue) :
m_idHasBeenSet(false),
m_sourceHasBeenSet(false),
m_subjectHasBeenSet(false),
m_targetHasBeenSet(false)
{
*this = jsonValue;
}
Subscription& Subscription::operator =(JsonView jsonValue)
{
if(jsonValue.ValueExists("Id"))
{
m_id = jsonValue.GetString("Id");
m_idHasBeenSet = true;
}
if(jsonValue.ValueExists("Source"))
{
m_source = jsonValue.GetString("Source");
m_sourceHasBeenSet = true;
}
if(jsonValue.ValueExists("Subject"))
{
m_subject = jsonValue.GetString("Subject");
m_subjectHasBeenSet = true;
}
if(jsonValue.ValueExists("Target"))
{
m_target = jsonValue.GetString("Target");
m_targetHasBeenSet = true;
}
return *this;
}
JsonValue Subscription::Jsonize() const
{
JsonValue payload;
if(m_idHasBeenSet)
{
payload.WithString("Id", m_id);
}
if(m_sourceHasBeenSet)
{
payload.WithString("Source", m_source);
}
if(m_subjectHasBeenSet)
{
payload.WithString("Subject", m_subject);
}
if(m_targetHasBeenSet)
{
payload.WithString("Target", m_target);
}
return payload;
}
} // namespace Model
} // namespace Greengrass
} // namespace Aws
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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|
\section{Introduction}
In the past decade a new wave of theoretical and experimental
interest in electron capture processes, involving two active
electrons, as double capture (DC), transfer ionization (TI) and
transfer excitation (TE), has shed light on the versatile effects
of electron correlation. New experimental techniques allow to
measure more than only total or single differential cross section
(SDCS). Fully differential cross sections (FDCS), which depend on
the momentum distribution of the escaped electron in TI give a
rather detailed view in the dynamical processes taking place. In
particular, in this paper we consider the reaction p+He
$\rightarrow$ H+He$^{2+}$+$e^-$.
Since the early publications \cite{OBK,Thomas} it became clear
that two principal mechanisms contribute to the transfer
ionization. This takes place via a capture of one electron with a
correlated (shake-off, SO) or sequential process (binary
encounter, BE), removal of the second electron. We use this
terminology in accord with single photon ionization of an atom
\cite{Briggs2000jpb, Knapp2002prl} in spite of quite different
transfer energies in both cases. Let us concentrate further on the
single transfer ionization, because this process is a subject of
this paper. Direct capture presumes the "usurpation" of one target
electron by the fast projectile proton, like it was described in
\cite{OBK}, and releasing of another electron due to the sudden
rearrangement of the field in the residual ion (typical SO). If
the fast proton is described by the plane wave in the lab frame,
and its scattering angles are very small (fractions of mrad), then
the OBK-mechanism \cite{OBK} presents the principal transition
matrix element alike to that for Electron Momentum Spectroscopy
\cite{EMS} (see also \cite{JETP}). In turn, it was shown that
latter one is very sensitive to angular and radial
electron-electron correlations in the target \cite{Takahashi}.
The captured electron always moves forward parallel to the
velocity vector of the proton projectile, i.e. its momentum
component is positive. If the electron-electron correlation in the
target is weak (say, only due to a mean field), the emitted
electron will be shaken off isotropically. In the opposite case of
strong angular correlations it moves predominantly in backward
direction ($k_{||}<0$) and we expect to see a backward peak in the
electron momentum distribution. A different process (analogue to
radiative electron capture) also resulting in a backward emitted
electron was suggested by Voitkiv and coworkers
\cite{Voitkiv2008prl,Schulz2012prl}. These calculations lack of
high differentiality, as the are neither in the scattering angle
dependent nor in the scattering plane. Therefore we will show our
data only in the longitudinal vs. transversal representation.
The sequential mechanism of TI presumes at least two successive
interactions of the fast projectile with both target electrons.
For its realization no electron-electron correlations are needed.
This mechanism in general is of the second order (and higher) in
the projectile-target interaction. However, features of the
capture processes allow to define transfer ionization already with
a first order amplitude \cite{Salim10}. After interaction of the
bound electron with the fast projectile proton it becomes also
fast. It can interact again with another electron or the target
nucleus on its way out (pure second order $Ne$- and $ee$-Thomas,
for example \cite{Thomas,Briggs}), but its movement keeps in
general the forward character $k_{||}>0$. So, the forward peak can
be connected with the BE mechanism; capture and ionization are
generally independent.
Of course, the above considerations are semiclassical, we shall
see an interplay of quantum mechanisms and coherent sum of
corresponding matrix elements, but we expect the general
forward-backward features to be present also in a full quantum
treatment.
We think, it is a time to defend the PWFBA, because first Born
theories are often believed to be inadequate for electron capture.
We would like to stress that it is not so. First, any FBA theory
works well until the higher Born terms become bigger in the region
of final state phase space considered. So for example at very
small scattering angles of a fast projectile ion (proton), the OBK
term is a leading one but if falls down rapidly with increasing
scattering angle, and the higher order terms begin to contribute.
But they do not contribute much at very small angles (see
calculations in \cite{Kim}). Second, the OBK matrix element, as it
was considered 80 years ago, now can include much better
correlated trial wave functions. This plays a crucial role for
transfer excitation and transfer ionization reactions (less for
charge transfer). At very small projectile scattering angles, the
corresponding SDCS curves for highly and loosely correlated target
ground functions start to differ substantially. Third, we have now
two main approaches for capture reaction: time-dependent
semiclassical and pure quantum mechanical. Within the
time-dependent approach the heavy fast projectile particle is
considered a source of a classical outer field. The interaction of
both, projectile and target ion, can easily be taken into account
as a phase-factor to the final (initial) wave function
\cite{Belcic}. In the pure quantum approach (see, for example,
\cite{Kim}) the interaction of heavy particles gives a
contribution to the FBA, and this term distorts noticeably the OBK
term both at very small (here the distortion is "positive", it
diminishes the peak value) and at larger scattering angles (here
this distortion is "negative", it increases the plateau). It was
shown that SBA terms can compensate this negative effect and
considerably improve the agreement between theory and experiment
\cite{Kim}. Physically it is clear that if the transferred
momentum and energy are relatively small (we are in a laboratory
frame, and this takes place at very small scattering angles), then
the velocity of atomic nucleus is practically zero, and it is
actually immovable during the scattering process. The nucleus
simply changes the initially directed path of the projectile due
to elastic scattering ("secondary" OBK). This is the main physical
role of this term at scattering angles close to zero. But its FBA
realization distorts this picture at larger angles, and the SBA
provides with necessary corrections.
In this paper we present experimental results and calculate fully
differential cross sections (FDCS) within the plane wave first
Born approximation (PWFBA) on proton-helium interaction at impact
energies of 300 and 630 keV. Both discussed above mechanisms, SO
and BE, contribute in this case.
Atomic units $\hslash = e = m_e = 1$ are used throughout unless
otherwise specified.
\section{Experiment}
To achieve the goals of this experiment all emitted particles have
to be measured in coincidence. Therefore we applied momentum
spectroscopy techniques, as reactions microscopes or COLTRIMS
(COLd Target Recoil Ion Momentum Spectroscopy)
\cite{Ullrich1997jpb,Doerner2000pr,Ullrich2003rpp}. The
experiments were performed at the Institut f\"ur Kernphysik at the
University of Frankfurt using the Van de Graaff accelerator. Using
3 sets of movable slits, the proton beam was collimated to a
divergence less than 0.15 mrad, an size of about 0.5 $\times$ 0.5
mm$^2$ at the overlap region with the gas jet. 15 cm upstream of
the target, a set of parallel electrostatic deflector plates
cleaned the primary beam from charge state impurities, deflecting
the primary beam slightly upwards. The H$^+$ beam was crossed
perpendicular with the helium gas jet. 15 cm downstream of the
target a second set of horizontal electrostatic deflector plates
separate the final charge state, thus only the neutral projectiles
H hit a position and time sensitive multichannel plate (MCP)
detector, placed 3 m downsteam the interaction point, yielding the
projectile deflection angle and the time zero of the collision.
The main part of the beam ($\approx$ 1 nA), which is still charged
was dumped in a Faraday cup.
The gas jet providing the target beam was generated by helium
gasexpanding through a 30 $\mu$m nozzle with a backing pressure of
20 bar and collimated in a two stage jet. A density of
$5\times10^{11}$ atoms/cm$^2$ and a diameter of 1.5 mm were
achieved. The active cooling by the supersonic expansion in
expansion direction combined with passive one in the perpendicular
direction by the geometry resulted in a 3 dimensional cold target
and a momentum uncertainty below 0.1 a.u.
At the intersection volume where proton and helium beam were
intersected, electrons and ions were created. A weak electrostatic
field of 4.8 V/cm was applied to project electrons and recoiling
ions onto two position and time sensitive detectors. To optimize
the resolution, a three dimensional time and space focusing
geometry \cite{Schoeffler2011njp,Mergel1995prl} was used for the
recoil ion arm of the spectrometer. The ion were detected by a
80~mm diameter micro channel plate (MCP) detector with delay-line
anode \cite{Jagutzki2002nima,Jagutzki2002nima2}. The time focusing
was realized using a field free drift tube \cite{WWil55}, while an
adjustable electrostatic lens was used to achieve space focusing.
This lens was optimized by minimizing the spatial width of the
lines on the detector from He$^+$ ions created by pure capture,
which have been recorded parallel to the transfer ionization
events (for an example see Fig. 1 in \cite{Schoeffler2009pra1} or
Fig. 1 in \cite{Kim}). A momentum resolution of 0.1 a.~u. was
achieved in all three directions. The electrons were guided by a
magnetic field (see \cite{Moshammer96nim}) of 15 and 25~Gauss and
accelerated over a length of 20~cm by the same electric field in a
time focusing geometry (40~cm additional field free drift tube)
onto a MCP detector of 120~mm active diameter. The overall
spectrometer geometry, especially the ion's part was simulated
using SIMION to gain the maximum resolution and efficiency.
We reached an overall acceptance of 4$\pi$ solid angle for recoil
ions up to a momentum of 10 a.u. and electrons up to 6 a.u. A
three-particle coincidence (H$^0$+He$^{2+}$+$e$) was applied to
record the data event by event. From the positions of impact on
the detectors and the time-of-flight we can derive the initial
momentum vectors of the recoil ion and the electron. The
projectile transverse momentum vectors were directly measured.
Checking energy and momentum conservation the background was
strongly suppressed during the off-line data analysis. Also the
overall resolution was good enough to measure the final electronic
state of the H and separate events where the hydrogen was found in
the ground state from where the electron was captured into an
excited state. Only these events, where the hydrogen is in the
ground state are presented in the following.
\section{Theory}
As stated above, we consider the He atom as a target for the TI
reaction. We follow definitions and notations given in
\cite{Salim10} and not repeat all conditions here. In the momentum
representation in the lab frame and at very small scattering angle
$\theta_p$ the symmetrized matrix element is given by
$$
\mathcal{T}_{FBA} = -4\pi\sqrt 2 \int \frac{d\vec x}{(2\pi)^3} \frac{
\widetilde{\phi}_{H}(x)} {\vert \vec v_p - \vec q - \vec
x\vert^2}[F(\vec q;0;\vec k) + F(\vec v_p - \vec x; -\vec v_p +
\vec q + \vec x; \vec k)
$$
$$
- 2F(\vec v_p - \vec x;0;\vec k)]=A1+A2+A3, \eqno (1)
$$
where
$$
{F(\vec y; \vec \eta; \vec k)} = \int e^{-i\vec y \vec r_1 - i\vec
\eta \vec r_2} \varphi_c^{-*}(\vec k, \vec r_2)\Phi_0 (\vec r_1,
\vec r_2) d\vec r_1 d\vec r_2, \eqno (2)
$$
$\vec v_p$ is the fast proton velocity, the transferred momentum
$\vec q=\vec p_H-\vec p_p$, $\vec k$ the electron momentum,
$\Phi_0({\vec r_1},{\vec r_2})$ the helium ground wave function,
and the Coulomb wave function of the final target ion
$$
\varphi_c^{-*}(\vec k, \vec r) =e^{-\pi\xi/2}\Gamma(1+i\xi)e^{-i{\vec k}{%
\vec r}}{_1}F_1(-i\xi,1;ikr+i{\vec k}{\vec r}); \quad \xi=-2/k.
$$
The FDCS is calculated by the formula
$$
\frac{d^2\sigma}{dk_{\perp}
dk_{||}}=\frac{m^2k_{\perp}}{(2\pi)^4}\int\limits_0^{\theta_{max}}\theta_pd\theta_p
\int\limits_0^{2\pi}d\phi_k |A1+A2+A3|^2, \eqno (3)
$$
with $m=1836.15$ being the proton mass. We display all vectors'
components for clarity: $\vec v_p=\{0,0,v_p\},\ \vec
q=\{mv_p\theta_p,0,q_{||}\},\ \vec
k=\{k_\perp\cos\phi_k,k_\perp\sin\phi_k,k_{||}\}$. We also remind
that $q_{||}={v_p}/{2}+{Q}/{v_p}$ with $Q=E_0^{He}-E^H-k^2/2$.
In (1) the term $A_1$ is the OBK amplitude, where any trial helium
wave function can be used. The amplitude $A_3$ can also be
attributed to SO. It describes the contribution of heavy particles
interaction and was discussed in the Introduction. The amplitude
$A_2$ is a typical PWFBA realization of the BE mechanism.
\section{Results and Discussion}
For calculations we use three trial helium wave functions. One is
the loosely correlated $1s^2$ Roothaan-Haartree-Fock (RHF)
function of Clementi and Roetti \cite{RHF} ($E_0^{He}$=-2.8617).
The two others are a highly correlated function of the type
$$
\Psi(r_1,r_2,r_{12})=\sum_{j=1}^{N}D_j\left[\exp(-\alpha_j
r_1-\beta_j r_2)+\exp(-\alpha_j r_2-\beta_j
r_1)\right]\exp(-\gamma_j r_{12}), \eqno (4)
$$
which was described in \cite{Chuka06} ($E_0^{He}$=-2.9037), and
the configuration interaction (CI) wave function of Mitroy
\cite{Mitroy} ($E_0^{He}$=-2.9031).
\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{figure1.eps}
\caption{(Color online) Experimental momentum distribution of the electron
for a) $E_p=$300 keV and b) $E_p=$630 keV. The projectile is moving in the
positive $k_{||}$ direction, i. e. from the left to the right.
The data are integrated over all other observables, i. e. the integral
over the shown distribution corresponds to the total transfer ionization
cross section for the H(n=1) state.}
\end{figure}
The experimental data at $E_p=300$ and $E_p=630$ keV, shown in
Figure 1, display a noticeable peak at backward (negative
$k_{||}$) direction and a less resolved peak at forward direction
(positive $k_{||}$). The forward peak structure has more intensity
at the lower projectile energy of 300 keV, as the
projectile-target interaction time is longer and therefore an
additional interaction, the electron knock-off, more likely to
occur.
As expected calculations with the loosely correlated wave function
[Figs. 2(c) and 2(d)] give practically no backward peak to the
electron's distribution. Both highly correlated helium wave
functions give very similar distributions [Figs. 2(a) and 2(b)],
which include both forward and backward peaks. However, visually
they are hard to compare with the experiment.
To avoid effects of color scales, we present additionally two
slices of these distributions at $E_p=300$ keV and fixed
$k_\perp$: $k_\perp$=0.2 in Fig. 3(a) and $k_\perp$=0.4 in Fig.
3(b). The experimental points are normalized to the theory's peak
maximum along the whole distributions. First, we clearly see that
both used correlated wave functions give practically the same
curves. Second, theory and experiment well coincide at negative
$k_{||}$, what clearly demonstrate that the PWFBA shake-off
amplitude is quite sufficient to describe the backward peak. This
requires of course, the use of highly correlated target wave
functions. Third, we see that the theory noticeably exceeds the
experimental points in the forward domain $k_{||}>0$. It is a
clear indication that the SBA calculations are needed here.
Unfortunately, we cannot provide these calculations at the moment.
We finally show a comparison of the total transfer ionization
cross section in PWFBA theory, using Mitroy helium wave function,
and experiment. In Figure 4 the agreement is quite satisfactory
over a wide range of the proton energies.
\section{Conclusions}
In conclusion, we presented highly differential theory (PWFBA) and
experimental data from a kinematical complete experiment on
transfer ionization in proton-Helium-collision at 300 and 630 keV.
The observed splitting into forward and backward emission
originates from two different contributions, the $A_2$-term
(binary encounter) and the $A_1+A_3$-term (shake-off). Comparison
of loosely and highly correlated wave functions for the initial
state confirms the high sensitivity of the experiment to the
subtle features of the initial state wave function. Better
agreement for the forward emitted electrons can be expected for
calculations in the second order. At the same time, backward
emitted electrons can be described within the first Born
approximation at high projectile energies.
\section{Acknowledgements}
We acknowledge financial support from the Deutsche
Forschungsgemeinschaft (DFG), Grant No. SCHO 1210/2-1. As well
this work partially supported by Russian Foundation of Basic
Research (RFBR), Grant No. 11-01-00523-a. All calculations were
performed using Moscow State University Research Computing Centre
(supercomputers Lomonosov and Chebyshev) and Joint Institute for
Nuclear Research Central Information and Computer Complex. The
authors are grateful to K. Kouzakov for inspiring discussions and help.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 2,227
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/118243\/maximum-likelihood-estimator-from-sample-distribution-n0-sigma2x-i2\/118249#118249","text":"# Maximum Likelihood estimator from sample distribution $N(0,\\sigma^2x_i^2)$\n\nLet independent random variable $Y_1,...,Y_n$ have respective distributions $N(0,\\sigma^2x_i^2)$, where $i=1,2,...,n$ are known constants such that $x_i\\neq 0$ for all $i=1,2,...,n$. Find the maximum likelihood estimator of $\\hat{\\sigma}^2$, of the unknown parameter $\\sigma^2$.\n\nMy attempt:\n\nThe likelihood function is $L(\\sigma^2)=\\prod_{i=1}^n\\frac{1}{\\sqrt{2\\pi\\sigma^2x_i^2}}exp(-\\frac{y_i}{2\\sigma^2x_i^2})=(2\\pi\\sigma^2)^{-\\frac{n}{2}}\\prod_{i=1}^n\\frac{1}{x_i}\\prod_{i=1}^nexp(-\\frac{y_i}{2\\sigma^2x_i^2})$. Since the $x_i$ are constants, I'll replace $\\prod_{i=1}^n\\frac{1}{x_i}$ with the constant $K$. Now I'm not exactly sure how I can rewrite the $\\prod_{i=1}^nexp(-\\frac{y_i}{2\\sigma^2x_i^2})$ term in a useful way. I thought of maybe saying $\\sum_i\\frac{y_i}{2\\sigma^2x_i^2}=\\frac{\\sum_i(\\frac{y_i}{x_i})\\prod_ix_i}{2\\sigma^2\\prod_ix_i}=\\frac{\\sum_i(\\frac{y_i}{x_i})}{2\\sigma^2}$, so then $L(\\sigma^2)=(2\\pi\\sigma^2)^{-\\frac{n}{2}}Kexp(-\\frac{\\sum_i(\\frac{y_i}{x_i})}{2\\sigma^2})$. Then I would maximize $ln(L(\\sigma^2))=-\\frac{n}{2}ln(2\\pi\\sigma^2)+ln(K)-\\frac{\\sum_i(\\frac{y_i}{x_i})}{2\\sigma^2}.$\n\nThis doesn't seem right though. Can anyone point out where I might have made a mistake or how I should go about this?\n\n\u2022 $\\displaystyle \\left(\\frac{Y_1}{x_1},\\frac{Y_2}{x_2}, \\cdots, \\frac{Y_n}{x_n}\\right)$ is a set of $n$ independent $N(0,\\sigma^2)$ random variables, and your book probably has a full description of the maximum-likelihood estimator of $\\sigma^2$ in terms of the $\\frac{Y_i}{x_i}$. So, apply this known solution to the scaled observations $\\frac{y_i}{x_i}, 1 \\leq i \\leq n$. Oct 8 '14 at 2:11\n\nI think you miss square of $y_i$ in the likelihood. Then the Likelihood is $\\Pi_{i=1}^n\\frac{1}{\\sqrt{2\\pi x_i^2 \\sigma^2}}e^{-\\frac{y_i^2}{2x_i^2 \\sigma^2}}=e^{-\\sum_{i=1}^n\\frac{y_i^2}{2x_i^2 \\sigma^2}} \\Pi_{i=1}^n\\frac{1}{\\sqrt{2\\pi x_i^2 \\sigma^2}}$. Taking log-likelihood and differentiating w.r.t. $\\sigma$, you get $$\\sum_{i=1}^n\\frac{y_i^2}{\\sigma^3 x_i^2}-\\frac{n}{\\sigma}=0$$.\nFrom this, you get $\\sigma^2=\\frac{\\sum_{i=1}^n\\frac{y_i^2}{x_i^2}}{n}$, which seems familiar...\n1. the density function $f_i$ of $Y_i \\sim \\mathcal{N}(0, \\sigma^2 x_i^2)$ is $$f_i (y_i) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2 x_i^2}} \\mathrm{exp}\\left\\{ - \\frac{1}{2} \\frac{(y_i - 0)^2}{\\sigma^2 x_i^2} \\right\\}$$\n2. e.g. $x_1 = -1$ and $x_2 = \\dots = x_n = 1$, so $$\\prod_{i=1}^n \\frac{1}{\\sqrt{x_i^2}} = \\frac{1}{1} \\cdot \\dots \\cdot \\frac{1}{1} = 1$$ but $$\\prod_{i=1}^n \\frac{1}{x_i} = \\frac{1}{-1} \\cdot \\frac{1}{1} \\cdot \\dots \\cdot \\frac{1}{1} = -1$$","date":"2021-10-22 19:35:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9676274061203003, \"perplexity\": 93.40276335496915}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585518.54\/warc\/CC-MAIN-20211022181017-20211022211017-00666.warc.gz\"}"}
| null | null |
package com.microsoft.azure.management.cosmosdb.v2015_04_08.implementation;
import retrofit2.Retrofit;
import com.google.common.reflect.TypeToken;
import com.microsoft.azure.CloudException;
import com.microsoft.rest.ServiceCallback;
import com.microsoft.rest.ServiceFuture;
import com.microsoft.rest.ServiceResponse;
import java.io.IOException;
import java.util.List;
import okhttp3.ResponseBody;
import retrofit2.http.GET;
import retrofit2.http.Header;
import retrofit2.http.Headers;
import retrofit2.http.Path;
import retrofit2.http.Query;
import retrofit2.Response;
import rx.functions.Func1;
import rx.Observable;
/**
* An instance of this class provides access to all the operations defined
* in Databases.
*/
public class DatabasesInner {
/** The Retrofit service to perform REST calls. */
private DatabasesService service;
/** The service client containing this operation class. */
private CosmosDBImpl client;
/**
* Initializes an instance of DatabasesInner.
*
* @param retrofit the Retrofit instance built from a Retrofit Builder.
* @param client the instance of the service client containing this operation class.
*/
public DatabasesInner(Retrofit retrofit, CosmosDBImpl client) {
this.service = retrofit.create(DatabasesService.class);
this.client = client;
}
/**
* The interface defining all the services for Databases to be
* used by Retrofit to perform actually REST calls.
*/
interface DatabasesService {
@Headers({ "Content-Type: application/json; charset=utf-8", "x-ms-logging-context: com.microsoft.azure.management.cosmosdb.v2015_04_08.Databases listMetrics" })
@GET("subscriptions/{subscriptionId}/resourceGroups/{resourceGroupName}/providers/Microsoft.DocumentDB/databaseAccounts/{accountName}/databases/{databaseRid}/metrics")
Observable<Response<ResponseBody>> listMetrics(@Path("subscriptionId") String subscriptionId, @Path("resourceGroupName") String resourceGroupName, @Path("accountName") String accountName, @Path("databaseRid") String databaseRid, @Query("api-version") String apiVersion, @Query("$filter") String filter, @Header("accept-language") String acceptLanguage, @Header("User-Agent") String userAgent);
@Headers({ "Content-Type: application/json; charset=utf-8", "x-ms-logging-context: com.microsoft.azure.management.cosmosdb.v2015_04_08.Databases listUsages" })
@GET("subscriptions/{subscriptionId}/resourceGroups/{resourceGroupName}/providers/Microsoft.DocumentDB/databaseAccounts/{accountName}/databases/{databaseRid}/usages")
Observable<Response<ResponseBody>> listUsages(@Path("subscriptionId") String subscriptionId, @Path("resourceGroupName") String resourceGroupName, @Path("accountName") String accountName, @Path("databaseRid") String databaseRid, @Query("api-version") String apiVersion, @Query("$filter") String filter, @Header("accept-language") String acceptLanguage, @Header("User-Agent") String userAgent);
@Headers({ "Content-Type: application/json; charset=utf-8", "x-ms-logging-context: com.microsoft.azure.management.cosmosdb.v2015_04_08.Databases listMetricDefinitions" })
@GET("subscriptions/{subscriptionId}/resourceGroups/{resourceGroupName}/providers/Microsoft.DocumentDB/databaseAccounts/{accountName}/databases/{databaseRid}/metricDefinitions")
Observable<Response<ResponseBody>> listMetricDefinitions(@Path("subscriptionId") String subscriptionId, @Path("resourceGroupName") String resourceGroupName, @Path("accountName") String accountName, @Path("databaseRid") String databaseRid, @Query("api-version") String apiVersion, @Header("accept-language") String acceptLanguage, @Header("User-Agent") String userAgent);
}
/**
* Retrieves the metrics determined by the given filter for the given database account and database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of metrics to return. The parameters that can be filtered are name.value (name of the metric, can have an or of multiple names), startTime, endTime, and timeGrain. The supported operator is eq.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @throws CloudException thrown if the request is rejected by server
* @throws RuntimeException all other wrapped checked exceptions if the request fails to be sent
* @return the List<MetricInner> object if successful.
*/
public List<MetricInner> listMetrics(String resourceGroupName, String accountName, String databaseRid, String filter) {
return listMetricsWithServiceResponseAsync(resourceGroupName, accountName, databaseRid, filter).toBlocking().single().body();
}
/**
* Retrieves the metrics determined by the given filter for the given database account and database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of metrics to return. The parameters that can be filtered are name.value (name of the metric, can have an or of multiple names), startTime, endTime, and timeGrain. The supported operator is eq.
* @param serviceCallback the async ServiceCallback to handle successful and failed responses.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the {@link ServiceFuture} object
*/
public ServiceFuture<List<MetricInner>> listMetricsAsync(String resourceGroupName, String accountName, String databaseRid, String filter, final ServiceCallback<List<MetricInner>> serviceCallback) {
return ServiceFuture.fromResponse(listMetricsWithServiceResponseAsync(resourceGroupName, accountName, databaseRid, filter), serviceCallback);
}
/**
* Retrieves the metrics determined by the given filter for the given database account and database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of metrics to return. The parameters that can be filtered are name.value (name of the metric, can have an or of multiple names), startTime, endTime, and timeGrain. The supported operator is eq.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<MetricInner> object
*/
public Observable<List<MetricInner>> listMetricsAsync(String resourceGroupName, String accountName, String databaseRid, String filter) {
return listMetricsWithServiceResponseAsync(resourceGroupName, accountName, databaseRid, filter).map(new Func1<ServiceResponse<List<MetricInner>>, List<MetricInner>>() {
@Override
public List<MetricInner> call(ServiceResponse<List<MetricInner>> response) {
return response.body();
}
});
}
/**
* Retrieves the metrics determined by the given filter for the given database account and database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of metrics to return. The parameters that can be filtered are name.value (name of the metric, can have an or of multiple names), startTime, endTime, and timeGrain. The supported operator is eq.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<MetricInner> object
*/
public Observable<ServiceResponse<List<MetricInner>>> listMetricsWithServiceResponseAsync(String resourceGroupName, String accountName, String databaseRid, String filter) {
if (this.client.subscriptionId() == null) {
throw new IllegalArgumentException("Parameter this.client.subscriptionId() is required and cannot be null.");
}
if (resourceGroupName == null) {
throw new IllegalArgumentException("Parameter resourceGroupName is required and cannot be null.");
}
if (accountName == null) {
throw new IllegalArgumentException("Parameter accountName is required and cannot be null.");
}
if (databaseRid == null) {
throw new IllegalArgumentException("Parameter databaseRid is required and cannot be null.");
}
if (this.client.apiVersion() == null) {
throw new IllegalArgumentException("Parameter this.client.apiVersion() is required and cannot be null.");
}
if (filter == null) {
throw new IllegalArgumentException("Parameter filter is required and cannot be null.");
}
return service.listMetrics(this.client.subscriptionId(), resourceGroupName, accountName, databaseRid, this.client.apiVersion(), filter, this.client.acceptLanguage(), this.client.userAgent())
.flatMap(new Func1<Response<ResponseBody>, Observable<ServiceResponse<List<MetricInner>>>>() {
@Override
public Observable<ServiceResponse<List<MetricInner>>> call(Response<ResponseBody> response) {
try {
ServiceResponse<PageImpl<MetricInner>> result = listMetricsDelegate(response);
List<MetricInner> items = null;
if (result.body() != null) {
items = result.body().items();
}
ServiceResponse<List<MetricInner>> clientResponse = new ServiceResponse<List<MetricInner>>(items, result.response());
return Observable.just(clientResponse);
} catch (Throwable t) {
return Observable.error(t);
}
}
});
}
private ServiceResponse<PageImpl<MetricInner>> listMetricsDelegate(Response<ResponseBody> response) throws CloudException, IOException, IllegalArgumentException {
return this.client.restClient().responseBuilderFactory().<PageImpl<MetricInner>, CloudException>newInstance(this.client.serializerAdapter())
.register(200, new TypeToken<PageImpl<MetricInner>>() { }.getType())
.registerError(CloudException.class)
.build(response);
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @throws CloudException thrown if the request is rejected by server
* @throws RuntimeException all other wrapped checked exceptions if the request fails to be sent
* @return the List<UsageInner> object if successful.
*/
public List<UsageInner> listUsages(String resourceGroupName, String accountName, String databaseRid) {
return listUsagesWithServiceResponseAsync(resourceGroupName, accountName, databaseRid).toBlocking().single().body();
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param serviceCallback the async ServiceCallback to handle successful and failed responses.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the {@link ServiceFuture} object
*/
public ServiceFuture<List<UsageInner>> listUsagesAsync(String resourceGroupName, String accountName, String databaseRid, final ServiceCallback<List<UsageInner>> serviceCallback) {
return ServiceFuture.fromResponse(listUsagesWithServiceResponseAsync(resourceGroupName, accountName, databaseRid), serviceCallback);
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<UsageInner> object
*/
public Observable<List<UsageInner>> listUsagesAsync(String resourceGroupName, String accountName, String databaseRid) {
return listUsagesWithServiceResponseAsync(resourceGroupName, accountName, databaseRid).map(new Func1<ServiceResponse<List<UsageInner>>, List<UsageInner>>() {
@Override
public List<UsageInner> call(ServiceResponse<List<UsageInner>> response) {
return response.body();
}
});
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<UsageInner> object
*/
public Observable<ServiceResponse<List<UsageInner>>> listUsagesWithServiceResponseAsync(String resourceGroupName, String accountName, String databaseRid) {
if (this.client.subscriptionId() == null) {
throw new IllegalArgumentException("Parameter this.client.subscriptionId() is required and cannot be null.");
}
if (resourceGroupName == null) {
throw new IllegalArgumentException("Parameter resourceGroupName is required and cannot be null.");
}
if (accountName == null) {
throw new IllegalArgumentException("Parameter accountName is required and cannot be null.");
}
if (databaseRid == null) {
throw new IllegalArgumentException("Parameter databaseRid is required and cannot be null.");
}
if (this.client.apiVersion() == null) {
throw new IllegalArgumentException("Parameter this.client.apiVersion() is required and cannot be null.");
}
final String filter = null;
return service.listUsages(this.client.subscriptionId(), resourceGroupName, accountName, databaseRid, this.client.apiVersion(), filter, this.client.acceptLanguage(), this.client.userAgent())
.flatMap(new Func1<Response<ResponseBody>, Observable<ServiceResponse<List<UsageInner>>>>() {
@Override
public Observable<ServiceResponse<List<UsageInner>>> call(Response<ResponseBody> response) {
try {
ServiceResponse<PageImpl<UsageInner>> result = listUsagesDelegate(response);
List<UsageInner> items = null;
if (result.body() != null) {
items = result.body().items();
}
ServiceResponse<List<UsageInner>> clientResponse = new ServiceResponse<List<UsageInner>>(items, result.response());
return Observable.just(clientResponse);
} catch (Throwable t) {
return Observable.error(t);
}
}
});
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of usages to return. The supported parameter is name.value (name of the metric, can have an or of multiple names).
* @throws IllegalArgumentException thrown if parameters fail the validation
* @throws CloudException thrown if the request is rejected by server
* @throws RuntimeException all other wrapped checked exceptions if the request fails to be sent
* @return the List<UsageInner> object if successful.
*/
public List<UsageInner> listUsages(String resourceGroupName, String accountName, String databaseRid, String filter) {
return listUsagesWithServiceResponseAsync(resourceGroupName, accountName, databaseRid, filter).toBlocking().single().body();
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of usages to return. The supported parameter is name.value (name of the metric, can have an or of multiple names).
* @param serviceCallback the async ServiceCallback to handle successful and failed responses.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the {@link ServiceFuture} object
*/
public ServiceFuture<List<UsageInner>> listUsagesAsync(String resourceGroupName, String accountName, String databaseRid, String filter, final ServiceCallback<List<UsageInner>> serviceCallback) {
return ServiceFuture.fromResponse(listUsagesWithServiceResponseAsync(resourceGroupName, accountName, databaseRid, filter), serviceCallback);
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of usages to return. The supported parameter is name.value (name of the metric, can have an or of multiple names).
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<UsageInner> object
*/
public Observable<List<UsageInner>> listUsagesAsync(String resourceGroupName, String accountName, String databaseRid, String filter) {
return listUsagesWithServiceResponseAsync(resourceGroupName, accountName, databaseRid, filter).map(new Func1<ServiceResponse<List<UsageInner>>, List<UsageInner>>() {
@Override
public List<UsageInner> call(ServiceResponse<List<UsageInner>> response) {
return response.body();
}
});
}
/**
* Retrieves the usages (most recent data) for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param filter An OData filter expression that describes a subset of usages to return. The supported parameter is name.value (name of the metric, can have an or of multiple names).
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<UsageInner> object
*/
public Observable<ServiceResponse<List<UsageInner>>> listUsagesWithServiceResponseAsync(String resourceGroupName, String accountName, String databaseRid, String filter) {
if (this.client.subscriptionId() == null) {
throw new IllegalArgumentException("Parameter this.client.subscriptionId() is required and cannot be null.");
}
if (resourceGroupName == null) {
throw new IllegalArgumentException("Parameter resourceGroupName is required and cannot be null.");
}
if (accountName == null) {
throw new IllegalArgumentException("Parameter accountName is required and cannot be null.");
}
if (databaseRid == null) {
throw new IllegalArgumentException("Parameter databaseRid is required and cannot be null.");
}
if (this.client.apiVersion() == null) {
throw new IllegalArgumentException("Parameter this.client.apiVersion() is required and cannot be null.");
}
return service.listUsages(this.client.subscriptionId(), resourceGroupName, accountName, databaseRid, this.client.apiVersion(), filter, this.client.acceptLanguage(), this.client.userAgent())
.flatMap(new Func1<Response<ResponseBody>, Observable<ServiceResponse<List<UsageInner>>>>() {
@Override
public Observable<ServiceResponse<List<UsageInner>>> call(Response<ResponseBody> response) {
try {
ServiceResponse<PageImpl<UsageInner>> result = listUsagesDelegate(response);
List<UsageInner> items = null;
if (result.body() != null) {
items = result.body().items();
}
ServiceResponse<List<UsageInner>> clientResponse = new ServiceResponse<List<UsageInner>>(items, result.response());
return Observable.just(clientResponse);
} catch (Throwable t) {
return Observable.error(t);
}
}
});
}
private ServiceResponse<PageImpl<UsageInner>> listUsagesDelegate(Response<ResponseBody> response) throws CloudException, IOException, IllegalArgumentException {
return this.client.restClient().responseBuilderFactory().<PageImpl<UsageInner>, CloudException>newInstance(this.client.serializerAdapter())
.register(200, new TypeToken<PageImpl<UsageInner>>() { }.getType())
.registerError(CloudException.class)
.build(response);
}
/**
* Retrieves metric defintions for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @throws CloudException thrown if the request is rejected by server
* @throws RuntimeException all other wrapped checked exceptions if the request fails to be sent
* @return the List<MetricDefinitionInner> object if successful.
*/
public List<MetricDefinitionInner> listMetricDefinitions(String resourceGroupName, String accountName, String databaseRid) {
return listMetricDefinitionsWithServiceResponseAsync(resourceGroupName, accountName, databaseRid).toBlocking().single().body();
}
/**
* Retrieves metric defintions for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @param serviceCallback the async ServiceCallback to handle successful and failed responses.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the {@link ServiceFuture} object
*/
public ServiceFuture<List<MetricDefinitionInner>> listMetricDefinitionsAsync(String resourceGroupName, String accountName, String databaseRid, final ServiceCallback<List<MetricDefinitionInner>> serviceCallback) {
return ServiceFuture.fromResponse(listMetricDefinitionsWithServiceResponseAsync(resourceGroupName, accountName, databaseRid), serviceCallback);
}
/**
* Retrieves metric defintions for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<MetricDefinitionInner> object
*/
public Observable<List<MetricDefinitionInner>> listMetricDefinitionsAsync(String resourceGroupName, String accountName, String databaseRid) {
return listMetricDefinitionsWithServiceResponseAsync(resourceGroupName, accountName, databaseRid).map(new Func1<ServiceResponse<List<MetricDefinitionInner>>, List<MetricDefinitionInner>>() {
@Override
public List<MetricDefinitionInner> call(ServiceResponse<List<MetricDefinitionInner>> response) {
return response.body();
}
});
}
/**
* Retrieves metric defintions for the given database.
*
* @param resourceGroupName Name of an Azure resource group.
* @param accountName Cosmos DB database account name.
* @param databaseRid Cosmos DB database rid.
* @throws IllegalArgumentException thrown if parameters fail the validation
* @return the observable to the List<MetricDefinitionInner> object
*/
public Observable<ServiceResponse<List<MetricDefinitionInner>>> listMetricDefinitionsWithServiceResponseAsync(String resourceGroupName, String accountName, String databaseRid) {
if (this.client.subscriptionId() == null) {
throw new IllegalArgumentException("Parameter this.client.subscriptionId() is required and cannot be null.");
}
if (resourceGroupName == null) {
throw new IllegalArgumentException("Parameter resourceGroupName is required and cannot be null.");
}
if (accountName == null) {
throw new IllegalArgumentException("Parameter accountName is required and cannot be null.");
}
if (databaseRid == null) {
throw new IllegalArgumentException("Parameter databaseRid is required and cannot be null.");
}
if (this.client.apiVersion() == null) {
throw new IllegalArgumentException("Parameter this.client.apiVersion() is required and cannot be null.");
}
return service.listMetricDefinitions(this.client.subscriptionId(), resourceGroupName, accountName, databaseRid, this.client.apiVersion(), this.client.acceptLanguage(), this.client.userAgent())
.flatMap(new Func1<Response<ResponseBody>, Observable<ServiceResponse<List<MetricDefinitionInner>>>>() {
@Override
public Observable<ServiceResponse<List<MetricDefinitionInner>>> call(Response<ResponseBody> response) {
try {
ServiceResponse<PageImpl<MetricDefinitionInner>> result = listMetricDefinitionsDelegate(response);
List<MetricDefinitionInner> items = null;
if (result.body() != null) {
items = result.body().items();
}
ServiceResponse<List<MetricDefinitionInner>> clientResponse = new ServiceResponse<List<MetricDefinitionInner>>(items, result.response());
return Observable.just(clientResponse);
} catch (Throwable t) {
return Observable.error(t);
}
}
});
}
private ServiceResponse<PageImpl<MetricDefinitionInner>> listMetricDefinitionsDelegate(Response<ResponseBody> response) throws CloudException, IOException, IllegalArgumentException {
return this.client.restClient().responseBuilderFactory().<PageImpl<MetricDefinitionInner>, CloudException>newInstance(this.client.serializerAdapter())
.register(200, new TypeToken<PageImpl<MetricDefinitionInner>>() { }.getType())
.registerError(CloudException.class)
.build(response);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,383
|
Centro Especial de Integración Audiovisual y Formación (CEIAF) es una empresa especializada en el campo de las adaptaciones de programas de televisión para personas ciegas y sordas, que dispone de metodologías para audiodescribir y subtitular contenidos audiovisuales. Realiza subtitulado para discapacitados auditivos. Asesoramiento empresarial en planes estratégicos y proyectos de accesibilidad. Es una empresa española integrada en AITE.
Enlaces externos
Página oficial de Centro Especial de Integración Audiovisual y Formación.
Página oficial de AITE.
CEIAF
CEIAF
CEIAF
CEIAF
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,351
|
\section{Introduction}
\label{sec:intro}
Origin of neutrino masses and nature of Dark Matter (DM) are among
the most compelling open questions in particle physics. In recent years,
models in which neutrinos acquire their masses at loop level have
received considerable attention (see
Ref. \cite{Bonnet:2012kz,Farzan:2012ev,Angel:2012ug,Sierra:2014rxa} for
a model-independent analysis). Within these models, the smallness of
neutrino masses can be understood (at least partially) by loop
suppression. If the new particles propagating in the loop are lighter
than a few TeV, the resulting scheme will be phenomenologically
interesting because in that case the new states can potentially be
produced at the LHC. If this turns out to be the case, the radiative neutrino mass model can be tested at man-made accelerators. This is a great advantages over
the ``canonical''
tree-level type-I seesaw model \cite{seesaw}, for which on-shell
production of the new states is inconceivable in any foreseeable
future in man-made accelerators.
Assuming that the only source of electroweak symmetry breaking is the vacuum expectation of the Higgs, $n$-loop contributions to neutrino masses can be estimated as
\begin{equation}
\label{eq:order-mag}
m_\nu\sim
\left(\frac{g^2}{16 \pi^2}\right)^n
\left(\frac{\langle H\rangle^2}{m_\text{New}}\right)
\left[1,\left(\log\frac{\Lambda}{m_\text{New}}\right)^n\right]\quad
\ ,
\end{equation}
where $m_\text{New}$ is the mass scale characterizing the new physical
degrees of freedom appearing in the loop and $\Lambda$ is the
ultraviolet (UV) cut-off scale of the model satisfying
$\Lambda\gg m_\text{New}$. Taking $m_{\text{New}}\sim 1$~TeV, $m_\nu \sim 0.1-1$~eV
\cite{Forero:2014bxa,GonzalezGarcia:2012sz,Fogli:2012ua}, $\Lambda/m_\text{New}\sim 10$ and $n=2$, we find that $g\sim 10^{-3}$. Increasing $n$, the required values of the couplings will of course increase. The same couplings also lead to Lepton Flavor Violating (LFV) processes. For $m_\text{NEW}<10$~TeV, null results of searches for LFV rare decays of the muon and the tau lepton yield strong bounds on the combinations of such couplings. For $n=2$, these bounds are naturally satisfied but for $n>2$, a special mechanism such as the flavor symmetries suggested in
\cite{Farzan:2012ev} have to be invoked to make neutrino masses consistent with LFV bounds. From this perspective, the two-loop neutrino mass models seem more natural and are favored over higher order loop models.
In order to explain the smallness of neutrino masses through
radiative schemes, one should make sure that lower---and therefore
dominant---loop contributions are absent. In \cite{Farzan:2012ev} based on general
considerations of topological structure of the loops and symmetries,
the requirements assuring the absence of lower order contributions have been systematically formulated. Here in this
paper, using the ``recipes and ingredients'' outlined in
\cite{Farzan:2012ev}, we reconstruct a model where neutrino masses are
generated at the two-loop level through what we call ``snail
diagrams''.
Our model respects a new $Z_2\times U(1)_\text{New}$ symmetry. These
symmetries stabilize two of the lightest particles with non-trivial
transformation under these discrete symmetries against decay. If these
stable particles are neutral, they may be considered as a candidate
for DM. In our model, a Dirac fermion, $\psi$ which is a singlet under
the electroweak symmetry plays the role of the DM. The DM couples to
left-handed leptons via a Yukawa coupling. The abundance of $\psi$ is
determined by thermal freeze-out scenario via annihilation to lepton
pairs. To avoid the severe bounds from LFV, we assume that $\psi$
couples exclusively to only one flavor.
An excess in the GeV range $\gamma$-ray has
been reported in Fermi-LAT data on signal from regions close to
galactic center. One of the solutions is dark matter of mass 10 GeV
annihilating into tau pair \cite{Celine}. Another possibility is annihilation into $b \bar{b}$ pair \cite{McCabe}. The dark matter origin of this signal has
been however questioned and alternative sources have been suggested \cite{Gabi}. We will
comment on the possibility of accommodating this scenario within our
model.
The paper is organized as follows. In section 2, we generally discuss two-loop contributions to neutrino masses based on the topology of the diagrams. In section 3, we introduce the content of the model. In section 4, we discuss lepton flavor violating effects. In section 5, we calculate the contribution to neutrino masses. In section 6, we discuss the annihilation of dark matter pair and possibility of accommodating the claimed gamma ray excess from the region close to the galactic center. In sections 7 and 8, we respectively discuss signatures at the LHC and contribution to anomalous magnetic dipole moment. Conclusions are summarized in section 9.
\section{Comments on two-loop neutrino masses: crab and snail
diagrams}
\label{sec:setup}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.9]{scalar-fermion-wave-functions.pdf}
\caption{\it Two-loop diagrams with one-loop wave function
renormalization of scalar ($a$) and fermion ($b$) fields.}
\label{fig:wave-func-diag}
\end{figure}
Two-loop diagrams contributing to neutrino masses have been
systematically discussed in \cite{Farzan:2012ev,Sierra:2014rxa}. Based on the
topologies of the two-loop diagrams, they can be classified in two
groups: $(1)$ Diagrams with a one-loop sub-diagram that can be
considered as a correction to one of the internal lines.
Figs. (\ref{fig:wave-func-diag}- $a)$ and (\ref{fig:wave-func-diag}- $b)$ show corrections to internal scalar and fermion lines, respectively. The ``bubble'' on the scalar
line may indicate a fermion loop, a scalar loop with trilinear scalar
vertices or a scalar loop with quartic scalar vertex.
Further details can be found in \cite{Farzan:2012ev}. $(2)$ Diagrams in which an
internal line interconnects the scalar and fermion lines coming from
the vertex connected to the external lines. These types of diagrams are
rather well-known and have been employed in the literature to
radiatively produce neutrino mass at the two-loop level. A pioneer
work using such diagram is the famous Cheng-Li-Babu-Zee model
\cite{Cheng:1980qt,Zee:1985id,Babu:1988ki}
In Ref. \cite{Farzan:2012ev}, it is argued that diagrams of type
(\ref{fig:wave-func-diag}-$a)$ contributing to the effective Weinberg
operator
\begin{equation}
\label{eq:weinberg-operator}
{\cal O}_5\sim
\left(L^T\,C\,i\tau_2\,H\right)\left(H^T\,i\tau_2\,L\right)\ ,
\end{equation}
can always be accompanied by a one-loop contribution to neutrino mass.
The reason is that if the symmetries of the Lagrangian allow the
one-loop internal sub-diagram, they will also allow a
renormalizable term with which the internal loop can be replaced.
Depending on where the two external Higgs lines are attached (vacuum
insertions $\langle H\rangle$), these renormalizable terms can be
$S_1\,S_2$, $S_1\, S_2\,H$ or $S_1\,S_2\,H^2$.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.9]{crab-snail-diag.pdf}
\caption{\it Generic crab and snail diagrams.
We have not specified on which fermionic line the chirality flip takes place.}
\label{fig:crab-plus-snail}
\end{figure}
On the contrary, the so-called rainbow diagrams generically depicted in Fig. (\ref{fig:wave-func-diag}-$b)$ are not necessarily accompanied by any one-loop
counterpart. The argument is based on the following fact. While a
term such as $S_1S_2H^2$ is renormalizable, its fermionic counterpart,
$F_1\,F_2\,H^2$, is not. Thus, depending on the electroweak structure
of the fermion lines attached to the internal loop ($F_1$ and $F_2$ in
Fig. \ref{fig:wave-func-diag}-$(b)$) and the way in which the Higgs
external lines are attached to the corresponding diagram, there might
or might not be a one-loop contribution.
For the sake of the following discussion, let us consider the diagrams
in Fig. \ref{fig:crab-plus-snail}: ``crab'' (diagrams $(a)-(c)$) and
``snail'' diagrams (diagram $(d)$). The internal loops in ``crab''
diagrams can be respectively replaced by renormalizable vertices
$F_4\,F_5$, $F_4\,F_2\,H$ and $F_1\,F_4\,H$. ``Crab'' diagrams are
therefore always accompanied by a leading one-loop contribution, and
are in that sense irrelevant. For ``snail'' diagrams, instead, there
is no such possibility because $F_1 F_2 H^2$, being non-renormalizable
cannot appear in the Lagrangian. This argument of course holds under
the assumption that neutrino masses are generated below the
electroweak symmetry breaking scale only from Weinberg operator in Eq.
(\ref{eq:weinberg-operator}). If we included a hypercharge $-2$
electroweak scalar triplet ($\Delta$), with scalar interactions
enabling a non-vanishing vacuum expectation value, $\langle
\Delta\rangle\neq 0$, the external Higgs lines (vacuum insertions
$\langle H\rangle$) could be replaced by a single triplet vacuum
insertion $\langle \Delta \rangle$. In that case the internal loop
could be replaced by the renormalizable vertex $\Delta\,F_1\,F_2$.
In what follows we build a model where the effective Weinberg operator
arises via a ``snail'' diagram.
\section{Snail models}
\label{sec:snail-model-DM}
In this section, we present a model that can provide a suitable Dirac
fermion DM and give mass to neutrinos via a two loop diagram. We first
introduce the symmetry structure and field content of the model and
then discuss why each assumption is made. In the next sections, we shall
discuss the contribution to neutrino mass, annihilation of DM pairs to
lepton pairs, effects on LFV and magnetic dipole moment of the
muon and signals at the LHC.
The model is based on an unbroken $Z_2 \times U(1)_{NEW}$ symmetry. The
SM particles are all even and neutral under this symmetry. The model
also enjoys an approximate lepton number symmetry, $U(1)_L$ softly
broken by a fermion mass mixing term. The field content of the model
is shown in table \ref{tabl}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
~& $SU(2)$ & $U(1)_Y$ & $U(1)_{L}$ & $U(1)_{NEW}$& $Z_2$ \\
\hline
$F_1$ & d & -1 & 1 & 1 & + \\
$F_2$ & d&- 1 & 1 & -1 & + \\
$F_3$ & d& 1 & 1 & 1 & + \\
$\psi$ &s & 0 & 1 & 1 &- \\
\hline
$S$ & s &0 &0 & -1 & + \\
$\Phi$ & d & -1 & 0& 0 & - \\
$\Phi^\prime$ & d & -1 & 0 & -1 & - \\
\hline
\end{tabular}
\caption{Field content of the model. By ``d" and ``s" in the second column we mean doublet and singlet, respectively. We have used the convention for hypercharge in which $Q=T^3+Y/2$. The first four fields ({
\it i.e.,} $F_1$, $F_2$, $F_3$ and $\psi$) are Dirac fermions and the last three lines ($S$, $\Phi$ and $\Phi^\prime$) are scalar fields. \label{tabl}}
\end{center}
\end{table}
The new fermions are all Dirac particles and their masses are of form
$$\sum_i m_{F_i} \bar{F}_iF_i+m_\psi \bar{\psi}\psi \ .$$
As a result, neutral and charged components of $F_i$ are degenerate.
We also include mass term of form
\be \label{mM} m_M (F_{2R}^a)^TcF_{3R}^b\epsilon_{a b}+m_M^\prime (F_{2L}^a)^TcF_{3L}^b\epsilon_{a b}+ {\rm H.c.}\ee
which is supposed to be the only source of lepton number violation.
The Yukawa couplings of the new particles symmetric under $Z_2 \times U(1)_{NEW} \times U(1)_L$ are
\be \label{Lagrangian}
\mathcal{L}_{Yukawa}= g_\alpha S^\dagger F_{1R}^\dagger L_\alpha+h_\alpha S F_{2R}^\dagger L_\alpha+Y_{R\alpha}\Phi^{\prime\dagger}\psi_R^\dagger L_\alpha+
\ee
$$ Y_1 \Phi^\dagger \psi_L^\dagger F_{1R}
+Y_2 \epsilon_{ab}\Phi^a \psi_L^\dagger F_{3R}^b + Y_1^\prime \Phi^\dagger \psi_R^\dagger F_{1L}
+Y_2^\prime \epsilon_{ab}\Phi^a \psi_R^\dagger F_{3L}^b+ {\rm H.c.} $$
The new scalars can have interactions between themselves and SM Higgs. We assume that only the SM Higgs obtains a VEV so $U(1)_{NEW}$ and the new $Z_2$ symmetries remain unbroken. The $Z_2$ and $U(1)_{NEW}$ forbid mass terms mixing the scalars such as $H^\dagger \Phi$ or $\Phi^\dagger \Phi^\prime$. We can however have couplings of form
$$(\lambda (H^a \Phi^b \epsilon_{ab})^2+{\rm H.c}) \ \ \ {\rm and} \ \ \ \lambda^\prime |H^\dagger \Phi|^2.$$ The $\lambda$ coupling after electroweak symmetry breaking will lead to a mass term of form $(\Phi^0)^2$ for the neutral component
of $\Phi^0\equiv (\phi_R+i\phi_I)/\sqrt{2}$. Thus, there will be a splitting between $\phi_R$ and $\phi_I$. We however take $\lambda$ to be real so these fields remain mass eigenstates. We will denote the masses of these components
with $m_I$ and $m_R$:
$$m_R^2-m_I^2= \lambda \langle H^0 \rangle^2.$$
The couplings of $\phi_R$ ($\phi_I$) to $F_1$ and $F_2$ are respectively given by $Y_1/\sqrt{2}$ ($iY_1/\sqrt{2}$)
and $Y_2/\sqrt{2}$ ($iY_2/\sqrt{2}$). Notice that $U(1)_{NEW}$ protects real and imaginary components of $S$ as well as the neutral component of $\Phi^\prime$ from such splitting.
The $\lambda^\prime$ coupling leads to a mass term of form $\lambda^\prime \langle H^0\rangle^2 |\phi^-|^2$. Taking $\lambda^\prime$ positive, $\phi^-$ can be heavier than $\phi_I$ and
$\phi_R$ so $\phi^-$ can decay to $\phi_R$ and/or $\phi_I$.
Imposing both the $Z_2$ and $U(1)_{NEW}$ symmetries opens a possibility of having two DM candidates.
The neutral components of $F_i$ cannot be suitable dark matter candidates in this model because, as mentioned above, charged components of $F_i^-$
are also degenerate with them and might lead to the presence of electrically charged DM. Thus, we take $F_i$ heavy enough to decay to $\psi$ and $\Phi$. In this case, $\phi_I$ which is the lightest
$U(1)_{NEW}$ neutral and $Z_2$-odd particle will be stable and contribute to the dark matter abundance. If $\phi_I$ and $\phi_R$ are quasi-degenerate ({\it i.e.,} $(m_R-m_I)/m_R<1/20$), their
contribution to DM abundance will be suppressed within thermal freeze-out scenario. The electroweak singlet $S$ can also kinematically be made stable and can therefore contribute to DM abundance.
The annihilations of $S$ will be then through the $g_\alpha$ and $h_\alpha$ couplings to $l \bar{l}$ pairs. The annihilation will be suppressed by $m_l^2/m_F^2 \ll 1$ where $m_F>few~100$ GeV, so within this scenario, the density of $S$ would overclose the universe.
Thus, we take $S$ heavy enough to decay into leptons and $F_i$.
We take the DM candidate to be the Dirac fermion, $\psi$. The Dirac field can annihilate to lepton and anti-lepton pair via $Y_{R \alpha}$ coupling with a cross section required within thermal freeze-out scenario.
Notice that $\Phi^\prime$ does not appear in the snail diagram. We have added this new scalar doublet to facilitate the annihilation of $\psi \bar \psi$ pair to lepton anti-lepton pairs via
the $Y_{R\alpha}$ coupling. Instead of the $Y_{R\alpha}$
coupling, we could introduce a coupling of form $Y_{L\alpha} \Phi''e_{R\alpha}^\dagger \psi_L$ where $\Phi''$ is a $SU(2)$ singlet with electric charge equal to that of the electron.
We have taken $Y_{R\alpha}$ coupling instead of $Y_{L\alpha}$ for definiteness. Replacing it with $Y_{L\alpha}$ does not change the discussion.
Similarly, we could include new colored and charged scalar(s) to introduce Yukawa couplings to quarks and hence annihilation of dark matter pair to quarks. Studying all these possibilities and their potential signature at the LHC is beyond
the scope of the present paper and will be done elsewhere. In summary, in our model DM is composed of $\psi$ along with a subdominant contribution from $\phi_I$.
The following remarks on the $U(1)_{NEW}$ symmetry are in order:
\begin{itemize}
\item The $U(1)_{NEW}$ not only protects the DM candidate from decay but it also protects the fermions (in particular $\psi$) from having Majorana mass.
If $\psi$ obtains even a tiny Majorana mass at loop level, it can be decomposed in terms of Majorana mass eigenstates $\psi_1 \equiv (\psi+\psi^c)/\sqrt{2}$
and $\psi_1 \equiv (\psi-\psi^c)/\sqrt{2}$ among which only the lighter one will survive and play the role of the dark matter. With Majorana dark matter,
$\sigma(\psi_1 \psi_1 \to l \bar{l})$ will be either p-wave suppressed or will be suppressed by $m_l^2/m_{\Phi^\prime}^2\ll m_{\psi}^2/m_{\Phi^\prime}^2$ and cannot account
for the observed DM abundance within the thermal freeze-out scenario.
\item Notice that we have assigned opposite $U(1)_{NEW}$ charges to $F_1$ and $F_2$ that appear in the vertices connected to the external $\nu_\alpha$ and $\nu_\beta$ lines.
Without $U(1)_{NEW}$, we could drop $F_2$ and have a lepton number violating mass term of form $F_1^TcF_3$ giving a neutrino mass contribution proportional to $g_\alpha g_\beta$. This will not however help us to make the model more economic because a mass matrix proportional to $g_\alpha g_\beta$ has only one nonzero mass eigenvalue which cannot account for the realistic neutrino mass structure with at least two nonzero values. To reconstruct the neutrino mass matrix, another field with nonzero coupling component in the direction perpendicular to $g_\alpha$ in the flavor space is required.
\item The $U(1)_{NEW}$ cannot be replaced with a $Z_2$ subgroup of it because $Z_2$ does not forbid Majorana mass for $\psi$. We could however invoke the $Z_3$ subgroup of $U(1)_{NEW}$ under which
$\psi_L \to e^{\pm i 2 \pi /3} \psi_L$ and $\psi_R \to e^{\mp i 2\pi/3} \psi_R$. For neutrino mass generation as well as DM consideration there is no significant difference between these two. The $Z_3$ symmetry allows terms such as $S^3$ but the $U(1)_{NEW}$ symmetry forbids them. The presence of such terms does not change our results. The reason why we have chosen $U(1)_{NEW}$ instead of $Z_3$ is that $U(1)_{NEW}$ can be eventually gauged to protect against symmetry breaking by quantum gravitational effects. Notice that only new particles are charged under $U(1)_{NEW}$. The
gauged $U(1)_{NEW}$ can provide a way to have self-interacting DM, which provide a better fit to small scale features.
A kinetic mixing of $U(1)_{NEW}$ with the photon can lead to a direct detection signal.
We will not however try to gauge $U(1)_{NEW}$ here.
\end{itemize}
\section{Lepton Flavor Violating rare decays\label{LFV}}
Before proceeding to discuss contribution to neutrino masses, dark matter abundance and effects at colliders, let us derive bounds on parameters from searches for LFV rare decays.
The $h_\alpha$ and $g_\alpha$ couplings in Eq. (\ref{Lagrangian}) lead to Lepton Flavor Violating (LFV) rare decays, $l_\alpha \to l_\beta \gamma$ at one loop level. Using formulas in
\cite{Lavoura}, we find that $g_\alpha$ coupling leads to
\begin{equation} \label{LLFFVV} \Gamma(l_\alpha \to l_\beta \gamma)=g_\alpha^2 g_\beta^2\frac{m_\alpha^5}{16 \pi} \frac{[S(t)]^2}{(16 \pi^2)^2 m_S^4} \end{equation}
where \begin{equation} \label{SSS} S(t)= \frac{t-3}{4(t-1)^2}+\frac{\log t}{2(t-1)^3}+\frac{-2t^2+7t-11}{12(t-1)^3}+\frac{\log t}{2(t-1)^4} \end{equation} in which
$t\equiv (m_{F_1^-}/m_S)^2$. $S(t)$ is a monotonously decreasing function with $S(0)=1/6$, $S(1)=1/24$ and $S(\infty ) =1/12t$ so, as expected from decoupling theorem, $\Gamma(l_\alpha \to l_\beta \gamma)$ is suppressed by $1/({\rm Max}(m_S^2,m_{F^-_1}^2))^2$. The effect of the $h_\alpha$ coupling is given by the same formula replacing $g_\alpha,g_\beta \to h_\alpha,h_\beta$ and $m_{F_1^-}\to m_{F_2^-}$. If $\Phi^\prime$ couples to more than one flavor, the $Y_{R \alpha}$ coupling can also lead to similar LFV effects. As mentioned before, to avoid LFV rare decays induced by $Y_{R\alpha}$, we assume $\Phi^\prime$ couples only to one flavor. In the following, we discuss constraints on $g_\alpha$ from LFV bounds.
The best present bounds on LFV rare decay branching ratios are ~\cite{pdg}
\be \label{mue} Br(\mu \to e \gamma)<5.7\times 10^{-13}~,\ee
\be \label{taue} Br(\tau \to e \gamma)<3.3\times 10^{-8} \ee
and
\be \label{taumu} Br(\tau \to \mu \gamma)<4.4\times 10^{-8}~.\ee
From Eq. (\ref{mue}), we find
\be \label{gegmu} g_eg_\mu \stackrel{<}{\sim} 10^{-3} \frac{{\rm Max}(m_S^2,m_{F^-_1}^2)}{{\rm TeV}^2} \ee
and from Eqs. (\ref{taue},\ref{taumu}), we find
\be \label{gegmugtau} g_eg_\tau, g_\mu g_\tau \stackrel{<}{\sim} \frac{{\rm Max}(m_S^2,m_{F^-_1}^2)}{{\rm TeV}^2} .\ee
Similar consideration and bound hold valid for the $h_\alpha$ coupling, replacing $m_{F_1^-}\to m_{F_2^-}$.
\section{Neutrino masses\label{NuM}}
\begin{figure}[t!]
\centering
\includegraphics[scale=1.3]{single-rainbow.pdf}
\caption{\it Diagram giving mass to neutrinos. $``\times " $ indicates the $m_M$ mass term insertion which violates lepton number conservation.
\label{rainbow}}
\end{figure}
For simplicity, let us set $Y_1^\prime=Y_2^\prime=0$. Discussion for nonzero $Y_1^\prime$ and $Y_2^\prime$ will be
similar. In this model, we have only one diagram contributing to neutrino mass. That is of form of snail diagram shown in Fig (2-d), where $S_1$, $S_2$ and $F_4$ should be respectively identified with $S$, $\Phi^0$ and $\psi$ of our model. Instead of using $\lambda \langle H\rangle^2 (\Phi^0)^2$ mass insertion approximation,
we can have mass eigenstates $\phi_I$ and $\phi_R$ (imaginary and real components of $\Phi^0$) propagating in the inner loop as shown in Fig. \ref{rainbow}. Going to mass basis $\phi_R$ and $\phi_I$, the contribution of these fields propagating in the inner loop will be respectively given by factors $(Y_1/\sqrt{2})(Y_2/\sqrt{2})[1/(p^2-m_R^2)]$ and $(iY_1/\sqrt{2})(iY_2/\sqrt{2})[1/(p^2-m_I^2)]$ so the sum of two contributions will be proportional to
$$\frac{Y_1Y_2(m_R^2-m_I^2)}{2(p^2-m_I^2)(p^2-m_R^2)}.$$
We use mass insertion approximation for $\langle F_3 F_2^T \rangle$ propagator: $k^2 m_M/[(k^2-m_{F_3}^2) (k^2-m_{F_2}^2)]$. Putting all these together we find that the two-loop snail diagram contribution to neutrino mass is given by
$$ (m_{\nu})_{\alpha \beta}= (g_\alpha h_\beta +g_\beta h_\alpha) m_M \frac{Y_1 Y_2}{2} (m_R^2-m_I^2)\int \frac{d^4p}{(2\pi)^4} \int \frac{d^4 k}{(2\pi)^4} $$
$$ \frac{1}{k^2-m_S^2} \frac{k\cdot \sigma}{k^2-m_{F_1}^2} \frac{( p+k) \cdot \bar{\sigma}}{(k+p)^2-m_{\psi}^2}\frac{1}{(p^2-m_R^2)(p^2-m_I^2)} \frac{k^2 }{(k^2-m_{F_2}^2)(k^2-m_{F_3}^2)} \ .$$
Without loss of generality, we can go to a basis where $g_\alpha$ takes the form of $(0,0,g)$. We still have the freedom to rotate $h_\alpha$ in the direction $(0,h_1,h_2)$. In this basis, the first row and column of $m_\nu$ vanishes so with this field content one of neutrino mass eigenvalues will be zero. The mass scheme will be therefore hierarchical but the mixing parameters and CP-phases can be reconstructed with proper choice of $g_\alpha$ and $h_\beta$. To obtain non-hierarchical scheme, we can add another singlet $S$ coupled to $L$.
Using Feynman parameters we find
$$
(m_{\nu})_{\alpha \beta}= \frac{(g_\alpha h_\beta +g_\beta h_\alpha)}{16} m_M Y_1 Y_2\frac{ (m_R^2-m_I^2)}{(16\pi^2)^2}
I(m_{F_1},m_{F_2},m_{F_3},m_S,m_\psi,m_I,m_R)$$
where $ I(m_{F_1},m_{F_2},m_{F_3},m_S,m_\psi,m_I,m_R)$ is defined as $$
\int_0^1dy\int_0^{1-y} {dx}\int_0^1 da_1 \int_0^{1-a_1} da_2 \int_0^{1-a_1-a_2} da_3\int_0^{1-a_1-a_2-a_3} da_4 ~\frac{1-x}{A}$$ in which $A$ is equal to
$$(a_1 m_{F_1}^2+a_2 m_{F_2}^2+a_3 m_{F_3}^2 +a_4 m_{S}^2)x(1-x)+(1-a_1-a_2-a_3-a_4)(xm_{\psi}^2+ym_I^2+(1-x-y)m_R^2).$$
Notice that $A$ is a positive definitive quantity over the whole integration range. Thus, the integration $ I$ is a finite quantity as expected.
$\psi$ is the lightest field propagating in the loops. Let us denote the mass of the heaviest field propagating in the loop by $m_{max}$. We can then write $ I(m_{F_1},m_{F_2},m_{F_3},m_S,m_\psi,m_I,m_R)=b/m_{max}^2$
where $b$ is a number. For $m_\psi/m_{max}$ (and therefore the rest of ratios) varying between $\sim 0.1$ to 1, the value of $b$ varies in the range $O(0.01)$-$O(0.1)$.
The neutrino mass can be then estimated as
\be \label{mnu-estimate} m_\nu \sim (0.01-0.1~{\rm eV}) Y_1 Y_2 \frac{g\times h}{10^{-1}\times 10^{-2}} \frac{m_M}{5 ~{\rm GeV}}\frac{(m_R^2-m_I^2)/m_{max}^2}{1/20}.\ee
Notice that $m_R^2-m_I^2\sim \lambda \langle H^0\rangle^2$. Taking $\lambda \sim 0.5$ and $m_{NEW} \sim ({\rm few ~TeV})$, it seems to be natural to have
$(m_R^2-m_I^2)/m_{R}^2\leq (m_R^2-m_I^2)/m_{NEW}^2 \stackrel{<}{\sim} 0.1$. As we will discuss in sec. \ref{annihilation}, $(m_R-m_I)/m_R$ should be smaller than $\sim 0.05$ to facilitate the coannihilation of
$\phi_I$ and $\phi_R$ ({\it e.g.,} $\phi_I \phi_R \to Z^* \to SM$) in the early universe and hence prevent over-closure of the universe by lighter component of $\phi_I$ and $\phi_R$.
The following points are in order:
\begin{itemize}
\item To make the estimate in Eq. (\ref{mnu-estimate}), we have taken $g_\alpha h_\beta\sim 10^{-3}$. As we saw see in section \ref{LFV}, for $m_{NEW}\sim 1$~TeV, the upper bounds on ${g_e g_\mu}$ and ${h_e h_\mu}$
from Br($\mu \to e \gamma$) are of order of $10^{-3}$ so we expect an observable effect in near future at searches for $\mu \to e \gamma$.
Within this model, saturating bounds on Br($\tau \to \mu \gamma$) or Br($\tau \to e \gamma$) can be possible only if $g_\alpha\sim 10^{-3} \ll h_\alpha\sim 1$ or
$g_\alpha\sim 10^{-3} \gg h_\alpha\sim 1$.
\item To arrive at Eq. (\ref{mnu-estimate}), we have used mass insertion approximation for the treatment of mass term mixing $F_2$ and $F_3$, $m_M$. Taking $m_M=5$ GeV and $m_{F_i}\sim $TeV, this approximation is valid. Taking smaller $m_M$ requires $Y_1,Y_2 \gg 1$ which leads to non-perturbativity.
\item In the range $m_{\Phi}\sim m_F \sim m_{max}\sim 1~{\rm TeV}-100~{\rm TeV}$ and $m_M\sim 5~{\rm GeV} (m_{max}^2/{\rm TeV}^2)$, we obtain desired values of $m_\nu$ satisfying bounds from LFV as well as collider searches and we still remain in the perturbativity range: $Y_1,Y_2<1$ and $m_M\ll m_F$.
The lower part of this range can be probed at second phase of the LHC, but the range $m_F,m_\phi>10$ TeV is out of the reach of the LHC.
\end{itemize}
\section{Annihilation to lepton pair \label{annihilation}}
As discussed in section \ref{sec:snail-model-DM}, we choose the main dark matter component to be $\psi$ which annihilates to a pair of leptons.
The annihilation cross section to a charged lepton pair of flavor $\alpha$ can be written
as
\be \label{annihi} \langle \sigma( \psi \bar \psi \to \ell_\alpha \bar\ell_\alpha) v \rangle = \frac{|Y_{R\alpha}|^4 }{32 \pi} \frac{ m_{\psi}^2}{
(m_{\psi}^2+(m_{\phi^{\prime -}})^2)^2}.
\ee
A similar equation can be rewritten for annihilation to a $\nu_\alpha \bar\nu_\alpha$ pair by replacing $m_{\phi^{\prime -}}$ with $m_{\phi^{\prime 0}}$.
To avoid large LFV effects, we assume that only one flavor component of $Y_{R\alpha}$ is nonzero.
Taking $\langle \sigma_{tot} v\rangle = 3 \times 10^{-26} ~{\rm cm}^3 {\rm sec}^{-1}$ (as predicted within the thermal freeze-out scenario) and typical values $m_\psi=300$ GeV and $m_{\phi^{\prime -}}=m_{\phi^{\prime 0}}=400$ GeV
we find $Y_R=0.55$. In general, we obtain
\be \label{YR}
m_{\phi^{\prime -}},~m_{\phi^{\prime 0}} \leq 1.4 Y_{R\alpha}^2 {\rm TeV} \ee
where equality corresponds to the limiting case of $m_\psi \to m_{\phi^{\prime -}}\simeq m_{\phi^{\prime 0}}$.
The large $Y_{R\alpha}$ coupling will not however affect the lepton or heavy meson decays because they are not heavy enough to emit $\psi$.
This large coupling can cause dips in the spectrum of very high energy cosmic neutrinos at ICECUBE due to scattering off the DM distributed all over the universe. The resonance energy is at
$E_{res} \sim (m_{\phi^{\prime 0}})^2/m_{\psi}\sim {\rm few} ~100~ {\rm GeV} $. For a given $m_{\phi^\prime}$, decreasing $m_\psi$, the value of $E_{res}$ and as a result the position of the dip shifts towards higher energies. One should however bear in mind that by decreasing $m_{\psi}/m_{\phi^\prime}$ the required $Y_R$ increases and eventually enters non-perturbative regime.
Data from the region close to galaxy center from Fermi-LAT shows a hint of GeV range gamma excess. One of the explanations is the annihilation of 10 GeV DM pairs to lepton pairs \cite{Celine}. It is tantalizing to try to accommodate this signal within our model.
Now, following Ref \cite{Lacroix:2014eea}, if we set $\langle \sigma( \psi \bar \psi \to l \bar l) v \rangle =0.86 \times 10^{-26} ~{\rm cm}^3 {\rm sec}^{-1}$ and
$m_{\psi}\sim 10 $ GeV, we obtain
$$Y_{R }=0.5 (m_{\phi^{\prime -}}/100~{\rm GeV}) (10 ~{\rm GeV}/m_{\psi})^{1/2}\ . $$
Notice that we have taken $\phi^\prime$ to be relatively light.
From the first run of the LHC there, there is already a lower bound of 325 GeV on the mass of new charged scalar such as $\phi^{\prime -}$ whose decay lead to the electron or the muon plus missing energy \cite{Aad}. Bounds on such scalar coupled to only tau is weaker: $m_{\phi^{\prime -}}>90$ GeV \cite{stau}. As a result, for annihilation to tau pair, the value of $m_{\phi^{\prime -}}$ satisfies the present bound. For heavier values of $\phi^{\prime -}$, we eventually enter non-perturbative regime. A more recent analysis of the gamma ray excess finds a better fit with $m_\psi \sim 50$ GeV and $\langle \sigma(\psi \bar\psi \to b \bar b)\rangle \sim 10^{-26}~ {\rm cm}^3 {\rm sec}^{-1}$ \cite{McCabe}. This can be achieved with a coupling of form $Y_b \bar{b}_R\psi \phi^{''}$ where $\phi^{''}$ is a colored and charged scalar singlet under $SU(2)$. From the LHC bounds, this scalar should be heavier than 620~GeV \cite{sb}. The annihilation cross section of $\psi \bar{\psi} \to b \bar{b}$ is given by Eq. (\ref{annihi}) replacing $\phi^\prime$ with $\phi^{''}$ and multiplying by a factor of three to account for the color degrees of freedom. To accommodate the signal with $m_\psi$ and $m_{\phi^{''}}\sim 700$~GeV, $Y_b$ should be of order of one. One should however bear in mind that DM origin of gamma ray excess has been questioned in a series of publication \cite{Gabi}.
As discussed before the lightest neutral component of $\phi$ ({\it i.e.,} $\phi_I$) can be another DM component if it is lighter than $\phi^-$. For $|m_R-m_I|\stackrel{<}{\sim} m_R/20$, coannihilation via
$\phi_I\phi_R \to Z^*\to SM$ will render its abundance negligible.
\section{Signature at the LHC\label{LHC}}
In this model, there are several fields with electroweak interactions that can be pair produced at the LHC provided that they are light enough. As discussed in sec. \ref{NuM}, $\Phi$
and $F_i$ fields propagating in the loops that contribute to $m_\nu$ can have masses in the range 1~TeV-100~TeV. As discussed in sect. \ref{sec:snail-model-DM}, we take $\Phi$ to be lighter than $F_1$ and $F_3$. As result, via large $Y_1$ and $Y_2$
couplings, the components of $F_2$ and $F_3$ will decay as $F_i^- \to \psi \phi^-$ and $F_i^0 \to \psi \phi^0_{I(R)}$. The $\psi$ particle as well as $\phi_I$ will appear as missing energy.
Via tree-level $Z^*$ exchange,
$\phi_R \to \phi_I \nu \bar{\nu}, ~\phi_I l \bar{l}$.
While $\Phi$ and $F_i$ particles can be too heavy to be produced at the LHC, as we saw in sect. \ref{annihilation}, there is an upper bound on the masses of the $\Phi^\prime$
components. Thus, if this model is realized in nature, it is guaranteed that the components of $\Phi^\prime$ will be pair produced at the second run of the LHC, leading to
the following signals:
\begin{itemize} \item Mono-lepton plus missing energy signal through
$u \bar{d} \to \phi^{\prime +} \phi^{\prime 0} \to (l^+\psi) (\nu \bar{\psi})$
and the charge conjugate processes.
\item Two-lepton plus missing energy signal through
$ u\bar{u},d \bar{d} \to \phi^{\prime +} \phi^{\prime -} \to (l^+\psi) (l^- \bar{\psi}).$
\item Missing energy through
$ u\bar{u},d \bar{d} \to \phi^{\prime 0} \bar{\phi}^{\prime 0} \to (\bar{\nu}\psi) (\nu \bar{\psi}).$
\end{itemize}
As discussed in section \ref{annihilation}, the present lower bounds on the masses of scalars whose decay lead to missing energy plus muon and electron \cite{Aad} and tau lepton \cite{stau}
are respectively 325~GeV and 90~GeV.
In fact, phenomenology of $\Phi^\prime$ doublet at the LHC (both production mechanism as well as signature of the decay product) is very similar to that of left-handed slepton doublet in the framework of Minimal Supersymmetric Standard Model (MSSM).
As mentioned before, we assume $\phi^\prime$ to couple mainly to only one flavor to avoid LFV rare processes. If this flavor happens to be the second generation, the signals at the LHC will
be cleaner.
In this case, we expect a contribution to $(g-2)_\mu$ which we elaborate on in the next section.
\section{Muon magnetic dipole moment}
In this model, there are several particles that couple to the muon and can give rise to $(g-2)_\mu$ at one loop level. Considering the bounds in Eq. (\ref{YR}) on the mass and coupling
of $\phi^\prime$, it can give largest contribution to $(g-2)_\mu$ if the $Y_{R\alpha}$ coupling is to the muon flavor.
The $Y_{R\mu}$ coupling leads to
$$\delta \frac{g-2}{2}= \frac{Y_{R\mu}^2}{16 \pi^2} \frac{m_\mu^2}{m_{\phi^{\prime -}}^2} K(r)$$
where
$$ K(r)=\frac{2 r^2+5r-1}{12(r-1)^3}-\frac{r^2 \log r}{2(r-1)^4}$$
in which $r=(m_\psi^2/m_{\phi^{\prime -}}^2)$.
Taking $m_{\phi^\prime}\sim 100~{\rm GeV}-1~{\rm TeV}$ and $Y_{R\alpha}\sim 1$ (see Eq. \ref{YR}), we find that
$(g-2)_\mu/2\sim 10^{-11}-10^{-12}$ which is well
below the current sensitivity limit \cite{pdg}.
\section{Conclusions}
\label{sec:conclusions}
Following the ``recipes'' developed in \cite{Farzan:2012ev}, we have built a model in which neutrinos receive Majorana mass via a two-loop diagram with topology
of ``snail diagram'' depicted in Fig. \ref{fig:crab-plus-snail}-d and in Fig. \ref{rainbow}. The particles propagating in the loops are new scalars and fermions charged under $SU(2)\times U(1)$. The field content is given in table \ref{tabl}. The lepton number is explicitly broken by fermion mass $m_M$ (see Eq. \ref{mM}) so the neutrino masses are proportional to $m_M$ as seen in Eq. (\ref{mnu-estimate}). Following the argument in Ref. \cite{Farzan:2012ev}, we confirm that the two-loop snail diagram is the leading contribution to neutrino mass. Within this model the neutrino mass scheme is predicted to be hierarchical with one vanishing mass eigenvalue. The model respects a global $U(1)_{NEW}\times Z_2$ symmetry which stabilizes two of new particles: $\phi_I$, the imaginary part of the neutral component of $\Phi$ and $\psi$, a singlet under electroweak group. We assume the mass splitting between $\phi_I$ and $\phi_R$ (the real component of $\phi^0$) is small enough to allow efficient co-annihilation in the early universe. $\phi_I$ is therefore only a sub-dominant component of dark matter. This assumption turns out to be natural within our model and does not need any fine-tuning.
The dominant component of dark matter is Dirac fermions $\psi$ that can annihilate to
a pair of standard model fermions via a Yukawa coupling involving new scalar $\Phi^\prime$.
In order to obtain the observed abundance of dark matter within freeze-out scenario ({\it i.e.,} $\langle \sigma(\psi \bar\psi \to f \bar
f)v \rangle \sim 1$~pb), the mass of ${\Phi^\prime}$ should be less than 1.5 TeV (see Eq. (\ref{YR})). This means the components of $\Phi^\prime$ can be eventually produced at the LHC via electroweak interactions and discovered through their signature of decay to standard model fermions plus missing energy. Moreover the corresponding Yukawa coupling should be of order of one. To avoid LFV rare decay, we assume $\Phi^\prime$ couples only to one flavor. If this flavor is the muon, the discovery potential of the LHC will be higher. The contribution to $(g-2)_\mu$ is then predicted to be one or two orders of magnitude below the present sensitivity. The coupling of the scalar singlet, $S$ to leptons ({\it i.e.,} $g_\alpha$ and $h_\alpha$) should involve more than one flavor to reconstruct the neutrino mass matrix structure. This in turn leads to LFV rare decays. From values of neutrino mass, we expect the $\mu \to e \gamma$ signal to be around the corner.
\section{Acknowledgments}
The author would like to specially thank Diego Aristizabal Sierra with whose collaboration the early stages of this work was done. She also thanks his daughter Sofia
Aristizabal for suggesting names for the ``snail'' and ``crab'' diagrams. The author also appreciates Majid Hashemi for useful remarks and discussions on possible signature at the LHC. She would like to thank NORDITA where this project started as well as Liege Univ where a part of this work was done. She
acknowledges partial support from the European Union FP7 ITN
INVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,094
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package maxsub
import (
"math"
"github.com/catorpilor/leetcode/utils"
)
func maxSubSum(nums []int) int {
return twoPasses(nums)
}
// twoPasses time complexity O(N), space complexity O(1)
func twoPasses(nums []int) int {
n := len(nums)
first := useKadane(nums, n)
preSum := 0
for i := range nums {
preSum += nums[i]
nums[i] = -nums[i]
}
preSum += useKadane(nums, n)
if first > 0 {
return utils.Max(first, preSum)
}
return first
}
// useKadane use kadanes's algorightms to find the max sum subarray
// but dit some modification, since we want non-empty subarray.
func useKadane(nums []int, n int) int {
var maxEndingHere int
maxSofar := math.MinInt32
for i := 0; i < n; i++ {
maxEndingHere += nums[i]
if maxEndingHere > maxSofar {
maxSofar = maxEndingHere
}
if maxEndingHere < 0 {
maxEndingHere = 0
}
}
return maxSofar
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,184
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Why Is There a Strange Bend on Saturn's Ring from NASA Image?
By Jessica F
Saturn's rings appear bent based on the fascinating image taken by the camera aboard the Cassini spacecraft
(Photo : NASA via Getty Images))
What's more bizarre than Saturn's perfectly formed rings? Recent image of the planet's rings taken by the Cassini spacecraft fascinates scientists and the public as the rings appeared warped or bent.
Bent rings? A lens effect from Saturn's atmosphere appears to warp them: https://t.co/1wxkLl2Ki1 @CassiniSaturn pic.twitter.com/GPPsZOAtGY
— NASA (@NASA) July 25, 2016
In the image, Saturn's A and F rings appeared to be warped, according to experts, it starts where the rings intersect with the planet's limb where the atmosphere acts like a big lens. The occurrence can be partly attributed to the light that is refracted or bent because of the cameras aboard Cassini.
"Saturn's atmosphere absorbs some of the light reflected by the rings as it passes through. But absorption is not the only thing that happens to that light. As it passes from space to the atmosphere and back out into space towards Cassini's cameras, its path is refracted, or bent," a NASA official said in a statement.
In short, the image of the bent Saturn rings is a simple optical illusion, according to a report. The result of this is an amazing image that shows the rings are bent or warped. The view looked towards the sunlit side of the rings and was taken by Cassini's narrow-angle camera on June 9 at a distance of approximately 1.1 million miles (1.8 million kilometers).
NASA, ESA and the Italian Space Agency Agenzia Spaziale Italiana (ASI) collaborate to execute the Cassini mission to explore the planet Saturn, its rings and moons. The Cassini mission is responsible for studies involving the Saturn's moon Titan, which reports say might have the right chemistry for life.
According to NASA, Cassini is scheduled to begin its Grand Finale orbits, diving between Saturn and its rings before plunging into the planet's atmosphere on Sept. 15, 2017. But not after proving the Earth tons of useful information and fascinating images like the bent rings of Saturn.
TagsSaturn, Cassini Mission, NASA Cassini, NASA, Saturn Rings, Saturn bent rings, Saturn moon, titan, refraction
Saturn's Moon Titan Could Have the Right Chemistry for Life
Saturn's Moon Titan Host to Alien Life? Chemistry Holds Key to Prebiotic Non-Water-Based Life
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{
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}
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\section{Introduction}
\label{sec:introduction}
In reinforcement learning (RL), an agent learns to selects actions to
navigate a state space and accumulate reward. In terms of theoretical
results, the majority of results address the tabular setting, where
the number of states and actions are finite and comparatively
small. However, tabular problems are rarely encountered in practical
applications, as state and action spaces are often large and may even
be continuous. To address these practically relevant settings, a
growing body of work has developed algorithmic principles and
guarantees for reinforcement learning in continuous spaces.
In this paper, we contribute to this line of work on reinforcement
learning in continuous spaces. We consider episodic RL where the joint
state-action space is endowed with a metric and we posit that the
optimal $Q^\star$ function is \emph{Lipschitz continuous} with respect
to this metric. This setup has been studied in several recent works
establishing worst case regret bounds that scale with the covering
dimension of the metric
space~\citep{song2019efficient,sinclair2019adaptive,touati2020zooming}. While
these results are encouraging, the guarantees are overly pessimistic,
and intuition from the special case of Lipschitz bandits suggests that
much more adaptive guarantees are achievable. In particular, while the
Lipschitz contextual bandits setting of~\citet{slivkins2014contextual}
is a special case of this setup, no existing analysis recovers his
adaptive guarantee that scales with the \emph{zooming dimension} of
the problem.
\paragraph{Our contribution.}
We give the first analysis for reinforcement learning in metric spaces
that scales with the zooming dimension of the instance instead of the
covering dimension of the metric space. The zooming dimension,
originally defined by~\citet{kleinberg2019bandits} in the context of
Lipschitz bandits, measures the size of the set of near-optimal
actions, and can be much smaller than the covering dimension in
favorable instances. For reinforcement learning, the natural
generalization is to measure near-optimality relative to the $Q^\star$
function; this recovers the definition of~\citet{kleinberg2019bandits}
and~\citet{slivkins2014contextual} for bandits and contextual bandits,
respectively as special cases. As a consequence, our guarantees also
strictly generalize theirs to the multi-step reinforcement learning
setting. In addition, our guarantee addresses an open problem
of~\citet{sinclair2019adaptive} by characterizing problems where
refined guarantees are possible.
Our result is based on a refined analysis of the algorithm
of~\citet{sinclair2019adaptive}. This algorithm uses optimism to select
actions and an adaptive discretization scheme to carefully refine a
coarse partition of the state-action space to focus (``zoom in'') on
promising regions. Adaptive discretization is essential for obtaining
instance-dependent guarantees, but the bounds
in~\citet{sinclair2019adaptive} do not reflect this favorable behavior.
At a technical level, the main challenge is that, unlike in bandits,
we cannot upper bound the number of times a highly suboptimal arm will
be selected by the optimistic strategy. Analysis for the bandit
setting uses these upper bounds to prove that the adaptive
discretization scheme will not zoom in on suboptimal regions, which is
crucial for the instance-dependent bounds. However, in RL, the
algorithm actually can zoom in on and select actions at suboptimal
regions, but only when there is significant error at later time
steps. Thus, in the analysis, we credit error incurred from a highly
suboptimal region to the later time step, so we can proceed as if we
never zoomed in on this region at all. Formally, this analysis uses
the \emph{clipped regret decomposition} of~\citet{simchowitz2019non}
as well as a careful bookkeeping argument to obtain the
instance-dependent bound.
\section{Preliminaries}
We consider a finite-horizon episodic reinforcement learning setting
in which an agent interacts with an MDP, defined by a tuple
$(\Scal,\Acal,H,\PP,r)$. Here
$\Scal$ the state space, $\Acal$ is the action space, $H \in \NN$ is
the horizon, $\PP$ is the transition operator and $r$ is the reward
function. Formally, $\PP: \Scal\times\Acal \to \Delta(\Scal)$ and
$r:\Scal\times\Acal \to [0,1]$ where $\Delta(\cdot)$ denotes the set
of distributions over its argument.\footnote{Deterministic rewards
simplifies the presentation but has no bearing on the final
results. In particular, we can handle stochastic bounded rewards
with minimal modification to the proofs.}
A (nonstationary) policy $\pi$ is a mapping from states to
distributions over actions for each time. Every policy has
non-stationary value and action-value functions, defined as
\begin{align*}
V_h^\pi(x)\defeq \EE_\pi\sbr{\sum_{h'=h}^H r_{h'}(x_{h'},a_{h'}) \mid x_h = x}, \qquad Q_h^\pi(x,a) \defeq r_h(x,a) + \EE\sbr{V_{h+1}^\pi(x') \mid x,a}.
\end{align*}
Here $\EE_\pi\sbr{\cdot}$ denotes that all actions are chosen by
policy $\pi$ and transitions are given by $\PP$. The optimal policy
$\pi^\star$ and optimal action-value function $Q^\star$ are defined
recursively as
\begin{align*}
Q_h^\star(x,a) \defeq r_h(x,a) + \EE\sbr{\max_{a'}Q^\star(x',a')\mid x,a}, \qquad \pi_h^\star(x) = \argmax_{a} Q_h^\star(x,a).
\end{align*}
The optimal value function $V_h^\star$ is defined analogously.
\iffalse
We play the MDP for $K$ rounds and for each round there are
$H$ steps. At round $k$ step $h$, we are given a state $x^k_h$ and
play an action $a^k_h$. We receive a reward of $r_h (x^k_h, a^k_h)$
and transition to step $h+1$ by taking a sample from the transition
probability $\PP_h(\cdot, x^k_h, a^k_h) \in \Delta(\Scal)$.
We would
like to find the policy $\pi$ that maps the state to action that yield
the maximum reward in expectation. We define the value function for a
policy $\pi$ as
\begin{align*}
V_h^{\pi}(x):= \EE_{\pi}[\sum_{h'=h}^{H} r_{h'}(x_h', \pi_{h'}(x_{h'})) | x_h = x ]
\end{align*}
where the expectation is taken over the transition probability under policy $\pi$. We define the $Q$ value function as
\begin{align*}
Q^{\pi}_h(x, a) := r_h(x, a) + \EE_{\pi}[V^{\pi}_{h+1}(x')|x,a]
\end{align*}
We use $V^\star$ to denote the value function and $Q^\star$ to denote the $Q$--function for the optimal policy $\pi^\star$.
\fi
The agent interacts with the MDP for $K$ episodes, where in episode $k$
the agent pick a policy $\pi_k$ and we generate the trajectory $\tau_k =
(x^k_1,a^k_1,r^k_1,x^k_2,a^k_2,r^k_2\ldots,x^k_H,a^k_H,r^k_H)$ where (1)
$x^k_1$ is chosen adversarially, (2) $a_h^k=\pi_k(x_h^k)$, (3)
$x^k_{h+1} \sim \PP(\cdot \mid x^k_h,a^k_h)$, (4) $r^k_h =
r(x^k_h,a^k_h)$. We would like to choose actions to maximize the
cumulative rewards $\sum_{h=1}^H r^k_h$.
Equipped with these definitions, we can state our performance
criterion. Over the course of $K$ episodes, we would like to
accumulate reward that is comparable to the optimal policy, formalized
via the notion of regret:
\begin{align*}
\mathrm{Reg}(K) \defeq \sum_{k=1}^K \rbr{V_1^\star(x_1^k) - \sum_{h=1}^H r_h^k}.
\end{align*}
In particular, we seek algorithms with regret rate that is sublinear
in $K$. Note that we have not assumed that $|\Scal|$ and $|\Acal|$ are
finite, and we also allow for the starting state $x_1^k$ to be chosen
adversarially in each episode.
\subsection{Metric spaces.}
Instead of assuming that $|\Scal|$ and $|\Acal|$ are finite, we will
posit a metric structure on these spaces. We recall the key
definitions for metric spaces. A space $Y$ equipped with a function
$\Dcal:Y \times Y \to \RR_+$ is a \emph{metric space} if $\Dcal$
satisfies (a) $\Dcal(y,y') = 0$ iff $y = y'$ (b) $\Dcal$ is symmetric,
and (c) $\Dcal$ satisfies the triangle inequality $\Dcal(x,y) \leq
\Dcal(x,z) + \Dcal(z,y)$. If these properties hold then $\Dcal$ is
called a \emph{metric}. For a radius $r>0$, we use the notation
$B(y,r)\defeq\{y' \in Y: \Dcal(y,y') < r\}$ to denote the open ball
centered at $y$ with radius $r$. For a subset $Y'\subseteq Y$ the
\emph{diameter} is defined as $\mathrm{diam}(Y') \defeq \sup_{y,y' \in
Y'}\Dcal(y,y')$.
We also use the standard notions of covering and packing to measure
the size of metric spaces.
\begin{definition}[Notions of size]
A \emph{covering} of $Y$ at scale $r$ (also called an $r$-covering) is
a collection of subsets of $Y$, each with diameter at most $r$, whose
union equals $Y$. The minimum number of subsets that form an
$r$-covering is the $r$-covering number, denoted $N_r(Y)$. A
\emph{packing} of $Y$ at scale $r$ (also called an $r$-packing) is a
collection of points $Z\subset Y$ such that $\min_{z\ne z' \in Z}
D(z,z') \geq r$. The maximum number of points that form an $r$-packing
is the $r$-packing number, denoted $N_r^{\text{pack}}(Y)$. An
\emph{$r$-net} of $Y$ is an $r$-packing $S\subset Y$ for which
$\{B(y,r)\}_{y \in S}$ covers $Y$.
\end{definition}
These definitions also apply to subsets of the metric space, which
will be important for our development. Also note that
$N_{2r}^{\text{pack}}(Y) \leq N_r(Y) \leq N_r^{\text{pack}}(Y)$.
\subsection{Main Assumptions.}
We now state the main assumptions that we adopt in our analysis. These
or closely related assumptions are standard in the literature on
bandits and reinforcement learning in metric
spaces~\citep{song2019efficient,sinclair2019adaptive,touati2020zooming,slivkins2014contextual}.
\begin{assum}
\label{assum:diameter}
$(\Scal \times \Acal, \Dcal)$ is a metric space with finite diameter $\mathrm{diam}(\Scal \times \Acal) = d_{max} < \infty$.
\end{assum}
\begin{assum}
\label{assum:lipschitz}
For every $h\in [H]$, $Q^{\star}_h$ is $L$-Lipschitz continuous with respect to $\Dcal$:
\begin{align}
\forall (x,a), (x',a'): \abr{ Q^\star_h(x,a) - Q^\star_h(x',a')} \le L\cdot\Dcal((x,a), (x',a')).\label{eq:q_lip}
\end{align}
Additionally $V_h^\star$ is $L$-Lipschitz with respect to the metric $\Dcal_X: (x,x') \mapsto \min_{a,a'}\Dcal((x,a),(x',a'))$:
\begin{align}
\forall x,x': \abr{ V^\star_h(x) - V^\star_h(x')} \le L\cdot\min_{a,a'}\Dcal((x,a), (x',a')).\label{eq:v_lip}
\end{align}
\end{assum}
Assumption~\ref{assum:diameter} is a basic regularity condition, while
the first part of Assumption~\ref{assum:lipschitz} imposes continuity
of the $Q^\star$ function. In particular, Lipschitz-continuity
characterizes how the metric structure influences the reinforcement
learning problem. These assumptions appear in prior work,
and we note that~\eqref{eq:q_lip} is strictly weaker than assuming
that $\PP$ is Lipschitz
continuous~\citep{kakade2003exploration,ortner2012online}.
The second part of Assumption~\ref{assum:lipschitz} reflects an
additional structural assumption on the problem, which is a departure
from previous work. In detail,~\eqref{eq:v_lip} posits that the
optimal value function $V_h^\star$ is $L$-Lipschitz with respect to a
metric defined only on the states that is derived from the original
one. This metric is dominated by the original one since for each
$(x,x',a)$ we have $\min_{a_1,a_2}\Dcal((x,a_1),(x',a_2)) \leq
\Dcal((x,a),(x',a))$, so this assumption is not directly implied
by~\eqref{eq:q_lip}. However, whenever $\Dcal$ is
sub-additive in the sense that $\Dcal((x,a),(x',a')) \leq
\Dcal_S(x,x') + \Dcal_A(a,a')$, then the assumption holds
trivially. Sub-additivity holds for most metrics of interest,
including those induced by $\ell_p$ norms for $p \geq 1$. As such,
this assumption is not particularly restrictive.
\subsection{Related work}
Reinforcement learning in the tabular setting, where the state and
action spaces are finite, is relatively
well-understood~\citep{azar2017minimax,dann2017unifying,zanette2019tighter}. Of
this line of work, the two most related papers are those of
of~\citet{jin2018q} and~\citet{simchowitz2019non}. Our results build
on the model-free/martingale analysis of~\citet{jin2018q}, which has
been used in recent work on RL in metric
spaces~\citep{song2019efficient,sinclair2019adaptive,touati2020zooming}. We
also employ techniques from the gap-dependent analysis
of~\citet{simchowitz2019non}. In particular, we use a version of their
``clipping'' argument, as we will explain in
Section~\ref{sec:analysis}.
Moving beyond the tabular setting, several papers study reinforcement
learning in metric spaces, originating with the results
of~\citet{kakade2003exploration}
(c.f.,~\citet{ortner2012online,ortner2013adaptive,song2019efficient,yang2019learning,sinclair2019adaptive,touati2020zooming}). Of
these, the most related result is that of~\citet{sinclair2019adaptive}
who study the adaptive discretization algorithm and give a worst-case
regret analysis, showing that the algorithm has a regret rate of
$K^{\frac{d+1}{d+2}}$ where $d$ is the covering dimension of the
metric space. Essentially the same results appear
in~\citet{touati2020zooming}, although the algorithm is slightly
different. However, none of these results give sharper
instance-dependence guarantees that reflect benign problem structure,
as we will obtain.
For the special case of (contextual) bandits, several
instance-dependent guarantees that yield improved regret rates
exist~\citep{auer2007improved,valko2013stochastic,kleinberg2019bandits,bubeck2011x,slivkins2014contextual,krishnamurthy2019contextual}. For
non-contextual bandits, the results and assumptions vary considerably,
but most results quantify a benign instance in terms of the size of
the set of near-optimal actions. The formulation that we adopt is the
notion of \emph{zooming dimension}, which measures the growth rate of
the $r$-packing number of the set of $O(r)$-suboptimal arms. This
notion has been used in several works on bandits and contextual
bandits in metric spaces, and we will recover some of these results as
a special case of our main theorem.
\section{Main Results}
Our main result is a regret bound that scales with the \emph{zooming
dimension}. We introduce this parameter with a sequence of
definitions. First, we define the $\mathrm{gap}$ function, which describes the
sub-optimality of an action $a$ for state $x$.
\begin{definition}[Gap]
For any $(x,a) \in S\times A$, for $h\in [H]$, the stage-dependent sub-optimality gap is
\begin{align*}
\mathrm{gap}_h(x,a) \defeq V^\star_h(x) - Q_h^{\star}(x,a).
\end{align*}
\end{definition}
We use the gaps to define the subset of the metric space that is near-optimal.
\begin{definition}[Near-optimal set]
\label{def:good_set}
We define near-optimal set as
\begin{align*}
\Pcal_{h,r}^{Q^\star} \defeq \cbr{(x, a) \in \Scal \times \Acal: \mathrm{gap}_h(x,a) \leq \rbr{\frac{2(H+1)}{d_{max}}+ 2L} r}.
\end{align*}
\end{definition}
Intuitively, $\Pcal_{h,r}^{Q^\star}$ is the set of state-action pairs
with gap that is $O(r)$ at stage $h$. The constant in the definition
is a consequence of our analysis, but it is quite similar to the
constant in the definition of~\citet{slivkins2014contextual} for
contextual bandits. In particular, he considers $d_{\max}=1, H=1,L=1$
and obtains a constant of $12$, while we obtain a constant of $6$ in
this case.
Finally, we define the zooming number and the zooming dimension.
\begin{definition}[Zooming number and dimension]
\label{def:zooming}
The $r$-zooming number is the $r$-packing number of the near-optimal
set $\Pcal_{h,r}^{Q^\star}$, that is
$N_{r}^{\text{pack}}(\Pcal_{h,r}^{Q^\star})$. The stage-dependent
zooming dimension is defined as
\begin{align*}
z_{h,c} \defeq \inf\cbr{ d>0 : N^{\text{pack}}_{r}(\Pcal_{h,r}^{Q^\star}) \le cr^{-d}, \forall r \in (0,d_{\max}]}.
\end{align*}
The zooming dimension for the instance as the largest among all stages $z_c = \max_{h\in[H]} z_{h,c}$.
\end{definition}
\begin{wrapfigure}{R}{0.4\textwidth}
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-1.5cm}}{}
\begin{center}
\definecolor{b1}{rgb}{0,0,1.0}
\definecolor{b2}{rgb}{0.2,0.2,1.0}
\definecolor{b3}{rgb}{0.4,0.4,1.0}
\begin{tikzpicture}
\draw[thick] (0,0) -- (4,0) node[anchor=north west] {$x$};
\draw[thick] (0,0) -- (0,3) node[anchor=south east] {$a$};
\draw[name path=C2,line width=0.1mm, b3] (0,0.25) .. controls (1,0.0) and (2,2.75) .. (4,1.75);
\draw[name path=B2,line width=0.1mm, b2] (0,0.5) .. controls (1,0.25) and (2,3) .. (4,2);
\draw[name path=A2,line width=0.1mm, b1] (0,0.7) .. controls (1,0.45) and (2,3.2) .. (4,2.2);
\draw[name path=A1,line width=0.1mm, b1] (0,0.8) .. controls (1,0.55) and (2,3.3) .. (4,2.3);
\draw[name path=B1,line width=0.1mm, b2] (0,1) .. controls (1,0.75) and (2,3.5) .. (4,2.5);
\draw[name path=C1,line width=0.1mm, b3] (0,1.25) .. controls (1,1) and (2,3.75) .. (4,2.75);
\tikzfillbetween[of=A1 and A2] {b1,opacity=0.8};
\tikzfillbetween[of=A1 and B1] {b2,opacity=0.6};
\tikzfillbetween[of=A2 and B2] {b2,opacity=0.6};
\tikzfillbetween[of=B1 and C1] {b3,opacity=0.4};
\tikzfillbetween[of=B2 and C2] {b3,opacity=0.4};
\draw[thick,|-|] (1.5, 1.91) -- (1.5, 1.41);
\node [align=left] at (2.8,1.25) {near optimal \\ actions for $x$};
\end{tikzpicture}
\end{center}
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.75cm}}{}
\caption{An example where the zooming dimension is $1$ while the
the covering dimension is $2$. }
\label{fig:zooming_example}
\end{wrapfigure}
Intuitively, the zooming dimension measures how the near-optimal region
grows as we change the sub-optimality level $r$. Importantly, we use
$r$ both to parametrize the radius in the packing number and the
sub-optimality. Thus, the zooming number captures how many $r$-separated points
can be packed into the $O(r)$ sub-optimal
region.
The more standard notion of complexity of a metric space is the
\emph{covering dimension}, defined as
\begin{align*}
d_c \defeq \inf\{d > 0, N_r^{\text{pack}}( \Scal \times \Acal) \leq cr^{-d}, \forall r \in (0,d_{\max}]\}.
\end{align*}
Examining the definitions, it is clear that we have $z_c \leq d_c$,
since the packing numbers are only smaller. However, in benign
instances where the sub-optimal region concentrates to a low
dimensional manifold, we may have $z_c < d_c$ (and possibly much
smaller), which will enable sharper regret bounds. An example is
illustrated in Figure~\ref{fig:zooming_example}, where the set of
near-optimal actions concentrates on a narrow band for each $x$. Thus
the entire space and hence the covering dimension is $2$-dimensional,
but the zooming dimension is $1$. More generally, if $\Scal$ is a
$d_S$ dimensional space and $\Acal$ is a $d_A$ dimensional space, then
the covering dimension could be $\Omega(d_S + d_A)$ while the zooming
dimension could be as small as $O(d_S)$.
With these definitions, we can now state the main theorem.
\begin{theorem}
\label{thm:main}
For any initial states $\{x_1^k : k\in [K]\}$, and any $\delta \in
(0,1)$, with probability at least, $1-\delta$ Adaptive Q-learning has
the following regret\footnote{Throughout the paper $\tilde{O}(\cdot)$ suppresses logarithmic dependence in its argument.}
\begin{align*}
\mathrm{Reg}(K) \le & \tilde{O} \left(H^{3/2} \inf_{r_0 \in (0, d_{max}]} \left( \sum_{h=1}^{H}\sum_{r=d_{max}2^{-i},r\ge r_0} N^{\text{pack}}_r(\Pcal_{h,r}^{Q^\star}) \frac{ d_{max}}{r} + \frac{Kr_0}{d_{max}} \right)\right)\\
& ~~~~~~~~ +\tilde{O}\rbr{H^2 + \sqrt{H^3K\log(1/\delta)}}.
\end{align*}
\end{theorem}
Before turning to a discussion of the theorem, we state some
corollaries. First, by optimizing $r_0$, we obtain a regret bound in terms of the zooming dimension.
\begin{corollary}
For any initial states $\{x_1^k : k\in [K]\}$, and any $\delta \in
(0,1)$, with probability at least $1-\delta$ Adaptive Q-learning has
$\mathrm{Reg}(K) \leq \tilde{O}\rbr{H^{5/2} K^{\frac{z_c+1}{z_c+2}}}$, for any
constant $c>0$.
\end{corollary}
Finally, we recover the regret rate of~\citet{slivkins2014contextual}
in the special case of contextual bandits.
\begin{corollary}[Contextual bandits]
\label{cor:cb}
If $H=1$, then Adaptive Q-learning has regret $\tilde{O}\rbr{K^{\frac{z_c+1}{z_c+2}}}$, which recovers the regret rate of~\citet{slivkins2014contextual}.
\end{corollary}
We now turn to the remarks:
\begin{itemize}
\setlength{\itemsep}{1pt}
\setlength{\parsep}{1pt}
\setlength{\parskip}{1pt}
\item Theorem~\ref{thm:main} gives a regret bound that depends on the
packing numbers of the near-optimal set
(Definition~\ref{def:good_set}). This bound should be compared with
the ``metric-specific'' regret guarantee
of~\citet{sinclair2019adaptive} or the ``refined regret bound''
of~\citet{touati2020zooming}. Both of these results have the same
form as ours with all terms in agreement, but with
$N_r^{\text{pack}}(\Scal\times\Acal)$ in the place of
$N_r^{\text{pack}}(\Pcal_{h,r}^{Q^\star})$. As
$\Pcal_{h,r}^{Q^\star} \subset \Scal\times\Acal$, our bound
is always sharper.
\item The more-interpretable bound is in terms of the zooming
dimension (Definition~\ref{def:zooming}), which highlights the
dependence on the number of episodes $K$. We obtain a regret rate of
$K^{\frac{z_c+1}{z_c+2}}$ for any constant $c>0$, which should be
compared with the non-adaptive rate $K^{\frac{d_c+1}{d_c+2}}$ that
scales with the covering
dimension~\citep{song2019efficient,sinclair2019adaptive,touati2020zooming}.\footnote{We always treat $c$ as a universal constant, so its dependence in the regret bounds is suppressed.} As
the zooming dimension can be smaller than covering dimension (recall
Figure~\ref{fig:zooming_example}), this bound demonstrates a
polynomial improvement over non-adaptive approaches.
\item Corollary~\ref{cor:cb} shows that our bound recovers the
guarantee from~\citet{slivkins2014contextual}, although his bound
does not require that~\eqref{eq:v_lip} holds. We give a more
detailed explanation on the necessity of~\eqref{eq:v_lip} in
Section~\ref{sec:analysis}. Nevertheless, the fact that we
essentially recover his bound suggests that our results are the
natural generalization to multi-step RL.
\item Finally, we remark that we can instantiate the result in the
tabular setting with finite $\Scal,\Acal$ by taking the metric to be
$\Dcal((x,a),(x',a')) = \one\{(x,a)\ne (x',a')\}$. In this case we
obtain a ``partial'' gap-dependent bound of the form:
\begin{align*}
\mathrm{poly}(H)\cdot\rbr{\sqrt{|\Scal|K} + \sum_{h=1}^H\sum_{x \in \Scal} \sum_{a: \mathrm{gap}_h(x,a) > 0} \frac{\log(K)}{\mathrm{gap}_h(x,a)}}.
\end{align*}
This is not a fully gap-dependent bound because of the
$\sqrt{|\Scal|K}$ term, but it does recover an intermediate result
of~\citet{simchowitz2019non}. In particular, this confirms that the
model-free methods can achieve a partial gap-dependent guarantee for
the tabular setting.
\end{itemize}
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.5em}}{}
\section{Algorithm}
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.5em}}{}
As we have mentioned, the algorithm we analyze is the Adaptive
$Q$-learning algorithm of~\citet{sinclair2019adaptive}. For
completeness, the pseudocode is reproduced in
Algorithm~\ref{alg:zooming}. The algorithm adaptively partitions the
state-action space to focus on the informative regions, and it uses
optimism to explore the space and drive the agent to regions with high
reward.
\begin{algorithm}[t]
\caption{Adaptive $Q$--learning}\label{alg:zooming}
\begin{algorithmic}[1]
\State For $h \in [H]$, initialize $\Pcal_h^1$ to be a single ball $B_h$ with radius $d_{max}$. $Q_h^1(B_h) \gets H$.
\For{each episode $k = 1,2,\ldots,K$}
\State Receive $x_1^k$.
\For{stage $h = 1,2,\ldots,H$}
\State $B_h^k = \argmax_{B\in \mathrm{rel}_h^k(x_h^k)} Q_h^k(B)$
\State Play action $a_h^k$ for some $(x_h^k, a_h^k) \in \mathrm{dom}_h^k(B_h^k)$
\State Receive $r_h^k, x_{h+1}^k$, update $t = n_h^{k+1}(B_h^k) = n_h^{k}(B_h^k) + 1$
\State $V_{h+1}^k(x_{h+1}^k) = \min \cbr{H, \max_{B \in \mathrm{rel}_{h+1}^k(x_{h+1}^k)} Q_{h+1}^k(B)}$.
\State $Q_h^{k+1}(B_h^k) = (1-\alpha_t) Q_h^k(B_h^k) + \alpha_t(r_h^k + b_t + V_{h+1}^k(x_{h+1}^k))$.
\If{$t_h^{k+1}(B_h^k) \ge \rbr{\frac{d_{max}}{r(B_h^k)}}^2$} split $B_h^k$: \label{lin:split}
\State Create a set of balls $\Bcal_h^k = \{r(B_h^k)\text{-net of } \mathrm{dom}_h^k (B_h^k)\}$.
\State Inherit the count and $Q_h^k$ from $B_h^k$. Set $\Pcal_h^{k+1} = \Pcal_h^{k} \cup \Bcal_h^k$.
\EndIf
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
During the execution, the algorithm creates many balls $B \subset
\Scal\times\Acal$ for each stage $h$. We use $\Pcal_h^k$ to denote the
set of balls created for stage $h$ up until episode $k$. Every ball
$B$ has a radius, denoted $r(B)$ and a \emph{domain}, denoted
$\mathrm{dom}_h^k(B)$. The domain is the set of points contained in this ball,
but not in any other active ball with smaller radius. Formally,
\begin{align*}
\mathrm{dom}_h^k(B) \defeq B \setminus \{\cup_{B' \in \Pcal_h^k: r(B') < r(B)} B'\}.
\end{align*}
For each ball, we also maintain a counter $t = n_h^k(B)$ which denotes
the number of times we have chosen state-action pairs in $B$ or its
ancestors. Parents and ancestors are defined via the splitting rule:
when a ball is split in line~\ref{lin:split}, the resulting balls are called the
children. Finally, we maintain a scalar $Q_h^k(B)$ which serves as an
upper bound on $\max_{(x,a) \in B} Q_h^\star(x,a)$.
In stage $h$ of episode $k$, we select the action for state $x_h^k$ as
follows: we consider all the smallest balls that contains $x_h^k$,
defined as ``relevant'' balls
\begin{align*}
\mathrm{rel}_h^k(x) \defeq \{B | \exists a, (x,a) \in \mathrm{dom}_h^k(B)\}.
\end{align*}
Among the relevant balls, the algorithm select the ball $B_h^k$ with
the highest $Q_h^k(B)$ value and plays an arbitrary action such that
$(x_h^k,a) \in B_h^k$. We increment the sample count $n_h^k(B_h^k)$ for
this ball and at the end of the episode, we update $Q_h^k(B_h^k)$ via
\begin{align*}
Q_h^{k+1}(B_h^k) &= (1-\alpha_t) Q_h^k(B_h^k) + \alpha_t(r_h^k + b_t + V_{h+1}^k(x_{h+1}^k))\\
V_{h+1}^k(x) &= \min\cbr{H, \max_{B\in \mathrm{rel}_{h+1}^k(x)} Q_{h+1}^k(B)}.
\end{align*}
where the $\alpha_t$ is the learning rate and $b(t)$ is the bonus added to ensure $Q_h^k$ is optimistic. Formally,
\begin{align*}
\alpha_t \defeq \frac{H+1}{H+t}, \qquad b_t \defeq 2\sqrt{\frac{H^3 \log(4HK/\delta)}{t}} + \frac{4Ld_{max}}{\sqrt{t}}.
\end{align*}
For all other balls at stage $h$, we set $Q_h^{k+1}(B) \gets
Q_h^k(B)$, with no update.
We split a ball $B$ as soon as $n_h^k(B) \geq
\rbr{\frac{d_{max}}{r(B)}}^2$. When splitting, we create a
set of new ``children'' balls with radius $r(B)/2$ that forms an
$r(B)/2$-net of $\mathrm{dom}_h^k(B)$. These balls inherit the count $n_h^k$
and the estimate $Q_h^k$ from the ``parent'' ball $B$, and we add them
to $\Pcal_h^{k+1}$. This splitting rule leads to the following
invariant
\begin{lemma}[Lemma 5.3 in~\cite{sinclair2019adaptive}]
For every $(h,k)\in [H]\times[K]$, we have
\begin{enumerate}
\setlength{\itemsep}{0em}
\setlength{\parsep}{0em}
\setlength{\parskip}{0em}
\item (Covering) The domains of balls in $\Pcal_h^k$ covers $\Scal \times \Acal$.
\item (Separation) For any two balls of radius $r$, their centers are at distance at least $r$.
\end{enumerate}
\end{lemma}
\paragraph{Computational considerations.}
As discussed in~\citet{sinclair2019adaptive}, this algorithm can be
implemented in a computationally efficient manner provided that the
metric space allows certain natural operations. Formally, we operate
in an oracle model, which allows us to query the metric to compute
$\mathrm{dom}(B), \mathrm{rel}(x)$, and to construct an $r$-net for any $r$ and any
subset of the metric space.
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.5em}}{}
\section{Proof sketch}\label{sec:analysis}
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.5em}}{}
In this section we describe the main steps of the proof, with details
deferred to the appendix.
It is worth reviewing prior regret analyses for episodic RL~\citep{jin2018q}. The arguments establish a regret
decomposition that relates the estimate $V_1^k$ to $V^{\pi_k}$, the
expected reward collected in episode $k$. The decomposition is
recursive in nature, involving differences between $Q_h^k$ and
$Q_h^\star$. These are controlled by the update rule and the design of
the learning rate. In particular, we can bound $Q_h^k - Q_h^\star$ by
an immediate ``surplus'' $\beta_t$ and the downstream value function
error. Formally for any ball $B$ with $(x,a) \in \mathrm{dom}_h^k(B)$
\begin{align}
\label{eq:old_q_bd}
Q_h^k(B) - Q_h^\star(x,a) \leq \one_{[t=0]}H + \sum_{i=1}^t\alpha_t^i(V_{h+1}^{k_i} - V_{h+1}^\star)(x_{h+1}^{k_i}) + \beta_t,
\end{align}
where $t = n_h^k(B), \alpha_i^t = \alpha_i
\prod_{j=i+1}^t(1-\alpha_j)$ and $\beta_t =
2\sum_{i=1}^t\alpha_i^tb_i$. Here $k_i$ is the index of the episode
where $B$ was selected for the $i^{\textrm{th}}$ time. Summing over
all episodes and grouping terms appropriately, we obtain
\begin{align*}
\sum_{k=1}^K (V_h^k - V_h^{\pi_k})(x_h^k) \leq \sum_{k=1}^K\rbr{H\one_{[n_h^k=0]} + \beta_{n_h^k} + \xi_h^k} + \rbr{1 + \nicefrac{1}{H}}\sum_{k=1}^K\rbr{V_{h+1}^k - V_{h+1}^{\pi_k}}(x_{h+1}^k),
\end{align*}
where $\xi_{h+1}^k$ is a stochastic term that can be ignored for this
discussion. Note that, as long as $V_h^k$ is optimistic (which we will
verify), this also provides a bound on the regret.
For the tabular setting,~\citet{jin2018q} use this regret
decomposition to obtain a worst-case bound. The leading term arises
from the ``surplus'' term $\beta_{n_h^k}$, which leads to a
$\mathrm{poly}(H)\sqrt{SAK}$ regret bound for the tabular setting. On the
other hand for our setting, the splitting rule implies that for any
ball $B$, we must have $n_h^k \leq
\rbr{\nicefrac{d_{\max}}{r(B)}}^2$. We can use this to obtain a bound
that depends on the number of active balls at each scale $r$ times
$d_{\max}/r$. If we could bound the number of active balls at scale
$r$ in terms of the packing number
$N_r^{\text{pack}}(\Pcal_{h,r}^{Q^\star})$, then we would obtain the
instance-dependent bound.
Unfortunately, this is not possible. In general, the algorithm will
activate balls outside of the near-optimal region, because we may have
to select a highly suboptimal ball many times to reduce downstream
over-estimation error. So indeed the number of active balls at scale
$r$ could be much larger than the packing of the near-optimal set.
We address this with the following key observation. If the surplus
$\beta_{n_h^k}$ is small compared to gap, and we choose this ball, it
must be the case that the downstream regret is quite large, otherwise
we would not have chosen this ball. If this is true, we can account
for the surplus by adding a small constant fraction of the future
regret. In otherwords, we can ``clip'' the surplus to zero once it is
proportional to the gap, and we only pay a constant factor in the
recursive term. This is the clipping trick developed
by~\citet{simchowitz2019non} to establish gap dependent bounds for
tabular MDP. Formally instead of~\pref{eq:old_q_bd}, we have the
following lemma.
\begin{lemma}[Clipped upper bound]\label{lem:clipupper}
For any $\delta \in (0,1)$ with probability at least $1-\delta/2$,
$\forall h\in [H]$,
\begin{align*}
Q_h^k(B_h^k) - Q_h^{\star}(x_h^k, a_h^k) &\le \rbr{1+\nicefrac{1}{H}}\rbr{\one_{[t=0]}H + \sum_{i=1}^{t} \alpha_t^i(V_{h+1}^{k_i} - V^\star_{h+1})(x_{h+1}^{k_i})} \\
& ~~~~~~~~ + \mathrm{clip}\sbr{\beta_t \mid \frac{\mathrm{gap}_h(x_h^k, a_h^k)}{H+1}},
\end{align*}
where $t = n_h^k(B), \alpha_i^t = \alpha_i
\prod_{j=i+1}^t(1-\alpha_j)$ and $\beta_t =
2\sum_{i=1}^t\alpha_i^tb_i$ and $\mathrm{clip}[\mu\mid\nu] \defeq \mu\one\{\mu \geq \nu\}$.
\end{lemma}
This bound should be compared with~\eqref{eq:old_q_bd}. On one hand
the recursive term is multiplied by $1+\nicefrac{1}{H}$, but, on the
other, we are able to clip the surpluses $\beta_t$. The former will
exponentiate but will asymptote to $e$, while the latter is crucial
for our instance dependent bounds.
Using this lemma, we can bound the difference between $V_h^k$ and
$V_h^{\pi_k}$.
\begin{lemma}[Clipped recursion, informal]\label{lem:cliprecur_main}
For any $\delta \in (0,1)$, with probability at least $1-\delta/2$, $\forall h\in [H]$,
\begin{align*}
\sum_{k=1}^K (V_h^k - V_h^{\pi^k})(x_h^k) \le & \sum_{k=1}^K \rbr{1+\nicefrac{1}{H}} \alpha_t^0 + \mathrm{clip}\sbr{\beta_{n_h^k} \mid \mathrm{gap}_h(x_h^k, a_h^k)/(H+1)} + \xi_{h+1}^k \\
&+ \rbr{1+\nicefrac{1}{H}}^2 \sum_{k=1}^K (V_{h+1}^k - V_{h+1}^{\pi^k})(x_{h+1}^k),
\end{align*}
where $\xi_{h+1}^k$ is conditionally centered random variable with
range $H$.
\end{lemma}
We bound $V_1^k - V_1^{\pi_k}$, and by optimism the regret, by
applying Lemma~\ref{lem:cliprecur_main} recursively.
The last step is to show that the sum of clipped surpluses can be
related to the zooming dimension. First note that for any ball $B$, its
ancestors must be played at least
$\nicefrac{1}{4}\rbr{\nicefrac{d_{\max}}{r(B)}}^2$ times before it
becomes activated. Since the ball inherits data from its ancestors, if
it becomes activated but only contains points with large gap, we can
always clip the surplus term. Thus all active balls $B$ that have
$r(B) \ll \min_{x,a \in B}\mathrm{gap}(x,a)$ do not contribute to the
regret.
Next, if a ball with radius $r$ contains a point where the gap is
small, we cannot appeal to clipping. However, by Lipschitzness, all
points in the ball must have small gaps, which means that this ball is
contained in the near optimal set at scale $r$. As above, the surplus for
each of these balls contributes at most $\nicefrac{d_{\max}}{r}$ to
the regret. Then, since all balls with radius $r$ are at least $r$
apart and we only incur regret for those entirely contained in the
near-optimal region, we obtain the bound that depends on
$N_r^{\text{pack}}(\Pcal_{h,r}^{Q^\star})$.
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-1em}}{}
\paragraph{Remarks on Assumption~\ref{assum:lipschitz}.}
We give some intuition on why our proof
requires~\eqref{eq:v_lip}, which is slightly stronger than what is
required for the zooming dimension analysis
of~\citet{slivkins2014contextual} for contextual bandits.
In~\citet{slivkins2014contextual}, the optimistic selection rule
ensures that the context-action pairs chosen by the algorithm have
small gap, but this is not true in the multi-step setting. In the RL
setting, we might select an action (in a ball) with a large gap
because the downstream regret is large. In this case, we can clip the
surplus, but we can only clip at the \emph{minimum} gap among all
$(x,a)$ pairs in the ball. To obtain a zooming dimension bound, we
must argue that this ball is contained in the near-optimal set, but
this requires that the value functions, and hence the gaps, are
Lipschitz. We recall that~\eqref{eq:v_lip} is implied
by~\eqref{eq:q_lip} if the metric is sub-additive.
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.5em}}{}
\section{Discussion}
\ifthenelse{\equal{arxiv}{arxiv}}{\vspace{-0.5em}}{}
In this paper, we give a refined analysis of the Adaptive Q-learning
algorithm of Sinclair, Banerjee and Yu (2019) for sample efficient
reinforcement learning in metric spaces. We show that the algorithm
has a regret bound that depends on the zooming dimension of the
instance, with rate $K^{\frac{z+1}{z+2}}$ when the zooming dimension
is $z$. This always improves on the worst-case bound that depends on
the covering dimension, and can be much better when the $Q^\star$
function concentrates quickly onto a low-dimensional set of
actions. The bound also recovers that
of~\citet{slivkins2014contextual} for contextual bandits in metric
spaces, under a slightly stronger assumption. The key technique is
the clipped regret decomposition of~\citet{simchowitz2019non}, which
we complement with a book-keeping argument. Our results show that
adaptivity to benign instances is possible in RL with metric spaces,
and partially mitigate the curse of dimensionality in such settings.
\subsection*{Acknowledgements}
We thank Wen Sun and Aleksandrs Slivkins for formative discussions
during the conception of this paper. We also thnk Max Simchowitz for
insightful discussions regarding the clipping technique.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 7,348
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Get Inspired: SunCommon's Climate Action Film Festival
Climate & Energy, News and Updates / By Sebbi Wu / February 7, 2020 July 1, 2022
Last night, more than 200 Vermonters braved the snowy weather to gather at the sold-out Burlington Premiere of the Climate Action Film Festival, organized by our friends at SunCommon.
The Festival features short films from around North America highlighting volunteer conservation work, community solar projects, carbon capture, and nonviolent direct action to block fossil fuels and raise awareness about the climate crisis. It's an impressive lineup, and the focus on climate solutions offers a much needed antidote to the feelings of doom and gloom that we often feel when confronted with one of humanity's greatest global challenges.
It's inspiring to see the way that communities across the globe, large and small, are taking meaningful action to address the challenges of climate change. The featured films illustrate the transformation power of grassroots activism and remind us why the work we do here at VPIRG – backed by our thousands of members and supporters across Vermont – is so critical. If we want our brave little state to be a leader on climate and other important issues, we all have to stand together and fight for our future.
If you couldn't make it to the Burlington premier, don't worry – it's not too late for you to attend one of the other screenings here in Vermont.
There's another full screening taking place at the Middlebury Marquis (65 Main Street, Middlebury, VT) from 6:00 p.m. to 8:00 p.m. next Wednesday, February 12th, as well as a number of other community events across the state featuring a selection of festival favorites and a discussion about effective local climate action.
Make sure you're signed up for our email list so we can notify you when there's a Climate Action Film Festival screening near you and head to www.actonclimatevt.org for more information on how you can support bold climate action in 2020.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,237
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Atelopus longibrachius és una espècie d'amfibi que viu a Colòmbia.
Està amenaçada d'extinció per la pèrdua del seu hàbitat natural.
Referències
Acosta-Galvis, A., Bolívar, W. & Lötters, S. 2004. Atelopus longibrachius. Llista Vermella d'Espècies Amenaçades de la UICN 2006. Consulta: 9 d'abril del 2008.
Longibrachius
|
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"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/jerrymahun.com\/index.php\/home\/open-access\/52-iv-least-squares-lite\/236-chapter-f-nonlinear-measurements?start=5","text":"### Article Index\n\nGiven the network and data in Figure F-5, determine:\n\n\u2022 adjusted coordinates of points J and K.\n\u2022 their standard errors\n\u2022 their standard errors\n Figure F-5 Combined Observations\n\nThere are seven observations and four unknowns: NJ, EJ, NK, EK. \u00a0The network has 7-4 = 3 DF.\n\n### a. Matrix structures\n\nMatrix structures are:\n\n### b. Initial approximations for points K and J\n\nThe computations are summarized and results shown:\n\n\u2022 Distance and direction for line BA by inverse\n\u2022 Solve angle ABK using Law of Cosines: 28\u00b028'09\"\n\u2022 Compute direction of BK using direction BA and angle ABK: 66\u00b033'27\"\n\u2022 Use direction BK and 1309.94' to compute coordinates of K: 5097.77' N, 3312.73' E\n\u2022 Solve angle JBA using Law of Cosines: 11\u00b056'48\"\n\u2022 Compute direction of BJ using direction BA and angle JBA: 26\u00b008'30\"\n\u2022 Use direction BJ and 871.35' to compute coordinates of J: 5358.86' N, 2494.82' E\n\n### d. Distances\n\nThe distances are computed from the fixed coordinates of A and B and the initial approximations of J and K.\n\n### e. Build [C] and [K] matrices\n\nNote: it is extremely important to minimize rounding errors by carrying enough digits in computations. Not doing so can either increase the number of iterations needed for a solution or could cause the solution to diverge.\n\nObs 1: Line KB\n\nObs 2: Line KJ\n\nObs 3: Line KA\n\nObs 4: Line JA\n\nObs 5: Line JB\n\nObs 6: Angle BKJ\n\nObs 7: Angle JKA\n\n### f. Solve U=[Q] x [CTWK]\n\nRather than detail the complete solution process, the mtrices and partial products are shown for each iteration.\n\n#### (1) First Iteration\n\nThe corrections aren't small enough so update the coordinates and repeat solution.\n\nUpdated coordinates:\n\nRepeat as corrections are not small enough.\n\n#### (2) Second Iteration\n\nUsing the updated coordinates, go back to Step d, recompute distances and matrices, and solve for [U]\n\nThe corrections still aren't small enough so update the coordinates and repeat solution.\n\nUpdated coordinates:\n\n#### (3) Third Iteration\n\nCorrections are acceptably small enough. Update coordinates one last time.\n\n### g. Compute statistics\n\n#### (1) Compute So and adjusted point uncertainties\n\nUsing last [C], [K] and [U] matricies, determine residuals from [V] = [C] x [U] - [K]\n\nUse [V] and [W] to compute So , then So and [Q] to compute standard deviations of the adjusted coordinates.\n\nAdd residuals to the original observations\n\nCompute standard errors for the adjusted observations. Example comps are shown for distance KB and angle BKJ.\n\nObs 1: distance KB\n\nUse first row of [C] and first column of [CT].\n\nObs 6: Angle BKJ\n\nUse sixth row of [C] and sixth column of [CT].\n\nRemaining standard errors are included in the following section.","date":"2023-02-07 17:44:07","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9064551591873169, \"perplexity\": 10113.844423667795}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500628.77\/warc\/CC-MAIN-20230207170138-20230207200138-00730.warc.gz\"}"}
| null | null |
{"url":"https:\/\/srdas.github.io\/MLBook\/Networks.html","text":"Chapter 9 Making Connections: Networks\n\n9.2 Small Worlds\n\nUseful introductory book on networks: http:\/\/www.cs.cornell.edu\/home\/kleinber\/networks-book\/\n\n9.4 Graphs\n\nWhat is a graph? It is a picture of a network, a diagram consisting of relationships between entities. We call the entities as vertices or nodes (set $$V$$) and the relationships are called the edges of a graph (set $$E$$). Hence a graph $$G$$ is defined as\n\n$$$G = (V,E)$$$\n\n9.4.1 Types of graphs\n\nIf the edges $$e \\in E$$ of a graph are not tipped with arrows implying some direction or causality, we call the graph an \u201cundirected\u201d graph. If there are arrows of direction then the graph is a \u201cdirected\u201d graph.\n\nIf the connections (edges) between vertices $$v \\in V$$ have weights on them, then we call the graph a \u201cweighted graph\u201d else it\u2019s \u201cunweighted\u201d. In an unweighted graph, for any pair of vertices $$(u,v)$$, we have\n\n$$$w(u,v) = \\left\\{ \\begin{array}{ll} w(u,v) = 1, & \\mbox{ if } (u,v) \\in E \\\\ w(u,v) = 0, & \\mbox{ if } (u,v) \\ni E \\end{array} \\right.$$$\n\nIn a weighted graph the value of $$w(u,v)$$ is unrestricted, and can also be negative.\n\nDirected graphs can be cyclic or acyclic. In a cyclic graph there is a path from a source node that leads back to the node itself. Not so in an acyclic graph. The term dag is used to connote a \u201cdirected acyclic graph\u201d. The binomial option pricing model in finance that you have learnt is an example of a dag.\n\nA graph may be represented by its adjacency matrix. This is simply the matrix $$A = \\{w(u,v)\\}, \\forall u,v$$. You can take the transpose of this matrix as well, which in the case of a directed graph will simply reverse the direction of all edges.\n\n9.6 igraph package\n\nlibrary(igraph)\n## Loading required package: methods\n##\n## Attaching package: 'igraph'\n## The following objects are masked from 'package:stats':\n##\n## decompose, spectrum\n## The following object is masked from 'package:base':\n##\n## union\ng = erdos.renyi.game(20,1\/10)\ng\n## IGRAPH U--- 20 23 -- Erdos renyi (gnp) graph\n## + attr: name (g\/c), type (g\/c), loops (g\/l), p (g\/n)\n## + edges:\n## [1] 1-- 5 8-- 9 2--10 6--11 7--11 1--12 6--12 10--12 11--12 6--13\n## [11] 7--15 12--15 1--16 4--16 8--16 3--17 1--18 1--19 10--19 14--19\n## [21] 16--19 5--20 9--20\nplot.igraph(g)\n\nprint(clusters(g))\n## $membership ## [1] 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 ## ##$csize\n## [1] 18 2\n##\n## $no ## [1] 2 g$no[[1]]\n## NULL\n\n9.7 Graph Attributes\n\n#GENERATE RANDOM GRAPH\ng = erdos.renyi.game(30,0.2)\nplot(g)\n\n#COMPUTE DEGREE DISTRIBUTION\ndd = degree.distribution(g)\ndd = as.matrix(dd)\nd = as.matrix(seq(0,max(degree(g))))\nplot(d,dd,type=\"l\")\n\n9.8 Dijkstra\u2019s Shortest Paths Algorithm\n\nThis is one of the most well-known algorithms in theoretical computer science. Given a source vertex on a weighted, directed graph, it finds the shortest path to all other nodes from source $$s$$. The weight between two vertices is denoted $$w(u,v)$$ as before. Dijkstra\u2019s algorithm works for graphs where $$w(u,v) \\geq 0$$. For negative weights, there is the Bellman-Ford algorithm. The algorithm is as follows.\n\nfunction DIJKSTRA(G,w,s)\nS = { }\n%S = Set of vertices whose shortest paths from\n%source s have been found\nQ = V(G)\nwhile Q notequal { } :\nu = getMin(Q)\nS = S + u\nQ = Q - u\nfor each vertex v in SUCC(u):\nif d[v] > d[u]+w(u,v) then:\nd[v] = d[u]+w(u,v)\nPRED(v) = u\n\n#DIJSKATRA'S SHORTEST PATHS ALGORITHM\ne = matrix(nc=3, byrow=TRUE, c(1,2,8, 1,4,4, 2,4,3, 4,2,1, 2,3,1, 2,5,7, 4,5,4, 3,5,1))\ne\n## [,1] [,2] [,3]\n## [1,] 1 2 8\n## [2,] 1 4 4\n## [3,] 2 4 3\n## [4,] 4 2 1\n## [5,] 2 3 1\n## [6,] 2 5 7\n## [7,] 4 5 4\n## [8,] 3 5 1\ng = graph.empty(5)\nplot(g)\n\nplot(g,edge.width=e[,3],edge.label=e[,3])\n\nget.shortest.paths(g,1)\n## $vpath ##$vpath[[1]]\n## + 0\/5 vertices:\n##\n## $vpath[[2]] ## + 3\/5 vertices: ## [1] 1 4 2 ## ##$vpath[[3]]\n## + 4\/5 vertices:\n## [1] 1 4 2 3\n##\n## $vpath[[4]] ## + 2\/5 vertices: ## [1] 1 4 ## ##$vpath[[5]]\n## + 5\/5 vertices:\n## [1] 1 4 2 3 5\n##\n##\n## $epath ## NULL ## ##$predecessors\n## NULL\n##\n## $inbound_edges ## NULL print(shortest.paths(g)) ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 5 6 4 7 ## [2,] 5 0 1 1 2 ## [3,] 6 1 0 2 1 ## [4,] 4 1 2 0 3 ## [5,] 7 2 1 3 0 print(average.path.length(g)) ## [1] 1.272727 el <- matrix(nc=3, byrow=TRUE,c(0,1,0, 0,2,2, 0,3,1, 1,2,0, 1,4,5, 1,5,2, 2,1,1, 2,3,1, 2,6,1, 3,2,0, 3,6,2, 4,5,2, 4,7,8, 5,2,2, 5,6,1, 5,8,1, 5,9,3, 7,5,1, 7,8,1, 8,9,4) ) el[,1:2] = el[,1:2]+1 #Note that the zero vertex option does not exist any more, so we added 1 g = add.edges(graph.empty(10), t(el[,1:2]), weight=el[,3]) plot(g) #GRAPHING MAIN NETWORK g = simplify(g) V(g)$name = seq(vcount(g))\n#l = layout.fruchterman.reingold(g)\nl = layout.circle(g)\nl = layout.norm(l, -1,1,-1,1)\n#pdf(file=\"network_plot.pdf\")\nplot(g, layout=l, vertex.size=10, vertex.label=seq(1,10), vertex.color=\"#ff000033\",\nedge.color=\"grey\", edge.arrow.size=0.75, rescale=FALSE,\nxlim=range(l[,1]), ylim=range(l[,2]))\n\n9.9 D3 plots\n\nD3 is a well known framework for plotting spring graphs. The following plot shows how one may use javascript in R, using the html widgets framework. See: http:\/\/www.htmlwidgets.org\/\n\nlibrary(networkD3)\n\nlinks$value = 1 nodes = data.frame(unique(c(links$target,links$source))) names(nodes) = \"name\" nodes$group = ceiling(3*runif(length(nodes$name))) forceNetwork(Links = links, Nodes = nodes, Source = \"source\", Target = \"target\", Value = \"value\", NodeID = \"name\", Group = \"group\", opacity = 0.8, fontSize = 75) 9.10 Centrality Centrality is a property of vertices in the network. Given the adjacency matrix $$A=\\{w(u,v)\\}$$, we can obtain a measure of the \u201cinfluence\u201d of all vertices in the network. Let $$x_i$$ be the influence of vertex $$i$$. Then the column vector $$x$$ contains the influence of each vertex. What is influence? Think of a web page. It has more influence the more links it has both, to the page, and from the page to other pages. Or think of a alumni network. People with more connections have more influence, they are more \u201ccentral\u201d. It is possible that you might have no connections yourself, but are connected to people with great connections. In this case, you do have influence. Hence, your influence depends on your own influence and that which you derive through others. Hence, the entire system of influence is interdependent, and can be written as the following matrix equation $$$x = A\\;x$$$ Now, we can just add a scalar here to this to get $$$\\xi \\; x = A x$$$ an eigensystem. Decompose this to get the principle eigenvector, and its values give you the influence of each member. In this way you can find the most influential people in any network. There are several applications of this idea to real data. This is eigenvector centrality is exactly what Google trademarked as PageRank, even though they did not invent eigenvector centrality. A = matrix(nc=3, byrow=TRUE, c(0,1,1, 1,0,1, 1,1,0)) print(A) ## [,1] [,2] [,3] ## [1,] 0 1 1 ## [2,] 1 0 1 ## [3,] 1 1 0 g = graph.adjacency(A,mode=\"undirected\",weighted=TRUE,diag=FALSE) res = evcent(g) print(names(res)) ## [1] \"vector\" \"value\" \"options\" res$vector\n## [1] 1 1 1\nres = evcent(g,scale=FALSE)\nres$vector ## [1] 0.5773503 0.5773503 0.5773503 A = matrix(nc=3, byrow=TRUE, c(0,1,1, 1,0,0, 1,0,0)) print(A) ## [,1] [,2] [,3] ## [1,] 0 1 1 ## [2,] 1 0 0 ## [3,] 1 0 0 g = graph.adjacency(A,mode=\"undirected\",weighted=TRUE,diag=FALSE) res = evcent(g,scale=FALSE) res$vector\n## [1] 0.7071068 0.5000000 0.5000000\nA = matrix(nc=3, byrow=TRUE, c(0,2,1, 2,0,0, 1,0,0))\nprint(A)\n## [,1] [,2] [,3]\n## [1,] 0 2 1\n## [2,] 2 0 0\n## [3,] 1 0 0\ng = graph.adjacency(A,mode=\"undirected\",weighted=TRUE,diag=FALSE)\nres = evcent(g,scale=FALSE)\nres$vector ## [1] 0.7071068 0.6324555 0.3162278 9.11 Betweenness Another concept of centrality is known as \u201cbetweenness\u201d. This is the proportion of shortest paths that go through a node relative to all paths that go through the same node. This may be expressed as $$$B(v) = \\sum_{a \\neq v \\neq b} \\frac{n_{a,b}(v)}{n_{a,b}}$$$ where $$n_{a,b}$$ is the number of shortest paths from node $$a$$ to node $$b$$, and $$n_{a,b}(v)$$ are the number of those paths that traverse through vertex $$v$$. Here is an example from an earlier directed graph. el = matrix(nc=3, byrow=TRUE, c(0,1,1, 0,2,2, 0,3,1, 1,2,1, 1,4,5, 1,5,2, 2,1,1, 2,3,1, 2,6,1, 3,2,1, 3,6,2, 4,5,2, 4,7,8, 5,2,2, 5,6,1, 5,8,1, 5,9,3, 7,5,1, 7,8,1, 8,9,4) ) el[,1:2] = el[,1:2] + 1 g = add.edges(graph.empty(10), t(el[,1:2]), weight=el[,3]) plot(g) res = betweenness(g) res ## [1] 0.0000000 18.5833333 18.2500000 0.8333333 5.0000000 20.0000000 ## [7] 0.0000000 0.0000000 0.0000000 0.0000000 g = erdos.renyi.game(30,0.1) d = seq(0,max(degree(g))) dd = degree.distribution(g) plot(g) #DIAMETER print(diameter(g)) ## [1] 6 #FRAGILITY print((t(d^2) %*% dd)\/(t(d) %*% dd)) ## [,1] ## [1,] 3.837209 #CENTRALITY res = evcent(g) res$vector\n## [1] 0.13020514 0.10654809 0.50328790 0.53703737 0.22421218 0.23555387\n## [7] 0.33641755 0.09718898 0.07088808 0.61028079 0.37861544 0.27615600\n## [13] 0.37620605 0.17105358 1.00000000 0.07332221 0.08635696 0.12932960\n## [19] 0.15630895 0.28404621 0.17887855 0.27369218 0.13102918 0.25669577\n## [25] 0.25669577 0.72508578 0.23833268 0.69685043 0.25944866 0.41435043\n\n9.12 Communities\n\nCommunity detection methods partition nodes into clusters that tend to interact together. It is useful to point out the considerable flexibility and realism built into the definition of our community clusters. We do not require all nodes to belong to communities. Nor do we fix the number of communities that may exist at a time, and we also allow each community to have different size.\n\nWith this flexibility, the key computational challenge is to find the \u201cbest\u201d partition because the number of possible partitions of the nodes is extremely large. Community detection methods attempt to determine a set of clusters that are internally tight-knit. Mathematically, this is equivalent to finding a partition of clusters to maximize the observed number of connections between cluster members minus what is expected conditional on the connections within the cluster, aggregated across all clusters. More formally, we choose partitions with high modularity $$Q$$, where\n\n$$$Q = \\frac{1}{2m} \\sum_{i,j} \\left[ A_{ij} - \\frac{d_i \\times d_j}{2m} \\right] \\cdot \\delta(i,j)$$$\n\n$$A_{ij}$$ is the $$(i,j)$$-th entry in the adjacency matrix, i.e., the number of connections in which $$i$$ and $$j$$ jointly participated, $$d_i=\\sum_j A_{ij}$$ is the total number of transactions that node $$i$$ participated in (or, the degree of $$i$$) and $$m = \\frac{1}{2} \\sum_{ij} A_{ij}$$ is the sum of all edge weights in matrix $$A$$.\n\nThe function $$\\delta(i,j)$$ is an indicator equal to 1.0 if nodes $$i$$ and $$j$$ are from the same community, and zero otherwise. $$Q$$ is bounded in [-1, +1]. If $$Q > 0$$, intra-community connections exceed the expected number given deal flow.\n\nConsider a network of five nodes $$\\{A,B,C,D,E\\}$$, where the edge weights are as follows: $$A:B=6$$, $$A:C=5$$, $$B:C=2$$, $$C:D=2$$, and $$D:E=10$$. Assume that a community detection algorithm assigns $$\\{A,B,C\\}$$ to one community and $$\\{D,E\\}$$ to another, i.e., only two communities. The adjacency matrix for this graph is given by matrix $$A$$ below.\n\nA = matrix(c(0,6,5,0,0,6,0,2,0,0,5,2,0,2,0,0,0,2,0,10,0,0,0,10,0),5,5)\nprint(A)\n## [,1] [,2] [,3] [,4] [,5]\n## [1,] 0 6 5 0 0\n## [2,] 6 0 2 0 0\n## [3,] 5 2 0 2 0\n## [4,] 0 0 2 0 10\n## [5,] 0 0 0 10 0\ng = graph.adjacency(A,mode=\"undirected\",weighted=TRUE,diag=FALSE)\nwtc = walktrap.community(g)\nres=membership(wtc)\nprint(res)\n## [1] 1 1 1 2 2\ng = graph.adjacency(A,mode=\"undirected\",weighted=TRUE,diag=FALSE)\nfgc = fastgreedy.community(g,merges=TRUE,modularity=TRUE,\nweights=E(g)$weight) res = membership(fgc) res ## [1] 1 1 1 2 2 g = graph.adjacency(A,mode=\"undirected\",diag=FALSE) wtc = walktrap.community(g) res = membership(wtc) print(res) ## [1] 2 2 2 1 1 print(modularity(g,res)) ## [1] 0.4128 #New functions in igraph for walktrap res = cluster_walktrap(g) print(res) ## IGRAPH clustering walktrap, groups: 2, mod: 0.41 ## + groups: ##$1\n## [1] 4 5\n##\n## $2 ## [1] 1 2 3 ## print(modularity(g,res$membership))\n## [1] 0.4128\n\n9.14 Risk Networks\n\n#RISK NETWORKS PROGRAM CODE\n\nlibrary(igraph)\n\n#FUNCTION FOR RISK INCREMENT AND DECOMP\nNetRisk = function(Ri,X) {\nS = sqrt(t(Ri) %*% X %*% Ri)\nRiskIncr = 0.5 * (X %*% Ri + t(X) %*% Ri)\/S[1,1]\nRiskDecomp = RiskIncr * Ri\nresult = list(S,RiskIncr,RiskDecomp)\n}\n\n9.14.1 Example\n\n#READ IN DATA\nna = dim(data)[2]-1 #columns (assets)\nnc = 20 #Number of controls\nm = dim(data)[1] #rows (first 1 is header, next n are assets, next 20 are controls, remaining are business lines, last line is weights)\nnb = m-na-nc-2 #Number of business lines\nX = data[2:(1+na),2:(na+1)]\nX = matrix(as.numeric(as.matrix(X)),na,na)\ncontrols = data[(2+na):(1+na+nc),2:(na+1)]\ncontrols = matrix(as.numeric(as.matrix(controls)),nc,na)\nRi = matrix(colSums(controls),na,1) #Aggregate risk by asset\nbus = data[(2+na+nc):(m-1),2:(na+1)]\nbus = matrix(as.numeric(as.matrix(bus)),nb,na)\nbus_names = as.matrix(data[(2+na+nc):(m-1),1])\nwts = data[m,2:(1+nb)]\nwts = matrix(as.numeric(as.matrix(wts)),nb,1)\/100 #percentage weights\n#TABLE OF ASSETS: Asset number, Asset name, IP address\ntab_assets = cbind(seq(1,na),names(data)[2:(na+1)],t(data[1,2:(na+1)]))\nwrite(t(tab_assets),file=\"DSTMAA_data\/tab_assets.txt\",ncolumns=3)\n#GRAPH NETWORK: plot of the assets and the links with directed arrows\nY = X; diag(Y)=0\nplot.igraph(g,layout=layout.fruchterman.reingold,edge.arrow.size=0.5,vertex.size=15,vertex.label=seq(1,na))\n\n9.14.2 Overall Risk Score\n\n#COMPUTE OVERALL RISK SCORE\n#A computation that considers the risk level of each asset (Ri)\n#The function S below is homogenous of degree 1, i.e., S(m*Ri) = m*S(Ri)\nS = sqrt(t(Ri) %*% X %*% Ri); print(c(\"Risk Score\",S))\n## [1] \"Risk Score\" \"11.6189500386223\"\nS\n## [,1]\n## [1,] 11.61895\n\n9.14.3 Risk Decomposition\n\n#COMPUTE RISK DECOMPOSITION\n#Exploits the homogeneity degree 1 property to compute individual asset\n#risk contributions, i.e., a risk decomposition.\n#Risk increment is the change in total risk score if any one asset's\n#risk level increases by 1.\nRiskIncr = 0.5 * (X %*% Ri + t(X) %*% Ri)\/S[1,1]\nRiskDecomp = RiskIncr * Ri\nsorted_RiskDecomp = sort(RiskDecomp,decreasing=TRUE,index.return=TRUE)\nRD = sorted_RiskDecomp$x idxRD = sorted_RiskDecomp$ix\nprint(\"Risk Contribution\"); print(RiskDecomp); print(sum(RiskDecomp))\n## [1] \"Risk Contribution\"\n## [,1]\n## [1,] 0.0000000\n## [2,] 0.0000000\n## [3,] 0.6885304\n## [4,] 0.8606630\n## [5,] 1.3770607\n## [6,] 0.6885304\n## [7,] 0.8606630\n## [8,] 1.3770607\n## [9,] 0.7745967\n## [10,] 0.0000000\n## [11,] 1.2049282\n## [12,] 1.2049282\n## [13,] 1.2049282\n## [14,] 0.5163978\n## [15,] 0.1721326\n## [16,] 0.0000000\n## [17,] 0.5163978\n## [18,] 0.1721326\n## [1] 11.61895\nbarplot(t(RD),col=\"dark green\",xlab=\"Node Number\",names.arg=idxRD,cex.names=0.75)\n\n9.14.4 Centrality\n\n#NODE EIGEN VALUE CENTRALITY\n#Centrality is a measure of connectedness and influence of a node in a network\n#accounting for all its linkages and influence of all other nodes. Centrality\n#is based on connections only and not risk scores, and measures the propensity\n#of a node to propagate a security breach if the node is compromised.\n#It is a score that is normalized to the range (0,1)\ncent = evcent(g)$vector print(\"Normalized Centrality Scores\") ## [1] \"Normalized Centrality Scores\" print(cent) ## [1] 1.0000000 0.4567810 0.4922349 0.3627391 0.3345007 0.1982681 0.3322908 ## [8] 0.4593151 0.5590561 0.5492208 0.5492208 0.5492208 0.5492208 0.3044259 ## [15] 0.2944982 0.5231594 0.4121079 0.2944982 sorted_cent = sort(cent,decreasing=TRUE,index.return=TRUE) Scent = sorted_cent$x\nidxScrit = sorted_crit$ix barplot(t(Scrit),col=\"orange\",xlab=\"Node Number\",names.arg=idxScrit,cex.names=0.75) 9.14.7 Cross Risk 9.14.8 Risk Scaling: Spillovers #CROSS IMPACT MATRIX #CHECK FOR SPILLOVER EFFECTS FROM ONE NODE TO ALL OTHERS d_RiskDecomp = NULL n = length(Ri) for (j in 1:n) { Ri2 = Ri Ri2[j] = Ri[j]+1 res = NetRisk(Ri2,X) d_Risk = as.matrix(res[[3]]) - RiskDecomp d_RiskDecomp = cbind(d_RiskDecomp,d_Risk) #Column by column for each asset } #3D plots library(\"RColorBrewer\"); library(\"lattice\"); library(\"latticeExtra\") cloud(d_RiskDecomp, panel.3d.cloud = panel.3dbars, xbase = 0.25, ybase = 0.25, zlim = c(min(d_RiskDecomp), max(d_RiskDecomp)), scales = list(arrows = FALSE, just = \"right\"), xlab = \"On\", ylab = \"From\", zlab = NULL, main=\"Change in Risk Contribution\", col.facet = level.colors(d_RiskDecomp, at = do.breaks(range(d_RiskDecomp), 20), col.regions = cm.colors, colors = TRUE), colorkey = list(col = cm.colors, at = do.breaks(range(d_RiskDecomp), 20)), #screen = list(z = 40, x = -30) ) brewer.div <- colorRampPalette(brewer.pal(11, \"Spectral\"), interpolate = \"spline\") levelplot(d_RiskDecomp, aspect = \"iso\", col.regions = brewer.div(20), ylab=\"Impact from\", xlab=\"Impact on\", main=\"Change in Risk Contribution\") 9.14.9 Risk Scaling with Increased Connectivity #SIMULATION OF EFFECT OF INCREASED CONNECTIVITY #RANDOM GRAPHS n=50; k=100; pvec=seq(0.05,0.50,0.05); svec=NULL; sbarvec=NULL for (p in pvec) { s_temp = NULL sbar_temp = NULL for (j in 1:k) { g = erdos.renyi.game(n,p,directed=TRUE); A = get.adjacency(g) diag(A) = 1 c = as.matrix(round(runif(n,0,2),0)) syscore = as.numeric(sqrt(t(c) %*% A %*% c)) sbarscore = syscore\/n s_temp = c(s_temp,syscore) sbar_temp = c(sbar_temp,sbarscore) } svec = c(svec,mean(s_temp)) sbarvec = c(sbarvec,mean(sbar_temp)) } #plot(pvec,svec,type=\"l\",xlab=\"Prob of connecting to a node\",ylab=\"S\",lwd=3,col=\"red\") plot(pvec,sbarvec,type=\"l\",xlab=\"Prob of connecting to a node\",ylab=\"S_Avg\",lwd=3,col=\"red\") 9.14.10 Too Big To Fail The change in risk score $${S}$$ as the number of nodes increases, while keeping the average number of connections between nodes constant. This mimics the case where banks are divided into smaller banks, each of which then contains part of the transacting volume of the previous bank. The plot shows how the risk score increases as the number of nodes increases from 10 to 100, while expected number of total edges in the network remains the same. A compromise vector is also generated with equally likely values $$\\{0,1,2\\}$$. This is repeated 5000 times for each fixed number of nodes and the mean risk score across 5000 simulations. #SIMULATION OF EFFECT OF INCREASED NODES AND REDUCED CONNECTIVITY nvec=seq(10,100,10); k=100; svec=NULL; sbarvec=NULL for (n in nvec) { s_temp = NULL sbar_temp = NULL p = 5\/n for (j in 1:k) { g = erdos.renyi.game(n,p,directed=TRUE); A = get.adjacency(g) diag(A) = 1 c = as.matrix(round(runif(n,0,2),0)) syscore = as.numeric(sqrt(t(c) %*% A %*% c)) sbarscore = syscore\/n s_temp = c(s_temp,syscore) sbar_temp = c(sbar_temp,sbarscore) } svec = c(svec,mean(s_temp)) sbarvec = c(sbarvec,mean(sbar_temp)) } plot(nvec,svec,type=\"l\",xlab=\"Number of nodes\",ylab=\"S\",ylim=c(0,max(svec)),lwd=3,col=\"red\") #plot(nvec,sbarvec,type=\"l\",xlab=\"Number of nodes\",ylab=\"S_Avg\",ylim=c(0,max(sbarvec)),lwd=3,col=\"red\") 9.15 Systemic Risk in Indian Banks 9.17 Shiny application The example above may also be embedded in a shiny application for which the code is provided below. The screen will appear as follows. The files below also require the data file systemicR.csv or an upload. #SERVER.R library(shiny) library(plotly) library(igraph) # Define server logic for random distribution application shinyServer(function(input, output) { fData = reactive({ # input$file1 will be NULL initially. After the user selects and uploads a\n# file, it will be a data frame with 'name', 'size', 'type', and 'datapath'\n# columns. The 'datapath' column will contain the local filenames where the\n# data can be found.\n\ninFile <- input$file if (is.null(inFile)){ data = read.csv(file=\"systemicR.csv\",sep=\",\") } else read.csv(file=inFile$datapath)\n})\n\nobserveEvent(input$compute, { output$text1 <- renderText({\ndata = fData()\nna = dim(data)[1] #columns (assets)\nRi = matrix(data[,1],na,1) #Aggregate risk by asset\nX = data[1:na,2:(na+1)]\nX = matrix(as.numeric(as.matrix(X)),na,na)\nS = as.numeric(sqrt(t(Ri) %*% X %*% Ri))\npaste(\"Overall Risk Score\",round(S,2))\n})\n\noutput$plot <- renderPlot({ data = fData() na = dim(data)[1] #columns (assets) bnames = names(data) Ri = matrix(data[,1],na,1) #Aggregate risk by asset X = data[1:na,2:(na+1)] X = matrix(as.numeric(as.matrix(X)),na,na) #GRAPH NETWORK: plot of the assets and the links with directed arrows Y = X; diag(Y)=0 g = graph.adjacency(Y) V(g)$color = \"#ffec78\"\nV(g)$color[degree(g)==max(degree(g))] = \"#ff4040\" V(g)$color[degree(g)==min(degree(g))] = \"#b4eeb4\"\nV(g)$size = Ri*8+10 plot.igraph(g,layout=layout.fruchterman.reingold,edge.arrow.size=0.5, vertex.label.color=\"black\",edge.arrow.width=0.8, vertex.label=bnames[1:na+1], vertex.label.cex=0.8) }, height = 550, width = 800) output$text2 <- renderText({\ndata = fData()\nna = dim(data)[1] #columns (assets)\nRi = matrix(data[,1],na,1) #Aggregate risk by asset\nX = data[1:na,2:(na+1)]\nX = matrix(as.numeric(as.matrix(X)),na,na)\nY = X; diag(Y)=0\nH = ((sum(degree(g)^2))\/na)\/((sum(degree(g)))\/na)\npaste(\"Fragility of the Network is \",round(H,2))\n})\n\noutput$plot2 <- renderPlotly({ data = fData() na = dim(data)[1] #columns (assets) Ri = matrix(data[,1],na,1) #Aggregate risk by asset X = data[1:na,2:(na+1)] X = matrix(as.numeric(as.matrix(X)),na,na) S = as.numeric(sqrt(t(Ri) %*% X %*% Ri)) RiskIncr = 0.5 * as.numeric((X %*% Ri + t(X) %*% Ri))\/S RiskDecomp = RiskIncr * Ri sorted_RiskDecomp = sort(RiskDecomp,decreasing=TRUE,index.return=TRUE) RD = as.numeric(as.matrix(sorted_RiskDecomp$x))\nidxRD = as.character(as.matrix(sorted_RiskDecomp$ix)) idxRD = paste(\"B\",idxRD,sep=\"\") xAx <- list( title = \"Node Number\" ) yAx <- list( title = \"Risk Decomposition\") plot_ly(y = RD,x = idxRD,marker = list(color = toRGB(\"dark green\")),type=\"bar\")%>% layout(xaxis = xAx, yaxis = yAx) # barplot(t(RD),col=\"dark green\",xlab=\"Node Number\",names.arg=idxRD,cex.names=0.75) }) output$plot3 <- renderPlotly({\ndata = fData()\nna = dim(data)[1] #columns (assets)\nRi = matrix(data[,1],na,1) #Aggregate risk by asset\nX = data[1:na,2:(na+1)]\nX = matrix(as.numeric(as.matrix(X)),na,na)\n\n#GRAPH NETWORK: plot of the assets and the links with directed arrows\nY = X; diag(Y)=0\ncent = evcent(g)$vector # print(\"Normalized Centrality Scores\") sorted_cent = sort(cent,decreasing=TRUE,index.return=TRUE) Scent = sorted_cent$x\nidxScent = sorted_cent$ix idxScent = paste(\"B\",idxScent,sep=\"\") xAx <- list( title = \"Node Number\" ) yAx <- list( title = \"Eigen Value Centrality\" ) plot_ly(y = as.numeric(t(Scent)),x = idxScent,marker = list(color = toRGB(\"red\")),type=\"bar\")%>% layout(xaxis = xAx, yaxis = yAx) # barplot(t(Scent),col=\"dark red\",xlab=\"Node Number\",names.arg=idxScent,cex.names=0.75) }) output$plot4 <- renderPlotly({\ndata = fData()\nna = dim(data)[1] #columns (assets)\nRi = matrix(data[,1],na,1) #Aggregate risk by asset\nX = data[1:na,2:(na+1)]\nX = matrix(as.numeric(as.matrix(X)),na,na)\n\nS = as.numeric(sqrt(t(Ri) %*% X %*% Ri))\nRiskIncr = 0.5 * as.numeric((X %*% Ri + t(X) %*% Ri))\/S\n\n#COMPUTE RISK INCREMENTS\nsorted_RiskIncr = sort(RiskIncr,decreasing=TRUE,index.return=TRUE)\nRI = sorted_RiskIncr$x idxRI = sorted_RiskIncr$ix\nidxRI = paste(\"B\",idxRI,sep=\"\")\n\nxAx <- list(\ntitle = \"Node Number\"\n)\nyAx <- list(\ntitle = \"Risk Increments\"\n)\nplot_ly(y = as.numeric(t(RI)),x = idxRI,marker = list(color = toRGB(\"green\")),type=\"bar\")%>%\nlayout(xaxis = xAx, yaxis = yAx)\n\n# barplot(t(RI),col=\"dark blue\",xlab=\"Node Number\",names.arg=idxRI,cex.names=0.75)\n\n})\n\n#CRITICALITY\n#Criticality is compromise-weighted centrality.\n#This is an element-wise multiplication of vectors $C$ and $x$.\n\noutput$plot5 <- renderPlotly({ data = fData() na = dim(data)[1] #columns (assets) Ri = matrix(data[,1],na,1) #Aggregate risk by asset X = data[1:na,2:(na+1)] X = matrix(as.numeric(as.matrix(X)),na,na) #GRAPH NETWORK: plot of the assets and the links with directed arrows Y = X; diag(Y)=0 g = graph.adjacency(Y) cent = evcent(g)$vector\n\ncrit = Ri * cent\nprint(\"Criticality Vector\")\nprint(crit)\nsorted_crit = sort(crit,decreasing=TRUE,index.return=TRUE)\nScrit = sorted_crit$x idxScrit = sorted_crit$ix\nidxScrit = paste(\"B\",idxScrit,sep=\"\")\n\nxAx <- list(\ntitle = \"Node Number\"\n)\nyAx <- list(\ntitle = \"Criticality Vector\"\n)\nplot_ly(y = as.numeric(t(sorted_crit\\$x)),x = idxScrit,marker = list(color = toRGB(\"orange\")),type=\"bar\")%>%\nlayout(xaxis = xAx, yaxis = yAx)\n\n# barplot(t(Scrit),col=\"orange\",xlab=\"Node Number\",names.arg=idxScrit,cex.names=0.75)\n})\n\n})\n\n})\n#UI.R\n\nlibrary(plotly)\nshinyUI(fluidPage(\n\ntitlePanel(\"Systemic Risk Scoring\"),\n\nsidebarLayout(\n\nsidebarPanel(\n# Inputs excluded for brevity\np('Upload a .csv file having header as Credit Scores and names of n banks. Dimensions of file will be\nfileInput(\"file\", label = h3(\"File input\")),\nactionButton(\"compute\",\"Compute Scores\"),\nhr(),\ntextOutput(\"text1\"),\ntextOutput(\"text2\"),\nhr(),\np('Please refer following Paper published for further details',\na(\"Matrix Metrics: Network-Based Systemic Risk Scoring.\",\nhref = \"http:\/\/srdas.github.io\/Papers\/JAI_Das_issue.pdf\"))\n),\n\nmainPanel(\ntabsetPanel(\ntabPanel(\"Network Graph\", plotOutput(\"plot\",width=\"100%\")),\ntabPanel(\"Risk Decomposition\", plotlyOutput(\"plot2\")),\ntabPanel(\"Node Centrality\", plotlyOutput(\"plot3\")),\ntabPanel(\"Risk Increments\", plotlyOutput(\"plot4\")),\ntabPanel(\"Criticality\", plotlyOutput(\"plot5\"))\n\n)\n)\n)\n))","date":"2022-05-23 02:09:26","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\": 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\": 6, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5030187964439392, \"perplexity\": 7609.346188430061}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662552994.41\/warc\/CC-MAIN-20220523011006-20220523041006-00466.warc.gz\"}"}
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This entry was posted on June 21, 2018 at 3:51 am and is filed under Global Warming,Global Freezing,Climate Change,Nature,Darwin,Assorted Modern Sciences of Dubious Honor, Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 6,832
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Q: If statement listing exceptions I'm having an issue with this codebat question:
The parameter weekday is True if it is a weekday, and the parameter vacation is True if we are on vacation. We sleep in if it is not a weekday or we're on vacation. Return True if we sleep in.
sleep_in(False, False) → True
sleep_in(True, False) → False
sleep_in(False, True) → True
Here's my solution:
def sleep_in(weekday, vacation):
if (True, False):
return False
else:
return True
And the result came back as:
sleep_in(False, False) → True False X
sleep_in(True, False) → False False OK
sleep_in(False, True) → True False X
sleep_in(True, True) → True False X
I'm confused as to why it's wrong.
In my solution, I stated:
if (True, False):
return False
And everything being True.
Can any one give me an idea where I am going wrong?
A:
We sleep in if it is not a weekday or we're on vacation.
This is best expressed as a straight-up boolean comparison.
def sleep_in(weekday, vacation):
return not weekday or vacation
Fair warning: this will evaluate to True if either of those parameters are truthy.
Speaking of "truthy", the experssion (True, False) is a tuple, and since it is a non-empty tuple, it will evaluate to True. For reference, here is a list of all expressions that evaluate to False. Anything outside of that list will evaluate to True in a boolean context.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
Given bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space
$V$, a \emph{transversal basis }is a basis of $V$ containing a single
distinguished vector from each of $B_{1},\dots,B_{n}$. Two transversal
bases are said to be \emph{disjoint} if their distinguished vectors
from $B_{i}$ are distinct, for each $i$ (here ``distinguished'' means that two copies of the same vector appearing in two $B_i$s are considered distinct). In 1989, Rota conjectured
(see \cite[Conjecture~4]{HR94}) that for any vector space $V$ over
a characteristic-zero field, and any choice of $B_{1},\dots,B_{n}$,
one can always find $n$ pairwise disjoint transversal bases.
Despite the apparent simplicity of this conjecture, it remains wide
open, and has surprising connections to apparently unrelated subjects.
Specifically, it was discovered by Huang and Rota \cite{HR94} that
there are implications between Rota's basis conjecture, the Alon--Tarsi
conjecture \cite{AT92} concerning enumeration of even and odd Latin
squares, and a certain conjecture concerning the supersymmetric bracket
algebra.
Rota also observed that an analogous conjecture could be made in the
much more general setting of \emph{matroids}, which are objects that
abstract the combinatorial properties of linear independence in vector
spaces. Specifically, a finite matroid $M=\left(E,\mathcal{I}\right)$
consists of a finite ground set $E$ (whose elements may be thought
of as vectors in a vector space), and a collection $\mathcal{I}$
of subsets of $E$, called independent sets. The defining properties
of a matroid are that:
\begin{itemize
\item{the empty set is independent (that is, $\emptyset\in\mathcal{I}$);}
\item{subsets of independent sets are independent (that is, if $A'\subseteq A\subseteq E$
and $A\in\mathcal{I}$, then $A'\in\mathcal{I}$);}
\item{if $A$ and $B$ are independent sets, and $\left|A\right|>\left|B\right|$,
then an independent set can be constructed by adding an element of
$A$ to $B$ (that is, there is $a\in A\backslash B$ such that $B\cup\left\{ a\right\} \in\mathcal{I}$).
This final property is called the \emph{augmentation property}.}
\end{itemize}
Observe that any finite set of elements in a vector space (over any
field) naturally gives rise to a matroid, though not all matroids
arise this way. A \emph{basis }in a matroid $M$ is a maximal independent
set. By the augmentation property, all bases have the same size, and
this common size is called the \emph{rank }of $M$. The definition
of a transversal basis generalises in the obvious way to matroids,
and the natural matroid generalisation of Rota's basis conjecture
is that for any rank-$n$ matroid and any bases $B_{1},\dots,B_{n}$,
there are $n$ disjoint transversal bases.
Although Rota's basis conjecture remains open, various special cases
have been proved. Several of these have come from the connection between
Rota's basis conjecture and the Alon--Tarsi conjecture,
which has since been simplified by Onn \cite{Onn97}. Specifically,
due to work by Drisko \cite{Dri97} and Glynn \cite{Gly10} on the
Alon--Tarsi conjecture, Rota's original conjecture for vector
spaces over a characteristic-zero field is now known to be true whenever
the dimension $n$ is of the form $p\pm1$, for $p$ a prime. Wild
\cite{Wil94} proved Rota's basis conjecture for so-called ``strongly
base-orderable'' matroids, and used this to prove the conjecture
for certain classes of matroids arising from graphs. Geelen and Humphries
proved the conjecture for ``paving'' matroids \cite{GH06}, and
Cheung \cite{Che12} computationally proved that the conjecture holds
for matroids of rank at most 4.
Various authors have also proposed variations and weakenings of Rota's
basis conjecture. For example, Aharoni and Berger \cite{AB06} showed
that in any matroid one can cover the set of all the elements in $B_{1},\dots,B_{n}$
by at most $2n$ ``partial'' transversals, and Bollen and Draisma
\cite{BD15} considered an ``online'' version of Rota's basis conjecture,
where the bases $B_{i}$ are revealed one-by-one. In 2017, Rota's
basis conjecture received renewed interest when it was chosen as the
twelfth ``Polymath'' project, in which amateur and professional
mathematicians from around the world collaborated on the problem.
Some of the fruits of the project were a small improvement to Aharoni
and Berger's theorem, and improved understanding of the online version
of Rota's basis conjecture \cite{Pol17}. See \cite{polyproposal}
for Timothy Chow's proposal of the project, see \cite{poly1,poly2,poly3}
for blog posts where much of the discussion took place, and see \cite{polywiki}
for the Polymath wiki summarising most of what is known about Rota's
basis conjecture.
One particularly natural direction to attack Rota's problem is to
try to find lower bounds on the number of disjoint transversal bases. Rota's basis conjecture
asks for $n$ disjoint transversal bases, but it is not completely
obvious that even two disjoint transversal bases must exist! Wild
\cite{Wil94} proved some lower bounds for certain matroids arising
from graphs, but the first nontrivial bound for general matroids was
by Geelen and Webb \cite{GW07}, who used a generalisation of Hall's
theorem due to Rado \cite{Rad42} to prove that there must be $\Omega\left(\sqrt{n}\right)$
disjoint transversal bases. Recently, this was improved by Dong and
Geelen \cite{DG18}, who used a beautiful probabilistic argument to
prove the existence of $\Omega\left(n/\log n\right)$ disjoint transversal
bases. In this paper we improve this substantially and obtain the
first linear bound.
\begin{thm}
\label{thm:new}For any $\varepsilon>0$, the following holds for
sufficiently large $n$. Given bases $B_{1},\dots,B_{n}$ of a rank-$n$
matroid, there are at least $\left(1/2-\varepsilon\right)n$ disjoint
transversal bases.
\end{thm}
Of course, since matroids generalise vector spaces, this also implies
the same result for bases in an $n$-dimensional vector space. We
also remark that for the weaker fact that there exist $\Omega\left(n\right)$
disjoint transversal bases, our methods give a simpler proof;
see \cref{rem:linear}.
In contrast to the previous work by Dong, Geelen and Webb, our approach
is to show how to build a collection of transversal bases in an iterative
fashion (reminiscent of augmenting path arguments in matching problems).
It is tempting to imagine a future path to Rota's basis conjecture
(at least in the case of vector spaces) using such an approach: by
improving on our arguments, perhaps introducing some randomness, it
might be possible to iteratively build a collection of $\left(1-o\left(1\right)\right)n$
transversal bases, and then it might be possible to use some sort
of ``template'' or ``absorber'' structure to finish the job. This
was precisely the approach taken in Keevash's celebrated proof of
the existence of designs \cite{Kee14}. Actually, it has been observed
by participants of the Polymath project (see \cite{poly1}) that Rota's
basis conjecture and the existence of designs conjecture both seem
to fall into a common category of problems which are not quite ``structured''
enough for purely algebraic methods, but too structured for probabilistic
methods.
\vspace{0.30cm}
\noindent{\bf Notation.} We will frequently want to denote the result of adding and removing single elements from a set. For a set $S$ and some $x\notin S$, $y\in S$, we write $S+x$ to mean $S\cup \{x\}$, and we write $S-y$ to mean $S\setminus \{y\}$.
\section{Finding many disjoint transversal bases}
In this section we prove \cref{thm:new}. It is convenient to think
of $B_{1},\dots,B_{n}$ as ``colour classes''.
\begin{defn}
Let $U=\left\{ \left(x,c\right):x\in B_{c},1\le c\le n\right\} $ be the set of
all coloured elements that appear in one of $B_{1},\dots,B_{n}$.
For $S\subseteq U$, let $\pi\left(S\right)=\left\{ x:\left(x,c\right)\in S\;\text{for some }c\right\} $
be its set of matroid elements. We say that a subset of elements of $U$
is a \emph{rainbow independent set} (RIS for short) if all its matroid
elements are distinct and form an independent set, and all their colours
are distinct.
\end{defn}
Note that an RIS with size $n$ corresponds to a transversal basis.
We remark that RISs are sometimes also known as \emph{partial transversals}.
Note that two transversal bases are disjoint if and only if their
corresponding RISs are disjoint as subsets of $U$.
Let $f=\left(1-\varepsilon\right)n/2$. The basic idea is to start
with a collection of $f$ empty RISs (which are trivially disjoint),
and iteratively enlarge the RISs in this collection, maintaining disjointness,
until we have many disjoint transversal bases.
Let $\mathcal{S}$ be a collection of $f$ disjoint RISs. We define the \emph{volume
}$\sum_{S\in\mathcal{S}}\left|S\right|$ of $\mathcal{S}$ to be the total number of
elements in the RISs in $\mathcal{S}$. We will show how to modify $\mathcal{S}$ to
increase its volume. We let $F=\bigcup_{S \in \mathcal{S}} S$ be the set of all currently used elements. One should think of $F$ as being the set of all elements which we cannot add to any $S\in\mathcal{S}$
without violating the disjointness of RISs in $\mathcal{S}$.
We stress that in the following two subsections we fix a collection $\mathcal{S}$ and define $F$ as above. All our definitions and claims are with respect to these $F$ and $\mathcal{S}$. We will show that under certain conditions the size of $\mathcal{S}$ can be increased, at which point one needs to restart the argument from the beginning with a new $\mathcal{S}$ (and a new $F$). This is made precise in \Cref{subsec:increasing}.
\begin{rem*}
We remark that it is actually possible to reduce to the case where each $B_c$ is disjoint, by making duplicate copies of all elements that appear in multiple $B_c$. So, instead of working with the universe $U$ of element/colour pairs, one can alternatively think of $U$ as being a collection of $n^2$ different matroid elements (each of which has a colour associated with it).
\end{rem*}
\subsection{\label{subsec:simple-swaps}Simple swaps}
Our objective is to increase the volume of $\mathcal{S}$. If an RIS $S\in\mathcal{S}$
is missing a colour $c$ and there is $x\in B_{c}$ independent to
the elements of $S$, such that $\left(x,c\right) \notin F$, then we can add $\left(x,c\right)$ to $S$
to create a larger RIS, increasing the volume of $\mathcal{S}$. We will want
much more freedom than this: we also want to consider those elements
that can be added to $S$ after making a small change to $S$. This
motivates the following definition.
\begin{defn}
\label{def:addable}Consider an RIS $S$ and a colour $b$ that does
not appear in $S$. Say an element $\left(x,c\right)\in U$ (possibly $(x,c)\in F$) is $\left(S,b\right)$-\emph{addable
}if either
\begin{itemize}
\item $S+\left(x,c\right)$ is an RIS, or;
\item There is $\left(x',c\right)\in S$ and $\left(y,b\right)\notin F$
such that $S-\left(x',c\right)+\left(y,b\right)+\left(x,c\right)$
is an RIS.
\end{itemize}
In the second case we say that $y$ is a \emph{witness} for the $\left(S,b\right)$-addability of $\left(x,c\right)$. For $(x',c) \in S$ and $(y,b) \notin F$ when $S-\left(x',c\right)+\left(y,b\right)$ is an RIS we say it
is the result of applying a \emph{simple swap }to $S$.
\end{defn}
If for some RIS $S\in\mathcal{S}$ missing a colour $b$ there is an $\left(S,b\right)$-addable
element $\left(x,c\right)\notin F$,
then we can increase the volume of $\mathcal{S}$ by adding $\left(x,c\right)$
to $S$, possibly after applying a simple swap to $S$. Note that we do not require $S\in \mathcal{S}$ for the definition of $(S,b)$-addability, though in practice we will only ever consider $S$ that are either in $\mathcal{S}$ or slight modifications of RISs in $\mathcal{S}$.
Our next objective is to show that for any $S$ missing a colour $b$,
either there is an $\left(S,b\right)$-addable element that is not
in $F$ (which would allow us to increase the volume of $\mathcal{S}$, as
above), or else there are \emph{many }$\left(S,b\right)$-addable
elements (which must therefore be in $F$). Although this will not
allow us to immediately increase the volume of $\mathcal{S}$, it will allow
us to transfer an element to $S$ from some other $S'\in\mathcal{S}$, and
this freedom to perform local modifications will be very useful.
Towards this end, we study which elements of $S$ can be used in a
simple swap.
\begin{defn}
Consider an RIS $S$ and consider a colour $b$ that does not appear
on $S$. We say that a colour $c$ appearing on $S$ is \emph{$\left(S,b\right)$-swappable}
if there is a simple swap yielding an RIS $S+\left(y,b\right)-\left(x',c\right)$,
with $\left(y,b\right)\notin F$ and $\left(x',c\right)\in S$. (For $S+\left(y,b\right)-\left(x',c\right)$ to be an RIS, we just need $\pi(S)+y-x'$ to be an independent set in our matroid.) We say that $y$ is a witness for
the $\left(S,b\right)$-swappability of $c$.
\end{defn}
(Basically, a colour is $\left(S,b\right)$-swappable if
we can replace it with a $b$-coloured element which is not in
$F$). For a colour $c$ we denote by $F_{c}=\left\{ x\in B_{c}:\left(x,c\right)\in F\right\}$ the set of matroid elements which appear in $\mathcal{S}$ with colour $c$.
\begin{claim}
\label{claim:many-good}For a nonempty RIS $S$ and a colour $b$ not
appearing in $S$, either there is an $\left(S,b\right)$-addable element $(y,b)\notin F$ or there are at least $n-\left|F_{b}\right|$ colours
which are $\left(S,b\right)$-swappable.
\end{claim}
\begin{proof}
For the purpose of contradiction, suppose that there is no $\left(S,b\right)$-addable element $(y,b)\notin F$, and that there are fewer than $n-\left|F_{b}\right|$ colours which are $\left(S,b\right)$-swappable. Let $S'\subseteq S$ be the set of all elements of
$S$ which have an $\left(S,b\right)$-swappable colour, so $\left|S'\right|<n-\left|F_{b}\right|$.
Also $\left|S'\right|<\left|S\right|$ because otherwise we would have $|S|<n-\left|F_{b}\right|$, so by the augmentation property there would be $y \in B_b \setminus F_b$ such that $S+(y,b)$ is an RIS (meaning that $(y,b)\notin F$ would be $\left(S,b\right)$-addable). Repeating this argument for $S'$ in place of $S$, there is $y\in B_{b}\backslash F_{b}$
such that $S'+\left(y,b\right)$ is an RIS. By repeatedly using the augmentation
property, we can add $\left|S-S'\right|-1$ elements of $S-S'$ to
$S'+\left(y,b\right)$. This gives an RIS of size $|S|$ of the form $S+\left(y,b\right)-\left(x',c\right)$
for some $\left(x',c\right)\in S-S'$. But this means $c$ is $\left(S,b\right)$-swappable, so $(x',c) \in S'$ by the definition of $S'$. This is a contradiction.
\end{proof}
Now we show that all elements of an $\left(S,b\right)$-swappable
colour which are independent to $\pi\left(S\right)$ are $\left(S,b\right)$-addable,
unless there is an \emph{$\left(S,b\right)$}-addable element not
in $F$. (Recall that $\pi\left(S\right)$ is the set of matroid elements
in $S$, without colour data.)
\begin{claim}
\label{claim:add-if-good}Consider an RIS $S$ with no element of
a colour $b$ and consider a colour
$c$ that is $\left(S,b\right)$-swappable with witness $y$. Either
$S+\left(y,b\right)$ is an RIS (thus, $\left(y,b\right)\notin F$
is $\left(S,b\right)$-addable), or otherwise for any $x\in B_{c}$
independent of $\pi\left(S\right)$, $\left(x,c\right)$ is $\left(S,b\right)$-addable with witness $(y,b)$.
\end{claim}
\begin{proof}
Let $\left(x',c\right)$ be the element with colour $c$ in $S$.
Consider some $x\in B_{c}$ independent to $\pi\left(S\right)$. Let $I=\pi\left(S\right)+x$ and $J=\pi\left(S\right)+y-x'$. By the augmentation property, there is an element of $I\backslash J$ that is independent of $J$; this element is either $x'$ or $x$. In the former case $S+\left(y,b\right)$
is an RIS. In the latter case, $S+\left(y,b\right)-\left(x',c\right)+\left(x,c\right)$ is
an RIS, showing that $\left(x,c\right)$ is $\left(S,b\right)$-addable.
\end{proof}
The following lemma gives a good illustration of how to use the ideas developed in this section to find many addable elements. It will be very useful later on.
\begin{claim}\label{claim:1-addability}
Let $S\in\mathcal{S}$ and let $b$ be a colour which does not appear in $S$. Then either we can increase the volume of $\mathcal{S}$ or there are at least $(n-|S|)\left(n-f\right)$ elements that are $\left(S,b\right)$-addable.
\end{claim}
\begin{proof}
If there is an element $\left(y,b\right)\notin F$ which is $\left(S,b\right)$-addable,
then we can directly add this element to $S$ (making a simple swap if necessary), increasing the volume
of $\mathcal{S}$. Otherwise, observe that $\left|F_{b}\right|\le\left|\mathcal{S}\right|=f$,
so by \cref{claim:many-good} there are at least $n-f$ colours that
are $\left(S,b\right)$-swappable. For
each such colour $c$, by the augmentation property, there are at least $n-|S|$ elements $x\in B_{c}$
independent to all the elements of $S$, each of which is $\left(S,b\right)$-addable
by \cref{claim:add-if-good}. That is to say, there are at least $(n-|S|)\left(n-f\right)$
elements which are $\left(S,b\right)$-addable, as claimed.
\end{proof}
In our proof of \cref{thm:new} we also make use of the following lemma. In the course of our arguments, when we need to find many addable elements with a given colour, it will allow us to ensure that these elements are actually distinct.
\begin{lem}
\label{lem:matching}Let $S$ be an RIS. Then for each $B_{b}$, we
can find an injection $\phi_{b}:S\to B_{b}$ such that for all
$\left(x,c\right)\in S$, $\phi_{b}\left(\left(x,c\right)\right)$
is independent of $\pi\left(S-\left(x,c\right)\right)$.
\end{lem}
\begin{proof}
Consider the bipartite graph $G$ where the first part consists of the elements of $S$ and
the second part consists of the elements of $B_{b}$, with an edge
between $\left(x,c\right)\in S$ and $y\in B_{b}$ if $y$ is independent
of $\pi\left(S-\left(x,c\right)\right)$. We use Hall's theorem
to show that there is a matching in this bipartite graph covering $S$. Indeed,
consider some $W\subseteq S$. By the augmentation property, there
are at least $\left|W\right|$ elements $y\in B_{b}$ such that $\pi\left(S-W\right)+y$
is an independent set, and again using the augmentation property,
each of these can be extended to an independent set of the form $\pi\left(S\right)+y-x$
for some $\left(x,c\right)\in W$. That is to say, $W$ has at least
$\left|W\right|$ neighbours in $G$.
\end{proof}
We thank the anonymous referees for pointing out that \cref{lem:matching} also follows from a result due to Brualdi \cite{brualdi69}.
\subsection{Cascading swaps}
Informally speaking, for any $S_{0}\in\mathcal{S}$ which is not a transversal
basis, we have shown that either we can directly augment $S_{0}$,
or there are many elements $\left(x_{1},c_{1}\right)\in U$ with which
we can augment $S_{0}$ after performing a simple swap. It's possible
that each such $\left(x_{1},c_{1}\right)$ already appears in some
other $S_{1}\in\mathcal{S}$, but if this occurs we need not give up: we can
transfer $\left(x_{1},c_{1}\right)$ from $S_{1}$ to $S_{0}$ and
then continue to look for elements $\left(x_{2},c_{2}\right)\in U$
with which we can augment $S_{1}-\left(x_{1},c_{1}\right)$ (again,
possibly with a swap). We can iterate this idea, looking for sequences
\[
S_{1},\dots,S_{\ell}\in\mathcal{S},\quad\left(x_{1},c_{1}\right)\in S_{1},\,\left(x_{2},c_{2}\right)\in S_{2},\dots,\left(x_{\ell},c_{\ell}\right)\in S_{\ell},\,\left(x_{\ell+1},c_{\ell+1}\right)\notin\bigcup_{S\in\mathcal{S}}S
\]
such that, after a sequence of simple swaps, each $\left(x_{i},c_{i}\right)$
is transferred from $S_{i}$ to $S_{i-1}$, and then $\left(x_{\ell+1},c_{\ell+1}\right)$
can be added to $S_{\ell}$. (We also need to ensure that the simple swaps we perform preserve disjointness of RISs in $\mathcal{S}$.) This transformation has the net effect
of adding an element to $S_{0}$ and keeping the size of all other
$S\in\mathcal{S}$ constant, thus increasing the volume of $\mathcal{S}$.
Crucially, because of the freedom afforded by simple swaps, each time
we expand our search to consider longer cascades, our number of options
for $\left(x_{\ell+1},c_{\ell+1}\right)$ increases. For sufficiently
large $\ell$, the number of options will be so great that there must
be suitable $\left(x_{\ell+1},c_{\ell+1}\right)$ not appearing in
any RIS in $\mathcal{S}$. In order to keep this analysis tractable, we will
only consider transformations that cascade along a single sequence
of RISs $S_0,\dots,S_{\ell}$; we will iteratively construct this
sequence of RISs in such a way that there are many possibilities $\left(x_{i},c_{i}\right)\in S_{i}$
relative to the number of possibilities $\left(x_{i-1},c_{i-1}\right)\in S_{i-1}$
in the previous step. The next definition makes precise the cascades
that we consider.
\begin{defn}\label{Defn_Cascade}
Consider a sequence of distinct RISs $S_{0},\dots,S_{\ell-1}\in\mathcal{S}$.
Say an element $\left(x_{\ell},c_{\ell}\right)\notin S_{0},\dots,S_{\ell-1}$
is \emph{cascade-addable with respect to} $S_{0},\dots,S_{\ell-1}$
if there is a colour $c_{0}$ and sequences
\[
\left(x_{1},c_{1}\right),\dots,\left(x_{\ell-1},c_{\ell-1}\right)\in U,\qquad y_{0}\in B_{c_{0}},\dots,y_{\ell-1}\in B_{c_{\ell-1}},
\]
such that the following hold.
\begin{itemize}
\item For each $1\le i\le\ell-1$, we have $\left(x_{i},c_{i}\right)\in S_{i}$;
\item $c_{0}$ does not appear in $S_{0}$, and $\left(x_{1},c_{1}\right)$
is $\left(S_{0},c_{0}\right)$-addable with
witness $y_{0}$;
\item for each $0\le i\le\ell-1$, $\left(x_{i+1},c_{i+1}\right)$ is $\left(S_{i}-\left(x_{i},c_{i}\right),c_{i}\right)$-addable
with witness $y_{i}$;
\item the colours $c_0,\dots,c_\ell$ are distinct.
\end{itemize}
We call $c_0, c_1, \dots, c_{\ell-1}$ \emph{a sequence of colours freeing $(x_{\ell}, c_{\ell})$}.
We write $Q\left(S_{0},\dots,S_{\ell-1}\right)$ for the set of all
elements outside $S_{0},\dots,S_{\ell-1}$ which are cascade-addable
with respect to $S_{0},\dots,S_{\ell-1}$.
\end{defn}
We remark that if $\ell=1$ then most of the conditions in the above definition become vacuous and an element being cascade-addable with respect to $S_0$ is equivalent to it being $(S_0, c_0)$-addable with a witness, for some colour $c_0.$
Observe
that if an element $\left(x_{\ell},c_{\ell}\right)$ is cascade-addable
then we can transfer it into $S_{\ell-1}$, as the final step in a
cascading sequence of simple swaps and transfers. The following lemma makes this precise.
\begin{claim}\label{Claim_Perform_Cascade}
Suppose that $\left(x_{\ell},c_{\ell}\right)$
is {cascade-addable }with respect to $S_{0},\dots,S_{\ell-1}$ and $c_0, c_1, \dots, c_{\ell-1}$ is a sequence of colours freeing $(x_{\ell}, c_{\ell})$.
Then there are $S_0'\dots S'_{\ell-1} \subseteq S_0\cup \dots \cup S_{\ell-1}\cup B_{c_0}\cup \dots \cup B_{c_{\ell-1}}$ such that replacing $S_0, \dots, S_{\ell-1}$ with $S'_0, \dots, S'_{\ell-1}$ in $\mathcal S$ results in a family $\mathcal S'$ of disjoint RISs of the same total volume as $\mathcal S$, in such a way that $S'_{\ell-1}+\left(x_{\ell},c_{\ell}\right)$ is an RIS.
\end{claim}
\begin{proof}
Let $\left(x_{1},c_{1}\right),\dots,\left(x_{\ell-1},c_{\ell-1}\right)\in U, y_{0}\in B_{c_{0}},\dots,y_{\ell-1}\in B_{c_{\ell-1}}$ be as in the definition of cascade-addability. For each $i=0, \dots, \ell-1$, let $(x'_i, c_{i+1})$ be the colour $c_{i+1}$ element of $S_i$ (which exists, because, from cascade-addability, $\left(x_{i+1},c_{i+1}\right)$ is $\left(S_{i}-\left(x_{i},c_{i}\right),c_{i}\right)$-addable
\emph{with a witness}). For each $i=1, \dots, \ell-2$, let $S_i'= S_i-(x_i,c_i)- (x'_i, c_{i+1})+ (y_i,c_i) + (x_{i+1}, c_{i+1})$. Let $S_0'=S_0- (x'_0, c_{1}) +(y_0,c_0) + (x_{1}, c_{1})$ and $S_{\ell-1}'= S_{\ell-1}-(x_{\ell-1},c_{\ell-1})- (x'_{\ell-1}, c_{\ell})+ (y_{\ell-1},c_{\ell-1})$.
Let $\mathcal S'$ be the family formed by replacing $S_0, \dots, S_{\ell-1}$ with $S'_0, \dots, S'_{\ell-1}$ in $\mathcal S$. It is easy to check that $\mathcal S'$ has the same total volume as $\mathcal S$, so it remains to check that it is a family of disjoint RISs.
For $i=1, \dots, \ell-2$, $S_i'$ is an RIS because it comes from $S_i-(x_i,c_i)$ by making the change in the definition of $(x_{i+1}, c_{i+1})$ being $(S_i-(x_i,c_i), c_i)$-addable with witness $y_i$ (and addability always produces an RIS by definition).
Similarly $S_{\ell-1}'+\left(x_{\ell},c_{\ell}\right)$ is an RIS.
To see that $S_0'$ is an RIS we use that $(x_1, c_1)$ is $(S_0,c_0)$-addable with witness $y_0$, and that $c_0$ does not appear in $S_0$, both of which come from the definition of cascade-addability.
It remains to show that the RISs $S_0', \dots, S_{\ell-1}'$ are disjoint from each other and the other RISs in $\mathcal S$. The elements $(y_i,c_i)$ occur in only one RIS $S_i'$ because they come from outside $F$ (since they are addability witnesses), and because their colours $c_0, \dots, c_{\ell-1}$ are distinct (from the definition of cascade-addability). The elements $(x_i, c_i)$ occur in only one RIS because they get removed from $S_{i}$ and added to $S_{i-1}$.
\end{proof}
The following lemma lets us build longer cascades.
\begin{claim}\label{Claim_Concatenate_cascade}
Suppose that $\left(x_{\ell},c_{\ell}\right)\in S_{\ell}$
is {cascade-addable }with respect to $S_{0},\dots,S_{\ell-1}$ and $c_0, c_1, \dots, c_{\ell-1}$ is a sequence of colours freeing $(x_{\ell}, c_{\ell})$. If $(x,c)$ is $(S_{\ell}-(x_{\ell}, c_{\ell}), c_{\ell})$-addable with a witness then either $(x,c)\in S_{0}\cup \dots\cup S_{\ell}\cup B_{c_0}\cup \dots\cup B_{c_{\ell}}$ or $(x,c)$ is {cascade-addable} with respect to $S_{0},\dots,S_{\ell}$.
\end{claim}
\begin{proof}
Suppose that $(x,c)\not \in S_{0},\dots,S_{\ell}, B_{c_0}, \dots, B_{c_{\ell}}$.
For the definition of $(x,c)$ being {cascade-addable}, all the conditions not involving $(x,c)$ and $(x_{\ell},c_{\ell})$ hold as a consequence of $\left(x_{\ell},c_{\ell}\right)\in S_{\ell}$
being {cascade-addable }with respect to $S_{0},\dots,S_{\ell-1}$.
It remains to check the conditions that $(x,c)\not\in S_{0},\dots,S_{\ell}$ and that
each of $c_0,\dots,c_\ell,c$ are distinct, both of which hold as a consequence of our assumption $(x,c)\not \in S_{0},\dots,S_{\ell}, B_{c_0}, \dots, B_{c_{\ell}}$.
\end{proof}
In the next lemma, we essentially show that given $S_{0},\dots,S_{\ell-1}$,
it is possible to choose $S_{\ell}$ in such a way that the number
of cascade-addable elements increases.
\begin{claim}
\label{claim:cascade-increase}Consider a sequence of distinct RISs
$S_{0},\dots,S_{\ell-1}\in\mathcal{S}$ with $1 \le \ell<f=\left|\mathcal{S}\right|$.
Then either we can modify $\mathcal{S}$ to increase its volume, or we can
choose $S_{\ell}\ne S_{0},\dots,S_{\ell-1}$ from $\mathcal{S}$ such that
\begin{equation}
\left|Q\left(S_{0},\dots,S_{\ell}\right)\right|\ge\frac{\left|Q\left(S_{0},\dots,S_{\ell-1}\right)\right|}{f-\ell}\cdot\left(n-f-\ell\right)-(\ell+1) n.\label{eq:recurrence}
\end{equation}
\end{claim}
\begin{proof}
If $Q\left(S_{0},\dots,S_{\ell-1}\right)$ contains an element $(x,c)$ not
in any $S\in\mathcal{S}$, then we can increase the volume of $\mathcal{S}$ with a
cascading sequence of simple swaps and transfers (using \cref{Claim_Perform_Cascade}, noting that if $\left(x_{\ell},c_{\ell}\right)\not\in F$, then we can add $\left(x_{\ell},c_{\ell}\right)$ to $S'_{\ell-1}$ in that lemma to get a larger family of RISs).
Otherwise, all the elements of $Q\left(S_{0},\dots,S_{\ell-1}\right)$ belong to some RIS $S\in \mathcal{S}\setminus\left\{ S_{0},\dots S_{\ell-1}\right\}$ (since $Q\left(S_{0},\dots,S_{\ell-1}\right)$ is defined to not contain any elements from $S_{0},\dots,S_{\ell-1}$).
Choose $S_{\ell}\in\mathcal{S}\setminus\left\{ S_{0},\dots S_{\ell-1}\right\}$ containing maximally many elements of $Q\left(S_{0},\dots,S_{\ell-1}\right)$. Since the $f-\ell$ RISs $S\in \mathcal{S}\setminus\left\{ S_{0},\dots S_{\ell-1}\right\}$ collectively contain all elements of $Q\left(S_{0},\dots,S_{\ell-1}\right)$, our chosen RIS $S_\ell$ must contain a proportion of at least $1/(f-\ell)$ of the elements of $Q\left(S_{0},\dots,S_{\ell-1}\right)$. In other words, if we let $Q=S_{\ell}\cap Q\left(S_{0},\dots,S_{\ell-1}\right)$, we have
\begin{equation}\label{eq:QBound}
\left|Q\right|\ge\frac{\left|Q\left(S_{0},\dots,S_{\ell-1}\right)\right|}{f-\ell}.
\end{equation}
Apply \cref{lem:matching} to $S_{\ell}$ to obtain an injection $\phi_{b}$, for every colour $b$.
Fix some $(x_{\ell},c_{\ell})\in Q$ and a sequence of colours $c_0, \dots, c_{\ell-1}$ freeing $(x_{\ell},c_{\ell})$. We prove a sequence of claims about how many elements are swappable/addable with respect to $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$, assuming we cannot increase the size of $\mathcal{S}$.
\begin{claim*}
{There are at least $n-f$ colours
which are $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-swappable.}
\end{claim*}
\begin{proof}
By \cref{claim:many-good}, either there is an $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-addable element $(y,c_\ell)\not\in F$, or there are at least $n-\left|F_{c_{\ell}}\right|\ge n-f$ colours
which are $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-swappable. In the former case, we can increase the volume of $\mathcal{S}$, by a cascading sequence of swaps and transfers (first consider $\mathcal S'$ from \cref{Claim_Perform_Cascade}, then move $(x_\ell,c_\ell)$ from $S_{\ell}$ to $S_{\ell-1}'$, then add $(y,c_\ell)$ to $S_\ell-(x_\ell,c_\ell)$).
\end{proof}
\begin{claim*}
{There are at least $n-f$ colours $c$ for which $\left(\phi_{c}\left(\left(x_{\ell},c_{\ell}\right)\right),c\right)$
is $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-addable.}
\end{claim*}
\begin{proof}
Let $c$ be a colour which is $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-swappable with witness $y$, as in the previous claim. If $y$ is independent
to $\pi\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right)\right)$, we
can increase the volume of $\mathcal{S}$ by adding it to $S_{\ell}$ after
a cascading sequence of swaps and transfers (first consider $\mathcal S'$ from \cref{Claim_Perform_Cascade}, then move $(x_\ell,c_\ell)$ from $S_{\ell}$ to $S_{\ell-1}'$, then add $(y,c_\ell)$ to $S_\ell-(x_\ell,c_\ell)$).
Otherwise, by \cref{claim:add-if-good} applied with $b=c_{\ell}$, $S=S_{\ell}-\left(x_{\ell},c_{\ell}\right)$, the element
$\left(\phi_{c}\left(\left(x_{\ell},c_{\ell}\right)\right),c\right)$
is $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-addable. Here we are using that $\left(\phi_{c}\left(\left(x_{\ell},c_{\ell}\right)\right),c\right)$
is independent from $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$ (which comes from the definition of $\phi_c$ in \cref{lem:matching}).
\end{proof}
\begin{claim*}
{There are at least $n-f-\ell$ colours $c\not\in\{c_0, \dots, c_{\ell-1}\}$ for which $\left(\phi_{c}\left(\left(x_{\ell},c_{\ell}\right)\right),c\right)$
is $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-addable.}
\end{claim*}
\begin{proof}
This ensues from the previous claim and the fact that the only requirement on $c$, besides addability, is that it is different from the $\ell$ colours in $\{c_0, \dots, c_{\ell-1}\}$.
\end{proof}
We now prove the following:
\begin{equation}\label{eq:QBound2}
|Q\left(S_{0},\dots,S_{\ell}\right)|\geq \left|Q\right|\left(n-\ell-f\right)-(\ell+1) n.
\end{equation}
From the last claim, we have $\left|Q\right|\left(n-\ell-f\right)$ elements of the form $\left(\phi_{c}\left(\left(x_{\ell},c_{\ell}\right)\right),c\right)$ which are all $\left(S_{\ell}-\left(x_{\ell},c_{\ell}\right),c_{\ell}\right)$-addable, with $c$ outside a sequence of colours freeing $(x_{\ell}, c_{\ell})$. Notice that these $\left(\phi_{c}\left(\left(x_{\ell},c_{\ell}\right)\right),c\right)$ are all distinct because $\phi_c$ is an injection.
By \cref{Claim_Concatenate_cascade}, each of these is cascade-addable with respect to $S_{0},\dots,S_{\ell}$, unless it appears in one of $S_{0},\dots,S_{\ell}$.
The total number of elements in $S_{0},\dots,S_{\ell}$ is at most $(\ell+1) n$, so we have found $\left|Q\right|\left(n-\ell-f\right)-(\ell+1) n$ cascade-addable elements with respect to $S_{0},\dots,S_{\ell}$, as required by \cref{eq:QBound2}.
The lemma immediately follows by combining \cref{eq:QBound} and \cref{eq:QBound2}.
\end{proof}
Now, we want to iteratively apply \cref{claim:cascade-increase} starting
from some $S_{0}\in\mathcal{S}$, to obtain a sequence $S_{0},S_{1},\dots,S_{h}\in\mathcal{S}$.
There are two ways this process can stop: either we find a way to
increase the volume of $\mathcal{S}$, in which case we are done, or else we
run out of RISs in $\mathcal{S}$ (that is, $h=f-1$). We want to show that
this latter possibility cannot occur by deducing from \cref{eq:recurrence}
that the $\left|Q\left(S_{0},\dots,S_{\ell}\right)\right|$ increase
in size at an exponential rate: after logarithmically many steps there
will be so many cascade-addable elements that they cannot all be contained
in the RISs in $\mathcal{S}$, and it must be possible to increase the volume
of $\mathcal{S}$.
A slight snag with this plan is that \cref{eq:recurrence} only yields
an exponentially growing recurrence if the ``initial term'' is rather
large. To be precise, let $C$ (depending on $\varepsilon$) be sufficiently
large such that
\begin{equation}
C\left(1+\varepsilon/2\right)^{\ell-1}\frac{1}{1-\varepsilon}-\ell-1\ge C\left(1+\varepsilon/2\right)^{\ell}\label{eq:C}
\end{equation}
for all $\ell\ge1$.
\begin{claim}
\label{claim:recurrence-estimate}For $S_{0},\dots,S_{h}$ as above,
suppose that $\left|Q\left(S_{0}\right)\right|\ge Cn$ or $\left|Q\left(S_{0},S_{1}\right)\right|\ge Cn$.
Then, for $0<\ell\le\min\left\{ h,\varepsilon n/2\right\} $, we have
\[
\left|Q\left(S_{0},\dots,S_{\ell}\right)\right|\ge C\left(1+\varepsilon/2\right)^{\ell-1}n.
\]
\end{claim}
\begin{proof}
We first establish a technical inequality. Recall that $f=(1-\varepsilon)n/2,$ so
\begin{equation}
\frac{n-f-\ell}{f-\ell} \ge \frac{n-(1-\varepsilon)n/2-n\varepsilon/2}{(1-\varepsilon)n/2}= \frac{1}{1-\varepsilon}.\label{eq:fneps}
\end{equation}
Now, let $Q_{\ell}=Q\left(S_{0},\dots,S_{\ell}\right)$. We proceed by
induction. First observe that if $|Q_0| \ge Cn$ then \cref{eq:recurrence}, \cref{eq:fneps} and \cref{eq:C} for $\ell=1$ imply $|Q_1|\ge Cn(n-f-1)/(f-1)-2n \ge(C/(1-\varepsilon)-2)n\ge Cn$, giving us the base case. If $\left|Q_{\ell}\right|\ge C\left(1+\varepsilon/2\right)^{\ell-1}n$,
then once again using \cref{eq:recurrence}, \cref{eq:fneps} and \cref{eq:C}, we obtain
\begin{align*}
\left|Q_{\ell+1}\right| & \ge\frac{C\left(1+\varepsilon/2\right)^{\ell-1}n}{f-\ell}\cdot\left(n-f-\ell\right)-(\ell+1) n\\
& =\left(C\left(1+\varepsilon/2\right)^{\ell-1}\frac{\left(n-f-\ell\right)}{f-\ell}-\ell-1\right)n\\
& \ge \left(C\left(1+\varepsilon/2\right)^{\ell-1}\frac{1}{1-\varepsilon}-\ell-1\right)n\\
& \ge C\left(1+\varepsilon/2\right)^{\ell}n.\tag*{\qedhere}
\end{align*}
\end{proof}
If we could choose $S_{0},S_{1}$ such that $\left|Q\left(S_{0}\right)\right|\ge Cn$
or $\left|Q\left(S_{0},S_{1}\right)\right|\ge Cn$, then \cref{claim:recurrence-estimate}
would imply that during the construction of $S_1,\dots,S_h$ we never run out of RISs in $\mathcal{S}$ (that is, $h<f-1$). Indeed, otherwise $Q(S_0,\dots,S_{\varepsilon n/2})$ would have size exponential in $n,$ which is impossible. Therefore, the process must stop at some point when we find a way to increase the volume of $\mathcal{S}.$ Provided we can again find suitable $S_0,S_1$ we can then repeat the arguments in this section, further increasing the volume of $\mathcal{S}$. After repeating these arguments enough times we will have obtained
$f=\left(1-\varepsilon\right)n/2\ge\left(1/2-\varepsilon\right)n$
disjoint transversal bases, completing the proof of \cref{thm:new}.
There may not exist suitable $S_{0},S_{1}\in\mathcal{S}$, but in the next
section we will show that if at least $\varepsilon n/2$ of the RISs
in $S$ are not transversal bases, then it is possible to modify $\mathcal{S}$
without changing its volume, in such a way that suitable $S_{0},S_{1}$
exist.
\begin{rem}
\label{rem:linear}With the results we have proved so far, we can already find linearly many disjoint transversal bases. Indeed, if $S_{0}$ is not a transversal basis (missing
a colour $b$, say), and the volume of $\mathcal{S}$ cannot be increased by
adding an element to $S_{0}$ (possibly after a simple swap), then
\cref{claim:1-addability} implies that there are at
least $n-f$ elements which are $\left(S_{0},b\right)$-addable,
meaning that $\left|Q\left(S_{0}\right)\right|\ge n-f$.
Take for example $\varepsilon=4/5$, meaning that $f\le n/10$ and $\left|Q\left(S_{0}\right)\right|\ge9n/10$. We can check
that \cref{eq:C} holds for all $\ell\ge1$ if $C=9/10$. That is to
say, as long as we have not yet completed $\mathcal{S}$ to a collection of
disjoint transversal bases, we can keep increasing its volume without
the considerations in the next section. This proves already
that it is possible to find linearly many disjoint transversal bases.
\end{rem}
\begin{rem}
It is not hard to add a term $(n-|S_{\ell}|)(n-f)$ to the right hand side of the inequality given by \cref{claim:cascade-increase} by considering also cascades along the sequence $S_0, \ldots, S_{\ell-1}$ of length strictly less than $\ell$. However, since this increase is only significant when $|S_{\ell}|$ is not close to $n$, which may never be the case, we omit it from our argument for the sake of readability.
\end{rem}
\subsection{Increasing the number of initial addable elements}\label{subsec:increasing}
Consider a collection $\mathcal{S}$ of $f=\left(1-\varepsilon\right)n/2$
disjoint RISs, at least $\varepsilon n/2$ of which are not transversal
bases. Recall the choice of $C$ from the previous section, and let $D=2C+4$, so that
$D\left(n-f-1\right)-2n\ge Cn$ for large $n$. We prove the following (for large $n$).
\begin{claim}
\label{claim:many-missing}We can modify $\mathcal{S}$ in such a way that at least one of the following holds.
\begin{enumerate}
\item [(a)]The volume of $\mathcal{S}$ increases;
\item [(b)]the volume of $\mathcal{S}$ does not change, and there is now $S_{0}\in\mathcal{S}$
missing at least $D$ colours;
\item [(c)]the volume of $\mathcal{S}$ does not change, and there are now distinct
$S_{0},S_{1}\in\mathcal{S}$ such that $S_{1}$ contains at least $D$ elements
that are $\left(S_{0},b\right)$-addable, for some colour $b$.
\end{enumerate}
\end{claim}
This suffices for our proof of \cref{thm:new}; indeed, if $S_{0}$ is missing at least $D$ colours, then by \cref{claim:1-addability}, either we can increase the volume of $\mathcal{S}$
or there are at least $D\left(n-f\right)\ge Cn$ elements which are
$\left(S_{0},b\right)$-addable for every $b$ not appearing in $S_0$, meaning that $\left|Q\left(S_{0}\right)\right|\ge Cn$.
If $S_{1}$ contains at least $D$ elements that are $\left(S_{0},b\right)$-addable,
then in the proof of \cref{claim:cascade-increase} with $\ell=1$ we have $|Q|\ge D$ so either we can increase the volume of $\mathcal{S}$ or $\left|Q\left(S_{0},S_{1}\right)\right|\ge D\left(n-f-1\right)-2n\ge Cn$ (recall \cref{eq:QBound2}).
Before proceeding to the proof of \cref{claim:many-missing},
we first observe that using \cref{claim:many-good} we can modify $\mathcal{S}$
to ensure that every $S\in\mathcal{S}$ that is not a transversal basis can
be assigned a distinct missing colour $b\left(S\right)$. To see this,
we iteratively apply the following lemma to $\mathcal{S}$.
\begin{lem}
\label{lem:different-missing}
Consider $f\le n/2$ and let $\mathcal{S}=\left\{ S_{1},\dots,S_{f}\right\} $
be a collection of disjoint RISs. We can either increase the size of $\mathcal{S}$ or we can modify $\mathcal{S}$ in such a way that the size of each $S_i$ remains the same, and in such a way that that there is a choice of disjoint colours $\{ b_1,\dots,b_f\}$ for which any $S_i$ that is not a transversal basis has no element of colour $b_i.$
\end{lem}
\begin{proof}
Suppose for some $i$ that we found distinct colours $b_1,\dots,b_{i-1}$
such that, for all $S_{j}$ which are not transversal bases, no element
of $S_{j}$ is of colour $b_{j}$. If $S_i$ is a transversal basis we choose an arbitrary unused colour as $b_i.$ Otherwise there is a colour, say $c$, not appearing in $S_{i}$. Then by \cref{claim:many-good} either we can increase the size of $\mathcal{S}$ or there are at least $n-f\ge n/2$ colours which are $(S_i,c)$-swappable. At least one of these colours does not appear in $\left\{ b_1,\dots,b_{i-1}\right\}$, since $i-1<f\le n/2$. Let $b$ be such a colour and set $b_i=b$. By performing a simple swap, we transform $S_i$ into a new RIS, still disjoint to all other $S_j \in \mathcal{S}$ and missing the colour $b$.
\end{proof}
Now we prove \cref{claim:many-missing}.
\begin{proof}[Proof of \cref{claim:many-missing}]
Recall that we are assuming there are at least $\varepsilon n/2$
RISs in $\mathcal{S}$ that are not transversal bases. Let $E$ be the largest
integer such that there are at least $M_{E}=\left(\varepsilon/\left(4D^{2}\right)\right)^{E}n$
RISs in $\mathcal{S}$ missing at least $E$ colours. We may assume $1\le E<D$.
By \cref{lem:different-missing} we may assume that each $S\in\mathcal{S}$ which is not a transversal basis
has a distinct missing colour $b\left(S\right)$. We describe a procedure
that modifies $\mathcal{S}$ to increase $E$.
We create an auxiliary digraph $G$ on the vertex set $\mathcal{S}$ as follows.
For every $S_{0}\in\mathcal{S}$ missing at
least $E$ colours, put an arc to $S_{0}$ from every $S_{1}\in\mathcal{S}$
such that $S_{1}$ contains at least $E+1$ elements that are $\left(S_{0},b\left(S_{0}\right)\right)$-addable.
Say an \emph{$\left(E+1\right)$-out-star} in a digraph is a set of
$E+1$ arcs directed away from a single vertex. Our goal is to prove that there are $M_{E+1}$ vertex-disjoint $\left(E+1\right)$-out-stars. To see why this suffices, consider an $\left(E+1\right)$-out-star (with centre $S_{1}$, say). We show how to transfer $E+1$ elements from $S_1$ to its out-neighbours, the end result of which is that $S_1$ is then missing $E+1$ colours. We will then be able to repeat this process for each of our out-stars.
For each of the $E+1$ out-neighbours $S_{0}$ of $S_{1}$ there are
at least $E+1$ elements of $S_1$ which are $\left(S_{0},b\left(S_{0}\right)\right)$-addable. Therefore, for each such $S_0$ we can make a specific choice of such an $\left(S_{0},b\left(S_{0}\right)\right)$-addable element, in such a way that each of these $E+1$ choices are \emph{distinct}. For each $S_0$ we can then transfer the chosen element from $S_1$ to $S_0$, possibly with a simple
swap. These simple swaps will not create any conflicts, because
any addability witness for any element in $S_{0}$ is in a colour
unique to that $S_{0}$ (by the property from \cref{lem:different-missing}). After this
operation, $S_i$ is now missing
at least $E+1$ colours.
It will be a relatively straightforward matter to find our desired out-stars by studying the digraph $G$. First we show that $G$ must have many edges.
\begin{claim*}
In the above auxiliary digraph, we may assume that every $S_{0}\in\mathcal{S}$ missing at least
$E$ colours has in-degree at least $\varepsilon n/D$.
\end{claim*}
\begin{proof}
By \cref{claim:1-addability} we can
assume that there are at least $E\left(n-f\right)$ elements which are $\left(S_{0},b\left(S_{0}\right)\right)$-addable. All these elements appear in various $S\in\mathcal{S}$ (otherwise we can increase the volume of $\mathcal{S}$).
Let $N^-(S_0)$ be the set of all $S_1$ such that there is an arc from $S_1$ to $S_0$ in $G$ (so $|N^-(S_0)|$ is the indegree of $S_0$). By definition, every $S\notin N^-(S_0)$ has at most $E$ elements which are $\left(S_{0},b\left(S_{0}\right)\right)$-addable. Moreover, observe that every $S\in \mathcal{S}$ has fewer than $D$ elements that are $\left(S_{0},b(S_0)\right)$-addable, or else (c) trivially occurs. It follows that
\begin{align*}
D|N^-(S_0)|+E(f-|N^-(S_0)|)&\ge E(n-f),
\end{align*}
so
\begin{align*}
|N^-(S_0)| & \ge\frac{E\left(\left(n-f\right)-f\right)}{D-E}\ge\frac{\varepsilon n}{D},
\end{align*}
as desired.
\end{proof}
We have proved that $G$ has at least $M_E \varepsilon n/D$ edges. Now we finish the proof by showing how to find our desired out-stars.
\begin{claim*}
$G$ has at least $M_{E+1}$ vertex-disjoint $\left(E+1\right)$-out-stars.
\end{claim*}
\begin{proof}
We can find these out-stars in a greedy fashion. Suppose that we have already found $t$ vertex-disjoint $\left(E+1\right)$-out-stars, for some $t< M_{E+1}$. We show that there must be an additional $\left(E+1\right)$-out-star disjoint to these. Let $G'$ be obtained from $G$ by deleting all vertices in the out-stars we have found so far. Each of these out-stars has $E+2$ vertices, so the number of arcs in $G'$ is at least
\begin{align*}
M_E \frac{\varepsilon n}D-t(E+2)\cdot2f&>M_E \frac{\varepsilon n}D-M_{E+1}(E+2)\cdot 2f \\
& = M_E \frac{\varepsilon n}D-\frac{M_{E}\varepsilon}{2D^2}\cdot(E+2)f\\
&\ge M_E \varepsilon\left(\frac n D-\frac f{D}\right)\\
& \ge M_E \varepsilon \cdot \frac f{D}\ge (E+1)f,
\end{align*}
where the last inequality holds for sufficiently large $n$, using the fact that $M_E$ is linear in $n$. This means that $G'$ (having at most $f$ vertices) has a vertex with outdegree at least $E+1$, which means $G'$ contains an $\left(E+1\right)$-out-star disjoint to the out-stars we have found so far.
\end{proof}
\end{proof}
\section{Concluding remarks}
In this paper we proved that that given bases $B_{1},\dots,B_{n}$
in a matroid, we can find $\left(1/2-o\left(1\right)\right)n$ disjoint
transversal bases. Although our methods do not extend past $n/2$,
we do not think that there is a fundamental obstacle preventing related
methods from going further. Indeed, by tracking the possible cascades
of swaps more carefully, it might be possible to find $\left(1-o\left(1\right)\right)n$
disjoint transversal bases, or at least to find $\left(1-o\left(1\right)\right)n$
disjoint partial transversals each of size $\left(1-o\left(1\right)\right)n$.
Although we cannot completely rule out the possibility that a full
proof of Rota's basis conjecture could be obtained in this way, we
imagine that more ingredients will be required. We are hopeful that
ideas used to prove existence of designs (see \cite{Kee14,GKLO16})
could be relevant, at least in the case of vector spaces.
Also, we remark that Rota's basis conjecture is reminiscent of some
other problems concerning rainbow
structures in graphs (actually, for a graphic matroid, Rota's basis
conjecture can be interpreted as a conjecture about rainbow spanning
forests in edge-coloured multigraphs). The closest one to Rota's basis
conjecture seems to be the Brualdi--Hollingsworth conjecture
\cite{BH96}, which posits that for every $(n-1)$-edge-colouring of the complete graph $K_n$, the edges can be decomposed into rainbow spanning trees. This conjecture has
recently seen some exciting progress (see for example \cite{Hor18,PS18,BLM18,MPS18}).
We wonder if some of the ideas developed for the study of rainbow
structures could be profitably applied to Rota's basis conjecture.
We also mention the following strengthening of Rota's basis conjecture due to Kahn (see \cite{HR94}). This is simultaneously a strengthening
of the Dinitz conjecture \cite{Dinitz} on list-colouring of $K_{n,n}$,
solved by Galvin \cite{Gal95}.
\begin{conjecture}
\label{conj:kahn}Given a rank-$n$ matroid and bases $B_{i,j}$ for
each $1\le i,j\le n$, there exist representatives $b_{i,j}\in B_{i,j}$
such that each of the sets $\{b_{1,j},\dots,b_{n,j}\}$ and
$\{b_{i,1},\dots,b_{i,n}\}$ are bases.
\end{conjecture}
The methods developed in this paper are also suitable for studying
\cref{conj:kahn}. In particular, the argument used to prove \cref{thm:new} can readily be modified to show the following natural partial result towards Kahn's conjecture.
\begin{thm}\label{thm:kahn}
For any $\varepsilon>0$ the following holds for sufficiently large $n$. Given a rank-$n$ matroid and bases $B_{i,j}$ for
each $1\le i\le n$ and $1\le j \le f=(1-\varepsilon)n/2$, there exist representatives $b_{i,j}\in B_{i,j}$ and $L\subseteq \{1,\dots,f\}$
such that each $\{b_{i,j}:i\in L\}$ is independent, and such that
$\{b_{i,1},\dots,b_{i,n}\}$ is a basis for any $i \in L$ and $|L| \ge (1/2-\varepsilon)n$.
\end{thm}
Note that if we are in the setting of \cref{conj:kahn} where bases are given for all $1 \le i,j\le n$ then the above theorem allows us to choose roughly which rows we would like to find our bases in.
Note also that if, for each fixed $j$, the bases $B_{1,j},\dots,B_{n,j}$ are all equal, then Kahn's conjecture reduces to Rota's basis conjecture. This observation also shows that \cref{thm:kahn} implies \cref{thm:new}.
It is not hard to adapt the proof of \cref{thm:new} to prove \cref{thm:kahn}. However, since it would require repeating most of the argument, we omit the details here. For interested readers we present the details in a companion note, which we will not publish but will make available on the arXiv \cite{arxiv:kahn}.
\medskip
\textbf{Acknowledgements.} We are extremely grateful to the anonymous referees for their careful reading of the paper and many useful suggestions.
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{
"redpajama_set_name": "RedPajamaArXiv"
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{
"redpajama_set_name": "RedPajamaC4"
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Twiningia bicolor är en insektsart som beskrevs av Ball 1909. Twiningia bicolor ingår i släktet Twiningia och familjen dvärgstritar. Inga underarter finns listade i Catalogue of Life.
Källor
Dvärgstritar
bicolor
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Le nom Grotte du Moulin désigne notamment :
France
Grotte du Moulin, Bize-Minervois, Aude, Languedoc-Roussillon.
Grotte du Moulin, grotte-abri ornée, Troubat, Hautes-Pyrénées, Occitanie.
Grotte du Moulin de Laguenay ou grotte de la Boissière, Lissac-sur-Couze, Corrèze.
Espagne
Grotte du Moulin (Espagne), traduction de l'espagnol Cueva del Molin, de Morcín, Asturies.
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There's a brand new gym opened in Cork called The Lions' Den Mallow and it's doing things no other gym is.
The owner James Ogden has many years experience in the business and seen the need for a new kind of inclusive gym in Cork.
Although it's only open 8 weeks it's already gained over 2.7k followers on it's Facebook page and over 800 members.
It really is a gym with a difference, the whole premises is wheelchair accessible along with it's showers. It's fully loaded with only top of the range equipment and highly qualified personal trainers.
Members can avail of the on site crèche while they work out and feel at ease knowing all staff members are garda vetted.
The Lions' Den Mallow offers gym classes for kids and teens, Cubs classes for 5 -10 year olds and Teen classes for 11 - 16 year olds. There's also 40 adult classes happening through the week .
Along with all that they also offer workouts for people with disabilities from athletes with injuries to MS suffers or clients who are paralysed. The Madonna Icare machine is a fully assisted and intelligent cross trainer suited for all needs.
There's an on site Physiotherapist, who does cranial and needling and a Sports Massage Therapist.
James is hoping to have the website up and running very soon but for now you can check them out on their Facebook page.
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"redpajama_set_name": "RedPajamaC4"
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Q: Transformation of $\mathbb R^2$ Let $T:\mathbb R^2\to \mathbb R^2$ be s.t. for all $x,y\in\mathbb R^2$, $\mid\mid T(x)-T(y)\mid\mid = \mid\mid x-y\mid\mid.$ Then, 2 questions:
*
*If $T(0)=0$, does it follow that $T$ must be linear?
*Show that $T$ is a translation, then a rotation, and then a reflection through the $x$ axis (in which any of those three could be the "empty" one).
I'm rather stumped by this problem.... for 1 think yes, proven perhaps by taking $x=0$? But I haven't actually been able to completely figure it out.
And I have no idea how to proceed with 2... perhaps looking at the matrices that generate such movements?
A: Let $e_1=[1,0]^t$ and $e_2=[0,1]^t.$ Let $M$ be the $2\times 2$ matrix whose first column is $T(e_1)$ and whose second column is $T(e_2)$. Show that if $T(0)=0,$ then $T(v)=Mv$ for all $v\in\Bbb R^2.$
Now, note that the matrix $M$ above must be invertible (why?), and in particular, has determinant $\pm1$ (why?). Show that: a $2\times2$ real matrix with determinant $1$ represents a (possibly "empty") rotation; a $2\times2$ real matrix with determinant $-1$ represents a reflection about a line through the origin, which can be obtained instead by a (possibly "empty") rotation followed by a reflection about the $x$-axis (why?).
For your second part, put $b=T(0)$ and let $S(v)=T(v)-b$ for all $v$. What can you say about $S$ in light of the first exercise? Use this to show that there is a vector $c$ and a matrix $M$ with determinant $\pm 1$ and a vector $c$ such that $$T(v)=M(v+c).$$
Edit: Let me expand on the first two paragraphs (since that's where the proof gets sticky).
Now, we're assuming that $$T(0)=0\tag{1}$$ and that for all $v,w\in\Bbb R^2$ we have $$\lVert T(v)-T(w)\rVert =\lVert v-w\rVert.\tag{2}$$ Putting $w=0$ in $(2)$, we have by $(1)$ that $$\lVert T(v)\rVert=\lVert v\rVert\tag{3}$$ for all $v\in\Bbb R^2$.
Letting $x\cdot y$ indicate the dot product of $x$ and $y$, it is easily proved that $$\lVert v\rVert^2=v\cdot v$$ for all $v\in\Bbb R^2$ (assuming that our norm is the Euclidean norm), so for all $x,y\in\Bbb R^2,$ we have by dot product properties that $$\begin{align}\lVert x-y\rVert^2 &= (x-y)\cdot(x-y)\\ &= x\cdot x-x\cdot y-y\cdot x+y\cdot y\\ &= x\cdot x-2(x\cdot y)+y\cdot y\\ &= \lVert x\rVert^2-2(x\cdot y)+\lVert y\rVert^2.\end{align}\tag{4}$$ Applying $(2),$ $(3)$, and $(4)$, we find that for all $v,w\in\Bbb R^2$, we have $$\begin{align}\lVert v\rVert^2-2\bigl(T(v)\cdot T(w)\bigr)+\lVert w\rVert^2 &= \lVert T(v)\rVert^2-2\bigl(T(v)\cdot T(w)\bigr)+\lVert T(w)\rVert^2\\ &= \lVert T(v)-T(w)\rVert^2\\ &= \lVert v-w\rVert^2\\ &= \lVert v\rVert^2-2(v\cdot x)+\lVert w\rVert^2,\end{align}$$ so in particular $$T(v)\cdot T(w)=v\cdot w.\tag{5}$$
Now, apply $(3),$ $(4),$ $(5),$ and dot product properties, so we see that for any $x,y,z\in\Bbb R^2$ we have $$\begin{align}\lVert z-x-y\rVert^2 &= \lVert z-x\rVert^2-2\bigl((z-x)\cdot y\bigr)+\lVert y\rVert^2\\ &= \lVert z-x\rVert^2-2(z\cdot y)+2(x\cdot y)+\lVert y\rVert^2\\ &= \lVert z\rVert^2-2(z\cdot x)+\lVert x\rVert^2-2(z\cdot y)+2(x\cdot y)+\lVert y\rVert^2\\ &= \lVert T(z)\rVert^2-2\bigl(T(z)\cdot T(x)\bigr)+\lVert T(x)\rVert^2-2(z\cdot y)+2(x\cdot y)+\lVert y\rVert^2\\ &= \lVert T(z)-T(x)\rVert^2-2(z\cdot y)+2(x\cdot y)+\lVert y\rVert^2\\ &= \lVert T(z)-T(x)\rVert^2-2\bigl(T(z)\cdot T(y)\bigr)+2\bigl(T(x)\cdot T(y)\bigr)+\lVert T(y)\rVert^2\\ &= \lVert T(z)-T(x)\rVert^2-2\Bigl(\bigl(T(z)-T(x)\bigr)\cdot T(y)\Bigr)+\lVert T(y)\rVert^2\\ &= \lVert T(z)-T(x)-T(y)\rVert^2.\end{align}\tag{6}$$ In particular, let $x,y\in\Bbb R^2,$ and put $z=x+y,$ so by $(6),$ we find that $$0=\lVert T(x+y)-T(x)-T(y)\rVert^2,$$ so $$0=\lVert T(x+y)-T(x)-T(y)\rVert$$ by nonnegativity, and so $$T(x+y)=T(x)+T(y)\tag{$\star$}$$ for all $x,y\in\Bbb R^2.$ You should be able to prove from here that $T$ is linear, so in particular is given by $$T(v)=Mv\tag{$\heartsuit$}$$ (where $M$ is given in the first paragraph).
Since $T$ is one-to-one (why?), then $M$ is invertible. Moreover, using the fact that $x\cdot y=x^ty$ for all $x,y\in\Bbb R^2,$ we have that $$\begin{align}x\cdot(M^tM-I)y &= x^t(M^tM-I)y\\ &= x^t(M^tMy-y)\\ &= x^tM^tMy-x^ty\\ &= (Mx)^tMy-x^ty\\ &= (Mx)\cdot(My)-x\cdot y,\end{align}$$ so by $(\heartsuit)$ and $(5)$ we have $$x\cdot(M^tM-I)y=0$$ for all $x,y\in\Bbb R^2.$ In particular, then, for any $y\in\Bbb R^2$, we have that $$\lVert(M^tM-I)y\rVert^2=(M^tM-I)y\cdot(M^tM-I)y=0,$$ whence $$(M^tM-I)y=0$$ for all $y\in\Bbb R^2,$ and so $M^tM=I.$ From this, we conclude that $\det(M)=\pm 1$. (Why?)
Can you take it from there?
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Q: Does recycle call Application_Start? Does Application_Start get called on recycling in IIS 7?
Thank you
A: No, it gets called on the first request after recycling when the application is actually loaded by IIS.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,902
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Q: Wordpress site - adding alt attributes to existing images in PHP I want to figure out how to add an alt attribute to pre-existing images on the site. I've read all the other posts I can find on StackOverflow and Wordpress.org, tested multiple functions, and haven't had much luck. Any help would be greatly appreciated.
In my functions.php file, this function exists, but it doesn't appear to do anything.
/* add alt attributes to all images */
function isa_add_img_title( $attr, $attachment = null ) {
$img_title = trim( strip_tags( $attachment->post_title ) );
$attr['title'] = $img_title;
$attr['alt'] = $img_title;
return $attr;
}
add_filter( 'wp_get_attachment_image_attributes','isa_add_img_title', 10, 2 );
Here is a previous function I've also implemented with no effect.
function addImageAlt($attribute){
$imgAttribute = trim( strip_tags($attribute->post_title) );
$totalAttribute["alt"] = $imgAttribute;
return $totalAttribute;
}
add_filter("wp_get_attachment_image", "addImageAlt", 10, 5);
Here is the template code, which is getting the image source
{
extract($atts);
if (!is_front_page()):
wp_enqueue_script('slick', get_template_directory_uri() . '/assets/js/slick.min.js', array( 'jquery' ), '20160816', true);
wp_enqueue_style('slick-css', get_template_directory_uri() . '/assets/css/slick.css');
endif;
$html = '<div class="brands-slider-wrapper">
<div class="brands-slider">';
$loop = new WP_Query(array( 'post_type' => 'zm_brands', 'posts_per_page' => $num_of_brands, 'order' => 'ASC', 'orderby' => 'meta_value' ));
if ($loop && $loop->post_count > 0) :
while ($loop->have_posts()) : $loop->the_post();
$imgsrc=wp_get_attachment_image_src(get_post_thumbnail_id(), 'brand-thumb');
$brand_thumbnail= $imgsrc[0] != "" ? $imgsrc[0] : get_template_directory_uri() . '/assets/images/default-music-icon.jpg';
$html.='<div class="slide-brand">
<div class="brand-logo-thumbnail">
<a href="'.get_permalink().'">
<img src='.$brand_thumbnail.' alt="'.basename($brand_thumbnail).'" title="'.get_the_title().'">
</a>
</div>
</div>';
endwhile;
endif;
$html .='</div>
</div>';
wp_reset_query();
return $html;
}
The images for the sliders on the website appear to be sourced differently.
$html .= '<div class="slide-instrument">
<div class="instrument-category-thumbnail auto-height">
<img src=' . $category_thumbnail . '" alt="' . $term->slug . '" title="' . $term->name . '">
</div>
<h4>' . $term->name . '</h4>
<a href="' . get_term_link($term->slug, $taxonomy) . '"><span>View More</span></a>
</div>';
A: You are using wp_get_attachment_image_src to fetch the image and above filters are different.
wp_get_attachment_image_src : this has been used for the image url only.
wp_get_attachment_image : this has been used for the image url along with the img tag so if you are going to use wp_get_attachment_image() function then your filter will be going to work.
According to me, you need to change your template code like this :
$html.='<div class="slide-brand">
<div class="brand-logo-thumbnail">
<a href="'.get_permalink().'">
'.wp_get_attachment_image(get_the_ID()).'
</a>
</div>
</div>';
I am assuming that you are geting the id in the get_the_ID()
I am assuming that your function is right with the filter.
Thanks
A: I figured this out by instead editing the application.js file, and adding this jquery function
$('img').attr('alt', function(){
var src = this.src.split('/').pop().split('.')[0];
return src;
});
This simply adds alt attributes to all images on the site, and is not necessarily the best way to do so, but it worked for me.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,112
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\section{Introduction}
In an attempt to explain the origin of galactic magnetic fields
through the amplification of primordial fields, several authors have
considered scenarios for generating magnetic fields in the early
universe \cite{fieldcreation}. In such scenarios, one attempts to
generate fields which will be sufficiently large after recombination
at least to seed galactic dynamos and at best to produce galactic
fields without dynamo amplification. It is generally assumed that
after a primordial field is generated in the early universe, it
becomes frozen into the cosmic plasma and redshifts by flux
conservation with the expansion of the universe ($B \propto a^{-2}$;
$a(t)$ is the cosmic scale factor). This assumption is usually
justified by noting that the cosmological plasma is highly conductive
and magnetic diffusion is insignificant.
In this paper, we show that this simple picture of magnetic field
evolution is incorrect: at certain epochs in the early universe,
particularly during recombination and neutrino decoupling, magnetic
field energy is converted into heat through the damping of
magneto-hydrodynamic (MHD) modes. The damping is caused by dissipation
in the fluid, which arises from the finite mean free path of photons
or neutrinos.
The physical process by which the MHD modes are damped is analogous to
that involved in the damping of density fluctuations around
recombination \cite{S}, and around neutrino decoupling
\cite{M68}. Studies of the damping of density fluctuations with no
magnetic fields present show that, in the diffusive regime (when the
scales of interest are much larger than the mean free path of photons
or neutrinos, $\l_{\rm mfp}$), the effective viscosity and heat
conductivity arising from the finite mean free path cause the damping
of acoustically oscillating density perturbations. Since with the
expansion of the universe the mean free path of the decoupling
particles grows faster than the wavelength of an oscillatory mode, all
modes whose wavelengths are smaller than the mean free path around
decoupling have previously been in the diffusion regime. The rate of
damping in this regime ensures that a wave is significantly damped
before the mean free path of the decoupling particles becomes
comparable to the wavelength of the mode. For this reason, the
investigation of damping in the diffusion regime yields a reasonable
estimate of the final damping scales of density fluctuations.
However, when magnetic fields are added to the fluid, the existence of
different MHD modes---Alfv\'en, fast magnetosonic, and slow
magnetosonic waves---adds complexity to the problem. We show that
while fast magnetosonic waves (which include sound waves) damp
efficiently in the diffusion regime by the described process, slow
magnetosonic and Alfv\'en waves may survive damping by diffusion. Slow
magnetosonic and Alfv\'en modes oscillate with frequencies which
depend on the strength of the background magnetic field and on its
direction relative to the mode's wave vector, and are in general
different from the frequency of sound waves of the same wavelength. In
the case of a weak background magnetic field or a large angle between
the background field and the wave vector, the frequency can be slow
enough for the damping by viscosity to overcome the oscillation,
producing behavior which resembles an overdamped oscillator and causes
the actual damping of the amplitude to be inefficient. The overdamped
slow magnetosonic and Alfv\'en modes therefore survive diffusion
damping. However, they undergo additional damping if, with the
expansion of the universe, they enter the so called free-streaming
regime, i.e. the mean-free-path grows to be much larger that the
wavelength of a mode. As a consequence, whereas fast magnetosonic
modes are damped mostly when radiation is diffusing, slow magnetosonic
and Alfv\'en modes are also significantly damped when radiation is
free-streaming. Therefore, when studying the damping of all MHD modes
in order to estimate their damping scales, it is necessary to
investigate both the free-streaming and the diffusion regimes even
before the final stages of the decoupling process.
The damping of MHD modes which causes the dissipation of magnetic
energy can be illustrated with the following picture: as long as there
exist spatially tangled magnetic fields, Lorentz forces accelerate the
fluid, setting up oscillations about a force-free field configuration;
the induced motions are damped by the effective viscosity of photons
or neutrinos; this causes the exponential decrease in the amplitude of
the oscillations and thus results in the straightening of magnetic
field lines towards a force-free configuration. After the cosmological
magnetic fields undergo this damping process they have little
structure on scales below a characteristic damping scale, and the
magnetic energy density in such primordial fields is much smaller than
that expected from the simple redshift argument above.
In this paper we follow the evolution of MHD modes and derive their
propagation velocities and damping rates both in the diffusion and
free-streaming regimes during the decoupling of photons and
neutrinos. The existence of highly relativistic particles with mean
free path much shorter than the wavelength of a MHD mode (e.g.,
photons and leptons) requires the use of relativistic MHD. In the
radiation diffusion regime, studied in \S 2, we develop a relativistic
description of viscous expanding fluids with magnetic fields, while in
the free-streaming case, \S 3, the effects of the photons or neutrinos
are included through heat exchange and a drag force which they exert
on the fluid. This procedure allows us to calculate, in \S 4, the
maximum damping lengths after the epochs of neutrino decoupling and
recombination. Our results may be applied to other astrophysical
environments where MHD waves propagate in a viscous fluid, since in
the derivation of the dispersion relations we leave the sources of
viscosity and heat conductivity unspecified.
\section{Damping of Magnetohydrodynamic Modes in the Radiation
Diffusion Regime}
When the mean free paths of all interacting particle species are
shorter than the wavelength of the MHD mode we are interested in
($\lambda \gg l_{\rm mfp}$), it is adequate to study the evolution of
a single fluid and account for the effect of the diffusing particles
by introducing shear viscosity, bulk viscosity, and heat conductivity
into the fluid equations \cite{W71}.
In order to calculate the damping of MHD modes following this
approach, we derive linearized relativistic MHD equations of an
expanding dissipative fluid. We start by reviewing the equations for a
non-ideal relativistic fluid in \S 2.1, and add the electromagnetic
contributions to the fluid equations in \S 2.2. In \S 2.3 we calculate
the propagation velocities and damping rates for all MHD modes. Our
results are applicable for general viscous relativistic and
non-relativistic plasmas, as long as the pressure is dominated by
radiation pressure.
Throughout the paper we assume that the magnetic field can be
decomposed into a large magnitude background component ${\bf B}_0({\bf
x},t)$, and a small perturbation, ${\bf b}({\bf x},t)$. We
additionally assume that the curl of the background component is
negligible when compared to the curl of the perturbations. These two
assumptions allow us to solve for the damping of MHD modes
analytically.
The use of scalar viscosities and heat conductivity implicitly
neglects any anisotropies in these quantities due to the presence of
the magnetic field. Further, since our equations are derived for an
isotropically, homogeneously, and adiabatically expanding plasma, the
background magnetic field, ${\bf B}_0$, is required to have vanishing
spatial average on sufficiently large scales, $\langle {\bf
B}_0\rangle =0$. In our derivation we also neglect gravitational
forces because the scales of interest are smaller than the Jeans mass
scale, and we assume the plasma to be infinitely conducting which is
an excellent approximation for most astrophysical plasmas and for the
early universe (see, e.g., Ref.~\cite{CO}).
\subsection{Relativistic Imperfect Fluids}
We consider the evolution of a non-ideal, relativistic fluid in a
homogeneously and isotropically expanding background using the
spatially flat Robertson-Walker metric $g_{\mu\nu}={\rm diag}(1, -a^2,
-a^2, -a^2)$ and comoving coordinates $x^{\mu}$. The time dependent
scale factor $a(t)$ provides the connection between proper (physical)
coordinates ${x'}^\mu$ and the comoving coordinates: ${x'}^0=x^0$ and
${x'}^i=ax^i$ (Greek indices run from 0 to 3 whereas Latin indices run
from 1 to 3).
The relativistic fluid is described by the energy-momentum tensor
\begin{equation} T^{\mu\nu} = T^{\mu\nu}_{ I} + \tau^{\mu\nu} +
T^{\mu\nu}_{{\rm EM}}, \label{eq:tensor}
\end{equation}
which is separated into three parts: the ideal fluid tensor
$T^{\mu\nu}_{I}$, the non-ideal fluid part $\tau^{\mu\nu}$, which accounts
for dissipation, and the electromagnetic energy-momentum tensor
$T^{\mu\nu}_{{\rm EM}}$ (added in \S 2.2). The equations of fluid dynamics can
be derived from energy-momentum conservation
\begin{equation}
T^{\mu\nu}{}_{;\nu} = 0. \label{eq:conserv_Tmn}
\end{equation}
In comoving coordinates Eq.~\ref{eq:conserv_Tmn} becomes
\begin{equation}
\frac{\partial T^{0\mu}}{\partial x^{\mu}} +
\dot{a}a\biggl(\sum_{i=1,3}T^{ii}\biggr) +
3\biggl(\frac{\dot{a}}{a}\biggr)T^{00} = 0, \label{eq:tmn_energy}
\end{equation}
and
\begin{equation}
\frac{\partial T^{i\mu}}{\partial x^{\mu}} +
5\biggl(\frac{\dot{a}}{a}\biggr) T^{i0} = 0, \label{eq:tmn_momentum}
\end{equation}
with the dot representing a derivative with respect to time
$x^0$.
The energy-momentum tensor for an ideal fluid is
\begin{equation}
T^{\mu\nu}_{I} = (\rho+p) U^{\mu} U^{\nu} - p g^{\mu\nu},
\end{equation}
where $\rho$, $p$, and $U^{\mu}$ are the total energy density,
the total pressure, and the four velocity of the fluid, respectively.
The non-ideal contributions to the fluid energy-momentum tensor can
be written as \cite{W72}
\begin{eqnarray}
\tau^{\mu\nu} &=& \eta\Bigl(U^{\mu ;\nu}
+ U^{\nu;\mu}
- U^{\mu} U^{\lambda} U^{\nu}{}_{;\lambda}
- U^{\nu} U^{\lambda} U^{\mu}{}_{;\lambda}\Bigr)
\nonumber \\
&&{} + \Bigl(\xi - \frac{2}{3} \eta\bigr) U^{\lambda}{}_{;\lambda}
\bigl(g^{\mu\nu} - U^{\mu} U^{\nu} \Bigr) \nonumber \\
&&{} + \kappa \biggl[U^{\mu} \biggl( \frac{\partial T}{\partial x_{\nu}}
- T U^{\nu}{}_{;\lambda} U^{\lambda}\biggr)
+ U^{\nu}\biggl( \frac{\partial T}{\partial x_{\mu}}
- T U^{\mu}{}_{;\lambda} U^{\lambda}\biggr)
- 2 U^{\mu} U^{\nu} \frac{\partial T}{\partial x_{\lambda}} U_{\lambda}
\biggr]. \label{eq:tau}
\end{eqnarray}
In this expression, $T$ stands for temperature and $\eta$, $\xi$, and
$\kappa$ are shear viscosity, bulk viscosity, and heat conductivity
respectively.
The effective viscosities and heat conductivity for either photons
or neutrinos are given by \cite{MS65,M68,W71}:
\begin{equation}
\eta={4\over 15}g{\pi^2\over 30}T^4 l_{\rm mfp}\, ,
\label{eq:shear}
\end{equation}
\begin{equation}
\xi=4g{\pi^2\over 30}T^4\biggl[{1\over 3}-\biggl({\partial
p\over\partial\rho}\biggr)_n\biggr] l_{\rm mfp}\, ,
\end{equation}
\begin{equation}
\kappa={4\over 3}g{\pi^2\over 30}T^3 l_{\rm mfp}\, ,
\label{eq:heat}
\end{equation}
where $n$ is the number density of the conserved particles in the
fluid and $g$ is the statistical weight of the diffusing particles.
The exact form of $\tau^{\mu\nu}$ is partially a matter of definition
since, in relativistic fluid mechanics, the fluid velocity can be
defined either by the flow of conserved particles \cite{W72} or by
the flow of energy \cite{LL75}. These definitions coincide in
non-relativistic fluid mechanics where the rest mass of particles
dominates the total energy. In our case, a relativistic one-fluid
approximation, the charged and strongly interacting particles
(protons, neutrons, electrons, etc.) which compose the fluid are all
perfectly coupled and have the same velocity as the conserved particle
number, the baryon number. The energy flow may differ from the
particle flow, however, due to the energy transported by the
imperfectly coupled neutrinos and photons. We choose to follow the
velocity of the charged particles (and, therefore, the flow of baryon
number) which appears explicitly in the magneto-hydrodynamic equations
below.
The conservation of particle number can be written as
\begin{equation}
n^{\mu}_{\,\,\,\, ;\mu}=0 \ , \label{eq:nmu_conserve}
\end{equation}
where $n^{\mu} =n U^{\mu}$ is the particle number four current with
$n$ the proper number density of particles. The particle number we
follow is the net baryon number, $n^b$, which is conserved for
temperatures below the electroweak transition.
We can now derive the linearized equations of ordinary
relativistic fluid dynamics in an expanding universe from
Eq.~\ref{eq:conserv_Tmn} -~\ref{eq:nmu_conserve}, by expanding the
fluid variables around their background values
\begin{eqnarray}
\rho({\bf x},t) &=& \rho_0(t)+\rho_1({\bf x},t), \\
p({\bf x},t) &=& p_0(t)+p_1({\bf x},t),\\
T({\bf x},t) &=& T_0(t)+T_1({\bf x},t),\\
n^b({\bf x},t) &=& n^b_0(t)+n^b_1({\bf x},t),\\
U^{\mu} &=& U^{\mu}_0+U^{\mu}_1 \ .
\end{eqnarray}
The four velocity is that of a stationary
fluid element (with respect to the comoving frame) plus a small
velocity perturbation
\begin{equation}
\quad\ U^{\mu}_0=\bigl(1,0,0,0\bigr),\quad
U^{\mu}_1=\bigl(0,{{\bf v}\over a}\bigr). \label{eq:u}
\end{equation}
We choose $U_1^{\mu}$ in this particular form so that the fluid
velocity in proper coordinates, ${\bf v}'=(\dot{a}/a){\bf x}'+{\bf
v}$, corresponds to an isotropic expansion plus an additional peculiar
velocity ${\bf v}$. We consider fluids in which the peculiar
velocities are much smaller than the speed of light, e.g. $|{\bf
v}|\ll 1$, where velocities are measured in units of the speed of
light. Although the fluid velocities are small, a relativistic
treatment is necessary to adequately account for the presence of
relativistic particles (e.g., photons and neutrinos).
Evaluating
equations~\ref{eq:conserv_Tmn}~and~\ref{eq:nmu_conserve} to
lowest order in the fluid variables, we obtain
\begin{equation}
\frac{\partial \rho_0}{\partial t} + 3\biggl(\frac{\dot{a}}{a} \biggr)\bigl(\rho_0+p_0\bigr)
= 9\xi \biggl(\frac{\dot{a}}{a} \biggr)^2 , \label{eq:entropy0}
\end{equation}
which represents conservation of entropy when $\xi =0$, and
\begin{equation}
\frac{\partial n_0^b}{\partial t} + 3\biggl({\frac{\dot{a}}{a}}\biggr)n_0^b = 0
, \label{eq:number0}
\end{equation}
represents conservation of baryon number. If we expand the energy
momentum tensor and baryon number four-current to first order in the
perturbation variables $U_1^{\mu}$, $\rho_1$, $p_1$, $T_1$, and
$n_1^b$, use Eq.~\ref{eq:tmn_energy}--\ref{eq:tmn_momentum}, and
subtract the zeroth order solution, we obtain
\begin{equation}
\frac{\partial\rho_1}{\partial t}
+ \bigl(\rho_0+p_0\bigr) \frac{1}{a}{\bf\nabla\cdot v}
+ 3\biggl({\frac{\dot{a}}{a}}\biggr)\bigl(\rho_1+p_1-2\xi{\bf\nabla\cdot
v}\bigr)
- \frac{\kappa}{a^2} \nabla^2 T_1
- \frac{\kappa}{a^2} \frac{\partial}{\partial t}\bigl(a T_0 {\bf\nabla\cdot
v}\bigr)
= 0, \label{eq:general_energy} \end{equation}
\begin{equation}
\frac{1}{a^4}\frac{\partial}{\partial t}\Biggl[ a^4\bigl(\rho_0+p_0\bigr)
{\bf v}
-\kappa a^3\biggl[\frac{\partial}{\partial t}\bigl(a T_0{\bf v}\bigr)
+ {\bf\nabla} T_1\biggr]
- 3\xi a^3\dot{a}{\bf v}\Biggr] + \frac{1}{a}{\bf\nabla} p_1
- \frac{\eta}{a^2} \nabla^2 {\bf v}
-\frac{1}{a^2}\bigl(\xi+\frac{1}{3}\eta\bigr){\bf\nabla}
({\bf\nabla\cdot v})
= 0, \label{eq:general_momentum}
\end{equation}
and
\begin{equation}
\frac{\partial n_1^b}{\partial t} + \frac{n^b_0}{a}{\bf \nabla\cdot v}
+ 3\biggl(\frac{\dot{a}}{a} \biggr)n_1^b= 0 . \label{eq:number1}
\end{equation}
These equations form a complete set describing the evolution of
a non-ideal fluid; Eq.~\ref{eq:general_energy} represents the
first law of thermodynamics in local form,
Eq.~\ref{eq:general_momentum} is the relativistic version of
Euler's equation, and Eq.~\ref{eq:number1} represents the
conservation of baryon number.
\subsection{Magneto-Hydrodynamics with Dissipation}
We now include the electromagnetic fields. In an inertial frame
(denoted by an overhat) the Maxwell tensor has the form
\begin{equation}
\hat{F}^{\mu\nu} = \left( \begin{array}{cccc}
0 & E_x & E_y & E_z \\
-E_x & 0 & B_z & -B_y \\
-E_y & -B_z & 0 & B_x \\
-E_z & B_y & -B_x & 0
\end{array} \right),
\end{equation}
where $E_i$ and $B_i$ are the electric and magnetic fields as
seen by an
observer in the inertial frame.
The Maxwell tensor
in comoving coordinates ($x^{\mu}$) can be derived
from the Maxwell tensor in inertial coordinates
($\hat{x}^{\mu}$) by using the transformation rules for
tensors
\begin{equation}
F^{\mu\nu}=\Lambda^{\mu}{}_{\lambda}\Lambda^{\nu}{}_{\sigma}\hat{F}^{\lambda\sigma}\ ,
\label{eq:transform}
\end{equation}
where
\begin{equation}
\Lambda^{\mu}{}_{\nu}={\partial x^{\mu}\over\partial \hat{x}^{\nu}}\ .
\end{equation}
The coordinate transformation which transforms the locally Minkowski metric
$\hat{g}^{\mu\nu}={\rm diag}(1,-1,-1,-1)$ into the Robertson-Walker metric
$g^{\mu\nu}={\rm diag}(1, -1/a^2, -1/a^2, -1/a^2)$ has
\begin{equation}
\Lambda^{\mu}{}_{\nu}={\rm diag}(1\, ,\, 1/a\, ,\, 1/a\, ,\, 1/a\, )\ .
\label{eq:lambda}
\end{equation}
Thus, in the comoving basis $F^{\mu\nu}$ is
\begin{equation}
F^{\mu\nu} = \left(\begin{array}{cccc}
0 & E_x/a & E_y/a & E_z/a \\
-E_x/a & 0 & B_z/a^2 & -B_y/a^2 \\
-E_y/a & -B_z/a^2 & 0 & B_x/a^2 \\
-E_z/a & B_y/a^2 & -B_x/a^2 & 0
\end{array} \right). \label{eq:fmn}
\end{equation}
The equations of motion for
the electromagnetic
fields are Maxwell's equations
\begin{equation}
F^{\mu\nu}{}_{;\nu} = 4\pi J^{\mu}
\end{equation}
and
\begin{equation}
{\partial\over\partial x^{\lambda}}F_{\mu\nu}+{\partial\over\partial x^{\nu}}F_{\lambda\mu}+{\partial\over\partial
x^{\mu}}F_{\nu\lambda}=0\ ,\label{eq:Maxwell}
\end{equation}
where $J^{\mu}$ is the electric four current.
In the limit of infinite
electrical conductivity, the electric field in the rest frame
of the charged particles vanishes
\begin{equation}
E^\mu = F^{\mu\nu} U_\nu = 0. \label{eq:mhd}
\end{equation}
This condition, evaluated in the comoving frame using
equations~\ref{eq:u} and~\ref{eq:fmn}, reads
\begin{equation}
{\bf E} = - {\bf v}\times{\bf B}. \label{eq:mhd1}
\end{equation}
We decompose the magnetic field
into its background value, ${\bf
B}_0$, and a small-amplitude perturbation, ${\bf b}({\bf x},t)$
\begin{equation}
{\bf B}({\bf x},t) = {\bf B}_0({\bf x}, t) + {\bf b}({\bf x},t),
\end{equation}
and impose
the following conditions
\begin{equation}
{\bf b}({\bf x},t) \ll {\bf B}_0({\bf x},t)\ ,
\end{equation}
\begin{equation}
\nabla\times{\bf B}_0({\bf x},t) \ll \nabla\times{\bf b}({\bf x},t)\ .
\end{equation}
to linearize our equations.
We can now derive the relevant Maxwell's equations to zeroth and first
order in the small quantities ${\bf v}$ and ${\bf b}$ by using
Eq.~\ref{eq:fmn},~\ref{eq:Maxwell},~and~\ref{eq:mhd1}. This yields
\begin{eqnarray}
\nabla\cdot{\bf b} & = & 0, \label{eq:maxw1} \\
\frac{1}{a^2} \frac{\partial}{\partial t} (a^2{\bf b})
& = & \frac{1}{a} \nabla\times ({\bf v}\times {\bf B}_0),
\label{eq:maxw2} \\
\frac{\partial}{\partial t}(a^2 {\bf B}_0) & = & 0. \label{eq:maxw3}
\end{eqnarray}
Equation~\ref{eq:maxw3} shows that the background field
${\bf B}_0$, by flux conservation, redshifts as~$1/a^2$ with the expansion
of the fluid.
To complete the system of equations needed to describe the evolution
of the fluid
in the presence of electromagnetic fields, we must add
the contribution from the
electromagnetic energy-momentum tensor $T^{\mu\nu}_{\rm EM}$
to the conservation of energy-momentum Eqs.~\ref{eq:tmn_energy}
and \ref{eq:tmn_momentum}.
The energy-momentum
tensor for electromagnetism is
\begin{equation}
T^{\mu\nu}_{\rm EM} = \frac{1}{4\pi}\Bigl\{ F^{\mu\sigma}
F^{\nu}{}_{\sigma} -
\frac{1}{4} g^{\mu\nu} F^{\sigma\rho}F_{\sigma\rho}\Bigr\},
\end{equation}
which in comoving coordinates becomes
\begin{equation}
T^{\mu\nu}_{\rm EM}= \frac{1}{4\pi}\left(
\begin{array}{cc}
A & {\bf S} \\
{\bf S} & \sigma^{ij}
\end{array}\right),
\end{equation}
with
\begin{equation}
A = \frac{\bigl({\bf E}^2+{\bf B}^2\bigr)}{2},
\quad {\bf S}=\frac{\bigl({\bf E\times B}\bigr)}{a}, \ \ {\rm
and }
\ \ \sigma^{ij}=\frac{1}{a^2}\Bigl(-E_iE_j-B_iB_j
+\frac{1}{2}\delta_{ij}\bigl({\bf E}^2+{\bf B}^2\bigr)
\Bigr).
\end{equation}
We can now evaluate the contributions from the electromagnetic stresses
to the conservation of entropy Eq.~\ref{eq:entropy0},
the first law of thermodynamics
in local form Eq.~\ref{eq:general_energy},
and the relativistic version of Euler's equation
Eq.~\ref{eq:general_momentum}.
The contribution to the left-hand-side of the zeroth order
Eq.~\ref{eq:entropy0} is
\begin{equation}
\frac{1}{8\pi a^4}\frac{\partial}{\partial t}(a^4 {\bf B}_0^2),\label{eq:mag_entropy0}
\end{equation}
while to first order in ${\bf v}$ and ${\bf b}$,
the electromagnetic stresses to be added to the left-hand-sides of
Eqs.~\ref{eq:general_energy}~and~\ref{eq:general_momentum} are
\begin{equation}
\frac{1}{4\pi}\biggl[\frac{1}{a^4} \frac{\partial}{\partial t}
\bigl(a^4{\bf b}\cdot
{\bf B}_0\bigr)
+ \frac{{\bf B}_0^2}{a} \bigl({\bf\nabla\cdot v}\bigr)
- \frac{1}{a} \bigl({\bf B}_0\cdot {\bf \nabla}\bigr)
\bigl({\bf v\cdot
B}_0\bigr)\biggr]\ . \label{eq:mag_energy}
\end{equation}
for Eq.~\ref{eq:general_energy}, and
\begin{equation}
\frac{1}{4\pi a^4} \frac{\partial}{\partial t} \biggl(a^4{\bf B}_0\times\bigl(
{\bf v} \times {\bf B}_0\bigr)\biggr)
+ \frac{1}{4\pi a} \biggl[{\bf B}_0{\bf\times}\bigl(
{\bf\nabla\times b}
\bigr)\biggr]\ , \label{eq:mag_momentum}
\end{equation}
for Eq.~\ref{eq:general_momentum}. After substituting Maxwell's
equations (Eq.~\ref{eq:maxw1}--\ref{eq:maxw3}) into
Eq.~\ref{eq:mag_energy}, we find that the electromagnetic contribution
to Eq.~\ref{eq:general_energy} is zero. Similarly,
Eq.~\ref{eq:mag_entropy0} is identical to zero by virtue of
Eq.~\ref{eq:maxw3} so that both Eq.~\ref{eq:entropy0}
and~\ref{eq:general_energy} are unmodified. The only coupling between
the field and the fluid to first order occurs through the velocity of
charged particles and the curl of the magnetic field.
Note that the first term of Eq.~\ref{eq:mag_momentum} is only
important in the relativistic limit. For a mode with frequency
$\omega$ and wavenumber $k$, Eq.~\ref{eq:maxw2} implies $\omega b\sim
k {\bar v} B$. Therefore, the relative contribution of the first term
in Eq.~\ref{eq:mag_momentum} compared to the second term is of order
$(\omega /k)^2$. Hence, the first term can only be neglected when the
group velocity of a mode, $\partial\omega / \partial k\approx \omega / k$, is much
smaller than the speed of light.
To first order in the quantities $T_1$, $n_1^b$, $\rho_1$, $p_1$, $\bf
v$, $\bf b$, equations~\ref{eq:general_energy}, \ref{eq:number1},
\ref{eq:maxw1}--\ref{eq:maxw3}, together with the equation obtained by
adding Eq.~\ref{eq:mag_momentum} to the left-hand-side of
Eq.~\ref{eq:general_momentum} describe magneto-hydrodynamics in an
expanding fluid. The following definitions help to rewrite our
equations into a more convenient form
\begin{equation}
\delta \equiv {T_1\over T_0}\ ,\quad \Delta\equiv {n_1^b\over n_0^b}\ , \label{eq:7}
\end{equation}
\begin{equation}
\tilde{\bf b}\equiv {{\bf b}\over \bigl(4\pi (\rho_r+p_r)
\bigr)^{1\over 2}}\ ,\quad
\tilde{\bf B}_0\equiv {{\bf B}_0\over \bigl(4\pi (\rho_r+p_r)
\bigr)^{1\over 2}}\ ,
\label{eq:8}
\end{equation}
\begin{equation}
\eta^{\prime}\equiv {\eta\over 3 (\rho_r+p_r)}\ ,\quad
\xi^{\prime}\equiv {\xi\over 3 (\rho_r+p_r)}\ ,\quad
\kappa^{\prime}\equiv {\kappa T\over 3 (\rho_r+p_r)}\ , \label{eq:9}
\end{equation}
where $\rho_r=g \pi^2 T_0^4/30 $ and $p_r=\rho_r/3$ are the
average energy density and pressure of relativistic particles,
and $g$ is the total statistical weight of relativistic
particles.
For a fluid comprised of baryons and relativistic particles
(e.g., photons,
neutrinos, $e^{\pm}$-pairs, etc.),
the energy density and pressure up to first order
in the small quantities are
given by
\begin{eqnarray}
\rho &=& \rho_0 + \rho_1
\nonumber\\
&=& \rho_r\bigl(1+4\delta\bigr) + \rho^b\bigl(1+\Delta\bigr) \, , \\
p &=& p_0 + p_1
\nonumber\\
&=& p_r\bigl(1+4\delta\bigr)\ ,\label{eq:pressure}
\end{eqnarray}
where $\rho_b = n^b_0 m_{\rm N}$ is the baryon energy density, and
$m_{\rm N}$ is the nucleon rest mass. In writing
Eq.~\ref{eq:pressure}, we assume that baryonic pressure is negligible
in comparison to radiation pressure. In this case, and when $\xi
=0$, Eq.~\ref{eq:entropy0} and Eq.~\ref{eq:number0} imply simple
redshift relations for the temperature $T_0\sim 1/a$ and the baryon
number density $n_0^b\sim 1/a^3$. We also define
\begin{equation}
R (t) \equiv {3\rho_b\over 4\rho_r} \label{eq:15}
\end{equation}
as a measure of the relative importance of baryon mass density with
respect to energy density in relativistic particles. For $R
\rightarrow 0$, both energy density and pressure are dominated by
relativistic particles, whereas for $R \gg 1$ the energy density is
dominated by the baryon rest mass and the pressure is dominated by
radiation.
In terms of the newly defined variables, the equations of
magneto-hydrodynamics become
\begin{equation}
\dot{\delta}+ 36 \xi^{\prime} \biggr(\frac{\dot{a}}{a}\biggr)^2 \delta + {1\over 3
a}{\bf\nabla\cdot v}
-6\biggl({\frac{\dot{a}}{a}}\biggr) {\xi^{\prime}\over
a}{\bf\nabla\cdot v}-{\kappa^{\prime}\over a^2}\nabla^2\delta
-{\kappa^{\prime}\over
a}{\partial\over\partial t}{\bf\nabla\cdot v}=0\ , \label{eq:finalfluid0}
\end{equation}
\begin{eqnarray}
\biggl({\partial\over\partial t}+36\xi'\biggr(\frac{\dot{a}}{a}\biggl)^2\biggr)
\biggl((1+R){\bf v}
- 3\kappa^{\prime}\bigl(\dot{\bf v}+{1\over
a}{\bf\nabla}\delta\bigr)
&-& 9\xi^{\prime}\Bigl({\frac{\dot{a}}{a}}\Bigr){\bf v}\biggr)
+ {1\over a}{\bf\nabla}\delta
-{3\eta^{\prime}\over a^2}
\biggl(\nabla^2
{\bf v}+{1\over 3}{\bf\nabla}\bigl({\bf\nabla\cdot v}\bigr)\biggr)
\nonumber \\
& &{} -{3\xi^{\prime}\over
a^2}{\bf\nabla}\bigl({\bf\nabla\cdot v}\bigr)
+ \tilde{\bf B}_0\times \biggl({\partial {\bf v}\over\partial
t}\times\tilde{\bf B}_0\biggr)
+{1\over a}\tilde{\bf
B}_0{\bf\times}\bigl({\bf\nabla\times\tilde{b}}\bigr)=0\ ,
\label{eq:finalfluid1}
\end{eqnarray}
\begin{equation}
{\partial\over\partial t}\tilde{\bf b}={1\over a}{\bf\nabla\times}\bigl({\bf
v\times\tilde{B}}_0\bigr)\ , \label{eq:finalfluid2}
\end{equation}
\begin{equation}
{\bf\nabla\cdot\tilde b}=0\ , \label{eq:finalfluid3}
\end{equation}
\begin{equation}
{\partial\over\partial t}\tilde{\bf B}_0=0\ . \label{eq:finalfluid4}
\end{equation}
\subsection{Dispersion Relations}
In order to calculate propagation velocities and damping rates, we
first derive the dispersion relations for the different MHD modes by
Fourier transforming all perturbative variables ($\delta$, $\Delta$,
$\bf b$, and $\bf v$, generically represented by $\Phi$ below) using
the convention
\begin{equation}
\Phi ({\bf x}, t) = \int\!d^3{\bf k}\, \Phi ({\bf k}, t)
\exp \bigl(i{\bf k}\cdot{\bf x}\bigr) \label{eq:F_transf}
\end{equation}
in which ${\bf k}$ is a constant comoving wave vector.
The time dependence of $\Phi ({\bf k}, t)$ is modified by the
expansion of the fluid which introduces a time variation into the
frequency and the amplitude of the modes. With this in mind it is
convenient to write
\begin{equation}
\Phi ({\bf k}, t) = \Phi_{\bf k}(t) \exp \Bigl[\int i\omega(t)\,
dt \Bigr]. \label{eq:time_depend}
\end{equation}
The decrease of the amplitude due to damping is included in the
exponential part through imaginary solutions for $\omega$ while the
explicit time dependence of $\Phi_{\bf k}(t)$ acounts only for the
effects of the expansion.
The system of equations resulting from the substitution of
Eq.~\ref{eq:F_transf} into
Eq.~\ref{eq:finalfluid0}--\ref{eq:finalfluid4} is solved separately
for the different MHD modes: Alfv\'en waves, for which the density and
the temperature of the fluid are uniform and the velocity of the fluid
is perpendicular to the background magnetic field (${\bf k}=k\hat{\bf
x}, {\bf B}_0= B_x\hat{\bf x}+B_z\hat{\bf z}, {\bf b}=b\hat{\bf y},
{\bf v}=v\hat{\bf y}$ and $\delta =0$); and magnetosonic waves, for
which the velocity of the fluid makes an arbitrary angle with the
background field (${\bf k}=k\hat{\bf x}, {\bf B}_0= B_x\hat{\bf
x}+B_y\hat{\bf y}, {\bf b}=b\hat{\bf y}, {\bf v}=v_x\hat{\bf x} +
v_y\hat{\bf y}$). Note that sound waves, which propagate along the
background field without affecting it (${\bf B}_0\,\|\, {\bf k}\,\|\,
{\bf v}$ and ${\bf b} =0$), are a special case of magnetosonic
waves. In all dispersion relations and their solutions, $\theta$
denotes the angle between the background magnetic
field and the wave vector.
The dispersion relation for magnetosonic modes allows two solutions:
slow magnetosonic modes and fast magnetosonic modes. Fast magnetosonic
modes are similar in nature to sound waves, while slow magnetosonic
modes are closer in character to Alfv\'en waves. This fact plays an
important role in the damping of magnetosonic waves, and is apparent
for weak magnetic fields ($\rho_B \ll \rho_{\rm fluid}$) where fast
magnetosonic modes oscillate with ${\bf v}$ almost along the direction
of ${\bf k}$ and involve oscillating density perturbations, while slow
magnetosonic modes oscillate almost perpendicularly to ${\bf k}$ and
have close to vanishing density perturbations. In the special case
${\bf k}\,\|\,{\bf B}_0$ fast magnetosonic waves become sound waves
and there are no slow magnetosonic solutions. (For a discussion of MHD
modes see, for example, Ref.~\cite{J62}.)
We obtain the dispersion relations by substituting
Eq.~\ref{eq:time_depend} into
Eqs.~\ref{eq:finalfluid0}--\ref{eq:finalfluid3}. The dispersion
relations are derived to first order in $\kappa'$, $\eta'$, $\xi'$,
which corresponds to the lowest non-trivial expansion in powers of
$l_{\rm mfp}/\lambda$ (where $\lambda$ is the wavelength of a
mode). We use the WKB approximation neglecting the time derivatives of
the Fourier amplitudes, $\partial \Phi_{\bf k}(t)/\partial t \ll
\omega$, and the time derivative of the frequency, $\partial \omega/\partial t \ll
-\omega^2$, which arises from the $\partial^2 {\bf v}/\partial t^2$ term in
Eq.~\ref{eq:finalfluid1}. This approximation is valid for modes with
oscillation frequencies much higher than the expansion rate, $
\omega \gg H $.
This procedure for Alfv\'en waves yields:
\begin{equation}
3\omega^3 \kappa'
+ i\omega^2\biggl(1+R + \tilde{B}_0^2\biggr)
+ \omega\biggl(\frac{\dot{a}}{a} R +3\eta'\biggl(\frac{k}{a}\biggr)^2\biggr)
- i\tilde{B}_0^2cos^2\theta\biggl(\frac{k}{a}\biggr)^2=0\ . \label{eq:Alfven}
\end{equation}
Complete dispersion relations for all MHD modes are, due to their
length, placed in Appendix~\ref{sec:diff_app}.
In the following two sections, we present the solutions for
oscillation frequencies and damping rates derived from the dispersion
relations in the following two limits: the oscillatory limit, when the
solution is oscillatory with an exponentially decaying amplitude; and
the overdamped limit, when the amplitude of modes exponentially
decrease without completing an oscillation.
\subsubsection{Oscillatory Limit}
The solutions to the dispersion relations, $\omega$, generally consist
of a real and an imaginary part, which represent the oscillation
frequency and the damping rate, respectively. In the oscillatory
limit, the dissipative effects are such that the fluid oscillates
many times as it damps, $ {\rm Re}\, \omega\ll {\rm Im}\, \omega$.
In this case, the dispersion relations can be solved by considering
all the viscosity and heat conductivity terms as perturbations on the
ideal fluid dispersion relation.
The solutions to the dispersion relations will be
conveniently expressed in terms of the speed of sound:
\begin{equation}
v_s = \sqrt{\biggl({\partial p\over\partial\rho}\biggr)_S}=
\frac{1}{\sqrt{3(1+R)}} \ ,
\end{equation}
and the relativistic Alfv\'en speed:
\begin{equation}
v_A = \frac{\tilde{B}_0} {\sqrt{1+R+\tilde{B}_0^2}}\ .
\end{equation}
The relativistic Alfv\'en speed includes the magnetic field energy
density in the denominator, which ensures that for strong magnetic
fields the Alfv\'en speed does not exceed the speed of light.
For all modes in the oscillatory limit we first solve the dispersion
relations for an ideal fluid and then compute the first order
contributions from the dissipative terms. For clarity, all the
solutions in their general form have been placed in
Appendix~\ref{sec:diff_app}. In this and the following section we give
the solutions for each MHD mode in the cosmologically relevant limit
of weak magnetic fields ($\tilde{B}_0 \ll 1$) and negligible redshift
terms ($\dot{a}/a$).
For weak magnetic fields, the leading terms in the frequencies for
fast magnetosonic waves do not depend either on the magnetic field
strength or on the direction of propagation, and are therefore the
same as the frequencies for sound waves:
\begin{equation}
\omega^{\rm FM}_{\rm osc}=\pm v_s\biggl(\frac{k}{a}\biggr)
+i\biggl(\frac{R^2}{2(1+R)^2}\kappa' +
\frac{2}{1+R}\eta' + \frac{3}{2(1+R)}\xi' \biggr)
\biggl(\frac{k}{a}\biggr)^2 + O(\tilde{B}_0^2).
\label{eq:fast_diff_damped}
\end{equation}
This reproduces the solution for propagation and damping of sound
waves given in Weinberg (1971). Similarly, the frequencies of slow
magnetosonic waves and Alfv\'en waves are identical to leading order
in $B_0$ and have the following form:
\begin{equation}
\omega^{\rm SM,A}_{\rm osc}=\pm v_A \cos\theta \biggl(\frac{k}{a}\biggr) + \frac{3}
{2}i{{\eta^{\prime}}\over {(1+R)}} \biggl(\frac{k}{a}\biggr)^2 + O(\tilde{B}_0^2)\
.\label{eq:freq_Alfven}
\end{equation}
The solutions show that, while for small magnetic fields the damping
of slow magnetosonic and Alfv\'en waves proceeds through shear
viscosity, fast magnetosonic waves are damped by shear and bulk
viscosity, as well as heat conductivity. Furthermore, fast
magnetosonic waves damp differently in different regimes: they damp
predominantly by heat conductivity when the matter density is larger
than the radiation density, and by shear viscosity when the radiation
density dominates. Note that after taking the time dependence of all
variables into account, the expansion of the fluid affects the
frequencies directly through $\dot{a}/a$ terms (see
Appendix~\ref{sec:diff_app}), and indirectly through the integral in
Eq.~\ref{eq:time_depend}. For instance, as in the case of sound waves
when $R \ll 1$, the oscillation frequency of a fast magnetosonic mode
with a given wavelength in a radiation dominated expansion is twice
the frequency of the same wavelength mode in a static background
metric.
\subsubsection{Overdamped Limit}
When dissipative effects become very strong, oscillations of MHD modes
are inhibited and the evolution of a given MHD mode is dominated by
the exponential decay of its amplitude with time. We seek solutions in
the extremely overdamped regime by expanding the equations in powers
of ${\rm Re}\,\omega_{\rm osc} / {\rm Im}\,\omega_{\rm osc}$, where
$\omega_{\rm osc}$ is the frequency of a wave derived in the
oscillatory limit.
In general, a dispersion relation expanded in powers of ${\rm
Re}\,\omega_{\rm osc} / {\rm Im}\,\omega_{\rm osc}$ has several
solutions distinguished in nature by their initial conditions. For
example, in the case of Alfv\'en waves, fast decaying solutions arise
from initial conditions such that when the velocities of the fluid are
damped away by shear viscosities the amplitude of the magnetic
perturbations vanish as well. In contrast, when initial conditions
generate slowly decaying Alfv\'en modes, the fluctuations are not
erased as the velocities damp to zero; after the damping of fluid
motions, the remaining magnetic forces tend to accelerate the fluid,
although inefficiently because of the strong viscous damping. Since
energy dissipation rates are proportional to the peculiar fluid
velocity, the timescale for dissipation of the magnetic field
perturbation of slow decaying modes may be extremely large.
While the amplitudes of the fast decaying modes damp at rates similar
to the ones calculated in the oscillatory regime (\S 2.3.1), the
amplitudes of slowly decaying modes decay at significantly different
rates. For weak magnetic fields ($\tilde{B}_0 \ll 1$), the decay rate
for the amplitude of overdamped slow magnetosonic modes is
\begin{equation}
\omega^{\rm SM}_{\rm od} = i \kappa^{\prime} v_A^2 \biggl(\frac{k}{a}\biggr)^2
+ i \frac{v_A^2 \cos^2\theta}
{3 \eta^{\prime}}\, ,
\label{eq:slow_diff_overd}
\end{equation}
and the decay rate for Alfv\'en modes is
\begin{equation}
\omega^{\rm A}_{\rm od} = i \frac{v_A^2 \cos^2\theta}{3 \eta^{\prime}}\, .
\label{eq:alf_diff_overd}
\end{equation}
Note that all modes with relativistic propagation velocities (${\rm
Re}\,\omega_{\rm osc} \sim k/a$) never enter the overdamped regime in
the diffusion limit. For this reason a discussion of overdamped
relativistic sound and fast magnetosonic waves is not necessary.
\section{Damping of Magnetohydrodynamic Modes in the Radiation
Free-Streaming Limit}
Slow magnetosonic and Alfv\'en modes which become overdamped during
the diffusion regime survive the damping and with the expansion of the
universe enter the free-streaming regime when the mean free path of
the decoupling particles grows to be larger than the wavelength of a
mode. In order to investigate the additional damping that these modes
undergo in the free-streaming regime, we study the general case of MHD
in an expanding fluid in the presence of a uniform background. Similar
to our analysis of the diffusion regime, we study the evolution of a
single dissipative fluid. However, in this case the fluid is comprised
of all the particles with mean free paths much shorter than the
wavelength of the MHD mode, while the decoupling particle species,
whose mean free path is now large as it decouples from the rest of the
fluid, represents a uniform background on the scales of interest. The
dissipation arises from occasional collisions of the fluid particles
with the relativistic background.
We generically define a drag coefficient $\alpha$
and a heat exchange coefficient $\gamma$ in the following way: the drag
force per unit volume on the fluid element from scattering with the
background particles is given by
\begin{equation}
{\bf f}\equiv -\alpha{\bf v}\rho_{\rm fluid}\ ,\label{eq:101}
\end{equation}
and the heat exchanged between the fluid element and the background is
\begin{equation}
\frac{\partial \rho_{\rm thermal}}{\partial t}\equiv -\gamma{T_1\over T_0}
\rho_{\rm thermal} \ .\label{eq:102}
\end{equation}
The exact form of these coefficients is obtained by calculating the
transfer of momentum and heat per scattering and averaging it over the
distribution of background and fluid particles. We presented the
coefficients later separately for neutrino decoupling and photon
decoupling.
In order to derive the free-streaming fluid equations, we use the
fluid equations developed in \S 2, as well as techniques for finding
solutions described therein. The heating rate from Eq.~\ref{eq:102} is
incorporated into Eq.~\ref{eq:general_energy}, while the drag force
from Eq.~\ref{eq:101} is added to Eq.~\ref{eq:general_momentum}
together with the magnetic field contribution from
Eq.~\ref{eq:mag_momentum}. Although the local thermodynamic
equilibrium between the fluid and the free-streaming component does
not hold in general, we consider the case in which the mean scattering
time between particles of the fluid component and the free-streaming
component is shorter than the characteristic expansion time scale of
the fluid. In this case the temperature and velocity of the background
is the same as the average temperature and velocity of the fluid. All
other assumptions, including non-relativistic fluid velocities, are
carried over from \S 2.
The resulting equations are:
\begin{equation}
{\partial\rho_1\over\partial t}+ {1\over a}(\rho_0+p_0){\bf\nabla\cdot v}+3\frac{\dot{a}}{a}(\rho_1
+p_1)=-\gamma\frac{T_1}{T_0}\rho_{\rm thermal}\ ,\label{eq:105}
\end{equation}
\begin{equation}
\frac{1}{a^4} {\partial\over\partial t}(a^4(\rho_0+p_0){\bf v})
+\frac{1}{a}{\bf\nabla}p_1 + {1\over 4\pi}\tilde{\bf
B}_0{\bf\times}\bigl(\frac{\partial {\bf v}}{\partial t} \times\tilde{B}_0\bigr)+{1\over
4\pi}\tilde{\bf
B}_0{\bf\times}\bigl({\bf\nabla\times\tilde{b}}\bigr)=-\alpha {\bf v}
\rho_0\ ,\label{eq:106}
\end{equation}
\begin{equation}
{\partial n^b_1\over\partial t}+3\frac{\dot{a}}{a} n^b_1+{n^b_0\over a}{\bf\nabla\cdot v}=0\
,\label{eq:107}
\end{equation}
and together with Maxwell's equations Eq.~\ref{eq:finalfluid2} -
Eq.~\ref{eq:finalfluid4} they form a complete set.
To derive dispersion relations for a given fluid, we have to specify
the energy density, matter density, and pressure, and substitute these
into the above set of equations. This is done in the rest of \S 3 for
two fluid combinations: a baryonic fluid with free-streaming photons;
and a fluid which consists of baryons and relativistic particles like
photons and $e^+ e^-$ pairs, in a background of free-streaming
neutrinos. All the dispersion relations as well as the solutions for
fast magnetosonic modes are given in the appendices. Here we present
the solutions to dispersion relations for slow magnetosonic modes and
Alfv\'en modes, the two modes that in the presence of weak magnetic
fields survive into the free-streaming regime before recombination.
\subsection{Neutrino Free-Streaming Limit}
Around neutrino decoupling the fluid consists of tightly coupled
baryons, photons, and $e^+ e^-$ pairs. The dominant component of the
pressure is radiation pressure
\begin{equation}
p_r=\frac{1}{3} \rho_r ,
\end{equation}
and, since $R\simeq 0$, the speed of sound is
\begin{equation}
v_s = \frac{1}{\sqrt{3}}\ .
\end{equation}
The energy density (equal to $\rho_{\rm thermal}$) has
contributions from all relativistic particles, counted in the number
of degrees of freedom $g_r$,
\begin{equation}
\rho_r = g_r\frac{\pi^2}{30}T^4.
\end{equation}
The heat exchange coefficient and drag coefficient defined by
Eq.~\ref{eq:102} and Eq.~\ref{eq:101} may be computed
by averaging the transfer of energy in each scattering between a
distribution of background particles and a distribution of fluid particles
\cite{JF}. For neutrino free-streaming
they have the form:
\begin{equation}
\gamma \simeq \sigma_{w} n_{w} \frac{\rho_\nu}{\rho_\gamma} =
\frac{g_{\nu}}{g_r l_{\nu}} \, ,
\label{eq:nu_alpha}
\end{equation}
and
\begin{equation}
\alpha \simeq \gamma\, ,
\label{eq:nu_gamma}
\end{equation}
where the second equation follows on dinensional grounds.
Here $\sigma_{w}$ is the cross section for scattering of neutrinos
with other weakly interacting particles, $n_{w}$ is the number density
of weakly interacting particles (scatterers), $g_{\nu}$ is the neutrino statistical
weigth, and $l_{\nu}$ is the neutrino mean free path.
Following the steps used in the diffusion regime (\S 2.3), we obtain
dispersion relations for the different MHD modes and present them in
Appendix~\ref{sec:nu_fs_app}.
\subsubsection{Oscillatory Limit}
The oscillation frequencies and damping rates for slow magnetosonic
and Alfv\'en modes are again obtained by first solving the dispersion
relations for an ideal fluid and then solving for the first order
dissipative terms. In terms of the previously defined Alfv\'en speed,
the solutions for small magnetic fields are the same for slow
magnetosonic and Alfv\'en modes and have the form:
\begin{equation}
\omega^{\rm SM,A}_{\rm osc} = \pm v_A \cos\theta \biggl(\frac{k}{a}\biggr) +
\frac{3i}{8}\alpha \, .
\end{equation}
The frequencies for oscillatory fast magnetosonic waves (including
sound waves) are presented in Appendix~\ref{sec:nu_fs_app}.
\subsubsection{Overdamped Limit}
As in the diffusion regime, the solutions in the extremely overdamped
regime (${\rm Re}\, \omega_{\rm osc} \ll {\rm Im}\, \omega_{\rm osc})$
are derived by expanding the equations in powers of ${\rm Re}\,
\omega_{\rm osc} / {\rm Im}\, \omega_{\rm osc}$. The overdamped
solution in the case of weak magnetic fields is, for slow
magnetosonic modes
\begin{equation}
\omega^{\rm SM}_{\rm od} = \frac{i}{4}\gamma v_A^2 +
\frac{4i v_A^2 \cos^2\theta}{3\alpha} \biggl(\frac{k}{a}\biggr)^2,
\label{eq:1400}
\end{equation}
and for Alfv\'en waves:
\begin{equation}
\omega^{\rm A}_{\rm od} = \frac{4i v_A^2 \cos^2\theta}{3\alpha}
\biggl(\frac{k}{a}\biggr)^2
\,. \label{eq:1401}
\end{equation}
As in the case of radiation diffusion (\S 2.3), relativistic fast
magnetosonic modes do not become overdamped in the free-streaming
regime.
\subsection{Photon Free-Streaming Limit}
Fluid equations for the modes in the free-streaming limit around
photon decoupling are somewhat different from the cases analyzed
so far. Namely, the only contribution to the energy density of the
perturbations is the thermal energy density of baryons,
\begin{equation}
\rho_{\rm thermal} = \frac{3}{2}(n_e+n_p)T = 3 n_b T,
\end{equation}
which enters Eq.~\ref{eq:105} but can be neglected in Eq.~\ref{eq:106}
because it is much smaller than the matter density. Here $n_e$ and
$n_p$ are electron and proton number densities respectively.
Furthermore, since the photons can be considered decoupled on the
free-streaming scales, the only pressure left to support the
oscillations is the pressure of the baryonic fluid itself:
\begin{equation}
p_b = (n_e + n_p) T = 2 n_b T.
\end{equation}
This yields the gradient of pressure in Eq.~\ref{eq:106} which depends
both on density and temperature fluctuations and is best expressed
through the sound speed:
\begin{equation}
\frac{1}{\rho} {\bf \nabla}p = \frac{3}{5} c_s^2 ({\bf \nabla}\Delta +
{\bf \nabla}\delta),
\end{equation}
where $c_s$ is the adiabatic baryonic speed of sound for a fully
ionized proton-electron fluid
\begin{equation}
c_s=\sqrt{\biggl({\partial p_b\over\partial\rho_b}\biggr)_S}= \sqrt{{10\over 3}{T\over
m_p}}\ . \label{eq:108}
\end{equation}
With these substitutions for the densities and the pressure, we obtain
dispersion relations for different MHD modes and present them in
Appendix~\ref{sec:ph_fs_app}.
The drag and heat exchange coefficients, which appear in the
dispersion relations and their solutions, are similarly obtained as in
the neutrino free-streaming case, and have the following form \cite{P65}:
\begin{equation}
\alpha \simeq \sigma_{\rm T} n_e \frac{\rho_\gamma}{\rho_b} =
\frac{1}{l_\gamma R},
\label{eq:ph_alpha}
\end{equation}
and
\begin{equation}
\gamma \simeq \frac{m_p}{m_e} \alpha.
\label{eq:ph_gamma}
\end{equation}
\subsubsection{Oscillatory Limit}
Unlike in the relativistic cases when the photon pressure dominates,
the structure of the non-relativistic equations with free-streaming
photons allows for oscillating magnetosonic modes with two different
propagation velocities and damping rates. These modes are commonly
referred to as adiabatic and isothermal, depending if heat transport is
slow or rapid compared to the oscillation time: a mode is adiabatic
when $\omega \gg \gamma$ and $\omega \gg \alpha$, and it is isothermal
when $\gamma \gg \omega \gg \alpha$. Alfv\'en modes have only one
solution since they do not include density or temperature fluctuations
and therefore are not affected by heat transport.
Again we derive the dispersion relations from
Eq.~\ref{eq:finalfluid2}--\ref{eq:finalfluid4} and
Eq.~\ref{eq:105}--\ref{eq:107} and place it in Appendix~\ref{sec:ph_fs_app}. Before
presenting the solutions for slow magnetosonic and Alfv\'en modes, it
is useful to introduce the non-relativistic Alfv\'en speed,
\begin{equation}
c_{A} = \frac{\tilde{B}_0} {\sqrt{R}}\ .
\end{equation}
The oscillation frequency and the damping rate for slow magnetosonic
waves in the adiabatic regime ${\rm Re}\,\omega\gg\gamma,\alpha$ are
given by:
\begin{equation}
\omega^{\rm SM}_{\rm osc}=\pm c_{A} \cos\theta \biggl(\frac{k}{a}\biggr)
+i\left({\alpha\over 2}+ {\gamma \over 5}{{\tilde
B}_0 \sin^2 \theta \over {c^2_s R}}\right) \, ,
\label{eq:116}
\end{equation}
and
\begin{equation}
\omega^{\rm SM}_{\rm osc}=\pm c_s \cos\theta \biggl(\frac{k}{a}\biggr)
+i\left({\alpha\over 2} + {\gamma \over 5}\right)\, ,
\label{eq:117}
\end{equation}
where the upper solution corresponds to weak magnetic fields such that
$c_s\gg c_A$ and the lower solution to strong magnetic fields with
$c_s\ll c_A$. It is important to remember that both of these solutions
are derived for a background magnetic field whose energy density is
much smaller than the energy density in photons. The condition for
adiabaticity is dependent on the strength of the magnetic field since
oscillation frequencies of magnetosonic waves are different for strong
and weak magnetic fields. Slow magnetosonic modes in the isothermal
regime, $\gamma\gg {\rm Re}\,\omega\gg\alpha$, have the solution
\begin{equation}
\omega^{\rm SM}_{\rm osc}=\pm c_{A} \cos\theta \biggl(\frac{k}{a}\biggr)
+i{\alpha\over 2},\label{eq:118}
\end{equation}
and
\begin{equation}
\omega^{\rm SM}_{\rm osc}=\pm \sqrt{3\over 5} c_s \cos\theta \biggl(\frac{k}{a}\biggr)
+i{\alpha\over 2},\label{eq:119}
\end{equation}
where again the upper solution is for $c_s\gg c_A$ and the
lower solution is for $c_s\ll c_A$.
The result for Alfv\'en waves is
\begin{equation}
\omega^{\rm A}_{\rm osc} =\pm c_{A} \cos\theta \biggl(\frac{k}{a}\biggr)
+i \frac{\alpha}{2}.\label{eq:120}
\end{equation}
The frequency of non-relativistic fast magnetosonic waves in the
adiabatic and isothermal limits are placed in
Appendix~\ref{sec:ph_fs_app}.
\subsubsection{Overdamped Limit}
The frequencies for slow magnetosonic waves corresponding to slow
exponential decay are:
\begin{equation}
\omega^{\rm SM}_{\rm od}=i{c_A^2 \cos^2\theta \over \alpha}
\biggl(\frac{k}{a}\biggr)^2\ , \label{eq:1300}
\end{equation}
and
\begin{equation}
\omega^{\rm SM}_{\rm od}=i{3c_s^2 \cos^2\theta \over 5\alpha}
\biggl(\frac{k}{a}\biggr)^2\ , \label{eq:1301}
\end{equation}
with the upper solution for $c_A\ll c_s$, and the lower solution for $c_A
\gg c_s$. The slowly decaying mode of overdamped Alfv\'en waves has
frequency
\begin{equation}
\omega^{\rm A}_{\rm od}=i{c_A^2 \cos^2\theta \over \alpha} \biggl(\frac{k}{a}\biggr)^2\ .
\label{eq:130}
\end{equation}
In contrast to relativistic MHD, some non-relativistic fast
magnetosonic modes enter the overdamped regime. Their damping rates
can also be found in Appendix~\ref{sec:ph_fs_app}.
\section{Damping of Magnetic Fields in the Early Universe}
In this section we discuss the implications of the damping of
magneto-hydrodynamic modes for the evolution of cosmological magnetic
fields. Magnetic fields generated in the early universe are likely to
be randomly oriented, spatially varying fields with small coherence
lengths, usually of the order of the horizon at the epoch when the
fields were created. We assume that the magnetic fields are created
with magnetic field energy below equipartition with the radiation
energy density, i.e. $\tilde{B}_0\ll 1$.
For an arbitrary magnetic field configuration, we
choose a separation of scales such that in a given volume the field
can be described as an approximately force-free background magnetic
field ${\bf B}_0$, and a spectrum of propagating modes ${\bf b}({\bf
k})$, where $|{\bf b}| \ll |{\bf B}_0|$. In this case we can decompose
the propagating modes into slow and fast magnetosonic, and Alfv\'en
modes with different wave vectors $\bf k$ and different phases.
Although the condition $|{\bf b}| \ll |{\bf
B}_0|$ may not be easily achieved for every field configuration, the
predicted evolution of propagating modes is indicative of the general
field evolution. In particular, the efficient viscous damping
discussed in this paper should cause the dissipation of magnetic energy in
generic field configurations.
We are interested in the evolution of individual MHD modes from before
the epoch of neutrino decoupling to recombination. For each epoch we
wish to determine the characteristic scales over which pre-existing
cosmic magnetic fields are damped. As previously described, the
evolution of fast magnetosonic waves is distinctively different from
the evolution of slow magnetosonic modes and Alfv\'en waves; therefore
they are discussed separately: fast magnetosonic waves in \S 4.1, and
slow magnetosonic and Alfv\'en waves in \S 4.2.
For the calculation of the damping scales we need the expressions for
the mean free path of the decoupling particles as well as the ratio of
the baryon density to the photon density. While the universe cools
from temperatures below the electroweak breaking scale ($T \sim$ 100
GeV) to neutrino decoupling ($T_{\nu} \sim$ MeV), neutrinos are the
particles with the longest mean free path and therefore the most
efficient momentum and heat transporters. The neutrino mean free path
at temperature $T$ can be written as
\begin{equation}
l_{\nu}(T)\simeq {1\over G_F^2T^2(n_l+n_q)}
\simeq 10^{11} {\rm cm} \,\biggl({T\over{\rm MeV}}\biggr)^{-5}
\biggl(\frac{g_l+g_q}{8.75}\biggr)^{-1}\,,
\label{eq:nu}
\end{equation}
where $n_l$ and $n_q$ are lepton and quark number
densities, $G_F$ is Fermi's constant, and $G_F^2T^2$ is a typical
weak interaction cross section. The quantities $g_l$ and
$g_q$ are the statistical weights of relativistic weakly interacting
leptons and quarks present at the epoch of interest.
At temperatures below the completion of the $e^+e^-$-annihilation
($T\sim 20$ keV), heat and momentum are most efficiently transported
by photons. The dominant process limiting the photon mean free path
during this period is Thomson scattering of photons off electrons
which gives the photon mean free path the following form:
\begin{equation}
l_{\gamma} (T) \simeq {1\over \sigma_{T}n_e}
\simeq 10^{22} {\rm cm} \,
\biggl({T\over 0.25{\rm eV}}\biggr)^{-3}
\biggl({\Omega_b h^2\over 0.0125}\biggr)^{-1}
X_e(T)^{-1}\,, \label{eq:gamma}
\end{equation}
where $X_e$ is the number of free electrons per baryon, $\sigma_{T}$
is the Thomson cross section, and $n_e$ is the electron density.
The damping of MHD waves is particularly efficient during the epochs
of neutrino decoupling and recombination, when efficient momentum
transfer and heat transport arise from the quickly growing mean free
path of the decoupling particles. Therefore, all the variables in the
above equations have been scaled to their values at those
epochs. Also, since at neutrino decoupling the dominant scattering
process is scattering of neutrinos off leptons, the appropriate values
for the parameters in Eq.~\ref{eq:nu} are $g_l=8.75$ ($e^{\pm}$ and
six neutrino species) and $g_q=0$. The remaining unspecified
parameter, the ionization fraction $X_e$, drops
within a short time from 1 to $\sim
10^{-5}$ during recombination which occurs approximately at
$T_{\gamma}^d \simeq 0.25$ eV.
The baryon mass density is negligible when compared to the radiation
energy density during neutrino decoupling ($R\approx 0$), while around
recombination it is approximately given by
\begin{equation}
R={3\rho_b\over 4\rho_r}
\approx 0.4 \biggl({T\over 0.25 {\rm eV}}\biggr)^{-1}
\biggl({\Omega_b h^2\over 0.0125}\biggr),
\label{eq:22}
\end{equation}
where $\Omega_b$ is the fractional contribution of
baryons to the closure density and $h$ is the present Hubble
constant in units of 100 km s$^{-1}$ Mpc$^{-1}$.
In writing Eq.~\ref{eq:22} we implicitly assume that neutrinos have
decoupled from the remaining particle species.
\subsection{Damping of Fast Magnetosonic Waves in the Early Universe}
The damping of all fast magnetosonic modes is to leading order
equivalent to the damping of sound waves if the energy density in the
large-scale magnetic field is much smaller than the energy density in
radiation. The damping occurs due to the diffusion of either neutrinos
prior to neutrino decoupling, or photons before recombination. From
the expressions in Eqs.~\ref{eq:nu} and \ref{eq:gamma}, which
represent mean free paths in proper units, it can be seen that in both
cases the comoving mean free path grows with the expansion of the
universe. As a consequence, MHD modes with wavelengths $\lambda_p = 2
\pi a(T)/k$ smaller than the mean free path $l_{\rm mfp} (T)$ at time
$t(T)$ were in the diffusion regime at some prior time in the early
universe, i.e. $\lambda_p(T') \gg l_{\rm mfp} (T')$ at $t(T')$ .
The amount of damping that fast magnetosonic modes undergo in the
diffusion regime can be calculated using the damping rates in
Eq.~\ref{eq:fast_diff_damped}. From the leading damping term, $2i
\eta^{\prime} (k/a)^2/(1+R)$, and the definitions in Eqs.~\ref{eq:9}
and \ref{eq:shear}, we see that the amplitude of the perturbation is
damped between time $t=0$ and $t$ by
\begin{equation}
\exp\biggl(-\int_0^t\, \frac{l_{\rm mfp}}{\lambda_p^2}
{\rm d}t^{\prime}\biggr).
\end{equation}
If we define a characteristic damping scale as the largest comoving
wavelength of an MHD mode whose initial amplitude has been damped by
at least one e-folding by time $t$, this damping scale approximately
corresponds to the comoving photon/neutrino diffusion length,
\begin{equation}
d^2 = \int_0^t {l_{\rm mfp}(t^\prime)\over
a^2(t^\prime)} {\rm d}t^\prime,
\end{equation}
which is the distance a photon/neutrino has random walked between time
$t=0$ and $t$.
In this section we present all the damping scales in comoving units,
in particular, comoving to the present epoch,
unless stated otherwise. The diffusion damping scale calculated for
fast magnetosonic waves prior to neutrino decoupling ($T \geq 1$ MeV)
is:
\begin{equation}
\lambda^{\rm FM}_{\nu}\simeq 2\times 10^{20} {\rm cm} \,
\biggl({T\over {\rm MeV}}\biggr)^{-{5\over 2}}
\biggl({g_{r}\over 10.75}\biggr)^{-{3\over 4}}
\biggl({g_{\nu}\over 5.25}\biggr)^{1\over 2}
\biggl({g_l+g_q\over 8.75}\biggr)^{-{1\over 2}}
\, .
\label{eq:30}
\end{equation}
Note that the damping scale at neutrino decoupling converted to
proper units approximately corresponds to the causal horizon at that
time, $\lambda_\nu ({\rm MeV}) \approx 5\times 10^{10} {\rm cm}\,$.
In a similar fashion we can compute the comoving damping scale of fast
magnetosonic waves due to the effects of the finite photon mean free
path at lower temperatures. Around recombination, where we assume
matter domination, this damping scale is
\begin{equation}
\lambda^{\rm FM}_{\gamma}\simeq 7\times 10^{25} \, {\rm cm}
\biggl({T\over {\rm 0.25 eV}}\biggr)^{-{5\over 4}}
\biggl({\Omega_b h^2\over 0.0125}\biggr)^{-{1\over 2}}
\bigl(\Omega_0h^2\bigr)^{-{1\over 4}}\,.
\label{eq:31}
\end{equation}
In this expression $\Omega_0$ is the total density in units of the critical
density at the present epoch.
Baryonic mass scales can be associated with the damping length scales
by defining
\begin{equation}
M \equiv {4\pi\over 3}\rho_b(T)\biggl({\lambda(T) a(T)\over 2}\biggr)^3\ ,
\end{equation}
where $\rho_b$ is the average baryon mass density at temperature
$T$. The baryonic mass scale associated with the damping
scale of fast magnetosonic waves by neutrino diffusion
around neutrino decoupling is
\begin{equation}
M^{\rm FM}_{\nu}\simeq 10^{-4} {\rm M}_{\odot} \,
\biggl({T_{\nu} \over {\rm MeV}}\biggr)^{-{15\over 2}}
\biggl({g_{r}\over 10.75}\biggr)^{-{9\over 4}}
\biggl({g_{\nu}\over 5.25}\biggr)^{3\over 2}
\biggl({g_l+g_q\over 8.75}\biggr)^{-{3\over 2}}
\biggl({\Omega_bh^2\over 0.0125}\biggr) \, ,
\label{eq:32}
\end{equation}
and by photon diffusion around recombination
\begin{equation}
M^{\rm FM}_{\gamma}\simeq 10^{13} {\rm M}_{\odot} \,
\biggl({T_{\gamma} \over {\rm 0.25 eV}}\biggr)^{-{15\over 4}}
\biggl({\Omega_b h^2\over 0.0125}\biggr)^{-{1\over 2}}
\bigl(\Omega_0h^2\bigr)^{-{3\over 4}} \, .
\label{eq:33}
\end{equation}
The above calculations for the damping scales are approximate in that
we assume that the damped modes are in the diffusion regime,
$l_{\nu,\gamma}\ll \lambda_p$. The diffusion approximation is not a
valid approximation late within the decoupling epochs. Around neutrino
decoupling, while our analysis uncovers the order of magnitude of the
damping mass scale, an improvement on the diffusion approximation
only, would not yield more accurate results since the calculated
damping scale approximately corresponds to the causal horizon at that
time. Around recombination, our treatment is analogous to the
calculations of the damping of sound waves in Ref.~\cite{W71}; in
writing Eq.~\ref{eq:31} we assume an instantaneous recombination while
the ionization fraction $X_e$ decreases gradually to zero during
recombination. More detailed treatments for the damping of sound waves
during recombination have been performed using the Boltzman equation
\cite{PY,P81,JW} or using a two-fluid model
\cite{BL,PV}. Since the dominant damping terms of fast
magnetosonic waves are the same as those of sound waves, values for
the sound wave damping scale calculated in more accurate models may be
used as better approximations to the fast magnetosonic damping scale.
A review of previous Silk scale calculations is given in Ref.~\cite{JW}.
\subsection{Damping of Slow Magnetosonic and Alfv\'en Waves in the Early
Universe}
Unlike the damping of fast magnetosonic waves, the damping of slow
magnetosonic and Alfv\'en waves in the early universe proceeds through
several different stages. We illustrate these stages by following an
Alfv\'en wave from after neutrino decoupling up to recombination.
Initially, in the diffusion regime where $\lambda \gg l_{\rm mfp}$, a
wave oscillates and damps in the same manner as described for fast
magnetosonic waves. The oscillation frequency and the damping rate are
shown in Eq.~\ref{eq:freq_Alfven}, and after using Eqs.~\ref{eq:9} and
\ref{eq:shear} they approximately become
\begin{equation}
\omega^{\rm A}_{\rm osc}\simeq\pm v_A\biggl(\frac{k}{a}\biggr)
\cos\theta+il_{\gamma}\biggl(\frac{k}{a}\biggr)^2
\, .\label{eq:301}
\end{equation}
The damping rate in this expression is the same as the damping rate
for fast magnetosonic modes, and is valid as long as $ v_A\cos\theta\gg
l_{\gamma}k/a$. The crucial difference, however, is that for a small
background magnetic field in the early universe, the oscillation
frequency of an Alfv\'en mode ($v_A k/a$) is much smaller than the
oscillation frequency of a fast magnetosonic mode with the same
wavelength ($v_s k/a$). While all fast magnetosonic modes of interest
satisfy the condition for damping in the oscillatory regime ($v_s \gg
l_{\gamma} k/a$), an Alfv\'en mode can become overdamped when, with
the expansion of the universe, the mean-free-path becomes large enough
for dissipative effects to overcome the oscillation ($v_A cos\theta
\simeq l_{\gamma}k/a$). One may
define a temperature dependent wavelength $\lambda_{\rm od}$,
\begin{equation}
\lambda_{\rm od}(T,B_0)\simeq {2\pi l_{\gamma}(T) \over
v_A\cos\theta}\ ,\label{eq:305}
\end{equation}
such that modes with proper wavelength $\lambda_p \ge \lambda_{\rm
od}$ are oscillatory while modes with $\lambda_p \le \lambda_{\rm od}$
are overdamped.
Overdamped modes are a superposition of fast and slowly decaying
overdamped modes, and the relative amplitudes depend on the phase of
the mode when it becomes overdamped. Fast decaying overdamped modes
damp at rates similar to the oscillatory modes, and therefore their
damping is equivalent to the damping of fast magnetosonic modes
discussed in the previous section. In this section, we follow the
significantly different evolution of the slowly decaying overdamped
modes which experience the least damping. The amplitude of a slowly
decaying overdamped Alfv\'en mode damps with a rate
given by Eq.~\ref{eq:alf_diff_overd}, which in terms of the photon
mean-free-path is
\begin{equation}
\omega^{\rm A}_{\rm od}\simeq i{v_A^2\cos^2\theta\over l_{\gamma}}\ .
\label{eq:302}
\end{equation}
Since the damping rate is
inversely proportional to the growing mean free path, the integrated
damping rate is much smaller than the integrated damping rate of fast
magnetosonic modes during the same period. As a result, the damping
in overdamped diffusion is inefficient and the damping scales of
Alfv\'en modes at the end of the diffusion regime are
smaller than the damping scales of fast magnetosonic modes.
The left-hand-side of Fig.~\ref{fig:scales-fig} (left of the dotted
line) illustrates the evolution of Alfv\'en waves in the diffusion
regime for $\cos\theta =1$ and a background magnetic field of
$\tilde{B}_0\approx 10^{-3}$ (corresponding to $B_0\approx 3\times
10^{-9}$ Gauss at present). In the diffusion regime there are two
important temperature dependent scales: the photon diffusion length
scale (dashed line), which is the scale damped by one e-folding by the
time the universe cools to temperature $T$ provided that modes are
still in the oscillatory regime; and the overdamping length scale
given by Eq.~\ref{eq:305} (solid line), which shows the temperature at
which a mode with comoving wavelength $\lambda_c$ enters the
overdamped regime. The modes which do not damp significantly in the
radiation diffusion regime before they become overdamped, preserve
their amplitude until they reach the free-streaming regime, apart from
small additional damping during the transition itself. On the graph,
these are all the modes with comoving wavelength larger than that
given by the intersection of the solid line with the dashed
line. Therefore, the intersection of these two lines roughly
represents the largest Alfv\'en mode that is damped by one e-folding
in the diffusion regime. Its position depends on the strength of the
background magnetic field and on the angle between the field and the
wave vector.
Some overdamped modes enter the free-streaming regime
before recombination (to the right of
the dotted line in Fig.~\ref{fig:scales-fig}) as the mean free path of
the decoupling particles grows with the expansion. The dissipation
coefficients in the free-streaming regime are inversely proportional
to the mean free path unlike those in the diffusion regime. This
implies that, when a wave enters the free-streaming regime, it is
initially overdamped and becomes oscillatory when the drag force and
the heat conduction decrease as the mean free path increases. It also
implies that modes in the free-streaming regime undergo most damping
while overdamped. The damping rates during free-streaming, derived
using Eq.~\ref{eq:ph_alpha} in Eqs.~\ref{eq:120} and \ref{eq:130},
are:
\begin{equation}
\omega^{\rm A}_{\rm od}\simeq i{c_A^2}
l_{\gamma} \biggl({\rho_b\over\rho_{\gamma}}
\biggr)\biggl(\frac{k}{a}\biggr)^2 \cos^2\theta\ ,\label{eq:303}
\end{equation}
when a wave is overdamped, and
\begin{equation}
\omega^{\rm A}_{\rm osc}\simeq
\pm c_A \biggl(\frac{k}{a}\biggr) \cos\theta
+i{1\over l_{\gamma}}{\rho_{\gamma}\over\rho_b}.
\label{eq:304}
\end{equation}
when it oscillates.
Using Eq.~\ref{eq:303} in Eq.~\ref{eq:time_depend}, we find that for
Alfv\'en waves during overdamped free-streaming, the largest comoving
wavelength whose amplitude is damped by one e-folding at time $t$ is:
\begin{equation}
(\lambda_{\gamma}^{\rm A}(t))^2 = \int_0^t c_A^2 \cos^2\theta\, R(t')
{l_{\gamma}(t^{\prime})\over a^2(t^{\prime})} {\rm
d}t^{\prime} = c_A^2 \cos^2\theta\, d_{\gamma}^2(t) R(t)\ ,\label{eq:904}
\end{equation}
where $d_{\gamma} (t)$ is the comoving photon diffusion length. As we
can see, the damping depends on the strength of the background
magnetic field and the angle between the field and the wave
vector. The above damping length evaluated at recombination is the
characteristic Alfv\'en wave damping length for the free-streaming regime, because
all modes stay overdamped before recombination regardless of the
strength of the magnetic field.
The damping of Alfv\'en waves with free-streaming photons is
illustrated on the right-hand side of Fig.~\ref{fig:scales-fig}. The
dashed line shows the damping scale from Eq.~\ref{eq:904}. The solid
line marks the transition from overdamped to oscillatory behavior,
defined in the same fashion as in the diffusion regime
(Eq.~\ref{eq:305}), with the overdamped region to the left of the
line. Since all the modes that cross the dashed line before
recombination get damped by one e-folding during overdamped
free-streaming, the length scale marked by this line at recombination
represents the free-streaming damping scale for Alfv\'en modes.
In our analysis we have assumed the WKB approximation.
This approximation does not formally hold during all the discussed
epochs for every mode. In particular, the approximation breaks down
for comoving wavelengths between the damping length scale for Alfv\'en
waves in the photon free-streaming regime, $\lambda_{\gamma}^{\rm A}$,
and the photon mean free path (i.e., between the dashed and dotted
lines on the right-hand-side of Fig.~\ref{fig:WKBfigure}).
Although the damping rates
calculated in the preceding section predict no damping in this region,
some damping is in principle possible since here WKB is not a good
approximation. In Fig.~\ref{fig:WKBfigure} we present the result of a
numerical integration of the magnetic field amplitude of Alfv\'en
waves in different wavelengths through the epoch of interest. The mode
with the largest wavelength which still damps before the epoch of
recombination ($T\simeq 0.25$eV) is consistent with our analytic
estimate. Note that in this calculation we assume that recombination
never occurs (i.e., $X_e \simeq 1$ for $T\le 0.25$eV in
Fig.~\ref{fig:WKBfigure}). We have extended the calculation into this
non-physical regime to learn when the results obtained by the WKB
approximation deviate significantly from the numerical results. From
the plot we see that significant discrepancies happen only with modes
that would damp after recombination if $X_e$ was kept fixed.
The damping of slow magnetosonic waves proceeds similarly to the
illustrated damping of Alfv\'en waves except for two
differences. First, slow magnetosonic waves in overdamped diffusion
damp at a slightly different rate than Alfv\'en waves because of the
additional damping expressed through the extra term in
Eq.~\ref{eq:slow_diff_overd}. This damping is relevant only for the
waves that propagate at very large angles with respect to the
background magnetic field. Second, the damping rates for slow
magnetosonic waves during free-streaming depend on whether the
non-relativistic sound speed is larger or smaller than the
non-relativistic Alfv\'en velocity
(Eq.~\ref{eq:1300}--Eq.~\ref{eq:1301}). The damping scale when $c_s >
c_A$ is the same as the damping scale for Alfv\'en waves, while the
damping scale for $c_s < c_A$ is unique to slow magnetosonic modes.
We can define damping length and mass scales for Alfv\'en and slow
magnetosonic waves analogously to the previous section, and consider
that the waves below these scales would have dissipated by the time of
neutrino/photon decoupling. The general evolution of all the damping
length scales with temperature, followed through the different damping
stages up to decoupling, can be found in
Appendix~\ref{sec:scales_app}. In this section we present the final
damping lengths at neutrino decoupling ($T \simeq 1 {\rm MeV}$) and at
recombination ($T \simeq 0.25 {\rm eV}$), for the background magnetic
field below $3\times 10^{-9}$ Gauss, which is the current estimate of
the upper limit on magnetic fields on Mpc scales (see,
e.g. Ref.~\cite{K94}).
The final damping scale of Alfv\'en waves for this range of the
background magnetic field strengths is determined by the
free-streaming damping length. At neutrino decoupling, most modes in
free-streaming are damped while overdamped, although the largest modes
are damped just as they begin to oscillate again. This determines the
comoving damping scale at neutrino decoupling to be:
\begin{equation}
\lambda_{\nu}^{\rm A} \simeq 10^{17} {\rm cm} \,
B_9 \cos\theta
\biggl({g_{r}\over 10.75}\biggr)^{-1}
\biggl({g_{\nu}\over 5.25}\biggr)^{1\over 3}
\biggl({g_l+g_q\over 8.75}\biggr)^{1\over 3} \, .
\label{eq:Alfven-length-nu}
\end{equation}
The background magnetic field is expressed through $B_9 = B_0/(3\times
10^{-9} {\rm Gauss})$, where $B_0$ is the background magnetic field
strength scaled to the present epoch. During photon decoupling, the
largest Alfv\'en mode damped by one e-folding during free-streaming is
damped while still in the overdamped regime, and its comoving wavelength is:
\begin{equation}
\lambda_{\gamma}^{\rm A} \simeq 2\times 10^{23} {\rm cm} \, B_9 \cos\theta
\biggl(\Omega_0 h^2\biggr)^{-{1\over 4}}
\biggl(\frac{\Omega_b h^2}{0.0125}\biggr)^{-{1 \over 2}}\, .
\label{eq:Alfven-length-gamma}
\end{equation}
The baryonic mass scales which correspond to the damping length scales at neutrino and photon decoupling are
\begin{equation}
M_{\nu}^{\rm A} \simeq 10^{-13} {\rm M}_{\odot} \, B_9^3 \cos^3\theta
\biggl(\frac{\Omega_b h^2}{0.0125}\biggr)
\biggl({g_{r}\over 10.75}\biggr)^{-3}
\biggl({g_{\nu}\over 5.25}\biggr)
\biggl({g_l+g_q\over 8.75}\biggr)
\label{eq:Alfven-mass-nu}
\end{equation}
and
\begin{equation}
M_{\gamma}^{\rm A} \simeq 10^{6} {\rm M}_{\odot}\,B_9^3 \cos^3\theta
\biggl(\Omega_0 h^2\biggr)^{-{3\over 4}}
\biggl(\frac{\Omega_b h^2}{0.0125}\biggr)^{-{1 \over 2}}\,.
\label{eq:Alfven-mass-gamma}
\end{equation}
The characteristic damping length scale for slow magnetosonic waves at
neutrino decoupling is the same as the damping length of Alfv\'en
waves (Eq.~\ref{eq:Alfven-length-nu}). At recombination however, the
slow magnetosonic damping scale depends on the value of
the non-relativistic sound speed relative to the non-relativistic Alfv\'en
speed. When $c_s>c_A$ the damping scale for slow magnetosonic waves
in the free-streaming regime is the same as the damping scale for Alfv\'en
waves
(Eq.~\ref{eq:Alfven-length-gamma}). On the other hand, when $c_s<c_A$
the slow magnetosonic damping scale is:
\begin{equation}
\lambda_\gamma^{\rm SM} \simeq 3\times 10^{21} {\rm cm} \, \cos\theta
\biggl(\Omega_0 h^2\biggr)^{-{1\over 4}}\, ,
\label{eq:slow_length_ph2}
\end{equation}
which is independent of the background magnetic field strength. The
condition $c_s > c_A$ is equivalent to $B_0\mathrel{{}^<_\sim} 5\times 10^{-11}
\,(\Omega_b h^2/0.0125)^{1/2} {\rm Gauss}\,$ for the strength of the
large scale magnetic field at the present epoch, or $\tilde{B}_0
\mathrel{{}^<_\sim} 2\times 10^{-5}\,(\Omega_b h^2/0.0125)^{1/2}$ and $B_9 \mathrel{{}^<_\sim}
1.7\times 10^{-2}\,(\Omega_b h^2/0.0125)^{1/2}$ for the two different
scalings of the magnetic field strength present in our equations.
The baryonic mass scales associated with the damping lengths of slow
magnetosonic modes are:
\begin{equation}
M_{\nu}^{\rm S} \simeq 10^{-13} {\rm M}_{\odot} \,
B_9^3 \cos^3\theta
\biggl(\frac{\Omega_b h^2}{0.0125}\biggr)
\biggl({g_{r}\over 10.75}\biggr)^{-3}
\biggl({g_{\nu}\over 5.25}\biggr)
\biggl({g_l+g_q\over 8.75}\biggr)\,,
\end{equation}
and
\begin{equation}
M_\gamma^{\rm SM} \simeq \left\{ \begin{array}{ll}
\displaystyle 10^{6} {\rm M}_{\odot} \,
B_9^3 \cos^3\theta
\biggl(\Omega_0 h^2\biggr)^{-{3\over 4}}
\biggl(\frac{\Omega_b h^2}{0.0125}\biggr)^{-{1 \over 2}}
& \mbox{for $c_s > c_A$} \\ [0.2in]
\displaystyle 1 {\rm M}_{\odot} \, \cos^3 \theta
\biggl(\Omega_0 h^2\biggr)^{-{3\over 4}}
\biggl(\frac{\Omega_b h^2}{0.0125}\biggr)
& \mbox{for $c_s < c_A$}
\label{eq:slow_mass_ph}
\end{array}
\right.
\end{equation}
\section{Conclusions}
In this paper we have studied the effects of dissipation on the
propagation of MHD modes in an expanding fluid composed of matter and
radiation. We have derived the propagation velocities and damping
rates for fast and slow magnetosonic, and Alfv\'en waves in the
radiation diffusion and radiation free-streaming regimes. The derived
damping rates have general applications in magnetized relativistic
and nonrelativistic
astrophysical environments. We have applied the damping rates to the
evolution of MHD modes in the early universe to show that cosmic
magnetic fields suffer significant damping from before neutrino
decoupling to the end of recombination.
Fast magnetosonic waves are damped by radiation diffusion on all
scales smaller than the radiation diffusion length in analogy to the
propagation of sound waves in a demagnetized plasma. The
characteristic damping scales are: the horizon scale at neutrino
decoupling, $M_{\nu}^{\rm FM} \approx 10^{-4} M_{\odot}$ in baryons,
and the Silk mass at recombination, $M_{\gamma}^{\rm FM} \approx
10^{13} M_{\odot}$ in baryons. In contrast to fast magnetosonic waves,
slow magnetosonic and Alfv\'en waves reach an overdamped regime during
which the damping is not very efficient; further significant damping
occurs once the radiation is free-streaming on the scale of the
perturbation. The maximum damping scales for slow magnetosonic and
Alfv\'en modes in general depend on the magnetic field strength and
the direction of propagation with respect to the background magnetic
field. At neutrino decoupling the damping scale is $M_\nu^{\rm A,SM}
\approx 10^{-13} M_{\odot} B_9^3 \cos^3\theta$, the same for both
types of modes. At recombination, if $B_9 < 1.7\times 10^{-2}$, this
scale is $M_\gamma^{\rm A,SM} \approx 10^{6} M_{\odot} B_9^3
\cos^3\theta$, and if $B_9 > 1.7\times 10^{-2}$, the damping scale for
slow magnetosonic modes is different: $M_\gamma^{\rm SM} \approx
1M_{\odot} \cos^3\theta$. The background magnetic field strength
scaled to the present epoch is expressed in terms of $B_9 =
(B_0/3\times 10^{-9} {\rm Gauss})$, the current observational limit on
the large scale magnetic field.
Our findings have various implications for cosmological magnetic
fields. The dissipation of magnetic field energy into heat during the
epoch of neutrino decoupling ensures that most magnetic field
configurations generated prior to neutrino decoupling satisfy big bang
nucleosynthesis constraints. The observed element abundances require
that the energy density in magnetic fields be less than one-third of
the photon energy density during nucleosynthesis \cite{KSV,COST}. Even
if processes prior to neutrino decoupling generate magnetic fields
with initial energy density comparable to the photon energy density,
neutrino damping causes the magnetic energy density to decrease
substantially relative to that of radiation by the time of
nucleosynthesis.
Further dissipation before recombination lessens considerably the
magnetic field energy in primordial magnetic fields available for
generating galactic magnetic fields or density perturbations. The
damping of fast magnetosonic modes implies that a sizable fraction of
the magnetic energy density is erased up to the Silk scale.
Finally, although Alfv\'en and slow magnetosonic modes also undergo
significant damping, their damping scales
depend on the strength
and the direction of the background magnetic field
and are generally smaller than the
damping scale for fast magnetosonic modes. As long as mode
coupling is not effective, which we expect to be true for
non-oscillating modes before recombination, magnetic energy density
can be stored in Alfv\'en and slow modes on scales well below the Silk
mass. The survival of these modes may be of significance to the
formation of structure on relatively small scales. In particular,
these modes may be responsible for fragmentation of early structures
as well as seeding early star formation. We will further discuss the
consequences of the cosmological evolution of MHD modes in a
subsequent paper.
\section*{Acknowledgments}
We thank B. Berger, J. Geddes, A. Konigl, R. Kulsrud, and R. Rosner for help
throughout this project.
We are particularly grateful to B. Chandran for alerting us to the
existence of overdamped magnetohydrodynamic modes.
This work was performed, in part, under the
auspices of the US Department of Energy by the Lawrence Livermore National
Laboratory under contract number W-7405-ENG-48 and DoE Nuclear Theory
grant SF-ENG-48. It was also supported by DOE at University of
Chicago, DOE and NASA at Fermilab.
We also acknowledge the hospitality of the Aspen Center
for Physics where some of this work was performed.
\def\nature{{\rm Nature}}
\def{\rm Nuc. Phys.}{{\rm Nuc. Phys.}}
\def\nucphysa{{\rm Nuc. Phys. A}}
\def{\rm Phys. Lett. B}{{\rm Phys. Lett. B}}
\def\physrevc{{\rm Phys. Rev. C}}
\def{\rm Phys. Rev. D}{{\rm Phys. Rev. D}}
\def\sovphysjetp{{\rm Soviet~Phys.~JETP}}
\def\ptpl{{\rm Progr.Theor.Phys.Lett}}
\def{\rm Prog.Theor.Phys.Suppl.}{{\rm Prog.Theor.Phys.Suppl.}}
\def{\rm Prog. Theor. Phys.}{{\rm Prog. Theor. Phys.}}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,097
|
/*
Gradebook from Names and Scores
I worked on this challenge by myself
This challenge took me 1 hours.
You will work with the following two variables. The first, students, holds the names of four students.
The second, scores, holds groups of test scores. The relative positions of elements within the two
variables match (i.e., 'Joseph' is the first element in students; his scores are the first value in scores.).
Do not alter the students and scores code.
*/
var students = ["Joseph", "Susan", "William", "Elizabeth"]
var scores = [ [80, 70, 70, 100],
[85, 80, 90, 90],
[75, 70, 80, 75],
[100, 90, 95, 85] ]
// __________________________________________
// Write your code below.
var gradebook = {};
gradebook['Joseph'] = {};
gradebook['Susan'] = {};
gradebook['William'] = {};
gradebook['Elizabeth'] = {};
gradebook['Joseph'].testScores = scores[0];
gradebook['Susan'].testScores = scores[1];
gradebook['William'].testScores = scores[2];
gradebook['Elizabeth'].testScores = scores[3];
gradebook.addScore = function(name, score) {
gradebook[name].testScores.push(score);
};
gradebook.getAverage = function() {};
average = function average(numbers) {
var sum = 0;
for (var i = 0; i < numbers.length; i++) {
sum += numbers[i];
};
return sum/numbers.length;
};
gradebook.getAverage = function(name) {
return average(gradebook[name].testScores);
};
// __________________________________________
// Refactored Solution
var gradebook = {};
for (var i = 0; i < students.length; i++) {
gradebook[students[i]] = {testScores: scores[i]};
};
gradebook.addScore = function(name, score) {
gradebook[name].testScores.push(score);
};
average = function average(numbers) {
var sum = 0;
for (var i = 0; i < numbers.length; i++) {
sum += numbers[i];
};
return sum/numbers.length;
};
gradebook.getAverage = function(name) {
return average(gradebook[name].testScores);
};
// __________________________________________
// Reflect
// What did you learn about adding functions to objects?
// I learned that you need to add functions by using dot notation.
//
// How did you iterate over nested arrays in JavaScript?
// I basically added the array's specific index as a new property and then iterated through that in the functions that I defined later.
//
// Were there any new methods you were able to incorporate? If so, what were they and how did they work?
// I didn't used any new methods.
//
// __________________________________________
// Test Code: Do not alter code below this line.
function assert(test, message, test_number) {
if (!test) {
console.log(test_number + "false");
throw "ERROR: " + message;
}
console.log(test_number + "true");
return true;
}
assert(
(gradebook instanceof Object),
"The value of gradebook should be an Object.\n",
"1. "
)
assert(
(gradebook["Elizabeth"] instanceof Object),
"gradebook's Elizabeth property should be an object.",
"2. "
)
assert(
(gradebook.William.testScores === scores[2]),
"William's testScores should equal the third element in scores.",
"3. "
)
assert(
(gradebook.addScore instanceof Function),
"The value of gradebook's addScore property should be a Function.",
"4. "
)
gradebook.addScore("Susan", 80)
assert(
(gradebook.Susan.testScores.length === 5
&& gradebook.Susan.testScores[4] === 80),
"Susan's testScores should have a new score of 80 added to the end.",
"5. "
)
assert(
(gradebook.getAverage instanceof Function),
"The value of gradebook's getAverage property should be a Function.",
"6. "
)
assert(
(average instanceof Function),
"The value of average should be a Function.\n",
"7. "
)
assert(
average([1, 2, 3]) === 2,
"average should return the average of the elements in the array argument.\n",
"8. "
)
assert(
(gradebook.getAverage("Joseph") === 80),
"gradebook's getAverage should return 80 if passed 'Joseph'.",
"9. "
)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,575
|
Published 26th Apr 2015
3,488 Views, 283 Comments
Wonderbolt Trials and Tribulations - White Comet
Rainbow Dash
Wonderbolts
Rainbow Dash finally gets the opportunity of a lifetime; a chance to join the Wonderbolts and fulfill her lifelong dream. She presumes that from this point on, it'll be smooth sailing. She was wrong...
Wonderbolt Trials and Tribulations written by White Comet
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Dash followed her fellow trainees up the stairs. The crew seemed very well organised, considering how there were significantly more bodies on the ship than usual.
After being directed through all the twists and turns of the inner confines of the ship, they made it up a stairwell directly onto the main deck. Looking at pictures of them and seeing them from afar just didn't do the vessel justice. Although the deck was a lot more cramped that Dash initially thought it would be. Nevertheless, there was easily enough room for everypony.
Dash found an empty space and sat down on the floor, eagerly awaiting instructions.
"What do you think they've got us doing this for?" Dahlia asked, sitting down next to Rainbow Dash. "They're not gonna ask us to pilot these things are they?"
"I doubt it," Dash replied, "These things probably cost more than a hundred million bits combined. What if we crashed them?"
"That's true I guess," Dahlia agreed.
As they were conversing, an abrupt buzz from the PA system caught everypony's attention. "Good afternoon, Wonderbolt candidates. This is Admiral Swift Sprint. I'm addressing you from the bridge behind you."
His directions immediately caused everypony to look behind them and up. They had been facing the wrong way this entire time? Sure enough, a pony was standing outside the bridge structure on a platform, holding an intercom device to his mouth.
"I'm sure you're eager to learn why you're all here. It's actually fascinating what you'll be getting involved with. You see, training for the Airship Force is also very intense, much like what you all are going through right now. However, we believe we have developed a method that will make training the next generation far quicker, to a higher standard, and giving them better experience that nopony will have even imagined from a fresh-faced airpony."
Dash thought about his words. New training method? Is this something they had potentially already seen? Not by the sounds of it, since he didn't mention it being used anywhere else yet…
"I'm sure you will have noticed the abundance of unicorn crew members as you came on board. Some even wearing uniforms relating to other branches of Equestria's military."
Dash only picked up on it now because he mentioned it. She turned her head to have a look around at some of the other ponies on the deck. She was some others doing the same. He was right, there was a lot of unicorns. From all sorts of branches.
"That's because we've found a way to use unicorn magic to act to replicate the actions and damage of a live battle, without the implications. Basically a very realistic wargame. And you're here today to take part in it."
Dash felt her heart rate increase at the thought. A wargame? Against actually qualified members of the REAF? There was no way they could stand a chance against that.
"We've tested it on a small scale, and it works remarkably well. But this will be our first time testing it on a full-size mock battle. Once setup is complete, the Canterlot and Trottingham will engage in a mock battle. You will take stations on your assigned ships, and will be expected to work alongside other members of the crew. One of you will even be lucky enough to take up the role of Captain for this exercise. But that will be picked by me for the Canterlot, and Captain Breeze on the Trottingham. You will be briefed on the specifics as and when needed. In the meantime, we'll get underway to the location of the test. Good luck!" The PA clicked and crackled as he put down the receiver.
"You heard the stallion," Spitfire's voice immediately sounded before anypony even had a chance to question what was being asked of them. "Those of you assigned to the Hoofington, make your way over there. Those of you assigned here, make your way up to the bridge of the Canterlot to see where you get assigned to."
Dash stood up and started walking towards the stairs. "Who do you think will get to be the Captain?" Dahlia asked Dash as they walked up.
"I don't know. It could almost be random if Swift Sprint gets to decide it. He doesn't know any of us," Dash answered back.
"Apart from you," Dahlia motioned her hoof at Dash as she replied, "I think everypony knows you by now Dash."
Dash rolled her eyes and smiled, "I guess."
As they piled their way onto the bridge, they were directed where to stand by another pony already on there. They stood perfectly in line, awaiting additional instructions.
Dash's ears twisted to the direction of certain hoofsteps that she heard walk alongside them. She looked with her eyes but didn't move her head. Admiral Swift Sprint walked around and stood tall in front of them. He didn't say anything for a while, he just took a good look at everypony.
So...this was Torque's father. So far he had maintained a very professional stance, not really giving Dash to gauge his personality. But she was able to pick out some resemblance to Torque.
"Okay," he suddenly broke the silence, "As I said, I have to choose somepony to take command of the ship during this exercise. So who would like the honour? Raise your hoof."
Dash heard a few hooves being raised, but she didn't dare raise her own. While she wouldn't protest being put in command, this was way too much out of her comfort zone. Sure, proving she could take command of one of these behemoths probably showed a higher degree of competency to the Wonderbolts, but she wasn't sure she could do that. As enticing as it sounded, the risk was just too high for the reward in Dash's opinion.
"...Alright then," Swift Sprint spoke, moving closer to the rows of trainees, then over to his left to look over everypony volunteering. He leant into a few that held their hooves high, getting a better look over them. "You, you, and you," he pointed his hoof at each one of them, "Front and centre," he quickly motioned with his hoof for them to stand behind him.
As the pegasi he selected made their way to the front of the bridge where he pointed, he continued examining the few remaining. Dash couldn't help but notice that he seemed to have stopped at a certain point. She quickly glanced over to look where he was. He was standing over Bright Skies, giving her a good long stare. Dash cringed but looked away. She could only imagine how Bright Skies was feeling. Her confidence had grown significantly since they met, but she wasn't sure she could handle somepony as intimidating as an admiral in the REAF.
"You," Swift Sprint ordered, motioning to the others. Bright Skies gulped and made her way to the front. She wasn't even holding her hoof up to volunteer, but he seemed to pick her anyway.
Dash heard him pick a few more pegasi out before reaching her. He seemed to be doing the same with her as he did with Bright Skies. Dash just looked him in the eye and held her nerve. She knew that he knew her. As Dahlia said earlier, everypony did by now.
"You," he ordered like he did with everypony else. It didn't take as long for him to decide on Dash like it did with Bright Skies though. "You, you....and you." He made his final choices and then turned around.
He looked over the line of ponies that he had selected. He then walked up to the first pony in front on his right, standing beside him. "Name?"
"H-High Tide, sir," The stallion in question answered.
"Why should I let you command my ship High Tide?" Swift Sprint asked directly and bluntly.
"W-well...you see...I've led my roommates in some of the tests here, and I think that gives me the experience necessary...to…take…" He slowly stopped talking as he noticed Swift Sprint was walking away.
He did the same again where he stopped right next to the next pony he wanted to ask. "Name?"
"Stinger, sir," the stallion immediately replied.
"Why should I let you command my ship Stinger?" He asked again in the same blunt and direct manner.
"I can promise you that we won't lose. With my skill and strategies, I'm sure we can beat them. And I'm really good at making predictions about what they might do which will help--" Swift Sprint placed a hoof over Stinger's mouth to shut him up, then continued walking up to the next pony.
"U-u-umm--"
"Why should I let you command my ship, U-h-umm?" Swift Sprint was almost sarcastic in his mocking of the mare's hesitance.
"Well...you see...I'm...good?"
Swift Sprint let her sentence sit for a few seconds to see if she would continue. But when it became apparent she wasn't, he raised an eyebrow.
"Umm...yeah...very, very good…" She said with an awkward smile.
Swift Sprint said nothing else and just continued to walk up to Bright Skies. Dash looked over as he approached her. She was staring straight and looking like she was just trying to keep it together.
"Bright Skies, sir," She spoke with confidence, but Dash could see her subtle trembling as he stood over her.
"Why should I let you command my ship Bright Skies?"
"Because I am a very quicker thinker, think on my hooves, and can always come up with a solution immediately sir." She answered concisely.
Swift Sprint raised an eyebrow but then flattened it again. He stood there in silence for several moments, looking down at Bright Skies. Bright Skies maintained her composure and continued to stare straight ahead.
"So you can come up with solutions immediately huh? Let me rephrase my question then." He craned his neck to lower his head down to meet her at eye level. "CAN you take command of my ship, Bright Skies?"
"Yes, sir!" Bright Skies answered almost instinctively, a voice filled with confidence. Ponies like Dash who had known her for a while could tell that it was fake confidence, but anypony that's never met her before would have a hard time picking up on it otherwise.
Swift Sprint once again remained silent for several moments. Continuing to stare Bright Skies in the eyes. Bright Skies wasn't sure how much more she could take. His intimidating glare was unbearable.
He suddenly blinked and smiled. "Well done," he said, removing his cap and placing it on Bright Skies head. "Congratulations, Captain."
Dash heard a big sigh of relief come from Bright Skies. Everypony immediately turned to look at Bright Skies. The cap was comically too big for her head, but she was sure the gesture from Swift Sprint had all the effect it needed regardless.
"So it looks like Bright Skies here will be taking command of the Canterlot. Is there anypony in particular that you would like to fill a role in your command Captain?"
Bright Skies couldn't help but blush and suppress a goofy smile the best she could as he referred to her as 'Captain'. She looked over to her right, leaning forward slightly to see Rainbow Dash. "Dash, I would like you to fly the ship."
Dash blinked, "Wh-what?"
"Helm is over there, Rainbow Dash," Swift Sprint added, pointing in the direction of the helm.
"And…" Bright Skies looked over the crowd, but it didn't help that she was particularly shorter than everypony. "Where's White Comet?"
A hoof raised itself from out of the crowd, helping Bright Skies pinpoint his location. Skies looked back up at Swift Sprint. "Comet's good at making calculations on the fly. Do you have some sort of department for calculating the trajectory of the cannons?"
Swift Sprint nodded, then looked over at Comet. "White Comet, you'll be in fire control. Head to the deck, we'll direct you from there."
The bridge remained quiet for a few seconds. Swift Sprint looked back at Bright Skies. "Anypony else?"
"No sir, that's it." Bright Skies confirmed.
"Alright then," Swift Sprint looked back at the remaining ponies. "I will assign you various roles around the ship and how to get to them. Don't be afraid to ask questions to me or anypony else here. The only stupid question is the one you don't ask. Dash, Comet, make your ways to your post." He gave his final words before walking in front of the remaining crew. "Now let's see...where should I put all of you…"
As he continued with choosing various roles for everypony, Dash trotted over to the position the helmpony sat in. The most apparent indication was the wheel that stuck out in the middle. That naturally steered the ship. But what all the other levers and buttons did she had no idea.
She took a seat in the chair. Only as she sat in the position overlooking all the controls was she beginning to realise how underprepared she was for this. Luckily, as far as she was aware, nopony else had ever to operate a warship before. She was on an even level.
"Hope my seat is comfy for you."
Dash turned her head to see a stallion in a REAF uniform standing beside her. An earth pony, with blonde fur and a brown mane. "You must be Rainbow Dash. I'm Ensign Tornado. The helmspony of the Canterlot." He extended his hoof to greet her.
"Pleased to meet you," Dash replied, returning his hoofshake.
"Guess you're after a crash course in piloting an airship?" He quickly guessed.
"Well...sort of," Dash reluctantly admitted.
"Daunting, I know," Tornado replied. "I'll see what I can do. So," he reached in front of her and placed a hoof on the wheel, "You've probably seen this before if you've watched any old war movies…"
"Uhh…" Dash's mind went blank, trying to recall some.
"The Mighty Four? Five Ponies Down? Iron Stallion? A Long Way To Griffinstone?" Tornado named a few.
"Yeah, yeah, I remember them," Dash replied, certain scenes in the films coming back to her.
"Right," Tornado continued, "So this controls the rudder. Left to turn left, right to go right. Simple. And that's about as simple as it gets. Now, these levers down here to your right," he looked down at them. Dash looked down with him, "these control the pitch of the of the ship. Push them forward, nose goes down. Pull them backwards, nose goes up. Now, that's with all the levers together. You can do more advanced manoeuvres if you pull that line of levers to your left."
Dash did as he instructed and pulled them hard to the left. They had a lot of resistance to them, but she heard a somewhat satisfying mechanical clunk when she managed it.
"What this does is allow you to change the pitch of each elevator, giving you more options on how you want to ascend or descend. But most of the time you'll leave it locked together," Tornado finished by pulling them back into the locked together position.
"Now, the fun stuff is down on your left. That's the telegraphs that control the engines."
Dash looked down to her left to see a similar arrangement of levers. "It's the same concept. All normally locked together, but pull it up and towards you and you'll be able to manage the power of each engine individually. The ship has six propellers, four on the back and two up front on the bow. Handy if you have any fancy aerobatics you want to do or need to avoid a shell on a certain part of the ship quickly."
Dash smirked, "Hopefully just the latter, if I can do it."
Tornado chuckled. "Push them forward to request more power, and pull them all the way back to you for idle and reverse. You can try it now if you like, the ponies down there know not to do anything yet."
Dash gripped the combined levers and pushed it forward. As she did, she saw some red needles moving on the dials in front of her.
"Those dials just to the left of where the column of the rudder wheel sticks out indicate your engine power. The red needle indicates your requested power, and the white one indicates the actual power the engine is currently producing. The can take a little while to spool up so it can help you plan ahead. It also helps let you know if there's anything wrong with the engines."
"Good to know, good to know," Dash commented, taking in all the instructions she was being given.
As she was looking over the controls, she suddenly heard a whistle from the other side of the bridge. "Prepare to launch. Set course for the designated mock combat zone." Swift Sprint ordered.
"That's our cue," Tornado followed up. "I'll get her started quickly." He leant over Dash and started flicking switches and pressing buttons. Dash looked around at the other controls briefly while he sorted it out.
"Okay," he stood back up, "slide the telegraph into the ahead one-third position and let's get started."
Dash pushed the sticks forward until the needles pointed at the 'one third' position. She heard some faint whining from outside the cabin and felt the ship begin to accelerate slowly.
"Now pull back on the wheel, but not too much," Tornado instructed, gently tapping his hoof on another one of the dials. "This indicates the pitch angle of the ship. You probably see a theme here. Red needle indicates your requested pitch, white indicates your actual. Try to keep it around five degrees while we're picking up speed."
"Right," Dash nodded, pulling back gently on the wheel in front of her and bringing the red needle to around five on the pitch indicator.
The ship continued to pick up speed, but Dash knew they couldn't keep going in a straight line forever. "Is there something I'm forgetting here?"
"Nope, I just haven't told you this part yet." Tornado replied, "Let me help you out." He leant in and started rotating some knobs which changed some numbers on the panel in front of her. He changed the number to one thousand then pressed a red button. "This tells the ponies in the engine room what level of buoyancy they should be aiming for with the ballon. I've just told them to set it to one thousand feet."
"Okay. So is there anything else I should do?" Dash asked.
"Nope. You'll feel it in your butt when she wants to take off. When she does, gently pull back to around ten degrees." Tornado instructed, tapping on the attitude indicator again for good measure.
"Right," Dash replied, concentrating straight ahead. But as she looked out the window, she looked out of all the other ones around her and realised something. For a position that's flying the entire airship, it's not exactly in the best place. She couldn't just about tell where the front of the ship was. As for the sides, no matter how much she leaned, she couldn't see where the sides of the ship started. And there was no rear view from this cabin at all. A pony with even the most finely-tuned spatial awareness would have trouble with this. No wonder the training took so long.
Just as Tornado said, she was starting to feel her seat pushing up against her. Taking that as her cue, she pulled back gently, and to her surprise, she felt the ship bite immediately.
"Wasn't expecting that huh?" Tornado asked with a chuckle, "It's a rather unconventional method for taking off, considering we can hover up and down vertically, but I just thought it might feel more familiar to you and help you settle in."
"It did. Thanks," Dash replied, smiling.
"Helm, take us up to five thousand feet and set course zero-seven-zero." Swift Sprint ordered.
"Aye Captain," Tornado replied.
Dash quickly glanced back at Swift Sprint in the captain's chair with Bright Skies standing next to him, then back forward, looking down at the console.
"So if you look down by your right hind hoof, you'll see a pedal."
Dash quickly glanced down as she listened to Tornado's instructions. "Got it," she replied, placing her hoof over the pedal.
"If you press your hoof onto the pedal, that'll hold the wheel in place."
Dash pressed down on the pedal and felt it click under her hoof.
Tornado placed a hoof on the wheel. "Now this part is essential, don't get these two confused. To turn the boat, you TURN the WHEEL." He explained, beginning to turn the wheel to the right to put them on the required course. Dash copied his action, helping him turn the wheel. "The stick the wheel's on does tilt left and right," he gently nudged it to show her, "but that doesn't steer the ship. That rolls it. And while rolling around in the sky must be easy for a pegasus like yourself, it ain't easy for hundreds of tons of steel and iron."
Dash smirked, "Noted. Should I adjust that altitude to say five thousand?"
Tornado nodded. Dash reached forward with her hoof and twisted the knobs until it said five thousand feet, then pressed the red button.
"While that number itself helps a lot to reach and maintain said altitudes, we have a lot of control over it regardless using this." He tapped the wheel. "We'd be here forever if I were to explain the whole theory behind it and how it fits in to the laws of physics, but the short and simple is that air isn't that dense, so buoyancy in air can be quite easily overcome by other forces, like with the propellers we control...if that makes sense…"
"I get it...well, enough anyway," Dash admitted.
Tornado smiled, "It'll be easier to understand once we start doing some manoeuvres. But it wouldn't be that simple if you were in a more dense medium. Say like a submarine submerged in water."
"You study a lot of physics then?" Dash joked.
"It's something we have to keep on top of to do the job we do. It helps when you know how to get the ship out of certain situations." Tornado replied, smiling. "We're approaching five thousand, start slowly levelling off."
Dash nodded and released the pedal from under her right hind hoof. She held her hooves behind the wheel and let it fall down in a fast but controlled manner. But she quickly gripped it again when she felt herself being twisted back in her seat.
"Easy," Tornado said, taking a step back to regain his footing. "She may not be as quick as your wings, but she still don't like being handled roughly."
"Sorry," Dash replied, keeping an eye on the altimeter and pitch indicator, slowly bringing it down one degree at a time until she got it onto five thousand. "There."
"Perfect. And we're only forty degrees away from being on the correct course. When you get to about fifteen, ten degrees away, start doing the same thing you just did with the pitch, but with the rudder instead."
Dash nodded and kept an eye on the course indicator. Once it was nearing the zero-seven-zero that Swift Sprint requested, she slowly started twisting the wheel to bring the rudder back to its centre. She used the same method, one degree at a time, nice and slow, so she didn't jerk the ship again. As she reached zero-seven-zero, she quickly put the rudder back to its middle position. And the ship stayed on course. "Yeah!" She couldn't help but praise herself for what probably seemed like a simple thing for Tornado, but a reasonably complicated learning experience for her.
"Congratulations, you can fly a REAF battle-airship. Something that normally takes three to five years of training to do." Tornado replied sarcastically, then looked over his shoulder. "Steer course zero-seven-zero Captain."
"Very good, all ahead standard," Swift Sprint ordered.
"All ahead standard, aye Sir," Tornado nodded, then looked over at Dash. "You heard him."
Dash nodded and pushed the telegraph forward until the needles reached the 'standard' position for all engines. "I mean...the actual actions don't seem that hard. It's just remembering where everything is and the order to do stuff in all the time that's the hard part." She picked up what he was saying before they were given orders.
"That's why it's mostly theory for the start of training. Plus normally you get to personalise your controls a little once you've earned your wings. I'm a rather short stallion, so you probably noticed a lot of mine is bunched quite close together."
"...Yeah…" Dash replied. She hadn't actually noticed before, but now that he mentioned it...there did seem to be a lot of empty space on the furthest reaches of the panel.
"I feel sorry for whoever one of you is manning the helm of the Trottingham...if anypony…" He continued, pointing out the right window at the Trottingham. "The Ensign at the helm of the Trottingham is a unicorn. I imagine he'd have his controls scattered everywhere, not being limited by his hooves and able to use magic instead."
Dash turned her head to look at the ship as well. She had seen a lot of rich ponies with their own private yachts and airships but had never fully visualised one being used as a weapon. And she never imagined it would be more like trying to sail a ship than flying. A skyship, sailing through the air. That's what it was like. She'd never seen a battle-airship up close before today, but she remembered going on a school trip once to Baltimare, where her class were lucky enough to see a battleship in the port. These didn't resemble much of sea-going battleships though. These battle-airships seemed more like something out of one of Discord's dreams.
While the spherical balloon filled with gas was still fairly obvious in its presence, it was coated in armour, unlike a regular airship. And while a sea-going battleship could only have its cannons on its deck, for obvious reasons, these ships seemed to have them stuck on wherever there was room. Although that obviously served a tactical purpose rather than lazy design work. While a sea-going battleship's target was on a flat plane on one dimension, a battle-airships target could also be above or below them. Making turrets with cannons at funny angles necessary.
Dash looked around the bridge, noting the unicorns around them that just appeared to be...waiting. There was one standing next to Swift Sprint. He was also the only unicorn she had noticed who was wearing a cap, indicating that he must be of a higher rank. He must be the chief of the whole operation.
"How do you think this will work?" Tornado asked Dash. "With this mock battle, I mean."
Dash broke out of her thought and looked at him. "I'm not really sure. We weren't even told about this 'till lunch," she replied with a chuckle. "But I'm guessing it's got something to do with magic seeing as they're all unicorns."
"I'd presume so. I'm actually kinda excited. We rarely get to fire these cannons except on training balloons and empty ground. And since Equestria's had no major conflict in hundreds of years, knowledge of these things could eventually dry up. Hell, the grandchildren of the last ponies to hear these cannons fire in a war are probably dead by now."
Dash nodded, understanding what he was saying. "But isn't that a good thing?"
Tornado paused for a moment, then shrugged. "I guess peace in the world is always a good thing. But that doesn't mean peace will last forever. I just hope Equestria's still up to the challenge when the peace is next disturbed. But I guess that's why we're recruiting you right?" He replied, winking.
"You bet," Dash replied, smiling.
As the journey continued on, Tornado helped familiarise Dash with the remainder of the controls and what the other dials meant. Dash wasn't sure she'd remember all of them, but she knew her brain had a knack for prioritising important information over the less useful stuff in front of her. But as much as she thought her job was tough, she spared a thought for some of the other posts her roommates were doing. It sounded like Comet would be doing some cannon calculations. Sure, he was quite good at stuff like that, but if she tried to do it, she wouldn't stand a chance of hitting anything. And she still found it almost funny that Bright Skies was the one that ended up in command of it all. The timidest of all of them. Granted, her confidence had improved a lot since she first got to know her, but she was still the last pony she would've picked to be the Captain.
"Oh look, fog," Tornado said and pointed out the forward window. Dash looked in the direction of his hoof to see a little bit of fog starting to form over the horizon. "That will take you on to the Crystal Mountains. A bit further east and we could fly to the Crystal Empire."
"All stop." Swift Sprint suddenly ordered.
"All stop, Helm-aye."
"All stop, sir."
Tornado and Dash spoke at the same time. Dash looked up at him and smiled as she pulled back on the telegraph. Tornado smiled back, "The lingo doesn't matter for you I guess. But it's something drilled into us religiously during training."
Dash felt the ship slowly continue to glide forward, gradually losing speed. It was surprising how long it took the ship to slow down though. Further proving just how much of a hoof-full flying something of this size and weight could be.
As the ship eventually slowed to a walking pace, Dash looked out the window to her right to see the Trottingham alongside them, also stopping.
"It's time. Get yourself set up," Swift Sprint's voice came from behind her, causing Dash to look over her shoulder at him. He was offering the mouthpiece for the intercom system to the unicorn standing beside him.
The unicorn took the mouthpiece with his magic and held it to his mouth, clicking the button in to speak. "All MAS team unicorns, stand by to synchronise." He addressed the ship over the PA and gave the mouthpiece back to Swift Sprint.
He closed his eyes, and his horn slowly lit up in the green aurora of his magic. It got brighter and brighter until a small green ball encompassed the tip of his horn.
Suddenly, a beam of magic shot towards another unicorn at the front of the bridge. It got instantly absorbed into the horn of the unicorn, and the same orb appeared on her horn tip, but in her own blue aura. And again without warning, a beam of magic shot out of her horn and headed somewhere to the decks below, passing through the floors and walls as if they weren't there.
Dash looked over at the Trottingham and saw a similar thing happening over there. Beams of magic shooting all over the place, with the purpose of reaching the horn of another unicorn. This must be part of the process.
She sat and watched for a few more minutes before the activity seemed to subside, leaving the ships with small glows from the horns of the unicorns. She didn't say anything. She just waited patiently for orders.
"Synchronisation complete Admiral. Ready when you are." The chief unicorn said to Swift Sprint.
"Before we start, let's do a test to make sure it's working as expected." Swift Sprint instructed, collecting the intercom again and pressing a few buttons above his head. "Trottingham this is Canterlot. We have confirmation that we're ready to rumble, we're just gonna fire a test shot as you to confirm it works as expected."
"Understood Canterlot. I'll have all unnecessary crew clear the port side just in case. Out." The Captain of the Trottingham replied through a crackly radio signal, then a buzz signalled the connection cut out.
Swift Sprint flicked another switch on the intercom's control panel above his head then spoke into it again. "Fire control, this is the bridge. Fire a test round at the Trottingham using one of the twelve's."
"Right away Sir." A voice came from the other end of the radio.
Dash heard some mechanical noises coming from in front of her. She looked straight ahead to see the furthest turret, the only turret she could see, turning towards the Trottingham.
"This is where it gets interesting…" Tornado commented beside her, "All three of the turrets on the main deck have three twelve inch cannons attached to them. They can do some damage."
Dash raised her eyebrow, then looked over at the Trottingham out the window. It had the same turret layout, two at the front, one at the back behind the structure, three in total. However, they only had two cannons each. "Is the Trottingham a different ship? It's got different cannons…"
Tornado nodded, "The Trottingham is a Stunner Class battleship. While it has three fewer cannons than us, its cannons are fourteen inches instead of twelve. And it trades armour for speed and agility. If it wants to beat is, it'll probably use some unconventional tactics. But considering the CO's have both never commanded an airship, I can see both of them using pretty…'unconventional' tactics to say the least."
Dash looked back at the turret turning, then back at the Trottingham again. She saw Swift Sprint and the chief unicorn walking towards the window.
She heard the mechanical noises stop. She looked back at the turret, which was now pointing towards the Trottingham. But it didn't seem to be doing anything.
Without warning, a loud bang came from the cannon, followed quickly by another explosion sound from outside. The sound of the turret shook the whole ship. If anypony was asleep, they definitely weren't now. Dash gripped the sides of her chair as she looked over at the Trottingham. A shot from the cannon at almost point-blank range definitely showed the damage it was capable of. The front left of the ship had almost completely gone. However, things weren't all as it seemed. There were faint multi-coloured lines around the contours of where the ship was before it was shot. And she could see a few ponies still standing or floating in what looked like empty space now.
"Everything looks good to me. How about your end?" Swift Sprint asked, tapping his forehead where a horn would be if he were a unicorn.
"I see no problems either." The unicorn replied.
"Perfect. Reset and get ready for real combat." He looked over at another crew pony standing by some other controls. "Send a message to the Trottingham, tell them to proceed to the starting position."
The crew pony nodded and picked up a mouthpiece like Swift Sprint was using.
Dash heard and saw the Trottingham beginning to pick up steam and move forward. But as it was moving, she also noticed that the damage was beginning to be repaired. It was fading back to what it was before it was shot. As if by...magic.
"So that's what they can do." Tornado started speaking to her, "I knew it would be using their magic to simulate damage or conditions, but I didn't know they were able to do it that well or repair it that seamlessly."
"It's amazing if it works," Dash agreed. "And nopony gets hurt either."
Swift Sprint turned back to his chair and looked at Bright Skies, still standing by his chair. He motioned for her to take a seat. "The bridge is yours, young lady. I'm simply an advisor for you now. Once both ships reach twenty-eight thousand feet, a flare from us will signal the start of the wargame. The loser is the first one to fall beneath five thousand feet. Now let's get up there and get this underway."
Bright Skies nodded, then looked over at the helm. "Dash, take us up to twenty-eight thousand feet."
"Twenty-eight thousand…" she twisted the controls until the numbers read twenty-eight thousand, then pressed the red button. "Yes, Captain." She struggled to say that to Bright Skies with a straight face.
Tornado pressed another button, "This'll help speed things up a bit. You can read what they all do there."
Dash had a brief look over at the buttons. She saw that they were a quick set of options for immediate orders, such as 'emergency dive', 'emergency rise', 'stationary rise/descend' and a few other options to help them out.
The Canterlot rose vertically until they reached their desired height. Dash overheard Swift Sprint going through some strategic advantages and disadvantages with Bright Skies as they were ascending. As much as this height was probably quite something for the unicorns and earth ponies on the ship, Dash just felt like she was sitting at home...except on a more uncomfortable chair, and an unshiftable oily-metallic smell.
A red flare suddenly shot off the bow of the Canterlot, marking the beginning of the wargame.
"Dash, all ahead flank," Bright Skies ordered confidently.
"Got it," Dash replied, not really thinking about what she was saying, just doing what she was told. She pushed the telegraph all the way forward and felt the ship begin to accelerate. The smokestacks were behind her, but she was sure they were producing a lot for this rate of acceleration.
There was a lot more noise both mechanical and general chatter around the ship. Dash didn't pay much attention to what was being said unless it was directed at her. She had a job to do and needed to concentrate.
"Rudder full to starboard, steer course one-six-zero."
"Aye," Dash replied, twisting the wheel to turn the ship.
Tornado quickly butted in, pulling the stick towards him. "At this speed, the ships gonna wanna roll. You'll need to balance it."
Dash nodded in acknowledgement, putting some pressure behind the stick to show that she had it. After feeling it, Tornado let go and let Dash manage it.
Even for a manoeuvre that seemed as simple as that, there was a lot to keep an eye on. Dash glanced out the window a few times to watch the turret traversing, but she kept her eyes on the gauges and dials more than anything else.
"Steady course one-six-zero," Dash said, levelling off the course. Although she overshot it slightly and had to correct.
Her attention was drawn when she heard cannon shots from the Trottingham. She looked out her left window at the Trottingham steaming towards them straight on. It took every ounce of her strength not to extend her wings and smack Tornado in the chest doing so. She knew that this wasn't real, but if this were a real battle, there's a lot of unicorns and earth ponies on the ship that didn't have the option she and other pegasi have of just jumping off and flying away. Even when they were being chased down by another ship with relentless cannons and devastating firepower...the thought was slightly terrifying.
Breaking her out of her thoughts, the Canterlot immediately returned fire from one of its forward turrets. Dash thought it was loud when it was miles away, but it rattled her brain inside her skull when they fired from this close.
The whistle of a shell flying past the Canterlot a few seconds later was enough to send a shiver down her spine. Again, if it were a real battle, that would've just served as a reminder of how close they came to potentially death…
She continued keeping an eye on her own role, but she kept finding herself getting distracted. But this time, it was for a good reason,. She heard the cheers of some of the crew on the bridge. She looked again to see that they had managed to score a hit on the Trottingham. Although structurally the Trottingham seemed unaffected, there was plenty of black smoke, and a few fires Dash could see on the deck. It must've hit a more armoured section of the ship. She couldn't tell if those were real fires or magic ones like the way the damage was simulated.
"Already scored a hit? That pony you put in fire control has good judgement...or he's very good at math." Swift Sprint commented.
"Fire control, have all cannons fire one more shot each then hold fire. Await my instructions." Bright Skies spoke into the intercom, then looked at Dash. "Dash, on my mark, I want you to bank the ship to port, about twenty degrees."
Dash blinked at what she was being asked to do. "W-what? I-uh...got it." She was confused by the move but trusted Bright Skies nonetheless. She evidently had a plan in mind.
She hit the back of her head on her chair at the same time she heard an explosion from behind her.
"We're hit!" Tornado cried, picking himself up off the floor. "Check the gauges. Make sure the engines are fine."
Dash looked down at the gauges. "They're all still reading the same." She replied.
"Good, let's hope they stay that way." Tornado commented, looking back out the window again.
They continued to hold their course, but their cannons went quiet as per Bright Skies ordered. All the while the Trottingham continued to close in, still firing volleys of shots.
"Shouldn't we fire back Si--...ma'am?" A crew pony asked on the bridge.
"Not yet." Bright Skies replied, the determined look on her face indicating that she had a plan in mind. Dash could tell from her expressions and voice that she was really getting into this.
The Trottingham was starting to get very close, too close for comfort as it made avoiding action difficult for the Canterlot. The Trottingham fired some more shots at them, one of them managing to hit the aft port side of the Canterlot.
"Now?" The crew pony asked again.
"No…" Bright Skies answered, but then a thought crossed her mind. She picked up the intercom. "Fire rear turret cannons only."
Dash saw a shot from the rear of the Canterlot fly past her window towards the Trottingham. It failed to hit, overshooting slightly. The next one hit the top of the gas balloon but hit at such a shallow angle it just bounced off.
"Point all fore turrets to starboard," Bright Skies bellowed over the intercom, then looked at Dash. "Steer us twenty degrees to starboard, drop five hundred feet."
Dash nodded and turned the wheel, not too sharply as they weren't turning far, and pushed it down. Also descending at a shallow angle.
The Trottingham took one more shot at the Canterlot before it was beginning to get too close for the turrets to track. It looked like it was going to fly over the top of the Canterlot.
Dash watched as the Trottingham steamed towards up, looking like it was making the appropriate adjustments to sail over them.
"Dash, roll us twenty degrees to port now!" Bright Skies suddenly ordered.
Dash didn't even acknowledge, but somehow instinctively grabbed the stick and yanked it to the left. The roll wasn't immediately apparent, but she could tell that the ship was tilting.
She watched out the right window as the Trottingham passed over the top of them and out over their starboard side…
...Where Bright Skies had ordered the gunners to train their cannons too!
"All turrets fire on the target!" Bright Skies yelled over the intercom, jumping out of her seat to see the effect.
Dash's roll had lined the cannons up almost perfectly, little adjustment was needed to get them on course. However, what she wasn't expecting was the force when all six guns were fired at once. The noise made her ears ring as she grabbed the wheel, trying to work out what was going on. The force had made the nose of the ship point a few degrees in the opposite direction, which she was trying to correct.
When her hearing did come back a few seconds later, her ears were immediately overpowered by the sounds of cheering.
"The Trottingham has taken serious engine damage. She's almost adrift in the sky." The chief unicorn in charge of the training exercise said.
Swift Sprint looked at Bright Skies and raised an eyebrow. "I suppose you could almost consider that a victory, Captain Skies," he commented, winking. "But you might need to finish them off."
Just as he was speaking, a shot fired from one of the rear turrets on the Trottingham and hit them straight amidships. Dash cringed as the ear-piercing sound of metal crushing metal met her ears as she was rocked around by the ship.
"Match our altitude, but keep our distance. Keep firing." Bright Skies continued to order.
Dash smirked as she started to slowly raise them up. She wasn't expecting Bright Skies to sound so confident in the orders she was issuing, but it was a pleasant surprise.
The Canterlot's cannons continued raining down on the Trottingham. Dash was starting to understand the firing pattern. Instead of firing all nine guns at once, the cannons would fire one at a time. By the time the ninth cannon had fired, the first had finished reloading. This allowed for a consistent stream of shells to be shot instead of leaving them unable to shoot back for at least thirty seconds.
A loud explosion (or at least louder than she was getting used to,) almost made her jump out of her seat. She craned her neck over to her right, looking up to try and get her best view from out the window without leaving the controls.
A visible tear had formed at the back of the Trottingham that was only getting bigger. The big fire coming from the back implied that the balloon was spilling gas. The armoured section surrounding the balloon was slowly becoming detached from the rest of the ship.
Dash watched as the Trottingham begun slowly losing altitude. Her thoughts were lost amidst the cheers of all the ponies on the bridge with her celebrating.
Suddenly, the ship started to glow and return to normal. Like it did earlier after the test fire.
"Trottingham confirmed shot down." The chief unicorn spoke to Swift Sprint.
Swift Sprint nodded, then looked over at Bright Skies sitting in his chair. "Well done Bright Skies. The way you predicted the movements of the Trottingham and aligned the turrets accordingly is nothing short of something I'd expect an actual commander to be able to do."
Bright Skies smiled sheepishly, trying her best to suppress blushing. She forgot her dark fur did an excellent job at that from time to time. "T-Thank you, sir!" She chirped, saluting him and holding the pose. Only for the cap to fall down in front of her head and cover her eyes.
Swift Sprint chuffed, then looked over at Dash. "Return to starting positions. Let's go again."
"Yes, sir!" Dash replied, commanding the controls to steer the ship in the right direction.
"You're starting to get the hang of this," Tornado commented, looking down at the controls to confirm Dash was still firmly in control.
"Well, a few cannon shots helps put the pressure on I guess," Dash chuckled and shrugged.
"Always helps," Tornado continued the sarcasm.
The ships returned back to their starting positions opposite each other, getting ready for another matchup. Bright Skies looked up at Swift Sprint, who looked back at her at the same time.
"Now while that was a perfect move you made last time, your opponent won't be naive enough to fall for that twice if they've made it this close to being a Wonderbolt. Don't ever lose respect for your opponent, because that's when you end up losing." He preached to Bright Skies.
"I understand," Bright Skies replied, nodding and taking in the words of advice he was giving her. She had no doubt in her mind whoever was commanding the Trottingham was getting some advice as well, so she knew she would have to keep her wits about her and figure out some strategies of her own.
Dash waited patiently, but readily. Her hooves where she needed them to be should she need to make a sudden move. And rightly so, as once again, a red flare popped into her vision off the bow of the Canterlot. Every instinct she had told her to smash the controls like her life depended on it, but she held herself back, knowing that she would need to wait for orders.
"All ahead flank, keep heading towards the Trottingham." Bright Skies finally ordered after what felt like an eternity to Rainbow Dash. In reality, it was almost instantaneous though.
"Yes, ma'am," Dash answered back with a nod, pushing the telegraph forward and feeling the ship begin to move. It still felt bizarre to be replying back to Bright Skies as if she was actually a commanding officer, but given how realistic their wargame was, it only made it all the more fitting.
She continued on their course, all the while hearing Bright Skies issue orders to the gunners to start firing. Dash saw a few shots fly from the forward turrets, and the Trottingham replied with its own.
"Maintain this distance, steer to port." Bright Skies ordered to Dash.
"Steering to port," Dash acknowledged, rotating the wheel.
"Now that can get a bit complicated," Tornado replied, leaning into the control panel. "One of the spotters from either there," he pointed at some of the ponies holding binoculars on the bridge, "or outside will start yelling angle on the bow readings at you."
"AOB, zero degrees," a voice came from a speaker built into the control panel behind the wheel.
"Like that," Tornado added. "What you do with that is twist this compass," he pointed to the compass in question then twisted it to the zero degrees position, "then press this buttooof--"
Dash reached her hoof out to stop him falling over at the Canterlot took shell fire from the Trottingham. He was almost instantly back up unassisted though. "Press the red button, and that puts a red line on your normal compass."
Dash nodded and pushed the red button, seeing a red arrow show on her compass alongside the black one that showed the direction she was currently steering in.
"The red arrow on the compass slowly moves over time to show where you should be trying to steer to maintain a good broadside shooting angle. But it's not a hundred percent accurate, so keep your head on a swivel and listen to what the spotters tell you."
"Got it," Dash replied, correcting her course.
She kept looking to her right, checking her position knowing the machinery wasn't entirely accurate as Tornado said. The Trottingham appeared to be turning to its starboard side, away from the Canterlot.
"AOB starboard one-O-five," another voice from one of the spotters on the bridge advised Dash.
Dash plugged in the information, slowly watching the red line begin to guide her. Although she wasn't sure if she would put her entire faith in it. Tornado said that the Trottingham was more agile than the Canterlot, and he wasn't kidding. If it closed in, it could probably run rings around them if it wanted to. However, with weaker armour, it probably wouldn't last long. There was a lot to consider before blindly making a move…
The seat didn't do a very good job of holding her in during the explosions. It had no support in it at all. She found her wings doing a better job of helping her keep her balance. The last blast gave her more to think about than getting rocked about though, as she noticed one of the engine power output dials begin to drop. And she felt different subtle hints through the wheel.
"Looks like that last hit damaged an engine," Dash commented, tapping on the dial on the console.
"Yeah," Tornado agreed, "It'll cause the ship to naturally want to turn starboard unless they can get it fixed. You're gonna need to steer away from it to compensate. Or change the power for each individual engine on the port side."
Dash nodded and immediately put her options into action. She didn't need any further explanation, she was confident she knew what to do. She tried to collate the two possibilities together but didn't anticipate controlling each engine individually being so complicated.
She kept an eye out the windows on her right to try and keep an eye on the Trottingham, but it was slowly fading out of view from where the window ended.
"AH!" Dash yelped as she was all but a leg thrown out to the left of her seat as an explosion ignited right below them. She turned her head in the faint whistling of the wind and saw a smashed window with shards of glass littering the floor, slowly dissolving as the unicorn magic no longer had an effect on them.
"Two bridge spotters injured." One of the unicorns parts of the illusion set up by the en-masse magic blurted out, pointing at two ponies standing at the very front of the bridge.
One of them shrugged, "Oh no, I've been injured," he spoke in a sarcastic tone, pretending to fall to the floor dramatically.
"I think I see the light," his shipmate followed on, clutching his head. The two of them couldn't help but exchange chuckles at their top-notch acting.
"Guess somepony has to take those two boneheads to sickbay," Tornado said, looking over at Dash. "I'll leave her in your capable hooves, good luck Rainbow Dash." He didn't wait for a response form Dash before running over to his 'injured' shipmates.
Dash blinked and gulped. Was she really trusted with this? Granted, Tornado probably wasn't the only pony who could take control if things got out of hoof. But she wasn't expected to be left alone. She looked over at Tornado with a concerned look as he pulled one of his 'injured' shipmates over his shoulders to carry them.
"Just remember what I taught you and trust your instincts. You'll do fine." Tornado encouraged her as he walked past, winking.
"Dash, can we expedite the turn? Our forward turrets can't aim at those angles." Bright Skies' request drew Rainbow Dash's attention.
"On it!" Dash replied, adjusting her controls. She didn't have time to watch Tornado leave the bridge as he left his prize seat in her command.
Dash steered tighter to the right to help the forward turrets hit their mark. She kept occasionally glancing over out the windows to see the Trottingham's position relative to theirs. But she had so much more to concentrate on.
"The Trottingham's going into a climb." Dash heard one of the outer spotters announce over the radio.
"Should I follow?" Dash asked Bright Skies without hesitation. Something she probably shouldn't have done, but it just came naturally to her.
"Yes Dash, bring the nose up and climb. Try and keep the forward guns within a firing arc." Bright Skies ordered in response.
"Got it!" Dash nodded, adjusting the controls.
Once she finished her adjustments, she looked out the window again to see the Trottingham climbing. Dash was trying her best to bring the Canterlot around, but as Tornado had told her before, the Trottingham was a different class of ship that her much better manoeuvrability than them.
Dash felt herself being pulled down by gravity into the back of her chair as the ship angled upwards. She heard the hums of the various ponies around her as they had to readjust their footing. The seat itself wasn't the comfiest thing in the world, to begin with, but being forced into it was even worse. The ship was also being continuously shaken from cannon fire landing on their hull. Dash wasn't sure of the damage she couldn't see, but she knew the fire landing in front of them was mostly bouncing off.
"Fire control, Bridge, why aren't we firing back?" Bright Skies asked over the radio.
"We can't get the turrets at a perfect angle because of our own and the targets relative positions. Do you want us to take luck shots Si--...ma'am?"
"Do it!" Bright Skies replied, slamming the radio down.
The loud bursts of the guns firing was a noise Dash was getting accustomed to as she watched the shells fly towards the Trottingham. While there was one that almost hit its mark, the others were miles apart.
Another volley of shots fired upwards towards them, but the results were the same. They all missed.
"Dash, level off." Bright Skies ordered.
"What?" Dash replied in shock, looking back at her. "But I thought we needed to catch them."
"We won't. Just level out. Once we're level, change speed and horizontal direction as much as possible." Bright Skies instructed.
"...Right!" Dash nodded, a smile crossing her face as she started to understand what Bright Skies was getting at. She remembered that the Canterlot was slower than the Trottingham, and climbed slower. Dash realised that the Trottingham deliberately tried to get them to follow them, to make them slow to a crawling speed and run rings around them with their superior agility.
Dash couldn't help but let out a chuff under her breath. She had to admit, if she was commanding, there was no way she would've saw that coming. Only once Bright Skies had pointed it out to her did she realise that was the Trottingham's game plan. Bright Skies was probably the only pony in the Wonderbolts trainees that could've worked that one out.
Once Dash levelled the ship out, she started doing erratic movements as instructed. She kept changing the vessels speed and heading, along with how sharp it was turning. She couldn't directly see what was happening above them, but she was getting constant updates from the spotters to their relative position.
"The Trottingham has levelled off above us. They're out of range for our turrets." Dash heard a spotter direct over the radio.
Dash glanced up above her only to cast her eyes on the metal plating that made up the ceiling above her head. Part of her wanted to jump out to see for herself, but she also knew that wouldn't go down too well with those examining her.
She continued to hear the shell fire fall around them. None of the shells hit their marks, but it was evident that range wasn't an issue for the Trottingham. That and it had the advantage of gravity being on its side.
"Rudder amidships Dash," Bright Skies suddenly ordered. "And reduce speed down one notch."
"...Got it," Dash hesitated before answering. She had no idea where Bright Skies was going with this, but she also knew that Bright Skies was too smart to fall for their tricks. She apparently had something in mind.
Dash held the course and speed of the ship, still trying her best to compensate for the damage. "The Trottingham is beginning to dive," one of the spotters called over the radio.
"Maintain course, standby to increase speed." Bright Skies relayed.
"Got it," Dash acknowledged with a nod, leaving a hoof on the telegraph.
She saw the trails of a few shells fly in front of them, one of them impacting on the front. Dash was some splinters of wood fly in front of the glass only for it to dissolve into remnants of unicorn magic.
"Trottingham is on an intercept course, but they're coming in at a blind spot. We can't return fire," another spotter instructed over the radio.
Bright Skies picked a radio up in her hoof. "Fire control, this is the bridge. Train our rear turret about ten degrees to starboard." She ordered before putting the radio down, not waiting for a response.
The Canterlot continued to take fire as the two ships closed, one even almost knocking Dash out of her chair as it exploded very near the bridge. It left her ears ringing for a little while, but not so severely that she couldn't hear one of the unicorns calling out that some more ponies had been 'injured'. But she was more concerned about what she grabbed to stop herself from falling off. The wheel. It didn't seem to move.
She composed herself then tried to twist the wheel again. It didn't budge. It was stuck. "Bright, the rudders jammed!" Dash yelled, desperately trying her hardest to get it unjammed.
Bright Skies flinched at the news. "All ahead flank, put some distance between us." She ordered.
"Yeah," Dash replied quickly, acknowledging the order, but she was still pulling and tugging on the wheel trying to get it to turn. She used her wing to push the telegraph forward.
Another explosion shook the ship even more violently, but Dash managed to maintain her seating. But she saw the needles for the starboard engines start to fall. "Entire starboard side is dead. We've got no steering, and if I keep the remaining engines on full power, we'll just spin in circles. Looks like we're dead in the air." Dash informed with an annoyed tone.
"Dammit," Bright Skies slammed a hoof on her chair. Then grabbed a radio. "Engine room this is the Bridge. Try and get the engines back, but keep us in the fight if you can."
"Yes, ma'am!" The pony on the other end of the radio replied just before Bright Skies put it down.
"Trottingham turning to port, they're going to pass over our bow," one of the spotters informed.
Bright Skies sighed and put a hoof on her forehead. "Train our forward turrets on them and fire if possible." She instructed in a frustrated tone. It sounded like nothing was going to plan for her at the moment.
Dash just sat and did nothing, staring at the dials to keep an eye on everything. With no steering and very little thrust control, there wasn't a lot she could do but wait. It seemed like game over for them now anyway.
She watched the Trottingham pass in front of them and over to their port side. It continued firing as it passed by, starting its turn back around.
A warning light displayed on Dash's instrument panel after a shot hit. It was next to a pressure gauge. It seemed to be measuring the pressure of the gas in the balloon that helped keep them afloat. Looks like a hit had managed to damage it to the point where they were losing gas. That loss of gas would eventually translate into a loss of altitude.
"Dash, all ahead flank on whatever engines remain on the port. Spin us around." Bright Skies yelled the order at Dash.
Dash didn't even acknowledge, she just reacted instantly by punching the telegraph.
The Canterlot spun around to the point where it's rear turret was pointing towards the Trottingham. It then fired all three cannons on the turret, which the Trottingham responded with. The shells whistled past each other before landing on their intended targets.
Dash noticed the pressure loss gauge start to indicate an even higher pressure loss, but before she could read it anymore, a pulse of multi-coloured unicorn magic slowly radiated through the bridge and seemingly through the rest of the ship. The smashed glass and splintered wood were instantly restored as the magic aurora passed through it.
"Canterlot's damage unrecoverable; Trottingham wins." The chief unicorn explained.
Swift Sprint took a step forward to Bright Skies and shrugged. "Can't win them all, unfortunately," he consoled, but the smile on his face implied that he wasn't exactly disappointed with her performance.
He leant across and picked up the radio. "All hooves on both ships, this is the admiral. For our final engagement, I'm going to set some conditions. The first being that the ships can no longer come within two thousand yards of each other and the second being that I'll have our unicorn friends here limit the magazine capacities to ten percent. So be wise with your shooting, and may the best crews win. We've drawn a bit of a crowd, so be sure to put on a good show."
Dash took a look out the window to see what he meant. Sure enough, a few ships had parked themselves around the perimeter of the section of sky they were using to watch them in their mock battles. She spotted around a dozen or so, maybe more. All of various shapes and sizes.
"You did well," Tornado's voice suddenly sounded behind her.
Dash looked back at Tornado. "I guess I did alright. Engines and rudder stopped working because of damage towards the end though," she explained.
Tornado shrugged, "Not your fault. Anyway, let's get back into the start position and put on a good show."
"I just noticed that," Dash replied, going through the motions with the controls to put the Canterlot back into its starting position. "Are we really that impressive to watch?"
"It's not just that, it's the magic. This whole simulated battle using unicorn magic thing? It's never been done before. We're the pioneers. And if it catches on, this could become a new standard in training."
"I guess it's very realistic," Dash gathered. "But what do you think of these conditions? Not a lot of ammo, and no getting very close."
"Not a lot of ammo gives us a slight advantage over the Trottingham since we have a larger magazine capacity. But the two thousand yard limit gives them an advantage for out-manoeuvring our shells." Tornado explained.
"But aren't the Trottingham's guns bigger? Can't it deal more damage?" Dash questioned.
"Only if they hit their target. They do no damage if they miss. Which is what makes the lowered ammo interesting."
"How many shells do we have now then?" She asked.
"At ten percent, I'd say only about eleven or twelve shells per gun for us. For the Trottingham, that's even lower. Maybe around nine or ten at most per gun."
"I guess it'll come down to whoever has the most accurate gunners then," Dash analysed, stopping at the start position.
"That's about right. You say that stallion is pretty good at this stuff? White Comet, was that his name?"
"Yeah, he's got an eye for stuff like that. Don't think he's ever done anything like this before though."
The Trottingham stopped in its position a few miles away in front of the Canterlot. Both facing each other, ready to go.
Swift Sprint cleared his throat then came away from the window and back towards Bright Skies. "Ask anypony, and they'll all have different theories about who has the better advantage or disadvantage. But I believe your best course of action would be to eliminate the Trottingham's advantage of agility. Get as close as you can and try to immobilise them." He advised Bright Skies.
"I think I've got a plan." Bright Skies replied to him, feeling said plan begin to piece together in her mind.
Swift Sprint smiled and nodded, "Good luck," he said his final words to her before grabbing the radio. "Commence engagement," he ordered the start of the battle then put the radio down. Taking a few steps back to watch the action unfold.
"All ahead flank, take us as close as possible to the Trottingham without crossing the two thousand yard boundary." Bright Skies ordered.
"Yes, ma'am!" Dash replied with enthusiasm in her voice as she pushed the telegraph forward into its fastest setting. She was feeling good about this one. She knew that they had immobilised the Trottingham in a similar fashion in the first round, so everypony knew what they had to do.
The turrets on Canterlot kept tracking the Trottingham, but they didn't make any attempts to fire. The Trottingham was also heading straight forward towards the Canterlot as well. As predicted, they were assumably going to try and utilise their agility advantage by getting as close as possible to the Canterlot. Exactly where the Canterlot wanted them to be.
Two shots from the Trottingham fired off their forward turrets and whistled towards the Canterlot. Bright Skies stood up from her chair and looked around the bridge. Seeing what she was looking for, she walked over to a table with a map and other planning equipment on it. She took a pencil and some scraps of paper in her wings and started taking notes as she walked back to her chair.
While the first shot missed, the second shot landed directly on top of one of the turrets. Being quite heavily armoured though, it seemed as if most of the explosion just bounced off it. The smoke did make it difficult for Dash to see ahead of her though.
The Trottingham started a turn to its right, firing another volley at the Canterlot. A few shots hit, but the Canterlot seemed to take them in its stride.
"Aren't we gonna fire back?" Somepony on the bridge asked.
"Not yet," Bright Skies replied, hastily scribbling some notes down on the paper she acquired. Swift Sprint took a look over her shoulder and smirked at what she was writing.
"Full port rudder," Bright Skies ordered as she finished making notes.
"You got it," Dash replied, twisting the wheel. The G-forces upset the footing of a few ponies as they started doing a very tight turn at very high speeds. The guns did their best to maintain their tracking, but the rate they were turning at meant that they couldn't traverse fast enough to compensate.
The Canterlot turned towards the Trottingham, which was still continuing its turn away from them. Seeing this, Bright Skies immediately thought of an idea. "Rudder to starboard, about five degrees. A slow, gradual turn."
"Got it," Dash acknowledged, adjusting the rudder accordingly.
"Fire a single volley from each turret in range when the turrets are at a good angle." Bright continued to order.
As the ship levelled out, the turrets managed to regain their firing angles. Seconds later, the loud BANG of the cannons firing rang throughout the ship. The shells flew through the air and impacted on the rear of the Trottingham. Through the smoke, Dash could make out some wood splinters and metal chunks breaking off the hull.
Before they could take in the damage done to the Trottingham, it fired a return volley. One of the shots landed in the front bow beyond what Dash could see, but the crunching noises that came from the front implied it wasn't good.
A loud hissing noise overpowered the breaking machinery noises, and the turrets slowly began to droop down. Looks like whatever hit managed to break something on the turret.
"Bridge, fire control, turret one is down. The system that holds pressure to hold the turrets up at an angle has been damaged." A voice over the radio called to Bright Skies.
Bright picked the radio up. "It is fixable?"
"We're trying," the voice replied.
"Keep me informed," Bright replied, placing the radio down.
The Canterlot continued to return fire with its remaining turrets regardless. Dash noticed that the rate of fire was considerably lower, probably down to the fact that they now had a limited amount of shots and wanted to make sure each one hit its target (or did its best to).
"Trottingham's speed is decreasing," one of the spotters called. "We must've damaged their engines!"
"Perfect. Dash, bring us alongside, as close as possible within limits." Bright Skies ordered.
"Yes, ma'am!" Dash replied, bring the ship forward with the controls.
The Canterlot steamed forward, its smokestacks kicking out a lot of smoke from its acceleration that was grasped by the wind and dragged away. Dash kept her eyes peeled on the Trottingham, slowly coming towards them. But as they got closer, Dash couldn't help but notice that they were getting much closer much quicker...a lot quicker...too quickly!
"Dammit," Dash cursed under her breath, realising that they were going to overshoot the Trottingham, passing on its starboard side. She adjusted the controls and did her best to compensate, but it was too little too late. Not only that, but it looked like the Trottingham had caught onto the possibility of this happening, so they had also slowed down an awful lot.
As the Canterlot sped past the Trottingham, Dash noticed its turrets training in their direction just before it went out of her sight. She knew what was coming, and all she could do was sit and wait…
The Canterlot shook violently back and forth as some loud explosions roared from behind them. The ship shook so violently that it almost winded Dash as she was blown forward against her instrument panel. Luckily she didn't affect the stability of the ship too much with whatever controls she might have accidentally pressed.
"The Trottingham is climbing!" A spotter called over the radio.
"Climb with them, and take us hard to port!" Bright ordered.
"Aye!" Dash replied, going for a flick of the wheel in the left direction. But it wouldn't budge. "Dammit, rudders jammed!" Dash informed.
"But engines are still good," Tornado butted in to advise her. "Reverse the port engines, and that should help the ship spin."
"Right," Dash nodded in acknowledgement, looking down at the telegraphs and adjusting them to do so. While at the same time, giving the orders to raise the altitude of the ship.
Dash felt the twisting effect start to happen a few seconds after she'd set the telegraphs to reverse the port engines. It felt a lot more unnatural than using the rudder. But it didn't appear to be too much slower than the rudder either. Looks like this was the best way for her to do it until they get the rudder fixed...if the battle lasted that long…
As the Canterlot turned and climbed, the Trottingham appeared to be trying to mirror its manoeuvre. However, the damage it sustained meant that it couldn't rise as fast as the Canterlot. That didn't stop it firing its guns though, as they were still in range.
"Return fire with the rear turret only," Bright ordered.
The odd order caused more than a couple ponies to raise their eyebrows. Nevertheless, she seemed to know what she was doing, and given that she wasn't formally trained by the REAF, everypony knew that her tactics would be a little on the unconventional side. She apparently had a plan in mind.
The crew did as Bright Skies ordered and only fired with the rear turret. Since they were teetering around the very limit of the range, around twenty-five hundred - thirty-five hundred yards, accuracy wasn't a problem for either ship.
"Maintain this angle to the Trottingham," Bright continued to order. "But keep me updated on the damage situation of our port side." Her voice was starting to show some strain in it as she continued to scribble down her notes about...whatever she was taking notes on…
While the tactics for the Trottingham looked relatively straightforward on paper, they were hard to follow when the ship was now basically limping along. Whenever the Trottingham tried to break away, the Canterlot simply shadowed it. Nevertheless, the Trottingham continued to pound shells into the Canterlot's port side.
Dash felt some mechanical clunking through her controls before she started to see some of the needles dropping on her gauges. "Two port side engines are down!" She exclaimed, trying her best to compensate. There wasn't much she could do with a jammed rudder though.
"Do your best to stay on course. We don't need to stay here much longer." Bright tried her best to encourage her. "I hope…" She muttered under her breath, continuing to scribble her notes. She seemed to be using the rocking of the ship from shell fire as a prompt to start writing.
The Canterlot's speed began to drop off, and it slowly began to veer closer towards the Trottingham. Dash instinctively dropped the speed on the remaining engines to stop them crossing the two thousand yard limit. She wasn't sure exactly how close they were, but they were pretty close.
"Three…" Bright Skies began counting under her breath, hearing another shell fire. "Two…"
The shell impacted on what must now be an utterly battered port side of the Canterlot. Dash heard a voice on the radio warning them about a fire starting on one of the decks.
"One…" Bright continued to count down as another shell impacted on the side. No sooner than she had the chance to process that, she heard another gunshot fire. "That's it!" She jumped out of her seat holding her hooves high. And utterly confusing everypony in the process.
The last shell barely skimmed the bottom and couldn't do much more damage than what was already done. To Dash's surprise, the ringing in her ears was beginning to subside...which must have meant the Trottingham stopped firing.
She looked over to her left at the ship out the window. Its guns were starting to retract into their resting position. Not aiming at them at all.
"They're out of ammo!" Bright Skies exclaimed, then a serious smile crossed her face. "Dash, spin us around. Do whatever's necessary to get both turrets pointed at the Trottingham."
"You got it!" Dash replied with an equal level of enthusiasm. She finally realised that Bright Skies had been playing one step ahead of the Trottingham all along. She deliberately put them in a position for the Trottingham to lay into them. It meant the Trottingham depleted its ammo. And by holding back from firing all of the Canterlot's guns, they had plenty of ammo to spare. The fact the Canterlot held together must have just been sheer luck. Or strategic positioning of putting their most armoured sections in the line of fire.
"Fire control, this is the bridge. When you get both turrets in range, fire everything you got!" Bright Skies happily exclaimed.
Since the starboard propellers were still working fine, spinning the Canterlot didn't take much effort. Dash adjusted the pitch and altitude to the best of her ability to give the turrets their best possible angle. Tornado had left Dash to it more or less and had gone to stand by the window. Dash smirked as he had obviously gone to take the best seat in the house to watch the fireworks.
The Trottingham had also spun around to face its rear turret towards the Canterlot. While it fired back, it was nothing compared to the barrage the Canterlot unleashed with its second forward turret and what was left in the magazine of the rear turret. The back of the Trottingham lit up like a Hearths Warming tree.
It didn't take long for the entire back of the Trottingham to be covered by smoke. The Canterlot just seemed to be firing blindly into the smoke hoping to hit something. Which considering that they were pretty much shooting point blank at this distance, was an easy shot to guess.
Not having much else to do but sit and watch, Dash put her hooves on her ears to at last give them some relief from all the constant bangs and booms of the gunshots and explosions they'd been exposed to. She swore she could feel her eardrums beating against her hooves.
Not a moment too soon, a wave of the signature multi-coloured unicorn magic slowly made its way through the bridge to repair all the damage from the fake battle. Glass was put back perfectly as it was, metal was bent back into shape, and wooden panels were magically put back together.
"Trottingham suffered severe boiler explosion. Canterlot wins!" The chief unicorn announced.
The cheers Dash overheard were almost as painful on her ears as the gunshots she'd just been hearing. But everypony had the right to be happy. They were the pioneers in this type of thing, and they'd won the inaugural test! But there was one pony they owed it all to…
Dash took flight out of her seat and flew towards Bright Skies, her hooves screeching to a stop on the floor just before she landed on Bright. "Bright, you did it!" Dash exclaimed, placing her hoof around her shoulders. "That was...INCREDIBLE!" She pulled Bright Skies into a hug and squeezed her tightly. She wasn't usually this huggy, but Bright deserved it. As much as she wouldn't have admitted it, there was no way Dash could've pulled that off herself.
Bright returned Dash's embrace, but only for a few seconds before Dash felt her forelegs to limp to her side. Dash slowly let Bright fall back into her seat. Bright had a goofy smile plastered all over her face and seemed to be in a world of her own. It was as if she had fainted.
Dash chuckled, "You're better at this leadership stuff than you think you are. If you showed that much confidence all the time, you'd no doubt be a Wonderbolt."
"Yeah...yeah...haha...that's...a good problem…" Bright mumbled incoherently and continued chuckling to herself hysterically. Dash left her alone to slump in the chair and relax. What she accomplished was probably overwhelming even her.
Dash saw Swift Sprint walking over to them. He merely kept his gaze locked on Bright Skies as he approached. But when he got there, he simply raised an eyebrow at the goofy smile combined with the absolute exhaustion on Bright Skies' face.
"Now that's what I like to see!"
Dash spun around to talk to Tornado instead, somepony she was hoping she impressed. And by the sounds of it, she did.
"I can see why they consider you guys Wonderbolt material; you did amazingly well for a bunch of ponies that had never done this before."
"Heheh, I'm a natural flyer at anything I guess," Dash replied with a shrug and a smile.
A loud bell sounded twice throughout the ship, causing the commotion that was everypony celebrating to gradually die down.
Swift Sprint stepped forward and picked up the radio microphone, adjusting a few settings before he began speaking. "All hooves, this is Admiral Swift Sprint. First, let me say congratulations to the crew of the Canterlot. A true and worthy victory winning two out of the three battles." He looked down at Bright Skies sitting on his chair to the side of him and smiled. "And also from the least likely of Captains, let's get a round of applause for young Bright Skies."
The bridge erupted with the sounds of hooves clapping together once again. Bright Skies could feel it through the seat of her chair as well. The ponies that were listening on the lower decks were cheering her on too. She felt stiff as a twig, she had had enough of being the centre of attention for today. But she smiled awkwardly and waved at everypony on the Bridge. Thankfully the flight suit she wore did an excellent job of hiding her blush.
"But really, a congratulations is in order from everypony," Swift Sprint continued. "Let me extend my thanks to our many unicorn friends from various forces we've had to accommodate to make this work. It's been very tight and snug around the quarters, but thankfully not for too long. And I also want to thank both crews for performing admirably and treating this exercise as if it were the real deal. And I'm sure I'm not the only pony on board who actually thought it was the real deal a few times."
His comments stirred a few chuckles from the ponies on the Bridge.
"And lastly, I would like to thank Captain Spitfire, the Wonderbolts, and the Wonderbolt trainees for taking part. They've proven the point that with a little guidance and pointing in the right direction, they were immediately able to get to work and almost function as if they had been through the same training programs you've all been through. This will provide valuable evidence to support the theory that this type of combat training far outweighs anything else we can currently offer. Now, trainees, Captain Spitfire would like to address you all. Please gather on the deck of the Canterlot so she can begin. You've been a great company, and I wish you the best of luck for the remainder of your training."
The radio clicked off, and Swift dropped it on its holster. "Helm set course for Canterlot."
"Aye, Admiral!" Tornado replied.
Dash looked over at Tornado, who immediately looked back at her. "Don't worry, I'll take it from here. You deserve the rest. Besides, I'm actually getting paid for this. You aren't," he replied with a wink.
Dash chuckled in response, "Thanks." She replied to him as she headed towards the stairs alongside Bright Skies.
"How was the helm?" Bright Skies asked as they descended the metallic stairs that made a slight dinging noise to them every time their hooves tapped on one of the steps.
"Stressful," Dash replied, "You've gotta think about what you wanna do and also know how to do it. It doesn't come as natural as flying with these wings does." She flicked her wings a little for emphasis. "If Tornado weren't there, I would've been lost."
"I don't think they would've put you at the helm of thousands of tons of metal without supervision," Bright stated the obvious.
"Says you!" Dash replied, raising an eyebrow. "You totally killed it as Captain. Where did that Bright Skies come from? She would be a Wonderbolt without a doubt."
"Oh ya know; fear, stress, anxiety, adrenaline…my brain goes a bit haywire when you throw all that together." She replied, chuckling awkwardly and playing with her mane.
"Fear? You didn't seem scared at all. Maybe a little anxious and hesitant, but anypony would be if they were suddenly told to take command of a battleship." Dash deduced as they walked onto the deck.
"I was terrified!" Bright Skies exclaimed. "I could barely think straight!"
"Right, because you totally sounded like a frightened little filly when you shouted orders out," Dash replied sarcastically, winking back at her.
"That's just what happens when I'm put in situations like that. I don't enjoy being in them. Seriously, my teeth hurt from grinding so much. And when Swift Sprint was questioning me about being Captain, he was so intimidating. I would've wet myself if I hadn't gone before we started."
Dash snorted, "You're overreacting. You're a natural. Just keep putting yourself in those positions, and it'll feel more natural over time."
Pegasi began to gather on the bridge, some had their uniforms more soiled than others. Dash presumed that it would've been from working within the depths of the machinery inside the ship. But what made her laugh was the state that Dahlia came out in. She was covered in soot from head to hoof.
"Have a good time?" Dash asked sarcastically.
"Tough," Dahlia replied unamused. "They had me working in the engine room. I felt like I was inside the sun."
"It's a good look for you," Bright Skies replied sarcastically.
Dahlia looked at her, raising an eyebrow. "Guess you're the fashion trend expert here now you've got a new accessory." She joked.
"Oh yeah," Bright Skies took the cap off and held it in her hooves, "He didn't say anything about it. I guess I'll give it back to him before we leave."
"Of course. I've got something I need to talk to him about too," Bright Skies added.
"I think we all do," the voice of White Comet added.
Dash turned in the direction of the voice behind her to see Comet, who had just found them in the crowd.
"What gun did you go to in the end?" Bright asked.
"The middle one, behind this one," Comet replied, motioning with his hoof to what he meant.
"You did some pretty good shooting. I wasn't counting the number of hits, but I'm sure you were on par with the other turrets."
Comet shrugged, "It's just simple maths and physics. I just got shown how the equipment works and the rest I figured out myself." He explained.
Dash chuckled as Comet seemed to have a much less stressful time than everypony else did. Although she was sure it didn't go that smoothly considering how much damage they took at various points. She was going to ask what Dahlia wanted to talk to Swift Sprint about, but she was ninety-nine percent sure that she wanted to ask him about Torque.
Just as she was thinking about it, she heard the muffled buzzing of a microphone coming to life over the speakers. She turned her head in the direction of the noise and saw Spitfire standing on the bow of the ship, with the rest of the Wonderbolts just behind her. She flew at a high enough level so everypony standing on the deck could see her.
"I just want to start by saying well done. I'm sure you're all exhausted, and there's not much else for me to say that the Admiral hasn't already said, so I'll make this quick. We're heading back to Canterlot now. We will be meeting to discuss the results of this exercise and to make our picks to see who goes forward from here on. Which we will be telling you when you make it back to HQ. So take some rest while we're in transit. Feel free to explore the ships a little more, but try not to get in the way of the crew. That's all for now, well done everypony."
With that, Spitfire flew back down to return the radio and everypony resumed their conversations. Dash turned to look at Bright Skies and Dahlia, but Dahlia had already started making her way towards the bridge of the Canterlot. Dash didn't say anything but just followed. Comet and Bright Skies followed suit.
The trio knew precisely where she was heading. They had nothing to say. And they couldn't deny that they were also curious if Torque's Dad knew anything about what happened to Torque. This kind of exercise would have been right up his street since he seemed to have all manners of knowledge about mechanics and engineering. Plus it sounded like he actually served in the REAF at some point.
Dahlia trotted gently up the stairs until she got right to the bridge. She stepped through the open door and looked around. Immediately spotting Swift Sprint sitting in the captain's chair. She trotted up to him but slowed her pace as she neared.
Swift Sprint noticed her approaching and turned to look at her. "Can I help you?" He asked, rather pleasantly.
Dahlia sighed and looked away, glancing over her shoulders to see Dash, Comet and Bright behind her. But she had put herself in the position to do the talking, so it looked like it was up to her to do it. "I just wanted to ask about...Torque...your son…"
Swift Sprint smiled, "I haven't seen him yet, I presumed that he was on the Trottingham. Is he okay?"
"He's fine, but...he wasn't on the Trottingham. In fact, he wasn't here at all...he quit yesterday."
Swift raised an eyebrow, "That's disappointing, but that sounds like his choice. What are you after from me?" His voice was very stoic. He sounded compliant, but he didn't seem thrilled about the questions Dahlia was implying.
Dahlia gulped, not sure how she should word it. "Well, he talked a little to me before he left, and... what he really wants to do is what you do. He wants to be in the REAF. But he said you were putting a lot of pressure on him, with some very high expectations. Which is why I think he left…" she trailed off, trying to think about what she was saying. Despite wanting answers, she hadn't really thought this through this far ahead.
"I guess what I'm trying to say is that you're being way too harsh on him and need to give him some sort of indication that you, the pony he idolises, are proud of him." She rambled, very fast and very loud...maybe a little too loud. She got a few glances from other ponies on the bridge.
An eerie silence fell between them for a few seconds before Swift inhaled sharply and bought himself to his hooves. "Outside," he motioned with his hoof, starting to walk towards the doorway.
"What?" Dahlia asked, confused.
"Do you want your answer or not?" He asked sharply, not ceasing in his path.
Dahlia didn't dare question him again and followed suit. The five of them all crept through the doorway, with Swift closing it behind him and holding it closed.
"You have some nerve questioning my parenting techniques considering I'm old enough to be your father young mare." Swift began in a stern tone.
Dahlia was taken back by his tone. She didn't mean to make him angry. Had she touched on a sore subject? "Well--" Dahlia tried to speak but was cut off.
"So let me make this clear to you; I am proud of him, and always will be regardless of what he chooses to do. But I'll be damned if I raise a son of mine into an underestimating, apathetic, naive, narcissistic, entitled brat."
The four of them all felt a funny feeling of deja vu as they were sure some of those words were what Torque used to describe Riptide when he finally did crack. Sounds like the apple doesn't fall too far from the tree.
"I act the way I do so it doesn't soften him up. He's tough, and always will be. He doesn't give up. He can solve problems on his own. He understands the value of things. He knows how hard you need to work to achieve your goals, something I'm sure you're all too familiar with." He tapped one of the medals on his jacket. "You think one of these just appears every time Daddy rustles my mane and says 'I'm so proud of you son'?" He said in a mocking tone. "The moment you get too complacent is the moment when you start losing. And I won't allow that to happen."
Dahlia opened her mouth to speak, but she couldn't find the words to come out of her mouth. While she didn't particularly agree with it...everything he was saying was right. And the damage was already done anyway, anything she did to argue it would be pointless. She guessed she just wanted to know the truth and knew where she could get a good answer.
"Anyway, that's my rant over. Now, there's some advice I want to give to the four of you, something very important to keep in mind for the rest of your Wonderbolt careers."
Dash, Comet, Dahlia and Bright's ears all perked up at the same time as he spoke.
"Don't EVER question your CO in public. If you've got something to say to them that you disagree on, speak to them in private. Your job is to execute the orders of your superior officer to maximum efficiency. When you start questioning them, it makes everypony else in earshot start to think as well. And all of a sudden, the unit stops functioning as it should."
At the end of his sentence, he opened the door to the bridge again. "Always keep that in mind...because if you don't, then one day it may cost you your lives." He spoke his final words before turning to walk away.
"Um, sir!" Bright stopped him before he walked away. He turned sharply and looked at her. Bright simply took his hat off her head and presented it to him.
Swift looked at the cap being presented to him. He took his hoof around it but didn't take it from Bright's grasp. He just held it in his hoof for a few seconds then pushed it back towards her. "Keep it."
"What?" Bright Skies questioned.
"You exceeded the expectations I would have even of actual REAF airponies young Bright Skies. You could be destined for great things. Let that be a reminder to you." He said nothing more and walked through the doorway.
The four of them stood in silence for a few moments. "Well...I guess you got your answer…" Comet said.
"Yeah...I guess I did…" Dahlia agreed softly, turning away from the bridge and heading back down the stairs. "I don't really know what I was looking for…but that's the answer I guess…"
Dash followed the pair of them back down the stairs. Her thoughts were distracted as she heard some very silent giggles from behind her. She glanced over her shoulders quickly to see Bright behind her smiling from end to end of her face and clutching the cap tightly into her chest as they walked down. Swift's comments had apparently left her feeling very giddy. But the boost to her confidence from a gesture like that would last for a while.
Like Dahlia, she found it nice to know some more background info about Torque's situation. But it didn't really do anything to help Torque. But who knows, maybe Dahlia's words got through to him, and they will see him again someday...maybe…
Dash landed perfectly in line with the formation on the runway, beginning to pound her hooves on the tarmac as they converted from flying in formation to marching in formation. A trick that was quite difficult to get right the first time, but plenty of practising meant everypony pretty much had it nailed at this point.
They marched all the way up the runway to the main HQ building. "Squad...HALT!" Spitfire ordered as they reached the end, making everypony come to an abrupt stop.
"At ease," Spitfire added, breaking formation and motioning to the rest of the Wonderbolts. They all gathered in front of the formation, collating some papers together and handing them to Spitfire.
This must be it. This must be where they decide who's made it through and who hasn't. Dash did her best to hold her nerve. She was nervous but excited. Very nervoucited, as Pinkie Pie would put it.
"Okay," Spitfire turned around to address everypony in the formation, shaking the papers in her hooves to line them up squarely with each other. "If I call your name, I want you to head inside." She instructed.
But to Dash's dismay, and probably everypony elses, she didn't give any further instructions than that. She just started calling names, and they obliged. But she didn't provide any context. If they were going inside, did that mean they passed? Or was it just to get them out the way so they can pack their things and head home? Whatever it meant, Dash was soon to find out.
"Riptide!" Spitfire called his name.
Dash looked over when she heard the hoofsteps of Riptide walking towards the building. Did he pass? Or was he out? She hadn't actually seen him throughout the whole exercise with the airships, so she had no idea how to measure how he performed.
As the last name was called, everypony waited patiently while they made their way into the building. The door was shut behind them as they entered.
Spitfire landed atop the steps up to the building, giving her a little bit of a pedestal to stand on. She looked around at all the ponies left, slowly shifting her glare before beginning to speak. "As for the rest of you...unfortunately it's not good news…"
Dash felt like she'd just taken a dagger through the heart as those words rolled off her tongue. All of this...for nothing? She did her best to stop her legs wobbling. She felt her bottom lip quiver. She wanted to cry. She felt like crying. But every ounce of her told her to stay strong in the face of--
"Instead it's great news!"
"Congratulations. You and the fifteen other pegasi you see standing on the runway with you now have passed."
Just the words Dash had been wanting to hear. But she still couldn't believe Spitfire had led them on like that! She felt her heart race at the news. She wanted to dart around the sky in all directions, but she knew her body wouldn't thank her for that after being so exhausted.
She heard everypony else let out relieved sighs and chuckles from the news. She looked around, seeing Bright Skies laying down on her back on the runway in disbelief. Dash ran towards her, seeing how happy she looked. "Bright, you did it! Can you believe it!?"
"We all did it!" Dahlia added, joining them for a three-way group hug.
"Whew…"
Dash looked over at Comet, who looked like somepony had just told him he'd won ten trillion bits.
"I...I thought I'd do well, but I never expected to make it this far…" He commented.
"Wait," Dahlia interrupted the celebrations, "If we're all here, then that means that Riptide…"
"...Wasn't good enough." Bright Skies finished her sentence for her.
"That's…I mean…" Dahlia shrugged, "it just seemed like Torque had got through to him and he started taking it seriously...too little too late I guess…"
"That's a harsh lesson," Comet added.
"Should we go and find him?" Bright Skies asked. "I bet he's devastated."
"We'll find him later, for now, let's just celebrate that we passed!" Dash jumped into the air, punching it with her forehooves.
Everypony on the runway celebrated between themselves for several moments. They celebrated as if they achieved the impossible. There was now only one more stage separating them from their dream. Dash was confident that there would be sore heads tomorrow morning.
"Are we done?" Spitfire deliberately bellowed to try and overpower the celebrations. The celebrations slowly died down, and everypony turned their attention back to her. "Good. Because while you're free to go for a couple of weeks, I've still got things to do. So please be off site by six PM." She turned to head back into the HQ, but with a sneaky smile on her face. "And enjoy your rest. It's well deserved."
Dash and her roommates made their way back into the HQ building. All the while chatting about their plans for their few weeks off before the final. Dash was sure she would meet some of them before then, but she had a certain pony she needed to pay a visit to first.
As they entered the room, they were expecting to see Riptide, but...no Riptide was in there.
"He didn't leave already did he?" Comet asked.
"Without saying goodbye?" Dahlia responded. "I doubt it."
But as she walked around his bed, her suspicions were all but confirmed. None of his things was there. It was as if he was never there…
"...Wow…" Bright Skies cut through the semi-speechlessness of everypony. "I didn't expect him to just take off. It must have really hurt him."
"Yeah, the room feels...kinda empty without him…" Dahlia replied, peeling her flightsuit off.
"As annoying as he was…pffft…" Dash covered her mouth with her hoof to hold in her laughter at what she saw on Dahlia.
"What's so funny?" Dahlia asked, finishing taking her flightsuit off and picking up on Dash's laughter.
"Have you looked in the mirror?" Dash answered back.
Dahlia raised an eyebrow as she heard some more snickers from Comet and Bright Skies. She walked over to the bathroom and took a look in the mirror. She could only smile and chuckle at what everypony else was laughing at. The bits on her face where her flightsuit started and ended had left her with what looked like tan lines, but with soot instead.
"Okay, that's pretty funny." Dahlia agreed with their sense of humour.
"I've got a few myself, mostly because when we were moving forward fast, the smoke from the cannons blasted the platform I was standing on," Comet added, appearing behind Dahlia and marking his own lines on his face.
"You got off light. That's just a dusting," Dahlia replied, turning the tap on and filling the sink with water.
"Lighter than you yeah. But my fur is white, so it stands out more." Comet deducted. "Your purple makes it a bit easier for you to hide it."
"Oh the horror," Dahlia replied sarcastically, lowered her head over the sink to wash her face.
"Leave the water in there when you're done. I want to do the same." Comet requested.
Dash packed her things as quickly as she could. While it may have been part of the adrenaline from officially making it to the final stage, she couldn't deny the fact that she was excited to see Soarin. They've hardly been able to spend any time together. And with the final coming up, it was likely that they would be seeing even less of each other on a personal level.
With her packing done, she threw her saddle bags over her back and quietly waited for everypony else to finish packing. It didn't take them much longer than her though, and they all headed for the door.
They trotted down the corridor and headed to reception, where the exit awaited them. "These past few weeks have been great guys. I'm sure we'll bump into each other sometime before that anyways." Dash said to send them off, waving at her friends.
"Well we're going the same way anyway, we can fly till we part ways?" Dahlia asked.
"Nah, I uh...gotta...go to the bathroom," Dash quickly made an excuse in her head. "You go on ahead. If I catch you, then I will."
"Fair enough," Dahlia replied, shrugging. "Let's have a group hug now then. Come on, one more time," she said, extending her forelegs wide so everypony could get in.
"Oh fine, one more," Dash replied sarcastically as she moved in with the rest of them. The four of them formed a tight circle in each other's embrace.
"Regardless of what happens, you guys have been great friends. And I want that to remain afterwards." Bright Skies explained.
"Of course we will," Dahlia replied, "I could never forget you guys, after all, we've been through."
"And I wasn't even in it to win it, yet here I am," Comet added.
They held the hug for a few seconds before they released their embraces. "Hit me up if you're ever in my neck of the woods, Dash!" Dahia advised.
"Sure, will do," Dash acknowledged, turning around. "See ya soon guys!" With that, she started walking off, back into the HQ.
It appeared as if her alibi worked. Nopony doubted her claim. However, she knew where she was going, and she was excited to get there.
She hadn't been up here in a while, but she still knew the way to Soarin's quarters very vividly. She approached his door and knocked on it. She couldn't get the smile off her face.
The door opened and there standing in the doorway was the very pony she was waiting to see. Every ounce of her just wanted to leap forward and hug him tight. But she kept her cool as he motioned with her hoof for her to come in.
Dash nodded and walked through the door, hearing Soarin close the door behind her. That was all she needed to slide her saddle bags off and leap towards him.
"Dash--mmph--" Soarin didn't even have a chance to get some words out before his lips were met with Dash's. The force of Dash colliding with him backed him up against a wall. But not that he was complaining. It had been far too long, and he was more than happy to be here with her again.
The passionate kiss lasted for only a few seconds, but it already had a significant impact. "Wow…" Soarin chuckled, finally able to start talking. "Now that's a great way to say hi. I was about to say you must be tired, but you've obviously got some energy left."
"Not much," Dash said with a smile, "But it's all for you."
"Well, let's not waste it then," Soarin replied, taking Dash under his wing and gently pulling her over to a sofa in his room. He set himself down first in the corner and let Dash lean in next to him. "How have you been? You made it through."
"I know, I almost can't believe it!" Dash replied, still excited every time she thought about it.
"Haha...yeah never doubted ya," Soarin replied, running his hooves through her mane.
Dash kept trying to think of things she wanted to say to him...but to be honest, she was completely happy just laying there, with him, in his embrace. It was quiet, but she found it soothing…
...But she also found it a little too quiet. Soarin was usually a lot more chatty than this. "So...what's up?" She asked.
"Hm?" Soarin looked down at her. "Sorry, I was a little spaced out there…"
Dash raised an eyebrow, "What's wrong Soarin? You seem a little...distracted."
"Oh, pfft," Soarin batted his hoof, "I was just thinking about the finals. I bet you probably are too."
"What about them?" Dash questioned.
"Well...to be honest, I've been thinking about it a lot, and...well, I'm struggling to find a way to back you up should I need to."
"Well if I'm so good you hopefully won't need to right?" Dash replied, winking.
"Yeah, but…" Soarin shimmied up the sofa, sitting upright but continuing to hold Dash in a hug with his wing. "What if it comes down to the wire? What if it ends up being the deciding factor?"
"Well…" Dash thought about it for a moment before responding. "I understand that you've got a job to do. And I won't hold it against you." She didn't like thinking this far ahead in the future, especially for something as serious as this because it just distracted her with the possibility of not getting the result she wanted. This was the first time she had really thought about it the way Soarin was putting it.
"You say that, but…" Soarin sighed and glanced away, "Let me put it this way Dashie...and be honest here." He held his head back up and looked at her. "Assuming things don't work out, would you really look at me the same way? Would you ever greet me again in the way you did a moment ago? Would you get so excited and giddy about coming to see me? Would you still enjoy my company knowing that I was the pony that got in-between you and the Wonderbolts?"
Dash opened her mouth to speak...but no words came out. What she wanted to say and what she had to say didn't match up. And it only caused her words to get jammed in her throat.
Soarin sighed, and his ears flopped down. "That pause says it all really…" he rested his head on his forehooves.
"Soarin…" Dash ears flopped down, and she glanced away. But then her ears came back up, and she glared a piercing glare at him. "That was very rude."
Soarin's ears perked up hearing her change of tone. "What?"
"Are you assuming I'm only sucking up to you to increase my chances of selection? That's not what this is about at all. I love you. And I mean it. And if you think otherwise...then honestly, I'd be offended." Dash blurted it all out.
Soarin remained silent for a few moments. He wasn't expecting an outburst like that at all. But if anything, it helped convince him that he was getting worked up over nothing.
He smiled and chuckled gently, twisting around to Rainbow Dash. "You're right. I'm sorry Dashie, I guess I was just overthinking. It's been a busy few weeks for me like you." He wrapped his forelegs around her shoulders and pulled her forward into a hug.
A relieved smile crossed Dash's face as she rested her chin on his shoulder. "It's okay, I understand." She replied, returning his embrace. "I'm sure you want to think about something else for a little while."
Soarin smiled. "Well, funny you mention that…" He pulled back to look at her face to face. "Me and a few other 'Bolts are having a little get together next Friday. Nothing serious, just dinner and drinks. If you wanna come then feel free…"
Dash's jaw hung ajar. "Did...you just invite me to hang out with you guys?"
"Well, like I said, it's just casual. So I thought you might like to come along. Ya know, get to know some of the others on a more informal level." Soarin explained.
"Who else will be there?" Dash asked. "Spitfire?"
Soarin shook his head, "No, Spitfire can't come. But Blaze is coming…" he chuckled nervously, "It was actually her idea to invite you…" He rubbed the back of his head as he admitted that.
"Wait, what?" Dash jumped backwards, "She knows about us!?"
"Well...kinda." Soarin admitted, "In fact, I probably would've never worked up the confidence to you if it weren't for her. I kinda owe her everything..."
"Really?" Dash replied, "I guess I owe her big time too. But geez, I didn't think anypony else knew," Dash rubbed the sides of her head, feeling the temperature start to rise in her cheeks. "Does anypony else know apart from Blaze?"
Soarin shook his head, "No, just them. Or at least if anypony else does know, they aren't letting on. But I don't think it'll remain a secret if you come." He analysed.
"Well...does it need to be a secret?" Dash asked. "Generally of course, but from the rest of the Wonderbolts?"
Soarin shrugged, "Blaze doesn't seem to mind. Quite the opposite in fact. I'm sure we'll get questions, and I probably won't hear the end of it, but…" he started chuckling, "I'm sure they'll all like you."
"Well, when you put it like that…" Dash rubbed her chin, "Of course I'd love to come. But seriously, I don't want you to end up doing something that you'll regret."
"Nah, don't worry about it," Soarin replied, winking. "I know it sounds cheesy, but...you complete me, Dashie. I'm bored without you. Sure, it'll be fun if I just went out with the 'Bolts, but honestly, I'll take any reason to spend more time with you."
"Bleugh," Dash made a fake vomiting motion, but then smiled. "Okay, I guess that's pretty sweet of you to say…" She admitted, feeling herself blush even more from a combination of what he said and what she was thinking about. "But Friday's still ages away. In the meantime, Spitfire told us we didn't have to clear out till six PM…" she said, running a hoof through his mane.
Soarin returned the gesture, understanding what she was implying. "That's more than enough time to make up for lost time... " he lowered his hoof down from her mane and bought her forward, bringing her into another long passionate kiss...
Author's Note:
Whew, that was way longer than I anticipated it to be!
I've been looking forward to writing this scene for a while, so I took my time with it to try and get it just right. It sounds like Dash and her roommates got a very exciting experience, something they weren't expecting at all. But Bright Skies overcame her fears and took to commanding a battle-airship like a fish in water. An experience that will last a lifetime.
Poor Riptide, he left without even saying goodbye. 2 friends being knocked out so close together can be a real kick in the ass to remind you that whilst you're all friends, it's a competition and there can only be one winner at the end of the day...
Sounds like Dash has been invited to meet some of the Wonderbolts with Soarin. It's extremely unlikely that they will be able to keep their relationship a secret from them if they go. Will Soarin regret that? Well, I suppose you'll have to read on to find out won't you
Fun fact: this chapter and the previous one were originally gonna be the same chapter. But I wanted to get something out before Christmas, so I cut it off. I'm glad I did, otherwise this would've turned into a 30,000 word behemoth of a chapter
And we've officially hit 200k words in this story! My god I wasn't expecting this to go on for so long. I wasn't even expecting half that Big thanks to you who have stuck with me and kept reading. I can now say we're somewhere around the 60%-75% completion mark, so hopefully not too much longer.
Viewing 1 - 50 of 4
Reiko #1 · Feb 18th, 2019 · · ·
Great chapter! Especially parts related to Bright Skies :)
White Comet
White Comet #2 · Feb 18th, 2019 · · ·
Thanks! She's my favorite character to write for, for various reasons that will be shown later.
Admiral Q Ponyform
Admiral Q Ponyform #3 · Mar 30th, 2019 · 1 · ·
The airship battle was awesome
redmar
redmar #4 · Mar 29th, 2021 · · ·
While the airship battle was good this chapter also makes clear that the author has completely forgotten that Rainbow Dash isn't the only pegasus from Ponyville to be selected for the tryouts at the start. After Rainbow Dash won he race against Thunderlane they comletely disappeared from the story which is one of the few negative points of this story.
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95,354 hits today, 1,560,712 yesterday
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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The "Cranken Rhyme" is a Cornish-language song known by farmer John Davey or Davy (1812–1891), who was one of the last people with some knowledge of the tongue. It was recorded by J. Hobson Matthews in his History of St. Ives, Lelant, Towednack, and Zennor, and is probably the latest known traditional Cornish verse.
Matthews records the song in a chapter on the Cornish language and the evidence for its late survival. It is not clear whether he ever met Davey, or if he was relying on second-hand testimony. Either way, the song is unknown from any other source, demonstrating that Davey had knowledge of some original Cornish in the late 19th century. Matthews himself thought the song to be merely a jumble of place-names, which Davey was reputed to be able to decipher. However, Robert Morton Nance respelled the song into a recognizable form and provided an English translation. It is evidently a bit of humour claiming that even the Penzance-Marazion road was more fertile than Cranken's stony fields.
References
Cornish-language literature
English poetry
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Le Pèlerinage de Lourdes è la XXXVII enciclica di Papa Pio XII.
Voci correlate
Apparizione mariana
Elenco delle encicliche
Enciclica
Nostra Signora di Lourdes
Papa Pio XII
Collegamenti esterni
Encicliche di Pio XII
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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All notable changes to this project will be documented in this file.
This project adheres to [Semantic Versioning](http://semver.org/).
## [2.2.1] - 2016-06-15
### Fixed
- Sending email with accents: https://github.com/sendgrid/sendgrid-nodejs/issues/239
- Thanks [eaparango](https://github.com/eaparango)!
## [2.2.0] - 2016-06-10
### Added
- Automatically add Content-Type: application/json when there is a request body
## [2.1.0] - 2016-06-08
### Added
- Cleaner request object initialization
- Ability to use http for testing
## [2.0.0] - 2016-06-06
### Changed
- Made the Request and Response variables non-redundant. e.g. request.requestBody becomes request.body
## [1.0.1] - 2016-04-08
### Added
- We are live!
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{
"redpajama_set_name": "RedPajamaGithub"
}
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The pop art style in interior design
Welcome to the world of bright colors, originality and beauty. The pop art style embodies all sorts of unique combinations and the most daring design ideas.
This style emerged in interior design in the 50-60-ies of the last century. Originally this style was preaching freedom from all rules, restrictions and uslovlennoe.
Pop art was at the peak of popularity all over the world a few decades ago, but now its use and recognition also felt. Pop art not just another style in interior design, it is a lifestyle which requires to live without rules and restrictions.
All the small elements of the interior in pop art style mattering a full and original artwork. This non-conservative style, it breathes the spirit of modernity and youth. Not strange why in pop art today is very often made out of a variety of youth cafes and clubs.
This style is not suitable for quiet family evenings at the fireplace, he more than has the fun, movement and noisy companies. Therefore, the style of pop art is not to everyone's taste, and not everyone can withstand it bright and bold tone.
The main features of pop art style, which made him known and popular throughout the world:
The use of abstract drawings, glossy, and recurring motifs;
A wide variety of accessories and small items of plastic and other lightweight materials;
Can often be blurred traditional boundaries of walls, floors and ceilings;
This style is suitable for processing both large and small rooms;
Bright walls, single or multi – tier ceiling light;
Room in the style of pop art may not have a certain color, it harmoniously combines all the colors of the world, both natural and synthetic;
Bright paintings, photographs and posters on the walls, the walls can also contain drawings;
Furniture pop art style is chosen according to the principle of minimalism, simplicity and convenience;
Niches, built-in wardrobes, bright bookshelves, all of this is often present in pop art, and rarely is made in different colors;
Elements, as a rule, cheap, but unusual: metal, leather, plastic, glass, etc.
Particularly valued things made by hand;
A striking feature of pop art- a variety of visual illusions: visual effects disappearing in the drawings, the curvature of space, etc.
Pop art charm and diverse style. He loves to shock the audience, but does it very neatly so that the room was turned into a garbage dump. Creativity and unstoppable flight of the creative imagination – all this is combined into a pop art form a beautiful harmonious composition.
In the style of pop art is made mostly living room and bedroom (often children), such rooms as kitchen and bathroom pop art to meet difficult.
Pop art is a modern, affordable style that's ideal for those who want to make repairs or build a house with his own hands. Modern technologies are also present in pop art. This is the style for the emotional and brave people, he breathes in the energy of youth, beauty and style.
20 Photos of the The pop art style in interior design
June 12, 2016 ≈ Living Room ≈ No Comments ≈
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Remodeling bathroom and a living room »
Oriental style interior design
Striking Inspiration for Home TheaterRooms
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Gray color in interior design
small family room ideas
small backyard landscaping ideas
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Le canton de Fleury-les-Aubrais est une circonscription électorale française, située dans le département du Loiret en région Centre-Val de Loire.
Le canton est créé sous la Révolution française en 1790 sous le nom de canton de Fleury ; il disparaît lors du redécoupage de la France en arrondissements après la suppression des districts en 1801 sous le Consulat, il est recréé sous la Cinquième République lors du redécoupage de 1973. À la suite du redécoupage cantonal de 2014, le nombre de communes composant le canton est passé de 2 à 7.
Histoire
Le canton est créé le sous la Révolution française, il porte alors le nom de canton de Fleury. Il est inclus dans le district d'Orléans. À la suite de la suppression des districts et de la création des arrondissements qui survient en 1801 (9 vendémiaire, an X) sous le Premier Empire, le canton de Fleury est supprimé.
Le canton de Fleury-les-Aubrais est recréé en 1973.
Un nouveau découpage territorial du Loiret (département) entre en vigueur à l'occasion des premières élections départementales suivant le décret du . Les conseillers départementaux sont, à compter de ces élections, élus au scrutin majoritaire binominal mixte. Les électeurs de chaque canton élisent au Conseil départemental, nouvelle appellation du Conseil général, deux membres de sexe différent, qui se présentent en binôme de candidats. Les conseillers départementaux sont élus pour au scrutin binominal majoritaire à deux tours, l'accès au second tour nécessitant 12,5 % des inscrits au . En outre la totalité des conseillers départementaux est renouvelée. Ce nouveau mode de scrutin nécessite un redécoupage des cantons dont le nombre est divisé par deux avec arrondi à l'unité impaire supérieure si ce nombre n'est pas entier impair, assorti de conditions de seuils minimaux. Dans le Loiret, le nombre de cantons passe ainsi de 41 à 21. Le nombre de communes du canton de Fleury-les-Aubrais passe de 2 à 7.
Le nouveau canton de Fleury-les-Aubrais est formé de communes des anciens cantons de Fleury-les-Aubrais (), de Neuville-aux-Bois () et de Chécy (). Il est entièrement inclus dans l'arrondissement de Orléans. Le bureau centralisateur est situé à Fleury-les-Aubrais.
Représentation
Représentation avant 2015
Résultats électoraux détaillés
Élections cantonales de 2004 : Michel Breffy (PS) est élu au avec 56,24 % des suffrages exprimés, devant Pierre Bauchet (UDF) (43,76 %). Le taux de participation est de 64,49 % ( sur ).
Élections cantonales de 2011 : Michel Breffy (PS) est élu au avec 68,67 % des suffrages exprimés, devant Michel Beato (FN) (31,33 %). Le taux de participation est de 44,48 % ( sur ).
Élections départementales de 2015
Représentation après 2015
Résultats détaillés
Élections de mars 2015
À l'issue du des élections départementales de 2015, trois binômes sont en ballottage : Fabienne Beaudoin et Roger Landon (FN, 30,75 %), Marie-Claude Donnat et Jean-Jacques Ratajski (Union de la Droite, 30,45 %) et Michel Breffy et Marie-Agnès Courroy (Union de la Gauche, 28,86 %). Le taux de participation est de 47,34 % ( sur ) contre 49,98 % au niveau départemental et 50,17 % au niveau national.
Au second tour, Michel Breffy et Marie-Agnès Courroy (Union de la Gauche) sont élus avec 35,9 % des suffrages exprimés et un taux de participation de 50,13 % ( pour et ).
Michel Breffy a quitté le PS et a adhéré à LREM.
Élections de juin 2021
Le premier tour des élections départementales de 2021 est marqué par un très faible taux de participation (33,26 % au niveau national). Dans le canton de Fleury-les-Aubrais, ce taux de participation est de 32,29 % ( sur ) contre 32,6 % au niveau départemental. À l'issue de ce premier tour, deux binômes sont en ballottage : Grégoire Chapuis et Marie-Agnès Courroy (DVG, 26,84 %) et Anthony Domingues et Marie-Claude Donnat (LR, 24,12 %).
Le second tour des élections est marqué une nouvelle fois par une abstention massive équivalente au premier tour. Les taux de participation sont de 34,36 % au niveau national, 32,79 % dans le département et 32,17 % dans le canton de Fleury-les-Aubrais. Grégoire Chapuis (PS) et Marie-Agnès Courroy (DVG) sont élus avec 52,74 % des suffrages exprimés ( pour et ).
Composition
Composition avant 2015
Le canton de Fleury-les-Aubrais, d'une superficie de , est composé de deux communes
.
Composition à partir de 2015
Le nouveau canton de Fleury-les-Aubrais comprend sept communes entières.
Démographie
Évolution démographique
Démographie avant 2015
En , le canton comptait .
Démographie depuis 2015
Âge de la population
La pyramide des âges, à savoir la répartition par sexe et âge de la population, du canton de Fleury-les-Aubrais en 2009 ainsi que, comparativement, celle du département du Loiret la même année sont représentées avec les graphiques ci-dessous.
La population du canton comporte 47,9 % d'hommes et 52,1 % de femmes. Elle présente en 2009 une structure par grands groupes d'âge légèrement plus jeune que celle de la France métropolitaine.
Il existe en effet de moins de pour de plus de , alors que pour la France l'indice de jeunesse, qui est égal à la division de la part des moins de par la part des plus de , est de 1,06. L'indice de jeunesse du canton est également supérieur à celui du département (1,1) et à celui de la région (0,95).
Notes et références
Notes
Références
Voir aussi
Loiret
Liste des arrondissements du Loiret
Liste des cantons du Loiret
Liste des communes du Loiret
Liste des conseillers départementaux du Loiret
Canton de Fleury-les-Aubrais
Canton dans l'ancien district d'Orléans
|
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The Bila Tserkva massacre was the World War II mass murder of Jews, committed by the Nazi German Einsatzgruppe with the aid of Ukrainian auxiliaries, in Bila Tserkva, Soviet Ukraine, on August 21–22, 1941. When the Jewish adult population of Bila Tserkva was killed, several functionaries complained that some 90 Jewish children were left behind in an abandoned building, and had to be executed separately. The soldiers reported the matter to four chaplains of the Heer, who passed along their protests to Field Marshal von Reichenau; it was the only time during World War II that Wehrmacht chaplains tried to prevent an Einsatzgruppen massacre, but Paul Blobel's verbal order was direct and decisive.
Description
In August 1941, General Walther von Reichenau, commander of the 6th Army of Nazi Germany, ordered his men to assist the Einsatzgruppen and their Ukrainian auxiliaries with killing the Jews of Bila Tserkva. Over the course of the following days, virtually the entire adult Jewish population of Bila Tserkva was shot. All that remained were the children and a few of the women, who were dumped off at a school to await execution.
Several soldiers were disturbed by the crying of the children and infants at the school, and asked their chaplains what to do. The two chaplains attached to the 295th Infantry Division, Catholic Father Ernst Tewes and Lutheran Pastor Gerhard Wilczek, visited the school. They were appalled by the condition of the frightened, hungry children. The chaplains asked the local army commander to free the children, but he refused. Tewes later reported he "turned out to be a convinced anti-Semitic". Joined by two other chaplains from the 295th Division, a series of protest letters were sent to people in positions of authority asking that the children of Bila Tserkva be spared. The chaplains won over staff officer Lieutenant-Colonel Helmuth Groscurth to their cause. He ordered a postponement of the planned massacre of the children. In areas near the front, the Einsatzgruppen were under Army command and so when Colonel Groscurth ordered the massacre to be delayed, the local Einsatzkommando leader had no choice but to comply. Ultimately, von Reichenau himself intervened and ordered the executions to go ahead. After receiving a protest letter from two of the chaplains, Reichenau wrote in response:
Tewes later recalled, "All those we wanted to save were shot. Because of our initiative it just happened a few days later than planned". SS-Obersturmführer August Häfner who saw the subsequent murders on 21 August 1941 testified at his own 1965 trial as follows:
The protests at Bila Tserkva were unique as being the only time during the war that Wehrmacht chaplains tried to prevent an Einsatzgruppen massacre. The American historian Doris Bergen wrote that all four chaplains involved in the protest were aware that Jewish adults were being killed and protested only when they learned that children were to be shot. Bergen further observed the "terrible irony" that a gesture of protest further served the genocidal aims of the regime; the soldiers who were troubled by the crying of the children waiting for their time to die felt that they had "dealt with" the issue by "doing something", namely appealing to Father Tewes and Wilczek, and they had no further role to play in this matter.
References
Bibliography
(originally published as )
Bila Tserkva
1941 in Ukraine
Massacres in 1941
Mass murder in 1941
Holocaust massacres and pogroms in Ukraine
August 1941 events
|
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using System.Web.Mvc;
using Twilio.AspNet.Mvc;
using Twilio.TwiML;
public class HelloWorldController : TwilioController
{
[HttpPost]
public ActionResult Index()
{
var voiceResponse = new VoiceResponse();
voiceResponse.Say("Hello! You will get an SMS message soon.");
voiceResponse.Sms("This is the ship that made the Kessel Run in fourteen parsecs?");
return TwiML(voiceResponse);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,516
|
{"url":"https:\/\/physics.stackexchange.com\/questions\/498476\/how-does-the-hubbard-hamiltonian-change-when-considering-a-peierls-distortion-b","text":"# How does the Hubbard hamiltonian change when considering a Peierls distortion (bipartite lattice)?\n\nThe following is the Hubbard contribution to the hamiltonian in the Hubbard-Tight Binding model.\n\n$$H_{hubbard}=U \\sum_i n_{i \\uparrow}n_{i\\downarrow}$$\n\nwhere $$n_{i \\sigma}=c_{i\\sigma}^\\dagger c_{i\\sigma}$$\n\nAnd the tight-binding hopping part:\n\n$$H_{TB}=t \\sum_{i\\sigma} (c_{i\\sigma}^\\dagger c_{i+1\\sigma}+ c_{i+1\\sigma}^\\dagger c_{i\\sigma})$$\n\nThe full hamiltonian is then given by:\n\n$$H=H_{TB}+H_{hubbard}$$\n\nIf I want to consider a Peierls distortion (which is considering dimerization, breaking the symmetry and now having two different sites A and B within the unit cell) the Tight Binding part is changed to:\n\n$$H_{Peierls}=t_1 \\sum_{i\\sigma} (a_{i\\sigma}^\\dagger b_{i\\sigma}+ b_{i\\sigma}^\\dagger a_{i\\sigma})+t_2 \\sum_{i\\sigma} (a_{i+1\\sigma}^\\dagger b_{i\\sigma}+ b_{i\\sigma}^\\dagger a_{i+1\\sigma})$$\n\nIn that case, how would the Hubbard hamiltonian change? (To be described in terms of these new operators $$a$$ and $$b$$).\n\nYour Hubbard portion is still just the on-site energy. So, since now you have the bipartite lattice, you get\n\n$$H_{Hubbard} = U\\sum_{j}\\left[a^\\dagger_{j\\uparrow}a^\\dagger_{j\\downarrow}a_{j\\downarrow}a_{j\\uparrow} + b^\\dagger_{j\\uparrow}b^\\dagger_{j\\downarrow}b_{j\\downarrow}b_{j\\uparrow}\\right]$$\n\n\u2022 Shouldn't there be a different U for each operator? such as U_a and U_b? \u2013\u00a0Caterina Aug 25 at 21:12\n\u2022 I mean, since a only acts on site A and b on site B, I would expect to have different on-site Us? Or am I wrong? \u2013\u00a0Caterina Aug 25 at 21:13\n\u2022 Keep in mind that $U$ is just the Coulomb repulsion between the two spins on a given lattice site. Since you are looking at a Peierls-deformed lattice, we are assuming that the atom species of the $A$ sub lattice is the same as the $B$ sub lattice. This means that the way the electrons interact within an individual atom is sub lattice-independent, which means that $U$ is the same for both \u2013\u00a0IcyOtter Aug 26 at 2:40\n\u2022 Thanks a lot! Makes complete sense. I just have another question. Do you know if there is an operator that describes 'on-site' repulsion in double bonds rather than on one atom. Because the Hubbard contribution is not as realistic as considering that the 2 electrons are more likely to be in between the 2 atoms rather than located at either A or B. \u2013\u00a0Caterina Aug 26 at 9:34\n\u2022 Well, the idea is that you are using the tight-binding Hamiltonian. This means the electrons do, in fact, spend most of their time around the atom. Otherwise, we can't really write the hopping Hamiltonian since the bonding would substantially modify the onsite energy states \u2013\u00a0IcyOtter Aug 27 at 13:36","date":"2019-12-14 07:05:54","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\": 8, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7190170884132385, \"perplexity\": 489.4765326744029}, \"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-2019-51\/segments\/1575540585566.60\/warc\/CC-MAIN-20191214070158-20191214094158-00389.warc.gz\"}"}
| null | null |
\section{The hidden assumption of quantum field theory}
\label{S1}
Consider a free relativistic neutral scalar field with action
\begin{eqnarray}
I_{\rm S}=\displaystyle{\int}d^4x \tfrac{1}{2}\left[\partial_{\mu}\phi\partial^{\mu}\phi-m^2\phi^2\right],
\label{1.1}
\end{eqnarray}
and wave equation, Hamiltonian, and equal time commutation relation of the form
\begin{align}
[\partial_{\mu}\partial^{\mu}+m^2]\phi=0,
\nonumber\\
H=\displaystyle{\int } d^3x \tfrac{1}{2}[\dot{\phi}^2+\bar{\nabla}\phi\cdot \bar{\nabla}\phi+m^2\phi^2],
\nonumber\\
[\phi(\bar{x},t),\dot{\phi}(\bar{x}^{\prime},t)]=i\delta^3(\bar{x}-\bar{x}^{\prime}).
\label{1.2}
\end{align}
With $\omega_k=+(\bar{k}^2+m^2)^{1/2}$ solutions to the wave equation obey
\begin{eqnarray}
\phi(\bar{x},t)=\int \frac{d^3k}{\sqrt{(2\pi)^3 2\omega_k}}[a(\bar{k})e^{-i\omega_k t+i\bar{k}\cdot\bar{x}}+a^{\dagger}(\bar{k})e^{i\omega_k t-i\bar{k}\cdot\bar{x}}],
\label{1.3}
\end{eqnarray}
and with $[a(\bar{k}),a^{\dagger}(\bar{k}^{\prime})]=\delta^3(\bar{k}-\bar{k}^{\prime})$ the Hamiltonian is given by
%
\begin{eqnarray}
H=\frac{1}{2}\int d^3k[\bar{k}^2+m^2]^{1/2} \left[a^{\dagger}(\bar{k})a(\bar{k})+a(\bar{k})a^{\dagger}(\bar{k})\right].
\label{1.4}
\end{eqnarray}
%
Given (\ref{1.4}) we can introduce a no-particle state $\vert \Omega\rangle$ that obeys $a(\bar{k})\vert \Omega \rangle=0$ for each $\bar{k}$, and can identify it as the ground state of $H$. This procedure does not specify the value of $\langle \Omega\vert \Omega\rangle$.
For the theory the associated c-number propagator obeys
\begin{align}
&(\partial_t^2-\bar{\nabla}^2+m^2)D(x)=-\delta^4(x),
\label{1.5}
\end{align}
so that
\begin{align}
& D(x)
=\int \frac{d^4k}{(2\pi)^4}\frac{e^{-ik\cdot x}}{(k^2-m^2+i\epsilon)}.
\label{1.6}
\end{align}
If we identify the propagator as a vacuum matrix element of q-number fields, viz.
\begin{eqnarray}
D(x)=-i\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle,
\label{1.7}
\end{eqnarray}
then use of the equal time commutation relation gives
\begin{align}
&(\partial_t^2-\bar{\nabla}^2+m^2)(-i)\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle=-\langle \Omega \vert\Omega \rangle \delta^4(x).
\label{1.8}
\end{align}
Comparing with (\ref{1.5}) we see that we can only identify $D(x)$ as the matrix element $-i\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle$ if the vacuum is normalized to one, viz. $\langle \Omega\vert \Omega\rangle=1$. Now if the normalization of the vacuum is finite we of course can always rescale it to one. However, that presupposes that the normalization of the vacuum is not infinite. We are not aware of any proof in the literature that the normalization of the vacuum is not infinite (either in this particular case or in general), and taking it to be finite is a hidden assumption. So in this paper we shall present a procedure for determining whether the normalization of the vacuum state is finite or infinite. The procedure is based on generalizing to quantum field theory what we know from quantum mechanics.
This paper is organized as follows. In Sec. \ref{S2} we discuss the quantum-mechanical harmonic oscillator. In Sec. \ref{S3} we present an example, the above second-order derivative scalar field theory, in which $\langle \Omega\vert \Omega\rangle$ is finite. And in Secs. \ref{S4} to \ref{S6} we present an example, a second-order plus fourth-order derivative scalar field theory, in which $\langle \Omega\vert \Omega\rangle$ is not finite. In Sec. \ref{S7} we show that by continuing the dynamical variables into the complex plane we can construct a second-order plus fourth-order derivative scalar field theory inner product that is finite. In Secs. \ref{S8} and \ref{S12} we provide an analogous path integral analysis of second-order plus fourth-order derivative theories and reach the same conclusions. In Sec. \ref{S9} we discuss the effect of interactions. In Sec. \ref{S10} we discuss fermion field theories. In Secs. \ref{S11} and \ref{S12} we discuss the relevance of our study to the construction of a consistent, unitary and renormalizable quantum gravity theory. In Sec. \ref{S13} we make some final comments.
\section{The quantum-mechanical simple harmonic oscillator}
\label{S2}
For a simple harmonic oscillator with Hamiltonian $H=\tfrac{1}{2}[p^2+q^2]$ and commutator $[q,p]=i$, there are two sets of bases, the wave function basis and the occupation number space basis. The wave function basis is obtained by setting $p=-i\partial/\partial q$ in $H$ and then solving the Schr\"odinger wave equation $H\psi(q)=E\psi(q)$. In this way we obtain a ground state with energy $E_0=\tfrac{1}{2}$ and wave function $\psi_0(q)=e^{-q^2/2}$. For occupation number space we set $q=(a+a^{\dagger})/\sqrt{2}$ and $p=i(a^{\dagger}-a)/\sqrt{2}$. This yields $[a,a^{\dagger}]=1$ and $H=a^{\dagger} a+1/2$. We introduce a no-particle state $\vert \Omega \rangle$ that obeys $a\vert \Omega \rangle =0$, with $\vert \Omega \rangle$ being the occupation number space ground state with energy $E_0=\tfrac{1}{2}$. However, in and of itself this does not fix the norm $\langle \Omega\vert \Omega\rangle$ of the no-particle state or oblige it to be finite.
To fix the $\langle \Omega\vert \Omega\rangle$ norm we need to relate the ground states of the two bases. With $a=(q+ip)/\sqrt{2}$ we set
\begin{align}
\langle q \vert a\vert \Omega \rangle= \frac{1}{\sqrt{2}}\left(q+\frac{\partial}{\partial q}\right)\langle q \vert \Omega \rangle=0,
\label{2.1}
\end{align}
and find that $ \langle q \vert \Omega \rangle=e^{-q^2/2}$. We thus identify $\psi_0(q)=\langle q \vert \Omega \rangle$. We now calculate the standard Dirac norm for vacuum, and obtain
\begin{align}
\langle \Omega \vert \Omega \rangle=\int_{-\infty}^{\infty}dq \langle \Omega \vert q\rangle\langle q\vert \Omega \rangle=
\int_{-\infty}^{\infty}dq \psi^*_0(q)\psi_0(q)
=\int_{-\infty}^{\infty} dq e^{-q^2}=\sqrt{\pi}.
\label{2.2}
\end{align}
We thus establish that the Dirac norm of the no-particle state is finite. And on setting $\psi_0(q)=e^{-q^2/2}/\pi^{1/4}$ we normalize it to one. That we are able to do this is because we know the form of the wave function $\psi_0(q)$.
While this procedure is both straightforward and familiar, it works because both the wave function basis approach and occupation number basis approach have something in common, namely that they are both based on an infinite number of degrees of freedom. For the occupation number basis we can represent the creation and annihilation operators as infinite-dimensional matrices labeled by $\vert \Omega\rangle$, $a^{\dagger}\vert \Omega\rangle$, $a^{\dagger 2}\vert \Omega\rangle$ and so on. For the wave function basis the coordinate $q$ is a continuous variable that varies between $-\infty$ and $\infty$. The two sets of bases are both infinite dimensional, one discrete and the other continuous. The advantage of the continuous basis is that it enables to us to express the normalization of the vacuum state as an integral with an infinite range, an integral that is then either finite or infinite.
\section{The quantum field theory oscillator}
\label{S3}
In the quantum field theory case we do not know the form of the wave function solutions to $H\vert \psi\rangle=E\vert \psi\rangle$, since we cannot realize the canonical commutator given in (\ref{1.2}) as a differential relation. Specifically, we cannot satisfy (\ref{1.2}) by setting $\dot{\phi}(\bar{x},t)$ equal to $-i\partial /\partial \phi(\bar{x},t)$ (though we could introduce a functional derivative $\dot{\phi}(\bar{x},t)=-i\delta /\delta \phi(\bar{x},t)$).
However, we can express the Hamiltonian in terms of creation and annihilation operators.
So what we can then do is reverse engineer what we did in the quantum-mechanical case. We thus introduce
\begin{align}
a(\bar{k})= \frac{1}{\sqrt{2}}[q(\bar{k})+ip(\bar{k})],\quad a^{\dagger}(\bar{k})= \frac{1}{\sqrt{2}}[q(\bar{k})-ip(\bar{k})],
\label{3.1}
\end{align}
so that
\begin{align}
&[q(\bar{k}),p(\bar{k}^{\prime})]=i\delta^3(\bar{k}-\bar{k}^{\prime}),\quad H=\frac{1}{2}\int d^3k[\bar{k}^2+m^2]^{1/2} [p^2(\bar{k})+q^2(\bar{k})],
\nonumber\\
&\phi(\bar{x},t)=\frac{1}{\sqrt{2}}\int \frac{d^3k}{\sqrt{(2\pi)^3 2\omega_k}}\left[[q(\bar{k})+ip(\bar{k})]e^{-i\omega_k t+i\bar{k}\cdot\bar{x}}+[q(\bar{k})-ip(\bar{k})]e^{i\omega_k t-i\bar{k}\cdot\bar{x}}\right].
\label{3.2}
\end{align}
These $q(\bar{k})$ and $p(\bar{k})$ operators bear no relation to any physical position or momentum operators. Their only role here is to enable us to convert the discrete infinite-dimensional basis associated with each $a(\bar{k})$ and $a^{\dagger}(\bar{k})$ into a continuous one. Specifically, we can realize the $[q(\bar{k}),p(\bar{k}^{\prime})]$ commutator by $p(\bar{k}^{\prime})=-i\partial/\partial q(\bar{k}^{\prime})$, with $H$ then becoming a wave operator. In this way for each $\bar{k}$ we obtain a solution to the Schr\"odinger equation of the form $\psi(\bar{k})=e^{-q^2(\bar{k})/2}/\pi^{1/4}$. We can define a no-particle vacuum that obeys $a(\bar{k})\vert \Omega\rangle$ for each $\bar{k}$. For each $\bar{k}$ we have
\begin{align}
\langle q(\bar{k})\vert a(\bar{k})\vert \Omega \rangle= \frac{1}{\sqrt{2}}\left[q(\bar{k})+\frac{\partial}{\partial q(\bar{k})}\right]\langle q(\bar{k})\vert \Omega \rangle=0,
\label{3.3}
\end{align}
so that $\langle q(\bar{k})\vert \Omega\rangle=e^{-q^2(\bar{k})/2}/\pi^{1/4}$, and thus
\begin{align}
\langle \Omega\vert \Omega \rangle=\Pi_{\bar{k}}\int d q(\bar{k})\langle \Omega\vert q(\bar{k}) \rangle\langle q(\bar{k})\vert \Omega \rangle
=\Pi_{\bar{k}}\int d q(\bar{k})\frac{e^{-q^2(\bar{k})}}{\pi^{1/2}}=\Pi_{\bar{k}}1=1.
\label{3.4}
\end{align}
Thus the vacuum for the full $H$ obeys $\langle \Omega\vert\Omega\rangle=1$, to thus have a finite normalization. In this way we establish that the vacuum state of the free relativistic scalar field is normalizable.
The general prescription then is to convert the occupation number space Hamiltonian into a product of individual occupation number spaces each with its own $\bar{k}$, and then determine whether the equivalent wave mechanics ground state wave functions constructed this way have a finite normalization in the conventional Schr\"odinger wave mechanics theory sense. If they do, then so does the full vacuum $\vert \Omega \rangle$ of the full $H$. If on the other hand the equivalent wave mechanics wave functions are not normalizable, then neither is the full $\vert \Omega \rangle$. Our results for the second-order scalar field theory are also presented in \cite{Mannheim2022}, and in this paper we apply them to the higher-derivative theories that are of interest to quantum gravity, showing that for them the standard Dirac norm $\langle \Omega \vert \Omega \rangle$ is not finite.
Once we are able to show that the vacuum state of the free theory is normalizable, this will remain true in the presence of interactions if the interacting theory is renormalizable. Specifically, if the free theory $D(x)=-i\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle$ is finite, which it will be if the free theory $\langle \Omega\vert \Omega\rangle$ is, then the interacting $D(x)=-i\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle$ propagator will equally be finite after renormalization. Consequently, the renormalized $\langle \Omega\vert \Omega\rangle$ will be finite too. Thus to establish the finiteness of the vacuum normalization of a renormalizable interacting theory, we only need to be able to make a creation and annihilation representation of the free theory. We discuss the role of interactions further in Sec. \ref{S9}.
As well as providing a procedure for determining whether or not $\langle \Omega\vert \Omega\rangle$ is finite, since the procedure enables is to express the free second-order-derivative Hamiltonian $H$ as an ordinary derivative operator, it does so for interactions as well. Specifically, from (\ref{3.2}) we can write $\phi(\bar{x},t)$ as a derivative operator, viz.
\begin{align}
&\phi(\bar{x},t)=\frac{1}{\sqrt{2}}\int \frac{d^3k}{\sqrt{(2\pi)^3 2\omega_k}}\left[\left[q(\bar{k})+\frac{\partial}{\partial q(\bar{k})}\right]e^{-i\omega_k t+i\bar{k}\cdot\bar{x}}+\left[q(\bar{k})-\frac{\partial}{\partial q(\bar{k})}\right]e^{i\omega_k t-i\bar{k}\cdot\bar{x}}\right].
\label{3.5}
\end{align}
Thus the insertion of (\ref{3.5}) into an interaction Hamiltonian of the form $H_I=\lambda \int d^3x \phi^4(\bar{x},t)$ enables us to write $H_I$, and thus $H+H_I$, as a derivative operator. While this procedure enables us to in principle set up the Schr\"odinger problem for $H+H_I$ as a wave mechanics problem, it is still quite a formidable one, just as interacting field theories always have been. However, since it would write the theory in terms of fourth-order derivatives, and since it has an interacting vacuum that is normalized to one and excited states that all have positive norm, it does provide an example of a theory with higher derivatives that is free of negative-norm ghost states. Thus we can anticipate (and in fact find) that the second-order plus fourth-order quantum field theory that we discuss below in Secs. \ref{S4} - \ref{S7} will be ghost free too.
Since the above analysis is driven by the fact that the dimension of the occupation number space is infinite, the analysis can be carried out for any bosonic field. However, because of the Pauli principle, the occupation number space basis for fermions of any given $\bar{k}$ is finite dimensional. Thus we have to treat fermions separately, and do so in Sec. \ref{S10}.
\section{Higher-derivative quantum field theories}
\label{S4}
Having presented an example of a theory whose vacuum state is normalizable, we now present an example for which $\langle \Omega\vert \Omega\rangle$ is not finite. The example is based on a second-order plus fourth-order derivative neutral scalar field theory with action and equation of motion
\begin{eqnarray}
I_S&=&\frac{1}{2}\int d^4x\bigg{[}\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}
\partial^{\nu}\phi-(M_1^2+M_2^2)\partial_{\mu}\phi\partial^{\mu}\phi
+M_1^2M_2^2\phi^2\bigg{]},
\nonumber\\
&&(\partial_t^2-\bar{\nabla}^2+M_1^2)(\partial_t^2-\bar{\nabla}^2+M_2^2)
\phi(x)=0,
\label{4.1}
\end{eqnarray}
with ${\rm diag}[\eta_{\mu\nu}]=(1,-1,-1,-1)$. While we now study this particular model just for illustrative purposes, we note that it actually arises in quantum gravity studies, and in Secs. \ref{S11} and \ref{S12} we shall explore the implications of this study for quantum gravity.
For (\ref{4.1}) the associated propagator obeys
\begin{align}
&(\partial_t^2-\bar{\nabla}^2+M_1^2)(\partial_t^2-\bar{\nabla}^2+M_2^2)D(x)=-\delta^4(x),
\nonumber\\
&D(x)=-\int \frac{d^4k}{(2\pi)^4}\frac{e^{-ik\cdot x}}{(k^2-M_1^2)(k^2-M_2^2)}
=-\int \frac{d^4k}{(2\pi)^4}\frac{e^{-ik\cdot x}}{(M_1^2-M_2^2)}\left[\frac{1}{(k^2-M_1^2)}-\frac{1}{(k^2-M_2^2)}\right].
\label{4.2}
\end{align}
The energy-momentum tensor $T_{\mu\nu}$, the canonical momenta $\pi^{\mu}$ and $\pi^{\mu\lambda}$, and the equal-time commutators appropriate to the higher-derivative theory are given by \cite{Bender2008b}
\begin{align}
T_{\mu\nu}&=\pi_{\mu}\phi_{,\nu}+\pi_{\mu}^{\phantom{\mu}\lambda}\phi_{,\nu,\lambda}-\eta_{\mu\nu}{\cal L},
\nonumber\\
\pi^{\mu}&=\frac{\partial{\cal L}}{\partial \phi_{,\mu}}-\partial_{\lambda
}\left(\frac{\partial {\cal L}}{\partial\phi_{,\mu,\lambda}}\right)=-\partial_{\lambda}\partial^{\mu}\partial^{\lambda}\phi- (M_1^2+M_2^2)\partial^{\mu}\phi,
\nonumber\\
\pi^{\mu\lambda}&=\frac{\partial {\cal L}}{\partial \phi_{,\mu,\lambda}}=\partial^{\mu}\partial^{\lambda}\phi,
\nonumber\\
T_{00}&=\tfrac{1}{2}\pi_{00}^2+\pi_{0}\dot{\phi}+\tfrac{1}{2}(M_1^2+M_2^2)\dot{
\phi}^2-\tfrac{1}{2}M_1^2M_2^2\phi^2
-\tfrac{1}{2}\pi_{ij}\pi^{ij}+\tfrac{1}{2}(M_1^2+M_2^2)\phi_{,i}\phi^{,i}
\nonumber\\
&=\frac{1}{2}\ddot{\phi}^2-\tfrac{1}{2}(M_1^2+M_2^2)\dot{
\phi}^2-\dddot{\phi}\dot{\phi}-[\partial_i\partial^i\dot{\phi}]\dot{\phi}
-\tfrac{1}{2}M_1^2M_2^2\phi^2
-\tfrac{1}{2}\partial_i\partial_j\phi\partial^i\partial^j\phi+\tfrac{1}{2}(M_1^2+M_2^2)\partial_i\phi\partial^i\phi,
\nonumber\\
&[\phi(\bar{0},t),\dot{\phi}(\bar{x},t)]=0, \qquad[\phi(\bar{0},t),\ddot{\phi}(\bar{x},t)]=0, \qquad [\phi(\bar{0},t),\dddot{\phi}(\bar{x},t])=-i\delta^3(x).
\label{4.3}
\end{align}
With the use of these commutation relations we find that
\begin{eqnarray}
D(x)=i\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle
\label{4.4}
\end{eqnarray}
indeed satisfies the first equation given in (\ref{4.2}), provided that is that $\langle \Omega\vert \Omega\rangle=1$ \cite{footnote1}.
To check whether $\langle \Omega\vert \Omega\rangle$ actually is finite, we need to express the scalar field Hamiltonian $H_S=\int d^3x T_{00}$ in terms of creation and annihilation operators and then construct an equivalent wave mechanics. Given that the solutions to (\ref{4.1}) are plane waves, we set
\begin{eqnarray}
\phi(\bar{x},t)=\int \frac{d^3k}{(2\pi)^{3/2}}\left[a_1(\bar{k})e^{-i\omega_1 t+i\bar{k}\cdot\bar{x}}+a^{\dagger}_1(\bar{k})e^{i\omega_1 t-i\bar{k}\cdot\bar{x}}+a_2(\bar{k})e^{-i\omega_2 t+i\bar{k}\cdot\bar{x}}+a^{\dagger}_2(\bar{k})e^{i\omega_2 t-i\bar{k}\cdot\bar{x}}\right].
\label{4.5}
\end{eqnarray}
where $\omega_1=+(\bar{k}^2+M_1^2)^{1/2}$, $\omega_2=+(\bar{k}^2+M_2^2)^{1/2}$. Given the commutators in (\ref{4.3}) we obtain
\begin{eqnarray}
&&[a_1(\bar{k}),a^{\dagger}_{1}(\bar{k}^{\prime})]=[2(M_1^2-M_2^2)(\bar{k}^2+
M_1^2)^{1/2}]^{-1}\delta^3(\bar{k}-\bar{k}^{\prime}),
\nonumber\\
&&[a_2(\bar{k}),a^{\dagger}_{2}(\bar{k}^{\prime})]=-[2(M_1^2-M_2^2)(\bar{k}^2+
M_2^2)^{1/2}]^{-1}\delta^3(\bar{k}-\bar{k}^{\prime}),
\nonumber\\
&&[a_1(\bar{k}),a_{2}(\bar{k}^{\prime})]=0,\quad[a_1(\bar{k}),a^{\dagger}_{2}(\bar{k}^{\prime})]=0,\quad[a^{\dagger}_1(\bar{k}),a_{2}(\bar{k}^{\prime})]=0,\quad
[a^{\dagger}_1(\bar{k}),a^{\dagger}_{2}(\bar{k}^{\prime})]=0,
\label{4.6}
\end{eqnarray}
with the Hamiltonian then taking the form
\begin{eqnarray}
H_S&=&\frac{1}{2}\int d^3k\bigg{[}2(M_1^2-M_2^2)(\bar{k}^2+M_1^2)\left[a^{\dagger}_{1}(\bar{k})a_1(\bar{k})
+a_{1}(\bar{k})a^{\dagger}_1(\bar{k})\right]
\nonumber\\
&-&2(M_1^2-M_2^2)(\bar{k}^2+M_2^2)\left[a^{\dagger}_2(\bar{k})a_{2}(\bar{k})
+a_{2}(\bar{k})a^{\dagger}_2(\bar{k})\right]\bigg{]}
\nonumber\\
&=&\int d^3k\bigg{[}2(M_1^2-M_2^2)(\bar{k}^2+M_1^2)a^{\dagger}_{1}(\bar{k})a_1(\bar{k})
-2(M_1^2-M_2^2)(\bar{k}^2+M_2^2)a^{\dagger}_2(\bar{k})a_{2}(\bar{k})
\nonumber\\
&+&\frac{1}{2}(\bar{k}^2+M_1^2)^{1/2}\delta^3(0)+\frac{1}{2}(\bar{k}^2+M_2^2)^{1/2}\delta^3(0)\bigg{]},
\label{4.7}
\end{eqnarray}
where $(2\pi)^3\delta^3(0)$ is a quantization box volume $V$.
We note that with $M_1^2-M_2^2>0$ for definitiveness, we see negative signs in both $H_S$ and the $[a_2(\bar{k}),a^{\dagger}_{2}(\bar{k}^{\prime})]$ commutator, while noting that despite this the zero-point energy is positive. We shall see below that the negative sign concerns will be resolved once we settle the issue of the normalization of the vacuum. To do that we now descend to the quantum-mechanical limit of the theory, the Pais-Uhlenbeck oscillator model.
\section{Higher-derivative quantum mechanics}
\label{S5}
In order to study the Pauli-Villars regulator, in \cite{Pais1950} Pais and Uhlenbeck (${\rm PU}$) introduced a fourth-order quantum-mechanical oscillator model with action and equation of motion
\begin{eqnarray}
I_{\rm PU}=\frac{1}{2}\int dt\left[{\ddot z}^2-\left(\omega_1^2
+\omega_2^2\right){\dot z}^2+\omega_1^2\omega_2^2z^2\right],\qquad \ddddot{z}+(\omega_1^2+\omega_2^2)\ddot{z}+\omega_1\omega_2z^2=0,
\label{5.1}
\end{eqnarray}
where for definitiveness in the following we take $\omega_1>\omega_2$. As constructed this action possesses three variables $z$, $\dot{z}$ and $\ddot{z}$. This is too many for one oscillator but not enough for two. The system is thus a constrained system. And so we introduce a new variable $x=\dot{z}$ and its conjugate $p_x$. And using the method of Dirac constraints obtain the time-independent Hamiltonian \cite{Mannheim2000,Mannheim2005}
\begin{eqnarray}
H_{\rm PU}=\frac{p_x^2(t)}{2}+p_z(t)x(t)+\frac{1}{2}\left(\omega_1^2+\omega_2^2 \right)x^2(t)-\frac{1}{2}\omega_1^2\omega_2^2z^2(t),
\label{5.2}
\end{eqnarray}
with two sets of canonical equal-time commutators of the form
\begin{align}
[z(t),p_z(t)]=i, \qquad [x(t),p_x(t)]=i.
\label{5.3}
\end{align}
The terms in $H_{\rm PU}$ are in complete parallel to the first four terms in the field theory $T_{00}$ given in (\ref{4.3}), with the PU oscillator model being the nonrelativistic limit of the relativistic scalar field theory, with the spatial dependence having been frozen out. Since canonical commutators only involve time derivatives, freezing out the spatial dependence will still give the full dynamical content of the relativistic theory. In fact we can set $i=[z,p_z]\equiv [\phi,\pi_0]=[\phi,-\dddot{\phi}-(M_1^2+M_2^2)\dot{\phi}]=i\delta^3(x)$, to thus parallel the commutators given in (\ref{4.3}).
On setting $p_z=-i\partial_z$, $p_x=-i\partial_x$ the Schr\"odinger problem for $H_{\rm PU}$ can be solved analytically, with
the state with energy $(\omega_1+\omega_2)/2$ having a wave function that is of the form \cite{Mannheim2007}
\begin{align}
\psi_0(z,x)=\exp[\tfrac{1}{2}(\omega_1+\omega_2)\omega_1\omega_2z^2+i\omega_1\omega_2zx-\tfrac{1}{2}(\omega_1+\omega_2)x^2].
\label{5.4}
\end{align}
While this wave function is well behaved at large $x$, it diverges at large $z$, and consequently as a wave function it is not normalizable.
To relate this wave function to the no-particle vacuum $\vert \Omega \rangle$ we second quantize the theory. And with the wave equation given in (\ref{5.1}), and with $\dot{z}=i[H_{\rm PU},z]=x$, $\dot{x}=p_x$, $\dot{p}_x=-p_z-(\omega_1^2+\omega_2^2)x$, $\dot{p}_z=\omega_1^2\omega_2^2z$, we obtain
\begin{eqnarray}
z(t)&=&a_1e^{-i\omega_1t}+a_1^{\dagger}e^{i\omega_1t}+a_2e^{-i\omega_2t}+a_2^{\dagger}e^{i\omega_2t},
\nonumber\\
p_z(t)&=&i\omega_1\omega_2^2
[a_1e^{-i\omega_1t}-a_1^{\dagger}e^{i\omega_1t}]+i\omega_1^2\omega_2[a_2e^{-i\omega_2t}-a_2^{\dagger}e^{i\omega_2t}],
\nonumber\\
x(t)&=&-i\omega_1[a_1e^{-i\omega_1t}-a_1^{\dagger}e^{i\omega_1t}]-i\omega_2[a_2e^{-i\omega_2t}-a_2^{\dagger}e^{i\omega_2t}],
\nonumber\\
p_x(t)&=&-\omega_1^2 [a_1e^{-i\omega_1t}+a_1^{\dagger}e^{i\omega_1t}]-\omega_2^2[a_2e^{-i\omega_2t}+a_2^{\dagger}e^{i\omega_2t}],
\label{5.5}
\end{eqnarray}
and a Hamiltonian and commutator algebra of the form \cite{Mannheim2000}
\begin{align}
H_{\rm PU}&=2(\omega_1^2-\omega_2^2)(\omega_1^2 a_1^{\dagger}
a_1-\omega_2^2a_2^{\dagger} a_2)
+\tfrac{1}{2}(\omega_1+\omega_2),
\label{5.6}
\end{align}
\begin{align}
[a_1,a_1^{\dagger}]&=\frac{1}{2\omega_1(\omega_1^2-\omega_2^2)}, \qquad
[a_2,a_2^{\dagger}]=-\frac{1}{2\omega_2(\omega_1^2-\omega_2^2)}.
\label{5.7}
\end{align}
We note the similarity to (\ref{4.7}) and (\ref{4.6}).
As constructed, (\ref{5.6}) and (\ref{5.7}) admit of two inequivalent realizations as one can take $a_1$ and either $a_2$ or $a_2^{\dagger}$ to annihilate the vacuum. Thus one can define a Hilbert space in which $a_1\vert\Omega\rangle=0$, $a_2\vert\Omega\rangle=0$, or one can define a separate and distinct Hilbert space in which $a_1\vert\Omega\rangle=0$, $a_2^{\dagger}\vert\Omega\rangle=0$. In the Hilbert space in which $a_1\vert\Omega\rangle=0$, $a_2\vert\Omega\rangle=0$, we note that even though the $a_2^{\dagger} a_2$ term appears in $H_{\rm PU}$ with a minus sign, there is a compensating minus sign in the $[a_2,a_2^{\dagger}]$ commutator. In consequence, in this realization all energy eigenvalues of $H_{\rm PU}$ are positive, with the no-particle state $\vert \Omega \rangle$ that both $a_1$ and $a_2$ annihilate being the state of lowest energy. However, in this Hilbert space the matrix element $\langle \Omega\vert a_2a_2^{\dagger}\vert \Omega\rangle$ is negative, the ghost problem of higher-derivative theories. This problem has been solved in the literature \cite{Bender2008a,Bender2008b}, and we shall return to it in detail below.
Alternatively, if one takes $a_2^{\dagger}\equiv b_2$ to annihilate the vacuum, we obtain a commutator $[b_2,b_2^{\dagger}]=1/[2\omega_1(\omega_1^2-\omega_2^2)]$ that is positive. However then the energy spectrum of the Hamiltonian becomes unbounded from below, the familiar Ostrogradski instability of higher-derivative theories.
While we thus have to deal with a negative-norm problem or a negative energy problem, we note that the two problems do not occur in one and the same Hilbert space. Thus in any given Hilbert space we at most only have to deal with one. And since an energy spectrum that is unbounded from below is not physical, we shall work solely in the Hilbert space in which $a_1\vert\Omega\rangle=0$, $a_2\vert\Omega\rangle=0$. In Sec. \ref{S12}, which provides a quick explanation of our results, we shall show that the $a_1\vert\Omega\rangle=0$, $a_2\vert\Omega\rangle=0$ and $a_1\vert\Omega\rangle=0$, $a_2^{\dagger}\vert\Omega\rangle=0$ realizations correspond to different and thus inequivalent Feynman $i\epsilon$ prescriptions.
In the Hilbert space in which both $a_1$ and $a_2$ annihilate the vacuum the energy spectrum is bounded from below, and the energy of the ground state is $(\omega_1+\omega_2)/2$. On solving the time-dependent Schr\"odinger equation the wave function of the ground state is $\psi_0(z,x)e^{-i(\omega_1+\omega_2)t/2}$, where $\psi_0(z,x)$is given in (\ref{5.4}). For this wave function the normalization of $\vert \Omega \rangle$ is then given by
\begin{align}
\langle \Omega\vert \Omega\rangle=\int_{-\infty}^{\infty} dz\int_{-\infty}^{\infty}dx\langle \Omega\vert z,x\rangle\langle z,x\vert\Omega\rangle=\int_{-\infty}^{\infty} dz \int_{-\infty}^{\infty}dx \psi_0^*(z,x)\psi_0(z,x).
\label{5.8}
\end{align}
With $\psi_0(z,x)$ diverging at large $z$, this normalization integral is infinite. Thus we see that through our knowledge of the form of the ground state wave function as given in (\ref{5.4}) we are able to determine the normalization of the PU theory vacuum and establish that it is infinite. We can thus anticipate and will immediately show in Sec. \ref{S6} that this is also the case for the second-order plus fourth-order scalar quantum field theory as well. Then in Sec. \ref{S7} we will discuss what to do about it, with there actually being a mechanism for obtaining a finite normalization \cite{Bender2008a,Bender2008b}, one that also takes care of the fact that according to (\ref{5.7}) $\langle \Omega\vert a_2a_2^{\dagger}\vert\Omega\rangle$ is negative.
Already in \cite{Mannheim2007} it was known that for the ground state of the second-order plus fourth-order quantum-mechanical PU oscillator model the vacuum $\langle \Omega\vert \Omega\rangle$ norm was infinite. However, it was not known whether the vacuum $\langle \Omega\vert \Omega\rangle$ norm of the second-order plus fourth-order quantum field theory was finite or infinite. Using the procedure developed in this paper we can now determine whether it is finite or infinite, and in Sec. \ref{S6} we show that $\langle \Omega\vert \Omega\rangle$ is in fact infinite. Then in Sec. \ref{S7} we show that there is another inner product, viz. the
$\langle L\vert R\rangle$ overlap of left-eigenstates and right-eigenstates of the Hamiltonian, that is finite.
\section{The nonnormalizable vacuum of higher-derivative field theories}
\label{S6}
To determine the second-order plus fourth-order scalar field theory vacuum normalization we first need to invert (\ref{5.5}). This yields
\begin{align}
a_1e^{-i\omega_1t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[-\omega_2^2z(t)-p_x(t)+i\omega_1x(t)+i\frac{p_z(t)}{\omega_1}\right],
\nonumber\\
a_1^{\dagger}e^{i\omega_1t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[-\omega_2^2z(t)-p_x(t)-i\omega_1x(t)-i\frac{p_z(t)}{\omega_1}\right],
\nonumber\\
a_2e^{-i\omega_2t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[\omega_1^2z(t)+p_x(t)-i\omega_2x(t)-i\frac{p_z(t)}{\omega_2}\right],\
\nonumber\\
a_2^{\dagger}e^{i\omega_2t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[\omega_1^2z(t)+p_x(t)+i\omega_2x(t)+i\frac{p_z(t)}{\omega_2}\right].
\label{6.1}
\end{align}
On generalizing to each $\bar{k}$ and setting $\omega_1(\bar{k}) =+(\bar{k}^2+M_1^2)^{1/2}$, $\omega_2(\bar{k}) =+(\bar{k}^2+M_2^2)^{1/2}$, we obtain
\begin{align}
a_1(\bar{k})e^{-i\omega_1(\bar{k})t}&=\frac{1}{2(M_1^2-M_2^2)}\left[-\omega_2^2(\bar{k})z(\bar{k},t)-p_x(\bar{k},t)+i\omega_1(\bar{k})x(\bar{k},t)+i\frac{p_z(\bar{k},t)}{\omega_1(\bar{k})}\right],
\nonumber\\
a_1^{\dagger}(\bar{k})e^{i\omega_1(\bar{k})t}&=\frac{1}{2(M_1^2-M_2^2)}\left[-\omega_2^2(\bar{k})z(\bar{k},t)-p_x(\bar{k},t)-i\omega_1(\bar{k})x(\bar{k},t)-i\frac{p_z(\bar{k},t)}{\omega_1(\bar{k})}\right],
\nonumber\\
a_2(\bar{k})e^{-i\omega_2(\bar{k})t}&=\frac{1}{2(M_1^2-M_2^2)}\left[\omega_1^2(\bar{k})z(\bar{k},t)+p_x(\bar{k},t)-i\omega_2(\bar{k})x(\bar{k},t)-i\frac{p_z(\bar{k},t)}{\omega_2(\bar{k})}\right],
\nonumber\\
a_2^{\dagger}(\bar{k})e^{i\omega_2(\bar{k})t}&=\frac{1}{2(M_1^2-M_2^2)}\left[\omega_1^2(\bar{k})z(\bar{k},t)+p_x(\bar{k},t)+i\omega_2(\bar{k})x(\bar{k},t)+i\frac{p_z(\bar{k},t)}{\omega_2(\bar{k})}\right].
\label{6.2}
\end{align}
Inverting (\ref{6.2}) gives
\begin{align}
z(\bar{k},t)&=a_1(\bar{k})e^{-i\omega_1(\bar{k})t}+a_1^{\dagger}(\bar{k})e^{i\omega_1(\bar{k})t}+a_2(\bar{k})e^{-i\omega_2(\bar{k})t}+a_2^{\dagger}(\bar{k})^{i\omega_2(\bar{k})t},
\nonumber\\
p_z(\bar{k},t)&=i\omega_1(\bar{k})\omega_2^2(\bar{k})
[a_1(\bar{k})e^{-i\omega_1(\bar{k})t}-a_1^{\dagger}(\bar{k})e^{i\omega_1(\bar{k})t}]+i\omega_1^2(\bar{k})\omega_2(\bar{k})[a_2(\bar{k})e^{-i\omega_2(\bar{k})t}-a_2^{\dagger}(\bar{k})e^{i\omega_2(\bar{k})t}],
\nonumber\\
x(\bar{k},t)&=-i\omega_1(\bar{k})[a_1(\bar{k})e^{-i\omega_1(\bar{k})t}-a_1^{\dagger}(\bar{k})e^{i\omega_1(\bar{k})t}]-i\omega_2(\bar{k})[a_2(\bar{k})e^{-i\omega_2(\bar{k})t}-a_2^{\dagger}(\bar{k})^{i\omega_2(\bar{k})t}],
\nonumber\\
p_x(\bar{k},t)&=-\omega_1^2(\bar{k})[a_1(\bar{k})e^{-i\omega_1(\bar{k})t}+a_1^{\dagger}(\bar{k})e^{i\omega_1(\bar{k})t}]-\omega_2^2(\bar{k})[a_2(\bar{k})e^{-i\omega_2(\bar{k})t}+a_2^{\dagger}(\bar{k})^{i\omega_2(\bar{k})t}].
\label{6.3}
\end{align}
From (\ref{6.3}) and the commutation relations given in (\ref{4.6}) it follows that
\begin{eqnarray}
&&[z(\bar{k},t),p_z(\bar{k}^{\prime},t)]=\delta^3(\bar{k}-\bar{k}^{\prime}),\qquad [x(\bar{k},t),p_x(\bar{k}^{\prime},t)]=\delta^3(\bar{k}-\bar{k}^{\prime}),
\nonumber\\
&&[z(\bar{k},t),x(\bar{k}^{\prime},t)]=0,\quad[z(\bar{k},t),p_x(\bar{k}^{\prime},t)]=0,\quad[p_z(\bar{k},t),x(\bar{k}^{\prime},t)]=0,\quad
[p_z(\bar{k},t),p_x(\bar{k}^{\prime},t)]=0.
\label{6.4}
\end{eqnarray}
Insertion of (\ref{6.2}) into the Hamiltonian given in (\ref{4.7}) then yields an equivalent, time-independent Hamiltonian
\begin{eqnarray}
H_S&=&\int d^3k\bigg{[}\frac{p_x^2(\bar{k},t)}{2}+p_z(\bar{k},t)x(\bar{k},t)+\frac{1}{2}\left[\omega_1^2(\bar{k})+\omega_2^2(\bar{k}) \right]x^2(\bar{k},t)-\frac{1}{2}\omega_1^2(\bar{k})\omega_2^2(\bar{k})z^2(\bar{k},t)\bigg{]}.
\label{6.5}
\end{eqnarray}
For each momentum state we recognize the quantum field theory Hamiltonian $H_S$ given in (\ref{6.5}) as being of precisely the form of the quantum-mechanical $H_{\rm PU}$ Hamiltonian that is given in (\ref{5.2}).
We can now proceed as in the second-order scalar quantum field theory discussed above and represent the commutators by
\begin{eqnarray}
&&\left[z(\bar{k},t), -i\frac{\partial}{\partial z(\bar{k}^{\prime},t)}\right]=\delta^3(\bar{k}-\bar{k}^{\prime}),\qquad \left[x(\bar{k},t),-i\frac{\partial}{\partial x(\bar{k}^{\prime},t)}\right]=\delta^3(\bar{k}-\bar{k}^{\prime}).
\label{6.6}
\end{eqnarray}
With the vacuum obeying $a_1(\bar{k})\vert \Omega\rangle=0$, $a_2(\bar{k})\vert \Omega\rangle=0$ for each $\bar{k}$, from (\ref{6.2}) we obtain
\begin{align}
&\langle z(\bar{k}),x(\bar{k})\vert a_1(\bar{k})\vert \Omega\rangle=\frac{1}{2(M_1^2-M_2^2)}\left[-\omega_2^2(\bar{k})z(\bar{k})+i\frac{\partial}{\partial x(\bar{k})}+i\omega_1(\bar{k})x(\bar{k})+\frac{1}{\omega_1(\bar{k})}\frac{\partial}{\partial z(\bar{k})}\right]\langle z(\bar{k}),x(\bar{k})\vert\Omega\rangle=0,
\nonumber\\
&\langle z(\bar{k}),x(\bar{k})\vert a_2(\bar{k})\vert \Omega\rangle=\frac{1}{2(M_1^2-M_2^2)}\left[\omega_1^2(\bar{k})z(\bar{k})-i\frac{\partial}{\partial x(\bar{k})}-i\omega_2(\bar{k})x(\bar{k})-\frac{1}{\omega_2(\bar{k})}\frac{\partial}{\partial z(\bar{k})}\right]\langle z(\bar{k}),x(\bar{k})\vert\Omega\rangle=0,
\label{6.7}
\end{align}
for each $\bar{k}$. From (\ref{6.7}) it follows that for each $\bar{k}$ we can identify each $\langle z(\bar{k}),x(\bar{k})\vert\Omega\rangle$ with the PU oscillator ground state wave function $\psi_0(z(\bar{k}),x(\bar{k}))$, which, analogously to (\ref{5.4}), is given by
\begin{align}
\psi_0(z(\bar{k}),x(\bar{k}))=\exp[\tfrac{1}{2}[\omega_1(\bar{k})+\omega_2(\bar{k})]\omega_1(\bar{k})\omega_2(\bar{k})z^2(\bar{k})+i\omega_1(\bar{k})\omega_2(\bar{k})z(\bar{k})x(\bar{k})-\tfrac{1}{2}[\omega_1(\bar{k})+\omega_2(\bar{k})]x^2(\bar{k})].
\label{6.8}
\end{align}
Consequently, the normalization of the vacuum is given by
\begin{align}
\langle \Omega\vert \Omega\rangle&=\Pi_{\bar{k}}\int_{-\infty}^{\infty} dz(\bar{k})\int_{-\infty}^{\infty}dx(\bar{k})\langle \Omega\vert z(\bar{k}),x(\bar{k})\rangle\langle z(\bar{k}),x(\bar{k})\vert\Omega\rangle
\nonumber\\
&=\Pi_{\bar{k}}\int_{-\infty}^{\infty} dz(\bar{k}) \int_{-\infty}^{\infty}dx(\bar{k}) \psi_0^*(z(\bar{k}),x(\bar{k}))\psi_0(z(\bar{k}),x(\bar{k})).
\label{6.9}
\end{align}
With each $\psi_0(z(\bar{k}),x(\bar{k}))$ diverging at large $z(\bar{k})$, we thus establish that the normalization of the field theory vacuum is infinite. Thus whatever is the normalization of the vacuum in the associated wave-mechanical limit translates into the same normalization in the quantum field theory.
\section{How to obtain a normalizable vacuum}
\label{S7}
In analyzing the second-order plus fourth-order scalar field theory we note that with a conventional Hermitian field $\phi(x)$, and thus with $a_1^{\dagger}(\bar{k})$ and $a_2^{\dagger}(\bar{k})$ being the Hermitian conjugates of $a_1(\bar{k})$ and $a_2(\bar{k})$, the $a_2^{\dagger}(\bar{k})a_2(\bar{k})$ product would be positive definite and the energy spectrum of $H_S$ as given in (\ref{4.7}) would initially be unbounded from below, this being the familiar Ostrogradski instability of higher-derivative theories with Hermitian fields. However, from (\ref{4.6}) we see that $\langle \Omega\vert a_2(\bar{k})a_2^{\dagger}(\bar{k})\vert \Omega \rangle$ would be negative. This would imply the potential presence of ghost states of negative norm, with it then not being the case that a product such as $a_2(\bar{k})a_2^{\dagger}(\bar{k})$ could be positive definite. If one accepts this then matrix elements of the $-2(M_1^2-M_2^2)(\bar{k}^2+M_2^2)a^{\dagger}_2(\bar{k})a_2(\bar{k})$ term in $H_S$ would be compensated for by the ghost signature, and the energy spectrum of $H_S$ would then be bounded from below. While this takes care of the unboundedness from below of the energy spectrum, it appears to do so at a high price, namely the potential presence of unitarity-violating ghost states. But if $a_2^{\dagger}(\bar{k})$ is the Hermitian conjugate of $ a_2(\bar{k})$ then $\langle \Omega\vert a_2(\bar{k})a_2^{\dagger}(\bar{k})\vert \Omega \rangle$ would have to be positive. Thus despite the dagger notation $a_2^{\dagger}(\bar{k})$ could not be the Hermitian conjugate of $a_2(\bar{k})$. Hence our starting assumption that $\phi(x)$ is Hermitian could not be valid. Consequently, the Hamiltonian that is built out of the $\phi(x)$ field could not be Hermitian either. And in fact we have actually established that it is not, since the diverging of $\psi_0(z(\bar{k}),x(\bar{k}))$ at large $z(\bar{k})$ means that in an integration by parts we could not drop surface terms, with the presence of such surface terms preventing Hermiticity or self-adjointness. With the eigenstates of the Hamiltonian not being normalizable, there not only are potential negative-norm states present, they are infinitely negative.
Surprisingly, it is this very inability to drop surface terms in an integration by parts that actually saves the theory \cite{Bender2008a,Bender2008b}. Specifically, we have seen that we are working with a Hamiltonian $H_S$ (and likewise $H_{\rm PU}$) that is not Hermitian. However, all the energy eigenvalues associated with $H_S$ and $H_{\rm PU}$ are real. Now Hermiticity is only sufficient for real eigenvalues, with the necessary condition \cite{bender2010PT,mannheim2018antilinearity} being that the Hamiltonian have an antilinear symmetry \cite{footnote2}. The theory thus falls into the class of $PT$ theories ($P$ is the linear parity operator and $T$ is the antilinear time reversal operator) developed by Bender and collaborators \cite{bender2007making,bender2019pt}. Critical to the $PT$ program is that the wave functions be normalizable in some domain in the complex plane, a domain known technically as a Stokes wedge. Since the wave functions are not normalizable with real $z$ or real $z(\bar{k})$, we have to continue $z$ and $z(\bar{k})$ into the complex plane in order to make them normalizable. Then the theory is well-defined, with, as we discuss in Sec. \ref{S8}, the domain of the measure needed for the Minkowski path integral accordingly also having to be continued into the complex plane in order to make it be well-defined too \cite{Bender2008b,mannheim2018antilinearity}. For the particular case of $\psi_0(z,x)$ and $\psi_0(z(\bar{k}), x(\bar{k}))$, replacing $z$ by $-iz$ and $z(\bar{k})$ by $-iz(\bar{k})$ would then make both $\psi_0(z,x)$ and $\psi_0(z(\bar{k}), x(\bar{k}))$ normalizable. (We have no need to modify $x$ or $x(\bar{k})$ since the wave functions already are well behaved when these quantities become large.)
To achieve the continuation of $z$ or $z(\bar{k})$ at the level of operators we effect similarity transformations as they preserve both energy eigenvalues and canonical commutators. We introduce
\begin{align}
S({\rm PU})=e^{\pi p_zz/2},\qquad S(S)=e^{\pi\int d^3x\pi_0(\bar{x},t)\phi(\bar{x},t)/2},
\label{7.1}
\end{align}
and obtain
\begin{align}
&S({\rm PU})zS({\rm PU})^{-1}=-iz\equiv y,\qquad S({\rm PU})p_zS({\rm PU})^{-1}=ip_z\equiv q,
\nonumber\\
&
S(S)z(\bar{k})S(S)^{-1}=-iz(\bar{k})\equiv y(\bar{k}),\qquad S(S)p_z(\bar{k})S(S)^{-1}=ip_z(\bar{k})\equiv q(\bar{k}).
\label{7.2}
\end{align}
\subsection{The PU case}
For the PU oscillator this leads to
\begin{align}
S({\rm PU})H_{\rm PU}S({\rm PU})^{-1}=\bar{H}_{\rm PU}&=\tfrac{1}{2}p_x^2(t)-iq(t)x(t)+\tfrac{1}{2}\left(\omega_1^2+\omega_2^2 \right)x^2(t)+\tfrac{1}{2}\omega_1^2\omega_2^2y^2(t),
\nonumber\\
[y(t),q(t)]&=i, \qquad [x(t),p_x(t)]=i.
\label{7.3}
\end{align}
%
Since the $\psi(y,x)$ wave functions are now well behaved at infinity, the $\bar{H}_{\rm PU}$ Hamiltonian is self-adjoint. However, because of the $-iq(t)x(t)$ term it is not Hermitian. Instead it is $PT$ symmetric. Specifically, with $p$ and $q$ being taken to be $PT$ even and $y$ and $x$ being taken to be $PT$ odd \cite{Bender2008b}, the $PT$ invariance of $\bar{H}_{\rm PU}$ and of the $[y,q]=i$ and $[x,p_x]=i$ commutators follows. Now when a Hamiltonian is not Hermitian the action of it to the right and the action of it to the left are not related by Hermitian conjugation. Thus in general one must distinguish between right and left eigenstates, both for the vacuum and the states that can be excited out of it. Thus we represent the $[y,q]=i$ and $[x,p_x]=i$ commutators by $q=-i\overrightarrow{\partial_y}$, $p_x=-i\overrightarrow{\partial_x}$ when acting to the right, and by $q=i\overleftarrow{\partial_y}$, $p_x=i\overleftarrow{\partial_x}$ when acting to the left. This then leads to right and left ground state wave functions of the form \cite{Bender2008b}
\begin{align}
\psi_0^R(y,x)&=\exp[-\tfrac{1}{2}(\omega_1+\omega_2)\omega_1\omega_2y^2-\omega_1\omega_2yx-\tfrac{1}{2}(\omega_1+\omega_2)x^2],
\nonumber\\
\psi_0^L(y,x)&=\exp[-\tfrac{1}{2}(\omega_1+\omega_2)\omega_1\omega_2y^2+\omega_1\omega_2yx-\tfrac{1}{2}(\omega_1+\omega_2)x^2].
\label{7.4}
\end{align}
Given these wave functions the vacuum normalization is given by \cite{Bender2008b}
%
\begin{align}
\langle \Omega^{L}\vert \Omega^R\rangle&=\int_{-\infty}^{\infty} dy\int_{-\infty}^{\infty}dx\langle \Omega^{L}\vert y,x\rangle\langle y,x\vert\Omega^R\rangle=\int_{-\infty}^{\infty} dy\int_{-\infty}^{\infty}dx\psi_0^L(y,x)\psi_0^R(y,x)
\nonumber\\
&=\int_{-\infty}^{\infty} dy\int_{-\infty}^{\infty}dx\exp[-(\omega_1+\omega_2)\omega_1\omega_2y^2-(\omega_1+\omega_2)x^2]
=\frac{\pi}{(\omega_1\omega_2)^{1/2}(\omega_1+\omega_2)},
\label{7.5}
\end{align}
%
with the vacuum state thus being normalizable. In the following we shall understand the wave functions to have been normalized to one, so that $\int dydx\psi_0^L(y,x)\psi_0^R(y,x)=1$ and $\langle \Omega^{L}\vert \Omega^R\rangle=1$ \cite{footnote3}.
With the above $PT$ assignments and with $\dot{y}=i[\bar{H}_{\rm PU},y]=-ix$, $\dot{x}=p_x$, $\dot{p}_x=iq -(\omega_1^2+\omega_2^2)x$, $\dot{q}=-\omega_1^2\omega_2^2y$, we set
\begin{align}
y(t)&=-ia_1e^{-i\omega_1t}+a_2e^{-i\omega_2t}-i\hat{a}_1e^{i\omega_1t}+\hat{a}_2
e^{i\omega_2t},
\nonumber\\
x(t)&=-i\omega_1a_1e^{-i\omega_1t}+\omega_2a_2e^{-i\omega_2t}+i\omega_1\hat{a}_1
e^{i\omega_1t}-\omega_2\hat{a}_2e^{i\omega_2t},
\nonumber\\
p_x(t)&=-\omega_1^2a_1e^{-i\omega_1t}-i\omega_2^2a_2e^{-i\omega_2t}-\omega_1
^2\hat{a}_1e^{i\omega_1t}-i\omega_2^2\hat{a}_2e^{i\omega_2t},
\nonumber\\
q(t)&=\omega_1\omega_2[-\omega_2a_1e^{-i\omega_1t}-i\omega_1a_2e^{-i
\omega_2t}+\omega_2\hat{a}_1e^{i\omega_1t}+i\omega_1\hat{a}_2e^{i\omega_2t}],
\nonumber\\
a_1e^{-i\omega_1t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[-i\omega_2^2y(t)-p_x(t)+i\omega_1x(t)+\frac{q(t)}{\omega_1}\right],
\nonumber\\
\hat{a}_1e^{+i\omega_1t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[-i\omega_2^2y(t)-p_x(t)-i\omega_1x(t)-\frac{q(t)}{\omega_1}\right],
\nonumber\\
ia_2e^{-i\omega_2t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[i\omega_1^2y(t)+p_x(t)-i\omega_2x(t)-\frac{q(t)}{\omega_2}\right],
\nonumber\\
i\hat{a}_2e^{+i\omega_2t}&=\frac{1}{2(\omega_1^2-\omega_2^2)}\left[i\omega_1^2y(t)+p_x(t)+i\omega_2x(t)+\frac{q(t)}{\omega_2}\right].\label{7.6}
\end{align}
In (\ref{7.6}) we have introduced $a_1$, $a_2$, $\hat{a}_1$ and $\hat{a}_2$, with the four creation and annihilation operators obeying $PTa_1TP=a_1$, $PTa_2TP=-a_2$, $PT\hat{a}_1TP=\hat{a}_1$, $PT\hat{a}_2TP=-\hat{a}_2$, so as to enforce the $PT$ assignments of $y$, $x$, $p_x$ and $q$. Comparing with (\ref{5.5}) we have $(a_1,a_2,a_1^{\dagger},a^{\dagger}_2)\rightarrow (a_1,ia_2,\hat{a}_1,i\hat{a}_2)$.
With (\ref{7.3}) and (\ref{7.6}) the Hamiltonian is given by
\begin{equation}
\bar{H}_{\rm PU}=2(\omega_1^2-\omega_2^2)\left(\omega_1^2\hat{a}_1a_1+\omega_2^2
\hat{a}_2a_2\right)+\tfrac{1}{2}(\omega_1+\omega_2),
\label{7.7}
\end{equation}
and the operator commutation algebra is given by
\begin{align}
&[a_1,\hat{a}_1]=\frac{1}{2\omega_1(\omega_1^2-\omega_2^2)},\quad
[a_2,\hat{a}_2]=\frac{1}{2\omega_2(\omega_1^2-\omega_2^2)},
\nonumber\\
&[a_1,a_2]=0,\quad[a_1,\hat{a}_2]=0,\quad[\hat{a}_1,a_2]=0,\quad
[\hat{a}_1,\hat{a}_2]=0.
\label{7.8}
\end{align}
With the $PT$ assignments of $a_1$, $a_2$, $\hat{a}_1$ and $\hat{a}_2$, we confirm the $PT$ invariance of (\ref{7.7}) and (\ref{7.8}).
In (\ref{7.7}) and (\ref{7.8}) the relative signs are all positive (we take $\omega_1>\omega_2>0$ for definitiveness), so these
equations define a standard positive energy, positive norm, two-dimensional harmonic oscillator system. Given the creation and annihilation operators the left and right vacua are defined by
\begin{align}
\langle \Omega^{L}\vert \hat{a}_1=0, \qquad \langle \Omega^{L}\vert \hat{a}_2=0,\qquad a_1\vert \Omega^R\rangle=0,\qquad a_2\vert \Omega^R\rangle=0.
\label{7.9}
\end{align}
By exciting modes out of the left and right vacua we can build excited states that have positive norm \cite{Bender2008a}, viz. $\langle n^L\vert m^R\rangle=\delta_{nm}$, and obey a completeness relation
\begin{eqnarray}
\sum \vert n_1^R\rangle\langle n_1^{L}\vert +\sum \vert n_2^R\rangle\langle n_2^{L}\vert =I.
\label{7.10}
\end{eqnarray}
Even though these norms are all positive, the insertion of (\ref{7.10}) into $-i\langle\Omega^{L}\vert T[y(t)y(0)]\vert \Omega^R \rangle$ (corresponding to $+i\langle\Omega^{L}\vert T[z(t)z(0)]\vert \Omega^R \rangle$) generates the relative minus sign in the nonrelativistic limit of the $-[1/(k^2-M_1^2)-1/(k^2-M_2^2)]/(M_1^2-M_2^2)$ propagator given in (\ref{4.2}), viz. $-[1/(\omega^2-\omega_1^2)-1/(\omega^2-\omega_2^2)]/(\omega_1^2-\omega_2^2)$. We thus establish the consistency and physical viability of the similarity transformed $PU$ oscillator theory.
\subsection{The relativistic case}
For $S(S)H_SS(S)^{-1}=\bar{H}_S$ we introduce creation and annihilation operators for $\bar{\phi}=S(S)\phi S(S)^{-1}=-i\phi(x)$ of the form
\begin{align}
\bar{\phi}(x)=\int \frac{d^3k}{(2\pi)^{3/2}}\left [-ia_1(\bar{k})e^{-i\omega_1(\bar{k})t+i\bar{k}\cdot \bar{x}}+a_2(\bar{k})e^{-i\omega_2(\bar{k}) t+i\bar{k}\cdot \bar{x}}-i\hat{a}_1(\bar{k})e^{i\omega_1(\bar{k}) t-i\bar{k}\cdot \bar{x}}+\hat{a}_2(\bar{k})e^{i\omega_2(\bar{k}) t-i\bar{k}\cdot \bar{x}}\right].
\label{7.11}
\end{align}
Comparing with (\ref{4.5}) we have $(a_1(\bar{k}),a_2(\bar{k}),a_1^{\dagger}(\bar{k}),a^{\dagger}_2(\bar{k}))\rightarrow (a_1(\bar{k}),ia_2(\bar{k}),\hat{a}_1(\bar{k}),i\hat{a}_2(\bar{k}))$.
Like $y(t)$, $\bar{\phi}(x)$ is $PT$ odd. As constructed the Hamiltonian is $PT$ even. (In general, even without Hermiticity one still has $CPT$ symmetry \cite{mannheim2018antilinearity}, but since the fields are neutral $C$ is separately conserved, so in this case $CPT$ defaults to $PT$.)
The field-theoretic $PT$-even Hamiltonian and $PT$-preserving commutation relations are given by \cite{Bender2008b}
\begin{eqnarray}
S(S)H_SS(S)^{-1}=\bar{H}_S&=&\frac{1}{2}\int d^3k\bigg{[}2(M_1^2-M_2^2)(\bar{k}^2+M_1^2)
\left[\hat{a}_{1}(\bar{k})a_1(\bar{k})+a_{1}(\bar{k})\hat{a}_1(\bar{k})\right]
\nonumber\\
&+&2(M_1^2-M_2^2)(\bar{k}^2+M_2^2)\left[\hat{a}_{2}(\bar{k})a_2(\bar{k})+a_{2}(\bar{k})\hat{a}_2(\bar{k})\right]\bigg{]},
\label{7.12}
\end{eqnarray}
and
\begin{eqnarray}
&& [\dot{\bar{\phi}}(\bar{x},t),\bar{\phi}(0)]=0,\qquad [\ddot{\bar{\phi}}(\bar{x},t),\bar{\phi}(0)]=0,\qquad [\dddot{\bar{\phi}}(\bar{x},t),\bar{\phi}(0)]=i\delta^3(x),
\nonumber\\
&&[a_1(\bar{k}),\hat{a}_{1}(\bar{k}^{\prime})]=[2(M_1^2-M_2^2)(\bar{k}^2+
M_1^2)^{1/2}]^{-1}\delta^3(\bar{k}-\bar{k}^{\prime}),
\nonumber\\
&&[a_2(\bar{k}),\hat{a}_{2}(\bar{k}^{\prime})]=[2(M_1^2-M_2^2)(\bar{k}^2+
M_2^2)^{1/2}]^{-1}\delta^3(\bar{k}-\bar{k}^{\prime}),
\nonumber\\
&&[a_1(\bar{k}),a_{2}(\bar{k}^{\prime})]=0,
\quad [a_1(\bar{k}),\hat{a}_{2}(\bar{k}^{\prime})]=0,
\quad [\hat{a}_{1}(\bar{k}),a_{2}(\bar{k}^{\prime})]=0,
\quad [\hat{a}_{1}(\bar{k}),\hat{a}_{2}(\bar{k}^{\prime})]=0.
\label{7.13}
\end{eqnarray}
With all relative signs being positive (we take $M_1^2>M_2^2$ for definitiveness), there are no states of negative norm or of negative energy. The discussion completely parallels that of the $PU$ oscillator model given above. We introduce
\begin{align}
y(\bar{k},t)&=-ia_1(\bar{k})e^{-i\omega_1(\bar{k})t}+a_2(\bar{k})e^{-i\omega_2(\bar{k})t}-i\hat{a}_1(\bar{k})e^{i\omega_1(\bar{k})t}+\hat{a}_2(\bar{k})
e^{i\omega_2(\bar{k})t},
\nonumber\\
x(\bar{k},t)&=-i\omega_1(\bar{k})a_1(\bar{k})e^{-i\omega_1(\bar{k})t}+\omega_2(\bar{k})a_2(\bar{k})e^{-i\omega_2(\bar{k})t}+i\omega_1(\bar{k})\hat{a}_1(\bar{k})
e^{i\omega_1(\bar{k})t}-\omega_2(\bar{k})\hat{a}_2(\bar{k})e^{i\omega_2(\bar{k})t},
\nonumber\\
p_x(\bar{k},t)&=-\omega_1^2(\bar{k})a_1(\bar{k})e^{-i\omega_1(\bar{k})t}-i\omega_2^2(\bar{k})a_2(\bar{k})e^{-i\omega_2(\bar{k})t}-\omega_1
^2(\bar{k})\hat{a}_1(\bar{k})e^{i\omega_1(\bar{k})t}-i\omega_2^2(\bar{k})\hat{a}_2(\bar{k})e^{i\omega_2(\bar{k})t},
\nonumber\\
q(\bar{k},t)&=\omega_1(\bar{k})\omega_2(\bar{k})[-\omega_2(\bar{k})a_1(\bar{k})e^{-i\omega_1(\bar{k})t}-i\omega_1(\bar{k})a_2(\bar{k})e^{-i\omega_2(\bar{k})t}+\omega_2(\bar{k})\hat{a}_1(\bar{k})e^{i\omega_1(\bar{k})t}+i\omega_1(\bar{k})\hat{a}_2(\bar{k})e^{i\omega_2(\bar{k})t}],
\label{7.14}
\end{align}
with $\bar{H}_S$ then taking the form
\begin{align}
\bar{H}_S=\int d^3k\bigg{[}\frac{p_x^2(\bar{k},t)}{2}-iq(\bar{k},t)x(\bar{k},t)+\frac{1}{2}\left[\omega_1^2(\bar{k})+\omega_2^2(\bar{k}) \right]x^2(\bar{k},t)+\frac{1}{2}\omega_1^2(\bar{k})\omega_2^2(\bar{k})y^2(\bar{k},t)\bigg{]}.
\label{7.15}
\end{align}
Introducing left and right vacua that obey
\begin{align}
\langle \Omega^{L}\vert \hat{a}_1(\bar{k})=0, \qquad \langle \Omega^{L}\vert \hat{a}_2(\bar{k})=0,\qquad a_1(\bar{k})\vert \Omega^R\rangle=0,\qquad a_2(\bar{k})\vert \Omega^R\rangle=0
\label{7.16}
\end{align}
for all $\bar{k}$, we find that
\begin{align}
\langle \Omega^{L}\vert \bar{H}_S\vert\Omega^R\rangle&=\int d^3k\bigg{[}\frac{1}{2}(\bar{k}^2+M_1^2)^{1/2}+\frac{1}{2}(\bar{k}^2+M_2^2)^{1/2}\bigg{]}\delta^3(0),
\nonumber\\
\langle \Omega^{L}\vert \Omega^R\rangle&=\Pi_{\bar{k}}\int_{-\infty}^{\infty} dy(\bar{k})\int_{-\infty}^{\infty}dx(\bar{k})\langle \Omega^{L}\vert y(\bar{k}),x(\bar{k})\rangle\langle y(\bar{k}),x(\bar{k})\vert\Omega^{R}\rangle
\nonumber\\
&=\Pi_{\bar{k}}\int_{-\infty}^{\infty} dy(\bar{k}) \int_{-\infty}^{\infty}dx(\bar{k}) \psi_0^L(y(\bar{k}),x(\bar{k}))\psi_0^R(y(\bar{k}),x(\bar{k}))=
\Pi_{\bar{k}}1=1.
\label{7.17}
\end{align}
We thus confirm that the vacuum normalization is both finite and positive, while the vacuum energy has the conventional zero-point infinity associated with an infinite number of modes. (This infinity occurs because $\bar{H}_S$ contains an infinite number of modes and not because $\langle \Omega^{L}\vert \Omega^R\rangle$ itself is infinite.) We thus establish the consistency and physical viability of the similarity-transformed higher-derivative scalar field theory. And we note that even though all the norms are positive, the insertion of (\ref{7.10}) into $-i\langle\Omega^{L}\vert T[\bar{\phi}(x)\bar{\phi}(0)]\vert \Omega^R \rangle$ (corresponding to $+i\langle\Omega^{L}\vert T[\phi(x)\phi(0)]\vert \Omega^R \rangle$) generates the relative minus sign in $-[1/(k^2-M_1^2)-1/(k^2-M_2^2)]/(M_1^2-M_2^2)$ \cite{Bender2008b}. Thus with one similarity transform into an appropriate Stokes wedge we solve both the vacuum normalization problem and the negative-norm problem.
At this point we can see the key aspect of our study. Ordinarily in quantum field theory it is taken as a given that one should use the Dirac inner product $\langle \Omega\vert \Omega\rangle$, viz. $\langle \Omega^R\vert \Omega^R\rangle$, for the vacuum. And also it is taken as a given that this inner product is finite. In this paper we have provided a procedure for checking whether this is in fact the case, and presented a second-order plus fourth-order derivative model in which it explicitly is not finite. For this particular model we have found a different inner product, viz. $\langle \Omega^{L}\vert \Omega^R\rangle$, that is finite. (For a Hamiltonian that is Hermitian $\vert \Omega^R\rangle=\vert \Omega\rangle$, $\langle \Omega^L\vert =\langle \Omega\vert$, and $\langle \Omega^L\vert \Omega^R\rangle=\langle \Omega\vert \Omega\rangle$.)
And thus in general one has to determine whether or not $\langle \Omega^R\vert \Omega^R\rangle$ is finite on case by case basis.
We should note that in general there is no requirement that a quantum field theory must use the Dirac inner product $\langle \Omega \vert \Omega \rangle$. What one does need of an acceptable inner product is that it be finite, positive and time independent. And as long as we can find one we have a well-defined Hilbert space. However, if the Hamiltonian is not Hermitian then the Dirac inner product obeys $\langle \Omega(t) \vert \Omega(t) \rangle = \langle \Omega(t=0) \vert e^{iH^{\dagger}t}e^{-iHt}\vert \Omega(t=0) \rangle \neq \langle \Omega(t=0) \vert \Omega(t=0) \rangle$, to thus not be time independent, and thus not acceptable.
In regard to time independence, we also note that in the nonrelativistic case the time derivative of the $\int d^3x \psi^*(x)\psi(x)$ probability is related to an asymptotic spatial surface term. The time independence of $\int d^3x \psi^*(x)\psi(x)$ is then secured if the asymptotic spatial surface term vanishes, i.e., the time independence of $\int d^3x \psi^*(x)\psi(x)$ is determined by asymptotic spatial boundary conditions. If it turns out that the theory is well behaved asymptotically for Hermitian fields then one can use the Dirac norm. If it is not well behaved asymptotically for Hermitian fields, then the insight of the $PT$ program is to look to see if instead there then might be some domain in the complex plane where the theory is well behaved asymptotically. If there is, then one can have a good Hilbert space, but the inner product will not then be the Dirac norm. If the theory has a $PT$ symmetry, then the norm will be the $PT$ theory norm, and when the Hamiltonian is not Hermitian the left-right inner product obeys $\langle \Omega^L(t) \vert \Omega^R(t) \rangle = \langle \Omega^L(t=0) \vert e^{iHt}e^{-iHt}\vert \Omega^R(t=0) \rangle= \langle \Omega^L(t=0) \vert \Omega^R(t=0) \rangle$
to thus be time independent. And this is the case for the second-order plus fourth-order theory.
With the finiteness or otherwise of the path integral also being determined by spatial boundary conditions, in Sec. \ref {S8} we now discuss our findings from the perspective of path integrals. In Sec. \ref{S12} we augment our path integral results with a quick and straightforward analysis of Feynman's $i\epsilon$ prescription for propagators.
\section{Path integrals and the normalization of the vacuum}
\label{S8}
We are interested in the path integral approach for both the PU oscillator and the second-order plus fourth-order field theory. We treat the field theory path integral in Sec. \ref{S12}. For the PU oscillator the Minkowski path integral ($PI$) associated with $I_{\rm PU}$ is of the form
\begin{align}
PI(MINK)=\int D[z]D[dz/dt]\exp\left[\frac{i}{2}\int_{-\infty}^{\infty} dt\left(\left(\frac{d^2 z}{dt^2}\right)^2-\left(\omega_1^2
+\omega_2^2\right)\left(\frac{d z}{dt}\right)^2+\omega_1^2\omega_2^2z^2\right)\right],
\label{8.1}
\end{align}
as initially integrated over paths with real $z$ and $x$.
Since the theory is fourth order we need four pieces of information to solve the equations of motion. The pieces that are the most convenient for path integral purposes are two initial and two final conditions, hence the path integral measure is over both $z$ and $dz/dt$. However, we had noted in Sec. \ref{S5} that the PU theory is a constrained theory. Thus we must treat the $z$ and $dz/dt$ path integrations as independent. We can do this directly as shown in the measure in (\ref{8.1}), or replace (\ref{8.1}) by
\begin{align}
PI(MINK,x,z)=\int D[z]D[x]\exp\left[\frac{i}{2}\int_{-\infty}^{\infty} dt\left(\left(\frac{dx}{dt}\right)^2-\left(\omega_1^2
+\omega_2^2\right)x^2+\omega_1^2\omega_2^2z^2\right)\right].
\label{8.2}
\end{align}
To make the path integral converge rather than just oscillate we first use the Feynman $i\epsilon$ prescription and replace $\omega_1^2$ and $\omega_2^2$ by $\omega_1^2-i\epsilon$ and $\omega_2^2-i\epsilon$. This yields
\begin{align}
PI(MINK,x,z)=\int D[z]D[x]\exp\left[\frac{1}{2}\int_{-\infty}^{\infty} dt\left(i\left(\frac{d x}{dt}\right)^2-i\left(\omega_1^2
+\omega_2^2\right)x^2+i\omega_1^2\omega_2^2z^2
-2\epsilon x^2+\epsilon\left(\omega_1^2+\omega_2^2\right)z^2
\right)\right],
\label{8.3}
\end{align}
as integrated over paths with real $x$ and real $z$. However, while the $x^2$ term is now damped the $z^2$ term is not. Consequently, as integrated with a real measure the path integral does not exist. Now the path integral is used to generate time-ordered Green's functions such as $D(x)=i\langle \Omega\vert T[\phi(x)\phi(0)]\vert\Omega\rangle$ (hence the $i\epsilon$ prescription). And thus these Green's functions will not be finite, with the vacuum in which the Green's function matrix elements are evaluated thus not being normalizable. Study of the Minkowski path integral thus gives us an alternate way to determine whether or not $\langle \Omega\vert \Omega\rangle$ is finite: the path integral with a real measure either exists or does not exist.
To make (\ref{8.2}) exist we need to damp the $z^2$ term, but not modify the $x^2$ term. Thus we continue $z$ into the complex plane and replace it by $y=-iz$, while leaving $x$ real. The path integral for Minkowski time then takes the form
\begin{align}
PI(MINK,x,y)=\int D[y]D[x]\exp\left[\frac{1}{2}\int_{-\infty}^{\infty} dt\left(i\left(\frac{d x}{dt}\right)^2-i\left(\omega_1^2
+\omega_2^2\right)x^2-i\omega_1^2\omega_2^2y^2
-2\epsilon x^2-\epsilon\left(\omega_1^2+\omega_2^2\right)y^2
\right)\right].
\label{8.4}
\end{align}
This puts us into a domain in the complex plane (known as a Stokes wedge) in which the path integral is now fully defined, and now the vacuum state is normalizable. This completely parallels the discussion of $\psi_0(z,x)$ that we gave in Sec. \ref{S5}.
\begin{figure}[htpb]
\centering
\includegraphics[scale=0.5]{contour.pdf}
\caption{Wick contour}
\label{wickcontour}
\end{figure}
However, our concern here could be missed in a Euclidean time path integral approach. Specifically, if we disperse in $t$ (assuming of course that we can, i.e., that the Cauchy-Riemann equations for complex $t$ are obeyed), we can, as shown in Fig. \ref{wickcontour}, write
\begin{align}
\int_{-\infty}^{\infty} +\int_{\infty}^{i\infty}+ \int_{i\infty}^{-i\infty}+ \int_{-i\infty}^{-\infty}={\rm pole~terms~plus ~cut~contributions},
\label{8.5}
\end{align}
i.e., along the real axis, then upper-half-plane quarter circle, then down the imaginary axis, and then lower-half-plane quarter circle. Assuming no pole, cut or circle contributions, and on setting $\tau=it$ and letting $I$ denote the action, from (\ref{8.1}) and (\ref{8.2}) we obtain
\begin{align}
I(MINK,z,x)&\equiv \int_{-\infty}^{\infty}idt\equiv -\int_{i\infty}^{-i\infty}idt=-\int_{-\infty}^{\infty} d\tau \equiv I(EUCL,z,x),
\nonumber\\
PI(EUCL,z,x)&=\int D[z]D[dz/d\tau]\exp\left[-\frac{1}{2}\int_{-\infty}^{\infty} d\tau\left(\left(\frac{d^2 z}{d\tau^2}\right)^2+\left(\omega_1^2
+\omega_2^2\right)\left(\frac{d z}{d\tau}\right)^2+\omega_1^2\omega_2^2z^2\right)\right]
\nonumber\\
&=\int D[z]D[x]\exp\left[-\frac{1}{2}\int_{-\infty}^{\infty} d\tau\left(\left(\frac{dx}{d\tau}\right)^2+\left(\omega_1^2
+\omega_2^2\right)x^2+\omega_1^2\omega_2^2z^2\right)\right].
\label{8.6}
\end{align}
Given the overall minus sign that multiplies the Euclidean action on every path, we see that with real $z$ and real $x=dz/d\tau$ the Euclidean path integral is well behaved. (The same is true of the analog relativistic second-order plus fourth-order scalar field theory path integral
\cite{Hawking2002}.) However, the Minkowski time path integral with a real measure is not. Thus we conclude that the pole and/or cut and/or circle contributions are not only not ignorable, they generate an infinite contribution. Hence their contribution in a Wick rotation cannot be ignored and the Euclidean time path integral does not correctly describe the situation.
In parallel, if we set $y=-iz$, then (\ref{8.6}) is replaced by
\begin{align}
I(MINK,y,x)&\equiv I(EUCL,y,x),
\nonumber\\
PI(EUCL,y,x)&=\int D[y]D[x]\exp\left[\frac{1}{2}\int_{-\infty}^{\infty} d\tau\left(-\left(\frac{dx}{d\tau}\right)^2-\left(\omega_1^2
+\omega_2^2\right)x^2+\omega_1^2\omega_2^2y^2\right)\right].
\label{8.7}
\end{align}
And since the $y$ and $x$ path integrations are independent, now it is the Euclidean time path integral that is not well defined. Thus with either $z$ or $y=-iz$, in neither case are the Minkowski time and Euclidean time path integrals simultaneously finite.
As we discuss in Sec. \ref{S12}, these same results carry over directly to the field theory case, and thus we see that even if finite, a Euclidean time path integral approach is only valid if the vacuum state of the theory (as determined in a Minkowski time analysis) is normalizable.
\section{Interactions}
\label{S9}
In developing Wick's contraction theorem in quantum field theory one needs to put the time-ordered product of Heisenberg fields $\phi(x)$, viz.
\begin{align}
\tau(x_1,...,x_n)=\langle \Omega \vert T[\phi(x_1)...\phi(x_n)]\vert \Omega \rangle,
\label{9.1}
\end{align}
into a form that can be developed perturbatively. To this end one introduces a set of in-fields $\phi_{in}(x)$ that satisfy free field equations with Hamiltonian $H_{in}$. And one also introduces an evolution operator $U(t)$ that evolves with the interaction Hamiltonian $H_I(t)$ according to
\begin{align}
i\frac{\partial U(t)}{\partial t}=H_I(t)U(t).
\label{9.2}
\end{align}
With this $U(t)$ we can relate $\phi(x)$ and $\phi_{in}(x)$ according to
\begin{align}
\phi(\bar{x},t)=U^{-1}(t)\phi_{in}(\bar{x},t)U(t).
\label{9.3}
\end{align}
If one introduces $U(t,t^{\prime})=U(t)U^{-1}(t^{\prime})$, then $U(t,t^{\prime})$ is given by
\begin{align}
U(t,t^{\prime})=1-i\int_{t^{\prime}}^tdt_1H_I(t_1)U(t_1,t^{\prime})=T\left[\exp\left(-i\int_{t^{\prime}}^tdt_1H_I(t_1)\right)\right].
\label{9.4}
\end{align}
Using these relations we obtain (see e.g. \cite{Bjorken1965})
viz.
\begin{align}
\tau(x_1,...,x_n)=\langle \Omega \vert U^{-1}(t)T\left[\phi_{in}(x_1)...\phi_{in}(x_n)\exp\left(-i\int_{-t}^tdt_1H_I(t_1)\right)\right]U(-t)\vert \Omega \rangle.
\label{9.5}
\end{align}
The contributions due to the $U(t)\vert \Omega \rangle$ and $\langle \Omega \vert U^{-1}(t)$ terms lead to
\begin{align}
\tau(x_1,...,x_n)&=\langle \Omega \vert T\left[\phi_{in}(x_1)...\phi_{in}(x_n)\exp\left(-i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle
\nonumber\\
&\times
\langle \Omega \vert T\left[\exp\left(i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle.
\label{9.6}
\end{align}
After inverting the last term we obtain the standard form \cite{Bjorken1965}
\begin{align}
\tau(x_1,...,x_n)&=\frac{\langle \Omega \vert T\left[\phi_{in}(x_1)...\phi_{in}(x_n)\exp\left(-i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle}
{\langle \Omega \vert T\left[\exp\left(-i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle}.
\label{9.7}
\end{align}
If one starts with (\ref{9.7}) it would appear that the normalization of the vacuum state is actually irrelevant since it would drop out of the ratio. And so it would not appear to matter if it did happen to be infinite. However, this is not the case since we could only go from (\ref{9.6}) to (\ref{9.7}) if $\langle \Omega \vert T\left[\exp\left(i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle$ is finite. And it would not be if the vacuum state is not normalizable. If we expand $\langle \Omega \vert T\left[\exp\left(i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle$ out as a power series in $H_I$ the first term is $\langle \Omega \vert \Omega \rangle$ as calculated in a free theory. Thus, as we had noted in Sec. \ref{S3}, for finiteness we need this term to be finite and need the power series expansion in $H_I$ to be renormalizable in order for $\langle \Omega \vert T\left[\exp\left(i\int_{-t}^tdt_1H_I(t_1)\right)\right]\vert \Omega \rangle$ to be finite. However, for a nonnormalizable vacuum the standard Wick expansion and Feynman rules are not valid. Since this concern is of relevance to radiative corrections to Einstein gravity we return to this point in Secs. \ref{S11} and \ref{S12} below.
\section{Fermions}
\label{S10}
For fermions we have to deal with anticommutators such as
\begin{align}
bb^{\dagger}+b^{\dagger}b=1.
\label{10.1}
\end{align}
Also, because of the Pauli principle we have
\begin{align}
b^2=0,\qquad b^{\dagger 2}=0.
\label{10.2}
\end{align}
We can represent (\ref{10.1}) and (\ref{10.2}) by matrices of the form
\begin{eqnarray}
b=\begin{pmatrix}
0& 0 \\
1 &0
\end{pmatrix},
\qquad
b^{\dagger}=
\begin{pmatrix}
0& 1 \\
0 &0
\end{pmatrix}.
\label{10.3}
\end{eqnarray}
Thus, unlike the infinite-dimensional matrix representation of the bosonic $a$ and $a^{\dagger}$ that obey $aa^{\dagger}-a^{\dagger}a=1$, the fermionic $b$ and $b^{\dagger}$ matrices are finite dimensional. Thus with a finite number of degrees of freedom, the fermion vacuum that obeys $b\vert \Omega\rangle=0$ has a finite $\langle \Omega\vert \Omega \rangle$ norm.
\section{Implications for radiative corrections in quantum Einstein gravity}
\label{S11}
As a quantum theory the standard second-order derivative Einstein gravitational theory with its $1/k^2$ propagator is not renormalizable since graviton loops cannot be made finite by counterterms that are themselves of the same generic form as the Einstein-Hilbert Ricci scalar action. Since these graviton loops generate higher-derivative gravity terms, one can construct a candidate theory of quantum gravity by augmenting the Einstein Ricci scalar action with a term that is quadratic in the Ricci scalar. This gives a much studied quantum gravity action of the generic form
\begin{eqnarray}
I_{\rm GRAV}=\int d^4x(-g)^{1/2}\left[6M^2R^{\alpha}_{\phantom{\alpha}\alpha}+(R^{\alpha}_{\phantom{\alpha}\alpha})^2\right].
\label{11.1}
\end{eqnarray}
This same action also appears in Starobinsky's inflationary universe model \cite{Starobinsky1979}.
On adding on a matter source with energy-momentum tensor $T_{\mu\nu}$, variation of this action with respect to the metric generates a gravitational equation of motion of the form
\begin{eqnarray}
-6M^2G^{\mu\nu}+V^{\mu\nu}=-\frac{1}{2}T^{\mu\nu}.
\label{11.2}
\end{eqnarray}
Here $G_{\mu\nu}$ is the Einstein tensor and $V_{\mu\nu}$ may for instance be found in \cite{Mannheim2006}, with these various terms being of the form
\begin{eqnarray}
G^{\mu\nu}&=&R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}g^{\alpha\beta}R_{\alpha\beta},
\nonumber\\
V^{\mu \nu}&=&
2g^{\mu\nu}\nabla_{\beta}\nabla^{\beta}R^{\alpha}_{\phantom{\alpha}\alpha}
-2\nabla^{\nu}\nabla^{\mu}R^{\alpha}_{\phantom{\alpha}\alpha}
-2 R^{\alpha}_{\phantom{\alpha}\alpha}R^{\mu\nu}
+\frac{1}{2}g^{\mu\nu}(R^{\alpha}_{\phantom{\alpha}\alpha})^2.
\label{11.3}
\end{eqnarray}
If we now linearize about flat spacetime with background metric $\eta_{\mu\nu}$ and fluctuation metric $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, to first perturbative order we obtain
\begin{eqnarray}
\delta G_{\mu\nu}&=&\frac{1}{2}\left(\partial_{\alpha}\partial^{\alpha}h_{\mu\nu}-\partial_{\mu}\partial^{\alpha}h_{\alpha\nu}-\partial_{\nu}\partial^{\alpha}h_{\alpha\mu}+\partial_{\mu}\partial_{\nu}h\right)-\frac{1}{2}\eta_{\mu\nu}\left(\partial_{\alpha}\partial^{\alpha}h-\partial^{\alpha}\partial^{\beta}h_{\alpha\beta}\right),
\nonumber\\
\delta V_{\mu\nu}&=&[2\eta_{\mu\nu}\partial_{\alpha}\partial^{\alpha} -2\partial_{\mu}\partial_{\nu}]
[\partial_{\beta}\partial^{\beta}h-\partial_{\lambda}\partial_{\kappa}h^{\lambda\kappa}],
\label{11.4}
\end{eqnarray}
where $h=\eta^{\mu\nu}h_{\mu\nu}$. On taking the trace of the fluctuation around a background (\ref{11.2}) we obtain
\begin{eqnarray}
[M^2+\partial_{\beta}\partial^{\beta}]\left(\partial_{\lambda}\partial^{\lambda}h-\partial_{\kappa}\partial_{\lambda}h^{\kappa\lambda}\right)=-\frac{1}{12}\eta^{\mu\nu}\delta T_{\mu\nu}.
\label{11.5}
\end{eqnarray}
In the convenient transverse gauge where $\partial_{\mu}h^{\mu\nu}=0$, the propagator for $h$ is given by
\begin{eqnarray}
D(h,k^2)=-\frac{1}{k^2(k^2-M^2)}=\frac{1}{M^2}\left(\frac{1}{k^2}-\frac{1}{k^2-M^2}\right).
\label{11.6}
\end{eqnarray}
As we see, in this case the $1/k^2$ graviton propagator for $h$ that would be associated with the Einstein tensor $\delta G_{\mu\nu}$ alone is replaced by a $D(h,k^2)=[1/k^2-1/(k^2-M^2)]/M^2$ propagator. And now the leading behavior at large momenta is $-1/k^4$. In consequence, the theory is thought to be renormalizable \cite{Stelle1977,Stelle1978}. But since $\langle \Omega\vert\Omega \rangle$ is not finite the proof of renormalizability has a flaw in it. Fortunately, the flaw is not fatal, and we rectify it below.
We recognize $D(h,k^2)$ as being of the same form as the second-order plus fourth-order scalar field theory propagator that was given in (\ref{4.2}), with $\phi$ being replaced by $h$ and with $M_1^2=M^2$, $M_2^2=0$. We can thus give $h$ an equivalent effective action of the form
\begin{eqnarray}
I_h&=&\frac{1}{2}\int d^4x\bigg{[}\partial_{\mu}\partial_{\nu}h\partial^{\mu}
\partial^{\nu}h-M^2\partial_{\mu}h\partial^{\mu}h\bigg{]}.
\label{11.7}
\end{eqnarray}
The action given in (\ref{11.7}) thus shares the same vacuum state normalization and negative norm challenges as the scalar field action given in (\ref{4.1}).
Thus if, as is conventional, we take $h$ to be Hermitian we would immediately encounter the negative-norm problem associated with the relative minus sign in (\ref{11.6}). However, since $M^2$ is Planck scale in magnitude, this difficulty can be postponed until observations can reach that energy scale. However, the lack of normalizabilty of the vacuum state has consequences at all energies and cannot be postponed at all. Specifically, with $\langle \Omega\vert\Omega \rangle$ being infinite we cannot even identify the propagator as $i\langle \Omega\vert T[h(x)h(0)]\vert \Omega \rangle$ since in analog to (\ref{1.8}) it will obey
\begin{align}
&(\partial_t^2-\bar{\nabla}^2)(\partial_t^2-\bar{\nabla}^2+M^2)D(h,x)=-\langle \Omega\vert\Omega \rangle\delta^4(x).
\label{11.8}
\end{align}
Consequently, we cannot make the standard Wick contraction expansion. And thus both the Feynman rules that are used presupposing that $\langle \Omega\vert\Omega \rangle$ is finite, and the renormalizability that is thought to then follow from them are therefore not valid. Additionally, with $\langle \Omega\vert\Omega \rangle$ being infinite, we cannot treat the Einstein theory with its $1/k^2$ propagator as an effective field theory that holds for momenta that obey $k^2 \ll M^2$.
However, as noted above, we can resolve all of these concerns by dropping the requirement that $h$ be Hermitian, and set it equal to $i\bar{h}$. Then, with the theory being recognized as a $PT$ theory, vacuum state normalization and negative-norm problems are resolved and the theory is consistent. Moreover, the propagator is given by $-i\langle \Omega^{L}\vert T[\bar{h}(x)\bar{h}(0)]\vert \Omega^R \rangle$ (corresponding to $+i\langle \Omega^{L}\vert T[h(x)h(0)]\vert \Omega^R \rangle$). And with the propagator still being given by (\ref{11.6}) as it satisfies $(\partial_t^2-\bar{\nabla}^2+M^2)(\partial_t^2-\bar{\nabla}^2)[-i\langle \Omega^{L}\vert T[\bar{h}(x)\bar{h}(0)]\vert \Omega^R \rangle]=-\delta^4(x)$, all the steps needed to prove renormalizability are now valid. At this point the only concern is that even though the $M^2$ field now has positive norm, it still remains in the spectrum and would eventually have to be observed.
As can be seen from (\ref{11.7}), the only reason that there is an $M^2$ term at all is because we are considering an action that has both second-order and fourth-order terms. With a pure fourth-order theory there would be no dimensionful parameter in the action and the theory would be scale invariant. If like the gauge theories of $SU(3)\times SU(2)\times U(1)$ this scale symmetry is also local, we would be led to conformal gravity, a metric theory of gravity in which the action is left invariant under local changes of the metric of the form $g_{\mu\nu}(x)\rightarrow e^{2\alpha(x)}g_{\mu\nu}(x)$, where $\alpha(x)$ is a local function of the coordinates. The conformal gravity theory has been advocated and explored in \cite {Mannheim2006,Mannheim2017} and references therein. And in \cite{hooft2015local} 't Hooft has also argued that there should be an underlying local conformal symmetry in nature.
In the conformal gravity theory an action that is to be a polynomial function of the metric has the unique form
\begin{eqnarray}
I_{\rm W}=-\alpha_g\int d^4x\, (-g)^{1/2}C_{\lambda\mu\nu\kappa}
C^{\lambda\mu\nu\kappa}
\equiv -2\alpha_g\int d^4x\, (-g)^{1/2}\left[R_{\mu\kappa}R^{\mu\kappa}-\frac{1}{3} (R^{\alpha}_{\phantom{\alpha}\alpha})^2\right],
\label{11.9}
\end{eqnarray}
where $\alpha_g$ is a dimensionless gravitational coupling constant, and $C_{\lambda\mu\nu\kappa}$ is the conformal Weyl tensor. The perturbative propagator has a $-1/k^4$ behavior at all $k^2$, and with its large $k^2$ behavior the theory is renormalizable \cite{Fradkin1985}. With a $-1/k^4$ propagator it would initially appear that there would be two massless particles at $k^2=0$. However, we cannot use the partial fraction decomposition given in (\ref{11.6}) as a guide since its $1/M^2$ prefactor is singular in the $M^2\rightarrow 0$ limit. Because of this singular behavior the $M^2=0$ Hamiltonian becomes of nondiagonalizable Jordan-block form and only has one massless eigenstate, with the other would-be massless eigenstate becoming nonstationary \cite{Bender2008b}.
To understand why we have lost an eigenstate let us consider two positive frequency mode solutions to
\begin{eqnarray}
[M^2+\partial_{\beta}\partial^{\beta}]\partial_{\lambda}\partial^{\lambda}h=0,
\label{11.10}
\end{eqnarray}
viz. modes with time dependence $e^{-i\omega_1t}$ and $e^{-i\omega_2 t}$, where $\omega_1=|\bar{k}|$, $\omega_2=+(\bar{k}^2+M^2)^{1/2}$. These modes are solutions to the $M^2\neq 0$ wave equation associated with (\ref{11.5}) (as are the two negative frequency solutions with time dependence $e^{i\omega_1t}$ and $e^{i\omega_2 t}$.) If we now let $M^2$ go to zero the two positive frequency mode solutions become equal and we only have one positive frequency mode solution. However, we cannot lose any solutions to the differential equation given in (\ref{11.10}) by setting $M^2=0$ in it since the wave equation is still fourth order. To find the other solution in the limit we consider
\begin{eqnarray}
\lim_{M^2\rightarrow 0}\frac{(e^{-i\omega_1t}-e^{-i\omega_2t})}{M^2}=\frac{ite^{-i\omega_1t}}{2|\bar{k}|}.
\label{11.11}
\end{eqnarray}
As we see, this second solution is not stationary and is thus not an eigenstate of the Hamiltonian.
Since we have lost a massless eigenstate the propagator should be constructed not as the $M^2\rightarrow 0$ limit of (\ref{11.6}) but as the manifestly ghost-free limit
\begin{eqnarray}
-\frac{1}{(k^2+i\epsilon)^2}=-\lim_{M^2\rightarrow 0} \frac{d}{dM^2}
\left(\frac{1}{k^2-M^2+i\epsilon}\right),
\label{11.12}
\end{eqnarray}
a limit that shows that there is only one $k^2=0$ pole not two. With the Hamiltonian not being diagonalizable, it could not be Hermitian. It does however have a $PT$ symmetry, with is ground state being normalizable. Conformal gravity is thus a fully consistent theory of quantum gravity, one which despite its fourth-order character only possesses one massless particle, not two \cite{footnote4}.
To understand why both second-order plus fourth-order theories and pure fourth-order theories must be ghost free consider the Dirac action for a fermion coupled to a background geometry of the form
\begin{eqnarray}
I_{\rm D}=\int d^4x(-g)^{1/2}\left[i\bar{\psi}\gamma^{c}V^{\mu}_c(x)(\partial_{\mu}+\Gamma_{\mu}(x))\psi -M\bar{\psi}(x)\psi(x)\right].
\label{11.13}
\end{eqnarray}
Here the $V^{\mu}_a(x)$ are vierbeins, $\Gamma_{\mu}(x)=-(1/8)[\gamma_a,\gamma_b](V^b_{\nu}(x)\partial_{\mu}V^{a\nu}(x)+V^b_{\lambda}(x)\Gamma^{\lambda}_{\nu\mu}(x)V^{a\nu}(x))$ is the spin connection, and $\Gamma^{\lambda}_{\nu\mu}(x)$ is the geometric Levi-Civita connection. We introduce the path integral $\int D[\psi]D[\bar{\psi}]\exp(iI_{\rm D})=\exp(iI_{\rm EFF})$. With $I_{\rm D}$ being linear in both $\psi$ and $\bar{\psi}$ the fermion path integration can be performed analytically, to thereby yield an effective action $I_{\rm EFF}$ whose leading term is of the form \cite{tHooft2010a}, \cite{Mannheim2017}
\begin{eqnarray}
I_{\rm EFF}&=&\int d^4x(-g)^{1/2}C\bigg{[}\frac{1}{20}\left[R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}(R^{\alpha}_{\phantom{\alpha}\alpha})^2\right]
-M^4+\frac{1}{6}M^2R^{\alpha}_{\phantom{\alpha}\alpha}\bigg{]},
\label{11.14}
\end{eqnarray}
where $C$ is a log divergent constant. We recognize $I_{\rm EFF}$ as containing none other than second-order and fourth-order gravitational terms. Now the $I_D$ action is a completely standard action of a fermion coupled to a background gravitational field, and as such it is not only ghost free, it would remain so if the fermion is given some internal quantum numbers and coupled to some gauge fields. Since the fermion path integral is equivalent to a one loop Feynman diagram and since one cannot change the signature of a Hilbert space in perturbation theory, higher-derivative gravity must be ghost free too. And if it were not, then the standard $SU(3)\times SU(2)\times U(1)$ model would not remain unitary when coupled to gravity.
Thus despite the negative-norm issue, higher-derivative gravity cannot contain any negative-norm states. And for the standard model not to be destabilized when coupled to gravity, the subsequent $g_{\mu\nu}$ path integration must be conducted with a measure $D[g_{\mu\nu}]$ that is continued into the complex plane, just as found with $PT$ theory.
Since, as noted in Sec. \ref{S8}, this continuation is required by the $i\epsilon$ prescription, we now comment on this prescription in more detail.
\section{$PT$ symmetry and the $i\epsilon$ prescription}
\label{S12}
The $i\epsilon$ prescription is central to both propagators and path integrals, and discussion of its role provides a quick explanation of our results. For the second-order plus fourth-order theory discussed in Sec. \ref{S4} the identification of the $D(k)$ propagator in (\ref{4.2}) as
\begin{align}
&D(k)=-\frac{1}{(k^2-M_1^2)(k^2-M_2^2)}
=- \frac{1}{(M_1^2-M_2^2)}\left[\frac{1}{(k^2-M_1^2)}-\frac{1}{(k^2-M_2^2)}\right]
\label{12.1}
\end{align}
is only formal since $D(k)$ is singular. To give it a meaning we need to define it via a contour integral and specify the appropriate complex $k_0$ plane contour. If, as is conventional, we take all of the operators in $H=\int d^3x T_{00}$ to be Hermitian, where as given in (\ref{4.3})
\begin{align}
T_{00}&=\tfrac{1}{2}\pi_{00}^2+\pi_{0}\dot{\phi}+\tfrac{1}{2}(M_1^2+M_2^2)\dot{\phi}^2-\tfrac{1}{2}M_1^2M_2^2\phi^2
-\tfrac{1}{2}\pi_{ij}\pi^{ij}+\tfrac{1}{2}(M_1^2+M_2^2)\phi_{,i}\phi^{,i},
\label{12.2}
\end{align}
we immediately find that because of the $-(1/2)M_1^2M_2^2\phi^2$ term the Hamiltonian $H$ is unbounded from below, the Ostrogradski instability that is characteristic of higher-derivative theories. Now the standard Feynman contour $k^2+i\epsilon$ prescription with
\begin{align}
&D(k)=-\frac{1}{(k^2-M_1^2+i\epsilon)(k^2-M_2^2+i\epsilon)}
=- \frac{1}{(M_1^2-M_2^2)}\left[\frac{1}{(k^2-M_1^2+i\epsilon)}-\frac{1}{(k^2-M_2^2+i\epsilon)}\right]
\label{12.3}
\end{align}
is chosen so that positive energy states propagate forward in time (viz. $\omega_1(\bar{k})=+(\bar{k}^2+M_1^2)^{1/2}$, $\omega_2(\bar{k})=+(\bar{k}^2+M_2^2)^{1/2}$ located below the real $k_0$ axis), while negative energy states propagate backwards ($-\omega_1(\bar{k})$, $-\omega_2(\bar{k})$ above the real $k_0$ axis); with this corresponding to an energy spectrum that is bounded from below while leading to negative residues that appear to correspond to states of negative norm. In contrast, having an energy spectrum that is unbounded from below would instead entail that negative energies in one of sectors ($\omega_2(\bar{k})$ say) are propagating forward in time (-$\omega_2(\bar{k})$ below the real $k_0$ axis), and positive energies are propagating backward in time ($\omega_2(\bar{k})$ above the real $k_0$ axis), and would require using an unconventional $i\epsilon$ prescription \cite{Bender2008b} of the form
\begin{align}
&D(k)=-\frac{1}{(k^2-M_1^2+i\epsilon)(k^2-M_2^2-i\epsilon)}
=- \frac{1}{(M_1^2-M_2^2)}\left[\frac{1}{(k^2-M_1^2+i\epsilon)}-\frac{1}{(k^2-M_2^2-i\epsilon)}\right].
\label{12.4}
\end{align}
While this possibility is unacceptable physically, it does have the feature that because of the way the singularities are traversed all pole residues are positive. Thus the two options are: bounded energies and negative residues, or unbounded energies and positive residues. These two realizations are inequivalent and correspond to different Feynman contours and different Hilbert spaces. As discussed in Sec. \ref{S5} and in \cite{Bender2008b} the first option corresponds to working in a Hilbert space in which $a_2$ effects $a_2\vert \Omega\rangle=0$, while the latter corresponds to working in a Hilbert space in which $a^{\dagger}_2\vert \Omega\rangle=0$. Thus in no Hilbert space do we have both negative energies and negative residues. Thus if we work in the Hilbert space in which $a_2\vert \Omega \rangle=0$ (which we do in this paper) and use the standard Feynman $i\epsilon$ prescription given in (\ref{12.3}), our results will not be affected by the Ostrogradski instability at all. Nor will it affect the continuation into the complex plane that resolves the negative-norm issue.
In analog to the quantum-mechanical (\ref{8.2}) and the field theory study given in \cite{Hawking2002} the Minkowski path integral associated with the field theory action given in (\ref{4.1}) is of the form
\begin{align}
PI(MINK)=\int D[\phi]D[\sigma_{\mu}]\exp\left[\frac{i}{2}\int_{-\infty}^{\infty} d^4x\left[\partial_{\nu}\sigma_{\mu}\partial^{\nu}\sigma^{\mu}-\left(M_1^2+M_2^2\right)\sigma_{\mu}\sigma^{\mu}+M_1^2M_2^2\phi^2\right]\right],
\label{12.5}
\end{align}
where $\sigma_{\mu}=\partial_{\mu}\phi$.
In order to damp out oscillations we choose the Feynman $i\epsilon$ prescription in which we replace $M_1^2$ and $M_2^2$ by $M_1^2-i\epsilon$, $M_2^2-i\epsilon$. For the path integral this yields
\begin{align}
PI(MINK)&=\int D[\phi]D[\sigma_{\mu}]\exp\bigg{[}\frac{1}{2}\int_{-\infty}^{\infty} d^4x\big{[}i\partial_{\nu}\sigma_{\mu}\partial^{\nu}\sigma^{\mu}-i\left(M_1^2+M_2^2\right)\sigma_{\mu}\sigma^{\mu}+iM_1^2M_2^2\phi^2
\nonumber\\
&-2\epsilon\sigma_{\mu}\sigma^{\mu}+(M_1^2+M_2^2)\epsilon\phi^2\big{]}\bigg{]}.
\label{12.6}
\end{align}
With $\phi$ and $\sigma_{\mu}$ being taken to be real and with $\sigma_{\mu}\sigma^{\mu}$ being taken to be timelike on every path, the $\sigma_{\mu}$ path integration is damped but the $\phi$ path integration is not.
For the unconventional $i\epsilon$ prescription in which we replace $M_1^2$ and $M_2^2$ by $M_1^2-i\epsilon$, $M_2^2+i\epsilon$ the path integral takes the form
\begin{align}
PI(MINK)&=\int D[\phi]D[\sigma_{\mu}]\exp\bigg{[}\frac{1}{2}\int_{-\infty}^{\infty} d^4x\big{[}i\partial_{\nu}\sigma_{\mu}\partial^{\nu}\sigma^{\mu}-i\left(M_1^2+M_2^2\right)\sigma_{\mu}\sigma^{\mu}+iM_1^2M_2^2\phi^2-(M_1^2-M_2^2)\epsilon\phi^2\big{]}\bigg{]},
\label{12.7}
\end{align}
and has no damping on the $\sigma_{\mu}$ path integration at all. The unconventional $i\epsilon$ prescription for the Feynman contour that leads to an unbounded from below energy spectrum thus cannot be associated with a well-defined path integral, and we cannot consider it at all.
Thus the only Feynman $i\epsilon$ prescription that can be relevant is the standard one with $M_1^2-i\epsilon$, $M_2^2-i\epsilon$. However even with this choice the $\phi$ path integration is not damped if $\phi$ is real. It becomes damped if we do not require $\phi$ to be real, but instead take it to be pure imaginary (though $({\rm Im}[\phi])^2> ({\rm Re}[\phi])^2$ would suffice). With $\bar{\phi}=-i\phi$ we replace (\ref{12.6}) by
\begin{align}
PI(MINK)&=\int D[\bar{\phi}]D[\sigma_{\mu}]\exp\bigg{[}\frac{1}{2}\int_{-\infty}^{\infty} d^4x\big{[}i\partial_{\nu}\sigma_{\mu}\partial^{\nu}\sigma^{\mu}-i\left(M_1^2+M_2^2\right)\sigma_{\mu}\sigma^{\mu}-iM_1^2M_2^2\bar{\phi}^2
\nonumber\\
&-2\epsilon\sigma_{\mu}\sigma^{\mu}-(M_1^2+M_2^2)\epsilon\bar{\phi}^2\big{]}\bigg{]},
\label{12.8}
\end{align}
With $\bar{\phi}$ and $\sigma_{\mu}$ being taken to be real and with $\sigma_{\mu}\sigma^{\mu}$ being taken to be timelike on every path, the path integral is now well defined and the theory is consistent.
In classical mechanics Poisson bracket relations are unaffected by symplectic transformations, while in quantum mechanics commutation relations are unaffected by similarity transformations. With path integrals being based on classical fields, the utility of making a symplectic transformation on the classical fields is that it can take a badly-behaved path integral with a real measure into a complex domain for the measure known as a Stokes wedge in which the path integral then is well behaved. Analogously, as described in Sec. \ref{S7}, a similarity transformation can take us into a complex domain in which nonnormalizable quantum wave functions become normalizable. In both the classical and quantum cases we thus seek appropriate complex domains that have good boundary behavior.
While one is always free to make symplectic transformations or similarity transformations, for standard positive Dirac norm Hermitian theories with energies that are bounded from below, these transformations contain no new information. For such Hermitian theories the quantum fields are Hermitian, the classical fields that appear in the path integral are their real eigenvalues, the path integral with a real measure and Feynman $i\epsilon$ prescription exists, the Dirac norm of the vacuum is finite, and the path integral can be associated with matrix elements of the form $i\langle \Omega \vert T[\phi(x)\phi(0)]\vert \Omega\rangle$.
However, it can happen that the path integral with real measure does not exist, but that it does exist for some appropriately chosen complex domain for the measure. In that case the Dirac norm of the vacuum is not finite. Also the Hamiltonian will not be Hermitian, but if all the poles in the propagator are real, the Hamiltonian will instead be $PT$ symmetric. In that case the propagator is given by $i\langle \Omega^L \vert T[(i\bar{\phi}(x))(i\bar{\phi}(0))]\vert \Omega^R\rangle
=-i\langle \Omega^L \vert T[\bar{\phi}(x)\bar{\phi}(0)]\vert \Omega^R\rangle$, where $\phi$ is transformed into $-i\phi=\bar{\phi}$. And the vacuum norm is given by $\langle \Omega^L \vert \Omega^R\rangle$, a norm that is finite, positive definite and time independent. Thus when we find for the second-order plus fourth-order derivative quantum field theory that propagator pole residues are negative or that the path integral with real measure does not exist, it does not mean that the theory does not exist, but that it has to be formulated in a Hilbert space with an inner product other than the standard Dirac one.
Since for the second-order plus fourth-order derivative theory the vacuum Dirac norm is not finite, the standard derivation of the Feynman rules from the Wick contraction procedure is invalid, as is then the renormalizabilty that would follow from these Feynman rules. Interestingly, since the pole structure of the propagator is not affected by a similarity transformation, even after the transformation we can still use the propagator given in (\ref{12.3}) with its $-1/k^4$ short-distance behavior. The only thing that changes is the quantum field matrix element that we identify it with. This leads to the same Feynman rules, only now validly derived from a Wick contraction procedure in which everything is finite, to thus provide for an a posteriori derivation of renormalizability, while deriving it now in a Hilbert space that possesses no states with negative norm. The renormalizability and unitarity of the second-order plus fourth-order field theory is thus established, just as needed for a consistent theory of quantum gravity.
\section{Final Comments}
\label{S13}
For a quantum field theory to be physically relevant it must be formulatable in a Hilbert space with an inner product that is time independent, finite and nonnegative. However, in and of itself, specifying an action and a set of canonical commutators is not enough to either fix the Hilbert space or specify the appropriate inner product. Ordinarily, one supplements these requirements with the additional (generally regarded as self-evident) requirements that the fields and the Hamiltonian of the theory be Hermitian, and that the inner product be the standard, presumed finite, Dirac $\langle n\vert n \rangle$ one. (In the axiomatic approach to quantum field theory \cite{Streater1964} the finiteness of $\langle \Omega \vert \Omega \rangle$ is one of the starting assumptions.) However, it is not automatic for an arbitrary theory that $\langle \Omega \vert \Omega \rangle$ be finite, and so one needs to check on a case by case basis. And in this paper we have presented a procedure for doing so. The procedure is based on using the occupation number space representation to construct an equivalent wave mechanics representation, from which we can then check for the normalizability of the vacuum state, and accordingly of the states that can be excited out of it. An alternative but equivalent approach is to check whether or not the Minkowski time path integral with a real measure exists. If it does not, then the standard Dirac inner product is not finite.
Using the occupation number space representation procedure we have found a case, a second-order plus fourth-order scalar field theory, in which the standard Dirac inner product $\langle n\vert n \rangle$ actually is not finite. In this example the Minkowski time path integral with a real measure diverges even though the Euclidean time path integral does not. In this case contributions from the Wick rotation contour cannot be ignored, and the use of a Euclidean time path integral can be misleading. Thus even if a Euclidean time path integral is well behaved, it only gives a good description of the theory if the Minkowski time path integral is well behaved too. Since $\langle \Omega\vert \Omega\rangle$ is not finite for the second-order plus fourth-order scalar field theory, use of the standard Feynman rules is not valid, with these rules not only leading to states with negative norm, they lead to states with infinite negative norm. This lack of finiteness means that the Hamiltonian is not self-adjoint when acting on these particular states.
However, the Hamiltonian of the second-order plus fourth-order scalar field theory is $PT$ symmetric, so we can use the techniques of the $PT$ symmetry program and continue the fields and the Hamiltonian in this theory into the complex plane. There is then a domain in the complex plane in which one can define an appropriate time-independent, positive and finite inner product, viz. the $\langle L\vert R\rangle$ overlap of left-eigenstates and right-eigenstates of the resulting Hamiltonian, with the resulting vacuum state then being normalizable, and with there being no states with negative or infinite $\langle L\vert R\rangle$ norm \cite{footnote5}. In this domain it is the Euclidean time path integral that diverges while the Minkowski time path integral does not. So again there are contributions from the Wick rotation contour. In this domain the second-order plus fourth-order scalar field theory is fully consistent, unitary and renormalizable, with this analysis being relevant to the construction of a consistent, unitary and renormalizable quantum theory of gravity.
\begin{acknowledgments}
The author wishes to thank Dr. C. M. Bender, Dr. J. Feinberg and Dr. O. Lechtenfeld for helpful comments.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,973
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Federal Courts/
The Onion Joins Free-Speech Case Against Police as Amicus
By Richard Dahl on October 06, 2022
Will the U.S. Supreme Court allow a man who was arrested for making fun of police on a Facebook page to sue? The man, Anthony Novak of Parma, Ohio, is trying to convince the court to take up his case and called upon an unusual party to assist him with an amicus brief: the satirical news platform The Onion.
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In a decision that could open the door to broader administrative restrictions in other areas, the U.S. Supreme Court limited the Environmental Protection Agency's powers to regulate greenhouse gas emissions from existing power plants. The 6-3 ruling in West Virginia v. Environmental Protection Agency split firmly across ideological lines. The agency went further than the power Congress granted it in the 1970 Clean Air Act by creating its own regulatory scheme to cap carbon emissions, the court ruled.
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Looking at Future Supreme Court Abortion Fights
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Even with almost two months to prepare, the Supreme Court decision in Dobbs v. Jackson Women's Health, which overturned Roe v. Wade and Casey v. Planned Parenthood, contained a lot of information to digest. While Justice Samuel Alito's majority opinion included relatively little new information from the leaked draft, the concurring opinions have opened the door to a lot of speculation. Below are four potential legal issues involving abortion that could make their way back up to the Supreme Court.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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{
"redpajama_set_name": "RedPajamaC4"
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Former Ossining Mayor Endorses Gearity For Post
by Miguel Hernandez
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OSSINING, N.Y. -- The Ossining Daily Voice accepts signed letters to the editor. Send letters to ossining@dailyvoice.com.
To the Editor
I was asked the other day who had my support in the upcoming primary race for mayor of the village of Ossining. Without any hesitation, I told the man who posed the question that my vote would go to village Trustee Victoria Gearity because she is focused on governing the village from the bottom up and not from the top down.
Her approach to solving the village's problems stands in stark contrast to her opponent's haughty, imperious. "my way or the highway" manner that interprets a citizens questions as threats to the established order. Victoria Gearity understands that democracy is a messy, necessary, affair and not merely an inconvenient obstacle to be run over on the path to progress.
Two weeks ago, the members of the town of Ossining Democratic Committee overwhelmingly voted to have Victoria Gearity as their candidate for mayor. It seems the principle of majority vote was not to the current mayor's liking, as it gets in the way of his ambition to go down in the Guinness Book of Records as the longest-serving mayor of a New York municipality. Apparently the current commander-in-chief wants to hold on to his title forever. Clearly, he does not understand that legacy, not longevity, is the hallmark of a true servant of the people.
On Sept. 9, 2014, I urge all registered Democrats in the village of Ossining to cast their precious vote for Victoria Gearity, a woman who respects this community's past, embraces its future, stands alongside you in moving this little village of ours forward and will graciously turn over the mayor's gavel to a new generation when you decide the time is right.
Miguel Hernandez Former Trustee and Mayor of the Village of Ossining
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{
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Theodore I Palaiologos or Palaeologus (Greek: Θεόδωρος Παλαιολόγος, full name: Theodoros Komnenos Doukas Angelos Palaiologos) ( – 24 April 1338) was Marquis of Montferrat from 1306 until his death.
Life
He was a son of Emperor Andronikos II Palaiologos and Irene of Montferrat. When his uncle John I died in 1305, the male line of the Aleramici Marquises of Montferrat became extinct. The March of Montferrat was passed to Irene's children. Patriarch Athanasius I of Constantinople blocked the candidacy of the elder son John, so Theodore went to Italy instead.
Theodore sailed to Genoa in 1306. In 1307 he married Argentina Spinola, daughter of Genoese magnate Opicino Spinola, Capitano del Popolo (co-ruler) of the Republic of Genoa. Spinola used his wealth to back Theodore's claim to Montferrat.
Theodore was opposed by Manfred IV of Saluzzo. Manfred was a cadet of the House of Savoy, and several Marquises of Montferrat had Savoyard wives. King Charles II of Naples also claimed parts of the March. He gradually overcame these foes and secured the whole March. In 1310 he received the imperial investiture from Emperor Henry VII.
Theodore died in Trino Vercellese in 1338. He was succeeded by his son John II Palaiologos.
Marriage and issue
Theodore and Argentina had:
John ΙΙ (1313–1372)
Yolande (1318–1342), who married Aimone, Count of Savoy
Writings
Theodore is known to have authored an original military manual, titled Les Enseignemens ou Ordenances pour un Siegneur qui a Guerres et Grans Gouvernemens a Faire, often referred to as Les enseignements. Originally composed in Greek in 1326-1327 while Theodore was in Constantinople, it exists now only in the medieval French translation of Jean de Vignay. The work is one of the most interesting medieval military manuals in that it is not dependent on Vegetius' De Re Militari or any other known classical text. It thus serves as an example of the military thinking of the late Byzantine and Medieval worlds.
Ancestors
External links
The French translation of Les enseignements
Les enseignements de Théodore Paléologue, Christine Knowles (ed.)
References
Sources
|-
1290 births
1338 deaths
Marquesses of Montferrat
Palaiologos dynasty
14th-century Byzantine people
14th-century Italian nobility
Medieval Greek military writers
14th-century Byzantine writers
Sons of Byzantine emperors
People from Trino
|
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{"url":"https:\/\/math.stackexchange.com\/questions\/1309445\/limit-of-left-fracx25x3x2x3-rightx","text":"# Limit of $\\left(\\frac{x^2+5x+3}{x^2+x+3}\\right)^x$\n\nWe have to evaluate:\n$$\\lim\\limits_{x\\to\\infty} \\left(\\frac{x^2+5x+3}{x^2+x+3}\\right)^x$$\n\nMy work:\n\nLet the desired limit equal a constant $L$.\n\nWhen I take $\\log$ of both sides, the exponent $x$ comes down. What do I do now? Where will we apply L'Hopital's rule? Can we do it without the rule also?\n\nThe answer is $e^4$.\n\n\u2022 In your next step, move the $x$ to the denominator as $1\/x$ to make a $0\/0$ fraction and use L'Hospital's. \u2013\u00a0T.J. Gaffney Jun 2 '15 at 16:24\n\u2022 It becomes too much complicated? And still no success. \u2013\u00a0Aditya Agarwal Jun 2 '15 at 16:29\n\n$$\\ln\\left(\\lim_{x\\to +\\infty}\\left(\\frac{x^2+5x+3}{x^2+x+3}\\right)^x\\right)=\\lim_{x\\to +\\infty}\\left(x(\\ln(x^2+5x+3)-\\ln(x^2+x+3))\\right)$$\n\n$$=\\lim_{x\\to +\\infty}\\frac{\\ln(x^2+5x+3)-\\ln(x^2+x+3)}{\\frac{1}{x}}$$\n\n$$\\stackrel{\\text{L'Hop}}=\\lim_{x\\to +\\infty}\\frac{\\frac{2x+5}{x^2+5x+3}-\\frac{2x+1}{x^2+x+3}}{-\\frac{1}{x^2}}$$\n\n$$=\\lim_{x\\to +\\infty}\\left(\\frac{x^2(2x+1)}{x^2+x+3}-\\frac{x^2(2x+5)}{x^2+5x+3}\\right)$$\n\n$$=\\lim_{x\\to +\\infty}\\frac{4x^2(x^2-3)}{(x^2+x+3)(x^2+5x+3)}$$\n\n$$=\\lim_{x\\to +\\infty}\\frac{4(1-\\frac{3}{x^2})}{(1+\\frac{1}{x}+\\frac{3}{x^2})(1+\\frac{5}{x}+\\frac{3}{x^2})}=4$$\n\n\u2022 But the answer is $e^4$ \u2013\u00a0Aditya Agarwal Jun 2 '15 at 16:36\n\u2022 @AdityaAgarwal The final number is the natural logarithm of your limit. \u2013\u00a0egreg Jun 2 '15 at 16:38\n\u2022 Just the answer I wanted! \u2013\u00a0Aditya Agarwal Jun 2 '15 at 16:43\n\nYou can do so: $$\\frac{x^2+5x+3}{x^2+x+3} = 1 + \\frac{4x}{x^2+x+3}\\sim 1 + \\frac 4x$$ at $x=\\infty$ and therefore $$\\lim_{x\\to\\infty}\\left(\\frac{x^2+5x+3}{x^2+x+3}\\right)^{\\!x} = \\lim_{x\\to\\infty} \\left(1+\\frac 4x\\right)^{\\!x} = e^4$$\n\n\u2022 Can you plz give a more rigorous proof? \u2013\u00a0Aditya Agarwal Jun 2 '15 at 16:33\n\u2022 @AdityaAgarwal It is rigorous! \u2013\u00a0egreg Jun 2 '15 at 16:38\n\u2022 You forgot the power $x$ in the first limit of the last line. \u2013\u00a0anderstood Jun 2 '15 at 16:45\n\u2022 @anderstood, thanks \u2013\u00a0Michael Galuza Jun 2 '15 at 16:46\n\u2022 As it is not the case in general that $f(x)\\sim g(x)$ implies $f(x)^x\\sim g(x)^x$, to make this rigorous requires more justification of the second to last equals sign, doesn't it? @egreg \u2013\u00a0Jonas Meyer Jun 2 '15 at 18:10\n\n$$\\left[\\lim_{x\\to\\infty}\\left(1+\\dfrac{4x}{x^2+x+3}\\right)^{\\dfrac{x^2+x+3}{4x}}\\right]^{\\lim_{x\\to\\infty}\\dfrac{4x^2}{x^2+x+3}}$$\n\nSet $\\dfrac{x^2+x+3}{4x}=n$ in the inner limit to find $=e$ as $x\\to\\infty,n\\to\\infty$\n\nand $\\lim_{x\\to\\infty}\\dfrac{4x^2}{x^2+x+3}=\\lim_{x\\to\\infty}\\dfrac4{1+\\dfrac1x+\\dfrac3{x^2}}=\\cdots$\n\nlook at how $\\frac{x^2 + 5x + 3}{x^2 + x + 3}$ behaves for large $x.$\n\nwe have $$\\frac{x^2 + 5x + 3}{x^2 + x + 3} = 1 + \\frac{4x}{x^2 + x+3} = 1 + 4x(x^2 + x+3)^{-1} = 1 + 4x\\left(x^{-2} -x^{-4}(x+3) + \\cdots \\right) = 1 + \\frac4x + \\cdots$$\n\ntherefore $$\\left(\\frac{x^2 + 5x + 3}{x^2 + x + 3} \\right)^x = \\left(1 + \\frac4x+\\cdots\\right)^x = e^4 \\text{ as } x \\to \\infty.$$\n\n\u2022 How did you change $(x^2+x+3)^{-1}$ to that? And how did you deduce that in the end? \u2013\u00a0Aditya Agarwal Jun 2 '15 at 16:35\n\u2022 @AdityaAgarwal., $\\frac1{x^2 + x+3} = (x^2 + x + 3)^{-1}.$ then use the binomial theorem $(big + small)^{-1} = big^{-1} - big^{-2}\\, small + \\cdots$ \u2013\u00a0abel Jun 2 '15 at 16:36\n\u2022 Nice use of expansion. The way I did it in my head was lazy division of $x$. Nearly hitting the 20k mark :). \u2013\u00a0Chinny84 Jun 2 '15 at 16:37\n\u2022 @Chinny84, thanks. good to see you active again. seen any nice nonlinear differential equation? \u2013\u00a0abel Jun 2 '15 at 16:38\n\u2022 Sadly not so much. not many exciting ones (just the multiply by $\\dot{y}$ and reduce :(. I am working on my own nonlinear ode from my thesis (I would share it on here with more accomplished mathematicians such as yourself, but I am being selfish ;) ). \u2013\u00a0Chinny84 Jun 2 '15 at 16:41","date":"2019-05-23 19:25:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.792952299118042, \"perplexity\": 1296.4098800737106}, \"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\/1558232257361.12\/warc\/CC-MAIN-20190523184048-20190523210048-00156.warc.gz\"}"}
| null | null |
{"url":"http:\/\/maxspywareremover.com\/standard-error\/what-is-the-standard-error-of-estimate-formula.php","text":"Home > Standard Error > What Is The Standard Error Of Estimate Formula\n\n# What Is The Standard Error Of Estimate Formula\n\n## Contents\n\nThe researchers report that candidate A is expected to receive 52% of the final vote, with a margin of error of 2%. The mean of these 20,000 samples from the age at first marriage population is 23.44, and the standard deviation of the 20,000 sample means is 1.18. It is simply the difference between what a subject's actual score was (Y) and what the predicted score is (Y'). You can use regression software to fit this model and produce all of the standard table and chart output by merely not selecting any independent variables. More about the author\n\nAdjusted R-squared can actually be negative if X has no measurable predictive value with respect to Y. To illustrate this, let\u2019s go back to the BMI example. The proportion or the mean is calculated using the sample. Later sections will present the standard error of other statistics, such as the standard error of a proportion, the standard error of the difference of two means, the standard error of recommended you read\n\n## Standard Error Formula Excel\n\nI did ask around Minitab to see what currently used textbooks would be recommended. Student approximation when \u03c3 value is unknown Further information: Student's t-distribution \u00a7Confidence intervals In many practical applications, the true value of \u03c3 is unknown. For example, the U.S. As with the mean model, variations that were considered inherently unexplainable before are still not going to be explainable with more of the same kind of data under the same model\n\nI was looking for something that would make my fundamentals crystal clear. Mini-slump R2 = 0.98 DF SS F value Model 14 42070.4 20.8s Error 4 203.5 Total 20 42937.8 Name: Jim Frost \u2022 Thursday, July 3, 2014 Hi Nicholas, It appears like The standard error can be computed from a knowledge of sample attributes - sample size and sample statistics. Standard Error Of Proportion Lane PrerequisitesMeasures of Variability, Introduction to Simple Linear Regression, Partitioning Sums of Squares Learning Objectives Make judgments about the size of the standard error of the estimate from a scatter plot\n\nTherefore, the standard error of the estimate is There is a version of the formula for the standard error in terms of Pearson's correlation: where \u03c1 is the population value of The margin of error and the confidence interval are based on a quantitative measure of uncertainty: the standard error. Related articles Related pages: Calculate Standard Deviation Standard Deviation . Or decreasing standard error by a factor of ten requires a hundred times as many observations.\n\nis a privately owned company headquartered in State College, Pennsylvania, with subsidiaries in the United Kingdom, France, and Australia. The Standard Error Of The Estimate Is A Measure Of Quizlet In the regression output for Minitab statistical software, you can find S in the Summary of Model section, right next to R-squared. Sampling from a distribution with a large standard deviation The first data set consists of the ages of 9,732 women who completed the 2012 Cherry Blossom run, a 10-mile race held However, as I will keep saying, the standard error of the regression is the real \"bottom line\" in your analysis: it measures the variations in the data that are not explained\n\n## Standard Error Of The Mean Definition\n\nTherefore, the predictions in Graph A are more accurate than in Graph B. http:\/\/blog.minitab.com\/blog\/adventures-in-statistics\/regression-analysis-how-to-interpret-s-the-standard-error-of-the-regression The standard error is important because it is used to compute other measures, like confidence intervals and margins of error. Standard Error Formula Excel Want to stay up to date? Standard Error Formula Statistics Sokal and Rohlf (1981)[7] give an equation of the correction factor for small samples ofn<20.\n\nSmaller is better, other things being equal: we want the model to explain as much of the variation as possible. http:\/\/maxspywareremover.com\/standard-error\/what-does-standard-error-of-estimate-mean.php doi:10.4103\/2229-3485.100662. ^ Isserlis, L. (1918). \"On the value of a mean as calculated from a sample\". Finally, confidence limits for means and forecasts are calculated in the usual way, namely as the forecast plus or minus the relevant standard error times the critical t-value for the desired The important thing about adjusted R-squared is that: Standard error of the regression = (SQRT(1 minus adjusted-R-squared)) x STDEV.S(Y). Standard Error Of Estimate Interpretation\n\nSimilar formulas are used when the standard error of the estimate is computed from a sample rather than a population. The sample mean will very rarely be equal to the population mean. They report that, in a sample of 400 patients, the new drug lowers cholesterol by an average of 20 units (mg\/dL). click site Our global network of representatives serves more than 40 countries around the world.\n\nThe regression model produces an R-squared of 76.1% and S is 3.53399% body fat. Standard Error Mean doi:10.2307\/2340569. If the model assumptions are not correct--e.g., if the wrong variables have been included or important variables have been omitted or if there are non-normalities in the errors or nonlinear relationships\n\n## This gives 9.27\/sqrt(16) = 2.32.\n\nIs there a textbook you'd recommend to get the basics of regression right (with the math involved)? Standard Error of the Estimate Author(s) David M. Therefore, the standard error of the estimate is There is a version of the formula for the standard error in terms of Pearson's correlation: where \u03c1 is the population value of Standard Error Of Regression Formula Also, the estimated height of the regression line for a given value of X has its own standard error, which is called the standard error of the mean at X.\n\nPopulation parameter Sample statistic N: Number of observations in the population n: Number of observations in the sample Ni: Number of observations in population i ni: Number of observations in sample To estimate the standard error of a student t-distribution it is sufficient to use the sample standard deviation \"s\" instead of \u03c3, and we could use this value to calculate confidence The accuracy of the estimated mean is measured by the standard error of the mean, whose formula in the mean model is: This is the estimated standard deviation of the navigate to this website How to compare models Testing the assumptions of linear regression Additional notes on regression analysis Stepwise and all-possible-regressions Excel file with simple regression formulas Excel file with regression formulas in matrix\n\nScenario 2. In each of these scenarios, a sample of observations is drawn from a large population. What is the Standard Error of the Regression (S)? ISBN 0-7167-1254-7 , p 53 ^ Barde, M. (2012). \"What to use to express the variability of data: Standard deviation or standard error of mean?\".\n\nHere are a couple of additional pictures that illustrate the behavior of the standard-error-of-the-mean and the standard-error-of-the-forecast in the special case of a simple regression model. Take it with you wherever you go. The standard error of the mean (SEM) (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all Each of the two model parameters, the slope and intercept, has its own standard error, which is the estimated standard deviation of the error in estimating it. (In general, the term\n\nDownload Explorable Now! The correlation between Y and X , denoted by rXY, is equal to the average product of their standardized values, i.e., the average of {the number of standard deviations by which III. I use the graph for simple regression because it's easier illustrate the concept.\n\nOf course, T \/ n {\\displaystyle T\/n} is the sample mean x \u00af {\\displaystyle {\\bar {x}}} . The mean age was 33.88 years.","date":"2018-04-26 03:49:30","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8193269371986389, \"perplexity\": 652.2718455568601}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125948064.80\/warc\/CC-MAIN-20180426031603-20180426051603-00613.warc.gz\"}"}
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\section{Introduction}
A wealth of information about star-forming galaxies is contained in their
optical spectra. The ratios of the fluxes of optical emission lines can be used
to understand the physical conditions of the gas and stars in star-forming
regions, including metallicities, temperatures, densities, and ionization
parameters. Near-infrared spectroscopy is used to study these properties in
high-redshift galaxies, as the strong rest-frame optical lines are
redshifted into the near-IR at $z > 1.5$. These lines include
[\ion{O}{2}]$\lambda$3727, H$\beta$, [\ion{O}{3}]$\lambda$$\lambda$5007,4959,
H$\alpha$, [\ion{N}{2}]$\lambda$6584, and
[\ion{S}{2}]$\lambda$$\lambda$6717,6731, and can be used to infer the oxygen
abundance, electron density and ionization parameter in \ion{H}{2} regions.
Results from a small existing sample indicate that $z\sim2$ galaxies have
intrinsically different rest-frame optical line ratios from those in the bulk of
nearby \ion{H}{2} regions and Sloan Digital Sky Survey (SDSS) star-forming
galaxies \citep{erb2006a,shapley2005,liu2008}. This difference appears in the
emission-line diagnostic diagram of [\ion{O}{3}]/H$\beta$ vs.
[\ion{N}{2}]/H$\alpha$, where high-redshift star-forming regions are found
offset from an extremely tight sequence formed by local objects
\citep{kauffmann2003b}. The offset may be caused by differences in the SFR,
average ionization parameter, shape of the stellar ionizing spectrum, \ion{H}{2}
electron density, or the increased contribution of an AGN or shocked gas to the
integrated spectra. Based on the analysis of SDSS local star-forming galaxies,
\citet{brinchmann2008} suggest that an elevated ionization parameter
due to high electron densities and non-zero escape fractions of hydrogen
ionizing photons is the primary factor.
Specifying the cause of this offset using direct measurements
from high-redshift galaxies will provide new insight
into the physical conditions and chemical abundances of the gas in which the
stars were being formed during the epoch when the star-formation activity of the
Universe was at its peak \citep{madau1996,steidel1999,bouwens2007}.
The optimal redshift range for this type of analysis using ground-based near-IR
spectroscopy is $z=2.0 - 2.5$. In this range, the largest number of strong and
weak nebular emission lines falls within the windows of atmospheric transmission
and bluewards of the bright thermal IR background at $\lambda \geq 2.35\mu $m.
However, $z\sim2$ UV-selected galaxies typically have faint magnitudes
($K_{Vega} > 20.0$ mag) and small angular sizes ($\leq 1^{\prime\prime}$). Thus,
it is difficult to obtain the high S/N spectra that are necessary for a truly
detailed view of high-redshift star-forming regions using anything but long
integration times with the largest telescope apertures. This problem can be
addressed by taking advantage of the boost provided by gravitational lensing,
which magnifies and offers more sensitive views of distant galaxies.
The best-studied high-redshift lensed galaxy is MS 1512-cB58
\citep[$z=2.73$,][]{yee1996}, a typical $\sim L^*$ Lyman break galaxy (LBG). The
high magnification \citep[$\sim30$,][]{willlew1996,seitz1998} of cB58 has
allowed for unusually sensitive analyses of the galaxy's rest-frame UV
\citep{pettini2000, pettini2002}, optical \citep{teplitz2000}, IR
\citep{siana2008}, and far-IR \citep{sawicki2001, baker2001} spectra and
photometry. In particular, rest-frame optical spectra were used to calculate
dust extinction, virial mass, chemical abundances, and SFRs. Recently, new
search techniques for strongly-lensed high-redshift galaxies have yielded
additional candidates which have been spectroscopically confirmed to lie at $z >
2$ \citep{allam2007, smail2007}. Three of these candidates, SDSS
J090122.37+181432.3 ($z = 2.26$, hereafter referred to as SDSS J0901+1814; Diehl
et al. 2009, in preparation), the Cosmic Horseshoe \citep[$z =
2.38$;][]{belokurov2007}, and the Clone \citep[$z = 2.00$;][]{lin2008} are at
redshifts even better-suited to rest-frame optical spectroscopy than cB58.
Analysis of the rest-frame optical spectra of these objects presents a unique
opportunity to study both \ion{H}{2} region physics and the difference between
local star-forming galaxies and those at high-redshift.
In this paper, we analyze Keck II NIRSPEC spectra for these three lensed
objects. In \S \ref{sec:obsdata} we describe the observations, data reduction,
and line flux measurements. We discuss several physical quantities derived from
these line fluxes in \S \ref{sec:physquant}. We investigate how our objects
compare to other objects at $z\sim2$ in \S \ref{sec:analysis}, and review the
physical quantities for these objects in order to explain the differences in the
physical conditions between high-redshift and local \ion{H}{2} regions. Finally,
we conclude in \S \ref{sec:conclusions}. Throughout, we assume a cosmology with
$\Omega_{M} = 0.27$, $\Omega_{\Lambda} = 0.73$, and $H_0 = 71$ km s$^{-1}$
Mpc$^{-1}$.
\section{Observations and Data Reduction}\label{sec:obsdata}
\subsection{NIRSPEC Observations\label{sec:nirspecobs}}
\begin{figure*}
\epsscale{1.}
\plotone{f1.eps}
\caption{Images of the Objects with the NIRSPEC slit overlaid, defining the
various apertures used in the paper. a) SDSS J0901+1814, from a SDSS $r$-band
image, b) The Clone, from a SDSS $g$-band image, and c) The Cosmic Horseshoe,
from a VLT/FORS2 $R$-band image (Lindsay King, private communication). The
NIRSPEC slit shown is 0\secpoint76 $\times 42^{\prime\prime}$, and the
separation between the different apertures are 12\secpoint52 for SDSS
J0901+1814, 3\secpoint95 for the Clone.\label{fig:spec1}, and 8\secpoint82 for
the Cosmic Horseshoe.}
\epsscale{1.}
\end{figure*}
We obtained near-IR spectra on 2008 February 19 using the NIRSPEC spectrograph
\citep{mclean98} on the Keck~II telescope. Our targets included three strongly
lensed high redshift galaxies: SDSS J090122.37, the Cosmic Horseshoe, and the
Clone \citep[Diehl et al. 2009;][]{belokurov2007,lin2008}. At $z\sim2$, several
strong rest-frame optical lines fall within windows of atmospheric transmission
in the near-IR. H$\alpha$ and [\ion{N}{2}] fall within the NIRSPEC 6 (similar to
$K$-band) filter for all three objects. H$\gamma$, [\ion{O}{3}]$\lambda$4363,
H$\beta$, and [\ion{O}{3}]$\lambda$$\lambda$4959,5007 fall in the NIRSPEC 5
(similar to $H$-band) filter for the Cosmic Horseshoe and SDSS J090122.37, and
NIRSPEC 4 for the Clone. [\ion{O}{2}] and [\ion{Ne}{3}] fall in the NIRSPEC 3
(similar to $J$-band) filter for the Cosmic Horseshoe and SDSS J090122.37, and
NIRSPEC 2 for the Clone. Exposure times for the different filters ranged from
$3\times600$ s to $9\times600$ s, and are listed in Table \ref{tab:obs}. All
targets were observed with a 0\secpoint76 $\times$ $42^{\prime\prime}$ long
slit. Conditions were photometric during the night and seeing ranged from
0\secpoint4 - 0\secpoint9. The spectral resolution as determined from sky lines
was $\sim10$\,\AA\, for the NIRSPEC-2, -3, -4, and -5 filters, and
$\sim15$\,\AA\, for the NIRSPEC-6 filter. For each object, we placed the slit
across two of the lensed images to look for variation in the line ratios between
them. The slit position angles (in degrees east of north) were $7.22^{\circ}$
for SDSS J090122.37, $10.73^{\circ}$ for the Cosmic Horseshoe, and
$219.97^{\circ}$ for the Clone, and were determined by the locations of high
surface brightness knots in the optical lensed images. The separation between
the different apertures are 12\secpoint52 for SDSS J090122.37, 8\secpoint82 for
the Cosmic Horseshoe, and 3\secpoint95 for the Clone. Images of the objects with
slits overlaid are presented in Figure \ref{fig:spec1}.
\begin{deluxetable*}{lccccccc}
\tablewidth{0pt} \tabletypesize{\footnotesize}
\tablecaption{Galaxies Observed with Keck~II
NIRSPEC\label{tab:obs}}
\tablehead{
\colhead{~~~~~~NAME~~~~~~} &
\colhead{R.A. (J2000)} &
\colhead{Dec. (J2000)} &
\colhead{$z_{{\rm H}\alpha}$} &
\colhead{~$r (mag)$~} &
\colhead{$g-r$$(mag)^a$} &
\colhead{Exposure (s)} &
\colhead{Band}
}
\startdata
SDSS J0901+1814 \dotfill & 09 01 22.37 & 18 14 32.35 & 2.2586 & 20.6 & 0.52, 0.21 & 3 $\times$ 600 & $J$ \\
& & & & & & 6 $\times$ 600 & $H$ \\
& & & & & & 6 $\times$ 600 & $K$ \\
Cosmic Horseshoe \dotfill & 11 48 33.14 & 19 30 3.20 & 2.3813 & 19.0 & 0.29, 0.25 & 3 $\times$ 600 & $J$ \\
& & & & & & 9 $\times$ 600 & $H$ \\
& & & & & & 8 $\times$ 600 & $K$ \\
Clone \dotfill & 12 06 2.09 & 51 42 29.52 & 2.0026 & 19.0 & 0.39, 0.30, & 3 $\times$ 600 & $J$ \\
& & & & & 0.34 & 3 $\times$ 600 & $H$ \\
& & & & & & 3 $\times$ 600 & $K$ \\
\enddata
\tablecomments{Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes and arcseconds.}
\tablenotetext{a}{These colors are listed for apertures 1, 2, and in the case of the Clone, the full aperture as shown in Figure \ref{fig:spec1}.}
\end{deluxetable*}
\subsection{Data Reduction and Optimal Background Subtraction\label{sec:dataredt}}
Data reduction was performed following the procedure described in
\citet{liu2008}, where the sky background was subtracted using an optimal method
on the two-dimensional spectral images \citep[][Becker, private
communication]{kelson03}. One notable exception to this procedure addressed the
matter of pattern noise. A large fraction of the exposures taken in 2008
February were affected by pattern noise, which consisted of a constant positive
or negative offset in the mean count level in every $8^{\mathrm{th}}$ row of the
upper-right hand quadrant of the NIRSPEC CCD images. Prior to the standard
reduction procedure, we removed this pattern noise in all 75 of the science
images.
Sky subtraction for the Cosmic Horseshoe observations required extra care, due
to the presence of continuum emission in the slit from the outskirts of the
lensing galaxy \citep[a luminous red galaxy at $z=0.444$;][]{belokurov2007}.
Indeed, the first stage of sky subtraction typically consisted of a simple
difference of adjacent, dithered exposures. The dithers adopted for this target
were (unfortunately) similar in size to the spacing between the positions of the
Horseshoe and lens images, leading to the subtraction of the lens continuum
(rather than blank sky) from the Horseshoe in some pairs of exposures. We
quantified and corrected for this effect using pairs of exposures in which no
oversubtraction occurred.
\begin{figure*}
\epsscale{1.}
\plotone{f2.eps}
\caption{NIRSPEC spectrum of SDSS J0901+1814. The top row contains the $J$, $H$,
and $K$-band spectra for aperture 1, while the bottom row shows those for
aperture 2. Apertures 1 and 2 are as labelled in Figure \ref{fig:spec1}. The
positions of the prominent emission lines are indicated. The dotted lower
spectrum indicates the 1$\sigma$ errors offset vertically by $-10\times10^{-18}$
erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ for clarity. Larger errors occur at the
position of the sky lines and at points of high atmospheric extinction.
\label{fig:spec3817}}
\epsscale{1.}
\end{figure*}
\begin{figure*}
\epsscale{1.}
\plotone{f3.eps}
\caption{NIRSPEC spectrum of the Cosmic Horseshoe. The top row contains the $J$,
$H$, and $K$-band spectra for aperture 1, while the bottom row shows those for
aperture 2. Apertures 1 and 2 are as labelled in Figure \ref{fig:spec1}. The
positions of the prominent emission lines are indicated. The dotted lower
spectrum indicates the 1$\sigma$ errors offset vertically by $-10\times10^{-18}$
erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ for clarity. Larger errors occur at the
position of the sky lines and at points of high atmospheric extinction.
\label{fig:specCH}}
\epsscale{1.}
\end{figure*}
\begin{figure*}
\epsscale{1.}
\plotone{f4.eps}
\caption{NIRSPEC spectrum of the Clone. The top row contains the $J$, $H$, and
$K$-band spectra for aperture 1, the second row shows those for aperture 2, and
the bottom row shows those from the full aperture. Apertures 1 and 2 are as
labelled in Figure \ref{fig:spec1}. The positions of the prominent emission
lines are indicated. The dotted lower spectrum indicates the 1$\sigma$ errors
offset vertically by $-30\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ for
the $J$-band spectra, and $-10\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$
for the $H$- and $K$-band spectra for clarity. Larger errors occur at the
position of the sky lines and at points of high atmospheric extinction.
\label{fig:specClone}}
\epsscale{1.}
\end{figure*}
\begin{figure*}
\epsscale{1.}
\plotone{f5.eps}
\caption{NIRSPEC summed spectra from both Cosmic Horseshoe apertures, for the
[SII] doublet. The dotted lower spectrum indicates the 1$\sigma$ errors offset
vertically by $-5\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ for clarity.
\label{fig:specCHSII}}
\epsscale{1.}
\end{figure*}
One-dimensional spectra were extracted from these two-dimensional reduced images
along with error spectra. For all three of our objects we extracted apertures
for each of the two lensed images covered. For the Clone, the lensed objects
almost overlapped, so we extracted a third, full aperture that extended along
the slit across both lensed images. These spectra were then flux-calibrated
using A-type star observations according to the method described in
\citet{shapley2005} and \citet{erb2003}.
\subsection{Line Flux Measurements\label{sec:lineflux}}
One-dimensional, flux-calibrated spectra are shown along with the error spectra
in Figures \ref{fig:spec3817}, \ref{fig:specCH}, and \ref{fig:specClone}. The
error spectrum represents the 1$\sigma$ uncertainty in the flux at each
wavelength. In addition, zoomed-in regions of select portions of the spectra are
featured in Figures \ref{fig:specCHSII} and \ref{fig:Hgamma}. The redshifts of
the lensed objects were measured by fitting a Gaussian profile to the H$\alpha$
emission feature. In all cases, the redshifts from the separate components of
the lensed image pairs agreed to within $\Delta z/(1+z)$ = 0.0001 ($\Delta$$v$ =
$27 - 30$ km s$^{-1}$).
The lines that we set out to measure were
[\ion{O}{2}]$\lambda$$\lambda$3726,3729, [\ion{Ne}{3}]$\lambda$3869, H$\gamma$,
[\ion{O}{3}]$\lambda$4363, H$\beta$, [\ion{O}{3}]$\lambda$$\lambda$4959,5007,
H$\alpha$, [\ion{N}{2}]$\lambda$6584, and
[\ion{S}{2}]$\lambda$$\lambda$6717,6732. Line fluxes were determined by fitting
Gaussian profiles using the IRAF task, SPLOT. The H$\beta$, H$\gamma$, and
[\ion{O}{2}]$\lambda$$\lambda$3726,3729 lines were measured individually, while
the pairs of [\ion{O}{3}]$\lambda$$\lambda$4959,5007, H$\alpha$ and
[\ion{N}{2}]$\lambda$6584, and [\ion{S}{2}]$\lambda$$\lambda$6717,6732 were
fitted simultaneously. With this method, we fixed the central wavelength and
FWHM based on the best-fit parameters for the brighter line and then obtained a
combined fit for the relative fluxes of the two lines.
In some cases, we had to measure specific lines through different methods
because of the presence of sky lines, or the weak fluxes of the lines. For the
Cosmic Horseshoe and, to a lesser extent, the Clone, sky systematics were an
issue. In both objects, the H$\beta$ line was detected, but its wavelength
coincided with that of a strong sky line. Accordingly, we treated the measured
H$\beta$ line flux as a lower limit. In \S \ref{sec:physquant}, we describe a
method for recovering the actual H$\beta$ flux in the face of sky systematics.
We measured a lower limit on the line flux for the H$\beta$ and
[\ion{S}{2}]$\lambda$6732 features of aperture 2 of SDSS J0901+1814 as these
lines were over-subtracted on one side during the sky subtraction process. We
placed upper limits on the [\ion{O}{3}]$\lambda$4363 line fluxes in all
apertures, and on the [\ion{Ne}{3}]$\lambda$3869 line fluxes in all but aperture
1 of SDSS J0901+1814 where it was detected. We summed the flux across both of
the [\ion{S}{2}] lines for aperture 1 of both SDSS J0901+1814 and the Cosmic
Horseshoe due to the low flux from the individual lines.
Uncertainties in line fluxes were estimated using a Monte Carlo approach. For
each aperture and filter, we generated five hundred artificial spectra. Fake
spectra were created by perturbing the flux at each wavelength of the true
spectrum by a random amount consistent with the 1$\sigma$ error spectrum. Line
fluxes were measured from these simulated spectra using the same procedure that
was applied to the actual data. The standard deviation of the distribution of
line fluxes measured from the artificial spectra was adopted as the error on
each line flux measurement. Emission-line fluxes and associated uncertainties
are given in Table \ref{tab:emi}.
\begin{deluxetable*}{cccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Emission Line Fluxes\label{tab:emi}$^a$}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{SDSS J0901+1814} & \colhead{SDSS J0901+1814} & \colhead{Cosmic Horseshoe} & \colhead{Cosmic Horseshoe} & \colhead{Clone} & \colhead{Clone} & \colhead{Clone}
}
\startdata
Aperture & 1 & 2 & 1 & 2 & 1 & 2 & Full \\
${\sigma_{\mathrm{H}\alpha}}^b$ & 131$\pm$5 & 112$\pm$4 & 69$\pm$4 & 58$\pm$3 & 71$\pm$8 & 73$\pm$6 & 80$\pm$4 \\
$\mathrm{F}_{[\mathrm{OII}]}$ & $<$9.0 & $<$10.0 & 18.8$\pm$0.9 & 20.0$\pm$1.0 & 29.0$\pm$3.0 & 47.0$\pm$3.0 & 85.0$\pm$5.0 \\
$\mathrm{F}_{[\mathrm{NeIII}]\lambda3869}$ & $<$8.0 & 19.0$\pm$3.0 & $<$4.0 & $<$4.0 & $<$17.0 & $<$16.0 & $<$28.0 \\
$\mathrm{F}_{[\mathrm{H}\gamma]}$ & - & - & 3.0$\pm$1.0 & 4.6$\pm$0.4 & 6.0$\pm$2.0 & 11.2$\pm$0.7 & 21.0$\pm$1.0 \\
$\mathrm{F}_{[\mathrm{OIII}]\lambda4363}$ & - & - & $<$1.5 & $<$1.6 & $<$2.6 & $<$2.4 & $<$4.3 \\
$\mathrm{F}_{\mathrm{H}\beta}$ & 5.0$\pm$2.0 & $>$$4^{c}$ & 8.0$\pm$1.0 & 7.0$\pm$1.0 & 15.0$\pm$2.0 & 25.0$\pm$2.0 & 33.0$\pm$4.0 \\
$\mathrm{F}_{\mathrm{H}\beta,\mathrm{inf}^d}$ & - & - & 12.6$\pm$0.3 & 16.2$\pm$0.3 & 15.4$\pm$0.6 & 36.0$\pm$1.0 & 53.0$\pm$1.2 \\
$\mathrm{F}_{[\mathrm{OIII}]\lambda4959}$ & 8.0$\pm$2.0 & 11.0$\pm$2.0 & 7.5$\pm$0.8 & 9.6$\pm$0.5 & 20.0$\pm$1.0 & 35.0$\pm$1.0 & 58.0$\pm$2.0 \\
$\mathrm{F}_{[\mathrm{OIII}]\lambda5007}$ & 18.0$\pm$4.0 & 39.0$\pm$3.0 & 21.9$\pm$0.6 & 31.2$\pm$0.6 & 56.0$\pm$1.0 & 106.0$\pm$1.0 & 173.0$\pm$2.0 \\
$\mathrm{F}_{\mathrm{H}\alpha}$ & 63.0$\pm$2.0 & 72.0$\pm$1.0 & 43.4$\pm$0.9 & 53.9$\pm$0.9 & 61.0$\pm$2.0 & 132.0$\pm$4.0 & 202.0$\pm$5.0 \\
$\mathrm{F}_{[\mathrm{NII}]\lambda6584}$ & 42.0$\pm$1.0 & 64.0$\pm$1.0 & 4.0$\pm$0.7 & 8.7$\pm$0.7 & 10.8$\pm$0.9 & 21.0$\pm$1.0 & 39.0$\pm$2.0 \\
$\mathrm{F}_{[\mathrm{SII}]\lambda6717}$ & 12.0$\pm$1.0$^{e}$ & 11.0$\pm$1.0 & 6.0$\pm$$2.0^{e,g}$ & 4.0$\pm$$1.0^{g}$ & 5.5$\pm$0.9 & 10.0$\pm$1.0 & 9.0$\pm$2.0 \\
$\mathrm{F}_{[\mathrm{SII}]\lambda6732}$ & 12.0$\pm$1.0$^{e}$ & $>$$6.4^{f}$ & 6.0$\pm$$2.0^{e,g}$ & 6.0$\pm$1.0$^{g}$ & 7.3$\pm$0.9 & 10.0$\pm$1.0 & 12.0$\pm$3.0 \\
\enddata
\tablenotetext{a}{Emission line flux and error in units of $10^{-17}$ ergs $\mathrm{s}^{-1}$ $\mathrm{cm}^{-2}$}
\tablenotetext{b}{One-dimensional H$\alpha$ velocity dispersion and error in units of km s$^{-1}$. }
\tablenotetext{c}{A sky line coincided with H$\beta$ for aperture 2 of SDSS J0901+1814, and the line flux value was found by summing only the positive flux, instead of fitting the line with a Gaussian.}
\tablenotetext{d}{These inferred H$\beta$ values are calculated from the observed broadband $g-r$ color and the value for H$\alpha$, as described in \S \ref{sec:dust}.}
\tablenotetext{e}{This value corresponds to the sum of the flux across both lines, with error.}
\tablenotetext{f}{A sky line coincided with [SII]$\lambda6732$ for SDSS J0901+1814, aperture 2, and the line flux value was found by summing only the positive flux, instead of fitting the line with a Gaussian.}
\tablenotetext{g}{The spectra from apertures 1 and 2 were summed for the Cosmic Horseshoe in order to better measure the [SII] line fluxes. The measured values were $\mathrm{F}_{[\mathrm{SII}]\lambda6717}$ = $8\pm2$ , and $\mathrm{F}_{[\mathrm{SII}]\lambda6732}$ = $8\pm2$.}
\end{deluxetable*}
\section{Physical Quantities}\label{sec:physquant}
The flux measurement of rest-frame optical emission lines allows us to probe the
physical state of the interstellar gas in the target objects. Because of the
magnification provided by gravitational lensing, we have detected lines with a
higher signal-to-noise ratio (S/N) than what is typically seen in unlensed
high-redshift objects. We can measure dust extinction from the ratios of the
Balmer lines of H$\alpha$, H$\beta$, and H$\gamma$, while the H$\alpha$ line
offers the highest S/N measure of the gas velocity dispersion, as well as the
SFR of the galaxy. The set of [\ion{O}{3}]$\lambda$$\lambda$4959,5007,
[\ion{O}{2}]$\lambda$$\lambda$3726,3729, [\ion{N}{2}]$\lambda$6584 and Balmer
emission lines provides multiple methods of determining the gas-phase oxygen
abundance. The ratio of [\ion{O}{2}] and [\ion{O}{3}] lines offers a probe of
the ionization parameter of the star-forming regions. Finally, we can use our
detection of the [\ion{S}{2}]$\lambda$$\lambda$6717,6732 lines and an upper
limit of the [\ion{O}{3}]$\lambda$4363 line to estimate the electron density and
electron temperature of the gas. Table~\ref{tab:phy} lists several
relevant line ratios and the physical quantities derived from them,
as described in the sections below.
\subsection{AGN Contamination}
We measured the ratio of [\ion{N}{2}]/H$\alpha = 0.65 \pm 0.03$ and $0.88 \pm
0.03$ for the two apertures of SDSS J0901+1814. Such high ratios, especially in
the second aperture, cannot be produced in star-forming regions from
photoionization by massive stars \citep{kewley2001a,kauffmann2003b}. We
therefore infer the presence of an AGN. With insights from the lens model for
this system, the AGN can be probed on smaller scales than is typically possible
at $z\sim2$. Such analysis, however, is outside the scope of the current work.
For our study of the majority of physical quantities, we focus on the Cosmic
Horseshoe and the Clone, where the line fluxes do not appear to be contaminated
by the presence of an AGN.
\begin{figure*}
\epsscale{1.}
\plotone{f6.eps}
\caption{NIRSPEC spectra of the H$\gamma$ lines for the target objects. The
dotted lower spectra indicate the 1$\sigma$ errors offset vertically by
$-5\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ for
clarity.\label{fig:Hgamma}}
\epsscale{1.}
\end{figure*}
\subsection{Velocity Dispersion}
The velocity dispersion is a measure of the dynamics of the gas bound to the
galaxy by gravity. We used H$\alpha$ line widths to estimate velocity dispersion
after correcting for the instrumental resolution. The velocity dispersions were
calculated by $\sigma$ = FWHM$/2.355 \times \frac{c}{\lambda}$, where the FWHM
is the full width at half maximum in wavelength after subtraction of the
instrumental resolution ($14 - 15$ \AA$\,$in the $K$-band) in quadrature. The
velocity dispersions we measured were $131\pm5$ km s$^{-1}$ and $112\pm4$ km
s$^{-1}$ for the apertures of SDSS J0901+1814, $69\pm4$ km s$^{-1}$ and $58\pm3$
km s$^{-1}$ for the apertures of the Cosmic Horseshoe, and $71\pm8$ km s$^{-1}$,
$73\pm6$ km s$^{-1}$, and $80\pm4$ km s$^{-1}$ for the two apertures and the
full aperture of the Clone. Uncertainties in velocity dispersion were estimated
using the same Monte Carlo approach that was applied to line fluxes.
\subsection{Dust Extinction\label{sec:dust}}
It is important to understand the effects of dust extinction on the various line
fluxes, as reddening by dust is highly wavelength dependent and thus can alter
the observed flux ratios of widely spaced emission lines from their intrinsic
values. Extinction in $z\geq 2$ star-forming galaxies
is commonly estimated from rest-frame UV colors and an
application of the \citet{calzetti2000} starburst
attenuation law. The Calzetti
law appears to provide a fairly accurate description
on average of the reddening and attenuation of the
UV stellar continuum in both nearby
and distant starburst galaxies \citep{reddy2004,reddy2006a}.
The degree of dust extinction in star-forming regions
can also be estimated from the Balmer lines of hydrogren,
because these strong optical lines have
intrinsic ratios that are well described by atomic theory. Under the assumption
of Case B recombination \citep{agn2} and for T = 10,000 K, the Balmer ratios are
set. Any deviations in the observed line ratios are then attributed to dust
extinction. \citet{calzetti2001} demonstrates that, in local
star-forming galaxies, the stellar continuum suffers
less reddening than the ionized gas, expressed as $E(B-V)_{\rm{star}}=0.44 E(B-V)_{\rm{gas}}$.
Furthermore, \citet{calzetti2001} suggests that
the reddening of the Balmer lines in nearby UV-selected starbursts
is better described by a foreground dust distribution and traditional Milky Way extinction
curve \citep{cardelli1989}. On the other hand, \citet{erb2006c} present evidence
that, in $z\sim 2$ star-forming galaxies,
$E(B-V)_{\rm{star}}\simeq E(B-V)_{\rm{gas}}$ and a \citeauthor{calzetti2000} starburst
extinction law applied to both UV-continuum and H$\alpha$ emission lines
gives rise to the best agreement between UV- and H$\alpha$-derived
SFRs. Our new observations of the Cosmic Horseshoe
and the Clone allow for a detailed comparison of different estimates
of extinction in high-redshift star-forming galaxies.
First, we used
the observed broadband $g-r$ color for our objects and an assumed intrinsic SED
to obtain an estimate of $E(B-V)$. The SED model assumed was a \citet{bc2003}
solar metallicity, constant SFR model with stellar age of 570 Myr, as
\citet{erb2006b} find a median age of 570 Myr for star forming regions of
galaxies at $z\sim2$. We used SDSS $g$ and $r$ magnitudes for the Cosmic
Horseshoe, and in the case of the Clone, the Apache Point Observatory SPIcam $g$
and $r$ magnitudes of the individual apertures presented in \citet{lin2008}. The
$E(B-V)$ values we calculated were 0.16, 0.13 for the Cosmic Horseshoe, and
0.28, 0.21, and 0.24 for the Clone.
Next, extinction was estimated
based on rest-frame optical emission lines.
In both the Cosmic Horseshoe and the Clone,
we have detected three of the most prominent Balmer lines: H$\alpha$, H$\beta$,
and (because of the magnification afforded by gravitational lensing) H$\gamma$.
Unfortunately, in both objects, H$\beta$ lines lay over prominent sky
lines with systematically negative residuals, yielding in some cases unphysically
low values of H$\beta$/H$\gamma$. \citet{agn2} gives the value of H$\beta$/H$\gamma$ under Case B
recombination as
2.14. Because reddening will only serve to increase this ratio, our measured
values of less than 2.14 in aperture 2 of the Horseshoe and the full
aperture of the Clone suggest that, in general, our measured
H$\beta$ values were systematically low as a result of the residual error in
subtraction of the coincident sky line. H$\gamma$ does not appear to
suffer from the same systematic sky residuals as H$\beta$,
and therefore we used a comparison of the observed and
intrinsic ratios of H$\alpha$/H$\gamma$ to infer
$E(B-V)_{\rm{gas}}$, assuming an intrinsic H$\alpha$/H$\gamma$
ratio of 6.12 \citep{agn2}. To increase the S/N of the
H$\gamma$ feature, we combine spectra from individual
apertures of each object to obtain an average
H$\alpha$/H$\gamma$ and $E(B-V)_{\rm{gas}}$. The observed
H$\alpha$/H$\gamma$ ratios imply values of $E(B-V)_{\rm{gas}}$
of $0.45\pm 0.04$ and $0.28\pm 0.04$ for the Cosmic Horseshoe and the
Clone, respectively.
The value of $E(B-V)_{\rm{gas}}$
for the Cosmic Horseshoe is almost three times
higher than that inferred from the rest-frame
UV continuum. While this difference at first appears
to support the prescription from \citet{calzetti2000},
regarding the relative extinction of starlight and ionized gas,
we argue in section~\ref{sec:sfr} that the $g-r$-based
estimate of $E(B-V)$ provides a more robust estimate
of the ionized gas extinction. In contrast,
both $g-r$ and
H$\alpha$/H$\gamma$ provide consistent estimates
of dust extinction in the Clone, i.e., $E(B-V)_{\rm{star}}\simeq E(B-V)_{\rm{gas}}$.
It is also worth noting that, over the wavelength range probed by
our NIRSPEC spectra, the \citet{calzetti2000} and
\citet{cardelli1989} extinction curves yield very similar values
for $E(B-V)_{\rm{gas}}$, as well as extinction-corrected
line ratios. In all subsequent discussion,
we adopt the \citet{calzetti2000} extinction
law and $E(B-V)$ values based on rest-frame
UV colors.
One important application of
our derived $E(B-V)$ values is for
inferring H$\beta$ fluxes, independent
of sky residuals. We follow a similar methodology
to that of \citet{lemoine2003},
and calculate the H$\beta$ flux based on the observed H$\alpha$ flux,
the intrinsic ratio of 2.86 between
H$\alpha$ and H$\beta$, and our estimated
$E(B-V)$ based on $g-r$ colors. Accordingly,
$F_{H\beta}= F_{H\alpha}/2.86\times 10^{-0.4 E(B-V)(\kappa_{\beta}-\kappa_{\alpha})}$.
In this expression, $\kappa_{\beta}$ and $\kappa_{\alpha}$ refer
to the extinction coefficients in the \citeauthor{calzetti2000} curve at the wavelengths of
H$\beta$ and H$\alpha$, respectively. Inferred H$\beta$ fluxes are listed
in Table~\ref{tab:emi} and used in the calculation of
physical quantities in subsequent sections.
\begin{deluxetable*}{cccccc}
\tabletypesize{\scriptsize}
\tablecaption{Calculated Physical Values\label{tab:phy}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{Cosmic Horseshoe} & \colhead{Cosmic Horseshoe} &
\colhead{Clone} & \colhead{Clone} & \colhead{Clone}
}
\startdata
Aperture & 1 & 2 & 1 & 2 & Full \\
$R_{23}$ & 3.82$\pm$0.13 & 3.78$\pm$0.11 & 6.88$\pm$0.35 & 5.25$\pm$0.18 & 5.97$\pm$0.18 \\
$R_{23}$$^a$ & 4.08$\pm$0.15 & 3.95$\pm$0.12 & 7.45$\pm$0.40 & 5.52$\pm$0.20 & 6.36$\pm$0.20 \\
$N2$ & -1.04$\pm$0.07 & -0.79$\pm$0.04 & -0.75$\pm$0.04 & -0.79$\pm$0.02 & -0.72$\pm$0.02 \\
$O3N2$ & 1.28$\pm$0.08 & 1.08$\pm$0.04 & 1.31$\pm$0.05 & 1.26$\pm$0.03 & 1.23$\pm$0.02 \\
12+log(O/H)$_{{R}_{23}}$ & 8.85$\pm$0.05 & 8.85$\pm$0.05 & 8.55$ \pm$0.05 & 8.70$\pm$0.05 & 8.63$\pm$0.05 \\
${12+\mathrm{log(O/H)}_{{R}_{23}}}^{a}$ & 8.82$\pm$0.05 & 8.84$ \pm$0.05 & 8.50$\pm$0.05 & 8.68$\pm$0.05 & 8.60$\pm$0.05 \\
12+log(O/H)$_{{{N}2}}$ & 8.31$\pm$0.18 & 8.45$\pm$0.18 & 8.47$ \pm$0.18 & 8.45$\pm$0.18 & 8.50$\pm$0.18 \\
12+log(O/H)$_{{O3N2}}$ & 8.32$\pm$0.14 & 8.38$\pm$0.14 & 8.31$ \pm$0.14 & 8.33$\pm$0.14 & 8.34$\pm$0.14 \\
$O_{32}$ & 1.56$\pm$0.09 & 2.00$\pm$0.12 & 2.61$\pm$0.26 & 2.98$\pm$0.20 & 2.73$\pm$0.17 \\
$O_{32}$$^a$ & 1.28$\pm$0.07 & 1.70$\pm$0.10 & 1.84$\pm$0.18 & 2.28$\pm$0.15 & 2.01$\pm$0.12 \\
$\mathrm{F}_{[\mathrm{SII}]\lambda6717}/\mathrm{F}_{[\mathrm{SII}] \lambda6734}$ & - & 0.7$\pm$0.2 & 0.8$\pm$0.1 & 1.0$\pm$0.1 & 0.7$\pm $0.1 \\
${\mathrm{L}_{\mathrm{H}\alpha}}^c$ & 17 & 17 & 7 & 7 & 7 \\
${\mathrm{L}_{\mathrm{H}\alpha}}^{a,c}$ & 26 & 26 & 15 & 15 & 15 \\
${\mathrm{L}_{\mathrm{H}\alpha}}^{b,c}$ & 66 & 66 & 17 & 17 & 17 \\
${\mathrm{SFR}_{\mathrm{H}\alpha}}^{d}$ & 73 & 73 & 32 & 32 & 32 \\
${\mathrm{SFR}_{\mathrm{H}\alpha}}^{a,d}$ & 113 & 113 & 64 & 64 & 64 \\
${\mathrm{SFR}_{\mathrm{H}\alpha}}^{b,d}$ & 289 & 289 & 75 & 75 & 75 \\
$E(B-V)_{{g-r}}$ & 0.16 & 0.13 & 0.28$^e$ & 0.21$^e$ & 0.24$^{e,f}$ \\
$E(B-V)_{\mathrm{H}\alpha/\mathrm{H}\gamma}$ & 0.45 & 0.45 & 0.28 & 0.28 & 0.28 \\
\enddata
\tablenotetext{a}{These values are corrected for reddening by the
broadband $g-r$ derived $E(B-V)$ values}
\tablenotetext{b}{These values are corrected for reddening by $E(B-V)$
values derived from H$\alpha$/H$\gamma$ values and assuming Case B
recombination}
\tablenotetext{c}{H$\alpha$ luminosity in units of $10^{42}$ ergs $
\mathrm{s}^{-1}$, corrected for gravitational lensing, and summed
across both apertures}
\tablenotetext{d}{SFR in units of $\mathrm{M}_{\sun}$ $
\mathrm{yr}^{-1}$, corrected for gravitational lensing and summed
across both apertures}
\tablenotetext{e}{$g$ and $r$ magnitudes taken from \citet{lin2008}}
\tablenotetext{f}{$g$ and $r$ magnitudes for the full aperture of the
Clone are taken from the flux of the individual apertures}
\end{deluxetable*}
\subsection{Star-Formation Rate}\label{sec:sfr}
SFRs were calculated from H$\alpha$ luminosities based on the calibrations of
\citet{kennicutt1998}, but including a normalization factor of 1.8 to convert
from a \citet{salpeter1955} initial mass function (used by Kennicutt) to a
\citet{chabrier2003} initial mass function. Lensing serves to increase the
measured H$\alpha$ luminosity, and thus the SFR, by the magnification factor of the
lensed image. In order to correct for lensing, we used the
published lensing solutions for the Cosmic Horseshoe
\citep{dye2008,belokurov2007} and the Clone \citep{lin2008}. The published
magnification factors apply to the entirety of the Einstein ring and not to the
individual knots that we targeted on our NIRSPEC apertures. We therefore used
the VLT $R$ and SDSS $r$-band images to find the fraction of the flux in the
total Einstein ring that fell onto the slit (0.11 for the Cosmic Horseshoe
and 0.30 for the Clone). Under the assumption that the
rest-frame UV flux in these bands is proportional to the flux in H$\alpha$, we
divided the measured H$\alpha$ luminosity (for both apertures) by the fraction
of flux that fell into the slit versus the total flux in the ring. This
calculation yielded the H$\alpha$ luminosity for the entirety of the ring, which
we could then convert to a lensed SFR and divide by the magnification to find
the unlensed SFR. For the Cosmic Horseshoe, the magnification factor for the
lens is $24\pm2$ \citep{dye2008}, and adopting this lensing model, we calculated
an unlensed SFR of 73 $M_{\sun}$ yr$^{-1}$ without correcting for reddening,
113 $M_{\sun}$ yr$^{-1}$ with the broadband $g-r$ color reddening correction and
289 $M_{\sun}$ yr$^{-1}$ for the H$\alpha$/H$\gamma$ reddening correction. For
the Clone, the magnification factor given for the lens is $27\pm 1$
\citep{lin2008}, and under this assumption, the unlensed SFR was 32 $M_{\sun}$
yr$^{-1}$ without correcting for reddening, 64 $M_{\sun}$ yr$^{-1}$ with the
broadband $g-r$ color reddening correction and 75 $M_{\sun}$ yr$^{-1}$ with the
H$\alpha$/H$\gamma$ reddening correction. Based on the quoted uncertainties
in the magnification factors and $E(B-V)$ values, the formal errors on the lensing-
and dust-corrected SFRs are $\sim 15$\%, yet these
errors do not fully reflect hard-to-quantify uncertainties in
the fraction of the H$\alpha$ flux that fell into our
slit and the underlying stellar absorption affecting the Balmer lines,
and systematics in the lensing models.
SFRs can also be estimated from rest-frame UV luminosities,
according to the relation presented in \citet{kennicutt1998}. We use the optical
photometry from \citet{belokurov2007} and \citet{lin2008} to obtain such
estimates for the Cosmic Horseshoe and the Clone, respectively, again converting
from a Salpeter to Chabrier IMF. We calculate an extinction-corrected $SFR_{UV}=
65-80\: M_{\sun}$ yr$^{-1}$ for the Cosmic Horseshoe, where the range reflects
the uncertainty in inferred $E(B-V)$ values ($\sim0.13
- 0.16$, see Table \ref{tab:phy}) and magnification factor.
The UV-derived SFR is significantly lower
than the SFR based on H$\alpha$ if the
H$\alpha$ extinction correction is calculated from H$\alpha$/H$\gamma$. Much
better agreement is found when the $E(B-V)$ value used to correct
H$\alpha$ is based on the $g-r$ color. We therefore conclude that,
in the case of the Cosmic Horseshoe, the $g-r$ color
is a more reliable tracer of the extinction of both
stars and gas than the H$\alpha$/H$\gamma$ ratio.
For the Clone, we find an
extinction-corrected $SFR_{UV}= 60-110\: M_{\sun}$ yr$^{-1}$, corresponding to
$E(B-V)$ values of $\sim0.21 - 0.28$. This UV-derived SFR agrees
very well with the extinction-corrected H$\alpha$ SFR
(regardless of whether $g-r$ or H$\alpha$/H$\gamma$
is used to extinction-correct H$\alpha$).
In conclusion, we present the extinction properties of
the Cosmic Horseshoe and the Clone, along with a comparison
of their H$\alpha$ and UV SFRs. These results
support the findings of \citet{erb2006c}, that $E(B-V)_{\rm{star}}\simeq E(B-V)_{\rm{gas}}$
in $z\sim 2$ star-forming galaxies.
\subsection{Metallicity}
Our NIRSPEC measurements include a large number of rest-frame optical emission
line fluxes, enabling the calculation of gas-phase metallicities for our
objects. The gas-phase metallicity reflects the integrated products of previous
star formation, modulated by gas inflow and outflow. Typically, oxygen is used
to probe gas metallicity in star-forming galaxies as it is the most
abundant heavy element, and is promptly
released into the interstellar medium following a burst of star
formation. Furthermore, the emission lines from the various
ionization states of oxygen are strong and easily measurable in the rest-frame
optical. In this paper, solar abundance is defined as 12 + log(O/H) = 8.66
\citep{allende2002,asplund2004}.
For local galaxies, the oxygen abundance can be inferred from the electron
temperature by comparing auroral lines (such as [\ion{O}{3}]$\lambda$4363) to
nebular emission lines (e.g. [\ion{O}{3}]$\lambda$$\lambda$4959,5007). Weak
auroral lines are difficult to measure in high-redshift galaxies, where the
ratios of various strong emission lines are used as a proxy for metallicity.
These ratios have been calibrated as oxygen abundance indicators in local
\ion{H}{2} regions. The magnification from the strong-lensing has made it
possible to obtain a large set of high quality measurements of emission lines in
relatively short exposure times. We therefore can compare three strong-line
ratios as oxygen abundance indicators: $R_{23}$ $\equiv$
([\ion{O}{2}]$\lambda$3727 + [\ion{O}{3}]$\lambda$$\lambda$4959,5007/H$\beta$),
$N2$ $\equiv$ log([\ion{N}{2}]/H$\alpha$), and $O3N2$ $\equiv$
log\{([\ion{O}{3}]$\lambda$5007/H$\beta$) /
([\ion{N}{2}]$\lambda$6584/H$\alpha$)\}. $N2$ and $O3N2$ have been calibrated using
direct O/H measurements for local \ion{H}{2} regions by Pettini \& Pagel (2004),
and we used the $R_{23}$ calibration from \citet{tremonti2004}, which has been
calibrated with photoionization models. The observed strong-line ratios and
inferred metallicities are listed in Table \ref{tab:phy}.
The $R_{23}$ indicator was introduced by \citet{pagel1979}, and is widely used
for measuring local metal abundances if the fluxes of [\ion{O}{3}] and
[\ion{O}{2}] are known. \citet{tremonti2004} provided an analytical fit to the
$R_{23}$ - metallicity relation for a set of local SDSS galaxies:
\begin{equation}
12 + \mathrm{log(O/H)} = 9.185 - 0.313x - 0.264x^2 - 0.321x^3
\end{equation}
\noindent where $x = \mathrm{log}\,R_{23}$. Since $R_{23}$ is double valued for
the larger values of the ratio, this formula is only valid for the high
metallicity branch of the $R_{23}$-abundance relation. This relationship has a
1$\sigma$ scatter of 0.05 dex for the sample of star-forming galaxies presented
in \citet{tremonti2004}, though this error may underestimate the true systematic
uncertainty \citep{kennicutt2003}. For $R_{23}$, we used the values for the
H$\beta$ flux that we inferred from the broadband color-derived value of
$E(B-V)$. $R_{23}$ is sensitive to reddening because of the large wavelength
difference between [\ion{O}{3}]$\lambda$5007 and [\ion{O}{2}]$\lambda$3727. The
observed [and extinction corrected] $R_{23}$ values that we calculated for our
objects were 3.82[4.07], 3.79[3.96] for the two apertures of the Cosmic
Horseshoe, and 6.86[7.41], 5.25[5.52], and 5.93[6.32] for the two apertures and
the full aperture of the Clone. The $R_{23}$ values for apertures 1 and 2 of the
Cosmic Horseshoe correspond to metallicities of 12+log(O/H) = 8.85[8.82] and
8.85[8.84], respectively. These are high relative to solar metallicity. For the
Clone, the inferred metallicities for apertures 1, 2 and the full aperture are
12+log(O/H) = 8.55[8.50], 8.70[8.68], and 8.64[8.60], respectively. These values
range from slightly subsolar to solar.
The $N2$ indicator is related to the oxygen abundance by:
\begin{equation}
12 + \mathrm{log(O/H)} = 8.90 + 0.57 \times \mathrm{N}2
\end{equation}
\noindent which is valid for 7.50 $<$ 12 + log(O/H) $<$ 8.75.
The relationship has a 1$\sigma$ scatter of $\pm$0.18 dex on the
y-intercept \citep{pettini2004}. We found values of $N2 = -1.04$
and $-0.79$ (12+log(O/H) = 8.31 and 8.45) for the Cosmic Horseshoe,
and $-0.75$, $-0.79$, and $-0.72$ (12+log(O/H) = 8.47, 8.45, and 8.50)
for the Clone. These values all indicate a subsolar gas metallicity.
We also measured values of the $O3N2$ indicator for our objects.
This indicator is related to oxygen abundance by:
\begin{equation}
12 + \mathrm{log(O/H)} = 8.73 - 0.32 \times \mathrm{O}3\mathrm{N}2
\end{equation}
\noindent which is valid for 8.12 $<$ 12+log(O/H) $<$ 9.05, with a 1$\sigma$
scatter of $\pm$0.14 dex \citep{pettini2004}. The errors shown for our oxygen
abundances are dominated by the systematic uncertainties from the calibration,
as opposed to measurement errors. For $O3N2$, we used the values of H$\beta$ that
were calculated from the $g-r$ color, as described in \S \ref{sec:dust}. The
values that we calculated are 1.28, 1.08 (12+log(O/H) = 8.32 and 8.38) for the
Cosmic Horseshoe, and 1.31, 1.26, and 1.23 (12+log(O/H) = 8.31, 8.33, and 8.34)
for the Clone. These values indicate subsolar metallicities in the gas. The $N2$
and $O3N2$ indicators do not need to be dereddened due to the close spacing of the
[\ion{N}{2}], H$\alpha$ lines and [\ion{O}{3}], H$\beta$ lines.
The $N2$ and $O3N2$ indicators yield consistent metallicities of $\sim0.5$ solar for
both the Cosmic Horseshoe and the Clone. However, the $R_{23}$ indicator points
to a significantly higher, and supersolar, metallicity in the Cosmic Horseshoe.
Such systematic discrepancies between $R_{23}$ and other indicators are well
known among local star-forming galaxies, and reflect the current limitations of
strong-line abundance indicators \citep{kewley2008, kennicutt2003}. For the
Clone, $R_{23}$ indicates a metallicity that is only slightly higher than, and
statistically consistent with, the values implied by $N2$ and $O3N2$. As we will
discuss in \S \ref{sec:offset}, the differences in metallicity for the Cosmic
Horseshoe and the Clone based on $R_{23}$ may reflect the relative positions of
these objects with respect to the emission-line excitation sequence of local
objects.
\subsection{Ionization Parameter}
The local ionization state in an \ion{H}{2} region is often characterized by the
ionization parameter, $U$, which is the ratio between the density of ionizing photons
and the density of hydrogen atoms. Commonly, ionization parameters are estimated
using $O_{32}$ = ([\ion{O}{3}]$\lambda$$\lambda$4959,5007 /
[\ion{O}{2}]$\lambda$$\lambda$3726,3729), and corrected for reddening. We
estimate extinction based on the observed $g-r$ colors. Our values of $O_{32}$,
both uncorrected [and corrected] for extinction, are 1.56[1.28], 2.00[1.70] for
the two apertures of the Cosmic Horseshoe, and 2.61[1.84], 2.98[2.28], and
2.73[2.01] for the two apertures and the full aperture of the Clone.
\citet{lilly2003} showed that a vast majority of local objects (from a Nearby
Field Galaxy Survey $B$-selected local galaxy sample from \citet{jansen2000})
have a value of $O_{32} < 1$, while the available data for objects at $z > 2$
\citep{pettini2001} indicate $O_{32} > 1$ \citep{pettini2001}.
The value of $O_{32}$, however, is dependent not only on ionization parameter,
but also, to a lesser extent, on metallicity \citep{kewley2002,brinchmann2008}.
\citet{kewley2002} provide formulae for relating observed values of
$O_{32}$ to ionization parameter, $\log(U)$, using photoionization models
spanning a range of metallicities.
We apply the relations for both $0.5$ and $1.0 Z_{\sun}$ models
to translate our extinction-corrected $O_{32}$ measurements into
$\log(U)$ values. The adopted models reflect the range of metallicities
inferred for the Cosmic Horseshoe and the Clone.
At higher metallicities, a given value of $O_{32}$ corresponds to a
higher ionization parameter, so the $0.5 Z_{\sun}$ models yield a lower
bound to the inferred ionization parameters. Assuming $Z=1.0 Z_{\sun}$,
we find $\log(U)$ ranging from -2.4 to -2.5 and -2.3 to -2.4 for the
apertures of the Cosmic Horseshoe and the Clone, respectively.
At $Z=0.5 Z_{\sun}$, the corresponding ionization parameters range from
$\log(U)=-2.7$ to -2.8 and -2.5 to -2.6, i.e., $0.2-0.3$ dex lower.
The $\log(U)$ values inferred for $0.5$ and $1.0 Z_{\sun}$
models are very similar to the high ionization parameters
(relative to local SDSS galaxies)
inferred by \citet{brinchmann2008} for the $z>2$ objects
in \citet{pettini2001}.
\begin{figure*}
\epsscale{1.}
\plotone{f7.eps}
\caption{$O_{32}$ versus $R_{23}$ diagram. The apertures of the Cosmic Horseshoe
and the Clone are plotted along with star-forming galaxies at intermediate
and high redshift. A selection of $0.47 < z < 0.92$ objects
from \citet{lilly2003} are shown in purple. High-redshift points
include five $z\sim3$ LBGs from \citet{pettini2001}, two lensed $z\sim 1.9$ objects
from \citet{lemoine2003}, and a composite spectrum of nine $z\sim 3.5$ LBGs
from \citet{maiolino2008}. Symbols are described in the legend. The values on
this plot are not corrected for dust extinction, however the red and
green arrows indicate, respectively, the small impact
of extinction corrections on the Cosmic Horseshoe and Clone data points.
Such dust corrections will shift points downwards and to the right.
The objects at $z\geq2$ are systematically displaced on average with respect
to the lower-redshift sample towards
larger values of $O_{32}$, at fixed values of $R_{23}$.
This empirical trend is evidence
for higher ionization parameters at fixed metallicity \citep{kewley2002}.
\label{fig:O32R23}}
\epsscale{1.}
\end{figure*}
\subsection{Electron Temperature and Density}
The [\ion{S}{2}]$\lambda$$\lambda$6717,6732 line ratio is sensitive to the
electron density of an ionized gas. We used this line ratio in our objects to
calculate the electron density for the ionized nebular gas. Similarly, the ratio
([\ion{O}{3}]$\lambda$$\lambda$4959, 5007 / [\ion{O}{3}]$\lambda$4363) is used
to measure the electron temperature, as [\ion{O}{3}] has an energy-level
structure that results in upper level emission lines with different excitation
energies, whose relative strengths thus depend strongly on temperature. We used
the IRAF procedure TEMDEN to calculate a range of densities and temperatures
based on these line ratios.
Upper limits on the electron temperatures were measured from the [\ion{O}{3}]
line ratios, since we were only able to measure upper limits for the
[\ion{O}{3}]$\lambda$4363 line flux. For the lowest densities ($\sim$100
cm$^{-3}$), the upper limits on the temperatures were 12,900 K and 12,200 K for
the two apertures of the Cosmic Horseshoe and 13,200 K, 9,700 K and 10,000 K for
the two apertures and the full aperture of the Clone, respectively. Changes in
the density over the range 100 to 5000 cm$^{-3}$ altered the calculated
temperatures only very slightly.
We calculated densities from the measured [\ion{S}{2}] line ratios in all
apertures except for aperture 1 of the Cosmic Horseshoe, which had insufficient
S/N. We also summed both apertures of the Cosmic Horseshoe to
obtain an average, higher S/N, [\ion{S}{2}] line ratio estimate for the galaxy.
In the calculation of densities we assumed a standard temperature of
10,000 K, consistent with our limits on electron temperatures. For
each object the $\pm 1\sigma$ range in electron density
was then derived from errors on the observed line ratios.
For the Cosmic Horseshoe, Aperture 2,
the $\pm 1\sigma$ range in electron density is $840-6900 \mbox{ cm}^{-3}$;
for the summed Cosmic Horseshoe apertures, the corresponding
range is $320-1600 \mbox{ cm}^{-3}$. For the Clone, apertures
1, 2, and full, we find ranges of $1110-2960\mbox{ cm}^{-3}$,
$530-1020 \mbox{ cm}^{-3}$, and $1270-2540\mbox{ cm}^{-3}$. Despite
the large uncertainties in the precise value of inferred electron density,
the empirical fact remains that
the high-redshift lensed targets exhibit systematically lower
doublet ratios from what is typically observed in emission-line galaxies
in the local universe \citep{liu2008}.
\section{Analysis}\label{sec:analysis}
\subsection{Comparison to the $z\sim2$ Population\label{sec:comparison}}
Recent studies have been undertaken to understand the physical conditions of
$z\sim2$ objects \citep{erb2006a,liu2008} and the observations presented here
enable comparisons of the Cosmic Horseshoe and the Clone with the larger
population of star-forming galaxies at similar redshifts. The Cosmic Horseshoe
and the Clone, based on their $ugr$ colors, satisfy the criteria for UV-selected
objects at $z\sim2$ \citep{steidel2004}. In the following analysis, we compare
the physical quantities calculated from the rest frame optical line fluxes to
those calculated in other high-redshift objects.
SFRs in high redshift objects have been measured in a variety of ways. We
calculated lensing-corrected values for the SFR of $\sim 113 \: M_{\sun}$
yr$^{-1}$ based on our method of dereddening the H$\alpha$ fluxes for the Cosmic
Horseshoe (i.e., from the $g-r$ color) and $64-75 \: M_{\sun}$ yr$^{-1}$ based on our method of
dereddening for the Clone (i.e., from both the $g-r$ color and H$\alpha$/H$\gamma$
ratio). The average SFRs for a population of $z\sim2$
galaxies from \citet{erb2006c} is $\langle SFR_{H\alpha} \rangle = 31 \pm 18$
$M_{\sun}$ yr$^{-1}$. Our lensed targets are therefore more actively star-forming
than average, but within the range spanned by the sample in \citet{erb2006c}.
However, we caution that our calculated
values depend on the magnification given by the particular lensing models that
we used. For comparison, based on the H$\alpha$ flux and magnification factor of
$\sim30$ \citep{teplitz2000, seitz1998} and with the assumption of a Chabrier
initial mass function and \citet{calzetti2000} dust extinction law with an
estimated $E(B-V)_{\rm{gas}} = 0.06$ \citep{siana2008}, the SFR for MS1512-cB58 is
$SFR(H\alpha) = 14$ $M_{\sun}$ yr$^{-1}$.
\citet{erb2006b} measure an average velocity dispersion for a sample of $z\sim2$
objects (with AGN removed) of $\langle \sigma \rangle = 108 \pm 5$ km s$^{-1}$,
and a standard deviation of 86 km s$^{-1}$. Our objects show a smaller velocity
dispersion than the average, but are within the sample's large standard
deviation. While SDSS J0901+1814, in particular aperture 2, displays AGN-like
line ratios, the velocity dispersions are very close to the average values for
other $z\sim2$ objects, indicating that SDSS J0901+1814 could be a narrow-line
AGN. The dust extinction values that we infer from the $g-r$ color
for the Cosmic Horseshoe are similar to
the mean $E(B-V)$ value of $0.16 \pm 0.01$ from \citet{erb2006b},
while the Clone appears to be dustier than average, based on
its $g-r$ color and H$\alpha$/H$\gamma$ ratio. Both objects are well within
the range of extinction values spanned by $z\sim 2$ star-forming galaxies.
Our values for $N2$ ranged from -0.72 to -1.04, within the interval spanned by
the sample of $z\geq2$ galaxies from \citet{erb2006a}. Values of $R_{23}$ span
from 4.0 to 7.4, resulting in a range of metallicities that are slightly
subsolar to slightly greater than solar, between 12+log(O/H) = 8.5 to 8.8.
Values for $R_{23}$ from \citet{pettini2001} for LBGs at $z\sim3$ are within
this range, between 4.0 and 12.3, including a value of $R_{23} = 8.3$ for MS
1512-cB58. \citet{lilly2003} plot star-forming galaxies at intermediate redshift
($0.47 < z < 0.92$) by their values for $O_{32}$ versus $R_{23}$, and we
reproduce their diagram in Figure \ref{fig:O32R23}, adding the Cosmic Horseshoe and
Clone. To place our new observations in the context of other, similar
measurements at high redshift, we also plot five LBGs at $z\sim3$ from \citet{pettini2001},
two lensed objects at $z\sim 1.9$ from \citet{lemoine2003},
and the composite $z\sim 3.5$ spectrum from \citet{maiolino2008},
based on nine individual LBGs. In this
plot, note that all of the values are as measured, and none have been
dereddened. The effect of correcting the points for
the Cosmic Horseshoe and Clone for dust extinction
is indicated with reddening vectors. This diagram shows how objects at $z\geq 2$ have
larger values of $O_{32}$ (indicative of high values of the ionization
parameter) for a given value of R$_{23}$ (and thus, metallicity),
relative to the sample from \citet{lilly2003}.
A common property of $z\geq 2 $ star-forming galaxies
is the kinematic evidence for star-formation feedback
in the form of large-scale gas outflows \citep{pettini2001,shapley2003}.
With measurements of the nebular, systemic redshifts for both the
Cosmic Horseshoe and Clone, as well as rest-frame UV absorption-line
spectra, it is possible to search for the signature
of outflows. Using Keck II Echelle Spectrograph and
Imager \citep[ESI;][]{sheinis2002} rest-frame ultraviolet spectra of the Cosmic
Horseshoe and the Clone, \citet{quider2009} and Quider et al. (in preparation)
analyze several of the strongest interstellar
absorption features for kinematic evidence of large-scale gas outflow.
In this analysis, redshifts measured for interstellar absorption
lines were compared with those of H$\alpha$ and UV stellar photospheric features.
For both targets, the slits were placed in the same position that was used for
the NIRSPEC observations, and the spectra from the two apertures were summed to
increase the signal-to-noise ratio. In the Cosmic Horseshoe,
the average outflow velocity for the low-ionization
interstellar absorption lines (e.g., \ion{Si}{2} $\lambda$1260,
the \ion{O}{1} + \ion{Si}{2} $\lambda$1303 doublet,
\ion{Si}{2} $\lambda$1527, \ion{Al}{2} $\lambda$1671)
is $\langle v_{out,low} \rangle = 146$ km s$^{-1}$,
and $\langle v_{out,high} \rangle = 167$ km s$^{-1}$ for
the high-ionization features (e.g., \ion{Si}{4} $\lambda\lambda $1394,1403,
\ion{Al}{3} $\lambda$1855, \ion{Al}{3}
$\lambda$1863). In the Clone, three low-ionization iron
lines (\ion{Fe}{2} $\lambda$2344, 2383, and 2600) were used
to measure interstellar kinematics. These three lines indicate similar
offsets, and the average outflow velocity is $\langle v_{out,low} \rangle = 154$
km s$^{-1}$. The outflow velocities observed in the Cosmic Horseshoe
and Clone are consistent with those calculated for star-forming
galaxies at $z\sim2$ \citep{steidel2004}. Also, for comparison, the outflow
velocity observed in MS 1512-cB58 is $\sim255$ km s$^{-1}$ \citep{pettini2002}.
The H$\alpha$ velocity dispersion, $\sigma$, allows us to calculate the virial
masses of our objects when combined with estimates of their sizes, using the
half-light radius, $r_{1/2}$. We use the equation given by \citet{pettini2001}
for virial mass, assuming an idealized case of a sphere of uniform density:
\begin{equation}
M_{\mathrm{vir}} = 1.2 \times 10^{10} M_{\sun} \frac{\sigma^2}{(100 \,\mathrm{km}\,\mathrm{s}^{-1})^2} \frac{r_{1/2}}{\mathrm{kpc}}
\end{equation}
\noindent The lensing models for both the Cosmic Horseshoe and the Clone yield
estimates of the intrinsic half-light radii, allowing us to calculate the
virial masses of the objects.
The lensing model for the Cosmic Horseshoe \citep{dye2008} produces a source
plane image with two objects, a ``northern" and a ``southern" source. These
objects are lensed into ring-like distributions in the image plane, with
slightly different morphologies. Accordingly, the relative contribution of the
two sources varies as a function of position in the combined Einstein ring. The
fainter, northern source, according to the \citet{dye2008} model, contributes
mostly to the portion of the Einstein ring that is not probed by our slit
position, although the light from this source does contribute a small fraction
of the emission in what we call aperture 1. The southern source, the more
prominent of the two, is the only contributor to the emission in aperture 2 and
constitutes the majority of the light in aperture 1. From the composite of the
reconstructed source plane image, we estimated a half-light radius of the
southern source of $\sim0\secpoint3$, which, at this redshift, and, for our
adopted cosmology, corresponds to a size of $r_{1/2, CH} = 2.5$ kpc. Thus, using
the velocity dispersion for the second aperture, since it corresponds only to
the southern source in the reconstructed image, we calculate a virial mass of
the Cosmic Horseshoe of $M_{vir,CH} = 1.0 \times 10^{10} M_{\sun}$.
For the Clone, the lensing models described in \citet{lin2008} and Allam et al.
(2009, in preparation) yield estimates of the half-light radius in the
reconstructed source plane of $\sim0\secpoint3$. At this redshift, the
corresponding physical radius is $r_{1/2, Clone} = 2.9$ kpc. The virial mass we
calculated for the Clone, using the velocity dispersion from the full aperture,
was $M_{vir,Clone} = 2.2 \times 10^{10} M_{\sun}$. Both the dynamical masses
calculated for the Cosmic Horseshoe and the Clone are similar to the dynamical
masses for objects at $z\sim2$ from \citet{erb2006b}.
While the line flux ratios for the Clone are very similar between the apertures,
the \citet{dye2008} lensing model might help to explain the difference in the
line flux ratios observed between the two apertures of the Cosmic Horseshoe. For
instance, as seen in Table \ref{tab:emi} and Figure \ref{fig:specCH}, the
[\ion{N}{2}] and H$\alpha$ lines vary in strength with relation to each other.
One possible hypothesis concerns the level of contribution from the ``northern"
source to the first aperture of the lensed ring. If the gas in the fainter,
northern source was at a lower metallicity, it would slightly lower the
[\ion{N}{2}]/H$\alpha$ line ratio of aperture 1 compared to aperture 2, which is
what is observed in our data. Future analysis will be undertaken with the
lensing model in order to disentangle the contribution from each object in the
source plane.
In summary, we have shown that the SFRs, extinction values, metallicities,
outflow velocities, and dynamical masses for our objects are relatively typical
of those found among UV-selected star-forming galaxies at $z\sim2$
\citep{steidel2004}. We conclude that the Cosmic Horseshoe and the Clone are a
representative sample of the luminous segment of the population of $z\sim2$ objects. Future
multi-wavelength observations of the broadband spectral energy distributions of
the lensed targets will allow us to characterize their stellar masses and ages
and therefore evolutionary states.
\subsection{The Offset in the [\ion{N}{2}]/H$\alpha$ vs. [\ion{O}{3}]/H$\beta$ diagram
for High-Redshift Galaxies}\label{sec:offset}
\begin{figure*}
\epsscale{1.}
\plotone{f8.eps}
\caption{\ion{H}{2} region diagnostic plot of log[\ion{N}{2}]/H$\alpha$ and
log[\ion{O}{3}]/H$\beta$ emission line ratios. The grey points and contours
represent SDSS local AGN and star forming galaxies that satisfy the criteria
described in \S \ref{sec:offset}, while the lensed objects are as labelled in
the legend. The
circled objects are those where the H$\beta$ flux values were inferred from the
$g-r$ color and H$\alpha$ fluxes. SDSS J0901+1814 points show large error bars
due to the presence of a sky line on the H$\beta$ line in both apertures. The
dashed line is an emprical demarcation between star-forming regions and AGN from
\citet{kauffmann2003b} based on the SDSS galaxies, whereas the dotted line is
the theoretical limit for star-forming galaxies from \citet{kewley2001a}. Local
star-forming galaxies and \ion{H}{2} regions exist in a well-defined excitation
sequence due to photoionization by massive stars, and these are found to the
left and below the curves. The apertures of the Clone, even with the inferred
values of H$\beta$, are on average offset from this excitation sequence.
\label{fig:BPT}}
\epsscale{1.}
\end{figure*}
Recently, \citet{shapley2005}, \citet{erb2006a}, and \citet{liu2008} have
presented evidence indicating a difference between \ion{H}{2} regions in
high-redshift galaxies and those in local galaxies. The physical conditions in
these regions are often probed by using the well known empirical diagnostic
diagram of Baldwin, Phillips, \& Terlevich (1981) (hereafter BPT). This diagram
separates star-forming galaxies and AGN according to the optical line ratios
[\ion{N}{2}]/H$\alpha$ and [\ion{O}{3}]/H$\beta$; star-forming galaxies fall in
a region of lower [\ion{N}{2}]/H$\alpha$ for a range of values of
[\ion{O}{3}]/H$\beta$. It has been demonstrated that a fraction of high redshift
star-forming galaxies at $z \sim 1 - 2$ lie offset from the local population of
\ion{H}{2} regions and star-forming galaxies, displaced towards higher
[\ion{N}{2}]/H$\alpha$ and [\ion{O}{3}]/H$\beta$ values. The strongly lensed,
$z\sim2$ objects are plotted on the BPT diagram in Figure \ref{fig:BPT} along
with local emission-line objects from the fourth data release (DR4) of the SDSS
\citep{york2000,adelman2006}. The SDSS objects were selected to have
[\ion{N}{2}], H$\alpha$, [\ion{O}{3}], and H$\beta$ line detections with S/N $>$
5. Also plotted on the diagram are curves designed to separate star forming
galaxies and AGN; one line is from \citet{kauffmann2003b} that demarcates
star-forming galaxies and AGN on an empirical basis, and the other from
\citet{kewley2001a} that represents a limit to the line-flux ratios that can be
produced for star forming regions from photoionization plus stellar population
synthesis models.
For both the Cosmic Horseshoe and the Clone, we plot two points
for each aperture, one for each of
method of calculating H$\beta$ fluxes. First, we plot upper limits on
[\ion{O}{3}]/H$\beta$ to reflect the fact that the measured H$\beta$ is a lower
limit due to sky-subtraction residuals. Next, we plot circled symbols
using the values of H$\beta$ flux inferred
from H$\alpha$ and the $g-r$ color as described in \S \ref{sec:dust}. While SDSS
J0901+1814 lies in the region occupied by local AGN, the two apertures of the
Clone both appear in the area above the local star-forming regions. For the
inferred H$\beta$ values, the line ratios for the Clone are shifted towards
higher [\ion{O}{3}]/H$\beta$ by $0.2 - 0.3$ dex, relative to star-forming SDSS
galaxies with similar [\ion{N}{2}]/H$\alpha$ ratios. Alternatively, the Clone is
shifted towards higher [\ion{N}{2}]/H$\alpha$ by 0.6 - 0.7 dex compared to SDSS
galaxies with similar [\ion{O}{3}] / H$\beta$ ratios.
The interpretation is less straightforward for the Cosmic Horseshoe.
Indeed, the [\ion{O}{3}]/H$\beta$ upper limit for its second aperture is also offset
from the local emission-line sequence, yet adopting the H$\beta$ value based
on the H$\alpha$ flux and $g-r$ color places this aperture on the top edge of
the locus of low-redshift galaxies. On the other hand,
using the H$\alpha$/H$\gamma$ ratio to estimate the H$\beta$ flux results
in an [\ion{O}{3}]/H$\beta$ ratio that is midway between the two
plotted symbols for this aperture, and, again, offset with
respect to the local emission-line sequence.
Our best estimate for aperture 2 of the Cosmic Horseshoe is that
its point on the BPT diagram lies somewhere between top of the local emission-line
sequence and the symbol indicating the upper limit for the aperture.
In contrast, the point for aperture 1 of the Cosmic Horseshoe
is consistent, or even below, the local emission-line sequence, depending on which
value of H$\beta$ is adopted. Identifying the origin of the emission-line
ratio variations within the Cosmic Horseshoe clearly requires a more detailed
investigation, in concert with the lensing model for this system. Such
analysis is outside the scope of the current work.
We have not corrected our
H$\beta$ fluxes for possible stellar absorption at this wavelength. Stellar
absorption would lead to an underestimate of the flux, and push the points
upward on the BPT diagram. However, as noted in \citet{shapley2005} and
\citet{erb2006a}, the stellar H$\beta$ absorption line should have an equivalent
width of only $W_{abs} \leq 5$ \AA \citep{charlot2002}. Given that the Clone has
$W_{H\beta} \geq 20-30$ \AA, and the Cosmic Horseshoe does not have significant continuum
detected in the spectral region near H$\beta$,
stellar absorption is not significant. Furthermore, we defer application
of this correction until the underlying stellar populations
for the Cosmic Horseshoe and the Clone have been better constrained
by deeper spectroscopy and multi-wavelength photometry.
While the Cosmic Horseshoe does not display the same clear offset as the Clone on the
BPT diagram, when considered as a {\it population}, the sample of $z\sim 2$
objects with line ratio measurements \citep[i.e., those presented in this paper
as well as those in][]{erb2006a} appears to be offset on average from local
galaxies. We have demonstrated that the Cosmic Horseshoe and Clone are typical
of $z\sim 2$ star-forming galaxies in many respects, and also
display several striking differences with respect to local galaxies
in terms of their physical properties. As such, we will use the
special insights afforded by gravitational lensing into the physical conditions
in the Cosmic Horseshoe and Clone to explain the properties of the $z\sim 2$
population as a whole. We recognize the simplification associated
with drawing conclusions based on only two objects, and the
fact that a significantly larger sample of galaxies with the complete
set of rest-frame optical emission lines is required to characterize
the full $z\sim 2$ population. The discussion of the Cosmic
Horseshoe and the Clone that follows represents the initial
step in the direction of such a statistical study.
An important point of contrast between local star-forming galaxies in the SDSS
and our high-redshift targets is the rate of star-formation activity.
\citet{shapley2005} report an aperture-corrected mean SFR for a local sample of
SDSS galaxies of 2.0 $M_{\sun}$ yr$^{-1}$. These authors also noted that offset
objects in the BPT diagram (both at high redshift and in the local SDSS sample)
seem to exhibit elevated SFRs relative to the local average, and this is further
demonstrated in a small sample of extreme SDSS galaxies in \citet{liu2008}. We
estimate lensing-corrected SFRs in the Clone and Cosmic Horseshoe that are even
higher than those presented in \citet{shapley2005}, more than an order of
magnitude larger than the average SDSS SFR.
The higher rate of star formation might lead to a larger reservoir of ionizing
photons, which is reflected in the ionization parameter. \citet{brinchmann2008}
demonstrated that the diagnostic line ratios in the BPT diagram are strongly
dependent on the ionization parameter. However, while previous discussion of the
offset has only provided possible explanations that could lead to this effect,
our measurement of a host of lines, such as
[\ion{O}{2}]$\lambda$$\lambda$3726,3729 in the magnified objects, allows us to
test this discussion more quantitatively. To date, only a small sample of
ionization parameter measurements have been obtained for objects at $z\geq2$
\citep{pettini2001, lemoine2003,maiolino2008}. In both the Cosmic Horseshoe and the Clone, our
measurements of $O_{32}$ indicate higher values of the ionization parameter than
seen in even the most extreme low-$z$ UV selected galaxies from
\citet{contini2002}, or a sample of star-forming galaxies at intermediate redshift
($0.47 < z < 0.92$) from \citet{lilly2003}. Figure \ref{fig:O32R23}
illustrates this point very clearly, in that, at fixed metallicity (i.e., $R_{23}$)
high-redshift galaxies are offset on average towards significantly higher
$O_{32}$ (i.e., ionization parameter). If $O_{32}$ values were considered in isolation, the differences
in $O_{32}$ between low- and high-redshift samples could simply be attributed
to metallicity differences. However, the addition of $R_{23}$ measurements indicates
that there is a real effect towards systematically higher ionization parameters
at high redshift.
Another idea discussed in \citet{brinchmann2008} and \citet{liu2008} is that the
higher ionization parameter that might lead to this offset on the BPT diagram is
due to a higher density in the \ion{H}{2} regions. The
[\ion{S}{2}]$\lambda$$\lambda$6717,6732 line ratio allowed us to measure the
density of the star forming regions in the Cosmic Horseshoe and the Clone, and
the low resulting ratios indicated high densities of about $\sim10^3$ cm$^{-3}$,
which is an order of magnitude higher than the values encountered in local
starbursts \citep{kewley2001b}. The ESI spectra analyzed by \citet{quider2009}
and Quider et al. (in preparation) also show another independent
measure of the density in the form of the
[\ion{C}{3}]$\lambda$1907/\ion{C}{3}]$\lambda$1909 line ratio. With this line
ratio, values of about 1.5 and above are in the low density regime, while values
from 0.2 and lower are in the high density regime. For the Cosmic Horseshoe,
deblending the two lines in a smoothed spectrum yields a line ratio of $1.1\pm 0.2$, and
for the Clone, the line ratio is $1.2\pm 0.2$. The values for the Cosmic Horseshoe and
the Clone indicate densities ranging from $5000-22000 \mbox{ cm}^{-3}$
and $3000-17000\mbox{ cm}^{-3}$, respectively. These are at least as high as those derived from the
[\ion{S}{2}] doublet \citep{agn2}. \citet{brinchmann2008} calculate that
densities of this order would account for an increase in the ionization
parameter, which might lead to the observed offset on the BPT diagram.
\section{Conclusions}\label{sec:conclusions}
We present NIRSPEC rest-frame optical spectra of three strongly-lensed $z\sim2$
galaxies. These include SDSS J0901+1814 ($z = 2.26$), which, due to its observed
line ratios, is possibly contaminated by an AGN; and two star-forming galaxies,
the Cosmic Horseshoe ($z = 2.38$) and the Clone ($z = 2.00$). The general
physical properties of the Cosmic Horseshoe and the Clone are representative of
the properties found for star-forming galaxies at the same redshift.
Specifically, we have measured the SFR from the H$\alpha$ luminosity, corrected
for reddening and magnification from the lensing, and found it to be high yet
typical of other measurements of high-redshift star formation (SFR = $\sim110$
$M_{\sun}$ yr$^{-1}$ for the Cosmic Horseshoe and $\sim70$ $M_{\sun}$ yr$^{-1}$ for the
Clone). We have also used the $R_{23}$, $N2$, and $O3N2$ methods to
calculate the metallicity of the \ion{H}{2} regions, and found that, while there
are differences among the metallicities calculated from these indicators, we can
still constrain the metalliticies to range from slightly sub-solar to solar. The
dynamical masses calculated from H$\alpha$ velocity dispersions and the
half-light radii of the reconstructed sources are on the order of $10^{10}$
$M_\sun$, which is typical of the dynamical masses of UV-selected star-forming
galaxies at $z\sim2$. Finally, ESI rest-frame UV spectra provide evidence for
the existence of outflowing gas with a velocity on the order of $\sim150-200$ km
s$^{-1}$ for the Cosmic Horseshoe and the Clone,
which are also similar to outflows seen in other high-redshift galaxies
\citep{pettini2001,steidel2004,adelberger2003}. In the future, it will be
valuable to model the stellar populations of our lensed targets using
multi-wavelength broadband photometry, and therefore obtain constraints on their
stellar masses and ages.
The combination of strong lensing and NIRSPEC observations has allowed us to
probe physical conditions that to date have been largely unexplored at $z\geq2$.
The measurements of the [\ion{S}{2}]$\lambda$$\lambda$6717,6732 line ratio
indicated high densities ($\sim10^3$ cm$^{-3}$) in these regions, in agreement
with the densities derived from
[\ion{C}{3}]$\lambda$1907/\ion{C}{3}]$\lambda$1909 line ratios for both objects.
Large ionization parameters were measured by using $O_{32}$, which further
indicates the high ionization state of the gas in high-redshift objects. The
high values for the ionization parameter, density, and SFR in both
the Cosmic Horseshoe and the Clone help in understanding why, on average, the
population of high-redshift objects are offset on the BPT diagram, a standard
diagnostic for star-forming regions \citep{shapley2005,erb2006a,liu2008}. The
exceptional data quality enables the measurement of quantities that previously
were only speculation for $z\sim2$ galaxies. These results offer more concrete
evidence of the different conditions under which star formation occurs in
galaxies at high redshift, yet a statistical sample is still required
to place the results on firmer ground. Future observations with the Multi-Object
Spectrometer for Infra-Red Exploration (MOSFIRE) instrument planned for the Keck
I telescope will allow for the assembly of a much larger sample of rest-frame
optical emission lines for this type of analysis.
\acknowledgments
We would like to thank Xin Liu, Anna Quider, Simon Dye, Thomas Diehl, and Huan Lin
for their assistance. We acknowledge Lindsay King for kindly providing the VLT/FORS2
$R$-band image of the Cosmic Horseshoe. A.E.S. acknowledges support from the
David and Lucile Packard Foundation and the Alfred P. Sloan Foundation. We wish
to extend special thanks to those of Hawaiian ancestry on whose sacred mountain
we are privileged to be guests. Without their generous hospitality, most of the
observations presented herein would not have been possible.
\bibliographystyle{apj}
|
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"redpajama_set_name": "RedPajamaArXiv"
}
| 8,730
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Zbrodnia w Bilczach – zbrodnia dokonana na polskich mieszkańcach wsi Bilcze w nocy z 13 na 14 kwietnia 1943.
Bilcze były wsią zamieszkaną w większości przez Ukraińców, z wyjątkiem 12-15 rodzin polskich. Część z nich, na wieść o napadach na Polaków, uciekła na początku kwietnia 1943 do Łucka. Według świadectw zebranych przez Władysława i Ewę Siemaszków w nocy z 13 na 14 kwietnia grupa ukraińskich nacjonalistów (ich ewentualna przynależność do formacji partyzanckich lub do OUN nie jest znana) napadła na pozostałe polskie zagrody, które zostały spalone. Dziesięciu Polaków, w tym jedno dziecko, zostało zabitych bagnetami i wrzuconych do studni. Następnego dnia ocalali z napadu Polacy wydobyli ze studni zwłoki zabitych oraz jedną żywą, ciężko ranną kobietę, po czym opuścili Bilcze.
Bibliografia
E. Siemaszko, W. Siemaszko, Ludobójstwo dokonane przez nacjonalistów ukraińskich na ludności polskiej Wołynia 1939-1945, Warszawa 2000, Wydawnictwo "von Borowiecky"; , t.I, s.53
Zbrodnie nacjonalistów ukraińskich w dawnym województwie wołyńskim
1943 w okupowanej Polsce
|
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"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,520
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We are one of Oxfordshire's leading specialist SEO agencies with a passion for helping businesses grow online using our unique blend of technical SEO analysis, expert content marketing, promotion & digital PR. We work with dozens of businesses throughout Oxfordshire, the Thames Valley into London and have clients in America and Australia. Variety Guaranteed!
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,969
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\section{Introduction}
Deep convolutional neural networks (CNNs) have been shown to be highly effective in image classification with large datasets, such as CIFAR-10/100~\citep{krizhevsky2009learning}, SVHN~\citep{netzer2011reading}, and ImageNet~\citep{deng2009imagenet}.
Improvements in computer hardware and network architectures have made it possible to train deeper and more complex networks.
Network grouping is an efficient technique to improve the accuracy in model learning.
The Inception architecture~\citep{szegedy2015going} was proposed to aggregate abundant features via multi-scale subnetworks.
In addition, dueling architecture~\citep{wang2015dueling} in deep reinforcement learning can explicitly exploit subnetworks to represent state value and action advantages.
Recently, the Residual Networks (ResNets)~\citep{he2015deep,he2016identity} can be regarded that a residual unit includes an identity shortcut and a residual subnetwork.
This approach can alleviate the vanishing gradient problem by bypassing the gradients without attenuation and thus can increase the network depth up to more than 100 layers.
As suggested in~\citep{abdi2016multi,huang2016deep,veit2016residual}, ResNets gains its superior performance by implicitly averaging many subnetworks.
The redundancy problem of ResNets has been raised in~\citep{huang2016deep, zagoruyko2016wide}.
Some studies primarily aimed at the improvement of the propagation in ResNet, thereby reducing the redundancy problem.
Stochastic Depth~\citep{huang2016deep} tackled this problem by randomly disabling residual units during training.
Wide Residual Networks~\citep{zagoruyko2016wide} addressed this problem by decreasing the depth and increasing the width of residual units for faster training.
Both of these network architectures are attempts to shorten the network and thereby improve information back-propagation during training.
Without shortening network, a recent work~\citep{he2016identity} analyzed various usages of rectified linear unit (ReLU) and batch normalization (BN) in ResNets for direct propagation, and proposed methods for identity mapping in residual units to improve training in very deep ResNets.
Some studies encouraged the direct feature reuse by replacing the element-wise addition in ResNets with concatenation.
FractalNet~\citep{larsson2016fractalnet} repeatedly combines many subnetworks in a fractal expansion rule to obtain large nominal network depth.
DenseNet~\citep{huang2016densely} is similar to FractalNet with the difference that DenseNet connects each layer to all of its preceding layers.
These approaches exhibit a behavior of mimicking deep supervision, which is important to the learning of discriminative features.
Some studies aimed at the improvement of the residual units by representing the residual function with many tiny subnetworks.
Inception-ResNet~\citep{szegedy2016inception} presented Inception-type residual units.
PolyNet~\citep{zhang2016polynet} replaces the original residual units with polynomial combination of Inception units for enhancing the structural diversity.
Multi-residual networks~\citep{abdi2016multi} and ResNeXt~\citep{xie2016aggregated} both aggregate residual transformations from many tiny subnetworks.
\begin{figure}[tbp]
\begin{center}
\includegraphics*[width=\linewidth]{Fig_copanet.pdf}
\end{center}
\caption{The concept of \textit{pathway encoding} in the proposed architecture. The category information is encoded on the route (red arrows) through which features propagate.}
\label{copanet}
\end{figure}
The idea behind the use of subnetworks is to simplify network for efficient training.
By explicitly factoring the network into a series of operations, features can be learned independently.
In this work, we embrace this observation and propose a novel deep architecture referred to as Competitive Pathway Network (CoPaNet).
Because the underlying mapping function can be decomposed into the maximum of multiple simpler functions and the residual learning~\citep{he2015deep} is a good strategy for approximating the mapping functions, the proposed competitive pathway (CoPa) unit was designed to comprise multiple parallel residual-type subnetworks followed by a max operation for feature competition.
Furthermore, identity cross-block shortcuts can be added to the CoPaNet to enhance feature reuse.
These strategies offer several advantages:
1. Feature redundancy can be reduced by dropping unimportant features through competition.
2. The competitive mechanism facilitates the network to modularize itself into multiple parameter-sharing subnetworks for parameter efficiency~\citep{srivastava2013compete}.
3. CoPaNet uses residual-type subnetworks and therefore inherits the advantage of ResNet for training very deep network.
4. With competitive mechanism and residual-type subnetworks, the CoPaNet explicitly exhibits the property of pathway encoding, as shown in Figure \ref{copanet}. Because the residual-type subnetwork can preserve feature identity such that the winning path can be traced back within the entire network.
That is, the \textit{routing pattern of propagating features} encodes category information.
5. The cross-block shortcuts encourage coarse feature reuse and implicit deep supervision.
CoPaNet was evaluated using several benchmark datasets such as CIFAR-10, CIFAR-100, SVHN, and ImageNet.
Our resulting models performed equally to or better than the state-of-the-art methods on the above-mentioned benchmark datasets.
\section{Related Work}
\subsection{Residual Networks (ResNets)}
ResNets~\citep{he2015deep} are motivated by the counterintuitive observation that the performance of neural networks actually gets worse when developed to a very great depth.
This problem can be attributed to the fact that the gradient vanishes when information back-propagates through many layers.
~\cite{he2015deep} proposed skipping some of the layers in convolutional networks through the implementation of shortcut connections, in the formulation of an architecture referred to as residual units. The original residual unit performs the following computation:
\begin{align*} x_{l+1}=ReLU(id(x_l )+f_l (x_l )) \, , \end{align*}
where $x_l$ denotes the input feature of the $l$-th residual unit, $id(x_l)$ performs identity mapping, and $f_l$ represents layers of the convolutional transformation of the $l$-th residual unit.
~\cite{he2016identity} further suggested to replace ReLU with another identity mapping, allowing the information to be propagated directly.
Thus, they proposed a pre-activation residual unit with the following form:
\begin{align*}x_{l+1}=id(x_l )+f_l (x_l ) \, . \end{align*}
Furthermore, the positions of BN and ReLU are changed to allow the gradients to be back-propagated without any transformation.
Their experimental results demonstrated the high efficiency of pre-activation residual units.
\subsection{Competitive Nonlinear functions}
Maxout Networks~\citep{goodfellow2013maxout} were recently introduced to facilitate optimization and model averaging via Dropout. The authors of this work proposed a competitive nonlinearity referred to as maxout, which was constructed by obtaining the maximum across several maxout hidden pieces. Maxout Networks can be regarded as universal approximators and can provide better gradient back-propagation than other activation functions.
Without down-sampling the features, Local Winner-Take-All (LWTA)~\citep{srivastava2013compete} was inspired by the characteristics of biological neural circuits.
Each LWTA block contains several hidden neurons and produces an output vector determined by local competition between hidden neurons activations.
Only the winning neuron retains its activation, whereas other hidden neurons are forced to shut off their activation.
In empirical experiments, both network architectures have been shown to have advantages over ReLU.
\begin{figure*}
\begin{center}
\includegraphics*[width=6in]{Fig_resunit.pdf}
\end{center}
\caption{The proposed CoPa unit and network architecture.}
\label{resunit}
\end{figure*}
\section{Competitive Pathway Network}
\subsection{Competitive pathway unit}
CoPaNet is an attempt to separate model into subnetworks through competition.
In the following, we refer to residual-type subnetworks as pathways.
In a CoPa unit, multiple pathways are compiled in parallel and features are selected by using a $\max$ operation.
A CoPa unit includes output $x_{l+1}$ with $K$ pathways, which can be formulated as follows:
\begin{align*}
x_{l+1}=\mathop{\max }_{k\in [1,\ldots , K]} z_l^k,
\end{align*}
where $z_l^k=id(x_l)+h_l^k (x_l)$, $x_l$ is the input feature, and $h_l^k (x_l)$ represents layers of transformations on the $k$-th pathway at the $l$-th CoPa unit. Figure \ref{resunit}a illustrates the CoPa unit (featuring two pathways) used in this paper.
Competitive pathways appear complex; however, the proposed CoPaNet is easy to train.
Notice that residual learning~\citep{he2015deep} is based on the hypothesis that underlying mapping function $H(x)$ is very hard to fit.
Nevertheless, the mapping function can be decomposed into two simpler functions: $H(x)=x+F(x)$.
~\cite{he2015deep} claimed that the residual function $F(x)$ is easier to approximate.
Motivated by the idea of streamlining the process of approximating the underlying mapping function, we first decompose the underlying mapping function into the maximum of two simpler functions, that is, $H(x)=\max\{f(x),g(x)\}$.
We then use residual learning~\citep{he2015deep} and let $f(x)=x+h^1 (x)$ and $g(x)=x+h^2 (x)$.\
The desired mapping function becomes $H(x)=\max\{x+h^1 (x),x+h^2 (x)\}$.
This illustrates the need for two parallel networks (one each for $h^1(x)$ and $h^2(x)$), each of which comprises several stacked layers in order to approximate discrete residual functions.
Because $f(x)$ and $g(x)$ are simpler, it would be easier to approximate $h^1 (x)$ and $h^2 (x)$ than the original residual learning~\citep{he2015deep}.
Our CoPa unit is different from maxout unit~\citep{goodfellow2013maxout}.
The original maxout unit is constructed to obtain the maximum across several elementary neurons.
Our method replaces the elementary neurons with generic functions, which are modeled by ResNets.
Further, the property of pathway encoding reveals in this architecture.
We consider a 2-pathway (denote as $h^1_l, h^2_l$) CoPaNet with three stacked CoPa units, as show in Figure~\ref{copanet}.
We denote that the output of the first CoPa unit is $y_1 = x + h^1_1(x)$ (if $h^1_1$ wins) where $x$ is the input feature.
The output of the second CoPa unit can be written as $y_2 = y_1 + h^1_2(y_1)$ (if $h^1_2$ wins).
The output of the third CoPa unit can be written as $y_3 = y_2 + h^2_3(y_2)$ (if $h^2_3$ wins).
The final output actually can be expressed as $y_3 = x + h^1_1(x) + h^1_2(y_1) + h^2_3(y_2)$.
This indicates that the final output is contributed by three winning subnetworks $h^1_1, h^1_2, h^2_3$ with reference to $x$.
Thus, the routing pattern can be revealed by propagating $x$ through the entire network.
Within a biological context, competitive mechanisms play an important role in attention~\citep{lee1999attention}.
Researchers formulated a biological computational model in which attention activates a winner-take-all competition among neurons tuned to different visual patterns.
In this model, attention alters the thresholds used to detect orientations and spatial frequencies.
This suggested that winner-take-all competition can be used to explain many of the basic perceptual consequences of attention~\citep{lee1999attention}.
\subsection{CoPaNet Architecture}
CoPaNets can be simply constructed by stacking CoPa units.
Let the opponent factor $k$ denote the number of pathway in a CoPa unit and the widening factor $m$ multiplies the number of features in convolutional layers.
That is, the baseline CoPa unit corresponds to $k = 2$, $m=1$; whereas ResNet corresponds to $k = 1$, $m=1$.
Figure \ref{resunit}b shows the architecture for CIFAR and SVHN as well as Table~\ref{params} detailed the deployment.
The residual shortcut in the proposed network performs identity mapping and the projection shortcut is used only to match dimensions (using 1$\times$1 convolutions) as ResNet~\citep{he2015deep,he2016identity}.
For each pathway, we adopted a ``bottleneck'' residual-type unit comprising three convolutional layers (1$\times$1, 3$\times$3, 1$\times$1).
Alternatively, we could select a ``basic'' residual-type unit comprising two convolutional layers (3$\times$3, 3$\times$3).
In practice, a ``bottleneck'' residual-type unit is deeper than a ``basic'' one, providing higher dimensional features.
In the proposed CoPaNet, we placed BN and ReLU after all but the last convolutional layer in every pathway.
\subsection{Cross-block Shortcut}
The cross-block shortcuts were motivated by DenseNet~\citep{huang2016densely} which reused features from all previous layers with matching feature map sizes.
In contrast to DenseNet~\citep{huang2016densely}, we propose a novel feature reuse strategy: to reuse the features from previous CoPa block (stacked by many CoPa units).
This is accomplished by adding identity shortcuts after pooling layers and concatenate with the output of the next block.
We refer to our model with the cross-block shortcuts as CoPaNet-R, as shown in Figure \ref{resunit}c.
\begin{table*}
\begin{center}
\caption{Network architectures for CIFAR/SVHN (left) and ImageNet (right). Parameters of competitive pathway units are presented in braces (see also Figures~\ref{resunit}b and c). Construction parameters for internal pathways are shown in brackets. The number of pathway is determined by the factor $k$ and the network width is determined by the factor $m$. The numbers in CoPaNet-26/50/101/164 denote the depths of neural network. For the sake of clarity, the final classification layer has been omitted. }
\includegraphics*[width=6in]{Table_params.pdf}
\label{params}
\end{center}
\end{table*}
\begin{table*}
\centering
\caption{Comparison of test error on CIFAR and SVHN. The value of $k$ denotes the number of hidden pieces or pathways used in a given competition. The symbol ``+'' indicates data augmentation (translation and horizontal flipping ).}
\begin{center}
\resizebox{1\linewidth}{!}{
\begin{tabular}{lcccccc}
\hline
Method& Dropout&Depth&Params&C10+&C100+& SVHN \\ \hline
Maxout Network ($k$=2)~\citep{goodfellow2013maxout}&${\surd}$& - & - & 9.38 & 38.57 & 2.47 \\
Network In Network~\citep{DBLP:journals/corr/LinCY13}&${\surd}$& - &0.98 M& 8.81 & 35.68 & 2.35\\
Maxout Network In Network ($k$=5)~\cite{chang2015batch}&${\surd}$& - &1.6 M& 6.75 & 28.86 & 1.81\\
Highway Network~\citep{srivastava2015training}& & - & - & 7.60 & 32.34 & -\\ \hline
ResNet~\citep{he2015deep}& & 110 & 1.7 M & 6.43 & - & -\\\hline
Stochastic Depth~\cite{huang2016deep} & & 110 & 1.7 M & 5.23 & 24.58 & 1.75\\
& & 1202 & 19.4 M & 4.91 & - & -\\ \hline
pre-activation ResNet~\citep{he2016identity} & & 164 & 1.7 M & 5.46 & 24.33 & -\\
& & 1001 & 10.2 M & 4.62 & 22.71 & -\\ \hline
Wide ResNet (width=8)~\citep{zagoruyko2016wide} & & 16 & 11.0 M & 4.27 & 20.43 & -\\
{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }{ }(width=10)&${\surd}$& 28 & 36.5 M & 3.89 & 18.85 & -\\\hline
DenseNet{ }{ }{ }{ }{ } (growth rate=24)~\citep{huang2016densely}& & 100 & 27.2 M & 3.74 & 19.25 & 1.59 \\
DenseNet-BC (growth rate=40)& & 190 & 25.6 M & 3.46 & \textbf{17.18} & - \\ \hline
CoPaNet ($k$=2, width=1) &${\surd}$& 164 & 1.75 M & 4.50 & 22.86 & 1.86\\
CoPaNet ($k$=2, width=2)& ${\surd}$ & 164 & 6.98 M & 4.10 & 20.48 & 1.83\\
CoPaNet ($k$=2, width=4) & ${\surd}$ & 164 & 27.9 M & 3.74 & 18.67 & 1.73\\ \hline
CoPaNet-R ($k$=2, width=2) &${\surd}$& 164 & 7.00 M & 3.55 & 20.29 & 1.72\\
CoPaNet-R ($k$=2, width=3)& ${\surd}$ & 164 & 15.7 M & \textbf{3.38} & 18.90 & \textbf{1.58}\\ \hline
\end{tabular}
}
\end{center}
\label{comparison}
\end{table*}
\section{Experiments}
We have tested the proposed CoPaNets and CoPaNets-R on several datasets, and compared the results with those of the state-of-the-art network architectures, especially ResNets.
\subsection{Training}
We constructed a CoPaNet-164, with a set number of pathways ($k=2$), and network width ($m=1,2,4$), detailed in Table~\ref{params}.
Furthermore, we constructed a CoPaNet-R-164, with a set number of pathways ($k=2$), and network width ($m=2,3$).
The networks were trained from scratch by using Stochastic Gradient Descent with 300 and 20 epochs for CIFAR and SVHN datasets, respectively.
The learning rate for CIFAR began at 0.1, divided by 10 at 0.6 and 0.8 fractions of the total number of training epochs.
The learning rate for SVHN began at 0.1, divided by 10 at 0.5 and 0.75 fractions of the total number of training epochs.
A batch size of 128 was used for all tests, except for $m=4$ when we used a batch size of 64.
On ImageNet, we trained from scratch for 100 epochs.
As shown in Table \ref{params}, we constructed several CoPaNets with 2 pathways for ImageNet.
The learning rate began at 0.1 and was divided by 10 after every 30 epochs.
The model was implemented using Torch7 from the Github repository \textit{fb.resnet.torch} (\url{https://github.com/facebook/fb.resnet.torch}).
Other settings were set exactly the same as those used for ResNet.
We adopted a weight decay of 0.0001 and momentum of 0.9 as in~\citep{he2015deep}.
Weights were initialized in accordance with the methods outlined by ~\cite{he2015delving}.
We also applied Dropout~\citep{srivastava2014dropout} after the average poolings except the last pooling, and it was deterministically multiplied by (1 - Dropout-rate) at test time.
The Dropout rate was set to 0.2 for CIFAR and SVHN as well as 0.1 for ImageNet.
The test error was evaluated using the model obtained from the final epoch at the end of training.
\begin{figure*}
\begin{center}
\includegraphics*[width=6in]{Fig_trainingcurve.pdf}
\end{center}
\caption{ (a) Training loss (dashed line) and test error (solid line) curves of the pre-activation ResNet-1001 (10.2M), CoPaNet-164 (1.75M), and CoPaNet-R-164 (1.75M). (b) Comparison of the parameter efficiency between pre-activation ResNets, CoPaNet, and CoPaNet-R.}
\label{curve}
\end{figure*}
\subsection{CIFAR-10}
The CIFAR-10 dataset consists of natural color images, 32$\times$32 pixels in size, from 10 classes, and with 50,000 training and 10,000 test images. Color normalization was performed as data preprocessing. To enable a comparison with previous works, the dataset was augmented by translation as well as random flipping on the fly throughout training.
As shown in Table \ref{comparison}, we obtained test error of 4.50\%, 4.10\%, and 3.74\% when using network width of $m=1$, $2$, and $4$, respectively.
We then compared CoPaNet-164 (1.75 M, $m=1$) to pre-activation ResNet-1001 (10.2 M), for which ~\cite{he2016identity} reported test error of 4.62\% (we obtained 4.87\% in our training procedure).
Figure \ref{curve}a presents a comparison of training and testing curves.
Furthermore, Our best result on CIFAR-10 was obtained by CoPaNet-R.
We obtained 3.38\% test error with only 15.7 M parameters.
\subsection{CIFAR-100}
The CIFAR-100 dataset is the same size and format as CIFAR-10; however, it contains 100 classes.
Thus, the number of images in each class is only one tenth that of CIFAR-10. Color normalization was performed as data preprocessing.
We also performed data augmentation (translation and horizontal flipping) on the CIFAR-100 dataset.
As shown in Table \ref{comparison}, we obtained the test error of 22.86\%, 20.48\%, and 18.67\% for network width of $m=1$, $2$, and $4$ with Dropout, respectively.
CoPaNet-164 (1.75 M, $m=1$) was compared to pre-activation ResNet-164 (1.7 M) for which ~\cite{he2016identity} reported test error of 24.33\%.
This puts the proposed network on par with pre-activation ResNet-1001 (10.2 M) which achieved test error of 22.71\%.
However, CoPaNet-R showed few benefits on CIFAR-100, and it obtained same level of accuracy.
\subsection{SVHN}
The SVHN dataset consists of color images of house numbers (32$\times$32 pixels) collected from Google Street View.
This includes 73,257 digits in the training set, 26,032 digits in the test set, and 531,131 in an extra set.
We used the entire training set and extra set for training.
We did not perform any data augmentation or preprocessing except for dividing the image intensity by 255.
As shown in Table \ref{comparison}, the CoPaNet-164 (1.75 M, width $m=1$) with test error of 1.86\%.
CoPaNet-R-164 (width $m=3$) achieved the state-of-the-art results (1.58\%) with only 15.7 M parameters.
\subsection{ImageNet}
The ImageNet 2012 dataset consists of 1000 classes of images with 1.28 millions for training, 50,000 for validation, and 100,000 for testing.
As shown in Table \ref{params}, we constructed two-pathway CoPaNet with various depths for ImageNet.
However, we reduce the number of feature maps to approximately 70\% in order to retain a similar number of parameters.
For a fair comparison, all results were achieved when the crop size was 224$\times$224.
Our results of single crop top-1 validation error showed better performance than ResNet, as shown in Figure~\ref{imagenet}.
These results reveal that CoPaNets perform on par with the state-of-the-art ResNets, while requiring fewer parameters.
CoPaNets performed worse than DenseNet with similar amounts of parameters.
The major reason could be that DenseNets were much deeper than CoPaNets.
\begin{figure}
\centering
\includegraphics*[width=3.6in]{Fig_ImageNet.pdf}
\caption{The comparison of top-1 validation error (single model and single crop with size 224$\times$224) across various number of parameters among ResNet, DenseNet, and CoPaNet. }
\label{imagenet}
\end{figure}
\section{Discussion}
\subsection{Parameter Efficiency}
The competitive mechanism modularizes the network into multiple parameter-sharing subnetworks and thus can improve parameter efficiency~\citep{srivastava2013compete}.
We trained multiple small networks with various depths on CIFAR-10+.
As shown in Figure~\ref{curve}b, both CoPaNet and its variant outperformed pre-activation ResNet.
The CoPaNet-R achieved better performance than CoPaNet.
When achieving the same level of accuracy, furthermore, CoPaNet requires around a half of the parameters of pre-activation ResNet.
\subsection{Number of Pathways}
Figure~\ref{pathway} demonstrates that CoPaNet has the capacity to exploit many pathways.
We trained several CoPaNets-56 (width $m=1$) for use on CIFAR-10+ using various numbers of pathways with the Dropout rate set to 0.2.
As shown in Figure \ref{pathway}, CoPaNet gains its benefit by increasing the number of pathways to handle complex dataset.
More pathways tend to lower test errors at the expense of more parameters.
Nonetheless, we adopted two pathways in our experiments to restrict the number of parameters.
\begin{figure}
\centering
\includegraphics*[width=3.6in]{Fig_numpath.pdf}
\caption{The influence of the number of pathways on performance in experiments based on CIFAR-10+. More pathways tends to lower test errors at the expense of more parameters.}
\label{pathway}
\end{figure}
\subsection{Pathway Encoding}
One paper~\citep{srivastava2014understanding} argued that ReLU network can also encode on subnetwork activation pattern, such as maxout and LWTA networks.
~\cite{srivastava2014understanding} discussed about the activation pattern of many filters in the same layer.
In contrast to~\cite{srivastava2014understanding}, we demonstrated the routing pattern that \textit{one feature map propagate through many stacked pathways (subnetworks)}.
We suppose that the routing patterns are similar within the same semantics and are different between distinct semantics, which is termed as pathway encoding.
As shown in Figure~\ref{pathwayencode}, we calculated the preference of routing patterns in a trained 2-pathway CoPaNet-164 (width $m=1$).
The preference of pathway was statistically estimated from the CIFAR-10 test set and can reveal the characteristics of the category.
We illustrates the routing patterns in the last block (comprising 18 CoPa units) which contained high-level features.
Each sub-figure showns the routing pattern of one feature map (4 representative feature maps were manually selected from the total of 180), and the color denoted the preference of pathways.
As shown in Figure~\ref{pathwayencode}a, a selected routing pattern can be regarded as encoding the non-living or living groups and the routing patterns are similar in the same group.
Figure~\ref{pathwayencode}b illustrates that the routing pattern may be encoding the flying concept such that the routing patterns of airplanes are similar to those of birds.
Notice that although airplanes belong to non-living group, there exists a special pattern resembling those of animals, including the bird, as shown in Figure~\ref{pathwayencode}c.
Furthermore, Figure~\ref{pathwayencode}d illustrates the diversity of routing patterns for different categories.
The similarity and diversity support our hypothesis that CoPaNet is able to use pathway encoding to well represent the object images of different groups.
\begin{figure*}
\begin{center}
\includegraphics*[width=6in]{Fig_pathwayencoding.pdf}
\end{center}
\caption{Routing patterns showing the preference of pathway selection in a trained 2-pathway CoPaNet-164 for the CIFAR-10 test dataset. Red color denotes a preference for the left pathway, blue color for the right pathway, and white color for no preference. The vertical axis denotes the $l$-th CoPa units, where $l$ indicates the depth. The category information can be represented by the routing pattern, which is referred to as \textit{pathway encoding} in the proposed work. Each sub-figure denotes the routing pattern that one feature map propagates through its preferred route in the network.
Routing patterns between (a) non-living vs. living, (b) non-flying vs. flying, (c) non-animal vs. animal, and (d) different categories are illustrated. Notice that the airplane category shows the routing pattern of ``bird'' in the ``animal'' group.}
\label{pathwayencode}
\end{figure*}
\subsection{Coarse Feature Reuse}
The CoPaNet-R architecture adds identity cross-block shortcuts to encourage feature reuse.
This facilitates that the last classification layer can reuse coarse features from all previous blocks.
Thus those shortcuts provide additional supervision because classifiers are attached to every CoPa blocks.
We trained a CoPaNet-R-164 (width $m=2$) on CIFAR-10+ and it achieved 3.55\% test error, as shown in Table~\ref{comparison}.
Figure~\ref{reuse} shows the $L^1$-norm of weights of the last classification layer.
In this figure, we can observe that the last classification layer uses features from early blocks.
The concentration towards the final block suggests that high-level features dominate in classification.
However, CoPaNet-R did not outperform CoPaNet on CIFAR-100 and ImageNet.
This may be due to the relatively few training samples for each class (500 samples per class in CIFAR-100 as well as around 1000 samples per class in ImageNet).
We conducted an experiment to demonstrate this effect.
We used a small CIFAR-10 dataset (1000 training samples per class) to train CoPaNet-164 and CoPaNet-R-164, both with width $m=2$, and achieved test errors of 12.58\% and 12.53\%, respectively.
There is no significant difference in this case.
With full training set (5000 training samples per class), CoPaNet-R has significant improvement compared to CoPaNet, as shown in Table~\ref{comparison}.
The coarse feature reuse may be effective only when the amount of training samples is large enough for each class.
\begin{figure}
\centering
\includegraphics*[width=4in]{Fig_Reuse.pdf}
\caption{The color-encoded $L^1$-norm of the weights of the last classification layer. Notice that the last classification layer concatenates outputs from all of the three CoPa blocks through cross-block shortcuts.}
\label{reuse}
\end{figure}
\section{Conclusions}
This paper proposes a novel convolutional neural network architecture, the CoPaNet.
It introduces a nice property that input features transmit through various routing patterns for different category information, called pathway encoding.
Empirical results demonstrate that the category information plays a role in selecting pathways.
We showed that CoPaNet inherits the advantages of ResNet which can scale up to hundreds of layers.
In our experiments, CoPaNet yielded improvements in accuracy as the number of parameters increased.
Moreover, CoPaNet requires fewer parameters to achieve the same level of accuracy as the state-of-the-art ResNet.
We further proposed a novel feature reuse strategy, CoPaNet-R: adding cross-block shortcuts in order to encourage the reuse of output from all previous blocks.
According to our experiments, CoPaNet-R can learn accurate models by exploiting the reuse of coarse features.
Our study showed that network partitioning, feature competition, and feature reuse can lead to performance improvements.
CoPaNet and its variant obtained the state-of-the-art or competitive results on several image recognition datasets.
Other studies showed that competitive networks have other beneficial properties such as mitigation of catastrophic forgetting~\citep{srivastava2013compete}.
In the future, we will try to adopt the trained CoPaNet to perform other tasks, such as object detection and segmentation.
\section*{Acknowledgement}
This work was supported in part by the Taiwan Ministry of Science and Technology (Grants MOST-106-2221-E-009-164-MY2 and MOST-105-2218-E-009-033).
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{"url":"https:\/\/www.aip.de\/de\/kalender\/events\/colloquia-de\/gunther-rudiger-manfred-kuker-aip?month:int=3&year:int=2020&orig_query=","text":"# G\u00fcnther R\u00fcdiger \/ Manfred K\u00fcker (AIP)\n\nThe Cross Helicity at the Solar Surface by Simulations and Observations\nWann Am 17.04.2012 von 15:00 bis 16:00 Special Seminar SH, H\u00f6rsaal vCal \/ iCal\n\nIn order to model the cross helicity at the solar surface, magnetoconvection under the presence of a vertical large-scale magnetic field is simulated with\u00a0 NIRVANA. The robust result of the calculations is that the cross helicity can well be represented by the observable correlation of the vertical flow and the vertical field. The characteristic helicity speed as the ratio of the eddy diffusivity and the density scale height results to 1.1 km\/s. This value coincides with the data of the Japanese HINODE satellite and the Swedish 1-m Solar Telescope. Both simulations and observations thus lead to 1012cm2\/s as the sofar unknown eddy diffusivity at the surface of the\u00a0 quiet Sun.","date":"2020-04-02 03:50:42","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8090256452560425, \"perplexity\": 3545.6995942355306}, \"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-2020-16\/segments\/1585370506580.20\/warc\/CC-MAIN-20200402014600-20200402044600-00196.warc.gz\"}"}
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{"url":"https:\/\/en.wikipedia.org\/wiki\/Horner's_method","text":"# Horner's method\n\nJump to navigation Jump to search\n\nIn mathematics, Horner's method (also known as Horner scheme in the UK or Horner's rule in the U.S.[1][2]) is either of two things:\n\nThe latter is also known as Ruffini\u2013Horner's method.[4]\n\nThese methods are named after the British mathematician William George Horner, although they were known before him by Paolo Ruffini[5] , six hundred years earlier, by the Chinese mathematician Qin Jiushao [6]and seven hundred years earlier, by the Persian mathematician Sharaf al-D\u012bn al-\u1e6c\u016bs\u012b.[7]\n\n## Description of the algorithm\n\nGiven the polynomial\n\n${\\displaystyle p(x)=\\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\\cdots +a_{n}x^{n},}$\n\nwhere ${\\displaystyle a_{0},\\ldots ,a_{n}}$ are real numbers, we wish to evaluate the polynomial at a specific value of ${\\displaystyle x}$, say ${\\displaystyle x_{0}}$.\n\nTo accomplish this, we define a new sequence of constants as follows:\n\n{\\displaystyle {\\begin{aligned}b_{n}&:=a_{n}\\\\b_{n-1}&:=a_{n-1}+b_{n}x_{0}\\\\&~~~\\vdots \\\\b_{0}&:=a_{0}+b_{1}x_{0}.\\end{aligned}}}\n\nThen ${\\displaystyle b_{0}}$ is the value of ${\\displaystyle p(x_{0})}$.\n\nTo see why this works, note that the polynomial can be written in the form\n\n${\\displaystyle p(x)=a_{0}+x(a_{1}+x(a_{2}+\\cdots +x(a_{n-1}+a_{n}x))).}$\n\nThus, by iteratively substituting the ${\\displaystyle b_{i}}$ into the expression,\n\n{\\displaystyle {\\begin{aligned}p(x_{0})&=a_{0}+x_{0}(a_{1}+x_{0}(a_{2}+\\cdots +x_{0}(a_{n-1}+b_{n}x_{0})))\\\\&=a_{0}+x_{0}(a_{1}+x_{0}(a_{2}+\\cdots +x_{0}b_{n-1}))\\\\&~~\\vdots \\\\&=a_{0}+x_{0}b_{1}\\\\&=b_{0}.\\end{aligned}}}\n\n## Examples\n\nEvaluate ${\\displaystyle f(x)=2x^{3}-6x^{2}+2x-1}$ for ${\\displaystyle x=3.}$\n\nWe use synthetic division as follows:\n\n x0\u2502 x3 x2 x1 x0\n3 \u2502 2 \u22126 2 \u22121\n\u2502 6 0 6\n\u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n2 0 2 5\n\n\nThe entries in the third row are the sum of those in the first two. Each entry in the second row is the product of the x-value (3 in this example) with the third-row entry immediately to the left. The entries in the first row are the coefficients of the polynomial to be evaluated. Then the remainder of ${\\displaystyle f(x)}$ on division by ${\\displaystyle x-3}$ is 5.\n\nBut by the polynomial remainder theorem, we know that the remainder is ${\\displaystyle f(3)}$. Thus ${\\displaystyle f(3)=5}$\n\nIn this example, if ${\\displaystyle a_{3}=2,a_{2}=-6,a_{1}=2,a_{0}=-1}$ we can see that ${\\displaystyle b_{3}=2,b_{2}=0,b_{1}=2,b_{0}=5}$, the entries in the third row. So, synthetic division is based on Horner's method.\n\nAs a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of ${\\displaystyle f(x)}$ on division by ${\\displaystyle x-3}$. The remainder is 5. This makes Horner's method useful for polynomial long division.\n\nDivide ${\\displaystyle x^{3}-6x^{2}+11x-6}$ by ${\\displaystyle x-2}$:\n\n 2 \u2502 1 \u22126 11 \u22126\n\u2502 2 \u22128 6\n\u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n1 \u22124 3 0\n\n\nThe quotient is ${\\displaystyle x^{2}-4x+3}$.\n\nLet ${\\displaystyle f_{1}(x)=4x^{4}-6x^{3}+3x-5}$ and ${\\displaystyle f_{2}(x)=2x-1}$. Divide ${\\displaystyle f_{1}(x)}$ by ${\\displaystyle f_{2}\\,(x)}$ using Horner's method.\n\n 2 \u2502 4 \u22126 0 3 \u2502 \u22125\n\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n1 \u2502 2 \u22122 \u22121 \u2502 1\n\u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u253c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n2 \u22122 \u22121 1 \u2502 \u22124\n\n\nThe third row is the sum of the first two rows, divided by 2. Each entry in the second row is the product of 1 with the third-row entry to the left. The answer is\n\n${\\displaystyle {\\frac {f_{1}(x)}{f_{2}(x)}}=2x^{3}-2x^{2}-x+1-{\\frac {4}{2x-1}}.}$\n\n### Floating-point multiplication and division\n\nHorner's method is a fast, code-efficient method for multiplication and division of binary numbers on a microcontroller with no hardware multiplier. One of the binary numbers to be multiplied is represented as a trivial polynomial, where (using the above notation) ai = 1, and x = 2. Then, x (or x to some power) is repeatedly factored out. In this binary numeral system (base 2), x = 2, so powers of 2 are repeatedly factored out.\n\n#### Example\n\nFor example, to find the product of two numbers (0.15625) and m:\n\n{\\displaystyle {\\begin{aligned}(0.15625)m&=(0.00101_{b})m=(2^{-3}+2^{-5})m=(2^{-3})m+(2^{-5})m\\\\&=2^{-3}(m+(2^{-2})m)=2^{-3}(m+2^{-2}(m)).\\end{aligned}}}\n\n#### Method\n\nTo find the product of two binary numbers d and m:\n\n1. A register holding the intermediate result is initialized to d.\n2. Begin with the least significant (rightmost) non-zero bit in m.\n2b. Count (to the left) the number of bit positions to the next most significant non-zero bit. If there are no more-significant bits, then take the value of the current bit position.\n2c. Using that value, perform a left-shift operation by that number of bits on the register holding the intermediate result\n3. If all the non-zero bits were counted, then the intermediate result register now holds the final result. Otherwise, add d to the intermediate result, and continue in step 2 with the next most significant bit in m.\n\n#### Derivation\n\nIn general, for a binary number with bit values (${\\displaystyle d_{3}d_{2}d_{1}d_{0}}$) the product is\n\n${\\displaystyle (d_{3}2^{3}+d_{2}2^{2}+d_{1}2^{1}+d_{0}2^{0})m=d_{3}2^{3}m+d_{2}2^{2}m+d_{1}2^{1}m+d_{0}2^{0}m.}$\n\nAt this stage in the algorithm, it is required that terms with zero-valued coefficients are dropped, so that only binary coefficients equal to one are counted, thus the problem of multiplication or division by zero is not an issue, despite this implication in the factored equation:\n\n${\\displaystyle =d_{0}\\left(m+2{\\frac {d_{1}}{d_{0}}}\\left(m+2{\\frac {d_{2}}{d_{1}}}\\left(m+2{\\frac {d_{3}}{d_{2}}}(m)\\right)\\right)\\right).}$\n\nThe denominators all equal one (or the term is absent), so this reduces to\n\n${\\displaystyle =d_{0}(m+2{d_{1}}(m+2{d_{2}}(m+2{d_{3}}(m)))),}$\n\nor equivalently (as consistent with the \"method\" described above)\n\n${\\displaystyle =d_{3}(m+2^{-1}{d_{2}}(m+2^{-1}{d_{1}}(m+{d_{0}}(m)))).}$\n\nIn binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2\u22121) is a right arithmetic shift, a (0) results in no operation (since 20 = 1 is the multiplicative identity element), and a (21) results in a left arithmetic shift. The multiplication product can now be quickly calculated using only arithmetic shift operations, addition and subtraction.\n\nThe method is particularly fast on processors supporting a single-instruction shift-and-addition-accumulate. Compared to a C floating-point library, Horner's method sacrifices some accuracy, however it is nominally 13 times faster (16 times faster when the \"canonical signed digit\" (CSD) form is used) and uses only 20% of the code space.[8]\n\n### Polynomial root finding\n\nUsing Horner's method in combination with Newton's method, it is possible to approximate the real roots of a polynomial. The algorithm works as follows. Given a polynomial ${\\displaystyle p_{n}(x)}$ of degree ${\\displaystyle n}$ with zeros ${\\displaystyle z_{n} make some initial guess ${\\displaystyle x_{0}}$ such that ${\\displaystyle x_{0}>z_{1}}$. Now iterate the following two steps:\n\n1. Using Newton's method, find the largest zero ${\\displaystyle z_{1}}$ of ${\\displaystyle p_{n}(x)}$ using the guess ${\\displaystyle x_{0}}$.\n\n2. Using Horner's method, divide out ${\\displaystyle (x-z_{1})}$ to obtain ${\\displaystyle p_{n-1}}$. Return to step 1 but use the polynomial ${\\displaystyle p_{n-1}}$ and the initial guess ${\\displaystyle z_{1}}$.\n\nThese two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials.[9]\n\n#### Example\n\nPolynomial root finding using Horner's method\n\nConsider the polynomial\n\n${\\displaystyle p_{6}(x)=(x-3)(x+3)(x+5)(x+8)(x-2)(x-7)}$\n\nwhich can be expanded to\n\n${\\displaystyle p_{6}(x)=x^{6}+4x^{5}-72x^{4}-214x^{3}+1127x^{2}+1602x-5040.}$\n\nFrom the above we know that the largest root of this polynomial is 7 so we are able to make an initial guess of 8. Using Newton's method the first zero of 7 is found as shown in black in the figure to the right. Next ${\\displaystyle p(x)}$ is divided by ${\\displaystyle (x-7)}$ to obtain\n\n${\\displaystyle p_{5}(x)=x^{5}+11x^{4}+5x^{3}-179x^{2}-126x+720}$\n\nwhich is drawn in red in the figure to the right. Newton's method is used to find the largest zero of this polynomial with an initial guess of 7. The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by ${\\displaystyle (x-3)}$ to obtain\n\n${\\displaystyle p_{4}(x)=x^{4}+14x^{3}+47x^{2}-38x-240}$\n\nwhich is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow. Horner's method is now used to obtain\n\n${\\displaystyle p_{3}(x)=x^{3}+16x^{2}+79x+120}$\n\nwhich is shown in green and found to have a zero at\u00a0\u22123. This polynomial is further reduced to\n\n${\\displaystyle p_{2}(x)=x^{2}+13x+40}$\n\nwhich is shown in blue and yields a zero of\u00a0\u22125. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing ${\\displaystyle p_{2}(x)}$ and solving the linear equation. As can be seen, the expected roots of \u22128, \u22125, \u22123, 2, 3, and 7 were found.\n\n#### Octave implementation\n\nThe following Octave code was used in the example above to implement Horner's method.\n\nfunction [y b] = horner(a,x)\n% Input a is the polynomial coefficient vector, x the value to be evaluated at.\n% The output y is the evaluated polynomial and b the divided coefficient vector.\nb(1) = a(1);\nfor i = 2:length(a)\nb(i) = a(i)+x*b(i-1);\nend\ny = b(length(a));\nb = b(1:length(b)-1);\nend\n\n\n#### Python implementation\n\nThe following Python code implements Horner's method.\n\ndef horner(x, *polynomial):\n\"\"\"Implement the Horner Scheme for evaluating a\npolynomial of coefficients *polynomial in x.\"\"\"\nresult = 0\nfor coefficient in reversed(polynomial):\nresult = result * x + coefficient\nreturn result\n\n\n#### C implementation\n\nThe following C code implements Horner's method.\n\ndouble HornerEvaluate (double x, double * CoefficientsOfPolynomial, unsigned int DegreeOfPolynomial)\n{\n\/*\nWe want to evaluate the polynomial in x, of coefficients CoefficientsOfPolynomial, using Horner's method.\nThe result is stored in dbResult.\n*\/\ndouble dbResult = 0.0;\nint i;\nfor(i = DegreeOfPolynomial; i >= 0; i--)\n{\ndbResult = dbResult * x + CoefficientsOfPolynomial[i];\n}\nreturn dbResult;\n}\n\n\nHere is a slightly optimized version using explicit fused Multiply\u2013accumulate operation, often execute faster than the above when running on a computer built with a processor supporting FMA instruction:\n\n\/\/ gcc -std=c11 -lm horner.c -o horner\n#include <math.h>\n\ndouble horner_fma(double x, const double *coeffs, size_t count)\n{\ndouble result = 0.0;\nfor (int idx = count-1; idx >= 0; idx--)\nresult = fma(result, x, coeffs[idx]);\nreturn result;\n}\n\n\n#### C# implementation\n\nThe following C# code implements Horner's method.\n\npublic double HornerEvaluate(int[] numbers, double x)\n{\ndouble result = 0;\n\nfor(int i = numbers.Length - 1; i >= 0; i--)\n{\nresult = result * x + numbers[i];\n}\n\nreturn result;\n}\n\n\n## Application\n\nHorner's method can be used to convert between different positional numeral systems \u2013 in which case x is the base of the number system, and the ai coefficients are the digits of the base-x representation of a given number \u2013 and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater. In fact, when x is a matrix, further acceleration is possible which exploits the structure of matrix multiplication, and only ${\\displaystyle {\\sqrt {n}}}$ instead of n multiplies are needed (at the expense of requiring more storage) using the 1973 method of Paterson and Stockmeyer.[10]\n\n## Efficiency\n\nEvaluation using the monomial form of a degree-n polynomial requires at most n additions and (n2\u00a0+\u00a0n)\/2 multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. (This can be reduced to n additions and 2n\u00a0\u2212\u00a01 multiplications by evaluating the powers of x iteratively.) If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately 2n times the number of bits of x (the evaluated polynomial has approximate magnitude xn, and one must also store xn itself). By contrast, Horner's method requires only n additions and n multiplications, and its storage requirements are only n times the number of bits of x. Alternatively, Horner's method can be computed with n fused multiply\u2013adds. Horner's method can also be extended to evaluate the first k derivatives of the polynomial with kn additions and multiplications.[11]\n\nHorner's method is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations. Alexander Ostrowski proved in 1954 that the number of additions required is minimal.[12] Victor Pan proved in 1966 that the number of multiplications is minimal.[13] However, when x is a matrix, Horner's method is not optimal[citation needed].\n\nThis assumes that the polynomial is evaluated in monomial form and no preconditioning of the representation is allowed, which makes sense if the polynomial is evaluated only once. However, if preconditioning is allowed and the polynomial is to be evaluated many times, then faster algorithms are possible. They involve a transformation of the representation of the polynomial. In general, a degree-n polynomial can be evaluated using only ${\\displaystyle {\\scriptstyle {\\left\\lfloor n\/2\\right\\rfloor +2}}}$ multiplications and n additions.[14]\n\n## Divided difference of a polynomial\n\nHorner's method can be modified to compute the divided difference ${\\displaystyle (p(y)-p(x))\/(y-x).}$ Given the polynomial (as before)\n\n${\\displaystyle p(x)=\\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\\cdots +a_{n}x^{n},}$\n\nproceed as follows[15]\n\n{\\displaystyle {\\begin{aligned}b_{n}&=a_{n},&\\quad d_{n}&=b_{n},\\\\b_{n-1}&=a_{n-1}+b_{n}x,&\\quad d_{n-1}&=b_{n-1}+d_{n}y,\\\\&{}\\ \\ \\vdots &\\quad &{}\\ \\ \\vdots \\\\b_{1}&=a_{1}+b_{2}x,&\\quad d_{1}&=b_{1}+d_{2}y,\\\\b_{0}&=a_{0}+b_{1}x.\\end{aligned}}}\n\nAt completion, we have\n\n{\\displaystyle {\\begin{aligned}p(x)&=b_{0},\\\\{\\frac {p(y)-p(x)}{y-x}}&=d_{1},\\\\p(y)&=b_{0}+(y-x)d_{1}.\\end{aligned}}}\n\nThis computation of the divided difference is subject to less round-off error than evaluating ${\\displaystyle p(x)}$ and ${\\displaystyle p(y)}$ separately, particularly when ${\\displaystyle x\\approx y}$. Substituting ${\\displaystyle y=x}$ in this method gives ${\\displaystyle d_{1}=p'(x)}$, the derivative of ${\\displaystyle p(x)}$.\n\n## History\n\nQin Jiushao's algorithm for solving the quadratic polynomial equation${\\displaystyle -x^{4}+763200x^{2}-40642560000=0}$\nresult: x=840[16]\n\nHorner's paper entitled \"A new method of solving numerical equations of all orders, by continuous approximation\"[17] was read before the Royal Society of London, at its meeting on July 1, 1819, with Davies Gilbert, Vice-President and Treasurer, in the chair; this was the final meeting of the session before the Society adjorned for its Summer recess. When a sequel was read before the Society in 1823, it was again at the final meeting of the session. On both occasions, papers by James Ivory, FRS, were also read. In 1819, it was Horner's paper that got through to publication in the \"Philosophical Transactions\".[17] later in the year, Ivory's paper falling by the way, despite Ivory being a Fellow; in 1823, when a total of ten papers were read, fortunes as regards publication, were reversed. But Gilbert, who had strong connections with the West of England and may have had social contact with Horner, resident as Horner was in Bristol and Bath, published his own survey of Horner-type methods earlier in 1823.\n\nHorner's paper in Part II of Philosophical Transactions of the Royal Society of London for 1819 was warmly and expansively welcomed by a reviewer in the issue of The Monthly Review: or, Literary Journal for April, 1820; in comparison, a technical paper by Charles Babbage is dismissed curtly in this review. However, the reviewer noted that another, similar method had also recently been published by the architect and mathematical expositor, Peter Nicholson. This theme is developed in a further review of some of Nicholson's books in the issue of The Monthly Review for December, 1820, which in turn ends with notice of the appearance of a booklet by Theophilus Holdred, from whom Nicholson acknowledges he obtained the gist of his approach in the first place, although claiming to have improved upon it. The sequence of reviews is concluded in the issue of The Monthly Review for September, 1821, with the reviewer concluding that whereas Holdred was the first person to discover a direct and general practical solution of numerical equations, he had not reduced it to its simplest form by the time of Horner's publication, and saying that had Holdred published forty years earlier when he first discovered his method, his contribution could be more easily recognized. The reviewer is exceptionally well-informed, even having sighted Horner's preparatory correspondence with Peter Barlow in 1818, seeking work of Budan. The Bodlean Library, Oxford has the Editor's annotated copy of The Monthly Review from which it is clear that the most active reviewer in mathematics in 1814 and 1815 (the last years for which this information has been published) was none other than Peter Barlow,one of the foremost specialists on approximation theory of the period, suggesting that it was Barlow, who wrote this sequence of reviews. As it also happened, Henry Atkinson, of Newcastle, devised a similar approximation scheme in 1809; he had consulted his fellow Geordie, Charles Hutton, another specialist and a senior colleague of Barlow at the Royal Military Academy, Woolwich, only to be advised that, while his work was publishable, it was unlikely to have much impact. J. R. Young, writing in the mid-1830s, concluded that Holdred's first method replicated Atkinson's while his improved method was only added to Holdred's booklet some months after its first appearance in 1820, when Horner's paper was already in circulation.\n\nThe feature of Horner's writing that most distinguishes it from his English contemporaries is the way he draws on the Continental literature, notably the work of Arbogast. The advocacy, as well as the detraction, of Horner's Method has this as an unspoken subtext. Quite how he gained that familiarity has not been determined. Horner is known to have made a close reading of John Bonneycastle's book on algebra. Bonneycastle recognizes that Arbogast has the general, combinatorial expression for the reversion of series, a project going back at least to Newton. But Bonneycastle's main purpose in mentioning Arbogast is not to praise him, but to observe that Arbogast's notation is incompatible with the approach he adopts. The gap in Horner's reading was the work of Paolo Ruffini, except that, as far as awareness of Ruffini goes, citations of Ruffini's work by authors, including medical authors, in Philosophical Transactions speak volumes: there are none - Ruffini's name only appears in 1814, recording a work he donated to the Royal Society. Ruffini might have done better if his work had appeared in French, as had Malfatti's Problem in the reformulation of Joseph Diaz Gergonne, or had he written in French, as had Antonio Cagnoli, a source quoted by Bonneycastle on series reversion (today, Cagnoli is in the Italian Wikipedia, as shown, but has yet to make it into either French or English).\n\nFuller[18] showed that the method in Horner's 1819 paper differs from what afterwards became known as 'Horner's method' and that in consequence the priority for this method should go to Holdred (1920). This view may be compared with the remarks concerning the works of Horner and Holdred in the previous paragraph. Fuller also takes aim at Augustus De Morgan. Precocious though Augustus de Morgan was, he was not the reviewer for The Monthly Review, while several others - Thomas Stephens Davies, J. R. Young, Stephen Fenwick, T. T. Wilkinson - wrote Horner firmly into their records, not least Horner himself, as he published extensively up until the year of his death in 1837. His paper in 1819 was one that would have been difficult to miss. In contrast, the only other mathematical sighting of Holdred is a single named contribution to The Gentleman's Mathematical Companion, an answer to a problem.\n\nIt is questionable to what extent it was De Morgan's advocacy of Horner's priority in discovery[5][19] that led to \"Horner's method\" being so called in textbooks, but it is true that those suggesting this tend themselves to know of Horner largely through intermediaries, of whom De Morgan made himself a prime example. However, this method qua method was known long before Horner. In reverse chronological order, Horner's method was already known to:\n\nHowever, this observation on its own masks significant differences in conception and also, as noted with Ruffini's work, issues of accessibility.\n\nQin Jiushao, in his Shu Shu Jiu Zhang (Mathematical Treatise in Nine Sections; 1247), presents a portfolio of methods of Horner-type for solving polynomial equations, which was based on earlier works of the 11th century Song dynasty mathematician Jia Xian; for example, one method is specifically suited to bi-quintics, of which Qin gives an instance, in keeping with the then Chinese custom of case studies. The first person writing in English to note the connection with Horner's method was Alexander Wylie, writing in The North China Herald in 1852; perhaps conflating and misconstruing different Chinese phrases, Wylie calls the method Harmoniously Alternating Evolution (which does not agree with his Chinese, linglong kaifang, not that at that date he uses pinyin), working the case of one of Qin's quartics and giving, for comparison, the working with Horner's method. Yoshio Mikami in Development of Mathematics in China and Japan published in Leipzig in 1913, gave a detailed description of Qin's method, using the quartic illustrated to the above right in a worked example; he wrote: \"who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe ... We of course don't intend in any way to ascribe Horner's invention to a Chinese origin, but the lapse of time sufficiently makes it not altogether impossible that the Europeans could have known of the Chinese method in a direct or indirect way.\".[22] However, as Mikami is also aware, it was not altogether impossible that a related work, Si Yuan Yu Jian (Jade Mirror of the Four Unknowns; 1303) by Zhu Shijie might make the shorter journey across to Japan, but seemingly it never did, although another work of Zhu, Suan Xue Qi Meng, had a seminal influence on the development of traditional mathematics in the Edo period, starting in the mid-1600s. Ulrich Libbrecht (at the time teaching in school, but subsequently a professor of comparative philosophy) gave a detailed description in his doctoral thesis of Qin's method, he concluded: It is obvious that this procedure is a Chinese invention....the method was not known in India. He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese.[23] Here, the problems is that there is no more evidence for this speculation than there is of the method being known in India. Of course, the extraction of square and cube roots along similar lines is already discussed by Liu Hui in connection with Problems IV.16 and 22 in Jiu Zhang Suan Shu, while Wang Xiaotong in the 7th century supposes his readers can solve cubics by an approximation method described in his book Jigu Suanjing.\n\n## Notes\n\n1. ^ Cormen et al. 2009, pp.\u00a041, 900, 990.\n2. ^ a b Weisstein, Eric W. \"Horner's Rule\". MathWorld.\n3. ^\n4. ^ See M\u00e9thode de Ruffini-Horner on French Wikipedia.\n5. ^ a b c\n6. ^ It is obvious that this procedure is a Chinese invention, in Libbrecht 2005, p.\u00a0178.\n7. ^ \"Al-Tusi_Sharaf biography\". www-groups.dcs.st-and.ac.uk. Retrieved 2018-03-17.\n8. ^ Kripasagar 2008, p.\u00a062.\n9. ^ Kress 1991, p.\u00a0112.\n10. ^ Higham 2002, Section 5.4.\n11. ^\n12. ^\n13. ^\n14. ^\n15. ^ Fateman & Kahan 2000\n16. ^ Libbrecht 2005, pp.\u00a0181\u2013191.\n17. ^ a b\n18. ^ Fuller 1999, pp.\u00a029\u201351.\n19. ^ a b c\n20. ^ Berggren 1990, pp.\u00a0304\u2013309.\n21. ^ Temple 1986, p.\u00a0142.\n22. ^ Mikami 1913, p.\u00a077.\n23. ^ Libbrecht 2005, p.\u00a0208.","date":"2018-10-23 23:46:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 69, \"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\": 0.6804033517837524, \"perplexity\": 2005.1988254360779}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2018-43\/segments\/1539583517495.99\/warc\/CC-MAIN-20181023220444-20181024001944-00188.warc.gz\"}"}
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\section{Introduction}
State of the art machine learning algorithms can achieve high accuracies in classification tasks but misclassify minor perturbations in the data known as as adversarial attacks
\cite{goodfellow2014explaining,papernot2016limitations,kurakin2016adversarial,carlini2017towards,brendel2017decision}. Adversarial examples have been shown to transfer across models which makes it possible to perform transfer-based (substitute model) black box attacks \cite{papernot2016transferability}. To counter adversarial attacks many defense methods been proposed with adversarial training being the most popular \cite{szegedy2013intriguing}. This is known to improve robustness to adversarial examples but also tends to lower accuracy on clean test data that has no perturbations \cite{raghunathan2019adversarial,zhang2019theoretically}. Many previously proposed defenses have also shown to be vulnerable \cite{carlini2017towards,athalye2018obfuscated,ghiasi2020breaking} thus leaving adversarial robustness an open problem in machine learning.
The 01 loss is known to be more robust to outliers than convex loss models \cite{xie2019,icml13optimize,bartlett04}. In addition to being robust to outliers the 01 loss is also robust to noise in the training data \cite{manwani2013noise,ghosh2015making}. Under this loss minimizing the empirical risk amounts to minimizing the empirical adversarial risk \cite{lyu2019curriculum,hu2016does} with certain assumptions of noise. Convex losses also fail to minimize the adversarial 01 loss on linear models \cite{bao2020calibrated}.
Motivated by the above robustness properties of 01 loss we propose a 01 loss dual layer neural network as a defense against adversarial attacks. Computationally 01 loss presents a considerable challenge because it is NP-hard to solve \cite{ben03}. Previous attempts \cite{nips01optimize,mixedint01,approx01,ijcnn01optimize,icml13optimize} lack on-par test accuracy with convex solvers and are slow and impractical for large image benchmarks. However, a recent stochastic coordinate descent method for linear 01 loss models \cite{xie2019} has shown to attain comparable accuracies to state of the art linear solvers like the support vector machine. Thus we extend the coordinate descent as an optimizer to train our network.
We compare the adversarial robustness of our model to an equivalent one that uses sigmoid activation and cross-entropy loss. This is the standard activation and loss that are widely used in neural networks today. We also compare our model to the binarized neural network \cite{galloway2017attacking,courbariaux2016binarized,rastegari2016xnor}. that also uses sign activations like our model but it has two differences. First its weights are also constrained to be binary +1 and -1, or 1 and 0 \cite{galloway2017attacking,courbariaux2016binarized,rastegari2016xnor}. Second, it is trained with gradient descent by approximating the sign activation whereas we take a direct coordinate descent approach.
Measuring adversarial robustness is not trivial and several best practices have been recommended \cite{carlini2019evaluating}. We incorporate several of them in our study. In particular we study (1) the robustness of models to random Gaussian noise, (2) the minimum distortion required to make a datapoint adversarial, and (3) the accuracy of substitute model black box attacks. We focus mainly on binary classification between classes 0 and 1 on the CIFAR10 image benchmark \cite{krizhevsky2009learning} where we make the following findings.
\begin{itemize}
\item All models are more robust to Gaussian noise than adversarial attacks, but our 01 loss network augmented with Gaussian noise during training has a higher accuracy on large distortions
\item The minimum distortion to make an image adversarial is higher for our 01 loss network compared to the standard sigmoid activated cross-entropy loss network and binarized networks
\item Substitute model black box attacks are far less effective on our 01 loss network compared to the standard sigmoid activated cross-entropy loss network and binarized networks
\item Compared to simple convolutional neural networks like LeNet \cite{lecun1998gradient} our model (without convolutions) has higher accuracies on adversarial examples from substitute model black box attacks when the distortions are high.
\end{itemize}
\section{Methods}
\subsection{Background}
The problem of determining the hyperplane with minimum number of misclassifications
in a binary classification problem is known to be NP-hard \cite{ben03}.
In mainstream machine learning literature this is called minimizing the 01 loss
\cite{kernel01} given in Objective~\ref{obj1},
\begin{equation}
\frac{1}{2n}\argmin_{w,w_0} \sum_i (1-sign(y_i(w^Tx_i+w_0)))
\label{obj1}
\end{equation}
where $w \in R^d$, $w_0 \in R$ is our hyperplane, and $x_i \in R^d, y_i\in \{+1,-1\}.\forall i=0...n-1$ are our training data. Popular linear classifiers such as the linear support
vector machine, perceptron, and logistic regression \cite{alpaydin} can be considered
as convex approximations to this problem that yield fast gradient descent solutions \cite{bartlett04}.
However, they are also more sensitive to outliers than the 01 loss \cite{bartlett04,icml13optimize,xie2019}
and more prone to mislabeled data than 01 loss \cite{manwani2013noise,ghosh2015making,lyu2019curriculum}.
\subsection{A dual layer 01 loss neural network}
We extend the 01 loss to a simple two layer neural network with $k$ hidden nodes and sign activation that we call the MLP01 loss. This objective for binary classification can be given as
\begin{equation}
\small
\frac{1}{2n}\argmin_{W, W_0, w,w_0} \sum_i (1-sign(y_i(w^T(sign(W^Tx_i+W_0))+w_0)))
\label{obj2}
\end{equation}
where $W \in R^{d\times k}$, $W_0 \in R^k$ are the hidden layer parameters, $w\in R^k, w_0\in R$ are the final layer node parameters, $x_i \in R^d, y_i\in \{+1,-1\}.\forall i=0...n-1$ are our training data, and $sign(v\in R^k)=(sign(v_0), sign(v_1),...,sign(v_{k-1}))$. While this is a straightforward model to define optimizing it is a different story altogether. Optimizing even a single node is NP-hard which makes optimizing this network much harder.
\subsection{Stochastic coordinate descent for 01 loss}
In Algorithm~\ref{mlp01} we sketch our coordinate descent for our 01 loss network that is based upon earlier work \cite{xie2019}. We initialize all parameters to random values from the Normal distribution with mean 0 and variance 1. We then randomly select a subset of the training data (known as a batch) and perform the coordinate descent analog of a single step gradient update in stochastic gradient descent \cite{bottou2010large}.
\begin{algorithm}[!h]
\caption{Stochastic coordinate descent for two layer 01 loss network}
\label{mlp01}
\textbf{Procedure: }
\begin{algorithmic}
\STATE 1. Initialize all network weights $W,w$ to random values from the Normal distribution $N(0,1)$.
\STATE 2. Set network thresholds $W_0$ to the median projection value on their corresponding weight vectors and $w_0$ to the projection value that minimizes our network objective.
\WHILE {$i < epochs$}
\STATE 1. Randomly sample a batch of data equally from each class
\STATE 2. Perform coordinate descent separately first on the final node $w$ and then a randomly selected hidden node $u$ (a random column from the hidden layer weight matrix $W$)
\STATE 3. In the coordinate descent we randomly pick a set of features and perform a single update to each one. For each update we determine the threshold that optimizes the 01 loss on the sampled data: we sort the projections $w^Tx_i$ and pick the optimal middle value between each consecutive pair in the projected values.
\STATE 4. After making the best update we evaluate the 01 loss on the full dataset and accept the change if it improves the loss.
\ENDWHILE
\end{algorithmic}
\end{algorithm}
When the gradient is known we step in its negative direction by a factor of the learning rate: $w=w-\eta\nabla(f)$ where $f$ is the objective. In our case since the gradient does not exist we randomly select $k$ features (set to 128 in our experiments), modify the corresponding entries in $w$ by the learning rate (set to 0.17) one at a time, and accept the modification that gives the largest decrease in the objective. Key to our search is a heuristic to determine the optimal threshold each time we modify an entry of $w$. In this heuristic we perform a linear search on a subset of the projection $w^Tx_i$ and select $w_0$ that minimizes the objective.
\begin{figure}[h]
\centering
\includegraphics[trim=80 50 0 70, clip, scale=.27]{nrows.png}
\caption{Train and test accuracy of our stochastic coordinate descent on CIFAR10 class 0 vs 1 with different batch sizes (denoted as nrows). \label{nrows}}
\end{figure}
In Figure~\ref{nrows} we show the effect of the batch size (as a percentage of each class to ensure fair sampling) on a linear 01 loss search on CIFAR10 between classes 0 and 1. We see that a batch size of 75\% reaches a train accuracy of 80\% faster than the other batch sizes. Thus we use this batch size in all our experiments going forward.
We also see that for this batch size the search flattens after 15 iterations (or epochs as given in the figure). We run 1000 iterations to ensure a deep search with an intent to maximize test accuracy. The problem with our search described above is that it will return different solutions depending upon the initial starting point. To make it more stable we run it a 100 times from different random seeds and use the majority vote for prediction. Full details of our training algorithms are in the Supplementary Material.
\subsection{Implementation, experimental platform, and image data}
\subsubsection{Implementation}
We implement our 01 loss network (MLP01) in Python and Pytorch \cite{pytorch}, the sigmoid activated cross-entropy loss network (MLP) in scikit-learn \cite{scikit}, and binarized neural network (BNN) with the Larq software suite \url{https://github.com/larq/larq}. We train MLP with stochastic gradient descent that has a batch size of 200, momentum of 0.9, and learning rate of 0.01. For BNN we use the approximate sign activation \cite{courbariaux2016binarized} that has been shown to give higher test accuracies than other variants and the original straight through estimator \cite{liu2018bi}.
\subsubsection{Computational platform}
We ran all experiments on Intel Xeon 6142 2.6GHz CPUs and NVIDIA Titan RTX GPU machines (for parallelizing multiple votes). Our MLP01 source code, supplementary programs, and data are available from \url{https://github.com/zero-one-loss/mlp01}.
\subsubsection{Data}
We experiment on the popular image benchmark CIFAR10 \cite{krizhevsky2009learning}
that has $32\times32$ color images with 50000 training and 10000 test. We extract data from classes 0 and 1 and experiment on binary classification between them. This gives us a total of 10000 training and 1000 test examples. We normalize each image by dividing each pixel value by 255.
\section{Results}
We refer to our 01 loss neural network as MLP01, the sigmoid activated cross-entropy loss network as MLP, and the binarized network as BNN. We use one hidden layer of 20 nodes in all networks. For each model we run it a 100 times with different random number generator seeds and return the majority vote as the prediction.
In addition to training each model on the training data, we also study three versions trained with augment Gaussian noise. In the augmentation we take each datapoint $x$ from the training set and add Gaussian noise to it: $x'=x+N(0,\sigma)$ where $N(0,\sigma)$ is a vector of the same dimension as $x$ and each entry is selected from the Normal distribution with mean 0 and standard deviation $\sigma$.
We refer to the accuracy on the test data as clean data test accuracy. An incorrectly classified adversarial example is considered a successful attack whereas a correctly classified adversarial is a failed one. Thus when we refer to accuracy of adversarial examples it is the same as $100-attack success rate$. The lower the accuracy the more effective the attack.
\subsection{Sensitivity to Gaussian noise}
We start with accuracy of models trained without and with noise on CIFAR10 class 0 vs. 1. In addition to evaluating the accuracy of each model on clean test data, we add noise to each test datapoint as $x'=x+N(0,\sigma)$ where $N(0,\sigma)$ is a vector of the same dimension as $x$ and each entry is selected from the Normal distribution with mean 0 and standard deviation $\sigma$. We consider $\sigma$ ranging from .004 to 1. The lower bound is the minimum distance of $\frac{1}{255}$ between two pixels and $1$ is the maximum distortion. If a pixel is negative or above 1 after adding noise we clip it to 0 and 1 respectively.
In Figure~\ref{noise} we see that noise does not affect the accuracy of models trained without and with noise upto distortion threshold of 0.125. After that all models begin to dip in accuracy with MLP01 model trained without noise showing the steepest descent. At the same time MLP01 trained with noisy augmentation of $\sigma=.2$ (denoted as mlp01\_ep2 in Figure~\ref{noise}) is also most robust to high levels of noise.
\begin{figure}[!h]
\centering
\includegraphics[scale=.425]{"Gaussian_noise"}
\caption{Accuracy of test data without and with random noise of different random Gaussian distortions. In addition to models trained on clean training data we consider two versions trained with augmented Gaussian noise with distortion thresholds of 0.1 and 0.2 (denoted with ep1 and ep2 respectivelt). \label{noise}}
\end{figure}
\subsection{Minimum adversarial distortion}
Determining the minimum distortion to an image such that it will fool a classifier is itself an NP-hard problem for ReLu activated neural networks \cite{katz2017reluplex,sinha2017certifiable} and tree ensemble classifiers \cite{kantchelian2016evasion}. Even approximating the minimum distortion for ReLu activated neural networks is NP-hard \cite{weng2018towards}. Recent heuristics such as Boundary attack \cite{brendel2017decision} and HopSkipJump attack \cite{chen2019hopskipjumpattack} attempt to find an adversarial example with the minimum distance to the clean correctly classified version.
We use both methods to evaluate the minimum distortion of all three models including their noise trained versions. Both methods can take long to finish with default parameters even for a single example. For example for a single image BNN takes 3 days on an exclusive CPU core. Thus we use all default parameters except for maxiter that we set to 100 so that the program finishes within our computing limitations. To confirm that this does not severely affect the relative distortions we ran both Boundary and HopSkipJump attacks with maxiter set to 10, 100, and 500 (which is the default) on a single image. We found the relative distortions between MLP and MLP01 to be the same across the three values.
Since both attack methods start with a random initialization we run each of them 10 times for a single example on each model and report the minimum value found. In Table~\ref{distortion} we report these values for a single random test datapoint from CIFAR10 classes 0 and 1 that is correctly classified by all models. We see that the minimum distortion of MLP01 is much higher than both MLP and BNN by both attack methods and under both $L_2$ and $L_\infty$ norms. We also see that HopSkipJump attack is more effective than Boundary attack and finds a smaller distortion.
\begin{table}[!h]
\caption{Minimum adversarial distortion of a single random test image \label{distortion}}
\centering
\begin{tabular}{lllllll} \hline
& \multicolumn{3}{c}{$L_2$ distance} & \multicolumn{3}{c}{$L_{\infty}$ distance} \\
& BNN & MLP & MLP01 & BNN & MLP & MLP01 \\
Boundary & 2.72 & 1.33 & 10.58 & 0.17 & 0.08 & 0.58 \\
HopSkipJump & 0.82 & 0.44 & 2.21 & 0.04 & 0.03 & 0.16 \\
\end{tabular}
\end{table}
In Table~\ref{distortion2} we report the HopSkipJump distortions for four more randomly selected images from CIFAR10 classes 0 and 1 that are correctly classified. For the first image both have comparable distortion but for the other three MLP01 is higher.
\begin{table}[!h]
\caption{Minimum adversarial distortion given by HopSkipJump of four random correctly classified test images \label{distortion2}}
\centering
\begin{tabular}{llllll} \hline
& \multicolumn{2}{c}{$L_2$ distance} & \multicolumn{2}{c}{$L_{\infty}$ distance} \\
& MLP & MLP01 & MLP & MLP01 \\
Image1 & .64 & .52 & .043 & .041 \\
Image2 & .75 & 2.42 & .06 & .15 \\
Image3 & .88 & 1.12 & .06 & .09 \\
Image4 & 1.15 & 4.86 & .09 & .28 \\ \hline
Average & .86 & 2.23 & .063 & 0.14 \\
\end{tabular}
\end{table}
Training our models with noise has an interesting effect on the distortions. In Table~\ref{distortion3} we see the minimum distortions of models trained with Gaussian noise with distortions of 0.1 and 0.2 (as described earlier). We report the distortions for the same image as in Table~\ref{distortion}. As we increase the noise threshold the MLP01 model's minimum distortion also rises whereas the other two models are stable or fluctuate.
\begin{table}[!h]
\caption{Minimum adversarial distortion of the single test image from Table~\ref{distortion} as given by HopSkipJump. Models are trained with Gaussian noise of increasing distortion shown by $\epsilon$. \label{distortion3}}
\centering
\begin{tabular}{lllllll} \hline
& \multicolumn{3}{c}{$L_2$ distance} & \multicolumn{3}{c}{$L_{\infty}$ distance} \\
& $\epsilon=.004$ &$\epsilon=.1$ & $\epsilon=.2$ & $\epsilon=.004$ &$\epsilon=.1$ & $\epsilon=.2$ \\
BNN & .51 & .47 & .53 & .025 & .021 & .025 \\
MLP & .39 & .35 & .36 & .023 & .021 & .022 \\
MLP01 & 2.52 & 2.61 & 3.32 & .196 & .215 & .22 \\
\end{tabular}
\end{table}
We make similar observations between MLP and MLP01 on four random examples as shown in Table~\ref{distortion4}. This suggests that perhaps training MLP01 with augmented noise examples increases their robustness.
\begin{table}[!h]
\caption{Minimum adversarial distortion of four random correctly classified test images from Table~\ref{distortion2} as given by HopSkipJump. Models are trained with Gaussian noise of increasing distortion shown by $\epsilon$. \label{distortion4}}
\centering
\begin{tabular}{lllllll} \hline
& \multicolumn{6}{c}{$L_2$ distance} \\
& \multicolumn{3}{c}{MLP} & \multicolumn{3}{c}{MLP01} \\
& $\epsilon=.004$ &$\epsilon=.1$ & $\epsilon=.2$ & $\epsilon=.004$ &$\epsilon=.1$ & $\epsilon=.2$ \\
Image1 & .58 & .47 & .47 & .69 & .61 & .47 \\
Image2 & .6 & .52 & .54 & 2.65 & 2.89 & 2.51\\
Image3 & .72 & .67 & .66 & 1.92 & 2.72 & 4.03\\
Image4 & .97 & .85 & .84 & 4.73 & 4.08 & 5.17 \\ \hline
Average & .72 & 63 & .63 & 2.5 & 2.58 & 3.05 \\ \hline \hline
& \multicolumn{6}{c}{$L_\infty$ distance} \\
& \multicolumn{3}{c}{MLP} & \multicolumn{3}{c}{MLP01} \\
& $\epsilon=.004$ &$\epsilon=.1$ & $\epsilon=.2$ & $\epsilon=.004$ &$\epsilon=.1$ & $\epsilon=.2$ \\
Image1 & .04 & .03 & .03 & .05 & .05 & .03 \\
Image2 & .04 & .03 & .03 & .18 & .2 & .18 \\
Image3 & .05 & .04 & .04 & .14 & .21 & .35 \\
Image4 & .07 & .05 & .05 & .28 & .29 & .35 \\ \hline
Average & .05 & .04 & .04 & .16 & .19 & .23 \\ \hline
\end{tabular}
\end{table}
\subsection{Substitute model black box attacks}
As further verification of the above distortions we perform substitute model black box attacks on all three models. In this method we try to approximate the target model with a substitute and then generate white box adversaries from the substitute to attack the target model. The success of this method relies upon transferability of adversarial examples between models. We use the standard adversarially augmented training algorithm of Papernot et. al. \cite{papernot2017practical} to train the substitute. In the Supplementary Material we provide full details of the algorithm.
This in fact is a powerful attack method that needs only predicted labels from the target (like Boundary and HopSkipJump) but requires much fewer queries. Once the substitute is trained it can produce adversaries for any input. Recent advances in transferability have made this method more effective and broken defenses based on adversarial training \cite{tramer2017ensemble,wu2020skip}. For the substitute model we use a three layer sigmoid activated cross-entropy loss network with 200 nodes in each hidden layer. We start with 200 random test data points from which we iteratively train the substitute model with augmented adversaries.
In Figure~\ref{blackbox} we see the accuracy of adversarial examples at the end of the $20^{th}$ epoch. We also show the accuracy of the three models on random Gaussian noise of the same distortions (from our earlier subsection above). Clearly the black box adversaries are far more effective than random noise indicating that the substitute model training was successful. In agreement with our distortions from Boundary and HopSkipJump above we see that MLP01 can correctly classify images of much higher distortion than BNN and MLP.
\begin{figure}[!h]
\centering
\includegraphics[scale=.425]{"blackbox"}
\caption{Accuracy of adversarial examples and test data with Gaussian noise (denoted as -GN for each model) for various distortion thresholds. The adversarial examples are far more effective than random noise. At distortion 0.125 both MLP and BNN have near 0\% accuracy whereas MLP01 has 40\%. \label{blackbox}}
\end{figure}
\subsection{Comparison to convolutional neural networks}
As a test against state of the art classification methods we compare our 01 loss network with 500 hidden nodes to two convolutional neural networks. First is LeNet \cite{lecun1998gradient} which is among the first convolutional networks to be proposed and second is SimpleNet500. In this model we use the same convolutional layers as LeNet followed by one layer of 500 nodes and then the final output node.
We employ the same substitute model training algorithm as in the above subsection. However instead of a dual hidden layer model we use a convolutional network as the substitute. In each convolutional block we have a $3\times3$ convolutional kernel followed by max pool and batch normalization. In the first, second, third, and fourth layer we have 32, 64, 128, and 256 kernels respectively following by a final layer for the output.
In Figure~\ref{cnn3} we see that our model has a comparable accuracy to the convolutional models on clean test data and low distortion thresholds. However, when we cross 0.03125 then MLP01 has the highest accuracy. At threshold 0.125 it is about 11\% higher than both LeNet and SimpleNet500.
\begin{figure}[!h]
\centering
\includegraphics[scale=.425]{"cnn3"}
\caption{Accuracy of adversarial examples for different distortion thresholds \label{cnn3}}
\end{figure}
\section{Discussion}
Binarized neural networks \cite{galloway2017attacking,courbariaux2016binarized,rastegari2016xnor} have weights and activations constrained to be near +1 and -1 (or 1 and 0) whereas our model weights are real numbers. The purpose of those networks is efficiency as opposed to robustness. Indeed we see in recent work that binarized networks offer marginal improvements in robustness to substitute model black box robustness on MNIST and none in CIFAR10 (see Tables 4 and 5 in \cite{galloway2017attacking} and Table 8 in \cite{panda2019discretization}).
We make the same observations here: BNN has similar distortions to MLP and similar accuracies on adversarial examples. Both BNN and MLP01 have sign activations yet MLP01 has higher distortions and higher adversarial accuracies. Perhaps this has to do with the optimization method. BNNs are trained with an approximation to the sign activation that is differentiable whereas we train with direct coordinate descent.
In separate work we study transferability between our 01 loss network and the standard sigmoid activated cross-entropy loss networks \cite{xue2020transferability}. There we show a lack of transferability between convex and 01 loss models in white box attacks and that both convex and 01 loss substitute model black box attacks are ineffective on our 01 loss network. However, in the work here we focus on the distortion thresholds of adversarial and Gaussian noise examples.
Interestingly the adversarial accuracy of our network is on-par with simple convolutional models that have the powerful advantage of convolutions. As future work 01 loss convolutions may be a promising avenue to obtain models with high clean test accuracy and high adversarial accuracy as well.
\section{Conclusion}
We show that our 01 loss neural network can correctly classify images with a higher distortion than both the sigmoid activated cross-entropy loss network and binarized neural networks.
|
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First published in Great Britain in 2018 by
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# Contents
Acknowledgements
Introduction: Global social work in a political context
Part One: The political context of contemporary social work
---
One | The political economy of social work
Two | Neoliberalism, social work and the state: retreat or restructuring?
Three | The privatisation of social work and social care
Part Two: Social work politics: past and present
Four | Social work's horrible histories: collusion and resistance
Five | Social work as a praxis for liberation: the case of Latin American reconceptualisation
Six | Refugees, migrants and social work
Seven | Social work, climate change and the Anthropocene
Part Three: Debating the politics of social work today
Eight | A new politics of social work?
Nine | The case for a social justice-based global social work definition
Conclusions: 'Making history'
References
# Acknowledgements
This book draws on our work with colleagues in a variety of radical social work networks across the global social work community. It would be impossible to thank them all individually, but we would like to acknowledge the support, friendship and comradeship of colleagues in the range of radical social work groups across the globe. These include the Social Work Action Network (SWAN) groups in Ireland, Greece, Canada and Britain. Similar groups in other global locations include the Boston Liberation Health Group, the New Approach Group in Hungary, the Progressive Welfare Network, Hong Kong, the Rebel Social Work Group in New Zealand/Aotearoa and the Arbeitskreis Kritische Soziale Arbeit (Critical and Radical Social Work Network) in Germany. We would also like to acknowledge the many inspiring representatives that we have met from the social work associations in Spain, Argentina and Brazil. In particular, we remember fondly the meeting we coordinated at the Global Social Work conference in Sweden to try and establish an international network together; the meeting was well attended and just about to start when the Brazilian colleagues arrived. Their presence was remarkable for two reasons: first, there were lots of them! Second, they all wanted to debate with us on the continuing relevance of Marxism to social work – not something we experience very often!
Our experience in social work has also been enhanced by the involvement of a range of service users and service-user organisations and colleagues in relevant campaigning organisations. Among the many organisations that we have worked with, we would like to thank Disabled People Against the Cuts, Recovery in the Bin, Black Triangle, Shaping Our Lives and Stand Up To Racism. In Britain, many social workers are active within both social work and their union. Helen Davies, Sharon Campion, Simon Cardy, Malcolm Jones, Bea Kay and Jimmie Main are active Unison members (and SWAN supporters) who regularly provide us with insights on working life on the social work front line.
As well as organisations, we would like to thank a number of colleagues who we have worked with in radical social work over the years: from the global community, the list includes Bessa Whitmore, Michael Reisch, Elaine Behring, Catherine MacDonald, Suzanne Dudziak, Jelka Zorn, Silvana Martinez, Fumihito Ito, Lúcia Garcia, Paul Michael Garrett, Dawn Belkin Martinez and Roberta Uchoa. Closer to home, we would like to thank Rich Moth, Linda Smith, Joe Greener, Rea Maglajlic, John Harris, Terry Murphy, Jeremy Weinstein, Natalie Farmer, Barrie Levine, Victoria Jupp-Kena, Lee Humber, Peter Beresford, Katherine MacKay, Des McDermott, Mark Baldwin, Chris Jones, Gurnam Singh, Maria-Inez Martinez, Alissa Ruane, Bob Williams-Finlay and the sadly missed Rona Woodward. Three people who deserve special mention for their inspiration and continuing commitment to radical social work theory and practice are Dora Teloni in Greece, Kerry Cuskelly in Ireland and Lam Chi Leung in Hong Kong – each represents everything that is good about radical social work and they have our utmost admiration for all they do!
We want to thank Policy Press for their continuing support of both the journal Critical and Radical Social Work (CRSW) and their commitment to publish radical social work texts. Thanks in particular to Julia Mortimer, Rebecca Tomlinson, Edwina Thorn, Catherine Grey and Isobel Bainton. We are grateful to Policy Press and CRSW for allowing us to use some material in Chapter Nine that originally appeared as an article by Vasilios, entitled 'Beyond the dichotomies of cultural and political relativism: arguing the case for a social justice-based "global social work" definition', Critical and Radical Social Work, 1(2): 183–200.
Finally, writing a book and taking part in various social movement activities takes time and can only be accomplished with the continuing support and co-participation of our partners. So we would like to record our thanks to Dorte Pape, Natalia Barrera Villota and Laura Penketh.
IF, VI, ML
# Introduction: Global social work in a political context
As we approach the end of the second decade of the 21st century, the world is facing a growing number of, seemingly intractable, problems. Where does social work sit in relation to these problems and how should social work respond? Let us map out the terrain upon which these problems rest.
What Michael Roberts (2016) calls the 'long depression' shows no signs of ending. Financial instability and low economic growth continue to have an impact on global economic performance. In response to this crisis, large units of capital and states across the globe have been involved in a concerted attempt to reduce living standards, for the majority, as a means of restoring profitability across the system. This has been enshrined in policies of 'austerity', the claim that we can no longer afford what is thought of as the 'post-war welfare settlement' and that, as a result, 'there is no alternative' to savage cuts and welfare restructuring.
The attempt to transform the 'welfare state' (in all their variety across the globe) is important for social work. It has a significant impact on those who social work services work with (who find their services cut and stigmatised). For front-line practitioners, it means fewer resources to support service users, but it has also been matched by attempts to restructure and narrow the social work task. How do we understand these processes and, as social workers, how do we respond?
The 10-year depression and 'austerity politics' have also had the effect of eroding public confidence in 'mainstream' politicians and state actors. This has been particularly acute with regard to the crisis of social democracy, which increasingly looks like a universal phenomenon. The politicians of the centre, across the globe, act as if little has changed. For those of the social-democratic Left, this means that they have continued to promote the (failed) policies of the first decade of the century – promoting 'globalisation', trade liberalisation and the marketisation of public services. However, these politicians are uncomprehending of the rejection of their politics as 'populist movements' (of both the Left and Right) grow to challenge the political centre. Over the last few years, we have witnessed new social movements challenging the power of the powerful but we have also seen the growth of right-wing political networks that promote economic and political nationalism, xenophobia, and racism (with particular hostility towards migrants and refugees). These developments present huge challenges for social work and, once more, raise questions about how we respond.
As the 21st century unfolds, it is becoming clear that present economic arrangements pose a huge threat to our climate and the future of the planet and its ecological system. Is this of any relevance to social work? How can we develop a social work practice that is attuned to the planetary crisis we face?
These themes – economic crisis, welfare restructuring, climate change and refugee rights – represent some of the most pressing issues facing the social work profession globally today. The shifting political context of the 21st century is the backdrop against which present-day understandings of the role, tasks and possibilities of social work take place. This book is our attempt to navigate our way through this complex maze and try to understand the challenges it presents for social work.
The book has developed out of our experiences of working alongside a range of radical social work groups in a number of countries across the globe. Over the last 20 years, we have learnt a great deal about the limits and possibilities for a more engaged and radical practice by listening at the feet of practitioners involved in groups like the Social Work Action Network (in Britain, Ireland, Greece and Canada), the Boston Health Liberation Group, the New Approaches group in Hungary, the Progressive Welfare Network in Hong Kong and the various official social work organisations across much of Latin America. In all these groups, practitioners continue to engage in radical practice in difficult political contexts, but their practice continues to reaffirm that radical social work represents some of the best examples of good social work practice.
This book is an attempt to synthesise some of the best from the radical social work tradition globally. We want to look at how radical social work ideas can help explain the current global situation that social workers find themselves in and we want to look at and generalise from examples of good radical practice in the field.
The book is divided into three parts. Part One looks at the political context of contemporary social work. Here, we look at what might be termed the political economy of social work. Chapter One looks at the contemporary economic crisis and the alternatives to the 'perpetual austerity' that is being promoted by sections of our governing classes. Understanding the nature of these developments is important for all concerned with the future of social work and social care, for three reasons. First, the dominant policy response to the economic crisis has been to shift the costs of bailing out the global banking system in 2008 onto the poorest sections of the working class (pensioners, the disabled, the low paid and the unemployed). Of course, these are the sections who make up the majority of social work service users.
Second, while the roots of the current crisis lie in the failings of neoliberalism, the dominant political and ideological response to that crisis has been more of the same in the form of increased privatisation and public sector cuts, leading one writer to speak of 'the strange non-death of neoliberalism' (Crouch, 2011). It is also clear that for some governments, the crisis has provided them with a golden political opportunity to shrink the welfare state to levels not seen for decades, not least in Britain, where the percentage of spending on welfare is now at its lowest level since the 1930s. Third, the unprecedented assault on the lives and living standards of disabled people and people on benefits throws up major questions of how to respond, particularly for social workers and social work.
Chapters Two and Three explore the relationship between social work, neoliberalism and the state. This is a relationship that is rarely discussed in social work books – though the radical social work literature of the 1970s did look at the development of what Jones (1983) called 'state-directed social work'. Here, we focus on the relationship between the development of capitalism, the state, social work and the conditions of welfare transformation. The attempt to fundamentally transform welfare states has led to the growth of privatised, for-profit, service providers. These providers are funded and regulated by the state but, as private companies, they are not subject to the same democratic accountability as state-provided services and their priority is not meeting people's needs, but protecting their profit margins. This is the development of what we term 'state-directed social work' within the neoliberal welfare system.
The first three chapters, then, set the general political and economic context for contemporary debates about social work. In Part Two, we look at politics in social work 'past and present'. Chapter Four, perhaps controversially, discusses one aspect of social work that is rarely mentioned, its 'horrible histories'. Here, we look at a number of cases where social work has acted as the tool of the state to undertake brutal, oppressive and abusive acts. We view social work as an inherently political activity and that means looking at, and learning from, those episodes when social work acted in ways that, most of us would think, breached our various professional 'codes of ethics'.
Chapter Five, by way of a contrast, examines the Latin American 'reconceptualisation' movement. This movement is little known in social work circles across much of the 'West', yet it represents, we suggest, one of the most progressive and inspiring examples of engaged radical social work practice. If we want to reinvigorate radical social work for the 21st century, then it is important that we learn the lessons from this great movement of the 1960s across the Latin American continent.
Chapters Six and Seven take two of the most pressing social issues of the present, looking at the social work response to the refugee crisis (Chapter Six) and to the planetary crisis created by climate change (Chapter Seven). In both chapters, we look at the causes of the crises and locate them in the broader drives of contemporary capitalism, but we finish by looking at what social work can, has and could do to address the political crises created by these contemporary political and social problems. Throughout Part Two of the book, we have used boxed sections to highlight examples of good practice or to reflect back on continuities in social work's history that often get lost.
In Part Three, we look at some contemporary debates about the politics of social work today. In Chapter Eight, we engage in friendly debate with colleagues in what is sometimes called the 'critical social work' tradition. Our case is that the insights of the radical social work tradition offer a clearer set of ideas and theories that help us understand, explain and engage in the modern world. In Chapter Nine, we look at ongoing debates about the 'definition of social work', something, we suggest, that brings us back to our starting point: social work is an inherently political activity and the profession will be stronger when this is explicitly acknowledged.
# Part One
The political context of contemporary social work
## ONE
## The political economy of social work
### Introduction: 'the 1930s in slow motion'
Writing in the early 1990s, the veteran British-Palestinian Marxist Tony Cliff described the new decade as being like 'a film of the 1930s in slow motion'. As in the 1930s, Cliff argued, there was a global recession that had resulted in millions of people becoming unemployed. Also, as in the 1930s, accompanying that recession was the emergence of fascist movements such as Jean-Marie Le Pen's Front National in France. Cliff's conclusion, however, was far from fatalistic: 'The 1930s was a decade of extremes.... The fact that the film of the 1930s returns, but in slow motion, means there is much greater opportunity to stop the film and direct it in the way we want' (Cliff, 2000: 81). Like the Marxist literary critic Walter Benjamin some 60 years previously, Cliff was sounding a kind of 'fire alarm' to his contemporaries, 'a warning bell attempting to draw attention to the imminent dangers threatening them, to the new catastrophes looming on the horizon' (Löwy, 2005: 16). More than two decades on and some 10 years after the onset of a global recession that began in the US sub-prime housing market before morphing into an economic and political crisis in the Eurozone area, there is a sense that the film has sped up, both economically and politically.
Economically, the recovery from the crisis of 2008 has been very weak indeed. Commenting on separate reports from the Organisation for Economic Co-operation and Development (OECD) and from the United Nations Conference on Trade and Development (UNCTAD) issued in autumn 2016, an editorial in The Guardian newspaper described them as being 'thick with cloud and short on silver lining'. Despite promises from politicians and policymakers across the globe that, following the crash, things would change:
nearly a decade later, what is most striking is how little has changed. In the US, the UK and the rest of the developed world, policymakers talk of the 'new mediocre', so tepid is economic performance. And in the developing world things look even worse. (The Guardian, 21 September 2016)
Echoing that theme in a speech in December 2016, the Governor of the Bank of England, Mark Carney, described the past 10 years as having been 'a lost decade' for the British economy; one would have to go back to the 1860s, Carney argued, to find a comparable period (Carney, 2016).
The effects of that 'long depression' (Roberts, 2016) have, of course, been experienced very differently by different sections of the global population. An Oxfam report published in early 2016 showed that 1% of the world's population now own more than the rest combined. Even more starkly, 62 people own as much as the poorest half of the world's population. 'An economy for the 1%' (Oxfam, 2016) showed that the wealth of the poorest half of the world's population – 3.6 billion people – has fallen by a trillion dollars since 2010. This 38% drop has occurred despite the global population increasing by around 400 million people during that period. Meanwhile, the wealth of the richest 62 has increased by more than half-a-trillion US dollars to US$1.76 trillion (Oxfam, 2016).
That inequality, coupled with the fact that the neoliberal promise that wealth would 'trickle down' has proved hollow for millions of people, has been one factor fuelling the political polarisation that has been a feature of the second decade of the 21st century. While rising racism and xenophobia have undoubtedly been contributory factors to a range of mid-decade political developments, including the 'Brexit' vote for Britain to leave the European Union (EU), the election of Donald Trump as President of the US and the rise of Far Right and neo-Nazi parties across Europe, most notably, Marine Le Pen's Front National in France. At least as important has been a sense of despair on the part of millions of people regarding their own future and that of their children and grandchildren. The polarisation, of course, has not all gone in one direction: the past few years have also seen the emergence of new social movements and political parties of the Left, for example, around presidential candidate Bernie Sanders in the US and Labour Party leader Jeremy Corbyn in Britain, as well as around Podemos in Spain and Syriza in Greece (though the extent to which the latter can still be described as a left-wing party is more questionable) (Watkins, 2016).
That economic instability both shapes and is shaped by wider political developments. Indeed, exploring the relationship between these two dimensions is at the heart of contemporary analyses of 'the new imperialism' (Harvey, 2004, 2005; Callinicos, 2009). At the time of writing, some of the more obvious features of the current situation include: war and counter-revolution in the Middle East; the biggest movement of refugees since the Second World War; and the emergence of a new Cold War, with 'hotspots' in areas like Ukraine.
Understanding the nature of these developments is important for all concerned with the future of social work and social care, for three reasons. First, while the response of national governments to the 'long depression' has varied to some degree from country to country and from continent to continent, the dominant policy response has been to shift the costs of bailing out the global banking system in 2008 onto the poorest sections of the working class (pensioners, the disabled, the low paid and the unemployed), in the now universally recognised term 'austerity'. These are also the sections of the population who are most likely to come into contact with social work services.
Second, as we shall argue later, while the roots of the current crisis lie in the inability of the neoliberal project of the last three decades to overcome deep-rooted structural problems within the global capitalist economy, the dominant political and ideological response to that crisis has been more of the same in the form of increased privatisation and public sector cuts, leading one writer to speak of 'the strange non-death of neoliberalism' (Crouch, 2011). It is also clear that for some governments, the crisis has provided them with a golden political opportunity to shrink the welfare state to levels not seen for decades, not least in Britain, where the percentage of spending on welfare is now at its lowest level since the 1930s. Again, this poses a real existential threat to social work – a profession that is, in the Western world at least, a child of the welfare state – a threat that needs to be understood and challenged.
Third, the unprecedented assault on the lives and living standards of disabled people and people on benefits throws up major questions of how to respond, particularly for social workers and social work organisations in countries such as Greece and Spain, where the cuts have gone deepest. Should the main emphasis be on defending the post-war welfare state or do the new forms of social support that have been thrown up in the course of the crisis, such as the Solidarity Committees in Greece, prefigure more popular forms of social work that should be nurtured and developed? Moreover, how should social workers respond to the growth of new forms of racism such as Islamophobia and anti-Roma racism?
These are some of the key questions that we will seek to address in this book. The focus of this chapter, however, will be a narrower one. One factor that has facilitated the imposition of austerity policies across Europe – and, in the process, undermined potential resistance to these policies – has been the widespread acceptance of the neoliberal argument that 'there is no alternative' to austerity. Acceptance of that view, albeit reluctant, has led a majority of the population in many countries to tolerate as inevitable cuts to services, cuts to benefits and, more generally, levels of hardship (eg including the return of food banks on a mass scale) that would have been unthinkable prior to the crisis. It is a view that we unequivocally reject. Providing a basis for that rejection, however, requires us to do what social work texts rarely do and attempt to engage with some of the economic arguments that have been put forward to explain the crisis and to suggest how it might be resolved. Unless social work academics and practitioners seek to critically engage with these dominant analyses, there is a danger that they will also be persuaded to accept – and collude with – policies that are profoundly harmful to some of the most disadvantaged sections of society, including users of social work services, but that, in reality, are neither theoretically nor morally justifiable. The remainder of this chapter will therefore address what we will call the political economy of social work.
### Explaining the crisis
In October 2013, a group of undergraduate students studying economics at the University of Manchester set up a Post-Crisis Economics Society (see: <http://www.post-crasheconomics.com>). The students' main demand was for a complete overhaul of the curriculum to include economic theories other than the neoclassical theories that made up the content of their teaching at Manchester (as at many other British universities). These theories, which have dominated mainstream economics for more than a century, assume that the capitalist system will always return to equilibrium – that goods will always find buyers – providing prices and wages are allowed to adjust without hindrance.
Not only, the students argued, had these theories failed to predict the great crash of 2008 (and had, in fact, even denied the very possibility of such a crash), but leading economics academics had played a key role in acting as cheerleaders for the policies and practices that had led to that crash. According to a spokesperson for the group, academic departments were 'ignoring the crisis' and by neglecting global developments and critics of the free market such as Keynes and Marx, the study of economics was 'in danger of losing its broader relevance'. Within a matter of months, similar groups had been formed in Cambridge and London, and by mid-2014, 41 groups had been set up in 19 different countries and had issued a common manifesto.
The post-crisis economics manifesto
Our critics have attempted to caricature our society as demanding "more Keynes and Marx". However, our argument is far broader: we are calling for an evidence based, pluralistic economics education.... The key point is that any reference to Marx is compartmentalised from the economic theory proper and his contribution is judged to be historical and now superseded. We argue that it would be far more valuable to use Marx's theories of crisis, exploitation, class struggle and the reserve army of unemployed as a lens through which to understand business cycles, income distribution and the labour market.... Teaching of Keynes in mainstream economics is not really Keynes.... Students at Manchester are not exposed to Keynes' theories first hand and are definitely not exposed to modern post-Keynesianism, which has developed and built on Keynes' framework substantially. Some would argue that poring over old texts is not the proper way to do a social science, and we agree to an extent. We only wish that particular thinkers' theories be taught insofar as they are relevant, and we think that these theories should be presented in their historical context where possible. It may be wise to teach thinkers from original texts if one wants to access their ideas rather than relying on watered-down impressions. (Post-Crash Economics Society, 2014)
The students were not alone in their critique of the neoclassical orthodoxy that had played such a major role in shaping the events of 2008. In a study of the financial crisis published in 2014, Martin Wolf, Chief Economics Commentator at the Financial Times and one of the UK's most respected economic spokespersons, acknowledged his own failure to predict the crash. This failing, he argued, was not because he was unaware of tensions developing within the global economy, but rather, he suggested, because:
I lacked the imagination to anticipate a meltdown of the Western financial system. I was guilty of working with a mental model of the economy that did not allow for the possibility of another Great Depression or even a 'Great Recession' in the world's most advanced economies. (Wolf, 2014: xvi)
Even more remarkable was the admission before a Congressional Committee by Alan Greenspan, chair of the US Federal Reserve for two decades and arguably therefore the most powerful player in the global economic establishment, that the slump had left him in 'a state of shocked disbelief':
He was questioned: 'In other words, you found that your view of the world, your ideology was not right, it was not working' (House Oversight Committee Chair, Henry Waxman). 'Absolutely, precisely, you know that's precisely the reason I was shocked, because I have been going for 40 years or more with very considerable evidence that it was working exceptionally well.' (Alan Greenspan, former chair of the Federal Reserve being questioned in the US Senate, cited in Roberts, 2012)
Despite this collapse of confidence and the inability of leading economists to explain the crisis, it is a remarkable fact that by 2016, eight years after the crisis began, the very same market fundamentalist policies that resulted in the 2008 slump continued to be vigorously promoted and enforced by bodies such as the European Central Bank and the International Monetary Fund, a fact that requires some explanation. In reality, it is the product of two factors, one theoretical and the other political.
The first reason for the continuing hegemony of neoliberal economic policies is quite simply that neither those in charge of national and global policy nor their academic advisers can envisage a theoretical alternative. When, for example, 60 leading US economists were asked in a 2012 survey 'If we are ever going to get out of this slump, what will it take?', the general view was 'don't know' (see <https://thenextrecession.wordpress.com/2012/10/17/the-dilemma-of-the-mainstream/>). Perhaps not surprisingly, then, despite the fact that austerity policies have manifestly failed in countries such as Greece (by 2015, Greek debt was higher and the economy smaller than when the policies were first implemented), not a single one of the other 27 EU members was prepared to support the Syriza government against the 'Troika' of the European Commission, the European Central Bank and the International Monetary Fund in its bid to end these policies in February 2015.
Second, the level of political resistance across Europe to the implementation of austerity measures has been uneven at best. In Southern countries, especially Greece and Spain, popular resistance has been at a relatively high level, whether through social movements, such as the Indignados movement and the Orange Tide social work organisation in Spain, or through the organised trade union movement in Greece, where 32 general strikes between 2010 and 2013 were an important contributory factor to the election of the left-wing Syriza party in January 2015. In Northern Europe, by contrast, resistance to austerity and to neoliberal policies has been more muted. This is not to say that there has been no resistance. Britain, for example, saw a 2-million strong strike in defence of retirement pensions on 30 November 2011. That struggle, however, like many others, ended abruptly in a shoddy climbdown by the official trade union leadership, a climbdown that British rank-and-file trade unionists lacked the confidence and the independent organisation to challenge.
It would be misleading, however, to suggest that there has been no questioning or criticism of the dominant orthodoxy. The depth of the crisis has prompted even some mainstream commentators and economists to revisit macroeconomic theories that look beyond the behaviours of individual 'rational consumers' and seek to make sense of the system as a whole, including the possibility that crises are an integral part of the workings of a capitalist economy. Martin Wolf, for example, has queried the neoclassical view that free-market economies always tend to equilibrium and that crises can only occur as a consequence of some external (or 'exogenous') shock:
Depressions are indeed one of the states that a capitalist economy can fall into. An economic theory that does not incorporate that possibility is as relevant as a theory of biology that excludes the risk of extinctions, a theory of the body that excludes the risk of heart attacks, or a theory of bridge-building that excludes the risk of collapse. (Wolf, 2014: xvii)
Such criticisms of neoclassical economics tend to be informed by the theories of one or other of the two great economic thinkers referred to earlier: Keynes and Marx. It is to a consideration of these theories that we shall now turn.
### Marx and Keynes: saving capitalism or ending capitalism?
The idea that crises are a feature of capitalist economies was one shared by the two giants of economic thought referred to by the Manchester students, namely, Marx and Keynes. While, however, Keynes believed – and earnestly desired – that such crises could be prevented or their effects ameliorated through government intervention, Marx, by contrast, argued that capitalism contained a long-term inbuilt tendency to crisis (Choonara, 2009; Fine and Saad-Filho, 2010). That tendency could be delayed by what he referred to as 'countervailing factors' (eg the exploitation of colonies) but would ultimately reassert itself. Given that Keynes was primarily concerned with saving capitalism while Marx devoted his life to its overthrow, it is perhaps not surprising that Keynes's ideas have tended to be more palatable not only to mainstream critics such as Wolf, but also to more radical critics of neoclassical economics, such as Joseph Stiglitz and Paul Krugman (Stiglitz, 2003, 2013; Krugman, 2008).
For Keynes, the principal cause of crisis is insufficient demand in the economy. The correct response to the onset of crisis, therefore, is for governments to replace that missing demand, either by increasing their own spending (eg through programmes of public works) or by cutting taxes so that consumers have more money to spend, which, in turn, will generate more demand, more jobs and so on. The fact that most governments responded to the crisis of 2008 by spending billions to bail out major banks and financial institutions led some to believe that what we were seeing was a return to the Keynesian policies of the post-war period. In reality, such government responses were prompted less by ideology than by the fact that the banks were simply (in a phrase much used at the time) 'too big to fail'. Indeed, as the experience of the merchant banker Lehman Brothers in 2008 showed, where one of these giants of finance capital did collapse, it deepened the crisis by pulling many others down with it and making things much worse (Harman, 2009).
US economist Nouriel Roubini's (2008) characterisation of the bailout, therefore, as 'socialism for the rich' was an accurate one, for the other side of the eye-watering sums given to the banks was austerity, privatisation and cuts in services for the mass of the population. In Chris Harman's (2009: 303) words: 'The new Keynesianism for capital was combined with a continuation of neoliberalism for those who worked for it'.
The mainstream view within economics continues to be that a combination of low interest rates and a dose of money printing ('quantitative easing') should be sufficient to get economies moving again. In reality, as Michael Roberts has argued, this has 'proved to be a chimera': 'Most of this extra credit or money has ended up in the stock and bond markets and in the cash reserves of the banks; very little has found its way to the so-called "real economy"' (Roberts, 2015).
The fact that neither zero interest rates nor quantitative easing have been sufficient to kick-start sluggish economies has led some Keynesian critics to argue that what is delaying the recovery is 'fiscal austerity' – spending cuts and deficit reduction – since its effect is to reduce demand in the economy. The solution, they argue, lies instead in a much greater fiscal stimulus – in other words, vastly increased government spending, with the deficit being addressed at some point in the future once the economy is healthy again. Leading British Keynesian theorist Simon Wrenn-Lewis, for example, has argued that the austerity policies of the UK Coalition government 'seriously blunted the recovery', taking at least 1% off growth in both 2010/11 and 2012. Arguing that both the Coalition government and the media have been profoundly mistaken in seeing deficit reduction as a priority, he argues that 'The key point is that deficit reduction should be left to a time when interest rates are high, so that they can be reduced to counteract the negative impact of austerity on demand' (Wrenn-Lewis, 2015).
Given the devastating effect that austerity policies are having on the lives of millions of people across the globe, the appeal of an argument which says that such policies are not only painful, but actually counterproductive, is not hard to see. In their opposition both to the politics of austerity and to rising inequality, adherents of both Marx and Keynes will often find themselves on the same side. Where they differ, however, concerns the extent to which policy reforms that remain within the framework of capitalism can overcome the deep-rooted problems facing the system.
First, Keynesianism is an 'underconsumptionist' theory in that it assumes that capitalism can escape crisis if the state intervenes to raise consumption the moment a recession seems likely to develop (Harman, 2009: 59). As the experience of the current crisis appears to show, however, no matter how much money governments throw at them, capitalists will only invest if they are guaranteed an acceptable rate of return on their profits. If not, then they will either simply save the money or alternatively 'invest' it in areas where they can make a quick return on their capital (such as the financial markets or even sub-prime mortgage markets, as they did prior to 2008) as opposed to investing it in the real economy. Such short-term investment, Michael Roberts (2016) argues, will not be sufficient to generate the kind of recovery or growth rates that were seen, for example, in the post-war period. The result is that even seven years after the beginning of the crisis, financial commentators in countries such as the UK were noting that the recovery, such as it was, was based not on investment or on increased exports, but rather on an upturn in consumer spending – hardly a sustainable basis.
Against the underconsumptionist arguments of the Keynesians, Roberts argues that what we are living through is a long-term crisis of profitability. Within the framework of capitalism, this will only be resolved either through the 'creative destruction' of large chunks of capital (eg through economic collapse or war) allowing the more profitable firms to expand, or by freeing up new funds for investment by forcing down the living standards of the mass of the people through cuts to wages, benefits and services (including the privatisation of public services). It is this latter approach that has underpinned the neoliberal project of the past 30 years and that also provides the rationale for current policies of austerity.
Jospeh Choonara makes a similar point:
Four decades of neoliberal attacks, including almost a decade of austerity, have failed to reverse this decline of profitability. The Keynesian solutions that are offered by most reformist politicians are not capable of transforming this situation. Versions of Keynesianism have been attempted both in the past and in the current crisis – for instance by the Abe government in Japan – to no avail. In order to stimulate the economy, money has to either come from capitalists, further undermining profitability, or from workers, in which case the measures are hardly progressive. Or the money can be borrowed, which tends to lead to greater financial instability. (Choonara, 2016)
These arguments also apply to what some have characterised as the latest, and perhaps least anticipated, return of Keynesianism. In their apparent rejection of free-market globalisation and embrace of national investment and protectionist policies, the influence of Keynes has been detected in the policies of Donald Trump, elected US president in 2016. One of Keynes' biographers, for example, Robert Skidelsky (2016), has praised 'Trumpism's positive potential' and argued that he is proposing 'a modern form of Keynesian fiscal policy'. Even putting to one side the odious racist and nationalist themes that underpinned Trump's election campaign, given current levels of global economic integration in the 21st century, it is difficult to see how Trump could pursue such protectionist policies without damaging the world economy even more.
The alternative, these Marxist economists argue, is not to bail out the system in the way in which the Keynesians suggest, but rather to replace it with a very different kind of economic system driven not by profit, but by the overarching requirement of meeting human need. Until then, however, all that the current rulers of the world seem able to offer their peoples is what the former British Prime Minister Cameron at a Lord Mayor's State Banquet in London in 2013 described as 'permanent austerity' (Cameron, 2013). We will conclude this chapter by considering what such austerity has meant in two of the countries most affected by it: Greece and Britain.
### The reality of austerity: Greece and Britain
It is difficult to compute the economic costs of the crisis of 2008. In Britain alone, for example, one 2011 estimate suggests that the costs to the Treasury of bailing out the banks was £123.93 billion, though, at several points, Treasury exposure (i.e. the amount guaranteed to be paid out) was 10 times that figure (Curtis, 2011). However, according to one leading economics academic and former Treasury adviser:
The costs of the crisis are not simply the costs to the Treasury, important though these may be.... I have estimated taking into account the losses of output that would have occurred since 2008 without the financial crisis caused by the reckless lending of the banks, and projecting the losses forward until the end of 2012, that the total cost to the economy – all of us – is around 11 to 13% of GDP [gross domestic product]. (Desmond, cited in Curtis, 2011)
While the precise figures may be a matter of dispute, what is not in doubt is that in Britain and elsewhere, the costs of a crisis resulting from what Desmond calls the 'reckless lending of the banks' have been borne not by the banks or financial institutions (in Britain, unlike Iceland, no bankers have been jailed). Rather, they have been borne by the poorest sections of society, including low-paid workers, the unemployed, the young and the disabled. The narrative that has justified this assault on the welfare state and the living standards of the poor has gone under the name of 'austerity'.
'Austerity' is the short-hand term for the economic, ideological and political strategy that has dominated Europe for the best part of a decade since 2008. Its attraction for governments is that it appears to provide a clear and simple explanation for the current crisis (excessive government spending, especially on welfare) and a solution to that crisis (cutting wages, reducing public spending and raising taxes). In almost every case, that solution has also involved 'structural reform', meaning greater market flexibility, pension reductions, the privatisation of public enterprises and so on.
The Keynesian assessment of the success or otherwise of austerity policy has been discussed earlier and is neatly summarised by Martin Wolf:
Austerity has failed. It has failed in the UK and it has failed in the Eurozone. Its failure was predictable and, by some at least, predicted. It turned a nascent recovery into stagnation. That imposes huge and unnecessary costs, not just in the short run, but in the long-term as well: the costs of investments unmade, of businesses not started and of hopes destroyed. (Wolf, 2013)
As a description of the devastating effect that austerity policies have had on the lives of millions across Europe, 'hopes destroyed' is perhaps something of an understatement.
#### Greece
Living in one of the weakest economies in the Eurozone, the Greek people have been made to pay a very high price for the bailout of Greek banks by the 'Troika' of the European Central Bank, the European Commission and the International Monetary Fund, not least because most of the money loaned goes straight back into the coffers of banks in Germany and other major European countries to repay earlier loans. Just what the bailout has meant for public health services was spelled out in a paper in the medical journal The Lancet in 2014 (Kentikelenis et al, 2014).
A key condition of the Troika bailout was that health expenditure in Greece be capped at 6% – lower than that of any other pre-2004 EU member country. Seeking to surpass even that target, in 2012, the government made huge cuts in hospital operating costs and pharmaceutical spending. As the former Minister of Health Andreas Loverdos admitted, 'the Greek public administration... uses butcher's knives [to achieve the cuts]'.
Predictably, the results have been devastating. One of the most affected areas has been prevention and treatment programmes for illicit drug users, at a time of rising levels of addiction. In 2009/10, the first year of the austerity programme, a third of all street work programmes were closed while the number of syringes and condoms distributed fell by 10% and 24%, respectively. Not surprisingly, there has been a huge increase in the number of new HIV infections – from 15 in 2009 to 484 in 2012 – and preliminary data also suggest a doubling of tuberculosis among this population (all figures from Kentikelenis et al, 2014).
Reductions to local budgets have led to cuts in public health practices such as mosquito-spraying, resulting in the re-emergence of locally transmitted malaria for the first time in 40 years. In addition, cuts to hospital budgets of 26% have resulted in greatly increased workloads for staff and longer waiting times. Cuts to pharmaceutical expenditure have meant that some drugs are now unobtainable because pharmacies have built up unsustainable debts and can no longer afford to stock them. A study cited by Kentikelenis and his colleagues found that in one province (Achaia), 70% of respondents said that they had insufficient income to purchase the drugs prescribed by their doctors. Pharmaceutical companies have reduced supplies because of unpaid bills and low profits.
Despite a rhetoric of 'maintaining universal access and improving the quality of care delivery', in reality, the introduction of charges (or the increase in existing charges) for out-patient visits and for prescriptions have created barriers to access. In addition, the difficulties of securing health care if you are not employed mean that an estimated 800,000 people now have no access to health care and depend on the voluntary clinics that have sprung up across Greece.
Mental health services have also been seriously affected. According to health economist David Stuckler, who has studied the impact of austerity policies on suicide rates across the globe, in terms of 'economic' suicides, 'Greece has gone from one extreme to the other. It used to have one of Europe's lowest suicide rates; it has seen a more than 60% rise' (Henley 2013). In general, each suicide corresponds to around 10 suicide attempts and – it varies from country to country – between 100 and 1,000 new cases of depression. In Greece, says Stuckler, 'that's reflected in surveys that show a doubling in cases of depression; in psychiatry services saying they're overwhelmed; in charity helplines reporting huge increases in calls' (Henley, 2013).
Ioakimidis and Teloni have explored the impact of these austerity policies on social services in what, they argue, was the only EU country where comprehensive social assistance, a safety net of last resort, did not exist. Even before the crisis, they suggest, 'Greek workers in the welfare sector and social workers were over-worked and underpaid' (Ioakimidis and Teloni, 2013). Nevertheless, in the years before 2009, EU funding allowed for the development of 'Home Care', with hundreds of projects at the local and regional levels, employing around 4,000 staff and supporting more than 100,000 service users (albeit at the cost of marginalising state-provided welfare). The underpinning philosophy of Home Care was to promote independence for service users, who included elderly and disabled people, by providing care and support directly in their own homes. As Ioakimidis and Teloni (2013: 40) note: 'In a country where adult care never really existed, it is not a surprise that the project immediately became very popular'. The crisis, however, put an end to all that. Home Care depended on EU funding and when that dried up, the then Conservative government discontinued the scheme. It was reintroduced by the socialist (PASOK) government in 2010 but in an extremely reduced form, based on means testing with strict eligibility criteria. The result, according to Ioakimidis and Teloni (2013: 42), was that by 2013:
[T]he number of Home Care projects has been significantly reduced, continuation of the scheme remains uncertain, and employees have not been paid for several months.... More importantly, more than half of the service users are now considered ineligible, leaving some of the most vulnerable groups without any support whatsoever. The project only targets pensioners, excludes disabled people and older people without pension support. Most of these people struggle to survive, receiving food from charity soup kitchens.
In January 2015, the left-wing Syriza party was elected to government with a commitment to ending austerity and renegotiating the bailout with the European banking institutions. However, after six months of what one leading player described as 'mental water-boarding' by the Troika of the European Bank, the European Commission and the International Monetary Fund, and despite massive popular support (reflected, above all, in a huge vote against the latest bailout terms in a popular referendum), the Syriza government caved in to the Troika's demands in July 2015 and implemented the required cuts and privatisations (Ovenden, 2015). One illustration of what the climbdown meant in practice was provided in October 2016, when Greek pensioners, protesting against the latest cut to their already meagre pensions, were tear-gassed in front of the Greek Parliament by riot police acting on the authority of the Syriza government.
#### Britain
If Greece entered the crisis with the least developed welfare state in Europe, then Britain, by contrast, would be seen by many as having one of the longest-established welfare states in the Western world and one of the most developed in respect of universal provision. It is a perception that requires some qualification. First, while for most of the post-war period, health-care provision in the UK in the form of the National Health Service (NHS) has been genuinely universal (in the sense that that treatment is free at the point of need), this has never been true of social care and social work, which have always been based on a mixture of means testing and discretion (Beresford, 2016). Second, the period since the early 1990s has seen the erosion of universal provision through the increasing marketisation and privatisation of the health service, particularly in England, by both Conservative and New Labour governments. Thus, by 2014, between a quarter and a half of all community services in England were run by Richard Branson's Virgin Care (see: www.virgincare.co.uk; see also White, 2016).
The imposition of austerity policies, especially since the election of Conservative–Liberal Democrat Coalition government in 2010, has further eroded that welfare state in three main ways. First, in place of what was once called 'social security', unprecedented cuts across all areas of benefits (especially disability benefits), the introduction of sanctions regimes that have contributed to up to 1 million people being forced to use food banks, and a spare room or 'bedroom' tax affecting around 600,000 people will increase the number of children living in poverty by 200,000, undermine families and force tens of thousands of people to uproot and move to other areas of the country or face eviction (The Independent, 2016). All of this has been underpinned by a brutal ideological offensive against people on benefits that has contributed to a significant increase in levels of disability hate crime, increased rates of depression and anxiety, and an increase in the rate of suicide among those on benefit.
Second, in social care, a combination of cuts of around 30% to local authority budgets since 2010, increasingly restrictive eligibility criteria for services and inadequate personal budgets have left millions without the support they need and increasingly dependent on the family, in particular, women family members.
Third, the Health and Social Care Act 2012 in England and Wales will remove the duty on the Secretary of State for Health to provide a comprehensive health service, while the requirement in the Act that up to 49% of services can be tendered out to 'any qualified provider' will rapidly lead to the privatisation of the NHS in England and Wales.
'Austerity' in the sixth richest country in the world
The young couple trudge into the church hall looking gaunt, meek and beaten. They have walked for three miles in heavy rain, their thin clothing so drenched it sticks to their skin as they huddle by a radiator for warmth. Across the room, a 34-year-old mother weeps into her hands as her shopping trolley is filled with food which will stop her two children going to bed that night with hunger pains. Over by the racks of second-hand jackets, shirts and jumpers for sale for 20p each, sit two jobless men without a penny in their pockets, pondering how the three-day emergency rations they've just been handed can last a week-and-a half.
Behind them a smartly dressed married couple who've never been on benefits before suffer the indignity of explaining how they have lost their jobs and home and cannot feed their three kids. (Excerpt from a 2013 report on food charities in the UK, cited in O'Hara, 2014)
### Conclusions
The present global political and economic situation – what the writer Tariq Ali has dubbed the 'new world disorder' – presents the social work profession with great challenges. The ways in which social workers across the globe are responding to these challenges will be the subject of later chapters. Before then, however, it is necessary to continue the task that we have begun in this chapter of mapping the ways in which social work has been transformed during the neoliberal era and the implications of that transformation for whatever emancipatory potential it retains. That will involve looking, first, at the changing relationship between social work and the state and then, in Chapter Three, at the impact of privatisation on social work and social care.
## TWO
## Neoliberalism, social work and the state: retreat or restructuring?
### Introduction
In this chapter, we will explore the relationship between social work, neoliberalism and the state. Such a focus is unusual in a social work text. Theoretical discussion of the state seldom figures in contemporary social work literature. There are a number of possible reasons for this. One may be the assumption that what is sometimes referred to as the 'retreat of the state' from welfare over the past three decades, as well as the development of globalisation, has rendered nation states less important than was previously the case. Another reason may be the influence within the critical social work literature of Foucauldian or post-structuralist perspectives, which see power as omnipresent, 'saturating' all relationships, with the state simply one source of power among many others. Probably the main reason for the neglect of the state within the mainstream literature, however, is the continuing influence of a view of social work as essentially a non-political project, an ethical or professional response to human need in particular societies in which the state provides, at most, a context in which this activity takes place.
All of these views will be challenged in the course of this chapter. Before then, however, it is necessary to make some preliminary observations. First, it is not the case that all currents within social work have ignored the state–social work relationship. The Marxist-influenced radical social work literature of the 1970s, while acknowledging the gains provided by post-war welfare states, nevertheless questioned the then dominant social-democratic view of such states as essentially benign, highlighting instead the repressive features of the welfare state and seeking to address in theory and in practice the challenges and contradictions of working 'in and against the state' (Bailey and Brake, 1975; LEWRG, 1979; Simpkin, 1983). Similarly, the movement for the 'reconceptualisation' of social work in Latin America from the 1960s onwards challenged the 'non-political' view of social work and state promoted by the dominant US-linked clinical perspectives:
The result was a qualitative leap, as the profession began to define not only new frameworks and purposes but also positioned social workers side by side with the masses, whether they were called the 'exploited', the 'marginalised' or the 'excluded'. (Rangel, 2005: 13)
Second, interest in the role of the state has re-emerged over the past decade in the wake of the return of capitalist crisis. As Bob Jessop (2016: 210) has observed in his magisterial study of the state, the fact that, in 2008, the global banking system was only saved from a financial catastrophe of its own making through massive intervention by supposedly irrelevant national states proved, if proof were needed, that 'the national state generally remains the addressee of last resort in appeals to resolve economic, political and social problems'. Since then, as he notes, 'The North Atlantic and Eurozone financial and economic crises, the state's role in crisis management, and serious fiscal and sovereign debt crises have revived interest in the limits of state power and in the challenges of global governance' (Jessop, 2016: 5).
Third, when imperialism is back with a vengeance on the global agenda and the world seems a more dangerous place than it has been for more than half a century, it is a strange time to talk of the 'retreat of the state'. On the contrary, in recent years, states across the globe have been demonstrating their seemingly unlimited capacity for brutal repression and mass destruction all the way from Guantanamo Bay to the Ukraine – above all, in wars throughout the Middle East that have led to the biggest movement of refugees since the Second World War (a movement that has, in turn, been repeatedly blocked by the actions of local states and European Union [EU] state institutions).
Nor, in fact, are such displays of state power inconsistent with the operations of neoliberalism. Guardian journalist Jack Shenker's observations concerning the relationship between neoliberalism and the state in Mubarak's Egypt in his superb study of the Arab revolutions and counter-revolutions have much wider application:
One of neoliberalism's great strengths is to appear as if it is the irresistible outcome of decentralised, common sense management decisions, the origins of which are so diffuse that they are impossible to identify accurately or resist. In reality the opposite is often true. (Shenker, 2016: 62)
As he goes on to argue:
Neoliberalism is a political project, and its implementation always involves a mass transfer of resources from the poor to the rich.... Without the state intervening to open up new markets and repress dissent among the citizenry, liberalization policies could never have been pushed through. (Shenker, 2016: 68–9)
As in Egypt, states, whether in the UK, Greece, South Africa or elsewhere, have played a key role in the reshaping of welfare to make it more amenable to the wider project of neoliberal capitalism. The nature of that reshaping, and, in particular, what has been involved in the transition from the welfare state to the neoliberal welfare state, will be addressed in the first part of this chapter. The next part of the chapter will look more specifically at the ways in which social work, both within the state and in the third sector, has been transformed in recent decades and the role that the state has played in this process. It is worth noting that very few of the major changes that have taken place in recent decades, such as the shift towards care management approaches in the 1990s in Britain, Japan and elsewhere, came from within the profession; for the most part, they were imposed by governments in the face of strong opposition from social work professionals. One factor undermining resistance to such top-down changes was an unwillingness by professional organisations to address the politics behind such 'reforms'.
Finally, and more positively, some of the most promising developments in global social work practice in recent years have emerged from within social movements as a response to the imposition of neoliberal or austerity policies, or in the face of the crises of war or environmental catastrophe. Elsewhere, we have described these as forms of 'popular social work', which contain a potential within them to go well beyond traditional social work responses to crises (see, eg, Lavalette and Ioakimidis, 2011a; Jones and Lavalette, 2013). These are exciting and significant developments. They do, however, raise the question of how we should view them and their relationship to state-provided social work services – as supplement, as alternative or as prefigurative examples of how social work could be in a differently organised society?
### Theorising neoliberalism
As the late Chris Harman (2007) argued, there has always been an ambiguity at the heart of the term 'neoliberalism':
Did it refer to a way of running the capitalist system that could be changed with a change in government policy, or did it refer to something intrinsic to the present phase of capitalism that only challenging the system as a whole could overcome?
If the former, then the problem was not capitalism per se, but rather a particular ideology or set of policies that should be replaced by an alternative set of policies. As an example of this view, Harman cited the French sociologist Pierre Bourdieu: 'The main issue is neoliberalism and the retreat of the state. In France neoliberal philosophy has become embedded in all the social practices and policies of the state' (Bourdieu, cited in Harman, 2007). This, Bourdieu continued, 'was the effect of a shared belief... which has created a climate favourable to the withdrawal of the state and submission to the values of the economy' (Bourdieu, cited in Harman, 2007).
To a greater or lesser extent, Bourdieu's view was shared by most of the leading figures in the anti-capitalist or global justice movement that emerged out of the protests against the World Trade Organization in Seattle in 1999. Issues such as globalisation, rising inequality or privatisation – some of the key components of neoliberalism – were seen by them as being 'the problem', rather than the logic of capitalism as a global system. As Harman noted, the logic of their position was to argue for a change of policy or direction at the top of society, often involving a return to the policies of the post-war years when levels of inequality were much lower and the welfare state was more extensive.
Such a view is open to two objections. First, it risks viewing 'welfare states' in the post-war period of 'the long boom' as essentially different from other state forms, neutral if not always benevolent, rather than being a response to a particular (and highly exceptional) phase of capitalist development. Even the award-winning British director Ken Loach's excellent documentary about the 1945–51 Labour government, The Spirit of '45, tends to present a largely uncritical portrayal of a government that, alongside its considerable achievements, sent in troops on 18 different occasions to break strikes, left the former bosses in charge of the newly nationalised industries and reimposed dental and prescription charges in 1951 (Miliband, 2009). As Marxist critics of the Attlee government such as John Saville and Ralph Miliband argued, whatever its achievements, even that government operated very clearly within the framework of capitalism and had no hesitation in putting the needs of capital before those of the working class.
A second objection to the arguments of those whose response to neoliberalism is to argue for a return to the policies of the post-war period is that these policies were the product of a very particular – and highly unusual – period of capitalism's development. As Michael Kidron (1968) argued in Western capitalism since the War, in the context of the Cold War, that boom was sustained by exceptionally high levels of arms spending that could not be maintained without undermining economic growth, both East and West. The neoliberal policies adopted by governments from the 1970s onwards were, above all, a response to the crisis of profitability exposed by the breakdown of the global economy in 1973. As Harman (2009: 240) argues, that response involved:
Imposing 'flexible labour markets' so as to get longer working hours and more intensive production (in Marx's terms absolute and relative surplus value) and to try to cut back on welfare expenditure. This was the rationale behind 'neoliberal' policies with counter reforms of welfare and the use of marketization and privatization measures to get workers competing with each other.
Or, as John Harris (2014: 8) has argued, 'neoliberalism's priority is the creation of conditions attractive to capitalist profitability in the global market'.
As the earlier quote from Bourdieu shows, it is common to describe the changes that have taken place in global welfare since the late 1980s in terms of 'the retreat of the state'. At one level, this is an accurate description of a process that has involved the state either contracting out welfare services to private providers or non-governmental organisations (NGOs), or simply withdrawing from areas of provision and leaving individuals, families and communities to get by as best they can. In another sense, however, the concept of 'retreat' is not helpful since it suggests that the state has become less important or is playing a less active role in welfare. In fact, as Harman (2007) argued:
The reality is that capital uses the state as much today as it did in the heyday of 'Keynesianism' – indeed, even more so, insofar as it is faced with more crises needing intervention. Neoliberalism as an ideology does not guide practice when it comes to this matter. The difference with the post-war decades is that capital is anxious to cut back on many of the positive reforms it granted in a more profitable era, and states respond accordingly.
None of this means that modern states can do without welfare. The three-part rationale for the British post-war welfare state developed by the Marxist historian John Saville in the mid-1950s still provides a useful framework for making sense of welfare regimes today. That welfare state, Saville argued, was best understood as a product of the interaction of three quite different factors:
The struggle of the working class against their exploitation; the requirements of industrial capital (a convenient abstraction) for a more efficient environment in which to operate and in particular the need for a highly productive labour force; and recognition of the property owners of the price that had to be paid for their political security. (Saville, 1957: 5–6)
All three of these factors continue to apply more than half a century later. So, for example, as we have argued elsewhere, the huge expansion of social work education in China by the ruling Communist Party in the 1990s and 2000s, with its strong emphasis on social work's role in promoting social stability, is best understood as a response to concerns within the Chinese ruling class over growing political and industrial unrest, particularly following the brutal crushing of the student protests in Tiananmen Square in 1989 (Ferguson, 2012).
Generally, in terms of welfare, what we are seeing is the state increasingly abandoning its role as 'provider' of welfare services but asserting its role as regulator and director of services. The state's role remains paramount but the direct provision of services is increasingly outsourced to a range of private and voluntary sector organisations, with 'welfare gaps' picked up by the informal sector (often service user families). In other words, we are witnessing a significant shift in the mixed economy of welfare and this is impacting upon social work services. In the past, several social work academics described social work services provided by the state as 'state social work' (eg Jones, 1983). As the state moves away from the direct provision of services, what we are seeing is the growth of 'state-directed social work', tightly regulated and controlled by state agencies but provided by a range of service provider organisations, including 'for-profits'.
The impact of the neoliberal turn on the organisation and provision of welfare services can usefully be periodised in two phases. In the first phase (the 1980s to approximately 2008), McDonald (2006: 37) suggests that the primary commonalities across states 'revolve around the linkage between employment policy and engagement with associated labour market programmes, the promotion of individual responsibility and increasingly conditional access to social support'.
Among other things, that involved: a shift from welfare to workfare, with work seen as the route out of poverty; more individualised forms of social support, including 'cash for care' and personalisation schemes; and increased conditionality. An example of the latter from Latin America is what the Economist magazine has called 'the world's favourite new anti-poverty device': conditional cash transfer programmes (CCTs). First tried out in Brazil as the Bolsa Familia, CCTs, as their name suggests, supply monetary benefits as long as recipients can demonstrate that they have met certain conditions, such as children's school attendance. They have proved popular with governments not only in Latin America, but also across much of the Global South. According to Lavinas, in 1997, only three Latin American countries had launched such programmes; a decade later, the World Bank reported that 'virtually every country' in the region had one, and others outside it were adopting them 'at a prodigious rate' (Lavinas, 2013). By 2008, 30 countries had them, from India, Turkey and Nigeria, to Cambodia, the Philippines and Burkina Faso; even New York City had put one in place (Lavinas, 2013). As with the neoliberal policies implemented by the Mubarak government in Egypt, the 'encouragement' by international financial bodies (in this case, the World Bank) for governments to adopt such schemes has been a crucial factor in their widespread use. Yet, while they have undoubtedly relieved the poverty experienced by some of the most marginalised in society, they suffer from a number of serious limitations, not least the fact that, as Lavinas (2013) argues:
[T]hey remain ad hoc instruments, unconstrained by legal and institutionalized principles of rights. The distinction is crucial: instead of being one dimension of a wider, universal system of social protection, such programmes enforce a principle of selectivity, targeting the poor as a residual category while insisting they assume individualized responsibility for their fates – thus working to diminish social solidarity and cohesion. The schemes are also designed to extend commodification, on the one hand disbursing monetary rewards to the poor in exchange for their participation as consumers, while on the other offering governments an alibi for scaling back provision of public goods. They thus pave the way for a retrenchment of welfare rather than its expansion.
The second phase of neoliberal restructuring of welfare has been the period since the onset of global economic crisis in 2008 until the present time. As we saw in Chapter One, global institutions and national states have made use of that crisis to promote an ideology and politics of 'austerity' and to fundamentally reconfigure the relationship between the state and its citizens. Jessop suggests that austerity can be studied in three ways. First, there are conjunctural austerity policies, usually presented as short-term responses to periods of economic difficulty, with the implication that once these difficulties have been overcome, then 'business as usual' can be resumed. This was how austerity policies were initially presented from 2010 onwards across Britain and much of the Eurozone. Gradually, however, that discourse has been replaced by a second one of enduring or permanent austerity. Such a politics, as Jessop (2015: 233) notes, 'Is intended to bring about a more lasting reorganisation of the balance of forces in favour of capital rather than to make policy adjustments to safeguard existing economic and political arrangements'. Thus, for example, the decision by the then British Chancellor of the Exchequer George Osborne in 2016 to reduce welfare spending as a proportion of gross domestic product (GDP) to levels not seen since the 1930s at the same time as reducing top rates of tax clearly reflects an ideological commitment to a smaller state rather than an economic requirement to balance the books.
Finally, there is what Jessop calls the austerity polity, which, he argues 'results from a continual and fundamental reorganization between the economic and the political in capitalist formations' (Jessop, 2015: 233). The experience of Greece in recent years at the hands of the Troika of the European Commission, the European Central Bank and the International Monetary Fund shows the extent to which parliamentary democracy can be dispensed with if it conflicts with or impedes the operations of wider neoliberal concerns. Former Greek Finance Minister Yannis Varoufakis, for example, has expressed his astonishment on hearing his German counterpart Wolfgang Schauble respond to the election of a Syriza government committed to challenging austerity by saying 'Elections cannot be allowed to change an economic programme of a member state!' (Varoufakis, 2016).
### Neoliberalism, the state and social work
What, then, has this reshaping of the economic and political polity meant for social work? Reference was made at the beginning of this chapter to the view that social work is a 'non-political' profession, an ethical response to human need. In reality, throughout its history, professional social work has been shaped by forces outside its own ranks, not least the state, arguably to a greater extent than almost any other profession. As John Harris (2008: 663) has argued in a review of five different 'moments' in the historical development of social work in Britain:
Rather than thinking about social work as a professional project that has orchestrated its own genesis and development, it is positioned at these different moments as an element in their respective welfare regimes, concerned with managing and regulating the sphere of the 'social'.
A number of internal and external factors, including a weak professional identity, a contested knowledge base and a lack of strong professional organisation, have meant that in the face of a ruling-class desire to regulate and control the behaviour of those sections of society perceived as deviant or dangerous, the organisation and content of the social work role has been shaped by the state to an even greater extent than professions such as medicine or teaching. As Jordan (1984: 114) has noted:
Social work has played various roles in contrasting systems of social provision, and has served a number of political ends. Its great virtue – that it is almost infinitely adaptable to circumstances – also makes it open to exploitation for any kind of policy objectives.
The state: friend or foe?
A proposed new law in Scotland would provide every child with a 'Named Person' to whom the child (or his or her family) could turn if he or she was experiencing difficulties. The Named Person would most likely be a health visitor, a school guidance teacher or sometimes the head teacher at the child's school. The new Act has been developed in consultation with organisations representing children in care and is seen by some of them as a way of offering additional protection and support to children. The Act has been fiercely opposed, however, by left-wing journalists, social work professionals and some children's organisations (as well as by some right-wing Christian groups) on the basis that it represents a massive extension of state power, the main effect of which will be to undermine families.
One result has been the 'horrible' histories of social work, which will be explored in Chapter Four. In our own times, states have intervened actively to reshape social work in the interests of neoliberal capitalism. As Harris (2014: 8) has argued, that reshaping has been based on three main propositions, which, he argues, have been played out in social work across the world:
• Markets are efficient and effective and should be introduced in as many and as wide a range of contexts as possible;
• Individuals should be responsible for themselves and run their own lives;
• Services in the public or voluntary sectors should be modeled on management knowledge and techniques drawn from the private business sector.
Here, some examples of what that reshaping has meant in specific countries will be considered.
#### China
Reference was made earlier to the expansion of social work education in China since the early 1990s. Several factors contributed to that expansion (Sigley, 2016) but the most important was arguably the government's need to promote 'social stability' in the wake of two key developments. The first of these, referred to earlier, was the crushing of the mass student movement in Tiananmen Square in 1989, and the second was the growing social divisions resulting from the social and political effects of the marketisation of the economy that began soon after that massacre. As in the West, it is helpful to identify two phases of neoliberal development. The period following the decision by the Chinese Communist Party in 1992 to inaugurate a 'socialist market economy' led to the wholesale privatisation of state-owned enterprises. The consequences of that shift were devastating for ordinary workers and peasants. According to the activist network China Labour Net:
More than 60 million workers in the state and collective sectors were sacked, a scale never seen in history.... At the same time, the flourishing private sector led 120 million poor rural residents to leave the land and roam the country as migrant workers in search of employment. The overwhelming number of them ended up working in private enterprises with wages so low that workers are barely able to sustain themselves, and with little social security. (Yu and Ruixe, 2010: 27–28)
Even these poorly paid jobs and that limited social security, however, disappeared with the onset of the global economic crisis of 2008. According to the writer Hsiao-Hung Pai (2012: 11):
Despite state proclamations of a 'rising' China, the reality is that China has been struck hard by the global recession and is as bitterly divided as the rest of the world. The 'iron rice bowl' no longer exists. More than 600,000 small and medium-sized firms closed in China in 2008, throwing millions out of work. The deepening slump has encouraged a reverse migration, back to the countryside.
Unsurprisingly, social upheaval on this scale has given rise to massive social problems – drugs, family breakdown, mental illness, homelessness and so on. No less important, it has also led to huge social unrest, including record numbers of workers taking strike action in recent years. The government's response has been to promote a 'harmonious society policy' within which social work has been accorded a central role. In an interesting review of the recent development of social work in China, Sigley (2016: 108) notes that:
[F]rom its inception the harmonious society campaign has emphasized 'social management' and recognized that NGOs and NPOs [Non-Governmental Organisations and Non-Profit Organisations] have a significant role to play in promoting social stability. The profession of social work, for example, was given an elevated position within the 'Harmonious Society' document, which described the development of a large cohort of professional social workers as 'an urgent task for socialist construction and the development of a harmonious society'.
As Sigley also notes, however, there is a potential tension between the long-standing practice and preference of party bureaucrats at both national and local levels to 'fix' social problems by political and administrative means, on the one hand, and the promotion of a profession that, even in its most conservative forms, encourages self-help and self-determination on the part of marginalised individuals and communities, on the other. Given the interest in more radical forms of social work in Hong Kong, Taiwan and mainland China expressed, for example, in well-attended conferences of the Progressive Social Welfare Association based in Hong Kong and the production of a manifesto for progressive social work (PSWN, 2011), it will be interesting to see how that tension plays out over the next few years.
#### South Africa
Social work in South Africa has its roots in apartheid ideology, born out of what was called the 'poor white problem' of the depression of the 1930s (Sewpaul, 2012). Thus, from its earliest beginnings until the first democratic elections in 1994, public sector social work functioned within apartheid structures and policies. (The 'architect of apartheid', Hendrick Verwoerd, was himself a social work academic.) As Sewpaul (2012) notes: 'Those working in non-governmental and faith-based organisations did not evade the horrendous arm of apartheid laws. A host of legislative and policy directives demanded that services be provided separately (and unequally) for the different race groups'.
The overthrow of apartheid provided the opportunity for the development of both a different kind of society and a different kind of welfare. After the democratic transition in 1994, South Africa declared itself to be a developmental state (Edigheji, 2006) and social welfare and social work moved towards the 'developmental approach' (Patel, 2005). The basis for this was the 'White Paper for social welfare' (RSA, 1997), which contains the principles, policies and programmes for developmental social welfare in South Africa. Therefore, both state departments and non-governmental welfare organisations (dependent on state subsidies for survival in most cases) shifted their focus towards the developmental approach, guided by various statutory imperatives such as the 'Financing policy', the 'Integrated service delivery model' and the 'White Paper for social welfare'. Social development also became the emphasis in social work education and curricula at all the various tertiary training institutions.
Progressive as this may sound, in fact, as Marjorie Mayo noted in a discussion of community development approaches in Bailey and Brake's (1975) seminal collection Radical social work, the concept of 'development' is inherently ambiguous:
[D]evelopment, progress, community and participation are all problematic terms – development and progress of what kind, for whose benefit in what type of community, composed graphically or in class terms, participating in what and with what degree of real power and influence? (Mayo, 1975: 130)
Answering these questions requires that we locate development approaches in their wider political and economic context. In post-apartheid South Africa from 1996 onwards, that context was explicitly one of neoliberalism:
The adoption of the neo-liberal Growth, Employment and Redistribution (GEAR) strategy in 1996... was a major shift from the Reconstruction and Development Programme, the major policy of the African National Congress at the time of the national democratic transition, aimed at ushering in liberation and a 'better life for all'.... The GEAR structural adjustment programme normalised corporate capitalist power in post-apartheid South Africa.... The popular developmental approach is thus framed within the context of GEAR and its neo-liberal ideologies. Such emphasis on the free-market, privatisation, personal responsibility, self-reliance and even development itself competes with a more politicised, radical and structural social work and does not augur well for social work with its professional commitment to social justice. (Ferguson and Smith, 2012: 978)
Thus, in a context of widespread poverty and income inequality that, according to economist Thomas Piketty (2014), is even higher than it was under apartheid, state-driven developmental social work increasingly means poor communities being expected to pull themselves up by their own bootstraps. In contrast to China, however, there is a long tradition in South Africa of powerful social movements, movements that played a key role in the overthrow of apartheid, as well as a highly organised working class. Radical forms of community work were often a feature of these movements. They, alongside new movements formed in the wake of the massacre of miners by the state at Marikana (under the command of a chief of police who was also a former social worker!) and around the issue of tuition fees, potentially provide a basis for more radical forms of social work practice and education than currently prevail (Alexander et al, 2012; Sewpaul, 2013; Motlalepule and Smith, 2017).
#### Britain
Britain was one of the first countries to engage in the neoliberal reshaping of social work and, arguably, it is there that the process has gone further than anywhere else. From the outset, it is a reshaping that has been driven from the top and one in which the state has played – and continues to play – a central role. The three key elements were put in place by the NHS and Community Care Act 1990 (Harris, 2003). These were, first, the deliberate creation of a market in social work and social care by re-designating local authorities as purchasers, not providers, of care, responsible for commissioning and purchasing services from private and voluntary sector organisations. Second, the social work role shifted from one in which social workers engaged directly with clients or service users through the use of a range of different methods (primarily casework, group work and community work) to one of assessment and care management in which social workers assessed clients' needs and then organised 'packages of care' to be provided by other agencies. Third, users of social work services were reframed not as clients, but as 'customers', who would shop around to find the most appropriate services for their needs from a social care market.
These processes, identified by Harris as marketisation, managerialisation and consumerisation, have transformed social work in the UK (as well as generating considerable resistance, to be considered later) (Harris, 2014). The British experience has been thoroughly documented elsewhere by different writers, so we will confine ourselves here to a few more general comments (Harris, 2003; Ferguson, 2008).
First, a dominant theme over the past decade has been the individualisation of services, primarily through the policy of individual budgets known as personalisation (or, in Scotland, self-directed support). While its origins lie in the disabled people's movement, the policy has largely been driven from the top but cleverly constructed around a discourse of 'choice and control' that has played to both worker and service user dissatisfactions with the bureaucratic, top-down and controlling nature of many state-provided welfare services in areas such a mental health and learning disability. At best, the experience of personalisation has been mixed (Beresford, 2014; Needham and Glasby, 2014; Pearson and Ridley, 2014). While some service users receiving the more generous direct payments have been positive about the policy, in many areas, personalisation has primarily been used as a means of reducing spending and undermining social forms of provision such as day centres, which, for all their limitations, have provided an important means of social support for people who are socially isolated. Here, as elsewhere, a consistent complaint from social workers is that despite a dominant rhetoric which suggests that personalisation will allow social workers to return to more relationship-based work, the opposite is frequently the case. In many cases, the emphasis on budgets and cost reductions, based on ever-tighter eligibility criteria, has led to an increase, not a reduction, in bureaucracy (Ryan, 2016).
A second key theme has been the reshaping of the social work role. Since the Thatcher years, the attitude of successive governments to the social work profession has been highly ambivalent. On the one hand, as different writers have noted, social work has often been excluded from key government initiatives, such as the Sure Start project under New Labour (Jordan and Jordan, 2000) or the Coalition government's 2011 Troubled Families programme. It has also been marginalised in such areas as mental health (Bamford, 2015). At the same time, the experience of the past decade suggests that the state continues to see the social work profession as having a role to play in managing individuals, families and communities perceived as problematic. Thus, considerable funding was invested by the then New Labour government in a social work reform process following the highly publicised death of a small child, Peter Connelly, in 2008 at the hands of his carers. More recently, the Conservative government has promoted a major reform of social work education in England, the aims of which are to align the value base of social work much more closely with neoliberal individualism, to dilute the social science knowledge base of the profession, to produce social workers who are much readier to blame clients for their own problems and to prepare sections of the profession for privatisation (SWAN, 2014). In the words of the minister who launched the reform programme:
In too many cases, social work training involves idealistic students being told that the individuals with whom they will work have been disempowered by society. They will be encouraged to see these individuals as victims of social injustice whose fate is overwhelmingly decreed by the economic forces and inherent inequalities which scar our society. (Gove, 2013)
Instead, Gove has argued, social problems such as poverty, drug misuse and homelessness should be seen as the result of individuals making 'wrong choices', with the role of social workers being to encourage them to make 'better choices'. It is, in other words, a wholly victim-blaming approach that downplays or ignores the extent to which people's lives are shaped by wider structural forces.
#### Greece
The final example from Greece is a rather different one. As discussed in Chapter One, Greece is the European country that has become the testing ground not only for the politics of austerity promoted by the 'Troika' of the European Commission, the European Central Bank and the International Monetary Fund, but also for strategies of resistance to that austerity. One form of resistance from below has been the creation of a nationwide network of solidarity committees, usually involving health and social work professionals working on a voluntary basis to address the care needs of those who have been abandoned by the statutory services. As the following boxed article indicates, these committees are important not only as a humanitarian response to unmet need (including support for refugees), but also as potential examples of 'prefigurative welfare', both more inclusive and more openly political (eg in challenging racism) than services provided (or not provided) by the state. At the same time, they raise important questions for radical social workers about the role of the state in welfare. Can such committees ever be a substitute for state-provided services? Given their limited resources, is there not a great danger of burnout on the part of those involved? Should the main energies of those on the Left not be directed towards defending the welfare state? It is perhaps not surprising that the Syriza government is enthusiastic about the committees given that it is that government that is now implementing the austerity policies demanded by the EU. Nevertheless, the committees are an inspiring example of what a radical social work response might involve.
Greece's solidarity movement: 'It's a whole new model and it's working'
'A long time ago, when I was a student,' said Olga Kesidou, sunk low in the single, somewhat clapped-out sofa of the waiting room at the Peristeri Solidarity Clinic, 'I'd see myself volunteering. You know, in Africa somewhere, treating sick people in a poor developing country. I never once imagined I'd be doing it in a suburb of Athens.'
Few in Greece, even five years ago, would have imagined their recession- and austerity-ravaged country as it is now: 1.3 million people – 26% of the workforce – without a job (and most of them without benefits); wages down by 38% on 2009, pensions by 45%, GDP by a quarter; 18% of the country's population unable to meet their food needs; 32% below the poverty line. And just under 3.1 million people, 33% of the population, without national health insurance.
So, along with a dozen other medics including a GP, a brace of pharmacists, a paediatrician, a psychologist, an orthopaedic surgeon, a gynaecologist, a cardiologist and a dentist or two, Kesidou, an ear, nose and throat specialist, spends a day a week at this busy but cheerful clinic half an hour's drive from central Athens, treating patients who otherwise would not get to see a doctor. Others in the group accept uninsured patients in their private surgeries.
'We couldn't just stand by and watch so many people, whole families, being excluded from public healthcare,' Kesidou said. 'In Greece now, if you're out of work for a year you lose your social security. That's an awful lot of people without access to what should be a basic right. If we didn't react we couldn't look at ourselves in the mirror. It's solidarity.'
The Peristeri health centre is one of 40 that have sprung up around Greece since the end of mass anti-austerity protests in 2011. Using donated drugs – state medicine reimbursements have been slashed by half, so even patients with insurance are now paying 70% more for their drugs – and medical equipment (Peristeri's ultrasound scanner came from a German aid group, its children's vaccines from France), the 16 clinics in the Greater Athens area alone treat more than 30,000 patients a month.
The clinics in turn are part of a far larger and avowedly political movement of well over 400 citizen-run groups – food solidarity centres, social kitchens, cooperatives, 'without middlemen' distribution networks for fresh produce, legal aid hubs, education classes – that has emerged in response to the near-collapse of Greece's welfare state, and has more than doubled in size in the past three years.
'Because in the end, you know,' said Christos Giovanopoulos in the scruffy, poster-strewn seventh-floor central Athens offices of Solidarity for All, which provides logistical and administrative support to the movement, 'politics comes down to individual people's stories. Does this family have enough to eat? Has this child got the right book he needs for school? Are this couple about to be evicted?'
As well as helping people in difficulty, Giovanopoulos said, Greece's solidarity movement was fostering 'almost a different sense of what politics should be – a politics from the bottom up, that starts with real people's needs. It's a practical critique of the empty, top-down, representational politics our traditional parties practise. It's kind of a whole new model, actually. And it's working.'
It also looks set to play a more formalised role in Greece's future under what polls predict will be a Syriza-led government from next week. When they were first elected in 2012 the radical left party's 72 MPs voted to give 20% of their monthly salary to a solidarity fund that would help finance Solidarity for All. (Many help further; several have transferred their entitlement to free telephone calls to a local project.) The party says the movement can serve as an example and a platform for the social change it wants to bring about. (Henley, 2015)
### Conclusion
This chapter highlights the limitations of a narrative of 'state withdrawal' as a means of understanding 'what is going on' in respect of social work and social care. In fact, the neoliberal state, far from having withdrawn from the field of social welfare and social work, is playing a more central role than ever in shaping both the content of the social work role and the tasks of social workers (as well as the behaviour of poor people, up to and including 'psycho-compulsion', the provision of cognitive-behaviour therapy for those seen as not being sufficiently motivated to seek work [Freidli and Stearn, 2015]). From Latin America to China, governments are actively reshaping the social work profession to address the challenges of 'permanent austerity', unprecedented levels of inequality and shrinking public services. Challenging that reshaping must start from the clear recognition that social work is a political activity and that these developments require a political response. Any other perspective will leave us disarmed in the face of an existential threat to the profession.
As we progress through the rest of the book we will discuss the way in which social workers across the globe are responding to this reshaping. Before then, however, it is necessary to address another central plank of the neoliberal reshaping of welfare in which states have also played a key role, namely, privatisation, and to explore the implications of this for global social work and those who rely on it.
## THREE
## The privatisation of social work and social care
### Introduction
The privatisation of what were previously state-controlled services has been a key plank of the neoliberal project since its beginnings in the 1970s. One of the first acts of the military junta that overthrew the democratically elected government of Salvador Allende in Chile in 1973 was to institute the wholesale privatisation of the Chilean economy under the supervision of the 'Chicago Boys', economists from Chicago University committed to market fundamentalism. Since then, privatisation has been a key element of neoliberal economic 'reform' everywhere, from the Structural Adjustment Programmes (SAPs) imposed on countries in the Global South from the 1990s onwards by the International Monetary Fund (IMF) and World Bank, to the more recent crippling austerity packages imposed on the people of Greece by the Troika of the European Commission, European Central Bank and IMF.
Social work and social care have not been immune from this process. While the privatisation of professional social work is at an earlier stage in most countries, it is clearly in the sights of governments keen to open up the profession to market forces and, in the process, create a more compliant workforce, less informed by social science theories or committed to the value of social justice (Murphy, 2016).
The first part of this chapter will explore the rationales – ideological, political and economic – for privatisation. Privatisation, like neoliberalism, is often viewed, first and foremost, as ideology and there is no question that the rhetoric of 'private good, public bad' has been enormously influential, taking on the mantle of a kind of common sense among global elites. Yet, an overemphasis on the ideological roots of privatisation can lead to an underestimation of the political and economic forces that underpin it, forces that will be explored in some detail later.
In the second part of the chapter, we will examine the ways in which the global social work and social care landscape has been changed as a result of the many privatisations and outsourcing over the last three decades of what were previously state-controlled services.
Finally, we will critically examine the claims of the neoliberal advocates of privatisation. Has it really led to increased choice and control, as they claimed it would? What has it meant for those who use social work and social care services and for those who work in them? What has its impact been on the quality of those services?
### Why privatisation?
The notion that something can exist and not be a source of profit is an affront to capitalist sensibilities. David Harvey's comments about the relationship between capital and nature can also be applied to the way in which social care services are viewed within neoliberal capitalism:
Nature is necessarily viewed by capital... as nothing more than a vast store of potential use values – of processes and things – that can be used directly or indirectly in the production and realization of commodity values.... While some aspects of nature are hard to enclose (such as the air we breathe and the oceans we fish in), a variety of surrogate ways can be devised usually with the help of the state to monetize and make tradeable all aspects of the commons of the natural world. (Harvey, 2014: 250)
As we shall see later, the creation of 'quasi-markets' in health and social care in Britain and other countries in the 1990s was precisely such as 'surrogate' way of making profit from people's poverty and ill-health.
Nevertheless, throughout much of the world for several decades after the Second World War, large parts of the economy, as well as health-care provision and social security, were in the hands of the state. Why were ruling classes prepared to tolerate the public ownership of key industries and health provision then but unwilling to countenance such ownership today? One early answer to that question came from the pen of Marxist historian John Saville, reference to whose arguments was made in Chapter Two.
Saville was clear that the welfare state was a gain for working-class people and that pressure from below was an important factor in leading to its creation. However, it would be a mistake, he argued, to see the welfare state purely as a response to working-class demands. No less crucial for its creation was what he referred to as 'the needs of capital'. One of these needs is for a healthy workforce. The Boer War more than half a century earlier had highlighted to the British ruling class the dangers of an undernourished proletariat, with nine out of 10 applicants for military service being rejected as unfit. Successful military and economic competition demanded the maintenance of a healthy workforce, something that the market, left to its own devices, clearly could not provide (for further discussion, see Ferguson et al, 2002).
By extension, the same argument applied to the creation of nationalised industries in the post-war period. Growing economic and military competition between the leading capitalist powers in the first decades of the 20th century required that the state become increasingly involved both directly and indirectly in the economy. At its most extreme, this involved the total subordination of the economy to the state in the 1930s, as in Hitler's Germany and Stalin's Russia. However, the phenomenon of greater or lesser state involvement in the management of the economy was a fairly universal phenomenon during the 1930s, through the Second World War and into the post-war period – and was something that capitalists were prepared to tolerate. Discussing the widespread nationalisations that took place in Britain and other countries during the 1940s and 1950s, Harman (2009: 116) writes:
For the state the purpose of nationalized industry was to enable domestic accumulation to match that undertaken by foreign rivals so as to be able to survive successfully in economic and/or military competition.... The state might plan production within the enterprises it owned but its planning was subordinated to external competition, just as the planning within any privately owned firm was.
Harman, like Saville more than half a century earlier, therefore challenged the view that either nationalisation or the provision of free health care in itself equalled socialism (eg as the leading Labour Party theoretician Antony Crosland [1956/2006] had argued in his book The future of socialism). As Saville (1957) put it:
As for the claim that the Welfare state is an early form of a socialist society, it must be emphasised that both in Western Europe and the United States social security schemes are placed firmly within the framework of a free enterprise economy and no one suggests that what is a natural development within a mature capitalist economy should be given new names.
Both Saville's and Harman's comments provide a useful corrective to those, both then and now, who see state ownership in itself as necessarily progressive or even socialist. Their arguments also challenge the widely held view that welfare states, in their different forms, are somehow 'non-capitalist' islands within a capitalist sea, a view most famously expressed in Gosta Esping-Andersen's (1989) celebrated distinction between 'commodified' and 'decommodified' sections of the economy, and, more recently, in David Harvey's (2004) argument that the privatisation of state-owned industries (include welfare provision) is an example of 'accumulation by dispossession', the plundering of the state sector to benefit the private sector. Both sets of arguments assume that, in some way, state-controlled industries represent some kind of challenge to private capitalism despite the fact that workers in these industries enjoyed no more control over production than those in private industry and very often the same bosses remained in place following nationalisation, a process now reversed in the former Soviet Union and Eastern Europe, where the former Communist Party bureaucrats are now often owners of privatised firms.
The reality of nationalisation was very different. As Cliff and Gluckstein (1998: 220) observe in their discussion of the post-1945 Labour government under Prime Minister Clement Attlee:
[M]odern capitalism required a well-developed infrastructure, including transport and power supplies, in order to function efficiently. Now this 'social overhead' capital tended to be costly, requiring long-term investment, and its benefits accrued to industry as a whole rather than the individual entrepreneur. For these reasons the infrastructure in most countries was developed and controlled by governments. In 1945 British capital was ready to break with tradition. It was unwilling to tie up large sums in rebuilding necessary but largely unprofitable areas of industry after the ravages of war. So the bulk of Attlee's nationalization programme did not worry capital in the least.
If the emphasis on privatisation is more than simply an ideological throwback, what, then, is its rationale and how should we respond? The answer, as we suggested in Chapter One, is that it was a key plank of the ruling-class response to the end of the long post-war boom in the early 1970s and the crisis of profitability that it exposed. It sought to address that crisis in several ways (Harman, 2009).
First, it was a pragmatic response to what the US economist James O'Connor (1991 [1973]) had previously described as 'the fiscal crisis of the state'. While global capitalism was expanding during the years of the long post-war boom, increased state expenditure, both on infrastructure (such as roads, airports, etc) and on health and social security, was tolerated or even encouraged. With falling profit rates and the return of global economic crisis in the early 1970s, however, that state expenditure began to be presented as a 'drain' on growth and pressure grew to cut state spending. In Britain, for example, the first round of cuts to welfare took place not under a Conservative government, but under the 1974–79 Labour government, at the behest of the IMF.
Second, the shift to privatisation was aimed at increasing competition between firms and also the exploitation of workers. There was a widespread perception in the 1970s that the managements of nationalised industries were complacent while their workers were lazy and underworked. As Harman (2007) notes:
The logic of this view was that breaking up state owned monopolies and opening them to the market would force their managers to be much harder on the workers, and would intimidate the workers into accepting worse conditions. Certainly the approach of privatisation often encouraged managers to force through methods to push up productivity. And once privatised, it was easy for companies to 'contract out' a range of activities, so breaking the links that tie weak groups of workers to potentially more powerful ones.
Ben Fine (1999: 42) similarly argues that 'privatisation has been an important way in which the relations between capital and labour have been reorganised' and is connected to so-called 'labour market flexibility'.
Third, privatisation plays an important ideological role in appearing to depoliticise welfare and other forms of what were previously state provision by shifting responsibility away from the state to private firms. As Harman (2007) observes, it may do this though outright privatisation or through the kind of surrogate measures referred to earlier by Harvey:
At the same time, privatisation is not absolutely necessary to create the illusion of the automatism of the market. The breaking up of state run institutions into competing units (NHS [National Health Service] trusts, foundation hospitals, city academies, 'self-governing' colleges, 'agencies') can try to achieve the same goal; so can 'market testing' within particular entities. So too can deregulation aimed to produce competition between different nationally based, and often still state owned, companies. This is what the European Union is trying to do to a whole range of industries such as electricity and postal services. The end result of privatisation in a country like Britain can be that whole sections of 'privatised' services such as electricity, water and rail can be run by foreign state owned companies.
As an illustration of what that 'illusion of automatism' means in practice, a series of industrial disputes in Glasgow in 2016 involved low-paid workers employed in Arms-Length External Organisations (ALEOs), delivering what were previously council-provided services but nominally independent of the council. In reality, such organisations are both funded by and controlled by the locally elected council (an example of state-directed social services). Their apparent independence, however, meant that elected councillors were able to claim that there was nothing that they could do to resolve the dispute (Kiernan, 2016).
In respect of Harman's second point, in a development that highlights the hollowness of privatisation rhetoric, the contract to deliver railway services in the West of Scotland in 2015 was awarded by the left-leaning Scottish National Party government to Abellio – a nationalised Dutch railway service (BBC, 2014)!
### Privatisation, social care and social work
How, then, has the global social work and social care landscape changed as a result of the privatisation and outsourcing agenda of the last three decades? As noted earlier, Chile under the Pinochet dictatorship was the initial test bed for neoliberal policies. Since then, however, many Latin American countries and large parts of Africa have been the recipients of IMF 'help' in the form of SAPs, which, as Patrick Bond et al (2006: 50) have noted, have typically 'meant the loss of state welfare programmes and in turn the need for civil society to pick up the pieces'. In his book The Egyptians, Jack Shenker (2016: 50) gives a graphic description of what such SAPs meant in the context of Hosni Mubarak's Egypt:
Through a series of laws passed through the 1990s and early 2000s, Egypt's public sector was divided into dozens of 'holding companies' which could then dispose of their assets on the open market. No longer were state institutions to be thought of as economic development projects, the success of which would be judged by whether or not they met the political and social goals of the state. Now they were merely an epic estate agent's inventory. Huge corporate tax exemptions were established.... New labour laws established fixed-term hiring as a norm and allowed employers to renew the temporary contracts of their staff indefinitely without offering them any longer-term employment rights; workers also faced new limits of their right to strike. Egypt's rulers were so enthused by structural adjustment that they went further and faster in implementing it than even the IMF had expected.... [B]y the end of Mubarak's reign no fewer than 336 public entities had been privatized.
As Shenker notes, it is a scenario that would be familiar to the peoples of dozens of countries in the Global South, from Jamaica to Jordan, who have been on the receiving end of IMF-driven SAPs over the past two decades. Since the Great Crash of 2008, however, citizens of European countries have experienced very similar programmes. Social work and social care were not the first areas to come under the privatisers' spotlight – other areas were far more lucrative – but from the early 1990s onwards, a sea-change took place in the shape and ownership of what one writer has called the 'public services industry' (Gosling, 2011). Here, the experience of three countries will be considered: the UK, Sweden and South Africa.
#### The UK
In contrast to the provision of health care, which, until recently, was funded through general taxation, free at the point of need and organised through a centralised NHS, social care services in the UK have always been much more fragmented, locally provided and often dependent on the payment of a charge (Beresford, 2016). Nevertheless, until the early 1990s, the state, in the form of local councils or the NHS, continued to be the main provider of social care services (albeit with the family also playing a central role). The picture has changed massively since then. The introduction of a 'quasi-market' in social care has meant that whereas in 1993, 95% of home care was provided directly by local councils, by 2012, that figure had fallen to 11%. Similarly, in 1979, 64% of residential care and nursing homes were provided either by local councils or by the NHS; by 2012, that figure had fallen to just 6% (Moriarty et al, 2014). Small wonder, then, that the privatisation of residential social care has been described by one commentator as 'arguably the most extensive outsourcing of a public service yet undertaken in the UK' (Gosling, 2011: 8).
The privatisation has taken two main forms: first, the outsourcing of local authority services to private or third sector providers; and, second, the promotion of individual budgets (personalisation or self-directed support). Both of these policy developments were accompanied with the familiar rhetoric of 'choice and control' for people using services. The reality has often been very different.
First, the introduction of competition to win contracts has led to a 'race to the bottom' in respect of the wages and conditions of staff in both private and third sector organisations, with worrying implications for the quality of service provided (Cunningham, 2008). Within the UK home care sector, more than 300,000 care workers are on zero-hours contracts, with very few employment rights, and frequently restricted to 15-minute visits to the vulnerable people in their care (White, 2016: 139) – hardly a basis for good-quality care.
Second, the introduction of the market into social care has changed the ethos and dynamic of the sector, with growing involvement by providers whose primary concern is profit rather than the needs of their service users. While the typical business is still a small family-owned firm operating in a converted house, big business and organised money are increasingly represented through chains of purpose-built homes so that the five largest chains accounted for nearly 20% of beds in 2015 (Burns et al, 2016). The British journalist Polly Toynbee cited an advertisement from Gravity International, an investment company specialising in 'sourcing and providing alternative investment products' with bases in Malaga, Dubai and Chester, offering an 18% return on investing in buildings that could be run as children's homes (Toynbee, 2014).
Third, as the global crisis of 2008 graphically illustrated, capitalism in its globalised form continues to be a system based on boom and slump. During periods such as the current 'long depression', market failures are commonplace, and the social care market is not exempt from such failures. The collapse in 2011 of Southern Cross, the biggest provider of residential care for older people in the UK, with 37,000 residents in 750 care homes, demonstrated all too clearly the dangers of leaving the provision of care for vulnerable people to market forces (Ferguson and Lavalette, 2014).
Even outside such periods of crisis, however, the claim that the market provides a better quality of care than publicly provided services does not stand up to scrutiny. A Care Quality Commission (CQC) report into learning disability services in England in 2012, for example, found that almost half of all locations inspected were non-compliant with the two outcomes of the care and welfare of people who use services and of safeguarding people who use services from abuse. However, NHS locations were nearly twice as likely to be compliant with both of the outcomes compared to independent providers, while residents tended to stay for much longer in independent that is, private, homes than in NHS facilities (CQC, 2012). A more recent report by the CQC found that 41% of community-based adult social care services, hospice services and residential social care services inspected since October 2014 were rated as inadequate or requiring improvement (CQC, 2015).
Despite these failures, however, the commitment of all the main political parties (with the partial exception of the Scottish National Party government in Scotland) to increasing the role of the market in social care and social work shows no sign of abating. The focus of the UK government in recent years has moved onto the privatisation of both children's services and social work education. Furthermore, the integration of health and social care services, which is a policy priority in all parts of the UK, is likely to speed up the process, especially as the Health and Social Care Act 2012 in England removes all barriers to the privatisation of the NHS.
#### Sweden
There is, apparently, a joke on the Swedish Left that everybody would like the Swedish model of welfare – and the Swedes would like it more than anyone else! The joke highlights the extent to which the post-war welfare state in Sweden, traditionally viewed as the epitome of genuinely universal welfare, has been eroded and transformed in recent years in ways that would be very recognisable to social workers in the UK and elsewhere. Jessica Jonsson (2015: 358) has summarised these broader changes in the following way:
The success of neoliberal ideology in Sweden has succeeded in gradually forcing the welfare state to retreat from its traditional and legitimate responsibilities for providing welfare and improving the living conditions of less privileged groups based on the traditional principles of the welfare state – that is, equal opportunities, social solidarity and social security for everybody. Following the neoliberal changes in the 1980s and the financial crises of the early 1990s, comprehensive neoliberal reforms have been continued both by Social Democratic (1998–2006) and right-wing governments (1990–1998, 2006–2014). Deregulations and re-regulations of the welfare state have led to new forms of 'governance' ... by introducing managerialism, privatisation and cuts in welfare provisions. Such changes have resulted in increasing socioeconomic inequalities, discrimination and social problems in Sweden.
The results are depressingly familiar. Thus, according to figures from the Organisation for Economic Co-operation and Development (OECD, 2015): relative poverty and income inequality have increased more in Sweden than in any other country studied by the OECD; the poverty rate in 2010 was more than twice as high as a decade previously; and children and young people, especially those from immigrant backgrounds, are increasingly marginalised (one factor contributing to the riots that erupted in Husby, a suburb of Stockholm, in 2013).
As in the UK, the state's retreat from welfare has meant a growing role for both the private sector and non-governmental organisations (NGOs). According to Shanks et al (2015):
The Swedish personal social services have... revealed evident signs of marketisation; the private sector has become an important provider of social care. For example, a majority of the residential care units for children and youth are today run by private companies in a care market.
Jonsson similarly identifies the transformation of care for older people, with a greatly increased role for the private sector, contributing, she argues, to a series of care scandals not dissimilar to those that have occurred in the UK. In an example of the 'hybridisation' identified by Moriarty and her colleagues, Swedish NGOs, she argues, are both acting as a bridge towards full-scale privatisation and also legitimising the idea of a market in social care, based not on universal rights, but on charity or on what you can afford to pay (Jonsson, 2015: 364–5).
As elsewhere, however, neoliberalism in Sweden is breeding resistance. Jonsson points to the emergence of new organisations of Swedish social workers like Nu bryter vi tysnaden ('Now We Break the Silence'). In language reminiscent of the UK-based Social Work Action Network manifesto, such initiatives, she notes:
Criticise the development of an increasing burden on social services and the reduction of social work to administrative actions, where social workers spend too much time on documentation and evaluation and which have destructive consequences both for the working conditions of social workers and for those people in need of social services. (Jonsson, 2015: 370–1)
Against this, she argues:
Social work as 'a profession for social change' and the improvement of the living conditions of less privileged groups has an important role to play in an alliance for change in resisting the neoliberal reorganisation and retreat of the welfare state in order to fight against inequalities and injustices. Social workers should actively participate in public debates about, and actions against, the retreat of the welfare state from its responsibilities for people. (Jonsson, 2015: 370–1)
#### South Africa
As noted earlier, the countries of the Global South were the test beds for many of the SAPs imposed by the IMF and World Bank during the 1980s and 1990s. Against a background of wholesale privatisation of these countries' public utilities, many hoped and expected that the road taken by post-apartheid South Africa would be a different one. After all, the magnificent victory in 1994 of the people of South Africa over the barbaric apartheid regime had been inspired by a Freedom Charter that, among other things, promised free education, free universal health care and an end to hunger. It is undoubtedly the case that major changes have taken place since then, both in terms of the extension of basic rights, such as the right to strike and to form political parties, and also economically, with millions more now benefitting from access to electricity, improved housing and social grants. Yet, with the abandonment by the African National Congress (ANC) government in 1996 of the redistributive Reconstruction and Development Programme put by COSATU, the main trade union federation, and its replacement by the Growth, Employment and Redistribution (GEAR) strategy, the country was firmly set on a neoliberal path (Ministry of Finance, 2006). Sewpaul (2013) has identified the main elements of GEAR as being:
• trade liberalisation;
• cutbacks in state expenditure on social spending;
• the privatisation of state assets;
• increased consumerism and the commodification of every facet of human life;
• the profit motive above human well-being and dignity; and
• efficiency, with the aim to do more for less.
One consequence of this policy was that by 2009, South Africa had overtaken Brazil as the most unequal country in the world. In relation to welfare, the dominant discourse in South Africa is the familiar one of the promotion of self-reliance, the importance of individuals, families and communities looking after themselves, and responsibility shifting from the state to the community, albeit wrapped up in the African concept of Ubuntu and a theoretical framework of social development. What is again striking, however, is the extent to which the core elements of neoliberal social work – marketisation, managerialism and consumerism – appear once more in a discourse that would be immediately familiar to social workers from Britain to Japan. Discussing the key policy document for financing welfare published in 2009, for example, Sewpaul (2013: 21) notes that it is 'Replete with managerial and market discourses. It speaks of business plans, contracts, affordability, efficiency, outputs, performance audits, outsourcing, venture financing, and service purchasing, thus effectively reconstructing the people we work with as customers'.
The complaint of one South African social worker cited by Sewpaul is strongly reminiscent of the voices of the experienced British social workers interviewed by Chris Jones (2001) in his influential paper 'Voices from the front line: state social workers and New Labour':
Government is looking at social work in a very commercialised way. And if you look at the theory and principles of social work – warmth, empathy, genuineness and the people-centred approach – I think we are becoming now forced to move away from this, we are actually forced to rationalise. We tend to focus now more on survival than actually servicing our clients.... Eventually your role will have to change completely... you might as well have become an accountant. (Cited in Sewpaul, 2013: 22)
### Privatisation, social work and social care: the balance sheet
The outsourcing of public utilities was justified by the free marketeers on the grounds of both efficiency and ethical superiority. Not only could the market deliver services more efficiently than the state, it was claimed, but in offering citizens (or customers, as they were now known) greater choice and control, it increased their freedom and 'empowered' them. Thirty years into the neoliberal era, such claims ring very hollow indeed.
In respect of efficiency, as Alan White (2016: 13) observes in his study of the 'shadow state' – the companies that now run much of Britain, such as Serco and G4S – the question has never really been answered: 'New Labour rarely if ever carried out detailed comparisons of outsourced provision with that performed in-house, nor were assessments of savings made: that lack of assessment has carried on to this day'. In fact, as White's study shows, these big companies were often far more effective at winning contracts than delivering them. Like George Monbiot's earlier (2000) study Captive state: The corporate takeover of Britain, White provides clear evidence that the record of these large companies in running public services, including social work and social care services, has been appalling. Two examples will illustrate the point.
In 2008, the Labour government in the UK asked ATOS, a French multinational company that specialises in information technology (IT) services, to reassess some 2.5 million disabled people on benefits for eligibility for benefits using a newly developed Work Capability Assessment (WCA). The strongest indication of how poorly ATOS carried out this test is shown by the fact that in 2014, when claimants were being assessed at the rate of 11,000 per week, some 35% of challenged decisions were being overturned on appeal at a cost of £70 million in 2013/14 to conduct the appeals. More important, however, has been the human cost. A study carried out by University of Liverpool academics in 2015 found that after taking into account the socio-economic background of different parts of the country, as well as long-term trends in mental health, a total of 590 additional suicides could be related to these assessments between 2010 and 2013. There were also an additional 279,000 extra cases of mental ill health and 725,000 more prescriptions for antidepressants (Barr, 2015).
Nor have other attempts by government to force people back into employment via outsourcing been any more successful. Action for Employment (A4E), founded in 1991 and with an annual turnover of more than £200 million, has been the largest provider of the government's welfare-to-work programme. The company was awarded a contract for the Pathways to Work scheme in 2008, with a target to get 30% of participants into employment. In February 2012, the Public Accounts Committee) heard that the success rate was 9%, which, as a number of commentators pointed out, was actually worse than doing nothing (White, 2016: 60). Committee Chair Margaret Hodge questioned why A4E had been awarded new contracts to deliver the Work Programme) despite this 'abysmal' performance. The decision to award the company new contracts is even more astonishing given that the company was also being investigated for fraud at this time, leading to the company's founder Emma Harrison (also Prime Minister David Cameron's 'families tsar') being forced to step down in 2012.
Even more concerning than poor performance, however, particularly in the field of elderly care, where the continuity of care is crucial to well-being, is market failure. Reference was earlier made to the catastrophic collapse in 2011 of Southern Cross, the biggest provider of residential care for older people in the UK. However, while much of the attention has focused on the viability of the big providers, many small care homes have been going under. Thus, between 2003 and 2010, almost 1,400 private care homes closed, often with less than four weeks' notice. According to a study by sector analysts LaingBuisson, the care home sector in the UK is closing more beds than it is opening for the first time since 2005, with a net loss of 3,000 across the UK last year. Homes most at risk are those dependent on residents paid for by local councils at rates far below those paid by self-funding residents: proprietors say that rates are actually below break-even point (LaingBuisson, 2015).
Outsourced services can also be criticised on the grounds of their lack of accountability and transparency. The two issues are closely linked. White discusses the experience of Tory-run Barnet Council in North London, which, in 2011, put around £600 million of services out to tender. A Guardian report gave a glimpse of what this 'radical experiment in privatisation' meant for local residents:
For those who live and work in Barnet, their local affairs are now handled remotely by people hundreds of miles away, who know nothing about them or the area. Payroll for what remains of council staff is done in Belfast, while for schools it's Carlisle. Pension queries go to Darlington. Benefits end up in Blackburn.... Got a complaint? Then you have to speak to someone you'll never see – that is, if you can speak to them at all. (Chakraborrty, 2014)
The privatisation was challenged by a vigorous campaign involving the local authority, trade unions and residents' groups. The full implications for accountability of transferring services from elected councils to private companies is evident in the following account of where supporters of the campaign challenged the council at a meeting of the new Board set up to implement the plan. The campaigners were asked to leave the meeting because 'confidential' issues were about to be discussed. When they asked what these issues were, they were told that they were 'too confidential' to be disclosed:
A row kicked off and the chair of the board walked out of the meeting.... [He was] caught on camera saying, when asked why he wasn't listening to the families that were present, 'This is a board meeting of a company. It is not a local authority meeting and therefore that right does not extend'. At which point one audience member said: 'That is exactly our problem with our services being outsourced'. (White, 2016: 145)
The story illustrates that for all their limitations, with state-controlled services, there is a degree of accountability and transparency. There are forums where elected representatives can be challenged, it is more difficult for them to hide behind 'commercial confidentiality' and, at the end of the day, elected representatives can be removed if they have failed to deliver on their manifestos. No such right of recall exists in relation to private companies, whether based locally or, as is increasingly the case with large providers of health and social care, in overseas tax havens.
### Conclusion
A common thread connects the different elements of the critique of privatisation offered in this chapter: the poorer quality of services; the competition that results in a 'race to the bottom' in health and social care services; the market failures; and the lack of accountability and transparency. It is so obvious that it should hardly require saying but here it is anyway: the primary, overriding purpose of private companies, whether it is in the manufacture of military equipment, pizzas or toilet rolls, is to make a profit, with all other considerations secondary. It is scarcely surprising, then, that when private companies move into the field of social care, their first concern is not with meeting people's needs. The point was made forcefully more than half a century ago by the architect of the NHS in Britain:
The danger of abuse in the health service is not in the way that ordinary people use the service. Abuse is always at the point where private commercialism impinges on the service – where an attempt is made to marry the incompatible principles of private profit with public service.
The solution is to decrease the dependence on private enterprise.
A free health service is a triumphant example of the superiority of the principles of collective action and public initiative against the commercial principle of profit and greed. (Bevan, 1952)
That is why, for all the deficiencies of state-provided welfare services, the struggle to keep such services public, while simultaneously pushing for greater democratisation, a stronger user voice and less reliance on biomedical models of health in areas such as mental health and learning disability, continues to be a priority whenever and wherever such services are threatened with privatisation.
# Part Two
Social work politics: past and present
## FOUR
## Social work's horrible histories: collusion and resistance
### Introduction
Reisch and Andrews (2002: 3) describe social work as a profession suffering from historical amnesia and assert that 'in an increasingly ahistorical culture, we are ignorant of those elements of our past that challenged the status quo'. Reisch and Andrews's concern is to recover examples of the profession's radical histories in the US. This is necessary work, and in Chapter Five, we explore the 'reconceptualisation movement' that was active in Latin America in the 1960s as an episode of radical social work that has been 'hidden' from our history.
In this chapter, however, we want to look at a different aspect of social work history. Our view is that the ahistorical culture in social work that Reisch and Andrews describe has also had the effect of suppressing discussion of what we term social work's 'horrible histories'. Mainstream social work histories often portray a benign profession that has gradually evolved and developed to support people in times of need, an inherently benevolent profession that constantly struggles for recognition (for a critique of this view, see Harris, 2008). At the heart of such approaches lies the belief that social problems have little to do with the way our societies are organised. Therefore, exploring social or political histories of social work does not give us enough information about the 'here and now' of the profession's mechanics. In fact, such curiosity may even harm the profession through unearthing awkward and unpopular aspects of the profession's past activities.
In contrast, in this chapter and Chapter Five, we argue that social work, as a contested profession, has a history that is complex and divided: within the profession, there are examples of social work that are inspiring as well as those that are, frankly, shameful. It is our contention that we need to learn from, and celebrate, those periods when social work rose to great heights and embraced the struggle for a more equal and just world, when social work was prepared to 'speak truth to power', and when the profession clearly aligned itself with the interests of the marginalised and excluded. Equally, we suggest that we must also be aware of and reflect upon those periods when social workers were involved in practices that reinforced oppression and exploitation in the interests of the powerful. These dark episodes emphasise what a social work in the interests of the powerful can mean for both social workers and those who use our services. They also show that social work's unwillingness to deal with the historical injustices affecting both the profession and the people we work with potentially damages the profession.
We start by outlining an example from Greece, where decades of political tension and the suppression of civil rights culminated in a seven-year military Junta (1967–74). Much of the resistance against the dictatorship came from young people, mostly school pupils and university students. When the military found it difficult to control 'unruly young people' and many teachers seemed to be too politicised to deal with the issue, the Greek Association of Social Workers (GASW) seemed willing to run to the state's assistance. The GASW initiated a public relations campaign and communicated to the dictator:
Social workers have not been utilised in schools yet. They can play a great role in this context. Not only providing early diagnosis of possible difficulties – something that even teachers can do – but identifying the family, social and physiological and biological reasons of such slowness. (GASW, 1968: 2)
The dictatorship's minister of welfare, in response, circulated a memo titled 'Arrangements for social workers and their professional utilization', suggesting that:
The Greek government during the general ongoing labour reform will look after all the remaining social work demands.... On the other hand, we really appreciate social workers' contribution in the constructive implementation of our social programs.... The ministry of social services will take all the appropriate measures for the advantageous and coherent organization of social workers in commission. (Ioakimidis, 2011: 515)
The GASW celebrated such collaboration, reassuring Colonel Papadopoulos, head of the military junta, that: 'The whole 660 currently qualified social workers are able to face and resolve issues of social adjustment' (GASW, 1968). In the same year (1970), the GASW stressed to the government that social workers: 'are better scientifically equipped than teachers in preventing the social tribulations [in schools]' (Ioakimidis, 2011: 515).
Many contemporary social workers would find such cordial communication between a vicious dictatorship and social workers' official representative body difficult to comprehend. The idea that social workers could collaborate with the military in order to suppress pro-democracy activism would sound mystifying and definitely would not fit with the presumed 'benign role' and the Kantian ethics promoted by mainstream social work discourses.
Some may want to focus on the 'poor morality' of the social work practitioners involved in the events just described and dismiss them as isolated and rare events. However, the disturbing social work histories we discuss in this chapter cannot be dismissed as mere historical quirks and neither can we distance ourselves from them by blaming the individual social workers involved while absolving the profession of any responsibility. Instead, we suggest that in order to understand these events, we have to take into consideration all those broad – and often contradictory – parameters that shape society and influence the actions of people, institutions and organised groups. Such a process can be difficult and painful but it is also necessary to help us answer crucial questions about the contradictory nature of the profession itself.
It is necessary to explore some of the darkest chapters of social work history through the political prism of the particular historical periods within which they occurred. For analytical purposes, we have categorised these histories in two broad and interrelated sections. In the first section, we explore the role of social work in the process of the top-down construction of the ideal-type family as envisaged and constructed by the ruling elites. The second section moves beyond the confines of the family and explores the involvement of social work in broader 'experiments' of social engineering with catastrophic consequences for whole communities.
### Children of nation, children of empire
Preoccupation with the nature, values and evolution of the institution of the family has been central to the history of social work. In capitalist societies, such a preoccupation has historically reflected the desire of the state to ensure the perpetuation of working-class families as a disciplined unit of production and consumption. Welfare services have been instrumental in this process.
The 'care and control' dichotomy of welfare states, discussed in previous chapters, finds no more complete and powerful expression than in the context of family services, for although in the most advanced capitalist states, the welfare system has been able to provide varying levels of care, the element of harsh social control of the poorest in society has always been present. In the West, these cases can be linked, most notably, with the rise of eugenic theories and their pseudo-scientific preoccupation with the creation of the 'superior race'. In the Global South, brutal and often genocidal 'family policies' were primarily linked to colonialism and the politics of assimilation.
#### Nazi Germany
The most notorious example of social work complicity is related to the practice of social, youth and community workers in Nazi Germany. Walter Lorenz (2004: 33) suggests that in the 1920s and 1930s, 'as social service staff came more directly under state control the position of value neutrality demonstrated its blindness to political misuse most catastrophically in Hitler's Germany'. Social policy in this context had a double purpose: on the one hand, it aimed at physically and socially segregating and exterminating those families and individuals 'unworthy' of being citizens of the Reich (see next section); on the other hand, it focused on educating the family and ensuring that all members had a clear understanding of the distinct role required by the state.
The expected roles for men in this context primarily included the functions of being a breadwinner, a good citizen and a good soldier. Women's role, on the other hand, was almost exclusively determined by their mission as mothers. The Nazi state expected women to give up on their careers, have several children, take care of the family and help create, physically and culturally, the perfect 'Aryan'. This was summarised by Goebbels' suggestion that 'The role of women is to be beautiful and to bring children into the world' (Haste, 2001: 74).
State policies ensuring the successful realisation of those aims were implemented through a system of rewards based on generous loans and moral commendations. Social workers and social pedagogues were directly involved in the process of monitoring the development of families and indoctrinating children. Young women were expected to stay healthy, exercise and devote their lives to the infamous concept of 'The 3 K's – Kinder, Küche, Kirche (Children, Cooking, Church)'.
The formal and informal education of children primarily focused on how to become good and obedient citizens. In its most sinister dimension, the education system indoctrinated children in the pseudo-science of eugenics, while encouraging them to join the Nazi Youth Organisations and prepare for the inevitability of war:
There were to be two basic educational ideas in [Hitler's] ideal state. First, there must be burnt into the heart and brains of youth the sense of race. Second, German youth must be made ready for war, educated for victory or death. The ultimate purpose of education was to fashion citizens conscious of the glory of country and filled with fanatical devotion to the national cause. (Zentner and Friedemann, 1991: 79)
Part of the education of the German youth, often facilitated by social workers and social pedagogues, included field visits to centres for the detention of people with disabilities. These visits were used as 'freak shows', where German youth could witness the 'realities' of racial hierarchy first hand. While Nazi authorities focused on the 'reconstruction' of the Aryan family, social services also focused on the removal of 'defective' children from the community. Johnson and Moorhead (2011) explain that this policy was developed for two reasons: first, as a result of the social eugenic policy of the day; and, second, to send a message to the community that the Nazi government was in control of the public and private lives of German citizens. Social workers were actively involved in these programmes and worked closely with authorities in identifying 'unworthy' children and facilitating their extermination (Kunstreich, 2003). Johnson and Moorhead (2011) suggest that:
As early as 1934, genetic health courts were created for the sole purpose of enforcing Nazi health laws and decrees. Documents from this era reveal that public health officials, doctors, teachers, and social workers were also required to report children who were deemed to have a disability or emotional problem.
According to Giles (1992, cited in Johnson and Moorhead, 2011), social workers and nurses had the responsibility to submit official documentation with regards to individuals that they considered unfit.
#### Spain under Franco
Although the politics of creating the ideal family was central to the mission of social work in most European countries, it was in the countries that experienced military rule that such a mission took on the character of an ideological 'crusade'. In fact, in many European countries, the idea of developing the social work profession was itself conceived and nurtured by military regimes. In Spain, the brief progressive example during the civil war (1936–39) offered women and children a glimpse of an alternative society based on the principles of solidarity, social justice and gender equality. These were the very principles that Franco's military regime, which emerged victorious after the civil war, attempted to obliterate. As early as 1937, Franco's regime recognised the importance of social services on the terrain of ideas. In winning 'hearts and minds', family policies and child protection were essential.
Spanish social services under Franco were initially modelled on the services in Nazi Germany. They required the unpaid labour of women, curtailed women's political rights, were constructed on the basis of absolute discipline and obedience, and adopted grotesque notions of racial purity. Following the Nazi blueprint meant that women in Franco's Spain had lost control and ownership of their bodies. Women were seen exclusively as mothers, whose prime mission was to conceive, deliver and care for 'the children of the nation'. Maintaining the purity of the race was of the utmost importance.
In Spain, unlike in Nazi Germany, the concept and celebration of 'race' was not obsessed with physical appearance, but primarily focused on the construction of a well-defined cultural, social and political national entity. In this case, the enemy was not so much the Jew or the disabled, but communism and modernity. This was epitomised in Castro Villacanas' (1948) description of national purity:
We want our friends, our servants and our fiancées to be Spanish. We want our children to be Spanish. In our holy Spain we want only our traditions. If you want to call this nationalism, this is fine with me. We do not want progress, the romantic, liberal, capitalist, bourgeois, Jewish, Protestant, atheistic and Masonic Yankee progress.
In fact, the Catholic Church played a key role in developing social services that would be fully compliant with these principles. The anti-clericalism of the Second Republic was replaced by anti-communism as the most important mission of the Spanish Catholic Church. This new crusade for Spanish Catholicism targeted the 'hydra of social and political revolution that had flourished with the Republic' (González Duro, 2008). Nearly all social work schools in Spain at the time were under the direct control of the Church. The curriculum and admissions process ensured that the chances of 'political contamination' among social work students and practitioners were minimal. National Catholicism, unconditionally subordinated to the fascist regime, was directly in charge of social services and social work education.
For the first decades of the dictatorship, social workers – whose desired profile was that of 'exemplary Catholic ladies' – were assigned paternalistic and assistance-oriented roles in relation to the relief and moral control of the poor (Sanz Cintora, 2001, cited in Martinez, 2017).
Martinez (2017: 74) suggests that:
The mission statement of the second school of social work in Spain, founded in Madrid at the beginning of the dictatorship, can serve to illustrate the ideological backlash in the field of social work brought about by the newly established political regime. According to this school's mission statement, social work [asistencia social] was 'a feminine area of study which aims [were] either a preparation of women for a service to society or an improvement of their education in order to become good and Christian mothers.
Apart from focusing on 'winning hearts and minds' and shaping the nationally pure 'New Spaniard', welfare services under Franco were actively involved in one of the darkest chapters in the modern history of Spain. Although the suppression that faced left-wing and republican families after the end of the civil war has been well documented, only recently have stories about the abduction and trafficking of babies begun to be openly discussed and investigated. The Spanish 'Pact of Forgetting' ('el pacto del olvido'), introduced in 1975 after the death of Franco, which banned research and investigations into the regime's atrocities, was not enough to prevent the hundreds of families whose children had mysteriously disappeared from hospitals and welfare institutions from demanding justice. Over a period of more than 30 years after the civil war, social services were implicated in an illegal mechanism set up by state and Church officials aimed at kidnapping children – mostly from left-wing and poor families – and offering them for illegal adoption to 'nationally minded' families (BBC, 2011b). At the epicentre of these revelations was the Catholic Church, which, over a period of half a century, had served as the closest ally of Francoism. The Catholic Church had constructed and controlled a nexus of welfare institutions ideologically obedient to Francoism, and social work education and practice itself emerged as a historical product of this period. Gómez and Buendia (2009: 2) suggest that such was the control of the Catholic Church over social work education that in the immediate post-civil war period, the main curriculum was divided into three main sections, all directly promoting religious moralism aligned with Franco's political agenda:
The curriculum consisted of three courses and each of these was structured on religious training. The first course, generically called 'Religious Education', revolved around the four disciplines: General Psychology, General Sociology, Social Service Methodologies, and the Practices of Social Services. The second course, also called Religious Education, was less generic and concerned with the Social Doctrine of the Church. It emphasised the subjects: Differential and Genetic Psychology, Social Structure, Individualized and Group Social Services, and the Practices of Social Services. The third and final course, again called Religious Education, established a moral code which was way beyond the professional code of ethics of today.
The ideological legacy of this period is recognised by Vázquez (1970: 40), who suggests that:
The initiative to professionalize social work began in the Catholic sector. Consequently, during many years, the content of the programs of study, the development of the schools of social service, and above all, the orientation given to professional activities have a marked confessional quality. To do justice to its origins, those who study Spanish social services cannot ignore the confessional quality of its genesis.
The notion of social work as a political reaction to social movements was not always directed towards families and children. In Spain during the Franco period, activists were routinely described as mentally ill and incarcerated in mental institutions. A recent comprehensive study into the pathologisation of political activism reveals how the whole concept of Francoist psychiatry was constructed on the notion of intellectual degeneration caused by communist ideas (González Duro, 2008); tellingly, the study was entitled 'The reds were not crazy'.
#### Greece
Similar questions have also been raised in Greece. Nearly 60 years after the end of the civil war, and although the sinister practice of 'child-gathering' (ie the removal of children from left-wing families) is acknowledged and documented, its extent and impact on thousands of children and families is still considered a taboo question. Research in Greek social work suggests that the profession was so immersed in the politics of 'child-gathering' that nearly all social work practitioners in the 1950s had, in one way or another, been involved in the notorious 'child colonies' (Ioakimidis, 2011).
Children from regions controlled by the Greek Left were moved to these institutions, which resembled 19th-century workhouses. They were separated from their families and were subjected to systematic brainwashing and torture. Mando Dalianis, in her unique longitudinal study that includes interviews with affected children over a period of 30 years, provided evidence of the horrific nature of those institutions (Mazower and Dalianis, 2000). As she explains, in the colonies, the children's experience was of a harsh set of activities aimed primarily at tackling communism, both on the battlefield and in the local communities, and only secondarily to respond to some immediate humanitarian needs:
Conditions in these villages in many ways resembled prison life and there was the same rigid sense of a division between 'inside' and 'outside' worlds separated usually by walls or guarded barbed-wire fences. They were run on quasi-military lines, often by former officers, who employed corporal punishment and made the children wear uniforms. Letters were censored, just like in prison, and the atmosphere was generally unfriendly. As in prison, there were no clocks or calendars, and the day was regulated by the ringing of a bell. The children were marched everywhere, even on occasional visits to the world outside, to the cinema or local park. Most teachers were indifferent or cruel to their charges, though there were some exceptions. (Mazower and Dalianis, 2000: 99)
It is important to highlight that, in these contexts, social work was conceived, developed and presented as a respected 'science of charity' in opposition to the principles of solidarity inspired by socialist movements. In fact, both in Spain (behind Republican lines during the civil war) and in Greece (during the years of National Resistance), societies had already experienced short but thriving periods of developing alternative social structures and institutions. These social alternatives were based on gender equality, collectivism, direct democracy and solidarity. In both cases, women had experienced a brief period of relative emancipation and active political involvement before being forced to return to the home and family, what was often portrayed as their 'natural kingdom'. In this sense, social work provided an ideological response to the grass-roots politics of social solidarity, one steeped in ideas of nationalism and women's oppression. These ideas were often pioneered by women of the local aristocracy. As Vervenioti (2002: 115) argues:
20 upper class women, the so-called commissioned Ladies of the Queen's Fund, pulled the strings of women. They, as biological and ideological reproducers, as 'cultural carriers' of the Greek nation, superseded the government in matters of internal and external affairs very effectively in the 'save the children' enterprise.... While Greek women had the legal status of minor, the right-wing women were crucial in establishing the specific form and agenda of the official Greek government.
The detachment of social work education from the university sector until 1994 ensured that training was the sole responsibility of charities linked to the 'Ladies of the Queen' and Church institutions. Until the middle of the 1960s, the vast majority of social work trainees had to attend at least one placement linked to children who required 'protection from communism and nationalist education'. The GASW, desperate to obtain a degree of professional recognition on a par with senior civil servants, was complicit with these practices throughout this period. During the 1967–74 military dictatorship, the GASW attempted to promote itself as the junta's major ally that could effectively ensure social control in school settings with rebellious pupils.
#### Colonial social work and indigenous children: Canada, Australia and Denmark
Whereas the dark histories of European social work were primarily concerned with the construction of the nationally minded, obedient and racially superior family, in countries of the capitalist periphery, the profession was exported as a potent colonial tool. The First International Conference of Social Work took place in Paris in 1928. The fact that it was attended by 2,500 delegates from 42 countries suggests that by the third decade of the 20th century, there was more than sufficient interest among welfare workers to 'internationalise' the profession. In a period overwhelmingly defined by the internal and external contradictions of colonialism, social work was not only influenced by colonial politics, but seemed to be the direct product of an era when 'the number of rulers officially calling themselves, or regarded by western diplomats as deserving the title of, "emperors" was at its maximum' (Hobsbawm, 1987: 56). As the character of colonisation in the 19th century and beginning of the 20th century shifted from a primarily 'mercantile' form to a more complex system of production and the unification of markets, the prominence of capital expansion meant that the old methods of military colonisation could be successfully replaced by means of market unification and cultural assimilation.
In the context of the partition of the world's wealth and territory among a handful of states, these calculations had a twofold objective: on the one hand, colonial powers were 'united' in their concern to firmly secure their grip over native populations through the suppression of their self-determination; on the other hand, they were locked into a vicious antagonism for global domination against each other. The latter escalated into two world wars and numerous regional conflicts. The former required a variety of 'soft' methods (assimilation, the suppression of local cultures, the reconfiguration of social institutions, bureaucratisation and the modernisation of local economies) and 'hard' methods (violent subordination, segregation, disenfranchisement and the politics of fear). The development of social work was invariably seen as a 'soft' approach to perpetuating colonial rule through social control and the reconfiguration of sociocultural institutions.
Although the emergence of the 'indigenisation' debate in recent years has generated some interesting arguments challenging the colonial nature of social work, not much attention has been paid to the ways in which social work played a key role in the oppressive politics of assimilation. In these contexts, social services actively attempted to suppress indigenous cultures and to forcibly extend settler values to native communities. Once again, the institution of the nuclear family was deemed to be the 'gold standard'. Individualistic approaches to child protection were also used as a quantifiable measure of assimilation success. Inevitably, these approaches failed spectacularly and led thousands of aboriginal families into misery and suffering. It would not be an exaggeration to claim that these policies were never meant to succeed anyway. By definition, the process of assimilation pre-designed conditions that would lead to the alienation, marginalisation and stigmatisation of native populations in order to justify more draconian state interventions. The infamous Canada Scoops, for example, a methodical process of child removal from aboriginal families, offers a sobering example of drastic assimilation under the guise of social care. According to The Aboriginal Justice Implementation Commission, within a period of nearly 20 years (from the early 1960s until the late 1980s) the child welfare system:
removed Aboriginal children from their families, communities and cultures, and placed them in mainstream society. Child welfare workers removed Aboriginal children from their families and communities because they felt the best homes for the children were not Aboriginal homes. The ideal home would instil the values and lifestyles with which the child welfare workers themselves were familiar: white, middle-class homes in white, middle-class neighbourhoods. Aboriginal communities and Aboriginal parents and families were deemed to be 'unfit.' As a result, between 1971 and 1981 alone, over 3,400 Aboriginal children were shipped away to adoptive parents in other societies, and sometimes in other countries. (The Aboriginal Justice Implementation Commission, 1999)
Likewise, in Australia, the Aboriginal Protection Act 1869 gave the colonial administration such extensive powers over the lives of indigenous children that communities experiencing assimilation through the child protection system have become known as 'stolen generations'. The oppressive role of Australian social workers has been evidenced in the recent Report of the National Inquiry into the Separation of Aboriginal and Torres Strait Islander Children from Their Families (NISATSIC, 1997), which acknowledges the fact that social workers routinely used unsubstantiated allegations of neglect in order to remove children from Aboriginal families. The report states that:
The children were still being removed in bulk, but it wasn't because they were part white. They had social workers that'd go around from house to house and look in the cupboards and things like that and they'd say the children were neglected (Molly Dyer evidence 219, speaking of the practice of the Victorian Aborigines Welfare Board in the 1950s). (NISATSIC, 1997: 28)
A lesser-known case of social work complicity with the brutality of colonial assimilation occurred in Greenland in the early 1950s. It involved welfare practitioners from Danish charity organisations who worked closely with the government of Denmark in attempting to 'modernise' Greenland through the creation of a new and re-educated generation exposed to the advancements of the Danish lifestyle. As part of this social engineering project, officially described as an 'experiment', several children were forcibly removed from their communities in Greenland and given to middle-class foster-families in Denmark. The Danish press were quick to celebrate the success of the 'experiment': 'The way of life here in Denmark is so different from what these children of nature are accustomed to but their ability to adapt is remarkable. Disagreements – caused by their reaction to civilisation – happen very rarely' (BBC, 2015b).
Scotland's lost children
In 2014, the Scottish newspaper the Daily Record reported on the horrific experience of the children of one family at the hands of the Scottish care system in the 1950s. George, Jimmy and Tommy Clark were taken from their family home in Greenock and put into local authority care after their parents were ruled incapable of looking after them. 'Boarded out' to a family in a remote part of Scotland, the brothers – then aged four, six and nine – say that for the next four years, they slept in a filthy chicken coop and were forced to steal dog food to survive.
According to the report, the brothers were now challenging the system to ensure that Scotland's so-called lost children were reunited with their siblings. For George Clark:
People were shocked by the film, 12 Years A Slave. But Scotland has its own shameful history of slavery too. Over 100,000 kids were shipped from children's homes to Australia, New Zealand and Canada until the late 1960s to work on farms and factories. There were also children like us, used as slaves and cheap labour, boarded out by local authorities across Scotland (www.dailyrecord.co.uk/news/scottish-news/secret-slaves-scotland-revealed-3300900).
The horrific experience of many of these children has been documented by the Golden Bridge Project (see: https://www.iriss.org.uk/resources/multimedia-learning-materials/golden-bridge) and also in the movie Oranges and Sunshine.
#### Britain's 'children of empire'
From the middle of the 19th century, British authorities and a number of well-known charities involved in broad social work activity were involved in the forced migration of children from Britain to various countries then part of the British Empire – especially Canada, Australia, New Zealand, South Africa and Zimbabwe (then Rhodesia). Initially, the motivation was a combination of 'child-saving' philosophies where, especially for a number of leading Christian charities (such as Barnardo's, Quarriers and the Catholic Emigration Association), the migration was portrayed as an 'opportunity' for pauper children to avoid the physical and moral hazards of the newly sprawling towns and cities of industrial Britain. For local government bodies (such as the Poor Law Unions), migration was favoured because the cost of transporting dependent children was far less than the potential cost of looking after them in workhouse institutions.
In the second half of the 19th century, large numbers of children were shipped to Canada in a process often dubbed 'philanthropic abduction'. The children were almost always under the age of 11, though some were much younger:
The child migrants were mostly aged between four and fourteen and were usually rounded up and accompanied by a Poor Law guardian or a representative of the organisation sending them. Sometimes there was no representative and the word 'care' didn't come in to it. (Bean and Melville, 1989: 2)
In 1880, 540 children were forced to migrate to Canada; by 1880, the number had risen to 2,104 per year (Kershaw and Sacks, 2008). 'At the peak of [this]... phase of child migration – 1870 to 1925 – at least 25 large philanthropic organisations were sending children to Canada' (Bean and Melville, 1989: 38). Canada remained an important destination until the practice ended in the late 1930s. An estimated 100,000 children had been shipped to Canada from Britain between 1869 and the late 1930s (Government of Canada, 2016).
The conditions that the children endured were often harrowing. The children were moved from the bustling streets of towns and cities in Britain and placed in isolated, rural farms in Canada, where they had to adapt to a vastly different climate and way of life. Most had families but migration brought separation from their family and support networks. They were forced to work long hours in gruelling labour, often in cruel and abusive conditions. This was state-sanctioned child abuse on a grand scale.
By the beginning of the 20th century (in the aftermath of the Boer War), the argument in favour of child migration had changed. Now Barnardo's described their child migrants as 'Bricks for Empire-Building' (Kershaw and Sacks, 2008: 9). By populating the Empire with 'good British stock', the Empire could ward off future rebellions and insurgencies (like those then recently witnessed in South Africa in the war against the Boer farmers). One of the first to adopt this new philosophy of migration was Thomas Sedwick, 'a social worker', who took 50 young boys to New Zealand in 1910. As they left Britain, he wrote to the King: 'The first party of town lads for colonial farms beg to convey to Your majesty the expression of our most dutiful and humble devotion to your Throne and person on our departure for the Dominion of New Zealand' (cited in Bean and Melville, 1989: 79).
In the 20th century, the process of child migration was linked to eugenic philosophies. As the Archbishop of Perth (cited in Humphries, 2011:12):
At a time when empty cradles are contributing woefully to empty spaces, it is necessary to look for external sources of supply. And if we do not supply from our own stock we are leaving ourselves all the more exposed to the menace of the teeming millions of our neighbouring Asiatic races.
The children shipped to Australia were checked to ensure that they had good eyesight, no flat feet and were of general 'healthy disposition':
Child migration to Australia... especially of those of solid Anglo-Saxon stock – was encouraged. Physical stamina was important, and children were prepared for living in the outback by sleeping outside in tents and learning to swim. They also underwent a series of medical tests.... They didn't take anyone wearing glasses or who was colour blind. (Kershaw and Sacks, 2008: 10)
In the late 1930s, the Fairbridge Society set up training schools and colleges in Southern Rhodesia – though their first migrants did not arrive until after the Second World War. Their stated aim was to 'Fill the empty spaces of Empire with selected children of sound stock from the over-crowded towns and cities of the United Kingdom' (cited in Bean and Melville, 1989: 98). In Southern Rhodesia, the intention was to select children who could eventually join the local white elite. To fill their allocated role in the social structure, the children went through significant selection. They undertook IQ tests, health checks and a background assessment of their social standing. The background checks were undertaken by trained social workers – members of the British Federation of Social Workers (Bean and Melville, 1989: 99).
After the Second World War, Australia became the most important destination for the migrant 'children of empire'. An estimated 10,000 children were shipped from Britain to Australia between 1945 and 1970 (BBC, 2006). As Chenery (2011) notes:
When bleak postwar Britain answered Australia's call for 'good white British stock' to build its population, it saw an opportunity to empty overflowing institutions of the innocent victims of poverty, illegitimacy and broken homes. In the child trafficking that became known as the child migration schemes it cost £5 a week to keep a child in care in Britain but just 10 shillings in Australia. Institutions that took children would be paid a subsidy for each one of them. All the reputable agencies – Barnardo's, the Salvation Army, the Fairbridge Society, National Children's Home, the Catholic and Anglican churches – colluded in sending children to the other side of the world for 'a better life'. They were thought a particularly attractive category of migrant, according to a 1945 prime ministerial brief to state premiers, 'on account of their easier assimilation, adaptability, long working life ahead and easier housing'.
The children were told that their parents were dead – or their mothers unfit – but that they were destined for better lives, with warm beds, full stomachs and a proper education. The reality was very different. As Bean and Melville (1989: 111) note 'The history of child migration to Australia is in many ways a history of cruelty, lies and deceit'. For many, it was also a gateway to hell as they ended up in institutions where physical and sexual abuse was standard. Those unlucky enough to be dispatched to institutions run by the Christian Brothers in Bindoon, Tardun and Clontarf, for example, were subjected to regular, routine abuse (Bean and Melville, 1989; Humphries, 2011). The story of the migrant children, sent across, first, the Empire and then the Commonwealth, by leading children's charities, with the knowledge and collusion of local authority children's departments, stands as one of the most shameful episodes in social work's history in Britain.
Margaret Humphreys and the Child Migrants Trust
In 1986, a woman from Adelaide, Australia, wrote to Margaret Humphreys (at the time, a social worker in Nottingham, England) claiming that, at the age of four, she had been sent to Australia, unaccompanied, on a boat from England. She claimed that this voyage was undertaken with the full knowledge of the UK authorities.
Humphreys was at first incredulous. However, after undertaking some research, she found the story to be true. In fact, Margaret Humphreys soon discovered that as many as 150,000 children had been deported from children's homes in Britain and shipped off to a 'new life' in distant parts of the Empire. The practice continued up until the late 1960s.
Many of the children were told that their parents had died, even though, for most, this was not true. Many of the parents were told that their children had been adopted in Britain – again, an outrageous lie. Siblings were separated and for numerous children, it led to a life of horrendous physical and sexual abuse in institutions in Western Australia and elsewhere.
Margaret Humphreys campaigned relentlessly for the rights of the forced migrant children and their families. In 1987, she set up the Child Migrants Trust – which continues to have a base in Australia and Britain.
Despite her 'discovery' of the children of empire in 1986, the children and families did not receive an official apology until 2010. It was only in 2017, over 30 years after the initial 'discovery', that the British government set an official inquiry into the child migrant trade.
### Social work and social Darwinism
As we discussed in the previous section, the political and professional position of social work as an agent of the state has implicated the profession in the development and implementation of oppressive family policies underpinned by theories of 'the nation', moralism and cultural assimilation. However, it is the profession's involvement in the politics of racial segregation and exclusion that has generated the most appalling and shocking social work histories. An inconvenient reality is the profession's love affair with eugenics and social Darwinism. Jones (1983: 46) has persuasively demonstrated that ideological proximity when saying that:
Drawing on social Darwinism, the COS [Charity Organisations Society] insisted that one's location in the social and economic world was in large measure a reflection of character, those at the bottom of the pile being the most deficient and disorganised.
Eugenics was so influential in social work at the beginning of the 20th century that a passing review of archived casework notes in the UK would almost certainly indicate several references to this pseudoscience. Ideological dependence on the state aside, these deterministic theories offered social work the illusion of scientific status and posture that was lacking in comparison to other professions.
Despite British social work's flirtation with Sir Francis Galton's grotesque ideas, it was in Nazi Germany where eugenics theories were put into practice in a systematic way and on an industrial scale. In Germany, the adoption of social Darwinism was neither merely symbolic nor peripheral to state policies. In fact, these ideas shaped the core of the Hitler's vision. The creation and reproduction of the Aryan race could not be achieved without the parallel segregation and eventual extermination of the inferior races. Annihilation of those 'unworthy' of life was seen as the absolute and final solution. Within a period of 10 years (1934–44), millions of people with physical or intellectual disabilities, Slavic minorities and Roma communities, and gay men and women were subjected to mass detention, hard labour, sterilisation and extermination. The catastrophic culmination of these policies was the Holocaust, which resulted in the killing of over 6 million Jews.
Social workers were heavily involved not only in advancing the ideas of social Darwinism, but also in actively utilising them. They were involved in the process of diagnosing racial inferiority and facilitating the detention of those deemed 'unworthy'. Lorenz (2006: 35) explains that:
The system relied on all welfare personnel filing case reports in which they listed the family histories of epilepsy or alcoholism [and] assessed the chances of rehabilitation of offenders or of children with learning difficulties. Sticking to their professional task with the air of value neutrality and scientific detachment they did not feel responsible for the consequences of their assessments.
Hope in the face of horror
Irena Sendlerowa was a social worker in Poland during the Second World War. She organised a small group of colleagues to smuggle Jewish babies and children out of the Warsaw ghetto between 1942 and 1943 and place them with Polish families. Irena's team smuggled the children out by hiding them in ambulances, taking them through the sewers or wheeling them out hidden in suitcases or boxes. The workers noted the names of the children on cigarette papers and sealed them in glass bottles, which were then buried. After the war, the bottles were dug up and attempts were made to reunite the children with their families – though most of them had perished in concentration camps (Connolly, 2007).
A similarly notorious example of social work's active and systematic collaboration with a regime promoting policies of racial segregation and social engineering is that of South Africa. Racist policies defining colonial rule in South Africa were typified and fully operationalised in the period of apartheid (1948–91). This was a complex, brutal and multilayered system of segregation designed to physically, politically, socially and culturally exclude non-white populations. Racial segregation also aimed at maximising labour exploitation through the consolidation of colonial rule. Non-white populations, deemed as inferior, were classified in different administrative categories according to the colour of their skin, were deprived of civic and political rights, and were allowed access only to substandard and segregated education, health and social services.
Mainstream white South African social work, which had largely accepted segregationist ideologies well before 1948, readily adopted and functioned within the context of racial separation culminating with the creation of apartheid. Legislation demanded that social workers, particularly in the public sector, provide services only to those designated as the same race as themselves while social work education was also defined by a differential university education based on race (Sewpaul, 2013). Smith's (2014: 313) analysis encapsulates the evolution of such historical and indefensible complicity in South African social work when suggesting that:
The origins of social work in South Africa are found within the forces of racist capitalism, social conflict and unequal power relations – shaped by the hegemonic ideologies of the various eras during the 20th century and even earlier colonial and imperialist origins. Through hegemonic discourses, social work generally supported the maintenance of the racist status quo and the capitalist mode of production, with individualist and liberal ideologies of freedom of choice and personal responsibility.
Oppressive social work practices influenced by the principles of social engineering have not, however, been restricted to countries with undemocratic or military regimes. The disturbing case of social workers' involvement in US sterilisation projects is another example. The eugenics movement in the US officially began in the late 19th century, reached its zenith in the 1920s and 1930s, fell out of favour during the Second World War, and then made a comeback in the 1950s (Anastas, 2011). As in the cases we reviewed earlier, the principles of US eugenics projects were inextricably linked to the effort to devise mechanisms of social control through the promotion of an ideal-type representing their own values and characteristics. Social problems such as poverty, crime and unemployment were seen as largely 'hereditary' within the inferior classes and therefore treated through practices aimed at preventing these classes from 'reproducing'. Despite the impression that the forced sterilisation of the poor in the US was a thing of the distant past, recent research suggests that in some states (most notably, in North Carolina), this practice lasted until well into the 1970s. In North Carolina, where sterilisation projects were widespread and, unlike other states, not confined to institutional settings only, more than 7,600 poor, vulnerable and minority citizens were permanently affected by this practice (Boggs, 2014). Evidence suggests that many sterilisations were recommended by social workers, who would base their 'diagnosis' solely on observations of home environments or poor school performance (Boggs, 2014):
The US National Association of Social Workers recently conceded that 'As early social workers researched and developed programs to provide solutions to alleviate American social ills, and encouraged greater investment in the social safety net, long held beliefs about the capacity of poor, mentally ill and other social "outsiders" colored a variety of government policies and the medical establishment'. (Anastas, 2011)
### Conclusion
In this chapter, we have attempted to present some of the most disturbing historical cases of social work complicity. What we suggest is that at key political junctures, many professional social work organisations (eg the GASW in Greece during the military junta or the National Association of Social Workers in the US during the McCarthy period) were more concerned to protect the interests and survival of the profession at the expense of any overarching commitment to social justice. Often masked under a narrative of eclectic positivism, the ideology of professionalism meant that there was a systematic depoliticisation of social work, which turned it into a work task for processing the poor, the disadvantaged, the political and the oppositional in the interests of the powerful.
In most of the cases we reviewed, social work as a profession was tightly controlled by the state. In the cases where social work associations sided with the oppressor and social work education was regulated by the state (or Church), there was very little room for individual social workers to create a viable movement for resistance. However, some did resist. Reducing horrible social work histories to textbook footnotes or justifying them as mere episodes of compromised judgement and poor morality blurs the crucial contribution of numerous, known or unsung, social workers who bravely fought for social justice. The contributions of African-American social worker Thyra Edwards, who travelled to Barcelona and joined the anti-fascist international brigades during the Spanish Civil War, or the story of Polish-Jewish practitioner Irena Sendlerowa, who saved nearly 2,500 children from the Warsaw ghetto between 1940 and 1943, would be incomplete if we did not consider the fact that many – if not most – of their contemporary colleagues sided with the fascist and Nazi forces at the time.
## FIVE
## Social work as a praxis for liberation: the case of Latin American reconceptualisation
### Introduction
In contrast to the horrible histories discussed in Chapter Four, in this chapter, we explore the Latin American reconceptualisation movement, a radical social work movement that was extraordinary in terms of both its intellectual contribution and its generation of transformative praxis. It is our contention that the reconceptualisation movement has been one of the most comprehensive, influential and far-reaching examples of emancipatory social work globally, yet it remains little known in social work circles across much of the world. The movement was an integral part of the 'radical triangle' (along with 'Liberation Theology' and 'Popular Education') that emerged in Latin America in the 1960s and 1970s and redefined the role and scope of the applied social and pedagogical professions. Although it is not easy to discuss in isolation any one component of the 'radical triangle', in this chapter, we will try to focus on the powerful influence of the reconceptualisation movement on the social work profession in Latin America and internationally.
Apart from its groundbreaking theory and practice, the reconceptualisation movement stands out from other radical social work movements for another reason too. Never have social work activists been targeted by state violence more aggressively and systematically than those involved in the Latin American reconceptualisation in the 1970s. Although records from international social work organisations suggest that the victimisation of social work activists has not been an uncommon phenomenon (IFSW, cited in SWAN-Greece, 2011a), the degree of suppression that Latin American social workers suffered in that period has been unparalleled in the history of the profession. The Argentinian Association of Social Workers, for example, suggest that in the first months following the 1976 military coup, 24 practitioners were murdered, while the records of the Association indicate that there are still 59 desaparecidos (missing) practitioners (Ministerio de Educación, 1984). Equally, data from the Chilean Association of Social Workers confirm the violent persecution, disappearance or assassination of 17 social workers following the coup in that country in 1973 (Colegio de Asistentes Sociales de Chile, 2008). Although the precise figures of murdered and violently persecuted social workers in this period are unknown due to the ruthlessness and arbitrariness of the military regimes, it would be safe to calculate the numbers of persecuted social workers into the hundreds.
One of them was Argentine Professor Luis M. Früm, who published his article in 1971 on 'Ideology in social work', which eloquently presented the contradictory ideological function of the profession:
Assuming that ideology guides our work, can we talk of 'reconceptualised' Social Services without reconceptualising the ideology that sustains it?... In this sense, the process of selecting a specific methodology is one that addresses the specific needs in real life; it could be referring to either a system which desires to be preserved or the specific need of a social class to transform reality. (Früm, 1971: 23)
Früm's observations in his article captured the most central task of the reconceptualisation movement: 'making sense of the ideological function of social work as a precondition for creating an alternative profession and eventually a just world' (Früm, 1971). Five years after the publication of this text, Früm was dismissed from his position as Chair of Social Work Methodology at his university, targeted by the military regime's infamous 'Dispensability of civil servants' laws. A few months after his dismissal, he went missing. His body was eventually found in a morgue in the area of Villa Mercedes, 'face down, blindfolded and hands tied with chains and a wire. Dressed in his pyjamas and shot in the neck' (El Diario Dela Republica, 2014).
Früm was probably the best known social worker murdered in this period. The most notable pioneers of the other two parts of the 'radical triangle' were also targeted during this purge. Catholic Priest Camilo Torres was executed in 1966 by the Colombian army and the critical pedagogue Paulo Freire lived in exile from 1964 until 1980, in both cases, as a direct consequence of their intellectual and political contributions.
The ferocity of the suppression and targeting of social workers and activists involved in the 'reconceptualisation' movement was, in a sense, a testimony to its wide popular influence (which transcended borders across the whole of Latin America). Between 1965 and 1975, the movement evolved from a rather marginal 'anti-paradigm' to the most dominant and influential social work approach in Chile, Argentina, Brazil, Uruguay, Colombia, Mexico and Puerto Rico. Its political legacy is still powerful across Latin America and visible in global social work.
It is not easy to present a linear history of reconceptualisation, for its development was dynamic, evolving through a series of interrelated responses to the contradictions of the 'modernisation' of social services across the region. Although, during roughly the same period, social work in much of the developing world had experienced the tense process of reforms coordinated by the United Nations (UN) and United States Agency for International Development (USAID), only in Latin America did social workers create a movement, structured and militant enough to earn the description of an 'anti-paradigm', a 'rupture' with mainstream social work (Barreix, 2003). The main reason behind the Latin American social work 'exception' can be found in the unique constellation of social, cultural and economic elements that gave rise to broader radical and popular movements.
Justice for Luis María Früm
In April 2015, Argentina's Federal Court found 11 former military officers guilty of crimes against humanity. Their historical crimes, committed during Videla's dictatorship, were mostly related to thousands of arbitrary and violent disappearances of pro-democracy activists. Among the defendants were the murderers of academic and radical social work pioneer Luis María Früm. The Argentinian Association of Social Workers welcomed the sentencing of Früm's murderers, describing it as 'Justice delivered after 38 years of pain'.
Luis María Früm, qualified as a social worker at the Institute of the Ministry of Social Assistance. Early in his career as a social worker and academic, he challenged the 'technocratic' basis of mainstream social work in Latin America, which, under the veil of political neutrality, assisted the perpetuation of neo-colonial macroeconomic policies. Früm became instrumental in the reconceptualisation movement in Argentina. This movement sought to create a social work paradigm that promotes the 'liberation of the oppressed communities and social classes'. He was recognised as one of the reconceptualisation movement's organic thinkers and his monographs on the ideology of social work were widely read among progressive academics across Latin America. He became a Professor and Director of the School of Social Work at the National University of San Luis. He used his professorial role as a means for developing genuinely participatory and democratic pedagogies in social work academia. He was also a founding member of the Conceptual, Referential and Operational Scheme (ECRO) group, the editorial collective behind the radical social work movement's most influential publications.
He continued his progressive academic and activist work even after the military Junta led by Colonel Videla suspended the constitution and suppressed democracy in Argentina. Luis María Früm was immediately targeted by the military and his activities were closely monitored by the intelligence service. In March 1976, he was dismissed from his academic position under the 'Dispensability of Civil Servants Act', a law designed to silence progressive civil servants. On 19 June 1976, a group of military officers showed up at Luis María Früm's family home in Villa Mercedes. When he opened the door, he was arbitrarily arrested and removed from his home by force. Several days later, he was found dead in a lake, 40km away from his home. He was shot in the neck and his body was blindfolded and handcuffed, still wearing his pyjamas.
Over 40 years since his murder, Luis María Früm's work and life are still celebrated and remembered by social workers across Latin America as an inspiring example of a social worker fighting for social justice even in the most dangerous and oppressive circumstances. Hundreds of social workers were murdered or 'disappeared' in Latin America in the 1960s and 1970s due to their activist and politically progressive work.
### 'The food of the minority is the hunger of the majority' (Galeano, 2009)
Until the late 1950s, social work in Latin America was a rather marginal, Church-related, activity. Although it claimed to be a 'profession', in reality, it was difficult to differentiate it from traditional philanthropy. In most countries of the region, the lack of social policies and unwillingness to create organised social services meant that social work only existed nominally as an organised profession. Social work in the socio-economic context of 'Hacientismo', the Latin American variation of landed social relations, was a world apart from the organised profession in the industrialised 'West'.
'Hacientismo' had its roots in the colonial socio-economic relations evolved during the Spanish Empire. The colonisation of the subcontinent by the Spanish Empire, already complete by the late 16th century, had effectively divided Latin America into four administrative regions: New Grenada (corresponding to modern-day Colombia, Ecuador, Panama and Venezuela), New Spain (Mexico, Central America and Spanish West Indies) and Río de la Plata (Argentina, Bolivia, Paraguay and Uruguay). Colonial Brazil was controlled by the Kingdom of Portugal.
Unsurprisingly, the colonial powers concentrated their efforts on the aggressive exploitation of natural resources. This was achieved through the creation of a system of which the main pillars were land appropriation and racial segregation. Large plantations were created, mostly in the central and northern zones of the cone, while in the southern areas the focus was on mining and the extraction of minerals. Slavery was of central importance to the colonial system. Unlike North America, in the southern cone, the colonial powers did not intend to physically exterminate the indigenous populations. Instead, the main priority was to incorporate them into the colonial production system as a labour force. Nevertheless, much of the indigenous population could not survive the European diseases brought to Latin America by settlers. In order to deal with labour shortages, the Spanish Empire resorted to extensive slave trading, forcibly transferring millions of Africans to Latin America and the Caribbean. The mix of African, indigenous and European populations resulted in very diverse cultural and racial communities. In order to retain their power, the new rulers created a complicated system of racial hierarchy, with varied levels of privilege according to race (Sistema de Castas). The Mestizos, people of mixed race and the majority group, although having fewer rights than the European-born ruling classes, enjoyed a bit more privilege than the indigenous or Afro-Latin American communities. The 'divide and rule' principle introduced in colonial times survived well into the 20th century, and its legacies are still evident in most Latin American countries.
Within this context of oppression, peasants (from the 17th century) started organising and articulated demands for agrarian reform. Unlike the northern part of the American continent, in Latin America, peasant rebellions moved beyond racial lines and often succeeded in creating effective grass-roots alliances. The rebellious spirit of the Latin American peasantry shaped the 'national liberation' movements of the 19th century that eventually defeated the Spanish Empire and divided the region into nation-states. Despite the defeat of colonial rule and the high hopes of the oppressed classes, the newly born states maintained the perpetuation of vast inequalities. Land and wealth remained in the hands of the privileged few while the majority of the population – and, in particular, the communities of indigenous and African origin – suffered from chronic marginalisation. Moreover, the emergence of the US as an imperial power established new forms of 'colonisation', favouring the use of economic means, interference in domestic affairs and the establishment of puppet regimes rather than direct military interventions.
In the 1950s, major advances in transportation and industrialisation in much of the formerly 'colonised world' had effectively created a new international division of power: 'a massive shift of industries producing for the world market from the first generation of industrial economies which had previously monopolized them to other parts of the world' (Hobsbawm, 1994: 362). Such shifts provided the foundations of what would later be called 'economic globalisation'. Despite the tectonic socio-political changes it caused, unlike the case of 19th-century Europe, the old 'latifundia' did not disintegrate. Instead, old oligarchies were incorporated into the new system, which combined the accumulation of land, colonial hierarchies and the process of rapid industrialisation. The result was that the poorest communities experienced the 'worst of both worlds': old ruling classes retained feudalistic social relations and repelled demands for agrarian reform, while in the newly created cities, inequality crushed the hopes of the working classes. Galleano (2009: 5) described this system of multiple oppressions in his iconic book Open veins of Latin America, which immediately became an indispensable 'primer' among radical social work students and practitioners:
The division of labor among nations is that some specialize in winning and others in losing. Our part of the world, known today as Latina America, was precocious: it has specialized in losing ever since those remote times when Renaissance Europeans ventured across the ocean and buried their teeth in the throats of Indian civilizations.... It continues to exist at the service of others' needs, as a source and reserve of oil and iron, of copper and meat, of fruit and coffee, the raw materials and foods destined for rich countries which profit more from consuming them than Latin America does from producing them.
In 1961, US President John Kennedy introduced his short-lived 'Alliance for Progress', a set of ambitious economic reforms and financial stimuli, supposedly aimed at creating growth and accelerating industrialisation in Latin America. Economic domination aside, a major objective of this agenda was to prevent the emergence of revolutionary movements like Fidel Castro's in Cuba, which had captured the imagination of the oppressed communities. The developmentalist reforms introduced by the 'Alliance for Progress' were destined to fail as apart from the flawed economics that characterised them, they were also deeply neo-colonial in nature. Based on old clichés, developmentalism identified the poorest communities not as victims of the system, but as the main reasons of stagnation. Overpopulation and the demographic 'time bomb' were highlighted as a hindrance to development, and it was inevitably the poorest who were presented as the main culprits for all the misfortunes of Latin America. Galleano (2009: 8) explains how 'Various US missions have sterilized thousands of women in Amazonia, although this is the least populated habitable zone on our planet'.
Unsurprisingly, these neo-colonial experiments, instead of fostering economic growth, contributed towards the growth of militant anti-imperialist movements. The developmentalist approach had underestimated the fact that Latin American societies had long lost faith in reforms and policies 'imported' by colonial powers. 'Modernisation' had not brought about the desired social change and the most oppressed communities directed their discontent into organised political action. The historically strong tradition of rebellion in the region gained momentum and eventually gave rise to well-organised movements seeking to 'emancipate' Latin American. These movements varied from well-organised guerrilla movements (such as the FARC in Colombia) to successful electoral alliances (such as the rise of Salvador Allende to President of Chile in 1970). What really characterised Latin America's 'age of rebellion' was its organic nature, which transcended cultural, geographical and often class boundaries. One has to remember that, almost simultaneously, peasants pushed for (and in the case of Guatemala's '10 years of Spring', achieved) meaningful land reform, indigenous communities joined communist parties en masse (Hobsbawm, 1994) and popular guerrilla movements were created and often led by members of the white metropolitan intelligentsia (as in the case of the Cuban Revolution, as well as FARC and M19 in Colombia). In many respects, the formidable force of Latin America's popular radicalisation had defied all known scripts, and unlike in other regions, social workers, educators and even priests swiftly swapped loyalties and found themselves in the forefront of these movements.
### Social work reconceptualisation: a child of its time
Like all transformative movements, reconceptualisation did not develop in isolation; it was the product of the broader political commotion that gave rise to radical politics across Latin America. The modernisation policies of the 1950s and 1960s required a broad range of well-trained personnel. Social workers, who had had a peripheral semi-professional role in the rudimentary social services of the region until then, became the 'chosen' profession assigned with the mission to operationalise the principles of community development on the ground. 'Developmentalist' approaches created a critical mass, organised and coherent enough to be described as a profession. For the first time in its history, social work was provided with space, status and resources comprehensive enough to lift the profession from the hitherto anaemic vocation of 'social assistance'. The (re)birth of the social work profession in the 1960s was largely based on Anglo-American positivism and it swiftly incorporated in its methodology technocratic tool kits promising immediate solutions to the chronic problems associated with the region's 'underdevelopment'. Social workers were employed in UN–US coordinated community projects, most of which emphasised community development, informal education and family planning. It needs to be noted that in the early years of this professional transformation, there was a great degree of optimism about the profession and what could be achieved.
Such transformation was also reflected in the rapid expansion of social work education. During this period, many social work programmes were organised across the region, benefiting from the support of the UN. Moreover, the provision of social work education moved to universities instead of the private and religious tertiary institutions that had monopolised the training of social workers until that point. The new generation of students and practitioners, educated within the context of technocratic reformism, were able to break free from the constraints of conservative philanthropy and engage with a wider range of courses and themes, such as political economy, community development, demographics and so on.
Ironically, the call for de-professionalisation, central to the radical arguments of the reconceptualisation movement, would never have developed without the pre-existing process of successful professionalisation introduced in the 1950s and early 1960s. The rapid expansion of social work education in the 1950's and 1960's meant that the class base of the profession shifted away from the traditional philanthropy of the middle class. Eventually more working class students entered the social work profession altering irrevocably the image of social work as a church-based, middle class, gendered vocation.
However, by the early 1960s, the initial optimism was gradually replaced by scepticism and disillusionment. Social work methodologies taught at the universities proved to be ineffective and irrelevant to the Latin American context as, in most cases, they were direct translations of US textbooks. Practitioners who were involved in the social assistance projects in both rural and urban areas witnessed first-hand the failures of a system supposedly designed to create growth and development.
While analysing the social work reconceptualisation movement, it is important to highlight the close proximity of and dialectical relationships between the three different professions (theology, education and social work). The radicalisation of all three activities evolved simultaneously and was a direct response to the shared realities of practitioners in the community intervention projects in impoverished regions. Teachers were sent into these community and welfare centres in order to contribute towards the eradication of illiteracy. Social workers were utilised in community development/family planning tasks, and priests engaged with the spiritual and material support of the poorest communities. The three professions coexisted in these centres and shared concerns, ideas and actions. Their criticisms mostly focused on the issue of the inequalities generated through the developmentalist approach. Crucially, they also recognised the ways in which reformism restrained the articulation of alternative and transformative models. Liberation theologian Gustavo Gutierrez (1971: 26) suggested that 'Developmentalism came to be synonymous with reformism and modernisation, that is to say synonymous with poor measures really ineffective in the long run and counterproductive to achieve real transformation'.
Increasingly, practitioners of the 'radical triangle' felt more confident in proposing political action as a legitimised method of intervention. The concept of 'neutrality', central to the modernising tool kits, was the first to be deconstructed in the process of social work radicalisation. Radical academics Alayon, Barreix and Cassineri (1971: 58) captured these tensions when they asked:
If the Alliance for Progress has been a political response to a political problem; if the developmentalist thesis was born in the shadow of this political approach, which crept into the Social Work curricula, then how could social workers develop alternative theoretical models linked to frontline realities and break free from the illusion of neutrality?
In order to break free from the limitations of mainstream community interventions and social services, indignation was not enough. A radical theory combined with radical practice was required and the major contribution of Latin American reconceptualisation was exactly this: it managed to transform anger against structural injustices into a concrete theory of practice. Central to emancipatory theory and practice developed in this period and used extensively by social workers and educators alike was the concept of 'conscientisation' (conscientização). This concept, largely influenced by the quest for 'class consciousness', emphasised tangible strategies that social workers and educators could employ in order to: (1) help expose the oppressive social structures that dehumanise people; (2) use dialogue, participatory research and active learning as tools for challenging these structures; and (3) highlight the importance of political mobilisation as a precondition for individual and collective liberation (Crivas, 1999):
Reconceptualized social work claims for itself activities relating to consciousness-raising, training, organization, and social mobilization, elements of a process through which it is possible to visualize the liberation of man.... The theoretical base of the movement of reconceptualization was viewed as follows: the first stage of development of the theoretical foundation of reconceptualization rested on a structural-functionalistic base and was known generically as the 'basic method'. The second stage was to implement those actions based on dialectical materialism, and which were consequently known as dialectic. (IASSW, 1977: 104)
Paulo Freire
Paulo Freire (1921–97) was a Brazilian radical educator and political activist whose work heavily influenced progressive theory and practice in social education and social work. Freire suggested that the fields of education and social welfare are never politically neutral. Instead, the perceived neutrality of these institutions only perpetuates a system of inequality and exclusion. According to Freire, educational and social welfare practice can become powerful tools of emancipation only when they work in collaboration with oppressed classes in order to develop a critical understanding of their own social realities and the potential for social action (a process he described as 'conscientisation'). To this end, he developed a creative methodology of practice that focused on critical thinking and action (praxis), which is used extensively by progressive social workers and social educators in Latin America and other developing regions. After the 1964 military coup in Brazil, Freire was detained and eventually forced to live in exile for 15 years. Upon his return to Brazil, he supervised one of the most ambitious adult literacy projects in Brazil.
### Rise and fall of reconceptualisation
The need for the articulation of alternative theories generated very rich debates within schools of social work. These debates were concurrent and often antagonistic, representing diverse political traditions. The breadth and diversity of these initial debates makes it rather difficult to pinpoint the exact time when social work's political engagement took the form of a more concrete movement. By convention, most social work historians (see Barreix, 2003) would suggest that the first issue of the magazine Hoy en el Servicio Social (Social Services Today) in 1965 (which, after the fourth issue, was renamed Hoy en el Trabajo Social [Social Work Today]) should be considered as the first major milestone in the creation of the 'reconceptualisation' movement. This is not to ignore the various events and actions preceding this publication, but, instead, to consider Social Services Today as the first systematic effort to coordinate dynamic activities and synthesise ideas towards developing an alternative social work theory and practice.
During the same year, two other major political developments that consolidated 'reconceptualisation' as an emerging movement also took place. The Latin American Social Service Seminar, an annual meeting point for radical social workers, was launched in Porto Alegre. In the same year, the social services undergraduate programme of the University of Uruguay became the first one to officially endorse a curriculum that allowed space for alternative social work theories and practices, followed by the Catholic University of Sao Paulo, which created the 'campo piloto', an unapologetically radical educational project that aimed at connecting social work with popular movements (Raichelis and Rosa, 1982). Social work academics, students and practitioners involved in these campaigns and publications would eventually be considered pioneers of Latin American social work. 'Generation 65' was considered a term synonymous with the reconceptualisation pioneers.
Social Work Today, until it was closed by the military dictatorship in Argentina, remained the most influential medium of communicating radical social work ideas and literature. Initially, most of the people involved in its editorial board had links with the Social Service Institute in Buenos Aires. The declared objective of the editorial board (known as ECRO) was not only to record and document the exciting political developments on the ground, but also to enrich social work theory with the concepts of critical pedagogy, popular culture and the 'philosophy of liberation'.
It needs to be noted that during the first period (1965–69) of reconceptualisation and despite great enthusiasm, the movement went through major theoretical debates and tensions. It was a movement that encompassed a variety of ideological currents and nurtured several strands of Marxism – reflecting the broader debates of the age – the ideas of Mao Zedong, Guevarism, dependency theory and concepts of popular education associated with Paulo Freire, and it was also influenced by variations of Peronism, the works of Althusser, Mounier and Oscar Lewis, and even more radical versions of Christianity (Gloria et al, 2008).
What united the activists involved in the movement at this early stage was their shared frustration at the inefficiency of developmentalist policies and the rejection of technocratic neutrality. The latter, as we discussed earlier, had created a type of practice so anodyne and detached from structural problems that ECRO would routinely describe it as 'aseptic social work practice' (ie 'ideologically sterile'). Despite the loud rejection of mainstream social work, at that early stage, it was totally unclear what could replace 'aseptic social work'. One of the main criticisms about this early period of reconceptualisation is that it tried to confront Western positivism often using the tools of positivism (Alayon and Molina, 2006). Although 'Generation 65' is characterised by evident ideological contradictions, it nonetheless allowed space for the deeper exploration of radical approaches to social work. In this sense, 'Generation 65' acted as a prelude for the much more militant wave of reconceptualisation that formed between 1969 and 1972.
The move towards the second phase of reconceptualisation is evident in the 'Araxa Document', named after the Brazilian city where the Brazilian committee of the International Social Services Conference attempted to summarise and clarify the intellectual contributions of progressive social workers thus far. The Araxa Document is not itself a radical manifesto; its main contribution lies in the fact that it offers a succinct review of scholarship and activities linked to the reconceptualisation movement. Such a systematisation of the rich, yet messy, ideological debates helped expose the limitations of the eclecticism that characterised the early stages of the movement (Barreix, 2003: 28). Such a contradictory mix of determinism with critical theory was evident in the document's list of conclusions, where it called upon social workers:
• to act in the process of creation, reformulation and adjustment of social policies;
• to facilitate public participation in this reformulation and planning process in order to ensure the appreciation of macro-analysis;
• to promote and participate in surveys in order to assess the policies that are being adopted; and
• to work with individuals or groups, providing a corrective, preventive and empowering method (see the Araxa Document, cited in Netto, 1994).
The call for people's involvement in re-imagining social services was a significant departure and helped create space for the consolidation of more radical voices in the reconceptualisation movement. By the end of the 1960s, some of these voices were presenting openly revolutionary arguments for a different social work: they suggested that social work had to not only work towards highlighting the structural inefficiencies of the capitalist system, but propose social work methods that could contribute towards overthrowing capitalism. The second period of reconceptualisation is characterised, ideologically, by a stronger emphasis on 'historical materialism', which allowed the reconceptualisation movement to move beyond the 'limitations of eclecticism' and focus on connecting social work practice with the politics of class liberation. Parra (cited in Servio, 2015) suggests that 'the movement boom between 1969 and 1972 was a consequence of the fact that conservative social workers and technocratic modernisers "resignedly or combatively" accepted the need for a renovation of the profession'.
In reality, the 'resignation of the more conservative voices in the movement' was not a reason for the strengthening of reconceptualisation; rather, it signalled that socialist social workers had won the 'battle of ideas' within the movement. This is the most productive period of reconceptualisation, both intellectually and methodologically. Barreix (2003) and Servio (2015) suggest that during this period, the influence of reconceptualisation was so wide that one could say that radical social work had become the 'new mainstream' across the whole South American cone. By the early 1970s, most major social work schools had adopted curricula influenced by Marxism of one variation or another, as well as the politics of liberation. Such was the intellectual engagement with reconceptualisation in social work schools that within 10 years, the scholarly output of radical social work writings had exceeded the total number of mainstream social work books and articles that existed since the inception of the profession (Alayon and Molina, 2006). Moreover, at a time when most international and regional social work associations were still confined within the Anglo-American zone of influence, the Latin American Association of Social Service Schools (founded in 1965) articulated a combatively anti-colonial approach.
However, as the movement was reaching its height, it was brought to an abrupt halt by the military regimes that seized power across the continent. As an integral part of broader political movements, the radical reconceptualisation movement was crushed by the extraordinary violence of the military dictatorships, which condemned Latin American societies to an extended period of authoritarianism. The US and local ruling classes had become increasingly wary of the influence of progressive and revolutionary politics in Latin America. By the late 1960s, it had become clear that the US strategy of 'soft colonisation' had failed. Social movements had become so powerful and popular that left-wing parties and alliances achieved unprecedented electoral successes. Local ruling classes demanded decisive action against what they perceived as a threat to their interests. To these developments, the US responded with a clandestine plan of intervention that provided intelligence, resources and military support to dictators across the continent (a Central Intelligence Agency (CIA) plan known with the code name 'Condor'). By 1976, democracy was suppressed in the largest part of the Latin American cone, including Brazil, Bolivia, Chile and Argentina. The rest of the continent, although technically not under military rule, also experienced a period of extensive violence and instability.
Social work was among the most demonised and persecuted professions as it was considered to be irreversibly Marxist. Most social work schools closed, several academics and practitioners involved in the reconceptualisation movement were arrested or murdered, and social work students, in the schools that had survived the purge, experienced a prolonged period of censorship and book-burning. In Chile, social work programmes were removed from university education altogether and social work books were banned on campuses (Castañeda and Salamé, 2014).
### Conclusion: the legacy and relevance of reconceptualisation
Although 'actually existing' reconceptualisation was short-lived due to the rise of military dictatorships, its legacy has been profound and long-lasting. Never in the global history of the social work profession had a grass-roots political movement influenced so extensively social work theory and practice. Liberation approaches, central to the reconceptualisation movement, have been enormously popular in the Spanish- and Portuguese-speaking world since the 1970s. They are still highly influential to social work education and practice in these countries, where political action is still considered a legitimate social work approach (Saracostti et al, 2012). Its influence and legacy can be witnessed in the Latin American contributions to global social work debates, which are always characterised by their structural and political analysis. One can find direct links between the legacy of reconceptualisation and 'orange tide', the contemporary anti-austerity social work movement in Spain (Ioakimidis et al, 2014). Albeit with some delay and a more modest reach, emancipatory social work theories nurtured in the reconceptualisation movement also inspired sections of North American and non-Spanish-speaking European social work (Belkin Martinez and Fleck-Henderson, 2014).
Reconceptualisation has been the most comprehensive and emphatic example of radical social work that succeeded in moving beyond the mere articulation of an 'alternative' narrative. Latin American social workers successfully created a 'complete' and holistic model, overcoming the division between theory and practice that dominates in much contemporary social work. In this sense, reconceptualisation also disproved arguments which suggest that radical social work is a 'theory without practice' (Servio, 2015).
This was possible through placing emphasis on the concept of 'praxis' as a dynamic intellectual-methodological intervention that is inextricably linked with the objective of liberation from all forms of oppression. Radical social workers in Latin America categorically opposed abstract theoretical schemata that were not informed by concrete front-line experiences. Equally, they were steadfast in rejecting social work models, either mainstream or 'activist', that solely focused on practice. Freire (1972: 156), while discussing the essentials of such holistic practice, highlighted the irreplaceable elements of:
Consistency between words and practice; boldness, which urges the witnesses [ie facilitators of 'praxis'] to confront existence as a permanent risk; radicalisation (not sectarianism) leading both the witnesses and the ones receiving that witness to increasing action; courage to love, which far from being accommodation to an unjust world, is rather the transformation of that world on behalf of the increasing liberation of human kind; and faith in the people.
What is particularly relevant to current social work debates grappling with postmodern and identity politics is the emphasis of the reconceptualisation movement on a concrete analysis. Despite the fact that reconceptualisation emerged as a diverse dialogue among disillusioned social workers coming from different traditions, it eventually became a movement that managed to synthesise these contributions into a concrete theory. This was a 'theory in action', responsive to the diverse experiences from practice but, at the same time, retaining a very clear focus on the broader historical and political context. Gagneten (1986) summarised these elements:
• Social workers should begin from popular culture and rediscover the needs and problematic conditions generated by economic oppression.
• The object of knowledge and innovation in social work should be focusing on unfair structures.
• People should be considered as historical and social beings with capacity to think and transform themselves as the main promoters of their welfare, well-being and history.
The emphasis of reconceptualisation in its latter stage was on the structural-political functions of capitalism and the ways in which this system deprives the oppressed classes of their humanity. Radical social work theory in Latin America allowed space for the rich and diverse experience of the people they worked with (workers, peasants and indigenous communities). Practitioners involved in reconceptualisation insisted that only an analysis that encompasses the shared experience of the oppressed classes and promotes unity in action could be transformed into a truly radical movement: 'To do otherwise is to absolutize and mythologize the relative; alienation then becomes unavoidable' (Freire, 1972: 157).
More than this, the reconceptualisation movement contributed significantly to the disentanglement of radical social work's Gordian knot, that is, progressive practice in the context of a capitalist state. Indeed, the question as to whether social work in a capitalist state was intrinsically and unavoidably oppressive had been of central concern to social workers involved reconceptualisation. As the reconceptualisation movement entered its second phase (1967–71), the period of intellectual maturity, a consensus was achieved that de-professionalising social work and rejecting it as an inherently oppressive profession was a simplistic, under-theorised and harmful approach (Barreix, 2003). Instead, radical social workers had to work with methods that exploited contradictions integral to the capitalist system – and this is what they did.
The fact that state social work had moved beyond the narrowness of philanthropic charity, through the modernisation agenda, meant that the old, religious, middle-class base of the profession was replaced by social workers with a diverse demographic and class background. It was exactly this shift in personnel and demographic that allowed social workers to assert the political identity of the profession. Radical social work analysis exploited this change and highlighted that in this new professional reality, social workers' class interests and life experiences were much closer to the people that they worked with rather than the state institutions that they worked for. Hence, the reconceptualisation movement called for social workers to reverse their political and occupation loyalties. Costa (1987: 120) explained such evolution, suggesting that:
The professional strategy seems to be, on the one hand the democratisation of the institution [social services] through client or subaltern class pressure and, on the other hand, social workers' conquest of autonomous space in order to ensure a professional practice addresses to the clientele of popular classes.
Moreover, the transition to industrialisation and dynamic change within state institutions seeking new roles and identities allowed social workers to exploit internal contradictions in state policy and achieve extraordinary autonomy. Latin American countries went through tectonic social, cultural and economic changes at the time. The breaking up of old communities in favour of rapid urbanisation was an uncharted territory for governments who required social workers to be in the forefront of dealing with the impact of these changes. Social workers, even when they started becoming 'ideologically unruly', were still very much necessary to the state as their community development projects were, in many cases, the only channel of interaction between the state and the population. The vastness of many Latin American countries, its uneven development and the inability of the state to monitor closely the work of professionals implementing state policies allowed increased autonomy in rural areas.
The 'peak years' of reconceptualisation coincided with the rise of left-wing or populist movements into positions of power, both at the regional and national level. Such an increase in the influence of progressive forces in the state apparatus meant that radical social workers discovered opportunities to directly engage with political action within the state. It is important to remember that Paulo Freire was a senior civil servant in the Ministry of Education when he developed (and practised) his critical pedagogy approaches. Eventually, he even served as a minister of education. In Chile, the electoral success of Salvador Allende allowed significant transformations to happen in social services.
The reconceptualisation movement has been the most important example of radical social work practice globally. Its extraordinary influence, reach and intellectual contribution to social work remains unsurpassed. It bridged the artificial dichotomy between radical theory and practice, demonstrating that meaningful transformation is only possible when social workers and communities are 'united in action'. Crucially, it exposed the ideological function of social services and the futility of neutrality, and proposed a 'preferential option for the poor and the oppressed'. The current resurgence of anti-capitalist movements in many parts of the world and the subsequent re-engagement of social work with radical approaches calls for a careful examination of the lessons and relevance of reconceptualisation. Reclaiming the radical history of our profession is an urgent task, being especially pertinent at a time when mainstream historiographies tend to:
ignore the conscious or compelled choices the profession made about its ideology goals, methods of interventions and public policy positions. They obscure the role that radical social work played in shaping many of the concepts that the profession and society now take for granted. (Reisch and Andrews, 2002: 4)
The legacy of reconceptualisation in contemporary Latin American social work practice
Decades of paramilitary violence and structural inequalities in Colombia have led to the displacement of approximately 5 million people across Colombia. This has disproportionately affected Afro-Colombians, indigenous people and peasants who have been forced off their land and moved into urban areas, where they face further oppression, violence and lack of opportunities. According to Amnesty International, in 2008 alone, 380,000 people were targeted and forced off their land. The despalzados (displaced) live in unacceptable conditions, without documents, jobs, access to electricity or medical care, and Doctors without Borders suggests that these families have borne witness to massacres, detentions and the disappearance of family members or their neighbours. They have been harassed by armed groups, 'taxed' for money and property, and, in some cases, forced to flee to save their children from forced recruitment.
As a result, in early 2011, new displaced communities moved into 'public spaces' in the centre of Bogota, including the large national park, which has been occupied by thousands of despalzados. The occupation of the parks served two main purposes: the first one was linked to the practical need to find space for temporary residence; and the second one emphasised on the need for collective political actions.
The response of the central and local governments was predictable and punitive. Police encircled the public areas in order to contain the expansion of camps within the public parks. They constantly harassed the displaced. Moreover, paramilitary groups, functioning as the state's long arm, also targeted indigenous leaders. By and large, the despalzados's leaders adopted a non-violent approach and accepted the involvement of social workers employed by the local council. This was mostly because most of the social workers involved came from Afro-Colombian and indigenous background themselves and they used approaches inspired by the reconceptualisation movement. Their practice focused on the following approaches.
Meeting urgent needs/mediation
Social workers worked with the displaced communities in order to address the urgent needs of the community. Part of the social work engagement was political in the sense that it exposed the state's dereliction of duty and failure to meet their statutory obligations. The state was forced to provide some help in the form of cash benefits, food stamps, shelter and medical care. Most importantly, the campaign in support of the displaced communities mobilised trade unions, students and other groups, who extended their solidarity and provided material support. On several occasions, social workers intervened when police action was threatened and mediated in order to construct the council's evacuation plans.
Trust building/participation and democracy
Despite their ordeal and ongoing tragedy, the despalzados have maintained a great sense of community. Large groups of people and families live close to each other, creating small networks of solidarity and self-help (psychological and practical). Social workers worked closely with the community in order to help them facilitate participatory processes, which include the organisation of open assemblies in the parks attended by the displaced and, in some cases, where trusting relationships have been built, representatives of local residential areas. During this process, social workers worked as organisers who attempted to make full use of their institutional capabilities (access to facilities, funds, etc) for the benefit of the community. On several occasions, this was followed by small cultural and sports events co-organised by the practitioners in direct collaboration with the communities. These events were 'owned' by the communities but would not have been organised had social workers not made use of their influence within the state system in order to obtain approval for the events.
Conscientisation and emancipation in social work practice
Social workers did not pretend that they possessed better expertise than the despalzados; they accepted that they constantly learn from the interaction with the displaced communities, creating 're-humanising' opportunities for both groups. Through Freirian interventions focusing on dialogue, mutual respect, democratic processes and cultural activities, both social workers and displaced communities work towards understanding the structural causes of their social problems. In doing so, a very important stage in this process is building broader social alliances. Oppression and social inequalities affect not only the groups of people occupying the national parks; all sections of Colombia's peasantry and working class have been affected by the violent conflicts, social inequalities, privatisations and neoliberal policies of recent decades. A number of activities within the parks and the public spaces have attempted to bring together all these groups in an effort to challenge alienation, indifference and individualism. Such active demonstrations of solidarity have had a twofold effect: 'rehumanising' communities that had experienced horrible violence and displacement; and creating a basis for a grass-roots alliance that was eventually transformed into a large, popular pro-peace movement. Eventually, indigenous and Afro-Colombian displaced communities played a key role in the pro-peace movement. In 2016, a peace agreement was ratified by the Colombian Parliament. The peace accord covered most of the demands of the displaced for relocation to their land, collective reparation and agrarian reform.
Source: Hinestroza and Ioakimidis (2011).
## SIX
## Refugees, migrants and social work
Give me your tired, your poor,
Your huddled masses yearning to breathe free. (Emma Lazarus, 1883)
### Introduction
Social work has a long history of work with and alongside refugees and migrants. Throughout the profession's history, there have been workers who have considered such work as essential and non-controversial – part of the profession's DNA, a reflection of its commitment to human rights and social justice. Equally, there have been practitioners wedded to notions of eugenics, happy to collude in the processing and regulating of vulnerable migrant communities.
In this chapter, we look at the contemporary 'refugee crisis'. Comprehending the scale and cause of the present crisis is necessary to inform social work theory and practice – and provides the basis for essential policy and practice demands that should be at the heart of any internationalist social work.
Eleanor Rathbone
Support for migrants and refugees has played an important, though often hidden, role in the history of social work activity. The activities of some social work pioneers in this field are worthy of consideration.
Eleanor Rathbone was a Liverpool-based social reformer and early social work pioneer. She was the founding chair of a large voluntary sector organisation called PSS (Person Shaped Support) (which is still active in the city) and was an advocate for women's rights. Support for refugee and asylum causes was an important part of her work. In 1936, she supported the British Provisional Committee for the Defence of Leon Trotsky, and signed a letter to the Manchester Guardian defending Trotsky's right to asylum, for example. In 1938, she campaigned for dissident Germans, Austrians and Jews from both countries to gain entry into Britain. Earlier, in 1937, she became a steering committee member of the Basque Children's Committee, which oversaw the transportation, arrival and dispersal of 4,000 Basque children fleeing fascist aggression during the Spanish Civil War. The children were housed across the UK in a number of 'colonies' that met their educational and material needs. An echo of this work can be found today in social work campaigns to support unaccompanied refugee children across much of Europe (Firth, 1986; Benjamin, 2012).
### The scale of the contemporary 'refugee crisis'
The last decade witnessed a developing, and unprecedented, refugee crisis across much of the globe. According to the United Nations High Commissioner for Refugees (UNHCR, 2015d), 59.5 million people worldwide were forcibly displaced at the end of 2014, 8.3 million more than the year before. Those who were displaced were overwhelmingly the victims of persecution, war, civil unrest and a range of human rights violations. This figure represented the highest number of displaced people on United Nations (UN) records, though these figures were dwarfed by those from 2015 and 2016. According to the Internal Displacement Monitoring Centre (IDMC), 2014 also saw 38 million people around the world forced to flee their homes by armed conflict and generalised violence, being pushed into internal displacement (moving to an alternative location within the borders of their normal country of residence); this figure is a 15% increase over the previous 12 months (IDMC, 2015). These figures represent the largest number of refugees since just after the end of the Second World War.
This is a crisis that has been fermenting for some time. The second decade of the 21st century witnessed a steady growth in refugee numbers:
Starting from 10.4 million at the end of 2011, the number increased to 10.5 million in 2012, to 11.7 million in 2013, and finally to 14.4 million by the end of 2014. By mid-2015, it had reached an estimated 15.1 million, its highest level in 20 years. Within three and a half years, then, the global refugee population grew by 4.7 million persons – some 45 per cent. (UNHCR, 2015e: 4)
Faced with this crisis, numerous politicians and much of the media across Europe and the 'advanced' economies of the world have framed the 'refugees and migrants... as a problem, rather than a benefit to host societies' (Berry et al, 2015: 5). Politicians and media outlets have described the refugees as 'economic migrants' (or, in the words of the British Prime Minister at the time, David Cameron, 'a bunch of migrants' [Freedland, 2016]), with the clear inference that they are attempting to enter the European Union (EU) to obtain ('better') work or access to welfare benefits.
Despite such claims, two clear facts stand out. First, in terms of country of origin, the top 10 refugee 'source countries' are those wracked by war, conflict and political unrest; at present, they are (in order) Syria, Afghanistan, Somalia, South Sudan, Sudan, Democratic Republic of Congo, Central African Republic, Myanmar, Eritrea and Iraq. People leave their homes and communities to seek refuge because they have no other reasonable choice. They make the voyage because they are desperate – not to get to Europe, Australia or the US, but to get away from the situation that they are in.
Second, the majority of the world's refugees do not enter Europe, the US, Australia or other parts of the 'economically advanced' world. At the end of 2014, the world's top refugee host was Turkey, followed by Pakistan, Lebanon, Iran, Ethiopia and Jordan. Proportionately, Lebanon hosts by far the largest number of refugees by population, 232 per 1,000 inhabitants. Worldwide, 86% of the refugees under the UNHCR's mandate live in developing countries (UNHCR, 2015d). To take the case of Syrian refugees, by mid-2015, an estimated 4.2 million had fled the country; overwhelmingly, they had moved to neighbouring countries, with Turkey (1.8 million), Lebanon (1.2 million), Jordan (628,800), Iraq (251,300) and Egypt (131,900) being the main hosts (UNHCR, 2015e: 4).
Nevertheless, between 2014 and 2016, much of the focus of the global refugee crisis was on Europe (despite its relatively small share of global refugee numbers). In a sense, the 'refugee crisis' only really started to be interpreted as such because refugees started to come to Europe, whereas for the previous few years, those fleeing Iraq and Syria had stayed in the region and were, to some extent, 'hidden from view'. According to the UNHCR, during 2014–16, Europe faced 'a maritime refugee crisis of historic proportions' (UNHCR, 2015c: 2). They estimate that in the first six months of 2015, 137,000 refugees and migrants attempted to enter the EU, a rise of 83% on the same period in 2014. The International Organisation for Migration argued that in the first two months of 2016, more than 102,547 migrants and refugees arrived in Greece by sea (IOM, 2016).
Arrival in Greece or other parts of Southern Europe is, for most refugees, simply a stage on their travels. Their end destination of choice is often Germany, Austria, Sweden (who, between them, took close to 90% of refugees arriving in Europe in 2015) and, for a small minority, the UK. Greece is part of the 26-country 'Schengen Area', which allows borderless travel between countries who have signed the agreement (Traynor, 2016). Yet, from late autumn 2015, countries started to close their borders, erect barriers and, effectively, suspend the Schengen Area in order to prohibit the movement of refugees. By mid-2016, the EU had entered into an agreement with Turkey to exchange supposedly 'legitimate' refugees from Syria while expelling (presumably 'illegitimate') refugees from other parts of the globe back to Turkey (for further processing) (Rankin, 2016). This was a decision that effectively trampled over internationally recognised refugee rights covered by the UN 1951 Convention and 1967 Protocol on Refugee Status (UNHCR, 1967 [1951]). The UNHRC was unequivocal in their declaration that such border restrictions intensified the refugee crisis (Chappell, 2016) and meant that European countries were unleashing a 'humanitarian crisis, largely of [their]... own making' (UNHCR, 2016).
The rise in migration across the Mediterranean, often in overcrowded boats, led to a sharp increase in the loss of life. In October 2013, a boat carrying hundreds of refugees and migrants from Libya to Italy sank near the island of Lampedusa, killing 368. Between January and March 2015, it is estimated that 479 refugees drowned or went missing, compared to 15 in the same period the year before (UNHCR, 2015c: 8). In the single month of April 2015, it was estimated that 1,308 refugees and migrants were lost at sea, compared to 42 in the same month the previous year (UNHCR, 2015c: 2): 'During the first six weeks of 2016, 410 people drowned out of the 80,000 people crossing the eastern Mediterranean... a 35-fold increase year-on-year from 2015' (Procaccini, 2016). In just one week at the end of May 2016, the UNHCR estimated that 'at least 880 people... died... as their vessels capsized in the Mediterranean, bringing the total fatalities along the dangerous crossing route to 2,510' in the year to that point (UN News Centre, 2016).
These figures are horrific but, again, deaths at sea do not only occur in the Mediterranean. The UNHCR estimated that almost 70,000 refugees, asylum-seekers and migrants – primarily from Ethiopia and Somalia – reached Yemen by sea in 2015. In the first 10 months of 2015, there were 88 recorded deaths of people travelling by sea between the Horn of Africa and Yemen. In early October 2015, 'a boat with migrants and refugees capsized in the Arabian Sea. Of the 68 passengers, only 33 survived' (UNHCR, 2015b). However, those who arrived in Yemen found themselves in the midst of an ongoing conflict, which led to a significant increase in the number of internally displaced people (IDP). In October 2015, the UNHCR estimated that Yemen's IDP population reached a 'record high' of 2,305,048 (UNHCR, 2015b).
Andreas Needham (2016) reported that mixed maritime movements in South-East Asia were three times more deadly than in the Mediterranean in 2015. Across the region, an estimated 33,600 refugees and migrants took to the sea to travel, including 32,600 in the Bay of Bengal and the Andaman Sea, and an estimated 1,000 who crossed the Straits of Malacca or attempted to get to Australia from Indonesia, Sri Lanka and Vietnam (UNHCR, 2015a). This included large numbers of Rohingya from Myanmar, who have faced state repression and officially sanctioned Islamophobia. According to Graham-Harrison (2015):
The Rohingya are faced with two options: stay and face annihilation, or flee.... Those who remain suffer destitution, malnutrition and starvation; severe physical and mental illness; restrictions on movement, education, marriage, childbirth, livelihood, land ownership; and the ever-present threat of violence and corruption.
The pace of departures picked up in 2015. It is estimated that up to 25,000 set off from the Bay of Bengal in the first three months of 2015 – double the levels in 2013 and 2014 – but approximately '12 of every 1,000 people who embark on mixed maritime movements from the Bay of Bengal do not survive the boat journey' (UNHCR, 2015a: 4).
Faced with this global crisis, the response of the most powerful states has been to further tighten their border controls. The dominant political model established across Europe, Australia and the US has been one of, what Carr (2012) calls, 'hard borders' against external migrants and refugees.
In Europe, the 'fortress' model has combined the internal liberalisation of movement (via the Schengen Agreement) with hard border controls on the periphery of the region. This has effectively pushed border controls onto the poorer, peripheral regions of the EU. On the EU borders, somewhere in the region of 400,000 border guards deployed through agencies such as Frontex (the European border agency) and Rapid Intervention Border Teams (called RABITs) mix with immigration officials and national police forces to control EU entry points. The EU border is covered by 'Satellite surveillance; naval patrols... a proliferation of immigration detention centres that extend across the continent and beyond; 'offshore' border controls and neighbourhood partnerships aimed at monitoring and trapping unwanted travellers before they even reach European territory' (Carr, 2012: 23).
Since the early 1990s, the US has built larger and more fortified fences on its Mexican border and has utilised surveillance technologies, border patrols and guards in an attempt to control Mexican and Latin American migrants. Those who are apprehended after crossing the border will be 'returned' to Mexico or held in US detention centres. According to Amnesty International (2009: 3): 'In 1996, immigration authorities had a daily detention capacity of less than 10,000.... Today [ie 2009] more than 30,000 immigrants are detained each day.... More than 300,000 men, women and children are detained by US immigration authorities each year'. The hard border policy in the US has created an atmosphere of growing hostility to migrants. Between 2010 and 2011, 164 anti-immigrant laws were passed by state legislatures across the US – indeed, only seven states (Alaska, Connecticut, Delaware, New Hampshire, Ohio, Wisconsin and Wyoming) did not pass anti-immigration Acts, with some passing as many as six Bills during this period (Gordon and Raja, 2012).
The UNHCR's Asylum trends report (2015e) notes that Australia received 8,960 asylum applications in 2014 – about 1% of all global applications. Refugees have attempted to reach Australia on boats from Indonesia, and mostly come from Afghanistan, Sri Lanka, Iraq, Iran or Myanmar (Burma). Hundreds have died making the dangerous journey. Yet, as they approach the Australian waters, they have been sent back out to sea in lifeboats (BBC, 2015a). Australia's 'hard borders' policy has included building offshore detention centres in remote Nauru and the Papua New Guinean island of Manus. In these centres, there have been repeated reports of violence against asylum seekers, including the sexual assault of children and the rape of men and women in detention, as well as other abuses, such as detainees being given out-of-date food and expired medication (Doherty, 2016).
The global refugee crisis and the closing of borders by powerful First World countries to vulnerable people fleeing war, a range of political, economic and environmental crises, and all manifestations of poverty and need pose important questions for social work. Given our strong value base and our commitment to international standards of practice, what can and should we do in the face of this situation? To begin to answer this question, we need to put the present crisis into a longer historical frame.
Social work with refugees and asylum seekers in England
Asylum Link Merseyside is a drop-in support centre for refugees in Liverpool, UK. The centre is supported by more than 90 volunteers. It runs English-language classes and serves lunch to around 220 people a day. There is access to free clothing for the homeless and cheap clothing for those needing support. There is limited housing and food for the destitute. Well Being programmes (bikes, allotments and football) and social events enable networking and solidarity. Individual casework support helps people navigate the complex legislation and UK Border Agency policy that determines refugees' and asylum seekers' rights to assistance.
Qualified social workers and social work students on placement work across the organisation's activities but are key providers of casework support. The work includes supporting people as they apply for help with accommodation or subsistence costs under section 95 of the Immigration and Asylum Act 1999, or help for those refused asylum and left destitute under section 4 of the Act.
The system is dehumanising, isolating and degrading, and destroys both dignity and self-respect. It disperses people around the country without any choice as to location. When they are given financial support, it is at a rate lower than existing benefit levels, sometimes via a 'payment card' that can only be used at certain shops. Using the cards is stigmatising and shaped by social policy responses steeped in notions of 'deterrence'.
Social workers working on section 4 applications have to navigate very strict criteria imposed by the UK Home Office. Applications can take weeks or months to be processed and people are often left homeless and destitute while decisions are made. A recent tactic employed by the state is for decisions to be drawn out, with constant requests for further information, one piece at a time, leading to further frustration and delay. All of this can have hugely damaging effects on individuals and lead to engagement with other areas of service delivery (eg mental health support, drug or alcohol addiction teams, etc).
Assisting people to fight for their liberty is at the heart of the caseworker's role at Asylum Link. They support people to appeal unsuccessful claims and access further statutory assistance. They also look to assist with any housing problems people face and offer to help with access to GPs and other health-care services. The role inevitably involves tapping into other local refugee and asylum support services. Asylum Link works closely with the Merseyside Refugee Support Network, Greater Manchester Immigration Aid Unit, Refugee Action and the British Red Cross to help secure a range of services unavailable at the centre. The charity also works with statutory services, including councils, the Home Office, the police, GPs, MerseyCare and other health providers. However, as an independent agency, Asylum Link also has to challenge the decisions made by statutory agencies where necessary, and, at times, relations can become strained between social workers working in different agency settings. Proposed new legislation will undoubtedly make things worse.
### Refugees, migrants and asylum seekers
One of the most pressing issues for social workers is to be clear about the language used around the refugee crisis. The language used to describe migrants, refugees and asylum seekers matters. To emphasise the point, let us take an example from the UK. Figures indicate that just as many UK citizens live outside the UK in the EU as there are EU citizens living in the UK, despite popular perceptions (Rettman, 2014). Just over 1 million people born in Britain live in Spain, 330,000 in France and 329,000 in Ireland. Yet, the British media and politicians never refer to these people as 'migrants' (and certainly not refugees or asylum seekers); they are more often referred to as 'ex-pats' – a term that has few negative connotations.
Political scientist Murray Edelman's (1985 [1964]) work on 'condensation symbols' looks at the way(s) that key words in political discourse become loaded with myth, emotion and symbolism. There is no doubt that the words 'migrant', 'refugee' and 'asylum seeker' are loaded with (often unspoken) meaning. As Carr (2012) notes, in the post-Second World War era, in the context of the Cold War, the term 'refugee' was often viewed positively, referring those fleeing the 'horrors' of the East European 'Communist' (or, in reality, state capitalist) societies. However, by the last quarter of the 20th century, with the ending of the Cold War and increasing numbers of refugees from the Global South seeking protection from wars and political turmoil in the West, the terminology shifted. These people were now increasingly:
Portrayed by politicians and the media as 'economic migrants' seeking to evade the continent's immigration restrictions. By the end of the Cold War, 'asylum seeker' was firmly enshrined in European political and media discourse as a sub-category of refugee whose legitimacy had yet to be proven and whose claims were often assumed to be suspect. These assumptions were often steeped in racial prejudices in which 'asylum seeker' became a code word for Third World immigrants in general. (Carr, 2012: 22)
Berry et al (2015) undertook a study of media language used to describe refugees entering the EU in 2015. They looked at the press coverage in five countries (German, Sweden, Italy, Spain and the UK). They found that:
The use of labels (migrant, refugee, immigrant etc.) varied markedly by country. Both Germany (91.0%) and Sweden (75.3%) overwhelmingly used the terms refugee (flüchtling(e)/flykting) or asylum seeker (asylsuchende(r)/asylsokande). In contrast migrant (migrante) was the most used term in Italy (35.8%) and especially the UK (54.2%).... In Spain, the dominant term was immigrant (immigrante) which was used 67.1% of time. (Berry et al, 2015: 7–8)
More recently, developing Carr (2012), we suggest that the terms 'migrant', 'refugee' and 'asylum seeker' are increasingly prefixed with the adjective 'Muslim' (although, in reality, many refugees are not Muslims and do not come from Muslim countries). Thus, the refugee crisis has emerged from and merged with the wars and turmoil created by Western interventions in the Middle East to reinforce Islamophobia and general hostility to Muslim migrants and Muslim communities more generally.
Let us look at the formal definitions. A refugee is someone who has fled armed conflict or persecution or who:
owing to a well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion, is outside the country of his nationality and is unable or, owing to such fear, is unwilling to avail himself of the protection of that country. (1951 UN Convention Relating to the Status of Refugees)
As such, it is recognised that refugees need international protection and are protected under international law by the 1951 Refugee Convention.
Thus, refugees who arrive in countries are subject to international conventions and domestic legislation based on the 1951 Refugee Convention, which 'commits agencies to provide services ranging from housing, health, employment, emergency aid and education' (Robinson, 2013: 2). The convention's basic principle is that refugees should not be expelled or returned to situations where their life and freedom would be under threat. Once someone has been recognised as a refugee, they are supposed to be given access to social housing and welfare benefits and helped to find a job and integrate into society. An asylum seeker is someone who is officially a refugee and has applied for asylum in another country. However, the legal process and assessment has not yet been completed; hence, they are seeking asylum.
A migrant is someone who moves from one place to another in order to live in another country for more than a year: 'It is estimated that some 232 million people or 3 per cent of the world's population live outside their country of birth' (Williams and Graham, 2014: il). Migrants include those who move to work or seek a better life: they may be international students, or those who move for family reasons; some will move to flee war, persecution or the effects of climate change; some will move because of the crushing effects of poverty; and some will move because their skills are in demand – or their personal wealth effectively allows them the freedom to settle where they wish. Of course, it is possible that people move because of a mixture of these reasons – that people fleeing the war in Syria face crushing poverty at home and want to better their own, and their families', prospects elsewhere.
Much popular discussion around the issue of migration assumes that people move through 'free choice'. Of course, some do. However, the overarching conditions shaping human movement are the social conditions created by capitalism and its impacts on our world.
'Popular' social work with refugees in Samos, Greece
The island of Samos, Greece, sits in the eastern Aegean Sea, separated from Turkey by the mile-wide Mycale Strait. Its 'normal' population is 32,977, yet this figure has been significantly increased because the island is one of the 'hot entry' points for refugees and migrants attempting to access 'fortress Europe' (Carr, 2012).
In the summer of 2015, approximately 1,500 refugees were arriving on Samos each day (Lee, 2015). Denied safe passage, they came by night packed in small rubber inflatables. Significant numbers did not make it, turning the sea into a graveyard. The official response to the migrant population processes them like criminals. The police are the main agency responsible. There is little support, minimal food and shelter, and no clothes or material goods on offer to replace those lost at sea or sold to pay for their hazardous journeys. They are housed behind a barbed-wire enclosed purpose-built 'reception centre', where they await papers that permit them to move on to Athens and an uncertain fate.
Faced with state indifference, a group of local activists, former social workers and volunteers formed 'Samos Refugees' to work with, and support, the migrants. Each morning, the group travel the island looking for those who have arrived overnight and pick them up from the beach. Until the arrival of some big non-governmental organisations (NGOs) in November 2015, this small group helped feed, clothe and support around 200–300 refugees a day.
The social work extends to playing with the children and talking to those traumatised by their perilous journey and the destruction of their homelands. They use every opportunity to involve the refugees in the daily tasks and to deepen their solidarities. All of their work is fundamentally informed by the key social work value of respect and an equally precious social work understanding that power and authority is never the friend of the poor and vulnerable.
So, each day, members of the group defy the authorities. They refuse to seek the permission of the police to be humane, and take food and other resources to those held behind the fences and in the police cells. Part of their work is political: involving refugees and migrants in various political events, and listening to the voice of the migrants and taking their advice over policy direction.
Samos Refugees vigorously maintains its independence. They refuse all overtures to collaborate with the 'international aid community'. However, the price of collaboration is high, for it entails keeping silent and making no criticism of the authorities. Finance is also a constant challenge, but through its website, it has so far been successful in securing funds for its work.
The project's commitment to working with and alongside the refugees, to involvement of refugees in decision-making and policy direction, to breaking down and challenging any notion of hierarchy between 'workers', 'volunteers' and 'refugees', marks Samos Refugees out as an example of high-quality popular social work working in this field (Chalalet and Jones, no date).
### Migrants, refugees and capitalism
Migration has been part of human history from its earliest times. However, the speed and significance of migratory movements has increased significantly with the spread and development of capitalism.
Capitalism was built on the forced dispersal, enslavement and internal displacement of people from the land to urban centres, or from regional homelands to agribusiness across the 'new world'. The transatlantic slave trade brutally and forcibly transported people to work in enslaved, 'unfree' labour in the plantations of the southern states of the US and the Caribbean (Blackburn, 1997). The trade in humans, in part, provided the wealth that funded the Industrial Revolution in Britain, France and the US, which, in turn, sucked 'free' labour (migrants) into the developing towns and cities from the surrounding rural areas (or, in the case of the US, from other countries), where they worked long, unregulated hours in arduous conditions. Of course, the reality of enslavement was far more barbaric than the brutality of urban life during the Industrial Revolution; nevertheless, both processes were rooted in the drives of capital accumulation in the early phase of capitalist development.
Within capitalism, various forms of 'unfree labour' have always existed (eg slavery or indentured labour) but capitalism is characterised as a system where labour is commodified – people meet their daily needs by selling their ability to work on the labour market – which is considered a system of 'free labour' (Miles, 1990).
The use of the term 'free labour' has two aspects to it. First, it means workers are not tied to any particular employer, they are 'free' to enter a contract to sell their labour power to anyone, and, of course, an employer is 'free' to employ those they consider can best meet their productive needs. Ideologically, this aspect of 'free labour' is utilised to give the impression that the employers and workers meet each other in the labour market as, apparent, equals. It also suggests that we all have a 'free choice' about who we work for, where and when.
However, things are not so straightforward. A second sense of 'free labour' is vitally important: workers within capitalism are also 'freed' of any other means of living or supporting themselves and their families (eg they do not have access to significant land to grow crops or raise animals, or they do not have independent wealth). Thus, they face a major problem: if they do not work, they face desperate poverty. Therefore, alongside their 'freedom' comes 'economic compulsion': their only means of supporting themselves and their dependants is by selling their labour power by undertaking waged labour.
The wage labourer enters a labour market that is dynamic and subject to all manner of changes. Marx and Engels (1848: 13), writing in The communist manifesto, describe capitalism as a system that 'constantly revolutionises the forces of production', with, as a result, 'uninterrupted disturbance of all social conditions, everlasting uncertainty... [where] All that is solid melts into air'. Furthermore, this is a system that constantly seeks to expand the 'market for its products' and therefore forces 'the bourgeoisie over the whole surface of the globe. It must nestle everywhere, settle everywhere, establish connexions everywhere' (Marx and Engels, 1848: 13).
What Marx and Engels highlight here is various ongoing tendencies within capitalism, the drive: to expand into new markets; to invest in new machines and technologies of production, in part, to establish cheaper ways of producing commodities; to search out cheaper sources of labour power; and to create new commodities to sell. As a result of these tendencies, the working class is constantly created and recreated – jobs that were once common can disappear, to be replaced with other types of employment and this, in turn, forces people to change jobs, to move location, to retrain or to be left redundant and abandoned to a more or less impoverished future.
Within capitalism, therefore, there are always 'push' and 'pull' factors encouraging some of us to move to look for opportunities to work and engage in the labour market. The 'push' factors vary according to time and place. In the earliest phases of capitalist development, social, economic and political developments forced people away from their traditional homes: land enclosures, famine, religious and political persecution, poverty, and political and economic turmoil forced people off the land and 'pulled' them to work in factories, mines, mills and railways in the newly expanding towns and cities. In the 19th century, some of the 'migrants' may not have moved far geographically (though some moved continents), but the social dislocation that people experienced was significant and traumatic.
The 'push–pull' pattern of 19th-century migration was repeated throughout the 20th century and continues to play a significant part in migratory movements today. In the face of grinding poverty, of disconnection from the land, of environmental catastrophes like flooding or drought, of famine and disease, of economic crisis and unemployment, or of war or political turmoil, people look to migrate. Sometimes they move to the local town or city in search of wage labour, sometimes they move to other parts of the country and sometimes they seek new beginnings in different lands.
Historically, of course, this means that the vast majority of us are migrants or the children of migrants. If we search our family backgrounds, then, at some point, some of our relatives have migrated with the hope that the move would improve their life and that of their dependants.
During much of the 19th century, migrants were viewed as an economic resource who would fuel economic development in the host location. The dominant philosophy of 'free market liberalism' equally applied to the 'free market in labour'. As a result, until the second half of the 19th century, there were few state immigration controls. However, this gradually changed. As the state started to take on a great organising role within society (the growth of what is sometimes referred to as the 'interventionist state'), it took on a more directing role within the economy and within social and political life. The state became more concerned with the organisation of its population, of its education and 'health', and such concerns were often imbued with eugenicist ideas of 'race' and nation: the notion that economic and imperial ambitions could be undermined by 'inferior' human stock – meaning, for example, the poor, the elderly and the disabled at 'home', as well as 'inferior migrants' coming from other lands.
Many of the first controls were introduced to restrict Chinese migration. In the aftermath of the Taiping Rebellion, large numbers of Chinese refugees looked to leave the country and were attracted to the US, Australia and Canada, both to work on the railroads and to participate in the 'gold rushes' of the time. However, the Chinese migrants faced hostility and increasing controls on their movement. Once implemented against Chinese workers, these laws soon spread and were applied to other groups.
In the US, the Page Act 1875 was the first attempt at the federal regulation of Chinese migrants, while the Chinese Exclusion Act was passed in 1882. These exclusions were then generalised and encoded in the Immigration Act 1903, which also included a clause to exclude 'anarchists'. In Australia, the Immigration Restriction Act 1855 targeted Chinese labour, though this was later superseded by the Immigration Restriction Act 1901, which formed the basis of the White Australia policy that sought to exclude all non-Europeans from Australia.
In Canada, the Chinese Immigration Act 1885 introduced a head tax on Chinese migrants and was the first attempt to control immigration to Canada. It was soon to be followed by similar legislation targeting Italians. In the UK, the first immigration controls (the Aliens Act 1905) were implemented to control Jewish migration as Jewish workers and peasants fled Tsarist pogroms in the Pale of Settlement. The legislation was implemented in response to agitation by the racist, anti-Semitic organisation the British Brothers League.
From their onset, then, immigration controls have discriminated on the basis of 'race', ethnicity and national origin; that purpose is built into their very DNA. The idea that it is possible to have immigration controls, controls on the movement of people, which do not reinforce discrimination and prejudice is an oxymoron.
Jane Addams and Ellen Gates Starr
Jane Addams and Ellen Gates Starr are well known as the founders of Hull House settlement. They founded Hull House in 1889, where it became an important focal point for newly arrived migrants to Chicago, USA. The settlement became an important centre for Greek, German, Jewish, Irish and Italian migrants, and helped ease the new migrants' integration into the broader Chicago community. According to Addams, the settlement was structured around three principles, known as the three Rs: residence, research and reform. The last of these, 'reform', emphasised a clear commitment to improving the lives of those who the settlement worked with through political reform and campaign work. The settlement offered lectures, educational classes, drama workshops and free meals for children, and engaged in a range of campaign activities in their locality. It is one of the earliest examples of social work with migrant communities (Addams, 1910).
### Social work with refugees
On World Social Work Day 2016, the two main international social work organisations (the International Federation of Social Workers and the International Association of Schools of Social Work [IASSW]) released a joint statement about the refugee crisis. The aim was to establish a five-point working plan to:
1. Coordinate social worker action in each of the affected countries to provide better understandings and responses to refugee needs during their journey, transfer and integration in the asylum countries.
2. Provide a focused strategy that supports vulnerable groups such as unaccompanied children and young people, older people, those with health issues, and trafficked persons.
3. Establish a comprehensive political advocacy strategy that reflects a 'ground-up' perspective on refugee needs, aspirations and solutions.
4. Develop social work models that support refugees in isolated or life-threatening situations where other forms of assistance are not available, including that there will be an increasing number of refugees who face closed borders or hostile host communities who will face the critical dilemma of staying where they are not welcome or returning to a war situation that may result in their death.
5. Enhance the skills of social workers working with others to constructively develop inclusive and cohesive societies (IASSW, 2016b).
In addition, the Social Care Institute for Excellence (SCIE, 2015) has established five principles for practice with refugees and asylum seekers, which reflect social work values and a rights-based approach to meeting social care needs. These principles are:
1. A humane, person-centred, rights-based and solution-focused response to the social care needs of asylum seekers and refugees.
2. Respect for cultural identity and the experiences of migration.
3. Non-discrimination and the promotion of equality.
4. Decision-making that is timely and transparent and involves people, or their advocates, as fully as possible in the process.
5. The promotion of social inclusion and independence.
Both the IASSW goals and the SCIE principles are admirable statements – though they perhaps underplay the negative role that some social workers play in the processing and assessment of vulnerable refugees (such as their involvement in the age assessment of child migrants [Moran and Gillett, 2014]).
The IASSW (2016b) statement goes on to note that:
Millions of people caught up in the crisis are working with social workers, some employed by the state or NGOs, and many who are volunteering. They have the skills to work effectively with very traumatised and distressed people. They also act as the catalyst in organising volunteers to welcome and assist the integration of refugees into new areas.
The IASSW makes an important point – one that stretches beyond social work. In the face of the refugee crisis, many in the media and large numbers of politicians emphasised the 'problems of large migratory flows', focusing on issues of control and numbers, and the social tensions that they predicted would 'inevitably' occur. Yet, the plight of refugees and migrants has generated significant levels of support in many countries. As Carr (2015) notes: 'What were once marginal forms of solidarity with refugees and migrants have moved into the public mainstream'. Richardson (2016) discusses the extent of refugee solidarity:
On 12 September 2015, SUTR [Stand Up To Racism] convened... a demonstration... at just two weeks notice. An estimated 80,000 people marched.... Across [Britain]... people have collected provisions in schools, colleges, workplaces, among faith groups and in local communities.... Across Europe ordinary families have offered refugees a warm welcome. Where the authorities have refused to allow asylum seekers in, activists have risked their own liberty by organising transport to ferry them across the borders.... In Greece... families cook[ed] extra food in order to provide a hot meal for refugees. Such solidarity has even been on display in some unlikely quarters. Those of us who are football fans are often on the defensive about the macho and offensive behaviour that frequently emanates from the stands. Yet even here the hoardings of football stadia have been draped with banners declaring 'Refugees Welcome Here'. Furthermore, a match between two Greek sides, AEL Larissa and Acharnaikos, was briefly halted when players from both sides sat down in protest at the treatment of Syrian refugees.
Social workers have not been immune to such solidarity drives. As the refugee crisis in Europe erupted in August 2015, large numbers of social workers joined campaigns for refugee rights. The Social Work Action Network (SWAN), drawing on longer traditions of solidarity with refugee struggles going back to networks like International Red Aid, which operated in the late 1920s and 1930s, worked with the UK-based anti-racist organisation Stand Up To Racism to organise convoys taking material aid and solidarity to refugees in both France and Greece. The convoys also joined political campaigns for refugee rights. This included SWAN having an official speaker at a demonstration inside the Calais Jungle demanding that refugees, and unaccompanied child refugees in particular, be allowed into the UK.
The SWAN convoys drew social work students and practitioners from across the UK to go to Calais, France, but it was impractical for many social workers to return to France on a regular basis. Out of the initial convoy, some SWAN supporters in Kent organised to travel back and forth to Calais on a more regular basis. Setting up Social Work First, practitioners travelled to the refugee camps and carried out needs assessments with the children. Schraer (2016) reported on the group's activities by quoting a practitioner who had taken part in the assessment process:
Our aim was to complete a single assessment on these children. Of course, they're not in the UK so they have no right to a statutory assessment, but we wrote an assessment of their needs... based on the format we would use at work.
In the camps, the network found that, within the dimensions of a single assessment, not a single need was being met.
Social Work Without Borders was a second network that formed in 2016. Drawing on experience of working in Leros, Greece, the network set itself up to safeguard women and children and challenge hostility to refugees in Europe (Wroe, 2016). As Wroe (2016) argues:
Inspired by refugee women speaking about the risks they face as they make their journey through Europe, I spent my last week creating a quiet space for women in the camp by converting a storage room into a small lounge. My observation is that it provided a release for women and girls whose presence is often overshadowed in the busy camps. By bringing refugee women together with female volunteers we created an environment in which safeguarding issues relating to these women could be identified, recorded and responded to.
These examples of campaigning social work models grew out of the refugee solidarity movement but established a distinctive role for social workers within it. These campaigns combine a focus on individual support and relationship building with a campaigning edge to address the political causes of the present crisis. In this, they drew on older traditions of radical social work practice in the refugee support field.
Mentona Moser
Mentona Moser was born in 1874 and was brought up in Southern Germany and Switzerland. In 1894, she travelled to London to study, where she came into contact with the Settlement Movement. In 1899, she worked at the Southwark Settlement, where she was confronted by extreme levels of poverty. This, and her growing contact with the workers' movement, had a radicalising impact on Moser. Over the following few years: she worked with disabled people, young women and families; she railed against 'snooping' inspectors and welfare bureaucrats; she went to work for the Swiss Association for Child Protection; and she was involved in setting up 'garden cities', workers' settlements and children's playgrounds in working-class Zurich (Moser, 1985, 1986). In 1919, she joined the Swiss Communist Party (Hering, 2003), and in the 1920s and 1930s, she became heavily involved in International Red Aid (Schilde, 2009). International Red Aid was active across most of Europe in the late 1920s and 1930s. It offered support to refugees and those fleeing political persecution. It was based on the centrality of 'solidarity' to those who were 'victims of the class struggle' (Schilling, 2015).
### Practical demands for an internationalist social work
The global definition of social work encapsulates a series of universal values applicable to all people in all societies. It asserts that 'Principles of social justice, human rights, collective responsibility and respect for diversities are central to social work' (IFSW, 2014). The present global refugee crisis poses some very particular questions for radical social workers internationally. We end this chapter by suggesting some practical social work demands that can shape social work intervention in this field.
#### Safe passage for refugees and migrants
In 2014, the Mare Nostrum operation, run by the Italian navy in the Mediterranean, saved an estimated 100,000 lives. Rather than welcoming the operation, the UK Home Secretary at the time, Theresa May, argued that it actively encouraged 'migrants' to make the perilous journey to Europe. In 2015, the EU cut funding for the Mare Nostrum programme and, as a result, there was, almost immediately, a sharp increase in deaths at sea.
The lives of thousands of vulnerable refugees fleeing war, human rights abuses, poverty and repression continue to be lost, needlessly, at sea, in the back of trucks, during travel on hazardous 'routes to safety' and in makeshift camps where they live in unacceptable conditions. The stretch of the Aegean Sea between Turkey and Greece is among the deadliest routes in the world for refugees and migrants: 'During the first six weeks of 2016, 410 people drowned... crossing the eastern Mediterranean. This amounts to a 35-fold increase year-on-year from 2015' (IOM, 2016).
As those seeking refuge travel, they are vulnerable to exploitation from those in authority, by border guards and by those who engage in 'people smuggling'. Politicians and media commentators often blame refugee deaths on 'traffickers', but this is too one-sided. Of course, when we see refugees killed in the back of overcrowded trucks or lost at sea after being crammed onto overcrowded and unsafe boats, it is clear that some traffickers are no more than gangsters out to exploit vulnerable people. However, there is a relationship between traffickers and refugees. Refugee networks pass on information about safer routes of passage and people to avoid if possible. Furthermore, there is a level of hypocrisy in governments blaming traffickers for refugee deaths:
no one would have to rely on traffickers if they were able to get to the places they want to go to by legitimate legal means. It is almost impossible for someone outside Europe to apply for asylum in Europe. The moment they book a flight they are likely to be identified and when they arrive at the airport the plane won't carry them because the carrier might be fined if they do. A few years ago there was a tabloid furore [in the UK] about Mugabe in Zimbabwe and how evil his regime was... most of which is true, but what was the UK government doing? It was making it harder for people who wanted to leave Zimbabwe to get to the UK. And that's what always happens whenever there's a country that seems like it's going to produce a lot of refugees – we simply make it more difficult for these people to obtain visas. (Carr, 2015)
As refugees take their lives in their hands and travel across the Mediterranean on boats that are often overcrowded and unsafe, they travel routes that are served by commercial passenger ships. In Lesbos and Samos, for example, refugees, if they survive the journey, will land on shore near ports where ships disembark passengers who have made the same journey in safety and paid a fraction of the price that refugees will have been charged to make their risky passage. The only way to ensure that the refugee crisis gets no worse, and to control the number of preventable deaths, is by introducing 'safe passage' measures:
• Effective access to coherent asylum procedures and assistance should be provided at entry points along migratory routes.
• Legal migration pathways need to be established.
• Dignified reception conditions must be offered to all.
#### Against incarceration
Across the so-called advanced world, governments are locking up refugees in detention centres. The UNHCR (2012a: 6, 12) are clear that:
The rights to liberty and security of person are fundamental human rights, reflected in the international prohibition on arbitrary detention, and supported by the right to freedom of movement.... Every person has the right to seek and enjoy in other countries asylum from persecution, serious human rights violations and other serious harm. Seeking asylum is not, therefore, an unlawful act. Furthermore, the 1951 Convention provides that asylum-seekers shall not be penalised for their illegal entry or stay, provided they present themselves to the authorities without delay and show good cause for their illegal entry or presence.
In 2015, 14,832 asylum seekers were locked up in detention centres in the UK. Approximately half of all asylum seekers find themselves detained during the asylum process, and this includes children, 154 of whom were detained in the UK between 2010 and 2015 (Refugee Council, 2015). In the US, Amnesty International is quite explicit that the government's use of detention centres 'as a tool to combat unauthorized migration falls short of international human rights law' (Amnesty International, 2009: 3), while in Australia, a report by the Australian Human Rights Commission concluded that 'immigration detention in [Australia] is harmful to the health and mental health of young children and youth' (Elliott and Gunasekera, 2016: 3). As such, the following should be recognised:
• Refugees, asylum seekers and migrants are not 'illegal'; they have committed no crime.
• Detention centres should be closed down as a breach of people's fundamental human rights.
#### Support for unaccompanied minors
Between September 2015 and February 2016, it was estimated that, on average, two children drowned every day trying to cross the Eastern Mediterranean, a figure that, if anything, is likely to be an underestimate given the numbers lost and never found (Procaccini, 2016). In the first five months of 2016, the United Nations Children's Fund (UNICEF) claimed that more than 7,000 unaccompanied children made the crossing from North Africa to Italy (The Guardian, 2016), while elsewhere, the UNHCR estimated that children accounted for 36% of those on the move (Procaccini, 2016). Figures from the EU's criminal intelligence agency suggest that at least 10,000 unaccompanied child refugees disappeared after arriving in Europe between 2013 and 2016 (Townsend, 2016).
Unaccompanied asylum-seeking children should have the same rights and entitlements to services as other young children. In the UK:
Local authorities have a statutory duty under the Children Act (1989) to safeguard and promote the welfare of all children 'in need'. Children 'in need' are those whose health and development would suffer without the provision of services. Unaccompanied children and young people, lacking the presence of parents or customary caregivers, are by definition children 'in need'. (Wade et al, 2005: 7)
Central to social work values is the need to protect and safeguard the needs and rights of children:
• Vulnerable, unaccompanied child refugees, 'children in need' as understood in the UK context, should have a right of entry to their country of choice.
• Refugee children should have a right to education, a right to protection and full access to welfare and social service provision.
#### The right to family reunification
The circumstances in which refugees flee their homes often lead to significant family disruption and separation. Family reunification is an essential part of bringing some form of normality back to refugees' lives. Pursuing the reunification of partners, parents with their children and siblings with each other should be viewed as a central social work task in our work with refugee communities. As the UNHCR (2012b: 3) states:
The reality for many refugees... is... that they have lost track of their families or have had to leave them behind. The family however plays an essential role to help persons rebuild their lives and can provide critical support to adapt to new and challenging circumstances. Restoring families can also ease the sense of loss that accompanies many refugees who, in addition to family, have lost their country, network and life as they knew it. Family support in this sense goes beyond any traditional and cultural understanding of a family but will include those who rely and depend on each other.
As such:
• Family reunification is a central human right that should be acted upon by governments.
#### The right to work
Refugee employment rights are enshrined in the 1951 Convention Relating to the Status of Refugees and its 1967 Protocol (referred to collectively as the 1951 Refugee Convention). As the Asylum Access and Refugee Work Rights Coalition (2014: 2) assert, 'access to safe and lawful employment is a fundamental human right... [which] applies to all persons, including refugees and asylum seekers'. They continue:
When permitted to engage in safe and lawful work, an individual may fulfil his or her basic survival needs and contribute to the needs of the family, community and the country in which they reside. The realization of the right is the means through which the individual may achieve a range of other civil, political, economic, social and cultural rights, fulfilling the human desire to feel useful, valued and productive. (Asylum Access and Refugee Work Rights Coalition, 2014: 2)
Refugees arrive with many skills and much to offer their host society. Yet, many states, for example, the UK, prohibit refugees from legally entering the labour market until their formal asylum status is established. They are forced into poverty and reliance on benefits, often paid at a rate lower than those offered to indigenous citizens:
• To maintain their human dignity, refugees should have the right to work.
#### Self-activity
Refugees are often portrayed as 'passive victims' of social and economic processes. Such a portrayal comes from media outlets, politicians, voluntary sector organisations and welfare institutions (Harrel-Bond, 1999). These perspectives deny refugee independence and agency. The ways in which refugees and asylum seekers are perceived as victims or threats has a direct bearing on their access to support, safety and services. Yet, there is a rich history of refugee self-activity and political campaign work that needs to be recognised and celebrated:
• Social workers need to recognise refugee agency and work and campaign alongside refugees in the co-design and co-production of suitable knowledge and practice (Moran and Lavalette, 2016).
#### No collusion with discriminatory laws
Immigration controls and laws discriminate against people because of their skin colour, religion, place or origin and identification as the 'other'. Non-discriminatory immigration controls are not possible – the concept is an oxymoron. Social work's commitment to anti-oppressive practice demands that the 'virtuous social worker', one who acts in accordance with our value base and system, is one who cannot collude with discriminatory laws and practices:
• Working in detention centres, working and colluding with government border control agencies, and processing refugees and asylum seekers in ways that infringe their human rights bring social workers into conflict with the profession's value base and question their 'fitness to practise'.
#### Open borders
Teresa Hayter (2000: 163) notes that 'Objections to immigration on economic and welfare grounds have little basis in reality... immigration controls exist mainly because of racism in the countries which apply them'. Here, Hayter makes two central points. First, the evidence of numerous studies is that immigration does not lead to increased unemployment among the 'indigenous population', nor does it undermine wages and working conditions, nor, indeed, does it place a significant strain on the economic system. In fact, refugees and migrants (when allowed to work) are overwhelmingly contributors to tax revenues and economic performance indicators in host nations. Per capita, they use fewer public services and consume fewer public resources than host nation citizens. This is because 'most international migrants are of working age (twenty to sixty-four years) [who] mak[e] up 74 per cent of the total' (Williams and Graham, 2014: il). Furthermore, there is little evidence to support a claim that migrants create unemployment. Indeed, they 'tend to take jobs which are shunned by the natives and therefore provide an essential means of enabling economies to function and expand' (Hayter, 2000: 159).
Hayter's (2000: 165) second point, therefore, is that 'immigration controls are inherently racist', and, indeed, by forcing refugees and asylum seekers into 'illegality', governments further fuel racism and prejudice within their countries. She concludes that 'To deny people the right to migrate is harsh and oppressive, and is leading to unbearable extremes of cruelty' (Hayter, 2000: 172):
• The solution is for a policy of 'open borders' and 'safe passage' that gives people the freedom of movement to settle anywhere they wish to.
RAPAR
RAPAR (Refugee and Asylum Seeker Participatory Action Research) was formed in 2001 in Manchester, UK (Moran and Lavalette, 2016). It is a social welfare and human rights organisation that brings academics, practitioners and refugees and asylum seekers together (as equal members of the organisation) to work on individual casework and community campaigns, and to undertake relevant research to shine a light on the predicament of forcibly displaced refugees. RAPAR is based on the notion of refugee and asylum-seeker agency. Those that work with the organisation become full and active members of the organisation, fighting for their rights and the rights of others.
There are three central values that shape the organisation's work. The first two are drawn from the work of Friere (1972): anti-assistentialism (the conscious rejection of dependency) and conscientisation (a commitment to transformative learning processes that arise from campaigning and participatory action research). The third, solidarity, is captured in the work of Galeano (2004 [1999]), where, in contrast to the 'vertical', hierarchical support offered by charity, he describes 'horizontal support' for equals that reveals our common humanity.
RAPAR provides basic legal training and support to refugees to fight and campaign on their own behalf, alongside volunteers from the social work, legal and welfare professions. It works with members around campaigns in their local communities for housing, access to education and other welfare services, and the service undertakes participatory action research to provide data to counter dominant ideas about refugees in British society.
### Conclusion
In the face of the contemporary 'refugee crisis', social work is faced with a fundamental question: whose side are we on? This is a question that can only be answered politically. The refugee crisis demands that we meet the needs of vulnerable people for social and material support and protection as they flee war, poverty, environmental degradation and oppression. Confronting the refugees, we have powerful states and elites who have imprisoned, impoverished, dehumanised and stigmatised the vulnerable. For a profession committed to social justice, equality and confronting both oppression and inequality, there should be no doubt: we stand with refugees. Our task is to work in solidarity with refugee communities, and, as a profession, we cannot and must not collude with agencies that undermine their social, political and civil rights.
## SEVEN
## Social work, climate change and the Anthropocene
The six-month period from January to June [2016] was the warmest half-year on NASA's [North American Space Agency's] global temperature record, with an average temperature that was 1.3 degrees Celsius (2.4 degrees Fahrenheit) warmer than the late nineteenth century. This follows 2015, which was the warmest year on record and among the warmest decade on record. The ongoing warming trend – as well as the increasing frequency and severity of high-humidity heat waves – is ultimately driven by rising concentrations of carbon dioxide and other greenhouse gases in the atmosphere. (Voiland, 2016)
### Introduction
One of the most pressing problems confronting humanity is the threat posed by human-induced climate change. This is an issue that social work is only slowly starting to address. In some parts of the world, for example, India, Canada and Australia, social workers are engaged in forms of social work for environmental justice. This is occasionally grouped under the rubric of 'green social work' – an eclectic mix of ideas and practices derived from a range of concerns about human impact on the global environment and global ecology.
The international definition of social work includes clear reference to the environment; yet, too often, this is interpreted unidirectionally as the impact of environments on people and their lives. It is increasingly clear that the impact of humans in their environment upon the planetary system is creating significant social problems that social workers will have to face in practice. As such, social work training and education programmes need to develop the professional knowledge base to consider and teach about climate change while developing and learning skills that could be fruitfully adapted to practice. This chapter engages with these important contemporary social work issues. We start by establishing the extent and roots of the crisis we face.
Medha Patkar, Indian social worker, environmental activist
In India, social workers have a long history of engagement in environmental campaigns, such as those to stop villages being flooded to build dams. Central to many of these campaigns has been the work and example of Medha Patkar. Medha Patkar was born in Mumbai, India, in 1954. In 1974, she gained her master's qualification in social work from the Tata Institute of Social Sciences (the providers of the first social work qualification in India in 1936), with a specialisation in community organising (The Viewspaper, 2008).
On qualification, she started to work alongside the indigenous peoples of the Narmada Valley. The Narmada river runs through Central and Western India, and in the 1950s and 1960s, the river and valley were identified as key sites for the Indian state's dam-building programme. The valley was to host:
30 major dams, 135 medium-sized dams and over 30,000 micro-harvesting (conservation) reservoirs. It was estimated these dams [would]... lead to direct displacement of 250,000 people living in the Narmada Valley and affect the livelihood of over one million people. (Pawar and Pulla, 2015: 79)
As the dams started to be built, local communities without the appropriate knowledge of their rights were often displaced to inappropriate locations, without adequate compensation and in the face of official indifference to their plight. In this context, Patkar utilised a range of creative and critical social work methods – derived from the Gandhian concepts of 'satyagraha' and 'asahakar' – to build a social movement (the Save Narmada Movement [Narmada Bachao Aandolan (NBA)]) in order to represent the indigenous peoples of the Narmada and fight with them for their rights.
Satyagraha is based on the 'power of truth', what Ghandi called a 'truth force'. Here, the idea is that campaigns and individuals search for truth, are prepared to 'speak truth to power' and act in ways that expose truth to the wider world. Asahakar means non-cooperation with those in power. Together, satyagraha and asahakar promote non-violent means of social action.
Patkar initially undertook action research to establish the extent to which people in the Narmada Valley knew about the development plans and their knowledge of their rights. She then used group-work methods to raise consciousness and promote social action (Mayani, 2009). The long-running campaign included: advocacy and rights-based approaches, where local, state and national politicians and civil servants were held to account; non-violent direct action against construction firms, financial institutions and representatives of the global financial institutions funding the projects; and establishing links between various oppressed and dispossessed groups and challenging perceptions of Dalit and indigenous communities.
There were marches and demonstrations, including (in 1991) a proposed 200km march that was stopped by police at the Gujarat–Madhya Pradesh border – leading to a 22-day hunger strike by a number of activists (including Patkar). In 1993, she went on hunger strike again to try and force the government to review the dam project. This strike resulted in her arrest after 14 days of fasting. This was not the only occasion that she was arrested and beaten by police (Jensen, 2004). The Save Narmada Movement that she established became India's largest social movement.
Her vision of social work is of: a value-driven profession committed to social justice, human rights and social change; professionals engaged and active in social movement activity; and the promotion of sustainable development and the protection of the environment as central social work goals. In 1991, Patkar and her NBA colleagues were given the Right Livelihood Award (the 'Alternative Nobel Peace Prize'), and in 1992, she won the Goldman Environmental Prize. Other awards that she has won include the Deena Nath Mangeshkar Award, the Mahatma Phule Award, the Green Ribbon Award for Best International Political Campaigner by the British Broadcasting Corporation (BBC), and the Human Rights Defender's Award from Amnesty International (The Viewspaper, 2008). In 2004, she was a keynote speaker at the World Social Forum in Mumbai. She stands out as a social work activist committed to environmental justice.
### The climate change problem
We are living in a world where unusual and extreme weather and climatic events are becoming more frequent – with devastating impacts on the environment and on humans caught in the maelstrom. In their end-of-year survey in 2015, the website Ecowatch identified 10 extreme events from the year that 'sounded the alarm' on climate change. In the Arctic, a heatwave in December caused temperatures in the North Pole to spike at 15 degrees Celsius above the norm for the season. In Paraguay, Uruguay, Brazil and Argentina, severe flooding led to the displacement of more than 150,000 people. Heavy rains in the Mississippi delta caused 'historically high' flooding in the US Midwest. A drought in South Africa was the worst in a generation, and the drought conditions across the south of the African continent meant that at least 29 million people faced food insecurity. In Yemen, a powerful and rare cyclone hit in November, impacting upon more than 1.1 million people and displacing 40,000. In the Philippines, a 'mega-typhoon' hit in October, affecting more than 1.2 million people. Across the Middle East, a dramatic heatwave caused temperatures in Iran to soar and temperatures spiked from Egypt to Syria. A deadly heatwave in Pakistan cost at least 2,000 lives while one in neighbouring India killed at least 2,500 people. Climbing temperatures in the Barents Sea, off the coast of Norway and Russia, are causing a poleward shift in fish communities, impacting upon wildlife as well as the indigenous communities that rely on them for their survival. Finally, and most alarming, for the first time in recorded history, global levels of carbon dioxide in the atmosphere averaged more than 400 parts per million (ppm) for an entire month – in March 2015 (Ecowatch, 2016).
The bleak picture continued to unfold in 2016. In May, a destructive wildfire burned through Fort McMurray in Canada. Windy, dry and unseasonably hot conditions created the conditions for the fire to spread and hold (Hansen, 2016). NASA's Goddard Institute for Space Studies (GISS), in their August 2016 review of global temperatures, found that people living in Siberia, the Middle East and North America faced extreme heatwaves during the summer of 2016. Siberia recorded temperatures more in keeping with the tropics, Kuwait recorded its hottest day ever (54.0 degrees Celsius), and in the US, several cities broke monthly temperature records, with, at one point, 124 million people living under extreme heat warnings (Voiland, 2016). In November, alarm was raised that the Brazilian city of Rio's coastal defences were insufficient to cope with rising water levels as four-meter waves lashed the city's beaches. Such storm surges are frequent; however, 'In the 1990s, storm surge disruptions occurred roughly once a year, but since 2010 they have hit Rio four or five times as frequently. There have already been four this year [2016], including two of the biggest ever seen' (Watts, 2016).
Three other events of note emphasise the range of ecological problems that the earth and its populations are facing. Gradual melting of winter snow helps feed water to farms, cities and ecosystems across much of the world. However, global warming is leading to declining snow accumulations across the globe. In 2015, the snowpack in California, which is suffering an ongoing drought as well as long-term warming, reached its lowest point in 500 years (Mankin et al, 2015). Since the late 19th century, temperatures in the Alps have been steadily rising, from an average yearly temperature of 9.6 degrees Celsius in the late 1800s to today's average of 10.8 degrees Celsius. About 40% of Europe's fresh water originates from the Alps, which stretch from Austria in the East to France in the West, dipping into parts of Italy and Monaco in the South. Climate change is threatening the area's water cycle (European Environment Agency, 2010). In the Himalayas and the Andes, the effects of climate change are working in seemingly opposite directions. In both regions, the glaciers are shrinking; in the Andes, there will be longer dry spells and less water. Current projections suggest a steady decline in water resources – to about a third of current levels – by the midpoint of the century. In the Himalayas, people will have to contend with flooding, with water discharge expected to increase by as much as 70% (Rüegg, 2016).
The 14-year drought (1998–2012) that engulfed the Eastern Mediterranean region (covering Cyprus, Jordan, Lebanon, Palestine, Syria and Turkey) is the worst drought in the region for 900 years. It has caused substantial hardship for poor and rural communities throughout the region (Cook et al, 2016). Satellite measurements have demonstrated that the waters of Australia's Great Barrier Reef have warmed by 0.2 degrees Celsius, on average, over the past 25 years. This warming has led to a decline in the amount of seafloor covered in coral and the 'death' (the complete ecosystem collapse) of much of the Great Barrier Reef (Slezak, 2016b): 'Bleaching caused by climate change has killed almost a quarter of its coral' in 2016 (Slezak, 2016a).
The root cause of all these extreme weather events and ecological changes is increasing global temperatures. The GISS has found that the earth's 2015 surface temperatures were the warmest since modern record keeping began in 1880 and that the planet's average surface temperature has risen about 1.0 degree Celsius since the late 19th century, a change largely driven by increased carbon dioxide and other human-made emissions into the atmosphere. 'Most of the warming', they noted, 'occurred in the past 35 years, with 15 of the 16 warmest years on record occurring since 2001' (GISS, 2016a). September 2016 recorded the hottest recorded September in 136 years, leading GISS (2016b) to note that 'The record-warm September means 11 of the past 12 consecutive months dating back to October 2015 have set new monthly high-temperature records'. Former NASA scientist James Hanson and his colleagues have gone further, arguing that current global temperature has increased to a level not seen for 115,000 years '[Current] Global temperature has just reached a level similar to the mean level in the prior interglacial (Eemian) period, when sea level was several meters higher than today' (Hansen et al, 2016).
Popular social work and Occupy Sandy, USA
In the face of a range of climate change disasters, social workers will have to respond to meet people's essential needs. These are not skills that are offered in most social work education programmes. In creating and establishing a relevant social work knowledge base, we would do well to engage and to learn from, what we have called elsewhere, 'popular social work' responses to disaster and extreme situations (Lavalette and Ioakimidis, 2011a). One such example would be the response of the US Occupy Movement to Hurricane Sandy, which struck the US in 2012.
Occupy Sandy was a grass-roots disaster relief network that emerged in New York in response to the devastation caused by Hurricane Sandy, which hit the area in October 2012 (Feuer, 2012). Its purpose was to provide mutual aid to communities affected by the storm. Occupy Sandy grew directly out of the Occupy Movement that began in Zuccotti Park, New York, in September 2011, itself part of the Global Justice Movement and influenced by such international movements as the Arab Spring and the Indignados movement in Spain.
From the end of 2011 and throughout 2012, Occupy spread across much of the globe, as 'a wave of protests swept across Asia, the Americas and Europe' (Buckley and Donadiooct, 2011). The movement set itself up to protest against the economic and political power of the '1%': the control of the economic and political system by the global elite at the expense of the '99%'. As Matchar (2011) noted:
It [Occupy] wants corporate money out of politics. It wants the widening gap of income inequality to be narrowed substantially. And it wants meaningful solutions to the jobless crisis. In short, it wants a system that works for the 99 percent.
In the immediate aftermath of Hurricane Sandy, the Occupy Movement in New York were among the first groups to respond by providing aid to those who were without electricity, food and essentials. They provided torches and hot food from a base they established in two churches. They then organised their internet networks to place calls for materials, and were soon inundated.
The group divided into hubs. Some worked social media, relaying calls for supplies or people with specialist trades or services that they could offer. Others worked in the distribution centres. Still more walked the streets and neighbourhoods providing aid and ensuring that people were safe. Feuer (2012), from the midst of the relief effort, reported:
Occupy Wall Street is capable of summoning an army with the posting of a tweet, and many of the volunteers last week were self-identifying veterans of the movement, although many more were not. Given the numbers passing through the churches, both fresh-faced amateurs and the Occupy managerial class – a label it would reject – were in evidence.
The Occupy Movement grew out of a critique of the priorities of global capitalism and a distrust of the state. The speed of its response to the Hurricane, the networks it both established and drew upon, the unconditionality of the support it delivered, and the philosophy of mutuality and solidarity upon which it was based are all significant features that social work should draw upon in thinking about the construction of a social work for environmental justice.
### The science of climate change
The basic science of climate change has been known about for a very long time. In the 1860s, John Tyndall found that carbon dioxide could let light through while trapping heat. In 1896, Svante Arrhenius found a link between the amount of carbon dioxide in the air and global temperature. Since then, scientists have gathered vast amounts of data that show how carbon dioxide, methane, water vapour and other 'greenhouse gases' allow radiation from the sun to pass through them but stop radiation from the earth rising back into space. The more greenhouse gases that are emitted into the atmosphere, the more radiation is trapped as heat, and, as a result, the atmosphere and planet warm up.
Without the greenhouse effect, the earth would be a very cold, inhospitable and uninhabitable place. Water makes the biggest contribution to the natural greenhouse effect, but the amount of water vapour is relatively stable and it does not play a great role in modern climate change. The most important greenhouse gases are carbon dioxide (CO2) and methane (CH4). There is far less CO2 than water in the atmosphere, but it is increasing rapidly – and it stays in the atmosphere for a long time (about 200 years). Methane has a much stronger warming effect than CO2 but it decomposes much more quickly (in about 12 years). Other gases like, Nitrous oxide (N2O) and a range of industrial gases add to the greenhouse effect but do not last as long.
Scientists have now established that over the last 12,000 years, the amount of CO2 in the atmosphere has been relatively stable, at between 260 and 280 ppm. However, the extensive, large-scale coal, oil and gas burning started by, and associated with, the Industrial Revolution launched the long-term concentration of greenhouse gases and has been central to the growth of atmospheric CO2. The US government National Oceanic and Atmospheric Administration have argued 'that carbon dioxide levels will not drop below the symbolic 400 parts per million (ppm) mark in our lifetimes – the highest concentration of CO2 since the Pliocene era 3m years ago)' (Milman, 2016).
There are significant carbon sinks that can, to some extent, trap the CO2 and offset the effect of greenhouse gases. The two main sinks are trees/plants and the oceans. However, we now produce more CO2 than the sinks can absorb and this fact is made worse by deforestation and by melting polar caps, both of which reduce the carbon sinks and result in the release of more of the earth's 'natural' CO2.
The second most significant greenhouse warming gas is methane. The amount of methane in the atmosphere has more than doubled in the past 250 years. It is estimated that it is responsible for about a fifth of global warming. There are two main sources of methane. Almost all natural gas is methane, which is produced by the fossil fuel industries. As Neale (2008: 16) notes: 'It leaks from coal-mines, oil-fields, gas-fields, gas pipelines and power stations'. Poor maintenance of Russian gas and oil pipelines is thought to be a significant producer, as is burn-off from oil wells and fracking processes (Pearce, 2016). The second source of methane is biological decay: organic waste in landfill sites, decaying plants and trees in wetlands, and as a by-product of intensive farming (eg it is produced in the guts of animals and by flooded paddy fields producing rice). The latest research suggests, however, that methane is also being produced by increasingly wet and warmer climatic conditions (eg melting polar caps releasing methane stores). As Pearce (2016) notes: 'climate change is starting to accelerate the processes that release methane into the atmosphere, potentially triggering a troubling positive feedback in which further warming could produce more methane and yet more warming'.
What Pearce notes here is the impact of 'feedbacks'. As we increase CO2 and methane levels, we are beginning to run into feedback effects. Two examples will explain how feedbacks work. Rising CO2 levels are now warming the Arctic. This begins to melt the permanent snow and ice. The snow and ice generally reflect the sun's heat back into the atmosphere (it is not absorbed by the earth's surface), but as they melt, they reveal sea, tundra and trees. These dark surfaces absorb heat and the Arctic warms up even more, causing the snow and ice to melt more quickly, which reveals more tundra, trees and sea, and so the cycle goes on. A second example comes from deforestation. As rainforests are cut back, large stores of carbon are released from the trees, and even more from the soil. The danger is that this will warm the area and cause drought. In turn, this kills more rainforest and the cycle continues. These two examples show that the more we warm the earth, the more we increase the speed of feedbacks, and the more they reinforce each other, creating the threat of 'runaway climate change', where the changes taking place to our climate cause other changes to the earth's environmental and ecological system that cause 'abrupt climate change' (Neale, 2008).
Despite the claims of well-funded 'climate change deniers', climate change is a scientific fact. Cook et al (2013: 1–2), examining 11,944 peer-reviewed academic papers between 1993 and 2011 matching the topics 'global climate change' or 'global warming', found that 'Surveys of climate scientists have found strong agreement (97–98%) regarding AGW [ie human induced global warming] amongst publishing climate experts.... Repeated surveys of scientists found that scientific agreement about AGW steadily increased from 1996 to 2009'.
On the basis of scientific evidence, there is no debate: human-induced climate change is happening now. From the snow-covered mountains of California and the Alps to the Great Barrier Reef, human-induced global warming is causing immense ecological damage and bringing greater numbers of extreme weather events. It means that the climate is getting warmer, more changeable and more extreme. As a result, there are – and will be – more floods, more droughts, more storms and more 'freak' weather episodes. In the next few decades, global warming, and the resultant rise in sea levels, will threaten coastal plains and cities, and some inhabited islands will be lost to the sea. It will lead to the extinction of numerous animal and plant species: 'Global populations of fish, birds, mammals, amphibians and reptiles declined by 58 per cent between 1970 and 2012 [and we are on track for]... a two-thirds decline in the half-century from 1970 to 2020' (WWF, 2016: 12). Taken together, this will make human life in general harder, and for vulnerable communities, harder still.
Heatwaves and floods
Climate change creates crises for people that require both a social policy and social work response. In Heat wave, Klinenberg (2002) looks at the impact of the Chicago heatwave of 1995: 700 people died as a consequence of the heatwave – leading Klinenberg to describe this as a 'silent killer' of a silenced people.
In France in the summer of 2003, an estimated 15,000 people died after a record-breaking heatwave. The period of extreme heat was the warmest for up to 500 years – an additional 5,000 people died across Western Europe during the same period (Met Office, 2003). The heatwave 'led to a shortage of space to store dead bodies in mortuaries. Temporary mortuaries were set up in refrigeration lorries' (Met Office, 2003). People died as a result of heatstroke, dehydration, sunburn, air pollution and drowning (as people tried to keep cool).
In 2014, the UK suffered a series of floods – an increasingly common event that reflects the impact of climate change on the country. The floods posed serious issues for those working in adult social care. Donovan (2014) outlined some of the social work responses in one region that was badly affected. The priority was initially to support vulnerable people. Initially, this meant social workers liaising with the range of social care agencies to assess the scale of need. More generally, as Donovan (2014) notes: 'social workers join[ed] the emergency services and district council housing teams as they went door-to-door in flood-hit areas to find out who needed help'. Social work support was provided in six areas:
1. Round-the-clock monitoring: of the developing environmental crisis and its impact on vulnerable communities.
2. Information provision: helping to provide information to people about sources of support, but also about suggested time-frames for the crisis.
3. Emergency residential care: coordinating with care providers to ensure that people had places of safety to go to.
4. Extra home-care services: support for those who remain in their homes, regular contact and oversight of medical, food, water and heat needs.
5. Emotional/trauma support: social workers played a key role supporting those directly affected – this work carried on after the floods had subsided.
6. Emergency aid: ensuring that people have clean water, food and heat.
However, this bleak picture is, if anything, an underestimation of the ecological crisis that we are facing.
### The Anthropocene
Over millions of years, the earth goes through periods of cooling and warming. The periods are caused by geological shifts, by fluctuations in the earth's orbit or by changes in sea movements. Over the last 11,500 years, the temperature of the earth has been relatively stable. This period – or geological epoch – is called the Holocene and the warm, stable climate facilitated the development of agriculture around the world, which, in turn, made possible the flourishing of human civilisation. Of course, during the Holocene (over the last 11,500 years), there have been colder and hotter periods – but the longer-term trends have been remarkably stable and the average global temperature during the whole epoch shows a variance of only 1 degree Celsius. The Holocene, argues Ian Angus (2016), is the only global environment that we are sure is safe for the complex, extensive civilisation that Homo sapiens have constructed. However, scientists are now suggesting that the earth's system is entering a new epoch called the 'Anthropocene' – a period in which human activity is disrupting the earth's system in fundamental ways, setting it on a dangerous and unpredictable trajectory and uncertain future.
The concentration of greenhouse gases that started to develop over 200 years ago is producing levels of atmospheric CO2 that are pushing global temperatures up significantly. However, many of the world's leading scientists at the International Geosphere Biosphere Programme (IGBP) have argued that this only captures part of the problem. The IGBP was set up in 1988 and brought thousands of scientists from across the globe together on a range of scientific projects linked to understanding the earth system. Gradually, they came to the conclusion that if we analyse physical and social-scientific phenomena separately, we can miss the extent to which each affects and impacts upon the other. The earth, they argue, needs to be studied as an integrated planetary system. As Steffen et al (2004: 1) argue:
[A] global perspective has begun to emerge in recent years and to form the framework for a growing body of research within the environmental sciences. Crucial to the emergence of this perspective has been the dawning awareness of two aspects of Earth System functioning. First, that the Earth itself is a single system within which the biosphere is an active, essential component. Secondly, that human activities are now so pervasive and profound in their consequences that they affect the Earth at a global scale in complex, interactive and apparently accelerating ways; humans now have the capacity to alter the Earth System in ways that threaten the very processes and components, both biotic and abiotic, upon which the human species depends.
In their path-breaking study Global change and the earth system (Steffen et al, 2004: 131, 132), the scientists looked at trends in 12 socio-economic indicators – population growth, gross domestic product (GDP), urban population, primary energy use, large dam construction, water usage, transportation, telecommunications, foreign direct investment, paper production, fertiliser consumption and international tourism – and 12 'earth system' indicators – CO2, nitrous oxide, methane, stratospheric ozone, surface temperature, marine fish capture, shrimp agriculture, tropical forest loss, domesticated land, coastal nitrogen, ocean acidification and terrestrial biosphere degradation – and found that in each of the 24 indicators, the 1950s marked a 'Great Acceleration' in the intensification and impact of human activity on the earth system:
[T]he profound transformation of Earth's environment that is now apparent... [is the result of human activity].... [T]his transformation has undergone a profound acceleration during the second half of the twentieth century. During the last 100 years the population of humans soared from little more than one to six billion and economic activity increased nearly 10-fold between 1950 and 2000.... Half of Earth's land surface has been domesticated for direct human use and nearly all of it is managed by humans in one way or another. Most of the world's fisheries are fully or over-exploited and little pristine coastline exists outside of the high latitudes. The composition of the atmosphere – greenhouse gases, reactive gases, aerosol particles – is now significantly different from what it was a century ago. The Earth's biota is now experiencing the sixth great extinction event, but the first caused by another species: Homo sapiens. The evidence that these changes are affecting the basic functioning of the Earth System, particularly the climate, grows stronger every year. Evidence from several millennia shows that the magnitude and rates of human-driven changes to the global environment are in many cases unprecedented. There is no previous analogue for the current operation of the Earth System. (Steffen et al, 2004: v)
The IGBP scientists' case is that human activity on the global environment is producing a transformation in the earth's system. They have suggested that it represents: 'The most rapid transformation of the human relationship with the natural world in the history of the species' (Steffen et al, 2004: 131).
The IGBP scientists moved the debate beyond 'simple' climate change to show the variety of ways in which human activity is fundamentally changing the earth system. For example, they note that nearly 50% of the land surface has been transformed by direct human action, with significant consequences for biodiversity, nutrient cycling, soil structure and biology, as well as climate. They argue that more than half of all accessible fresh water is appropriated for human purposes and that coastal wetlands have been significantly affected by human activities, with the loss of 50% of the world's mangrove ecosystems. Additionally, they note that up to 50% of marine fish stocks for which information is available are fully exploited, 15–18% are overexploited and 9–10% have been depleted or are recovering from depletion (Steffen et al, 2004: 258–9).
On the basis of extensive scientific data, the IGBP scientists argue that the earth system moved into a new stage, what they have termed the 'Anthropocene', in the middle of the 20th century. The move from the Holocene to the Anthropocene is very significant. It is not just another stage in human history, or an indicator that humans are having an impact on the world. Rather, the scientists are indicating that the earth has entered a new geological epoch where the dominant conditions on earth have changed significantly. The conclusion for the planet, its species and for humanity is startling. It has created, they argue, a 'planetary emergency' that will, within the next 100 years, see catastrophic ecological impacts on earth and its species move from the bounds of the possible to the probable.
What the scientists are suggesting is that as a result of human activity, the earth has left its natural geological epoch; forces unleashed by contemporary human societies now rival the great forces of nature itself and the planet is entering an unknown future (they call this 'planetary terra incognita'). In a 2015 update of their 2004 report, they concluded that nine out of the 12 'earth system' indicators that they identified had now moved beyond the bounds of Holocene variation, in other words, the data were no longer within the range of variability that we have had for the previous 12,000 years (Angus, 2016: 46). To emphasise the point, Friedrich et al (2016) have argued that 'Global mean surface temperatures are rising in response to anthropogenic greenhouse gas emissions... [to the extent that]... within the 21st century, global mean temperatures will very likely exceed maximum levels reconstructed for the last 784,000 years'. The conclusion, they suggest, is that the earth could be on course for global warming of more than 7 degrees Celsius within a lifetime. This is far higher than any previous estimates of potential global temperatures (Johnston, 2016).
The conclusions for the planet, its species and humanity are frightening. If human activity continues unabated on its present path, by the end of the present century, vast parts of the globe will be too hot to live in, freshwater resources will be depleted, coastal areas of habitation will be flooded by rising oceans and animal and flower species will increasingly die out. As the scientists point out, this is a planetary emergency and a threat to civilisation as we know it.
Water
Social workers in various parts of the world find themselves working in and with communities where access to water is becoming an important issue. According to Gonzalez and Yanes (2015), water is a trillion-dollar industry but – as a result of population growth, industrial production and ecological change – an increasingly scarce one that may well become the source of military and political conflict in the years to come.
In Palestine, for example, social workers in Palestinian villages have been part of campaigns to stop privatised Israeli water companies from diverting supplies away from the communities and farms that the water aquifers have served for centuries. Since the war of 1967, when Israel took control of the Jordan Valley, the River Jordan and the Golan Heights, Gonzalez and Yanes suggest that the 'manipulation of thirst' has been a strategy employed by the Israeli state to heap pressure on the Palestinians of the West Bank and Gaza.
### Capitalist ecocide
The scientists at the IGBP rightly identify the impact that human activity has had, and continues to have, on the earth system. However, that human activity does not exist in a vacuum; rather, it exists within a concrete socio-economic system. Our current relationship with the environment and the earth system is shaped by the needs and drives of modern capitalism. As Moore (2015: 2) argues: '"The economy" and "the environment" are not independent of each other. Capitalism is not [simply] an economic system; it is not [only] a social system; it is [also] a way of organizing nature'.
In 1848, Marx and Engels wrote the Communist manifesto, where they heap praise on the way in which capitalism had 'revolutionised the forces of production'. Capitalism is a dynamic system of commodity production that has fundamentally altered our relationships to each other and the world around us. It has harnessed many of the forces of nature and utilised them in the relentless drive for profit maximisation. In the process, it has created the potential for us to meet human need in innumerable ways, for example, it is certainly possible for us to eradicate poverty and to ensure that everyone has enough food to eat each day, that people have appropriate living conditions and sound homes, and that everyone has access to appropriate jobs and forms of labour. However, under capitalism, these possibilities remain unfulfilled. The unplanned and chaotic nature of capitalist production is geared to meeting not the needs of the many, but the profits, desires and goals of the few. Furthermore, the drives of capitalist competition mean that the earth's resources are exploited, used and abused – without due consideration to issues of sustainability and ecological destruction – as sources of profit and wealth generation. According to Heede (2014), just 90 global companies have caused two thirds of man-made global warming emissions.
The exploitation of natural resources and the burning of fossil fuels during the earliest phase of industrial capitalism marked the starting point of human-induced climate change. The spread and impact of capitalism across the globe in the post-Second World War era (during the period of the long post-war boom from 1947 to 1971) – often under the label of 'national development' or 'communism' as much as 'free market capitalism' – marked the point where we entered the Anthropocene.
To satisfy its unquenchable thirst for profits, capital has spread across the world – it is now a truly global system – and as part of this process, it has burnt ever-greater quantities of fossil fuels, produced greater quantities of greenhouse gases, dammed waterways, mined and altered water deltas, deforested and domesticated land, pumped chemicals into the soil, altered river flows, and exhausted much of the ocean's fish stocks – all in the interests of profit maximisation. It is also a system that is riven by wars, and as the 20th century progressed, chemical, gas and nuclear weapons – alongside the impact of more traditional weaponry and military equipment – have all left their mark on humans and on the earth system (McCarthy, 2016).
This is not to suggest that capitalist companies have deliberately and consciously set out to inflict such harm on the planet – climate change and the move to the Anthropocene have been the unintended by-product and consequence of the carbon economy within which capitalism is embedded. However, capitalism is a system of 'many competing capitals', of ruthless competition, in which individual units of capital put their interests for profit above all else. This means, however, that any attempt to control, for example, the emission of greenhouse gasses that is perceived to interfere with the process of accumulation is likely to be resisted. This has been referred to as the 'ecological tyranny of the bottom line: when protecting humanity and plants might reduce profits, corporations will always put profits first' (Angus, 2016: 114).
To emphasise the point, the magazine Industry Week, whose strapline is 'advancing the business of manufacturing', produces a regular list of the top 1,000 companies. In 2015, they noted that 'Petroleum & Coal Industries dominate the IndustryWeek 1000 list not only by capturing seven of the top 10 spots, but by landing 135 companies on the list, 45 more than the No. 2 industry' (Panchak and Szilagyi, 2015). The top six global corporations (in terms of revenue) were (in order): (1) China Petroleum and Chemical Corp; (2) Royal Dutch Shell; (3) Exxon Mobile; (4) Petrochina; (5) BP; and (6) Oil and Natural Gas (India) (Panchak and Szilagyi, 2015).
In the term used by Ian Angus (2016: 203), the modern world represents a form of 'capitalist ecocide' – uncontrolled and unregulated capitalism is driving the planet towards a frightening and uncertain ecological and planetary future. This also indicates that unfettered capitalism is a barrier hindering the actions that need to be undertaken to protect the plant. Ultimately, we need to establish a social system that works in harmony with the earth system, but in the short term, we need to bring pressure to bear on governments and corporations to ensure that we can mitigate, to some extent, the worst outcomes of climate change.
If the scientists at the IGBP are right, we have already entered the Anthropocene. That cannot be reversed, but what we do will determine what the Anthropocene is like. As Angus (2016: 207) argues, we need to slow capitalism's 'ecocidal drive', act immediately to drastically cut greenhouse gas emissions, commit to renewable energies and move towards the rapid elimination of industries – such as arms production and factory farming – that create human and planetary harm. He suggests the following six 'essential immediate changes':
• the drastic and enforceable reduction in the emission of greenhouse gases;
• the development of clean energy sources;
• the provision of an extensive free public transport system;
• the progressive replacement of trucks by trains;
• the creation of pollution clean-up programmes; and
• the elimination of nuclear energy and war spending.
These demands are, by necessity, political. However, as a profession committed to human emancipation, human need, human rights and social justice, the social work profession needs to engage with contemporary discussions of climate change, environmental justice and the Anthropocene because they are, at heart, about the fundamental change affecting the lives, needs and demands of people and communities.
Two food projects in the UK: Fareshare and the Food Cycle
In Britain, it is estimated that 3 million tons of food is wasted each year while 8.4 million people, the equivalent of the entire population of London, are struggling to afford to eat (Cafiero et al, 2016). This is a reflection of increasing food prices (between 2005 and 2013, food prices increased by 43.5% in the UK), increases in general household costs, stagnant wages and irregular working, changes to the social security system, and growing poverty and inequality. In response to these developments, two projects in Liverpool, UK, attempt to combine a commitment to issues of food waste, climate change and food poverty into their practice.
Fareshare is a national organisation that saves food destined for waste and sends it to charities and community groups, who transform it into nutritious meals for vulnerable people. The food that is redistributed is fresh, of good quality and 'in date' surplus from the food industry. In 2015, Fareshare redistributed enough food for 18.3 million meals across the UK. They distributed food to breakfast clubs for children in poor areas, to homeless hostels, to community cafes and to domestic violence refuges – places that provide support to vulnerable communities, as well as lunch and dinner.
The Food Cycle in Liverpool is part of a national food waste network that aims to use 'waste food' to serve nutritious three-course meals to those who find themselves in food poverty. The project in Toxteth, Liverpool, is based in a local church (though the organisation has no religious affiliations) and opens as a community cafe one night each week. The group has one full-time worker whose job is to work with local food shops and supermarkets to get access to 'waste food', undertake fundraising and apply for funding, raise awareness about food poverty and the environmental impact of food waste, and work alongside local campaigns to highlight food issues. The cafe opens each Wednesday and is open to all – offering a free three-course meal to those in need.
### Social work in the Anthropocene
The international definition of social work continues to identify the profession as one that should work towards 'enhancing human well-being', 'promoting social justice' and 'sustainability', and working with 'people in their environment'. The scale of the environmental crisis discussed earlier means that we are likely to face increasing situations where people's lives will be disrupted by extreme weather events and their communities will be threatened by the impact of pollution, or where they may struggle to find appropriate food and water resources. In these circumstances, more people will be forced to leave their homes and become – for the short or long term – climate refugees (Demirbilek, 2016). Yet, to date, most social work academics and professionals have not fully considered how these environmental justice themes can be embedded within professional practice.
Part of the reason for this, as Gray and Coates (2013: 357) argue, reflects the roots of social work in the Enlightenment philosophies that shaped the modern era:
social work as a profession began at a time when science was revered and Western societies adhered to notions of a mechanistic universe, industrial and agricultural innovation, and the spread of capitalism. The impact of human actions on the environment was largely beyond consideration within a worldview where people were largely understood to be in control of their own destiny.
While this is true, it is also the case that many early social workers were concerned with the impact of urban environments on poor communities and that community-based approaches to social work have always had a concern with 'environmental issues' such as bad housing, pollution and poor living and urban conditions. Furthermore, in much of the Global South, such concerns have never fully moved off the social work agenda. In India, for example, social workers have a long and honourable history of engagement with communities and social movements fighting for their social and environmental rights in areas designated as ripe for damming (Jensen, 2004). Nevertheless, as more radical and 'structurally oriented' approaches to social work have been marginalised within the profession, so the space for consideration of the impact of global climate change on people and communities has narrowed. Indeed, as Philip and Reisch (2015: 472) argue, social workers 'tend to regard the environment as the context for practice instead of a dynamic component of people's lives'. Such realities have led Hawkins (2010: 68) to point out that:
Social work must extend [its]... mission to include environmental justice, the human right to live in a clean, safe, and healthy environment. The world's most poor, vulnerable, and oppressed people often live in the most degraded environments and have no control over resources. The important connections between social work, sustainability, human rights, and environmental justice in our contemporary world need to be more clearly articulated.
Dominelli (2011, 2012) argues that the profession needs to develop its research base because, given its professional framework, it is 'well placed to contribute to climate change policy discussions and interventions' (Dominelli, 2011: 430), while McKinnon (2008: 266) argues that 'social work has the opportunity to be part of the solution rather than an uninvolved bystander to the emerging environmental predicaments'.
In the work of McKinnon (2008), Hawkins (2010), Dominelli (2011, 2012), Gray and Coates (2013), Philip and Reisch (2015) and Demirbilek (2016), we are starting to see social work theorists addressing the social consequences of climate change and theorising over our 'knowledge base that links social work practice with universal human rights, environmental justice, and sustainable development [and that] will help them to envision the world as a more just and humane place' (Hawkins, 2010: 79). However, what might a social work with environmental justice at its heart look like?
Social work in Kasimode (Chennai, India) in the aftermath of tsunami
Kasimode is a large, overcrowded slum that sits as part of the northern dock area of Chennai, the capital of Tamil Nadu, India. It covers 1 square mile and is home to over 20,000 people (Lavalette, 2009). India is a country of vast inequalities. In the suburbs of Chennai, there are large gated communities where English is increasingly the language of choice. The wealth and opulence in these areas is comparable to that which you would see in the affluent parts of any city in the world. Yet, Chennai is also a city dominated by slums, by people sleeping on the streets or at their place of work, and by desperate people begging for survival. The World Bank estimates that 456 million Indians (42% of the total Indian population) now live under the global poverty line of US$1.25 per day. This means that a third of the global poor now reside in India.
In Kasimode, the people travel to work in the city or work on the fishing boats that are based there. Yet, despite the fact that most people work, they live in desperate poverty. The fish are sold for a few rupees on the landing areas before being driven away by 'merchants' to factories and expensive restaurants in the city. Kasimode is a place of immense poverty: the living and sanitary conditions in the slum are desperate; the electricity regularly fails; the streets are rarely swept; there are few government services in the area; and there is one poorly equipped school and one tiny community centre.
Kasimode was hit by the devastating tsunami that struck the Bay of Bengal on 26 December 2004. Although less than 20 people were killed, over half the houses were destroyed and washed away. There is now significant evidence to suggest that both the Indian government and the state authorities of Tamil Nadu knew that the tsunami was approaching on that fateful day – yet they did nothing. Indeed, some harbour suspicions that the authorities hoped that the tsunami would 'deal' with some of their 'problem communities'. The authorities and police force regularly attack slum areas. They destroy houses and turn families out onto the streets. The former slum areas are then handed over to property developers for housing, office or factory development.
In the days that followed the tsunami, the relief efforts in the poorest areas were totally inadequate and large amounts of international aid 'disappeared' before it could get to the front line. In these circumstances, it is not surprising that there is a very deep-seated suspicion of anything to do with the Indian state – for the Indian poor, it is obvious that the state does not work in their interests.
Given the hardship of daily life, you might think that the people of Kasimode would be too consumed with the struggle for survival to engage with political life. However, these are not a defeated people. The slum hosts a range of social and political movements fighting for the rights of slum dwellers – and local social workers are embedded within these campaigns.
In the central community centre, social workers – drawing on broadly Ghandian principles – work to support the community in ways that meet their material needs and support their political campaigns fighting for their rights. Social workers have been: involved in setting up a local food co-op, which brings good-quality food to local residents; supporting the fish workers, who are in the process of setting up their own union co-op to set up a small smoking factory; involved in various campaigns to fight for Dalit rights, the rights of disabled slum residents and of the poor more generally.
### Social work for environmental justice in practice
By way of a conclusion to this chapter, this final section looks at some examples from practice where social workers are starting to work with communities to focus on environmental justice issues.
#### Disasters
As we enter the Anthropocene, we are increasingly likely to face extreme weather and climatic events that will impact on people and their communities. According to Dominelli (2012), social work has an important role to play in the aftermath of a 'natural' disaster or in the aftermath of an extreme weather event as the 'organisers and managers' of aid and relief efforts. There is no doubt that social work skills can, and have been, useful in such situations – as in the flood situations in the UK discussed earlier, for example. She notes examples where social workers have travelled to disaster zones in the aftermath of hurricanes, floods, earthquakes and droughts, where they establish shelters for people who are forced to evacuate their homes,, manage food distribution, address issues of trauma and provide aid. In these circumstances, social workers fill the need for emergency management that focuses on people instead of the needs of insurance companies. There is no doubt that social work skills can play a vital role in such circumstances. Nevertheless, while social workers have some useful skills, these need to be developed and enhanced – and taught on relevant training programmes.
As we think about building our knowledge base in this area, we need to recognise that 'natural disasters' also shine a light onto the kind of society within which we live and force us to address wider issues (Lavalette and Ioakimidis, 2011a). Disasters pose immediate questions: who has been killed and who has been saved? How can we best meet the immediate needs of survivors? Is the existing social and political structure a help or a hindrance to the relief efforts? We should not be surprised to recognise that different groups in society are likely to answer these questions in different ways.
In The shock doctrine, Naomi Klein (2007a) offers various examples of where in the face of such 'natural disasters', multinational corporations and powerful economic and political interests see the events as an opportunity to promote 'economic restructuring' and cement their position within the hierarchy of social relations in the society in question. She gives some examples: 'After a Tsunami wipes out the coasts of Southeast Asia, the pristine beaches are auctioned off to tourist resorts.... New Orleans' residents, scattered from Hurricane Katrina, discover that their public housing, hospitals and schools will never be reopened' (Klein, 2007b).
By way of contrast, Rebecca Solnit suggests that in the midst of disasters, we get a glimpse of what she terms a 'paradise built in hell'. She argues that while:
Disasters are... terrible, tragic, grievous... [and] not to be desired... [they] drag us into emergencies that require that we act, and act altruistically, bravely, and with initiative in order to survive or save neighbours.... The positive emotions that arise in those unpromising circumstances demonstrate that social ties and meaningful work are deeply desired, readily improvised, and intensely rewarding. (Solnit, 2009: 6–7)
There is no simple answer to whether a society suffers a post-disaster 'social shock' or a 'paradise built in hell', but in a range of recent disasters – from Hurricane Katrina (2005), to the Haitian earthquake (2010), to Hurricane Sandy (2012) – social work intervention has been at both the official level (through state agencies and non-governmental organisations [NGOs] working in the relief field) and through the work of popular social work interventions where social movement activists and grass-roots social workers have combined to meet the needs of communities and individuals in the most desperate circumstances (Ioakimidis and Lavalette, 2016). As Herbert-Boyd (2007: 5) notes:
Crises force individuals and [social work] professionals to examine their role in the social order, and they can be a catalyst for effective community development, progressive social policy and social change. Conversely, they can lead to conservatism, fear and the maintenance of the status quo.
As we develop our environmental justice social work practice tools, we will benefit from working closely with, and listening to, social movements and communities as they develop responses to disaster and crisis situations.
#### Advocacy
Representing individuals and communities and working alongside them to promote their rights is central to advocacy-based models of social work. In April 2016, the Texas-based Energy Transfer Partners initiated a US$3.7 billion project to establish an 11,720-mile pipeline (the Dakota Access pipeline) to transport fracked crude oil from the Bakken oilfield in North Dakota to a refinery near Chicago. It is proposed that the pipeline will carry 470,000 barrels of crude oil per day (Levin, 2016).
Fracked oil threatens the environment in numerous ways and continues to feed the fossil fuel economy and its attachment to greenhouse gas production. The pipeline emphasises the priorities of fossil fuel capitalism and threatens significant environmental and ecological damage. The proposed route went through lands of historical significance to the Standing Rock Lakota Sioux Nation, including sacred burial grounds, and was seen as a direct threat to both the tribe's water supply and the Missouri river. By the end of 2016, 90 Native American nations had converged on Standing Rock and the Native Americans had garnered significant global support for their campaign to stop the fracking, stop the pipeline and protect the Sioux lands and water sites.
The pipeline, therefore, was a threat to the environment, gave further support to the fossil fuel economy and trampled over indigenous rights in North Dakota. The response of the Native American nations was to launch legal challenges, invite the United Nations (UN) to investigate abuses of indigenous people's rights, launch several peace camps and undertake a range of protests to block the construction as it moved towards the sacred burial lands (Levin, 2016; Levin and Woolf, 2016). In response, US federal forces attacked protestors (who called themselves 'water protectors') with dogs, arrested hundreds and imprisoned them in cages.
Social workers and youth workers were involved in the Standing Rock movement. Youth workers were centrally involved in organising the marches and protests (Levin and Woolf, 2016). Social work educators were involved in offering advice regarding legal challenges, and practitioners associated with the Boston Liberation Health Group were on the ground offering support to those affected by the police action and the protest events.
#### Pollution
According to the United Nations Children's Fund (UNICEF), 300 million children live in areas with 'extreme air pollution', where toxic fumes are six times international guidelines (Rees, 2016). The UNICEF report suggests that these figures contribute to 600,000 child deaths a year, that poor children are most at risk and that the data emphasise that 'climate change threatens the well-being of children' (Rees, 2016: 10).
The increasing volume of cars and attachment to fossil fuels and other greenhouse gases is at the heart of the pollution crisis. In poor parts of the world, communities are often reduced to burning rubbish and waste, which produces 'highly hazardous' toxins. From a children's rights and human rights perspective, dealing with pollution is an essential part of establishing good community and individual health and well-being.
In Kasimode in Chennai, India, social work practitioners work alongside the community to improve living conditions for the community. Kasimode is a large urban slum near the city's docks. The conditions are very poor – housing is overcrowded, water and sanitation facilities are basic or non-existent, and waste and refuge disposal is very haphazard and infrequent. The practitioners are shaped by broadly Gandhian approaches to community engagement. They work through a local community centre on a range of projects to promote the interests of children and families. This includes clean-up campaigns of the local environment and community organising attempts to address waste and sanitation issues. There is a very clear understanding that such campaigns improve people's health, well-being and life expectancy while also improving the environment.
#### Food
Each year, roughly one third of all the food produced in the world is wasted. This amounts to about 1.3 billion tons of food waste per year. This means that vast resources used in food production are used in vain, and that the greenhouse gas emissions caused by the production of food that gets lost or wasted are also 'emissions in vain'. The per capita food waste by consumers in Europe and North America is 95–115kg/year, while this figure in sub-Saharan Africa and South/South-east Asia is only 6–11kg/year (Gustavsson et al, 2011). As Gustavsson et al (2011: 1) argue:
The issue of food losses is of high importance in the efforts to combat hunger, raise income and improve food security in the world's poorest countries. Food losses have an impact on food security for poor people, on food quality and safety, on economic development and on the environment.
The Real Junk Food Project is a UK national network of pay-as-you-feel (PAYF) cafes. The community cafes intercept food that would otherwise go to waste from supermarkets, restaurants and a number of other sources, and turn it into healthy meals for anyone and everyone on a PAYF donation. The network self-identifies not as a 'food poverty' network, but as a 'food waste' network. Part of their mission is to engage with communities and raise awareness about the ecological and social costs of food waste.
#### Water
Observational records and climate projections provide abundant evidence that freshwater resources are vulnerable and have the potential to be strongly impacted by climate change, with wide-ranging consequences for human societies and ecosystems. (Bates et al, 2008: 3)
Water rights are an issue of growing concern across the globe. Fresh water is drained, dammed, polluted, diverted and fought over, and poor communities often find themselves dealing with the consequences. In 2016, community activists in Flint in the US had to take emergency action to deal with a water crisis. In April 2014, the city authorities began using water from the Flint River without treating it for corrosive pollutants. The result was that lead appeared in the public water supply. Ingesting lead is dangerous for anyone, but it is especially damaging to children and pregnant women, with long-term effects including hearing problems, delayed foetus growth and harm to the brain, kidneys and other essential organs.
Research shows, however, that good nutrition and calcium can help mitigate the impact on children with elevated lead levels. Given this, community activists worked with the local farmers' market to try and ensure that Flint's poor communities were supplied with good-quality fruits and vegetables, and activists from Michigan State University set up cooking classes to teach participants to prepare meals with the lead-fighting foods:
Various other community groups... joined the lead-fighting effort... the market hosted a program sponsored by the National Basketball Players Association (NBPA) and the non-profit FlintNOW. On the last day of school, 8,000 Flint public school children received 'nutrition backpacks' containing a mini basketball, information about nutrition and lead, and three $5 gift certificates to the Flint farmers' market. Additional funding from the Detroit Pistons [the city basketball team] matched each $5 gift certificate with another $5 – bringing the total to $30 worth of free food. (Hardman, 2016)
### Conclusion
To date, social work has not paid as much attention to the impact of climate change on the environment and on humanity as it should have. Climate change is creating a range of environmental and social problems that will most significantly impact our most vulnerable communities. Social workers will increasingly have to deal with the consequences of environmental degradation by meeting people's needs in new, complex and unpredictable circumstances. We need to develop our knowledge base, we need to develop our practice skills and we need to embrace the demands of environmental justice in our theory and practice.
# Part Three
Debating the politics of social work today
## EIGHT
## A new politics of social work?
### Introduction
Radical social work is often associated with an understanding of society based on Marxist ideas, above all, the idea that we live in a society in which the central division (though clearly not the only division) is class. While that tradition has been important in shaping radical social work theory and practice both in the 1970s and currently, it has not been the only source of critical thinking within social work. Feminist and anti-racist approaches, for example, have also been influential, as have approaches based on identity politics more generally (essentially, the idea that only those experiencing a particular form of oppression can either define it or fight against it). While postmodern and post-structuralist approaches have had less influence in social work than in other academic disciplines, in Australia, Canada and, especially, the US, they have shaped much of the critical social work literature.
In previous publications, we have sought to provide an assessment of different strands of critical social work thought including postmodernism, post-structuralism and identity politics, as well as exploring the roots of oppression and alienation. Rather than repeating the arguments presented there, we would refer readers to these earlier writings (Ferguson and Lavalette, 1999; Ferguson et al, 2002; Ferguson, 2008; Lavalette, 2011). Rather, our intention here is to look at what are often new takes on older questions.
As an example, two prominent social work academics, Mel Gray and Stephen Webb, have suggested that current developments in political theory and political philosophy, notably, the work of Nancy Fraser, Axel Honneth and Alain Badiou, 'have the potential to galvanize a new politics of social work by innovatively reworking agendas on social justice and solidarity, of political possibility, and transformative ideas relating to universal emancipation and freedom' (Gray and Webb, 2013: 4). They go on to provide a list of thinkers whom they see as having a contribution to make to the formation of what they call a 'New Social Work Left', which includes, inter alia, Slavoj Zizek, Jacques Ranciere, Chantal Mouffe, Jean-Luc Nancy, Antonio Negri, Peter Hallward, Costas Douzinas, Gianni Vattimo, Susan Buck-Morss, Alberto Toscano, Roberto Esposito and Giorgio Agamben.
Some of these names also pop up in another recent collection of writings that aims to revisit anti-discriminatory and anti-oppressive practice, drawing primarily on the work of Michel Foucault but as the editors also note:
Although the work of Foucault is forefronted, there are a range of other theoreticians manifest in this book including Derrida, Badiou, Butler and even Althusser who, as a Marxist, might have been considered too 'modern' for such a post-structuralist/postmodernist text. This theoretical focus enables theoretical depth, similar to an oil-painting with its many-layered profundity. (Cocker and Hafford-Letchfield, 2014: vii)
There are undoubtedly positive aspects to these developments. First, the insistence of Gray and Webb, among others, that social work is a political project is welcome contrast to a mainstream social work literature that frequently eschews politics and presents social work as primarily a therapeutic, practical or technical enterprise. Second, with all their differences, what the thinkers listed share is a spirit of resistance to neoliberal capitalism, a spirit fuelled, in part, by a global anti-capitalist movement that emerged out of the protests against the World Trade Organization in Seattle in 1999 (Held and McGrew, 2002; Callinicos, 2003; George, 2004). Third, engaging with new ideas is crucial if the critical and radical social work tradition is to thrive and develop. Theory and practice cannot be set in stone.
However, these developments are not without difficulties. For a start, rather than 'many-layered profundity', there is, to put it at its mildest, a real danger of eclecticism in lumping together thinkers who often start from very different theoretical and political premises. Gray and Webb, for example, hold up as representative of the new politics a conference that took place in London in 2009 entitled The idea of communism (Douzinas and Žižek, 2010). While it is true that all of those involved were united in seeking an alternative to neoliberal capitalism, they did so from very different starting points. They included, for example: Terry Eagleton, a fairly orthodox classical Marxist, one of whose recent publications was entitled Why Marx was right; the maverick Slovenian philosopher Slavoj Žižek, whose political positions in recent years have included support for Donald Trump in the 2016 US presidential election and, more recently, a highly ambivalent response to the issue of refugees fleeing war and famine; and Alain Badiou, a leading French philosopher, who began his political life as a Maoist and, despite all the evidence to the contrary, still appears to view the Chinese Cultural Revolution as a key emancipatory event in human history. While these thinkers share some areas in common, their differences are at least as important. While fresh ideas are always to be welcomed, it is difficult to see quite how a 'new politics of social work' that has any coherence or consistency can emerge from such an eclectic (and frequently highly abstract) group of thinkers. That said, insofar as there is a theoretical convergence between them (and, here, thinkers such as David Harvey and Terry Eagleton are notable exceptions), it tends to be around a set of themes sometimes labelled 'post-Marxist'.
Post-Marxism is a current of thought that came into existence in the 1980s, though its origins may be said to lie in the ideas developed in 1970s' France by Jacques Derrida and Michel Foucault. Its founding text, which we shall discuss in some detail later, is often identified as Hegemony and socialist strategy by the Argentinian scholar Ernesto Laclau and his partner and collaborator Chantal Mouffe (Laclau and Mouffe, 1985).
It would be inaccurate and unfair to characterise all those included in Gray and Webb's list, let alone many others working in the critical social work tradition, as 'post-Marxist'. Some have never claimed to be Marxist, some continue to see themselves as Marxist, and some left Marxism behind a long time ago. Others, such as Alain Badiou, reject the label but share several of its key assumptions. As his biographer comments:
If Badiou both rejects any direct articulation of politics with economics and tolerates a degree of reliance on the state, in what sense does his project still merit the Marxist label? Badiou recognizes no single subject of History, no global historical movement, no priority of the mode of production – not even the ultimate political primacy of class struggle per se. (Hallward, 2003: 239)
His conclusion is that 'Badiou's work must figure as part of the "eclectic", antisystemic trend of much Western social and cultural theory since the early 1970s' (Hallward, 2003: 239).
Whatever their differences, some of the themes shared by several of the writers cited earlier – a view of Marxism as essentialist or reductionist; a rejection of any 'privileged' role for the working class; and a search for alternative agents of change, including social movements and intellectuals – are also the core themes of post-Marxism. These ideas also form a kind of 'common sense' among many of those working within the critical social work tradition.
Given the likelihood that many social work practitioners (and many social work academics) will be unfamiliar with these thinkers and their concerns, it is tempting to dismiss these debates as irrelevant to the realities of practice. This is an understandable response and we are certainly not suggesting that the influence of post-Marxist thinking on social work theory and practice is the central issue confronting the social work community at the present time. That said, ideas matter and do filter their way down to the front line of practice, often through social work education or more indirectly. One of us recalls visiting a children and families agency in the east of Scotland some years ago where a social work practice educator confidently informed a student that recent findings in neuroscience had proved beyond doubt that the first three months of a child's life were crucial and that social work intervention with children or parents after that period was largely futile (for a more balanced account of the findings of neuroscience and their implications for social work, see Wastell and White, 2017).
In addition, Laclau and Mouffe's ideas have recently taken on an additional if unexpected political and ideological significance as an important influence on the thinking and strategy of the leaderships of Syriza and Podemos, the new radical Left parties in Greece and Spain (Hancox, 2015). For both of these reasons, these ideas merit our consideration. The first part of this chapter will therefore outline some of the key themes of Hegemony and socialist strategy. The second part of the chapter will offer a critique of these ideas, while the final part will assess the extent to which post-Marxist perspectives more generally can provide a firm foundation for a New Social Work Left.
### Post-Marxism: key themes
In a recent reassessment of the putative heirs of the Italian Marxist Antonio Gramsci, Perry Anderson has provided a useful outline of the ideas of Laclau and Mouffe, who are among those who claim to be working in the Gramscian tradition:
Together, in 1985, they published Hegemony and Socialist Strategy, bringing post-structuralism boldly to bear on the Marxist tradition, in political sympathy with what had been Eurocommunism, but in theoretical outlook now declaratively post-Marxist. Reviewing the history of the Second and Third Internationals, they concluded that both had remained trapped in the illusion that ideologies corresponded to classes, and historical development led by economic necessity to the triumph of socialism. What neither had been able to resolve was the existence, not only of divisions within the working class as the carrier of this supposed necessity in the shape of the revolutionary subject of history, but also of non-capitalist classes that did not form part of the working class.... The way forward was now to scrap all residues of class essentialism and drop any idea of a war of movement. Rather than interests giving rise to ideologies, discourses created subject-positions, and the goal today should not be socialism but a 'radical democracy', of which socialism would – since capitalism bred relations of undemocratic subordination – remain a dimension, not the other way round.
Here, Anderson provides a succinct summary of some of the key themes of post-Marxism: first, a rejection of what Laclau and Mouffe call 'class essentialism', above all, the notion that the working class has any 'privileged' role as an agent of societal transformation; second, a denial of any link between the dominant ideas in society (eg racist ideas) and material interests and conflicts between classes (instead, there is 'the autonomy of discourse'); and, third, a view that the goal of the progressive Left should not be the total transformation of society in the direction of socialism (however conceived), but rather the radical extension of democracy. Each of these themes will be considered in turn.
#### Class essentialism
First, there is the rejection of what Laclau and Mouffe refer to as 'classism' or 'class struggle essentialism' and its replacement by antagonisms based around gender, race and other divisions:
What is now in question is a whole conception of socialism which rests upon the ontological centrality of the working-class, upon the role of the Revolution with a capital 'r', as the founding moment of the transition from one society to another, and upon the illusory prospect of a perfectly unitary and homogeneous collective will that will render pointless the moment of politics. (Laclau and Mouffe, 2014 [1985]: xxii)
The idea that the working class, insofar as it even exists, no longer has any role to play, let alone a central or 'privileged' role, in changing society has been repeated so often over the past four decades, not only by right-wing ideologues, but also by thinkers on the Left such as Eric Hobsbawm and Guy Manning, that it has taken on the status of a truism. In a discussion of agency, Webb and Gray, for example, remark almost in passing that 'This absence of agency is a structural effect conditioned by the disappearance of a politically influential working-class' (Webb and Gray, 2013: 215).
Now, it is undoubtedly true that the shape and composition of the working class has changed considerably in recent decades. It is also true that the level of class struggle in some countries, notably, Britain, is at a historically low level. However, is it really accurate to characterise this as 'disappearance' (Ferguson et al, 2002)? In Britain, for example, the economy has experienced structural change since the end of the Second World War, with a decline in the manufacturing sector and an increase in the service sector. Jobs in the service industries have increased by 45%, from 14.8 million in 1978 to 21.5 million in 2005, while those in manufacturing have fallen 54%, from 6.9 million to 3.2 million over the same period (Smith, 2007).
It is a mistake, however, to see this as a 'decline', let alone a disappearance, in the working class. For one thing, even in Britain, one in seven workers is still employed in manufacturing industry. As Smith (2007) notes:
These workers often work in large and well-organised workplaces like engineering, car manufacturing and food production. Although their numbers have fallen, those workers who are still employed have become more and more productive and in some senses more powerful. Take, for instance, the UK car industry. Over the last 30 years there has been a huge fall in the numbers of workers employed in the industry. However, car production has barely fallen. At the height of UK car production in the 1970s Britain produced about 1.7 million cars a year. By 2005 it had only fallen to 1.6 million a year. New technology means that one car worker can produce eight times what their predecessors could 30 years before.
It seems more accurate, then, to talk about a restructuring of the working class in response to the needs of capital. For example, the need for a highly skilled and educated workforce in the post-war era has resulted in a massive expansion of education in most West European countries, resulting in the employment of more teachers and lecturers. That investment requires, in turn, a reasonably comprehensive health service to ensure that these workers remain productive. Furthermore, in order to address the social and emotional needs of these workers and their families (as well as ensuring that they do not behave in disruptive or damaging ways), a social work and social care workforce is required – hence the massive growth in the service sector in many countries since the Second World War (Ferguson, 2014).
In turn, this has resulted in what the American Marxist Harry Braverman (1974), in his classic Labour and monopoly capital, called 'the degradation of work in the 20th century'. In other words, groups of workers such as civil servants, teachers, lecturers and social workers who, at one time, were seen as 'middle class' and enjoyed both considerable status and autonomy in carrying out their work are increasingly subject to the same working conditions and forms of supervision as other workers (Ferguson and Lavalette, 1999; Harris, 2003). That process has intensified in recent years as a result of the imposition of the New Public Management practices associated with neoliberalism. The consequence is that groups of white-collar workers who previously thought of themselves as being superior to manual workers will now often belong to trade unions and see themselves as part of the working class. So, for example, the British Social Attitudes Survey for 2016 found that the majority of Britons identify as working class even if they have stereotypically middle-class jobs. The survey found that although just 25% of people now work in routine and manual occupations, 60% of Britons regard themselves as working class. Nearly half of people in managerial and professional occupations identify as working class (Butler, 2016).
On a global scale, Filmer's detailed analysis of global employment in the mid-1990s found that out of a total of 2,474 million people who participated in the global non-domestic labour market (and excluding members of 'the new middle class' – managers, supervisors, etc), some 700 million could be defined as workers (working for a wage or salary), with one third employed in manufacturing and two thirds in services (health, education, transport, etc). Taking their dependants and retired workers into account, this suggests a global working class of between 1.5 billion and 2 billion (Harman, 2009: 331–2). In fact, more recent figures suggest that the figure is now around 3 billion (Panitch and Albo, 2014) – hardly a disappearing class!
None of this, of course, is to deny that there are serious and legitimate questions to be asked about the capacity of the working class in the 21st century to play a central role as an agent of social transformation. Nor does it mean that workers will therefore automatically challenge capitalism or even have progressive ideas. In the next section of this chapter, we will explore the relation between people's material position and their ideas. What the preceding discussion suggests, however, is that, on a global scale, far from having disappeared, the working class is now bigger than it has ever been, both geographically and also in including within its ranks millions of white-collar workers who would previously not have considered themselves working class. As Jack Shenker has shown in his eye-witness account of the revolution against Hosni Mubarak's neoliberal Egyptian state in 2011, that class continues to play a central role in the struggle for a different world, even if that struggle is too often marked (as in Egypt) by terrible setbacks as well as by great victories:
Workers were at the forefront of the days of fighting that had enabled the revolution to reach this point: those killed by Mubarak's troops on the days of rage included a welder, a plumber, a cement factory mechanic, a stone mason, a shoemaker and a microbus driver. The Egyptians most directly afflicted by structural adjustments had brought protest from the margins into the core of the capital. Now, as the eighteen days rumbled on, they swept revolution back out again from Tahrir [Square, in central Cairo] to workplaces up and down the country; on the final days before Mubarak crumbled, workers in the transport, petroleum, steel, fertilizer, cement, textile, printing, military production and many, many other industries came out on strike to apply a final dose of pressure on a broken dictatorship, pressure which sealed Mubarak's fate. (Shenker, 2016: 226)
#### 'All the world is discourse'
In the previous section, we argued that, empirically, not only does the working class exist, but it is also numerically larger than it has ever been in its history. There is, however, a second question that is no less important, namely: even accepting that the working class has not disappeared, is that class capable of becoming a revolutionary force, an agent of social transformation? To what extent, in Marx's own formulation, is the working class as a class in itself capable of becoming a class for itself? How, in other words, can the working class move from its current non-revolutionary consciousness to a point where a majority – or, at the very least, substantial numbers – of workers see the need to overthrow capitalism?
Within the Marxist tradition, the ideas that people hold are seen as being shaped by two factors: first, their material circumstances and day-to-day experience; and, second, the ideas that are promoted by the ruling class, the main aim of which is to reinforce the status quo, for example, by suggesting that the world we live in is the best of all possible worlds or by diverting discontent onto scapegoats such as minority ethnic groups or homosexuals. The best exploration of this relationship between ideas and class society is contained in Antonio Gramsci's discussion of what he called 'contradictory consciousness', which we shall address more fully later.
Post-Marxists see things very differently. So, not only do Laclau and Mouffe deny any relationship between people's material circumstances and the ideas they hold, but they also appear to deny any reality outside language or discourse. The editors of a recent social work text on anti-discriminatory practice define discourse as meaning 'A body of statements organized in a regular and systematic way which claim to represent the "truth" or promote dominant narratives in order to provide the rationale and justification for following subsequent rules' (Cocker and Hafford-Letchfield, 2014: 75). As the reference to dominant narratives suggests, in practice, the term is often used by writers within the critical social work tradition to explore and critique ideas and practices that maintain forms of oppression, such as the biomedical model or assumptions about the role of women within the family.
While critical discourse analysis can be useful in terms of deconstructing dominant assumptions and terminologies, as well as exposing the assumptions that underpin them (see, eg, Cowden and Singh's [2007] discussion of the language of 'user involvement'), from the perspective of developing an anti-oppressive social work practice – and, more widely, challenging social injustice – it suffers from three main limitations. First, there are philosophical objections, specifically, the question of the relationship between language and material reality. For Marx and Engels, arguing against the idealist philosophers of their own time, material reality was primary. One can abolish the 'idea' of gravity but that will not prevent things from falling down! As we have seen, however, the post-Marxists, like the Young Hegelians almost two centuries earlier, reverse this relationship. Thus, Laclau and Mouffe (2014 [1985], p 93) argue that 'Our analysis rejects the distinction between discursive and nondiscursive practices. It affirms... that every object is constituted as an object of discourse, insofar as no object is given outside every discursive condition of emergence'.
In his critique of Hegemony and socialist strategy, Norman Geras (1987: 66) has demonstrated the absurdity of this kind of thinking:
However frequently these may be denied, either in high philosophical argument or in popular assertion, a pre-discursive reality and an extra-theoretical objectivity form the irreplaceable basis of all rational enquiry, as well as the condition of meaningful communication across and between differing viewpoints. This foundation once removed, one simply slides into a bottomless, relativist gloom, in which opposed discourses or paradigms are left with no common reference point, uselessly trading blows. The most elementary facts of existence become strictly unthinkable without the aid of more or less elaborate theoretical sophistries. Was not the pre-human world 'an objective field constituted outside of any discursive intervention' – or did it have to await the appearance of humanity to 'construct' it? And even today, 'several currents of contemporary thought' notwithstanding, are there not realities of nature, both external and human, which are not merely 'given outside' every discourse... but the material precondition of them all?
John Molyneux makes a similar point in the context of a discussion of 'political correctness' and the overriding emphasis that some sections of the Left place on the use of 'correct' language:
While it is true that the development of language gave an enormous boost to the development of consciousness and thought, and that the nature of language exercises an important influence on what is thought and what is 'thinkable', it cannot be true that there is no consciousness or thought prior to language or animals would be unable to hunt, cats would not find their way home, chimps could not engage in elementary tool use and babies would not be able to learn language. Nor is it true that language constructs or determines consciousness from nothing. If it were, the project of language reform would itself be inconceivable. There is an ongoing complex interaction between external material conditions, physical and psychological human needs, human social relations and human thought and language. Within this interaction social being – the combination of circumstances, needs and social relations – remains primary. (Molyneux, 1993)
Second, seeing ideas – and politics – as completely autonomous from any class or material base offers no way of understanding where these ideas come from, why particular ideas dominate at particular times or, most importantly, how ideas can change – or, rather, as Geras (1987) observes, it does provide a way of understanding but one that is far from new and one that could hardly be further from the Marxist tradition that Laclau and Mouffe claim to be developing:
The suggestion... that there is no stronger relation between socialism and the working class than there is between socialism and anybody else is an idealism... run wild [which displays a] cavalier disregard... in the name of discourse, for material realities, relationships and needs.
Third, the alternative to what Geras describes as this 'shamefaced idealism' does not need to fall back on vulgar Marxist notions of 'false consciousness', or Stalinist versions of 'banking' models in which ruling-class ideas are simply poured into the heads of workers and passively accepted. A much more nuanced and complex analysis of where ideas come from, how they can be changed and their relationship with material reality was provided by the Italian Marxist Antonio Gramsci in his concept of contradictory consciousness. Gramsci distinguished between two different types of consciousness that chaotically coexist in the heads of working-class people. On the one hand, there is what he called 'common sense', based on the ideas that are churned out on a daily basis by the tabloid press and by mainstream politicians. Typical examples would be the idea that excessive welfare spending is the cause of economic crises, or that immigrants are responsible for overcrowded health services. In reality, these are examples of what Marx called 'the ideas of the ruling class', ideas that legitimise the capitalist status quo and divide workers. Alongside these, however, are very different ideas that spring from people's direct experience, particularly in the workplace: the understanding, for example, that it is only by acting collectively that it is possible to stand up to the employer or to gain improvements in working conditions, or learning through the experience of being on strike that the media do not always tell the truth and that the police are not our friends. Gramsci called such ideas 'good sense':
The active man-in-the-mass has a practical activity, but has no clear theoretical consciousness of his practical activity, which nonetheless involves understanding the world in so far as it transforms it. His theoretical consciousness can indeed be historically in opposition to his activity. One might almost say that he has two theoretical consciousnesses (or one contradictory consciousness): one which is implicit in his activity and which in reality unites him with all his fellow workers in the practical transformation of the real world; and one, superficially explicit or verbal, which he has inherited from the past and uncritically absorbed. (Forgacs, 1998: 333)
Gramsci's primary concern was with how 'good' sense could be purged of the stains of 'common sense', in other words, of how the working class could develop a genuinely collective consciousness free of racist, sexist and other backward ideas. For that to happen, he argued, based on his time as editor of a workers' newspaper during Italy's two 'red years' of 1919–20, two things were required.
The first was the experience of collective struggle, in the course of which, workers would become aware of the role of the media, the employers and the state, on the one hand, and their own collective power, on the other. Such struggles did not automatically lead to a change in ideas, but they created the conditions in which such a change could take place. Second, for political as well as ideological purposes, a revolutionary organisation was necessary, made up not of left-wing academics (though some choose to join), but rather of 'organic intellectuals', workers who through experience, study and discussion had developed a revolutionary worldview that enabled them to act as more effective leaders in their workplace or community.
Many examples could be given of the ways in which, in the course of collective struggle, when so-called ordinary people begin to develop an awareness of their own power, reactionary ideas – sexist, racist and homophobic – begin to change. Here, two will illustrate the point.
The 2014 movie Pride is set during the British miners' strike of 1984. It tells the true story of the formation of the first Lesbians and Gays Support the Miners group (LGSM), the relationship they developed with the striking mining community in South Wales, and the national impact they had. LGSM was formed after two gay activists decided to do a collection for the miners at the Gay Pride march in 1984. Having set about raising money, the group tried to make contact with the National Union of Miners but encountered an embarrassed silence. Determined not to be rebuffed, the group randomly chose a mining village in Wales and contacted its strike committee. So began a relationship that culminated in the astonishing and inspirational sight of a Welsh miners' brass band leading Pride in 1985 and the National Union of Miners voting in support of gay rights at the Trade Union Congress and Labour Party conferences the same year.
The second more recent example is from the early stages of the Egyptian revolution (or 'Arab Spring') of 2011 and shows how even the most deeply ingrained prejudices, in this case, religious prejudices, can be broken down when people are involved in a common struggle for a better world. In her eye-witness account, journalist and academic Anne Alexander (2011) described how just weeks after a New Year's Day church bombing rocked the port city of Alexandria, Egypt's religious tensions were set aside as the country's Muslims and Christians joined forces at anti-government protests:
Making my way to Tahrir Square during the anti-Mubarak protests, a striking piece of graffiti caught my eye. Scrawled on the concrete pillar of a flyover was the symbol of a Muslim crescent embracing the Christian cross and the words: 'We are all against the regime'. During the big 'Day of Departure' protest in Tahrir Square last Friday, Coptic Christian protesters made a human chain around their Muslim brothers and sisters as they performed the noon prayers. Two days later, 'Martyrs' Sunday' was celebrated by Egyptians of both religions as an affirmation of national unity in struggle. On that day, a crowd had gathered in front of a sound system by the Mogamma government building on the eastern side of the square. A man in a jacket began to speak as chanting died away. 'He was someone from the church, a priest,' someone in the crowd told me. I was told: 'We are all in this together. Muslims and Christians.' Other people were listening and nodding. 'One hand, one hand,' the crowd roared.
Such changes in people's ideas and consciousness are not simply fortunate by-products of struggle; rather, as Marx and Engels (1845) recognised in 'The German ideology', they are an essential part of the process through which people become fitted to build a new world:
Both for the production on a mass scale of this communist consciousness, and for the success of the cause itself, the alteration of men on a mass scale is necessary, an alteration which can only take place in a practical movement, a revolution; this revolution is necessary, therefore, not only because the ruling class cannot be overthrown in any other way, but also because the class overthrowing it can only in a revolution succeed in ridding itself of all the muck of ages and become fitted to found society anew.
#### Intersectionality?
As well as the post-Marxist ideas discussed earlier, another important concept that has recently gained some traction in contemporary social work theorisation is the idea of 'intersectionality', a concept explored by Sarah Carr in the collection by Cocker and Hafford-Letchfield (2014). Carr's (2014) piece is primarily a focused critique of bureaucratic, administrative and managerialist approaches to anti-oppressive social work practice. Carr's (2014: 141–3) point is, rightly, that oppression has a detrimental impact on people's lives and cannot be reduced to a functional tick-box exercise carried out as part of the 'proceduralist culture' that shapes the routine of much social work engagement. Carr (2014: 143–4) is also critical of those perspectives that portray oppression as a permanent given or a fixed state – with 'permanent oppressors and perpetual victims'. In contrast, she argues that it can, and must, be challenged and that this must involve those that suffer at the forefront of structures of discrimination and oppression.
On all these points, we have little disagreement with Carr. The second decade of the 21st century has witnessed renewed struggles around the impact of oppression on people's lives, with campaigns against structural racism and police brutality (where key roles have been played by organisations such as Black Lives Matter and Stand Up To Racism), against the impact of austerity (where, in Britain, organisations such as Disabled People Against Cuts and Recovery in the Bin have been central) and for women's equality (with organised Slut Walks, Reclaim the Night marches and a renewed focus on abortion rights). As radical social workers, we celebrate these movements and the challenges they present to social work theory and practice. As we have emphasised in many of our writings over the years, radical social work has much to learn from immersing itself into social movement activity and confronting the issues, problems and dilemmas that such activity generates (eg Ferguson et al, 2005; Lavalette and Ferguson, 2007; Lavalette and Ioakimidis, 2011a).
However, Carr's piece claims that theories of intersectionality can clarify and enhance our understanding of oppression, and this is something that we wish to challenge. Notions of intersectionality have developed out of the work of black feminist writers Kimberle Crenshaw (1989, 1991) and Patricia Hill Collins (2009, 2013). The concept was developed by them as a critique of postmodern approaches to difference and a critique of some white feminist practice that, they argued, ignored and excluded the specific experience of black women in the movement. Audre Lorde (2000: 289), for example, writes that:
By and large within the women's movement today, white women focus upon their oppression as women and ignore differences of race, sexual preference, class and age. There is a pretence to homogeneity of experience covered by the word sisterhood that does not in fact exist.
The attraction of intersectionality is that it points to a range of oppressions and to a common cause – to build unity against oppressive structures. In the work of Patricia Hill Collins, it also seems (in contrast to postmodern theories) to re-emphasise the importance of the socioeconomic analysis of inequality (Carr, 2014).
At the level of description, it can also usefully describe the reality of the issues that some people face. So, for example, for a social worker, it can be a useful reminder that the needs of a migrant, black, disabled woman for a particular service might be significantly different from other service users who are black but not disabled, for instance. However, the danger here, in the terms that Carr (2014) poses the issue, is that the 'tick-box' approach just becomes further extended within 'proceduralist practice'.
In any case, this descriptive use of intersectionality is not what Crenshaw meant when she used the term. Crenshaw first developed the concept after she used a metaphor of different roads (different oppressions) that 'crashed' when they met at the 'intersections'. In her case, she described her oppression as a woman and as a member of the black community, but at the 'intersection', when the two oppressions crashed into each other, the whole became greater than the sum of the parts. In other words, it was not merely an accumulation of oppressions, layered one upon another. Rather, the intersection produced something more: an integrated and compounded oppression as a black woman (where she was on the receiving end of racism from white women and sexism from black men). This is a powerful reminder that there is no necessary solidarity within oppressed groups – the way in which many French feminists have colluded with the French state's oppression of Muslim hijab-wearing women in their country is perhaps the most obvious recent example.
Here is a demand to consider the impact of class, race, gender, disability, sexuality, poverty and the way these interact to shape people's experiences. However, in its usage, there are two important consequences deriving from this. The first is that, as Carr (2014) notes, this approach often sits within Foucauldian ideas of power and its distribution, where every individual is inescapably part of a multiplicity of oppressive relationships – what Patricia Hill Collins (2013) calls a 'matrix of domination'.
For Foucault, power is not something that is primarily concentrated in the hands of capital or the state (as classical Marxists would argue): 'Neither the caste which governs, nor the groups which control the state apparatus, nor those who make the most important economic decisions direct the entire network of power that functions in a society' (Foucault, 1981: 95). Rather, power is distributed throughout society and exists in all social and interpersonal relationships. This has some very important strategic conclusions. Foucault (1981: 96) argues, for example, that there is, by necessity, a 'multiplicity of points of resistance' and so:
there is no single locus of great Refusal, no soul of revolt, source of all rebellions, or pure law of the revolutionary. Instead there is a plurality of resistances, each of them a special case... by definition, they can only exist in the strategic field of power relations.
The consequence is that each of the oppressions and intersections come laden with notions of discrimination, penalty, individual privilege and interpersonal domination. As Patricia Hill Collins (2013: 234) argues: 'each one of us derives varying amounts of penalty and privilege from the multiple systems of oppression that frame our lives'.
However, rather than draw us together, to consider what we have in common and how we can work together collectively to confront and overcome a system of oppression, such perspectives tend to infinite division and separation as 'difference' and micro-relations of power are emphasised at the expense of commonality and solidarity. Here, the collective confrontation of structural inequalities and oppressions is replaced with individual solutions and demands that we each 'check our privilege' that is derived (perhaps unconsciously) for our social position in an intersectional world.
The second theme that comes from the literature on intersectionality is an emphasis on individual 'experience' and subjects' 'lived reality'. Within social work, one of the great advances of recent years has come about from the demands of the service user movements that their voice be heard and given full consideration within social work processes, and that 'knowledge through experience' is a vital source for discovering, tackling and addressing all manner of institutional practices that reinforce discrimination. However, theories of intersectionality remain locked in broadly postmodern concerns with essentialist knowledge that prioritises individual experiences. The problem with this is that it removes our ability to make judgements between competing 'experiences'. Here is one rather extreme example. In 2013, the then Mayor of London (now UK Foreign Secretary) Boris Johnson wrote an article telling us that we should 'stop bashing the rich':
I neither resent nor disapprove of [London's super-rich]... quite the reverse.... we should stop bashing the rich.... On the contrary, the latest data suggest that we should be offering them humble and hearty thanks. It is through their restless concupiscent energy and sheer wealth-creating dynamism that we pay for an ever-growing proportion of public services.... And yet they are brow-beaten and bullied and threatened with new taxes, by everyone from the Archbishop of Canterbury to Nick Clegg [at the time the Deputy Prime Minister of the UK]. (Johnson, 2013)
Johnson's rather ludicrous point was that the super-rich were a picked-upon minority – a point not even worth responding to. However, the more serious point in our discussion here is that if the super-rich self-define as a 'picked-upon minority' bashed by government, media and the population at large, how can we disagree if we prioritise individual 'experience' as the key definer of oppression?
In contrast, classical Marxist approaches to oppression see it as rooted in the systems of structured inequality that shape capitalist (and, more generally, class) societies. Thus, individual experiences are important, and shed light on the impact of oppression and discrimination, but this does not, in itself, tell us where that oppression originated, its social function and how it has been 'socially constructed' within the specific context of modern class society. This does not mean that classical Marxism ignores oppression. From the writing of Marx and Engels onwards, the classical Marxist tradition has attempted to offer analysis of the roots of women's oppression, of racism and of disability discrimination, for example, and has tied these theoretical approaches into a political struggle to counter oppression and fight for liberation. It is the question of liberation that brings us back to the question of agency.
### 'The point is to change it!': the question of agency
Finally, we return to the question of agency. In a world dominated by 'capitalist realism' in which, as Frederic Jameson (2003) has observed, people often find it easier to envisage the end of the word than the end of capitalism, the fact that there is a resurgence of interest in utopian thinking, driven by the idea that 'another world is possible', is to be welcomed. Envisioning a different world, however, is not enough: the key question is how do we get there and what role, if any, can a critical and radical social work play in this process? What force, or forces, in society have the interest, power and capability to challenge the seemingly impregnable citadels of neoliberal capitalism? These are questions that Gray and Webb seek to address in the concluding part of their search for a 'new politics of social work':
The question of a militant political agent – Foucault's subject of resistance – antagonistic to the supposed inevitability and universality of capitalism – sits at the very centre of this seemingly bleak state of affairs.... The question is just as pressing for social work as it is for any other agency of progressive change. Indeed, any attempt to construct a new politics of social work must confront the question directly. (Gray and Webb, 2013b: 215)
Indeed it must. Unfortunately, their attempts to confront that question do not take us very far. Positively, they are highly critical of the contribution of both postmodernism and identity politics:
Some social work researchers are fascinated and seduced by the aura of postmodernism. Let us be clear, postmodern politics is not the slightest bit concerned with equality, justice and poverty. Postmodernism is not cool. The vagaries found in the postmodern social work literature, and often associated with 'identity politics', celebrations of diversity and the Othering of difference is a self-defeating exercise that can only lead to a political blind alley. (Gray and Webb, 2013b: 218)
Less positively, they offer no real alternative. Thus, the possibility that a radical alternative might inhere in Marxism is dismissed with a 1978 quote from the Polish ex-Marxist philosopher Leszek Kolakowski suggesting that 'Marxist literature, although plentiful in quality, has a depressing air of sterility and helplessness' (cited in Webb and Gray, 2013b, p 218). This is lazy thinking. For a start, Kolakowski's view of Marxism, understandably given his experience of life and persecution under 'actually existing socialism' in both Poland and the Soviet Union, was that brutal Stalinism was the logical outcome of Marxism. In similar fashion, Geras (1987, emphasis in original) notes that, while based on very different experiences, 'Laclau and Mouffe go so far as to conflate the whole of Marxism with its Stalinist, or authoritarian, forms by writing sometimes as though democracy was just external to it'.
While the equation of authoritarian Stalinism with Marxism was a standard feature of Cold War thought (and a very convenient one for the ideologues of Western capitalism), in reality, within the classical Marxist tradition, democracy and socialism have been seen as inextricably linked. As Geras (1987) argues in his critique of Laclau and Mouffe, there have always been those on the Left who:
knew – what the authors have evidently forgotten – and sought to strengthen and extend, the principles and sources within the Marxist tradition which, against both the forms and the pretensions of 'actually existing socialism', spoke insistently of socialist democracy: in Marx, from his earliest philosophical writings to the Paris Commune; in Trotsky, from the pluralist arguments of Our Political Tasks to the fight against Stalinism; in Luxemburg – incandescent – practically everywhere; and in Lenin, and elsewhere besides. The cover of darkness, either intellectual or contemporary political, was not a necessary excuse.
Insofar as they do offer an alternative agency, it seems to be based on a combination of a 'politics of refusal', protest groups, the creation of community spaces and envisioning an alternative to capitalism. All of these are important but they hardly constitute an adequate vehicle for challenging a global system that is threatening to destroy within decades the very conditions for life on this planet (Klein, 2015).
### Conclusion: 'Anything goes'
In a brutal assessment of the propositions of Laclau, Mouffe and post-Marxism in general, Perry Anderson (2016) has written:
The linguistic turn of the theory, in common with its late-twentieth-century vogue in general, proposed a discursive idealism severing significations from any stable connexion with referents. Here the result was to detach ideas and demands so completely from socio-economic moorings that they could in principle be appropriated by any agency for any political construct. Inherently, the range of articulations knows no limit.
Similarly, a philosophy that not only fails to address the material realities of the lives of those who rely on social work services – realities of poverty, inequality and oppression – but also ignores or denies the ways in which these realities shape their lives, their behaviour and their consciousness, is of little value to a social work that is seeking to make sense of these realities in order to change them.
## NINE
## The case for a social justice-based global social work definition
The social work profession promotes social change, problem solving in human relationships and the empowerment and liberation of people to enhance well-being. Utilising theories of human behaviour and social systems, social work intervenes at the points where people interact with their environments. Principles of human rights and social justice are fundamental to social work. (IFSW, 2002, cited in BASW, 2012)
### Social work and social justice: current definitions and debates
Part of the current ideological and political discussion about social work is reflected in the crucial question 'How do we define what we do?'. In fact, the different ways in which international organisations, states and movements have engaged with this debate reveal the ideological tensions that divide the social work project. This is not simply a theoretical or abstract debate: defining social work has an impact on what happens in practice. Prescriptive, 'clinical' or apolitical interpretations of social work have profound effects on the realities on the ground. When they stress individualistic and moralistic interpretations of social work, they tend to reduce social work to a technical activity in the most formal ways. Official definitions that feed into law, academic curricula and job descriptions can create artificial professional boundaries limiting space for relationship-based and politically engaged social work. The 2002 International Federation of Social Work (IFSW) and International Association of Schools of Social Work (IASSW) global definition (see opening quote), in particular, which has been the most widely recognised one, usually results in direct changes in regional and country-specific legislation regarding the nature, description and professional boundaries of social work practice.
Definitions do matter: the experience of social work in Colombia
In 2016, the Colombian Ministry of Education made use of the United Nations Educational, Scientific, and Cultural Organization (UNESCO) definition of social work in order to reclassify social work as a health profession, thus challenging its social science basis and openly demanding that curricula be devoid of sociology and discussions about social movements (Ministerio de Educación Nacional, 2016). Colombian social work has historically been influenced by the tradition of liberation theology, and Colombian practitioners and educators were central to the development of the reconceptualisation movement in Latin America. Moreover, social work in the country has been important in the development of a peace and social justice movement demanding an end to the decades-long conflict. The decision to reclassify the profession as a technical activity, using the UNESCO definition, aimed at rooting out the most radical and political traditions of social work. After weeks of mobilisation, sit-ins, nationwide protests and an important wave of solidarity from other Latin American countries and the IFSW, the Colombian government backtracked and reinstated the social science basis of social work. The movement developed organically, adopting and quickly uniting most social work departments across the country under the banner 'Mi trabajo es SOCIAL' ('My work is SOCIAL').
In 2014, the IFSW and IASSW adopted an 'updated' global definition after a consultation that lasted for almost five years. We will examine later the political and professional debates that underpinned this change. In our view, the latest definition has been a step backwards in terms of its aspirations for a social justice-based social work as it includes mutually conflicting terms and concepts (see later) that eventually render the definition unworkable and meaningless. The adoption of an eclectic and mixed social work definition has rather obscured the profession's commitments to social justice and can potentially weaken efforts to promote a genuinely anti-oppressive and emancipatory practice.
It is our contention that social work practitioners, service users and students need to engage with this discussion meaningfully and contribute towards a definition of social work that reflects the profession's commitment to social justice. In this chapter, we assess the main aspects of the current debate, focusing on the concepts and contradictions that have informed discussion. In particular, there is an emphasis on the importance of understanding the contested history of social work and the main tension between 'social care' and 'social control' as a way of comprehending the complexity and significance of this debate.
### Exposing and challenging the structural causes of personal problems
Social work as an activity is located in a rather unique position: at the point of interaction between the state and the most vulnerable people in society. Therefore, front-line practitioners in much of the world routinely witness some of the most brutal effects of structural inequalities on people who, voluntarily or involuntarily, interact with social services. Despite the fact that social workers deal on a daily basis with the brutality of a socio-political system based on a profoundly illogical and disastrous distribution of wealth, this knowledge and experience has not always fed into our theory and practice. Most importantly, in societies where discussions about poverty and inequality are customarily silenced, speaking truth to power about clients' experiences needs to be the basis of an ethical, value-based and politically engaged social work practice.
Modern capitalist societies are characterised by one of the most overwhelming paradoxes in human history. On the one hand, tremendous advances and innovations in technology, industry and research have made it possible for humanity to generate unparalleled wealth. The global annual output of wealth is nowadays sufficient to feed, educate and provide good standards of life for the whole population on this planet. A research report commissioned by Oxfam (2013) revealed that the annual income of the richest 100 people is enough to end global poverty four times over. On the other hand, the main characteristic – and tragedy – of modern capitalism is the immense concentration of such wealth in the hands of a minuscule minority of individuals and businesses. In 2006, a study by the World Institute for Development Economics Research revealed that the bottom half of the world's population owns almost 1% of all global wealth (Davies et al, 2006), while Oxfam (2013) indicates that this trend is not confined to rich countries:
In the UK inequality is rapidly returning to levels not seen since the time of Charles Dickens. In China the top 10% now take home nearly 60% of the income. Chinese inequality levels are now similar to those in South Africa, which is now the most unequal country on earth and significantly more unequal than at the end of apartheid. Even in many of the poorest countries, inequality has rapidly grown.
As this report indicates, even though inequality is not a phenomenon unheard of in the history of societies, its extent, pace and ferocity in the age of neoliberal capitalism is unprecedented. Apart from the startlingly unequal distribution of resources at a global level, one has to consider the widening inequalities within individual country contexts.
The impact of structural inequality on individuals and communities alike is nowadays very well documented. Recent epidemiological studies (Wilkinson and Pickett, 2009; Stuckier and Basu, 2013) have confirmed that the more unequal societies are, the more likely it will be for the poorest people to die younger, suffer mental health problems, be affected by chronic illnesses, be exposed to criminal activities, experience racism and face exclusion from education. In short, neoliberal capitalism, a brutal system based on exploitation and the unequal distribution of resources, is responsible for most of the reasons for people to interact – whether voluntarily or involuntarily – with social services. Social work clients arrive at social services not as a result of their own individual inadequacies, weakness or misfortunes. Neither is it because they are idle, malicious and greedy people who want to benefit from an overgenerous 'nanny state' – as the dominant neoliberal rhetoric misleadingly claims. Poverty, inequality and unemployment are unambiguous determinant factors forcing the most vulnerable people into exclusion and alienation. Nevertheless, as we argue later, much of mainstream social work has turned a blind eye towards this reality.
Pathologisation, stigmatisation and surveillance have been the norm rather than the exception in much of the history of top-down welfare (Jones and Novak, 1999). Omitting to highlight the structural causes of 'private ills' leaves space for 'blaming the victims' rather than addressing the roots of the problem. A genuinely non-stigmatising and anti-oppressive social work practice needs to be organically rooted to the principles of social justice. Therefore, it is imperative that the global definition of social work reflects the struggle towards alleviating the structural causes of 'private ills' and does not legitimise the perpetuation of social injustice towards the people we work with. As we explain later, in the history of social work, there has been considerable tension between those social work approaches that have endorsed the need for social justice, on the one hand, and 'top-down' bureaucratic approaches whose main concern has been with the policing of poor families and individuals, on the other.
### 'Nunca Mas!' ('Never again!'): exposing oppressive practice and the dark history of social work
In 2010, the disciplinary committee of the College of Social Service Professionals in Argentina decided to expel and take legal action against a social work practitioner on the basis of his politically unethical practice during the period of General Videla's military junta in that country (1976–83) (see Alayón, 2010). The practitioner had collaborated with the regime in promoting one of the most horrible examples of oppressive and punitive social work practice. While working at the Central Hospital of Neuquén, he gained access to injured students and other activists who had participated in the movements for democratic change. Under the pretext of professionalism and claiming confidentiality, he 'caseworked' the patients, collecting information about their political actions. He then routinely passed all this sensitive information to the junta's intelligence services (Intelligence Battalion 601). Several of the activists involved in the case were arrested and tortured. The Argentinian Federation of Professional Associations of Social Service Workers (FAAPSS) justified its decision to expel and take legal action against this individual by stating that 'It is important to promote the politics of memory, truth and justice reflecting the commitment of Argentinian social workers to the process of social emancipation for the peoples of Latin America' (Martins, cited in Alayón, 2010). Such swift and powerful action against social workers who had collaborated with the regime and engaged with oppressive practice is part of a broader movement in Argentina that demands justice through the exploration of truth with reference to the 'dark' years of the military junta; the movement has adopted the short but powerful motto 'Nunca Mas!' ('Never again!').
The open debate within Argentinian social work is rather unique in the context of the profession. Social work as a profession has long suffered from 'selective amnesia' (Reisch and Andrews, 2002). Literature about the history of the profession is scarce and shallow, emphasising the almost 'inherent' benevolence of social workers and their sacred commitment to 'do good'. Mainstream histories about the origins of social work are monotonously structured around the brilliance of charitable individuals, known as the 'pioneers', who were committed to developing expertise in order to pull 'dysfunctional' people and families out of their misery (for a critique of this approach, see Lavalette and Ioakimidis, 2011b). These kinds of historiographies are filled with didactic cliches about the kind-hearted 'do-gooders' and the readily available tool kits of social workers that seem appropriate to nearly all known 'dysfunctions' and 'problems'. They hardly ever mention the fact that social work is the direct product of political calculations and that the creation of the profession has always been bound to contradictory and politically opposing ideologies. References to the broader political context, the class nature of the pioneers' activities and the internal contradictions within the profession are almost non-existent (Jones, 2012). Moreover, the existence of a rich radical social work tradition that developed in different parts of the world, and that rejected top-down professionalism in favour of developing social alliances that could bring about social change, is wiped out of most mainstream social work histories. The direct outcome of such selective amnesia seems to be the creation of a distorted, unrealistic and disabling self-imaginary, detached from social realities. Historically, such lack of self-awareness has rendered social work a questionable activity in much of the world.
Tragically, the refusal to understand the political context within which social work operates has led to some of the darkest chapters of the profession's history. Despite the fact that research on the 'dark side' of social work is scarce, the findings available are strong enough to suggest that behind the pretext of 'political neutrality', social work has been involved in some of the most terrible cases of political oppression. The examples that we discussed in Chapter Four of this book are only a few illustrations of social work's systematic involvement in political practices contrary to the ethos of the profession's public declarations and statements. Unless we reflect on our history (even the grimmest chapters of it) and acknowledge the clear political nature of social work activity, we are incapable of developing a global social work distinctly and unconditionally committed to social justice. It is imperative that any global definition of social work reflects this dynamic process of politically repositioning the profession towards social justice and clearly opposes epistemologically vague, abstract or apolitical notions of social work. Otherwise, we risk the possibility of re-experiencing some of the ugliest cases in social work history.
### Recognising and protecting practitioners who fight for social justice
Ultimately, it is the behaviour, ideology and practices of front-line practitioners that determine whether social work is a 'worthy' or 'unworthy' activity, 'relevant' or 'irrelevant' to the service users and broader society. Since social work does not operate in a political vacuum, it is the way in which social workers endorse or challenge oppressive and unjust policies that resolves the historic contradiction between 'social care' and 'social control'. In short, the decisive question that social workers often need to answer is 'Which side are you on?'.
Fortunately, alongside the cases of oppression mentioned earlier, social work is entitled to boast some brilliant examples of steadfast commitment to social justice. These examples have been present since the very beginning of social work history, thus forming a long-standing radical tradition. In the UK, in opposition to the elitist and punitive agenda of the Charity Organisation Society – which transformed charitable activity to 'scientific' social work – a reformist tradition emerged around the 'settlement movement' (in the late 19th century). Social workers and volunteers involved in this movement placed emphasis on the material context of poverty and deprivation, hence challenging the elitist belief that the poor are inferior and inherently incapable of leading meaningful lives (Ferguson, 2008). Furthermore, the strengthening of trade unionism in the first three decades of the 20th century and its subsequent suppression by the ruling classes across Europe informed the creation of one of the most extensive networks of grass-roots politically committed social work. The Red Aid was an umbrella organisation that originated in Germany in 1919, and by the Second World War, its operations covered many European countries, including Austria, Poland, France, Greece, Sweden and the UK. The Red Aid not only emphasised the class divisions and inequalities prevalent in the capitalist world, but also adopted a clear militant stance towards alleviating the structural causes of inequality and poverty (Schilde, 2003). Most importantly, it provided support to groups and individuals who suffered from the criminalisation of political activism and were excluded from state-controlled welfare organisations. In practice, the Red Aid tried to combine multilevel support to political prisoners, refugees and their families with mobilising the 'masses' against oppression and injustice. Many of the 'interventions' and methods that mainstream social work discovered nearly 60 years later (such as 'advocacy and legal representation', women's rights and the use of contraceptives, political campaigning, non-judgemental moral and material support, and social education) originated from this radical tradition. These activities were based on class solidarity and were in opposition to the ethos and traditions of ruling-class charity. As Mentona Mosser, the founder of the first social work school in Switzerland and leading member of the Red Aid, explained, 'The bourgeoisie is never so repulsive as in those cases when they are doing charity work, "stinking" charity work' (cited in Hering, 2003: 90), while Clara Zetkin described the social work activities of the Red Aid as 'the aid squad of the class struggle' (cited in Schilde, 2003: 142).
Another large social justice-based tradition that has greatly influenced social work in parts of the world (Latin America in particular) is that of liberation theology. Suffice it to say that this tradition has also been excluded from mainstream social work textbooks and historiographies. Liberation theology emerged in the 1950s, blending the influence of the Christian faith in the region with the optimism for social emancipation of the oppressed embedded in the Marxist tradition (see Gerassi, 1973). In much of Latin America, the old colonial systems were not overthrown after independence, but only replaced by local corrupt and authoritarian elites, manipulated by imperialist powers. As the peoples of Latin America were disillusioned by the perpetuation of inequality, devastation and extensive exploitation, a distinct current of grass-roots social solidarity shook the conservative Catholic Church. This radical current focused its theory and praxis on two main elements: (1) the material and spiritual support of the oppressed; and (2) political 'conscientisation', a necessary stage in the process of broad political emancipation. The former was expressed through grass-roots community work, based on genuine class solidarity – not charity – while the latter took the form of open confrontation (often violent) against authoritarian states and large landowners. For liberation theologians, the idea that the poor had to humbly accept the harsh realities of their lives, hoping for spiritual redemption, was simply unacceptable and had to be challenged by any means available. Such an ethos was reflected in the famous 'preferential option for the poor approach' articulated by Gutierez (1971). This revolutionary tradition was tremendously popular in several Latin American countries (including Brazil, El Salvador, Paraguay, Argentina, Colombia and Chile) and led to the political transformation of social work (Norwood-Evans, 1992). The 'reconceptualisation' movement in these countries is still highly influential and has historically generated powerful examples of genuine anti-oppressive practice, even when a wave of US-sponsored military coups devastated most of Latin America – Brazil (1964), Peru (1968), Bolivia (1971), Chile (1973), Uruguay (1973) and Argentina (1976) (see Allayon, 2005). Radical social work theory and practice in Latin America has emphasised the importance of a holistic view of the challenges that people experience in their lives, the primacy of understanding the material and political context of social work practice (conscientisation), the need to develop social alliances and partnerships based on democratic values and mutual understanding, and, finally, the importance of direct action in order to alleviate the structural causes of oppression. Paulo Freire, the leading educational theorist, specifically advocated for a social work practice based on political action (praxis) rather than professional elitism within social work. His remarks are timely and important in the context of debates around the global definition of social work:
In this sense, therefore, the social worker, as much as the educator, is not a neutral agent, either in practice or in action. One of the inclinations that we sometimes have – and this is an offense, an illegality, that we imbibe in our technological society – is to think that the social worker is a very specialized person, a technician, who works in a compartmentalized technical area, and who has a sort of protection within this area, a sort of aggregate of rights, as a particular social group, to stand apart from the political battles of society. For me, this is impossible. It is an error. Social workers are compromised if they become convinced that they possess a technical expertise that is more to be defended than is the work of other workers. (Freire, 1990: 5)
At present, in the context of austerity policies across much of Europe, social workers have not only participated en masse in demonstrations in defence of the welfare state and social services, but also engaged with acts of direct social activism and political disobedience against government policies that violate human rights (Ioakimidis and Teloni, 2013). In this process, social work networks and national associations opposing the politics of austerity have used the existing definition of social work as a tool in their campaigning work. In several cases, the politically progressive content of the definition as well as the 'statement of ethical principles' adopted by the IFSW has provided the necessary justification and legitimisation for activist social workers to pursue radical practice even when they faced criminalisation and intimidation by their employers and the state.
Two notable examples of such practice come – not coincidentally – from countries 'bailed out' by the notorious international Troika of lenders (the European Union [EU], European Central Bank and International Monetary Fund [IMF]). One of the most malicious characteristics of the austerity imposed in several European countries is its absolute disregard of constitutionally protected human rights. Such disregard of the post-war 'social contract' is safeguarded by crude authoritarianism and justified on the basis of the 'law of necessity', often used in extreme circumstances in order to circumvent constitutional processes (see Pillay, 2011). Using such legal argumentation, in 2011, the Hungarian government passed a law making homelessness punishable by a fine or prison (BBC, 2011a). At a time when violent austerity had pushed more than 10,000 Hungarians into homelessness in Budapest alone, the government decided to outlaw poverty rather than dealing with the causes of it. Norbert Ferenzc, a social worker who belonged to a local radical social work network, played an active role in the campaign against state violence targeting the most vulnerable in society. He actively participated in demonstrations against the municipal ordinance that classified 'dumpster diving' as a misdemeanour, linked to the aforementioned law. His involvement in this campaign was considered by the police as 'incitement', a felony punishable by up to three years in prison. Ferenzc was eventually arrested and stood trial (SWAN, 2011b).
His arrest triggered an overwhelming international movement of solidarity among social workers. Thousands of practitioners, students and academics signed a petition demanding his immediate release. At this point, it is critical to highlight that legal and ethical justification of both Ferenzc's action and the international campaign in his support was based on the mandate to social justice stated in official documents of international social work organisations. The IFSW/IASSW definition and the Code of Ethics were effectively utilised in order to demonstrate that Ferenzc acted appropriately and in accordance with the guidelines of international social work organisations. The solidarity petition that was initiated by the Social Work Action Network and was signed by thousands of social workers stated that:
As a social worker, Norbert was following the International Federation of Social Work's definition of social work as an activist occupation that confronts social injustices. The Hungarian Code of Ethics for Social Workers also makes it clear that practitioners have a duty and a responsibility to inform the public of the growth of poverty and inequality and the state's responsibility to address these problems. We demand that all charges against Mr. Norbert Ferencz are dropped, as he was merely following his professional Code of Ethics. The Code states that 'social workers [should] facilitate change through their activities and professional stance' (Point 11) and that 'it is the social workers' responsibility, as well as a right and duty of the undersigned professional organizations, to call the attention of decision makers and the general public to their respective responsibility for the emergence of poverty and suffering as well as for their obstruction of the alleviation thereof'. (SWAN, 2011b)
Such arguments proved effective and the Hungarian Supreme Court eventually acquitted Norbert Ferenzc, in a case that clearly demonstrated the importance of using a social justice-based social work definition as a tool for defending social workers who fight injustices and work in partnership with service users.
Another similar case emerged in Greece almost a year after the arrest of Norbert Ferenzc. In December 2011, the Greek government, desperate to raise necessary funding for the recapitalisation of the banking sector (through a bailout), circulated a decree introducing a draconian 'urgent' property tax payable through electricity bills (Ioakimidis and Teloni, 2013). At a time of vast unemployment, such a tax particularly targeted the most vulnerable and those living below the breadline unable to pay any further taxes. This tax was flat and variations in income were not taken into consideration: even people on benefits had to pay. The fact that it was payable through an electricity bill guaranteed that people unable to afford this tax would face electricity cut-offs. The government, realising that the law was clearly unconstitutional and would be dropped at the Supreme Court, decided to use social workers to sugar-coat this measure. Social work practitioners were thus instructed to participate in committees that would 'assess' poor households in order to decide whether access to electricity should be cut off or not. In reality, the already-set targets in the national budget ensured that there was no space for meaningful assessments. Such an unethical use of social workers was met with fierce opposition. The Greek Association of Social Workers expressed opposition to this law (GASW, 2011), while the Greek Social Work Action Network (SWAN-Greece) and the trade union of local authority workers (POE-OTA) called for social workers to disobey the decree and consciously refuse to participate in these committees (Ioakimidis and Teloni, 2013). Once again, the global definition of social work and the IFSW ethical statement were utilised by social workers in order to justify their, potentially punishable, acts of civil disobedience against unethical and oppressive policies. The Greek Association of Social Workers circulated a statement vehemently rejecting the government's instructions while making reference to the global commitment of social work to social justice, 'the importance of the campaigning role under austerity and the profession's commitment to social change, social justices and universal social welfare' (GASW, 2011). Likewise, SWAN-Greece (2011; emphasis added) explained that:
the international social work codes of ethics, recognizing that human suffering is constructed upon socially unjust policies and acknowledging that the idea of 'political neutrality' within social work is deceptive, clearly define the ethical commitment of social workers towards exposing and fighting against socially unjust policies and practices.
After the coordinated response of local authority social workers and the POE-OTA trade union, the government was forced to retreat. Even though the urgent tax went ahead, the idea of 'assessment committees' invoking social workers as means of implementing this policy was scrapped.
Across Europe, while similar examples of political action against the dismantlement of social services have occurred, the largest movement in defence of universal social welfare has emerged in Spain. Known as the Orange Tide and popularising the slogan 'No Cuts! Don't Shut Up!', the Orange Tide brings together thousands of social workers, welfare officers and service users. Once again, striking social workers based their struggle on their obligation to defend human rights and dignity, which derives, inter alia, from the current definition of social work (IFSW-Europe, 2012). The cases presented in this section clearly suggest that a withdrawal of a global definition unconditionally committed to social justice will not only weaken the profile of the profession, but also harm front-line social workers who experience prosecution due to their ethical and value-based practice.
### Supporting indigenous movements and challenging the conservatism of cultural relativism
In the run-up to the Global Social Work Congress in Australia in 2014 that ratified the revised definition of social work, heated debate took place within the IASSW board. In one of these meetings, board members placed particular emphasis on a sentence stating that 'Underpinned by theories of social work, social sciences, humanities and indigenous knowledges, social work engages people and structures to address life challenges and enhance well-being'. While most members of the board – which is dominated by North American and European delegates – celebrated the innovation of acknowledging the 'importance of indigenous knowledges', some African board members seemed baffled and clearly thought otherwise. One of them stood up and in a state of frustration and disbelief asked the following question: 'So, what you are trying to say... is that you represent the social sciences and I am the indigenous knowledge?'. This anecdote provides a striking example of the hugely paradoxical nature of the 'indigenisation question' within social work. The epistemological and political complexity of this debate has allowed for the emergence and reproduction of some very conservative and patronising arguments masked by the obscurity of the fashionable term 'indigenisation'.
In recent years, there has been a resurgence of interest around this theme, mostly deriving from the broad region defined in the IASSW as the 'Asia-Pacific' (see, eg, Gray et al, 2008; Yunong and Xiong, 2008). Historically, the indigenisation paradigm originated from national liberation movements and the concept of 'Third Worldism', which gained momentum in international politics in the early 1970s (Atal, 1981). Much of the debate revolved around the issue of professionalisation and, in particular, concentrated on the question of 'transferability'. With the notable exception of Midgley (1981), who provided a groundbreaking analysis of the imperialist nature of international social work, mainstream literature at the time followed a largely developmentalist approach, attempting to provide narrow technical answers to the rather political question of 'transferability' (Shawky, 1972; Resnick, 1976; Walton and Abo El Nasr, 1988).
In the early 2000s, the domination of postmodern theories in the social sciences reinvigorated the indigenisation debate. Aided by the obscurantism of postmodern theorising, the 21st-century debate on indigenisation placed an almost exclusive emphasis on the centrality of culture. As Gray et al (2008: 8) suggest: 'Not only is social work itself a cultural construction but whenever we are attempting to make social work responsive to local contexts we are fashioning it in a cloth that is culturally embedded'.
One of the main deceptions of the indigenisation debate is its use of a language of 'resistance' even though, in reality, it promotes an implicitly conservative agenda. On the one hand, indigenisation seems to be dismissing nearly everything that derived from 'modernity' or that originated from the West, thus heavy-handedly conflating mainstream social work, for example, with the radical tradition of resistance and social movements – equating, in other words, modernity with oppression and exploitation. On the other hand, the unconditional glorification of local indigenous traditions and systems often presupposes the acceptance of rigidly unequal and oppressive hierarchies. Significantly, even though China and Japan (whose representatives appear to be active proponents of the indigenisation agenda) account for almost half of the IASSW membership base, the presence of Latin American schools is nearly non-existent. Consequently, the region with the richest history of the mobilisation of the indigenous population seems to be excluded from the current debates. One reason for that may be that the Latin American approach to liberation of indigenous peoples is diametrically opposite to the conservatism of the 'Asia-Pacific'. This is clearly reflected in the manifesto of the EZLN, the iconic indigenous organisation of the Chiapas, which instead of prioritising 'cultural differences' argues for the development of critical alliances and common struggles with the other oppressed groups in Mexican society:
We are going to continue fighting for the Indian peoples of Mexico, but now not just for them and not with only them, but for all the exploited and dispossessed of Mexico... we are going to go about building, along with those people... a program of struggle, but a program which will be clearly of the left, or anti-capitalist, or anti-neoliberal, or for justice, democracy and liberty for the Mexican people. (EZLN, 2005)
Furthermore, the official 'Asia-Pacific' line of argumentation seems to undermine the importance of 'social change and social justice', focusing on the need for 'stability of society, harmony in relations, unique culture and traditions, responsibility' (Angten and Paulsen, 2008). Some have gone so far as to anecdotally claim that the concepts of social justice and human rights do not even translate into local languages (such as Mandarin) and others prioritise spirituality. Henrickson (cited in Angten and Paulsen, 2008) explains that 'We have agreed that spirituality needs to be part of the way social work understands itself and its purpose'.
These approaches are, in essence, conservative and seem to be representing an elitist understanding of social work in the context of Asia-Pacific. Their conservatism derives from the fact that they: (1) prioritise idealist understandings of the material context that affects the lives of individuals and communities, masking in this way the fierce inequalities and oppression that exists in the Asia-Pacific countries; and (2) do not seem to be representative of popular indigenous movements that struggle to transform these oppressive structures in unison with other social groups (like the indigenous movements in Latin America).
The question of spirituality is a complex one and, indeed, within specific contexts, bears relevance to influential faith-based approaches. It is worth focusing, however, on the cases of China and Japan, whose large membership base in international social work organisations allows them to elevate the idea of 'harmony and stability' over 'social change and justice'.
Over the last decade, the rapid development of the Chinese economy and the gradual endorsement of an 'open market' economy did relatively little to alleviate poverty and inequality in the country. In fact, urbanisation has led to massive waves of internal migration readily available to be used as cheap labour. By 2002, the number of rural–urban migrants exceeded 100 million and the emergence of a new rich entrepreneurial class, largely benefiting from cronyism and corruption, provided an excellent case study of the various processes that generate wealth inequality (Knight, 2013). While urban China experiences considerable wealth generation – fuelling a sharp rise in inequality – rural China is still trapped in poverty. The World Bank (2010) estimated that in 2010, almost 486 million people lived on less than US$2.5 a day. The recent natural disasters that affected rural China exposed the extent of poverty and underdevelopment, and forced the local press to recognise that 'The poverty and fragility that the earthquake has revealed are still pressing problems in China's rural areas and show that the country needs to expend more efforts to reach its target' (see: <http://english.people.com.cn/90882/8222866.html>).
Mainstream social work in China seems to be developing rapidly following a 'top-down' bureaucratic approach. The creation of social work has been effectively decided and designed at the higher levels of the state apparatus rather than demanded by the poorest and most vulnerable in society. According to Wang (2012), it was the 6th Plenary session of the Chinese CP [Communist Party] central committee that decided to construct 'a large strong team of social work professionals', the purpose of which was to ensure stability and harmony in society. Here, the concept of stability is presented in opposition to the popular struggles for social transformation towards a socially just society. In a country where poverty and inequality are still rife and past efforts to promote social change have been violently crushed, invoking the concepts of harmony and 'spirituality' in order to mask existing class tensions is inextricably linked to the effort to develop a depoliticised, government-controlled social work.
Interestingly, although never mentioned in mainstream social work platforms, let alone the international social work organisations, in China and Hong Kong, a dynamic current of radical social work practitioners and academics already exists, reflecting the class divisions and tensions of Chinese society. Their power and militancy was well represented when, during the 2010 Global Social Work Conference in Hong Kong, striking social workers and trade unionists formed picket lines near the conference venue and, concurrently with the main conference, organised a well-attended counter-conference entitled 'Reclaiming Progressive Social Work' (Ferguson and Lavalette, 2012). Afterwards, the group of radical social workers involved in this conference produced a manifesto stating that:
we must launch a campaign for progressive social work and social welfare, and insist upon protecting the spirit of social work that is to promote human rights, justice, democracy and equality. We must resist any oppression against social welfare and its service users in order to establish a society that respects human rights and secures social justice. (PSWN, 2011)
Similarly, in Japan, an ESRC-funded series of seminars in 2009–10 found that:
In contrast to some recent influential British literature which has portrayed Japan as a more equal, and therefore 'happier' society, Japanese colleagues highlighted the wide range of social problems to which neoliberal policies have contributed including very high suicide rates, family breakdown, social isolation and withdrawal. (Ferguson et al, 2011)
Moreover, the same report highlighted the impact of neoliberalism and managerialism on Japanese social work, with colleagues explaining that recent reforms have led to the erosion of social workers' autonomy, marking a significant turn towards neoliberalism.
There is much evidence to suggest that despite the fanfare that surrounds the primacy of harmony in the 'Asia-Pacific' region, scratching the surface would reveal a similar context of structural problems, class divisions and political contradictions rife in social work across the globe. A global definition that neutralises the commitment of social work to social change, emancipation and social justice will harm and disarm social work practitioners in this region in ways very similar to those of the 'West'.
### Concluding remarks
The evolution of the definition debates throughout the history of the profession accurately reflects social work's ideological tensions and varying political agendas. The Global Definition adopted by the IFSW/IASSW in 2014 (see the following box), with its new emphasis on 'social cohesion', 'indigenous knowledges' and a let-out clause at the end that allows the definition to be 'expanded' as appropriate, is clearly intended as a compromise, generating an eclectic definition that can be interpreted in various 'convenient' ways. The current debate is reflective of political contradictions embedded in social work throughout its history. This chapter has focused on the contrasting political traditions and histories of social work and rejects the argument that social work can be a politically neutral activity. In fact, the history of social work suggests that in times of fierce inequality and sharp class divisions, 'neutrality' is, in fact, a conscious political stance: that of siding with the oppressors. Consequently, in the current climate of neoliberal domination and the erosion of civil rights, the adoption of a social justice-based definition could have provided front-line practitioners with the necessary tools and protection in order to defend the values and ethical commitments of the profession.
Different traditions, contrasting definitions
The following two definitions represent very different political traditions, scope and aspirations. The first one is the current global definition adopted in 2014. As we suggest in this chapter, the theoretical eclecticism underpinning this definition intentionally obscures the political dimension of social work. In contrast, the definition proposed by the Latin American Association for Teaching and Research in Social Work emphasises the importance of the 'social question' and the 'socio-historical' context of social work.
Global definition of social work (IFSW, 2014)
Social work is a practice-based profession and an academic discipline that promotes social change and development, social cohesion, and the empowerment and liberation of people. Principles of social justice, human rights, collective responsibility and respect for diversities are central to social work. Underpinned by theories of social work, social sciences, humanities and indigenous knowledge, social work engages people and structures to address life challenges and enhance wellbeing.
The above definition may be amplified at national and/or regional levels.
Definition proposed by the Latin American Association for Teaching and Research in Social Work, with the Brazilian Association of Teaching and Research in Social Work (2012–)
Social Work as a profession is positioned within the context of citizens and their interaction with the State; [these relationships] are shaped by the different socio-historical settings of the professional activity. It develops a social praxis and a set of socio-educational actions that focus on life's material and social reproduction from the perspective of social transformation. It is committed to democracy and the fight against social inequalities, by strengthening autonomy, participation and the safeguarding of citizenship for the achievement of human rights and social justice.
# Conclusions: 'Making history'
In consideration of all these qualities, I would speak about one last quality of an intellectual and political nature that progressive social workers ought to cultivate, to develop, to perfect, in their practice and that is an understanding of what is historically possible. As I perceive history, it is not something that happens necessarily but something that will be made, can be made, that one can make or refrain from making. I recognize, therefore, the role of the subjective in the process of making or of being made by history. And this then gives me a critical optimism that has nothing to do with, on the one hand, a critical pessimism and an immobilizing fatalism; and on the other hand, nothing to do with history marching on without men, without women, that considers history outside. No, history is not this; history is made by us and we make it, we are made and remade by it. (Freire, 1988)
### Introduction
In this book, we have explored the ways in which the roles and activities of social workers have been shaped by political and economic forces. Too often, such shaping has been a top-down process driven by political actors with their own interests and agendas and in opposition to the core values of the profession. What is also the case, however, is that there have been periods when social workers, often in alliance with wider social movements, have sought to create new forms of theory and practice based on opposition to oppression and to the economic and social forces that constrain people's lives. The radical social work movement that emerged in several different parts of the world in the early 1970s was one example. The reconceptualisation movement in Latin America, discussed in Chapter Five of this book, was another, in which the ideas of Paulo Freire, quoted earlier, played an important role.
Freire's injunction to 'make history' is particularly important for social workers at the present time. We began this book by noting some of the similarities between the world of the 1930s and the world today. These similarities, and particularly the rise of extreme right-wing political forces across much of Europe, should fuel a sense of urgency on the part of all those opposed to racism and xenophobia. However, our purpose in noting these similarities is not to create what Freire calls an 'immobilizing fatalism'; rather, it is to argue that this time round, the outcome can be different, depending on how we respond.
In 2004, two of us were involved in writing Social work and social justice: A manifesto for a new engaged practice (Jones et al, 2004). Our aim in writing the manifesto was to challenge the increasing influence of neoliberalism on social work practice and to begin to identify the 'resources of hope' out of which a new and more politically engaged social work could be created. More than a decade later and against the background of the political developments referred to earlier, the position of social work is, in some respects, more difficult and more challenging than it was back in 2004. However, what is also true is that the resources of hope are still there if only we look for them. It is still the case that out of the opposition to the dogmas and the priorities of neoliberal capitalism, a new social work radicalism is emerging. Practitioners across the globe are forging new forms of theory and practice that, though still fragile, contain the seeds of a different kind of social work. Here, we shall discuss some examples of these new developments.
### Boston Health Liberation Group
The US has a long, if often neglected, history of social work radicalism, which suffered during the Reagan–Thatcher years in the same way as did radical social work elsewhere (Reisch and Andrews, 2002). In response to the inequality and scapegoating of the poor that have characterised the neoliberal era, however, and shaped by a range of social movements, from the post-Seattle anti-capitalist movement to Occupy Wall Street, the past few years have seen a revival of radical social work theory and practice in the US. One of the best examples of this is the Boston Liberation Health Group. In addition to campaigning activity, the Group has developed a model of practice and documented its use across a range of social work settings (Martinez and Fleck-Henderson, 2014). Its guiding principles are summarised by Dawn Belken Martinez as being:
• Holistic: situating individuals in their full matrix of personal structural, ideological and institutional determinants.
• Critical: refusing to accept neoliberalism and refusing to accept the notion that social work ought to subordinate itself to its social agenda.
• Empowering: seeking to liberate clients and social workers from the confusing belief that current conditions are inevitable and beyond our power to change, and seeking to support their becoming active allies of individuals and movements working for social change.
• Hopeful: rescuing memory of and valuing 'the collective human capacity to create change' (Reisch, 2013: 68; see also Martinez and Fleck-Henderson, 2014).
### The Social Work Action Network
Founded at a 300-strong conference at Liverpool University in 2006, the Social Work Action Network's (SWAN's) activities have taken four main forms. First, as one recent mainstream historian of social work has noted, 'SWAN has held some very successful annual conferences, attracting numbers far beyond the reach of the British Association of Social Workers or the College of Social Work' (Bamford, 2015). These conferences, held each year in different universities across the UK, have provided the main forum for discussing and debating national policy responses to issues affecting social work, such as austerity, privatisation and racism. They are typically attended by between 300 and 400 people, with a key feature being the role played by service users, both as platform speakers and as delegates. The conference also hosts the SWAN annual general meeting, which elects a national steering committee to coordinate activities between conferences.
Second, SWAN has been involved in a number of campaigns at both national and local levels. Nationally, these have included challenging the scapegoating in 2008 of social workers involved in the case of Baby P (Peter Connelly), a small child who died at the hands of his carers (Ferguson and Lavalette, 2009; Jones, 2014). More recently, SWAN has played an important role in critiquing the proposed 'reforms' of social work education by the current Conservative government, reforms that would undermine both the academic and ethical bases of the profession and would lead to the creation of a social work elite, an 'officer corps' more well-disposed to privatisation (SWAN, 2014). Locally, SWAN groups have engaged in a range of campaigns, including the defence of asylum seekers, opposition to the privatisation of children's services and challenging cuts to mental health services.
Third, 2013 saw the launch of Critical and Radical Social Work: An International Journal, published by an academic publisher. Although not formally linked to SWAN, the fact that the two co-editors were also founder members of SWAN and that many members of the Editorial Board are leading SWAN activists means that, in practice, the links are close. The journal has now established a wide readership and is providing a forum for the development of new thinking in critical and radical social work, with contributions from every continent, including Latin America.
Finally, one of the most exciting developments since 2006 has been the creation of SWAN groups in several countries other than the UK. There are now active SWAN groups in Ireland, Northern Ireland, Greece, Japan, Canada and, most recently, Denmark. Members of the Greek group have played a very active role in Patras and on various Greek islands supporting refugees during the current crisis, while in 2015, SWAN-Ireland members successfully challenged government attempts to reduce pay and conditions for newly qualified workers. SWAN activists have contributed to crucial global debates and intervened in international conferences and events. Most notably, during the Global Conference in 2014, SWAN members successfully moved a historical motion emphasising the Right of Return for Palestinian refugees. During the 2016 Global Social Work Conference in Korea, radical social workers were instrumental in supporting direct action from the Korean Solidarity Against Disability Discrimination group, who were violently removed from the stage in the opening ceremony. The intervention of SWAN activists helped ensure that the service user campaigning group was given space to address the conference and shed light on the impact of neoliberalism on disabled people in Korea (SWAN, 2016).
### Sweden
As discussed in an earlier chapter, neither the Swedish welfare system nor Swedish social work has been immune from the impact of neoliberalism. A recent study of 'rationalization, responsibility and resistance' among social workers in that country begins by quoting one experienced worker as saying:
Those of us who've been around for a while and know how we used to work, we keep getting into conflicts with social workers of this new dawn.... [T]oday, if I allow myself to categorize, you may divide social workers into occupational professionals and organizational professionals. And it's the organizational professionals who are rewarded. Because they follow rules, guidelines and never go outside the box... you do your documentation and shut up. (Lauri, 2016: 1)
What is less known outside Sweden, however, is the level of resistance that there appears to have been to these developments. According to Lauri, during 2011 and 2012, several different networks of critical social workers were formed, including Critical Organized Social Workers (KAOSA), Social Workers for Social Action (SFSA) and Breaking the Silence (NBVT). He uses the latter as an example of these new networks:
Founded in Stockholm, this group voiced protest over the disintegration of the social security system, and the deteriorating situation for clients that they were witnessing. They also pointed to a problematic organization of social work, lack of resources and untenable working conditions. Moreover, in the summer of 2012, hundreds of social workers gathered in the city of Gothenburg to hand over a manifesto to local politicians, Calling for Social Welfare, protesting against the heavy workloads and recurring budget cuts that they were experiencing, claiming that under current circumstances they could not deliver qualitative help, nor ensure their clients' legal rights. (Lauri, 2016: 1)
According to Lauri, there were media reports of similar protests in several other Swedish cities.
### The New Approach Group, Hungary
On Friday, 4 November 2011, Hungarian social worker Norbert Ferencz was sentenced to three years' probation for charges of 'incitement against the public peace and a call for general dissent'. His 'crime' was to participate in a demonstration of social workers aimed specifically against a Budapest municipal ordinance that classified taking food from rubbish bins ('dumpster diving') as a misdemeanour.
Ferencz's defence was that he was simply following the Hungarian social work Code of Ethics, which stipulates that social workers have a responsibility, right and a duty to call to the attention of decision-makers and the general public the emergence of poverty and any obstruction of the alleviation thereof. The judge did consider this as a mitigating circumstance in the case.
What emerged during the campaign was that Ferencz was also a member of a group called New Approach, whose aims and objectives are remarkably similar to those of SWAN. According to the group's manifesto:
The New Approach to community work and radical social work is based on the idea of combining workshops and action groups, and also the renewal of social work codes of ethics.
This dual function is located in a long-term goal:
• Workshops: we want to provide space for discussing issues concerning the social sphere, development of action strategies.
• As an action group we are committed to the profession and the public's attention is drawn to the situation of those excluded. We seek to be a professional community that is not afraid to stand up for those in need. (New Approach, 2011)
A European-wide campaign led by both SWAN and the European Federation of Social Workers in defence of Ferencz succeeded in averting a custodial sentence and also led to links being established between New Approach and radical social work organisations elsewhere in Europe.
### The Progressive Welfare Network, Hong Kong
Another group of front-line workers that has been particularly active in social movements in recent years and has played an important role in developing more radical forms of practice is located in Hong Kong and linked to the Progressive Social Work Network. These workers played an important role both in the Occupy Hong Kong movement in 2011 and an even more central role in the Umbrella democracy movement of 2014/15. In 2010, they organised a well-attended Progressive Social Welfare Conference, which was addressed by SWAN activists, and in 2014, they organised the first-ever South East Asian Progressive Social Work Conference, with delegates from Hong Kong, Taiwan, Mainland China and Japan. Once again, the flavour of their approach is captured in their manifesto for progressive social welfare, published in 2014:
For a decade, social welfare in Hong Kong has been severely challenged. The government's neoliberal approach to welfare has led it to adopt a residualist welfare system, thereby undermining social welfare as a powerful tool for securing human rights and justice. Without long-term planning, welfare spending has been steadily decreasing. The so-called flexible planning and funding mechanism has rendered social welfare to a sporadic, ad hoc services [sic]. As a result, there is a widening rich–poor gap, intensifying social stratification, and worsening of [sic] quality of life for the grassroots. As the important role of Hong Kong's social welfare system is undermined, autonomy of social services, the core values of promoting social justice in social work have also been challenged to an unprecedented degree [sic]. (SWAN, 2011a)
### The Orange Tide, Spain
Better known, in part, because of its active support by IFSW General Secretary Rory Truell, and the fact that its leaders have won international awards, is the Orange Tide, a movement in Spain combining social workers and the people who use services. Across Spain, they regularly gather in orange T-shirts on the streets with music and dance to the message of 'No Cuts to Social Services'. Spanish newspapers and television have widely reported on this social action and the Orange Tide has become a part of mainstream media. While the government clearly has a programme of cutting social services at a time when they are needed, the cuts are arguably not as drastic as they would otherwise have been because of the visibility of the Orange Tide (Truell, 2014).
### South Africa
Indignation over the impact of austerity and deteriorating working conditions for social workers culminated in one of the most inspiring social work mobilisations in Africa in 2016. In September that year, over 20,000 South African social work practitioners, educators and students took to the streets of Pretoria, demanding better working conditions and more resources available for social services. In a direct reference to historical struggles of the South African people and the development of the more recent student movement, social workers were dressed in white T-shirts and exclusively sung political songs such as 'Senzeni Na' and 'Thupa'. The demonstrators reached the Union Buildings in Pretoria and handed in a list of demands to the government. This demonstration has been among the largest social work mobilisations in the history of the profession worldwide (Nathane and Smith, forthcoming).
### Peace movement: Colombia and Cyprus
Radical social workers have been at the forefront of the struggle for social justice and peace in countries experiencing war or post-conflict transitions. The most notable examples in recent years include practitioners from Colombia and Cyprus. Colombian social workers, influenced by the principles of liberation theology and the reconceptualisation movement, have been working directly with indigenous and peasant communities affected by state and paramilitary violence. These practitioners and academics have not only engaged with grass-roots community work supporting the victims of violence, but, most importantly, joined the political movements that emphasise the need for agrarian reform, collective reparations, autonomy for indigenous communities, social justice and amnesty for guerrilla fighters as a prerequisite for peace (IASSW, 2016a).
In recent years, progressive social work academics and practitioners from both sides of the Cypriot separation line have worked towards breaking free from the sectarian and nationalist politics of the profession and demanding the reunification of the island. Greek Cypriot and Turkish Cypriot social workers have organised meetings in the United Nations (UN)-controlled buffer zones, organised discussions with victims of the conflict and families of missing persons, and, through events and mobilisations, highlighted the importance of raising the 'social question' as an integral part of the peace process. These practitioners have expressed their concern about the lack of attention that has been given to the importance of social justice and universal welfare as means of achieving viable peace, and have declared that:
in the current conjuncture, we need to address the long-standing issue passed on from generation to generation: the issues of past violence, such as disappearances and other forms of violence, requires immediate attention at a moment where Social Services are facing the combination of major social questions in society (poverty, new psychological and mental problems and social ills), and austerity-based recipes that curb various social programs and downsize social services departments in time of increasing need. We must simultaneously address the shape of a future social policies and welfare state of a reunited Cyprus in a manner properly connects this to the broader issues of the ongoing and future struggles for equality. (SSCC, 2016)
This statement was followed by concrete actions, including bi-communal meetings, participation in pro-peace mobilisations and the inclusion of trade unions in their discussions about the role of social work and the welfare state in post-solution Cyprus.
To this list, we could add the Green Social Work network in Australia, which is developing social work theory, practice and campaigns around environmental and climate change issues, and the Rebel Social Work group in New Zealand/Aotearoa, which is actively campaigning against cuts to services. These are just a few examples of the types of radical social work initiatives that have developed over the past decade which emphasise that radical social work remains a 'living' and 'viable' set of ideas and practices that focus on creating 'another social work'.
### The new social work radicalism – key elements
So, what, then, are some of the common features of this new radicalism?
#### Against the market in social work – people before profit
As John Harris (2014: 7) has noted:
In many countries, neoliberalism continues to strive for unquestioned acceptance of the superior wisdom of the private sector, its first-rate way of doing things and the transferability of knowledge to other contexts, despite the financial and economic crash of 2008 and subsequent austerity politics.
In reality, as we have argued throughout this book, the three processes that Harris identifies as underpinning neoliberal social work – marketisation, managerialisation and consumerisation – have been responsible for much of what has gone wrong over the past two decades, with social workers from Scotland to Japan complaining that 'what we're doing is not social work' (ESRC, 2011).
Opposition to the market in social work is therefore rooted in two factors: first, a value-based belief that social work policy and practice should primarily be concerned with addressing human need rather than being driven by market considerations of competition, efficiency and profit; and, second, an awareness that neoliberal forms of social work, including their domination by computer-based assessment and recording processes (White et al, 2009), have seriously undermined much of what would be universally recognised as good practice.
As we argued in Chapter Two, however, a rejection of market forces in social work does not entail an uncritical acceptance of state-provided services. Rather, we need to fight for social services that are publicly funded but are also accountable, democratic and based on the views and wishes of those who use them.
#### Reclaiming relationships
One casualty of neoliberalism has been relationship-based social work. While it is undoubtedly the case that an overemphasis on the worker–client relationship in the past sometimes became a way of not addressing the structural factors shaping the lives of social work clients (for examples, see Mayer and Timms, 1970; see also, therefore, Case Con, the title of the magazine of radical social workers in the UK in the 1970s), one strength of the personal social services at their best is that they can provide a space where people can explore past or present harms or losses in the context of a supportive relationship – or as the Hungarian social worker George Konrad put it at the end of his great novel The Case Worker: 'Let all those come who want to: one of us will talk, the other will listen; at least we shall be together' (Konrad, 1970: 172).
That involves time, space to think and reflect, and also good supervision – all casualties of excessive caseloads, cuts to services, the neoliberal drive for efficiency and productivity, and the undermining of workplace-based union organisation.
#### Reclaiming community development
Another casualty of neoliberal social work in the UK and elsewhere has been community development, once a key part of social work responses to poverty and deprivation in Britain and the US. Within social work education in other countries, however, such as South Africa and Brazil, community work continues to form part of the curriculum. Community development approaches, particularly those that emphasise community self-help and 'pulling yourself up by your own bootstraps', can, of course, be at least as conservative as individualised approaches. However, more radical forms of community work, as well as community work placements for students that involve direct involvement with social movements, offer real possibilities for addressing structural inequalities and for highlighting the link between private troubles and public issues (Smith and Ferguson, 2016).
#### Developing radical theory for radical practice
The claim that social work is a purely practical activity that 'anyone can do' and that requires little theoretical input is one that is heard all too frequently, usually from the mouths of mainstream politicians who are irritated by social workers' insistence on talking about such uncomfortable issues as poverty and oppression. Social work is, of course, a practical activity; having a sophisticated understanding of the roots of capitalist exploitation is of little value if one is unable to relate effectively to the small child experiencing abuse at the hands of her parents or the young man terrified of the voices that he is hearing in his head. However, what these examples point to is precisely the complexity of the social work task: the need to fuse an understanding of 'what is going on' in a particular situation, based on good relationship and assessment skills, with skills of intervention and use of self. Similarly, if we are to avoid joining in the general scapegoating of disabled people, people on benefits or the Muslim community, we need a theoretical understanding of racism, disablism and ideology more generally That is why we need to insist on the importance of maintaining the social science base of social work education. It is also why publications such as the journal Critical and Radical Social Work are crucial as a means of deepening the critical-theoretical base of the profession. That said, radical and critical social work theory cannot and should not be developed in isolation from the experience, knowledge and practice wisdom of workers and service users; rather, it should seek to inform and generalise from that experience. 'Pure theory' of the type advocated by some adherents of the post-Marxist approaches discussed in Chapter Nine has little to contribute to a critical understanding of the role that social workers can play in challenging oppression.
#### Building coalitions of workers, academics, service users, movement activists and trade unions
Against those who would deny that social workers have any role to play in the struggle against oppression and for a more equal society, in this book, we have emphasised their capacity and potentiality to do so. Clearly, however, given the strength of the forces ranged against all of us who seek to build a better world (or simply to defend existing welfare states), no one should be under any illusions regarding the nature of the contribution that a rather weak and disorganised profession can make – all the more important, then, that social workers form alliances with trade unions, other campaigning organisations and organisations of disabled people and service users to defend existing services and to promote alternatives. As an example, in the UK, SWAN has linked in with the British Association of Social Workers, Disabled People Against Cuts and the service user organisation Shaping Our Lives to form Social Workers and Service Users Against Cuts in order to campaign more effectively against the effects of austerity policies.
### A global movement
A limitation of the radical social work movement of the 1970s is that it was mainly confined to English-speaking countries, notably, Britain, Canada and Australia. The new movement is much more global, for three reasons. First, the rise of the Internet and various forms of social media have made it much easier to gain an awareness of what is happening in other countries, to share information and to maintain contact across continents. Second, international social work organisations are playing a much more prominent role than was previously the case, with conferences such as the biennial International Association of Schools of Social Work (IASSW) conference offering opportunities for international networking. As an example, a meeting to launch the new journal Critical and Radical Social Work at the IASSW conference in Stockholm in 2012 was attended by around 70 delegates from many different countries, including a large delegation from Latin America. Third, and most importantly, the global dominance of neoliberal ideology and economic policy for three decades and the subordination of welfare services to market requirements has meant that the issues facing social workers across the globe are often very similar. The similarities between the experience of social workers in the UK and those in Japan as a result of the imposition of care management approaches was one of the main findings of an Economic and Social Research Council-funded seminar series in 2011.
Strengthening these global networks and learning from the experience of practitioners, academics, students, service users and campaigners in other countries (including the examples of 'popular social work' that we discussed in Chapter Six) is a priority. However, this is not just about sharing information; it is also about showing solidarity. We all benefit from showing solidarity. Social workers in Britain and Greece who have been involved in supporting and welcoming asylum seekers have found that their colleagues and the wider public are often grateful for the opportunity to sign petitions, give clothes or contribute money to help asylum seekers and refugees. One poll in the UK, for example, found that in the period following the widely publicised death of a three-year-old Syrian-Kurdish refugee child, Aylan Kurdi, on a Greek beach in September 2015 as he and his family crossed the Mediterranean to seek sanctuary in Europe, more than a third of people in the UK had supported refugees in one or other of the forms mentioned earlier.
That is why we should re-inscribe solidarity as a core social work value. Reasserting our common humanity is not only the most effective way of challenging the racists and xenophobes who would divide us; it also challenges the narrow selfish individualism that is at the cold heart of neoliberalism and provides a basis for a different kind of social work. In a discussion of the ways in which the individualism that neoliberalism creates has created an epidemic of loneliness and mental ill-health in many countries, the British journalist George Monbiot has argued:
This does not require a policy response. It requires something much bigger: the reappraisal of an entire worldview. Of all the fantasies human beings entertain, the idea that we can go it alone is the most absurd and perhaps the most dangerous. We stand together or we fall apart. (Monbiot, 2016)
A social work that emphasises our social nature as human beings and our basic need for connectedness, that finds new ways to translate that into practical ways of working, and that challenges the political, social and economic forces that would isolate us and turn us against each other has a part to play in developing that new worldview. That means, in the words of our Hungarian colleagues quoted earlier, becoming 'a professional community that is not afraid to stand up for those in need'.
# References
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I Came By (titulada: Yo estuve aquí en Hispanoamérica y Pasaba por aquí en España) es una película británica de suspenso escrita, coproducida y dirigida por el cineasta británico-iraní Babak Anvari. Estrenada el 19 de agosto de 2022 en los cines del Reino Unido, y el 31 del mismo mes en la plataforma Netflix a nivel internacional, está protagonizada por George MacKay, Percelle Ascott, Kelly Macdonald y Hugh Bonneville.
Argumento
Toby Nealey (George MacKay) es un graffitero de 23 años que irrumpe en las casas de personas de clase alta y deja el mensaje "I Came By" en las paredes. Su último objetivo es el juez retirado Héctor Blake (Hugh Bonneville). A pesar de que el magistrado se muestra socialmente progresista en público, Toby cree que es un hipócrita. Cuando entra en su casa, descubre un estudio de cerámica y un hombre secuestrado en el sótano.
Atormentado por lo que halló, Toby le cuenta a su amigo y ex cómplice Jay (Percelle Ascott), quien está concentrado en su vida personal, al estar su novia embarazada. Al no saber como proceder, Toby hace un llamado anónimo a la policía, que tras revisar la casa de Blake, no encuentra nada.
Lizzie (Kelly Macdonald), la madre viuda de Toby, que está preocupaba por el hecho de que su hijo no tiene un rumbo en su vida, lo echa de la casa después de una fuerte discusión. Esa noche, Toby decide volver a irrumpir en la residencia para así liberar al prisionero, pero es descubierto y asesinado por Blake con un bate de cricket. Luego usa el horno en su estudio para incinerar el cuerpo, y tira las cenizas por el inodoro.
Jay, que es reacio a revelar sus actividades vandálicas con Toby, roba del buzón de Blake una carta y la implanta en el cuarto de su amigo para incitar a la policía a investigar nuevamente al juez. Encuentran la prisión del sótano, pero la hace pasar por una sala de pánico, y es arrestado por obstruir la investigación, pero usa sus conexiones para ser liberado rápidamente.
Lizzie, convencida de que Blake tiene que ver con la desaparición de su hijo, lo comienza a seguir. Una noche lo ve llevar a un hombre a su casa: Omid, un solicitante de asilo iraní gay, ofreciéndose a ayudarlo a obtener la residencia permanente en el Reino Unido. El juez le cuenta la historia de que su padre tomó a un chico indo-persa como amante, lo que llevó al suicidio a su madre; además de que un joven Hector agredió lo agredió y casi asesinó. Mientras contaba la historia, Blake drogó a Omid, quien a pesar de esto pudo escapar de la casa, para ser salvado por Lizzie. Sin embargo, no acudió a la policía debido a su precaria situación residencial. Al día siguiente, Blake rastrea, amenaza, encarcela y asesina a Omid.
Cada vez más desesperada, Lizzie le pide a Jay que la ayude a entrar en la casa de Blake. Jay se niega porque es responsable de su hijo recién nacido y porque considera que al ser negro y tener antecedentes, podría resultar más perjudicado si lo atrapan. Debido a esto, Lizzie entra sola, solo para ser atrapada, asesinada e incinerada.
Jay se culpa por la muerte de la madre de su mejor amigo, y su relación con su novia Naz (Varada Sethu) se desmorona debido a falta de comunicación entre ambos. Cierto tiempo después, Naz le informa a Jay que Hector Blake asistiría a la celebración del aniversario de una universidad.
Después de que el juez abandona el evento, Jay lo sigue a una finca en el campo. Tras atrapar, golpear y atar a Blake, Jay encuentra un prisionero: el mismo hombre que Toby descubrió en primer lugar, y lo libera. Llama a la policía y huye de la escena habiendo escrito "I Came By" en la pared.
Elenco
Hugh Bonneville como Hector Blake
Percelle Ascott como Jameel "Jay" Agassi
George MacKay como Toby Nealey
Kelly Macdonald como Lizzie Nealey
Varada Sethu como Naserine "Naz" Raheem
Antonio Aakeel como Faisal
Marilyn Nnadebe como Oficial Hunter
Yazdan Qafouri como Omid
Recepción
La película recibió críticas mixtas tras su estreno. En Rotten Tomatoes, la película cuenta con una calificación del 69% de la crítica, mientras que en Metacritic, tiene una puntuación de 57 sobre 100.
John Nugent de la revista Empire Magazine elogió la actuación de Bonneville, pero criticó el mensaje político de la película, describió la cinematografía como «plana» y al producto final como «desordenado». Noel Murray elogió de manera similar a Bonneville en el periódico The Los Angeles Times y señaló que el cambio de la narrativa a diferentes personajes mantuvo la película «impredecible», pero también comentó que perjudicó el ritmo y la tensión. Brian Tallerico del portal RogerEbert.com comentó que la película fue demasiado ambiciosa y que tenía «el problema opuesto de tantas miniseries en las plataformas de streaming, en el sentido de que tiene una temporada de televisión repleta de ideas en su tiempo de ejecución». Clarisse Loughrey del periódico The Independent criticó de una manera similar la falta de enfoque en los mensajes de la película, afirmando que era «un trabajo bien intencionado que luchaba por encontrar su voz».
Referencias
Enlaces externos
Películas de Regency Enterprises
Películas de Film4 Productions
Películas ambientadas en Londres
Películas ambientadas en Inglaterra
Películas de 2022
Películas en inglés
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installBiocDataPackage.sh "bsgenome.celegans.ucsc.ce10-1.4.0"
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"Music for All Ages: Space, Science and Science Fiction," presented by the Rockville Concert Band. Sunday, April 7 at 3 p.m.
Conducted by John Saint Amour, music director. No tickets required; $5 suggested donation.
Murder at the Mansion: "A Wedding to Die For," presented by A Taste For Murder Productions. 8 p.m. April 12 and 13; 2 p.m. April 14.
From the groom to the garter, Madison planned the perfect wedding. Will someone have to die to keep it from spiraling into chaos? Doors open 90 minutes before the performance.
Complimentary desserts and cash bar with beer, wine and soda.
Show-only tickets cost $35 per person or $60 per couple. A buffet by Mission BBQ is available 90 minutes before the show for $18 per person. Buffet and show tickets are $53 per person or $96 per couple.
Tina and Prince – A Royal Tribute. Saturday, April 20 at 7 p.m.
On the eve of the anniversary of Prince's death, pay tribute to the man, the myth and the legend, performing alongside Queen of Soul, Tina Turner.
For tickets or more information, visit www.rockvillemd.gov/theatre, call the box office at 240-314-8690, or email theatre@rockvillemd.gov. F. Scott Fitzgerald Theatre and Glenview Mansion are located at 603 Edmonston Drive.
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Begin Reading
Table of Contents
About the Author
Copyright Page
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This book is dedicated with love to my husband, Michael, and our daughters, Jamie and Misha. Without the three of you, I would never have been a class mom.
* * *
To: Parents
From: JDixon
Date: 9/4
Subject: Getting to know me—your new class mom
Hello, parents of Miss Ward's kindergarten class!
My name is Jennifer Dixon and I have "volunteered" to be your class mom for this coming year. Since this is a thankless job, don't expect warm fuzzy emails like you probably got in preschool. Wake up! You're in kindergarten on the mean streets of William H. Taft Elementary School, and it's time to face a few facts. The main one is that I'm in charge and I have some strong suggestions to make this an easy year for all of us, especially me.
First and foremost, read the school's @#$%& weekly email! It may seem boring, but it actually gives good information and keeps me from having to answer questions like "When is curriculum night?" (See below, by the way, for the answer to that one.)
Second, when I ask for something, volunteer! If I say we need doughnuts, say, "How many?" not "Can I bring cups?" I don't want to have to assign stuff, so please be among the first to email me back. I will be testing you on this very soon because curriculum night is... (see below).
And finally, if there is an event... show up! They may seem dull and tedious but let's never forget we are here to build a community. It can't just be the same five people showing up, because they'll get sick of each other.
Important Dates to Commit to Memory
My Birthday: April 18. No gifts necessary, but I do enjoy Starbucks.
Curriculum Night: Tuesday, September 27—6:30 to 8:30 p.m.
This will be your first chance to see the effects of alcohol on your fellow parents. I'll be asking for refreshment contributions sometime next week.
Parent Dialogue Coffee—October 7—Location: The Lounge—no idea what the "dialogue" is, but don't be surprised if it's about coffee.
Parent Social, K–6th Grades: January 18—6 to 8 p.m.—Location: Cafeteria. It's been my experience that these are a bit awkward, kind of like a 7th-grade dance. Go at your own risk.
Miss Ward has also requested that you drop off a photo of your child before the first day of school. Let me repeat that... before the first day of school. I'm not sure, but I think she plans to use them in some sort of Wiccan ritual to "cleanse" the classroom.
That's it for now.
Any questions?
Jennifer
P.S.: By the way, if any of you consider yourselves "crafty," let me know. This year all gifts for the teacher have to be homemade, so for this and only this, I am open to suggestions.
* * *
I click Send on my laptop, sit back in my chair, and grimace.
"Well, that should give them something to think about," I say to absolutely no one.
Rubbing my tired eyes, I wonder for the fiftieth time that day why I ever agreed to be a class mom again.
My first instinct had been the right one.
"Absofuckinglutely NOT," I told Nina Grandish when she asked me. Nina is the reigning high priestess of the school's PTA. In spite of that, she is my best friend. "It's the worst job I've had since I worked customer service at Allstate."
"Please!" she begged. "I really need you."
"Nope. I don't have time."
"Yes, you do. Vivs and Laura have already gone back to school."
"I'm starting my mud-run training."
"That's unlikely," Nina scoffed.
"I'm thinking of getting a dog for Max and I'll be busy with that."
"No, you're not. You hate dogs. Come on! Think of all the experience you bring to the job."
"Oh, wow," I said. "Thanks for reminding me how much older I am than all the other parents."
"Not older," Nina cooed, "wiser."
And I am, by almost fifteen years. The nineties were a bit of a lost decade for me. After a blistering four years at the University of Kansas (Go, Jayhawks!) I found myself with a super-useful degree in art history and not a chance in hell of finding a job with it. So, I decided to hit the road and see a bit of the world. Some people go to Paris to look at great art; some go to Rome to look at great architecture. Me? I went to Amsterdam to see a great band. INXS was just starting to ride their wave of international success, thanks to the album X. Luckily for me, they weren't so famous that they would only date supermodels. I got picked out of the audience thanks in part to the "no bra" phase I was going through, and lo and behold I ended up a groupie.
You know that Cameron Crowe movie Almost Famous, where the girls are called Band-Aids and they travel with the band and keep the musicians', um, morale up? It was kind of like that but not nearly as glamorous. I was with INXS for a little over a year, then moved on to a folksinger named Greg Brown. Yeah, I had never heard of him either, but he could certainly draw a crowd, albeit an unwashed one. In those three years away, I somehow ended up with two kids, one of whom may or may not have been fathered by Michael Hutchence. Thanks to his untimely death in 1997, poor Vivs may never know. But Laura's sperm donor was most definitely Greg Brown's banjo player. I'm 65 percent sure.
To quote the poet Steve Perry of Journey, "They say that the road ain't no place to start a family." So I took my two kids from two different fathers and made my way home to Kansas City.
Actually, by that time my parents had fulfilled a lifelong dream and moved to Overland Park, which is a fancy suburb of KC. I was sad that I didn't get a chance to say good-bye to our old house, but thrilled that I had such a nice place to bring Vivs and Laura home to.
Let's just say I had a bucket load of explaining to do to Kay and Ray Howard, my extremely Catholic parents, when I landed on their swank new doorstep with Laura and Vivs both still in diapers. My mother's face went from confused to horrified to delighted so quickly that I thought she was having a stroke.
Luckily, they are more the forgiving kind of religious people and less the judgmental kind. So after a few dozen Hail Marys and one excruciating afternoon at Our Lady of Unity doing the Stations of the Cross, I moved in with them and started what I now call the normal years. With their help, I raised the girls, worked for a while at Allstate, and, yes, was class mom seven endless years in a row. It's a record that I believe still stands at William Taft. I hope it's not what ends up as the most noteworthy thing in my obituary, but you never know.
It was while working at Allstate that I met the man who would become Baby Daddy #3 and Husband #1, Ron Dixon. By the way, I still have had only one husband. I just think it's funny to introduce him that way. Ron called to file a complaint with the people with whom he thought he was in good hands. As was my job, I took the call and tried to talk him out of canceling his policy. Ron has an amazing voice. Even when he's complaining, it sounds like he just swallowed liquid velvet. I could have listened to him all day. It was around the time he called me a soulless bitch that I decided I wanted to meet him. To this day, he thinks I took all my disgruntled phone callers to lunch.
What can I say? I had him at hello. I'm not unattractive, considering my age and the mileage I've put on my body, and Ron happened to be single, having just gone through a soul-sucking divorce. In fact, when he called the insurance company he was trying to put in a force majeure claim for a fallen tree that had clearly been hit by a car. I later found out that the tree was the victim of domestic abuse, having been plowed over by his ex.
As a member of the sisterhood, I take exception to men always calling women crazy, but in this case I can say unequivocally that Ron's ex-wife, Cindy, is nuts. Not fear-for-your-life nuts, just garden-variety nuts. The biggest problem is that you never know in what form the nuttiness is going to rear its ugly head. Like one day, a few months after Ron and I moved in together, six Costco-sized crates of diapers appeared on our doorstep with a card from Cindy saying, "Get the message?" I figured she was either calling us babies or suggesting that we have a baby. Ron said she was telling us we are full of shit.
Ron is a good fit for me. He's what my father would call a solid guy, both physically and emotionally. He's about five eleven (although he tells people he's six feet, for reasons that are not quite clear to me) and fit without looking bulked up, and he has short dark hair that is thinning at the temples. He's not what I had typically found attractive in the past—I mean, he doesn't even have a tattoo—but he has immense charisma and just about the kindest face on the planet. Combine that with the voice, and I was a goner the moment I saw him. Our courtship was short and sweet, because when it's right, it's right, and why screw around? And thanks to crazy Cindy's fear of vomit, they never had any children. So when he dropped the B bomb on me on our first anniversary I shouldn't have been surprised.
We were having dinner at Garozzo's, and over penne Victoria he casually mentioned that he would really like to have a baby. I stifled my first thought (Well, good luck with those labor pains!) and told him of course we would try. I pretty much counted on my aging womb to keep anything from happening, but wouldn't you know it? I had one good egg left. And thank goodness for that, because Max is the dessert of my parenting life.
So now at the ripe old age of forty-six, I have two girls in college and one boy starting kindergarten. And I'm the oldest mom in the grade. Oh sorry, the wisest.
* * *
"Max! Get down here. Your toast is getting cold."
I sit back down at my kitchen-counter office and slam out an email to my class parents that I hope they read before drop-off this morning.
* * *
To: Parents
From: JDixon
Date: 9/6
Subject: Questions answered
Dear Parents,
Wow. When I said, "Any questions?" at the end of my previous email, it was what is known as a rhetorical question, as in one that doesn't need a response. Oh, well. Allow me to answer them in the order they were received.
1) No, I'm not kidding.
2) Yes, I'm serious.
3) No, beer making is not a craft.
4) The date of curriculum night is on the original email—just look.
5) No, you can't be fired from a job you volunteered for.
Thank you for the feedback.
Jennifer
* * *
Max rounds the corner to the kitchen wearing an outfit that bears no resemblance to the one I picked out for him.
"Wow. Love the red pants. Aren't they part of a costume?"
"Yup. Pac-Man."
"And the purple top?"
"Nana gave it to me, remember?"
"I do. Are you sure you want to wear them together on the first day of school?"
"Ya. I want to stand out."
"Well, mission accomplished." I quietly thank the heavens that he didn't wear the matching Pac-Man hat.
Max smiles and takes a bite of his toast. Ever since he was old enough to pick out his own clothes, he has exhibited a, shall we say, unique taste in fashion. You never know what ensemble he is going to come up with. Sometimes I think he uses a blindfold and a dart to put his outfits together. I'd left him a pair of khakis and a white polo shirt on his bed in the hopes he would embrace his new school uniform, but I guess he didn't.
Ron comes in from his run all sweaty. I love him this way.
"Hey!" I grab his butt. "I need you ready to go in ten minutes if you want in on the first-day-of-school fun."
"I'm ready now." He grins and dashes up the stairs.
"Mom, what's my teacher's name, again?"
"Miss Ward."
"Is she nice?"
"I haven't met her yet, but she seems nice from her email."
"I hope she likes purple."
"Who doesn't?" I smile. "You know you have to start wearing the uniform tomorrow, right?"
He nods, his mouth full of toast.
"I'm going to pick you up at noon and we can go out for lunch anywhere you like."
"Can Dad come?"
"I don't think so. He needs to be at the Fitting Room." I'm referring to the sporting goods store Ron owns.
Before I can even yell, "What the hell is taking you so long?," my husband is showered and ready. It really must be nice to be a man. I'm not a very high-maintenance woman, but I do need more than six minutes to shower and look presentable.
"Who's ready for kinderga—"
Ron stops mid-word as he takes in what Max is wearing.
"Is that what you're wearing, Max?" he asks.
I give him a look across the kitchen table that says, "Don't be an asshole about this."
"Ya. I want to stand out."
"I thought there was a dress code." Ron looks at me.
"Not on the first day." I shoot him another warning look. They usually work. I must be misfiring. "And it's only a half day today, so let's get going!"
THE FIRST DAY OF SCHOOL. Yup, all caps, bold and italicized. That's how epic it is in my mind. Everyone is so clean and excited! Backpacks are fresh, sneakers are squeaky, and pencils are sharp. Take this same snapshot mid-November and it's a whole different story.
We make our way down the well-worn hallways of Vivs's and Laura's old stomping ground, William H. Taft Elementary School. When we get to room 147, we find the prettiest and preppiest person I have ever seen standing at the classroom door greeting people. She has long blond hair, which is kept back by a pink headband. She is wearing light-pink-checkered pants and a white blouse with ruffles. I hope she owns a smock.
As we approach, she hits us with a dazzling smile and holds out her hands.
"Is this Max? Oh, my goodness, Max, I have been so excited to meet you! Is that a new shirt? Purple is my favorite color!"
Well, color me impressed. Miss Ward is a real charmer. She has obviously studied the pictures we all sent in at the end of the summer. Max hasn't said a word, but wears the goofy smile of a man smitten. So does Ron when I look over at him.
"Hi, Miss Ward, so nice to meet you. I'm—"
"No, no!" Miss Ward interrupts me. "This is not about Max's parents. It's all about Max today. Come on in and find your name on your desk, sweetie." She ushers Max into the room and he eagerly follows without a backward glance.
Ron and I look at each other. I shrug.
"It's all about Max."
As we head out of the school, Ron asks what I'm up to.
"I'm going to meet my new trainer."
He looks at me skeptically.
"I know what you're thinking, but after that debacle at your store I feel like I need to step up my workouts."
"Or, just, you know, start them." He smiles and gives my shoulder a squeeze.
Here's the thing. Ron's sporting goods store is one of the biggest in KC. A few months ago, they hosted a mini mud run to promote our governor's "Get Fit" initiative. When he mentioned that he needed participants, I volunteered. That was my first mistake. I thought I was in shape, thanks to my twice-weekly visits to our neighborhood Curves, which I had joined shortly after Max was born. So when I got to Ron's store that day and saw the course setup I was, like, "No problemo." That was my second mistake.
Let's just say that the upper-body strength you get from hauling a toddler around for a few years doesn't exactly prep you to climb a rope or swing from monkey bars or even drop to your belly and crawl through mud, although that was the easiest part.
It was weeks before I could show my face down at the store again. I mean, it's not great when the wife of the owner breaks down and cries because she can't get over the wall. Plus I was sore for days in areas I didn't know existed.
"Who'd you get to train you?" Ron asks when I don't acknowledge his dig. I can tell he is annoyed that I hadn't consulted him on the decision.
"Someone my mother recommended. He comes to your home and works you out. I figured I'd finally start using Ron's Gym and Tan." That's my nickname for the home gym Ron has set up in our basement.
Ron gives a fake gasp. "You mean you're going to give up Curves?" He's never been a fan. Ron's kind of a gym snob.
"See you later." I give him a sly smile and head to my minivan. "Hot new trainer's awaiting."
Ron frowns. "Hot? You didn't say he was hot."
I laugh as I open my car door. I actually have no idea what he looks like. But with a name like Garth, I have high hopes.
* * *
My mom has actually told me very little about Garth, just that he used to be a trainer at the local Lucille Roberts gym. He had to stop for a while and is now getting back into it. He's very cheap for an in-home personal trainer—$30 an hour. I just hope this isn't a case of "You get what you pay for."
I pull up to the house and see a white Prius parked in my driveway. My new trainer is ten minutes early. Me likey. As I get out of my car, he does the same, and I get my first look at the man I will be spending two hours a week with.
I wish I could tell you that everything turned to slow motion and "Dream Weaver" started playing in my head as he whipped his hair around and flashed me a dazzling smile, but that would be lying.
Garth is about 5'6" and mostly bald, and he looks like he's in his midfifties. He reminds me a bit of the actor Michael Chiklis from The Shield.
As I rearrange my expectations in my head, he walks over and guess what? He does have a dazzling smile! It makes me like him immediately.
"Hi, Jennifer, I'm Garth." He shakes my hand and nearly crushes it.
"Ow. Hi, Garth. That's a good grip you've got there."
"Oh, good gravy, I'm sorry," he says and lightens up his vise grip immediately. "I always forget to take it down a notch for the girls."
"No problem. Clearly I need to toughen up a bit."
"Well, that's what I'm here for." He smiles and follows me to the front door.
"Can I get you something to drink?" I ask while throwing my purse on the hall table.
"Nope. Thanks. I always bring my own." He proudly holds up a gallon jug full of water. Clearly Garth is old school, and fancy water bottles are not his style.
"Umm, why don't I show you our workout area and then I'll run up and change."
"Sounds good." Garth smiles again. "After you, my good woman."
As I lead him down to the basement, I wonder just how old school he is. Not for nothing, but I've been at Curves for five years. That's some pretty advanced stuff.
Ron's Gym and Tan is located in a corner of our basement, right next to the laundry room. It consists of a treadmill, a bench press, free weights, a mat, and one of those big exercise balls.
"This is fantastic!" Garth declares, and it only takes me a second to realize he is not kidding.
"Really?" I ask. "Do we need any other equipment?"
"No. This is perfect. Why don't you get changed, and I'll lay out a workout plan." He actually sounds excited.
"Okay. I'll be right back."
As I run up the stairs to my bedroom, I wonder what I've gotten myself into.
* * *
I'll admit I have a somewhat acerbic way of presenting myself, but I had no idea how many parents I could offend with just one email. Actually, it wasn't that many, but it only takes one to stir the pot. Nina calls me just as I am getting out of the shower after my workout.
"Oh, my God, what did you say in your class email?" she screams.
"Just the usual stuff. Why?" I toss my wet towel in the hamper and head toward my closet.
"I just got off the phone with Asami Chang and she is pi-issed!"
"About what?" I ask, rifling through my T-shirts.
"She says it was an inappropriate way to address kindergarten parents."
"So?"
"So, was it?"
"Probably. But I can't believe anyone took it seriously."
Nina sighs. "That's what I thought. But you know your, um... humor is sometimes lost on people. Asami wants you to step down as class mom and let her take over."
"Well, I think she is absolutely right. I am not fit to liaise with parents." I make a mental note to send Asami a basket of fruit.
"Not so fast, funny girl. You promised me you would do this."
"Yes, well, the people have spoken. I'm not wanted on the voyage."
"I want you on the voyage. I think it will be good for you to meet some people, and I know Max loves it."
"Ooooh. Good one, bringing Max into it. What about what's-her-name?"
"Asami Chang. I'll deal with her. So we're good?"
"Define 'good.'"
"And you'll tone down your emails?"
"Not a chance."
Nina laughs. "There's my girl. How was your new trainer?"
"Interesting," I say. "Different from Curves, that's for sure."
"Different, good, or different, bad?"
"Well, I certainly haven't done a burpee in a long time."
Nina cracks up. "A burpee? What the hell is that?"
"I'd really have to show you. One thing's for sure, my ass is going to be sore tomorrow."
"Well, that's a good thing. Okay, I gotta run. Remember, play nice with the parents!"
I hang up and pull on my jeans. I admire Nina for the way she successfully navigates both sides of the fence. She is the perfectly perfect ideal of what the PTA president should be, but she can also slum it with hoi polloi. She is this cute little five-foot-tall dynamo with skin the color of cappuccino and a very short Afro that she keeps threatening to take "native." She's like the bunny that never runs out of batteries. I don't know where she gets her energy. Being president of the PTA is not a job for sissies. It's a full-time, relentless piece of crap that very few people would want to inflict upon themselves. But year after year Nina manages to squeeze it into what I know is a full schedule running her graphic design business.
She and I met about ten years ago at a bicycle shop. It was so random. I was looking at cycling gloves and she was getting a new tire rim. A man walked into the store and announced to no one in particular that he had a loose nut. I swear to God, at the exact same moment we both said, "Well, you should see a doctor about that." And that was it, soul mates for life.
Nina is a single mom, but you'd never know it. She is totally on top of things and never complains about being alone although I can tell she still carries a torch for Sid, the father of her daughter, Chyna. He left her two weeks before Chyna was born and basically fell off the face of the earth, but she still hopes he will come back. I'm not sure I understand why because he sounds like a total skeeze. But the heart wants what it wants, so she has kept the candle burning for lo these twelve years. I've tried to set her up with a few guys—mostly customers from my hubby's sporting goods store—but no one has caught her fancy. I guess it's hard to measure up to the stellar example that was Sid.
Chyna is just like her mom—petite, dynamic, and full of shit. I can't wait till she's old enough to babysit for me.
We bonded as single moms, but even after I hooked up with Ron we stayed close. In fact, Vivs and Laura used to tag-team babysit for Chyna.
* * *
To: Parents
From: JDixon
Date: 9/18
Subject: curriculum night party
Hello, fellow parents,
Now that the awkwardness of last week's attempted coup on my class mom fiefdom is behind us (no hard feelings, Asami; I understand your people's need for power), let's get on to some serious business, like who is bringing the wine.
September 27th (aka curriculum night) is fast upon us. It's my favorite night of the year, because it answers burning questions such as "Who has the hottest husband?" and "Who spent a little too much money at the ice cream truck this summer?" Plus, I want everyone to think that Miss Ward's class is the place where people PAR-TAY! To that end, we need some provisions.
2 kegs (I'll bring the funnel)
Jell-O shots (lime and cherry, please!)
"Special" brownies—Wolffe family, I'm counting on you for these.
If you're still reading and haven't yet speed-dialed Principal Jakowski, here are a few other things we MAY need.
Mini quiches (the microwavable kind)
Small cheese platter
Small veggie platter
Yummy cookies or brownies
Cups, small plates, cocktail napkins
Sparkling and flat water
Red and white wine
The phone lines are now open, so run, don't walk, to your keyboard and volunteer to bring something. Don't be shy!
Thanks in advance for what I'm sure will be an overwhelming heed to the call. Response times will be noted.
Jennifer
* * *
* * *
Just as I close my laptop, my two favorite men come in the back door.
"Mom! The tent is up!" Max yells even though I am sitting right there.
"Already? Wow. Are you guys sure you want to do this?" My question is really for Ron—he's the one with the fifty-year-old spine.
"Camping out is a time-honored tradition among Dixon men," my husband says.
Max nods solemnly. I know he is all in on this camping adventure, but it's hard to take him seriously when he's wearing a sombrero and poncho.
"And besides," Ron adds, "we have the Kodiak Canvas Flex-Bow Deluxe out there. We could go to base camp with that baby, right, buddy?"
I roll my eyes. I know it's what he does for a living, but I still can't believe how jacked up Ron gets about any type of sports gear. Max, on the other hand, is putting on his game face. He's not really an outdoor sporty kind of kid, but he's trying to be one for his dad's sake. I worry about that sometimes. They are planning to camp out in the backyard this Friday.
I shrug. "Okay. Just don't be surprised if it's a bit chilly out there. You guys should have done this in August."
"August, Shmaugust," Ron scoffs. "We're Dixon men. Besides, we'll be sleeping in the Nemo Nocturne 15." He looks to me for a reaction, but I really don't have one.
"Well, I'll leave the back door open that night, just in case." I wink at my boy. I'm not sure, but I think he looks relieved.
* * *
To: JDixon
From: Sasha Lewicki
Date: 9/18
Subject: curriculum night party
I am out of the office until September 20.
Thank you,
Sasha
To: JDixon
From: Shirleen Cobb
Date: 9/19
Subject: curriculum night party
Dear Jennifer,
You didn't mention anything about food allergies. My son, Graydon Cobb, is VERY allergic to peanuts, dairy, wheat, grass, wheatgrass, chocolate, and airborne dust. Please don't allow any of these things in the classroom.
Shirleen Cobb
To: Shirleen Cobb
From: JDixon
Date: 9/19
Subject: curriculum night
Dear Shirleen,
Since curriculum night is for parents only, I wasn't going to worry about food allergies, but from your note I can see that Graydon's situation is very serious and he could hive up at any moment. Just how big is the bubble he comes to school in?
Jennifer
* * *
Why, oh why, is it always the mother with the most allergic kid who is, herself, a nut? I mean, I get it, allergies are serious. Life-threatening, even. They're nothing to joke about. But when did this all happen? When did peanut butter become the grade-school equivalent of anthrax? When I was in second grade, I sat beside a kid named Alan Ervine who smelled like peanut butter all the time. I'm convinced he dabbed it behind his ears like cologne. No one in our classroom had a problem with it. The banishing of PB is a problem for us because PBJ sandwiches are the only ones Max will eat. In the name of peanut butter, someone needs to figure this thing out. I would, but you know how busy I am being class mom.
* * *
To: Parents
From: JDixon
Date: 9/21
Subject: Hello? Did anyone read my last email?
Dear Miss Ward's class,
Shocked? Appalled? No, "disappointed" best describes my feelings after the less than adequate response to my call for help. Only two people got back to me. Sasha Lewicki, sending an out-of-office autoreply, was the first, with an impressive turnaround time of 11 seconds. And Jackie Westman stopped me in the parking lot to say she'd bring cups. Listen, people, we are going to be in that classroom for TWO HOURS. Don't you think we're at least going to need water, to say nothing of alcohol? So get your fingers on the keyboard and start volunteering to bring stuff pronto.
Geez!
Jennifer
P.S. Response times will be noted.
* * *
I click Send. This is the part of being class mom I hate the most—begging people to do stuff for the classroom. Everyone always thinks someone else is going to volunteer, and the class mom gets stuck with all of it.
"Well, not this time, my little kindergarten parenteers," I say to my reflection in the computer screen. "This is the year I shame you all into participating. Mwa ha ha!"
"Are you talking to yourself again?"
I jumped at the unexpected sound of my husband's voice.
"What are you doing here? I thought the Dixon men were going to tough it out in the backyard."
"We were until a squirrel jumped on top of the tent. Max freaked out, so I brought him in."
I could see Ron's disappointment.
"He's in his bed?"
"Fast asleep. And look at us. Ten p.m., and nothing to do."
He saunters over to where I'm sitting on the bed with my computer.
"Who says I have nothing to do?"
Ron takes my computer and puts it on the dresser.
"I do," he says as he leans in for a kiss. "I want to see what Garth's thirty bucks an hour is getting me."
"Well, not much yet except for a bag of sore muscles." I dodge his kiss and roll to the other side of the bed. Ron follows me.
"What are you doing?" I ask.
"What does it look like?" He stalks me across the bed.
"It looks like you're not getting the hint," I snipe back.
He sits up with a mix of hurt and curiosity on his face.
"What's wrong?"
That's actually a good question. My handsome husband wants to have sex with me and I'm being kind of a bitch. But here's the thing. I was really looking forward to this night. I adore Ron, but sometimes it's nice to have a little break. I was so juiced up to be all alone in our Cali-king bed—no one beside me snoring or stealing the blankets. And now it's not happening and I'm pissed off. No, I'm disappointed, but it reads the same as pissed off sometimes.
"I just wasn't expecting you, that's all." I know my explanation is weak.
"You weren't expecting me?" He pushes himself off the bed. "Would you rather I go back out to the tent?"
Well, actually I would, I think but do not say. Instead, I get up and walk to the door.
"I'm going to check on Max."
As I leave, even I am wondering where the hell that all came from.
* * *
To: Parents
From: JDixon
Date: 9/23
Subject: Well done
Dear Miss Ward's class,
Thank you so much for finally responding to my call for help. Who knew so many of you have "special" brownie recipes?
Okay, this is how it is going to shake out.
Mini quiches—Dixons, Elders
Cheese platter—Changs (please include crackers)
Veggie platter—Wolffes
Wine—Batons (who are French so we're expecting some good stuff)
Cookies—Kaplans
Sparkling and flat water—Zalis
Brownies—Fancys
Plates/napkins—Aikenses
Cups—Westmans
The rest of you are off the hook for this party but don't suppose a slow response time is going to get you out of supplying snacks at some point.
Please drop everything off BEFORE 6:30 on curriculum night. Miss Ward wants it all out before her presentation starts.
Okay. That's it. Move along.
xo
P.S. Response times were weak, people, WEAK! I'm not going to embarrass everyone by posting them THIS TIME. Just know I'm keeping a list. A list you really don't want to be on.
* * *
* * *
I look at my watch and realize I have exactly four minutes until Garth is ten minutes early for our workout. I like how consistent he is. As I turn to get my workout clothes on, I'm reminded of how hard he worked me in our last session. Everything hurts just a little bit. Not enough to debilitate me, but enough that I'm aware of what my body has been doing. It's as though Curves never even happened for me! I intend to write them a strongly worded letter about their false promises.
I'm a little on edge, because tonight is parents' night at school. It's my first face-to-face as class mom, and I'm nervous. I know I've done it before, but this time it's different. This time I actually give a crap what the other parents think of me. Don't ask me why, but I do. Oh, to once again be twenty-six and so full of your own sense of what is right that you can give a virtual finger to the establishment.
The doorbell interrupts my thoughts and I run down to let Garth in.
* * *
"What has gotten into you today?" Garth enthuses as I complete yet another set of burpees.
I shrug. "Nervous energy, I guess." I'm really in a zone.
"What are you nervous about?"
"It's curriculum night at my son's school."
"So?" says Garth. Clearly he has never been to one.
"Well..." I start.
"Tell me while you're doing crunches," Garth suggests.
I get down on the mat on my back. Garth is sitting on the big exercise ball. I start to do my ab work.
"It's just that I sent out an email at the start of the school year and it was supposed to be funny but I guess it confused and offended some people. Things like that didn't used to bother me but now they do." I'm basically grunting out my explanation as I crunch my core.
"Eighteen, nineteen, twenty. Okay, rest. Why does it bother you now?" Garth asks from the ball.
I lie there and think for a minute. "Well, I guess—"
"Tell your story crunching," he interrupts.
"Jesus, okay!" I grunt. "I guess I care more now because of Ron. When Vivs and Laura were small and I was a single mom, I think I felt I had something to prove. I was also young and stupid." I flop to my back.
"One more set, but take thirty," Garth says.
I roll on my side and look up at him.
"I realize now that the things I do and say and write have a direct reflection on Ron and Max, too. I didn't really think about Vivs and Laura when I was waging war against the world. But I'm starting to see why they were always upset with me."
"Sounds like you're growing up," Garth says, with more than a bit of wisdom in his words.
"About time, I guess." I smile and start crunching.
I am just finishing printing out the class lists I'm going to hand out at curriculum night when I get the call.
"Mrs. Dixon?"
"Hey, Ashley. Are you on your way?"
"I can't babysit tonight. My mom says I need to focus on school work and that you should just get over it."
I roll my eyes. Gotta love Ashley. She is the most inappropriate babysitter ever. Reminds me of me.
"Uh, kinda leavin' me hanging tonight, girlie. We have something at Max's school."
"Yeah, I know, but my mom told me to tell you I'm sick. Wait, I think I was supposed to say that first."
"Okay, well, you tell your mom I said thanks."
"Okay, bye."
Damn. Ashley is so good with Max. Too bad I'm going to have to fire her seventeen-year-old ass.
"Ron!"
"Jesus, what? I'm right here."
Things have been a little tense with us since the camping-out night. I can tell he's still waiting for some kind of explanation. I'm still waiting to think of one.
"Ashley just canceled, so one of us has to stay home. I vote for me."
"And I vote for me," Ron countered. "And I know somewhere in the prenup we established that a tie goes to me."
I want to argue, but I know he's right. As class mom, I have to be there to press the flesh, kiss some babies, and talk about world peace. Oh, wait, that's POTUS. I just have to be there.
"Max!"
"Mommy, I'm right here. Why are you yelling?" He is standing behind me wearing a pirate hat and a feather boa.
"Sorry, just a habit. Ashley is sick or doing her homework or something, so Dad is going to stay home with you while I go meet your teacher."
"Okay. Can I watch TV?"
"I'd say your chances are pretty good."
"Yes! Say hi to Miss Ward for me. I love her."
"You do?"
"Yup."
"Okay, then."
I kiss Max five times, grab the two platters of mini quiches I said I would bring, and head out the door.
* * *
Miss Ward's classroom must be seen to be believed. Think Pee-wee's Playhouse and then vomit Disney crap all over it. There isn't one inch of space that isn't covered in colorful, um, stuff.
I almost don't recognize Miss Ward, who is sitting on her desk putting on lipstick. She is wearing a purple leather miniskirt and a pink low-cut V-neck sweater that couldn't possibly be any tighter. Her blond hair is in a messy ponytail. The best part of her getup is the thigh-high black boots. They're not quite stilettos, but they make her look like a life-sized Bratz doll.
As I approach her I can't help but wonder what happened to Sister Mary Perfect.
"Hi, Miss Ward."
She leaps up and hugs me.
"Jenny! I'm so happy you're here! Are those mini quiches? Yum. What time is everyone coming? I've been ready for an hour."
"They should be here any minute." I put my trays on a table next to an impressive platter of sushi. "Wow, did you bring this?" I ask, a bit too loudly
"No. Nadine Lewicki's mom sent it. Wasn't that sweet of her?"
"Very." I'm actually impressed. I didn't think Sasha read my emails, what with all the out-of-office replies. Not that I asked for sushi, but still.
"Max says Nadine has never been in class. Is she okay?"
Miss Ward seems taken aback by the question.
"Well, it's not something I can really talk about. But her mom and I are in close contact."
"It seems like her mom works a lot. Oh, by the way, Max wanted me to tell you hello. He says he loves—"
"Um... Jenny?" Miss Ward suddenly has the "I'm the teacher, take me seriously" look on her face. "Can you just respect that tonight is a getting-to-know-you party and not the time to get into personal issues about your child?"
I open my mouth and close it again. I am speechless, and believe me when I tell you that does not happen often. But it wasn't until that moment that I noticed the crazy eyes. Miss Ward has crazy eyes. I recognize them from Ron's ex-wife, Cindy. It doesn't make her a bad person, but it's definitely noteworthy.
"I'm sorry. You're right. I'll save my thoughts for conference day."
At this point, other parents start to trickle in and I get busy playing hostess.
"Are you the class mom?" a breathy voice from behind me demands.
I turn, and a large woman with short red hair is standing there, huffing and puffing like she's just run from the parking lot. She is wearing an orange ribbed sweater and a brown skirt. Pinned to her sweater is a big button that says, "It's No Joke."
"Yes, hi, I'm Jennifer Dixon, and you must be Shirleen Cobb."
She looks shocked. "How did you know?"
"The pin. Allergies. 'It's no joke,'" I say solemnly.
"Well, exactly. That's what I want to talk to you about. I think you need to—"
Thankfully, I am saved from finding out what I "need" to do by another parent, this one asking me where to put the brownies. I excuse myself from Shirleen and show a skinny blonde dressed all in black where to put her goodies.
And so it begins, my first evening with my fellow kindergarten parents. I am definitely the most, shall we say, marinated of the group. Most of the couples look to be in their early thirties.
As I scan the room, I notice a really tall couple over by the birthday calendar. I think it's nice when tall people find each other. Short people, too, although I feel sorry for their kids because, let's face it, they don't have a shot. I see the blond woman with the brownies talking to another woman also dressed entirely in black, their heads bent together as they take in Miss Ward's ensemble. The two men at their sides must be their husbands. One of them is very good looking. Hmm... I notice he's checking out Miss Ward, too. The other husband seems to be picking wax out of his ear.
Across the room, a couple is looking at the turtle tank. They have their backs to me, and I can't help but notice that the guy sure can fill out a pair of jeans nicely. As I'm admiring his butt, he turns around and I get a look at his face. Ho-ly shit! Goose bumps spread across my body as I recognize him. Don Burgess. He's such a fox. The words come to my head before I can stop them, because back in high school you never heard one without the other. "Don Burgess he's such a fox." It's like it was his full name. Today it would be a hashtag. #donbur gesshessuchafox.
Every high school has a Don Burgess—the one guy that every girl wants to be with and every guy wants to be. But Don took his coolness to another level. It was effortless. His jeans never looked new, but never looked old, either. He drove a lime green Dodge Charger and brushed his hair with his fingers. Best of all, he was sooo rock 'n' roll. Not in a skeezy way, just a super-cool way. The permanent expression on his face let you know he was in on some cosmic joke that you would never be groovy enough to understand. If he smiled at you in the hallway, it was as though the angels had come down from heaven for that brief moment and filled you with light. If he talked to you, forget it. The first time for me was junior year, when he bumped into me in the cafeteria and said, "Whoops! Sorry, Jen." I felt as though I'd been hit by a truck. He knew my name! You'd think he had asked me to prom by the way I reacted. I was high on life for a week.
"Don't I know you?"
I look up and he is standing right in front of me—dirty blond hair, green eyes, and just the right amount of stubble. Holy crap. Why does age look so damn good on some men?
I let out a nervous giggle, like the total tool that I am.
"Hey, Don, it's me, Jen Burgess. Uh. I mean, Jen Howard." I can't even imagine how many shades of purple I am right now.
"Jen! No freakin' way!"
I giggle again and attempt to moderate my pulse.
"Wow. You grew up gorgeous!"
He leans in to give me a hug, and a whiff of Polo by Ralph Lauren immediately transports me back to the hallways of East High. That was his scent, and it always lingered for a good thirty seconds after he had passed by. Now it was on my clothes. The normal me would be thinking how weird/pathetic it is that he still wears his high school cologne, but clearly normal me is nowhere to be seen. I back away and take a stab at acting cool.
"Do you have a child in this class?" I find asking the obvious is always nice.
"Yep. Lulu. Who's yours?"
"Max." I start looking around to see if anyone is watching me talk to Don Burgess (he's such a fox). Like anyone would care.
"So, are you married?"
"I am. My husband's at home with Max. Our babysitter canceled on us so I had to come alone. But he totally would have been here." Shut up, diarrhea mouth, I think to myself.
"Cool. Ali—that's Ali over there." He points to the woman still looking at the turtles. She waves.
"We're not married, but we co-parent Lulu. It works, I guess." He shrugs and gives me his trademark Don Burgess he's such a fox smile.
I feel dizzy, like I've been out in the sun for too long. I realize I'd better keep moving; otherwise, I risk making a complete fool of myself.
"Well, I'm paid to mingle..." I say, and start to walk away.
"Hey remember the P.E. laundry room?"
Of course I remember the P.E. laundry room! I want to scream, but instead I respond with that witty giggle I have recently adopted and, get this... a thumbs-up. Could I be any dorkier?
I get back to the job at hand and leave my high school crush to charm someone else. I glance around, and over by the book nook I notice a shortish woman with mousy brown hair and a guy who is a total hunk. They can't be together. I don't want to generalize, but let's face it, hot goes with hot, average goes with average, and so on down the line unless there is a large amount of money involved. But that would mean she has the money. I wonder if that's why Miss Ward is all tarted up tonight. Between Don Burgess and this guy, my time would have been well spent putting on some lipstick.
After Miss Ward's presentation about all the "super-fun" things the kids are going to learn this year, it is time for me to give my spiel. I stand up on a chair for maximum effect.
"Hi, everyone. I'm Jennifer Dixon, your class mom. Perhaps you have seen my emails?" I smile at the hunk, but notice that he's holding hands with his Mrs. She must be loaded. Don is standing behind him and gives me a thumbs-up.
As I continue, I get a few grins, but mostly blank stares, plus one toxic glare from Asami Chang. Tough room. Luckily I was planning to make my remarks brief. I'm much braver in writing than I am in person.
"I have brought copies of the class list for everyone. I also sent it to you in a PDF. If you have any questions, you can always call or email me. All my contact information is at the bottom."
At this point, I look up and see Nina walk in, wearing a megawatt smile and a fabulous aqua blouse that really brings out the blue in her eyes. I feel like I need to end with a bang to impress the woman who put me in this power position, so I add, "Here's to a great year!"
As everyone turns to the refreshment table, Nina walks over.
"Way to whip them into a frenzy."
"Yeah, well, I'm better in print."
"Have you made any friends yet?"
"Shirleen Cobb and I had a moment."
"Ah, the allergy mom."
"And, oh my God!" I drop my voice to a whisper. "The hottest guy in my high school is a parent in this class! I just made a total idiot of myself talking to him."
"Which one is he?" She looks around. "That guy?" She motions toward the guy with the rich wife.
"No. He's over by the food. Dirty blond hair and jeans."
Nina spots him and raises her eyebrows in approval.
"Not my flavor, but he's pretty damn cute."
"Are you kidding me? He's totally hot."
She shrugs. "I can see how he would have been gorgeous in high school."
For some reason, this irritates the crap out of me.
"Shouldn't you be bothering your own classroom instead of checking on mine?"
Nina laughs. "Nope. It's my job as PTA president to visit everyone. Oooh! Is that sushi? Nice touch." She makes a beeline for the California rolls.
As she leaves, the two women in black approach me minus their husbands. They both have long straight hair and are dressed in turtleneck sweaters, black jeans, and over-the-knee boots. They are both as skinny as I believe a woman can be and still be called a woman. Their only difference is one is about six inches taller than the other, and one is blond, the other brunette. In my head I immediately dub them Dr. Evil and Mini-Me. Dr. Evil speaks first.
"Hi, I'm Kim Fancy, Nancy's mom."
Seriously, her daughter is Nancy Fancy. Who would do that?
"Hi, Kim. I think Nancy and Max sit at the same table."
"Oh, really? Nancy never said anything."
"Well, neither did Max, but their names are on their seats."
I turn to Mini-Me.
"I'm Jen."
"I'm Kit's mom, JJ Aikens."
"O-KK," I joke. Nothing. Mini-Me just stares.
"When are you planning the parent cocktail night?" she asks.
"Sorry? The what?"
Dr. Evil looks at me like I have two heads.
"Every year the class mom organizes a cocktail party so the parents can get to know each other."
"Really? Is that new?"
"Uh, no," Mini-Me says condescendingly. "We started it in preschool."
"Well, okay. I'll start thinking about it." In my head I'm counting the ways I'm going to exact my revenge on Nina. Mini-Me interrupts my thoughts.
"We both have really busy schedules, so the sooner you can pick a date, the better."
"Okay. Good to know."
"Can't wait!" Dr. Evil says, and they both smile and walk away. I make a mental note to get the backstory on them.
I turn away, looking to rip Nina a new one, when I bump into the extremely tall couple.
"Hi, Jen? I'm Peetsa and this is my husband, Buddy."
"Pizza?"
"Yup. Just like the food. We're Zach Tucci's parents."
"Oh, my gosh, Max talks about Zach T. all the time."
"Same here," Buddy says, and then he blushes. "I mean, Zach talks about Max."
I think I like these two despite their intimidating height. Buddy has jet black hair that he wears brushed back, dark eyes, and a nose that might just be a little too small for his face. Peetsa can only be described as a classic Italian beauty. Think Sophia Loren, but with smaller boobs and lips.
"We should set up a play date," I suggest.
"Your place or mine?" Buddy says, and immediately blushes again. Oh, jeez, this guy is going to be fun to tease.
Peetsa rolls her eyes. "He means we'd be happy to host. By the way, we love your emails. It's the first time I've ever laughed out loud reading something from a class mom."
"Thanks. I try to have fun with them. I'll host the play date. Max just got a remote-control helicopter and he's dying to show someone."
"Is Max your only child?" Peetsa asks me.
"No, I have two older daughters."
"Oh, do they go here?"
"Well, they used to. Do you have any besides Zach?"
"Our daughter Stephanie is in sixth grade. Kinda crazy how long we waited between kids, huh?"
I decide to hold off telling her just how not crazy six years between kids sounds to me.
"I'll call you tomorrow to set up a play date. By the way"—she leans in to me—"is it me, or does our teacher look like a hooker?"
It's official. I love this woman.
"Maybe she has a paying gig after this?" I offer.
"Buddy can't take his eyes off that sweater."
"What?" Buddy blushes. "It's a nice color."
Peetsa laughs and pulls him away.
I turn to grab something to eat, and bump into two women, one tall with short blond hair and the other even taller with really short brown hair, scoping out the brownies. Encouraged by my last parent encounter, I put on my friendly face and say hello.
"Hi. I'm Jen."
They both smile.
"Hi, I'm Carol, Hunter's mom," says Blondie.
"And I'm Kim, Hunter's other mom," says Short Brown Hair.
I don't know why, but the two-mommy thing takes me by surprise. Unfortunately, when that happens, my ability to rebound isn't stellar. I start to babble.
"Oh, wow! How great. Good for you! We should have a play date. Does Hunter like helicopters? Or do you guys try to keep him away from gender-specific toys?"
SHUT UP! SHUT UP! SHUT UP! I scream in my head. Boy, this night is one for the record books.
They look at each other and start laughing.
"Breathe, Jen," says Kim... or Carol. I've already forgotten who is who. "We know you're cooler than this. Unless someone else writes your emails."
"Sorry." I blush. "I am cooler than this!"
"Want to tell us how some of your best friends are gay?" asks Kim or Carol with a smile. I start to laugh.
"Yes, thanks. That was next on my list of stupid things to say."
"Don't worry. We've heard worse. But a play date sounds good. Should we email you?"
"That would be great. See you later and again, I'm sorry."
As they walk away, the blond one says loudly: "What's that you say, Jen? You had a lesbian experience in college? How wonderful!"
Nina sidles up to me. I'm actually sweating now.
"Look at you, making friends," she coos, as only Nina can.
"Oh, my God. Why didn't you walk over here two minutes ago?"
"And miss the train wreck that was you meeting Hunter's two mommies? No way."
I shake my head, trying to push the whole thing out of my mind. I turn to Nina.
"Hey, what can you tell me about the turtleneck twins over there?" I nod toward the book nook, where Dr. Evil and Mini-Me are pretending to look at Wild About Books while obviously assessing every parent in the room, kind of like I'm doing.
Nina rolls her eyes and picks a sesame seed out of her teeth.
"Well, the taller one, Kim, moved here two years ago from New York—or Manhattan, as she always calls it just in case we think she means any other part of New York."
"Someone should tell her there's a Manhattan, Kansas. That'll shut her up."
Nina laughs. "The shorter one is JJ Aikens—I've known her for years. She used to be seminormal, but ever since Kim turned up all she wants to be is another Kim. I think she's even convinced herself that she's from New York, too."
"You mean Manhattan," I correct her.
"Sure I do."
"Well, that explains a thing or two. Kim's husband was checking out Miss Ward."
"Oh, hell, Jen, who wasn't? I haven't had sex in so long, Miss Ward is looking good to me."
"Do you know that tall couple?" I ask her.
"Peetsa and Buddy? Sure. Their daughter and Chyna have been in the same class on and off for years."
I am about to say something, but Miss Ward is waving an empty garbage bag at me, which I guess is her way of asking me to start cleaning up.
Nina sighs. "Well, I'm off to Mr. Greely's class to check in. No sushi there, I can promise you that."
As I'm grabbing cups and plates, I find myself glancing around the room, looking for Don Burgess. Our eyes meet over the large recycling bin.
"Hey, are you trying to take my job away?" There's that smile again.
"Sorry?" This time I manage to suppress the giggle.
"Waste management. That's what I do." He hands me some used napkins to put in my trash bag.
"Really? That's..." I stop because I don't know what to say. That's interesting? That's cool? That's disappointing? Who knew Suchafox would end up a garbageman?
"I know." He laughs at the look on my face. "It's not what you'd think I'd be doing, but there's a lot of money in garbage."
"Well, that's great. Good for you. Helping to keep KC clean."
"I do what I can," he says. As he walks away, he turns back to me and says the last thing I thought I would ever hear Don Burgess say:
"Don't forget to recycle those cups!"
* * *
When I get home that night, I find Max and Ron in our bed, both fast asleep. Max is wearing his Spider-Man costume as pajamas, of course. As I pick him up, I realize he's almost too big to carry, and my heart hurts a little bit. I remember when the girls got too heavy for me. It's just one more step away from you that they take without even knowing it. As I lay him down in his race-car bed, I put my nose into his hair and take a deep breath. He never lets me do that when he's awake.
I sit down on the bed and look around Max's room. A stranger would think that two completely different children share it. One is a diehard sports fan, evidenced by all the posters of Chiefs and Royals superstars papering the walls. The other is a techie/wizard fan who has a flair for fashion, shown through all the gadgets and drawings littered about.
When I get back to the bedroom, Ron is sitting up, squinting at his iPad.
"How'd it go?" he asks.
"It was a real barn burner. You would have loved it."
"Really?"
"No." I stretch out on the bed. "What did you guys do?"
"We had hot dogs and watched hockey on ESPN Classics in bed. Perfect manly-man night."
"For our perfectly unmanly little boy," I add.
"He loved it."
"Ron, please stop trying to make him love sports. If he does, he does. If he doesn't, it's not the end of the world."
"I'm just trying to show him all the options. I've given up on football and soccer, but I have high hopes for hockey."
I shake my head.
"You know that kid Zach T. he always talks about?"
Ron shrugs. "The one he ate lunch with that day?"
"No, that's Zach B."
"The one he says picks his nose?"
"No, that's Zach E. Zach T. is the one who is really into gadgets. I met his parents tonight. They seem like nice people."
Ron gasps. "Don't tell me you actually met someone you like!"
"Ha, ha. Their names are Peetsa and Buddy."
"Pizza?"
"Just like the food."
"God, can you imagine going through high school with that name? 'Hey, Pizza! Can I have a slice?' I would have tortured that poor girl."
"I'm sure she's heard them all."
I'm about to tell him about Don Burgess he's such a fox when the phone rings. I roll onto my side to answer it.
"Hello."
"Hi, Mom." It's my older daughter.
"Hey, Vivs. What's up, baby?"
"Not much."
"How's school?"
"Good."
Vivs is my show pony. She came out of the womb just knowing the right thing to say and do, which is pretty much a miracle considering Michael Hutchence and I were less than stable people at the time of conception. There were a couple of dark years during puberty when Vivs thought she was Marilyn Manson and, I'm not going to lie, I was terrified of her. But we got through it. Currently she is a junior at KU, majoring in human ecology, whatever the hell that is.
"What's going on?"
"Nothing."
"Jesus, Vivs, really? Do I have to drag it out of you? What's up? Do you need money?"
"Mom! No. I just called to say hi. Jeez. Why so bitchy?"
"Sorry. I just got back from curriculum night at Max's school."
Vivs laughs. She knows how much I hate school functions.
"How's he doing?"
"He's good, but I think Ron traumatized him tonight by making him watch a hockey game."
"He loved it!" Ron yells.
"Still on the sports thing, huh?"
"Yeah." I sigh. "How is Raj?"
"He's okay."
Raj is Vivs's on-again off-again boyfriend. I can't really get a handle on what their problem is but they seem to break up and make up an awful lot. Ron and I really like him. He's an engineering major at KU, which means he will actually leave school with a useful degree, which is more than I can say for either of my daughters.
"Have you heard from Laura?" she asks.
"Not for a couple of days. You?"
"Yeah, I had lunch with her today. She's freaking out because she was snapped making out with two different guys on the same night. Someone posted the pictures on Instagram and now she looks like the campus slut. She didn't call you?"
"Oh, my God. No, she didn't."
Ron raises an eyebrow at me.
"I'll call her tomorrow."
"Don't tell her I said anything."
"I won't. Thanks for giving me a heads-up."
"K. Love you."
"Love you too, baby." I hang up.
"Oh, my God!"
"What happened?" Ron asks.
"I guess Laura's taking a page out of her old mom's handbook." I can't help but smile.
"What, is she pregnant?"
"Nooo, that comes after college. Haven't you been paying attention?"
"Obviously not."
I roll on my side to face him.
"It seems she made out with two different guys at a party and someone took pictures and posted them on Instagram."
"Our Laura? Didn't-have-a date-until-she-was-seventeen Laura?"
"The very same. Poor baby. Vivs says she's freaking. I'll call her tomorrow."
* * *
To: Parents
From: JDixon
Date: 10/25
Subject: Parent/teacher conferences
Hello, fellow parents!
Ever wondered what Miss Ward really thinks of your child? Oh, I know it's all smiles and compliments on curriculum night, but prepare yourselves for brutal honesty as we sally forth into parent/teacher conference season.
And best of all, I'M IN CHARGE OF THE SCHEDULE! Oh, I'm just drunk on the power I have now! Who wants the coveted 12:30 Thursday spot that allows you to take a long weekend? Cuz there's a super-cute coat sitting in the window at Macy's with my name on it... first person to get that for me wins their pick of times.
OR we could do it the old-fashioned way and have you email me 2 times that work for you from the list below and I will "try my best" (wink) to accommodate you.
Thursday, Nov. 17, p.m.
12:30, 1:00, 1:30, 2:00, 2:30, 3:00, 3:30
Friday, Nov. 18, a.m.
8:00, 8:30, 9:00, 9:30, 10:30, 11:00, 11:30, 1:00, 1:30
Let me know as soon as you can, so I can get on with my life. Early birds will be rewarded with primo spots.
Dixon out!
To: JDixon
From: Sasha Lewicki
Date: 10/25
Subject: Parent/teacher conferences
I am out of the office until October 27.
Thank you,
Sasha
To: Jen Dixon
From: Carol Alexander
Date: 10/25
Subject: Parent/teacher conferences
Hi, Jen,
Any time on Thursday works for us. Kim and I have a big lesbian conference to go to on Friday. Want to come with us? The theme this year is Dykes or Bikes: Which Would You Rather Ride?
Kidding, of course. Except for the part about any time on Thursday.
Thanks,
Carol
To: Jen Dixon
From: Jill Kaplan
Date: 10/25
Subject: Parent/teacher conferences
Hi, Jen,
I'm Rachel Kaplan's mother. I'm so sorry I didn't say hello at curriculum night, but I wasn't feeling very social. My asshat of a husband wouldn't come with me and I was too pissed to be friendly to anyone.
Anyhoo, I will take any time Friday morning. God knows Steve doesn't care. He thinks it's women's work to meet the teacher. You may wonder why we even had a kid if he isn't going to participate. I ask myself that all the time.
Let's have coffee sometime!
Jill
To: JDixon
From: AGordon
Date: 10/25
Subject: Parent/teacher conferences
Dear Jen,
I missed meeting you on curriculum night, but Don mentioned that he spoke with you. Did you really know each other in high school? Small world. I'm not sure if he has responded to this email, but I'm pretty sure we are good for any time after 12:30 on Friday. That's when we usually do the handoff.
Thanks,
Ali Gordon
* * *
Huh. It seems Don and his baby mama don't live together. He said they were co-parenting Lulu. I guess they're divorced.
* * *
To: JDixon
From: JJ Aikens
Date: 10/25
Subject: Parent/teacher conferences
Hi,
There is only one option for us and that's 12:30 on the 18th. Sorry. No flexibility whatsoever.
Thanks,
JJ
To: JDixon
From: Kim Fancy (Nancy's mom)
Date: 10/25
Subject: Parent/teacher conferences
Hi, Jen,
We have scheduled a trip back to Manhattan, so we have to go first on Thursday. We absolutely must make our flight in order to be at Lincoln Center for the ballet Thursday night.
Thank you,
Kim
To: JDixon
From: Asami Chang
Date: 10/25
Subject: Parent/teacher conferences
Jennifer,
We will take the first spot on Thursday—12:30. And I won't be buying you a coat for it, I can assure you. I have notified the head of the Parents' Association that you are soliciting bribes.
Asami
To: JDixon
From: Peetsa Tucci
Date: 10/25
Subject: Parent/teacher conferences
Hi Jen,
Thanks again for the play date. Zach had a great time. I think a remote-control helicopter is definitely on his Christmas list now!
As for the conferences, we are around, so plug us in anywhere.
See you for trick-or-treating.
Peetsa
xo
* * *
Peetsa's email makes me realize that I haven't heard from Miss Ward about having a Halloween party in the classroom. Max is crazy about dressing up and has been planning for weeks to wear his costume to school that day.
* * *
To: Miss Ward
From: JDixon
Date: 10/25
Subject: Halloween party???
Hi, Miss Ward,
I know you're busy, but I never heard from you about planning a Halloween party for the kids. Do you want me to pull something together?
Thanks,
Jennifer
* * *
Much to my surprise, her response is almost immediate.
* * *
To: JDixon
From: PWard
Date: 10/25
Subject: Halloween party???
Hi, Jenny,
I didn't see a request from you, but just so you know, I only have parties when I think the kids deserve them—not because Hallmark tells me it's a holiday.
By the way, when will you have my conference schedule ready?
Keep up the good work!
Peggy
* * *
Well what do you know. Miss Ward is able to come up with the one answer I would have never thought of.
I start tackling the conference schedule almost immediately. I knew that first slot on Thursday would be a popular one, because there's no school the next day: It's a great opportunity to take a long weekend somewhere. I have three people who insist that it is imperative they have it, so obviously none of them is getting it. I know, I'm a bitch. If any of them had asked for the spot instead of demanded it, it would have been theirs. As my mother always says, ask me to do anything. Don't tell me to do anything. As I type, I count the number of enemies I'm about to make. Ah, heavy is the head that wears the crown.
* * *
To: JDixon
From: NGrandish
Date: 10/27
Subject: Threats and bribes
Jen,
I've heard thru the grapevine that you are asking for a coat in exchange for a preferred conference time. Please don't do this. Go for something smaller, like jewelry or maybe a watch. The evidence is easier to hide.
Kisses,
Nina
To: Parents
From: JDixon
Date: 10/30
Subject: Parent/teacher conference schedule
Hi, folks,
If you've ever seen The Hunger Games, you'll have an idea of what I went through trying to make everyone happy with these conference times.
Here is how it shook out. If you don't like what you get, well, good luck finding someone who cares.
And remember, we have a half day on Thursday the 17th and no school on Friday the 18th. Please, please don't send your kids to school on Friday unless you want them to have an introduction to custodial services.
Over and out.
Jen
Conference Schedule:
Thursday, Nov. 17
12:30 Lewicki
1:00 Fancy
1:30 Aikens
2:00 Chang
2:30 Alexander
3:00 Brown
3:30 Kaplan
Friday, Nov. 18
8:00 Cobb
8:30 Dixon
9:00 Westman
9:30 Baton
10:30 Tucci
11:00 Elder
11:30 Wolffe
1:00 Gordon/Burgess
1:30 Zalis
* * *
I push Send knowing what a shit storm I have just delivered, and wait for the fallout. Of course Sasha Lewicki's out-of-office reply is the first to arrive, but it is followed very quickly by emails from Dr. Evil and Mini-Me.
* * *
To: JDixon
From: Kim Fancy
Date: 10/30
Subject: Conference schedule
Hi, Jen,
I'm sorry if I didn't make myself clear in my previous email. I simply must have the first conference time because we are flying to Manhattan! Please switch me with Sasha Lewicki.
Kim
To: JDixon
From: JJ Aikens
Date: 10/30
Subject: Conference schedule
Jen,
I'm really shocked by how you ignored all our requests, especially since you asked us to tell you what we wanted. I mean, Kim and I both asked for the first time slot, assuming you would give it to at least one of us. Kim has a flight to Manhattan. She has tickets at Lincoln Center. So you see how important it is that she get the first spot. I'm sure the Lewickis won't mind switching when you ask them.
Thank you,
JJ
To: Kim Fancy, JJ Aikens
From: JDixon
Date: 10/30
Subject: Conference schedule
Hi, Kim and JJ,
The schedule stands as is. If you would like to reach out to the Lewickis yourself, be my guest. They were the first to respond and I'm pretty clear on where I stand with prompt replies.
By the way, to my knowledge there are no airports in Manhattan. I hope someone didn't sell you a fake plane ticket!
Jennifer
To: Sasha Lewicki
From: JJ Aikens
Date: 10/30
Re: Conference schedule
Hi, Sasha,
How are you? My name is JJ Aikens and my daughter Kit is in class this year with your daughter, Nadine. I'm sorry we didn't get a chance to meet on curriculum night, but I'd love to get the girls together for a play date. JJ's usually playing with Nancy Fancy, but maybe we could set something up for November 20? Speaking of which, can I switch conference times with you on the 18th? I really need that first spot. Well, I don't, but Kim does (you know Kim Fancy, don't you? She moved here from Manhattan a couple of years ago). She needs to jump on a plane back to the city. We told the class mom, but she seems to just do things any way she wants. I don't think she's set a date for the class cocktail party yet, not to mention she didn't even bother to organize a class Halloween party!
Anyway, let me know as soon as you can. And let's do coffee soon.
JJ
To: JJ Aikens
From: Sasha Lewicki
Date: 10/30
Subject: Conference schedule
I am out of the office until November 3.
Thank you,
Sasha
* * *
I'm not sure why I hate Halloween so much, but I really do. It might have something to do with my parents' lack of enthusiasm for "that heathen holiday," but who knows? Having kids means you have to jump on the bandwagon and pretend that dressing up in ridiculous outfits and going door to door begging strangers for candy is A-OK.
I blessed the day Vivs and Laura outgrew trick-or-treating, and I lived happily with the knowledge that I would never have to do it again. Yeah. That's what I get for making plans. Luckily, I live with the only person more in love with Halloween than a five-year-old boy. I've gone out with Ron and Max other years, but tonight they are flying solo. My only job is to give kids a hard time when they come to our house.
As I'm filling our trick candy bowl (the green skeleton hand comes down when you reach inside to get treats, mwa-ha-ha-ha!), the doorbell rings.
"Trick or treat, Mrs. Dixon!"
It's Zach T. and Peetsa. Zach T. is dressed as a mailman. Peetsa has a witch's hat on and a bottle of wine in her hand.
"Hey, guys! You look great, Zach. Let's see a trick."
He looks at me and frowns. Peetsa starts to laugh.
"Don't you know any tricks? I need to see a trick before I can give a treat," I say as I wink at her.
Zach thinks for a minute. "Well, I can burp the alphabet. Is that a trick?"
"Yeah, it is! Let's hear it."
As Zach burps away, I hug Peetsa and take the bottle of wine she offers.
"You read my mind," I whisper.
"T burp U burp V burp W burp X burp Y burp Z."
"Nicely done." I offer him the candy bowl. "Now can you go upstairs and see what's taking Max so long?"
"Sure!" He runs to the stairs.
"Oh, my God, do you make every kid do that?" Peetsa asks.
"Damn right. Nothing is free. They should learn that early in life."
I'm so glad Peetsa agreed to help me give out candy tonight. Buddy is doing home duty at their house and Ron is taking the boys out, so it's a perfect girls' night in. I head to the kitchen and pull out some wineglasses. As I go to open the bottle, I notice it has a screw top. I whistle.
"Whew! Hope you didn't break the bank with this one."
"Excuse me, but that's actually a really good bottle."
I wave the screw top at her.
"A lot of vineyards don't use cork anymore for some of their vintages. That, my friend, is a fine Australian Shiraz stolen from Buddy's wine closet."
"I love that you're so into wine."
Peetsa takes the glass from me.
"Well, Buddy's the real oenophile. I just reap the benefits. Cheers."
Just then I hear what sounds like thunder coming down the stairs, and a mailman, a ninja, and a football player appear before my eyes.
"Wow! You guys look awesome! Peetsa, this is my first husband, Ron."
Ron rolls his eyes. "She thinks that never gets old." He shakes hands with my new friend.
Peetsa laughs. "Well, it's pretty funny the first time you hear it."
Max strikes a pose.
"Ninja!" he yells.
"Show Mrs. Tucci your trick, Max."
"Mom, no one ever asks for a trick," Max whines.
"You never know," I say. "Come on, just show her."
Max grimaces but indulges me. He holds the sword sideways in both hands, brings it to knee level, and awkwardly jumps over it, through his arms. Peetsa and Ron clap. Zach hoots his approval.
"Love that!" exclaims Peetsa. "You will get so much more candy if you bust that move out at every house."
"Do it while Zach burps the alphabet!" I chime in.
"What trick should I do?" my husband murmurs as he kisses me on the cheek. He is decked out in a Kansas City Chiefs uniform.
"Maybe you could rough up the quarterback a little later." I give him a good pat on the butt.
"Deal." He smiles. "Okay guys, grab your bags and let's hit it. You girls have fun."
"Ninja!" Max yells as they head out the door.
* * *
As Peetsa and I settle in to the two chairs I have moved close to the front door, I take my first sip of wine.
"Oh, wow. That is good."
"Mm-hmm." Peetsa washes down a bite of pizza with a big gulp.
Just then the doorbell rings.
"Oh, God, here we go." I open the door and standing in front of me are three princesses a little older than Max.
"Yes?" I ask. "Can I help you?"
Silence.
"Trick or treat!" says the mother standing behind them. She's wearing a crown.
"What's your trick, Princess?" I ask with a big smile.
Nothing.
"They're shy," says the mom by way of explanation for their muteness.
"Hmmm. I really need a trick to give a treat."
Now they're all staring at me. The mother is giving me a look that says, "Really, you're going there?" I stand my ground. Then one of the little girls pipes up, "I can do a cartwheel."
"That is a great trick!" I encourage her. "Let's see it."
She puts down her crown-shaped candy bag, walks down the steps to our lawn, hikes up her dress, and does a perfect cartwheel. I clap and cheer and offer the candy bowl to all three girls.
"Bye, girls. Have a fun night." Just as I'm closing the door, I see the mother give me the finger.
Peetsa is doubled over laughing.
"Oh, my God! You are horrible. There should be a warning on your door."
Over the next hour, our conversation is interrupted about thirty times by the doorbell. After something like the sixteenth ring, I lose my enthusiasm for torture and just hand out the candy. Except when I open to see a group of teenagers just standing there holding bags. No costumes, no trick-or-treat. This is my pet peeve. I'm sorry, but door-to-door candy gathering is for children, not semi-adults looking for sugar.
"What's up?" I ask.
"Uh, trick or treat?" one of them tries.
"Great! What's your trick?"
"My what?"
"Your trick. The thing you do to get the treat."
"Uh, we just came to get candy. Do you got any?"
"Not for people who don't do a trick." I smile.
"Do you mean, like, magic or something?"
"Sure. Do you know any magic tricks?"
"Uh..."
"Dude, let's go," says another teen. "She's a bitch."
I look at the kid who just spoke. There's something...
"Robbie Pritchard? Is that you?"
"Oh, shit," they all say.
The Pritchards lived next door to my parents for years.
"Did you just call me a bitch?" I ask calmly.
At that, they all turn and run off the porch like a pack of scared dogs.
"I'll be calling your mother later," I yell after them.
Peetsa just shakes her head. "I see you bring the same enthusiasm to everything that you do to being our class mom."
"I know. I'm the worst."
"You're not the worst! I love your emails. I couldn't wait to meet you. But I'm not surprised you piss some people off."
"I didn't even want to do it, but Nina knew it was the only way I was going to... Wait, who's pissed off?"
Peetsa looks at her wineglass like it might have the answer.
"Umm... I don't know, I've just heard a few people talking."
I jump up.
"Who? What did they say? You have to tell me. I live for this stuff."
Peetsa laughs. "You really are crazy."
"I'm not! I've just been through this before, and I'm determined to get parents to lighten up."
"Anyone who is even remotely cool totally gets your humor. You know who the stick-up-the-asses are—Asami, JJ, Kim Fancy, Ravital Brown..."
"Zach B.'s mom?" I ask, a little hurt.
Peetsa nods. "But I think she just doesn't understand your sarcasm. She told me her husband has to explain all your emails to her."
"Huh. Well, maybe I can win her over. Who else?"
"That's all I know of. It's a small but vocal bunch."
"More wine?" I ask. I head to the kitchen, wondering how I can make an ESL person understand what snarky means.
"Sure, thanks. Hey, how's your daughter?"
"Which one?"
"The slutty one."
"Peetsa!" I scream in shock. "Please! We prefer 'loose' or 'sleazy.'"
We both crack up.
"She's doing fine. The nice thing about her generation is that they move on pretty quickly. Two days after her drama, some poor girl was snapped eating two hotdogs at once, so the spotlight is off Laura."
"I still can't believe you have two kids in college. Max must have been the shock of your life!"
I pour more wine for both of us and we head back to the chairs by the door.
"Not really. Ron wanted to have a kid."
"Didn't he already have two?" she asks, confused.
I inwardly cringe. This is the part I hate explaining to people.
The doorbell rings just as we sit down. Wow, sometimes you really are saved by the bell.
Peetsa jumps up.
"I've got it."
She opens the door and I hear two little voices sing, "Trick or treat."
"Do you have a trick to show me?" Peetsa asks them. I've taught her well.
Then a familiar voice says, "Hey, don't we go to the same school?"
I nearly spit my wine out. Suchafox is at my front door! I jump up from my chair and sprint to stand beside Peetsa.
"Hey, guys!" I say a little too loudly. Lulu is dressed like a zombie bride, the other little girl is a nurse, and Don is wearing a cowboy hat. He looks perfect, of course. The butterflies in my stomach start doing a happy dance.
"Jen! No way. Do you guys live here?"
"I do." Oh cripes, that giggle is back. "Peetsa is helping me out tonight. Do you guys know each other?"
"I definitely saw you at curriculum night," Peetsa says to Don. She has a goofy smile on her face. "I'm Peetsa, Zach T.'s mom."
"Peetsa?" Don asks, and once again I get a little glimpse into the hell that must be her world thanks to her unique name.
"Just like the food," I offer. "P., this is Don Burgess. We went to high school together. Isn't that crazy?" I giggle.
"Very," Peetsa affirms.
I turn to him. "Do you live in this neighborhood?"
"No, we live west of here, but Lulu wanted to trick-or-treat with Rachel. It's nuts. I don't see you for, like, thirty years and now I see you all the time."
"I know, right?" I giggle and offer the little girls the candy bowl.
"Wait!" says Peetsa. "What about the trick?" She turns to Don. "They're supposed to do a trick to get a treat."
"Really?" His look says it's the dumbest idea he's ever heard.
"Oh, P.!" Giggle. "Lighten up on the trick part, will you?" I smile at Don and shake my head like I don't know what the hell she's so uptight about. "Go ahead, girls."
I avoid Peetsa's glare as Lulu and Rachel dig into the candy bowl. They screech when the skeleton hand comes down to grab them and, after taking a handful of candy, they march down our front steps and onto the walkway. I glance toward Don and, for the briefest moment, I wonder what it would be like to kiss him. The thought is gone as quickly as it came, but I can't help feeling a little uncomfortable. Peetsa interrupts my wayward thoughts.
"So you guys were friends in high school?"
"Well, we knew each other, but we didn't really hang out," I answer quickly.
"But there was one pretty significant moment in the P.E. laundry room." He gives me a wry smile.
"Oh really?" She arches an eyebrow at me. "Tell me everything."
Don lets out a guffaw. "And this is where I leave you!" He starts down the stairs to where the little girls are waiting for him. "See you around."
I watch him walk away with what I'm sure is a ridiculous smile on my face. When I turn to go inside, Peetsa is staring at me.
"What?" I play dumb.
"That's my question. What the hell was that all about?"
"He's a father in our class." I shrug.
"Yeah, so is Buddy, but you don't act like that when he's around. What was that giggle?"
I lead her back into the house while I explain.
"I had a huge crush on him in high school."
"And apparently you still do." She imitates my annoying giggle.
"P.! Be nice. He's still so cute."
"He must have been really something in high school."
"He was such a fox."
Peetsa bursts out laughing.
"God, I haven't said that about anyone in years." We sit back down on our chairs. "So what happened in the P.E. laundry room?"
The doorbell rings again. I jump up to get it, thrilled to avoid Peetsa's question. Maybe Suchafox forgot something!
I'm only a bit disappointed to see it's not Don, but our sons standing there.
"Hey! How did it go?" I ask as they charge in with full bags of candy.
"Great! The Gibsons were giving out whole chocolate bars and we went twice. They didn't even notice." They start to dump their candy out on the living room floor.
Ron comes in, helmet in hand, looking shell-shocked.
"They ran the whole time. I didn't even let them have candy."
"Come have some wine, babe."
"No, thanks. I need a shower."
He heads upstairs and Peetsa gives Zach a ten-minute warning.
"Wheels up in ten, kiddo. Do your trading and put your stuff back in your bag."
"Okay!" Zach yells from six feet away.
Peetsa looks at me and frowns.
"What were we talking about?"
I grab the chance to change the subject.
"What do you really think of Miss Ward?"
"No, wait! P.E. laundry room..."
I wince. "Some other time, okay?"
She looks surprised. "Oh! Okay."
"So, Miss Ward, what do you think?"
She shrugs. "I don't know. Zach loves her, that's for sure."
"So does Max! I just get a really weird feeling from her."
"Well, you'd know. You spend the most time with her."
"That's just it. You'd think I would, because of the class mom thing. But honestly, I never see her and she told me at the beginning of the year that she doesn't want me bothering her with"—I make finger quotes—"class-related stuff."
Peetsa's eyes pop.
"She said that?"
"Better. She wrote it in an email. That's the only way she talks to me, and usually it's only to tell me to do something like organize the parent/teacher conferences."
"Is that normal? I've never been a class mom."
"There's really no normal. Every teacher is different. In the lower grades, they tend to want you helping out in the classroom as much as possible. But Miss Ward hasn't asked me once."
"Well, it's only been two months. But it was kind of weird that the kids didn't have a Halloween party. Did she say anything to you about that?"
I look at her over my wineglass.
"She said she doesn't celebrate Hallmark holidays."
Peetsa does a spit take.
"Oh, well, that makes sense," she says, wiping wine off her chin. "Who would want to celebrate that great Hallmark holiday, Halloween?"
"I'm just wondering, when does she give a party?"
"Arbor Day?" Peetsa snickers.
"Groundhog Day!" I chime in.
We laugh as Peetsa gets up.
"Zach, let's hit the road," she calls to her son. "Hey, thanks for tonight. This was fun."
"Best Halloween I've ever had," I say. And I actually mean it.
* * *
In the bathroom, getting ready for bed, I mull over my lingering thought about kissing Don. I've never had a moment like that before, and it's making me uncomfortable on several levels. It was so out of left field. I mean, I adore my husband and we still have a pretty great sex life even after ten years. Since the day we met, I've never even thought of being with another man... unless you count my Bruce Springsteen fantasies. While I brush my teeth, I close my eyes and try to put it out of my mind. One thing's for certain, Ron Dixon is getting some tonight.
* * *
To: Parents
From: JDixon
Date: 11/18
Subject: Party time!
Hi, gang,
Long time no blah blah!
Big big news! Our children are going to have a compliment party ("You're cute"; "No, you're cute!"). Apparently they're encouraged to write down compliments to each other and put them in a jar. When the jar is full, voilà, a compliment party.
Miss Ward has chosen to let them make ice cream sundaes on the Wednesday before Thanksgiving (November 23) so she can send them off for the long weekend on a sugar high. I'll bring the ice cream, but we need the following toppings and supplies:
chocolate sauce
caramel sauce
sprinkles
gummy bears
plain M&Ms
whipped cream
marshmallow topping
spoons, bowls, and napkins
and of course the Batons will bring wine
Please remember, NO NUTS!!! (See, Shirleen? I didn't forget)
This is a wonderful opportunity for all of you to get into my good graces by volunteering early and often to bring something. As always, response times will be noted.
No, no. Don't thank me. It's my reward for being class mom.
Jen
* * *
* * *
As I click Send, I look at my watch and realize I had better get my ass in gear if I'm going to be on time for my teacher conference with Miss Ward. I wouldn't miss this for the world. I have been shut down by her two separate times while trying to make casual conversation, so I can't imagine what having a real discussion about Max is going to be like.
I run upstairs to my bedroom and notice how out of breath I am. When are all my hard work and conditioning finally going to pay off? It's only five months until the mud run, and failure is not an option. I'd better talk to Garth about ramping up my cardio.
I throw open my closet door, pretending that I'm actually going to choose something other than my official mom uniform of Levi's jeans, white T-shirt, and gray (or black) sweater. I have twenty different combos of basically the same outfit, and that isn't going to change anytime soon. My only indulgence is a pair of black Prada short boots. I have had them for nine years and they probably receive better care than any of my kids. I only switch it up when I absolutely have to or when Ron makes a comment like "Didn't you wear that yesterday... and the day before?" He's actually the one who coined the phrase "mom uniform." He thought he was sending me a subtle message, but I took it as a compliment. I also have an evening mom uniform for special nights out. It's form-fitting black pants or pencil skirt, and a black button-down shirt. I have been mistaken for a waiter on more than one occasion. Once, we were at a fund-raiser at city hall, and Don Cheadle from Ocean's Eleven asked me to get him a refill.
Occasionally I will switch up the jeans-and-sweater look, but not today. Today I need all the comfort I can get. I never know which Miss Ward is going to show up, so I have to be at my most relaxed and nonjudgmental.
Ron is meeting me at the school, so I grab my purse and the car keys and head out the door.
Have I mentioned that I love my car? It is a totally tricked-out Honda Odyssey minivan, and let me tell you, I feel like the king of the road in it. Ron thought I was crazy for wanting the "I give up" car, as he calls it, but he'd never had kids before and didn't realize how vital automatic sliding back doors would be to our existence. He also didn't realize that I would be running Mom's taxi service for Max and his friends for the foreseeable future. Now he understands, and he even drives the minivan... sometimes... when he thinks no one will see him.
This is actually my third minivan, and by far my favorite. I've only had it for a couple of months, so Max and his friends haven't had time to crap it up yet. I'll try to keep the "no eating" rule as long as possible, but eventually I know I will have to let him have a snack while we are driving somewhere, and then it's all over. We traded in the last minivan without ever knowing what the hell that smell was, although I have a sneaking suspicion it was a combo of yogurt, urine, and the remnants of my first bottle of kombucha.
When I pull into the school parking lot, I see Ron waiting out front for me. I lock up the van and run to meet him. We are just on time.
As we walk down the hall to room 147, I try to remember who is scheduled to meet with Miss Ward before us. I know I considered putting Gordon/Burgess either before or after us just so I could get a glimpse of Suchafox, but the timing didn't work out.
When we arrive, Shirleen Cobb and her husband are coming out of the classroom, laughing, with Miss Ward right behind them. Everyone seems happy happy happy, which I take as a good sign. But when they see us, they stop laughing. Not a good sign for us.
Shirleen comes right over to me, as if we were in midconversation.
"Graydon cannot eat ice cream. It makes him terribly gassy even though he loves it, poor lamb. You need to think of a different party treat."
I glance sideways at Ron and smile. Welcome to my world!
"Shirleen, the ice cream party was not my idea. I'm just following orders. But I want you to know I thought of Graydon and am planning to bring Tofutti just for him."
Shirleen sizes me up and gives me a nod. "Good."
And off she stalks with her husband in tow.
"Always a pleasure," I murmur.
Ron leans into me. "You didn't introduce me."
"You're welcome."
Miss Ward beckons us into the classroom. She is wearing her hair pulled back into a tight bun, and her navy blue pantsuit is a flattering fit without being inappropriate. She is all business today.
"Hi, Jenny. And you must be Ron. Come on in and have a seat."
She motions us toward two children's chairs. Why do teachers do this? We are grownups with grown-up knees and grown-up butts. Would it be so hard to pull in a couple of adult-sized chairs for conference day?
As we navigate our way down into the chairs—holding on to each other for dear life—Miss Ward sternly consults a folder that says "Dixon, Max" in purple marker. When she looks up, she smiles.
"Well, what can I say? Max is a wonderful boy. He is kind and polite and really well liked by everyone in class."
Ron and I smile at each other. He takes my hand and squeezes it. My eyes get a bit teary.
"However"—Miss Ward consults her notes—"Max is the only child in the class who doesn't seem to enjoy P.E., although he loves recess. Any idea why that is?"
I look knowingly at Ron. He just shrugs.
"No idea," he says.
"Well, it's nothing to worry about. Just something I noticed." She smiles. "He is doing very well in math. Here is some of his recent work." She slides a few sheets of paper across the mini table and explains what the class has been learning. The papers have purple smiley faces at the top of them. I guess that's her equivalent of a grade. I'm tempted to ask what the kids who aren't doing well get at the top of their pages, but I decide I'd rather not know.
"Here is the book we're reading aloud right now." She hands over a book that I read when I was in first grade: The Dragons of Blueland, by Ruth Stiles Gannett.
"Hey, I remember this book," I say to no one in particular.
"Max is great when we are in listening mode, but when it comes to talking about the book, he never raises his hand. Any idea why that is?"
Ron looks up, surprised.
"Uh, okay. I mean, I read to him at night but..."
"Maybe stop after every page or so and ask him some questions or get his opinion on what you've just read."
I think about the books we are reading Max right now. Exactly what questions can you ask a child after reading Hop on Pop? Ron just nods in agreement—or defeat.
"I think Max just needs a confidence boost."
Confidence boost? He wears red pants, for God's sake. How much more confident can he be?
"That's pretty much all I have to say." Miss Ward stands. "Do you have any questions?"
Here's your hat, what's your hurry? I think, but do not say. Ron and I struggle to get out of the mini chairs.
"Uh, I'm sure I do, but I can't think of any right now," says Ron. I feel bad for him. He's not used to the crazy abruptness that is Miss Ward.
"Anything planned for the class that I should know about?" I ask.
"Jenny, yes! Thank you for reminding me. We are taking a class trip to the dump in two weeks and I'll need three parents to help chaperone."
"The dump?" I can't hide my incredulity.
"Well, it's really the Kansas City Recycling Center, but believe me when I tell you the kids get much more excited when I call it the dump."
I can't argue with her logic. I know Max loves going to the dump, but the real one. She is going to have a mutiny on the school bus when they pull up to a recycling plant.
"Okay. I'll send out an email and get some volunteers."
"I don't need to know the details, Jenny." Miss Ward ushers us to the door. As we are walking out, she inexplicably bursts into a peal of laughter. It actually makes both of us crack up, because it's so out of nowhere. Out in the hall, the Westmans look at us in surprise.
"Sounds like you guys had fun," Jackie says to me.
And that's when I remember the Cobbs walking out of the classroom before our meeting and laughing. Hmm... a little home-court intimidation. Crazy like a fox is our Miss Ward.
* * *
To: Parents
From: JDixon
Date: 11/15
Subject: Party time and trash talk
What is it, my birthday???
You guys are awesome! I got all the volunteers I needed without having to send a follow-up begging email. My class parents are growing up so quickly!
Kudos to Sasha Lewicki's out-of-office reply for once again taking the top response time of 22 seconds, but hot on her heels was Jill Kaplan at 1:47. The rest of you lollygagged a bit, but hey, you got there in the end. Here's the lineup:
chocolate sauce—Kaplans
caramel sauce—Zalises
sprinkles (chocolate AND rainbow!)—Elders
gummy bears—Gordon/Burgess
plain M&Ms—Alexanders
whipped cream—Browns (guess we just learned a little more about the Browns, huh?)
marshmallow topping—Fancys
spoons, bowls, and napkins—Aikenses
cups—Eastmans
And the Batons will bring wine.
Expect some seriously sugared-up kiddos when you pick them up on Wednesday.
On another note, Miss Ward has planned a class trip to the Kansas City recycling center for Tuesday, November 22 (FYI, she's telling them they're going to the dump). We will need 3 parents to help chaperone the trip. So if you've had all your shots, don't be shy! Get those fingers tapping and volunteer.
GO TYPE NOW!
Jennifer
* * *
I mentally start to guess who will be among the brave and crazy enough to sign up. Of course, Sasha Lewicki's out-of-office reply is right on top of things.
* * *
To: JDixon
From: Sasha Lewicki
Date: 11/15
Subject: Party time and trash talk
I am out of the office until November 30.
Thank you,
Sasha
To: JDixon
From: Peetsa Tucci
Date: 11/15
Subject: Party time and trash talk
I'll go if you will.
xo
P.
* * *
I groan, although Peetsa is the one thing that would make a class trip to the dump tolerable.
* * *
To: JDixon
From: Ravital Brown
Date: 11/15
Subject: Party time and trash talk
This is not a joke, right? I am never sure. If it's not a joke, I will go. I mean, I don't like garbage, but I think Zach would like me to come.
Thank you,
Ravital
To: JDixon
From: Don Burgess
Date: 11/15
Subject: Party time and trash talk
Hi, Jen,
Not sure if I told you, but I'm the manager at the recycling center, so I can act as a chaperone. You and Peetsa should come. Our plant is pretty impressive.
Cheers,
Don
* * *
When Don said he worked in waste management, I didn't even think of the recycling center. Well, that seals it. I email Peetsa to tell her she's the lucky winner of a day of refuse. And while I'm at it, I assure myself that I was totally going to go anyway, even before I got the email from Suchafox.
"We're home!" Vivs and Laura scream in unison.
"What?" I exclaim. I drop the turkey I'm washing in the sink and run to hug my girls.
"I wasn't expecting you until the end of the day." I take them into my arms and squeeze my hardest. I can't believe they are both taller than me.
"Well, Vivs decided to ditch her last two classes, so we left at, like, ten." Laura can't help sounding like a tattletale.
"I didn't ditch." Viv scowls. "I was going to be the only one there. The teachers will probably thank me."
I'm about to make a crack about our hard-earned money when I realize something. "Hey, where's Raj?"
"He's not coming."
"What? Why?"
"Because he's an asshole, Mom," Vivs snaps, and walks upstairs, her long brown hair swinging behind her.
"Oh, God, what now?" I ask Laura.
She shrugs. "I have no idea. She wouldn't talk about it in the car. And she drove like a crazy person, Mommy! We got here in an hour."
I hug her. "How are you, my sweet girl?"
"I'm good!" She brightens. "I still love where I'm living. My roommates are so fun."
"And your classes? Ever get a chance to make it to any?"
"Occasionally." She smirks at me.
Laura has blossomed into such a beauty. Her oversized eyes and lips made her look a bit like a bug when she was younger, but she has finally grown into her face and has figured out how to tame her blond curls. She looks nothing like me or her sister.
"Where's Maximilian Swell?" she asks.
I smile at the nickname. Much improved over "Maxipad," which is the first one she gave him.
"He and Ron are out getting cranberries. Hey, can you grab an apron and start pulling bread apart for the stuffing?"
"Sure. Just let me go to the bathroom." Laura heads upstairs to the room she shares with Vivs when they're home. Two minutes later, both girls come down and put on aprons. I love that they just do this without a lot of nagging on my part. It reminds me of how strict I was when I raised them. They had set chores every day and were never allowed to be sitting down if I was still working. I've really slacked off with Max. Vivs gives me shit about it all the time.
As they pull apart two loaves of white bread, I go back to the turkey I had unceremoniously dumped in the sink. Vivs is taking her bad mood out on the bread and ripping it apart with gusto.
"Want to talk about it?" I ask.
"Talk about what?" She looks up, annoyed.
"Your bad mood."
"I'm not in a bad mood, Mom, I'm just preoccupied."
Unfortunately, Laura decides to jump in.
"Oh, and I'm just lucky to be alive, because you were"—she makes quotes with her hands, which are full of bread—"'preoccupied' while you were driving."
"Oh, my God, you are such a baby." Vivs slams her hands down on the counter.
"Girls! What the hell? Is this how we're going to start Thanksgiving weekend? Laura, let it go about the driving. She got you here. And you." I look directly at Vivs. "Either tell us what's going on, or snap out of it."
Vivs looks down.
"I'm sorry. It's just Raj did something so stupid and I'm really pissed at him."
She pauses, as if trying to decide something.
"He asked me to marry him."
"What?" screams Laura. "Are you kidding me? That's what you're upset about? A hot guy wants to marry you?"
"Shut up. You have no idea what you're talking about," Vivs shoots back.
Laura throws her hands up in the air.
"Somebody please give me a problem like that," she says to the air around her.
"Mom!" Vivs whines.
"Okay, stop. Jeez, you guys. You've only been home for half an hour."
I grab a towel to dry my hands and wish like hell it was time for a glass of wine. It's three o'clock somewhere, isn't it?
"So he proposed and you said..."
"No," Vivs answers quickly. "I said no. Of course I said no."
"Why would you say no?" Laura asks without an ounce of guile.
"Uh, because I'm not white trash living in the backwoods of Kentucky," Vivs says.
I look at Laura, who is still confused. I put my arm around her.
"Don't you think twenty-one is a little young to get married?" I ask her.
She looks at me thoughtfully.
"Not if you're in love. I would."
We both wait to see if she's kidding. She is not. This is definitely something to revisit later.
I turn to Vivs. "So, is your age the only reason you said no? I mean, do you love him?"
Vivs sits down at the kitchen table and takes a deep breath.
"I do. I mean, I think I do. How do you know?"
"Oh, God." I snort. "Don't ask me. It took me years to figure it out."
"Great, Mom. Thanks. Good talk." Vivs's voice is dripping sarcasm.
I roll my eyes at her. These are the mom moments that I love and dread all at the same time. I want to say the right thing, give her the right advice, but I'm not a hundred percent sure what that right advice is.
"Well, I think you have to look beyond the dizzy infatuation you have right now and think about who you want to share the best and worst times of your life with. The passion will fade—it has to, or you'd never get anything done." Vivs smiles at this. "But if you end up with your best friend, then you've made the right decision."
Vivs raises her eyebrows. "So Ron's your best friend?"
"Well, Nina really is, but Ron's definitely a close second, or maybe third. The point is, you want someone you can stand being around forty years from now."
"But how do you know that?" Vivs screams, exasperated.
"You know that when you get to know yourself better!" I raise my voice in frustration.
At this Norman Rockwell moment of holiday joy, Ron and Max walk in the back door to the kitchen.
"Uh, hi?" Ron asks with more than a little trepidation.
"Sissy!" Max jumps ahead of his dad and into Vivs's lap, giving her a big hug.
"Buddy!" Vivs squeezes him back.
"Hey, what about me, Maxilla?" Laura walks over and scoops up her little brother.
"I got a helicopter and I was a ninja for Halloween," Max chirps.
"Tell me something I don't know, brotha!" Laura puts him in a fireman's hold and takes him into the living room.
"Hi, Ron!" she calls over her shoulder.
"Hey, Laurs," Ron answers, still standing in the doorway holding a grocery bag. He closes the door and carefully walks to the counter, like he's casing a minefield. He kisses my head and bends to hug Vivs. "Anything I should know about?"
"Just girl talk." I give him my sweet smile.
"Ron, how did you know that Mom was the one?" Vivs blurts out.
Okay, girl and boy talk.
Ron looks leery but to his credit decides just to answer the question.
"How did I know? Umm... Well, I didn't at first."
I look up, surprised.
"No offense, honey, but you were a lot to take on. No man in his right mind wants to live with a woman who gives him so much shit all the time."
Shit? I think to myself. He doesn't know what shit is! I'll give him... Oh.
Ron continues. "But after a while I realized I was much happier taking shit from all three of you than I ever was with anyone else. So I knew."
"But he was forty-three at the time," I needlessly remind everyone in the room. "And he'd already had a full life with crazy Cindy."
Vivs ignores my babbling and looks directly at Ron.
"What if you had met Mom when you were twenty-one?"
I can tell Ron needs some context at this point, so I chime in.
"Raj asked Vivs to marry him. She said no and now he's not coming for Thanksgiving. There. You're all caught up."
"So, you're asking would I have married your mom if I'd met her thirty years ago?"
"Yes," Vivs says.
"Probably not, but I feel like that would have been a huge mistake. I can't imagine my life with anyone else. Can you imagine your life with anyone else?"
"Frankly, yes I can," Vivs answers, a little too quickly.
"Well, then," says Ron, starting to unpack the grocery bag, "you have your answer."
Vivs goes back to tearing up bread and I thank God for the millionth time that I married the right man.
* * *
"What time are Nana and Poppy getting here?" Laura asks over breakfast on Thanksgiving morning. Ron has done his usual great job with bacon and eggs. He really doesn't cook at all, but he manages this one meal without too much mess.
"They're going to church this morning and then they'll be over," I say while chewing toast.
"They go to church a lot," Max observes.
"Okay, while I have you all here, this is how the day will play out." I go into drill sergeant mode. "Vivs, you have to make sure the turkey is stuffed and in the oven by noon."
"Check." Vivs gives me a salute.
"Ron, the potatoes and turnips have already been made, so all you need to do is put them in the microwave when I tell you to. Repeat, when I tell you to."
"Jeez, a guy heats things up late one time and gets branded for life," Ron gripes.
I ignore him and move on. "Laura, you are on gravy-and-special-peas duty. You know the routine."
"Check," says Laura. "Make gravy, hide gravy from Nana, let Nana make gravy, then swap out Nana's gravy for mine. Easy."
"You say that now, but I think Nana's on to us. Watch her carefully."
"What's wrong with Nana's gravy?" Max asks. "I love it."
"You've never had it," Laura assures him. "You've only ever had my gravy."
"Lucky," Ron says, and gives him a solemn nod.
Let me just say that my mom has a good heart and no taste buds. Everything she makes has way too much seasoning. The problem's getting worse as she gets older. I feel so sorry for my dad. Thank God, he has terrible sinus issues, so I don't think he notices too much.
"And don't forget to dig out the gravy boat from hell," I remind Laura. "It's in the linen closet behind the old towels."
"Jeez, do we have to keep calling it that?" Ron sounds defeated.
"What else would you call it?" I ask him. It was a wedding gift from Ron's ex. She sent us a gravy boat shaped like a turkey, where the neck is the handle and the gravy comes out the ass. I insist on using it at least twice a year.
He shakes his head. "I was thinking just 'gravy boat.' but whatever."
"Max," I continue, "you are in charge of collecting leaves for the table. I want to see lots of different colors, okay?"
"Okay!" he says, clearly feeling very important.
"Okay," I repeat. "I will set the table and make sure the pies get put in the oven once the turkey is out. Any questions?"
I get a lot of blank stares.
"Right, then. Let's get to it. No TV until your work is done."
"I don't want to watch TV," Max says.
"I was talking to Daddy."
As everyone scatters to do their chores, I clean up the breakfast mess and then sit down to check my email. Amid the usual Pottery Barn, Shopbop, and Amazon notices is a note from none other than Miss Ward.
* * *
To: JDixon
from: PWard
Date: 11/25
Subject: I'm feeling thankful!
Dear Jenny,
As I sit here in my apartment on this beautiful Thanksgiving Day with a bottle of wine and all four Twilight movies to enjoy, I just want to thank you for your hard work and friendship so far this year. I really think we make a great team! Having said that, I need you to be more on the ball with the field trips. We can't let what happened at the recycling center happen again. Agreed? Great.
Have a nice dinner!
Peggy
* * *
* * *
Ouch. I really can't believe she went there.
Peetsa and I chaperoned Tuesday's field trip and, as predicted by me, the kids had a mass meltdown when they figured out that they weren't going to the "real dump."
Thank goodness for Suchafox! He was waiting for us by the front door of the plant, and when he heard the sobbing from inside the bus he came onboard and took charge. Within five minutes, he managed to convince the kids how lucky they were to be here instead of the dump, citing the really bad smell and the giant rats. Luckily, the kids bought this hook, line, and sinker—especially when he said he would show them how to turn a plastic water bottle into a pair of jeans. Suddenly they couldn't wait to take the tour. Lulu looked so proud.
"P.E. laundry room," Peetsa mumbled to me as we walked the plant floor. I burst out laughing. Miss Ward gave us a stern look.
At the end of the tour, Don was as good as his word. We sat the kids down in a cafeteria-type room and showed them a video about plastic being made into fabric. Peetsa sat with her son and I noticed Miss Ward sneak out the side door with her cell phone as soon as the movie started. I settled myself in the back of the crowd and leaned back in my chair. That's when it happened.
Don came and sat beside me.
"Remember being so happy when we got to watch a movie in class?" he whispered.
Calm down, Jen.
"Yeah," I responded with my usual wit.
He rested his arms on his knees and leaned in so close I could see the light hairs on his ears.
"So..." He smiled.
"So..." I smiled back, trying to remember the last time I sat this near to a man who wasn't my husband.
"How do you think it's going?" he whispered.
"I think it's going great." I wondered whether we were talking about the same thing. I mean, I was sitting within inches of my high school crush. How much greater could it be?
"Do you think the kids liked the plant? Lulu was really nervous about everyone getting bored."
"No, they loved it. I loved it." I hoped my smile reassured him.
"Mom!" Max was standing in front of me.
"What?" I jumped away from Don. The movie was still playing and the kids were quietly watching. I focused on my son. "What's up?"
"Graydon's not here."
Don and I looked at each other and sprang up. Thirty minutes of panic ensued while we put out an APB on Graydon, who had taken it upon himself to look for the bathroom shortly after the movie started. He had left the recycling center and gone to the main building, where the offices were, after judging it the "safest bathroom option." Luckily, he didn't run into any peanut butter or airborne dust on his journey, or Shirleen would have had my hide. As it is, I think he got a rash from the soap in the bathroom.
I apologized to Miss Ward profusely and was classy enough not to point out that she was nowhere to be found for the first twenty minutes of Graydon's disappearance.
And now she throws it at me, on Thanksgiving? I think about a retaliatory email—something biting about drinking wine at ten o'clock in the morning and being a Twi-hard, but in the spirit of the day, I decide to take the high road. Instead, I text Don Happy Thanksgiving and thank him again for all his help with the kids on the field trip. And I am just a little bit thrilled when he texts me right back, Any time!
* * *
"Ah, that was a fine meal, Mother." Ron pushes his chair back from the Thanksgiving table and pats his nonexistent belly. Sometimes he likes to pretend we're an old couple from the 1950s. It's one of those things that was cute the first five or six times but has since worn thin. I don't have the heart to tell him he needs some new material.
I have to say, dinner was delicious. Thanksgiving is one of those idiot-proof meals. Just make sure the turkey isn't overdone, and you're halfway there.
"You really outdid yourself, girlfriend," Nina calls from across the table. "Those peas were unbelievable."
Nina and Chyna always join us for holiday dinners, because we are the closest thing to family that they have in KC. Nina's parents both died in a horrible boating accident on vacation in the Bahamas. Nina was only eight years old and saw the whole thing from the beach. She went to live with her grandmother in Topeka after that. She doesn't talk about it much, but she did tell me that when her grandmother died, a lawyer handed her a check for $326,342 as some sort of compensation from the mega-resort where her parents were killed. Apparently, her grandmother had been holding on to it so Nina wouldn't "spend it on clothes or drugs," as the will put it. She was twenty-five and completely alone in the world. It was the perfect time for the dashing Sid to sweep her, and her money, off her feet.
"The secret is nutmeg," I tell her.
"And about a pound of butter," my mother chimes in. "Sweetheart, I swear you put in more and more every year. Your grandmother would be horrified."
"But not because of the taste," I counter. "Granny would be shocked because butter is expensive."
"That's true," my mom agrees. "My goodness, that woman was tight with a dollar. She'd make my father drive twenty miles away to save ten cents on paper towels."
Nina smiles. I can tell she enjoys the family banter, since she never really had any of her own.
"How is Garth doing?" my mother asks as she starts stacking all the dishes she can reach without getting up. Ron and my dad have wandered back to the television, and the girls and Max have gone to the basement to play Xbox.
"He's good. I really like him. He's such a nice guy."
"What's he doing today?"
I pause for a moment. "He didn't say."
"Well, you should have invited him for dinner today," my mom admonishes me.
"Mom, I barely know him! He barely knows me. It would have been weird to just randomly invite him to Thanksgiving."
My mom shakes her head. "I thought I taught you better than that, Jennifer Rose."
"You never taught me to invite strangers to dinner."
"He's not a stranger and he doesn't have any family in town."
"How do you know?"
"Because he told me. How much do you know about him?"
"Not much." I shrug. "We don't really talk."
"Well, maybe you should." My mother pushes her chair back and yells at a surprisingly loud volume: "Okay! Everyone is on cleanup crew except Jen and Nina." There are a lot of groans from the living room and basement.
"Come on. Many hands make light work."
Everyone drags themselves into the dining room.
"Do what Nana says or she'll make you say the rosary afterward," I warn everyone. Suddenly the pace picks up noticeably.
"Oh, we'll be saying the rosary anyway," my mother assures me. That elicits even more groans.
Nina and I grab the rest of the wine and our glasses and head into the living room.
"Your mama just gave you a spanking, my friend." Nina smirks.
"I know, right? What's her deal with Garth?"
Nina shrugs. "So how is my class mom doing?"
"Oh, you know, living the dream." I start to tell her about the drama surrounding scheduling the parent/teacher conferences.
"I don't know if Kim Fancy ever got that stupid time slot. I'm hoping Sasha Lewicki stuck with it."
"Who?" Nina asks from behind her wineglass.
"Sasha Lewicki. Her daughter is Nadine?"
"Never heard of her."
"Oh, my God. Does that mean I actually know someone you don't?" I'm sort of kidding, but not really. Nina makes it a point to know everyone because, as she says, "You never know where you'll find a good lead." As a freelance graphic designer, she is always looking for new clients. How she does it without pissing people off is beyond me.
"They must be new." She lowers her voice. "Still crushing on the old flame?"
"Trying not to. But it's kind of fun, you know?"
"Yes, ma'am, I do know." She laughs.
"Really?"
She nods. "There is a dad in Chyna's class who would be shocked to know the things I think about him."
"Stop it!" I never would have guessed.
She nods. "But he never looks twice at me. You need to be careful. Old flames like to reignite."
"He's not an old flame. I had a crush on him. It wasn't mutual."
"Uh-huh." She looks down at her belly and groans. "Why do I always overeat when I come here? Look at this shit!" She shows me a handful of stomach.
"Wanna try Garth? He comes to your house."
"Maybe in the new year. I've got another month of overeating to get through first."
* * *
To: Parents
From: JDixon
Date: 12/5
Subject: Miss Ward's class is having a bash!
To Be Sung to the Tune of "Santa Claus Is Coming to Town"
You better watch out. You better not cry.
You better not pout. I'm telling you why.
Miss Ward's class is having a bash!
I'm making a list of what we will eat.
10:30 in the morning is when we'll all meet.
Miss Ward's class is having a bash.
We'll need some bagels and cream cheese
Some fruit and doughnuts, too.
Some water, juice, tea, coffee,
And some bottles of Yoo-hoo hoo!
So, volunteer soon and don't you be late!
You don't want to make the list of people I hate.
Miss Ward's class is having a bash!
December 22, people. It is in the classroom right after the concert.
The lines are now open, so run, don't walk to your nearest computer and sign up to bring something.
Cheerio!
Jen
To: JDixon
From: SLewicki
Date: 12/5
Subject: Miss Ward's class is having a bash!
Hi,
I will be out of the office until December 8.
Thank you,
Sasha
To: JDixon
From: PTucci
Date: 12/5
Subject: Miss Ward's class is having a bash!
Hi, Jen,
We will bring bagels.
I'm surprised Miss Ward is letting you have a Christmas party. I thought she doesn't celebrate "Hallmark" holidays?
xo
P
To: PTucci
From: JDixon
Date: 12/5
Subject: Miss Ward's class is having a bash!
Hey, P,
Well, you would be correct about that. But if you read my email carefully, there is no mention of this being a Christmas party. It's just a party that takes place after the holiday concert. This was a big negotiating point with Miss Ward, believe me!
Are we still on for girls' night Wednesday?
xo
Jen
To: JDixon
From: SCobb
Date: 12/5
Subject: Miss Ward's class is having a bash!
Jen,
Not that you asked, but I will bring soy butter for Graydon's gluten-free bagel.
Shirleen
To: JDixon
From: DBurgess
Date 12/5
Subject: Miss Ward's class is having a bash!
Hey, Jen,
I'll cover doughnuts for you.
Cheers,
Don
To: JDixon
From: JWestman
Date: 12/5
Subject: Miss Ward's class is having a bash!
I'll bring cups.
Thanks,
Jackie
To: JDixon
From: KFancy
Date: 12/5
Subject: Miss Ward's class is having a bash
Hi, Jen,
Silly me, I thought this email was going to be about the parent cocktail party!
We will be happy to bring coffee to the party. Cream and sugar too, I'm guessing?
While I have you, let's look at December 17 as a good night for the adult party. I know it is only 12 days away, but I'm happy to host it. I just need you to send the invite.
It will be a great way for all of us to launch into the holiday season, don't you think?
Kim
* * *
"Suffering cats, that woman is persistent!"
"Who?" asks Ron. We are in bed doing what every red-blooded American couple does at night—watching TV and checking our email.
"Kim Fancy. She's been bugging me to have a class cocktail party since curriculum night."
Ron shrugs. "So?"
"So? What do you mean, 'so'?"
"So what's the big deal? It might be nice to hang out with other parents."
"We do. We hang out with Peetsa and Nina."
"No, you hang out with them." Ron mutes the TV. "Is it too much work for you?"
"No. She's going to host it. But she wants me to send out an email inviting people."
"And that's a problem because..."
I really hate when Ron gets like this. He's supposed to be on my side till death do us part. But sometimes he goes all logical on me instead.
"Because... I don't want to! It's her party. If it's a class event, then I feel like it should happen when I'm ready. Not because she wants it." As I'm saying this, I know it sounds ridiculous.
Ron just looks at me, then unmutes the TV.
As I watch Law & Order: SVU and brood, my phone buzzes. To my delight/surprise, it's a text from Don.
Hey there. I'm bringing doughnuts to the Christmas party. Is it okay if they have sprinkles? I'm not too familiar with food allergies and that Graydon's mom seems pretty strict.
I laugh.
"What?" Ron asks.
"Just one of my class parents asking a question."
He turns back to the TV, and I respond to Suchafox.
Neither am I, but I'm relatively sure that sugar is okay.
I hit Send, then start another message.
Hey, if there was a parent cocktail party before Christmas, would you and Ali go?
He replies immediately.
Can't speak for Ali, but I'd be there.
"Huh."
"What?" Ron asks again.
"Nothing," I mumble and start to type with purpose.
* * *
To: Parents
From: JDixon
Date: 12/5
Subject: And one for the grown-ups
Hi again,
Why should the kids have all the fun? Kim and David Fancy would like to invite the parents of Miss Ward's class to a cocktail party at their home on December 17 from 7 to 10 p.m. All you need to bring is your good cheer!
Address: 9314 West 146th Place in Overland Park. You can RSVP directly to the Fancys.
Hope everyone can make it!
Jen
To: JDixon
From: SLewicki
Date: 12/5
Subject: And one for the grown-ups
Hi,
I will be out of the office until December 8.
Thank you,
Sasha
* * *
* * *
I have never been a big fan of organized "girls' nights." For years, every night was girls' night with Vivs and Laura. I was so busy with work and my daughters that I didn't have time for friends. Then Nina came along and she made it seem like a really good idea to go out and get drunk once in a while. So we have made it a bit of a tradition. Once a month or so, we go for margaritas at Luna Azteca. Usually it's just the two of us, but tonight I have invited Peetsa to join Nina and me. I figured it's a safe match because they already know each other.
I pull up to Peetsa's house in Ron's old blue Camry. I left the minivan for him so he can pick up Max and his friends from Scouts.
I don't even have to honk my horn. As soon as I pull in, she bolts out her front door and comes loping down the driveway. She practically rips the car door off getting in. I'm guessing she's trying to get away from the bitter cold night.
"Drive, drive, drive!" she gasps.
I throw it in reverse and do my best Mario Andretti out of there.
"What the hell?" I ask.
"Zach was in the bathroom. I had to get out before he figured out I was leaving."
"Won't he freak out when he sees you're gone?"
"No. It's the leaving that makes him crazy. Once I'm gone, it doesn't matter."
"Laura used to freak out whenever I left. Preschool was the worst."
"Max doesn't give you a hassle?" she asks, with more than a little envy in her voice.
I shake my head.
"As long as he can watch a little TV, he doesn't care who is watching him."
"Honest to God, I don't know what people did before TV. And I really don't get those anti-TV parents. I mean, good for you if you can amuse your kids and make dinner and fold laundry all at the same time. But I don't choose to make my life that hard, know what I mean?"
I laugh at her. I've never seen Peetsa so whipped up about something.
"What?" she asks.
"I've just never really thought about it." I shrug. "Who's so anti-TV?"
"Oh! Didn't I tell you? Miss Ward is. She emailed me the other day to tell me she thinks Zach watches too much TV."
"How the hell would she know?" The traffic light in front of us changes to red and I have to brake a little too hard. We both lean forward and then snap back into our seats.
"Apparently every morning at circle time she asks the kids to tell her what they did the night before. I guess Zach always says he watched TV. I mean, don't they all watch TV? Does Max?"
I don't know what to say. I'm definitely not anti-TV, but we do limit it to thirty minutes a night, mainly because Ron wants to spend time with Max when he gets home from the store. And when you factor in dinner, bath, and reading there isn't much time left. That's why more than half an hour is such a treat for him. But I don't say any of this. Instead I say:
"Of course! He loves TV." Which isn't a lie at all.
We pull up to Luna Azteca. It's freezing out so we hurry into the restaurant.
I see that the owner, Mr. Barrera, has gone with a tropical Santa motif for his Christmas decorations this year. Nothing says ho-ho-ho like Santa in red flowered shorts hanging ten on a light-up surfboard.
Nina is already sitting at a table and waves us over.
"There she is," I say to Peetsa, and we head to the table.
"Hey there!" I give Nina a hug.
"You guys know each other, right?"
They smile at each other.
"Yup." Nina nods. "We go way back."
And just like that, I feel uncomfortable. God, am I so petty and jealous that I can't have two people I adore have a history without me? What am I, in seventh grade? While I'm chewing on this, Nina says, "I started without you," motioning to the empty margarita glasses in front of her.
"Well then, we'd better catch up." Peetsa winks at me.
Just then the waiter appears with another drink for Nina.
"Two more for my friends, Jonathan," she says to the waiter, with a bit of a slur. "And keep 'em coming."
Hmm. Nina, drunk. Not like I haven't seen it before, but generally not before eight p.m.
"Hey, slow down there." I'm half joking and actually a little embarrassed. I don't want Peetsa to think my best friend is a lush.
"Let's get some guacamole, too, please," I yell to the waiter's back.
Nina definitely needs to get some food in her.
"So, what's up, girls? Anything new?" Nina asks, chewing her ice.
"Well, Max came home from school and told me Miss Ward has a great pair of legs. I'm trying to figure out which—"
"I found Sid," Nina blurts out.
"What?" I say, a little too loudly.
"Who's Sid?" Peetsa asks me.
"I didn't know you were still looking for him. What the hell, Neens?"
Nina avoids my glare. "I really wasn't, but I have a permanent Google search on his name, and this morning I got an alert."
"Who's Sid?" Peetsa asks again.
"You can do that?" I ask, impressed.
"Yes." Nina sounds annoyed. "Everyone can do that. He posted a picture on Facebook and it came up in the search." The ice chewing is getting more intense.
Peetsa slaps both hands down on the table.
"Who is Sid?" she asks for the third and sounds like final time.
"Sorry, P. Sid is Chyna's father."
Peetsa looks at Nina. "Your ex?"
"We were never married, but yeah, my ex. He took off just before Chyna was born." Nina throws back the rest of her margarita in one swallow.
At this magic moment, Jonathan returns.
"Here are your drinks, ladies!" he practically sings. "Guacamole is on the way. Do you want to hear the specials?" His smile fades as he reads the mood of the table. He wisely backs away. "Just call me when you're ready to order."
"He's fucking married," Nina spits out. "He's married. With kids." She puts her hands over her face.
"Oh, my God. Where is he living?" I have so many questions, but that's the first one that comes to my head.
"San Jose."
"California?"
"No, Pennsylvania."
Peetsa jumps in before I can make a snide remark back.
"Wait a sec. How long have you been looking for him? Twelve years?"
Nina and I both nod.
"Why wouldn't his Facebook page or something else have popped up before now?"
"He didn't have one," Nina answers between ice crunchings.
Jonathan our waiter slithers in and puts down the guacamole and a basket of tortilla chips. He raises his eyebrow to me, as if to say, "Order now?" I shake him off.
"He just got a Facebook page? A little late to the party, isn't he?" Peetsa looks incredulous.
"He was never the sharpest tool in the shed." I scoop up some guacamole and pop it in my mouth.
Nina gives me a sad look. "He really wasn't," she agrees. Her eyes start to water, but she lets out a laugh instead. "But, God, was he hot." She wipes her cheek.
"Well, I would hope so," says Peetsa. "I'd hate to think you fell for a guy who is both stupid and ugly."
This makes us all crack up.
"So, what's on his page?" I ask when we settle down.
"All he has right now is a picture of a three-year-old sitting beside a newborn baby and a lot of messages that say, "'Welcome to Facebook!'"
"How do you know he's married?" asks Peetsa.
"It's in his status. It also says he works for some tech company." She turns to me in disbelief. "He's got a job! In high tech!" Her eyes well up again.
"What is he, the janitor?" I mumble. Nina starts to cry.
I'm not sure how long I'm going to be able to fake my sympathy. Sid is such a pant load, and seeing someone as dynamic as Nina fall to pieces over him makes me crazy.
"Hey, let's get Jonathan over here," I suggest. "If we're going to drown our sorrows, I need some food."
I wave to Jonathan, and Nina blows her nose with her napkin.
By the end of the night, Peetsa and Nina are three sheets to the wind; I have designated myself the driver since I nursed one margarita the whole evening.
After leaving Jonathan a very nice tip (I tend to tip well because I was a waitress in college and it's a pretty crappy job), we head out to the car. The freezing cold is like a slap in all our faces and I think helps to sober the girls up a little. I drop Nina first and watch from the car as she wobbles to her front door and figures out how to use her key.
"Wow," says Peetsa as we watch her stumble in. "Is she going to be all right?"
"I hope so. I don't envy the hangover she'll have tomorrow, but maybe she'll be a little less emotional about Sid."
"I've never seen this side of her." P. shakes her head.
"Everyone has their kryptonite." I shrug. "Nina's is Sid."
I back the car out of the driveway and head for Peetsa's house.
"What's yours?" she asks me.
"My kryptonite? Rock stars."
She laughs. "I would not have guessed that about you."
"Well, my friend, let me tell you a little story about Vivs's father."
Garth has me doing jumping jacks downstairs in Ron's Gym and Tan.
I've noticed that, since having Max, I don't do them with the same abandon I used to. Things just aren't as tight down there, if you know what I mean.
When I asked him to step up my cardio, I had visions of jogging on a treadmill. But this works, too, I guess. I definitely don't huff and puff when I run up the stairs anymore.
"... ninety-eight, ninety-nine, one hundred. Good job, Jen! Nice form. Get some water."
"I have to cut it short today," I tell him between gasps and gulps of water. "We have a party to go to tonight, and I'm actually going to get a manicure."
"Oh, fun! I love holiday parties."
I smile at Garth. He's always so enthusiastic. When I tell him I need to get my car washed, he'll say, "Isn't a clean car just the best?" I wish I could rub some of his positive energy off on Nina. She has gone to the dark place since our night out. When I can get her on the phone, all she does is moan. I went to her house yesterday to bring her Chick-fil-A (her favorite) and she was a mess. She even asked me to take Chyna for the weekend, which I gladly did under the pretense that I needed a babysitter for both Friday and Saturday nights. It's half true: I'm going to leave Max with Chyna when we go to the Fancys' shindig tonight. I didn't really need her last night, but Ron and I took the opportunity to go see an early movie. It had real people and sex and swearing. I had forgotten movies like that existed.
I'm roused from my thoughts by Garth snapping his fingers in front of my face.
"Hello? Where did you go? I said get some water, not some shut-eye!"
"Sorry. Just thinking about tonight."
"Well, give me twenty push-ups and tell me all about it."
I drop to my knees on the mat and get into position.
"Oh, I think we can skip the girly push-ups."
I give him a frantic look.
"Come on. Let's try some real push-ups. Give me as many as you can."
I put myself into plank position and ease into it.
"One, two..."
"I'll count. You just work. So, where's the party?"
"At the home of some parents in Max's class."
Garth nods. "I'm guessing they're not friends."
"Why do you say that?"
"Well, if they were your friends you'd say the party is at a friend's house."
I don't know why I think this is so funny, but I can't stop laughing. Clearly, the endorphins are getting me high.
"Well, I hope you're wearing something sleeveless, giddy girl. Your arms are looking great."
"You know what? I think I will. What about you? Got a hot date tonight?"
Garth smiles. "I wish! I think my dating days are over."
I think about my mother's reprimand at Thanksgiving.
"Well, if you don't have any plans for Christmas, we'd love to have you for dinner. My parents will be here, too."
"Oh, that's so kind of you. Can I let you know?"
"Of course." I'm a little surprised that I didn't get a yes immediately, but whatever.
Garth extends his hand to me. "Okay, that's it for today. Have a great time tonight, hot mama."
"From your lips," I mutter.
* * *
"So, what is this couple's name, again?" Ron asks as we are driving to the Fancys' neighborhood.
"Kim and David. Daughter is Nancy."
"And what does he do for a living?"
"No idea," I reply. "They're from New York, so something in finance maybe? Wow. Is that their house?" I point to the one at the end of the street. Sure enough, it matches the address I have written on a piece of paper in front of me.
"I'm guessing finance," Ron says.
The Fancy house is impressive. It takes up what looks like two standard-sized lots and has two driveways. That's a lot even for Overland Park.
I look down at what I'm wearing. I have abandoned the mom uniform for tonight and gone with a black sleeveless silk top, black skirt, and some ridiculous black heels that Laura made me buy when we were Black Friday shopping. I thought all black would help me blend in, but now I'm wondering if I should have worn my wedding dress. It's the fanciest thing I own.
"I've always wanted to see the inside of this house," Ron muses as we park on the street.
"You've seen it before?" I'm surprised.
"Cindy's parents live a block away. I used to take a lot of walks in the neighborhood to escape the insanity. I remember when they were just building this place."
"Who owned it back then?" I ask.
"Some AT&T bigwig." Ron shrugs.
"Oh, God," I sigh. "We should have brought a nicer bottle of wine."
* * *
By the time we get up to the door, we have gained two more couples—Kim and Carol Alexander and the hunky dad with his loaded wife from curriculum night, Jean-Luc and Mary Jo Baton. I have no proof yet that he married her to get his green card, but I hope to by the end of the evening.
We all nod and smile hello; Kim rings the doorbell, which chimes out "We Three Kings."
A twenty-something man opens the door wearing a crisp white shirt with a black bow tie and black pants.
"Merry Christmas. Welcome. May I take your coats?" he asks without a trace of a smile.
We walk into a foyer that is almost as big as my living room. The ceilings have to be sixteen feet high. There are tasteful Christmas decorations decking the halls. You know the kind: white lights and tree boughs. No blow-up Santa for the Fancys. I feel like I've walked onto the set of a very classy holiday movie. Or a Pottery Barn photo shoot.
We unload our outer things on the unsmiling doorman and I look for a place to put the wine I have brought. I see a table laden with colorfully wrapped wine bottles, so I add ours to the pack with a sigh.
As we walk from the foyer into the living room, another guy in a bow tie offers us champagne and says, "Be sure to stop by the library to see the jewelry."
Ron grabs two glasses and hands me one.
"What jewelry?" he asks me.
"No idea," I say, and swallow my drink in one gulp.
"Slow down, slugger. It's a long night," Ron warns me with a smile.
"Champagne has no effect on me," I assure him. "I could drink twenty of these."
"I know for a fact that's not true." Ron hands me his champagne. "But I'm going to enable you just this once, because I know you don't really want to be here."
We walk into a gorgeously spread-out living room with a roaring fireplace at one end and lots of comfortable seating. In the middle of the room is a large wooden table overflowing with cheeses and breads and olives and vegetables, all placed around a beautiful wreath. Again, very catalogue-like.
The living room is crowded with people I know and some I don't, milling around taking in the general splendor of their surroundings. Yet another bow tie walks up to us with a large platter of shrimp.
"Shrimp? Have you visited the jewelry in the library?" He smiles and walks away.
"They must have some jewelry collection," Ron notes. "I'll bet they have security in there, too. Want to take a look?"
I nod, still chewing my shrimp.
As we wander toward what might or might not be the library, I scan the room for Peetsa and Buddy. Suddenly, I lock eyes with Dr. Evil. She looks incredibly skinny in a tight black dress with long sleeves and a high neck. Hanging from her ears are gorgeous drop earrings made of black pearls and diamonds. She is walking over to us with her husband, the dashing David, who looks every inch the master of the house in a burgundy velvet jacket and black pants.
"Jen. How great that you came. Thank you so much for organizing this party."
Is she kidding?
"Kim, this is my husband, Ron. Ron, Kim and David are our hosts."
The men shake hands and then David leans in to kiss me on the cheek. I'm not expecting it so it turns into an awkward kiss on the mouth.
"Oh, hi, um... We were just admiring your beautiful home."
"Amazing what a million dollars will get you out here in the sticks, isn't it?" She smiles with everything but her eyes.
David looks like Max does when being forced to say thank you. "Have you seen the jewelry yet?"
"No. We keep hearing about it, but we haven't made our way to the library yet." David nods and walks away. His attention has been drawn just over my shoulder. So has Dr. Evil's.
I turn to see what is so interesting and am just in time to take in the impressive sight of Miss Ward showing off her curves in a slinky red dress.
Wow. This is by far the hottest outfit I have ever seen her wear. The dress is knee length and tight, with triangle sections cut out around her waist and a V going down the back. Cindy Crawford in her heyday wouldn't have been able to pull this off. Her blond hair is long and loose around her shoulders.
"Jen, did you invite Peggy?" Dr. Evil asks me.
"No. I just sent the email to the class. Maybe one of the moms told her about it."
"Or one of the dads." Ron grins widely in Miss Ward's general direction.
Dr. Evil turns a polite death glare on him.
"Did you do it, Ron?"
Ron looks at me, and I can tell he's a little scared. His mouth says no, but only a weird squeak comes out. I feel bad for him, but I'm also secretly glad he is getting the full Fancy. I know he thinks I'm just exaggerating half the time, and of course I am, but it doesn't mean there isn't some truth in there. Just because they think you're crazy, it doesn't mean you're nuts. Put that on my gravestone.
I grab Ron's hand and steer him away from our hostess.
"Why would she care if Miss Ward is here?" Ron whispers.
"No idea." I finish another glass of champagne and look for the refill bow tie.
We mill through the living room for the next hour or so, talking to parents as we go. Asami Chang is all smiles for a change. Maybe she's enjoying the champagne, too. Ravital Brown runs up to me with her husband in tow.
"Jen, I want my husband to meet you. Rob, this is the woman who writes the crazy emails."
Rob shakes my hand.
"Wow, you really had my wife confused that first day. I had to talk her out of calling the principal." He smiles.
"But now I get it!" Ravital says triumphantly. "I know it's all a joke."
"Well, not all of it, I hope!" I laugh along with them.
I introduce them to Ron and they immediately start a conversation about their respective jobs. I tune them out and look around.
It's an impressive turnout. The Westmans are here, and I can't help but wonder if Jackie brought cups. I see the Elders, the Wolffes, and the Kaplans as I scan the room. I am curious whether Sasha Lewicki is here, but then, I wouldn't know even if she were standing right in front of me, since I have never seen her.
Everyone really cleans up nicely, especially Suchafox, whom I spy across the room. He has traded in his jeans for a black suit and tie. I'm surprised when I see him making his way over to me and giving me the old once-over. I feel myself blushing from head to toe.
"You look so pretty," he says, and kisses my cheek. Immediately I feel like I'm cheating on Ron. I don't want him to think anything is going on, but why would he? David Fancy kissed me on the lips, and my husband didn't even blink. My silly crush is giving me a guilty conscience.
"Hey, there. Where's Ali?" I ask, as though she's the one I really want to see.
He shrugs. "I guess she didn't come."
I grab my sixth glass of champagne from a roving bow tie and finally ask Don the question that has been burning in my mind since curriculum night.
"Are you guys not together?"
"Here you are!" Ron comes to stand beside me, with what can only be described as the worst timing ever.
"Hey there. I'm Ron Dixon." He sticks his hand out to shake Don's.
"Don Burgess. Nice to meet you." Killer smile.
"Oh, right, the high school crush!" Ron says, and my mind flashes to all the ways I'm going to murder him later. What would possess him to say that out loud?
"Really?" Don raises his eyebrows at me. "I had no idea."
"One of many." I assure him. I'm guessing I'm as red as Miss Ward's dress by now.
"Did you tell him about the P.E. laundry room?" Don jokes.
"What do you think?" I ask; then we both crack up. Not that anything hilarious happened in there, but I guess the champagne is kicking in.
"What happened in the P.E. laundry room?" Ron isn't laughing.
"Long story, sweetie. I'll tell you later."
"So, is your wife here?" Ron asks.
"We're not married, but no, Lulu's mom isn't here."
Ron nods. "I've been there. Divorce sucks."
"We were never married," Don offers up.
"Oh, sorry, man. I shouldn't have assumed."
"No worries. It's all good."
Wow. Thanks to my husband, I know more than I ever would have had the guts to find out on my own. Ron makes a great wingman.
While we're chatting, I start to notice people walking around with small gold shopping bags. Ooh! I love me a gift bag. I'm guessing whatever is in there is more expensive than the wine we brought. I see Peetsa with one.
"What'd you get?" I ask, with only a bit of a slur.
"A necklace." She sighs. "I think I'll give it to my mother. Did you buy something?"
"Buy something?" I ask.
"Some jewelry. They're selling it in the library."
"Who is 'they'?" I ask, making a beeline for the next room. Peetsa and Ron trail after me.
As I walk into the library, I can't believe my eyes. There are two large tables of jewelry—earrings, necklaces, bracelets, and rings all nicely displayed. JJ Aikens is behind the tables with another woman I don't know, and they are working diligently to keep the sales going.
I try to put the pieces together using my inebriated brain. Kim Fancy pushes me to have a parents' cocktail party. She is eager to host it, but doesn't want to send out the invites. I send out the invites, which makes it look like I asked her to host it. We all get here, and she's pushing jewelry? I did not see this coming.
I walk up and stand beside Shirleen Cobb, who's trying on earrings.
"Those look pretty," I offer.
"Well, they should be for the price. Good lord." She takes them off and looks me up and down. "I thought this was a Christmas party, not a shakedown."
As she stomps away, I see JJ out of the corner of my eye.
"Hey, JJ. Where did all this stuff come from?"
"Oh, hi, Jen. Didn't think you were coming. This stuff is jewelry designed by Kim's friend Delia from Manhattan."
Someone hands me another glass of champagne. It's Ron.
"Say something nice," he whispers in my ear. I ignore him.
"How much are these earrings?" I hold up the pair Shirleen was so disgusted by. They are gold disk clip-ons.
"Those are two hundred and seventy-five dollars."
"Are they real gold?"
"Gold plated." She smiles. "Aren't they gorgeous? You would look great in those." She looks at Ron. "Are you Jen's husband? You should totally get her those for Christmas."
"Umm" is all Ron can think of to say.
"Really. You should. And you can even say you got them in Manhattan, because technically that's where they're from." JJ is working hard for what I'm sure is some sort of commission.
Just then we hear a loud crash from the other room. I hand JJ the earrings, and Ron and I hurry toward the noise. We are greeted by the sight of Nancy Fancy in her pajamas sitting on the floor with the Christmas wreath centerpiece and some cheese and bread around her. Kim Fancy glides over from the other side of the room.
"What happened?"
"I'm sorry, Mommy. I just wanted a piece of cheese."
Kim looks up at her guests with an embarrassed smile.
"Sorry, everyone. It seems Nancy still needs to learn how to ask for a snack."
There is some light laughter and then, out of nowhere, Miss Ward comes sprinting across the room like a streak of red paint.
"Oh, my goodness, sweetie, are you okay?" She kneels down beside the very embarrassed little girl.
"I'm okay," Nancy says in a tiny voice.
"Well, let me check you all over." Miss Ward starts to tickle Nancy, who starts to giggle.
"Thanks, Miss Ward," she says, and then gives her a hug.
"Awww," sings the Greek chorus of parents standing around.
Kim steps in, gives Miss Ward a strained smile, and helps Nancy get up. She whispers something in the little girl's ear that makes her nod and scoot out of the room.
"And that's our show for tonight," Kim says with a smile that once again doesn't reach her eyes.
I turn to Ron, who's staring at Miss Ward with a frown.
"What?"
"I might be crazy, but wasn't the V in the back of her dress?"
I look at Miss Ward, who is just getting herself off the floor and showing some impressive cleavage.
"Why, I believe you are correct, Mr. Observant."
Peetsa and Buddy join us as people get back to their conversations.
"Wasn't the V down the back of her dress earlier?" Buddy asks Ron.
"I said the same thing." Ron laughs, and they high-five.
"Really, guys?" I roll my eyes. "Buddy, close your eyes. What color are my pants?"
"Uh, black?" He blushes.
"I'm wearing a skirt."
"You have to admit it's kind of weird," Peetsa says. "Why would she turn her dress around?"
"Why does she do anything?" I'm frustrated and a little off balance from the champagne.
"Maybe someone took it off for her," Buddy mumbles to Ron and they high-five again. I take this as a sign that it's time to go home. I really want to take my pantyhose off.
"I'm going to pee and then we should get out of here before I say something I'm going to regret."
I'm not on the sturdiest of legs as I walk toward the foyer powder room, which I had noticed on the way in. I open the door and find Don Burgess zipping up his pants.
"Whoops, sorry." As I turn to back out, I trip on my heels and fall sideways toward Don. He grabs me with both arms.
"Wow, you're really falling for me," he jokes.
He stands me back upright, but doesn't let go. Either he doesn't think I can stand up on my own or he's enjoying this mini embrace as much as I am. For a moment we stand face to face and that urge to kiss him rears its ugly head again. But this time I feel the pull from him as well. The energy between us is heady—or maybe that's the Polo. Oh, my God, is this really going to happen?
"Everything okay?" Ron is standing looking at what I can imagine is a very compromising situation.
"You bet," Don answers quickly. "Jen tripped and I was just helping her steady herself."
He gives my arms a squeeze and lets me go. Ron walks right between us and takes my arm.
"Thanks, man, I've got her."
Don gives us a thumbs-up and walks back toward the foyer.
"Are you okay?" Ron's raised eyebrow tells me he's not pleased.
"I still have to pee," I admit. At the same time I take off my heels and hand them to him.
"Here, hold these so I don't fall again."
He takes the shoes, points me in the direction of the bathroom, and closes the door. I hike up my skirt, sit down, and take the whiz of a lifetime.
"Holy shit holy shit holy shit," I whisper to myself. Did that really just happen? I start to giggle. Suddenly I'm seventeen all over again and thrilled that Suchafox just spoke to me. This is too much for my inebriated brain to process. I flush, turn to the sink, and splash cold water on my face. One look in the mirror tells me that my waterproof mascara is actually not. I have two lovely black tears streaking down my cheeks.
"Crap." I grab one of the fancy towels and do my best to clean up my face. I really need to go home.
Heading out to the foyer, I see Ron, Peetsa, Buddy, and Don chatting with their coats on. Ron is holding mine plus my shoes.
"Feel better?" he asks, helping me into my coat.
I don't say anything, but I smile and put my arm through his. We say our good nights and I manage to escape without making eye contact with Don.
* * *
I'm trying to find the words to describe how I feel. It's like someone has stuffed a wool sock in my mouth and pulled a rubber glove over my head. Ugh. Champagne hangovers are the worst. Why do I always forget that?
As I attempt to get up the courage to roll out of bed and relieve my bladder, the door bursts open and Ron and Max come charging in.
"Good afternoon, Mommy!" Ron sings at the top of his lungs.
Max jumps up to give me a crushing hug.
"Mom, you slept for so long! It's already lunchtime."
My stomach starts to turn at the thought of food. Ron hands me a giant cup of coffee.
"I put a little hair of the dog in there for you." He smiles and sits on the bed.
The thought of putting more alcohol into my system makes me want to throw up, but I give him a grateful smile.
"You put dog hair in Mom's coffee? Where did you get it from? Wait, are we getting a dog?" Max screams the last part, and my head nearly splits in two. Oh, my God, is this going to be a long day.
"No dog, sorry, champ. Let's let Mom get dressed and we can go pick up lunch, okay?" Ron starts to pull him off the bed.
"Dad says you're going to want Burger King for lunch, but I said no way because you always say we don't eat garbage."
"Well, today it looks like we will." I give Max the most enthusiastic smile I can muster.
There's nothing like a greasy burger when you have a hangover. That's pretty much all I took away from my four years at KU.
It isn't until I'm sitting on the toilet that the events of last night come crashing back to me. I hold my head in my hands and replay the scene outside the powder room. Boy, I am not the girl I used to be. The old Jen would have grabbed that man and kissed the shit out of him. But the girl who slept her way around Europe wasn't married, nor was she in love. And I really do love my husband. I don't want to be attracted to someone else. And yet, here I am.
I go back to my bed and call Nina. No answer, as usual. She's still in the dark place. I pause a moment, then dial Peetsa, but hang up before she answers. I've got to talk to someone about this, but I'm not sure how Peetsa will react. Damn it, Nina! Why can't you get your shit together so I can lose mine?
* * *
We are chowing down on burgers and fries in the kitchen when I realize I'm feeling a bit better. Max can't believe his luck. He's actually having fast food and it isn't someone's birthday.
"Can we do this every Sunday?" he asks hopefully.
"Not a chance," Ron says, although I'm sure he would be okay with it if I weren't around to say no.
I glance over at my kitchen-counter office and play with the idea of checking my email. I'm wondering how much crap I'm going to get for last night's party turning into an episode of Home Shopping Network. Maybe none. I mean it's not like I invited everyone... Oh, wait. While I'm wondering who painted "Sucker" on me, I boot up my iMac with the twenty-seven-inch screen (go big or go home, baby!) and check my email.
Holy shit storm.
* * *
To: JDixon
From: AChang
Date: 12/13
Subject: Class-less party
Jen,
I can't say I'm surprised that you chose to turn a lovely class party into a jewelry sale. It's pretty much what we have all come to expect from you as class mom. I know some people enjoy your "wacky" emails and generally lax attitude toward the job, but last night was the last straw. Do you even realize that people felt obligated to buy that overpriced junk? I only hope the money is going to charity. But I have to say there will be no more charity for you. I plan to take this straight to Principal Jakowski first thing tomorrow.
Asami
* * *
Well, that was expected. I'll bet she was composing this in her head while smiling at me over her champagne glass.
* * *
To: JDixon
From: SCobb
Date: 12/13
Subject: Class party
Jennifer,
I'm surprised at you. What would make you think anyone wanted to go to a party so they could spend money? Such a tacky idea.
Shirleen
* * *
I'm surprised she didn't mention the fact that there were no gluten-free hors d'oeuvres.
* * *
To: JDixon
From: CAlexander
Date: 12/13
Subject: About last night...
Jen,
I'm glad you organized the party last night, but was selling the jewelry really necessary? Being asked to fork out money for some really crappy stuff kind of put a damper on the whole night.
The lesbians are disappointed.
Carol (and Kim)
* * *
Ouch. That one hurt. Now the cool moms think I'm a jerk. I scroll through the rest of the emails—ten in total—and pretty much get the same message from each of them. Then I get to the one from Kim Fancy, which is addressed to the whole class.
* * *
To: Parents
From: KFancy
Date: 12/13
Subject: Oh, what a night!
Hello, friends,
I hope everyone enjoyed themselves last night. It was certainly a pleasure to have you all to our home.
I want to send a special thanks to our class mom, Jen Dixon, for organizing such a unique and fun gathering. Good job, Jen!
See everyone at the holiday concert!
Love,
Kim
* * *
I'm contemplating how to respond to the class when one last email catches my eye and the Whopper I inhaled earlier does a flip-flop in my stomach.
* * *
To: JDixon
From: DBurgess
Date 12/13
Subject: Last night
Jen,
Nice bumping into you in the bathroom! Let's have coffee sometime.
Don
* * *
For reasons I can't explain, I immediately grab for my cell phone and text Don instead of answering his email.
Thanks for catching me! So, do you mean coffee or, you know, COFFEE?
Only a few seconds go by before he texts me back.
Which answer gets me coffee?
I giggle. I'm not sure how to answer that one, so I don't.
* * *
To: Parents
From: AChang
Date: 01/05
Subject: I'm your new class mom
Happy New Year,
Most of you know me, but for those who don't, I'm Asami Chang and I will be taking over the duties of class mom from Jennifer Dixon.
It will be my pleasure to get our class back on track after a bumpy fall.
First of all, these emails will be content driven, not a forum for me to tell jokes and solicit bribes. Second, I welcome comments and input. I may be in charge, but you definitely have a say.
Below are things you need to know:
January 18 is picture day. Make sure your children's uniforms are clean and pressed. I'm concerned by the lack of emphasis most of you place on neatness. Please pay attention to this. I find a daily bath is very helpful. I've also noticed a lot of messy hair. I plan to stand beside the photographer with a brush and I will be using it. I suggest a ponytail for the girls with long hair, or perhaps a braid. I'm happy to braid hair that day. I will assume if you send your daughter to school with her hair down, you want me to braid it.
I'm organizing a class coffee meeting after pickup on January 12. We will meet at Homer's Coffee House at 8:30 a.m. Please be prompt.
Miss Ward informs me our children will be taking a field trip to the Quindaro Underground Railroad Museum on February 28. I will need one mother to go with me. Please don't volunteer if you already went on the recycling field trip. And be prepared to watch the kids closely and not socialize.
And now here are a few messages from the school administration...
* * *
I have to stop reading. My blood is boiling. I thought I was over being pissed off, but as I unclench my back teeth, I can see this clearly isn't the case.
How is it acceptable for Asami Chang to give grooming alerts and threaten kids with braids? All I ever did was try to make people lighten up.
Principal Jakowski's words are still burned into my brain. He had accosted me outside the gym just before the kids' holiday concert and basically fired me from being class mom.
"I'm sorry, Jen, but some of the parents feel you use the position to push your own agenda."
"What agenda? I don't have an agenda."
"Is it true you solicited bribes in return for better conference times?"
"That was a joke! Do you really think I'd let someone buy me a coat?"
"What about asking for Starbucks gift cards?" he asked.
"Oh, my God, I don't even remember doing that, but I'm sure I didn't mean it."
"And of course there is the jewelry incident at the class cocktail party."
"Why does everyone think that was my idea?"
"Mrs. Fancy said you asked her to throw the party. I mean, you did send out the invitations."
"She wanted to have the party," I explained. "I was just following orders."
"Well, according to her, you asked for the party on that specific date and her jewelry friend was booked to come to town for a visit. She felt she had to let her display her things as compensation for the party being held during her visit."
Clearly, this man will believe anything.
"I'm still having a hard time seeing how that's my fault."
"Yes, well, some of the parents have also complained that you make racist remarks in your emails."
"What?" I screamed. I couldn't believe what I was hearing. "That is not true!"
Principal Jakowski pulled out a piece of paper and handed it to me. It was one of my first emails to the class. I had been organizing the curriculum night class get-together. He had helpfully highlighted the offending phrase.
No hard feelings, Asami. I understand your people's lust for power.
I laughed before I could stop myself, and then put my hand over my mouth.
"Okay, I can see how that might look, but I was just trying to make light of the situation. I mean, the woman was trying to have me kicked out after one week on the job! Nina Grandish and I already talked about this."
Of course, Nina! She'd talk some sense into this guy. I suggested the principal call the head of the Parents' Association.
"I already have. I had to make three calls and text her twice before she called me back. She didn't seem interested in addressing the situation at all. In fact, she asked me to handle it. It seems she's been under the weather. Anyway, now I'm forced to get involved and I want you to know we take racism very seriously in this school."
At that very moment, I realized that I was fighting for a job I'd never wanted in the first place. Principal Jakowski was handing me a Get Out of Jail Free card and I was trying to get back in. Was I nuts?
"You want me to step down? Fine. Merry Christmas." I stomped into the gym just as the first group of kids was starting their performance.
As the sixth-grade class sang "Holly Jolly Christmas" (my God, that song is repetitive!) I stewed. They don't want me? Fine. Nina agrees with them? Fine. I'd fix her wagon. I'd fix all their wagons. Visions of prom night in the movie Carrie kept me busy while each class filed in and sang Christmas carols.
I really wished Ron had been able to make it to the concert. He would have talked me off the ledge. But he was in the middle of his Christmas savings bonanza down at the store. I'd promised him I would record the show on my phone.
So when Miss Ward took the stage to introduce the kindergarteners, I brought my focus back to the gym. Max had been very secretive about what the class was planning, so of course I pushed Record and braced myself. God bless that Miss Ward; she never disappoints.
The kids filed onto the stage in what are best described as tacky tourist costumes. Max was wearing a tropical shirt, a baseball cap, cargo shorts, and sandals with brown socks. Some of the kids wore cameras around their necks and zinc oxide on their noses. Naturally, they all looked adorable, but I couldn't imagine what Miss Ward had in store for us. She explained that they wanted to remind everyone that people celebrate Christmas all over the world, not just in places that have snow (because people not remembering that is one of the bigger problems we face in the world?) And then they sang... get ready for it... "Kokomo," by the Beach Boys! Not the Kodak moment I was imagining for my five-year-old son's first Christmas concert, but I laughed at the sheer cuteness of it. I shelved my mass murder plot, at least for a little while.
At the class party afterward, I was grateful to be surrounded by kids and parents the whole time, as it kept me from welling up about the shame spiral I was in. Fired! As class mom! How was I going to tell Max?
Out of the corner of my eye, I saw Don Burgess with Ali and Lulu, looking at the class's homemade snow globes. Part of me wanted to run to him and sob my broken heart out. Ron, while being my rock, would ultimately give me the "I'm the master of my own misery" speech. Somehow I knew Suchafox wouldn't do that. He'd take me in those solid arms and tell me the people who got me fired were all assholes. Or would he? I'll never know because, thank God, I didn't act on that impulse. But I did receive a text from him later that day.
Nice job on the holiday party. Sorry I didn't get a chance to talk to you. You seemed kind of down. Everything ok?
I didn't think my rage/humiliation was noticeable. I need a better game face.
Yeah, just the holiday blues, I guess. Thanks for noticing.
And not ten seconds later...
Well, I'm around if you want to talk.
Hmm... maybe that's what he meant by coffee. I thanked him and wished him a Merry Christmas. I was too bummed out to even attempt to be flirty.
I tried to do what Taylor Swift does and just shake it off, but between the class mom thing and my wayward thoughts about Don, a pall was cast over my whole holiday season. Christmas Day was a complete disaster. Nina and Chyna came, of course, along with my parents, the girls, Raj (back in favor for now), and, to everyone's surprise, Garth. I forgot that I had even asked him until he gave me his answer on Christmas Eve.
Nina still wasn't in great shape and I was in no mood to prop her up. I mean, she had basically thrown me under the bus. I thought it was gutsy of her to show up at all, but realized she probably did it for Chyna. She spent a lot of time by the Christmas tree, drinking wine. The only person she spoke to at any great length was Garth, although I couldn't imagine what they'd have to talk about.
It was nice to see Raj back in the mix. No ring on Vivs's finger, but no hostility between them, either. Apparently they were at an impasse.
And I guess I am, too. I log out of my email and sigh. Asami has won. The only thing left for me to do is get Max ready for picture day.
Oh, yes, that is exactly what I will do.
* * *
As I plot my picture day revenge in my head, I run up to get changed for my first workout of the New Year. T minus four months until the mud run, and I know Garth is going to step it up.
My phone buzzes as I'm putting my T-shirt on. I figure it's Peetsa, but it's actually Don sending me a picture of a beautiful baby. What the...? I type back to him.
What's up with the baby?
It's me. Cute, wasn't I?
Yes, you were. What the hell happened? You're hideous now.
Ha ha, puberty happened. We all can't be natural beauties like you.
I get goose bumps when I read this. I'm a natural beauty, am I?
Well, it's a curse, this beauty of mine. I spend all day fending off advances from strange men.
Hey, I'm not that strange.
I laugh out loud as I put the phone down to tie my shoe. It buzzes again almost immediately.
What are you up to? Can you meet for coffee?
There's that coffee offer again.
Sorry, I can't. I'm going to work out.
I could help you work out.
I'll bet you could, I think to myself, then snap back to reality. Why am I even encouraging this? Flirty texting is just a gateway to adultery. I know that's what Nina would tell me. But it's also harmless and kind of fun.
I'm sure you could, but I have a hot trainer waiting for me in my basement. Gotta go!
I push Send and turn off my phone in case I'm lured into any more sparring.
As I walk down to Ron's Gym and Tan, I note that Garth has finally taken my suggestion and let himself in. He's busy setting up some kind of obstacle course for me.
"Hey, Garth, happy New Year." I walk over and hug him.
"You too! What did you guys end up doing?"
"The only thing you can do when you have kids." I shrug. "You do fake midnight at nine o'clock and then head to Club White Sheets."
"Where is Club White Sheets? Is it that new place over on Grand?"
I start to laugh. "It's bed, Garth. Club White Sheets is bed."
He looks confused for a second, then bursts out laughing. His laugh makes me laugh ever harder and soon we're rolling on the floor.
Garth gains control first.
"Okay. Enough. While you're on the ground, why don't we start with twenty push-ups and fifty sit-ups, just to get warmed up?"
"Spoilsport." I scowl, but flip over and give him twenty perfect man push-ups. I surprise both of us.
"Nice, Jen. Wow! New year, new you."
"I can't believe I just did that," I pant.
I flip over and launch into my sit-ups with newfound confidence. By forty I'm hurting, but I make it to fifty before I collapse, panting.
"So much for the new me," I say.
"Don't be silly. You've come a long way, baby! I'm proud of you. Now, up off your caboose and let's get going."
* * *
Garth manages to have me sweating like a pig by the end of the hour. As I'm walking him to the door, he casually asks about Nina.
"What about her?" I ask a little too briskly.
"Weeelll," Garth drawls. I think I might have scared him off. "I had a really nice time talking to her and I'm wondering if she's mentioned me at all."
"Jesus, Garth, I don't know. I'll pass her a note in science class."
As soon as the words are out of my mouth, I feel bad. I can see the hurt look in his eyes.
"I'm sorry. I didn't mean that. I just haven't spoken to Nina since Christmas. She has been going through a bit of a rough patch thanks to her ex, and she hasn't been answering her phone."
Garth waves my comments aside.
"No worries. I was just wondering how she's doing. We had talked about her possibly designing a website for me."
"Well, she's really good at it, that's for sure. If she talked about it with you, I'm sure she's already coming up with ideas. It may take her some time to get out of her funk and get working, though."
"I get it," he says. But I wonder if he really does.
After Garth leaves, I check my phone and see it has blown up in the past hour, thanks to Asami's email. Everybody wants the 411 on my class mom demise. Peetsa, Ravi, and Kim Alexander all express their concern with a "WTF," and Shirleen Cobb is not at all happy about having to teach the new class mom all about Graydon's many needs. Only Suchafox sees a silver lining.
Maybe now you'll have time for coffee with me.
I really need to make sure we're talking about the same type of coffee. But before that, I'd better figure out which one I want.
* * *
I stroll up to the school the next day to wait for Max to come out, and I see what I have been dreading—all the mothers from my class standing around in little klatches, talking and drinking Starbucks. Normally I would be one of them with my grande skim chai latte, but because of my class mom shame, I have been avoiding this scene since the new year started. For the past couple of days, I have had Ron leave the store to do pickup, but today I decide to face the music.
As I walk toward the front of the school, I can't help but feel like everyone is talking about me. I know I'm just being paranoid. I walk up to Peetsa and Ravital Brown.
"Oh, my God, Jen, we were just talking about you. Where the hell have you been?"
"Did you write that email from Asami as a joke?" asks Ravi. "That was so funny. Best one yet."
I gave them both a hug. I've missed them.
"No, sadly—this time the joke's on me."
"So it's true?" Peetsa gasps. "One of Hunter's moms tried to tell me, but I wouldn't believe her."
"Well, believe it."
"Jeez. Are you okay?" she asks. "I can't believe you didn't tell me."
"You were away and I didn't want to bring you down over the holidays."
I tell them about my conversation with Principal Jakowski and they react exactly how I would expect my besties to react. Outraged! Infuriated! Bent on revenge! Until I mention the accusation of racism.
"I mean, come on, it was a joke!" I say, exasperated.
They look at me and then at each other and then at the ground.
"What am I missing?" I ask.
Ravi takes the bullet.
"Well, I have to admit when Asami asked me what I thought of the phrase 'your people's lust for power,' I said I thought it was a little off. I mean, now that I know you, I see you were being funny, but at the time I didn't know what to think." She looks at her shoes.
"Hey, we all love your emails," Peetsa adds, "but people are really sensitive about racism. I know you meant it as a joke, but maybe the class email isn't the best place for it."
I look at both of them and am about to say something when a swarm of kids runs out the front door. In the sea of winter jackets, Max is easy to spot in his leopard-print coat. He's carrying Zach B. on his back.
"Hey, Mom. Zach B. is riding me like I'm a horse."
"Well, you do look like an animal in that coat." I smile.
I look up at Peetsa and Ravi, who are hugging their boys.
"Ravi, I'm sorry if I offended you. Really. It was a thoughtless thing to write."
Ravital shakes her head.
"Trust me, I wasn't that offended."
"Oh, God, I'm really going to miss your emails," Peetsa moans.
"What, you don't like personal-hygiene tips in your class emails?" I ask. "I found that very helpful. A bath! Who would have thought?"
As we are walking to our cars, Peetsa asks me about Nina.
"You know, I haven't spoken to her since Christmas. She and Chyna came over for dinner, but she was still not herself."
"Wow, when she goes to the dark place she really pitches a tent." Peetsa shakes her head. "What did she say about the class mom stuff?"
"I haven't talked to her about it." I shrug. "Jakowski told her what he was planning and she said she didn't want to get involved."
"That's cold."
"I know, right?" Suddenly I feel vindicated. "I really think she could have put the kibosh on this whole thing if she had just taken her head out of her ass for five minutes."
Peetsa is shaking her head and laughing.
"Too harsh?" I ask.
She puts her thumb and finger up to show me an inch.
I buckle Max into his car seat and as I slide into the minivan I check my phone. I'm rewarded with a text from Don.
You look nice today. Very fit.
I immediately look up to see if he's watching me.
How the hell would you know?
I caught a glimpse as I was pulling up to get Lulu.
Oh. Well, thanks. What are you up to?
Taking Lulu to dance class. You?
Max has Scouts.
So... still no coffee?
Not today!
But there is the hope of someday?
Absolutely.
I put the phone in my purse and start the minivan. Texts from Don have become kind of a regular thing. I'm enjoying the sparring but can't help but feel that I'm doing something wrong, like picking my nose in public. Then that feeling gets me pissed off, because I'm just having fun and it feels good to have the attention of someone besides Ron and it doesn't mean anything and can't I just have a friend who is a guy, dammit? Welcome to the cocktail party in my head.
I pull out of the school's parking lot and decide to not think about it any more today.
* * *
For the next couple of weeks my life goes back to its normal, dull housewifey routine, although without the class mom crap to annoy me I find I have a bit of spare time on my hands.
"Idle hands are the devil's workshop," I can hear my mother say. I never fully understood what that meant until I realize that I'm spending most of my spare time either flirty-texting with Don or concocting a plan to sabotage Asami's picture day efforts. My thoughts go from the benign (tell the kids not to smile, or else) to the macabre (light a fire in school so the sprinklers go off and soak everyone's hair. Braid that, Asami), but I don't want to do anything that will hurt or upset the kids, so my options are limited. I consider asking Don to be my accomplice, but realize I want to take either all the credit or all the blame, depending on how things go down.
Garth is pushing me harder than ever, and I have to say I'm pleased with the results. My usual post-holiday five pounds didn't materialize, so I'm looking and feeling better than I ever have.
"Any plans for the long weekend?" Garth asks as I'm finishing my final set of lunges across the basement floor.
"Nothing much. You?"
"Well, nothing yet, but I thought if you were up for it we could go to Wichita on Saturday. This charity I work with is having a scaled-down indoor mudder at Hartman Arena."
"An indoor mudder?" I take a swig of water. "They do that?"
"Actually, this is the first one."
"Sounds dirty and smelly."
"I'll be disappointed if it isn't," Garth assures me. "I think you need to get a look at what you'll be facing. You've only ever seen it on YouTube. I just want you to get a sense of the scale. What we're going to see still isn't a full mudder, but it's the best I can do in Kansas in February."
"It's not a bad idea. Can I bring Max?"
"Sure!" says Garth as he gives me a huge smile. "Aren't road trips just the best?"
* * *
I'm standing with Peetsa and Ravi, waiting for the kids to get out of school. It's warm for January so we don't have our heads and faces covered as we usually do. I see Don Burgess standing with Kim or Carol Alexander, and I wave. Don holds up his phone and gives me a shrug. He's wondering why I haven't answered his text from this morning. I told him I was going to Wichita with my trainer this weekend and he's been bugging me for details.
Are you going to have coffee with him?
I take my phone out of my pocket and type a quick response.
I never disclose my coffee-drinking plans.
The girls and I are talking about our plans for Martin Luther King weekend. Peetsa tells us she and Buddy are packing up the kids and taking them skiing at Buddy's parents' place at Snow Creek. When I mention my big road trip to Wichita to observe the mudder, they are pretty impressed. Ravi says she doesn't have any plans so I ask if Zach B. wants to join us on our road trip. Selfishly, I know it will go better if Max has a buddy.
"He'd love to," she says, and then her face lights up. "Oh, my goodness, does that mean I'll have a Saturday to myself?"
"And a Friday night, if you'll let him sleep over."
Just then the bell rings and the kindergarteners start pouring out of the school. Normally they are a pretty wild bunch, but today I notice a lot of heads down and even a few kids crying. When I locate Max's leopard coat, I can see that he looks very unhappy. When he sees me, his little face crumples and he holds out his arms.
"What is it, baby?" I get down on my knees to hug him. Peetsa and Ravi are doing the same thing with their kids. They both give me a "What the hell?" look.
"Max, sweetie, what happened?" I pull his head away to look him in the face.
"He's dead. We saw him."
"Who's dead?" I'm thinking the class fish.
"Martin Luther King. Someone shot him with a gun and they put him in a box just like Rufus." Rufus was our pet guinea pig. He died last year of natural causes and he's buried in a shoebox underneath the wild rhubarb that grows in the backyard.
"Oh, sweetie, I'm so sorry. Where did you see the picture?" I ask as I mentally compose a hate letter to Miss Ward.
"Mrs. Chang showed us."
"What?" Peetsa, Ravi, and I say all at once.
The boys cry louder.
I really want to go back into the school and find out what the hell happened, but I can't leave Max like this.
"Did he really die?" Max asks through his sobs. "He was so nice and helpful."
I can tell this is going to open up the death discussion again, and I'm just not up for it. Memories of Max dealing with Rufus's death come flooding back to me. He cried for days. Ron was at a loss, so he brought home a book that someone at his store recommended called Something Is Wrong with Grandma. It's supposed to help kids understand and deal with death, but all it did was convince Max that something was wrong with his grandma. It took him months to get over his fear that my mother was going to keel over any second.
"You know, he died a long time ago, and it was very sad. But he did so many amazing things in his life and when you think about it, he now has a whole day for people to remember how good he was."
"Where did they bury him?" Max asks me. I give Ravi and Peetsa a desperate look, because how the hell would I know?
Ravi comes to the rescue: "I think he's buried in Atlanta, right near where he grew up." That sounds about right. I give her a grateful smile.
Peetsa looks at all three boys.
"Did Mrs. Chang show you a picture of his grave?" she asks, trying to get a clearer idea of what they saw.
"No, it was a picture of him lying in a box with his eyes closed," Zach T. says. His eyes start to water.
Oh, good God. No wonder they're traumatized. Showing a picture of a dead body in a casket to five-year-olds. I turn to Max.
"Hey, can you sit with Mrs. Tucci in her car for a minute? I want to go talk to Miss Ward."
"Why don't I take all three of you to our place for hot chocolate?" Peetsa offers.
The boys nod and smile. Proof once again that chocolate solves just about all of life's problems.
"Want me to come with you?" Ravi asks me.
"Sure. P., we'll be over in a little while."
"Sounds good." Peetsa waves as she hustles the boys to her car.
Ravi and I head into the school and march right down to room 147.
"You can do the talking," she says as we reach the door.
"Count on it." I wink at her.
As we enter Miss Ward's colorful classroom, I can see we are not the first parents to arrive. Dr. Evil is leaning over the front of Miss Ward's desk and speaking in low but severe tones. As we walk in I hear Kim say, "... and I'm sick of it."
"Hi. Sorry to interrupt, but we have some really upset little boys on our hands." I look directly at Kim Fancy. "Was Nancy upset, too?"
"About what?" Kim's slight scowl tells me she's both annoyed and confused.
"Hi, Jen. Is this about the Martin Luther King presentation?" Miss Ward asks, as if she's asking how the weather is.
"Uh, yes. Max and his friends came out of school really freaked out from seeing a picture of a dead body."
"A what?" Kim and Miss Ward ask at once.
"Weren't you here for Asami's presentation?"
"No." Miss Ward actually looks contrite. "I, um, had some papers to grade, so I went to the teachers' lounge while she did it. When I got back, she told me she had already dismissed them."
"Seriously? You let a parent dismiss the kids?" I'm a little surprised. I'm also wondering what kind of papers a kindergarten teacher needs to grade.
"Well, I wouldn't normally, but she seemed to have things under control. You say she showed them a dead body? Whose?"
"Martin Luther King's," I say, exasperated. "He was in his coffin. Max is completely traumatized. He came running out of school crying."
Miss Ward and Kim look at each other. Kim shakes her head and walks out of the classroom. What the hell?
"Well, I will certainly talk to Asami about it and find out what happened," Miss Ward assures me. She pauses and smiles sardonically.
"Jenny, it's so funny to have you complaining about her. She complained about you constantly."
"Yes, it must be hilarious for you." I turn quickly and almost hit Ravi as I'm walking out. I totally forgot she was with me.
As we head down the hall, Ravi seems to read my thoughts. "I can't believe she left the class alone with a parent. Is that normal?"
"Depends on whose world you live in." I sigh.
Saturday morning at 7:50 sharp, Garth arrives at my house, his usual ten minutes early. I'm just clearing breakfast away for Max and Zach B. They are bleary-eyed from their sleepover and I predict a car nap in the not too distant future.
"Hey, Garth. Want some coffee for the road?"
"No, thanks. Brought my own." He holds up a Starbucks cup. I immediately think of Don and laugh to myself.
"What's your poison?"
"Grande triple-shot latte with extra foam." He smiles and cheers me.
"Well, that will put some punch in your pumpkin." I cheers him back with my mug. "We'll be ready in five."
I'm halfway up the stairs when I yell over my shoulder.
"Okay, boys, lock and load. Wheels up in five. Bring a couple of pillows. Let's move it, monkeys!"
I hear Zach B. say, "Your mom talks weird."
I check my phone and find a text from Don:
Have fun in Wichita.
I smile. This is the weirdest relationship I have ever had. We text all the time and know everything about each other's lives, but we never meet up for that much-talked-about coffee.
We decide to take my bitchin' minivan so we have room to stretch out and the boys can watch a movie. Don't judge me. I wish I were the type of mom who has endless ideas for car games and the energy to play them, but I am not. What I do have is an endless supply of DVDs that I pull out for any car ride longer than forty-five minutes, because that is Max's breaking point.
The boys snuggle up to their pillows in their car seats as we take off and about ten minutes into The Lego Movie, they are passed out.
"So what is the charity you work with?" I ask Garth as I steer the van onto I-35 South. The weekend morning traffic is light.
"The Wounded Warrior Project."
"You know, my mom volunteers for them. She hosts a Proud Supporter event every year with her church group."
"I know. That's where I met her." Garth seems to smile at the memory.
"Were you helping out at the pancake breakfast or something?" I sneak a look at him.
"Something like that."
"Wait, are you a vet?"
He nods. "I did two tours in Afghanistan."
"When?" I say a little too loudly. Shit! I check the rearview to make sure the boys are still sleeping.
"Oh, 2004 to 2006."
"Were you in combat?"
"Well, I wasn't there for the weather."
"Did you get hurt?"
He shrugs.
"I took some shrapnel in my left side. I got off easy compared to some of my friends."
"Holy shit! I can't believe I didn't know this about you. Why didn't you tell me?"
"It never came up."
We both stare at the windshield, watching miles of highway slip beneath us before I speak again.
"Do you mind talking about it?"
He chuckles.
"No, not at all. What do you want to know?"
I think about that for a minute and self-edit the inappropriate questions that race to the tip of my tongue.
"Um, what do you guys miss most when you were over there?"
"It's different for everyone," he muses. "Everyone misses home in one way or another. Could be your family, your bed, wearing jeans, normal food. For me, it was Campbell's tomato soup."
"What?" I start to laugh. "Tomato soup?"
Garth nods. "Don't ask me why, but the whole time I was away I craved tomato soup. When I came back, I couldn't get enough of it."
I shake my head. "Too funny. I wonder what I would miss."
"Whatever it is, I guarantee it won't be what you'd expect it to be."
As we are passing Emporia, which is just about halfway, I ask Garth if he needs a bathroom break.
"That would be great," he says.
I pull off the highway and head to the first gas station I see. While Garth finds the restroom, I fill the tank. Of course, the lack of car movement makes the boys wake up, and they ask to go to the bathroom, too. And get a snack. And get water. And start the movie again. By the time I take a pee break and we're back on the road, a half hour has passed.
"What time does this thing start?" I ask, glancing at the clock on the dashboard.
"Noon. We're good for time." Garth reclines his seat a bit and gets comfortable. "So, have you talked to Nina?" he asks as casually as he can.
"Nope."
His disapproving stare almost gives my cheek a tan. My shoulders sag.
"I know I need to. I hate not talking to her."
"So what's the problem? Pick up the phone."
"I will. It's just way past time and I don't even know what to say to her."
"Well, neither of you has been a particularly good friend at this point, so maybe start with 'I'm sorry.'"
I start to argue, but don't have the heart. He's right. Nina and I have never before gone this long without talking.
"I'll call her when we get home later."
"You'll feel better," Garth says and shuts his eyes. "Mind if I take a catnap?"
"Go for it. I'll probably sleep on the way back."
"Guess that means I'm driving, then." Garth smiles to himself.
Before long, he's out.
* * *
As we pull into the parking lot of Hartman Arena, I can already see it's a big event. We park nowhere near the entrance and have to hike to the doors. The boys are practically bouncing out of their shoes. Neither has ever been to Wichita.
Inside, Garth is greeted like he's a regular at these events. Everyone waves or says hello or pats him on the back.
"Wow, I feel like I'm with the most popular boy in school," I say to him.
He rolls his eyes. "Not quite. But if you go anywhere often enough, people are going to get to know you."
We stop at a cluster of tables with a huge Wounded Warrior banner hanging above them. A clean-cut, good-looking older man gives us a big smile over the crowd around him.
"Garth, man, how are you?"
"I'm great, Jack. How are you doing?"
"I'm good." He looks at the boys and me.
"I'm Jack."
"Hi, I'm Jen." I go to shake his hand and see that his right arm is gone. He offers his left hand and I'm a bit thrown. I rebound from the awkwardness with my usual grace and style.
"Oh, sorry, uh, hi." I change my handshake position to a wave. "This is my son, Max, and his friend Zach."
The boys are staring with their eyes and mouths wide open.
"Where's your arm?" asks my chip off the old block.
"I lost it," Jack says solemnly.
"Where did you lose it?" Max is almost whispering.
Jack puts his one hand on his hip.
"Well, if I knew that, don't you think I'd go get it?"
Max starts to giggle and then so does Zach B.
Garth steps in.
"Sorry, where are my manners? Jen, Jack and I served together in Afghanistan."
"Nice to meet you." I give him a grateful look.
"You, too. How do you know this jarhead?"
I turn to look at Garth and raise my eyebrow.
"'Jarhead.' I like that."
Garth gives me a fake scowl.
"I've been training Jen for the Kansas City Mud Run in April."
"Well, you're in good hands. He is one tough mudder himself. Are you signed up for today?"
"Nope," I say. "We're just spectators and supporters of the cause."
"Well, you better get in there. It's about to start. Garth, why don't you guys sit in our section?" Jack offers.
"Great, thanks." Garth steers me toward the arena. "Come on, guys, let's get some seats."
* * *
The indoor mud run is really impressive. The course takes up the entire floor of Hartman Arena, which is the size of a professional hockey rink.
Every inch has been carefully designated for a different obstacle, and I'd be lying if I said the sight didn't scare the crap out of me.
"This is so cool!" Max screams over the loud music pumping through the arena speakers. He and Zach B. are taking it all in with huge grins on their faces. I grab their hands and lead them up into the stands. We take our seats by a handful of people who, judging by the number of missing body parts, are veterans, and their families. It's then that I finally get a good look at the course. Before I start to freak out, Garth is talking me through it step by step.
"So, it starts with a run up to the top of the arena and back down again. Then you climb a rope over a ten-foot wall, followed by a long crawl through the mud over there that has barbed wire stretched across the top." He points to the other side of the building.
"After that, you have to carry a huge log about five hundred yards with a group of people, then run up that thing that looks like a half-pipe wall. You need to get a lot of speed for that. You'll want to quit halfway up, but good momentum should take you to the top. Climb over it, then jump into the freezing-cold water."
I wince. I don't even like lukewarm showers.
"Get out of the water and go over to the wall with the pegs. Grab two rings and hook them onto the pegs to get yourself across the moat. That's probably the toughest part. You need a lot of upper-body strength. After that, it's pretty smooth sailing. Normally, you'd have to jump through fire just before the ice bath, but the fire department wouldn't give them an indoor permit, so it's just running up and down a few dirt mounds before you cross the finish line."
My heart is thumping like a jackrabbit's, my mouth is dry, and I'm relatively certain I have soiled myself. I can't speak.
Garth starts to laugh.
"Hey, what's up? You could do this. You could totally do this."
"In what universe can I jump through fire?"
"Well, it's not really something we can practice, but you will find that in the heat of the moment—no pun intended—your adrenaline will carry you anywhere you need to go."
He puts his arm across my shoulders and gives me a hug.
"You will do this, I promise."
I look at Max, who is watching the arena floor and chomping on a hot dog—courtesy of Jack, who has just joined us—and suddenly it is very clear to me.
I will do this.
* * *
On the way home, I let Garth take the wheel and I nap. I have a terrible dream that I'm competing in the mudder but I only have one arm. I wake with a sudden jerk as I let go of the ring I'm holding on to on the pegboard.
"Holy crap!"
Garth looks at me. "Did you have a mud-mare?"
"I think I did," I say, rubbing my eyes. I check out the backseat; the boys are watching yet another movie. Max has hit his screen time for the month, that's for sure. "Is that normal?"
"Oh, yeah, especially while you're training. What was it about?"
"I was competing, but I only had one arm. When I got to the pegboard I had to let go, because I couldn't move the ring to the next peg." I shake my head. "That was brutal."
"You should see the guys do it who actually only have one arm. Insane upper-body strength." There is a lot of admiration in Garth's voice. I'm thinking I've only scratched the surface of my trainer's surprising emotional depth.
* * *
When we get home, after dropping an exhausted Zach B. off, my prince of a husband takes charge of Max, which gives me time to wash the day off my body. After my shower, I wrap a towel around my head, grab my robe, and head to my side of the bed. This is it. I'm calling Nina until she picks up.
I've settled in for a long session of redialing, but she picks up on the first ring.
"Hi." She is whispering.
"Hi. Why are you whispering?" I'm whispering, too, for no reason.
"Chyna just fell asleep on the couch. She was at a gymnastics tournament today and she's beat." I can tell Nina is moving into another room so she can talk.
"How did she do?" I ask. This is so weird.
"Third place overall for her age group. First place on the beam."
"Wow, that's great. Tell her I said way to go."
"I will."
And... silence. I take a deep breath.
"Look, Neens..."
"Hang on. Before you say anything, let me tell you how bad I feel about the way I've been acting. I'm really sorry. After I found Sid, I basically took a nosedive into depressed oblivion and I've just resurfaced."
"Oh, Neens, I'm sorry, too. I just didn't know what to do for you. And then when you didn't help with my class mom situation, I just got really pissed."
"Your what?"
"It's nothing. Just that thing with Asami and Jakowski, you know..."
"What are you talking about? What happened?"
"Holy shit, you really don't know?"
I briefly tell her all the gory details of my class mom downfall and how Principal Jakowski had apparently called and consulted her, but she told him she wasn't interested in handling it and he should.
There is silence at the other end of the call. I think we've been cut off.
"Hello?"
"I'm here," Nina says, but she sounds like her mind is elsewhere. "I don't remember him even calling. Shit, this is worse than being blackout drunk."
I laugh, but it comes out as a half laugh, half sob. Sort of like a hiccup.
Nina laughs. "What the hell was that?"
"I'm just so happy to be talking to you." I sniffle.
"Yeah, well, I'm out of the bell jar so tell me everything."
I give her the rundown on my ouster as class mom and also update her on Vivs and Laura. Suddenly something occurs to me.
"So, what ever happened with Sid? You seem kind of over it."
"Oh, I'm way over it. What an asshat."
"Tell me something I don't know."
"Yeah, yeah, I know. It took me a while, but believe me, I'm here now. He actually hit on me."
"You saw him?"
"No, on Facebook. Hang on, I'll send it to you." I hear some shuffling and Nina's voice at a distance saying, "Girl, you are not going to believe it."
My phone buzzes as Nina says more clearly. "Okay, I just copied the IMs and sent them to you."
"Hang on." I look at my phone and push Messages. I see what Nina has sent.
"Holy shit, how long is this?"
"Long. Just scroll. The first part is just us catching up and him explaining why he left. Not apologizing, mind you, just explaining that he freaked out and realized he wasn't ready for a kid. That he thought it would be better for me if he left because he knew he wasn't going to be able to help me."
"Wait." I'm scrolling and trying to listen to her at the same time. "He's talking about not wanting to meet Chyna..."
"Oh, yeah. He doesn't want to upset her life or his new kids' lives. They aren't even his kids! He married a pregnant widow."
"Eww," I say involuntarily, wondering just how desperate a woman would have to be to allow Sid into her kids' lives.
"I know. Even I can see the eww in it now."
I keep scrolling, trying to skim all the bullshit about his new life and new job at the high-tech company, which I find out is more of a low-tech company that makes the hydraulic mechanisms for office chairs, or "computer chairs," as the company calls them. Explains the name of the company, Compu-lift. Apparently, Sid is a "tester," which means he sits on his ass all day making sure the chairs go up and down.
"Huh. Sure that job isn't too much for him?" I say into the phone.
Nina laughs. "Get to the good part."
I keep scrolling and find where he talks about how tough marriage is.
"How the hell would he know? If that baby is a newborn he can't be married more than a few months."
"Seven."
"Oh, my God. Who is this poor woman he married?"
"According to Sid, her dad owns the company where he works. He told me they're grooming him to take over."
I snort. "So he married the boss's daughter. Good to know he uses his brain for something more than testing chairs."
"What part are you at now?" Nina asks.
"Um... you're asking if he will come to see Chyna."
"Okay, read from there. This is when he makes the jump from scumbag to douchebag."
I don't want to tell her that jump was made years ago, so I just read.
Sid: I just think it would really mess Chyna up for me to walk into her life out of nowhere.
Nina: I'm sure she'd be happier finally getting to meet her father.
Sid: Doesn't she have a father? Like one of the guys you hooked up with after me.
Nina: I never hooked up with anyone else.
Sid: Yeah, right.
Nina: I was a heartbroken single mother of a baby. I didn't want to be with anyone.
Sid: Even after all these years?
Nina: I've been busy.
Sid: Busy missing me?
Nina: Busy raising a child, working, and volunteering at school. I don't have time for that shit.
Sid: You had time for it with me.
Nina: Oh, please.
Sid: Remember the weekend we rented that cottage?
Nina: No.
Sid: Come on. We didn't wear clothes the whole weekend. That's the shit I remember. The shit I miss. I've been thinking about it a lot lately.
I stop reading.
"Gross!"
"What part?" says Nina's disembodied voice.
"He's reminding you of that weekend you rented the cottage."
"The one that cost me a thousand dollars. How fucking stupid is he to bring that up?"
"I'm assuming that's a rhetorical question."
"Hardy-har. Keep reading."
I return to the IM stream.
Nina: But not enough to come back.
Sid: Do you want me to come back?
Nina: For a long time I did.
Sid: What about now?
Nina: Now? Why would you come back now? You have a new family.
Sid: I could come for a visit.
Nina: I thought you didn't want to meet Chyna.
Sid: I could come back and see you.
Nina: Why would you do that?
Sid: I don't know. Just see what's up.
Nina: What do you mean?
Sid: You seem lonely, baby. Maybe I could cheer you up a bit.
"Oh my God!" I scream at my phone.
"Can you believe him? I was like, where the hell did that come from?"
"I hope you shut him down."
"Not well enough. I just said no, thanks, and haven't answered any of his messages since."
"When was the last time you heard from him?"
"Today. He sends me messages almost every day."
"What do they say?"
Nina sighs. "It's a lot of 'What did I do, baby? Are you mad at me? I still want to come for a visit' and shit like that. I should just block him, but I'm sort of enjoying torturing him a bit."
"You aren't torturing him enough!"
"But I really don't want any contact with him. I think ignoring him is good enough."
I'm not convinced, but it's not my battle.
"My turn," Nina says. "Whatever happened with your fantasy man? What was his name, again?"
"Don." I smile. Finally I can talk to someone about this!
I tell her all about how fun he is to flirty-text with and read her some of my favorites.
"And in this one he asks me... Nina? Are you still there?"
I hear breathing on the other end of the line, and then Nina practically takes my head off.
"Jen, you need to shut this shit down immediately!"
"What? Are you serious?"
"Yeah, I'm serious."
"Neens, they don't mean anything. They're totally harmless."
"Oh, really? Do you show them to Ron? Are you and he getting a good old laugh about all those coffee double entendres?"
"No. But I would show them to him. They're just funny."
"So you wouldn't mind if Ron had a little banter going on with an old girlfriend of his?"
"Jesus, he wasn't my boyfriend," I mutter. But I give her question some thought. Would I mind if I saw a stream like this on Ron's phone?
"I see what you're saying, and I get how it looks, but I know what my feelings are and I have no intention of cheating on Ron."
Even I can hear how lame I sound, but all Nina says is "Yeah, well, it's always fun until someone loses a husband. Just be careful."
I really don't want to fight with her, so I promise that I will.
We make plans for lunch the next day and, after a lot of "I love you"s and "I'm sorry"s, we hang up. I feel better than I have in weeks.
* * *
To: Miss Ward's Class
From: AChang
Date: 01/17
Subject: Picture Day
Hello, Parents,
It has come to my attention that some of the children did not react well to my PowerPoint presentation on the life and death of Martin Luther King. In hindsight, showing him dead in a coffin may have been too much.
Just a quick reminder that tomorrow is picture day. I know that a long weekend can sometimes make people forget. I hope you have already ironed your child's uniform. I want us to be the cleanest class at William Taft. A bath tonight would really help with that.
I will be there tomorrow to make sure your children are neat and smiling.
Regards,
Asami
To: PWard
From: JDixon
Date: 01/17
Subject: Picture Day
Hi, Miss Ward,
Do you need any help wrangling the kids tomorrow when they're getting their pictures taken? I'd be happy to help.
Thanks,
Jen
To: JDixon
From: PWard
Date: 01/17
Subject: Picture Day
Jen,
Sorry, but that's a question for the class mom. You can reach out to her.
Thanks,
Peggy
* * *
Damn. I was hoping to avoid that. Ugh. This is going to hurt.
* * *
To: AChang
From: JDixon
Date: 01/17
Subject: Picture Day
Hi, Asami,
Happy New Year. Hope you are enjoying your new job. Your first email was very informative.
I was just wondering if you need any help wrangling the kids tomorrow while they are getting their pictures taken. I would be happy to keep an eye on them in the hallway before they go in, or bring a snack.
Let me know.
Jen
To: JDixon
From: AChang
Date: 01/17
Subject: Picture Day
Jen,
Well, I certainly didn't expect to hear from you. I really don't think I will need help, but if you are having trouble disengaging from your power seat and you want to come in for a bit, I guess it would be fine. Please bring the children a snack.
Asami
* * *
A snack, yes! I shut my laptop and turn my chair to look at my kitchen. I got me some baking to do.
* * *
Never say never. This is my new motto. I said I would never have another kid after Laura, and lo and behold I did. I said I would never read Fifty Shades of Grey, but after seeing Nina's reaction to it, I did. And I said I would never again make my mother's Sticky Chewy Five Napkin Brownies, but here I am putting the second batch into the oven.
These babies are killer. The recipe calls for, among other things, nine eggs, two cups of sugar, a whole pound of butter, toffee, chocolate chips, and whipping cream. Totally unhealthy, but they taste like heaven and my kids love them. The only downside is they are a leeetle messy to eat. If I'm being honest, you need a lot more than five napkins to keep yourself together. A container of Wet Ones is more like it.
Asami asked for a snack, and a snack she will get. Yes, that's right. I'm going to give a bunch of five- and six-year-olds the equivalent of a mud pie to eat before they get photographed. I'm that small and petty. Our new class mom should never have reminded me to bathe my child.
* * *
On picture day, I sally forth to school around ten a.m., armed with the Sticky Chewy Five Napkin Brownies, a roll of paper towels, and, of course, a gluten- and nut-free snack for Graydon.
I head to room 147, where I see that Asami has all her hair-styling equipment lined up on a tray and ready for action.
The children are seated at their tables, listening with rapt attention as Miss Ward explains the best way to smile for a picture.
"When they ask you to smile, try to think of a funny joke. That way, when you smile it will be with your whole face and not just your mouth."
She then proceeds to show the kids what a just-a-mouth smile looks like. She looks ridiculous, but the kids love it. They are laughing and doing it to each other.
"Okay, settle down. Now, who knows a joke, so I can show you a smile with my whole face?"
Sixteen hands shoot up, including Max's, and I can't help but wonder what jokes he knows. There is a chorus of "Me, please, me please, oh me please!" as Miss Ward takes her time deciding.
"Zach T. What have you got?"
Zach T. beams with excitement as he stands up.
"Knock, knock."
"Who's there?" "Europe."
"Europe who?"
"No, you're a poo!" Zach yells out, and the class explodes with laughter as Miss Ward's face bursts into a bright smile. Even Asami is laughing.
"Good one, Zach. Put a marble in the compliment jar."
As Zach proudly steps up I hear my universal name being called.
"Mom!" Max runs up and hugs me like he didn't just see me an hour and a half ago. I love it. Way too soon, he will find it embarrassing when I come to his classroom, but for now it's still a treat.
"Hi, buddy." I give him a big squeeze. He still smells of last night's bath and I take an extra whiff before I let him go.
"Jennifer, I'm glad you're finally here" is all the greeting I get from Asami. "The photographer is set up two doors down the hall. The children will get their individual shots done and when they're all finished, the photographer will take the group shot. I will be with the photographer and you will be with the class, sending the kids to me one at a time. Got that?"
"And good morning to you, Asami!" I reply.
"Did you bring a snack?"
I hold up my shopping bag.
"What are those for?" she asks, pointing to the paper towels.
"Just in case the kids get messy." I hope to God she doesn't ask me what I brought. "I even have something special for Graydon," I add by way of distracting her.
But it isn't necessary. She just nods to me and picks up her hair supplies. As she heads out the door, she has one parting instruction.
"Don't give them their snack until they come back from getting their picture taken."
"Got it!" I say, a bit too enthusiastically.
Miss Ward has been watching the whole exchange. She raises her eyebrows at me.
"Wow, Dixon goes to China, huh?"
Why is that okay for her to say, but my "your people" comment is still offensive? Seriously, where's the line? Do they keep moving it?
"Okay, class, Max's mom is going to let you know when it's your turn to get your picture taken. The rest of the time, we will be practicing our letters. I want everyone to get out your workbooks and start working on capital 'M's."
There is a brief commotion as the kids get their books from their cubbies; then they all settle down. I have to say, I still think Miss Ward is crazy, but damn if she doesn't run a tight ship. She walks over and hands me a piece of paper.
"Here's a class list. May as well go in alphabetical order." She heads to the door.
"Class, Max's mom is in charge. Please listen to her."
"Wait, where are you going?" I ask and I can hear a little panic in my voice. I don't want to be left alone with sixteen kids. I can barely handle my one.
"Just to the bathroom. I'll be right back." As she walks out, I see she's carrying her cell phone. Who does she need to call right now?
I consult the list and revel in the irony that it is the one I typed up for her at the beginning of the school year. When I look up, sixteen pairs of eyes are gazing at me, gauging my level of commitment to keeping order.
"Okay, Kit, you're up first. The rest of you, back to your letters."
Kit Aikens jumps up like she has just won bingo and skips out the door. Damn those lucky kids with "A" last names, always first for everything. I eye the rest of the group to let them know I'm not going to take any nonsense, and they all get back to work..
I take this opportunity to head to the long table at the back of the classroom and take out my weapon of mass destruction—the Sticky Chewy Five Napkin Brownies. Oh, they smell divine. I have a small twinge of guilt as I look at how nice all the kids look today, but when Kit Aikens walks back into the room with tears in her eyes and her beautiful blond curls tamed into a braid, I realize I'm on the side of right.
"Hey, Kit, come on back here. You can have a brownie while you work." I look at the list. "Hunter, you're up next."
Hunter dashes out the door with the reckless abandon that only a six-year-old can supply. Kit joins me at the back table.
"Do you like brownies?" I ask with a smile. I feel like I'm pushing drugs.
She nods and takes one hungrily. I hand her a paper towel. "You might need one of these."
When Hunter returns, his hair all combed back and spit-shined, I send Nick Baton out and invite Hunter back for a snack.
It's all going as planned until a severely braided Nancy Fancy gets back from her photo shoot with Miss Ward in tow. By this time eight of the sixteen kids are covered in chocolate. It's smeared on their faces, in their hair, on their clothes. Miss Ward walks in and just stares, her mouth gaping. I decide to just keep it moving.
"Nancy, come on back and have a snack. Lulu, it's your turn to go next door."
As Lulu walks out, Miss Ward is still standing in the doorway, taking in the splendor of my work in progress. She waits a good minute before she slowly walks to the back of the room, grabs a brownie with her perfectly manicured hands, and stuffs it in her mouth.
"Good brownies, Jenny." She walks back to her desk, licking her fingers.
I'm a little shocked. I never thought I'd have an ally in Miss Ward. Maybe Asami is getting on her nerves, too.
By the time Isabel Zalis comes back for her brownie, the class looks pretty comical. All the girls' hair has been put into some sort of braid, and all the boys have had their hair wet-combed out of their face. They have all done a stellar job with the brownies. Even poor Suni Chang, who did her best to stay neat, ended up with brownie on her nose. It looks as though someone has filled a room with 1930s-style gangsters and Pippi Longstockings and splattered mud all over them. The one exception is, of course, Graydon Cobb, whose hair is too short to grease back and who didn't have a brownie. Weirdly, it works. This is going to be one cute class photo.
* * *
To: Miss Ward's Class
From AChang
Subject: Class photos
Date: Jan. 19
Dear Kindergarten Parents,
I take full responsibility for yesterday's class picture fiasco. I assure you that under my watch the children were spic-and-span and their individual photos will look very sharp. However, because of a certain parent's unfortunate choice of snack, the class picture turned into a dog's breakfast. I have been assured, by the photographer, that it is "quite cute," especially since Miss Ward covered herself in chocolate to blend in with the class. We will see. Perhaps we can all pitch in and hire the photographer to come and reshoot the class picture.
Onward.
Asami Chang
To: AChang
From: SLewicki
Subject: Class photos
Date: Jan. 19
Hi,
I will be out of the office until January 31.
Thanks,
Sasha
To: AChang
From: SCobb
Subject: Class photos
Date: Jan. 19
Asami,
What snack? Was Graydon given chocolate? Do I have to give you his list of allergies again?
Shirleen
To: AChang
From: AGordon
Subject: Class photos
Date: Jan. 19
Asami,
I believe I can speak for most of the mothers of girls when I say the snack was the least offensive thing that happened yesterday. When I picked Lulu up, she was very upset that you braided her hair even though we had just washed and blown it out for picture day. She told me all the girls felt the same way. What were you thinking? When you sent us the emails about bathing our children and telling us you would be standing by to do hair, I thought you were kidding. I would agree to put in money for a reshoot, but only so we could get rid of the braids.
Ali
* * *
I'm a good daughter. At least, that's what I tell myself as I cross the bridge into Kansas City, Kansas, to find organic prunes for my mother. There is only one grocery store within a twenty-mile radius that carries the kind she likes. Apparently they act as a laxative for my dad, who, according to my mother, "can't get the train out of the tunnel." She is more than capable of driving to get them herself, but honest to God, I don't think she wants to spend the gas money. She's getting more like my grandmother every day.
It's actually a splendid day for a little road trip. It's mid-February—the thirteenth, to be exact. It's still cold, but the roads are dry and the sun is shining.
God, I love my KCK—that's Kansas City, Kansas, for those of you not from the Wheat State. It's where I grew up and what I know best. I remember when I was a kid my dad would take me and my friends up to Sauer Castle at night and scare the crap out of us with goofy stories that seemed so scary at the time. He'd talk about a guy with a crazy cat who lived in the castle and wasn't allowed out; then he'd pretend to see the guy in the window. We would all scream and laugh at the same time.
But now we live in Overland Park, Kansas, essentially a suburb of Kansas City, Missouri, which is generally known as Kansas City. The two KCs are spitting distance from each other, but sometimes I feel like a traitor for moving across the bridge.
Max is spending the day at the store with Ron, so I have a little extra time on my hands. Plus the place with the magic prunes is right beside this really cool coffee shop, called Grab a Java, that I love and hardly ever get a chance to go to. It's the kind of place where bearded lumber-sexuals and their female counterparts hang out. I feel hip just walking in there. It was the first place I ever ate avocado toast. I consider texting Don to see if he wants to meet me there, you know, for coffee, but ever since my conversation with Nina I have been trying not to instigate anything. Now I'm just a reactor.
I'm feeling pretty good about myself these days. The fallout from Brownie-gate was almost nonexistent. My sabotage efforts, though not in vain, turned out to be unnecessary. Asami took almost all the heat because of the hair debacle. And once again, as predicted by me, the chocolate-smudged class photo was absolutely adorable.
Physically, I'm feeling great. I'm at peak performance level for a woman of my age and commitment to exercise. That's what Garth tells me, anyway. I've cut back on my wine since January and plan to stay semidry until after the mud run. It's not like I have a drinking problem, but I am trying to eat and drink clean to help make my body a more efficient machine. My only indulgence is one cup of coffee a day, which is why I'm humming Katy Perry's "Roar" when I pull into the parking lot of Rupert's Fine Foods. I can already smell the Grab a Java brewing.
After picking up a shitload of prunes, some Ezekiel bread, coconut water, and kale, I head next door craving the double breve I'm going to revel in. As I'm walking, some yelling down the street grabs my attention. I look toward the sound and about fifty yards away are two women, a blonde and a brunette, standing beside a black SUV yelling at each other. The blonde is dressed all in black and the other seems to have a white jacket on.
I'm not much of a rubbernecker, but for some reason I'm intrigued. The words aren't clear, but both women seem to be giving as good as they get. Then, much to my surprise, the brunette hauls off and slaps the blonde across the face and boy it's a resounding smack. What can I say? We grow our women tough here in KCK!
I walk into Grab a Java and head to the counter, wondering under what circumstances I would slap another woman. Asami comes to mind.
Grab a Java is its usual groovy self. Today's barista is a nymphlike little pixie with cropped jet-black hair and a stud in her lip. The chalkboard sign tells me her name is Jack. Of course it is. No girl who looks like that is ever named Susan.
"Hey." I nod. She nods back. Very hip.
"Double breve, please."
She nods again. I look around the tiny shop. It has a rustic charm. Metal and wood tables are scattered around the room, as are barrels filled (not really) with coffee beans. The walls are black chalkboard and present the menu of drinks and food—limited but good. Did I mention the avocado toast? All kinds of quips are also scattered around the room; my favorite is "Dear Karma, I have a list of people you missed." It's surprisingly quiet for a Saturday—only three people hunched over their computers with their headphones on, a guy writing music notes on a piece of paper, and an older man reading the paper with a dog sitting at his feet.
"Double breve." Jack speaks her first words to me. "Four twenty-five."
I pay and toss the change into a jar labeled "Tipping—Not Just for Cows." Normally I would stay and savor my coffee—being here is like a little vacation—but my mother is probably waiting to stew up the prunes for my dad, so I jump back into the Odyssey and pull onto the street. First, though, I take a selfie in front of Grab a Java and text it to Don. So much for being the reactor.
I notice the battling women are still standing by the side of the road. I take a peek as I go by and lock eyes with Kim Fancy. Five things go through my mind immediately.
1. Hey! There's Dr. Evil.
2. I wonder if she knows about Grab a Java.
3. Who is she with?
4. Huh, I wonder what they were arguing about?
And finally,
5. Holy shit! One of them bitch-slapped the other.
I'm way past them by the time that final thought enters my mind. I try to remember who slapped who. They were both on the street, but I'm pretty sure the one in white did the smacking so that would be Dr. Evil. Well, no surprise there.
As I'm crossing back into KCMO, my cell phone rings. It's Nina. I put it on speaker.
"You are not going to believe what I just saw!"
"What's going on? Where are you? I need to talk to you."
"I'm driving home. Want to meet me?"
"Sure, but I'm hungry, so can we meet at the place with the signs?"
"I can be there in, like, ten minutes."
"Well, slow your ass down, 'cause I won't be there for twenty."
"K. See you there."
I laugh and slap the steering wheel. The caffeine is clearly kicking in.
* * *
The place with the signs is Nina's and my favorite little diner. It's actually called Stu's Diner, but that name just doesn't do it justice. Not only are there overstuffed red leather booths and an old-school jukebox that doesn't play anything released after 1977, but also the walls of the restaurant are covered in funny signs that the owner (not named Stu, oddly enough) has collected from across the country. If he couldn't steal the actual sign, he would take a picture and replicate it when he got home. Over the years, customers have sent him pictures of signs for him to hang as well. You can go there twenty times in a year and always find something new to read. Oh, and they happen to have the best apple pie in three counties.
The tiny place is packed, but as I walk in I spy a free table in the corner under a sign that says:
UNATTENDED CHILDREN WILL BE GIVEN ESPRESSO AND A FREE PUPPY.
I commandeer the corner and wave to Stephanie, the waitress on duty.
I don't know how long she has been working here, but she reminds me of the character Flo from the old sitcom Alice. Flo was a tall thin drink of water with a head of relentlessly bright red hair done up in a bouffant. She had a sassy southern accent and was always telling her boss to "kiss my grits" as she smacked her gum. Steph doesn't have a southern accent, but the rest rings pretty true.
"Be with you in a jif, hon!" she yells to me across the diner. Not one person lifts their head in surprise. Everybody knows Steph.
I take out my phone and check my messages. An IM from Nina saying she is "five away," a picture from Ron showing Max doing the flexed arm hang at the store, and a text from Don asking if he can join me. I realize that he thinks I'm still at Grab a Java. I IM Nina Here, send Ron a kiss and a hug, and text Don Sorry, no. I was just giving you a coffee update, to which I get an immediate sad-face reply. When I check my email, wouldn't you know there's one from Kim Fancy.
* * *
To: JDixon
From: KFancy
Subject: Was that you?
Date: February 13
Hi, Jen,
Was that you driving through KCK this morning? You should have stopped. Peggy and I were just meeting for a cup of coffee at that weird little place beside the grocery store. We were discussing the spring carnival.
See you soon.
Kim
* * *
I stare at my phone. Holy shit! I can't believe it. She slapped Miss Ward? What the hell?
Nina makes her entrance at this opportune moment. I wave enthusiastically at her. I'm practically jumping out of my chair.
"Hey, girl—" Nina starts.
"Shut up and sit down! You are not going to believe the gossip I have for you!"
"What?" Nina looks momentarily confused.
"Okay, so I'm over in KCK getting my mom some things at the organic grocery store she loves."
"The one by Grab a Java?" Nina asks unnecessarily.
"Yes."
"Did you stop in for a breve?"
"Neens, stop interrupting."
"Sorry," she grumbles. "I need coffee."
"Anyway as I'm going into Grab a—"
"What's up, girls?" Steph's voice makes me jump. "Apple pie's almost gone, if that's what you came in for."
"I'll have coffee and scrambled egg whites and wheat toast, no butter," Nina orders.
Steph nods and looks at me.
"I'll take the pie."
She nods again. As she is walking away, she points to the wall.
"Did ya see the new one?"
We both look at where she's pointing. It's a large piece of plywood with orange letters:
PLEASE DON'T THROW CIGARETTE BUTTS ON THE FLOOR. THE COCKROACHES ARE GETTING CANCER.
Nina laughs. "Nice one, Steph!"
"Came in from Tucson," she yells from behind the counter.
Nina looks back at me. "Okay, so you went shopping..."
I lean in.
"No. I went to the organic grocery store across the river to get my mom some prunes."
"Uh huh. Prunes." Nina seems distracted. I see Steph coming with her coffee, so I sit back and wait.
"Here you go, honey. Pie and toast will be up in a minute." She looks at me. "Ice cream or Cool Whip?"
"Neither, thanks."
I turn back to Nina. She is savoring her first sip with her eyes closed.
"Oh, my God, do I need this. You won't believe what I did last night. I—"
"Whoa, whoa, whoa. Me first. I have to tell you what I just saw."
"Seriously? I called you," Nina reminds me.
I sigh in frustration.
"Okay, I'll say mine and you say yours and we'll decide whose is better. Me first. I saw Kim Fancy slap Miss Ward across the face!"
Nina's eyes widen. "Oh, man, that is good."
"What's yours?" I ask, pretty confident that I have won.
"I had sex with Garth last night."
I look at her evenly. "You totally win."
Nina nods knowingly and takes a huge gulp of her coffee.
"Holy shit! How? When? Why?" I have more questions than I know what to do with.
Nina is about to answer when Steph descends upon us with eggs and toast and pie. She also puts the check down.
"More coffee, hon?" she asks.
Nina nods gratefully.
"Okay. Spill it."
She sighs. "It was our third date."
"Third date? He never said a word." I'm shaking my head. It always amazes me how men can keep a secret. You ask them not to say anything, and they actually don't. We women could learn a thing or two from that.
"We didn't want to freak you out, so we decided to keep it on the down low for a while."
"Okay, whatever. How did it start? Did you start his website?"
She looks at me with surprise. "You heard about that?"
I nod. "Garth mentioned it, but didn't tell me any of this."
"Well, we sort of connected at your Christmas dinner. I was still in my Sid funk, and he was just really nice to talk to, you know?"
I smile. I certainly do know.
Steph swoops in with a refill and is gone.
"So after that, he would call once in a while to check in, and we just started having these great phone conversations, first about his website and then about everything else. I told him all about Sid and my parents and grandmother—about how I raised Chyna by myself. He told me about being in Afghanistan—girl, he saw some serious shit over there. He told me about his breakdown at the gym and how he still goes to therapy for post-traumatic stress."
I can't believe my ears. How does she know more about Garth than I do?
"He told you all that?" I ask.
"Well, I asked him about it. We've been talking a lot."
"Talked yourselves right into bed," I say, with a bit too much snark.
Nina raises her beautifully arched eyebrow at me.
"Sorry. I just can't believe I didn't know any of this. Wait, did you have phone sex?"
Nina snorts coffee out of her nose and starts to giggle.
"No. Oh, my God, no. Only you would ask that. No, we were talking one evening and realized we had both skipped dinner, so we decided to meet at Garozzo's. We had a great time together. Did you know he doesn't drink alcohol or eat pasta?"
"No, but I'm not surprised. He's in great shape."
"I hear that!" Nina says, and I think she's blushing. I sit back in my chair, trying to process all this new information. The tectonic plates of my world have had a true rattling this morning.
"By the time we had our second date, it felt like our twentieth," Nina feels the need to add.
"Uh-huh. Are you trying to rationalize your sluttiness to me or to you?"
"To you." Nina doesn't bat an eye. "I'm all good with my sluttiness."
"Are you charging him for your, um, services?"
"Yes!" She smirks. "But I'm giving him the special friends discount."
My phone buzzes and I sneak a glance at it. Don has texted me an emoji of a turd drinking coffee. I turn my phone face down.
Nina's crystal blue eyes are staring at me. "Something important?"
"Nope. So, was Chyna home?" I deftly pivot.
"No, thank God. That's why I wanted to meet you. When did you tell the girls about Ron?"
I scoop the last of the scrumptious apple pie into my mouth and scowl. The girls' first time meeting Ron is not one of my favorite memories.
We had been dating for about a month before I even told him I had kids. I wish I had taken a picture of the look on his face. Here he thought he was dating this hot (his word, not mine), single thirty-something who had never been married and who seemed relatively normal. After pleasuring him in the front seat of his car one night, I casually mentioned that I was harboring two small fugitives in my home. He took it relatively well. At least, he didn't run screaming in the other direction.
It was another month before I let him meet them. Shrinks these days will tell you to wait a year, but that wisdom wasn't available to me, so I went with my gut. (Actually, I'm sure it was available somewhere, but I'm generally lazy when it comes to researching stuff like that.)
I invited Ron to dinner one Saturday when my parents were away on a spiritual retreat or, as I like to call it, a booty call with the Lord. There was no way I was going to bombard him with two kids and Kay and Ray.
The girls knew that I had been going on dates with someone, but they also thought I was taking a pottery class at the local Our Name Is Mud. It was the only way I could get out more than twice a week. Ron and I were at that euphoric beginning of a relationship where we couldn't keep our hands off each other, and we were having a lot of car sex. When I told the girls I was going to introduce them to the man I was dating, they reacted like the polar opposites they are. Vivs rolled her eyes and said, "Well, this should be good." Where does her sarcasm come from? Laura started jumping up and down, beyond excited, asking if he was going to be our new dad. I thought she was joking, so in my infinite wisdom I told her yes, he absolutely was, as long as they didn't blow it for me. But guess what? She wasn't joking, and she didn't think I was, either.
Ron arrived promptly at six with gift bags for the girls, who were ten and twelve at the time. I could tell he was nervous, and not just because he had pit stains the size of pizzas under the arms of his gray polo shirt. He was very jittery and kept looking around the kitchen as if someone was going to jump him. I kissed him, handed him a beer, and told him to relax.
"They're just little girls," I assured him with as much conviction as I could muster. I really wasn't sure how the evening was going to play out. Good thing I didn't have high expectations, because it ended up an unmitigated disaster.
The first to appear was Laura. She was wearing her prettiest dress and had tried to put her hair in a bun, without much success.
"Sweetie, I'd like you to meet my friend Ron. Ron, this is Laura, my little one."
"Hi, Laura. So nice to meet you." Ron stuck out his hand for a formal greeting, which Laura bypassed in favor of a huge hug.
"Welcome to our family," she said sincerely.
I guess I should have seen that coming.
"Thank you." Ron looked a little confused, but to his credit he went with it. "It's so nice to meet you, Laura. I love your hair."
Laura looked surprised and pleased. "Really? I just did it all by myself."
"Well, I helped," was how Vivs announced she had arrived in the kitchen. She had chosen to wear all black, which included the expression on her face.
"Ron, this is Vivs. Vivs, this is my friend Ron."
Ron just smiled this time, but it was Vivs who formally put her hand out. Ron shook it.
"My grandpa says you can learn a lot about a man by his handshake," Vivs informed him. "Yours is wet." She wiped her palm on her black pants.
"Oh. Sorry about that," Ron mumbled.
"Vivs!" Laura frowned at her sister. "Maybe he just washed his hands."
"Who's hungry?" I jumped in before it all went south.
"I am!" Ron said, a little too enthusiastically. "How about you guys?" He looked toward the girls. Laura nodded like a bobblehead. Vivs ignored him and turned to me.
"What are we having?"
"Lasagna."
She made a face like it was the worst thing I could be serving, even though it was one of her favorites. I gave her my stone-cold stare.
"Cut it out," I said quietly.
Ron picked up the bags he had dropped by the door.
"Hey, I brought you guys something."
He handed the girls identical lime-green gift bags tied with pink ribbon.
Laura stepped forward and took the bag shyly.
"Thank you so much. I love it."
"You haven't even looked at it yet." Vivs rolled her eyes and held out her hand to take her bag from Ron.
"Thank you."
"It's hard to buy for people you've never met, so if you don't like it you can exchange it," he assured them. I gave him a "You're doing great" smile.
The girls simultaneously opened their bags to find matching pink Gap sweatshirts and a large Hershey kiss. The perfect gift for Laura and the absolute last thing my newly goth devil child would want.
Laura gave an overexaggerated gasp. "Oh, pink is my favorite color! Thank you so much." She immediately put the sweatshirt on and gave Ron another hug. Meanwhile, Vivs and I were in an evil stare-down. Her big brown eyes registered contempt, and my eyes said, "I dare you to say anything but thank you."
I laugh when I think about it now, but at the time I was convinced that I would never see Ron again. But he showed up the next night just before dinner with a black Gap sweatshirt for Vivs. That didn't win her over completely, but it was a glimpse into the thaw of what was certainly the brief ice age of her tween years.
I look up and see Nina waiting for an answer.
"I guess it was about two months. We had a rough start with the girls, remember? Vivs and that goth phase?"
"Oh, my God, the one she couldn't quite commit to?" Nina and I crack up at the memory of Vivs acting all dark and dangerous until a Backstreet Boys song came on the radio. Then she would forget herself and start singing her head off. When our laughter turns to sighs, I look directly at Nina.
"Are you sure about this?"
"About what? Garth?"
"About Garth, Chyna, everything. I feel like you just got out of the Dumpster. Are you ready to jump back into something? I mean, you barely know him."
"Are you kidding me? Do you know how long it's been since I've had sex? I know I was hung up on Sid, but shit, no one is more ready than me."
"Well, I'm glad for you, but I think you should wait to introduce him to Chyna."
Nina nods and finishes her eggs.
"Now, tell me"—she licks her lips—"who bitch-slapped who?"
* * *
That evening, as I'm making Max's favorite dinner, skillet tacos, I ponder the events of my day. As if the one-two punch of what I saw in KCK and Nina's news wasn't enough, I'd still had an entire day of errands to tackle. I got the minivan washed, took a load of stuff to the dry cleaners, replaced the battery in Ron's favorite watch, talked to our local kids' gym about Max's sixth-birthday party next month, and spent a half hour on the phone with Peetsa analyzing the Dr. Evil/Miss Ward smackdown. Her theory? That Miss Ward has been making eyes at the dashing David Fancy, and Dr. Evil was just protecting her territory. I have to say, it has some legs. If Miss Ward flirted with Ron, I'd definitely be scaring her straight with a few choice words, but physical violence? Not unless she slept with him. But in that case, I'd be saving most of my rage for Ron.
I sigh as I take a sip of cooking wine, so named because it's the wine I drink while I'm cooking. I look at the clock and realize the boys should be home any minute. Skillet tacos are ready and Chyna is on her way over to babysit. Ron and I are going out to celebrate Valentine's Day. It isn't until tomorrow, but Ron likes to take me out for what he calls Scoundrel's Night. Apparently it's the night before Valentine's, when men take their mistresses to dinner. He thinks it's sexy, and who am I to argue? It's also a lot cheaper, and you aren't locked into one of those stupid theme menus that every restaurant seems to think is necessary on February 14.
I grab my wine and sit down at the kitchen-counter office to check my emails. Hmm... Overstock.com is having a Valentine's Day sale. Nothing says "I love you" like discount furniture. There is a note from my mom thanking me again for the prunes, which apparently did the trick for my dad. There is an email from Laura with the itinerary for our family ski trip to Utah in March, and, to my great surprise and horror, I see an email from Asami Chang. I take a deep breath and click on it.
* * *
To: JDixon
From: AChang
Subject: A question...
Date: February 13
Hello, Jen,
When you were class mom, did you ever hear from Sasha Lewicki, aside from her out-of-office reply?
Asami
* * *
I have to laugh. I want to tell her Sasha and I are best friends and see each other every weekend. But I don't, because I'm trying to turn over a new leaf and embrace the Asami. I'm also slightly curious about why she's asking, so my reply is friendly and open.
* * *
To: AChang
From: JDixon
Subject: A question...
Date: February 13
Hi, Asami,
No, I never did, but she sent sushi to the curriculum night party, so I
know she exists. LOL!
Jen
* * *
I know, I know: weak joke when you have to put LOL at the end, but since there is no definitive proof that Asami has a sense of humor, I thought I'd spell it out for her. Her reply is almost instantaneous.
* * *
To: JDixon
From: AChang
Subject: A question...
Date: February 13
Jen,
I don't think she does.
Asami
* * *
I blink three times and stare at the screen. What the hell does that mean? Sasha Lewicki doesn't exist? I'm working through this thought when the Dixon men come tramping through the door. They spent the morning at the store and the afternoon attempting to ice-skate at the indoor rink. Ron wanted to take Max to the frozen pond, but I suggested that if it was too cold he wouldn't like it and the whole experience would be over before it even started. If Ron took him to the indoor rink and plied him with hot chocolate, he would definitely be more cooperative.
"Mom!" Max yells needlessly.
"Hey! How was your day?" I ask as I pull off his leopard-print jacket, orange hat, and soaking-wet striped mittens. I notice that he has a black scarf tied over his lime-green pants. It looks like a skirt.
"It was awesome! I made it all the way around once by myself." Max's cheeks are rosy and his eyes are shining. My heart bursts with love for this little munchkin. I look to Ron for confirmation and he nods.
"Next stop, hockey." He grins.
"Or I could do what that guy was doing in the middle of the ice—remember, Dad?"
Max starts to spin around in the middle of the kitchen.
"Figure skating," Ron mouths to me, and I have to look away so I won't laugh at the disappointment on his face.
"Looks cool. Now go wash your hands. Dinner's ready. Skillet tacos just for you."
"Ninja!" Max yells, and runs to the bathroom off the kitchen.
"How was your day, babe?" Ron gives me a quick kiss on the lips and heads to the fridge.
It's such a simple question, but with so many possible answers. I decide to keep it brief.
"Well, let's see. My best friend slept with my trainer."
Ron registers only mild interest. Why don't men ever react the way you want them to?
"Really? I didn't know they were a thing." His head is in the fridge so I can barely hear him.
"I just found out myself. I'm a little freaked out."
Ron turns around with one of Max's Danimals in his hand.
"Why?"
"I think Garth might be a player."
"A player?" Ron cracks up. "I don't think so." He downs the Danimals in one gulp.
"Why not?"
He keeps laughing.
"Well, to be a player you've got to have game. And that man just does not have it. I mean, he's a nice guy, but there's no way he's playing Nina."
"I hope you're right. I don't think she could take another heartbreak."
"What heartbreak? They slept together once."
"Sometimes once is all it takes for a woman to fall in love. A man, too, by the way."
Ron looks at me skeptically. "Did you fall in love the first time we had sex?"
"Well, no. But it was three sweaty minutes in the back of your car. All we did was burst the dam of lust that had built up. The first time we made love in a bed, I was pretty swept away."
"So location has something to do with falling in love."
"Oh, my God. Are you even listening to me?"
Just then, the back door opens and Chyna walks in. At the same time Max returns with clean hands and an empty stomach, so I know that our discussion is over. Ron turns and heads into the living room and I'm relieved, because clearly I am arguing myself into a corner and making no sense to anyone.
"Chyna! Sweetie. How are you?" I give her a hug.
She smiles and hugs me back nice and tight.
"I'm good."
She looks so much like her mom that I often wonder if she has even one drop of Sid's DNA.
"How's your mom?" l ask as I fill a plate for Max and place it in front of him.
"Really good. She's been in such a great mood lately."
"I've noticed that, too. Hey, can you sit with Max while he eats? I have to get ready."
"Sure thing. Hey, Max, whatcha eating?" She sits down beside him.
"Skillet tacos." Max answers with his mouth full, of course. "Want some?"
"Yeah, I do!" Chyna knows she can help herself to anything in our house. We have an open-fridge policy.
While they eat, I dash upstairs and find Ron in the shower. I head into the bedroom and check my phone. Two texts from Don. One is a selfie outside the Starbucks near school and the other says:
Do you have time for a Valentine's Day coffee tomorrow?
I actually do have time tomorrow, but I hold off texting him back. Not sure where the line is these days, but I think that would definitely be crossing it.
* * *
J. Gilbert's is the best steakhouse in Overland Park. Their dry-aged steaks are phenomenal and the restaurant has a comfortable old-school feel with its mahogany furniture and crisp white table linens. There isn't one waiter under the age of fifty, and they are formal to the point of being rude.
But the waiters are worth tolerating because J. Gilbert's happens to serve the most delicious onion rings I have ever tasted. They are pretzel-coated and served with three dipping sauces that are so good I don't know which one to have first. Ron knows it's my favorite fancy place, so he surprises me once in a while. Tonight it genuinely is a surprise, because we were just here for New Year's Eve.
"Twice in two months? Are you cheating on me?" I narrow my eyes at him over my menu.
"Actually, New Year's was kind of a bust, if you remember, so I thought we deserved a do-over." Ron gives my hand a squeeze across the table.
He's being kinder than I deserve. I was still in my class mom funk on New Year's Eve and I was determined not to have a good time.
Mission accomplished, by the way. Not only did I have a shitty time, I was able to suck the fun away from everyone at our table, which was filled with Ron's favorite customers and their spouses. It's one of my superpowers, along with growing a person in my stomach and peeling labels off beer bottles intact. It was definitely not my finest moment as the wife of a successful sports-store owner, but in the moment I felt more than justified in forcing my pity party on everyone.
So on the most romantic of holidays (ahem), I'm thrilled to embrace this do-over night with the love of my life and an excellent bottle of '94 Turley Zinfandel (clearly I'm taking a little break from that clean eating and drinking). We order dinner, then sit back and enjoy our first few sips of wine. What a perfect night.
Just as I am getting my first hint of a buzz on, my eyes are pulled across the room to a couple being seated on the other side of the restaurant. Both tall and thin, him with short salt-and-pepper hair and her with long brown hair flowing down her back.
Well, well, well, if it isn't the dashing David and Kim Fancy, celebrating Scoundrel's Night at the same place we are. My mind starts to click through the events of this morning—the bitch slap and all—and I try to piece together why these two would be out for a romantic dinner.
"Jen!"
"What?" Ron's voice snaps me out of my reverie.
"What are you staring at?" He looks annoyed.
"Sorry, babe, I just noticed the Fancys sitting across the room and wondered what they're doing here."
Ron shrugs.
"Same as us, probably."
I doubt it, but say nothing. Instead, I ask him to tell me all about his ice-capades with Max this afternoon.
"Man, he really loved it. Good call on the indoor rink, by the way." He raises his glass to me in salute.
"Did he really go around the rink by himself?"
"I was right behind him, but yes, he did." I can tell Ron is proud. "It didn't take him long at all to find his legs. Now, if I can just get a stick in his hands..."
I give him an encouraging smile while he unfolds his long-term plan to get Max to the NHL. As I take a sip of my insanely delicious wine (seriously, if you can ever find a bottle, you will not be disappointed) I glance back over to the Fancy table. They are sitting across from each other, but are both leaning in. Kim seems to be doing a lot of talking while the dashing David just nods and listens. Is she reprimanding him for his affair? Is she telling him that she smacked his mistress around? That he better not dare step out on her again or there will be hell to pay? Damn, I wish my lip-reading skills were better. Or, you know, existent. The waiter, walking to our table, interrupts my view.
"Petite filet for the lady and porterhouse for you, sir. Enjoy." He turns on his heel and walks away just as a second waiter arrives with our side dishes and of course, my onion rings. We dig in.
Dinner is so good I forget about the Fancys across the room. Ron regales me with tales from the trenches of retail and has me howling over an incident with a woman who wanted to return a tennis racket because she said it didn't improve her game.
"How long had she been using it?"
"About a year." Ron shakes his head. "The grip was worn down and everything. She threatened to call the Better Business Bureau if I didn't give her a refund."
"So what did you do?"
"I told her this wasn't Costco and that there was no proof she had even bought the racket at our store, but I'd be happy to sell her a new one with a forty percent discount."
"That was generous."
"What are you going to do? Even a bad customer is still a customer."
I drain my wineglass and sigh contentedly.
"Thank you, my darling, for this do-over dinner. I love you so very much."
Ron grins. "Now, that's the booze talking, but you are so very welcome."
As we stand to leave, I look around to see that we are just about the last table in the restaurant. I love when that happens. You get into a cocoon of conversation and the entire world disappears around you.
Ron heads to the bathroom and I check my phone. There's a text from Don.
???
I get a tight feeling in my stomach. How could I even think about having coffee with another man after such an amazing evening with my husband? He doesn't deserve that. I text back immediately.
Nope. Sorry. Very busy day.
I put the phone back in my purse as Ron joins me.
"Everything okay at home?" he asks, assuming I was checking in on Max. Jesus, strike two. I really need to get my head out of my ass.
I nod to him and can only hope I'm right.
* * *
Ron has a tight hold on me as we walk through the parking lot. Half a bottle of wine is a lot for me, and I'm a little unsteady. A car pulls up beside us.
"We meet again." Kim Fancy's voice floats from inside a silver Mercedes. "We saw you guys in the restaurant, but you seemed so deep in conversation that we didn't want to disturb you."
"We're celebrating Scoundrel's Night," I say with a slight slur. "How 'bout you?"
I hear a snort from the dashing David, who is behind the wheel.
"Nice," he says to Ron appreciatively.
"I don't get it." Kim sounds annoyed.
Ron decides to explain.
"We're celebrating Valentine's a day early. Same with you guys?"
Kim Fancy lets out a very un-Fancy-like guffaw.
"God, no. Tomorrow I expect to be going somewhere much nicer than this place."
And once again I'm reminded why I don't like Kim Fancy.
* * *
To: Miss Ward's Class
From: AChang
Date: 2/20
Subject: Class trip
Hello, parents,
I'm sorry to interrupt your week-long celebration of Mr. Lincoln's birthday, but I have not yet heard from any of you regarding chaperoning the class trip to the Underground Railroad Museum on February 28. I have, however, heard from many of you about the absence of a Valentine's party in the classroom. Miss Ward, who does not like to celebrate Valentine's Day, was not in favor of a party. I'm sure you parents were more disappointed than the children.
Anyway, I would like to know who is volunteering for the class trip. Even if you have been on a previous class trip, you can volunteer again.
Thank you,
Asami
* * *
I'm finding Asami's class mom emails more and more intriguing. She obviously caught shit for not having a Valentine's Day party. I knew it was never going to happen, given Miss Ward's aversion to "Hallmark holidays." But seeing her beg for volunteers (oh yes, I would say opening it up to me and Peetsa is her way of begging) is interesting. I think Asami is learning the hard way that (creds to the great Erma Bombeck) the grass is always greener over the septic tank. She will get no more grief from me. She is clearly getting enough from everyone else. Looks like karma found its target after all.
* * *
I can barely look at Garth as we work out on Monday. I know, and he knows I know, and neither of us has said anything. It's very distracting, picturing Garth naked and sweaty on top of Nina. I blanch for the fiftieth time in the past hour and try to keep my focus on the task at hand, which is using all my body weight to try and push Garth over.
"Come on Jen, push! Push me like you hate me!"
That's it. I double over laughing. I can't help myself.
"Is that how you sweet-talked Nina?" I'm panting and laughing at the same time.
Garth gives me a pensive look. "Why, yes, it was, but I said it with a growl. It's pretty sexy when I growl."
I stand up and hug him—for the first time, I think. "I'm guessing it must be."
I grab a towel from the nearby dirty laundry basket and wipe off my face and arms.
"I'm really happy for you guys. It's a big move for Nina."
"For me, too," Garth says, and his sincerity makes me want to know more.
"When was the last time you had a girlfriend?"
"A girlfriend? Probably ten years."
"That's quite a dry spell." Have I mentioned I enjoy stating the obvious?
Garth laughs. "That's putting it mildly. But it wasn't exactly dry. I kept company with some very nice ladies over the years."
I guess that's his polite way of saying he slept around, which doesn't sit well with me at all.
"I hope that's not what Nina is."
I can see the surprise register on his face.
"I don't know what she is yet. But I know I like her very much and I'm pretty sure she likes me."
I chew my cheek and debate whether I should say more or shut up. Sadly, saying more wins out.
"Well, you should know that she hasn't 'kept company'"—I make air quotes—"with anyone since Sid left, so, whatever your feelings are, please keep that in mind."
And then I see it in his eyes. I have managed to piss off the nicest guy in the world.
He takes a deep breath.
"Jen, I know Nina is your best friend, but I think you should just keep out of it, at least until we figure it out." He turns and starts putting equipment away.
Over his shoulder, he mumbles, "I think we're done for the day."
I really don't like the way this is playing out.
"Garth, wait." I'm not sure what I want to say, but I don't want to leave it like this. I need to lighten things up.
"I'll kick your ass if you hurt her." I smirk.
He smiles and shrugs. "And what happens if she hurts me?"
Since that hasn't even occurred to me, I don't have a ready answer, so I say the first thing that pops in my head.
"Then she'll be the one getting the ass kicking."
* * *
Garth and I head upstairs and say good-bye. I walk to the kitchen for some water and do a double take when I see Vivs sitting at the table playing cards with Max.
"Aren't you supposed to be in New York?" I know she and Raj had planned a romantic getaway to the Big Apple. Well, actually New Jersey, where Raj has an uncle who lives in Teaneck. But they had plans to hang out in the Village with some friends.
"Don't ask," Vivs sing-songs. She slaps a card down and yells, "Uno!"
Max jumps up from his chair "No, Sissy! Not gonna happen." And he slams down a multicolored card. "I'm changing it to red! Take that, sucka!"
"Whoa! Excuse me, what did you say?" I seriously can't believe that just came out of Max's mouth.
"I said, 'Take that, sucka,'" he repeats, with a little less enthusiasm. "Graydon says it all the time."
"Well, just because Graydon says it doesn't mean you can. It's just not a nice thing to say to someone." I consider calling Shirleen to share this little nugget about her perfect son. Maybe later. I turn to Vivs. "How did you get here?"
"I drove with Laura."
"She's here too?" I can't remember the last time the girls were home for Presidents' Week.
Vivs puts down her final card. "Red Maxazillion!" she yells, and then whispers, "Take that, sucka."
Max giggles and starts gathering up the cards. I give Vivs a disapproving scowl.
"So where's Laura, upstairs?"
"She's at Town Hall helping this guy set up for his show tonight." Vivs tosses this at me as though I should already know. I'm so not in the mood for this.
"What guy? What show? You girls think you tell me things, but you don't. And why aren't you in New York?"
Vivs sighs and gives me her "I'm barely tolerating you" look.
"I'm not in New York because Raj blew me off to help work on the plans for the new addition to the library at school and Laura has been dating this bass player and his band is playing Town Hall tonight. There, all caught up." She turns back to Max to start another game of Uno.
There is so much mind-splintering information in that one sentence that it takes my almost-fifty-year-old brain a few seconds to process it all. I pick the most shocking thing to attack first.
"Laura is dating a musician?" Oh, my God. Where did I go wrong?
"She only started seeing him about a week ago. He's nice. The band sucks, though."
"How come she can say 'sucks,' but I can't say 'sucka'?" Max jumps in.
"Max, can you go play with your helicopter while I talk to Sissy, please." It's a command more than a request.
"But we just started a new game."
"We can play later, bud. I'm here till Saturday."
Max seems satisfied with this promise and runs up to his room.
Now that I'm alone with Vivs, I grab a bottle of water and a yogurt from the fridge and join her at the table.
"Okay," I sigh, "from the beginning, please. What is Raj doing?"
Vivs sighs and shuffles the Uno cards with vigor.
"He got asked to work on the plans for the new library at school. So he bailed on New York."
"Isn't it a big deal for him to be asked to do that?"
"Yeah, but we've been planning this trip since Christmas. He chose work over me. That's just not a good sign for our future."
I nearly burst out laughing. For someone so smart, Vivs can be so obtuse, not to mention self-absorbed.
"Sweetie, he didn't pick work over you. He picked a once-in-a-lifetime opportunity over a trip that can happen pretty much any time. Surely to God you see that."
Vivs finishes shuffling and puts the cards back in the box.
"I guess I do. I mean, I do. I'm just bummed. And I made him feel guilty about it, so now I feel worse. Am I the biggest bitch on the planet?"
Not the biggest, I think but do not say.
"Not even close." I stand and walk around to her side of the table to give her a hug.
"Just make sure you apologize to him. I mean, it's pretty huge, what he's doing."
She nods. "Maybe I'll make him some cookies. Or you know he loves the Five Napkin Brownies."
I decide to change the subject before I get roped into making them.
"So tell me about Laura's new guy."
She shrugs. "I really don't know that much. His name is Travis, he's a creative writing major, he's my age, and he plays in this band that's pretty popular on campus, although I don't know why. Their music is so average. The lead singer is this hot Asian guy. I'm surprised she didn't go after him."
I can't imagine Laura "going after" anyone.
"What's the name of the band?"
"Sucker Punch."
"Eww." I can't stop my reaction.
"I know, stupid name, stupid songs. I don't know what people like about them. But Travis seems okay. I promised Laurs I'd go with her to see them tonight. I'm dreading it."
"How bad can they be? They have a gig at Town Hall," I offer.
"Yeah. On a Monday."
She has a point.
"Wanna come?"
"Seriously?" I can't believe my daughter is asking me to hang out at a concert.
"Sure, why not? You can tell me if he's any good on bass."
Bass players were never my specialty, but the thought of going to hear a rock band is mildly pleasing. I haven't revisited the shame of my youth since Max was born. I wonder if I can convince Ron to go.
"Maybe. Let me talk to Ron and see if I can get a babysitter."
I pick up my phone and notice a text from Don.
Hey, is everything okay? I didn't mean to freak you out by asking to go for coffee on Valentine's Day.
Truth be told I wasn't freaked out, but it was a bit of a wake-up call for me. I had made a promise to myself that night that I would dial down the flirt.
You didn't. I'm just crazy busy.
Well, my offer for coffee stands.
Good!
Back in reactor mode where I belong, I start to make plans for tonight.
* * *
I secure Chyna to babysit, and reach out to see if anyone else wants to join us for dinner and a "show." That's how I present it, anyway. The Tuccis are in, and it only takes a little begging to convince Nina that she and Garth should give their binge-watching of some series on Netflix a rest and come out to have fun. I feel like I need to see Garth and Nina together to get a sense of what's really going on. I have a sneaking suspicion that I'm the only one creating drama in their relationship.
As I'm cleaning up the lunch dishes, Laura comes in the back door and rushes to give me a hug.
"Mom!"
"Hey, sweetie. Such a nice surprise that you guys are home." I hug her back.
"I know! It was kind of a last-minute decision."
"I heard." I give her an expectant look.
"Did Vivs tell you about Travis?" She is blushing.
"Yes. He sounds really cool. Can't wait to hear him play."
"Wait, are you coming tonight?" I can't tell if she is excited or horrified.
"Yes! Vivs invited me and I invited Peeta and Buddy and Nina and Garth."
"Oh, my God, Mom! Are you kidding me?" Yup, horrified.
"No, it'll be fun!"
"Not for me! I don't want Travis to have to meet all of you." She's whining now.
"Don't be silly. He doesn't even have to know we're there."
"Oh, yeah, right. I'm going to kill her. Vivs!" she suddenly screams at the top of her lungs.
"Laura! Stop yelling. We're going to come and see Sucker Punch play"—try saying that with a straight face—"and have a fun night out. If you don't want us to meet Travis, fine. We'll meet him another time. But we're coming. I haven't seen a live band in ages."
Laura is still sporting a pout, but she doesn't say anything else except "Fine," and stomps up the back stairs, no doubt to lay her wrath at Vivs's doorstep. Better Vivs than me. I need to find a place for all of us to eat before the concert.
* * *
After dinner at our local Bonefish Grill—picked because of its proximity to Town Hall—we all stroll over to the venue. I have been eyeing Nina and Garth all night and I've come to the conclusion that I will learn nothing from watching them. They don't give anything away except the idea that they are a very happy couple.
We all get in line to buy tickets and discover that, believe it or not, Sucker Punch is not the headliner. That distinction goes to an all-girl Led Zeppelin cover band called Lez Zeppelin. Sucker Punch is one of two opening acts.
Ron has his arm around me as we walk through the door, give the tattooed gentleman $40, and get our hands stamped. Our hands stamped! It's been too many years since I've had the telltale sign of a night out imprinted on the back of my hand. I can't help feeling a bit giddy.
We stop by the mandatory coat check and entrust our garments to a painted lady whose T-shirt says "Call Me Maybe." She doesn't give us a ticket in return, and I can't help but wonder if I'll ever see my black sheepskin jacket again.
Town Hall is exactly what it sounds like—an all-purpose venue used for everything from Christian revivals to Zumba classes. It is a very large rectangular room with a stage at one end and a balcony that runs along the other three sides. It has the feel of a school gymnasium but without the smell of utility balls. You can pretty much hold any event here, but you cannot serve alcohol. It's a municipal rule that has been challenged many times over the years but has yet to be changed.
The joint is packed with the unwashed youth of KC. Okay, they probably aren't unwashed so much as trying to look that way. I don't really get the whole ripped-jeans, holes-in-the-T-shirt, bedhead look that seems to be so popular, but then again, I'm sure my mom didn't get the slutty-biker-chick look that I sported as a teen.
There are no seats, of course, so we all just stand around waiting for something to happen. Soon enough, a group of kids, dressed pretty much like their audience, ambles onto the stage with guitars and start plugging into the amplifiers. There is some ungodly noise as they get tuned up and I realize my heart is racing. Man, does this bring me back. I turn to Ron with what I'm sure is the goofiest smile I've ever displayed.
He raises an eyebrow. "What?"
"Nothing. I'm just... it's fun to be here, that's all."
He smiles and nods like he's humoring a mental patient. I look down the line at the rest of our crew, but don't see anyone quite as excited as I am. Peetsa is plugging her ears, and Buddy is yelling something in Garth's ear. I notice Vivs has joined us and is talking to Nina. I walk over to them.
"Is this Sucker Punch?" I yell.
"No. Next band. This is Grope. High school kids."
Just then Grope's lead singer steps up to the mic.
"Good evening, Kansas City!" he screams, and immediately gets feedback from the monitors.
"Whoa. Sorry about that. Uh, we're Grope and we're here to get you rockin'. One two three four..."
And it begins. Grope starts rattling out a song about going to hell. It's actually not too bad, considering how young they are. I look to my right and see my friends nodding their heads to the beat. Okay. Not a disaster. Grope's second song is a ballad—a sensitive song about a bitch who did him wrong. The melody is good, but these guys really need help with their lyrics. "I was trashed so I crashed at her hash-pad." What?
They get the crowd going with their third and final song—a pretty decent cover of "London Calling," by the Clash. Always leave 'em wanting more, Grope. Well done.
After a quick changeover, Sucker Punch hits the stage. Four guys—all adorable—take their places with much more panache than Grope. Vivs is right, the lead singer is a very good-looking Asian boy/man. He's obviously in his twenties, but has a young look. He reminds me of John Cho, the guy from Harold and Kumar.
I look at the bass player and see a thin blond guy with a very cute smile and a great ass. I know it's not something I should be noticing about my daughter's boyfriend, but I do still have a pulse, and may I say he is wearing his skintight khakis very well.
The lead singer introduces the band, then says, "Let's hear it one more time for Grope."
The audience gives a generous round of applause and then Sucker Punch kicks off their set with a slow song that eventually builds to a frantic beat. I don't love the song, but I can tell these guys are talented musicians. Travis is totally in tune with the drummer and doesn't grandstand like the Grope bass player did. I can already tell he's a nice guy. I feel my hand being squeezed and I see Laura has joined us in the audience.
"What do you think?" she screams in my ear, so loudly that I feel reverb.
"They are awesome!" I scream back. She looks relieved.
We listen to all three of their songs together. Laura knows every word, and my heart aches just a bit. I remember being that girl.
When Sucker Punch finishes their set, the crowd shows its appreciation by asking for an encore. I'm a little surprised. I mean, they weren't that good, and Lez Zeppelin's fans have been waiting patiently through two boy bands.
Laura is beaming.
"Travis is going to be so happy. They're not allowed to play another song, but he was hoping people would want one."
Peetsa, Buddy, and Ron join Laura and me to give their two cents.
"This is so fun!" Peetsa enthuses. She is still yelling, even though the music has stopped.
"Great bands!" Buddy adds. I don't believe either of them.
"Where is everyone else?" I realize half our party is MIA.
"They all went to watch from the balcony," Peetsa explains.
"Mom, do you want to come and meet the band?" Laura asks. I'm floored. I didn't think she even wanted me here. I guess I've passed some unspoken test.
"You mean go backstage? You don't have to ask me twice!"
Laura winces. Whoops. Not cool, I guess.
"I mean, sure," I rebound. "You guys wanna come?" I ask the Tuccis.
"I think we'll hang here and catch the Led Zeppelin band thing," Buddy says, and Peetsa nods her agreement. I give them a knowing smile and follow Laura to the door beside the stage.
Town Hall isn't exactly Wembley Stadium, but backstage still has the slightly seedy electric undercurrent of exclusivity. I'm in my wheelhouse. The only things missing are beefy security guards, a bevy of skanks, and a pass around my neck. And alcohol. But whatever, I'm here and I'm feeling great.
Laura leads me through a common area with sofas to a hallway by the back door where Sucker Punch is packing up their instruments. Laura runs to Travis and throws herself at him. Good lord! I guess I never taught her the art of playing hard to get. Actually, I never practiced that art, either.
When they come up for air, she is rapturous.
"You were amazing! You sounded so good."
Travis seems pleased.
"Did you see when I gave you our sign?"
She beams and starts making out with him again.
I realize I should stop staring at them. I turn and see the rest of the band just packing up their stuff and joking around. The lead singer is wiping off his guitar. Up close, he really doesn't look that much like John Cho, but he reminds me of someone.
"You guys were great," I offer him.
He gives me a puzzled look.
"Uh, thanks. Are you with Lez Zeppelin?"
"Me? No." I start to giggle like an idiot. "No, Laura's my daughter." I point to the makeout session.
"Oh, cool. She's cool."
I'm about to say something stupid like "Just like her mom" when I hear a voice that makes the hairs on the back of my neck stand up like quills on a porcupine.
"Jeen!"
We both turn and see Asami Chang walking toward us. Immediately three things pop into my head:
1. What the hell is she doing here?
2. She looks good in those leather pants.
3. Why did she call me Jeen?
"Hey, Auntie!" The lead singer scoots around me and gives Asami a hug.
Auntie?
"You were wonderful!" Asami is gushing and fussing over the man/boy, who I now understand to be her nephew and named Jeen. I realize she hasn't even noticed me. Time to put a buzz kill on her night.
"Asami? Hi!"
They look over at me.
"Is this your friend, Auntie? I was just talking to her."
The look on Asami's face is pretty comical as she tries to fit all the pieces together.
"Jen. What are you doing here?" I can't tell if she's shocked or pissed off.
"Here to see the band, just like you." I walk toward her.
And then something completely unexpected and magical happens: Asami smiles. It's a genuine "I'm happy the stick is finally out of my ass" smile. It changes her face completely.
"Weren't they wonderful? Did you meet Jeen? Jeen, this is Mrs. Dixon."
"Just Jen." I smile and shake his hand.
"Jeen is my nephew," she explains needlessly. She turns to him. "I like this band of yours!"
"Thanks." Jeen smiles. I can tell he's happy to have her approval.
"So different from the string quartet!"
He laughs. "Yeah. Can't get much more different."
Just then the grinding beat of "Whole Lotta Love" starts up, signaling that the headliners have taken the stage. I can't hear lyrics, just the muted bass and drumbeats.
I feel like I'm having a bit of an out-of-body experience. Asami has a nephew who left a string quartet to play in a rock band called Sucker Punch and she's okay with it. Who is this woman?
"Has your father heard you play yet?"
I don't get to hear the answer to this question, because Laura suddenly appears in front of my face.
"Mom!"
"What? Sorry. I ran into a friend."
"Come meet Travis!"
With difficulty, I turn away from the Asami show and walk toward the object of my daughter's unbridled desire.
Travis is hunched over his bass case. As we walk up, he straightens, hitches up his pants, and runs a hand through his hair. He's nervous. Good.
"Mom, this is Travis."
I put on my best mom smile and stick out my hand.
"Hi, Travis. You guys really rocked it tonight."
"Thanks, Mrs. Dixon. Nice to meet you."
His handshake is warm and firm, and he looks directly in my eyes. Not bad, Travis. You've been raised well.
"How many shows have you guys done together?" I ask.
"This is our third since Jeen came on board. We're still working out the kinks."
"Not at all. I thought you guys were pretty tight."
Travis just smiles. I'm sure he's thinking, How the hell would you know, old lady? To which I'm thinking: Well, I'll tell you how I know, buddy boy. Ever heard of a little band called INXS?
"Mom, what were you talking to Jeen about?" Laura interrupts the cocktail party in my head.
"What? Oh, nothing. I know his aunt." I look back over and see Asami talking to Jeen and the drummer. Man, peel an onion and you get a lot of layers.
Travis and the band finish packing up their equipment to the slow thumping beat of Lez Zeppelin playing what I think is "All of My Love." I can't help feeling like I'm missing the best band of the night. I go over to where Laura is standing.
"Do you need a ride home, sweetie?"
She blushes, which I take to mean either no or "I don't know yet." Travis answers for her.
"I think we're going to hang for a while." He puts his arm around her.
I've seen enough PDA to last me a while.
"Okay. Have fun, guys. Travis, it was nice to meet you. I'm sure I'll see you again."
"I hope so." He shakes my hand again. Call me lame, but that stuff goes a long way with me.
As I'm heading to find the rest of my posse, Asami catches up with me.
"Your nephew is very talented," I tell her. "And he looks so much like you!"
"Thank you. He is a classically trained cellist, but we always knew he wouldn't stick with it. This really suits him."
I nod, and as we walk toward the front of the house, the music gets louder.
"Are you going to watch the band?" she asks.
"Not sure. I have to find my group. Is your husband here?"
She looks down. "No. I came by myself to see Jeen."
"Well, I'm here with the Tuccis and Nina Grandish. You can hang with us if you like." I can't believe the words coming out of my mouth. Someone should check the weather in hell.
"That would be really nice, thanks." I think we're both surprised that she has accepted my offer.
I whip out my cell and text Ron to ask where he is.
He responds pretty quickly: Up in the balcony to your right. I look up and see him waving at me. Nina and Garth are with him, but there's no sign of the Tuccis.
I lean over to Asami and yell, "Come on. They're upstairs."
By the time we navigate the crowd and the questionably up-to-code stairway, Peetsa and Buddy are with the rest of the group. I'm getting a lot of funny looks as we join them.
"Look who I bumped into backstage," I yell just as the band wraps up "The Rain Song."
There are nods and smiles all around, but it is quite possibly one of the most awkward moments of my life, and that's saying something.
We all watch Lez Zeppelin for a while. These girls are amazing. Even if you don't happen to enjoy the musical stylings of Robert Plant and Jimmy Page, you can't help but appreciate what talented musicians these women are. The crowd is going wild, and even my little nerd group is grooving.
As they wind down the show with "Stairway to Heaven," Asami leans in and asks if I would be able to chaperone the class trip next week.
"Didn't you get any volunteers?"
She shakes her head. "Not one."
I actually feel sorry for her.
"Well, it's hard sometimes. Just keep at them. I had to do a lot of begging."
"Being class mom is a lot tougher than I thought."
"Miss Ward doesn't help much, either."
Asami's eyes widen.
"I know! I can never tell what she wants."
"Don't beat yourself up about it. She's a tricky one."
I'm getting tired of yelling, and my throat is really dry. I turn to the group and suggest we head out before the stampede toward the door starts. They agree and we all maneuver our way down the deathtrap-like stairs, surprisingly get the correct coats from Miss Call Me Maybe, and go out into the cold, clear February night. As Ron strides away to get our car, Peetsa comes up and gives me a hug.
"We're heading home." She then adds in a whisper: "I want a full report in the morning." I give her an extra squeeze.
"We are, too," Nina says from the comfort of Garth's arms.
"Thanks, you guys, for coming. I know it meant a lot to Laura. Neens, I'll call you in the morning. Garth, see you Thursday." I blow them both kisses.
I notice Asami is still lingering, so I offer her a ride home.
"No, thanks. I have my car. I just wanted one more minute with you."
"What's up? I'll chaperone the trip if you want me to." I'm starting to shiver.
"Thank you, that would be great." She pulls her collar up around her neck to stay warm. "But I just wanted to apologize to you for pushing you out as class mom. You didn't deserve it."
Wow, an apology from Asami. I'm floored. I see Ron pull up in the car, so I only have a minute.
"Well, I kind of did after the 'your people' comment. I really was just trying to be funny, but I know it was inappropriate and I'm sorry I hurt your feelings.."
"I accept your apology." She nods and turns to walk away. "Good night, Jen."
"Good night, Asami."
And that, children, is the story of how two mortal enemies found peace. And to think it all started with a sucker punch.
* * *
To: Miss Ward's Class
From: JDixon
Date: 3/5
Subject: I'm back, babies!
Dear Parents,
Big, big news from room 147! Asami Chang and I have decided being your class parent is a two-mom job, so we will be splitting duties. Asami will handle the detail-oriented stuff that, let's face it, I stink at, and I will be in charge of communicating with all of you. So if you're allergic to snark, you better stop reading now.
First, shame on all of you for not volunteering for the fabulous trip to the Quindaro Underground Railroad Museum. Asami is too nice to say anything, but I have put a black mark beside each of your names. The kids all had such a wonderful time after the initial disappointment when they found out there wasn't an actual train ride involved.
Moving forward, there are a lot of opportunities for you to get back in my good graces. First up, the William Taft Annual Book Fair! It's happening March 10. The Parents' Association needs 3 volunteers from each class to make sure no brawls start over who gets the last copy of Captain Underpants.
More breaking news from room 147! Just after spring break, we will be having an Easter/Passover party! According to Miss Ward, these are not, repeat not Hallmark holidays and are therefore worthy of celebration. We will be decorating eggs and Jill Kaplan has volunteered to show the kiddos how to make charoset, which is a very yummy Seder treat. So, to that end, here's a list of what we'll need. Remember, there is no shame in offering to bring more than one thing.
3 dozen hard-boiled eggs
stickers
small chocolate eggs
a live rabbit (not really, just making sure you're still paying attention)
apples
cinnamon
grape juice
fruit for Graydon
water
cups (already got you signed up, Jackie)
wet wipes
We will provide the PAAS egg-dyeing kits.
Thank you for your cooperation. Response times will be noted and demerits will be given.
Don't thank me, I'm just glad to be back.
Jen (and Asami in spirit)
* * *
I feel good as I click Send. I've missed having my little rant sessions to the class. I'm sure there will be a lot of surprised responses to the news that Asami and I are co–house parenting.
It was actually her idea. A few days after the concert at Town Hall, she called and asked if I could meet her for coffee. I was a little unsure because, let's face it, it's one thing to have a cease-fire with someone, but quite another to all of a sudden become girlfriends.
But I was needlessly concerned. As we sat down with our lattes on one of the couches at the Starbucks right by the school, Asami, true to form, came right out with it.
"Jen, I don't expect we will ever be friends, but I think we would make a good team."
"A good team of what?" I really wasn't sure where she was going with this.
"Class mothers. You have your strengths and I have mine. Together we could get the job done really well, don't you think?"
I looked at her skeptically. Was this Asami's idea of humor?
"Do you mean this year? Be class mothers together this year?"
"Yes, exactly. You can write your silly letters and get people to participate, and I will make sure everything else runs smoothly."
I ignored the implication that things didn't run smoothly under my regime and gave Asami's idea some real thought. I didn't take long to realize it was a pretty good one, and it actually appealed to me.
"Sure, why not?"
"Honestly, I don't get your jokes, but people seem to really enjoy the wit in your emails."
I smiled at the backhanded compliment and raised my cup to her.
"Here's to strange bedfellows."
"Indeed." Asami raised her cup, too.
So here I am, back in the saddle with half the workload and permission to write ridiculous emails. I feel like I should go buy a lottery ticket and keep the lucky train rolling. But I can't, because I have to start doing the heavy lifting for Max's birthday party this weekend. So I grab my keys and sprint to the minivan to avoid the freezing rain we are being treated to.
We are having his party at a place called Emerald City Gym. It's one of those great play-zone places that have all kinds of fun things for the kids to do. The staff sets up and cleans up, so all we have to do is show up. I can't help but think it must be the worst job ever to wrangle kids at one of these birthday parties. I'm planning on tipping well.
Max has invited all the boys in his class, because "girls are gross" according to Graydon Cobb, who has apparently become the oracle of Miss Ward's kindergarten class. Whatever pearls come out of his mouth are repeated at dinner tables throughout the greater Kansas City area, including mine.
"Jingle bells, Batman smells, Robin laid an egg," Max cheerfully sang one evening. "Graydon made that up. He's so funny."
I guess we can add plagiarism to Graydon's list of talents.
Party City is my first stop this morning to pick up goody-bag crap. If I ruled the world, there would be no such thing as a goody bag. There would be an implied contract between the inviter and the invitee: I give you food and cake and some kind of activity, and you bring me a present. Why do I have to then give you a present? Isn't that what the party is?
When Vivs and Laura were small, you just gave candy at the end of the party—probably in a clear plastic bag with a ribbon... maybe. These days the gift bag is just that, a gift, and it's supposed to be something that is in theme with the rest of the party. The pressure is crippling, I tell you.
Max's theme this year is ninja warriors, so as I walk the aisles of the store I keep that in mind. Little did I know that the ninjas have a whole aisle to themselves. Banzai! This party is practically going to plan itself. I pick up ninja masks and figurines and fake swords for the gift bags, along with two ninja piñatas, plus cups, tablecloths, plates, and napkins.
After I pay and load my bags in the back of the minivan, I check my phone for emails and messages and I am surprised to see ten replies to the class email I sent out an hour ago. As usual, Sasha Lewicki's out-of-office reply leads the pack. Is that woman ever in the office? I guess she's at home taking care of her sick daughter. At least, I hope she is. Thank goodness every other reply is from an actual human.
* * *
To: JDixon
From SCobb
Date: 3/5
Subject: I'm back, babies!
Jennifer,
I just googled charoset and it has nuts in it! Are you trying to kill my son?
I will bring fruit for Graydon.
Shirleen
To: SCobb
From JDixon
Date: 3/5
Subject: I'm back, babies!
Shirleen,
Please, this is not my first trip to the rodeo. The charoset recipe Jill is using has no nuts.
Jen
To: JDixon
From CAlexander
Date: 3/5
Subject: I'm back, babies!
Welcome back, Jen!
You can put all the eggs in our basket (get it?).
Also, Kim can do the book fair. What time should she be there?
Thanks,
Carol
To: JDixon
From RBrown
Date: 3/5
Subject: I'm back,babies!
Jen,
I hope this isn't a joke and you are really back. I'm so excited!
We can bring eggs for everyone. Also Zach is really excited about Max's party on Saturday.
See you then!
Ravi
To: JDixon
From DBurgess
Date: 3/5
Re: I'm back, babies!
Hey, Jen,
Way to go, getting your old job back . You'll have to fill me in. You know, a man could die of thirst trying to get a cup of coffee out of you. Just sayin'...
BTW, Lulu will bring in eggs for everyone.
Cheers,
Don
* * *
Poor Don—he must be so confused. After a five-month bout of flirty texting, I have pulled way back. He still texts about once a week, trying to meet up, but I'm either make-believe busy or actually busy—truthfully, more the latter than the former. I may never know what he really meant by "coffee," but I'm okay with that.
* * *
To: JDixon
From JJ Aikens
Date: 3/5
Subject: I'm back, babies!
Jen,
Well, this is quite a turn of events. I guess I should say welcome back, but I think you know it would be insincere. So I'll just say we will bring eggs for everyone to decorate.
JJ
* * *
Oh, mother of God. Am I being punked? I scroll through the other emails to find that yup, everyone wants to bring eggs. What are the chances? I definitely need to think of a better system. I peck out an email on my phone.
* * *
To: Miss Ward's Class
From: JDixon
Date: 3/5
Subject: Egg-cellent response!
Me again,
Thanks for the great and may I say timely responses! Shirleen, your charming email came in second place after Sasha Lewicki's out-of-office reply, with a winning time of 58 seconds. Well done.
It may surprise you to know that just about everyone offered to bring eggs. I guess the yolk's on me! Since we don't need 10 dozen eggs, I'm going to assign some of you other things to bring. See the list below, and thanks again for getting back to me so quickly.
Eggs—Alexanders, Burgess/Gordon (two dozen each, please—hard-boiled)
Stickers—Aikens Family
Cups—Westmans
Apples—Browns
Grape juice—Kaplans
Water—Changs
Fruit—Cobbs
Cinnamon—Zalises
Wet wipes—Wolffes
And the Batons will bring wine.
Please bring all supplies on the morning of April 4 when you drop your kids off. I know, I know, it's a month away. I'll send a reminder email a few days before.
As for the book fair, thanks to Kim Alexander and Peetsa Tucci for volunteering. Is there anyone else dying to straighten up books for 3 hours? No? Okay. Guess it's on me, then.
Jen
* * *
* * *
After I drop the party supplies at Emerald City Gym, I stop by the Upper Crust bakery to give them the good news that I will need a ninja cake that feeds twenty. Not a Teenage Mutant Ninja Turtles cake, a real ninja cake. I think I have a 50/50 chance of actually getting the right cake.
My cell phone rings as I run to the minivan. The rain has stopped, but the temperature is now below freezing so I nearly wipe out on the icy tarmac as I yank open the car door. I sit down and grab my cell from my purse.
"Hello," I say, a bit too loudly.
"Jen, it's Asami. Do you have time to meet me for coffee?"
"Anything wrong?"
"No, I just want to talk to you about something."
"Is it about the class email? Because if you don't want to bring water, I can bring it."
"No, water is fine. Do you have time to meet or not?"
Now, that's the Asami I know and love. I look at the clock on my dashboard and calculate how much I still have left to do before I pick up Max.
"How about two o'clock at Starbucks by school?" I can't imagine why she needs to see me, but whatever it is can't take more than half an hour.
"See you there." She hangs up before I can say anything.
* * *
Asami is already waiting when I walk in at two o'clock on the dot, secretly patting myself on the back for once again stuffing ten pounds of potatoes into a five-pound bag.
After I grab a chai latte, I join her on the couch. She has certainly made a brave choice of headgear on this cold, wet day. She is sporting a green hunter's hat with eyes on the front and a lid that looks like a duckbill.
"So what's up?" I take a sip and ease my coat off.
As usual, Asami comes right to the point. She leans toward me with purpose.
"I think I told you once that I believe there is no such person as Sasha Lewicki, and now I'm more convinced than ever."
Really? This is what I had to rush here for? I never pegged Asami for the conspiracy-theory type.
"How can you even think that? I mean, I know I've never met her, but I heard that Miss Ward goes to her house, like, three times a week and tutors Nadine."
"According to whom?" Asami raises a very defined eyebrow at me.
"Uh, I can't remember who told me. Why?"
"Because I went to the address listed in her school file, and guess what? No one lives there. It's one of those old abandoned row houses off Mission Street near the Walmart."
I really don't know what to say. Do I want to ask how she got access to the Lewickis' school file in the first place?
"Maybe they moved. Or maybe it's a misprint. There could be a lot of explanations." I can't believe I'm the voice of reason in this conversation.
"Maybe, but then I Googled Sasha Lewicki, and guess what? All I found was some doctor who works at Kaiser Permanente in California and a girl at Boston College who puts inappropriate pictures of herself on Instagram." She pauses for what I guess is effect or drama, then says, "There is no Sasha Lewicki in Kansas City."
I want to say "So what?" but I can see that Asami is really whipped up about this. I take a long sip of chai in the hopes of finding some answers for her.
"So what's your endgame here, Asami? What exactly are you trying to do?"
"I'm trying to prove that these people are made up."
"By who? And for what reason?" I can't keep the irritation out of my voice.
"Well, that's what we need to find out." Asami sits back for the first time during our conversation and crosses her arms.
I'm still not convinced there is anything to give a royal rip about, but I take a few breaths to absorb her information. Okay, we have a kid in class that no one has ever seen; a mother who only answers emails with an out-of-office autoreply, but manages to contribute to class parties; and there is a wrong address in the school records. If she is made up, someone has taken the time to plan this ruse admirably.
"Have you talked to Miss Ward? Asked her about it?"
"Of course I have." Asami seems insulted. "She brushed me off by saying she couldn't talk about another student."
I remember having a similar experience when I casually asked Miss Ward about Nadine at the beginning of the year. At the time, I wasn't fishing; I was just making conversation. I look at my watch. We have about ten minutes.
"Okay, let's say Nadine and Sasha are made up. So what? It doesn't affect the class dynamic at all. Why do you care?"
"It just bothers me. It's like a loose end that is just... dangling there." Asami waves her hand in front of my face. "Plus I can't stop feeling like someone is having a good laugh at our expense, and I do not like to be laughed at."
If you don't like to be laughed at, you should seriously rethink your hat choices, I think.
"Okay, so what would you like me to do?"
"I want you to help me get to the bottom of this. See if we can force this person to show herself."
"Right. And how am I supposed to do that?"
"I was hoping you'd have an idea. You've got that slick, cagey mind."
I chew my lip and consider the backhanded compliment my co–class parent just launched at me. God, she is hard to take seriously with that hat on. I start to put my coat on, and she does the same.
"Let me think about it," I say as we walk to the door. "My slick, cagey mind needs time to brew."
We run to our respective cars in a lame attempt to hold on to the warmth from Starbucks, and drive the quarter mile to school. We park and get out and Asami joins me as I walk to where we wait for the kids. As we approach, I can imagine Peetsa and Ravi checking me out with my new bestie, but to my surprise I find them locked in conversation with none other than Shirleen Cobb.
"Hey, girls, what's up?" I say by way of inserting myself into the exchange.
"Jennifer, I'm glad you're here. Does Graydon say inappropriate things to Max?" Shirleen asks.
"Inappropriate?" I look to Ravi and Peetsa for any clue as to what we have walked into.
"Yes, inappropriate. Surely you, of all people, know what that means."
And I thought I was the snarky one.
"Well, to be honest, he did tell Max not to invite girls to his birthday party because they're gross."
Judging by the openmouthed stares I'm receiving, this is the wrong answer. When will I ever learn that the only thing you say to other parents is how wonderful their child is? Even if they ask you for the truth, they really don't want it.
"I don't see anything inappropriate about that. In fact, it is extremely age appropriate."
"And that's why I never said anything to you about it." I look around the circle. "Why are we talking about this?"
I feel like Peetsa is about to say something, but Shirleen jumps in.
"As a matter of fact, Zach told my son that his mother said that Graydon says too many inappropriate things and he shouldn't listen to him."
"Which Zach?" Asami and I say at the same time.
"That's what I'm trying to find out." She turns to Peetsa and Ravi. "But I don't seem to be getting anywhere."
"It must have been Zach E., Shirleen," Peetsa says, in her most appeasing voice. I'm impressed by her ability to throw Trudy Elder under the bus without even blinking. "I certainly hear all about Graydon from my Zach, but I would never say anything like that to him."
Ravi nods solemnly, but doesn't say anything.
"Well, I guess I should go have a word with Trudy." Shirleen turns to leave our weird little circle of friends, but then pauses for a final comment.
"I sure hope you girls will always come to me if there's a problem with Graydon."
With that, she stalks away.
"Because clearly you are open to the criticism," I say when I know she's out of earshot.
My besties laugh. So does Asami.
Just then the bell rings and our cuties come trudging out looking the way most kids do by this time of winter—exhausted and disheveled. I spot Max's leopard-print jacket in the crowd and wave to him. He has accessorized today with a headband that has brown felt antlers on top. He is walking arm in arm with Zach T., and they both look upset.
"Hey, sweetie." I give him a quick hug. "Are you okay?"
"I'm fine." His voice tells me he's anything but.
I look to Peetsa to see if she has any idea what they're so glum about. She shrugs and takes Zach's bag from him.
"See you guys tomorrow," she says.
"Don't forget to ask your mom!" Zach yells to Max as he walks away, holding his mother's hand.
"Ask me what?" I look down at him.
"You know it's five days till my birthday, right?"
"Five days till your birthday party," I correct him as we walk to the car, "a week till your birthday."
"Did you get the cake yet?" he asks, handing me his backpack.
"Well, I ordered it. We won't pick it up until Saturday."
As I buckle him into his booster seat, he lets out a very big sigh for a little boy. I buckle myself in and turn on the car. I check my phone before I pull out and see I have a text from Don.
You're having a party without me?
I guess word is out about Max's shindig.
We are. Six-year-old boys only. Girls are gross.
What about 48-year-old boys?
They're gross, too.
LOL
I shake my head and put my phone in my purse.
"How was school today?" I ask over my shoulder.
"Good."
"What did you do?"
"Nothing."
"Who did you play with?"
"I don't know." He suddenly looks at me as though he has just realized I'm there. "Hey, Mom, can we invite Jack to my party?"
"Jack?" I frown. "Who is Jack?"
"You know, Garth's friend from when we went to Wichita that day."
"Why would you want to invite Jack, sweetie? We barely know him."
"Mom, please can you ask him? Please?"
"Well, I'll need to know why first."
Max frowns. "Graydon doesn't believe we know a guy with one arm. He called me and Zach B. liars."
"He did?" I start planning my phone call to concerned parent Shirleen Cobb.
Max nods. I look in the rearview mirror and see him squeeze his eyes shut tight. I can tell he's upset.
"Did that make you sad?"
He nods and looks out the window.
"Did you cry, buddy?"
He lets out a long dramatic sigh. "I did for a minute. And then I manned up."
I suppress a smile, knowing that last part is courtesy of his dad.
"So can we ask him, Mom? I want to show Graydon I'm not lying."
This is what I find hard about parenting. The petty, small, vindictive part of me wants to say, "Hell, yes, we'll get him there and teach Graydon Cobb a lesson or two about calling people liars." But the rational adult side of me knows my son needs to learn to rise above this crap. Plus it sounds like he was doing some pretty serious bragging about meeting a man with a handicap. I'm not too happy about that.
* * *
I tell Garth the whole story over jumping jacks on Thursday morning.
"Honest to God, just when I think I've heard it all from that kid," I puff.
Garth chuckles. "You know, Jack is such a nice guy, I'm sure if I asked him he would stop by. Shut this kid Graydon up."
I stop jumping and try to catch my breath.
"Please don't. But I'd love it if you would stop by."
"Really?" He seems surprised.
"Of course. You're practically family now. Plus, I'm dying to see you eat a piece of cake."
Garth laughs.
"I will if it's chocolate. I can't stay away from that."
"I noticed." I smirk. "Come with Nina. It'll be fun."
"Okay, thanks for the invite. I'll talk to her. Now let's work your legs a little more."
We get back to business and soon I'm a sweaty mess. Garth tells me that he's going to move our workouts outside as soon as the snow melts. I'm a little nervous to leave the comforts of Ron's Gym and Tan, but I think it will be nice to change the scenery up a bit.
Saturday morning, I jump out of bed like the woman on a mission that I am—for today, anyway. Max's birthday party starts at eleven, and I have a ton to do. I glance at the rumpled sheets on the other side of the bed and see that Ron is already up and at 'em as well. I'm impressed. I grab the To Do list I scribbled in the middle of the night when I couldn't sleep. Most of it is illegible but I can get the gist of it.
—Remind my parents where the party is, again
—Put gift bags in the car
—Pick up cake
—Pick up balloons
—Decorate party room
Normally I would farm some of this out to Vivs and Laura, but they are barely going to make it to the party. They are driving up from school with Raj and Travis and not leaving until ten this morning, because apparently Travis "needs his sleep." Yes, Travis is still in the picture. I'm thinking this might be love for my little girl. Never mind that the great Steve Perry of Journey says, "Lovin' a music man ain't always what it's supposed to be," Laura is smitten and she's going to roll the dice with Travis. God, I envy her. She has no idea what it's like to get your heart broken.
I quickly shower, don the mom uniform, and trot down to the kitchen for a much-needed cup of joe. No sign of Ron, but I find a note taped to the Keurig telling me he has gone running. I pop a pod into the machine and grab the milk. It's ridiculous how much I need this first cup.
As I'm savoring my maiden sip of the day, Max comes into the kitchen, rubbing his eyes. He walks over for a hug.
"Good morning, sweetie pie." I give him a big squeeze. "Happy birthday-party day."
His eyes spring open so quickly it's comical. "It's my party today?" He starts to jump up and down, doing his happy dance, which looks a little like the way Martin Short dances as Ed Grimley. "Can we go now?"
I can't help but laugh as it is only 8:30 in the morning.
"Well, we've got a few things to do before we go. First, breakfast. You want to have a lot of energy for the gym. How about an egg in the hole?"
"Okay." He sounds a bit disappointed. I know he's dying to get the party started. Just then Ron comes in the back door, looking like the Unabomber with sunglasses and his hood up.
"Man, is it chilly out there!" He stomps around for a bit, then shrugs his running jacket off.
"Dad, are you ready for the party?" Max asks.
"Absolutely, buddy. Just give me ten seconds."
"Coffee?" I hold up my mug.
"Please." He kisses me, then takes a huge chug from his water bottle.
I throw a pod in for him, then start on Max's breakfast.
"I have a few errands to run before the party. Can you get Max ready and bring him to Emerald City around ten forty-five?"
"No problem. Want me to bring anything?"
"I think I've got it all. Just make sure the guest of honor is there."
"Who's the guest of honor?" Max asks. He has an orange juice mustache.
"You, silly." I place the egg in a hole in front of him and he digs in with gusto.
The kitchen-counter office beckons me, so I have a seat and fire up the iMac to check my emails and send one to Asami.
* * *
To: JDixon
From: KHoward
Date: 03/10
Subject: Max's party
Hello, darling,
Can you remind me where Max's party is again? I keep forgetting the name of it. Also send me the address so I can put it in the Garmin.
Love,
Your Mother
To: JDixon
From: SCobb
Date 3/10
Subject: Today
Jennifer,
I'm just writing to make sure this party will be a safe environment for Graydon. Have they done a sweep for nuts and dust? Or should I send him with his mask?
Shirleen
* * *
I roll my eyes. I'd feel sorrier for that child if he wasn't so much like his mother. I send my mom the address, tell Shirleen that a mask is always a good idea (that'll teach Graydon to call my son a liar), and then send Asami the news she has been waiting for.
* * *
To: AChang
From: JDixon
Date: 3/10
Subject: Operation Who Is She?
Hi, Asami,
I think I've figured out a way to smoke out Sasha. Let's meet at Starbucks before pickup on Monday.
Jen
* * *
I close the laptop and look at my watch. Yeesh! Nine fifteen already. I'd better get my ass in gear. I run up the stairs two at a time ('cause that's how I roll these days) and holler to Ron, who's in the shower.
"I'm heading out. Remember, get Max dressed and over to Emerald City by ten forty-five."
"You got it!" he yells back. I can smell his musky shampoo from the hallway.
I charge back down the stairs and grab my keys from the hook by the kitchen door.
"I'm on my way, buddy. Dad's going to bring you to the party."
"Where are you going?" Max looks up from Ron's iPad.
"I'm going to set up for the most awesome ninja party ever." I give him a kiss on the head. "See you in a while."
"Ninja!"
* * *
I have to give myself snaps. Max's party is a bona fide hit. Seven boys from his class and three from his Cub Scout Troop all arrive in various forms of ninja dress and start running around Emerald City like it's on fire. I told everyone it's a drop-off party, but there are always mothers who want to stay. Today it seems everyone wants to hang out, so I'm treated to time with Peetsa, Ravi, Hunter's two moms, Shirleen, Trudy Elder, Jackie Westman, and, to my delight, Jean-Luc Baton, whom I have not seen since curriculum night. Still gorgeous, I'm happy to report. I have to say that as much as I love my husband and happen to think he's the bee's knees, nothing can put a spring in your step quite like a good-looking man in your midst. Jean-Luc is the only dad here besides Ron, so they're huddled in a corner, talking about work no doubt. The moms are sitting sipping coffee and looking happy to be doing nothing for a minute.
The party room looks great, thanks to my decorations and the ministrations of Brandon and Kayla, my Emerald City ambassadors. They were able to make magic with my Party City purchases and the black, green, and gold balloons I picked up this morning.
A flurry at the door catches my eye: my girls have arrived with their beaux. Does anyone say "beau" anymore? I picked it up when I was ten and reading Anne of Green Gables and it stuck with me.
"Mom, sorry we're late. Seventy was a parking lot getting into the city." Laura gives me a quick hug.
"That's okay, baby. You haven't missed much." I look past her and wave to Vivs and Raj. "Where's Travis?"
"Parking the car. Where should I put these?" She holds up a couple of presents wrapped in ninja paper.
"Over in the corner. Do me a favor and circulate with your sister—introduce yourselves to people. I have to go talk to my guy about the food."
As if on cue, Brandon walks in.
"I think the boys are getting hungry. Should I bring the food out in about ten minutes?"
"Perfect. Thanks."
"Would you like something besides pizza for them?" He nods toward the mothers.
Since I hadn't been expecting eight moms, I hadn't made any arrangements for them.
"Um... how about a couple of large Caesar salads with chicken?"
Brandon nods and heads off to the kitchen.
I walk toward Vivs and Raj and give them both a welcome hug.
"Thanks for driving in for this, you guys."
"Like I'd miss Maxi's birthday party!" Vivs snorts.
"Are grown-ups allowed to play out there, too?" Raj wonders.
"Why? You want to go play, little boy?" Vivs teases him.
He blushes. "Well, it looks like fun."
"I think you can, Raj, just take your shoes off and watch out for rug rats." I smile.
"Oh, rats don't bother me. When I was a kid, I spent every summer in India." He heads off, yelling for Travis to join him.
Vivs and I look at each other and burst out laughing. The moment is cut short when Max comes running into the party room. I see Ron has convinced him to wear plain old jeans and a T-shirt to his party. I'm thinking that might be why he storms over to me crying angry tears.
"I hate this party!" he yells. "Why did we have to have it here?"
"What on earth are you talking about?" I get down on one knee and hold his arms.
"I want another party somewhere else." He's sobbing now, and has a bubble of snot coming out of his nose. I look up and notice the whole room has gone quiet. I so love an audience when I'm having a bad moment with my kid. I give Peetsa a pleading look, and she nods.
"Does anyone need to use the bathroom?" She sounds like she's asking a group of preschoolers. Most of the moms get up with her, but Trudy and Shirleen stay put. I turn back to Max.
"Max, calm down and tell me what happened," I say in the nicest voice I can, considering I'm dying inside. Vivs and Laura have moved to stand behind Max in an attempt to make this moment more private. Ron has joined me on the floor.
"We... we... were at the mini r-r-ropes course." He's having trouble getting his words out. "And G-g-graydon wanted to do the grown-up r-r-ropes course..." He takes a deep breath as I rub his arms. "And they said no and Graydon said this is the worst party he has ever been to." Max starts up a fresh round of sobs.
"He what?" I can see Shirleen charging over to where I'm kneeling with my son. Max moves into the safety of my arms.
"Shirleen, it's okay. I'm handling it." I'm hoping my look conveys how much I don't want her to interfere. Apparently it doesn't.
"Where is Graydon now, Max?" He shrugs. She stomps out of the party room.
"Is Graydon going to get in trouble?" Max asks, wiping his nose on his sleeve.
"I think he might, buddy," Ron says.
"Good," Max says firmly. "He says mean things."
"Why don't you hang in here, sweetie? The pizza is coming out soon."
I hug him and get up off my knees to see that Nina and Garth have arrived, gifts for Max in hand.
"Hey, you guys! I'm so glad you could come. Max, look, Aunt Nina and Garth are here." I send him over to give them a hug, which gives me the opportunity to take a huge "serenity now" breath. Ron puts his arm around my shoulder and gives me a squeeze.
"Best mom ever," he whispers in my ear.
Hardly, I think, but I take the compliment with a smile.
Brandon and Kayla arrive with the pizza just as the boys are coming back from the play area and the mothers return from their fake visit to the bathroom. Last to arrive is Shirleen, followed by Graydon.
"Jennifer, there has been a misunderstanding. Graydon didn't say it was the worst party he's ever been to."
"Yes, he did." Max comes running back to me.
"He's lying," says Graydon. "He lies a lot."
Whoa. Hold on there, bubble boy. You are in dangerous territory. I'm about to say something, but Max decides to come to his own defense.
"I don't lie," he says. "Almost never," he adds with conviction.
"You lied about the one-armed man," Graydon counters.
At this point, the entire room's focus is on these two little boys. Even Shirleen is hanging back.
"I didn't lie about him! He's real. Zach B. saw him, too."
Zach B. looks up and nods. He seems pleased to have been brought into the drama.
"Well, I don't believe you." Graydon folds his arms in front of his chest in an exact imitation of his mother. I almost burst out laughing.
"Wait a minute." Garth steps in. I want to wave him off, but stop myself.
"Do you mean this one-armed guy?" He reaches into the gift bag he's holding, pulls out a picture frame, and turns to the picture toward Graydon. From where I'm standing, I can see it's a photo of Jack, from the day of the indoor mudder. He has his one arm around both boys, and they are all smiling.
"Yes!" Max yells as if he has won the lottery... or a good game of Lego Indiana Jones. He runs to hug Garth and takes the picture. "See, Graydon? I told you I wasn't lying." He is beaming.
"We told you." Zach B. puts his arm around Max.
Graydon walks closer to the boys for a better look at the picture. After studying it for about ten seconds, he hands it back to Max.
"Cool. Mom, what can I eat here?"
If I was hoping for an apology, I was going to be disappointed. That "cool" was all the ground Graydon was ever going to give. I look at Garth.
"You are my hero," I mouth to him. He nods and winks. Nina blows me a kiss.
The kids have already moved on and are scarfing down pizza like it's the first and last time they will ever have it. I'm sitting with the moms, enjoying chicken Caesar salad and a delicious cup of smugness. Shirleen sits down beside me.
"Well, I'm sure glad the boys hashed that out. It's good to get to the truth, I always say."
Clearly, she doesn't notice the slice of humble pie in front of her. There are many ways I could respond, but I go with a nod and a smile.
As the famous anorexic Karen Carpenter once said, "Rainy days and Mondays always get me down." Man alive, we have had a lot of precipitation these past few weeks. It's either rain or sleet or snow or hail that has me scraping off or digging out the minivan every day. Oh, well. At least the mail is getting delivered... or so they say.
On this particular freezing rainy Monday I'm meeting Asami (please, not the duck hat; please, not the duck hat) to unveil my master plan to get to the bottom of the Sasha and Nadine Lewicki mystery/Asami obsession. As I'm driving to Starbucks, my cell phone rings. I hit the Answer button on the steering wheel.
"Jen's Nail Salon; how can I help you?"
"On a scale of one to ten, what is your mood right now?" It's Ron.
"Is ten the best or the worst?" I ask.
"Whatever makes you happier."
"Well, then, I guess I'd say I'm about a five, and you can do the math."
"I just got offered tickets to watch the Roller Warriors play tonight."
"Okay... the roller whats?" I can hear Ron's eyes rolling on the other end of the line.
"Kansas City's roller derby team."
"You like roller derby?" I'm wondering how he got through the vetting process without this little nugget of information coming out.
"I'm trying to make a deal to supply all their skates for the next five seasons. The manager offered me tickets, so I think it would be good for me to go."
"Why are you even asking me? Of course it would be good for you to go."
"I kind of want to bring Max."
"In God's name, why?"
"I think he'll really enjoy it. It's a lot of fun to watch."
I turn into the Starbucks parking lot and find a spot.
"I'll have to take your word for it." I smile to myself, thinking how many questions Max is going to have.
"So, when you pick him up from school, can you drop him at the store? I'll get him dinner and we'll go from here."
"Sure, babe. I'll see you in about an hour."
"Perfect, thanks."
I press the button on the steering wheel again to hang up and have a good laugh. Watch this be the sport Max falls in love with.
Starbucks smells like roasted coffee and something else... a croissant being baked? Asami and her hat are already on the love seat, patiently waiting for me.
I've had quite enough caffeine already today, so I opt for a calm tea and one of those yummy cake pops. I choose the one with nuts, reasoning that it is healthier.
As I settle in across from Asami, I can tell she's dying to know what I have come up with, so I don't waste time with pleasantries.
"So, remember when the FBI did that huge Mafia sting a bunch of years ago?"
"No."
"Well, a bunch of Mafia guys had gone into hiding and the FBI couldn't find them anywhere. So they sent out letters to the Mafiosos' last known addresses, saying they had won some really big prize but they had to show up at a certain warehouse to claim it." I'm pretty sure I'm fudging some of the facts.
"Did they go?"
"Yes. A lot of them did and were arrested. Maybe if Sasha thinks she has won money or something, she will show up to get it."
Asami's face is pinched in irritation.
"What's wrong?" I tear into my cake pop.
"Well, it's just so... simple. It's such an obvious plan. I'm mad I didn't think of it myself."
It isn't that obvious, I think but do not say. "So, do you want to try it?"
"Of course! What should we say she has won?"
"How about a car? Or fifty thousand bucks?"
"Perfect. You should send the letter out today."
"Me? No, no. This is all you, partner."
"Don't you want to see who shows up?"
"I look forward to you telling me all about it," I assure Asami.
I sit back and we sip our respective drinks for a minute.
"So is your daughter still dating the boy in Jeen's band?" Asami is trying to make actual conversation with me. I'm touched.
"Travis, yes. They seem very fond of each other. How is Jeen doing?"
"Fine. He really loves school."
"That's great."
Awkward pause while I realize we have both run out of small talk. That's fine. It's time to go, anyway.
"I'm heading out to get Max. Good luck with your Sasha hunt."
Asami turns her determined eyes to me and nods. Whoever Sasha Lewicki is, I can't help but feel sorry for her.
On my way out the door, I bump into Suchafox heading in. I haven't seen him in ages—not even at pickup. He gives me a quick hug and the Polo wafts over me.
"So you really do drink coffee, just not with me!" he says lightheartedly.
"Ha, ha. Yes! How are you?"
"Good. Do you have time for a chat?"
I look at my phone. "No, I've got pickup in five minutes; don't you?"
His face falls a bit. "No, Ali does pickup now."
"Oh" is all I can think of to say. I'd love to know more, but I hate not being there when Max comes out. I was five minutes late to Scouts once and he was convinced I was never coming back for him.
"Gotta run. See you soon." I dash out to the minivan and as I'm getting in I turn to see Don still watching me from the door. He looks really sad. I make a mental note to reach out to him later.
* * *
Don't ever tell Ron this, but I love a night to myself. After dropping a very confused but excited Max off at the store, I stop to pick up some groceries, then drive home. I know exactly how I want to spend my evening. An uninterrupted bath is unspeakable luxury when you are a mom. The knowledge that no one is going to bother me is intoxicating. I fill the deep claw-foot tub in our master bath with the hottest water I can stand and add an entire bag of Dr. Teal's Epsom salts. It's not exactly "Calgon, take me away," but they do have a nice lavender scent. I light a few candles, get myself a generous glass of Oregon pinot noir, and connect my iPhone to the Bluetooth speaker. I put my phone beside the bed and think about texting Don to see if he's okay, but decide to do it later. I don't want anything distracting me from my bliss.
As you know, my heart belongs to rock and roll, but when I'm trying to relax I like to switch it up with a little smooth jazz on Pandora. Anita Baker starts to sing.
So here I sit, and you couldn't find a happier person on the planet at this moment.
When I first hit the water, it feels like a thousand tiny needles piercing my body, but once I submerge, the pain dissolves with the salt.
I close my eyes and start to visualize the mud course. I take myself through each phase—the run, the wall, the pegs, the steep ramp, the freezing water, the ring of fire, and finally the crawl through the mud to the finish line. Two months to go, but I wish it were tomorrow. I'm so ready. I'm so ready....
My eyes snap open. The bath water is cold and I'm shivering. I must have dozed off. I stand up quickly and throw one leg over the tub while I reach for a towel. I must lean too far because the next thing I know, my foot on the floor is slipping forward and my other leg isn't out of the bath yet. I'm forced into a split that crashes my vagina into the side of the tub. I feel a snap in my groin, followed by eye-watering pain.
"Shit, fuck, ow, crap, goddammit!" I roll onto my side and back and force my other leg out of the tub.
"Son of a bitch, mother fuck." I start to cry. I can't believe how much my groin hurts. I have no idea how I'm going to get up. I'm shivering, so I pull the towel over me for warmth and put my head down on the fluffy white bath mat. When I've calmed down I start crawling to the bathroom door. I need to get to a phone and call someone. The bottom half of my body is useless, so I rely on my arms and pull myself along the floor.
Is it shallow that even in my excruciating pain, I can't help but notice how strong my upper body is? Go me! Argghhhh! My groin is throbbing.
I slither to the door of the bathroom, grab the knob, and yank it open enough to crawl through. God, I'm cold! Bone cold, as my mom would say. I've never fallen asleep in the bathtub, and now I know why it's not recommended.
I can only imagine the spectacle that Ron is treated to when he walks in a minute later as I'm making my way to my cell phone by the bed, naked, towel barely around me, crying and swearing at the same time.
"Jesus, Jen!" He runs to my side and is down on the carpet with me in a nanosecond.
"I f-f-fell in the t-t-tub," I manage to sob out before I collapse into his body.
"Where's Mommy?" I hear Max yelling from downstairs.
"Oh, my God. He can't see this. He'll have nightmares for years."
"Hang on, buddy. She's in the bathroom," Ron calls to him. "Get your PJs on and brush your teeth." He looks at me. "What hurts? Did you break anything? Do you need an ambulance?"
"I don't think so. It's just my groin." I'm shivering uncontrollably.
Ron appraises the situation and decides to scoop me up and onto the bed. He grabs a sweatshirt out of my drawers and helps me put it on. I welcome the feel of my pillows as I lean back and Ron pulls the duvet over me. My vagina is still throbbing, but my shivers are starting to die down.
"I'll be right back." He kisses my head and runs out of the room. I can hear him talking to Max in a low voice and then going downstairs. When he returns, he has one of the ice packs I keep at the ready in the freezer, a glass of water, and two ibuprofen.
"Can I take a look?"
I nod and help him pull the duvet aside. He hisses when he sees my upper legs. I look down: there is already the beginning of a bruise right where my leg meets my hip. It's hard to tell if my vulva is also bruised, because I've gone native down there, but it feels like it is.
"Shit," I say, and my eyes fill with tears again. I shove the pills in my mouth and take a gulp of water. After I swallow, a sob escapes my throat.
"Shhh. It's okay. Let's get the ice on it." Ron places the pack gently at the top of my leg. I wince at first, but soon the cold pack brings relief to the ache.
"Is Max okay?" I sniffle. "How was roller derby?"
"Max is watching TV and roller derby was fine. Tell me what happened here." Ron's velvet voice is rich with concern.
"I was taking a bath and I fell asleep. When I woke up, I was cold, so I got out too fast and I slipped with one leg out and landed on my bun," I say, using the word for vagina that came into our house via the wisdom of Graydon Cobb. (Max was thrilled to come home one day and tell me that Graydon says boys have hot dogs and girls have buns.) "I felt my groin snap and the pain was mind-blowing."
"Oh, babe, I'm so sorry. I think you should go for an X-ray tomorrow."
I nod, but suddenly realize I'm too tired to talk anymore. My eyes close without any effort on my part.
"Do you have Garth tomorrow? I should call and cancel."
I hear Ron's words, but they take a few seconds to sink in. Garth, training, mudder. Mudder!
"Oh my God," I groan.
"What?"
"All my training." I take a deep breath and start crying yet again.
"Hey, hey, hey." Ron rubs my head. "It's okay."
"Why are you crying?" says a little voice from the door.
I quickly cover up the ice pack and look around Ron at Max. He's wearing his "Where's Waldo?" PJs, hat, and glasses.
"Come here, sweetie." I hold out my arms and he walks into them while still standing beside the bed.
"Why are you crying? Did you miss us?"
Only a child would see it like that.
"I did miss you, but I also got a little boo-boo on my thigh and it hurts a bit."
"Do you want me to kiss it and make it better?"
"That's okay, Daddy already did. How was roller derby?"
Max stands up, eyes shining.
"It was so funny! These ladies roller-skate around in a circle and try to knock each other over. I didn't like it at first, but Daddy says it's all pretend and they don't really get hurt when they fall."
I raise my eyebrow at Ron. He shrugs.
"The other team was called the Roller City Rats."
"Yeah, Max loves that name." Ron smirks.
"Well, I'm glad you had fun, but it's way past your bedtime. Give me a hug and let Daddy put you to bed so I can rest my boo-boo." I hold out my arms again.
Max snuggles into me. "But I hate when Daddy puts me to bed," he whines. "He always falls asleep before I do."
"Not tonight, buddy!" Ron assures him as he swings Max over his shoulder.
"Good night, baby." I sigh and lie back down on my pillow. I'm asleep before Ron comes back to bed.
Four hours at Overland Park Regional Medical Center later, I learn I haven't cracked or broken anything. I have a badly pulled groin muscle and a beauty of a bruise. As stupid as this sounds, I'm a bit disappointed. I mean, if I'm going to go through something this painful, I'd like to have bragging rights to a cracked pelvis, not a bruised front bum. Dr. Sintay, the man on duty this morning, tells me I'm lucky I didn't have a more serious injury. "The bathroom is a very dangerous place, Mrs. Dixon."
Especially after my husband has been in there, I think but do not say.
Ron brings me home with my filled prescription of painkillers; after I assure him I'll be fine, he heads off to the store. He has left me propped up on the comfy couch with ice on my crotch and the remote in my hand. Who could ask for more? Peetsa is taking Max to her place after school, so I am a lady of leisure for a few hours. I pick up the phone and call Garth.
"What did the doctor say?" is how he greets me.
"No breaks or cracks, just a pulled ligament and a really bad bruise."
"Oh, Lordy, thank goodness. Did he say when you could start training again?"
"Nothing for three weeks; then we'll see. That will give us three until the race."
Silence.
"What is it?"
"Jen, we may want to rethink the mud run for April."
"Are you crazy? It's just a bruise. I'm fine!" I'm practically yelling.
"Take it easy." Garth's voice is annoyingly calm. "Hear me out."
I let out an audible sigh and give the phone the finger. "Okay, what?"
"You have done great work. I can't believe how far we've come over the past six months."
"Garth, don't handle me, please. Just spit it out." I'm speaking more harshly than I mean to, but my patience has already been taxed by twenty minutes of medical attention spread out over four hours at the hospital.
"Okay, here it is. If you're benched for the next three weeks, you will not be ready for the mud run. You think you'll be back where you left off, but you won't—you'll be out of shape. It won't take you long to get back to peak performance, but I'd rather you not risk another injury. There is another run in Springfield in August. We can aim for that."
He's right. I know he's right.
"But I feel so ready. That's exactly what I was thinking in the bath last night. I was wishing the race was today."
"Well, be thankful that it isn't. We can reevaluate in a month, but right now I say we take it off the table."
"I'll agree to reevaluate in a month," I concede, stubborn to the end. "What can I do right now, while I'm recovering?"
"Rest, Jen. That's the best medicine. Don't jump the gun, or you'll be on your butt for another month. I'll come and see you tomorrow." I hear a voice screaming in the background. "And Nina says you're a spaz."
"Ha! Tell her to call me later. I'm going to take a nap." I stifle a yawn.
"Good girl. See you tomorrow."
I hang up with Garth and snuggle down into the comfy couch. Just as I'm nodding off, my cell phone buzzes. It's a text from Asami.
Can you meet?
Oh, God, I'm so not in the mood for her brand of crazy right now. I decide to ignore it.
The phone buzzes twice more in quick succession.
Hello?
Can you meet?
I growl and type a quick reply.
Not feeling well. Let's try next week.
Of course, Asami is back at me in seconds. She must have twenty fingers.
It's important.
I know I can't do anything today, but I might be able to see her tomorrow, so I text her as much.
Where?
My house at 1.
I don't get another text, but I can imagine Asami firmly nodding to herself.
* * *
The next day at one sharp, Asami rings my doorbell. Nina and Garth had come earlier to bring me lunch (Taco Bell, my favorite binge) so I asked them to let her in on their way out. I hear an exchange of greetings and a little mumbling; then Asami joins me in the living room.
"I had no idea you had an accident." I think I hear genuine concern in her voice, and I'm touched.
"Yup. I slipped getting out of the bathtub." The more I say it, the more of an idiot I feel like.
"Did you yell, 'I've fallen and I can't get up'?" Asami asks. It takes me a moment to realize this is her idea of a joke. I smile.
"Good one. So what's up?"
"Well..." She sits in the chair across from me and undoes her coat. No duck hat today, sadly. I would have loved for Nina to see it. "I hate to kick you when you're down, so to speak, but your idea didn't work."
"My idea about..." I leave the question hanging. I know what she's talking about, but I need to make her pay for her joke.
She frowns at me.
"Your idea of telling Sasha Lewicki that she had won something, so she would show up and claim it. It didn't work. No one showed up."
"You sent her an email?"
Asami nods.
"Using a fake address?"
She nods again.
"And told her she won..."
"Ten thousand dollars."
"Really? Where did you tell her she could collect it?"
"The food court at the mall. In front of the Wok and Roll."
No wonder it didn't work.
"Why there?"
"I wanted a crowd around in case something unexpected happened."
"Like what?"
Asami shrugs.
"I don't know. I just wanted witnesses."
I shift on the couch and wince in pain. I should really take another painkiller but I'll hold off until Asami leaves, although this conversation might be more enjoyable if I were a bit high.
"How long did you wait?"
"Five hours."
"And no one showed. Did you see anyone you know?"
"I saw a lot of people from school."
"Did you talk to anyone?"
"Not really." Asami looks a little embarrassed. "Some people waved, though."
"Well, it's very possible that she did show up and saw you and maybe some other people from the class, and she got spooked. Or she did show up, but you know her by another name. Who did you see?"
Asami closes her eyes and thinks for a moment.
"I saw Principal Jakowski, Peetsa, Kim Fancy and her daughter, um... and Zach Elder's mother."
"Trudy."
"Right, yes. Trudy. That was pretty much it."
"Huh. Well, I don't know what to say. I'm sure she'll show herself at some point." I stifle a yawn. "Sorry. I'm still a little dopey."
Asami jumps up the way Max does when I tell him dinner's ready.
"Oh, I'm sorry. I should go. Thanks for hearing me out." I can tell she's a bit bummed by this setback in her investigation.
She pulls on her coat and starts toward the door.
"Can I get you anything before I go?"
"No, I'm fine, but thank you. And don't worry, Asami. There's probably a really simple explanation for this whole thing."
Asami gives me the firm nod that I now associate only with her and leaves me to my painkillers.
* * *
The aftermath of my bathroom hijinks brings forth some good news and some bad news. The bad news was that Dr. Sintay says I definitely should not go on our spring break family ski trip out west. The good news? I'm staying home alone for a week. Well, not completely alone. Nina will move in with me, and Chyna will take my place on the slopes.
I know I should be more upset about this, but I can't seem to muster the sadness. I love the place we always go—a little hidden gem in Utah called Solitude that totally lives up to its name. I swear there has never been a lift line even if we are skiing the busiest week of the year. It's populated mostly by locals who want to avoid the craziness of Park City, and I absolutely love going there... usually.
But since the accident, all I can think of is the sheer felicity of time to myself. What, oh what, will I do to fill the days that are usually taken up with errands and housecleaning and laundry and Max and Ron and their various time-sucking wants and needs? Don't for one minute think I don't absolutely adore my life as a wife and mom. But even the best lives need a vacation and, let's face it, renting a house with your family at a ski resort is not a vacation. It's basically moving your life from one location to another. Unless someone else is making the beds, doing the laundry, and cooking, it's just the same old life with the added inconvenience of not knowing where anything is in the kitchen.
My convalescence is going more slowly than I thought it would, so I am completely useless when it comes to packing Ron and Max for the trip. Max is only capable of loading up the toys he wants, and Ron hasn't packed a bag since the day he said "I do." It's nothing short of torture for me to sit in bed while he ransacks the storage bins and throws things willy-nilly into suitcases.
"Sweetie, you may want to have Max try on some of his ski clothes before you pack them."
Since this is about the thirtieth time I have butted into his business, Ron is done with me. He pauses on his way out the door.
"I love you, but if you say one more thing to me about packing, I'm going to hide your pain meds."
I don't tell him I switched to Aleve two days ago because the painkillers were interfering with my wine drinking. I stay mute for the rest of the morning, knowing full well that Max's toothbrush and toothpaste will never make it into the suitcase.
And now I sit on my super-comfy couch with six days stretching out in front of me like a red carpet of possibilities. Scrumptious smells are coming from my kitchen, where Nina is cooking dinner, and a fine glass of wine is within reach. If it weren't for the constant ache down there, life would be pretty perfect.
I grab my laptop from the side table and open it up. There is a joke email from Peetsa with the subject line "15 Ways to Make Everything Awkward," the usual spate of crap from the Gap, Zappos, Pottery Barn, and Weight Watchers, and a note from my mother.
* * *
To: JDixon
From: KHoward
Date: 3/26
Subject: How are you feeling?
Honey,
How are you feeling? Better, I hope. I would call, but ever since I woke you up in the middle of the day I'm worried I'll do it again and I know how you need your sleep. But please call me whenever you want, oh but not for the next few days. Dad and I are heading out of town to our post–St. Patrick's Day drying-out retreat. Half of our congregation is still loaded.
Feel better, sweetheart.
Love,
Your Mother
* * *
I'm just about to close my computer when an email pops up from Miss Ward.
* * *
To: JDixon
From: PWard
Date: 3/26
Subject: Spring break
Hello, Jenny,
Haven't seen you in a while and then someone told me about your accident. Hope you are feeling better.
On April 12, we will be taking a class trip to the Elbow Chocolate factory. I think the children will enjoy seeing how chocolate bunnies are made.
I will need 3 mothers to go with us. Can you and Asami send out a note?
Thanks,
Peggy
* * *
Well, this is pretty odd, considering we had our Easter celebration just before the break. But mine is not to reason why. The clown car that is the inner workings of Miss Ward's mind will remain an enigma to me. Oh, and by the way, I'll be skipping that field trip, thank you very much. Putting me in a chocolate factory is like putting an alcoholic in a whiskey distillery. I have a serious addiction and the only way I can keep it in check is complete abstinence. There were some dark (chocolate) days when I first returned from Europe and realized full-on what my life was going to be like (a) living with my parents and two small daughters and (b) working at a crappy job. Things looked pretty bleak, so I turned to my one constant source of sunshine... chocolate. I got quite a taste for the good stuff while I was chasing bands overseas. Ever have a Milka Bar? INXS used to insist on having them in the green room at their concerts. My mouth waters just thinking about it. I am not exaggerating when I say I ate that sweet devil for breakfast, lunch, and dinner every day for about six months. I couldn't stop, nor did I want to. Chocolate made me happy—fat and happy, and then just fat. I remember vividly the day I took Vivs and Laura for their flu shots. I asked the nurse to give me one, too. She looked me up and down and said, "Let me ask the doctor. I'm not sure it's okay in your condition. How far along are you?"
That was it. I went home, threw out a good ten pounds of Milka Bars a friend and fellow band stalker had sent me, and have almost never touched chocolate again. I think that's one of the reasons I hate Halloween—all that chocolate coming into my house unprotected and available.
I need to email the class immediately to start trolling for volunteers.
* * *
To: Miss Ward's Class
From: JDixon
Date: 3/26
Subject: Chocolate, anyone?
Hello, classmates!
Happy spring break! Hope you are all having a nice week off. Obviously, I'm still on duty as your trusty class mom. It's a 24/7 job that knows no rest. You're welcome.
Miss Ward wanted Asami and me to let you know our kids will be going on a field trip to the Elbow Chocolate factory on April 12. Yes, I realize we have already had our class Easter party, but Miss Ward has arranged for them to learn how to make chocolate bunnies so that next year they will be ready! Anyhoo, I need 3 parent volunteers to help chaperone the trip.
I know many of you have yet to volunteer for a field trip and I have no problem calling you out in the parking lot at pickup if I have to. My advice is to jump on this bandwagon. It's chocolate, for God's sake. It doesn't get much better than that.
Response times will be noted.
I remain forever your girl,
Jen (and Asami in spirit)
* * *
I close my laptop and sigh. I wonder how the ski trip is going. It's only been a couple of days, but I miss my crew and wonder how they're getting on without the shoemaker's elf (that's me). I'm fairly certain Vivs and Laura are helping Ron with everything and not just playing house with their boyfriends. They had the fear of God put in them at a very early age. I worry most about Max, because I know he's going to miss me. I tucked a few love notes into his suitcase for him to find randomly, just to let him know I'm thinking of him. I would have done the same for Ron but my poor baby has hidden-note PTSD from his crazy ex-wife. Cindy used to put a couple of dozen notes into his bag whenever he went away for a boys' weekend or a sporting-goods expo. He had to open each note at a certain time on a certain day, and she would call to make sure he was doing it. Every note ended the same way: "Don't you dare cheat on me. I will know if you do. All my love, Cindy."
"Dinner's ready." Nina sashays over to the comfy couch to help me get up. I can walk, but getting up and down is still painful.
"Sure you don't want to eat in the living room?" she asks.
"Nah." I grab her arm, and together we hoist me up. "I need a change of scenery." As I stand, my 'gines starts to throb and my eyes water from the pain. But I suck it up, make my way to the table, and ease onto one of the padded dining room chairs. Dinner looks great.
"What is this? It smells delicious."
"Curried chicken, mango chutney, green beans with pesto and parmesan, and basmati rice." Nina sits at the head of the table and raises her wineglass to me.
"You found all that in my kitchen?"
"You have a lot of great stuff in your cupboards. You guys must get a lot of gift baskets, because you have all these little jars of gourmet ingredients that I know damn well you didn't buy yourself."
"Like what?" I'm trying not to be insulted.
"Uh, caviar, pralines, chili pepper jam, vacuum-packed Israeli dates, cornichons." Nina counts them off on her fingers.
"Okay, okay." I take a bite of the chicken and savor it. "Oh, my God. This is so good." I raise my wineglass. "Here's to the chef."
Nina clinks my glass.
"And thank you for staying with me. I really owe you."
Nina waves my gratitude aside. "Are you kidding? I'm loving this. I hope you don't mind if Garth comes over sometimes."
"Not at all. I'm hoping he'll start me on some stretches or something. I have quite a little food baby, thanks to not working out for two weeks." I pat my stomach for effect.
Nina gives me the "You're nuts" look and continues eating. I decide now is a good time to spring an idea on her that I have had brewing for a couple of weeks.
"So, what's the word on Sid?"
She raises her eyebrows.
"Um... not much. I finally blocked him on Facebook. I'm guessing he got the message. Why?"
"Well, I've been thinking that he hasn't suffered nearly enough for being such a world-class douchebag to you."
Nina shrugs. "I'm over it." Jeez, doesn't anyone hold a grudge anymore? She is living proof that nothing makes you forget an old love faster than a new love.
"Well, I care. That guy needs to suffer for his sins."
Nina sits back with her wine and a curious look on her face.
"And how do you propose he suffer?" She picks a piece of rice out of her teeth.
"We spam him." I grab my wine and take a sip.
"We what?"
"Spam him. Sign him up for every stupid spam email possible."
"Seriously?"
I can tell by Nina's face she doesn't think much of my idea.
"Think about it. There is nothing more annoying than having your In box jammed with hundreds of messages from every website in the world. I propose we sign him up for everything from Jehovah's Witnesses to the Kardashian fan page."
Nina starts to giggle. "Or a Green Bay Packers fan site. He hates that team so much."
"Okay, good. Now you're talking. What else does he hate?"
"He hates ABBA."
"The band?"
"Yup."
"Who the hell hates ABBA? Okay, what else?"
"Um... Oh, God, I haven't really thought about this."
"Take your time—"
"Richard Simmons! He hates Richard Simmons—the workout guy. And scary movies—he was always such a wimp. Do they have a website for that?"
"Darling, they have a website for everything."
Nina grabs my computer from the comfy couch and we pass the next hour listing everything Sid doesn't like and finding websites we can sign him up for. When Nina runs out of memories, I just start signing him up to get emails from local politicians and the NRA. He's not going to know what hit him. Is what we are doing small and petty? Yes. Is it an abuse of the Internet? Absolutely. Do I feel bad about it? Not one bit. I hope Sid chokes on his In box.
While we are executing our attack, FaceTime rings, and my two favorite men pop up on my screen. They both have red faces and look exhausted.
"Hey there!"
"Hi, Mommy! How are you feeling?"
"I'm better. How was skiing today?"
"It was cold. Are you feeling better enough to come skiing?"
"Not quite, sweetie, sorry. How long did you ski? Did you go on any blue runs?"
"Nope. Dad says that's tomorrow. But I don't want to ski tomorrow. I was so cold," he starts to whine.
"How cold is it up there?" I direct my question to Ron.
"Today is supposed to be the worst day. It will be high twenties tomorrow. Thirties at the bottom of the mountain," he assures me, but I'm skeptical. Ron always has a tough-it-out mentality when it comes to Max. I personally hate skiing when it's really cold, and if I were there I wouldn't have made Max go out. It's one of the fundamental differences between Ron's parenting style and mine. I am much more prone to baby my baby.
"Where are the kids?" I ask, to keep the conversation on a positive note.
"They're all out for dinner, except Chyna. She's running Max's bath."
Just then a disembodied voice yells out, "Max, have we found your toothbrush yet?" I debate telling them that I know exactly where it is, but decide against saying anything. According to my mother, it's not nice to gloat.
"Go get in your bath, Maxi. That will warm you up for tomorrow."
"Okay. Bye!" He jumps off Ron's lap and away from my screen.
Ron looks surprised. "That was easy."
"Only till he gets out of the bath and realizes I'm not on the computer anymore."
He lets out a big sigh. "He really misses you. We all do."
I'd be flattered, but I know that most of what they miss is everything I do for them. I'm not being cynical. I just know my customers.
"I miss you guys, too. It's so damn quiet here. How did everyone ski today?"
"Great! Well, except for Travis. I don't think he's ever skied before."
"Oh, no! Did he take a lesson?"
"Actually, Vivs is a pretty good teacher. She got him up and going, and then we each took an hour with him on the bunny hill. Max loved that he wasn't the slowest one on the hill. He insisted on teaching Travis pizza and french fries."
He's referring to the way instructors teach little kids how to snowplow and slalom. I can just picture him.
"Please make sure he's bundled up tomorrow. You'll never keep him out there if he's cold."
"I will. I promise. How's your 'gines?"
"Getting there. I'm walking pretty well, but getting up and down is still a bitch. I think Garth is going to give me some light stretches to do later this week."
"Just take it easy, please."
"I will. Give the girls my love." I blow a kiss to the screen. "Love you."
"Love you, too," Ron answers, and then the screen goes black.
* * *
"Just a little more, Jen. Take a deep breath. You're doing great."
Garth and I have reached a new level of intimacy. He is doing something called Thai massage on me. We are currently on the rug in my living room; Garth is sitting snugly behind me with his arms wrapped around my arms, which are wrapped around my torso. Allegedly, he is stretching me using his own body to enhance the stretch, but I can't help feeling like this is a joke he and Nina cooked up.
I take yet another deep breath as Garth gently releases me from the "stretch." I actually feel really good, so I try to override the feelings of weirdness that keep popping up. It doesn't help that Nina is sitting right in front of us, watching and no doubt having threesome fantasies.
"Wow. That felt great. Thank you." I look at Nina. "Has he done this to you?"
She smirks. "He does me a little differently."
Garth blushes and stands up. "Okey-dokey. That should really help with your stiffness, Jen."
"Does it help with your stiffness, Garth?" I ask with as straight a face as I can muster. Nina bursts out laughing.
"You two are lethal. My gosh, five days together and you have your pay-per-view special all scripted."
"You should hear what we say when you're not here," I kid.
"No, thanks. I don't think I could take it and I was in the military."
Nina stands, too. "What time do you think they're going to get here?" She's asking about our intrepid skiers, who are due back this afternoon.
"Ron says..." I grab for my phone and check my texts. "ETA is around six, assuming they land on time. Do you guys want to stay for dinner? We can order in."
"No, thanks," they say in unison. Nina continues: "I want to get Chyna home and unpacked. But you guys should be able to have a major leftover binge with all the stuff I've cooked this week."
"You are a rock star. I can't thank you enough." I really mean it. Nina and I had so much fun—actually, just the right amount of fun. I needed this week of convalescence, but now I'm totally ready to have my Dixon men back.
You can imagine how thrilled I am to be touring the Elbow Chocolate factory on this fine April morning. Oh, I got my three volunteers from the email I sent out, so I figured I was home free. That is, until Trudy Elder called me this morning to back out... something about Zach having Coxsackie. Oh, isn't that convenient. Hey, lady, some of us just suck it up and honor our commitments.
So here I stand in a hell that involves a still-sore groin, the futility of making chocolate bunnies, and the torture of those sweet sirens (milk and dark) being created right in front of my face. I have said, "No, thank you," at least ten times to samples passed my way. I wish there was a Nobel Prize for self-denial.
The kids are having a ball watching chocolate turn into bunnies. I guess I suffer from a tragic lack of curiosity, because I have never wondered how they make the hollow ones, but it's actually knowledge I am now happy to have.
Our guide this morning is Jacques (effectively morphed from his real name, Jack, to make our chocolatier experience that much more exotic) and he has already showed us how to pour just the right amount of chocolate into the bunny mold and then twist it around until all the sides are evenly covered and there is no excess chocolate, all while flirting with Miss Ward in a disturbingly obvious way. I mean, jeez, there are kids watching! Enough with the double entendres, Jacques.
Joining me on my chocolate journey this morning are Jill Kaplan and JJ Aikens, who was much friendlier than I expected when I showed up instead of Trudy.
She and I are standing on the periphery of the chocolate inner circle and have a distant but effective view of the Jacques show.
"Now, who wants a solid bunny tail to bite into?" he asks the children in an accent that can only have been acquired by watching Maurice Chevalier movies and spending time in North Dakota. We see him wink at Miss Ward, and she giggles.
"Can you believe her?" JJ says out of the side of her mouth, in case there are any lip-readers in the vicinity.
"Well, they've definitely got a little something going on," I counter in a low voice.
"Uh, yeah, they do. How do you think we got this private tour?"
"I'm sorry, what?"
"I hear she's sleeping with him!" JJ side-talks to me again.
"Are you sure?" I turn to see if she is kidding.
"Well, I wasn't there, but Kim told me."
"Was she there?"
I'm rewarded with a barking laugh from JJ. That's a first.
"You never know." JJ sounds a little bitter.
I'm trying to play catch-up with all this new info being launched my way. Jacques and Miss Ward are having a fling, and Kim Fancy knows all about it. Kim Fancy who (and I wouldn't believe it if I hadn't seen it) slapped Miss Ward across the face two months ago. I need to know more. I regard JJ and wonder just how much truth serum she has taken today.
"I haven't seen you and Kim together much lately," I fish with small bait.
JJ looks at me like I'm a drink of water in the desert. "Thank you! I know. I feel very pushed aside."
"Pushed aside by who?" I fake genuine concern. "Miss Ward?"
"What? No," she scoffs. "By her training."
"Her training?"
"Yeah. She's spent the last few months training to do a mud run this month."
Mud run? I get a little sick to my stomach.
"She's obsessed with it. She's working with a trainer and everything."
This news hits me like a punch in the gut. Why, I do not know. I mean, who cares if Kim Fancy is doing the same mud run that I am? It doesn't make me any less special.
Yes it does! screams the cocktail party in my head.
I see Jill Kaplan waving frantically at me from the other side of the chocolate river.
"We better join them," I say to JJ.
As we walk toward where the kids are going to decorate their bunnies, I pull myself together. Jacques has lined up fifteen small bunnies, one for each of the kids to personalize and take home. There are candy eyes, candy bows, candy hats, and even candy carrots for them to have fun with. I notice Max trying to put the candy carrot on the bunny's nose like he would make a snowman nose, but he can't get it to stick, so he eats it.
"Hey, no eating the accessories, buddy," I admonish him.
"What's the assesories?"
"The decorations." There is a lull in the constant chatter as the kids busy themselves with their task. I glance up to see Miss Ward and Jacques standing in the corner whispering while they watch.
"Could she be any more obvious?" JJ sidles up to me to continue her bitch session. I find myself wondering if Miss Ward did indeed whore herself out for a free class tour of the chocolate factory and decide that if she did, she's a better woman than I am.
"You know, I've been training for a mud run, too," I tell JJ as a way to change the subject. "I wonder if it's the same one Kim is doing."
"You are?" JJ's tone is a little more incredulous than I would like.
"Yeah. I've been training since September."
"Wow, I had no idea mud runs are so in vogue." She seems baffled that she would be out of the loop on something popular.
"Oh, yes. All the middle-aged housewives are doing it."
"Really?"
I smile. "No, I'm just kidding. Do you know which race Kim is planning to run?"
"Uh, it's here in April. That's all I know. Is that the one you are doing?"
"I think so. I had an accident a few weeks ago and it put me on the disabled list. I get to start training again next week." As I say this, it starts to throb down there and I realize starting to work out again is going to be a bitch. I'm still tender, plus I'm out of shape. Garth is going to have an "I told you so" field day.
"Maybe I should do it, too." JJ jostles me out of my thoughts.
My God, does this woman ever do anything on her own? She is a professional bandwagon-jumper-on, if there is such a thing.
"Absolutely. Give it a go," is all I say to her.
"Less chitchat, please, moms." Miss Ward has extracted herself from Jacques and joined us without JJ or me noticing. I ignore the reprimand.
"Looks like you and Jacques have made a love connection," I comment.
Miss Ward makes the face I make when I have smelled bad cheese.
"He's not my type at all."
I look at JJ and she rolls her eyes.
"Okay, it's time to get the kids into their coats and back on the bus." Miss Ward pulls her purse over her shoulder.
As the little ones load onto the bus, clutching their bunnies, each one wrapped in cellophane, I give Max a hug and thank JJ and Jill for chaperoning.
My phone buzzes while I'm starting up the trusty minivan. God, it feels good to sit down! It's a text from Ron.
Want a night off from cooking? Let's go to Garozzo's for dinner.
Oh, he read my mind.
Max too?
Already got Chyna to sit with him, so it's just you and me, babe.
Sounds like just what I need.
Sometimes I can't believe how lucky I got. I did kiss a lot of frogs, but it was worth it to end up with this prince of a man.
My phone buzzes again.
You look great today.
Dinner and flattery? Hm... something's up.
Thanks! You always look great.
Seriously? I didn't think you noticed.
How can I not?
Glad you like what you see.
Typical Ron. He has no idea how cute he is.
I do indeed.
You're making me smile.
I want to do more than that.
Aren't you frisky today!
You are, too.
What do you want to do about it?
I text him my sign for boobs since they're his favorite part of my body. The poor man hasn't had sex since my accident. He's been so patient. Then I have an idea.
How about we take a little bathroom break at Garozzo's tonight?
Really?
Really. You've waited long enough.
Wow. I don't know what to say.
Say you'll see me at 7, dummy!
Okay.
I laugh as I start the minivan and pull into traffic. Ron and I used to flirt-text all the time. It's been too long. Tonight should be fun. I haven't had bathroom sex in ages.
* * *
Ron has a last-minute work crisis, so I end up meeting him at the restaurant. He is waiting for me in his car and jumps out when I pull up. He seems a bit preoccupied, so to cheer him up I give him a huge hug and let him know I've gone commando tonight.
"Really? Wow. That's unexpected." He grins like a goof.
"I thought I'd make it as convenient as possible." I squeeze his butt.
"It?" He squeezes mine back.
"Yes, it."
"Well, this is a nice surprise."
"Really?" I stop walking and look at him. "We planned it earlier."
"We did? When?"
"Today. You were flirty-texting me." I reach into my purse to show him my phone and when I look up I see Don Burgess coming around the corner. He stops short when he sees us. Complete and utter horror washes over me as I realize what I'm going to see when I look at my texts. Don. Ron. Shit. I seriously need to think about getting glasses.
"What are you talking about? Let me see." He reaches for my cell and I consider making a run for it. But in the end I know this will all go better if I just 'fess up.
Don is frozen in place, looking very confused. I'm guessing he's trying to assess the situation.
Ron finishes reading and looks first at me and then notices Don.
"Don. What's up? Having dinner here tonight?" Ron's voice is way too calm for my comfort.
"Uh, yeah." Don, to his credit, looks confused and uncomfortable. "You guys, too?"
"We sure are. Who are you eating with?" Ron crosses his arms and takes a wide stance, like he's a bouncer.
"Okay, okay, let's not make this awkward," I say, as though it isn't already. I stand between the two of them. Don and Ron. Jeez, what are the odds?
"Did you come here to have sex with my wife in the bathroom?" Ron's velvet voice is about an octave lower than usual.
Don's face shows no sign of the panic that is ripping through me.
"Um, no. I came here to tell her I couldn't have sex with her in the bathroom."
"What?" Ron and I say at the same time. I'm a little insulted. He'd be lucky to have sex with me anywhere, especially a bathroom. I realize this wouldn't be a smart thing to say right now, so I force out a laugh.
"It's all a big misunderstanding." I put my hand on Ron's arm. "It's actually really funny when you think about it."
"It would be funny if this guy hadn't shown up thinking he was going to screw my wife." Ron yanks his arm away and glares at Don.
"Hey, whoa," says Don. "That's not why I came." He puts his hands up and starts to walk backward. Not a great idea as he only has about two feet before he falls over a planter filled with super-tunias and lands flat on his back.
"Are you okay?" I start toward him, but Ron holds me back.
"He's fine." He leads us both into the restaurant. I mouth, "I'm sorry," to Don as I'm pulled through the door.
"Hi, Mr. and Mrs. Dixon."
Irina greets us from behind her podium. I call her the hostess with the moistest because her hands are always wet. She is one of our favorite people at Garozzo's, and normally I would ask how her kids are and take a few moments to chat, but tonight I don't dare do anything except give her a quick smile and nod.
Ron takes the wheel.
"Irina, can we get the table in the back by the small window?"
She gives him a puzzled look, but only says, "Sure thing, follow me."
She gathers up a couple of menus and leads us through the half-full restaurant to what is generally known as the crappiest table at Garozzo's because it has the distinct honor of being both by the bathroom and near the place where the waiters congregate to place their orders. If Ron is trying to punish me, mission accomplished.
As we sit down, I open my mouth to start explaining, but Ron, who still has my phone, is busy scrolling through my texts with Don. Oh, shit. He finally looks up at me.
"What the fuck is going on?"
Uh-oh. Ron never drops the f-bomb. This is bad.
"Sweetie, I'm sorry. You saw the texts. You know it was a mistake."
"For you, it was. But he thought it was real. Why the fuck would he think you'd invite him to screw you at a restaurant?" He pauses. "Are you having an affair?"
"No!" I say as emphatically as I can. "No. No. Never. I would never do that."
"Oh, come on, Jen!" Ron snaps at me. I look over his shoulder and notice we're drawing looks from some of the customers. "You've been texting with this guy since the beginning of the school year."
"Yes, but it's just texting. Stupid, mindless texting that means nothing." I'm trying to remember just how bad the flirting got.
"Why would you write 'Do you mean coffee or COFFEE' in capital letters? If some woman texted that to me, I'd think she was coming on to me."
"Would you like to hear the specials?" Our waitress makes an untimely entrance.
"Can I have a glass of red wine?" I ask.
"Me, too." Ron rubs his hands over his eyes.
She nods and walks away. I look at Ron, and he takes a deep breath.
"I just meant to be funny, I wasn't coming on to him. You need to believe that."
Ron shakes his head and looks at the table.
The wine arrives, and we both take a huge gulp. I can tell Ron doesn't know what to say, so I keep going.
"You know, it probably wouldn't have happened if you hadn't told him I had a crush on him in high school."
"Oh, please, he already knew that."
"Trust me, he had no idea."
"What about your big hookup in the P.E. laundry room?"
"We didn't hook up!" I suddenly realize I've never told him the story. "I walked in on him going down on the girls' volleyball coach."
Ron raises his eyebrows.
"Really?"
"I was in detention and they used to make us help out the custodial staff. I was given a bunch of uniforms to wash. So I went to the P.E. building, walked in the laundry room, and got an eyeful of Don having a box lunch."
"Wow. Did they see you?"
"Unfortunately, yes."
"Did the teacher get fired?"
"I don't know. The only real conversation Don and I ever had in high school was when he asked me not to say anything. He said they were in love, if you can believe it."
Ron smirks, and I think I can see a thaw coming. But then his brows come together again.
"But none of this explains why he came here tonight." He runs his hands through his hair and scratches really hard. "I mean, Jesus. He thought he was going to have sex with you. There must be something else."
"Well, technically, he says he came here not to have sex with me, so..."
Ron scowls at me. I heave a deep breath trying to slow my heart beat down. We drink our wine in silence until Ron finally says, "I've lost my appetite." He gets up and puts some money on the table.
"I'll see you at home." As he walks away, I can barely see him through the tears welling up in my eyes.
* * *
I finish both glasses of wine at the table by myself. I have a headache from trying not to cry, so when I get to the minivan I let loose and sob for a good ten minutes. Ron has never walked out on a fight before. That's usually my job—I'm the runner, he's the chaser. I have no idea how to make things right. I'm really hoping he just needs time to cool down.
I drive home in a fog. My stomach is queasy and my head is still pounding despite my sobfest—most likely because I haven't had anything to eat since two o'clock. As I pull up to our house, I see about five cars in the driveway; all are familiar, and I'm immediately panic-stricken. I leap out of the minivan and run to the front door, where I'm greeted by a burst of laughter coming from my living room. WTF?
When I walk into the room, my family and friends are sitting in a circle playing Apples to Apples. My mother looks up at me.
"Well, finally. We thought you guys would never get here."
"What are you all doing here?" I ask with no social grace whatsoever.
"Ask your husband," my mom suggests.
"Is he here?"
"He's supposed to be with you." Nina stands and walks toward me. "What's wrong?" She knows me better than anyone. Not to mention my eyes probably look like I've gone five rounds with Muhammad Ali.
"He left the restaurant before me, that's all. What are you guys doing here?" I ask again.
"We came to talk to you," Laura says as she starts to clean up the card game.
"About what? What's wrong? Why are you two not at school?"
"We're heading back tonight, don't have a cow." Vivs frowns. "Why isn't Ron here, is the real question. He called this powwow."
"I need some water," I say, and head to the kitchen. What the hell are all these people doing in my house? I'm so not in the mood for this. Chyna walks in as I'm chugging my drink.
"Max is on your bed watching TV," she tells me. "Do you want me to stay with him?"
"Would you mind, sweetie? I need to deal with what's going on in my living room."
"Sure, good luck." She gives me what looks like a pitying smile and heads back upstairs.
I start for the living room, but my mother cuts me off in the hallway and drags me back to the kitchen. She backs me up against the counter and looks at every square inch of my face. Finally she speaks, in the kindest voice I think I have ever heard her use.
"What's wrong, honey?"
"Oh, Mom." I dissolve into tears and throw my arms around her.
"Shhh. Okay, okay. Let it out."
When I can finally calm down enough to get a coherent sentence out, I briefly tell her the whole sordid tale, starting with when I first saw Don at curriculum night and ending with this evening's fiasco.
"I thought it was just fun, you know? It made me feel giddy. But I never ever would have taken it anywhere. I love Ron and I'm happier than most married people I know. Why would I put any of that at risk just to have a laugh? What is wrong with me?"
"Oh, sweetheart, there's nothing wrong with you. You're just mourning your youth."
Great, my mother's calling me old. I say as much to her.
"Jennifer Rose, that is not what I said. Of course you're not old, but you're also not that twenty-two-year-old girl traipsing through Europe throwing her cat around."
"Mom!"
"No, listen to me." She leads me to the kitchen table and pulls out two chairs for us.
"You did it all and had a great time, and don't ever tell your father this, but I'm glad you had a little fun. We got married right out of high school, and he's the only man I have ever kissed, let alone... you know." She looks at her lap.
I let that little overshare sink in for a minute.
"Are you saying you regret only being with Dad?"
"No, I'm saying it's okay to realize that the really fun part of your life just might be behind you. Be sad that it's over, and then move on. Those years aren't coming back, no matter how young a little flirting makes you feel. And believe me, there's a whole lot of good stuff still to come."
I look at my mother and try to see her not as my mom, but as a woman. It's not something we often do as daughters. Then something occurs to me.
"So, did you ever have a little flirtation with someone?"
She gives me a worldly grin. "Now, sweetheart, you don't think I went to bingo all those years because I loved playing it, do you?"
I crack up at this and continue to laugh until I start crying again. Ugh.
My mom hands me a paper napkin from the holder on the table and waits for me to calm down.
When I do, I ask her to tell me why everyone is here.
"Oh, my goodness. I forgot about them. We should get back in there."
She holds her hand out for me, the way she used to when I was a child. I happily take it.
"But why are they here?" I ask again.
"You'll see. It was Ron's idea."
At the mention of his name, my stomach does a high dive. I wonder when he'll be home.
That question is answered as we enter the living room; he's sitting with my dad, my daughters, Nina, and Garth, quietly waiting for something to happen.
"So, what's going on?" I say to no one and everyone.
My mother, who has joined my father on the couch by this time, nods to Ron, who nods back; then she stands up and takes a piece of paper out of her bra. Nice, Mom. Way to keep it classy.
"Jennifer, I just want to start by saying we are so proud of you. I mean, let's face it, you weren't exactly on the fast track to success after your little trip to Europe."
That elicits a good laugh from the crowd.
"But you pulled yourself together and have created a beautiful life."
"Thanks, Mom." My eyes start to well up again, and so do hers. She hands the paper to my father, sits down, and blows her nose into the tissue that permanently lives in her sleeve.
My father stands and clears his throat.
"We have really enjoyed watching you get yourself back into shape and, honey, you really look terrific. Not that you weren't always a beauty, but you have a real nice healthy glow these days."
"Um, thanks, Dad." Why are they saying all these nice things? My birthday isn't for, like, two weeks.
Nina goes next.
"Girl, you know how I love to bust you about your training, but I know you've really put your heart and soul into it. I wish I had the discipline to go the distance like you have. You're my hero." She raises her glass to me in salute.
My eyes are misty with tears that just won't stop. I can't believe all the love I'm receiving. It's too much, considering the emotional tsunami I have already been through today.
"My turn!" Vivs stands up and makes everyone laugh.
"Mom, you are without a doubt one of the most insane people I know. Nobody I know has a mom like you. You taught Laurs and me to be strong, independent women, but you also forced us to learn how to do laundry and cook. I swear I'm the only one of my friends who knows how to sew a button."
"Well, that's just shameful," my mother interjects. The lack of domestic capability among the younger generation really grinds her gears.
"Anyway, we love you and we are really proud of you, no matter what."
No matter what?
As I'm pondering this last line, Laura takes the wheel.
"You have been the best mother ever. Even when you did things like hit us with your shoe when you were driving—"
"That was one time!" I feel the need to defend myself.
"I know! I'm just saying that even though you lost your temper and were really mean sometimes—"
"Laura, stop talking." Vivs takes her by the arm and pulls her down.
"She knows I love her," Laura protests, but remains seated.
Garth stands up beside Nina and gives me that great megawatt smile I remember so well from the first time I met him.
"Jen, you are a rock star. You took a chance on an old has-been trainer like me, and I have enjoyed the last six months so very much. Becoming your friend has just been icing on an already frosted cake."
"Thanks, Garth, I feel the same way."
"But here's the thing."
Ah! The thing. Finally we get to the thing.
"I don't think you should attempt a mud run for the next few months."
I'm surprised I didn't put this together sooner. I mean, my dad talked about what a nice body I have now. When would that ever come up in a happy-birthday speech?
Everyone is watching to see what my reaction will be. If they had caught me on another night, I might have argued with them, but tonight the fight is all out of me, so I just shrug.
"I don't really know what to say. I can see you've put a lot of thought into this, and clearly you have had meetings behind my back."
"Just one," Laura assures me.
I wink at her; then I look straight at Garth.
"Really?"
He smiles sheepishly. "We'll do this, I promise. I just don't think you're ready. You were, believe me. But missing these last three weeks has put you off schedule for the one in April. I've told you before, August will be your shining moment."
"But in the meantime..." Ron stands up. I haven't been able to make eye contact with him this whole time. "I want you to think about doing the mini event at my store." His monotone feels like a slap in the face after all the love I've been getting.
"You're doing that event again?"
He nods. "The governor's office called us last week and asked if we would participate again."
I look at Garth, who gives me a thumbs-up.
"Back to the scene of the crime, huh?"
"This time you'll crush it," Vivs assures me. Everyone nods in agreement.
I'm suddenly overwhelmed by the events of the night. I can't help it. I start to cry yet again.
Laura and Vivs get out of their chairs and rush over to awkwardly hug me. How did they know that's just what I needed?
* * *
On my way to bed that night, I tiptoe into Max's room to kiss him good night and find him still awake.
"Can't you sleep, buddy?" I sit down on his bed and brush the hair from his eyes.
"No. I think I ate too much bunny," he mumbles.
I suppress a smile.
"I'm surprised Chyna let you eat it."
"She didn't really know," he whispers. "She thinks I just ate the ears."
"Well, that's not cool. Just because I'm not home doesn't mean you can break the rules."
"I know." He yawns. "I'm sorry. Will you lie down with me?"
I know I should go into the bedroom and try to talk to Ron, but I really don't feel like being iced again. After everyone left, he went upstairs without a word. And anyway, who could refuse such an offer? I cram myself into the race-car bed and Max snuggles into me.
* * *
To: Ms. Ward's Class
From: JDixon
Date: 4/8
Subject: Ye Olde Parent/Teacher Conferences
Good morrow, good friends!
I was going to do the whole email in olden speak but I'm already bored with it, so I'm switching to acerbic.
Can you believe it's conference time again? I feel like we just went through this whole rigmarole. I mean, my God, how much more can we talk about our kids?
Apparently Miss Ward has a lot to say, so we will be squeezing our butts back into those little chairs come April 27 and 28.
My plan is to use the same schedule as I did in September. I have attached it below. If you have a problem, keep it to yourself or tell Asami. She has a much more sympathetic ear.
Conference Schedule:
Thursday, April 27
12:30 Lewicki
1:00 Fancy
1:30 Aikens
2:00 Zalis
2:30 Alexander
3:00 Kaplan
Friday, April 28
8:00 Cobb
8:30 Dixon
9:00 Westman
9:30 Baton
10:30 Tucci
11:00 Elder
11:30 Wolffe
1:00 Gordon/Burgess
1:30 Chang
2:00 Brown
By the way, on April 29 there will be a mini mud run at my husband's store (the Fitting Room, on Drummond St.) to help raise awareness for the governor's "Get Fit" campaign. If anyone would like to participate, email me and let me know. I can bring five people.
That's it. Move along. Nothing to see here.
Jen (and Asami in spirit)
* * *
* * *
I can tell Garth is taking it a bit easy on me, and frankly, I'm glad. How sad is it that it takes six months to get into shape and basically six days to fall out of shape?
He's keeping away from anything that might tax my groin area, which unfortunately doesn't exclude burpees. After five, I cry uncle and he gives me a breather. We have already done push-ups and sit-ups and a bit of jump rope cardio, but I had to stop because the blood was pumping a little too enthusiastically through my downtown area. We're only twenty minutes into the workout and I'm already done.
"Looking good, Jen."
"Oh, please, Garth! I'm like a newbie. When was the last time I quit after five burpees?"
"Give yourself a break. We have two and a half weeks to get you back in fighting shape, and it's not going to happen in one day. I want to work your cardio a little more, so why don't we fast-walk on the treadmill? I'll give you a bit of an angle so it feels like a hill."
I sigh and hoist myself off the floor of Ron's Gym and Tan. I haven't been down here in a few weeks and I forgot all the little changes I made to the décor over the winter. The is now a red Nike poster with black letters that says, "If no one thinks you can, then you have to." It was a Christmas gift from Peetsa and I love it. I also put in a decorative basket of towels for when I sweat, and a pitcher of water, which sometimes has lemons in it and sometimes cucumbers. I usually light a Bay Breeze Yankee Candle, too. All in all, it's a nice place to work out.
"What are you doing for your birthday?" Garth asks as he pushes the buttons on our treadmill.
"I'm not sure." I'm fast-walking but I sneak a look at Garth. "Why? Is Ron planning something?"
I sound a bit desperate. Things at home have been pretty tense for the past week. I hope Max isn't picking up on it. I mean, we still eat dinner together and hang out, but Ron has a force field around him when it comes to me. We talked the morning after the weird intervention thing, when he found me sleeping in Max's bed. Ron said he just needed time and space and would I just please give it to him, which I have, but it's been really hard. I just want to say over and over how sorry I am and how much I love him, but he won't give me the chance.
"Not that I know of," Garth says in answer to my birthday plans question, and I'm sure he's right. He slows down the treadmill and looks at me thoughtfully.
"Are you okay with just doing the store's mini mudder?"
"Totally okay." I huff and puff. "I'm so out of shape, I'll be lucky to get through it."
"I'm going to make sure you are more than ready."
* * *
After Garth leaves, I sit down at the kitchen-counter office with ice on my groin to check my emails.
I have a lot of responses, so I don't even bother opening Sasha Lewicki's. I don't really need to know when she will be out of the office until.
* * *
To: JDixon
From: AChang
Date: 4/8
Subject: Ye Olde Parent/Teacher Conference
Jennifer,
I have not seen you in a while. I assume you have recovered from your accident.
Thank you for taking care of the conference schedule. Can you meet me for tea before pickup tomorrow? I have a new theory about you-know-who.
Asami
* * *
Oh, for the love of God, woman, give it up. I'm going to have to shut her down. I email that I will meet her at Starbucks tomorrow at two. She needs a reality check.
* * *
To: JDixon
From: KFancy
Date: 4/8
Subject: Ye Olde Parent/Teacher Conference
Hi, Jen,
Well, it's lucky I didn't plan a trip back to Manhattan, or I would have been in trouble again. My conference time is fine.
By the way, I would love to participate in the mud run at your husband's store. What a cute idea. It will be nice and easy after I do the real KC mud run the week before. Will you be doing it too?
Thanks,
Kim
* * *
I figured she would take the bait. Kim Fancy is just the incentive I need to truly rock the store's mud run.
* * *
To: JDixon
From MJBaton
Date: 4/8
Subject: Ye Olde Parent/Teacher Conference
Dear Jen,
Our conference time is great and Jean-Luc would like to participate in the mud run at your husband's store. Would that be okay?
Thanks,
Mary Jo
* * *
Jean-Luc Baton wearing shorts and working out? Uh, yes, please. Then I open Shirleen Cobb's response and it gives me the only good laugh I've had in days.
* * *
To: JDixon
From: SCobb
Date: 4/8
Subject: Ye Olde Parent/Teacher Conference
Jennifer,
Conference time is fine. I would like to have helped you and your husband out with the mud run thingy but I have been training at Curves for about two months and I don't want to do anything that might interfere with my progress.
Shirleen
* * *
Much to my surprise, no one had a problem with their conference time and I filled all five spots for the mini mud run. Besides Kim and Jean-Luc, Hunter's two moms signed up, and so did Ali Gordon.
When I see her email, it reminds me that I owe Don one. The morning after Flirty-Text-Gate, he wrote me a very nice note explaining that he showed up at Garozzo's to let me know that he thinks I'm awesome, but not in that way. Apparently he's been trying to get back together with Ali and all his romantic focus has been on that. He said he really had wanted to meet me for coffee all those times but just to talk to me about her. He told me he loved our texting banter, but never thought of it going beyond that.
* * *
When I got your invitation for sex, I was surprised and flattered. I mean, really flattered. But I knew there was no way it was going to happen. I wanted to talk to you about it in person and not just leave you hanging alone at the restaurant. You're a great girl and I was worried I had done something to lead you on. Now that I know the texts weren't meant for me, it all makes sense.
I hope you and Ron were able to laugh it off. He didn't look too happy, but I'm sure once you explained it to him, he was fine. If not, I'd be happy to talk to him and set him straight.
Cheers,
Don
* * *
Is he kidding me? This is the guy who said he wanted to help me work out. If that's not flirting, then someone hand me a dictionary. That's such a guy's way out. Oh, you didn't want to have sex with me? Yeah, me either. It was just banter. Right, Don, hold on to that.
I'm not going to lie. Finding out that my little crush was possibly only one-sided all along was a real punch in the boob. I know I said the texts didn't mean anything and they were just for fun, but the sad truth is that, once again, Don Burgess is not interested in me. At least this time we're friends—or we were friends; I'm not sure what we are now.
I write Don back a note saying all is well and wasn't that hilarious and blah blah blah. I wish him luck with Ali and say I'll see him around.
Good-bye, Suchafox! It was fun while it lasted.
* * *
April is my favorite month, and not just because it's my birthday. I love the way the air smells of mud created by the ground thawing. It's one of the first signs of spring and always makes me think of my childhood.
I take a deep breath before I head into Starbucks for my Asami intervention. I spot her standing in line to order, so I walk up beside her and say, "Okay, what's your new theory?"
I kind of like the way my relationship with Asami works. There's no preamble, no fake kisses and chitchat. We just get right down to it.
"It's Miss Ward," Asami blurts.
Oh, Jesus, this is going to be worse than I thought.
"My treat today," she continues. "What would you like?"
"Wow, thanks. I'll have a tall Peach Tranquility. I'll go grab the couch for us."
I settle in and check my phone for messages, hoping for something from Ron, but no luck. He's still being chilly. When Asami joins me, she places a giant cookie between us. Chocolate may be a no-no for me but I never say no to a chocolate chip cookie.
"Help yourself," she says as she takes off her sweater.
Who is this woman? Or maybe this has been the real Asami all along and I just never saw it. I decide I need to be kind but firm about her crazy witch-hunt.
"I have to say something to you and I hope you hear me," I begin. "I really think you're barking at the moon. I know it's a bit of a mystery, who this Sasha Lewicki woman really is, but in the grander scheme of things, who cares? Is it hurting Suni in any way? Is it affecting the quality of your day-to-day life? Probably not, so why don't you just drop it?"
I silently give myself props for my nice little speech. I see that Asami's frown has formed a small "v" on her forehead and her mouth is poised in an "o." I take this moment to break off a bit of the cookie and pop it in my mouth, but find to my horror that it's filled with raisins, not chocolate chips. There are few things in life more disappointing. I would spit it out if that was socially acceptable.
Asami still hasn't said anything, but she is looking at me.
"I'm sorry. I hope I didn't hurt your feelings. I just think there are more important things to worry about."
She nods. "You're right, there are. I'm not sure why I'm so focused on it."
"Well, it will definitely go down as one of the great mysteries of room 147," I proclaim. "That and how Mary Jo Baton landed Jean-Luc for a husband."
She smiles at my little joke. I think there's hope for Asami's sense of humor after all.
* * *
I gratefully lower myself into bed and place a big bag of ice on my groin. It only bothers me after a long day like the one I just had. Usually Ron would want to fool around tonight, because, just like every man in the world, he thinks sex is the perfect gift to give your wife on her birthday. But the way things have been, I'm really not sure. He was very sweet this morning when he and Max gave me breakfast in bed, but then I didn't hear from him all day. It was actually fine. I was unexpectedly busy. The girls surprised me on FaceTime by singing the Beatles' version of "Happy Birthday" with Travis on bass and Raj on tambourine. Later I had lunch with Nina, Peetsa, and Ravi at the place with the signs (Stu's Diner), where we pigged out on homemade chili and Steph gave me a whole apple pie to take home. Stu's only had one new sign—a small one hanging over the front door. It said,
"If I Wanted to Listen to an Asshole, I'd Fart"
I spent the rest of the afternoon getting my hair cut and blown out, and topped the day off having a really fun dinner with my folks and Max and Ron at Minsky's. They have the best pizza in KC. Some might say Waldo's is better, but we're a Minsky's family from way back. We always order the same thing—a Papa Minsky's with pepperoni, Italian sausage, salami, and roasted red peppers. You don't want to be sleeping with any of us on a Minsky's night, let me tell you.
So this is forty-eight. All things considered, I'll take it over twenty-eight any day. Especially my twenty-eight, which saw me living with my parents and raising two small kids. As the great Billy Joel once said, "It's some kind of miracle that I survived." This is definitely my time to remember.
I wake up way before my seven a.m. alarm. I'm beyond excited, but before I jump out of bed I force myself to do my morning bed stretches and affirmations. This is something Garth made me start while I was injured. Here goes.
"My mind and body are in perfect balance. I am unlimited."
Nice, right? Loving myself isn't really up my alley, but I find this very empowering. It's better than the mantra I used to have, which was "Get your fat ass out of bed."
I know today is just a store-sponsored mini mud run, but to me it's the Olympic Decathlon and the Super Bowl rolled into one and placed in a large bag of chips. My nerves are crack-a-lacking. I'm going back to the scene of my crushing defeat—my complete and utter breakdown in the face of physical challenge. That was a bad bad day. The only thing that could have made it worse would have been shitting myself while trying to get over the wall.
But today, that wall is mine. "My mind and body are in perfect balance. I am unlimited."
I hoist my fat ass out of bed and drag it to the shower. Max is still sleeping. I can only assume Ron is already at the store supervising the setup. I don't have to be there until nine.
God, I wish I had slept longer; I was up so late. I lean against the shower wall for support. As luck would have it, Ron and I chose last night to finally hash out Textgate. Things had certainly been lightening up between us, but we hadn't had a real discussion about it. After I put Max to bed, I joined Ron in our bedroom and caught him reading something on my phone.
"Is that my phone?" I tried not to sound too indignant, because we've always had an open-phone policy in our marriage. But seeing him scrolling without even asking kind of set me on edge.
"Yeah, it is," he answered without any guilt in his voice. "I was rereading those texts you had with Don."
Okay, so we're doing this. I girded my loins and jumped in.
"See anything you missed the first time around?" I asked.
"Yeah, a lot. You guys were really chatting it up."
"Just about stupid stuff." I walked over and sat on the bed beside him.
"I can see that." He continued to look at my phone and not me.
I touched his arm. "Ron, I'm sorry. I really am."
"I still don't understand why you felt the need to have such a back-and-forth with this guy. Am I not interesting enough?"
I sighed. How could I say, "It's not you, it's me," without sounding trite?
"Sweetie, this is all on me. You are more than enough of everything I could possibly want in life. But according to my mother, I'm having a bit of a midlife crisis."
Finally Ron looked at me.
"What's the crisis?"
"Uh, I'm forty-eight, my best years are behind me, and I'm going to be a grandmother."
Alarmed, he sat up. "What? Who's pregnant?"
"Well, no one yet, but it's coming just like everything else."
"Jesus, Jen, you nearly gave me a heart attack."
"Sorry. It's just what I think about."
"What else do you think about?" He seemed leery of my answer.
I lay back on the bed and closed my eyes. Cripes, what don't I think about?
"I think about how I look just a little less attractive every day. I think that when I'm sixty, Max will just be finishing high school. I wonder if I should have had a career instead of a bunch of jobs. I wonder why you love me and when you might stop. I worry that I'm not a good enough wife, daughter, mother, and friend. And I worry that if this is it, this is my whole life, will it be enough?"
There was a long pause, and then my husband said, "That's it?"
It took me a moment to realize he was joking. I started to belly-laugh. He lay down beside me.
"So this is why you started flirting with an old boyfriend?"
"He was never my boyfriend. But..." I was trying to nail down what had been driving me this whole time.
"But... it made you feel young?"
Ding ding ding! Ron for the win.
"I guess in a way it did. I mean, he knew me before college, before kids... before you."
"Well, he knew the young you, but not the best you, as far as I'm concerned. I don't know if I would have liked seventeen-year-old Jen as much as forty-seven-year-old Jen."
"Forty-eight," I corrected him.
"Right, forty-eight. I'm sorry you're having a midlife crisis about getting old, but you need to see yourself through our eyes."
"Our eyes?"
"Mine and Max's. We love you and think you're amazing. That ski trip was no fun without you, and not just because no one made skillet tacos or us."
I started to say something, but Ron cut me off.
"Let me finish. You are everything to us... to me. But if we aren't enough for you, then that scares me."
I sat up on the bed. "You are! You are! I love my life with you guys and with the girls. It's just hard getting older. I'm not the prettiest girl at the party anymore, and I need to adjust to that."
Ron sat up beside me and pulled me into his arms.
"You will always be the prettiest girl at my party. Don't ever doubt it."
Corny, right? But it was music to my ears, and the makeup sex really burst the dam of tension between us. I'm so glad we had it out. I just wish it hadn't been the night before the mini mud run, because now I'm physically and mentally wiped.
Out of the shower I grab my cell phone and check the weather. Sunny, with a high of 67 degrees: perfect.
I put on a pair of Lululemon cropped yoga pants, my favorite workout bra, and one of the Fitting Room T-shirts Ron had made for the event. I run a brush through my hair and decide a ponytail will be my best bet.
I'm humming the Rocky theme as I run down to the kitchen and whip up some scrambled egg on Ezekiel bread with ketchup—my breakfast of champions.
It's 7:30 and I'm ready to go. Shit. I need a distraction, so I go into Max's bedroom and rumble around until he wakes up from the noise.
"Hi, Mommy," he says through a yawn.
"Hey, buddy." I curl up in his race-car bed with him and snuggle.
"Is your run today?" he asks.
"Yup."
"Are you going to win?"
"I will win just by finishing the course."
He grabs my face so I'm looking right at him.
"Mommy. Winning is winning." He sounds like Ron.
"No, sweetie, winning is doing your best." I pull him into a hug.
"Want to hear my song about winning?"
"Sure." I stifle a yawn. "Lay it on me."
"Winning, winning, winning, winning, winning," he sings softly, to the tune of absolutely nothing recognizable. I shut my eyes and sigh with happiness.
"Mommy!"
I open my eyes and something has changed. The light in the room is different, and Max smells like cheese.
"What time is it?" I ask.
"I don't know." He goes over to his iPad Mini and opens it. "It's eight-five-five."
"What the fu... dge." I scramble off the bed. "Did I fall asleep?"
"Yeah, when I was singing. So I went down and made my own breakfast without using the oven or the microwave." He sounds so proud of himself.
Boy, nothing good ever comes from me dozing off. I start pulling clothes out of the dresser and throwing them on his bed. "Sweetie, we have to get going. Can you get yourself dressed?"
"But Teenage Mutant Ninja Turtles is on."
"Max, you knew you were going to miss it today. Please get dressed and meet me in the kitchen."
"I don't feel like it." He's pouting and whining now.
"Max, please! This is my big day. I need full cooperation. Lock and load, let's go." I start pulling his PJs off.
"No! Stop it! Don't! Hands are not for hitting!" he yells.
"I'm not hitting you. I'm undressing you. Stop fussing around!" I'm inches from losing it. "What is wrong with you?"
And as I say it, I know.
"What did you make yourself for breakfast?"
"Cheez Whiz," he grumbles.
He's hungry.
"Buddy, how about you get dressed and I'll let you have two Pop-Tarts for breakfast. Then you can watch a DVD in the car on the way to the store."
Threats and bribes are the only two ways I know how to parent. Luckily, this bribe works and before you can say, "Jen's a crappy mom," Max and I are in the minivan and speeding downtown to the store.
As we pull into the parking lot, it looks like the circus has come to town. There is a big orange tent where people are gathering. A large blue, orange, and white banner has been hung on the building welcoming everyone to the Governor's Get Fit Mini Mud Run sponsored by the Fitting Room.
"Max, look at that!" I yell so he can hear me over the headphones. The sight is impressive enough to tear him away from his movie.
"Whoa!" he yells. "Cool."
And indeed it is cool. Ron's team has done an amazing job putting together a fierce-but-not-too-fierce obstacle course, which covers half the parking lot and the adjoining field. It takes me a minute to register that, much to my horror, a fire obstacle has been included. They've really upped the ante from last year.
There is no room for cars in the lot, so I have to park down the street. Max and I rush back to the parking lot and wend our way through the crowd to where Ron is registering participants and getting liability waivers signed.
"Hey, sorry we're late."
"Are you late?" Ron doesn't even look up. I can tell he is overwhelmed by the turnout. I know it's a big day for him, but it's a big one for me, too.
I leave Max to hang with his dad and the rest of the team behind the desk. In the parking lot, I bump into Hunter's two moms. Kim and Carol are dressed in matching shorts and homemade T-shirts that say "Team Hunter." I give them both a hug.
"You guys look great!"
"Thanks. I can't believe that obstacle course!" says Kim or Carol. "I'm a little intimidated."
"It's bigger than the one they had last year," I tell them.
"Did you do it last year?"
"Uh, you could say that." I really don't want to relive the shame of last year's failed attempt. I spot Garth, Nina, and Chyna over by the door to the store and excuse myself.
"How are you feeling?" Garth asks after I hug all of them.
"I'm a little frazzled. I fell back asleep and woke up, like, fifteen minutes ago," I tell them. "Chyna, sweetie, want to make twenty bucks the hard way?"
She smiles and walks toward the tent where Max is.
"I already told her that's what she's here for," Nina assures me. "How are you and Ron doing?"
"We finally talked about it last night."
Nina raises her perfect eyebrows. "Boy, when Ron says he needs time and space, he isn't joking."
"I know, right?"
"So, are you forgiven?"
"You could say that." I blush, thinking once again about our makeup sex. "I think he was determined to prove that my reality is better than any fantasy."
Nina smiles. "Pulled out all the stops, did he?"
I lean in and tell her she'll get all the details later. Right now I need to focus.
I turn to Garth.
"As soon as I saw the course, my groin started to hurt. Is that normal?"
"You'll be fine. It's a great course. I'm looking forward to it."
"And I'm looking forward to seeing you in action." Nina gives his arm a squeeze.
"Me, too," I purr.
"All right. Cut it out right now. Jeez." Garth is blushing at our now ongoing joke.
Hot Dad Jean-Luc and Kim Fancy are having a tête-à-tête over by the Gatorade stand, which is right in my line of vision. Kim is wearing a long-sleeved black unitard that makes her already thin body seem emaciated. Jean-Luc is looking mighty fine in thigh-length running shorts and a hoodie. I look down at my T-shirt and leggings and realize I should have put a little more thought into my outfit.
I wave to them both and start to walk over. Don Burgess's baby mama, Ali, falls into step with me; I completely forgot she'd signed up. I wonder if she knows about the texting drama, or if she would care.
"Hey there. Thanks for letting me do this," she says, panting a little. "My New Year's resolution was to dump my bad habits and get back into shape. That course looks really scary."
She stops walking to catch her breath; I wonder how well she's going to do, given she can't even keep up a brisk walk without huffing and puffing.
"Did Don come out to watch you?"
"Oh, God, I hope not," Ali replies, then puts her hand over her mouth. "Sorry. That was an overshare."
"Not at all. Do you not like when he watches you do things?"
She shakes her head. "It's hard to explain."
I frown to show her I have no idea what she's talking about, but just as she's about to continue I see Kim and Jean-Luc coming toward us, so I touch her arm and ask her to hang on.
"Thanks so much for coming!" I enthuse to them. "Kim and Carol are by the tent, so it looks like room 147 is here to represent!"
"I'll say," a voice behind me chimes in.
I turn, and of all people in the world, you'll never guess who's there. Well, maybe you will, but it shocks the shit out of me.
Miss Ward is standing there in a pink warm-up jacket and shorts. Her blond hair is done in two braids and she's wearing a white do-rag.
There are cries of "Miss Ward!" and "Peggy!" as everyone in the group greets her in their own way.
"Here to cheer us on?" I ask.
"I'm here to do more than that. I'm going to do the course."
"Good for you." Jean-Luc seems disproportionately happy to hear this news. I, however, have gone into silent panic mode. I flash back to when I told Ron I was going to invite some people to do the course. We were in the kitchen, cleaning up the dinner dishes.
"Uh, I'm not sure." He seemed less than thrilled.
"Or maybe I won't."
"It's just that we've had a huge response this year. I put a poster up in the store two weeks ago, and I already have a hundred and seventy-five people signed up to participate. I'm just wondering how many can do it in three hours."
"Well, if they're anything like I was last year, you can bet on thirty seconds before they collapse in defeat."
Normally that would have made him smile, but because we were still in the ice age, all I got was the back of his head.
"So you'd rather I didn't ask anyone?"
"No, you can, but just keep it to five max, okay?"
"Okay."
Not wanting to rock the boat, I took his request very seriously. And now here's Miss Ward, assuming she can just tag along. I take her arm and lead her away from the other parents.
"Did you sign up through the store?" I ask, as politely as I can.
"No." She frowns. "I saw it in your email."
"Well, you didn't RSVP, and I said we could only have five spots. I don't know if they can squeeze you in." How did she see the email?
"But I did RSVP, Jen. In fact, I was probably the first to RSVP."
I give her a quizzical look. Her gaze is steady.
"I'm always the first one to respond."
And in that moment, I feel a huge shift in my equilibrium. I'm off balance for a nanosecond but quickly steady myself. Oh. My. God. I take a breath.
"Sasha Lewicki, I presume."
"In the flesh."
Pieces of the puzzle start to click into place. Miss Ward is the only one who has ever seen Sasha and her daughter, Nadine; she isn't on the class email list and yet always knows what the emails say. Oh, my God, Asami was right! She is going to lord this over me for the rest of my life.
I'm about to launch into a hundred questions, the first one being "In God's name, why?," but I'm interrupted by the loudspeaker announcement that participants should start making their way to the course.
Suddenly Garth is at my elbow. "I think we should watch a few people before we do it." I nod and allow him to lead me away. I'm completely floored, but I remember my manners and invite the group from 147 to follow us.
"Are you okay?" Garth asks as we head to the starting line.
"Yes, why?"
"You just have a really strange look on your face."
"Just nervous, I guess." I could tell him about the big mystery I just solved, but I don't think he'd get it.
We all gather in line to do the course. I turn and look at my group. Kim and Carol have joined us and are giving each other a pep talk. Dr. Evil has her game face on and Ali is biting her nails. She is the one I can relate most to. Jean-Luc Baton is doing some last-minute stretches, but is momentarily distracted when Miss Ward takes off her warm-up jacket. I'm with him. Her boobs truly defy gravity. I take a moment to admire them before I turn my attention to the course.
"This is going to be fun," Garth enthuses as we watch the first people take off. They are letting groups of two go every two minutes. The first up are two middle-aged women who I may or may not recognize from Curves. They can't get over the wall, so they do something that never would have occurred to me: they walk around it.
The course is pretty basic, but that doesn't mean it's easy. Just like last year, the six-foot wall is the first obstacle. Then you have to carry a tire about fifty yards, then run a hundred yards, and then crawl under a net through a long mud patch. After that comes a set of monkey bars, and a fake hill to climb and then slide down the other side. Then you run through some sprinklers to get wet, and jump over the line of fire and race to the finish line.
Adrenaline is pulsing through my veins as Garth and I get closer to the starting line. I want to scream, "I'm not ready yet!," but the truth is, I am. At this point I just want to get it over with. Especially that damn wall. I look around for Ron, hoping he realizes that I'm about to go, but I only see Chyna and Max waving at me from the sidelines.
I turn to Team 147. I see Kim Fancy and Miss Ward/Sasha Lewicki have paired up to do the course together. Huh. I thought they were mortal enemies, what with the slap and all. Competition makes for strange bedfellows. I shake my head to clear it. I can't lose focus.
"Good luck, you guys!" I say to everyone. They all smile and give a thumbs-up. As I'm taking a last look around for Ron, some guy at the starting line tells Garth and me that we're next. Garth takes my hand.
"You got this, Jen." He winks at me, and then we're off.
We run to the six-foot wall. Scenes from last year flash through my mind, and I immediately fall into my old, bad habit of trying to hoist myself over using my arms.
"Use your legs!" Garth yells to me from the top of the wall. Jesus, he's already up there?
I remember what he's taught me about saving my arm strength and using my legs. I grab the top of the wall and, keeping my arms straight, I frog-jump up the wall with my legs until I can hook my foot over the top and follow it with my body. I jump down the other side and wince with pain.
Garth jumps down, too.
"All good?" he asks. I nod.
We run toward the tires and each pick one up. It's heavy but not unmanageable—more awkward than anything. I can't run with it, but I walk as quickly as I can and am more than happy to see the place where you drop them. Garth, of course, is carrying two.
"Doing great," he puffs as he throws his tires on the pile and we take off on the hundred-yard run. "How's your groin?"
It's actually hurting, but there's no way I'm telling him that.
"Good. Fine," I pant.
At the end of our sprint is the mud patch. We get down on our bellies and crawl through what seems like five miles of muck but in reality is only about fifty yards. I'm exhilarated as I come to the end. I jump to my feet and high-five Garth, who naturally is right there waiting for me.
The monkey bars are next. Without thinking, I jump up and grab the first one; immediately, my hands slip off and I land on my butt.
"Wipe your hands on the grass or you'll never get a grip!" Garth yells while sailing across with ease. I rub my hands on the grass trying to get as much mud off as possible. When I grab the bars again, my hands hold tight and I scramble across.
Garth and I run up the manmade hill and then slide down the other side, which is all mud. As we do, I realize we are coming to the end. We run through the line of sprinklers to wet ourselves down and then hold hands as we jump over the fire line. I don't even feel the heat.
As we jog to the finish line, I see my whole heart waiting for me. Mom, Dad, Max, the girls, their boyfriends, Nina, and, right in front, Ron holding an ice pack.
I run straight into the group with my arms open wide in the hopes of embracing all of them at once. There is laughter and tears and congratulations and screams of "You're getting me dirty!" from my mother. I gratefully take the ice pack from Ron and hold it between my legs.
"I love you so much," I tell him with tears in my eyes.
"I know." He hugs me.
We have to make room for the other racers, so we herd ourselves over to the side and chat while we cheer on the rest of Team 147 as they cross the finish line.
Kim and Carol are first. They hold hands while they run, smiling the whole way. They seem to have a really nice marriage. As they complete the course, Carol lifts Kim into her arms and swings her around in a big hug. Well, now we know who's the bitch and who's the butch. Another mystery solved!
Next I see Jean-Luc running with Ali, both covered in mud but only one of them looking like he just finished a Men's Health photo shoot. Beside him, Ali is struggling to keep pace and looks happy to see the finish line. As they cross, Jean-Luc picks her up and swings her around. Is this some kind of ritual I wasn't made aware of?
Last but not least from our little team come Kim Fancy and Miss Ward—covered in mud and running like they are racing each other. I'm not kidding. They are neck and neck and running so close together that one of them could easily take the other out with a good shove. I'm thinking Kim Fancy for the win, but they actually cross the finish line together, clasp hands, and run toward the side of the building where the Porta Potties are located.
"When you gotta go, you gotta go," my mom says.
"I want to do it again," I say to Ron. "It went by so fast." I hug him and reach my arm out for Max. "You guys are the best cheering section in the world." I turn to my family and friends. "I can't believe you all came for this." They are all talking to each other and completely ignore me.
"I've got to get back to work." Ron extracts himself from Max's and my hug. "Are you going to hang out, or what?"
"The first thing I have to do is pee. I'll come and find you." I hand him my ice pack and take off toward the side of the building. I'm so happy, I'm practically flying. But when I turn the corner of the building, I stop short, because I'm treated to the sight of Miss Ward and Kim Fancy over by the side of the Porta Potties, making out! I mean, really going at it. Kim has our kindergarten teacher backed against the toilet, and Miss Ward has one leg wrapped around her like a rope. They don't see me, so I immediately backtrack around the corner and wonder if I have just imagined the whole thing. A quick second glance confirms that I have not. This is too much to take in. Miss Ward is Sasha Lewicki and she's Kim Fancy's secret lover? I feel like I'm in a movie. A really dirty one.
I still have to pee, but I decide to use the bathroom inside the store. I walk back through the parking lot, only semi-aware that people are still doing the obstacle course. My mind is trying to put together all I know about Miss Ward and Kim Fancy. How long has this been going on? Did Miss Ward not have an affair with the dashing David Fancy? I had been so sure about that, especially after the Christmas party–turned–jewelry show. I mean, Miss Ward had obviously taken her dress off at some point and then put it on backward. I always assumed she had hooked up with the dashing David in the bathroom for a quickie and that was why Kim slapped her, back in February. That slap! What was that all about? A lovers' quarrel? Foreplay?
I walk into the store, grateful for the relative quiet. This is a terrific event, but I'm not sure what it does for Ron's business. Everyone is outside having fun, not inside buying things.
I wave to Kendra, the salesgirl behind the counter, and head to the bathroom, which is on the left side of the store right near the jockstraps and sports bras. I lock myself in because I need to think, plus it's nice to sit down for a minute. I feel like I have been at Mach 10 with my hair on fire ever since I woke up for the second time this morning.
Usually, I do some of my best thinking on the toilet, but today all I come up with is more questions. So I flush and wash my hands. A glance in the mirror shows I have the battle scars of a mud warrior all over me and I feel all kinds of cool as I head back to the parking lot. I look at my watch and can't believe it is only 10:15. All this has happened in an hour? It makes me think of the army slogan, "We do more before nine a.m. than most people do all day."
I find Max and Chyna at the Gatorade table, passing out cups to people.
"Mommy, Garth is looking for you," Max informs me, handing me a cup of orange Gatorade.
"Okay, thanks. Are you guys having fun?"
"Dad says I can play in the mud when everyone is done and swing on the monkey bars."
"He did, huh? You are so lucky."
"He said I could try to do the course, too," Chyna tells me. "If it's okay with you."
"Of course. But maybe get Garth to help you. Any idea where he and your mom are?"
"I think they went to talk to the guy from the governor's office."
"Where is he?"
"Sitting with the guy making all the announcements." She points to a table set up near the starting line.
"Max, want to come with me or stay with Chyna?"
"Mommy, I'm working."
"Righty-o! I'll be back in a little while."
On my walk over to the table, I am stopped at least a dozen times by friends and customers all congratulating me on the success of the day, as though I had anything to do with it. I promise to pass along their compliments to Ron, if I ever see him again. Finally, I spy the announcer's table and Garth chatting up some guy in a suit. I wait until he sees me, then I wave him over. I'm not in the mood to press any more flesh.
"There you are! I was looking for you. You must have really had to pee."
"You have no idea," I respond. "Where's Nina?"
"She's getting something from the car. How are you feeling?" He puts his arm around me.
"I feel great! That was a breeze. I wanted to go right back and do it again. How did I look?"
"Like a girl who's ready for a bigger challenge, that's for sure. I'm so proud of you. You must be on such a high."
Well, I was, I think but do not say. The whole Miss Ward/Kim Fancy drama has really pulled me off track. Garth is right—I should be walking on air. Instead, all I want is to find Nina and tell her everything. But Garth doesn't need to know any of this.
"I am. It was amazing and I owe it all to you." I give him a hug.
I can tell Garth is happy. "And we aren't finished yet. August, baby. That's when the real deal will happen. We need to keep training all summer."
"I'm ready," I tell him. I spot Nina walking toward us. "I'm going to steal your girl for a few minutes." He smiles and waves me on.
I grab Nina's arm and redirect her to a bench near the street.
"Looks like you and Ron—"
I cut her off immediately.
"I have to tell you something, but you can't tell anyone else."
"Okay." Nina seems more dubious than curious.
"Swear on Chyna's life that you won't say anything."
"Uh, no. But I'll swear on our friendship, if that makes you feel any better."
I regard her for a moment, then nod.
"Good enough." I take a deep breath. "I just saw Miss Ward and Kim Fancy making out by the Porta Potties." I wait for a reaction, and Nina doesn't disappoint. Her eyes nearly pop out of her head.
"Are you shitting me?"
"Nope."
"Didn't you tell me one of them slapped the other?"
"Yup."
"Are you sure they were making out? Maybe they were just hugging."
"With their tongues?"
"Oh, my God, seriously?"
"Yup."
"Was it hot?"
"No!" I laugh. Only Nina.
"I'm just sayin'—they're both good-looking women. Would a guy think it was hot?"
"Probably. Not really the point I was getting at, though."
"This is seriously good gossip."
"You promised you wouldn't say anything," I remind her.
"I won't. Wait, can I tell Garth?"
"Of course." I know I'll be giving Ron a play-by-play later.
Nina looks past me toward the parking lot.
"We should get back. Garth is giving me the 'Mayday' sign."
While Nina goes to rescue Garth, I check in with Max, who is still manning the Gatorade table with Chyna.
"Hey, guys. Having fun?"
"Mom, I'm hungry. Can Chyna and I go to McDonald's?" I can see Max is losing interest in his job, and the golden arches next door are acting like a bat signal on him. Let's see, Pop Tarts for breakfast and now McDonald's for lunch. I hope Child Protective Services isn't watching.
"I guess so," I say. "But I don't have my purse with me."
"I have money from my mom," Chyna assures me.
"Okay. Well, don't eat too much if you're going to do the course later."
"We won't," they say together, and head off.
The event is going strong. People are still lined up to do the course, and the announcer has started to play great motivational tunes like "We Will Rock You" by Queen. I smile to myself, because I'm happy for Ron. This is a real win for him.
I pass Ali Gordon limping to her car and realize we'd never finished our conversation.
"Are you okay?" I ask her.
"Yeah, I'm fine, it's just my calves are starting to seize up. I think I should have trained a bit for this."
"As someone who did nothing but train, I'm going to agree with you. Need some help?" I lean toward her.
She gives a resigned laugh and puts her arm over my shoulder. "I guess I do, thanks."
I support her around the waist and we start walking. I'm generally a nice and helpful person, but I'm not going to pretend I don't have an ulterior motive. I want to see if I can get Ali to spill the beans about her and Don. I'm in no shape to be clever or crafty, so I just come straight out with it.
"So, what's your deal with Don?"
She sags a bit more into me.
"I'm sorry I said that earlier. I just get frustrated sometimes."
"Why?"
She sighs. "Look, he's a good guy. He's just really..." She's searching for a word, but I don't have any suggestions.
"He's Peter Pan," she finishes.
"He dresses in green and flies around the city?" It's a joke, but apparently not to Ali.
"I mean, he has Peter Pan syndrome. He doesn't want to grow up."
"Really? I don't get that from him at all."
"Why would you? You haven't had to raise a kid with him."
I think about that for a second.
"Is he not involved with Lulu's life? I mean, he shows up for all the school stuff."
"Yeah, that's a recent development. Now that she walks and talks and communicates, all of a sudden he wants to be Dad. When she was a baby, forget it."
"That's rough. Why did you guys have a kid to begin with?"
She gives a bitter guffaw.
"The condom broke."
"No!" I gasp. "Oh, my God, that actually happens? I always think of it as an urban myth like the alligators that live in the New York sewers."
This she does laugh at. "Nope. Not a myth. We had only been dating a few months when it happened. That's my car." She points to a beige Hyundai parked on the corner. She takes her arm from around my neck and fishes her key fob out of her fanny pack. But I'm not letting her leave just yet.
"And you wanted to keep the baby, of course."
"We both did. I mean, Don was forty-two at the time and I was in my late thirties, so we thought why not, you know? He wanted to get married, but I held off on that, thank God. I was like, 'What's the rush?' Don was really into it until Lulu was born, and then reality set in."
"Oh, God," is all I can think of to say.
"He freaked out." She sighs and leans against her car. "He was actually jealous of all the attention Lulu got. He really didn't get it. We started fighting all the time, so I told him if he couldn't accept that he wasn't the child anymore then he should just stay away."
"And did he?"
"Yes!" she yells. "Can you believe it? I didn't see him for, like, three years."
"What an asshat." I can't help but think how lucky I am to have Ron "I'll change the diaper" Dixon.
"My parents live in Des Moines, so they weren't much help. I was on welfare for a while. I mean, it really sucked."
"When did he come back?" This conversation is making me remember how tough it was carting Vivs around Europe while I was pregnant with Laura. Now there's a book I should write.
"You won't believe me."
"Try me."
"We met speed dating."
"What?"
She nods her head with wide eyes.
"Of all the gin joints, right? When he sat down in front of me I nearly spat my drink out."
"What did you say to him?" I'm completely spellbound.
"I actually said, 'Of all the gin joints.'" She shrugs. "We agreed to meet afterward, and he claimed he wanted back in Lulu's life."
"So you let him?"
"Not right away. I mean, the guy was AWOL for three years. If he'd wanted back in so badly, he could have called or emailed anytime. I always wonder what would have happened if I hadn't gone to that stupid dating thing."
I give her another way to look at it.
"But it's kind of romantic. Like it was meant to be."
"I guess. He's been around a lot. It's been great having help with Lulu, especially money help. And, I don't know if you know this, but he's really good with kids."
"I actually witnessed that firsthand on a field trip."
"He's a natural. It makes me sad for him that he missed all those years with Lulu."
"Do you guys date other people?"
"I'm not sure what he does, but I haven't had a date in two years. That's why I want to get in shape, you know? Stop looking like something the cat dragged in."
I definitely knew where she was coming from. After I had Max, "dumpy" was the only real way to describe me. That's why I joined Curves. I consider suggesting it to Ali.
"You're hardly something the cat dragged in," I assure her. "Have you guys ever considered getting back together?"
Ali opens her car door and groans loudly as she eases herself into the driver's seat. "Don talks about it, but I don't know. I'm still getting over being mad at him."
I nod.
She starts her car. "Thanks for the help."
"Epsom salt bath and Advil," I advise. "For the pain."
You'd think with all my newfound knowledge I would have been out there like a bumblebee pollinating all of William Taft Elementary with news of Miss Ward, Kim Fancy, and the phantom Sasha Lewicki. But I didn't. I sat on all of it, which if you ask me shows Herculean restraint. Ron thinks I'm just scared of the fallout, and maybe I am. This isn't just "I saw her bingeing on ice cream at Ben and Jerry's" gossip. This is information that has the potential to hurt people and change lives. I keep thinking about Nancy Fancy and how she will have to deal with everything if this comes out. And poor little Nadine Lewicki! Oh, wait, she's not real.
I know I should at least let Asami know she was right all along and put her out of her misery, but I don't know what she'll do with the information, so I simply go on with my life.
The mud run at Ron's store was a huge success. It was even on the front page of the Kansas City Star, albeit below the fold. The picture they used was of Garth just getting out of the mud. You can actually see my elbow in the shot! The lieutenant governor called Ron personally to thank him. They want him to start thinking about next year and maybe doing a cross-promotion with one of the TV stations.
It's mid-May, so Vivs and Laura are done with school for the year, but both have opted to stay on campus and work rather than come home. I can't say I'm surprised. Love is definitely in the air. They are both coming home for Max's last day of school, which to my shock and awe is only two weeks away. Normally, the kids would be in classes until the third week of June, but apparently the school board has scheduled good old William Taft for a facelift, so they're shelving those oh-so-critical last two weeks when the kids do nothing but play games and go on field trips.
I realize I should email Miss Ward to find out if she wants to have a year-end party. The last day of school isn't a Hallmark holiday, is it?
I get up from folding laundry and head to the kitchen-counter office to send her an email. When I log on I notice that Shirleen Cobb, Nina, and Miss Ward have all emailed me. Aren't I the popular one today! Just for shits and giggles, I read Shirleen's first.
* * *
To: JDixon
From: SCobb
Date: 5/23
Subject: Play date
Jen,
Apparently Graydon would like to have a play date with Max. I'd like to have it at my house so I can monitor what Graydon eats. How about this Saturday?
Shirleen
* * *
Yes, Graydon and Max are friends again. The whole "You're a liar" incident is long forgotten... by some. I'm just glad Shirleen wants to host. I already have more gum than I can chew. Worrying about Graydon's list of dos and don'ts just might push me over the edge. I email her back telling her it's fine and move on.
* * *
To: JDixon
From: NGrandish
Date: 5/23
Subject: Hey
Breaking news from the principal's office! Call me.
Xo
* * *
Just then my phone buzzes and I see that Nina has texted me the same thing. I think I know what it is. She has been making noise about stepping down as PTA president, but I always assumed she was all hat and no horse. Wow, I guess she finally did it. I'll call her later.
* * *
To: JDixon
From: PWard
Date: 5/24
Subject: Today
Hi, Jenny,
I know we're coming up to the long weekend, but could you meet me in the classroom after school today at 3?
Thanks,
Peggy
* * *
Finally! The universe has rewarded my patience and cowardliness. Miss Ward wants a meeting. I'll seriously die if Kim Fancy is there, too.
I type back a quick affirmative reply to Miss Ward, then text Peetsa and ask if she can take Max home with her after school.
I just have time to run up and take a mini shower (no hair washing), put on a fresh mom uniform, and head to Starbucks before my meeting. Peetsa has responded that she will take Max with her and Zach while they go grocery shopping for their Memorial Day barbecue, which we're invited to. I really owe her one. Taking two six-year-olds grocery shopping is only slightly easier than herding cats.
"You look good, Mama," I say to her at pickup. It's a sunny spring day and Peetsa has busted out a short pleated skirt and a light blouse for the occasion. She looks so pretty.
"I thought you weren't going to be here." She squints at me.
"Miss Ward asked to meet with me."
"About what?"
I shrug. "No idea." I hate lying to her. Fortunately, Ravi comes up to us, and the subject turns to summer and what we're going to do with the boys. We're trying to formulate some kind of shared schedule when the school bell rings. The good weather has infected the kids and there is extra noise and activity as they pour out of the building.
I wave Max over and give him a hug. He smells like dirt, so I can tell they were outside a lot today.
"Hey, buddy, do you mind going with Zach T. for a while? I need to meet with Miss Ward."
"Are you going to talk about me?" He looks worried.
"Nope. We're going to talk about the super big end-of-the-year party we're going to have for you guys." Peetsa raises her eyebrows at me, and I shrug.
"Can we get a bouncy castle?" Max asks.
"And cotton candy," Zach T. adds.
"It's not a carnival, guys, it's just a party. I'll see you later." I wave to Ali and Lulu as I walk toward the school. My phone rings just as I am walking in. It's Nina.
"I'm sorry, I meant to call you earlier. Did you do it? Did you quit?"
"I was going to, but I'm having second thoughts. But I did overhear something that you'll be interested in."
"What?" I pause in the lobby because I don't want my voice echoing down the empty hall on my way to room 147.
"Miss Ward resigned."
"What?"
"Yup. Apparently she told Jakowski she's leaving. I don't know anything else."
"I'm actually on my way to a meeting with her right now." I whisper even though there is no one around.
"Oh, my God. Go find out what's going on and call me right back." Nina hangs up before I can even say good-bye.
I put my phone in my purse and hurry to the classroom, where I find Miss Ward sitting at her desk, humming a kids' song and organizing some papers. When she sees me, she jumps up much the way she did on curriculum night, oh so many months ago.
"Jenny!" She hugs me. "Thanks for coming. Sit down. I have some news for you." She hops up to sit on her desk, leaving me with either one of the kids' tables or one of their chairs. I choose a table and just hope it holds me.
"What's up?" I ask as casually as I can.
"I wanted to let you know that I have resigned."
"Really?" I act genuinely surprised, or at least I hope I do. "Why?"
I think she's going to confess that she got caught inventing a fake student and parent and they asked her to leave. But once again, she surprises me.
"The private school I used to work at in New Jersey really wants me to come back, and they've made me a great offer."
"Really? Wow. Good for you," is all I can think of to say. And then something occurs to me. "You could have just emailed me the news. Why did you want to meet?" Bold, I know. But I'm not walking away without answers.
She looks at me slyly. "I thought you might have some questions for me."
"I do, actually. Did you want to have an end-of-year party? I meant to email you about it."
"Sure." She waves her hand at me. "Anything you want. I'll leave it to your imagination. Anything else?"
I sigh. She clearly wants me to ask.
"Why did you make up Sasha and Nadine Lewicki?" There, it's out.
She gives me a smile that lets me know she thinks she has won some kind of standoff.
"Well, I was new here, and I wanted to keep tabs on the class without anyone knowing it. I loved your emails, by the way. You're very funny."
"Thanks. Is that the only reason?" I ask.
"I've had some bad experiences with class parents in the past. Mothers can be so catty sometimes."
I look at her, but don't say anything. She shrugs.
"What other reason would there be?"
I'm feeling very unsatisfied. It's like having an itch in the middle of your back that you can't quite reach.
"I don't know. You went to a lot of trouble just to spy on your own class. I would have cc'd you on the emails if you wanted."
"Yes, but they wouldn't have been the same, now would they? I wanted the real flavor of the class."
"Oh, well, okay. Good. I'm glad it worked for you. I'm just happy there isn't a terribly neglected sick little girl with a workaholic mother out there."
She laughs. "I know, right? Hello, Child Services?"
I'm positive there's more to the story, but I've heard all I want to.
"Okay, so I'll arrange for the party on the last day of school. Maybe we can play freeze tag on the grass behind the school."
"I don't need to know the details, Jenny. I'm sure it will be great."
This is my cue to leave. As I open the door, Miss Ward is right behind me, pulling me into an awkward hug.
"Thanks, Jenny. You were a good friend this year."
I really don't know what to say to that, so I just hug her back and walk away.
On my way back to the car, I dial Nina. She answers on the first ring.
"What happened?"
"Well, you were right—she's leaving."
"Did she tell you why?"
"She said her old school in New Jersey wants her back and made her a great offer."
"Do you believe her?"
"I do. Why would she make that up? The bigger question is, does Kim Fancy know, and how is she taking it?"
"Are you friendly with her at all?"
"Not even a bit," I admit.
I unlock the minivan and slide into the driver's seat.
"So is this public knowledge?" Nina asks.
"Well, she didn't say to keep it quiet, so I guess it is. I'm feeling a little bad. She might have been crazy, but Miss Ward was a good teacher. Maxi is going to miss her."
"Oh, he'll fall in love with his next teacher. They always do."
We make plans to see each other over the long weekend. As I pull out of the parking lot, I see a red Grand Cherokee with the hood up on the side of the road. I slow down to see if they need help and regret the impulse immediately when I see JJ Aikens. I roll down my window.
"Hey, JJ, need any help?"
She looks up, confused, then walks from her car over to mine.
"No, thanks." She sighs. "I'm just waiting for Triple A. I told my husband the transmission was going on this thing."
"Do you have Kit with you? I can drop her somewhere if you like."
"It's okay. She's playing on my phone and it shouldn't be too much longer." She looks at her watch.
She seems disproportionately sad, even for someone with a broken-down car. I decide to tell her the big news about our teacher to take her mind off her troubles.
"Miss Ward just told me she's leaving the school and going back to New Jersey."
JJ doesn't seem a bit surprised. But what happens next really startles me.
"So is Kim," she says in a weird, squeaky voice. She then screws up her face and bursts into tears.
"She is?" I say a bit too loudly. This makes JJ cry harder.
I throw the minivan into Park and get out. I walk over to her and put my arm around her shoulder.
"I'm so sorry. I know you guys are good friends."
She shrugs off my semi-hug and looks at me like I'm the crazy one.
"Do you think I'm sad? I'm not sad, I'm mad!" She wipes her apparent tears of anger away and looks me straight in the eye. "You have no idea what I have put up with from her."
I'm so confused. "What are you talking about?"
"I can't tell you. I'm not supposed to tell anyone."
"Tell anyone what?" I lob out there to see if she'll bite.
She shakes her head. "For almost three years I've kept her secrets and supported her stupid ideas." She is speaking more to herself than me. "Did you know she's always trying to have stuff delivered, because I guess people in New York don't go out and do their own shopping?"
"You mean Manhattan," I say, trying to lighten her up.
"Manhattan," JJ sneers. "New Jersey is more like it."
"She's from Jersey?" I ask, genuinely surprised.
JJ looks panic-stricken. "Oh, my God, don't tell her I told you."
"Told me what?" I wink at her. "But is she really?"
JJ wipes her nose with her hand and nods.
"A place called Edgewater." She shrugs. "It's supposed to be nice."
"So why all the talk about Manhattan?"
"She wanted to make herself sound more important. I didn't even know the truth until this year when that woman showed up."
I frown. "You mean the jewelry designer?"
"No, Miss Ward!"
"She knew Miss Ward from New Jersey?"
There's that panic-stricken look again.
"Oh, my God. Don't say anything. No one's supposed to know."
I'm trying to remain cool and calm, but my heart is racing.
"Why would it be a secret that they knew each other in New Jersey?"
"I'm not supposed to tell anyone," JJ whispers.
Yeah, you're a real gatekeeper,
As bad timing goes, Triple A showing up right then counts as a winner. JJ is immediately pulled out of our conversation cocoon when Dusty (according to his name tag) walks up and asks what the problem is.
As JJ takes him to the front of the car, I debate whether I should stay and push my luck, or cut and run with what I have.
The decision is made when JJ calls out to me asking if I can drive her and Kit home.
"I'm happy to," I answer with a smile. I jump in my driver's seat and pray for lots of traffic.
* * *
"I don't think I've ever been to your house," I tell JJ. I know damn well I haven't.
"I'm off Trail's End, right by the mall."
"Gotcha." I pull into the street and calculate that with traffic I will have about fifteen minutes to glean as much information from her as possible, unless we get lucky and there's a three-car pileup. Kit is sitting in Max's car seat happily watching one of his movies with the headphones on.
"So..." I lead off, hoping JJ will pick up the ball.
She turns her whole body to me.
"You have to promise you'll never tell anyone what I told you," she begs.
"You've only told me that Miss Ward and Kim knew each other from New Jersey. I won't say anything, but I think I'd have a hard time finding someone who cares."
"It's not that they know each other, it's how they know each other."
We're at a stoplight, so I turn and give her my best "confused" look, which isn't hard because I am still pretty confused.
JJ gives an exasperated sigh.
"Look, if I tell you, you can't tell anyone."
"I won't."
"Swear on Max's life."
Funny, I had just asked Nina to swear on her kid's life, and she had refused. I'm not sure it's wise to bet Max's life on my ability to keep my yap shut, but I'm too damn curious not to.
"Okay. I swear."
"On Max's life."
I grimace.
"On Max's life." Now I know I won't tell anyone. I look at JJ expectantly.
"Well, from what I know, Kim moved here to get away from Peggy."
I try to keep my eyes on the road but I have to look over to see if she's serious.
"When I first met Kim at preschool, we hung out all the time because we had to be there to help the kids with separation. You weren't there, were you?"
Am I that forgettable?
"I didn't do preschool for Max."
I want to slap the judgmental look she gives me right off her face.
"Kim told me back then that they'd moved for David's job, but also because he'd had an affair and they needed a fresh start. They seemed to really be trying. They used to take these romantic weekends. We watched Nancy a few times."
"How does Miss Ward fit in to all this?"
"When we first got the letter from the school telling us who the kindergarten teacher was, Kim totally wigged out." JJ laughs humorlessly at the memory. "She told me this was the woman David had the affair with and she had obviously followed them here. She was really unhinged and I felt bad for her. She even went to Principal Jakowski and tried to have her fired before school started."
"I'm surprised she didn't manage to," I say, thinking about how relentless Kim Fancy can be when she wants something.
"Turn left here," JJ directs. We are now on Trail's End, fairly close to our destination.
"She couldn't, though. Have you ever seen Miss Ward's résumé?"
I shake my head no.
"She has a doctorate in early childhood education from Columbia and she started her career pioneering a 'Mandarin for toddlers' program for the state of New Jersey. There was no way Jakowski was going to turn her away."
"That's crazy," is all I can think of to say.
"That's my house at the end of the block, with the blue mailbox," JJ tells me. Shit! I'm not ready to let her go yet. I pull into her driveway and put the minivan in Park.
"So, is this like a stalking situation? She followed them here to boil a bunny on their stove?"
"What?" JJ clearly doesn't get the reference to my favorite movie of all time, Fatal Attraction.
"Nothing. So she moves here supposedly to win David back, and what? Try to get him to leave Kim?"
"That's what I thought, but it turns out it was Kim who had the affair, and Kim who she came here to win back." She shakes her head in disbelief. "And now she has."
"When did you find out all this?"
"Two weeks ago. Until then I thought David had had the affair with her, and I felt so bad for Kim to have to see this woman. She bad-mouthed her constantly, but then would go and have"—she makes air quotes—"'meetings' with her. I thought she was trying to save her marriage. But then I walked in on them making out in Kim's backyard. I think it was just after that event at your husband's store."
We both sit in silence for a moment. Miss Ward moves to Kansas to follow her heart and ultimately wins the girl. Huh. It's kind of romantic, in a crazy-bitch sort of way. Then I frown.
"So what's going to happen to Nancy and David?"
JJ shrugs. "No idea. Do you think he would go back to New Jersey with them?"
"I can't even pretend to know what these people would do." I laugh. Boy, nothing like this ever happened when I was class mom for Vivs and Laura.
"God, it feels good to talk about this," JJ admits. "I've been keeping a lot of secrets this year." She starts gathering up her bags and tells Kit to get ready. "Thanks so much for the ride."
"Any time."
"It was nice to talk to you. I used to have a lot of friends, but after Kim moved here I became kind of obsessed with her and I lost most of them." She opens the minivan door and gets out, but pauses before she closes it.
"Maybe I was a little in love with her, too."
And with that slight overshare, she closes the door and heads up her front walkway with Kit.
"Remember," she turns and yells, although I can barely hear her through my closed car windows. "You can't tell anyone."
* * *
To: The Soon-to-Be 1st Grade Parents
From: JDixon
Date: 6/8
Subject: HAGS! As the kids would say
Dear Formerly Miss Ward's Class,
I just want to give a third and final reminder to everyone that tomorrow is the last day of school for our kindergarteners and everyone is invited to our field day/picnic, which takes place from 10 a.m. to 2 p.m. on the west side of the field behind the school.
Thanks to everyone who volunteered to bring food and drinks. I'm happy to report that we have a new winner in our speedy response category. Since Sasha Lewicki's automated reply has left the class, Ravi Brown leads the pack with an impressive 58 seconds, but you should know that all responses came in at under ten minutes. I couldn't be prouder.
I also want to thank everyone for pitching in these last two weeks and helping Principal Jakowski manage our class. Miss Ward's untimely departure could have been a disaster, but we all pulled together and made it work. Special shout-out to Ali Gordon and Don Burgess. You guys are amazing with kids! You should be teachers.
Hope to see you tomorrow, but if I don't, then HAGS (which I think is a truly horrible acronym for Have a Great Summer, but the kids seem to love it).
Over and out for good!
Jen (and Asami in spirit)
* * *
* * *
And there it is, my last official email as class mom. After tomorrow, I'm a free woman. Well, "free" is a relative term, I guess, since I'm still a mom and I now have an active little boy home with me all day, every day. Thank goodness for play dates. I already have Max scheduled through July Fourth weekend.
These last two weeks have been what my mother would call a mare's nest. Miss Ward never came back after Memorial Day weekend, if you can believe it. She must have been planning all along to leave then. I should have known from that hug.
The children were very confused when they wandered into room 147 on the Tuesday after the long weekend and found a note from Miss Ward written on the Smart Board. Only about a third of them could actually understand it. According to Max, Suni Chang saved the day by reading it aloud to the class. The note explained that her work here was done, that they were all going to be wonderful first-graders and she had to move on to a new group of kindergarteners who needed her... just like Mary Poppins. She wrote that! Just like Mary Poppins. Go fly a kite, Miss Ward.
The school had to scramble to make sure her class was covered for the final two weeks. Most of the moms took a turn co-teaching with Principal Jakowski, who showed no indication that he had ever spent any real time in a classroom. My shout-out to Ali and Don was from the heart. They came in together, which I was surprised and happy to see, and the kids loved them. They created all these amazing learning games that were really fun (all this according to Max, whose favorite was something called What's in the Bag?). I can't help but wonder if my chat with Ali kick-started a little something between them. Probably not, but it's more fun for me to think that it did.
I found it a bit out of character for Miss Ward to leave without giving the kids any kind of closure. I've said it before, she was a bit of a wack-job but a great teacher. The kids were sad that they didn't get a chance to say good-bye to her or their beloved classmate Nancy Fancy, who (big shocker!) didn't come back, either. I can only assume they're all happily ensconced somewhere in Manhattan or New Jersey.
I kept the promise I made on Max's life, but about a week after JJ Aikens swore me to secrecy, she started telling anyone who would listen. It was quite the topic at the klatch, let me tell you. And, like any good story, it got better with each telling. My personal favorite was when Shirleen Cobb said she heard that Miss Ward and the Fancys were making porn together. When I asked her what kind of porn, she said, "The un-American kind."
The only secret that actually stayed a secret was the real identity of Sasha Lewicki. Of course, I did tell Asami (I mean, who else would really care?), and she could not have been more magnanimous when she said, "I told you so." I have to give her full props. She totally called it.
Right now I have to dash to Party City to get water balloons for one of the games we'll be playing at the picnic tomorrow. That party will be my final duty as class mom and, I have to admit, the thought chokes me up a little bit.
I debate sharing this with Nina when I see her at the picnic tomorrow. Ever since she re-upped as PTA president, she has been dropping little hints about me being a class mom again. She hasn't come right out and asked me yet, but I know she will. And when she does, I'm pretty sure I know what I'll tell her.
"Absofuckinglutely not!"
* * *
To: The parents of Mrs. Peele's 1st grade class
From: JDixon
Date: 8/30
Subject: I'm your class mom!
Dear Parents,
For those of you who don't know me, my name is Jennifer Dixon, and it is my pleasure (wink) to be your class mom for this coming year.
To the parents who were with me last year in kindergarten, all I can say is you've made it through boot camp already. You know the rules and you can stop reading this now. See you on curriculum night, which is (see below).
To the rest of you, make sure to read every damn word of this email...
* * *
Acknowledgments
First and foremost, I'd like to acknowledge Beinstock/UTA super-agent Paul Fedorko, who convinced me that all my whining about being a class parent might actually make a good book. Thank you for not only encouraging me but also shaming me by saying "Danielle Steel isn't too busy to write and she has more kids than you."
While writing Class Mom, I had no idea that the five most dreaded words a friend of the author can hear are "Will you read my book?" Thankfully, I had a bevy of generous souls who were more than willing to read various drafts and give me feedback, and for that I am grateful. The most loyal of these was my paid professional friend Gabrielle Maertz. If Gabby hadn't laughed in all the right places, I never would have kept writing beyond the first forty pages. Others who gave me their time and wisdom: My forever sister/friends Maria Crocitto, Nancy Bennet and Cindy Vervaeke. Alison Cody who lived the whole nightmare with me. Jessica Aguirre who printed out all 360 pages and lugged it around until she finished it, God bless her. Paige Baldwin, who told me that Jen needed a goal. Sheri Impemba who was the first to read it and ask me if she could let her friends read it too. Jan Weiner who let me use a very sweet story about her son Caden, and Caroline Rhea, who gave me the funniest line in the book ("I'll let you decide which one it is).
Thank you to Serena Jones, my editor at Holt, who had the scary task of telling me that the fruits of my two-year labor of love needed rewrites. There is an old joke that goes, "How many writers does it take to change a lightbulb?" Answer: "It doesn't need changing!" But indeed it did, and Serena's razor-sharp insights and gentle nudging made Class Mom a much better story.
And finally, thank you to the Starbucks at Eighty-Eighth and Broadway for allowing me to use your café as my writing room and for always having the PB-and-J lunchbox and iced green tea that fueled my creativity.
About the Author
Laurie Gelman was born and raised in the Great White North. She spent twenty-five years as a broadcaster in both Canada and the United States before trying her hand at writing novels. Laurie lives in New York City with her husband, Michael Gelman, and two teenage daughters. Class Mom is her first book. You can sign up for email updates here.
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Contents
Title Page
Copyright Notice
Dedication
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
Acknowledgments
About the Author
Copyright
CLASS MOM. Copyright © 2017 by Laurie Gelman. All rights reserved. For information, address Henry Holt and Co., 175 Fifth Avenue, New York, N.Y. 10010.
www.henryholt.com
Cover design by Nicolette Seeback
The Library of Congress has cataloged the print edition as follows:
Names: Gelman, Laurie author.
Title: Class mom: a novel / Laurie Gelman.
Description: First edition. | New York: Henry Holt and Company, 2017.
Identifiers: LCCN 2016052495 | ISBN 9781250124692 (hardcover) | ISBN 9781250124708 (electronic book)
Subjects: LCSH: Middle-aged women—Fiction | Middle-aged mothers—Fiction. | Domestic fiction. | CYAC: Schools—Fiction.
Classification: LCC PS3607.E465 C57 2017 | DDC 813/.6—dc23
LC record available at <https://lccn.loc.gov/2016052495>
First Edition: August 2017
Our e-books may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at (800) 221-7945, extension 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.
This is a work of fiction. All of the characters, organizations, and events portrayed in this novel either are products of the author's imagination or are used fictitiously.
## Contents
1. Title Page
2. Copyright Notice
3. Dedication
4. Chapter 1
5. Chapter 2
6. Chapter 3
7. Chapter 4
8. Chapter 5
9. Chapter 6
10. Chapter 7
11. Chapter 8
12. Chapter 9
13. Chapter 10
14. Chapter 11
15. Chapter 12
16. Chapter 13
17. Chapter 14
18. Chapter 15
19. Chapter 16
20. Chapter 17
21. Chapter 18
22. Chapter 19
23. Chapter 20
24. Chapter 21
25. Chapter 22
26. Chapter 23
27. Chapter 24
28. Acknowledgments
29. About the Author
30. Newsletter Sign-up
31. Copyright
## Guide
1. Cover
2. Table of Contents
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The Vice President of the Federal Republic of Nigeria, Yemi Osinbajo says he prefers listening to Davido instead of Wizkid.
During a radio show on MAX FM, the VP was asked to pick who he prefers between Wizkid or Davido and he answered saying Davido.
It would be recalled that some days ago, PDP's presidential candidate, Atiku Abubakar called Davido "the reigning king of music", when he spoke about the type of music he liked.
#VIDEO--- Vice President #YemiOsinbajo chooses #Davido ..
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\section{Introduction}
\subsection{Background}
The goal
of {\em position-based cryptography} is to use the geographical position
of a party as its only ``credential".
For example, one would like to send a
message to a party at a geographical position~$\text{\sl pos}$ with the
guarantee that the party can decrypt the message only if he or she
is physically present at~$\text{\sl pos}$.
The general concept of position-based cryptography was introduced by Chandran, Goyal, Moriarty and Ostrovsky~\cite{CGMO09}; certain specific related tasks have been considered before under different names (see below and Section~\ref{sec:RelatedWork}).
A central task in position-based cryptography is the problem of {\em
position-verification}. We have a {\em prover}~$P$ at position~$\text{\sl pos}$,
wishing to convince a set of {\em verifiers} $V_0,\ldots,V_k$
(at different points in geographical space) that~$P$ is indeed at that
position~$\text{\sl pos}$. The prover can run an interactive protocol with the
verifiers in order to convince them. The main technique for such a
protocol is known as distance bounding~\cite{BC93}. In this technique,
a verifier sends a random nonce to~$P$ and measures the time taken for
$P$ to reply back with this value. Assuming that the speed of
communication is bounded by the speed of light, this technique gives
an upper bound on the distance of~$P$ from the verifier.
The problem of secure positioning has been studied before in the
field of wireless security, and there have been several proposals
for this task (\cite{BC93,SSW,VN04,B04,CH05,SP05,ZLFW06,CCS06}).
However, \cite{CGMO09} shows that there exists no
protocol for secure positioning that offers security in the presence
of {\em multiple colluding} adversaries. In other words, the set of
verifiers cannot distinguish between the case when they are
interacting with an honest prover at~$\text{\sl pos}$ and the case when they
are interacting with multiple colluding dishonest provers, none of
which is at position~$\text{\sl pos}$. Their impossibility result holds even
if one makes computational hardness assumptions, and it also rules out
most other interesting position-based cryptographic tasks.
In light of the strong impossibility result, \cite{CGMO09} considers a setting that assumes restrictions on the parties' storage capabilities, called the Bounded-Retrieval Model (BRM) in the full version of \cite{CGMO09}, and constructs
secure protocols for
position-verification and for position-based key exchange (wherein the
verifiers, in addition to verifying the position claim of a prover,
also exchange a secret key with the prover). While these protocols
give us a way to realize position-based cryptography, the underlying setting is relatively hard to justify in practice.
This leaves us with the question: are there any other
assumptions or settings in which position-based cryptography is
realizable?
\subsection{Our Approach and Our Results}
In this work, we study position-based cryptography in the {\em
quantum} setting. To start with, let us briefly explain why moving
to the quantum setting might be useful. The impossibility result
of~\cite{CGMO09} relies heavily on the fact that an adversary can
locally store all information she receives {\em and} at the same time
share this information with other colluding adversaries, located
elsewhere. Recall that the positive result of~\cite{CGMO09} in the
BRM circumvents the impossibility result by assuming that an adversary
{\em cannot} store all information he receives. By considering the
quantum setting, one may be able to circumvent the impossibility
result thanks to the following observation. If some information is
encoded into a quantum state, then the above attack fails due to the
no-cloning principle: the adversary can either store the quantum state
or send it to a colluding adversary (or do something in-between, like
store part of it), but \emph{not both}.
However, this intuition turns out to be not completely accurate. Once
the adversaries pre-share entangled states, they can make use of
quantum teleportation~\cite{BBCJPW93}. Although teleportation on its
own does not appear to immediately conflict with the above intuition, we
show that, based on techniques by Vaidman~\cite{Vaidman03},
adversaries holding a large amount of entangled quantum states can
perform \emph{instantaneous nonlocal quantum computation}, which in particular implies that they
can compute any unitary operation on a state shared between them,
using only local operations and {\em one} round of classical mutual
communication. Based on this technique, we show how a coalition of
adversaries can attack and break any position-verification scheme.
Interestingly, sharing entangled quantum systems is vital for
attacking the position-verification scheme.
We show that there exist schemes that are secure in the
information-theoretic sense, if the adversary is not allowed to
pre-share or maintain entanglement.
Furthermore, we show how to construct secure protocols for
several position-based cryptographic tasks: position-verification,
authentication, and key exchange.
This leads to an interesting open question regarding the amount of
pre-shared entanglement required to break the positioning scheme: the
case of a large amount of pre-shared states yields a complete break of
any scheme while having no pre-shared states leads to
information-theoretically secure schemes. The threshold of pre-shared
quantum systems that keeps the system secure is yet unknown.
\subsection{Related Work}\label{sec:RelatedWork}
To the best of our knowledge, quantum schemes for position-verification have first been considered by Kent in 2002 under the name
of ``quantum tagging''. Together with Munro, Spiller and Beausoleil, a
patent for an (insecure) scheme was filed for HP Labs in 2004 and
granted in 2006~\cite{KMSB06}. Their results have not appeared in the
academic literature until 2010~\cite{KMS10}. In that paper, they
describe several basic schemes and describe how to break them using
teleportation-based attacks. They propose other variations (Schemes
IV--VI in~\cite{KMS10}) not suspect to their teleportation attack and
leave their security as an open question. Our general attack shows
that these schemes are insecure as well.
Concurrent and independent of our work and the work on
quantum tagging described above, the approach of using quantum
techniques for secure position-verification was proposed by Malaney
~\cite{Mal10a, Mal10b}. However, the proposed scheme is merely claimed
secure, and no rigorous security analysis is provided. As pointed out
in~\cite{KMS10}, Malaney's schemes can also be broken by a
teleportation-based attack.
Chandran et al.\@ have proposed and proved secure a quantum scheme for
position-verification~\cite{CFGGO10}. However, their proof implicitly assumed that the adversaries have no pre-shared entanglement; as shown
in~\cite{KMS10}, their scheme also becomes insecure without this assumption.
In a subsequent paper~\cite{LL11}, Lau and Lo use similar ideas as
in~\cite{KMS10} to show the insecurity of position-verification
schemes that are of a certain (yet rather restricted) form, which
include the schemes from~\cite{Mal10a,Mal10b} and~\cite{CFGGO10}.
Furthermore, they propose a position-verification scheme that resists
their attack, and they conjecture it secure. While these protocols
might be secure if the
adversaries do not pre-share entanglement, our attack shows that all
of them are insecure in general.
In a recent note~\cite{Kent10}, Kent considers a different model for
position-based cryptography where the prover's position is \emph{not} his only credential, but he is assumed to additionally
share with the verifiers a classical key unknown to the adversary. In
this case, quantum key distribution can be used to expand that key ad
infinitum. This classical key stream is then used as authentication
resource.
The idea of performing ``instantaneous measurements of nonlocal
variables'' has been put forward by Vaidman~\cite{Vaidman03} and was
further investigated by Clark et al.~\cite{CCJP10}. The concept of
instantaneous nonlocal quantum computation presented here is an
extension of Vaidman's task.
After the appearance and circulation of our work, Beigi and
K\"onig~\cite{BK11} used the technique of port-based teleportation by
Ishizaka and Hiroshima~\cite{IH08,IH09} to reduce the amount of
entanglement required to perform instantaneous nonlocal quantum
computation (from our double exponential) to exponential.
In~\cite{GLM02}, Giovannetti et al.~show how to measure the distance
between two parties by quantum cryptographic means so that only
trusted people have access to the result. This is a different kind
of problem than what we consider, and the techniques used there
are not applicable in our setting.
\subsection{Our Attack and Our Schemes in More Detail}
\paragraph{Position-Verification - A Simple Approach.}\label{sec:SimpleApproach}
Let us briefly discuss the $1$-dimensional case in which we have
two verifiers $V_0$ and $V_1$, and a prover~$P$ at position~$\text{\sl pos}$
that lies on the straight line between~$V_0$ and~$V_1$. Now, to verify
$P$'s position, $V_0$~sends a BB84 qubit $H^\theta\ket{x}$ to~$P$, and
$V_1$~sends the corresponding basis~$\theta$ to~$P$. The sending of
these messages is timed in such a way that $H^\theta\ket{x}$ and
$\theta$~arrive at position~$\text{\sl pos}$ at the same time. $P$ has to
measure the qubit in basis~$\theta$ to obtain~$x$, and immediately
send~$x$ to both~$V_0$ and~$V_1$, who verify the correctness of~$x$ and if
it has arrived ``in time''.
The intuition for this scheme is the following.
Consider a dishonest prover $P_0$ between $V_0$ and~$P$, and a
dishonest prover $P_1$ between $V_1$ and $P$.
(It is not too hard to see that additional dishonest
provers do not help.) When $P_0$ receives the BB84 qubit, she does
not know yet the corresponding basis $\theta$. Thus, if she measures
it immediately when she receives it, she is likely to measure it
in the wrong basis and $P_0$ and $P_1$ will not be able to
provide the correct $x$. However, if she waits until she knows the
basis~$\theta$, $P_0$ and $P_1$ will be too late in sending $x$ to~$V_1$ in time.
Similarly, if she forwards the BB84 qubit to $P_1$, who receives
$\theta$ before $P_0$ does, then $P_0$ and $P_1$ will be too
late in sending $x$ to~$V_0$.
It seems that in order to break the scheme, $P_0$ needs to
store the qubit until she receives the basis~$\theta$ and at the same
time send a copy of it to~$P_1$. But such actions are excluded by the
no-cloning principle.
\paragraph{The Attack and Instantaneous Nonlocal Quantum Computation.}
The above intuition turns out to be wrong. Using pre-shared
entanglement, $P_0$ and $P_1$ can perform quantum teleportation
which enables them (in some sense) to act coherently on the complete
state immediately upon reception. Combining this fact with the observation by
Kent et al.~\cite{KMS10} that the Pauli-corrections resulting
from the teleportation commute with the actions of the honest prover
in the above protocol shows that colluding adversaries can perfectly
break the protocol.
Much more generally, we will show how to break \emph{any}
position-verification scheme, possibly consisting of multiple
(and interleaved) rounds. To this end, we will show how to perform \emph{instantaneous
nonlocal quantum computation}. In particular, we prove that any unitary operation
$U$ acting on a composite system shared between players can be
computed using only a single round of mutual classical communication.
Based on ideas by Vaidman~\cite{Vaidman03}, the
players teleport quantum states back and forth many times in a clever
way, {\em without} awaiting the classical measurement outcomes from
the other party's teleportations.
\paragraph{Position-Verification in the No Pre-shared Entanglement (No-PE\xspace) Model.}
On the other hand, the above intuition is correct in the {\em no pre-shared entanglement} (No-PE\xspace) model,
where the adversaries are not allowed to have pre-shared entangled
quantum states prior the execution the protocol, or, more generally, prior the execution of each round of the protocol in case of multi-round schemes.
Even though this model may be somewhat unrealistic and
artificial, analyzing protocols in this setting serves as stepping
stone to obtaining protocols which tolerate adversaries who pre-share and maintain
some {\em limited} amount of entanglement.
But also, rigorously proving security in the restrictive (for the adversary) No-PE\xspace model is
already non-trivial and requires heavy machinery. Our proof uses the {\em strong complementary information trade-off} (CIT) due to Renes and Boileau~\cite{RB09},
and it guarantees that
for any strategy, the success probability of $P_0$ and $P_1$ is
bounded by approximately~$0.89$. By repeating the above simple scheme
sequentially, we get a secure multi-round positioning scheme with
exponentially small soundness error.
We note that when performing sequential repetitions in the No-PE\xspace model,
the adversaries must enter each round with no entanglement; thus, they are not allowed to generate entanglement in one round, store it, and use it in the next round(s).
\paragraph{Position-based authentication and key-exchange in the No-PE\xspace Model.}
Our position-based authentication scheme is based on our position-verification
scheme. The idea is to start with a ``weak''
authentication scheme for a $1$-bit message $m$:
the verifiers and $P$ execute the secure position-verification
scheme; if $P$ wishes to authenticate $m=1$, then $P$ correctly
finishes the scheme by sending $x$ back, but if $P$ wishes to
authenticate $m=0$, $P$ sends back an ``erasure'' $\bot$
instead of the correct reply $x$ with some probability $q$ (which
needs to be carefully chosen). This authentication
scheme is weak in the sense that turning $1$ into $0$ is easy for
the adversary, but turning a $0$ into a $1$ fails with constant
probability.
The idea is to use a suitable {\em balanced} encoding of the actual
message to be authenticated, so that for any two messages, the
adversary needs to turn many $0$'s into $1$'s. Unfortunately, an
arbitrary balanced encoding is not good enough. The reason is
that we do not assume the verifiers and the honest $P$ to be
synchronized. This asynchrony allows the adversary to make use of honest $P$ who
is authenticating one index of the encoded message, in order to
authenticate another index of the modified encoded message towards the
verifiers.
Nevertheless, we show that the above approach does work for carefully chosen codes.
We show that, for instance, the bit-wise encoding which maps
$0$ into $00...0\,11...1$ and $1$ into
$11...1\,00...0$ is such a code.
Our solution borrows some ideas from~\cite{RW03, KR09, CKOR10} on authentication based on weak secrets. However, since in our setting we cannot do "liveness" tests (to check that the verifier is alive in the protocol), the techniques from~\cite{RW03, KR09, CKOR10} do not help us directly.
Given a position-based authentication scheme, one can immediately
obtain a position-based key-exchange scheme simply by (essentially)
executing an arbitrary quantum-key-distribution scheme
(e.g.~\cite{BB84}), which assumes an authenticated classical
communication channel, and authenticate the classical communication
by means of the position-based authentication scheme.
\subsection{Organization of the paper}
In Section~\ref{sec:preliminaries}, we begin by introducing
notation, and presenting the relevant background from quantum
information theory. In Section~\ref{sec:setup}, we describe the
problem of position-verification and define our standard quantum
model, as well as the No-PE\xspace model in more detail.
A protocol for computing any unitary operation using local operations and one round
of classical communication is provided and analyzed in Section~\ref{sec:nonlocal}, and
in Section~\ref{sec:Impossibility} we conclude that there does not exist any
protocol for position-verification (and hence, any protocol for
position-based cryptographic tasks) in the standard quantum model.
We present our position-verification protocol in the No-PE\xspace model in
Section~\ref{sec:basicSP}. Section~\ref{sec:auth+KE} is devoted to
our position-based authentication protocol and showing how to
combine the above tools to obtain position-based key exchange.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Notation and Terminology}\label{sec:Notation}
We assume the reader to be familiar with the basic concepts of quantum
information theory and refer to~\cite{NC00} for an excellent
introduction; we merely fix some notation.
\paragraph{Qubits.}
A {\em qubit} is a quantum system $A$ with a 2-dimensional state space~$\H_A = \Bbb{C}^2$.
The {\em computational basis} $\set{\ket{0},\ket{1}}$ (for a qubit) is given by $\ket{0} = {1 \choose 0}$ and $\ket{1} = {0 \choose 1}$, and the {\em Hadamard basis} by $H\set{\ket{0},\ket{1}} = \set{H\ket{0},H\ket{1}}$, where $H$ denotes the 2-dimensional {\em Hadamard matrix}, which maps $\ket{0}$ to $(\ket{0}+\ket{1})/\sqrt{2}$ and $\ket{1}$ to $(\ket{0}-\ket{1})/\sqrt{2}$.
The state space of an $n$-qubit system $A = A_1\cdots A_n$ is given by
the $2^n$-dimensional space $\H_A = (\Bbb{C}^2)^{\otimes n} = \Bbb{C}^2 \otimes \cdots \otimes \Bbb{C}^2$.
Since we mainly use the above two bases, we can simplify terminology and notation by identifying the computational basis $\set{\ket{0},\ket{1}}$ with the bit $0$ and the Hadamard basis $H\set{\ket{0},\ket{1}}$ with the bit $1$. Hence, when we say that an $n$-qubit state $\ket{\psi} \in (\Bbb{C}^2)^{\otimes n}$ is measured in basis $\theta \in \set{0,1}^n$, we mean that the state is measured qubit-wise where basis $H^{\theta_i}\set{\ket{0},\ket{1}}$ is used for the $i$-th qubit.
As a result of the measurement, the string $x \in \set{0,1}^n$ is observed with probability $|\bra{\psi}H^\theta\ket{x}|^2$, where $H^\theta = H^{\theta_1} \otimes \cdots \otimes H^{\theta_n}$ and $\ket{x} = \ket{x_1}\otimes\cdots\otimes\ket{x_n}$.
An important example of a $2$-qubit state is the {\em EPR pair}, which is given by $\ket{\Phi_{AB}} = (\ket{0}\zero + \ket{1}\one)/\sqrt{2} \in \H_A \otimes \H_B = \Bbb{C}^2 \otimes \Bbb{C}^2$ and has the following properties: if qubit $A$ is measured in the computational basis, a uniformly random bit $x \in \set{0,1}$ is observed and qubit $B$ collapses to $\ket{x}$. Similarly, if qubit $A$ is measured in the Hadamard basis, a uniformly random bit $x \in \set{0,1}$ is observed and qubit $B$ collapses to $H\ket{x}$.
\paragraph{Density Matrices and Trace Distance.}
For any complex Hilbert space $\H$, we write ${\cal D}(\H)$ for the set of all {\em density matrices} acting on $\H$.
We measure closeness of two density matrices $\rho$ and $\sigma$ in ${\cal D}(\H)$
by their {\em trace distance}: $\delta(\rho,\sigma) := \frac12
\mathrm{tr}|\rho-\sigma|$. One can show that for any physical processing of two quantum states described by
$\rho$ and $\sigma$, respectively, the two states behave in an
indistinguishable way except with probability at most
$\delta(\rho,\sigma)$. Thus, informally, if $\delta(\rho,\sigma)$ is
very small, then without making a significant error, the two quantum
states can be considered equal.
\paragraph{Classical and Hybrid Systems (and States).}
Subsystem $X$ of a bipartite quantum system $XE$ is called {\em classical}, if the state of $XE$ is given by a density matrix of the form
$
\rho_{XE} = \sum_{x \in \cal X} P_X(x) \proj{x} \otimes \rho_{E}^x \, ,
$
where $\cal X$ is a finite set of cardinality $|{\cal X}| = \dim(\H_X)$, $P_X:{\cal X} \rightarrow [0,1]$ is a probability distribution, $\set{\ket{x}}_{x \in \cal X}$ is some fixed orthonormal basis of $\H_X$, and $\rho_E^x$ is a density matrix on $\H_E$ for every \mbox{$x \in \cal X$}. Such a state, called {\em hybrid} state (also known as {\em cq-}state, for {\em c}lassical and {\em q}uantum), can equivalently be understood as consisting of a {\em random variable} $X$ with distribution $P_X$ and range $\cal X$, and a system $E$ that is in state $\rho_E^x$ exactly when $X$ takes on the value $x$. This formalism naturally extends to two (or more) classical systems $X$, $Y$ etc.\@ as well as to two (or more) quantum systems.
\paragraph{Teleportation.}
The goal of teleportation is to transfer a quantum state from one location to another by only communicating classical information. Teleportation requires pre-shared entanglement among the two locations.
Specifically, to teleport a qubit $Q$ in an arbitrary (and typically unknown) state $\ket{\psi}$ from Alice to Bob, Alice performs a
Bell-measurement on $Q$ and her half of an EPR-pair, yielding a
classical measurement outcome $k \in \set{0,1,2,3}$.
Instantaneously, the other half of the corresponding EPR pair, which is held by Bob, turns into
the state $\PC_k^\dag\ket{\psi}$, where $\PC_0, \PC_1, \PC_2, \PC_3$ denote the
four Pauli-corrections $\set{\Bbb{I},X,Z,XZ}$, respectively, and $\PC^\dag$
denotes the complex conjugate of the transpose of $\PC$. The classical information $k$ is then
communicated to Bob who can recover the state $\ket{\psi}$ by performing $\PC_k$ on his EPR half.
Note that the operator $\PC_k$ is Hermitian and unitary, thus $\PC^\dag_k=\PC_k$ and $\PC_k\PC^\dag_k=\Bbb{I}$.
\subsection{Some Quantum Information Theory}
The {\em von Neumann entropy} of a quantum state $\rho \in {\cal D}(\H)$ is given by $\Hone(\rho) := -\mathrm{tr}\bigl(\rho \log(\rho)\bigr)$, where here and throughout the article, $\log$ denotes the binary logarithm.
$\Hone(\rho)$ is non-negative and at most $\log(\dim(\H))$.
For a bi-partite quantum state $\rho_{AB} \in {\cal D}(\H_A \otimes \H_B)$, the {\em conditional} von Neumann entropy of $A$ given $B$ is defined as $\Hone(\rho_{AB}|B) := \Hone(\rho_{AB}) - \Hone(\rho_B)$.
In cases where the state $\rho_{AB}$ is clear from the context, we may write $\Hone(A|B)$ instead of $\Hone(\rho_{AB}|B)$.
If $X$ and $Y$ are both classical, $\Hone(X|Y)$ coincides with the classical conditional Shannon entropy.
Furthermore, in case of conditioning (partly) on a classical state, the following holds.
\begin{lemma}\label{lemma:Average}
For any tri-partite state $\rho_{ABY}$ with classical~$Y$: $\Hone(A|BY) = \sum_y P_Y(y) \Hone(\rho_{AB}^y|B)$.
\end{lemma}
Lemma~\ref{lemma:Average}
along with the concavity of $\Hone$ and Jensen's inequality
implies that for classical $Y$: $\Hone(A) \geq \Hone(A|Y) \geq 0$.
The proof of Lemma~\ref{lemma:Average} is given in Appendix~\ref{app:technical}.
The following theorem is a generalization of the well-known Holevo bound~\cite{Holevo73} (see also~\cite{NC00}), and follows from the {\em monotonicity of mutual information}.
Informally, it says that measuring only reduces your information.
Formally, and tailored to the notation used here, it ensures the following.
\begin{theorem
\label{thm:Holevo}
Let $\rho_{AB} \in {\cal D}(\H_A \otimes \H_B)$ be an arbitrary bi-partite state, and let $\rho_{AY}$ be obtained by measuring $B$ in some basis to observe (classical) $Y$. Then
$
\Hone(A|Y) \geq \Hone(A|B)
$.
\end{theorem}
For classical $X$ and $Y$, the Fano inequality~\cite{Fano61} (see also~\cite{CT91}) allows to bound the probability of correctly guessing $X$ when having access to $Y$. In the statement below and throughout the article, $\h:[0,1]\to[0,1]$ denotes the {\em binary entropy function} defined as $\h(p) = -p \log(p) - (1-p)\log(1-p)$ for $0 < p < 1$ and as $\h(p) = 0$ for $p = 0$ or $1$, and $\h^{-1}:[0,1]\to[0,\frac12]$ denotes its inverse on the branch $0 \leq p \leq \frac12$.
\begin{theorem}[Fano inequality]\label{thm:Fano}
Let $X$ and $Y$ be random variables with ranges $\cal X$ and $\cal Y$, respectively, and let $\hat{X}$ be a guess for $X$ computed solely from $Y$. Then $q:= \P[\hat{X} \!\neq\! X]$ satisfies
$$\h(q) + q \log(|{\cal X}|-1) \geq \Hone(X|Y)\ .$
In particular, for binary $X$: $q \geq \h^{-1}(\Hone(X|Y))$.
\end{theorem}
\subsection{Strong Complementary Information Tradeoff}
The following entropic uncertainty principle, called {\em strong complementary information tradeoff} (CIT) in~\cite{RB09}
and generalized in~\cite{BCCRR10}, is at the heart of our security proofs.
It relates the uncertainty of the measurement outcome of a system $A$ with the uncertainty of the measurement outcome when the complementary basis is used instead, and it guarantees that there can coexist at most one system $E$ that has full information on {\em both} possible outcomes. Note that by the {\em complementary} basis $\bar{\theta}$ of a basis $\theta = (\theta_1,\ldots,\theta_n) \in \set{0,1}^n$, we mean the $n$-bit string $\bar{\theta} = (\bar{\theta}_1,\ldots,\bar{\theta}_n) \in \set{0,1}^n$ with $\bar{\theta}_i \neq \theta_i$ for all $i$.
\begin{theorem}[CIT]\label{thm:CIT}
Let $\ket{\psi_{AEF}} \in \H_A \otimes \H_E \otimes \H_F$ be an arbitrary tri-partite state, where $\H_A = (\Bbb{C}^2)^{\otimes n}$. Let the hybrid state $\rho_{XEF}$ be obtained by measuring $A$ in basis $\theta \in \set{0,1}^n$, and let the hybrid state $\sigma_{XEF}$ be obtained by measuring $A$ (of the original state $\ket{\psi_{AEF}}$) in the complementary basis $\bar{\theta}$. Then
\begin{equation*}
\Hone(\rho_{XE}|E) + \Hone(\sigma_{XF}|F) \geq n \ .
\end{equation*}
\end{theorem}
CIT in particular implies the following (the proof is given in Appendix~\ref{app:technical}):
\begin{corollary}\label{cor:CIT}
Let $\ket{\psi_{AEF}} \in \H_A \otimes \H_E \otimes \H_F$ be an arbitrary tri-partite state, where $\H_A = (\Bbb{C}^2)^{\otimes n}$.
Let $\Theta$ be uniformly distributed in $\set{0,1}^n$ and let $X$ be the result of measuring $A$ in basis $\Theta$. Then
\begin{equation*}
\Hone(X|\Theta E) + \Hone(X|\Theta F) \geq n\ .
\end{equation*}
\end{corollary}
\section{Setup and The Task of Position Verification}\label{sec:setup}
\subsection{The Security Model}\label{subsec:model}
We informally describe the model we use for the upcoming sections,
which is a quantum version of the Vanilla (standard) model introduced
in~\cite{CGMO09} (see there for a full description). We also describe
our {\em restricted model} used for our security proof, that we call
the no pre-shared entanglement (No-PE\xspace) model. We consider entities
$V_0,\dotsc, V_k$ called {\em verifiers} and an entity $P$, the
(honest) {\em prover}. Additionally, we consider a coalition $P$ of
{\em dishonest provers} (or {\em adversaries})
\smash{$P_0,\ldots,P_\ell$}. All entities can perform arbitrary
quantum (and classical) operations and can communicate quantum (and
classical) messages among them.
For our positive results, we consider a restricted model, which prohibits entanglement between the dishonest verifiers. Specifically, the {\em No-PE\xspace model} is such that the dishonest provers enter every new round of communication, initiated by the verifiers, with no pre-shared entanglement. That is, in every round, a dishonest prover can
send an entangled quantum state only {\em after} it receives the verifier's
message, and the dishonest provers cannot maintain such an entangled state in order to use it in the next round.
As mentioned in the introduction, considering this simple
(but possibly unrealistic) model may help us in obtaining protocols that
are secure against adversaries with {\em limited} entanglement.
For simplicity, we assume that quantum operations and communication
are noise-free; however, our results generalize to the more realistic
noisy case, assuming that the noise is low enough. We require that the
verifiers have a private and authenticated channel among themselves, which
allows them to coordinate their actions by communicating before,
during or after protocol execution. We stress however, that this
assumption does
not hold for the communication between the verifiers and~$P$: $P$
has full control over the destination of messages communicated between
the verifiers and~$P$ (both ways). In particular, the
verifiers do not know per-se if they are communicating with the honest
or a dishonest prover (or a coalition of dishonest provers).
The above model is extended by incorporating the notion of {\em time}
and {\em space}. Each entity is assigned an arbitrary fixed position
$\text{\sl pos}$ in the $d$-dimensional space~$\Bbb{R}^d$, and we assume that
messages to be communicated travel at fixed velocity $v$ (e.g.\ with
the speed of light), and hence the time needed for a message to travel
from one entity to another equals the Euclidean distance between the
two (assuming that~$v$ is normalized to~$1$). This timing assumption holds for honest
and dishonest entities. We assume on the other hand that local
computations take no time.
Finally, we
assume that the verifiers have precise and synchronized clocks, so
that they can coordinate exact times for sending off messages and
can measure the exact time of a message arrival. We do not
require $P$'s clock to be precise or in sync with the verifiers.
However, we do assume that $P$ cannot be reset.
This model allows to reason as follows.
Consider a verifier $V_0$ at position $\text{\sl pos}_0$, who sends
a challenge $ch_0$ to the (supposedly honest) prover claiming to be
at position $\text{\sl pos}$. If $V_0$ receives a reply within time
$2\d(\text{\sl pos}_0,\text{\sl pos})$, where $\d(\cdot,\cdot)$ is the Euclidean
distance measure in $\Bbb{R}^d$ and thus also measures the time a message
takes from one point to the other, then $V_0$ can conclude that he
is communicating with a prover that is within distance
$\d(\text{\sl pos}_0,\text{\sl pos})$.
We stress that in our model, the honest prover $P$ has no advantage over the dishonest provers beyond being at its position $\text{\sl pos}$. In particular, $P$ does not share any secret information with the verifiers, nor can he per-se authenticate his messages by any other means.
Throughout the article, we
require that the honest prover $P$ is {\em enclosed} by the
verifiers $V_0,\ldots,V_k$ in that the prover's position $\text{\sl pos} \in
\Bbb{R}^d$ lies within the tetrahedron, i.e., convex hull, $\Hull(\text{\sl pos}_0,\dotsc,\text{\sl pos}_k) \subset \Bbb{R}^d$
formed by the respective positions
of the verifiers.
Note that in this work we consider only {\em stand-alone security}, i.e.,
there exists only a single execution with a single honest prover, and we
do not guarantee concurrent security.
\subsection{Secure Position Verification}
A position-verification scheme
should allow a prover $P$ at position $\text{\sl pos} \in \Bbb{R}^d$ (in
$d$-dimensional space) to convince a set of $k+1$ verifiers
$V_0,\ldots,V_k$, who are located at respective positions
$\text{\sl pos}_0,\dotsc,\text{\sl pos}_k \in \Bbb{R}^d$, that he is indeed at position
$\text{\sl pos}$. We assume that $P$ is enclosed by
$V_0,\ldots,V_k$.
We require that the verifiers jointly accept if an honest prover $P$ is at
position $\text{\sl pos}$, and we require that the verifiers reject with
``high''
probability in case of a dishonest prover
that is not at position $\text{\sl pos}$. The latter should hold even if the
dishonest prover consist of a {\em coalition} of collaborating
dishonest provers $P_0,\ldots,P_{\ell}$ at arbitrary
positions
$\text{$apo\hspace{-0.1ex}s$}_0,\dotsc,\text{$apo\hspace{-0.1ex}s$}_\ell \in \Bbb{R}^d$ with $\text{$apo\hspace{-0.1ex}s$}_i \neq \text{\sl pos}$ for
all~$i$. We refer to~\cite{CGMO09} for the general formal definition
of the completeness and security of a position-verification scheme. In
this article, we mainly focus on position-verification schemes of the
following form:
\begin{definition}\label{def:1roundSP}
A 1-round \textbf{position-verification} scheme ${\sf PV}=({\sf Chlg},{\sf Resp},{\sf Ver})$
consists of the following three parts.
A
challenge generator ${\sf Chlg}$, which outputs a list of challenges
$(ch_0,\ldots,ch_k)$ and auxiliary information~$x$; a response
algorithm ${\sf Resp}$, which on input a list of challenges outputs a
list of responses $(x'_0,\ldots,x'_k)$; and a verification algorithm
${\sf Ver}$ with ${\sf Ver}(x'_0,\ldots,x'_k,x) \in \set{0,1}$.
${\sf PV}$ is said to have {\bf perfect completeness} if
${\sf Ver}(x'_0,\ldots,x'_k,x) = 1$ with probability~1 for
$(ch_0,\ldots,ch_k)$ and $x$ generated by~${\sf Chlg}$ and
$(x'_0,\ldots,x'_k)$ by~${\sf Resp}$ on input $(ch_0,\ldots,ch_k)$.
\end{definition}
The algorithms ${\sf Chlg}$, ${\sf Resp}$ and ${\sf Ver}$ are used as described in Figure~\ref{fig:1roundSP} to verify the claimed position of a prover $P$. We clarify that in order to have all the challenges arrive at $P$'s (claimed) location $\text{\sl pos}$ {\em at the same time}, the verifiers agree on a time $T$ and each $V_i$ sends off his challenge $ch_i$ at time $T-\d(\text{\sl pos}_i,\text{\sl pos})$.
As a result, all $ch_i$'s arrive at
$P$'s position $\text{\sl pos}$ at time $T$. In Step~\ref{step:check}, $V_i$
receives $x'_i$ {\em in time} if $x'_i$ arrives at $V_i$'s position
$\text{\sl pos}_i$ at time $T+\d(\text{\sl pos}_i,\text{\sl pos})$. Throughout the article, we use
this simplified terminology. Furthermore, we are sometimes a bit
sloppy in distinguishing a party, like $P$, from its location
$\text{\sl pos}$.
\begin{figure}[htb]
\begin{framed}
\small
Common input to the verifiers: their respective positions $\text{\sl pos}_0,\dotsc,\text{\sl pos}_k$, and $P$'s (claimed) position $\text{\sl pos}$.
\begin{enumerate}\setlength{\parskip}{0.1ex}\setcounter{enumi}{-1}
\item $V_0$ generates a list of challenges $(ch_0,\ldots,ch_k)$ and auxiliary information $x$ using ${\sf Chlg}$,
and sends $ch_i$ to $V_i$ for $i = 1,\ldots,k$.
\item Every $V_i$ sends $ch_i$ to $P$ in such a way that all $ch_i$'s
arrive at the same time at $P$'s position~$\text{\sl pos}$.
\item $P$ computes $(x'_0,\ldots,x'_k):= {\sf Resp}(ch_0,\ldots,ch_k)$ as soon as all the $ch_i$'s arrive, and he sends $x'_i$ to $V_i$ for every $i$.
\item\label{step:check} The $V_i$'s jointly accept if and only if all $V_i$'s receive $x'_i$ in time and ${\sf Ver}(x'_0,\ldots,x'_k,x) = 1$.
\end{enumerate}
\end{framed}
\caption{Generic 1-round position-verification scheme. }
\label{fig:1roundSP}
\end{figure}
We stress that we allow ${\sf Chlg}$, ${\sf Resp}$ and ${\sf Ver}$ to be {\em quantum}
algorithms and $ch_i$, $x$ and $x'_i$ to be quantum information. In our constructions, only $ch_0$ will actually be quantum; thus, we will only require quantum
communication from $V_0$ to $P$, all other communication is
classical. Also, in our constructions, $x'_0 = \ldots = x'_k$, and ${\sf Ver}(x'_0,\ldots,x'_k,x) = 1$ exactly if $x'_i = x$ for all~$i$.
\begin{definition}
A 1-round position-verification scheme ${\sf PV} = ({\sf Chlg},{\sf Resp},{\sf Ver})$ is
called {\bf $\varepsilon$-sound} if for any position $\text{\sl pos} \in
\Hull(\text{\sl pos}_0,\dotsc,\text{\sl pos}_k)$, and any coalition of dishonest provers
\smash{$P_0,\ldots,P_{\ell}$} at arbitrary positions
$\text{$apo\hspace{-0.1ex}s$}_0,\dotsc,\text{$apo\hspace{-0.1ex}s$}_\ell$, all $\neq \text{\sl pos}$, when executing the
scheme from Figure~\ref{fig:1roundSP} the verifiers accept with
probability at most $\varepsilon$. We write ${\sf PV}^\varepsilon$ for such a
protocol.
\end{definition}
In order to be more realistic, we must take into consideration physical limitations of the equipment used,
such as measurement errors, computation durations, etc. Those allow a dishonest prover which
resides arbitrarily close to $P$ to appear as if she resides at $\text{\sl pos}$.
Thus, we assume that all the adversaries are at least $\Delta$-distanced from $\text{\sl pos}$, where $\Delta$ is determined
by those imperfections. For sake of simplicity, this $\Delta$ is implicit in the continuation of the paper.
A position-verification scheme can also be understood as a
(position-based) {\em identification} scheme, where the
identification is not done by means of a cryptographic key or a
password, but by means of the geographical location.
\section{Instantaneous Nonlocal Quantum Computation}\label{sec:nonlocal}
In order to analyze the (in)security of position-verification schemes,
we first address a more general task, which is interesting in its own
right: {\em instantaneous nonlocal quantum computation}\footnote{This is an extension of the task of ``instantaneous measurement of
nonlocal variables'' introduced by Vaidman~\cite{Vaidman03}.}.
Consider the following problem, involving two parties Alice and
Bob. Alice holds $A$ and Bob holds $B$ of a tripartite system $ABE$
that is in some unknown state $\ket{\psi}$. The goal is to apply a
known unitary transformation $U$ to $AB$, but {\em without} using any
communication, just by local operations. In general, such a task is
clearly impossible, as it violates the non-signalling principle. The
goal of instantaneous nonlocal quantum computation is to achieve
almost the above but without violating non-signalling. Specifically,
the goal is for Alice and Bob to compute, without communication, a state
$\ket{\varphi'}$ that coincides with $\ket{\varphi} = (U \otimes \mathbb{I})
\ket{\psi}$ up to {\em local} and {\em qubit-wise} operations on $A$ and $B$,
where $\mathbb{I}$ denotes the identity on $E$. Furthermore, these local and
qubit-wise operations are determined by {\em classical} information that
Alice and Bob obtain as part of their actions. In particular, if Alice
and Bob share their classical information, which can be done with {\em one}
round of simultaneous mutual communication, then they can transform
$\ket{\varphi'}$ into $\ket{\varphi} = U \ket{\psi}$ by local
qubit-wise operations. Following ideas by Vaidman~\cite{Vaidman03}, we
show below that instantaneous nonlocal quantum computation, as
described above, is possible if Alice and Bob share sufficiently many
EPR pairs.
In the following, let $\H_A$, $\H_B$ and $\H_E$ be Hilbert spaces where the former two
consist of $n_A$ and $n_B$ qubits respectively, i.e.,
$\H_A = (\Bbb{C}^2)^{\otimes n_A}$ and $\H_B = (\Bbb{C}^2)^{\otimes n_B}$. Furthermore, let $U$ be a unitary matrix acting on $\H_A \otimes \H_B$. Alice holds system $A$ and Bob holds system~$B$ of an arbitrary and unknown state $\ket{\psi} \in \H_{ABE} = \H_A \otimes \H_B \otimes \H_E$. Additionally, Alice and Bob share an arbitrary but finite number of EPR pairs.
\begin{theorem}\label{thm:local}
For every unitary~$U$ and for every $\varepsilon > 0$, given sufficiently
many shared EPR pairs, there exist local operations ${\cal A}$~and~${\cal B}$,
acting on Alice's and Bob's respective sides, with the
following property. For any initial state $\ket{\psi} \in \H_{ABE}$, the joint execution ${\cal
A} \otimes {\cal B}$ transforms~$\ket{\psi}$ into~$\ket{\varphi'}$
and provides classical outputs~$k$ to Alice and~$\ell$ to Bob,
such that the following holds except with probability~$\varepsilon$.
The state $\ket{\varphi'}$ coincides with $\ket{\varphi} = (U \otimes \mathbb{I})
\ket{\psi}$ up to local qubit-wise operations on $A$~and~$B$ that are determined
by~$k$ and~$\ell$.
\end{theorem}
We stress that ${\cal A}$ acts on $A$ as well as on Alice's shares of
the EPR pairs, and the corresponding holds for ${\cal
B}$. Furthermore, being equal up to local qubit-wise operations on
$A$ and $B$ means that $\ket{\varphi} = (V^A_{k,\ell} \otimes
V^B_{k,\ell} \otimes \mathbb{I}) \ket{\varphi'}$, where
$\{V^A_{k,\ell}\}_{k,\ell}$ and $\{V^B_{k,\ell}\}_{k,\ell}$ are fixed
families of unitaries which act qubit-wise on $\H_A$ and $\H_B$,
respectively. In our construction, the $V^A_{k,\ell}$ and
$V^B_{k,\ell}$'s will actually be tensor products of one-qubit Pauli operators.
As an immediate consequence of Theorem~\ref{thm:local}, we get the following.
\begin{corollary}
For every unitary $U$ and for every $\varepsilon > 0$, given sufficiently
many shared EPR pairs, there exists a nonlocal operation $\cal AB$ for Alice and Bob which consists of local operations and {\em one} round of mutual communication, such that for any initial state $\ket{\psi} \in \H_{ABE}$ of the tripartite system $ABE$, the joint execution of $\cal AB$ transforms $\ket{\psi}$ into $\ket{\varphi} = (U \otimes \mathbb{I})
\ket{\psi}$, except with probability~$\varepsilon$.
\end{corollary}
For technical reasons, we will actually prove the following extension
of Theorem~\ref{thm:local}, which is easily seen equivalent. The
difference to Theorem~\ref{thm:local} is that Alice and Bob are
additionally given classical inputs: $x$ to Alice and $y$ to Bob, and
the unitary $U$ that is to be applied to the quantum input depends on
$x$ and $y$. In the statement below, $x$ ranges over some arbitrary
but fixed finite set $\cal X$, and $y$ ranges over some arbitrary but
fixed finite set~$\cal Y$.
\begin{theorem}\label{thm:local+}
For every family $\set{U_{x,y}}$ of unitaries and for every $\varepsilon > 0$, given sufficiently many shared EPR pairs, there exist families $\{{\cal A}_x\}$ and $\{{\cal B}_y\}$ of local operations, acting on Alice's and Bob's respective sides, with the following property. For any initial state $\ket{\psi} \in \H_{ABE}$ and for every $x \in \cal X$ and $y \in \cal Y$, the joint execution ${\cal A}_x \otimes {\cal B}_y$ transforms the state $\ket{\psi}$ into $\ket{\varphi'}$ and provides classical outputs $k$ to Alice and $\ell$ to Bob, such that the following holds except with probability $\varepsilon$. The state $\ket{\varphi'}$ coincides with $\ket{\varphi} = (U_{x,y} \otimes \mathbb{I}) \ket{\psi}$ up to local qubit-wise operations on $A$ and $B$ that are determined by $k$ and $\ell$.
\end{theorem}
The solution works by teleporting states back and forth in a
clever way~\cite{Vaidman03}, but {\em without} communicating the
classical outcomes of the Bell measurements, so that only local
operations are performed. Thus, in the formal proof below, whenever we
say that a state is teleported, it should be understood in this
sense, i.e., the sender makes a Bell measurement resulting in some
classical information, and the receiver takes his shares of the EPR
pairs as the received state, but does/can not (yet) correct it.
\begin{proof}
To simplify notation, we assume that the joint state of $A$ and $B$ is pure, and thus we may ignore system $E$. However, all our arguments also hold in case the state of $A$ and $B$ is entangled with $E$.
Next, we observe that it is sufficient to prove Theorem~\ref{thm:local+} for the case where $B$ is ``empty'', i.e., $\dim \H_B = 1$ and thus $n_B = 0$. Indeed, if this is not the case, Alice and Bob can do the following. Bob first teleports $B$ to Alice. Now, Alice holds $A'=AB$ with $n_{A'}=n_A+n_B$, and Bob's system has collapsed and thus Bob holds no quantum state anymore, only classical information. Then, they do the nonlocal computation, and in the end Alice teleports $B$ back to Bob. The modification to the state of $B$ introduced by teleporting it to Alice can be taken care of by modifying the set of unitaries $\{U_{x,y}\}$ accordingly (and making it dependent on Bob's measurement outcome, thereby extending the set $\cal Y$). Also, the modification to the state of $B$ introduced by teleporting it back to Bob does not harm the requirement of the joint state being equal to $\ket{\varphi} = U_{x,y} \ket{\psi}$ up to local qubit-wise operations.
Hence, from now on, we may assume that $B$ is ``empty'', and we write $n$ for~$n_A$.
Next, we describe the core of how the local operations ${\cal A}_x$ and ${\cal B}_y$ work.
To simplify notation, we assume that ${\cal X} = \set{1,\ldots,m}$.
Recall that Alice and Bob share (many) EPR pairs. We may assume that the EPR pairs are grouped into groups of size $n$; each such group we call a {\em teleportation channel}. Furthermore, we may assume that $m$ of these teleportation channels are labeled by the numbers $1$ up to $m$, and that another $m$ of these teleportation channels are labeled by the numbers $m+1$ up to $2m$.
\begin{enumerate}
\item
Alice teleports $\ket{\psi}$ to Bob, using the teleportation channel that is labeled by her input $x$.
Let us denote her
measurement outcome by $k_\circ \in \set{0,1,2,3}^{n}$.
\item
For every $i \in \set{1,\ldots,m}$, Bob does the following. He applies the unitary $U_{i,y}$ to the $n$ qubits that make up his share of the EPR pairs given by the teleportation channel labeled by $i$. Then, he teleports the resulting state to Alice using the teleportation channel labeled by $m+i$. We denote the corresponding measurement outcome by $\ell_{\circ,i}$.
\item
Alice specifies the $n$ qubits that make up her share of the EPR pairs given by the teleportation channel labeled by $m+x$ to be the state $\ket{\varphi'}$.
\end{enumerate}
Let us analyze the above. With probability $1/4^n$, namely if $k_\circ = 0\cdots0$, teleporting $\ket{\psi}$ to Bob leaves the state unchanged. In this case, it is easy to see that the resulting state $\ket{\varphi'}$ satisfies the required property of being identical to $\ket{\varphi} = U_{x,y} \ket{\psi}$ up to local qubit-wise operations determined by $\ell_{\circ,x}$, and thus determined by $x$ and
$\ell_\circ = (\ell_{\circ,1},\ldots,\ell_{\circ,m})$.
This proves the claim for the case where $\varepsilon \geq 1 - 1/4^{n}$.
We show how to reduce $\varepsilon$. The crucial observation is that if in the above procedure $k_\circ \neq 0\cdots0$, and thus $\ket{\varphi'}$ is not necessarily identical to $\ket{\varphi}$ up to local qubit-wise operations, then
$$
\ket{\varphi'} = V_{\ell_{\circ,x}} U_{x,y} V_{k_\circ}\ket{\psi} = V_{\ell_{\circ,x}} U_{x,y} V_{k_\circ} U_{x,y}^\dagger \ket{\varphi} \, ,
$$
where $V_{\ell_{\circ,x}}$ and $V_{k_\circ}$ are tensor products of Pauli matrices.
Thus, setting $\ket{\psi'}:= \ket{\varphi'}$, $x' := (x,k_\circ)$ and $y' := (y,\ell_\circ)$, and
$U'_{x',y'} := U_{x,y} V_{k_\circ} U_{x,y}^\dagger V_{\ell_{\circ,x}}$, the state~$\ket{\varphi}$ can be written as $\ket{\varphi} = U'_{x',y'} \ket{\psi'}$.
This means, we are back to the original problem of applying a unitary, $U'_{x',y'}$, to a state, $\ket{\psi'}$, held by Alice, where the unitary depends on classical information $x'$ and $y'$, known by Alice and Bob, respectively. Thus, we can re-apply the above procedure to the new problem instance. Note that in the new problem instance, the classical inputs $x'$ and $y'$ come from larger sets than the original inputs $x$ and $y$, but the new quantum input, $\ket{\psi'}$, has the same qubit size, $n$. Therefore, re-applying the procedure will succeed with the same probability~$1/4^n$.
As there is a constant probability of success in each round, re-applying the above procedure sufficiently many times to the resulting new problem instances guarantees that except with arbitrary small probability, the state $\ket{\varphi'}$ will be of the required form at some point
(when Alice gets $k_\circ=0\cdots0$).
Say, this is the case at the end of the $j$-th iteration. Then, Alice stops with her part of the procedure at this point, keeps the state $\ket{\varphi'}$, and specifies $k$ to consist of $j$ and of her classical input into the $j$-th iteration (which consists of $x$ and of the $k_\circ$'s from the prior $j-1$ iterations).
Since Bob does not learn whether an iteration is successful or not, he has to keep on re-iterating up to some bound, and in the end he specifies $\ell$ to consist of the $\ell_\circ$'s collected over all the iterations.
The state~$\ket{\varphi'}$ equals $\ket{\varphi} = U_{x,y} \ket{\psi}$ up to local qubit-wise operations that are determined by $k$~and~$\ell$.
\end{proof}
Doing the maths shows that the number of EPR pairs needed by Alice and
Bob in the scheme described in the proof is double exponential in
$n_A+n_B$, the qubit size of the joint quantum system.
In recent subsequent work~\cite{BK11}, Beigi and K\"onig have used a
different kind of quantum teleportation by Ishizaka and
Hiroshima~\cite{IH08,IH09} to reduce the amount of entanglement needed
to to perform instantaneous nonlocal quantum computation to
exponential in the qubit size of the joint quantum system. It remains
an interesting open question whether such an exponentially large
amount of entanglement is necessary.
In Appendix~\ref{sec:nparties}, we explain how to perform
instantaneous nonlocal quantum computation among more than two
parties.
\section{Impossibility of Unconditional Position Verification}\label{sec:Impossibility}
In this section we show that no position-verification scheme is secure against
a coalition of quantum adversaries in the Vanilla model.
For simplicity, we consider the one-dimensional case, with two
verifiers $V_0$ and~$V_1$, but the attack can be generalized to higher
dimensions and more verifiers.
We consider an arbitrary position-verification scheme in our model (as
specified in Section~\ref{subsec:model}). We recall that in this
model, the verifiers must base their decision solely on {\em what} the
prover replies and {\em how long} it takes him to reply, and the
honest prover has no advantage over a coalition of dishonest provers
beyond being at the claimed position\footnote{In particular, the
prover does not share any secret information with the verifiers,
differentiating our setting from models as described for example in
\cite{Kent10}.}. Such a position-verification scheme may be of the
form as specified in Figure~\ref{fig:1roundSP}, but may also be made up
of several, possibly interleaved, rounds of interaction between the
prover and the verifiers.
For the honest prover $P$, such a general scheme consists of steps
that look as follows.
$P$ holds a local quantum register $R$, which is set to some default value at the beginning of the scheme.
In each step, $P$ obtains a system $A$ from $V_0$ and a system $B$ from $V_1$, and $V_0$ and $V_1$ jointly keep some system $E$. Let $\ket{\psi}$ be the state of the four-partite system $ABRE$; it is determined by the scheme and by the step within the scheme we are focussing on.
$P$ has to apply a
fixed%
\footnote{$U$ is fixed for a fixed scheme and for a fixed step within the scheme, but of course may vary for different schemes and for different steps within a scheme. }
known unitary transformation $U$ to $ABR$, and send the
(transformed) systems $A$ and $B$ back to $V_0$ and $V_1$ (and keep
$R$). Note that after the transformation, the state of $ABRE$ is given
by $\ket{\varphi} = (U \otimes \mathbb{I}) \ket{\psi}$, where $\mathbb{I}$ is the
identity acting on $\H_E$. For technical reasons, as in
Section~\ref{sec:nonlocal}, it will be convenient to distinguish
between classical and quantum inputs, and therefore, we
let the unitary $U$ depend on classical information $x$ and $y$,
where $x$ has been sent by $V_0$ along with $A$, and $y$ has been sent
by $V_1$ along with $B$.
We show that a coalition of two dishonest provers $P_0$ and
$P_1$, where $P_0$ is located in between $V_0$ and $P$ and $P_1$
is located in between $V_1$ and $P$, can perfectly simulate the
actions of the honest prover $P$, and therefore it is impossible for
the verifiers to distinguish between an honest prover at position
$\text{\sl pos}$ and a coalition of dishonest provers at positions different
from $\text{\sl pos}$. The simulation of the dishonest provers perfectly imitates
the {\em computation} as well as the {\em timing} of an honest $P$. Since
in our model this information is what the verifiers have to base their
decision on, the general impossibility of
position-verification in our model follows.
Consider a step in the scheme as described above, but now from the
point of view of $P_0$ and $P_1$. Since $P_0$ is closer to
$V_0$, he will first receive $A$ and $x$; similarly, $P_1$ will
first receive $B$ and $y$. We specify that $P_1$ takes care of and
maintains the local register $R$. If the step we consider is the
{\em first} step in the scheme, the state of $ABRE$ equals
$\ket{\psi}$, as in the case of an honest $P$. In order to have an
invariant that holds for all the steps, we actually relax this
statement and merely observe that the state of $ABRE$, say
$\ket{\psi'}$, equals $\ket{\psi}$ up to local and qubit-wise
operations on the subsystem $R$, determined by classical information
$x_\circ$ and $y_\circ$, where $P_0$ holds $x_\circ$ and $P_1$
holds $y_\circ$. This invariant clearly holds for the first step in the scheme,
when $R$ is in some default state, and we will show that it also holds
for the other steps.
By Theorem~\ref{thm:local+}, it follows that without communication,
just by instantaneous local operations, $P_0$ and $P_1$ can
transform the state $\ket{\psi'}$ into a state $\ket{\varphi'}$ that
coincides with $\ket{\varphi} = (U_{x,y} \otimes \mathbb{I}) \ket{\psi}$ up to
local and qubit-wise transformations on $A$, $B$ and $R$, determined
by classical information $k$ (known to $P_0$) and $\ell$ (known to
$P_1$). Note that the initial state is not $\ket{\psi}$, but rather
a state of the form $\ket{\psi'} = (V_{x_\circ,y_\circ} \otimes
\mathbb{I})\ket{\psi}$, where $x_\circ$ is known to $P_0$ and $y_\circ$ to
$P_1$. Thus, Theorem~\ref{thm:local+} is actually applied to the
unitary $U'_{x',y'} = U_{x,y} V_{x_\circ,y_\circ}^\dagger$, where $x'
= (x_\circ,x)$ and $y' = (y_\circ,y)$. Given $\ket{\varphi'}$ and $k$
and $\ell$, $P_0$ and $P_1$ can exchange $k$ and $\ell$ using {\em one}
mutual round of communication and transform
$\ket{\varphi'}$ into $\ket{\varphi''}$ that coincides with
$\ket{\varphi}$ up to qubit-wise operations only on $R$, and send $A$
to $V_0$ and $B$ to $V_1$. It follows that the state of $ABE$ and the
time it took $P_0$ and $P_1$ for the computation and communication
is identical to that of an honest $P$, i.e., $P_0$ and $P_1$ have
perfectly simulated this step of the scheme.
Finally, we see that the invariant is satisfied, when moving on to the
next step in the scheme, where $P_0$ and $P_1$ receive new $A$ and
$B$ (along with new classical $x$ and $y$) from $V_0$ and $V_1$,
respectively. Even if this new round interleaves with the previous
round in that the new $A$ and $B$ etc.\@ arrive {\em before}
$P_0$ and $P_1$ have finished exchanging (the old) $k$ and $\ell$,
it still holds that the state of $ABRE$ is as in the case of honest
$P$ up to qubit-wise operations on the subsystem $R$. It follows
that the above procedure works for all the steps and thus that $P_0$
and $P_1$ can indeed perfectly simulate honest $P$'s actions
throughout the whole scheme.
\section{Secure Position-Verification in the No-PE\xspace model}\label{sec:basicSP}
\subsection{Basic Scheme and its Analysis}
In this section we show the possibility of secure position-verification in the No-PE\xspace model.
We consider the
following basic 1-round position-verification scheme, given in Figure~\ref{fig:BasicSP}. It is based on the BB84
encoding.
\begin{figure}[htb]
\small
\begin{protocol}
\begin{enumerate}\setlength{\parskip}{0.1ex}\setcounter{enumi}{-1}
\item $V_0$ chooses two random bits $x,\theta \in \set{0,1}$ and privately sends them to~$V_1$.
\item $V_0$ prepares the qubit $H^{\theta}\ket{x}$ and sends it to $P$, and $V_1$ sends the bit $\theta$ to $P$, so that $H^{\theta}\ket{x}$ and $\theta$ arrive at the same time at $P$.
\item When $H^{\theta}\ket{x}$ and $\theta$ arrive, $P$ measures $H^{\theta}\ket{x}$ in basis $\theta$ to observe $x' \in \set{0,1}$, and sends $x'$ to $V_0$ and $V_1$.
\item $V_0$ and $V_1$ accept if on both sides $x'$ arrives in time and $x' = x$.
\end{enumerate}
\end{protocol}
\caption{\!\mbox{Position-verification scheme $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ based on the BB84 encoding.}}
\label{fig:BasicSP}
\end{figure}
We implicitly specify that parties abort if they receive any
message that is inconsistent with the protocol, for instance
(classical) messages with a wrong length, or different number of
received qubits than expected, etc.
\begin{theorem}\label{thm:BasicSP}
The 1-round position-verification scheme
$\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ from Figure~\ref{fig:BasicSP} is $\varepsilon$-sound with
$\varepsilon = 1- \h^{-1}(\frac12)$, in the No-PE\xspace model.
\end{theorem}
Recall that $\h$ denotes the binary entropy function and $\h^{-1}$ its inverse on the branch $0 \leq p \leq \frac12$.
A numerical calculation shows that \smash{$\h^{-1}(\frac12) \geq 0.11$} and thus $\varepsilon \leq 0.89$. A particular attack for a dishonest prover \smash{$P$}, sitting in-between $V_0$ and $P$, is to measure the qubit \smash{$H^{\theta}\ket{x}$} in the {\em Breidbart} basis, resulting in an acceptance probability of $\cos(\pi/8)^2 \approx 0.85$. This shows that our analysis is pretty tight.
\begin{proof}
In order to analyze the position-verification scheme it is convenient to
consider an equivalent {\em purified} version, given in
Figure~\ref{fig:BasicEPR}. The only difference between the original
and the purified scheme is the preparation of the bit~$H^{\theta}\ket{x}$.
In the purified version, it is done by preparing
$\ket{\Phi_{AB}}=(\ket{0}\zero + \ket{1}\one)/\sqrt{2}$ and measuring~$A$ in basis~$\theta$.
This way of preparation changes the point in time when $V_0$~measures~$A$,
and the point in time when $V_1$~learns~$x$.
This, however, has no
influence on the view of the (dishonest or honest) prover, nor on
the joint distribution of $\theta$, $x$ and $x'$, and thus neither
on the probability that $V_0$ and $V_1$~accept. It therefore
suffices to analyze the purified version.
\begin{figure}[htb]
\small
\begin{protocol}
\begin{enumerate}\setlength{\parskip}{0.1ex}\setcounter{enumi}{-1}
\item $V_0$ and $V_1$ privately agree on a random bit $\theta \in \set{0,1}$.
\item $V_0$ prepares an EPR pair $\ket{\Phi_{AB}} \in \H_A \o \H_B$, keeps qubit~$A$ and sends~$B$ to~$P$, and $V_1$~sends the bit~$\theta$
to~$P$, so that $B$~and~$\theta$ arrive at the same time at~$P$.
\item When $B$ and $\theta$ arrive, $P$ measures $B$ in basis $\theta$ to observe $x' \in \set{0,1}$, and sends $x'$ to $V_0$ and $V_1$.
\item Only now, when $x'$ arrives, $V_0$ measures $A$ in basis
$\theta$ to observe $x$, and privately sends~$x$ to~$V_1$.
$V_0$ and $V_1$ accept
if on both sides $x'$~arrives in time and $x' = x$.
\end{enumerate}
\end{protocol}
\caption{EPR version of $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$. }
\label{fig:BasicEPR}
\end{figure}
We first consider security against two dishonest provers $P_0$ and
$P_1$, where $P_0$ is between $V_0$ and $P$ and $P_1$ is between
$V_1$ and $P$. In the end we will argue that a similar argument holds
for multiple dishonest provers on either side.
Since $V_0$ and $V_1$ do not accept if $x'$ does not arrive in time
and dishonest provers do not use pre-shared entanglement in the
No-PE\xspace-model, any potentially successful strategy of $P_0$ and $P_1$
must look as follows. As soon as $P_1$ receives the bit $\theta$
from $V_1$, she forwards (a copy of) it to $P_0$. Also, as soon as
$P_0$ receives the qubit $A$, she applies an arbitrary quantum
operation to the received qubit $A$ (and maybe some ancillary
system she possesses) that maps it into a bipartite state $E_0 E_1$
(with arbitrary state space $\H_{E_0} \o \H_{E_1}$), and $P_0$ keeps
$E_0$ and sends $E_1$ to $P_1$. Then, as soon as $P_0$ receives
$\theta$, she applies some measurement (which may depend on $\theta$)
to $E_0$ to obtain $\hat{x}_0$, and as soon as $P_1$ receives $E_1$,
she applies some measurement (which may depend on $\theta$) to $E_1$ to
obtain $\hat{x}_1$, and both send $\hat{x}_0$ and $\hat{x}_1$
immediately to $V_0$ and $V_1$, respectively. We will argue that
the probability that $\hat{x}_0 = x$ {\em and} $\hat{x}_1 = x$ is
upper bounded by $\varepsilon$ as claimed.
Let $\ket{\psi_{A\, E_0 E_1}} \in \H_A \otimes \H_{E_0} \otimes
\H_{E_1}$ be the state of the tri-partite system $A\, E_0 E_1$ after
$P_0$ has applied the quantum operation to the qubit $B$. Note that
in the No-PE\xspace model, the quantum operation and thus $\ket{\psi_{A\, E_0
E_1}}$ does not depend on $\theta$.\footnote{We stress that this independency breaks down if $P_0$ and $P_1$ may start off with an entangled state, because then $P_1$ can act on his part of the entangled state in a $\theta$-dependent way, which makes the overall state dependent of $\theta$. }
Recall that~$x$ is obtained by measuring $A$ in either the computational (if
$\theta = 0$) or the Hadamard (if $\theta = 1$) basis. Writing $x$,
$\theta$, etc.\ as random variables $X$, $\Theta$, etc., it follows
from CIT (specifically Corollary~\ref{cor:CIT}) that $ \Hone(X|\Theta
E_0) + \Hone(X|\Theta E_1) \geq 1 \, . $ Let $Y_0$ and $Y_1$ denote
the classical information obtained by $P_0$ and $P_1$ as a result
of measuring $E_0$ and $E_1$, respectively, with bases that may depend
on~$\Theta$. By the (generalized) Holevo bound Theorem~\ref{thm:Holevo}, it
follows from the above that
$$\Hone(X|\Theta Y_0) + \Hone(X|\Theta Y_1) \geq 1\ ,$$
therefore $\Hone(X|\Theta Y_i)\geq\frac12$ for at least one $i \in \set{0,1}$.
By Fano's inequality (Theorem~\ref{thm:Fano}), we can conclude that the
corresponding error probability $q_i = \P[\hat{X}_i \!\neq\! X]$ satisfies
\(
\textstyle\h(q_i) \geq \frac12 .
\)
It thus follows that the failure probability
$$q = \P[\hat{X}_0 \!\neq\! X \vee \hat{X}_1 \!\neq\! X]
\geq \max\set{q_0,q_1} \ge \h^{-1}(\frac12)\ ,$$ and the probability
of $V_0$ and $V_1$ accepting, \smash{$\P[\hat{X}_0 \!=\! X \wedge
\hat{X}_1 \!=\! X] = 1 - q$}, is indeed upper bounded by $\varepsilon$ as
claimed.
It remains to argue that more than two dishonest provers in the No-PE\xspace
model cannot do any better. The reasoning is the same as
above. Namely, in order to respond in time, the dishonest provers that
are closer to $V_0$ than $P$ must map the qubit $A$---possibly
jointly---into a bipartite state $E_0E_1$ {\em without knowing
$\theta$}, and jointly keep $E_0$ and send $E_1$ to the dishonest
provers that are ``on the other side'' of $P$ (i.e., closer to
$V_1$). Then, the reply for $V_0$ needs to be computed from $E_0$~and~$\theta$
(possibly jointly by the dishonest provers that are closer to
$V_0$), and the response for $V_1$ from $E_1$ and $\theta$. Thus, it
can be argued as above that the success probability is bounded by
$\varepsilon$ as claimed.
\end{proof}
\subsection{Reducing the Soundness Error}\label{sec:SPreduce}
In order to obtain a position-verification scheme with a negligible
soundness error, we can simply repeat the 1-round scheme
$\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ from Figure~\ref{fig:BasicSP}. Repeating the scheme
$n$ times {\em in sequence}, where the verifiers launch the next
execution only after the previous one is finished, reduces the
soundness error to $\varepsilon^n$. Recall that in the No-PE\xspace modeL defined
in Section~\ref{subsec:model}, the
adversaries must start every round without pre-shared
entanglement. Therefore, the security of the sequentually repeated
scheme follows immediately from the
security of the 1-round scheme.
\begin{corollary}
In the No-PE\xspace model, the $n$-fold sequential repetition of $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ from
Figure~\ref{fig:BasicSP} is $\varepsilon^n$-sound with $\varepsilon = 1-
\h^{-1}(\frac12)$.
\end{corollary}
In terms of round complexity, a more efficient way of repeating
$\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ is by repeating it {\em in parallel}: $V_0$ sends $n$
BB84 qubits $H^{\theta_1}\ket{x_1},\ldots,H^{\theta_n}\ket{x_n}$ and
$V_1$ sends the corresponding bases $\theta_1,\ldots,\theta_n$ to $P$
so that they all arrive at the same time at $P$'s position, and $P$
needs to reply with the correct list $x_1,\ldots,x_n$ in time. This
protocol is
obviously more efficient in terms of round complexity and appears to
be the preferred solution. However, we do not have a proof for the
security of the parallel repetition of $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$.
\subsection{Position Verification in Higher Dimensions}\label{sec:3D}
The scheme $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ can easily be extended into higher
dimensions.
The scheme for $d$ dimensions is a generalization of the scheme $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ in
Figure~\ref{fig:BasicSP}, where the challenges of the verifiers
$V_1$, $V_2$, $\ldots$, $V_d$ form a {\em sum sharing} of the basis
$\theta$, i.e., are random $\theta_1,\theta_2, \ldots, \theta_d \in \set{0,1}$
such that their modulo-2 sum equals $\theta$.
As specified in Figure~\ref{fig:1roundSP}, the state
$H^\theta\ket{x}$ and the shares $\theta_i$ are
sent by the verifiers to $P$ such that they arrive at $P$'s
(claimed) position at the same time. $P$ can reconstruct~$\theta$
and measure $H^\theta\ket{x}$ in the correct basis to
obtain $x' = x$, which he sends to all the verifiers who check if
$x'$ arrives in time and equals $x$.
We can argue security by a reduction to the scheme in 1 dimension.
For the sake of concreteness, we consider 3~dimensions. For 3 dimensions, we need a set of (at least) 4
non-coplanar verifiers $V_0, \dotsc, V_3$, and the prover $P$ needs
to be located inside the tetrahedron defined by the positions of the
4 verifiers.
We consider a coalition of dishonest provers
$P_0,\ldots,P_{\ell}$ at arbitrary positions but different to
$P$. We may assume that
$P_0$ is closest to $V_0$. It is easy to see that there exists a
verifier~$V_j$ such that $d(P_0,V_j) > d(P,V_j)$. Furthermore, we
may assume that $V_j$ is not $V_0$ and thus we assume for
concreteness that it is $V_1$. We strengthen the dishonest
provers by giving them $\theta_2$ and $\theta_3$ for free from the
beginning. Since, when $\theta_2$ and $\theta_3$ are given, $\theta$
can be computed from $\theta_1$ and vice versa, we may assume that
$V_1$ actually sends $\theta$ as challenge rather than~$\theta_1$.
But now, $\theta_2$ and $\theta_3$ are just two random bits,
independent of $\theta$ and $x$, and are thus of no help to the
dishonest provers and we can safely ignore them.
As $P_0$ is further away from $V_1$ than $P$ is, $P_0$ cannot afford to store $H^\theta\ket{x}$ until he has learned $\theta$. Indeed, otherwise $V_1$ will not get a reply in time. Therefore, before she learns $\theta$, $P_0$ needs to apply a quantum transformation to $H^\theta\ket{x}$ with a bi-partite output and keep one part of the output, $E_0$, and send the other part, $E_1$ to $P_1$. Note that this quantum transformation is independent of $\theta$, as long as $P_0$ does not
share an entangled state with the other dishonest provers (who might know $\theta$ by now).
Then, $\hat{x}_0$ and $\hat{x}_1$, the replies that are sent to $V_0$ and $V_1$, respectively, need to be computed from $\theta$ and $E_0$ alone and from $\theta$ and $E_1$ alone. It follows from the analysis of the scheme in one dimension that the probability that both $\hat{x}_0$ and $\hat{x}_1$ coincide with $x$ is at most $\varepsilon = 1- \h^{-1}(\frac12)$.
\begin{corollary}
The above generalization of $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ to $d$ dimensions is
$\varepsilon$-sound in the No-PE\xspace model with $\varepsilon = 1- \h^{-1}(\frac12)$.
\end{corollary}
\section{Position-Based Authentication and Key-Exchange}\label{sec:auth+KE}
In this section we consider a new primitive: position-based
authentication. In contrast to position-verification, where the goal of
the verifiers is to make sure that entity $P$ is at the claimed
location $\text{\sl pos}$, the verifiers want to make sure that a given
message $m$ originates from an entity $P$ that is at the claimed
location $\text{\sl pos}$. We stress that it is not sufficient to first
execute a position-verification scheme with $P$ to ensure that $P$ is
at position $\text{\sl pos}$ and then have $P$ send or confirm $m$, because a
coalition of dishonest provers may do a {\em man-in-the-middle}
attack and stay passive during the execution of the positioning
scheme but modify the communicated message $m$.
Formally, in a position-based authentication scheme the prover
takes as input a message $m$ and the verifiers $V_0,\ldots,V_k$ take
as input a message $m'$ and the claimed position $\text{\sl pos}$ of $P$, and
we require the following security properties.
\begin{itemize
\item{\em $\varepsilon_c$-Completeness: } If $m = m'$, $P$ is honest and at the claimed position $\text{\sl pos}$, and if there is no (coalition of) dishonest prover(s), then the verifiers jointly accept except with probability $\varepsilon_c$.
\item{\em $\varepsilon_s$-Soundness:} For any
$\text{\sl pos} \in \Hull(\text{\sl pos}_0,\dotsc,\text{\sl pos}_k)$ and for any coalition of
dishonest provers $P_0,\ldots,P_\ell$ at locations all different
to $\text{\sl pos}$, if $m \neq m'$, the verifiers jointly reject except with probability $\varepsilon_s$.
\end{itemize}
We build a position-based authentication scheme based on our
position-verification scheme. The idea is to incorporate the message to be
authenticated into the replies of the position-verification scheme. Our
construction is very generic and may also be useful for turning
other kinds of identification schemes (not necessarily
position-based schemes) into corresponding authentication schemes.
Our aim is merely to show the existence of such a scheme; we do not
strive for optimization.
We begin by proposing a weak position-based
authentication scheme for a 1-bit message $m$.
\subsection{Weak 1-bit authentication scheme}
Let ${\sf PV}^\varepsilon$ be a 1-round position-verification scheme between
$k+1$ verifiers $V_0, \ldots, V_k$ and a prover $P$.
For simplicity, we assume that, like for the scheme $\SP_{\text{\rm\tiny \!BB84}}^\varepsilon$ from Section~\ref{sec:basicSP}, $x$ and $x'_0,\ldots,x'_k$ are classical, and ${\sf Ver}$ accepts if $x'_i = x$ for all $i$, and thus we understand the output of ${\sf Resp}(ch_0,\ldots,ch_k)$ as a single element~$x'$ (supposed to be~$x$).
We require
${\sf PV}^\varepsilon$ to have perfect completeness and soundness $\varepsilon < 1$.
We let $\bot$ be some special symbol.
We consider the weak authentication scheme given in Figure~\ref{fig:weakAuth} for a 1-bit message $m \in \set{0,1}$. We assume that $m$ has already been communicated to the verifiers and thus there is agreement among the verifiers on the message to be authenticated. The weak authentication scheme works by executing the 1-round position-verification scheme ${\sf PV}^\varepsilon$, but letting $P$ replace his response $x'$ by $\bot$ with probability $q$, to be specified later.
\begin{figure}[htb]
\small
\begin{protocol}
\begin{enumerate}\setlength{\parskip}{0.1ex}\setcounter{enumi}{-1}
\item
$V_0$ generates $(ch_0,\ldots,ch_k)$ and $x$ using ${\sf Chlg}$ and sends $ch_i$
and $x$ to $V_i$ for $i=1,\ldots,k$.
\item
Every verifier $V_i$ sends $ch_i$ to $P$ in such a way that all
$ch_i$s arrive at the same time at $P$.
\item When the $ch_i$s arrive, $P$ computes the authentication tag $t$ as follows and sends it back to all the verifiers.
\\ If $m = 1$ then $t := {\sf Resp}(ch_0,\ldots,ch_k)$, and if $m = 0$
then $t := \bot$ with probability $q$ and \\ $t :=
{\sf Resp}(ch_0,\ldots,ch_k)$ otherwise.
\item If different verifiers have received different values for $t$, or it didn't arrive in time, the verifiers abort.
\\ Otherwise, they jointly accept if $t = x$ or both $m = 0$ and $t =
\bot$.
\end{enumerate}
\end{protocol}
\caption{Generic position-based weak authentication scheme ${\sf wAUTH}^\varepsilon$ for 1-bit message~$m$. }
\label{fig:weakAuth}
\end{figure}
We analyze the success probability of an adversary authenticating a bit $m' \in \set{0,1}$. We consider the case where there is no honest prover present (we call this an {\em impersonation attack}), and the case where an honest prover is active and authenticates the bit $m \ne m'$ (we call this a {\em substitution attack}).
The following properties are easy to verify and follow from the security property of ${\sf PV}^\varepsilon$.
\begin{lemma}\label{lemma:wAUTH}
Let $P$ be a coalition of dishonest provers not at the claimed
position and trying to authenticate message $m' = 1$. In case of an
impersonation attack, the verifiers accept with probability at most
$\varepsilon$, and in case of a substitution attack (with $m = 0$), the
verifiers accept with probability at most $\delta = (1-q) + q\varepsilon = 1-q(1-\varepsilon) <
1$.
\end{lemma}
On the other hand, $P$ can obviously authenticate $m' = 0$ by
means of a substitution attack with success probability 1; however,
informally, $P$ has bounded success probability in authenticating
message $m' = 0$ by means of an impersonation attack unless he uses
the tag $\bot$. (This fact is used later to obtain a strong
authentication scheme.)
Let us try to extend the above in order to get a strong authentication scheme.
Based on the observation that by performing a substitution attack on ${\sf wAUTH}^\varepsilon$,
it is easy to substitute the message bit $m = 1$ by $m' = 0$ but
non-trivial to substitute $m =0$ by $m' = 1$,
a first approach to
obtain an authentication scheme with good security might be to apply
${\sf wAUTH}^\varepsilon$ bit-wise to a {\em balanced encoding} of the message.
Such an encoding should ensure that for any distinct messages $m$ and $m'$, there are many positions in which the encoding of $m'$ is $1$ but the encoding of $m$ is $0$.
Unfortunately, this is not good enough. The reason is that $P$ and
the verifiers are not necessarily synchronized. For instance, assume
we encode $m = 0$ into $c = 010101...01$ and $m' = 1$ into $c' =
101010...10$, and authentication works by doing ${\sf wAUTH}^\varepsilon$
bit-wise on all the bits of the encoded message. If $P$ wants to
substitute $m = 0$ by $m' = 1$ then he can simply do the following.
He tries to authenticate the first bit $1$ of $c'$ towards the
verifiers by means of an impersonation attack. If he succeeds, which
he can with constant probability, he simply authenticates the
remaining bits $01010...10$ of $c'$ by using $P$, who is happy to
authenticate all of the bits of $c = 010101...01$. Because of
this issue of $P$ bringing $P$ and the verifiers out of sync, we
need to be more careful about the exact encoding we use.
\subsection{Secure Position-Based Authentication Scheme}
We specify a special class of codes, which is strong enough for our purpose.
\begin{definition}\label{def:codes}
Let $c\ \in \{0,1\}^{N}$. A vector $e \in
\{-1,0,1\}^{2{N}}$ is called an {\bf embedding} of $c$ if by
removing all the $-1$ entries in $e$ we obtain $c$.
Furthermore, for two strings $c, c' \in \{0,1\}^{{N}}$ we say
that $c'$ $\mathbf \lambda${\bf-dominates}
$c$ if for all embeddings $e$ and $e'$ of
$c$ and $c'$ (at least) one of the following holds: (a) the number
of positions $i\in \{1,\ldots,2{N}\}$ for which $e_i'=1$ {\em
and} $e_i < 1$ is at least $\lambda$, or
(b) there exist a consecutive sequence of indices $I$ such that
the set $J=\Set{i\in I}{ e'_i > -1}$ has size $|J|\ge 4\lambda$
and it contains at least $\lambda$ indices $i \in J$ with $e_i=-1$.
\end{definition}
For instance, let
$c = 00...0\,11...1$ and
$c' = 11...1\,00...0$, where the blocks of $0$'s and $1$'s are of length $N/2$.
It is not hard to see that the two codewords $N/4$-dominate each other.
However, $\tilde{c}' = 0101...01$ does not dominate
$\tilde{c} = 1010...10$, since $\tilde{c}'$ can be embedded into $\ddag
0101...01\ddag\ddag...\ddag$ and $\tilde{c}$ into
$1010...10\ddag\ddag...\ddag$, where here and later we use $\ddag$ to
represent $-1$.
\begin{definition}
A code $C$ is {\bf $\mathbf \lambda$-dominating}, if any two codewords in $C$ $\lambda$-dominate each other.
\end{definition}
We note that the requirement for $\lambda$-dominating codes can be relaxed
in various ways
to allow a greater range of codes.
Let ${\sf wAUTH}^\varepsilon$ be the above weak authentication scheme satisfying
Lemma~\ref{lemma:wAUTH}. In order to authenticate a message
$m \in \set{0,1}^\mu$ in a strong way (with $\lambda$ a security parameter), an encoding $c$ of $m$ using a
$\lambda$-dominating code $C$ is bit-wise authenticated by means of
${\sf wAUTH}^\varepsilon$, and the verifiers perform statistics over the number of
$\bot$s received.
The resulting authentication scheme is given in
Figure~\ref{fig:genAuth}; as for the weak scheme, we assume that the
message $m$ has already been communicated.
\begin{figure}[htb]
\small
\begin{protocol}
\begin{enumerate}\setlength{\parskip}{0.1ex}\setcounter{enumi}{-1}
\item $\P$ and the verifiers encode $m$ into a codeword $c=(c_1,\dotsc,c_{N})\in C$, for a $\lambda$-dominating code~$C$.
\item\label{it1:repeat} For $j = 1,\ldots,{N}$, the following is repeated in sequence.
\begin{enumerate}[\ref{it1:repeat}.1]\setlength{\parskip}{0.1ex}
\item $P$ authenticates $c_j$ by means of ${\sf wAUTH}^\varepsilon$. Let $t_i$ be the corresponding tag received.
\item\label{it1:check}
If $j > 4\lambda$, the verifiers compute $n_\bot(j) = |\Set{i\in\set{j-4\lambda,\ldots,j}}{c_i=0 \wedge t_i = \bot}|$.
\end{enumerate}
\item
If any of the ${\sf wAUTH}^\varepsilon$ executions fails, or if $n_\bot(j) >
8q \lambda$ for some round $j>4\lambda$, the verifiers jointly
\\ reject. Otherwise, $m$ is accepted.
\end{enumerate}
\end{protocol}
\caption{A generic position-based authentication scheme ${\sf AUTH}$. }
\label{fig:genAuth}
\end{figure}
\begin{theorem}
The generic position-based authentication scheme ${\sf AUTH}$ (Figure~\ref{fig:genAuth}) is
$Ne^{-2q\lambda}$-complete.
\end{theorem}
\begin{proof}
An honest prover which follows the above scheme can fail only if for
some round $r$, $n_\bot > 8q\lambda$. Using the Chernoff
bound~\cite{Chernoff52}, the probability of having $n_\bot >
8q\lambda$ at a specific round $r$, is upper bounded by
$e^{-2q\lambda}$. Using the union bound for every possible round
$j$, we can bound the failure probability with $Ne^{-2q\lambda}$.
\end{proof}
Before we analyze the security of the authentication scheme, let us
discuss the possible attacks on it. We
treat $P$ as a single identity, however $P$ represents a
collaboration of adversaries. Similarly, we refer
the $k+1$ verifiers as a single entity, $V$. We point
out that we do not assume that honest $P$ and $V$ have synchronized
clocks. Therefore, we allow $P$ to arbitrarily schedule and
interleave the ${N}$ executions of ${\sf wAUTH}^\varepsilon$ that $V$
performs with the ${N}$ executions that $P$ performs. The
only restriction on the scheduling is that $P$ and $V$
perform their executions of ${\sf wAUTH}^\varepsilon$ in the specified order.
This means that at any point in time during the attack when $P$ has executed ${\sf wAUTH}^\varepsilon$ for the bits $c_1,\ldots,c_{j-1}$ and $V$ has executed ${\sf wAUTH}^\varepsilon$ for the bits $c'_1,\ldots,c'_{j'-1}$ and both are momentarily inactive (at the beginning of the attack: $j = j' = 1$), \smash{$P$} can perform one of the following three actions. (1) Activate $V$ to run ${\sf wAUTH}^\varepsilon$ on $c'_{j'}$ but not activate $P$; this corresponds to an impersonation attack. (2) Activate $V$ to run ${\sf wAUTH}^\varepsilon$ on $c'_{j'}$ and activate $P$ to run ${\sf wAUTH}^\varepsilon$ on $c_j$; this corresponds to a substitution attack if $c_j \neq c'_{j'}$. (3) Activate $P$ to run ${\sf wAUTH}^\varepsilon$ on $c_j$ but not activate $V$; this corresponds to ``fast-forwarding'' $P$.
We note that \smash{$P$}'s choice on which action to perform
may be adaptive and depend on what he has seen so far.
However, since $V$ and $P$ execute ${\sf wAUTH}^\varepsilon$ for each position within $c$ independently, information gathered from previous executions of ${\sf wAUTH}^\varepsilon$ does not improve $P$'s success probability to break the next execution.
It is easy to see that any attack with its (adaptive) choices of
(1), (2) or (3) leads to embeddings $e$ and $e'$ of $c$ and $c'$,
respectively. Indeed, start with empty strings $e=e'=\emptyset$ and
update them as follows. For each of $P$'s rounds, update $e$ by $e\ddag$ and $e'$ by
$e'c'_{j'}$ if $P$ chooses (1), update $e$ by $ec_j$ and $e'$ by
$e'c'_{j'}$ if he chooses (2), and update $e$ by $ec_j$ and $e'$ by
$e'\ddag$ if he chooses (3). In the end, complete $e$ and $e'$ by
padding them with sufficiently many $\ddag$s to have them of length
$2{N}$. It is clear that the obtained $e$ and $e'$ are indeed
valid embeddings of $c$ and $c'$, respectively.
\begin{theorem}\label{cor:CantAuthDom}
For any $\varepsilon > 0$ and $0 < q < (1-\varepsilon)/8$, the generic
position-based authentication scheme ${\sf AUTH}$
(Figure~\ref{fig:genAuth}) is
$2^{-\Omega(\lambda)}$-sound in the No-PE\xspace model.
\end{theorem}
\begin{proof}
Let $m$ and $m' \neq m$ be the messages input by $P$ and the verifiers, respectively, and let $c$ and $c'$ be their encodings. Furthermore, let $e$ and $e'$ be their embeddings, determined (as explained above) by $P$'s attack.
By the condition on the $\lambda$-dominating code $C$ we know that one of the
two properties (a) or (b) of Definition~\ref{def:codes} holds. If (a) holds, the number of positions
$i \in \set{1,\ldots,2{N}}$ for which $e'_i = 1$ and $e_i \in
\set{-1,0}$ is $\lambda$. In this case, by construction of the
embeddings, in his attack $P$ needs to authenticate (using
${\sf wAUTH}^\varepsilon$) the bit $1$ at least $\lambda$ times (by means of an
impersonation or a substitution attack). By Lemma 2, the success
probability of $P$ is thus at most $\delta^{\lambda}$, which is
$2^{-\Omega(\lambda)}$.
In the case where property (b) holds, there exists a consecutive sequence of indices $I$ such that the set $J=\Set{i\in I}{ e'_i > -1}$ has size $|J|\ge 4\lambda$
and contains at least $\lambda$ indices $i \in J$ with $e_i=-1$.
For
any such index $i \in J$ with $e_i=-1$, $P$ needs to authenticate (using
${\sf wAUTH}^\varepsilon$) the bit $e'_i$ by means of an impersonation attack, while
he may use $\bot$ for (at most) a $8q$-fraction of those $i$'s.
However, by the $\varepsilon$-soundness of ${\sf PV}^\varepsilon$, if we require $\varepsilon
< 1 - 8q$, the probability of $P$ succeeding in this attack is
exponentially small in~$\lambda$.
\end{proof}
A possible choice for a dominating code for $\mu$-bit messages is the {\em balanced repetition code} $C_{\mbox{\tiny\rm $\ell$-BR}}^\mu$, obtained by applying the code
$C_{\mbox{\tiny\rm $\ell$-BR}} = \{00..011..1, 11..100..0\}\subset \{0,1\}^{2\ell}$ bit-wise.
\begin{lemma}
For any $\ell$ and $\mu$, the balanced repetition code $C_{\mbox{\tiny\rm $\ell$-BR}}^\mu$
is $\ell/4$-dominating.
\end{lemma}
\begin{proof}
Let $c,c' \in \set{0,1}^{2\ell\mu}$ be two distinct code words from $C_{\mbox{\tiny\rm $\ell$-BR}}^\mu$, and let $e$ and $e'$ be their
respective embeddings. Note that $c$ is made up of blocks of $0$'s and $1$'s of length $\ell$.
Correspondingly, $e$ is made up of blocks of $0$'s and $1$'s of length $\ell$, with $\ddag$'s inserted at various positions. Let $I_1,\ldots,I_{2\mu}$ be the index sets that describe these 0 and 1-blocks of $e$. In other words, they satisfy: $I_j < I_{j+1}$ element-wise, $|I_j| = \ell$, and $\Set{e_i}{i \in I_j}$ equals $\set{0}$ or $\set{1}$. Furthermore, the sequence of $e_i$'s with $i \in I_1 \cup \ldots \cup I_{2\mu}$ equals $c$, and as such, for any odd $j$, one of $I_j$ and $I_{j+1}$ is a 0-block and one a 1-block.
Let $\phi : \set{1,\ldots,\mu} \to \set{1,\ldots,2\mu}$ be the function such that $I_{\phi(k)}$ is the $k$-th 1-block in $I_1,\ldots,I_{2\mu}$.
The corresponding we can do with $c'$ and $e'$, resulting in blocks $I'_1,\ldots,I'_{2\mu}$ and function $\phi'$. For any $j$, we define $cl(I'_j)$ to be the smallest "interval" in $\set{1,\ldots,4\mu\ell}$ that contains $I'_j$.
For 1-blocks $I_j$ and $I'_{j'}$, we say that $I_j$ {\em overlaps} with $I'_{j'}$ if $|I_j \cap cl(I'_{j'})| \geq 3\ell/4$.
We make the following case distinction.
{\em Case 1:} $I_{\phi(k')}$ does not overlap with $I'_{\phi'(k')}$ for some $k'$.
If all the indices in $I_{\phi(k')} \setminus cl(I'_{\phi'(k')})$ are larger than those in $cl(I'_{\phi'(k')})$,
then $e'_i = 1$ for all $i \in I'_{\phi'(1)} \cup \ldots \cup I'_{\phi'(k')}$ but $e_i < 1$ for at least $\ell/4$ of these $i$'s.
A similar argument can be used when all these indices are smaller than those in $cl(I'_{\phi'(k')})$.
If neither of the above holds, then $e'_i = 1$ for all $i \in I'_{\phi'(k')}$ but $e_i < 1$ for at least $\ell/4$ of these $i$'s. Hence, property (a) of Definition~\ref{def:codes} is satisfied (with parameter $\ell/4$).
{\em Case 2:} $I_{\phi(k)}$ overlaps with $I'_{\phi'(k)}$ for every $k$.
Since $c$ and $c'$ are distinct, and by the structure of the code, there must exist two subsequent 1-blocks $I_{\phi(k)}$ and $I_{\phi(k+1)}$ such that the number of 0-blocks between $I_{\phi(k)}$ and $I_{\phi(k+1)}$ is strictly smaller than the number of 0-blocks between the corresponding 1-blocks $I'_{\phi'(k)}$ and $I'_{\phi'(k+1)}$.
If there is no 0-block between $I_{\phi(k)}$ and $I_{\phi(k+1)}$ and (at least) one 0-block between $I'_{\phi'(k)}$ and $I'_{\phi'(k+1)}$ then by the assumption on the overlap, at least half of the indices $i$ in the 0-block $I'_{\phi'(k)+1}$ satisfy $e_i = \ddag$.
If there is one 0-block between $I_{\phi(k)}$ and $I_{\phi(k+1)}$ and two 0-blocks between $I'_{\phi'(k)}$ and $I'_{\phi'(k+1)}$ then at least a quarter of the indices $i \in I'_{\phi'(k)+1} \cup I'_{\phi'(k)+2}$ satisfy $e_i = \ddag$.
In both (sub)cases, property (b) of Definition~\ref{def:codes} is satisfied (with $\lambda = \ell/4$).
\end{proof}
Plugging in the concrete secure positioning scheme from
Section~\ref{sec:3D}, we obtain a secure realization of
position-based authentication scheme in $\Bbb{R}^d$, in the No-PE\xspace
model.
\subsection{Position-Based Key Exchange}
The goal of a position-based key-exchange scheme is to have the
verifiers agree with honest prover $P$ at location $\text{\sl pos}$ on a key
$K \in \set{0,1}^L$, in such a way that no dishonest prover has
any (non-negligible amount of) information on $K$ beyond its
bit-length $L$, as long as he is not
located at $\text{\sl pos}$.%
\footnote{The length $L$ of the key may depend on the course of the scheme. In particular, an adversary may enforce it to be $0$. }
Formally, we require the following security
properties.
\begin{itemize
\item{\em $\varepsilon_c$-Completeness: } If $P$ is honest and at the claimed position $\text{\sl pos}$, and if there is no (coalition of) dishonest prover(s), then $P$ and $V_0,\ldots,V_k$ output the same key $K$ of positive length, except with probability~$\varepsilon_c$.
\item{\em $\varepsilon_s$-Security:} For any position $\text{\sl pos} \in
\Hull(\text{\sl pos}_0,\dotsc,\text{\sl pos}_k)$ and for any coalition $P$ of
dishonest provers at locations all different to $\text{\sl pos}$, the hybrid
state $\rho_{K E}$, consisting of the key $K$ output by the
verifiers and the collective quantum system of $P$ at the end of
the scheme, satisfies $\delta(\rho_{K E},\rho_{\tilde{K}} \otimes \rho_{E}) \le \varepsilon_s$, where \smash{$\tilde{K}$} is chosen independently and at random of the same bit-length as $K$.
\end{itemize}
Note that the security properties only ensure that the {\em verifiers}
can be convinced that $P$ has no information on the key they obtain;
no such security is guaranteed for $P$.
Indeed,
$P$ can always honestly execute the scheme with $P$,
acting as verifiers. Also note that the security properties do
not provide any guarantee to the verifiers that $P$ has obtained the
\emph{same} key that was output by the verifiers,
in case of an active attack by $P$,
but this feature can always be achieved e.g.\ with the help of a position-based authentication
scheme by having $P$ send an authenticated hash of his key.
A position-based key-exchange scheme can easily be obtained by
taking any quantum key-distribution (QKD) scheme that requires
authenticated communication, and do the authentication by means of a
position-based authentication scheme, like the scheme from the
previous section. One subtlety to take care of is that QKD schemes
usually require {\em two-way} authentication, whereas position-based
authentication only provides authentication from the prover to the
verifiers. However, this problem can easily be resolved as follows. Whenever
the QKD scheme instructs $V_0$ (acting as Alice in the QKD scheme)
to send a message $m$ in an authenticated way to $P$ (acting as
Bob), $V_0$ sends $m$ without authentication to $P$, but in the
next step $P$ authenticates the message $m'$ he has received
(supposedly $m' = m$) toward the verifiers, who abort and output an
empty key $K$ in case the authentication fails.
Using standard
BB84 QKD, we obtain a concrete
position-based key-exchange scheme.
The security of that scheme follows from the security of the BB84
protocol~\cite{LC99,BBBMR00-STOC, SP00,Mayers01,BHLMO05,Renner05} and of the
position-based authentication scheme.
\section{Conclusion and Open Questions} \label{sec:conclusion}
Continuing a very recent line of
research~\cite{Mal10a,Mal10b,CFGGO10,KMS10,Kent10}, we have given a general
proof that information-theoretic position-verification quantum schemes
are impossible, thereby answering an open question about the security
of schemes proposed in~\cite{KMS10} to the negative. On the positive
side, we have provided schemes secure under the assumption that
dishonest provers do not use pre-shared entanglement. Our results
naturally lead to the question: How much entanglement is needed in
order to break position-verification protocols? Can we show security
in the bounded-quantum-storage model~\cite{DFSS05} where adversaries
are limited to store, say, a linear fraction of the communicated
qubits?
\section*{Acknowledgments}
We thank Charles Bennett, Fr\'ed\'eric Dupuis and Louis Salvail for interesting
discussions. HB would like to thank Sandu Popescu for explaining
Vaidman's scheme and pointing~\cite{CCJP10} out to him.
\small
\bibliographystyle{alpha}
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{"url":"https:\/\/xmlamsterdam.com\/gordon-welchman-lstyt\/336c70-determinant-of-permutation-matrix","text":"and is the identity Since the identity matrix is diagonal with all diagonal entries equal to one, we have: $\\det I=1.$ We would like to use the determinant to decide whether a matrix is invertible. A product of permutation matrices is again a permutation matrix. Now with all this information the determinant can be easily calculated. \ufffd*\ufffd7\ufffdN\ufffdD\ufffd\"\/\ufffd\ufffd|+\ufffd\ufffd\ufffd\ufffd&(H)Mnl9\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdQ>\/\ufffd\ufffd\ufffdB=\ufffdd9\ufffd\ufffd\u00fb\ufffd\ufffd\ufffdmj:\ufffd\ufffd|\ufffd \ufffd\ufffd:10s\ufffd5x}41\ufffd\ufffdO\ufffd\ufffdWB\ufffd\ufffd\ufffd\ufffd\ufffd7\ufffd(s[\ufffdG\u03b6J\ufffdR\ufffd\ufffd\ufffdIp \ufffdk\ufffd\ufffdNv]{\ufffd\ufffd%\ufffdB\ufffd\ufffdr\ufffd%w\ufffd{\ufffd)\ufffd\ufffd@\ufffdQ\ufffd\ufffd\ufffd6II\ufffdZ\ufffd,\ufffdR\ufffd8\ufffdRI4y!\u02a9\ufffd\ufffd5\u069b\ufffd'\u0305\ufffdb\ufffd\ufffdg\u016f(\ufffd4@\ufffd\ufffd~v\ufffd{R \ufffd\ufffd\/\ufffd\u064d \ufffd\ufffds\ufffd\ufffd\ufffd\u02a4F\ufffd\ufffd\ufffdx0dP\ufffd\ufffd8\ufffd\ufffd\ufffd:\ufffdC\ufffd\ufffd[\ufffd\ufffd \ufffd\u043b\ufffdG\ufffdN\ufffd\ufffd\u044a\ufffdDd\ufffd\ufffd\ufffd\ufffd\u03edDD\ufffdSL\ufffdid\ufffdev\ufffd\ufffdC\ufffd9K\ufffdz\ufffd\ufffd\ufffd\ufffdM\ufffd2L\ufffd\ufffd\ufffd\"\ufffd\ufffd\ufffd\u430a\\\ufffd\ufffd\ufffd0\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd|\ufffd\ufffdC\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdy$\ufffd:\ufffd\ufffd$\ufffd; \ufffd\ufffdg\ufffdr[\ufffd\ufffd_\ufffd\ufffdw \ufffdca\ufffdd\ufffd%E\ufffdJ\ufffd$\ufffd\ufffd\u073b\ufffdn~\u01f3%\ufffd:b\ufffd\ufffd\u0683$ Unlimited random practice problems and answers with built-in Step-by-step solutions. Determinant of a 2x2 matrix. And we prove this formula with the fact that the determinant of a matrix is a multi-linear alternating form, meaning that if we permute the columns or lines of a matrix, its determinant is the same times the signature of the permutation. A permutation s The permutation matrices of order two are given by, A permutation matrix is nonsingular, and the determinant is always . Analysis. If a matrix order is n x n, then it is a square matrix. The determinant of a matrix is a special number that can be calculated from a square matrix. Definition:the signof a permutation, sgn(\u03c3), is the determinant of the corresponding permutation matrix. Before we look at determinants, we need to learn a little about permutations. column therefore contains precisely a single 1 with 0s everywhere else, and every While such systems may have a unique solution or So this is the determinant of a permutation. Now what is the determinant? permutation corresponds to a unique permutation matrix. New York: Springer-Verlag, p.\u00a0889, 2004. If we derive a formula for the determinant of a 4\u00d74 matrix, it will have 24 terms, each term a product of 4 entries according to a permutation on 4 columns. endobj The #1 tool for creating Demonstrations and anything technical. Join the initiative for modernizing math education. Infinite determinants, i.e. Here, we consider only permutations of finite sets. Knowledge-based programming for everyone. _vT\ufffd\ufffd\ufffd88\ufffd\ufffdH5:\ufffdaz6\ufffdRH\ufffd\ufffd\ufffdC\ufffd\ufffd$\ufffdi\ufffd3P\ufffd\ufffd\ufffde!A~\ufffdr]\u00cfXZ\ufffd\ufffd^[\ufffd\ufffd\ufffdF\ufffd\ufffd\ufffd\ufffd\ufffdW\ufffdvTT\ufffd9+e\ufffd#\ufffdb0\ufffdi\ufffd\ufffd)\u0628\ufffdT\ufffd4t\ufffd[\ufffd\ufffd&\ufffd\ufffds?\ufffd \ufffd\ufffdN?\ufffd \ufffdt\ufffd4Dj\/\ufffd\ufffd\ufffd#\ufffd\ufffd\ufffd\ufffd\u079d\ufffd\ufffd\ufffd\ufffd~Oe\ufffdI\ufffd\ufffd\ufffd%h\ufffd\ufffdg\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdJ7\ufffd\ufffd\ufffd!\ufffd$\ufffd&\ufffd\ufffd\ufffdg\ufffd\ufffd\ufffd&\"Q \ufffd\ufffd\ufffd\ufffd\ufffd]\ufffdA\ufffdw\ufffd\u03e4. Determinant of a 4\u00d74 matrix is a unique number which is calculated using a particular formula. The proof of the following theorem uses properties of permutations, properties of the sign function on permutations, and properties of sums over the \u2026 This quickly becomes impractical, but let's prove it anyways. Now, P is back to standing for permutation. p\ufffd\\*a~\ufffdk\ufffd\ufffd\ufffd\ufffd\ufffdp p\ufffd\ufffd\ufffd\ufffd\ufffd(|n\u0332\"\ufffd\ufffd+p\ufffd$\ufffd(u>\ufffd\ufffdfL;\ufffd:\ufffdeNbM\ufffdl\ufffd terms [since there are n! A 5\u00d75 matrix gives a formula with 120 terms, and so on. Baltimore, MD: Johns Hopkins, p. 109, 1996. We will now look at an application of inversions of permutations with respect to matrix determinants. For example, a permutation of the set $$\\{1,2,3\\}$$ could be 3, 1, 2. Every row and Property 2 tells us that The determinant of a permutation matrix P is 1 or \u22121 depending on whether P exchanges an even or odd number of rows. J\ufffd#\ufffdN\ufffd\ufffd\ufffd\ufffd\ufffdz ^~:\ufffd=\ufffdl|k\ufffd\ufffd\ufffd \ufffd\ufffd\/ \ufffd$\ufffd\ufffdW\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdjS\ufffd\ufffd5\ufffd\/\ufffd\ufffdp\ufffd\ufffd \ufffd\ufffd6b_\ufffd\ufffdF\ufffdO\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdw\ufffdF\ufffdD\ufffd 7 \ufffd\ufffd5\ufffd\ufffdb\ufffd\ufffd\ufffd2\ufffd\ufffd\ufffd5\ufffd6\ufffd\ufffd\ufffd\ufffd\u0141wK\ufffd\ufffdIh\ufffdL \ufffd\ufffdo|8\ufffdc\ufffd[P'\ufffd+\ufffd\ufffd\ufffd\ufffdk\\Kf\ufffdM\ufffd\ufffdtp\ufffdg\ufffdK\\\ufffdR\u00fc\ufffd\ufffd\ufffd^:\ufffd*R\ufffd\ufffdb,\ufffdq\ufffdJ\ufffd\ufffd8\ufffdJ\ufffdZ\ufffdd\ufffdO\ufffd94&r\ufffd\ufffd\ufffd\ufffdr27\ufffdD\ufffd\ufffd\ufffd:\ufffdCV\ufffd\ufffdc\ufffde\ufffd\ufffdZ\ufffd\u0669\ufffdS\ufffd\ufffdN\ufffd\ufffd+h:\ufffdcO\ufffd\ufffdA\ufffdn9\ufffd\ufffd&\ufffd\u0281|\ufffd ^\/\ufffd\ufffd\ufffd\ufffdE^\ufffdL\ufffdAyq\ufffdO\ufffd\ufffd{\ufffd\ufffd\ufffd\ufffdj\ufffdDO\ufffd5h\ufffd\ufffd\ufffd\ufffd\ufffdO\ufffd\ufffdn\ufffdF\ufffd\u01c9\ufffd\ufffd\u022e\ufffd.\ufffdBdD\ufffd\ufffd]R\ufffd27\ufffd\ufffd2\ufffda\ufffd\ufffdc\ufffd\ufffd\ufffdIh\ufffdp8\ufffd From MathWorld--A Wolfram Web Resource. A Matrix is an array of numbers: A Matrix (This one has 2 Rows and 2 Columns) The determinant of that matrix is (calculations are explained later): 3\u00d76 \u2212 8\u00d74 = 18 \u2212 32 = \u221214. However, the permutation matrices provide only a subset of possible solutions. This is because of property 2, the exchange rule. Hints help you try the next step on your own. matrix. \ufffd \ufffd\/\ufffd\ufffd\"\ufffd\ufffd\/R0\u050c\ufffdI \ufffd\ufffd~\ufffd\ufffdJ x\ufffd\ufffd\ufffdn\ufffd\ufffd\ufffd}\ufffdB\ufffdKi\ufffdb8W\ufffdl \u04b4 Z$E\ufffdC\ufffd\ufffdD[De\ufffd!\ufffdx\ufffd_\ufffds\ufffd\ufffdHzd\ufffd\ufffdm\ufffd \ufffdy8\ufffd3\ufffd~\ufffd\ufffd\ufffd\ufffd\ufffdX\ufffd\ufffdUZ\ufffd\ufffd\ufffdX\ufffd;\ufffdVV\ufffd\u0714v\ufffd\ufffd{\ufffd\u04eb\"\ufffdJ\ufffdzL\ufffd\u98f2\ufffd:?\ufffd\ufffdN\ufffd\ufffdj_\ufffd \ufffd\ufffd3\ufffd\ufffdB\ufffd\ufffdJ_\ufffdy\ufffd\ufffd\ufffd\ufffd\ufffdT\ufffd\ufffdV\ufffd\ufffd\u072cJ\ufffdk\ufffd)T^Z\ufffdz\ufffd]\ufffd#\ufffd\ufffdJ\ufffdY\ufffd\ufffdas\ufffd\ufffdP M{\ufffd\ufffd https:\/\/mathworld.wolfram.com\/PermutationMatrix.html. \ufffd>\ufffd$\ufffd@(\ufffd\ufffd$W\ufffd\ufffd),&\ufffd\ufffd\ufffd2\ufffd6\ufffd\ufffd\u061b)I\u027c\ufffd\ufffd u\ufffdr\u0437V\ufffd\ufffd)\ufffd5\ufffd)\ufffdQ\ufffdw\ufffd6UYf?\ufffdMU\ufffd\ufffd\\C \ufffd\ufffdIk50\ufffd A permutation matrix is a matrix obtained by permuting the rows of an identity \/0h\ufffdD\ufffd\/\ufffd\ufffdL\\I\ufffd@\ufffd3\ufffd\ufffdJ\ufffd\ufffd\u0177S\ufffd\ufffd=Hj+(\ufffd\ufffd\u0230\ufffd\ufffd\ufffd\ufffd\ufffd W E\ufffd0S\ufffdCG\ufffdb\ufffdV\ufffdc\ufffd\ufffdwl\ufffd\ufffd\ufffd\ufffd\u07e0(FNGP\ufffd\ufffd\ufffdhJJ&pkD\ufffdJ\ufffd\u0335\ufffd\ufffdR\ufffd\ufffd\ufffd?\ufffd_H\ufffd\ufffd\u052e>\ufffd\ufffd\ufffd\u059b\ufffd Permutation matrices include the identity matrix and the exchange matrix. Interpreting the 1s in an permutation The sum is over the set of all possible permutations . Proper isomorphism between upper and lower ones. Summary A generalized permutation matrix is square ma-trix with at most one nonzero element in each row and column. The determinant of P is just the sign of the corresponding permutation (which is +1 for an even number of permutations and is \u22121 for an odd number of permutations). In particular, the determinant of the identity matrix I is 1. detONE: 7. We\u2019ll form all n! Permutations A permutation of the set S = f 1; 2;:::;n g is a rearrangement of its elements. A permutation matrix is a matrix obtained by permuting the rows of an identity matrix according to some permutation of the numbers 1 to . Operations on matrices are conveniently defined using Dirac's notation. Practice online or make a printable study sheet. Determinants. Applied to a matrix , gives with rows interchanged according to the permutation We\u2019ll add those that correspond to \\even permutations\" \u2026 We noted a distinction between two classes of T\u2019s. permutations of n integers, see Corollary 1.1 here for details if needed]. If the determinant of P is 2, then the asked Dec 5, 2019 in Trigonometry by Rozy ( 41.8k points) The determinant of a matrix with a zero row or column is zero The following property, while pretty intuitive, is often used to prove other properties of the determinant. The coefficient must be the same as the determinant of this matrix, for exactly the same reason. Weisstein, Eric W. \"Permutation Matrix.\" Loosely speaking, a permutation of a set is a specific arrangement of the elements of the set. a) A matrix A is singular if and only if detA = 0. \ufffd\ufffd\ufffdLb\ufffdy\ufffdf\ufffd|Hi+s2\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd[\ufffd\ufffd\ufffd3\ufffds\ufffd;\ufffd\ufffd'9LG\ufffd\ufffdR\u0487c\/i\ufffd\ufffd\ufffdL\ufffd~\ufffda\ufffdT\ufffd1\ufffd\ufffd:U\ufffd7\ufffd\ufffd)Eb\ufffd\ufffd\ufffdE\ufffd\ufffd\ufffdUv\ufffdC\ufffdp\ufffdZ(\ufffdC\ufffd\ufffd9YcB=\ufffd\ufffd*\ufffd1u\ufffd\ufffd\ufffd\ufffdXr\ufffd\ufffd\ufffd\ufffd\ufffdV09; \ufffd;\uf8be\ufffdLc1t\u0688q\ufffd\ufffd\ufffdR\ufffdl'\ufffd\u010bD \ufffd.\ufffd+N\ufffd\ufffd1j\ufffd\ufffd\ufffdl\u06ae\ufffd{f\u02d6+0\ufffd\u0506\ufffdr~Mq\ufffdy;o\ufffd\ufffd\ufffd\ufffd\ufffd?a x*\ufffdse\ufffd-t93\ufffd{\ufffd\ufffdi\ufffd\ufffd3\ufffdd\ufffd}\u039eF\ufffd\ufffdi\ufffdF\ufffd:\ufffdb\ufffdma(N@ v\ufffd\ufffd\u05a6w25\ufffd*\ufffdJ\ufffd6\ufffd\u0605\ufffd' f\ufffdTE\ufffd\ufffdj\ufffd2\ufffd\ufffdN\u05d4,\ufffd\u04c5#\ufffd \ufffd~!\ufffd\ufffd\ufffd\ufffd\ufffd\ufffdj\ufffd%\ufffd(\ufffd\ufffd(\ufffdD\ufffdq\ufffd\ufffdK \ufffdo\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffdRE\ufffd\"j}F\ufffd\ufffd\ufffd\ufffd\ufffdY(M\ufffd{Who\ufffd\u0216\u0453H\"\ufffd\u0550\ufffd=cra~\ufffd\ufffdZ\ufffd \ufffdC\ufffd\ufffd\ufffd\ufffd\ufffd(\ufffdB\ufffd\ufffdO'\ufffd)\ufffdag\ufffdb\ufffda\ufffd\ufffd\ufffdr\ufffd{ \ufffd$\ufffdI?\ufffdS1\ufffd\/kKr\ufffd\ufffdPR\ufffd\ufffd9 n[nq3\ufffdS.U\ufffd Mathematics, 4th ed applied to a unique solution or there are main. 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Matrix as rooks gives an allowable configuration of nonattacking rooks on an chessboard now with this!","date":"2021-10-25 07:26:55","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\": 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\": 0.9525019526481628, \"perplexity\": 827.7061596459332}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587655.10\/warc\/CC-MAIN-20211025061300-20211025091300-00213.warc.gz\"}"}
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The article focuses on carbon offsetting, a strategy to mitigate global warming as well as to help businesses control their energy costs. It mentions the benefits offered by carbon offsets to the environment and how it is done to keep the ecosystem balance. It also stresses the disadvantages of the said strategy which include providing multiple opportunities for abuse resulting in investigations of their implementation.
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Utopia University, known in Chinese as Datong or Tatung University, was a private university in Shanghai. It was established in March 1912 by a group of former Tsinghua faculty members led by Hu Dunfu, and became one of the most reputable private universities in China. After the founding of the People's Republic of China, the Communist government closed Utopia along with many other private universities in 1952. Its departments, faculty members, and students were divided and merged into various universities in Shanghai.
Founding
In April 1911, the American government established Tsinghua College (now Tsinghua University) in Beijing, using part of indemnity payment made by the Qing dynasty in the aftermath of the Boxer Rebellion. In summer, a group of eleven Chinese faculty members at Tsinghua, most of whom were from the Jiangnan region, established the Lida Society () to promote education in China. Hu Dunfu was the head of the society, and other members included Ping Hailan (), Zhu Xiangwan (), and Gu Yangwu (). In November 1911, the eleven members of Lida resigned from Tsinghua after having a disagreement with the American administrators at the college, and left Beijing for Shanghai. Hu Dunfu, the first dean of Tsinghua who had graduated in mathematics from Cornell University, insisted that students of Tsinghua should study more science and engineering, but American teachers, backed by the American ambassador, maintained that they should focus more on the English language and the literature, history and geography of America. They were also angered by the stark contrast in treatment between American and Chinese teachers in Tsinghua; the former were paid nearly ten times as much as the latter, so they were determined to found a university that was academically independent and truly belonged to the Chinese people.
In Shanghai, Hu Dunfu was invited by Ma Xiangbo, his former teacher at the Catholic school Collège Saint-Ignace, to be the dean of Fudan Public School (now Fudan University), which was founded by Ma in 1905. However, Hu soon left Fudan after a student strike at the school. In March 1912, Hu and his colleagues of the Lida Society founded Utopia School () in Nanshi, Shanghai, with Hu as president. The inaugural class had 91 students.
Republican era
In September 1922, Utopia School was certified by the government as a private university, and changed its name to Utopia University. It had 600 students at the time. In the 1920s, Utopia developed into one of the best private universities in China. It was divided into three schools, for literature, science, and business, which were further divided into nine departments. It also established an affiliated secondary school. The campus occupied an area of 90 mu, and had 15 buildings. A 1920s inspection report of six major private universities in Shanghai by the Chinese Ministry of Education praised Utopia and University of Shanghai as the most outstanding. Nationally, Utopia and Nankai University in Tianjin were considered the best private universities in China.
In 1937, the Second Sino-Japanese War erupted, and the Imperial Japanese Army attacked Shanghai in August. During the battle Japanese bombing destroyed many of Utopia's buildings. After the Japanese occupied the Chinese-held areas of Shanghai, Utopia was forced to relocate to the French Concession. It moved several times between temporary accommodations, until settling down on Xinzha Road in September 1939. By this time the university and its affiliated high school had 2,000 students. It also established another secondary school.
The university expanded further after the Second World War, becoming the largest private university in Shanghai. By 1948, it had more than 2,700 university students, and 2,500 students enrolled at its two secondary schools. It had four schools (the school of engineering had been established in addition to the original three) and fourteen departments. Among Utopia's students, 39 later became academicians of the Chinese Academy of Sciences or the Chinese Academy of Engineering.
Demise
In 1949, the Communist Party of China defeated the Kuomintang government in the Chinese Civil War and established the People's Republic of China. In October 1952, the new Communist government closed Utopia along with many other private universities. Its business school was merged into Shanghai University of Finance and Economics, its school of literature became part of the newly established East China Normal University, the school of science was merged into Fudan University, and the school of engineering was divided and became part of Tongji University, Shanghai Jiao Tong University, and the new East China Institute of Chemical Technology (now East China University of Science and Technology). Utopia's two affiliated secondary schools remained, and are now known as Datong High School and Wusi (May Fourth) High School.
Hu Dunfu, Utopia's founder, went to Taiwan along with the defeated Kuomintang government. He attempted to reestablish Utopia University in Taiwan, but was unsuccessful.
References
Defunct universities and colleges in Shanghai
1912 establishments in China
1952 disestablishments in China
Educational institutions established in 1912
Educational institutions disestablished in 1952
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Q: Implementation of an algorithm for simultaneous diagonalization I am trying to write an implementation of an algorithm for the simultaneous diagonalization of two matrices (which are assumed to be simultaneously diagonalizable). However, the algorithm does not seem to converge. The algorithm is described in SIAM J. Matrix Anal. Appl. 14, 927 (1993).
Here is the first part of my code to set up a test case:
import numpy as np
import numpy.linalg as lin
from scipy.optimize import minimize
N = 3
# Unitary example matrix
X = np.array([
[-0.54717736-0.43779416j, 0.26046313+0.11082439j, 0.56151027-0.33692186j],
[-0.33452046-0.37890784j, -0.40907097-0.70730291j, -0.15344477+0.23100467j],
[-0.31253864-0.39468687j, 0.05342909+0.49940543j, -0.70062586+0.05835082j]
])
# Generate eigenvalues
LA = np.diag(np.arange(0, N))
LB = np.diag(np.arange(N, 2*N))
# Generate simultaneously diagonalizable matrices
A = X @ LA @ np.conj(X).T
B = X @ LB @ np.conj(X).T
This should generate two 3x3 matrices which are simultaneously diagonalizable, since they are constructed this way via X. The following code block then defines a few helper functions:
def off2(A, B):
"""Defines the distance from the matrices from
their diagonal form.
"""
C = np.abs(A) ** 2 + np.abs(B) ** 2
diag_idx = np.diag_indices(N)
C[diag_idx] = 0
return np.sum(C)
def Rijcs(i, j, c, s):
"""Function R(i, j, c, s) from the paper, see
Eq. (1) therein. Used for plane rotations in
the plane ij.
"""
res = np.eye(N, dtype=complex)
res[i, i] = c
res[i, j] = -np.conj(s)
res[j, i] = s
res[j, j] = np.conj(c)
return res
def cs(theta, phi):
"""Parametrization for c and s."""
c = np.cos(theta)
s = np.exp(1j * phi) * np.sin(theta)
return c, s
With these definitions, the algorithm can be implemented:
tol = 1e-10
Q = np.eye(N, dtype=complex)
while True:
off = off2(A, B)
# Print statement for debugging purposes
print(off)
# Terminate if the result is converged
if off <= tol * (lin.norm(A, "fro") + lin.norm(B, "fro")):
break
for i in range(N):
for j in range(i + 1, N):
def fij(c, s):
aij = A[i, j]
aji = A[j, i]
aii = A[i, i]
ajj = A[j, j]
bij = B[i, j]
bji = B[j, i]
bii = B[i, i]
bjj = B[j, j]
x = np.array(
[
[np.conj(aij), np.conj(aii - ajj), -np.conj(aji)],
[aji, (aii - ajj), -aij ],
[np.conj(bij), np.conj(bii - bjj), -np.conj(bji)],
[bji, (bii - bjj), -bij ]
]
)
y = np.array(
[
[c ** 2],
[c * s],
[s ** 2]
]
)
return lin.norm(x @ y, 2)
# 5
result = minimize(
lambda x: fij(*cs(x[0], x[1])),
x0=(0, 0),
bounds=(
(-0.25 * np.pi, 0.25 * np.pi),
(-np.pi, np.pi)
),
)
theta, phi = result['x']
c, s = cs(theta, phi)
# 6
R = Rijcs(i, j, c, s)
# 7
Q = Q @ R
A = np.conj(R).T @ A @ R
B = np.conj(R).T @ B @ R
As you can observe from the print statement, the "distance" of A and B from diagonal form does not really converge. Instead, the values printed range from 0.5 up to 3 and oscillate up and down. Is there a bug in this code and if so, where exactly is it?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,209
|
Q: How do I change the default for the "df" command in Unix/Linux? I am in the Unix/Linux Terminal under a specific directory that is owned by another person. In the terminal, I'm trying to change a default for the "df" command. When I type in the "df" command, I see the amount of disk free space listed in KILOBYTES. I want to change the default so that when I type in "df" alone, I will see the disk free space in MEGABYTES. I already know that when I type "df -m" or "df -h" I will see it in MEGABYTES, but I strictly just want to type in "df" so that I can see the values in MEGABYTES. If I can get a step by step on how to make this possible on changing the default for the "df" command to MEGABYTES any any screenshots as to what I can do, I would love that. I was told that command line flags also could affect how this command could work, so I would also like clarification on that as well. I need literally step for step on everything on this.
A: Add an alias somewhere in an initialization script:
alias df='df -m'
So you could add that in ~/.bash_profile or ~/.bashrc or such if you're using bash.
A: create an alias in your bash.rc file like:
alias df='df -m'
A: If it's coreutils df another way is to export the desired size in DF_BLOCK_SIZE or BLOCKSIZE:
export DF_BLOCK_SIZE=1048576
df
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,447
|
{"url":"https:\/\/mailman.ntg.nl\/pipermail\/ntg-context\/2021\/101077.html","text":"# [NTG-context] \\doassign variable - updates not picked up\n\nWerner Hennrich wehe at gmx.at\nFri Jan 22 09:48:58 CET 2021\n\nHello everyone,\n\nI'm using ConTeXt\u00a0 (ver: 2020.09.20 23:02 LMTX) and have to have\nconditional page headers based on specific content criteria. To achive\nthis I'd like to get some global variables in TeX, change their values\nseveral times throughout the document and have my macro for the header\nread them and act accordingly (like skip the header at all, change a\ncolor, etc).\n\nAs a basis for this I established the following pattern in an iolated\ntest case:\n\n}\n\\stopsetups\n\\starttext\npage: 1\n\\page\npage: 2\n\\page\npage: 3\n\\page\npage: 4\n\\page\npage: 5\n\\stoptext\n\nThis works as expected and the header-macro prints the updated value.\nBut when I put the same pattern in my big environment-file, I can see\nthe variable being updated with different values (console), but the\nheader always puts out just the variable's initial value - so the\nPeeking into console, log & tuc didn't raise any idea yet as to what is\ngoing wrong...\n\nAny idea what is happening and how I can fix this?\n\nAny hints on how I can track this down?\n\nThanks a lot,\nWerner Hennrich, Vienna AT\n\n-------------- next part --------------\nAn HTML attachment was scrubbed...\nURL: <http:\/\/mailman.ntg.nl\/pipermail\/ntg-context\/attachments\/20210122\/b5413c6c\/attachment.htm>","date":"2021-05-11 11:32:19","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\": 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\": 0.5061808824539185, \"perplexity\": 11748.179358519528}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991982.8\/warc\/CC-MAIN-20210511092245-20210511122245-00306.warc.gz\"}"}
| null | null |
{"url":"http:\/\/www.physicsforums.com\/showthread.php?t=312173","text":"# Integration by finding limits?\n\nby Ruckstar033\nTags: integration, limits\n P: 290 $$\\int_a^b{f(x)}\\, \\text{d}x = \\lim_{n \\to \\infty}{\\sum_{i=1}^n{f(x_i)} \\Delta x$$ Is the definition of an definite integral. If she said to find the limit then it seems like she doesn't want you to estimate with a large n value, but to actually find the limit of the sum.","date":"2014-09-16 19:32:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9226223230361938, \"perplexity\": 359.93800357924994}, \"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-2014-41\/segments\/1410657119220.53\/warc\/CC-MAIN-20140914011159-00033-ip-10-196-40-205.us-west-1.compute.internal.warc.gz\"}"}
| null | null |
Written By: Luanne Rice
Narrated By: Laural Merlington
Publisher: Brilliance Audio
Abandoned by the father of her child, May Taylor supports herself by her work as a gifted wedding planner with her best friend and great-aunt, carrying on the timeless traditions established by her grandmother and mother. She is a devoted mother to her daughter, Kylie - a very special child. For Kylie sees and hears things no one else can. Her unique visions will lead May to a love she never expected and a life she never imagined. Martin Cartier is a professional hockey player and sports legend. Women everywhere respond to his charm and good looks. But Martin's polished exterior barely hides a core of rage and bitterness. Haunted by his father's negligence and the sudden death of his daughter, he strives to win at all costs. Against all odds, May, Martin, and Kylie become a family. Gradually, May's compassion and Kylie's insight lead Martin to reveal the painful secrets of his past. But when May begins a search for Martin's estranged father, her efforts nearly destroy their relationship. Then Martin faces a threat to his career that his drive and strength cannot overcome. Only Kylie sees the way home - and only May can lead him there.
mac neil
LOVED this one. Got me from the beginning.
by Luanne Rice
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,459
|
\section{Introduction}
Gamma-Ray Bursts (GRBs) are energetic cosmological events, emitting
$\ga10^{51}$ ergs in $\gamma$-ray. A leading model for the central engine
of GRBs is the formation of a black hole (BH) and an accretion disk,
following the gravitational collapse of a massive star whose main sequence mass
($M_{\rm ms}$) is at least as large as $25M_{\odot}$ (for reviews, see
Woosley \& Bloom 2006; Nomoto et al. 2007).
A relativistic flow generated by neutrino annihilation (Woosley 1993; MacFadyen \& Woosley 1999)
or magnetic activity (Brown et al. 2000; Proga et al. 2003)
is proposed to trigger a GRB.
A link between (a class of) GRBs and Type Ic supernovae
(SNe Ic) has been established observationally.
The most convincing cases for the supernovae associated with
GRBs (hereafter GRB-SNe) have been provided by spectroscopic detection of
supernova features in an optical afterglow of a GRB or at the position
consistent with a GRB.
Three nearby GRB-SNe detected in this way are found to be similar to one another.
The category includes GRB 980425/SN 1998bw (the proto-typical GRB-SN;
Galama et al. 1998),
GRB 030329/SN 2003dh (Hjorth et al. 2003; Kawabata et al. 2003;
Matheson et al. 2003; Stanek et al. 2003),
and GRB 031203/SN 2003lw (Malesani et al. 2004; Thomsen et al. 2004).
Optical observations of these GRB-SNe are well explained by an explosion of a
carbon-oxygen (CO) star,
which has evolved from a massive star ($M_{\rm ms} \sim 40M_{\odot}$) and
has lost its H- and He-envelopes during the hydrostatic evolutionary phase
(Iwamoto et al. 1998; Woosley et al. 1999; Nakamura et al. 2001a;
Mazzali et al. 2003, 2006).
The kinetic energy ($E_{\rm K}$) of the expansion is large,
$E_{51} \equiv E_{K}/10^{51}$ ergs $\ga 10$ (note that $E_{51} \sim 1$
for canonical supernovae).
They eject $\sim 0.3 - 0.7M_{\odot}$ of $^{56}$Ni
(which powers the SN luminosity by the decay chain $^{56}$Ni $\to$ Co $\to$ Fe).
Hereafter, the mass of $^{56}$Ni is denoted by $M$($^{56}$Ni).
Recently, another example of the association has been reported (Della Valle et al. 2008;
Soderberg et al. 2008), i.e., GRB 081007/SN Ic 2008hw, while
the observed properties of this SN have not been modeled yet.
Despite the similarity within the well studied cases mentioned above,
GRB-SNe do seem to have diverse properties.
Peak magnitudes of so-called supernova bumps seen in GRB optical afterglows show
diversity (Zeh, Klose, \& Hartmann 2004; Woosley \& Bloom 2006),
highlighted by sub-luminous (possible) SNe in
GRBs 040924 and 041006 (Soderberg et al. 2006).
A few GRBs show no evidence for the supernova bump (Hjorth et al. 2000; Price et al. 2003).
Non-detection of SN features in two nearby GRBs 060505 and
060614 has been reported,
placing the upper limit to brightness of possible underlying
SNe $\sim 100$ times fainter than SN 1998bw
(Della Valle et al. 2006; Fynbo et al. 2006; Gal-Yam et al. 2006).
In spite of the observational constraints, the explosion mechanism of GRBs and GRB-SNe are still unknown.
In particular, how properties of the central engine are related to
the bulk expansion of the stellar materials, observed as a supernova
mainly in visual light,
is still under debate.
A possibility is that a supernova is induced by a disk wind
generated by viscous heating (MacFadyen \& Woosley 1999;
Narayan, Piran, \& Kumar 2001).
Although there were numerical calculations for
the wind/jet-driven explosions (Khokhlov et al. 1999;
MacFadyen, Woosley, \& Heger 2001;
Maeda \& Nomoto 2003; Nagataki et al. 2003; Maeda 2004,
Nagataki, Mizuta, \& Sato 2006; Tominaga 2007a; Tominaga et al. 2007b;
and Tominaga 2009), it has not been clarified what fundamentally determines
the properties of resulting SNe and in what ways.
Also, the numerical investigations have been restricted
in the parameter space.
Aiming to overcome these problems, this study is complementary to the
past numerical studies.
Our goal in this paper is to express theoretically expected
features of SNe resulting from the wind/jet-driven explosion,
as a function of rates of the mass and energy
($\dot M_{\rm w}$ and $\dot E_{\rm w}$, where
the subscript "w" denotes "wind", or $\dot M_{\rm iso}$ and
$\dot E_{\rm iso}$, referring to the isotropic equivalent values) generated and
injected from the central system (i.e., a black hole plus a disk)
into the surrounding stellar mantle
(in this paper, stellar "mantle" refers to the stellar materials above
the central remnant, i.e., the outermost layer of the Fe core,
and the Si- and CO-layers).
Our strategy is the following.
(1) We first clarify what are main ingredients of
the wind/jet-driven SN explosion (\S 2).
We develop a simplified description for the shock propagation and
nucleosynthesis in the explosion (\S 3).
We especially focus on the production of $^{56}$Ni,
addressing how two proposed sites for the $^{56}$Ni production,
a shocked stellar mantle and a disk wind (MacFadyen \& Woosley 1999;
see Maeda \& Nomoto 2003 for a review), can be distinguished.
(2) We then compare the results with observations of SNe
associated with a GRB, in order to
constrain $\dot M_{\rm iso}$ and $\dot E_{\rm iso}$ in these SNe (\S 4).
The required values for $\dot M_{\rm iso}$ and $E_{\rm iso}$ then
should be regarded as conditions that any models for the
central engine should satisfy. In other word,
by constraining $\dot M_{\rm iso}$ and $\dot E_{\rm iso}$,
we aim to provide useful constraints in studying
the properties of the central engine of GRBs.
\section{Models}
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth]{f1.eps}
\caption{A schematic picture of a wind/jet-driven SN explosion.
(a) A situation before the launch of the outward shock wave.
The wind/jet is assumed to be continuously injected
from the central remnant with properties denoted by the
subscript "w" at radius $R_{\rm w}$.
When materials initially at $M_{r}$ in the presupernova progenitor
fall to $R_{\rm w}$, their properties are denoted by
the subscript "m". At this point, the mass of the central remnant
(the mass within $R_{\rm w}$) is $M_{r}$.
(b) A situation after the launch of the outward shock wave. The shock wave
arrives at radius $R_{\rm sw}$, and temperature behind the shock is
denoted by $T$. $^{56}$Ni is produced both in the
shocked stellar materials between $R_{\rm sw}$ and $R_{\rm w}$ and within the
wind/jet itself.}
\label{fig1}
\end{figure*}
We consider a situation that a supernova explosion
is driven by outflow of materials (e.g., a disk wind; MacFadyen \& Woosley 1999)
from the vicinity of a central remnant (likely a black hole).
This energy input can be different from the relativistic jet
producing a GRB; The wind can either be relativistic or non-relativistic
at its injection from the central remnant into the surrounding stellar mantle.
A schematic picture of the problem considered in this paper
(and most of the past numerical studies) is shown in Figure 1.
Throughout this paper,
we adopt $25M_\odot$ and $40M_\odot$ progenitor models from Nomoto
\& Hashimoto (1988). Following the treatment of the past numerical
studies, the wind/jet (we
hereafter frequently call it simply a wind) is injected by hand at a
certain radius ($R_{\rm w}$ in this paper).
\subsection{Properties of the wind/jet}
In a one-dimensional hydrodynamic problem,
three independent variables must be specified (two for thermodynamic variables
and one for a hydrodynamic variable) as initial/boundary conditions.
For simplicity, we assume that the wind is dominated by the kinetic energy.
This does not drastically defeat our results, since it is expected that
the thermal energy deposited at the root of the wind, near the central remnant,
is quickly converted to the kinetic energy well below $R_{\rm w}$.
Thanks to this simplification, only two independent
variables at $R_{\rm w}$ are required to determine the hydrodynamic evolution of the system.
The choice of the independent variables can be arbitrary, thus we take
the energy injection rate ($\dot E_{\rm w}$) and the mass injection rate
($\dot M_{\rm w}$).
The Lorentz factor of the wind/jet at the injection (at $R_{\rm w}$)
is expressed as a function of these
two independent variables, i.e.,
$\Gamma_{\rm w} \sim 1 + \dot E_{\rm w}/(\dot M_{\rm w} c^2)$,
where $c$ is the speed of light.
Hereafter, quantities expressing the properties of the wind
at injection (at $R_{\rm w}$) are denoted by the subscript "w".
Trivially, the wind is initially highly relativistic only if
$\dot E_{{\rm w}, 51}/\dot M_{{\rm w}, \odot} \gg 1000$,
where $\dot E_{{\rm w}, 51} \equiv \dot E_{\rm w}/(10^{51}$ erg s$^{-1})$
and $\dot M_{{\rm w}, \odot} \equiv \dot M_{\rm w}/(M_{\odot}$ s$^{-1})$.
Our analysis is based on a one-dimensional radial flow;
The collimation of the wind is taken into account with a
geometrical factor $f_{\Omega} (\le 1)$.
$f_{\Omega}$ relates the wind
intrinsic properties and isotropic equivalents as
\begin{eqnarray}
\dot E_{\rm w} & = & f_{\Omega} \dot E_{\rm iso} \ , \\
\dot M_{\rm w} & = & f_{\Omega} \dot M_{\rm iso} \ , \\
E_{\rm w} & = & f_{\Omega} E_{\rm iso} \ ,
\end{eqnarray}
where $E_{\rm w}$ is the intrinsic total energy (i.e., the injected energy
integrated over time), and the subscript "iso" refers to isotropic
equivalent values.
$f_{\Omega}$ is a measure of
the collimation angle of the wind at $R_{\rm w}$:
When the wind is a spherically symmetric flow, $f_{\Omega} = 1$.
The wind is more narrowly collimated at the injection for smaller $f_{\Omega}$.
For the temporal history of the wind properties, we focus on the situation that the
wind is injected at $R_{\rm w}$, with $\dot E_{\rm w}$ and $\dot M_{\rm w}$
constant in time for $t < t_{\rm w}$, and then terminated at $t = t_{\rm w}$, so that
\begin{equation}
E_{\rm w} = \dot E_{\rm w} t_{\rm w} \ ({\rm or} \ {\rm equivalently,} \
E_{\rm iso} = \dot E_{\rm iso} t_{\rm w}) \ .
\end{equation}
Some of the following results, however, do not rely on this assumption.
\subsection{Processes involved}
\subsubsection{Collapse to Explosion}
The first function we have to consider is the dynamical effect
(\S 3.1) of the wind to the collapsing stellar mantle;
the overlying stellar mantle continues to collapse onto the central
remnant, and the wind does not always have a sufficiently large momentum to
overcome the ram pressure of the infalling materials (Fig. 1a).
The importance of the dynamical effect to determine the outcome of a wind/jet-driven
supernova explosion was pointed out by Maeda (2004) and numerically
examined by Maeda (2004) and Tominaga et al. (2007b).
Some numerical calculations have included the dynamical effect self-consistently (e.g.,
Maeda \& Nomoto 2003; Maeda 2004; Tominaga et al. 2007b; Tominaga 2009),
while others have not (e.g., Maeda et al. 2002; Nagataki et al. 2003).
\subsubsection{Production of $^{56}$Ni}
Once the shock wave is launched, it propagates outward into the stellar mantle
(Fig. 1b). The kinetic energy is now converted to the thermal energy
following the shock wave propagation.
The temperature behind the shock wave is initially so high
that the nuclear burning converts the initial stellar composition (mostly
oxygen and silicon) mainly to $^{56}$Ni (\S 3.2.1).
The temperature decreases as the shock wave moves outward; once the temperatures drops
below $\sim 5 \times 10^{9}$ K, then the efficiency of the production
of $^{56}$Ni decreases rapidly.
At the same time, the temperature of
the injected wind itself may also be sufficiently high so that a fraction
of this material may be
converted to $^{56}$Ni (\S 3.2.2). In most of the past numerical studies,
the production of $^{56}$Ni in shocked stellar mantle
has been investigated in detail.
The ejection of $^{56}$Ni in the wind has been examined by
a different kind of numerical simulations involving the innermost
part of the collapsing star (e.g., MacFadyen \& Woosley 1999; Nagataki et al. 2007).
However, in such simulations, it is practically
difficult to follow the dynamics of the
shock wave propagating into the stellar mantle; Thus, it has not been yet clear
how the amount of
$^{56}$Ni synthesized in the wind is connected to properties of the
progenitor star and of the resulting supernova.
\subsubsection{Geometry of the ejecta}
Finally, the properties of the wind affect the shape of the supernova ejecta,
as the shock wave propagates through the stellar mantle.
This has been directly examined in the past numerical studies, but restricted in
the parameter space. In this paper,
we derive a simple estimate of the shape, in terms of the properties of the wind
(\S 3.3).
\section{Wind/jet-driven explosions}
\subsection{Collapse to Explosion}
In this section, we examine dynamical effect of the wind
on the collapsing stellar mantle (Fig. 1a).
We follow analysis similar to that given by Fryer \& M\'esz\'aros (2003)
for a standard neutrino-driven delayed explosion for (canonical) supernovae.
We evaluate outcome of the
interaction between the wind and the infalling materials
at radius $R_{\rm w} \sim 8 \times 10^7$ cm (for $M_{\rm ms} = 40M_{\odot}$)
and $1.2 \times 10^8$ cm ($M_{\rm ms} = 25M_{\odot}$),
where the presupernova enclosed mass ($M_r$) is $1.4M_{\odot}$.
As time goes by, materials initially at larger $M_r$ collapses
to the radius $R_{\rm w}$, and add to the mass of the central remnant ($M_{\rm BH}$).
During a whole period before the launch
of the shock, the temporal evolution of the system is thus specified by the
central remnant mass $M_{\rm BH}$ which monotonically increases as a function of time
(i.e., larger $M_{\rm BH}$ for later time).
If the trajectory of the infalling materials is that of free fall, the
density ($\rho_{\rm m}$) and velocity ($v_{\rm m}$) of the infalling
materials at $R_{\rm w}$ are written as
\begin{eqnarray}
\rho_{\rm m} & \sim & \rho_{\rm presn}
\left(\frac{R_{\rm presn}}{R_{\rm w}}\right)^{3/2} \ , \\
v_{\rm m} & \sim & \left(\frac{2 G M_r}{R_{\rm w}}\right)^{1/2} \ .
\end{eqnarray}
The subscript "presn" is used for the pre-collapse initial
values for the material at the mass coordinate $M_r$.
The subscript "m" refers to quantities of the same material
(at $M_r$) when it collapses to $R_{\rm w}$ (Figure 1a).
Figure 2 shows snap shots of the density structure
for different $M_{\rm BH}$ for $M_{\rm ms} = 40 M_{\odot}$.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{f2.eps}
\caption{Density Structure (left vertical axis label)
following the gravitational collapse,
at $M_{\rm BH} = 2.4 M_{\odot}$ (thick-solid line), $3.4 M_{\odot}$
(medium-solid), and $13 M_{\odot}$ (thin-solid). Also shown is the
postshock temperature ($T_{9} \equiv T/10^{9}$ K; right vertical
axis label) as a function of the shock radius, for $M_{\rm BH} =
2.4 M_{\odot}$, $E_{{\rm iso}, 51} = 30$, and
$\dot E_{\rm iso} = 10$ (dashed line) or $1$ (dotted line).
The progenitor model ($M_{\rm ms} = 40 M_{\odot}$)
is from Nomoto \& Hashimoto (1988).
The position within which the postshock temperature is above
$5 \times 10^{9}$ K (horizontal thin-dashed line),
for $M_{\rm BH} = 2.4 M_{\odot}$,
is marked by the open circle ($\dot E_{{\rm iso}, 51} = 10$)
and by the open square ($\dot E_{{\rm iso}, 51} = 1$).
}
\label{fig2}
\end{figure}
The outward shock wave is launched
if the ram pressure of the wind
($P_{\rm w}$) overcomes that of the infalling materials ($P_{\rm m}$), i.e.,
\begin{equation}
P_{\rm w} \sim {\Gamma_{\rm w}}^2 \rho_{\rm w} {v_{\rm w}}^2
\ge P_{\rm m} = \rho_{\rm m} v_{\rm m}^2
\equiv \rho_{\rm m} \bar g \frac{G M_{\rm BH}}{R_{\rm w}} \ .
\end{equation}
Here $M_{\rm BH}$ is the mass of the BH
when the material at $M_r$ ($= M_{\rm BH}$) collapses to $R_{\rm w}$.
Although a numerical constant $\bar g = 2$ in the above estimate,
we find $\bar g = 1/2$ yields a better
representation of a set of numerical simulations (Maeda 2004;
Tominaga et al. 2007b). We assume $\bar g = 1/2$ throughout the paper.
Equation (7) can be expressed in terms of $\dot E_{\rm w}$ and $\dot M_{\rm w}$,
which are related to other properties of the wind/jet as follows.
\begin{eqnarray}
\dot E_{\rm w} & \sim & A_{\rm w} \Gamma_{\rm w} (\Gamma_{\rm w} - 1)
\rho_{\rm w} v_{\rm w} c^2 \ , {\rm and}\\
\dot M_{\rm w} & = & A_{\rm w} \Gamma_{\rm w} \rho_{\rm w} v_{\rm w} \ ,
\end{eqnarray}
where $A_{\rm w}$
is the area subtended by the wind.
The asymptotic expressions for the momentum balance are
derived by substituting equations (8) and (9) into equation (7);
\begin{eqnarray}
\sqrt{\dot E_{\rm w} \dot M_{\rm w}} & \equiv &
f_{\Omega} \sqrt{\dot E_{\rm iso} \dot M_{\rm iso}}
\ge \frac{1}{\sqrt{2}}
A_{\rm w} \rho_{\rm m} \bar g \frac{G M_{\rm BH}}{R_{\rm w}} \nonumber\\
& & {\rm for} \ \Gamma_{\rm w} \sim 1 \ , \\
\dot E_{\rm w} & \equiv & f_{\Omega} \dot E_{\rm iso}
\ge c A_{\rm w} \rho_{\rm m} \bar g \frac{G M_{\rm BH}}{R_{\rm w}} \nonumber\\
& & {\rm for} \ \Gamma_{\rm w} \gg 1 \ .
\end{eqnarray}
The RHS's of equations (10) and (11), except for $A_{\rm w}$ depending on $f_{\Omega}$,
are completely determined by the progenitor structure, as a function of $M_{\rm BH}$
(Fig. 2). The requirement for $\dot E_{\rm w}$ and $\dot M_{\rm w}$
in terms of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$
is shown in Figure 3 as a function of $M_{\rm BH}$ (i.e., as a function of
time). Hereafter, we use a notation $X_{n} \equiv X/10^{n}$ in CGS unit.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{f3.eps}
\caption{The requirement for $\dot E_{\rm iso}$
to initiate the explosion as a function of $M_{\rm BH}$.
For $\Gamma_{\rm w} \sim 1$, the case with
$\dot M_{\rm iso} = 0.1M_{\odot}$ s$^{-1}$ is shown for presentation
(note that the product $\dot E_{\rm iso} \dot M_{\rm iso}$ should exceed
the value corresponding $P_{\rm m}$ if $\Gamma_{\rm w} \sim 1$; equation 10).
Three horizontal dashed lines at $\dot E_{{\rm iso}, 51} = 50$, $10$, and $7$
are shown for illustrating purpose (arrows indicate the position where
the outward shock wave is launched);
Let us assume $M_{\rm ms} = 40 M_{\odot}$, $\Gamma_{\rm w} \sim 1$ and $\dot M_{\rm iso} =
0.1 M_{\odot}$ s$^{-1}$. If $\dot E_{{\rm iso}, 51} = 50$ or $10$,
the outward shock wave is launched at the intersections between the Si- and
CO-layers at $M_{\rm BH} \sim 2.4 M_{\odot}$. On the other hand, if
$\dot E_{{\rm iso}, 51} = 7$, the momentum can not exceed the infalling
ram pressure until nearly the whole CO star collapses to the central remnant
at $M_{\rm BH} \sim 13 M_{\odot}$.
}
\label{fig3}
\end{figure}
If $\Gamma_{\rm w} \sim 1$, it is necessary that the
product $\dot E_{\rm iso} \dot M_{\rm iso}$
should be larger than a specific value given by equation (10), which
corresponds to $P_{\rm m}$ as a function of $M_{\rm BH}$, in order to initiate
the explosion. For $\Gamma_{\rm w} \gg 1$, $\dot E_{\rm iso}$ should exceed the value
given by equation (11), and the requirement becomes independent from $\dot M_{\rm iso}$.
In other ward, once the temporal evolution of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$ is given,
the outward shock wave is launched when $\dot E_{\rm iso} \dot M_{\rm iso}$
(or $\dot E_{\rm iso}$ if $\Gamma_{\rm w} \gg 1$) is higher than the
specific value representing $P_{\rm m}$ (Fig. 3).
Although $P_{\rm m}$ overall decreases as a function of $M_{\rm BH}$
following the density decrease,
jumps are seen at the edges of
the characteristic hydrostatic burning layers.
Assuming freefall, $P_{\rm m} \propto \rho_{\rm presn}
R_{\rm presn}^{3/2} M_{\rm BH}$. Within each layer,
$\rho_{\rm presn}$ drops slowly as a function of $M_{\rm BH}$ and $R_{\rm presn}$,
making $P_{\rm m}$ nearly constant as a function of $M_{\rm BH}$.
On the other hand, at the edges,
$\rho_{\rm presn}$ decreases suddenly, leading to rapidly decreasing $P_{\rm m}$.
As a result, local minimum values appear at the intersection of different layers (Fig. 3).
It infers that the explosion is likely initiated at one of these intersections,
but not within each layer.
This behavior of $P_{\rm m}$ provides an interesting implication.
For illustration purpose,
let us take the case with $M_{\rm ms} = 40M_{\odot}$
and $\Gamma_{\rm w} \sim 1$,
assuming that $\dot E_{{\rm iso}, 51}$ and $\dot M_{\rm iso}$ are
constant in time.
If $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \ga 1$,
then the explosion is initiated below the Fe/Si interface ($M_{\rm BH} < 2.4M_{\odot}$).
If $0.8<\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} < 1$, the explosion
position jumps to the Si/CO interface ($M_{\rm BH} \sim 3.4M_{\odot}$).
Then, if $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \la 0.8$, almost whole CO layer collapses onto the central remnant.
Thus, we conclude that a small difference of the wind properties can
lead to totally different outcome, with the critical value
$\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \sim 0.8 - 1$
(for $\Gamma_{\rm w} \sim 1$) or $\dot E_{{\rm iso}, 51} \sim 70$
(for $\Gamma_{\rm w} \gg 1$).
The critical value depends on $M_{\rm ms}$, and
is smaller for a less massive progenitor
(Fig. 3).
\subsection{Production of $^{56}$Ni}
Once the ram pressure of the wind ($P_{\rm w}$)
overcomes the ram pressure of the infalling material
($P_{\rm m}$), an outward shock wave is launched and explosive
nucleosynthesis takes place.
In this section, we examine production of $^{56}$Ni
as the shock wave passes through the stellar mantle (Fig. 1b).
We first discuss production of $^{56}$Ni in
a stellar mantle heated by the shock wave (\S 3.2.1),
then we comment on production of $^{56}$Ni within the wind/jet (\S 3.2.2).
\subsubsection{Shocked stellar mantle}
The shocked mantle can become the predominant site for
the $^{56}$Ni synthesis, if the shock wave sweeps up a
large amount of the stellar mantle, at least
$\sim 0.1 M_{\odot}$. This inevitably results in
a non-relativistic shock wave,
even if the wind/jet at its emergence from the central system
at $R_{\rm w}$ is highly relativistic.
Let us denote the average velocity and the mass
of the expanding material swept up by the shock wave, by
$\Gamma_{\rm shock}$ and $M_{\rm shock}$, respectively, then
\begin{equation}
\Gamma_{\rm shock} \sim 1 + 6 \times 10^{-4}
\frac{\dot E_{{\rm w}, 51} t}
{M_{{\rm shock}, \odot}} \ ,
\end{equation}
at time $t$ (in second). In order to accelerate
$\sim 0.1 M_{\odot}$ of the stellar mantle materials
to $\Gamma_{\rm shock}=100$,
the explosion energy is required to be higher than $10^{55}$ ergs
We can thus use non-relativistic approximation for
the postshock temperature ($T$) as a function of the radius of the shock wave
($R_{\rm sw}$). For a radiation-dominated
fireball expanding into a uniform medium with the density $\bar \rho$
(Maeda \& Nomoto 2003),
\begin{equation}
T_9 \sim \left\{
\begin{array}{ll}
5.7 \dot E_{{\rm iso}, 51}^{1/6} R_{{\rm sw}, 8}^{-1/3} {\bar \rho_{6}}^{1/12}
& \quad \mbox{for $t<t_{\rm w}$} \ , \\
24 E_{{\rm iso}, 52}^{1/4} R_{{\rm sw}, 8}^{-3/4} & \quad \mbox{for $t \ge t_{\rm w}$} \ .
\end{array}\right.
\end{equation}
This expression explicitly utilizes the assumption that
$\dot E_{\rm iso}$ is constant in time for $t \le t_{\rm w}$
and zero afterward.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{f4.eps}
\caption{The isotropic
mass of $^{56}$Ni synthesized in the shocked stellar mantle
($M_{\rm T_9\ge5}$; solid contours),
for $M_{\rm BH} =2.4M_{\odot}$ and $M_{\rm ms} = 40M_{\odot}$
($13.8M_{\odot}$ CO star).
The dashed aslant line shows $\dot E_{{\rm iso}, 51} =
\dot E_{{t_{\rm w}, 51}}$ ($E_{\rm iso}$) as is defined by equation (14).
Note that the possible value of $\dot E_{\rm iso}$
that leads to $M_{\rm BH} = 2.4 M_{\odot}$ is a function of $\dot M_{\rm iso}$
(see the main text).
Three cases are shown for illustration, assuming the
temporally constant wind injection: If $\Gamma \gg 1$, the required
value for $\dot E_{\rm iso}$ is independent from $\dot M_{\rm iso}$, and
it is required that $\dot E_{{\rm iso}, 51} \sim 70$ (otherwise the outward shockwave
cannot be launched at $M_{\rm BH} = 2.4 M_{\odot}$.)
If $\Gamma_{\rm w} \sim 1$, it is required that the product $\dot E_{\rm iso} \dot M_{\rm iso}$
should exceed a certain value, i.e., $\dot E_{\rm iso} \dot M_{\rm iso} \sim 1$
for the outward shock wave being launched at $M_{\rm BH} = 2.4 M_{\odot}$
(e.g., $\dot E_{{\rm iso}, 51} \sim 10$ if $\dot M_{{\rm iso}, \odot} = 0.1$,
and $\dot E_{{\rm iso}, 51} \sim 1$ if $\dot M_{{\rm iso}, \odot} \sim 1$).
}
\label{fig4}
\end{figure}
Since $M_{\rm BH}$ increases monotonically with time, the density
structure of the stellar mantle is specified for given $M_{\rm BH}$ (Fig. 2).
Therefore, we can estimate the enclosed mass within a sphere in which
the post-shock materials attain a certain temperature
(Fig. 2; see also figure 7 of Maeda \& Nomoto 2003).
$^{56}$Ni is synthesized in the region ($M_{\rm T_9\ge5}$) where $T_9 \ge 5$.
Figure 4 shows $M_{\rm T_9\ge5}$ [$= M$($^{56}$Ni) $/f_{\Omega}$; the isotropic
equivalent value for $M$($^{56}$Ni)]
for $M_{\rm BH} = 2.4M_{\odot}$ and $M_{\rm ms} = 40M_{\odot}$.
$M_{\rm T_9\ge5}$ depends not only on $E_{\rm iso}$ and $\dot E_{\rm iso}$,
but also on $M_{\rm BH}$.
On the other hand, as concluded in \S 3.1, there is a condition in terms of
$\dot E_{\rm iso}$ and $\dot M_{\rm iso}$, to initiate the explosion at given $M_{\rm BH}$:
This indirectly relates $\dot M_{\rm iso}$ to $M_{\rm BH}$.
In order to obtain $M_{\rm BH} = 2.4 M_{\odot}$ and $M_{\rm ms} = 40 M_{\odot}$,
it is required that $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \ga 1$
(for $\Gamma_{\rm w} \sim 1$) or that $\dot E_{{\rm iso}, 51} \ga 70$
(for $\Gamma_{\rm w} \gg 1$).
Vertical dotted lines in Figure 4 explicitly describe the requirement:
(1) for $\Gamma \gg 1$, $\dot E_{{\rm iso}, 51} = 70$,
(2) for $\Gamma \sim 1$ and $\dot M_{{\rm iso}, \odot} = 0.1$,
$\dot E_{{\rm iso}, 51} = 10$, and
(3) for $\Gamma \sim 1$ and $\dot M_{{\rm iso}, \odot} = 1$,
$\dot E_{{\rm iso}, 51} = 1$.
Interestingly, the dependence of $M$($^{56}$Ni) on $\dot E_{\rm iso}$
and $E_{\rm iso}$ (Fig. 4) changes the behavior at a specific
aslant line in the $\dot E_{\rm iso} - E_{\rm iso}$ plane.
The line is obtained by equalizing the two RHS's of equations (13)
and defined as
\begin{equation}
\dot E_{{t_{\rm w}}, 51} (E_{\rm iso})
\equiv \left(\frac{2.9}{T_9}\right)^{-10/3} {\bar \rho_6}^{-1/2}
{E_{{\rm iso}, 51}}^{2/3} \ ,
\end{equation}
with $T_9 = 5$, i.e., the temperature necessary for the production
of $^{56}$Ni. For given $E_{\rm iso}$, $M$($^{56}$Ni) is larger for
larger $\dot E_{\rm iso}$ as long as $\dot E_{\rm iso} \le \dot E_{t_{\rm w}}$,
but saturated for $\dot E_{\rm iso} > \dot E_{t_{\rm w}}$.
The dividing energy injection rate, $\dot E_{t_{\rm w}} (E_{\rm iso})$,
appears, because of the different behavior of the post-shock temperature
before and after $t = t_{\rm w}$ (Maeda 2004).
The mass of $^{56}$Ni is determined by the radius of the outward shock wave
when the temperature drops down to $T_{9} = 5$.
If $\dot E_{\rm iso} \le \dot E_{t_{\rm w}}$ (for given $E_{\rm iso}$),
then the temperature behind the shock wave drops down to $T_{9} = 5$
during the phase when the energy source is still active (i.e.,
$t < t_{\rm w}$). Thus, larger $\dot E_{\rm iso}$ corresponds to
the larger amount of energy contained behind the shock wave,
leading to the larger radius of the shock wave when the condition
$T_{9} = 5$ is satisfied (e.g., see Fig. 2).
On the other hand, the condition
$\dot E_{\rm iso} > \dot E_{t_{\rm w}}$ corresponds to the
case in which all the energy from the central source is liberated
in short time scale such that the temperature is still high
($T_{9} > 5$) at $t = t_{\rm w}$. The following evolution of
the shock wave, which determines $M_{\rm T_9\ge5}$,
is controlled by $E_{\rm iso}$, not by $\dot E_{\rm iso}$.
It has been suggested that $M$($^{56}$Ni) is larger for larger $E_{\rm iso}$
(e.g., Nakamura et al. 2001b), but this holds true
only if the energy is generated and liberated almost
promptly (i.e., $t_{\rm w} \to 0$ and $\dot E_{\rm iso} \to \infty$).
For example, if $\dot E_{{\rm iso}, 51} = 1$, then
$M$($^{56}$Ni) does not depend on the total energy $E_{\rm iso}$ (or $E_{\rm w}$)
(as long as $E_{{\rm iso}, 51} \ga 0.2$) (Fig. 4).
\subsubsection{$^{56}$Ni in the wind/jet}
Mass of the wind is
\begin{equation}
M_{\rm w} = \frac{\dot M_{\rm w}}{\dot E_{\rm w}} E_{\rm w}
\sim \frac{E_{\rm w}}{c^2 (\Gamma_{\rm w} - 1)} \ ,
\end{equation}
which must be smaller than the mass of the accreted
materials, i.e., $M_{\rm w} < M_{\rm BH} - 1.4M_{\odot}$.
Composition of the materials within the disk wind is largely uncertain,
because it depends on the thermal history of the wind
in the vicinity of the central remnant (i.e., well below $R_{\rm w}$).
The deep understanding of the composition of the wind
material requires numerical calculations or analytic investigation
of the innermost part of the collapsing star (e.g., MacFadyen \& Woosley 1999;
Pruet, Thompson, \& Hoffman 2004; Nagataki et al. 2007).
However, it is still possible to make a rough estimate, in order to
see in what regions in the $\dot E_{\rm iso} - \dot M_{\rm iso}$ plane
the disk wind is important.
Assuming that the energy of the wind is initially thermal energy-dominated
near the central remnant, the typical entropy of the wind material
$s \equiv S/(k_{\rm B}/m_{\rm u})$
(where $k_{\rm B}$ and $m_{\rm u}$ are the Boltzmann constant and
atomic unit mass, respectively) is written by
$\dot E_{\rm iso}$ and $\dot M_{\rm iso}$ as follows:
\begin{equation}
s \sim \left\{
\begin{array}{ll}
22 (\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot})^{3/8}
& \quad \mbox{for $\Gamma_{\rm w} \sim 1$} \ , \\
11 \dot E_{{\rm iso}, 52}^{3/4}
& \quad \mbox{for $\Gamma_{\rm w} \gg 1$} \ .
\end{array}\right.
\end{equation}
We find that $s \gg 1$ in the parameter range
of our interest; $s$ can be as small as about unity,
only when either $\dot E_{\rm iso} \dot M_{\rm iso}$ or $\dot E_{\rm iso}$
is very small, but in such a case the whole star collapses onto the central remnant (\S 3.1).
Thus, when $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$ are large enough to
result in a supernova explosion, the wind material
should experience the strong $\alpha$-rich freezeout, leaving mainly $^{4}$He,
not $^{56}$Ni. The strong $\alpha$-rich freezeout is consistent
with numerical calculations of the collapsing star (Nagataki et al. 2007).
In what follows, we take $M$($^{56}$Ni) $ = 0.2 M_{\rm w}$ in the wind
which is typical for the strong $\alpha$-rich freezeout
(e.g., Pruet et al. 2004;
but see \S 5 for further discussion on the uncertainty).
\subsection{Geometry of the ejecta}
The geometry of the bulk supernova materials provide
a useful constraint on the model, since recently more and
more observational data have been available to
address the ejecta geometry (see Wang \& Wheeler 2008 for a review of
spectropolarimetry;
for recent spectroscopy, see Maeda et al. 2008; Modjaz et al. 2008).
Here we give a rough estimate of how the
geometry depends on the wind
parameters. Note that the geometry of the bulk supernova materials
is different from the geometry of the wind at the injection
measured by $f_{\Omega}$; Even if the wind is initially collimated
($f_{\Omega} < 1$), the bulk expansion of the stellar mantle,
as is induced by the wind, can be
less collimated or even quasi-spherical.
The shock breakout time ($t_{\rm sb}$) of the wind/jet
is estimated by
\begin{eqnarray}
t_{\rm sb} & \sim & R_{\rm CO}/v_{\rm shock} \ ,\\
v_{\rm shock} & \equiv &
\left(\frac{2 \dot E_{\rm iso} t_{\rm sb}}{(M_{\rm CO} - M_{\rm BH})}\right)^{1/2} \ .
\end{eqnarray}
Here $R_{\rm CO}$ and $M_{\rm CO}$ are the radius and the mass of the
CO star.
For $M_{\rm ms} = 40M_{\odot}$, these values are $R_{\rm CO} \sim 10^{10}$ cm
and $M_{\rm CO} \sim 13.8M_{\odot}$.
As the wind/jet pushes the stellar mantle, the wind/jet loses
its energy by depositing its energy into the surroundings.
If the energy injection is terminated before
it breaks through the progenitor surface ($R_{\rm CO}$),
a large fraction of the energy of the wind/jet are transferred
to the stellar mantle. This results in a quasi-spherical explosion
of the bulk of the stellar mantle, even if the wind/jet is initially collimated.
This happens if the following condition is satisfied.
\begin{equation}
t_{\rm w} \sim \frac{E_{\rm iso}}{\dot E_{\rm iso}} < t_{\rm sb} \sim
\left(\frac{{R_{\rm CO}^2} (M_{\rm CO} - M_{\rm BH})}{2 \dot E_{\rm iso}}\right)^{1/3} \ .
\end{equation}
On the other hand, if $t_{\rm w} > t_{\rm sb}$, then
a large amount of materials are ejected toward the jet
direction as compared to the equatorial direction, resulting in a
strongly jetted explosion.
\section{Comparison to Observations}
The expected SN properties can be expressed by
$\dot E_{\rm iso}$ and $\dot M_{\rm iso}$ and the temporal evolution
of these, once the other parameters are
specified ($M_{\rm ms}$, $E_{\rm iso}$ or $E_{\rm w}$, and $f_{\Omega}$).
For comparison to the observation presented below,
we examine the simplest case, in which $\dot E_{\rm iso}$ and
$\dot M_{\rm iso}$ are constant in time for $t < t_{\rm w}$ and
zero afterward (i.e., $E_{\rm iso} = \dot E_{\rm iso} t_{\rm w}$).
This is a situation examined in most of the previous numerical studies,
except for MacFadyen et al. (2001) and
Maeda \& Nomoto (2003) who examined the case where the energy injected
by the jet (wind) is connected to the accretion rate to the central
remnant.
First, the momentum balance determines when the outward shock wave is launched,
yielding $M_{\rm BH}$ at this time as a function of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$
(\S 3.1; Figure 3). Then, the analysis presented in
\S 3.2.1 gives us the mass of $^{56}$Ni, as this is given as a function of $M_{\rm BH}$,
$\dot E_{\rm iso}$, and $E_{\rm iso}$ (Figure 4). The mass of $^{56}$Ni in the disk
can be roughly evaluated using the result of \S 3.2.2 (equation 15). Finally,
the typical geometrical feature can be derived as a function of
$\dot E_{\rm iso}$, $E_{\rm iso}$, $M_{\rm BH}$ (\S 3.3; equation 19).
Using the expressions derived in \S 3,
we characterize properties of a SN explosion
as a function of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$
for a given progenitor model ($M_{\rm ms}$).
Other parameters are $f_{\Omega}$,
$E_{\rm w}$ (or $t_{\rm w}$), but these can be
set without large ambiguity by observations for
some GRB-SNe of special interest.
\subsection{SN 1998bw: the origin of $^{56}$Ni}
SN Ic 1998bw is a prototypical SN associated with a GRB.
For SN 1998bw, intensive observational data in the optical
wavelength are available:
modeling these observations
(by one-dimensional radiation transfer calculations;
Iwamoto et al. 1998; Woosley et al. 1999; Nakamura et al. 2001a)
suggests that $E_{\rm iso} \sim 30$ and $M_{\rm ms}
\sim 40 M_{\odot}$.
Adding to this, there exists intensive study for this object
using multi-dimensional radiation transfer calculations
(Maeda 2006a; Maeda et al. 2006bc; Tanaka et al. 2007).
The study suggests that the intrinsic explosion energy is smaller than
the isotropic value by a factor of about 3, inferring that
$f_{\Omega} \sim 0.3$ is a good approximation.
Figure 5a shows the result for $E_{{\rm iso}, 51} = 30$,
$f_{\Omega} = 0.3$, and $M_{\rm ms} = 40M_{\odot}$.
Here, we try to constrain the properties of the wind
generated by the activity of the central engine of SN 1998bw, i.e.,
$\dot E_{\rm iso}$ and $\dot M_{\rm iso}$ (or equivalently, $\dot E_{\rm w}$ and
$\dot M_{\rm w}$).
For SN 1998bw, observations constrain three quantities, the ejecta
mass, $M$($^{56}$Ni), and the shape of the ejecta:
(1) The ejecta contain a large amount of the CO-core materials,
so that $M_{\rm BH} < 10M_{\odot}$, as inferred by the optical light curve
and spectra (\S 3.1).
(2) $M$($^{56}$Ni) $\sim 0.4M_{\odot}$ to explain its peak luminosity (\S 3.2).
(3) The ejecta are suggested to be aspherical, but still a large amount
of materials are ejected into the equatorial direction as inferred especially
by spectra at $\sim 1$ year since the explosion
(Patat et al. 2001; Mazzali et al. 2001; Maeda et al. 2002).
This indicates that the expanding supernova ejecta are quasi-spherical,
rather than extremely bipolar (\S 3.3)
The three conditions are simultaneously satisfied
if $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \ga 1$
and $\dot E_{{\rm iso}, 51} \ga 20$ (for $\Gamma_{\rm w} \sim 1$), or
$\dot E_{{\rm iso}, 51} > 70$ (for $\Gamma_{\rm w} \gg 1$)
(region A in Figure 5b).
In these cases, $^{56}$Ni is predominantly
produced at the shocked stellar mantles.
The wind origin for $^{56}$Ni is disfavored for SN 1998bw.
The wind contribution exceeds the shocked mantle contribution only if
$\dot E_{{\rm iso}, 51} \la 5$ and
$\dot M_{{\rm iso}, \odot} \ga 0.2$ (i.e., region C of Figure 5b).
However, this combination of the parameters results in $t_{\rm w}/t_{\rm sb} > 1$
and an essentially bipolar explosion, as is inconsistent with the observation.
In short, the wind should be massive and long-lived (i.e., large $\dot M_{\rm iso}$
and large $t_{\rm w}$; see equation 15),
in order to produce a large amount of $^{56}$Ni within the wind
(as large as $\sim 0.4 M_{\odot}$). However, such an explosion
with a long-lived energy injection results in the extremely bipolar
explosion, since the outward shock wave can reach the stellar surface
before the energy injection is terminated: This argues against the
wind-origin of $^{56}$Ni in SN 1998bw.
\begin{figure*}
\centering
\includegraphics{f5.eps}
\caption{(a) Estimate of $M$($^{56}$Ni) in the $\dot E_{\rm iso} - \dot M_{\rm iso}$ plane.
The progenitor is the $13.8M_{\odot}$ CO star ($M_{\rm ms} = 40M_{\odot}$), taken
from Nomoto \& Hashimoto (1988).
The other two parameters are set as $E_{{\rm iso}, 51} = 30$ and $f_{\Omega} = 0.3$,
representing the bright GRB-SN 1998bw.
$M$($^{56}$Ni)/$M_{\odot}$ is shown for the shocked stellar mantle (red contours)
and the wind (blue contours).
$M_{\rm BH}$/$M_{\odot}$ is shown by the black-dotted contours.
The shaded region denoted by "Forbidden" is excluded
by the requirement that the ejected mass must not exceed the
available accreted mass budget. The dark shaded region denoted by "Collapse"
corresponds to whole collapse of the CO star.
The green curve shows the line where $t_{\rm w} \sim t_{\rm sb}$.
(b) Expected characteristics of GRB-SNe for $M_{\rm ms} = 40M_{\odot}$.
In each region, "L" denote a rough estimate of the SN peak luminosity
normalized by that of SN 1998bw.}
\label{fig5}
\end{figure*}
\subsection{Diversity of GRB-SNe}
Figure 5b shows the expected characteristics of GRB-SNe for
$E_{\rm iso} = 30$, $f_{\Omega} = 0.3$, and $M_{\rm ms} = 40M_{\odot}$.
A variety of features are predicted for the wind-driven supernovae
depending on $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$.
If $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \ga 1$ ($\Gamma_{\rm w} \sim 1$)
or $\dot E_{{\rm iso}, 51} \ga 70$ ($\Gamma_{\rm w} \gg 1$),
then the resulting SNe should be similar
to SN 1998bw in the ejected mass.
The features are further divided as follows (regions A, B, and C in
Figure 5b):
\begin{description}
\item {\bf (A) SN 1998bw-like -- GRB-SNe 1998bw, 2003dh, and 2003lw: }
In the parameter region A, the wind-driven supernova is similar to
GRB-SN 1998bw, in virtually all the observed characteristics.
The luminosity is similar to
that of SN 1998bw, since $M$($^{56}$Ni) $\sim 0.3 - 0.6M_{\odot}$.
This category can account for GRB-SNe 1998bw, 2003dh, and 2003lw.
The region is relatively large in terms of $\dot E_{\rm iso}$ and
$\dot M_{\rm iso}$, and the features of supernovae are insensitive to
$\dot E_{\rm iso}$ and $\dot M_{\rm iso}$;
This can explain why these three GRB-SNe are
similar in the optical properties.
\item {\bf (B) Sub-luminous SN 1998bw-like -- GRBs 040924 and 041006?: }
In the parameter region B, the expected supernovae are similar to SN 1998bw in the ejected mass
and the geometry, but the difference is seen in $M$($^{56}$Ni). This indicates
that the supernovae has similar optical properties
with SN 1998bw, with the diversity in the luminosity, covering
$\sim 0.3 - 1$ times that of SN 1998bw. This diversity arises
because, in this parameter region,
$\dot E_{\rm iso}$ is smaller than $\dot E_{t_{\rm w}}$, and
thus $M$($^{56}$Ni) is dependent on $\dot E_{\rm iso}$ (\S 3.2.1)
[note that $\dot E_{\rm iso}$ is larger than $\dot E_{t_{\rm w}}$ in region A,
and thus $M$($^{56}$Ni) is independent from $\dot E_{\rm iso}$ unlike region B].
This parameter region B can account for
sub-luminous SNe found as bumps in optical afterglows
of some GRBs (e.g., GRBs 040924 and 041006).
\item {\bf (C) Bipolar SN 1998bw-like: }
In the parameter region C,
a resulting SN is similar to SN 1998bw in its luminosity and ejecta mass,
but the ejecta are more highly beamed than SN 1998bw. We have never directly
observed such peculiar GRB-SNe.
If such a GRB-SN is observed in future,
it will provide a strong evidence of the diverse property of
the central engine, in terms of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$.
Here, $^{56}$Ni is mainly produced within the wind materials.
The bipolar explosion results from the central energy source
being active for a long time period (c.f., equation 19; e.g.,
low $\dot E_{\rm iso}$ smaller than $\dot E_{\rm w}$).
As such, (1) the mass of $^{56}$Ni produced within the shocked stellar mantle
is small, and (2) the wind should be massive to initiate the explosion
(to provide the sufficiently large momentum in the non-relativistic regime).
As a result, the wind contribution dominates over the shocked stellar mantle in
the production of $^{56}$Ni for the (extremely) bipolar explosion case.
\end{description}
If $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \la 1$ ($\Gamma_{\rm w} \sim 1$)
or $\dot E_{{\rm iso}, 51} \la 70$ ($\Gamma_{\rm w} \gg 1$),
then the resulting SNe should intrinsically different from
SN 1998bw in the ejected mass (\S 3.1; regions D and E in Figure 5b). At the low end of
$\dot E_{\rm iso}$ or $\dot M_{\rm iso}$, a supernova can
never be triggered.
\begin{description}
\item {\bf (D) Bipolar SN with the ejected mass $\la 1 M_{\odot}$ -- GRBs 060505 and 060614?: }
In the parameter region D, only a small amount of materials ($< 1M_{\odot}$)
near the surface of the CO layer are ejected.
The supernova ejecta are essentially bipolar in this case.
$^{56}$Ni is mainly originated in the wind.
Thus, $M$($^{56}$Ni) $\propto \dot M_{\rm w}/\dot E_{\rm w}$, ranging from
$M$($^{56}$Ni) $< 10^{-5}M_{\odot}$ to $\sim 0.3M_{\odot}$.
The expected luminosity is thus diverse, depending on $\dot E_{\rm iso}$
and $\dot M_{\rm iso}$.
This may correspond to non-detection of supernova features in
GRBs 060505 and 060614 (which may also be explained by region E below).
\item {\bf (E) No supernova -- GRBs 060505 and 060614?: }
In the parameter region E, the wind injection can never set the outward shock wave.
This case corresponds to whole collapse of the progenitor CO star without
a SN.
\end{description}
\section{CONCLUSIONS AND DISCUSSION}
In this paper, we discussed theoretically
expected characteristics of a supernova driven by the wind/jet.
We found that the resulting supernova features can be categorized
as a function of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$.
Thus, it is possible to constrain the nature of the central engine of GRBs
by observations of associated supernovae.
Results of the past numerical studies can be understood as a limiting case.
For example, Nakamura et al. (2001b) showed that $M$($^{56}$Ni) is
larger for larger $E_{\rm iso}$. In this paper, we have clarified that
it holds true only for the limiting case where $t_{\rm w} \to 0$ and
thus $\dot E_{\rm iso} \to \infty$.
For an another example,
Tominaga et al. (2007b) showed that
$\dot E_{\rm iso}$ determines $M$($^{56}$Ni). In this paper,
we have shown that this behavior appears as a limiting case where
$\Gamma \gg 1$, and the dependence is different when $\Gamma \sim 1$.
For SN 1998bw, we find that observations are reproduced only if
$\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot} \ga 1$
and $\dot E_{{\rm iso}, 51} \ga 20$ (if $\Gamma_{\rm w} \sim 1$), or
$\dot E_{{\rm iso}, 51} \ga 70$ (if $\Gamma_{\rm w} \gg 1$) (region A in Fig. 5
b).
We favor the shocked stellar mantle as the main site of the production of
$^{56}$Ni.
Furthermore, the observed diversity of supernovae associated (or not associated) with GRBs
can be accounted for by the diverse properties of the wind from the central remnant
($\dot E_{\rm iso}$ and $\dot M_{\rm iso}$).
It is shown that the different wind properties
can potentially explain the diversity of supernovae
associated with GRBs.
Further diversity can arise from different progenitors ($M_{\rm ms}$).
Irrespective of $M_{\rm ms}$, the expected features of the wind-driven supernovae
can be categorized as we did for $M_{\rm ms} = 40 M_{\odot}$ (\S 4), i.e.,
regions A -- E.
The positions of the boundaries between different regions, as well as
maximum $M$($^{56}$Ni) at the high end of $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$,
are dependent on $M_{\rm ms}$.
For $M_{\rm ms} = 25M_{\odot}$, the boundary between regions A and D
is located at $\dot E_{{\rm iso}, 51} \dot M_{{\rm iso}, \odot}
\sim 0.05$ (for $\Gamma_{\rm w} \sim 1$) or $\dot E_{{\rm iso}, 51} \sim 20$
(for $\Gamma_{\rm w} \gg 1$),
smaller than for $M_{\rm ms} = 40M_{\odot}$ by about one order of magnitude (Fig. 3).
Using $f_{\Omega} = 0.3$ and $E_{{\rm iso}, 51} = 30$,
maximum $M$($^{56}$Ni)
is $\sim 0.24M_{\odot}$ even if $M_{\rm BH} = 1.4M_{\odot}$.
Thus, $M_{\rm ms} = 25M_{\odot}$ never yield bright SN 1998bw-like SNe.
We therefore confirmed that the prototypical GRB-SN 1998bw and the similar SNe
2003dh and 2003lw must have originated from a massive progenitor ($M_{\rm ms} \sim 40 M_{\odot}$)
from the nucleosynthesis argument.
Also, the diversity arising from $M_{\rm ms}$ may be
important to explain the sub-luminous supernovae
possibly associated with GRBs 040924 and 041006.
These can be explained by $M_{\rm ms} \sim 40 M_{\odot}$
with smaller $\dot E_{\rm iso}$ and $\dot M_{\rm iso}$ than for
SN 1998bw. Alternatively, it is also possible that
the central engine provides $\dot E_{\rm iso}$ and
$\dot M_{\rm iso}$ similar to SN 1998bw, but the progenitor mass is smaller than SN 1998bw
(i.e., region A for $M_{\rm ms} = 25 M_{\odot}$).
It is thus important to derive $M_{\rm ms}$ for these sub-luminous cases,
by spectroscopic and light curve modeling,
to distinguish these possibilities.
On the other hand, the diversity in terms of $M_{\rm ms}$ is not important
in interpreting no detection of supernovae associated with GRBs 060505 and 060614.
If they are the outcome of the core collapse of
a massive star, the properties of the wind/jet
should fall in the low end of $\dot E_{\rm w}$ and $\dot M_{\rm w}$ (region E),
or region D with $\dot M_{{\rm iso}, \odot} \la 10^{-3}$.
If this is not the case, we should have detected associated supernovae.
Note that Tominaga et al. (2007b) showed that the wind/jet-driven explosion
with small $\dot E_{\rm iso}$ "can" account for the non-detection of
supernovae in these GRBs, while this study
has shown that $\dot E_{\rm iso}$ and/or $\dot M_{\rm iso}$ "must" be small
in the central engine of these GRBs.
These GRBs highlight our suggestion that we can constrain the properties of
the central engine by observations (non-detection, in this case) of
associated supernovae.
The present analysis is based on one-dimensional calculations.
We predict that the different categories (A--E) are divided
rather sharply in terms of $\dot M_{\rm iso}$ and $\dot E_{\rm iso}$,
while in reality this may well be smoothed by the
jet-accretion interaction (Maeda \& Nomoto 2003; Tominaga et al. 2007).
Although the jet-accretion interaction should create a non-radial flow,
key conclusions in the present paper are not defeated.
For example, our model can explain the qualitative behaviors found
in the past numerical study, and can reach at least rough quantitative agreement
for the important quantity like $M$($^{56}$Ni) (for limited parameter space
as investigated by numerical study; see also Maeda \& Nomoto 2003).
The largest uncertainty involved in the present analysis is in the treatment
of nucleosynthesis within the wind material, specifically, the assumption that
20\% of the wind materials become $^{56}$Ni. Surman, McLaughlin, \& Hix (2006) concluded
that the $^{56}$Ni mass fraction could be as large as $\sim 50$\% for the wind/jet
with the entropy in the range between 10 and 30. This does not affect our conclusion
for $\Gamma_{\rm w} \gg1$, as the mass of the wind materials is anyway
small in this case. The only conclusion possibly affected by this uncertainty
is the supernova feature in the region B. From equation 16, we see that the entropy
of the wind materials falls into the range $s = 10 - 30$ if
$\dot E_{{\rm iso},51} \dot M_{{\rm iso}, \odot} \sim 1$.
This line crosses the vicinity of region B, and lead to the contribution of
the wind materials for the $^{56}$Ni production comparable to that of
the shocked mantle in this region. As such, $M$($^{56}$Ni) is this region
may have a significant contribution from the wind materials.
\section*{Acknowledgments}
This research has been supported by World Premier
International Research Center Initiative (WPI Initiative),
MEXT, Japan, and partly by the Grant-in-Aid for Young Scientists
of the JSPS/MEXT (20840007).
N.T. is supported by the JSPS postdoctoral fellowship.
The authors would like to thank Ken'ichi Nomoto, Hideyuki Umeda,
Masaru Shibata, and Yuichiro Sekiguchi for useful discussion.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Windows Installer 4.5 for 64 bit How can i install windows installer for 64 bit? I am getting the following error while building the setup file in InstallShield Limited Edition.
A: As for your errors, they can be caused by something as simple as having the directory open in an explorer window (try closing it a rerun). And have a look at this old IS KB Article
Another possible cause is your Setup PreRequisite file, see this link
"Error appears (in Installshiled 12) if during editing a .prq in Setup Prerequisite Editor select the check-box "Requires Windows Installer engine and/or .NET Framework to be installed first" and after deselect this option. After that in prq-file appears empty section "dependencies", which incorrectly, probably, processed by the builder. It is necessary to remove this section (in any text editor) to avoid an error. "
With regards to your warning:- Have you setup the software id tag ? the following is taken from here Especially read the part I've put in bold.
To include a software identification tag in your installation:
*
*In the View List under Installation Information, click General Information.
*In the Software Identification Tag area of the view, modify the values of the settings as needed.
*The Use Software Identification Tag setting lets you specify whether you want to include a tag in your installation. Select Yes, which is the default value, and then configure the other settings in the Software Identification Tag area as needed.
When you use tagging in your project, InstallShield adds the tag to two new components that it creates, and it associates the components with one of your project's features. The components are:
*
*ISO19770_LocalTag, which has a destination of INSTALLDIR
*ISO19770_SystemTag, which has a destination of CommonAppDataFolder
Use the Setup Design view if you want to associate these components with a different feature in your project. For more information, see Component-Feature Associations.
At build time, if the following conditions are true, InstallShield includes the software identification tag with the installation that it builds:
*
*Yes, the default value, is selected for the Use Software Identification Tag setting in the General Information view.
*The Unique ID, Tag Creator, and Tag Creator ID settings in the General Information view have values.
Note that if tagging is enabled but you have not entered values in one or more of the three aforementioned tag identification settings, InstallShield generates a build warning to inform you that the tag could not be included in your release. To resolve this warning, configure the settings in the Software Identification Tag area of the General Information view as needed.
If you configure your project to include a software identification tag and you also configure the release in the Releases view to use a .pfx file to digitally sign your release, InstallShield digitally signs the tag at build time. Note that the .NET Framework 2.0 or later must be installed on your build machine in order to sign a tag file.
A: Leaving the stuff below, but looking closer at your screenshot it looks like it's claiming certain files aren't in folders where it's looking... I don't think that has anything to do with installing the file to your system as your system isn't Vista, XP, AND Server 2008 (at least I would assume you aren't running three OS at the same time lol). I could be wrong, but this sounds like an import problem... it looks like it's trying to import those files as files your user needs to install your program... you need to get those 3 installers and then import them (package them, whatever this program you are using does) as prerequisites. That SHOULD solve the problem.
--------------Probably not applicable but if above is incorrect----------
Simple suggestions... unlikely, but my usual troubleshooting steps when all else fails.
Check your file paths.
Double check that you ARE installing the version for 64bit.
Your harddrive isn't full is it?
Turn off virus scanners.
Try installing in safe mode.
Run a virus scan (use something good like Malwarebytes).
Check the Windows error log.
If you downloaded it on a different device try downloading it on the computer you want to install it to.
Download the installer on a different device.
Run as Admin or from a new user account.
Change the folder you are running the installer out of... try running the installer right out of c:\
Install it through Windows Updates (I'm just assuming Win 7 has them, I avoid windows update).
Make sure you have all the .NET frameworks installed (I've seen odd things happen when you don't)
Run the installer on a different computer to make sure it works.
Found this through Google... http://winhlp.com/node/40 the bottom of the page shows some software that can contribute to this error... it may say 'server' but I'm sure that even if the computer isn't networked these can still cause the problem...
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: angular 9 execute subscribe in code behind synchronously I need to run a method with 2 parameters, each parameter is gotten through some form of subscribe function. the first is the collection which is gotten through the url from angular's page routing. The second is the dokument, this is the firebase's firestore document.
export class FirebaseDocument implements OnInit {
collection: string;
dokument: any;
//== CONSTRUCTORS
constructor(
private route: ActivatedRoute,
private _db: AngularFirestore
) {}
//== Initialize
ngOnInit() {
console.log("__loading page component");
this.route.params.subscribe(params => {
this.collection = params["collection"];
});
console.log(this.collection);//collection populated correctly
//load the document from AngularFirestore
console.log("loading the document from firebase");
let itemsCollection = this._db.collection(url).valueChanges();
//subscribe to get the dok of the first document in the collection
itemsCollection.subscribe(docArr => {
this.dokument = docArr[0];
console.log(this.dokument);//dokument is populated
});
console.log(this.dokument);//dokument is undefined
this.doMultiParameterMethod(this.collection, this.dokument);
}
}
this.collection populates perfectly fine;
this.dokument is only populated inside the subscribe method
I need this to be populated by the time the next line is run. the console.log(this.dokument);
I have been dumbstruck by this because essentially the same code is used by the 2 subscribe methods but they don't behave the same way.
A: Sometimes a subscribe can be synchronous. This happens when the Observable is a ReplaySubject a BehaviorSubject or an Observable which has a shareReplay() pipe. (probably other options as well.
This will make the observable immediately fire on subscription. However, you should never count on this behavior, and always continue within your subscribe.. Or use pipes like mergeMap and create other observables which you can access in your template using the async pipe.
In your case. The this.route.params is obviously a 'replaying' Observable from which you get the latest value after subscribing. Otherwise you would have to wait for the params to change again until you get a value.
Your Database call cannot return an immediate response, because it's essentially a network request.
In your example code, you can update it to this, and use the async pipe in your template
export class FirebaseDocument implements OnInit {
readonly collection$: Observable<string> = this.route.params.pipe(
map((params) => params.collection)
);
readonly doc$: Observable<any[]> = this.db.collection(this.url).valueChanges().pipe(
shareReplay({ refCount: true, bufferSize: 1 })
);
constructor(private route: ActivatedRoute, private db: AngularFirestore) {}
ngOnInit() {
// don't forget to unsubscribe
combineLatest([
this.collection$,
this.doc$
]).subscribe((collection, document) => {
this.doMultiParameterMethod(collection, document);
});
}
}
A: Maybe you should make the Observable a Promise, in your case would be the following :
export class FirebaseDocument implements OnInit {
collection: string;
dokument: any;
//== CONSTRUCTORS
constructor(
private route: ActivatedRoute,
private _db: AngularFirestore
) {}
//== Initialize
ngOnInit() {
console.log("__loading page component");
this.route.params.subscribe(params => {
this.collection = params["collection"];
});
console.log(this.collection); //collection populated correctly
this.getDokument().then(docArr => {
this.dokument = docArr[0];
this.doMultiParameterMethod(this.collection, this.dokument);
});
}
getDokument(): Promise<any> {
let itemsCollection = this._db.collection(url).valueChanges();
return new Promise((resolve, reject) => {
itemsCollection.subscribe((response: any) => {
resolve(response);
}, reject);
});
}
}
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,374
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\section{Introduction}
Feed-forward deep neural networks (DNNs) are function approximators wherein weighted combinations inputs are filtered through nonlinear activation functions that are organized into a cascade of fully connected (FC) hidden layers. In recent years DNNs have become the tool of choice for many research areas such as machine translation and computer vision
The objective function for training a DNN is highly non-convex, leading to numerous obstacles to global optimization \cite{choromanska2015loss}, notably proliferation of saddle points \cite{dauphin2014identifying} and prevalence of local extrema that offer poor generalization off the training sample \cite{chaudhari2016entropy}. These observations have motivated regularization schemes to smooth or simplify the energy surface, either explicitly such as weight decay \cite{krogh1991simple} or implicitly such as dropout \cite{srivastava2014dropout} and batch normalization \cite{ioffe2015batch}, so that the solutions are more robust, \emph{i.e. } better generalized to test data.
Training algorithms face many numerically difficulties that can make it difficult to even find a local optimum. One of the well-known issues is so-called vanishing gradient in back propagation (chain rule differentiation) \cite{chapter-gradient-flow-2001}, \emph{i.e. } the long dependency chains between hidden layers (and corresponding variables) tend to drive gradients to zero far from the optimum. This issue leads to very slow improvements of the model parameters, an issue that becomes more and more serious in deeper networks \cite{glorot2010understanding}. The vanishing gradient problem can be partially ameliorated by using non-saturating activation functions such as rectified linear unit (ReLU) \cite{lecun2015deep}, and network architectures that have shorter input-to-output paths such as ResNet \cite{he2016deep}.
The saddle-point problem has been addressed by switching from deterministic gradient descent to stochastic gradient descent (SGD), which can achieve weak convergence in probability \cite{bottou2016optimization}.
Classic proximal-point optimization methods such as the alternating direction method of multipliers (ADMM) have also shown promise for DNN training \cite{taylor2016training,Zhang_2016_CVPR}, but in the DNN setting their convergence properties remain unknown.
{\bf Contributions:} In this paper,
\begin{enumerate}[label*=\arabic*.]
\item We propose a novel Tikhonov regularized multi-convex formulation for deep learning, which can be used to learn both dense and sparse DNNs;
\item We propose a novel block coordinate descent (BCD) based learning algorithm accordingly, which can guarantee to globally converge to stationary points with R-linear convergence rate of order one;
\item We demonstrate empirically that DNNs estimated with BCD can produce better representations than DNNs estimated with SGD, in the sense of yielding better test-set classification rates.
\end{enumerate}
Our Tikhonov regularization is motivated by the fact that the ReLU activation function is equivalent to solving a smoothly penalized projection problem in a higher-dimensional Euclidean space. We use this to build a Tikhonov regularization matrix which encodes all the information of the networks,
\emph{i.e. } the architectures as well as their associated weights. In this way our training objective can be divided into three sub-problems, namely, (1) Tikhonov regularized inverse problem \cite{willoughby1979solutions}, (2) least-square regression, and (3) learning classifiers. Since each sub-problem is convex and coupled with the other two, our overall objective is multi-convex.
Block coordinate descent (BCD) is often used for problems where finding an exact solution of a sub-problem with respect to a subset (block) of variables is much simpler than finding the solution for all variables simultaneously \cite{nesterov2012efficiency}. In our case, each sub-problem isolates block of variables which can be solved easily (\emph{e.g. } close-form solutions exist).
One of the advantages of our decomposition into sub-problems is that the long-range dependency between hidden layers is captured within a sub-problem whose solution helps to propagate the information between inputs and outputs to stabilize the networks (\emph{i.e. } convergence). Therefore, {\em it does not suffer from vanishing gradient at all.}
In our experiments, we demonstrate the effectiveness and efficiency of our algorithm by comparing with SGD based solvers
\subsection{Related Work}
{\bf (1) Stochastic Regularization (SR) \emph{vs. } Local Regularization \emph{vs. } Tikhonov Regularization:} SR is a widely-used technique in deep learning to prevent the training from overfitting. The basic idea in SR is to multiple the network weights with some random variables so that the learned network is more robust and generalized to test data. Dropout \cite{srivastava2014dropout} and its variants such like \cite{kingma2015variational} are classic examples of SR. Gal \& Ghahramani \cite{gal2015modern} showed that SR in deep learning can be considered as approximate variational inference in Bayesian neural networks
Recently Baldassi \emph{et al. } \cite{baldassi2015subdominant} proposed smoothing non-convex functions with local entropy, and latter Chaudhari \emph{et al. } \cite{chaudhari2016entropy} proposed Entropy-SGD for training DNNs. The idea behind such methods is to locate solutions locally within large flat regions of the energy landscape that favors good generalization. In \cite{chaudhari2017deep} Chaudhari \emph{et al. } provided the mathematical justification for these methods from the perspective of partial differential equations (PDEs)
In contrast, our Tikhonov regularization tends to smooth the non-convex loss {\em explicitly, globally, and data-dependently}. We deterministically learn the Tikhonov matrix as well as the auxiliary variables in the ill-posed inverse problems. The Tikhonov matrix encodes all the information in the network, and the auxiliary variables represent the ideal outputs of the data from each hidden layer that minimize our objective. Conceptually these variables work similarly as target propagation \cite{bengio2014auto}.
{\bf (2) SGD \emph{vs. } BCD:}
In \cite{bottou2016optimization} Bottou \emph{et al. } proved weak convergence of SGD for non-convex optimization. Ghadimi \& Lan \cite{ghadimi2013stochastic} showed that SGD can achieve convergence rates that scale as $O\left(t^{-1/2}\right)$ for non-convex loss functions if the stochastic gradient is unbiased with bounded variance, where $t$ denotes the number of iterations.
For non-convex optimization, the BCD based algorithm in \cite{xu2014globally} was proven to converge globally to stationary points. For parallel computing another BCD based algorithm, namely Parallel Successive Convex Approximation (PSCA), was proposed in \cite{razaviyayn2014parallel} and proven to be convergent.
{\bf (3) ADMM \emph{vs. } BCD:}
Alternating direction method of multipliers (ADMM) is a proximal-point optimization framework from the 1970s and recently championed by Boyd \cite{boyd2011distributed}. It breaks a nearly-separable problem into loosely-coupled smaller problems, some of which can be solved independently and thus in parallel. ADMM offers linear convergence for strictly convex problems, and for certain special non-convex optimization problems, ADMM can also converge \cite{nishihara2015general,wang2015global}.
Unfortunately, thus far there is no evidence or mathematical argument that DNN training is one of these special cases. Therefore, even though empirically it has been successfully applied to DNN training \cite{taylor2016training,Zhang_2016_CVPR}, it still lacks of convergence guarantee.
Our BCD-based DNN training algorithm is also amenable to ADMM-like parallelization. More importantly, as we prove in Sec. \ref{sec:convergence}, it will converge globally to stationary points with R-linear convergence.
\section{Tikhonov Regularization for Deep Learning}
\subsection{Problem Setup}
{\bf Key Notations:} We denote $\mathbf{x}_i\in\mathbb{R}^{d_0}$ as the $i$-th training data, $y_i\in\mathcal{Y}$ as its corresponding class label from label set $\mathcal{Y}$, $\mathbf{u}_{i,n}\in\mathbb{R}^{d_n}$ as the output feature for $\mathbf{x}_i$ from the $n$-th ($1\leq n\leq N$) hidden layer in our network, $\mathbf{W}_{n,m}\in\mathbb{R}^{d_n\times d_m}$ as the weight matrix between the $n$-th and $m$-th hidden layers, $\mathcal{M}_n$ as the input layer index set for the $n$-th hidden layer, $\mathbf{V}\in\mathbb{R}^{d_{N+1}\times d_N}$ as the weight matrix between the last hidden layer and the output layer, $\mathcal{U}, \mathcal{V}, \mathcal{W}$ as nonempty closed convex sets, and $\ell(\cdot,\cdot)$ as a {\em convex} loss function.
{\bf Network Architectures:} In our networks we only consider ReLU as the activation functions. To provide short paths through the DNN, we allow {\em multi-input ReLU} units which can take the outputs from multiple previous layers as its inputs.
\begin{wrapfigure}{r}{0.35\linewidth}
\vspace{-27pt}
\begin{center}
\includegraphics[width=\linewidth]{network.eps}\vspace{-10pt}
\caption{\footnotesize Illustration of DNN architectures that we consider in the paper.}
\label{fig:network}
\end{center}
\vspace{-15pt}
\end{wrapfigure}
Fig. \ref{fig:network} illustrates a network architecture that we consider, where the third hidden layers (with ReLU activations), for instance, takes the input data and the outputs from the first and second hidden layers as its inputs. Mathematically, we define our multi-input ReLU function at layer $n$ for data $\mathbf{x}_i$ as:
\begin{align}\label{eqn:mir}
\hspace{-2mm}\mathbf{u}_{i,n} = \left\{
\begin{array}{ll}
\mathbf{x}_i, & \mbox{if} \, n=0 \\
\max\left\{\mathbf{0}, \sum_{m\in\mathcal{M}_n}\mathbf{W}_{n,m}\mathbf{u}_{i,m}\right\}, & \mbox{otherwise}
\end{array}
\right.
\end{align}
where $\max$ denotes the entry-wise max operator and $\mathbf{0}$ denotes a $d_n$-dim zero vector. Note that multi-input ReLUs can be thought of as conventional ReLU with skip layers \cite{he2016deep} where $\mathbf{W}$'s are set to identity matrices accordingly.
{\bf Conventional Objective for Training DNNs with ReLU:} We write down the general objective\footnote{For simplicity in this paper we always presume that the domain of each variable contains the regularization, \emph{e.g. } $\ell_2$-norm, without showing it in the objective explicitly.} in a recursive way as used in \cite{Zhang_2016_CVPR} as follows for clarity:
\begin{align}\label{eqn:obj}
& \min_{\mathbf{V}\in\mathcal{V}, \tilde{\mathcal{W}}\subseteq\mathcal{W}} \sum_i\ell(y_i, \mathbf{V}\mathbf{u}_{i,N}), \, \mbox{s.t.} \; \mathbf{u}_{i,n} = \max\left\{\mathbf{0}, \sum_{m\in\mathcal{M}_n}\mathbf{W}_{n,m}\mathbf{u}_{i,m}\right\}, \mathbf{u}_{i,0}=\mathbf{x}_i,
\forall i, \forall n,
\end{align}
where $\tilde{\mathcal{W}}=\{\mathbf{W}_{n,m}\}$. Note that we separate the last FC layer (with weight matrix $\mathbf{V}$) from the rest hidden layers (with weight matrices in $\tilde{\mathcal{W}}$) intentionally, because $\mathbf{V}$ is for learning classifiers while $\tilde{\mathcal{W}}$ is for learning useful features. The network architectures we use in this paper are mainly for extracting features, on top of which any arbitrary classifier can be learned further.
Our goal is to optimize Eq. \ref{eqn:obj}. To that end, we propose a novel BCD based algorithm which can solve the relaxation of Eq. \ref{eqn:obj} using Tikhonov regularization with convergence guarantee.
\subsection{Reinterpretation of ReLU}
The ReLU, ordinarily defined as $\mathbf{u}=\max\{\mathbf{0}, \mathbf{x}\}$ for $\mathbf{x}\in\mathbb{R}^d$, can be viewed as a projection onto a convex set (POCS) \citet{bauschke1996projection}, and thus rewritten as a simple smooth convex optimization problem,
\begin{align}\label{eqn:relu}
\max\{\mathbf{0}, \mathbf{x}\} \equiv \arg\min_{\mathbf{u}\in\mathcal{U}}\|\mathbf{u}-\mathbf{x}\|_2^2,
\end{align}
where $\|\cdot\|_2$ denotes the $\ell_2$ norm of a vector and $\mathcal{U}$ here is the nonnegative closed half-space. This non-negative least squares problem becomes the basis of our lifted objective.
\subsection{Our Tikhonov Regularized Objective}
We use Eq. \ref{eqn:relu} to lift and unroll the general training objective in Eq. \ref{eqn:obj} obtaining the relaxation:
\begin{align}\label{eqn:our_obj}
\min_{\tilde{\mathcal{U}}\subseteq\mathcal{U}, \mathbf{V}\in\mathcal{V}, \tilde{\mathcal{W}}\subseteq\mathcal{W}} & f(\tilde{\mathcal{U}}, \mathbf{V}, \tilde{\mathcal{W}}) \stackrel{\Delta}{=} \sum_i\ell(y_i, \mathbf{V}\mathbf{u}_{i,N}) + \sum_{i, n}\frac{\gamma_n}{2}\left\|\mathbf{u}_{i,n} - \sum_{m\in\mathcal{M}_n}\mathbf{W}_{n,m}\mathbf{u}_{i,m}\right\|_2^2, \\
\mbox{s.t.} \hspace{9mm} & \mathbf{u}_{i,n} \geq \mathbf{0}, \mathbf{u}_{i,0}=\mathbf{x}_i, \forall i, \forall n\geq 1, \nonumber
\end{align}
where $\tilde{\mathcal{U}}=\{\mathbf{u}_{i,n}\}$ and $\gamma_n\geq0, \forall n$ denote predefined regularization constants. Larger $\gamma_n$ values force $\mathbf{u}_{i,n}, \forall i$ to more closely approximate the output of ReLU at the $n$-th hidden layer.
Arranging $\mathbf{u}$ and $\gamma$ terms into a matrix $\mathbf{Q}$, we rewrite Eq. \ref{eqn:our_obj} in familiar form as a Tikhonov regularized objective:
\begin{align}\label{eqn:f}
\min_{\tilde{\mathcal{U}}\subseteq\mathcal{U}, \mathbf{V}\in\mathcal{V}, \tilde{\mathcal{W}}\subseteq\mathcal{W}} f(\tilde{\mathcal{U}}, \mathbf{V}, \tilde{\mathcal{W}}) \equiv \sum_i\left\{\ell(y_i, \mathbf{V}\mathbf{P}\mathbf{u}_{i}) + \frac{1}{2}\mathbf{u}_i^T\mathbf{Q}(\tilde{\mathcal{W}})\mathbf{u}_i\right\}.
\end{align}
Here $\mathbf{u}_i, \forall i$ denotes the concatenating vector of all hidden outputs as well as the input data, \emph{i.e. } $\mathbf{u}_i=[\mathbf{u}_{i,n}]_{n=0}^N, \forall i$, $\mathbf{P}$ is a predefined constant matrix so that $\mathbf{P}\mathbf{u}_i=\mathbf{u}_{i,N}, \forall i$, and $\mathbf{Q}(\tilde{\mathcal{W}})$ denotes another matrix constructed by the weight matrix set $\tilde{\mathcal{W}}$.
\begin{prop}\label{lem:Tikhonov}
$\mathbf{Q}(\tilde{\mathcal{W}})$ is positive semidefinite, leading to the following Tikhonov regularization: $$\mathbf{u}_i^T\mathbf{Q}(\tilde{\mathcal{W}})\mathbf{u}_i\equiv(\boldsymbol{\Gamma}\mathbf{u}_i)^T(\boldsymbol{\Gamma}\mathbf{u}_i)=\|\boldsymbol{\Gamma}\mathbf{u}_i\|_2^2, \exists \boldsymbol{\Gamma}, \forall i,$$
where $\boldsymbol{\Gamma}$ is the Tikhonov matrix.
\end{prop}
\begin{defi}[Block Multi-Convexity \cite{xu2013block}]\label{def:bmc}
A function $f$ is {\em block multi-convex} if for each block variable $\mathbf{x}_i, \forall i$, $f$ is a convex function of $\mathbf{x}_i$ while all the other blocks are fixed.
\end{defi}
\begin{prop}\label{lem:multi-convex}
$f(\tilde{\mathcal{U}}, \mathbf{V}, \tilde{\mathcal{W}})$ is block multi-convex.
\end{prop}
\section{Block Coordinate Descent Algorithm}\label{sec:BCD}
\subsection{Training}
Eq. \ref{eqn:our_obj} can be minimized using alternating optimization, which decomposes the problem into the following three convex sub-problems based on Lemma \ref{lem:multi-convex}:
\begin{itemize}
\item Tikhonov regularized inverse problem: $\min_{\mathbf{u}_i\in\mathcal{U}} \ell(y_i, \mathbf{V}\mathbf{P}\mathbf{u}_{i}) + \frac{1}{2}\mathbf{u}_i^T\mathbf{Q}(\tilde{\mathcal{W}})\mathbf{u}_i, \forall i.$
\item Least-square regression: $\min_{\forall\mathbf{W}_{n,m}\in\tilde{\mathcal{W}}} \frac{\gamma_n}{2}\sum_{i}\left\|\mathbf{u}_{i,n} - \sum_{m\in\mathcal{M}_n}\mathbf{W}_{n,m}\mathbf{u}_{i,m}\right\|_2^2$;
\item Classification using learned features: $\min_{\mathbf{V}\in\mathcal{V}} \sum_i \ell(y_i, \mathbf{V}\mathbf{P}\mathbf{u}_{i})$.
\end{itemize}
All the three sub-problems can be solved efficiently due to their convexity. In fact the inverse sub-problem alleviates the vanishing gradient issue in traditional deep learning, because it tries to obtain the {\em estimated} solution for the output feature of each hidden layer, which are dependent on each other through the Tikhonov matrix. Such functionality is similar to that of target (\emph{i.e. } estimated outputs of each layer) propagation \cite{bengio2014auto}, namely, propagating information between input data and output labels.
Unfortunately, a simple alternating optimization scheme cannot guarantee the convergence to stationary points for solving Eq. \ref{eqn:our_obj}. Therefore we propose a novel BCD based algorithm for training DNNs based on Eq. \ref{eqn:our_obj} as listed in Alg. \ref{alg:bcd}. Basically we sequentially solve each sub-problem with an extra quadratic term. These extra terms as well as the convex combination rule guarantee the global convergence of the algorithm (see Sec. \ref{sec:convergence} for more details).
Our algorithm involves solving a sequence of quadratic programs (QP), whose computational complexity is cubic, in general, in the input dimension \cite{nesterov1994interior}.
In this paper we focus on the theoretical development of the algorithm, and consider fast implementations in future work
\begin{algorithm}[t]\footnotesize
\SetAlgoLined
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
\Input{training data $\{(\mathbf{x}_i, \mathbf{y}_i)\}$ and regularization parameters $\{\gamma_n\}$}
\Output{network weights $\tilde{\mathcal{W}}$}
\BlankLine
Randomly initialize $\tilde{\mathcal{U}}^{(0)}\subseteq\mathcal{U}, \mathbf{V}^{(0)}\in\mathcal{V}, \tilde{\mathcal{W}}^{(0)}\subseteq\mathcal{W}$;
Set sequence $\left\{\theta_t\right\}_{t=1}^{\infty}$ so that $0\leq\theta_t\leq 1, \forall t$ and sequence $\left\{\sum_{k=t}^{\infty}\frac{\theta_k}{1-\theta_k}\right\}_{t=1}^{\infty}$ converges to zero, \emph{e.g. } $\theta_t = \frac{1}{t^2}$;
\For{$t=1,2,\cdots$}{
$\mathbf{u}_i^* \leftarrow \arg\min_{\mathbf{u}_i\in\mathcal{U}} \ell(y_i, \mathbf{V}^{(t-1)}\mathbf{P}\mathbf{u}_{i}) + \frac{1}{2}\mathbf{u}_i^T\mathbf{Q}(\tilde{\mathcal{W}}^{(t-1)})\mathbf{u}_i + \frac{1}{2}(1-\theta_t)^2\|\mathbf{u}_{i} - \mathbf{u}_{i}^{(t-1)}\|_2^2, \forall i$;
$\mathbf{u}_{i}^{(t)}\leftarrow\mathbf{u}_{i}^{(t-1)} + \theta_t(\mathbf{u}_i^* - \mathbf{u}_{i}^{(t-1)}), \forall i$;
$\mathbf{V}^*\leftarrow\arg\min_{\mathbf{V}\in\mathcal{V}} \sum_i \ell(y_i, \mathbf{V}\mathbf{P}\mathbf{u}_{i}^{(t)}) + \frac{1}{2}(1-\theta_t)^2\|\mathbf{V}-\mathbf{V}^{(t-1)}\|_F^2$;
$\mathbf{V}^{(t)}\leftarrow\mathbf{V}^{(t-1)} + \theta_t(\mathbf{V}^* - \mathbf{V}^{(t-1)})$;
$\tilde{\mathcal{W}}^* \leftarrow \arg\min_{\tilde{\mathcal{W}}\subseteq\mathcal{W}} \sum_i\frac{1}{2}[\mathbf{u}_i^{(t)}]^T\mathbf{Q}(\tilde{\mathcal{W}})\mathbf{u}_i^{(t)} + \frac{1}{2}(1-\theta_t)^2\sum_n\sum_{m\in\mathcal{M}_n}\|\mathbf{W}_{n,m} - \mathbf{W}_{n,m}^{(t-1)}\|_F^2$
$\mathbf{W}_{n,m}^{(t)}\leftarrow\mathbf{W}_{n,m}^{(t-1)} + \theta_t(\mathbf{W}_{n,m}^* - \mathbf{W}_{n,m}^{(t-1)}), \forall n, \forall m\in\mathcal{M}_n, \mathbf{W}_{n,m}^*\in\tilde{\mathcal{W}}^*$;
}
\Return $\tilde{\mathcal{W}}$;
\caption{Block Coordinate Descent (BCD) Algorithm for Training DNNs}\label{alg:bcd}
\end{algorithm}
\subsection{Testing}\label{ssec:testing}
Given a test sample $\mathbf{x}$ and learned network weights $\tilde{\mathcal{W}}^*, \mathbf{V}^*$, based on Eq. \ref{eqn:our_obj} the ideal decision function for classification should be $y^*=\arg\min_{y\in\mathcal{Y}} \left\{ \min_{\mathbf{u}} f(\mathbf{u}, \mathbf{V}^*, \tilde{\mathcal{W}}^*) \right\}.$ This indicates that for each pair of test data and potential label we have to solve an optimization problem, leading to unaffordably high computational complexity that prevents us from using it.
Recall that our goal is to train feed-forward DNNs using the BCD algorithm in Alg. \ref{alg:bcd}. Considering this, we utilize the network weights $\tilde{\mathcal{W}}^*$ to construct the network for extracting deep features. Since these features are the approximation of $\tilde{\mathcal{U}}$ in Eq. \ref{eqn:our_obj} (in fact this is a feasible solution of an extreme case where $\gamma_n=+\infty, \forall n$), the learned classifier $\mathbf{V}^*$ can never be reused at test time. Therefore, we
retain the architecture and weights of the trained network and replace the classification layer (\emph{i.e. } the last layer with weights $\mathbf{V}$) with a linear support vector machine (SVM).
\subsection{Experiments}
\subsubsection{MNIST Demonstration}
\begin{wrapfigure}{r}{0.35\linewidth}
\vspace{-18pt}
\begin{center}
\includegraphics[width=\linewidth]{parallel.eps}\vspace{-3mm}
\caption{\footnotesize The network architecture for algorithm/solver comparison.}
\label{fig:structure}
\end{center}
\vspace{-20pt}
\end{wrapfigure}
To demonstrate the effectiveness and efficiency of our BCD based algorithm in Alg. \ref{alg:bcd}, we conduct comprehensive experiments on MNIST \cite{lecun1998mnist} dataset using its $28\times 28 = 784$ raw pixels as input features. We refer to our algorithm for learning {\em dense} networks as ``BCD'' and that for learning {\em sparse} networks as ``BCD-S'', respectively. For sparse learning, we define the convex set $\mathcal{W}=\{\mathbf{W} \mid \|\mathbf{W}_k\|_1\leq 1, \forall k\}$, where $\mathbf{W}_k$ denotes the $k$-th row in matrix $\mathbf{W}$ and $\|\cdot\|_1$ denotes the $\ell_1$ norm of a vector. All the comparisons are performed on the same PC. We implement our algorithms using MATLAB GPU implementation without optimizing the code.
We compare our algorithms with the six SGD based solvers in Caffe \cite{jia2014caffe}, \emph{i.e. } SGD \cite{bottou2012stochastic}, AdaDelta \cite{zeiler2012adadelta}, AdaGrad \cite{duchi2011adaptive}, Adam \cite{kingma2014adam}, Nesterov \cite{sutskever2013importance}, RMSProp \cite{tieleman2012lecture}, which are coded in Python. The network architecture that we implemented is illustrated in Fig.~\ref{fig:structure}. This network has three hidden layers (with ReLU) with 784 nodes per layer, four FC layers, and three skip layers inside.
Therefore, the mapping function from input $\mathbf{x}_i$ to output $\mathbf{y}_i$ defined by the network is:
\begin{align}
& f(\mathbf{x}_i) = \mathbf{V}\mathbf{u}_{i,3}, \, \mathbf{u}_{i,3} = \max\{\mathbf{0}, \mathbf{x}_i+\mathbf{u}_{i,1}+\mathbf{W}_{3,2}\mathbf{u}_{i,2}\}, \nonumber \\
& \mathbf{u}_{i,2}=\max\{\mathbf{0}, \mathbf{x}_i+\mathbf{W}_{2,1}\mathbf{u}_{i,1}\}, \, \mathbf{u}_{i,1}=\max\{\mathbf{0}, \mathbf{W}_{1,0}\mathbf{x}_i\}. \nonumber
\end{align}
For simplicity without loss of generality, we utilize MSE as the loss function, and learn the network parameters using different solvers with the same inputs and random initial weights for each FC layer.
\begin{figure}
\centering
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[width=\textwidth]{obj.eps}
\centerline{\footnotesize (a)}
\end{minipage}
~
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[width=\textwidth]{test_err.eps}
\centerline{\footnotesize (b)}
\end{minipage}
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[width=\textwidth]{run_time.eps}
\centerline{\footnotesize (c)}
\end{minipage}
~
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[width=\textwidth]{sparse_nonzero.eps}
\centerline{\footnotesize (d)}
\end{minipage}
\vspace{-5mm}
\caption{\footnotesize {\bf (a)} Illustration of convergence for BCD and BCD-S. {\bf (b)} Test error comparison. {\bf (c)} Running time comparison. {\bf (d)} Sparseness comparison for BCD and BCD-S.}\label{fig:error}
\vspace{-4mm}
\end{figure}
Without fine-tuning the regularization parameters, we simply set $\gamma_n=0.1, \forall n$ in Eq. \ref{eqn:our_obj} for both BCD and BCD-S algorithms. For the Caffe solvers, we modify the demo code in Caffe for MNIST and run the comparison with carefully tuning the parameters
to achieve the best performance that we can. We report the results within 100 epochs by averaging three trials, because at this point the training of all the methods seems convergent already. For all competing algorithms, in each epoch the entire training data is passed through once to update parameters. Therefore, for our algorithms each epoch is equivalent to one iteration, and there are 100 iterations in total.
{\bf Convergence:} Fig. \ref{fig:error}(a) shows the change of training objective with increase of epochs for BCD and BCD-S, respectively. As we see both curves decrease monotonically and become flatter and flatter eventually, indicating that both algorithms converge. BCD-S converges much faster than BCD, but its objective is higher than BCD. This is because BCD-S learns sparse models that may not fit data as well as dense models learned by BCD.
{\bf Testing Error:} As mentioned in Sec. \ref{ssec:testing}, here we utilize linear SVMs and last-layer hidden features extracted from training data to retrain the classifier. Based on the network in Fig. \ref{fig:structure} the feature extraction function is $\mathbf{u}_{i,3}=\max\{\mathbf{0}, \mathbf{x}_i+\max\{\mathbf{0}, \mathbf{W}_{1,0}\mathbf{x}_i\}+\mathbf{W}_{3,2}\max\{\mathbf{0}, \mathbf{x}_i+\mathbf{W}_{2,1}\max\{\mathbf{0}, \mathbf{W}_{1,0}\mathbf{x}_i\}\}\}$.
To conduct fair comparison, we retrain the classifiers for all the algorithms, and summarize the test-time results in Fig.~\ref{fig:error}(b) with 100 epochs. Our BCD algorithm which learns dense architectures, same as the SGD based solvers, performs best, while our BCD-S algorithm works still better than the SGD competitors, although it learns much sparser networks. These results are consistent with the training objectives in Fig. \ref{fig:error}(a) as well.
{\bf Computational Time:} We compare the training time in Fig. \ref{fig:error}(c). It seems that our BCD implementation is significantly faster than the Caffe solvers. For instance, our BCD achieves about 2.5 times speed-up than the competitors, while achieving best classification performance at test time.
{\bf Sparseness:} In order to compare the difference in terms of weights between the dense and sparse networks learned by BCD and BCD-S, respectively, we compare the percentage of nonzero weights in each FC layer, and show the results in Fig. \ref{fig:error}(d). As we see, expect the last FC layer (corresponding to parameter $\mathbf{V}$ as classifiers) BCD-S has the ability of learning much sparser networks for deep feature extraction. In our case BCD-S learns a network with $2.42\%$ nonzero weights\footnote{Since we will retrain the classifiers after all, here we do not take the nonzeros in the last FC into account.}, on average, with classification accuracy $1.34\%$ lower than that of BCD which learns a network with $97.15\%$ nonzero weights. Potentially this ability could be very useful in the scenarios such as embedding systems where sparse networks are desired.
\subsubsection{Supervised Hashing}
To further demonstrate the usage of our approach, we compare with \cite{Zhang_2016_CVPR}\footnote{MATLAB code is available at \url{https://zimingzhang.wordpress.com/publications/}.} for the application of supervised hashing, which is the state-of-the-art in the literature. \cite{Zhang_2016_CVPR} proposed an ADMM based optimization algorithm to train DNNs with relaxed objective that is very related to ours. We train the same DNN on MNIST as used in \cite{Zhang_2016_CVPR}, \emph{i.e. } with 48 hidden layers and 256 nodes per layer that are sequentially and fully connected (see \cite{Zhang_2016_CVPR} for more details on the network). Using the same image features, we consistently observe marginal improvement over the results (\emph{i.e. } precision, recall, mAP) reported in \cite{Zhang_2016_CVPR}. However, on the same PC we can finish training within 1 hour based on our implementation, while using the MATLAB code for \cite{Zhang_2016_CVPR} the training needs about 9 hours. Similar observations can be made on CIFAR-10 as used in \cite{Zhang_2016_CVPR} using a network with 16 hidden layers and 1024 nodes per layer.
\section{Convergence Analysis}\label{sec:convergence}
\subsection{Preliminaries}\label{ssec:preliminaries}
\begin{defi}[Lipschitz Continuity \cite{erikssonapplied}]\label{defi:Lipschitz}
We say that function $f$ is {\em Lipschitz continuous} with Lipschitz constant $L_f$ on $\mathcal{X}$, if there is a (necessarily nonnegative) constant $L_f$ such that
$$|f(x_1)-f(x_2)|\leq L_f|x_1-x_2|, \forall x_1 , x_2 \in \mathcal{X}.$$
\end{defi}
\begin{defi}[Global Convergence \cite{lanckriet2009convergence}]\label{def:gl_conv}
Let $\mathcal{X}$ be a set and $x_0\in\mathcal{X}$ a given point, Then an Algorithm, $\mathcal{A}$, with initial point $x_0$ is a point-to-set map $\mathcal{A}: \mathcal{X}\rightarrow\mathcal{P}(\mathcal{X})$ which generates a sequence $\{x_k\}_{k=1}^{\infty}$ via the rule $x_{k+1}\in \mathcal{A}(x_k), k=0,1,\cdots$. $\mathcal{A}$ is said to be {\em global convergent} if for any chosen initial point $x_0$, the sequence $\{x_k\}_{k=0}^{\infty}$ generated by $x_{k+1}\in \mathcal{A}(x_k)$ (or a subsequence) converges to a point for which a necessary condition of optimality holds.
\end{defi}
\begin{defi}[R-linear Convergence Rate \cite{nocedal99}]\label{defi:rate}
Let $\{x_k\}$ be a sequence in $\mathbb{R}^n$ that converges to $x^*$. We say that convergence is {\em R-linear} if there is a sequence of nonnegative scalars $\{v_k\}$ such that $\|x_k-x^*\|\leq v_k, \forall k$, and $\{v_k\}$ converges Q-linearly to zero.
\end{defi}
\begin{lemma}[3-Point Property \cite{Baldassarre}]\label{lem:3-point}
If function $\phi(\mathbf{w})$ is convex and $\hat{\mathbf{w}}=\arg\min_{\mathbf{w}\in\mathbb{R}^d}\phi(\mathbf{w})+\frac{1}{2}\|\mathbf{w}-\mathbf{w}_0\|_2^2$, then for any $\mathbf{w}\in\mathbb{R}^d$,
$$
\phi(\hat{\mathbf{w}})+\frac{1}{2}\|\hat{\mathbf{w}}-\mathbf{w}_0\|_2^2\leq\phi(\mathbf{w})+\frac{1}{2}\|\mathbf{w}-\mathbf{w}_0\|_2^2-\frac{1}{2}\|\mathbf{w}-\hat{\mathbf{w}}\|_2^2.
$$
\end{lemma}
\subsection{Theoretical Results}
\begin{defi}[Assumptions on $f$ in Eq. \ref{eqn:our_obj}]\label{defi:assumption}
Let $f_1(\tilde{\mathcal{U}})\stackrel{\Delta}{=}f(\tilde{\mathcal{U}}, \cdot, \cdot), f_2(\mathbf{V})\stackrel{\Delta}{=}f(\cdot, \mathbf{V}, \cdot), f_3(\tilde{\mathcal{W}})\stackrel{\Delta}{=}f(\cdot, \cdot, \tilde{\mathcal{W}})$ be the objectives of the three sub-problems, respectively. Then we assume that $f$ is lower-bounded and $f_1, f_2, f_3$ are Lipschitz continuous with constants $L_{f_1}, L_{f_2}, L_{f_3}$, respectively.
\end{defi}
\begin{prop}\label{prop:1}
Let $x,y,\hat{x}\in\mathcal{X}$ and $y=(1-\theta)x+\theta\hat{x}$. Then $\frac{1}{2}\|\hat{x}-y\|_2^2 = \frac{1}{2}\left(1-\theta\right)^2\|\hat{x}-x\|_2^2.$
\end{prop}
\begin{lemma}\label{lem:phi}
Let $\mathcal{X}$ be a nonempty closed convex set, function $\phi:\mathcal{X}\rightarrow \mathbb{R}$ is convex and Lipschitz continuous with constant $L$, and scalar $0\leq\theta\leq 1$. Suppose that $\forall x\in\mathcal{X}, \hat{x}=\arg\min_{z\in\mathcal{X}}\phi(z)+\frac{1}{2}\|z-z_0\|_2^2$ and $z_0=y=(1-\theta)x+\theta\hat{x}$. Then we have
$$\frac{1-\theta}{\theta}\|y-x\|_2^2 \leq \phi(x)-\phi(y) \leq L\|y-x\|_2 \Rightarrow \|y-x\|_2\leq \frac{L\theta}{1-\theta}.$$
\end{lemma}
\begin{proof}
Based on the convexity of $\phi$, Prop. \ref{prop:1}, and Lemma \ref{lem:3-point}, we have
\begin{align}
& \phi(x)-\phi(y) \geq\phi(x)-\left[\left(1-\theta\right)\phi(x)+\theta\phi(\hat{x})\right]=\theta\left[\phi(x) - \phi(\hat{x})\right] \nonumber \\
&\geq \theta\left[\frac{1}{2}\|x-\hat{x}\|_2^2 + \frac{1}{2}\|\hat{x} - z_0\|_2^2 - \frac{1}{2}\|x-z_0\|_2^2\right] = \theta\left(1-\theta\right)\|x-\hat{x}\|_2^2 = \frac{1-\theta}{\theta}\|y-x\|_2^2, \nonumber
\end{align}
where $\|y-x\|_2^2=0$ if and only if $\hat{x}=x$ (equivalently $\phi(x)=\phi(y)$); otherwise $\|y-x\|_2^2$ is lower-bounded from 0 provided that $\theta\neq 1$.
Based on Def. \ref{defi:Lipschitz}, we have $\phi(x)-\phi(y) \leq L\|y-x\|_2$.
\end{proof}
\begin{thm}\label{thm:BCD}
Let $\left\{\left(\tilde{\mathcal{U}}^{(t)}, \mathbf{V}^{(t)}, \tilde{\mathcal{W}}^{(t)}\right)\right\}_{t=1}^{\infty}\subseteq\mathcal{U}\times\mathcal{V}\times\mathcal{W}$ be an arbitrary sequence from a closed convex set that is generated by Alg. \ref{alg:bcd}. Suppose that $0\leq\theta_t\leq 1, \forall t$ and the sequence $\left\{\sum_{k=t}^{\infty}\frac{\theta_k}{1-\theta_k}\right\}_{t=1}^{\infty}$ converges to zero. Then we have
\begin{enumerate}
\item $\left(\tilde{\mathcal{U}}^{(\infty)}, \mathbf{V}^{(\infty)}, \tilde{\mathcal{W}}^{(\infty)}\right)$ is a stationary point;
\item $\left\{\left(\tilde{\mathcal{U}}^{(t)}, \mathbf{V}^{(t)}, \tilde{\mathcal{W}}^{(t)}\right)\right\}_{t=1}^{\infty}$ will converge to $\left(\tilde{\mathcal{U}}^{(\infty)}, \mathbf{V}^{(\infty)}, \tilde{\mathcal{W}}^{(\infty)}\right)$ globally with R-linear convergence rate.
\end{enumerate}
\end{thm}
\begin{proof}
1. Suppose that for $\tilde{\mathcal{U}}^{(\infty)}$ there exists a $\triangle\tilde{\mathcal{U}}\neq \emptyset$ so that $f_1(\tilde{\mathcal{U}}^{(\infty)}+\triangle\tilde{\mathcal{U}}) = f_1(\tilde{\mathcal{U}}^{(\infty)})$ (otherwise, it conflicts with the fact of $\tilde{\mathcal{U}}^{(\infty)}$ being the limit point). From Lemma \ref{lem:phi}, $f_1(\tilde{\mathcal{U}}^{(\infty)}+\triangle\tilde{\mathcal{U}}) = f_1(\tilde{\mathcal{U}}^{(\infty)})$ is equivalent to $\tilde{\mathcal{U}}^{(\infty)}+\triangle\tilde{\mathcal{U}} = \tilde{\mathcal{U}}^{(\infty)}$, and thus $\triangle\tilde{\mathcal{U}}=\emptyset$, which conflicts with the assumption of $\triangle\tilde{\mathcal{U}}\neq \emptyset$. Therefore, there is no direction that can decrease $f_1(\tilde{\mathcal{U}}^{(\infty)})$, \emph{i.e. } $\nabla f_1(\tilde{\mathcal{U}}^{(\infty)})=\mathbf{0}$. Similarly we have $\nabla f_2(\mathbf{V}^{(\infty)})=\mathbf{0}$ and $\nabla f_3(\tilde{\mathcal{W}}^{(\infty)})=\mathbf{0}$. Therefore, $\left(\tilde{\mathcal{U}}^{(\infty)}, \mathbf{V}^{(\infty)}, \tilde{\mathcal{W}}^{(\infty)}\right)$ is a stationary point.
2. Based on Def. \ref{defi:assumption} and Lemma \ref{lem:phi}, we have
\begin{align}
&\sqrt{\sum_{\mathbf{u}_{i,n}\in\tilde{\mathcal{U}}}\left\|\mathbf{u}_{i,n}^{(t)}-\mathbf{u}_{i,n}^{(\infty)}\right\|_2^2 + \left\|\mathbf{V}^{(t)}-\mathbf{V}^{(\infty)}\right\|_F^2 + \sum_{\mathbf{W}_{n,m}\in\tilde{\mathcal{W}}}\left\|\mathbf{W}_{n,m}^{(t)}-\mathbf{W}_{n,m}^{(\infty)}\right\|_F^2} \nonumber \\
\leq & \sum_{\mathbf{u}_{i,n}\in\tilde{\mathcal{U}}}\left\|\mathbf{u}_{i,n}^{(t)}-\mathbf{u}_{i,n}^{(\infty)}\right\|_2 + \left\|\mathbf{V}^{(t)}-\mathbf{V}^{(\infty)}\right\|_F + \sum_{\mathbf{W}_{n,m}\in\tilde{\mathcal{W}}}\left\|\mathbf{W}_{n,m}^{(t)}-\mathbf{W}_{n,m}^{(\infty)}\right\|_F \nonumber \\
= & \hspace{0mm}\sum_{\mathbf{u}_{i,n}\in\tilde{\mathcal{U}}}\left\|\sum_{k=t}^{\infty}\mathbf{u}_{i,n}^{(k)}-\mathbf{u}_{i,n}^{(k+1)}\right\|_2 + \left\|\sum_{k=t}^{\infty}\mathbf{V}^{(k)}-\mathbf{V}^{(k+1)}\right\|_F + \hspace{0mm} \sum_{\mathbf{W}_{n,m}\in\tilde{\mathcal{W}}}\left\|\sum_{k=t}^{\infty}\mathbf{W}_{n,m}^{(k)}-\mathbf{W}_{n,m}^{(k+1)}\right\|_F \nonumber \\
\leq & \sum_{k=t}^{\infty}\left[\sum_{\mathbf{u}_{i,n}\in\tilde{\mathcal{U}}}\left\|\mathbf{u}_{i,n}^{(k)}-\mathbf{u}_{i,n}^{(k+1)}\right\|_2 + \left\|\mathbf{V}^{(k)}-\mathbf{V}^{(k+1)}\right\|_F + \sum_{\mathbf{W}_{n,m}\in\tilde{\mathcal{W}}}\left\|\mathbf{W}_{n,m}^{(k)}-\mathbf{W}_{n,m}^{(k+1)}\right\|_F\right] \nonumber \\
\leq & \sum_{k=t}^{\infty}\left[\sum_{\mathbf{u}_{i,n}\in\tilde{\mathcal{U}}} \frac{L_{f_1}\theta_k}{1-\theta_k} + \frac{L_{f_2}\theta_k}{1-\theta_k} + \sum_{\mathbf{W}_{n,m}\in\tilde{\mathcal{W}}}\frac{L_{f_3}\theta_k}{1-\theta_k}\right] = O\left(\sum_{k=t}^{\infty}\frac{\theta_k}{1-\theta_k}\right). \nonumber
\end{align}
By combining this with Def.~\ref{def:gl_conv} and Def. \ref{defi:rate} we can complete the proof.
\end{proof}
\begin{cor}
Let $\theta_t=\left(\frac{1}{t}\right)^p, \forall t$. Then when $p>1$, Alg. \ref{alg:bcd} will converge globally with order one.
\end{cor}
\begin{proof}
\begin{align}
\sum_{k=t}^{\infty}\frac{\theta_k}{1-\theta_k} = \sum_{k=t}^{\infty}\frac{1}{k^p-1} & \leq \int_{t^p-1}^{\infty}\frac{1}{x}d(x+1)^{\frac{1}{p}}=\frac{1}{p}\int_{t^p-1}^{\infty}\frac{1}{x}(x+1)^{\frac{1}{p}-1}dx \nonumber \\
& \stackrel{\because p>1}{\leq}\frac{1}{p}\int_{t^p-1}^{\infty}x^{\frac{1}{p}-2}dx = (p-1)^{-1}(t^p-1)^{\frac{1}{p}-1}.
\end{align}
Since the sequence $\left\{(t^p-1)^{\frac{1}{p}-1}\right\}_{t=1}^{\infty}, \forall p>1$ converges to zero sublinearly with order one, by combining these with Def.~\ref{defi:rate} and Thm.~\ref{thm:BCD} we can complete the proof.
\end{proof}
\section{Conclusion}
In this paper we first propose a novel Tikhonov regularization for training DNNs with ReLU as the activation functions. The Tikhonov matrix encodes the network architecture as well as parameterization. With its help we reformulate the network training as a block multi-convex minimization problem. Accordingly we further propose a novel block coordinate descent (BCD) based algorithm, which is proven to converge globally to stationary points with R-linear converge rate of order one. Our empirical results suggest that our algorithm does converge, is suitable for learning both dense and sparse networks, and may work better than traditional SGD based deep learning solvers.
\newpage
{\small
\bibliographystyle{ieee}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,332
|
{"url":"http:\/\/math.stackexchange.com\/questions\/389037\/derivative-of-trig-functions-intuition-help","text":"Derivative of Trig Functions (Intuition Help?)\n\nLooking for some intuition help here.\n\nI have the following exercise and these are the steps I take: $$y = \\sin\\left(\\frac{1}{x}\\right)$$\n\n$$u=\\frac{1}{x}$$\n\n$$y = \\sin u,\\;\\;\\frac{dy}{du} = \\cos u= \\cos\\left(\\frac{1}{x}\\right)$$\n\n$$u=x^{-1};\\;\\frac{du}{dx} =-x^{-2}=-\\frac{1}{x^2}$$\n\n$$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}=cos\\frac{1}{x}\\times-\\frac{1}{x^2}$$ This is incorrect but intuitively I want to multiply it this way. $$cos\\frac{1}{x}\\times-\\frac{1}{x^2}=cos-\\frac{1}{x^3}$$\n\nBut the correct answer is: $$-\\frac{cos-\\frac{1}{x}}{x^2}$$\n\nHelp me absorb the why so I can intuitively solve problems like these.\n\n-\nSorry, not sure I understand the question: you're struggling with the intuition behind calculating $\\frac{d}{dx}\\sin(\\frac{1}{x})$? \u2013\u00a0 AWertheim May 12 '13 at 2:49\nThe intuition of multiplying cos1\/x times -1\/x^2 \u2013\u00a0 ItsMitch May 12 '13 at 2:58\n\nHere is where parentheses come in handy:\n\nYou found, correctly, $\\dfrac{dy}{du}$ and $\\dfrac{du}{dx}$.\n\nBut the scope of $\\cos$ is restricted to its argument: $\\left(\\dfrac 1x\\right)$ ONLY:\n\nThe FUNCTION $\\dfrac{dy}{du} = \\cos\\left(\\dfrac 1x\\right)$ is multiplied by the function $\\dfrac{du}{dx} = -\\dfrac 1{x^2}$. That is not what you did. You multiplied argument of the $\\cos$ function by the function $\\dfrac{du}{dx} = -\\dfrac{1}{x^2}$.\n\n$$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}=\\left[cos\\left(\\frac{1}{x}\\right)\\right]\\times\\left(-\\frac{1}{x^2}\\right)$$ $$= -\\frac 1{x^2} \\cos\\left(\\frac 1x\\right) = -\\dfrac{\\cos\\left(\\frac 1x\\right)}{x^2}$$\n\n-\nBut why do I enclose it in brackets? \u2013\u00a0 ItsMitch May 12 '13 at 2:56\nBecause $\\cos$ is applied to the argument $\\frac 1x$ ONLY. The FUNCTION $\\dfrac{dy}{du} = \\cos\\left(\\dfrac 1x\\right)$ is multiplied by the function $\\dfrac{du}{dx} = -\\dfrac 1{x^2}$ \u2013\u00a0 amWhy May 12 '13 at 3:03\nThank you very much! Where could I see more of these trig argument \"laws\" or rules? \u2013\u00a0 ItsMitch May 12 '13 at 3:14\nFor example, If we have $f(x) = cos x,$ and we have $g(x) = x$, then $f(x)\\cdot g(x) = \\cos(x)\\cdot (x) = x\\cos(x)$ \u2013\u00a0 amWhy May 12 '13 at 3:15\n@amWhy: Even nice follw ups! +1 \u2013\u00a0 Amzoti May 13 '13 at 0:23\n\nFirst, you are taking short cuts when you are writing your equations that simply confuse things. Also, LaTeX has \\sin and \\cos. Start over.\n\n\\begin{aligned} \\text{Let}\\quad f(x) &= \\sin\\frac{1}{x}.\\\\ \\text{Let}\\quad u &= \\frac{1}{x}.\\\\ f(x) &= \\sin u\\\\ \\frac{df}{dx} &= \\frac{df}{du} \\frac{du}{dx} \\quad\\text{by the Chain rule}\\\\ \\frac{df}{du} &= \\cos u\\\\ \\frac{du}{dx} &= -\\frac{1}{x^2} \\\\ \\frac{df}{dx} &= \\frac{df}{du} \\frac{du}{dx} \\\\ &= \\cos u \\cdot [-\\frac{1}{x^2}] \\\\ &= \\cos \\frac{1}{x} \\cdot [-\\frac{1}{x^2}]\\\\ &= -\\frac{1}{x^2} \\cos \\frac{1}{x}. \\end{aligned}\n\nBy the way, you are panicking and falling into the \"Universal Distributive Law of Freshmen\"; you think that all operations distribute over each other, and all operations commute. This is what has caused legions of students to replace $\\sqrt{1+x^2}$ with $1 + x$. You wrote:\n\n$$\\cos\\frac1x \\cdot \\frac{1}{x^2} = \\cos\\frac{1}{x^3}.$$\n\nEven though $\\cos x$ looks like a product, it isn't. Had you been using a programming language, you'd see that COS(X)*Y is not the same as COS(X*Y). Try any of these cases with actual numbers and a pocket calculator; you'll see what I mean.\n\nWhen you panic like that, always take a deep breath and start over. And, never write a function next to its derivative and equate them.\n\n-\nThank you. Could you point me to a quick tutorial on LaTex and the correct way to write equations so I do not confuse more people? \u2013\u00a0 ItsMitch May 12 '13 at 3:20\nI should have looked at this too: meta.math.stackexchange.com\/questions\/107\/\u2026 \u2013\u00a0 Eric Jablow May 12 '13 at 3:24","date":"2014-09-01 13:49:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 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\": 0.9999675750732422, \"perplexity\": 1811.1843516465967}, \"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-2014-35\/segments\/1409535919066.8\/warc\/CC-MAIN-20140901014519-00101-ip-10-180-136-8.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
The fusion of two nuclei occurs at small impact parameters when the overlap between their wave functions
is big enough to allow the strong interaction to overcome the Coulomb repulsion. Heavy-ion fusion reactions have
numerous applications, like the study of high spin states in yrast and super-deformed bands \cite{jan91} or the formation
of Heavy and Super Heavy Elements (SHE) \cite{hof98}.
Induced by beams of unstable nuclei, this mechanism will also allow to produce
very exotic species and allow for the study of isospin equilibration in the fused system.
The fusion process can be schematically divided in three steps: {\it (i)} an approach phase during which
each nucleus
feels only the Coulomb field of its partner
and which ends up when the nuclear interaction starts to dominate, {\it (ii)} a rapid equilibration of the energy
and the angular momentum transfered from the relative motion to the internal degrees of freedom, leading to
the formation of a Compound Nucleus (CN) and {\it (iii)} a statistical decay of the CN.
Lots of theoretical and experimental efforts \cite{das98} are made to understand step {\it (i)}. These studies
focus on an energy range located around the fusion barrier. At these energies
the fusion is controlled by quantum tunneling
which is strongly influenced by the couplings between the internal degrees of freedom and the relative
motion of the two colliding partners.
Although the cooling mechanisms involved in {\it (iii)} are well known and consist mainly in light particle
and $\gamma$-ray emission in competition with fission for heavy systems, the initial conditions of the statistical decay depend
on the equilibration process {\it (ii)} which is still subject to many debates nowadays. Indeed, step {\it (ii)} is characterized by
an equilibration of several degrees of freedom like the shape \cite{bon80}
or the charge \cite{bon81} which can be
accompanied by the emission of preequilibrium particles.
Such emission decreases the excitation energy
and the angular momentum.
The latter quantities are crucial and must be determined precisely because they have a major influence
on the CN survival probability and therefore on the synthesis of very exotic systems
such as the SHE.
In this paper we study the equilibration of the charges in fused systems, its interplay with other
macroscopic degrees of freedom like the shape and the rotation, and its implications on the statistical decay.
To probe theoretically and experimentally
this way to fusion, we use the preequilibrium isovector Giant Dipole Resonance (GDR)
\cite{cho1,bar1,bar2,bar3,sim1}.
Giant Resonances
are interpreted as the first quantum of collective vibrations involving
protons and neutrons fluids.
The Giant Monopole Resonance can be described as a breathing mode, an
alternation of compression and dilatation of the whole nucleus.
The GDR corresponds to a collective oscillation of the protons against the neutrons. The Giant
Quadrupole Resonance consists in a nuclear shape oscillation between prolate and oblate
deformations. Many other resonances have been
discovered \cite{Wo87,Har01}. In particular
Giant Resonances have
been observed in hot nuclei formed by fusion \cite{New81,gaa}.
This demonstrates the survival of ordered vibrations in very excited systems,
which are known to be chaotic, even if some Giant Resonance
characteristics like the width are affected by the
temperature \cite{bra89,cho95}. Moreover, the strong couplings between various collective modes
which occur for Giant Resonances built on the ground state \cite{sim2,fal} are still present in fusion
reactions \cite{sim1,cho03}. It might therefore be possible to use the Giant Resonances properties to probe
the nuclear structure of the composite system on its way to fusion.
The choice of the preequilibrium GDR, that is, a GDR excited in step {\it (ii)} before
the formation of a fully equilibrated CN, is motivated by the fact that its
properties strongly depend on the structure of the state on which it is built, for instance the deformation \cite{bon81}.
The idea is to form a CN with two N/Z asymmetric reactants. Such a reaction may lead
to the excitation of a dipole mode because of the presence of a net dipole moment in the
entrance channel. This dipole oscillation should occur {\it before} the charges are fully equilibrated,
that is, during the preequilibrium phase in which the system keeps a memory of the entrance channel
\cite{bon81,cho1,sur89,bar1,bar2,bar3,das,sim1}. In addition, for such N/Z asymmetric reactions, an enhancement of the
fast GDR $\gamma$-ray emission is expected as compared to the "slower" statistical $\gamma$-ray
yield \cite{bar1,bar2,bar3,sim1}. This is of particular interest since the properties of these GDR $\gamma$-rays characterize
the dinuclear system which precedes the hot equilibrated CN. The first experimental
indications on the existence of such new
phenomenon have been reported in \cite{fli,cin,pie03fus,pie05,amo04} for fusion reactions and in
\cite{cam95,san,pap,amo,pie03deep,pap03,amo04,pap05} in the case of deep inelastic collisions.
The paper is organized as follows:
In Sec. II we study
the properties of the preequilibrium GDR using the Time-Dependent Hartree-Fock (TDHF) formalism.
In Sec. III we show how an N/Z asymmetric entrance channel may increase the fusion-evaporation
cross-sections. Finally, we conclude in section IV.
\section{TDHF study of the preequilibrium Giant Dipole Resonance}
At the early time of the fusion reaction, the system keeps the memory of the entrance channel.
We call this stage of the collision the {\it preequilibrium phase} which ends
when all the degrees of freedom are equilibrated in the compound system and when the
statistical decay starts.
One of these degrees of freedom is the isospin, which measures the asymmetry between
protons and neutrons.
When the two nuclei have different N/Z ratios, the proton
and neutron centers of mass of the total system do not coincide.
As shown in \cite{cho1,das}, there is a non zero force
between the two kind of nucleons which tends to restore the initial isospin asymmetry.
In such a case, an oscillation of protons against neutrons on the way to fusion might occur,
that is, the so-called preequilibrium GDR \cite{bon81,sur89,cho1,bar1,bar2,bar3,sim1}.
In fusion reactions the shape of the system changes drastically during the preequilibrium phase.
Studies of the dynamics in the fusion reaction mechanism requires
sophisticated calculations to extract the preequilibrium GDR characteristics
(energy, width...) and in turn, on the way to fusion. To achieve this goal, we choose
to use, as in the pioneer work of Bonche and Ng\^o on charge equilibration \cite{bon81},
the TDHF approach because it is a fully microscopic theory
which takes into account the quantal nature of the single particle dynamics.
Moreover in the present study we will restrict ourself to the observation of one-body observables
(e.g. the density $\rho (r)$) which are supposed to be well described
by such a mean field approach.
However it is clear that an important challenge is to develop methods going beyond mean field
which is beyond the scope of this paper.
In this section we present quantum calculations on preequilibrium giant collective
vibrations using the TDHF theory. We shall start with a brief
description of the TDHF theory in Sec. \ref{subsec:tdhf}. Then we examine the role of
various relevant symmetries in the entrance channel, namely the N/Z and mass symmetries
(Sec. \ref{delta}-\ref{couplings}).
Finally, in Sec. \ref{compar} we shall compare our results with the experimental data obtained by
Flibotte {\it et al.} \cite{fli}.
\subsection{TDHF approach \label{subsec:tdhf}}
In the TDHF approach \cite{har,foc,dir30,vau,eng75,bon,neg},
each single particle wave function is propagated
in the mean field generated by the
ensemble of particles. The mean field approximation does not take into account
the dissipation due to
two-body interactions \cite{gon,won,lac,jui}.
However TDHF takes care of one-body mechanisms
such as Landau spreading and evaporation damping \cite{cho2}.
Quantum effects
induced by the single particle dynamics like shell effects or modification of the moment of inertia \cite{sim04}
are accounted for properly.
The main advantage of TDHF is its fully microscopic treatment of the N-body dynamics
with the same effective interaction as the one used for the calculation of the Hartree-Fock (HF) ground sates of the collision partners.
The consistency of the method for the structure of nuclei and the nuclear reactions
increases its prediction power and its availability to study the interplay between exotic structures and reaction mechanisms.
Moreover the TDHF equation is strongly non linear
which is of great importance for reactions around the barrier because it includes couplings between
relative motion and internal degrees of freedom of the collision partners.
Also TDHF provides a good description of collective motion and can even
exhibit couplings between collective modes \cite{sim2}.
In fact the TDHF theory is optimized for the prediction of expectation values
of one-body observables
and gives their exact evolution in the extreme case
where the residual interaction vanishes.
However, the TDHF prediction of multipole moments in nuclear collision,
for instance, may differ from the correct evolution
because of the omission of the residual interaction.
An improvement of the description would be given by
the inclusion of the effect of the residual interaction on the dynamics,
which would increase considerably the computational time
and is beyond the scope of this paper.
The TDHF theory describes the evolution of the one-body density matrix $\rho (t)$
of matrix elements $\<{\mathbf{r}} sq|{\hat{\rho}}|{\mathbf{r}'} s'q'\>=\sum_i \, \varphi_i^*({\mathbf{r}'} s'q') \, \varphi_i({\mathbf{r}} sq)$,
where $ \varphi_i({\mathbf{r}} sq) = \<{\mathbf{r}}sq|i\>$ denotes the
component with a spin $s$ and isospin $q$ of the occupied single particle
wave-function $\varphi_i$.
This evolution is determined by a non linear Liouville-von Neumann equation,
\begin{equation}
i\hbar \frac{\partial }{\partial t}{\rho} -\left[{h}(\rho ),{\rho} \right]=0
\label{eq:tdhf}
\end{equation}
where ${h}(\rho )$ is the matrix associated to the self consistent mean-field Hamiltonian.
We have used the code built
by P. Bonche and coworkers \cite{kim} with an effective Skyrme interaction \cite{sky56}
and SLy4$d$ parameters \cite{kim}. In its actual version, TDHF does not account for pairing interactions.
\subsection{N/Z asymmetric reactions \label{delta}}
As far as the dipole motion in the preequilibrium phase is concerned,
it is obvious that the main relevant asymmetry responsible for such a motion
is a difference in the charge-to-mass ratio between the collision partners \cite{cho1}.
The associated experimental signature is an enhancement of the $\gamma$-ray emission
in the GDR energy region of the compound system \cite{fli,cin,pie03fus,pie05,amo04}
which is attributed to a dipole oscillation.
Several informations about the fusion path can be extracted from such a dipole oscillation
and its corresponding $\gamma$-ray spectrum.
For numerical tractability we start our study of the fusion process
with a light
system:
$^{12}$Be+$^{28}$S$\rightarrow ^{40}$Ca.
We first deduce the $\gamma$-ray spectrum from the dipole motion.
Then we study the effects of the deformation of the compound system,
and of the impact parameter on this motion.
\subsubsection{The preequilibrium GDR $\gamma$-ray spectrum \label{GDRspec}}
We first consider a central collision at an energy of $1$~MeV/nucleon in the center of mass.
The expectation value of the dipole moment $\hat{Q}_D$ is defined
by
\begin{equation}
Q_D = \<\hat{Q}_D\> = \frac{NZ}{A} (X_p-X_n)
\label{eq:qd}
\end{equation}
where
$X_p = \sum_p \frac{\<\hat x_p\>}{Z}$
and
$X_n = \sum_n \frac{\<\hat x_n\>}{N}$ are the positions of the proton and neutron centers
of mass respectively.
The expectation value of the conjugated dipole moment $\hat{P}_D$ is then associated to the relative
velocity between protons and neutrons, and is defined by the relation
\begin{equation}
P_D = \<\hat{P}_D\> = \frac{A}{2NZ}(P_p-P_n)
\label{eq:pd}
\end{equation}
where
$P_p = \sum_p \<\hat p_p\>$
and
$P_n = \sum_n \<\hat p_n\>$
are the total proton and neutron moments respectively.
These definitions ensure the canonical commutation relation $\left[\hat{Q}_D,\hat{P}_D\right]=i\hbar$.
The time evolutions of $Q_D$ and $P_D$ are plotted in Figs.~\ref{asym}-c and \ref{asym}-b respectively.
The trajectories in both the ($Q_D$,t) and ($P_D$,t) planes exhibit oscillations which we
attribute to the preequilibrium GDR.
We also note that $P_D(t)$ oscillates in phase quadrature with $Q_D(t)$ and that those
oscillations are damped due to
the one-body dissipation.
Consequently, the plot of $P_D$ as a function of $Q_D$ shown in Fig.~\ref{asym}-a is a spiral.
The GDR period extracted from these plots is around $107$ fm/c, which corresponds to an
energy of $\sim 11.6$ MeV.
\begin{figure}
\begin{center}
\epsfig{figure=./spiralenoverz.eps,width=8cm}
\caption{
Time evolution of the expectation value of the dipole moment, $Q_D$, and its conjugated moment, $P_D$,
in the reaction $^{12}$Be+$^{28}$S$\rightarrow ^{40}$Ca
at an energy of 1 MeV/nucleon in the center of mass and at zero impact parameter.}
\label{asym}
\end{center}
\end{figure}
During the collision and before the equilibrium is reached, a fast rearrangement
of charges occurs within the composite system \cite{bon81}, generating the $\gamma$-ray emission.
We extract the preequilibrium GDR $\gamma$-ray spectrum from the Fourier
transform of the acceleration of the charges \cite{jac,bar3}
\begin{equation}
\frac{dP}{dE_\gamma}(E_\gamma)=\frac{2\alpha}{3\pi}\frac{|I(E_\gamma)|^2}{E_\gamma}
\label{fourier}
\end{equation}
where $\alpha$ is the fine structure constant and
$$I(E_\gamma)=\frac{1}{c}\int_0^\infty \!\!\! dt \,\, \frac{d^2Q_D}{dt^2}\exp\left(i\frac{E_\gamma t}{\hbar}\right).$$
The spectrum obtained from Eq.~\ref{fourier} is plotted in Fig.~\ref{spectre} (solid line).
In order to have a spectrum without spurious peaks coming from the finite
integration time, we multiply the quantity $\frac{d^2Q_D}{dt^2}$ by a gaussian
function $\exp\left(-\frac{1}{2}(\frac{t}{\tau})^2\right)$ \cite{mar05}.
In addition, this function plays a role of a filter in the time domain.
This filter prevents the signal to be affected by the interaction
between the nucleus and the emitted nucleons which have
been reflected on the box \cite{rei06}.
We choose $\tau = 320$ fm/c in our calculations.
This ensures the fact that the spectra are free of spurious
effects coming from the echo. However this procedure
adds a width $\Gamma\sim \frac{\hbar}{\tau} \sim 0.6$ MeV.
This is a drawback if one is concerned with detailed spectroscopy.
However, in this paper, we are only interested by
the gross properties of the preequilibrium GDR
in order to study the fusion mechanisms.
As we can see in Fig.~\ref{spectre}, the
preequilibrium GDR energy is $E^p_{GDR} = 11.64$ MeV, which corresponds to the previous value
deduced from the GDR oscillation period.
\begin{figure}
\begin{center}
\epsfig{figure=./tf.eps,width=8cm}
\caption{preequilibrium GDR $\gamma$-ray spectrum calculated in the reaction
$^{12}$Be+$^{28}$S $\rightarrow$ $^{40}$Ca (solid line) at an energy of 1 MeV/nucleon
in the center of mass
and $\gamma$-ray spectrum of a GDR built on the ground state of
$^{40}$Ca (dotted line).}
\label{spectre}
\end{center}
\end{figure}
The energy of the preequilibrium GDR is much lower than the one of the GDR built on the spherical ground state of the $^{40}$Ca. This situation will be now explored into more details.
\subsubsection{Deformation effect \label{defor}}
To better characterize the preequilibrium GDR, it is necessary to compare it with the
usual GDR built upon the CN ground state \cite{sur89}.
This GDR is generated by applying
an isovector dipole boost with a velocity $k_D$ on the $^{40}$Ca HF ground
state $|\psi(t)\> = \exp\left({-ik_D\hat Q_D}\right)|HF\>$ yielding
an oscillation of $Q_D(t)$ and $P_D(t)$ in phase quadrature as we can see
in Fig.~\ref{ca}. The period of the oscillation is around 80 fm/c which is lower
than in the fusion case and corresponds to a higher energy ($E_{GDR}=15.5$ MeV) as it is shown
in the associated GDR $\gamma$-ray spectrum in Fig.~\ref{spectre} (dotted line).
The lower energy obtained for the fusing system reveals a strong prolate deformation
\cite{sim1,bon81,sur89,bar3}. The two mechanisms (fusion reaction and dipole boost)
are expected to generate
a GDR with quite different dynamical properties. This can be seen in the density plot projected in
the reaction plane shown in Fig.~\ref{dens}, which shows that in the case of a fusion reaction,
the CN relaxes its initial prolate elongation along the collision axis with a
time which is larger than the typical dipole oscillation period of the
GDR.
\begin{figure}
\begin{center}
\epsfig{figure=./spirale_ca40.eps,width=8cm}
\caption{
GDR built upon the HF ground state in $^{40}$Ca and excited by an isovector dipole boost:
evolution of the expectation value of the associated dipole moment, $Q_D$, and its conjugated moment, $P_D$,
as a function of time.}
\label{ca}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{figure=./dens.eps,width=8.5cm}
\caption{Density plots projected on the reaction plane for different times in the case of the fusion reaction.
Lines represent isodensities.}
\label{dens}
\end{center}
\end{figure}
Deformation effects can be studied all along the fusion path \cite{bon80,sur89}.
The quadrupole
deformation parameter $\epsilon$ is defined by a scaling
of the axis from a spherical to
a deformed shape along the $x$-axis
\begin{eqnarray}
R_x & = & R_0(1+\alpha) \nonumber \\
R_{yz} & = & R_0(1-\epsilon)
\label{def_epsilon}
\end{eqnarray}
where $\alpha$ is defined by the conservation of the volume of the nucleus $R_xR_yR_z=R_0^3$, which leads to
\begin{equation}
\alpha = \frac{(2-\epsilon)\epsilon}{(1-\epsilon)^2}\, \, \, .
\label{eq:alpha}
\end{equation}
If one neglects high order terms in $\epsilon$,
we get the usual value $\alpha \simeq 2\epsilon$.
The deformation parameter is related to the expectation values of the monopole and quadrupole
moments $\hat{Q}_{0}$ and $\hat{Q}_{2}$ which are expressed by
\begin{eqnarray}
Q_{0} =\<\hat{Q}_{0}\> & = & \frac{1}{\sqrt{4\pi}} \int \!\!\! d{\mathbf r} \,\, \rho({\mathbf r}) r^2 \label{mono} \\
Q_{2} =\<\hat{Q}_{2}\> & = & \sqrt{\frac{5}{16\pi}}\int \!\!\! d{\mathbf r} \,\, \rho({\mathbf r}) r^2 \left(3\frac{x^2}{r^2}-1\right).
\label{momexpres}
\end{eqnarray}
We can write $Q_{2}$ as a function of $Q_{0}$
\begin{equation}
Q_{2}= -\frac{\sqrt{5}}{2}Q_{0}+3\sqrt{\frac{5}{16\pi}} \int \!\!\! d{\mathbf r} \,\, \rho({\mathbf r}) x^2.
\label{eq:Q2Q0}
\end{equation}
Eqs.~\ref{def_epsilon} and \ref{mono} lead to
\begin{equation}
\int \!\!\! d{\mathbf r} \,\, \rho({\mathbf r}) x^2 = (1+\alpha)^2 \frac{\sqrt{4\pi}}{3} Q_{0}.
\label{eq:rhoQ0}
\end{equation}
Using Eqs.~\ref{eq:alpha}, \ref{eq:Q2Q0}, \ref{eq:rhoQ0} and $\epsilon<1$, we get
\begin{equation}
\epsilon(t)=1-\left(1+\frac{2Q_2(t)}{\sqrt{5}Q_0(t)}\right)^{-\frac{1}{4}}
\label{eq:def}
\end{equation}
which, at first order in $\epsilon$, becomes
\begin{equation}
\epsilon(t) = \frac{Q_{2}(t)}{2\sqrt{5}Q_{0}(t)}.
\label{eq:def2}
\end{equation}
\begin{figure}
\begin{center}
\epsfig{figure=./deformation.eps,width=8cm}
\caption{Time evolution of the deformation, $\epsilon$, in $^{40}$Ca
formed in the $^{12}$Be+$^{28}$S fusion reaction
at an energy of 1 MeV/nucleon in the center of mass.
The time axis origin is chosen when the maximum of the fusion barrier is reached.
The average preequilibrium deformation $\epsilon_p$ obtained from
the GDR energy (see Eq.~\ref{eq:defGDR}) is represented by a dashed line.
}
\label{deformation}
\end{center}
\end{figure}
In Ref. \cite{sim1} we used Eq.~\ref{eq:def2} to characterize the average deformation.
In Fig.~\ref{deformation} we present the time evolution of the deformation, $\epsilon(t)$,
obtained from the more general expression of $\varepsilon$ given in Eq.~\ref{eq:def}.
We consider a $^{40}$Ca formed in the $^{12}$Be+$^{28}$S fusion reaction at an energy of 1 MeV/nucleon
in the center of mass.
The important point here is that the deformation does not relax and strongly affects the frequency of
the oscillations. A lower energy is expected for the longitudinal collective
motion $ E^p_{GDR}$ in the fused system as compared to the one simulated in a
spherical $^{40}$Ca \cite{bon81,sur89,bar1,bar2,bar3,sim1}. Following a macroscopic model for the dipole oscillation,
we expect the energy of the GDR to evolve with the deformation along
the $x$-axis (collision axis) as
\begin{equation}
\frac{E^p_{GDR}}{E_{GDR}} = \frac{R_0}{R_x}= (1-\epsilon_p)^2
\label{eq:defGDR}
\end{equation}
where $\epsilon_p$ is the average deformation during the preequilibrium stage.
The frequency of the GDR along the deformation axis fulfills this relation
with $\epsilon_p\simeq 0.13$ in excellent agreement with the observed deformation
in Fig.~\ref{deformation}.
\begin{figure}
\begin{center}
\epsfig{figure=./energy.eps,width=8cm}
\caption{{\it a)} Energy of the preequilibrium GDR obtained from the first oscillation of the dipole moment
and {\it b)} the deformation parameter, $\epsilon$, obtained from Eq.~\ref{eq:defGDR} (dashed line) and from
Eq.~\ref{eq:def} (solid line), as a function of the center of mass energy.}
\label{energy}
\end{center}
\end{figure}
We have also investigated the effect of the center of mass energy $E_{CM}$ on the
preequilibrium GDR energy and on the deformation parameter (see Fig.~\ref{energy}).
The GDR energy exhibits small
variations (less than 1 MeV) with the center of mass energy (Fig.~\ref{energy}-a).
For $E_{CM}<40$ MeV, the increase of $E^p_{GDR}$ with $E_{CM}$ is attributed to the formation
of a dinuclear system with a slow neck dynamics at low energy \cite{bar3}.
The presence of the neck is in fact expected to slow down the charge equilibration
process, and then to increase the GDR period.
For $E_{CM}>40$ MeV,
Fig.~\ref{energy}-a a decrease of $E^p_{GDR}$
when $E_{CM}$ increases. As illustrated in Fig.~\ref{energy}-b, this is associated to a larger quadrupole
deformation when the collision is more violent.
Consequently, the higher the center of mass energy,
the more prolately deformed the CN.
In Fig.~\ref{energy}-b, the deformation is estimated from
Eq.~\ref{eq:defGDR} (dashed line) and from
Eq.~\ref{eq:def} (solid line) at the first maximum
after one oscillation of $\epsilon(t)$
(e.g. at $t\sim 225$ fm/c in the case of $E_{CM}=1$ MeV/u
as we can see in Fig.~\ref{deformation}).
We also observe in this energy domain a good agreement
between the deformations calculated with both methods.
This lowering of the GDR energy due to
deformation is not specific to nuclear physics. Indeed, an energy splitting
of the isovector dipole mode has been observed in fissioning atomic clusters
due to a strong prolate deformation of the fission phase \cite{cal}.
In such systems, the use of LASERs with the "pulse and probe" technique
is expected to give access to the deformation and also to the fission time \cite{din05}.
\subsubsection{non central collisions}
To better mimic the situation of a fusion reaction, we extended our calculations to
non-zero impact parameters. In fact, a non central collision may
excite collective rotational states in the deformed preequilibrated CN.
This rotation may be coupled to the preequilibrium GDR \cite{sur89}. In particular, the interplay of
dipole vibration and deformation can be affected by the rotation. In addition to the
center of mass coordinates with $x$ along the beam axis and $y$ perpendicular to the
reaction plane, we define a new coordinate system $x'$, $y'$, $z'$, where $x'$ is the
deformation axis, and $y=y'$ is the rotation axis (see Fig.~\ref{coordinates}). In
the head-on collision example studied previously, those two frames are the same.
For symmetry reasons, the dipole oscillation cannot occur along the $z=z'$ and $y=y'$ axis.
\begin{figure}
\begin{center}
\epsfig{figure=./coordinates.eps,width=8cm}
\caption{Description of the two frames used in non central collisions.}
\label{coordinates}
\end{center}
\end{figure}
For non-central collisions, the oscillation is only forbidden along the $y=y'$ axis \cite{bon81,sur89}.
In this case the amplitude of the oscillation along $x'$ slightly decreases with the
impact parameter. This decrease becomes significant at rather large impact parameters as we
can see in Fig.~\ref{param} where we have plotted the amplitude of the first oscillation
of the dipole moment along $x'$ (solid line) as a function of the impact parameter.
This decrease is accompanied by an oscillation of the dipole moment along the $z'$ axis
with a smaller amplitude which increases with the impact parameter $b$.
Both amplitudes are of the same order when $b\sim 5$ fm.
\begin{figure}
\begin{center}
\epsfig{figure=./param.eps,width=8cm}
\caption{Amplitude of the first oscillation of the dipole moment along $x'$ (solid line) and along $z'$ (dashed line)
as a function of the impact parameter, $b$, in the $^{12}$Be+$^{28}$S fusion reaction
at an energy of 1 MeV/nucleon in the center of mass.}
\label{param}
\end{center}
\end{figure}
The oscillation along the $z'$ axis results from a weak symmetry breaking due to the
rotation of the system \cite{sim1}. In order to demonstrate this, let us start with the time-dependent
Schr\"odinger equation in the laboratory frame $\mathcal{R}$:
$ i\hbar |\dot{\psi}\> = \hat{H} |\psi\>.$
In the rotating frame $\mathcal{R}'$, the expression of the wave function is
$ |\psi '\> = \hat{R}(\alpha) |\psi\>$ where $\hat{R}(\alpha) = e^{-i\alpha(t)\hat{J}_y}$
is a rotation matrix, $\hat{J}_y$ is the generator of the rotations around $y$ and $\alpha(t)$
is the angle between the two frames (see Fig.~\ref{coordinates}). We express the
Schr\"odinger equation as
$- \hbar \dot{\alpha} \hat{J}_y \hat{R}^{-1}|\psi '\> + i \hbar \hat{R}^{-1} |\dot{\psi} '\> = \hat{H} \hat{R}^{-1}|\psi '\> $
and we get \cite{sim1}
\begin{equation}
i\hbar |\dot{\psi}'\> =\left(\hat{R}\hat{H}\hat{R}^{-1}+\hbar \dot{\alpha}\hat{J}_y\right)|\psi '\>.
\label{schrod}
\end{equation}
Eq.~\ref{schrod} is the Schr\"odinger equation expressed
in the rotating frame $\mathcal{R}'$ of the CN
and $\hat{H}' = \hat{R}\hat{H}\hat{R}^{-1}+\hbar \dot{\alpha}\hat{J}_y$
is the Hamiltonian expressed in this frame.
The last term induces a motion along the $z'$ axis from a dipole vibration along $x'$.
It is quantified by the dipole moment along $z'$ which is plotted
as a dashed line in Fig.~\ref{param}.
This is a clear manifestation of couplings between rotational and vibrational motions in nuclei.
In this subsection we have shown that an N/Z asymmetry in the entrance
channel generates a dipole oscillation during the preequilibrium phase of a fusion reaction.
In the next one we will see that, due to polarization effects, such a motion also occurs
in N/Z symmetric systems although with a smaller amplitude.
\subsection{N/Z symmetric reactions \label{noDelta}}
\begin{figure}
\begin{center}
\epsfig{figure=./qd_sym.eps,width=8cm}
\caption{Time evolution of the total dipole moment for the $^{8}$Be+$^{32}$S$\rightarrow^{40}$Ca reaction
at an energy of 1 MeV/nucleon in the center of mass. At time $t=0$ fm/c, the distance between the centers of mass of the nuclei is 92.8 fm. The arrow indicates the time when the fusion barrier is reached. The dashed line gives the result of the adiabatic model (cf. Eq.~\ref{eq:dnp}).}
\label{sym}
\end{center}
\end{figure}
We now examine the situation of a central collision involving
two $N = Z$ nuclei using the example of $^8$Be+$^{32}$S at $E_{CM}=1$ MeV/nucleon
($^8$Be is bound with a strong prolate deformation in Hartree-Fock calculations with
the SLy4$d$ force). As we can see in Fig.~\ref{sym},
the amplitude of the dipole oscillations is significantly reduced as compared to the
N/Z asymmetric case (cf. Fig.~\ref{asym}-c). In this latter system ($^{12}$Be+$^{28}$S),
the dipole oscillations are generated by the N/Z asymmetry, whereas in the
$^{8}$Be+$^{32}$S reaction, they are only due to the mass
asymmetry of the two collision partners.
Indeed, a mass asymmetry induces a
difference in the isovector polarization in the collision partners.
This polarization is
due to Coulomb repulsion between protons of
the colliding nuclei {\it before} the fusion starts \cite{bon81}.
\begin{figure}
\begin{center}
\epsfig{figure=./polar.eps,width=8cm}
\caption{Schematic representation of the isovector polarization due to Coulomb repulsion
between protons that occurs before fusion. The protons are represented by a solid line and the neutrons by a dotted line. $X_i$ is the position of the center of mass of the nucleus $i$.}
\label{fig:polar}
\end{center}
\end{figure}
To show it, let us use an adiabatic approach in which we consider that the
polarization of a nucleus at a distance $X=X_2-X_1$ between the centers of mass
is generated by the Coulomb field of its collision partner.
$X_i$ is the position of the center of mass of the nucleus $i$.
The distance between the proton and neutron centers
of mass in nucleus $i$ is supposed to be small as
compared to $X$ (see Fig.~\ref{fig:polar}).
The equality between the external Coulomb field and
the restoring force between protons and neutrons leads
to a dipole moment in the nucleus $i$
$$Q_{D_i}(t) \simeq (-1)^i \frac{N_i Z_i Z_j e^2\hbar^2}{A_i {E_{GDR}}_i^2 m X(t)^2}$$
where $i\neq j = 1$ (for $^{32}$S) or 2 (for $^{8}$Be).
The GDR energy is calculated in each collision partner
from the dipole response frequency following a small amplitude dipole boost.
We get $E_{GDR}=23.0$ MeV for $^{32}$S and $E_{GDR_x}=17.2$ MeV
for $^{8}$Be along its deformation axis which is chosen to be aligned
with the collision axis.
The dipole moment in the total system becomes
\begin{equation}
Q_D(t) = \frac{N_1Z_2-N_2Z_1}{A}X(t)+Q_{D_1}(t)+Q_{D_2}(t).
\label{eq:dnp}
\end{equation}
The first term of the right hand side of Eq. \ref{eq:dnp} is usually dominant
for a N/Z asymmetric reaction \cite{bar2}.
However, it vanishes for a N/Z symmetric one.
In this case, one is left with the sum of the dipole moments of the partners.
This simple adiabatic model (dashed line in Fig.~\ref{sym})
gives the good trend of the total dipole moment up to the vicinity of the contact point.
After the fusion starts, the dipole moment increases and oscillates in the
preequilibrium system. The adiabatic model is too simple
to describe this phenomenon.
In fact, due to the polarization,
the nuclear interaction acts first on neutrons and then is expected to modify
strongly the dipole moment at the initial stage of the fusion \cite{bon81}.
The consequence of this polarization in a mass asymmetric system is a
dipole oscillation which can be interpreted, as previously, in term of an excitation
of a preequilibrium GDR. However, the GDR excitation is very small
as compared to the N/Z asymmetric case.
Of course, for a mass and N/Z symmetric reaction no preequilibrium GDR are
allowed for symmetry reason \cite{sim1}.
As we will see in the next section, the special case of an N/Z asymmetric and
mass symmetric system exhibits some interesting behaviors as far as the collective
motions are concerned.
\subsection{mass asymmetry and isoscalar vibrations \label{couplings}}
In this subsection we study the couplings between the isovector dipole motion
and isoscalar vibrations in the preequilibrium phase and their dependence on
the mass asymmetry in the entrance channel.
The dipole motion can be coupled to isoscalar vibrations through the non linearity
of the TDHF equation \cite{sim2,sim1,cho03}.
The presence of such isoscalar vibrations in the preequilibrium system depends on the
structure of the colliding partners and on their mass asymmetry.
For instance a mass symmetric system has a stronger quadrupole deformation
at the touching point than a mass asymmetric one. In such a system a quadrupole vibration
might appear.
Let us start this study with the time evolution of the instantaneous dipole period
\cite{sim1} which is very sensitive to couplings with isoscalar vibrations.
We define this period as being
twice the time to describe half a revolution in the spiral diagram representing the evolution
of the system in the $(P_D,Q_D)$ space.
The resulting evolution is plotted in Fig.~\ref{periode} for two N/Z
asymmetric central collisions:
\begin{itemize}
\item the mass asymmetric $^{12}$Be+$^{28}$S reaction at $E_{CM}=1$~MeV/nucleon.
\item the mass symmetric $^{20}$O + $^{20}$Mg
reaction at $E_{CM}=1.6$~MeV/nucleon.
\end{itemize}
The center of mass energy has been chosen to obtain the same
$E_{CM}/V_B$ ratio
for both reactions ($V_B$ is the Coulomb barrier).
\begin{figure}
\begin{center}
\epsfig{figure=./periode.eps,width=8cm}
\caption{Time evolution of the GDR period for $^{20}$O+$^{20}$Mg at 1.6 MeV/nucleon
(solid line) and for $^{12}$Be+$^{28}$S at 1~MeV/nucleon (dashed line).
Both energies are in the center of mass.}
\label{periode}
\end{center}
\end{figure}
The mean values of the GDR period obtained for the two
reactions are different. For the mass symmetric reaction, this value
is $\simeq 170$ fm/c, whereas in the mass asymmetric case it is
$\simeq 105$ fm/c, in good agreement with the one obtained from
Fig.~\ref{asym} (107 fm/c). This difference is attributed to a larger
deformation of the CN in the mass symmetric case which, in average,
is $\epsilon \sim 0.2$ (from Eq. \ref{eq:def2}), as compared to the mass asymmetric
system ($\epsilon \sim 0.13)$.
Note that it is not appropriate to use Eq.~\ref{eq:defGDR}, to calculate
the deformation from the observed GDR energy frequency for $^{20}$O+$^{20}$Mg
since it is valid only for small deformations.
The dipole moment time evolution for those two reactions
(Figs.~\ref{asym} and \ref{spiraleOMg}), shows that unlike $^{20}$O+$^{20}$Mg,
the oscillations in the $^{12}$Be+$^{28}$S system
are dominated by a single energy. This is consistent
with the evolution of the GDR period in Fig.~\ref{periode} which is
rather constant in the mass asymmetric case whereas it
exhibits strong oscillations in the mass symmetric one. This anharmonicity
can also be seen in the GDR $\gamma$-ray spectrum of the $^{20}$O+$^{20}$Mg
reaction plotted in Fig.~\ref{OMgspectra}. Indeed, one can clearly
identify two peaks in this spectrum at 7.7 MeV and 10.8 MeV.
\begin{figure}
\begin{center}
\epsfig{figure=./spiraleOMg.eps,width=8cm}
\caption{Evolution of the expectation value of the dipole moment, $Q_D$, and its conjugated moment, $P_D$,
in the reactions $^{20}$O+$^{20}$Mg$\rightarrow ^{40}$Ca at an energy of 1.6 MeV/nucleon in the
center of mass.}
\label{spiraleOMg}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{figure=./tfOMg.eps,width=8cm}
\caption{GDR $\gamma$-ray spectrum calculated in the $^{20}$O+$^{20}$Mg$\rightarrow ^{40}$Ca reaction
at an energy of 1.6 MeV/nucleon in the center of mass.}
\label{OMgspectra}
\end{center}
\end{figure}
To better understand what is the origin of the differences
between the two systems, we have
calculated the evolutions of the monopole $Q_0$ and quadrupole $Q_2$
moments defined by Eqs.~\ref{mono} and \ref{momexpres} respectively.
Those evolutions are plotted in Fig.~\ref{t_q0_q2}-a for $^{20}$O+$^{20}$Mg.
We first note that $Q_2$ is always positive, that is,
the compound system keeps a prolate deformation. In addition,
$Q_0$ and $Q_2$ exhibit strong oscillations with the same period $\sim165$ fm/c.
Therefore, we conclude that they have the same origin which
is interpreted as a vibration of the density around a prolate shape \cite{sim1}.
This mode is only excited in the mass symmetric channel:
the evolutions of $Q_0(t)$ and $Q_2(t)$ for the mass asymmetric reaction
($^{12}$Be+$^{28}$S)
at 1 MeV/nucleon in the center of mass (thick lines in Fig.~\ref{t_q0_q2}-b)
do not show any significant oscillation of these moments.
Evolutions of $Q_0(t)$ and $Q_2(t)$ at 1.6 MeV/nucleon in the center of mass
are also plotted (thin lines in Fig.~\ref{t_q0_q2}-b). They do not
exhibit any significant oscillation neither. Therefore, the vibrations observed
in Fig.~\ref{t_q0_q2}-a are not attributed to a difference in the collision energy
but to the mass asymmetry in the entrance channel.
The monopole and quadrupole oscillations
modify the properties of the dipole mode in a time dependent way
\cite{sur89,bar2}.
Let us consider a harmonic oscillator for the dipole motion
with a time dependent rigidity constant.
This is a way to simulate
the non linearities of TDHF. Indeed, the observed oscillation of the
density modifies the restoring force between protons and neutrons.
This is due to the fact that the density enters in the
mean field potential of the TDHF equation (Eq.~\ref{eq:tdhf}). This restoring
force is lower along the deformation axis of a prolately deformed
nucleus than in the perpendicular axis. Thus, variations of the density
profile in the TDHF equation can be modeled by a corresponding variation
of the rigidity constant $k(t)$. In such a model,
the evolution of the dipole moment is given
by the differential equation $ \ddot Q_D(t)+(k(t)/\mu) Q_D(t)=0$ where
$\mu=\frac{NZ}{A}m$ is the reduced mass of the system.
We note $\omega_0$ the average pulsation related
to the rigidity constant given by $k(t)/ \mu=\omega_0^2(1+\eta \cos{\omega t})$,
where $\omega$ is the pulsation of the density oscillation deduced from
Fig.~\ref{t_q0_q2}-a and $\eta$ is a
dimensionless constant which quantifies the coupling between the GDR and
the other collective mode associated to the density vibration. We thus have
\begin{equation}
\ddot Q_D(t)+\omega_0^2[1+\eta \cos{\omega t}]Q_D(t) = 0.
\label{eq:mathieu1}
\end{equation}
This equation is the so called Mathieu's equation \cite{sim1}.
It is interesting to show how we can get this equation from a more microscopic equation like the TDHF one
(Eq.~\ref{eq:tdhf}) in a one dimensional framework.
Following the way of ref. \cite{sch}, the Wigner transform of Eq.~\ref{eq:tdhf}
for a local self consistent potential $V$ is
\begin{equation}
\frac{\partial f }{\partial t} + \frac{p}{m} \frac{\partial f}{\partial x} = \frac{2}{\hbar}\sin\left(\frac{\hbar}{2}\frac{\partial^V }{\partial x}\frac{\partial^f }{\partial p}\right)V f
\label{WF}
\end{equation}
where $f(x,p,t)=\int \!\! ds\,\, \exp(-ip.s/\hbar)\,\rho(x\!+\!\frac{s}{2},x\!-\!\frac{s}{2},t)$
is the Wigner transform of the density matrix $\rho(x_1,x_2,t) = \<x_1|\hat{\rho}(t)|x_2\>$.
The upper indices on the derivative operators in Eq.~\ref{WF} stand for the function on
which the operator acts. We have of course $f = f_p+f_n$ where $f_p$
and $f_n$ are the Wigner transforms of the proton and neutron density matrices respectively.
We now apply the Wigner Function Moment (WFM) method to get a closed system
of dynamical equations for the dipole and its conjugated moments.
We calculate the integrals on the phase space of Eq.~\ref{WF} with
the weights $x \tau$ on the one hand, and $p \tau$ on the other hand ($\tau\!\!=\!\!1$
for protons and $-1$ for neutrons). The distance $D$ between proton and neutron centers of mass can be written as
$D=\int \!\! dx \, dp\,\, x\, (f_p -f_n)$ and we get
$$
\dot D + \int \!\!\! dp \,\,\frac{p}{m} \,\int\!\!\! dx\, \,x \,\frac{\partial}{\partial x} \,(f_p - f_n) $$
$$
= \frac{2}{\hbar} \int \!\!\! dx\,dp\,\, x \,\sin\left(\frac{\hbar}{2}\frac{\partial^V }{\partial x}\frac{\partial^f }{\partial p}\right)V \,(f_p - f_n)
$$
where the time dependence has been omitted for simplicity.
The right hand side term is the integral of multiple $p$-derivatives of $f$ so it vanishes because $f_p$, $f_n$ and all their $p-$derivatives
vanish for $|p|\rightarrow \infty$. With $P$ being the relative momentum between
protons and neutrons $P = \int dp\, dx \,p \,(f_p-f_n)$ we get
\begin{equation}
\dot D = \frac{P}{m}.
\label{int1}
\end{equation}
We now calculate the integral of Eq.~\ref{WF} with the weight $p\tau$. Noting
the matter density $n(x,t) = \int dp \,f(x,p,t)$ and the kinetic energy density
${\mathcal A}(x,t) = \frac{1}{m} \int dp\, p^2 \,f(x,p,t)$ we have
$$
\dot{P}+\int \!\!\! dx \,\, \frac{\partial}{\partial x} \left({\mathcal A}_p-{\mathcal A}_n\right)=-\int \!\!\! dx \,\, \frac{\partial V}{\partial x}\left(n_p-n_n\right).
$$
Using ${\mathcal A}=0$ for $|x| \rightarrow \infty$ we have
\begin{equation}
\dot{P} = -\int \!\!\! dx \,\, \frac{\partial V}{\partial x}\left(n_p-n_n\right).
\label{int2}
\end{equation}
Eqs.~\ref{int1} and \ref{int2} are the system of dynamical equations of
motion we were looking for. It is important to stress that this system of equations
is obtained without approximation for a local potential. To go further, we need
an explicit form of the potential. If we consider for instance a harmonic oscillator
$V=kx^2/2$, we obtain the dipole moment evolution equation:
$ m\ddot{D} = -k\, D$ with the solution $D = D_0 \cos{\omega_0 \,t}$, where $\omega_0=\sqrt{k/m}$.
If a breathing mode occurs at a pulsation $\omega$, then the density $n(x,t)$ oscillates with the pulsation $\omega$:
$n(x,t) = n_0(x)\left[1+\lambda(x)\cos{\omega t}\right]$.
Since the potential is self consistent, it also presents oscillations which are a function of $\cos{\omega t}$: $V(x,t)\equiv V(x,\cos{\omega t})$.
We assume for this potential the separable form
$V(x,t)=V_0(x)\left(1+{\mathcal F}\left[{\cos{\omega t}}\right]\right)$,
where $V_0(x)$ is the potential when no breathing mode is excited.
Using a harmonic picture for $V_0$, that is, $V_0(x) = \frac{1}{2}m\omega_0 x^2$,
we get from Eqs.~\ref{int1} and \ref{int2} the equation for the dipole moment
$Q_D = \frac{NZ}{A}D$:
\begin{equation}
\ddot{Q}_D(t)+\omega_0^2\left(1+{\mathcal F}[{\cos{\omega t}}]\right)Q_D(t) = 0.
\label{WTresult}
\end{equation}
We finally see that the Mathieu's equation (Eq.~\ref{eq:mathieu1}) appears
to be an approximation of Eq.~\ref{WTresult} where only the linear part of
the function ${\mathcal F}(\xi)\simeq \eta \xi$ is conserved.
\begin{figure}
\begin{center}
\epsfig{figure=./t_q0_q2.eps,width=8cm}
\caption{Evolution with time of the monopole ($Q_0$, solid line) and quadrupole ($Q_2$, dashed line)
moments in the reactions $^{20}$O+$^{20}$Mg$\rightarrow ^{40}$Ca at 1.6 MeV/nucleon (a) and
for $^{12}$Be+$^{28}$S at 1 MeV/nucleon (thick lines) and 1.6 MeV/nucleon (thin lines) (b). Both energies are in the center of mass.}
\label{t_q0_q2}
\end{center}
\end{figure}
We have solved the Mathieu's equation numerically with
a set of parameters suitable for our problem.
The pulsation of the density oscillation is extracted from Fig.~\ref{t_q0_q2}-a and we get $\omega \simeq 7.5$ MeV$/\hbar$.
For the pulsation of the GDR we choose the main peak at $\omega_{GDR}\simeq 7.7$ MeV/$\hbar$ (see Fig.~\ref{spiraleOMg}).
It is related to the pulsation $\omega_0$ by the relation $\omega_0 = r \omega_{GDR}$.
The constants $r$ and $\eta$ are tuned to reproduce approximatively
the TDHF results period.
The parameter $r$ is expected to be close to $1$ but not exactly $1$
because of the presence of the oscillating term which may slightly change the
mean value of the dipole pulsation. The solution of the Mathieu's equation
oscillates with a time-dependent period which reproduces the TDHF case quite well
with $r \simeq 1.1$ and $\eta\simeq 0.5$ (see Fig.~\ref{mathieu}).
\begin{figure}
\begin{center}
\epsfig{figure=./mathieu.eps,width=8cm}
\caption{Time evolution of the GDR period calculated for the reaction $^{20}$O+$^{20}$Mg$\rightarrow$ $^{40}$Ca
at an energy of 1.6 MeV/nucleon in the center of mass
(solid line) and its modelization by the Mathieu's equation (dashed line).}
\label{mathieu}
\end{center}
\end{figure}
In a recent paper \cite{cho03}, following the formalism developed
in a study of non linear vibrations \cite{sim2}, we related $\eta$
to a matrix element of the residual interaction coupling collective states.
As a consequence, the excitation of collective modes such as the quadrupole
and monopole vibrations
is coupled to the preequilibrium GDR. Such vibrations occur only in the
mass symmetric reaction we studied. The effects of this coupling are a reduction
of the GDR energy (estimated around 10 per cent in this case) and an additional
spreading of the resonance line shape due to the modulation of the dipole frequency.
\subsection{comparison with experiments \label{compar}}
As a test case, we have performed TDHF calculations of the reactions studied
by Flibotte {\it et al.} \cite{fli}. In this paper, two systems have been investigated:
an N/Z asymmetric one ($^{40}$Ca+$^{100}$Mo) and an N/Z quasi-symmetric one
($^{36}$S+$^{104}$Pd) at a center of mass energy of 0.83~MeV/nucleon.
These systems have been chosen because they lead to the
same composite system ($^{140}$Sm).
The corresponding dipole evolutions obtained from TDHF
are plotted in Fig.~\ref{spiral_fli_asym} for the N/Z asymmetric reaction and
in Fig.~\ref{spiral_fli_sym} for the N/Z quasi-symmetric one.
A dipole oscillation is observed in both reactions but with a stronger amplitude
in the N/Z asymmetric one.
\begin{figure}
\begin{center}
\epsfig{figure=./spirale_fli_asym.eps,width=8cm}
\caption{Evolution of the expectation value of the dipole moment, $Q_D$, and its conjugated moment, $P_D$,
in the case of the N/Z asymmetric reaction
$^{40}$Ca+$^{100}$Mo at a center of mass energy of 0.83 MeV/nucleon.}
\label{spiral_fli_asym}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{figure=./spirale_fli_sym.eps,width=8cm}
\caption{Evolution of the expectation value of the dipole moment, $Q_D$, and its conjugated moment, $P_D$,
in the case of the N/Z quasi-symmetric reaction
$^{36}$S+$^{104}$Pd at a center of mass energy of 0.83 MeV/nucleon.}
\label{spiral_fli_sym}
\end{center}
\end{figure}
The preequilibrium GDR $\gamma$-ray spectra for those reactions are
calculated using Eq.~\ref{fourier} and plotted in Fig.~\ref{fourier_fli}-a.
The area under the peak associated to the N/Z asymmetric reaction (solid line) is considerably
larger than the one under the N/Z quasi-symmetric one (dashed line).
To estimate the importance of the preequilibrium $\gamma$-ray emission
with respect to the statistical decay and its role on the fusion process, we have calculated the spectrum
associated to the first chance statistical $\gamma$-ray decay.
It is obtained from the $\gamma$-ray emission probability in all directions per energy
unit assuming an equilibrated CN \cite{sno,bar3,bri}. Its expression is
\begin{equation}
\frac{dP}{dE_\gamma} = \frac{4\alpha}{\pi m c^2}\frac{ \Gamma_{GDR}}{ \Gamma_{CN}}\frac{NZ}{A}\frac{E_\gamma^4\,\, e^{-\frac{E_\gamma}{T}}}{\left(E_\gamma^2-E_{GDR}^2\right)^2+\Gamma_{GDR}^2E_\gamma^2}
\label{stat}
\end{equation}
where $m$ is the nucleon mass, $\Gamma_{GDR}$ and $E_{GDR}$ are the width and the energy of
the statistical GDR respectively, and $T$ is the temperature of the equilibrated CN. At first order, the
energy of the GDR does not depend on the temperature \cite{gaa}.
We use the values $E_{GDR}=15$ MeV and $\Gamma_{GDR}=7$ MeV.
Following the same method as the one employed in Ref. \cite{bar3}, we
approximate the CN width $\Gamma_{CN}$ with the total neutron width
\begin{equation}
\Gamma_{CN} \simeq \Gamma_n = \frac{2 m r_0^2 A^{\frac{2}{3}}}{\pi \hbar^2}T^2 e^{-\frac{B_n}{T}}
\label{eq_Gamma_n}
\end{equation}
where $B_n = 8.5$ MeV is the neutron binding energy and $r_0=1.2$ fm.
The temperature $T$ is calculated from the equation
\begin{equation}
T=\sqrt{\frac{E^*}{aA}}
\label{temperature}
\end{equation}
where $a \simeq 1/10$ MeV$^{-1}$ is the level density parameter
and $E^*=71$ MeV is the excitation energy.
The resulting spectrum is plotted in
Fig.~\ref{fourier_fli}-a (dotted line).
We note that the N/Z
asymmetric preequilibrium spectrum is comparable in intensity to the first
step statistical one.
This fact has already been pointed out by Baran {\it et al.} \cite{bar3}
who got a similar spectrum for the N/Z asymmetric
reaction with a semiclassical approach.
Another important conclusion which can be drawn from Fig.~\ref{fourier_fli}-a
is the lowering of the GDR $\gamma$-ray energy for the non
statistical part as compared to the statistical one
which is attributed to the deformation of the nucleus (see sec. \ref{defor}).
This phenomenon is also reported by Baran {\it et al.} \cite{bar3}.
In fact we get from Fig.~\ref{fourier_fli}-a a position of the peak
of about 7.5 MeV for the preequilibrium GDR while Baran {\it et al.} obtained
$\sim$9 MeV.
On the experimental side,
the $\gamma$-ray spectra are dominated by a statistical background
decreasing exponentially. In addition to this background,
the GDR creates a bump located around the GDR energy (Fig.~1 of Ref. \cite{fli}).
To get rid of the statistical background, the authors of \cite{fli} linearized
the $\gamma$-ray spectra by dividing them by a theoretical statistical background.
The resulting spectra are plotted in Fig.~2 of Ref. \cite{fli}.
This procedure is used by the authors to determine the preequilibrium to statistical ratio
for the GDR component. However it cannot be used to determine the positions in energy
of the peaks because the division by an exponential background induces
a shift in energy which is different for both contributions
(statistical and preequilibrium)
if they are not centered around the same energy, as expected from Fig.~\ref{fourier_fli}-a.
We modified the procedure as follows. First, we assume
that no preequilibrium $\gamma$-ray is emitted
in the N/Z quasi-symmetric reaction $^{36}$S+$^{104}$Pd.
We then subtract the total $\gamma$-ray spectrum
associated to the quasi-symmetric reaction from the N/Z asymmetric one.
These two spectra are plotted in Fig.~1 of Ref. \cite{fli}.
The result of this subtraction is the preequilibrium component of the GDR
in the reaction $^{40}$Ca+$^{100}$Mo, and is plotted in Fig.~\ref{fourier_fli}-b.
The error bars are both statistical and systematic due to the graphical extraction of the data.
Below 5 MeV the systematic error is to high to get relevant data.
Focusing on the energy position of the preequilibrium component,
we note a good agreement between TDHF predictions and experimental data.
To conclude, we extracted from existing data, for the first time,
an experimental observation of the lowering of the preequilibrium
GDR predicted by our TDHF calculations.
This analysis shows that the preequilibrium GDR is, indeed, a powerful
experimental tool to study the fusion path.
Another application of N/Z asymmetric fusion reactions
is proposed in the next section.
\begin{figure}
\begin{center}
\epsfig{figure=./tf_fli.eps,width=8cm}
\caption{
{\it a) } preequilibrium GDR $\gamma$-ray spectrum calculated in the reactions
$^{40}$Ca+$^{100}$Mo (solid line) and $^{36}$S+$^{104}$Pd (dashed line).
The dotted line represents the first chance statistical $\gamma$-ray decay spectrum.
{\it b) } Experimental data resulting from the subtraction
of the $\gamma$-ray spectra obtained by Flibotte {\it et al.} \cite{fli}
in the reactions $^{40}$Ca+$^{100}$Mo and $^{36}$S+$^{104}$Pd.
}
\label{fourier_fli}
\end{center}
\end{figure}
\section{Fusion/Evaporation Cross Sections of heavy nuclei}
As mentioned in \cite{bar3}, the emission of a preequilibrium GDR $\gamma$-ray
decreases the excitation energy hence the initial temperature of
the nucleus reaching the statistical phase. The emission of preequilibrium
particles, which can be controlled in our example by the N/Z asymmetry, is
thus a new interesting cooling mechanism for the formation of Heavy and
Super Heavy Elements. For such nuclei, the statistical fission
considerably dominates the neutron emission and the survival
probability of the CN becomes very small.
SHE must be populated at low excitation energy. Firstly, because the
smaller the excitation energy,
the smaller the fission probability. Secondly, because the shell corrections decrease
with excitation energy \cite{ign79}. These corrections are responsible
for the stability of the transfermiums nuclei ($Z>100$) in their ground state.
The quantum stabilization
decreases quite rapidly with excitation energy until the fission barrier vanishes.
Those two reasons are strong motivations to
study the cooling mechanisms
involved in the preequilibrium phase of the CN formation.
In the following, we expose one cooling mechanism responsible for the predicted
enhancement of the survival probability in the case of a N/Z asymmetric reaction.
As an illustration, we treat only the $\gamma$-emission part of the preequilibrium
GDR decay. Although it may play an important role, we do not treat the
preequilibrium neutron emission for two reasons:
\begin{itemize}
\item Only the direct neutron decay of giant resonances can be assessed
in TDHF. Then, we would be able to describe
only a small part of this neutron emission,
the other parts being the sequential and statistical decays.
Missing the sequential decay would be a strong limitation of the description.
\item We would need not only the number of emitted neutrons, but also their energy.
Consequently, huge spatial grid would have to be used in order to perform
a spatial Fourier transform of the single particle wave functions, which is
out of range of three dimensional TDHF codes
because of computational limitations.
\end{itemize}
Let us define $P_{E^*_{init.}}(E^*)$ the survival probability at an excitation
energy $E^*$ of a CN which started its statistical decay at the energy $E^*_{init}$.
We also note $P_{surv}^S$ and $P_{surv}^A$ the final survival probabilities of
the CN formed by N/Z symmetric and asymmetric reactions respectively.
\begin{figure}
\begin{center}
\epsfig{figure=./stat.eps,width=8cm}
\caption{Schematic representation of the CN population during the statistical decay
in the case of an N/Z symmetric collision (a) and an N/Z asymmetric reaction (b).}
\label{stat_decay}
\end{center}
\end{figure}
Fig.~\ref{stat_decay}-a illustrates schematically the evolution of the survival
probability ({\it x}-axis) when the excitation energy decreases ({\it y}-axis)
in a case of an N/Z symmetry in the entrance channel.
In this case, no $\gamma$-ray emission is expected in the preequilibrium phase
and the initial excitation energy is always maximum $E^*_{init}=E^*_0$,
where $E^*_0=Q+E_{cm}$ is the excitation energy when no preequilibrium particles
are emitted, $E_{cm}$ is the center of mass energy and $Q=(M_1+M_2-M_{CN})c^2$.
During the statistical decay, the excitation energy decreases mainly through
neutron emission, but at the same time the survival probability of the compound nucleus
decreases too.
For instance, when the excitation energy reaches $E^*_1=E^*_0-E_{GDR}$, the survival
probability $P_1=P_{E^*_0}(E^*_1)$ at this energy might be small.
At the end of the decay, when the excitation energy is zero, the survival probability becomes
$P_{surv}^S = P_{E^*_0}(0) = P_1 P_{E^*_1}(0). $
Fig.~\ref{stat_decay}-b shows the
same for an N/Z asymmetric reaction. In this last case, the nucleus can emit a
preequilibrium GDR $\gamma$-ray with a probability $P_\gamma$.
The nuclei which
emit such a $\gamma$-ray begin the statistical decay at a lower energy $E_{init}=E^*_1$,
whereas those which did not emit a $\gamma$-ray still starts their decay at $E_{init}=E^*_0$.
The probability for the latter case is $1-P_\gamma$.
The survival probability at the end of the decay then reads
$P_{surv}^A = \left[\left(1-P_\gamma\right) P_1+ P_\gamma\right] P_{E^*_1}(0)$.
The ratio of the survival probabilities between the N/Z symmetric and asymmetric cases is
\begin{equation}
\frac{P_{surv}^A}{P_{surv}^S} = 1+ \frac{P_\gamma}{P_1} \left(1-P_1\right).
\label{enhancement}
\end{equation}
We now use a simple model to get an estimate of this quantity.
It is clear that, to get a quantitative predictions of survival probabilities,
the studied mechanism has to be included in more elaborated statistical models,
which is beyond the scope of this paper.
The probability $P_{\gamma}$ can be calculated by integrating Eq.~\ref{fourier}
over the energy range. This can be
done for example with a TDHF calculation or using the classical electrodynamic formulae
from Ref.~\cite{jac}. Following these formulae, we approximate the probability to
emit a preequilibrium GDR $\gamma$-ray per interval of energy by
\begin{eqnarray}
\frac{dP_\gamma}{dE} & = & \frac{2 e^2 Q_{D}(0)^2}{3\pi (\hbar c)^3}\left(E_1^2+\frac{{\Gamma_{GDR}}^2}{4}\right) \nonumber \\
& & \frac{E_1^2 E}{\left[(E-E_1)^2+\frac{{\Gamma_{GDR}}^2}{4}\right]\left[(E+E_1)^2+\frac{{\Gamma_{GDR}}^2}{4}\right]} \nonumber
\end{eqnarray}
where $E_1=\sqrt{E_{GDR}^2-\frac{{\Gamma_{GDR}}^2}{4}}$
is the ``shifted'' energy of the damped harmonic motion
and ${\Gamma_{GDR}}$ is the damping width of the preequilibrium GDR.
The initial value of the dipole moment, $Q_D(0)$, can be estimated from Eq. \ref{eq:dnp}
at the touching point and neglecting the polarization of the collision partners \cite{bar2}.
We get
$$ Q_D(0)\simeq \frac{R_1+R_2}{A} \left( Z_1N_2-Z_2N_1\right)$$
where $R_i$ is the radius of nucleus $i$.
To determine $P_{E^*_0}(E^*_1)$, we need to solve a system of six
equations: Eqs.~\ref{eq_Gamma_n}, \ref{temperature} and
\begin{equation}
\frac{dE^*}{dt} = - \frac{\Gamma_n(t)}{\hbar}(B_n+T(t))
\label{Eex}
\end{equation}
\begin{equation}
\frac{dP}{dt} = - \frac{\Gamma_f(t)}{\hbar}P(t)
\label{Psurv}
\end{equation}
\begin{equation}
\Gamma_f(t) = \frac{\hbar \omega_0 \omega_s}{2\pi \beta} e^{-\frac{B_f(t)}{T(t)}}
\label{eq_Gamma_f}
\end{equation}
\begin{equation}
B_f(t) \equiv B_f[E^*(t)] = B_f(0) e^{-\frac{E^*}{E_d}}
\label{Bf}
\end{equation}
Eq.~\ref{Eex} gives the evolution of the excitation energy, assuming as in \cite{bar3}
that the CN width
can be identified to the neutron width.
This implies that we neglect the statistical gamma emission.
This choice is justified by the fact that the statistical neutron emission
is much more probable than the gamma emission in the excitation
energy domain of interest where the fission dominates,
which is above the neutron emission threshold $B_n$.
Eq.~\ref{Psurv} gives the evolution of the survival probability against fission $P$. Eq.~\ref{eq_Gamma_f}
gives the evolution of the fission width. The parameters $\omega_0$ and $\omega_S$ are the oscillator
frequencies of the two parabolas approximating the potential $V(x)$ in the first minimum
and at the saddle point respectively. The variable $x$ is related to the distance between
the mass centers of the nascent fission fragments (see \cite{ari}) and $\beta = 5\times 10^{21}$s$^{-1}$
is the reduced friction. Eq.~\ref{Bf} gives the evolution of the fission barrier $B_f$.
For SHE, this barrier has only a quantum nature and vanishes at high excitation energy.
$E_d\simeq 20$ MeV is the shell damping energy \cite{ari}. We consider that a CN with an excitation
energy between $E^*_1$ and $E^*_0$ decays only by fission or neutron emission.
We take here the example of the reaction $^{124}$Xe+$^{141}$Xe$\rightarrow ^{265}$Hs$^*$
at the fusion barrier ($E_{cm}=B_{fus}$), that is, an excitation energy $E^*_0=54$ MeV. With
an energy and a width of the GDR of 13 MeV and 4 MeV respectively, the preequilibrium $\gamma$-ray emission
probability is $P_\gamma \simeq 0.05$.
For the statistical decay we take $B_f[E^*=0]\simeq 8.5$
MeV, $B_n = 6.5$ MeV and $\omega_0 \simeq \omega_S \simeq 1$ MeV/$\hbar$.
We also get a survival probability $P_{E^*_0}(E^*_1) \simeq 0.01$
which is small as compared to $P_\gamma$.
Following Eq.~\ref{enhancement}, the enhancement of the total survival probability
due solely to the N/Z asymmetry in the entrance channel becomes
$P_{surv}^A/P_{surv}^S \sim 6$.
To conclude, we see that such an effect may be useful for the formation of Heavy and
Super Heavy Elements. Indeed, based on our conclusions, very asymmetric N/Z collisions induced
by radioactive ion beams that are coming online in several laboratories, should allow the synthesis
SHE with a larger cross sections than are obtainable with beams of stable isotopes.
\section{conclusion}
In this paper we have performed TDHF calculations to study in some details the properties of the
preequilibrium GDR that can be excited before the formation of a fully equilibrated CN.
We have shown that this probe can be used to better understand
the early stage of the fusion path, and more precisely the charge equilibration.
We have clarified the role of the N/Z and/or mass asymmetries on the GDR excitation.
The energy of the preequilibrium GDR is expected to decrease with excitation energy, an effect
attributed to a strong prolate shape associated to the fused system.
We presented the first experimental indication of this shift in energy.
The calculations for an N/Z asymmetric collisions at non zero impact parameters have been performed and
revealed couplings between the dipole oscillations and the CN rotation. Other couplings
between vibrational modes for mass symmetric reactions have also been studied.
Finally we suggest that the use of N/Z asymmetric fusion reactions is a good choice to synthesize
Heavy and Super Heavy Elements. In that case, the preequilibrium GDR $\gamma$-ray emission cooling
mechanism might be well suited to reach the statistical phase with a low excitation energy yielding
a larger survival probability against fission. The availability of radioactive beams with large
N/Z asymmetry and sufficient intensities for these kind of studies will be extremely useful to
check experimentally our predictions in the near future.
\begin{acknowledgments}
This paper is dedicated to the memory of P. Bonche, the author of the TDHF code we used.
We are grateful to M. Di Toro for a useful reading of the manuscript.
We also thank V. Baran, M. Colonna, D. Lacroix, D. Boilley
and J. P. Wieleczko for several fruitful discussions, and P. Schuck for providing a pertinent reference.
\end{acknowledgments}
|
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| 2,664
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The male lead of a smash hit West End musical says Derry audiences are in for a treat when the production comes to Derry next week.
Mike Sterling plays the role of former Argentine dictator Juan Perón the in the musical, which charts the rise of his wife Eva from peasant to spiritual leader of the nation.
"It is a real love story," explained Mike, who is best known for his long running role as The Phantom in 'The Phantom of the Opera'.
"Their relationship starts off like a sort of contract or a business arrangement but he falls in love with her.
"Whatever people think of dictators this guy is clever, he studies everything.
The production is being brought to Derry straight from its West End run with all the production values that entails.
"They have really gone to town with the set, the orchestra, the cast- it's just incredible," said Mike.
"This type of production is not seen very often.
"Eva is played by Lucy O'Byrne who is really incredible and she is just 27-years-old.
"She takes Eva from a 15-year –old girl to a 33-year-old woman.
"And that journey is really something.
And Mike says he looking forward to bringing the production to the North West.
"This will be my first time in Derry but I'm very much looking forward to seeing what it has to offer.
Evita performs at the Millennium Forum from Tuesday October 9 to Saturday October 13. Tickets are now available from the Box Office. Telephone 71 264455 or visit millenniumforum.co.uk for bookings.
|
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{"url":"https:\/\/www.physicsforums.com\/threads\/can-we-sum-out-the-vacuum-state-0-rangle-langle-0.774629\/","text":"# Can we sum out the vacuum state $|0\\rangle\\langle 0|$ ?\n\n1. Oct 6, 2014\n\n### ccnu\n\nFor example, when we write down the operator definition of quark fragmentation matrix element:\n$\\Phi_{ij} = \\sum_X \\int d^4 x e^{ikx}\\langle 0|\\psi_i(x)|P,X\\rangle\\langle P,X|\\bar{\\psi}_j(0)|0\\rangle$.\nCan we rewrite is as:\n$\\Phi_{ij} = \\sum_X \\int d^4 x e^{ikx}\\langle P,X|\\bar{\\psi}_j(0)|0\\rangle\\langle 0|\\psi_i(x)|P,X\\rangle = \\sum_X \\int d^4 x e^{ikx}\\langle P,X|\\bar{\\psi}_j(0)\\psi_i(x)|P,X\\rangle$?\n\n2. Oct 6, 2014\n\n### Orodruin\n\nStaff Emeritus\nNo, this is not allowed. The entity $\\left|0\\rangle\\langle 0\\right|$ is a projection operator onto the vacuum state and not the unit operator.\n\nKnow someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook","date":"2018-01-17 20:30:59","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\": 0.7931057214736938, \"perplexity\": 3066.0269740080157}, \"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-2018-05\/segments\/1516084886964.22\/warc\/CC-MAIN-20180117193009-20180117213009-00652.warc.gz\"}"}
| null | null |
AJW
Supermarket sweep for mysterious Mario Lanza artist
November 24, 2011 christoff3000AJW, Beacon Hotel, beermat artist, Mario Lanza, morrison's, Sedgley5 Comments
Following on from a visit to Sedgley's Beacon Hotel earlier this year, it seems local legend AJW, renowned for leaving beermats in pubs featuring a drawing of 1950's star Mario Lanza, has been getting hungry for more exposure.
Anthony Arnold, a regular reader of Sedgleyscene, was amazed to discover the mysterious artist's latest calling card drawn on an empty box of chicken pies in the somewhat unglamorous surroundings of a local Morrison's supermarket toilet.
He says: "On Monday, November 21, I found a drawing of Mari Lanza . It had been drawn on the back of one of those frozen chicken packages. It's got the usual picture and signature AJW and the Facebook.com under the picture and a Yahoo address down the side. No message except 'Super Mario!' written on it. I found it on top of the thingy machine in the gents toilet at Morrison's, Wednesbury. I thought he only did them on beer mats – don't know if it's the real thing or what."
Could this be the latest calling card from the mysterious AJW?
Impressed by his discovery, Anthony took a picture, which is posted here. All the trademark signs are there – the line drawing of Mario Lanza, together with this anonymous artist's Facebook page and email addresses. However, Tony is unsure as to whether this latest offering is the real deal and is appealing to Sedgleyscene readers to help.
The mysterious drawings, which have been appearing on beermats in pubs since the 1970s, have continued to baffle local residents.
No-one seems to know why each inscription by this phantom artist – who simply calls his or herself AJW – bears a dedication to Mario Lanza, a famous American tenor singer and actor who died in 1959.
This enduring mystery has led to a Facebook page being set up in this mysterious character's honour which now has 172 members. It seems AJW also likes to get around, leaving calling cards in pubs and other public buildings across the Black Country and Birmingham – often with personal dedications to members of staff.
Some avid followers of AJW also believe there could be several artists at work – especially as the original phantom would now be more than 80 years old – but no-one has been able to confirm this.
Can you help Sedgleyscene and Anthony confirm this latest discovery? Get in touch with us at sedgleyscene@gmail.com We'd love to hear from you.
|
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| 263
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#ifndef CTKEASYNCMASTERTHREAD_P_H
#define CTKEASYNCMASTERTHREAD_P_H
#include "ctkEAInterruptibleThread_p.h"
/**
* Used to execute synchronuous tasks. This is not done
* in the main thread because we need to be able to
* interrupt the thread waiting on a synchronuous task
* (for example on a timeout). ctkEAInterruptibleThread
* provides this capability.
*/
class ctkEASyncMasterThread : public QObject
{
Q_OBJECT
public:
ctkEASyncMasterThread();
void syncRun(QRunnable* command);
void stop();
public Q_SLOTS:
void runCommand();
private:
ctkEAInterruptibleThread thread;
QRunnable* command;
QMutex mutex;
QWaitCondition waitCond;
};
#endif // CTKEASYNCMASTERTHREAD_P_H
|
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"redpajama_set_name": "RedPajamaGithub"
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| 3,635
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Bayot may refer to:
People
Robert Bayot
Miguel Bayot
Margaret-Ann Bayot
Adolphe Jean-Baptiste Bayot
Other uses
Bayot language
Bayot, a Cebuano word for effeminate men
|
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| 4,294
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{"url":"https:\/\/physicshelpforum.com\/threads\/evaporating-water-with-a-boiler-type-vessel.15396\/","text":"# Evaporating Water with a Boiler Type Vessel\n\n#### tommyb1982\n\nI am attempting to evaporate water to the atmosphere with a boiler type vessel. Normal operating pressure of the vessel would be 5 psig. How much water could potentially be evaporated? Also, what issues come to mind with this concept?\n\n#### Woody\n\nIf you are evaporating to Atmosphere, there is no limit to the amount of water that could be evaporated.\nHowever if you are limited to 5 psi over atmospheric, there will be a limit to how fast you can evaporate the water.\n\nThe actual limit will depend on lots of details you have not supplied;\n\nThe energy rate available from your source of heat\nThe amount of Water,\nThe size of the Vent from your Boiler to Atmosphere,\netc...\n\n#### benit13\n\nYou can estimate the rate of evaporation using the equation for latent heat transfer:\n\n$$\\displaystyle \\dot{m} = \\frac{\\dot{Q}}{h_{liq-vap}}$$\n\nwhere m is the evaporation rate (in kg\/s), Q is the rated power output of your boiler (in W) and $$\\displaystyle h_{liq-vap}$$ is the specific enthalpy of liquid-to-vapour phase change of water.\n\nAt room temperature and pressure, $$\\displaystyle h_{liq-vap} = 2265$$ kJ\/kg, but it's actually a function of temperature and pressure, so it's probably a little bit lower than this. You should recalculate this constant at the boiling point of water at that pressure (which I think is about 108 deg. C at 134 kPa).","date":"2019-12-14 18:56:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7070016264915466, \"perplexity\": 717.430160000273}, \"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-2019-51\/segments\/1575541288287.53\/warc\/CC-MAIN-20191214174719-20191214202719-00325.warc.gz\"}"}
| null | null |
I Am Kuba
Pop | I Am Cuba is a Czech singer who writes songs himself and tries to combine them in a straightforward melodic pop with an independent sound. "Influenced me Bee Gees, Daft Punk and Damien Rice," indicates the area which would be at home to move efficiently. Among its advantages are also versatile talent in playing musical instruments - able to play the guitar, piano, percussion and violin .
{gt}
Vergangene Konzerte
© 2023 GoOut, s.r.o., Tschechien
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\label{intro}
Surface parameterization refers to the process of bijectively mapping a complicated surface to a simple canonical domain. In recent years, the use of parameterizations has been widespread in computer graphics and geometry processing. The applications of parameterizations include surface registration, texture mapping, mesh editing, remeshing, morphing, detail synthesis, mesh compression and medical visualization. For instance, in surface registration, which aims to find a one to one correspondence between two surfaces, it is common to parameterize the surfaces to simpler domains, such as the unit disk $\mathbb{D}$ or the unit sphere $\mathbb{S}^2$, to simplify the computation. This approach has been widely used in medical imaging for obtaining the surface registrations between anatomical structures, such as the cortical surfaces \cite{gu04,lui07,lam13} and the hippocampal surfaces \cite{lui10}. In texture mapping \cite{tutte02}, the geometric details and material properties are usually modeled as texture images. With the aid of the parameterizations of 3D meshes to the planar domain, the texture can be glued to the meshes. Besides, parameterizations are applied to solve PDEs on complicated 2D domains \cite{krichever04,krichever05,lui08}. The abovementioned applications reflect the importance of obtaining a good parameterization.
Numerous studies have been devoted to surface parameterizations. For simply-connected open surfaces, it is common to parameterize the surfaces to a unit disk. To achieve a meaningful parameterization, the distortion of certain geometric quantities, such as distance, area, and angle, should be minimized. By the Riemann mapping theorem, a simply-connected open surface is conformally equivalent to the unit disk $\mathbb{D}$. In other words, the existence of a disk conformal parameterization is theoretically guaranteed. Therefore, it is desirable to look for a disk conformal parameterization.
Conformal parameterization of disk-type surfaces has been a difficult topic in surface parameterization theory for a long time. Different research groups have developed brilliant algorithms to tackle the problem. Despite the effectiveness of the state-of-the-art approaches, there are still opportunities for further enhancements in the computational time and the conformality of the parameterizations. Firstly, as most of the latest algorithms are nonlinear, the computation is quite inefficient. This becomes an obstacle for practical applications in which a large number of surfaces are involved. Secondly, the conformality distortion is still far from negligible. The distortion affects the accuracy of the parameterizations, and thus hinders practical applications. There are two sources of the conformality distortion. One of the sources is the discretization of the surfaces and the operators in different algorithms. Since the surfaces are usually represented as triangulated meshes, the operators are discretized. Although the conformality of the parameterizations is theoretically guaranteed in the continuous case, certain numerical angular distortions inevitably exist for the discrete case under any algorithms. Another source is the limitations of the algorithms themselves due to different assumptions and conditions in the algorithms. In this work, we aim to develop a numerical method for disk conformal parameterizations that overcomes the above mentioned obstacles. First, we propose to speed up the computation by linearizing the algorithm as much as possible. To enhance the accuracy, we then propose a simple two-step iteration to correct the conformality distortion with the aid of quasi-conformal theories. Experimental results suggest that our proposed method outperforms other state-of-the-art approaches.
The rest of the paper is organized as follows. In Section \ref{previous}, we review the previous works in the literature related to our work. In Section \ref{contribution}, we outline the contributions of our work. Our proposed method is explained in details in Section \ref{main}. In Section \ref{implementation}, we describe the numerical implementation details of our proposed method. In Section \ref{experiment}, we show and analyze the experimental results of our proposed method. The paper is concluded in Section \ref{conclusion}.
\begin{table}[t]
\centering
\begin{tabular}{ |C{45mm}|c|c|c|c| }
\hline
Methods & Boundary & Bijectivity & Complexity\\ \hline
Shape-preserving \cite{floater97} & Fixed & Yes & Linear \\ \hline
MIPS \cite{hormann00} & Free & Yes & Nonlinear \\ \hline
ABF/ABF++ \cite{sheffer00,sheffer05} & Free & Local (no flips) & Nonlinear\\ \hline
LSCM \cite{desbrun02,levy02} & Free & No & Linear\\ \hline
Mean-value \cite{floater03} & Fixed & Yes & Linear\\ \hline
Circle patterns \cite{kharevych05} & Free & Local (no flips) & Nonlinear \\ \hline
Spectral conformal \cite{mullen08} & Free & No & Linear \\ \hline
Double covering \cite{jin05} & Free & No & Nonlinear \\ \hline
Discrete Ricci flow \cite{jin08}& Fixed & Yes & Nonlinear \\ \hline
IDRF \cite{yang09} & Fixed & Yes & Nonlinear \\ \hline
Yamabe Riemann map \cite{luo04} & Fixed & Yes & Nonlinear \\ \hline
Holomorphic 1-form \cite{gu02} & Fixed & Yes & Nonlinear \\ \hline
\end{tabular}
\caption{Several previous works on conformal parameterization of disk-type surfaces.}
\label{previouswork}
\end{table}
\section{Previous Works} \label{previous}
In this section, we describe some previous works closely related to our work.
Mesh parameterization has been extensively studied by different research groups. The goal is to map a complicated 3D or 2D surface to a simple parameter domain, such as the unit sphere $\mathbb{S}^2$ or the unit disk $\mathbb{D}$. Several surveys of mesh parameterization methods can be found in \cite{floater02,floater05,sheffer06,hormann07}.
Parameterizations inevitably create different kinds of geometric distortions. Thus, there is always a trade-off between the different types of distortions. This results in various criteria in determining the least distorted parameterizations. One of the criteria is the geodesic distance distortion. A parameterization that preserves distances is called \emph{isometric}. However, it is well known in differential geometry that distance preserving planar parameterizations only exist for developable surfaces, or equivalently, surfaces with zero Gaussian curvature \cite{docarmo76}. Hence, finding an isometric parameterization is impossible in most cases.
Another criteria is the area distortion. A parameterization without any area distortion is said to be \emph{authalic}. Desbrun et al. \cite{desbrun02} reported a discrete authalic parameterization method by introducing the Chi energy. Zou et. al. \cite{zou11} proposed an area preserving algorithm using Lie advection. Zhao et. al. \cite{zhao13} reported a method using optimal transport to achieve an area preserving mapping. However, as authalic parameterizations allow extreme angular distortions, they are less applicable in applications \cite{floater02}
On the contrary, \emph{conformal} parameterization is more favorable as it preserves angles and hence the local geometry. For this reason, numerous studies have been devoted to conformal parameterizations. In particular, since conformal maps are equivalent to harmonic maps for genus-0 surfaces \cite{schoen97}, the problem can be turned into finding a harmonic map. The study of the discretization of harmonic maps is originated from \cite{pinkall93,eck95}. Pinkall and Polthier \cite{pinkall93} introduced a discretization of the Dirichlet energy for computing piecewise linear minimal surfaces. Eck et al. \cite{eck95} proposed a discrete harmonic mapping to approximate the continuous harmonic maps using finite element method. Later, different conformal parameterization algorithms are proposed. Unlike the conformal parameterizations of genus-0 closed surfaces \cite{angenent99,haker00,gu02,gu04,hurdal04,lai13,lam13}, the presence of the surface boundary leads to more variations in the conformal parameterizations of simply-connected open surfaces. Floater \cite{floater97} introduced a method for making shape-preserving parameterizations of surface triangulations. Hormann and Greiner \cite{hormann00} presented the Most Isometric Parameterization of Surfaces method (MIPS) for disk-like surfaces. Sheffer and De Sturler \cite{sheffer00} proposed the Angle Based Flattening (ABF) method, which constructs a parameterization by minimizing a functional that punishes the angular distortion. In \cite{sheffer05}, Sheffer et. al. reported the ABF++ method, which is an extension of the ABF method that overcomes its drawbacks. In \cite{levy02}, L\'{e}vy et al. proposed a parameterization method by approximating the Cauchy-Riemann equations using the least-squares method. Desbrun et al. \cite{desbrun02} introduced the intrinsic parameterizations which minimize the distortion of different intrinsic measures of the surface patches. In \cite{floater03}, Floater derived a generalization of barycentric coordinates to improve methods for parameterizations. Kharevych et. al. \cite{kharevych05} introduced an approach for conformal parameterizations based on circle patterns. Gu and Yau \cite{gu02} constructed a basis of holomorphic 1-forms and integrated holomorphic differentials to obtain a conformal parameterization. Luo \cite{luo04} developed the combinatorial Yamabe flow for conformal parameterizations. Jin et. al. \cite{jin05} proposed a method for disk conformal parameterizations using the double covering \cite{gu03} followed by the spherical conformal mapping \cite{gu04}. Later, Mullen et. al. \cite{mullen08} reported a spectral approach to discrete conformal parameterizations. In \cite{jin08}, Jin et. al. proposed the discrete surface Ricci flow algorithm for conformal parameterizations. Yang et. al. \cite{yang09} generalized the discrete Ricci flow to improve the flexibility and robustness of the above method. The properties of the abovementioned conformal parameterization algorithms for disk-type surfaces are summarized in Table \ref{previouswork}.
As shown in Table \ref{previouswork}, for simply-connected open surfaces, there are two major types of conformal parameterizations, namely, 1) Free boundary parameterizations and 2) Fixed boundary parameterizations. Free boundary parameterizations do not restrict the shape of the boundary of the planar parameterizations. As there are more flexibilities on the boundary, less conformality distortions will be caused. However, because of the absence of boundary constraints, the planar parameterizations of two surfaces are usually completely different in shapes. This hinders the comparisons between different surfaces. For practical applications, it is desirable to obtain a planar parameterization with a more regular boundary. In this case, fixed boundary parameterization is preferred. The boundary of the mesh is usually restricted to a convex domain. Particularly, it is common to enforce the boundary to be a unit circle. Because of this extra constraint, the conformality distortion for fixed boundary parameterizations is unavoidably larger. This reflects the significance of finding a fast and accurate disk conformal parameterization algorithm.
\section{Contribution} \label{contribution}
The contributions of our work on disk conformal parameterizations are divided into three directions. Firstly, we improve the conformality distortion of the parameterizations by introducing a ``north pole-south pole'' iterative scheme. After obtaining an initial disk harmonic map, the conformality distortion of the inner region is corrected on the upper half plane by a composition of quasi-conformal maps. Then, we extend $\mathbb{D}$ to $\overline{\mathbb{C}}$ through a reflection, and correct the distortion near the boundary of $\mathbb{D}$ by a composition of quasi-conformal maps. Secondly, as every single step of our method is linear and the iteration converges shortly, our proposed algorithm is more computationally efficient than other state-of-the-art algorithms. Thirdly, our proposed method for disk conformal parameterizations is bijective. The bijectivity is ensured by the property of the Beltrami differential of the composition map. In summary, we propose an algorithm for disk conformal parameterization of genus-0 open surfaces with
\begin{enumerate}
\item improved conformality;
\item faster computation; and
\item guaranteed bijectivity.
\end{enumerate}
\section{Proposed Method} \label{main}
In this section, we describe our proposed method for the disk conformal parameterizations of simply-connected open surfaces. The disk conformal parameterizations are achieved with the aid of an efficient iterative algorithm. In \cite{lam13}, the North Pole-South Pole iterative scheme was introduced for the fast spherical conformal parameterizations of genus-0 closed surfaces. The main idea of the iterative scheme is to improve the conformality distortions near the north pole and the south pole of the spherical parameterizations step by step. In the ``north pole'' step, a genus-0 closed surface is mapped to a unit sphere using a highly efficient method. The conformality distortion near the south pole of the sphere is small while the distortion near the north pole is relatively large. After that, in the ``south pole'' step, the conformality distortion near the north pole is corrected, with the region around the south pole kept fixed. In other words, to achieve a globally conformal parameterization, one can try to ensure the conformality of one part first, and then obtain the conformality of the other part in the second step, with the aid of the conformal part obtained before. Motivated by this idea, we introduce a ``North Pole-South Pole'' iterative scheme for disk conformal parameterizations of simply-connected open surfaces. In our case, instead of the actual geometric poles of the unit disk, by the ``north pole'' and the ``south pole'' we mean the two regions of a disk at which we handle the conformality distortion one by one. Table \ref{features} highlights the features of and the comparisons between the Fast Spherical Conformal Parameterization \cite{lam13} and our proposed method.
Our proposed method consists of three steps: 1) initialization, 2) ``north pole'' step, 3) ``south pole'' iteration. The three steps will be described in the subsequent three subsections respectively.
\begin{table}[t]
\centering
\begin{tabular}{ |C{22mm}|C{40mm}|C{40mm}| }
\hline
Features & Fast Spherical Conformal Parameterization \cite{lam13} & Our proposed method\\ \hline
Type of surfaces & Genus-0 closed surfaces & Simply-connected open surfaces\\ \hline
``North pole'' step & Use the stereographic projection and work on $\mathbb{C}$ & Use the Cayley transform and work on the upper half plane\\ \hline
``South pole'' step & South pole stereographic projection & Reflection along the unit circle \\ \hline
Boundary adjustment & No & Yes \\ \hline
Output & Unit Sphere & Unit disk\\ \hline
Bijectivity & Yes & Yes\\ \hline
\end{tabular}
\caption{Features of the Fast Spherical Conformal Parameterization \cite{lam13} and our proposed method.}
\label{features}
\end{table}
\subsection{Initialization using the discrete harmonic map}
In the first step of our proposed method, we look for an initial map for the disk parameterizations. Among the existing algorithms, we use the disk harmonic map \cite{gu08} as an initialization since it is computationally efficient and easy to implement. We first briefly describe the harmonic map theory.
A map $f:M \to N$ between two Riemann surfaces is said to be \emph{conformal} if there exists a positive scalar function $\lambda$ such that $f^*ds_N^2 = \lambda ds_M^2$. It is easy to observe that every conformal map preserves angles.
The \emph{harmonic energy functional} for $f: M\to \mathbb{S}^2$ is defined as
\begin{equation}
E(f) = \int_M |\nabla f|^2 dv_M.
\end{equation}
In the space of mappings, the critical points of $E(f)$ are called \emph{harmonic mappings}. For genus-0 closed surfaces, conformal maps are equivalent to harmonic maps. For more details, please refer to \cite{schoen94,schoen97}. By Riemann mapping theorem, every simply-connected open surface $M$ can be conformally mapped onto $\mathbb{D}$. Also, a conformal map between two simply-connected open surfaces can be uniquely determined, provided that three-point correspondences are given.
In \cite{gu08}, Gu and Yau described a simple method to compute the disk harmonic map $f: M \to \mathbb{D}$ for a disk-type surface $M$. $f$ can be computed by solving the following Laplace equation:
\begin{equation}\label{Laplace}
\begin{cases}
\Delta_{M} f(u) = 0 & \text{ if }u \in M\setminus \partial M\\
f|_{\partial M} = g
\end{cases}
\end{equation}
where $g:\partial M \to \partial \mathbb{D}$ is given by the arc length parameterization.
In the discrete case, the Laplace equation $ \Delta_{M} f(u) = 0$ in Equation (\ref{Laplace}) becomes a sparse symmetric positive definite linear system. The boundary vertices $\{v_i\}_{i=0}^{n-1}$ are mapped to the unit circle according to the ratio of the edge lengths:
\begin{equation}
f(v_i) = (\cos \theta_i, \sin \theta_i),
\end{equation}
where $l_{[v_i,v_{i+1}]}$ denotes the length of the edge $[v_i,v_{i+1}]$ and
\begin{equation}
\left\{\begin{split}
s &:= \sum_{i=0}^{n-1} l_{[v_i,v_{i+1}]}\\
s_i&:= \sum_{j=0}^{i-1} l_{[v_j,v_{j+1}]}\\
\theta_i &:= 2 \pi \frac{s_i}{s}.
\end{split}\right.
\end{equation}
Hence, the Laplace Equation (\ref{Laplace}) can be efficiently solved. Note that the harmonic parameterization with fixed boundary condition is generally not conformal and induces conformality distortions. However, in practice, the harmonic disk parameterization is still a good enough mapping to serve as an initialization. The conformality distortions of the interior region and the boundary region of the disk will be corrected by the steps introduced in Subsection \ref{north} and \ref{south} respectively.
\subsection{Improvement of conformality on the upper half plane} \label{north}
\begin{figure*}[t]
\centering
\includegraphics[width=0.7\textwidth]{beltrami.png}
\caption{An illustration of how the conformality distortion can be determined by the Beltrami coefficient. The picture is adapted from \cite{fletcher06}.}
\label{beltrami}
\end{figure*}
There are two drawbacks of the disk harmonic map algorithm \cite{gu08}. One of the drawbacks is that the conformality distortion is often quite large due to the restrictive circular boundary constraints. Secondly, the bijectivity is usually lost with bad triangulations. Foldings or overlaps may exist if there are extremely irregular triangles. We aim to alleviate these two drawbacks in this subsection and the following subsection.
To improve the conformality distortion of the initial disk harmonic map, our strategy is to compose the map with a \emph{quasi-conformal map}. Quasi-conformal maps are the generalizations of conformal maps, which are orientation preserving homeomorphisms between Riemann surfaces with bounded conformality distortions, in the sense that their first order approximations take small circles to small ellipses of bounded eccentricity \cite{gardiner00}. Mathematically, $f:\mathbb{C} \to \mathbb{C}$ is a \emph{quasi-conformal map} if it satisfies the Beltrami equation
\begin{equation}\label{beltramieqt}
\frac{\partial f}{\partial \bar{z}} = \mu(z) \frac{\partial f}{\partial z}
\end{equation}
for some complex-valued functions $\mu$ with $\| \mu \|_\infty<1$. $\mu$ is called the \emph{Beltrami coefficient} of $f$. Beltrami coefficient measures the conformality distortion of a map. In particular, $f$ is conformal around a small neighborhood of $p$ if and only if $\mu(p) = 0$.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{north-south-iteration.png}
\caption{An illustration of the (a) ``north pole" iteration and (b) ``south pole" iteration.}
\label{iterationillustration}
\end{figure*}
From $\mu(p)$, we can determine the angles of the directions of maximal magnification and shrinking and the amount of them as well. Specifically, the angle of maximal magnification is $arg(\mu(p))/2$ with magnifying factor $1 + |\mu(p)|$. The angle of maximal shrinking is the orthogonal angle $(arg(\mu(p)) - \pi)/2$ with shrinking factor $1 - |\mu(p)|$. Thus, the Beltrami coefficient $\mu$ gives us all the information about the properties of the map (See Figure \ref{beltrami}).
The maximal dilation of $f$ is given by:
\begin{equation}
K(f) = \frac{1+\|\mu\|_\infty}{1-\|\mu\|_\infty}.
\end{equation}
The Beltrami coefficient of a composition of quasi-conformal maps is related to the Beltrami coefficients of the original maps. Suppose $f: \Omega \to f(\Omega)$ and $g: f(\Omega) \to \mathbb{C}$ are two quasi-conformal maps with Beltrami coefficients $\mu_f$ and $\mu_g$ correspondingly. The Beltrami coefficient of the composition map $g \circ f$ is given by
\begin{equation}\label{composition_formula}
\mu_{g \circ f} = \frac{\mu_f+\frac{\overline{f_z}}{f_z} (\mu_g \circ f)}{1+\frac{\overline{f_z}}{f_z} \overline{\mu_f} (\mu_g \circ f)}.
\end{equation}
Quasi-conformal map can also be defined between two Riemann surfaces. In this case, \emph{Beltrami differential} is used. A Beltrami differential $\mu(z)\frac{\overline{dz}}{dz}$ on a Riemann surface $S$ is an assignment to each chart $(U_\alpha, \phi_\alpha)$ of an $L_\infty$ complex-valued function $\mu_\alpha$, defined on local parameter $z_\alpha$ such that
\begin{equation}
\mu_\alpha \frac{d \overline{z_\alpha}}{dz_\alpha} = \mu_\beta \frac{d \overline{z_\beta}}{dz_\beta},
\end{equation}
on the domain which is also covered by another chart $(U_\beta, \phi_\beta )$. Here, $\frac{dz_\beta}{dz_\alpha} = \frac{d}{dz_\alpha}\phi_{\alpha\beta}$ and $\phi_{\alpha\beta} = \phi_\beta \circ \phi_\alpha$. An orientation preserving diffeomorphism $f: M \to N$ is called quasi-conformal associated with $\mu(z)\frac{\overline{dz}}{dz}$ if for any chart $(U_\alpha, \phi_\alpha)$ on $M$ and any chart $(U_\beta, \psi_\beta)$ on $N$, the mapping $f_{\alpha\beta} := \psi_\beta \circ f \circ f_\alpha^{-1}$ is quasi-conformal associated with $\mu_\alpha \frac{d \overline{z_\alpha}}{dz_\alpha}$. Readers are referred to \cite{gardiner00,fletcher06} for more details about quasi-conformal theories.
Beltrami differential is closely related to the bijectivity of a map. Specifically, if $f:M_1\to \mathbb{D}$ is a $C^1$ map satisfying $\|\mu_f\|_{\infty} <1$, then $f$ is bijective. This can be explained using the jacobian of $f$. The jacobian $J_f$ of $f$ is given by
\begin{equation}
J_f = \left| \frac{\partial f}{\partial z} \right|^2 (1-|\mu_f|^2).
\end{equation}
Since $\|\mu_f\|_{\infty} <1$, we have $\left| \frac{\partial f}{\partial z} \right|^2 \neq 0$ and $(1-|\mu_f|^2)>0$. Therefore, the Jacobian is positive everywhere. Since $\mathbb{D}$ is simply-connected and $f$ is proper, we can conclude that $f$ is a diffeomorphism. In fact, $f$ is a universal covering map of degree 1. Hence, $f$ must be bijective.
Intuitively, with a composition of two maps with the same Beltrami differential, the conformality distortion is cancelled out. In our case, let $\mu$ be the Beltrami differential of $f$, where $f: M \to \mathbb{D}$ is obtained by solving Equation (\ref{Laplace}). We proceed to look for a quasi-conformal map $g:\mathbb{D} \to \mathbb{D}$ with the same Beltrami differential $\mu$. The composition map, $\widetilde{f} := g\circ f^{-1}: M\to \mathbb{D}$, is then conformal. This can be explained by the following theorem:
\begin{theorem}\label{composition}
Let $f:M_1\to M_2$ and $g:M_2 \to M_3$ be quasi-conformal maps. Suppose the Beltrami differential of $f^{-1}$ and $g$ are the same. Then the Beltrami differential of $g\circ f$ is equal to 0. Hence, $g\circ f: M_1\to M_3$ is conformal.
\end{theorem}
\begin{proof}
Note that $\mu_{f^{-1}} \circ f = -(f_z / |f_z|)^2 \mu_f$. Since $\mu_{f^{-1}} =
\mu_g$, we have
\begin{equation}
\mu_f+\frac{\overline{f_z}}{f_z} (\mu_g \circ f) = \mu_f+ \frac{\overline{f_z}}{f_z} (\mu_{f^{-1}} \circ f)
= \mu_f+\frac{\overline{f_z}}{f_z} (-\frac{f_z}{\overline{f_z}}) \mu_f = 0.
\end{equation}
Hence, by the composition formula,
\begin{equation}
\mu_{g \circ f} = \frac{\mu_f+\frac{\overline{f_z}}{f_z} (\mu_g \circ f)}{1+\frac{\overline{f_z}}{f_z} \overline{\mu_f}(\mu_g \circ f)} = 0.
\end{equation}
Thus, $g \circ f$ is conformal.\qed
\end{proof}
Motivated by the above theorem, we can fix the conformality distortion of the initial $f$ by a quasi-conformal map $g:\mathbb{D}\to \mathbb{D}$. More specifically, suppose the Beltrami differential of $f$ is $\mu_f$, we compute another quasi-conformal map $g:\mathbb{D}\to \mathbb{D}$ with the same Beltrami differential. According to Theorem \ref{composition}, the composition map $g\circ{f}$ is conformal. The distortion of $f$ is therefore corrected.
Of course, one crucial issue is to efficiently compute $g$. Lui et al. \cite{lui13} proposed a linear algorithm, called the \emph{Linear Beltrami Solver({\bf LBS})}, to reconstruct a quasi-conformal map $g(x,y) = u(x,y) + i v(x,y)$ from its Beltrami coefficient $\mu_g = \rho + i\ \eta $ on rectangular domains in $\mathbb{C}$. Lam et al. \cite{lam13} extended this algorithm on triangular domains in $\mathbb{C}$. The brief idea of {\bf LBS} is as follows.
From the Beltrami Equation (\ref{beltramieqt}), we have
\begin{equation}
\mu_g = \frac{(u_x - v_y) + i(v_x + u_y)}{(u_x + v_y) + i(v_x - u_y)}.
\end{equation}
By direct computation, it remains to solve
\begin{equation}\label{eqt:BeltramiPDE}
\nabla \cdot \left(M \left(\begin{array}{c}
u_x\\
u_y \end{array}\right) \right) = 0\ \ \mathrm{and}\ \ \nabla \cdot \left(M \left(\begin{array}{c}
v_x\\
v_y \end{array}\right) \right) = 0
\end{equation}
where
\begin{equation}
M = \left( \begin{array}{cc} \frac{(\rho -1)^2 + \eta^2}{1-\rho^2 - \eta^2} & -\frac{2\eta}{1-\rho^2 - \eta^2}\\
-\frac{2\eta}{1-\rho^2 - \eta^2} & \frac{1+2\rho+\rho^2 +\eta^2}{1-\rho^2 - \eta^2} \end{array}\right).
\end{equation}
In the discrete case, solving the above elliptic PDEs (\ref{eqt:BeltramiPDE}), with certain boundary conditions on $u$ and $v$, can be discretized into solving a sparse symmetric positive definite linear system. Readers are referred to \cite{lui13,lam13} for details. For convenience, we denote the quasi-conformal map associated with the Beltrami differential $\mu$ by $\mathbf{LBS}(\mu)$.
In our case, we need to find a quasi-conformal map $g:\mathbb{D}\to \mathbb{D}$. This involves suitably allocating the boundary vertices on $\mathbb{S}^1$. In other words, instead of having a fixed boundary condition, we need to search for an optimal boundary correspondence $g|_{\partial \mathbb{D}}$ to reduce conformality distortions. This makes the problem nonlinear and causes computational difficulties. To alleviate this issue, our strategy is to transform the current domain to another domain, such that the problem can be linearized. More specifically, we use the Cayley transform to map the unit disk onto the upper half plane. The Cayley transform is a bijective conformal map. Mathematically, the Cayley transform $W: \mathbb{D} \to \mathbb{H}=\{x+iy | y\geq0; x,y \in \mathbb{R}\}$ is defined by
\begin{equation}
W(z) = i\frac{1+z}{1-z}.
\end{equation}
Our problem is then transformed to finding a quasi-conformal map $h:\mathbb{H} \to \mathbb{H}$ whose Beltrami differential is equal to $\mu_{(W\circ f)^{-1}}$. According to Theorem \ref{composition}, the composition map $h\circ W \circ f: M\to \mathbb{H}$ is conformal.
Note that under the Cayley transform, the boundary of the disk is mapped onto the real axis $y=0$. In other words, to enforce a circular boundary, we only need to enforce that $h$ maps the real axis to the real axis. Equivalently, we only need to restrict $v=0$ on $\partial \mathbb{H}$ while solving equation (\ref{eqt:BeltramiPDE}) and put no restriction on $u$. This allows us to compute $h$ by solving two separate elliptic equations for $u$ and $v$.
In the discrete case, the surface $M$ is represented discretely by a triangulated mesh. The initial parameterization $f$ projects $M$ onto a triangulated mesh $\Omega$ of $\mathbb{D}$. $W$ maps $\Omega$ to a big triangle in $\mathbb{H}$. The three vertices of the big triangles are $W(p_1)$, $W(p_2)$ and $W(p_3)$, where $[p_1,p_2,p_3]$ is a triangular face of $\Omega$ enclosing the point $z=1\in \mathbb{D}$ (see Figure \ref{iterationillustration}(a)). To compute the desired quasi-conformal map $h$, we solve the Beltrami's equation, subject to the constraints that the three vertices of the big triangles are fixed and that vertices on the real axis slide along the real axis:
\begin{equation}
\begin{cases}
h = \mathbf{LBS}(\mu_{(W \circ f)^{-1}})\\
h(W(p_i)) = W(p_i) \text{ for } i=1,2,3\\
\textbf{Im} (h(W(z))) = 0 \text{ for any }z \in \partial \mathbb{D}.
\end{cases}
\end{equation}
The above can be formulated as two sparse symmetric positive linear systems, which can be solved efficiently using the conjugate gradient method.
After that, we map the upper half plane (or the big triangle in the discrete case) back to the unit disk using the inverse Cayley transform
\begin{equation}
W^{-1}(z) = \frac{z-i}{z+i}.
\end{equation}
As we enforce the boundary vertices to be on the real axis under the map $h$, the corresponding boundary vertices under the composition map
\begin{equation}\label{def_g}
g:= W^{-1} \circ h \circ W \circ f
\end{equation}
will still be on $\partial \mathbb{D}$. This preserves the circular boundary condition of disk conformal parameterizations. Also, since $W^{-1}$ is conformal, the composition map $g$ is our desired disk conformal parameterization.
As a remark, the three vertices of the big triangle $\Omega$ correspond to the three vertices closest to $z=1$ on $\mathbb{D}$. Since the point $z=1$ here serves like the north pole in the stereographic projection in \cite{lam13}, we regard this step as the ``north pole'' step. It should be noted that the conformality distortion of the inner region of $\mathbb{D}$ is significantly improved by the composition of quasi-conformal maps, as explained in Theorem \ref{composition}. Also, the additional freedom on the boundary vertices slightly alleviates the conformality distortion near the boundary, although not perfectly. The conformality distortions near the boundary will be further adjusted in the next subsection. Besides, by the composition formula in Theorem \ref{composition}, we have $\|\mu_g\|_{\infty}<1$. Hence, $g$ is a diffeomorphism.
In summary, Figure \ref{iterationillustration}(a) gives a geometric illustration of the algorithm proposed in this subsection. To fix the conformality distortion at the inner region, our strategy is to compute a quasi-conformal map of the unit disk with the same Beltrami coefficient as the initial parameterization. In order to linearlize the computation of the quasi-conformal map, the unit disk is mapped to $\mathbb{H}$ or a big triangle in the discrete case by the Caley transform $W$. The problem is then reduced to solving the Beltrami's equation using $\mathbf{LBS}$ on the big triangle, which involves two sparse symmetric positive definite linear systems. The desired quasi-conformal map and hence the desired disk conformal parameterization can be obtained by an inverse Caley transform $W^{-1}$.
\subsection{Correction of boundary conformality distortion by reflection} \label{south}
After improving the conformality of the inner region, the next step is to correct the distortion near the boundary. Note that as we enforce the boundary of the parameter domain to be a unit circle (instead of a free boundary), this constraint causes conformality distortions near the boundary.
In \cite{lam13}, for spherical conformal parameterizations, the ``north pole'' iteration corrects the conformality distortion near the south pole of the sphere, and then the ``south pole'' iteration improves the conformality near the north pole, with the southernmost region fixed. Since the conformality distortion around the south pole is corrected by the ``north pole'' iteration, conformality distortions will not be induced by fixing the southernmost region in the ``south pole'' iteration. We extend this idea to our case. For our case, recall that the conformality of the inner region is significantly improved by the ``north pole'' step as described in subsection \ref{north}. Thus, the innermost region of the disk is the ``south pole'' we desire. Motivated by \cite{lam13}, we fix this least distorted region and correct the conformality distortion near the boundary by a quasi-conformal map.
Similar to the last subsection \ref{north}, we will make use of the $\mathbf{LBS}$ \cite{lui13} to construct the suitable quasi-conformal map. As observed in the last subsection, the composition of a quasi-conformal map can significantly reduce the distortion at the inner region but not near the boundary. Since our goal is to handle the distortion on the boundary, the aforementioned idea in the last subsection cannot be directly applied. To settle this problem, our strategy is to enlarge the domain, so that boundary region becomes the inner region of a much bigger domain. More precisely, we conformally reflect the unit disk along the circular boundary, such that the new domain of interest becomes the whole complex plane $\overline{\mathbb{C}}$. In the discrete case, the new domain is a big triangle whose three vertices are the reflected vertices of the triangle near the origin. The boundary region is now located at the inner region of the new big triangle (see Figure \ref{iterationillustration}b). Conformality distortion can be corrected by finding the appropriate quasi-conformal map of the big triangle.
Mathematically, Lui et. al. \cite{lui12} introduced an extension of a diffeomorphism on $\mathbb{D}$ to $\overline{\mathbb{C}}$ through a reflection as follows.
\begin{theorem} \label{extend}
Let $f:\mathbb{D} \to \mathbb{D}$ be a diffeomorphism of the unit disk fixing 0 and 1 and satisfying the Beltrami equation $f_{\bar{z}} = \mu_f f_z$ with $\mu_f$ defined on $\mathbb{D}$. Then an extension of $f$ from $\mathbb{D}$ to $\overline{\mathbb{C}}$ given by
\begin{equation}\label{f_extend}
\tilde{f}(z) =
\begin{cases}
f(z) & \text{ if }|z| \leq 1 \\
\frac{1}{\overline{f(1/\bar{z})}} & \text{ if } |z| > 1
\end{cases}
\end{equation}
satisfies the Beltrami Equation $\tilde{f}_{\bar{z}} = \tilde{\mu}_{\tilde{f}} \tilde{f}_z$ on $\overline{\mathbb{C}}$, where
\begin{equation}\label{mu_extend}
\tilde{\mu}_{\tilde{f}}(z) =
\begin{cases}
\mu_f(z) & \text{ if }|z| \leq 1 \\
\frac{z^2}{\bar{z}^2}\overline{\mu_f(1/\bar{z})} & \text{ if } |z| > 1.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
See \cite{lui12}.\qed
\end{proof}
In other words, by appropriately defining the Beltrami coefficient, one can extend a quasi-conformal map of $\mathbb{D}$ to a quasi-conformal map of $\overline{\mathbb{C}}$.
Now we are ready to propose the ``south pole'' step for correcting the conformality distortion near the boundary of the disk parameterization. Using the formula of reflection in Equation (\ref{f_extend}), we construct a copy of the points on $\mathbb{D} \setminus \partial \mathbb{D}$ outside $\mathbb{D}$ by the correspondence
\begin{equation}
z \in \mathbb{D} \setminus \partial \mathbb{D} \longleftrightarrow \frac{1}{\bar{z}} \in \overline{\mathbb{C}} \setminus \mathbb{D},
\end{equation}
and extend the map $g$ in Equation (\ref{def_g}) to the extended map $\tilde{g}: \overline{\mathbb{C}} \to \overline{\mathbb{C}}$ defined by
\begin{equation}
\tilde{g}(z) = \begin{cases}
g(z) & \text{ if }z \in \mathbb{D} \\
\frac{1}{\overline{g(1/\bar{z})}} & \text{ if } z \in \overline{\mathbb{C}} \setminus \mathbb{D}.
\end{cases}
\end{equation}
Recall that the conformality distortion of the innermost region of the disk parameterization is small after the ``north pole'' step. Since the outermost region of the new domain corresponds to the innermost region of $\mathbb{D}$ after the reflection (Equation (\ref{mu_extend})), the conformality distortion of the outermost region of the new domain is also negligible. With this characteristics, we apply the {\bf LBS} \cite{lui13} to compose the map $\tilde{g}$ by a quasi-conformal map $\widetilde{Q}:\overline{\mathbb{C}} \to \overline{\mathbb{C}}$ with the Beltrami differential $\tilde{\mu}_{{\widetilde{g}}^{-1}}$, leaving the outermost region of the new domain fixed:
\begin{equation}
\begin{cases}
\widetilde{Q} = \mathbf{LBS}(\tilde{\mu}_{{\widetilde{g}}^{-1}})\\
\widetilde{Q}(z) = z \text{ for } |z| \gg 1.
\end{cases}
\end{equation}
The conformality of $\tilde{g}$ will be significantly improved, according to the composition formula of Beltrami differentials in Equation (\ref{composition_formula}). In particular, the conformality distortion near the boundary of the original disk will be alleviated. Besides, by Theorem \ref{extend}, the region corresponding to $\mathbb{D}$ will be exactly mapped onto $\mathbb{D}$ under the map $\widetilde{Q}$. In other words, the boundary of the region is guaranteed to be a perfect circle in the continuous case. This results in a disk conformal parameterization
\begin{equation}
\varphi:= \widetilde{Q} \circ \widetilde{g}|_M ,
\end{equation}
with the conformality distortions at both the boundary region and the inner region corrected. Moreover, as the composition of quasi-conformal maps results in a zero Beltrami differential, we have $\|\mu_{\varphi}\|_{\infty}<1$. Hence, $\varphi$ is guaranteed to be bijective.
In the discrete case, the domain obtained using the reflection formula in Theorem \ref{extend} is a big triangle $\Omega$, where the outermost triangular faces of $\Omega$ corresponds to the innermost triangular face near the origin in $\mathbb{D}$ (see Figure \ref{iterationillustration}b). The computation of $\tilde{Q}$ is reduced to finding a quasi-conformal map of $\Omega$, which is solved efficiently using $\mathbf{LBS}$. Now, as the Beltrami differential is piecewise constant on each triangular face $T$ in $\mathbb{D}$, we cannot directly apply Equation (\ref{mu_extend}) to obtain the Beltrami differential $\tilde{\mu}_{{\widetilde{g}}^{-1}}(\widetilde{T})$ on the reflected triangular faces $\widetilde{T}$ on $\Omega\setminus \mathbb{D}$. Instead, we approximate $\tilde{\mu}_{{\widetilde{g}}^{-1}}(\widetilde{T})$ by
\begin{equation}
\tilde{\mu}_{{\widetilde{g}}^{-1}}(\widetilde{T}) = \frac{(\overline{z_1}^2/z_1^2 + \overline{z_2}^2/z_2^2 + \overline{z_3}^2/z_3^2)}{3} \overline{\mu_{g^{-1}}(T)},
\end{equation}
where $T = [z_1,z_2,z_3]$. This approximation unavoidably introduces numerical errors. Hence, the boundary of the inner region, which is the original disk, may not be transformed to a perfect circle under the composition. In this situation, we project the image boundary to the unit circle. That is, for any vertex $z\in \partial \mathbb{D}$,
\begin{equation}
z \mapsto \frac{z}{|z|}.
\end{equation}
Then we repeat the ``south pole'' step to extend the unit disk to the big triangle using the reflection formula in Equation (\ref{f_extend}) and perform the composition of quasi-conformal maps again until convergence.
\section{Numerical Implementation} \label{implementation}
In this section, we describe the numerical implementation of our proposed method for the disk conformal parameterization in details.
Firstly, we introduce the discretization of the disk harmonic map \cite{gu08} that we use as an initial map. Let $K$ be the triangulation of a genus-0 open surface $M$. Denote the edge spanned by two vertices $u,v$ by $[u,v]$. The discrete harmonic energy of $f:K \to \mathbb{D}$ is given by
\begin{equation}
E(f) = \sum_{[u,v] \in K} k_{uv} ||f(u)-f(v)||^2.
\end{equation}
Here $k_{uv} = \cot \alpha + \cot \beta$, where $\alpha,\beta$ are the angles opposite to the edge $[u,v]$. This is known as the cotangent formula.
From the above, the Laplace-Beltrami operator is discretized as
\begin{equation}
\Delta_M f(v_i) = \sum_{v_j \in N(v_i)} k_{v_i v_j} (f(v_j)-f(v_i))
\end{equation}
where $N(v_i)$ is the set of the vertices of the one-ring neighbors of the vertex $v_i$. Hence, the Laplace equation (\ref{Laplace}) becomes a sparse linear system in the form
\begin{equation}
Az = b
\end{equation}
where $A$ is a square matrix, $A(i,j) = k_{v_i v_j}$, $A(i,j) = -\sum_{v_j \in N(v_i)} k_{v_i v_j}$ subject to the arc-length parameterized boundary constraint. The linear system can be efficiently solved using the conjugate gradient method. Therefore, the initialization of our proposed method can be computed efficiently.
One important mathematical quantity in our proposed method is the Beltrami differential. In the discrete case, the computation of the Beltrami differentials between surfaces in $\mathbb{R}^3$ can be simplified to the computation of Beltrami coefficients on $\mathbb{C}$. This simplification is done as follows.
As the surfaces are represented as triangulated meshes, it is convenient to define the Beltrami coefficients on the triangular faces. Suppose $f=(u + i v) :K_1 \subset \mathbb{R}^2 \to K_2 \subset \mathbb{R}^2$ is an orientation preserving piecewise linear homeomorphism between two planar triangulated meshes. To compute the associated Beltrami coefficient $\mu_f$, which is a complex-valued function defined on each triangular face of $K_1$, we approximate the partial derivatives on every face $T_1$ on $K_1$.
Suppose $T_1$ on $K_1$ corresponds to another triangular face $T_2$ on $K_2$. The approximation of $\mu_f$ on $T_1$ can be achieved using the coordinates of the six vertices of $T_1$ and $T_2$. Specifically, suppose $T_1 = [a_1+i\ b_1, a_2+ i\ b_2, a_3 + i\ b_3]$ and $T_2 = [w_1, w_2, w_3]$, where $a_1,a_2,a_3,b_1,b_2,b_3 \in \mathbb{R}$, and $w_1,w_2,w_3 \in \mathbb{C}$. We approximate the Beltrami coefficient
\begin{equation}
\mu_f(z) = \frac{\partial f}{\partial \bar{z}} \left/ \frac{\partial f}{\partial z}\right.
\end{equation}
on $T_1$ by
\begin{equation}
\mu_f(T_1) = \frac{\frac{1}{2} \left(Dx + i\ Dy \right) \left( \begin{array}{c} w_1 \\ w_2 \\ w_3 \end{array} \right) }{\frac{1}{2} \left(Dx - i\ Dy \right) \left( \begin{array}{c} w_1 \\ w_2 \\ w_3 \end{array} \right) },
\end{equation}
where
\begin{equation}
Dx = \frac{1}{2 Area(T_1)} \left( \begin{array}{c} b_3-b_2 \\ b_1-b_3 \\ b_2-b_1 \end{array} \right)^t \ \ \mathrm{and}\ \
Dy = -\frac{1}{2 Area(T_1)} \left( \begin{array}{c} a_3-a_2 \\ a_1-a_3 \\ a_2-a_1 \end{array} \right)^t.
\end{equation}
More explicitly, the approximation of the Beltrami coefficient $\mu_f$ on $T_1$ is given by
\begin{equation}
\mu_f(T_1) = \frac{\left((b_3-b_2)w_1 + (b_1-b_3)w_2 + (b_2-b_1)w_3\right) - i\ \left((a_3-a_2)w_1 + (a_1-a_3)w_2 + (a_2-a_1)w_3\right)}{\left((b_3-b_2)w_1 + (b_1-b_3)w_2 + (b_2-b_1)w_3\right) + i\ \left((a_3-a_2)w_1 + (a_1-a_3)w_2 + (a_2-a_1)w_3\right)}.
\end{equation}
The above method can be extended to compute the Beltrami differential of a quasi-conformal map $g: K_1 \subset \mathbb{R}^2 \to K_2 \subset \mathbb{R}^3$. In this case, to compute the Beltrami differential between the corresponding triangular faces, we use a rigid motion $R$ to translate each triangular face of $K_2$ onto $\mathbb{R}^2$. Then we can use the abovementioned method to compute the Beltrami coefficient of the map $R \circ g$ on the triangular face $T_1$ on $K_1$. Since rigid motions are conformal, we have
\begin{equation}
\mu_R = 0.
\end{equation}
Hence, by the composition formula in Equation (\ref{composition_formula}),
\begin{equation}
\begin{split}
\mu_{R \circ g} &= \frac{\mu_g+\frac{\overline{g_z}}{g_z} (\mu_R \circ g)}{1+\frac{\overline{g_z}}{g_z} \overline{\mu_g} (\mu_R \circ g)} \\
&= \frac{\mu_g+0}{1+0} \\
&= \mu_g.
\end{split}
\end{equation}
That is, the Beltrami differential of $g$ is equal to that of $R \circ g$. Therefore, the extension introduced above for the computation of Beltrami differentials is valid.
With the above extension, we can easily obtain the Beltrami differentials of the mappings in the ``north pole'' step and the ``south pole'' iteration in Subsection \ref{north} and \ref{south} respectively. Recall that Beltrami differentials are associated with quasi-conformal maps. Now, we look for an efficient method to compute the quasi-conformal map $f$ associated with a given Beltrami differential $\mu_f$. To achieve this, we apply the {\bf LBS} \cite{lui13} to reconstruct a quasi-conformal map from a given Beltrami differential with the three vertices of the big triangule fixed. We briefly explain the key idea for the discretization of the {\bf LBS} \cite{lui13}.
Note that the quasi-conformal map associated with a given Beltrami differential can be obtained by solving equation (\ref{eqt:BeltramiPDE}). {\bf LBS} aims to discretize equation (\ref{eqt:BeltramiPDE}) and reduce it to a linear system.
For each vertex $v_i$, let $N_i$ be the collection of neighborhood faces attached to $v_i$. Let $T = [v_i,v_j, v_k]$ be a face and $w_l = f(v_l)$ where $l=i,j$ or $k$. Suppose $v_l = g_l + i\ h_l$ and $w_l = s_l + i\ t_l$ ($l=i,j,k$). Assume further that the Beltrami differential of the face $T$ is denoted by $\mu_f(T) = \rho_T + i\ \eta_T$. It can be proved that equation (\ref{eqt:BeltramiPDE}) can be discretized into the following linear system:
\begin{equation}\label{eqt:linearB12}
\begin{split}
\sum_{T\in N_i}\frac{1}{Area(T)} \left\{ (h_j-h_k)[\alpha_1(T) a_T + \alpha_2(T) b_T]+(g_j-g_k)[\alpha_2(T) a_T + \alpha_3(T) b_T] \right\} = 0\\
\sum_{T\in N_i}\frac{1}{Area(T)} \left\{ (h_j-h_k)[\alpha_1(T) c_T + \alpha_2(T) d_T]+(g_j-g_k)[\alpha_2(T) c_T + \alpha_3(T) d_T] \right\} = 0
\end{split}
\end{equation}
where
\begin{equation}
\left( \begin{array}{cc} \alpha_1(T) & \alpha_2(T)\\
\alpha_2(T) & \alpha_3(T) \end{array}\right) = \left( \begin{array}{cc} \frac{(\rho_T -1)^2 + \eta_T^2}{1-\rho_T^2 - \eta_T^2} & -\frac{2\eta_T}{1-\rho_T^2 - \eta_T^2}\\
-\frac{2\eta_T}{1-\rho_T^2 - \eta_T^2} & \frac{1+2\rho_T+\rho_T^2 +\eta_T^2}{1-\rho_T^2 - \eta_T^2} \end{array}\right),
\end{equation}
and $a_T$, $b_T$, $c_T$ and $d_T$ are certain linear combinations of the $x$-coordinates and $y$-coordinates of the desired quasi-conformal map $f$. Hence, we can obtain the $x$-coordinate and $y$-coordinate function of $f$ by solving the linear system in equation (\ref{eqt:linearB12}). For more details, please refer to \cite{lui13}.
As the computations of the Beltrami differentials and the associated quasi-conformal maps (using {\bf LBS} \cite{lui13,lam13}) are both linear, the ``north pole'' step and the ``south pole'' iteration are highly efficient. Hence, our proposed method significantly speeds up the computation of the disk conformal parameterizations. The detailed implementation of our proposed method is described in Algorithm \ref{algorithm}.
\begin{algorithm}[h]
\KwIn{A simply-connected open mesh $M$, an energy threshold $\epsilon$.}
\KwOut{A bijective disk conformal parameterization $\varphi:M \to \mathbb{D}$.}
\BlankLine
Denote the boundary of $M$ as $\partial M = [v_0,v_1,...,v_n]$. Compute the edge lengths $l_{[v_i,v_{i+1}]}$ for $i=0,1,...,n$, where $v_{n+1} := v_0$\;
Obtain an initial disk parameterization $f:M \to \mathbb{D}$ by
$$
\begin{cases}
\sum_{v \in N(u)} k_{uv} (f(u)-f(v)) = 0 & \text{if } u\notin \partial M \\
f(v_i) = (\cos \theta_i, \sin \theta_i) & \text{if } u = v_i \in \partial M
\end{cases}
$$where $s:= \sum_{i=0}^{n-1} l_{[v_i,v_{i+1}]}, s_i:= \sum_{j=0}^{i-1} l_{[v_j,v_{j+1}]}$ and $\theta_i := 2 \pi s_i/s$\;
Apply the Cayley transform $W:\mathbb{D} \to \mathbb{H}$ defined by $$W(z) = i\frac{1+z}{1-z}$$\;
Compute the Beltrami differential $\mu_{(W \circ f)^{-1}}$ of the map $(W \circ f)^{-1}$\;
Compute the quasi-conformal map $$h = \mathbf{LBS}(\mu_{(W \circ f)^{-1}})$$ with the boundary vertices $(W \circ f)(v_i)$ restricted on the real axis\;
Project the upper half plane to the unit disk by the inverse Cayley transform $$W^{-1}(z) = \frac{z-i}{z+i}.$$ Denote $\varphi:=W^{-1} \circ h \circ W \circ f$\;
\Repeat{$\text{mean}(|\mu_{\varphi^{-1}}|) - \text{mean}(|\nu|)<\epsilon$}{
Update $\nu$ by the Beltrami differential $\mu_{\varphi^{-1}}$ of the map $\varphi^{-1}$\;
By reflection, extend $\varphi^{-1}$ and $\mu_{\varphi^{-1}}$ on $\mathbb{D}$ to $\widetilde{\varphi}^{-1}$ and $\tilde{\mu}_{{\widetilde{\varphi}}^{-1}}$ on a big triangular domain $B$ using Equation (\ref{f_extend}) and Equation (\ref{mu_extend}). For each face $T=[z_1, z_2, z_3]$ on $\mathbb{D}$, define
$$\tilde{\mu}_{{\widetilde{\varphi}}^{-1}}(\widetilde{T}) = \frac{(\overline{z_1}^2/z_1^2 + \overline{z_2}^2/z_2^2 + \overline{z_3}^2/z_3^2)}{3} \overline{\mu_{\varphi^{-1}}(T)}$$\;
Compute the quasi-conformal map $$\widetilde{Q} = \mathbf{LBS}(\tilde{\mu}_{{\widetilde{\varphi}}^{-1}})$$ with the outermost vertices of $B$ fixed\;
Update $\varphi$ by the restriction $\widetilde{Q} \circ \widetilde{\varphi}|_M$\;
Project the boundary of $\varphi(M)$ onto the unit circle\;
}
\caption{Fast disk conformal parameterization}
\label{algorithm}
\end{algorithm}
\section{Experimental Results} \label{experiment}
In this section, we demonstrate the effectiveness of our proposed method using various 3D simply-connected open meshes. The meshes are freely available on the AIM@SHAPE Shape Repository \cite{aim@shape}. The algorithm is developed using MATLAB on Windows 7 platform. All experiments are performed on a PC with a 3.40 GHz CPU. In our experiments, the error threshold in Algorithm \ref{algorithm} is set to be $\epsilon=10^{-5}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{face_mesh.png}
\includegraphics[width=0.35\textwidth]{face_disk_mesh.png}
\includegraphics[width=0.35\textwidth]{face.png}
\includegraphics[width=0.35\textwidth]{face_disk.png}
\end{center}
\caption{A human face and its disk conformal parameterization using our proposed method. Top: the triangulations. Bottom: the mean curvature texture maps.}
\label{fig:face}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{lion_mesh.png}
\includegraphics[width=0.35\textwidth]{lion_disk_mesh.png}
\includegraphics[width=0.35\textwidth]{lion.png}
\includegraphics[width=0.35\textwidth]{lion_disk.png}
\end{center}
\caption{A Chinese lion head and its disk conformal parameterization using our proposed method. Top: the triangulations. Bottom: the mean curvature texture maps.}
\label{fig:lion}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{brain_mesh.png}
\includegraphics[width=0.35\textwidth]{brain_disk_mesh.png}
\includegraphics[width=0.35\textwidth]{brain.png}
\includegraphics[width=0.35\textwidth]{brain_disk.png}
\end{center}
\caption{A human brain and its disk conformal parameterization using our proposed method. Top: the triangulations. Bottom: the mean curvature texture maps.}
\label{fig:brain}
\end{figure}
Figure \ref{fig:face}, Figure \ref{fig:lion} and Figure \ref{fig:brain} respectively show a human face mesh, a Chinese lion head mesh, a human brain mesh, and their disk conformal parameterizations obtained by our proposed method. The histograms of the norms of the Beltrami differentials are shown in Figure \ref{fig:histogram}. It is apparent that the peaks of the norms are close to $0$, which implies that the conformality distortions are small. Besides, from the energy plots shown in Figure \ref{fig:energy}, it can be observed that our proposed iterative method converges shortly. Hence, our method is very efficient.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.32\textwidth]{face_hist.png}
\includegraphics[width=0.32\textwidth]{lion_hist.png}
\includegraphics[width=0.32\textwidth]{brain_hist.png}
\end{center}
\caption{Histograms of the norm of Beltrami differentials $|\mu|$ of our proposed method for a human face mesh, a Chinese lion head mesh and a human brain mesh. The small norms with the peak at $|\mu| \approx 0$ indicate that the conformality distortion is very small. Left: Human face. Middle: Chinese lion head. Right: Human brain.}
\label{fig:histogram}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.32\textwidth]{face_energy.png}
\includegraphics[width=0.32\textwidth]{lion_energy.png}
\includegraphics[width=0.32\textwidth]{brain_energy.png}
\caption{Energy plots of mean($|\mu|$) of our proposed method for a human face mesh, a Chinese lion head mesh and a human brain mesh. For all of the meshes, the iterations converge shortly. Left: Human face. Middle: Chinese lion head. Right: Human brain.}
\label{fig:energy}
\end{figure}
To quantitatively assess the quality of our proposed method, three different factors are considered, including the computational time, the mean of the norm of the Beltrami differentials, and the standard deviation. The computational time evaluates the efficiency of our proposed method, the mean of the norm of the Beltrami differential checks if our proposed method is of small conformality distortions in general, and from the standard deviation we can see whether there exists any region with extremely large conformality distortions. As we aim at a bijective disk conformal parameterization, we compare our proposed method with four state-of-the-art algorithms that guarantee bijectivity and enforce a circular boundary. One of the methods is the discrete Ricci flow (RF) algorithm proposed by Jin et. al. \cite{jin08}. Another method is the inversive distance Ricci flow (IDRF) algorithm introduced by Yang et. al. \cite{yang09}. The Yamabe Riemann map algorithm by Luo \cite{luo04} and the holomorphic 1-form algorithm by Gu and Yau \cite{gu02} are also considered. The statistics of the performance of our proposed method and the four mentioned algorithms are listed in Table \ref{table_disk_para}.
\begin{table}
\centering
\begin{center}
\begin{tabular}{ |C{15mm}||c||c||c|c| }
\hline
Surfaces & No. of faces & Our Method & RF \cite{jin08} & IDRF \cite{yang09}\\
\cline{3-5}
&& \multicolumn{3}{ c| }{Time (seconds) / mean($|\mu|$) / sd($|\mu|$)}\\ \hline
Human face & 49982 & 6.86/0.0101/0.0284 & 13.01/0.2421/0.1563 & fail \\ \hline
Sophie & 41587 & 4.88/0.0056/0.0083 & 23.86/0.1476/0.0792 & 29.43/0.0057/0.0086 \\ \hline
Max Planck & 99515 & 6.44/0.0102/0.0109 & 21.54/0.1431/0.0818 & 30.86/0.0103/0.0106 \\ \hline
Mask & 62467 & 8.63/0.0043/0.0051 & 13.46/0.2379/0.1425 & fail\\ \hline
Nicolo da Uzzano & 50042 & 5.71/0.0136/0.0314 & 10.34/0.2992/0.1434 & fail\\ \hline
Julius Caesar & 433956 & 72.69/0.0032/0.0100 & 108.72/0.1033/0.0689 & 173.55/0.0033/0.0094\\ \hline
Bimba & 48469 & 2.61/0.0217/0.0254 & 10.07/0.2947/0.1430 & fail\\ \hline
Human brain & 96811 & 4.90/0.0250/0.0217 & 22.87/0.1861/0.1007 & 32.30/0.0249/0.0220 \\ \hline
Hand & 105860 & 6.89/0.0194/0.0168 & 35.38/0.0550/0.0281 & 39.63/0.0211/0.0212\\ \hline
Chinese lion & 34421 & 2.23/0.0240/0.0271 & 8.03/0.2029/0.1024 & 10.11/0.0238/0.0265 \\ \hline
Lion vase & 98925 & 4.19/0.0236/0.0257 & 33.45/0.3687/0.1726 & fail\\ \hline
\end{tabular}
\end{center}
\bigskip
\begin{center}
\begin{tabular}{ |C{15mm}||c||c||C{33mm}|C{33mm}| }
\hline
Surfaces & No. of faces & Our Method & Yamabe Riemann map \cite{luo04} & Holomorphic 1-form \cite{gu02} \\
\cline{3-5}
&& \multicolumn{3}{ c| }{Time (seconds) / mean($|\mu|$) / sd($|\mu|$)}\\ \hline
Human face & 49982 & 6.86/0.0101/0.0284 & fail & 52.12/0.0111/0.0292 \\ \hline
Sophie & 41587 & 4.88/0.0056/0.0083 & 34.32/0.0057/0.0085 & 57.36/0.0058/0.0083 \\ \hline
Max Planck & 99515 & 6.44/0.0102/0.0109 & 34.34/0.0103/0.0106 & 92.74/0.0103/0.0109 \\ \hline
Mask & 62467 & 8.63/0.0043/0.0051 & fail & fail\\ \hline
Nicolo da Uzzano & 50042 & 5.71/0.0136/0.0314 & fail & 50.97/0.0143/0.0275\\ \hline
Julius Caesar & 433956 & 72.69/0.0032/0.0100 & 175.01/0.0033/0.0094 & 462.63/0.0033/0.0096\\ \hline
Bimba & 48469 & 2.61/0.0217/0.0254 & fail & 56.12/0.0230/0.0250\\ \hline
Human brain & 96811 & 4.90/0.0250/0.0217 & 31.52/0.0249/0.0220 & 89.88/0.0251/0.0217\\ \hline
Hand & 105860 & 6.89/0.0194/0.0168 & 44.25/0.0211/0.0212 & 104.04/0.0224/0.0269\\ \hline
Chinese lion & 34421 & 2.23/0.0240/0.0271 & 14.82/0.0238/0.0265 & 40.43/0.0244/0.0271\\ \hline
Lion vase & 98925 & 4.19/0.0236/0.0257 & fail & 137.93/0.0271/0.0282\\ \hline
\end{tabular}
\end{center}
\caption{Performance of our proposed method and four state-of-the-art algorithms.}
\label{table_disk_para}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{hand2_mesh.png}
\includegraphics[width=0.35\textwidth]{hand2_disk_mesh.png}
\includegraphics[width=0.35\textwidth]{hand2.png}
\includegraphics[width=0.35\textwidth]{hand2_disk.png}
\end{center}
\caption{A hand mesh and its disk conformal parameterization using our proposed method. Top: the triangulations. Bottom: the mean curvature texture maps.}
\label{fig:hand}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{our_bdy.png}
\includegraphics[width=0.4\textwidth]{idrf_bdy.png}
\end{center}
\caption{Comparison of the norm of the Beltrami differentials between our proposed method and the IDRF algorithm \cite{yang09}. The colormaps show the norm of the Beltrami differentials on each triangular face. Left: Our proposed method. Right: the IDRF algorithm \cite{yang09}. }
\label{fig:boundary}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{vaselion.png}
\includegraphics[width=0.4\textwidth]{vaselion_disk.png}
\end{center}
\caption{A lion vase mesh and its disk conformal parameterization using our proposed method. The features of the lion vase mesh are well preserved by the disk parameterization. This demonstrates the conformality of the disk parameterization by our proposed method. }
\label{fig:vaselion}
\end{figure}
Our proposed method is highly efficient and accurate. For a 3D mesh with 100k faces, the time taken by our method is usually less than 10 seconds. Our method is capable of handling different types of meshes. Besides the typical human face meshes, our method also works for meshes with irregular shapes and bad triangulations. For instance, our proposed method successfully computes the disk conformal parameterization of a human hand mesh with long fingers (see Figure \ref{fig:hand}). As a remark, in all our experiments, the resulting disk conformal parameterizations are bijective. There exists no flips or overlaps in the disk conformal parameterizations obtained by our proposed method.
For a more detailed comparison, it is easy to see from Table \ref{table_disk_para} that the computational time of our method is shorter than that of the four aforementioned algorithms. Firstly, the computational time of our proposed method is 60\% shorter than that of the discrete Ricci Flow algorithm \cite{jin08} on average. For the conformality, it is apparent from the mean and the standard deviation of the norm of Beltrami differentials that our method surpasses the RF algorithm \cite{jin08}. The mean of the norm by our proposed method is over 85\% smaller than that by the RF algorithm \cite{jin08} on average. This shows that our proposed method has a better conformality. Also, the standard deviation by our proposed method is 80\% lower than that by the RF algorithm \cite{jin08} on average, which illustrates that the dispersion of the conformality distortion of our proposed method is much smaller.
Then, for the comparison with the Inversive Distance Ricci Flow algorithm \cite{yang09}, it is noteworthy that the computational time of our proposed method is 75\% shorter than that of the IDRF method \cite{yang09} on average. Besides, the conformality of our method is comparable to (and sometimes better than) that of the IDRF algorithm \cite{yang09}. In particular, the conformality near the boundary of our method is usually much better (see Figure \ref{fig:boundary}). This demonstrates the effectiveness of our proposed method in correcting the conformality distortion at the boundary region of the disk parameterization. More importantly, our method is applicable for a wider class of meshes. The IDRF algorithm \cite{yang09} may sometimes fail. For instance, the IDRF algorithm \cite{yang09} fails for the lion vase mesh shown in Figure \ref{fig:vaselion} while our proposed method works well. The features of the lion vase mesh, such as the circular patterns around the boundary and the texture of the hair, are well preserved. These results reflect the advantages of our proposed method.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.4\textwidth]{our_disk.png}
\includegraphics[width=0.4\textwidth]{idrf_disk.png}
\includegraphics[width=0.4\textwidth]{our_zoom.png}
\includegraphics[width=0.4\textwidth]{idrf_zoom.png}
\end{center}
\caption{Comparison between the disk parameterizations obtained by our proposed method and by the IDRF algorithm \cite{yang09}. The colormaps show the norm of the Beltrami differentials on each triangular face. It is observed that there is a region with exceptionally large conformality distortion in the center of the disk by the IDRF algorithm \cite{yang09}. Top: the disk parameterizations obtained by our proposed method (left) and by the IDRF algorithm \cite{yang09}(right). Bottom: The zoom in of the center of the disks.}
\label{fig:puncture}
\end{figure}
When compared with the Yamabe Riemann map algorithm \cite{luo04}, our method also demonstrates a significant improvement in the computational time. More explicitly, our proposed method is 80\% faster than the Yamabe Riemann map algorithm \cite{luo04} on average. Our method is also applicable to a wider class of meshes as the Yamabe Riemann map algorithm \cite{luo04} fails for a number of meshes, especially for meshes with bad triangulations. For the remaining cases, the conformality of our method is comparable to (and sometimes better) than that of the Yamabe Riemann map algorithm \cite{luo04}.
For the differences between our proposed algorithm and the holomorphic 1-form algorithm \cite{gu02}, it is noteworthy that our proposed algorithm is 90\% faster than the holomorphic 1-form algorithm \cite{gu02} on average. The conformality of our method is comparable to (and often better) than that of the holomorphic 1-form algorithm \cite{gu02}. Besides, our method is more stable than the holomorphic 1-form algorithm \cite{gu02} as the holomorphic 1-form algorithm \cite{gu02} occasionally fails.
In addition, Figure \ref{fig:puncture} contrasts another feature between our proposed method and the IDRF algorithm \cite{yang09}. For the four abovementioned algorithms, one of the triangular faces has to be punctured at the beginning and filled at the end. For the region around the puncture, the conformality distortion is exceptionally large. On the contrary, the parameterization obtained by our proposed method is of small conformality distortion and is free of such unnaturally distorted regions. This again demonstrates the advantage of our proposed method.
Besides the four aforementioned methods, we also compare our proposed method with the double covering algorithm \cite{jin05} for simply-connected open surfaces. In \cite{jin05}, Jin et. al. suggested to glue two copies of the same surface along the boundaries to form a closed symmetric surface, and then computed the spherical conformal mapping \cite{gu04} of the new surface, and obtained the disk parameterization by applying the stereographic projection on the hemisphere. It is noteworthy that the spherical conformal mapping \cite{gu04} uses the Gauss map as initialization. In practice, it fails for most glued surfaces because the Gauss map is often undefined on the sharp ``boundaries'' of the glued surfaces. Instead of using the Gauss map, we use another method to obtain the initial spherical map. We apply the disk harmonic map \cite{gu08} and project the surface onto the southern hemisphere by the stereographic projection, and then glue the surface with a copy of it in the northern hemisphere to form the initial sphere. The energy threshold of the spherical conformal mapping \cite{gu04} is set to be $10^{-3}$. The comparison of the two algorithms is shown in Table \ref{table_double}.
\begin{table}
\centering
\begin{tabular}{ |C{15mm}|c|c|c|c| }
\hline
Surfaces & No. of faces & Our Method & Double covering \cite{jin05}\\
\cline{3-4}
&& \multicolumn{2}{ c| }{Time (seconds) / mean($|\mu|$) / sd($|\mu|$) / $\sum_{\text{boundary}} |1-|z|^2|$}\\ \hline
Human face & 49982 & 6.86/0.0101/0.0284/3.5305e-14 & 82.20/0.0679/0.0770/0.0036\\ \hline
Sophie & 41587 & 4.88/0.0056/0.0083/4.3854e-14 & 31.62/0.0517/0.0390/0.0023\\ \hline
Max Planck & 99515 & 6.44/0.0102/0.0109/1.2768e-14 & 115.53/0.0326/0.0214/0.0022\\ \hline
Mask & 62467 & 8.63/0.0043/0.0051/1.0203e-13 & 74.88/0.1104/0.0461/0.0178\\ \hline
Nicolo da Uzzano & 50042 & 5.71/0.0136/0.0314/3.5749e-14 & 99.71/0.0917/0.0624/0.0577\\ \hline
Julius Caesar & 433956 & 72.69/0.0032/0.0100/1.3922e-13 & 467.42/0.1313/0.0361/0.0014\\ \hline
Bimba & 48469 & 2.61/0.0217/0.0254/1.8763e-14 & 80.94/0.0638/0.0389/0.0625\\ \hline
Human brain & 96811 & 4.90/0.0250/0.0217/9.3259e-15 & fail\\ \hline
Hand & 105860 & 6.89/0.0194/0.0168/2.4314e-14 & fail\\ \hline
Chinese lion & 34421 & 2.23/0.0240/0.0271/2.3204e-14 & 50.37/0.0477/0.0439/0.0034\\ \hline
Lion vase & 98925 & 4.19/0.0236/0.0257/1.8430e-14 & 180.30/0.0959/0.0602/0.0101\\ \hline
\end{tabular}
\caption{Performance of our proposed method and the double covering method \cite{jin05}.}
\label{table_double}
\end{table}
As illustrated by Table \ref{table_double}, the computational time of our proposed method is much shorter. Our proposed method is over 12 times faster than the double covering algorithm \cite{jin05} on average. Also, the conformality of our proposed method is significantly better. The mean and the standard deviation of the norms of the Beltrami differentials of our proposed method are 80\% and 60\% smaller than those of the double covering algorithm \cite{jin05} on average respectively. As a remark, the boundary of the disk parameterization obtained by the double covering algorithm \cite{jin05} is usually different from a perfect circle, while our proposed method always guarantees a circular boundary. This can be demonstrated from the last quantity in the table. The quantity measures the deviation of the boundary from a perfect circle, which is defined as $\sum_{\text{boundary}} |1-|z|^2|$. If the quantity is equal to zero, it means the boundary is a perfect circle. As shown in the table, our method successfully creates disk conformal parameterizations onto a perfect disk, while the double covering approach cannot.
\section{Conclusion and Future Work} \label{conclusion}
In this paper, we have presented a novel algorithm for the disk conformal parameterization of simply-connected open surfaces. Our method consists of three major steps. In the first step, we compute the initial disk parameterization using the disk harmonic map \cite{gu08}. Then, we project the unit disk to the upper half plane and compose the map with a quasi-conformal map to correct the conformality distortion at the inner region of the disk. After that, we extend the unit disk to a big triangular domain in $\mathbb{C}$ using a reflection, and compose the extended map with a quasi-conformal map with the extended Beltrami differential. Finally, by projecting the boundary to the unit disk if necessary and repeating the previous step, a disk conformal parameterization is obtained. It is noteworthy that the bijectivity of the parameterization is ensured by the composition formula of quasi-conformal maps. Experimental results have illustrated the effectiveness of our proposed method, with significant improvements in the computational time, the conformality and the stability. In the future, we will investigate the possibility of extending the proposed method for enhancing the parameterizations of high genus surfaces.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,616
|
The House unanimously approved a resolution on Thursday urging the Obama administration to fight efforts to give a United Nations agency more control over the Internet.
Proposals to give the UN's International Telecommunication Union (ITU) more control over the governance of the Internet could come up at a conference in Dubai in December. The move is reportedly backed by China, Russia, Brazil, India and other U.N. members.
The Obama administration has already announced its strong opposition to such proposals.
"Today's unanimous vote sends a clear and unmistakable message: the American people want to keep the Internet free from government control and prevent Russia, China and other nations from succeeding in giving the U.N. unprecedented power over Web content and infrastructure," said Rep. Mary Bono Mack (R-Calif.), who sponsored the resolution. "We cannot let this happen."
The proposals could give the U.N. more control over cybersecurity, data privacy, technical standards and the Web's address system. They could also allow foreign, government-owned Internet providers to charge extra for international traffic and allow for more price controls.
The Internet is currently governed under a "multi-stakeholder" approach that gives power to a host of nonprofits, rather than governments.
The resolution urges the administration to "promote a global Internet free from government control and preserve and advance the successful multistakeholder model that governs the Internet today."
Sen. Marco Rubio Marco Antonio RubioDems plot aggressive post-Mueller moves, beginning with McGahn Senate Republicans tested on Trump support after Mueller Cuban negotiator says Trump's efforts to destabilize Cuba's government will fail MORE (R-Fla.) has introduced a counterpart resolution in the Senate.
Google applauded the House vote in a blog post by Vint Cerf, one of the founders of the Internet who is now Google's "Chief Internet Evangelist."
"In the lead-up to the December conference, the future of the Internet is at stake, and I hope that other countries will adopt publicly similar positions," Cerf wrote.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,935
|
using System;
using System.Data;
using System.Data.SqlClient;
using Csla;
using Csla.Data;
namespace ParentLoadROSoftDelete.Business.ERLevel
{
/// <summary>
/// E03_SubContinentColl (read only list).<br/>
/// This is a generated base class of <see cref="E03_SubContinentColl"/> business object.
/// </summary>
/// <remarks>
/// This class is child of <see cref="E02_Continent"/> read only object.<br/>
/// The items of the collection are <see cref="E04_SubContinent"/> objects.
/// </remarks>
[Serializable]
public partial class E03_SubContinentColl : ReadOnlyListBase<E03_SubContinentColl, E04_SubContinent>
{
#region Collection Business Methods
/// <summary>
/// Determines whether a <see cref="E04_SubContinent"/> item is in the collection.
/// </summary>
/// <param name="subContinent_ID">The SubContinent_ID of the item to search for.</param>
/// <returns><c>true</c> if the E04_SubContinent is a collection item; otherwise, <c>false</c>.</returns>
public bool Contains(int subContinent_ID)
{
foreach (var e04_SubContinent in this)
{
if (e04_SubContinent.SubContinent_ID == subContinent_ID)
{
return true;
}
}
return false;
}
#endregion
#region Find Methods
/// <summary>
/// Finds a <see cref="E04_SubContinent"/> item of the <see cref="E03_SubContinentColl"/> collection, based on item key properties.
/// </summary>
/// <param name="subContinent_ID">The SubContinent_ID.</param>
/// <returns>A <see cref="E04_SubContinent"/> object.</returns>
public E04_SubContinent FindE04_SubContinentByParentProperties(int subContinent_ID)
{
for (var i = 0; i < this.Count; i++)
{
if (this[i].SubContinent_ID.Equals(subContinent_ID))
{
return this[i];
}
}
return null;
}
#endregion
#region Factory Methods
/// <summary>
/// Factory method. Loads a <see cref="E03_SubContinentColl"/> object from the given SafeDataReader.
/// </summary>
/// <param name="dr">The SafeDataReader to use.</param>
/// <returns>A reference to the fetched <see cref="E03_SubContinentColl"/> object.</returns>
internal static E03_SubContinentColl GetE03_SubContinentColl(SafeDataReader dr)
{
E03_SubContinentColl obj = new E03_SubContinentColl();
obj.Fetch(dr);
return obj;
}
#endregion
#region Constructor
/// <summary>
/// Initializes a new instance of the <see cref="E03_SubContinentColl"/> class.
/// </summary>
/// <remarks> Do not use to create a Csla object. Use factory methods instead.</remarks>
[System.ComponentModel.EditorBrowsable(System.ComponentModel.EditorBrowsableState.Never)]
public E03_SubContinentColl()
{
// Use factory methods and do not use direct creation.
var rlce = RaiseListChangedEvents;
RaiseListChangedEvents = false;
AllowNew = false;
AllowEdit = false;
AllowRemove = false;
RaiseListChangedEvents = rlce;
}
#endregion
#region Data Access
/// <summary>
/// Loads all <see cref="E03_SubContinentColl"/> collection items from the given SafeDataReader.
/// </summary>
/// <param name="dr">The SafeDataReader to use.</param>
private void Fetch(SafeDataReader dr)
{
IsReadOnly = false;
var rlce = RaiseListChangedEvents;
RaiseListChangedEvents = false;
var args = new DataPortalHookArgs(dr);
OnFetchPre(args);
while (dr.Read())
{
Add(E04_SubContinent.GetE04_SubContinent(dr));
}
OnFetchPost(args);
RaiseListChangedEvents = rlce;
IsReadOnly = true;
}
#endregion
#region DataPortal Hooks
/// <summary>
/// Occurs after setting query parameters and before the fetch operation.
/// </summary>
partial void OnFetchPre(DataPortalHookArgs args);
/// <summary>
/// Occurs after the fetch operation (object or collection is fully loaded and set up).
/// </summary>
partial void OnFetchPost(DataPortalHookArgs args);
#endregion
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,859
|
\section{The Virgo Cluster}
The Virgo Cluster, alongside its counterpart in Fornax,
is the nearest sizeable galaxy cluster. It is close enough for
detailed morphological studies to be possible even for low
luminosity dwarf galaxies.
It is an irregular, poor cluster of Bautz-Morgan type III
(Abell 1975) and Abell richness class 0. As such, it
allows detailed studies of the dwarf population
in an environment substantially different from the Local Group
and other nearby groups.
A seminal study was carried out by Binggeli, Sandage and
Tammann, who generated the Virgo Cluster Catalog consisting
of 1277 galaxies classified as certain members and a further
574 possible members over an area of 140~deg$^2$ (Binggeli et al. 1985).
Membership was assigned by visual inspection,
essentially based on the larger angular sizes of the
cluster galaxies compared with the background population.
Their dwarfs conformed to a moderately steep luminosity function
(Sandage et al. 1985, Binggeli et al. 1988).
Various detailed studies of cluster members have been performed
subsequently, including the dwarf population (e.g. Ferguson \&
Sandage 1989, Binggeli \& Cameron 1993, Durrell 1997,
Young \& Currie 1998).
Of particular note, Impey, Bothun \& Malin (1988) performed
a survey for large angular size low surface brightness galaxies
in a single Schmidt field centred on the M87 cluster core,
using a photographic stacking technique. They identified 137
galaxies having central surface brightnesses in the range 23 to
26~B~mag~arcsec$^{-2}$, of which 27 were new detections.
\section{The Bristol -- AAO Virgo Cluster Survey}
The properties of the faint end of the galaxy luminosity function
are poorly constrained outside the Local Group, both in terms
of the numbers and characteristics of the galaxies. Some studies
have found evidence for very steep luminosity functions in both
cluster (e.g. De Propris et al. 1995, Smith, Driver \&
Phillipps 1997, Trentham 1997, 1998a) and field (Loveday 1997,
Morgan, Smith \& Phillipps 1998) environments. At the very lowest
luminosities, dwarf spheroidal
galaxies have been identified in nearby groups (e.g. Caldwell
et al. 1998), extending the number of known dSphs and allowing
a comparison of their properties with Local Group members.
The importance of very low luminosity galaxies in understanding
galaxy formation (e.g. Frenk et al. 1996, Kauffmann,
Nusser \& Steinmetz 1997) and evolution (Gallagher \& Wyse 1994,
Caldwell et al. 1998, Trentham 1998b) demands that progress
is made in identifying and studying extremely low luminosity dwarfs
in new environments (see Phillipps et al. 1998a, and references
therein).
In order to extend surveys of galaxies to fainter surface
brightnesses than previous surveys
and over a full ten degree square region of the cluster, the
Bristol--Anglo-Australian Observatory survey is using multiple
exposures with the United Kingdom Schmidt Telescope (UKST) on Kodak
Tech Pan emulsion through an R band filter
(Schwartzenberg, Phillipps \& Parker 1995, Schwartzenberg \&
Phillipps 1995, 1997, Schwart\-zenberg 1996, Phillipps et al. 1998b,
Jones et al. 1998).
Six individual exposures of each Virgo field of 1 -- $1\frac{1}{2}$
hour duration are digitally stacked to give a total integration
time of 7 hours.
The Tech Pan emulsion combines a high efficiency (approaching
10\%, Parker et al. 1998, Phillipps \& Parker 1993) and fine grains
(providing a high imaging resolution of 5~microns and a high uniformity).
The four UKST fields of the Virgo survey area are shown in
Figure~\ref{fig-1}. Field
coordinates have been selected so as to cover a 100~deg$^2$
region centred on the M87 cluster core (``Cluster~A'' of
Binggeli, Sandage \& Tammann 1987). The data therefore survey
both the region
of high density in the core and the lower density regions at
the periphery. The survey area extends south as far as the M49
cluster (``Cluster B'') and includes the ``M Cloud'' west of
the M87 cluster. The central area in the grid has been imaged
on all 24 films, in effect providing a further 1~mag depth gain in
this region and offering the opportunity of a deeper survey in a
restricted area.
Table~\ref{tab-1} presents details of the four fields.
\begin{figure}[!htb]
\vspace*{-30mm}
\plotone{fig1.eps}
\vspace*{-7mm}
\caption{
The four UKST fields and the Virgo Cluster. The boundaries
of the four Schmidt fields are drawn superimposed on the
distribution of cluster members given in the Virgo Cluster Catalog
of Binggeli, Sandage \&
Tammann (1985). The full 6~deg $\times$ 6~deg extent of the fields
are shown; in practice the scanned regions are slightly smaller.
The grid of fields is centred on the M87 core of the cluster.
The M49 condensation lies at the southern boundary of the
surveyed region. The small squares show the fields used in the
initial survey of Section~5.
}\label{fig-1}
\vspace*{10mm}
\end{figure}
\begin{table}
\caption{The four quadrants of the surveyed area.} \label{tab-1}
\begin{center}
\begin{tabular}{lcccl}
\hspace*{3mm} Field & \multicolumn{2}{c}{Central coordinates} &
No. exposures & Total integ. \\
& R.A. (1950.0) & Dec. (1950.0) & used & \hspace{4.5mm} time \\[1mm]
\tableline \\[-2.5mm]
Northwest & \hspace*{1.4mm}12$^{\rm h}$ 16$^{\rm m}$ & ~+15.4$^\circ$ & 6 &
\hspace{4mm} 6.3 hr \\[1mm]
Southwest & 12~~ 16~ & +11.2 & 6 & \hspace{4mm} 6.0 \\[1mm]
Northeast & 12~~ 34~ & +15.4 & 6 & \hspace{4mm} 6.3 $\;^{1}$ \\[1mm]
Southeast & 12~~ 34~ & +11.2 & 6 & \hspace{4mm} 7.1 \\[1mm]
\end{tabular}
\mbox{ }\\[3mm]
$\;^{1}$ Estimated (one exposure still awaited).
\end{center}
\end{table}
The photographic study compares very favourably with any CCD surveys
currently feasible. The very large solid
angles of the Schmidt fields, and the very long integration times,
overcome the modest telescope aperture (1.2m) and the low quantum
efficiency of the emulsion compared with CCDs. It has been
possible to perform a very deep survey of a 10~deg $\times$ 10~deg region
in the central part of the Virgo Cluster in the equivalent of
only four nights' observing on the Schmidt. An equivalent survey
to the same depth using CCDs might reasonably take several weeks,
dependent on the details of detector and telescope. To date, the
photographic observations are $96\%$ complete.
\section{Data Reduction and Analysis}
The photographic data are digitised using the SuperCOSMOS
plate measuring machine at the Royal Observatory Edinburgh.
Only the best quality films are used for stacking: adding poor
quality films does not provide a useful gain in depth
(Bland-Hawthorn, Shopbell \& Malin, 1993).
With 10~micron wide pixels, corresponding to 0.67~arcsec on the sky,
data extents are very large, causing
particular data processing difficulties.
The individual exposures are sky-subtracted using a pixel-by-pixel
subtraction of a median-filtered version of the data. The use of a
3~arcmin square spatial filter maximises the ability to detect small,
faint images but limits the survey, at least at the present time,
to small galaxy images -- for example, the larger galaxies in the
survey of Impey, Bothun and Malin (1988) would be partially
removed.
All six films of a given part of sky have their intensities normalised
to the same scale, thus correcting for factors such as differences in
exposure time, atmospheric transmission or the details of the
development process.
Coaddition of the six exposures is accomplished using median
stacking, which efficiently overcomes problems such as film defects,
dust particle images or satellite trails that affect single
films. Image detection is performed with a connected-pixel
algorithm using a detection threshold
$\mu_{lim} = 25.5$~R~mag~arcsec$^{-2}$ and a minimum image
area above this isophote of $A_{lim} = 11\:\mbox{arcsec}^2$.
These parameters ensure that each detection will have a signal-to-noise
ratio of least 10 and magnitudes $R \leq 22$~mag.
\section{Identifying Virgo Dwarf Spheroidals}
Whilst the detection of galaxies in the direction of the
Virgo Cluster may not present a particular challenge in itself,
the identification of cluster members is significantly more
difficult. Nearby clusters cover large angular scales and
consequently the cluster population is swamped by the numerically
dominant background population (see, for example, the review
by Trentham 1998b). In the absence of redshift information
(for example because of the practicalities of performing spectroscopy for
very large samples of galaxies extending over wide areas, or because
of the difficulties of obtaining spectra for low surface brightness
galaxies), membership for the dwarf population must often be assigned on
morphological grounds.
Binggeli, Sandage \& Tammann (1985) were able to assign membership for
dwarf galaxies in the cluster on the basis of the galaxies' visual
appearance: the dwarfs are characterised by their low surface
brightnesses for their sizes.
In the present survey, even though
the galaxies are more extreme than those of the Virgo Cluster
Catalog, it is still possible to isolate samples of cluster
galaxies likely to suffer only a small degree of
contamination by the background population. A comparison of Virgo
photographic data with deep CCD data from a South Galactic Pole
field shows that the background contamination is as small as several
percent for galaxies having central surface brightnesses in the
range $\mu_0 = 22.0$ to 24.5~R~mag~arcsec$^{-2}$ and exponential
scale lengths $a \ge 3$~arcsec, even when allowance is made for
differences in resolution between the two data sets. These issues
are discussed in greater detail by Schwartzenberg (1996)
and by Jones et al. (1998).
The galaxies selected have properties comparable to those that
Local Group dwarf spheroidals would have if they were at the
distance of the Virgo Cluster. They have exponential scale lengths
$h\geq 260$~pc and absolute magnitudes $M_{\rm R} = -11$ to $-16$
(for an assumed distance modulus of 31.3~mag, equivalent to
$H_0 = 70$~km$\:\mbox{s}^{-1}\:\mbox{Mpc}^{-1}$).
Local Group dwarf spheroidals have sizes between $h=90$ and 400~pc and
absolute magnitudes $M_{\rm R} = -9$ to $-14$. The Virgo objects
generally have fainter surface brightnesses than the galaxies of
Binggeli, Sandage and Tammann (1985) and are smaller than those
of Impey, Bothun and Malin (1988).
\section{The Initial Survey}
A preliminary study of two subfields in the Southeast quadrant
of the full survey area
has been carried out. The SuperCOSMOS scans of the quadrant
were provided as nine sections, each $7680 \times 7680$~pixels
in size, two of which were used for the initial survey
(Schwartzenberg et al., 1995).
Excluding their edges, both fields are 1.3~deg square, providing
a total area of 3.2~deg$^2$. The fields, listed in Table~\ref{tab-2},
were selected to sample the core of the cluster and a region of lower
density 3.1~deg to the southeast. The core field included M87,
although due to the raised background light levels, the region
immediately around M87 was excluded from the study.
The data were further subdivided into $2280 \times 2280$~pixel
subregions for data processing because of computer hardware
limitations. Fuller details are given by Schwartzenberg (1996)
and Phillipps et al. (1998b).
\begin{table}
\label{tab-2}
\caption{Details of the fields studied in the initial survey.}
\begin{center}
\begin{tabular}{llc}
\hspace*{3mm} Property & Core field & Outer field \\[1mm]
\tableline \\[-2.5mm]
Field area & ~1.58 deg$^{2}$ & ~1.61 deg$^{2}$ \\[1mm]
Field centre: & & \\[1mm]
\hspace*{4mm} R.A.(1950)& $12^{\rm h} \; 28.2^{\rm m}$ &
~$12^{\rm h} \; 33.9^{\rm m}$ \\[1mm]
\hspace*{4mm} Dec.(1950) & +12$^{\circ}\; 36'$ & +09$^{\circ}\; 49'$ \\[1mm]
\end{tabular}
\end{center}
\end{table}
\begin{figure}[!htb]
\vspace*{-50mm}
\plotfiddle{fig2a.eps}{90mm}{0}{15}{15}{-170}{0}
\plotfiddle{fig2b.eps}{0mm}{0}{15}{15}{-50}{25}
\plotfiddle{fig2c.eps}{0mm}{0}{15}{16}{70}{50}
\vspace*{-8mm}
\caption{
Examples of candidate Virgo Cluster dwarf spheroidal galaxies
from the initial survey of 3.2~deg$^{2}$. The images have been
produced by coadding six UKST exposures on Tech Pan film.
None of the galaxies appears in either the Virgo Cluster
Catalog or the sample of Impey, Bothun \& Malin (1988).
Each frame is 1.0~arcmin in width. North is at the top.
}\label{fig-2}
\vspace*{10mm}
\end{figure}
A total of 56~000 images were detected over the two fields.
From these a subsample of galaxies was selected having central surface
brightnesses in the range $\mu_0 = 22.0$ to 24.5~R~mag~arcsec$^{-2}$
and scale lengths $a \ge 3$~arcsec after fitting exponential
light profiles to the data.
Through a detailed comparison of the numbers of Virgo galaxies with
the South Galactic Pole field population at each
point in the magnitude -- surface brightness plane, it is
possible to remove the background contamination.
The overall contamination of the sample is expected to be
8~per cent. After this background subtraction, the sample contains
1570 galaxies across the two fields.
A luminosity function can be constructed
by binning the background-subtracted galaxy densities by
magnitude. The luminosity function of this sample is found
to be steep, with a formal faint end slope of $\alpha = -2.2$
(Phillipps et al. 1998b), and as such is comparable with the
steep functions found in some more distant clusters (e.g.
Driver et al. 1994, De Propris et al. 1995,
Smith, Driver \& Phillipps 1997, Wilson et al. 1997, Trentham
1997, 1998a). At the bright end, galaxy numbers are consistent with
those of Sandage, Binggeli and Tammann (1985).
The dwarf density in the cluster core field is actually smaller
than that in the outer field (430 galaxies deg$^{-2}$ against
560~deg$^{-2}$). The dense environment in the cluster core, and
particularly the presence of M87, may have a direct effect on the
low surface brightness galaxy population, either through the
removal of these galaxies or suppressing their formation. A similar
effect has been found in the core of the Coma Cluster by
Thompson \& Gregory (1993).
\section{The Full Cluster Survey}
Progress is underway on reducing the full 100~deg$^{2}$ Virgo
survey area. To date,
23 of the required 24 exposures have been taken.
It is intended that the full survey will measure the background
population using data identical in format to those from Virgo,
reducing systematic errors which might be introduced during the
subtraction of the background galaxy numbers.
Work is therefore progressing on the Virgo Northwest field
and on a background field centred at right ascension 10h 40m,
declination 0$^\circ 00'$ (1950).
The photometric calibration has been accomplished using R band
observations on the Anglo-Australian Telescope obtained for
other projects.
For data reduction, each field is subdivided into 16 subregions,
each $6800 \times 6800$ pixels.
Data are stored as 4-byte numbers per pixel, and therefore
each 1/16-th section is 185~Mbyte in size.
New computer hardware resources provide
sufficient memory to reduce each $6800 \times 6800$ pixel section
as a whole, avoiding any need to break these sections into a large
number of smaller regions as was done in the initial survey.
This has the advantage of much simpler data handling than
would be needed if large numbers of small sections were
used. Similarly, establishing astrometric reference frames
for a very large number of different sections would be
prohibitive.
As in the case of the initial survey of 3.2~deg$^2$,
emphasis is being put on searches for small angular
size low surface brightness dwarfs. However,
the feasibility of performing a parallel survey with a different
sky background
subtraction is being investigated, in order to allow searches for
large angular size low surface brightness galaxies of the type
studied by Impey, Bothun and Malin (1988), but over a wider area
than was available for their study.
\section{Conclusions}
The new survey is already providing large samples of candidate
Virgo Cluster dwarf spheroidal galaxies suitable for more detailed
study. The galaxies have absolute magnitudes as faint as
$M_{\rm R} = -11$ to $-16$ (for
$H_0 = 70$~km$\:\mbox{s}^{-1}\:\mbox{Mpc}^{-1}$).
They will be important for addressing
questions relating to the properties and evolution of
the lowest luminosity galaxies in an environment very different
from that in nearby galaxy groups.
An initial survey of 3.2~deg$^2$ has found a high density of
these galaxies and evidence for a steep luminosity function.
The survey is continuing over an area of 100~deg$^2$.
The very large numbers of dwarf spheroidals should reduce the errors
in the faint end of the luminosity function and help to define
its shape. They will enable clear differences to be found at
these very faint limits between the cluster core and periphery
to significantly better accuracy than has been possible for any
other cluster before now. A parallel survey of the Fornax Cluster
is planned for which several films are already available.
\acknowledgments
We wish to thank the UKST and Super{\small COSMOS} staff in
providing their usual excellent service.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,512
|
Oak Photo Art
Whatever Wherever Whenever- We're here to make sure your Favorite Days are captured forever so that the Memories live on...
Action Portraits & Sports
Oak Photo Art Sports & Leisure
Capturing Images to Remember
Portraits, Head Portraits and Action Portraits
We capture portrait images that tell stories in art form. These images are made primarily for wall art. For each image you purchase you receive one to share online so that they can be accessible for all your loved ones to appreciate.
Our Action portraits are captured to be displayed as wall art rather than photojournalistic images for some news source working on a deadline. We believe that everyone is an action person.
Head Portraits are our version of what other Photographers call Headshots. They are much more intense and capture the essence of the subject.
Events and Sports
We capture images that tell your story for years to come. The images are captured to become wall art so that others can appreciate them with you.
Film and Theater
We capture images that record film production and Theater work. This is primarily for personal records although the images have been used for promotion, publication and even recreation of the project.
What We Are Known for:
Oak Photo Art is known for capturing everything from Sports Events to everyday life. Some examples are: Weekly School Sports, Championships, fund raisers, reunions, weddings and church events. Our most significant Sports Events covered were the 5th and 6th Ebix Charity Challenges, hosted by John Isner with Andy Rodick and The Bryan Brothers as guest stars.
Our clients depend on us for outstanding action portraits, sports coverage and now portrait and wall art. Our creative sessions are a memorable experience that last for years to come.
Mission Statement.
Our Mission is to Capture images of our clients that will tell their stories, turn them into memories and become personal and family treasures.
oakphotoart@gmail.com
Copyright © 2020 Oak Photo Art
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,887
|
define(['knockout', 'knockback'], function(ko, kb) {
var viewOptions = {
events : {
"domReady" : "onReady",
},
onReady : function() {
},
initialize : function() {
var self = this;
$(this.el).ready(function() {
self.on("domReady");
});
},
render : function() {
var self = this;
$(this.el).load(window.app.viewFolder + "/"+ this.viewName +".html", function() {
self.onReady("domReady");
});
}
};
var view = Backbone.View.extend(viewOptions);
return {
extend : function(options) {
options = options || {};
if (options.events) {
options.events = _.extend(viewOptions.events, options.events);
}
options._super = view;
return view.extend(options);
}
}
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,075
|
Bayou Gauche és una població dels Estats Units a l'estat de Louisiana. Segons el cens del 2000 tenia una població de 1.770 habitants.
Demografia
Segons el cens del 2000, Bayou Gauche tenia 1.770 habitants, 598 habitatges, i 508 famílies. La densitat de població era de 38,9 habitants/km².
Dels 598 habitatges en un 44,8% hi vivien nens de menys de 18 anys, en un 74,4% hi vivien parelles casades, en un 8,4% dones solteres, i en un 14,9% no eren unitats familiars. En l'11,4% dels habitatges hi vivien persones soles el 4,7% de les quals corresponia a persones de 65 anys o més que vivien soles. El nombre mitjà de persones vivint en cada habitatge era de 2,96 i el nombre mitjà de persones que vivien en cada família era de 3,19.
Per edats la població es repartia de la següent manera: un 29,1% tenia menys de 18 anys, un 7,6% entre 18 i 24, un 32,5% entre 25 i 44, un 24,2% de 45 a 60 i un 6,6% 65 anys o més.
L'edat mediana era de 34 anys. Per cada 100 dones de 18 o més anys hi havia 97,6 homes.
La renda mediana per habitatge era de 51.667 $ i la renda mediana per família de 52.212 $. Els homes tenien una renda mediana de 45.982 $ mentre que les dones 23.966 $. La renda per capita de la població era de 22.170 $. Cap de les famílies i l'1,7% de la població estaven per davall del llindar de pobresa.
Poblacions més properes
El següent diagrama mostra les poblacions més properes.
Referències
Entitats de població de Louisiana
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,751
|
import copy
from preprocessor_exceptions import *
class PreprocessorBase(object):
functions = {}
@classmethod
def expose(cls, name):
"""
Decorator function to expose a function as a preprocessor function
Example usage:
@PreprocessorBase.expose('MyAwesomeFunction')
def myfunc(preprocessor, parameter):
return '[%s]' % (parameter,)
Given the following template (YAML):
key: value
key2: {'Rb::MyAwesomeFunction': 'a great value'}
The preprocessor will return:
{'key': 'value',
'key2': '[a great value]'}
"""
def decorator(f):
cls.functions[name] = f
return f
return decorator
class Preprocessor(object):
def __init__(self, datasource_collection, region):
self.datasource_collection = datasource_collection
self.region = region
def process(self, template):
"""
Go through template, look for {'Rb::FunctionName': <parameters>} dictionaries, calling the Rainbow function
FunctionName to process them.
:type template: dict
:param template: input dictionary
:return: a copy of the template dictionary with all the Rb:: function calls processed
"""
template = copy.deepcopy(template)
if isinstance(template, dict):
if len(template) == 1 and type(template.keys()[0]) is str and template.keys()[0].startswith('Rb::'):
k, v = template.items()[0]
if k.startswith('Rb::'):
function = k[4:]
if not function in PreprocessorBase.functions:
raise InvalidPreprocessorFunctionException(
'Rainbow Function (Rb::) %s not found in %r' % (function,
PreprocessorBase.functions.keys(),))
else:
return PreprocessorBase.functions[function](self, v)
else:
for k, v in template.iteritems():
template[k] = self.process(v)
return template
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,281
|
About SBEDF ›
Christie's Dreams Seafood Wins Startup St. Bernard Competition presented by the Meraux Foundation and SBEDF
St. Bernard Parish, La. (March 15, 2016) — Judges announced Christie's Dreams Seafood the winner of the second annual Startup St. Bernard held on March 14, 2016. Christie's Dreams Seafood took home a $110,000 prize package of cash and in-kind services designed to help propel their business to the next level. The St. Bernard Parish-based company was one of five finalists to pitch their business at the live "Shark Tank" style public event at the Meraux Foundation's Arlene Meraux River Observation Center at Docville Farm.
Christie's Dreams Seafood is an existing local business that entered Startup St. Bernard to help launch a new product line — St. Bernard Parish branded seafood. The company plans to freeze and package fresh shrimp and soft shell crabs locally caught in St. Bernard Parish waters using a flash freezing method that guarantees premium quality. In the initial stages the business plans to sell the product at local farmers markets throughout Louisiana and to their existing wholesale customers in Maryland, Virginia, and Pennsylvania. Long term, Christie's Dreams Seafood will create a web site to sell St. Bernard branded seafood throughout the United States.
I would like to congratulate Christie's Dream Seafood on winning Startup St. Bernard," said Rita Gue, president of the Meraux Foundation, which created and hosted the event. "Their product will help export the St. Bernard brand across the country!"
Other finalists included Atlas Mechanical Design, a new business focused on providing 3D printing solutions; Girls Gone Vegan, a bakery that specializes in gluten free and vegan desserts; Kayak-iti-Yat, an eco-tourism business offering fishing-kayak rentals as well as an ecology and seafood industry-centered guided kayak tours; and The Coffee House, a startup drive-through coffee shop in Arabi that plans to expand operations to include an indoor and outdoor cafe where food will be served and the community can gather.
"It was difficult to see only one of our finalists walk away with the lucrative Startup St. Bernard prize package, said Andrew Jacques, Executive Director of the St. Bernard Economic Development Foundation. However, Christie's Dreams Seafood represents one of St. Bernard's finest assets, world class seafood. I look forward to watching their business and brand grow exponentially."
Prior to today's pitch event, the five finalists were selected out of a pool of entries. Startup St. Bernard was open to any business that agreed to locate their base of operations to St. Bernard Parish, help improve St. Bernard Parish, and become part of the St. Bernard Parish entrepreneur ecosystem. Finalists were identified by an independent scoring committee based on the criteria published on the entry form, including the quality of their business plan and impact on St. Bernard Parish.
The finalists' pitches were judged by an esteemed panel of businesspeople, including: Peter Bodenheimer, COO of Plantbid, partner in Flatstack and Launch Pad; Joel Dondis, Founder of Sucré; Robby Showalter, Chief Financial Officer of Boasso America Corporation; Allen Square Jr., Founder of Square Button, member of New Orleans Startup Fund; and Ann Tuennerman, Founder of Tales of the Cocktail® . WYES-TV's Tom Gregory emceed the event.
The $110,000 prize package was a combination of cash and in-kind services donated by the Meraux Foundation, Park Investments, Ltd., Palms Truck Stop, Sidney Torres, III, Mrs. Rita Gue, Mr. Floyd Gue, Nunez Community College Foundation, Southern Insurance Agency LLC, Taurus Properties, Palmisano Contractors, Sheriff Jimmy Pohlmann, Mr. Terry and Mrs. Daryl Tedesco, BERG Services, State Senator JP Morrell, Casey Hunnicutt, 40 Arpent Brewing, Caraway LeBlanc, Coast Roast. Gulf Coast Bank & Trust Company St. Bernard Entrepreneurial Center, evanSchmidt Design, LLC, Ragusa Consulting, John Brian White, Esq., InHouse Consultants, Inc., and AdamsIP, LLC, Sherman Strategies, LLC, Baldwin Haspel Burke & Mayer, LLC | Valued at $5,000, Person Huff CPA Group, Ditto's, Shine Productions, Outfront Media, St. Bernard Chamber of Commerce, St. Bernard Tourism Commission, Global Staffing Solutions, Bill Haines, Crave, and 40 Arpent Brewing.
About the Arlene and Joseph Meraux Charitable Foundation
The Meraux Foundation is a private 501(c)(3) nonprofit organization that was established by Arlene Meraux to benefit the community of St. Bernard Parish by leveraging its landholdings. Today, the Foundation's board of directors is presided over by Arlene's niece, Rita Gue, and consists of members Floyd Gue, Bill Haines, Chris Haines, and Sidney Torres, III. Guided by Arlene's vision, the Board is implementing an innovative strategy to create lasting change and build a better St. Bernard. The Meraux Foundation is a founding sponsor and organizer of Startup St. Bernard.
About St. Bernard Economic Development Foundation
St. Bernard Economic Development Foundation is 501(c)(3) nonprofit organization founded in 2005 to promote economic development and commercial growth of St.Bernard Parish and to lead the initiation of partnering organizations seeking to improve the quality of life for St. Bernard Parish citizens. To achieve this mission, SBEDF primarily engages in business attraction, retention, and expansion. Additionally, workforce development and policy development are pursued as key initiatives to drive economic development activities. The Board of Directors consists of 28 voting members and 8 non-voting honorary members and public officials, all of which serve as volunteers to the organization. SBEDF's work is made possible through the general support of its investors, St. Bernard Parish Government, St. Bernard Port, Harbor and Terminal District, the Arlene and Joseph Meraux Charitable Foundation and Cox Communications at the Champion Level and the Law Offices of Sidney Torres, III and Associated Terminals at the Pillars Level.
About New Orleans Entrepreneur Week
New Orleans Entrepreneur Week (NOEW) is the annual festival celebrating business, innovation, and advanced thinking in New Orleans. Produced by The Idea Village, NOEW engages a dynamic global network of entrepreneurs, investors, corporations, nonprofits, students, and professionals to support growing companies and elevate entrepreneurial success through eight days of discussion, debate, education, competition and celebration. This year, the event will take place March 11-18, 2016 and 10,000+ are expected to be engaged through over 75 different events.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,698
|
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