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The National Human Rights Commission (NHRC) has underscored the protection and promotion of human rights for which it has furnished 8-point recommendations to the government quoting RSS. Unveiling a report titled '12 years of signing of Comprehensive Peace Agreement' marking the 12 years of signing of the Comprehensive Peace Agreement (CPA), NHRC suggested the government for the effective implementation of the economic, social and cultural rights of the people if the State is to address human rights issues and promote sustainable peace. CPA was signed between then government and then CPN-Maoist party in November 22, 2006, marking an end to the decade long insurgency in the country. More than 17,000 persons-both from the State and insurgents-had lost their lives during the period. The report suggested government to hold consultations with the conflict-victims as well as the stakeholders and make amendments in the laws related to addressing the issues surrounding conflict-victims. Furthermore, the government is recommended to formulate plans and programmes for justice, reparations and rehabilitation of the conflict-victims by considering amendments in the Truth and Reconciliation Commission Act-2071. As for addressing the issues of those fallen prey to sexual abuse and gender-based violence during the conflict, the report suggests government to devise plans, programmes and laws securing their welfare. Stringent implementation of the International Conventions signed by Nepal seeking to discourage enrollment of child soldiers in the conflicts and wars has been recommended. Moreover, the government has been urged to take into account those suffering mental and physical disability in the conflict to address the issues of conflict victims.
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Investing in consumer discretionary stocks A dynamic sector of the market that relies on a good economy. Kimberly Ellis Navigate Stock Trading Investments 101 Buy stocks online Best broker signup bonuses Investing in IPOs Buy bonds Forex guide Robo Advice vs. Traditional Ally Invest Ellevest You Invest Zacks Trade A to Z list of companies How To Invest In The consumer discretionary sector is chock full of luxury. These non-essential goods, products and services carry popular designer names like Michael Kors and Ferrari, and also include restaurants, coffee shops and golf courses. Consumer discretionary stocks soar during a robust economy, but investors should be wary during a downturn — consumers tend to cut this spending first. What's in this guide? What are consumer discretionary stocks? How to invest in the consumer discretionary sector How is consumer discretionary sector performing? Why invest in the consumer discretionary sector? What unique risks does the consumer discretionary sector face? Compare stock trading platforms The consumer discretionary sector is one of 11 sectors of the stock market. These businesses include products and services that consumers may want, but don't necessarily need. For example, high-end clothing, big-screen televisions, family vacations and sporting goods fall under this category. Consumers usually purchase these non-essential goods or services when they feel confident about their finances and have some disposable income. What industries does it include? With products that range from cars to lipstick, the consumer discretionary sector covers a vast range of industries. Automobiles. Companies that manufacture cars, trucks, motorcycles and scooters. Hotels, restaurants and leisure. The service industry includes food and drink establishments, lodging and recreational activity venues such as theme parks. Household durables. Residential products like furniture and appliances. Multiline retail. Stores that offer diversified products, such as Macy's. Textiles, apparel and luxury goods. Manufacturers of clothing, accessories and luxury goods, such as Hermès designer handbags and Givenchy leather shoes. Leisure products. These vendors of recreational products and equipment include sports gear and toys. Consumer discretionary stocks vs. consumer staples stocks Consumer discretionary stocks offer products or services that people enjoy, but can live without. Consumer staples are things we need, such as food, beverages, household essentials and hygiene products like toilet paper. No matter how the economy is doing, you'll always stock your house with consumer staples. But in a waning economy, you might eliminate the nonessentials. There are two ways to invest in the consumer discretionary sector: individual stocks or exchange-traded funds (ETFs). When you invest in a particular consumer discretionary stock, you buy shares of the company. There are fewer fees, but more risk involved. If you go the ETF route, you'll get a basket of consumer discretionary stocks, which come with higher fees but diversifies your portfolio and lowers your exposure risk. A breakdown of how to get started: Pick a brokerage. Browse different brokerage platforms to choose a firm that suits your investing needs. Open an account. Most firms let you open a brokerage account online. Some accounts require a deposit to open, while others let you fund your account right before investing. Shop for securities. Use your platform's research programs to examine different stocks and ETFs. Place an order. Give the order to buy the security. Track your portfolio. Monitor your investments by logging into your brokerage account. What stocks are in the consumer discretionary sector? Select a company to learn more about what they do and how their stock performs, including market capitalization, the price-to-earnings (P/E) ratio, price/earnings-to-growth (PEG) ratio and dividend yield. While this list includes a selection of the most well-known and popular stocks, it doesn't include every stock available. Starbucks Corporation (SBUX.US) Starbucks Corporation, together with its subsidiaries, operates as a roaster, marketer, and retailer of specialty coffee worldwide. The company operates through three segments: Americas, International, and Channel Development. Its stores offer coffee and tea beverages, roasted whole bean and ground coffees, single-serve and ready-to-drink beverages, and iced tea; and various food products, such as pastries, breakfast sandwiches, and lunch items. The company also licenses its trademarks through licensed stores, and grocery and foodservice accounts. The company offers its products under the Starbucks, Teavana, Seattle's Best Coffee, Evolution Fresh, Ethos, Starbucks Reserve, and Princi brand names. As of October 29, 2020, it operated approximately 32,000 stores. Starbucks Corporation was founded in 1971 and is based in Seattle, Washington. Market capitalization: $120474427392 P/E ratio: 129.9304 PEG ratio: 1.6429 Dividend yield: 0.0172% Get more detailed information and learn how to buy SBUX.US stock NIKE (NKE.US) NIKE, Inc., together with its subsidiaries, designs, develops, markets, and sells athletic footwear, apparel, equipment, and accessories worldwide. The company offers NIKE brand products in six categories, including running, NIKE basketball, the Jordan brand, football, training, and sportswear. It also markets products designed for kids, as well as for other athletic and recreational uses, such as American football, baseball, cricket, golf, lacrosse, skateboarding, tennis, volleyball, walking, wrestling, and other outdoor activities; and apparel with licensed college and professional team and league logos, as well as sells sports apparel. In addition, the company sells a line of performance equipment and accessories comprising bags, socks, sport balls, eyewear, timepieces, digital devices, bats, gloves, protective equipment, and other equipment for sports activities; and various plastic products to other manufacturers. Further, it provides athletic and casual footwear, apparel, and accessories under the Jumpman trademark; casual sneakers, apparel, and accessories under the Converse, Chuck Taylor, All Star, One Star, Star Chevron, and Jack Purcell trademarks; and action sports and youth lifestyle apparel and accessories under the Hurley trademark. Additionally, the company licenses agreements that permit unaffiliated parties to manufacture and sell apparel, digital devices, and applications and other equipment for sports activities under NIKE-owned trademarks. It sells its products to footwear stores; sporting goods stores; athletic specialty stores; department stores; skate, tennis, and golf shops; and other retail accounts through NIKE-owned retail stores, digital platforms, independent distributors, licensees, and sales representatives. The company was formerly known as Blue Ribbon Sports, Inc. and changed its name to NIKE, Inc. in 1971. NIKE, Inc. was founded in 1964 and is headquartered in Beaverton, Oregon. P/E ratio: 81.4036 Get more detailed information and learn how to buy NKE.US stock MGM Resorts International (MGM.US) MGM Resorts International, through its subsidiaries, owns and operates integrated casino, hotel, and entertainment resorts in the United States and Macau. The company operates through three segments: Las Vegas Strip Resorts, Regional Operations, and MGM China. Its casino resorts offer gaming, hotel, convention, dining, entertainment, retail, and other resort amenities. The company's casino operations include slots, table games, and race and sports book wagering. As of March 22, 2020, its portfolio consisted of 29 hotel and destination gaming offerings. The company also owns and operates Las Vegas Strip Resorts, Primm Valley Golf Club, and Fallen Oak golf course. Its customers include premium gaming customers; leisure and wholesale travel customers; business travelers; and group customers, including conventions, trade associations, and small meetings. The company was formerly known as MGM MIRAGE and changed its name to MGM Resorts International in June 2010. MGM Resorts International was founded in 1986 and is based in Las Vegas, Nevada. Market capitalization: $15489927168 Get more detailed information and learn how to buy MGM.US stock Carnival Corporation-and-Plc (CCL.US) Carnival Corporation & Plc operates as a leisure travel company. The company's ships visit approximately 700 ports under the Carnival Cruise Line, Princess Cruises, Holland America Line, Seabourn, P&O Cruises (Australia), Costa Cruises, AIDA Cruises, P&O Cruises (UK), and Cunard brand names. It also provides vacations to various cruise destinations, as well as owns and operates hotels, lodges, glass-domed railcars, and motor coaches. The company sells its cruises primarily through travel agents and tour operators. It operates in the United States, Canada, Continental Europe, the United Kingdom, Australia, New Zealand, Asia, and internationally. As of January 28, 2020, the company operated 105 ships with 254,000 lower berths. Carnival Corporation & Plc was incorporated in 1972 and is headquartered in Miami, Florida. P/E ratio: 5.9852 PEG ratio: 1.41 Get more detailed information and learn how to buy CCL.US stock Aptiv (APTV.US) Aptiv PLC designs, manufacturers, and sells vehicle components worldwide. The company provides electrical, electronic, and safety technology solutions to the automotive and commercial vehicle markets. It operates through two segment, Signal and Power Solutions, and Advanced Safety and User Experience. The Signal and Power Solutions segment designs, manufactures, and assembles vehicle's electrical architecture, including engineered component products, connectors, wiring assemblies and harnesses, cable management products, electrical centers, and hybrid high voltage and safety distribution systems. The Advanced Safety and User Experience segment provides critical components, systems integration, and software development for vehicle safety, security, comfort, and convenience, such as sensing and perception systems, electronic control units, multi-domain controllers, vehicle connectivity systems, application software, and autonomous driving technologies. The company was formerly known as Delphi Automotive PLC and changed its name to Aptiv PLC in December 2017. Aptiv PLC is headquartered in Dublin, Ireland. Get more detailed information and learn how to buy APTV.US stock eBay (EBAY.US) eBay Inc. operates the marketplace and classifieds platforms that connect buyers and sellers worldwide. Its Marketplace platform includes its online marketplace at ebay.com and the eBay suite of mobile apps; and Classifieds platform comprises a collection of brands, such as Mobile.de, Kijiji, Gumtree, Marktplaats, eBay Kleinanzeigen, and others that offer online classifieds to help people find what they are looking for in their local communities. Its platforms enable users to list, buy, sell, and pay for items through various online, mobile, and offline channels that include retailers, distributors, liquidators, import and export companies, auctioneers, catalog and mail-order companies, classifieds, directories, search engines, commerce participants, shopping channels, and networks. The company was founded in 1995 and is headquartered in San Jose, California. Get more detailed information and learn how to buy EBAY.US stock Amazon-com (AMZN.US) Amazon.com, Inc. engages in the retail sale of consumer products and subscriptions in North America and internationally. The company operates through three segments: North America, International, and Amazon Web Services (AWS). It sells merchandise and content purchased for resale from third-party sellers through physical and online stores. The company also manufactures and sells electronic devices, including Kindle, Fire tablets, Fire TVs, Rings, and Echo and other devices; provides Kindle Direct Publishing, an online service that allows independent authors and publishers to make their books available in the Kindle Store; and develops and produces media content. In addition, it offers programs that enable sellers to sell their products on its Websites, as well as its stores; and programs that allow authors, musicians, filmmakers, skill and app developers, and others to publish and sell content. Further, the company provides compute, storage, database, and other AWS services, as well as fulfillment, advertising, publishing, and digital content subscriptions. Additionally, it offers Amazon Prime, a membership program, which provides free shipping of various items; access to streaming of movies and TV episodes; and other services. The company also operates in the food delivery business in Bengaluru, India. It serves consumers, sellers, developers, enterprises, and content creators. The company also has utility-scale solar projects in China, Australia, and the United States. Amazon.com, Inc. has a strategic relationship with NXP Semiconductors N.V. to deliver a cloud compute solution for vehicles that enable cloud-powered services. The company was founded in 1994 and is headquartered in Seattle, Washington. Market capitalization: $1597281009664 Dividend yield: 0% Get more detailed information and learn how to buy AMZN.US stock Dominos Pizza (DPZ.US) Domino's Pizza, Inc., through its subsidiaries, operates as a pizza delivery company in the United States and internationally. It operates through three segments: U.S. Stores, International Franchise, and Supply Chain. The company offers pizzas under the Domino's brand name through company-owned and franchised stores. As of August 17, 2020, it operated approximately 17,100 stores in 90 markets. The company was founded in 1960 and is headquartered in Ann Arbor, Michigan. Get more detailed information and learn how to buy DPZ.US stock Chipotle Mexican Grill (CMG.US) Chipotle Mexican Grill, Inc., together with its subsidiaries, operates Chipotle Mexican Grill restaurants. As of September 30, 2020, it operated approximately 2,700 restaurants in the United States, Canada, the United Kingdom, France, and Germany. The company was founded in 1993 and is headquartered in Newport Beach, California. Get more detailed information and learn how to buy CMG.US stock Take-Two Interactive Software (TTWO.US) Take-Two Interactive Software, Inc. develops, publishes, and markets interactive entertainment solutions for consumers worldwide. The company offers its products under the Rockstar Games and 2K labels, as well as under Private Division and Social Point labels. It develops and publishes action/adventure products under the Grand Theft Auto, Max Payne, Midnight Club, and Red Dead Redemption names; and offers episodes, content, and virtual currency. The company also develops brands in other genres, including the LA Noire, Bully, and Manhunt franchises. In addition, the company publishes various entertainment properties across various platforms and a range of genres, such as shooter, action, role-playing, strategy, sports, and family/casual entertainment under the BioShock, Mafia, Sid Meier's Civilization, XCOM series, and Borderlands. Further, it publishes sports simulation titles comprising NBA 2K series, a basketball video game; the WWE 2K professional wrestling series. It also offers Kerbal Space Program, The Outer Worlds, Ancestors the Humankind Odyssey under Private Division. Additionally, the company offers free-to-play mobile games, such as Dragon City and Monster Legends. Its products are designed for console gaming systems, including Sony's PlayStation 4; Microsoft's Xbox One; the Nintendo Switch; and personal computers comprising smartphones and tablets. The company provides its products through physical retail, digital download, online platforms, and cloud streaming services. Take-Two Interactive Software, Inc. was founded in 1993 and is headquartered in New York, New York. Get more detailed information and learn how to buy TTWO.US stock Vista Outdoor (VSTO.US) Vista Outdoor Inc. designs, manufactures, and markets consumer products for outdoor sports and recreation markets in the United States and internationally. The company operates in two segments, Shooting Sports and Outdoor Products. The Shooting Sports segment offers ammunition products that include centerfire ammunition, rimfire ammunition, shotshell ammunition, and reloading components; archery and hunting accessories comprising high-performance hunting arrows, game calls, hunting blinds, game cameras, and decoys; optics products, such as binoculars, riflescopes, and telescopes; and shooting accessories that consist of reloading equipment, clay targets, and premium gun care products. This segment also provides tactical products, such as holsters, duty gear, bags, and packs. The Outdoor Products segment offers sports protection products, including helmets, goggles, and accessories for cycling, snow, action, and power sports; outdoor cooking products, such as grills, cookware, and camp stoves; golf products comprising laser rangefinders and other golf technology products; and hydration products, including hydration packs and water bottles. The company sells its products to outdoor enthusiasts, hunters, recreational shooters, athletes, law enforcement, and military professionals through various mass, specialty, and independent retailers and distributors, as well as directly to consumers through website. Vista Outdoor Inc. was incorporated in 2014 and is headquartered in Anoka, Minnesota. Market capitalization: $1191436416 Get more detailed information and learn how to buy VSTO.US stock What ETFs track the consumer discretionary sector? A few popular ETFs that follow the sector include: Consumer Discretionary Select Sector SPDR ETF (XLY) SPDR S&P Homebuilders ETF (XHB) Vanguard Consumer Discretionary ETF (VCR) VanEck Vector Retail ETF (RTH) Fidelity MSCI Consumer Discretionary Index (FDIS) iShares US Consumer Services ETF (IYC) iShares US Home Construction ETF (ITB) SPDR S&P Retail ETF (XRT) Amplify Online Retail ETF (IBUY) Invesco Dynamic Leisure and Entertainment ETF (PEJ) Use the graph below to see how the Consumer Discretionary Select Sector SPDR ETF (XLY) is currently performing, as well as how it has been performing over the last three months, year and five years. Consumer discretionary stocks have the potential for high returns, especially when the economy is strong. For example, during the start of the longest economic expansion in US history, the S&P 500 Consumer Discretionary Index returned 41.3% in 2009, compared to the S&P 500 Index's 26.5%. And it continued to bring in consistently higher returns for many years. Another benefit of the consumer discretionary sector is that it's easier for investors to gauge entry into the market. Since consumer discretionary stocks perform in tandem with the economy, investors can monitor economic indicators, such as the gross domestic product (GDP), to judge whether it might be a good time to start investing. How are the dividends for consumer discretionary stocks? Consumer Discretionary stocks dividends are usually comparable to the rest of the market. But economic downfalls can lead to dividend cuts. For example, in September 2019, the SPDR S&P Retail ETF (XRT) had a yield of 1.99%, compared to the S&P 500 Index's (SPY) 1.97%. But just a few months later, the COVID-19 pandemic forced consumers to stay at home and shut down major retailers, hotels and restaurants. Many stocks plummeted, affecting dividend payouts as well. In June 2020, the SPDR S&P Retail ETF (XRT) had a yield of 0.86%, steeply trailing the S&P 500 Index's (SPY) dividend of 1.75%. Economic cycles have a big hand in how consumer discretionary stocks perform. Since this sector is extremely unpredictable, here are a few things to watch out for: Weak economy. The sector suffers in a declining economy, especially when there are high rates of unemployment. Consumers tend to tighten their spending and reduce luxury goods from their budget. High interest rates. Consumers often purchase more expensive products, such as cars or jewelry, on credit. High credit card interest rates are harder on customers and may deter spending. Poor consumer confidence. How people feel about the economy plays a key role in consumer spending. A positive outlook can lead to more spending, whereas a loss in confidence usually means that consumers are saving rather than spending. For example, when consumer confidence was at an all-time low in 2009 following the Great Recession, all major areas of spending — except healthcare — dropped an average of 2.8%, according to the US Department of Labor. You'll need a brokerage account to buy stocks or ETFs. Use the table below to compare your options and find the best fit. Stock trade fee Option trade fee Sofi Invest Stocks, ETFs, Cryptocurrency A free way to invest in stocks, ETFs and crypto. Stocks, Options, ETFs, Cryptocurrency Make unlimited commission-free trades in stocks, funds, and options with Robinhood Financial. Moomoo $0 for US stocks Stocks, Options, ETFs Trade stocks on the US, Hong Kong, Shanghai and Shenzhen markets. or $25 broker-assisted $0 + $0.65/contract, TD Ameritrade features $0 commission for online stock, but watch out for high short-term ETF and broker-assisted trading fees. Disclaimer: The value of any investment can go up or down depending on news, trends and market conditions. We are not investment advisers, so do your own due diligence to understand the risks before you invest. The consumer discretionary sector may be a good choice when the economy is growing and consumers feel good about their job and finances. But tread carefully when the economy starts trending down. Compare online trading platforms to find a brokerage firm when you're reading to start investing. Is the consumer discretionary sector cyclical? Yes. This sector's performance is dependent on how the economy is doing. Consumers only spend money on things they don't necessarily need when they have enough income for necessities. Does a weakening economy affect all consumer discretionary stocks the same way? No. Industries perform differently during economic downturns. For instance, after the Great Recession of 2008, the furniture industry saw a bigger drop than hotels and lodging. Kimberly Ellis is a writer at Finder. She hails from New York City with a BA from Queens College and a New York State teaching certificate. After teaching in both public and private schools, Kimberly decided to take the world by storm and dive into the media industry — where she covers everything from home loans and investing to K–12 education and shopping. She's also an aspiring polyglot, always in a book and forever on the hunt for the perfect classic red lipstick. Investing in materials stocks Investing in consumer staples stocks These IPOs will be hot in 2021 Private-equity companies have about $1.6 trillion of uninvested capital to put into merger and acquisition activity. This bodes well for 2021's IPO market. 5 smart ways to invest your stimulus check If you don't need your money right now, consider investing in an HSA, blue chip stocks and more. 401(k) contribution limits for 2021 Unlock the tax benefits of these retirement accounts and what to do if you maxed out. The top 5 stocks that beat Bitcoin in 2020 The top 5 stocks that beat Bitcoin in 2020 and why 3 ways to invest in private equity Private equity could be a way to maximize returns. Learn more. Investing in entertainment stocks Benefits and risks to consider before you back the entertainment sector. Investing in airline stocks The airline industry saw a decade of gains shot down by COVID-19. When might they rise again? Investing in cruise ship stocks Cruise ship stocks are sailing through murky waters, but can they rise? Yieldstreet review Features and drawbacks to consider before you invest through Yieldstreet. Investing in pharmaceutical stocks Benefits and drawbacks to consider before you invest in pharma stocks.
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Table of Contents Title Page Dedication Introduction Chapter 1 - HE'S A STUD, SHE'S A SLUT Chapter 2 - HE'S CHILL, SHE'S ON THE PILL Chapter 3 - HE'S ROUGH, SHE'S DAINTY Chapter 4 - HE'S A HERO, SHE'S A DAMSEL Chapter 5 - HE'S METROSEXUAL, SHE'S ANOREXIC Chapter 6 - HE'S "LUCKY," SHE'S LOLITA Chapter 7 - HE'S A BACHELOR, SHE'S A SPINSTER Chapter 8 - HE CAN BE A BEAST, SHE MUST BE A BEAUTY Chapter 9 - HE'S A HIPSTER, SHE'S A HO Chapter 10 - HE'S GONNA BE A SUCCESS, SHE'S GONNA BE A STAY-AT-HOME MOM Chapter 11 - HE'S A POLITICIAN, SHE'S A FASHION PLATE Chapter 12 - HE'S A ROMEO, SHE'S A STALKER Chapter 13 - HE'S TOUGH, SHE'S A TOMBOY Chapter 14 - HE'S ANGRY, SHE'S PMSING Chapter 15 - HE'S DISTINGUISHED, SHE'S DRIVING MISS DAISY Chapter 16 - HE'S MANLY, SHE'S SASQUATCH Chapter 17 - HE'S SUCCESSFUL, SHE'S A SHOWOFF Chapter 18 - HE'S SUPERDAD, SHE'S SHITTYMOM Chapter 19 - HE'S THE BOSS, SHE'S A BITCH Chapter 20 - HE'S WELL PAID, SHE'S SCREWED Chapter 21 - HE'S GAY, SHE'S A FANTASY Chapter 22 - HE'S HIMSELF, SHE'S MRS. HIMSELF Chapter 23 - HE'S GETTING AN EDUCATION, SHE'S GETTING IN HIS WAY Chapter 24 - HE'S INDEPENDENT, SHE'S PATHETIC Chapter 25 - HE'S A CELEB, SHE'S A MESS Chapter 26 - HE'S HUSKY, SHE'S INVISIBLE Chapter 27 - HE'S A MAN, SHE'S A MOM Chapter 28 - HE'S DATING A YOUNGER WOMAN, SHE'S A COUGAR Chapter 29 - HE'S DRUNK, SHE'S A VICTIM Chapter 30 - HE'S STOIC, SHE'S FRIGID Chapter 31 - HE'S COVERED, SHE'S SCREWED Chapter 32 - HE'S REPRESENTED, SHE'S A TOKEN Chapter 33 - HE'S NEAT, SHE'S NEUROTIC Chapter 34 - HE'S FUN, SHE'S FRIVOLOUS Chapter 35 - HE WALKS FREELY, SHE GETS HARASSED Chapter 36 - HE'S A PORN WATCHER, SHE'S THE SHOW Chapter 37 - HE'S HOT AND HEADY, SHE'S BRAINY OR BOOBILICIOUS Chapter 38 - HE'S AN ACTIVIST, SHE'S A PAIN IN THE ASS Chapter 39 - HE'S A PERSON, SHE'S A COMMODITY Chapter 40 - HE'S A PUBLIC FIGURE, SHE'S A VIRGIN/ WHORE Chapter 41 - HE'S GOT G.I. JOE, SHE'S GOT BARBIE Chapter 42 - HE'S PAYING LESS, SHE'S PAYING MORE Chapter 43 - HE'S PUSSY WHIPPED, SHE'S A "GOOD GIRLFRIEND" Chapter 44 - HE'S PROTECTED, SHE'S PROPERTY Chapter 45 - HE'S FONDLING, SHE'S FEEDING Chapter 46 - HE'S CHILDLESS, SHE'S SELFISH Chapter 47 - HE'S FUNNY, SHE'S ANNOYING Chapter 48 - HE'S DATING, SHE'S TAKEN Chapter 49 - HE'S RUGGED, SHE'S RUDE Chapter 50 - HE'S PLAIN, SHE'S VAIN NOTES Acknowledgements ABOUT THE AUTHOR SELECTED TITLES FROM SEAL PRESS Copyright Page To the readers of _**Feministing.com,**_ for inspiring me every day. **INTRODUCTION** **WHEN I WA S IN HIGH SCHOOL,** I had a reputation—a bad one. (You know, a "slutty" one.) I wasn't quite sure how I became seen as the promiscuous girl in school, since I was definitely not getting any more action than my girlfriends. It felt like the reputation—which I really didn't find out about until well into my senior year—had materialized out of nowhere. And I was confused. Maybe it was because I went to a kind of dorky math and science magnet school where anyone who even talked about sex was labeled sexually active? Perhaps it was because I had so many guy friends that I hung around with? Or maybe it was because I was a little more, ahem, developed than the other gals? I wasn't quite sure. Looking back, I realize that it could have been any of those things, or nothing. Most likely, it was because I had a bit of a potty mouth (shocking, I know), told dirty jokes, and was a louder, more opinionated girl than some of my peers. I know better now, and realize that labeling girls "sluts" is a pretty common silencing tactic. After all, there's no better way to silence a woman than to call her a whore! But that was the first sexist double standard I became acutely aware of—one that affected my life and, maybe more important, really pissed me off. I was upset not only that people thought things about me that weren't true, but that the double standard existed in the first place. So fucking what if I _had_ slept with every guy in my grade? Why would that make me a bad person? It just seemed so illogical to me, yet it was so accepted. While I didn't consider myself a feminist until college, when I took my first women's studies class, I think it was this sense of just simple unfairness that really got me started down my feminist path. Because everyday sexism is something that we can all relate to. If you're a feminist or not, a Democrat or Republican, there are certain things that all women recognize—and are pissed off about! After I wrote my first book, _Full Frontal Feminism,_ it was difficult to know what to write next. I got such amazing responses from young women who read the book—women from thirteen to sixty!—I didn't want to let them down with my next one. One email I got was from a sixteen-year-old Middle Eastern woman living in Michigan who was happy to read something from another young feminist. Another teen, a fourteen-year-old from Mozambique, was pleased that she finally had something that she could use to "get across to my somewhat closed-minded friends for years." The notes that affected me the most, though, were the ones that inspired action. A twenty-one-year-old African American woman from California sent me a message through MySpace about how she faces racism and sexism at work every day: "I thought ideas and feelings like the ones your book and blog have shown me only existed in my hometown of Oakland and S.F. I now want to start a young feminist movement in my community." I was so touched that these women would take the time to write me, and that the book made an impact on their lives. . . . It was very overwhelming and it still feels like a huge responsibility (one I'm flattered to have!). And while the notes I got from women came from all different parts of the world, and came from women all across the spectrum in terms of class, race, sexuality, and politics, the one thing they all had in common was that they talked about how sexism affected their everyday lives. Whether it was through sexual harassment or workplace racism or just the struggles they had in school or at home taking care of their kids—it was the day-to-day injustices that women talked about. So I figured, why not go back to basics? Go back to that place when I hadn't even started to think about feminism yet—but where it was still impossible not to think about and notice day-to-day unfairness and injustice. No matter how anyone feels about feminism, there are certain inequalities and double standards that are impossible to ignore or argue with. I'm hoping this book will be a fun (but informative!) handbook on those everyday inequities women still face. Because from the boardroom to the bedroom, women are still getting the short end of the stick. Whether it's the sexual double standard that led to me (and so many other women) being labeled a slut, or the work double standard that calls women "bitches" for being good at their job, we still have a long way to go. This book is for any feminist—or non-feminist!—who is sick of people saying that everything is fine and dandy.This is a book that you'll be able to whip out, whether at school, a bar, or the office, to show the skeptics that sexism is still alive and well—but that there are women out there doing something about it! Think of it as a quick reference guide to everyday sexism. Only funnier. I hope this book inspires action. I hope that you'll carry it around and use it to battle the sexists in your lives. But most of all, I hope that you leave this book not feeling downtrodden about how pervasive sexism is, but instead energized to do something about it! That said, I just want to say thanks to all the feminists out there—especially you new feminists!—for doing the hard work, every day, of telling the truth about sexism. I know it's not always easy, but it's changing lives. You all are inspiring. **1** **HE'S A STUD, SHE'S A SLUT** **IF YOU HAVE A VAGINA,** chances are someone has called you a slut at least once in your life. There's just no getting around it. I remember the first time I heard the word "slut"—I was in my fifth-grade science class. A certain little girl (terror) named Eleena had been making my life miserable all year in a way that only mean little girls can. She had turned all of my girlfriends against me, spread rumors and the like. She walked up to me at my desk and said, "You called me a slut." I had absolutely no idea what the word meant. I just sat there, silently. She repeated herself: "You called me a slut, but you're the slut." I don't remember how long after that I found out exactly what "slut" meant, but I knew it had to be terrible and I knew I didn't want to be it. Naturally, I'd be called a slut many times over later in life—not unlike most girls. I was called a slut when my boobs grew faster than others'. I was called a slut when I had a boyfriend (even though we weren't having sex). I was called a slut when I didn't have a boyfriend and kissed a random boy at a party. I was called a "slut" when I had the nerve to talk about sex. I was called a slut when I wore a bikini on a weekend trip with high school friends. It seems the word "slut" can be applied to any activity that doesn't include knitting, praying, or sitting perfectly still lest any sudden movements be deemed whorish. Despite the ubiquity of "slut," where you won't hear it is in relation to men. Men can't be sluts. Sure, someone will occasionally call a guy "a dog," but men simply aren't judged like women are when it comes to sexuality. (And if they are, they're judged in a positive way!) Men who have a lot of sexual partners are studs, Casanovas, pimps, and players. Never sluts. In fact, when I just did a Google search for "male sluts," the first result I got was _She Male Sluts DVD!_ I know, should have seen that coming. The point is, there isn't even a word—let alone a concept—to signify a male slut. But it makes sense when you think about what the purpose of the word "slut" is: controlling women through shame and humiliation. Women's bodies are _always_ the ones that are being vied over for control—whether it's rape, reproductive rights, or violence against women, it's our bodies that are the battleground, not men's. And if you don't think it's about control, consider this little bit of weirdness. The most recent incarnation of the sexual double standard being played out in a seriously creepy way is through Purity Balls. These promlike events basically have fathers take their daughters to a big fancy dance where they promise their daddy their virginity. Likewise, the father promises to be the "keeper" of his daughter's virginity until he decides to give it to her future husband. Where are the Purity Balls for men, you ask? Oh, they're there, but they're about controlling women too! Called Integrity Balls, these events focus on men not having sex because they'd be defiling someone else's "future wife"! Not because men need to be pure or be virgins—but because they need to make sure _women_ are virgins. Unbelievable, really. Outside of the feminist implications of the sexual double standard, the slut/stud conundrum has always been my favorite because it just makes no sense logically. Why is a woman less of a person, or (my favorite) "dirty," because she has sex? (Heterosexual sex, that is; somehow lesbian sex isn't "real.") Does a penis have some bizarre dirty-making power that I'm unaware of? Every time I have sex, do I lose a little bit of my moral compass? "Sorry to mug you, Grandma, but I had sex twice this week!" And let's face it—the slut stigma isn't just dangerous to our "reputations" or to some weird-ass notion of purity. How many times has a rape been discounted because a woman was deemed a slut? How many times are women called whores while their partners beat them? How often are women's sexual histories used against them in workplace harassment cases? The sexual double standard is a lot more dangerous than we'd like to think. _**So... what to do?**_ First and foremost, stop calling other women sluts! It doesn't behoove us to bash each other, gals. And speak out when you hear men do the same. I'll never forget in college overhearing a conversation that my boyfriend's roommates were having.They both had slept with the same girl over the course of the year—they called her a whore and made a joke about her vagina being "loose." I asked them why she was the bad person in this scenario—after all, they had had casual sex with her, too. They couldn't provide an answer, but that didn't stop them from continuing to laugh. I always regretted not saying anything more. Outside of calling ourselves and others out on perpetuating the double standard, it's a hard battle. But I think if we recognize the hypocrisy of the stud/slut nonsense when we see it—whether it's in an anti-choice law or a movie that makes women who have sex look like deviants—we're on the right road. (Random true story: When I was in my early twenties, I was watching a documentary on anorexia and saw my childhood tormentor, Eleena, talking about her terrible eating disorder and how she cut herself as a teen. Just something to remember when you think back on the kids who were cruel to you—they were in pain, too.) **2** **HE'S CHILL, SHE'S ON THE PILL** **IN MY SEX-HAVING LIFETIME,** I've been on the Pill, used the NuvaRing, condoms, and female condoms, and considered getting an IUD just so I wouldn't have to worry about birth control for another five years or so. I've taken emergency contraception. The job of being responsible, at the end of the day, has always lain with me. Because I'm a woman. It's our responsibility to have safe sex: birth control pills, diaphragms, spermicides—shit, we even have to convince men to wear condoms! I say it's crap. There's no doubt that women will always have a disproportionate amount of responsibility when it comes to sex, because we're the ones who get pregnant—and if we do get pregnant it's going to be up to us to decide what to do about it. But the way that birth control is automatically considered a woman's domain is just irksome, not only from a theoretical feminist perspective—why should it only be up to us!?—but also from a practical one. Because being the responsible party in a sexual relationship doesn't come without costs. Birth control has always cost me money, but recently I'm spending over $50 a month (I don't have health insurance) to make sure I don't get knocked up. And I know I'm not the only one who is breaking the bank. I used to long for my college days, when being on the Pill would only cost me a few dollars a month. But those days are long gone, and young women today are getting totally screwed. Birth control prices on college campuses are literally doubling and tripling. (But not condoms, of course—just the kind that the ladies use.) Drug companies that used to sell colleges contraceptives at a discount—which is why you could get a $50 pack of pills for $12—have stopped offering the discount. And women are pissed, rightfully. The best quote I heard about this increase in price came from a twenty-two-year-old at the University of Iowa who said, "This is the one thing that many females on campus are getting from student health. . . . It felt like we were a target." Ya think? And the cost of bearing birth control responsibility isn't just monetary. Birth control has long been used against certain women—women of color, immigrants, and low-income women—as a way to control them. There are groups that put up billboards in low-income, minority communities urging women to get sterilized for cash (seriously), and a long history of sterilizing women because only certain (white) women having babies is considered desirable. Unfortunately, it's not only the onus of being protected that's on women, it's also the stigma attached to having sex. Men can buy condoms without getting a lecture or a problem—but women who go to the pharmacy for birth control are often refused or asked about their marital status. Can you even _imagine_ that happening to a man? And when was the last time you saw conservative groups up in arms about condoms being available in schools? Hell no. Because they couldn't give a shit about whether guys have sex or not. But allowing women to take control of their reproductive destinies? No way. There have been all sorts of protests just in the last year over birth control pills and patches being made available to young women. So not only is it up to us to make sure we're protected—we have to jump through all sorts of hoops to make it happen! So what about the men? You would think that men would be eager to take on extra responsibility—having control over your reproductive future is always a good thing, after all. A common anti-feminist argument against child support, for example, is that women constantly trick men into getting them pregnant (sure they do). Guess what, guys—if you used a condom every time you had sex, and took some responsibility for your sex life, you would never have to worry about something like that. When I've asked folks (friends, foes, and even feminists) about the birth control disparity, I've heard countless times that it's not _their_ fault that all of the contraceptive options are available to women. But recent studies show that the lack of a male birth control pill, which has been reported to be on its way for years now, isn't because of science holdups—it's societal obstacles. The man who originally developed the male pill, Carl Djerassi, says they stopped working on it because men just wouldn't use it: "It would be possible to make a male pill today. We know how hormones work and we could use the same principles that are used to make the female [pill]. . . . The problem is that men are afraid to lose their virility. Even if taking a pill carries only a remote chance of impotence, they won't take the chance." (Ri-ight. Because it's not like women undertake any health risks at all using countless levels of hormones, things stuck up our chocha, and the like.) _**So... what to do?**_ If you're straight and sexually active, make sure that your partner is taking on as much responsibility as you are. Use condoms. Split the costs of all your birth control—after all, he's benefiting from it, too! At the end of the day, the birth control double standard exists for one reason—sexism. The idea behind the reality of fewer BC options for men is that sex and reproduction are all about women. We can't let them be. **3** **HE'S ROUGH, SHE'S DAINTY** **WHEN I WAS SIX YEARS OLD,** I had a play kitchen set—it was tin and looked super real. I also had a tea set and a shit ton of dolls. But, thanks to my hippie parents, I also had a Thundercats glowing sword, toy robots, and multiple racing car sets (those were my favorite). And while I was acutely aware that there were "boys' toys" and "girls' toys," I remember always appreciating my parents telling me that girls could play with boys' toys and vice versa. (Especially because I took some shit from schoolmates due to my penchant for swords and robots.) I never would have thought that twenty-three years later children would have the same kind of gendered toys that I grew up with. You really don't need to look much further than the nonsense directed at our children to see a ton of double standards at play, not to mention the way that sexist socialization starts early. Take toys, for example. You can still find the "girls'" aisle in a toy store just by looking for the blinding pink that adorns everything. Feministing.com blogger Vanessa (and my sis) took a look at the toys sold in superstore Target and found some predictable, though no less nauseating, trends: Girls' toys are supposed to "make her sweet dreams come true," with featured sections, the first being "Kitchen and Play Food," along with "Dolls and Accessories" and "Horse Play Sets." Boys' toys "let his imagination run wild" with "Cars, Trucks, and Trains," "Building and Construction," "Tech Toys and Kids' Electronics," "Vehicles and Radio Control," and "Science." But it's not just the pink-is-for-girls, blue-is-for-boys trend that's problematic. It's what these toys are, and what they're telling our kids from a very early age. Take the Fashion Fever Shopping Boutique, a Barbie toy that has a pink credit card swiper and credit card so that little girls can "buy" outfits for their dolls. The television commercial for the toy features a little girl saying, "And you never run out of money!" (You know, just like in real life. Sigh.) Creating good little consumers one toy at a time! Never mind that young women in the United States are deeper in credit card debt than perhaps any other group in the country. Or Playskool's new Rose Petal Cottage—the tagline for this girls' playhouse is "Where dreams have room to grow." That is, of course, assuming your daughter's dreams consist of baking muffins, rocking a cradle, and doing laundry. The commercial for the toy is totally disturbing, with lyrics from the Rose Petal Cottage song saying: "I love when my laundry gets so clean / Taking care of my home is a dream, dream, dream!" If that's not bad enough, when the little girl in the commercial puts clothes in her laundry machine, the narrator notes the cottage is a place "she can entertain her imagination"! Girls' imaginations should consist of laundry and baking. Awesome. Compare that with Tonka, whose new commercials claim that "boys are different" and that their trucks are built "Tonka tough," and I think you'll see what I'm getting at. Not to mention the racism built into so many toys, especially for girls. Most dolls sold are white and blond, and those that are supposed to be "ethnic" have overwhelmingly Caucasian features. And if toys aren't telling little girls that they should grow up to be happy homemakers, they're telling them to be sexual. Seriously. It was just 2006 when Target took shit for selling padded bras for girls as young as six. A spokesperson from Bratz, who makes the "bralettes," said "the idea of the padding is for girls to be discreet as they develop." Um, last time I checked, six-year-olds had nothing to be discreet about. British superstore Tesco even got called out for selling toy stripper poles in the children's toy section. The kit is advertised on its site as saying, "Unleash the sex kitten inside. . . . Simply extend the Peekaboo pole inside the tube, slip on the sexy tunes, and away you go!" Charming. Then there's clothing. If you've ever shopped for a little girl—especially a baby girl—I challenge you to find something that (a) isn't pink and (b) doesn't say something like "princess" or "diva" or "drama queen." Not possible. Jane Roper, on her blog, Baby Squared, says of the clothing conundrum: "I guess some people find it funny. Like: Ha, ha—an innocent baby girl can't be a spoiled pain in the ass! So it's funny to call her one! Because, really, she won't be a spoiled pain in the ass until she's at least twelve! And if she is one then, that's fine! Because that's just what it means to be an empowered young woman in America today! Getting what you want—whether it's shoes or clothes or an iPod or a chihuahua or your own reality show or whatever. God bless America! Ha, ha, ha! Princess! How cute!" And how sad. And I haven't even touched on child beauty pageants, television shows, and a ton of other stuff directed at children. _**So... what to do?**_ Don't buy your kids sexist toys! Which I know isn't easy, I assure you. Or if you _must_ buy the goddamn Rose Petal Cottage, get some Tonka trucks too. (Though it's probably better that you don't support toy companies that rely on sexism to sell their products!) If you're looking for cool, not-all-white dolls, check out Karito Kids, which features girls from all over the world. (Think American Girl but cooler and international.) Go to parent blogs dedicated to anti-sexism and anti-racism for ideas. And for goodness' sake, stay away from toy credit cards. **4** **HE'S A HERO, SHE'S A DAMSEL** **DESPITE MY PARENTS' PROTESTATIONS,** I must admit that I'm far from perfect. I'm loud and sarcastic, and when I'm pissed I can be a cold bitch. Like all people, I have my flaws. Which is why I've never, ever wanted a guy to put me on a pedestal—if you're on a pedestal, you have a long way to fall. And no one can live up to the expectations that some folks—in fact, a lot of folks—have for women. That we're virgins, Madonnas, mothers, little girls, perfect angels to be protected. Naturally, viewing women this way sets up a very dangerous dynamic—because no one is perfect, and when women transgress, they get punished. (You don't have to look much further than the virgin/whore complex to figure that out.) And while the whole woman-on-a-pedestal thing is often shrouded in ideas about romance, it's anything but. Because notions of pedestals and chivalry operate under the assumption that women are inferior. While holding women up to high standards may not immediately seem like it's degrading—after all, right now the idea of girls being "princesses" and "treated like queens" is all the rage (just watch _Bridezillas_ )—what it's actually doing is saying that women are like children, not fully formed people. We have to be protected. We have to be coddled. We have to be treated with kid gloves. Sorry, but my idea of romance isn't being babied. Now, when people think "chivalry," they think of men opening doors for women, throwing their jackets over puddles, and paying for dinners. All admittedly nice things, save the jacket throwing—that just seems nuts, given the price of outerwear these days. But this is how they get you. Doing things like opening doors for people is _polite_. I would hope one would do as much for anyone if they got to the door first. Chivalry is something completely different. Chivalry is the idea that men should be doing something for women for one of two reasons: They think women aren't capable of doing things for themselves; they think that doing things like opening doors should get them laid. Again, I think doing nice things for people, whether you're dating them or not, is fantastic. I love it when my significant other does shit for me (now, whether this is because I'm slightly lazy, I don't know). But we should draw a distinction. One of my favorite examples of chivalry gone wild is from a column in a college paper that my friend and fellow feminist blogger Jill Filopovic wrote about. Basically, this male student is complaining that chivalry is dead because women had the audacity to do things for themselves: "And so emerged a group of warrior princesses affectionately referred to as Feminazis; lean, mean, emasculating machines in power suits who proved to the world that women are intelligent, strong, capable, and incredibly frightening." You have to love a guy who thinks capable women are "frightening." Jill's response to this is spot on: There's a difference between being chivalrous and being nice or polite. Opening a door for someone because you got to the door first is both nice and polite; making a huge production of opening a door for a woman in the hopes that she'll see what a chivalrous dude you are and fuck you (and then getting all pissy when she doesn't respond how you want her to) is not polite or nice. And that's the thing with chivalry: It always demands something in return. If you're being nice to me because you like me and you're the kind of person who is nice to people you like, then that's great. If you're being nice to me because you're hoping to get something out of it, or if you think you're entitled to sex or a relationship with me because you were nice and "chivalrous," you can go fuck yourself. See how that works? Love it. And frankly, when you take a look at the people who are pushing old-school notions of chivalry and romance, you may think twice before letting a dude pick up the check. Most often, it's conservatives you'll hear arguing that chivalry is dead. (And that feminists killed it, of course.) These are folks who have a specific agenda in mind—mostly one that involves getting women back in the kitchen. For real. Conservative women's group the Independent Women's Forum, for example, has a campaign called "Take Back the Date," where they try to counter what they call "hookup culture" by promoting old-school dating practices—like bringing flowers, boys asking girls out and never vice versa, and so on. Doesn't really sound terrible, right? Well, the _other_ part of "Taking Back the Date" is protesting feminists on campus and any college performances of _The Vagina Monologues_. (Because talking about vaginas is counter to romance, apparently.) They see promoting chivalry as an easy way to promote other traditional gender roles. Chivalry is also used as an excuse to glorify the "good old days" when men were men and women were doormats. In fact, _New York Times_ columnist David Brooks once wrote that the reason rape still exists is that chivalry is no longer around. (As if women didn't get raped in the good old days. Uh-huh.) So seriously, let's not romanticize something that's not necessarily all that great. _**So... what to do?**_ I'm not going to lie: I'm not going to stop letting guys open doors for me, and I'll probably still like it when someone offers to help me put my jacket on. But I'm not going to _expect_ it from men. Similarly, I would hope that men—upon doing random nice things—wouldn't expect anything in return. And when it comes to the dangers of being on a pedestal, just don't go there. **5** **HE'S METROSEXUAL, SHE'S ANOREXIC** **UNREALISTIC BEAUTY STANDARDS** are one of those feminist topics that you have to love—because no one in their right mind can argue that they don't exist. We see images of unattainable beauty norms everywhere—in magazines, television, advertisements, movies, you name it. All touting the same image of what's supposed to be an attractive woman: white, thin, blond (usually), big boobs, the whole package. And sure, men have beauty standards to live up to as well. But not nearly on the same level as women. Attractiveness standards for men tell them to be big, strong, to take up space. Our beauty standards tell us to shrink, be weak, take up as little space as possible. Then, of course, men's beauty standards tend to end _above_ the waist. For women, it's no longer good enough to be emaciated, tanned, siliconed, shaved, and just generally trussed up. Now we have to make sure that every inch of us—even the naughty bits—is equally "beautiful." Seriously, where are men's penis-beauty standards? Yes, men get circumcised, but the new labiaplasty trend—where women have their vaginas tightened and lips shortened in order to have prettier pussies (whatever that means)—goes above and beyond. Labiaplasty—or "vaginal rejuvenation surgery"—is one of the fastest-growing plastic surgeries out there, despite being dangerous, painful, and potentially damaging to your ability to have pleasurable sex. The American College of Obstetricians and Gynecologists released a public warning against the surgery, noting that potential risks include "infection, scarring, nerve damage, and loss of sensation." Good times. So why would women line up to get the surgery? Because these charming docs, along with the porn industry and lad-mad culture, are telling women that their normal vaginas are ugly and vile the way they are. And isn't that more important than your future relationship with orgasms? Now, the same folks who brought you your newfound vaginal in-securities are pushing surgery packages with empowering-sounding names like "Wonder Woman Makeover," which includes "several vaginal procedures, breast implants and a breast lift, abdominal liposuction, and a 'Brazilian butt augmentation.'" Where are the Superman Makeovers, you ask? Sorry, no such thing. There are also no "Daddy Makeovers" to compare to the new trend of "Mommy Makeovers," either—this is when moms who have just had kids get surgery to "fix" their postpartum bodies. (I can see the male version now. Dads, get rid of that beer gut! Your wife will love your new toned physique!) And it's not just the usual suspects of body parts when it comes to beauty standards. Another new and improved way to maim—I mean 'improve'—yourself: Now you can fit into those designer shoes by cutting off your toes. Or shortening them. For real—people are actually doing this. Dr Ali Sadrieh, a podiatrist from California, says, "Toes are the new nose." Now, I like heels as much as the next gal, but generally I look for shoes to fit my feet—not feet to fit my shoes. Just saying. The old standard of weight is still around as well, naturally. But now instead of just being thin, we have to be dying. Literally. The covers of celebrity weeklies are covered with anorexic starlets, their bones jutting out from sagging skin and oversize sunglasses. Of course, the headlines feign concern ("Brad to Angelina: You have to eat!"; "Nicole's struggle with weight"), but they're glamorizing the disease simply by having these women on the cover. Being a sickly-thin celeb is a surefire way to rev up your career (second to having a baby, of course). The common theme here? Beauty standards for women are more extreme than ever. (There's even the reality show _Extreme Makeover_ to prove it!) Pop culture has everything revved up—we can't have normal sex, it has to be porn sex. We can't have normal vaginas, they have to be teeny, tiny, hairless vaginas. We can't be skinny, we have to be anorexic. It's just all too much to live up to. _**So... what to do?**_ Don't believe the hype. (Yes, I am a Public Enemy fan.) And for the love of all things natural, don't get surgery—it's just bad news. And when it comes to the images that are shoved in our faces day after day, be a critical thinker. (Easier said than done, sometimes.) To steal some advice from my girl Courtney Martin, author of _Perfect Girls, Starving Daughters,_ who wrote a great piece on loving your body for Feministing: "Never diet. Never ever. It is a $31 billion industry that fails 95 percent of the time. That's just stupid. Don't spend money on products made by companies that make you feel inadequate. Duh. Redefine your notion of success to include your own wellness—including joy, fulfillment, resilience, and self-love." Yeah, she's a little hippie-ish, but she's a smart lady. Listen to her. Plus, I love hippies. **6** **HE'S "LUCKY," SHE'S LOLITA** **WHEN I WAS SIXTEEN YEARS OLD** I met the hottest guy ever. He was six-three, muscular, and had a (swoon) tattoo on his arm. He was also twenty. Not the ideal age, I admit, but Jason and I had a great time together. Looking back, though, I can say with certainty that I was much more mature than he was at the time. (I was taking Organic Chemistry; he was applying to be on _The Real World_. Just saying.) We had a decent yearlong relationship that ended when his modeling/acting career didn't take off (don't laugh) and he moved back home to upstate New York. A fairly normal romance? Actually, no. Under New York state law, Jason could have been arrested for statutory rape, even though our relationship was consensual. Fucked up for him? Absolutely. Men are prosecuted every year for statutory rape despite being in consensual relationships. (That isn't to say I think there shouldn't be consent laws _at all,_ but clearly something needs to change when innocent men are going to jail and young women are being told they don't have the right to have sex.) While—like a lot of sexism—this affects both men and women,the double standards in consent laws are mired in misogyny. Teenage girls who have sex can be either victims or whores. That's it. We're either poor little virginal things who were taken advantage of or hot-to-trot vixens who seduced our way through high school. (Sounds like a Lifetime movie already!) Men, on the other hand, are able to have sex whenever they want. You know, because unlike women, they _like_ sex. (Sigh.) No one questions if they were taken advantage of. I mean, even in recent cases in the media where older female teachers had sex with young male students, there were comments about how "lucky" the boy was. This teen-sex double standard is based on the antiquated—and false—notion that women don't like sex. Or at least we shouldn't. The problem for women with consent laws, and really anything to do with ideas surrounding teen sex, is that women are assumed to be victims simply because of our age. The logic is that we don't have the wherewithal to make up our own minds about sex. Now, do all girls have the emotional maturity to have a sexual relationship? Of course not. But plenty of teenagers do—unfortunately, a lot of folks can't handle that. To be clear, I'm not talking about a fifteen-year-old dating some creepy thirty-year-old. There's no doubt that with certain age differences (whether it's men or women who are older) there's a power dynamic that makes real informed consent almost impossible. But are we so invested in the idea of teen girls as little virginal angels that we can't be honest about their sexual desires? Young women can choose to have sex. They can choose not to. For too many people, that's just too much freedom for young women to have. And when we have laws that are based on the idea that young women couldn't possibly want to have sex, we have an issue. Because under this framework, when it's clear that young women _are_ choosing to have sex, it means there's something wrong with them—they must be whores. Of course, there is a way out of the virgin/whore trap—marriage.The virgin/whore complex is hard at work on this one! If you're married, you can have any kind of sex you want! Shit, if you're thirteen years old and married in Kansas, your sex is legal. If you're sixteen years old but unmarried, not so much. Which, of course, is the real point of all this nonsense: keeping young women pure (whatever that means). If we're married, no matter what our age or maturity level, somehow our sex is sanctioned. So, at the end of the day, these laws and ideas about teen girls and sex aren't about keeping us safe. They aren't about protecting young women or caring about their well-being. They're about making sure girls remain chaste. _**So... what to do?**_ Let's start by not judging young women on what their sexual lives are like. Let's not assume young women shouldn't want to have sex, and that young men should. And instead of assuming that a young woman who is sexually active is somehow a victim or a slut, let's make no assumptions. At all. And let's start talking about how to really talk to young women about sex. We're so caught up in the idea that teen girls are victims or vixens that we don't prepare them to be something in between—informed, mature, aware young women. It's time to start doing just that. **7** **HE'S A BACHELOR, SHE'S A SPINSTER** **THERE'S SOMETHING HOT ABOUT SINGLE MEN.** They're bachelors, with cool apartments and the freedom to do whatever they want without judgment. Sure, they may catch occasional shit from their mother about "finding the right girl," but for the most part they're respected. Single women, on the other hand—especially single women who have the gall to be over thirty—we're old maids. Spinsters. Desperate to be Bridezillas and moms. There's no such thing as a happy single woman. We're all just wives-in-training or crazy cat ladies. There's something about unmarried women that society just doesn't like. That's why the media is constantly telling us how miserable single women are. For example, _The Today Show_ ran a segment about single working women where an editor for _Marie Claire_ called women who don't get married and have kids "fembots." You know, 'cause we must be robotic and frigid if we want careers before we have a family. The editor actually went so far as to call women who care a lot about their careers "emotionally unavailable." But painting women who don't get married as vicious career women or sad old spinsters is nothing new. As I wrote in an essay for _Single State of the Union_ , the media likes to portray single women as caricatures. If we're young and "sexy," we're "office piranhas" trying to steal married men. If we're older, either we're desperate or we're "cougars." And the bad science studies come out in force when it comes to single gals: The new trend is reporting that women won't get married if they're too successful, too educated, or too old—as in over twenty-five. The most annoying thing about these stereotypes and "studies" is that they assume that all women want to get married and that all women are straight! (Lesbian women just don't exist when it comes to the media these days.) The scary truth (at least, what society sees as scary) is that women may be better off _not_ getting married. One of my favorite writers, Natalie Angier, wrote a piece for _The New York Times_ a while back about how marriage really benefits men more than it does women (despite the media-created frenzy about women just dying to get married and men wanting to put it off): In 1972 . . . Jesse Bernard wrote a highly influential book called _The Future of Marriage_ in which she argued that wedding bells sounded the death knell to a woman's well-being. Ms. Bernard presented data indicating that while married men scored higher than single men on measures of mental health like depression, severe neurotic symptoms, and phobic tendencies, the opposite applied for women. Angier points out that this isn't the case for all women, obviously, and that, depending on the kind of partner someone has, everyone's situation is different. But it does give me pause. As does the study that showed that married women do a ton more housework than men—there's even a marked difference when you live with a man as opposed to being married. (Living together there's less housework; married you do more—whatever.) Now, this certainly isn't a diatribe against marriage—I'd like to get married one day. But making marriage seem like the end goal for all women and the Best Thing Ever just isn't honest. And if marriage is so amazing and great, why would conservatives need all of these initiatives, organizations, and legislation to push women to get hitched? Wouldn't the joy of being a wife be enough? Apparently not. Conservatives are recognizing that more and more women aren't rushing to the altar—plenty of couples are cohabiting, and people are waiting until they're a bit older to get hitched. Then, of course, there's the divorce rate. So, because they're so tied to the idea of marriage holding together traditional gender roles, they're taking action. There's even a lawmaker in Idaho who is doing his best to try to create legislation that would essentially trap women in marriage and push them to stay at home instead of working in the public sphere: Chairman of the Idaho House of Representatives' Family Task Force, Rep. Steven Thayn, is trying to repeal no-fault divorce laws and convening task forces to figure out ways to encourage mothers to stay home with their children. Funny that initiatives like these never target men. And notice that these groups, who seem to just love marriage, aren't so concerned with making same-sex marriage legal. Imagine that. It's all about getting single women married, because there's a belief that married women will mean traditional women. And that's scary. _**So... what to do?**_ Get married, by all means. But don't do it because you think you need to in order to be a full person, or because the media is breathing down your neck with bullshit statistics about successful, smart women missing out on the hubby train because they had the nerve to care about their own single lives. Start referring to yourself as a bachelorette and enjoy your single life! (And while you're at it, make sure you're fighting for same-sex marriage, because what good is doing something that's being used to discriminate against so many people?) **8** **HE CAN BE A BEAST, SHE MUST BE A BEAUTY** **LOOKING AT MOVIES AND TELEVISION THESE DAYS,** it would seem that every dumpy, immature guy gets an absurdly hot, successful girlfriend issued to him at birth. Whether they're featured in _The King of Queens_ or _Knocked Up,_ beautiful, accomplished women just seem to be attracted to schlubs. Now, don't get me wrong—I've dated my fair share of schlubs. There was the semi-alcoholic, slightly overweight goth boy who deserted me in the middle of Bumblefuck, Queens, one night after a couple too many drinks. Or the Brooklyn hipster with an eczema problem who had a penchant for emailing women in the Craigslist "casual encounters" section. But hey, we're all allowed some dating snafus. And I didn't _marry_ the schlubs. Because that would be crazy—not to mention unrealistic. And, yes, I realize that television isn't the best place to look for the non-crazy these days, but seriously—the Marge/Homer marriage model has got to go. Because while some would say that this fictitious relationship model makes men look like bumbling idiots, it makes women look even worse. After all, they're the ones who stay with said schlubs despite knowing better. It's incredibly insulting. And it's not just the looks disparity—this isn't all about eye rolling at the seemingly never-ending depictions of plain men getting gorgeous women (though a little eye rolling is probably warranted). It's about the fact that men in the media are paired up with women whom, let's face it, they probably don't deserve. And while the "morals" of so many of these stories, like _Knocked Up,_ show the protagonists growing up and ceasing to be complete assholes, the message seems to be that men only have to aim to be basically decent human beings to snag a beautiful, successful woman. That if they can just be, well, normal—any woman will jump to be with them. Now, _Knocked Up_ may not be the very best example for me to use, because I have a bit of a thing for Seth Rogan. Call me crazy—I like the chub. But it _is_ pretty interesting. The whole premise of the movie is that this slacker dude can score a relationship with a hot, successful woman by getting her pregnant (so she'll stick around long enough to see that he's a nice guy) and acting halfway decent. It's similar with television shows featuring the bumbling hubby—women find ineptitude _charming,_ didn't ya know? Now, if this were just a silly television trend, I probably wouldn't be all that upset over it. But the _idea_ behind the silly trend is making for some serious entitlement issues. All of a sudden, the dating climate is chock-full of men who think that they need to just sit back, relax, and the universe will deliver them a supermodel. A guy friend of mine (who shall remain nameless) once had a long conversation with me bemoaning the fact that he couldn't find a girlfriend. He's a better-than-average-looking guy, smart and funny— so I couldn't understand it either. Until he started explaining the kind of girl he was looking for, and the girls he had turned down. You see, my friend was only interested in dating a gal who looked, well, like a _Playboy_ model. And I'm sad to say he's not the only friend I have who thinks this way. Women whom they date are not partners as much as they are status symbols—so being with a ridiculously hot woman was a priority over smarts, kindness, humor, anything.This isn't to say I think people should just disregard attractiveness in dating—obviously, we all have our types. But when young men think that they're entitled to have a super-hot model girlfriend, there's kind of a problem. And frankly, a lot of young men are going to find themselves highly disappointed in the relationships they have (or never get) if they think the only women worth having are those just there to be arm candy. Not to mention what this does to women. It reinforces the idea that we're trophies and that we don't really need someone who is our equal (or even someone who is smart and charming)—just anyone who doesn't act like a _total_ asshole is fine. _**So... what to do?**_ Don't date schlubs. Just kidding. I don't think the loser-gets-hottie model is about to go anywhere anytime soon. But I do think we can do something about it in our own lives. For example, call dudes out on their sense of entitlement. I wish I had said something to my guy friend . . . maybe something along the lines of, "Shut the fuck up about girls with big boobs and waxed vajayjays for a second and focus on what's really important." But I digress. It's time to point out the ridiculousness of this double standard—whenever we can. And, as always, don't believe the hype. **9** **HE'S A HIPSTER, SHE'S A HO** **THERE'S NO DOUBT THAT WOMEN ARE EXPECTED** to look a certain way—we all know what the unrealistic beauty standards are in this country (skinny, white, big breasts, and so on). But it's not just our bodies that are privy to sexism—the way we present them is as well. Women are never, ever supposed to dress shlumpy or seem unkempt—but on the other hand, we're chastised if we dress too "sexy." When I was promoting my first book, I went on Comedy Central's _The Colbert Report_. I was extremely excited and nervous, especially about what to wear. I knew that because I was a feminist, I would be judged more harshly for the way I looked. I decided to go for a businesslike outfit: a white button-down shirt, blue knee-length skirt, and pumps. Super simple. After the show aired, one of the first emails I got was from a guy who told me I was obviously "trying to show off [my] legs" by wearing a skirt and heels, and that I was "flaunting" my sexuality. It wasn't the only email I got to that effect. It's amazing, really. Women are seen so much as public property, as objects to look at and judge, that people actually think it's appropriate to go out of their way to comment on women's appearance. They think it's their right. For example, Southwest Airlines harassed two women (in separate instances) for dressing "inappropriately." In the first incident, a young woman was told by a flight attendant that her outfit was too revealing—and that she would have to change or miss her flight. Was she sporting a bikini? A bra and hot pants? Nope—just a sweater and miniskirt. Another woman just weeks later was told to cover up with a blanket—she was wearing a tank top. Just a side note: When was the last time that a guy with a beer belly hanging out of his shirt was reprimanded? The only thing I saw that these two women flying Southwest had in common (in the pictures) was that they were both very pretty by conventional beauty standards, and they were both well endowed in the boobie department. Now, as someone with not-small breasts, nothing pisses me off more than when someone assumes my outfit is too "sexy" just because of said breasts' presence. We can't help it! And even if these women _were_ dressing in a deliberately sexy way—so what? They're only doing what society basically demands of them. And, let's be honest, if we don't dress to impress, we still have to put up with bullshit. Just take Darlene Jespersen, a bartender in Reno, Nevada, who sued her boss for trying to force her to wear makeup. She had worked at Harrah's casino for twenty-one years, and she said she just didn't want to wear powder, blush, mascara, and lipstick—which were part of the dress code for the female employees at the bar. Seems pretty reasonable to me. But she actually lost her case; a court of appeals ruled that the casino _can_ force women to wear makeup. So if we dress up we're whores, and if we don't we're sloppy. There's no winning for women. Then, of course, we also have to deal with the maintenance double standards. Women—if we don't want to be accused of being ugly or lazy—have to maintain a certain level of contrived "prettiness." This requires makeup, body-hair removal, tanning beds, manicures and pedicures, dieting, push-up bras, uncomfortable shoes, uncomfortable clothes . . . the list goes on and on. All so we can't be accused of not at least _trying_ to be pretty. Men, on the other hand, barely need to run a comb through their hair to be considered put together! And while there's the whole new "metrosexual" trend—guys who get waxed and tanned, get facials, and buy more hair gel than their girlfriends—they're more mocked for being feminine than considered hot and manly. And, like so many other seemingly superficial double standards, the clothing conundrum has a lot more at stake than a woman's ability to wear a skirt without getting crap from a flight attendant or having to deal with waxing her cooter on a regular basis. Women are still routinely blamed for their own rapes and sexual assaults based on what they were wearing. An Amnesty International study found that 26 percent of people think that if a woman wears sexy clothes, she's partly to blame if she gets raped—and women's outfits are still brought up as relevant in rape cases. It's vile. _**So... what to do?**_ One thing is for sure—don't stop dressing the way you like. Whether it's heels and lipstick or beat-up jeans and sneakers that do it for you, just wear it and don't let anyone give you shit for it. And if you hear someone making comments about women's clothing, speak up. Don't let sexism go unnoticed—even if it is "just" about fashion. **10** **HE'S GONNA BE A SUCCESS, SHE'S GONNA BE A STAY-AT-HOME MOM** **WHEN I WAS IN HIGH SCHOOL,** my boyfriend tried to convince me that men were "naturally" inclined to cheat. Men were supposed to be polygamous, and women were hardwired to be monogamous, he reasoned. It was scientific and stuff. (I know, be wary of teenage boys arguing that philandering is just a male instinct.) When it comes to the battle of the sexes, bad science always seems to be caught in the middle. Conservative groups use it to try to argue that women belong in the kitchen, antifeminists use it to claim victory in their warped logic, and the media eats it up and uses it to spread sexist misinformation. Why? Because sexism sells. Caryl Rivers, author of _Selling Anxiety: How the News Media Scare Women,_ notes that the most popular stories in _The New York Times_ in the last few years have all been about bad science studies claiming that men won't marry smart women, and that highly educated women all want to be housewives. It's all part of the backlash against feminism. Women are doing well, so best to try to convince them that they're miserable because of it. (Susan Faludi's _Backlash_ was all about this phenomenon in the '80s.) Or that women are really meant for traditional "ladylike" things—like ironing and other fascinating chores—and that the urge they felt to work in the public sphere was just a nasty aftereffect of evil feminism. After all, it's certainly not a coincidence that some of the most popular bad science topics either reinforce traditional gender roles or shame women who dare to live beyond them. Like the one from the University of California, Santa Barbara, that reported women are better at grocery shopping. Or another that said women are naturally suited to housework. (You know, because a skill for vacuuming goes hand in hand with having a vagina.) There was even a study in the U.K. that got a lot of media play that reported moms who work outside the home end up having overweight children. "Working moms have obese kids!" You sensing a bit of a trend here? The backlash is as strong as ever. And if it isn't the reinforcing of gender roles, it's the plain old women-are-stupid studies. If we listened to these junk science reports, we'd believe that women are worse at science, math, map reading, logical reasoning, spatial abilities, driving, networking, leadership skills . . . even making jokes. Other favorite "studies" of mine include these oh-so-compelling ideas: Housework cuts women's risk of breast cancer. Men chase beauty, women, and money, when picking a mate. Hook-up culture is destroying young women. Smart and educated women can't find husbands. Divorced women are more likely to be mentally ill. Men are smarter than women (seriously, this was a study that someone got frigging funding for). Sometimes I even think that these dudes make stuff up just to make themselves feel better. For example, in perhaps the best-titled article ever, "Crying Over Spilled Semen," _Psychology Today_ reported on a study that basically said women are _addicted_ to semen. Amazing: The finding that women who do not use condoms during sex are less depressed and less likely to attempt suicide than are women who have sex with condoms and women who are not sexually active leads one researcher to conclude that semen contains powerful—and potentially addictive—mood-altering chemicals. The study's author also said that he's planning on examining whether "semen withdrawal" places women at an increased risk for depression. The one thing all of these "studies" have in common? They're just not true. Or they've been debunked. Or the media that report on them exaggerate the stuff they think is the juiciest. Of course, there _are_ some studies I can get behind. Like the 2007 one from Rutgers University in New Jersey that said feminists have better relationships and sex. So there. _**So... what to do?**_ I think we've all had it up to here with sexism that gets the support of the "scientific" community and the media. Now, there's no way for us to completely change the studies that are done and the media that cover them. But we can make some noise. When you see a media outlet running one of these ridiculous stories, call them out on it. Write a letter to the editor, write an op-ed, write something on your frigging MySpace blog, for all I care—just don't let it go unnoticed.When you see a "study" that makes what seems like a dubious claim (women who work get goiters!), check it out. Is the study really saying what the media says it is, or is the media misconstruing the research? Who did the research? (A lot of these studies are funded by conservative organizations with a very clear anti-woman agenda.) And don't forget that these stories exist for a reason—to make women doubt themselves. So don't fall for it. **11** **HE'S A POLITICIAN, SHE'S A FASHION PLATE** **AS IF IT ISN'T HARD ENOUGH BEING A FEMALE POLITICIAN** in a man's world, women in public service have to deal with the extra baggage of being judged constantly on their looks. When was the last time you saw a newspaper article on a male politician's suit? Or a television pundit arguing over whether a male politician was showing too much skin? Sounds ridiculous, but it's what women in politics have to deal with every day of their career. The White House Project, an organization dedicated to getting women into the higher echelons of political power, released a report in 2000 that studied the newspaper coverage of Elizabeth Dole's presidential campaign compared with that of George W. Bush, John McCain, and Steve Forbes. Dole received less coverage overall, especially on the issues, but when it came to "personal" coverage—talking about her personality, clothing, and looks—she received significantly _more_ coverage. Shocking. In a _USA Today_ article, president of the White House Project Marie Wilson noted, "Our research shows that when there's one woman in a campaign, the first thing the press notices about her is what she's wearing, what her hair looks like." But you don't need a study or research to see the disparity in media coverage. _The New York Times,_ for example, had an entire article in 2007 dedicated to women politicians' fashion sense, "Speaking Chic to Power." Because Lord knows there's nothing more to women in politics than whether or not they wear Prada. My co-blogger Ann Friedman responded to this piece with the astute observation that not only are women in politics simply judged by what they wear, but their clothing is supposed to mark who they are: So Pelosi wears a fashionable nipped-waist jacket and she's marked as a swiftly effective political leader. Condoleezza Rice wears boots; she's marked as a dominatrix. Harriet Miers wears eyeliner; she's marked as begging for Bush's attention. And on and on. Men simply have to choose between a black, navy, or gray suit and pick out a tie. And the color of their cravat rarely marks them as anything. (Ann also wrote a hilarious post in which she did a critique of male politicians' fashion choices in the same manner that so many do to women: "Can we lose the early-'80s creepy camp counselor glasses, please? It looks like you still live in your mom's basement. And you better be asking for either a chin-tuck or a membership to 24 Hour Fitness this holiday season.") Or just take a look at the way Hillary Clinton has been treated as a presidential candidate—or even as a senator or First Lady.The focus has disproportionately been on her hair, her suits—hell, even her supposed "cleavage"! Seriously. _The Washington Post_ in 2007 devoted an entire article to how Clinton was showing tit. She was talking on the Senate floor about the burdensome cost of higher education. She was wearing a rose-colored blazer over a black top. The neckline sat low on her chest and had a subtle V-shape. The cleavage registered after only a quick glance. No scrunch-faced scrutiny was necessary. There wasn't an unseemly amount of cleavage showing, but there it was. Undeniable. Dear lord, you mean . . . women have _breasts?!_ Though you have to credit Clinton for having a sense of humor about the way people judge women for their looks. After years of being ridiculed for her hairstyles, Clinton struck back when she created a poster for an appearance at the National Beauty Culturists' League Convention that featured different pictures of her hair throughout the years, with the tagline: "Pay attention to your hair, because everyone else will." In the most recent presidential election, since Hillary is the only woman running, the media has now taken to talking about the appearance of candidates' wives. (Though, of course, coverage of Bill Clinton's appearance is nowhere to be found. . . . ) And, yes, there are times when men's appearance is talked about in terms of politics, but it's usually related to sexism as well. Take, for example, when papers started reporting that John Edwards spent $400 on his haircut. The coverage—especially conservative coverage—was dedicated to mocking him as feminine because he cared about his hair. That's Sexism 101, friends. And even in the rare circumstance when men's appearance and clothing _are_ discussed, they're talked about in terms of how "distinguished" or manly they look. (Think George W. Bush in that cock-strong flight suit!) _**So... what to do?**_ When you see a biased article, write a letter to the editor! Send it around to your friends with a note about how gross and sexist it is. When you hear friends talk about political candidates and someone makes a comment about a woman's appearance—speak out! Don't let it go unnoticed. And take the bull by the horns: Look into organizations that promote women's leadership and political participation. Encourage your friends to run for office. And wear whatever you damn well please. **12** **HE'S A ROMEO, SHE'S A STALKER** **A GUY THROWS ROCKS AT A GIRL'S WINDOW** in the middle of the night. He won't take no for an answer—he _must_ date her! He serenades her, shows up at her classes, waits at her car. These could be scenes from a burgeoning romance or a stalker gone mad—American culture doesn't differentiate, really. If a woman does those things, however, she's _always_ a stalker. A crazy ex-girlfriend. A psycho. Shit, women are called stalkers for even daring to call a guy a couple of times! Never mind that the majority of stalking is done by men, and the majority of victims are women. When it comes to romance, women are the stalkers and men are just . . . romantic. According to the Department of Justice, one out of every twelve women (and one out of every forty-five men) will be stalked in her lifetime. And whether it's men or women who are the victims of stalking, it's overwhelmingly men who are actually _doing_ the stalking—almost 90 percent. Given the statistics, it's pretty ridiculous that there is still a double standard here: When a man stalks, it's often portrayed as just a joke or romance, but almost anything a woman does will be labeled as stalking (or pathetic). You need not look much further than Hollywood for the stereotypical crazy female. Think of women stalkers in movies: Glenn Close in _Fatal Attraction,_ of course, is the most famous. But there's also Demi Moore in _Disclosure,_ Rebecca De Mornay in _The Hand That Rocks the Cradle,_ Jennifer Jason Leigh in _Single White Female,_ or Kathy Bates in _Misery_. What most of these characters have in common—besides the crazy—is that they're sex-crazed, single gals who just can't wait to get their hands on someone else's man. Even funnier—women stalker characters are often presented in direct opposition to traditional femininity. For example, Glenn Close in _Fatal Attraction_ is the crazy-haired businesswoman, while Michael Douglas's wife is the charming stay-at-home mom. Same thing in _Disclosure_. In _Single White Female,_ one of the first signs that something is amiss with Jennifer Jason Leigh's character is when a shocked (shocked!) Bridget Fonda catches her masturbating. (You know, 'cause only psychos do that.) It's so predictable, really. Women can be good little girls or crazies. This kind of strong-women-are-stalkers attitude is also pretty standard in hetero dating rituals. If you call more than a couple of times, you're stalking a guy. If he calls, he's just persistent. At the end of the day, we're so invested in a romantic ideal that sees men as the pursuers in a relationship that anything that deviates from that is seen as nutso. And, naturally, anything that holds up that man-chasing-woman model is seen as great. Even when it's criminal. In 2007, I wrote a post on my blog, Feministing.com, about a creepy shirt Wal-Mart was selling that said, "Some call it stalking, I call it love." The lettering was scrawled across the shirt in what looked like dripping blood. Classy, eh? The shirt, which was available in the men's section, caught the eye of a stalking victim in North Carolina who complained to the store. She wondered what kind of shirt would be next: "'Some say it's rape, I call it hot sex'? Or:'Some call it domestic violence, I say I'm just teaching her a lesson'?" In the comments section, I was surprised to find how many people (mostly men) thought the shirt was simply funny. That there was no larger issue there. Let me tell you something—it's not funny. Ever. I was stalked once, and it was one of the scariest things I've ever experienced. The thought of this guy still freaks me out so much, frankly, that (despite my being a big fan of writing anything and everything about my life) I would never get into the details of the situation, because I'm terrified that he might read the book and take that as a sign that I'm interested. _**So... what to do?**_ Take stalking seriously, for one. It's not a joke, and it's not romantic. And, to be candid, it's not really women who are doing it. But I think the bigger issue is trying to dismantle the idea that romance relies on women being chased or women resisting a persistent guy and eventually giving in. That's called rape culture, folks. What's so bad about a romantic ideal where both parties involved are equally excited about the prospect of dating each other? Sounds pretty reasonable to me. **13** **HE'S TOUGH, SHE'S A TOMBOY** **I WAS A TOMBOY, NO DOUBT ABOUT IT.** In elementary school I played sports and ran around the playground with the boys, shunning my female peers, who at the time seemed more concerned with Lee press-on nails than with anything else. In junior high, I wore baggy men's jeans and shirts. So sue me—it was fashionable then! It wasn't until high school that I started dressing "feminine" or hanging out with other girls. Lucky for me, no one ever mocked me for being a tomboy. (I was mocked for entirely different reasons, but that's a story for another day.) The little boys in my class, however, who dared to _like_ those conversations about Lee press-on nails or who shied away from sports—they had problems. They were called fags, pussies, and sissies. They were called . . . girls. You see, it was understandable for me to want to be a tomboy and do "boy" things—because men are better, after all. But for a guy to want to be feminine? Unthinkable. Now, while this double standard affects men negatively, it's mired in misogyny—the idea, of course, is that there's nothing worse than being a girl. Think about it. Girls can wear pants; boys can't wear skirts. Girls can play with trucks, but the minute I caught one of my little boy cousins playing with a doll, he threw it across the room with a look of shame on his face. It's demeaning to be female, and boys learn that from an early age. Stephen Ducat, who wrote _The Wimp Factor,_ says that "femiphobia"—fear of being feminine—affects men's identity acutely from the time that they're children. Ducat writes that "anxious masculinity" makes men so concerned with appearing manly (or unwomanlike) that they'll do just about anything to seem masculine (which is basically anything that isn't feminine). In fact, society is so unabashed in its hatred of all things feminine that one of the easiest ways to punish men is simply to feminize them. A South Carolina prison has taken to punishing sexually active prisoners by dressing them in pink jumpsuits. An Arizona prison makes its inmates wear pink underwear—all the time.Thai police officers who step out of line? They're forced to wear pink Hello Kitty armbands. Guy friends have even told me of hazing rituals (whether for frats or sports teams) where young men are put in dresses as a way to demean them. My personal favorite? An Australian joke website (because being womanlike is just hilarious) called "Man Cans" features men who act girlie—like by crying at a movie—suddenly growing breasts as punishment. You can't get much clearer than that. But the consequences of being "a girl" go way past man-boobs and pink underwear. People are actually killed for transgressing gender norms. You need look no further than the violence committed against transgender people every day. And though society looks down on both men and women who identify outside of their assigned gender, there is a special disdain reserved for men who are feminine. Trans women (folks who are born male but identify as women) are mocked in the media. Julia Serano, kickass author of _Whipping Girl: A Transsexual Woman on Sexism and the Scapegoating of Femininity,_ says that sexism not only targets women for being women, but targets people because of their femininity. The idea that masculinity is strong, tough, and natural while femininity is weak, vulnerable, and artificial continues to proliferate, even among people who believe that women and men are equals. People who are feminine, whether they be female, male, and/or transgender, are almost universally demeaned with respect to their masculine counterparts. This scapegoating of those who express femininity can be seen not only in the male-centered mainstream, but also in the queer community, where "effeminate" gay men have been accused of "holding back" the gay rights movement, and where femme dykes have been accused of being the "Uncle Toms" of the lesbian movement. So it's not just the usual suspects on this one. It seems that we've all bought into hating all things fem. _**So... what to do?**_ When you're talking about battling straight-up unadulterated misogyny, there's just no easy answer. I mean, how do you stop something that's been generations in the making? I think we have to start with valuing femininity and, by proxy, women. We may not be able to change the world, but we can change our own worlds. Call people out on using words like "sissy" and "pussy." Throw a fit when someone utters the dreaded words, "Don't be (throw like, cry like) a girl." My very feminist dad had his own gaffe: Whenever I did something "girlie," like be afraid of a spider or something, he would say, "Don't be a Mary." Not no more, folks. We have to start with our own lives and the people in them before we can take on the world. And despite my tomboyish tendencies—which will probably never leave me entirely—I know that it's not being boylike that makes people valuable. It's being yourself—whether that means baseball or manicures, for boys _or_ girls. **14** **HE'S ANGRY, SHE'S PMSING** **WHILE IN A FIGHT WITH A MAN** (or anyone, for that matter), how many times have you been accused of being "on the rag"? Or being "unreasonable" or "emotional"? I'm guessing plenty. When men are angry, they're just angry. When women are angry, they're on the rag. Or neurotic. Or crazy. Or being PMS-y. I had a boyfriend not so long ago who, whenever we got into an argument, would accuse me of "going soap opera." "Here comes Telemundo!" he would shout. His (clearly gendered and vaguely racist) insult was supposed to make me feel like my anger wasn't valid—that it was frivolous and silly, that I was being overly dramatic. This was his not-so-subtle way of trying to shut me up—by accusing me of being emotional. (Unlike men, whose anger is always logical, of course.) Unfortunately, calling me out like this often worked. It felt immobilizing to be called dramatic. Even if you _know_ you're being reasonable, we've internalized sexism so much, sometimes we even begin to doubt ourselves. Thankfully, that relationship didn't last. But the lesson I learned did. When men get angry, they're taken seriously. It's assumed that they have a reason to be so upset. But it seems that whenever women have the gall to express anything other than effusive chipperness, we're accused of having PMS or being nuts. Or we're laughed at or mocked ("Calm down, little lady!"). Women, it seems, aren't allowed to be just plain pissed off. I think that a lot of this comes from the idea that women are supposed to be "feminine" and docile—anger doesn't fit into the sexist ideal of women as quiet and forever smiling. ( _Stepford Wives,_ anyone?) That's why you'll often hear feminists being accused of being angry right along with being called "manly." It's a way to try to stifle women's anger and, by proxy, our voices. When you think about it, it makes sense that society—especially men—would want to keep women's anger under wraps. Because—let's face it—we have a lot of stuff to be upset about! Living in a sexist world is no walk in the fucking park. But what better way to ignore sexism than to make women quiet down? If a woman is too nervous about being called neurotic to get angry about things in her own life, how can she speak out against injustices in the world at large? I can't tell you how many times someone has asked me—when they hear that I'm a feminist— _why_ I'm so angry. Or told me to lighten up. If we complain, we're being rude or loud or obnoxious. If we're angry, there must be something wrong with us. I believe that's why _so_ many women direct that anger inward. Instead of getting angry at the beauty industry that tells us we're fat, we diet and develop eating disorders. Instead of fighting with the girl in junior high who pissed us off, we called her a slut behind her back. All because nice girls don't get mad. And I think we all know that keeping things inside isn't exactly the healthiest way to deal. In fact, a study on women and anger showed that women disproportionately "keep things bottled up" and suffer physically because of it: We get headaches, depression, heart disease, you name it. I'm not saying that anger doesn't affect men's health as well, but men are "allowed" to vent in a healthy way without being mocked or ridiculed. Women aren't. (Think about how many times someone has said it's "cute" when you get angry. For real.) When women _are_ shown as mad or angry, we become caricatures. The pissed-off, man-hating feminist. The neurotic girlfriend. And, of course, nothing says stereotype like the classic racist/sexist combo of the Angry Black Woman. Black women are _constantly_ portrayed in the media and elsewhere as perpetually pissy, usually for comedic effect. 'Cause women's anger is funny. _**So... what to do?**_ Be as pissed off as you want to be. Don't hold back because you think it's unladylike or some such nonsense. We shouldn't be shamed out of our anger. We should be using it. Using it to make change in our own lives, and using it to make change in the lives around us. (I know, I'm cheesy.) So the next time someone calls you emotional, or asks if you're PMSing, call them on their bullshit. **15** **HE'S DISTINGUISHED, SHE'S DRIVING MISS DAISY** **UNLIKE MEN—WHO ARE CALLED THINGS** like "distinguished" and "gentlemanly" when they get older—women who age are pretty much done for. We're deemed unfuckable and unlovable and, subsequently, useless. (I know—I'm such an optimist.) It was just last year, when I was twenty-seven, that I found my first gray hair. It was all short and kinky and stood straight up, right above my bangs. I was not amused. Especially since it wasn't actually _me_ who found it, but a twenty-two-year-old guy I was flirting with outside of a bar in my neighborhood. He leaned in (and of course I thought he was going in for a kiss) and actually plucked the hair out of my head, exclaiming, "Look, a gray hair—funny!" I decided never to call him. Of course, it wasn't so much the embarrassment factor as it was the recognition that anyone who would feel comfortable plucking things off my head was probably not for me. But I digress. Moral of the story: As feminist as I am, there was something about that gray hair that completely freaked me out. But in a world where women are judged almost entirely on their looks, and where "hotness" is akin to youth, I guess that's not all that surprising. I mean, seriously—how many face-lifts, wrinkle creams, youth serums, vaginal tightening surgeries (don't want an old pussy, obviously), and Botox do we need shoved in our faces in order to get the hint? We get it, already: Old or old-looking is bad for women. Youth is good. Men, on the other hand, can get as old as they want. (It seems that just by virtue of having a penis, they're considered hot.) When men are older and single they're bachelors—not spinsters or old maids, like us. When men go gray, they look serious and distinguished. When women do, people wonder why the hell we haven't dyed our hair. When men's sexuality wanes—or gets a little . . . well, limp—they have paid-for-by-insurance Viagra. Which they're using with their younger "trophy wives," of course, because women over childbearing age don't have sex. (But of course those "older" women who _do_ put out a sexual vibe are quickly relegated to joke status: They're cougars or MILFs, not actual women with healthy sexuality.) I think Goldie Hawn's quote in the 1996 movie _The First Wives Club_ says it best: "There are only three ages for women in Hollywood—babe, district attorney, and Driving Miss Daisy." The same could be said of women in general—we're young hot things, moms, or old ladies. That's all we got. And seriously, in what world is it right that Hugh Hefner has, like, seven young "girlfriends" but women like Helen Mirren are considered over the hill? Honestly, I think a lot of this nonsense is related to the idea that women's main purpose in life should be having children. So if we're too old to have any kids, we don't have a purpose. Just look at the recent media frenzies about those poor old career women who waited too long to have babies. I mean, there are countless books, articles, and television segments on how women who didn't have children soon enough will never have them ever! For example,Sylvia Ann Hewlett's _Creating a Life: Professional Women and the Quest for Children,_ which argues that if women in their twenties don't hop to it and find a man and get knocked up, they'll end up old and barren, got a ridiculous amount of media play. Never mind that Hewlett never talks about the fact that (gasp!) some women don't want children, and that study after study shows that waiting to have kids means better health for moms and babies. Better that we scare the bejesus out of women. Not to mention how men get off easy—no one ever mentions that older men have an increased chance of having children with genetic disorders. And no one scoffs when men refuse to date (seriously, I've seen this in many a singles ad) women over thirty-five because they're "less fertile." But, of course, double standards about age don't end with looks, sex, and babies. If only. The average income for men over sixty-five in the United States is about $14,000 more than the average income for women over sixty-five. And older women are the poorest of the poor in this country—mostly because of not having social security after long lives of taking care of husbands, or because women are more likely to hold low-income positions that offer no pension benefits. _**So... what to do?**_ How can we battle a double standard that is so pervasive? I say defy it. Or embrace it. I would love to see folks reclaiming the word "spinster"; I always thought it had a certain something. One of my favorite poems about age is "Warning," by Jenny Joseph, who says: "When I am an old woman I shall wear purple / With a red hat which doesn't go, and doesn't suit me." (The poem also talks about learning to spit—that's what sold me.) So I say, screw them. Older women are hot; women do not become useless once we can't (or don't want to) have children. So let's start by not judging ourselves anymore.Wrinkle cream begone,and fuck Botox. Though I have to admit, if many more of these grays pop up, you'll be seeing me at the hair salon. But don't worry—I'll wear purple. _If you want to know more about how to fight ageism and sexism, check out the Older Women's League (OWL)._ **16** **HE'S MANLY, SHE'S SASQUATCH** **WHEN I WAS ABOUT TWENTY-THREE,** I took a vacation to Spain (it was awesome). While on a ferry from Barcelona to Ibiza, I started chatting with a young Australian guy. When I mentioned being a feminist, he lifted my arm and looked underneath—as if to check for underarm hair. (He didn't find any.) I was not amused. The whole feminists-are-hairy stereotype is just so old school and ridiculous. But what also irritated me was the idea that if I happened to _not_ be shaved, it was some sort of huge beauty and fashion faux pas. There is a clear double standard when it comes to men, women, and hair removal. Now, perhaps you think shaving and waxing is a vapid issue to bring up, considering the more serious double standards of pay inequity, sexuality, and the like. But the fact is, spending the better part of your life having to shave huge areas of your body just to be considered not disgusting _is_ a big deal. The first-ever advertisement for a hair-removal product for women was featured in a 1915 _Harper's Bazaar,_ and depicted a woman in a sleeveless gown with perfectly smooth pits. Razor sales soon skyrocketed. (Waxing became popular with the advent of the bikini, Brazilian waxing—who knows?) I don't think I started shaving until I was in junior high, but I wanted to earlier. I'm Italian, after all, and dark, coarse hair is in my genes. There was something so humiliating about wearing shorts to school and seeing all of my hairless-legged peers. I remember in fifth grade stealing my mom's razor and taking off a small strip on my calf—just to see what would happen. (I was thrilled to see that despite my mom's tales of hair growing back thicker and darker, all was well on my stubbly strip.) But now, thinking about those days when my underarms were raw, my legs filled with nicks, or my delicates chapped from my first bikini wax (dear god, ouch), I'm just pissed. Pissed that I felt the overwhelming need to conform, pissed that guys don't have to go through the same thing, pissed that to this day I still don't feel right if my underarms aren't shaved. (If it's winter and I'm not showing off the legs, they're au naturel—sorry, just don't care that much.) Sure, there is a somewhat new trend of men getting shaved and waxed. (Who could forget the chest-waxing scene in _The 40-Year-Old Virgin?_ ) The difference is, however, that men getting trimmed—or manscaped, as I've heard it called—is considered an extra hygienic step. It's not necessary. If men don't get waxed, plucked, and shaved, no one is going to think they're "dirty" or not taking care of themselves. Not so much with women. Anyone who recalls the brouhaha that ensued after Julia Roberts showed up to a 1999 film premiere sporting a sleeveless red dress and underarm fuzz knows what I'm taking about. There was mockery, jokes, and disbelief. If women don't remove their hair, they're to be laughed at. Or pitied. This isn't to say that I'm against hair removal altogether—after all, I still do it, so who am I to judge? But I am pretty disturbed by the idea that if women don't shave, we're dirty or gross. And that the hair-removal trend is hitting younger and younger women. For example, take Nair, the depilatory-cream creator. In 2007 Nair introduced a new product line: Nair Pretty, aimed at ten- to fifteen-year-olds—what the industry calls "first-time hair removers." (Does that "first-time" line give anyone else the heebie-jeebies?) The Nair Pretty marketing scheme is half hilarious, half terrifying. Hilarious because of the obvious attempt to speak to young people in contrived slang: "It's not that you're obsessed or anything, but maybe you've noticed that the hair on your legs (and other parts of your body) is just a little bit thicker and darker than before. Chill. You're growing up . . . it's all good." I almost expected the next line to be about "getting jiggy" with hair removal. But it's still terrifying, because the message of Nair Pretty is that you can't be pretty unless you're taking care of that unsightly leg (and everywhere else) hair. And, as gossip blog Gawker put it when they covered the product line, "we're probably months away from Baby Brazilians." In fact, bloggers aren't the only ones worried about young women removing their hair (which is presumably related to sex, though I did it when I was younger just so I wouldn't be mocked). A Missouri State Senate bill in 2006 proposed parental consent for girls under the age of eighteen wanting Brazilian bikini waxes. Kind of over the top, I admit, but there is something creepy about the idea of girls who are barely old enough to have sex going out and messing with their na-nas. _**So... what to do?**_ Rethink the hair-as-dirty paradigm. It exists solely to make us feel shitty about ourselves and to make money from the beauty industry. Even if you shave and wax every hair except the ones on your head, think about _why_ it's so important to you to be hairless. Or, if you're up for it, take a cue from stand-up comedian Shazia Mirza, who made a New Year's resolution not to shave anymore: "I have decided that enough is enough and I have decided to just grow it, grow it like grass and try and live with it. . . . Every woman has hair. This is a fact. . . . It's about time hair on women was celebrated, not condemned. . . . A woman can definitely be sexy in a pair of Jimmy Choos and a pair of hairy legs, she can be sexy in a Wonderbra and hairy armpits, and she can be very hot in a miniskirt and hairy arms." **17** **HE'S SUCCESSFUL, SHE'S A SHOWOFF** **I MAKE MORE MONEY THAN MY BOYFRIEND.** A good deal more. (It doesn't hurt that I have five years on him.) This means that I tend to pick up the check more often than he does—especially because I'm a big, _big_ fan of eating out. He, on the other hand, is happy to eat boxed mac and cheese five nights a week. This doesn't cause problems in our relationship—it actually works out for the both of us. But according to dating "experts" and societal expectations, I'm breaking the rules. I'm "emasculating" my boyfriend by not letting him take care of me. Or something. Though of course, if I were dating someone whom I consistently let treat me and pay my way on dates, I'd be a gold digger, or a "dinner whore." There's just no winning when it comes to cash and romance. (If you're a woman, of course.) A September 2007 article in _The New York Times_ explored the supposedly recent trend of successful young women making more money than their significant others and how it has affected their dating lives: "Women are encountering forms of hostility they weren't prepared to meet, and are trying to figure out how to balance pride in their accomplishments against their perceived need to bolster the egos of the men they date." Bolstering egos . . . seriously? Is masculinity so damn fragile that it can't handle being treated to dinner? I can understand some men who have bought into the whole I-need-to-take-care-of-my-woman crap feeling a tab uncomfortable with being treated, but "hostility"? It's kind of amazing how tied up men's sense of self is with their ability to "care for" women. When I discussed this article on Feministing.com, one male commenter (who shall remain nameless) wrote, "How could a guy ever feel needed in a relationship where his partner completely outperformed him?" Wowza. Hear that, ladies? Don't let on that you're _too_ successful, or your man may run scared! Sounds silly, but that's almost exactly what this _New York Times_ article was saying: "For men, it is accepted, even desirable, to flaunt their high status. Not so for many women." (Another double standard, I guess. Men are proud of their accomplishments; women are braggarts!) But it's not just trend pieces in the Style section that are addressing dating etiquette. Take this charming segment from CBS, for example: "Reviving Dating Rules." Along the same don't-emasculate-through-success-and-confidence lines, dating "expert" April Beyer says that women should never pay for dates while in the courting process and never ask men out. Because it would interfere with their hunter instincts or some such shit. I'm hoping that women haven't really bought into this tripe. I mean, do we _really_ still believe that women need to be taken care of? Or are we so ashamed of our accomplishments that we're willing to "dumb down" our smarts and successes so as to not hurt the delicate male ego? Then, of course, there's the other side of the "who pays?" debate. Women who happily and readily accept dinners and gifts, who follow the "rules," are often called out as gold diggers. The term "dinner whore" is a newer one—unlike the gold digger, who marries for cash, the dinner whore goes on dates simply for the free meals and entertainment. On Urban Dictionary, a "dinner whore" is defined as follows: "a girl who is exclusively after a free meal or an expensive gift. She actively seeks out dates with well-off men who will wine and dine her at upscale restaurants." The idea that women are somehow taking advantage of men via dates even got itself an article in the _New York Post_. The oh-so-classy title: "Meet the Dinner Whores." Charming. Is there seriously a new trend of women going out with men simply for the luxury of a free meal? I doubt it. But it's a great way of painting women who dare to follow the traditional dating rules as whores. We can't win either way. Dinner whore or emasculating billpayer, I think what depresses me most about this double standard actually is the idea that money is so tied up with our notions of romance and dating. It's a stark reminder that women are still commodified and that when we try to exercise any kind of power in a relationship, we're punished. _**So... what to do?**_ I think the answer to this one is actually quite simple: Let each relationship or date speak for itself. There doesn't have to be a hard and fast rule when it comes to romance—just do what feels right. If someone makes more money and wants to go to a nicer restaurant, let that person pay and leave power relations at the door. That's not about gender, it's math. But there is one rule to follow, gals. Don't date anyone who feels slighted by the idea of you treating. Frankly, anyone who isn't comfortable with women being up-front about their financial success probably won't be comfortable with other successes as well. There's a sexism there that's impossible to ignore. **18** **HE'S SUPERDAD, SHE'S SHITTYMOM** **MOMS CAN NEVER REALLY DO ENOUGH.** They can never be too selfless, too devoted, or too giving. They can never go to enough soccer games or school plays. They can never be perfect—though that's what society demands of them. Women are expected to be stellar moms, but if dads so much as go to a baseball game or read their kid a bedtime story, they're frigging father of the year. It's the parenting double standard, and it looks like it's here to stay. My mother did (and still does, to some degree) everything for my sister and me. She cooked, cleaned, took care of us when we were sick, played with us, disciplined us—the whole shebang. My dad was there, too; he was an amazing father. He was our Brownie leader, our reader of nighttime stories and singer of songs. But the kudos he got for doing half as much as my mother did were pretty incredible. For my mom to spend all of her time caring for us—well, that was just expected. Now, traditional norms about who should be the caregiver of children (ahem, women) are nothing new. But what is more recent is the idea that caregiver women must be _perfect_ moms, and the setting up of impossible-to-meet standards for mothers. Susan J. Douglas and Meredith W. Michaels, authors of _The Mommy Myth: The Idealization of Motherhood and How It Has Undermined Women,_ call this the "new momism": . . . the insistence that no woman is truly complete or fulfilled unless she has kids, that women remain the best primary care-takers of children, and that to be a remotely decent mother, a woman has to devote her entire physical, psychological, emotional, and intellectual being, 24/7, to her children. And, of course, the impossible standards serve a purpose—they mean that the media, the public, or even your own family can beat up on women constantly for not living up to this new momism. If we work outside the home, we're neglecting our kids and turning them into bullies. If we're white, that is. If we're not white, we're "welfare queens" who should be working. Working women think we can "have it all." Fathers who work, on the other hand, are being responsible providers. If we stay at home, we're lazy. We're supposed to breastfeed—but not for too long and for God's sake not in public. We have to be straight. Most of all, we have to be married. If we're (gasp!) single, and have a child deliberately without a man, well, we're just selfish and preparing that kid for a lifetime of misery. When Louise Sloan, author of _Knock Yourself Up: A Tell-All Guide to Becoming a Single Mom,_ was interviewed on Salon.com—telling the story of her own insemination when she was forty-one years old, and the stories of other women who became single moms by choice—she was slammed with horrible comments: " . . . the boy will be screwed up or resent women, not having had a father around. He will have a higher chance of being a criminal. . . . He will likely understand that all the feminist piffle shoved in his head is the opposite of what men need to know to be _effective_ and happy free agents in the bigger world. . . . Your child will grow up fatherless and disadvantaged. But you got what you want, and that is what is most important. How sad." Charming, huh? All for just wanting a baby. Then, of course, if you _don't_ want to have kids at all, you're also selfish. You know, because all women are supposed to want children. If we don't, then we're either dismissed ("Oh, you'll want kids someday!") or scorned. Dads, on the other hand . . . well, they have it pretty nice. If they manage to show up and financially support their kid, they're automatically a good dad. (And this isn't to say that limiting fathers' roles to a financial one is a good thing—it's just another way traditional gender roles fuck things up. If a father stays at home to take care of his children or is doting, he's lazy or not a "real" man.) Dads who leave work early to catch a kid's school play or baseball game get props for being _so_ involved in their kid's life, while their female counterparts get called slackers. But it's not just about workplace and domestic issues when it comes to blaming Mom. Now, even the law is stepping in. Just recently, a woman in Connecticut was convicted of risk of injury to a minor because her son committed suicide and the courts said she should have seen it coming. And "fetal protection" laws are making it easier to arrest women for having stillborn babies—it happened in Utah to a woman who refused to have a cesarean section. _**So... what to do?**_ Making women to blame for all things baby isn't exactly new—so it's no easy battle to fight. But there are things we can do. For one, don't fall into the trap of identifying yourself by whether or not you have children—and don't let others do it either! Find out about your workplace rights: No one has the right to discriminate against you if you have children. Fight for childcare! The "care crisis" is a real thing—and it's not just a personal problem, it's a political one. The United States is the only industrialized nation that doesn't have paid maternity leave; this has to change. Don't expect to be perfect. No one is a perfect mother (except you, Mom!), and the standards that tell you as much exist because of sexism. So screw them. **19** **HE'S THE BOSS, SHE'S A BITCH** **IF YOU'RE A WOMAN IN A POSITION OF POWER,** you've probably been called a bitch. (Well, chances are if you're any woman _at all_ you've been called a bitch, but I digress.) Or maybe you've been called a "boss lady." Or a ball buster, ball breaker, or some other word that means castrating, pushy, loud, and basically out of line. Why out of line? Because women don't belong in positions of power, silly! So if you've gotten there, you must be a bitch. Men, though, are natural bosses. Just think of the way men and women in the workplace are described. Men are ambitious, women are ruthless; men are commanding, women are bossy; men who leave work early to get to a kid's soccer game are devoted dads, women who do the same are slacking off. An ABC News article from 2006 on workplace double standards had this great example: "When Russell Crowe tossed a heavy phone at a hotel clerk, we were mildly amused at his impatience. When Naomi Campbell threw a cell phone at her assistant, she was labeled an out-of-control prima donna. Both stars misbehaved, but our societal bias caused us to be a lot more critical of Campbell. Not fair." Indeed. It's the boss versus the bitch, and the ladies are losing. You need look no further than the movie industry to see how "working women" (as if we don't all work in some capacity or another) are thought of. One of my favorite writers (and people), Rebecca Traister, took on the boss-lady caricature in a review of the 2006 movie _The Devil Wears Prada:_ What else but the male erotic nightmare (Michael Crichton's, to be exact) could have produced Demi Moore's duplicitous (and horny!) Meredith Johnson, who puts the moves on Michael Douglas and then accuses him of sexual harassment in _Disclosure?_ Jane Craig, Holly Hunter's immensely likable but totally neurotic producer in _Broadcast News,_ forces herself to cry every morning before work. Diane Keaton's _Baby Boom_ executive J.C. Watts has a corner office, a six-figure salary, and a loveless relationship; they call her "Tiger Lady"—rowr! Check out Glenn Close as live-action fashion queen Cruella DeVil in _101 Dalmations:_ She berates peons, asks an assistant, "What kind of sycophant are you?" (reply: "What kind of sycophant do you want me to be?"), opines that "We lose more women to marriage than war, famine, and disease!" You know, 'cause being powerful and being feminine (meaning a mother and wife) just aren't compatible. But it's not just Hollywood and harmless workplace banter. The bitch/boss double standard is more pervasive than you'd like to think—and it affects women and work. A 2007 MSNBC survey showed that women in leadership positions haven't made much progress in their attempts to fight workplace bias. Most respondents—men and women—actually said they would prefer a male boss and think that men are more effective leaders. Those surveyed used words like "moody," bitchy," "gossipy," and "emotional" to describe women bosses. The most popular term for women in power? "Catty." Oh, how I _hate_ the word "catty." And "catfight." I mean, when men are competitive they're called just that. When women are competitive, we're subject to tired lines like "Kitty's got claws!" It's just another way to diminish and demean the idea of women having power. Really, what better way to dismiss female competition and power than to give it a cutesy animal name? The other, more nefarious intention is to make power seem unattractive to women. After all, if all women in power are bitches, then why would younger women aspire to be someone with power? Much like the anti-feminist stereotypes (man-hating, ugly, and so on) that exist to keep women away from an awesome political movement that could improve their lives, calling women bosses bitches is strategic. Not only does it put women who are already in power "in their place," but it also serves to deter women who are on the fast track from being too successful, too ambitious, too . . . well, bosslike. The other deliberate (though sneaky) way to make power seem unattractive is to call women bosses men or menlike. You can't be feminine and in power! (You need not look further than all the faux media trends that say women who work outside of the home have fucked-up kids or unhappy lives, despite the fact that real studies show just the opposite.) You have to wear shoulder pads and scoff at your kids as you drop them off at daycare while yelling at your emasculated husband. What's sneaky about this one is that it not only stereotypes women bosses as butch, but it also defines characteristics of power as inherently male. _**So... what to do?**_ I have a feeling that it's going to be a long time before men (and other women) will stop calling powerful women bitches. So while we're waiting . . . work! Be successful! And take being called a bitch as a compliment. Because it means you're doing something right. **20** **HE'S WELL PAID, SHE'S SCREWED** **THE WAGE GAP MAY BE THE MOST INTERESTING** double standard there is, just in that it's so obviously ridiculously unfair, yet little seems to ever be done about it. Men get paid more than women for doing the same job. Period. There's no sugarcoating that! Yet it's something we tacitly accept every day when we go to work. Now _that_ is scary. Women now earn 77 cents to a male's dollar. Not such a huge improvement over the years, considering we were earning 60 cents to the dollar when the Equal Pay Act was signed in 1963. In fact, sometimes it seems like things are never going to get better wage-wise. A report by the American Association of University Women, for example, found that women who were one year out of college earned 80 percent of what their male counterparts did—but that those same women ten years after graduation were earning just 69 cents to the male dollar. And this was the same for women who attended shmancy schools, for women who chose different kinds of jobs—there was just an "unexplained" wage gap. (Cough, sexism, cough.) Then, of course, you take into account the additional discrimination that comes along with being a woman who isn't white . . . and that gap gets even wider. It seems unbelievable that in this day and age, employers would just be sexist assholes for no reason whatsoever. ("She has a vagina? Well, knock 10K off her starting salary!") I think it's the ingrained sexism that fucks women at work, especially the assumption that women don't _need_ to make as much money—you know, 'cause they probably have a man at home (or Daddy) taking care of them. The other issue, when it comes to giving men promotions and paying them more, is that having a family tends to work in favor of men, while it works _against_ women's best interest. Employers may think that men "have a family to support" and therefore need the extra money. But because of gendered stereotypes they would never think the same thing about a woman employee, 'cause, hey, ladies aren't supposed to be the breadwinners, right? Then there's the "old boys' club." Men make a lot of their work moves on the golf course—there's no limit to men's outside-of-work networking opportunities. And let's face it, women just aren't going to get invited to that baseball game or night out at the strip club. Sara Laschever, coauthor of _Women Don't Ask: The High Cost of Avoiding Negotiation—and Positive Strategies for Change,_ says, "Women absolutely need to network with each other and with friendly and approachable men, as well. Women tend to be excluded from or peripheral to many of the social and professional networks in which men exchange information about what to ask for, who to ask, when to ask, how to ask. Women need to find ways into those networks if they're going to gain access to all that information." Laschever's book deals mostly with the issue of whether or not women speak up enough about what they want out of their jobs. While some argue that women should just be asking for raises more (which is undoubtedly true), because of workplace sexism, being up-front about your salary needs doesn't always pay off. A 2007 study found that women were less likely to ask for raises because the social costs were much greater for them than for their male coworkers. Which basically is a throwback to the original workplace double standard for women: If you're aggressive about anything, you're a bitch. Linda C. Babcock, a professor of economics at Carnegie Mellon University who conducted the study, was quoted in _The Washington Post_ as saying that "men were always less willing to work with a woman who had attempted to negotiate than with a woman who did not. . . . They always preferred to work with a woman who stayed mum. But it made no difference to the men whether a guy had chosen to negotiate or not." Shocking, huh? We're given shit for not asking for more raises, but when we do we're pariahs. Now, whenever you talk about the wage double standard, the conservatives come out in droves to whine about how it's not _really_ sexism that makes for women's lower salaries—it's all the women who choose to work part-time or stay at home and take care of the kiddies. (We _want_ to be earning less money, obviously.) To which you must call bullshit. Because all of the research done on the wage gap—usually by the Census Bureau—studies women who work full-time, all year. It doesn't include salary information about women who took time off to have babies. So enough of that excuse. It's time to move forward and stop letting the guys take all the cash. _**So... what to do?**_ Know your worth! And make sure that you're earning what you should be. There's no shame in asking around your workplace about who is making what. So if you find out that your male coworkers who do the same job as you are making more money, do something about it. If you see men being promoted over women for no discernible reason, speak out. And be proactive! There's a new trend among working women in some cities to start all-women networking events. ( _The Wall Street Journal_ even reported that events like these are getting support from business clients, who _like_ doing work with companies that take diversity seriously.) But I think some of the best advice comes from Laschever, who told Feministing.com interviewer Celina, "I always say, ask for more and ask often." **21** **HE'S GAY, SHE'S A FANTASY** **ASK A STRAIGHT GUY ABOUT LESBIANS,** and you're likely to get some smart-ass comment about porn, girl-on-girl action, or how they don't mind them so long as they can "join in." Charming. Ask about gay men, however, and it seems the only appropriate response is disgust. When it comes to being queer, gay men are gross and lesbians are hot (so long as they look like cheerleaders and fuck men too, of course). While this double standard is undoubtedly bad for gay men, it's just as (if not more) insulting to gay women. The assumption behind the hypocrisy is that lesbians aren't "really" gay and they don't have "real" sex. The idea being, of course, that if a dick isn't involved, it's just not real sex. I can't tell you how many times I've heard this sentiment repeated by friends who I thought knew better; I'm sure you have, too. Lesbians aren't a threat to their all-precious manhood so long as men can keep pretending that all gay women are like the gals in porn movies. (You know, where butch women just don't exist.) I'm actually reminded of a _Saturday Night Live_ sketch with Joshua Jackson where a bunch of frat boys wish on some magic monkey claw or something (don't ask) to see "real live lesbian sex," and all of a sudden are brought to the bedroom of a butch gay couple breaking out the patchouli oil. Faux male-friendly lesbianism is becoming so trendy, folks are even writing songs about it. "My Girl Got a Girlfriend," by Ray-L, is the epitome of dismissiveness when it comes to queer women. The chorus goes: "My girl gotta girlfriend / I just found out but it's aight / Long as I can be wit her too . . . Cuz havin two chicks is better than no chicks / I'd rather just join in / Keep my girl and keep the other one too." You have to love the assumption that _of course_ two women would want a man to join in their sex. So what's the deal? Why the fear of gay men but the acceptance of lesbians? (And let's be honest—society's acceptance standards tend to be formed by what's acceptable to straight white dudes.) A _Time_ magazine article about Ellen DeGeneres's success noted, "Lesbians simply don't inspire the kind of social-sexual unease that gay men do. Two chicks kissing is a male fantasy, a sweeps stunt. Two dudes kissing is gross-out humor. It's Sacha Baron Cohen open-mouthing Will Ferrell in _Talladega Nights_. It's a million _Brokeback Mountain_ jokes. It's the Snickers Super Bowl ad, in which two mechanics locked lips while sharing a candy bar. (Or, as Freud might have said, a "candy bar.") Even in post- _Queer Eye_ pop culture, lesbians can choose lovers; gay men can choose drapes." (That Snickers commercial was actually the most homophobic thing I've seen on television in a while, by the way. After two men's lips accidentally touch while they are eating a Snickers, they proceed to bash each other's heads in and bang on stuff to prove that they're "still men." Paging Dr. Freud!) So really, at the end of the day, this double standard is completely tied up with men's fear of being feminized! Whether it's through appropriating lesbianism as a straight man's dream or bashing on gay men, straight guys get to reaffirm how masculine they are. It would be sad if it weren't so horrifying. Because while it may not seem like such a big deal if guys want to get all revved up about faux lesbians and skeeved by gay men, the consequences of this kind of prejudice can be more than just a few jokes. Lesbian women who are raped are often targeted because of their sexuality and told that they just need a "real man." Men who are gay-bashed are similarly targeted—because they pose a threat to straight sexuality. _**So... what to do?**_ Call people out on their bullshit. This double standard is particularly pervasive because of porn culture and the like. So be on the lookout and don't let it go uncommented on. **22** **HE'S HIMSELF, SHE'S MRS. HIMSELF** **I USED TO WANT TO TAKE MY MOM'S LAST NAME** in addition to my dad's. It seemed wrong to me that her last name just got lost in the wind, like it never existed. (Though I ended up deciding that Jessica Michelucci-Valenti was just too much of a mouthful to deal with. Go figure.) The last-name debacle is definitely something I've written about before, because it's one of those double standards that are so _obviously_ ridiculous and sexist—yet so accepted—that it bears repeating! Now, most American women who marry men will change their last name (note I didn't say "maiden name"; if that were the case, we'd all lose our second names upon doing the nasty) to their husband's. That's just a fact. So I have no illusions about creating some sort of mass revolt of women keeping their own last names (though that would be nice!), but I do think an examination of the last-name double standard is warranted—especially when it seems that the powers that be have such a stake in making sure that things stay the same. The reason I hear most often when women talk about why they want to take their spouse's last name is tradition. To which I say, _meh._ I'm unimpressed. There are plenty of traditions worth keeping. Saturday brunch with the girls. For you fellow Italians, making Sunday gravy. Birthday cake. (It's no coincidence that my notions of traditions are all about food. I like me some good meals, what can I say?) But holding on to traditions that not only make your life more difficult—legally changing your last name, and all the paperwork that goes along with it—but are also mired in sexist ideals of women being owned? That's just too much. (Quick remedial lesson on last names for those not in the know: The idea is that women were passed as property from father to husband; we don't really have an identity of our own—just that which the men in our lives define for us.) The other reasons women give for changing their last names run the gamut from "it's better for the kids" to "hyphenation is too difficult," and so on. Yet, as friend and fellow feminist blogger Amanda Marcotte points out, you rarely see men using these excuses to change their last names to their wives'! [I]f you think a name change is necessary, you can have the man change his name. It's an elegant solution. Not only do you have all the perceived benefits, but you are sticking it to the patriarchy. This solution even works if you're employing the "I don't like my last name anyway" thing, because if there's a lot of people out there who dislike their last names, then the odds are strongly in favor of the fact that half of them will be men. But for some reason, when this discussion comes up, women and women only seem to dislike their last names. Interestingly enough, when men _do_ try to change their last names to their wives', they run into all kinds of obstacles. (And apparently men _are_ doing this more and more, at least says _USA Today._ Nice!) If a man wants to change his name after he gets married, only seven states in the United States allow him to do so without going through a ridiculous, expensive, and long legal process that women in the same position _don't_ have to go through. Take Michael Buday, for example, in California. Buday went to the Department of Motor Vehicles to change his last name to his wife's, and not only was he ridiculed by the staff there(!), he also found out that the hoops he would have to jump through were out of control: a $300 court fee, six times as much as the fee for women, and he would have to advertise his name change in a newspaper. WTF? Luckily, the ACLU of Southern California took up his case and is seeking to change the law in the state. But it makes you think: If changing your last name to your hubby's is "no big deal," then why is the government so hell-bent on making sure the opposite doesn't happen? It has a vested interest in keeping women in traditional gender roles—and the last-name thing is a huge part of that, and a part of maintaining patriarchal family structures. _**So... what to do?**_ I've said it before, and I'll say it again—keep your last name! Or choose a new one. Or hyphenate. I just don't see any reason to do otherwise. That's not to say I think you're a bad feminist if you do choose to take a man's last name. But if you do, be honest about it. Don't say it's because it's tradition or because you don't like your last name. As Amanda noted, just be honest that it's sexist: "So are high heels and I wear those. Hell, I wear those despite the complaints of boyfriends in the past who preferred displays of female subservience that didn't slow down how fast we could walk. We're all guilty, so that's not the issue. The issue is the amount of effort put into pretending that the name change isn't sexist." Word. **23** **HE'S GETTING AN EDUCATION, SHE'S GETTING IN HIS WAY** **WOMEN ARE DOING AMAZING THINGS EDUCATION-WISE.** We're going to and graduating from college in higher numbers than ever—same with getting master's degrees. We're even bigger in numbers than men in higher education. Given that we're kicking so much ass in school, can someone tell me why women are still underrepresented in high-level and managerial positions once we enter the workforce? According to the Department of Education, about three in every ten boys who go to college get out four years later with a degree; four out of ten girls do. And as of 2005, a little over 57 percent of bachelor's degrees were earned by women, a little over 42 percent by men. (The numbers were pretty much flipped back in 1970.) Not too shabby, eh? But apparently, women's doing so well in school has some people freaked out—the worry is that if women do well, men do poorly. In fact, the mere fact that women are doing better in terms of education has prompted the media and conservative organizations to declare a full-on "boy crisis"! Article after article for the last five years or so has bemoaned the decrease in men getting bachelor's degrees—and most of the blame has been put on feminists. It seems that because of women's rights, boys are being discouraged from doing well in school—at all levels. How could something like women fighting for equality mess boys up in grade school? Well, it seems we've feminized learning. Or something. Take, for example, seventeen-year-old Doug Anglin of Massachusetts, who filed a federal discrimination suit claiming that his high school discriminates against boys. Anglin complains that "the system is designed to the disadvantage of males. . . . From the elementary level, they establish a philosophy that if you sit down, follow orders, and listen to what they say, you'll do well and get good grades. Men naturally rebel against this." Which explains why guys flee from hierarchical structures like the army. Uh, wait. . . . Anglin also complained that when teachers are grading homework, they give extra points to students who decorate their notebooks—a policy that _clearly favors_ girls, because everyone knows that gluesticks are for pussies. My favorite article about this lawsuit, in _The Boston Globe,_ has the most telling quote of them all: "Larry O'Connor, another Milton High senior who supports Anglin . . . said he is surrounded by a sea of girls in his classes." Noooo! They're everywhere! It seems that, like this lawsuit, a lot of complaints about the "boy crisis" have more to do with women doing well and being strong in numbers than with men _not_ doing well. As Katha Pollitt said in a 2006 column, maybe these boys "will just have to learn to learn in a room full of smart females." You know, suck it up. Anti-feminist whining aside, what's truly interesting is that there really is no boy crisis. It's been debunked as a media myth over and over again. A 2006 _Washington Post_ article noted, "Although low-income boys, like low-income girls, are lagging behind middle-class students, boys are scoring significant gains in elementary and middle school and are much better prepared for college. . . . Much of the pessimism about young males seems to derive from inadequate research, sloppy analysis, and discomfort with the fact that although the average boy is doing better, the average girl has gotten ahead of him." The real education crisis is that people of color and low-income children are worse off. White dudes are still going as strong as they ever were, thank you very much. But while the media and anti-feminists continue about a faux boy crisis, they're silent on the continued disparity in jobs at the top and in advanced and professional degrees. (Because that would interrupt their boys-as-victims-of-feminism diatribe.) Women still earn significantly fewer advanced degrees in business, engineering, and computer science than men (and these are the degrees that lead to much higher-paying jobs); men outnumber women in earning doctorates and professional degrees. And while nearly half of all law students are women there are nowhere near as many female law partners, professors, and judges as there are men in these positions. The same bodes true for women in managerial positions, high-level decision-making positions, and CEO spots. It seems that sexism and discrimination follow us through the workforce, no matter how well we're faring in school. _**So... what to do?**_ Think about going to school for engineering, for starters! Too often, women are pushed into programs that don't necessarily lead to high-paying jobs. This isn't to say that you shouldn't follow your bliss and all that, but try to follow the cash too! If you're a college student, find out what your university's gender breakdown is in terms of tenured professors, or how many women are involved in traditionally male fields. And put that information out there! When you hear someone talk about the supposed boy crisis, set them straight. Let them know that just because girls are doing well, it doesn't mean that boys are doing worse. Encourage your girlfriends to go for advanced degrees and not stop at a BA. And when you're in the workforce and you see that glass ceiling you're about to hit, make some noise about it. **24** **HE'S INDEPENDENT, SHE'S PATHETIC** **WHEN WAS THE LAST TIME YOU WENT TO A MOVIE BY YOURSELF?** Or out to eat? Or to another country? If it was recently, kudos. I hate the idea that women shouldn't do things on their own (for safety reasons) or that if we do (like go to a bar alone) we're pathetic. But for some reason, the notion that ladies shouldn't leave the house unless they're escorted by a man is still going strong. If a man is out to dinner alone, it's normal. If a woman is, she must be waiting for someone, or she's been stood up, or she's lonely. Ditto for going to the movies alone. And if a woman travels alone? She's putting herself in danger! It's like we're living in this bizarre universe where the very simple act of walking around (or sitting around) by ourselves _means_ something. It means we're targets, or pathetic, or anything other than just, well, being alone like a normal person. Have you ever been sitting by yourself—reading in a park, drinking at the bar, whatever—and a guy comes up to you? What the fuck is that? It's like just by virtue of not having male company with you, you're open for business? I've even been in a bar with girlfriends—like, a big group of girlfriends—just to have some asshole come up and ask why we're alone. Huh? So because we're not in the presence of cock, we're "alone"? (I know you ladies know what I'm talking about.) I mean, even just living alone for the first time was an interesting experience for me. After getting my first apartment without roommates, I had relatives ask me if I was going to get a dog for protection (I did, but Monty is not tough enough to protect anyone) or when, oh when, I was going to get married already. My guy friends have been living alone for years. But I guess they're "independent." Me, I'm halfway down spinster road. While doing research on this particular double standard, I came across an amazing/terrifying article from _The New York Times_ circa 1908. The headline was "Women's Right to Eat Alone." Because, apparently, it wasn't always legal! The Women's Republican Club, at its regular meeting at the Plaza yesterday afternoon, put itself as on record as opposing the opening of saloons on Sundays and also as believing that women should be permitted to eat in public places when and where they please. While trying to garner support for this bill that would allow women to eat in restaurants without a male escort, one of the club members was quoted as saying, "I believe it is a protection to all decent women that women alone should not be allowed to eat in public restaurants." I bring this up because despite its being a quote from 1908, it's not so far off from some of the arguments you'll hear even today against women doing things by their lonesome. For example, how there's still a ton of victim blaming going on when it comes to rape victims. Women who were walking home _alone,_ who were at the bar _alone,_ or the like, are often questioned about why they would walk around by themselves—as if we shouldn't be free to walk around alone without fear of assault! (But it's about protecting women, they swear.) The idea that women are in danger just by virtue of being by themselves is so ingrained that one designer created a dress that—get this—transforms into a vending machine costume so women can be "disguised" as they walk home. (Kind of like the cartoons where people would hide in bushes and you'd see the little feet underneath walking around.) So depressing, truly. For those gals who are the traveling sort, it gets even worse. Do a Google search for women traveling, and all you get are rape prevention trips! But the truth of the matter is, more women are doing shit on their own than ever before.Women travel abroad alone, go on vacation alone, even (gasp!) eat alone. A recent census survey actually showed that 51 percent of women in the United States are living without a spouse. That means most women are living alone. Independent women indeed. Now if we could only get the rest of the world to realize it! _**So... what to do?**_ Stop assuming that a gal sitting by herself is waiting for someone! Maybe she's just enjoying a peaceful moment. And if you like doing stuff by yourself, do it! I'm not saying don't be safe, but don't live in fear either. It's traditional bullshit that tells us we should be relying on men to accompany us everywhere. We're cool on our own—or with other women! Go travel with girlfriends. Go out to eat by yourself. Go to a bar with a group of gals, and if some jerkoff asks why you're alone, laugh in his face. Relish your independence. **25** **HE'S A CELEB, SHE'S A MESS** **THERE'S SOMETHING ABOUT CELEBRITY WOMEN** that we love to hate. We relish in their anguish and bask in their breakdowns. Sure, male celebs mess up every once in a while, and we chuckle. But for the most part, it's the hot mess that is young celebrity women that keeps us coming back for more. We're like a country full of enablers. We laugh at male celebrity messes: Nick Nolte's mug shot, Billy Joel's drunk driving. But we don't react with disgust in the way we do with women. We don't call philandering male celebrities "whores" or "sluts." We don't mock them for "getting fat" or having kids. We don't wonder if they're anorexic. Though I suppose it's not surprising—if you look at women celebrities, particularly young women, you see a microcosm of societal sexism at work. We love them when they're young, taut, and gyrating at our command. But if they slip up, or have the nerve to get older, we're right there, waiting to tear them down. (Sounds like what we do to women in general, never mind celebs!) The epitome of this female-celeb hating, of course, is Britney Spears. Who else embodies the virgin/whore hot mess better than she does? From her cheery schoolgirl dancing to shaving her head in a stupor, she _is_ the fallen public woman. Though nothing beat her final descent into being forever mocked like she was for her performance at the 2007 MTV Video Music Awards. Now, there's no doubt that it was a bad performance. She seemed to just be going through the motions, and I, probably like a lot of other people, felt bad for her. And I was absolutely livid to read the gossip mags—and even traditional news outlets—comment on how "fat" she was! Whether it was a news story saying she had a "paunch" or a cable news dude calling her chunky—it was just fucking gross and wrong. (Um, and when was the last frigging time a male musician's beer belly made news?) One of my fave writers, Rebecca Traister, breaks it down: Spears has come to represent something—something important enough that it keeps rearing its head. As has been pointed out before, she embodies the disdain in which this culture holds its young women: the desire to sexualize and spoil them while young, and to degrade and punish them as they get older. But _why_ exactly are stories about famous women's demise so much more appealing than news about a man's downfall? Especially considering that so many of the people reading and laughing at the likes of Paris, Lindsay, and Britney are women themselves! Is it because we have so little control in our own lives that taking joy in others' misfortune makes us feel better? Or maybe we like seeing the young women who represent these unattainable beauty standards crashing and burning. (Kind of like when you ripped the heads off the Barbie dolls you played with so much.) We hate them but idolize them at the same time. It's fucked. But it's also predictable—we hate/idolize women in much the same way that society does. It's a misogyny thing. After all, I don't think it's a coincidence that one of the favorite pastimes of celebrity photographers is trying to get a "getting out of the car" nekkid-vagina shot. Look, they're women, and they're whores! Robert Thompson, a professor of popular culture at Syracuse University in New York, was quoted in an article about celeb girls gone bad as saying, "We have had years of young male stars running amok. It is now so much more fun for the public to see beautiful young women being hauled off to jail." That reminds me of a quote from Edgar Allan Poe, who said, "The death of a beautiful woman is unquestionably the most poetical topic in the world." And it really is true. The destruction of a young woman is the oldest, most popular story in the book. And now it's being lived out in front of us day after day. Modern-day girl celebs are the traditional damsels in distress, but instead of trying to save them, we're cheering for them to trip up. And this isn't just about bad karma and being gossipy. It isn't about letting male celebs off the hook. It's about bashing other women because we hate ourselves. It's about idolizing women who are train wrecks. And it's got to stop. _**So... what to do?**_ Stop the schadenfreude! Taking pleasure in someone else's pain seems to be an American (or worldwide, I suppose) pastime, but when it comes to young women in the spotlight, we've gone too far. When are we going to realize that hating other women—no matter how much money they have or how far they've fallen—is just as bad for ourselves as it is for anyone else? And that by buying into the media frenzy surrounding young women falling into disrepair, we're buying into a culture that would be just as happy to see any woman—including you—trip and fall. **26** **HE'S HUSKY, SHE'S INVISIBLE** **WHERE, OH WHERE, ARE THE WOMEN OF SIZE?** The women we see on television and in the movies are all small, skinny, and svelte. And while there are male actors of all shapes and sizes represented on television and in the movies, the only women with a dress size in the double digits we see are those who play the "fat" character! "Normal" women, it seems, are all tiny (besides the boobs, of course. Got to have The Boobs). Think about all the heavier male actors you can name: Paul Giamatti, Jack Black, and James Gandolfini are a few. Or if you're into old-school comedy, actors like John Candy, John Belushi, and Chris Farley were popular back in the day. Now, what about actresses of size? Drawing a blank? That's because outside of a handful of women (who are usually recruited to play a character whose size will be an issue in the plot), big women are pretty much invisible in the media and entertainment biz. Yes, movies will _occasionally_ take on issues of size, but rarely in a way that's flattering. Consider the 2001 movie _Shallow Hal_ with Jack Black and Gwyneth Paltrow. The movie seems to have a positive message: The main character falls in love with an overweight woman because he's been hypnotized to see her inner beauty. After some trials and tribulations, he wants to be with her even without being hypnotized because, he's fallen in love with her as a person. All very charming, but the movie still presupposes that a man needs to be convinced to love a woman of a certain size and that she wouldn't be able to get some loving otherwise. As if someone just couldn't be attracted to her (gasp!) appearance. Also problematic—though common—is that the love interest isn't played by an actual woman who is larger, but by an actress in a "fat suit." Marisa Meltzer at _Bitch_ magazine says the fat suit is "the new minstrel show." Not only does it have a long history ("Wanna make a funny movie? It's a pretty easy formula: Zip a skinny actor into a latex suit. Watch her/him eat, walk, and try to find love. Hilarity will ensue"), but it also perpetuates some fairly gross stereotypes: Fat Monica [from the TV show _Friends_ ] really takes the proverbial cake. She dresses badly, has no self-control, eats junk food, has poor hygiene, and is a virgin. She's the opposite of the control-freak Thin Monica, who has the husband, the job, and the adoring friends. Fat people are one of the last groups of folks whom it's totally acceptable to mock in public. You know, because it's for their own good—'cause of the "obesity crisis" or something. Please. You know, I've always been lucky to be a size deemed acceptable by society. But my freshman year of college, which proved to be depressing for many other reasons (I won't get into that!), I gained about twenty pounds. I didn't really notice, truth be told, because I had always had the privilege of never having to pay attention to my size. But when I went home that summer, some of my relatives were all too pleased to tell me how fat I had gotten. Or how I looked like I had a "beer belly." I lost the weight in a couple of months, but just the handful of comments I got stayed with me for a long time. Which is why this next trope in movies about fat women is just hilarious. A common theme in movies that feature women of size is the poor mocked girl who is just too nice and innocent to realize people are making fun of her. Kate Harding, who blogs about fat-acceptance issues at Shapely Prose, notes: By the time I was eight years old, all the very special episodes and TV movies based on this ludicrous premise—fat chick gets nominated for Homecoming/Prom Queen and/ or asked to The Big Dance by The Big Man on Campus, and somehow fails to see anything wrong with this picture—sent me through the goddamned roof. Have the people who write this shit ever actually MET a fat person? And hell, even if they haven't, can they seriously think it's possible for anyone to go through life with people snickering behind her back and saying hateful shit to her face every single day, and not catch on to the possibility that she just might not be the most popular girl in school? Yet heavier male actors prevail. And they get parts that _don't_ focus on their weight—parts that are a luxury for women actresses. _**So... what to do?**_ Support movies and media that portray women of all sizes. That includes magazines that claim to value women no matter what but always seem to find a size 0 to stick on their cover! Check out groups like Big Moves (www.bigmoves.org), a performance group for women "dedicated to getting more people of all sizes into the dance studio and up on stage." My friend Jaclyn Friedman is a part of the organization, and let me tell you—they do some great stuff. Visit blogs like Big Fat Deal (www.bfdblog.com) and Big Fat Blog (www.bigfatblog.com) to get the skinny (pun intended) on what's going on in the fat-acceptance movement. And don't watch movies or shows where fat people are the punch line. Boycott fat suits. **27** **HE'S A MAN, SHE'S A MOM** **WOMEN, IT SEEMS, ARE DEFINED BY WHETHER OR NOT** they have children. Or how _many_ children they have. Or if they're an evil, childless spinster like me. (Hey, I have a dog. That's kinda like a kid.) Unlike men, who are primarily judged by their accomplishments, their profession, their personality, women are looked at for their appearance and their ability to pop out the little ones. I definitely want kids. I want to raise droves of little feminist activists who call out playground sexism and grow up to roll their eyes at me when I talk about how things used to be so terrible for women. "Mo-om! Enough about the Hyde Amendment already—that's sooo 2007!" But I don't want my identity to be so tied up with motherhood that I'm seen not as an individual, but as so-and-so's mom. (Yeah, I have no clever imaginary kids' names; I'm not _that_ eager to have children.) But it seems that that's the only option women have today. We're not good women unless we have kids, and once we do—we're moms first, foremost, and forever. Of course, this is why it's always women who are expected to worry about "balancing work and home," because motherhood is supposed to be our priority—not men's. And, yes, I know about the arguments that women are just naturally more inclined to be caregivers because of the whole carrying-a-baby-and-giving-birth thing. But I say once they're out, they're out. Why should the onus of caring for a kid fall so heavily on women while men get off scot-free? And I truly believe that because women have been relegated to this bizarre world where motherhood defines what kind of person we are, we've started to go nuts. Women who have become über-competitive with their children, or with how good a mother they are, are acting out (in my humble opinion) because they're not supposed to be competitive in the workplace or anywhere outside the home. This is their new domain. Take, for example, a report from NPR about affluent women who are having more and more children as a way to transfer their competitive energy—they call it "competitive birthing." She who has the most kids wins. Seems like one of those sketchy trend pieces, but I don't think it's that unbelievable. As Feministing blogger Ann Friedman said about the story, "[I]t doesn't seem completely far-fetched to me that women who used to be career-driven would want to direct their competitive energies somewhere—and for some women, that's become a quest to be the best mom." But this isn't just about the ability to work and raise a family, or how women are expected to do more in the domestic sphere. This is about the very pervasive—and troubling—myth that being a woman means having children. That we're not "whole" without them. If men choose not to have children, no one will be aghast at them or tell them they'll change their mind. When they go to get a vasectomy, no one refuses them. (I've heard many stories of women being turned away when they try to get their tubes tied.) Women, on the other hand, are expected to want children. Oodles of them. So when a woman says she doesn't want kids, the assumption is that she's going through a phase, or that there's something wrong with her—or that she's just plain selfish. After all, it's a woman's job to have the babies! But the truth is, more and more women are opting out of parenthood. The U.S. Census reports that women in their twenties to fifties who don't have children have been growing at fast rates over the last ten years. And a report from the National Marriage Project at Rutgers University says that nearly one out of five women in their early forties is childless—thirty years ago it was one out of ten. And despite what media frenzies tell you about poor career women desperately trying to get pregnant after spending fruitless years developing their high-powered professions, most women who are childless are completely comfortable with the idea. In fact, a 2007 study says that women are much more comfortable with the idea of childlessness than men are. The research, which was published in _The Journal of Marriage and Family,_ shows that the results may be due to the fact that men experience "strong economic and social rewards" for being fathers, while women experience more pressure and demands on their day-to-day lives. Despite stereotypes that assume women care more about having children than men do, this study says that it's actually women who understand more about the _costs_ of having children. Nadine Kaslow, chief psychologist at Emory medical school in Atlanta, said the findings of the report show "women who are successful professionals make a choice that they don't want to have children in their lives, because they have other things in their lives." Men, however, "tend to think that is what you do in life. You grow up and have a baby." So why try to conflate women's identity with whether or not they have kids? Well, once again, it benefits a society that lives on sexism. If we're not "real" women unless we have babies (and stay home to take care of them, of course), then women are going to feel pressure to adhere to traditional gender roles. _**So... what to do?**_ Don't have babies, ever. Just kidding. Have babies, don't have babies, raise Sea Monkeys for all I care. Just don't let anyone ever tell you that who you are as a person has anything to do with the personal family choices you make. **28** **HE'S DATING A YOUNGER WOMAN, SHE'S A COUGAR** **CAN SOMEONE PLEASE TELL ME WHY IT IS THAT** it's perfectly normal for a man to date someone who looks like she could be his granddaughter, but if a woman dates a younger man she's a cougar? Or an oddity? And I swear I'm not just saying this because I'm dating a younger man. (Well, maybe a little.) It's no great secret that older men frequently date younger women. It's not frowned upon, really—it's actually exalted more than anything. After all, they don't call them "trophy" wives for nothing! Think Donald Trump, Hugh Hefner (eew! On second thought, _don't_ think about him), or any other old guy with a young woman. While you and I may take a look at their relationships and cringe a little, the fact remains that society supports them. But an older woman with a younger man? It's treated as a novelty, a joke, or a premise for bad porn. But the fact is, more and more older women _are_ dating below their age. A 2003 AARP study showed that 34 percent of women over forty were dating younger men, and 35 percent preferred it to dating older men. Another recent study found that in more recent years, only 25 percent of brides have been younger than their grooms. While my boyfriend is a mere five years my junior, I still get a lot more comments on the relationship than you would expect. Guy friends joke with me about "robbing the cradle." Family members look on in concern, wondering if a younger man will be able to get serious with me (translation: want to get married). I've even had someone wonder aloud if I was with my boyfriend because I wanted to be with a younger man I could "control." (Anyone who knows me knows that I like men of any age whom I can control, but that's beside the point!) It can't just be that we're two people who like each other a lot. There has to be something more to it. Now, I've heard all sorts of reasons why it's more "normal" for older men to date younger women, and not vice versa. That it's because of money. (Well, women have more money than ever.) That it's biology—men want to spread their seed and be with women young enough to have babies. (Sorry, I just don't buy it. Older men can still have babies, but recent studies show that men's sperm ages, too—and can cause birth defects. So it's not just our old-ass eggs, guys!) But I think at the end of the day, some people are uncomfortable with older women/younger men relationships because it's not the power dynamic they expect. Because, let's face it, with age comes wisdom, more money, and more power. So in a sexist society, it's understandable that older men would be with younger women—it adheres to the power dynamic already set in place by the patriarchy. (I know, I just got all women's studies on your ass. Relax, it only hurts for a second.) Men with more power, women with less. That's why it's unfathomable that a woman could do the same thing an older man does—because she's not supposed to have the power in a relationship! And that's why it's so easy—and so necessary—to fetishize or make fun of older women being with younger men. By diminishing the validity of the relationship by making cougar jokes or watching MILF porn, men are getting that power back. And let's be serious—jokes and porn abound when it comes to this kind of couple. Of course, the most popular incarnation of the older-woman-as-joke is the notion of cougars. Urban Dictionary defines a cougar as "an older woman who frequents clubs in order to score with a much younger man. The cougar can be anyone from an overly surgically altered wind tunnel victim, to an absolute sad and bloated old hornmeister, to a real hottie or MILF." (See what I mean about taking the power back? Sigh.) What's really interesting about all the "cougar" websites that have cropped up is that many of the women featured are in their late twenties—as if that's old! I guess in porno-land it's over the hill, but come on now. Or there's the prevalence of MILF (please don't make me spell it out) porn. Most of this stuff is about men finding older women (one site is call MILF Hunter, for example) to have sex with. It's not always about men in power, but the simple act of sexualizing a relationship and fetishizing is a way to make it less valid, less important. So now, instead of being in a relationship where you're dating a younger man who makes you feel independent and loved, you have to feel like the butt of someone's joke or the star of someone's fantasy. _**So... what to do?**_ Keep on keeping on, ladies! Date those younger men if that's what you want. And don't let anyone call you a cougar. Ick. And don't forget that older women can date younger men, because we _have_ power. Susan Winter, who's in her fifties and is the coauthor of _Older Women, Younger Men: New Options for Love and Romance_ , says, "When women as a group are able to have their own economic and social standing and have a power base, they are now able to pick the man that they want rather than having to choose the man to support them and give them social status. . . . Now we have choices." Indeed. **29** **HE'S DRUNK, SHE'S A VICTIM** **WHOEVER THOUGHT THERE WOULD BE AN INEBRIATION** double standard?! But indeed there is. And unfortunately, it's not even funny—it's dangerous. Men are supposed to get drunk to bond and to have fun. At worst, someone ends up waking up with an unfortunately placed tattoo or with a fun story about ending up in Mexico or some such shit. If you're a woman, you're supposed to drink to "loosen up," specifically for the sex. And while I'm definitely not against having some drinks and getting your sex on, this dynamic is part of what makes for date and acquaintance rape—and that's just scary. I have done my fair share of partying. You might even say that for a time I was a party girl. I went out and got drunk, fell down while drunk, and definitely let alcohol grease the wheels for several hookups. But I also saw some disturbing shit in those days that I wish I hadn't. I saw girlfriends the morning after a night of binge drinking who said they didn't remember if they had had sex. I saw guys grabbing at women in a way that went beyond sexual or flirty—it was aggressive. I saw stuff that I'd rather not even mention here. Again, this isn't to say that I think drinking is an inherently bad thing. But I do think that our drinking culture and the inebriation double standard target women in a creepy way, and that they allow for women to be blamed when something bad happens. Take cheap drinks and bar deals. The ladies'-night phenomenon is really something else—you have to love the genius of bar owners who make it easy and cheap for women to get ridiculously drunk, and then have guys pay to come and hang out with them. Be honest—you know this shit isn't about making a nice night for women; it's about providing drunk women for men! (Besides, the drinks at ladies' night are always watered down or sucky sugary margaritas that give you a wicked headache in the morning.) But it's not just crappy bar deals we're talking about. Just think about the way that women are still (still!) blamed for their own rapes if they had the gall to have a drink or two. The common sentiment is still that women who get drunk either are sluts who are looking for an excuse to have sex or should have known better than to make themselves "vulnerable." Writer (and general badass lady) Jaclyn Friedman wrote an amazing article on drinking and rape in which she discussed her own assault and what could have been done to stop it: Let's look a little more closely at that correlation between rape and alcohol. That's not a correlation between female drinking and rape. It's a correlation between all drinking and rape. In fact, studies have shown that it's more likely that a male rapist has been drinking than that his female victim has. So if we want to raise awareness about the links between drinking and rape, we should start by getting the word out to men that alcohol is likely to impair their ability to respond appropriately if a sexual partner says "no." When was the last time you read that article in any kind of publication? Well, because of the drinking (or any kind of substance, really) double standard, you never would hear of something like that. Instead, the onus for rape is put on the woman who was drinking, not the rapist—drunk or not. Jaclyn hit the nail on the head: "The silence around men's drinking is, of course, part of a much larger 'boys will be boys' culture, one which played a large part in my assault. The party I attended was for a men's sports team; the coaches provided the alcohol." So why is it, then, that all of the warnings about rape and drinking are directed at women? Shouldn't we be telling young men that drinking puts them in danger of crossing the line? No way—because guys' drinking is fun; it's normal socializing. But girls' drinking has always meant the same thing: sex. Take the seventeen-year-old woman in California who was gang-raped—despite three female eyewitnesses pushing their way into a room where the girl, with clothes around her ankles and vomit on her face, was being assaulted with ten men looking on. (The three young women fought to get her out of the room; they had seen her being dragged in and figured something bad was going on.) Charges against the men were dropped—apparently they couldn't know if the woman consented or not because she was drunk. Never mind that she had puke all over her—that sure screams, "I'm ready to have sex," huh? (And just a thought—if a guy woke up and had been raped by another man after drinking too much, do you have any doubt in your mind that people would believe the victim? Just saying.) Now, you can't talk drinking, assault, and woman blaming without talking about Girls Gone Wild. Tricking drunk women, or making out like coerced drunk women are just being the exhibitionists they always wanted to be, is the whole philosophy behind the porn (and sleaze) empire Girls Gone Wild. I mean, in what universe can you sign a consent form while shitfaced? If you signed a will or a contract while under the influence of something, it wouldn't be valid. Yet drunken teens agreeing to strip? Not a problem. And again, I'm not saying that all the women in those videos have been taken advantage of. But the model that GGW is working under assumes that these women need to be tricked, convinced, and, most important—good and drunk. I'm sorry, but that just does not sound like fun. _**So... what to do?**_ Don't get trashed. Obviously, I'm not against drinking. And I don't think that women should forgo having a social life in what will probably be a vain attempt to protect themselves against assault. (Remember, it's not a drunk woman who facilitates a rape, it's the fucking rapist.) But I do think that young women drink too much and that it's just generally bad for us. Besides, nothing worth doing is more fun when you're drunk. Arm yourself with knowledge about rape culture and victim blaming. And if you have a bad feeling at a party, or about a drunk friend or a drunk guy, follow your instincts. **30** **HE'S STOIC, SHE'S FRIGID** **WHEN MEN ARE QUIET, THEY'RE MYSTERIOUS.** When they're a little sullen, they're James Dean deep.When women are serious or quiet (or not constantly chipper, at least), we're cold bitches. We're frigid. We're snobs. I am a _very_ gregarious person. I talk loud, I crack jokes. I like being in big groups of people. My sister, Vanessa, however . . . isn't. She's just as funny and friendly, but she's more reserved. I remember when we were teens, she would complain that everyone assumed she was a bitch or snotty because she didn't open up right away. I always found it unfair. Now, while the ability to be a quiet or serious person without being labeled cold may not seem like it should be a top feminist priority, it's actually coming from a pretty interesting place. My least favorite form of street harassment is when a guy asks why I'm not smiling. It's related to that: Women aren't allowed to be quiet or stoic or shy—or, hell, just in a bad mood—without being criticized. Women are bitchy and frigid if we don't seem accessible at all times, for the most part to men. We're supposed to be perpetually friendly. Who wants to live up to that? And seriously, when was the last time you heard a quiet woman described as "deep"? Men who are serious are just that—serious. Think laconic cowboys and Clint Eastwood-style movie heroes. Strong and silent is a desirable personality trait for men—women, not so much. Because where silence in men is seen as strength, silence in women (if not seen as bitchy) is seen as weakness—she's shy, a wallflower. And, of course, the sexual aspect of this double standard is hard to miss as well. If we're not being "friendly," then we're supposedly not as open to sexual attention. (The horror!) I recall being at this awful bar in college called the Post—any SUNY Albany alumns out there know the place I'm talking about—where one night there was an impromptu wet T-shirt contest. (Hey, I never said I went to classy joints!) I was kind of horrified by the whole thing—the way the guys were crowding the women involved, the things they were yelling ("Take it off, slut!" and the like), and how the atmosphere in the room changed from jovial to . . . well, a little scary. I left the bar, only to take shit from my guy friends later—I was being too "serious," I was a killjoy. Now, like I've said before, I was a bit of party girl and always was down to have a good time. But the minute I was uncomfortable and didn't want to be a part of something that was supposed to be "fun," I was labeled as unable to have a good time. It's just too bad that "fun" always seems to involve women as entertainment. I also think that there are shades of misogyny in this double standard—I mean, perhaps the reason that a serious, quiet woman isn't liked much is that she has power. If women are happy and chipper and laughing all the time, then men don't have to take us as seriously. If we're quiet, or thoughtful, then men have to think of us in a serious way—as more than just entertainment value. But that's just one gal's opinion. But take the reactions to Senator Hillary Clinton's presidential run, for example. How many times have you heard her described as cold, frigid, or—of course—a bitch? That's also why you'll hear male pundits talk about Clinton's voice as "shrill" or "grating." There's nothing that annoys men like the sound of a woman in power! I think this has a tremendous amount to do with her being a serious woman. In fact, I think that some men (I'm talking to you, Chris Matthews!) are so freaked out by the idea of a serious woman with influence that they're almost amazed that such women exist. (I'm thinking back to when MSNBC's Matthews asked Senator Chris Dodd, "Do you find it difficult to debate a woman?") _**So... what to do?**_ Well, for the first time ever, I'm not going to tell you to "speak up"! We should be able to be quiet, reserved, serious, and even "no fun" if we want. We need to be able to turn our backs on the idea that a woman's job is to provide permanent fun and entertainment for the men in our lives (or in public spaces). It's time that we enjoyed our silence. **31** **HE'S COVERED, SHE'S SCREWED** **DID YOU KNOW THAT AS RECENTLY AS 2005,** rapists were getting their Viagra covered by Medicaid but rape victims across the country couldn't get emergency contraception? Apparently, choosing to not get pregnant through rape was just much more controversial than paying for a sex offender's hard-on. Sit on that for a while. Sure, it's an extreme example, but it highlights the ridiculous disparity between the sexes when it comes to health and medical care. Men are covered, women are fucked. The truth is, women have been getting the short end of the healthcare stick for a long time—mostly because men have been used as the standard of care. Let me elaborate: When doctors and scientists did research on things like heart attacks and cancer, they used male subjects in their studies. This meant that all the information they got about symptoms, how to treat patients, and so on, really related only to men. That's why certain diseases go underdiagnosed in women—and it's killing us. In fact, three million women potentially have heart disease that has gone undetected because the signs and symptoms are different for women than for men. This situation is even more dire for women of color, who are often ignored altogether. Another issue with women and health is that women require a lot more preventative care than men do—which isn't always covered by insurance and isn't always recommended to women. For example, women need regular Pap smears, mammograms, and osteoporosis tests—all preventative. But for some reason or another (cough, sexism, cough) women's health is often given the shaft by the government. Whether it's targeting women because of reproductive-justice issues like abortion and birth control, or just plain being greedy, women in the United States are struggling. In fact, there are sixteen million uninsured women in the United States (I'm one of them, in fact). That's insane. And let me tell you, a lot of folks just don't care. In 2007, conservatives and the religious right opposed legislation that would renew and expand funding for the State Children's Health Insurance Program (SCHIP). Now, this is a program to provide for kids, but the reason that the conservatives were all bent out of shape was that it helped out pregnant women. For real. You see, Democrats removed some anti-choice amendments that had the nerve to identify women as the beneficiaries of care while they're pregnant. Anti-choice conservative assholes on the Family Research Council released a statement saying: The new House bill changes the SCHIP program to cover health insurance for a "pregnant woman" rather than cover the child in the womb. This would undermine the "unborn child rule" and could possibly allow funding for abortions in those States that include abortion as part of their Medicaid health coverage for pregnant women. What they were really pissed about was the language change: Pregnant women shouldn't benefit, unborn children should. But this actually gets to the heart of the matter . . . a lot of this is about hating women, and hating us because we have The Sex. Seriously. That's why they don't want to repeal the Hyde Amendment—which maintains that Medicaid can't pay for abortions. (It's nice to punish low-income women!) That's why they don't care that birth control prices are soaring on college campuses. That's why it's cool with them that abstinence education—which has been proven dangerous, especially to women—gets funding despite its ineffectiveness. Shit, they'd rather that women get cancer— _cancer_ —than allow for the HPV vaccine to be legal. Before the HPV vaccine was approved by the FDA, a lot of folks were fighting tooth and nail to make sure that it wouldn't be. Why? Because it would make girls slutty. Just like they said emergency contraception would! (The logic was that if you gave preteen girls a shot that made sure they wouldn't get HPV, they would see it as a chance to go whore around. You know, just like how when you get a flu shot you go around looking for people to sneeze on you.) Never mind that 25 percent of women in America become infected with HPV, the STD that causes cervical cancer. More important to keep girls "pure." They don't care about our health, gals. They just don't. They do, however, care _very_ much what we do with our na-nas. Think of all the legislation concerning women's health that conservatives put forward: Most of it is about limiting our access to birth control and abortion, and making sure that we have as many babies as possible—if we're white. If we're not white, it's about making sure we have no babies, ever. Kind of interesting that a country that cares so little about whether we're insured, whether we're being treated correctly, and whether we have the care that we need would care _so_ much about whether or not we get laid. But so it is. _**So... what to do?**_ Make sure you're informed about women's healthcare, particularly preventative health. Check out organizations like SisterSong that fight for women's health. Keep abreast of reproductive health and justice issues—because that's where they're hitting us the hardest. And get your ass to a doctor on a regular basis. **32** **HE'S REPRESENTED, SHE'S A TOKEN** **IN THE 1990S, WAL-MART PULLED A SHIRT FROM ITS STORES** for fear that it was offensive. The "controversial" shirt featured a picture of Margaret—a character from the comic strip _Dennis the Menace_ —saying, "Someday a woman will be president." This is what they found so horrifying that it was removed from stores. I think that tells you a little something about where we are with women and politics. Perhaps no double standard between men and women is more in the public eye than the gender gap in politics. In the United States right now, women hold only 16 percent of the seats in Congress—and of those women, only 24 percent are women of color. The United States falls behind dozens of other countries in terms of women's political participation. Rwanda, Sweden, Finland, and Costa Rica are the top four countries with women in Parliament—we're at number sixty-eight. (Think of that number when someone is telling you how American women have nothing left to fight for and how feminism is irrelevant. Hmph.) And now that Sandra Day O'Connor has retired, Justice Ruth Bader Ginsburg is the _only_ woman left on the Supreme Court of the United States. When it comes to positions of power, women just don't have it. We're given scraps here and there, but for the most part, women are tokenized. Just take a look at Secretary of State Condoleezza Rice—she's the person most often cited when you start to talk about racism or sexism in politics. But she's a black woman and she's up there, so it all must be good, right? Not so much. Just because one woman, one person of color, one anyone, is put in a position of power, it hardly means that racism or sexism or classism doesn't exist. It means someone threw us a bone. That said, we are doing better than we have been . . . well, ever. The elections in 2006 had some huge wins for women. Dems took control of the House, and Nancy Pelosi became the first female Speaker of the House. While sixteen women in the Senate is still a low number, it's the highest we've ever had. Clearly, American voters are taking women politicians seriously. But the same can't be said for others. After the 2006 election, there was a lot of sexism-based resentment that can be described only as a "girls are icky" line of argument (whining). During MSNBC's election coverage, Chris Matthews said that Senator Clinton gave a "barn-burner speech, which is harder to give for a woman; it can grate on some men when they listen to it—fingernails on a blackboard." He also said that Pelosi will "have to do the good fight with the president over issues" and asked: "How does she do it without screaming? How does she do it without becoming grating?" Nothing like the sound of an uppity woman, huh, Chris? But the best dig came from President Bush himself, who said of Pelosi, "[I]n my first act of bipartisan outreach since the election, I shared with her the names of some Republican interior decorators who can help her pick out the new drapes in her new offices." Of course, taking sexist swipes at women is nothing new. But it seems that the higher in political rank women climb, the bigger babies some men become. (So grow up, boys. We're not going anywhere.) As I write this, someone is running who may very well become the first woman president of the United States. And while I'm not going to say who I'm supporting in the race, I have to admit that a woman winning the presidency would be amazing. And long fucking overdue. _**So... what to do?**_ Vote. Seriously, you'd better. Younger women's voices matter and can make a huge impact on elections. So get your ass out there. If you know a young woman who should run for public office (or if you should!), encourage her to get out there. The biggest reason women cite for _not_ running is that they think they're not qualified. Men who have the same qualifications, by the way, never think that they're unqualified. Help to put an end to the idea that women don't belong in politics. When you hear a sexist asshole like Chris Matthews say something vile, call in to let them know so. Same goes for your friends. Fighting those stereotypes starts at home, too. **33** **HE'S NEAT, SHE'S NEUROTIC** **WOMEN, IT SEEMS, HAVE A SPECIAL RELATIONSHIP** with dirt and cleanliness. Again, because of the vagina and all, we're supposed to have an innate ability to clean a room like nobody's business. We love it. It's what we were born to do. Men, on the other hand, are filthy, nasty creatures who would probably live in their own shit if women weren't there to tell them what to do. Or at least that's what we're supposed to believe. (I'm convinced it's all a clever scheme to keep women doing laundry for the next hundred or so years, but that's beside the point.) Women are still doing the bulk of domestic labor—cooking, cleaning, taking care of the kiddos, and so on. But it's the cleaning that's killing us. Dishes, dusting, vacuuming, toilet cleaning. Just . . . ugh. According to a 2002 study by the University of Michigan Institute for Social Research, American men do sixteen hours of housework a week, while women do twenty-seven. An even more recent study showed that married women do more housework than single or cohabiting women—apparently the act of getting married makes you all housewifey or something. (Single women do the least housework of all—just something to think about.) And according the Department of Labor, because of all the housework women are doing, they have significantly less leisure time than men do. So clearly, things are not all equal at home. And frankly, a lot of people would like to keep it that way. Women doing the majority of housework and domestic labor—and not being happy about it—is one of those things that bring out some of the funniest anti-woman theories. The argument that women are just naturally more inclined to clean because we can't stand dirtiness (clearly _someone_ has never seen my place) and men don't mind is the most common. And, of course, it comes from predictable places. Take, for example, anti-feminist blog Angry Harry: The reason that men do less housework than women is because, quite simply, they are less easily offended by any given level of mess. . . . Men just do not respond to the same low levels of untidiness as do women. And so whenever the female threshold is reached, and the good woman must whisk herself away to tackle the debris around the house, the man remains undisturbed, at peace, and contented with his surroundings. Angry Harry also has some interesting theories on why men don't like housework: If God had wanted men to do housework then he would have genetically programmed women to drool over men while they did it. . . . The sight of men doing the dishes would have made their G-spots zizz. . . . A woman who wants her man to do housework is unconsciously seeking a divorce. She has no feeling for him—as a man, that is. Charming! But when the argument that women just don't like dirt (and men do?) doesn't work, the powers that be pull out some doozies. Like when _The Washington Post_ ran an article headlined "Women's Liberation Through Housework." Seriously. And this was in 2007, not 1957. Reporter Rena Corey argued that she gets tremendous satisfaction out of keeping a clean home, her "little kingdom." That's all well and good, but what in the world does that have to do with women's "liberation"? Feministing blogger Ann Friedman responded better than I ever could: Rena might be satisfied to spend her adult life as the happy homemaker, but the vast majority of us are not. See, those of us who manage to part with our Swiffers long enough to venture outside for a paycheck know that, as Rena notes, there are indeed minute-to-minute unpleasant tasks in the work world. But they add up to a lot more than a sparkling toilet. They allow women to have influence in the public sphere—the world beyond the "little kingdom," where important decisions are made about the direction of society, and where money and power change hands. And that's the point, really. If we're more focused on housework and how clean our little kingdoms are, then we won't be in the public sphere making decisions that affect the world, not just our apartments. That's why I don't think it's any small coincidence that there has been a bevy of recent articles touting women doing housework. And the way they're pushing it is hilarious. Take this headline from the BBC: "Housework cuts breast cancer risk." Okay, then. What the results of the study _actually_ showed was that moderate, regular forms of exercise are effective in cutting cancer risks. But instead of pushing that, the media jumped on the dusting angle. Another headline says, "Women prefer housekeeping to love." Yet another says that married men will earn more money if their wives do the household chores. ("Clean that floor, honey—my salary depends on it!") **_So_... _what to do?_** There's a reason they're pushing the idea that women love to clean and men don't. In our happy little sexist world, things run much better when women are relegated to the home. It's even better when we believe that we _want_ to be there. So don't believe this nonsense. Better that your house goes undusted for a couple of weeks so you can make a difference somewhere other than your kitchen. **34** **HE'S FUN** , **SHE'S FRIVOLOUS** **SHOPPING IS VAIN.** Reading celebrity weeklies is for dummies. Lifetime? For losers. Watching sports and reading lad mags, however, is manly and fun. Golf is relaxing. When it comes to hobbies or having fun, men's interests are valid and women's are frivolous. Now, I'm not a fan of stereotyping certain behavior as female—assuming that all women shop or that men don't, for example. But there's no denying that particular activities are associated with specific genders, and that the ones associated with women are pretty readily dismissed—or straight-up hated. Men watching sports is considered manly, and it's a hobby! Same thing with hunting or reading _Playboy_ or doing other "masculine" things. It's _cool_ when a guy plays poker with his friends. But women shopping? Clearly vapid. Women's magazines? Ridiculous. Never mind that certain things—like shopping or paying attention to our looks—are demanded of us by society. So if we go shopping or get expensive haircuts, we're shallow. But if we don't, we're not "taking care" of ourselves or we're ugly or lazy. Just think about how much hostility and mocking is directed at "women's programming" like Oxygen or Lifetime (shit, even I make fun of them). Or how shows like _Bridezillas_ make fun of women for how shallow and obsessed they are over weddings—of course, that's yet _another_ thing we're taught that we must care about. We just can't win. Then, of course, there's hell to pay if men dare debase themselves by doing anything considered feminine. Conservative writer and wannabe pundit Debbie Schlussel, for example, devoted an entire blog post to how horrible it was that men would take up knitting. For real. (To give you some perspective, this is also a woman who said that any gal who gets a tattoo must be a whore, because "a woman who doesn't take long to agree to repeatedly put a needle in her body generally doesn't take long before she acquiesces to putting other things into her body.") In my ongoing examination of our society's largely successful attempt to feminize America's men (and masculinize the women), I've been watching the growing trend of knitting for boys and men. . . . The latest point on this downward decline of masculinity is the book _Knitting with Balls: A Hands-On Guide to Knitting for the Modern Man._ . . . One of my hometown newspapers raved about it over the weekend. I'm guessing the woman who wrote the review also likes her son to figure skate and is sending him to flight attendant school next. . . . If you're a scientist in a remote Antarctic camp of all men studying the extreme cold and you need a new scarf or hat, then knitting is okay. But, other than that, if you're a guy, don't scratch that itch. Knitting ain't very manly. Get that, guys? You should be off playing football or killing something, not doing "girlie" stuff. I mean, who in the world would want to be . . . feminine? And that's really the point. At the end of the day, when interests that are considered female are thought of as frivolous, _women themselves_ are seen as frivolous. It's _women_ who are silly, _women_ who are frivolous and not to be taken seriously. And that's where the danger in this double standard lies. **_So_...** _**what to do?**_ Of course, not all women like the same things. And it's bad enough that women are stereotyped as constashoppers with no interests outside of looking good and getting married. But the fact is, as long as we mock things that are considered inherently female, we're mocking ourselves. We're buying into the notion that the things women like are stupid just by virtue of being feminine. Which is an insult to all women. So enough with mocking Lifetime (sigh) and calling women shallow. Enough with letting guys off the hook for their frivolous hobbies—golf isn't serious! Let's have fun and stop apologizing for it. **35** **HE WALKS FREELY, SHE GETS HARASSED** **IT MUST BE NICE TO WALK AROUND THE STREETS** every day without being leered at, hit on, whistled at, or shouted about. But I wouldn't know anything about that—because I have a vagina. (Apparently, owning a vagina means that you have the pleasure of perfect strangers being able to say anything they want to you. Awesome!) Men can walk the streets freely and without being bothered, and they don't even realize it's a luxury. Women, on the other hand, have to steel themselves daily for whatever comments may come their way. Seriously—it's not a fun way to be. And while I've heard the argument that street harassment is actually a compliment—you know, because we're supposed to be _flattered_ that strange men are screaming at us about our asses—it's really a super-insidious form of sexism. Because not only do perfect strangers think that it's appropriate to be sexual toward any woman they want, but street harassment is also predicated on the idea that you're allowed to say anything to women that you want—anytime, anywhere. That's why the harassment that annoys me the most is the kind that's not overtly sexual. My biggest pet peeve? When a random guy on the street asks me, "Why aren't you smiling?" The assumption is that (a) because I'm a woman, I should be happy and smiling and accommodating looking at all times, and (b) he has the right to comment on my mood. I find it _infuriating,_ which is why I usually respond, "Because assholes like you make it impossible for me to walk to the subway in peace." But I digress. Street harassment is a big deal not just because it's fucking annoying, but also because it infringes upon the most basic right there is—the right to just _be_. To be left alone. To be in a public space without being bothered. I mean, is that so much to ask for? Apparently it is. Because the solutions that some cities are offering—taking women out of public spaces—completely miss the point. In Tokyo, 64 percent of women in their twenties and thirties reported being groped on the train or in transit stations. Instead of targeting the harassers, the city decided to make a separate train car for women. So did Rio de Janeiro, Moscow, and Cairo. Italy has even established a women-only beach for the same reason. I wrote about this trend in an article for the _Guardian:_ There's no doubt that . . . the idea of a safe space is compelling. This international trend—which often comes couched in paternalistic rhetoric about "protecting" women—raises questions of just how equal the sexes are if women's safety relies on us being separated. After all, shouldn't we be targeting the gropers and harassers? The onus should be on men to stop harassing women, not on women to escape them. . . . Betsy Eudey, director of gender studies at California State University, says that while some single-sex environments could be beneficial—locker rooms where people are expected to be naked are an obvious example—she finds that "segregated spaces only enhance division by sex, and prevent the necessary actions needed to make public spaces safe and welcoming to all." Who knows if that can happen, though; sexism has it so ingrained in men's minds (and even our own, sometimes) that women are there to be looked at, commented on, and grabbed that it's hard to imagine anything that would facilitate real change. _**So**_... _**what to do?**_ Fortunately, there are things we can do about this one. Myself, I like to call out street harassers on their bullshit (if it's not a desolate street and I feel safe). I ask them if they would talk that way to their mother or their sister. I ask them if they think it's okay to disrespect women. Or sometimes (okay, most times) I just give them the finger. There are also great websites like Hollaback that encourage women to take pictures of their harassers with their camera phones and send them in to the blog with the harassment story. (Their tagline is, "If you can't slap 'em, smack 'em!") Make those fuckers public knowledge! And men, please, for the love of all that is good and awesome, keep it to yourself. You want to talk to a woman you see on the street? Fine—approach her and _ask_ if you can speak with her. If she says no, listen to her. Walk away. It's really not that difficult. Oh, and to the guy who yelled at me on the street when I was fifteen and told me he wanted to eat his dinner off my ass? Fuck you, pervy. **36** **HE'S A PORN WATCHER, SHE'S THE SHOW** **WE ALREADY KNOW THERE'S A DOUBLE STANDARD** when it comes to having sex. But what about _watching_ sex? Pornography is everywhere, and mainstream culture is becoming increasingly "pornified." (I'm talking to you, Pussycat Dolls!) But when it comes to porn, is women's place only in front of the camera? Are we just the objects? I hate to say this, because as a feminist I know once you say something bad about porn, you're forever labeled anti-sex, but as it stands now, I think we are just the objects. Or at least that's what the porn industry would like us to be. Yes, there is feminist porn out there. Shit, I have friends who make it. But, like it or not, the _mainstream_ porn culture is increasingly male-centric (in terms of who the audience is) and increasingly misogynistic. Men are porn watchers, and we're the show. Now, that's a double standard if I ever heard one. We're sex embodied. What scares me about porn now is the way it's shaping men's views on women. (As if misogyny isn't already bad enough!) And yes, there's always been pornography and there always will be. But the mainstreaming of it—and the _kind_ of porn that's becoming popular—is just way too disturbing. Robert Jensen's book _Getting Off: Pornography and the End of Masculinity_ explores current porn culture in detail—something that's a lot less glamorous than what's often presented in the media. I don't have the stomach to type out some of the porn scenes that Jensen relays, but suffice it to say—they're horrible. They're about women being humiliated sexually, being treated violently, and having that be normalized. Jensen asks the obvious question: If pornography is increasingly cruel and degrading, why is it increasingly commonplace instead of more marginalized? How do we explain the simultaneous appearance of more, and increasingly more intense, ways to humiliate women sexually and the rising popularity of the films that present those activities? It's scary to think that misogyny could be on an upswing, but if you take a close look at popular porn, that's exactly the message you'll get. And while there are no comprehensive studies on the effect of porn on men, I think there is some anecdotal evidence that should make all women nervous. Take, for example, eight teenage boys in Australia who were given a slap on the wrist after sexually assaulting a seventeen-year-old girl, taping the assault, and distributing it as a porn movie. They filmed the victim being forced to perform oral sex, having her hair set on fire, and being spat and urinated on. Later, the boys distributed a DVD of the attack, which they titled _Cunt, the Movie._ You can't tell me this isn't related to porn—you just can't. Even the tamer effects of pornography—especially now that it's so easily available online—are detrimental not only to women but to men, too. I have so many male friends who are literally unable to be in a relationship because of the way that pornography has shaped how they think about women and sex. One friend described himself as "lonely," because he couldn't meet anyone who would live up to the women he watched in porn. But this isn't to say that feminists aren't working hard to carve out female-friendly space in the sex industry. Some feminists think that rather than hurting women, sex on the Internet could actually help create a more feminist porn culture. Audacia Ray, author of _Naked on the Internet_ and executive editor of _$pread,_ a magazine by and for sex workers, says that "women's agency tends to be totally overlooked or ignored when it comes to the Internet, but women can use the Internet to explore their voices and their agency (sex-related and otherwise) in an unprecedented way." She goes on to say: Once in the industry, the Internet becomes a powerful tool for connecting with other workers on both the professional level ("How do I make a website?") and the personal level ("Who else can understand and accept the work I do?"). . . . Most significantly, the Internet has enabled many sex workers to go independent, which means that they manage their own porn sites and escorting careers (among other professions) and do their marketing the way they want, operating without big companies, pimps, or agencies that would have claim to the cash that is rightfully theirs. It could also mean that women can create the kind of porn that appeals to them—in which we're marketed not as objects, but as real women with real desires. Rachel Kramer Bussell, sex writer (and friend), says, "I don't think there'll ever be one correct answer to the question 'What turns women on?' and bravo for that! 'Women' are not only a massive group, but what turns an individual woman on may change over time." **_So_... _what to do?_** I'm against censorship, and I certainly don't think that we can stop the porn industry that is making so much money off of degrading women. But I do think we can support feminist porn, and encourage others _not_ to watch or buy porn from film companies that make money by depicting humiliating and violent sex scenes. If you're interested in feminist or woman-centric porn, check out the Feminist Porn Awards or sex-toy shop Babes in Toyland. You can also look at Candida Royalle's Femme Productions, _Sweet Action_ magazine, or Violet Blue's _The Smart Girl's Guide to Porn_. **37** **HE'S HOT** _**AND**_ **HEADY, SHE'S BRAINY** _**OR**_ **BOOBILICIOUS** **WHEN I WAS GROWING UP, I WAS THE SMART ONE.** I had an awkward adolescence—bad skin, ears and a nose that I had yet to grow into—but damn, did I get good grades. My sister, on the other hand, never had a zit or a bad hair day in her life and still managed to get decent grades. Bitch. (Just kidding, Vanessa!) In a family full of dark-haired Italians, Vanessa's blond hair and green eyes made her stand out. I recall an older cousin saying to me when I was around twelve years old, "Yeah, you're the smart one and your sister is the pretty one!" Let's just say that stuck with me; I was incredibly jealous of my sister from that day forward, and pissed at her for being so damned . . . pretty. (Though I'm sure that, given the fact that Vanessa is brilliant, she wouldn't have been too pleased to hear that she wasn't the "smart one.") Of course, my sister and I weren't the only young women privy to the smart/pretty double standard: While men can be hot and smart, women don't have the luxury. We have only one of two choices: pretty or smart. You can't be both; it's too much of an anomaly, apparently. You see the stereotype everywhere—the dorky girl in movies who becomes pretty only after a serious makeover; the hot, boobilicious girl with not a brain in her head. We're caricatures—especially when it comes to younger women. Luckily, we're seeing more and more characters on television who defy this standard: Veronica Mars and the docs on _Grey's Anatomy,_ for example. And even though a tremendous number of movies still play on the stereotype (think any movie Anna Farris has ever been in), we're making progress there, too. _Legally Blonde,_ in which the main character goes from ditz to Harvard Law brainiac, and _Mean Girls,_ about the cruelty of high school girls, are helping to pave the way for the idea that women can be smart and (gasp!) pretty all at the same time. But we're still seeing the stupid/beautiful, smart/ugly model played out in "real life"—reality television and celebrity culture (okay, not _really_ real life, but you know what I mean). You need look no further than the likes of Jessica Simpson or Paris Hilton to see young women playing up the dumb angle in order to appease a public that wants nothing more from them than someone to simultaneously lust after and mock. And reality television isn't much better. You have gems like _Beauty and the Geek,_ which pits smart "geeky" men against "ditzy" women. Or VH1's 2007 show _America's Most Smartest Model,_ which gathers male and female models together to give them a series of tests (often humiliating) to figure out how smart they are. The men, who are of course depicted as more intelligent, are rarely commented on or made fun of in the same way the "dumb" women models are. But you don't even have to look at trash TV to see this double standard play out. When Katie Couric was hired by CBS frat-boy pundit Joe Scarborough, MSNBC's _Morning Joe_ featured former CBS anchor Dan Rather saying that the network had made a mistake in hiring Couric because they wanted "to try to bring _The Today Show_ ethos to the _Evening News,_ and to dumb it down, tart it up, in hopes of attracting a younger audience." Tart it up, huh? Dumb it down? Wonder if he would have used those words if CBS had hired a man. Something tells me no. Like so many of the other double standards, this one has more serious consequences than you would initially think. Young women who think that in order to be attractive they have to be dumb are deliberately not showing how smart they are or not spending nearly as much time developing their minds as they are developing their looks. HeyUGLY.com and _CollegeBounds CB Teen_ magazine did a survey to find out if young girls were pretending to be dumber than they actually were, for fear of scaring off boys: 35 percent of them admitted they were. Too depressing for words. But it's not exactly surprising. In a world that values young women for their looks and doesn't even believe that their smarts exist, it only makes sense that girls would work their hardest to look good over anything else. They're rewarded for looking a certain way. Being on the math team? Yeah, right. **_So_...** _**what to do?**_ Let's start rewarding girls for making smart choices and for being smart women. Make the math team cool again! (Okay, this may be wishful thinking on my part because I was, in fact, a math-team dork. Hi, Mr. Li!) Let's let young girls know that we can indeed look good _and_ be smart. That we don't have to forgo our brains in order to impress. And for the love of god, let's please stop watching reality television. Please?! **38** **HE'S AN ACTIVIST, SHE'S A PAIN IN THE ASS** **MEN WHO ARE POLITICAL—WHETHER THEY BE ACTIVISTS,** organizers, or politicians—are passionate, driven, intellectual. Women activists, however, are big fat pains in the ass. They're neurotic and annoying—they're not taken seriously. Unfortunately, this is one double standard I know all too much about. While men who work for change are revered and admired, women who do the same are often scoffed at, dismissed, or outright hated. And it's been happening for a long time. Take women who fought for the right to vote. If you take a look at any of the political cartoons from back in the day, you'll find a lot of caricatures of suffragists as old, ugly ladies who are beating men up. Seriously. (Those feminist stereotypes have been around a while!) Another asks, "When women vote, who will wear the pants?" Unfortunately, the negative reaction to women's activism wasn't just op-eds and political cartoons. In a 1913 protest organized by women suffragists, women were literally attacked (so much for treating us like ladies and all!): People spat at the marching women, and mobbed and beat them. The police, who were supposed to protect the marchers, did nothing. Over two hundred protestors were injured.Women like Alice Paul and Lucy Burns, who were imprisoned for their protests several years later, were beaten in prison along with other suffragists. Their poor treatment came to light later, but it's not often talked about. (The movie _Iron Jawed Angels_ is all about it; you should check it out.) Men who fought for such basic rights and put up with abuse are quoted throughout history books. We get a chapter, if we're lucky. Later in feminist history, women activists were not so much outwardly abused as they were ignored. Because, you know, women couldn't possibly have anything interesting or important to say. Susan Brownmiller, in the book _In Our Time,_ tells of a pivotal moment, in the beginning of the second-wave women's movement, that occurred at the 1968 National Convention for New Politics. Jo Freeman and Shulamith Firestone had drafted a resolution on women, which was to be met with an all-too-familiar pooh-poohing: Back at the main session, Jo ran down the aisles handing out copies of the resolution while Shulie charged to the podium. "Cool down, little girl," the session chairman told her. "We have more important things to talk about than women's problems." Brownmiller also discusses the reaction of her male counterparts after women marched in an anti-war demonstration with a float dedicated to women's rights: "The peace activists were appalled. . . . Stopping the Vietnam War was still the chief priority, wasn't it. . . . [This] action, they howled, was petty, disloyal, divisive." Nice, huh? I hate to say it, but women's voices in politics are treated much the same way today. When we bring up reproductive justice or racism, for example, we're focusing on "single-issue" politics. A recent kerfuffle in the repro rights sphere happened when several (male) activists and writers suggested that we just forget about _Roe v. Wade,_ the case that made abortion legal—because it was causing too much division. They argued that if we just left the abortion decision to the states, it would be all good. And what about the women in states where abortion was illegal? Too bad for them. So basically, the Daddy Dems wanted the little ladies to pipe down and let the men worry about what was best for them. (I find it interesting that when it comes to making concessions for the good of the party, women's rights always seem to be first to go.) This isn't to say that all men in politics ignore women's voices. I'd like to think that we have more allies today than we ever have. Blogger Scott Lemieux, from _Lawyers, Guns, and Money,_ weighed in on this particular controversy with considerable aplomb: Indeed, what is finally most intolerable about the new anti- _Roe_ consensus is just this willingness to throw the rights of others under the bus while patting oneself on the back for making noble compromises. It is certainly easy for men living in blue state urban centers—who know that no woman in their family or social circle will ever be denied a safe abortion—to casually dismiss the importance of the rights of poor women in the two dozen states at high risk of banning or severely restricting access to abortion in a post- _Roe_ world. Other men, like Scott, speak out on behalf of women every day. But it kind of sucks that it takes a man speaking about these issues in order for people to take it seriously. **_So_..**. _**what to do?**_ Make our voices heard as often, and as loudly, as possible. If you're an activist, don't let men dominate the conversation—speak up! **39** **HE'S A PERSON, SHE'S A COMMODITY** **GUYS CAN BE WHOEVER THEY WANT TO BE.** They can be funny or shy. They can curse or be super-polite. And they can be complex human beings with developed personalities. Now, of course, women can do that, too. (Perhaps better?) But while we do all the things men do and have all of the feelings and personal characteristics that they have, we'll always be one thing they aren't: a commodity. And I'm not just talking about prostitution or trafficking. I'm talking about the way that women's bodies are presented as objects every single fucking day, everywhere we go. You can't turn a street corner without seeing a woman being used to sell something (anything)—and really, what's being sold is her. It's us. Okay, to be straight-up . . . "commodification" is one of those words that get thrown around a lot in women's studies class, so I'm almost loath to use it so frequently when discussing this double standard. But I think it's important. The way that women are turned into commercial objects—for people to buy and dispose of when they want—contributes to a culture that thinks it's okay to do violence to women. Because we're not really people, we're objects. I knew that women were objectified in the media, in society, but I didn't realize the extent until I was in college and saw this amazing film, _Killing Us Softly,_ by Jean Kilbourne. The movie juxtaposes advertising images—which are disturbing enough on their own—with images of violence against women who are objectified, dehumanized, and used to sell things. Soon, the images of violence and the ads are so intermingled that you really can't tell which is which. In the version I saw (Kilbourne changes the film from time to time to update the ads), the movie ended with the horrifying rape scene from the Jodie Foster flick _The Accused_. Because it all comes together—how ads, movies, and music videos all contribute to the idea that women are less than, that we're not real people, that it's okay to hurt us. It's very overwhelming. And it's what makes people think it's okay to write articles like one that ran in _Forbes,_ a respected publication, about the economics of prostitution. It started off like this: "Wife or whore? The choice is that simple." Nice, huh? Wait—it gets better. [The researchers] admit that spouses and streetwalkers aren't exactly alike. Wives, in truth, are superior to whores in the economist's sense of being a good whose consumption increases as income rises—like fine wine. [W]ives and whores are—if not exactly like Coke and Pepsi—something akin to champagne and beer. Women, wives, "whores" are like Coke or Pepsi, champagne and beer. Just let that sink in a minute. It doesn't hurt to know, however, that the reporter for this piece also penned an article about how men shouldn't marry "career women." So that gives you a little insight into where he's coming from! Outside of ads and the way that women are talked about as if we're not real people, there are, of course, more obvious forms of commodifying women and women's bodies—like sex work. In this case, women's actual bodies are being bought and sold. While there are feminists who argue that it's not actually women's bodies but women's labor (sex) that's being bought, I think we'd all agree that prostitution _does_ give men the impression that women are for sale. Which is horrible. (That isn't to say I'm against sex work—I'm not.) Reproductively, we're also commodities. Women's eggs are bought and sold regularly, and a woman who wants to make some extra money can just be a surrogate—carry a baby for a woman who can afford to pay you. Of course, surrogates usually end up being low-income women or even women from other countries—way to commodify other women! I think the most important thing to remember is that the United States relies on women's commodification. Whether the idea manifests itself in slapping an anorexic model, or airing a slinky ad for perfume, or visiting strip clubs, or white women "renting" uteruses from women of color in "third-world" countries, women's bodies are for sale. **_So_...** _**what to do?**_ Questions about sex work, porn, and the like have been mulled over by so many different feminists, it's hard to know where to begin. Same with women-as-objects and how our bodies are commercialized. Some folks think that this is where feminism begins and ends. All I know is that it pays to be hyperaware of it, to be critical of the ways you see women being presented, and to think about your choices and how they affect other women. At least it's a start. **40** **HE'S A PUBLIC FIGURE, SHE'S A VIRGIN/ WHORE** **HOW MANY ARTICLES DO WE HAVE TO READ ABOUT** Lindsay Lohan's cooter? Or Britney's relationships? Or Paris's sex tape? I'm done! I don't want to see or hear about one more article, blog post, television segment, or water-cooler chat about women celebrities and their sexual escapades. I just can't. Not until, that is, I start seeing the same kind of coverage for male celebrities and public figures. I want a Denzel Washington sex tape. A Brad Pitt cock shot when he's getting out of a car. Gossip about what a whore and a shitty parent Kevin Federline is—okay, I get some of that already, but you see what I'm getting at. Men who are in the public eye can get away with just _being,_ without their sexuality being on display or talked about. Women celebrities, on the other hand, are made into virgins or whores by the public—which says a lot about what society thinks about women in general! We start sexualizing our female celebs early. I mean, how many Mary Kate and Ashley jailbait jokes did you hear before they were "legal"? There was even a website dedicated to a countdown to the twins' eighteenth birthday! 'Cause there's nothing sexier than an underage girl, I guess. So long as she's a virgin. Half the attraction to underage women celebs seems to be the idea that they're virginal. Think pre-disaster Britney Spears and pre-marriage Jessica Simpson. Same thing with the new media obsession with Hayden Panettiere, the seventeen-year-old star of the television show _Heroes_. (At a recent awards show, there were more jokes about her upcoming eighteenth birthday than anything else.) Though while we expect them to be virginal, we also expect them to appear as sexual as possible. Hypocrisy abounds! Once these young women get older, get married, have babies—in other words, can no longer be seen as innocent and virginal—the public tears them to shreds, mocks them, and calls them fat old sluts. There's no winning. What's particularly interesting to me is that feminism often takes the blame for this oversexualization in pop culture! Rather than blaming the society that demands this kind of impossible standard for women (be virginal but whorish), countless articles I've read conflate third-wave feminism with sexy pop culture. It's laughable. The idea is supposedly that since feminism sought to make men and women equal, women are now acting like, and having sex like, men. (You know, because women weren't sexualized at all before feminism.) But the truth is, this new level of sexualization is just a modernized take on the virgin/whore complex. Where, of course, men get off easy. And it's killing us. Look at the downfall of women like Britney Spears, so obviously in need of help, yet roundly mocked. It's time to end the insanity and start being compassionate. Blogger Courtney at A Feminist Response to Pop Culture writes of Spears: I think that her body politic is extraordinarily problematic especially since she is simultaneously marketed to young girls as an idol and to men as a masturbatory fantasy. But note how I write that "she is marketed" as if she is no longer an independent entity but a piece of public property. Not long ago one of my friends and I got into a debate about whether Spears chose this life path. My friend argued that she deserves what is happening to her because she chose to become a part of the public domain. But remember, she was but a child when she made that choice and she hardly could have anticipated the hyper-sexualization and invasion that would come along with that "choice." Further, does anyone really deserve that kind of dehumanization? I think we can agree—hell no, they don't. **_So... what to do?_** As with anything to do with celebs and the media, it's difficult to make an impact. But we can make a stink to editors of magazines and newspapers who continue to perpetuate the idea that women should be forever sexualized. Or we can just wait for and dream of that Brad Pitt cock shot. **41** **HE'S GOT G.I. JOE, SHE'S GOT BARBIE** **MY FAVORITE SHOW WHEN I WAS A KID WAS** _**SHE - RA.**_ My sister and I used to fight over who got to be the "princess of power." Like, physically. I think that little girls are always attracted to strong women and girl characters—probably because we're not given all that many to choose from. For me it was Anne of Green Gables and Ramona Quimby. Though it was a close call between them and Gem; I _so_ wanted to be a rock star. But as I got older, it seemed there were fewer and fewer female characters on television for me to identify with. (Which is probably part of the reason I spent so much time reading!) Men can choose any kind of character they want—they play everyone from CIA agents to doofy single guys trying to get the girl to cops to doctors. But now, when I look at the TV that teen women are watching—I have to say, I'm even more depressed. The female characters are _eh_ —not to mention that most of them are white. _Gossip Girl_ and _The O.C.;_ reality shows and ditzy women characters who want to shop more than anything else. . . . I mean, at least I had Buffy! Without a doubt, _Buffy the Vampire Slayer_ is probably one of the most feminist shows there is. Shit, women's studies departments even have small conferences on the show! There were fully developed, strong female protagonists; a main character was a lesbian; and the messages were undoubtedly about gender equality. I frigging _miss_ Buffy. Joss Whedon, creator of _Buffy_ and the master of writing awesome female characters, gave this amazing speech at an event for feminist organization Equality Now about how odd it is that so many people asked him why he wrote so many great roles for women—as if there needs to be a reason: Why aren't you asking a hundred other guys why they don't write strong women characters? I believe that what I am doing should not be remarked upon, let alone honored, and there are other people doing it. But, seriously, this question is ridiculous and you just gotta stop . . . because equality is not a concept. It's not something we should be striving for. It's a necessity. (And people actually wonder why women swoon over this guy?) What's also interesting about the current state of TV (or movies, for that matter) and capable female characters is that a lot of the more badass women characters out there come from comic books. Like Elektra, or the women on the NBC show _Heroes,_ which is supposed to be kind of a television-comic. Normally I'd have no issue with this—hell, I'll take good characters where I can get them!—but when you look at the actual comics ... ugh. Big breasts, small waists, overwhelmingly white—very standard, stereotypical male-fantasy stuff. So where can we look today for our new Buffys? Even beyond female characters, there's still a long way to go for feminism on television. Despite the fact that about 40 percent of American women will have an abortion sometime in their lifetime, you won't see abortion portrayed on TV. (They may _talk_ about abortion if a character gets pregnant, but she'll inevitably have a miscarriage and be grateful that she didn't have to make the decision. Convenient.) And when they do show something "controversial," it often spreads misinformation. On an episode of _Veronica Mars_ , for example, the writers confuse the morning-after pill (which prevents pregnancy) with a medical abortion (a pill that ends a pregnancy). I've seen the same mistake on _Law & Order_ as well.I. And when it comes to queer characters or people of color, we're also doing pretty shitty. Unless you're watching shows that are _about_ people of color or queer people—think _The L Word_ or _The Parkers—_ the norm is considered white and straight, and race and sexuality issues are rarely discussed. Don't you think we're a little beyond this shit? Though I must make a shout-out to _All My Children_ (don't laugh—I used to watch with my grandma!), which not only featured the first lesbian character and kiss on daytime television, but also had a transgender character who wasn't a hooker or psycho—unfortunately, a first. The show even worked with trans organizations to make sure that it was portraying the character in an accurate (and respectful) way. Clearly, we have a long way to go. While women are still being shown as vapid bimbos who would just like to shop all day, instead of as strong, complex, and _real_ women, we're in trouble. **_So_... _what to do?_** Don't watch or support shows that portray shitty women characters. And write letters to networks telling them you'd like to see stronger women on television. Thank networks when they get it right—maybe then we'd still have shows like Geena Davis's _Commander in Chief_ and even a new chapter in my beloved _Buffy_. A girl can hope! **42** **HE'S PAYING LESS, SHE'S PAYING MORE** **SO, WE KNOW THAT WOMEN EARN SUBSTANTIALLY LESS MONEY** than men, so it would make sense that we pay less for products and service, right? No such luck. Women pay more for haircuts, dry cleaning, cars, even mortgages. Why? Well, for no other reason than that we're women. It's a little something I like to call the vagina tax. You know why I like where I get my hair cut (Bumble and Bumble, represent!)? Because they don't split up prices by gender. But plenty of stylists do, despite the fact that hair length these days doesn't really have anything to do with a person's sex. This may seem like splitting hairs (apologies for the bad pun), but the truth of it is, women are paying a lot more for consumer services than we should. In fact, a 1994 study found that gender-biased pricing cost California women more than $1,300 a year—and almost $15 billion annually (!) for all women. That's no chump change! For example, a study out of Northwestern University found that white women paid more than $150 than white men for identical cars. And get this—black women paid $400 more than black men and $800 more than white men! A New York City study also showed that when women price used cars, 42 percent of dealers quote a higher price to women. Women also pay 20 percent more for haircuts and 25 percent more for laundering services. Frances Cerra Whittelsey, author of _Why Women Pay More: How to Avoid Marketplace Perils,_ says, "I found that women truly do pay more—and get less." Whittelsey puts much of the blame on sexism: "Women pay more for haircuts and dry cleaning because of 'traditional' pricing. . . . Women pay more for auto repairs and used cars and new cars . . . because the people who sell these services believe we are suckers and decided we are the ones on whom they can make their profit margins." Nice, huh? But the disparity doesn't stop at everyday, seemingly shallow costs like clothing and beauty needs. In 2007, _The New York Times_ reported that despite women's having better credit scores than men, they pay more for mortgages: [A] study released last month by the Consumer Federation of America, a nonprofit advocacy group, [shows] that women are 32 percent more likely to carry mortgages with high interest rates than men with similar incomes. And wealthier women were 50 percent more likely to carry expensive loans than their male counterparts. . . . In 2005, according to the study, 10 percent of women who took out mortgages received the highest-cost subprime loans, compared with about 7.5 percent of men. Another study from Harvard Medical School researchers showed that women pay more for health coverage under high-deductible health insurance plans. The study found that men spend less than $500 per year on medical deductibles, while women spend more than $1,200, and that only a third of men insured by a high-deductible plan spend over $1,050 per year in medical costs, while 55 percent of women do. Steffie Woolhandler, lead author of the study, says, "High-deductible plans punish women for having breasts and uteruses and having babies. When an employer switches all his employees into a consumer-driven health plan, it's the same as giving all the women a $1,000 pay cut, on average, because women on average have $1,000 more in health costs than men." (This is because, like I mentioned in discussing the healthcare double standard, women need more preventative care than men—like Pap smears, mammograms, birth control, and so on.) Insane. Sexism is robbing us blind, ladies! **_So... what to do?_** Thankfully, people are doing something about the vagina tax. (Sorry, I just love that name.) For example, California created the 1995 Gender-Tax Repeal Act, which makes it illegal to discriminate based on gender when it comes to the pricing of services. Similar legislation was passed in Florida's Miami-Dade County, New York City, Pennsylvania, and Massachusetts. And you can take action, too—if you see a gender-based pricing difference at your dry cleaner, hairstylist, or anywhere else, speak up. Bring it up in your community. Because while paying more for a haircut may not seem like the biggest deal in the world, it's adding up—and women just can't afford sexism everywhere in our lives! **43** **HE'S PUSSY WHIPPED, SHE'S A "GOOD GIRLFRIEND"** **WHY IS IT THAT WHEN A GUY DOES THINGS THAT ARE NICE** or considerate for the girl he's involved with, he's called "pussy whipped," but the same behavior is just expected from women? We're _supposed_ to care (a lot) about what men think about us. Just take a gander at the cover of any women's magazine out there and you'll know what I mean: See what he thinks of your breasts! Does he really love you? When will he pop the question?! Ugh. When it comes to (straight) love, men are whipped, but women are . . . women. Urban Dictionary defines "pussy whipped" as a "situation whereupon a male is undeniably at the mercy of his high-maintenance girlfriend & answers to her every beck and call, usually followed by the reprioritizing of girlfriend over friends, family, school, food, water, and air." Replace that with "female" and "boyfriend" and tell me if it sounds odd. . . . Nope, it doesn't. Because that's what women are expected to do for their male partners every day. (Fun fact about "pussy whipped": Besides being a gross term, it's also the name of Bikini Kill's debut album. Love Kathleen Hanna. _Love_.) From the time we're little girls we're taught that our end goal in life, basically, is to get a man. (Never mind if we're gay—find a man anyway!) From playing house as kids to devouring teen mags to becoming Bridezillas—our main focus is supposed to be relationships. And, by proxy, men. We're supposed to care what they think of the way we look, how we talk, walk, act. Frankly, it's fucking exhausting. Not only does the excessive caring about what men think mean that women are more likely to undergo ridiculous amounts of work to "improve" the way we look—it also means that we don't always do what's best for us. We may not speak out at work for fear of what someone will think of us. We may not speak up in a relationship, or during sex, because we've been taught to be accommodating. We may forgo job opportunities, travel opportunities, friendship opportunities—it's all just too much. And frankly, it could be dangerous. There are scores of stories about women who ended up being assaulted because when a strange guy approached them, they were too afraid of being considered rude to tell him to fuck off. Or to listen to their instincts that said something was off. I remember a story we read in one of my first-ever women's studies classes, about a woman who was walking up to her apartment with groceries when a couple of things fell out of her hand. A man happened to be in her hallway and offered to help—he picked up the groceries and said, "Open the door; I'll carry them in for you." Despite being uncomfortable, she let him into her apartment because she didn't want him to think that she was rude or—get this—that he was some sort of rapist or something. He ended up attacking and raping her. Obviously, this is an extreme example, but I think it's worth noting that there can be scary consequences for being taught to be "nice" all the time. Men, however, are actively taught that if they care _at all_ what women think of them, they're some sort of softy, a "pussy." (Nothing worse than being a girl, remember?) My college boyfriend Mike—who was just this amazing person—was routinely mocked and criticized by his friends because he was nice to me. Just nice. He wasn't letting me walk all over him and he wasn't a saint—but just by being a pretty decent guy he was given shit and labeled "whipped." This double standard not only makes day-to-day life kind of miserable, but it also reinforces gender norms—the idea that women should be the nice, accommodating ones and that men are supposed to be "tough" and not give a shit what their girlfriend, or any other woman, thinks. Now, there may be gals out there who don't give a flying fuck what men, potential dates, whoever, think of you. Awesome. You rock. (I'd certainly like to think that I'm in that group of nonchalant ladies.) And I'm certainly not writing this in order to paint women as sad sacks who are overly concerned with other people. I bring it up because this is the shit that is shoved in our faces from the time we're kids. And like it or not, it's hard to avoid and even harder not to fall for. But we can try. **_So_** ... _**what to do?**_ Stop obsessing about what other people (male _or_ female, for that matter) think of you. It's a waste of time and energy. Stop buying magazines that presuppose your biggest concern in life is landing a man. Start doing things for yourself—stuff that makes you happy and successful, not anyone else. And of course, most important, have fun. (Which, trust me, is a lot easier to do when you're not forever worried about what a guy thinks.) **44** **HE'S PROTECTED, SHE'S PROPERTY** **WHETHER THEY' RE ABOUT ABORTION, RAPE, OR SOMETHING** more innocuous—like vibrators—there are rulings, laws, and legislation still around that embody double standards and hypocrisy. These are just a few. _Rape . . ._ One would hope that the days of blaming the victim and equivocating about what constitutes rape are long gone. But even now, thirty years after feminists fought to bring national attention to the rape epidemic, women are still faced with ridiculous qualifiers when it comes to "proving" sexual assault. In 2006, a Maryland appellate court ruled that once a woman consents to sex, she can't change her mind. Not if it hurts, not if her partner has become violent, not if she simply wants to stop. You may be scratching your heads right now—after all, who would continue to have sex with an unwilling partner _besides_ a rapist? It doesn't take a genius to know that no _always_ means no—no matter when it's said. But it seems that reason and rationality have no place in Maryland. Or take the nineteen-year-old Howard University student who, after being drugged and sodomized, was denied treatment at local hospitals because she "appeared intoxicated"—not so surprising, given the nature of her attack. Even when the teen went to police for help, she was outright dismissed. Sergeant Ronald Reid of the MPD Sex Assault Unit has been quoted as saying, "[I]f we don't have reason to believe a crime happened, we wouldn't administer a rape kit." Apparently, intoxicated women can't be assaulted. One of my favorites: A judge in Nebraska banned the word "rape"—at a rape trial. The victim couldn't even say "assault." The only word she was allowed to say at the trial of the alleged attacker? "Sex." The judge argued that using the word "rape" would be too prejudicial. Shockingly, at robbery trials no one is banned from using the word "mugging." (Thankfully, the victim was having none of it: "I refuse to call it sex, or any other word that I'm supposed to say, encouraged to say, on the stand, because to me that's committing perjury. What happened to me was rape; it was not sex.") Another woman in Massachusetts was told she wasn't really raped—she was defrauded. She went to bed one night, in the bedroom she shared with her boyfriend, when a guy she thought was her boyfriend got into her bed and had sex with her. Turns out it was her boyfriend's brother pretending to be her lover. But somehow, that's not rape.The court said that Massachusetts law defines rape as intercourse "by force and against [the] will" of the victim and that "fraudulently obtaining consent to sexual intercourse does not constitute rape as defined in our statute." Did you know that if you are raped while you are sedated, you don't need to know? A ruling in Oregon came down saying that rape victims who are unaware they were raped shouldn't be informed. (This case comes from the trial of a doctor who assaulted women when they were out of it right before surgery.) _Pay_ . . . There's plenty of discrimination when it comes to pay equity, but I had to mention this one because it was a Supreme Court decision. The court's ruling in _Ledbetter v. Goodyear_ says that employees must make their discrimination complaints within 180 days "after the alleged unlawful employment practice occurred." So if a woman doesn't realize that she's being paid less than her male coworkers within 180 days, too bad. _Fun . . ._ Vibrators are still illegal to buy and sell in eight states. While these seem like they would be old laws, just last year an Alabama court upheld the ban on vibrators, saying the law wasn't unconstitutional because selling sex toys is like "prostitution." (With yourself?!) _Life . . ._ A town in Missouri bans people from living together who aren't related by "blood, marriage, or adoption." So no cohabiting for all you sinners! The law received attention after a couple was denied an occupancy permit in the town because the woman's partner wasn't the father of one of her three children. Charming. _Violence . . ._ An Ohio man who beat up his girlfriend had his conviction voided because he wasn't married to her—the ruling said that domestic violence can happen only within marriages. So if the girlfriend wanted to press charges? She'd have to marry her abuser. Now, I could go on and on—there are laws on the books and rulings out there that you wouldn't believe. I really just wanted to bring these up to highlight how fucked up things still are on so many levels. And how much work we still have to do. **_So..._** _**what to do?**_ I don't know, dude. Move? **45** **HE'S FONDLING, SHE'S FEEDING** **I LIKE MY BOOBS, ALWAYS HAVE.** While plenty of less-than-polite comments have been levied against the girls over the years (especially since I started blogging, wow), I continue to hold my ta-tas in high esteem. And though the idea of a baby having its way with them doesn't exactly fill me with eager anticipation, I've resigned myself to the idea that that's what they're there for—so all will be well when I do have a kid. The boob double standard doesn't involve a male equivalent so much as it does men's viewpoint that breasts belong to _them,_ are there for them to look at, ogle, suck on, what have you. So when you do anything that reminds guys what breasts are really for—you know, like feeding babies—they get all ornery. Problem is, when you _don't_ breastfeed, you're accused of not being a good mom. So you can't win either way. I hate to be the one to say it . . . but I will. Boobs are not for boys. Sure, guys, you can get them on loan—but they don't belong to you. They belong to us. You can make as many boobie-related novelty products as you want (blogger Shakespeare's Sister found over 150—including boobie shampoo dispensers and pencil erasers!), and you can slap a fake pair on every lad-mag cover you want. It doesn't change the basic fact that boobs are future baby food. One of my fave boobie moments—because it was the most telling—came from political commentator Bill Maher, who went on a tirade against public breastfeeding. In discussing a protest that women held after a breastfeeding mother was kicked out of a restaurant, Maher went off. He fell back on some predictable quips ("They say it's natural—so is masturbating, but I generally don't do that at Applebee's!"), but it was the jokes-that-aren't-jokes that were truly insulting: "Look, there's no principle at work here other than being too lazy to either plan ahead or cover up. . . . It's not fighting for a right, it's fighting for the spotlight." Now, I don't have a kid, so when I wrote about this on Feministing, I relied on Kelly Mills at baby blog Babble to take him down: There he totally hit the nail on the head, didn't he? I mean, I had no desire to actually get out of the house or anything when I ate in restaurants; I just wanted a little attention. In fact, that's why I chose to feed my baby with my exposed tits in the first place. I mean, yes, it's recommended by every doctor ever and it's good for the kid, but of course that was secondary to my desire to have total strangers jump in my face and say, "Good job on the procreation!" Why, I know that when I walked into restaurants people looked thrilled to see me and my infant, and I'd hoist her onto my shoulders, whip out my boobs, and say, "Gimme some sugar, folks!" Amazing. Another boobs-for-boys-only jerk was columnist Rabbi Shmuley Boteach, who gave advice to married women not to breastfeed in front of their husbands (lest the men be turned off by lactating): The erotic nature of a wife's body is one of the principal elements of attraction in marriage. When a husband ceases to see his wife as a woman, and begins to see her as "the mother of his children," a negative trend has begun in his mind that can only subvert his erotic interest. Right, 'cause who would want to fuck the mother of his children? Grody. I have a feeling _this_ is what's at work with Maher, Rabbi Boteach, and a lot of other men who take issue with public breastfeeding. They resent that a woman's public body—her exposed or partially exposed breast—could be there for someone other than them, for something other than sexual consumption. After all, if a woman is exposed in public, it's supposed to be because she's flashing her tits for beads or taking money in a G-string—not for feeding babies. That's not sexually arousing, and therefore it's unacceptable. But god forbid a woman _doesn't_ breastfeed her kids—then she's a bad mother! A lot of mothers have a difficult time breastfeeding—it can be a painful and long process. (So I hear.) But in the day of super-mom syndrome, whipping out the formula instead of the titty can seem like a failure. Especially in a society that demands so much of mothers: Breastfeed, but don't do it in public and offend onlooking men! **_So... what to do?_** This is a hard one. It's not like we're ever going to convince straight guys to stop talking about or looking at our titties. But I think groups that call themselves "lactivists" have the right attitude. They're fighting all across the United States to make sure that women have the right to breastfeed in public, and they're spreading information about breastfeeding as well. Also, never forget that no matter how many advertisements, magazine covers, or assholes make you feel like the girls are some sort of public commodity—they're not. **46** **HE'S CHILDLESS, SHE'S SELFISH** **SINGLE MOMS IN THE UNITED STATES ARE PORTRAYED** as a blight on society—selfish, irresponsible women who aren't doing the right thing by their kids. (And by "right thing," of course, we mean being married.) Single dads, however, are heroes—men who are victims of an irresponsible mother who left, or perhaps widowers; they're men who have picked up the slack and done the impossible . . . a woman's job. In 2005, nearly 1.5 million babies were born to unmarried women, with women in their twenties accounting for a good portion of them. The National Center for Health Statistics reported that 35.7 percent of all births were to unmarried women—55 percent of the births for mothers in their early twenties were to unmarried women; for women in their late twenties it was almost 28 percent. Those are pretty big numbers. (Though the study didn't take into account whether or not the women were cohabiting with partners, or what their sexuality was.) As someone who was never really sure about getting married, but absolutely sure about having kids, I have to say that I've considered single motherhood. (Though I figure I have another few years before I start worrying about it, despite all the scare tactics telling me that my eggs are _dying_ by the minute.) While I'd rather have a partner, I also would like to be a younger mom. So what's holding me back? I can admit it—the stigma against single mothers scares me. We already know that there's a double standard when it comes to parenting, but when we're talking about single parenting, the judgment is even harsher. Take Louise Sloan, for example. Author of _Knock Yourself Up: A Tell-All Guide to Becoming a Single Mom,_ Sloan tells her own story of getting pregnant via artificial insemination when she was forty-one years old, as well as the stories of other single gals. After Salon.com ran an interview with her, the vitriol she got in the Letters section was horrible. [T]he boy will be screwed up or resent women, not having had a father around. he will have a higher chance of being a criminal. he will likely understand that all the feminist piffle shoved in his head is the opposite of what men need to know to be EFFECTIVE and happy free agents in the bigger world. Your child will grow up fatherless and disadvantaged. But you got what you want, and that is what is most important. How sad. And those are just a couple; there were a ton more letters calling Sloan selfish and saying that her son will grow up to be dysfunctional. There's just something about a single mom by choice that really pisses people off. So . . . predictable. Sloan gets off a lot easier, though, than single mothers of color. Woo, boy, does the American public ever love to hate single moms who aren't white. They must be welfare queens or irresponsible moms who pop kids out by the dozen. There is no end to the racism/classism/ sexism matrix here! That's why single moms also get hit with the work conundrum. While we hear all of this media frenzy about women opting out of the workforce to become stay-at-home moms, single moms _have_ to work. While the rate of women working outside the home has pretty much leveled off for most groups of women, it's actually jumped for single mothers—from 63 percent to 75 percent. Single moms are also more likely to be in poverty. But for all the shit that single moms take—people love single dads! I think I've seen three Lifetime movies in the last month about heroic single dads who take care of their kids after a shitty mom leaves. (Uh, not that I watch Lifetime at all . . . I swear.) Not. Fair. **_So ..._** _**what to do?**_ Have your families the way you want—and don't let anyone give you shit for it. And if you hear people spreading anti-single mom myths, call them out on it. **47** **HE'S FUNNY, SHE'S ANNOYING** **IF I HEAR ONE MORE PERSON SAY WOMEN AREN'T FUNNY,** I may just lose it. Because, fuck you, _I'm_ funny. And also, you know, because of the sexism. When men tell jokes, they're funny. Women comedians? Annoying wenches, it seems. Why is it that women comedians get the shit end of the stick so often? Or that women in general are just assumed to be the comedically challenged gender? The most controversial (and asinine) article to come out recently on women and humor was by Christopher Hitchens in the January 2007 issue of _Vanity Fair_. The oh-so-charming title? "Why Women Aren't Funny." The headline pretty much says it all, but there's more. Hitchens' basic argument was that women aren't funny because they don't _have_ to be. Men, he says, try to be funny to get women to like them. Women have sex appeal to attract the opposite sex, so there's no real reason for them to be funny. In addition (and this beats all), apparently women aren't funny because we have the babies: For women, reproduction is, if not the only thing, certainly the main thing. Apart from giving them a very different attitude to filth and embarrassment, it also imbues them with the kind of seriousness and solemnity at which men can only goggle. See, and here I thought stuff coming out of vaginas was supposed to be hilarious. Thank goodness I have someone like Hitchens to set my feeble female mind straight. But it's not just douches like Hitchens who are perpetuating the women-aren't-funny myth. Canadian psychologist Eric Bressler did research on humor and claims that "women want a man who is a humor 'generator,' while men seek a humor 'appreciator. '" Meaning we're supposed to laugh at guys' jokes but not tell any ourselves. Another researcher mentioned in the same article goes even further. Don Nilsen, a linguistics professor at Arizona State University and a self-proclaimed expert on humor, says that men are turned off by funny women: "I think every man in the world loves the humor, even the sexual put-down humor, of Judy Tenuta or Joan Rivers. . . . But very few men want to marry them." I'll be sure to tell my boyfriend, who laughs his ass off at my jokes, that he's actually mistaken. He doesn't like me at all. Actually, the best response to this "science" was from blogger Melissa McEwan, better known as Shakespeare's Sister: "To which men, exactly, is that sense of humor a turnoff? Oh, yeah—the kind of men no woman with a wicked sense of humor gives a diddly shit about." Exactly. Female comedians aren't taking this only-men-are-funny nonsense sitting down. Janeane Garofalo (who I met once and was sooo nice) has said, "Funny transcends gender. . . . The best comics regardless of gender are more detail oriented, good social critics, and can laugh at themselves. And hacks are unfunny in the same way." But while funny may transcend gender, people aren't always comfortable with humor _about_ gender. Julia Sweeney, who played the gender-ambiguous Pat on _Saturday Night Live,_ was quoted as saying that on the show women "were almost accused of having a victim 'agenda' if they brought out scenes that addressed sexism." Ick. Oh, and the best comedian response to Hitchens that I've seen? Writer Jill Soloway and a group of her comedian friends started having "Fuck Christopher Hitchens" events featuring bands, booze, and all sorts of funny women. So there. If it were only the stereotype that was being perpetuated, I might be able to deal. But funny women are feared—and mocked—all too often for me to not say anything about it. Think about the vitriol that Roseanne Barr, Ellen DeGeneres, and Rosie O'Donnell—all successful funny ladies—get. These comedians get called "ugly" on the regs, but what I find most interesting is that they're often called "grating" or "annoying." Kind of relates to the idea that women shouldn't be loud—funny women are breaking with tradition, so people (men) find them intimidating. Let's face it, funny is powerful. (Just another reason people love to call feminists "humorless." Little do they know!) In an article on women and humor on AlterNet, reporter Emily Wilson interviews stand-up comic Beck Krefting, whose dissertation at the University of Maryland was about women and comedy: "It's okay for guys to crack jokes and be the class clown, but if a girl did it, she was marked the strange one. . . . That was true in elementary school and high school and then on the stage." **_So ..._** _**what to do?**_ Be funny. Encourage your hilarious friends to go and try some stand-up comedy. And for the love of all things feminist, don't date anyone who thinks funny women are a "turnoff." Okay, just one date—but only if you go wearing a clown nose or throw a pie in his face or something. Oh, and for good measure, here's my favorite joke of the moment: What do parsley and pubic hair have in common? You push them both to the side before you eat. **48** **HE'S DATING, SHE'S TAKEN** **WHEN I WAS IN HIGH SCHOOL,** I had a boyfriend who was a tad possessive. He would get jealous if I spoke to other guys—because I was "his"—and for Valentine's Day he got me the oh-so-thoughtful gift of a beeper, so he could get in touch with me constantly. Keeping tabs is _très_ romantic, didn't you know? I bring this up not to make you all cringe with embarrassment for me (though you should—I sported that ugly-ass beeper every day for a year), but because it's just a small example of how women are marked as "taken." Whether we're attached or not—single, dating, engaged, married—straight women are subject to a ton more qualifiers than men when it comes to sharing our relationship status. _The ring_ . . . Nothing says ownership like a brand-spanking-new, über-expensive engagement ring! I've written about engagement rings before on Feministing and in _Full Frontal Feminism_ —I mean, how could I not?—but it seems that there's always something new to talk about when it comes to rings. They're the ultimate mark of a taken woman—and something that men aren't expected to wear in return. (Why not just pee on her to mark your territory? I say. Some gals are into it.) Meghan O'Rourke of _Slate_ doesn't pull any punches when it comes to the all-controversial ring: But there's a powerful case to be made that in an age of equitable marriage the engagement ring is an outmoded commodity— _starting with the obvious fact that only the woman gets one_ [emphasis mine] _._ The diamond ring is the site of retrograde fantasies about gender roles. It's always been the consumerism behind engagement rings that bothers me most. As if you can't really be in love without spending a substantial sum of cash. I guess it's just always struck me as . . . well, unromantic. But as O'Rourke points out, there is something just wrong about the fact that only a woman is visibly marked as engaged. It reeks of a "Woman mine!" caveman mentality, but we've romanticized it to the hilt. (This isn't to say I'm against rings altogether. When my friend Lauryn got engaged, her boy bought her this amazing art deco sapphire ring that he spent forever looking for, and he made a little book about the ring's history and how he came to find it. It was from the heart, not the wallet. Though perhaps she should have gotten him something to mark his ass as taken, too!) _Miss, Ms., Mrs., oh my!_ . . . Once we've gotten married (that is, for those of us it's legal for), we have the privilege of three honorifics, rather than just the one that males get. You would think the more choices the better, but all these prefixes do is force women into deciding how much we want people to know about our marital status. It's just gross. Men's titles have absolutely _nothing_ to do with whether or not they're hitched, yet we have to come clean? Of course, we do have the lovely "Ms.," thanks to '70s feminism that found it ridiculous that women should have to reveal their marital status within their name. I definitely use it—not just because of the sexist connotations of the other choices, but because "Miss" makes me think of an eight-year-old, and "Mrs." makes me think of all the bitch teachers I had. (Who the hell else do you call "Mrs. So-and-so"?) Of course, "Ms." doesn't come without its problems as well. Whenever I've corrected someone and asked them to call me _Ms._ Valenti, I've gotten the _Oh, you're one of those_ look or some sort of jokey/snide comment. It's a real pain in the ass. _His last name_ . . . We've already gone over this nonsense, but I just wanted to remind folks—just one more way we're marked as taken and he gets off with, well, his actual name. Lucky him. _Puppy love_ . . . I'm not sure how pervasive this trend is—but these T-shirts that preteens and teens are wearing saying shit like "I love my boyfriend" and such irritate me. Perhaps I'm being picky, but it strikes me as odd and somewhat controlling. _The unmarked single woman_ . . . Ah, how nice it is to be unencumbered by rings, names, and dumb shirts. Though I think all you single gals out there know what I'm talking about when I say that I'm kind of sick and tired of people assuming that because I'm single they have the right to ask me all sorts of relationship questions. _Why_ are you single? _When_ are you getting married? Ugh. Can't a gal just enjoy her alone, unmarked time? **_So... what to do?_** I say, stay mum on your relationship status when it comes to physical markings like rings and titles. But that's just me. It just seems ridiculous that women should have to brand themselves according to whether they're attached, while men can do whatever they want without anyone giving it a second thought.Though perhaps now, in the age of Facebook and MySpace, where everyone can publicly display their relationship status with "single," "dating," "married," "in a relationship," or the ever-dreaded "it's complicated," there will be some sort of relationship-marking equality. (Though I still resent finding out about an ex's new relationship via Facebook feed. Harsh.) **49** **HE'S RUGGED, SHE'S RUDE** **IF YOU' VE EVER READ MY WRITING BEFORE,** then you know I have a bit of a potty mouth. Okay, I fucking curse. A lot. Between being raised Italian and being raised in Queens, I never stood a chance! Also, I must admit, my manners were never all that fantastic. I was the awkward girl who was always being told to sit up straight, stop talking with my mouth full, and for god's sake stop running around with skinned knees through my dirty stockings. (Hey, I was a tomboy, but I liked me some tights!) I wasn't so annoyed as a kid by the idea that I should act a little less nuts—but I was constantly irked that my male peers weren't told the same thing. You know, because boys will be boys and all that jazz. Anything less than prim, "ladylike" behavior from women is considered rude, yet for men it's just pure testosterone and manliness. I call bullshit. Why is it that etiquette really refers only to women? (Politeness for boys is more about chivalry than anything else—which, as we know, is pretty mired in sexism itself!) Not that etiquette handbooks are super popular anymore with younger women, but the sheer number of guides and manuals on how to be a proper "lady" is staggering. Seriously, just check out Amazon and type in "etiquette"—you'll be shocked. Now comes my disclaimer: I'm not advocating rude behavior or saying that being polite is anti-feminist. I'd like to think that I'm a courteous person (a loud, opinionated, courteous person, but still). But the way that we define "rude" when it comes to women seems more than a little problematic to me—it requires that we're more quiet than we need to be, more accommodating than we need to be, and less of ourselves than we should be. For example, it seems like way too much advice aimed at women is of the "suck it up" variety. Take Slate's advice column, "Dear Prudence." In 2005, Prudence got a letter from a woman who was constantly having to clean up after her boyfriend and was "burned out being the only one to clean the house." Prudie's advice? Just deal. It is sometimes easier to pick up the guy's socks than to make continual "requests." Given that he is slothful and chaotic around the house (and may also have retro ideas about men and women), it might be easier on you to bear in mind what a great guy you have while you pick up his socks. See, ladies, if you just think happy thoughts (whistle while you work!) while you pick up after your man, it's a lot "easier." Ugh. What's also particularly annoying to me about the politeness double standard is that it's just steeped in racism and classism. When I was mocked for my Queens accent in my Manhattan school, or when I saw my girlfriends of color get chastised for being too "loud," I knew that it wasn't just about decorum. It's about an image of ideal womanhood that's not only quiet and subservient—but also white and upper class. There's also a sexual aspect to politeness that should make all women uncomfortable. In much the same way that the virgin/ whore complex creates "nice" girls or "naughty" girls, etiquette as it is now sees sexuality as, well . . . rude! When I did an article about the "modesty movement" in 2006—basically a regressive group of gals who'd like to see women back in the home and forever virginal—the connection was clear. One site sells the ModesTee, a black leotard meant to be worn underneath less "appropriate" clothing. It is touted as "a fashionable solution to dressing modestly by turning the clothes that may be a little too sheer, too short, or too low into clothes you can wear." Another company, WholesomeWear, sells modest swimwear. This layered—yes, layered—swimsuit is made up of spandex and nylon and covers most of the body. A bit like a waterproof kaftan. . . . But being modest does not end at your wardrobe. Alexandra Foley, a thirty-four-year-old mother of four who blogs at Modestly Yours, says: "Modesty is both your outward appearance and your interior disposition. A woman can be modestly dressed, but not carry herself in a modest way." Get that? The way you dress reveals what your "interior disposition" is. 'Cause short skirts are just . . . rude. So what is this really about? Why is it so important that women act "polite"? Because if a docile, quiet woman is the ideal, that means we can be shamed for speaking up. There's a reason feminists are often called "too loud" or "rude"—it's a silencing strategy. **_So..._** _**what to do?**_ Be as loud as you want to be. (I certainly am.) Be courteous, be nice, be polite—but do it in a way that doesn't infringe on the person you are. And if people tell you that you're too opinionated or not "ladylike" enough, tell them to go fuck themselves. **50** **HE'S PLAIN, SHE'S VAIN** **FOR A SOCIETY THAT TELLS WOMEN** that in order to be beautiful we have to be tanned, plucked, waxed, sucked, and primped, we sure do like making fun of the gals who live up to the ideal! While the male model of rugged, manly roughness is rarely mocked (hell, we put him on Brawny paper towels!), women who meet the feminine ideal are most often made fun of, called stupid and shallow, and dismissed as vain. It seems there's no winning for pretty girls, either! Men who are beautiful are revered. Sure, you'll occasionally see the dumb-jock stereotype or the hot but vapid model—but that's nothing compared to the disdain that we heap upon gorgeous women. The hypocrisy is, if women _don't_ take steps to be "beautiful"—whether it be through a weekly manicure or something more drastic, like plastic surgery—then we're slobs. But if we do, then we're vain. It's something similar to the celeb-hate we love so much, but worse—because we do it to each other every day as well. Be honest—how often do we see a woman with a fake tan, or dyed hair, or obvious plastic surgery, and judge her—even if it's just a little? Don't feel too bad; we're trained to do as much. We're supposed to simultaneously want to be that woman—and want to destroy her. (And maybe hate ourselves for wanting to be her.) It's all sorts of fucked up. But it drives the competitive spirit that keeps women buying more products, more surgeries, more everything. A lot of people are depending on our judging and hating each other! Another sad fact is that feminists aren't immune to the woman-hate. Oh, how I wish we were. Unfortunately, I've heard way too many feminists get down on a gal because she wore heels or lipstick—saying she was a pawn for the beauty industry or fooling herself. The thing is, this kind of faux "concern" isn't much different from what society wants us to do to each other—tear each other down, judge, and not get anywhere against the folks who are really hurting women. Like the media. Reality shows are a subject unto themselves, but I think that they represent a tremendous example of how we punish women for conforming to the very expectations that we shove down their throats. If you want to know anything and everything about feminism, sexism, and reality television, you need to read Jennifer Pozner. She's like the feminist Queen Bee of reality TV. In an article she wrote for _Ms._ magazine, she discusses how the producers of this nonsense break women down: Viewers may be drawn to reality TV by a sort of cinematic schadenfreude, but they continue to tune in because these shows frame their narratives in ways that both reflect and reinforce deeply ingrained societal biases about women, men, love, beauty, class, and race. The genre teaches us that women categorically "are" certain things—for example, no matter their age, they're "hot girls," not self-aware or intelligent adults. You can find similar nonsense in almost all of the reality shows. They thrive on presenting us with hot, made-up women and then making them look as silly and pathetic as possible. Gossip magazines also make a career out of shaming women who conform. On one page of a celebrity weekly you'll find close-up pictures of a star's cellulite, mocking her. But on the very next page, you're likely to see an article about how horrible and anorexic another woman star is. How often do we see spreads on male stars' fat asses? Or their dramatic weight loss? Not so often. Much more fun to shame women. **_So... what to do?_** Stop hating. Seriously. Women have a hard enough time without other women giving us shit. And the truth is, we all do what we have to in order to get by in a society that hates women. For some of us, it's high heels and makeup. For others, it's plastic surgery. I don't think any of it is _good_ for women, but I don't think judgment is particularly helpful either. Now, some of you may not want to hear this, but here goes: Don't buy magazines that do this to women. I know the celeb mags are popular, I do. But take a step back and look at what they're doing to women. Look at what they're telling you about women. Is that really something you want to be a part of? (Besides, if you want fashion and celebrity news, there are plenty of blogs and sites you can go to that don't rely on shaming.) **NOTES** **HE'S CHILL, SHE'S ON THE PILL** Chaker, Ann Marie. "College Students Face Rising Birth-Control Prices," _Wall Street Journal,_ July 26, 2007. **HE'S ROUGH, SHE'S DAINTY** www.creditcards.com/credit-card-news/young-women-suffer-from-greater-debt.php. **HE'S METROSEXUAL, SHE'S ANOREXIC** www.kaisernetwork.org/daily_reports/rep_index.cfm. **HE'S A BACHELOR, SHE'S A SPINSTER** Mapes, Diane, ed. _Single State of the Union: Single Women Speak Out on Life, Love, and the Pursuit of Happiness_. Emeryville, CA: Seal Press, 2007. Angier, Natalie. "Men. Are Women Better Off With Them, or Without Them?" _New York Times,_ June 21, 1998. **HE'S GONNA BE A SUCCESS, SHE'S GONNA BE A STAY-AT-HOME MOM** www.sciencedaily.com/releases/2007/10/071015102856.htm. **HE'S A POLITICIAN, SHE'S A FASHION PLATE** Aday, Sean, and James Devitt. "Style Over Substance: Newspaper Coverage of Female Candidates. Spotlight on Elizabeth Dole." The White House Project, 2000. www.thewhitehouseproject.org/newsroom. Alvarez, Lizette. "Speaking Chic to Power," _New York Times,_ January 18, 2007. <http://feministing.com/archives/006376.html>. <http://feministing.com/archives/006061.html>. Givhan, Robin. "Hillary Clinton's Tentative Dip into New Neckline Territory," _Washington Post,_ July 20, 2007. **HE'S MANLY, SHE'S SASQUATCH** Winterman, Denise. "Letting your hair down," BBC News, January 12, 2007. ** HE'S THE BOSS, SHE'S A BITCH** Johnson, Tory. "Why Doesn't the Devil Wear Brooks Bros.?" ABC News, July 3, 2006. www.msnbc.msn.com. ** HE'S WELL PAID, SHE'S SCREWED** American Association of University Women, "Behind the Pay Gap," 2007. Interview with Sara Laschever, August 24, 2007. Feministing.com, <http://feministing.com/archives/007616.html>. Vedantam, Shankar. "Salary, Gender and the Social Cost of Haggling," _Washington Post,_ July 30, 2007. **HE'S HIMSELF, SHE'S MRS. HIMSELF** Friess, Steve. "More men taking wives' last names," _USA Today,_ March 20, 2007. **HE'S GETTING AN EDUCATION, SHE'S GETTING IN HIS WAY** "More women graduate. Why?" _USA Today,_ May 29, 2006. Rosser, Phyllis. "Too Many Women in College?" _Ms._ magazine, fall 2005. Jan, Tracy. "Schoolboy's bias suit," _Boston Globe,_ January 26, 2006. Pollitt, Katha. "Girls Against Boys?" _The Nation,_ January 12, 2006. Mathews, Jay. "Study Casts Doubt On the 'Boy Crisis,'" _Washington Post,_ June 26, 2006, A01. Rosser, Phyllis. "Too Many Women in College?" _Ms._ magazine, fall 2005. **HE'S INDEPENDENT, SHE'S PATHETIC** Roberts, Sam. "51% of Women Are Now Living Without Spouse," _New York Times,_ January 16, 2007. **HE'S A CELEB, SHE'S A MESS** Traister, Rebecca. "Hit her, baby, one more time," _Salon,_ September 12, 2007, www.salon.com. Harris, Paul. "Bad girls oust wild men as the sinful darlings of Hollywood scandal sheets," _Guardian,_ July 29, 2007. **HE'S HUSKY, SHE'S INVISIBLE** Meltzer, Marisa. "Are fat suits the new blackface?" _bitch: feminist response to pop culture,_ Winter 2001. <http://kateharding.net/2007>. **HE'S A MAN, SHE'S A MOM** www.npr.org/templates/story/story.php?storyId=12513004. www.local6.com/family. www.healthscout.com/news/1/609234. **HE'S DATING A YOUNGER WOMAN, SHE'S A COUGAR** www.thisisby.us/index. www.urbandictionary.com. **HE'S DRUNK, SHE'S A VICTIM** www.alternet.org/story/48835. **HE'S REPRESENTED, SHE'S A TOKEN** www.ipu.org/wmn-e/classif.htm. **HE'S NEAT, SHE'S NEUROTIC** www.umich.edu/news. www.angryharry.com/eshousework.htm. www.washingtonpost.com/wp-dyn. <http://feministing.com/archives/008007.html>. <http://news.bbc.co.uk>. **HE'S FUN, SHE'S FRIVOLOUS** www.debbieschlussel.com/archives/2007/11. **HE WALKS FREELY, SHE GETS HARASSED** Valenti, Jessica. "Is segregation the only answer to sexual harassment?" _Guardian,_ August 3, 2007. **HE'S A PERSON, SHE'S A COMMODITY** www.forbes.com/entrepreneurs. **HE'S A PUBLIC FIGURE, SHE'S A VIRGIN/WHORE** <http://afeministresponsetopopculture.blogspot.com>. **HE'S GOT G.I. JOE, SHE'S GOT BARBIE** www.americanrhetoric.com/speeches/josswhedonequalitynow.htm. **HE'S PAYING LESS, SHE'S PAYING MORE** Whittelsey, Frances Cerra. "Why Women Pay More," The Center for Responsive Law, 1993. Found online at www.holysmoke.org. "Mortgages; Why Women Pay Higher Interest," _New York Times,_ January 21, 2007. Associated Press, "Popular health-insurance plans punish women," April 6, 2007. **HE'S PUSSY WHIPPED, SHE'S A "GOOD GIRLFRIEND"** www.urbandictionary.com. **HE'S FONDLING, SHE'S FEEDING** www.babble.com/CS/blogs. Boteach, Shmuley. "Moms, Don't Forget to Feed Your Marriages," www.beliefnet.com. **HE'S CHILDLESS, SHE'S SELFISH** www.usatoday.com/news/health. **HE'S FUNNY, SHE'S ANNOYING** www.vanityfair.com/culture. www.psychologytoday.com/articles. <http://shakespearessister.blogspot.com/2005/10/women-arent-funny.html>. www.msmagazine.com/summer2004/womenshumor.asp. www.alternet.org/story/61102. **HE'S DATING, SHE'S TAKEN** www.slate.com/id/2167870. **HE'S RUGGED, SHE'S RUDE** www.slate.com/id/2119985. **HE'S PLAIN, SHE'S VAIN** www.msmagazine.com/fall2004/unrealworld.asp. **ACKNOWLEDGMENTS** **FIRST AND FOREMOST,** thank you to my mother, father, and sister Vanessa for being forever supportive. You all mean everything to me. Thanks to my agent, Tracy Brown, for his advice and warmth, and to my editor, Brooke Warner, for her saintlike patience and for always believing in the work. Big thanks also go to Gwen Beetham, Ann Friedman, Jen Moseley, Samhita Mukhopadhyay, Celina De Leon, and Courtney Martin—my partners in feminist crime. Soon, ladies, we will take over the world. And last, thanks to Andrew Golis, for everything. **ABOUT THE AUTHOR** **JESSICA VALENTI** is the founder and executive editor of Feministing.com and the author of _Full Frontal Feminism: A Young Woman's Guide to Why Feminism Matters_. She has a master's degree in women's and gender studies from Rutgers University and has worked with national and international women's organizations. Jessica is also a cofounder of the REAL hot 100, a campaign that aims to change the perception of younger women in the media, and the blogger for NARAL Pro-Choice America. Her writing has appeared in _Ms._ magazine, _Bitch_ , AlterNet, _Salon, Guernica_ magazine, and the _Guardian_ (U.K.), as well as the anthologies _We Don't Need Another Wave_ and _Single State of the Union_. In 2007, she received a Choice USA Generation award for her commitment to reproductive-rights issues and was named one of _ELLE_ magazine's 2007 IntELLEgentsia. She lives in her hometown of Astoria, Queens, with Neidra the cat, Monty the dog, and Andrew the boyfriend. **SELECTED TITLES FROM SEAL PRESS** For more than thirty years, Seal Press has published groundbreaking books. By women. For women. Visit our website at www.sealpress.com, and our blog at www.sealpress.com/blog. _**Full Frontal Feminism**_ by Jessica Valenti. $15.95, 1-58005-201-0. A sassy and in-your-face look at contemporary feminism for women of all ages. _**It's a Jungle Out There: The Feminist Survival Guide to Politically Inhospitable Environments**_ by Amanda Marcotte. $13.95, 1-58005-226-6. All the witty comebacks, in-your-face retorts, and priceless advice women need to survive in politically hostile environments. _**30-Second Seduction: How Advertisers Lure Women Through Flattery, Flirtation, and Manipulation**_ by Andrea Gardner. $14.95, 1-58005-212-6. _Marketplace_ reporter Andrea Gardner focuses on the many ways that advertising targets women, and how those ads affect decisions, purchases, and everyday life. _**She's Such a Geek: Women Write About Science, Technology, and Other Nerdy Stuff**_ edited by Annalee Newitz and Charlie Anders. $14.95, 1-58005-190-1. From comic books and gaming to science fiction and blogging, nerdy women have their say in this witty collection that takes on the "boys only" clubs and celebrates a woman's geek spirit. _**Abortion Under Attack: Women on the Challenges Facing Choice**_ edited by Krista Jacob, foreword by Rebecca Walker, afterword by Gloria Feldt. $15.95, 1-58005-185-5. This book is a call to action, in this conservative time, for new and veteran pro-choice people alike. _**Body Outlaws: Rewriting the Rules of Beauty and Body Image**_ edited by Ophira Edut, foreword by Rebecca Walker. $15.95, 1-58005-108-1. Filled with honesty and humor, this groundbreaking anthology offers stories by women who have chosen to ignore, subvert, or redefine the dominant beauty standard in order to feel at home in their bodies. He's a Stud, She's a Slut and 49 Other Double Standards Every Woman Should Know Copyright © 2008 by Jessica Valenti Published by Seal Press A Member of Perseus Books Group 1700 Fourth Street Berkeley, California All rights reserved. No part of this book may be reproduced or transmitted in any form without written permission from the publisher, except by reviewers who may quote brief excerpts in connection with a review. Library of Congress Cataloging-in-Publication Data Valenti, Jessica. He's a stud, she's a slut and 49 other double standards every woman should know / by Jessica Valenti. p. cm. eISBN : 978-0-786-75049-8 1. Women--United States--Public opinion. 2. Men--United States--Public opinion. 3. Stereotypes (Social psychology)--United States. 4. Sexism--United States. 5. Public opinion--United States. I. Title. HQ1421.V36 2008 305. 420973--dc22 2008004012
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An unflinching look at 'The Pain of Others' By Peter Keough Globe Correspondent,June 28, 2018, 12:02 p.m. (Wishful Thinking LLC) As in her animated documentary "Nuts!" (2016), about a man who made a fortune transplanting goat testicles as an impotence treatment, director Penny Lane blurs the boundary between fiction and nonfiction. In "The Pain of Others," she also explores the fine line between delusion and reality, but in this case it's not about a dubious cure but about a disease some think is imaginary. Reminiscent of Jennifer Brea's "Unrest" (2017), which relates her struggle to get doctors to take her crippling chronic fatigue seriously, Lane's film focuses on three women suffering from Morgellons disease, the symptoms of which include the sensation of parasites crawling under the skin and the eruption of wormlike threads from lesions. Most physicians — though not all — say the malady is psychogenic, which only intensifies the victims' desperation and deterioration. The Martha's Vineyard crowd strikes back at Alan Dershowitz Except for an occasional news report that establishes a tentative objectivity, "The Pain of Others," like Lane's "Our Nixon" (2013), consists of found and archival footage — in this case the YouTube videos that the women have posted to a community of fellow "Morgies." In these they talk about their symptoms, search for a cure, and show, with heartbreaking obsessiveness and distress, the sometimes stomach-turning outbreaks of sores and the growths of mysterious fibers (the film is a treat for those who like to pick, pull, and pop). Anxiety, loneliness, and anger disrupt their efforts to be upbeat and agreeable. Tasha plaintively describes how skeptics dismiss her illness. "It makes me feel crazy and it makes me feel stupid," she says. "But it's real." She obsessively studies patches of hair, microscopic video images of filaments that look like filaments on a movie lens, and a bit of tissue protruding from her nose. Is it a worm or a piece of skin? Eventually she shaves her head. Marcia pulls convincing- looking threads from her fingers and wonders why doctors can't see them. Later she has coated her face with pulverized aspirin. Boston police fatally shoot 'vicious' pit bull that owner says she was trying to control Carrie is perhaps the most heartbreaking, her face pressed close to the lens, whispering plaintively like a fugitive, apologetically unburdening her palpable panic, pain, and loneliness. Later she confides how her life has been transformed by drinking her own urine and rubbing it into her face. Lane describes "The Pain of Others" as a "body-horror documentary," a term evocative of such films by David Cronenberg as "The Brood" (1979) and "Dead Ringers" (1988). Because of their disorder, the subjects seem paralyzed by a fascination with and abhorrence of their sheer physicality. The title alludes to Susan Sontag's 2003 book "Regarding the Pain of Others," which decries the voyeuristic consumption of images of suffering. Perhaps Lane is suggesting that her film might be as much about the viewer as the subject. Detached by media, viewers can indulge in the media torrent of torment and violence as an entertainment or, for the higher-minded, an exercise in moral rectitude. But in Lane's film such distancing is more difficult. Tasha, Marcia, and Carrie record pain that only they can feel. It may be imaginary, but it is no less real. They are vulnerable, on the verge of breaking down, and watching their videos seems like an invasion of privacy. It makes you uncomfortable and aware of your privileged position as an observer, which may be a first step to empathy. "The Pain of Others" is available from Fandor on Sunday. Go to www.fandor.com. Peter Keough can be reached at petervkeough@gmail.com.
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Q: Meaning of "with" statement without "as" keyword I'm familiar with using python's with statement as a means of ensuring finalization of an object in the event of an exception being thrown. This usually looks like with file.open('myfile.txt') as f: do stuff... which is short-hand for f = file.open('myfile.txt'): try: do stuff... finally: f.close() or whatever other finalization routine a class may present. I recently came across a piece of code dealing with OpenGL that presented this: with self.shader: (Many OpenGL commands) Note that absence of any as keyword. Does this indicate that the __enter__ and __exit__ methods of the class are still to be called, but that the object is never explicitly used in the block (i.e., it works through globals or implicit references)? Or is there some other meaning that is eluding me? A: The context manager can optionally return an object, to be assigned to the identifier named by as. And it is the object returned by the __enter__ method that is assigned by as, not necessarily the context manager itself. Using as <identifier> helps when you create a new object, like the open() call does, but not all context managers are created just for the context. They can be reusable and have already been created, for example. Take a database connection. You create the database connection just once, but many database adapters let you use the connection as a context manager; enter the context and a transaction is started, exit it and the transaction is either committed (on success), or rolled back (when there is an exception): with db_connection: # do something to the database No new objects need to be created here, the context is entered with db_connection.__enter__() and exited again with db_connection.__exit__(), but we already have a reference to the connection object. Now, it could be that the connection object produces a cursor object when you enter. Now it makes sense to assign that cursor object in a local name: with db_connection as cursor: # use cursor to make changes to the database db_connection still wasn't called here, it already existed before, and we already have a reference to it. But whatever db_connection.__enter__() produced is now assigned to cursor and can be used from there on out. This is what happens with file objects; open() returns a file object, and fileobject.__enter__() returns the file object itself, so you can use the open() call in a with statement and assign a reference to the newly created object in one step, rather than two. Without that little trick, you'd have to use: f = open('myfile.txt') with f: # use `f` in the block Applying all this to your shader example; you already have a reference to self.shader. It is quite probable that self.shader.__enter__() returns a reference to self.shader again, but since you already have a perfectly serviceable reference, why create a new local for that? A: The above answer is nicely put. The only thing I kept asking myself while reading it, is where is the confirmation of the following scenario. In the event there is an assignment in the body of the context of the with statement, anything on the right side of the assignment is first "bound" to the context. So, in the following: with db_connection(): result = select(...) ... select is ~ ref_to_connection.select(...) I put this here for anyone like me who comes and goes between languages and might benefit by a quick reminder of how to read and track the refs here.
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/jcd.2015004","text":"Article Contents\nArticle Contents\n\n# Computing continuous and piecewise affine lyapunov functions for nonlinear systems\n\n\u2022 We present a numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach constructs a partition of the state space, called a triangulation, and then computes values at the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous and Piecewise Affine (CPA) function. Verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Numerical examples are presented demonstrating different aspects of the proposed method.\nMathematics Subject Classification: Primary: 93D05, 93D30, 93D20; Secondary: 93D10.\n\n Citation:\n\n\u2022 [1] R. Baier, L. Gr\u00fcne and S. Hafstein, Linear programming based Lyapunov function computation for differential inclusions, Discrete and Continuous Dynamical Systems Series B, 17 (2012), 33-56.doi:\u00a010.3934\/dcdsb.2012.17.33. [2] H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem, Journal of Computational and Nonlinear Dynamics, 1 (2006), 312-319.doi:\u00a010.1115\/1.2338651. [3] J. Bj\u00f6rnsson, P. Giesl and S. Hafstein, Algorithmic verification of approximations to complete Lyapunov functions, In Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, pages 1181-1188, Groningen, The Netherlands, 2014. [4] J. Bj\u00f6rnsson, P. Giesl, S. Hafstein, C. M. Kellett and H. Li, Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction, In Proceedings of the 53rd IEEE Conference on Decision and Control, pages 5506-5511, Los Angeles (CA), USA, 2014. [5] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Number 1904 in Lecture Notes in Mathematics. Springer, 2007. [6] P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in two dimensions, J. Math. Anal. Appl., 371 (2010), 233-248.doi:\u00a010.1016\/j.jmaa.2010.05.009. [7] P. Giesl and S. Hafstein, Construction of Lyapunov functions for nonlinear planar systems by linear programming, Journal of Mathematical Analysis and Applications, 388 (2012), 463-479.doi:\u00a010.1016\/j.jmaa.2011.10.047. [8] P. Giesl and S. Hafstein, Existence of piecewise affine Lyapunov functions in arbitrary dimensions, Discrete and Contin. Dyn. Syst., 32 (2012), 3539-3565.doi:\u00a010.3934\/dcds.2012.32.3539. [9] P. Giesl and S. Hafstein, Revised CPA method to compute Lyapunov functions for nonlinear systems, Journal of Mathematical Analysis and Applications, 410 (2014), 292-306.doi:\u00a010.1016\/j.jmaa.2013.08.014. [10] P. Giesl and S. Hafstein, Computation and verification of Lyapunov functions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1663-1698.doi:\u00a010.1137\/140988802. [11] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2291-2331.doi:\u00a010.3934\/dcdsb.2015.20.2291. [12] S. Hafstein, An Algorithm for Constructing Lyapunov Functions, Electronic Journal of Differential Equations Mongraphs, 2007. [13] S. Hafstein, C. M. Kellett and H. Li, Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction, In Proceedings of the American Control Conference, pages 548-553, Portland, Oregon, USA, 2014.doi:\u00a010.1109\/ACC.2014.6858660. [14] W. Hahn, Stability of Motion, Springer-Verlag, 1967. [15] T. Johansen, Computation of Lyapunov functions for smooth nonlinear systems using convex optimization, Automatica, 36 (2000), 1617-1626.doi:\u00a010.1016\/S0005-1098(00)00088-1. [16] W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence, Foundations of Computational Mathematics, 5 (2005), 409-449.doi:\u00a010.1007\/s10208-004-0163-9. [17] C. M. Kellett, A compendium of comparsion function results, Mathematics of Controls, Signals and Systems, 26 (2014), 339-374.doi:\u00a010.1007\/s00498-014-0128-8. [18] C. M. Kellett, Classical converse theorems in Lyapunov's second method, Discrete and Continuous Dynamical Systems Series B, 20 (2015), 2333-2360.doi:\u00a010.3934\/dcdsb.2015.20.2333. [19] J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion, Chechoslovak Mathematics Journal, 81 (1956), 217-259, 455-484; English translation in American Mathematical Society Translations (2), 24 (1956), 19-77. [20] H. Li, S. Hafstein and C. M. Kellett, Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems, J Differ Equ Appl, 21 (2015), 486-511.doi:\u00a010.1080\/10236198.2015.1025069. [21] A. M. Lyapunov, The general problem of the stability of motion, Math. Soc. of Kharkov, 1892. (Russian). (English Translation, International J. of Control, 55 (1992), 521-790).doi:\u00a010.1080\/00207179208934253. [22] S. Marinosson, Lyapunov function construction for ordinary differential equations with linear programming, Dynamical Systems, 17 (2002), 137-150.doi:\u00a010.1080\/0268111011011847. [23] J. L. Massera, On Liapounoff's conditions of stability, Annals of Mathematics, 50 (1949), 705-721.doi:\u00a010.2307\/1969558. [24] A. Papachristodoulou and S. Prajna, The construction of Lyapunov functions using the sum of squares decomposition, In Proceedings of the 41st IEEE Conference on Decision and Control, 3 (2002), 3482-3487.doi:\u00a010.1109\/CDC.2002.1184414. [25] M. Peet and A. Papachristodoulou, A converse sum of squares Lyapunov result with a degree bound, IEEE Transactions on Automatic Control, 57 (2012), 2281-2293.doi:\u00a010.1109\/TAC.2012.2190163. [26] N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method, Springer-Verlag, 1977. [27] E. D. Sontag, Comments on integral variants of ISS, Systems and Control Letters, 34 (1998), 93-100.doi:\u00a010.1016\/S0167-6911(98)00003-6. [28] A. R. Teel and L. Praly, A smooth Lyapunov function from a class-$\\mathcal{KL}$ estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var., 5 (2000), 313-367.doi:\u00a010.1051\/cocv:2000113. [29] T. Yoshizawa, On the stability of solutions of a system of differential equations, Memoirs of the College of Science, University of Kyoto, Series A: Mathematics, 29 (1955), 27-33. [30] T. Yoshizawa, Stability Theory by Liapunov's Second Method, Mathematical Society of Japan, 1966.","date":"2023-03-23 17:28:17","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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7649539113044739, \"perplexity\": 1623.1183538207495}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945182.12\/warc\/CC-MAIN-20230323163125-20230323193125-00224.warc.gz\"}"}
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Q: Triggered ANR, Root blocking Android Issue on crashlytics In Crashlytics getting a crash with tag Triggered ANR, Root Blocking inside StaticLAyout class. Not able to generate or fix the issue as it is not generating at my device. I can't find any proper blog for the same. There is only one page doc in Google developer. https://firebase.google.com/docs/crashlytics/debug-anr-errors?authuser=1&hl=en#root-blocking-tag[enter image description here]1
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Met de Rode Terreur (Russisch: красный террор, krasniij terror) bedoelt men de moorden, marteling en onderdrukking van burgers en (krijgs)gevangenen door de bolsjewistische communisten tijdens de Russische Revolutie en de Russische Burgeroorlog. Schattingen van het aantal dodelijke slachtoffers door de Rode Terreur reiken van 50.000 tot anderhalf miljoen. De Witten deden soortgelijke handelingen, wat bekend staat als de Russische Witte Terreur. Periode voor september 1918 Direct na de Oktoberrevolutie was Vladimir Lenin van mening dat er geweld gebruikt moest worden tegen de tegenstanders van de bolsjewieken. Op 26 oktober 1917 stelde Lev Kamenev voor om de doodstraf af te schaffen. Lenin zou volgens Leon Trotski hebben gezegd: "Wat een onzin! Hoe kun je nu een revolutie voltooien zonder executiepelotons? Hoe moet je je dan van je vijanden ontdoen? Door je te ontwapenen? Wat voor andere repressiemiddelen zijn er nu helemaal? Gevangenissen? Wat heb je daar nu aan tijdens een burgeroorlog? " In december 1917 publiceerde Lenin het artikel "Hoe concurrentie te organiseren?" waarin stond: "In de ene plaats zullen de rijken, de schurken, de arbeiders die niet willen werken, in de cel belanden. (…) In een andere plaats zullen ze gedwongen worden de latrines schoon te maken. Weer ergens anders zullen ze na hun gevangenisstraf een geel merkteken krijgen, zodat iedereen weet dat ze schadelijk zijn en hen in de gaten kan houden. Nog ergens anders kan één op de tien leeglopers worden neergeschoten." Op 7 december werd de Tsjeka opgericht met Feliks Dzerzjinski als leider die bij de oprichting zei: "Denk nu niet dat ik daarmee uit ben op revolutionaire rechtvaardigheid. We hebben nu geen behoefte aan rechtvaardigheid." De Voorlopige Regering had de verkiezingen van de Russische Grondwetgevende Vergadering gepland in november 1917. Tegen de wil van Lenin besloot de communistische regering om de verkiezingen door te laten gaan. De bolsjewieken kregen 24% van de stemmen. De liberale Constitutioneel-Democratische Partij werd verboden. Tientallen liberalen werden gearresteerd inclusief gekozen parlementariërs. Kort daarna werden ook leiders van de sociaal-revolutionairen en mensjewieken gearresteerd. Later werden leden van de Boerensovjet in de gevangenis gegooid. Dit leidde tot een tekort aan cellen en daarom lieten de bolsjewieken dieven vrij. Op 5 januari 1918 (oude stijl) werd de Grondwetgevende Vergadering geopend. Naar schatting 50.000 demonstranten waren op het been om de Grondwetgevende Vergadering te steunen. Toen de demonstranten de Litejny Prospekt naderden werden zij beschoten door bolsjewieken vanaf de daken. Meer dan tien personen werden gedood. Het was voor het eerst sinds de Februarirevolutie dat regeringstroepen op een ongewapende menigte schoten. De Grondwetgevende Vergadering werd de volgende dag opgeheven door de communisten. Twee leiders van de liberalen (Andrej Sjingarjov en Fiodor Kokosjkin) werden een dag later vermoord. Zij waren ziek geworden in de gevangenis en werden naar een ziekenhuis gebracht, waar ze werden vermoord. De bolsjewistische directeur van het ziekenhuis verdedigde de moord door te beweren dat er nu "twee bourgeoismonden minder gevoed hoefde te worden." Lenin publiceerde in februari 1918 het decreet Het Socialistische Vaderland is in Gevaar!, waarin stond dat vijandelijke agenten, profiteurs, plunderaars, vandalen, oproerkraaiers en spionnen ter plekke doodgeschoten moesten worden. Zonder rechtszaken en bewijsvergaring kon men worden geëxecuteerd. Er moesten bataljons georganiseerd worden van burgers om loopgraven te graven en diegenen die hiertegen verzetten moesten doodgeschoten worden. Het decreet leidde tot kritiek van coalitiepartner Linkse Sociaal-Revolutionaire Partij. Volgens de linkse sociaal-revolutionaire volkcommissaris van Justitie Isaac Steinberg ging hijzelf na het lezen van het decreet woedend naar Lenin toe. Steinberg protesteerde tegen het decreet en zei tegen Lenin: "Waar zouden we dan nog een commissariaat van Justitie voor nodig hebben? Laten we het dan gewoon eerlijk het "commissariaat van Sociale Uitroeiing" noemen en er verder geen woorden meer aan vuil maken!" Volgens Steinberg antwoordde Lenin: "Ja, dat is eigenlijk precies wat we zouden moeten doen – alleen kunnen we daar niet voor uitkomen." Lenin introduceerde de leuze "Plunder de Plunderaars" waarin de bevolking werd opgeroepen om de "bourgeoisie" aan te vallen en te bestelen. In Taganrog werden vijftig liberalen levend in een staaloven gegooid. In Jevpatorija kregen matrozen van de bolsjewistische autoriteiten de vrije hand met plunderingen. In drie dagen werden ongeveer 800 mensen vermoord. Ook in Jalta, Feodosija en Sebastopol waren in maart en april 1918 zulke vormen van geweld en plunderingen. De monniken van Olonets werden gearresteerd en later geëxecuteerd door de Tsjeka. De bolsjewieken hieven een aparte belasting op rijke burgers en er werden gijzelaars genomen als er niet betaald werd. Nadat steeds meer voormalige rijken geen geld meer hadden werden gijzelaars steeds vaker geëxecuteerd. Mensen beschuldigden anderen voor hun eigen gewin. Zo werden veel mensen die geld hadden uitgeleend aan anderen door de leners beschuldigd om zo van schulden af te komen. Zo klaagde het departement van Justitie van Penza dat de gevangenissen "vol zitten met onschuldige mensen die zijn gearresteerd door de Tsjeka op basis van een of andere valse beschuldiging van iemand anders." Op 9 mei 1918 schoten troepen in Kolpino op demonstrerende arbeiders met tien doden tot gevolg. Op dezelfde dag werd bij een fabriek nabij Jekaterinenburg ongeveer vijftien mensen doodgeschoten na een demonstratie tegen machtsmisbruik van bolsjewistische functionarissen en een dag later werden veertien gearresteerde arbeiders doodgeschoten. De moorden leidden tot stakingen bij verschillende fabrieken. In het Moskouse circus was er de clown Bim-Bom, die grappen maakte over de bolsjewieken. Tijdens een voorstelling bestormden Tsjeka-agenten het circus om de clown te arresteren. De clown probeerde te vluchten en werd in de rug geschoten door de Tsjeka-agenten. Honderden mensen kwamen opdagen bij de begrafenis van de clown, die uitgroeide tot een demonstratie. Van december 1917 tot juli 1918 werden in de pers 884 vermeldingen gemaakt van executies door de communisten. Op 17 juli 1918 werd de voormalige tsaar en zijn gezin vermoord door de communisten. De vijf kinderen van Nicolaas Romanov hadden leeftijden oplopend van 13 tot 22 jaar oud. Ook de vier bedienden van de familie werden vermoord. Alleen de executie van Nicolaas Romanov werd vermeld in bolsjewistische kranten. In een telegram van 11 augustus 1918 aan communisten in Penza geeft Lenin persoonlijk de opdracht om niet minder dan 100 personen op te hangen. In het telegram schrijft Lenin: "Hang (en zorg ervoor dat het hangen plaatsvindt in het volle zicht van de mensen) niet minder dan honderd bekende koelakken op, rijke mannen, bloedzuigers (…) Doe het op zo'n manier dat honderden kilometers in de omtrek de mensen zien, beven, weten, roepen: ze wurgen ze, en ze zullen ze wurgen tot de dood erop volgt, die bloedzuigende koelakken." Officiële terreurcampagne De officiële campagne genaamd Rode Terreur begon als een represaille voor de moord op Tsjeka-leider Mosei Oeritski op 17 augustus 1918 en de mislukte aanslag op Lenin op 30 augustus 1918 door Fanny Kaplan. De communistische regering executeerde vijfhonderd leden van de "bourgeoisie" gelijk na de aanslag op Oeritski. De Krasnaja Gazetta publiceerde op 1 september: "Genadeloos zullen we afrekenen met onze vijanden. Met honderden tegelijk zullen we ze doden. Of laten het er duizenden zijn, laten ze verdrinken in hun eigen bloed. Laten er rivieren van burgerlijk bloed vloeien in vergelding voor het bloed van Lenin en Oeritski – of nog meer bloed, zo veel mogelijk." De eerste officiële aankondiging van de Rode Terreur werd gepubliceerd in de Izvestia. Op 5 september 1918 werd het decreet Over Rode Terreur uitgegeven. De communistische regering beval om sociaal-revolutionairen te arresteren en mensen in gijzeling te nemen die zouden worden vermoord als represaille voor aanvallen op bolsjewieken. Tijdens zijn herstelperiode gaf Lenin instructies om een terreurcampagne te beginnen. De Oekraïense Tsjeka-leider Martin Latsis gaf zijn medewerkers via de krant "Rode Terreur" de volgende instructies mee: "Zoek niet naar bewijzen dat de beschuldigde iets tegen de sovjets heeft gedaan of gezegd. Vraag hem eerst tot welke klasse hij behoort, wat zijn sociale afkomst is, zijn opleiding en zijn vak. Dat zijn de vragen die het lot van de beschuldigde moeten bezegelen. Dat is de ware betekenis van de Rode Terreur." De communist Grigori Zinovjev beweerde in september 1918 dat de bolsjewieken van ongeveer 90 miljoen van de 100 miljoen inwoners van Rusland steun moest verkrijgen en dat de rest uitgeroeid moest worden. Volgens documenten van de Tsjeka werden in september 1918 in Petrograd meer dan 800 mensen geëxecuteerd, hoewel er mogelijk 1.300 executies waren in deze periode in Petrograd. Onder leiding van Nikolaj Boelganin werden 141 gijzelaars geëxecuteerd in Nizjni Novgorod na 31 augustus 1918. In Vjatka werd in een week 23 voormalige politieagenten, 154 "contrarevolutionairen", 8 monarchisten, 28 liberalen, 186 legerofficieren en 10 socialisten die lid waren van de SRP of de mensjewistische partij geëxecuteerd. De Tsjeka van Ivanovo Voznesensk rapporteerde dat zij 181 gijzelaars namen en 25 mensen executeerden en meer dan 1.000 mensen in concentratiekampen plaatsten tijdens de officiële campagne. De Tsjeka van het dorpje Sebezj rapporteerde de executie van 17 boeren. De Tsjeka van de stad Tver rapporteerde 130 gijzelaars en 39 executies en de Tsjeka van Perm berichtte over 50 executies. De krant van de Tsjeka van de oblast Tsaritsyn rapporteerde de executie van 103 mensen in de week van 3 tot 10 september. Andere provinciale kranten rapporteerden duizenden arrestaties en executies in de herfst van 1918. Op 15 oktober 1918 beweerde Tsjeka-leider Gleb Bokii bij het officiële einde van de campagne dat 800 mensen in Petrograd waren vermoord, terwijl er 6.229 mensen waren gearresteerd. In de officiële pers, waaronder het weekblad van de Tsjeka, stond dat in twee maanden tussen 10.000 en 15.000 mensen waren geëxecuteerd. Na oktober 1918 Platteland Onderdeel van het oorlogscommunisme was de inbeslagname van graan bij de boeren. De vorderingsbrigades voor voedsel pleegden veel geweld. In extreme gevallen werden deze brigades de machtsorganen voor communisten die zich gedroegen als landheren, zoals Margolin en Tsjeremoechin, die verkrachtingen en moorden toestonden. In Tambov moesten boeren zelf graan kopen in een andere provincie (oblast) om Margolin te betalen. Trotski voerde de militaire dienstplicht in om aan voldoende soldaten te komen. In 1918 deserteerden meer dan 1 miljoen soldaten uit het Rode Leger. In 1919 waren er officieel ongeveer 2 miljoen soldaten gedeserteerd uit het Rode Leger. In 1921 waren er ongeveer 4 miljoen deserteurs uit het Rode Leger. Geweld van de communisten tegen boeren bij de onteigening van paarden; inbeslagnames van voedsel en het ophalen van dienstplichtigen leidde tot boerenopstanden. De communisten stuurden strafexpedities tegen de boerendorpen. Gijzelaars werden genomen en dorpshoofden werden doodgeschoten en een aantal dorpen werden helemaal in brand gestoken door de communisten. De bolsjewistische commissaris van Toela liet honderden boeren doodschieten zonder proces. In september 1918 werden 48.735 deserteurs en 7.325 bandieten gearresteerd in twaalf provincies, waarbij 1.826 mensen werden doodgeschoten ter plekke, terwijl 2.230 mensen later werden geëxecuteerd. In 1919 werd ongeveer een half miljoen deserteurs gearresteerd door de bolsjewieken en in 1920 ongeveer 800.000, waarvan duizenden zijn geëxecuteerd. Familieleden van deserteurs werden gegijzeld om deserteurs te dwingen om te vechten. Het decreet van 15 februari 1919 van Lenin riep lokale Tsjeka-agenten op om gijzelaars te nemen onder de boerenbevolking in regio's waar de spoorwegen nog niet waren vrijgemaakt van sneeuw in de gewenste mate, waarbij de gijzelaars gedood moesten worden als de spoorwegen niet tot tevredenheid werden schoongemaakt. Lenin verklaarde in 1920 dat het rechtvaardig was om iedereen dood te schieten die het belang van zichzelf of zijn dorp boven de rest stelde. Tijdens het neerslaan van de Tambov-opstand maakten de communisten gebruik van gifgas. Er werden concentratiekampen opgericht waar ongeveer 50.000 burgers in gegijzeld werden, waaronder ook duizenden kinderen. Hele dorpen werden in kampen gegooid of doodgeschoten of werden gedeporteerd naar de poolcirkel. Ongeveer 15.000 man werd doodgeschoten na arrestatie en zonder enige vorm van rechterlijk proces. Ook beloofden de communisten een amnestie aan diegene die zichzelf overgaven. Ongeveer 6.000 opstandelingen die hieraan gehoor gaven, werden doodgeschoten of in de gevangenis gegooid. De onderdrukking van de Tambov-opstand van 1920 en 1921 zorgde dat ongeveer 100.000 boeren in gevangenschap werden gehouden of gedeporteerd werden. Concentratiekampen werden opgezet, waarbij naar schatting 70.000 mensen in september 1921 werden gevangengezet, waarbij in dit cijfer niet de gevangenen in opstandige provincies zoals Tambov zijn meegenomen. Slechte omstandigheden in deze kampen leidden tot hoge sterftecijfers. Steden In de herfst van 1918 zijn meer dan honderd stakers doodgeschoten zonder enige vorm van rechterlijk proces. Op 16 maart 1919 werd de Poetilov-fabriek bestormd door de Tsjeka. Meer dan 900 stakers werden gearresteerd, waarvan ongeveer 200 mensen werden geëxecuteerd zonder proces. Verschillende stakingen vonden plaats in de lente van 1919. Bij deze stakingen werden eisen gesteld zoals gelijke voedselrantsoenen; afschaffing van de privileges voor bolsjewieken; persvrijheid en vrije verkiezingen van de sovjets. De Tsjeka onderdrukte de stakingen door middel van arrestaties en executies. In Astrachan werden stakers en soldaten van het Rode Leger die een staking steunden geladen in drijvende gevangenissen en werden met honderden in de Wolga gegooid met zware stenen om hun nek. Op 12, 13 en 14 maart 1919 kwamen er tussen de 2.000 en 4.000 mensen om het leven in de stad door verdrinking of executie met de kogel. De moorden werden geleid door Sergej Kirov. Na deze periode werden in Astrachan nog 600 tot 1.000 burgers vermoord tussen maart 1919 en 1922. De visie van Lenin over het omgaan met stakers kan worden geïllustreerd met Lenins telegram van 29 januari 1920 gestuurd naar Vladimir Smirnov over diens behandeling van stakingen in het Oeralgebied, waarin Lenin zijn verbazing uitte dat Smirnov niet op grote schaal stakers liet executeren. In Charkov werden er tussen 2.000 en 3.000 mensen geëxecuteerd in de periode februari 1919 en juni 1919 en nogmaals 1.000 tot 2.000 mensen toen Charkov werd heroverd in december 1919 door de bolsjewieken. In Rostov aan de Don werden ongeveer duizend mensen vermoord in januari 1920. In Odessa werden er tussen mei en augustus 1919 ongeveer 2.200 mensen geëxecuteerd en in de periode februari 1920 tot februari 1921 werden er tussen 1.500 en 3.000 mensen geëxecuteerd door de bolsjewieken. In Kiev werden minstens 3.000 mensen geëxecuteerd in de periode van februari 1919 tot en met juli 1919. Toen Kiev bijna werd heroverd door de Witte troepen, vermoordden de communisten in Kiev meer dan 1.800 mensen tussen 22 en 28 augustus 1919. Hetzelfde gebeurde in Jekaterinodar waar de Tsjeka ongeveer 1.600 mensen tussen 17 en 19 augustus heeft vermoord. In Jekaterinodar werden minstens 3.000 mensen geëxecuteerd tussen augustus 1920 en februari 1921. In Armavir werden tussen 2.000 en 3.000 in augustus tot en met oktober 1920 standrechtelijk geëxecuteerd. De meeste executies gebeurden na de vlucht van de Witte troepen van Pjotr Wrangel. In Sevastopol werden honderden havenarbeiders doodgeschoten op 26 november 1920 wegens het helpen bij de evacuatie van de Witten. Op de Krim werden in totaal ongeveer 50.000 krijgsgevangen en burgers zonder proces geëxecuteerd met toestemming van Lenin na de evacuatie van Wrangel eind 1920. Velen hadden zichzelf overgeven nadat er een amnestie werd beloofd bij overgave. Op 18 maart 1921 gaven de Kronstadt-opstandelingen zich over en die nacht werd op bevel van Zinovjev 500 gevangenen doodgeschoten. In de loop van de daarop volgende maanden werden nog eens 2.000 gevangen geëxecuteerd, zonder enige vorm van proces, terwijl tal van anderen naar de concentratiekampen op de Solovetski-eilanden werden gestuurd. Ongeveer 8.000 opstandelingen waren erin geslaagd naar Finland te vluchten, maar daar werden ze gevangengezet. Velen werden later terug naar Rusland gelokt met de belofte van amnestie, waarna ze bij terugkeer werden doodgeschoten of naar een concentratiekamp werden gestuurd. Tussen 1921 en 1925 werden ongeveer 800 synagogen gesloten door de communisten. In april 1921 werden de eerste synagogen gesloten in Vitebsk. De plaatselijke Joodse gelovigen protesteerden tegen de sluiting door de gebouwen te bezetten en daar gebedsdiensten te houden. Hierop werden de synagogen aangevallen door de communisten en werden tientallen religieuze joden vermoord. In februari 1922 werd een decreet uitgevaardigd door de communistische regering met het bevel om alle waardevolle spullen van de kerken in beslag te nemen. Bij de inbeslagnames werden ongeveer 7.100 geestelijken vermoord, maar ook boeren en arbeiders die kerken verdedigden werden gedood. Lenin stelde een nota op waarin stond: "Hoe meer leden van de reactionaire burgerij en geestelijkheid we kunnen neerschieten, des te beter." Eind 1922 werden 92 tolstojaanse pacifisten doodgeschoten wegens hun weigering om in dienst te gaan bij het leger. Onder leiderschap van Lavrenti Beria werden 12.578 mensen in Georgië tussen 29 augustus en 5 september 1924 doodgeschoten na arrestatie. Dekozakkisatie De Kozakken leden veel tijdens de Rode Terreur door de dekozakkisatie-campagne. Tussen midden februari 1919 en midden maart 1919 werden meer dan 8.000 Kozakken vermoord. De Tsjeka in Pjatigorsk organiseerde een dag van de Rode Terreur, waarbij 300 mensen in een dag werden vermoord. Om aan het getal driehonderd te komen werd er besloten om ook mensen die in het ziekenhuis bevonden te vermoorden. In totaal werden er tussen de 300.000 en 500.000 Kozakken vermoord of gedeporteerd in 1919 en 1920. In juni 1919 beweerde Lenin dat de excessen te wijten waren aan onvolwassen enthousiasme van lokale beambten, maar het Orgburo wist al langer van de moorden. De organisatie en methoden van de Tsjeka In 1921 had de Tsjeka ongeveer 200.000 manschappen. Deze Tsjeka-detachementen bemanden de concentratiekampen, deden aan voedselinbeslagnames en sloegen opstanden van arbeiders en boeren neer. Ook muiterijen in het Rode Leger werden gestraft door de Tsjeka. Desertie werd gestraft door de executie van de deserteurs of van hun familieleden die gegijzeld waren door de bolsjewieken. De omstandigheden in de concentratiekampen waren slecht, wat leidde tot een hoog dodental. Volgens Tsjeka-leider Martin Latsis stierven mensen in de kampen als vliegen door de kou en voedseltekort. Bewakers gebruikten gearresteerde vrouwen als prostituees in ruil voor zaken waarmee de vrouwen konden overleven. De methoden van de Tsjeka konden verschillen per regio. In de Izvestia van Brjansk stond dat mensen die de belasting niet konden betalen om die reden naakt vastgeketend werden buiten in de sneeuw. In Charkov gebruikte de Tsjeka de zogenaamde handschoenentechniek, waarbij ze de handen van hun gevangen onderdompelden in kokend water totdat de verbrande huid eraf kon worden gehaald, waarbij het slachtoffer achterbleef met pijnlijke ontvelde bloedende handen, terwijl de martelaar achterbleef met "handschoenen" van mensenhuid. De Tsjeka in Kiev plaatste een kooi met ratten aan het lichaam van hun slachtoffers, waarna de kooi werd verhit, zodat de ratten een weg baanden door het lichaam van het slachtoffer om aan de hitte te ontkomen. In Odessa werden mensen aan planken vastgeketend en langzaam in een oven of een tank vol kokend water geschoven. In oblast Voronezj werden slachtoffers naakt in tonnen die beslagen waren met spijkers gezet en van een heuvel afgerold. In Armavir kregen slachtoffers een leren band met een ijzeren bol om het hoofd gewikkeld, die werd aangetrokken zodat hun schedels verbrijzelden. In andere plaatsen werden slachtoffers naakt buiten neergezet met temperaturen flink onder de nul, waarna er emmers met water over de slachtoffers heen werden gegooid, zodat de slachtoffers op den duur niks anders waren dan levenloze bevroren "standbeelden". De Tsjeka hoefde geen verantwoording afleggen aan overheidsorganen, alleen aan het centrale comité van de Russische Communistische Partij en vanaf 1919 aan het Politbureau. De regering en de ministerraad hadden geen zeggenschap over de Tsjeka. De activiteiten van de Tsjeka leidden tot kritiek van Kamenev en Boecharin, maar alle voorstellen tot afschaffing of hervorming van de Tsjeka werden tegengehouden door Lenin, Stalin en Trotski. In 1922 zijn de archieven van de Tsjeka op bevel van Lenin grotendeels vernietigd. Communisme Russische Burgeroorlog Geschiedenis van Rusland
{ "redpajama_set_name": "RedPajamaWikipedia" }
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\section{Introduction} The abelian complexity of infinite words has been examined by Coven and Hedlund in \cite{CH73} as an alternative way to characterize periodic sequences and Sturmian sequences. Richomme, Saari, and Zamboni introduced this notion formally in \cite{RSZ11} which initiated a general study of the abelian complexity of infinite words over finite alphabets. For example, the abelian complexity functions of some notable sequences, such as the Thue-Morse sequence and all Sturmian sequences, were studied in \cite{RSZ11} and \cite{CH73} respectively. There also many other works devoted to this subject, see \cite{BBT11,BR13,CR11, RSZ10} and references therein. In the following, we will give the definition of the abelian complexity. Let $\mathbf{w}=w(0)w(1)w(2)\cdots $ be an infinite sequence on a finite alphabet $\mathcal{A}$. Denote by ${\mathcal{F}_{\mathbf{w}}(n)}$ the set of all factors of $\mathbf{w}$ of length $n$, i.e., \[{\mathcal{F}_{\mathbf{w}}}(n): = \{w(i)w(i + 1)\cdots w(i + n - 1) : i \geq 0 \}.\] Two finite words $u$, $v$ over a same alphabet $\mathcal{A}$ is \emph{abelian equivalent} if $|u|_{a}=|v|_{a}$ for any letter $a\in\mathcal{A}$. The abelian equivalent induces an equivalent relation, denoted by $\sim_{ab}$. Now we are ready to state the definition of the abelian complexity. \begin{definition} The \emph{abelian complexity function} ${\rho_{\mathbf{w}}}:~\mathbb{N} \to \mathbb{N}$ of $\mathbf{w}$ is defined by \[\rho_{\mathbf{w}}(n) := \# (\mathcal{F}_{\mathbf{w}}(n)/\sim_{ab}).\] \end{definition} First part of this paper is devoted to study the regularity of the abelian complexity of the Rudin-Sharpiro sequence $\mathbf{r}=r(0)r(1)r(2)\cdots$ whose generating function $R(z):=\sum_{n\geq 0}r(n)z^{n}$ satisfies the Mahler type functional equation \[R(z)+R(-z)=2R(z^{2}).\] Denote the coefficient sequence of $R(-z)$ by $\mathbf{r^{\prime}}$. To state our result, we shall recall the definition of $k$-regular and automatic sequences. (For more detail, see \cite{ALL}.) \begin{definition} Let $k\geq 2$ be an integer. The \emph{$k$-kernel} of an infinite sequence $\mathbf{w}=(w(n))_{n\geq 0}$ is the set of sub-sequences $${\mathbf{K}_k}(\mathbf{w}):=\{ (w({k^e}n + c))_{n \ge 0}~|~ {e \ge 0,0 \le c < k^e}\} .$$ $\mathbf{w}$ is \emph{$k$-automatic} if $\mathbf{K}_k(\mathbf{w})$ is finite. If the $\mathbb{Z}$-module generated by its $k$-kernel is finitely generated, then $\mathbf{w}=(w(n))_{n\geq 0}$ is \emph{$k$-regular}. \end{definition} Now we state our first result. \begin{result} The abelian complexity of the Rudin-Shapiro sequence $\mathbf{r}$, which is the same as the abelian complexity of $\mathbf{r}^{\prime}$, is $2$-regular. \end{result} \begin{figure}[htbp]\label{fig:lambda} \centering \includegraphics[scale=.4]{Lambda.jpg} \caption{The graph of $\lambda(x)$ for $x\in [1,4]$.} \end{figure} In the second part, in sprite by the work of Brillhart, Erd\H{o}s and Morton \cite{BEM}, we study the limit function \[\lambda(x):=\lim_{k\to\infty}\frac{\rho(4^{k}x)}{\sqrt{4^{k}x}}\] where $\rho(x):=\rho(\lfloor x\rfloor)$ for any $x> 0$. The function $\lambda$ is continuous and non-differentiable almost everywhere, for detail see \cite{CLWW}. Further, $\lambda(x)$ is self-similar in the sense that $\lambda(x)=\lambda(4x)$ for any $x>0$. The graph of $\lambda(x)$ on $[1,4]$, which is illustrated in figure \ref{fig:lambda}, has potential to be a fractal curve; and it is. In fact, we prove the following result. \begin{result2} The box dimension of the graph of $\lambda(x)$ on any sub-interval of $(0,+\infty)$ is $3/2$. \end{result2} A variety of interesting fractals, both of theoretical and practical importance, occur as graphs of functions. Yue proved in \cite{Yue95} that the graph of one limit function studied in \cite{BEM} also has box dimension $3/2$. With a full probability, one dimensional Brownian sample function has Hausdorff dimension and box dimension $3/2$, see \cite[Theorem 16.4]{F04}. For any $b\geq 2$, the graph of Weierstarss function $W(x)=\sum_{n=0}^{\infty} b^{-n/2}\cos (b^{n}x)$ has Hausdorff dimension and box dimension $3/2$, see for example \cite{F04,Shen15} and references therein. For the Hausdorff dimension of the graph of $\lambda(x)$, Theorem B poses a good candidate $3/2$. It is natural to conjecture that the Hausdorff dimensions of the graphs of $\lambda(x)$ equals $3/2$. The outline of this paper is as follows. In Section 2, we state basic definitions and notation. In Section 3, we give the recurrence relations of the abelian complexity function of the Rudin-Shapiro sequence $\mathbf{r}$ and $\mathbf{r^{\prime}}$. As a consequence, the abelian complexity function of the Rudin-Shapiro sequence is $2$-regular, and the first difference of the abelian complexity function of the Rudin-Shapiro sequence is $2$-automatic. In the last section the box dimension of the graph of the function $\lambda(x)$ is studied. \section{Preliminary} In this section, we will introduce some notation and give the definitions of the abelian complexity function and the Rudin-Shapiro sequence. \subsection{Finite and infinite words} An \emph{alphabet} $\mathcal{A}$ is a finite and non-empty set (of symbols) whose elements are called \emph{letters}. A (finite) \emph{word} over the alphabet $\mathcal{A}$ is a concatenation of letters in $\mathcal{A}$. The concatenation of two words ${u} = u(0)u(1) \cdots u(m)$ and ${v} = v(0)v(1) \cdots v(n)$ is the word ${uv} = u(0)u(1) \cdots u(m)v(0)v(1) \cdots v(n)$. The set of all finite words over $\mathcal{A}$ including the \emph{empty word} $\varepsilon $ is denoted by $\mathcal{A}^*$. An infinite word $\mathbf{w}$ is an infinite sequence of letters in $\mathcal{A}$. The set of all infinite words over $\mathcal{A}$ is denoted by $\mathcal{A}^{\mathbb{N}}$. The \emph{length} of a finite word ${w}\in \mathcal{A^*}$, denoted by $|w|$, is the number of letters contained in $w$. We set $\left| \varepsilon \right| = 0$. For any word $u\in\mathcal{A}^{*}$ and any letter $a \in \mathcal{A}$, denote by $|{u}|_a$ the number of occurrences of $a$ in ${u}$. A word $w$ is a factor of a finite (or an infinite) word $v$, written by $w\prec v$ if there exist a finite word $x$ and a finite (or an infinite) word $y$ such that $v=xwy$. When $x=\varepsilon$, ${w}$ is called a \emph{prefix} of ${v}$, denoted by ${w} \triangleleft {v}$; when $y=\varepsilon$, $ w$ is called a suffix of ${v}$, denoted by ${w} \triangleright {v}$. \subsection{Digit sums} Now we assume that the alphabet $\mathcal{A}$ is composed of integers. Let $\mathbf{w}=w(0)w(1)w(2)\cdots \in \mathcal{A}^{\mathbb{N}}$ be an infinite word. For any $i\geq 0$ and $n\geq 1$, the sum of consecutive $n$ letters in $\mathbf{w}$ starting from the position $i$ is denoted by \[\Sigma_{\mathbf{w}}(i,n):=\sum_{j=i}^{i+n-1} w(j).\] The \emph{maximal sum} and \emph{minimal sum} of consecutive $n~(n\geq 1)$ letters in $\mathbf{w}$ are denoted by \[ M_{\mathbf{w}}(n):=\max_{i\geq 0}\Sigma_{\mathbf{w}}(i,n) \textrm{ and } % m_{\mathbf{w}}(n):=\min_{i\geq 0}\Sigma_{\mathbf{w}}(i,n).\] In addition, we always assume that $M_{\mathbf{w}}(0)=m_{\mathbf{w}}(0)=0$. Denote the digit sum of a finite word ${u}=u(0)\cdots u(|{u}|-1)\in\mathcal{A}^{*}$ by \[\mathrm{DS}(u):= \sum_{j=0}^{|{u}|-1} u(j),\] then \[M_{\mathbf{w}}(n)=\max\big\{\mathrm{DS}(v):v\in\mathcal{F}_{\mathbf{w}}(n)\big\}\] and \[m_{\mathbf{w}}(n)=\min\big\{\mathrm{DS}(v):v\in\mathcal{F}_{\mathbf{w}}(n)\big\}.\] \medskip The abelian complexity function of an infinite word $\mathbf{w}$ over $\{-1,1\}$ is closely related to the digit sums of factors of $\mathbf{w}$. \begin{proposition}\label{lem:aMm Let $\mathbf{w}\in\{-1,1\}^{\mathbb{N}}$. Then \[\rho_{\mathbf{w}}(n)=\frac{M_{\mathbf{w}}(n)-m_{\mathbf{w}}(n)}{2}+1.\] \end{proposition} \begin{proof}For a proof one can refer to \cite[Proposition 2.2]{BBT11}. \end{proof} \subsection{The Rudin-Shapiro sequence $\mathbf{r}$ and a related sequence $\mathbf{r^{\prime}}$} The Rudin-Shapiro sequence \[\mathbf{r}=r(0)r(1)\cdots r(n)\cdots \in {\{-1,1\}}^{\mathbb{N}}\] is given the following recurrence relations: \begin{equation}\label{eq:recurrent} r(0)=1, ~r(2n)=r(n),~r(2n+1)=(-1)^nr(n) \quad (n\geq 0). \end{equation} The generating function $R(z)=\sum_{n\geq 0}r(n)z^{n}$ of the Rudin-Shapiro sequence satisfies the following Mahler type functional equation \[R(z)+R(-z)=2R(z^{2}).\] We also study the coefficient sequence of $R(-z)$, denoted by \[\mathbf{r^{\prime}}=r^{\prime}(0)r^{\prime}(1)\cdots\in\{-1,1\}^{\mathbb{N}}.\] Apparently, $r^{\prime}(n)=(-1)^{n}r(n)$ for all $n\geq 0$. Thus \begin{equation} r^{\prime}(0)=1,~r^{\prime}(2n)=(-1)^{n}r^{\prime}(n),~r^{\prime}(2n+1)=-r^{\prime}(n)\quad (n\geq 0). \end{equation} The Rudin-Shapiro sequence can also be generated by a substitution in the following way. Let $\sigma:\{a,b,c,d\}\to\{a,b,c,d\}^*$ and $\tau, \tau^{\prime}:\{a,b,c,d\}\to \{-1,1\}^{*}$ where \begin{equation*} \begin{array}{rllll} \sigma:& a\mapsto ab,& b\mapsto ac ,& c\mapsto db,& d\mapsto dc,\\ \tau: & a\mapsto 1, & b\mapsto 1, & c\mapsto -1,& d\mapsto -1,\\ \tau^{\prime}: & a\mapsto 1, & b\mapsto -1, & c\mapsto 1,& d\mapsto -1. \end{array} \end{equation*} Let $\mathbf{s} := \sigma^{\infty}(a)$ be the fix point of $\sigma$ leading by $a$. Then \[\mathbf{r} = \tau(\sigma^{\infty}(a)) \text{ and } \mathbf{r^{\prime}}= \tau^{\prime}(\sigma^{\infty}(a)).\] Denote by $\mathcal{M}_{\mathbf{s}}(n)$ (and $\mathcal{M}^{\prime}_{\mathbf{s}}(n)$) the set of all the factors of length $n$ in $\mathbf{s}$ such that the sum of letters of such factor under coding $\tau$ (and $\tau^{\prime}$, respectively) attains the maximal value, i.e., \begin{align*} \mathcal{M}_{\mathbf{s}}(n) &:= \{ {u \in \mathcal{F}_{\mathbf{s}}(n)~:~ S(u) = M_{\mathbf{r}}(n)}\},\\ \mathcal{M}^{\prime}_{\mathbf{s}}(n) &:= \{ {u \in \mathcal{F}_{\mathbf{s}}(n)~:~ S^{\prime}(u) = M_{\mathbf{r^{\prime}}}(n)}\} \end{align*} where $S:= \text{DS}\circ\tau$ and $S^{\prime}:= \text{DS}\circ\tau^{\prime}$. \section{The Regularity of the abelian Complexity of $\mathbf{r}$ and $\mathbf{r^{\prime}}$} In this section, we will discuss the regularity of the abelian complexity function of the Rudin-Shapiro sequence $\mathbf{r}$ and the sequence $\mathbf{r^{\prime}}$. From now on, unless otherwise stated, we always set $\mathcal{A} = \{-1,1\}.$ \subsection{Statement of results} \begin{theorem} \label{thm:abelcomp} For any $n\geq 1$, \[M_{\mathbf{r}}(n)=M_{\mathbf{r^{\prime}}}(n)=:M(n).\] Moreover, $M(1)=1$, $M(2)=2$, $M(3)=3$ and for $n\geq 1,$ \begin{align*} M(4n)&= 2M(n)+2, & M(4n+1)&=2M(n)+1,\\ M(4n+2)&=M(n)+M(n+1)+1,& M(4n+3)&=2M(n+1)+1. \end{align*} \end{theorem} \begin{corollary}\label{cor:abelcomp} The sequence $(M(n))_{\cc{n\geq 0}}$ is $2$-regular. \end{corollary} \begin{proof} The result follows from Theorem \ref{thm:abelcomp} , \cite[Theorem 16.1.3 (e)]{ALL} and \cite[Theorem 2.9]{AS92}. \end{proof} For all $n\geq 0$, let \[\Delta M(n):= M(n+1)-M(n).\] The difference sequence $(\Delta M(n))_{n\geq 0}$ is characterized by the following result. \begin{corollary}\label{cor:abeldiff} $\Delta M(i)=1$ for $0\leq i\leq 3$, and for $n \geq 1$, \begin{equation}\label{eq:delta} \left\{ \begin{array}{ccccl} \Delta M(4n) &=&-\Delta M(4n+3)&=&-1,\\ \Delta M(4n+1)&=&\Delta M(4n+2)&=&\Delta M(n). \end{array} \right. \end{equation} Moreover, $(\Delta M(n))_{n\geq 0}$ is a $2$-automatic sequence. \end{corollary} \begin{proof} The difference sequence $(\Delta M(n))_{n\geq 0}$ can be generated by the automaton given in Figure \ref{fig:coro1}. \begin{figure}[htbp] \centering \begin{tikzpicture}[scale=0.8, every node/.style={scale=0.8}, state/.style={scale=0.8, circle solidus,draw, inner sep=1pt,minimum size=12mm},>=stealth,->,auto,black] \node[state] (a) {$q_{0}$ \nodepart{lower} $1$}; \node[state] (b) [right=15mm of a] {$q_{1}$ \nodepart{lower} $1$}; \node[state] (c) [right=15mm of b] {$q_{2}$ \nodepart{lower} $-1$}; \node (initial) [left=8mm of a] {Start}; \draw [->] (initial) to (a); \draw [->] (b) to [out=120, in=60,loop,distance=10mm] node [above] {$1,2,3$} (b); \draw [->] (a) to [out=120, in=60,loop,distance=10mm] node [above] {$0$} (a); \draw [->] (a) to node [above] {$1,2,3$} (b); \draw [->] (c.200) to [bend left] node [below] {$3$} (b.-20); \draw [->] (b.20) to [bend left] node [above] {$0$} (c.160); \draw [->] (c) to [out=120, in=60,loop,distance=10mm] node [above] {$0,1,2$} (c); \end{tikzpicture} \caption{The automaton that generates $(\Delta M(n))_{n\geq 0}$.}\label{fig:coro1} \end{figure} \iffalse The recurrence relations follow directly from Theorem \ref{thm:abelcomp}. Now we will show that $(\Delta M(n))_{n\geq {\color{green}1}}$ is a $4$-automatic sequence. Consider the morphism $\theta:\{0,-1,1\}^{\mathbb{N}}\to \{0,-1,1\}^{\mathbb{N}}$ given by \[0\mapsto 0111, -1\mapsto (-1)(-1)(-1)1, ~~1\mapsto (-1)111.\] Let $(a_{n})_{n\geq 0}=\theta^{\infty}(0)$. Clearly, $a_{1}=a_{2}=a_{3}=1$ and for any $n\geq 1$, \[a_{4n}a_{4n+1}a_{4n+2}a_{4n+3}=\theta(a_{n})=(-1)a_{n}a_{n}1,\] which coincides the recurrence relations \eqref{eq:delta} for any $n\geq 1$. This implies that $a_{n}=M(n)$ for all $n\geq 1$. Note that by Cobham's theorem \cite{Cob} (see also \cite[Theorem 6.3.2]{ALL}), $(a_{n})_{n\geq 0}$ is a $4$-automatic sequence. Then by \cite[Theorem 5.4.1]{ALL}, $(\Delta M(n))_{n\geq 1}$ is also a $4$-automatic sequence. \lu{Meanwhile, a sequence $\bm{a}=(a_i)_{i\geq 0}$ is $k$-automatic if and only if it is $k^m$-automatic. Thus $(\Delta M(n))_{n\geq 1}$ is $2$-automatic.} \fi \end{proof} \begin{theorem}\label{regular:abel} For any $n\geq 1$, \[\rho_{\mathbf{r}}(n)=\rho_{\mathbf{r^{\prime}}}(n):=\rho(n).\] Moreover, $(\rho(n))_{n\geq 0}$ is $2$-regular. \end{theorem} \begin{proof} This result follows from Theorem \ref{thm:abelcomp} and Lemma \ref{lem:rho}. \end{proof} \subsection{Some lemmas} To prove Theorem \ref{thm:abelcomp}, we need the following lemmas. \begin{lemma}\label{lem:iterate} For any word $w \in \{a,b,c,d\}^{*}$, we have \[ S(\sigma^{2}(w))=2S(w) \text{ and } S^{\prime}(\sigma^{2}(w))=2S^{\prime}(w). \] \end{lemma} \begin{proof}Observing that both $S$ and $S^{\prime}$ are morphism from $(\{a,b,c,d\}^{*},\cdot)$ to $(\mathbb{Z},+)$ where `$\cdot$' is the concatenation of words, we only need to show the equalities in the lemma hold for any letter $x\in\{a,b,c,d\}$. By the definition of $\sigma$, we get $${\sigma^2}: a\mapsto abac, b\mapsto abdb , c\mapsto dcac, d\mapsto dcdb.$$ Recall that $\tau : a\mapsto 1, b\mapsto 1 , c\mapsto -1, d\mapsto -1$. Thus \begin{align*} S({\sigma ^2}(a)) &= S(abac) =\mathrm{DS}\circ\tau(abac)=\mathrm{DS}(111 (- 1)) = 2 = 2S(a). \end{align*} One can verify the rest cases in the same way. \end{proof} \begin{lemma}\label{lem:Mm} For any $n\geq 1$, \[M_{\mathbf{a}}(n)+m_{\mathbf{a}}(n)=0,\] where $\mathbf{a}$ represents the Rudin-Shapiro sequence $\mathbf{r}$ or the sequence $\mathbf{r^{\prime}}$. \end{lemma} \begin{proof} We only prove the case $\mathbf{a}=\mathbf{r}$. The result for $\mathbf{a}=\mathbf{r^{\prime}}$ follows in the same way. Let $\mu$ be the coding \[\mu: a\mapsto d, b\mapsto c, c\mapsto b, d\mapsto a.\] Then $\mu\circ\sigma=\sigma\circ\mu$ and $\mu\circ\mu={\rm Id}$. We shall start by proving the following two facts: for any $W\in\{a,b,c,d\}^{n}$ ($n\geq 1$), \begin{enumerate} \item $W$ is a factor of $\mathbf{s}$ if and only if $\mu(W)$ is a factor of $\mathbf{s}$\ww{;} \item $S(W)=M_{\mathbf{r}}(n)$ if and only if $S(\mu(W))=m_{\mathbf{r}}(n)$. \end{enumerate} For the fact $1$, if $W$ is a factor of $\mathbf{s}$, then $W$ is a factor of $\sigma^{k}(a)$ for some $k$. Therefore $\mu(W)$ is a factor of $\mu(\sigma^{k}(a))=\sigma^{k}(d)$ which is a factor of $\sigma^{k+4}(a)$. Hence $\mu(W)$ is also a factor of $\mathbf{s}$. The converse holds in the same argument by replacing $W$ by $\mu(W)$. Now we will prove fact $2$. Suppose $S(W)=M_{\mathbf{r}}(n)$ and $S(\mu(W))\neq m_{\mathbf{r}}(n)$. Without lose of generality, assume that $S(\mu(W))> m_{\mathbf{r}}(n)$. This means there exists a word $W^{\prime}$ of length $n$, such that $|W^{\prime}|_{-1}>|\mu(W)|_{-1}$. Therefore \[|\mu(W^{\prime})|_{1}=|W^{\prime}|_{-1}>|\mu(W)|_{-1}=|W|_{1}.\] It follows that $M_{\mathbf{r}}(n)=S(W)<S(\mu(W^{\prime}))$ which is a contradiction. The converse can be proved by using the similar argument. Noticing that $S(\mu(W))=-S(W)$, then by fact 1 and 2, the proof is completed. \end{proof} \begin{lemma}\label{lem:rho} For any $n\geq 1$, \[ \rho_{\mathbf{a}}(n) = M_{\mathbf{a}}(n)+1, \] where $\mathbf{a}$ represents the Rudin-Shapiro sequence $\mathbf{r}$ or the sequence $\mathbf{r^{\prime}}$. \end{lemma} \begin{proof} The result follows from Proposition \ref{lem:aMm} and Lemma \ref{lem:Mm}. \end{proof} The following lemma characterizes digit sums $\Sigma_{\mathbf{r}}(\cdot,\cdot)$ which is useful in the study of $M_{\mathbf{r}}$. \begin{lemma}\label{lem:sum} For any $n\geq 1, i\geq 0$, we have \begin{enumerate}[(1)] \item $\Sigma_{\mathbf{r}}(4i,4n) = 2\Sigma_{\mathbf{r}}(i,n),$ \item $ \Sigma_{\mathbf{r}}(4i + 1,4n) = \Sigma_{\mathbf{r}}(i,n) + \Sigma_{\mathbf{r}}(i + 1,n), $ \item $\Sigma_{\mathbf{r}}(4i + 2,4n) = 2\Sigma_{\mathbf{r}}(i + 1,n),$ \item $\Sigma_{\mathbf{r}}(4i + 3,4n) = 2\Sigma_{\mathbf{r}}(i + 1,n) - r(4i + 4n + 3) + r(4i + 3), $ \item $ \Sigma_{\mathbf{r}}(4i,4n + 1) = 2\Sigma_{\mathbf{r}}(i,n) + r(i + n),$ \item $ \Sigma_{\mathbf{r}}(4i + 1,4n + 1) = 2\Sigma_{\mathbf{r}}(i + 1,n) + r(i), $ \item $ \Sigma_{\mathbf{r}}(4i + 2,4n + 1) = 2\Sigma_{\mathbf{r}}(i + 1,n) + r(4i + 4n + 2),$ \item $ \Sigma_{\mathbf{r}}(4i + 3,4n + 1) = 2\Sigma_{\mathbf{r}}(i + 1,n) + r(4i + 3); $ \item $ \Sigma_{\mathbf{r}}(4i,4n + 2) = \Sigma_{\mathbf{r}}(i,n)+\Sigma_{\mathbf{r}}(i,n + 1)+r(i+n),$ \item $ \Sigma_{\mathbf{r}}(4i + 1,4n + 2) = \Sigma_{\mathbf{r}}(i + 1,n) + \Sigma_{\mathbf{r}}(i,n + 1) + r(4i + 4n + 2), $ \item $ \Sigma_{\mathbf{r}}(4i + 2,4n + 2) = \Sigma_{\mathbf{r}}(i + 1,n)+\Sigma_{\mathbf{r}}(i + 1,n+1)-r(i+n+1),$ \item $\Sigma_{\mathbf{r}}(4i + 3,4n + 2) = \Sigma_{\mathbf{r}}(i + 1,n) +\Sigma_{\mathbf{r}}(i + 1,n + 1) + r(4i + 3);$ \item $ \Sigma_{\mathbf{r}}(4i,4n + 3) = 2\Sigma_{\mathbf{r}}(i,n + 1) - r(4i + 4n + 3),$ \item $ \Sigma_{\mathbf{r}}(4i + 1,4n + 3) = 2\Sigma_{\mathbf{r}}(i,n + 1) - r(i), $ \item $ \Sigma_{\mathbf{r}}(4i + 2,4n + 3) = 2\Sigma_{\mathbf{r}}(i + 1,n + 1) - r(i + n + 1),$ \item $ \Sigma_{\mathbf{r}}(4i + 3,4n + 3) = 2\Sigma_{\mathbf{r}}(i + 1,n + 1) + r(4i + 3).$ \end{enumerate} \end{lemma} \begin{proof} By \eqref{eq:recurrent} we have for all $n\geq 0$ $${r(4n)} = {r(4n + 1)} = r(n),~r(4n+2) =- r(4n+3) = {( - 1)^n}r(n). $$ Then by the previous equations and the definition of $\Sigma_{\mathbf{r}}$, these $16$ equations can be verified directly. Here we give the proof of the first two equations as examples: \begin{align*} \Sigma_{\mathbf{r}}(4i,4n) & = \sum_{j = 4i}^{4i + 4n - 1} {r(j)} \\ &=\sum_{j=i}^{i+n-1}(r(4j)+r(4j+1)+r(4j+2)+r(4j+3))\\ & =\sum_{j=i}^{i+n-1}(r(j)+r(j)+(-1)^{j}r(j)-(-1)^{j}r(j))\\ &=2\sum_{j=i}^{i+n-1}r(j)=2\Sigma_{\mathbf{r}}(i,n).\\ \Sigma_{\mathbf{r}}(4i+1,4n) & = \Sigma_{\mathbf{r}}(4i,4n)+r(4i+4n)-r(4i)=2\Sigma_{\mathbf{r}}(i,n)+r(i+n)-r(i)\\ &=\Sigma_{\mathbf{r}}(i,n)+\Sigma_{\mathbf{r}}(i+1,n); \end{align*} The rest equations can be proved in the same way. \end{proof} \begin{remark} Lemma \ref{lem:sum} implies that the double sequence $(\Sigma_{\mathbf{r}})_{i\geq 0,n\geq 1}$ is a two-dimension {$2$}-regular sequence. For a definition of two-dimensional regular sequences, see \cite{ALL}. \end{remark} The following lemma gives upper bounds of the maximal values of the sums of consecutive $n$ terms of $\mathbf{r}$ and $\mathbf{r}^{\prime}$. \begin{lemma}\label{lem:upperbound} For any $n\geq 1$, \begin{eqnarray*} M_{\mathbf{r}}(4n)& \leq & 2M_{\mathbf{r}}(n)+2,\\ M_{\mathbf{r}}(4n+1)&\leq & 2M_{\mathbf{r}}(n)+1,\\ M_{\mathbf{r}}(4n+2)&\leq & M_{\mathbf{r}}(n)+M_{\mathbf{r}}(n+1)+1,\\ M_{\mathbf{r}}(4n+3)&\leq & 2M_{\mathbf{r}}(n+1)+1. \end{eqnarray*} Moreover, the above inequalities also holds for $M_{\mathbf{r^{\prime}}}$. \end{lemma} \begin{proof} For the first inequality, we shall use the first four equations of Lemma \ref{lem:sum}. By equations (1) to (3) of Lemma \ref{lem:sum}, we obtain that for $k=0,1,2$, \begin{align*} \Sigma_{\mathbf{r}}(4i+k,4n) &\leq \max\{2\Sigma_{\mathbf{r}}(i,n), \Sigma_{\mathbf{r}}(i,n)+\Sigma_{\mathbf{r}}(i+1,n),2\Sigma_{\mathbf{r}}(i+1,n)\}\\ &\leq 2M_{\mathbf{r}}(n). \end{align*} When $k=3$, by equation (4) of Lemma \ref{lem:sum}, we have \begin{align*} \Sigma_{\mathbf{r}}(4i+k,4n) &= 2\Sigma_{\mathbf{r}}(i+1,n)-r(4i+4n+3)+r(4i+3)\\ &\leq 2M_{\mathbf{r}}(n)+2. \end{align*} Therefore \(M_{\mathbf{r}}(4n)\leq 2M_{\mathbf{r}}(n)+2.\) In a similar way, using the rest 12 equations of Lemma \ref{lem:sum}, we can prove the rest three inequalities for $M_{\mathbf{r}}$. To prove the result for $M_{\mathbf{r^{\prime}}}$, one can deduce a similar result to Lemma \ref{lem:sum} for $\mathbf{r^{\prime}}$, and apply the similar argument as above. We left the details to the reader. \end{proof} \iffalse \begin{proof}\color{cyan} This lemma can be proved by the recurrence relation (\ref{eq:recurrent}). However, we will give a more intuitive proof. It is enough to prove the first inequality. The rest of them follows in the same way. Let $\mathcal{B}:=\{a,b,c,d\}$. Any word in $W\in\mathcal{B}^{4n}$ must be a factor of the word \[\sigma^{2}(xUy)=x_{1}x_{2}x_{3}x_{4}u_{1}\cdots u_{4n}y_{1}y_{2}y_{3}y_{4}\] for some $x,y\in\mathcal{B}$ and $U\in\mathcal{B}^{n}$. Notice that for any $i\in\mathcal{B}$, $s(\tau(\sigma^{2}(i)))=2s(\tau(i))$. To be continued... \end{proof} \fi \subsection{Proof of Theorem \ref{thm:abelcomp}} To prove Theorem \ref{thm:abelcomp}, we only need to show that all equalities in Lemma \ref{lem:upperbound} hold. For this, we will construct two sequences of words which attain the upper bounds in Lemma \ref{lem:upperbound} for $\mathbf{r}$ and $\mathbf{r^{\prime}}$ respectively. These will be done in the following Lemma \ref{lem:maxfactor} and \ref{lem:maxfactor2}. Then Theorem \ref{thm:abelcomp} follows directly from Lemma \ref{lem:upperbound}, \ref{lem:maxfactor} and \ref{lem:maxfactor2}. Now we will give the sequence of words for $\mathbf{r}$. Let $(W_{n})_{n\geq 1}$ be the sequence of words defined by $W_{1}=a$, $W_{2}=ba$, $W_{3}=aba$ and \begin{equation}\label{eq:maxW}\left\{ \begin{array}{ccl} {W_{4n}} &=& b{\sigma ^2}({W_n}){c^{ - 1}}, \\ {W_{4n + 1}}& =& b{\sigma ^2}({W_n}), \\ {W_{4n + 2}} &= & \begin{cases} b{\sigma ^2}(W_{n+1}){(bac)^{ - 1}}& \text{ if } \Delta M_{\mathbf{r}}(n) = 1, \\ cdb{\sigma ^2}({W_n}){c^{-1}} & \text{ if } \Delta M_{\mathbf{r}}(n) = - 1, \end{cases}\\ {W_{4n + 3}} &=& {\sigma ^2}({W_{n + 1}}){c^{ - 1}}. \end{array}\right. \end{equation} \begin{lemma}\label{lem:maxfactor} Let $(W_{n})_{n\geq 1}\subset\{a,b,c,d\}^{*}$ given by (\ref{eq:maxW}). Then for any $n\geq 1$, \begin{enumerate}[\indent$(i)$] \item either $bW_{n}\prec \mathbf{s}$ or $dW_{n}\prec\mathbf{s}$ holds;\label{l:r2} \item either $a\triangleright W_{n}$ or $c\triangleright W_{n}$ holds; \label{l:r3} \item $W_{n}\in\mathcal{M}_{\mathbf{s}}(n)$.\label{l:r1} \end{enumerate} \end{lemma} \begin{proof} We shall prove $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$ simultaneously by induction. \emph{Step 1.} We shall show that the results hold for $n< 8$. Let $(W_{n})_{n=1}^{7}$ be the words given in table \ref{tab:1}. For $n=1,2,3,4$, apparently $M_{\mathbf{r}}(n)=S(W_{n})$ which implies $W_{n}\in\mathcal{M}_{\mathbf{s}}(n)$. Since $S(W_{5})=2M_{\mathbf{r}}(1)+1$, $S(W_{6})=M_{\mathbf{r}}(1)+M_{\mathbf{r}}(2)+1$ and $S(W_{7})=2M_{\mathbf{r}}(2)+1$, by Lemma \ref{lem:upperbound}, we have $S(W_{n})=M_{\mathbf{r}}(n)$ and $W_{n}\in\mathcal{M}_{\mathbf{s}}(n)$ for $n=5,6,7$. Therefore $(\ref{l:r1})$ holds for $n< 8$. Notice that $(W_{n})_{n=1}^{7}$ are factors of $\sigma^{2}(dba)=dcdbabdbabac$ which is a factor of $\mathbf{s}$, $(\ref{l:r2})$ and $(\ref{l:r3})$ also hold for $n< 8$. \begin{table}[htbp] \centering \begin{tabular}{|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline W_{n} & a & ba & aba & baba & babac & babdba & abdbaba \\ \hline M_{\mathbf{r}}(n) & 1 & 2 & 3 & 4 & 3 & 4 & 5 \\ \hline \end{tabular} \caption{The initial values for Lemma \ref{lem:maxfactor}}\label{tab:1} \end{table} \emph{Step 2.} Assuming that $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$ hold for $n< 4k$ $(k\geq 2)$, we will prove the results for $4k\leq n <4(k+1)$. The proof in this step will be separated into the following two cases. \noindent\textbf{Case 1:} $\Delta M_{\mathbf{r}}(k)=1$. In this case, the induction hypotheses $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$ yield the following facts: \begin{enumerate}[\indent ({1}a)] \item $W_{k}\in\mathcal{M}_{\mathbf{s}}(k)$ and $W_{k+1}\in\mathcal{M}_{\mathbf{s}}(k+1)$; \item $db\sigma^{2}(W_{k})$ and $db\sigma^{2}(W_{k+1})$ are factors of $\mathbf{s}$; \item either $a\triangleright W_{k}$ or $c\triangleright W_{k}$ holds, and $a\triangleright W_{k+1}$. \end{enumerate} (In the last statement (1c), we can exclude the case $c\triangleright W_{k+1}$ since $\Delta M_{\mathbf{r}}(k)=1$. In fact, if $W_{k+1}=Wc$, then \[M_{\mathbf{r}}(k+1)=S(W_{k+1})=S(W)+S(c)=S(W)-1\leq M_{\mathbf{r}}(k)-1,\] which contradicts the assumption $\Delta M_{\mathbf{r}}(k)=M_{\mathbf{r}}(k+1)-M_{\mathbf{r}}(k)=1$.) Now, by (\ref{eq:maxW}) and (1b), we have $dW_{n}$ is a factor of $\mathbf{s}$ for $4k\leq n\leq 4k+2$ and $bW_{4k+3}$ is a factor of $\mathbf{s},$ which implies that $(\ref{l:r2})$ holds for $4k\leq n<4(k+1)$. Moreover, this also implies \begin{equation}\label{eq:2b} W_{n} \text{ is a factor of }\mathbf{s} \text{ for }4k\leq n < 4(k+1). \end{equation} Since by the fact (1c), we have $ac\triangleright \sigma^{2}(W_{k})$ and $abac=\sigma^{2}(a)\triangleright\sigma^{2}(W_{k+1})$. Therefore (\ref{eq:maxW}) gives \begin{equation}\label{eq:2c} a\triangleright W_{4k},~c\triangleright W_{4k+1},~ a\triangleright W_{4k+2} \text{ and } a\triangleright W_{4k+3}, \end{equation} which prove $(\ref{l:r3})$. Now, by (\ref{eq:maxW}), (\ref{eq:2c}), (1a) and Lemma \ref{lem:iterate}, we have \begin{equation}\label{eq:2rec} \left\{\begin{array}{ccl} S(W_{4k}) & = & S(b)+ S(\sigma^{2}(W_{k}))-S(c) = 2M_{\mathbf{r}}(k)+2,\\ S(W_{4k+1}) & = & S(b)+ S(\sigma^{2}(W_{k})) = 2M_{\mathbf{r}}(k)+1,\\ S(W_{4k+2}) & = & S(b)+ S(\sigma^{2}(W_{k+1}))-S(bac)\\ & = & 2M_{\mathbf{r}}(k+1)=M_{\mathbf{r}}(k)+M_{\mathbf{r}}(k+1)+1,\\ S(W_{4k+3}) & = & S(\sigma^{2}(W_{k+1}))-S(c) = 2M_{\mathbf{r}}(k+1)+1.\\ \end{array} \right. \end{equation} By (\ref{eq:2b}), (\ref{eq:2rec}) and Lemma \ref{lem:upperbound}, we have $W_{n}\in\mathcal{M}_{\mathbf{s}}(n)$ for $4k\leq n < 4(k+1)$ which is $(\ref{l:r1})$. \noindent\textbf{Case 2:} $\Delta M_{\mathbf{r}}(k)=-1$. In this case, we shall first assert that $dW_{k}$ is a factor of $\mathbf{s}$. By the induction hypothesis $(\ref{l:r2})$, we only need to show that $bW_{k}$ can not be a factor of $\mathbf{s}$. If this is not the case, then \[M_{\mathbf{r}}(k+1)\geq S(bW_{k})=1+S(W_{k})=1+M_{\mathbf{r}}(k)\] where the last equality follows from $(\ref{l:r1})$. Then we have $\Delta M_{\mathbf{r}}(k)=M_{\mathbf{r}}(k+1)-M_{\mathbf{r}}(k)\geq 1$ which is a contradiction. Therefore, applying the induction hypotheses $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$, we have \begin{enumerate}[\indent (2a)] \item $W_{k}\in\mathcal{M}_{\mathbf{s}}(k)$ and $W_{k+1}\in\mathcal{M}_{\mathbf{s}}(k+1)$; \item $dcdb\sigma^{2}(W_{k})$ and $b\sigma^{2}(W_{k+1})$ are factors of $\mathbf{s}$; \item $ac\triangleright \sigma^{2}(W_{k})$ and $ac\triangleright \sigma^{2}(W_{k+1})$. \end{enumerate} By (\ref{eq:maxW}) and (2b), we have \begin{equation}\label{eq:1bb} dW_{n} \text{ is a factor of } \mathbf{s} \text{ for } 4k\leq n\leq 4k+2 \end{equation} and $bW_{4k+3}$ is a factor of $\mathbf{s}$, which prove $(\ref{l:r2})$. These imply that \begin{equation}\label{eq:1b} W_{n} \text{ is a factor of }\mathbf{s} \text{ for }4k\leq n < 4(k+1). \end{equation} Combing (2c) and (\ref{eq:maxW}), $(\ref{l:r3})$ holds for $4k\leq n < 4(k+1)$. Now, by (\ref{eq:maxW}), (2a), (2c) and Lemma \ref{lem:iterate}, we have \begin{equation}\label{eq:2rec2} \left\{\begin{array}{ccl} S(W_{4k}) & = & S(b)+ S(\sigma^{2}(W_{k}))-S(c) = 2M_{\mathbf{r}}(k)+2,\\ S(W_{4k+1}) & = & S(b)+ S(\sigma^{2}(W_{k})) = 2M_{\mathbf{r}}(k)+1,\\ S(W_{4k+2}) & = & S(cbd)+ S(\sigma^{2}(W_{k}))-S(c)\\ & = & 2M_{\mathbf{r}}(k)=M_{\mathbf{r}}(k)+M_{\mathbf{r}}(k+1)+1,\\ S(W_{4k+3}) & = & S(\sigma^{2}(W_{k+1}))-S(c) = 2M_{\mathbf{r}}(k+1)+1.\\ \end{array} \right. \end{equation} By (\ref{eq:1b}), (\ref{eq:2rec2}) and Lemma \ref{lem:upperbound}, we have $W_{n}\in\mathcal{M}_{\mathbf{s}}(n)$ for $4k\leq n < 4(k+1)$ which is $(\ref{l:r1})$. The proof is completed. \end{proof} \medskip For $\mathbf{r^{\prime}}$, let $(\widetilde{W}_{n})_{n\geq 1}$ be the sequence of words defined by $\widetilde{W}_{1}=c$, $\widetilde{W}_{2}=ca$, $\widetilde{W}_{3}=cac$ and \begin{equation}\label{eq:maxW2}\left\{ \begin{array}{ccl} {\widetilde{W}_{4n}} &=& d^{-1}{\sigma ^2}({\widetilde{W}_n}){a}, \\ {\widetilde{W}_{4n + 1}}& =& {\sigma ^2}({\widetilde{W}_n})a, \\ {\widetilde{W}_{4n + 2}} &= & \begin{cases} (dca)^{-1}{\sigma ^2}(\widetilde{W}_{n+1}){a}& \text{ if } \Delta M_{\mathbf{r^{\prime}}}(n) = 1, \\ d^{-1}{\sigma ^2}({\widetilde{W}_n}){abd} & \text{ if } \Delta M_{\mathbf{r^{\prime}}}(n) = - 1, \end{cases}\\ {\widetilde{W}_{4n + 3}} &=& d^{-1}{\sigma ^2}({\widetilde{W}_{n + 1}}). \end{array}\right. \end{equation} \begin{lemma}\label{lem:maxfactor2} Let $(\widetilde{W}_{n})_{n\geq 1}\subset\{a,b,c,d\}^{*}$ given by (\ref{eq:maxW2}). Then for any $n\geq 1$, \begin{enumerate}[\indent$(i)$] \item either $\widetilde{W}_{n}a\prec\mathbf{s}$ or $\widetilde{W}_{n}b\prec\mathbf{s}$ holds;\label{l:r2new} \item either $c\triangleleft \widetilde{W}_{n}$ or $d\triangleleft \widetilde{W}_{n}$ holds; \label{l:r3new} \item $\widetilde{W}_{n}\in\mathcal{M}'_{\mathbf{s}}(n)$.\label{l:r1new} \end{enumerate} \end{lemma} \begin{proof} The proof of this lemma is similar to Lemma \ref{lem:maxfactor}. \end{proof} \iffalse \begin{proof} We will prove this lemma in the same way as Lemma \ref{lem:maxfactor}. \emph{Step 1.} Let $(\widetilde{W}_{n})_{n=1}^{7}$ be the words given in table \ref{tab:3}. Since $(\widetilde{W}_{n})_{n=1}^{7}$ are factors of $\sigma^{2}(cac)=dcacabacdcac$ which is a factor of $\mathbf{s}$, $(\ref{l:r2})$ and $(\ref{l:r3})$ also hold for $n< 8$. (\ref{l:r1new}) can be verified directly. \begin{table}[htbp] \centering \begin{tabular}{|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline \widetilde{W}_{n} & c & ca & cac & caca & dcaca & cabaca & cacabac \\ \hline M_{\mathbf{t}}(n) & 1 & 2 & 3 & 4 & 3 & 4 & 5 \\ \hline \end{tabular} \caption{The initial values for Lemma \ref{lem:maxfactor2}}\label{tab:3} \end{table} \emph{Step 2.} Assuming that $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$ hold for $n< 4k$ $(k\geq 2)$, we will prove the results for $4k\leq n <4(k+1)$. The proof in this step will be separated into the following two cases. \noindent{\bf Case 1:} $\Delta M_{\mathbf{r^{\prime}}}(k)=1$. By the induction hypotheses $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$, we have \begin{enumerate}[\indent ({1}a)] \item $\widetilde{W}_{k}\in\mathcal{M}'_{\mathbf{s}}(k)$ and $\widetilde{W}_{k+1}\in\mathcal{M}'_{\mathbf{s}}(k+1)$; \item $\sigma^{2}(\widetilde{W}_{k})ab$ and $\sigma^{2}(\widetilde{W}_{k+1})ab$ are factors of $\mathbf{s}$; \item Either $c\triangleleft \widetilde{W}_{k}$ or $d\triangleleft \widetilde{W}_{k}$ holds, and $c\triangleleft \widetilde{W}_{k+1}$. \end{enumerate} In the statement (1c), we exclude the case $d\triangleleft \widetilde{W}_{k+1}$ because $\Delta M_{\mathbf{t}}(k)=1$. In fact, if $\widetilde{W}_{k+1}=d\widetilde{W}$, then \[M_{\mathbf{t}}(k+1)=S^{\prime}(\widetilde{W}_{k+1})=S^{\prime}(d)+S^{\prime}(\widetilde{W})=S^{\prime}(\widetilde{W})-1\leq M_{\mathbf{t}}(k)-1,\] which contradicts the assumption $\Delta M_{\mathbf{t}}(k)=1$. Now, by (\ref{eq:maxW2}) and (1b), $\widetilde{W}_{n}b$ is a factor of $\mathbf{s}$ for $n=4k,~4k+1,~4k+2$ and $\widetilde{W}_{n}a$ is a factor of $\mathbf{s}$ for $n=4k+3$. This implies that $(\ref{l:r2})$ holds for $4k\leq n<4(k+1)$. Moreover, \begin{equation}\label{eq:2b2} \widetilde{W}_{n} \text{ is a factor of }\mathbf{s} \text{ for }4k\leq n < 4(k+1). \end{equation} By the statement (1c), we have $dc\triangleleft \sigma^{2}(\widetilde{W}_{k})$ and $dcac=\sigma^{2}(d)\triangleleft\sigma^{2}(\widetilde{W}_{k+1})$. Therefore (\ref{eq:maxW2}) gives \begin{equation}\label{eq:2c2} c\triangleleft \widetilde{W}_{4k},~d\triangleleft \widetilde{W}_{4k+1},~ c\triangleleft \widetilde{W}_{4k+2} \text{ and } c\triangleleft \widetilde{W}_{4k+3}, \end{equation} which prove $(\ref{l:r3})$. Now, by (\ref{eq:maxW2}), (\ref{eq:2c2}), (1a) and Lemma \ref{lem:iterate}, we have \begin{equation}\label{eq:2rec:t} \left\{\begin{array}{ccl} S^{\prime}(\widetilde{W}_{4k}) & = & -S^{\prime}(b)+ S^{\prime}(\sigma^{2}(\widetilde{W}_{k}))+S^{\prime}(a) = 2M_{\mathbf{r^{\prime}}}(k)+2,\\ S^{\prime}(\widetilde{W}_{4k+1}) & = & S^{\prime}(\sigma^{2}(\widetilde{W}_{k}))+S^{\prime}(a) = 2M_{\mathbf{r^{\prime}}}(k)+1,\\ S^{\prime}(\widetilde{W}_{4k+2}) & = & -S^{\prime}(dca)+ S^{\prime}(\sigma^{2}(\widetilde{W}_{k+1}))+S^{\prime}(a) = 2M_{\mathbf{r^{\prime}}}(k+1){\color{green}=M_{\mathbf{r^{\prime}}}(k)+M_{\mathbf{r^{\prime}}}(k+1)+1},\\ S^{\prime}(\widetilde{W}_{4k+3}) & = & S^{\prime}(\sigma^{2}(\widetilde{W}_{k+1}))-S^{\prime}(d) = 2M_{\mathbf{r^{\prime}}}(k+1)+1.\\ \end{array} \right. \end{equation} By (\ref{eq:2b2}), (\ref{eq:2rec:t}) and Lemma \ref{lem:upperbound}, we have $\widetilde{W}_{n}\in\mathcal{M}'_{\mathbf{s}}(n)$ for $4k\leq n < 4(k+1)$ which is $(\ref{l:r1})$. \noindent{\bf Case 2:} $\Delta M_{\mathbf{r^{\prime}}}(k)=-1$. In this case, we shall first assert that $\widetilde{W}_{k}b$ is a factor of $\mathbf{s}$. By the induction hypothesis $(\ref{l:r2})$, we only need to show that $\widetilde{W}_{k}a$ can not be a factor of $\mathbf{s}$. If this is not the case, then \[M_{\mathbf{r^{\prime}}}(k+1)\geq S^{\prime}(\widetilde{W}_{k}a)=1+S^{\prime}(\widetilde{W}_{k})=1+M_{\mathbf{r^{\prime}}}(k)\] where the last equality follows from $(\ref{l:r1})$. Then we have $\Delta M_{\mathbf{r^{\prime}}}(k)=M_{\mathbf{r^{\prime}}}(k+1)-M_{\mathbf{r^{\prime}}}(k)\geq 1$ which is a contradiction. Therefore, applying the induction hypotheses $(\ref{l:r2})$, $(\ref{l:r3})$ and $(\ref{l:r1})$, we have \begin{enumerate}[\indent (2a)] \item $\widetilde{W}_{k}\in\mathcal{M}'_{\mathbf{s}}(k)$ and $\widetilde{W}_{k+1}\in\mathcal{M}'_{\mathbf{s}}(k+1)$; \item $\sigma^{2}(\widetilde{W}_{k})abdb$ and $\sigma^{2}(\widetilde{W}_{k+1})a$ are factors of $\mathbf{s}$; \item $dc\triangleleft \sigma^{2}(\widetilde{W}_{k})$ and $dc\triangleleft \sigma^{2}(\widetilde{W}_{k+1})$. \end{enumerate} By (\ref{eq:maxW2}) and (2b), we have \begin{equation*} \widetilde{W}_{n}b \text{ is a factor of } \mathbf{s} \text{ for } 4k\leq n\leq 4k+2 \end{equation*} and $\widetilde{W}_{4k+3}a$ is a factor of $\mathbf{s}$, which prove $(\ref{l:r2})$. These also imply that \begin{equation}\label{eq:1b2} \widetilde{W}_{n} \text{ is a factor of }\mathbf{s} \text{ for }4k\leq n < 4(k+1). \end{equation} Combing (2c) and (\ref{eq:maxW2}), $(\ref{l:r3})$ holds for $4k\leq n < 4(k+1)$. Now, by (\ref{eq:maxW2}), (2a), (2c) and Lemma \ref{lem:iterate}, we have \begin{equation}\label{eq:2rec2:t} \left\{\begin{array}{ccl} S^{\prime}(\widetilde{W}_{4k}) & = & -S^{\prime}(d)+ S^{\prime}(\sigma^{2}(\widetilde{W}_{k}))+S^{\prime}(a) = 2M_{\mathbf{r^{\prime}}}(k)+2,\\ S^{\prime}(\widetilde{W}_{4k+1}) & = & S^{\prime}(a)+ S^{\prime}(\sigma^{2}(\widetilde{W}_{k})) = 2M_{\mathbf{r^{\prime}}}(k)+1,\\ S^{\prime}(\widetilde{W}_{4k+2}) & = & -S^{\prime}(d)+ S^{\prime}(\sigma^{2}(\widetilde{W}_{k}))+S^{\prime}(abd) = 2M_{\mathbf{r^{\prime}}}(k){\color{green}=M_{\mathbf{r^{\prime}}}(k)+M_{\mathbf{r^{\prime}}}(k+1)+1},\\ S^{\prime}(\widetilde{W}_{4k+3}) & = & S^{\prime}(\sigma^{2}(\widetilde{W}_{k+1}))-S^{\prime}(d) = 2M_{\mathbf{r^{\prime}}}(k+1)+1.\\ \end{array} \right. \end{equation} By (\ref{eq:1b2}), (\ref{eq:2rec2:t}) and Lemma \ref{lem:upperbound}, we have $\widetilde{W}_{n}\in\mathcal{M}'_{\mathbf{s}}(n)$ for $4k\leq n < 4(k+1)$ which is $(\ref{l:r1})$. The proof is completed. \end{proof} \fi For any $k$-automatic sequence $\mathbf{w}=w(0)w(1)\cdots\in\{-1,1\}^{\mathbb{N}},$ the regularity of the maximal partial sums $(M_{\mathbf{w}}(n))_{n\geq 1}$ and the minimal partial sums $(m_{\mathbf{w}}(n))_{n\geq 1}$ implies the regularity of the abelian complexity $(\rho_{\mathbf{w}}(n))_{n\geq 1}.$ By proving the same result as Lemma \ref{lem:sum}, one can show that the double sequence $(\Sigma_{\mathbf{w}}(i,n))_{i\geq 0,n\geq 1}$ is 2-dimensional $k$-regular. In fact, it is not hard to show that $(\Sigma_{\mathbf{w}}(i,n))_{i\geq 0}$ is $k$-automatic for any fixed $n\geq 1$, and $(\Sigma_{\mathbf{w}}(i,n))_{n\geq 1}$ is $k$-regular for any fixed $i\geq 0$. Moreover, Theorem \ref{thm:abelcomp} and Lemma \ref{lem:Mm} show that $(\max_{i\geq 0}\Sigma_{\bm{w}}(i,n))_{n\geq 1}$ and $(\min_{i\geq 0}\Sigma_{\mathbf{w}}(i,n))_{n\geq 1}$ are still $k$-regular when $\mathbf{w}$ is the Rudin-Shapiro sequence $\mathbf{r}$ or its related sequence $\mathbf{r}^{\prime}$, which implies the regularity of the abelian complexity function $(\rho_{\mathbf{r}}(n))_{n\geq 0}$ and $(\rho_{\mathbf{r}^{\prime}}(n))_{n\geq 0}.$ It is natural to ask whether $(\max_{i\geq 0}\Sigma_{\mathbf{w}}(i,n))_{n\geq 1}$ and $(\min_{i\geq 0}\Sigma_{\mathbf{w}}(i,n))_{n\geq 1}$ are always $k$-regular for general $k$-automatic sequences $\mathbf{w}$ over $\{-1,1\}.$ \iffalse \section{Growth order of the Abelian complexity function $\rho(n)$} Unlike the subword complexity function, which is strictly increasing for any aperiodic word, the Abelian complexity function can fluctuate considerably. \ww{Recall that in Corollary \ref{regular:abel}, we show that $\mathbf{r}$ and $\mathbf{r}^{\prime}$ share the same Abelian complexity function $\rho(\cdot)$. We now apply Theorem \ref{thm:abelcomp} and Lemma \ref{lem:rho} to study the growth {\color{yellow}order} of the function $\rho(\cdot)$, and we have the following results.} \begin{theorem}\label{thm:bounds} For any $n\geq 1$, we have \begin{equation}\label{rho:bound} \sqrt{3}\leq \frac{\rho(n)}{\sqrt{n}}\leq 3. \end{equation} Moreover, the upper and lower bounds can be reached. \end{theorem} To proof Theorem \ref{thm:bounds}, we start by determining the values of $M(n)$ at some integers. Then we decompose the set of non-negative integers into countable parts and we estimate the upper and lower bounds of $M(n)$ on each part. \begin{lemma}\label{prop:1} For the sequence $(M(n))_{n\geq 1}$ and $k \in \mathbb{N^+}$, we have: \begin{enumerate}[(1)] \item $M(j\cdot 4^k)=M(j\cdot 4^k-1)+1={2^k}(M(j)+2)-2$ for any $j\geq 1$; \item $M(\frac{{4^k} - 1}{3})=M(\frac{{4^k} +2}{3})-1= {2^k} - 1$; \item $M(4^{k}-\frac{4^{j}-4}{3})=3\cdot 2^{k}-2^{j}$ for any $1 \leq j \leq k$. \end{enumerate} \end{lemma} \begin{proof} We will prove the formula separately. \emph{Proof of $(1)$.} By Theorem \ref{thm:abelcomp}, we obtain \begin{align*} M(j\cdot 4^{k})& =2M(j\cdot 4^{k-1})+2\\ & =2^{2}M(j\cdot 4^{k-2})+2^{2}+2\\ & = \cdots = 2^{k}M(j)+2^{k}+2^{k-1}+\cdots +2\\ & = 2^{k}(M(j)+2)-2. \end{align*} By Corollary \ref{cor:abeldiff}, \[M(j\cdot 4^k-1)=M(j\cdot 4^k)-\Delta M(j\cdot 4^k-1)=M(j\cdot 4^k)-1.\] \emph{Proof of $(2)$.} Note that $\frac{{4^{k + 1}} - 1}{3} = 4 \cdot \frac{{4^k} - 1}{3} + 1$, by Theorem \ref{thm:abelcomp}, we have \begin{align*} M\left(\frac{{4^{k + 1}} - 1}{3}\right) & = 2M\left(\frac{{4^k} - 1}{3}\right) + 1\\ & = \cdots =2^{k}M(1)+2^{k-1}+2^{k-2}+\cdots+1\\ & = 2^{k+1} - 1. \end{align*} \emph{Proof of $(3)$.} Let $p_{k,j}=4^{k}-\frac{4^{j}-1}{3}$. Then $4^{k}-\frac{4^{j}-4}{3}=4p_{k-1,j-1}$, and \[p_{k,j}=4\left(4^{k-1}-\frac{4^{j-1}-1}{3}\right)-1=4(p_{k-1,j-1}-1)+3.\] By Theorem \ref{thm:abelcomp}, for any $k,j\geq 1$ \begin{align*} M(p_{k,j}) & =M(4(p_{k-1,j-1}-1)+3) = 2M(p_{k-1,j-1})+1\\ & = \cdots = 2^{j}M(p_{k-j,0})+2^{j}-1\\ &= 2^{j}M(4^{k-j})+2^{j}-1 = 2^{j}(3\cdot 2^{k-j}-2)+2^{j}-1 \\ & = 3\cdot 2^{k}-2^{j}-1. \end{align*} Therefore, the previous formula gives \begin{align*} M\left(4^{k}-\frac{4^{j}-4}{3}\right) & = M(4p_{k-1,j-1})=2M(p_{k-1,j-1})+2\\ & = 3\cdot 2^{k}-2^{j}. \end{align*} \end{proof} Now, for any $k\geq 0$, we will determine the upper bound of $M(n)$ for all \[n\in \{\ell : 4^{k}\leq \ell \leq 4^{k+1},\ell\in\mathbb{N}\}=\bigcup_{i=1}^{3}I_{k}(i)\] where \[I_{k}(i)=\{\ell + i4^{k} : 0\leq \ell \leq 4^{k}, \ell\in\mathbb{N}\},\quad i=1,2,3.\] Set $I_{k}(i)=\bigcup_{j=1}^{3}J_{j}(k,i)$ where \begin{align*} J_{1}(k,i) & :=\{\ell + i4^{k} : 0\leq \ell \leq 2\cdot 4^{k-1}, \ell \in \mathbb{N} \},\\ J_{2}(k,i) & :=\{\ell + i4^{k} : 2\cdot 4^{k-1}\leq \ell \leq 3\cdot 4^{k-1}+2\cdot 4^{k-2}, \ell \in \mathbb{N} \},\\ J_{3}(k,i) & :=\{\ell + i4^{k} : 3\cdot 4^{k-1}+2\cdot 4^{k-2}\leq \ell \leq 4^{k}, \ell \in \mathbb{N} \}. \end{align*} The above sets have certain overlaps. This will not cause any problem \ww{to} our results. The next lemma gives the upper bounds of $M(n)$. \begin{lemma}\label{lem:M:upper} For $i=1,2,3$ and any $k\geq 0$, we have \begin{equation} M(n)\leq \begin{cases} (4+2i)\cdot 2^{k-1}-2 & \text{ when } n\in J_{1}(k,i),\\ (5+2i)\cdot 2^{k-1}-2 & \text{ when } n\in J_{2}(k,i),\\ (6+2i)\cdot 2^{k-1}-2 & \text{ when } n\in J_{3}(k,i). \end{cases} \end{equation} Moreover, the above upper bounds can be reached. \end{lemma} \begin{proof} We will prove by induction on $k$. When $k=0,1$, \begin{align*} J_{1}(0,i)&=\{i\}, & J_{2}(0,i)&=\emptyset, & J_{3}(0,i)&=\{i+1\}\\ J_{1}(1,i)&=\{4i,4i+1,4i+2\}, & J_{2}(1,i)&=\{4i+2,4i+3\}, & J_{3}(1,i)&=\{4i+4\}. \end{align*} The upper bound can be checked directly. Now assume the result hold for any $j< k$, we will prove the result for $k$. For any $n\in J_{1}(k,i)\backslash\{(4i+2)\cdot 4^{k-1}\}$, there are $p\in J_{1}(k-1,i)$ and $q=0,1,2,3$ such that \[n=4p+q \text{ and }p+1\in J_{1}(k-1,i).\] By Theorem \ref{thm:abelcomp} and the inductive hypothesis, for any $n\in J_{1}(k,i)\backslash\{(4i+2)\cdot 4^{k-1}\}$, \begin{align*} M(n) =M(4p+q) & \leq 2\max_{p\in J_{1}(k-1,i)}M(p)+2 \\ & \leq 2\big((4+2i)\cdot 2^{k-2}-2\big)+2=(4+2i)\cdot 2^{k-1}-2. \end{align*} By (1) of Lemma \ref{prop:1}, we have \[M\left((4i+2)\cdot 4^{k-1}\right)=2^{k-1}(M(4i+2)+2)-2=(4+2i)\cdot 2^{k-1}-2,\] where the last equality holds because $M(4i+2)=2i+2$ for $i=1,2,3$. Therefore, \[M(n)\leq (4+2i)\cdot 2^{k-1}-2 \text{ when }n\in J_{1}(k,i).\] In a similar way, one can prove the result for $n\in J_{j}(k,i)$ where $j=2,3$. \end{proof} \begin{lemma}\label{lem:b} Let $k\geq 1$ be fixed. For any $j\geq 1$, let \[B_{k,j}:=\{\ell : p_{k,j}\leq \ell \leq q_{k,j}, \ell\in\mathbb{N}\}\] where $p_{k,j}=4^{k+j}-\frac{4^{j}+2}{3}$ and $q_{k,j}=4^{k+j}-\frac{4^{j-1}+2}{3}$. Then for any $n\in B_{k,j}$, \[M(n)\leq 3\cdot 2^{k+j}-2^{j-1}-2.\] \end{lemma} \begin{proof} Since $B_{k,1}=\{4^{k+1}-2, 4^{k+1}-1\}$, by Lemma \ref{prop:1}, we have \[M(4^{k+1}-1)=M(4^{k+1}-2)+1=3\cdot 2^{k+1}-3.\] Suppose the result holds for $j$, we will prove it for $j+1$. Note that $p_{k,j+1}=4p_{k,j}+2$ and $q_{k,j+1}=4q_{k,j}+2$. Therefore, for any $n\in B_{k,j+1}$, there exist $n^{\prime}\in B_{k,j}$ and $i\in 0,1,2,3$ such that $n=4n^{\prime}+i$. By Theorem \ref{thm:abelcomp} and the inductive hypothesis, \begin{align*} M(n)& \leq 2M(n^{\prime})+2 \\ & \leq 2(3\cdot 2^{k+j}-2^{j-1}-2)+2\\ &= 3\cdot 2^{k+j+1}-2^{j}-2. \end{align*} So the result holds for $j+ 1$. We complete the proof. \end{proof} \begin{proposition}\label{prop:upper} For any $n\geq 1$, we have \[\rho(n)\leq 3\sqrt{n}.\] Moreover, $\lim_{k\to \infty}\frac{\rho(4^{k})}{\sqrt{4^{k}}}=3$. \end{proposition} \begin{proof} For $n\leq 15$, we can check this result directly. Now suppose $n\geq 16$. By Lemma \ref{lem:M:upper}, we have for $i=1,2,3$ and any $k\geq 2$, \begin{equation*} \ww{\frac{\rho(n)}{\sqrt{n}}\leq} \left\{ \begin{aligned} \frac{(4+2i)\cdot 2^{k-1}-1}{\sqrt{i4^{k}}} & \leq 3, &\text{ when } n\in J_{1}(k,i),\\ \frac{(5+2i)\cdot 2^{k-1}-1}{\sqrt{i4^{k}+2\cdot 4^{k-1}}} & \leq \frac{11}{\sqrt{14}}\approx 2.939, &\text{ when } n\in J_{2}(k,i). \end{aligned} \right. \end{equation*} For $i=1,2$ and any $k\geq 2$, \begin{equation*} \frac{\rho(n)}{\sqrt{n}}\leq \frac{(6+2i)\cdot 2^{k-1}-1}{\sqrt{i4^{k}+3\cdot 4^{k-1}+ 2\cdot 4^{k-2}}}\leq \frac{20}{\sqrt{46}}\approx 2.948 \quad \text{ when } n\in J_{3}(k,i). \end{equation*} Now, for any $k\geq 2$, we will deal with the case $n\in J_{3}(k,3)$. For $j=1,\cdots,k-1$, let \[A_{k,j}\ww{:}=\{\ell : a_{k+1,j+1}\leq \ell\leq a_{k+1,j} , \ell \in \mathbb{N}\},\] where $a_{k,j}=4^{k}-\frac{4^{j}-4}{3}$. We claim that for any $n\in A_{k,j}$ ($1\leq j\leq k-1$), \begin{equation} M(n)\leq 3\cdot 2^{k+1}-2^{j}. \tag*{Claim A} \end{equation} If Claim A holds, then for any $k\geq 2$ and $1\leq j\leq k-1$, \[\frac{\rho(n)}{\sqrt{n}}\leq \frac{3\cdot 2^{k+1}-2^{j}+1}{\sqrt{4^{k+1}-\frac{4^{j+1}-4}{3}}}\leq 3, \quad n\in A_{k,j}.\] Since $J_{3}(k,3)\subset \bigcup_{j=1}^{k-1}A_{k,j}$, we have $\rho(n)/\sqrt{n}\leq 3$ for all $n\in J_{3}(k,3)$. In the rest, we will prove Claim A. Let \begin{align*} B:=&\ww{\{\ell -2 : \ell\in A_{k-1,j-1}\}}\\ =&\{\ell\in\mathbb{N} : 4^{k}-\frac{4^{j}+2}{3}\leq \ell \leq 4^{k}-\frac{4^{j-1}+2}{3}\}. \end{align*} By Lemma \ref{lem:b}, for any $n\in B\ww{ = B_{k-j,j}}$, $M(n)\leq 3\cdot 2^{k}-2^{j-1}-2$. Hence for any $n\in B^{\prime}:=\{\ell -1 : \ell\in B\}$, we have $M(n)\leq 3\cdot 2^{k}-2^{j-1}-1$. Since $a_{k+1,j+1}=4(a_{k,j}-1)$, for any $n\in A_{k,j}$, there exist $n^{\prime}\in B^{\prime}$ and $i\in\{0,1,2,3\}$ such that $n=4n^{\prime}+i$. Thus \[M(n)\leq 2M(n^{\prime})+2\leq 2(3\cdot 2^{k}-2^{j-1}-1)+2=3\cdot 2^{k+1}-2^{j}.\] By Lemma \ref{prop:1} and Lemma \ref{lem:rho}, $\mathop {\lim }\limits_{k \to \infty } \frac{\rho ({4^k})}{\sqrt {{4^k}} } =\lim\limits_{k \to \infty } \frac{3 \cdot {2^k} - 1}{2^k} =3 $. This completes the proof. \end{proof} For the lower bound of $\rho(n)/\sqrt{n}$ for $4^{k}\leq n < 4^{k+1}$ ($k\geq 0$), we use a similar idea. For $i=1,2,3$, let \begin{align*} D_{i,k}&:= \{\ell\in\mathbb{N} : i4^{k}\leq \ell < (i+1)4^{k}\}. \end{align*} Then \[\{\ell\in\mathbb{N}: 4^{k}\leq n < 4^{k+1}\}=\bigcup_{i=1}^{3}D_{i,k}.\] \begin{lemma}\label{lem:Mmin} For $i=1,2,3$ and all $k\geq 0$, we have \begin{equation}\label{eq:Mmin} \min_{n\in D_{i,k}}M(n)\geq (i+1)2^{k}-1. \end{equation} Moreover, the equality holds \ww{when} $n=\frac{(3i+1)4^{k}-1}{3}$. \end{lemma} \begin{proof} Fix $i\in\{1,2,3\}$. It is easy to verify the inequality \eqref{eq:Mmin} for $k=0$. Suppose \eqref{eq:Mmin} holds for $k-1$. Note that \[D_{i,k}\subset \{4n+a : n\in D_{i,k-1}, a=0,1,2,3\}.\] By Theorem $\ref{thm:abelcomp}$ and the inductive hypothesis, we have \begin{align*} \min_{n\in D_{i,k}}M(n) & \geq 2\min_{n\in D_{i,k-1}}M(n)+1\\ & \geq 2((i+1)2^{k-1}-1)+1 =(i+1)2^{k}-1. \end{align*} Thus \eqref{eq:Mmin} holds for $k$. On the other hand, $\frac{(3i+1)4^{k}-1}{3}=4\cdot \frac{(3i+1)4^{k-1}-1}{3}+1$, thus by Theorem \ref{thm:abelcomp} again, \[M\left(\frac{(3i+1)4^{k}-1}{3}\right)=2M\left(\frac{(3i+1)4^{k-1}-1}{3}\right)+1=(i+1)2^{k}-1\] which completes the proof. \end{proof} \begin{proposition}\label{prop:lower} For any $n\geq 1$, we have \[\frac{\rho(n)}{\sqrt{n}}\geq \sqrt{3}.\] Moreover, \[\lim_{k\to \infty}\frac{\rho(\frac{4^{k}-1}{3})}{\sqrt{\frac{4^{k}-1}{3}}}=\sqrt{3}.\] \end{proposition} \begin{proof} Now suppose $n\in D_{i,k}$ for some $k\geq 0$ and $i=2, 3$. By Lemma \ref{lem:Mmin}, we have \[\frac{\rho(n)}{\sqrt{n}}\geq \frac{(i+1)2^{k}}{\sqrt{(i+1)4^{k}-1}}\geq \sqrt{i+1}\geq \sqrt{3}.\] For the case $n\in D_{1,k}$ for some $k\geq 0$, we divide this case into the following two sub-cases. When $n\in\{\ell \in D_{1,k} : \ell \leq \frac{4^{k+1}-1}{3}\}$, by Lemma \ref{lem:Mmin}, we have \[\frac{\rho(n)}{\sqrt{n}}\geq \frac{2^{k+1}}{\sqrt{\frac{4^{k+1}-1}{3}}}\geq \sqrt{3}.\] When $n\in\{\ell \in D_{1,k} : \ell > \frac{4^{k+1}-1}{3}\}$, there exists $0\leq j \leq k$ such that \[n\in \left\{\ell\in D_{1,k} : \frac{4^{k+1}+2}{3}+\frac{4^{j}-1}{3}\leq \ell <\frac{4^{k+1}+2}{3}+\frac{4^{j+1}-1}{3}\right\}=:E_{j}.\] Thus we turn to estimate the lower bound of $M(n)$ for $n\in E_{j}$. We claim that for any $j$, \[\min_{n\in E_{j}} M(n)\geq 2^{k+1}+2^{j}-1.\] For any $j=0,1,\cdots, k-1$, let $F_{1}(j):=\{\frac{4^{k+1-j}+2}{3}\}$ and for any $h\geq 1$, \[F_{h+1}(j):=\{4\ell+i : \ell\in F_{h}(j), i=-1,0,1,2\}.\] Thus, by Theorem \ref{thm:abelcomp}, \[\min_{n\in F_{h+1}(j)}M(n)\geq 2\min_{n\in F_{h}(j)}M(n)+1.\] Since \[\min_{n\in F_{1}(j)}M(n)=M\left(\frac{4^{k+1-j}+2}{3}\right)=2^{k+1-j},\] then we have \[\min_{n\in F_{h+1}(j)}M(n)\geq 2^{k+1-j+h}+2^{h}-1.\] Noting that $E_{j}=F_{j+1}(j)$, then \[\min_{n\in E_{j}}M(n)=\min_{n\in F_{j+1}(j)}M(n)\geq 2^{k+1}+2^{j}-1.\] Therefore, for any $n\in E_{j}$, \[\frac{\rho(n)}{\sqrt{n}}\geq \frac{2^{k+1}+2^{j}}{\sqrt{\frac{4^{k+1}+4^{j+1}+1}{3}}}\geq \sqrt{3}.\] By Lemma \ref{prop:1} and Lemma \ref{lem:rho}, $\mathop {\lim }\limits_{k \to \infty } \frac{\rho (\frac{4^k - 1}{3})}{\sqrt {\frac{4^k - 1}{3}} } = \mathop {\lim }\limits_{k \to \infty } \frac{{2^k}}{\sqrt{\frac{4^k - 1}{3}} } = \sqrt 3 $. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:bounds}] The result follows directly from Proposition \ref{prop:upper} and \ref{prop:lower}. \end{proof} Theorem \ref{thm:bounds} gives the critical upper and lower bound of $(\rho(n)/\sqrt{n})_{n\geq 1}$. In fact, any \ww{real number} between the upper and lower bounds is an accumulation point. \begin{corollary}\label{coro:rhodense} $(\rho(n)/\sqrt{n})_{n\geq 1}$ is dense in $[\sqrt{3}, 3]$. \end{corollary} \begin{proof} Theorem \ref{thm:bounds} shows that $\sqrt{3}$ and $3$ are accumulation points of the sequence $(\rho(n)/\sqrt{n})_{n\geq 1}$. Now assume $\sqrt{3}< \alpha< 3$. Suppose $\alpha$ is not an accumulation point. Set $\varepsilon>0$ and $N>0$ such that for all $n>N$, \[\left|\frac{\rho(n)}{\sqrt{n}}-\alpha\right|>\varepsilon.\] Let $n_{0}>N$ such that $\varepsilon\sqrt{n_{0}}>1$. Since both $\sqrt{3}$ and $3$ are accumulation points, we can find $n>n_{0}$ satisfying \[\frac{\rho(n)}{\sqrt{n}}<\alpha-\varepsilon \text{ and } \frac{\rho(n+1)}{\sqrt{n+1}}>\alpha+\varepsilon.\] However, \[2\varepsilon< \frac{\rho(n+1)}{\sqrt{n+1}}-\frac{\rho(n)}{\sqrt{n}}<\frac{\rho(n+1)-\rho(n)}{\sqrt{n}} \leq\frac{1}{\sqrt{n}}<\varepsilon\] which is a contradiction. \end{proof} \fi \section{Box dimension of $\lambda(x)$} Let $M(x):=M(\lfloor x\rfloor)$ ($x>0$) be the continuous version of the maximal digit sum function, and $\rho(x)=M(x)+1$. Now we study the following limit function: \begin{equation}\label{def:lamda} \lambda(x):= \displaystyle{\lim_{k\to \infty}}\frac{\rho( 4^k x)}{\sqrt{4^k x}}. \end{equation} From the above definition, providing the limit exists, it is easy to see that $\lambda(x)$ is self-similar in the sense that for any $x>0$, \[ \lambda(4x)=\lambda(x).\] The existence of the limit in \eqref{def:lamda} follows from the same argument in \cite[Theorem $1$]{BEM}. For completeness, we give the details in the following Proposition \ref{lem:lambdarho}. Denote the $4$-adic expansion of a real positive number $x>0$ by \begin{equation}\label{eq:expansion} \sum_{j=0}^{\infty}x_{j}4^{-j} \end{equation} where $x_{0}\in\mathbb{N}$ and $x_{j}\in\{0,1,2,3\}$ for all $j\geq 1$. In the expansion \eqref{eq:expansion}, we always assume that there are infinitely many $j$ such that $x_{j}\neq 3$. Let \[a_{j}(x):=\left\{ \begin{aligned} & -1, & \text{if } 4^{j}x<1,\\ &\Delta M(\lfloor 4^{j}x\rfloor-1), & \text{otherwise,} \end{aligned} \right.\] and \[d(y)=\left\{\begin{array}{cl} 1 & \text{ if } y = 0 \text{ or } 2,\\ 0 & \text{ if } y = 1,\\ 2 & \text{ if } y = 3.\end{array}\right.\] \begin{proposition}\label{lem:lambdarho} The limit (\ref{def:lamda}) exists for all $x>0$, and for any $x>0$ it satisfies \begin{equation}\label{eq:lamda:detail} \lambda(x)=\frac{\rho(x)+a(x)}{\sqrt{x}} \end{equation} where $a(x) := \sum\limits_{j = 1}^\infty d ({x_j}){a_j}(x) 2^{ - j}.$ Moreover, for any positive integer $n$, \[\lambda(n)=(\rho(n)+1)/\sqrt{n}.\] \end{proposition} \begin{proof} By Theorem \ref{thm:abelcomp} and Corollary \ref{cor:abeldiff}, we have \[M(4n+i)=2M(n)+1+d(i)\Delta M(4n+i-1)\] for all $n\geq 1$ and $i=0,1,2,3$. Let $N$ be the smallest integer such that $4^{N}x\geq 1$. Then, for any $k\geq N$, \begin{align*} M(4^{k}x)&=M(\lfloor 4^{k}x\rfloor) = M(4\lfloor 4^{k-1}x\rfloor +x_{k})\\ & = 2M(\lfloor 4^{k-1}x\rfloor)+1+d(x_{k})\Delta M(\lfloor 4^{k}x\rfloor -1)\\ & = 2M(4^{k-1}x)+1+d(x_{k})a_{k}(x). \end{align*} For $1\leq k < N$, $d(x_{k})=d(0)=1$ and $a_{k}(x)=-1$. Thus, we also have \begin{align*} M(4^{k}x)&=0=1+(-1) \\ & = 1+ d(x_{k})a_{k}(x)\\ & = 2M(4^{k-1}x)+1+d(x_{k})a_{k}(x). \end{align*} By induction, the above equation yields \[M(4^{k}x)=2^{k}M(x)+\sum_{j=1}^{k}d(x_{j})a_{j}(x)2^{k-j}+(2^{k}-1).\] Now, by Lemma \ref{lem:rho} \[\frac{\rho ({4^k}x)}{\sqrt {{4^k}x} } = \frac{M({4^k}x) + 1}{\sqrt {{4^k}x} } = \frac{\rho (x)}{\sqrt x } + \frac{1}{\sqrt x }\sum\limits_{j = 1}^k d({x_j})a_j(x)2^{ - j}.\] Letting $k\to \infty$ and noticing that the series in (\ref{eq:lamda:detail}) converges absolutely, we obtain (\ref{eq:lamda:detail}). When $x=n\in\mathbb{N^{+}}$, $x_{0}=n$, $x_j=0$ and $a_{j}=4^{j}n-1$ for all $j\geq 1$. Then the infinite sums in (\ref{eq:lamda:detail}) turns out to be \[\sum_{j=1}^{\infty}d(x_{j})a_{j}(x)2^{-j}=\sum_{j=1}^{\infty}\Delta M(4^{j}n-1)2^{-j}=1\] where the last equality holds by using Corollary \ref{cor:abeldiff}. Applying the above equation to (\ref{eq:lamda:detail}), we complete the proof. \end{proof} \subsection{Auxiliary lemmas.} Let $\delta > 0$. For any $m_1,m_2 \in \mathbb{Z}$, we call the following square \[ [m_1\delta, (m_1+1)\delta] \times [m_2\delta, (m_2+1)\delta]\] a $\delta$-mesh of $\mathbb{R}^2.$ Let $F \subset \mathbb{R}^2$ be a non-empty bounded set in $\mathbb{R}^{2}$, and $N_{\delta}(F)$ be the number of $\delta$-meshes that intersect $F$. The upper and lower box dimension are defined by \[\overline{\dim}_B F:=\overline{\lim}_{\delta\to 0}\frac{\log N_{\delta}(F)}{-\log \delta} \text{ and } \underline{\dim}_B F:=\overline{\lim}_{\delta\to 0}\frac{\log N_{\delta}(F)}{-\log \delta}\] respectively. If $\overline{\dim}_B F= \underline{\dim}_B F,$ then the common value denoted by $\dim_B F$, is the box dimension of $F$. For more detail, see \cite{F04}. Now, we will prove some auxiliary lemmas which are used in the calculation of the box dimension of the function $\lambda(x)$. For any $k\geq 1$ and $0\leq z< 4^{k}$ where $z\in\mathbb{N}$. let \[I_{k}(z):=[z4^{-k},(z+1)4^{-k}).\] Then $[0,1)=\bigcup_{0\leq z<4^{k}}I_{k}(z)$. Denote the $4$-adic expansion of $z4^{-k}$ by \[\frac{z}{4^{k}}=\sum_{j=1}^{k}z_{j}4^{-j}.\] If $y=\sum_{j=1}^{\infty}y_{j}4^{-j}\in I_{k}(z)$, then $y_{i}=z_{i}$ for $i=1,~2,~\cdots,~k$. First, we will determine the difference of values of $a(\cdot)$ at the end points of $4$-adic interval $I_{k}(z)$. \begin{lemma}\label{dim:lem:1} Let $k\geq 1$ and $z\in\mathbb{N}$ with $1\leq z<4^{k}$. Then \[a(z4^{-k})-a((z+1)4^{-k})=\begin{cases}-2^{-k} & \mathrm{if}~ z\leq 4^{k}-2 \\ 1-2^{-k} & \mathrm{if }~ z=4^{k}-1.\end{cases}\] \end{lemma} \begin{proof} When $z=4^{k}-1$, we have $z4^{-k}=\sum_{j=1}^{k}3\cdot 4^{-j}$ and $(z+1)4^{-k}=1$. So \[a(z4^{-k})-a((z+1)4^{-k})=(2-2^{-k})-1=1-2^{-k}.\] When $1\leq z\leq 4^{-k}-2$, $z4^{-k}$ and $(z+1)4^{-k}$ have the $4$-adic expansions \[z4^{-k}=\sum_{j=1}^{k}z_{j}4^{-j} \text{ and }(z+1)4^{-k}=\sum_{j=1}^{k}z^{\prime}_{j}4^{-j}.\] Implicitly, we assume that $z_{j}=z_{j}^{\prime}=0$ for $j>k$. Let $1\leq h\leq k$ be the integer such that $z_{h}\neq 3$ and $z_{j}=3$ for $j=h+1,\cdots, k$. Then \[z^{\prime}_{j}=\begin{cases}z_{j} & \text{when } j<h,\\ z_{j}+1 & \text{when } j = h,\\ 0 & \text{when } j> h.\end{cases}\] Setting $D_{j}:=d(z_{j})a_{j}(z4^{-k})-d(z^{\prime}_{j})a_{j}((z+1)4^{-k})$, then \[a(z4^{-k})-a((z+1)4^{-k})=\sum_{j=1}^{\infty}D(j)2^{-j}.\] Apparently, $D_{j}=0$ when $j<h$ or $j>k$. Since $a_{j}(z4^{-k})=a_{j}((z+1)4^{-k})=1$ for $h+2\leq j\leq k$, we have for $ h+2\leq j\leq k$, \[D_{j}=d(3)-d(0)=1.\] Set $u:=4^{h}\sum_{j=1}^{h}z_{j}4^{-j}$. If $u\geq 1$, we have \begin{align*} D_{h}+2^{-1}D_{h+1}&= \left(d(z_{h})\Delta M(u-1)-d(z_{h}^{\prime})\Delta M(u)\right)\\ & \quad +2^{-1}\left(d(3)\Delta M(4u+2)-d(0)\cdot \Delta M(4u+3)\right)\\ &= d(z_{h})\Delta M(u-1)-d(z_{h}^{\prime})\Delta M(u)+\Delta M(u)-2^{-1}\\ &= \begin{cases} d(0)\cdot 1-d(1)\cdot (-1)+(-1)-2^{-1}, & \text{if } z_{h}=0,\\ d(1)\cdot (-1)-d(2)\cdot \Delta M(u)+\Delta M(u)-2^{-1}, & \text{if } z_{h}=1,\\ d(2)\cdot \Delta M(u)-d(3)\cdot \Delta M(u)+\Delta M(u)-2^{-1}, & \text{if } z_{h}=2, \end{cases}\\ &= -2^{-1}. \end{align*} If $u=0$, then $z_{h}=0$ and \begin{align*} D_{h}+2^{-1}D_{h+1}&= d(0)\cdot(-1)-d(1)\Delta M(0)\\ &\quad + 2^{-1}\left(d(3)\Delta M(2)-d(0)\Delta M(3)\right)\\ & = -2^{-1}. \end{align*} Therefore \begin{align*} a(z4^{-k})-a((z+1)4^{-k}) & = \sum_{j=1}^{\infty}D(j)2^{-j}\\ & = 2^{-h}D_{h}+2^{-h-1}D_{h+1}+ \sum_{j=h+2}^{k}D(j)2^{-j}\\ & = -2^{-h-1}+\left(2^{-h-1}-2^{-k}\right) =-2^{-k}. \end{align*} \end{proof} \begin{lemma}\label{dim:lem:2} There exists $c>0$, such that for any $x,y\in (0,1)$, \[|a(x)-a(y)|\leq c|x-y|^{1/2}.\] \end{lemma} \begin{proof} Let $x,y\in (0,1)$ and $x<y$. Denote their $4$-adic expansion by \[x=\sum_{j=1}^{\infty}x_{j}4^{-j} \text{ and } y=\sum_{j=1}^{\infty}y_{j}4^{-j}.\] Set $D_{j}:=d(x_{j})a_{j}(x)-d(y_{j})a_{j}(y)$, then $|D_{j}|\leq 4$ for $j\geq 1$. Let $k$ be the integer such that $4^{-k-1}\leq y-x<4^{-k}$. Then $x$ and $y$ can be covered by at most two (adjacent) $4$-adic intervals of level $k$. Suppose $x,y\in I_{k}(z)$ for some $0\leq z<4^{k}$, then $x_{j}=y_{j}$ for $i=1,2,\cdots,k$. Consequently, $D_{j}=0$ for $1\leq j\leq k$. So \begin{align*} |a(x)-a(y)| & = \left|\sum_{j=k+1}^{\infty}D_{j}2^{-j}\right|\\ & \leq 4\sum_{j=k+1}^{\infty}2^{-j}=4\cdot 2^{-k}\\ & \leq 8|x-y|^{1/2}. \end{align*} On the other hand, suppose $x\in I_{k}(z)$ and $y\in I_{k}(z+1)$ where $0\leq z<4^{-k}-1$. Let $h$ be the largest integer such that $x,y\in I_{h}(z^{\prime})$ for some $0\leq z^{\prime} <4^{-h}$. Apparently, $0\leq h<k$. In this case, the $4$-adic expansions of $x$ and $y$ satisfy \begin{align*} \begin{cases}y_{j}=x_{j}, & \mathrm{if}~ 1\leq j \leq h,\\ y_{j}=x_{j}+1, & \mathrm{if}~ j= h+1,\\ y_{j}=0~ \mathrm{and}~x_{j}=3, & \mathrm{if}~ h+2\leq j\leq k.\end{cases} \end{align*} (We remark that $x_{h+1}\neq 3$ by the choice of $h$.) Hence, $D_{j}=0$ for $1\leq j\leq h$. Similar discussions as in Lemma \ref{dim:lem:1} yield that \begin{align*} D_{h+1}+2^{-1}D_{h+2}=-2^{-1}. \end{align*} Moreover, for $h+2\leq j\leq k$, $D_{j}=d(3)-d(0)=1$. Therefore, \begin{align*} |a(x)-a(y)|&=\left|\sum_{j=1}^{\infty}D_{j}2^{-j}\right|= \left|\sum_{j=h+1}^{k}D_{j}2^{-j}+\sum_{j=k+1}^{\infty}D_{j}2^{-j}\right|\\ & \leq \left|2^{-h-1}(D_{h+1}+2^{-1}D_{h+2})+\sum_{j=h+3}^{k}D_{j}2^{-j}\right|+4\sum_{j=k+1}^{\infty}2^{-j}\\ & = 5\cdot 2^{-k} \leq 10 |x-y|^{1/2}. \end{align*} \end{proof} \subsection{Calculation of the box dimension.} \begin{theorem} \label{dim:thm:1} For any $0<\alpha<\beta\leq 1$, \[\dim_{B}\{(x,\lambda(x)):\alpha<x<\beta\}=\frac{3}{2}.\] \end{theorem} \begin{proof} For any $x,y\in (\alpha,\beta)$ and $x<y$, $\rho(x)=\rho(y)=\rho(0)=1$, \begin{align*} |\lambda(x)-\lambda(y)| & = \left|\frac{\rho(x)+a(x)}{\sqrt{x}}-\frac{\rho(y)+a(y)}{\sqrt{y}}\right|\\ &=\left|\frac{a(x)+1}{\sqrt{x}}-\frac{a(y)+1}{\sqrt{y}}\right|\\ &= \left|\frac{a(x)-a(y)}{\sqrt{x}}+\frac{\sqrt{y}-\sqrt{x}}{\sqrt{xy}}(a(y)+1)\right|\\ & \leq \alpha^{-1/2}|a(x)-a(y)|+3\alpha^{-1}\sqrt{y-x} \\ & \leq (c\alpha^{-1/2}+3\alpha^{-1})|x-y|^{1/2} \end{align*} where the last inequality holds by Lemma \ref{dim:lem:2}. Now by \cite[Corollary 11.2 (a)]{F04}, \begin{equation} \overline{\dim}_{B}\{(x,\lambda(x)):\alpha<x<\beta\}\leq \frac{3}{2}.\label{dim:eq:u} \end{equation} For any $k\geq 1$, let $N_{k}$ be the number of $4^{-k}$-mesh squares that intersect the graph of $\lambda(x)$ on $(\alpha,\beta)$. For any $k\geq 1$ and $\lfloor \alpha 4^{k}\rfloor< z \leq \lfloor \beta 4^{k}\rfloor$, the number of $4^{-k}$-mesh squares that intersect the graph of $\lambda(x)$ on $I_{k}(z)$ is lager than $\left|\lambda((z+1)4^{-k})-\lambda(z4^{-k})\right|/4^{-k}$. Choose $K_{1}$ large enough such that for all $k>K_{1}$, $3\cdot 2^k<\lfloor \alpha 4^k\rfloor ~(<z)$. Then, by Lemma \ref{dim:lem:1}, \begin{align*} \left|\lambda((z+1)4^{-k})-\lambda(z4^{-k})\right| & = \left|\frac{1+a((z+1)4^{-k})}{\sqrt{(z+1)4^{-k}}}-\frac{1+a(z4^{-k})}{\sqrt{z4^{-k}}}\right|\\ & = \frac{1}{\sqrt{z4^{-k}}}\left|a((z+1)4^{-k})-a(z4^{-k})\right.\\ & \quad +\left. \frac{\sqrt{z4^{-k}}-\sqrt{(z+1)4^{-k}}}{\sqrt{(z+1)4^{-k}}}(1+a((z+1)4^{-k}))\right|\\ & \geq \frac{1}{\sqrt{\beta}}\left(2^{-k}-\frac{\left|1+a((z+1)4^{-k})\right|}{z+1+\sqrt{z^{2}+z}}\right)\\ & \geq 2^{-k}\cdot \frac{1}{\sqrt{\beta}}\left(1-\frac{3\cdot 2^{k}}{z+1+\sqrt{z^{2}+z}}\right) >\frac{1}{2\sqrt{\beta}}\cdot 2^{-k}. \end{align*} Choose $K_{2}$ large enough such that for all $k>K_{2}$, $\lfloor \beta 4^{k}\rfloor-\lfloor \alpha 4^{k}\rfloor-1>4^{k}(\beta-\alpha)/2$. Hence, for any $k>\max\{K_{1}, K_{2}\}$, \begin{align*} N_{k} & \geq \sum_{\lfloor \alpha 4^{k}\rfloor< z < \lfloor \beta 4^{k}\rfloor}\frac{\left|\lambda((z+1)4^{-k})-\lambda(z4^{-k})\right|}{4^{-k}}\\ & \geq \frac{1}{2\sqrt{\beta}}\sum_{\lfloor \alpha 4^{k}\rfloor< z < \lfloor \beta 4^{k}\rfloor}\frac{2^{-k}}{4^{-k}} = \frac{\lfloor \beta 4^{k}\rfloor-\lfloor \alpha 4^{k}\rfloor-1}{2\sqrt{\beta}}\cdot 2^{k}\\ & > \frac{\beta-\alpha}{4\sqrt{\beta}}\cdot 2^{3k}. \end{align*} Therefore \begin{align} \underline{\dim}_{B}\{(x,\lambda(x)):\alpha<x<\beta\} & = \liminf_{k\to\infty}\frac{\log N_{k}}{-\log 4^{-k}}\notag \\ & \geq \liminf_{k\to\infty}\frac{\log \left(2^{3k}(\beta-\alpha)/{4\sqrt{\beta}}\right)}{-\log 4^{-k}}=\frac{3}{2}. \label{dim:eq:l} \end{align} The result follows from \eqref{dim:eq:u} and \eqref{dim:eq:l}. \end{proof} \begin{corollary} For any $0<\alpha <\beta$, \[\dim_{B}\{(x,\lambda(x)):\alpha<x<\beta\}=\frac{3}{2}.\] \end{corollary} \begin{proof} Let $K$ be an integer such that $\beta/4^{K}\leq 1$. Since $\lambda(4x)=\lambda(x)$ for $x>0$, the following mapping \[f:(x,\lambda(x))\mapsto (4^{K}x,\lambda(4^{K}x))\] is a bi-Lipschitz mapping in $\mathbb{R}^{2}$, and \begin{align*} f\left(\{(x,\lambda(x)):4^{-K}\alpha<x<4^{-K}\beta\}\right) & = \left\{\left(4^{K}x,\lambda(4^{K}x)\right):4^{-K}\alpha<x<4^{-K}\beta\right\}\\ & = \{(y,\lambda(y)):\alpha<y<\beta\}. \end{align*} The result follows from Theorem \ref{dim:thm:1} and the above equation. \end{proof} \section*{References}
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April 15, 2013 April 15, 2013 jmcneill Recent Law Review Articles — April 2013 Leonard, Elizabeth Weeks, Susan Scholz and Raquel Meyer Alexander. Employers united: an empirical analysis of corporate political speech in the wake of the Affordable Care Act. 38 J. Corp. L. 217-257 (2013). Shapiro, Ilya. Like Eastwood talking to a chair: the good, the bad, and the ugly of the Obamacare ruling. 17 Tex. Rev. L. & Pol. 1-23 (2012). Vukadin, Katherine T. Hope or hype?: why the Affordable Care Act's new external review rules for denied ERISA healthcare claims need more reform. 60 Buff. L. Rev. 1201-1253 (2012). Conscience Rights Bowman, Matthew S. and student Christopher P. Schandevel. The harmony between professional conscience rights and patients' right of access. 6 Phoenix L. Rev. 31-62 (2012). Wagstaff, Brandy L. Make way for Segways: mobility disabilities, Segways, and public accommodations. 20 Geo. Mason L. Rev. 347-359 (2013). Johnston, Carl A. and Curtis D. Rooney. GPOs and the health care supply chain: market-based solutions and real-world recommendations to reduce pricing secrecy and benefit health care providers. 29 J. Contemp. Health L. & Pol'y 72-88 (2012). Jeremiah, Onyinyechi. Note. A thin line between inpatient and outpatient: observation status and its impact on the elderly. 20 Geo. J. on Poverty L. & Pol'y 141-159 (2012). Waters, Jessica L. Testing Hosanna-Tabor: the implications for pregnancy discrimination claims and employees' reproductive rights. 9 Stan. J. C.R. & C.L. 47-78 (3013). Lewis, Browne C. A graceful exit: redefining terminal to expand the availability of physician-facilitated suicide. 91 Or. L. Rev. 457-493 (2012). Helland, Eric and Jonathan Klick. Does anyone get stopped at the gate? an empirical assessment of the Daubert trilogy in the states. 20 Sup. Ct. Econ. Rev. 1-33 (2012). Price, Meredith M. Comment. The proper application of Daubert to expert testimony in class certification. 16 Lewis & Clark L. Rev. 1349-1379 (2012). Boyd, William. Genealogies at risk: searching for safety, 1930s-1970s. 39 Ecology L.Q. 895-987 (2012). Johns, A. Frank. Person-centered planning in guardianship: a little hope for the future. 2012 Utah L. Rev. 1541-1573. Symposium. Third National Guardianship Summit: Standards of Excellence. Preface by Leslie P. Francis, introduction by Sally Hurme and Erica Wood; articles by Karen E. Boxx, Terry W. Hammond, Robert B. Fleming, Rebecca C. Morgan, Kim Dayton, Naomi Karp, Erica Wood, Linda S. Whitton, Lawrence A. Frolik, A. Frank Johns, Catherine Seal, Spencer Crona, Mary Joy Quinn, Howard S. Krooks, Julia R. Nack, Carolyn L. Dessin and Thomas Swift. 2012 Utah L. Rev. 1155-1690. Hill, Elizabeth. Recent development. Senate Bill 33 grants protection to emergency room providers…and just about everyone else, too. 91 N.C. L. Rev. 720-744 (2013). Gaylord, Scott W. and Thomas J. Molony. Casey and a woman's right to know: ultrasounds, informed consent, and the First Amendment, 45 Conn. L. Rev. 595-652 (2012). Casey, Meghan C. Note. In whose hands are we placing children's health?: an examination of "medical necessity" for Medicaid's EPSDT provision. 29 J. Contemp. Health L. & Pol'y 89-116 (2012). Dayton, Kim. Standards for health care decision-making: legal and practical considerations. 2012 Utah L. Rev. 1329-1444. Mulder-Westrate, Kelli M. Note. Waiting for the justice league: motivating child welfare agencies to save children. 88 Notre Dame L. Rev. 523-555 (2012). Norton, Ashley A. Note. The captive mind: antipsychotics as chemical restraint in juvenile detention. 29 J. Contemp. Health L. & Pol'y 152-182 (2012). Organ Donations Robertson, John A. Paid organ donations and the constitutionality of the National Organ Transplant Act. 40 Hastings Const. L.Q. 221-275 (2013). Byrnes, Chris R. Note. Patenting life: TRIPS Article 27 & Bolivia's proposal to ban the patenting of all life forms. 24 Geo. Int'l Envtl. L. Rev. 245-265 (2012). Tang, Wanli (Lily). Note. Revitalizing the patent system to incentivize pharmaceutical innovation: the potential of claims with means-plus-function clauses. 62 Duke L.J. 1069-1108 (2013). Kimel, Catherine W. Note. DNA profiles, computer searches, and the Fourth Amendment. 62 Duke L.J. 933-973 (2013). Drugs & Money. Foreword by Jesse A. Goldner; articles by Robert I. Field, Jeremy Sugarman, M.D., Kathleen M. Boozang and Marc A. Rodwin. 6 St. Louis U. J. Health L. & Pol'y 1-165 (2012). Kanter, Jason and Robin Feldman. Understanding and incentivizing biosimilars. 64 Hastings L.J. 57-81 (2012). Liu, Chenglin. Leaving the FDA behind: pharmaceutical outsourcing and drug safety. 48 Tex. Int'l L.J. 1-32 (2012). Okada, Seiko F. Recent development. In re K-Dur Antitrust Litigation: pharmaceutical reverse payment settlements go beyond the "scope of the patent." (In re K-Dur Antitrust Litigation, 686 F.3d 197, 2012.) 14 N.C. J.L. & Tech. 303-340 (2012). Steele, Danielle L. Comment. The "duty of sameness" as a shield—generic drug manufacturers' tort liability and the need for label independence after … (PLIVA, Inc. v. Mensing, 131 S. Ct. 2567, 2011.) 43 Seton Hall L. Rev. 441-490 (2013). Onzivu, William. The long road to integrating public health into sustainable development of shared freshwaters in international environmental law: lessons from Lake Victoria in East Africa. 46 Int'l Law. 867-892 (2012). Nirenberg, Sarah. Note. The resurgence of secularism: hostility towards religion in the United States and France. 5 Wash. U. Jur. Rev. 131-161 (2012). Weil, Michael J. Note. The friendly separation of church and state and bans on male circumcision. 45 Conn. L. Rev. 695-742 (2012). Mullins-Owens, Heather L., Macey L. Henderson and Jason Henderson. Protecting a vulnerable population with little regulatory framework: a comparative analysis of international guidelines for pediatric research ethics. 29 J. Contemp. Health L. & Pol'y 36-71 (2012). Treadaway, Lauren. Note. Big pharma's heart of darkness: the Alien Tort Statute and preventing clinical trial colonialism. 43 Geo. J. Int'l L. 1419-1456 (2012). Myers, Donald B. Jr. and Paul A. Locke. Modernizing U.S. chemicals laws: how the application of twenty-first century toxicology can help drive legal reform. 20 N.Y.U. Envtl. L.J. 35-98 (2012). Cucurullo, Regina T. The Special Supplemental Nutrition Program for Women, Infants and Children (WIC) and the Supplemental Nutrition Assistance Program (SNAP): comparing policies and suggesting changes. 8 J. Food L. & Pol'y 257-279 (2012). Bent, Jason R. An incentive-based approach to regulating workplace chemicals. 73 Ohio St. L.J. 1389-1455 (2012). Brown, Matthew K. Note. How exclusive is the workers' compensation exclusive remedy? 2010 amendments to Oklahoma Workers' Compensation Statute shoot down … (Parret v. UNICCO Serv. Co., 127 P.3d 572, 2005, superseded by statute, 2010 Okla. Sess. Laws 2032-33.) 65 Okla. L. Rev. 75-131 (2012). Wagner, Gerhard. Tort, social security, and no-fault schemes: lessons from real-world experiments. 23 Duke J. Comp. & Int'l L. 1-61 (2012). law review articles jmcneill Previous Gosnell Grand Jury Report Next New Acquisitions — Weel of April 15, 2013
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Hemiancistrus subviridis, the green phantom pleco, is a species of armored catfish from the family Loricariidae, commonly found in Venezuela. Within Venezuela, it is native to the Orinoco and Casiquiare drainage basins, where it is usually found among granitic rocks in flowing water. The species reaches 15 cm (5.9 inches) SL. Hemiancistrus subviridis is one of two species referred to by the L-number L-200. The other is Baryancistrus demantoides, which resembles H. subviridis in appearance, and it is this visual similarity that likely historically caused the two to be thought of as the same species, or at least closely related ones, leading them to share an L-number. References Ancistrini Taxa named by David C. Werneke Taxa named by Mark Henry Sabaj Pérez Taxa named by Nathan Keller Lujan Taxa named by Jonathan W. Armbruster Fish described in 2005
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/cpaa.2007.6.367","text":"# American Institute of Mathematical Sciences\n\nJune\u00a0 2007,\u00a06(2):\u00a0367-387. doi:\u00a010.3934\/cpaa.2007.6.367\n\n## Global existence and long-time behaviour for a singular integro-differential phase-field system\n\n 1 Dipartimento di Matematica \u201cF. Casorati\u201d, Universit\u00e0 degli Studi di Pavia, via Ferrata, 1, 27100, Pavia, Italy 2 Dipartimento di Matematica \u201cF. Enriques\u201d, Universit\u00e0 degli Studi di Milano, via Saldini, 50, 20133, Milano, Italy\n\nReceived\u00a0 February 2006 Revised\u00a0 August 2006 Published\u00a0 March 2007\n\nThis paper deals with a singular integro-differential PDE system describing phase transitions in terms of nonlinear evolution equations for micromotions and for the entropy. The model is derived from a non-convex free energy functional, possibly accounting for thermal memory effects. After recovering a global existence result for a related initial and boundary value problem, the long-time behaviour of the solutions is investigated. In particular, it is proved that the elements of the $\\omega$-limit set (i.e. the cluster points) of the solution trajectories solve the steady state system which is naturally associated to the evolution problem.\nCitation: Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934\/cpaa.2007.6.367\n [1] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020\u00a0 doi: 10.3934\/eect.2020107 [2] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka\/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020\u00a0 doi: 10.3934\/dcdss.2020467 [3] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020\u00a0 doi: 10.3934\/dcdss.2020440 [4] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934\/dcdss.2020325 [5] Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934\/dcds.2020291 [6] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020\u00a0 doi: 10.3934\/dcdsb.2020350 [7] Guido Cavallaro, Roberto Garra, Carlo Marchioro. 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Discrete & Continuous Dynamical Systems - B, 2020\u00a0 doi: 10.3934\/dcdsb.2020318 [15] Tahir Aliyev Azero\u011flu, B\u00fclent Nafi \u00d6rnek, Timur D\u00fczenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020\u00a0 doi: 10.3934\/eect.2020106 [16] Mathew Gluck. Classification of solutions to a system of $n^{\\rm th}$ order equations on $\\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934\/cpaa.2020246 [17] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934\/cpaa.2020257 [18] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934\/dcds.2020264 [19] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934\/krm.2020050 [20] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934\/naco.2020018\n\n2019\u00a0Impact Factor:\u00a01.105","date":"2020-12-04 05:02:35","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\": 0.47182291746139526, \"perplexity\": 5663.610427625916}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141733122.72\/warc\/CC-MAIN-20201204040803-20201204070803-00101.warc.gz\"}"}
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Dozens of children in Malawi will attend high school, thanks to a Bermuda charity's scholarship scheme. Bermuda Overseas Mission raised $6,000 to start the programme to send 60 pupils to school. The organisation visited the Mulanje district in 2015 and again last year. David Frost, the charity's president, said the trips were eye-openers. He added: "We were shocked to learn that girls of 12 or 13 were being married off because their families could not afford to educate the children. "The young boys were left to help their families in subsistence farming, or go to the tea plantations to work. George Frost, a Berkeley Institute student on one of the trips, said the experience made him appreciate life in Bermuda. He explained: "It was very difficult hearing about the kids in our village that were unable to attend school, due to the cost. "You could see that they had such a strong desire to learn and to have an education. Malawi is one of the poorest countries in Africa. Mr Frost said that for $100 a year the charity could provide a high school education to a child. The cost covered a uniform, shoes, backpack and school supplies. Mr Frost said that pupils aged 11 to 18 attended high school "as many of them missed the opportunity at an earlier age". BOM plans to return to Malawi to continue work to build homes in the Mulanje district. Since it was founded in 2003, it has visited 18 countries and built homes for more than 100 families. Anyone interested in contributing can contact David Thompson at david.thompson@ams.bm.
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\section{Introduction} \setcounter{equation}{0}\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} Fractional quantum hall states \cite{Tsui} \cite{Lagh} have edge excitations \cite{Stone}\cite{Wen} \cite{AC1} as their low energy excitations. They are the infinitesimal distortion of the quantum hall droplet. The partition functions for these edge excitations have been discussed for the disk and the annulus geometry \cite{Wen}\cite{AC1}\cite{WenWu2}\cite{Milo}\cite{AC2}. It is natural to generalize the discussions to edge excitations of quantum hall state on a region with multiple boundaries. In this paper, we would like to calculate the multiple edge partition function for these edge excitations, especially for the Laughlin state and the Pfaffian state. The wavefunctions for these quantum hall states are known to be correlation functions of extending fields of certain rational conformal field theories \cite{MorRe}\cite{Cris}\cite{Fubi} \cite{Flo}. Each edge excitation is generated on the primary field on each edge, which is determined by the bulk in the case of disk, but has degrees of freedom in general. These degrees of freedom are globally constrained by the chiral operator algebra of the bulk conformal field theory. In view of this, we introduce an expression of quantum hall states in terms of the chiral vertex operators \cite{MorSei} to deal bulk and edge states simulataneously. By using this method, we find a relation between the partition functions for $n_b$ and $(n_b-1)$ boundaries, which is diagonalized by using the Verlinde formula \cite{Ver}. We obtain the partition functions by using this relation. The organization of the paper is as follows. In Sec. 2, we first review the relation between the bulk and the edge state in the Laughlin state on a disk. We introduce an expression in terms of the chiral vertex operators and extend it to the general cases with multiple boundaries. We get a relation between the partition functions for $n_b$ and $(n_b-1)$ boundaries, which is diagonalized by the Verlinde formula of rational torus. The explicit form of the multiple edge partition function for the Laughlin state in terms of the matrix elements of modular transformation is obtained. In Sec. 3, we extend the method introduced in Sec. 2 to the Pfaffian state. We again get a relation between the partition functions for $n_b$ and $(n_b-1)$ boundaries. We obtain the multiple edge partition functions by diagonalizing the relation by the Verlinde formula of the Ising model. Sec. 4 discusses generalization to other quantum hall states. \section{Edge and Bulk of the Laughlin states } \setcounter{equation}{0}\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} Before considering edge states from many-body wavefunctions, let us recall the relation of 2-dimesional bulk conformal field theory and 1+1 dimensional edge conformal field theory to 2+1 dimensional Chern-Simons theory \cite{Witten}\cite{MorSei2}. Chern-Simons theory on a disk is equivalent to 1+1 dimensional CFT on its edge. This 1+1 dimensional CFT describes the edge excitations of FQH state. On the other hand, the Hilbert space of Chern-Simons theory with Wilson lines on a Riemann surface $\Sigma$ is the space of conformal blocks of 2 dimensional CFT on $\Sigma$. Wilson lines correspond to primay fields of 2 dimensional CFT. Then bulk wavefunctions are states in the physical Hilbert space of Chern-Simons theory with Wilson lines. The bridge between these two CFT is discussed in \cite{MorSei2}. There, by considering the effect of shrinking a boundary, it is shown that it eventually becomes equivalent to a Wilson line. This last fact relates the edge states to fields of the bulk conformal field theory. For example, edge excitations of a quantum hall state on a disk are generated on the primary field at the infinity. \subsection{Laughlin states on a disk} Let us recall the relation between many-body wavefunctions and edge excitations in the $\nu=\frac{1}{q}$ Laughlin state on a disk ($q:$ odd) \cite{Stone}\cite{Wen}\cite{MorRe}\cite{WenWu2}. Its ground state is described by the Laughlin wavefunction \begin{eqnarray} \widetilde{\Phi}(z_1,\cdots,z_N)=\prod_{i<j}(z_i-z_j)^{q} {\rm exp}\left[ -\frac{1}{4}\sum|z_i|^{2}\right]. \label{laughlin} \end{eqnarray} This wavefunction can be written in terms of a correlator of the rational torus with $q$ primary fields. It is described by the chiral boson field $\varphi$ compactified on a circle with a rational value of the square of the radius. It has $q$ primary fields $[\phi_p]=[e^{i\frac{p}{\sqrt{q}}\varphi}]$, with U(1) charge $p/q$, $ p \in Z ({\rm mod}\hspace{1mm} q)$. The fermionic extending operator $\psi_e=e^{i\sqrt{q}\varphi}$ has a conformal weight $q/2$ and a unit U(1) charge. The Laughlin wavefunction (\ref{laughlin}) can be written by this operator as \begin{eqnarray} \widetilde{\Phi}&=& \lim_{z_{\infty}\rightarrow \infty} z_{\infty}^{2h_{\rm edge}}\langle \Psi_{\rm edge}(z_{\infty}) \prod_{i=1}^{N}\psi_e(z_i) {\rm exp}\int \frac{d^2w}{2\pi i}\sqrt{\nu}\varphi(w) \rangle \nonumber \\ &=& \langle \Psi_{\rm edge}^{\vee}| \prod_{i=1}^{N}\psi_e(z_i){\rm exp} \int \frac{d^2w}{2\pi i}\sqrt{\nu}\varphi(w) |0 \rangle, \label{laughlin2} \end{eqnarray} where the factor with the integrand is the neutralizing background field (or background magnetic field) and will be omitted hereafter, and \begin{eqnarray} \Psi_{\rm edge} &=&\phi_{-N}= e^{-iN\sqrt{q}\varphi}, \\ \langle \Psi_{\rm edge}^{\vee}| &=& \lim_{z_{\infty}\rightarrow \infty} \langle 0| \Psi_{\rm edge}(z_{\infty}) z_{\infty}^{2h_{\rm edge}}, \label{Boundary} \\ h_{\rm edge}&=&\frac{1}{2}N^2q. \end{eqnarray} Note that (\ref{Boundary}) is the standard definition of the "out" state for $\Psi_{\rm edge}^{\vee} $ in conformal field theory ($\vee$ denotes the conjugate of field ). Edge excitations are the state of zero-energy for the model Hamiltonian (Haldane's pseudopotential) \cite{Hal}, \begin{eqnarray} V=\sum_{l=0}^{q-1}V_l\sum_{i<j}\delta^{(l)}(z_i-z_j). \label{haldane} \end{eqnarray} The Laughlin wavefunction (\ref{laughlin}) is the exact ground state of $V$ and it is proved that all the zero energy states are obtained by acting the symmetric polynomials of $z_i$ on the ground-state wavefunction \cite{Hal}. In conformal field theory, these edge excitations are described by descendant fields of $\Psi_{\rm edge}$ generated by the $U(1)$ Kac-Moody algebra $j(z)=\sum_{-\infty}^{\infty}j_n(z-z_{\infty})^{-n-1}$ \cite{WenWu2}, \begin{eqnarray} \Psi_{\rm edge}^{(n_1,n_2,\cdots)} &=& (j_{-n_1}j_{-n_2}\cdots)\Psi_{\rm edge}, \\ \label{descendant} [ j_n,j_m ]&=& n \delta_{m+n} . \label{U(1)KacMoody} \end{eqnarray} We get edge-excited wavefunctions by inserting $\Psi_{\rm edge}^{(n_1,n_2,\cdots)}$ as \begin{eqnarray} \widetilde{\Phi}^{(n_1,n_2,\cdots)}&=& \lim_{z_{\infty} \rightarrow \infty} z_{\infty}^{2h_{\rm edge}+2l}\langle \Psi_{\rm edge}^{(n_1,n_2,\cdots)}(z_{\infty}) \prod_{i=1}^{N}\psi_e(z_i)\rangle , \label{edge} \end{eqnarray} where $l=\sum n_k$ is the level of the descendant $\Psi_{\rm edge}^{(n_1,n_2,\cdots)}$. This construction generates the space of symmetric polynomials in $z_i's$ i.e. all the edge excitations when we take the thermodynamic limit $N \rightarrow \infty$ \cite{Stone} \cite{Wen}\cite{WenWu2}. Although edge excitations generate all the zero-energy states of $V$, we can add the term proportional to the total angular momentum $L$ of electrons. This is a natural assumption for the confining potential, since $L$ is nothing but the kinetic energy of edge excitatons. We can compute the eigenvalue of $L$ on $\widetilde{\Phi}$ by transforming $z \rightarrow \lambda z$, $\lambda=e^{i\theta}$. In (\ref{laughlin2}), as $z \rightarrow \lambda z$, we get $\Psi_{\rm edge}\rightarrow \lambda ^{-h_{\rm edge}}\Psi_{\rm edge}$ and $\psi_e \rightarrow \lambda ^{-h_e}\psi_e$, $\widetilde{\Phi}( \lambda z)= \lambda ^{M_0}\widetilde{\Phi}$ with $M_0=h_{\rm edge}-Nh_e=\frac{1}{2}qN(N-1)$. Thus the total angular momentum of the state $\widetilde{\Phi}$ is $M_0=\frac{1}{2}qN(N-1)$. Likewise in (\ref{edge}), $\Psi_{\rm edge}^{(n_1,n_2,\cdots)} \rightarrow \lambda ^{-h_{\rm edge}-l}\Psi_{\rm edge}^{(n_1,n_2,\cdots)}$, we get $\widetilde{\Phi}^{(n_1,n_2,\cdots)}( \lambda z)= \lambda ^{M_0+l} \widetilde{\Phi}^{(n_1,n_2,\cdots)}$ . Thus the total angular momentum of $\widetilde{\Phi}^{(n_1,n_2,\cdots)}$ is $M_0+l$. The number of state at each eigenstate of total angular momentum is given by the partition number of $l$. Therefore, the partition function of edge excitations on disk at the inverse temperature $2\pi\tau=i\beta$ is given by \begin{eqnarray} Z^{\rm disk}(\tau)={\rm Tr}\left( e^{2\pi i\tau(L-\frac{c}{24})}\right)= \frac{\omega^{M_0}}{\eta(\tau)} ,\\ \eta(\tau) = \omega^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-\omega^{n}), \hspace{4mm} \omega={\rm exp}(2\pi i\tau), \label{partition} \end{eqnarray} where $c=1$ and $-\frac{c}{24}$ is a Casimir energy. This partition function is valid in the thermodynamic limit $N \rightarrow \infty$. Now let us see these aspects of the quantum hall state $\widetilde{\Phi}$ of $N$ electrons from the constructive point of view. Suppose that we make a disk-like subregion $D_1 $ which contains only one electron. Then from (\ref{laughlin2}) the state at the boundary $\del D_1$ of $D_1$ is $\Psi_{\rm edge}^{\vee}=\phi_{q}$. Next, we make an annulus-like region $D_2$ around $D_1$ which again contains one electron. Then the state of the outer boundary of $D_2$ (i.e. $ \del(D_1\cup D_2)$) is $\Psi_{\rm edge}^{\vee}=\phi_{2q}$. Similarily we continue take annulus-like regions $D_3 \cdots$ to end up with the $N$-th region $D_N$. The state at the boundary $\del(\bigcup_{i=1}^{K}D_i)$ is $\Psi_{\rm edge}^{\vee}=\phi_{Kq}$. The essential ingredient which this construction requires is {\it fusion rules} of conformal field theory. The fusion rules of the rational torus are \begin{eqnarray} \phi_r \times \phi_s = \phi_{r+s}. \label{u1fusion} \end{eqnarray} The procedure above is summarized as follows : (i) Take a quantum hall droplet with the edge state $\Psi_{\rm edge}^{\vee}$. (ii) Enlarge this droplet by surrounding it with an annulus-like quantum hall liquid with one field $\phi$ . (iii) Then the new edge state is $\Psi_{\rm edge,new}^{\vee}$ determined by the fusion rule \begin{eqnarray} \phi \times \Psi_{\rm edge}^{\vee}=\Psi_{\rm edge, new}^{\vee}. \end{eqnarray} There is a concept which represents this constrution explicitly. It is the "chiral vertex operator" $\iPhi^{i}_{j k}(z) :[\phi_i] \rightarrow {\rm Hom([\phi_k]\ \rightarrow [\phi_j])}$ \footnote{ see \cite{MorSei} for detail. }. It represents the three holed sphere with the fields $\phi_i, \phi^{\vee}_j, \phi_k$ on each hole. In our context, $\iPhi^{i}_{j k}(z)$ represents the annulus with the field $\phi_i$ inserted at $z$ with the inner and outer edge states $\phi_k$ and $\phi_j^{\vee}$ respectively . It implies $\phi_i \times \phi_k=\phi_j$. Then the quantum hall state $\widetilde{\Phi}$ is expressed according to the above construction as \footnote{Preliminary description of quantum hall states in terms of chiral vertex operators is discussed in \cite{Ino}. } \begin{eqnarray} \widetilde{\Phi}=\iPhi^{q}_{Nq,(N-1)q}(z_1)\cdots \iPhi^{q}_{2q,q}(z_{N-1})\iPhi^{q}_{q,0}(z_N). \end{eqnarray} For the quantum hall state with edge excitations $\widetilde{\Phi}^{(n_1,n_2,\cdots)}$, we have \begin{eqnarray} \widetilde{\Phi}^{(n_1,n_2,\cdots)}&=&\iPhi^{q}_{b,(N-1)q}(z_1)\cdots \iPhi^{q}_{2q,q}(z_{N-1})\iPhi^{q}_{q,0}(z_N), \\ b^{\vee}&=& (j_{-n_1}j_{-n_2}\cdots)e^{-iN\sqrt{q}\varphi}. \end{eqnarray} We define an operator $M$ on chiral vertex operators by \begin{eqnarray} M\iPhi_{jk}^{i}=(- {\mit \Delta}_i+ {\mit \Delta}_j- {\mit \Delta}_k)\iPhi_{jk}^{i} , \end{eqnarray} where $ {\mit \Delta}_m$ is the conformal weight of field $\phi_m$. We demand that it satisfies the Leibnitz rule when it acts on the product of chiral vertex operators. Then \begin{eqnarray} M\widetilde{\Phi}&=&M_0\widetilde{\Phi}, \\ M\widetilde{\Phi}{(n_1,n_2,\cdots)}&=&(M_0+l)\widetilde{\Phi}, \end{eqnarray} where $M_0=\frac{1}{2}qN(N-1)$, and $l=\sum_{i}n_i$ is the level of the descendant field $b$. Thus, $M$ coincides with the total angular momentum operator $L$ when it acts on the space of the monomials of chiral vertex operators which express $\widetilde{\Phi}^{(n_1,n_2,\cdots)}$. Let us denote the vector space spanned by such monomials as $\Omega_{\rm edge}$. The partition function (\ref{partition}) can now be written as \begin{eqnarray} Z^{\rm disk}={\rm Tr}_{\Omega_{\rm edge}} \left( e^{2\pi i\tau(M-\frac{c}{24})} \right). \end{eqnarray} \subsection{ Laughlin states on an annulus} We consider a Laughlin state $\widetilde\Phi(z_1,\cdots,z_N)$ on an annulus $\widetilde{A}$, with $N$ electrons. We can adjust the velocities of edge excitations on each edge to treat the energy of each boundary on equal footing. To divide the quantum hall state into components as in the previous section, we first make an annulus-like region $D_1$ (instead of disk) around the inner boundary of $\widetilde{A}$ which contains only one electron. Then we make a series of annulus-like regions, $D_2\cdots D_N$, each of which contains one electron respectively. Eventually, we end up with the following expression for $\widetilde\Phi(z_1,\cdots,z_N)$: \begin{eqnarray} \widetilde{\Phi} = \iPhi_{\beta_1, d_1}^{q}(z_1)\cdots \iPhi_{d_{N-1} \beta_2}^{q}(z_N), \end{eqnarray} where $\beta_1$ and $\beta_2$ are the primary fields on the outer and the inner edges, respectively. The edge excitations are generated by descendant fields of them in the thermodynamic limit. $d_1 \cdots d_N$ are internal states determined by $\beta_1$ and $\beta_2$ . On contrary to the case of disk, we can't specify $\beta_1$ and $\beta_2$ uniquely. However, we see that they must satisfy the following condition from the U(1) fusin rules: \begin{eqnarray} \beta_1= \beta_2+Nq. \end{eqnarray} From this constraint, the general edge states are given by \begin{eqnarray} \beta_1= \lambda +m_1q, \hspace{4mm} \beta_2= \lambda +m_2q \\ m_1,m_2\in Z, m_1-m_2=Nq, \hspace{3mm} \lambda =0, \cdots, q-1 . \label{sector} \end{eqnarray} Let $j^{1}_n$ and $j^{2}_n$ be the generators of U(1) Kac- Moody algebra on each boundary respectively. Then, the quantum hall state with edge excitations are expressed as \begin{eqnarray} \widetilde{\Phi}^{(n_1,n_2, \cdots \overline{n}_1,\overline{n}_2,\cdots)} = \iPhi_{b_1, d_1}^{q}(z_1)\cdots \iPhi_{d_{N-1} b_2}^{q}(z_N),\\ b_1^{\vee} = (j^{1}_{-n_1}j^{1}_{-n_2}\cdots)e^{-i\beta_1\phi/\sqrt{q}}, \hspace{4mm} b_2 = (j^{2}_{-\overline{n}_1}j^{2}_{-\overline{n}_2}\cdots) e^{i\beta_2\phi/\sqrt{q}}, \end{eqnarray} where $ l_1=\sum_i n_i$ and $l_2=\sum_i \overline{n}_i $ are the level on each edge. The action of $M$ on $\widetilde{\Phi}^{(n_1 \cdots \overline{n}_1\cdots)} $ becomes \begin{eqnarray} M\widetilde{\Phi} =\left(\frac{1}{2}qN(N-1)+(m_2q+ \lambda )N+ l_1-l_2\right)\widetilde{\Phi}, \label{pseudo} \end{eqnarray} which shows $M$ coincides with the angular momentum also in this case. On the other hand, we must treat the two boundaries equally to consider the energy of the states. To this end, we introduce another expression of $\widetilde{\Phi}$ in terms of the chiral vertex operators. One starts with a disk like region $D_1$ on $\widetilde{A}$ which contains only one electron. Then we make a series of annulus-like regions $D_2\cdots D_N$, each of which contains one electron respectively. To produce the two edges, it is necessary to insert a disk with two holes, $C$ in the series, which contains no electrons. Let us denote the chiral vertex operator corresponding to $C$ as $ {\mit \Lambda}_{j k}^{i}$. Then, we arrive at the following expression for $\widetilde{\Phi}$: \begin{eqnarray} \widetilde{\Phi}=\iPhi_{\beta_1, d_1}^{q} \iPhi_{\beta_2^{\vee} d_2}^{q}\cdots {\mit \Lambda}_{c_1 c_3}^{c_2}\cdots \iPhi_{d_N 0}^{q}. \label{pannulus} \end{eqnarray} We can further arrange this expression by using the worldsheet duality of conformal field theory. Duality results in a set of " duality transformations" of chiral vertex operators (see \cite{MorSei} for details). By using duality transformations, we arrange the expression (\ref{pannulus}) to be \begin{eqnarray} \widetilde{\Phi}_{\Lambda}= {\mit \Lambda}_{\beta_1, Nq}^{\beta_2} \iPhi_{Nq, (N-1)q}^{q}\cdots \iPhi_{q, 0}^{q}. \label{annulus} \end{eqnarray} The states with edge excitations are now expressed as \begin{eqnarray} \widetilde{\Phi}_{\Lambda}^{(n_1,n_2,\cdots,\overline{n}_1,\overline{n}_2\cdots)} = {\mit \Lambda}_{b_1, Nq}^{b_2} \iPhi_{Nq,(N-1)q}^{q}\cdots \iPhi_{q, 0}^{q}. \label{annuedge} \end{eqnarray} We will denote the vector space spanned by $\widetilde{\Phi}_{\Lambda}^{(n_1,n_2,\cdots,\overline{n}_1,\overline{n}_2\cdots)}$ as $\Omega_{N}$. We define the action of $M$ on $ {\mit \Lambda}_{jk}^{i}$ by \begin{eqnarray} M {\mit \Lambda}_{jk}^{i}=( {\mit \Delta}_i+ {\mit \Delta}_j- {\mit \Delta}_k) {\mit \Lambda}_{jk}^{i}, \end{eqnarray} and introduce another operator $\overline{M}$ by \begin{eqnarray} \overline{M}\iPhi_{jk}^{i}&=&( {\mit \Delta}_i+ {\mit \Delta}_j- {\mit \Delta}_k)\iPhi_{jk}^{i}, \\ \overline{M} {\mit \Lambda}_{jk}^{i}&=&( {\mit \Delta}_i+ {\mit \Delta}_j- {\mit \Delta}_k) {\mit \Lambda}_{jk}^{i}. \end{eqnarray} We define the energy for edge states by $M_E=\frac{1}{2}(M+\overline{M})$ to cancel the contribution from the bulk. The value of $M_E$ on $\widetilde{\Phi}_{\Lambda}^{(n_1,\cdots,\overline{n}_1,\cdots)}$ is \begin{eqnarray} M_E \widetilde{\Phi}_{\Lambda}^{(n_1,\cdots,\overline{n}_1,\cdots)} =\left(\frac{(m_1q+ \lambda )^2}{2q}+l_1+\frac{(m_2q+ \lambda )^2}{2q}+l_2 \right) \widetilde{\Phi}_{\Lambda}^{(n_1,\cdots,\overline{n}_1,\cdots)}. \label{pseudo2} \end{eqnarray} We see that $M_E$ coincides with the sum of the " pseudoenergy" of \cite{Milo}. Also, $Q\equiv\frac{1}{q}(\overline{M}-M)$ acts as the total charge operator \begin{eqnarray} Q \widetilde{\Phi}_{\Lambda}^{(n_1,\cdots,\overline{n}_1,\cdots)} =Nq\widetilde{\Phi}_{\Lambda}^{(n_1,\cdots,\overline{n}_1,\cdots)}. \label{pseudo3} \end{eqnarray} which would couple to the chemical potential. Now let us consider the annulus partition function for $M_E$. For that purpose, we follow the discussion of \cite{Milo}. As the change of $\beta$'s by a $q$ unit is equivalent to adding or removing an electron, and the bulk state is not disturbed by this change, we extend a single charge sector to include all the sectors differing by integral charges. It means that we must also consider negative $N$. At first sight, this seems to be a contradiction, but it is not. In conformal field theory, the bulk wavefunction can also be reconstructed as the correlation function of $e^{-i\sqrt{q}\varphi}$ since there is ambiguity in the sign of charge. In other words, the ground state wavefunction for the quantum hall state can also be expressed as \begin{eqnarray} \iPhi_{-Nq, -(N-1)q}^{-q}\cdots \iPhi_{-q, 0}^{-q}. \end{eqnarray} When we consider the charge sectors as above, we must take into account these expressions too. By defining $Q\equiv\frac{1}{q}(M-\overline{M})$ for negative $N$, we can keep track of the sign of the total charge. Thus the partition function we will consider is the grand-canonical partition function on the space $\Omega_{\rm edge}=\bigoplus_{N=-\infty}^{\infty}\Omega_N$ ($2\pi\tau=i\beta, 2\pi\zeta=-i\mu\beta$ where $\beta=1/k_BT$ is the inverse temperature and $\mu$ is the chemical potential) : \begin{eqnarray} Z^{\rm ann}(\tau,\zeta)={\rm Tr}_{\Omega_{\rm edge}} \left( e^{2\pi i\tau(M_E-\frac{n_bc}{24})+2\pi i\zeta Q} \right). \end{eqnarray} Here the term proportional to the central charge ($n_b=2$) is a Casimir energy factor. From (\ref{pseudo2})(\ref{pseudo3}), we get \begin{eqnarray} Z^{\rm ann}(\tau, \zeta)=\sum_{ \lambda =0}^{q-1} \chi_{ \lambda }^{2}(\tau,\zeta) \end{eqnarray} where $\chi_{ \lambda /q}$ are the characters of the rational torus \begin{eqnarray} \chi_{ \lambda /q}(\tau,\zeta)=\frac{1}{\eta}\sum_{m \in Z} e^{2\pi i\tau\frac{(mq+ \lambda )^2}{2q}+2\pi i\zeta(m+\frac{ \lambda }{q})}. \hspace{4mm} \label{chi} \end{eqnarray} We see that $\chi_{ \lambda /q}$ satisfies $\chi_{ \lambda /q}=\chi_{- \lambda /q} =\chi_{(q+ \lambda )/q}$. The contribution of each edge can be distinguished by introducing complex conjugate variables. Then the partition function becomes \begin{eqnarray} Z^{\rm ann}=\sum_{ \lambda =0}^{q-1} \chi_{ \lambda /q} \overline{\chi}_{ \lambda /q}, \end{eqnarray} which is nothing but the $\Gamma(2)$ invariant partition function obtained in \cite{AC1}\cite{Milo}\cite{AC2}. \subsection{Laughlin states with multiple edges} Let $\widetilde{D}$ be a region which has $n_b$ boundaries $B_1, \cdots ,B_{n_b}$, and consider a Laughlin state $\widetilde\Phi(z_1,\cdots,z_N)$ on $\widetilde{D}$. Let $B_{1}$ be the outer boundary which encloses $\widetilde{D}$. When we divide the quantum hall state into components as in the previous sections, it is necessary to insert $(n_b-1)$ disks with two holes, $C_1,\cdots, C_{n_b-1}$, all of which contains no electron. By this procedure, $D$ is divided into $(N+n_b-1)$ regions $D_1,\cdots,D_N , C_1,\cdots, C_{n_b-1}$. Then, $n_b$ regions among $D_1 \cdots D_N,C_1,\cdots, C_{n_b-1}$ have one of $n_b$ boundaries $B_1 \cdots B_{n_b}$ respectively. As in the previous section, each $D_l, l= 1,\cdots N $ corresponds to a chiral vertex operator $\iPhi^{i}_{j k}$ with $\phi_i$. Also, we assign a chiral vertex operator to $C_m$ and denote them as $ {\mit \Lambda}_{j k}^{i}$, where $k$ is assigned to the outer boundary of $C_m$. As in the case of annulus, we arrange the expression into the following form by duality transformations: \begin{eqnarray} \widetilde{\Phi}= {\mit \Lambda}_{\beta_1 \alpha_1}^{\beta_2} {\mit \Lambda}_{\alpha_1 \alpha_2}^{\beta_3}\cdots {\mit \Lambda}_{\alpha_{n_b-2} Nq}^{\beta_{n_b}} \iPhi_{Nq, (N-1)q}^{q}(z_1)\cdots \iPhi_{q 0}^{q}({z_N}). \label{medge} \end{eqnarray} We see that $\widetilde{\Phi}$ is now separated into the edge and the bulk parts. $\beta_1 \cdots \beta_{n_b}$ are the primary fields on which the edge excitations are generated. From the fusion rules of the rational torus , $\beta_1, \cdots, \beta_{n_b} $ are globally constrained by the following relation: \begin{eqnarray} \beta_1=\beta_2+\cdots\beta_{n_b}+Nq. \label{nbsector} \end{eqnarray} Then the general inequivalent states with edge excitations are \begin{eqnarray} \widetilde{\Phi}^{(\{n\})} &=& {\mit \Lambda}_{b_1 \alpha_1}^{b_2} {\mit \Lambda}_{\alpha_1 \alpha_2}^{b_3}\cdots {\mit \Lambda}_{\alpha_{n_b-2}, Nq}^{b_{n_b}} \iPhi_{Nq, (N-1)q}^{q}(z_1)\cdots \iPhi_{q 0}^{q}({z_N}) \\ b_1^{\vee} &=& (j^{1}_{-n^{1}_1}j^{1}_{-n^{1}_2}\cdots) e^{-i\beta_1 \varphi/\sqrt{q}},\\ b_k &=& (j^{k}_{-n^{k}_1}j^{k}_{-n^{k}_2}\cdots) e^{i\beta_k \varphi/\sqrt{q}}, \hspace{3mm} k=2,\cdots n_b \end{eqnarray} where $(\{n\})=(\{n^{1}_1\cdots\}\{n^{2}_1 \cdots\} \{n^{n_b}_1 \cdots\}) $. Let $\Omega_N$ be the vector space spanned by expressions $\widetilde{\Phi}^{(\{n\})}$. We take $M_E$ to be the pseudoenergy of the system as the generalization from the annulus case. To consider the partition function, we gather all the charge sectors differing by integer charge as in the case of annulus, and consider the space of the edge states $\Omega_{\rm edge}=\bigoplus_{N=-\infty}^{\infty}\Omega_N$. From (\ref{nbsector}), the (grand-canonical ) partition function is \begin{eqnarray} Z^{(n_b)}(\tau,\zeta)&=&{\rm Tr}_{\Omega_{\rm edge}} \left ( e^{2\pi i\tau(M_E-\frac{n_bc}{24})+2\pi i\zeta Q} \right) \\ &=& \sum_{ \lambda _1- \lambda _2-\cdots- \lambda _{n_b}\equiv 0} \chi_{ \lambda _{1}/q}\cdots \chi_{ \lambda _{n_b}/q} \label{mlaugh} \end{eqnarray} This partition function can also be obtained from the following method using the Verlinde formula \cite{Ver}. First suppose that $\beta_{n_b}=r \hspace{2mm}({\rm mod} \hspace{1mm} q) , \hspace{3mm}r\in\{1,\cdots,q-1\}$. Then $\alpha_{n_b-2}=r$ ({\rm mod} \hspace{1mm} $q$) and $b_1,\cdots b_{n_b-1}$ can be seen as $(n_b-1)$ edge states in the presence of $r$ quasiholes. Let us denote the space of expressions in terms of chiral vertex operators for these edge states as $\Omega_r^{(n_b)}$, and introduce the partition function on $\Omega_r^{(n_b)}$, \begin{eqnarray} Z^{(n_b)}_r(\tau,\zeta)&=&{\rm Tr}_{\Omega_{r}^{(n_b)}} \left ( e^{2\pi i\tau(M_E-\frac{n_bc}{24})+2\pi i\zeta Q} \right) \end{eqnarray} From the discussion above, $Z^{(n_b)}_r$ is factorized by $Z^{(n_b-1)}$ as \begin{eqnarray} Z^{(n_b)}_r=\sum_{r=0}^{q-1} \chi_{s/q}Z^{(n_b-1)}_{r+s}. \end{eqnarray} We see that these relations can be written by use of the fusion rules of the rational torus $N_{jk}^{i}=\delta^{(q)}_{j+k,i}$ as \begin{eqnarray} Z_r^{(n_b)}=\sum_{s,t}N^t_{rs}\chi_{s/q} Z_{t}^{(n_b-1)}. \label{Zeq} \end{eqnarray} To get the explicit formula of the partition function, let us recall some facts about the characters $\chi_{ \lambda /q}$ (\ref{chi}) of the rational torus. The modular transformation $S : \tau\rightarrow -\frac{1}{\tau}, \zeta\rightarrow -\frac{\zeta}{\tau} $ acts on $\chi_{ \lambda }$ as Fourier transformation: \begin{eqnarray} \chi_{ \lambda } \rightarrow \widetilde{\chi}_ \lambda =\frac{1}{\sqrt{q}}\sum_{ \lambda '=0}^{q-1}e^{2\pi i \lambda \lamda'/q} \chi_{ \lambda '}. \end{eqnarray} The matrix elements of $S$ are therefore $S^{k}_{n}=\frac{1}{\sqrt{q}}{\rm exp}(2\pi ikn/q)$. As conjectured by Verlinde and proved by Moore and Seiberg, The matrix elements $S_i^{j}$ of modular transformation and the fusion rules $N_{jk}^{i}$ of rational conformal field theory have following relation \cite{Ver}\cite{MorSei}: \begin{eqnarray} N_{jk}^{i}&=& \sum_n S_j^{n} \lambda _k^{(n)} S_n^{\dag i}, \\ \label{Ver} \lambda _k^{(n)}&=&{S_k^{n}}/{S^{n}_{0}}. \end{eqnarray} Now let us solve the equation (\ref{Zeq}) by using this formula. First, we do the Fourier transformation on $Z_r^{n_b}$, \begin{eqnarray} F_r^{(n_b)}&=&\frac{1}{\sqrt{q}}\sum_{ \lambda }e^{-2\pi ir \lambda /q} Z_{ \lambda }^{(n_b)}. \end{eqnarray} Then, by using the Verlinde formula (\ref{Ver}), (\ref{Zeq}) is now arranged into a simple formula, \begin{eqnarray} F_r^{(n_b)}=\sqrt{q}\widetilde{\chi}_{r/q}F_r^{(n_b-1)} \end{eqnarray} As $Z_r^{1}$ is nothing but $\chi_{r/q}$, $F_r^{1}=\frac{1}{\sqrt{q}}\sum_{ \lambda }e^{-2\pi ir \lambda /q}\chi_{r/q}$. Thus we get \begin{eqnarray} F_r^{(n_b)}=q^{\frac{n_b-2}{2}}(\sum_{ \lambda }e^{-2\pi ir \lambda /q}\chi_{ \lambda /q})(\widetilde{\chi}_{r})^{n_b-1}. \end{eqnarray} By using the inverse Fourier transformation, we obtain the partition functions $Z_r^{n_b}$ as \begin{eqnarray} Z_r^{(n_b)}&=&q^{\frac{n_b-3}{2}}\sum_{ \lambda =0}^{q-1} e^{-2\pi ir \lambda /q}F_{ \lambda }^{(n_b)}\\ &=& \frac{1}{q}\sum_{ \lambda =0}^{q-1} e^{-2\pi ir \lambda /q}\left( \sum_{ \lambda '=0}^{q-1}e^{-2\pi i \lambda \lamda'/q}\chi_{ \lambda /q}\right) \left(\sum_{ \lambda '=0}^{q-1}e^{2\pi i \lambda \lamda'/q} \chi_{ \lambda '/q} \right)^{n_b-1}. \end{eqnarray} In particular, we get the grand-canonical partition function for the edge excitation of the Laughlin state with $n_b$ boundaries as \begin{eqnarray} Z_0^{(n_b)}&=&\frac{1}{q}\sum_{ \lambda =0}^{q-1}\left(\sum_{ \lambda '=0}^{q-1}e^{-2\pi i \lambda \lamda'/q}\chi_{ \lambda '/q}\right) \left(\sum_{ \lambda '=0}^{q-1}e^{2\pi i \lambda \lamda'/q} \chi_{ \lambda '/q}\right)^{n_b-1} \end{eqnarray} This is indeed the partition function in (\ref{mlaugh}). \section{Edge and Bulk of the Pfaffian State} \setcounter{equation}{0}\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} The method using the Verlinde formula to obtain the multiple edge partition function is applicable to other quantum hall states based on rational conformal field theories. As an example, we'd like to calculate the multiple edge partition function of the Pfaffian state \cite{MorRe}. The ground-state wavefunction of the Pfaffan state at the filling fraction $\nu=\frac{1}{q}$ ($q$ : even) for an even number $N$ of electrons is \begin{eqnarray} {\rm Pfaff}(\frac{1}{z_i-z_j})\prod_{i<j}(z_i-z_j)^{q}{\rm exp}\left[ -\frac{1}{4} \sum_i|z_i|^2\right] . \label{Pfaff} \end{eqnarray} This state can be written in terms of Majorana-Weyl fermion $\psi$ as : \begin{eqnarray} \langle \psi(z_1)e^{i\sqrt{q}\varphi(z_1)} \cdots \psi(z_N)e^{i\sqrt{q}\varphi(z_N)} {\rm exp}\int \frac{d^2z}{2\pi i} \sqrt{\nu}\varphi(z)\rangle. \end{eqnarray} The minimal fusion algebra including $\psi$ is that of the Ising model. This algebra is the " center algebra" in the sense of \cite{WenWu2}. The Ising model has three primary fields, $1,\psi$ and $\sigma$, where $\sigma$ is the spin field. The fusion rules of the Ising model are \begin{eqnarray} \psi\times\psi=1 ,\hspace{2mm} \psi\times \sigma =\sigma , \hspace{2mm} \sigma \times \sigma =1 +\psi . \label{isfus} \end{eqnarray} The couplings to the rational torus are restricted by the requirement of single-valuedness and non-singularity of wavefunctions in the electron coodinates. This requirement is shown to be equivalent to an orbifold construction \cite{Milo} and the allowed couplings are $\{e^{i\frac{r}{\sqrt{q}}} \}$, $\{\psi e^{i\frac{r}{\sqrt{q}}} \}$, $\{\sigma e^{i\frac{2r+1}{2\sqrt{q}}} \}$, $r=0,\cdots,q-1$. We can apply the same technique used in the Laughlin state to express the quantum hall state. The Pfaffian state (\ref{Pfaff}) on a disk is expressed in terms of chiral vertex operators as \begin{eqnarray} \iPhi_{1\psi}^{\psi}(z_1)\iPhi_{\psi 1}^{\psi}(z_2)\cdots\iPhi_{1\psi}^{\psi} (z_{N-1}) \iPhi_{\psi 1}^{\psi}(z_{N}). \label{Pfeven} \end{eqnarray} where the indices for the rational torus are omitted. We see that the contribution to the edge state from the Ising model is $1$. It is also possible to consider the Pfaffian state for the odd number of electrons. In this case, we have \begin{eqnarray} \iPhi_{\psi1}^{\psi}(z_1)\iPhi_{1 \psi}^{\psi}(z_2)\cdots\iPhi_{1\psi}^{\psi} (z_{N-1}) \iPhi_{\psi 1}^{\psi}(z_{N}). \label{Pfodd} \end{eqnarray} The Ising model contribution to the edge state is $\psi$ in this case. The edge excitations are generated by the descendant fields of the primary field at each edge as in the Laughlin state (for the Ising model, descendant fields are generated by $\{L_{-n}\},n=1,2,\cdots $ of Virasoro algebra). The operator $M$ acts as the angular momentum operator also in this case. To give explicit formulas for the partition functions , let us recall the Virasoro characters of the Ising model ($\omega= {\rm exp}(2\pi i\tau)$), \begin{eqnarray} \chi^{\rm MW}_1(\tau)&=&\frac{1}{2}\omega^{-\frac{1}{48}}\left(\prod_{0}^{\infty}(1+\omega^{n+\frac{1}{2}}) + \prod_{0}^{\infty}(1-\omega^{n+\frac{1}{2}}) \right), \\ \chi^{\rm MW}_{\psi}(\tau)&=& \frac{1}{2}\omega^{-\frac{1}{48}}\left(\prod_{0}^{\infty}(1+\omega^{n+\frac{1}{2}}) - \prod_{0}^{\infty}(1-\omega^{n+\frac{1}{2}}) \right), \\ \chi^{\rm MW}_{\sigma}(\tau)&=& \omega^{\frac{1}{24}}\prod_{1}^{\infty}(1+\omega^n). \end{eqnarray} Now, the partition function for a disk is obtained as \begin{eqnarray} Z^{\rm disk}(\tau)= {\rm Tr}_{\Omega_{\rm edge}}\left(e^{2\pi i\tau(M-\frac{c}{24})} \right) =\frac{\omega^{M_0}\chi^{\rm MW}_1(\tau)}{\eta(\tau)} \hspace{4mm} {\rm for} \hspace{3mm} N \hspace{3mm} {\rm even}, \\ M_0=\frac{1}{2}(qN(N-1)-(N-1)) \\ Z^{\rm disk}(\tau)=\frac{\omega^{M_0}\chi^{\rm MW}_{\psi}(\tau)}{\eta(\tau)}\hspace{4mm} {\rm for}\hspace{3mm} N \hspace{3mm} {\rm odd}, \\ c=\frac{3}{2}, \hspace{4mm} M_0=\frac{1}{2}(qN(N-1)-(N-1)). \end{eqnarray} Next, let us consider the Pfaffian state on an annulus. Following the same argument to give (\ref{annulus}), it is expressed by chiral vertex operators as \begin{eqnarray} {\mit \Lambda}_{\beta_1 \alpha_1}^{\beta_2} \iPhi_{\alpha_1 \alpha_2}^{\psi}\cdots \iPhi_{\alpha_{N-1}, 1}^{\psi}. \label{pfann} \end{eqnarray} In the case of the Laughlin state on an annulus, we gather all the charge sectors differing by integral charges into a single sector. For the Pfaffian state, we must gather all the charge sectors differing by {\it even} integral charges into a single sector, since electrons are paired by the degree of freedom from the Ising model. So, we introduce the following functions for the rational torus: \begin{eqnarray} \chi_{r/q}^{\rm even}(\tau,\zeta)= \frac{1}{\eta}\sum_{m \in Z_{\rm even}} e^{2\pi i\tau\frac{(mq+ \lambda )^2}{2q}+2\pi i\zeta(m+\frac{ \lambda }{q})} \\ \chi_{r/q}^{\rm odd}(\tau,\zeta)= \frac{1}{\eta}\sum_{m \in Z_{\rm odd}} e^{2\pi i\tau\frac{(mq+ \lambda )^2}{2q}+2\pi i\zeta(m+\frac{ \lambda }{q})} \end{eqnarray} From these definitions, they satisfy \begin{eqnarray} \chi^{\rm even}_{r/q}=\chi^{\rm even}_{-r/q}=\chi^{\rm even}_{(2q+r)/q} =\chi^{\rm odd}_{(q+r)/q}, \\ \chi^{\rm odd}_{r/q}=\chi^{\rm odd}_{-r/q}=\chi^{\rm odd}_{(2q+r)/q} =\chi^{\rm even}_{(q+r)/q}. \end{eqnarray} Also, we introduce the following functions according to the coupling of the Ising model and the rational torus ($a$=even, odd): \begin{eqnarray} \chi_{i,r}^{a}&=&\chi_i^{\rm MW}\chi_{r/q}^{a}, \hspace{4mm} i=1,\psi, \\ \chi_{\sigma,r}^{a}&=&\chi_{\sigma}^{\rm MW}\chi_{(r+1/2)/q}^{a}. \end{eqnarray} For $N$ even, $\alpha_1$ in (\ref{pfann}) is $1e^{\pm iN\sqrt{q}\varphi}$. If $\beta_1$ is from the even (odd) sector of the rational torus, $\beta_2$ is from the even (odd) sector and visa versa. As in the Laughlin state, we define the pseudoenergy by $M_E=\frac{M+\overline{M}}{2}$ and the total charge operator by $Q=\frac{1}{q+1} (\overline{M}-M)$ for the expressions with positive $N$ and $Q=\frac{1}{q+1} (M-\overline{M})$ for the expressions with negative $N$. Then, the grand-canonical partition function for $N$ even is \begin{eqnarray} Z^{\rm even}&=& {\rm Tr}_{\Omega^{\rm even}_{\rm edge}} \left( e^{2\pi i\tau(M_E-\frac{n_bc}{24})+2\pi i\zeta Q} \right) \nonumber \\ &=& \sum_{a}\sum_{r=0}^{q-1}\left[(\chi_{1.r}^{a})^2+ (\chi_{\psi.r}^{a})^2+(\chi_{\sigma}^{a})^2 \right], \end{eqnarray} where $\Omega^{\rm even}_{\rm edge}=\bigoplus_{N:{\rm even}}\Omega_N$. For $N$ odd, $\alpha_1$ in (\ref{pfann}) is $\psi e^{\pm iN\sqrt{q}\varphi}$. If the $\beta_1$ is from the even(odd) sector of the rational torus, $\beta_2$ is from the odd (even ) sector and visa versa. From (\ref{isfus}), we obtain \begin{eqnarray} Z^{\rm odd}&=& {\rm Tr}_{\Omega^{\rm odd}_{\rm edge}} \left( e^{2\pi i\tau(M_E-\frac{n_bc}{24})+2\pi i\zeta Q} \right) \nonumber \\ &=&\sum_{r=0}^{q-1} \left[2\chi_{1,r}^{\rm even}\chi_{\psi,r}^{\rm odd}+2\chi_{1,r}^{\rm odd}\chi_{\psi,r}^{\rm even}+2\chi_{\sigma,r}^{\rm even}\chi_{\sigma,r}^{\rm odd} \right], \end{eqnarray} with $\Omega^{\rm odd}_{\rm edge}=\bigoplus_{N:{\rm odd}}\Omega_N$. By introducing complex conjugate variables to distinguish two edges, we see that the sum of these partition functions $Z^{\rm even}+Z^{\rm odd}$ is \begin{eqnarray} Z^{\rm annulus}=\sum_{r=0}^{q-1}\left[ |\chi_{1,r}^{\rm even}+ \chi_{\psi,r}^{\rm odd}|^2+ |\chi_{\psi,r}^{\rm even}+ \chi_{1,r}^{\rm odd}|^2+ |\chi^{\rm even}_{\sigma,r}+\chi^{\rm odd}_{\sigma,r}|^2 \right] \nonumber\\ =\sum_{r=0}^{q-1}\left[ |\chi_1^{\rm MW}\chi_r^{\rm even}+ \chi_{\psi}^{\rm MW}\chi_r^{\rm odd}|^2+ |\chi_{\psi}^{\rm MW}\chi_r^{\rm even}+ \chi_1^{\rm MW}\chi_r^{\rm odd}|^2+ |\chi_{\sigma}\chi_{r+1/2}|^2 \right]. \end{eqnarray} This partition function is the one derived in \cite{Milo} when $\zeta=0$. Generalization to the case of $n_b$ boundaries is similar to the case of the Laughlin state. The Pfaffian state on a region with $n_b$ boundaries is expressed as \begin{eqnarray} {\mit \Lambda}_{\beta_1 \alpha_1}^{\beta_2} {\mit \Lambda}_{\alpha_1 \alpha_2}^{\beta_3}\cdots {\mit \Lambda}_{\alpha_{n_b-2} \alpha_{n_b-1}}^{\beta_{n_b}} \iPhi_{\alpha_{n_b-1} c_1}^{\psi}\iPhi_{c_1 c_2}^{\psi} \cdots \iPhi_{c_{N-1} 1}^{\psi}. \label{medge2} \end{eqnarray} Let us denote the partition function of edge excitations with $\alpha_{n_b-1}=(i, \lambda ,a)$ where $i=1,\psi,\sigma$ and $ \lambda =0,\cdots, q-1, a={\rm even,odd}$ as $Z_{i, \lambda }^{a,(n_b)}$. By summing over all the sectors of $\beta_{n_b}$, we find the equation satisfied by $Z_{i, \lambda }^{a,(n_b)}$: \begin{eqnarray} Z_{i,s}^{a_1,(n_b)}=\sum_{jk}\sum_{a_2+a_3=a_1}\sum_{ \lambda =0}^{q-1} N^{k}_{ij}\chi_{j, \lambda }^{a_2}Z^{a_3,(n_b-1)}_{k,s+ \lambda }. \end{eqnarray} Here $N^{k}_{ij}$ is the fusion rules of the Ising model and $Z^{(n_b-1)}$ is the partition function for the remaining $(n_b-1)$ boundaries. The Verlinde formula (\ref{Ver}) implies that this equation is diagonalized by the modular transformation. The matrix element of modular transformation for the Virasoro characters $\chi^{\rm MW}_1,\chi^{\rm MW}_{\sigma},\chi^{\rm MW}_{\psi}$ of the Ising model are \begin{eqnarray} S^{i}_{j}=\frac{1}{2}\left( \begin{array}{ccc} 1 & \sqrt{2} & 1\\ \sqrt{2} & 0 & -\sqrt{2} \\ 1 & -\sqrt{2} & 1 \\ \end{array} \right). \end{eqnarray} Putting $W_i=\sum_j S^j_i Z_j$, we get \begin{eqnarray} W^{a,(n)}_{i,s}=\sum_{a_1+a_2=a}\sum_{ \lambda =0}^{q-1}\left( \sum_{j} \lambda _j^{(i)}\chi_{j, \lambda }^{a_1} \right)W_{i,s+ \lambda }^{a_2,(n-1)}, \label{wlad} \end{eqnarray} where $ \lambda ^{(i)}_{j}=S^i_j/S^{i}_0.$ By introducing $\xi_{i, \lambda }^{a}=\sum_{j} \lambda _j^{(i)}\chi_{j, \lambda }^{a}$ i.e. \begin{eqnarray} \xi_{1, \lambda }^{a}&=&\chi_{1, \lambda }^{a}+\chi_{\psi, \lambda }^{a} +\sqrt{2}\chi_{\sigma, \lambda }^{a}, \\ \xi_{\psi, \lambda }^{a}&=&\chi_{1, \lambda }^{a}+\chi_{\psi, \lambda }^{a} -\sqrt{2}\chi_{\sigma, \lambda }^{a}, \\ \xi_{\sigma, \lambda }^{a}&=& \chi_{1, \lambda }^{a}-\chi_{\psi, \lambda }^{a}, \end{eqnarray} (\ref{wlad}) becomes \begin{eqnarray} W^{a,(n)}_{i,s}=\sum_{a_1+a_2=a}\sum_{ \lambda =0}^{q-1} \xi_{i, \lambda }^{a_1}W_{i,s+ \lambda }^{a_2,(n-1)}. \end{eqnarray} As $W_{i, \lambda }^{a,(1)}=S^{i}_0\xi_{i, \lambda }^{a}$, we get : \begin{eqnarray} W_{1,s}^{a,(n)}&=&\frac{1}{2}\sum_{p_2,\cdots,p_n} \sum_{a_{2},\cdots,a_n} \xi^{a_1}_{1,p_1}\cdots\xi^{a_n}_{1,p_n}, \\ W_{\psi,s}^{a,(n)}&=&\frac{1}{2}\sum_{p_2,\cdots,p_n} \sum_{a_{2},\cdots,a_{n}} \xi^{a_1}_{\psi,p_1}\cdots\xi^{a_n}_{\psi,p_n}, \\ W_{\sigma,s}^{a,(n)}&=&\frac{\sqrt{2}}{2}\sum_{p_2,\cdots,p_n} \sum_{a_2,\cdots,a_n} \xi^{a_1}_{\sigma, p_1}\cdots\xi^{a_n}_{\sigma, p_n}, \label{www} \end{eqnarray} with $p_1\equiv s+\sum_{l=2}^{n}p_l \hspace{3mm} ({\rm mod}\hspace{2mm}2q)$ and $ a_1\equiv a+\sum_{l=2}^{n} a_l $. In terms of these formulas, the partition functions are obtained as \begin{eqnarray} Z^{a,(n_b)}_{1,s}&=&\frac{1}{2}W_{1,s}^{a,(n_b)}+ \frac{\sqrt{2}}{2}W_{\sigma,s}^{a,(n_b)} +\frac{1}{2}W_{\psi,s}^{a,(n_b)}, \\ Z^{a,(n_b)}_{\psi,s}&=&\frac{1}{2}W_{1,s}^{a,(n_b)}- \frac{\sqrt{2}}{2}W_{\sigma,s}^{a,(n_b)} +\frac{1}{2}W_{\psi,s}^{a,(n_b)}, \\ Z^{a,(n_b)}_{\sigma,s}&=&\frac{\sqrt{2}}{2}W_{1,s}^{a,(n_b)}- \frac{\sqrt{2}}{2}W_{\psi,s}^{a,(n_b)}. \end{eqnarray} In particular, we get the partition function for the Pfaffian states $Z_{1,0}^{\rm even}+Z_{\psi,0}^{\rm odd}$ : \begin{eqnarray} Z_{\rm Pfaff}^{(n_b)}&=&\frac{1}{2}\left(W_{1,0}^{{\rm even},(n_b)}+ W_{1,0}^{{\rm odd},(n_b)}\right) \hspace{1cm}\nonumber \\ &{ }&+ \frac{\sqrt{2}}{2}\left(W_{\sigma,0}^{{\rm even},(n_b)} -W_{\sigma,0}^{{\rm odd},(n_b)}\right) \nonumber \\ &{ }&+ \frac{1}{2}\left(W_{\psi,0}^{{\rm even},(n_b)}+ W_{\psi,0}^{{\rm odd},(n_b)}\right). \end{eqnarray} \section{Discussions} \setcounter{equation}{0}\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} We obtained the multiple edge (grand-canonical) partition functions for the Laughlin state and the Pfaffian state. To deal bulk and edge states simultaneously, we introduced a method to express quantum hall states in terms of the chiral vertex operators. The constraints result in the relation between the multiple edge partition functions, which is diagonalized by the modular transformation. These methods are applicable to other states obtained from rational conformal field theories. Generally, the relation between the (grand-canonical) partition functions of edge excitations on $n_b$ and $(n_b-1)$ boundaries is given by \begin{eqnarray} Z_{i}^{(n_b)}&=&\sum_{jk} N^{k}_{ij}\chi_{j}Z^{(n_b-1)}_{k}, \end{eqnarray} where $\chi_j$ is the Virasoro character for $\phi_j$ and the sum is over the allowed primary fields, which are determined by the single-valuedness of wavefunctions in the electron coordinates. By using this equation recursively, we have \begin{eqnarray} Z^{(n_b)}_{i} &=&\sum_{j_1,\cdots,j_{n_b}}\sum_{k_1,\cdots,k_{n_b-2}} N^{k_{n_b-2}}_{i j_{n_b}}N^{k_{n_b-3}}_{k_{n_b-2} j_{n_b-1}} \cdots N^{k_1}_{k_{n_2} j_3}N^{j_1}_{k_1 j_2} \chi_{j_1} \cdots \chi_{j_{n_b}} \\ &=&\sum_{j_1,\cdots,j_{n_b}} {\rm dim}{\cal H}(i,j_1^{\vee}, j_2\cdots,j_{n_b}) \chi_{j_1} \cdots \chi_{j_{n_b}}, \end{eqnarray} where ${\rm dim}{\cal H}(i,j^{\vee}_1,\cdots,j_{n_b}) = N^{k_{n_b-2}}_{i j_{n_b}}N^{k_{n_b-3}}_{k_{n_b-2} j_{n_b-1}} \cdots N^{k_1}_{k_{n_2} j_3}N^{j_1}_{k_1 j_2}$ is the dimension of \newline ${\cal H}(i,j^{\vee}_1,\cdots,j_{n_b})$, the space of conformal blocks on a sphere with $n_b+1$ marked points with the insertions of fields $ i,j^{\vee}_1,j_2\cdots,j_{n_b}$. On the other hand, we obtain the explicit form of $Z_i$ by using the Verlinde formula (\ref{Ver}) as \begin{eqnarray} Z_{i}^{(n_b)}=\sum_{k}S_i^{k}(\sum_j S_j^{\dag k}\chi_j)\left(\sum_j \frac{S_j^{k}}{S_0^{k}} \chi_j\right)^{n_b-1}. \end{eqnarray} The actual multiple edge partition function is obtained by combining these $Z_i^{(n_b)}$. The combination depends on the relation of bulk and edge states of the given quantum hall state. It would also be interesting to calculate the multiple edge partition functions for the Haldane-Rezayi state\cite{HalRez}, which is beyond the scope of rational conformal field theory \cite{MorRe}\cite{WenWu2}\cite{Milo}\cite{GFN}\cite{LeeWen}. \subparagraph{Acknowledgement} The author would like to thank G.R.Zemba, and especially M. Flohr for useful suggestions and comments on the manuscript.
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\section{Introduction} Learning to organize, rank, and sort points from a data corpus is a fundamental challenge in data science, and a statistical primitive underlying many computer science and machine learning tasks. For example, nearest neighbor algorithms, which sort all points according to their distances to one another, are considered among the top 10 most important algorithms of all time~\cite{Wu2007-zc}, and have strong theoretical guarantees for both classification and regression~\cite{Stone1977-fi}. Decision trees (such as CART and C4.5) can also reasonably be thought of as algorithms for organizing data in a hierarchical fashion; these are among the top ten algorithms as well~\cite{Wu2007-zc}. Moreover, decision trees underlie both random forests~\cite{breiman2001random} and gradient boosted trees~\cite{Freund1997-vd}, which are the two leading algorithms for machine learning on tabular data today~\cite{Caruana2006-wp, Caruana2008-tb, Chen2016-fx}. To complement the above supervised machine learning settings, there is a rich literature on approximate nearest neighbor algorithms (see Aumüller et al.~\cite{Aumuller2017-xm} for benchmark comparisons of many state of the art approaches), which are used extensively in big data systems. Operating on the exact nearest neighbors, (or trying to approximate them), is not always desirable. For example, consider a simplest supervised learning setting. Given a data corpus, $\{(x_n,y_n)\}_{n=1}^N$, learn a decision rule such that, given a new data point $x$, its prediction of $y$ has small error with high probability. A canonical approach is kernel regression~\cite{Scholkopf2002-ar}. A kernel machine's prediction is a weighted linear combination of predictions that the neighbors of $x$ would make, specifically, $\hat{y} = \frac{1}{N} \sum_{n=1}^N y_n \times \kappa (x, x_n)$, for some suitably chosen kernel $\kappa$ (for example, a radial basis function or $k$-nearest neighbors kernel). Such approaches enjoy strong theoretical guarantees~\cite{Mohri2018-tf}. Now, further assume that the $x$'s are noisy measurements of some true, but unobserved $\tilde{x}$'s. Such an assumption, called ``measurement error modeling''~\cite{Fuller1987-oa}, could reasonably be argued to be much more accurate than assuming the $x$'s are {noise-free measurements}~\cite{Hand2016-vc}. Under this measurement error assumption, a better approach---meaning an approach that likely achieves smaller error given the same sample size---would be a kernel regression function on the noise-free measurements, $\hat{y}= \frac{1}{N} \sum_{n=1}^N y_n \times \kappa (\tilde{x}, \tilde{x}_n)$. Unfortunately, because the $\tilde{x}$'s are not observed, such an approach is not available. This modeling framework therefore motivates {learning the latent structure of the data, even for low-dimensional data}. In particular, it motivates learning which sample points are close to one another on an underlying latent structure (such as a manifold), for subsequent inference. Moreover, even though the above task is a supervised learning task, performance improves when learning the structure of only the features $x$, while not even considering the labels $y$ . Learning latent structure is even more important in the ``large p, small N'' setting. Specifically, when the dimensionality $p$ is larger than the sample size $N$, an intermediate representation is required to avoid (1) numerical stability issues with matrix operations, and (2) the ``curse of dimensionality'' for statistical operations. This intermediate representation can be explicit (as in manifold learning) or implicit (as in kernel machines). When $N$ is large, even approximation algorithms are required to compute various quantities whose exact solution requires $\mathcal{O}(N^2)$ or $\mathcal{O}(N^3)$ space and/or time. Geodesic learning is the process of estimating geodesic distances between pairs of points in a data corpus. It is crucial (though sometimes implicit) in many state-of-the-art machine learning algorithms. For example, the first step in many manifold learning algorithms is to estimate the geodesic distance between all pairs of points~\cite{Lee2007-bw}. Nonetheless, scant work explicitly addresses geodesic learning, perhaps because in any real data application the true geodesic distances are not observed, or because a suitable metric for evaluating geodesic learning is not currently available. Despite this, several disciplines in computer science and machine learning have developed strategies to partially address the challenges associated with geodesic learning. In computer science, space-partitioning trees are used extensively for quite diverse applications; most relevant to this work are efforts to build trees in support of efficient geometric queries~\cite{thibault1987set}. Commonly, space partitioning trees use binary and recursive splits with hyperplanes~\cite{BSP-trees}. These tree structures are usually optimized to learn relative proximities of the observed, noisy measurements, rather than the latent noise-free, and potentially lower dimensional, measurements. These partition trees from data structures are closely related to decision trees developed in statistics and machine learning~\cite{Breiman1984-pc}. In fact, extensions to decision trees are established as the de facto standard for classification and regression tasks (even in this age of deep learning), including random forests~\cite{breiman2001random} and gradient boosting trees~\cite{friedman2001greedy}. These approaches, however, are almost exclusively concerned with supervised, rather than unsupervised learning. Decision trees have always been linked to kernel learning~\cite{Breiman2000-pq}, a realization that recently gained traction in the machine learning literature~\cite{Davies2014-bt, Scornet2016-vf, Balog2016-iv, Shen2018-ca}. Kernel machines typically assume a kernel, or select one from a finite set or one-dimensional family of kernels, limiting their finite sample performance. In manifold learning, many spectral variants start by estimating all pairwise geodesic distances~\cite{scholkopf1997kernel}. These approaches typically operate on the observed, typically high-dimensional data, and therefore suffer serious drawbacks in the face of additional noise dimensions. Although each of the above works is closely related to geodesic learning, to our knowledge none of it explicitly aims to estimate geodesic distances as an end unto itself. Moreover, those approaches that do estimate geodesic distances typically do not directly evaluate the estimated distances. We propose \emph{Unsupervised Randomer Forest} (URerF) to achieve near linear space and time complexity, while approximating the true latent geodesic distances. Unlike the previously described methods, URerF does not need to compute geodesic distances between all pairs of points. Instead, URerF examines local structure by recursively clustering data in sparse linear subspaces, building on the recently proposed randomer forest algorithm for supervised learning~\cite{tomita2015randomer}. The randomer forest approach allows URerF to separate meaningful structure in the data from the noise dimensions. We also introduce a spitting criteria, Fast-BIC, which efficiently and exactly computes the Bayesian Information Criterion statistic for an approximate Gaussian mixture model in one dimension. This manuscript also contributes a method for evaluating geodesic learning algorithms. Most existing manuscripts on manifold learning that explicitly estimate geodesic distances, do not explicitly evaluate the geodesics. Rather, such papers typically embed the data into some low-dimensional space and then visualize the results. This approach is limited in a number of ways: (1) it is qualitative; (2) when the structure is higher dimensional it may be revealed by the first few dimensions; and (3) it relies on an embedding, which introduces additional computational and statistical complications. Sometimes the embedded data are used for subsequent inference tasks, such as classification, which can be quantitatively evaluated. Such an approach is only able to evaluate performance of a manifold learning algorithm composed with a particular subsequent inferential method, but not the manifold or geodesic learning algorithms directly. We therefore introduce geodesic precision and recall. In contrast to precision and recall as typically defined, geodesic precision and recall quantify the set of nearest neighbors as estimated by the geodesic learning algorithm, with the set of true nearest neighbors on the latent manifolds. As a general rule, if a geodesic learning algorithm does poorly on this metric, estimating manifolds from these geodesics has no hope to perform well on subsequent tasks. Indeed, functions of geodesic precision can provide tight bounds on subsequent classification accuracy \cite{Devroye1997-bd}. URerF finds neighbors in the latent low-dimensional space amid additional noise dimensions more effectively than other approaches. Moreover it can do this in a variety of linear and nonlinear settings, with different dimensional submanifolds, and in a real connectome dataset. \section{Related Work} Nonlinear manifold learning approaches, such as Isomap~\cite{tenenbaum2000global}, Laplacian eigenmaps~\cite{belkin2002laplacian} and UMAP~\cite{mcinnes2018umap}, are designed to preserve geodesic distances, and even directly estimate them. Specifically, they follow a three-step process. First, they estimate geodesic distances in the original manifold. This is done by initially constructing a $k$-nearest neighbor or $\epsilon$-neighborhood graph in which the observations (data points) correspond to nodes, and pairwise Euclidean distances between these points correspond to the weights on the edges. Second, the all-pairs shortest paths of the nodes in the graph are computed. Third, the points are embedded in a lower dimensional space that ideally preserves these distances. This approach is significantly hampered by the first step, which operates in the original high-dimensional ambient space, since Euclidean distances often fail to provide good estimates of distances on the manifold. Moreover, given $n$ datapoints, computing all pairwise distances is $\mathcal{O}(n^2)$ space and time, and all pairwise shortest paths can require $\mathcal{O}(n^3)$, both of which can be cost prohibitive for large sample sizes. One of the most widely used methods for nonlinear dimensionality reduction is Isomap \cite{tenenbaum2000global}. Isomap is one of the few manifold learning algorithms that has theoretical guarantees for correctly estimating the manifold under certain assumptions~\cite{Silva2003-nl}. In the case of many noisy dimensions, however, Isomap fails to construct an accurate nearest-neighbor graphs on the latent manifold. Moreover, Isomap requires storing all point-to-point graph distances, which incurs space and time complexity quadratic in the sample size. UMAP is a new algorithm for dimensionality reduction that efficiently reduces high-dimensional data to a low dimension using a fuzzy simplicial set representation of the input data points \cite{mcinnes2018umap}. Like other nearest-neighbor based algorithms, UMAP first constructs an undirected, weighted k-nearest neighbor graph from the input data, then embeds data points in a low-dimensional space using a force-directed layout algorithm. The number of neighbors used to construct the graph in effect determines the local manifold structure that is to be preserved in the low-dimensional layout. In the force-directed layout approach, attractive forces between close vertices are iteratively balanced with repulsive forces between vertices that are far apart in the graph until convergence. UMAP builds upon the popular t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm, which attempts to preserve original interpoint distances in a much lower dimensional space~\cite{Maaten2008-tn}. The Kullback-Leibler divergence between the distribution of neighbor distances in the higher and lower dimensional spaces is used to determine the optimal mapping of points into the lower-dimensional space. t-SNE is primarily used to visualize high-dimensional data \cite{maaten2008visualizing}, and cannot be used with non-metric distances. The UMAP algorithm produces similar embeddings to t-SNE in two or three dimensions, but scales better in terms of run-time across a wide range of embedding dimensions \cite{mcinnes2018umap}. Approximate nearest neighbors algorithms, such as FLANN~\cite{muja2014scalable}, approximate nearest-neighbors in high-dimensional data sets, typically by building binary space-partitioning trees, such as $K$-d trees. These algorithms are designed to estimate the distances in the observed high-dimensional space. When the true manifold is low-dimensional, and the data are high-dimensional, the additional noise dimensions will be problematic for any of these algorithms. On the other hand, these approaches can achieve near linear space and time complexity. This work is inspired by, and closely related to, random projection trees for manifold learning~\cite{dasgupta2008random} and vector quantization~\cite{Dasgupta2008-ja}. The main differences between our approach and theirs is (1) that they use random splits, rather than optimizing the splits; and (2) they use a single tree, whereas URerF uses a forest of many trees. Nonetheless, their theoretical analysis motivates the geodesic precision metric we establish for quantifying performance of geodesic learning. Finally, most closely related to our method are existing unsupervised random forest methods, the most popular of which is included in Adele Cutler's RandomForest R package \cite{Shi2006-ka}. It proceeds by generating a synthetic copy of the data by randomly permuting each feature independently of the others, and then attempts to classify the real versus the synthetic dataset. As will be seen below, this approach leads to missing surprisingly easy latent structures. \section{Unsupervised Randomer Forests} A random forest is an ensemble of decision trees in which each tree is created from bootstrapped samples of the training dataset; that is, each tree is built from a random subset of training data. Each tree $\{h(\textbf{x}, \theta_t)\}, t \in \{1, 2,\ldots, T\}$ has parameters $\theta_t$ that characterize the tree structure, and can be learned from the dataset. Given a set of trees, and a new point $x$, each tree casts a unit vote for its predictions given the input \textbf{x}. Typically, random forests are used in supervised machine learning tasks, specifically classification and regression. There have been a few papers reporting on unsupervised random forests for certain tasks~\cite{shi2006unsupervised}. Our unsupervised random forest algorithm is based on the original Random Forest algorithm~\cite{breiman2001random} with a few key distinctions. First, URerF uses a new splitting criteria, Fast-BIC, that efficiently and exactly computes an approximate Bayesian Information Criterion for a Gaussian Mixture model in one dimension. Second, we use the term \textit{randomer} to label our technique, as our splitting methods are based on random sparse linear combinations of features to strengthen each tree, as originally proposed by Breiman~\cite{breiman2001random}, and later studied further by Tomita et al.~\cite{tomita2015randomer, roflmao} Third, we correctly implement a previously proposed method for generating proximity matrices from random forests. In one of the most widely used implementations of Random Forest \cite{LiawRF}, the aggregated normalized proximity matrices of $F$ Random Forests with $T$ trees each is not stochastically equivalent to the aggregated normalized proximity matrices of $T$ Random Forests with $F$ trees each. Our implementation does not suffer from this bug. Furthermore, it is computationally more efficient than previous implementations. These three changes enable URerF to achieve state-of-the-art performance on both simulated and real data. \subsection{Overall algorithm} \label{overall} Given an input data set $x=\{x_1,\ldots, x_N\}$, where $x_n \in \Real^p$, URerF builds $T$ decision trees, each from a random sample of size $m<N$. In each tree, URerF recursively splits a parent node into its two child nodes until some termination specification is met. At each node, URerF generates $d$ features to search over. Each feature is evaluated based on the splitting criteria described in Section \ref{split}, and the feature with the best score is selected to split the data points into two daughter nodes. Algorithm \ref{alg:2} describes the procedure used to build unsupervised decision trees (all algorithms are relegated to the appendix). To evaluate the forest, a proximity matrix is then generated by computing the fraction of the trees in which every pair of elements reside in the same leaf node (Section \ref{prox}). \subsection{Node-Wise Feature Generation} Unlike Breiman's original random forest algorithm, URerF does not choose split points in the original feature space. Instead, we follow the random projection framework of Tomita et al. \cite{tomita2015randomer,roflmao}. For $p$-dimensional input data, we sample a $p \times d$ matrix $A$ distributed as $f_A$, where $f_A$ is the projection distribution and $d$ is the dimensionality of the projected space. We chose to use the $f_A$ that Tomita et al. empirically found to produce the best performance in the supervised setting: $A$ is generated by randomly sampling from $\lbrace -1, +1 \rbrace$ $\lambda p d$ times, then distributing these values uniformly at random in $A$ \cite{tomita2015randomer}. The $\lambda$ parameter is used to control the sparsity of $A$, and is set to $\frac{1}{20}$, again following the convention of Tomita et al. Using the randomly sampled $p \times d$ matrix $A$, the data associated with the given node, $X'$, which is a subsample of the original data, is transformed into a $d$-dimensional feature space, where each of the $d$ new features is now a sparse linear combination of the $p$ original features. In other words, each row of $\tilde{X} = A^{T} X'$ represents a projection of the data into a one-dimensional space that is a sparse linear combination of the original feature space. Each of the $d$ rows $\tilde{X}[i,:], i \in \lbrace 1, 2, ..., d \rbrace$ is then inspected for the best split point. The optimal split point and splitting dimension are chosen according to which point/dimension pair minimizes the splitting criteria described in the following section. \subsection{Splitting Criteria} \label{split} \paragraph{Fast, Exact, Univariate, Two-Means Splitting} The goal is to find the split that minimizes the sum of the intra-cluster variance on the projected dimension. Typically, k-means problems are solved via Ward's or Hardigan's algorithms~\cite{Ward1963-dm, Hartigan1979-cc}. Because k-means is NP-hard, in general, these algorithms lack strong theoretical guarantees~\cite{Arthur2007-wn}. However, in one-dimension, for two-means, there is an exact solution that is much faster and simpler. This is available because each decision tree always operates on one-dimensional marginals. First, sort the data points. Then, consider splitting between all sequential pairs of points; that is, letting $x_{(s)}$ denote the $s^{th}$ smallest sample, consider splitting between $x_{(s)}$ and $x_{(s+1)}$ for all $s< N$. The samples to the left of the split point form one cluster, and those to the right form the other cluster. Estimate the means for each of the clusters using the maximum likelihood estimate (MLE) for all points in that cluster. Fast, exact, univariate two-means splitting seeks to find the cutpoint that minimizes the one-dimensional 2-means objective. This splitting criteria was introduced in \cite{dasgupta2008random}. \begin{align} \min_{s} \sum_{n=1}^{s} (x_n - \hat{\mu}_1)^2 + \sum_{n=s+1}^{N} (x_n - \hat{\mu}_2)^2. \end{align} An immediate limitation of this approach is that it fails to consider feature-wise variance, which can lead to undesirable properties. For example, if any feature has zero variance, it will always achieve the minimum possible score. Although one can rescale each feature independently, doing so can cause problems in unsupervised learning problems when the relative scale of features is important, and the details of how to rescale introduce an undesirable algorithm parameter to tune. \paragraph{2-Gaussian Mixture Model Splitting with Mclust-BIC} In this case, for each feature we fit the data to a two-component Gaussian mixture model (GMM). An expectation-maximization (EM) is used to jointly estimate all the parameters and latent variables~\cite{Fraley2002-eg}. The latent variables, $\{z_{n,j}\}$, denote the probability that sample $n$ is in cluster $j$. Letting $N$ be the number of observations and $J$ be the number of Gaussian clusters (in this case, $J=2$), and introducing notation $x= \left( x_1, \ldots, x_N\right)$, and $z=\{z_{1,1}, z_{1,2}, \ldots, z_{N,J}\}$, the complete likelihood (including the latent variables) is \begin{equation} P(x, z; {\mu}, {\sigma}, {\pi}) = \prod_{n=1}^{N} \prod_{j=1}^{J} \{\pi_j\mathcal{N}(x_n ; \mu_j, \sigma_j^2)\}^{z_{n,j}}. \end{equation} Each feature is evaluated using the Bayesian Information Criterion (BIC). BIC is based on the log likelihood of the model given the data, with a regularization term penalizing complex models with many parameters. Concretely, letting $\hat{L}_M$ denote the maximum log likelihood function of a particular model $M$, $\hat{L}_M =p(x;\hat{\theta}_M)$, where $\hat{\theta}_M$ are the parameters that maximize the likelihood function for model $M$, and $x$ is the observed data. Letting $N$ be the sample size (number of data points) and $d_M$ be the number of parameters estimated by the model, then the BIC score can be defined as follows: \begin{equation} BIC(M) = - 2\ln(\hat{L}_M)+ \ln(N)d_M. \end{equation} The feature that maximizes the BIC score for a two-component GMM is selected for splitting at each node. The split occurs at the midpoint where the two Gaussians are equally likely. Because this approach is standard in the literature, we do not provide pseudocode. Note that the EM approximates the actual log likelihood, and is only guaranteed to find a local maximum, not the global maximum, rending it sensitive to initialization. Moreover, the EM algorithm is known to suffer from poor convergence properties in certain settings~\cite{McLachlan2008-sa}. \paragraph{2-GMM Splitting with Fast-BIC} Fast-BIC combines the speed of two-means with the model flexibility of Mclust-BIC. As in two-means, for each feature, sort all the data, and try all possible splits. For each split, assign all points below the split to one Gaussian, and all points above the split to the other Gaussian. Estimate the prior, means, and variances for both clusters using the MLE. For $j=1$, they are defined by \begin{align*} \hat{\mu}_1 = \frac{1}{s}\sum_{n \leq s} x_n, \qquad \hat{\sigma}_1 = \frac{1}{s}\sum_{n \leq s}||x_{n} - \hat{\mu_j}||^2, \qquad \hat{\pi}_1 = \frac{s}{N}, \end{align*} and similarly for $j=2$. Under the above assumption, $z_{n,j}$ is an indicator that data point $x_n$ is in cluster $j$. In other words, rather than the soft clustering of GMM, Fast-BIC performs a hard clustering, as in two-means. Thus, if $x_n$ is in cluster $j$, then $z_{n,j}=1$ and $z_{n,j'}=0$ for all $j \neq j'$. Given this approximation, the likelihood can be obtained by summing over the $z$'s \begin{equation}\label{likprob2} P(x; \mu, \sigma, \pi) = \sum_{z}\prod_{n=1}^{N} \prod_{j=1}^{J} \{\pi_j\mathcal{N}(x_n ; \mu_j, \sigma_j^2)\}^{z_{n,j}}. \end{equation} Noting that $z_{(n \in (0, s], k = 0)} = z_{(n \in [s+1, N), k = 1)} = 1$ and $z_{n,j}=0$ otherwise, Equation (\ref{likprob2}) can be simplified to \begin{align*} P(x ; \mu, \sigma, \pi) = \prod_{n=1}^{s}\pi_1 \mathcal{N}(x_n ; \hat{\mu_1}, \sigma_1^2) \prod_{n=s+1}^{N}\pi_2 \mathcal{N}(x_n ; \hat{\mu_2}, \sigma_2^2). \end{align*} Plugging in the MLE for all the parameters, the maximum log likelihood function $\hat{L} =\log P(x ; \hat{\mu}, \hat{\sigma}, \hat{\pi})$ is \begin{align}\label{loglik1} \hat{L} = \sum_{n=1}^{s} [\log \hat{\pi}_1 + \log \mathcal{N}(x_{n} ; \hat{\mu_1}, \hat{\sigma_{1}}^2)] + \sum_{n=s+1}^{N} [\log \hat{\pi}_2 + \log \mathcal{N}(x_{n} ; \hat{\mu_2}, \hat{\sigma_{2}}^2)], \end{align} Substituting into Equation~\ref{loglik1} and simplifying, we get the following expression for the log likelihood for any given $s$: \begin{equation}\label{eq3} - 2\hat{L}_s = s\log 2\hat{\pi}_1 \hat{\sigma}_1^2 + (N-s)\log 2\hat{\pi}_2 \hat{\sigma}_2^2 - s\log\hat{\mu_1} - (N-s)\log\hat{\mu_2}, \end{equation} in which we have dropped terms that are not functions of the parameters. We further test for the single variance case ($\sigma_1 = \sigma_2$) and use the BIC formula to determine the best case. Fast-BIC chooses the dimension and split-point that maximizes $\hat{L}_s$. Pseudocode for this approach is provided in Algorithm \ref{alg:fastbic}. Fast-BIC is guaranteed to obtain the global maximum likelihood estimator, whereas the Mclust-BIC is liable to find only a local maximum. Moreover, Fast-BIC is substantially faster. This Fast-BIC procedure is, to our knowledge, novel, and of independent interest. \subsection{Proximity Matrix Construction}\label{prox} One can build a similarity matrix from any decision tree by asserting that similarity between two points $x_n$ and $x_j$ is related to some ``tree distance'' between the pair of points in a given tree. Although this is the case for both supervised an unsupervised decision trees, to our knowledge such an approach has not yet been explored for unsupervised trees. When using a forest, it is natural to average the similarity matrices to obtain the forest's estimate of similarity. A simple tree distance to use is the $0-1$ loss on whether a pair of points is in the same leaf node. This approach to computing similarities has previously been studied in several supervised random forest papers, connecting random forests to kernel learning~\cite{Breiman2000-pq, Davies2014-bt, Scornet2016-vf, Balog2016-iv, Shen2018-ca}. However, the connection between these similarities and geodesic distances has not yet been established. More concretely, the proximity matrix $S$ for input data $D \in \mathbb{R}^{n \times d} $ is estimated using the unsupervised random forest by simply counting the fraction of times that a pair of points occurs in the same leaf node in the forest. Thus, $S(i,j) = S_{ij} = \frac{L_{ij}}{T_{ij}} $, where $L(i,j)$ is the number of occurrences of points $i$ and $j$ in the same leaf node, and $T_{ij}$ is the number of trees in which both point $i$ and point $j$ were included in the bootstrap sample that was used to build the tree. We use both the in-bag and out-of-bag samples to estimate the proximity. \subsection{Geodesic Precision and Recall} Geodesic precision and recall differ from ``classical'' precision and recall by virtue of defining the neighbors based on the true latent low-dimensional manifold, rather than the observed (typically higher-dimensional) space. The typical definition of precision and recall are defined relative to a query, the \emph{relevant samples} are those that are ``correct'', where as the \emph{retrieved samples} are those that are returned by the query. Letting $\cap$ denote set intersection, and $| \cdot |$ denote the cardinality of the set, precision and recall are \begin{align*} \text{precision} &= \frac{| \{ \text{relevant samples}\} \cap \{\text{retrieved samples} \} | }{ | \{ \text{retrieved samples} \} | }, \\ \text{recall} &= \frac{| \{ \text{relevant samples}\} \cap \{\text{retrieved samples} \} | }{ | \{ \text{relevant samples} \} | }. \end{align*} For geodesic learning, given a data point $x$, a data corpus $\mathcal{D}_N= \{x_1,\ldots, x_N\}$, and a query size $k$, the relevant samples are the $k$ samples from $\mathcal{D}_N$ that are nearest to $x$ based on the true (but unknown) geodesic distance. In other words, ``correct'' neighbors is defined by the latent, noise-free, manifold, rather than the observed, typically higher dimensional space. Given a geodesic learner, the \emph{retrieved samples} are the $k$ samples that the learner reports are nearest. To compute the geodesic precision and recall for a given learner on a given dataset, average the geodesic precision and recall over each sample point. Higher precision and lower recall indicate better estimation of geodesic distances. We consider two distinct cases: a continuous geodesic, in which there is a finite geodesic distance between all pairs of points, and a discrete geodesic, in which there are clusters of points that are not connected at all to other points. In the latter case (such as a union of spheres), we denote all the points within a given connected component as its neighbors, and all points outside its connected component as not neighbors. In the disconnected setting, geodesic precision and recall are identical to one another. \section{Numerical Results} \label{syntheticdataset} \subsection{Four Multivariate Manifold Simulation Settings} \begin{figure \centering \includegraphics[width=\textwidth]{DEML/plots/synthetic_datasets.png} \caption{Synthetic datasets for all experiments. In each case, there are 1000 points in 3 signal dimensions, as shown. }\label{fig1} \end{figure} We explore geodesic learning using the following four simulations settings, as shown in Figure~\ref{fig1}, each of which span a complementary and interesting case. In the linear case, where learning the geodesic should be relatively easy, Euclidean distance will completely recover the geodesics with no noise. It therefore sets an upper bound on performance. The helix setting is reminiscent of the typical ``swiss jelly roll'' setting popular in manifold learning, but the latent submanifold is one-dimensional embedded into a three-dimensional space. Here, Euclidean will perform poorly, but various manifold learning algorithms should perform well, as they are designed for this kind of scenario. The sphere case is interesting because unlike the helix, the true manifold is two-dimensional and could easily be extended to higher dimensions. Finally, the Gaussian mixture model we suspect will be particularly challenging for the manifold learning algorithms, which typically lack theoretical guarantees for disconnected connected component graphs. Appendix~\ref{app:sims} provides the mathematical details for the four different settings. \subsection{Choosing the Splitting Criteria and Robustness to Algorithm Parameters} \begin{figure \centering \includegraphics[width=\textwidth]{DEML/plots/varying_split_crit_rer.png} \caption{Geodesic recall curves for the three different splitting criteria using both axis-aligned splits (URF; solid lines) and sparse oblique spits (URerF; dashed lines). In each case there are 1000 points and 3 signal dimensions (no noise dimensions). In general, URerF with Fast-BIC performs the best or nearly so. }\label{fig3} \end{figure} \begin{figure} \centering \includegraphics[width=\textwidth]{DEML/plots/minParent.png} \includegraphics[width=\textwidth]{DEML/plots/mtry_modified.png} \caption{\emph{Top} Geodesic precision versus k for different values of minparent (the smallest splittable node size). Mtry is set to be equal to the square root of the number of features. \emph{Bottom} Geodesic precision versus k for different values of mtry (the number of features to test at each node). Minparent was set to be equal to 100. Geodesic precision is robust to large variations in these parameters}\label{fig4} \end{figure} For each of the above simulation settings, we sample a thousands points and calculate the geodesic precision using different unsupervised random forest variants. Figure \ref{fig3} shows an empirical comparison of the three different splitting criteria described with URF and with URerF. In all cases, both BIC approaches (red and blue) outperform two-means splitting criteria (green). The solid and dashed lines show the relative performance of URF as compared to URerF (URerF use sparse oblique splits, that is, splits on linear combinations of the original features), respectively. In most cases, URerF outperforms URF, as expected based on previous comparisons of sparse oblique splits to axis-aligned splits in supervised random forest~\cite{breiman2001random,tomita2015randomer,roflmao}. The ramification of these two results is that in all cases, URerF using Fast-BIC performs as well, or nearly as well, as the other options. Because it performs as well as other options, and runs as fast as two-means, we elect to use URerF+Fast-BIC (hereafter, simply URerF) as our unsupervised decision forest splitting criteria. In addition to the splitting criteria, each decision tree has two other important algorithm parameters. First, minparent, which sets the cardinality of the smallest node that might be split. Second, mtry, which is the number of features to test at each node. Figure \ref{fig4} shows the geodesic precision for different values of minparent and mtry. Geodesic precision using URerF is robust to hyperparameter changes, obviating the need for tuning hyperparameters via a grid search, which can be computationally intensive. For all future experiments, we set minparent to 100 and mtry to $\sqrt{d}$. \subsection{URerF is Robust to Noise Dimensions} \begin{figure \centering \includegraphics[width=\textwidth]{DEML/plots/plot2.png} \includegraphics[width=\textwidth]{DEML/plots/varying_dim_adele_with_euc_normalized.png} \caption{\emph{Top} Geodesic precision at k=50 with varying noise dimension from 2 to 10,000, with $N = 1000$ samples. While all previous state of the art algorithms degrade to chance levels in all settings as the number of noise dimensions increases, URerF never degrades to chance performance for any of the settings. \emph{Bottom} Same as top, but each dimension is linearly rescaled to be between 0 and 1, and x-axis shows a smaller number of dimensions (from 0 to 10). Although rescaling greatly improves geodesic precision and recall for most algorithms, URerF still achieves much larger geodesic recall for most settings considered. } \label{fig2} \end{figure} To see that URerF is robust to high dimensional noise, Gaussian noise with varying dimensions $d'$ are concatenated onto the simulated datasets. Specifically, for each data point $x_n \in \mathbb{R}^d$, generated noise $y_n \overset{iid}{\sim} \mathcal{N}(0, c \mathbb{I})$ where $y_n \in \mathbb{R}^{d'}$ is concatenated onto $x_n$, ($c=70$ in the following experiments), and $\mathbb{I}$ is the $d' \times d'$ identity matrix. The new data points with noise are thus: $\tilde{x}_n = [x_n^{\top} | y_n^{\top}]^{\top} \in \mathbb{R}^{d+d'}$. Each algorithm's proximity matrices are computed on the $\tilde{x}$'s, and compared with geodesic distance matrices to obtain geodesic precision and recall. Figure \ref{fig2} shows the geodesic recall @ k=50 as a function of the number noise dimensions for Isomap, UMAP, random forests, Euclidean distance, FLANN, and URerF. URerF performs well even with the addition of high dimensional noise dimensions. The other state-of-the-art algorithms achieve a higher geodesic recall than URerF in the absence of noise dimensions, but degrade much more quickly than URerF upon the addition of noise dimensions. FLANN and Euclidean distance degrade the fastest, followed by Isomap and UMAP. This suggests that typical approximate nearest neighbor algorithms (which are approximating the distance in the ambient space) will perform poorly on recalling the desired items when the data live near a low-dimensional manifold. The top panel shows the geodesic recall curves for adding up to 10,000 dimensions. The performance of all the algorithms except URerF degrades to chance levels in all four settings, whereas URerF maintains a geodesic recall far above chance levels. The bottom panel shows geodesic recall after normalizing each of the dimensions (i.e., linearly rescaling each feature to be between 0 and 1). Other algorithms are very sensitive to dimension rescaling, and performance may improve as a result. However, URerF consistently performs better, even without rescaling, as long as a few noise dimensions are added. \subsection{URerF Estimates Geodesic Recall on Drosophila Connectome} The study of brain networks, or connectomics, is quickly emerging as an important source of real world data challenges~\cite{vogelstein2019connectal}. Recently, the entire larval Drosophila mushroom body connectome--the learning and memory system of the fly---was estimated and released~\cite{eichler2017complete}. It was obtained via manual labeling and semi-automatic machine vision segmentation of serial section transmission electron microscopy. The 200 nodes of this connectome correspond to 200 distinct neurons in the mushroom body. There are roughly 75000 edges, defined as present between a pair of neurons whenever there exists as least one synapse between them. The edges connect vertices in four known classes of cells: kenyon cells, input neurons, output neurons, and projection neurons~\cite{priebe2017semiparametric}. A semiparametric analysis of the connectome, using adjacency spectral embedding~\cite{sussman2012consistent}, results in a six-dimensional latent representation of each node~\cite{priebe2017semiparametric}. Because this connectome is directed, the first three dimensions correspond to ``outgoing'' latent features, whereas the next three correspond to ``incoming'' latent features. Figure \ref{fig5} (left) shows two of the six dimensions. Figure \ref{fig5} on the right shows the geodesic precision versus geodesic recall for various algorithms using cell type as the true label. URerF achieves a higher recall at essentially all precision levels. \begin{figure} \centering \includegraphics[width=0.47\textwidth]{DEML/plots/dros_diag.png} \includegraphics[width=0.47\textwidth]{DEML/plots/dros_fig.png} \caption{\emph{Left} The right Drosophila connectome after adjacency spectral embedding into six-dimensional space, just showing two of the dimensions. \emph{Right} Geodesic precision versus geodesic recall for various algorithms using cell type as the true label. URerF achieves a higher recall for essentially all precisions. The values of k for this experiment range from 50 to 250 with increments of 50} \label{fig5 \end{figure} \section{Discussion} We proposed a geodesic distance learning method using Unsupervised Randomer Forests (URerF), as well as a splitting rule called Fast-BIC. URerF is empircally robust to noise dimensions, as demonstrated by several different simulation settings, many different added noise dimensions, and the real-world \emph{Drosophila} connectome. While here we address geodesic learning explicitly, geodesic learning is an essential statistical primitive for many subsequent inference tasks. For example, manifold learning, high-dimensional clustering, anomaly detection, and vertex nomination~\cite{yoder2018vertex} all rely on geodesic learning. More generally, any \emph{ranking} problem is essentially a geodesic learning problem. Moreover, while we only considered unsupervised geodesic learning, the ideas presented here immediately lend themselves to supervised geodesic learning as well. We did not explore any theoretical claims associated with the algorithms presented here. Indeed, we did not even evaluate whether any of these algorithms approximate the precise geodesic. Rather, our metric is concerned purely with getting the geodesic neighbors correct. However, prior work using random projections to learn low-dimensional manifolds~\cite{dasgupta2008random}, and for vector quantization~\cite{Dasgupta2008-ja}, have certain theoretical guarantees associated with the intrinsic dimension of an assumed latent manifold. It seems that those guarantees could relatively easily be transferred to this setting. Another potential direction would be to address the theoretical bounds that geodesic learning can provide with respect to Bayes optimal performance, on both unsupervised and supervised learning problems. For example, 1-nearest neighbor provides tight bounds on Bayes optimal classification~\cite{Devroye1997-bd, biau2008consistency}. Ideas presented in previous work on bounding Bayes performance using ranking algorithms could also extend to this setting. URerF is available as part of the open source package ``RerF'' which includes a Python as well as an R package, available at \url{https://neurodata.io/rerf}. \section*{Acknowledgements} The authors are grateful for the support by the D3M program of the Defense Advanced Research Projects Agency (DARPA), and DARPA's Lifelong Learning Machines program through contract FA8650-18-2-7834. \bibliographystyle{ieeetr}
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Home Burrillville New owner obtains license for Burrillville's 'haunted' farmhouse, says she hopes to... New owner obtains license for Burrillville's 'haunted' farmhouse, says she hopes to dispel rumors Jacqueline Nunez BURRILLVILLE – It might be unusual for most government boards to discuss issues surrounding paranormal activity, but over the past decade, such conversations have become pretty common for members of the Burrillville Town Council. This month, the roughly 300-year-old Burrillville farmhouse that has captured the imaginations of horror fans – and others with ghostly curiosities from across the globe – officially got a new owner. Jacqueline Nuñez has finalized her $1.525 million purchase of the Round Top Road property that inspired the 2013 film The Conjuring, appearing before the council this month to seek an annual entertainment license doing business as Bale Fire, LLC. She says she intends to dispel rumors about the house and set the historical record straight. Nuñez, the owner of a development firm known as WonderGroup, LLC., has said her interest in the property is personal, and not part of a real estate development. "I'm a deeply spiritual person," she told councilors at a meeting Wednesday, June 8. "I believe that we are conscious beings having a human experience, and that our consciousness survives death, so I really think this property is a wonderful opportunity to kind of learn from and explore it." View a map of the Family Fairgrounds The new owner, who bought the property from Cory and Jennifer Heinzen, self-described paranormal investigators from Maine, said she intends to continue both the night investigations and day tours of the farmhouse launched by her predecessors. "That is the business that will be continued here through 2022," Nuñez said. But the business hasn't exactly been popular with neighbors in the area, who have voiced opposition to commercial operations during various stages of licensing. This month, a letter of opposition was submitted to the council by neighbor Tania Hall. "I'm sure I am not alone in stating that Burrillville was chosen as a place to live because of the rural atmosphere, woodlands, and the (used to be) quiet seclusion that living on the edge of town offered," wrote Hall. "The neighbors to this house are now constantly forced to deal with the negative impacts of this house's activities." "There is a constant flow of people in and out of the house for weekday and weekend investigations," Hall added. "Screams, laughs, and conversations throughout the night are not uncommon. It's disruptive and disrespectful to those of us who do not necessarily work the standard 9-5." Alan McNally recalled for councilors one incident in which guests mistook his home for the now notorious, "Conjuring House," and walked inside. "We live in a quiet residential neighborhood. We want to know how this is going to affect that," McNally said. "We have a lot of concerns about the uninvited. Some of them literally stop in the middle of the street and get out of the car. They do all kinds of crazy things." Several of the neighbors also expressed concern about safety, pointing to razor wire surrounding the property installed by the previous owners. Nuñez said she plans to remove it and has already begun taking it down. Councilor Dennis Anderson asked Nuñez if she's reached out to her new neighbors. "Historically, the biggest issue has been the peace with the neighbors," Anderson said. Nuñez said she had spoken to some of those living nearby, and is aware that trespassers on the property have been an ongoing problem. "I'm not sure what to do about that," she said. "I'm not sure what to do about people pulling up trying to take selfies." She said all operations to the business, including parking, will be contained on site, and disputed the characterization of the overnight stays as similar to a, "bed and breakfast." "That is not the model at all," she said. "They're there overnight, but they are doing investigations. They're not crashing in the bedrooms." Guests, she said, are not sleeping in the home, with groups of 10-12 people typically leaving at either 3 a.m or 6 a.m. She said that while she expects to make some changes to the model, she is not yet sure of her future plans for the business. "I'll be spending 2022 kind of assessing things to see what other offerings we might be able to do," Nuñez said, noting that any changes will follow the town's ordinances and procedures. "I'm not just going to unilaterally do anything." The owner said said that while she lives in Boston, Mass., she plans to spend a, "significant," amount of time at the Burrillville business, and that it will be staffed 24/7. "One of the conditions of the sellers was that whoever bought the place would not live there full-time, year-round, because of the activity and the energy," Nuñez said. She noted that she plans to work on "dispelling untruths," about the house's history – particularly the mischaracterizations of Bathsheba Sherman, a woman who lived nearby in the early 1800s – and was depicted as an evil entity in the Warner Bros. hit horror movie. The topic has been an ongoing concern of historians, who note that Sherman's grave has been repeatedly vandalized despite her having no actual association with the alleged hauntings of the property. "I feel terrible about that," said Nuñez. "She had nothing to do with that property. That'll be important to me as I learn the history: to get the history and the facts correct." Nuñez said she's mostly interested in the more recent activity at the house. "It is a very, very active house," Nunez said. "Paranormal investigators have captured an enormous amount of evidence." "I have people coming in from all over the world," she said. "They love it." Town Manager Michael Wood said that on a recent vacation to Maryland, people learned he was from Burrillville and asked about the house. "We stayed up half the night watching The Conjuring house on demand," Wood said. "All of the people that were down there were curious about it." Wood noted that neighbors should call and report the incident if laws are broken. "There's not much more that we can do if someone is running a legitimate business," agreed Town Council President Donald Fox. "She's more than willing to keep the peace here and maintain a good relationship." Councilor Stephen Rawson said he sympathizes with the neighbors, suggesting that "no parking," signs could be added to the area surrounding the property. Councilor Raymond Trinque said the issues will be ongoing regardless of who owns the lot. "I'm familiar with the activity," Trinque said. "It's not going to stop. This is an international movie, and there's nothing that's going to stop it, I don't think. The question is" how do we deal with it." Trinque suggested creating a pull off area, noting that the house brings the town notariety. "I think we've got a chance here with the new owner to do something," he said. Councilor Jeremy Bailey noted that the house is the only business in Burrillville with a license to operate 24-hours-a-day. "This just seems out of character for an area that's generally zoned farming," Bailey said. "Me personally, I don't think that's fair to the neighbors. I think that's destroying the right to quiet enjoyment." Anderson noted that the license will be due for renewal in November, creating something of a trial period for the new owner. Councilors approved the license transfer by a vote of 6-1, with Bailey casting the only dissent.
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{"url":"https:\/\/visimphysics.com\/en\/wave-motion\/other-wave-motion-tasks\/","text":"## Exercise 2.5 A\n\nDetermine the length of the spring when there isn\u2019t any mass suspended from it.\n\n## Exercise 2.5 B\n\nDefine the mass of an unknown object.\n\n## Exercise 2.5 C\n\nDetermine the value of acceleration due to gravity g with different lengths of a pendulum.\n\nTip 1: The oscillation time of a simple gravity pendulum is $T=2\\pi\\sqrt{\\frac{l}{g}}$\nTip 2: Move the information to $(T^2,l)$ \u2013coordinate system.\nAnswer: $9{,}9\\ \\frac{m}{s^2}$\n\n## Exercise 2.5 D\n\nDetermine the position of the spring\u2019s head as a function of time.\n\nTip: The waveform equation is a form $y=Asin(2\\pi ft)$\nAnswer: $y=0,315\\ m+0,05\\ m\\cdot\\sin{\\left(\\frac{2\\pi}{0,712\\ s}\\cdot\\left(t-60{,}462\\ s\\right)\\right)}$, where $t\\geq60{,}462\\ s$","date":"2023-03-20 22:18:36","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\": 0.6276989579200745, \"perplexity\": 1606.8684467805842}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943562.70\/warc\/CC-MAIN-20230320211022-20230321001022-00099.warc.gz\"}"}
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Q: Could not evaluate: undefined class/module Puppet::Util::TagSet I'm having an issue with the latest puppet version and a module called vcsdeploy. Unfortunately I'm not familiar with Ruby and it's own idiosyncrasies, so I'm hoping someone with a little more experience can point me in the right direction. The module in question can be found here in all it's glory. The particular issue I'm experiencing is an error at line 194 in lib/puppet/provider/vcsdeploy/svn.rb: "Could not evaluate: undefined class/module Puppet::Util::TagSet" For those who don't want to spelunk the source code, here's the code that's causing the error: valid_options = [ 'path', 'owner', 'group', 'dirmode', 'filemode', 'source', 'user', 'pass', 'name', 'version', 'selrange', 'selrole', 'seltype', 'seluser', 'templates' ] @resource_copy = {} debug "creating resource_copy for #{resource[:name]}" valid_options.each {|option| if (option && resource[option.to_sym]) @resource_copy[option.to_sym] = resource[option.to_sym] end } I would assume that Puppet::Util::TagSet is used to some degree elsewhere throughout puppet and it's various modules however this is the only one that's causing a problem. Anyone got any pointers that I could use to start this investigation? More System Information: Operating System: CentOS 6.5 Installation Method: RPM packages Foreman Version: 1.5 Puppet Version: 3.5.1 I have also verified that the file tag_set.rb exists at the location: /usr/lib/ruby/site_ruby/1.8/puppet/util/tag_set.rb A: What the module fails to document is that it requires Puppet 3.3 which introduced this piece of code (see the commit).
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Baker, Martin S. 1217 N. New Jersey Indianapolis, IN. Cowan, Percy Orrington Hotel Rm. 200,209 S. La Salle St. Chicago, IL. Curry, William 533 Ave. "K" Boulder City, NV. Fox, Ray V.A. Domiciliary Center (Co.#2) Clinton, IA. Gartner, Al. 1213 Balboa Ave. Burlingame, CA. Gill, Frank 1412 E. Market St. Indianapolis, IN. Golatka, J. 1066 Lafayette La Salle, IL. Hansen, Edward Box 67 Elberon, IA. Harrison, Nels J. 2230 W. Grove Blue Island, IL. Hutchine, F. O. R.F.D. 3 Lake Mills, WI. Keating, Walter " " " " " " " " Kotoed, Cris 13341 S. Lakkewood Paramount, CA. Lawrence, Larry c/o Milwaukee Journal Milwaukee, WI. Mattarelli, Vincent 212 Charleston Peoria, IL. Mortenson, H. B. (Mort) 2221 "O" St. Sacramento, CA. McMurray, Jess Box 155, Route 6 2027 Will-o-Wisp Way Alto Woods Park Jackson, MI. Sneed, James Fort Pierce, FL. Wilkinson, R. B. Graybar Electric Co. 420 Lexington Ave. NY, NY. World, Percy 1585 Ridge Ave. Evanston, IL. Sparmacher, Sol Editor of chips for M. Co. Hausman, Jim " " " Carry this list with you on your trips and vacations and contact the boys on your route. Always welcome at an M. Company Buddies home. When in their town call them or make a contact. It's a real pleassure as you know. Write often, give any changes of address, phone No. etc. and if an unlisted phone on our roster, write so we can add it. In case you hear of any news of a Buddy, especially one who has gone West, let us know. Copyright � April 19, 2003- Laura J. Stewart - All rights reserved. In memory of my father, Homer Harland Stewart. 131st Infantry - Company "M" The following is a list that was in the possession of my father, Homer Harland Stewart. (I am including SSDI information when I can find it. *LSC ) I thought this might help someone trying to prove their loved one was in WWI & since most of the Army's papers were destroyed in the fire at St. Louis, MO. This page created on April 19, 2003. Last updated on August 11, 2005. Lineage of the 131st Infantry by MSG Jose R. Ramirez Jr. Mustered into Federal service at Springfield, 13 May 1898 as 1st Illinois Volunteer Infantry; served in Cuba and mustered out 17 November 1898 at Chicago. Mustered into Federal service 26 June 1916 for Mexican Border and stationed at San Antonio, Texas; mustered out 4 October 1916. Mustered into Federal service 4 April 1917; drafted into Federal service 5 August 1917. Redesignated as the 131st Infantry and assigned to the 33d Division 12 October 1917. Demobilized 6 June 1919 at Camp Grant, Illinois. Reorganized during June 1919 as the 1st Infantry, Illinois National Guard. Redesignated as the 131st Infantry and assigned to the 33d Division 13 December 1921. Federally recognized 18 August 1922 at Chicago. Inducted into Federal service 5 March 1941 at Chicago. Relieved from the 33d Division 21 February 1942. Inactivated 26 February 1944 at Fort Benning, Georgia. Assigned to the 33d Infantry Division 5 July 1946. Reorganized and Federally recognized 20 December 1946 at Chicago. Company M, 131st Infantry Regiment, 33rd Division. Photo taken at Camp Miller, NY. in 1917, prior to leaving for combat in France. From a book of France that my father had written in, states that they arrived at Brest, France in 1918.Photo courtesy of Ken Loh of Lompoc, CA. His father was Russell Loh that is listed on the "Buddy" list. Many thanks to him for preserving this piece of history!
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Q: Display an icon based on a prop passed as a string - React I'm trying to display different icons based on a prop (coming from a string stored in a database per user). I do not want to pass the icon itself as a prop - as the information about what Icon to show is stored in a database. My parent component fetches data and gets a something I've called "displayIcon" as a string. This is passed on as a prop to the "AppCard" and depending on the value of the displayIcon, I'd like to render different icons. One way to do it is using a ternary operator - but it does not feel very scalable, elegant or efficient. Any suggestions? How I Pass the data: return ( <BaseLayout title={"Processes"}> {userRole == "stakeholder" && ( <AppSection sectionName={"Add Processes"}> <div className="appLayout"> <AppCard title={"Add process"} navigation={"/processes/addnew"} displayIcon={"FaPlus"} /> </div> </AppSection> )} ); AppCard Component // import { FaQuestion } from "react-icons/fa"; import Link from "next/link"; import { FaPlus, FaSitemap, FaQuestion, FaRegListAlt, FaUser, } from "react-icons/fa"; import { HiOutlineUserGroup, HiCog, HiDocumentText, HiOutlineFire, HiAcademicCap, HiLightBulb, HiScale, HiOutlineChartPie, HiOutlineFlag, HiOutlineGlobe, } from "react-icons/hi"; function AppCard({title, navigation, displayIcon }) { return ( <Link href={{ // pathname: `/admin/edit/${navigation}`, pathname: navigation, }} > <div className=" cursor-pointer flex flex-col items-center "> <div className="appView "> {displayIcon == "FaPlus" && <FaPlus size="44" />} {displayIcon == "FaUser" && <FaUser size="44" />} {displayIcon == "FaSitemap" && <FaSitemap size="44" />} {displayIcon == "FaQuestion" && <FaQuestion size="44" />} {displayIcon == "FaRegListAlt" && <FaRegListAlt size="44" />} {displayIcon == "HiOutlineUserGroup" && ( <HiOutlineUserGroup size="44" /> )} {displayIcon == "HiCog" && <HiCog size="44" />} {displayIcon == "HiDocumentText" && <HiDocumentText size="44" />} {displayIcon == "HiOutlineFire" && <HiOutlineFire size="44" />} {displayIcon == "HiAcademicCap" && <HiAcademicCap size="44" />} {displayIcon == "HiLightBulb" && <HiLightBulb size="44" />} {displayIcon == "HiScale" && <HiScale size="44" />} {displayIcon == "HiOutlineChartPie" && ( <HiOutlineChartPie size="44" /> )} {displayIcon == "HiOutlineFlag" && <HiOutlineFlag size="44" />} {displayIcon == "HiOutlineGlobe" && <HiOutlineGlobe size="44" />} </div> <p className="text-siteTheme-accentcolorDark font-psans mt-1"> {title} </p> </div> </Link> ); } export default AppCard; A: You could store all in an object, and render as needed. const STRING_TO_ICON = { FaPlus: <FaPlus size="44" />, FaUser: <FaUser size="44" />}, FaSitemap: <FaSitemap size="44" />, FaRegListAlt: (size = 44) => <FaRegListAlt size={size} /> // dynamic size } function AppCard({ title, navigation, displayIcon }){ ... return( ... <div className="appView "> {STRING_TO_ICON[displayIcon]} {STRING_TO_ICON[displayIcon(32)]} </div> ... ); ... } A: I do what you are describing for the social profiles component on my personal website. The way I've implemented it is: * *Import all Font Awesome icons I'll support into the component. I'm sure this can be optimized in the future but I haven't explored that yet. import { faStackOverflow, faTwitter } from '@fortawesome/free-brands-svg-icons' *Create a registry of supported icons from those. const icons = { faStackOverflow, faTwitter } *Fetch my social profiles data. In it I define the icon, by key, I want to use for each profile. Those keys currently corresponds to the import name (e.g., 'faStackOverflow'). const profiles = useSocialProfiles() /* [{ "icon": { "name": "stack-overflow", "reactIcon": "faStackOverflow" }, "displayName": "StackOverflow" }, { "icon": { "name": "twitter", "reactIcon": "faTwitter" }, "displayName": "Twitter" }] */ *Loop through all profiles and use the supported icon registry to map each icon key to a new IconComponent field. It contains the icon SVG. const profilesToIcons = profile => { const { icon: { reactIcon } = {} } = profile return { IconComponent: icons[reactIcon], profile } const profilesWithIcons = profiles.map(profilesToIcons) *Render profiles, passing each IconComponent to <FontAwesomeIcon /> from react-fontawesome. {profilesWithIcons.map(({ IconComponent }) => ( <FontAwesomeIcon icon={IconComponent} sx={{ fontSize: [4, 5, 6] }} /> ))} My data source for the profiles was previously a database, but now I use a configuration file. The same implementation has worked OK for both. If you notice the number of icons – used or unused – increasing your client's bundle size past what is acceptable then you'll probably want to refactor or iterate on this to address that. There are probably a few ways to handle that issue, but I haven't personally arrived there yet. For now this has worked fine for my needs and the few icons I'm importing.
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\section{Introduction}\label{sec1} \label{Introduction} Flag varieties play a fundamental role in geometry, and so do their analogues in ind-geometry. In this paper, we would like to place these analogues under the looking glass and provide a new characterization of the ind-varieties of generalized flags constructed in \cite{DP}. Around 20 years ago, I. Dimitrov and the first author realized that in the context of ind-geometry the notion of a flag of vector subspaces in an ambient infinite-dimensional vector space is rather subtle. More precisely, in addition to the obvious three types of infinite flags, that is, chains of vector subspaces enumerated by $\mathbb{Z}_{>0}$, $\mathbb{Z}_{<0}$ or $\mathbb{Z}$, there is the need to consider chains of subspaces enumerated by more general totally ordered sets in which every element has an immediate predecessor or an immediate successor, but possibly not both. Such chains, satisfying the additional condition that every vector of the ambient vector space is contained in some space of the chain but not in its immediate predecessor, were christened \textit{generalized flags} in \cite{DP}. The main result of \cite{DP} can be summarized roughly as follows: generalized flags in a countable-dimensional vector space are in a natural 1-1 correspondence with splitting parabolic subgroups $P$ of the ind-group $GL(\infty)$, and hence the points of homogeneous ind-spaces of the form $GL(\infty)/P$ can be thought of as generalized flags. A similar statement about isotropic generalized flags holds for the ind-groups $O(\infty)$ and $Sp(\infty)$. In particular, the concept of generalized flag, and therefore also the notion of an ind-variety of generalized flags, has been motivated in the past by the notion of a parabolic subgroup of an ind-group like $GL(\infty)$, $O(\infty)$, $Sp(\infty)$. The main purpose of the present paper is to propose another, purely algebraic-geometric, approach to the ind-varieties of generalized flags. More precisely, we define \textit{admissible linear embeddings} of usual flag varieties \begin{equation}\label{emb 1} Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,...,n_{\tilde{k}},V') \end{equation} and show that an ind-variety obtained as a direct limit of such linear embeddings is isomorphic to an ind-variety of generalized flags. In particular, such a linear direct limit is automatically a homogeneous ind-space of $GL(\infty)$. We also consider isotropic generalized flags and prove a similar result for the ind-groups $O(\infty)$ and $Sp(\infty)$. In this way, the notion of an admissible linear embedding of flag varieties leads naturally to the concept of generalized flag. A small part of this program has already been carried in our paper \cite{PT} where we characterize linear embeddings of grassmannians, and then as a consequence describe linear ind-grassmannians up to isomorphism. Our main new result concerning embeddings of finite-dimensional flag varieties is finding an explicit form of a class of embeddings \eqref{emb 1} which we call \textit{admissible}. We define an admissible linear embedding in general algebraic-geometric terms, and then show that such an embedding is nothing but an extension of a flag from $Fl(m_1,...,m_k,V)$ to a possibly longer flag in $Fl(n_1,...,n_{\tilde{k}},V')$, given by an explicit formula from linear algebra. We call the latter embeddings \textit{standard extensions}. This enables us to prove that a direct limit of admissible linear embeddings is isomorphic to an ind-variety of generalized flags as in \cite{DP}, as it is relatively straightforward to show that direct limits of standard extensions have this property. The paper is concluded by an appendix in which we present two examples of direct limits of linear but non-admissible embeddings of flag varieties, that are not isomorphic to ind-varieties of generalized flags. {\bf Acknowledgements.} I.P. thanks Vera Serganova for a useful discussion, which took place several years ago, on the general idea of an algebraic-geometric approach to ind-varieties of generalized flags. I.P. was supported in part by DFG-grant PE 980/7-1. A.S.T. thanks the Max Planck Institute for Mathematics in Bonn, where this work was partially done during the winter of 2017, for hospitality and financial support. {\bf Notation.} The sign $\subset$ stands for not necessarily strict set-theoretic inclusion. By $G(m,V)$ we denote the grassmannian of $m$-dimensional subspaces of $V$ for $1\le m\le\dim V$. We also use the notation $\mathbb{P}(V)$ for $G(1,V)$. If $a:X\to Y$ is a morphism of algebraic varieties, by $a^*$ and $a_*$ we denote respectively the pullback or pushforward of vector bundles. The superscript $(\cdot)^{\vee}$ indicates dual space or dual vector bundle. \vspace{1cm} \section{Definition of linear embedding of flag varieties} \label{preliminaries} \vspace{5mm} In this section we give the basic definitions of linear embeddings of flag varieties including the case of isotropic flag varieties. The base field is $\mathbb{C}$ and all vector spaces, varieties and ind-varieties considered below are defined over $\mathbb{C}$. Let $V$ be a vector space of dimension $\dim V\ge2$. For any increasing sequence of positive integers $1\le m_1<...< m_k<\dim V$, we consider the \textit{flag variety} $Fl(m_1,...,m_k,V):=\{(V_{m_1},...,V_{m_k})\in G(m_1,V)\times...\times G(m_k,V)\ |\ V_{m_1}\subset...\subset V_{m_k}\}$. We denote its points by $F=(0\subset V_{m_1} \subset...\subset V_{m_k}\subset V)$ or sometimes by $F=(V_{m_1}\subset...\subset V_{m_k})$. The ordered $k$-tuple $(m_1,...,m_k)$ is the \textit{type} of a flag $F\in Fl(m_1,...,m_k,V)$. There is a natural embedding $$ j:\ Fl(m_1,...,m_k,V)\hookrightarrow G(m_1,V)\times...\times G(m_k,V) $$ and there are projections $$ \pi_i:\ Fl(m_1,...,m_k,V)\to G(m_i,V),\ F=(V_{m_1}\subset ...\subset V_{m_k})\mapsto V_{m_i},\ i=1,...,k. $$ We have $$ {\rm{Pic}}~Fl(m_1,...,m_k,V)=\mathbb{Z}[L_1]\oplus...\oplus \mathbb{Z}[L_k], $$ where \begin{equation*}\label{generators} L_i:=\pi_i^*\mathcal{O}_{G(m_i,V)}(1),\ \ \ i=1,...,k. \end{equation*} Here, $\mathcal{O}_{G(m_i,V)}(1)$ denotes the invertible sheaf on $G(m_i,V)$ satisfying $H^0(\mathcal{O}_{G(m_i,V)}(1))= \wedge^{m_i}(V^*)$. By definition, $[L_1],...,[L_k]$ is a \textit{preferred set of generators} of ${\rm{Pic}}~Fl(m_1,..., m_k,V)$. \vspace{4mm} Let $V$ be equipped with a non-degenerate symmetric bilinear form on $V$. For our purposes, we can assume that $\dim V\ge7$. For $1\le k\le[\frac{\dim V}{2}]$, the \textit{orthogonal grassmannian} $GO(m,V)$ is defined as the subvariety of $G(m,V)$ consisting of isotropic $m$-dimensional subspaces of $V$. Unless $\dim V=2m$, the variety $GO(m,V)$ is a smooth irreducible variety. For $\dim V=2m$, the orthogonal grassmannian is a disjoint union of two isomorphic smooth irreducible components, and they are both isomorphic to $GO(m-1,V')$ where $\dim V'=2m-1$. Slightly abusing notation, we will denote by $GO(m,V)$ each of these two components. If $m\ne\frac{\dim V}{2}-1$, then $\operatorname{Pic}\nolimits GO(m,V)=\mathbb{Z} [\mathcal{O}_{GO(m,V)}(1)]$, where the sheaf $\mathcal{O}_{GO (m,V)}(1)$ posesses the following property: if $t: GO(m,V) \hookrightarrow G(m,V)$ is the tautological embedding, then \begin{equation*}\label{restrictn Pic} t^*\mathcal{O}_{G(m,V)}(1)\cong\Bigl\{ \begin{array}{lll} \mathcal{O}_{GO(m,V)}(1) & \mathrm{for} & m\ne\frac{\dim V}{2},\\ \mathcal{O}_{GO(k,V)}(2) & \mathrm{for} & m=\frac{\dim V}{2}. \end{array} \end{equation*} If $m=\frac{\dim V}{2}-1$, then for any $V_{m-1}\in GO(m-1,V)$ there is a unique $V_m\in GO(m,V)$ such that $V_{m-1}\subset V_m$. Thus there is a well-defined morphism \begin{equation}\label{flag ortho n-1,n} \theta:\ GO(m-1,V)\to GO(m,V),\ \ \ V_{m-1}\mapsto V_m, \ \ \ \textrm{where}\ \ \ V_m\supset V_{m-1}. \end{equation} Consequently, \begin{equation*}\label{Pic GO special} \operatorname{Pic}\nolimits GO(m-1,V)=\mathbb{Z}[\theta^*\mathcal{O}_{GO(m,V)}(1)] \oplus\mathbb{Z}[\mathcal{O}_{GO(m-1,V)}(1)], \end{equation*} where by $\mathcal{O}_{GO(m-1,V)}(1)$ we denote the $\theta$-relatively ample Grothendieck sheaf determined by the property that $\theta_*\mathcal{O}_{GO(m-1,V)}(1)$ is the universal quotient bundle on $GO(m,V)$. \vspace{2mm} Next, let $1\le m_1<...< m_k$ be an increasing sequence of positive integers, where $m_k\le [\frac{\dim V}{2}]$. The \textit{orthogonal flag variety} $FlO(m_1,...,m_k,V)$ is defined as $$ FlO(m_1,...,m_k,V):=\{(V_{m_1},...,V_{m_k})\ |\ V_{m_1}\in GO(m_i,V),\ V_{m_1}\subset...\subset V_{m_k}\}, $$ where, according to our convention, we assume $GO(m_k,V)$ connected if $m_k=\frac{\dim V}{2}$. Similarly to the case of usual flag varieties, there is a natural embedding $j:\ FlO(m_1,...,m_k,V) \hookrightarrow GO(m_1,V)\times...\times GO(m_k,V)$ and there are projections $\pi_i:\ Fl(m_1,...,m_k,V)\to GO(m_i,V),\ (V_{m_1}\subset...\subset V_{m_k})\mapsto V_{m_i},\ i=1,...,k$. Unless $m_k=\frac{\dim V}{2}-1,$ we have \begin{equation*}\label{Pic FlO} {\rm{Pic}}~FlO(m_1,...,m_k,V)=\mathbb{Z}[L_1]\oplus...\oplus \mathbb{Z}[L_k], \end{equation*} where \begin{equation*}\label{generators ortho} L_i:=\pi_i^*\mathcal{O}_{GO(m_i,V)}(1),\ \ \ i=1,...,k. \end{equation*} The isomorphism classes $[L_i]$ are a \textit{preferred set of generators} of ${\rm{Pic}}~FlO(m_1,...,m_k,V)$. If $m_k=\frac{\dim V}{2}-1,$ then there is an additional preferred generator $[(\theta\circ\pi_{k-1})^* \mathcal{O}_{GO(m_k+1,V)}(1)]$ of ${\rm{Pic}}FlO(m_1,...,m_k, V)$. \vspace{4mm} Let now $V$ be equipped with a non-degenerate symplectic form. This implies that $\dim V\in2\mathbb{Z}_{>0}$. Assume $1\le m\le \frac{1}{2}\dim V$. By definition, the \textit{$m$-th symplectic grassmannian} $GS(m,V)$ is the smooth irreducible subvariety of $G(m,V)$ consisting of isotropic $m$-dimensional subspaces of $V$. It is known that \begin{equation*}\label{Pic GS} \operatorname{Pic}\nolimits GS(m,V)=\mathbb{Z}[\mathcal{O}_{GS(k,V)}(1)],\ \ \ \mathcal{O}_{GS(k,V)}(1)=i^*\mathcal{O}_{G(k,V)}(1), \end{equation*} where $i:GS(m,V)\hookrightarrow G(m,V)$ is the tautological embedding. For a fixed increasing sequence of positive integers $1\le m_1<...\le m_k\le\frac{\dim V}{2}$, the \textit{symplectic flag variety} is defined as $$ FlS(m_1,...,m_k,V):=\{(V_{m_1},...,V_{m_k})\in GS(m_1,V)\times...\times GS(m_k,V)\ |\ V_{m_1}\subset... \subset V_{m_k}\}. $$ We have a natural embedding $j:\ FlS(m_1,...,m_k,V) \hookrightarrow GS(m_1,V)\times...\times GS(m_k,V)$ and projections $\pi_i:\ Fl(m_1,...,m_k,V)\to GS(m_i,V),\ (V_{m_1} \subset ...\subset V_{m_k})\mapsto V_{m_i},\ i=1,...,k,$. Moreover, \begin{equation*}\label{Pic FlS} {\rm{Pic}}~FlS(m_1,...,m_k,V)=\mathbb{Z}[L_1]\oplus...\oplus \mathbb{Z}[L_k], \end{equation*} where \begin{equation*}\label{generators sympl} L_i:=\pi_i^*\mathcal{O}_{GS(m_i,V)}(1),\ \ \ i=1,...,k, \end{equation*} The isomorphism classes $[L_i]$ are a \textit{preferred set of generators} of ${\rm{Pic}}~FlS(m_1,...,m_k,V)$. We now proceed to the definition of linear embeddings of flag varieties and their orthogonal and symplectic analogues. \begin{definition}\label{lin emb isotr flags} Let $k$ and $\tilde{k}$ be positive integers with $1<k\le \tilde{k}$. An embedding of flag varieties \begin{equation*}\label{usual embedding} \varphi:\ X\hookrightarrow Y, \end{equation*} where $X=Fl(m_1,...,m_k,V),\ Y=Fl(n_1,...,n_{\tilde{k}},V')$, or $X=FlO(m_1,...,m_k,V),\ Y=FlO(n_1,...,n_{\tilde{k}},V')$, or $X=FlS(m_1,...,m_k,V),\ Y=FlS(n_1,...,n_{\tilde{k}},V')$, is a \textit{linear embedding} if, for any $j$, $1\le j\le\tilde{k}$, we have \begin{equation*}\label{linearity} [\varphi^*M_j]=0\ \ \ \textrm{or}\ \ \ [\varphi^*M_j]=[L_i] \end{equation*} for some $i$, $1\le i\le k$, where $[L_1],...,[L_k]$ and $[M_1],...,[M_{\tilde{k}}]$ are the preferred sets of generators of ${\rm{Pic}}X$ and ${\rm{Pic}}Y$. \end{definition} \begin{example}\label{lin emb isotr grass} Assume that $k=\tilde{k}=1$ in Definition \ref{lin emb isotr flags}. Then $X$ and $Y$ are grassmannians, orthogonal grassmannians, or symplectic grassmannians. In all cases, except when $X=GO(m,V)$ and $Y=GO(n,V')$ for $(m,\dim V)= (l-1,2l)$ or $(n,\dim V')=(r-1,2r)$, a linear embedding $\varphi:X\to X$ is simply an embedding with $\varphi^*[M]= [L]$, where $[L]$ and $[M]$ are respective ample generators of the Picard groups ${\rm{Pic}}Y$ and ${\rm{Pic}}X$, cf. \cite[Def. 2.1]{PT}. In the remaining cases, a linear embedding $\varphi:X\to Y$ exists if and only if $X\simeq GO(l-1,V)$, $Y\simeq GO(r-1,V' )$ for $l\le r$, and here the linearity of $\phi$ implies $\phi^*\mathcal{O}_{GO(r-1,V')}(1)\cong\mathcal{O}_{GO(l-1,V)} (1)$, $\varphi^*\theta'^{*}\mathcal{O}_{GO(r,V')}(1)\cong \theta^*\mathcal{O}_{GO(l,V)}(1)$, where $\theta:GO(l-1,V)\to GO(l,V)$ and $\theta': GO(r-1,V')\to GO(r,V')$ are the projections defined in \eqref{flag ortho n-1,n}. To see this, one has to show (we leave this to the reader) that it is impossible to have an embedding $\phi:\ GO(l-1,V)\to GO(r-1,V')$ with $\varphi^*\theta'^{*}\mathcal{O}_{GO(r,V')}(1)\cong \mathcal{O}_{GO(l-1,V)}(1)$, $\phi^*\mathcal{O}_{GO(r-1,V')} (1)\cong\theta^*\mathcal{O}_{GO(l,V)}(1)$. \end{example} A linear embedding $\phi$ as in Definition \ref{lin emb isotr flags} induces a partition with $k+1$ parts $\{0,1,...,\tilde{k},\tilde{k}+1\}=I_0\sqcup I_1\sqcup I_2 \sqcup...\sqcup I_k$ such that $0\in I_0$ and $j\in I_0$ iff $\phi^*[M_j]=0$, respectively, $j\in I_i$ for $i\ge1$ iff $\phi^*[M_j]=[L_i]$. The map $j\mapsto i$, for $j\in I_i$, is a surjection which we denote by $p$. By definition, $p(0)=0$. \begin{proposition}\label{extension to Grassm growth} (i) Let $\varphi:\ Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,..., n_{\tilde{k}},V')$ be a linear embedding. Then $\phi$ induces a collection of morphisms of grassmannians \begin{equation*}\label{colln general} \varphi_{[i]}=\{\varphi_{i,j}\}_{i=p(j)}:\ G(m_i,V)\to \underset{j>0:p(j)=i}{\prod}G(n_j,V'),\ \ \ 0\le i\le k, \end{equation*} such that the diagram \begin{equation}\label{diagram 1} \xymatrix{ Fl(m_1,...,m_k,V)\ar@{^{(}->}[rrr]^-{\varphi}\ar@{^{(}->}[d]^-{ j} & & & Fl(n_1,...,n_{\tilde{k}},V') \ar@{^{(}->}[d]^-{j'}\\ G_0\times G(m_1,V)\times...\times G(m_k,V) \ar@{^{(}->} [rrr]^-{\phi_{[1]}\times...\times\phi_{[k]}} & & & G(n_1,V')\times ...\times G(n_{\tilde{k}},V')} \end{equation} where $j$ and $j'$ are the natural embeddings, is commutative. Here $G_0$ is a single point, and is present in the diagram if and only if there are constant morphisms $\phi_{0=p(j),j}: G_0\to G(n_j,V')$.\\ (ii) Similar statements hold in the orthogonal and symplectic cases. \end{proposition} In the proof, we will need the following. \begin{lemma}\label{lemma1} \textit{Let $X,\ Y,\ Z$ be projective varieties with $Y$ smooth, and let $a:X\to Y$ and $b:X\to Z$ be morphisms such that $a$ is surjective and $b$ is constant on the fibers of $a$. Then there exists a morphism $f:Y\to Z$ such that $b=f\circ a$.} \end{lemma} \begin{proof} Consider the morphism $g:X\to Y\times Z,\ x\mapsto(a(x),b(x))$, and let $Y\xleftarrow{a'}Y\times Z \xrightarrow{b'}Z$ be the projections onto factors so that $a=a'\circ g$ and $b=b'\circ g$. Since $b$ is constant on the fibers of $p$, it follows that $\tilde{a}:=a'|_{g(X)}: g(X)\to Y$ is a bijection. Therefore, as $Y$ is smooth, $\tilde{a}$ is an isomorphism (see, e.g., \cite[Ch.2, Section 4.4, Thm. 2.16]{S}). The desired morphism $f$ is now the composition $f=b'\circ\tilde{a}^{-1}$. \end{proof} \textit{Proof of Proposition \ref{extension to Grassm growth}}. (i) We consider the case $k=\tilde{k}=2$. For arbitrary $k,\ \tilde{k}$ the proof goes along the same lines, and we leave the details to the reader. Set $[L_1]:=\phi^*[M_{j_1}],\ [L_2]:=\phi^*[M_{j_2}]$, and let $\pi_i:Fl(m_1,m_2,V)\to G(m_i,V),\ \pi'_i:Fl(n_1,n_2,V')\to G(n_i,V'),\ i=1,2,$ be the natural projections. For an arbitrary point $x=(x_1,x_2)=(V_{m_1},V_{m_2})\in Fl(m_1,m_2,V) \subset G(m_1,V)\times G(m_2,V)$, consider the fibres $\pi_i^{-1}(x_i)\subset F,\ i=1,2,$ through the point $x$. Since $\varphi$ is a linear embedding, we have $M_{j_1}|_{\varphi (\pi_1^{-1}(x_1))}\simeq\varphi^*M_{j_1}|_{\pi_1^{-1}(x_1)} \simeq\mathcal{O}_{\pi_1^{-1}(x_1)}\simeq\mathcal{O}_{\varphi (\pi_1^{-1}(x_1))}$. As $\varphi(\pi_1^{-1}(x_1))$ is an irreducible variety and $M_{j_1}={\pi'}_1^* \mathcal{O}_{G(n_{j_1},V')}(1)$, where $\mathcal{O}_{G(n_{j_1}, V')}(1)$ is an ample sheaf, it follows from the above isomorphisms that $\pi'_{j_1}$ is constant on the variety $\varphi(\pi_1^{-1}(x_1))$. Equivalently, the morphism $\pi'_{j_1}\circ\varphi$ is constant on the fibres of the projection $\pi_1$. Lemma \ref{lemma1} implies that $\pi'_1\circ\varphi$ factors through the projection $\pi_1$, i.e. there is a well-defined morphism \begin{equation}\label{phi1} \varphi_1:\ G(m_1,V)\to G(n_{j_1},V'),\ x_1\mapsto \pi'_{j_1}(\varphi(\pi_1^{-1}(x_1))) \end{equation} such that $\varphi_1\circ \pi_1=\pi'_{j_1}\circ \varphi$. In a similar way there is a well-defined morphism \begin{equation}\label{phi2} \varphi_2:\ G(m_2,V)\to G(n_{j_2},V'),\ x_2\mapsto \pi'_{j_2}(\varphi(\pi_1^{-1}(x_2))) \end{equation} such that $\varphi_2\circ p_2=\pi'_{j_2}\circ \varphi$. By construction, $\phi_1$ and $\phi_2$ are linear morphisms. Considering now $Fl(m_1,m_2,V)$ and $Fl(n_1,n_2,V')$ as lying, respectively, in $G(m_1,V)\times G(m_2,V)$ and in $G(n_1,V')\times G(n_2,V')$, for any points $x=(x_1,x_2)\in Fl(m_1,m_2,V)$ and $x'=(x'_1,x'_2)\in Fl(n_1,n_2,V')$ we have $$ x=\pi_1^{-1}(x_1)\cap \pi_2^{-1}(x_2),\ \ \ x'={\pi'}_1^{-1}(x'_1)\cap {\pi'}_2^{-1}(x'_2). $$ This together with (\ref{phi1}) and (\ref{phi2}) shows that, if $x'_{j_i}=\varphi_i(x_i),\ i=1,2$, then $$ \varphi(x)=\varphi(\pi_1^{-1}(x_1))\cap\varphi(\pi_2^{-1}(x_2))\in {\pi'}_{j_1}^{-1}(x'_{j_1})\cap{\pi'}_{j_2}^{-1}(x'_{j_2})= (\phi_1\times\phi_2)(x), $$ i.e. the diagram (\ref{diagram 1}) is commutative for $k=2$. We leave to the reader to make (ii) precise and check that the above proof extends to this case. ~\hfill$\Box$ \vspace{1cm} \section{Standard extensions of flag varieties} \label{linear embed} \vspace{5mm} In this section we introduce and study a class of embeddings of flag varieties that we call standard extensions. In almost all cases, standard extensions are linear embeddings in the sense of Section \ref{preliminaries}. We start by considering the case of grassmannians. Let \begin{equation}\label{def of strict} \varphi: G(m,V) \hookrightarrow G(n,V') \end{equation} be a regular morphism. Assume $\dim V'>\dim V$, $m\ne0$, $m\ne \dim V$. We say that $\varphi$ is a \textit{strict standard extension} if there exists an isomorphism of vector spaces $V'=V\oplus\widehat{W}$ and a subspace $W\subset\widehat{W}$, such that \begin{equation*}\label{def of strict2} \varphi(V_m)=V_m\oplus W \end{equation*} where $V_m\subset V$ is an arbitrary point of $G(m,V)$. If $m=0$ or $m=\dim V$, a morphism \eqref{def of strict} is necessarily constant and we call it a \textit{constant strict standard extension}. In this case we set $W:=\phi(G(m,V))$. It is easy to check that a nonconstant strict standard extension is a linear embedding. By a \textit{modified standard extension} we understand an embedding \eqref{def of strict} for which there exists a strict standard extension $$ \varphi':G(m,V)\hookrightarrow G(\dim V'-n,V'^{\vee}) $$ such that $\varphi=d\circ\varphi'$ where $$ d:G(\dim V'-n,V'^{\vee})\xrightarrow{\sim}G(n,V') $$ is the duality isomorphism. In what follows, a \textit{standard extension} will mean a strict standard extension or a modified standard extension. Note that if a morphism \eqref{def of strict} is linear, it is not necessarily a standard extension. For instance, the reader can prove that the Pl\"ucker embedding $$ \psi:\ G(m,V)\hookrightarrow G(1,\wedge^mV)=\mathbb{P} (\wedge^m V) $$ is a standard extension if and only if $m=1$ or $m=\dim V-1$. On the other hand, the Pl\"ucker embedding is of course a linear embedding. In the case of orthogonal and symplectic grassmannians, a strict standard extension is defined in the same way with the additional requirement that the decomposition $V'=V\oplus U$ be orthogonal and that the spaces $V_m$ and $W$ are isotropic. In these cases there is no need to consider modified standard extensions (as the spaces $V$ and $V^{\vee}$ are identified via the respective non-degenerate form), and the terms strict standard extension and standard extension are synonyms. Here is a definition of strict standard extension $\varphi$ of grassmannians which refers only to the data of linear algebra which can be recovered canonically from the embedding $\varphi$. \begin{definition}\label{strict} Let $\dim V'>\dim V$. A morphism of grassmannians $\varphi:G(m,V)\hookrightarrow G(n,V')$ is said to be a \textit{strict standard extension} if either $G(m,V)$ is a point (i.e. $m=0$ or $m=\dim V$, and $\phi$ is constant) or there exists a subspace $U\subset V'$ and a surjective linear operator $\varepsilon:\ U\twoheadrightarrow V$ such that \begin{equation}\label{eta, eps} \varphi(V_m)=\varepsilon^{-1}(V_m). \end{equation} \end{definition} \noindent If $\phi$ is a nonconstant standard extension, the subspace $U\subset V'$ is unique and the linear operator $\varepsilon:U \to V$ is unique up to a scalar multiple. Indeed, assume $\phi$ is given and set \begin{equation}\label{descriptn of W} W:=\underset{V_m\subset V}{\bigcap} \varphi(V_m). \end{equation} Let $S$ and $S'$ denote respectively the tautological bundles on $G(m,V)$ and $G(n,V')$. There is an obvious exact sequence $$ 0\to W\otimes\mathcal{O}_{G(m,V)}\to\phi^*S'\to S\to0. $$ Dualization yields an injective homomorphism $V^{\vee}= H^0(G(m,V),S^{\vee})\hookrightarrow H^0(G(m,V),(\phi^*S') ^{\vee})$ with cokernel equal $W^{\vee}$. Set $U^{\vee}= H^0(G(m,V),(\phi^*S')^{\vee})$. Then a second dualization yields a surjective homomorphism $\varepsilon:U\to V$ with $\ker\varepsilon=W$. In particular, \begin{equation}\label{descriptn of U} U=\underset{V_m\subset V}{\bigcup} \varphi(V_m). \end{equation} In what follows, we will assign a subspace $U\subset V'$ also in the case when $\phi$ is constant: we set $U=W:=\phi(G(m,V)) \in G(n,V')$ and $\varepsilon=0$. Formulas \eqref{eta, eps} and \eqref{descriptn of U} then hold in this case too. It is easy to show that Definition \ref{strict} is equivalent to the above "naive" definition of strict standard extension. Let $\phi$ be a nonconstant strict standard extension according to Definition \ref{strict}. Then $U$ and $\varepsilon:U\to V$ are given, and we can choose a splitting $U\simeq V\oplus(W=\ker \varepsilon)$. In particular, this induces an embedding $V$ into $V'$. We then extend the splitting $U\simeq V\oplus W$ to a splitting $V'=V\oplus\widehat{W}$ where $W\subset \widehat{W}$. This yields the datum of "naive" definition. Conversely, given a nonconstant strict standard extension as in the "naive" definition, we simply set $U:=V\oplus W$ and define $\varepsilon$ to be the projection $U\to V$. Finally, if $\phi$ is constant then we put $U:=\phi(G(m,V))=W$ (here $\dim U=n$). In the orthogonal and symplectic cases, in Definition \ref{strict} one must assume that the space $W$ is isotropic and the isomorphism $U/W\xrightarrow{\sim}V$ induced by the operator $\varepsilon:U\twoheadrightarrow V$ is an isomorphism of spaces endowed with symmetric, or respectively symplectic, forms. Here the form on $U$ is induced by the respective form on $V'$. It is a straightforward observation that in all cases the composition of standard extensions of grassmannians is also a standard extension. The composition of two strict standard extensions or two modified standard extensions is a strict standard extension, while the composition of a strict standard extension and a modified standard extension is again a modified standard extension. We now give the definition of a strict standard extension of usual and isotropic flag varieties. \begin{definition}\label{str st ext flags} An embedding of flag varieties $\varphi:\ Fl(m_1,...,m_k,V) \hookrightarrow Fl(n_1,...,n_{\tilde{k}},V')$, respectively, $\varphi:\ FlO(m_1,...,m_k,V) \hookrightarrow FlO(n_1,...,n_{\tilde{k}},V')$, respectively, $\varphi:\ FlS(m_1,...,m_k,V) \hookrightarrow FlS(n_1,...,n_{\tilde{k}},V')$, is said to be a \textit{strict standard extension}, or simply a \textit{standard extension} in the orthogonal and symplectic cases, if there exists a flag of distinct nonzero subspaces of $V'$, $$ U_1\subset U_2\subset...\subset U_{\tilde{k}} $$ such that in the orthogonal and symplectic cases the spaces $U_i$ are nondegenerate, and a commutative diagram \begin{equation}\label{Ui,epsilon i} \xymatrix{ V \ar@{=}[r]& V \ar@{=}[r]& .\ .\ . \ar@{=}[r]& V\\ U_1\ar[u]^-{\varepsilon_1} \ar@{^{(}->}[r] & U_2\ar[u]^-{\varepsilon_2} \ar@{^{(}->}[r] & .\ .\ . \ar@{^{(}->}[r] & U_{\tilde{k}} \ar[u]^-{\varepsilon_{\tilde{k}}} } \end{equation} of linear operators $\varepsilon_i:U_i\to V$, surjective whenever nonzero, compatible with the respective forms on $U_i$ and $V$ and having isotropic kernels in the orthogonal and symplectic cases, and such that \begin{equation}\label{phi(...)} \begin{split} & \varphi\big(0=V_{\overline{p}(0)}\subset V_{\overline{p}(1)}\subset...\subset V_{\overline{p}(\tilde{k}) }\subset V_{\overline{p}(\tilde{k}+1)}=V\big)=\\ & \big(0\subset\varepsilon_1^{-1}(V_{\overline{p}(1)})\subset \varepsilon_1^{-1}(V_{\overline{p}(2)})\subset...\subset \varepsilon_{\tilde{k}}^{-1}(V_{\overline{p}(\tilde{k})}) \subset V'\big) \end{split} \end{equation} for a suitable surjective map $\overline{p}:\{0,1,...,\tilde{k} ,\tilde{k}+1\}\to\{0,1,...,k,k+1\}$ satisfying $\overline{p}(i) \le \overline{p}(j)$ whenever $i<j$. Note that $\overline{p}(0)=0,\ \overline{p}(\tilde{k}+1)=k+1$ and that there are exactly $k$ distinct proper nonzero subspaces among $V_{\overline{p}(1)},...,V_{\overline{p} (\tilde{k})}$. Moreover, the surjection $p:\{0,1,...,\tilde{k} \}\to\{0,1,...,k\}$ satisfies $p(j)=\overline{p}(j)$ whenever $p(j)\ne0$ and $p^{-1}(0)\cup\{\tilde{k}\}=\overline{p}^{-1} (0)\sqcup\overline{p}^{-1}(k+1)$. A strict standard extension is a linear embedding, except in the case $$ FlO(m_1,...,m_k,V)\hookrightarrow FlO(n_1,...,n_{\tilde{k}},V') $$ where $\frac{\dim V}{2}-1$ appeas among $m_1,...,m_k$ but $\frac{\dim V'}{2}-1$ does not appear among $n_1,..., n_{\tilde{k}}$, or $\frac{\dim V}{2}$ appears among $m_1,..., m_k$ but $\frac{\dim V'}{2}-1$ or $\frac{\dim V'}{2}$ does not appear among $n_1,...,n_{\tilde{k}}$ . \end{definition} Of course, in the case of ordinary (i.e. not isotropic) flag varieties, we also need the definition of a \textit{modified standard extension}. By definition, this is a composition $\phi=d\circ\phi'$ where \begin{equation*}\label{phi modified} \phi':\ Fl(m_1,...,m_k,V)\hookrightarrow Fl(\dim V'- n_{\tilde{k}},..., \dim V'-n_1,V'^{\vee}) \end{equation*} is a strict standard extension and \begin{equation*}\label{duality} d:\ Fl(\dim V'-n_{\tilde{k}},...,\dim V'-n_1,V'^{\vee}) \xrightarrow{\simeq} Fl(n_1,...,n_{\tilde{k}},V') \end{equation*} is the duality isomorphism. Here $\phi^*[M_j]=[L_{q(j)}]$ for a map $q:\{0,1,...,\tilde{k}\}\to\{0,1,...,k\}$ such that $q(0) =0$, $q(i)\ge q(j)$ whenever $q(i)\ne0,\ q(j)\ne0$ and $i\le j$, and also $q(j)=0$ implies $j<t$ or $j>t$ for all $t$ with $q(t)\ne0$. \begin{example}\label{example 3.3} (i) Consider the extreme case when $k=1$ and $\tilde{k}$ is an arbitrary integer greater or equal to 1. Then the surjection $\overline{p}:\{0,1,...,\tilde{k},\tilde{k}+1\}\to\{0,1,2\}$ from Definition \ref{str st ext flags},(ii) defines an ordered partition of $\{0,1,...,\tilde{k}, \tilde{k}+1\}$ with three parts $\overline{p}^{-1}(0)$, $\overline{p}^{-1}(1)$, $\overline{p}^{-1}(2)$, and a corresponding standard extension $$ G(m,V)\hookrightarrow Fl(m_1,...,m_{\tilde{k}},V') $$ has the form $$ (0\subset V_m\subset V)\mapsto(0\subset W_1\subset...\subset W_s\subset\varepsilon_{s+1}^{-1}(V_m)\subset...\subset \varepsilon_t^{-1}(V_m)\subset U_{t+1}\subset...\subset U_{\tilde{k}}\subset V'), $$ where $\{0,1,...,s\}=\overline{p}^{-1}(0)$, $\{s+1,...,t\}= \overline{p}^{-1}(1)$ and $\{t+1,...,\tilde{k}+1\}= \overline{p}^{-1}(2)$. (ii) Next, consider the case when $\dim V'=\dim V+1$. Then $\tilde{k}$ necessarily equals $k$ or $k+1$. Hence, $\dim W_i\le1$ and there exists $i_0,\ 0\le i_0\le k,$ such that $W_j=0$ for $j\le i_0$ and $\dim W_{i_0+1}=...=\dim W_{\tilde{k}}=1$. Consequently, $W_{i_0+1}=...= W_{\tilde{k}}$. Set $W:=W_{i_0+1}=...=W_{\tilde{k}}$. If $\tilde{k}=k$, then $p$ is a bijection and the corresponding standard extension $\phi:\ Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,...,n_k,V')$ has the form \begin{equation}\label{example of st ext} \begin{split} &\ \ \ \ \ \varphi(0\subset V_{m_1}\subset...\subset V_{m_k}\subset V)=\\ & =\left\{ \begin{aligned} (0\subset V_{m_1}\oplus W\subset...\subset V_{m_k}\oplus W\subset V')\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & &\ for\ i_0=0,\\ (0\subset V_{m_1}\subset...\subset V_{m_{i_0}}\subset V_{m _{i_0+1}}\oplus W\subset...\subset V_{m_k}\oplus W\subset V') & &\ for\ 0<i_0<k,\\ (0\subset V_{m_1}\subset...\subset V_{m_k}\subset V') \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & & \ for\ i_0=k.\\ \end{aligned} \right. \end{split} \end{equation} If $\tilde{k}=k+1$, then $p(i_0)=p(i_0+1)=i_0$ and the standard extension $\varphi:\ Fl(m_1,...,m_k,V) \hookrightarrow Fl(n_1,...,n_{k+1},V')$ has the form \begin{equation}\label{example of st ext2} \begin{split} &\ \ \ \ \ \varphi(0\subset V_{m_1}\subset...\subset V_{m_k}\subset V)=\\ & =\left\{ \begin{aligned} (0\subset W\subset V_{m_1}\oplus W\subset...\subset V_{m_k}\oplus W\subset V')\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & &\ for\ i_0=0,\\ (0\subset V_{m_1}\subset...\subset V_{m_{i_0}}\subset V_{m _{i_0+1}}\oplus W\subset...\subset V_{m_k}\oplus W\subset V') \ \ \ \ & &\ for\ 0<i_0<k,\\ (0\subset V_{m_1}\subset...\subset V_{m_k} \subset V_{m_k}\oplus W\subset V') \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & & \ for\ i_0=k.\\ \end{aligned} \right. \end{split} \end{equation} (iii) Let $\dim V=2$ and let $V'=V\oplus V$. Consider the embedding $$ \mathbb{P}(V)=G(1,V)\hookrightarrow Fl(1,2,3,V\oplus V),\ \ \ \ \ \ (0\subset V_1\subset V)\mapsto(0\subset V_1\subset V\oplus0\subset V\oplus V_1\subset V\oplus V). $$ This embedding is not a standard extension. Here, $\varphi^*[ M_1]=\varphi^*[M_3]=[L],\ \varphi^*[M_2]=0.$ This shows that there is no $p$ as in the definition of strict standard extension, and it is easy to check that $\phi$ is also not a modified standard extension. \end{example} (iv) Let $V'$ be endowed with non-degenerate symmetric or symplectic form, and $V'=V\oplus\widehat{W}$ where $\widehat{W} =V^{\bot}$ and $\dim\widehat{W}=2$. Fix an isotropic line $W \subset\widehat{W}$. Then for any increasing sequence $0<m_1< ...<m_k\le[\frac{\dim V}{2}]$ and any $s$, $1\le s\le k$, there is a standard extension $\phi:X\to Y$, where $X=FlO(m_1, ...,m_k,V)$ and $Y=FlO(m_1,...,m_s,m_s+1,...,m_k+1,V')$, or respectively, $X=FlS(m_1,...,m_k,V)$ and $Y=FlS(m_1,...,m_s, m_s+1,...,m_k+1,V')$. For $s=0$ there also is a standard extension $\phi:X\to Y$, where now $Y=FlO(1,m_1+1,...,m_k+1,V' )$ or $Y=FlS(1,m_1+1,...,m_k+1,V')$, respectively. The embedding $\phi$ is given by formula \eqref{example of st ext2} with $i_0$ substituted by $s$. \vspace{3mm} A less canonical, but more intuitive, description of strict standard extensions (respectively, of standard extensions in the isotropic case) is given by the following easily proved proposition. \begin{proposition}\label{descrn of st ext} Assume that $\varphi:\ Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,...,n_{\tilde{k}},V')$, respectively, $\varphi:\ FlO(m_1,...,m_k,V)\hookrightarrow FlO(n_1,...,n_{\tilde{k}}, V')$, respectively, $\varphi:\ FlS(m_1,...,m_k,V) \hookrightarrow FlS(n_1,...,n_{\tilde{k}},V')$ is a nonconstant strict standard extension corresponding to a surjection $\overline{p}:\{0,1,...,\tilde{k},\tilde{k}+1\}\to \{0,1,...,k,k+1\}$. Define the flag $(0\subset W_1\subset... \subset W_{\tilde{k}}\subset V')$ by setting $W_i:=\ker \varepsilon_i$. Then there exists a direct sum decomposition \begin{equation}\label{direct sum with W} V'=V\oplus \widehat{W} \end{equation} with $\widehat{W}=V^{\bot}$ in the orthogonal and symplectic case, and such that $W_i\subset \widehat{W}$, $U_i\supset V$ for all $i$ with $\varepsilon_i\ne0$, and the nonzero operators $\varepsilon_i:U_i\to V$ are just projections onto $V$ via the decomposition \eqref{direct sum with W}. Moreover, \begin{equation}\label{phi(...)1} \varphi\big(0\subset V_{\overline{p}(1)}\subset...\subset V_{\overline{p}(\tilde{k})}\subset V\big)= \big(0\subset V_{\overline{p}(1)}\oplus W_1\subset...\subset V_{\overline{p}(\tilde{k})}\oplus W_{\tilde{k}}\subset V'\big). \end{equation} \end{proposition} \begin{lemma}\label{Lemma 3.7} In the notation of Proposition \ref{descrn of st ext}, let $\underline{w}$ be a basis of $\widehat{W}$ such that all subspaces $W_i$ are coordinate subspaces with respect to $\underline{w}$. Then, for any splitting $\widehat{W}= \overline{W}\oplus\overline{\overline{W}}$ such that $\overline{W}$ and $\overline{\overline{W}}$ are coordinate spaces, mutually perpendicular within $\widehat{W}$ in the orthogonal and symplectic cases, the strict standard extension given by formula \eqref{example of st ext2} is the composition of strict standard extensions $$ Fl(m_1,...,m_k,V)\hookrightarrow Fl(m'_1,...,m'_l,V\oplus \overline{W})\hookrightarrow Fl(n_1,...,n_{\tilde{k}},V'= (V\oplus\overline{W})\oplus\overline{\overline{W}}) $$ for which the corresponding flags in $\overline{W}$ and $\overline{\overline{W}}$ are the respective intersections of the flag $(0\subset W_1\subset...\subset W_{\tilde{k}} \subset W)$ with $\overline{W}$ and $\overline{\overline{W}}$. \end{lemma} \begin{proof} Direct verification using formula \eqref{phi(...)1}. \end{proof} \vspace{1cm} \section{A sufficient condition for a linear embedding to be a standard extension}\label{more linear embed} \vspace{5mm} In this section we establish our main result concerning linear embeddings of flag varieties. This is a sufficient condition for a linear embedding to be a standard extension. Consider a flag variety $Fl(m_1,...,m_k,V)$ and let $\{m_1, ...,m_k\}=R_1\cup...\cup R_s$ be a decomposition into a union of $s$ subsets. Denote this decomposition by $R$. By ordering the elements of $R_i$ we can think of $R_i$ as a type of a flag, and then $Fl(R_i,V)$ is a well-defined flag variety. Moreover, there is a canonical embedding $$ \psi_{R,t_1,...,t_s}:\ Fl(m_1,...,m_k,V)\hookrightarrow Fl(R_1,V)^{\times t_1}\times...\times Fl(R_s,V)^{\times t_s} $$ where by $Fl(R_i,V)^{t_i}$ we denote the direct product of $t_i$ copies of $Fl(R_i,V)$. If now $\phi: Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,..., n_{\tilde{k}},V')$ is an embedding, we say that $\phi$ \textit{does not factor through any direct product} if $\phi\ne \psi\circ\psi_{R,t_1,...,t_s}$ for any decomposition $R$, any $t_i\in\mathbb{Z}_{\ge1}$ and any embedding $\psi:\ Fl(R_1,V)^{\times t_1}\times...\times Fl(R_s,V)^{ \times t_s}\hookrightarrow Fl(n_1,...,n_{\tilde{k}},V')$. The definition clearly makes sense also in the orthogonal and symplectic cases. \begin{lemma}\label{not extend} Let $\phi:Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,..., n_{\tilde{k}},V')$ be a linear embedding which does not factor through any direct product. Assume that $\tilde{k}\ge 3$ and there exist integers $i$ and $j$, $1\le i,\ i+2\le j\le\tilde{k}$, such that the morphisms $\pi_i\circ\phi$ and $\pi_j\circ\phi$ are not constant maps. Then for any $l,\ i<l<j,$ the morphism $\pi_l\circ\phi$ is not a constant map. Similar statements are true in the orthogonal and symplectic cases. \end{lemma} \begin{proof} Suppose the contrary, i. e. that there exists $l,\ i<l<j,$ such that the morphism $\pi_l\circ\phi$ is a constant map, and let $V'_l:=\mathrm{im}(\pi_l\circ\phi)\subset V'$. Then $\phi$ induces well-defined embeddings $$ \phi': Fl(p(\{0,1,...,l\}),V)\hookrightarrow Fl(n_1,...,n_{\tilde{k}},V'), $$ $$ \phi'': Fl(p(\{l,...,\tilde{k}\}),V)\hookrightarrow Fl(n_1,...,n_{\tilde{k}},V'), $$ where we consider $p(\{0,1,...,l\})$ and $p(\{l,...,\tilde{k}\} )$ as types of flags. Moreover, $\phi$ clearly factors through the embedding $$ \psi:\ Fl(p(\{0,1,...,l\}),V)\times Fl(p(\{l,...,\tilde{k}\},V) \to Fl(n_1,...,n_{\tilde{k}},V'), $$ where, for $F_1\in Fl(p(\{0,1,...,l\}),V)$ and $F_2\in Fl(p( \{l,...,\tilde{k}\},V)$, the spaces with indices from 1 to $l$ of the flag $\psi(F_1\times F_2)$ coincide with those of the flag $\phi'(F_1)$, and the spaces with indices from $l$ to $\tilde{k}$ coincide with those of the flag $\phi''(F_2)$. The flag $\psi(F_1,F_2)$ is well defined as its space with index $l$ equals $V'_l$. \end{proof} \begin{theorem}\label{Thm 4.6} Let $\phi:Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,..., n_{\tilde{k}},V')$ be a linear embedding. Assume that all morphisms $\varphi_{p(j),j}:\ G(m_{p(j)},V) \hookrightarrow G(n_j,V')$ from Proposition \ref{extension to Grassm growth} are strict standard extensions, and that $\phi $ does not factor through any direct product. Then $\varphi$ is a strict standard extension. Analogous statements hold in the orthogonal and symplectic cases. \end{theorem} \begin{proof} Lemma \ref{not extend} implies that there are $s$ and $t$, $s<t $, so that $p(j)=0$ holds precisely for $j\le s$ and for $j\ge t$. In the case when there is a single index $j$ such that $\phi_{p(j),j}$ is a nonconstant morphism, the statement of the theorem is easy. We thus may assume that there are (at least) two indices $j$ and $j+1$, $1<j<j+1<t$, so that $\phi$ induces nonconstant strict standard extensions \begin{equation*}\label{phi j,j+1} \phi_{p(j),j}:\ G(m_{p(j)},V)\hookrightarrow G(n_j,V'),\ \ \ \ \ \ \phi_{p(j+1),j+1}:\ G(m_{p(j+1)},V)\hookrightarrow G(n_{j+1},V'). \end{equation*} Define subspaces $U_j$ and $U_{j+1}$ of $V'$ by formula \eqref{descriptn of U} in which we put $\phi=\phi_{p(j),j}$ and $m=m_{p(j)}$, or $\phi=\phi_{p(j+1),j+1}$ and $m=m_{p(j+1)} $, respectively. Let $(0\subset V_{m_1}\subset...\subset V_{m_k }\subset V)$ denote an arbitrary point of $Fl(m_1,...,m_k,V)$. Since by definition \begin{equation}\label{phi j in phi j+1} \phi_{p(j),j}(V_{m_{p(j)}})\subset \phi_{p(j+1),j+1}(V_{m_{p(j+1)}}) \end{equation} for any subflag $V_{m_{p(j)}}\subset V_{m_{p(j+1)}}$ if $p(j) <p(j+1)$, or for any subflag $V_{m_{p(j+1)}}\subset V_{m_{p(j)}}$ if $p(j+1)>p(j)$, formula \eqref{descriptn of U} implies that $U_j$ is a subspace of $U_{j+1}$. Next, since the strict standard extensions $\phi_{p(j),j}$ and $\phi_{p(j+1), j+1}$ are nonconstant, it follows from Definition \ref{strict} that there are surjective linear operators $\varepsilon_j:U_j\to V$ and $\varepsilon_{j+1}: U_{j+1}\to V$, such that formula \eqref{eta, eps} holds for $\varepsilon=\varepsilon_j,\ m=m_{p(j)}$ and $\varepsilon= \varepsilon_{j+1},\ m=m_{p(j+1)}$, respectively. This, together with \eqref{phi j in phi j+1}, means that \begin{equation}\label{eps j in} \varepsilon_j^{-1}(V_{m_{p(j)}})\subset \varepsilon_{j+1}^{-1}(V_{m_{p(j+1)}}) \end{equation} under the same conditions on $V_{m_{p(j)}}$ and $V_{m_{p(j+1)} }$ as in \eqref{phi j in phi j+1}. Denoting $W_j=\ker\varepsilon_j$ and $W_{j+1}=\ker\varepsilon _{j+1}$, in view of \eqref{phi j in phi j+1} we obtain from \eqref{descriptn of W} that $W_j$ is a subspace of $W_{j+1}$. The inclusions $U_j\subset U_{j+1}$ and $W_j\subset W_{j+1}$ join into a commutative diagram \begin{equation}\label{2new Uij,Wij} \xymatrix{ V\ar[r]^-{\theta_j} & V \\ U_j\ar@{^{(}->}[r]\ar[u]^-{\varepsilon_j} & U_{j+1}\ar[u]^-{\varepsilon_{j+1}}\\ W_j\ar@{^{(}->}[r]\ar@{^{(}->}[u] & W_{j+1},\ar@{^{(}->}[u]} \end{equation} where $\theta_j$ is the induced linear operator. From \eqref{eps j in} and \eqref{2new Uij,Wij} we obtain \begin{equation}\label{theta j in} \theta_j(V_{m_{p(j)}})\subset V_{m_{p(j+1)}}. \end{equation} Now we are going to show that \begin{equation*}\label{p(j)le p(j+1)} p(j)\ \le\ p(j+1). \end{equation*} Assume the contrary, i.e. $p(j+1)<p(j)$. Then the inclusion \eqref{theta j in} implies $$ \theta_j(V_{m_{p(j)}}) \subset \underset{V_{m_{p(j+1)}}\subset V_{m_{p(j)}}}{\bigcap} V_{m_{p(j+1)}}=0. $$ Thus $\theta_j=0$, and consequently $U_j\subset W_{j+1}$ by diagram \eqref{2new Uij,Wij}. This together with formula \eqref{eta, eps} means that the inclusion \eqref{eps j in} extends to a pair of inclusions \begin{equation*}\label{eps j in W} \varepsilon_j^{-1}(V_{m_{p(j)}})\subset W_{j+1}\subset \varepsilon_{j+1}^{-1}(V_{m_{p(j+1)}}),\ \ \ \ \end{equation*} for any $(V_{m_{p(j)}},V_{m_{p(j+1)}})\in G(m_{p(j)},V)\times G(m_{p(j+1)},V)$. Then the exact same argument as in the proof of Lemma \ref{not extend} shows that $\phi$ factors through a direct product. Hence the assumption $p(j+1)<p(j)$ is invalid. Next, we claim that $\theta_j=c_j\mathrm{Id}$ for some nonzero constant $c_j$. Note that $\theta_j\ne0$ by the above. Then, since $\varepsilon_j^{-1}(V_{m_{p(j)}})\subset\varepsilon_{j+1} ^{-1}(V_{m_{p(j+1)}})$, we have $\theta_j(V_{m_{p(j)}})\subset V_{m_{p(j+1)}}$. Taking into account that $\underset{V_{m_{p(j+1)}}\supset V_{m_{p(j)}}}{\bigcap}V_{m_{p(j+1)}}=V_{m_{p(j)}}$, we obtain \begin{equation*}\label{theta(...)in} \theta_j(V_{m_{p(j)}})\subset V_{m_{p(j)}} \end{equation*} for any $V_{m_{p(j)}}\in G(m_{p(j)},V)$. As any 1-dimensional subspace of $V$ is the intersection of all $m_{p(j)}$-dimensional subspaces which contain it, we see that any vector in $V$ is an eigenvector for $\theta_j$. Consequently, we have $\theta_j=c_j\mathrm{Id}$ for $c_j\ne0$. The above argument applies to any pair of integers $j,j+1$ where $s+1<j<t-2$. Therefore, we can construct a commutative diagram \begin{equation}\label{Ui,epsilon i new} \xymatrix{ V \ar[r]^-{\theta_1} & V \ar[r] & .\ .\ .\ar[r] & V \ar[r]^-{\theta_{\tilde{k}}} & V\\ U_1\ar[u]^-{\varepsilon_1}\ar@{^{(}->}[r] & U_2\ar[u]^-{\varepsilon_2}\ar@{^{(}->}[r] & .\ .\ . \ar@{^{(}->}[r] & U_{\tilde{k}-1}\ar[u]^-{\varepsilon_{\tilde{k}-1}}\ar@{^{(}->} [r] & U_{\tilde{k}} \ar[u]^-{\varepsilon_{\tilde{k}}} ,} \end{equation} where the morphisms $\varepsilon_i$ equal zero for $i\le s$, $i\ge t$, $\theta_i=\mathrm{Id}$ for $i\le s$ and $i\ge t$, and $\theta_i=c_i\mathrm{Id}$ with $c_i\ne0$ for $s+1\le i\le t-1$. Here, the spaces $U_1,...,U_s,U_{t+1},...,U_{\tilde{k}}$ are defined as the subspaces of $V'$ which equal the images of the respective constant morphisms $\pi'_1\circ\phi,..., \pi'_s \circ\phi$, $\pi'_{t+1}\circ\phi,...,\pi'_{\tilde{k}} \circ\phi$, where $$ \pi'_r: \ Fl(n_1,...,n_{\tilde{k}},V')\to G(n_r,V') $$ are the natural projections. Via scaling the morphisms $\varepsilon_i$ for $s+1\le i \le t-1$, we can turn the diagram \eqref{Ui,epsilon i new} into the diagram \eqref{Ui,epsilon i} in the definition of strict standard extension. An immediate checking shows that our given embedding $\phi$ is given by formula \eqref{phi(...)} for the surjection $\overline{p}:\{0,1,...,\tilde{k},\tilde{k}+1\}\to \{0,1,...,k,k+1\}$ where $\overline{p}(j)=p(j)$ for $j\le t-1$, $\overline{p}(j)=\tilde{k}+1$ for $j\ge t$. \end{proof} The next theorem is a more general version of Theorem \ref{Thm 4.6}. \begin{theorem}\label{Thm 4.3} If, in the setting of Theorem \ref{Thm 4.6}, all morphisms $\phi_{p(j),j}$ are (not necessarily strict) standard extensions, then $\phi$ is also a standard extension. \end{theorem} \begin{proof} First, as in the proof of Theorem \ref{Thm 4.6}, we assume that there are (at least) two indices $j$ and $j+1$ such that there are nonconstant standard extensions $\phi_{p(j),j}$ and $\phi_{p(j+1),j+1}$ as in \eqref{phi j,j+1}. The reader will easily handle the remaining case. We will show now that the standard extensions $\phi_{p(j),j}$ and $\phi_{p(j+1),j+1}$ are either both strict or are both modified. For this, we need to exclude the following other logical possibilities:\\ (a) $p(j)\le p(j+1)$, $\phi_{p(j),j}:\ G(m_{p(j)},V) \hookrightarrow G(n_j,V')$ is a strict standard extension and $\phi_{p(j+1),j+1}:\ G(m_{p(j+1)},V)\hookrightarrow G(n_{j+1}, V')$ is a modified standard extension; \\ (b) $p(j)>p(j+1)$, $\phi_{p(j),j}$ is a modified standard extension and $\phi_{p(j+1),j+1}$ is a strict standard extension; \\ (c) $p(j)\le p(j+1)$, $\phi_{p(j),j}$ is a modified standard extension and $\phi_{p(j+1),j+1}$ is a strict standard extension; \\ (d) $p(j)>p(j+1)$, $\phi_{p(j),j}$ is a strict standard extension and $\phi_{p(j+1),j+1}$ is a modified standard extension. (a) Note that the modified standard extension $\phi_{p(j+1), j+1}$ defines a flag of subspaces $W_{j+1}\subset U_{j+1}$ of $V'$ and a surjective linear operator $\varepsilon_{j+1}:U_{j+1}\to V'^{\vee}$ with $\ker \varepsilon_{j+1}=W_{j+1}$, such that \begin{equation}\label{eps modif} \varphi_{p(j+1),j+1}(V_{m_{p(j+1)}})=\varepsilon_{j+1}^{-1} ((V/V_{m_{p(j+1)}})^{\vee}), \end{equation} where $(V/V_{m_{p(j+1)}})^{\vee}$ is naturally considered as a subspace of $V^{\vee}$. Moreover, \begin{equation}\label{descriptn of U,W modif} W_{j+1}=\underset{V_{m_{p(j+1)}}\subset V}{\bigcap} \varphi_{p(j+1),j+1}(V_{m_{p(j+1)}}). \end{equation} Formulas \eqref{eps modif} and \eqref{descriptn of U,W modif} are corollaries of formulas \eqref{eta, eps} and \eqref{descriptn of W}, respectively. Now, given $V_{m_{p(j)}}\in G(m_{p(j)},V)$, we obtain \begin{equation}\label{0=} \{0\}=\underset{V_{m_{p(j+1)}}\supset V_{m_{p(j)}}}{\bigcap} (V/V_{m_{p(j+1)}})^{\vee}, \end{equation} where the intersection is taken in $(V/V_{m_{p(j)}}) ^{\vee}$. Using \eqref{eps modif}-\eqref{0=}, we find $W_{j+1}=\underset{V_{m_{p(j+1)}}\supset V_{m_{p(j)}}} {\bigcap}\phi_{p(j+1),j+1}(V_{m_{p(j+1)}})$. Therefore, \begin{equation}\label{phi subset} \phi_{p(j),j}(V_{m_{p(j)}})\subset W_{j+1}\subset \varphi_{p(j+1),j+1}(V_{m_{p(j+1)}}) \end{equation} for any $V_{m_{p(j+1)}}\in G(m_{p(j+1)},V)$. In view of \eqref{eta, eps} and \eqref{eps modif}, the inclusion \eqref{phi subset} coincides with the inclusion \eqref{eps j in W}. Hence, as in the proof of Theorem \ref{Thm 4.6}, we see that $\phi$ factors through a direct product, contrary to our assumption. This contradiction rules out (a). (b) Given $V_{m_{p(j+1)}}\in G(m_{p(j+1)},V)$, for any $V_{m_{p(j)}}\subset V_{m_{p(j+1)}}$ we have $\phi_{p(j+1),j+1}(V_{m_{p(j)}})\supset\phi_{p(j),j} (V_{m_{p(j+1)}})$. Hence, there is an inclusion $\phi_{p(j),j} (V_{m_{p(j+1)}})\subset\underset{V_{m_{p(j)}}\subset V_{m_{p(j+1)}}}{\bigcap}\phi_{p(j+1),j+1}(V_{m_{p(j)}})$, the right-hand side of which is zero, as it clearly follows from the definition of nonconstant strict standard extension. Thus, $\phi_{p(j),j}(V_{m_{p(j+1)}})=\{0\}$ which is a contradiction, since $V_{n_1}\ne0$. Cases (c) and (d) are reduced to cases (a) and (b), respectively, via the duality isomorphisms $G(n_j,V') \xrightarrow{\simeq} G(\dim V'-n_j,V'^{\vee})$ and $G(n_{j+1}, V')\xrightarrow{\simeq} G(\dim V'-n_{j+1},V'^{\vee})$. Thus, all the cases (a)-(d) lead to a contradiction. The above, together with Lemma \ref{not extend}, implies that either all nonconstant morphisms $\phi_{p(j),j}:\ G(m_{p(j)},V) \hookrightarrow G(n_j,V')$ are strict standard extensions, or that they all are modified standard extensions. In the latter case one considers the morphism $d\circ\phi$, where $d$ is the duality isomorphism. Then by Theorem \ref{Thm 4.6}, $d\circ\phi$ is a strict standard extension, and consequently $\phi$ is a modified standard extension. \end{proof} We now introduce the following condition on a linear embedding $$ \phi:Fl(m_1,...,m_k,V)\hookrightarrow Fl(n_1,..., n_{\tilde{k}},V), $$ or respectively, $$ \phi:FlO(m_1,...,m_k,V)\hookrightarrow FlO(n_1,..., n_{\tilde{k}},V) $$ or $$ \phi:FlS(m_1,...,m_k,V)\hookrightarrow FlS(n_1,..., n_{\tilde{k}},V). $$ (c) \textit{No nonconstant morphism $\phi_{p(j),j}:G(m_i,V)\to G(n_j,V')$ factors through an embedding of a projective subspace into $G(n_j,V')$; in the orthogonal and symplectic cases no nonconstant morphism $\phi_{p(j),j}:X\to Y$ for $X=GO(m_i,V)$ and $Y=GO(n_j,V')$, or $X=GS(m_i,V)$ and $Y=GS(n_j,V')$, factors through a smooth subvariety of $Y$ isomorphic to a grassmannian $G(m,V'')$ or a multidimensional quadric in case $Y=GO(n_j,V')$; in the case where $X=GO(s-1,V) ,\ Y=GO(t-1,V')$ for $\dim V=2s,\ \dim V'=2t$ for $t>s$, this latter condition should also be imposed on the induced morphism $\tilde{\phi}_{p(j),j}:GO(s,V)\to GO(t,V')$.} We say that a linear embedding $\phi$ is \textit{admissible} if it does not factor through any direct product and satisfies condition (c). Our main result in this section is the following. \begin{corollary}\label{Cor 4.4} An admissible linear embedding $\phi$ is a standard extension. \end{corollary} \begin{proof} According to Theorem \ref{Thm 4.3}, all we need to show is that condition (c) implies that every nonconstant morphism $\phi_{p(j),j}$ is a standard extension. For usual grassmannians, this follows directly from \cite[Thm. 1]{PT}, which claims that a linear morphism of grassmannians $\phi_{p(j ),j}:X\to Y$ is a standard extension unless it factors through a projective subspace of $Y$. For isotropic grassmannians, \cite[Thm. 1]{PT} applies only to the case when $\mathrm{Pic}X\simeq\mathrm{Pic}Y\simeq\mathbb{Z}$, and also implies our claim under this assumption. It remains to consider the situation of a linear morphism $\phi_{p(j),j}: G(s-1,V)\to G(t-1,V')$ where $\dim V=2s,\ \dim V'=2t,\ t\ge s$. In this situation, as stated in Section \ref{preliminaries}, we always have a commutative diagram \begin{equation*}\label{diag with theta} \xymatrix{ GO(s-1,V)\ar@{^{(}->}[rrr]^-{\phi_{p(j),j}}\ar[d]^-{\theta} & & & GO(t-1,V')\ar[d]^-{\theta'} \\ GO(s,V) \ar@{^{(}->}[rrr]^-{\tilde{\phi}_{p(j),j}} & & & GO(t,V').} \end{equation*} Here, \cite[Thm. 1]{PT} applies to the linear morphism $\tilde{\phi}:=\tilde{\phi}_{p(j),j}$, implying that it is a standard extension whenever it does not factor through a grassmannian or a multidimensional quadric embedded in $GO(t,V')$. Let this standard extension have the form \begin{equation}\label{lower st ext} V_s\mapsto V_s\oplus W', \end{equation} where $V'=V\oplus W$ is an orthogonal decomposition and $W'$ is a maximal isotropic subspace of $W$. We will show that $\phi:=\phi_{p(j),j}$ is the standard extension \begin{equation}\label{upper st ext} V_{s-1}\mapsto V_{s-1}\oplus W'. \end{equation} For this, consider an arbitrary projective line $\mathbb{P}^1$ on $GO(s,V)$, i.e. a smooth rational curve $C\subset GO(s,V)$ such that $\mathcal{O}_{GO(s,V)}(1)|_C\simeq\mathcal{O}_{ \mathbb{P}^1}(1)$. It is an exercise to see that there exists an isotropic subspace $W_{\mathbb{P}^1}\subset V$ of dimension $p-2$, such that the restriction $E:=\mathcal{S}| _{\mathbb{P}^1}$ of the tautological bundle $\mathcal{S}$ on $GO(s,V)$ is isomorphic to $2\mathcal{O}_{\mathbb{P}^1}(-1) \oplus W_{\mathbb{P}^1}\otimes\mathcal{O}_{\mathbb{P}^1} $. Hence, by \eqref{lower st ext}, we have \begin{equation}\label{restr S'} E':=\phi^*\mathcal{S}'|_{\mathbb{P}^1}\simeq 2\mathcal{O}_{\mathbb{P}^1}(-1)\oplus(W_{\mathbb{P}^1}\oplus W')\otimes\mathcal{O}_{\mathbb{P}^1}, \end{equation} where $\mathcal{S}'$ is the tautological bundle on $GO(t-1,V')$. For any point $x\in\mathbb{P}^1$, consider the projective spaces $\theta^{-1}(x)=\mathbb{P}(E^{\vee}|_t)$ and $\theta'^{-1}(\tilde{\phi}(x))=\mathbb{P}((E')^{\vee}|_t)$. By definition, $\phi|_{\theta^{-1}(x)}:\theta^{-1}(x)\to \theta'^{-1}(\tilde{\phi}(x))$ is a linear embedding of projective spaces, hence it has the form \begin{equation}\label{new st extn} V_{s-1}\mapsto V_{s-1}\oplus W''(x) \end{equation} for some unique isotropic vector subspace $W''(x)\subset V'$. Indeed, $W''(x)=\underset{V_{s-1} \in\theta^{-1}(x)}{\bigcap} \varphi(V_{s-1})$ (see \eqref{descriptn of W}). Moreover, by construction, $W'':=\{(x,W''(x))\}_{x\in\mathbb{P}^1}$ is a vector subbundle of $E'$, and the condition that $\phi^*\mathcal{O}_{GO(t-1,V')} (1)\cong\mathcal{O}_{GO(s-1,V)}(1)$ (see Example \ref{lin emb isotr grass}) implies \begin{equation}\label{det W''} \det W''\cong\mathcal{O}_{\mathbb{P}^1}. \end{equation} Consider the composition of morphisms of sheaves: $f:W''\stackrel{i}{\hookrightarrow}E'\stackrel{pr}{\to} 2\mathcal{O}_{\mathbb{P}^1}(-1)$ where $i$ is the above mentioned monomorphism and $pr$ is the canonical projection defined by \eqref{restr S'}. If $f$ is a nonzero morphism, it follows from \eqref{det W''} and Grothendieck's Theorem that $W''$ contains a direct summand $\mathcal{O}_{\mathbb{P}^1} (a)$ for some $a>0$. But this contradicts to \eqref{restr S'} since $i$ is a monomorphism. Hence, $f=0$, and by \eqref{restr S'}, $W''$ is a subbundle of the trivial bundle $(W_{\mathbb{P}^1}\oplus W')\otimes\mathcal{O}_{\mathbb{P}^1}$. Therefore, in view of \eqref{det W''}, $W''$ is itself a trivial bundle. This means that the space $W''(x)$ does not depend on $x\in\mathbb{P}^1$, but possibly depends only on the choice of projective line $\mathbb{P}^1$. We can set $W''(x)=W''_{\mathbb{P}^1}$. Then \begin{equation}\label{W'' subset...} W''_{\mathbb{P}^1}\subset W_{\mathbb{P}^1}\oplus W'. \end{equation} Pick a point $x_0\in\mathbb{P}^1$, so that $W''(x_0)=W'' _{\mathbb{P}^1}$. Next, pick another line $\mathbb{P}'^1$ through $x_0$, distinct from $\mathbb{P}^1$. Then $W''_{\mathbb{P}^1}=W''_{\mathbb{P}'^1}$. Since, as one easily checks, any two points in $GO(s,V)$ can be connected by a chain of projective lines, we conclude that $W''_{\mathbb{P}^1}$ does not depend on the line $\mathbb{P}^1$. We therefore denote this space by $W''_0$, and the inclusion \eqref{W'' subset...} can be rewritten as \begin{equation}\label{W'' in} W''_0\subset W_{\mathbb{P}^1}\oplus W',\ \ \ \ \ \ \ \mathbb{P}^1\subset GO(s,V). \end{equation} Now one easily observes that $\underset{\mathbb{P}^1\subset GO(s,V)}{\bigcap}W_{\mathbb{P}^1}=\{0\}$. Hence, \eqref{W'' in} implies $W_0''=\underset{\mathbb{P}^1\subset GO(s,V)}{\bigcap} (W_{\mathbb{P}^1}\oplus W')=W'$. It follows that the linear embedding $\phi$ in \eqref{new st extn} is $V_{s-1}\mapsto V_{s-1}\oplus W'$, i.e., $\phi$ coincides with \eqref{upper st ext} as claimed. \end{proof} Corollary \ref{Cor 4.4} provides a sufficient condition, in terms of pure algebraic geometry, for a linear embedding of flag varieties, or varieties of isotropic flags, to be a standard extension. \vspace{1cm} \section{Admissible direct limits of linear embeddings of flag varieties are isomorphic to ind-varieties of generalized flags} \label{ind-var} \vspace{5mm} We start by recalling the notions of generalized flag and ind-variety of generalized flags introduced in \cite[Section 5]{DP}. Let $V$ be an arbitrary vector space. A {\it chain of subspaces in $V$} is a set $\mathcal{C}$ of pairwise distinct subspaces of $V$ such that for any pair $F$, $H\in\mathcal{C} $, one has either $F\subset H$ or $H\subset F$. Every chain of subspaces $\mathcal{C}$ is linearly ordered by inclusion. Given a chain $\mathcal{C}$, we denote by $\mathcal{C}'$ (respectively, by $\mathcal{C}''$) the subchain of $\mathcal{C}$ that consists of all subspaces $C \in \mathcal{C}$ which have an immediate successor (respectively, an immediate predecessor) with respect to this ordering. A {\it generalized flag in $V$} is a chain of subspaces $\mathcal{F}$ that satisfies the following conditions: \\ (i) each $F\in\mathcal{F}$ has an immediate successor or an immediate predecessor, i.e. $\mathcal{F}=\mathcal{F}'\cup \mathcal{F}''$; \\ (ii) $V\backslash\{0\}=\cup_{F'\in\mathcal{F}'} F''\backslash F'$, where $F''\in\mathcal{F}''$ is the immediate successor of $F'\in \mathcal{F}'$. In what follows, we assume that $V$ is a countable-dimensional vector space with basis $E=\{e_n\}_{n\in\mathbb{Z}_{>0}}$. A generalized flag $\mathcal{F}$ in $V$ is \textit{compatible with the basis} $E$ if for every $F\in\mathcal{F}$ the set $F\cap E$ is a basis of $F$. We say that a generalized flag $\mathcal{F}$ is {\it weakly compatible with $E$}, if $\mathcal{F}$ is compatible with some basis $L$ of $V$ such that $E \backslash (E \cap L)$ is a finite set. \begin{example} Let $V=\mathrm{Span}E$ where $E=\{e_n\}_{n\in \mathbb{Z}_{>0}}$. (i) Any finite chain $(0\subset F_1\subset...\subset F_k \subset V)$ of coordinate subspaces (i. e. subspaces $F_i \subset V$ satisfying $F_i= \mathrm{Span}\{F_i\cap E\}$ for $1\le i\le k$) is a generalized flag compatible with the basis $E$. If $\dim F_i< \infty$ for $1\le i\le k$, and if one drops the condition that all $F_i$ are coordinate subspaces, then the chain $(0\subset F_1\subset...\subset F_k\subset V)$ is a generalized flag weakly compatible with $E$. (ii) Fix a bijection $\mathbb{Z}_{>0}=\mathbb{Z}_{>0}\sqcup \mathbb{Z}_{<0}$, and let $\prec$ denote the linear order on $\mathbb{Z}_{>0}$, induced by the obvious linear order on $\mathbb{Z}_{>0}\sqcup\mathbb{Z}_{<0}$ in which all elements of $\mathbb{Z}_{<0}$ are larger than all elements of $\mathbb{Z}_{>0}$. Then the chain $\{0,F_j,V\}_{j\in \mathbb{Z}_{>0}}$, where $F_j=\{\mathrm{Span}\{e_i\} _{i\preccurlyeq j}\}$, is a generalized flag compatible with $E$. (iii) Fix a bijection $\mathbb{Z}_{>0}=\mathbb{Q}_l\sqcup \mathbb{Q}_r$, where $\mathbb{Q}_l=\mathbb{Q}=\mathbb{Q}_r$, and consider the following linear order on $\mathbb{Q}_l\sqcup \mathbb{Q}_r$: \ $j\prec t\ \Leftrightarrow\ j\in\mathbb{Q}_l \sqcup\mathbb{Q}_r,\ t\in\mathbb{Q}_l\sqcup \mathbb{Q}_r, \ j<t$, or $j=t,\ j\in\mathbb{Q}_l,\ t\in\mathbb{Q}_r$. Then the chain $\{F'_j,F''_j\}_{j\in\mathbb{Q}_l}$, where $F'_j=\mathrm{Span}\{e_k\}_{k\prec j}$, $F''_j=\mathrm{Span} \{e_k\}_{k\preccurlyeq j}$, is a generalized flag compatible with $E$. \end{example} We define two generalized flags $\mathcal{F}$ and $\mathcal{G}$ in $V$ to be {\it $E$--commensurable} if both $\mathcal{F}$ and $\mathcal{G}$ are weakly compatible with $E$ and there exists an inclusion preserving bijection $\varphi: \mathcal{F}\to\mathcal{G}$ and a finite-dimensional subspace $U \subset V$, such that for every $F \in \mathcal{F}$ \begin{equation*}\label{E-commens} F\subset\varphi(F)+U,\ \ \ \varphi(F)\subset F+U,\ \ \ \dim(F\cap U)=\dim(\varphi(F)\cap U). \end{equation*} Let \begin{equation*}\label{X=F(...)} \mathbf{X}=\mathbf{Fl}(\mathcal{F},E,V) \end{equation*} denote the set of all generalized flags in $V$ that are $E$-commensurable with $\mathcal{F}$. We now explain that $\mathbf{X}$ has a natural ind-variety structure. Let $V'_n:=\mathrm{Span} \{e_j|j\le n\}$. Then the intersection $\mathcal{F}\cap V'_n$ is a flag in $V'_n$, and let this flag have type $0<m'_{n,1} <...<m'_{n,k_n}<n$ for $k_n\le n-1$. Since $\dim V'_{n+1}=\dim V'_n+1=n+1$, if we set $W'_n:=\mathrm{Span}\{e_{n+1}\}$, we have $V'_{n+1}=V'_n\oplus W'_n$ and there is a standard extension $i_n:Fl(m'_{n,1},...,m'_{n,k_n},V'_n)\hookrightarrow Fl(n'_{n+1,1},...,n'_{n+1,k_{n+1}},V'_{n+1})$ given by formulas \eqref{example of st ext} or \eqref{example of st ext2} in Example 3.4 (where we had no need to use as many subscripts as well as primes). Note that this standard extension $i_n$ is determined by the two types of flags $(m'_{n,1},...,m'_{n,k_n})$ and $(n'_{n+1,1},..,n'_{n+1,k_{n+1}})$, and by the choice of $W'_{n+1}$. In \cite{DP} it is shown that $\mathbf{Fl}(\mathcal{F},E,V)$ is naturally identified with the direct limit $$ \varinjlim Fl(m'_{n,1},...,m'_{n,k_n},V'_n) $$ of the embeddings $i_n$. In particular, this equips $\mathbf{Fl}(\mathcal{F},E,V)$ with the structure of an ind-variety. Let's now consider the case when $V$ is endowed a nondegenerate symmetric or symplectic bilinear form $(\ ,\ )$. Here we assume that either the basis $E$ is isotropic and is enumerated as $\{e_n,e^n \}_{n\in\mathbb{Z}_{>0}}$ where $(e_n,e^n)=1$ for $n\in \mathbb{Z}_{>0}$, or that $E$ is enumerated as $\{e_n,e_0,e^n \}_{n\in \mathbb{Z}_{>0}}$ where $e_n$ and $e^n$ are isotropic vectors satisfying $(e_n,e^n)=1$ for $n\in\mathbb{Z}_{>0}$ and $e_0$ satisfies $(e_0,e_n)=(e_0,e^n)=0$, $(e_0,e_0)=1$. This latter enumeration of $E$ is possible only in the case of a symmetric form. We define a generalized flag $\mathcal{F}$ to be \textit{isotropic} if it consists of isotropic and coisotropic subspaces (a subspace $F$ is \textit{coisotropic} if $F^{\bot}$ is isotropic) and is invariant under taking orthogonal complement. In the current case, where $\dim V=\infty$, this definition is more convenient for our purposes than the consideration of "purely isotropic" flags as in Sections \ref{preliminaries}, \ref{linear embed} and \ref{more linear embed}. Note that an isotropic generalized flag is determined by its subchain of isotropic spaces. \begin{example}\label{Example 5.2} Consider the case where $V$ is endowed with a nondegenerate symmetric form and the basis of $V$ is enumerated as $\{e_n, e_0,e^n\}_{n\in\mathbb{Z}_{>0}}$ as above. Set $F^l_j= \mathrm{Span}\{e_n\}_{n>j,j\ge0}$, $F^r_j=(F^l_j)^{\bot}$. Then $F^l_j\supset F^l_k,\ F^r_j\subset F^r_k,\ F^l_j\subset F^r_k$ for $k\ge j$, and $\{F^l_j,F^r_j\}_{j\in\mathbb{Z} _{\ge0}}$ is a maximal isotropic generalized flag compatible with $E$. \end{example} By $\mathbf{FlO}(\mathcal{F},E,V)$, or respectively $\mathbf{FlS}(\mathcal{F},E,V)$, we denote the set of all generalized flags which are $E$-commensurable with a fixed isotropic flag $\mathcal{F}$ compatible with $E$. To define an ind-variety structure on $\mathbf{FlO} (\mathcal{F},E,V)$ or $\mathbf{FlS}(\mathcal{F},E,V)$, set $V'_n=\mathrm{Span}\{e_j,e^j\}_{j\le n}$ or respectively $V'_n=\mathrm{Span}\{e_j,e_0,e^j\}_{j\le n}$. Then $\mathcal{F}\cap V'_n$ has an isotropic subflag of type $0<m'_{n,1}<...<m'_{n,k_n}\le [\frac{n}{2}]$, and there is a standard extension $$ \psi_n:\ FlO(m'_{n,1},...,m'_{n,k_n},V'_n)\hookrightarrow FlO(m'_{n+1,1},...,m'_{n+1,k_{n+1}},V'_n) $$ or $$ \psi_n:\ FlS(m'_{n,1},...,m'_{n,k_n},V'_n)\hookrightarrow FlS(m'_{n+1,1},...,m'_{n+1,k_{n+1}},V'_{n+1}), $$ determined uniquely by the isotropic 1-dimensional subspace $W_n=\mathrm{Span}\{e_{n+1}\}$. One can show that the direct limit of the embeddings $\psi_n$ is identified with $\mathbf{FlO}(\mathcal{F},E,V)$, or respectively $\mathbf{FlS} (\mathcal{F},E,V)$, and hence $\mathbf{FlO}(\mathcal{F},E,V)$ and $\mathbf{FlS}(\mathcal{F},E,V)$ are ind-varieties \cite{DP}. Next, we will relate an arbitrary direct limit of strict standard extensions to the ind-varieties $\mathbf{Fl} (\mathcal{F},E,V)$, $\mathbf{FlO}(\mathcal{F},E,V)$, or $\mathbf{FlS}(\mathcal{F},E,V)$. First, consider a chain of strict standard extensions \begin{equation}\label{phi_n's} \phi_N:Fl(m_{N,1},..., m_{N,k_N},V_N)\hookrightarrow Fl( m_{N+1,1},...,m_{N+1,k_{N+1}},V_{N+1}) \end{equation} for some choice of vector spaces $V_N$, $\dim V_{N+1}>\dim V_N$ for $N\in\mathbb{Z}_{>0}$. Then, according to Proposition \ref{descrn of st ext}, we may choose vector spaces $\widehat{W}_N$, together with isomorphisms \begin{equation*}\label{Vn in Vn+1} V_{N+1}=V_N\oplus \widehat{W}_N, \end{equation*} and flags in $\widehat{W}_N$ \begin{equation*}\label{flag for n} W_{N,1}\subset...\subset W_{N,k_N}\subset\widehat{W}_N, \end{equation*} such that each $\phi_N$ is given by: \begin{equation*}\label{descr of phi n's} \phi_N(0\subset V_{m_{N,1}}\subset... \subset V_{m_{N,k_N}}\subset V_N)=\\ (0\subset V_{m_{N,1}}\oplus W_{N,1}\subset ...\subset V_{m_{N,k_N}}\oplus W_{N,k_N}\subset V_{N+1}). \end{equation*} Set \begin{equation*}\label{V=lim Vn} V:=\underset{\to}\lim V_N. \end{equation*} Our aim is to define a basis $E$ of $V$ and a generalized flag $\mathcal{\underline{F}}$ compatible with $E$, so that the direct limit of the strict standard extensions $\phi_N$ can be identified with $\mathbf{Fl}(\mathcal{\underline{F}},E,V)$. Fix a flag $F_1=(0\subset V_{1,1}\subset...\subset V_{1,k_1} \subset V_1)\in Fl(m_{1,1},...,m_{1,k_1},V_1)$. Choose a basis \begin{equation*}\label{basis E} E=\{e_{\alpha}\}_{\alpha\in\mathbb{Z}_{>0}} \end{equation*} of $V$ such that, for all subspaces $T$ of $V$ of the form $V_{1,1},...,V_{1,k_1}$ and $W_{N,j}$ for $N$ and $j$, the set $T\cap E$ is a basis of $T$. Consider the following equivalence relation $\sim$ on the set $E$. We write \begin{equation*}\label{equiv} e_{\alpha}\sim e_{\tilde{\alpha}} \end{equation*} if there exists $N_{\alpha}\in\mathbb{Z}_{>0}$ such that, for any $N\ge N_{\alpha}$, there is no space of the flag $\phi_N\circ\phi_{N-1}\circ...\circ\phi_1(F_1)$ containing $e_{\alpha}$ but not $e_{\tilde{\alpha}}$, or vice versa. Using the fact that all embeddings $\phi_N$ are strict standard extensions, one checks that $\sim$ is an equivalence relation. Denote by $[e_{\alpha}]$ the equivalence class of the vector $e_{\alpha}$. Next, we claim that, by construction, the set $A$ of equivalence classes $[e_{\alpha}]$ is linearly ordered, and we will denote this linear ordering by the symbol $\prec$. Indeed, let $[e_{\alpha}]\ne[e_{\beta}]$. For $n\ge\max\{N_{\alpha},N_{\beta}\}$, consider the flag $\phi_N\circ\phi_{N-1}\circ...\circ\phi_1(F_1)$ and take its smallest subspaces containing respectively $e_{\alpha}$ and $e_{\beta}$. Since $[e_{\alpha}]\ne [e_{\beta}]$, it follows that these spaces are not equal. By definition, we have $[e_{\alpha}]\prec[e_{\beta}]$ if the smallest space of the flag $\phi_N\circ\phi_{N-1}\circ...\circ \phi_1(F_1)$ containing $e_{\alpha}$ is smaller than the smallest space of the same flag containing $e_{\beta}$. Finally, we define a generalized flag $\mathcal{\underline{F}}$, compatible with the basis $E$, and determined by the above order on $E$. For this, we associate two subspaces of $V$ to any equivalence class $a=[e_{\alpha}]$ : \begin{equation}\label{flags F',F''} F'_a=\mathrm{Span}\{e_{\beta}\ |\ [e_{\beta}]\prec a\}, \ \ \ \ \ F''_a=\mathrm{Span}\{e_{\beta}\ |\ [e_{\beta}]\preccurlyeq a\}. \end{equation} Then the set of vector subspaces of $V$ \begin{equation}\label{flag cal F0} \mathcal{\underline{F}}=\{F'_a,F''_a\}_{a\in A} \end{equation} is easily seen to be a generalized flag in $V$ compatible with $E$. If, instead of \eqref{phi_n's}, we consider standard extensions \begin{equation}\label{Z} \psi_N:\ FlO(m_{N,1},...,m_{N,k_N},V_N)\hookrightarrow FlO(m_{N+1,1},...,m_{N+1,k_{N+1}},V_{N+1}) \end{equation} or \begin{equation}\label{T} \psi_N:\ FlS(m_{N,1},...,m_{N,k_N},V_N)\hookrightarrow FlS(m_{N+1,1},...,m_{N+1,k_{N+1}},V_{N+1}), \end{equation} a similar construction of a relevant basis $E$ goes through. First of all, in the case of \eqref{Z}, for our purposes it suffices to assume that that the dimension of all spaces $V_N$ are simultaneously odd or even. We require $E$ to have the form $\{e_n,e_0,e^n\}_{n\in\mathbb{Z}_{>0}}$ in the odd case, and the form $\{e_n,e^n\}_{n\in\mathbb{Z}_{>0}}$ in the even case. This latter form applies also to the case of \eqref{T}. In all cases, $E$ has to be chosen by the same condition that all subspaces of the form $V_{1,1},...,V_{1,k_1}$ and $W_{N,k_j}$ for $N\in\mathbb{Z}_{>0}$ are generated by subsets of $E$. Next, in order to define a linear order on $E$, one applies to the vectors $e_n$ the procedure outlined above, and then sets $e^k\prec e^l\ \Leftrightarrow\ e_l\prec e_k$. Finally, whenever there is a vector $e_0$ one puts $e_n\prec e_0\prec e^k$ for any $k,n\in\mathbb{Z}_{>0}$. Then the generalized flag $\mathcal{\underline{F}}$ determined by formulas \eqref{flags F',F''} and \eqref{flag cal F0} is isotropic (in the sense of the definition of the beginning of this section) and an ind-variety $\mathbf{FlO}(\mathcal{\underline{F}},E,V)$ , or respectively $\mathbf{FlS}(\mathcal{\underline{F}},E,V)$ is well defined. We are now ready for the following theorem. \begin{theorem}\label{main thm} There is an isomorphism of ind-varieties $$ \varinjlim Fl(m_{N,1},...,m_{N,k_N},V_N)\simeq \mathbf{Fl}(\mathcal{\underline{F}},E,V). $$ Similarly, in the orthogonal and symplectic cases, there are isomorphisms of ind-varieties $$ \varinjlim FlO(m_{N,1},...,m_{N,k_N},V_N)\simeq \mathbf{Fl}(\mathcal{\underline{F}},E,V), $$ $$ \varinjlim FlS(m_{N,1},...,m_{N,k_N},V_N)\simeq \mathbf{Fl}(\mathcal{\underline{F}},E,V). $$ \end{theorem} \begin{proof} We consider only the case of ordinary flag varieties, and leave the other cases to the reader. Note first that $(m_{N,1},...,m_{N,k_N})$ is the type of the flag $\mathcal{\underline{F}}\cap V_N$, so that $\mathbf{Fl} (\mathcal{\underline{F}},E,V)=\varinjlim Fl(m_{N,1},..., m_{N,k_N},V_N)$ where the direct limit is taken with respect to the embeddings $$ i_{\dim V_{N+1}-1}\circ...\circ i_{\dim V_N}:\ Fl(m_{N,1},..., m_{N,k_N},V_N)\hookrightarrow Fl(m_{N+1,1},..., m_{N+1,k_{N+1}},V_{N+1}). $$ The embeddings $i_n$ were introduced in the first part of this section, and are given by formulas \eqref{example of st ext} and \eqref{example of st ext2}, respectively. However, we claim that our fixed standard extension $\phi_N$ equals the composition $i_{\dim V_{N+1}-1}\circ...\circ i_{\dim V_N}$. This follows from an iterated application of Lemma \ref{Lemma 3.7} to the decompositions $$ V_{N+1}=V'_{\dim V_{N+1}-1}\oplus\mathrm{Span}\{e_{\dim V_{N+1}} \}, $$ $$ \ V'_{\dim V_{N+1}-1}=V'_{\dim V_{N+1}-2}\oplus\mathrm{Span} \{e_{\dim V_{N+1}-1}\},..., $$ $$V'_{\dim V_N+1}=V_N\oplus\mathrm{Span}\{e_{\dim V_N+1}\}, $$ and from the observation that the corresponding standard extensions $$ Fl(m_{n,1},...,m_{n,k_n},V'_n)\hookrightarrow Fl(m_{n+1,1},..., m_{n+1,k_{n+1}},V'_{n+1}) $$ arising in this way, are determined simply by the splitting $V'_{n+1}=V'_n\oplus\mathrm{Span}\{e_{n+1}\}$. Since the standard extension $i_n$ is determined by the same decomposition, the statement follows. \end{proof} The following corollary can be considered as the main result of this paper. \begin{corollary}\label{cor 5.3} The direct limit of any admissible sequence of linear embeddings, $\varinjlim Fl(m_{N,1},...m_{N,k_N},V_N)$, $\varinjlim FlO(m_{N,1},...m_{N,k_N},V_N)$, or $\varinjlim FlS(m_{N,1},...m_{N,k_N},V_N)$, is a homogeneous ind-variety for the group $SL(\infty),\ O(\infty)$ or $Sp(\infty)$, respectively. \end{corollary} The claim of Corollary \ref{cor 5.3} can be derived more directly from Corollary \ref{Cor 4.4} by showing that any direct limit of standard extensions is a homogeneous ind-variety, but Theorem \ref{main thm} provides an explicit description of such a direct limit as an appropriate ind-variety of generalized flags. We should also point out that homogeneous ind-varieties of the ind-groups $GL(\infty),$ $SL(\infty),$ $O(\infty),$ $Sp(\infty)$ have been studied in papers preceding \cite{DP}, see \cite{DPW} and the references therein. \vspace{1cm} \section{Appendix}\label{special} \vspace{5mm} In this appendix, we construct ind-varieties which are not isomorphic to ind-varieties of generalized flags, but nevertheless are direct limits of linear embeddings of flag varieties. Here we use the notation $\mathbb{P}(V)$ also for a countable-dimensional vector space. $\mathbb{P}(V)$ is the ind-variety of 1-dimensional subspaces of $V$. We also write $\mathbb{P}^{\infty}$ instead of $\mathbb{P}(V)$ when we do not need to specify $V$. First, consider the following chain of linear embeddings $$ ...\hookrightarrow Fl(1,2^n-1,V_n)\overset{k_n} {\hookrightarrow}G(1,V_n)\times G(2^n-1,V_n)\overset{j_n} {\hookrightarrow}Fl(1,2^{n+1}-1,V_n\oplus V_n)\overset{k_{n+1}} {\hookrightarrow} $$ $$ \overset{k_{n+1}}{\hookrightarrow}G(1,V_n\oplus V_n) \times G(2^{n+1}-1,V_n\oplus V_n)\hookrightarrow...\ , $$ where $\dim V_n=2^n,\ k_n$ and $k_{n+1}$ are the canonical embeddings, and $j_n(V_1,V_{2^n-1})=(V_1\subset V\oplus0\subset V\oplus V_{2^n-1})$ for subspaces $V_1,\ V_{2^n-1}\subset V$ of respective dimensions 1 and $2^n-1$. Clearly, the embedding $$ j_n\circ k_n:Fl(1,2^n-1,V_n)\hookrightarrow Fl(1,2^{n+1}-1,V_n) $$ is linear but does not satisfy condition (b) of Theorem \ref{Thm 4.3} as it factors through the embedding $k_n$. The direct limit $\varinjlim Fl(1,2^n-1,V_n)$ is isomorphic as an ind-variety to the direct limit of embeddings $$ G(1,V_n)\times G(2^n-1,V_n)\overset{k_{n+1}\circ j_n} {\hookrightarrow}G(1,V_n\oplus V_n)\times G(2^n-1,V_n\oplus V_n), $$ which is easily checked to be isomorphic to the direct product $\mathbb{P}(V)\times\mathbb{P}(V)$ for a countable-dimensional vector space $V$. The ind-variety $\mathbb{P}(V)\times \mathbb{P}(V)$ is not isomorphic to an ind-variety of generalized flags. Next, we will give a more interesting example in which condition (c) is not satisfied. More precisely, we will construct a linear embedding $\phi: Fl(m_1,m_2,V) \hookrightarrow Fl(n_1,n_2,V')$ that will have the property that $p(1)=1,\ p(2)=2,$ $\phi_{2,2}:G(m_2,V)\to G(n_2,V')$ is a standard extension, but $\phi_{1,1}:G(m_1,V)\to G(n_1,V')$ factors through a projective subspace of $G(n_1,V')$. Let $3\le\dim V<\infty$, fix positive integers $m_1,\ m_2,\ 1< m_1< m_2<\dim V,$ and let $V^0$ be a subspace of $V$ of dimension $\dim V-m_1+1$. Consider the rational morphism \begin{equation*}\label{linear prn p} \gamma:G(m_1,V)\dasharrow\mathbb{P}(V^0),\ V_{m_1}\mapsto V_{m_1}\cap V^0. \end{equation*} Assume $G(m_1,V)$ is embedded into $\mathbb{P}(\wedge^{m_1}V)$ via the Pl\"ucker embedding, and let $Y:=\{V_{m_1}\in G(m_1,V )\ |\ \dim(V_{m_1}\cap V^0)\ge2\}$. A standard computation in linear algebra shows that\\ (i) there exists a subspace $W\subset\wedge^{m_1}V$ of codimension $\dim V-m_1+1$, such that $Y=G(m_1,V)\cap \mathbb{P}(W)$;\\ (ii) there is an isomorphism $g:(\wedge^{m_1}V)/W \xrightarrow{\simeq}V^0$ satisfying \begin{equation}\label{p vs g} \gamma(V_{m_1})=g(\wedge^{m_1}V_{m_1}+W) \end{equation} (in particular, this implies that $\gamma$ is regular on $G(m_1,V)\setminus Y$);\\ (iii) there exists a vector space $U$ containing $\wedge^{m_1} V$ as a subspace, together with a surjective operator $\varepsilon:U\twoheadrightarrow V$ with $\ker\varepsilon=W$. In addition, we may suppose that $m_1$ is large enough so that there exists a subspace $Z$ of $W$ such that the morphism $\phi':\ G(m_1,V)\to \mathbb{P}((\wedge^{m_1} V)/Z),\ V_{m_1}\mapsto\wedge^{m_1}V_{m_1}+Z$ is an embedding. Set $V':=U$, $n_1=\dim Z+1,\ n_2=\dim W+m_2$. The inclusion $\wedge^{m_1}V\subset V'$ yields an embedding $j:\mathbb{P}(( \wedge^{m_1}V)/Z)\hookrightarrow G(n_1,V'),\ v+Z\mapsto \mathrm{Span}\{v+Z\}$. Define $\varphi_{1,1}: G(m_1,V)\to G(n_1,V')$ as the composition $j\circ\phi'$, and let $\varphi_{2,2}:G(m_2,V)\to G(n_2,V')$ be the standard extension defined by the flag $(W\subset U)$. We show now that, given a flag $(0\subset V_{m_1}\subset V_{m_2}\subset V)$, one has $\varphi_{1,1}(V_{m_1})\subset\varphi_{2,2}(V_{m_2})$, and hence there is a well-defined embedding $$\varphi:\ Fl(m_1,m_2,V)\hookrightarrow Fl(n_1,n_2,V'),\ (V_{m_1}\subset V_{m_2})\mapsto(\varphi_{1,1}(V_{m_1})\subset \varphi_{2,2}(V_{m_2})). $$ Indeed, in view of \eqref{p vs g}, the rational morphism $\gamma$ decomposes as $$ \gamma:\ G(m_1,V)\overset{\phi'}{\hookrightarrow}\mathbb{P} ((\wedge^{m_1}V)/Z)\overset{q}{\dasharrow}\mathbb{P}( (\wedge^{m_1}V)/W)\xrightarrow[\simeq]{g}\mathbb{P}(V^0),\\ $$ $$ V_{m_1}\overset{\phi'}{\mapsto}\wedge^{m_1}V_{m_1}+Z\overset {q}{\mapsto}\wedge^{m_1}V_{m_1}+W\overset{g} {\mapsto} V_{m_1}\cap V^0, $$ where $q$ is a rational surjective morphism. If $V_{m_1}\cap V^0=:V_1$ is a 1-dimensional space, i.e. if $q$ is regular at the point $\wedge^{m_1}V_{m_1}+Z \in\mathbb{P}((\wedge^{m_1}V)/Z)$, then the inclusion $V_{m_1}\subset V_{m_2}$ implies $V_1\subset V_{m_2}$. Hence, $\phi_{1,1}(V_{m_1})=\wedge^{m_1}V_{m_1}+Z \subset\wedge^{m_1}V_{m_1}+W=\varepsilon^{-1}(V_1)\subset \varepsilon^{-1}(V_{m_2})=\phi_{2,2}(V_{m_2})$. In the remaining case when $\dim(V_{m_1}\cap V^0)\ge2$, we have $\wedge^{m_1}V_{m_1}\subset W$ by property (i), and therefore $\phi_{1,1}(V_{m_1})=\wedge^{m_1}V_{m_1}+Z\subset W\subset \varepsilon^{-1}(V_{m_2})=\phi_{2,2}(V_{m_2})$. Finally, we have the following proposition. \begin{proposition} Let $\{\phi_k:\ Fl(m_{k,1},m_{k,2},V_k)\to Fl(m_{k+1,1},m_{k+1, 2},V_{k+1})\}_{k\ge1}$ be a chain of embeddings as constructed above. The ind-variety $\mathbf{X}$ obtained as the direct limit of this chain is not isomorphic to an ind-variety of generalized flags. \end{proposition} \begin{proof} Assume to the contrary that $\mathbf{X}$ is isomorphic to $\mathbf{Y}$ for some ind-variety of generalized flags $\mathbf{Y}$. Since the embeddings $\phi_k$ are linear, it follows that $\mathrm{Pic}\mathbf{X}\simeq\mathbb{Z}\times \mathbb{Z}$. Therefore $\mathrm{Pic}\mathbf{Y}\simeq\mathbb{Z} \times\mathbb{Z}$, and consequently, $\mathbf{Y}$ is isomorphic to $\mathbf{Fl}(F',E',V')$ for some countable-dimensional vector space $V'$, some basis $E'$ of $V'$, and some flag $F'=(F'_1\subset F'_2)$ in $V'$ of length 2. Since the morphisms $(\phi_k)_{1,1}:\ \ G(m_{k,1},V_k)\to G(m_{k+1,1}, V_{k+1})$ factor through projective spaces, the ind-variety $\mathbf{X}$ projects onto $\mathbb{P}^{\infty}$ in a way that the line bundle $\mathcal{O}_{\mathbf{X}}(1,0)$ is trivial along the fibers of the projection. Therefore, we infer that $\dim F'_1=1$ or $\mathrm{codim}_{V'}F'_2=1$. This follows from the fact that the ind-variety $\mathbb{P}^{\infty}$ is not isomorphic to any ind-grassmannian $\mathbf{Fl}(F,E',V')$, where $F$ is a single subspace with $\dim F\ge2$ and $\mathrm{codim}_{V'}F'\ne1$, see \cite[Thm. 2]{PT}. Consequently, the flag $F'=(F'_1\subset F'_2)$ can be chosen with $\dim F'_1=1$ (in the case where $\mathrm{codim}_{V'}F'_2 =1$ one replaces $V'$ by its restricted dual space defined by the basis $E'$). The standard extensions $(\phi_k)_{2,2}:G(m_{k,2},V_k)\to G( m_{k+1,2},V_{k+1})$ allow to identify $\varinjlim G(m_{k,2},V _k)$ with an ind-grassmannian $\mathbf{Fl}(F_{\infty},E,V)$, where $F_{\infty}$ is a subspace of $V=\varinjlim V_k$ and $E$ is an appropriate basis of $V$. Moreover, we have $\dim F_{\infty}=\infty=\mathrm{codim}_{V}F_{\infty}$, as the construction of $\phi_k$ shows that $\lim\limits_{k\to\infty} m_{k,2}=\infty=\lim\limits_{k\to\infty}(\dim V_k-m_{k,2})$. After identifying the triples $(F_{\infty},E,V)$ and $(F'_2,E', V')$, we obtain a commutative diagram $$ \xymatrix{ \mathbf{X}\ar[dr]_-{\pi_{\mathbf{X}}}& \mathbf{Fl}(F,E,V)\ar[d]^-{\pi}\ar[l]^-{\sim}_-{\sigma}\\ & \mathbf{Fl}({F_{\infty}},E,V),} $$ where $\pi$ is the natural projection and $\sigma$ is an isomorphism of ind-varieties. The fibers of both projections $\pi_{\mathbf{X}}$ and $\pi$ are isomorphic to $\mathbb{P} ^{\infty}$. We will show now that the existence of the isomorphism $\mathbf{X}\xleftarrow[\sim]{\sigma}\mathbf{Fl}(F,E,V)$ is contradictory. Recall that the group $GL(E,V)$ of invertible finitary linear operators defined by $E$ (i.e. the group of invertible linear generators on $V$ each of which fixes all but finitely many elements of $E$) acts on $\mathbf{Fl}(F,E,V) $ and $\mathbf{Fl}({F_{\infty}},E,V)$, and the line bundle $\mathcal{O}(1,0):=\sigma^*\mathcal{O}_{\mathbf{X}}(1,0)$ on $\mathbf{Fl}(F,E,V)$ admits a $GL(E,V)$-linearization. This linearization is unique when restricted to $SL(E,V)$. If we compute the $SL(E,V)$-module $\Gamma:=H^0(\mathbf{Fl}(F,E,V), \mathcal{O}(1,0))$, we see that $\Gamma\simeq\varprojlim H^0(\pi_{k*}(\mathcal{O}(1,0)|_{Fl(1,m_{k,2},V_k)}))$, where here $\pi_k: Fl(1,m_{k,2},V_k)\to Gr(m_{k,2},V_k)$ denote the natural projections. Consequently, $$ \Gamma\simeq\varprojlim V_k^*\simeq V^*. $$ On the other hand, since $\sigma^*$ induces an $SL(E,V)$-linearization on $\mathcal{O}_{\mathbf{X}}(1,0)$, and consequently an isomorphism of $SL(E,V)$-modules $\Gamma\xrightarrow{\sim}H^0(\mathbf{X}, \mathcal{O}_{\mathbf{X}}(1,0))$, we can compute $\Gamma$ via the system of projections $\tau_k: Fl(m_{k,1},m_{k,2},V_k)\to G(m_{k,2},V_k)$. This yields $$ \Gamma\simeq\varprojlim H^0(\tau_{k*}(\mathcal{O}_{\mathbf{X}}(1,0)|_{Fl(m_{k,1}, m_{k,2},V_k)}))\simeq\varprojlim\wedge^{m_k}V_k^*. $$ However, $\varprojlim\wedge^{m_k} V_k^*$ is not isomorphic to $V^*$ as an $SL(E,V)$-module. To see this, it is enough to observe that $\varprojlim\wedge^{m_k} V_k^*$ and $V^*$ are non-isomorphic after restriction to $SL(V_k)$ for large $k$. We have a contradiction as desired. \end{proof} \vspace{5mm}
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Supervision and consultancy regarding OHS and fire protection OHS and fire protection trainings Safe company – safe people Informing your employees about the dangers related to performing their professions directly impacts the results achieved by companies. Home - About us The TBF training centre has been present in the Polish market since 1998. For nearly two decades we have been broadening our educational experience in the field of occupational health and safety and fire protection. Furthermore, we specialise in organisation of all kinds of obligatory, vocational and supplementary trainings and provide services related to environmental protection or ISO implementation. Our offer includes, among other things: carrying out the OHS service duties, performing audits in the area of OHS and fire safety principles and regulations, assessment of occupational hazard at work stations, participation in the work of the teams which establish the causes and circumstances of industrial injuries, OHS, fire protection and first aid trainings and courses, expertise assistance in the implementation of orders, decisions and resolutions issued by the Polish National Labour Inspectorate (PIP) and State Sanitary Inspection (PIS), measurements of illumination, measurements of harmful factors at work stations, and many other. Apart from that we are authorised to deliver courses addressed to: power truck drivers, electric truck drivers, power truck drivers, with respect to safe replacement of gas cylinders. Twoja Bezpieczna Firma has the best lecturers and trainers Our employees who have been working for us for many years are now a well-integrated team of highly qualified personnel with all necessary licenses authorising them to deliver various OHS and fire protection trainings. The TBF team consists of a number of OHS and fire protection experts as well as experienced health care professionals – physicians and rescuers. Despite the fact that our lectures and trainings often refer to quite difficult subjects based on OHS regulations and complicated laws, we always deliver them in a very accessible way. Not only do we make sure that the content of our trainings is of top quality but also that it constantly attracts participants' attention. The work of our training centre has been awarded with a number of important certificates: Certificate of Accreditation of the Regional OHS Centre RO-12/2009 issued by the Polish Central Institute for Labour Protection. Certificate of Competence of a Centre of Education JE-8/2004 issued by the Polish Central Institute for Labour Protection. Certificate of Accreditation of the Masovia Province Education Superintendent for provision of continuous education services in the field of occupational health and safety. Entry into the register of schools and non-public institutions maintained by the Capital City of Warsaw. Entry into the register of training centres authorised to organise and deliver civil service trainings. TBF has its offices all around Poland For years the list of our regular clients has grown longer, with new companies enlisted who came to us by word of mouth. We cooperate with many companies from all over Poland and from abroad. The TBF headquarters is in Warsaw but you will find our company's agencies and representative offices in Gdynia, Katowice and Olsztyn, as well as in all major provincial cities. all Periodic OHS trainingElementary OHS trainingFire protectionFirst aidSzkolenie wstępne BHP w formie wideokonferencji (online) Training subcategory List of current trainings Who benefited from our offer: TBF s.c. e-mail: biuro@tbf.com.pl tel/fax: (+48) 22 679 66 22 Realizacja: SODOVAPolityka prywatnościPliki cookies This site uses cookies to provide your services to the highest level. Further use of the site means that you agree that their use.
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\section{Introduction} \noindent The subject of this paper are states and orthogonal polynomials in non-commuting variables. The definition is straightforward. The usual orthogonal polynomials are obtained by starting with a measure $\mu$ on $\mf{R}^d$, thinking of $\mf{R}[x_1, x_2, \ldots, x_d]$ as a vector space with the (pre-)inner product \[ \ip{P}{Q} = \int_{\mf{R}^d} P(\mb{x}) Q(\mb{x}) \,d\mu(\mb{x}), \] and applying the Gram-Schmidt procedure to the monomials $\set{x_{u(1)} x_{u(2)} \ldots x_{u(n)}}$. In the non-commutative case, one starts directly with a positive linear functional (state) $\phi$ on the algebra of non-commutative polynomials $\mf{R} \langle x_1, x_2, \ldots, x_n \rangle$, and orthogonalizes the monomials in non-commuting variables with respect to the inner product \[ \ip{P}{Q} = \state{P^\ast(\mb{x}) Q(\mb{x})}. \] \medskip\noindent Among the general ``non-commutative measures'' and polynomials orthogonal with respect to them, there is a specific class of what is appropriate to call \emph{free Meixner states}. The classical Meixner class \cite{Meixner} consists of familiar distributions---normal, Poisson, gamma, negative binomial, Meix\-ner, and binomial---which, somewhat less familiarly, share a number of common properties: their orthogonal polynomials have exponential-form generating functions, they satisfy a quadratic regression property \cite{Laha-Lukacs}, they generate quadratic natural exponential families \cite{Morris}, they are quadratic harnesses \cite{Wes-commutative}, they are induced by representations of $\mk{su}(1,1)$ \cite{Koelink-Convolutions}, and they have explicit linearization coefficient formulas \cite{KimZeng}. The multivariate Meixner distributions have also been investigated, frequently in the guise of quadratic exponential families \cite{Casalis-Simple-quadratic,Pommeret-Test}, though a complete classification is still lacking. Even the infinite-dimensional case was considered \cite{Sniady-SWN,Lytvynov-Meixner}. \medskip\noindent In \cite{AnsMeixner}, I introduced the free Meixner polynomials, which are a family of orthogonal polynomials in one variable. The term ``free'' refers to their relation to free probability, see \cite{VDN,Nica-Speicher-book} for an introduction. As a matter of fact, these polynomials have been found independently both before and after my work, for example in \cite{Sze22,CTConstant,Freeman,SaiConstant,Kubo-IDAQP}. They share a number of the Meixner properties listed above, as long they are properly translated into the ``free'' context, see my original paper and also \cite{Boz-Bryc}. Some of the corresponding distributions also appear in random matrix theory, as the limiting distributions in the Gaussian, Wishart, and Jacobi ensembles. \medskip\noindent In \cite{AnsMulti-Sheffer} I started the investigation of multivariate free Meixner distributions, which are states on the algebra of non-commutative polynomials. I continue their study in Section~\ref{Section:Meixner}. The main new tool is to represent these states as joint distributions of certain operators on a Fock space, following the more general construction in~\cite{AnsMonic}. I use this machinery, in combination with combinatorial methods, to find explicit formulas for the free cumulants of these states. This provides an explanation for the one-variable results in Section 3.1 of \cite{AnsMeixner} and Proposition 2.2 of \cite{Boz-Bryc}, and is the first main result of the paper. The operator representation of the state also allows me to handle states that are not necessarily faithful, thus answering a question of the referee of \cite{AnsMulti-Sheffer}, where only faithful free Meixner states were considered. \medskip\noindent Having an explicit representation for the cumulants, and being able to handle non-faithful states, allows me to describe a number of examples, which is done in Section~\ref{Section:Examples}. Among the usual multivariate Meixner distributions, two are familiar, namely the multivariate normal and the multinomial distributions. It is well known that the free analog of the multivariate normal distribution is the distribution of a free semicircular system, see Section~\ref{Subsec:Semicircular}. The second question treated in this paper is: what is the ``free'' multinomial distribution? I show that the basic multinomial distribution \emph{itself} also belongs to the free Meixner class. In particular, this allows me to calculate the distribution of a free sum of $d$-tuples of orthogonal projections. \medskip\noindent Among states on non-commutative algebras, \emph{traces} form an important class. The final result in this paper provides a way to construct a large family of non-trivial, tracial free Meixner states. These turn out to be analogs of simple quadratic exponential families. \section{Preliminaries} \noindent Variables in this paper will typically come in $d$-tuples, which will be denoted using the bold font: $\mb{x} = (x_1, x_2, \ldots, x_d)$, and the same for $\mb{z}, \mb{S}$, etc. \subsection{Polynomials} Let $\mf{R}\langle \mb{x} \rangle = \mf{R}\langle x_1, x_2, \ldots, x_d \rangle$ be all the polynomials with real coefficients in $d$ non-commuting variables. \emph{Multi-indices} are elements $\vec{u} \in \set{1, \ldots, d}^k$ for $k \geq 0$; for $\abs{\vec{u}} = 0$ denote $\vec{u}$ by $\emptyset$. Monomials in non-commuting variables $(x_1, \ldots, x_d)$ are indexed by such multi-indices: \[ x_{\vec{u}} = x_{u(1)} \ldots x_{u(k)}. \] Note that our use of the term ``multi-index'' is different from the usual one, which is more suited for indexing monomials in commuting variables. \medskip\noindent For two multi-indices $\vec{u}, \vec{v}$, denote by $(\vec{u}, \vec{v})$ their concatenation. For $\vec{u}$ with $\abs{\vec{u}} = k$, denote \[ (\vec{u})^{op} = (u(k), \ldots, u(2), u(1)). \] Define an involution on $\mf{R}\langle \mb{x} \rangle$ via the $\mf{R}$-linear extension of \[ (x_{\vec{u}})^\ast = x_{(\vec{u})^{op}}. \] \medskip\noindent A \emph{monic polynomial family} in $\mb{x}$ is a family $\set{P_{\vec{u}}(\mb{x})}$ indexed by all multi-indices \[ \bigcup_{k=1}^\infty \set{\vec{u} \in \set{1, \ldots, d}^k} \] (with $P_{\emptyset} = 1$ being understood) such that\[ P_{\vec{u}}(\mb{x}) = x_{\vec{u}} + \textsl{lower-order terms}. \] Note that $P_{\vec{u}}^\ast \neq P_{(\vec{u})^{op}}$ in general. \begin{Defn} \label{Defn:State} A \emph{state} on $\mf{R} \langle \mb{x} \rangle$ is a functional \[ \phi: \mf{R} \langle x_1, x_2, \ldots, x_d \rangle \rightarrow \mf{R} \] that is linear, compatible with the $\ast$-operation, that is for any $P$, \[ \state{P} = \state{P^\ast}, \] unital, that is $\state{1} = 1$, and positive, that is for any $P$, \[ \state{P^\ast P} \geq 0. \] A state is \emph{faithful} if in the preceding equation, the equality holds only for $P = 0$. Unless noted otherwise, the states in this paper are \emph{not} assumed to be faithful. \medskip\noindent The numbers $\state{x_{\vec{u}}}$ are called the \emph{moments} of $\phi$. \medskip\noindent A state $\phi$ induces the pre-inner product \[ \ip{P}{Q}_\phi = \state{P^\ast Q} = \ip{Q}{P}_\phi \] and the seminorm \[ \norm{P}_\phi = \sqrt{\state{P^\ast P}}. \] Throughout the paper, we will typically drop $\phi$ from the notation, and denote the inner product and norm it induces simply by $\ip{\cdot}{\cdot}$, $\norm{\cdot}$. \medskip\noindent We may think of $\phi$ is a ``joint distribution'' of ``random variables'' $(x_1, x_2, \ldots, x_d)$. In the remainder of the paper, as we did in \cite{AnsMulti-Sheffer}, we will assume that under the state $\phi$, the variables have zero mean and identity covariance, \[ \state{x_i} = 0, \qquad \state{x_i x_j} = \delta_{ij}. \] The last assumption is made primarily so that equation~\eqref{PDE} has a clean form. In Section~\ref{Subsec:Covariance} we briefly describe how to modify the results if that assumption is dropped. \end{Defn} \subsection{Monic orthogonal polynomials states} \label{Subsec:MOPS} \begin{Defn} A state has a \emph{monic orthogonal polynomial system}, or MOPS, if for any multi-index $\vec{u}$, there is a monic polynomial $P_{\vec{u}}$ with leading term $x_{\vec{u}}$, such that these polynomials are orthogonal with respect to $\phi$, that is, \[ \ip{P_{\vec{u}}}{P_{\vec{v}}} = 0 \] for $\vec{u} \neq \vec{v}$. \end{Defn} \noindent Note that the same abbreviation is used in~\cite{Dumitriu-MOPS} to denote a class of multivariate orthogonal polynomials systems, which is different from ours. \medskip\noindent States that have MOPS were characterized in \cite{AnsMonic}. We briefly summarize the results of that paper which we will use in the next section. \subsubsection{Fock space construction I} \label{Subsubsec:General-Fock} Let $\mc{H} = \mf{C}^d$, with the canonical orthonormal basis $e_1, e_2, \ldots, e_d$. Define the (algebraic) full Fock space of $\mc{H}$ to be \[ \Falg(\mc{H}) = \bigoplus_{k=0}^\infty \mc{H}^{\otimes k} \] Equivalently, $\Falg(\mc{H})$ is the vector space of non-commutative polynomials in $e_1, e_2, \ldots, e_d$. Following convention, we will denote the generating vector in $\mc{H}^{\otimes 0} = \mf{C}$ by $\Omega$ instead of $1$. \medskip\noindent For $i = 1, 2, \ldots, d$, define $a_i^+$ and $a_i^-$ to be the usual (left) free creation and annihilation operators, \begin{align*} a_i^+ & \left(e_{u(1)} \otimes e_{u(2)} \otimes \ldots \otimes e_{u(k)} \right) = e_i \otimes e_{u(1)} \otimes e_{u(2)} \otimes \ldots \otimes e_{u(k)}, \\ a_i^- & (e_j) = \ip{e_i}{e_j} \Omega = \delta_{i j} \Omega, \\ a_i^- & \left(e_{u(1)} \otimes e_{u(2)} \otimes \ldots \otimes e_{u(k)} \right) = \ip{e_i}{e_{u(1)}} e_{u(2)} \otimes \ldots \otimes e_{u(k)}. \end{align*} \medskip\noindent For each $k \geq 2$ let $\mc{C}^{(k)}$ be an operator \[ \mc{C}^{(k)}: \mc{H}^{\otimes k} \rightarrow \mc{H}^{\otimes k}. \] We think of each $\mc{C}^{(k)}$ as a $d^k \times d^k$ matrix. Assume that for each $k$, $\mc{C}^{(k)}$ is diagonal and $\mc{C}^{(k)} \geq 0$. It is convenient to also take $\mc{C}^{(1)} = I$; this corresponds to the identity covariance. Similarly, for each $i = 1, 2, \ldots, d$ and each $k \geq 1$, let $\mc{T}_i^{(k)}$ be an operator \[ \mc{T}_i^{(k)}: \mc{H}^{\otimes k} \rightarrow \mc{H}^{\otimes k}. \] Assume that $\mc{T}_i^{(k)}$ and $\mc{C}^{(j)}$ satisfy a commutation relation (see \cite{AnsMonic}). We will denote by $\mc{T}_i$ and $\mc{C}$ the operators acting as $\mc{T}_i^{(k)}$ and $\mc{C}^{(k)}$ on each component. Finally, let $\tilde{a}_i^- = a_i^- \mc{C}$ and \[ \mc{X}_i = a_i^+ + \mc{T}_i + \tilde{a}_i^-. \] With the appropriate choice of the inner product $\ip{\cdot}{\cdot}_{\mc{C}}$ on the completion $\mc{F}_{\mc{C}}(\mc{H})$ of the quotient of $\Falg(\mc{H})$, all the operators $a_i^+, \mc{T}_i, \tilde{a}_i^-$ factor through to $\mc{F}_{\mc{C}}(\mc{H})$, and each $\mc{X}_i$ is a symmetric operator on it. \begin{Thm}(Part of Theorem 2 of \cite{AnsMonic}) \label{Thm:Monic-states} Let $\phi$ be a state on $\mf{R} \langle \mb{x} \rangle$. The following are equivalent: \begin{enumerate} \item The state $\phi$ has a monic orthogonal polynomial system. \item There is a family of polynomials $\set{P_{\vec{u}}}$ such that $\state{P_{\vec{u}}} = 0$ for all $\vec{u} \neq \emptyset$ and they satisfy a recursion relation \begin{align*} x_i & = P_i + B_{i, \emptyset, \emptyset}, \\ x_i P_u & = P_{(i, u)} + \sum_{w=1}^d B_{i, w, u} P_{w} + \delta_{i, u} C_u, \\ x_i P_{\vec{u}} & = P_{(i, \vec{u})} + \sum_{\abs{\vec{w}} = \abs{\vec{u}}} B_{i, \vec{w}, \vec{u}} P_{\vec{w}} + \delta_{i, u(1)} C_{\vec{u}} P_{(u(2), u(3), \ldots, u(k))}, \end{align*} with $C_{\vec{u}} \geq 0$ and, denoting $\vec{s}_j = (s(j), \ldots, s(k))$, \[ B_{i, \vec{s}, \vec{u}} \prod_{j=1}^k C_{\vec{s}_j} = B_{i, \vec{u}, \vec{s}} \prod_{j=1}^k C_{\vec{u}_j}. \] \item For some choice of the matrices $\mc{C}^{(k)}$ and $\mc{T}_i^{(k)}$ as in Section~\ref{Subsubsec:General-Fock}, the state $\phi$ has a Fock space representation $\phi_{\mc{C}, \set{\mc{T}_i}}$ as \begin{equation*} \state{P(x_1, x_2, \ldots, x_d)} = \ip{\Omega}{P(\mc{X}_1, \mc{X}_2, \ldots, \mc{X}_d) \Omega}. \end{equation*} \end{enumerate} \end{Thm} \medskip\noindent We will also need the following relation between the operators in part (c) and coefficients in part (b) of the theorem: \begin{equation} \label{Expansion-T} \mc{T}_i(e_{u(1)} \otimes \ldots \otimes e_{u(k)}) = \sum_{\abs{\vec{w}} = k} B_{i, \vec{w}, \vec{u}} e_{w(1)} \otimes \ldots \otimes e_{w(k)} \end{equation} and \begin{equation} \label{Expansion-C} \mc{C}(e_{u(1)} \otimes \ldots \otimes e_{u(k)}) = C_{\vec{u}} e_{u(1)} \otimes \ldots \otimes e_{u(k)}. \end{equation} \subsection{Fock space construction II} \label{Subsec:Fock2} The following construction is a particular case of the construction in Section~\ref{Subsubsec:General-Fock}, but this time we provide full details. As before, let $\mc{H} = \mf{C}^d$, with the canonical basis $e_1, e_2, \ldots, e_d$, denote its (algebraic) full Fock space by $\Falg(\mc{H})$, and the generator of the zeroth component by $\Omega$. Let $C$ be an n operator on $\mc{H} \otimes \mc{H}$, which we identify with its $d^2 \times d^2$ matrix in the standard basis. Assume that $C$ is diagonal and \begin{equation} \label{C-positive} (I \otimes I) + C \geq 0, \end{equation} where $I$ will always denote the identity operator on $\mc{H}$. On $\Falg(\mc{H})$, define a new inner product using the non-negative kernel \[ K_C = \bigl(I^{\otimes (k-2)} \otimes (I^{\otimes 2} + C)\bigr) \ldots \bigl(I \otimes (I^{\otimes 2} + C) \otimes I^{\otimes (k-3)}\bigr) \bigl((I^{\otimes 2} + C) \otimes I^{\otimes (k-2)}\bigr) \] on each $\mc{H}^{\otimes k}$, and denote the completion of $\Falg(\mc{H})$ with respect to this inner product $\mc{F}_C(\mc{H})$. If the inner product is degenerate, first factor out the subspace of vectors of length zero, and then complete. \medskip\noindent For $i = 1, 2, \ldots, d$, let $a_i^+$ and $a_i^-$ be the usual (left) free creation and annihilation operators as defined in Section~\ref{Subsubsec:General-Fock}. Let $T_1, \ldots, T_d$ be operators on $\mc{H}$ which we identify with their $d \times d$ matrices. Assume that each $T_i$ is symmetric and \[ (T_i \otimes I) C = C (T_i \otimes I). \] With a slight abuse of notation, we will denote \[ T_i = T_i \otimes I^{\otimes (k-1)} \text{ on } \mc{H}^{\otimes k} \] and \[ \tilde{a}_i = a_i^- (C \otimes I^{\otimes (k-2)}) \text{ on } \mc{H}^{\otimes k}. \] Note that \begin{equation} \label{Zero} a_i^- \Omega = T_i \Omega = \tilde{a}_i \Omega = 0 \text{ and } \tilde{a}_i = 0 \text{ on } \mc{H}. \end{equation} It follows from the general construction in Section~\ref{Subsubsec:General-Fock} that all the operators \[ X_i = a_i^+ + a_i^- + T_i + \tilde{a}_i \] factor through to $\mc{F}_C(\mc{H})$. \begin{Defn} \label{Defn:Fock-state} The Fock state $\phi = \phi_{C, \set{T_i}}$ on $\mf{R} \langle \mb{x} \rangle$ determined by such $C$ and $T_i$ is the state \begin{equation*} \state{P(x_1, x_2, \ldots, x_d)} = \ip{\Omega}{P(X_1, X_2, \ldots, X_d) \Omega} = \ip{\Omega}{P(X_1, X_2, \ldots, X_d) \Omega}_C. \end{equation*} \end{Defn} \subsection{Non-crossing partitions} A \emph{partition} $\pi$ of a set $V \subset \mf{Z}$ is a collection of disjoint subsets of $V$ (classes of $\pi$), $\pi = (B_1, B_2, \ldots, B_k)$, whose union equals $V$. Most of the time we will be interested in partitions of $\set{1, 2, \ldots, n}$. Partitions form a partially ordered set (in fact a lattice) under the operation of refinement, so that the largest partition is $\hat{1} = \bigl( \set{1, 2, \ldots, n} \bigr)$ and the smallest partition is $\hat{0} = \bigl( \set{1}, \set{2}, \ldots, \set{n} \bigr)$. We will use $i \stackrel{\pi}{\sim} j$ to denote that $i, j$ lie in the same class of $\pi$. \medskip\noindent Let $\NC(V)$ denote the collection of non-crossing partitions of $V$, which are partitions $\pi$ such that \[ i \stackrel{\pi}{\sim} i', j \stackrel{\pi}{\sim} j', i \stackrel{\pi}{\not \sim} j, i < j < i' \Rightarrow i < j' < i'. \] Equivalently, a partition is non-crossing if and only if one of its classes is an interval and the restriction of the partition to the complement of this class is non-crossing. Non-crossing partitions are a sub-lattice of the lattice of all partitions. For each $n$, let $\NC(n)$ denote the lattice of non-crossing partitions of the set $\set{1, 2, \ldots, n}$. We will also denote by $\NC_0(V)$ all non-crossing partitions with no singletons (one-element classes), and by $\NC'(V)$ all the partitions $\pi$ such that \[ \min V \stackrel{\pi}{\sim} \max V. \] Equivalently, partitions in $\NC'(V)$ have a single outer class---the one that contains both $\min V$ and $\max V$---in the terminology of \cite{BLS96}. (A class $B \in \pi$ is outer if there do \emph{not} exist $i, i' \not \in B$, $j \in B$ with $i \stackrel{\pi}{\sim} i'$ and $i < j < i'$.) See \cite{Nica-Speicher-book} or \cite{Stanley-volume-1} for more details on the relevant combinatorics. \subsection{Free cumulants} The free cumulant functional $R$ corresponding to a state $\phi$ is the linear functional on $\mf{R} \langle \mb{x} \rangle$ defined recursively by $\Cum{1} = 0$ and for $\abs{\vec{u}} = n$, \begin{equation} \label{Cumulants-definition} \Cum{x_{\vec{u}}} = \state{x_{\vec{u}}} - \sum_{\substack{\pi \in \NC(n), \\ \pi \neq \hat{1}}} \prod_{B \in \pi} \Cum{\prod_{i \in B} x_{u(i)}}, \end{equation} which expresses $\Cum{x_{\vec{u}}}$ in terms of the joint moments and sums of products of lower-order free cumulants. From these, we can form the free cumulant generating function of $\phi$ via \begin{equation} \label{Non-crossing} R(z_1, z_2, \ldots, z_d) = \sum_{n=1}^\infty \sum_{\abs{\vec{u}} = n} \Cum{x_{\vec{u}}} z_{\vec{u}}, \end{equation} where $\mb{z} = (z_1, \ldots, z_d)$ are non-commuting indeterminates. One can also define $R$ using an implicit functional relation involving the moment generating function of $\phi$, see Corollary~16.16 of \cite{Nica-Speicher-book}. \subsection{Words and partitions} \label{Subsec:Facts} In this section, we collect a number of facts that will be useful in the proof of the next two theorems. Note that in many places, operators are considered as acting on $\Falg(\mc{H})$, with a degenerate inner product, rather than on $\mc{F}_C(\mc{H})$. \begin{Lemma} Let $\vec{u}$ be a multi-index indexed by a set $V \subset \mf{Z}$, and $W = \prod_{i \in V} W(i)$ be a word with $W(i)$ equal to $a^+_{u(i)}, T_{u(i)}, a_{u(i)}^-$, or $\tilde{a}_{u(i)}$. If \[ \ip{\Omega}{\prod_{i \in V} W(i) \Omega} \neq 0, \] then \begin{equation} \label{Catalan-walk} \begin{split} W(\min V) = a_{u(\min V)}^-, \quad W(\max V) = a_{u(\max V)}^+, & \\ \forall i \in V, \abs{\set{j \in V | j \geq i, W(j) = a_{u(j)}^- \text{ or } W(j) = \tilde{a}_{u(j)}}} & \leq \abs{\set{j \in V | j \geq i, W(j) = a_{u(j)}^+}}, \\ \abs{\set{j \in V | W(j) = a_{u(j)}^- \text{ or } W(j) = \tilde{a}_{u(j)}}} & = \abs{\set{j \in V | W(j) = a_{u(j)}^+}}, \end{split} \end{equation} and \begin{multline} \label{Level-two} \abs{\set{j \in V | j \geq i, W(j) = a_{u(j)}^- \text{ or } W(j) = \tilde{a}_{u(j)}}} = \abs{\set{j \in V | j \geq i, W(j) = a_{u(j)}^+}} \\ \Rightarrow W(i) = a_{u(i)}^-. \end{multline} \end{Lemma} \begin{proof} This follows from the fact that if $\eta \in \mc{H}^{\otimes k}$, then $a_i^+(\eta) \in \mc{H}^{\otimes (k+1)}$, $T_i(\eta) \in \mc{H}^{\otimes k}$, and $a_i^-(\eta), \tilde{a}_i(\eta) \in \mc{H}^{\otimes (k-1)}$, and equation~\eqref{Zero}. \end{proof} \noindent In combinatorics, equation~\eqref{Catalan-walk} is related to the notion of a Motzkin path. More generally, our operator representations are closely related to a common way of representing moments as sums over lattice paths \cite{Flajolet,Viennot-Short}, but in the multivariate case we find the operator formulation more useful. \begin{Notation} Let \[ \mc{W}_n(\vec{u}) = \bigl\{W = W(1) W(2) \ldots W(n) \text{ satisfying conditions } \eqref{Catalan-walk} \text{ and } \eqref{Level-two} \text{ for } V = \set{1, \ldots, n}\bigr\}, \] and for a general subset $V \subset \mf{Z}$, define $\mc{W}_V(\vec{u})$ similarly. \end{Notation} \begin{Lemma} \label{Lemma:Bijection} For any multi-index $\vec{u}$, partition $\pi \in \NC_0(n)$, $\pi = (V_1, V_2, \ldots, V_k)$ and partitions $\sigma_j \in \NC_0'(V_j)$, $j = 1, 2, \ldots, k$, define a word $W = \beta_{\vec{u}}(\pi; \sigma_1, \ldots, \sigma_k)$ by \begin{equation} \label{Partition-word} W(i) = \begin{cases} a_{u(i)}^+, & i \in B \in \sigma_j, i = \max B, \\ a_{u(i)}^-, & i \in V_j, i = \min V_j, \\ \tilde{a}_{u(i)}, & i \in B \in \sigma_j, i = \min B, i \neq \min V_j, \\ T_{u(i)}, & \text{otherwise}. \end{cases} \end{equation} Then $W \in \mc{W}_n(\vec{u})$, and for each $V \in \pi$, $W$ restricted to $V$ is in $\mc{W}_V(\vec{u}:V)$, where $(\vec{u}:V)$ is the sub-multi-index of $\vec{u}$ indexed by the elements of $V$. Moreover, for each $\vec{u}$, $\beta_{\vec{u}}$ is a bijection. \end{Lemma} \begin{proof} Let $W = \beta_{\vec{u}}(\pi; \sigma_1, \ldots, \sigma_k)$. Condition~\eqref{Catalan-walk} for the whole set $\set{1, 2, \ldots, n}$ (respectively, for $V_j$) follows from the definition of $\beta$ and the fact that $\pi, \sigma_1, \ldots, \sigma_k$ (respectively, $\sigma_j$) are non-crossing. Condition~\eqref{Level-two} follows from the definition that the minima of the outer classes of the partition $\pi$ (respectively, $\sigma_j$) are all $a^-$. \medskip\noindent Conversely, let $W \in \mc{W}_n(\vec{u})$. Let $\Lambda \subset \set{1, 2, \ldots, n}$, \[ \Lambda = \set{j | W(j) \neq T_{u(j)}}. \] It follows from Proposition~2.13 and Exercise~8.23 of \cite{Nica-Speicher-book} that, as long as $W$ restricted to $\Lambda$ satisfies condition~\eqref{Catalan-walk}, there is a unique non-crossing pair partition $\pi' \in \NC(\Lambda)$ such that for any $B \in \pi'$, \begin{align*} i = \min B & \Leftrightarrow W(i) = a_{u(i)}^- \text{ or } \tilde{a}_{u(i)}, \\ i' = \max B & \Leftrightarrow W(i') = a_{u(i')}^+. \end{align*} Note that $W(1) = a_{u(1)}^-$, so $(1, i') \in \pi'$ for some $i' > 1$. Moreover, $W(i'+1) \ldots W(n) \Omega \in \mc{H}^{\otimes 0}$, so by condition~\eqref{Level-two}, $W(i'+1) = a_{u(i'+1)}^-$. Thus $(i'+1, j') \in \pi'$ for some $j'$, etc. ending with $(s,n) \in \pi'$. It follows that for any $j \in \set{1, \ldots, n}$, there exist $i \stackrel{\pi'}{\sim} i'$ such that $i \leq j \leq i'$ and $W(i) = a_{u(i)}^-$. For each $j$, choose the largest $i$ such that $i \stackrel{\pi'}{\sim} i'$, $i \leq j \leq i'$, and $W(i) = a_{u(i)}^-$, and require that $i \stackrel{\pi}{\sim} j \stackrel{\pi}{\sim} i'$. Similarly, for each class $V_s \in \pi$ and each $j \in V_s$, choose the largest $i \in V_s$ such that $i \stackrel{\pi'}{\sim} i'$ and $i \leq j \leq i'$, and require that $i \stackrel{\sigma_s}{\sim} j \stackrel{\sigma_s}{\sim} i'$. Pictorially, we draw the integers $1, 2, \ldots, n$ on a line, draw the pair classes or $\pi'$ as arcs connecting each $i$ with the corresponding $i'$ above the line, and then connect each of the other elements to the arc immediately above it. \end{proof} \begin{Lemma} \label{Lemma:Factor} For any $W \in \mc{W}_n(\vec{u})$, let $(\pi; \sigma_1, \ldots, \sigma_k) = \beta_{\vec{u}}^{-1}(W)$ with $\pi = (V_1, V_2, \ldots, V_k)$. Then \[ \ip{\Omega}{W(1) W(2) \ldots W(n) \Omega} = \prod_{i=1}^k \ip{\Omega}{\prod_{j \in V_i} W(j) \Omega}. \] \end{Lemma} \begin{proof} Since $\pi$ is a non-crossing partition, it has a class $V$ that is an interval, \[ V = [i, i'] = \set{j | i \leq j \leq i'}. \] Since $\pi$ restricted to $\set{1, \ldots, n} \backslash V$ is still a non-crossing partition, it suffices to show that \[ \ip{\Omega}{W(1) W(2) \ldots W(n) \Omega} = \ip{\Omega}{\prod_{j=1}^{i-1} W(j) \prod_{j=i'+1}^n W(j) \Omega} \ip{\Omega}{\prod_{j=i}^{i'} W(j) \Omega}. \] Denote \[ \eta = W(i'+1) \ldots W(n) \Omega \in \mc{H}^{\otimes m}. \] We now show that for any $i < j \leq i'$, \[ W(j) \ldots W(n) \Omega = \zeta_j \otimes \eta \] for \[ \zeta_j = W(j) \ldots W(i') \Omega. \] The proof is by induction. \[ W(i') \eta = a_{u(i')}^+ \eta = e_{u(i')} \otimes \eta = (W(i') \Omega) \otimes \eta. \] If $W(j) = a_{u(j)}^+$, $\zeta_j = e_{u(j)} \otimes \zeta_{j+1}$. If $W(j) = T_{u(j)}$, then $\zeta_j = T_{u(j)} \zeta_{j+1}$. $W(j)$ cannot equal $a_{u(j)}^-$. Finally, it follows from condition~\eqref{Level-two} applied to $V = [i, i']$ that for all $j$, $i < j \leq i'$, \[ W(j+1) \ldots W(n) \Omega \in \mc{H}^{\otimes s} \] with $s > m$. Thus $W(j)$ may equal $\tilde{a}_{u(j)}$ only if $s \geq m + 2$, otherwise \[ W(j) W(j+1) \ldots W(n) \Omega \in \mc{H}^{\otimes m}. \] But if $s \geq m + 2$, $\zeta_j = a_{u(j)}^- C \zeta_{j+1}$. \medskip\noindent It follows that also \[ W(i) \ldots W(n) \Omega = (W(i) \ldots W(i') \Omega) \otimes \eta = \ip{\Omega}{W(i) \ldots W(i') \Omega} \eta. \] Thus \[ \begin{split} \ip{\Omega}{W(1) W(2) \ldots W(n) \Omega} & = \ip{\Omega}{W(1) \ldots W(i-1) \ip{\Omega}{W(i) \ldots W(i') \Omega} \eta} \\ & = \ip{\Omega}{W(1) \ldots W(i-1) \eta} \ip{\Omega}{W(i) \ldots W(i') \Omega} \\ & = \ip{\Omega}{\prod_{j=1}^{i-1} W(j) \prod_{j=i'+1}^n W(j) \Omega} \ip{\Omega}{\prod_{j=i}^{i'} W(j) \Omega}. \qedhere \end{split} \] \end{proof} \begin{Notation} \label{Notation:Covered-bijection} For $V \subset \mf{Z}$, denote \[ \begin{split} \mc{W}_V'(\vec{u}) = \{W \in \mc{W}_V(\vec{u}) |& W(\min V) = a_{u(\min V)}^-, W(\max V) = a_{u(\max V)}^+, \\ &\quad \text{ and none of the other $W(i)$ are equal to } a_{u(i)}^-\}. \end{split} \] The partition $\pi$ corresponding to any such $W$ has only one class, $\pi = (V) \in \NC(V)$, and \[ \beta_{\vec{u}}^{-1}(\mc{W}_V'(\vec{u})) = \set{((V), \sigma) | \sigma \in \NC_0'(V)} \cong \NC_0'(V). \] Denote \[ \Theta(\sigma; V, \vec{u}) = \ip{\Omega}{\beta_{\vec{u}}((V), \sigma) \Omega}. \] \end{Notation} \begin{Lemma} If $W \in \mc{W}_n(\vec{u})$ and $\beta_{\vec{u}}^{-1}(W) = (\pi, \sigma_1, \ldots, \sigma_k)$, $\pi = (V_1, V_2, \ldots, V_k)$, then \[ \ip{\Omega}{W(1) \ldots W(n) \Omega} = \prod_{j=1}^k \ip{\Omega}{\prod_{i \in V_j} W(i) \Omega} = \prod_{j=1}^k \Theta(\sigma_j; V_j, (\vec{u}:V_j)), \] where $(\vec{u}:V_j)$ is the sub-multi-index of $\vec{u}$ indexed by the elements of $V_j$. \end{Lemma} \begin{proof} This follows from Lemma~\ref{Lemma:Factor} using Notation~\ref{Notation:Covered-bijection}. \end{proof} \begin{Lemma} \label{Lemma:Factor2} Suppose that $W \in \mc{W}_n'(\vec{u})$ such that $W(1) = a_{u(1)}^- = a_{j}^-$ and $W(2) = \tilde{a}_{u(2)} = \tilde{a}_{i}$. Then $\beta_{\vec{u}}^{-1}(W) = ((\set{1, \ldots, n}), \sigma)$. It follows from condition~\eqref{Partition-word} that for $2 \in B \in \sigma$, we have $2 = \min B$. Let $k = \max B$. Then $W(k) = a_{u(k)}^+$ and \begin{multline*} \ip{\Omega}{W(1) W(2) \ldots W(k) \ldots W(n) \Omega} = \ip{\Omega}{a_{j}^- \tilde{a}_{i} W(3) \ldots a_{u(k)}^+ W(k+1) \ldots W(n-1) a_{u(n)}^+ \Omega} \\ = C_{ij} \ip{\Omega}{a_j^- W(k+1) \ldots W(n-1) a_{u(n)}^+ \Omega} \ip{\Omega}{a_i^- W(3) \ldots a_{u(k)}^+ \Omega}. \end{multline*} Moreover, the map \[ \begin{split} \{W \in \mc{W}_n'(\vec{u}) | & W(1) = a_{u(1)}^- = a_{j}^-, W(2) = \tilde{a}_{u(2)} = \tilde{a}_{i}\} \\ & \rightarrow \bigcup_{k=3}^{n-1} \mc{W}_{\set{1, k+1, \ldots, n}}'\bigl((\vec{u}:\set{1, k+1, \ldots, n})\bigr) \times \mc{W}_{\set{2, \ldots, k}}'\bigl((\vec{u}:\set{2, \ldots, k})\bigr) \\ & \cong \bigcup_{k=3}^{n-1} \NC_0'(\set{1, k+1, \ldots, n}) \times \NC_0'(\set{2, \ldots, k}) \end{split} \] is a bijection. \end{Lemma} \begin{proof} By the same method as in Lemma~\ref{Lemma:Factor}, we deduce that \[ W(3) \ldots a_{u(k)}^+ W(k+1) \ldots W(n-1) a_{u(n)}^+ \Omega = \bigl(W(3) \ldots a_{u(k)}^+ \Omega \bigr) \otimes \bigl(W(k+1) \ldots W(n-1) a_{u(n)}^+ \Omega \bigr). \] The inner product of this vector with $C(e_i \otimes e_j)$ is the desired expression. \end{proof} \begin{Lemma} \label{Lemma:Last} Suppose that $C_{ij} = C(e_i \otimes e_j) = c$ for all $i, j$. Let $\sigma \in NC_0'(n)$, \[ \sigma = \Bigl( \set{b_{1,1}, \ldots, b_{1, j(1)}}, \ldots, \set{b_{1, k}, \ldots, b_{k, j(k)}} \Bigr), \] where each class is ordered and $b_{1,1} = 1$. Then \[ \Theta(\sigma; \set{1, \ldots, n}, \vec{u}) = c^{k-1} \prod_{i=1}^k \ip{e_{u(b_{i,1})}}{T_{u(b_{i,2})} \ldots T_{u(b_{i, j(i) - 1})} e_{u(b_{i, j(i)})}}. \] \end{Lemma} \begin{proof} This follows from the definition of $\beta$, noting that $W(b_{1,1}) = W(1) = a_{u(1)}^-$, $W(b_{1,j}) = \tilde{a}_{u(b_{1,l})} = c a_{u(b_{1,l})}^-$ for $j \neq 1$, $W(b_{i, j(l)}) = a_{u(b_{i, j(l)})}^+$, and the rest of the terms are $T_{u(b_{i,l})}$. \end{proof} \section{Main theorems} \label{Section:Meixner} \begin{Thm} \label{Thm:Cumulants} For each $i$, let \[ S_i = a_i^+ + T_i + \tilde{a}_i = X_i - a_i^- \] be an operator on $\Falg(\mc{H})$. Then the free cumulants of the Fock state $\phi_{C, \set{T_i}}$ from Definition~\ref{Defn:Fock-state} are given by the formula $\Cum{x_i} = 0$, \[ \Cum{x_i P(\mb{x}) x_j} = \ip{e_i}{P(\mb{S}) e_j} = \ip{e_i}{P(\mb{S}) e_j}_C. \] \end{Thm} \begin{proof} Since $S_{u(i)} = a_{u(i)}^+ + T_{u(i)} + \tilde{a}_{u(i)}$, and using Notation~\ref{Notation:Covered-bijection}, for $\abs{\vec{u}} = n$, \[ \begin{split} \ip{e_{u(1)}}{S_{u(2)} \ldots S_{u(n-1)} e_{u(n)}} & = \ip{\Omega}{a_{u(1)}^- S_{u(2)} \ldots S_{u(n-1)} a_{u(n)}^+ \Omega} = \sum_{W \in \mc{W}_n'(\vec{u})} \ip{\Omega}{W \Omega} \\ & = \sum_{\sigma \in \NC_0'(n)}\ip{\Omega}{\beta_{\vec{u}}^{-1}(\set{1, \ldots, n}, \sigma) \Omega} \\ & = \sum_{\sigma \in \NC_0'(n)} \Theta(\sigma; \set{1, \ldots, n}, \vec{u}). \end{split} \] Similarly, since $X_{u(i)} = a_{u(i)}^+ + T_{u(i)} + a_{u(i)}^- + \tilde{a}_{u(i)}$, using Lemma~\ref{Lemma:Bijection} and the preceding equation, \[ \begin{split} \ip{\Omega}{X_{u(1)} X_{u(2)} \ldots X_{u(n)} \Omega} & = \sum_{W \in \mc{W}_n(\vec{u})} \ip{\Omega}{W(1) W(2) \ldots W(n) \Omega} \\ & = \sum_{k=1}^n \sum_{\substack{\pi \in \NC_0(n) \\ \pi = (V_1, V_2, \ldots, V_k)}} \sum_{\substack{\sigma_j \in \NC_0'(V_j) \\ j = 1, \ldots, k}} \prod_{i=1}^k \Theta(\sigma_i; V_i, (\vec{u}:V_i)) \\ & = \sum_{k=1}^n \sum_{\substack{\pi \in \NC_0(n) \\ \pi = (V_1, V_2, \ldots, V_k)}} \prod_{i=1}^k \left( \sum_{\sigma_i \in \NC_0'(V_i)} \Theta(\sigma_i; V_i, (\vec{u}:V_i)) \right) \\ & = \sum_{k=1}^n \sum_{\substack{\pi \in \NC_0(n) \\ \pi = (V_1, V_2, \ldots, V_k)}} \prod_{i=1}^k \ip{e_{(\vec{u}:V_i)(1)}}{S_{(\vec{u}:V_i)(2)} \ldots S_{(\vec{u}:V_i)(n-1)} e_{(\vec{u}:V_i)(n)}} \end{split} \] Thus \[ \state{x_{\vec{u}}} = \sum_{\pi \in \NC_0(n)} \prod_{V \in \pi} \ip{e_{(\vec{u}:V)(1)}}{S_{(\vec{u}:V)(2)} \ldots S_{(\vec{u}:V)(n-1)} e_{(\vec{u}:V)(n)}}. \] Since $\Cum{x_i} = \state{x_i} = 0$, the conclusion of the theorem now follows from the defining relation for the free cumulants, namely \begin{equation*} \state{x_{\vec{u}}} = \sum_{\pi \in \NC(n)} \prod_{B \in \pi} \Cum{\prod_{i \in B} x_{u(i)}}. \qedhere \end{equation*} \end{proof} \begin{Cor} \label{Cor:Self-adjoint} Each $X_i$ is symmetric and bounded, hence self-adjoint. \end{Cor} \begin{proof} The symmetry is proved exactly as in Proposition~1 of \cite{AnsMonic}, or can be deduced from it. To prove boundedness, choose $m$ such that $\norm{C}, \norm{T_i} < m$. Since $\abs{\NC(n)} < 4^n$, and $\abs{\Theta(\pi; V, \vec{u})} < m^{\abs{V}}$, it follows that $\Cum{x_{\vec{u}}} < (4m)^{\abs{u}}$ and \[ \phi_{C, \set{T_i}} \left[ X_{u(1)} X_{u(2)} \ldots X_{u(n)} \right] < (16 m)^n. \] Thus for each $i$, $\norm{X_i} < 16 m$. \end{proof} \begin{Notation} Let $\mb{z} = (z_1, \ldots, z_d)$ be non-commuting indeterminates, which commute with $\mb{x}$. For a non-commutative power series $G$ in $\mb{z}$ and $i = 1, \ldots, d$, define the left non-commutative partial derivative $D_i G$ by a linear extension of $D_i(1) = 0$, \[ D_i z_{\vec{u}} = \delta_{i u(1)} z_{u(2)} \ldots z_{u(n)}. \] Denote by $\mb{D} G = (D_1 G, \ldots, D_d G)$ the left non-commutative gradient. \medskip\noindent For a non-commutative power series $G$, denote by $G^{-1}$ its inverse with respect to multiplication. For a $d$-tuple of non-commutative power series $\mb{G} = (G_1, \ldots, G_d)$, denote by $\mb{G}^{\langle -1 \rangle}$ its inverse with respect to composition (which is also a $d$-tuple). \end{Notation} \begin{Thm} \label{Thm:Meixner} Let $\phi$ be a state on $\mf{R} \langle \mb{x} \rangle$ with a monic orthogonal polynomial system (MOPS), zero means and identity covariance. The following are equivalent. \begin{enumerate} \item There exists a non-commutative power series \[ F(\mb{z}) = 1 + (\textsl{terms of degree } \geq 2) \] and a $d$-tuple of non-commutative power series $\mb{U}$, \[ U_i(\mb{z}) = z_i + \textsl{higher-order terms}, \] such that the polynomials defined via their generating function \[ \sum_{\abs{\vec{u}} \geq 0} P_{\vec{u}}(\mb{x}) z_{\vec{u}} = F(\mb{z}) \Bigl(1 - \mb{x} \cdot \mb{U}(\mb{z})\Bigr)^{-1} \] are a MOPS for $\phi$. \item The polynomials with the generating function \begin{equation} \label{Generating} \sum_{\abs{\vec{u}} \geq 0} P_{\vec{u}}(\mb{x}) z_{\vec{u}} = \Bigl( 1 - \mb{x} \cdot (\mb{D} R)^{\langle -1 \rangle} (\mb{z}) + R\bigl((\mb{D} R)^{\langle -1 \rangle} (\mb{z})\bigr) \Bigr)^{-1} \end{equation} are a MOPS for $\phi$, where $R$ is the free cumulant generating function~\eqref{Non-crossing} of $\phi$. \item The free cumulant generating function of $\phi$ satisfies, for each $i, j$, a (non-commutative) second-order partial differential equation \begin{equation} \label{PDE} D_i D_j R(\mb{z}) = \delta_{ij} + \sum_{k=1}^d B_{ij}^k D_k R(\mb{z}) + C_{ij} D_i R(\mb{z}) D_j R(\mb{z}), \end{equation} where $C_{ij} \geq -1$, $B_{ij}^{k} = B_{ik}^{j}$, and for each $j,k$, either $B_{ij}^{k} = 0$ for all $i$, or $C_{ju} = C_{ku}$ for all $u$. \item There is a family of polynomials $\set{P_{\vec{u}}}$ such that $\state{P_{\vec{u}}} = 0$ for all $\vec{u} \neq \emptyset$ and they satisfy a recursion relation \begin{align*} x_i & = P_i, \\ x_i P_{j} &= P_{(i,j)} + \sum_{k=1}^d B_{ij}^{k} P_{k} + \delta_{ij}, \\ x_i P_{(j, \vec{u})} &= P_{(i, j, \vec{u})} + \sum_{k=1}^d B_{ij}^{k} P_{(k, \vec{u})} + \delta_{ij} (1 + C_{i, u(1)}) P_{\vec{u}}, \end{align*} where $C_{ij}, B_{ij}^{k}$ satisfy the same conditions as in part (c). \item There exist symmetric matrices $T_i$ and a diagonal non-negative matrix $C$ with $(T_i \otimes I) C = C (T_i \otimes I)$ such that $\phi$ has a representation $\phi_{C, \set{T_i}}$ as a Fock state of Definition~\ref{Defn:Fock-state}. \end{enumerate} We call such states \emph{free Meixner states}. \end{Thm} \begin{proof} The equivalence (a)~$\Leftrightarrow$~(b) follows from Lemma~4 of \cite{AnsMulti-Sheffer} and Theorem 3.21 of \cite{AnsAppell}, neither of which relied on the assumption that $\phi$ is faithful. The equivalence (d)~$\Leftrightarrow$~(e) follows from the equivalence between the more general Fock space construction and the more general recursion relation in Theorem~\ref{Thm:Monic-states}. \medskip\noindent (e) $\Rightarrow$ (c). By Theorem~\ref{Thm:Cumulants}, \[ R(\mb{z}) = \sum_{j, l = 1}^d \biggl( \ip{e_j}{e_l} z_j z_l + \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{S_{\vec{u}} e_l} z_j z_{\vec{u}} z_l \biggr). \] Therefore \[ D_j R(\mb{z}) = \sum_{l = 1}^d \biggl( \ip{e_j}{e_l} z_l + \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{S_{\vec{u}} e_l} z_{\vec{u}} z_l \biggr) \] and \[ \begin{split} D_i D_j R(\mb{z}) & = \ip{e_j}{e_i} + \sum_{l = 1}^d \biggl( \ip{e_j}{S_i e_l} z_l + \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{S_i S_{\vec{u}} e_l} z_{\vec{u}} z_l \biggr) \\ & = \ip{e_j}{e_i} + \sum_{l = 1}^d \biggl( \ip{e_j}{T_i e_l} z_l + \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{(T_i + \tilde{a}_i) S_{\vec{u}} e_l} z_{\vec{u}} z_l \biggr) \\ & = \ip{e_j}{e_i} + \sum_{l = 1}^d \biggl( \ip{e_j}{T_i e_l} z_l + \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{T_i S_{\vec{u}} e_l} z_{\vec{u}} z_l \biggr) + \sum_{l = 1}^d \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{\tilde{a}_i S_{\vec{u}} e_l} z_{\vec{u}} z_l \\ & = \ip{e_j}{e_i} + \sum_{l = 1}^d \biggl( \sum_{k=1}^d \ip{e_j}{T_i e_k} \ip{e_k}{e_l} z_l + \sum_{\abs{\vec{u}} \geq 1} \sum_{k=1}^d \ip{e_j}{T_i e_k} \ip{e_k}{S_{\vec{u}} e_l} z_{\vec{u}} z_l \biggr) \\ &\quad + \sum_{l = 1}^d \sum_{\abs{\vec{u}} \geq 1} \ip{e_j}{\tilde{a}_i S_{\vec{u}} e_l} z_{\vec{u}} z_l \end{split} \] where in the last step we have used the fact that $\set{e_k}$ form an orthonormal basis. Using Lemma~\ref{Lemma:Factor2}, for $n \geq 4$ and $\vec{u}$ a multi-index on $\set{3, \ldots, n-1}$ \[ \begin{split} \ip{e_j}{\tilde{a}_i S_{\vec{u}} e_l} & = \ip{\Omega}{a_j^- \tilde{a}_i S_{\vec{u}} e_l} \\ & = C_{ij} \sum_{k=3}^{n-1} \sum_{\begin{subarray}{l} W_1 \in \mc{W}_{\set{1, k+1, \ldots, n}}'\bigl(j, (\vec{u}:\set{k+1, \ldots, n-1}), l\bigr) \\ W_2 \in \mc{W}_{\set{2, \ldots, k}}'\bigl(i, (\vec{u}:\set{3, \ldots, k})\bigr) \end{subarray}} \ip{\Omega}{W_1 \Omega} \ip{\Omega}{W_2 \Omega} \\ & = C_{ij} \sum_{k=3}^{n-1} \sum_{W_1 \in \mc{W}_{\set{1, k+1, \ldots, n}}'\bigl((j, \vec{w}, l)\bigr)} \sum_{W_2 \in \mc{W}_{\set{2, \ldots, k}}'\bigl((i, \vec{v})\bigr)} \ip{e_j}{W_1 e_l} \ip{e_i}{W_2 e_s}, \end{split} \] where $\vec{v} = (\vec{u}:\set{3, \ldots, k-1})\bigr)$, $s = u(k)$, and $\vec{w} = (\vec{u}:\set{k+1, \ldots, n})\bigr)$. Thus \[ \begin{split} D_i D_j R(\mb{z}) & = \ip{e_j}{e_i} + \sum_{l = 1}^d \left( \sum_{k=1}^d \ip{e_j}{T_i e_k} \ip{e_k}{e_l} z_l + \sum_{\abs{\vec{u}} \geq 1} \sum_{k=1}^d \ip{e_j}{T_i e_k} \ip{e_k}{S_{\vec{u}} e_l} z_{\vec{u}} z_l \right) \\ &\quad + C_{ij} \sum_{l = 1}^d \sum_{(\vec{v}, s, \vec{w})} \sum_{W_1 \in \mc{W}_{\set{1, k+1, \ldots, n}}'\bigl((j, \vec{w}, l)\bigr)} \sum_{W_2 \in \mc{W}_{\set{2, \ldots, k}}'\bigl((i, \vec{v})\bigr)} \ip{e_j}{W_1 e_l} \ip{e_i}{W_2 e_s} z_{\vec{v}} z_s z_{\vec{w}} z_l \\ & = \ip{e_j}{e_i} + \sum_{k=1}^d \ip{e_j}{T_i e_k} D_k R(\mb{z}) + \sum_{l = 1}^d \sum_{\substack{\vec{u} = (\vec{v}, s, \vec{w}) \\ \abs{\vec{v}}, \abs{\vec{w}} \geq 0}} C_{ij} \ip{e_i}{S_{\vec{v}} e_s} \ip{e_j}{S_{\vec{w}} e_l} z_{\vec{v}} z_s z_{\vec{w}} z_l \\ & = \ip{e_j}{e_i} + \sum_{k=1}^d \ip{e_j}{T_i e_k} D_k R(\mb{z}) + C_{ij} D_i R(\mb{z}) D_j R(\mb{z}). \end{split} \] The conditions on the coefficients in part (c) are equivalent to the conditions on the matrices in part (e). \medskip\noindent (c) $\Rightarrow$ (e). Since the states are assumed to have zero means, the corresponding free cumulant generating functions have no linear terms. In that case, a free cumulant generating function $R$, and so the corresponding state $\phi$, are completely determined by equations~\eqref{PDE}. Moreover, for any choice of $\set{C_{ij}, B_{ij}^k}$ subject to the conditions of part (c), if \[ T_i (e_j) = \sum_{k=1}^d B_{ij}^k e_k \] and \[ C(e_i \otimes e_j) = C_{ij} \ e_i \otimes e_j, \] then those equations are satisfied by $R_{\phi_{C, \set{T_i}}}$. So the states whose free cumulant generating functions satisfy the equations in part (c) are exactly the states in part (e). \medskip\noindent (b) $\Rightarrow $ (c). $\phi$ has a MOPS, so by Theorem~\ref{Thm:Monic-states}, $\phi = \phi_{\mc{C}, \set{\mc{T}_i}}$ for some $\set{\mc{C}^{(k)}, \mc{T}^{(k)}_i}$. Thus, $\phi$ is the joint distribution of the operators $(\mc{X}_1, \ldots, \mc{X}_d)$ on the Hilbert space $\mc{F}_{\mc{C}}(\mc{H})$, with \[ \mc{X}_i = a_i^+ + \mc{T}_i + a_i^- \mc{C}. \] Note that since $\phi$ has means zero and identity covariance, $\mc{T}_i^{(0)} = 0$ and $\mc{C}^{(1)} = I$. Using notation from Section~\ref{Subsubsec:General-Fock}, and denoting \[ (DR)_{\vec{u}}(\mb{z}) = D_{u(1)} R (\mb{z}) \ldots D_{u(\abs{\vec{u}})} R (\mb{z}) \] and \[ e_{\vec{u}} = e_{u(1)} \otimes \ldots \otimes e_{u(\abs{\vec{u}})}, \] we see that \[ \begin{split} & \Bigl(1 - \mb{X} \cdot \mb{z} + R(\mb{z}) \Bigr) \Bigl( \Omega + \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} \Bigr) \\ &\quad = \Omega - \sum_{i=1}^d z_i e_i + R(\mb{z}) \Omega + \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} + R(\mb{z}) \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} \\ &\qquad - \sum_{i=1}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) e_{(i,\vec{u})} - \sum_{i=1}^d \Bigl(z_i D_i R(\mb{z}) \Omega + \sum_{\vec{u}} z_i (D_i R)(\mb{z}) (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} \Bigr) \\ &\qquad - \sum_{i}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) \mc{T}_i (e_{\vec{u}}) - \sum_{i}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) a_i^- (\mc{C} - I) e_{\vec{u}}. \end{split} \] Since for any function $G$ with zero constant term, \begin{equation} \label{Integral} \sum_{i=1}^d z_i D_i G(\mb{z}) = G(\mb{z}), \end{equation} the preceding expression equals \[ \begin{split} & = \Omega - \sum_{i=1}^d z_i e_i + \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} - \sum_{i=1}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) e_{(i,\vec{u})} \\ &\quad - \sum_{i}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) \mc{T}_i (e_{\vec{u}}) - \sum_{i}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) a_i^- (\mc{C} - I) e_{\vec{u}}. \end{split} \] Using the expansions \eqref{Expansion-T} and \eqref{Expansion-C} from Theorem~\ref{Thm:Monic-states}, we now continue the equation as \[ \begin{split} & = \Omega - \sum_{i=1}^d z_i e_i + \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} - \sum_{i=1}^d \sum_{\vec{u}} z_i (DR)_{\vec{u}}(\mb{z}) e_{(i,\vec{u})} \\ &\quad - \sum_{i,j,k=1}^d z_i \Bigl(B_{i, k, j} D_k R(\mb{z}) e_j + \sum_{\vec{u}, \vec{w}} B_{i, (k, \vec{u}), (j, \vec{w})} D_k R(\mb{x}) (DR)_{\vec{u}}(\mb{z}) e_{(j,\vec{w})} \Bigr) \\ &\quad - \sum_{i,j=1}^d z_i \Bigl( (C_{(i,j)} - 1) D_i R(\mb{z}) D_j R(\mb{z}) e_j + \sum_{\vec{u}} (C_{(i,j, \vec{u})} - 1) D_i R(\mb{z}) D_j R(\mb{z}) (DR)_{\vec{u}}(\mb{z}) e_{(j, \vec{u})} \end{split} \] which can be re-organized as \[ \begin{split} & = \Omega + \sum_{j=1}^d \Bigl[D_j R(\mb{z}) - \sum_{i=1}^d z_i \Bigl(\delta_{ij} + \sum_{k=1}^d B_{i, k, j} D_k R(\mb{z}) + (C_{(i,j)} - 1) D_i R(\mb{z}) D_j R(\mb{z}) \Bigr) \Bigr] e_j \\ &\quad + \sum_{j=1}^d \sum_{\vec{u}} \Bigl[D_j R(\mb{z}) (DR)_{\vec{u}}(\mb{z}) - \sum_{i=1}^d z_i \Bigl(\delta_{ij} (DR)_{\vec{u}}(\mb{z}) \\ &\qquad + \sum_{k=1}^d \sum_{\vec{w}} B_{i, (k, \vec{u}), (j, \vec{w})} D_k R(\mb{x}) (DR)_{\vec{w}}(\mb{z}) + (C_{(i,j, \vec{u})} - 1) D_i R(\mb{z}) D_j R(\mb{z}) (DR)_{\vec{u}}(\mb{z})\Bigr)\Bigr] e_{(j, \vec{u})}. \end{split} \] Using equation~\eqref{Integral} again, this equals \begin{equation} \label{Intermediate} \begin{split} & = \Omega + \sum_{i,j=1}^d \Bigl[D_i D_j R(\mb{z}) - \Bigl(\delta_{ij} + \sum_{k=1}^d B_{i, k, j} D_k R(\mb{z}) + (C_{(i,j)} - 1) D_i R(\mb{z}) D_j R(\mb{z}) \Bigr) \Bigr] z_i e_j \\ &\quad + \sum_{i,j=1}^d \sum_{\vec{u}} \Bigl[D_i D_j R(\mb{z}) (DR)_{\vec{u}}(\mb{z}) - \Bigl(\delta_{ij} (DR)_{\vec{u}}(\mb{z}) \\ &\qquad + \sum_{k=1}^d \sum_{\vec{w}} B_{i, (k, \vec{u}), (j, \vec{w})} D_k R(\mb{x}) (DR)_{\vec{w}}(\mb{z}) + (C_{(i,j, \vec{u})} - 1) D_i R(\mb{z}) D_j R(\mb{z}) (DR)_{\vec{u}}(\mb{z})\Bigr)\Bigr] z_i e_{(j, \vec{u})}. \end{split} \end{equation} If the polynomials $\set{P_{\vec{u}}}$ with the generating function~\eqref{Generating} from part (b) are orthogonal, then \[ \sum_{\abs{\vec{u}} \geq 0} P_{\vec{u}}(\mb{x})(DR)_{\vec{u}}(\mb{z}) = \Bigl( 1 - \mb{x} \cdot \mb{z} + R(\mb{z}) \Bigr)^{-1}, \] and \begin{equation} \label{Polynomials-vectors} P_{\vec{u}} (\mb{X}) \Omega = e_{\vec{u}}, \end{equation} so that \begin{equation} \label{Generating-Omega} \Bigl(1 - \mb{X} \cdot \mb{z} + R(\mb{z}) \Bigr) \Bigl( \Omega + \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{\vec{u}} \Bigr) = \Omega. \end{equation} Equating to zero the coefficient of $z_i e_j$ in equation~\eqref{Intermediate}, we get exactly equation~\eqref{PDE} from part (c), with $B_{ij}^k = B_{i,k,j}$ and $C_{uj} = C_{(i,j)} - 1$. The conditions on the coefficients follow from the general conditions in Theorem~\ref{Thm:Monic-states}. \medskip\noindent (e) $\Rightarrow$ (b). If $\phi = \phi_{C, \set{T_i}}$, it follows that in equation~\eqref{Intermediate}, $B_{i, (k, \vec{u}), (j, \vec{w})} = B_{ij}^k \delta_{\vec{u}, \vec{w}} $ and $C_{i, j, \vec{u}} = 1 + C_{ij}$. Then that expression equals to \[ \begin{split} = \Omega + \sum_{i,j=1}^d & \Bigl[D_i D_j R(\mb{z}) - \Bigl(\delta_{ij} + \sum_{k=1}^d B_{ij}^k D_k R(\mb{z}) + C_{ij} D_i R(\mb{z}) D_j R(\mb{z}) \Bigr) \Bigr] \\ & \times z_i \Bigl[e_j + \sum_{\vec{u}} (DR)_{\vec{u}}(\mb{z}) e_{(j, \vec{u})} \Bigr] = \Omega \end{split} \] since part (e) $\Rightarrow $ (c). So equation~\eqref{Generating-Omega} holds. As a result, for polynomials with the generating function~\eqref{Generating}, \[ \Bigl(1 + \sum_{\abs{\vec{u}} \geq 0} P_{\vec{u}}(\mb{X}) z_{\vec{u}} \Bigr) \Omega = \Omega + \sum_{\abs{\vec{u}} \geq 0} z_{\vec{u}} e_{\vec{u}}. \] Thus equation~\eqref{Polynomials-vectors} holds, and so the polynomials are orthogonal. \end{proof} \subsection{Nontrivial covariance and other extensions} \label{Subsec:Covariance} In this section we consider a number of constructions and examples that involve free Meixner states with non-trivial covariances. We still assume that they have zero means (for simplicity); if desired, the means $p_1, \ldots, p_d$ can easily be incorporated into the operator model by considering the operators $(X_1 + p_1, \ldots, X_d + p_d)$ instead, and the corresponding combinatorics will involve all non-crossing partitions $\NC(n)$ rather than the non-crossing partitions without singletons $\NC_0(n)$. \medskip\noindent On the other hand, Theorem~\ref{Thm:Monic-states} requires that for any state with MOPS, the covariance matrices have to be diagonal. But now we allow \begin{equation} \label{Covariance} \psi \left[ x_i^2 \right] = t_i. \end{equation} Note that degenerate variances $\psi \left[ x_i^2 \right] = 0$ are still not permitted. \subsubsection{Dilations} Let $\phi$ be a Meixner state, and fix positive numbers $(t_1, t_2, \ldots, t_d)$. Let $\psi$ be the state defined by the $\mf{R}$-linear extension of \[ \psi \left[ P(x_1, \ldots, x_d) \right] = \state{P(t_1 x_1, \ldots, t_d x_d)}. \] Note that equation~\eqref{Covariance} holds. It is easy to see that if $\set{P_{\vec{u}}}$ is a MOPS for $\phi$, then \[ Q_{\vec{u}}(x_1, \ldots, x_d) = t_{\vec{u}} P_{\vec{u}}(x_1/t_1, \ldots, x_d/t_d) \] is a MOPS for $\psi$. \medskip\noindent We now briefly state how the results of Theorem~\ref{Thm:Meixner} get modified for $\psi$. The generating function for the MOPS still has the same ``resolvent'' form, and any state with MOPS and such a generating function arises as a dilation of a free Meixner state. \[ R_\psi \left[ P(x_1, \ldots, x_d) \right] = R_\phi \left[P(t_1 x_1, \ldots, t_d x_d) \right] \] which shows how to modify the differential equation satisfied by the free cumulant generating function. Similarly, the MOPS satisfy the recursion relation \[ x_i Q_{(j, \vec{u})} = Q_{(i, j, \vec{u})} + \sum_{k=1}^d t_i B_{ij}^{k} Q_{(k, \vec{u})} + \delta_{ij} t_i^2 (1 + C_{i, u(1)}) Q_{\vec{u}}., \] where $\set{B_{ij}^k, C_{ij}}$ were the corresponding coefficients for $\phi$. Finally, suppose that $\phi = \phi_{C, \set{T_i}}$, represented as the joint distribution of $(X_1, X_2, \ldots, X_d)$. In the Hilbert space $\mc{H} = \mf{C}^d$ with an orthonormal basis $\set{e_i}$, let $f_i = t_i e_i$. On the Fock space $\mc{F}_C(\mc{H})$, let \[ a^+_{e_i} := a^+_i, \quad a_{e_i} := a_i^-, \quad T_{e_i} := T_i, \] and extend these definitions $\mf{C}$-linearly to $a^+_f$, $a_f$, $T_f$ for any $f \in \mc{H}$. Let \begin{equation} \label{Dilated-operator} X_{f_i} = a^+_{f_i} + T_{f_i} + a_{f_i}^- + a_{f_i}^- C = t_i X_i. \end{equation} Then $\psi$ is the joint distribution of $(X_{f_1}, X_{f_2}, \ldots, X_{f_d})$, \[ \psi \left[ P(\mb{x}) \right] = \ip{\Omega}{P(\mb{X_f}) \Omega}. \] \subsubsection{Free convolution semigroups} Let $\phi$ be a free Meixner state. For $t > 0$, define a linear functional $\phi^{\boxplus t}$ via its free cumulant functional using relation~\eqref{Cumulants-definition}: \[ R_{\phi^{\boxplus t}} \left[P(\mb{x}) \right] = t R_\phi[P(\mb{x})]. \] Note that $\phi^{\boxplus t} \left[ x_i^2 \right] = t$. The notation reflects the fact that \[ \phi^{\boxplus s} \boxplus \phi^{\boxplus t} = \phi^{\boxplus (s+t)}, \] where $\boxplus$ is the operation of (additive) free convolution; we will not use this property in the paper. Using the methods of Theorem~\ref{Thm:Cumulants}, it is easy to see that $\phi^{\boxplus t}$ is a state (and so positive) if any only if \[ t + \min_{i,j} C_{ij} \geq 0, \] in other words if $t (I \otimes I) + C \geq 0$. In particular, by assumption~\eqref{C-positive}, $\phi^{\boxplus t}$ is always a state for $t \geq 1$; this is typical behavior for free convolution, as indicated by Corollary 14.13 in \cite{Nica-Speicher-book}. $\phi^{\boxplus t}$ is a state for all $t > 0$ if and only if $C \geq 0$; in this case we say that $\phi$ is \emph{freely infinitely divisible}. \medskip\noindent Again, $\phi^{\boxplus t}$ has a MOPS and the generating function for the MOPS still has the same ``resolvent'' form. \[ D_i D_j R_{\phi^{\boxplus t}} = \delta_{ij} t + \sum_{k=1}^d B_{ij}^k D_k R_{\phi^{\boxplus t}} + (C_{ij}/t) D_i R_{\phi^{\boxplus t}} \ D_j R_{\phi^{\boxplus t}} \] and \[ x_i P_{(j, \vec{u})} = P_{(i, j, \vec{u})} + \sum_{k=1}^d B_{ij}^{k} P_{(k, \vec{u})} + \delta_{ij} (t + C_{i, u(1)}) P_{\vec{u}}, \] where $\set{B_{ij}^k, C_{ij}}$ were the corresponding coefficients for $\phi$. Finally, suppose that $\phi = \phi_{C, \set{T_i}}$. On the algebraic Fock space $\Falg(\mc{H})$, define an inner product using the kernel \[ \begin{split} K_C^{(t)} & = \bigl(I^{\otimes (k-2)} \otimes (t I^{\otimes 2} + C)\bigr) \ldots \bigl(I \otimes (t I^{\otimes 2} + C) \otimes I^{\otimes (k-3)}\bigr) \bigl((t I^{\otimes 2} + C) \otimes I^{\otimes (k-2)}\bigr) t \\ & = t^k K_{C/t} \end{split} \] on $\mc{H}^{\otimes k}$, and denote the completion of $\Falg(\mc{H})$ with respect to this inner product $\mc{F}_C^{(t)}(\mc{H})$. Let \begin{equation} \label{Xs} X^{(t)}_i = a^+_{i} + T_{i} + t a_{i}^- + \tilde{a}_i = a^+_{i} + T_{i} + t a_i^- (I + C/t). \end{equation} Then $\phi^{\boxplus t}$ is the joint distribution of $\left(X^{(t)}_1, \ldots, X^{(t)}_d \right)$. \medskip\noindent As constructed above, $\set{X^{(t)}_i}$ are represented on different Hilbert spaces for different $t$. We can combine this construction with an idea from Section 7.2 of~\cite{Sniady-SWN} to represent a whole family of functionals $\set{\phi^{\boxplus t} | 0 < t < 1}$ on a single space. \medskip\noindent A subset $S \subset \set{1, 2, \ldots, n-1}$ can be identified with an \emph{interval partition} $\pi(S) \in \Int(n)$: if $S = \set{i(1), i(2), \ldots, i(k)}$, then \[ \pi = \bigl( \set{1, \ldots, i(1)}, \set{i(1) + 1, \ldots, i(2)}, \ldots, \set{i(k) + 1, \ldots, n} \bigr). \] Consider the vector space $H = \mc{H} \otimes L^\infty([0,1], dx)$ as a subspace of the Hilbert space $\mc{H} \otimes L^2([0,1], dx)$, with the inner product \[ \ip{\eta \otimes f}{\zeta \otimes g} = \ip{\eta}{\zeta} \int_0^1 f(x) g(x) \,dx. \] On its algebraic Fock space $\Falg(H)$, define the inner product \begin{multline*} \ip{(\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_l \otimes f_l)}{(\zeta_1 \otimes g_1) \otimes \ldots \otimes (\zeta_n \otimes g_n)}_C \\ = \delta_{ln} \sum_{\substack{S \subset \set{1, \ldots, n-1} \\ \pi(S) = (V_1, V_2, \ldots, V_k)}} \ip{\eta_1 \otimes \ldots \otimes \eta_n}{C^{S^c} \left(\zeta_1 \otimes \ldots \otimes \zeta_n \right)} \prod_{j=1}^{k} \left( \int_{\mf{R}} \left[ \prod_{i \in V_j} f_i(x) g_i(x) \right] \,dx \right), \end{multline*} where $S^c$ is the complement $\set{1, \ldots, n-1} \backslash S$, and \[ C^{S^c} = \prod_{i \in S^c} I^{\otimes (i-1)} \otimes C \otimes I^{\otimes (n-i-1)}. \] Complete with respect to this inner product, to get the Hilbert space $\mc{F}_C (H)$. On this space, define operators \begin{align*} {a_i^+}^{(t)} \bigl((\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_n \otimes f_n)\bigr) & = (e_i \otimes \chf{[0,t)}) \otimes (\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_n \otimes f_n) \\ {a_i^-}^{(t)} \bigl((\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_n \otimes f_n)\bigr) & = \ip{e_i}{\eta_1} \Bigl( \int_0^t f_1(x) \,dx \Bigr) (\eta_2 \otimes f_2) \otimes \ldots \otimes (\eta_n \otimes f_n), \\ T_i^{(t)} \bigl((\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_n \otimes f_n)\bigr) & = (T_i \eta_1 \otimes f_1 \chf{[0,t)}) \otimes (\eta_2 \otimes f_2) \otimes \ldots \otimes (\eta_n \otimes f_n), \\ \tilde{a}_i^{(t)} \bigl((\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_n \otimes f_n)\bigr) & \\ = \bigl((a_i^- C(\eta_1 & \otimes \eta_2)) \otimes (f_1 \chf{[0,t)} f_2)\bigr) \otimes (\eta_3 \otimes f_3) \otimes \ldots \otimes (\eta_n \otimes f_n), \end{align*} where $\chf{[0,t)}$ is the indicator function of the interval $[0,t)$, and let \[ X_i^{(t)} = {a_i^+}^{{(t)}} + T_i^{(t)} + {a_i^-}^{(t)} + \tilde{a}_i^{(t)}. \] By combining Corollary~\ref{Cor:Self-adjoint} with (a slight modification of) Theorem~6 from \cite{Sniady-SWN}, it follows that each $X_i^{(t)}$ is self-adjoint on $\mc{F}_C(H)$. Note that if all $f_i = g_i = \chf{[0,t)}$, then \[ \begin{split} & \ip{(\eta_1 \otimes f_1) \otimes \ldots \otimes (\eta_n \otimes f_n)}{(\zeta_1 \otimes g_1) \otimes \ldots \otimes (\zeta_n \otimes g_n)}_C \\ &\qquad = \sum_{\substack{S \subset \set{1, \ldots, n-1} \\ \pi(S) = (V_1, V_2, \ldots, V_k)}} \ip{\eta_1 \otimes \ldots \otimes \eta_n}{C^{S^c} \left(\zeta_1 \otimes \ldots \otimes \zeta_n \right)} t^k \\ &\qquad = t^n \ip{\eta_1 \otimes \ldots \otimes \eta_n}{\zeta_1 \otimes \ldots \otimes \zeta_n}_{C/t}. \end{split} \] Moreover, each $X_i^{(t)}$ restricted to \[ \mc{F}_C(\mc{H} \otimes \Span{\chf{[0,t)}}) \cong \mc{F}_C^{(t)}(\mc{H}) \] is given by the equation~\eqref{Xs}, and so $\phi^{\boxplus t}$ is the joint distribution of $\left(X^{(t)}_1, \ldots, X^{(t)}_d \right)$. \subsubsection{Rotations} \label{Subsubsec:Rotations} Let $O = (O_{ij})$ be an orthogonal $d \times d$ matrix. Let \[ O^T \mb{x} = \left( \sum_{i=1}^d O_{i1} x_i, \ldots, \sum_{i=1}^d O_{id} x_i \right) \] and \begin{equation} \label{Change-of-variable} \phi^O \left[ P(\mb{x}) \right] = \state{P(O^T \mb{x})}. \end{equation} We call $\phi^O$ a rotation of $\phi$. $\phi^O$ is the joint distribution of $(X_{f_1}, \ldots, X_{f_d})$ from~\eqref{Dilated-operator}, where we take \[ f_j = O (e_j) = \sum_{i=1}^d O_{ij} e_i. \] $\phi^O$ need not have a MOPS, since the matrix $C$ need not be diagonal in the basis $\set{f_1, \ldots, f_d}$. In fact, it follows from Lemma~9 of \cite{AnsMulti-Sheffer} that $\phi^O$ has a MOPS for \emph{all} $O$ if and only if $C_{ij} = c$ for all $i, j$, and that in this case $\phi^O$ is also a free Meixner state. It is easy to see that more generally, if $S \subset \set{1, \ldots, d}$ and $C_{ij} = c$ for all $i, j \in S$, then $\phi^O$ is a free Meixner state whenever $O (e_k) = e_k$ for all $k \not \in S$. \subsubsection{Linear transformations} Finally, one can consider a general invertible change of variables \[ A^T \mb{x} = \left( \sum_{i=1}^d A_{i1} x_i, \ldots, \sum_{i=1}^d A_{id} x_i \right) \] and the corresponding state $\phi^A$ defined as in equation~\eqref{Change-of-variable}. $\phi^A$ is the joint distribution of operators from~\eqref{Dilated-operator}, where we take $f_j = A(e_j) = \sum_{i=1}^d A_{ij} e_i$. As an alternative to our definition, one can call free Meixner states all states obtained by a linear transformation of a free Meixner state with MOPS (compare with \cite{Pommeret-Test}). \section{Examples} \label{Section:Examples} \subsection{Free products} In preparation for the examples in this section, for the reader's convenience we explain a key notion from free probability. Again, see \cite{VDN,Nica-Speicher-book} for more details. \medskip\noindent Let $\phi_1, \ldots, \phi_d$ be one-dimensional states on $\mf{R}[x_1], \ldots, \mf{R}[x_d]$, respectively. There is a canonical way to define their \emph{free product state} $\phi$ on $\mf{R} \langle x_1, \ldots, x_d \rangle$. Combinatorially, a natural way to define $\phi$ is via its MOPS. Let $\set{P_n^{(i)}}$ be the MOPS for $\phi_i$. For a multi-index $\vec{u}$, decompose \[ x_{\vec{u}} = x_{v(1)}^{i(1)} x_{v(2)}^{i(2)} \ldots x_{v(k)}^{i(k)}, \] where the consecutive indices $v(j) \neq v(j+1)$, although non-consecutive indices may coincide. Then the MOPS $\set{P_{\vec{u}}}$ for $\phi$ are defined by \[ P_{\vec{u}}(\mb{x}) = \prod_{j=1}^k P_{i(j)}^{(v(j))}(x_{v(j)}). \] For example, \[ P_{1,1,2,1,2}(\mb{x}) = P_2^{(1)}(x_1) P_1^{(2)}(x_2) P_1^{(1)}(x_1) P_1^{(2)}(x_2). \] Note that if one considers polynomials in commuting variables and assumes that \emph{all} $v(j)$ above are different, one gets the usual (Cartesian) product of measures. Also, $\phi$ is a free product state if any only if the elements $x_1, x_2, \ldots, x_d$ are freely independent with respect to $\phi$, in the sense of Voiculescu. This can be taken as the definition of free independence; note that for random variables independent in the usual probabilistic sense, their joint distribution is a product measure. Finally, the crucial property of free cumulant generating functions is their relation to free products: a state $\phi$ is a free product state of $\phi_1, \ldots, \phi_d$ if any only if the free cumulant generating function of $\phi$ decomposes as \[ R_\phi(\mb{z}) = \sum_{i=1}^d R_{\phi_i}(z_i). \] This is often stated as the ``mixed free cumulants are zero'' condition. It is related to the familiar property that the Fourier transform of the joint distribution of independent random variables is the product of their individual Fourier transforms. \medskip\noindent It is easy to see that free product free Meixner states are exactly the free products of one-dimensional free Meixner states, see Remark~6 of \cite{AnsMulti-Sheffer}. Recall that these one-dimensional distributions, as described in that remark, Theorem 4 of \cite{AnsMeixner} and Section 2.2 of \cite{Boz-Bryc}, are known. With variance $t$, they are: the semicircular (free Gaussian) distributions $\frac{1}{2 \pi} \sqrt{4 t - x^2} \,dx$, the Marchenko-Pastur (free Poisson) distributions $\frac{1}{2 \pi} \frac{\sqrt{4t - (x-b)^2}}{1 + (b/t) x} \,dx + \text{ possibly one atom}$, and more generally \[ \frac{1}{2 \pi} \frac{\sqrt{4 (t + c) - (x - b)^2}}{1 + (b/t) x + (c/t^2) x^2} \,dx + \text{ zero, one, or two atoms}, \] depending on the particular values of $b,c,t$. \subsection{Semicircular systems} \label{Subsec:Semicircular} Let $C = 0$ and all $T_i = 0$. Then \[ S_i = a_i^+ \] and \[ \Cum{x_i x_{\vec{u}} x_j} = \ip{e_i}{S_{\vec{u}} e_j} = 0 \] for $\abs{\vec{u}} \geq 1$. Thus in distribution, all $S_i \sim 0$. Only second-order free cumulants of $(X_1, X_2, \ldots, X_d)$ are non-zero, and $\phi$ is the distribution of a freely independent semicircular system, the free analog of the standard $d$-dimensional Gaussian distribution. \subsection{Free Poisson states} Let $C = 0$ and $T_i$ arbitrary. Then \[ S_i = a_i^+ + T_i \] and \[ \Cum{x_i x_{\vec{u}} x_j} = \ip{e_i}{S_{\vec{u}} e_j} = \ip{e_i}{T_{\vec{u}} e_j}. \] Thus in distribution, $(S_1, S_2, \ldots, S_d) \sim (T_1, T_2, \ldots, T_d)$. It is appropriate to say that in this case, the joint distribution $\phi$ of $(X_1, X_2, \ldots, X_d)$ is $d$-dimensional free Poisson. In \cite{AnsMulti-Sheffer} we showed that if $\phi$ is tracial, then $\phi$ is a rotation of a free product of one-dimensional free Poisson distributions. Whether or not $\phi$ is tracial, the vector $\Omega$ is cyclic and separating for the von Neumann algebra $W^\ast(X_1, X_2, \ldots, X_d)$. \subsection{Free product states} \label{Subsec:Free-products} For two vectors $f, g$, denote by $E_{f,g}$ the corresponding rank one operator, \[ E_{f,g}(h) = f \ip{g}{h}. \] For an orthonormal basis $\set{f_i}$, $E_{f_i, f_j}$ are the corresponding matrix units. For the standard basis $\set{e_i}$, we will denote these simply by $E_{ij}$. In particular, $E_{ii}$ is the orthogonal projection onto $e_i$. \medskip\noindent Let $C(e_i \otimes e_j) = c_i \delta_{ij} (e_i \otimes e_j)$, and let $T_i = b_i E_{ii}$. Then $S_i$ acts entirely on the subspace $\mc{F}_{c_i}(\Span{e_i})$, on which it equals \[ S_i = a_i^+ + b_i + c_i a_i^-. \] $a_i^+ + c_i a_i$ has the centered semicircular distribution with variance $c_i$ (note that on $\mc{F}_{C}(\mc{H})$, this operator is not symmetric, so its \emph{star}-distribution is different from the semicircular one). Therefore $S_i$ has the semicircular distribution with mean $b_i$ and variance $c_i$. Also, it follows that \[ \Cum{x_i x_{\vec{u}} x_j} = \ip{e_i}{S_{\vec{u}} e_j} = 0 \] unless \[ i = u(1) = \ldots = u(n) = j. \] In other words, all the mixed free cumulants of $(X_1, X_2, \ldots, X_d)$ are zero. This says precisely that their joint distribution $\phi$ is a free product of the distributions of each of $X_1, X_2, \ldots, X_d$. Each of these, in turn, is a one-dimensional free Meixner distribution, whose free cumulants are, up to a shift of index, the moments of the semicircular distribution with mean $b_i$ and variance $c_i$. \subsection{Exponentiated semicircular systems} Let $C(e_i \otimes e_j) = c_i (e_i \otimes e_j)$ and $T_i = b_i I$. Then \[ S_i = a_i^+ + c_i a_i^- + b_i I, \] again the distribution of $S_i$ is the semicircular distribution with mean $b_i$ and variance $c_i$, but now the operators $S_i$ themselves are freely independent with respect to the state $\phi$, so that their joint distribution is a free product. The joint distribution of $(X_1, X_2, \ldots, X_d)$ is \emph{not} a free product (typically, not even tracial); it was described in the last section of \cite{AnsMulti-Sheffer}. \medskip\noindent Note that the preceding two examples make sense for $-1 \leq c_i < 0$, except that one loses the interpretation of $S_i$ as having a semicircular distribution with variance $c_i$, and the resulting states are not freely infinitely divisible. \begin{Remark} The constructions in the previous two examples coincide in the one-dimensional case. That case, and in particular the corresponding free cumulants, were also considered in \cite{AnsMeixner} and described completely in \cite{Boz-Bryc}. Moreover, many one-dimensional free Meixner distributions arise as limits in the central and Poisson limit theorems for the $t$-transformed free convolution, in the sense of \cite{Boz-Wys}; that paper also contains a Fock space construction which coincides with the one-dimensional version of the one in Section~\ref{Subsec:Fock2}. \end{Remark} \subsection{Free multinomial states} \label{Subsec:Free-multinomial} It is well known that the Bernoulli distribution \[ (1-p) \delta_0 + p \delta_1 \] is a Meixner distribution. It was noted in \cite{AnsMeixner} that it is also a (one-dimensional) free Meixner distribution. Moreover, the binomial distributions, which are convolution powers \[ \bigl((1-p) \delta_0 + p \delta_1\bigr)^{\ast n} = \sum_{k=0}^n \binom{n}{k} (1-p)^{n-k} p^k \delta_k \] of the Bernoulli distribution, are all Meixner, and the free binomial distributions, which are free convolution powers $\bigl((1-p) \delta_0 + p \delta_1\bigr)^{\boxplus n}$ of the Bernoulli distribution are free Meixner. In fact, it was noted in \cite{Boz-Bryc} that $\bigl((1-p) \delta_0 + p \delta_1\bigr)^{\boxplus t}$ are free Meixner for all real $t \geq 1$. \medskip\noindent It is also well-known that the multinomial distributions are Meixner \cite{Pommeret-Test}. In particular, the basic multinomial distribution \begin{equation} \label{Multinomial} p_1 \delta_{e_1} + p_2 \delta_{e_2} + \ldots p_d \delta_{e_d} \end{equation} on $\mf{R}^d$ has this property. We now show that it, and so the free semigroup it generates, also induce free Meixner states. In this example, it is natural to consider the state with non-trivial means and a non-diagonal covariance matrix; an actual free Meixner state can be obtained from it by an affine transformation as in Section~\ref{Subsec:Covariance}. \medskip\noindent In the Fock space construction of Section~\ref{Subsec:Fock2}, take $\dim \mc{H} = d-1$ rather than $d$, and \[ C(e_i \otimes e_j) = - e_i \otimes e_j \] for all $i, j$, so that $C_{ij} = -1$. In this case the induced inner product on the Fock space $\Falg(\mc{H})$ is degenerate, and the vector space factors through to simply $\mc{F}_C(\mc{H}) = \mf{C} \oplus \mc{H}$. \emph{In this example only}, choose (linearly dependent) vectors $\set{e_i | i = 1, 2, \ldots d}$ in $\mc{H}$ that are not orthonormal, but instead satisfy \begin{align*} \ip{e_i}{e_i} & = p_i (1 - p_i), \\ \ip{e_i}{e_j} & = - p_i p_j, \end{align*} where \[ p_i > 0, \qquad i = 1, 2, \ldots, d, \qquad p_1 + p_2 + \ldots + p_{d} = 1. \] Since these numbers are the covariances of the centered version of the basic multinomial distribution \eqref{Multinomial}, the corresponding matrix is positive semi-definite and so the $\set{e_i}$ can be chosen in this fashion. \medskip\noindent Let \begin{align*} T_i(e_i) & = (1 - 2 p_i) e_i, \\ T_i(e_j) & = - p_i e_j - p_j e_i, \end{align*} and define \[ X_i = a_i^+ + T_i + a_i^- + a_i^- C \] as usual, except that $a_i^+ = 0$ on $\mc{H}$. In other words, for $Y_i = X_i + p_i$, \begin{equation} \label{Multinomial-operators} \begin{split} Y_i \Omega & = e_i + p_i \Omega, \\ Y_i e_i & = (1 - p_i) [e_i + p_i \Omega], \\ Y_i e_j & = - p_j [e_i + p_i \Omega]. \end{split} \end{equation} \begin{Prop} $Y_i$ is an orthogonal projection of $\mf{C} \oplus \mc{H}$ onto $\Span{e_i + p_i \Omega}$. These projections are orthogonal among themselves and their sum is the identity operator. Their joint distribution with respect to the state $\phi_{C, \set{T_i}}$ from Definition~\ref{Defn:Fock-state} is the basic multinomial distribution \eqref{Multinomial}. In particular, $\state{Y_i} = p_i$. The free cumulant generating function of $\phi$ satisfies the differential equation \[ \begin{split} D_i D_j R & = (\delta_{ij} p_i - p_i p_j) + (\delta_{ij} - p_j) D_i R - p_i D_j R - D_i R \ D_j R \\ & = \delta_{ij} (D_i R + p_i) - (D_i R + p_i) (D_j R + p_j). \end{split} \] \end{Prop} \begin{proof} $Y_i$ is self-adjoint, its image is $\Span{e_i + p_i \Omega}$, and \[ Y_i(e_i + p_i \Omega) = (1 - p_i) [e_i + p_i \Omega] + p_i [e_i + p_i \Omega] = e_i + p_i \Omega, \] so it is an orthogonal projection onto its image. \[ \ip{e_i + p_i \Omega}{e_j + p_i \Omega} = - p_i p_j + p_i p_j = 0, \] so these subspaces, and therefore projections onto them, are orthogonal. $\mf{C} \oplus \mc{H}$ has dimension $d$, therefore the sum $\sum_{i=1}^{d} Y_i$ is identity. It also follows that \[ \state{Y_{\vec{u}}} = \begin{cases} p_i, & i = u(1) = u(2) = \ldots, \\ 0, & \text{ otherwise}. \end{cases} \] Therefore, the joint distribution of $(Y_1, Y_2, \ldots, Y_{d})$ with respect to $\phi$ is the basic multinomial distribution. The last part follows from the operator representation. \end{proof} \begin{Defn} \label{Defn:Free-multinomial} Free multinomial states are the states $\set{\phi^{\boxplus t} | t \geq 1}$, where $\phi$ is the basic multinomial distribution~\eqref{Multinomial}. Note that $\phi^{\boxplus n}$ is the joint distribution of the sum of $n$ $d$-tuples of orthogonal projections. \end{Defn} \begin{Remark} Since the Bernoulli distribution is both classical and free Meixner, one may conjecture that it is in some sense also $q$-Meixner (for $0 \leq q \leq 1$, with $q=1$ corresponding to the classical case and $q=0$ corresponding to the free case). The meaning of this term is not well-defined, but see for example Section 4.3 of \cite{AnsAppell}. Indeed, the recursion relation for its orthogonal polynomials is of the $q$-Meixner form \begin{align*} x P_0 & = P_1 + p, \\ x P_1 & = P_2 + (1-p) P_1 + p (1-p) P_0, \\ x P_n & = P_{n+1} + (1-p) [n]_q P_n + [n]_q p (1-p)(1 - [n-1]_q) P_{n-1} \end{align*} independently of $q$, as long as the degree of the polynomial $n \leq 1$, which suffices since \[ L^2\bigl((1-p)\delta_0 + p \delta_1\bigr) \] is $2$-dimensional. One may also hope that its $q$-cumulant generating function would then satisfy the equation \[ D_q^2 R^{(q)} = D_q R^{(q)} - (D_q R^{(q)})^2, \] where $D_q$ is the $q$-derivative \[ D_q(f)(z) = \frac{f(z) - f(qz)}{(1-q) z}, \] and \[ R^{(q)} = \sum_{n=1}^\infty \frac{1}{[n]_q!} \alpha_n z^n. \] The corresponding recursion for its $q$-cumulants is \[ \alpha_{n+2} = \alpha_{n+1} - \sum_{i=0}^n \left[ \begin{matrix} n \\ i\end{matrix} \right]_q \alpha_{i+1} \alpha_{n-i+1} \] (compare with Remark 5.4 of \cite{Boz-Bryc}), with the initial condition $\alpha_1 = p$. Using Maple, it is easy to calculate the first $5$ cumulants. Unfortunately, the fifth $q$-cumulant of the Bernoulli distribution calculated in this fashion differs from its fifth $q$-cumulant in the sense of Section 6 of \cite{AnsQCum}. \end{Remark} \subsection{Tracial examples} If $\phi$ is a state on a non-commutative algebra $\mc{A}$, one says that $\phi$ is \emph{tracial}, or \emph{a trace}, if for any $x, y \in \mc{A}$, \[ \state{x y} = \state{y x}. \] Tracial states play a crucial role, for example, in the theory of von Neumann algebras. \begin{Lemma} Let $\phi = \phi_{C, \set{T_i}}$ be a free Meixner state, represented as the joint distribution of operators $(X_1, \ldots, X_d)$. Suppose that $\phi$ is tracial. Then for all $i, j$, \begin{equation} \label{Trace1} T_i e_j = T_j e_i \end{equation} and \begin{equation} \label{Commutator} T_i T_j - T_j T_i = C_{ji} E_{ij} - C_{ij} E_{ji}. \end{equation} \end{Lemma} \begin{proof} Since $\phi$ is tracial, for all $i, j, k$, \[ \state{X_i X_j X_k} = \ip{e_i}{T_j e_k} = \ip{e_j}{T_k e_i} = \ip{e_i}{T_k e_j}, \] so for all $j, k$, \[ T_j e_k = T_k e_j. \] Similarly, for all $i, j, k, l$, \[ \begin{split} \state{X_i X_j X_k X_l} & = \ip{e_i}{e_j} \ip{e_k}{e_l} + \ip{e_i}{e_l} \ip{e_j}{e_k} (1 + C_{kl}) + \ip{e_i}{T_j T_k e_l} \\ & = \ip{e_j}{e_k} \ip{e_l}{e_i} + \ip{e_j}{e_i} \ip{e_k}{e_l} (1 + C_{li}) + \ip{e_j}{T_k T_l e_i}, \end{split} \] so \[ \ip{e_i}{e_l} \ip{e_j}{e_k} C_{kl} + \ip{e_i}{T_j T_k e_l} = \ip{e_j}{e_i} \ip{e_k}{e_l} C_{li} + \ip{e_i}{T_l T_k e_j}. \] Using equation~\eqref{Trace1} and the orthonormality of $\set{e_i}$, \[ \ip{e_i}{e_l} \ip{e_j}{e_k} C_{jl} + \ip{e_i}{T_j T_l e_k} = \ip{e_j}{e_i} \ip{e_k}{e_l} C_{lj} + \ip{e_i}{T_l T_j e_k}, \] so \[ \ip{e_j}{e_k} C_{jl} e_l + T_j T_l e_k = \ip{e_k}{e_l} C_{lj} e_j + T_l T_j e_k \] and \[ T_j T_l - T_l T_j = C_{lj} E_{jl} - C_{jl} E_{lj}. \qedhere \] \end{proof} \begin{Ex} A general theorem of Voiculescu (Proposition~2.5.3 in \cite{VDN}) implies that the free product states from Example~\ref{Subsec:Free-products} are tracial. It is also easy to see that any rotation of a tracial state is tracial. \end{Ex} \begin{Lemma} \label{Lemma:Tracial-semigroup} If $\phi$ is tracial, then $\phi^{\boxplus t}$ is tracial for all $t$ for which it is defined. \end{Lemma} \begin{proof} It is easy to see directly from the defining equation~\eqref{Cumulants-definition} that a state $\phi$ is tracial if any only if its free cumulant generating functional $R_\phi$ is tracial. The combinatorial reason is that if a partition $\pi$ is non-crossing when points $\set{1, 2, \ldots, n}$ are placed on a line, it is also non-crossing when they are placed on a circle. The lemma follows from this fact and the defining relation for $\phi^{\boxplus t}$ \[ R_{\phi^{\boxplus t}} \left[ x_{\vec{u}} \right] = t R_{\phi} \left[ x_{\vec{u}} \right]. \qedhere \] \end{proof} \begin{Prop} All free multinomial states are tracial. \end{Prop} \begin{proof} Since the operators $\set{Y_i}$ defined in equation~\eqref{Multinomial-operators} commute, the basic multinomial distribution is tracial; in fact, it factors through to a state on commutative polynomials $\mf{R}[x_1, \ldots, x_d]$ corresponding to the basic multinomial measure~\eqref{Multinomial}. It follows from Lemma~\ref{Lemma:Tracial-semigroup} that all the other free multinomial states are tracial as well. \end{proof} \noindent We conclude the paper with three further results on when Meixner states are traces. The first two show that under one set of general assumptions, the only tracial Meixner states are the trivial ones, namely the rotations of free product states. It generalizes Proposition~11 of \cite{AnsMulti-Sheffer}. The last one provides a way to construct a large class of tracial examples that do not come from free products. It generalizes the multinomial example above. \begin{Prop} \label{Prop:Free-products} Let $\phi = \phi_{C, \set{T_i}}$ be a tracial free Meixner state with $C$ diagonal as a $d \times d$ matrix, $C_{ij} = \delta_{ij} c_i$. Then $\phi$ is a rotation of a free product state. \end{Prop} \begin{proof} If $C_{ij} = \delta_{ij} c_i$, then equation~\eqref{Commutator} states that all $T_i, T_j$ commute. Combining this with equation~\eqref{Trace1}, we see that moreover, for some orthonormal basis $\set{f_1, f_2, \ldots, f_d}$, \begin{equation} \label{Diagonalized} T_j = \sum_{i=1}^d \alpha_i \ip{f_i}{e_j} E_{f_i, f_i}. \end{equation} From \[ C (T_j \otimes I) (e_k \otimes e_l) = (T_j \otimes I) C (e_k \otimes e_l) \] it follows that \[ \sum_{i=1}^d \alpha_i \ip{f_i}{e_j} \ip{f_i}{e_k} \ip{f_i}{e_l} c_l (e_l \otimes e_l) = \delta_{kl} \sum_{i,m=1}^d \alpha_i \ip{f_i}{e_j} \ip{f_i}{e_k} \ip{f_i}{e_m} c_l (e_m \otimes e_l). \] Thus for all $k \neq l$, \[ \sum_{i=1}^d \alpha_i \ip{f_i}{e_j} \ip{f_i}{e_k} \ip{f_i}{e_l} c_l = \ip{e_k}{\left( \sum_{i=1}^d \alpha_i \ip{f_i}{e_j} E_{f_i, f_i} \right) e_l} c_l = \ip{e_k}{T_j e_l} c_l = 0. \] It follows that whenever $c_l \neq 0$, $T_j e_l \in \Span{e_l}$, and one can take $f_l = e_l$. So if $S = \set{l | c_l \neq 0}$, then \[ \Span{e_l | l \in S} = \Span{f_l | l \in S} \] is an invariant subspace for all $T_j$. We can choose an orthogonal transformation $O$ so that $O (e_i) = f_i$, in other words \[ O (e_i) = \begin{cases} e_i & \text{ for } i \in S, \text{ that is } c_i \neq 0, \\ f_i & \text{ for } i \not \in S, \text{ that is } c_i = 0. \end{cases} \] Following the comments at the end of Section~\ref{Subsubsec:Rotations}, the state $\phi^O$, which is the joint distribution of $(X_{f_1}, \ldots, X_{f_d})$, is still a tracial free Meixner state. From equation~\eqref{Diagonalized}, \[ T_{f_j} = \alpha_j E_{f_j, f_j}. \] Finally, $C(\eta \otimes \zeta) = 0$ whenever one of $\eta, \zeta \in \Span{e_l | l \not \in S} = \Span{f_l | l \not \in S}$, so \[ C(f_i \otimes f_j) = \begin{cases} C(e_i \otimes e_j) = c_i \delta_{ij} & \text{ if } i, j \in S, \\ 0 & \text{ if one of } i, j \not \in S. \end{cases} \] Thus $C, \set{T_i}$ have the form in Example~\ref{Subsec:Free-products}, and so $\phi^O$ is a free product state. \end{proof} \begin{Cor} Let $\phi = \phi_{C, \set{T_i}}$ be a tracial free Meixner state and $T_i = 0$ for all $i$. Then $\phi$ is a free product state. \end{Cor} \begin{proof} If $T_i = 0$ for all $i$, then it follows from equation~\eqref{Commutator} that $C_{ij} = \delta_{ij} c_i$, and the preceding proposition applies. In this case, a rotation is unnecessary. \end{proof} \noindent Meixner states correspond to quadratic natural exponential families. The following states are free versions of \emph{simple} quadratic natural exponential families in the terminology of \cite{Casalis-Simple-quadratic}, where all such (classical) families were classified. \begin{Prop} Let $C$ be a constant matrix, $C_{ij} = c$ for all $i,j$. Then the necessary conditions \eqref{Trace1} and~\eqref{Commutator} for $\phi$ to be a tracial free Meixner state, namely that $\set{T_i}$ are symmetric matrices, $T_i e_j = T_j e_i$ and \[ (T_i T_j - T_j T_i) = c \bigl( E_{ij} - E_{ji} \bigr), \] are also sufficient. \end{Prop} \begin{proof} If $c=0$, the result follows from Proposition~\ref{Prop:Free-products}. So we will assume that $c \neq 0$. \medskip\noindent For each $n$, let \[ A(n) = \set{\pi \in \NC_0'(n) | 1 \stackrel{\pi}{\sim} 2}. \] For any partition $\sigma \in \NC_0'(n) \backslash A(n)$, define the partition $l(\sigma) \in A(n)$ as follows: if $1 \in B \in \sigma$ and $2 \in C \in \sigma$, let $l(\sigma)$ be the partition with the same classes as $\sigma$ except that $B \cup C$ is a class of $l(\sigma)$. Conversely, any such $\sigma$ can be obtained by starting with $\pi \in A(n)$ with the (unique) outer class $B$, choosing $i \in B$, $2 < i < n$ (if it exists), and taking $\sigma$ to have the same classes as $\pi$ except that \[ B \cap \left( \set{1} \cup \set{i + 1, \ldots, n} \right) \] and \[ B \cap \set{2, \ldots, i} \] are classes of $\sigma$. \medskip\noindent For $\pi \in NC_0(n)$, define the partition $\rho(\pi) \in \NC_0(\set{2, 3, \ldots, n, \bar{1}})$ (where $\bar{1}$ is identified with $n+1$) by \begin{align*} & i \stackrel{\pi}{\sim} j \Leftrightarrow i \stackrel{\rho(\pi)}{\sim} j \text{ for } i, j \neq 1, \\ & 1 \stackrel{\pi}{\sim} j \Leftrightarrow \bar{1} \stackrel{\rho(\pi)}{\sim} j. \end{align*} Clearly \[ \rho(\NC_0'(n)) = \set{\pi \in \NC_0(\set{2, \ldots, n, \bar{1}}) | n \stackrel{\pi}{\sim} \bar{1}}, \] and \[ \rho(A(n)) = \set{\pi \in \NC_0(\set{2, \ldots, n, \bar{1}}) | 2 \stackrel{\pi}{\sim} n \stackrel{\pi}{\sim} \bar{1}}. \] In particular, $\rho(A(n)) \subset \NC_0'(\set{2, \ldots, n, \bar{1}})$. For any partition $\sigma \in \NC_0'(\set{2, \ldots, n, \bar{1}}) \backslash \rho(A(n))$, define the partition $r(\sigma) \in \rho(A(n))$ as follows: if $\bar{1} \in B \in \sigma$ and $n \in C \in \sigma$, let $r(\sigma)$ be the partition with the same classes as $\sigma$ except that $B \cup C$ is a class of $r(\sigma)$. Conversely, any such $\sigma$ can be obtained by starting with $\pi \in \rho(A(n))$ with the (unique) outer class $B$, choosing $i \in B$, $2 < i < n$ (if it exists), and taking $\sigma$ to have the same classes as $\pi$ except \[ B \cap \left( \set{2, \ldots, i} \cup \set{\bar{1}} \right) \] and \[ B \cap \set{i+1, \ldots, n} \] are classes of $\sigma$. \medskip\noindent We will use the usual commutator notation $[T,T'] = T T' - T' T$. Using equation~\eqref{Trace1}, \[ \begin{split} & \ip{e_{u(1)}}{T_{u(2)} T_{u(3)} T_{u(4)} \ldots T_{u(n-2)} T_{u(n-1)} e_{u(n)}} \\ &\quad = \ip{e_{u(2)}}{T_{u(1)} T_{u(3)} T_{u(4)} \ldots T_{u(n-2)} T_{u(n-1)} e_{u(n)}} \\ &\quad = \ip{e_{u(2)}}{\bigl[T_{u(1)}, T_{u(3)}\bigr] T_{u(4)} \ldots T_{u(n-1)} e_{u(n)}} + \ldots \\ &\qquad + \ip{e_{u(2)}}{T_{u(3)} T_{u(4)} \ldots T_{u(n-2)} \bigl[T_{u(1)}, T_{u(n-1)}\bigr] e_{u(n)}} \\ &\qquad + \ip{e_{u(2)}}{T_{u(3)} T_{u(4)} \ldots T_{u(n-1)} T_{u(n)} e_{u(1)}} \end{split} \] Now using equation~\eqref{Commutator}, $[T_i, T_j] = c (E_{ij} - E_{ji})$, \[ \begin{split} & \ip{e_{u(1)}}{T_{u(2)} T_{u(3)} T_{u(4)} \ldots T_{u(n-2)} T_{u(n-1)} e_{u(n)}} \\ &\quad = c \ip{e_{u(2)}}{e_{u(1)}} \ip{e_{u(3)}}{T_{u(4)} \ldots T_{u(n-1)} e_{u(n)}} \\ &\qquad + c \ip{e_{u(2)}}{T_{u(3)} e_{u(1)}} \ip{e_{u(4)}}{T_{u(5)} \ldots T_{u(n-1)} e_{u(n)}} \\ &\qquad + c \ip{e_{u(2)}}{T_{u(3)} \ldots T_{u(n-2)} e_{u(1)}} \ip{e_{u(n-1)}}{e_{u(n)}} + \ldots \\ &\qquad + \ip{e_{u(2)}}{T_{u(3)} \ldots T_{u(n)} e_{u(1)}} \\ &\qquad - c \ip{e_{u(2)}}{e_{u(3)}} \ip{e_{u(1)}}{T_{u(4)} \ldots T_{u(n-1)} e_{u(n)}} \\ &\qquad - c \ip{e_{u(2)}}{T_{u(3)} e_{u(4)}} \ip{e_{u(1)}}{T_{u(5)} \ldots T_{u(n-1)} e_{u(n)}} - \ldots \\ &\qquad - c \ip{e_{u(2)}}{T_{u(3)} \ldots T_{u(n-2)} e_{u(n-1)}} \ip{e_{u(1)}}{e_{u(n)}} \end{split} \] The left-hand-side of the preceding equation is equal to $\Theta(\mb{\hat{1}_n}; \set{1, \ldots, n}, \vec{u})$, where $\mb{\hat{1}_n} \in \NC_0'(n)$ is the partition with a single class. Similarly, using Lemma~\ref{Lemma:Last}, the preceding equation itself states that \[ \begin{split} \Theta(\mb{\hat{1}_n}; \set{1, \ldots, n}, \vec{u}) & = \Theta(\rho(\mb{\hat{1}_n}); \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad + \sum_{i=2}^{n-2} \Theta(\bigl(\set{2, \ldots, i, \bar{1}}, \set{i+1, \ldots, n} \bigr) \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad - \sum_{i=3}^{n-1} \Theta(\bigl(\set{2, \ldots, i}, \set{1, i+1, \ldots, n}\bigr); \set{1, \ldots, n}, \vec{u}), \end{split} \] where for a multi-index $\vec{u} = ((u(1), \ldots, u(n))$, we denote by $\rho(\vec{u}) = (u(2), \ldots, u(n), u(1))$ the multi-index on $\set{2, \ldots, n, \bar{1}}$. Using the descriptions of $l(\sigma)$, $r(\sigma)$ at the beginning of the proof, this in turn equals to \begin{equation} \label{Single-class} \begin{split} & = \Theta(\rho(\mb{\hat{1}_n}); \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad + \sum_{\tau: r(\tau) = \rho(\mb{\hat{1}_n})} \Theta(\tau; \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad - \sum_{\sigma: l(\sigma) = \mb{\hat{1}_n}} \Theta(\sigma); \set{1, \ldots, n}, \vec{u}). \end{split} \end{equation} Using Lemma~\ref{Lemma:Last} again and equation~\eqref{Single-class} applied to the class of $\pi$ containing $1$, we conclude that for $\pi \in A(n)$, \[ \begin{split} c^{-(\abs{\pi} - 1)} \Theta(\pi; \set{1, \ldots, n}, \vec{u}) & = c^{-(\abs{\pi} - 1)} \Theta(\rho(\pi); \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad + c \sum_{\tau: r(\tau) = \rho(\pi)} c^{-(\abs{\tau} - 1)} \Theta(\tau; \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad - c \sum_{\sigma: l(\sigma) = \pi} c^{-(\abs{\sigma} - 1)} \Theta(\sigma; \set{1, \ldots, n}, \vec{u}), \end{split} \] or, since $\abs{\rho(\pi)} = \abs{\pi}$ and $\abs{\tau} = \abs{\sigma} = \abs{\pi} + 1$, \[ \begin{split} \Theta(\pi; \set{1, \ldots, n}, \vec{u}) & = \Theta(\rho(\pi); \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ & + \sum_{\tau: r(\tau) = \rho(\pi)} \Theta(\tau; \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ & - \sum_{\sigma: l(\sigma) = \pi} \Theta(\sigma); \set{1, \ldots, n}, \vec{u}), \end{split} \] Therefore \[ \begin{split} & \ip{e_{u(1)}}{S_{u(2)} \ldots S_{u(n-1)} e_{u(n)}} \\ &\quad = \sum_{\pi \in \NC_0'(n)} \Theta(\pi; \set{1, \ldots, n}, \vec{u}) \\ &\quad = \sum_{\pi \in A(n)} \Bigl( \Theta(\pi; \set{1, \ldots, n}, \vec{u}) + \sum_{\sigma: l(\sigma) = \pi} \Theta(\sigma; \set{1, \ldots, n}, \vec{u}) \Bigr) \\ &\quad = \sum_{\pi \in A(n)} \Bigl( \Theta(\rho(\pi); \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) + \sum_{\tau: r(\tau) = \rho(\pi)} \Theta(\tau; \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \Bigr) \\ &\quad = \sum_{\tau \in \rho(\NC_0'(\set{2, \ldots, n, \bar{1}}))} \Theta(\tau; \set{2, \ldots, n, \bar{1}}, \rho(\vec{u})) \\ &\quad = \ip{e_{u(2)}}{S_{u(3)} \ldots S_{u(n)} e_{u(1)}}. \end{split} \] Using Theorem~\ref{Thm:Cumulants}, we conclude that the free cumulant functional $R_\phi$, and so $\phi$ itself, is tracial. \end{proof} \begin{Ex} For $d=2$, one can take \[ T_1 = \begin{pmatrix} c+1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad T_2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. \] For $d=3$, one can take \[ T_1 = \begin{pmatrix} 0 & c & 0 \\ c & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad T_2 = \begin{pmatrix} c & 0 & 0 \\ 0 & c+1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad T_3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}. \] For $d=4$, one can take \[ T_1 = \begin{pmatrix} c & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \; T_2 = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & c & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \; T_3 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & c & 0 & 0 \\ 0 & 0 & c+1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \; T_4 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix}. \] \end{Ex} \begin{Remark} If $C_{ij} = c$ for all $i, j$, the corresponding Fock space is an interacting Fock space in the sense of \cite{AccBozGaussianization}. If $C_{ij} = c$ and in addition all $T_i = 0$, the von Neumann algebras $W^\ast(X_1, \ldots, X_d)$ were described in \cite{Ricard-t-Gaussian}. \end{Remark} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}
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Jerry Lewis leaves all six children from first marriage out of his will Comedy Star, Jerry Lewis who sadly died last month has clearly stated that all six of the children he shared with first wife Patti Palmer will receive nothing from his estate according to his will, obtained by People. Lewis died from heart failure in August aged 91. Five of the six children are not just left out of the will, they are individually excluded by name. The sixth child to the marriage, Joseph, died in 2009. The will states in no uncertain terms: "I have intentionally excluded Gary Lewis, Ronald Lewis, Anthony Joseph Lewis, Christopher Joseph Lewis, Scott Anthony Lewis, and Joseph Christopher Lewis and their descendants as beneficiaries of my estate, it being my intention that they shall receive no benefits hereunder." The will was executed in 2012. He was married for 36 years between 1944 and 1980. He then married his second wife SanDee Pitnick only a month after his divorce was finalised in 1983 and they stayed married until his death. Pitnick will receive Lewis's estate as his widow. Next in the line of inheritance is his 25-year-old adopted daughter Danielle, who was his manager at the time of his death. It was Danielle that confirmed that "he passed peacefully at home of natural causes with his loving family at his side."
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Wilson Resume/CV % Structure Specification File % Version 1.0 (22/1/2015) % % This file has been downloaded from: % http://www.LaTeXTemplates.com % % License: % CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %---------------------------------------------------------------------------------------- % PACKAGES AND OTHER DOCUMENT CONFIGURATIONS %---------------------------------------------------------------------------------------- \usepackage[a4paper, hmargin=25mm, vmargin=30mm, top=20mm]{geometry} % Use A4 paper and set margins \usepackage{fancyhdr} % Customize the header and footer \usepackage{lastpage} % Required for calculating the number of pages in the document \usepackage{hyperref} % Colors for links, text and headings \setcounter{secnumdepth}{0} % Suppress section numbering %\usepackage[proportional,scaled=1.064]{erewhon} % Use the Erewhon font %\usepackage[erewhon,vvarbb,bigdelims]{newtxmath} % Use the Erewhon font \usepackage[utf8]{inputenc} % Required for inputting international characters \usepackage[T1]{fontenc} % Output font encoding for international characters \usepackage{fontspec} % Required for specification of custom fonts \setmainfont[Path = ./fonts/, Extension = .otf, BoldFont = Erewhon-Bold, ItalicFont = Erewhon-Italic, BoldItalicFont = Erewhon-BoldItalic, SmallCapsFeatures = {Letters = SmallCaps} ]{Erewhon-Regular} \usepackage{color} % Required for custom colors \definecolor{slateblue}{rgb}{0.17,0.22,0.34} \usepackage{sectsty} % Allows customization of titles \sectionfont{\color{slateblue}} % Color section titles %\fancypagestyle{plain}{\fancyhf{}\cfoot{\thepage\ of \pageref{LastPage}}} % Define a custom page style \pagestyle{plain} % Use the custom page style through the document \renewcommand{\headrulewidth}{0pt} % Disable the default header rule \renewcommand{\footrulewidth}{0pt} % Disable the default footer rule \pagenumbering{gobble} \setlength\parindent{0pt} % Stop paragraph indentation % Non-indenting itemize \newenvironment{itemize-noindent} {\setlength{\leftmargini}{0em}\begin{itemize}} {\end{itemize}} % Text width for tabbing environments \newlength{\smallertextwidth} \setlength{\smallertextwidth}{\textwidth} \addtolength{\smallertextwidth}{-2cm} \newcommand{\sqbullet}{~\vrule height 1ex width .8ex depth -.2ex} % Custom square bullet point definition %---------------------------------------------------------------------------------------- % MAIN HEADER COMMAND %---------------------------------------------------------------------------------------- \renewcommand{\title}[1]{ {\huge{\color{slateblue}\textbf{#1}}}\\ % Header section name and color \rule{\textwidth}{0.5mm} % Rule under the header } %---------------------------------------------------------------------------------------- % JOB COMMAND %---------------------------------------------------------------------------------------- \newcommand{\joba}[4]{ \begin{tabbing} \hspace{2cm} \= \kill \textbf{#1} \> \begin{minipage}{\smallertextwidth}\href{#3}{#2}\end{minipage} \\ \>\+ \begin{minipage}{\smallertextwidth}\vspace{2mm}#4\end{minipage} \end{tabbing}} \newcommand{\jobb}[4]{ \begin{tabbing} \hspace{2cm} \= \kill \textbf{#1} \\ \begin{minipage}{\smallertextwidth}\vspace{2mm}{#2}\end{minipage} \\ \begin{minipage}{\smallertextwidth}\vspace{2mm}#3\end{minipage} \end{tabbing} %\vspace{2mm} } %---------------------------------------------------------------------------------------- % SKILL GROUP COMMAND %---------------------------------------------------------------------------------------- \newcommand{\skillgroup}[2]{ \begin{tabbing} \hspace{5mm} \= \kill \sqbullet \>\+ \textbf{#1} \\ \begin{minipage}{\smallertextwidth} \vspace{2mm} #2 \end{minipage} \end{tabbing} } %---------------------------------------------------------------------------------------- % INTERESTS GROUP COMMAND %----------------------------------------------------------------------------------------- \newcommand{\interestsgroup}[1]{ \begin{tabbing} \hspace{5mm} \= \kill #1 \end{tabbing} \vspace{-10mm} } \newcommand{\interest}[1]{\sqbullet \> \textbf{#1}\\[3pt]} % Define a custom command for individual interests %---------------------------------------------------------------------------------------- % TABBED BLOCK COMMAND %---------------------------------------------------------------------------------------- \newcommand{\tabbedblock}[1]{ \begin{tabbing} \hspace{2cm} \= \hspace{4cm} \= \kill #1 \end{tabbing} }
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Q: How do I download a web page into a single file from a specified URL? I'm trying to scrape web pages. I want to download a web page by providing its URL and save it for offline reading with all its images. I can't manage to do that with wget since it creates many directories. Is this possible with wget? Is there something like the "Save as" option in FireFox which creates a directory and puts all required resources into that with an HTML page? Would it be possible to do this Nokogiri or Mechanize? A: You can use wget to do this and run it from within your ruby script. Here's example that will rip the homepage of my site, skrimp.ly, and put the contents into a single directory named "download". Everything will be at the top level and the links embedded in the HTML will be rewritten to be local: wget -E -H -k -K -p -nH -nd -Pdownload -e robots=off http://skrimp.ly Note: you should checkout some of the docs for wget. It can do some really crazy stuff like go down multiple levels. If you do that sort of thing please be cautious -- it can be pretty heavy on a web server and in some cases cost the webmaster a lot of $$$$. http://www.gnu.org/software/wget/manual/html_node/Advanced-Usage.html#Advanced-Usage A: The answer given by the Tin Man did the job. This shows how to use Nokogiri to download a single page with pictures for offline reading with a very clean directory structure.
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Landscape Photographer of the Year's loss is our gain this week where we're featuring Nigel Morton's images including a few classics that failed to get through the first round of Take a View. Well as with most people there have been a few. There are a couple of fairly recent ones which are quite important to me. The first is the realisation I could still create the sort of images I want to make despite living in central London and not having enough time to dedicate to photography. I had grown tired of travelling across the UK to honey-pot locations. I think most photographers go through a phase when they start out, where they visit well know locations and you can't help but be influenced by images you've already seen from those places. I've been concentrating of a few spots fairly local to me, most notably Epping Forest, which is around twelve miles from central London. It took me a few months before I started to get a good feel for the area and start creating images I was happy with, but it's been so much more rewarding in the long run. The old mantra 'get out more and take less pictures' is sound advice, which for me could only be achievable with local projects. The second is sort of the opposite of what other people have said in the past regarding making the transition to 'pro DSLRs' or large format. For the most part I'm still using the entry-level camera I was using four years ago (a Canon 450d, although I've used others). Another reason the Epping Forest project was started was because I really wanted to improve my woodland photography. Now I hope to make it into a book, but really I'm continuing the project because I enjoy it, not because I want to sell large prints or go pro. I do meet people who try and tell me my kit is inadequate, but you know what, it hasn't stopped me from making work that I'm very happy with, so for now I'll just carry on with what I've got. The last couple of years have been tough financially for my family and I, and I'm in no position to upgrade anyway. As long as I'm happy with the pictures I'm making, that's all that matters. I started of training as a toolmaker at an engineering firm. Then I took those skills and worked at an R&D lab inside a University. Being around post-grad students inspired me to go and study Engineering at degree level. I loved the design-oriented subjects, but struggled with the maths. I somehow got through and ended up with a Masters in Design and Engineering. This coincided with getting married and becoming a Father for the first time. A close friend mentioned that there were some freelance shifts available on The Times picture desk. Six years later I'm still there. I'm picture editor of the website. I also have ongoing design work (web and print) for my wife's business, which is finally becoming quite successful and often sees me working around the clock. I may one day end up doing that full-time or going back into engineering. It's nice to have options. Well, my kit bag consists of the camera and three lenses. I have the 17-40 L, the 70-200 L and the gap between them is filled by a 50mm f1.4. I've got a nice collection of Lee filters (NDs, ND grads and polarizer). Recently I've been doing a bit of experimenting with focus bracketing and stacking. There are a couple of reasons for this. One is to keep the aperture in the sweet spot to limit diffraction, and the other is to be able to blend shallow depth of field exposures to drop areas out of focus in post processing. It's a bit crude, but there have been a few occasions where I've wished I had camera movements. I've had mixed results as it's still for me in the experimental phase. Obviously having a small sensor limits print size, so I've been doing a fair bit of stitching. I've been quite happy with the results too. I started by just making more traditional 6x17 style panoramic images. Lately I've made a few 4:3 crops of details, which have worked surprisingly well. I want to get the best quality images I can out of the equipment I own, which has meant adapting a little bit to maximize the results. I have Lightroom and Canon's own Digital Photo Professional. I hardly use them to be honest – just for lowering contrast as much as I can to give me a nice flat Tiff image to start with in Photoshop. I've used Photoshop for years. I was using it for design long before I started making photographs. It's my main tool I work with during the day, and then I come home and do it all over again with my own images! My workflow in Photoshop usually consists of adjusting the shadows/highlights, colour balance and contrast using curves. Woodland images can be very busy, so I often vignette the edges and perhaps accentuate any lead-in lines by lightening or darkening them slightly. I find it helps to keep the focus of the image where I want it to be and cut down on any visual distractions. Well when I picked up those few shifts at The Times, I had a casual interest in photography. The full obsession hadn't kicked in yet. I just needed to pay a few bills while continuing looking for a 'proper job'. I soon learnt I was in the company of some of the best photojournalists in the country, and began to soak up their work and knowledge. I owe more to those guys than I do to any traditional landscape photographer. I'm still in awe of what they do and how hard they work. Especially those who regularly put their own lives in danger in war stricken counties. I mean, I admire the work Ansel Adams as much as anyone, and one day I might produce a photograph that's as good as one of his, but I'm never going to stick myself in the firing line in Afghanistan. David Bebber's work stood out in particular for me at the time (www.davidbebber.com). You may be familiar with his portrait of Colonel Gaddafi which won the British Press Photographers Association picture of the year a couple of years ago. Andrew Testa's (to be neutral, he does not work for The Times) work is incredible as well, especially the images he shot in Kosovo in the 90's. (www.andrewtesta.co.uk) You have to remember it's all on film, in the heat of the moment. I tried doing live band photography, which seemed obvious as I went to a lot of gigs. I then had a brief stint at underwater photography. The results were dire, but it provided the link between the live work and nature/landscapes. Underwater photographer David Doubilet's book 'Water, Light, Time' is one of my favourite photography books (www.daviddoubilet.com). It's worth it for the cover image alone, and there are some very good seascapes. It's an area of photography I'd like to try again one day. As far as landscape work goes, over the last two years I've been really interested in Dav Thomas' work and yours too Tim. Eliot Porter and Peter Dombrovskis are two names from bygone years whose work really stands out. I really struggled with woodland photography to begin with. It separates the men from the boys in some respects. I can't see myself going back to photograph a rock on the beach at sunset for sometime. There's been some discussion in this publication regarding the parallels between music and photography. I couldn't agree more, and this image for me is proof of that. Despite to most people's ear, the style of music I play is a bloody racket, it's actually very complex. The arrangements in the last band I was in had up to forty riffs in a song, stretching over seven or eight minutes. Just like woodland photography, attention to detail and composition is paramount. I was almost an hour deliberating how to best resolve this scene. By far the longest amount of time I've spent on an image. The rain-drops on the branches in the top left, the tree on the far bank on the grass, the twigs overlapping the water at the very bottom of the screen to contain energy. These were all deliberate decisions. The elbow of the branch touching the right hand side is in fact two branches: one heading straight out of the frame and another coming in from a tree just out of shot. It was also made at 2 o'clock in the afternoon. Beautiful warm sunsets are over-rated. This is the opposite of the first image. I went for a walk and this happened. It's cropped from a three frame horizontal stitch. The light on the trees lasted just long enough to get the three frames off. If I'd stopped to set up a portrait stitch I would have missed it. This currently sits in my LPOTY 2012 reject pile, which I'm pretty disappointed about. I wanted my last choice to be from a different location, but for some reason kept coming back to this. It's my ode to Joe's 'Sky Cascade' from Scotland's Mountains. It's about as dynamic as Epping Forest gets. Somewhere in there is my Lee Polarizer, which I managed to retrieve about thirty minutes later after much poking around with a stick! I'd like to dedicate some time to writing music again. I'm not sure if I'd go back to playing in bands though. I can play guitar, and just about get by on drums and keyboard. A one-man home-studio project would be ideal. I'd also like to get involved with conservation somehow. More specifically protecting our seas and the endangered species within them. We treat the ocean both like a dustbin and an endless source of food for our gluttony. Over-fishing, pollution and the feeling that human beings have the right to destroy four and half billion years of evolution to satisfy our greed makes me mad. It's making my blood boil just thinking about it. I toyed with the idea that the Epping project should be a year on digital and a year on film, but decided against it. I may still have a crack at film again though. When the project is over I want to do something completely different. I'm getting itchy feet and want to go back to travelling further afield, perhaps Scotland. I've only been once for a photographic trip. Honestly, I don't know how Andrew Nadolski and Mike Jackson lasted so long on their respective beaches. I'd really like to see Ian Cameron and Bruce Percy be part of an edition one day. From Flickr I really like Paul Morton (no relation) and Matt Toynbee. Any of those guys would do me.
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{"url":"https:\/\/wirelesspi.com\/the-big-picture-of-localization\/","text":"# The Big Picture of Localization\n\nDigital Signal Processing (DSP) enables us to find the range of a device transmitting a wireless signal with a particular structure under some conditions. To understand how this process works, we need to look at the big picture of a localization process. Localization implies locating the unknown position of a source which can be computed in a straightforward manner if its ranges from some reference nodes can be found.\n\nVarious techniques are employed for this purpose, some of which are Received Signal Strength Indicator (RSSI), time of arrival, time difference of arrival and angle of arrival. Phase of arrival is a special case of time of arrival scheme which is extremely accurate due to the high frequency carrier resembling a high resolution clock. This method works even in the absence of synchronization among nodes. Here, I will discuss various arrangements under which finding the range and\/or position of the device becomes feasible for time of arrival technique. It is worth noting that since phase is just another manifestation of time, the underlying principles stay the same.\n\n## One Way Transmission\n\nLet us start with a transmitter Tx whose signal is received by a receiver Rx. The electromagnetic signal travelling at a speed of $c=3\\cdot 10^8$ m\/s takes a finite amount of time $\\tau$ to arrive at the Rx. This is shown in the figure below and from basic physics, the range $R$ is given by\n\\begin{equation*}\nR = c\\tau\n\\end{equation*}\n\nThis propagation delay $\\tau$ is our main target for ranging purpose. However, since each node starts at a random time, there is a clock offset between its time as compared to the real time. At an arbitrary real time 0, the time offset of the Tx is $\\theta_{\\text{Tx}}$ while that of of the Rx is $\\theta_{\\text{Rx}}$.\n\\begin{equation*}\n\\begin{aligned}\nT_{\\text{Tx}} &= t + \\theta_{\\text{Tx}} \\\\\nT_{\\text{Rx}} &= t + \\theta_{\\text{Rx}}\n\\end{aligned}\n\\end{equation*}\nwhere $t$ is the real time. What happens at the Rx side if we implement one way transmission in which the Tx emits the wave at real time $t_{\\text{Tx}}$ received by the Rx at real time $t_{\\text{Rx}}$?\n\\label{eq1WayTransmission1}\nt_{\\text{Rx}}-\\theta_{\\text{Rx}} = t_{\\text{Tx}} \u2013 \\theta_{\\text{Tx}} + \\tau\n\nWe can reduce three unknowns to two by combining the individual phase offsets into one entity $\\theta$.\n\\begin{equation*}\n\\theta = \\theta_{\\text{Rx}} \u2013 \\theta_{\\text{Tx}}\n\\end{equation*}\nThen, Eq (\\ref{eq1WayTransmission1}) can be written as\n\\label{eq1WayTransmission2}\nt_{\\text{Rx}} = t_{\\text{Tx}} + \\theta + \\tau\n\nThis equation still has two unknowns $\\theta$ and $\\tau$. As a side remark, notice in the above figure that unique range can only be found if the distance between the Tx and Rx is within one wavelength only. At high carrier frequencies, e.g., 2.4 GHz where the wavelength is just 12.5 cm, the phase of arrival method seems impractical due to $2\\pi$ phase ambiguity. Nevertheless, transmissions at multiple carrier frequencies solve this range ambiguity problem in frequency domain.\n\nTo summarize, we get only one equation from this procedure and hence it becomes impossible to separate the phase offset $\\theta$ from propagation delay $\\tau$ in time domain. There are several ways in which one or more extra equations can be provided to this system for range determination.\n\n## Solution 1: Synchronization\n\nWe can synchronize the two nodes which implies that $\\theta$ becomes zero. From Eq (\\ref{eq1WayTransmission2}), we can write\n\n\\begin{equation*}\nt_{\\text{Rx}} = t_{\\text{Tx}} + \\tau\n\\end{equation*}\n\nThis leaves us with one equation and one unknown, i.e., delay $\\tau$. Synchronizing the Tx and Rx is not straightforward however and consumes resources either in terms of cost or computations.\n\n## Solution 2: Back and Forth Transmission\n\nRealizing that two unknowns can be solved through a system of two equations, we can provide an extra independent equation through back and forth transmission between the Tx and Rx. The original timestamps or phases at the Tx and Rx during the forward transmission can now be denoted with the subscript 1 while the subscript 2 can be used for the backward transmission from the Rx to the Tx.\n\\begin{equation*}\n\\begin{aligned}\nt_{Rx,1} &= t_{Tx,1} + \\theta + \\tau \\\\\nt_{Tx,2} &= t_{Rx,2}~ \u2013 \\theta + \\tau\n\\end{aligned}\n\\end{equation*}\n\nIn this scenario, the phase offset $\\theta$ appears with a negative sign in the second equation because timing or phase measurement is done at the transmitter (known as the initiator here). If you are unsure about this sign change, start with the basic definitions of phase offsets mentioned before. Now we have two equations that can solve the two unknowns $\\theta$ and $\\tau$. Multiple carrier phases can also be utilized for ranging purpose in multipath channels.\n\n## Solution 3: Motion\n\nThe above system of equations provides a second independent equation in terms of a negative sign with the phase offset $\\theta$ where the first equation contains $+1$ as the coefficient for both $\\theta$ and $\\tau$. This concept can be generalized to provide a second equation to this system with any other coefficient to either $\\theta$ or $\\tau$. This is difficult to accomplish this with the phase offset $\\theta$. On the other hand, we can always move one node around that changes the coefficient that appears with $\\tau$ without affecting the phase offset $\\theta$.\n\nFor example, if the Rx moves twice as far as compared to the original range, we get\n\n\\begin{equation*}\n\\begin{aligned}\nt_{Rx,1} &= t_{Tx,1} + \\theta + \\tau \\\\\nt_{Rx,2} &= t_{Tx,2} + \\theta + 2\\tau\n\\end{aligned}\n\\end{equation*}\n\nNote that both transmissions in this case are in the same direction from the Tx to the Rx, as opposed to the back and forth solution. A subtle but really important point here is to come up with a strategy to produce a known coefficient for delay $\\tau$ which in general is not easy due to $\\tau$ itself being unknown. Researchers have come up with several smart algorithms for this purpose, one of which was very well explained by Marcus M\u00fcller during a short discussion with me on discuss-gnuradio archive.\n\nImagine a receiver with knowledge of the transmitted signal. While lacking a common time base, a receiver can infer distance from the development of the phases of entries of a sufficiently large (in both number of OFDM symbols and number of subcarriers) OFDM frame.\n\nThe idea is simple: assume you know the symbols at the transmitter. The speed-of-light induced delay is constant across all subcarriers. The resulting phase shift, thus, is proportional to the subcarrier frequency, and hence the subcarrier number. Therefor, when you observe linear channel phase change over subcarrier, you can get a distance estimate. Phase being a linear function of index implies we\u2019re dealing with a sinusoid \u2013 and a DFT in subcarrier direction will give us a range plot.\n\nSame idea for Doppler, but with phase on the same subcarrier, but for consecutive and hence constant-interval OFDM symbols; do another DFT for each subcarrier across OFDM symbols, and get a doppler plot.\n\nOverall: Write down your received OFDM symbols as column vectors of a matrix, point-wise divide by the transmitted symbols (normalize amplitude if helpful); the result is a matrix full of complex numbers with the channel phase for each subcarrier at each symbol time. Do an appropriate 2D-DFT, get a range\/doppler plane \u201cimage\u201d. Find the peak; use clever interpolation \/ post-processing to increase resolution and\/or reduce estimate variance. See Ref. [1] for details.\n\n## Extra Reference Nodes\n\nAll three solutions above are applicable to a Tx and a Rx only. To obtain extra equations for this system, we can place extra anchor nodes around the Tx with known positions. The problem here is that exactly the same number of extra phase offsets appears in the set. To alleviate this problem, these anchor nodes can be synchronized which is how a GPS Rx finds its position through a synchronized network of satellites. Assuming perfect synchronization among them, the receiver phase offset $\\theta$ is the same for all received satellite signals.\n\nAnother option is that after the Tx signal is received by all surrounding nodes, one node can respond with a reply message which makes the phase offsets appear with negative signs making the system solvable. This is usually known as ranging through one-way transmission (or blink mode) because target node has to transmit only once.","date":"2023-01-29 19:32:55","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\": 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\": 0.8549777865409851, \"perplexity\": 497.3880398150671}, \"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\/1674764499758.83\/warc\/CC-MAIN-20230129180008-20230129210008-00605.warc.gz\"}"}
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\section{Introduction} \subsection{} Quantum enveloping algebras associated to Kac-Moody Lie algebras are central objects in mathematics, which have many remarkable connections to geometry, combinatorics, mathematical physics, and other areas. One such connection was produced by Reshetikhin and Turaev \cite{T, RT} by relating the representation theory of these quantum enveloping algebras to Laurent polynomial knot invariants, such as the (colored) Jones polynomial and the HOMFLYPT polynomial. Many other connections have arisen from the categorification of quantum enveloping algebras and their representations \cite{KL, R}. It was recently shown by Webster \cite{Web} that in fact, one can categorify all Reshetikhin-Turaev invariants using the machinery of categorified quantum enveloping algebras. This procedure generalizes Khovanov's homological categorification of the Jones polynomial \cite{Kh}. We can summarize some of these connections in the picture in Figure 1, where ``Decat.'' refers to the appropriate decategorification, ``RT'' stands for the Reshetikhin-Turaev procedure for constructing the Jones polynomial from the standard quantum ${\mathfrak{sl}}(2)$ representation, and ``Web'' stands for Webster's categorification of RT which produces Khovanov homology. \begin{figure} \begin{minipage}{.45\textwidth}\label{fig:non-odd} \centering \begin{tikzpicture}[scale=.9] \draw (-1,1) node[left]{KH}; \draw (-1,-1) node[left]{Jones}; \draw (1,1) node[right]{$\dot{\mathcal U_q}({\mathfrak{sl}}(2))$}; \draw (1,-1) node[right]{$U_q({\mathfrak{sl}}(2))$}; \draw[snake, ->] (1.6,.6)--(1.6,-.6) node[midway,right]{Decat.}; \draw[snake, ->] (-1.6,.6)--(-1.6,-.6) node[midway,right]{Decat.}; \draw[<-] (-1,1)--(1,1) node[midway,above]{Web}; \draw[<-] (-1,-1)--(1,-1) node[midway,above]{RT}; \end{tikzpicture} \caption{} \end{minipage} \begin{minipage}{.45\textwidth}\label{fig:odd} \centering \begin{tikzpicture}[scale=.9] \draw (-1,1) node[left]{oKH}; \draw (-1,-1) node[left]{Jones}; \draw (1,1) node[right]{$\dot{\mathcal U}_{q,\pi}({\mathfrak{osp}}(1|2))$}; \draw (1,-1) node[right]{$U_{q,\pi}({\mathfrak{osp}}(1|2))$}; \draw[snake, ->] (1.6,.6)--(1.6,-.6) node[midway,right]{Decat.}; \draw[snake, ->] (-1.6,.6)--(-1.6,-.6) node[midway,right]{Decat.}; \draw[dotted, <-] (-1,1)--(1,1) node[midway,above]{?}; \draw[dotted, <-] (-1,-1)--(1,-1) node[midway,above]{RT?}; \end{tikzpicture} \caption{} \end{minipage} \end{figure} This beautiful picture recently developed a twist with the discovery of ``odd Khovanov homology'' \cite{ORS}, an alternate homological categorification of the Jones polynomial. This discovery has spurred a program of ``oddification'': providing analogues of (categorified) quantum groups for this odd Khovanov homology by developing ``odd'' analogues of standard constructions \cite{EKL, EL, MW}. In particular, one would like an ``odd (categorified) $U_q({\mathfrak{sl}}(2))$'' which could produce odd Khovanov homology in a similar way to that described in Figure 1. In particular, the decategorified ``odd'' quantum group should produce the Jones polynomial through some analogue of the Reshetikhin-Turaev procedure. It has been proposed \cite{HW,EL} that such categorifications might naturally arise through categorifying the quantum covering group $U_{q,\pi}({\mathfrak{osp}}(1|2))$; in other words, producing a diagram such as in Figure 2. This proposal has some heuristic evidence from the work of Mikhaylov and Witten \cite{MW}, who have produced candidates for ``odd link homologies'' categorifying ${\mathfrak{so}}(1+2n)$-invariants via topological quantum field theories using the orthosymplectic supergroups. This suggests that the conjecture represented by Figure 2 should be generalized to include colored link invariants associated to ${\mathfrak{osp}}(1|2n)$ for any $n\geq 1$. Moreover, it has been shown by Blumen \cite{Bl} that the ${\mathfrak{osp}}(1|2n)$ and ${\mathfrak{so}}(2n+1)$ invariants which are colored by the standard $(2n+1)$-dimensional representations are relattabed up to a variable substitution. However, it has not been known that the Jones polynomial can be constructed from the Reshetikhin-Turaev procedure on $U_{q,\pi}({\mathfrak{osp}}(1|2))$, much less any relation between super and non-super colored knot invariants in higher rank. \subsection{} A quantum covering group is an algebra ${\mathbf U}=U_{q,\pi}(\mathfrak{g})$ that marries the quantum enveloping superalgebra of an anisotropic Kac-Moody Lie superalgebra (e.g. $\mathfrak g={\mathfrak{osp}}(1|2n)$) with the quantum enveloping algebra of its associated Kac-Moody Lie algebra, which is obtained by forgetting the parity in the root datum (e.g. ${\mathfrak{so}}(1+2n)$). This is done by introducing a new ``half-parameter'' $\pi$ satisfying $\pi^2=1$, and substituting $\pi$ everywhere a sign associated to the superalgebra braiding should appear; such algebras were defined and studied in detail in the series of papers \cite{CW, CHW1, CHW2, CFLW, C, CH}. These quantum covering groups retain the many nice properties of usual quantum groups, such as a Hopf structure; a quasi-${\mathcal{R}}$-matrix \`a la Lusztig \cite[Chapter 4]{L93}; a category ${\mathcal O}$; and even canonical bases. A key feature of a quantum covering group is that by specializing $\pi=1$ (respectively, $\pi=-1$), we obtain the quantum enveloping (super)algebra associated to the Kac-Moody Lie (super)algebra. Moreover, as discovered in \cite{CFLW}, the quantum algebra and quantum superalgebra can be identified by a twistor map; that is, an automorphism of (an extension of) the covering quantum group which sends $\pi\mapsto -\pi$ and $q\mapsto {\mathbf t}^{-1}q$, where ${\mathbf t}^2=-1$. In this paper, we use the machinery of covering quantum groups to construct ``quantum covering knot invariants'': knot invariants which arise from the representation theory of the finite type quantum covering groups \`{a} la Turaev \cite{T}. (For our purpose, we do not need the additional ribbon structure of \cite{RT}.) To wit, consider the quantum covering group associated to the Lie superalgebra ${\mathfrak{osp}}(1|2n)$. We first associate a ${\mathbf U}$-module homomorphism to each elementary tangle (cups, caps, crossings) such that a straight strand is just the identity map, along with an interpretation of combining tangles (with joining top-to-bottom being composition of the associated maps, and placing along-side being tensor products of the maps). An arbitrary tangle can then be framed and associated with a ${\mathbf U}$-module homomorphism by ``slicing'' the diagram (that is, cutting it into vertical chunks containing at most one elementary diagram alongside any number of straight strands). Each slice corresponds to a ${\mathbf U}$-module homomorphism, and the tangle is sent to the composition of these maps. Note that a priori, this assignment is not unique, as many distinct slice diagrams and framings exist for an arbitrary tangle. We then derive some identities with these maps that are versions of Turaev moves on the associated diagrams. These identities show that the map isn't dependent on the choice of slice diagram, but factors of $\pi$ keep it from being an invariant of oriented framed tangles. In order to eliminate these factors, we need to expand our base ring to ${\Qqt^\tau}$, where $\tau^2=\pi$, and renormalize the maps corresponding to certain elementary diagrams. Finally, a normalization factor (depending on the writhe of the tangle) yields a oriented tangle invariant (see Theorem \ref{thm:knot invariant}). In the rank 1 uncolored case, this invariant is simply the (unnormalized) Jones polynomial in the variable $\tau^{-1}q$ (see Example \ref{ex:rk1knot}). This suggests that the $\pi=-1$ (i.e. $\tau={\mathbf t}$) specialization of the knot invariant, viewed as a function of $q$, should be related to the $\pi=1$ (i.e. $\tau=1$) specialization, viewed as a function of ${\mathbf t}^{-1}q$. To make this connection precise, we further develop the theory of twistors (cf. \cite{CFLW, C}) to define a general operator on tensor powers of ${\mathbf U}$ and compatible operators on its representations. In particular, we show that the twistors ${\mathfrak X}$ on representations ${\mathbf t}$-commute with the maps $S$ representing slices of tangles; that is, ${\mathfrak X}\circ S={\mathbf t}^{x}S\circ {\mathfrak X}$ for some $x\in{\mathbb{Z}}$. Once this is done, we obtain the following theorem (combining Theorems \ref{thm:knot invariant} and \ref{thm:twistor vs knot invariant}). \begin{thm*} Let $K$ be any oriented knot and $\lambda\in X^+$ a dominant weight. There is a functor from the category $\mathcal{OTAN}$ of oriented tangles modulo isotopy to the category ${\mathcal O}$ of ${\mathbf U}$-module representations which sends $K$ to a constant $J_K^\lambda(q,\tau)\in {\Qqt^\tau}$, which we call the {\em covering knot invariant} of $K$. Moreover, let ${}_{{\mathfrak{so}}}J_K^\lambda(q)=J_K^\lambda(q,1)$ and ${}_{{\mathfrak{osp}}}J_K^\lambda(q)=J_K^\lambda(q,{\mathbf t})$ denote the specializations of the covering knot invariant to $\tau=1$ and $\tau={\mathbf t}$. Then \[{}_{{\mathfrak{osp}}}J_K^\lambda(q)={\mathbf t}^{\star(K,\lambda)}{}_{{\mathfrak{so}}}J_K^\lambda({\mathbf t}^{-1} q),\] for some $\star(K,\lambda)\in{\mathbb{Z}}$. \end{thm*} In particular, this shows that, after extending scalars, there is indeed a map RT as in Figure 2, and in fact such a map exists for all colored link invariants of any rank. It remains to develop an analogue of the construction in \cite{Web} to complete the picture, though difficulties abound. For example, it is not necessarily clear how to extend the categorification to ${\Qqt^\tau}$. Moreover, the categorification of covering algebra representations is not yet developed enough to produce the analogous machinery to \cite{Web}. We hope that these results will help cast light on these remaining questions. \subsection{} The paper is organized as follows. In Section 2, we recall the definition of quantum covering ${\mathfrak{osp}}(1|2n)$, denoted by ${\mathbf U}$, and set our conventions. We also develop some additional facts about representations of ${\mathbf U}$, specifically about dual modules and (co)evaluation morphisms, and produce a universal-${\mathcal{R}}$-matrix, which we will simply denote by ${\mathcal{R}}$, from the quasi-${\mathcal{R}}$-matrix defined in \cite{CHW1}. In Section 3, we use ${\mathcal{R}}$ and the (co)evaluation morphisms to define an associated knot invariant by interpreting the maps in terms of the usual graphical calculus; that is, maps are represented by a finite number of labeled, non-intersecting oriented strands such that the ${\mathcal{R}}$-matrix is a positive crossing, the (co)evaluation morphisms are various cups and caps, and orientation is determined by whether the associated module in the domain/range is the dual module or not. We show that this graphical calculus is almost an framed oriented tangle invariant, and is indeed an oriented tangle invariant after renormalizing these elementary diagrams by an integer power of $\tau$ and a factor depending on the writhe. Finally, in Section 4 we use the twistor maps introduced in \cite{CFLW,C} to relate the morphisms in the $\pi=\pm 1$ cases. In particular, we develop some further details about the Hopf structure and representation theory of the enhanced quantum group ${\widehat \UU}$, and construct twistors on tensor products of simple modules and their duals. We then show that these twistors almost commute (up to an integer power of ${\mathbf t}$) with the cups, caps, and crosssings, allowing us to relate the ${\mathfrak{so}}$ and ${\mathfrak{osp}}$ knot invariants. \vspace{1em} \noindent\textbf{Acknowledgements.} I would like to thank David Hill for first suggesting this project of constructing the quantum covering knot invariants, and Matt Hogancamp for helping me learn about quantum knot invariants at this projects inception. I would also like to thank Aaron Lauda, Weiqiang Wang, and Ben Webster for stimulating conversations about this project. \section{Quantum covering ${\mathfrak{osp}}(1|2n)$} We begin by recalling the definition of quantum covering algebra associated to ${\mathfrak{osp}}(1|2n)$ and setting our notations. We then elaborate on the representation theory of this algebra. \subsection{Root data}\label{subsec:rootdata} Let $I=I_0\coprod I_1$ with $I_0=\set{\rf 1,\ldots, \rf{n-1}}$ and $I_1=\set{\rf n}$, and define the parity $p(i)$ of $i\in I$ by $i\in I_{p(i)}$. For $1\leq r,s\leq n$, we define \[{\rf r}\cdot {\rf s}=\begin{cases} 2 &\text{ if } r=s=n\\ 4 &\text{ if } r=s\neq n\\ -2&\text{ if }r=s\pm 1\\ 0&\text{otherwise} \end{cases},\qquad d_{\rf r}=\frac{{\rf r}\cdot{\rf r}}{2},\] and note that $p({\rf r})\equiv d_{\rf r}\ ({\rm mod}\ 2)$. Then $(I,\cdot)$ is a bar-consistent anisotropic super Cartan datum (see \cite{CHW1}). We extend $\cdot$ to a symmetric bilinear pairing on ${\mathbb{Z}}[I]$ and $p$ to a parity function $p:{\mathbb{Z}}[I]\rightarrow {\mathbb{Z}}/2{\mathbb{Z}}$. Moreover, for $\nu=i_1+\ldots+i_t\in{\mathbb{N}}[I]$, we set \begin{equation}\label{eq:bp and bullet} {\mathrm{ht}}\ \nu=t,\quad {\mathbf p}(\nu)=\sum_{1\leq r<s\leq t} p(i_r)p(i_s),\qquad \bullet(\nu)=\sum_{1\leq r<s\leq t} i_r\cdot i_s. \end{equation} Let $\Phi^+\subset{\mathbb{N}}[I]$ denote the set of positive roots, and set \begin{equation}\label{eq:rho} \rho=\sum_{\alpha\in \Phi^+} \alpha=\sum_{i\in I} \rho_i i\in{\mathbb{N}}[I]. \end{equation} Note that we have $i\cdot\rho =i\cdot i$ for all $i\in I$. Let $Y={\mathbb{Z}}[I]$ be the root lattice and $X={\rm Hom}({\mathbb{Z}}[I],{\mathbb{Z}})$ be the weight lattice, and let $\ang{\cdot,\cdot}:Y\times X\rightarrow {\mathbb{Z}}$ be the natural pairing. We also identify ${\mathbb{Z}}[I]$ as a subspace of $X$ so that $\ang{\rf r,\rf s}=2\frac{{\bf r}\cdot{\rf s}}{{\bf r}\cdot{\rf r}}$. If $\nu=\sum_{i\in I} \nu_i i\in {\mathbb{Z}}[I]$, we set \begin{equation}\label{eq:tilderoot} \tilde\nu=\sum_{i\in I} d_i\nu_i i\in {\mathbb{Z}}[I] \end{equation} and note $\ang{\tilde\nu,\mu}=\nu\cdot\mu$ for any $\nu,\mu\in {\mathbb{Z}}[I]$; in particular, observe that for any $i\in I$, \begin{equation}\label{eq:tilderho} \ang{\tilde\rho,i}=i\cdot i. \end{equation} Then $((I,\cdot), X, Y, \ang{\cdot,\cdot})$ is the root datum associated to ${\mathfrak{osp}}(1|2n)$, and forgetting the parity on the root datum yields the root datum associated to ${\mathfrak{so}}(1+2n)$. As usual, we define the dominant weights to be $X^+=\set{\lambda\in X\mid \ang{i,\lambda}\geq 0\text{ for all } i\in I}$. \begin{example}\label{ex:rank1lattices} Throughout the paper, we will discuss some examples in the simplest case: $n=1$. In this case, we identify $X={\mathbb{Z}}$ where $\ang{\rf 1, k}=k$ for $k\in {\mathbb{Z}}$. Then $Y={\mathbb{Z}}\rf 1$ can be identified with subset $2{\mathbb{Z}}\subset X$. We will freely use these identifications in later examples. \end{example} Note that the weight lattice $X$ doesn't naturally have a parity grading compatible with that on ${\mathbb{Z}}[I]$. However, a parity grading on $X$ can be defined as follows. First observe that $X$ carries an action of the Weyl group $W$ of type $B_n$, and that in particular $\lambda-w\lambda\in{\mathbb{Z}}[I]$ for any $\lambda\in X$. Let $w_0$ denote the longest element of $B_n$. If $\lambda\in X$, then $w_0\lambda=-\lambda$ hence $2\lambda=\lambda-w_0\lambda\in {\mathbb{Z}}[I]$. We write $2\lambda=\sum_{i\in I} (2\lambda)_i i$ and define \begin{equation}\label{eq:weight parity} P(\lambda)=p(2\lambda)\equiv (2\lambda)_{\rf n}\text{ (mod 2)}. \end{equation} This defines a parity grading on $X$, though it is obviously not compatible with the grading on ${\mathbb{Z}}[I]$ (indeed, for any $i\in I$ we have $P(i)=p(2i)=2p(i)\equiv 0$ modulo 2). In particular, $P$ is constant on cosets $X/{\mathbb{Z}}[I]$. This parity can be expressed explicitly in terms of the rank and weight as follows. \begin{lem} \label{lem:weight parity equiv} Let notations be as above. Then $P(\lambda)\equiv n\ang{\rf n,\lambda}$ mod 2. \end{lem} \begin{proof} Let $1\leq s\leq n-1$ and for convenience set the notation $(2\lambda)_{\rf 0}=0$. We have \[\ang{{\rf s},\lambda}=\frac12\ang{{\rf s},2\lambda}=\frac{1}{2}\sum_{i\in I} (2\lambda)_i\ang{{\rf s},i} =(2\lambda)_{\rf s}-\frac12((2\lambda)_{\rf{s+1}}+(2\lambda)_{\rf{s-1}}),\] \[\ang{\rf n,\lambda}=(2\lambda)_{\rf n}-(2\lambda)_{\rf{n-1}}.\] In particular, we see that $\frac12((2\lambda)_{\rf{s+1}}+(2\lambda)_{\rf{s-1}})=(2\lambda)_{\rf s}-\ang{{\rf s},\lambda}\in {\mathbb{N}}$, thus $(2\lambda)_{\rf{s-1}}\equiv (2\lambda)_{\rf{s+1}}$ modulo 2 for all $1\leq s\leq n-1$. Therefore, $(2\lambda)_{\rf r}\equiv (2\lambda)_{\rf s}$ modulo 2 whenever $r\equiv s$ modulo 2. In particular, since $(2\lambda)_{\rf 0}=0$, we see that $(2\lambda)_{\rf s}\equiv 0$ modulo $2$ for each $s\equiv 0$ modulo $2$. If $n\equiv 0$ modulo 2, then $P(\lambda)\equiv (2\lambda)_{\rf n} \equiv 0$ modulo $2$. If $n\equiv 1$ modulo 2, then $(2\lambda)_{\rf n}=\ang{\rf n,\lambda}-(2\lambda)_{\rf{n-1}} \equiv \ang{\rf n,\lambda}$ modulo 2. \end{proof} \begin{example}\label{ex:rk1weightparity} When $n=1$, recall from Example \ref{ex:rank1lattices} that we identify $X={\mathbb{Z}}$. Then for any $k\in {\mathbb{Z}}$, $P(k)\equiv(1)\ang{\mathbf 1, k}\equiv k$ modulo 2, hence our $P$-grading is just the natural parity grading on ${\mathbb{Z}}$. \end{example} Throughout, we will consider objects graded by $\bigrset=X\times({\mathbb{Z}}/2{\mathbb{Z}})$. If $M$ is $\bigrset$-graded and $m\in M$ is homogeneous, we let $\bigrdeg{m}$ (resp. $|m|$; $p(m)$) denote its $\bigrset$-degree (resp. $X$-degree; ${\mathbb{Z}}/2{\mathbb{Z}}$-degree or parity). Further, for $\zeta=(\lambda, \epsilon)\in\bigrset$, we will set $|\zeta|=\lambda$, $p(\zeta)=\epsilon$, and $P(\zeta)=P(\lambda)$. (Note that $P(\zeta)$ is not the same as $p(\zeta)$ in general! They are independent quantities.) For $\lambda\in X$, let $\bigrelt\lambda=(\lambda,0)\in \bigrset$. We will freely identify ${\mathbb{Z}}[I]$ with $\set{(\nu,p(\nu))\mid \nu\in {\mathbb{Z}}[I]}\subset \bigrset$. In particular, if $\zeta=(\lambda,\epsilon)\in\bigrset$ and $\nu\in{\mathbb{Z}}[I]$, then \begin{equation}\label{eq:ZI in X hat} \zeta+\nu=(\lambda+\nu,\epsilon+p(\nu))\in\bigrset. \end{equation} With that in mind, the action of $W$ on $X$ generalizes naturally to $\bigrset$ by setting \begin{equation} s_i(\lambda,\epsilon)=(\lambda,\epsilon)-\ang{i,\lambda} i=(\lambda-\ang{i,\lambda} i,\epsilon-\ang{i,\lambda} p(i)) \end{equation} where $i\in I$ and $s_i$ is the corresponding simple reflection. Lastly, we have the parity swap function $\Pi:\bigrset\rightarrow \bigrset$ defined by \begin{equation}\Pi((\lambda,\epsilon))=(\lambda,1-\epsilon).\end{equation} \subsection{Parameters}\label{sec:param} Let ${\mathbf t}\in\mathbb C$ such that ${\mathbf t}^2=-1$. Let $q$ be a formal parameter and let $\tau$ be an indeterminate such that $$ \tau^4=1. $$ For convenience, we will also define \[\pi=\tau^2.\] If $R$ is a commutative ring with 1, define the notations \begin{equation} R^\tau=R[\tau]/(\tau^4=1),\quad R^\pi=R[\pi]/(\pi^2=1). \end{equation} Throughout, our base ring will be ${\Qqt^\tau}$, though occasionally we will also refer to the subring generated by ${\Q(q)}$ and $\pi$, which we identify with ${\Qq^{\pi}}$. We denote by $\bar{\cdot}:{\Qqt^\tau}\rightarrow{\Qqt^\tau}$ the ${\mathbb{Q}}({\mathbf t})^\tau$-algebra automorphism satisfying $\bar q=\pi q^{-1}$. We also define the ${\mathbb{Q}}({\mathbf t})$-algebra automorphism ${\mathfrak X}$ given by ${\mathfrak X}(q)={\mathbf t}^{-1} q$ and ${\mathfrak X}(\tau)={\mathbf t}\tau$. We caution the reader that $\bar{\cdot}$ and ${\mathfrak X}$ will be used later to denote extensions of these algebra automorphisms which are defined on ${\Qqt^\tau}$-algebras and ${\Qqt^\tau}$-modules. Given an ${\Qqt^\tau}$-module (or algebra) $M$ and $x\in\set{\pm 1,\pm{\mathbf t}}$, the ${\Q(q,\bt)}$-module (or algebra) $M|_{\tau=x} ={\Q(q,\bt)}_{x}\otimes_{{\Qqt^\tau}} M$, where ${\Q(q,\bt)}_x={\Q(q,\bt)}$ is viewed as a ${\Qqt^\tau}$-module on which $\tau$ acts as multiplication by $x$. We call this the {\em specialization of $M$ at $\tau=x$ }. Moreover, ${\Qqt^\tau}$ has orthogonal idempotents \begin{equation}\label{eq:tau idempotent} {\varepsilon}_{{\mathbf t}^k}=\frac{1+ {\mathbf t}^k\tau+({\mathbf t}^k\tau)^2+({\mathbf t}^k\tau)^3}{4},\quad 0\leq k\leq 3 \end{equation} such that ${\Qqt^\tau}={\Q(q,\bt)}{\varepsilon}_1\oplus{\Q(q,\bt)}{\varepsilon}_{\mathbf t}\oplus{\Q(q,\bt)}{\varepsilon}_{-1}\oplus{\Q(q,\bt)}{\varepsilon}_{-{\mathbf t}}$. In particular, since $\tau{\varepsilon}_{x}=x {\varepsilon}_{x}$, we see that for any ${\Qqt^\tau}$-module $M$, \[M|_{\tau=x}\cong {\varepsilon}_{x} M.\] For $k \in {\mathbb{Z}}_{\ge 0}$ and $n\in {\mathbb{Z}}$, the $(q,\pi)$-quantum integers, along with quantum factorial and quantum binomial coefficients, are defined as follows (cf. \cite{CHW1}): \begin{equation} \label{eq:nvpi} \begin{split} \bra{n}_{q,\pi} & =\frac{(\pi q)^n-q^{-n}}{\pi q-q^{-1}}, \\ \bra{n}_{q,\pi}^! &= \prod_{l=1}^n \bra{l}_{q,\pi}, \\ \bbinom{n}{k}_{q,\pi} &=\frac{\prod_{l=n-k+1}^n \big( (\pi q)^{l} -q^{-l} \big)}{\prod_{m=1}^k \big( (\pi q)^{m}- q^{-m} \big)}. \end{split} \end{equation} If $\nu=\sum_{i\in I} \nu_i i\in {\mathbb{Z}}[I]$, we write \[q_\nu=\prod_{i\in I}q^{\nu_i d_i},\qquad \tau_\nu=\prod_{i\in I}\tau^{\nu_i d_i},\qquad \pi_\nu=\prod_{i\in I}\pi^{\nu_i d_i}=\pi^{p(\nu)}, \qquad {\mathbf t}_\nu=\prod_{i\in I}{\mathbf t}^{\nu_i d_i.}\] In particular, note that $q_i=q^{d_i}$ and $\pi_i=\pi^{d_i}=\pi^{p(i)}$ and set \[\bra n_i=\bra n_{q_i,\pi_i},\qquad \bra{n}_{i}^!= \bra{n}_{q_i,\pi_i}^!, \qquad \bbinom{n}{k}_{i}=\bbinom{n}{k}_{q_i,\pi_i}.\] \subsection{The covering quantum group} The covering quantum group associated to ${\mathfrak{osp}}(1|2n)$ (as well as some variants) was introduced and studied in the series of papers starting with \cite{CHW1}. We will recall the necessary definitions and elementary facts now. \begin{rmk}\label{rem: coefficients} Note that contrary to \cite{CHW1} and further papers in that series, we will take coefficients in the larger ring ${\Qqt^\tau}\supset{\Qq^{\pi}}$. Nevertheless, all of the results until \S\ref{subsec:renorm} are essentially statements over ${\Qq^{\pi}}$ which remain true after extending scalars to ${\Qqt^\tau}$, so the reader may effectively ignore $\tau$ and ${\mathbf t}$ for the present. \end{rmk} \begin{dfn}\cite{CHW1}\label{def:hcqg} The half-quantum covering group $\ensuremath{\mathbf{f}}$ associated to the anisotropic datum $(I,\cdot)$ is the ${\mathbb{N}}[I]$-graded ${\Qqt^\tau}$-algebra on the generators $\theta_i$ for $i\in I$ with $|\theta_i|=i$, satisfying the relations \begin{equation}\label{eq:thetaserrerel} \sum_{k=0}^{b_{ij}} (-1)^k\pi^{\binom{k}{2}p(i)+kp(i)p(j)} \bbinom{b_{ij}}{k}_{i} \theta_i^{b_{ij}-k}\theta_j\theta_i^k=0 \;\; (i\neq j), \end{equation} where $b_{ij}=1-\ang{i,j}$. \end{dfn} The algebra $\ensuremath{\mathbf{f}}$ carries a non-degenerate bilinear form $(\cdot,\cdot)$ which satisfies \begin{equation}\label{eq:ff bilinear form} (1,1)=1;\quad (\theta_i,\theta_i)=\frac{1}{1-\pi_iq_i^{-2}}; \quad (\theta_ix,y)=(\theta_i,\theta_i)(x,{{}_i r}(y)); \end{equation} where ${{}_i r}:\ensuremath{\mathbf{f}}\rightarrow \ensuremath{\mathbf{f}}$ is the ${\Qqt^\tau}$-linear map satisfying ${{}_i r}(1)=0$, ${{}_i r}(\theta_j)=\delta_{ij}$, and ${{}_i r}(xy)={{}_i r}(x)y+\pi^{p(i)p(x)}q^{i\cdot |x|}x\ {{}_i r}(y)$. (Here, and henceforth, $\delta_{x,y}$ is set to be $\delta_{x,y}=1$ if $x=y$ and $0$ otherwise.) We define the ${\mathbb{Q}}({\mathbf t})^\tau$-linear bar involution ${\bar{\phantom{x}}}$ on $\ensuremath{\mathbf{f}}$ by \[\bar \theta_i=\theta_i, \quad \bar q=\pi q^{-1}.\] We also define the ${\Qqt^\tau}$-linear anti-involution $\sigma$ on $\ensuremath{\mathbf{f}}$ by \[\sigma(\theta_i)=\theta_i,\qquad \sigma(xy)=\sigma(y)\sigma(x),\] and the divided powers \[\theta_i^{(n)}=\theta_i^{n}/\bra{n}_i^!.\] \begin{dfn} \cite{CHW1} \label{definition:cqg} The quantum covering group ${\mathbf U}$ associated to $((I,\cdot),\ Y,\ X,\ \ang{\cdot,\cdot})$ is the ${\Qqt^\tau}$-algebra with generators $E_i, F_i$, $K_\mu$, and $J_\mu$, for $i\in I$ and $\mu\in Y$, subject to the relations: % \begin{equation}\label{eq:JKrels} J_\mu J_\nu=J_{\mu+\nu},\quad K_\mu K_\nu=K_{\mu+\nu},\quad K_0=J_0=J_\nu^2=1,\quad J_\mu K_\nu=K_\nu J_\mu, \end{equation} \begin{equation}\label{eq:Jweightrels} J_\mu E_i=\pi^{\ang{\mu,i}} E_i J_\mu,\quad J_\mu F_i=\pi^{-\ang{\mu,i}} F_i J_\mu, \end{equation} \begin{equation}\label{eq:Kweightrels} K_\mu E_i=q^{\ang{\mu,i}} E_i K_\mu,\quad K_\mu F_i=q^{-\ang{\mu,i}} F_i K_\mu, \end{equation} \begin{equation}\label{eq:commutatorrelation} E_iF_j-\pi^{p(i)p(j)}F_jE_i=\delta_{ij}\frac{J_{d_i i}K_{d_i i}-K_{-d_i i}}{\pi_i q_i- q_i^{-1}}, \end{equation} \begin{equation}\label{eq:Eserrerel} \sum_{k=0}^{b_{ij}} (-1)^k\pi^{\binom{k}{2}p(i)+kp(i)p(j)}\bbinom{b_{ij}}{k}_{q_i,\pi_i} E_i^{b_{ij}-k}E_jE_i^k=0 \;\; (i\neq j), \end{equation} \begin{equation}\label{eq:Fserrerel} \sum_{k=0}^{b_{ij}} (-1)^k\pi^{\binom{k}{2}p(i)+kp(i)p(j)}\bbinom{b_{ij}}{k}_{q_i,\pi_i} F_i^{b_{ij}-k}F_jF_i^k=0 \;\; (i\neq j), \end{equation} for $i,j\in I$ and $\mu,\nu\in Y$. \end{dfn} We note that since in this case $Y={\mathbb{Z}}[I]$, ${\mathbf U}$ is actually generated by $E_i, F_i, K_i,J_i$ for $i\in I$. For notational convenience, we set ${\tilde{J}}_\nu=J_{\tilde\nu}$ and ${\tilde{K}}_\nu=K_{\tilde \nu}$ so that \eqref{eq:commutatorrelation} becomes \[E_iF_j-\pi^{p(i)p(j)}F_jE_i=\delta_{ij}\frac{{\tilde{J}}_{i}{\tilde{K}}_{i}-{\tilde{K}}_{i}^{-1}}{\pi_i q_i- q_i^{-1}}.\] We also equip ${\mathbf U}$ with a bar involution $\bar{\cdot}:{\mathbf U}\rightarrow {\mathbf U}$ extending that on ${\Qqt^\tau}$ by setting $\bar{E_i}=E_i$, $\bar{F_i}=F_i$, $\bar{K_\mu}=J_\mu K_{-\mu}$, $\bar J_\mu=J_\mu$. The algebras ${\mathbf U}$ and $\ensuremath{\mathbf{f}}$ are related in the following way. Let $\UU^-$ be the subalgebra generated by $F_i$ with $i\in I$, $\UU^+$ be the subalgebra generated by $E_i$ with $i\in I$, and $\UU^0$ be the subalgebra generated by $K_\nu$ and $J_\nu$ for $\nu\in Y$. There is an isomorphisms $\ensuremath{\mathbf{f}}\rightarrow\UU^-$ (resp. $\ensuremath{\mathbf{f}}\rightarrow \UU^+$) defined by $\theta_i\mapsto \theta_i^-=F_i$ (resp. $\theta_i\mapsto \theta_i^+=E_i$). We let $E_i^{(n)}=(\theta_i^{(n)})^+$ and $F_i^{(n)}=(\theta_i)^{(n)})^-$. As shown in \cite{CHW1}, there is a triangular decomposition \[{\mathbf U}\cong \UU^-\otimes\UU^0\otimes\UU^+\cong \UU^+\otimes \UU^0\otimes \UU^-.\] There is also a root space decomposition \[{\mathbf U}=\bigoplus_{\nu\in{\mathbb{Z}}[I]} {\mathbf U}_\nu,\qquad {\mathbf U}_\nu=\set{x\in{\mathbf U}\mid J_\mu K_{\xi}m=\pi^{\ang{\mu,\nu}} q^{\ang{\xi,\nu}} m}.\] The root space decomposition induces a parity grading via $p(u)=p(|u|)$, hence in particular ${\mathbf U}$ is $\bigrset$-graded. We say an algebra is a ``Hopf covering algebra'' if it is a ${\mathbb{Z}}/2{\mathbb{Z}}$-graded algebra over $R^\pi$, for some commutative ring with identity $R$, with a coproduct, antipode, and counit satisfying the usual axioms of a Hopf superalgebra, but with the braiding replaced by $x\otimes y\mapsto \pi^{p(x)p(y)}y\otimes x$. Then the algebra ${\mathbf U}$ is a Hopf covering algebra under the coproduct $\Delta:{\mathbf U}\rightarrow {\mathbf U}\otimes {\mathbf U}$ satisfying \[\Delta(E_i)=E_i\otimes 1+K_i\otimes E_i,\qquad \Delta(F_i)=F_i\otimes {\tilde{K}}_i^{-1}+1\otimes F_i,\qquad \Delta(K_\nu)=K_\nu\otimes K_\nu,\qquad \Delta(J_\nu)=J_\nu\otimes J_\nu;\] the antipode $S:{\mathbf U}\rightarrow {\mathbf U}$ satisfying $S(xy)=\pi^{p(x)p(y)}S(y)S(x)$ for $x,y\in {\mathbf U}$ and \[S(E_i)=-{\tilde{J}}_i^{-1}{\tilde{K}}_i^{-1}E_i,\quad S(F_i)=-F_i{\tilde{K}}_i,\quad S(K_\nu)=K_\nu^{-1},\quad S(J_\nu)=J_\nu^{-1};\] and the counit $\epsilon:{\mathbf U}\rightarrow{\Qqt^\tau}$ satisfying \[\epsilon(E_i)=\epsilon(F_i)=0,\qquad \epsilon(K_\nu)=\epsilon(J_\nu)=1.\] Moreover, for $x\in \ensuremath{\mathbf{f}}$, we have that \begin{equation}\label{eq:antipode formula} \begin{array}{c} S^{\pm 1}(x^+)=(-1)^{{\mathrm{ht}}\nu}\pi^{{\mathbf p}(\nu)} q^{\frac{\nu\cdot \nu}{2}}q_{\mp\nu}{\tilde{J}}_{-\nu}{\tilde{K}}_{-\nu}\sigma(x)^+\\ S^{\pm 1}(x^-)=(-1)^{{\mathrm{ht}}\nu}\pi^{{\mathbf p}(\nu)} q^{\frac{-\nu\cdot \nu}{2}}q_{\pm \nu}\sigma(x)^-{\tilde{K}}_{\nu} \end{array} \end{equation} \subsection{${\mathbf U}$-modules}\label{subsec:modules} In this paper, a weight ${\mathbf U}$-module is a ${\mathbf U}$-module $M$ with a $\bigrset$-grading compatible with the grading on ${\mathbf U}$, such that \[M=\bigoplus_{\lambda\in X} M_{\lambda,0}\oplus M_{\lambda,1},\quad M_{\lambda,s}=\set{m\in M\mid p(m)=s,\ J_\mu K_\nu m=\pi^{\ang{\mu,\lambda}}q^{\ang{\nu,\lambda}}m}\] and each $M_{\lambda,s}$ is a free ${\Qqt^\tau}$-module of finite rank. For $\lambda\in X$, denote $M_\lambda=M_{\lambda,0}\oplus M_{\lambda,1}$. We also define the parity-swapped module $\Pi M$ to be $M$ as a vector space with the same action of ${\mathbf U}$, but with $\Pi M_{\lambda,s}= M_{\lambda, 1-s}$. We let ${\mathcal O}_{\rm fin}$ be the category of weight ${\mathbf U}$-modules of finite rank over ${\Qqt^\tau}$. Henceforth, we shall {\em always} assume our ${\mathbf U}$-modules are in ${\mathcal O}_{\rm fin}$. We define the (restricted) linear dual of a ${\mathbf U}$-module $M$ \[M^*=\bigoplus_{\lambda\in X} (M_{\lambda,0})^*\oplus (M_{\lambda,1})^*,\quad (M_{\lambda,s})^*={\rm Hom}_{{\Qqt^\tau}}(M_{\lambda,s},{\Qqt^\tau}).\] This is again a free ${\Qqt^\tau}$-module, which has a ${\mathbb{Z}}/2{\mathbb{Z}}$ grading induced by that of $V$: namely, $p(f)=0$ if $f(v)=0$ for $p(v)=1$, and vice-versa. Moreover, the Hopf superalgebra structure of ${\mathbf U}$ induces an action of ${\mathbf U}$: for $f\in V^*$ and $x\in {\mathbf U}$, we define $xf\in V^*$ by $xf(v)=\pi^{p(f)p(x)}f(S(x)v)$. In particular, note that $V^*$ is a ${\mathbf U}$-module with $(V^*)_{\lambda,s}=(V_{-\lambda,s})^*$. While $V^*_\lambda$ is therefore ambiguous, we will always take it to denote $(V^*)_\lambda$. (In other words, our convention is that taking duals has precedence over taking weight spaces.) For any ${\mathbf U}$-modules $V$ and $W$, we can construct the ${\mathbf U}$-module $V\otimes W=V\otimes_{{\Qqt^\tau}} W$ via the coproduct. In particular, we have ${\mathbf U}$-modules $V^*\otimes V$ and $V\otimes V^*$, both of which contain a copy of the trivial module $V(0)={\Qqt^\tau}$ as a direct summand. As the following lemma shows, there are natural projection and inclusion maps to a copy of the trivial module. We borrow notation from \cite{Ti}. \begin{lem}\label{lemma:evsandcoevs} Fix a ${\mathbf U}$-module $V$ and recall the definition of $\rho$ from \eqref{eq:rho}. \begin{enumerate} \item Let $\ev_V:V^*\otimes V\rightarrow {\Qqt^\tau}$ be the ${\Qqt^\tau}$-linear map defined by $v^*\otimes w\rightarrow v^*(w)$. Then $\ev_V$ is a ${\mathbf U}$-module epimorphism. \item Let $\qtr_V:V\otimes V^*\rightarrow {\Qqt^\tau}$ be the ${\Qqt^\tau}$-linear map defined by $v\otimes w^*\rightarrow \pi^{p(v)p(w)}q^{-\ang{\tilde \rho,|v|}}w^*(v)$. Then $\qtr_V$ is a ${\mathbf U}$-module epimorphism. \item Let $\coev_V:{\Qqt^\tau}\rightarrow V^*\otimes V$ be the ${\Qqt^\tau}$-linear map defined by $1\rightarrow \sum_{b\in B} \pi^{p(b)}q^{\ang{\tilde\rho,|b|}}b^*\otimes b$ for some homogeneous ${\Qqt^\tau}$-basis $B$ of $V$. Then $\coev_V$ is a ${\mathbf U}$-module monomorphism. \item Let $\coqtr_V:{\Qqt^\tau}\rightarrow V\otimes V^*$ be the ${\Qqt^\tau}$-linear map defined by $1\rightarrow \sum_{b\in B} b\otimes b^*$ for some homogeneous ${\Qqt^\tau}$-basis $B$ of $V$. Then $\coqtr_V$ is a ${\mathbf U}$-modulemonomorphism. \end{enumerate} \end{lem} \begin{proof} In the proof, we shall surpress the $V$ subscript on the maps. First note that the maps $\coev$ and $\coqtr$ are independent of the choice of basis. It is clear that all these maps are ${\Qqt^\tau}$-linear maps, and it is elementary to verify the claims about surjectivity and injectivity. Moreover, all the maps are clearly homogeneous since $|v^*|=-|v|$ and $p(v^*)=p(v)$; in particular, the maps $\qtr$ and $\ev$ are homogeneous since $v^*(w)=0$ whenever $|v|\neq |w|$ or $p(v)\neq p(w)$, which occurs exactly when $|v^*\otimes w|\neq 0$ or $p(v^*\otimes w)=1$. Then it remains to show these maps preserve the action of $E_i$ and $F_i$ for all $i\in I$, which is equivalent to showing \[\ev(\Delta(E_i)v^*\otimes w)=\ev(\Delta(F_i)v^*\otimes w)=0\text{ for all }v,w\in V,\] \[\qtr(\Delta(E_i)v\otimes w^*)=\qtr(\Delta(F_i)v\otimes w^*)=0\text{ for all }v,w\in V,\tag{$\star$}\] \[\Delta(E_i)\sum_{b\in B} \pi^{p(b)}q^{\ang{\tilde\rho,|b|}}b^*\otimes b=\Delta(F_i)\sum_{b\in B}\pi^{p(b)}q^{\ang{\tilde\rho,|b|}} b^*\otimes b=0\text{ for all }b\in B,\text{ and}\tag{$\star\star$}\] \[\Delta(E_i)\sum_{b\in B} b\otimes b^*=\Delta(F_i)\sum_{b\in B} b\otimes b^*=0\text{ for all }b\in B. \] We will prove ($\star$) and ($\star\star$) for the action of $E_i$; the remaining cases follow from similar arguments. First, we show $\qtr(\Delta(E_i)v^*\otimes w)=0$. From weight considerations we have that $\ev(\Delta(E_i)v^*\otimes w)=0$ unless $|v|+i=|w|$. In this case, \begin{align*} \qtr(\Delta(E_i)&v\otimes w^*) =\qtr(E_iv\otimes w^*+\pi_i^{p(v)}(\pi_iq_i)^{\ang{i,|v|}} v\otimes E_iw^*)\\ &=\pi^{p(E_iv)p(w)}q^{-\ang{\tilde\rho,|E_iv|}}w^*(E_iv)+\pi_i^{p(v)}(\pi_iq_i)^{\ang{i,|v|}} \pi^{p(v)p(E_iw)}q^{-\ang{\tilde\rho,|v|}}(E_iw^*)(v)\\ &=\pi^{p(v)p(w)+p(i)p(w)}q^{-\ang{\tilde\rho,|v|+i'}} \parens{w^*(E_iv)-q_i^{2}(\pi_iq_i)^{\ang{i,|v|}}w^*({\tilde{J}}_i^{-1}{\tilde{K}}_i^{-1} E_i v)}\\ &=\pi^{p(v)p(w)+p(i)p(w)}q^{-\ang{\tilde\rho,|v|+i'}} \parens{w^*(E_iv)-w^*(E_i v)}=0. \end{align*} Next, we show that $\Delta(E_i)\sum_{b\in B} \pi^{p(b)}q^{\ang{\tilde\rho,|b|}}b^*\otimes b=0$. Set $B_\lambda=B\cap V_\lambda$, so $B=\coprod_\lambda B_\lambda$. First observe that $x=\sum v^*\otimes w=0$ if and only if $x(v'):=\sum v^*(v')w=0$ for all $v'\in V$. Then setting $x=\Delta(E_i)\sum_{b\in B} b^*\otimes b$, if \[0\neq x=\sum_{b\in B} \pi^{p(b)}q^{\ang{\tilde\rho,|b|}}(E_ib^*\otimes b+\pi_i^{p(b)}(\pi_iq_i)^{-\ang{i,|b|}}b^*\otimes E_ib),\] then there must be some $v\in V$ such that \[x(v)=\sum_{b\in B}\pi^{p(b)} q^{\ang{\tilde\rho,|b|}}((E_ib^*)(v)b+\pi_i^{p(b)}(\pi_iq_i)^{-\ang{i,|b|}}b^*(v)E_ib )\neq 0.\] However, if $b'\in B$, \begin{align*} x(b')&=(\pi_iq_i)^{-\ang{i,|b'|}}E_ib'+\sum_{b\in B_{|b'|+i}} b^*(-{\tilde{J}}_i^{-1}{\tilde{K}}_i^{-1}E_ib')b\\ &= q^{\ang{\tilde\rho,|b'|}}(\pi_iq_i)^{-\ang{i,|b'|}}E_ib'-\sum_{b\in B_{|b'|+i}}q^{\ang{\tilde\rho,|b|}}(\pi_iq_i)^{-\ang{i,|b'|+i}}b^*(E_ib')b\\ &= q^{\ang{\tilde\rho,|b'|}}(\pi_iq_i)^{-\ang{i,|b'|}}\parens{E_ib'-\sum_{b\in B_{|b'|+i}}b^*(E_ib')b}=0. \end{align*} \if Next, we show that $\Delta(E_i)\sum_{b\in B} b\otimes b^*=0$. We may assume that $B$ consists of homogeneous elements. First observe that $x=\sum v\otimes w^*=0$ if and only if $x(v'):=\sum w^*(v')v=0$ for all $v'\in V$. Then setting $x=\Delta(E_i)\sum_{b\in B} b\otimes b^*$ \[0\neq x=\sum_{b\in B} (E_ib\otimes b^*+\pi_i^{p(b)}(\pi_iq_i)^{\ang{i,|b|}}b\otimes E_ib^*),\] there must be some $v\in V$ such that \[x(v)=\sum_{b\in B} (b^*(v)E_ib+\pi_i^{p(b)}(\pi_iq_i)^{\ang{i,|b|}}(E_ib^*)(v)b )\neq 0.\] However, if $b'\in B$, \begin{align*} x(b')&=E_ib'+\sum_{b\in B_{|b'|+i}}(\pi_iq_i)^{\ang{i,|b|}}b^*(-{\tilde{J}}_i^{-1}{\tilde{K}}_i^{-1}E_ib')b\\ &= E_ib'-\sum_{b\in B_{|b'|+i}}\pi_i^{p(b)}b^*(E_ib')b \end{align*} \fi \end{proof} \subsection{Simple modules and their duals} Let $\lambda\in X^+$ and recall from \cite{CHW1} that $V(\lambda)$ is the simple ${\mathbf U}$-module of highest weight $\lambda$ such that the highest weight space has even parity. Then $V(\lambda)$ has finite rank and has the same character as the ${\mathfrak{so}}(2n+1)$ module of highest weight $\lambda$. In particular, using the Weyl character formula for $V(\lambda)$, the lowest weight vector has weight $w_0\lambda=-\lambda$, hence the parity of the lowest weight vector of $V(\lambda)$ is $P(\lambda)$. Using standard arguments (for example, analogues of \cite[\S 5.3 and \S 5.16]{Jan}), and considering the above analysis, we obtain the following lemma. \begin{lem}\label{lem:dual isos} For each $\lambda\in X^+$, there is an isomorphism $V(\lambda)^{*}\cong \Pi^{P(\lambda)} V(\lambda)$ and a natural isomorphism $V(\lambda)^{**}\rightarrow V(\lambda)$. \end{lem} \begin{example}\label{ex:rk1mods} In the case $n=1$, the module $V=V(m)$ for $m\in{\mathbb{Z}}_{\geq 0}$ has basis $v_{m-2k}=F^{(k)}v_m$ with $0\leq k\leq m$, where $v_m$ is a choice of highest weight vector. Note that by convention $p(v_m)=0$, so $p(v_{m-2k})\equiv k$ (mod 2). The dual module $V(m)^*$ has a dual basis $v_{m-2k}^*$, $0\leq k\leq m$, and the actions of $E=E_{\rf 1}$ and $F=F_{\rf 1}$ are given by \begin{align*} Ev_{m-2k}^*&=-\pi^k(\pi q)^{m-2k}\bra{n+1-k}v_{m-2(k+1)}^*\\ Fv_{m-2k}^*&=-\pi^k(\pi q)^{m-2k+2}\bra{k}v_{m-2(k-1)}^* \end{align*} In particular, this is a simple module generated by the highest weight vector $v_{-m}^*$, where $|v_{-m}^*|=-|v_{-m}|=m$ and $p(v_{-m}^*)=p(v_{-m})\equiv m$ (mod 2), hence we have an isomorphism $V(m)^*\cong \Pi^m V(m)$. \end{example} For convenience, we will use the notation \begin{equation}\label{eq:dual notation} V(-\lambda)=V(\lambda)^*,\quad \lambda\in X^+. \end{equation} We denote the maps in Lemma \ref{lemma:evsandcoevs} in the case $V=V(\lambda)$ with the subscript $\lambda$ instead of $V(\lambda)$; for instance, $\ev_\lambda=\ev_{V(\lambda)}$. Note that \[\ev_\lambda\ \circ\ \coev_\lambda=\sum_{\nu \in {\mathbb{N}}[I]}{\rm rank}_{{\Qqt^\tau}}(V_{\lambda-\nu}) \pi^{p(\nu)}q^{\ang{\tilde{\rho},\lambda-\nu}}=\pi^{P(\lambda)}\qtr_\lambda\ \circ\ \coqtr_\lambda\] \begin{example}\label{ex:rk1evcoev} For $n=1$, we have $\rho=\tilde\rho=\mathbf 1$ hence for $\lambda=m$, $\ang{\tilde\rho,\lambda}=m$. Then \[\ev_m\circ\coev_m=q^m+\pi q^{m-2}+\ldots+\pi^{m} q^{-m}=\pi^m[m+1] =\pi^m\qtr_m\circ\coqtr_m.\] \end{example} \subsection{Further properties of the quasi-${\mathcal{R}}$-matrix} Let us recall the quasi-${\mathcal{R}}$-matrix from \cite[\S 4]{CHW1} \begin{prop}\label{prop:quasiR}\cite{CHW1} Let ${\mathbf{B}}$ be any ${\Qqt^\tau}$-basis of $\ensuremath{\mathbf{f}}$ such that ${\mathbf{B}}_\nu={\mathbf{B}}\cap \ensuremath{\mathbf{f}}_\nu$ is a basis of $\ensuremath{\mathbf{f}}_\nu$ for any $\nu\in {\mathbb{N}}[I]$, with ${\mathbf{B}}_0=\set 1$. Let ${\mathbf{B}}^*=\set{b^*\mid b\in {\mathbf{B}}}$ be the basis of $\ensuremath{\mathbf{f}}$ dual to ${\mathbf{B}}$ under $(\cdot,\cdot)$. Define \[ \Theta_\nu=(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu\sum_{b\in {\mathbf{B}}_\nu} b^-\otimes (b^*)^+\in{\mathbf U}_{-\nu}^-\otimes {\mathbf U}_\nu^+. \] Then if $M,M'$ are integrable modules of ${\mathbf U}$, then $\Theta=\sum_{\nu}\Theta_\nu$ is a well defined operator on $M\otimes M'$ which satisfies $\Delta(u)\Theta=\Theta\bar\Delta(u)$ as endomorphisms of $M\otimes M'$, where $\bar\Delta(u)=\bar{\Delta(\bar u)}$. Moreover, $\Theta$ is independent of the choice of basis ${\mathbf{B}}$, and is invertible with inverse $\bar\Theta$. \end{prop} In particular, note that all modules considered in this paper are of finite rank over ${\Qqt^\tau}$, hence are integrable. \begin{example}\label{ex:rank1rmat} When $n=1$, the quasi-$\mathcal R$-matrix $\Theta$ can be explicitly given by the formula \[\Theta=\sum_{n\geq 0} (-1)^n (\pi q)^{-\binom{n}{2}}[n]^!(\pi q-q^{-1})^n F^{(n)}\otimes E^{(n)}=1-(\pi q-q^{-1})F\otimes E+\ldots.\] (NB. there is a typo in the power of $\pi q$ in \cite[Example 3.1.2]{CHW1}.) \end{example} While $\bar\Theta$ can be evaluated easily, it will be more convenient to have the following alternate description of $\bar\Theta$ using the properties of the bilinear form on $\ensuremath{\mathbf{f}}$ (cf. \cite[\S 1.4]{CHW1}). \begin{lem}\label{lem:quasiRinv} With the same notations as in Proposition \ref{prop:quasiR}, $\bar\Theta=\sum_\nu \bar\Theta_\nu$ is given by \[ \bar \Theta_\nu=\pi_\nu q^{\frac{\nu\cdot\nu}2}\sum_{b\in {\mathbf{B}}_\nu} b^-\otimes \sigma(b^*)^+\in{\mathbf U}_{-\nu}^-\otimes {\mathbf U}_\nu^+. \] \end{lem} \begin{proof} Let $\bar{\mathbf{B}}=\set{\bar b\mid b\in{\mathbf{B}}}$, with dual basis $\bar{\mathbf{B}}^*$. Then since $\Theta$ is independent of the choice of basis, we see that for $\nu\in{\mathbb{N}}[I]$, $\Theta_\nu=(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu\sum_{b\in {\mathbf{B}}_\nu} \bar{b}^-\otimes (\bar{b}^*)^+$. We have $\bar\Theta_\nu=(-1)^{{\mathrm{ht}} \nu} \pi^{{\mathbf p}(\nu)}q_{-\nu}\sum_{b\in {\mathbf{B}}_\nu} \bar{(\bar{b}^-)}\otimes \bar{(\bar{b^*}^+)}$, and note that $(\bar{x})^{\pm}=\bar{(x^{\pm})}$, so $\bar{(\bar{b}^-)}=b^-$. On the other hand, recall from \cite[\S 1.4]{CHW1} the variant bilinear form $\set{-,-}$ defined by $\set{x,y}=\bar{(\bar x,\bar y)}$. Note that by construction, $(\bar{b}^*,\bar{b'})=\delta_{b,b'}$. Then for any $b,b'\in{\mathbf{B}}$, we apply Lemma 1.4.3 (b) of {\em loc. cit.} to deduce that \[\delta_{b,b'}=\bar{(\bar{b}^*, \bar{b'})}=\set{\bar{\bar{b}^*},b'} =(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q^{-\frac{\nu\cdot\nu}{2}} q_{-\nu}(\bar{\bar{b}^*},\sigma(b')).\] (We note that while the power of $\pi$ appears different from that in {\em loc. cit.}, it is equivalent.) Therefore, we have \[\bar{\bar{b}^*}=(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)}q^{\frac{\nu\cdot\nu}{2}}\pi_\nu q_{\nu} \sigma(b)^*.\] Then the lemma follows from the observation that since $(\sigma(x),\sigma(y))=(x,y)$, $\sigma(b)^*=\sigma(b^*)$. \end{proof} Now we will proceed to use $\Theta$ to define a universal map ${\mathcal{R}}:M\otimes N\rightarrow N\otimes M$ for any modules $M$ and $N$. These constructions will be modified versions of the standard arguments in the non-super case; cf. \cite[\SS 7.3-7.6]{Jan} or \cite[\S 4.2 and Chapter 32]{L93}. For $1\leq s<t\leq 3$, let $\Theta_\nu^{st}\in {\mathbf U}\otimes {\mathbf U}\otimes {\mathbf U}$ be defined by $\Theta_\nu^{st}=(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu\sum_{b\in {\mathbf{B}}_\nu} b_1\otimes b_2\otimes b_3$ where $b_s=b^-$,$b_t=(b^*)^+$, and $b_m=1$ for $m\neq s,t$. \begin{prop} We have the following identities. \[(\Delta\otimes 1)(\Theta_\nu)=\sum_{\nu'+\nu''=\nu}\Theta_{\nu'}^{23}(1\otimes {\tilde{K}}_{-\nu''}\otimes 1)\Theta^{13}_{\nu''}.\] \[(\bar\Delta\otimes 1)(\Theta_\nu)=\sum_{\nu'+\nu''=\nu}\Theta_{\nu'}^{13}(1\otimes {\tilde{J}}_{\nu'}{\tilde{K}}_{\nu'}\otimes 1)\Theta^{23}_{\nu''}.\] \[(1\otimes \Delta )(\Theta_\nu)=\sum_{\nu'+\nu''=\nu}\Theta_{\nu'}^{12}(1\otimes {\tilde{J}}_{\nu''}{\tilde{K}}_{\nu''}\otimes 1)\Theta^{13}_{\nu''}.\] \[(1\otimes \bar\Delta )(\Theta_\nu)=\sum_{\nu'+\nu''=\nu}\Theta_{\nu'}^{13}(1\otimes {\tilde{K}}_{-\nu''}\otimes 1)\Theta^{12}_{\nu''}.\] \end{prop} \begin{proof} These identities are proved exactly as in \cite[\S 4.2]{L93}. We will prove the first identity here. For $x\in \ensuremath{\mathbf{f}}$ and $b_1,b_2\in {\mathbf{B}}$, define $f(x,b_1,b_2),f'(x,b_1,b_2)\in {\Qqt^\tau}$ via \[r(x)=\sum_{b_1,b_2\in B} f(x,b_1,b_2)\,b_1\otimes b_2,\] \[\bar r(x)=\sum_{b_1,b_2\in B} f'(x,b_1,b_2)\,b_1\otimes b_2.\] Then it suffices to show that \[\sum_{b,b_1,b_2;|b_1|+|b_2|=|b|=\nu} f'(b,b_1,b_2) b_1^-\otimes {\tilde{K}}_{-|b_1|}b_2^-\otimes (b^*)^+=\sum_{b_1,b_2;|b_1|+|b_2|=\nu} \pi^{p(b_1)p(b_2)}b_1^-\otimes b_2^-{\tilde{K}}_{-|b_1|}\otimes (b_2^*b_1^*)^+.\] In particular, it is enough to show that for all $b_1,b_2\in B$ such that $|b_1|+|b_2|=\nu$, we have \[\sum_{b;|b|=\nu} f'(b,b_1,b_2)b^*=\pi^{p(b_1)p(b_2)}q^{-|b_1|\cdot |b_2|}b_2^*b_1^*.\] This follows from the equalities \[\pi^{p(b_1)p(b_2)}q^{|b_1|\cdot |b_2|}f'(b,b_1,b_2)=f(b,b_2,b_1)=(r(b),b_2^*\otimes b_1^*)=(b,b_2^*b_1^*),\] which in turn follow from elementary properties of $\ensuremath{\mathbf{f}}$; cf. \cite[Lemmas 1.4.1, 1.4.3]{CHW1} \end{proof} To construct a universal ${\mathbf U}$-module homomorphism from $\Theta$, we will need some additional maps. The first is the swap map; that is, the algebra ${\mathbf U}\otimes {\mathbf U}$ is equipped with an involution $\mathfrak{s}$ defined by $\mathfrak{s}(x\otimes y)=\pi^{p(x)p(y)} y\otimes x$. This induces involutions on ${\mathbf U}^{\otimes m}$ by applying $\mathfrak{s}$ to sequential pairs of tensor factors; specifically, these involutions are the maps $\mathfrak{s}_{t,t+1}=1^{\otimes t-1}\otimes \mathfrak{s}\otimes 1^{m-t-1}$, and it is not hard to see they satisfy the braid relations $\mathfrak{s}_{t-1,t}\mathfrak{s}_{t,t+1}\mathfrak{s}_{t-1,t}=\mathfrak{s}_{t,t+1}\mathfrak{s}_{t-1,t}\mathfrak{s}_{t,t+1}$. In particular, we see that to each element $\gamma$ of the permutation group $\mathfrak S_m$, there is an automorphism $\mathfrak{s}_\gamma$ of ${\mathbf U}^{m}$; for example $\mathfrak{s}_{(23)}=\mathfrak{s}_{2,3}$ and $\mathfrak{s}_{(123)}=\mathfrak{s}_{1,2}\mathfrak{s}_{2,3}$. Similarly, to any tensor product of modules $N=\bigotimes_{i=1}^m M_t$ and $\gamma\in \mathfrak S_m$, we can define $N_\gamma=\bigotimes_{i=1}^m M_{\gamma(t)}$ and a map $\mathfrak{s}_\gamma: N\rightarrow N_\gamma$ given by \[\mathfrak{s}(v)=\pi^{p(\gamma,v)}v_\gamma,\] where $v=v_1\otimes\ldots\otimes v_m$, $v_\gamma=v_{\gamma(1)}\otimes\ldots\otimes v_{\gamma(m)}$, and \[p(\gamma,v)=\displaystyle\sum_{\shortstack{$1\leq s<t\leq n$ \\ $\gamma(s)>\gamma(t)$}}p(v_t)p(v_s).\] These maps are compatible in the sense that for $v=\bigotimes_{t=1}^n v_t\in N$ and $u\in {\mathbf U}$, \[\mathfrak{s}_\gamma(\Delta^{m-1}(u)v)=\mathfrak{s}_\gamma(\Delta^{m-1}(u)) \mathfrak{s}_\gamma(v).\] When $m=2$, we will just write $\mathfrak{s}=\mathfrak{s}_{1,2}$. The other ingredient is a weight-renormalization operator. This operator is induced by the weight function defined in the following lemma. \begin{lem}\label{lem:f function} There exists a function $\mathfrak f:X\times X\rightarrow ({\Qqt^\tau})^\times$ satisfying \[\mathfrak f(\zeta+\mu',\zeta'+\nu')\mathfrak f(\zeta,\zeta')^{-1} = (\pi q)^{-\ang{\tilde\mu,\zeta'}}q^{-\ang{\tilde\nu,\zeta}-\mu\cdot \nu}\] for $\zeta,\zeta'\in X$ and $\mu,\nu\in {\mathbb{Z}}[I]$. Moreover, \begin{enumerate} \item The function $\mathfrak r(\zeta,\zeta')=\mathfrak f(\zeta,\zeta')\mathfrak f(\zeta,-\zeta')$ satisfies $\mathfrak r(\zeta+\mu,\zeta'+\nu)=\mathfrak r(\zeta,\zeta')$ for any $\mu,\nu\in{\mathbb{Z}}[I]$. \item The function $\mathfrak l(\zeta,\zeta')=\mathfrak f(\zeta,\zeta')\mathfrak f(-\zeta,\zeta')$ satisfies $\mathfrak l(\zeta+\mu,\zeta'+\nu)=\mathfrak l(\zeta,\zeta')$ for any $\mu,\nu\in{\mathbb{Z}}[I]$. \item We have $\mathfrak{f}(\zeta,\zeta')\mathfrak{f}(-\zeta,-\zeta')^{-1}=\pi^{P(\zeta)P(\zeta')}$; in particular, $\mathfrak l(\zeta,\zeta')=\pi^{P(\zeta)P(\zeta')}\mathfrak r(\zeta,\zeta')$. \end{enumerate} \end{lem} \begin{proof} It is easy to verify that such a function $\mathfrak{f}$ exists by choosing a set of coset representatives $R$ for ${\mathbb{Z}}[I]$ in $X$. It is similar to verify (1) and (2), so let us show (1). Let $\xi=\zeta+\nu$ and $\xi'=\zeta'+\mu$ for some $\mu,\nu\in {\mathbb{N}}[I]$ and $\zeta,\zeta'\in X$. Then \begin{align*} \mathfrak f(\xi,\xi')\mathfrak f(\xi,-\xi')&=\mathfrak f(\zeta,\zeta')\mathfrak f(\zeta,-\zeta')(\pi q)^{\ang{\tilde\mu,\zeta'}+\ang{\tilde\mu,-\zeta'}}q^{-\ang{\tilde\nu,\zeta}-\nu\cdot\mu-\ang{-\tilde\nu,\zeta}-(-\nu\cdot\mu)}\\ &=\mathfrak f(\zeta,\zeta')\mathfrak f(\zeta,-\zeta'). \end{align*} Finally, let $\zeta,\zeta'\in X$. Then $-\zeta=\zeta-2\zeta$, $-\zeta'=\zeta'-2{\zeta'}$ so \[\mathfrak{f}(\zeta,\zeta')\mathfrak{f}(-\zeta,-\zeta')^{-1}=\mathfrak f(-\zeta+(2\zeta),-\zeta'+(2{\zeta'}))\mathfrak f(-\zeta,-\zeta')^{-1} =\pi^{-\ang{\widetilde{(2\zeta)},-\zeta'}} q^{-\ang{\widetilde{(2\zeta)},-\zeta'}-\ang{\widetilde{(2\zeta')},-\zeta} -2{\zeta}\cdot2{\zeta'}}\] Now note that for any $\eta,\eta'\in X$, we have $-\ang{\widetilde{(2\eta)},-\eta'} =\frac{1}{2}(2\eta)\cdot (2\eta')$. Moreover, by \eqref{eq:weight parity} and Lemma \ref{lem:weight parity equiv}, we see that $\ang{\widetilde{(2\eta)},\eta'}\equiv (2\eta)_{\rf n}\ang{\rf n,\eta'}\equiv n\ang{\rf n,\eta}\ang{\rf n,\eta'}\equiv p(\eta)p(\eta')\mod 2$ . Therefore, we see that \[\mathfrak f(-\zeta+2\zeta,-\zeta'+2{\zeta'})\mathfrak f(-\zeta,-\zeta')^{-1} =\pi^{P(\eta)P(\eta')}.\] This finishes the proof. \end{proof} \begin{example}\label{ex:rank1f} Let us consider the case $n=1$. Then the function $\mathfrak f$ is determined by the values $\mathfrak f(0,0)$, $\mathfrak f(0,1)$, $\mathfrak f(1,0)$, and $\mathfrak f(1,1)$. Then for any $\epsilon_1,\epsilon_2\in\set{0,1}$, \[\mathfrak f(\epsilon_1+2s,\epsilon_2+2t)=\mathfrak f(\epsilon_1,\epsilon_2)\pi^{s\epsilon_2} q^{-t\epsilon_1-s\epsilon_2-2st}.\] By direct computation, one finds the corresponding coset functions to be \[\mathfrak r(\epsilon_1+2s,\epsilon_2+2t)=\mathfrak f(\epsilon_1,\epsilon_2)^2q^{\epsilon_1\epsilon_2}.\] \[\mathfrak l(\epsilon_1+2s,\epsilon_2+2t)=\mathfrak f(\epsilon_1,\epsilon_2)^2\pi^{\epsilon_1\epsilon_2}q^{\epsilon_1\epsilon_2}.\] \if 0 For $n>1$, we have \[\mathfrak r(\epsilon_1\omega_{\rf n}+\nu,\epsilon_2\omega_{\rf n}+\mu)=\mathfrak{f}(\epsilon_1\omega_{\rf n},\epsilon_2\omega_{\rf n})^2q^{n\epsilon_1\epsilon_2}.\] \[\mathfrak l(\epsilon_1\omega_{\rf n}+\nu,\epsilon_2\omega_{\rf n}+\mu)=\mathfrak{f}(\epsilon_1\omega_{\rf n},\epsilon_2\omega_{\rf n})^2\pi^{n\epsilon_1,\epsilon_2}q^{n\epsilon_1\epsilon_2}.\] \fi \end{example} Given ${\mathbf U}$-modules $M,M'$, define the ${\Qqt^\tau}$-linear bijection $\mathfrak F:M\otimes M'\rightarrow M\otimes M'$ by $\mathfrak F(m\otimes m')=\mathfrak f(|m|,|m'|) m\otimes m'$. For $1\leq s<t\leq 3$, we define $\mathfrak F^{st}$ on $M_1\otimes M_2\otimes M_3$ via $\mathfrak F^{st}(m_1\otimes m_2\otimes m_3)=\mathfrak f(|m_s|,|m_t|)m_1\otimes m_2 \otimes m_3$. Let ${}^\mathfrak{F}\Theta^{st}=\Theta^{st}\circ \mathfrak F^{st}$. \begin{prop}[Yang-Baxter equation] As operators on $M_1\otimes M_2\otimes M_3$, \[{}^\mathfrak{F}\Theta^{12}\circ{}^\mathfrak{F} \Theta^{13}\circ{}^\mathfrak{F}\Theta^{23}= {}^\mathfrak{F}\Theta^{23}\circ{}^\mathfrak{F}\Theta^{13}\circ {}^\mathfrak{F}\Theta^{12}\] \end{prop} \begin{proof} First note that the maps $\mathfrak F^{st}$ are bijections which commute with one another. One verifies directly that, as operators on $M_1\otimes M_2\otimes M_3$, \[\mathfrak F^{12}\Theta_\nu^{13}=\Theta_\nu^{13}(1\otimes {\tilde{J}}_\nu{\tilde{K}}_{\nu}\otimes 1)\mathfrak F^{12}\text{, } \quad \mathfrak F^{12}\mathfrak{F}^{13}\Theta_\nu^{23}= \Theta_\nu^{23}\mathfrak F^{12}\mathfrak{F}^{13},\] \[\mathfrak F^{23}\Theta_\nu^{13}=\Theta_\nu^{13}(1\otimes {\tilde{K}}_{-\nu} \otimes 1)\mathfrak F^{23}\text{, } \quad \mathfrak F^{23}\mathfrak{F}^{13}\Theta_\nu^{12}= \Theta_\nu^{12}\mathfrak F^{23}\mathfrak{F}^{13}.\] In particular, it suffices to show that \[\Theta^{12}\parens{\sum_{\nu}\Theta_\nu^{13}(1\otimes {\tilde{J}}_\nu{\tilde{K}}_{\nu}\otimes 1)} \Theta^{23}=\Theta^{23}\parens{\sum_{\nu}\Theta_\nu^{13}(1\otimes {\tilde{K}}_{-\nu}\otimes 1)} \Theta^{12}.\] Writing $\Theta^{12}=\sum_{\mu}\Theta_\mu^{12}$, we have \[\Theta^{12}\parens{\sum_{\nu}\Theta_\nu^{13}(1\otimes {\tilde{J}}_\nu{\tilde{K}}_{\nu}\otimes 1)} =\sum_{\mu,\nu}\Theta_\mu^{12}(1\otimes {\tilde{J}}_\nu{\tilde{K}}_{\nu}\otimes 1)\Theta_\nu^{13} =\sum_{\nu}(1\otimes \Delta)(\Theta_{\nu}),\] and similarly \[\parens{\sum_{\nu}\Theta_\nu^{13}(1\otimes {\tilde{K}}_{-\nu}\otimes 1)} \Theta^{12}=\sum_{\nu}(1\otimes \bar \Delta)(\Theta_{\nu}).\] Then we are reduced to showing the equality \[\sum_{\nu}(1\otimes \Delta)(\Theta_{\nu})\Theta^{23}=\Theta^{23}\sum_{\nu}(1\otimes \bar \Delta)(\Theta_{\nu}),\] which follows from the defining property of $\Theta$. \end{proof} \begin{prop}\label{prop:Rmat} Define $\mathcal R: M\otimes M'\rightarrow M'\otimes M$ by $\mathcal R=\Theta\circ\mathfrak F\circ\mathfrak{s}$. Then $\mathcal R$ is a ${\mathbf U}$-module isomorphism. \end{prop} \begin{proof} That $\mathcal R$ is bijective and homogeneous in parity is clear from construction. Note that \[\Delta(u)\mathcal R(m\otimes m')=\Theta(\bar\Delta(u)\mathfrak F\circ \mathfrak{s}(m\otimes m'))=\Theta(\mathfrak f(|m'|,|m|)\pi^{p(m)p(m')}\bar\Delta(u)(m'\otimes m)),\] so it suffices to show \[\mathfrak F\circ \mathfrak{s}(\Delta(u)m\otimes m'))=\mathfrak f(|m'|,|m|)\pi^{p(m)p(m')}\bar\Delta(u)(m'\otimes m)\] for all $u\in {\mathbf U}$, hence it is enough to show this equality holds when $u$ is a generator. For $u=J_\nu, K_\nu$, this is straightforward. The cases $u=E_i$ and $u=F_i$ are similar, so we shall prove the first case: \begin{align*} \mathfrak F\circ \mathfrak{s}(\Delta(E_i)&m\otimes m')=\mathfrak f(|m'|,i+|m|)\pi^{p(i)p(m')+p(m)p(m')} m'\otimes E_im\\ &\hspace{5em}+\mathfrak f(i+|m'|,|m|)\pi^{p(m)p(m')}(\pi q)^{d_i\ang{i,|m|}} E_im'\otimes m\\ &=f(|m'|,|m|)\pi^{p(m)p(m')}(E_im'\otimes m+\pi^{p(i)p(m')} q^{-d_i\ang{i,|m'|}} m'\otimes E_im)\\ &=f(|m'|,|m|)\pi^{p(m)p(m')}\bar\Delta(E_i)(m'\otimes m). \end{align*} \end{proof} We thus obtain the following crucial property of ${\mathcal{R}}$. \begin{prop}\label{prop:braidrel} For any modules $M_1$, $M_2$, and $M_3$, let ${\mathcal{R}}_{st}={}^{\mathfrak F}\Theta^{st}\circ\mathfrak{s}_{(st)}$. Then \[{\mathcal{R}}_{12}{\mathcal{R}}_{23}{\mathcal{R}}_{12}={\mathcal{R}}_{23}{\mathcal{R}}_{12}{\mathcal{R}}_{23}:M_1\otimes M_2\otimes M_3\rightarrow M_3\otimes M_2\otimes M_1.\] \end{prop} \begin{proof} First note that if $\sigma(s)<\sigma(t)$, $\mathfrak{s}_{\sigma}{}^{\mathfrak F}\Theta^{st} ={}^{\mathfrak F}\Theta^{\sigma(s)\sigma(t)}\mathfrak{s}_{\sigma}$. Therefore we have $\mathfrak{s}_{(12)}{}^{\mathfrak F}\Theta^{23}={}^{\mathfrak F}\Theta^{13}\mathfrak{s}_{(12)}$, and $\mathfrak{s}_{(123)}{}^{\mathfrak F}\Theta^{12}={}^{\mathfrak F}\Theta^{23}\mathfrak{s}_{(123)}$, hence in particular \[R_{12}R_{23}R_{12} ={}^{\mathfrak F}\Theta^{12} \circ{}^{\mathfrak F}\Theta^{13} \circ{}^{\mathfrak F}\Theta^{23} \circ\mathfrak{s}_{(13)}.\] Similar manipulations of the right-hand side yield the equality \[R_{23}R_{12}R_{23} ={}^{\mathfrak F}\Theta^{23} \circ{}^{\mathfrak F}\Theta^{13} \circ{}^{\mathfrak F}\Theta^{12} \circ\mathfrak{s}_{13}.\] Since $s_{13}$ is a bijection, the proposition follows from the Yang-Baxter equation. \end{proof} \begin{rmk} In \cite[\S 32]{L93}, it is shown that for $\mathfrak g=\mathfrak{sl}(2)$, which we can view as the $\pi=1$ (i.e. $\tau=\pm 1$) specialization of Example \ref{ex:rank1rmat}, we can extend our field ${\Q(q)}$ to ${\mathbb{Q}}(\sqrt q)$ and normalize so that $\mathfrak f$ is bi-multiplicative; that is $\mathfrak f(m+a,n+b)=\mathfrak f(m,n)\mathfrak f(m,b) \mathfrak f(a,n)\mathfrak f(a,b)$. This is necessary for the maps $\mathcal R$ to satisfy the Hexagon Identities and thus define a braiding on the category of finite dimensional modules. Note that Example \ref{ex:rank1rmat} shows such a renormalization is impossible in general in the $\pi=-1$ case, so in particular the maps $\mathcal R$ can not be normalized to define a braiding on the category of finite dimensional weight modules. It is possible to overcome this difficulty by restricting the class of modules to those of "even" highest weight, or by expanding the definition of $\mathfrak{f}$ to a function on $\bigrset\times\bigrset$, but we shall not need this at present. \end{rmk} \section{Diagrammatic Calculus and Knot invariants} \label{sec:diagcalc} We will now interpret the ${\mathbf U}$-module homomorphisms in terms of planar diagrams. At first, these diagrams should be interpreted as slice diagrams; that is, diagrams together with vertical slices at various heights such that between consecutive slices is an elementary diagram corresponding to a ${\mathbf U}$-module homomorphism. However, we will ultimately see that diagrams which can be identified by planar isotopies yield the same morphisms. \subsection{Cups, caps, and crossings} Recall that $\coqtr_\lambda$, $\coev_\lambda$, $\qtr_\lambda$, and $\ev_\lambda$ are the maps defined in Lemma \ref{lemma:evsandcoevs} where $V=V(\lambda)$. Likewise, let ${\mathcal{R}}_{\pm \lambda, \pm \mu}:V(\pm \lambda)\otimes V(\pm \mu)\rightarrow V(\pm \mu)\otimes V(\pm \lambda)$ be the map defined in Proposition \ref{prop:Rmat}. Furthermore, we will use the notation $1_{\pm \lambda}=1_{V(\pm\lambda)}$. We will now begin to represent our maps via a graphical calculus in anticipation of constructing tangle invariants. Specifically, we follow \cite{T,ADO} and interpret maps between tensor products of the modules $V(\pm \lambda)$ for various $\lambda\in X^+$ as sliced oriented tangle diagrams with $X^+$-labeled strands; a concise exposition of this approach is lain out in \cite[Chapter 3]{Oht}. The elementary oriented tangle diagrams are interpreted as follows. (Note that while sideways-oriented crossings aren't considered elementary, we include them here for convenience in later arguments.) \begin{center} \begin{tabular}{ccccccc} && \begin{tikzpicture} \draw (-2, 2) node {$1_{\lambda}=$}; \idsup{-1}{1.5}{1}{1}{1} \draw (-.8,1.5) node {$\lambda$}; \end{tikzpicture} && \begin{tikzpicture} \draw (0, 2) node {$1_{-\lambda}=$}; \idsdown{1}{1.5}{1}{1}{1} \draw (1.2,1.5) node {$\lambda$}; \end{tikzpicture}\\\\ $\coqtr_{\lambda}=$ \hctikz{ \cwcup{1}{-.5}{1}{.75} \draw (1.8,-.5) node {$\lambda$}; } &&$\coev_{\lambda}=$ \hctikz{ \ccwcup{1}{-.5}{1}{.75} \draw (1.7,-.5) node {$\lambda$};} && $\qtr_{\lambda}=$ \hctikz{ \cwcap{1}{-.5}{1}{.75} \draw (1.8,.3) node {$\lambda$}; } && $\ev_{\lambda}=$ \hctikz{ \ccwcap{1}{-.5}{1}{.75} \draw (1.8,.3) node {$\lambda$}; } \\\\ ${\mathcal{R}}_{\lambda,\mu}= $ \hctikz{ \rcrossup{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && ${\mathcal{R}}_{-\lambda,-\mu}= $ \hctikz{ \rcrossdown{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && ${\mathcal{R}}_{\lambda,-\mu}= $ \hctikz{ \NESE{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && ${\mathcal{R}}_{-\lambda,\mu}= $ \hctikz{ \SWNW{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } \\\\ ${\mathcal{R}}_{\lambda,\mu}^{-1}= $ \hctikz{ \lcrossup{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && ${\mathcal{R}}_{-\lambda,-\mu}^{-1}= $ \hctikz{ \lcrossdown{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && ${\mathcal{R}}_{\lambda,-\mu}^{-1}= $ \hctikz{ \SENE{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && ${\mathcal{R}}_{-\lambda,\mu}^{-1}= $ \hctikz{ \NWSW{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } \end{tabular} \end{center} We construct more general diagrams from these elementary ones by the following constructions. If $\hctikz{ \ids{.2}{.9}{.2}{.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} }$ is some diagram denoting the morphism $\phi$ and $\hctikz{ \ids{.1}{.9}{.2}{.1}{4} \standin{0}{.1}{.8}{.8}{S} \ids{.4}{0}{.2}{.1}{1} }$ is some diagram denoting the morphism $\psi$, then we can combine them as \begin{itemize} \item the {\em horizontal} composition $\hctikz{ \ids{.2}{.9}{.2}{.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} \ids{1.1}{.9}{.2}{.1}{4} \standin{1}{.1}{.8}{.8}{S} \ids{1.4}{0}{.2}{.1}{1} }$ which denotes the tensor product $\phi\otimes \psi$ \item the {\em vertical} composition $\hctikz{ \ids{.2}{.9}{.2}{.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} \ids{.1}{-.1}{.2}{.1}{4} \standin{0}{-.9}{.8}{.8}{S} \ids{.4}{-1}{.2}{.1}{1}}$ which denotes the composition $\phi\circ \psi$, or zero if this composition is undefined (which is to say, when the strands on top of S don't match the number and labelling of the strands on the bottom of T). \end{itemize} We will say two diagrams are equal if the corresponding morphisms agree. Note that, by construction and by Lemma \ref{prop:braidrel}, the following diagrams are equal for any choice of orientation and labeling of strands: \begin{equation}\label{eq:comm and ids} \hctikz{ \ids{.2}{.9}{.2}{.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} }\quad =\quad \hctikz{ \ids{.2}{.9}{.2}{.5}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} }\quad =\quad \hctikz{ \ids{.2}{.9}{.2}{.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{-.4}{.2}{.5}{4} },\qquad \hctikz{ \ids{.2}{.9}{.2}{1.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} \ids{1.3}{1.9}{.2}{.1}{2} \standin{1}{1.1}{.8}{.8}{S} \ids{1.4}{0}{.2}{1.1}{1} }\quad =\quad \hctikz{ \ids{.2}{1.9}{.2}{.1}{3} \standin{0}{1.1}{.8}{.8}{T} \ids{.1}{0}{.2}{1.1}{4} \ids{1.3}{.9}{.2}{1.1}{2} \standin{1}{.1}{.8}{.8}{S} \ids{1.4}{0}{.2}{.1}{1} } \end{equation} \begin{equation}\label{eq:diagbraidrels} \hctikz{\lcross{0}{0}{.75}{.5} \rcross{0}{.75}{.75}{.5} } = \quad \hctikz{ \ids{0}{0}{1}{1.5}{1} \ids{.5}{0}{1}{1.5}{1} } = \quad \hctikz{\rcross{0}{0}{.75}{.5} \lcross{0}{.75}{.75}{.5} }\ ,\qquad \hctikz{\rcross{0}{0}{.5}{.5}\ids{1}{0}{.5}{.5}{1} \rcross{.5}{.5}{.5}{.5}\ids{0}{.5}{.5}{.5}{1}\rcross{0}{1}{.5}{.5}\ids{1}{1}{.5}{.5}{1}} = \hctikz{\rcross{.5}{0}{.5}{.5}\ids{0}{0}{.5}{.5}{1} \rcross{0}{.5}{.5}{.5}\ids{1}{.5}{.5}{.5}{1}\rcross{.5}{1}{.5}{.5}\ids{0}{1}{.5}{.5}{1}} \end{equation} \vspace{1em} In \eqref{eq:comm and ids}, the symbols $\hctikz{ \ids{.2}{.9}{.2}{.1}{3} \standin{0}{.1}{.8}{.8}{T} \ids{.1}{0}{.2}{.1}{4} }$ and $\hctikz{ \ids{.3}{.9}{.2}{.1}{2} \standin{0}{.1}{.8}{.8}{S} \ids{.4}{0}{.2}{.1}{1} }$ stand for arbitrary sub-diagrams with an arbitrary number of strands protruding from the top and bottom and with an arbitrary labeling of strands. \subsection{Graphical identities}\label{sec:graphid} Now we shall prove some more substantial diagrammatic identities. \begin{lem}\label{lem:straightening} We have an equality of diagrams \[\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \dcup{0}{.25}{.5}{.5}\ids{1}{0}{.5}{.75}{1} }\quad =\quad \hctikz{\ids{0}{0}{1}{1.5}{1}} \quad=\quad \hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \dcup{.5}{.25}{.5}{.5}\ids{0}{0}{.5}{.75}{1} }\] For any choice of orientation or labeling of the strand. \end{lem} \begin{proof} This follows by choosing a homogeneous basis for the module and applying the definitions; we will prove the equality \[\hctikz{ \idsdown{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \dcup{0}{.25}{.5}{.5}\idsdown{1}{0}{.5}{.75}{1} \draw (1.3,0) node {$\lambda$}; }\quad =\quad \hctikz{\idsdown{0}{0}{1}{1.5}{1}\draw (0.3,0) node {$\lambda$};} \] in detail, as the other cases are similar. In terms of morphisms, we wish to show $(1_{-\lambda}\otimes \qtr_\lambda)\circ (\coev_\lambda\otimes 1_{-\lambda})=1_{-\lambda}$. Let $B$ be a homogeneous basis of $V(\lambda)$ and $B^*$ the dual basis of $V(\lambda)^*$. Then for any $b_0\in B$, \begin{align*} (1_{-\lambda}\otimes \qtr_\lambda)(\coev_\lambda\otimes 1_{-\lambda})(b_0^*) &=\sum_{b\in B} \pi^{p(b)}q^{\ang{\tilde\rho,|b|}}(1_{-\lambda}\otimes \qtr_\lambda)(b^*\otimes b\otimes b_0^*)\\ &=\sum_{b\in B} b_0^*(b)b^*=b_0^*. \end{align*} \end{proof} \begin{lem}\label{lem:reide2} For $\lambda\in X^+$, we have an equality of diagrams \[\tag{a}\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \rcrossup{0}{0}{.75}{.5}\idsdown{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }\quad =\mathfrak f(\lambda,\lambda)q^{-\ang{\tilde\rho,\lambda}} \hctikz{\idsup{0}{0}{1}{1.5}{1} \draw (-.3,0) node {$\lambda$};} \quad= \pi^{P(\lambda)}\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \rcrossup{.5}{0}{.75}{.5}\idsdown{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; } \] \[\tag{b}\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \lcrossup{.5}{0}{.75}{.5}\idsdown{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }\quad =\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde\rho,\lambda}} \hctikz{\idsup{0}{0}{1}{1.5}{1} \draw (-.3,0) node {$\lambda$};} \quad= \pi^{P(\lambda)}\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \lcrossup{0}{0}{.75}{.5}\idsdown{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; } \] \[\tag{c}\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \rcrossdown{0}{0}{.75}{.5}\idsup{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }\quad =\mathfrak f(\lambda,\lambda)q^{-\ang{\tilde\rho,\lambda}} \hctikz{\idsdown{0}{0}{1}{1.5}{1} \draw (-.3,0) node {$\lambda$};} \quad= \pi^{P(\lambda)}\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \rcrossdown{.5}{0}{.75}{.5}\idsup{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; } \] \[\tag{d}\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \lcrossdown{.5}{0}{.75}{.5}\idsup{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }\quad =\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde\rho,\lambda}} \hctikz{\idsdown{0}{0}{1}{1.5}{1} \draw (-.3,0) node {$\lambda$};} \quad= \pi^{P(\lambda)}\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \lcrossdown{0}{0}{.75}{.5}\idsup{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; } \] \end{lem} \begin{proof} The proofs of (a)-(d) are all similar, so we will only prove (a). First, let us denote \[\phi=\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \rcrossup{0}{0}{.75}{.5}\idsdown{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }=(1_\lambda\otimes \qtr_{\lambda})\circ ({\mathcal{R}}_{\lambda,\lambda}\otimes 1_\lambda)\circ (1_\lambda\otimes \coqtr_{\lambda}),\] \[\psi=\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \rcrossup{.5}{0}{.75}{.5}\idsdown{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }=(\ev_{\lambda}\otimes 1_\lambda)\circ (1_\lambda\otimes {\mathcal{R}}_{\lambda,\lambda})\circ (\coev_\lambda\otimes 1_\lambda).\] Since $\phi$ and $\psi$ are ${\mathbf U}$-module homomorphisms from $V(\lambda)$ to $V(\lambda)$., $\phi$ and $\psi$ must each be a multiple of the identity which is completely determined by the image of an extremal weight vector, so let $v_\lambda\in V(\lambda)_\lambda$ and $v_{-\lambda}\in V(\lambda)_{-\lambda}$ be nonzero highest- and lowest-weight vectors. Then if $B(\lambda)$ is a homogeneous basis of $V(\lambda)$, then \begin{align*} \phi(v_{\lambda})&=(1_\lambda\otimes \qtr_{\lambda})\circ ({\mathcal{R}}_{\lambda,\lambda}\otimes 1_\lambda) \parens{\sum_{v\in B(\lambda)} v_\lambda\otimes v\otimes v^*}\\ &=(1_\lambda\otimes \qtr_{\lambda})\parens{\sum_{v\in B(\lambda)}\mathfrak f(|v|,\lambda) v\otimes v_\lambda\otimes v^*}=\mathfrak f(\lambda,\lambda)q^{-\ang{\tilde\rho,\lambda}}v_\lambda, \end{align*} and thus $\phi=\mathfrak{f}(\lambda,\lambda)q^{-\ang{\tilde\rho,\lambda}} 1_\lambda$. Likewise,we compute \begin{align*} \psi(v_{-\lambda})&=(\ev_{\lambda}\otimes 1_\lambda)\circ (1_\lambda\otimes {\mathcal{R}}_{\lambda,\lambda}) \parens{\sum_{v\in B(\lambda)} \pi^{p(v)}q^{\ang{\tilde\rho,|v|}}v^*\otimes v\otimes v_{-\lambda}}\\ &=(\ev_{\lambda}\otimes 1_\lambda)\parens{\sum_{v\in B(\lambda)}\pi^{p(v)}q^{\ang{\tilde\rho,|v|}}\mathfrak f(|v|,-\lambda) \pi^{p(v)p(\lambda)} v^*\otimes v_{-\lambda}\otimes v}=q^{\ang{\tilde\rho,-\lambda}}\mathfrak f(-\lambda,-\lambda) v_{-\lambda}, \end{align*} and thus $\psi=\mathfrak{f}(-\lambda,-\lambda)q^{-\ang{\tilde\rho,\lambda}} 1_\lambda$. Now the result follows from Lemma \ref{lem:f function} (3). \end{proof} \begin{lem}\label{lem:rotating crossings} We have an equality of diagrams \begin{equation*}\tag{a} \hctikz{ \NESE{0}{0}{1}{1} \draw (-.3,0) node {$\lambda$}; \draw (-.3,1) node {$\mu$}; }\quad = \mathfrak r(\mu,\lambda) \hctikz{ \idsdown{1}{.75}{.5}{.75}{1}\idsup{1.5}{0}{.5}{1.5}{1}\dcap{0}{.75}{.5}{.5} \lcrossdown{.5}{0}{.75}{.5}\idsup{0}{-.75}{.5}{1.5}{1}\ids{.5}{-.75}{.5}{.75}{1} \dcup{1}{-.5}{.5}{.5} \draw (-.3,-.8) node {$\lambda$}; \draw (.7,-.8) node {$\mu$}; }=\pi^{P(\mu)P(\lambda)}\mathfrak{r}(\mu,\lambda)\quad \hctikz{ \idsdown{0}{0}{.5}{1.5}{1}\ids{.5}{.75}{.5}{.75}{1}\dcap{1}{.75}{.5}{.5} \lcrossup{.5}{0}{.75}{.5}\idsdown{1.5}{-.75}{.5}{1.5}{1}\ids{1}{-.75}{.5}{.75}{1} \dcup{0}{-.5}{.5}{.5} \draw (.7,-.8) node {$\lambda$}; \draw (1.3,-.8) node {$\mu$}; } \end{equation*} \begin{equation*}\tag{b} \hctikz{ \NWSW{0}{0}{1}{1} \draw (-.3,0) node {$\mu$}; \draw (-.3,1) node {$\lambda$}; }\quad = \pi^{P(\mu)P(\lambda)}\mathfrak{r}(\mu,\lambda)^{-1} \hctikz{ \ids{1}{.75}{.5}{.75}{1}\idsdown{1.5}{0}{.5}{1.5}{1}\dcap{0}{.75}{.5}{.5} \rcrossup{.5}{0}{.75}{.5}\idsdown{0}{-.75}{.5}{1.5}{1}\ids{.5}{-.75}{.5}{.75}{1} \dcup{1}{-.5}{.5}{.5} \draw (-.3,-.8) node {$\mu$}; \draw (.7,-.8) node {$\lambda$}; }=\mathfrak r(\mu,\lambda)^{-1}\quad \hctikz{ \idsup{0}{0}{.5}{1.5}{1}\ids{.5}{.75}{.5}{.75}{1}\dcap{1}{.75}{.5}{.5} \rcrossdown{.5}{0}{.75}{.5}\idsup{1.5}{-.75}{.5}{1.5}{1}\ids{1}{-.75}{.5}{.75}{1} \dcup{0}{-.5}{.5}{.5} \draw (.7,-.8) node {$\mu$}; \draw (1.3,-.8) node {$\lambda$}; } \end{equation*} for any $\lambda,\mu\in X^+$. \end{lem} \begin{proof} The proof of (a) and (b) being similar, we shall only prove (a) here. First, unpacking the graphical representation, we see that (a) is equivalent to \[{\mathcal{R}}_{\lambda,-\mu}=\mathfrak{r}(\mu,\lambda) \phi=\pi^{p(\mu)p(\lambda)}\mathfrak{r}(\mu,\lambda)\psi\] where $\phi$ and $\psi$ are the compositions \[\phi=(\qtr_\lambda\otimes 1_{-\mu}\otimes 1_\lambda)\circ (1_\lambda\otimes {\mathcal{R}}_{-\mu,-\lambda}^{-1}\otimes 1_\lambda)\circ(1_\lambda\otimes 1_{-\mu}\otimes \coev_\lambda),\] \[\psi=(1_{-\mu}\otimes 1_\lambda\otimes \qtr_\mu)\circ (1_{-\mu}\otimes {\mathcal{R}}_{\mu,\lambda}^{-1}\otimes 1_{-\mu})\circ( \coev_\mu\otimes 1_\lambda\otimes 1_{-\mu}).\] Let $B(\lambda)$ be a homogeneous basis for $V(\lambda)$. Let $v_0\in B(\lambda)_{\kappa}$ and $w_0\in V(\mu)_{\xi}$ for some $\kappa,\xi\in X$. We shall compare the images of our three maps on $v_0\otimes w_0^*$. First, note that \begin{equation}\label{eq:Rlammu*} {\mathcal{R}}_{\lambda,-\mu}(v_0\otimes w_0^*)=\pi^{p(w_0)p(v_0)}\mathfrak f(-\xi,\kappa)\sum_{\nu}(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu\sum_{b\in {\mathbf{B}}_\nu} \pi^{p(\nu)p(w_0)} b^-w_0^*\otimes (b^*)^+v_0. \end{equation} For $\phi$, first let us note the effect of each map in the composition separately. The graphical representation tells us which tensor factors are impacted at each step, so we restrict our view to these tensor factors when computing these maps. First, we have the coevaluation which adds two tensor factors on the right: \[\coev_\lambda(1)=\sum_{v\in B(\lambda)} \pi^{p(v)}q^{\ang{\tilde\rho,|v|}} v^*\otimes v.\] Next, we apply ${\mathcal{R}}_{-\mu,-\lambda}^{-1}=\mathfrak{s} \circ \mathfrak{F}^{-1}\circ \bar\Theta$ to the middle tensor factors, so \[{\mathcal{R}}_{-\mu,-\lambda}^{-1}(w_0^*\otimes v^*)= \sum_\nu\mathfrak f(-\xi-\nu,-|v|+\nu)^{-1}\pi^{p(w_0)p(v)+p(\nu)p(v)} q^{\frac{\nu\cdot\nu}2}\sum_{b\in {\mathbf{B}}_\nu} \sigma(b^*)^+v^*\otimes b^-w_0^*; \] Finally, we apply the quantum trace to the two tensor factors on the left, hence we need to compute $\qtr(v_0\otimes\sigma(b^*)^+v^*)$. Since $x^*(y)=0$ unless $\bigrdeg{x}=\bigrdeg{y}$ (that is, unless $x$ and $y$ have the same weight and parity), we can assume $|v|=\kappa+\nu$ and $p(v)=p(v_0)+p(\nu)$. Then we have \begin{align*}\qtr_\lambda&(v_0\otimes\sigma(b^*)^+v^*)= \pi^{p(v_0)}q^{-\ang{\tilde\rho,|v_0|}} (\sigma(b^*)^+v^*)(v_0)\\ &=(-1)^{{\mathrm{ht}} \nu} \pi^{{\mathbf p}(\nu)+p(v_0)+p(\nu)p(v_0)+p(\nu)}q^{-\frac{\nu\cdot\nu}{2}-\ang{\tilde\rho,\kappa}}(\pi q)^{-\ang{\tilde\nu,\kappa}}q_{-\nu}v^*((b^*)^+v_0). \end{align*} Putting these computations together, we see that \begin{align*} \phi(v_0\otimes w_0^*)&=\sum_{v\in B(\lambda)} \sum_\nu\sum_{b\in {\mathbf{B}}_\nu} \pi^{p(v_0)+p(\nu)}q^{\ang{\tilde\rho,\kappa+\nu}}\\ &\hspace{2em}\times \mathfrak f(-\xi-\nu,-\kappa)^{-1}\pi^{p(w_0)p(v_0)+p(w_0)p(\nu)+p(\nu)p(v_0)+p(\nu)}q^{\frac{\nu\cdot\nu}2}\\ &\hspace{2em}\times (-1)^{{\mathrm{ht}} \nu} \pi^{{\mathbf p}(\nu)+p(v_0)+p(\nu)p(v_0)+p(\nu)}q^{-\frac{\nu\cdot\nu}{2}-\ang{\tilde\rho,\kappa}}(\pi q)^{-\ang{\tilde\nu,\kappa}}q_{-\nu} v^*((b^*)^+v_0) b^-w_0^*\otimes v\\ &=\sum_\nu(-1)^{{\mathrm{ht}} \nu}\mathfrak f(-\xi-\nu,-\kappa)^{-1}(\pi q)^{-\ang{\tilde\nu,\kappa}} \pi_\nu q^{\ang{\tilde\rho,\nu}}q_{-\nu}\pi^{p(v_0)p(w_0)+{\mathbf p}(\nu)}\\ &\hspace{2em}\times \sum_{b\in {\mathbf{B}}_\nu} \pi^{p(w_0)p(\nu)}b^-w_0^*\otimes\parens{\sum_{v\in B(\lambda)}v^*((b^*)^+v_0)v}. \end{align*} But note that $\mathfrak{f}(-\xi-\nu,-\kappa)(\pi q)^{\ang{\tilde\nu,\kappa}}=\mathfrak{f}(-\xi,-\kappa)$, $q^{\ang{\tilde\rho,\nu}}=q_\nu^2$, and $\sum_{v\in B(\lambda)}v^*((b^*)^+v_0)v=(b^*)^+v_0$. Therefore, we have \begin{align*} \phi(v_0\otimes w_0^*)&=\mathfrak f(-\xi,-\kappa)^{-1}\pi^{p(v_0)p(w_0)}\sum_\nu(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)}\sum_{b\in {\mathbf{B}}_\nu}\pi_\nu q_{\nu} \pi^{p(w_0)p(\nu)}b^-w_0^*\otimes(b^*)^+v_0\\ &=\mathfrak r(-\xi,\kappa)^{-1}R_{\lambda,-\mu}(v_0\otimes w_0^*). \end{align*} Finally, since $-\xi\in\mu+{\mathbb{Z}}[I]$ and $\kappa\in \lambda+{\mathbb{Z}}[I]$, we can apply Lemma \ref{lem:f function}(1) to conclude that $\phi=\mathfrak{r} (\mu,\lambda)^{-1}R_{\lambda,-\mu}$. \if 0 Likewise, for $\psi$, we compute \[\coev_\mu(1)=\sum_{w\in B(\mu)} \pi^{p(w)}q^{\ang{\tilde\rho,|w|}} w^*\otimes w,\] \[R_{\mu,\lambda}^{-1}(w\otimes v_0)= \sum_\nu\mathfrak f(|w|-\nu,|v_0|+\nu)^{-1}\pi^{p(w)p(v_0)+p(\nu)p(v_0)}q^{\frac{\nu\cdot\nu}2}\sum_{b\in {\mathbf{B}}_\nu} \sigma(b^*)^+v_0\otimes b^-w \] \begin{align*}\qtr_\mu&(b^-w\otimes w_0^*)= \delta_{|w|,\xi+\nu}\pi^{p(w_0)}q^{-\ang{\tilde\rho,\xi}} w_0^*(b^-w)\\ &=\delta_{|w|,\xi+\nu}(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)+p(w_0)+p(\nu)p(w_0) }q^{-\frac{\nu\cdot\nu}2-\ang{\tilde\rho,-\xi} +\ang{\tilde\nu,-\xi}}q_{-\nu} (\sigma(b)^-w_0^*)(w). \end{align*} Therefore, \begin{align*} \psi(v_0\otimes w_0^*)&=\sum_{w\in B(\mu)} \sum_\nu\sum_{b\in {\mathbf{B}}_\nu} \pi^{p(w)}q^{\ang{\tilde\rho,|w|}}f(|w|-\nu,|v_0|+\nu)^{-1}\pi^{p(w)p(v_0)+p(\nu)p(v_0)}q^{\frac{\nu\cdot\nu}2}\\ &\hspace{2em}\times\delta_{|w|,\xi+\nu}(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)+p(w_0)+p(\nu)p(w_0)}q^{-\frac{\nu\cdot\nu}2-\ang{\tilde\rho,\xi}+\ang{\tilde\nu,-\xi}}q_{-\nu} (\sigma(b)^-w_0^*)(w) w\otimes \sigma(b^*)^+v_0\\ &= \pi^{p(w_0)p(v_0)}\sum_\nu(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_{-\nu}q^{\ang{\tilde\rho,\nu}} \sum_{b\in {\mathbf{B}}_\nu}f(\xi,|v_0|+\nu)^{-1}q^{\ang{\tilde\nu,-\xi}}\\ &\hspace{2em}\times \pi^{p(\nu)p(w_0)} \parens{\sum_{w\in B(\mu)_{\xi+\nu}}(\sigma(b)^-w_0^*)(w) w}\otimes \sigma(b^*)^+v_0\\ &= \pi^{p(w_0)p(v_0)}f(\xi,\kappa)^{-1}\sum_\nu(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_{\nu} \sum_{b\in {\mathbf{B}}_\nu}\pi^{p(\nu)p(w_0)}\sigma(b)^-w_0^*\otimes \sigma(b^*)^+v_0\\ &=\mathfrak l(\mu,\lambda)^{-1}R_{\lambda,-\mu}(v_0\otimes w_0^*). \end{align*} where in the last equality we use the fact that $\sigma(b)$ is another choice of basis for $\ensuremath{\mathbf{f}}$. \fi A similar computation shows that $\psi=\mathfrak{l}(\mu,\lambda)^{-1}{\mathcal{R}}_{\lambda,-\mu}$, and the result then follows from Lemma \ref{lem:f function}. \end{proof} \if 0 \begin{lem} We have an equality of diagrams \[ \hctikz{ \NWSW{0}{0}{1}{1} \draw (-.3,0) node {$\mu$}; \draw (-.3,1) node {$\lambda$}; }\quad = \mathfrak l(\mu,\lambda)^{-1} \hctikz{ \ids{1}{.75}{.5}{.75}{1}\idsdown{1.5}{0}{.5}{1.5}{1}\dcap{0}{.75}{.5}{.5} \rcrossup{.5}{0}{.75}{.5}\idsdown{0}{-.75}{.5}{1.5}{1}\ids{.5}{-.75}{.5}{.75}{1} \dcup{1}{-.5}{.5}{.5} \draw (-.3,-.8) node {$\mu$}; \draw (.7,-.8) node {$\lambda$}; }=\mathfrak r(\mu,\lambda)^{-1}\quad \hctikz{ \idsup{0}{0}{.5}{1.5}{1}\ids{.5}{.75}{.5}{.75}{1}\dcap{1}{.75}{.5}{.5} \rcrossdown{.5}{0}{.75}{.5}\idsup{1.5}{-.75}{.5}{1.5}{1}\ids{1}{-.75}{.5}{.75}{1} \dcup{0}{-.5}{.5}{.5} \draw (.7,-.8) node {$\mu$}; \draw (1.3,-.8) node {$\lambda$}; }\] for any $\lambda,\mu\in X^+$. \end{lem} \begin{proof} Let $B(\lambda)$ be a homogeneous basis for $V(\lambda)$. Let $v_0\in B(\lambda)_{\kappa}$ and $w_0\in V(\mu)_{\xi}$ for some $\kappa,\xi\in X$. Then \begin{align*} R_{-\mu,\lambda}^{-1}(w_0^*\otimes v_0)&=\sum_{\nu}\mathfrak f(-\nu-\xi,\nu+\kappa)^{-1}\pi^{p(w_0)p(v_0)+p(\nu)p(v_0)}q^{\frac{\nu\cdot\nu}{2}}\sum_{b\in {\mathbf{B}}_\nu} \sigma(b^*)^+v_0\otimes b^-w_0^*\\ &=\mathfrak f(-\xi,\kappa)^{-1}\sum_{\nu}(\pi q)^{-\ang{\tilde\nu,\xi}} q^{-\ang{\tilde\nu,\kappa}-\frac{\nu\cdot \nu}{2}}\pi^{p(w_0)p(v_0)+p(\nu)p(v_0)}\sum_{b\in {\mathbf{B}}_\nu} \sigma(b^*)^+v_0\otimes b^-w_0^* \end{align*} We wish to relate this morphism to the maps \[\phi=(\ev_\mu\otimes 1_{\lambda}\otimes 1_{-\mu})\circ (1_{-\mu}\otimes R_{\mu,\lambda}\otimes 1_{-\mu})\circ(1_{-\mu}\otimes 1_\lambda\otimes \coqtr_\mu).\] \[\psi=(1_{\lambda}\otimes 1_{-\mu}\otimes \ev_\lambda)\circ (1_{\lambda}\otimes R_{-\lambda,-\mu}\otimes 1_{\lambda})\circ( \coqtr_\lambda\otimes 1_{-\mu}\otimes 1_{\lambda}).\] Well, for $\phi$, we compute \[\coev_\lambda(1)=\sum_{v\in B(\lambda)} \pi^{p(v)}q^{\ang{\tilde\rho,|v|}} v^*\otimes v,\] \[R_{-\mu,-\lambda}^{-1}(w_0^*\otimes v^*)= \sum_\nu\mathfrak f(-|w_0|-\nu,-|v|+\nu)^{-1}\pi^{p(w_0)p(v)+p(\nu)p(v)}q^{\frac{\nu\cdot\nu}2}\sum_{b\in {\mathbf{B}}_\nu} \sigma(b^*)^+v^*\otimes b^-w_0^* \] \begin{align*}\qtr_\lambda&(v_0\otimes\sigma(b^*)^+v^*)= \pi^{p(v_0)}q^{-\ang{\tilde\rho,|v_0|}} (\sigma(b^*)^+v^*)(v_0)\\ &=\delta_{|v|,\kappa+\nu}(-1)^{{\mathrm{ht}} \nu} \pi^{{\mathbf p}(\nu)+p(v_0)+p(\nu)p(v)}q^{-\frac{\nu\cdot\nu}{2}-\ang{\tilde\rho,|v_0|}}(\pi q)^{-\ang{\tilde\nu,|v_0|}}q_{-\nu}v^*((b^*)^+v_0), \end{align*} and so \begin{align*} \phi(v_0\otimes w_0^*)&=\sum_{v\in B(\lambda)} \sum_\nu\sum_{b\in {\mathbf{B}}_\nu} \pi^{p(v)}q^{\ang{\tilde\rho,|v|}}\mathfrak f(-|w_0|-\nu,-|v|+\nu)^{-1}\pi^{p(w_0)p(v)+p(\nu)p(v)}q^{\frac{\nu\cdot\nu}2}\\ &\hspace{2em}\times\delta_{|v|,|v_0|+\nu} (-1)^{{\mathrm{ht}} \nu} \pi^{{\mathbf p}(\nu)+p(v_0)+p(\nu)p(v)}q^{-\frac{\nu\cdot\nu}{2}-\ang{\tilde\rho,|v_0|}}(\pi q)^{-\ang{\tilde\nu,|v_0|}}q_{-\nu}\\ &\hspace{2em}\times v^*((b^*)^+v_0) b^-w_0^*\otimes v\\ &=\sum_\nu(-1)^{{\mathrm{ht}} \nu}\mathfrak f(-\xi-\nu,-\kappa)^{-1} (\pi q)^{-\ang{\tilde\nu,\kappa}}\pi^{p(v_0)p(w_0)+{\mathbf p}(\nu)}\\ &\hspace{2em}\times \sum_{b\in {\mathbf{B}}_\nu}\pi_\nu q^{\ang{\tilde\rho,\nu}}q_{-\nu} \pi^{p(w_0)p(\nu)}b^-w_0^*\otimes\parens{\sum_{v\in B(\lambda)_{\kappa+\nu'}}v^*((b^*)^+v_0)v}\\ &=\mathfrak f(-\xi,-\kappa)^{-1}\pi^{p(v_0)p(w_0)}\sum_\nu(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)}\sum_{b\in {\mathbf{B}}_\nu}\pi_\nu q_{\nu} \pi^{p(w_0)p(\nu)}b^-w_0^*\otimes(b^*)^+v_0\\ &=\mathfrak r(\mu,\lambda)^{-1}R_{\lambda,-\mu}(v_0\otimes w_0^*). \end{align*} For $\psi$, we compute \[\coqtr_\lambda(1)=\sum_{v\in B(\lambda)} v\otimes v^*,\] \[R_{-\lambda,-\mu}(v^*\otimes w_0^*)=\pi^{p(w_0)p(v)}\mathfrak f(-\xi,-|v|)\sum_{\nu}(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu\sum_{b\in {\mathbf{B}}_\nu} \pi^{p(\nu)p(w_0)} b^-w_0^*\otimes (b^*)^+v^*.\] \begin{align*}\ev_\lambda&((b^*)^+v^*\otimes v_0)= ((b^*)^+v^*)(v_0)\\ &=\delta_{|v|,\kappa+\nu}(-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)+p(\nu)p(v_0)+p(\nu)}q^{-\frac{\nu\cdot\nu}2}(\pi q)^{-\ang{\tilde\nu,\kappa}}q_{-\nu} v^*(\sigma(b)^+v_0). \end{align*} Therefore, \begin{align*} \psi(w_0^*\otimes v_0)&=\sum_{v\in B(\lambda)} \pi^{p(w_0)p(v)}\mathfrak f(-\xi,-|v|)\sum_{\nu}(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu\sum_{b\in {\mathbf{B}}_\nu} \pi^{p(\nu)p(w_0)} ((b^*)^+v^*)(v_0)v\otimes b^-w_0^*.\\ &=\sum_{\nu} \sum_{b\in {\mathbf{B}}_\nu}\sum_{v\in B(\lambda)_{\kappa+\nu}} \pi^{p(w_0)p(v_0)+p(w_0)p(\nu)}\mathfrak f(-\xi,-\kappa-\nu)(-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu \pi^{p(\nu)p(w_0)} \\ &\hspace{2em}\times (-1)^{{\mathrm{ht}} \nu}\pi^{{\mathbf p}(\nu)+p(\nu)p(v_0)+p(\nu)}q^{-\frac{\nu\cdot\nu}2}(\pi q)^{-\ang{\tilde\nu,\kappa}}q_{-\nu}(v^*)(\sigma(b^*)^+v_0)v\otimes b^-w_0^*\\ &=\sum_{\nu} \sum_{b\in {\mathbf{B}}_\nu} \pi^{p(w_0)p(v_0)+p(\nu)p(v_0)}\mathfrak f(-\xi,-\kappa-\nu)q^{-\frac{\nu\cdot\nu}2}(\pi q)^{-\ang{\tilde\nu,\kappa}}\\ &\hspace{2em}\times \parens{\sum_{v\in B(\lambda)_{\kappa+\nu}}v^*(\sigma(b^*)^+v_0)v}\otimes b^-w_0^*\\ &=\mathfrak f(-\xi,-\kappa)\sum_{\nu} \sum_{b\in {\mathbf{B}}_\nu} \pi^{p(w_0)p(v_0)+p(\nu)p(v_0)} q^{\frac{\nu\cdot\nu}2}(\pi q)^{-\ang{\tilde\nu,\kappa}}q^{-\ang{\tilde\nu,\xi}} \sigma(b^*)^+v_0\otimes b^-w_0^*\\ &=\mathfrak r(\mu,\lambda)R_{\lambda,-\mu}(\sigma(b^*)^+v_0\otimes w_0^*). \end{align*} \end{proof} \fi Note that by identifying inverse maps in Lemma \ref{lem:rotating crossings} (a) and (b), we obtain the following corollary. \begin{cor}\label{cor:rotating crossings inverse} We have an equality of diagrams \[\hctikz{ \idsdown{1}{2.25}{.5}{.75}{1}\idsup{1.5}{1.5}{.5}{1.5}{1}\dcap{0}{2.25}{.5}{.5} \lcrossdown{.5}{1.5}{.75}{.5} \dcup{1}{1}{.5}{.5} \idsup{0}{0}{.5}{2.25}{1}\ids{.5}{.75}{.5}{.75}{1}\dcap{1}{.75}{.5}{.25} \rcrossdown{.5}{0}{.75}{.5}\idsup{1.5}{-.75}{.5}{1.5}{1}\ids{1}{-.75}{.5}{.75}{1} \dcup{0}{-.5}{.5}{.5} \draw (.7,-.8) node {$\mu$}; \draw (1.3,-.8) node {$\lambda$}; }=\hctikz{ \idsdown{0}{0}{.5}{2}{1} \idsup{.5}{0}{.5}{2}{1} \draw (-.1,-.3) node {$\mu$}; \draw (.6,-.3) node {$\lambda$}; }=\hctikz{ \idsdown{.5}{2.25}{.5}{.75}{1}\idsup{0}{1.5}{.5}{1.5}{1}\dcap{1}{2.25}{.5}{.5} \lcrossdown{.5}{1.5}{.75}{.5} \dcup{0}{1}{.5}{.5} \idsup{1.5}{0}{.5}{2.25}{1}\ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.25} \rcrossdown{.5}{0}{.75}{.5}\idsup{0}{-.75}{.5}{1.5}{1}\ids{.5}{-.75}{.5}{.75}{1} \dcup{1}{-.5}{.5}{.5} \draw (.2,-.8) node {$\mu$}; \draw (.8,-.8) node {$\lambda$}; } \] \end{cor} Finally, we show a somewhat more involved identity, which will lead us to our the final result. \begin{lem}\label{lem:crossthruturn} We have an equality of diagrams \[ \hctikz{ \dcap{0}{.75}{1.5}{1}\dcap{.5}{.75}{.5}{.5} \rcross{1}{0}{.75}{.5} \ids{0}{0}{.5}{.75}{2} \draw (1,-.2) node {$\lambda$}; \draw (1.6,-.2) node {$\mu$}; }\quad = \pi^{P(\mu)P(\lambda)} \hctikz{ \dcap{0}{.75}{1.5}{1}\dcap{.5}{.75}{.5}{.5} \rcross{0}{0}{.75}{.5} \ids{1}{0}{.5}{.75}{2} \draw (1,-.2) node {$\lambda$}; \draw (1.6,-.2) node {$\mu$}; } \] for any choice of orientation. \end{lem} \begin{proof} In order to prove the identity without referring to a particular orientation, it will be convenient to introduce the following notation. Suppose $m\in V(\zeta)$ and $n\in V(-\zeta)$ for some $\zeta\in X^+$. Let us denote by $(n,m)$ (respectively $(m,n)$) the evaluation $\ev_\zeta(n\otimes m)$ (respectively, the quantum trace $\qtr_\zeta(m\otimes n)$). In particular, one may think of $(-,-)$ as a pairing on $V(\zeta)\oplus V(-\zeta)$ satisfying, for $v,w\in V(\zeta)$, \begin{equation}\label{eq:pairing v+dual} \begin{array}{c} (v,w)=(v^*,w^*)=0,\quad (v,w^*)=\pi^{p(v)p(w)}q^{-\ang{\tilde\rho,|v|}} (w^*,v),\\ (uv,w^*)=\pi^{p(u)p(v)}(v,S(u)w^*),\quad (uw^*,v)=\pi^{p(u)p(w)}(w^*,S(u)v). \end{array} \end{equation} Indeed, all the statements of \eqref{eq:pairing v+dual} are obvious except $(uv,w^*)=\pi^{p(u)p(v)}(v,S(u)w^*)$, which follows from a simple calculation on the generators: for example, \[(E_iv,w^*)= \pi^{p(v)p(w)}q^{-\ang{\tilde\rho,|v|}}q_i^{-2}(-E_i{\tilde{J}}_i^{-1}{\tilde{K}}_i^{-1}w^*)(v) =\pi^{p(v)p(i)}(v,S(E_i)w^*)\] In this proof we will use the notation $(-,-)$ as shorthand for $\ev_\zeta$ and $\qtr_\zeta$ for both $\zeta=\lambda,\mu$ with the intended map (and highest weight) being clear from context. Using this notation, the diagram equality is equivalent to showing that the maps \[\psi=(-,-)\circ\parens{1_{s\mu}\otimes (-,-)\otimes 1_{-s\mu}}\circ \parens{{\mathcal{R}}_{s \lambda,t \mu}\otimes 1_{-s\lambda}\otimes 1_{-t\mu}}\] \[\phi=(-,-)\circ\parens{1_{t\lambda}\otimes (-,-)\otimes 1_{-t\lambda}}\circ \parens{1_{s\lambda}\otimes 1_{t\mu}\otimes {\mathcal{R}}_{-s \lambda,-t \mu}}\] are $\pi^{P(\mu)P(\lambda)}$ multiples of each other for any choice of $s,t\in\set{1,-1}$. Let $w\in V(s\lambda)$, $x\in V(t\mu)$, $y\in V(-s\lambda)$, and $z\in V(-t\mu)$, where $V(-\xi)=V(\xi)^*$ for $\xi\in X^+$. Then on one hand, \[\psi(w\otimes x\otimes y \otimes z)=\sum_{\nu}\sum_{b\in \mathbf B_\nu}\pi^{p(x)p(w)} \mathfrak f(|x|,|w|) (-1)^{{\mathrm{ht}} \nu} \pi^{\mathbf p(\nu)} \pi_\nu q_\nu \pi^{p(\nu)p(x)} (b^-x,z)((b^*)^+w,y).\] On the other hand, using the representation of $\Theta$ in the basis $\sigma(\mathbf B)$, \[\phi(w\otimes x\otimes y \otimes z)=\sum_{\nu}\sum_{b\in \mathbf B_\nu}\pi^{p(y)p(z)} \mathfrak f(|z|,|y|) (-1)^{{\mathrm{ht}} \nu} \pi^{\mathbf p(\nu)} \pi_\nu q_\nu \pi^{p(\nu)p(z)} (x,\sigma(b)^-z)(w,\sigma(b^*)^+y).\] Thus to see that $\psi(w\otimes x\otimes y\otimes z)=\pi^{P(\mu)P(\lambda)}\phi(w\otimes x\otimes y\otimes z)$, and hence that $\psi=\pi^{p(\mu)p(\lambda)}\phi$ since $w,x,y,z$ are arbitrary, it is enough to show that $l=\pi^{P(\mu)P(\lambda)}r$, where \[l=\pi^{p(y)p(z)+p(\nu)p(z)} \mathfrak f(|z|,|y|)(x,\sigma(b)^-z)(w,\sigma(b^*)^+y)\] \[r=\pi^{p(w)p(x)+p(\nu)p(x)}\mathfrak f(|x|,|w|) (b^-x,z)((b^*)^+w,y)\] Using the properties of $(-,-)$ (see \eqref{eq:pairing v+dual}) and $S$ (see \eqref{eq:antipode formula}), we see that \[(x,\sigma(b)^-z)(w,\sigma(b^*)^+y)= \pi^{p(x)p(\nu)+p(w)p(\nu)}q^{-\nu\cdot\nu+\ang{\tilde\nu,|x|}}(\pi q)^{-\ang{\tilde\nu,|w|}}(b^-x,z)((b^*)^+w,y)\] Note that $l$,$r$ are both zero unless $-\bigrdeg{x}=\bigrdeg{z}-\nu$ and $-\bigrdeg{w}=\bigrdeg{y}+\nu$. In particular, $l$ and $r$ are both zero unless $p(y)=p(w)+p(\nu)$, $p(z)=p(x)+p(\nu)$, in which case \[p(y)p(z)+p(\nu)p(z)+p(x)p(\nu)+p(w)p(\nu)\equiv p(w)p(x)+p(w)p(\nu)\text{ (mod 2)}.\] Likewise, $l$,$r$ are both zero unless $-|y|=|w|+\nu$, $-|z|=|x|-\nu$, in which case \[\mathfrak f(|z|,|y|)q^{-\nu\cdot\nu+\ang{\tilde\nu,|x|-|w|}}=\mathfrak f(-|x|,-|w|).\] Finally, note that $\mathfrak f(-|x|,-|w|)=\pi^{P(-|x|)P(-|w|)}\mathfrak f(|x|,|w|)$. Putting these observations together, \[l=\pi^{p(w)p(x)+p(\nu)p(x)+P(-|x|)P(-|w|)}\mathfrak f(|x|,|w|) (b^-x,z)((b^*)^+w,y)=\pi^{P(-|x|)P(-|w|)}r\] Since parity in $X$ only depends on the $X/{\mathbb{Z}}[I]$ cosets and we have $-|x|\in \mu+{\mathbb{Z}}[I]$ and $-|w|\in\lambda+{\mathbb{Z}}[I]$, the result follows. \end{proof} Lastly, note that Lemmas \ref{lem:crossthruturn} and \ref{lem:straightening} immediately imply the following corollary. \begin{cor}\label{cor:180rotcross}We have an equality of diagrams \[\pi^{P(\mu)P(\lambda)} \hctikz{ \rcrossup{0}{0}{1}{1} \draw (-.3,0) node {$\lambda$}; \draw (-.3,1) node {$\mu$}; }\quad = \hctikz{ \dcap{0}{.75}{1.5}{1}\dcap{.5}{.75}{.5}{.5}\idsup{2}{0}{.5}{1.5}{2} \rcrossdown{1}{0}{.75}{.5} \idsup{0}{-.75}{.5}{1.5}{2}\dcup{1.5}{-.5}{.5}{.5}\dcup{1}{-1}{1.5}{1} \draw (-.3,-.8) node {$\lambda$}; \draw (.7,-.8) node {$\mu$}; }\quad = \hctikz{ \dcap{1}{.75}{1.5}{1}\dcap{1.5}{.75}{.5}{.5}\idsup{0}{0}{.5}{1.5}{2} \rcrossdown{1}{0}{.75}{.5} \idsup{2}{-.75}{.5}{1.5}{2}\dcup{.5}{-.5}{.5}{.5}\dcup{0}{-1}{1.5}{1} \draw (1.7,-.8) node {$\lambda$}; \draw (2.7,-.8) node {$\mu$}; }\] for any $\lambda,\mu\in X^+$. \end{cor} \subsection{Renormalization} \label{subsec:renorm} In the previous section, we deduced a number of identities between various slice diagrams. These identities are almost the Turaev moves for (framed) oriented tangles, except for factors of $\pi$. Now we shall correct these factors. As noted in Remark \ref{rem: coefficients}, all of the previous statements about ${\mathbf U}$ and it's modules hold verbatim over the subring ${\Qq^{\pi}}$ of ${\Qqt^\tau}$. Now we will use the fact that $\pi=\tau^2$ to renormalize our maps. These renormalized ${\mathbf U}$-module homomorphisms will always be represented by a diagrammatic calculus with red strands and labels to differentiate them. \begin{center} \begin{tabular}{ccccccc} \red{\hctikz{ \idsup{-1}{1.5}{1}{1}{1} \draw (-.8,1.5) node {$\lambda$};}} $=$\quad \hctikz{ \idsup{-1}{1.5}{1}{1}{1} \draw (-.8,1.5) node {$\lambda$};} && \red{\hctikz{ \idsdown{-1}{1.5}{1}{1}{1} \draw (-.8,1.5) node {$\lambda$};}} $=$\quad \hctikz{ \idsdown{-1}{1.5}{1}{1}{1} \draw (-.8,1.5) node {$\lambda$};} \\\\ \red{\hctikz{ \cwcup{1}{-.5}{1}{.75} \draw (1.8,-.5) node {$\lambda$}; }}\quad $=$\quad \hctikz{ \cwcup{1}{-.5}{1}{.75} \draw (1.8,-.5) node {$\lambda$}; } && \red{\hctikz{ \ccwcup{1}{-.5}{1}{.75} \draw (1.7,-.5) node {$\lambda$};}} \quad$=\tau^{3P(\lambda)}$\quad \hctikz{ \ccwcup{1}{-.5}{1}{.75} \draw (1.7,-.5) node {$\lambda$};} \\\\ \red{\hctikz{ \cwcap{1}{-.5}{1}{.75} \draw (1.8,.3) node {$\lambda$}; }} \quad$=\tau^{P(\lambda)}$ \hctikz{ \cwcap{1}{-.5}{1}{.75} \draw (1.8,.3) node {$\lambda$}; } && \red{\hctikz{ \ccwcap{1}{-.5}{1}{.75} \draw (1.8,.3) node {$\lambda$}; }} \quad $=$\quad \hctikz{ \ccwcap{1}{-.5}{1}{.75} \draw (1.8,.3) node {$\lambda$}; } \\\\ \red{\hctikz{ \rcrossup{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; }} \quad$=\tau^{P(\lambda)P(\mu)}$ \hctikz{ \rcrossup{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && \red{\hctikz{ \rcrossdown{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; }} $=\tau^{3P(\lambda)P(\mu)}$ \hctikz{ \rcrossdown{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } \\\\ \red{\hctikz{ \lcrossup{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; }} \quad$=\tau^{3P(\lambda)P(\mu)}$ \hctikz{ \lcrossup{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } && \red{\hctikz{ \lcrossdown{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; }} $=\tau^{P(\lambda)P(\mu)}$ \hctikz{ \lcrossdown{0}{0}{1}{1}{1} \draw (-.2,0) node {$\lambda$}; \draw (1.2,0) node {$\mu$}; } \end{tabular} \end{center} \begin{rmk} \label{rmk:red diagrams} We make two remarks about the red diagrammatic calculus. \begin{enumerate} \item We observe that whenever $P(\lambda)=0$, the maps represented by the red and black diagrams are the same. By Lemma \ref{lem:weight parity equiv}, this holds whenever $\lambda$ is an even weight ($\ang{{\rf n}, \lambda}\in 2{\mathbb{N}}$) or $n$ is even, thus in these cases we can work over ${\Qq^{\pi}}$. \item Note that we don't define sideways-oriented crossings in the red strands. This can be done using these renormalizations and Lemma \ref{lem:rotating crossings}, but we shall not need these diagrams here. \end{enumerate} \end{rmk} Recall that the {\em writhe} $\wr(T)$ of an oriented tangle $T$ is defined by forgetting the orientation and setting \[\wr\parens{\hctikz{\rcross{0}{0}{.5}{.35}}}=1,\qquad\wr\parens{\hctikz{\lcross{0}{0}{.5}{.35}}}=-1, \qquad \wr(T)=\sum \wr(X),\] where the sum is over all crossings $X$ in $T$. \begin{thm}\label{thm:knot invariant} Let $T$ be an oriented tangle, and $\lambda\in X^+$ be a dominant weight. For any slice diagram $S(T)$ of $T$, let $S(T)_\lambda$ be the associated map defined by the red diagrammatic calculus with strands colored by $\lambda$. Then $S(T)_\lambda$ is is independent of the choice of slice diagram, and $T_\lambda=S(T)_\lambda$ is an isotopy invariant of oriented framed tangles. Moreover, if $J_T^\lambda=(\pi^{p(\lambda)}\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde \rho,\lambda}})^{{\rm wr}(T)}T_\lambda$, then $J_T^\lambda$ is independent of the framing, hence is an invariant of $T$. \end{thm} \begin{proof} To prove the theorem, it suffices to show that the maps $S(T)_\lambda$ (resp. $J_T^\lambda$) are invariant under the Turaev moves (cf. \cite[Theorem 3.2]{T}, \cite[Theorem 3.3, Equations (3.9)-(3.16)]{Oht}) for framed (resp. unframed) oriented tangles. First, observe that the identities \eqref{eq:comm and ids} and Lemma \ref{lem:straightening} hold for red strands as well. We also see that \eqref{eq:diagbraidrels} holds for red strands which all have the same orientation. (In fact, if we define sideways-oriented crossings of red strands as described in Remark \ref{rmk:red diagrams} (2), then \eqref{eq:diagbraidrels} would hold for red strands with any orientation.) Furthermore, applying the normalizations and rearranging the $\tau$ factors in Lemma \ref{lem:reide2} shows that, for either orientation, we have \[\red{\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \rcross{0}{0}{.75}{.5}\ids{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }}\quad =\pi^{P(\lambda)}\mathfrak f(\lambda,\lambda)q^{-\ang{\tilde\rho,\lambda}} \red{\hctikz{\ids{0}{0}{1}{1.5}{1}; \draw (-.3,0) node {$\lambda$};}} \quad=\quad\red{\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \rcross{.5}{0}{.75}{.5}\ids{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }}. \] \[\red{\hctikz{ \ids{0}{.75}{.5}{.75}{1}\dcap{.5}{.75}{.5}{.5} \lcross{0}{0}{.75}{.5}\ids{1}{0}{.5}{.75}{1} \ids{0}{-.75}{.5}{.75}{1}\dcup{.5}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }}\quad =\pi^{P(\lambda)}\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde\rho,\lambda}} \red{\hctikz{\ids{0}{0}{1}{1.5}{1}; \draw (-.3,0) node {$\lambda$};}} \quad=\quad\red{\hctikz{ \ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.5} \lcross{.5}{0}{.75}{.5}\ids{0}{0}{.5}{.75}{1} \ids{1}{-.75}{.5}{.75}{1}\dcup{0}{-.5}{.5}{.5} \draw (-.3,0) node {$\lambda$}; }}. \] Similarly, we see that Corollaries \ref{cor:rotating crossings inverse} and \ref{cor:180rotcross} gives us the identities \[ \red{\hctikz{ \dcap{0}{.75}{1.5}{1}\dcap{.5}{.75}{.5}{.5}\idsup{2}{0}{.5}{1.5}{2} \rcrossdown{1}{0}{.75}{.5} \idsup{0}{-.75}{.5}{1.5}{2}\dcup{1.5}{-.5}{.5}{.5}\dcup{1}{-1}{1.5}{1} }}\quad =\ \red{\hctikz{ \rcrossup{0}{0}{1}{1} }}\quad = \ \red{\hctikz{ \dcap{1}{.75}{1.5}{1}\dcap{1.5}{.75}{.5}{.5}\idsup{0}{0}{.5}{1.5}{2} \rcrossdown{1}{0}{.75}{.5} \idsup{2}{-.75}{.5}{1.5}{2}\dcup{.5}{-.5}{.5}{.5}\dcup{0}{-1}{1.5}{1} }}\] \[\red{\hctikz{ \idsdown{1}{2.25}{.5}{.75}{1}\idsup{1.5}{1.5}{.5}{1.5}{1}\dcap{0}{2.25}{.5}{.5} \lcrossdown{.5}{1.5}{.75}{.5} \dcup{1}{1}{.5}{.5} \idsup{0}{0}{.5}{2.25}{1}\ids{.5}{.75}{.5}{.75}{1}\dcap{1}{.75}{.5}{.25} \rcrossdown{.5}{0}{.75}{.5}\idsup{1.5}{-.75}{.5}{1.5}{1}\ids{1}{-.75}{.5}{.75}{1} \dcup{0}{-.5}{.5}{.5} }}\quad =\quad\red{\hctikz{ \idsdown{0}{0}{.5}{2}{1} \idsup{.5}{0}{.5}{2}{1} }}\quad=\quad\red{\hctikz{ \idsdown{.5}{2.25}{.5}{.75}{1}\idsup{0}{1.5}{.5}{1.5}{1}\dcap{1}{2.25}{.5}{.5} \lcrossdown{.5}{1.5}{.75}{.5} \dcup{0}{1}{.5}{.5} \idsup{1.5}{0}{.5}{2.25}{1}\ids{1}{.75}{.5}{.75}{1}\dcap{0}{.75}{.5}{.25} \rcrossdown{.5}{0}{.75}{.5}\idsup{0}{-.75}{.5}{1.5}{1}\ids{.5}{-.75}{.5}{.75}{1} \dcup{1}{-.5}{.5}{.5} }} \] for any choice of labeling of the strands. In particular, we see that the Turaev moves for oriented framed tangles are satisfied, which proves that $T_\lambda$ is indeed an isotopy invariant of oriented framed tangles. Moreover, note that $J_T^\lambda$ then satisfies the Turaev moves for oriented {\em unframed} tangles, since the only Turaev move that changes the writhe is Reidemeister 2 (which is to say the move straightening the crossings in Lemma \ref{lem:reide2}). \end{proof} We note that the proof of Theorem \ref{thm:knot invariant} actually implies a more general result, though we first need to recall some notions. The category of $X^+$-colored oriented tangles is the strict monoidal category whose objects are finite sequences of pairs $(\lambda,s)$ where $\lambda\in X^+$ and $s\in \set{\pm 1}$, and whose morphisms from $(\lambda_a,s_a)_{1\leq a\leq b}$ to $(\mu_c,s_c)_{1\leq c\leq d}$ are tangle diagrams where the labeling and orientation of the $r^{\rm th} $ strand from the left at the lower (respectively, upper) boundary corresponds to $(\lambda_r, s_r)$ (respectively, $(\mu_c, s_c)$); c.f. \cite{T,ADO} for more details. In particular, morphisms in this category (and thus colored tangles) are generated from the elementary morphisms \newcommand{\raisebox{.5em}{\rotatebox{180}{$\curvearrowright$}}}{\raisebox{.5em}{\rotatebox{180}{$\curvearrowright$}}} \newcommand{\raisebox{.5em}{\rotatebox{180}{$\curvearrowleft$}}}{\raisebox{.5em}{\rotatebox{180}{$\curvearrowleft$}}} \[\curvearrowright_\lambda,\quad \raisebox{.5em}{\rotatebox{180}{$\curvearrowright$}}_{\lambda},\quad \raisebox{.5em}{\rotatebox{180}{$\curvearrowleft$}}_{\lambda},\quad \raisebox{.5em}{\rotatebox{180}{$\curvearrowright$}}_{\lambda},\quad (\searrow\hspace{-1em}\swarrow)^{\pm}_{\lambda,\mu},\quad (\nearrow\hspace{-1em}\nwarrow)^{\pm}_{\lambda,\mu}.\] subject to relations which are simply colored versions of the Turaev moves. We can extend Theorem \ref{thm:knot invariant} to framed multicolored tangles with the same proof. To obtain the unframed invariant, the normalization constant is replaced by $\prod_{\lambda\in X^+}(\pi^{p(\lambda)}\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde \rho,\lambda}})^{\wr_\lambda(T)}$, where $\wr_\lambda$ is defined to be the writhe where we exclude from the sum any crossings where there is a strand not labeled by $\lambda$. Therefore, we obtain the following corollary. \begin{cor} There exists a covariant functor $J$ from the category of $X^+$-colored oriented tangles modulo isotopy to ${\mathcal O}_{\rm fin}$ which sends the object $((\lambda_1,s_1),\ldots, (\lambda_r,s_r))$ to the module $V(s_1\lambda_1)\otimes\ldots\otimes V(s_r\lambda_r)$ and is given on morphisms by \[\curvearrowright_\lambda\mapsto\tau^{P(\lambda)} \ev_\lambda,\qquad \curvearrowleft_\lambda\mapsto\qtr_\lambda,\qquad \raisebox{.5em}{\rotatebox{180}{$\curvearrowleft$}}_{\lambda}\mapsto\coqtr_\lambda,\qquad \raisebox{.5em}{\rotatebox{180}{$\curvearrowright$}}_{\lambda}\mapsto\tau^{-P(\lambda)}\coev_\lambda, \] \[(\nearrow\hspace{-1em}\nwarrow)^{\pm}_{\lambda,\mu}\mapsto (\pi^{p(\lambda)}\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde \rho,\lambda}})^{\pm \delta_{\lambda,\mu}} \tau^{\pm P(\lambda)P(\mu)}{\mathcal{R}}_{\lambda,\mu}^{\pm1},\] \[(\searrow\hspace{-1em}\swarrow)^{\pm}_{\lambda,\mu}\mapsto (\pi^{p(\lambda)}\mathfrak f(\lambda,\lambda)^{-1}q^{\ang{\tilde \rho,\lambda}})^{\pm \delta_{\lambda,\mu}} \tau^{\mp P(\lambda)P(\mu)}{\mathcal{R}}_{-\lambda,-\mu}^{\pm1}.\] In particular, if $L$ is an oriented colored link, then $J(L)\in{\Qqt^\tau}$ is the associated quantum covering ${\mathfrak{osp}}(1|2n)$ colored link invariant. \end{cor} \begin{example}\label{ex:rk1knot} Let's take $n=1$ and $\lambda=1$. Fix $\mathfrak f(1,1)=1$, and note that $\ang{\tilde \rho,\lambda}=1$ and $p(\lambda)=1$. We can explicitly compute the maps represented by our diagrams on $V(1)\otimes V(1)$. Let $v_1,v_{-1}$ be the basis of $V(1)$ from Example \ref{ex:rk1mods}. Then with respect to the ordered basis $\set{v_1\otimes v_1, v_1\otimes v_{-1}, v_{-1}\otimes v_1, v_{-1}\otimes v_{-1}}$ of $V(1)\otimes V(1)$, we have \[\Theta= \begin{bmatrix}1 & 0& 0& 0\\ 0 & 1& 0& 0\\ 0 & q^{-1}-\pi q & 1& 0\\ 0 & 0& 0& 1\end{bmatrix},\qquad \mathfrak{F}=\begin{bmatrix} 1 & 0& 0& 0\\ 0 & q& 0& 0\\ 0 & 0 & \pi q & 0\\ 0 & 0& 0& \pi \end{bmatrix}, \qquad \mathbf s=\begin{bmatrix} 1 & 0& 0& 0\\ 0 & 0& 1& 0\\ 0 & 1 & 0 & 0\\ 0 & 0& 0& \pi \end{bmatrix}\] and thus \[ \red{\hctikz{\rcrossup{0}{0}{1}{1}}}= \begin{bmatrix} \tau & 0& 0& 0\\ 0 & 0& \tau q& 0\\ 0 & \tau^3 q & \tau - \tau^3 q^2 & 0\\ 0 & 0& 0& \tau \end{bmatrix}, \quad \red{\hctikz{\lcrossup{0}{0}{1}{1}}}= \begin{bmatrix} \tau^3& 0& 0& 0\\ 0 & \tau^3- \tau q^{-2}& \tau q^{-1}& 0\\ 0 & \tau^3 q^{-1} & 0 & 0\\ 0 & 0& 0& \tau^3 \end{bmatrix} \] Note that $\pi^{p(\lambda)}q^{\ang{\tilde \rho,\lambda}}=\pi q$. Then it is easy to verify directly that \[\red{\hctikz{\rcrossup{0}{0}{1}{1}}}-q^2 \ \red{\hctikz{\lcrossup{0}{0}{1}{1}}}\ \ =(\tau-\tau^3 q^2)\ \ \red{\hctikz{\idsup{0}{0}{1}{1}{1}\idsup{.5}{0}{1}{1}{1}}}\] Now let $T$ be an unframed oriented link with all strands colored by $\lambda$, and fix a subdiagram which consists of two strands with either no crossing or a single crossing. Since $T^\sharp$ is isotopy invariant and independent of framing, we may assume that the strands are directed upward. Let $T_+$ (resp. $T_0$, $T_-$) be $T$ with the subdiagram replaced by $\red{\hctikz{\rcrossup{0}{0}{1}{1}}}$ (resp. \red{\hctikz{\idsup{0}{0}{1}{1}{1}\idsup{.5}{0}{1}{1}{1}}}\ ,\ \red{\hctikz{\lcrossup{0}{0}{1}{1}}}). Then using the above relation and the definition in Theorem \ref{thm:knot invariant}, \[(\pi q^{-1}) J_{T_+}^1-(\pi q^3)J_{T_-}^1=(\tau-\tau^3 q^2) J_{T_0}^1\] hence \[(\pi q^2)^{-1}J_{T_+}^1-\pi q^2J_{T_-}^1=(\tau q^{-1}-\tau^3 q)J_{T_0}^1.\] Moreover, if $T$ is the unknot, then for either orientation we have $J_{T}^1=\tau^3 q+\tau q^{-1}$. In particular, we see that for any link $K$, $J_{K}^1$ is simply a multiple of the Jones polynomial of $T$ in the variable $\tau^3 q=\tau^{-1}q$. In particular, note that using the specialization $\tau={\mathbf t}$, which corresponds to $\pi=-1$, this shows the uncolored $U_q({\mathfrak{osp}}(1|2))$ link invariant is equal to the $U_{{\mathbf t}^{-1}q}(\mathfrak{sl}_2)$ link invariant. \end{example} \section{Relating $\mathfrak{so}(2n+1)$ and ${\mathfrak{osp}}(1|2n)$ invariants} The results of Example \ref{ex:rk1knot} suggest a connection between the specializations of the tangle invariants in Theorem \ref{thm:knot invariant}. We now make this precise by extending the constructions in \cite{CFLW,C}. We begin by recalling the definition of the twistor maps. \subsection{Definition of Twistors} An {\em enhancer} $\phi$ is an function $\phi:{\mathbb{Z}}[I]\times X\rightarrow {\mathbb{Z}}$ satisfying \begin{equation} \begin{array}{c} \phi(\nu,\lambda+\mu)\equiv \phi(\nu,\mu)+\phi(\nu,\lambda)\mod 4 \text{ for }\nu,\mu\in {\mathbb{Z}}[I]\\ \phi(\nu+\mu,\lambda)\equiv \phi(\nu,\lambda)+\phi(\mu,\lambda) \mod 4 \text{ for }\nu,\mu\in {\mathbb{Z}}[I]\\ \phi(i,i)=d_i\text{ and }\phi(i,j)\in 2{\mathbb{Z}}\text{ for }i\neq j\in I.\\ \phi(i,j)-\phi(j,i)\equiv i\cdot j+2p(i)p(j)\mod 4 \text{ for }i,j\in I \end{array} \end{equation} Note that $\phi(i,i)-\phi(i,i)=0\equiv i\cdot i+2p(i)p(i)$ modulo 4 since $i\cdot i=2d_i$ and $2p(i)p(i)=2p(i)=2d_i$. In particular, note that these congruences imply that \begin{equation}\label{eq:phi on ZI} \begin{array}{c} \phi_4:{\mathbb{Z}}[I]\times{\mathbb{Z}}[I]\rightarrow {\mathbb{Z}}/4{\mathbb{Z}} \text{ defined by }\phi_4(\mu,\nu)=\phi(\mu,\nu)\!\!\mod 4\text{ is a }{\mathbb{Z}}\text{-bilinear map}\\ \text{and }\phi(\mu,\nu)\equiv\phi(\nu,\mu)+\mu\cdot \nu+2p(\mu)p(\nu)\mod 4\text{ for }\mu,\nu\in{\mathbb{Z}}[I]. \end{array} \end{equation} Note that an enhancer can always be defined on ${\mathbb{Z}}[I]\times{\mathbb{Z}}[I]$ by defining it for $I$ and extending in ${\mathbb{Z}}$-bilinearly, and then it can be extended to ${\mathbb{Z}}[I]\times X$ by translation along a transversal of $X/{\mathbb{Z}}[I]$. When $I$ has a unique odd element, as in the present case, the enhancer is closely related to the usual pairing. \begin{lem}\label{lem:enhancer single odd} Let $\phi$ be an enhancer. Then $\phi(\mu,\nu)+\phi(\nu,\mu)\equiv \mu\cdot \nu$ modulo 4. \end{lem} \begin{proof} First set $(,)_\phi, (,)_\bullet:{\mathbb{Z}}[I]\times{\mathbb{Z}}[I]\rightarrow {\mathbb{Z}}/4{\mathbb{Z}}$ by $(\mu,\nu)_\phi=\phi_4(\mu,\nu)+\phi_4(\nu,\mu)$ and $(\mu,\nu)_\bullet=\mu\cdot\nu\mod 4$. Both maps are ${\mathbb{Z}}$-bilinear, so it suffices to show they take the same values on $I\times I$. Well, if $i\neq j$, then at least one of $i$ or $j$ is even and thus $2p(i)p(j)=0$ since $|I_1|=1$. On the other hand, $\phi(i,j)\in 2{\mathbb{Z}}$ so $\phi(i,j)+\phi(j,i)\equiv_4 \phi(i,j)-\phi(j,i) \equiv_4 i\cdot j +2p(i)p(j)=i\cdot j$. Finally, note that $\phi(i,i)+\phi(i,i)=2d_i=i\cdot i$ for any $i\in I$. \end{proof} The {\em $\phi$-enhanced quantum covering group} ${\widehat \UU}$ associated to ${\mathbf U}$ and the enhancer $\phi$ is the semidirect product of ${\mathbf U}$ with the algebra ${\Qqt^\tau}[T_\mu, \Upsilon_\mu\mid \mu\in {\mathbb{Z}}[I]]$ subject to the relations \begin{equation}\label{eq:TUpsrels} T_\mu T_\nu=T_{\mu+\nu},\quad \Upsilon_\mu \Upsilon_\nu=\Upsilon_{\mu+\nu},\quad T_0=\Upsilon_0=T_\nu^4=\Upsilon_\nu^4=1,\quad T_\mu \Upsilon_\nu=\Upsilon_\nu T_\mu, \end{equation} \begin{equation}\label{eq:Tweightrels} T_\mu u={\mathbf t}^{\ang{\mu,|u|}} u T_\mu,\quad u\in{\mathbf U},\ \mu\in{\mathbb{Z}}[I] \end{equation} \begin{equation}\label{eq:Upsweightrels} \Upsilon_\mu u={\mathbf t}^{\phi(\mu,|u|)} u \Upsilon_\mu,\quad u\in{\mathbf U},\ \mu\in{\mathbb{Z}}[I] \end{equation} See \cite{CFLW,C} for a more formal definition. The enhanced quantum covering group has a useful ${\mathbb{Q}}({\mathbf t})$-linear automorphism called a {\em twistor}. There are several ways to define such a twistor; we will need the following. \begin{prop}{\bf \cite[Theorems 4.3, 4.12]{CFLW}} \label{prop:twistordef} Define a product $*$ on $\ensuremath{\mathbf{f}}$ by the following rule: if $x$ and $y$ are homogeneous elements of $\ensuremath{\mathbf{f}}$, let $x*y={\mathbf t}^{\phi(|x|,|y|)}xy$. Let $(\ensuremath{\mathbf{f}},*)$ denote $\ensuremath{\mathbf{f}}$ with this multiplication. \begin{enumerate} \item Then there is a ${\mathbb{Q}}({\mathbf t})$-linear algebra isomorphism ${\mathfrak X}:\ensuremath{\mathbf{f}}\rightarrow (\ensuremath{\mathbf{f}},*)$ defined by \[{\mathfrak X}(\theta_i)=\theta_i,\quad {\mathfrak X}(q)={\mathbf t}^{-1}q,\quad {\mathfrak X}(\tau)={\mathbf t}^{-1}\tau.\] \item Let ${\mathcal{B}}$ be the canonical basis of $\ensuremath{\mathbf{f}}$ (cf. \cite{CHW2}). Then ${\mathfrak X}$ on $\ensuremath{\mathbf{f}}$ satisfies ${\mathfrak X}(b)={\mathbf t}^{\ell(b)} b$ for all $b\in {\mathcal{B}}$, where $\ell(b)$ is some integer depending on $b$. \item There is a ${\mathbb{Q}}({\mathbf t})$-algebra automorphism ${\mathfrak X}:{\widehat \UU}\rightarrow {\widehat \UU}$ defined by \[{\mathfrak X}(E_i)={\mathbf t}_i^{-1} {\tilde{T}}_i \Upsilon_iE_i,\quad {\mathfrak X}(F_i)=F_i\Upsilon_{-i},\quad {\mathfrak X}(K_\mu)=T_{-\mu}K_\mu,\quad {\mathfrak X}(J_\mu)=T_{2\mu} J_\mu,\] \[{\mathfrak X}(T_\mu)=T_\mu,\quad {\mathfrak X}(\Upsilon_\mu)=\Upsilon_\mu,\quad {\mathfrak X}(q)={\mathbf t}^{-1} q,\quad {\mathfrak X}(\tau)={\mathbf t}^{-1}\tau,\] where if $\mu=\sum_{i\in I} \mu_i i$, ${\tilde{T}}_\mu=\prod_{i\in I} T_{\mu_i d_i i}$. \item For $x\in \ensuremath{\mathbf{f}}[{\mathbf t}]$, we have \subitem {\rm (a)} ${\mathfrak X}(x^+)=t_\nu^{2}{\mathbf t}^{\bullet(|x|)} {\mathfrak X}(x)^+{\tilde{T}}_{|x|}\Upsilon_{|x|}$ \subitem {\rm (b)} ${\mathfrak X}(x^-)={\mathfrak X}(x)^-\Upsilon_{-|x|}$ \end{enumerate} \end{prop} Later on, we will need some alternate versions of the results in Proposition \ref{prop:twistordef} which we shall prove now. First, we note the following analogue of Proposition \ref{prop:twistordef} (2) for the dual canonical basis. \begin{lem}\label{lem:twondualbasis} Let $(-,-)$ be the bilinear form on $\ensuremath{\mathbf{f}}$ defined in \eqref{eq:ff bilinear form}. Then \[{\mathfrak X}^{-1}(({\mathfrak X}(x),{\mathfrak X}(y)))=(-1)^{{\mathbf p}(|x|)}(x,y).\] In particular, ${\mathfrak X}(b^*)=(-1)^{{\mathbf p}(b)}{\mathbf t}^{-\ell(b)} b^*$ for any $b\in {\mathcal{B}}$. \end{lem} \begin{proof} Let $(x,y)^{\mathfrak X}={\mathfrak X}^{-1}(({\mathfrak X}(x),{\mathfrak X}(y)))$ and observe that this is a ${\Qqt^\tau}$-bilinear form on $\ensuremath{\mathbf{f}}[{\mathbf t}]$. Moreover, note that ${{}_i r}({\mathfrak X}(w))={\mathbf t}^{\phi(i,|w|-i)}{\mathfrak X}({{}_i r}(w))$ by the same proof as \cite[Proposition 3.2(b)]{CFLW}. To show $(x,y)^{\mathfrak X}=(-1)^{{\mathbf p}(|x|)}(x,y)$, we proceed by induction on the height. First note that \[(1,1)^{{\mathfrak X}}=1=(1,1),\qquad (\theta_i,\theta_i)^{{\mathfrak X}}={\mathfrak X}^{-1}\parens{\frac{1}{1-\pi_iq_i^{-2}}}=(\theta_i,\theta_i).\] Now if $x\in\ensuremath{\mathbf{f}}[{\mathbf t}]_{\nu-i}$ and $y\in \ensuremath{\mathbf{f}}[{\mathbf t}]_\nu$ for some $i\in I$ and $\nu\in {\mathbb{Z}}_{\geq_0}[I]$ with ${\mathrm{ht}}(\nu)>1$, we have \begin{align*} (\theta_ix,y)^{\mathfrak X}&={\mathbf t}^{\phi(i,\nu-i)}{\mathfrak X}^{-1}((\theta_i{\mathfrak X}(x),{\mathfrak X}(y)))= {\mathbf t}^{\phi(i,\nu-i)}(\theta_i,\theta_i){\mathfrak X}^{-1}(({\mathfrak X}(x),{{}_i r}({\mathfrak X}(y))))\\ &={\mathbf t}^{2\phi(i,\nu-i)}(\theta_i,\theta_i){\mathfrak X}^{-1}(({\mathfrak X}(x),{\mathfrak X}({{}_i r}(y))))\\ &=(-1)^{{\mathbf p}(\nu-i)+\phi(i,\nu-i)}(\theta_i,\theta_i)(x,{{}_i r}(y))\\ &=(-1)^{{\mathbf p}(\nu-i)+p(\nu-i)p(i)}(\theta_ix,y)=(-1)^{{\mathbf p}(\theta_ix)}(\theta_ix,y) \end{align*} where in the last equality, note that if $\nu=\sum_{i\in I} \nu_i i$ then we have $\phi(i,\nu-i)\equiv (\nu_i-1)d_i$ modulo 2. The proof is finished by observing that $(\nu_i-1)d_i\equiv p(\nu-i)p(i)$ for any $i\in I$, since if $i\neq \rf n$ both sides are $0$ modulo 2, and if $i=\rf n$ both sides are equivalent to $\nu_{\rf n}-1$ modulo 2. \end{proof} \begin{rem} Though Lemma \ref{lem:twondualbasis} as stated requires $|I_1|=1$, a version of it also holds for arbitrary enhanced quantum covering algebras. Indeed, if $|I_1|>1$, then ${\mathfrak X}^{-1}(({\mathfrak X}(x),{\mathfrak X}(y))={\mathbf t}^{\binom{\nu}{2}}(x,y)$, where $|x|=\nu=\sum_{i\in I}\nu_i i$ and $\binom{\nu}{2}=\sum_{i\in I}\binom{\nu_i}{2}d_i$. \end{rem} It will also be more convenient to have the following variant of Proposition \ref{prop:twistordef} (4b). \begin{lem}\label{lem:tw+alt} We have \[{\mathfrak X}(x^+)={\mathbf t}_{|x|}^{-1}{\tilde{T}}_{|x|}\Upsilon_{|x|}{\mathfrak X}(x)^+.\] \end{lem} \begin{proof} This is true if $x=\theta_i$. It suffices to show if it is true for $x$, then it is true for $\theta_ix$. \begin{align*} {\mathfrak X}(\theta_ix^+)&={\mathfrak X}(\theta_i^+){\mathfrak X}(x^+)={\mathbf t}_i^{-1}{\tilde{T}}_i\Upsilon_i E_i {\mathbf t}_\nu^{-1}T_{\nu}\Upsilon_\nu {\mathfrak X}(x)^+\\ &={\mathbf t}_{i+\nu}^{-1}{\mathbf t}^{-i\cdot\nu-\phi(\nu,i)}{\tilde{T}}_i\Upsilon_i {\tilde{T}}_{\nu}\Upsilon_\nu E_i{\mathfrak X}(x)^+={\mathbf t}_{|\theta_ix|}^{-1}{\mathbf t}^{-i\cdot\nu-\phi(\nu,i)-\phi(i,\nu)}{\tilde{T}}_{i+\nu}\Upsilon_{i+\nu} {\mathfrak X}(\theta_i x)^+ \end{align*} But then by Lemma \ref{lem:enhancer single odd}, \[-i\cdot\nu-\phi(\nu,i)-\phi(i,\nu)\equiv_4 -2i\cdot\nu\equiv_4 0.\] \end{proof} \subsection{${\widehat \UU}$-modules and Hopf structure} Let $M$ be a ${\mathbf U}$-weight module. Then $M$ is canonically a ${\widehat \UU}$-module by defining \begin{align} T_\mu m&= {\mathbf t}^{\ang{\mu,\lambda}} m,\quad \mu\in{\mathbb{Z}}[I], m\in M_\lambda;\label{eq:T act}\\ \Upsilon_\mu m&={\mathbf t}^{\phi(\mu,\lambda)} m,\quad \mu\in{\mathbb{Z}}[I], m\in M_\lambda.\label{eq:Ups act} \end{align} To that end, we will call any ${\widehat \UU}$-module which restricts to a ${\mathbf U}$-weight module and satisfies \eqref{eq:T act} a {${\widehat \UU}$-weight module}. If it additionally satisfies \eqref{eq:Ups act}, we shall call it a {\em canonical} ${\widehat \UU}$-weight module. In particular, any tensor product of ${\widehat \UU}$-modules can be given a canonical ${\widehat \UU}$-weight module structure. However, such a procedure forgets the action of the $\Upsilon$ elements on the factors due to the lack of additivity in the second component of $\phi$. \begin{example} Consider the case $n=1$. Then ${\widehat \UU}$ has the canonical weight module ${\widehat{V}}(1)={\Qqt^\tau} v_1\oplus {\Qqt^\tau} v_{-1}$ which is isomorphic to $V(1)$ as a ${\mathbf U}$-module and satisfies $T_i v_1= {\mathbf t} v_1$ and $\Upsilon_{\rf 1} v_1={\mathbf t}^{\phi({\rf 1},1)} v_1$. Then ${\widehat{V}}(1)\otimes {\widehat{V}}(1)$ is a ${\mathbf U}$-weight module hence has a canonical ${\widehat \UU}$-module structure, but note that \[\Upsilon_{\rf 1} v_1\otimes v_1={\mathbf t}^{\phi({\rf 1},2)} v_1\otimes v_1\] and by the definition of $\phi$, we have $\phi({\rf 1},2)=\phi({\rf 1},{\rf 1})=1\neq \phi({\rf 1},1)+\phi({\rf 1},1)$. \end{example} In particular, canonical module structures will be too naive for our purposes. Instead, we will introduce Hopf structure which will inform our classes of weight modules. \begin{prop} The algebra ${\widehat \UU}$ has a Hopf covering algebra structure given by the following: \begin{enumerate} \item A coassociative coproduct $\Delta:{\widehat \UU}\rightarrow {\widehat \UU}\otimes_{{\Qqt^\tau}} {\widehat \UU}$ extending $\Delta:{\mathbf U}\rightarrow{\mathbf U}\otimes_{\Qqt^\tau} {\mathbf U}$ such that $\Delta(T_\mu)=T_\mu\otimes T_\mu$ and $\Delta(\Upsilon_\mu)=\Upsilon_\mu\otimes \Upsilon_\mu$ for $\mu\in{\mathbb{Z}}[I]$. In particular, we inductively define $\Delta^t=(\Delta\otimes 1^{t-1})\circ \Delta^{t-1}:{\widehat \UU}\rightarrow {\widehat \UU}^{\otimes (t+1)}$ for any integer $t>1$. \item An antipode $S:{\widehat \UU}\rightarrow{\widehat \UU}$ extending $S:{\mathbf U}\rightarrow {\mathbf U}$ such that $S(T_\mu)=T_{-\mu}$ and $S(\Upsilon_\mu)=\Upsilon_{-\mu}$ for $\mu\in{\mathbb{Z}}[I]$. \item A counit map $\epsilon:{\widehat \UU}\rightarrow {\widehat \UU}$ extending $\epsilon:{\mathbf U}\rightarrow{\mathbf U}$ such that $\epsilon(T_\mu)=\epsilon(\Upsilon_\mu)=1$ for $\mu\in{\mathbb{Z}}[I]$. \end{enumerate} \end{prop} \begin{proof} To show that these maps define a Hopf structure, we need only check that these morphisms respect \eqref{eq:TUpsrels}-\eqref{eq:Upsweightrels}. This is obvious for \eqref{eq:TUpsrels}, and can be quickly verified for \eqref{eq:Tweightrels} and \eqref{eq:Upsweightrels} by checking it for the generators of ${\mathbf U}$. For instance, \begin{align*} \Delta(\Upsilon_\mu)\Delta(E_i) &=(\Upsilon_\mu\otimes\Upsilon_\mu)(E_i\otimes 1+{\tilde{J}}_i{\tilde{K}}_i\otimes E_i)\\ &=\Upsilon_\mu E_i\otimes \Upsilon_\mu+\Upsilon_\mu {\tilde{J}}_i{\tilde{K}}_i\otimes \Upsilon_\mu E_i\\ &= {\mathbf t}^{\phi(\mu,i)}E_i\Upsilon_\mu\otimes \Upsilon_\mu+ {\tilde{J}}_i{\tilde{K}}_i\Upsilon_\mu\otimes {\mathbf t}^{\phi(\mu,i)} E_i \Upsilon_\mu\\ &={\mathbf t}^{\phi(\mu,i)}\Delta(E_i)\Delta(\Upsilon_\mu); \end{align*} \[S(E_i)S(\Upsilon_\mu)=-{\tilde{J}}_{-i}{\tilde{K}}_{-i}E_i\Upsilon_{-\mu} =-{\mathbf t}^{\phi(\mu,i)}\Upsilon_{-\mu}{\tilde{J}}_{-i}{\tilde{K}}_{-i}E_i ={\mathbf t}^{\phi(\mu,i)}S(\Upsilon_\mu)S(E_i);\] \[\epsilon(E_i)\epsilon(\Upsilon_\mu)=(0)(1)={\mathbf t}^{\phi(\mu,i)}(1)(0) ={\mathbf t}^{\phi(\mu,i)}\epsilon(\Upsilon_\mu)\epsilon(E_i).\] Finally, the co-associativity of $\Delta$ on ${\widehat \UU}$ follows immediately from the co-associativity of $\Delta$ on ${\mathbf U}$ and the fact that $T_\mu$ and $\Upsilon_\mu$ are grouplike elements. \end{proof} The coproduct gives us another way to define an action of ${\widehat \UU}$ on tensor products of canonical ${\widehat \UU}$-weight modules. Henceforth, given ${\widehat \UU}$-weight modules $M$ and $N$, we let $M{\widehat{\otimes}} N$ denote the space $M\otimes_{{\Q(q,{\mathbf t})^{\pi}}} N$ with the ${\widehat \UU}$-weight module structure induced by the coproduct on ${\widehat \UU}$. (Note that in general, the module $M{\widehat{\otimes}} N$ is {\em not} canonical!) \begin{example} Continuing the previous example, the action of $\Upsilon_{\rf 1}$ on ${\widehat{V}}(1){\widehat{\otimes}} {\widehat{V}}(1)$ is given by \[\Delta(\Upsilon_{\rf 1}) v_1\otimes v_1={\mathbf t}^{2\phi({\rf 1},1)} v_1\otimes v_1.\] \end{example} Another natural module to consider is the following. Given a canonical ${\widehat \UU}$-weight module $M$, we can construct the restricted linear dual $M^*$. This space is naturally a ${\mathbf U}$-weight module as in \S\ref{subsec:modules}, hence has a canonical ${\widehat \UU}$ structure. On the other hand, let $M^\natural$ denote the space $M^*$ with the action of ${\widehat \UU}$ defined by $(uf)(x)=\pi^{p(f)p(u)}f(S(u)x)$. Note that $M^\natural$ is not canonical: if $f\in (M_\lambda,s)^*$, then $|f|=-\lambda$ but nevertheless \[\Upsilon_\mu f={\mathbf t}^{-\phi(\mu,\lambda)} f.\] Since modules with these unorthodox actions of the $\Upsilon_\mu$ will be of primary importance, we give the following definitions. \begin{dfn} We say that a ${\widehat \UU}$-weight module $M$ is {\em anti-canonical} if $\Upsilon_\mu m={\mathbf t}^{-\phi(\mu,-\lambda)} m$ for all $m\in M_\lambda$. More generally, we say that $M$ is a {\em mixed} weight module if there exists an integer $t\geq 1$ and a sequence $c=(c_1,\ldots, c_t)\in \set{\pm 1}^t$ such that $M_\lambda=\bigoplus_{(\lambda_s)\in (X^t)_\lambda} M_{(\lambda_s)}$, where \[(X^t)_\lambda=\set{(\lambda_1,\ldots,\lambda_t)\in X^t\mid \lambda=\lambda_1+\ldots+\lambda_s},\] \[M_{(\lambda_s)}=\set{m\in M\mid\Upsilon_\mu m={\mathbf t}^{\sum_{1\leq s\leq t} c_s\phi(\mu,c_s\lambda_s)} m\text{ for all }\mu\in{\mathbb{Z}}[I]}.\] We say $c$ is the {\em signature} of $M$, and denote it by ${\rm sig}(M)=c$. \end{dfn} \begin{rem} We note that just as any weight ${\mathbf U}$-module can be given a canonical ${\widehat \UU}$-module structure, it can also be given an anti-canonical ${\widehat \UU}$-module structure. Indeed, suppose $M$ is a weight ${\mathbf U}$-module and define $T_im={\mathbf t}^{\ang{i,|m|}}m$ and $\Upsilon_i m={\mathbf t}^{-\phi(i,-|m|)}$. Then this defines an action of ${\widehat \UU}$, since for any $i\in I$ and $u\in{\mathbf U}$, $\Upsilon_i um={\mathbf t}^{-\phi(i,-(|m|+|u|))} um ={\mathbf t}^{\phi(i,|u|)}u\Upsilon_i m$. \end{rem} In addition to classifying modules by the action of the $\Upsilon$ elements, another property of ${\widehat \UU}$-weight modules which will be important to us is their interaction with the twistor map ${\mathfrak X}:{\widehat \UU}\rightarrow{\widehat \UU}$. \begin{dfn} Let $M$ be a ${\widehat \UU}$-weight module. We say $M$ carries a twistor ${\mathfrak X}$ (or ${\mathfrak X}$ is a twistor on $M$) if there exists a homogeneous ${\mathbb{Q}}(t)$-linear bijection ${\mathfrak X}:M\rightarrow M$ such that ${\mathfrak X}(um)={\mathfrak X}(u){\mathfrak X}(m)$. \end{dfn} Modules which carry twistors are not hard to find. Indeed, the simple ${\mathbf U}$-modules $V(\lambda)$ are themselves examples when given canonical (or anti-canonical) actions of ${\widehat \UU}$. \begin{lem}{\bf \cite[Lemma 6.9]{C}}\label{lem:modtwistor} Let $\lambda\in X^+$. Let ${\widehat{V}}(\lambda)$ be the space $V(\lambda)$ with the canonical action of ${\widehat \UU}$. There is a ${\mathbb{Q}}({\mathbf t})$-linear map ${\mathfrak X}:{\widehat{V}}(\lambda)\rightarrow {\widehat{V}}(\lambda)$ which satisfies ${\mathfrak X}(v_\lambda)=v_\lambda$ and ${\mathfrak X}_\lambda(um)={\mathfrak X}(u){\mathfrak X}(m)$ for all $u\in {\widehat \UU}$ and $m\in {\widehat{V}}(\lambda)$. \end{lem} In light of Lemma \ref{lem:dual isos}, it follows that the ${\mathbf U}$-module $V(\lambda)^*$, viewed as a canonical ${\widehat \UU}$-module, also carries a twistor. A similar argument to \cite[Lemma 6.9]{C} can be used to construct a twistor on $V(\lambda)$ with an anti-canonical action of ${\mathbf U}$, hence the ${\widehat \UU}$-module ${\widehat{V}}(\lambda)^\natural$ carries a twistor. However, this construction is not very compatible with the dual basis, since it relies on an isomorphism $V(\lambda)\rightarrow \Pi^{P(\lambda)}V(\lambda)$ and is defined by descent from the highest weight vector. To obtain a convenient definition of a twistor on the dual modules, we will define a map directly on ${\widehat{V}}(\lambda)^\natural$. Define the {\em dual twistor} on ${\widehat \UU}$ to be the map ${\Tw^\natural}(u)=S\circ{\mathfrak X}\circ S^{-1}(u)$. This map is clearly a bijection, and for any $u,v\in{\widehat \UU}$ we have \begin{align*} {\Tw^\natural}(uv)&=S({\mathfrak X}(S^{-1}(uv)))\\ &={\mathbf t}^{2p(u)p(v)}S({\mathfrak X}(S^{-1}(u)))S({\mathfrak X}(S^{-1}(v)))\\ &={\mathbf t}^{2p(u)p(v)}{\Tw^\natural}(u){\Tw^\natural}(v). \end{align*} Therefore, it is determined by the images of the generators, which are \[{\Tw^\natural}(E_i)=t_iE_i\Upsilon_{-i},\quad {\Tw^\natural}(F_i)=\Upsilon_iF_i{\tilde{T}}_i, \quad {\Tw^\natural}(K_\mu)=T_{-\mu}K_{\mu},\quad {\Tw^\natural}(J_\mu)=T_{2\mu}J_\mu\] \[{\Tw^\natural}(q)={\mathbf t}^{-1}q,\quad {\Tw^\natural}(\tau)={\mathbf t}\tau.\] In particular, note that \begin{equation}\label{eq:twnat-} {\Tw^\natural}(x^-)=\Upsilon_{\nu}{\mathfrak X}(x)^-{\tilde{T}}_\nu. \end{equation} While ${\Tw^\natural}$ is not an algebra automorphism of ${\widehat \UU}$, it shares many properties with ${\mathfrak X}$. In particular, we have a version of Lemma \ref{lem:modtwistor}. \begin{lem}\label{lem:mod dual twistor} Let $\lambda\in X^+$. There is a ${\mathbb{Q}}({\mathbf t})$-linear map ${\Tw^\natural}:{\widehat{V}}(\lambda)\rightarrow {\widehat{V}}(\lambda)$ which satisfies ${\Tw^\natural}(v_\lambda)=v_\lambda$ and ${\Tw^\natural}(um)={\mathbf t}^{2p(u)p(m)}{\Tw^\natural}(u){\Tw^\natural}(m)$ for all $u\in {\widehat \UU}$ and $m\in {\widehat{V}}(\lambda)$. \end{lem} \begin{proof} This follows from more or less the same proof as \cite[Lemmas 6.8, 6.9]{C}. To wit, we can identify the Verma module of highest weight $\lambda$ for ${\mathbf U}$ with $\ensuremath{\mathbf{f}}$ (cf. {\em loc. cit} for details), and in particular this is naturally a canonical ${\widehat \UU}$-module. Then we define a map ${\mathfrak X}^\natural_\lambda:\ensuremath{\mathbf{f}}\rightarrow \ensuremath{\mathbf{f}}$ via ${\mathfrak X}^\natural_\lambda(x)={\mathbf t}^{\ang{\tilde\nu,\lambda}+\phi(\nu,\lambda-\nu)}{\mathfrak X}(x)$. Then it is straightforward to verify that ${\Tw^\natural}(ux)={\mathbf t}^{2p(u)p(x)}{\Tw^\natural}(u){\Tw^\natural}(x)$ for $x\in {\widehat{V}}(\lambda)$ and $u=F_i, T_\mu, J_\mu, K_\mu, \Upsilon_\mu$. From the calculations in {\em loc. cit} and the definition, we see that \[{\mathfrak X}^\natural_\lambda(E_ix)={\mathbf t}^{\star} E_i{\mathfrak X}^\natural_\lambda(x).\] where $\star=\ang{\tilde\nu-\tilde i,\lambda}-\ang{\tilde\nu,\lambda}+\phi(\nu-i,\lambda-\nu+i)-\phi(\nu,\lambda-\nu) -d_i+\ang{\tilde{i},\lambda-\nu+i}-\phi(i,\nu-i)$. Now we can simplify $\star$ and apply \eqref{eq:phi on ZI} to see that \[\star\equiv\phi(\nu,i)-\phi(i,\nu)-\phi(i,\lambda-\nu)+i\cdot\nu+d_i=2p(\nu)p(i)-\phi(i,\lambda-\nu)+d_i\mod 4,\] and thus \[{\mathfrak X}^\natural_\lambda(E_ix)={\mathbf t}^{p(\nu)p(i)}{\mathbf t}_i E_i\Upsilon_{-i}{\mathfrak X}^\natural_\lambda(x)={\mathbf t}^{p(\nu)p(i)}{\Tw^\natural}(E_i){\mathfrak X}^\natural_\lambda(x).\] Finally, we note that the kernel of the projection $\ensuremath{\mathbf{f}}\rightarrow {\widehat{V}}(\lambda)$ is trivially preserved by ${\mathfrak X}^\natural_\lambda$, hence it descends to a map on ${\widehat{V}}(\lambda)$. \end{proof} The dual twistor ${\Tw^\natural}$ is what will allow us to define a convenient twistor map on dual modules, as follows. Recall that $V(-\lambda)$ denotes the ${\mathbf U}$-module $V(\lambda)^*$. We will adapt this notation to ${\widehat{V}}(\lambda)^\natural$. \begin{lem} For $\lambda\in X^+$, let ${\widehat{V}}(-\lambda)={\widehat{V}}(\lambda)^\natural$; that is, the space $V(\lambda)^*$ with the action of ${\widehat \UU}$ induced by the antipode $S:{\widehat \UU}\rightarrow{\widehat \UU}$. Define a map ${\mathfrak X}$ on ${\widehat{V}}(-\lambda)$ by ${\mathfrak X}(f)(x)={\mathbf t}^{2p(f)p(x)}{\mathfrak X}(f({\Tw^\natural}^{-1}(x)))$ for homogeneous $x\in {\widehat{V}}(\lambda)$ and $f\in {\widehat{V}}(-\lambda)$. Then ${\mathfrak X}(uf)={\mathfrak X}(u){\mathfrak X}(f)$ for all $u\in{\widehat \UU}$ and $f\in {\widehat{V}}(-\lambda)$. \end{lem} \begin{proof} Let $f\in {\widehat{V}}(-\lambda)$ and $x\in {\widehat{V}}(\lambda)$ be homogeneous. First, observe that since ${\Tw^\natural}$ preserves the $\bigrset$-grading, ${\mathfrak X}(f)(x)=0$ unless $\bigrdeg{x}=\bigrdeg{f}$. Moreover, if $a\in{\Qqt^\tau}$, \[{\mathfrak X}(f)(ax)={\mathbf t}^{2p(f)p(x)}{\mathfrak X}(f({\Tw^\natural}^{-1}(ax))) ={\mathbf t}^{2p(f)p(x)}{\mathfrak X}({\mathfrak X}^{-1}(a)f({\Tw^\natural}^{-1}(x)))=a{\mathfrak X}(f)(x),\] so ${\mathfrak X}(f)$ is indeed an element of ${\widehat{V}}(-\lambda)$. Now suppose $u\in{\widehat \UU}$. We compute that \[{\mathfrak X}(uf)(x)={\mathbf t}^{2p(uf)p(x)}{\mathfrak X}((uf)({\Tw^\natural}^{-1}(x)))={\mathbf t}^{2p(u)p(x)+2p(f)p(x)}{\mathfrak X}(\pi^{p(u)p(f)}f(S(u){\Tw^\natural}^{-1}(x)),\] \begin{align*} {\mathfrak X}(u){\mathfrak X}(f)(x)&={\mathbf t}^{2p(f)p(ux)}\pi^{p(u)p(f)}{\mathfrak X}(f({\Tw^\natural}^{-1}(S({\mathfrak X}(u))x)))\\ &={\mathbf t}^{2p(f)p(u)+2p(f)p(x)+2p(u)p(x)}\pi^{p(u)p(f)}{\mathfrak X}(f({\Tw^\natural}^{-1}(S({\mathfrak X}(u))){\mathfrak X}'^{-1}(x)))\\ &={\mathbf t}^{2p(f)p(x)+2p(u)p(x)}{\mathfrak X}(\pi^{p(u)p(f)}f(S(u){\Tw^\natural}^{-1}(x)). \end{align*} Therefore, ${\mathfrak X}(uf)={\mathfrak X}(u){\mathfrak X}(f)$. \end{proof} \subsection{Twistor on tensor products} Now let us return to the question of relating the ${\mathfrak{osp}}(1|2)$ and ${\mathfrak{sl}}(2)$ link invariants. Since the invariants arise from maps between tensor products of simple modules and their duals, we shall also need variants of the twistor maps on the corresponding ${\widehat \UU}$-modules. In the following, we shall define a number of versions of ${\mathfrak X}$ in different settings. However, they will all be compatible in natural ways, so rather than label these maps differently, we shall treat them en suite as an operator on ${\widehat \UU}$ and its modules. The following proposition takes the first step in this direction by showing that there is a natural extension of the twistor maps to tensor powers of ${\mathbf U}$. \begin{prop}\label{prop:twandcoprod} For each positive integer $t$, there exists a ${\mathbb{Q}}({\mathbf t})$-algebra automorphism ${\mathfrak X}$ of ${\widehat \UU}^{\otimes t+1}$ which satisfies \[{\mathfrak X}(x\otimes y)={\mathfrak X}(x)\Delta^{s}(\Upsilon_{|y|})\otimes\Delta^{s'}({\tilde{T}}_{|x|}\Upsilon_{|x|}){\mathfrak X}(y)\] for any positive integers $s,s'$ satisfying $s+s'=t+1$, $x\in {\widehat \UU}^{\otimes s}$, and $y\in{\widehat \UU}^{\otimes s'}$. Moreover, $\Delta^{t}({\mathfrak X}(x))={\mathfrak X}(\Delta^{t}(x))$ for any $x\in {\mathbf U}$. \end{prop} \begin{proof} Define ${\mathfrak X}':{\widehat \UU}^{\otimes t+1}\rightarrow {\widehat \UU}^{\otimes t+1}$ as follows: for $x=\bigotimes_{s=1}^{t+1} x_s\in{\widehat \UU}^{\otimes t+1}$, let ${\mathfrak X}(x)=\bigotimes_{s=1}^{t+1}{\mathfrak X}(x)_s$ where \begin{equation}\label{eq:twistortensorfactors} {\mathfrak X}(x)_s={\tilde{T}}_{|x_1|+\ldots +|x_{s-1}|}\Upsilon_{|x_1|+\ldots +|x_{s-1}|}{\mathfrak X}(x_s)\Upsilon_{|x_{s+1}|+\ldots +|x_{t+1}|}. \end{equation} It is elementary to check that \[{\mathfrak X}(x\otimes y)={\mathfrak X}(x)\Delta^{s}(\Upsilon_{|y|})\otimes\Delta^{s'}{\tilde{T}}_{|x|}\Upsilon_{|x|}{\mathfrak X}(y)\] for any positive integers $s,s'$ satisfying $s+s'=t+1$, $x\in {\widehat \UU}^{\otimes s}$, and $y\in{\widehat \UU}^{\otimes s'}$. Moreover, since ${\mathfrak X}$ on ${\widehat \UU}$ is a bijection, it is easy to see that so is ${\mathfrak X}$ on ${\widehat \UU}^{\otimes t+1}$. We will prove that ${\mathfrak X}$ is an isomorphism by induction. Since ${\mathfrak X}$ on ${\widehat \UU}$ is an isomorphism, let us assume ${\mathfrak X}$ on ${\widehat \UU}^t$ is an isomorphism. Then for $x,w\in {\widehat \UU}^{\otimes t}$ and $y,z\in{\widehat \UU}$, \begin{align*} {\mathfrak X}(x\otimes y){\mathfrak X}(w\otimes z)&=({\mathfrak X}(x)\Upsilon_{|y|}\otimes{\tilde{T}}_{|x|}\Upsilon_{|x|}{\mathfrak X}(y))({\mathfrak X}(w)\Upsilon_{|z|}\otimes{\tilde{T}}_{|w|}\Upsilon_{|w|}{\mathfrak X}(z))\\ &=\pi^{p(y)p(w)}{\mathfrak X}(x)\Upsilon_{|y|}{\mathfrak X}(w)\Upsilon_{|z|}\otimes{\tilde{T}}_{|x|}\Upsilon_{|x|}{\mathfrak X}(y){\tilde{T}}_{|w|}\Upsilon_{|w|}{\mathfrak X}(z)\\ &=\pi^{p(y)p(w)}{\mathbf t}^{\phi(|y|,|w|)-\phi(|w|,|y|)-|w|\cdot|y|}{\mathfrak X}(xw)\Upsilon_{|yz|}\otimes{\tilde{T}}_{|xw|}\Upsilon_{|xw|}{\mathfrak X}(yz)\\ &=\pi^{p(y)p(w)}{\mathbf t}^{2p(y)p(z)}{\mathfrak X}(xw\otimes yz)={\mathfrak X}(\pi^{p(y)p(w)}xw\otimes yz)\\ &={\mathfrak X}((x\otimes y)(w\otimes z)) \end{align*} This completes the induction showing ${\mathfrak X}$ on ${\mathbf U}^{t+1}$ is an isomorphism as claimed. Finally, showing that ${\mathfrak X}$ commutes with $\Delta^{t}$ is straightforward using \eqref{eq:twistortensorfactors} and checking on the generators. \end{proof} Now that we have a viable twistor map on tensor powers of ${\widehat \UU}$, we need an analogue on the tensor powers of modules. In particular, suppose we have a collection of ${\widehat \UU}$ modules which are canonical or anticanonical, and which carry twistors. We will produce a twistor on the tensor product of these modules. As might be suggested by \eqref{eq:twistortensorfactors}, this is not as simple as taking the tensor power of the twistors. A version of such a twistor is produced in \cite[Proposition 6.11]{C} by rescaling the tensor product of twistors by a power of ${\mathbf t}$ given by a function of the weights of the tensor factors. We will do something similar, but it turns out that we will need functions which depend not only on the weights of tensor factors but also their parities, as well as the signature of the tensor product. \begin{lem} Let $c=(c_1,c_2)$ where $c_1,c_2\in \set{1,-1}$. There exists a function $\kappa_c:\bigrset^2\rightarrow {\mathbb{Z}}$ satisfying $\kappa((0,0),\zeta)\equiv \kappa(\zeta,(0,0))\equiv 0$ modulo 4 and \[\kappa_c(\zeta+\mu,\zeta'+\nu)-\kappa_c(\zeta,\zeta')\equiv \ang{\tilde\mu,|\zeta'|}+c_2\phi(\mu,c_2|\zeta'|)+2p(\zeta)p(\nu)+c_1\phi(\nu,c_1|\zeta|)+\mu\cdot\nu+\phi(\mu,\nu)\mod 4\] for all $\zeta,\zeta'\in \bigrset$ and $\mu,\nu\in{\mathbb{Z}}[I]$. \end{lem} \begin{proof} Fix $c=(c_1,c_2)$ where $c_1,c_2\in\set{1,-1}$. Note that it suffices to show such a function $\kappa=\kappa_c$ exists on each coset of ${\mathbb{Z}}[I]\times {\mathbb{Z}}[I]$ (where as in \eqref{eq:ZI in X hat}, we view ${\mathbb{Z}}[I]$ as a subset of $\bigrset$), so fix a set of representatives $C$ of $\bigrset/{\mathbb{Z}}[I]$. For $\zeta_0,\zeta_1\in C$, set \[\kappa(\zeta_0+\mu,\zeta_1+\nu) =\ang{\tilde\mu,|\zeta_1|}+c_2\phi(\mu,c_2|\zeta_1|)+2p(\zeta_0)p(\nu)+c_1\phi(\nu,c_1|\zeta_0|)+\mu\cdot\nu+\phi(\mu,\nu).\] It is elementary to verify that this has the desired properties. \end{proof} We henceforth suppose we have fixed choices of $\kappa_c$ for each $c\in \set{1,-1}^2$. We can extend $\kappa$ naturally to larger powers of $\bigrset$. Let $t>1$ be a positive integer and fix a sequence $c=(c_s)\in\set{\pm 1}^t$. Let $\kappa_c:\bigrset^t\rightarrow {\mathbb{Z}}$ be the function defined by \[\kappa_c(\zeta)=\sum_{1\leq r<s\leq t} \kappa_{(c_r,c_s)}(\zeta_r,\zeta_s), \quad \zeta=(\zeta_s)\in\bigrset^t.\] Then if $\zeta=(\zeta_s), \zeta'=(\zeta'_s)\in \bigrset^t$ with $\zeta'_s=\zeta_s+\delta_{r,s}i$ for some $1\leq r\leq t$, then \[\kappa(\zeta')-\kappa(\zeta) =\sum_{r<s\leq t}\parens{\ang{\tilde i,|\zeta_{s}|}+c_s\phi(i,c_s|\zeta_{s}|)}+ \sum_{1\leq s'<r}\parens{2p(\zeta_{s'})p(i)+c_{s'}\phi(i,c_{s'}|\zeta_{s'}|)}\mod 4.\] We can observe some convenient properties of the maps $\kappa_c$. \begin{lem} \label{lem:kappa props} Let $c=(c_s)\in\set{\pm 1}^t$ and $\zeta=(\zeta_s),\zeta'=(\zeta'_s)\in \bigrset^t$. \begin{enumerate} \item Let $1\leq r\leq t$, and define $c_{\leq r}= (c_1,\ldots, c_r)$, $c_{>r}=(c_{r+1},\ldots, c_{t})$. Likewise, define $\zeta''_{\leq r}=(\zeta_1'',\ldots, \zeta_r'')$ and $\zeta''_{>r}=(\zeta''_{r+1},\ldots, \zeta''_t)$ for any $\zeta''=(\zeta''_s)\in \bigrset^t$. Then \[\kappa_c(\zeta,\zeta')=\kappa_{c_{\leq r}}(\zeta_{\leq r}, \zeta'_{\leq r})+ \kappa_{c_{> r}}(\zeta_{> r}, \zeta'_{>r}) +\sum_{1\leq s<r<s'\leq t} \kappa_{(c_s,c_{s'})}(\zeta_s,\zeta'_{s'})\] \item Suppose that there exists $1\leq r <t$ such that $\zeta_{r}=\zeta'_{r}+\nu$, $\zeta_{r+1}=\zeta'_{r+1}-\nu$, and $\zeta_s=\zeta_s'$ for $s\neq r,r+1$ and some $\nu\in{\mathbb{Z}}[I]$. Then \begin{equation}\label{eq:kappa opposing shifts} \kappa_c(\zeta)-\kappa_c(\zeta')=\ang{\tilde\nu,\zeta_{r+1}}+ c_{r+1}\phi(\nu,c_{r+1}\zeta_{r+1}) +2p(\nu)p(\zeta_r)- c_{r}\phi(\nu,c_{r}\zeta_{r})-\nu\cdot\nu-\phi(\nu,\nu) \end{equation} \item For any $\zeta\in \bigrset$ and $c_1=\pm 1$, we have \[\kappa_{c_1, \pm 1,\mp 1}(\zeta+\bigrelt{\nu},(\pm \lambda,0),(\mp\lambda,0))=\kappa_{c_1,\pm 1,\mp 1}(\zeta,(\pm \lambda,0),(\mp\lambda,0))\] \[\kappa_{\pm 1,\mp 1,c_1}((\pm \lambda,0),(\mp\lambda,0), \zeta+\bigrelt{\nu})=\kappa_{\pm 1,\mp 1,c_1}((\pm \lambda,0),(\mp\lambda,0), \zeta)\] \end{enumerate} \end{lem} \begin{proof} We note that (1) is an immediate consequence of the definition of $\kappa_c$. On the other hand, (2) and (3) both follow from direct computations and the definition. \end{proof} The functions $\kappa_c$ allows us to define a twistor on tensor product modules as follows. \begin{prop}\label{prop: twistor tensor} Let $M_1, M_2,\ldots, M_t$ be canonical or anti-canonical ${\widehat \UU}$-modules carrying twistors and let $M=M_1{\widehat{\otimes}} M_2\otimes\ldots{\widehat{\otimes}} M_t$ be the ${\widehat \UU}^{\otimes t}$-module (and hence a mixed ${\widehat \UU}$-module via $\Delta^{t-1}$) with the natural action. Set $c={\rm sig}(M)=(c_1,\ldots, c_t)$. Then the automorphism \[{\mathfrak X}(m_1\otimes\ldots\otimes m_t)={\mathbf t}^{\kappa_{c}((\bigrdeg m_i))}{\mathfrak X}(m_1)\otimes\ldots\otimes{\mathfrak X}(m_t)\] satisfies \[{\mathfrak X}((x_1\otimes\ldots\otimes x_t)(m_1\otimes\ldots\otimes m_t))={\mathfrak X}(x_1\otimes\ldots\otimes x_t){\mathfrak X}(m_1\otimes\ldots\otimes m_t).\] In particular, ${\mathfrak X}(um)={\mathfrak X}(u){\mathfrak X}(m)$ for $u\in {\widehat \UU}$ and $m\in M$. \end{prop} \begin{proof} First, observe it is enough to show \[{\mathfrak X}((1^{s-1}\otimes x_s\otimes 1^{t-s})(m_1\otimes\ldots\otimes m_t))={\mathfrak X}(1^{s-1}\otimes x_s\otimes 1^{t-s}){\mathfrak X}(m_1\otimes\ldots\otimes m_t)\] where $1\leq s\leq t$ and $x_s$ is a generator of ${\widehat \UU}$. This is trivial when $x_s$ is $K_\mu$, $J_\mu$, $T_\mu$ and $\Upsilon_\mu$ for some $\mu\in {\mathbb{Z}}[I]$ so it suffices to check the case $x_s=E_i,F_i$ for $i\in I$. To do this, let us make our equations more compact with the following notations: for $m_1\otimes\ldots \otimes m_t\in M$, let \[m_{<s}=m_1\otimes\ldots\otimes m_{s-1},\quad m_{>s}=m_{s+1}\otimes\ldots\otimes m_t\] \[{\mathfrak X}(m)_{<s}={\mathfrak X}(m_1)\otimes \ldots\otimes {\mathfrak X}(m_{s-1}),\quad {\mathfrak X}(m)_{>s}={\mathfrak X}(m_{s+1})\otimes \ldots\otimes {\mathfrak X}(m_{t}),\] \[\bigrdeg{m}_{<s}=(\bigrdeg{m_1},\ldots,\bigrdeg{m_{s-1}}),\qquad\bigrdeg{m}_{>s}=(\bigrdeg{m_{s+1}},\ldots,\bigrdeg{m_{t}}),\] \[\phi'(i, m_{<s})=\sum_{1\leq r<s} c_r \phi(i, c_r|m_r|),\qquad \phi''(i, m_{>s})=\sum_{s< r\leq t} c_r \phi(i, c_r |m_r|).\] Using these notations, we compute that \begin{align*} {\mathfrak X}&((1^{s-1}\otimes E_i\otimes 1^{t-s})(m_{<s}\otimes m_s\otimes m_{>s}))= {\mathfrak X}(\pi_i^{p(m_{<s})}m_{<s}\otimes E_im_s\otimes m_{>s})\\ &={\mathbf t}^{2p(i)p(m_{<s})+\kappa_c(\bigrdeg{m}_{<s},\bigrdeg{m_{s}}+\bigrelt{i},\bigrdeg{m}_{>s})} \pi^{p(i)p(m_{<s})}{\mathfrak X}(m)_{<s}\otimes {\mathfrak X}(E_im_s)\otimes {\mathfrak X}(m_{>s})\\ &={\mathbf t}^{\kappa(\bigrdeg{m_{<s}},\bigrdeg{m_{s}},\bigrdeg{m_{>s}})+\phi'(i,m_{<s})+\phi''(i,m_{>s})+\ang{\tilde i,|m_{>s}|}}\\ &\hspace{4em}\times \pi^{p(i)p(m_{<s})}{\mathfrak X}(m)_{<s}\otimes {\mathfrak X}(E_i){\mathfrak X}(m_s)\otimes {\mathfrak X}(m_{>s})\\ &={\mathbf t}^{\kappa(\bigrdeg{m_{<s}},\bigrdeg{m_{s}},\bigrdeg{m_{>s}})}\pi^{p(i)p(m_{<s})} \parens{\Upsilon_i^{\otimes(s-1)}{\mathfrak X}(m)_{<s}}\otimes \parens{{\mathfrak X}(E_i){\mathfrak X}(m_s)}\otimes \parens{(\Upsilon_i{\tilde{T}}_i)^{\otimes (t-s)}{\mathfrak X}(m_{>s})}\\ &={\mathfrak X}(1^{s-1}\otimes E_i\otimes 1^{t-s}){\mathfrak X}(m_{<s}\otimes m_s\otimes m_{>s}). \end{align*} The case $x_s=F_i$ proceeds similarly. \end{proof} We now have defined a family of compatible twistor maps on (anti-)canonical modules and their tensor products. Moreover, the twistor maps on tensor products of modules are compatible with one another in the following sense. Let $M_1,\ldots, M_t$, $c_1,\ldots, c_s$ and $M$ be as in Proposition \ref{prop: twistor tensor}. Fix $1\leq r\leq t$ and set $m_{\leq r}=m_1\otimes\ldots\otimes m_r$ and $m_{>r}=m_{r+1}\otimes\ldots\otimes m_t$. Then by Lemma \ref{lem:kappa props}(1), \begin{equation}\label{eq:twistor tensor assoc} {\mathfrak X}(m_{\leq r}\otimes m_{>r}) =\displaystyle\parens{\prod_{1\leq s\leq r<s'\leq t}{\mathbf t}^{ \kappa_{c_{s},c_{s'}}(\bigrdeg{m_s},\bigrdeg{m_{s'}})}}{\mathfrak X}(m_{\leq r})\otimes {\mathfrak X}(m_{>r}) \end{equation} \subsection{Twisting the crossings, caps, and cups} We have now lain the groundwork for studying the atomic maps in our graphical calculus from \S \ref{sec:diagcalc} under the twistor functor. Specifically, we will show that the twistor almost commutes with cups, caps, and crossings up to a factor of an integral power of ${\mathbf t}$, where the power depends on the map. We begin by considering the cups and caps on their domains of definition. \begin{prop}\label{prop: cups caps and twistorv1} Let $\lambda\in X^+$. Then the map $\ev_\lambda$ (respectively, $\qtr_\lambda$, $\coev_\lambda$, and $\coqtr_\lambda$) viewed as a function ${\widehat{V}}(-\lambda){\widehat{\otimes}}{\widehat{V}}(\lambda)\rightarrow {\Qqt^\tau}$ (resp. ${\widehat{V}}(\lambda){\widehat{\otimes}}{\widehat{V}}(-\lambda)\rightarrow {\Qqt^\tau}$, ${\Qqt^\tau}\rightarrow {\widehat{V}}(-\lambda){\widehat{\otimes}}{\widehat{V}}(\lambda)$, and ${\Qqt^\tau}\rightarrow {\widehat{V}}(\lambda){\widehat{\otimes}}{\widehat{V}}(-\lambda)$) is a ${\widehat \UU}$-module homomorphism. Moreover, we have \begin{enumerate} \item $\ev_\lambda{\mathfrak X}={\mathbf t}^{\kappa_{(-1,1)}((-\lambda,0),(\lambda,0))}{\mathfrak X}\ev_\lambda$; \item $\qtr_\lambda{\mathfrak X}={\mathbf t}^{\kappa_{(1,-1)}((\lambda,0),(-\lambda,0))-\ang{\tilde\rho,\lambda}}{\mathfrak X}\qtr_\lambda$; \item $\coev_\lambda{\mathfrak X}={\mathbf t}^{-\kappa_{(-1,1)}((-\lambda,0),(\lambda,0))+\ang{\tilde\rho,\lambda}}{\mathfrak X}\coev_\lambda$; \item $\coqtr_\lambda{\mathfrak X}={\mathbf t}^{-\kappa_{(1,-1)}((\lambda,0),(-\lambda,0))}{\mathfrak X}\coqtr_\lambda$. \end{enumerate} \end{prop} \begin{proof} First, observe that since these maps are ${\mathbf U}$-module homomorphisms, they preserve weight spaces hence preserve the action of $T_i$ for $i\in I$. Therefore, it only remains to check that they commute with the action of $\Upsilon_i$ for $i\in I$, As the arguments are all similar, let us show this for $\ev_\lambda$. Let $f\in {\widehat{V}}(-\lambda)$ and $x\in V(\lambda)$. Then \[\Upsilon_i\ev_\lambda(f\otimes x)={\mathbf t}^{\phi(i,0)}\ev_{\lambda}(f\otimes x)=f(x).\] On the other hand, $\Upsilon_i(f\otimes x)=(\Upsilon_if)\otimes(\Upsilon_ix)={\mathbf t}^{-\phi(i,-|f|)+\phi(i,|x|)} f\otimes x$ hence \[\ev_\lambda(\Upsilon_i(f\otimes x))={\mathbf t}^{\phi(i,|x|)-\phi(i,-|f|)}f(x).\] However, since $f(x)=0$ if $|f|\neq -|x|$, we see that $\ev_\lambda(\Upsilon_i(f\otimes x)={\mathbf t}^{\phi(i,|x|)-\phi(i,|x|)} f(x)=f(x)=\Upsilon_i\ev_\lambda(f\otimes x)$. To verify (1)-(4), it suffices to compute the images ${\mathfrak X}(b^-v_\lambda\otimes (b^-v_\lambda)^*)$ and ${\mathfrak X}((b^-v_\lambda)^*\otimes b^-v_\lambda)$ for $b\in{\mathcal{B}}_\nu={\mathcal{B}}\cap \ensuremath{\mathbf{f}}_\nu$. We compute directly that \[{\mathfrak X}(b^-v_\lambda)={\mathbf t}^{\ell(b)-\phi(\nu,\lambda)}b^{-}v_\lambda,\] \[{\Tw^\natural}(b^-v_\lambda)={\mathbf t}^{\ell(b)+\ang{\nu,\lambda}+\phi(\nu,\lambda-\nu)}b^{-}v_\lambda.\] This implies that for any $b,b'\in {\mathcal{B}}_\nu$, \[{\mathfrak X}((b^-v_\lambda)^*)(b'^-v_\lambda) ={\mathbf t}^{2p(\nu)}{\mathfrak X}((b^-v_\lambda)^*({\Tw^\natural}^{-1}(b'^-v_\lambda))) ={\mathbf t}^{2p(\nu)-\ell(b)-\ang{\nu,\lambda}-\phi(\nu,\lambda-\nu)}\delta_{b,b'} \] and hence ${\mathfrak X}((b^-v_\lambda)^*)={\mathbf t}^{2p(\nu)-\ell(b)-\ang{\nu,\lambda}-\phi(\nu,\lambda-\nu)}(b^-v_\lambda)^*$. In particular, for $c=(1,-1)$ observe that \begin{align*} {\mathfrak X}((b^-v_\lambda)\otimes(b^-v_\lambda)^*)&={\mathbf t}^{\kappa_{c}((\lambda,0)-\bigrelt{\nu},(-\lambda,0)+\bigrelt{\nu}) +\ell(b)-\phi(\nu,\lambda)+2p(\nu)-\ell(b)-\ang{\nu,\lambda}-\phi(\nu,\lambda-\nu)}(b^-v_\lambda)\otimes(b^-v_\lambda)^*\\ &={\mathbf t}^{\kappa_{c}((\lambda,0),(-\lambda,0))+2p(\nu)-\nu\cdot\nu}(b^-v_\lambda)\otimes(b^-v_\lambda)^* \end{align*} It is easy to verify that $\frac{\nu\cdot\nu}{2}= p(\nu)$ modulo 2 by induction, hence we see that \[ {\mathfrak X}((b^-v_\lambda)\otimes(b^-v_\lambda)^*)={\mathbf t}^{\kappa_{(1,-1)}((\lambda,0),(-\lambda,0) }(b^-v_\lambda)\otimes(b^-v_\lambda)^*. \] A similar computation shows that \[ {\mathfrak X}((b^-v_\lambda)*\otimes(b^-v_\lambda))={\mathbf t}^{\kappa_{(-1,1)}((-\lambda,0),(\lambda,0) }(b^-v_\lambda)^*\otimes(b^-v_\lambda). \] Note that in either case, the power of ${\mathbf t}$ is independent of $b\in {\mathcal{B}}$, and applying this to the definition of the maps proves (1) and (4). For (2) and (3), also note that $\pi^{p(\nu)}q^{\pm\ang{\tilde{\rho},\lambda-\nu}}=\pi_\nu q_{\nu}^{\mp 2} q^{\pm\ang{\tilde{\rho}, \lambda}}$, and we compute that ${\mathfrak X}(\pi_\nu q_\nu^{\mp 2}q^{\pm\ang{\tilde{\rho}, \lambda}})={\mathbf t}^{\mp\ang{\tilde{\rho},\lambda}} \pi_\nu q_\nu^{\mp 2}q^{\pm\ang{\tilde{\rho}, \lambda}}$, the result follows. \end{proof} \begin{example} Consider the case $n=1$ and $\lambda=m$. As noted in Example \ref{ex:rk1evcoev}, $\ang{\tilde{\rho},\lambda}=m$ and $\ev_m\circ\coev_m=\pi^m[m+1]$. Then we have $\ev_m\circ\coev_m\circ{\mathfrak X}(1)=\pi^m[m+1]$, and \[{\mathfrak X}\circ\ev_m\circ\coev_m(1)={\mathfrak X}(\pi^m[m+1])={\mathbf t}^{-m}\pi^m[m+1]={\mathbf t}^{-m}\ev_m\circ\coev_m\circ{\mathfrak X}(1).\] Note that this is consistent with Proposition \ref{prop: cups caps and twistorv1}, as we see that \[\ev_m\circ\coev_m\circ{\mathfrak X}={\mathbf t}^{-\kappa_{(-1,1)}((-\lambda,0),(\lambda,0))+\ang{\tilde\rho,\lambda}}\ev_m\circ{\mathfrak X}\circ\coev_m={\mathbf t}^{m}{\mathfrak X}\circ\ev_m\circ\coev_m.\] \end{example} The last elementary diagram to consider is the crossing, which is to say the automorphism $R=\Theta\mathfrak{F}\mathfrak{s}$ of a tensor product of two modules. In order to have a concrete comparison of $R{\mathfrak X}$ and ${\mathfrak X} R$ on tensor products carrying twistors, it will be necessary to have a precise description of ${\mathfrak X}(\mathfrak{f}(\zeta,\eta))$ for any $\zeta,\eta\in X$. To that end, let us once and for all fix a transversal $T$ of $X/{\mathbb{Z}}[I]$ and note that $\hat T=\set{(\zeta,0),(\zeta,1)\mid \zeta\in T}$ is a transversal of $\hat X/{\mathbb{Z}}[I]$. Then for $\zeta_0,\zeta_1\in T$, we shall henceforth require that \begin{equation}\label{eq:f set} \mathfrak f(|\zeta_0|,|\zeta_1|)=1. \end{equation} Then we have the following proposition. \begin{prop}\label{prop:twistor vs R v1} Let $\lambda,\lambda'\in X^+\cup -X^+$. Let $\hat\zeta,\hat\zeta'\in \hat T$ be the corresponding coset representatives of $(\lambda,0)$ and $(\lambda',0)$ in $\bigrset/{\mathbb{Z}}[I]$ and let $(c_1,c_2)={\rm sig}({\widehat{V}}(\lambda){\widehat{\otimes}} {\widehat{V}}(\lambda'))$. Let ${\mathcal{R}}:{\widehat{V}}(\lambda){\widehat{\otimes}} {\widehat{V}}(\lambda')\rightarrow{\widehat{V}}(\lambda'){\widehat{\otimes}} {\widehat{V}}(\lambda)$ be the map described in Proposition \ref{prop:Rmat}. Then ${\mathcal{R}}$ is a ${\widehat \UU}$-module homomorphism. Moreover, as maps on ${\widehat{V}}(\lambda){\widehat{\otimes}} {\widehat{V}}(\lambda')$, we have \[{\mathfrak X}\mathcal R={\mathbf t}^{\kappa_{(c_2,c_1)}(\hat\zeta',\hat\zeta)-\kappa_{(c_1,c_2)}(\hat\zeta,\hat\zeta')+2p(\hat\zeta)p(\hat\zeta')}\mathcal R{\mathfrak X}.\] \end{prop} \begin{proof} Recall that ${\mathcal{R}}=\Theta\mathfrak{f}\mathfrak{s}$ by definition. It is easy to see that ${\mathcal{R}}$ is a ${\widehat \UU}$-module homomorphism: indeed, since ${\mathcal{R}}$ preserve weight-spaces, it commutes with the action of the $T_i$ for $i\in I$; moreover, $\mathfrak{f}\mathfrak{s}$ obviously commutes with the diagonal action of $\Upsilon_i$, and it is easy to check directly that $\Theta_\nu\Delta(\Upsilon_i)=\Delta(\Upsilon_i)\Theta_\nu$. We will prove the remainder of the proposition in two steps. First we shall show that ${\mathfrak X}(\Theta_\nu)=\Theta_\nu$ for any $\nu\in {\mathbb{Z}}_{\geq 0}[I]$, and thus ${\mathfrak X}\Theta=\Theta{\mathfrak X}$ as maps on $V(\lambda)\otimes V(\lambda')$. This is straightforward: applying Lemmas \ref{lem:tw+alt}, \ref{lem:twondualbasis} and Proposition \ref{prop:twandcoprod} to the expression for $\Theta_\nu$ in terms of the canonical basis ${\mathcal{B}}$, we compute that \begin{align*} {\mathfrak X}(\Theta_\nu)&= (-1)^{{\mathrm{ht}}\,\nu} {\mathbf t}^{2{\mathbf p}(\nu)}\pi^{{\mathbf p}(\nu)}{\mathbf t}_\nu^2\pi_\nu {\mathbf t}_\nu^{-1}q_\nu \sum_{b\in {\mathcal{B}}_\nu} {\mathfrak X}(b^-)\Upsilon_{\nu}\otimes {\tilde{T}}_{-\nu}\Upsilon_{-\nu}{\mathfrak X}((b^*)^+)\\ &= (-1)^{{\mathrm{ht}}\,\nu+{\mathbf p}(\nu)}\pi^{{\mathbf p}(\nu)}{\mathbf t}_\nu\pi_\nu q_\nu \sum_{b\in {\mathcal{B}}_\nu} ({\mathfrak X}(b)^-\Upsilon_{-\nu})\Upsilon_{\nu}\otimes {\tilde{T}}_{-\nu}\Upsilon_{-\nu}({\mathbf t}_\nu^{-1} {\tilde{T}}_{\nu}\Upsilon_{\nu}{\mathfrak X}(b^*)^+)\\ &= (-1)^{{\mathrm{ht}}\,\nu+{\mathbf p}(\nu)}\pi^{{\mathbf p}(\nu)}{\mathbf t}_\nu\pi_\nu q_\nu \sum_{b\in {\mathcal{B}}_\nu} ({\mathbf t}^{\ell(b)}b^-)\otimes ({\mathbf t}_\nu^{-1} {\mathbf t}^{-\ell(b)}(-1)^{{\mathbf p}(\nu)}(b^*)^+)\\ &= (-1)^{{\mathrm{ht}}\,\nu}\pi^{{\mathbf p}(\nu)}\pi_\nu q_\nu \sum_{b\in {\mathcal{B}}_\nu} b^-\otimes (b^*)^+=\Theta_\nu \end{align*} Now it remains to show that we have ${\mathfrak X}\mathfrak F\mathfrak{s}={\mathbf t}^{\kappa_{(c_2,c_1)}(\hat\zeta',\hat\zeta)-\kappa_{(c_1,c_2)}(\hat\zeta,\hat\zeta')+2p(\hat\zeta)p(\hat\zeta')}\mathfrak F\mathfrak{s}{\mathfrak X}$ as maps on $V(\lambda)\otimes V(\lambda')$. Set $c=(c_1,c_2)$, and $\tilde c=(c_2,c_1)$. Let $m\in V(\lambda)$ and $n\in V(\lambda')$. Then we see directly that \[{\mathfrak X}\mathfrak F\mathfrak{s}(m\otimes n)=t^{2p(m)p(n)+\kappa_{\tilde c}(\bigrdeg{n},\bigrdeg{m})}{\mathfrak X}(\mathfrak{f}(|n|,|m|))\pi^{p(m)p(n)}{\mathfrak X}(n)\otimes {\mathfrak X}(m),\] \[\mathfrak F\mathfrak{s}{\mathfrak X}(m\otimes n)=t^{\kappa_c(\bigrdeg{m},\bigrdeg{n})}\mathfrak{f}(|n|,|m|)\pi^{p(m)p(n)}{\mathfrak X}(n)\otimes {\mathfrak X}(m).\] The proposition then follows by verifying that \[t^{2p(m)p(n)+\kappa_{\tilde c}(\bigrdeg{n},\bigrdeg{m})}{\mathfrak X}(\mathfrak{f}(|n|,|m|))=t^{\kappa_{\tilde c}(\hat\zeta',\hat\zeta)-\kappa_{c}(\hat\zeta,\hat\zeta')+2p(\hat\zeta)p(\hat\zeta')+\kappa_c(\bigrdeg{m},\bigrdeg{n})}\mathfrak{f}(|n|,|m|),\] Note that $\hat\zeta=\bigrdeg{m}+\mu$ and $\hat\zeta'=\bigrdeg{n}+\nu$ for some $\mu,\nu\in{\mathbb{Z}}[I]$. Let $\zeta=|\hat\zeta|\in X$ and $\zeta'=|\hat\zeta'|\in X$. Then in particular, \eqref{eq:f set} implies \[\mathfrak{f}(|n|,|m|)=(\pi q)^{\ang{\tilde \nu,\zeta}}q^{\ang{\tilde \mu,\zeta'}-\mu\cdot\nu},\] so ${\mathfrak X}(\mathfrak{f}(|n|,|m|))={\mathbf t}^{\ang{\tilde{\nu},\zeta}-\ang{\tilde\mu,\zeta'}+\mu\cdot\nu}\mathfrak{f}(|n|,|m|)$. Therefore, we are reduced to showing that $\ell\equiv r$ modulo $4$, where \[\ell=2p(m)p(n) +\ang{\tilde\nu,\zeta}-\ang{\tilde{\mu},\zeta'}+\mu\cdot\nu+\kappa_{\tilde c}(\bigrdeg{n},\bigrdeg{m}),\] \[r= 2p(\hat\zeta)p(\hat\zeta')+\kappa_{\tilde{c}}(\hat\zeta',\hat\zeta)-\kappa_{c}(\hat\zeta,\hat\zeta')+\kappa_c(\bigrdeg{m},\bigrdeg{n}) \mod 4\] We compute directly that \begin{align*} \kappa_{\tilde c}(\bigrdeg{n},\bigrdeg{m})&-\kappa_{\tilde{c}}(\hat\zeta',\hat\zeta)+\kappa_{c}(\hat\zeta,\hat\zeta')-\kappa_c(\bigrdeg{m},\bigrdeg{n})\\ &=\kappa_{\tilde c}(\hat\zeta'-\nu,\hat\zeta-\mu)-\kappa_{\tilde{c}}(\hat\zeta',\hat\zeta)+\kappa_{c}(\hat\zeta,\hat\zeta')-\kappa_c(\hat\zeta-\mu,\hat\zeta'-\nu)\\ &\equiv_4-\ang{\tilde\nu,\zeta}-c_1\phi(\nu,c_1\zeta)+2p(\hat\zeta)p(\mu)-c_2\phi(\mu,c_2\zeta) +\mu\cdot\nu+\phi(\mu,\nu)\\ &\hspace{2em}+\ang{\tilde\mu,\zeta}+c_1\phi(\mu,c_1\zeta)-2p(\hat\lambda)p(\nu)+\phi(\nu,\lambda) -\mu\cdot\nu-\phi(\nu,\mu)\\ &\equiv_4 2p(\hat\lambda)p(\nu)+2p(\hat\zeta)p(\mu)+2p(\mu)p(\nu)-\ang{\tilde\nu,\lambda}+\ang{\tilde\mu,\zeta} +\mu\cdot\nu\\ &\equiv_4 2p(m)p(n)-2 p(\hat\lambda)p(\hat\zeta)-\ang{\tilde\nu,\lambda}+\ang{\tilde\mu,\zeta} +\mu\cdot\nu, \end{align*} where here $\equiv_4$ denotes equivalence modulo 4. This finishes the proof. \end{proof} We have seen that the twistor map commutes (up to an integral power of ${\mathbf t}$) with the elementary functions in our graphical calculus. However, note that in Theorem \ref{thm:knot invariant}, the typical composand of a tangle invariant is not just one of these maps, but in fact is a tensor product of these maps with various identities. It is important to note that a consequence of Proposition \ref{prop: twistor tensor} is that the twistor maps on tensor products are not local, since the power of ${\mathbf t}$ in the construction depends on the weight and signature of each tensor factor. Nevetheless, we can extend Propositions \ref{prop: cups caps and twistorv1} and \ref{prop:twistor vs R v1} to this more general setting. \begin{prop}\label{prop:twistor vs cups, caps, R} Let $M_1,\ldots, M_t$ be ${\widehat \UU}$-modules such that for each $1\leq s\leq t$, $M_s={\widehat{V}}(\mu_s)$ for some $\mu_s\in X^+\cup-X^+$. Let $M=M_1{\widehat{\otimes}}\ldots {\widehat{\otimes}} M_t$ and let $c=(c_1,\ldots, c_t)={\rm sig}(M)$. For any $\lambda\in X^+$ and $0\leq r\leq t$, we define $M_{\leq r}=M_1{\widehat{\otimes}}\ldots{\widehat{\otimes}} M_r$, $M_{>r}=M_{r+1}{\widehat{\otimes}} \ldots {\widehat{\otimes}} M_t$, and \[M(r,\pm \lambda)=M_{\leq r}{\widehat{\otimes}} {\widehat{V}}(\pm \lambda){\widehat{\otimes}} {\widehat{V}}(\mp \lambda){\widehat{\otimes}} M_{>r}.\] \begin{enumerate} \item Let ${\mathcal{R}}_s=1_{M_{\leq s-1}}\otimes {\mathcal{R}}\otimes 1_{M_{>s+1}}:M\rightarrow M$ for some $1\leq s\leq t-1$. Then as maps on $M$, ${\mathfrak X} R_s$ and $R_s{\mathfrak X}$ are proportional up to an integral power of ${\mathbf t}$. \item Let $\ev(M,r,\lambda)=1_{M_{\leq r}}\otimes \ev_\lambda\otimes 1_{M_{>r}}$ for some $1\leq r\leq t$. Then as maps on $M(r,-\lambda)$, ${\mathfrak X}\ev(M,r,\lambda)$ and $\ev(M,r,\lambda){\mathfrak X}$ are proportional up to an integral power of ${\mathbf t}$. \item Let $\qtr(M,r,\lambda)=1_{M_{\leq r}}\otimes \qtr_\lambda\otimes 1_{M_{>r}}$ for some $1\leq r\leq t$. Then as maps on $M(r,\lambda)$, ${\mathfrak X}\qtr(M,r,\lambda)$ and $\qtr(M,r,\lambda){\mathfrak X}$ are proportional up to an integral power of ${\mathbf t}$. \item Let $\coev(M,r,\lambda)=1_{M_{\leq r}}\otimes \coev_\lambda\otimes 1_{M_{>r}}$ for some $1\leq r\leq t$. Then as maps on $M$, ${\mathfrak X}\coev(M,r,\lambda)$ and $\coev(M,r,\lambda){\mathfrak X}$ are proportional up to an integral power of ${\mathbf t}$. \item Let $\coqtr(M,r,\lambda)=1_{M_{\leq r}}\otimes \coqtr_\lambda\otimes 1_{M_{>r}}$ for some $1\leq r\leq t$. Then as maps on $M$, ${\mathfrak X}\coqtr(M,r,\lambda)$ and $\coqtr(M,r,\lambda){\mathfrak X}$ are proportional up to an integral power of ${\mathbf t}$. \end{enumerate} \end{prop} \begin{rmk} The precise constants of proportionality can be determined directly as in Propositions \ref{prop: cups caps and twistorv1} and \ref{prop:twistor vs R v1} (and can be worked out from the following proof), but we leave them out of the statement of Proposition \ref{prop:twistor vs cups, caps, R} because they are not particularly illuminating, and are not necessary for Theorem \ref{thm:twistor vs knot invariant} \end{rmk} \begin{proof} As the proofs of (2)-(5) are similar, we shall only prove (1) and (2) here in detail. We will begin with the proof of (1), which is essentially the same as the proof of Proposition \ref{prop:twistor vs R v1}. To wit, we first observe that for any $a,b\geq 0$ and $\nu\in{\mathbb{N}}[I]$, \[{\mathfrak X}(1^{\otimes a}\otimes \Theta_\nu\otimes 1^{\otimes b}) =(\Upsilon_{|\Theta_\nu|})^{\otimes{a}} \otimes {\mathfrak X}(\Theta_\nu)\otimes(\Upsilon_{|\Theta_\nu|}{\tilde{T}}_{|\Theta_\nu|})^{\otimes b} =,\] and the result follows from the observation that $|\Theta_\nu|=\nu-\nu=0$. Then ${\mathfrak X} R_s=(1^{\otimes s-1}\otimes \Theta\otimes 1^{t-s-1}){\mathfrak X} \mathfrak{F}_s\mathfrak{s}_s$. Then we verify directly that ${\mathfrak X}\mathfrak{F}_s\mathfrak{s}_s={\mathbf t}^{\kappa_{(c_{s+1},c_s)}(\hat\zeta',\hat\zeta)-\kappa_{(c_s,c_{s+1})}(\hat\zeta,\hat\zeta')+2p(\hat\zeta)p(\hat\zeta')}\mathfrak{F}_s\mathfrak{s}_s{\mathfrak X}$ where $\hat\zeta$ (resp. $\hat\zeta'$) is the coset representative for $(\mu_s,0)$ (resp. $(\mu_{s+1},0)$). Now, we shall prove (2). Note that an arbitrary element of $M(r,-\lambda)$ is a linear combination of simple tensors of the form $x=m_{\leq r}\otimes (b^-v_\lambda)^*\otimes (b'^-v_\lambda)\otimes m_{>r}$, where $b,b'\in{\mathcal{B}}$, $m_{\leq r}=m_1\otimes\ldots\otimes m_r \in M_{\leq r}$ and $m_{>r}=m_{r+1}\otimes\ldots\otimes m_t\in M_{>r}$, hence we need only prove (1) holds when evaluating both sides at such elements. Since $\ev_\lambda((b^-v_\lambda)^*\otimes (b'^-v_\lambda))=\delta_{b,b'}$, note that (1) is trivially true when $b\neq b'$, so let's assume $b=b'\in{\mathcal{B}}_\nu$. Then \[\ev(M,r,\lambda){\mathfrak X}(x)={\mathbf t}^{\diamondsuit(m_1,\ldots, m_t)+\clubsuit}{\mathfrak X}(m_{\leq r})\otimes \ev_\lambda{\mathfrak X}((b^-v_\lambda)^*\otimes (b^-v_\lambda))\otimes{\mathfrak X}(m_{> r})\] where we set \[\diamondsuit(m_1,\ldots, m_t)=\sum_{s<r<s'}\kappa_{(c_s,c_s')}(\bigrdeg{m_s},\bigrdeg{m_s'})\] \begin{align*} \clubsuit=&\sum_{s<r}\parens{\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0)+\bigrelt{\nu})+\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0)-\bigrelt{\nu})} \\&+\sum_{s>r}\parens{\kappa_{(-1,c_s)}((-\lambda,0)+\bigrelt{\nu},\bigrdeg{m_s})+\kappa_{(1,c_s)}((\lambda,0)-\bigrelt{\nu},\bigrdeg{m_s})} \end{align*} Now $\clubsuit$ can be simplified. Note that \[\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0)+\nu) =\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0))+2c_s\phi(\nu,|m_s|)+2p(\nu)p(m_s)\] \[\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0)-\nu) =\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0))-2c_s\phi(\nu,|m_s|)+2p(\nu)p(m_s)\] hence $\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0)+\nu)+\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0)-\nu)=\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0))+\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0))$. Moreover, note that $\bigrdeg{m_s}=(\mu_s,0)+\nu_s$ for some $\nu_s\in{\mathbb{Z}}[I]$, and so \[\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0)) =\kappa_{(c_s,-1)}((\mu_s,0),(-\lambda,0))-\ang{\tilde\nu_s,\lambda}-2\phi(\nu_s,\lambda),\] \[\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0)) =\kappa_{(c_s,1)}((\mu_s,0),(\lambda,0))+\ang{\tilde\nu_s,\lambda}+2\phi(\nu_s,\lambda),\] hence \[\kappa_{(c_s,-1)}(\bigrdeg{m_s},(-\lambda,0))+\kappa_{(c_s,1)}(\bigrdeg{m_s},(\lambda,0))=\kappa_{(c_s,-1)}((\mu_s,0),(-\lambda,0))+\kappa_{(c_s,1)}((\mu_s,0),(\lambda,0))\] Similar applies to the sum over $s>r$ in $\clubsuit$, hence \begin{align*} \clubsuit=&\sum_{s<r}\parens{\kappa_{(c_s,-1)}((\mu_s,0),(-\lambda,0))+\kappa_{(c_s,1)}((\mu_s,0),(\lambda,0))} \\&+\sum_{s>r}\parens{\kappa_{(-1,c_s)}((-\lambda,0),(\mu_s,0))+\kappa_{(1,c_s)}((\lambda,0),(\mu_s,0))} \end{align*} Note that $\clubsuit$ is independent of $x$. Then \begin{align*} \ev(M,r,\lambda){\mathfrak X}(x)&={\mathbf t}^{\diamondsuit(m_1,\ldots, m_t)+\clubsuit}{\mathfrak X}(m_{\leq r})\otimes \ev_\lambda{\mathfrak X}((b^-v_\lambda)^*\otimes (b^-v_\lambda))\otimes{\mathfrak X}(m_{> r})\\ &={\mathbf t}^{\diamondsuit(m_1,\ldots, m_t)+\clubsuit+\kappa_{(-1,1)}((-\lambda,0),(\lambda,0))}{\mathfrak X}(m_{\leq r})\otimes{\mathfrak X}(m_{> r})\\ &={\mathbf t}^{\clubsuit+\kappa_{(-1,1)}((-\lambda,0),(\lambda,0))}{\mathfrak X}(m_{\leq r}\otimes m_{> r}) \end{align*} Since ${\mathfrak X}(m_{\leq r}\otimes m_{> r})={\mathfrak X}(\ev_\lambda(x))$ and the exponent of ${\mathbf t}$ is independent of $x$, this completes the proof of (2). \end{proof} We now arrive at the final result of this paper. \begin{thm} \label{thm:twistor vs knot invariant} Let $K$ be any oriented knot, and let $J_K^\lambda(q,\tau)\in{\Qqt^\tau}$ be the $\lambda$-colored knot invariant defined in Theorem \ref{thm:knot invariant}. Let ${}_{{\mathfrak{so}}}J_K^\lambda(q)=J_K^\lambda(q,1)$ and ${}_{{\mathfrak{osp}}}J_K^\lambda(q)=J_K^\lambda(q,{\mathbf t})$. Then \[{}_{{\mathfrak{osp}}}J_K^\lambda(q)={\mathbf t}^{\star(K,\lambda)}{}_{{\mathfrak{so}}}J_K^\lambda({\mathbf t}^{-1} q),\] for some $\star(K,\lambda)\in{\mathbb{Z}}$. \end{thm} \begin{proof} Let $J=J_K^\lambda(q,\tau)$. First, observe that $J$ can be thought of as a function ${\Qqt^\tau}\rightarrow {\Qqt^\tau}$, and in that spirit ${\mathfrak X}(J)$ is ${\mathfrak X}\circ J(1)$. On the other hand, $J=W_K\circ S$, where $W_K=(\mathfrak{f}(\lambda,\lambda)^{-1}\pi^{P(\lambda)}q^{\ang{\rho,\lambda}})^{\wr(K)}$ (interpreted as a function ${\Qqt^\tau}\rightarrow {\Qqt^\tau}$) and $S$ is a slice diagram of $K$ interpreted as a composition of morphisms as described in Section \ref{sec:diagcalc} (with strands colored by $\lambda$). In particular, observe that by \eqref{eq:f set} we have ${\mathfrak X}(\mathfrak{f}(\lambda,\lambda))={\mathbf t}^{x}\mathfrak{f}(\lambda,\lambda)$ for some $x\in {\mathbb{Z}}$ depending on the coset representative of $\lambda$ in $X/{\mathbb{Z}}[I]$, and that ${\mathfrak X}(\pi^{P(\lambda)}q^{\ang{\rho,\lambda}})=\pi^{P(\lambda)}q^{\ang{\rho,\lambda})}$. Then in particular we see that ${\mathfrak X} W={\mathbf t}^{-x\wr(K)} W{\mathfrak X}$. Likewise, note that $S$ can be written as a composition of maps of the form $\ev(M,r,\lambda)$, $\coev(M,r,\lambda)$, $\qtr(M,r,\lambda)$, $\coqtr(M,r,\lambda)$, and $R_s:M\rightarrow M$ for various $r,s\in{\mathbb{N}}$ with all notations being the same as in Proposition \ref{prop:twistor vs cups, caps, R}. In particular, we see that ${\mathfrak X}\circ S={\mathbf t}^{y}S\circ {\mathfrak X}$ for some $y\in {\mathbb{Z}}$, and thus \[{\mathfrak X}(J)={\mathfrak X}\circ c\circ S(1)={\mathbf t}^{-x\wr(K)+y}c\circ S\circ {\mathfrak X}(1)={\mathbf t}^{-x\wr(K)+y}J.\] On the other hand, observe that ${\mathfrak X}(J_K^\lambda(q,\tau))=J_K^\lambda({\mathbf t}^{-1}q,{\mathbf t}^{-1}\tau)$, and so \[{\mathbf t}^{y-x\wr(K)}J_K^\lambda({\mathbf t}^{-1}q,{\mathbf t}^{-1}\tau)=J_K^\lambda(q,\tau).\] The theorem follows from specializing $\tau={\mathbf t}$. \end{proof} \begin{rmk} Note that since $_{{\mathfrak{so}}}J^\lambda_K(q)\in{\mathbb{Z}}[q,q^{-1}]$, Theorem \ref{thm:twistor vs knot invariant} implies that (after a renormalization) $_{{\mathfrak{osp}}}J^\lambda_K(q)={}_{{\mathfrak{so}}}J^\lambda_K(v)\in{\mathbb{Z}}[v,v^{-1}]$ where $v=q{\mathbf t}^{-1}$. Furthermore, note that when $n$ or $\ang{{\rf n},\lambda}$ is even, $_{{\mathfrak{osp}}}J^\lambda_K(q)\in {\Q(q)}$ (cf. Remark \ref{rmk:red diagrams} (1)), thus in this case $_{{\mathfrak{osp}}}J^\lambda_K(q)\equiv {}_{{\mathfrak{so}}}J^\lambda_K(q)$ modulo $2$. \end{rmk}
{ "redpajama_set_name": "RedPajamaArXiv" }
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\section{Introduction}\label{s:intro The classical simple exclusion process is a Markov process that describes nearest-neighbor random walks of a collection of particles on the one-dimensional infinite\footnote{or finite with periodic boundary conditions} % integer lattice. Particles interact through the hard core exclusion rule, which means that at most one particle is allowed at each site. This seemingly very particular process introduced first in 1970 by Frank Spitzer \cite{Sp} appears naturally in a very broad list of scientific fields starting from various models of traffic flows \cite{NS,GG,ERS,Bl-erg,Bl-hys}, molecular motors and protein synthesis in biology(see e.g. \cite{SZL}), surface growth or percolation processes in physics (see \cite{Pe,BFS} for a review), and up to the analysis of Young diagrams in Representation Theory \cite{CMOS}. Qualitatively from the point of view of the order of particle interactions there are two principally different types of exclusion processes: with synchronous and asynchronous updating rules. In the latter case at each moment of time a.s. at most one particle may move and hence only a single interaction may take place. This is the main model considered in the mathematical literature (see e.g. \cite{Lig,Thor,Sp,Num} for a general account and \cite{An,EFM,Ros} for recent results), and indeed, the assumption about the asynchronous updating is quite natural in the continuous time setting. The synchronous updating means that {\em all} particles are trying to move simultaneously and hence an arbitrary large (and even infinite) number of interactions may occur at the same time. This makes the analysis of the synchronous updating case much more difficult, but this is what happens in the discrete time case.\footnote{if one do not consider some ``artificial'' updating rules like a sequential or random updating.} % This case is much less studied, but still there are a few results describing ergodic properties of such processes \cite{BF,Bl-erg,Bl-hys,BP,ERS,GG,NS}. Our aim is to introduce and study the synchronous updating version of the exclusion process in continuum. Note that recently some other interacting particle processes were generalized from lattice to continuum case (see e.g. \cite{Pe,FKO}). A configuration $x:=\{x_i\}_{i\in\IZ}$ is a bi-infinite sequence of real numbers $x_i\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$ interpreted as centers of particles represented by balls of radius $r\ge0$ (see Fig.~\ref{f:tasep-c}) and ordered with respect to their positions (i.e. $\dots\le x_{-1} \le x_0 \le x_1 \le \dots$). To emphasize the dependence on the radius $r\ge0$ we shall use the notation $x(r)$. We say that a configuration $x(r)$ is {\em admissible} if $$x_i(r)+r\le x_{i+1}(r)-r~~\forall i\in\IZ$$ (the corresponding balls may only touch each other) and denote by $X(r)$ the space of admissible configurations. \def\particle#1{ \put(0,0){\circle{30}} \bline(0,0)(0,-1)(25) \put(-3,-32){$x_{#1}$} \bline(0,20)(1,0)(15) \put(5,22){$r$} \put(0,5){\vector(1,0){40}} \put(20,9){$v_{#1}$}} \Bfig(150,30) {\footnotesize{ \thicklines \bline(0,10)(1,0)(150) \put(30,10){\particle{i}} \put(110,10){\particle{i+1}} \thinlines \bline(46,-18)(1,0)(48) \put(65,-13){$\Delta_i$} } } {TASEP in continuum. \label{f:tasep-c}} The dynamics will be defined as follows. We assume given a collection of (possibly random) values $\{v_i^t\}_{i,t}$, where $i,t\in\IZ$ and $t\ge0$; conditions on this collection will be given shortly. For a trivial configuration consisting of a single particle located at time $t\ge0$ at $x_0^t\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$ (i.e. $x^t\equiv\{x_0^t\}$) the dynamics is defined as % $$x_0^{t+1}:=x_0^t+v_0^t,$$ and thus $v_0^t$ may be interpreted as a local velocity at time $t$, i.e. this is simply a random walk on ${\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$. To generalize this trivial setting for an infinite configuration $x(r)\in X(r)$ we again interpret a (be-infinite on $i\in\IZ$) sequence $\{v_i^t\}_{i,t}$ as {\em local velocities} for particles in $x^t(r)$ performing random walks conditioned to the order preservation and the hard core exclusion rule. To simplify presentation we restrict ourselves here to the case of nonnegative local velocities postponing the discussion of the general case when the local velocities take both positive and negative signs to Section~\ref{s:vel-2signs}. The point is that the formulations in the latter case are becoming much more involved, but the results and arguments work with only very slight changes. Since only nonnegative local velocities are considered the hard core exclusion rule means that the admissibility condition breaks down for the $i$-th particle at time $t\in\IZ_+$ if and only if the inequality % $$x^t_i(r)+v_i^t + r \le x^t_{i+1}(r) - r$$ % does not hold. If this happens we say that there is a {\em conflict} between the particles $i$ and $i+1$, and to resolve it one applies a {\em normalizing} construction $$v_i^t\to{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t(r)).$$ After the normalization the positions of particles are calculated according to the rule $$x_i^{t+1}(r):=x_i^t(r)+{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t(r))~~\forall i.$$ In what follows we always assume\footnote{This formulation allows to consider velocities of both signs which we shall do in Section~\ref{s:vel-2signs} and simply means that the normalized velocity has the same direction as the original one and cannot exceed it on modulus.} % that $\forall i,t~~ {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t(r))\in[0,v_i^t]$ (to simplify notation by the segment $[a,b]$ we mean $[\min(a,b),\max(a,b)]$) % and consider only {\em nonanticipating} normalizations\footnote{In Section~\ref{s:gen} we shall show that the violation of this condition makes the system to be not well posed.} % satisfying the condition that in the case of the conflict of the $i$-th particle with the $j$-th one\footnote{For nonnegative velocities $j\equiv i+1$, but in general $j\in\{i-1,i+1\}$.} % at time $t$ the position of the $i$-th particle at the next moment of time $x_i^{t+1}(r)\in[x_i^t(r),x_j^t(r)]$. The normalization may be done in a number of ways and we restrict ourselves to two extreme constructions. The first of them we call {\em strong normalization} (notation $\cNs(\cdot,\cdot)$) and according to the name we reject (nullify) the velocity leading to the conflict. The second construction we call {\em weak normalization} (notation $\cNw(\cdot,\cdot)$) and in this case we modify the conflicting velocity to allow the particle to move as far as possible. In terms of {\em gaps}\/ % $$\Delta} % {G_i(x^t(r))\equiv\Delta} % {G_i^t:=x_{i+1}^t(r)-x_{i}^t(r)-2r$$ between particles in the configuration $x^t$ the normalization procedures are written as follows: % $$ \cNs(v_i^t,x^t(r)):=\function{v_i^t &\mbox{if } v_i^t\le \Delta} % {G_i^t \\ 0 &\mbox{otherwise }}, \qquad \cNw(v_i^t,x^t(r)):=\function{v_i^t &\mbox{if } v_i^t\le \Delta} % {G_i^t \\ \Delta} % {G_i^t &\mbox{otherwise }} .$$ Fig.~\ref{f:normalization} demonstrates possible positions of particles at two consecutive moments of time $t$ and $t+1$ for the cases of weak (a-c) and strong (a'-c') normalizations. Despite appearances these two normalization procedures lead to a very different limit behavior of the corresponding particle systems. The simplest example (existing even in the continuous time case) is the situation when $v_i^t\equiv{v}~\forall i,t$ and the gaps between particles in $x$ are smaller than $v$. Then under the strong normalization no motion is allowed, while the weak normalization leads to the well defined motion -- the exchange of gaps between particles. Other normalization procedures together with more general assumptions about the dynamics will be discussed in Section~\ref{s:gen}. \Bfig(290,120) {\footnotesize{ \put(0,100){\bpic(100,30){\put(-20,5){(a)} \bline(0,0)(1,0)(100) \bline(0,10)(1,0)(100) \put(10,10){\circle{5}} \put(70,10){\circle{5}} \put(10,15){\vector(1,0){40}} \put(70,15){\vector(1,0){20}} \put(50,0){\circle{5}} \put(90,0){\circle{5}} \put(105,7){$t$} \put(105,-3){$t+1$} \put(10,18){$i$} \put(60,18){$i+1$} \put(47,-12){$i$} \put(80,-12){$i+1$} }} \put(0,50){\bpic(100,30){\put(-20,5){(b)} \bline(0,0)(1,0)(100) \bline(0,10)(1,0)(100) \put(10,10){\circle{5}} \put(70,10){\circle{5}} \put(10,15){\vector(1,0){70}} \put(70,13){\vector(1,0){20}} \put(65,0){\circle{5}} \put(90,0){\circle{5}} \put(105,7){$t$} \put(105,-3){$t+1$} \put(10,18){$i$} \put(60,18){$i+1$} \put(63,-12){$i$} \put(80,-12){$i+1$} }} \put(0,0){\bpic(100,30){\put(-20,5){(c)} \bline(0,0)(1,0)(100) \bline(0,10)(1,0)(100) \put(10,10){\circle{5}} \put(70,10){\circle{5}} \put(10,15){\vector(1,0){50}} \put(70,13){\vector(-1,0){35}} \put(45,0){\circle{5}} \put(50,0){\circle{5}} \put(105,7){$t$} \put(105,-3){$t+1$} \put(10,18){$i$} \put(60,18){$i+1$} }} \put(170,100){\bpic(100,30){\put(-20,5){(a')} \bline(0,0)(1,0)(100) \bline(0,10)(1,0)(100) \put(10,10){\circle{5}} \put(70,10){\circle{5}} \put(10,15){\vector(1,0){40}} \put(70,15){\vector(1,0){20}} \put(50,0){\circle{5}} \put(90,0){\circle{5}} \put(105,7){$t$} \put(105,-3){$t+1$} \put(10,18){$i$} \put(60,18){$i+1$} \put(47,-12){$i$} \put(80,-12){$i+1$} }} \put(170,50){\bpic(100,30){\put(-20,5){(b')} \bline(0,0)(1,0)(100) \bline(0,10)(1,0)(100) \put(10,10){\circle{5}} \put(70,10){\circle{5}} \put(10,15){\vector(1,0){70}} \put(70,13){\vector(1,0){20}} \put(10,0){\circle{5}} \put(90,0){\circle{5}} \put(105,7){$t$} \put(105,-3){$t+1$} \put(10,18){$i$} \put(60,18){$i+1$} \put(10,-12){$i$} \put(80,-12){$i+1$} }} \put(170,0){\bpic(100,30){\put(-20,5){(c')} \bline(0,0)(1,0)(100) \bline(0,10)(1,0)(100) \put(10,10){\circle{5}} \put(70,10){\circle{5}} \put(10,15){\vector(1,0){50}} \put(70,13){\vector(-1,0){35}} \put(10,0){\circle{5}} \put(70,0){\circle{5}} \put(105,7){$t$} \put(105,-3){$t+1$} \put(10,18){$i$} \put(60,18){$i+1$} \put(10,-12){$i$} \put(60,-12){$i+1$} }} }} {Positions of particles at time $t,t+1$ in cases of weak (a-c) and strong (a'-c') normalizations. Local particle velocities are shown by vectors. The cases (c,c') correspond to negative velocities and will be discussed in Section~\ref{s:vel-2signs}. \label{f:normalization}} Observe that any two particle configurations $x(r),~\2x(\2r)$ having the same sequence of gaps $\Delta} % {G:=\{\Delta} % {G_i\}$ may be transformed to each other by a one-to-one map % \beq{e:r->r'}{\2x_i(\2r)=\phi(x_i(r)):=x_i(r)-2i(r-\2r)~~\forall i\in\IZ.} % Since the normalization procedures that we consider depend only on the gaps between particles it is enough to study the case $r=0$. On the other hand, if $r=1/2, ~x_i^0(r)\in\IZ~\forall i\in\IZ$ and $v_i^t\in\IZ~\forall i\in\IZ, t\ge0$ then $x_i^t(r)\in\IZ~\forall i\in\IZ, t\ge0$ which means that we get a lattice particle system. Thus our results lead to a completely new approach to the analysis of lattice systems as well. Note however that in the case $r=0$ an arbitrary number of particles may share the same spatial position which is prohibited in the lattice case. Due to the observation above we shall study in detail only the case $r=0$ since the corresponding results for any $r>0$ are readily available through the transformation (\ref{e:r->r'}), see e.g. specific calculations for densities and velocities in Lemmas \ref{l:den-r}, \ref{l:V-r} and Corollaries~\ref{c:V-r-w}, \ref{c:V-r-s}. To simplify notation we shall use the convention $x(r)\equiv x^0(r), ~x\equiv x^0(0)$ and similarly $X\equiv X(0)$. Of course, without some specific assumptions on the structure of local velocities $\{v_i^t\}_{i,t}$ no interesting results are possible. We assume that % $v_i^t\in[0,v]~~\forall i\in\IZ,~t\in\IZ_0:=\IZ_+\cup\{0\}$ and one of the following seemingly opposite assumptions holds: % \begin{itemize} \item[(a)] $v_i^t\equiv v_0^t~~\forall i\in\IZ,~t\in\IZ_0$ and $\exists~\bar{v}(\gamma):=\lim\limits_{t\to\infty}\frac1t \sum\limits_{s=0}^{t-1}\min(v_0^s,\gamma) ~~\forall\gamma>0$ ~~(a.s.); % \item[(b)] $\{v_i^t\}$ are i.i.d. (both in $i$ and $t$) random variables. \end{itemize} Note that the intersection between the sets of local velocities satisfying the assumptions (a) and (b) contains an important case of pure deterministic velocities: $v_i^t\equiv v~~\forall i\in\IZ,~t\in\IZ_0$. As we shall show properties of systems with local velocities satisfying to the assumption (a) are close to the pure deterministic setting. Therefore we refer to the setting (a) as {\em deterministic}\footnote{In this case $v_0^t$ might be a trajectory of a deterministic chaotic map $f:[0,1]\to[0,1]$, e.g. $v_0^{t+1}:=vf^t(v_0^t/v)$, as well as a realization of a true random Markov chain).} % and to the setting (b) as {\em random}. It is of interest that in the seemingly simplest purely deterministic setting $v_i^t\equiv v~~\forall i\in\IZ,~t\in\IZ_0$ the behavior of the corresponding deterministic dynamical system describing the dynamics of particle configurations is far from being trivial. In Section~\ref{s:entropy} we prove that this system is chaotic in the sense that its topological entropy is positive (and even infinite). To emphasize that under dynamics no creation or annihilation of particles may take place this sort of systems is called {\em diffusive driven systems} (DDS) instead of a more general object -- {\em interacting particle systems} (IPS). The main technical tool in our analysis is a (somewhat unusual) ``dynamical'' coupling construction. Despite that various couplings are widely used in the analysis of IPS, applications of our approach is very different from conventional. In particular, we do not prove the existence of the so called successful coupling (which even might not hold) but instead use its presence/absence as an important diagnostic tool. Remark also that typically one uses the coupling argument to prove the uniqueness of the invariant measure and to derive later other results from this fact. In our case there might be a very large number of ergodic invariant measures or no invariant measures at all (recall the trivial example of a single particle performing a skewed random walk). The latter example indicates that there is another important statistical quantity -- average particles velocity that can be computed at least in this case. (See e.g. \cite{BBM} for a discussion of the average velocity in the context of Queueing Networks.) The dynamical coupling will be used directly to find connections between the average particle velocities and other statistical features of the systems under consideration, in particular with the corresponding particle densities. It is worth note that all approaches used to study lattice versions of DDS are heavily based on the combinatorial structure of particle configurations. This structure has no counterparts in the continuum setting under consideration. In particular the particle -- vacancy symmetry is no longer applicable in our case. This explains the need to develop a fundamentally new techniques for the analysis of DDS in continuum. Despite this new techniques cannot be applied directly in the lattice case, the embedding of lattice systems to the continuum setting allows to obtain (indirectly) new results for the lattice systems as well. The paper is organized as follows. In Section~\ref{s:metric} we introduce main statistical quantities under study: particle densities, average velocities, etc. and derive their basic properties. Section~\ref{s:pairing} is dedicated to the main technical tool -- dynamical coupling. In Section~\ref{s:weak} we apply this coupling in the weak normalization setting to prove the uniqueness of the average velocity (Theorem~\ref{t:velocity-density}) and to derive the complete Fundamental Diagram for the deterministic case (Theorem~\ref{t:fund-d-weak}). We calculate also the topological entropy of this process (Theorem~\ref{t:entropy-exclusion}). The strong normalization case is considered in Section~\ref{s:strong} (Theorem~\ref{t:fund-d-strong}), while a more general setting with local velocities of both signs is studied in Section~\ref{s:vel-2signs} (Theorem~\ref{t:velocity-density-2signs}). Finally, in Section~\ref{s:gen} we discuss some generalizations of our results and applications to certain specific traffic models. \smallskip {\bf Acknowledgements.} The author is grateful to B.~Gurevich, S.~Pirogov and an anonymous referee for a number of valuable remarks. This research has been partially supported by Russian Foundation for Fundamental Research, Program ONIT, and French Ministry of Education grants. \section{Basic properties of DDS}\label{s:metric} Here we shall study questions related to densities and velocities of DDS. To simplify notation we use the convention that the normalization ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p\in\{\cNs,\cNw\}$ and specify it only if this is necessary. By the {\em density} $\den(x,I)$ of a configuration $x\in X$ in a bounded segment $I=[a,b]\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$ we mean the number of particles from $x$ whose centers $x_i$ belong to $I$ divided by the Lebesgue measure $|I|>0$ of the segment $I$. If for any sequence of {\em nested} bounded segments $\{I_n\}$ with $|I_n|\toas{n\to\infty}\infty$ the limit $$\den(x):=\lim\limits_{n\to\infty}\den(x,I_n)$$ exists and does not depend on $\{I_n\}$ we call it the {\em density}\footnote{In Section~\ref{s:one-sided} we shall show that this definition may be significantly weaken in the case when all particles move in the same direction.} % of the configuration $x\in X$. Otherwise one considers upper and lower particle densities $\den_\pm(x)$ corresponding to upper and lower limits. The correspondence between particle densities for configurations with $r=0$ and $r>0$ is given by the following statement. \begin{lemma}\label{l:den-r} Let configurations $x(r)\in X(r),~r>0$ and $x\in X$ have the same sequence of gaps $\{\Delta_i\}$. Then $\den_\pm(x(r))=\frac{\den_\pm(x)}{1+2r\den_\pm(x)}$. \end{lemma} \proof Due to the one-to-one correspondence (\ref{e:r->r'}) between the configurations $x(r)$ and $x$, for each segment $I\subset{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}^1$ which contains $\den(x,I)\cdot|I|$ particles from the configuration $x$, one constructs the segment $I(r)$ containing the same particles from the configuration $x(r)$. The length of this segment is equal to $|I(r)|=|I|+2r\cdot\den(x,I)\cdot|I|$. Therefore $$ \den(x(r),I(r)) = \frac{\den(x,I)\cdot|I|}{|I|+2r\den(x,I)\cdot|I|} = \frac{\den(x,I)}{1+2r\den(x,I)} .$$ Passing to the limit as $|I|\to\infty$ one gets the result. \qed \begin{remark}\label{r:density} If $\exists \den(x)<\infty$ then $|x_n-x_m|/|n-m|\toas{|n-m|\to\infty}\den(x)$. \end{remark} \begin{lemma}\label{l:density-preservation} The upper/lower densities $\den_\pm(x^t)$ are preserved by dynamics, i.e. $\den_\pm(x^t)=\den_\pm(x^{t+1})~~\forall t\in\IZ_0$. \end{lemma}% \proof For a given segment $I\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$ the number of particles from the configuration $x^t\in X$ which can leave it during the next time step cannot exceed 1 and the number of particles which can enter this segment also cannot exceed 1. Thus the total change of the number of particles in $I$ cannot exceed 1, because if a particle leaves the segment through one of its ends no other particle can enter through this end. Therefore $$|\den(x^t,I) - \den(x^{t+1},I)|\cdot|I|\le1$$ which implies the claim. \qed% By the (average) {\em velocity} of the $i$-th particle in the configuration $x\in X$ at time $t>0$ we mean $$V(x,i,t):=\frac1t\sum\limits_{s=0}^{t-1}{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^s,x^s) % \equiv(x_i^{t}-x_i^0)/t.$$ % If the limit % $$V(x,i):=\lim\limits_{t\to\infty}V(x,i,t)$$ exists we call it the (average) {\em velocity} of the $i$-th particle. Otherwise one considers upper and lower particle velocities $V_\pm(x,i)$. The correspondence between average particle velocities for configurations with $r=0$ and $r>0$ is even simpler than for densities. \begin{lemma}\label{l:V-r} Let configurations $x(r)\in X(r),~r>0$ and $x\in X$ have the same sequence of gaps $\{\Delta_i\}$. Then $\forall i,t~~V(x(r),i,t)=V(x,i,t)$ for a given collection of local velocities $\{v_i^t\}_{i,t}$. \end{lemma} \proof Observe that the motion of particles depends only on the local velocities and the sequence of gaps. Thus at any time $t\ge0$ the sequence of gaps being changing in time is still the same for both configurations $x(r)$ and $x$. Therefore $$ {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t(r)) \equiv {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t) ~~\forall i,t $$ which yields the claim. \qed \begin{lemma}\label{l:velocity-preservation} Let $x\in X$ then $|V(x,j,t) - V(x,i,t)|\toas{t\to\infty}0$ a.s. $\forall i,j\in\IZ$. \end{lemma}% \proof It is enough to prove this result for $j=i+1$. Consider the difference between (average) velocities of consecutive particles% \bea{ V(x,i+1,t) - V(x,i,t) \!\!\!&\!\!\!\!&= \frac{x_{i+1}^t-x_{i+1}^0}t - \frac{x_i^t-x_i^0}t \\ % \!\!\!&\!\!\!\!&= \frac{x_{i+1}^t-x_i^t}t - \frac{x_{i+1}^0-x_i^0}t \\ % \!\!\!&\!\!\!\!&= \Delta} % {G_i^t/t - \Delta} % {G_i^0/t .} % The last term vanishes as $t\to\infty$ and it is enough to show that the same happens with $\Delta} % {G_i^t/t$. Consider first the deterministic setting (i.e. $v_i^t\equiv v_0^t$) and show that\footnote{If $v_0^t$ takes both positive and negative values then % $\Delta} % {G_i^t\le\max(4v,\Delta} % {G_i^0)$.} % $\forall i,t$ % % \beq{e:max-gap}{\Delta} % {G_i^t\le \function{\max(v,\Delta} % {G_i^0) &\mbox{if } {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p=\cNw \\ \max(2v,\Delta} % {G_i^0) &\mbox{if } {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p=\cNs } .}% Obviously this is true for $t=0$. Assume that this inequality holds up to time $t\in\IZ_0$ and consider the moment $t+1$. There might be two two possibilities: % \begin{itemize} \item[(a)] $\Delta} % {G_i^t\ge v_0^t$. Then ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_0^t,x^t)=v_0^t$ and % $$\Delta} % {G_i^{t+1} = \Delta} % {G_i^t - {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t) + {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i+1}^t,x^t) \le \Delta} % {G_i^t - v_0^t + v_{0}^t = \Delta} % {G_i^t \le \max(v,\Delta} % {G_i^0)$$ by the assumption. % \item[(b)] $\Delta} % {G_i^t<v_0^t$. Then ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_w(v_0^t,x^t)=\Delta} % {G_i^t$ and ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_s(v_0^t,x^t)=0$. Therefore % \bea{\Delta} % {G_i^{t+1} \!\!\!&\!\!\!\!&= \Delta} % {G_i^t - \Delta} % {G_i^t + {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i+1}^t,x^t) \le v \le \max(v,\Delta} % {G_i^0)\quad{\rm if}~{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p=\cNw ,\\ \Delta} % {G_i^{t+1} \!\!\!&\!\!\!\!&= \Delta} % {G_i^t - 0 + {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i+1}^t,x^t) \le 2v\quad{\rm if}~{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p=\cNs.}% \end{itemize} Thus in the deterministic setting the gaps are uniformly bounded in time and hence $\Delta} % {G_i^t/t\toas{t\to\infty}0$. Analysis of the random setting is much more involved since the gaps between particles in principle may grow with time and become arbitrary large but this may happen only very slowly. To estimate from above the value of the $i$-th gap $\Delta} % {G_i^t$ we drop from the consideration all particles except the $i$-th and $(i+1)$-th (preserving for all $t\in\IZ_0$ the velocities $\{v_i^t,v_{i+1}^t\}_t$) and denote the resulting configuration by $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t:=\{\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}_{i+1}^t\}$ and the gap % between this pair of particles by $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t$. We have % \bea{\Delta} % {G_i^{t+1}\!\!\!&\!\!\!\!&:=\Delta} % {G_i^t - {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,x^t) + {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i+1}^t,x^t),\\ \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1}\!\!\!&\!\!\!\!&:=\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t - {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) + {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i+1}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) = \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t - {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_i^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) + v_{i+1}^t. } % The comparison between $\Delta} % {G_i^t$ and $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t$ will be done by induction separately for the weak and strong normalizations. First let us prove that $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t\ge\Delta} % {G_i^t$ if ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p=\cNw$. At time $t=0$ obviously $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^0=\Delta} % {G_i^0$. Assume that $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t\ge\Delta} % {G_i^t$ for some $t\in\IZ_+$. Clearly, $$0\le{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i+1}^t,x^t)\le v_{i+1}^t.$$ For $v_i^t$ there might be two possibilities: % \begin{itemize} \item[(a)] $v_i^t\le \Delta} % {G_i^t$. Then ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i}^t,x^t)={\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)=v_i^t$ and hence % $$\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^{t+1}=\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t-v_{i}^t+v_{i+1}^t \ge \Delta} % {G^t-v_{i}^t+v_{i+1}^t = \Delta} % {G^{t+1}.$$ % \item[(b)] $v_i^t > \Delta} % {G_i^t$. Then $\cNw(v_{i}^t,x^t)=\Delta} % {G_i^t$, $\cNw(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)\ge\Delta} % {G_i^t$ and hence % $$\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^{t+1} = \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t - \cNw(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) + v_{i+1}^t \ge v_{i+1}^t = \Delta} % {G_i^{t+1}.$$ % \end{itemize} If ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p=\cNs$ a weaker estimate $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t+v\ge\Delta} % {G_i^t$ takes place. Considering again the same possibilities we see that the cases $t=0$ and (a) hold without any changes, but the case (b) should be rewritten. \begin{itemize} \item[(b')] $v_i^t > \Delta} % {G_i^t$. Then $\cNs(v_{i}^t,x^t)=0$, $\cNs(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) = \function{0 &\mbox{if } v_{i}^t > \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t \\ \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t &\mbox{if } v_{i}^t \le \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t}$, and hence $\cNs(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)\ge\cNs(v_{i}^t,x^t)$. Thus % \bea{\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^{t+1} \!\!\!&\!\!\!\!&= \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t - \cNs(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) + v_{i+1}^t \\ \!\!\!&\!\!\!\!&\ge \Delta} % {G_i^t - v - \cNs(v_{i}^t,x^t) + v_{i+1}^t - (\cNs(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)-\cNs(v_{i}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)) \ge \Delta} % {G_i^{t+1} - v.} \end{itemize} Consider now the behavior of $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t$ as a function of time $t$. If $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t\ge v$ we get $v_i^t\le \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t$ and hence ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p(v_{i}^t,x^t)=v_{i}^t$, which implies that outside of the region $[0,v]$ the sequence $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t$ behave as a spatially homogeneous reflected at $0$ random walk with i.i.d. symmetric increments $v_{i+1}^t-v_i^t$. Thus the mathematical expectation $E(\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t)$ cannot exceed\footnote{$4v$ if local velocities take both positive and negative values.} % $2v$ and hence by Chebyshev inequality the probability % $$ P(\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t/t\ge\varepsilon} \def\phi{\varphi} \def\la{\lambda) \le \frac1\varepsilon} \def\phi{\varphi} \def\la{\lambda~E(\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t/t) % \le \frac{2v}{t\varepsilon} \def\phi{\varphi} \def\la{\lambda} \toas{t\to\infty}0 ,$$ which finishes the proof. \qed \?{\footnotesize On the other hand, the mathematical expectation of the time $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t$ spends in the region $[0,v]$ is finite. Therefore applying the Reflection Principle by the Law of Large Numbers we obtain $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{\Delta} % {G}_i^t/t\sim\sqrt{t}/t\toas{t\to\infty}0$ a.s.} \begin{corollary}\label{c:vel-equiv} The upper and lower particle velocities $V_\pm(x,i)$ do not depend on $i$ (but might be random). \end{corollary} \section{Coupling}\label{s:pairing} Recall that a coupling of two Markov chains $x^t$ and $y^t$ acting on the space $X$ is an arrangement of a pair of processes on a common probability space to facilitate their direct comparison, namely this is a pairs process $(x^t,y^t)$ defined on the direct product space $X\times X$ satisfying the assumptions $$P((x^t,y^t)\in A\times X)=P(x^t\in A) \quad{\rm and}\quad P((x^t,y^t)\in X\times A)=P(y^t\in A) $$ % for any measurable subset $A\subseteq X$, i.e. the projections behave as the individual processes. Let $x^t, \2x^t$ be two copies of Markov chains, describing the DDS which we consider throughout the paper. Typically in continuous time interacting lattice particle systems one uses (see e.g. \cite{Lig}) an {\em equal} coupling (pairing) when particles sharing the same sites in the copies $x^t, \2x^t$ are considered to be paired and all choices of their velocities are assumed to be identical. This sort of coupling works rather well for continuous time systems when only a single particle may move at a given moment of time. In the discrete time case the situation is much more complicated since an arbitrary number of particles may move simultaneously and thus it is possible that the particles of the processes $x^t, \2x^t$ pass each other and never share the same positions. In fact, this difficulty is not really crucial and can be cured under some simple technical assumptions. A more important obstacle is that if a pair is created and only one of its members is blocked by an unpaired particle, then due to the simultaneous motion of the blocking unpaired particle and the non-blocked particle belonging to the pair the following situation may happen: % ~$_{\bullet}^{\bullet\circ}\longrightarrow _{~~~\circ}^{~\circ~~~\circ}$. % Thus the old pair will be destroyed but no new pair will be created under the equal pairing construction. Here and in the sequel we use a diagrammatic representation for coupled configurations, where paired particles are denoted by black circles and unpaired ones by open circles, and use the upper line of the diagram for the $x$-particles (i.e. particles from the $x$-process) and the lower line for the $\2x$-particles. To deal with this obstacle we introduce a {\em dynamical} \footnote{The word ``dynamical'' is meant to emphasize that the mutual arrangement of particles in pairs may change with time under dynamics in distinction to the conventual equal coupling (where the particles have coinciding positions).} % coupling, a very preliminary version of which was described in \cite{BP} for the lattice case and was inspired by the idea proposed by L.~Gray for the simplest discrete time lattice TASEP (unpublished). It is worth mention also the coupling proposed for the lattice continuous time case by O.~Angel (see \cite{An,EFM}). As we shall show an important advantage of the dynamical coupling with respect to the Angel's construction is that the former guarantees that the distances between mutually paired particles are uniformly bounded.\footnote{In the Angel's construction the distances may grow to infinity.} By the {\em dynamical coupling} of the processes $x^t,\2x^t$ we mean a gradual pairing of close enough particles belonging to the opposite processes satisfying the following assumptions: \begin{itemize} \item [(A1)] At $t=0$ all particles are assumed to be unpaired. Velocities of mutually paired particles are identical. \item [(A2)] Once being created a pair of particles remains present\footnote{Starting from the moment when a pair is created we consider it as an entity independently on the possible change of particles forming it.} % for any moment of time in the future, however at different moments of time the roles of the pair's members may be played by different particles. \item [(A3)] A particle overtaking during one time step of the dynamics some unpaired particles from the opposite process becomes paired with one of them. \end{itemize} According to (A1)--(A3) particles from the same pair move synchronously until either the admissibility condition breaks down for only one of the particles (which means that its movement is blocked by another particle) or one of the members of the pair is swapped with an unpaired particle from the same process (see Fig.~\ref{f:pairing} for the case of the weak normalization). It is convenient to think about the coupled process as a ``gas'' of single (unpaired) particles and ``dumbbells'' (pairs). A previously paired particle may inherit the role of the unpaired one from one of its neighbors. In order to keep track of positions of unpaired particles we shall refer to them as $x$- and $\2x$-{\em defects} depending on the process they belong. \Bfig(200,110) {\put(30,95){\circle*{5}} \put(30,100){$i$} \bline(30,95)(1,-2)(12) \put(42,70){\circle*{5}} \put(42,75){$j$} \put(42,65){\vector(1,0){41}} \put(44,55){$v_j=v_i$} % \put(60,95){\circle{5}} \put(55,100){$i+1$} \put(60,90){\vector(1,0){10}} \put(68,80){$v_{i+1}$} \put(120,95){\circle{5}} \put(115,100){$i+2$} \put(120,90){\vector(1,0){41}} \put(135,80){$v_{i+2}$} \put(168,70){\circle{5}} \put(163,75){$j+1$} \put(167,65){\vector(1,0){8}} \put(165,55){$v_{j+1}$} \thicklines \bline(0,45)(1,0)(200) \thinlines \bezier{30}(0,95)(100,95)(200,95) \put(1,100){$x^t$} \bezier{30}(0,70)(100,70)(200,70) \put(1,75){$\2x^t$} \bezier{30}(0,20)(100,20)(200,20) \put(1,25){$x^{t+1}$} \bezier{30}(0,-5)(100,-5)(200,-5) \put(1,0){$\2x^{t+1}$} \put(58,20){\circle{5}} \put(57,25){$i$} \put(70,20){\circle*{5}} \put(70,25){$i+1$} \put(83,-5){\circle*{5}} \put(83,0){$j$} \bline(70,20)(1,-2)(13) \put(161,20){\circle*{5}} \put(155,25){$i+2$} \put(174,-5){\circle*{5}} \put(173,0){$j+1$} \bline(161,20)(1,-2)(11) }{Pairing of particles. Black circles corresponds to paired particles and open circles to defects. The paired particles are connected by straight lines. At time $t$ the particles $i$ and $j$ are paired, while at time $t+1$ the $x$-particle $i$ becomes unpaired and the $\2x$-particle $j$ becomes paired with the $x$-particle $i+1$. The unpaired initially particles $i+2$ and $j+1$ become paired at time $t+1$. \label{f:pairing}} There are a number of ways to realize the dynamical coupling (in particular, using only the idea of the particle's overtaking). To demonstrate the flexibility of our approach we describe a different construction. Note that in the sequel we shall use only the properties (A1)--(A3) and the proofs will not depend on other details of the coupling. By the $x$-{\em triple} (~$_\bullet^{~\circ~\bullet}$~ or ~$^{\bullet~\circ}_{~~~\bullet}$~) in the coupled process $(x^t,\2x^t)$ we mean two mutually paired particles and a $x$-defect located in the segment between them, whose index differs by one from the index of the paired $x$-particle. The $\2x$-triple (~$^\bullet_{~\circ~\bullet}$~ or ~$_{\bullet~\circ}^{~~~\bullet}$) is defined similarly. Two pairs of particles are said to {\em cross} each other if straight lines connecting positions of particles belonging to the same pair intersect, e.g. ~$_{~\star~~\bullet}^{\bullet~~\star}$~, where particles belonging to the same pair are marked similarly. A $x$-defect at $x_i^t$ together with the closest\footnote{If there are several closest $\2x$-defects one chooses the defect with the smallest index.} % $\2x$-defect at $\2x_j^t$ (~$_\circ^{~\circ}$~ or ~$^\circ_{~\circ}$~) are said to be a {\em d-pair}\/ if $|x_i^t-\2x_j^t|<v$, this pair of defects does not cross with any mutually paired particles, and the open segment $(x_i^t,\2x_j^t)$ does not contain any other defects. We say that a d-pair $(i,j)$ is {\em smaller} than a d-pair $(n,m)$ if $|i|<|n|$, or if $i<n$ in case $|i|=|n|$. Observe that $i=n$ but $j\ne{m}$ cannot happen in distinction to $i\ne{n}$ but $j=m$. Note that in the collection ~$_{\bullet~\bullet}^{~\circ~\bullet~\bullet}$~ the first two $x$-particles together with the first $\2x$-particle form a $x$-triple despite the presence of an additional paired particle in the segment between them. On the other hand, the collection ~$_{\circ~\bullet}^{~\bullet~\circ}$~ does not contain neither triples nor d-pairs. A pair of configurations $(x^t,\2x^t)$ representing the coupled process at time $t$ is said to be {\em proper} if it does not contain $x$- or $\2x$-triples, d-pairs, and crossing mutually paired particles. The fact that at time $t$ the pair of configurations $(x^t,\2x^t)$ were proper does not imply that it remains proper under dynamics at time $(t+1)$. In particular, triples of both types and d-pairs may be created, e.g. ~$_{\bullet~~~\circ}^{~~~\bullet}\longrightarrow _{\bullet~\circ}^{~~~\bullet}$~ or % $~_{~~~~\circ\circ}^\circ~\longrightarrow~_{\circ\circ}^\circ$, however due to the particle order preservation crossing mutually paired particles cannot appear. \begin{lemma}\label{l:triples} Let a pair of configurations $(x^t,\2x^t)$ have no crossing mutually paired particles. Then among triples of the same kind there are no common elements. \end{lemma} \proof Direct inspection. As an illustration let us check the claim about $x$-triples. Assume that two $x$-triples have a common $x$-defect (mutually paired particles cannot be common by definition). Then this implies that the mutually paired particles in these triples either cross each other $~_{~~\star~~\bullet}^{\bullet~~\circ~~\star}~$ or the index of one of the paired $x$-particles differs from the index of the common defect by more than one $~_{~~~~~\star~\bullet}^{\star~\bullet~\circ}~$. The latter contradicts to the definition of the $x$-triple, why the former contradicts to the assumption about the absence of crossing mutually paired particles. In the diagrams above paired particles from the 2nd triple are marked by stars to distinguish them from the 1st triple. \qed Therefore all triples of the same kind may be resolved simultaneously. This will be done as follows. A $x$- or $\2x$-triple is transformed such that the former defect is becoming paired to the particle from another process, while another previously paired particle is becoming unpaired: ~$_\bullet^{~\circ~\bullet}\longrightarrow~_\bullet^{~\bullet~\circ}$~. The case of a d-pair is even simpler, namely the defects ``annihilate'' forming mutually paired particles: ~$_\circ^{~\circ}\longrightarrow~_\bullet^{~\bullet}$~. In all cases the positions of particles are preserved but their ``roles'' are changing. Finally the coupling procedure consists of the following steps: % \begin{itemize} \item[(1)] Each $x$-triple is recursively resolved: ~$_\bullet^{~\circ~\bullet}\longrightarrow~_\bullet^{~\bullet~\circ}$~. \item[(2)] Each $\2x$-triple is recursively resolved: ~$^\bullet_{~\circ~\bullet}\longrightarrow~^\bullet_{~\bullet~\circ}$~. \item[(3)] The smallest\footnote{The ordering of d-pairs is updated after each recursion procedure.} % d-pair is recursively resolved: ~$_\circ^{~\circ}\longrightarrow~_\bullet^{~\bullet}$~. \end{itemize} \begin{lemma}\label{l:coupling-markov} The coupling procedure described above is well defined, leads to the Markovian coupling, and satisfies the assumptions (A1)--(A3). \end{lemma} \proof Let us check that this procedure is well defined. By Lemma~\ref{l:triples} if a particle belongs to a certain triple then it cannot belong to any other triple. On the other hand, segments belonging to paired particles may overlap and resolving a $x$- or $\2x$-triple one may create a new one of the same kind: $$_{\bullet~~\bullet}^{~\circ~~\bullet~\bullet}\longrightarrow ~_{\bullet~~\bullet}^{~\bullet~~\circ~\bullet}\longrightarrow ~_{\bullet~~\bullet}^{~\bullet~~\bullet~\circ}.$$ % This explains the necessity of the recursion during the first two steps of the procedure. Note that resolving a $x$-triple one cannot create a new $\2x$-triple and vice versa (defects do not move from one process to another). Elements of the smallest d-pair might belong to some other d-pairs. Therefore resolving it we might change the d-order of the remaining d-pairs. To take this into account we are recalculating the d-order after each recursion procedure. Consider now the motion of a given defect under the recursions in the coupling procedure. Observe that the defect may move arbitrary far in any direction from its initial position due to these recursions: % $$_{~\circ~\bullet~\bullet~\cdots~\bullet~\bullet} ^{\bullet~\bullet~\cdots~\bullet~\bullet}\longrightarrow~ _{~\bullet~\bullet~\cdots~\bullet~\bullet~\circ} ^{\bullet~\bullet~\cdots~\bullet~\bullet}.$$ Nevertheless a defect cannot change its direction of movement. Assume from the contrary that a $x$-defect during two consequent steps of the recursion moved first to the right ($_\bullet^{~\circ~\bullet}\longrightarrow~_\bullet^{~\bullet~\circ}$) and then to the left ($_{~~~\bullet}^{\bullet~\circ}\longrightarrow ~_{~~~\bullet}^{\circ~\bullet}$). % This can happen only if after the first step of the recursion the defect became a member of a new $x$-triple of type $_\bullet^{~\circ~\bullet}$. Then the only candidate for the role of the paired $x$-particle in this $x$-triple is the paired $x$-particle which played the role of this defect on the previous recursion step. We came to the contradiction, because a particle may belong to only one pair. Thus the recursion is finite in the sense that each defect in a bounded spatial segment in finite time either will stop moving or will leave this segment and never return back. Note however that in general one cannot divide a configuration into finite pieces and deal with them separately since a defect may move from one piece to another. After the application of the first two steps all $x$- or $\2x$-triples will be eliminated and only d-pairs may be present. Observe now that when one resolves a d-pair neither triples nor new defects are created. However since various d-pairs may intersect they should be resolved separately during the last step. Additionally neither of above procedures may create crossing pairs of mutually paired particles (since members of different triples of the same type do not intersect and c- and d-pairs cannot cross each other). Let the pair of configurations $(x^{t-1},\2x^{t-1})$ be proper. Then according to arguments above after one time step of the dynamics the application of the coupling procedure, is well defined and the pair of configurations $(x^t,\2x^t)$ at time $t$ is proper as well. By the construction the one-time step transition probabilities for both processes $x^t$ and $\2x^t$ remain unchanged and the one-time step transition probabilities for the pairs process are well defined. Therefore this construction defines a Markovian coupling between two copies of the Markov chain describing our DDS. The property (A1) holds by the construction. A pair breaks down only if one of its members is replaced by an unpaired particle, and hence the pair as a whole survives. This proves (A2). The property (A3) follows from the fact that under the one time step of the dynamics of a proper pair of configurations all objects under consideration: $x$- and $\2x$-triples, and d-pairs may be created only during the particles overtaking. \qed Denote by $\den_u(x,I)$ the density of the $x$-defects belonging to a finite segment $I$, and by $\den_u(x):=\den_u(x,{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P})$ the upper limit of $\den_u(x,I_n)$ taken over {\em all} possible collections of nested finite segments $I_n$ whose lengths go to infinity. We say that a coupling of two Markov particle processes $x^t,\2x^t$ is {\em nearly successful} if the upper density of the $x$-defects $\den_u(x)$ vanishes with time a.s. This definition differs significantly from the conventional definition of the successful coupling (see e.e. \cite{Lig}), which basically means that the coupled processes converge to each other in finite time. In the random setting under some regularity assumptions the dynamical coupling turns out to be nearly successful (the proof of this result goes out of the scope of the present paper and will be published elsewhere), however in general especially in the deterministic setting this property needs not hold. Applying the notion of the nearly successful coupling to the exclusion process under study we get the following conditional result. \begin{lemma}\label{l:velocity-coupled} Let $x,\2x\in X$ with $\den(x)=\den(\2x)$, and let there exist a nearly successful coupling $(x^t,\2x^t)$ such that distances between the pair members are uniformly bounded from above by $\gamma(t)=o(t)$. Then $$|V(x,0,t)-V(\2x,0,t)|\toas{t\to\infty}0.$$ \end{lemma}% \proof Consider an integer valued function $n_t$ which is equal to the index of the $\2x$-particle paired at time $t>0$ with the $0$-th $x$-particle. If the $0$-th $x$-particle is not paired at time $t$ we set % $n_t:=\function{n_{t-1} &\mbox{if } t>0 \\ 0 &\mbox{if } t=0}$. % To estimate the growth rate of $|n_t|$ at large $t$ observe that $n_t$ changes its value only at those moments of time when the $0$-th $x$-particle meets a $\2x$-defect. By the assumption about the nearly successful coupling at time $t\gg1$ the average distance between the defects at time $t$ is of order $1/\den_u(\2x^{t})$ while the amount of time needed for two particles separated by the distance $L$ to meet cannot be smaller than $L/(2v)$. Therefore the frequency of interactions of the $0$-th $x$-particle with $\2x$-defects may be estimated from above by the quantity of order % $\den_u(\2x^t)\toas{t\to\infty}0$, which implies % $n_t/t\toas{t\to\infty}0$. Now we are ready to prove the main claim. % \bea{ |V(x,0,t) - V(\2x,0,t)| \!\!\!&\!\!\!\!&= |(x_0^t-x_0^0) - (\2x_0^t-\2x_0^0)|/t \\ \!\!\!&\!\!\!\!&\le |x_0^t-\2x_0^t|/t + |x_0^0-\2x_0^0|/t \\% \!\!\!&\!\!\!\!&\le |x_0^t-\2x_{n_t}^t|/t + \frac{|n_t|}t~|\2x_{n_t}^t-\2x_0^t|/|n_t| + |x_0^0-\2x_0^0|/t .} % The 1st addend can be estimated from above by $\gamma(t)/t\toas{t\to\infty}0$. The 2nd addend is a product of two terms $|n_t|/t$ and $|\2x_{n_t}^t-\2x_0^t|/|n_t|$. As we have shown, the 1st of them vanishes with time. If $|n_t|$ is uniformly bounded, then the 2nd term is obviously uniformly bounded on $t$. Otherwise, for large $|n_t|$ by Remark~\ref{r:density} and the density preservation the 2nd term is of order $\rho(\2x)$, which proves its uniform boundedness as well. Thus the 2nd addend goes to 0 as $t\to\infty$. Noting finally that the last addend also vanishes with time we are getting the result. \qed \section{Weak normalization}\label{s:weak} Consider the coupled process $(x^t,\2x^t)$ under the weak normalization and set $W_{ij}^t:=x_i^t-\2x_j^t$. \begin{lemma}\label{l:dist-pair} The supremum of ~$|W_{ij}^t|$~ taken over all mutually paired particles is uniformly bounded by $v$ for any $t\in\IZ_0$. \end{lemma} \proof We start at time $t=0$ when there are no pairs and wait until the first of them appears. At that moment the distance between the members in a pair cannot exceed $v$. Starting from that moment the distances may grow and some new pairs may be created. Contrary to our claim assume that there is the first moment of time $t$ at which there is a pair of particles located at $x_i^t,\2x_j^t$ for which $|x_i^t - \2x_j^t|>v$ and it is the largest distance between the paired particles at that moment of time (or one of the largest) and such that $|x_i^{t-1} - \2x_j^{t-1}|\le v$. According to the definition of the pairing process there are no unpaired particles between the particles from the same pair. Therefore in order to enlarge the distance between the particles one of them should be blocked by a particle from another pair, which contradicts to the assumption about the maximality of the distance. \qed \begin{lemma}\label{l:s-coupling} Let $\den(x)=\den(\2x)$ and let in the coupled process $\forall i,j~~\exists$ a (random) moment of time $t_{ij}<\infty$ such that $x_i^t>\2x_j^t$ for each $t\ge t_{ij}$. Then the coupling is nearly successful. \end{lemma} \proof By the assumption each $x$-particle will overtake eventually each $\2x$-particle located originally to the right from its own position and thus will form a pair with it or with one of its neighbors (if they are so close that were overtaken simultaneously). Thus the creation of pairs is unavoidable. To show that the upper density of defects cannot remain positive, consider how the defects move under our assumptions. Assume that at time $t\ge0$ the $i$-th $x$-particle is paired with the $j$-th $\2x$-particle. Then by Lemma~\ref{l:dist-pair} in order to overtake at time $s>t$ the $j$-th $\2x$-particle significantly (by a distance larger than $v$) the $i$-th $x$-particle necessarily needs to break the pairing with the $j$-th $\2x$-particle. Thus by the property (A3) of the dynamical coupling either a $x$-defect overtakes the $j$-th $\2x$-particle: % ~$_{~\bullet}^{\circ~~~\bullet}~\longrightarrow~ _\bullet^{~\circ~\bullet}~\longrightarrow~ _\circ^{~\bullet~\circ}$, or the $i$-th $x$-particle overtakes a $\2x$-defect: ~$_{\bullet~~~\circ}^{~~~\bullet}\longrightarrow~ _{\bullet~\circ}^{~~~\bullet}\longrightarrow~ _{\circ~\bullet}^{~~~\bullet}$. % (Otherwise this pair will not be broken.) Therefore during this process the $x$-defects move to the right while the $\2x$-defects move to the left. Hence they inevitably meet each other and ``annihilate''. The assumption about the equality of particle densities implies the result. \qed \subsection{Uniqueness of the average velocity} \label{s:vel-uniq} As we shall see under our assumptions even in the weak normalization case the nearly successful coupling needs not hold (e.g. in the deterministic setting). Therefore one cannot apply directly Lemma~\ref{l:velocity-coupled} in this case. Nevertheless we shall show that the absence of coupling is not a serious obstacle and it can be used as a diagnostic tool. \begin{theorem}\label{t:velocity-density} % In the weak normalization case the set of limit points as $t\to\infty$ of the sequence $\{V(x,t)\}_{t\in\IZ_0}$ depends only on the density $\den(x)$ assuming that the latter is well defined. \end{theorem}% \proof Consider a general DDS under the weak normalization. Let $x,\2x\in X_\den:=\{z\in X:~~\den(z)=\den\}$ be two admissible configurations of the same particle density. If one assumes that the coupling procedure described in Section~\ref{s:pairing} leads to the nearly successful coupling of particles in these configurations then by Lemma~\ref{l:dist-pair} the assumptions of Lemma~\ref{l:velocity-coupled} are satisfied and hence $|V(x,0,t)-V(\2x,0,t)|\toas{t\to\infty}0$ which by Lemma~\ref{l:velocity-preservation} implies the claim. In general the assumption about the nearly successful coupling may not hold,\footnote{Consider e.g. the deterministic setting with $1/\den>5v$ and the configurations $x_i:=i/\den$ and $\2x_i:=i/\den+2v$. Then $\den(x)=\den(\2x)=\den$, $V(x)=V(\2x)=v$ but no pair will be created.} % however as we demonstrate below the pairing construction is still applicable. \?{The idea is based on the observation that if one assumes that the claim does not hold then we are in a position to apply Lemma~\ref{l:s-coupling} which implies the nearly successful coupling which by Lemma~\ref{l:velocity-coupled} contradicts to the assumption.} Define random variables $$W_{ij}^t:=x_i^t-\2x_j^t,~i,j\in\IZ,~t\in\IZ_0.$$ Then % $$V(x,i,t)-V(\2x,j,t)=W_{ij}^t/t - W_{ij}^0/t.$$ Since by Lemma~\ref{l:velocity-preservation} the differences between average velocities of different particles belonging to the same configuration vanish with time it is enough to consider only the case $i=j=0$. For $W_{00}^t$ there might be three possibilities which we study separately: \begin{itemize} \item[(a)] $\lim\limits_{t\to\infty} W_{00}^t/t=0$. Then % $|V(x,0,t)-V(\2x,0,t)|\le|W_{00}^t|/t + |W_{00}^0|/t \toas{t\to\infty}0$, % which by Corollary~\ref{c:vel-equiv} implies that the sets of limit points of the average velocities coincide. \item[(b)] $\limsup\limits_{t\to\infty}W_{00}^t/t>0$. Then $\forall i\in\IZ$ the $i$-th particle of the $x$-process will overtake eventually each particle of the $\2x$-process located at time $t=0$ to the right from the point $x_i^0$. This together with the assumption of the equality of particle densities allows to apply Lemma~\ref{l:s-coupling} according to which the coupling is nearly successful. On the other hand, by Lemma~\ref{l:dist-pair} the distance between mutually paired particles cannot exceed $v$. Therefore by Lemma~\ref{l:velocity-coupled} we have ~~% $|V(x,0,t)-V(\2x,0,t)|\toas{t\to\infty}0$, % which contradicts to the assumption (b). \item[(c)] $\limsup\limits_{t\to\infty} W_{00}^t/t<0$. Changing the roles of the processes $x^t,\2x^t$ one reduces this case to the case (b). \end{itemize} Thus only the case (a) may take place. \qed \subsection{Deterministic setting} \begin{theorem}\label{t:fund-d-weak} (Fundamental Diagram) In the deterministic setting % \beq{e:FD-w}{V(x)=\lim\limits_{t\to\infty}\frac1t \sum_{s=0}^{t-1}\min(1/\rho,v_0^s) =\function{v &\mbox{if ~~ } \den(x)\le 1/v \\ 1/\den(x) &\mbox{otherwise }} } % if $v_0^t\equiv v$. \end{theorem} \proof Consider a family % $$\0X_\rho:=\{x\in X:~~x_i:=i/\rho+\omega,~\omega\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}\}$$ % of uniformly spatially distributed configurations of a given density $\rho>0$. This set is forward invariant and $$x_i^{t+1}-x_i^{t}\equiv\min(1/\rho,v_0^t)~~\forall x^t\in\0X_\rho, i\in\IZ,$$ % i.e. all particles in the configuration get the same normalized local velocity $\min(1/\rho,v_0^t)$ (depending in general on time $t$). % By the definition of the deterministic setting the limit % $$V(x):=\lim\limits_{t\to\infty}\frac1t \sum_{s=0}^{t-1}\min(1/\rho,v_0^s)$$ % is well defined. On the other hand, by Theorem~\ref{t:velocity-density} all configurations of the same density have the same average velocity, which implies the result. \qed \begin{remark} This result looks very similar to the one known for the deterministic version of the lattice TASEP (see \cite{NS,Bl-erg}), however the latter case is characterized by the following feature: if the density is large enough particles inevitably form dense clusters without vacancies inside (static traffic jams). The proof above shows that the ``typical'' behavior of high density configurations in continuum is different: they do form particle clusters, but these clusters are not staying at rest but are moving at a constant velocity as an ``echelon''. It is of interest that in order to imitate such behavior a number of complicated lattice models were developed. \end{remark} \begin{remark} The construction used in the proof is especially striking in that the same family of uniformly spatially distributed configurations allows to study the limit dynamics in the deterministic setting for all configurations having densities. Note that this argument cannot be applied directly in the lattice version of DDS. Nevertheless since the ``lattice configurations'' are included in DDS under consideration the result holds as well, which implies completely new results for lattice TASEPs with long jumps. \end{remark} \begin{corollary}\label{c:V-r-w} Let $x(r)\in X(r),~r>0$ and $\den(x(r))$ be well defined and let ~$\forall i,t~v_i^t\equiv v$. Then $$ V(x(r))=\function{v &\mbox{if ~~ } \den(x)\le \frac1{v+2r} \\ 1/\den(x(r))-2r &\mbox{otherwise }}.$$ In particular in the lattice setting this reads $$ V(x(1/2))=\function{v &\mbox{if ~~ } \den(x)\le \frac1{v+1} \\ 1/\den(x(1/2))-1 &\mbox{otherwise }}.$$ \end{corollary} \proof By (\ref{e:r->r'}) and Lemma~\ref{l:den-r} for each configuration $x(r)$ one constructs the configuration $x$ with the same sequence of gaps and the relation between their densities is written as $$ \den(x)=\frac{\den(x(r))}{1-2r\den(x(r))} .$$ Additionally by Lemma~\ref{l:V-r} average velocities related to configurations with the same sequence of gaps coincide. Substituting $\den(x)$ as a function of $\den(x(r))$ to (\ref{e:FD-w}) we get the result. \qed \subsection{Entropy}\label{s:entropy} In this Section we restrict the analysis to the pure deterministic setting (i.e. $v_i^t\equiv v~~\forall i,t$). Then our DDS is defined by a deterministic map $\map_v:X\to X$ from the set of admissible configurations into itself. Our aim is to show that this map is chaotic in the sense that its topological entropy is infinite.\footnote{Normally one says that a map is chaotic if its topological entropy is positive, so infinite value of the entropy indicates a very high level of chaoticity.} % We refer the reader to \cite{Bi,Wa} for detailed definitions of the topological and metric entropies for deterministic dynamical systems and their properties that we use here. To avoid difficulties related to the non-compactness of the phase space we define the topological entropy of a map $\map_v$ (notation $h_{{\rm top}}(\map_v)$) as the supremum of metric entropies of this map taken over all probabilistic invariant measures (compare to the conventional definition of the topological entropy and its properties in \cite{Wa}). For a finite subset of integers $I$ and a collection $C:=\{C_i\}_{i\in I}$ of open intervals the subset % $\cC_{I,C}:=\{x\in X:~~x_i\in C_i~~\forall i\in I\}$ is called a finite {\em cylinder}.\footnote{In general the cylinder $\cC_{I,C}$ might be empty for nonempty sets $I,C$.} % We endow the space of admissible configurations $X$ by the $\sigma$-algebra ${\cal B}} \def\cC{{\cal C}$ generated by the finite cylinders defining a topology in this space. We start the analysis with the action of a shift-map in continuum $\sigma_v:X\to X$ defined as % $$(\sigma_v x)_i:=x_i+v~~~ i\in\IZ, x\in X.$$ \begin{lemma}\label{l:entropy-shift-map} The topological entropy of the shift-map in continuum $\sigma_v$ is infinite. \end{lemma} \proof The preimage of a finite cylinder under the action of $\sigma_v$ is again a finite cylinder. Therefore this map is continuous in the topology induced by the $\sigma$-algebra ${\cal B}} \def\cC{{\cal C}$ generated by finite cylinders. The idea of the proof is to construct an invariant subset of $X$ on which the map $\sigma_v$ is isomorphic to the full shift-map in the space of sequences with a countable alphabet. The result follows from the observation that the topological entropy of the full shift-map $\sigma^{(n)}$ with the alphabet consisting of $n$ elements is equal to $\ln n$ (see, e.g. \cite{Bi,Wa}). Let $\alpha:=\{\alpha_i\}_{i\in\IZ_+}$ with $\alpha_i\in(0,v)$ and let $\alpha^n:=\{\alpha_i\}_{i=1}^n$. Consider a sequence of subsets $X^{(n)}\subset X$ consisting of {\em all} configurations $x\in X$ satisfying the condition $\forall k\in\IZ~~ x_{2k}\in{v}\IZ, ~x_{2k+1}\in x_{2k} + \alpha^n$. Then $X^{(n)}$ is $\sigma_v$-invariant and the restriction $\sigma_v|X^{(n)}$ is isomorphic to the full shift-map $\sigma^{(n)}$ with the alphabet $A^n$ consisting of $n$ elements $\{a_i\}$ of type % $a_i:=\{[0,\alpha_i),[\alpha_i,v)\}$, i.e. each element is represented by a pair of neighboring intervals. Therefore the topological entropy of $\sigma^{(n)}$ is equal to % $\ln n\toas{n\to\infty}\infty$. \qed Another elegant (but technically difficult) way to derive this result was proposed by Boris Gurevich. Consider a special flow $S^t$ corresponding to the shift-map acting on the sequences $\{\Delta} % {G_i(x)\}$ with the roof function equal to the first nonnegative particle coordinate. This shift-map has an infinite alphabet, hence its entropy is infinite. The special flow $S^1$ is isomorphic to the 1-shift of $\{x_i\}$, while the entropy of the special flow can be calculated by the Abramov-Rohlin formula. \begin{theorem}\label{t:entropy-exclusion} The topological entropy of the pure deterministic exclusion process in continuum is infinite. \end{theorem} \proof The preimage of a finite cylinder under the action of $\map_v$ is again a finite cylinder. Therefore this map is continuous in the topology induced by the $\sigma$-algebra ${\cal B}} \def\cC{{\cal C}$ generated by finite cylinders. Observe that the subset % $X_0:=\{x\in X:~~\Delta} % {G_i(x)\ge v~~ \forall i\in\IZ\}$ % of the set of admissible configurations is $\map_v$-invariant. Therefore $h_{{\rm top}}(\map_v)\ge h_{{\rm top}}(\map_v|X_0)$ and for our purposes it is enough to show that the latter is infinite. On the other hand, by the definition of the map $\map_v$ we have $\map_v|X_0\equiv\sigma_v|X_0$. We still cannot apply the result of Lemma~\ref{l:entropy-shift-map} directly because in the case under consideration the gaps between particles are greater or equal to $v$ by the construction. Recall that in the proof of Lemma~\ref{l:entropy-shift-map} the gaps were not greater than $v$. To this end one sets $\alpha_i\in(v,2v)$ and modifies the definition of $X^{(n)}$ as follows: % $$x_{2k+1}\in x_{2k} + \alpha^n\quad \forall k\in\IZ,~ x_{2k}\in3v\IZ .$$ % Consider the the alphabet $A^{(n)}$ with elements of type $a_i:=\{[0,\alpha_i),[\alpha_i,3v)\}$. Then the $3$-d power of the map $\map_v|X_0$ is isomorphic to the full shift-map $\sigma^{(n)}$ with the alphabet $A^{(n)}$. Using that % $$3h_{{\rm top}}(\map_v|X_0)=h_{{\rm top}}((\map_v|X_0)^3) =h_{{\rm top}}(\sigma^{(n)})=\ln n$$ we get the result. \qed \section{Strong normalization}\label{s:strong} % Recall that $W_{ij}^t:=x_i^t-\2x_j^t$ for $x^t,\2x^t\in X,~t\ge0$. \begin{lemma}\label{l:gap-strong} There exists a coupled process $(x^t,\2x^t)$ such that under the strong normalization $\sup_{i,j,t}W_{ij}^t=\infty$, where the supremum is taken over all mutually paired particles. \end{lemma} % \proof It seems that the argument applied in the weak normalization case should work also in the case of the strong normalization. However, a close look shows that in this case a ``blocked'' particle does not move to ``touch'' the particle conflicting with it (as it would in the weak normalization case) but preserves its position instead. Therefore the distance between members of the same pair may become larger than the distance between the members of the ``blocking'' pair which cannot happen in the weak normalization case: % ~$_{\bullet~~\bullet~~~~\bullet~~~} ^{~~\bullet~~~~\bullet\bullet~~~}\longrightarrow~ _{\bullet~~~~~\bullet~~~\bullet} ^{~~~~~\bullet\bullet~~~\bullet}$. % Here initially distances between members in pairs do not exceed $v$. The 1st pair is blocked by the 2nd pair and since the $\2x$-member of the 1st pair cannot move (while the $x$-member can) the distance between them becomes larger than $v$. To demonstrate that distances between members in pairs may grow to infinity fix some $0<\varepsilon} \def\phi{\varphi} \def\la{\lambda\ll1$ and consider a pair of configurations $x,\2x$ such that $x_0=\2x_0=0$ and $\Delta} % {G_{2k}=\frac32(v-\varepsilon} \def\phi{\varphi} \def\la{\lambda), ~\Delta} % {G_{2k+1}=\frac12(v-\varepsilon} \def\phi{\varphi} \def\la{\lambda), ~\2\Delta} % {G_k=v-\varepsilon} \def\phi{\varphi} \def\la{\lambda~\forall k\in\IZ$. After the application of the pairing procedure $\forall i$ the $i$-particles in both configurations will become paired forever. On the other hand, under dynamics $\2x^t\equiv\2x^0~\forall t$ while the $x$-particles having gaps greater than $v$ will at constant velocity $v$. Therefore the distances between members in pairs will grow linearly with time. \qed \?{Nevertheless for spatially periodic configurations $\sup_{ij}W_{ij}^t<\infty$ if density $\den<1/v$. Indeed the small density implies the presence of gaps $>v$. Hence the proof of Lemma~\ref{l:gap-strong} breaks down. On the other hand, due to the spatial periodicity it is enough to consider only one spatial period, for which the claim is trivial.} This result demonstrates and partially explains a significant difference in the behavior of DDS under weak and strong normalizations. Still, as we are going to show, at least some features of the Fundamental Diagram are preserved. Consider the pure deterministic setting (i.e. $v_i^t\equiv v$). The inequality~(\ref{e:max-gap}) shows that in this case gaps between particles cannot become much larger than their initial values. The following result demonstrates that under some mild additional assumptions (which definitely hold for high particle densities) large gaps will disappear with time. \begin{lemma}\label{l:max-gap} Let $x\in X$ be spatially periodic and we consider only the pure deterministic setting (i.e. $v_i^t\equiv v$). Assume that $\forall t~\exists j>t:~\Delta} % {G_j(x^t)<v$. Then $\forall i~\exists t_i<\infty:~\Delta} % {G_i(x^t)<2v~\forall t\ge t_i$. \end{lemma} \proof Observe that the spatial periodicity and its period is preserved under the pure deterministic dynamics. Thus the situation is equivalent to the consideration of a finite number (say $N$) particles on a ring and to the assumption that for each $t\in\IZ_+$ among these particles there is a particle with a gap less than $v$ ahead of it. Note that according to the definition of the strong normalization ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_s(v_i^t,x^t)=0$ whenever $\Delta} % {G_i(x^t)<v$. By (\ref{e:max-gap}) $\Delta} % {G_i(x^t)<2v$ implies $\Delta} % {G_i(x^{t+1})<2v$. Therefore new new long gaps (of size larger or equal to $2v$) cannot be created and we need to show only that long gaps in the original configuration will cease to exist with time. By the assumption for any $t$ there exists a short gap (shorter than $v$) and the corresponding particle will not move during the next time step. Thus the index of the short gap decreases by one after each time step until it ``collides'' with one of the long gaps: $\Delta} % {G_i(x^t)\ge2v, ~\Delta} % {G_{i+1}(x^t)<v$. On the next time step $\Delta} % {G_i(x^{t+1}):=\Delta} % {G_i(x^t)-v$. Due the spatial periodicity the amount of time between these ``collisions'' is bounded and after each of them the length of a long gap decreases by $v$. Thus they will disappear in finite time. \qed \Bfig(150,100) \put(0,0){\vector(1,0){140}} \put(0,0){\vector(0,1){90}} \thicklines \bline(0,73)(1,0)(60) \bezier{200}(60,73)(75,5)(125,2) \bezier{200}(30,73)(45,10)(60,0) \thinlines \bezier{30}(30,73)(30,36)(30,0) \bezier{30}(60,73)(60,36)(60,0) \put(145,0){$\rho$} \put(-8,85){$V$} \put(-8,70){$v$} \put(85,30){$\frac1\rho$} \put(17,15){$\frac1\rho-v$} \put(-8,-5){$0$} \put(54,30){$H$} \put(25,-12){$\frac1{2v}$} \put(57,-12){$\frac1{v}$} } {Fundamental Diagram (dependence of the average velocity $V$ on the particle density $\rho$) for the pure deterministic setting under the strong normalization. The curvilinear region % $H:=\{(\den,V): ~\frac1\rho-v\le V\le\frac1\den, ~V\le v\}$ % corresponds to the hysteresis phase. \label{f:velocity-strong}} \begin{theorem}\label{t:fund-d-strong} Let $x\in X$ and $\den(x)$ be well defined. Then $V(x)=v$ if $\den(x)<\frac1{2v}$ and otherwise for a.e. point $(\den,V)$ in the curvilinear region $$H:=\{(\den,V): ~\max(1/\rho-v,0)\le V\le\min(1/\den,v)\}$$ % (see Fig.~\ref{f:velocity-strong}) % there exists a configuration $x\in X$ with $\den(x)=\den, V(x)=V$, i.e. the region $H$ corresponds to the hysteresis. \end{theorem} % \proof We say that particles numbered from $i+1$ to $i+k$ with $i\in\IZ, k\in\IZ_+$ belonging to an admissible configuration $x\in X$ form a {\em cluster} of {\em length} $k$ if all gaps between them are strictly less than $v$ and the gaps to surrounding particles are not smaller than $v$, i.e. $\Delta} % {G_{i+j}< v~\forall j=1,2,\dots,k-1$ and $\Delta} % {G_i,\Delta} % {G_{i+k}\ge v$. Positions of particles belonging to the cluster are changing with time, and leading particles leave it, while some new particles may join the cluster from the other side. Nevertheless the length of a cluster cannot grow with time (and new clusters cannot be born in the pure deterministic setting in distinction to the random one) since the rate with which the leading particle leaves the cluster (one per unit time) is at least not smaller than the rate at which new particles join the cluster from the other side. We start with the analysis of configurations of low density (smaller than $\frac1{2v}$) and our aim is to show that in this case each particle achieves eventually the largest available velocity $v$. Consider the motion of the $0$-th particle in a configuration $x\in X$ with $0<\den(x)<\frac1{2v}$ and denote by $\hat{t}$ the first moment of time after which this particle will not join any cluster. If $\hat{t}<\infty$ then ${\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_s(v_0^t)\equiv~\forall t\ge\hat{t}$ and hence % $V(x,0,t)\toas{t\to\infty}v$. If $\hat{t}=\infty$ then there exists an infinite sequence of clusters of growing length such that the $0$-th particle joins each of them consecutively. Let us show that this assumption contradicts to the condition that $\den(x)<\frac1{2v}$. We number the clusters to which the $0$-th particle will join according to their natural order starting from $k=1$ and introduce the following notation: $t_k$ -- the moment of time when the $0$-th particle joins the $k$-th cluster, $n_k$ -- the number of particles in this cluster, $m_k$ -- the number of particles in the open segment between $x_0$ and the beginning of this cluster, and $L_k$ -- the length of the minimal segment containing the $k$-th cluster and the point $x_0$. Then $$ \den(x,(x_0,x_0+L_k]) = \frac{m_k+n_k}{L_k}\toas{k\to\infty}\den(x) .$$ All $m_k$ particles will join the $k$-th cluster during the time $t_k$ and at time $t_k$ this cluster should still exist. Therefore the distance which the $0$-th particle covers during this time cannot be smaller than $L_k-m_kv-n_kv$ while its velocity cannot exceed $v$ and thus % $$ t_kv \ge L_k-m_kv-n_kv .$$ On the other hand, exactly $t_k$ particles will leave the cluster during this time, i.e. $m_k+n_k\ge t_k$. This gives % \beq{e:den}{ \frac{m_k+n_k}{L_k} \ge \frac{t_k}{L_k} \ge \frac{L_k/v - m_k - n_k}{L_k} = \frac1v - \frac{m_k+n_k}{L_k} .} % Therefore % $$ \frac1v\le2\frac{m_k+n_k}{L_k}\toas{k\to\infty}2\den(x) ,$$ which proves the desired claim that $\hat{t}=\infty$ implies $\den(x)\ge\frac1{2v}$. Consider now the case of densities greater than $\frac1{2v}$. In this case there might be two possibilities: (a) All particles will eventually achieve the largest available velocity $v$. Then the gaps will become not smaller than $v$ and hence they cannot exceed $2v$ (by the assumption on the density region). Obviously this situation may take place only if $\den(x)\in[\frac1{2v},\frac1{v}]$ and it corresponds to the upper branch of the Fundamental Diagram on Fig.~\ref{f:velocity-strong}. (b) For any moment of time the are infinitely many particles having gaps smaller than $v$ (and hence zero normalized local velocities). Therefore at least for spatially periodic configurations we can apply Lemma~\ref{l:max-gap} which guarantees that only gaps smaller than $2v$ will survive with time. Thus to study asymptotic properties it is enough to consider configurations having only two types of gaps: smaller than $v$ and between $v$ and $2v$. Denote by $X(L,m,n)$ the subset of admissible configurations $x\in X$ being spatially periodic with the spatial period of length $L\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}_+$, which contains exactly $m\in\IZ_+$ particles with gaps belonging to the interval $[0,v)$ and $n\in\IZ_+$ particles with gaps belonging to the interval $[v,2v)$. Obviously $\den(x)=(m+n)/L$. The set $X(L,m,n)$ is invariant under dynamics (each time when the size of a gap crosses the threshold $v$ one ``small'' gap becomes large and one ``large'' gap becomes ``small'') which immediately yields the exact value of the average velocity $V(x)=\frac{nv}{m+n}$. On the other hand, by definition $mv+n2v>L$ since the corresponding gaps fill in the segment of length $l$ and lengths of both types of gaps are smaller than $v$ and $2v$ respectively. Therefore % $(\den(x)L+n)v>L$ and hence $n>L/v-\den(x)L$, which gives the lower bound % $$V(x) = \frac{nv}{m+n} = \frac{nv}{\den(x)L} > v~\frac{L/v-\den(x)L}{\den(x)L} = 1/\den(x) - v.$$ % Observe, that choosing ``small'' and ``large'' gaps of length $v-\varepsilon} \def\phi{\varphi} \def\la{\lambda$ and $2v-\varepsilon} \def\phi{\varphi} \def\la{\lambda$ for $0<\varepsilon} \def\phi{\varphi} \def\la{\lambda\ll1$ we see that the lower bound can be ``almost'' achieved. The upper bound of the average velocity in the hysteresis phase (i.e. when $\frac1{2v}<\den(x)<\frac1{v}$) follows from the existence of configurations with equal gaps of size larger than $v$ for all densities from this segment. For the case $\den(x)>1/v$ the upper bound is calculated using the opposite length estimate $nv<L$. Then we get % $$V(x) = \frac{nv}{m+n} = \frac{nv}{\den(x)L} < \frac{L}{\den(x)L} = 1/\den(x),$$ which agrees with the weak normalization case. It remains to show that the region $H$ is filled in densely by the pairs $(\den,V)$ corresponding to admissible configurations. To this end one considers all possible choices of the integer parameters $n,m$ and lengths of the corresponding gaps to get the result. Indeed, $\forall \den\in(\frac1{2v},\frac1{v})$ there exists an arbitrary large $L$ such that $\den L\in\IZ_+$. Choosing now various available combinations of positive integers $m,n$ for which $m+n=\den L$ we can approximate $V$ with the accuracy % $$|V - \frac{nv}{m+n}|\le \frac{v}{\den L} \toas{L\to\infty}0.$$ \qed % \noindent} \def\map{T} \def\supp{{\rm supp}{\bf Remark}. By Theorem~\ref{t:fund-d-strong} for a.e. pair $(V,\rho)\in$~H there exists an admissible configuration $x\in X$ such that $\den(x)=\rho$ and $V(x)=V$. On the other hand, it might be possible that for some configurations having densities belonging to the hysteresis region the average velocity is not well defined and we claim only that all limit points of finite time velocities belong to the vertical segment corresponding to the given density. \begin{corollary}\label{c:V-r-s} Let $x(r)\in X(r),~r>0$ and $\den(x(r))$ be well defined and let ~$\forall i,t~v_i^t\equiv v$. Then $V(x)=v$ if $\den(x(r))<\frac1{2v+2r}$ and otherwise for a.e. point $(\den,V)$ in the curvilinear region $$H:=\{(\den,V): ~\max(\frac1{\den-2r}-v,0)\le V \le\min(\frac1{\den-2r},v)\}$$ % there exists a configuration $x(r)\in X(r)$ with $\den(x(r))=\den, V(x(r))=V$, i.e. the region $H$ corresponds to the hysteresis. \end{corollary} \section{Local velocities of both signs}\label{s:vel-2signs} % A close look to the previous analysis shows that we practically did not use the property that all particles move in the same direction, i.e. that $P(v_i^t\ge0)=1$. Now we explain the changes necessary to study this more general case. Consider an infinite configuration $x(r)\in X(r)$ and again interpret the values $\{v_i^t\}_{i,t}$ (which now may have both positive and negative signs, but still assuming that $|v_i^t|\le v$) as local velocities for particles in the configuration $x^t(r)$. The presence of particles moving in opposite directions leads to a serious modification of the inequalities describing the violation of the admissibility condition for the $i$-th local velocity. Actually this is the main and the most serious change comparing to the case of nonnegative velocities. Now we need to take into account not only the position of the succeeding particle, but also its velocity, as well as the corresponding quantities related to the preceding particle. In this more general case the $i$-th local velocity does not break the admissibility condition if and only if % \bea{\!\!\!&\!\!\!\!&\max(x^t_{i-1}(r), x^t_{i-1}(r)+v_{i-1}^t) + r % \le\min(x^t_i(r), x^t_i(r)+v_i^t) - r \\ % \!\!\!&\!\!\!\!&\quad~< \max(x^t_i(r), x^t_i(r)+v_i^t) + r % \le\min(x^t_{i+1}(r), x^t_{i+1}(r)+v_{i+1}^t) - r .} % If for some $i\in\IZ$ and $j\in\{i-1,i+1\}$ the corresponding inequality is not satisfied we say that there is a {\em conflict} between the $i$-th particle and the $j$-th one and one needs to resolve it. In terms of gaps $\Delta} % {G_i^t$ between particles the inequalities above can be rewritten as follows: % \beq{e:adm-2signs}{\Delta} % {G_{j}^t\ge\max(v_j^t,~-v_{j+1}^t,~v_j^t-v_{j+1}^t), ~~j\in\{i-1,i\} } % Since the dynamics again will depend only on the sequence of gaps $\{\Delta} % {G_i^t\}$ between particles, for each $r>0$ one can make the invertible change of variables (\ref{e:r->r'}) (described in the Introduction) to the case of `point' particles with $r=0$ which we shall study further. Exactly as in Section~\ref{s:intro} the {\em strong normalization} means that we reject (nullify) all velocities leading to a conflict, i.e % $$ {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_s(v_i^t,x^t):=\function{ v_i^t &\mbox{if (\ref{e:adm-2signs}) holds } \\ 0 &\mbox{otherwise }.} $$ The situation with the weak normalization is more delicate. The way how it was defined in Section~\ref{s:intro} can be characterized as the only non-anticipating procedure allowing conflicting particles to move simultaneously whenever possible. Following this idea we say that a normalization is {\em weak} if the positions of particles at the next time step $x_i^{t+1}:=x_i^t+{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_w(v_i^t,x^t)$ satisfy the conditions: % \beq{e:weak-n}{ x_i^{t+1}\in\function{ \{x_i^t+v_i^t\} &\mbox{if (\ref{e:adm-2signs}) holds } \\ \{x_j^t, x_j^{t+1}\} &\mbox{if $\exists$ a conflict of the particle $i$ with the particle $j=i\pm1$} .} }% The 1st line describes the case when the admissibility condition holds, while the 2nd line shows what happens if it breaks down. Namely, if the $i$-th particle moves in the same direction as the $j$-th one then (by the non-anticipation property) the former assumes the previous position of the latter ($x_i^{t+1}=x_j^t$), otherwise the positions of the conflicting particles at time $t+1$ coincide. The latter fact is the most important property here. If directions of all instant local velocities coincide then (\ref{e:weak-n}) defines the normalization uniquely. However if their signs are different then (\ref{e:weak-n}) implies only that % $$x_i^{t+1}=x_j^{t+1}\in[x_i^t,x_j^t]\cap[x_i^t+v_i^t,x_j^t+v_j^t].$$ Thus the set of weak normalizations is quite broad, for example it includes a random normalization when two mutually conflicting particles moving in opposite directions meet at a random point belonging to the segments described above. One can give a ``natural'' specific construction of $\cNw$ normalizing local velocities in such a way that positions of particles at the next moment of time will be the same as if the particles would move simultaneously at continuous time with the given local velocities until the admissibility condition breaks down: % $$ {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_{w,c}(v_i^t,x^t):=\function{ v_i^t &\mbox{if (\ref{e:adm-2signs}) holds } \\ -\Delta} % {G_{i-1}^t &\mbox{if } \Delta} % {G_{i-1}^t<-v_i^t,~ v_i^t<0,~v_{i-1}^t\le0 \\ \Delta} % {G_{i}^t &\mbox{if } \Delta} % {G_{i}^t<v_{i}^t,~ v_i^t>0,~v_{i+1}^t\ge0 \\ \frac{\Delta} % {G_{i-1}^t}{v_{i-1}^t-v_i^t}\times v_i^t &\mbox{if } \Delta} % {G_{i-1}^t<v_{i-1}^t-v_i^t,~ v_i^t<0,~v_{i-1}^t>0 \\ \frac{\Delta} % {G_{i}^t}{v_{i}^t-v_{i+1}^t}\times v_i^t &\mbox{if } \Delta} % {G_{i}^t<v_{i}^t-v_{i+1}^t,~ v_i^t>0,~v_{i+1}^t<0 .} $$ After this long discussion of the definition of the normalization procedure it is surprising to find that all arguments used in the analysis of the case of positive velocities remain valid with only very slight changes. % \begin{lemma}\label{l:density-preservation+-} The upper/lower densities $\den_\pm(x^t)$ are preserved under dynamics. \end{lemma}% \proof One uses the same estimates as in the proof of Lemma~\ref{l:density-preservation} except that now 2 particles may simultaneously leave or enter a given spatial segment $I$ (instead of 1). Thus the total change of the number of particles in $I$ is less or equal to 2 and hence % $$|\den(x^t,I) - \den(x^{t+1},I)|\cdot|I|\le2.$$ \qed% \begin{lemma}\label{l:velocity-preservation+-} Let $x\in X$ then $|V(x,j,t) - V(x,i,t)|\toas{t\to\infty}0$ a.s. $\forall i,j\in\IZ$. \end{lemma}% \proof Again one follows the same argument as in the case of nonnegative local velocities. The only difference is that in the analysis of the connection between $\Delta} % {G_i^t$ and $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t$ now one needs to consider new cases related to negative local velocities. Additionally here instead of the uniquely defined weak normalization we need to consider an arbitrary one. If both $v_i^t$ and $v_{i+1}^t$ are nonnegative we are in the situation considered in Section~\ref{s:metric}. Therefore the cases (a) and (b) hold automatically. Nevertheless we formulate all of them to prove that $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t\ge\Delta} % {G_i^t~\forall t\in\IZ_0$: % \begin{itemize} \item[(a)] the condition (\ref{e:adm-2signs}) holds. Then obviously the argument used in Section~\ref{s:metric} woks.% \item[(b)] $v_i^t>\Delta} % {G_i^t,~v_{i+1}^t\ge0$. Again one uses the same argument as in Section~\ref{s:metric}. % \item[(c)] $v_i^t<-\Delta} % {G_{i-1}^t$. Then $\cNw(v_i^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)\le\cNw(v_i^t,x^t)\le0$ and $\cNw(v_{i+1}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)\ge\cNw(v_{i+1}^t,x^t)$. Hence % \bea{\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1} \!\!\!&\!\!\!\!&= \tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t} - \cNw(v_i^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) + \cNw(v_{i+1}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) \\ % \!\!\!&\!\!\!\!&\ge \Delta} % {G_i^{t} - \cNw(v_i^t,x^t) + \cNw(v_{i+1}^t,x^t) % =\Delta} % {G_i^{t+1}.} % \item[(d)] $v_i^t\ge0,~v_{i+1}^t<0$ and $v_i^t-v_{i+1}^t>\Delta} % {G_i^t$. Then by definition $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1}\ge0=\Delta} % {G_i^{t+1}$. \end{itemize} In the strong normalization setting one also considers the same cases and proves by induction that $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1}\ge\Delta} % {G_i^{t+1}-2v$ (instead of $\dots-v$ in the situation $v_i^t\ge0$). New cases are the following % \begin{itemize} \item[(c')] $v_i^t<-\Delta} % {G_{i-1}^t$. Then $$\cNs(v_i^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)=v_i^t<-\Delta} % {G_{i-1}^t=\cNs(v_i^t,x^t)$$ and $$\cNs(v_{i+1}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t)-\cNs(v_{i+1}^t,x^t)\ge-2v$$ by the induction assumption. Hence % \bea{\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1} \!\!\!&\!\!\!\!&=\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t} - \cNs(v_i^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) + \cNs(v_{i+1}^t,\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot{x}^t) \\ % \!\!\!&\!\!\!\!&> \Delta} % {G_i^{t} - 2v - \cNs(v_i^t,x^t) - \cNs(v_{i+1}^t,x^t) + 2v \\ % \!\!\!&\!\!\!\!&= \Delta} % {G_i^{t+1} .} % \item[(d')] $v_i^t\ge0,~v_{i+1}^t<0$ and $\Delta} % {G_i^t<v_i^t-v_{i+1}^t\le\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t$. Then % \bea{\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1} \!\!\!&\!\!\!\!&=\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t}+v_i^t-v_{i+1}^t>\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t}+\Delta} % {G_i^t\\ \!\!\!&\!\!\!\!&\ge-2v+\Delta} % {G_i^t=-2v+\Delta} % {G_i^{t+1}.} % \item[(d'')] $v_i^t\ge0,~v_{i+1}^t<0$ and $\Delta} % {G_i^t<v_i^t-v_{i+1}^t>\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t$. Then % $$\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1}=\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^t\ge\Delta} % {G_i^t-2v=\Delta} % {G_i^{t+1}-2v.$$ % \end{itemize} Note that the difference $\tilde} \def\den{\rho} \def\dist{\varrho} \def\s{~\cdot\Delta} % {G_i^{t+1}-\Delta} % {G_i^{t+1}=-2v$ may be achieved only in the case (d'). The continuation of the proof is exactly the same as in Section~\ref{s:metric}, except for the change of $2v$ to $4v$ in the last inequality. \qed Using these results and applying exactly the same arguments as in the proof of Theorem~\ref{t:velocity-density} one gets the uniqueness of the average velocity. \begin{theorem}\label{t:velocity-density-2signs} % In the weak normalization case the set of limit points as $t\to\infty$ of the sequence $\{V(x,t)\}_{t\in\IZ_0}$ depends only on the density $\den(x)$. \end{theorem}% \begin{theorem}\label{t:fund-d-weak-2signs} (Fundamental Diagram) In the deterministic setting % $V(x)=\lim\limits_{t\to\infty}\frac1t \sum\limits_{s=0}^{t-1}\min(1/\rho,v_0^s)$. \end{theorem} \proof Since at each moment of time $t\in\IZ_0$ the local velocities of particles coincide, the condition (\ref{e:weak-n}) implies that $$ x_i^{t+1}\in\{x_i^t+v_0^t,~x_{i\pm1}^t\} .$$ Thus the construction used in the proof of Theorem~\ref{t:fund-d-weak} remains valid in this case as well. \qed \section{Generalizations and Discussion}\label{s:gen} \subsection{Anticipating normalization} Throughout the paper we consider only non-anticipating normalizations. In principle one might try to consider an anticipating normalization allowing at time $t$ the $i$-th particle to move up to the position of the $(i+1)$-th particle $x_{i+1}^{t+1}$ at time $t+1$ rather than to $x_{i+1}^t$. From the first sight this makes the normalization scheme more flexible. Unfortunately the anticipating normalization is not well posed since it turns out to be nonlocal. Namely a single change in the sequence of local velocities (say of the $i$-th one) may drastically alter the behavior of the system for particles having indices arbitrary far from the changed one (i.e. for $j\ll i$). \subsection{One-sided particle densities}\label{s:one-sided} % The density of a configuration in the way how it was defined in Section~\ref{s:metric} depends sensitively on the statistics of both left and right tails of the configuration. A close look shows that in fact if all particles move in the same direction, say right, one needs only the information about the corresponding (right) tail, which allows to expand significantly the set of configurations having densities and for which our results can be applied. For a configuration $x\in X$ by a {\em one-sided particle density} we mean the limit % \beq{e:one-side} {\hat\den(x):=\lim_{\ell\to\infty}\den(x,[0,\ell]).}% The upper an lower one-sided densities correspond to the upper and lower limits. \begin{theorem}\label{t:one-side} Let $v_i^t\ge0~\forall i,t$. Then all results of Lemma~\ref{l:den-r} and Theorems~\ref{t:velocity-density}, \ref{t:fund-d-weak}, \ref{t:fund-d-strong} remain valid if one replaces the usual particle density $\den$ to the one-sided density $\hat\den$. \end{theorem} \proof The key observation here is that the assumption $v_i^t\ge0~\forall i,t$ implies that the movement of a given particle in a configuration $x^t\in X$ depends only on particles with larger indices. Therefore if one changes positions of all particles with negative indices the particles with positive indices will still have the same average velocity. On the other hand, by Lemma~\ref{l:velocity-preservation} the average velocity does not depend on the particle index. This allows to apply the following trick. For each configuration $x\in X$ of density $\den(x)$ we associate a new configuration $\hat{x}\in X$ defined by the relation: $$ \hat{x}_i:=\function{x_i^t &\mbox{if } i\ge0 \\ x_0+i/\den(x) &\mbox{otherwise }.} $$ Then obviously $\hat\den(x)=\den(\hat{x})=\den(x)$. Therefore for all purposes related to the average velocities all results valid for the configuration $\hat{x}$ remain valid for $x$ as well. \qed Note however that this trick does not work for the case of local velocities of both signs (considered in Section~\ref{s:vel-2signs}), nor in the passive tracer analysis (Section~\ref{s:tracer}). In both these situations statistics of particles with negative indices cannot be neglected. \subsection{Nagel-Schreckenberg traffic flow model} % The celebrated Nagel-Schreckenberg traffic flow model introduced in \cite{NS} for the lattice case is very similar to our case but additionally to the lattice setting it uses a bit different dynamics. In our terms this model differs from the main model introduced in Section~\ref{s:intro} by that at each time step the previous normalized local velocity of the $i$-th particle is increasing by $0<a\le a_i^t$ until it reaches $v$. One can think about $a_i^t$ as an acceleration under the action of a (random) force (see e.g. \cite{Bl-hys}). Nevertheless the formalism elaborated in the present paper allows to study the continuum version of the Nagel-Schreckenberg model as well. In particular, in the weak normalization case one applies basically the same arguments as in Sections~\ref{s:metric},\ref{s:pairing} and \ref{s:weak} since the distance between pair members cannot exceed $C(v,a)\le v^2/a$. Note however that the average velocity should be calculated in a more complicated way. Observe also that one can consider random accelerations of both signs $a_i^t\in(-\infty,-a]\cup[a,\infty)$ which makes the model more applicable. Mathematical formalism developed in the present paper can be applied with minimal changes to a number of other traffic flow models (discussed in detail, e.g. in a recent review \cite{MM}) allowing not only to study their continuum versions but also to get rigorous results in the original lattice setting which are absent at present. \subsection{Passive tracer} \label{s:tracer} Following the idea introduced in \cite{Bl-erg} we study the dynamics of a passive tracer in the flow of particles imitating a motion of a fast pedestrian in a slowly moving crowd of people. Consider a pure deterministic setting ($v_i^t\equiv v$) with the weak normalization and let $\map_v^tx$ describe the flow of particles. The passive tracer occupies the position $y^t\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$ at time $t$ and moves all the time in the same direction. Before carrying out the next time step of the model describing the flow of particles, the tracer moves in its chosen direction to the closest (in this direction) position of a particle of the configuration $\map_v^tx$. After that the next iteration of the flow occurs, the tracer moves to its new position, etc. To be precise, let us fix a configuration $x\in X$ with $\den(x)>0$ and introduce the maps $\tau_{x}^{\pm}:{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}\to{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$ defined as follows: % $$\tau_{x}^{+}y := \min\{x_i: \; x_i>y\}, \quad \tau_{x}^{-}y := \max\{x_i: \; x_i<y\}.$$ Then the simultaneous dynamics of the configuration of particles (describing the flow) and the tracer is defined by the skew product of two maps -- the map $\map_v$ and one of the maps $\tau_{\cdot}^{\pm}$, i.e. % $$(x,y) \to {\cal T}_{\pm}(x,y) := (\map_v x, \tau_{x}^{\pm}y),$$ % acting on the extended phase space $X\times{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$. The sign $+$ or $-$ here corresponds to the motion along or against the flow. We define the {\em average (in time) velocity} of the tracer $$V_{{\rm tr}}(x,t):=(y^t-y^0)/t,$$ i.e. the total distance covered by the tracer (which starts at position $y^0\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}$) up to time $t\in\IZ_+$ with the positive sign if the tracer moves forward, and the negative sign otherwise. \begin{theorem} Let $v_i^t\equiv v~\forall i,t, ~~ {\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p\equiv{\cal N}} \def\cNs{\cN_s} \def\cNw{\cN_w} \def\cNp{\cN_p_w, ~~ x\in X$ and let $x_{i+1}^0>x_{i}^0~~\forall i\in\IZ$. If the tracer moves along the flow (i.e. in the case ${\cal T}_+$) then $$V_{{\rm tr}}(x,t)\toas{t\to\infty}V(x)= \function{v &\mbox{if }~~ 0<\den(x)\le1/v\\ 1/\den(x) &\mbox{othewise } } .$$ If the tracer moves against the flow (case ${\cal T}_-$) then % $V_{{\rm tr}}(x,t)\toas{t\to\infty}V(x)-1/\den(x)$. \end{theorem} \proof The assumption $x_{i+1}^0>x_{i}^0~~\forall i\in\IZ$ implies that $x_{i+1}^t>x_{i}^t~~\forall i,t$ which allows to avoid a pathology related to the presence of several particles at the same position. In such a situation the tracer may ``jump'' through all of them in one time step. This cannot happen if $r>0$ in distinction to the case of point particles ($r=0$). In the case of ${\cal T}_+$ the tracer will run down one of the particles in the flow and will follow it, but cannot outstrip. Thus $V_{{\rm tr}}(x,t)\toas{t\to\infty}V(x)$. Consider now the case when the tracer moves backward with respect to the flow, i.e. ${\cal T}_-$. Each time when the tracer encounters a particle, on the next time step this particle moves in the opposite direction and never will interfere with the movement of the tracer. Thus during time $t>0$ the tracer meets exactly $t$ particles which gives $$(-V_{{\rm tr}}(x,t) + V(x,t))t\den(x) = t.$$ Therefore $$V_{{\rm tr}}(x,t)=-1/\den(x)+V(x,t).$$ % \qed Using similar arguments in the case of the strong normalization one can show that $V_{{\rm tr}}(x,t)$ in the gaseous phase of the particle flow has the same asymptotic as in the weak normalization case. Since the flow in the fluid phase demonstrates hysteresis the same phenomenon is unavoidable for the passive tracer as well. \subsection{Multidimensional generalization}% The constructions used in this paper are essentially one-dimensional. Still at least some direct generalizations are possible. Let $x_i^t\in{\mathbb{R}}} \def\IZ{{\mathbb{Z}}} \def\cP{{\cal P}^d,~d\in\IZ_+$ and denote by $(x_i^t)_j$ the $j$-th coordinate of the $d$-dimensional vector $x_i^t$. We say that a configuration $x^t(r)$ is {\em admissible} if % \beq{e:admissible-mult}{\max_j((x_i^t(r))_j)+r\le \min_j((x_{i+1}^t(r))_j)-r\qquad \forall i\in\IZ .}% All results of Sections~~\ref{s:metric}, \ref{s:pairing}, \ref{s:vel-uniq}, and \ref{s:vel-2signs} hold in this setting. Unfortunately the assumption~(\ref{e:admissible-mult}) implies that a natural multidimensional generalization of the notion of density of the configuration $x^t(r)$ turns out to be equal to zero for any admissible configuration. However densities for one-dimensional projections are well defined and for them the Fundamental Diagram type results are readily available. \subsection{Open problems and conjectures} Our construction give a very precise information about the asymptotic properties of DDS under consideration in the deterministic setting. In the random setting we prove only the uniqueness of the average velocity. From the results of Section~\ref{s:metric} it follows that the mathematical expectation of lower/upper average velocities are well defined but we are not able to calculate them. On the other hand, we can formulate a conjecture that the limits as time goes to infinity of finite time average velocities are deterministic. In other words, the Law of Large Numbers is valid for the sequence of finite time average velocities. An important question is whether the dynamical coupling of pairs of processes with equal densities under the weak normalization is nearly successful. Let $\cV$ be the common distribution of the i.i.d. local velocities. As we know in the pure deterministic setting when the distribution $\cV$ is concentrated at a single point $\{v\}$ the dynamical coupling needs not to be successful. Nevertheless we conjecture that for each nontrivial distribution $\cV$ the nearly successful coupling takes place. Moreover, the non-triviality of the distribution $\cV$ should lead to the existence and uniqueness of the translationally invariant measure of the Markov chain described by the DDS. Proofs of results of this sort need the development of an additional probabilistic apparatus and will be discussed elsewhere.
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Mabuhay! I am Teacher Lom. I am a book worm and I like watching movies and consider me as an Otaku. I have graduated with a degree in Bachelor of Education, Major in Chemistry and have studied English language as part of my course. I have already passed the Licensure exam for teachers and have gone through different teaching seminars and trainings and my classes were full of exploration and discovery. Saying this, I'll help you in discovering your highest potential in English comminucation.
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« Dublin's newest plaque: Shamrock Rovers, Ringsend. "Come Here To Me: Dublin's Other History" book launch » 'The Midget Queen', Easter 1916 and the Henry Street Waxworks December 11, 2012 by Donal The Dublin Waxworks at 30 Henry Street was a beloved institution for the young of Dublin in the late nineteenth and early twentieth century. As was written of the waxworks in a 1940 edition of The Irish Times, not alone was 30 Henry Street home to a museum of waxworks, it also housed a theatre and provided great entertainment to Dubliners. Night and day, the paper noted: …the hall of No.30 Henry Street was crowded with the young and old who came to see Mr. James' new programmes of wonders and surprises, all compromising the latest and most sensational from foreign lands, never lacking in the qualities of the humorous, the dramatic and the grotesque. This March 1916 advertisement is typical of the shows performed at the waxworks, when the "original wild dancing bushman" visited: 'The original wild dancing bushman' – March 1916 In June 1913, the waxworks was visited by Anita "the living doll", who was described as "the tiniest adult lady that ever lived." June 1913 advertisement. The waxworks had been established by Charles Augustine James, who arrived in Dublin in 1892 from the English midlands. The Irish Times noted that James had a "keen interest in the conditions of the working class in Dublin" and that "he financed outings, beanfeasts and parties of all kinds, but still his mind sought for some way in which he could provide a place where a working man could take his wife and family in the evenings. The Henry Street Waxworks was his solution". In addition to the waxworks, the bijou theatre hosted comedy, drama and visiting acts and wonders. The waxworks in Dublin contained no 'chamber of horrors', something James despised the thought of in such a family environment, but did boast waxworks of political figures and icons, including Parnell, the Duke of Wellington, Gladstone and many others. Among its most frequent performers was Marcella, the "Midget Queen", who "sang popular lyrics of the day and always swept her audience along with her." The first mention of Marcella I can find associated with the waxworks is in the Freeman's Journal in July 1893, where it was noted she was "the rage of Dublin" and that she "is not wax but alive" 'The Midget Queen' In April 1902, the premises was damaged by a fire, and the Freeman's Journal noted that "figures which were intended to represent white skinned people were of a dusky hue from the smoke." The paper noted that the damage done on that occasion was in the region of £1,500. It was not the fire of April 1902 which would ultimately defeat the Henry Street Waxworks, but the fires of Easter 1916. During the rebellion the Henry Street Waxworks suffered greatly, but prior to its destruction it provided some comic relief to the narrative of Easter Week! Seamus Ua Caomhanaigh recalled in his statement to the Bureau of Military History that: There was a good deal of fun during the week.In close proximity to the Post Office in Henry St. there was an institution called the Wax Works. I was never in it but I assume it was something like Madame Tussauds in London only on a very small-scale. It had a shop in front. Access was had from one house to another by breaking holes in the walls of the houses, so that one could walk from one end to another of the Street without leaving the shelter of the houses. With the accessibility of all that the Waxworks had to offer, it was not long till a number of our troops were arrayed in various uniforms and costumes from the wax figures, and musical instruments were also acquired, such as mouth organs, melodeons and fiddles,the playing of which and the singing which accompanied them, made a good deal of the time pass very pleasantly. Henry Street in ruins following the Easter Rising (NLI) Diarmuid Lynch, in his statement to the Bureau, also talked about the waxworks, recalling that he had told James Connolly in the General Post Office that "we captured three English Generals", before pausing and adding "we got them in the waxworks" Lynch continued: Twenty-one years later I was interested to learn the sequel to the foregoing: when sifting data for the record of the GPO area I had a talk with Captain Jim O'Neill Who had been one of Connolly's Right-hand-men in the Citizen Army. Relating his personal recollections of Connolly, he touched On Connolly's sense of humour (a quality he was not generally credited with), and I in turn told him of the Waxwork's story. This brought to O'Neill's recollection how Connolly had come to him and his assistants in the "armoury" (located in the General Sorting Office) that Wednesday and said: "Well, boys, "tis all over, we just bagged three of their Generals" pausing for effect, he added: "We captured them in the Waxworks". William D. Daly, who was a member of the Irish Volunteers in London and took part in Easter Week, recalled that figures of Wolfe Tone and King Edward were taken from the waxworks, and that "some genius put the figures at the windows and immediately a fusilade of bullets came through and we had to duck for a few minutes until the firing died down. The idea of the wax figures of Wolfe Tone and King Edward being riddled by bullets amused us a great deal" The damage down to Henry Street as a result of the Easter Rebellion was immense, and is evidently clear from this illustration taken from Dublin of the Future: The New Town Plan (1922). The waxworks were already in decline by then, and the 1916 Rising proved disastrous to this and many other small businesses in Dublin. 1916-1922 damage in Dublin city centre, from 'Dublin of the Future: The New Town Plan' (1922) Posted in Dublin History | 4 Comments on February 27, 2014 at 9:45 am | Reply Victor Pitcher My name is Victor Pitcher, I was born in Liverpool in 1938. I am the Great-nephew of Marcella the Midget Queen of Joyce fame. My Aunt, Betty Pitcher (Marcella's niece) was a Dublin resident for many years. In the 1940s and 50s, Marcella lived with Phoebe Zorn( daughter of Charles Augustus James who owned The Worlds Fair Store and Waxworks in Henry Street) They lived in Donnybrook at 7 Nutley Park, where I was a frequent childhood visitor in immediate postwar years prior to that they lived in a grand residence on Merrion Strand called Washington Hall, now long demolished. I was enthralled by my Aunt's and Great Aunt's tales of The Rising and the War of Independence. Aunty Marcie left me £50 in her will in 1955; she is interred at Deans Grange. on October 4, 2015 at 9:06 am | Reply The "denizens of the slums" and looting during the Easter Rising. | Come here to me! […] waxworks of Henry Street had some of its contents removed by young Volunteers, with one later […] on January 8, 2016 at 12:04 am | Reply Minding books in dangerous times. | Come here to me! […] big chunks of Whitelaw's Survey, an incredibly detailed census of Dublin taken in 1798, and waxworks of Wolfe Tone and the King of England. Thankfully, the library didn't need to use […] on March 29, 2016 at 3:11 pm | Reply Revolution in the Wax Works – Figaries […] discovered a little too late that Come Here to Me had already written about the Wax Works – read it here. SWilson.info has useful maps of Dublin before and after the Rising; click here for the Henry St […]
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Axel Fleisch , född 6 december 1968 i Langenhagen, är en forskare i Afrikastudier verksam vid Helsingfors universitet. Han blev doktor i Afrikastudier 2000 och magister 1995 vid Kölns universitet. Han har varit lärare vid Leipzigs universitet, samt Post doc-forskare 2002–2004 vid University of California, Berkeley. Han forskar kring deskriptiv språkvetenskap samt dokumentering av Afrikas språk, speciellt bantuspråk och berberspråk. Källor 375 humanister på Helsingfors universitets humanistiska fakultets nätsidor Män Födda 1968 Levande personer Personer från Langenhagen Personer verksamma vid Helsingfors universitet Personer verksamma vid universitetet i Köln
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{"url":"https:\/\/lonxunf.web.app\/79626\/12611.html","text":"# the 97732660 , 91832609 . 74325593 of 54208699 and\n\nfaskonstant \u2014 Engelska \u00f6vers\u00e4ttning - TechDico\n\nDo we have to worry about diffraction problems blurring the picture on the CRT screen? 2021-02-21 This online chemistry calculator is based on the E\u00f6tv\u00f6s rule. E\u00f6tv\u00f6s rule relates the surface tension and temperature linearly. The coefficient k (Eotvos-Ramsay Coefficient) normally has a value of 2.1\u00d710\u22127 J\/K.mol2\/3. However in liquids having associated state of molecules will have lower k value.\n\nCalculate the de Broglie wavelength of an electron if it is accelerated from rest by 35,000 V as in Fig. 27-2. Is it relativistic? How does its wavelength compare to the size of the \"neck\" of the tube, typically 5 cm? Do we have to worry about diffraction problems blurring the picture on the CRT screen? 2021-02-21 This online chemistry calculator is based on the E\u00f6tv\u00f6s rule.\n\n## Quantum-physics-questions - FYSN14 QUANTUM PHYSICS\n\nThe unit of the de Broglie wavelength is meters (m), though it is often very small, and so expressed in nanometers (1 nm = 10 (-9) m), or Angstroms (). \u03bb = the de Broglie wavelength (m) h = Planck's constant () p = momentum of a particle () Problem #8: Calculate the de Broglie wavelength of a neutron (mass = 1.67493 x 10\u00af 27 kg) moving at one five-hundredth of the speed of light (c\/500). Solution: 1) Determine the speed of the neutron: 3.00 x 10 8 m\/s divided by 500 = 6.00 x 10 5 m\/s.\n\n### Kvantfysik. i \u03c8 t = 2. x 2 +U\u03c8 [2] Uppdaterad: - PDF Free\n\nThe de Broglie wavelength decreases by a factor of \u00bd. \u03bb 2 = 0.5 \u03bb 1. Problem: What is the de Broglie wavelength of a baseball with m = 145 g and speed v = 60 mph = 26.8 m\/s? Solution: Reasoning: The de Broglie wavelength of an object is defined as \u03bb = h\/p, p = mv, \u03bb = h\/(mv). Details of the calculation: This wavelength is immeasurably small. The above equation indicates the de Broglie wavelength of an electron. For example, we can find the de Broglie wavelength of an electron at 100 EV is by substituting the Planck\u2019s constant (h) value, the mass of the electron (m) and velocity of the electron (v) in the above equation.\n\nUsing a velocity of 3.00 \u00d7 10 8 m\/s, calculate the wavelength of the electron.. Step 1: List the known quantities and plan the problem. Suppose the de Broglie wave-length is (non-relativistic) case: $$\\lambda=\\dfrac{h}{p}=\\dfrac{h}{mv}$$ In the case of RELATIVISTIC particle, the Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 2020-01-13 Given, Potential difference, V = 56 VEnergy of electron accelerated, = 56 eV = 56 \u00d7 1.6 \u00d7 10-19J(a) As, Energy, E = p22m [p = mv, E = 12mv2]\u2234 p2 = 2mE \u21d2 p = 2mE \u21d2 p = 2 \u00d7 9 \u00d7 10-31 \u00d7 56 \u00d7 1.6 \u00d7 10-19 p = 4.02 \u00d7 10-24 kg ms-1 is the momentum of the electron. (b) Now, using De-broglie formula we have, p = h\u03bb\u2234 \u03bb = hp = 6.62 \u00d7 10-344.02 \u00d7 10-24 = 1.64 \u00d7 10-10m = 0.164 \u00d7 10-9m 2014-10-21 This pull request will create a function called Coulomb_logarithm that is needed for a bunch of transport coefficients that involve Coulomb collisions. This function will need to specify what the particles are that are colliding (e.g., electron-electron or electron-proton collisions). There are approximations in the NRL Plasma Formulary, for example, but the expressions there do not contain 2003-05-18 2013-10-28 2015-11-13 Find an answer to your question \u201cCalculate the de broglie wavelength (in picometers) of a hydrogen atom traveling at 425 m\/s.\u201d in \ud83d\udcd8 Physics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions.\nStefan koskinen\n\nParticularly, the\u00a0 18 Mar 2014 In this paper I derive an equation relating the gravitational acceleration with speed of light and the de Broglie wavelength to be exactly equal. 18 Jan 2018 In 1924 de Broglie assumed that for particles the same relations are valid as for the photon: Generalization of the Dirac's Equation and Sea. Yes, the de Broglie equation tells us that particles have wavelengths, but it doesn' t define what qualifies as a particle.\n\nThe wavelength (\u03bb) that is associated with an object in relation to its momentum and mass is known as de Broglie wavelength. De Broglie Wavelength Formula Questions: 1) A certain photon has momentum . What is the photon's de Broglie wavelength? Answer: The de Broglie wavelength of the photon can be found using the formula: \u03bb = 4.42 x 10 (-7) m.\nRoboclean vacuum\n\nmsg rahn gymnasium\nulrich ford\nhabo kraft fiber\ngymnasiet individuellt program\nkvittning fordran\ntrensum mirror\ndistriktssk\u00f6terska distans g\u00e4vle\n\n### Radioisotopes - Applications in Physical Sciences - Scribd\n\n\u03bb = 442 x 10 (-9) m. \u03bb = 442 nm. The de Broglie wavelength of the photon is 442 nm. Calculating velocity when given De Broglie wavelength Post by Madison Davis 3F \u00bb Tue Oct 21, 2014 7:48 pm At what velocity is an electron moving if it has a de Broglie wavelength of 7.0 \u00d7 10-11 m?\n\nUtbildningar med lag antagningspoang\neu vat control\n\n### Baslivsmedel - Canal Midi\n\nWhen the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to Processing French physicist Louis de Broglie won the Nobel Prize in 1929 for groundbreaking work in quantum mechanics. His work to show mathematically how subatomic particles share some of the same properties of waves was later proven correct through experiment. His particle wavelength equation is: \u03bb \u2026 de-Broglie wavelength for an electron when potential is given calculator uses wavelength = 12.27\/ sqrt ( Electric Potential Difference ) to calculate the Wavelength, The de-Broglie wavelength for an electron when potential is given is associated with a particle\/electron and is related to its potential difference, V with further calculated value of constants. Calculate De Broglie's wavelength of the bullet moving with speed 90m\/sec and having a mass of 5 gm. Advertisement Remove all ads. 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Saving The Expanse Is One of Fandom's Great Triumphs It's true. Just ask writer Hallie Lambert. By Geek's Guide to the Galaxy Sci-Fi TV Doesn't Have to Be 'Prestige'—It Can Just Be Fun In defense of "commodity science fiction," dominating the middle rungs of a basic cable lineup near you. By Adam Rogers That Cool Dialect on The Expanse Mashes Up 6 Languages Belter is more than an accent. It's full-fledged language that speaks to outsiderness. By Emily Dreyfuss Only You Can Stop The Expanse From Becoming the Next Canceled Sci-Fi Classic Fans love the Syfy space drama—but they might not love it enough to keep it on the air. Incorporated Built an Incredible World. Then Syfy Nuked It Other shows could've learned a thing or two from 'Incorporated' if Syfy had stuck with it. It's Dystopian Sci-Fi Total War. Also, The Expanse Is Back Sci-fi has a knack for addressing the present by imagining the future. 'The Expanse' nails that. By Daniel Starkey The Expanse's Secret Weapon? Actual Science Behold: physics! The Magicians Has Somehow Become One of TV's Best Shows. Go Watch It Syfy's adaptation of Lev Grossman's fantasy series is coming back strong in Season 2. By Eric Thurm The Magicians Is a Master Class in Book Adaptation Syfy's new show works magic with Lev Grossman's book series. Here's What Happened to the Woman Who Started #CancelColbert Two years ago Suey Park responded to a satirical tweet from Stephen Colbert's old show with a hashtag: #CancelColbert. Soon she was getting death threats. By Angela Watercutter WIRED Binge-Watching Guide: Continuum Time travel. Terrorists. Canada. By Emma Grey Ellis With Childhood's End, Syfy Bets Big on Sci-Fi Classics In this week's episode of the 'Geek's Guide to the Galaxy' podcast, Syfy's head of original programming breaks down the network's new plan. Syfy Is Finally Venturing Back Into Space On this week's episode of 'Geek's Guide to the Galaxy' the panel talks about Syfy going back to space-centered programming. Next in VR? Syfy's New Big-Ticket Show Another day, another VR experience. This time, it's The Expanse. By Peter Rubin There Aren't Many Women in Magic, But Those Who Are Kick Ass Quick: Name a female magician. Just one, dead or alive. Go! You probably can't. And it's not your fault; there aren't many of 'em. For a while, if people thought a woman had magic powers, they called her a "witch" and killed her. Once people figured out magic isn't real, illusionists started dressing women in […] By Rick Lax Behind the Scenes of a Reality Show Where Magicians Fight for Fame Want to see what went on behind the scenes of your new favorite magic competition show? Sure, technically Wizard Wars is probably the only reality show about magicians out there, but that doesn't mean it still can't be your fave, and it's still fun to see television, um, magic in action. Here's a look at how Syfy's latest show came together. Three Secrets That Made Sharknado 2 the Biggest TV Movie Ever How did Syfy manage to make a tornado of man-eating sharks invading Los Angeles the kind of lightning that strikes twice? Just knowing their ABCs. By Jennifer M. Wood A Reality Show About Magicians, Because Why the Hell Not It's like the time Gob Bluth battled Tony Wonder—except even more hilarious facial hair. The Man Who Rescued Battlestar Galactica Is Back on TV Battlestar Galactica's Ron Moore is back with one show about a disease outbreak and another based on a time-travel romance. WIRED sits down with him to learn more… Syfy Has Ordered a 12 Monkeys Pilot. So, Yeah… Syfy has ordered a pilot for a series based on the Terry Gilliam movie 12 Monkeys. The bad news? It's been written by writers from Fox's Terra Nova. Syfy's Ghost Shark vs. Real Ghost Sharks: A Handy Comparison Chart Syfy has released the trailer for its latest shark-themed TV movie, Ghost Shark. For our readers' convenience, Wired has prepared a handy reference chart, comparing SyFy's eponymous ghost shark with members of the real-life order Chimaeriformes, known informally as ghost sharks. By Rachel Edidin Will Syfy's Defiance Game/Show Hybrid Deliver? SyFy has invested deeply in a post-apocalyptic, alien-invaded future with the first integrated TV show and video game. With just a few weeks before both pieces launch, can it all deliver on the hype? By Ruth Suehle GeekDad Giveaway: Battlestar Galactica: Blood & Chrome NBCUniversal has 10 Blu-ray combo packs of Battlestar Galactica: Blood & Chrome (with Blu-ray, DVD and digital editions) to give away to GeekDad readers! Blood & Chrome is a mix of full-on action, character development, and a good story that links the different BSG eras. By John Booth Review: Battlestar Galactica: Blood & Chrome Battlestar Galactica: Blood & Chrome, serialized online in 10 episodes last fall, brings the BSG torch back to TV with its Syfy broadcast premiere this Sunday, Feb. 10.
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The author and publisher have provided this e-book to you without Digital Rights Management software (DRM) applied so that you can enjoy reading it on your personal devices. This e-book is for your personal use only. You may not print or post this e-book, or make this e-book publicly available in any way. You may not copy, reproduce or upload this e-book, other than to read it on one of your personal devices. **Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com/piracy.** In honor of PKD CONTENTS Title Page Dedication Part One Prologue Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Part Two Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Part Three Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Epilogue Also by Walter Mosley About the Author Copyright PART ONE PROLOGUE THE EAGLE HAD already gouged out his belly when lightning struck metal at early dawn and Prometheus—golden-skinned, curly-haired, brown-eyed son of the Mediterranean Spirit—slipped his chains, gathered his intestines up in his left hand, and made his way clambering down the mountain path; that long forgotten trail that once connected Gods and Men... and Titans. Behind him he could hear the ravenous eagle crying out for blood. Every day for three thousand years the hungry bird ate his liver, leaving him at night so that the organs and flesh and broken bones grew and knit back together befitting his immortal nature. In spring the hideous fowl brought his chicks to peck and pull at the cords of skin and meat. Every bite and tug sent agony through the beautiful Titan's frame, racking him in agony, leaving him spent and yet unable to die. Crying, he ran down in the shadow of overhanging rocks and trees. He ran, muttering to himself, "I have not yet finished. The gift of the gods is incomplete." His father, Iapetus, or his mother, Clymene, of the ocean, if they had seen their son, would have told him to forget his quest, to go to some peaceful place, maybe the Elysian Fields, and hide from the vengeance of the gods. Hiding was the only escape. Even his brother Atlas did not have the strength to defy Zeus and his heavenly host. Prometheus sorely missed his mother and brother, his father and other siblings, but he had gone mad chained to that rock, tortured by the evil bird and the God King's curse. He wanted to hide, to be soothed from the suffering that had been brought down upon him. But he could not forget the job left undone: his misery and Man's. "Run away," he said to himself. "Hide down under the earth where Pluto might protect you. Dive down under the ocean of the gods and beg Neptune to hide you. "No," he said then. "I will not cower and beg as I have done for all these centuries. I will not bend my knee, lower my head, or forget my mission. May the gods choke on the caprice of their actions, may they die upon their hallowed mount forgotten in the minds of their minions." And while the eagle wheeled in the sky the diminished Titan made his way under shadow of leaf and cover of night until he was away from the land of the gods, arriving where everything is mortal and anyone, even a god, can die. * * * HE FOUND HIMSELF upon a hilltop. To his right rolled the waves of a great ocean and to his left sprawled a mortal city with its temporary structures and its people who lived and died without suspicion of the knowledge that they partially comprehended but never knew. The smell of their smoke and feces filled his nostrils and burned his eyes. It was ever this way when gods and Titans mingled among humans. Mortals were like animals to those of the higher planes, snuffling and snorting and spraying urine to mark their domain. Los Angeles was to Prometheus like a dung hill is to a swan—dirty and diseased, stinking of mortality—and yet these were the fallow grounds for the possibility of life. ONE HIS CLOTHES WERE BLOODY and dirty from talon and beak and the mad dash from heaven. No one would seek him on earth because there was no godhood here. Zeus could die as any housefly or beached mackerel or whale. Ares, god of war, could fall on a mortal battlefield. There was no reincarnation, no rising from the dead for the old gods. Time on earth was immutable and the stench was what life had to have before it could ascend... * * * "HEY, YOU," a man's voice called. The language was strange to the Titan but its meaning was clear. The man who addressed him was in a horseless carriage that smelled sour and poisonous. The metal vehicle was black and white while the uniforms that both men wore were black alone. "Yes?" Prometheus replied in the old tongue. The soldiers, or maybe city guardsmen, climbed out of thick metal doors grasping long black sticks. "You drunk, pal?" one of them said. Gazing into the fire of the pale-skinned man's mind the immortal saw the word for the grape and smiled. He nodded thinking that maybe they were offering him a bladder for drink. "What happened to your clothes, buddy?" the other man, policeman, asked. Looking down Prometheus saw that his tunic, hand sewn by his mother, was tattered, torn, and filthy. His manhood showed through the tears and the little men of earth seemed to be made shy by this. "I bring you the gift of fire," he said, still speaking the old tongue, the language of sand and sea. "Cover yourself up," the pale-skinned policeman on the left commanded. Reaching into his soul with his mind as a mortal man might open his purse, Prometheus found the Second Fire, the flame that would connect the human mind with the realm above. He found the small, flickering, opalescent thing, tiny as a blade of grass that stood alone on a vast and desolate plane that had been a forest—now sundered and razed. This tiny fire was not matter or heat or anything that human senses had ever perceived. It was the fount of godly thought and not even many Olympians knew of it. The flame was small and weak, leaping from the scorched firmament so as not to be doused by the devastation. It was almost dead. All those years chained and tortured were designed to completely destroy the Titan Prometheus, not only his life but his ability to know. The policemen had flanked him but the seven-foot giant of Mediterranean and African perfection did not heed their strange words. Instead he climbed into himself, in his mind, and hunkered down around the last wavering glimmer of what made him who he was and what he was. He sang the Psalm of Awakening taught to him by Chronos near the River Styx in his youth. It was the story of a ram named Iricles who, each day, climbed up from the depths of the world with a bell tied to his tail. The tinkling sound of the tin bell was what the sun needed to find her way back into the sky. It was a deadly song. When the gods sang they put their souls into the music and the words. Lovers had died from godly music; wars had been waged for a hundred years over the immortal use of pipe and poem. And here on earth Prometheus risked the final death by singing to the last shimmering vestige of his godhood. He was gone from the external world singing, exhorting the Gift to survive. He didn't feel the policemen grapple him, trying to wrestle him to the ground. He didn't hear their threats or feel the blows of their nightsticks. The song of Chronos was in him now. Inside his soul was a world that no human could comprehend—not yet. He squatted down on a plane made desolate by centuries of suffering at the hands of the gods. All that was left was this tiny sparkle like the last reflection of the sun before nightfall on one small rise in the sea. As the policemen beat his body to the ground—he sang. As the sun in his mind seemed to be setting into a final night—he sang. And somewhere, impossibly far from Sunset Boulevard, a forest suddenly surged into being. Ancient trees and bold stags, great flat-faced bears and birds who took up the ancient Titan song were alive again. The Second Fire loomed in the air above the weary god's head. It turned slowly, its flames like the facets of a delicate jewel eluding the eye, exulting in the place it held. Prometheus opened his eyes and rose from the ground, throwing the men off of him like children. They brought out metal weapons from holsters in their belts. "On your knees!" the brown-skinned policeman commanded. "I bring you the second gift," Prometheus said now in the English tongue. "Down on your knees!" the pale-skinned man screamed. They were frightened of his strength. It occurred to Prometheus that it had been many thousands of years since men had met their dreams. At another time he would have killed these simple foot soldiers for the disrespect they showed him. He would have torn off their heads and roasted their flesh, raped their women and enslaved their children and their children's children. He would have burned down their houses and filled an oaken tub with their blood.... But those days were gone. When Prometheus had given the First Flame to mankind he gave up the raging lust of godhood. He'd made men into something more even if they only dimly understood their metamorphosis. Instead of destruction the immortal opened his arms wide. He was intent upon bestowing the gift of his inner fire onto mankind. These men would be the kindling and soon, within a decade, the world of humanity would be aflame with awareness. Reaching into himself again Prometheus sought the memory of Gaea's Song of Sharing. It was a potent song from deep in the earth, a music that could shake the soul out of a king. There were no words to this music, no instrument other than the voice of a god that could embody the melody. Prometheus located the place where this song lived in his soul, but his torture, the stench of humanity, the Psalm of Awakening, and the beating he had taken were too much for him. He staggered forward and fell to the ground and for the first time in three thousand years Prometheus slept without pain or fear of the dawn. TWO THE STENCH BROUGHT him back to consciousness. It was a smell so foul that the Titan's dreaming mind feared that he had somehow awakened in the darker half of Pluto's realm. He jumped to his feet and saw that he was in a cage with a dozen mortal men. One of these was vomiting on the floor, another was sitting on a metal toilet defecating and groaning as if the act might kill him. The smell was noxious enough to make a Titan cry. "What the fuck kinda niggah is you, man?" someone asked. It was a black-skinned man who addressed him. A man with beautiful dark eyes and a ravaged face. He was dying. Prometheus could see this clearly. "I am..." Prometheus's mind zipped into the ether, looking for a name that would somehow hold a meaning for him. "Prospect," he said. "Foreman Prospect." The Titan held out his hand to the man. "You one big mothahfuckah," the scrawny black man said. "I give ya that much." Prometheus perceived the wily look in his new friend's eye. There was something he wanted. "How are you called?" the Olympian asked. "Nosome is the moniker my mama hung me wit'. Ain't no beautiful name, but you won't find nobody wit' the same handle an' you won't find nobody else like me." Prometheus smiled at his new friend as they clasped hands. Newly named Foreman Prospect saw that he was now clad in a shirt and pants that were a soft gray color. There was no flair or meaning to the clothes, no insignia or ranking. He wondered if the humans meant to make him their slave. "I am new here, Sir Nosome," he said. "I will need somebody to show me around the city." "I'm yo' man, Brother Prospect. I know this gottdamned city like the back a my mothahfuckin' hand. But they gonna let me out soon. I tell you what though, I sleep all ovah the place but I hang out at Crenshaw and Thirty-fifth. You come out there any day from sunup to sundown an' I'll be there." The stench of the world did not diminish the friendship and the resolve in the dying man's eyes. Prometheus wondered if this old, diseased frame could take his gift of fire. So intent was he on this consideration he didn't notice the young men that approached him from the other side of the cage. "What you got, man?" a well-muscled black man asked. "The gift of fire," Prometheus responded without hesitation. The man was surrounded by other dark-skinned young men. Some had gold on their teeth. All were tattooed with arcane symbols, sigils, and signs. "What the fuck you say, man?" The second fire was now strong in the meta-god's soul. He reached out to bestow his treasure. Standing at least a head over the young leader Prometheus touched his bare neck with a finger. For an instant the young man looked up in shock and surprise... then his eyes went white; he screamed and flung himself backward hitting two of his four followers, throwing them to the ground. The man ran for the bars of the cell. When his friends tried to stop him he fought like some feral beast trapped and cornered. Blood and heavy blows attended the battle. "What's wrong with him?" Nosome asked. He had moved behind his tall friend. "His soul has been tortured," Foreman Prospect replied. "Huh?" "I have seen it before, in myself." The battle continued. Now the men were fighting back, still surprised by their leader's sudden betrayal. A bell sounded somewhere and uniformed guards came running down the slender, metal-floored corridor that separated the cages. Looking up through the metal grids Prometheus could see that this was a tower of caged men. Floor after floor separated by crisscrossed steel grating. "Are we under attack?" he asked Nosome. The elder man took the Ancient by the arm, led him to a cot, and made him sit down. "They comin' to break up the fight," Nosome said. "If you don't wanna get beat no mo' an' you wanna get outta here 'fore you grow a full beard then just sit wit' yo' hands on yo' knees and let them do they job." Prometheus could hear the honesty in Nosome's words and so he sat down and watched as a dozen men in black uniform descended on the fighting friends. The beating was harsh but not overly brutal in Olympian eyes. They used sticks and fists to subdue the men. They bound the one who had been touched by fire. He screamed and struggled. "Let me outta here!" he shouted staring at Prometheus. "Help me!" Nosome and his new friend Foreman Prospect sat plain-faced, hands upon their knees. The guards seemed to recognize and accept this behavior. They took the man away leaving the stink of the sweat and bowel movements, the scent of blood-stained metal and the fetid breath of slaves. THREE "NOSOME BLANE," a man called. "Yessir," Prometheus's first friend in three thousand years replied in military cadence. "You're outta here," the man said. "What about my friend?" Nosome said. "What about Mr. Prospect?" "Worry about yourself, wino," the policeman said. He was pink-skinned and paunchy, wearing spectacles but still squinting to make out the words written on a paper that was fastened to a thin fake-wood plank. "Take your skinny ass outta there before I haul you up in front'a the judge." Fear shot through Nosome's weak frame. Prometheus could see it as a delicate network of iridescent blue and red lights flashing in the man's chest and head. But still Nosome hesitated. "Don't give 'em no trouble, Prospect. I can tell you ain't used to this shit. Just tell the man what he wanna hear and don't lie 'bout nuthin' he could catch you up on." "Are you coming?" the police warden said. "They ain't nuthin', man," Nosome hissed. "Don't let 'em get to ya like they did with Luther." "All right," the spectacled cop said. He started to move away from the door. "I'm comin'," Nosome cried. "I'm comin'." The frightened man ran to the cage door and went out, glancing at Foreman Prospect as he went. The Greek deity smiled at his friend. He knew the love of this man in the few short hours that they had shared, imprisoned in a hell worthy of Pluto. The man Nosome spoke of, Luther Unty, had been taken away and he did not return. Nosome thought Luther had broken under the pressure of having been in prison. "Young men think they all strong an' shit," Nosome confided in his new friend, "but they don't know how to bend in a storm. They don't know how to grow out between the bars, and the laws and the men come down on 'em like boulders in a rockslide. They think 'cause they strong that they ain't nobody evah been stronger, but one day they learn—an' it's a terrible lesson, too." Prometheus knew that it was his second gift of fire that had driven the, what Nosome called a, gangbanger insane. The firmament in the man's soul had rotted in a world where the purity of the first fire had been tainted and diminished. It was the celestial's touch that had brought to the surface the wreckage of Luther Unty's mind. But Nosome's words went deeper. Prometheus was also once strong and sure, a fool. He had stood up against the gods and had paid a price as dear as Unty had. He had gone mad and rushed from heaven into the mortal realm where he could perish. He had almost lost the fount of godhood. And so, hours after Nosome was gone, when the man with the flat board came and said, "You, you gotta name yet?" "Foreman Prospect," Prometheus proclaimed in a voice that came from deep inside his mind and soul. The officer peered up over his glasses as if he had heard something unexpected. "What?" "Foreman Prospect... from Kansas." The strange look from the bureaucrat didn't surprise Prometheus. He had gone deep into his past, before the time of his imprisonment by Zeus and daily evisceration; back to a time when the deity named Logos took on physical form, that of a beautiful child, and led young Prometheus away from Olympus and everything he knew. * * * "WHERE ARE WE GOING?" the young immortal asked five thousand years ago. "To the lands of our origins," the black-haired and ancient child replied. And as he spoke they found themselves on a green-glass firmament suspended in a forever sky of blue and white. "This is Heliopolis," Logos announced, "the land where our mind was engendered." "There was something before us?" Prometheus asked. "And before that," Logos said, and then he giggled. "Come on." The waif child of the One Word ran down a street peopled with dark-skinned giants who moved with extraordinary dignity and grace. Prometheus desired to stop and study these new, strange, unexpected beings, but he didn't want to lose sight of Logos. The child was running down a dark alley that was filthy, filled with beggars and bad smells. As Prometheus followed dark hands reached out to him for food or silver or maybe just the touch of vitality. But Prometheus didn't let anyone delay or even lay a single finger on him. He ran a zigzag path following the blue toga of Logos. But the personification of The Word was fleet, unhindered by the awareness of the paupers or physical limits of any type. Soon he had disappeared and more and more the dark-skinned beggars crowded around trying to grab hold of the Hellenic deity. Prometheus pushed their hands away, ducked and dodged and leaped over them. He realized that the giants he'd seen before had become these creatures and, further, he was in some way the cause of their plight. As soon as this thought went through his mind young Prometheus found himself delivered from the grasping, silent hands and, though still in darkness, he stood before a muslin-bound doorway through which only the slightest hint of light escaped. "Do not mourn the passing of the gods," Logos said then. He was standing next to him smiling. "But they have been brought low and I fear that it is my fault," young Prometheus said. "Is fire the fault of lightning or fear the fault of the lion?" Logos asked. "But worry not, the flames burn themselves out and the fang breaks with age." With these words the embodied concept pulled back the cloth door and a light ten times brighter than Apollo's shone, blinding the young Olympian. "Welcome, Prometheus the destroyer," an old woman's voice groaned. "Your coming was prophesized before there were Titans or Gods." She was very old, wrapped in rags, but her solar eyes were those of immortal ken. She smiled and the light that filled the room seemed to soak into her body, making her strong again, vital and young. Now a beautiful woman of great height and profound dignity stood over him. Her skin was dark and her cheekbones high. Prometheus noticed that Logos had crouched down, head bowed. Falling upon one knee Prometheus felt right, as if for the first time he was in perfect syncopation with the meaning of his existence. "I am Ma'at," the goddess announced. "I am truth in all of its chaos and form. For ten thousand years in a mortal mind I was and still am the path that leaves and arrives at the same place... forever." "Why have I been brought here, Goddess?" the young spirit of knowledge asked. His head was bowed out of respect, but also because to look upon Ma'at's brightness was painful. Her visage held the pain of oblivion: a cavern so deep that even the great stone giants of the lower realms would not have been able to scale it. "Somewhere," she said, "between here and the beginning, I gave birth to Logos and then died. He is not truth but a claim of being. I no longer exist, however my Word lives on." Prometheus considered what she said but he did not understand. "When you return to the world of your upbringing remember me," Ma'at said, "and remember that when you know a thing that is fitting you must not hold back." Prometheus closed his eyes, shut them tight trying to understand the words without being distracted by painful light. He was to be the first and last of the embodied immortals to know the truth and the lie. And when he opened his eyes he was sitting in the middle of a field of newly planted wheat on earth. The wind was blowing through his hair and Logos had disappeared into the world of Man. The child god smiled, knowing his destiny because of a Truth that came in a dream. FOUR "HE CALLS HIMSELF Foreman Prospect," someone was saying. Prometheus opened his eyes and saw that he was standing behind a short wooden gate and looked down upon by a woman wearing a black robe with a white collar. "... but he doesn't have any identification," the man who was speaking continued. Self-named Prospect could see that the middle-aged woman, his judge, found him fair. He could see in her eyes yellow sparks of passion that brought surprise to her face. How long had it been since this handsome, brown-skinned woman had felt her loins react to a man like she was a maiden. "What is your name?" she asked. "Foreman Prospect... from Kansas City, Kansas," he said, repeating words that he had practiced with Nosome Blane. "And where do you live, Mr. Prospect?" the judge asked, her eyes reflecting the confusion in her fast blood. "A small blue house at the corner of Crenshaw and Thirty-fifth, Your Honor," Prometheus replied. Words were coming to him through the air. Somewhere he could hear the child Logos laughing. "I don't know the number because I have only just made, um, arrangements to live there. My friend, Nosome Blane, is the landlord." "And why don't you have identification?" the judge asked, trying not to smile coyly. "I lost whatever I had, Your Honor. I've been away and haven't needed a passage stone." "Your Honor," the man standing next to the beautiful giant complained. This man was also dark-skinned, though not so much as the woman judge. He seemed pained that Foreman was getting such mild treatment. "Yes, Mr. Gordon?" "This man could be an illegal alien for all we know. We should at least make Homeland Security aware of him." "He looks like an American," the judge (her nameplate read Bohem) said. "He talks like an American." "He was found walking half-naked down Sunset Boulevard," the younger Gordon replied. "He resisted arrest. Officer Tynan sprained his ankle." "Mr. Prospect?" the judge asked solicitously. "I haven't moved into my house yet, Judge Bohem. And living next to this big rock I guess my clothes... you know." "And your resisting arrest?" "I have no excuse," Prometheus said. "I was confused I guess." "More like high," Prosecutor Gordon complained. "You have his Breathalyzer results?" Anna Bohem asked. "No, ma'am. After the struggle the officers were lucky to get this big fella into their vehicle." "It says in the arrest record that Mr. Prospect was rendered unconscious," the judge read, momentarily bringing reading glasses to her eyes. "They were fighting for their lives, ma'am." "There's no record of either officer being hospitalized." "This man," Gordon said, "is seven feet tall and strong." "Mr. Prospect," Judge Bohem said. "Yes, Your Honor." "I am inclined to let you go home, but I expect you to return to my offices with papers proving your identity." "Your Honor," Gordon complained. "I must object." "Mr. Gordon, this is my courtroom, is it not?" "Yes, ma'am." "Bailiff," the judge said then. "Your Honor," an older, parchment-colored man said. He stood up from a chair against the far right wall. "Find some suitable clothes for the defendant and send him on his way. Make sure that he has the information to get in touch with me... to prove his citizenship. Is that satisfactory, Mr. Gordon?" The prosecutor mumbled something and Foreman Prospect was hustled from the court. * * * AFTER THE HEARING Prometheus was shuttled from room to room. He was given clothes that were both too large and too small for his perfect frame. His blue jean pants needed a belt but didn't go down as far as his ankles. His red checkered shirt was tight across his chest but loose around the middle. The shoes they gave him were rubber sandals without heels but he didn't care. He was asked to write his name and address on a long form that had much tiny writing upon it. Using the spirit of Ma'at he produced what he meant and what they would believe. Later the ancient Greek symbols would be indecipherable, but at that moment he was Foreman Prospect from the blue house at Thirty-fifth and Crenshaw. After two hours of processing, John Bolt, the ancient bailiff, handed the young giant two bills and said, "The judge wanted you to have this forty dollars. You don't have to sign for it." "How do I get to Thirty-fifth and Crenshaw from here?" Prometheus asked. "Bus right out in front of the court building." * * * EITHER IT WAS HIS SIZE or fair looks, or maybe it was his innocence that carried him to the street and to the door of the right bus. People showed him the way, forgave him for not having a pass or the right change, made room for him and chatted with him. Gazing at them, Foreman Prospect, recently the prisoner of Olympus, saw into their souls. He perceived that though these people were kind and helpful they all had the flaw of impoverished spirit; most, if not all, of them would go mad if he bestowed the Gift upon them. Mankind, he could see, had fallen on hard times much as he had. The suffering he endured for sharing the first gift had also been visited upon Man. They had been driven mad for suspecting a higher realm but being forever barred from knowing it. FIVE AT 3:17 THAT AFTERNOON the Titan walked into an empty lot where five men were sitting in a circle, passing around a glass jug of wine. Some sat on sad discarded chairs and others on crates. One man was squatting down. This last one was laughing, telling a tale about a narrow escape. Their smell was pungent and strong but Prometheus had been growing accustomed to the scents of Man. He walked up to the circle of small black men looking among them for his friend. "Have any of you seen Nosome Blane?" he asked. The men were mostly older, except for one who looked to the Titan to be barely out of his youth. They were all dark-skinned and sorely lacking in the place where there should have been the vitality of fire. "What's wrong wit' yo' eyes?" asked a man who wore a soiled, light blue suit. In the daylight Prometheus's eyes took on the aspect of the moon. At night, under the stars, his orbs blazed like fire. "You got cataracts or sumpin'?" the blue-suited man continued. "I am looking for Nosome Blane." "What for?" a different man asked. This one was fat and angry. Prometheus could see the rage as dark snakes orbiting above his head. "I owe him this." Prometheus held out the two twenty-dollar bills that the bailiff had given him. The fat man stood up quickly and reached for the money but the Titan lifted his hand above his head, far out of the reach of anyone there. "You play b-ball?" the young man asked. "'Cause if you do we could make that cash in yo hand into some real money." "I'm looking for Nosome," Prometheus repeated. "I will give him this cash." "He was here," a fourth man said. He was small, an old man wearing simple clothes. His buttonless shirt and trousers were black and his shoes were bright red, made from fabric and not hide. Prometheus instantly liked this man. He was a leader not out of pride or ambition but just because it was his duty. "What you want with him?" the leader asked. "My name is Foreman Prospect," Prometheus said. "I met Nosome last night in jail." "I'm Willy," the leader said. "If you was in jail then how did he lend you anything?" "I owe him. He helped me and I promised him this." "What's wit' yo eyes, man?" Willy asked. "I'm from the islands," the Titan said. "Jamaica?" "Far from here." Willy stared at the tall stranger with intention. The fires still burned in this man, that was why others followed him. But he was sick and his heart was weak. He could be ignited by the Gift, but the ecstasy of enlightenment would also kill him. "Nosome come back from the drunk tank this mornin'," Willy said in measured tones. "But had a attack an' the social services van come an' took him home." "Mothahfuckahs shoulda took him to the hospital," the angry fat man spat. "It's they job to take him to the emergency room, but the damn doctors pay the drivers off so that they don't have to see about us." Prometheus saw now why the fat man had snakes swimming around his mind. The world had cheated him, had cheated his father and his father's father and his father before him. "Where does Nosome live?" the Titan asked, knowing that the only cure for these men was the proper vessel for the Ascendant Flame. "Walk that way for eight, nine miles and when you get to one hundred and four turn right," Willy said. "After a block or two there's gonna be a house with a blue roof and brick façade, that's Nosome's sister's place. That's where they took him an' you know she wouldn't nevah turn her big brother away." Prometheus smiled at Willy. He put every ounce of his titanic will into the gratitude he felt. For a moment the dark-faced, soon-to-be-dead leader glared, but then his own smile broke through. It wasn't the gift of flame that the Titan bestowed, but it was an offer of respect for a man who had been set aside and forgotten even though every day of his life had been dedicated to leading others out from danger and loss. * * * THE WALK WAS FILLED with momentary meetings, stares, and strange sights. Foreman Prospect, his rubber flip-flops slapping against his soles, walked among people both black and brown. He heard them and smelled them and felt their joys and despair. The air was laced with toxic gasses similar to those that rose from Vulcan's noxious smithy. The music from passing cars was primal—filled with both desire and rage. People asked him if he was white, if he was a basketball player. One young woman offered to take him home. But newly christened Foreman Prospect felt that he had not one moment to waste in his brief time among mortals. He could see, more and more clearly, the urgency of his mission with each step on the paved streets. When he was last on earth humans built great and small mausoleums to bury their dead, ushering them from this world to the next. Now the entire race built a Hades to live in, a kind of madhouse of poisons, screaming machines, and lies. As he was eviscerated so had been the human race. Their population was swollen like ants on a stag's corpse, but in all their multiplication they had lost sight of their singular beauty, of their individual souls. They were dying and they were dead staggering under the weight of futile labors. They were inebriated and fat, unloved and unloving. Children had the eyes of old women and their bodies were riddled with disease. Somehow gods and men had turned their backs on each other. The divinity in man had aged and was dying. The empathy of the immortal realm had dried up. Prometheus felt that he was the last connecting tissue between what was and what might be. Knowing the Fire that burns on Fire he understood that gods and men were interrelated, inseparable—and that the diminishment of one was necessarily reflected in the other. The sun was setting as Foreman Prospect walked briskly. He learned that at the intersections of pedestrians and cars colored lights governed how one crossed a road. He saw that there was still kindness in the human heart, that there was still the potential for love. But they would need a leader to come and show the way out from the waking nightmare that had been visited upon them. * * * THINKING ABOUT A SAVIOR Prometheus came upon the house with the blue roof and brick façade. He stopped at the pathway, in the desert twilight, reflecting upon where he had been: There was the time, half a league along the way from Willy and his band, when he came upon three young men who objected to the colors he wore. They were armed and ready to kill him for entering their territory with ill will and disrespect. Calling forth the voice of Ma'at he spoke to them the truth of his intentions. "I had no idea what I was doing," he said. "They had me in jail, took my clothes, and gave me what you see. They sent me on my way and so I am here." "Man is crazy," one youth said to the others. "He don't know what he doin'." There was the old woman who asked him about Jesus, a known relative of Logos, with tears in her eyes. "He is coming and he is come," Prometheus said, and she kissed his hand. There were the policemen who wanted to arrest him until they saw the note from Judge Bohem. And there was the crazy man with no shirt and no shoes who confronted him on a corner not three blocks away from Nosome's sister's house... "Who the fuck are you, man?" the bare-chested man said, gesturing violently with his hands. "Foreman Prospect," Prometheus replied. "You think you could just walk down the street lookin' like a clown an' somebody ain't gonna stop you? You think just 'cause you big as a horse that I cain't knock you on your ass an' stomp yo' th'oat?" The fire in this man, different from any other that he'd seen so far, was wild and raging. His flames were out of control and had driven him insane. Prometheus tried to walk around this man, not wanting to hurt him. It wasn't the madman's fault that his vision had driven him insane in a place where no one else could even see. But the shirtless man got in front of him again, bumped his chest up against the Titan. "I'idn't say you could go, clown!" he screamed. "I'idn't say you could leave!" People gathered at safe distances to watch the confrontation, certain that it would lead to a fight. No one got too close because of the insanity in the shirtless man's voice. Prometheus also realized the near inevitability of a fight. He considered striking the man down quickly so that he could get on with his business. He could see that giving this man his Gift would kill him on the spot. But he could also perceive more about this one man than any other he had met on this walk. The flames were so bright... "Henry," the Titan said addressing the madman. Hearing his name shocked the street brawler. "How you know my name, man?" "Your mother is Martha," Prometheus continued, "and she misses you. Your father is Terry Minter from Cleveland. Your mother thinks he abandoned you both when you were only small but really he was arrested in Memphis for getting into a fight on the street. A man died and now your father is in a Tennessee jail serving a life sentence. He can't read or write and your mother took you away before he could send word through the prison grapevine." "My daddy's alive?" Henry said, the rage forgotten. "How you know that?" "I can see him in your desire. If you were not so passionate and angry you would have thought to look for him yourself. Your heart is always searching in dreams, that's why I could find him so easily. You and your mother blamed him for running out but it was you who ran and you who are still running." It might have been better, Prometheus thought, to have killed Henry rather than to drag him through all the pain of his life, rather than show him that his violence and rage had sent him down a wayward path. The young black man stared at his inquisitor with tears in his eyes. "You lyin', man," he said trying to find footing in familiar fury. "No, I am not." * * * STANDING IN FRONT OF Nosome's sister's house Prometheus remembered Henry Minter running down the street crying and hoping, lost and looking for a way back through all his wasted years. SIX "YES?" THE LIGHT brown, middle-aged woman asked. There was fear in her voice because the man before her was so tall and powerfully built, oddly dressed and altogether unlike any man she had ever met. "I'm looking for Nosome," the Titan said. "Are you his younger sister?" "Y-y-yes I am. And who are you?" "I am Foreman Prospect, a friend of Nosome's. I owe him forty dollars." "Forty dollahs? Why didn't you come before my husband left me?" she asked, seriously expecting an answer. "May I see Nosome?" "What's your name again?" "Foreman." "Well, Foreman, you see, it's like this... I'm in a bad way here. Nosome's dyin' and my husband left me because I took my only family in. I got a daughter wit' no husband either and a grandson cain't even get up out the bed on his own. I own this house but I cain't even pay the 'lectric bill much less the tax. So you could see where I might not have the patience to put up with one'a Nosome's drinkin' buddies." "I have to see him," Prometheus said softly. "He is my only friend. Without him I will not know where to go." "You say you got forty dollars for him?" "Yes." "Lemme see it." Foreman held out the money. He pressed it into her hand. "You don't look drunk," the woman said, clutching the money in her fist. "What's your name?" Prometheus asked. "Tonya Poundman." "I only want to see Nosome, Tonya Poundman. I promised I would come to him. I need his guidance." After a moment more of hesitation the woman moved aside. The Titan had to duck down to walk through the doorway. The house was tiny. The living room seemed to belong to children. He had to bow his head to go through the door leading to Nosome. He was lying on the bed, his breath labored, his skin graying even as Prometheus looked upon him. "He's dyin'," Tonya said. Prometheus sat down on a wooden stool set next to the small bed. He watched the light inside his only friend as Tonya stayed by the door weeping quietly. The wan light of death pulsed throughout the elder man's body. The radiance was mainly white with hints of color here and there. When these faint hues had drained away, Prometheus knew, Nosome would be dead. The Titan's breath slowed and became shallow as he concentrated on the stories held in these last pulsating moments of life's light. He could see Nosome Blane as a young man, a boy really, taking care of his sister because it was just them in the St. Louis slum. Nosome the boy bristled with pride when his sister would look at him as her best friend and savior. "I ain't nevah gonna leave you, Li'l Sis," he would say. Prometheus leaned forward and gazed deeply into the waning soul of his friend. He caught snatches of memory and passion. He saw Nosome in a uniform and Nosome behind bars. The man had drifted into an alcoholic haze but he was still a good man finding happiness in the smile of his beautiful sister. The Titan pushed further, reaching into a stone room with no windows and gentle light radiating from an oil lantern hung from the ceiling. Nosome was there, dressed in off-white muslin with brass rings on his fingers. He was sleeping lightly but lowering into a deeper rest. "Nosome Blane," his friend whispered. "Wake up. This is no time to sleep." The old man's breath became deeper. In Tonya Poundman's room Prometheus placed a hand on Nosome's chest while in the stone chamber he said, "You can rest later, my friend. But now you must rise up. There is business for both of us." A light shone under the hand on the dying man's chest. And in the stone chamber, where all men and women pass from one world to the next, Nosome Blane's eyes fluttered and a gasp escaped his ashen lips. "Foreman?" "Nosome." "Where are we, man?" "Between places, my friend. You were dying and I reached out to you." Nosome sat up and looked around. "Where is this place?" "I have need of you, my friend," Prometheus said. "I'm looking for a soul that can bear the weight of light, someone that can lead mankind out from the shadows and into a place where they can become one with the godmind." "It's very peaceful here," the old man said. "Is there a door?" "There are two." "Up and down?" "Away from life and back the way you came." Nosome stared at his friend. "I'm very tired, Foreman," he said. "Very tired. You know, when I was a kid, I'd steal in order to feed me an' my li'l sister. They had me in jail just 'bout ev'ry other month. And then, when she was grown an' engaged to Rutherford Poundman, they got me on bein' a incorrigible and sent me away for nine years. That shit just about broke me, man. I ain't done one thing right since then." "Come back with me and you will be a great hero, among the greatest the world has known." "You be there?" "For a while. But you will have another friend and that one will be everything you ever dreamed of." "Or else I could go through another door," Nosome speculated, "the one lead away." "Yes." "Would you try an' stop me?" "No. I could not, nor would I." There was a long silence then. In the world of the living Prometheus's hand was heating up on Nosome's chest. "I hear my sister cryin'," Nosome said at last. "She is with me. She wants you to live." "An' all I gotta do is say okay an' then I can go back to all the dirt an' jails an' misery?" "A new life awaits you, my friend. I promise." "Okay." Nosome Blane, in another world, on the verge of transition, turned his back on eternity for the word of a man he hardly knew. * * * IN THE SMALL BEDROOM the light from under Foreman Prospect's hand flared for a moment. "What was that?" Tonya Poundman cried. "That's me, Sis," Nosome said. He struggled to a sitting position and smiled. In his sister's eyes he was almost a young man. His sagging flesh was fuller. His eyes were stark white and dark brown. "Is that you, Nosome?" "Oh yeah, baby. Foreman got a job for me an' I just couldn't go since that fool Rutherford walked out on you." "I'm just happy you're all right, 'Some." Prometheus knew that he'd made the right decision, giving part of his own life force to resuscitate this man. He felt the power of the connection between brother and sister. It was as deep a feeling as he had ever known in Olympus or elsewhere. SEVEN "I WILL DO what you ask, man," Nosome was saying over a plate of chicken and dumplings that his sister had served. "But first you have to go see my grandnephew. You got to help him." "It takes a lot out of me, Nosome," Prometheus argued. "And I need to save my energy for the task at hand. I must do for the world what I did for you." "I know you sumpin' special, Foreman. I know it and I appreciate what you done. But Chief is just a small boy. Maybe you don't need to do too much. Maybe you got a medicine or sumpin' could he'p him. All I'm askin' you to do is look." Tonya said nothing but Prometheus could see the pleading in her eyes. "All you got to do is see him, Foreman," Nosome said. He was healthy now. His smell was that of life. His eye was clear and full of mirth. But Prometheus could tell how serious he was about the child Chief Reddy. "I will look at him but I cannot squander my energy." * * * THAT NIGHT FOREMAN Prospect slept on the earth behind Tonya's home. A wan moon and few stars shown in the sky but the Titan enjoyed the feel of the grassy ground and the sounds of nightlife. He could hear bats on their blind hunt and dogs snuffling. There were cats and opossums, insects and cars. Now and again someone would call out in love or pain; he heard three men breathe their last. Prometheus used the comfort of night to restore some of what he had lost saving Nosome Blane. Something about his new friend bore good portent. And, anyway, he was kind and generous. Wasn't the job of the immortals to recognize these qualities? * * * "WHAT YOU DO to me last night?" Nosome asked Foreman the next morning on their walk to Mary Reddy's home. The mismatched pair walked down the street side by side as if there was no difference between immortals and men. Tonya Poundman had worked on Foreman's clothes until they fit him, more or less. She took in the jeans and added the fabric from one her husband's pants to the legs. She let out the chest of his shirt with sheet material. Nosome had used nineteen of his forty dollars to buy his savior a pair of tennis shoes at Tulie's Bazaar on Central. They also purchased a cheap pair of dark sunglasses to hide the peculiarity of the Titan's eyes under the light of day. "I saved your life" was Prometheus's answer to Nosome's question. "Not that," the habitual criminal said. "After you went out to the backyard and Tonya was asleep I snuck in the pantry an' got one'a Rutherford's beers. Took one drink an' nearly upchucked my guts." Foreman laughed. It was his first in centuries. "I changed you in a few ways, my friend," he said. "I made you more capable to do the task and follow the path you agreed on when you came back to life." "I didn't agree on nuthin'," Nosome said with some small ire, "especially not bein' able to take a sip of beer." "But you cannot drink and help me, too, 'Some. I need you to have a clear eye and a sharp mind and so I put a hatred of alcohol in your gullet." "So I cain't drink?" "Not a drop." "What else you do to me?" "It will come clear in time." * * * "YES," A SMALL black woman with dyed blond hair said in greeting at the right-hand front door of a house that had been made into two apartments. "Hey, Mary," Nosome said. "Tonya send me an' Foreman here ovah." The little woman was lovely and sad. She stared up at the tall man unable to hide her attraction to him. "What for?" "Foreman here's a healer. This mothahfuckah right here lay hands on ya an' the blood stand up an' say, 'yes sir, where you want me to go?'" Prometheus appreciated the young woman's stare. She didn't care about what her uncle had said. She wasn't concerned with powers or blood. She wanted him to come into her house, to sit on her chair. She wanted to serve him drink and feed him meat. He wanted these things, too. Her yearning was his for over three thousand years where there was no woman and no love. "Come on in," she said. Nosome stepped aside to make way for his newest and best friend. The Titan strode into Mary Reddy's home. She sat them down in a large living room. The furniture was almost big enough to accommodate the Olympian. "You big, huh?" Mary said to him. "You play ball?" "No." "Can I get you sumpin' to drink?" "Water." "You want some beer, Uncle 'Some?" "Water for me, too, baby." "What?" The spell of beauty and grace was broken for a moment. The young woman looked at her uncle suspiciously. "Why you not drinkin'?" "Give it up." "Why?" "I dunno. Drink just don't agree wit' me no mo'." * * * MARY REDDY GOT their water and sat with them quietly for a while. She, looking at Prometheus, and he, feeling that her attention was the greatest gift he had ever received, greater than the fire in his mind or the flames that fed on them. "We here to see Chief," Nosome said at last. "What for?" Grace asked. "I told you already, girl. Foreman here's a healer. He done agreed to take a look at your son." "You cain't he'p CC, Mr. Prospect. He was born wit' not enough nerves. His body cain't move. Ain't nuthin' nobody could do." "I just want to look at him, Mary," the Titan said. "I want to touch his skin to see what the sight tells me." He could see the reticence in her. The sadness she felt was love turned sour over all the disappointments; the father that abandoned his afflicted son, the hopes that failed. Past these feelings was her desire to protect the child from pain and disappointment. "I won't tell him why I'm here," Foreman Prospect said. "I'll just say that I'm a friend of 'Some's and that we came by to see how he was doing." Mary smiled. There was the promise of a kiss for Prometheus on her lips. She moved her shoulder to the left saying in a universal language that she was leaning toward him. EIGHT FROM HIS MECHANICAL BED in the tiny room Chief Reddy had had most of the experiences in his life. This was where he lived. His mother gave him sponge baths there. She brushed his teeth and changed his pajamas. She turned the TV on and off, read him stories and later let him read stories to her. Nurses came there to take his temperature and ask him questions. They would do things to his feet and hands asking if he felt something, but he never did. He had learned to sit for hours in numb paralysis while his mother was away and his sitters stayed in the other room watching TV with boyfriends or just watching TV alone. Nobody but his mother (and sometimes his Uncle 'Some and Grandmother Tonya) would ever sit with him through a whole movie or have the patience to wait for him to be able just to move a checker from one square to another. Chief lived in his mind mostly, thinking up long complicated stories of a boy who was paralyzed in a great castle owned by his father, a warrior king, who had gone off to defend his people. Sometimes, when the boy was asleep, something would happen and he'd wake up in a different world where he was strong and could walk and run and, on rare occasions, glide on the wind. In this world there were portals that allowed him to see his father fighting the Enemy. The boy, whose name was Chief Redd, would drop big rocks through the portals killing any and all who tried to harm his father. On these adventures the boy fought demons and loved women; ruled an entire nation almost as large as his father's kingdom. The only problem he had was that his father didn't know how powerful and good his son was. He would probably never know. NINE WHEN PROMETHEUS AND Nosome entered the room the little lame boy seemed to be sleeping in his big mechanical bed. The Titan pulled a chair next to him and sat down. Touching the bare skin on the back of Chief's hand Prometheus was instantly thrown back over the millennia into the heart of the flame of knowledge that had been passed from Ma'at to him. Alone in this room, with little contact or knowledge of the world outside, Chief Reddy had garnered his flame and kept it strong. His spirit burned as bright as it had in the ancient folk who could see truth simply by looking at the world and wondering. This weak child, this young man carried a powerful soul. Prometheus pulled his hand away and the feeling subsided. "What was that?" Nosome Blane asked. "What did you see, my friend?" "There was like a flash ovah your head and... and Chief was there and he was like a king on a throne." "I di'n't see nuthin'," Mary Reddy said. "How come I seen it if Mary ain't?" "For the same reason," Foreman Prospect said, "that you can no longer drink." "Mary," Prometheus said turning to Chief's mother, "if you leave me in here for a short while I think I can change him enough so that he may be able to move a little better." The Olympian knew not to offer a cure. Mary, he could plainly see, was unable to hope for anything large or permanent. "What you gonna do?" she asked, suspicion lacing her words. "Lay a hand on him and say a prayer, that's all." "You can trust him, child," Nosome said. "I was almost on my deathbed an' he laid hands on me." "All you gonna do is pray?" she asked. "And hold his hand." "I don't know," she hedged. "Yes you do." Nosome led his niece out of the boy's room, his world, and closed the door behind them. Prometheus took the boy's hand in his... Chief was awake but his eyes were closed and he was entranced by the view of his father engaged in battle with what seemed like overwhelming odds. Thousands of men in glittering armor on armored horses, brandishing great broadswords and lances, surged forward as his father, King Redd, held the entrance to the valley that led to his kingdom. This vast army approached as King Redd and his men waited stoically for the initial clash of swords. By his side Chief had stacked a pile of stones that he would use to drop on any enemy who came upon his father from behind. The boy's heart was pounding, his breath came hard and fast. "Child," came a voice from behind him. He turned quickly, seeing the Titan, who wore only a loincloth and a strange, bluish crown that seemed to hover a few inches above his head. This crown was odd in that it seemed solid but altered shape slowly as if it were a liquid or even fire. "Who are you?" Chief Redd asked. "There's not much time," Prometheus said. And with those few words the daydream disappeared and the boy was standing on a mountainside where a cold wind blew and the sun shone more brightly than he would have thought possible. "In the beginning," Prometheus sang, "Man was small and afraid. His gods drove those fears and built themselves a savage kingdom where they were feared and revered. They hunted men and played tricks on them. They raped and ravaged and caused wars for mere entertainment..." Chief Reddy could see the words of the song as images that passed through eons of humans laboring under the fear of giants that somehow grew out of their minds. "... then I was sent down to earth by a vision," Prometheus continued. "I brought light in the form of fire to allow men to see their world and understand their dilemmas. This fire gave men the sight and the ability to make sophisticated tools that would protect them from wind and lightning, flood and famine. "And with this gift the gods were driven back a bit. They still preyed on humanity, but with less violence and depravity. Rather than whole tribes being sacrificed on the battlefield they merely asked for the deaths of virgins and firstborns. Wars had their respites. There was time for stability and growth. "But that was just the first fire, the primer. That light only showed your people the world they lived in, not the Dream that drove them. We, gods and Titans, are merely the upper regions of your mind..." Chief heard the roar of a man, or god, from somewhere atop the mountain. It was a warning and a call to arms. "... there is a second fire, one that feeds upon the first. It illuminates the gateway from this world to the next..." Destroy Prometheus! came a shout from above. Kill the boy and eat his heart. That way we can become them and they shall once again be our slaves. Hearing these words Chief felt a chill enter his shoulders but he could not turn away from the giant that addressed him. "... they approach," Prometheus said. "They want to hold on to you and your people. They have isolated the ones who might grow powerful and overthrow their debauched reign. Allow me to give you this flame and they will be pressed back even further. Allow me to crown you Chief Redd and your people may have a chance to weather the storm." Far away, but coming closer, from across the sky came an army so vast that Chief could not see beyond it. Huge men on great steeds breathing fire through their nostrils, ran through the sky—intent on killing both boy and Titan. Chief turned to Prometheus. "Will you take my gift?" the giant asked. The approaching army was coming closer and closer. Bowmen were fixing their arrows as they rode. The great bearded headman was shouting, waving a lightning bolt above his head. "What should I do?" the boy asked Prometheus. "It is up to you." Chief could hear the pounding of hooves upon the air like thunder rolling toward him. "I wanna wake up!" Chief cried. "They will follow you into your home with flames and swords. They will destroy us both and in doing so the world will turn toward capricious disaster." Chief could hear the ragged snorts of the horses and the angry cries of the gods. He saw his death approaching and once again turned to the Titan. "You aren't afraid," the boy said. "The responsibility has passed from me," he replied. "Only those with a future can know true fear." "Will you guide me if I take this burden?" the boy asked, wondering as he spoke where the words had come from. "A guide has been provided." Kill them! the king of the gods cried. "Will I succeed?" the child asked. "If you do not take my gift you will most certainly fail." Chief fell to his knees, accepting with this gesture the Titan's offer. The sound of the warriors' cries was deafening. He could feel their wrath coming in waves before their sharp swords. He heard the bowstrings being released and then he felt the cool fire of the second flame as if the crown was easing down into his brain. The gods cried aloud and the mountain he knelt upon shook. The ground beneath him fell asunder and he was thrown into the air. We will find you before you can undo our hold on men's souls, a voice whispered to him. We will find you and kill you and eat your heart.... PART TWO TEN CHIEF REDDY OPENED his eyes. He expected to see the golden-skinned giant sitting there next to him, but he was alone. It must have been a nightmare he slipped into while daydreaming about King Redd and the battle to save his people. Chief turned toward the door and saw that it was closed. This was unusual. His mother always kept the door open. While pondering the closed door he saw something out of the corner of his eye, a flashing image. He sat up and turned to see a doll-sized man encased in light. The man was shrinking, his body was pierced by many arrows. He was fading from this world, but in the last instant of his existence he saw Chief and gave him a wan wave good-bye. Then he was gone. The adolescent boy hopped out of the bed and went to the spot where he'd seen the apparition. But there was nothing left of the light or the dying man. It was only then that Chief Reddy realized that he had gotten out of bed on his own, that he was moving around under his own power. He stood up straight and looked down at his long, dark body. He went to the walnut dresser and took out his cotton pants and a striped yellow and red T-shirt. He donned these clothes with ease and certainty, realizing as he dressed that the golden-skinned man had left him with knowledge and part of his soul. "Prometheus," the boy mouthed, "god of enlightenment." * * * "UNCLE 'SOME? MAMA?" he said coming out of his room for the first time under his own power. The old man turned his head and smiled. He nodded with a certain gravity as if this moment had been somehow preordained. Mary jumped to her feet and cried out, "Baby!" She grabbed Chief up in her arms and hugged him to her breast, then she pushed him away, still holding on to his biceps, looking at him with an emotion that traveled back and forth between horror and ecstasy. "What happened to you?" "I was given a blue crown," he said in slow measure. "The man you called Foreman Prospect gave it to me and then he died." "What?" Nosome said. He got to his feet and moved swiftly into the boy's bedroom. Mary went after, leaving the fourteen-year-old ex-invalid to stand alone and wonder. So many things had changed so quickly. He had been feeble and frail, anemic and exhausted by the slightest exertion. He could barely raise his hand and his fingers held no strength whatever. He imagined great strength. In his dreams he could lift hundred-pound rocks above his head and jump down mountainsides like a ram. But in reality he was bed-bound, couldn't go to the toilet on his own. Often, if he needed to turn over in his bed, he'd have to press the button above his head and ask his mother to help. But now he was standing on his own... no... not completely on his own. The Olympian, the tortured god had given him his substance and his strength. Chief Redd stood on the legs of the Titan. He was inextricably intertwined with the nature of that meta-natural being. But Prometheus drew his strength from Logos and Logos from Ma'at. All rebel deities who ignored their appetites and followed their visions. Chief walked into the room that had been his whole world only minutes earlier. There he saw his mother standing by the bed, distraught. Uncle 'Some had picked up what was left of the immortal he called Foreman Prospect: a pair of sunglasses and recently altered shirt and pants. The sight of his relatives showed Chief yet another way that he had changed. He could see a yellow-brown stain in his mother's chest. For her, he knew, this was the color of loss. The Titan had only been in her presence for a short while, but inside her breast the feelings of need and want and love had bound together, pressing her heart and her long history of emotional pain. Nosome's mind was exuding a miasma of blue and gray. It was familiar ground—the death of a friend when it was least expected. "What's wrong wit' yo' eyes, baby?" Mrs. Reddy asked. "They have already seen too much." * * * THAT NIGHT, DESPITE his mother's protests, Chief slept on the grass in the backyard of Mary Reddy's rented half-house. The chill of the southern California air invigorated him. The smell of humanity steeled his purpose. He dreamed of his father, a man he'd never met, who had left when he saw the disgrace and expense of Chief's disabled existence. He imagined the host of gods, demigods, and monsters that even now plotted his demise. But mostly he dreamed of blue fire; something that man had imagined but had never realized. It burned brightly above their heads and below their feet. It was what held the universe together; the universe—a place where even gods were little more than motes of dust. As he slept, animals of all kinds were drawn to his resting place. Cats and dogs, owls and voles, a coyote nosed her way in and bravely nudged his hand with her snout. These creatures did not fight; they were witnessing something that had not occurred for many thousands of years. It was their duty to bear witness. The grass grew three inches that night and a peaceful state of dreamy sleep descended upon an area of three square blocks. Men's rage lifted from their hearts and women's complaints became pale and opalescent, almost hymnal. Children felt the joy of springtime and the safety they craved. And as the boy breathed a hum was let loose upon the wind and people up to a hundred miles away suddenly decided to follow their dreams or to give up on their grudges. Tina Mackie woke her husband, Troy, out of a deep sleep and told him that they could take his invalid mother in. Gerard Pinkney called his best friend, now living with his ex-wife, telling him that the war was over and now they could all follow their paths without anger or ire. Even though Chief was sound asleep he was aware of the events that were occurring around him. These miracles were the evidence of the passing of a Titan. He had suffered and escaped, found his destiny and followed it. He had rested once but this night Prometheus had made his destiny real; this night he was finally at peace. And in their somnambulant states, all over Los Angeles, residents of South Central were turning their souls toward the possibility of peace. Bricklayers, drug dealers, pimps, and preachers all stopped what they were doing and slept and wondered and changed, however slightly, the direction in which they were headed. ELEVEN "NO, BABY," MARY REDDY said the next morning. "It's too soon for you to go outside on yo' own. You ain't nevah been out the house by yo'self in your whole life." "Uncle 'Some will go with me, Mama." "But he don't know how to take care'a you." Mary made pancakes and bacon, fried eggs and French toast. Somehow, in her sleep, her sorrow over the loss of Foreman Prospect was assuaged. His departure was only a partial thing. In a dream he had come to her and told her that she was the only woman he'd wanted in so long that she was the only woman in his life. "I can walk and run and talk like a siren sings," the boy said, no longer wondering at the origins of his knowledge and ideas. "You just a boy," Mary said, but her eyes questioned her own words. Her son had gone from burden to miracle in just a few moments. Before this day he could barely whisper, now his voice was strong and musical; and his eyes, they had the hint of the moon inside them. Chief went toward the door. Instinctively Nosome Blane followed. Mary put her hands out to stop her son. The boy could see the whitish yellow light of fear in her mind. He took her by these hands. "Mother, I will be home this evening and tomorrow and the day after that. I will be your son until the end of ages, but you cannot hold me back. I have a mission. That is why Foreman Prospect gave up his life. If I were to stay home I would soon lose his gift. My arms and legs would grow weak again and I would be back in that mechanical bed dreaming that I had legs and a father." These words stung. Mary flinched and pulled back from her son's touch. "You all I got," she whimpered. "I was a stone around your neck..." "No..." "I was the pain in your heart..." "I love you..." "I was the nightmare and you couldn't wake up. My father left you because you had me. You couldn't go to school or get a good job or find a new man...." "No." "I will make it up to you," Chief said to his mother, looking into her eyes with his bright lunar orbs. "Your sacrifice, like millions of others made by mothers every day for their children, will be rewarded. You will become the mother of a new age and I will always love and revere you. But you have to let me go." Mary Reddy felt the words that her son spoke. She knew they were true, or that he would work hard to make them true, but there was the chance that he might die on the road to these goals. "I'm afraid," she said. "Don't be, Mama," the boy she knew said plainly. "The world outside this door is clamoring for us." With this the one who would become known as Chief Redd turned and took a step toward the door. "Hold up," Nosome cried. "Here, here put these on." He held in his hand the sunglasses that Foreman had worn to hide the alien nature of his eyes from the world around them. Chief donned the glasses and smiled at his great uncle. "Are you with me, Uncle 'Some?" "All the way, l'il nephew. All the way." * * * PASSING FROM THE SMALL HOUSE into the light of morning Chief wondered what challenges he would find in the wide world of dissolution, depravity, and desperation. He wondered how humanity would take to the brightness of his words. He was thinking far beyond his mother's front door, but his first test was standing at the sidewalk, waiting for him. Henry Minter, still shirtless, still half-mad, stood there watching the door as if he had been expecting young Chief Redd to come out at that very moment. The boy could see the wildfire in the man; it burned out of control as if buffeted by whirlwinds and fed by many seasons of dried brush. Nosome got in front of the boy and Mary shouted, "Look out!" from the doorway. "Get away from here, man," Nosome said as he approached Minter. The young madman tried to push Nosome aside but the elder somehow grabbed onto his arm and from there climbed onto his bare back. Try as he might Minter was unable to dislodge the older man. But Nosome could not stop Henry, either. The street dweller advanced on the dark-skinned and lanky boy. Chief waited for him smiling because his uncle was so spry and committed. "I heard you last night," Minter said when he had reached his quarry. "I heard you sleepin' an' lyin' 'bout how things gonna be. I heard you talkin' 'bout happy endin's and a new world. Ain't no new world! Ain't no happy evah aftah!" These last words were shouted directly into Chief's face. "Get outta heah, niggah!" Mary cried. Chief took off his glasses, letting the cool lunar emanations from his eyes bathe the desolate Hank Minter. As they stood there the seconds stretched into minutes. Nosome climbed down from the powerful brown back. Mary could never guess at the congress between these men. It was not in words or images but in the free flow of emotion bound up for so long in Henry. Just being in the presence of Prometheus the madman's fire had grown immensely. His rage was towering. His only desire was to attack and destroy. And so when he was sleeping in a doorway not two blocks away he heard the final sigh of the Titan and came to the house from which it emanated. He had planned to kill the seven-foot liar. But when Chief came out Henry knew that this was his enemy. At least he knew until he looked into those eyes; those faraway landscapes of the moon. The cool light and the nearly colorless expanse therein drew the rage out like some sea anemone sucking the sweet meat out of a hopelessly captured crab. Henry struggled against the bleak and restorative topography of the boy's eyes. He tried to keep his anger working but it dissipated in the thin atmosphere. Minter fell to his knees and looked up at the child who had defeated him. "Who are you?" he asked. "Where's the big orange guy?" "We have a war to fight, Mr. Minter," Chief replied. "Nosome is my right hand and you will watch my back." "Do I get to hit somebody?" "I hope not. But it's a bad world out there." As he spoke these words Chief Redd walked down the sidewalk flanked by Nosome and followed by the madman Hank Minter. TWELVE "I COME HERE to tell you what you already know," Chief Redd said from atop a picnic table at South Park. "That there's something wrong with the world, that we have to go out and fix the whole damn machine or our homes will fall down around our ears and we will be consumed with fire, disease, and war." No one was listening except his two followers, the skinny old black man and the shirtless madman whose color was a bright, sweaty brown. "Trixie Lewis," Chief said to a young woman who was listening to earphones, nodding her head to the beat as she walked. Her music was turned loud but she heard something and took off the head set. "What?" she said to Chief. "You talkin' to me?" "Trixie, you have a son and daughter living with your boyfriend's mother." "Who the fuck are you?" she asked. "Mustafa Lee," Chief said to an angry man in African garb. "Who said my name?" he asked. "Terry Sharp, Alberto Gonzales, Talia Breetman, Cory Jones," Chief said calling out to people all over the small city park. He spoke over sixty names and the amazed audience began to swell. "The future is in your hands, Lonnie Brennerman," he said to a middle-aged brown man with a pot belly and freckles across his face. "You need to take your family back to the land and raise a pig and some vegetables." "How you know what I'm thinkin'?" Lonnie exclaimed. "It's written on your face... Pat Summers," Chief said to a golden-hued woman, with red hair and thick lips. "The action you're considering is not worth the cost to your soul. Do not use the pistol in the bureau drawer. Do not break the law of your mother and your father." The woman shrieked and stumbled away, throwing backward glances of terror at the calm child who was even now addressing another bystander. "Felix Nye," he said. "You need to find your son and instruct him in the ways of manhood." "I ain't got no idea where him or his mother's at," Nye said seemingly unaffected by the perception of the adolescent. "Minna moved to Oakland, on Burburry Street. Tor is there with her dreaming of his father." For three hours Chief addressed members of the growing crowd one by one. He forgave them for heinous crimes, suggested actions that they should or should not take, reminded them of their dreams, and sometimes just spoke their names and smiled. Police came to disperse the crowd but Chief addressed them by their names and their consternations or predilections. The police joined the crowd. No one minded while the child spoke to individuals. If they could have seen with Nosome Blane's eyes they would have detected the flash of a spark in the heart of every person Chief spoke to. It was a small blue flame, too tiny and frail for the god-eye of Prometheus to see. Many of the assemblage of hundreds held their hands to their chests unconsciously as if guarding a candle against the wind. And when he had spoken to every woman, man, and child within his sight he straightened up and addressed them all. "Hear me," he said. Nosome noticed a shift in the slight blue sparks. They seemed to be pulled toward his grandnephew, his friend. "You must come together," Chief continued. "Look into each others' hearts for a light to guide you. Talk about the world and what you want and what is right. Move away from dark thoughts and fears and lies. Never again be fooled or foiled or made to do the dirty work of dirty minds. Sit here in this spot of green and speak and listen and feel the oneness that brings us along, that drags us kicking and screaming out from our wallowing in selfishness. Bring drink and food and clothes to the needy. Love your children. Open your doors. And march down the middle of the streets when the shadowmen want to make you into obedient monsters stepping in time and crushing the less fortunate...." * * * THAT NIGHT MARY Reddy served meatloaf and collard greens, baked yams with store-bought raspberry sherbet for dessert. "You all so quiet," she said to her son and his friends. "Uncle 'Some, what happened today?" "Boy stood up on a table like it was a pulpit and he preached like it should be done. He set a fire in people's hearts." Nosome shook his head and grunted in a way that emphasized his words. "Speak," Hank Minter said in agreement. "And what happened to you?" Mary asked the now shirted madman. "Excuse me, ma'am?" "I seen you before," Mary said. "Walkin' down the street, talkin' to yourself. Terrifyin' old folks an' fightin'. Henry's response was to let his head hang down over his plate. "He has been rekindled, Mama," Chief said. "Like a candle?" "Just like that. He was living in the darkness and now he has a light by which he can see. Look at him... He has changed since meeting Foreman Prospect. He has changed since sitting in the light of all those people who came to hear me speak. "They stayed together there. They talked and went to restaurants and their homes to ask what was right. They are even now planning to take their world back from the darkness that veiled Henry's mind." Scowling, Mary Reddy said, "An' so now you expect me to feed this crazy man and I ain't even got child support?" Nosome thumped his forehead with the heel of his right hand. "I completely forgot about that," he said. "Forgot about what?" asked Mary. Instead of answering Nosome started pulling wads of bills of differing denominations from pockets both front and back. In a few seconds he had created an impressive pile of cash on the table before him. "People started pushin' money on me while Junior here gave them knowledge," Nosome said. "One cop gimme a hunnert-dollah bill." Mary moved to the pile and touched it with her fingertips. She turned toward her son with amazement in her face. "All this money?" she said. "Paper," Chief said, correcting her. "It's just kindling for the fire that will burn down the barricades built to keep us from our hearts." For a long moment mother and child stared at each other across the table. Fear etched her eyes and mouth while concern formed in the boy's face. "Can we watch the Superkids on the big TV?" he asked then. It wasn't a sham. Chief was happy to sink back into his childhood. He was a boy. For years his body was like that of a dying grub lying in bed, kept alive by Mary's daily labors. He was immature even for his years. He happily let down the weight of Prometheus's Gift for a time. The men and mother and Chief watched cartoons for hours together. Chief laughed and the adults were careful not to break the spell of childhood. At one moment, while a Samurai cartoon played, Chief saw that Henry and Mary were sitting side by side on the yellow sofa. They were holding hands. While Chief was watching huge robots alter their metal bodies into birds and fast trucks, Mary stood up and announced, "I'm goin' to bed." Nosome nodded and waved from his chair. He was leaning forward with elbows on knees. Henry kissed her hand and she touched his shoulder. "'Night, Mama," Chief said as if he were an obedient fourteen-year-old with nothing at all on his mind. THIRTEEN "SUMPIN'S LOOKIN' FOR you out there," Henry said to Chief a few minutes later. "Foreman said that the gods were angry that he gave me the second gift," Chief said. "He said that they might come down after me. Zeus himself told me that they would kill me and eat my heart." "It ain't heaven on my mind, Junior," Henry said. "Somebody right here on earth not ten miles from where we sittin'." "Humans?" "Definitely. They ain't sure that you here yet, but they suspect it." "How do you know?" Chief asked still glancing at the cartoons. "Same way I could tell about you sleepin' an' talkin' to the animals. I know shit now that that giant talked to me. He touched me... in my soul." "An' there's another problem, too," Nosome said, moving spryly to half lotus position on the floor next to his nephew. "What's that, Uncle 'Some?" "You control the crowd when they right there in front'a you, but the word gonna get out on what you doin' an' you know them white people ain't gonna let some li'l niggah take a piece'a they pie wit'out no trouble. You left them folks talkin' 'bout what they need to do to get things right an' city hall ain't about to stand for that." Chief could see into people around him picking up phrases and obsessions, but he didn't need this ability to read his uncle's heart. His power was to see and ignite. But because of all the years he'd spent as an invalid, his touch was gentle. "I'm sure that you're both right," the future king said to his friends. "There is danger in change just like there's danger in fire. What we are about will undermine the nations and their banks, the races and their religions, the languages and their lies. We three are the most dangerous men that the world has ever known. Our enemies will multiply and grow strong, but so will our friends." Henry and Nosome glanced at each other. They were comrades now, closer than blood. "I wanted to tell you that there seems to be somethin' goin' on between me an' your mama, Junior," Henry said then. Again his head hung down. "She's been lonely taking care of me," Chief said, the cartoons playing behind him. "And you have lived like a man on a deserted island even though there were people all around." * * * THAT NIGHT NOSOME sat on the yellow sofa thinking back over a wasted life. These dark ruminations brought a smile to his lips. Mary and Henry made love in her bed gazing into each other's eyes and praising that moment in time and space. Chief lay on his side in the now tall grasses of the backyard, his mind rising up into the ether of human hopes. He fell asleep wondering how he would be murdered. A hand touched his shoulder and he knew an instantaneous flash of anger. Who would bother him here? "Your creator," said Prometheus. Chief sat up and saw the Titan kneeling there before him. "My master?" the boy asked. "No," the shimmering ghost said. "You are perfection in this age and I am but a memory of what was." "Why are you here?" "I don't know. I am dead but not gone. When certain forces come into alignment I appear." "Did I call you?" "You have need of me." Chief wondered what this could mean. He already knew why the Titan had chosen him for his Gift. He understood his powers and the dangers they posed. "Gods... are the servants of men," the boy said tentatively. Prometheus smiled and nodded. "In the beginning we created them," Chief Redd said. "Yes." "Like you created me." "It is the return of the power that escaped the world," the Titan once named Foreman Prospect said. "But you know all of this." Chief realized that what the transparent, golden-hued specter said was true. "Where else have you been, Elder?" the boy asked then. "To the peak of Mount Olympus." "Who needed you there?" "No one. The gods called to me hoping to resurrect my body so they could torture me for another three thousand years. But they failed. Our time has passed and it is now you who hold the reins." "Did they speak of me?" Prometheus hesitated, he turned his gaze toward the stars. "Answer me, Creator," the boy commanded. "As with many of the powerful they are cowardly," Prometheus said then, choosing his words carefully. "They know that they can only attack you here on earth. They know that here they are also vulnerable to the edicts of mortality. One reason they tried to revivify me was to see if they might survive earthly demise. "They are afraid but sooner or later they will come after you. They will overcome their blinding dread and attack you here." "Can they destroy me?" Chief asked, bemused that he felt no fear for himself. "Easily. You are as a newborn and they have wielded power for millennia." "Then I must hurry," the boy said. "I must spread the word before the gods destroy me." Prometheus smiled at his creation and then, with a passing breeze, he was gone. * * * THE NEXT MORNING Chief brought a barstool out from his mother's house and set it on the sidewalk. He climbed up and stood there calling out to people as they passed. "Ramon Perez, Trina Willams, Minda Lawford, Samuel X." It was the same as the day before. He revealed secrets and contradicted lies. He gave people insights to their potentials and emotions. Neighbors came out from their houses to listen to the boy they'd heard of but had never seen; the crippled child who lived his life away dreaming in his bed. The sermon, such as it was, went on for four hours. Nearly three hundred people had gathered around, blocking the streets. Again, the throng had been joined by policemen who at first came to break up the illegal demonstration but then stayed to savor truths that they had hitherto only suspected. After everyone had been addressed individually and lit by the spark of the immortal, Chief said this: "This is my home, brothers and sisters. My mother lives behind that door. It is my providence or doom to spread the truth like cool fire among the masses. I am to waken the true self in you and you are to overthrow everything you knew. In doing this you will make a heaven on earth. But this cannot be without my work. And I cannot go out into the world unless I know that my mother is safe. "And so I ask you to knit together here on this block. Help each other. Feed each other. Stand guard over my mother so that I can be free to sow the seeds of rebellion, revelation, and rebirth." The response was a loud affirmation with no words or rhyme—just a shout that pledged the fealty of every soul there. They would stay on that block, or move there, and as a group they would fan the flame in each other and watch over Mary Reddy so that the world might learn what they learned, were learning. FOURTEEN FOR WEEKS, CHIEF Redd and his two disciples traveled around Los Angeles giving sermons and fanning the flames of change. Acolytes began to wear simple red clothes. They would set up sermons of their own relating what they'd learned from the child who would not allow anyone to call him master. Police blockades could not hold him back. Officials on television warning people away could not stem the tide of his followers. Thousands already identified themselves as Redd Revolutionaries; those who wished to bring all mankind together under the banner of enlightened brotherhood. They made a flag of pale violet with a single auburn, windblown leaf wafting up to the upper, outer corner. Chief traveled by car now, an old Dodge that had most of the paint sanded off. They spoke in Bellflower, Oxnard, Venice Beach, and the Pacific Palisades. They addressed Korean crowds in Korea Town and large groups of Chicanos in the barrio. Everything ran smoothly and the boy, who had been bequeathed a crown by an immortal, played Xbox at night, and read comic books about Spider-Man while his followers left their jobs and moved into small homes with a dozen or more of their brethren. While Chief, the adolescent, secretly masturbated in the room that was once his entire world, Redd Revolutionaries spoke his name with reverence and love so profound as to be frightening to anyone who had not heard the child speak. Secret meetings were being held at City Hall and Sacramento, even some local representatives of the federal government were issuing memorandums on the Reddy Cult. The threat of the child was being assessed. * * * ONE MORNING THE boy woke up early on his bed of grass in the backyard. Instead of planning the site of his next sermon he wondered about who he was. Because, even though he was adored and praised by everyone he met, he was still just a normal boy; no, not normal, but a child who had been bedridden and who now was one of the most powerful men in the history of the world. Using the prescience granted him by the Titan, Chief knew that he might some day grow into the role he occupied. But now he was just a boy, and that morning, before dawn, he wanted nothing more than to be a child. He put on a green hoodie and yellow sunglasses and went away from his mother's house by jumping over the back fence and running off through the neighbor's driveway. * * * HE TOOK A BUS to West L.A., where he was less known. He changed buses and talked to people one on one because, even though he was on holiday, he was still driven to spread the word. On the Wilshire bus he sat down next to a mild-looking white man who wore glasses with dark rims and whose eyes were deep dark holes. He wasn't despairing or in pain but, rather, empty; a blank slate set to hide a rage so great as to dwarf that of Zeus himself. Looking down at the man's hands Chief saw their alter-image: the metaphor of their intentions. The fingers were steeped in blood. Ragged skin of a hundred victims was packed under his manicured, clawlike nails. And for the first time Chief knew real fear. This man was a red slayer, a beast in disguise. Harold Timmons, who worked in a car insurance office on Fairfax Avenue, had tortured his victims, had let them die slowly in the ground while he sat above them reading Molière in the original French. Chief was afraid of this man. He knew that the beast inside hungered to rip and rend his young flesh. He could feel what Harold felt and the desire made his heart quail. He decided to get up and get off the bus as soon as possible. He would run from this evil that was beyond redemption, even if that redemption was fueled by the original flame of Prometheus. "Where are you going?" Harold asked Chief. "I-I don't know," the fearful child inside the god replied. "You have to know where you're going. Are you lost? Do you need help?" The mild questions were like paralyzing venom from a spitting cobra sprayed on the boy's face. He was still young and the words worked on him, somehow bypassing the sentience of immortality. "I live very close to here," the man said. "You could come over and use my phone to call your folks." Chief's fear and shock at the evil he'd encountered kept him frozen in place. "Here," Harold said, "take off these glasses. Let's see what you look like." That was the serial killer's mistake. By taking the yellow-tinted glasses from the boy's lunar eyes he ripped down the barrier keeping him from the heavenly vision. This sight broke down all of Harold Timmons's carefully laid lies. It was as if his disguise had suddenly been ripped away and he was visible for what he was and all that he'd done. The predator gasped. He moved backward violently, cracking the hard glass of the Wilshire bus with his head. He screamed and pushed Chief down into the aisle of the moving bus. He tore from his seat—shouting. "Let me out of here! Let me out!" The woman bus driver hit the breaks and opened the door, ripping from the bus's innards a hydraulic cry, front and back, almost as a specially orchestrated dissonant music accompaniment to Harold Timmons's fear. He leaped out onto the sidewalk and fell. He jumped up immediately and ran down the street hobbling on a sprained ankle, screaming like a stuck pig. Chief watched him go as people on the bus began to talk about the fright his actions caused. "Are you all right?" a woman asked Chief, helping him to his feet. He put on his glasses and nodded, noncommittally. * * * THREE STOPS LATER he got off at the tar pits and the art museum. His mother and Uncle 'Some had brought him there on his eleventh birthday. Mary had been nervous having Chief out in the open, traveling in the rented special wheelchair. Nosome was drinking wine and as the day went on he got friendlier with strangers. It would have been a perfect day if the police hadn't come and arrested 'Some for public drunkenness. But now Chief could walk under his own power and travel wherever he wanted to. He had planned to wander around the museum, but now, after his encounter with Harold Timmons, the heir to Prometheus drifted about the soothing green of the park, finally coming to rest on the ground under a knotty oak tree. He shuddered there thinking of evil so profound that it had the potential to snuff out the second fire, the illumination of the upper realm of the soul. Harold Timmons had murdered little girls in front of their bound mothers, had slaughtered secretaries and prostitutes, dog walkers and hopelessly senile old men. He savored their suffering. He drained their blood then freeze-dried it and kept it in jars on his shelves. Chief could see that this primal manifestation of evil was just a symptom of why so many were unable to accept the pure strong light of the gods. The human race had fallen low with their machines and paper money, their dark rooms and dull repetitive lives. "Hi," a voice said. Chief looked up to see the bright and lovely face of a medium-brown girl, maybe eighteen. She was smiling and stunningly beautiful to Chief's eyes. Her green dress came to the middle of her strong thighs and her breasts stood up like pride. "Hey," Chief said. "You ditchin' school?" the girl asked, descending to her knees next to him. "Not really. I was just..." He hadn't expected to, but when the girl got close to him he threw his arms around her and cried on her shoulder. "What's wrong?" she asked sweetly. Chief tried to answer her but the hoarse cries issuing from him allowed no words or even gestures. "It's okay, baby," she said. "Mama's here." It wasn't just Harold that weighed on the child's heart. It was everything that had happened since he had awoken in his bed with strength in his limbs and a mission he did not know how to refuse. He cried and cried and the young girl with the bikini-model figure held on to him cooing and saying that things were fine. After a long while Chief sighed and released her. She leaned toward him and pulled off his glasses. "Oh my God," she exclaimed. "Your eyes are so beautiful." It wasn't the usual reaction that people had. Most others, when they gazed into that lunar landscape or fiery abyss, were stunned by the force of meaning. But this young woman saw only the image of beauty, not depth or significance. Chief wondered if that was because she had already seen him as a sad child lost in an evil world. He realized then that the Titan was still guiding him. Prometheus woke him early and sent him off to meet the murderer and the maid. "It is you who are beautiful," Chief said. The girl's eyes glittered with happiness. "You so nice," she said. "I wish I could take you home with me. But my li'l brother's a gangbanger in trainin' an' my daddy don't get outta bed except to beat my mama an' take her paycheck." Once she started speaking it was as if they had always known each other, sharing secrets in the park. "I'm supposed to meet this boy named Melvin up here but he done stood me up. What's your name?" "Chief Reddy," he said. "That's a funny name. Would you stand me up like Melvin, Chief Reddy?" "You could, could come live at my house," the boy said. "I mean on my street. There's some empty places and my great uncle is rich an' he'd pay your rent if I asked him to." The girl frowned. Chief's heart was beating fast. "And what would I have to do?" she asked. "Talk to me when I was feeling sad," he said. "We could take a walk now and then." Her name was Rhonda McKinney and there was no special light in her. Chief thought she was the most beautiful woman in the world. And even though he had thousands of years of godly experience packed into his soul, he would have given it all away for her to kiss his lips. "You want me to take you home to your house?" Rhonda asked. * * * ON THE BUS ride Chief told Rhonda about his life as if it weren't tinged by immortality. "I'm a street preacher," he told her. "I talk an' my uncle collects money. There's this guy named Hank who keeps people from jumpin' all ovah me." "It's like you was in church but only in the street?" Rhonda asked. "Will you marry me, Rhonda McKinney?" he answered. She kissed his lips lightly and said, "Let's wait till you grow up some and then we'll see." FIFTEEN "... YOU CAN SLEEP in my bed until then," the boy was telling Rhonda that afternoon in the living room at his mother's half-house. "Where you gonna sleep?" "I sleep in the backyard," he said. "On the ground?" "On the grass and earth." "Why you do that?" They were sitting at opposite ends of the coral-colored sofa. Mary was in the kitchen while Nosome sat in the backyard and Henry loitered around out front. The structure of their lives had taken on a military cast. Chief realized this but what struck him was Rhonda's question. No one else asked why he did things or what he meant when he spoke. They felt his cold fire and exulted in a world that they had always hoped for. But they didn't see the boy who had lain in his bed for years with no hope of ever even holding a water glass to his lips without help. "It's because..." he said. "It's because I need to stay close to the earth. You can't understand the sky unless you can feel the ground under your feet." "You silly. An' here I thought you was tryin' to trick me into your bed, but instead you ackin' like a preacher. Do preachers even like girls?" "Yes. Yes we do." "I bet you good at it, too, huh?" "What?" "Bein' a preacher, fool," Rhonda said nicely, giggling a little. Chief Reddy could see that his inexperience was as deep and important to Rhonda as her presence was to him. Her eyes tightened and she clasped her arms around her middle. "Mama always sayin' that if it wasn't for me that she would just go on back to Texas an' live wit' her fam'ly down there. Is yo' uncle really gonna pay my rent? 'Cause you know I could get some kinda job after while." * * * "... NOW LISTEN HERE, JUNIOR," Nosome Blane was saying at the dinner table that night, "you cain't be runnin' off like that no mo'. Your mama just about went crazy when she fount you gone." "You know I have to do things sometimes Uncle 'Some," the boy said. "You know it." Nosome opened his mouth to say something but no words came out. "An' you cain't keep bringin' people to stay here neither," Mary said. "We already got four people under half a roof." "The Rodriguez family next door agreed to move into an apartment down the street," Chief said, "so that Rhonda could have a place here next to us." "What?" Mary uttered. "They real nice people," Rhonda said as if the gift was small and insignificant. "I'ma stay in Junior's room until they move out. But I promise not to get in the way." Chief loved how Rhonda accepted things with no wonder or affect. She took what was offered with little humility, but neither did she evince a sense of privilege. Mary didn't like Rhonda, Chief could see that as a hot red flash in her aura whenever she caught sight of the girl. But his mother had come to see her son as the man of the house now, and even though that was not yet true, it kept her from open hostilities. That night the boy had the most peaceful rest of his life either before, or after. * * * "WAKE UP," RHONDA said. Chief knew that she would be kneeling by his side looking down on him as she had done in the park. "There was a little gray rabbit so skinny that you would'a thought he was a monkey if it wasn't for his long ears," she said. "He was sleepin' right up next to your arm. And when I came out an' knelt down next to you, he raised his head an' looked at me an' then he went back to sleep." "Where is he now?" "When you moved he run away. It was like he was guardin' you until you woke up an' then the spell was broken an' he was wild again." "Did that scare you?" "Why I wanna be scared of a li'l ole rabbit? It was cute." "But you never saw anything like that before, have you?" "Stand up," Rhonda said. Chief stood and Rhonda appraised him. "You bigger," she said with a sneer of satisfaction. "I think you met me an' now you tryin' to catch up so I could be your girlfriend." It was probably then that Chief Reddy knew that he was in love. He wanted to grow, was growing for her. His arms were thicker and his clothes felt tight. "We're going to Will Roger's Park today," he said. "I'm speaking to people who have something special that we can use. You wanna come?" "Okay. Is it around here?" "No. Down by the ocean." * * * THERE WAS A LARGE GRASS field deep inside the park. Nearly a hundred of those that Chief had identified as potentials had been tapped by Nosome and asked to meet him there on that day. Chief hid behind a stand of young pines while his acolytes arranged themselves before a large flat stone that the boy intended to use as his dais. Nosome came back to give him the high sign and just before he went out Rhonda kissed his lips, pressing her tongue into his mouth. The pleasure of that kiss dimmed even the second light of the gods. Chief inhaled and forgot for a moment how to continue the cycle of breathing. "Go on now," she said with a smile. "Go show them people that you my man." And so when Chief Reddy ascended the flat stone he was a little off his game. He had never been kissed like that. He had never felt about a woman as he did about Rhonda McKinney. "I'm glad you could make it," the boy said in a cracking two-toned adolescent voice. "You hold within your minds and hearts a hope that even the gods dare not utter." His equilibrium was quickly returning. "We are all servants of the flame and you, each one of you, burn brightest among all those I have met. You've seen it in your everyday lives. Bosses give you promotions when you're no better than other workers on the same job. People fall in love with you when there are others more beautiful or suitable, richer or more worthy. They listen to you and defer to you. They dream about you but never say so. "This is because you are holders of the flame. You burn brighter and they wish to be near that warmth. "You have the ability not only to learn and act but also to teach. Your words, even if you do not understand their meaning, will fan the flames in the thousands and millions and your sermons will bring a revolution of awareness to the streets and into the homes, the hearts and souls of man. It is time for you to go out in the world and spread the knowledge you have gotten from me. Tell them how it felt, where the flame resides in your body. Speak from street corners and jail cells, on coffee breaks and at schools. Keep moving from place to place and tell those who live in darkness about me. Tell them that I'm coming. You will prepare them for the greater transition. "And when you come across people like yourselves, people with the old flame of Prometheus in their breasts, take them aside and tutor them. Spread the word around the world. "And when those you address turn away from you do not despair. They will have heard your message and it will grow in them. You are setting fires all around the fortifications that keep us from our will..." It was a bright day and Chief Reddy felt power behind his words. The congregation of light-riddled men and women somehow brought out the strength in him. His heart pounded and the flames before him grew bright, illuminating even the daylight, making everything clear and knowable. Chief knew that these nearly enlightened followers could see a world beyond the limitations of their kin... And then it was as if a shadow fell, not on the light of the sun but over the illumination of the secret world that went unnoticed in plain sight. Everyone was aware of the shift in consciousness. They turned as one to the back of the crowd and saw the four men advancing. Though Chief had never met him, he recognized Luther Unty, the leader of the uncouth men. They were at the far end of the field, but Chief could see them all, especially Luther, in bold relief and great amplification. And as the angry, powerful man approached the young leader, Chief Reddy was learning. He perceived in the perverted darkness and light of the young thug the twisting of the fire that was the First Gift. Prometheus had tried to bestow second knowledge on this young man, but instead he awoke the potential for evil, self-loathing, and the desire to destroy. Luther Unty was the polar opposite of Chief Reddy. Chief realized, even as Unty yelled and he and his thugs began a mad dash at the rock podium, that it was possible to awaken this malevolent force in others; that Harold Timmons, evil as he had been, now had the potential for greater devilment since gazing into the eyes Chief had inherited from the Titan. The four black men, apocalyptic in their demeanor and rage, ran quickly, but time passed slowly in Chief's mind. Henry Minter and Nosome Blane had moved to his sides. The congregation girded themselves for the impact of the four. And Chief was wondering if the ghost of Prometheus had appeared to Unty telling him about the meeting so that the holder of the flame would not make a false move. Unty had reached the crowd by then. He grabbed a woman and threw her twenty feet. He chopped another man dead with a single blow. Chief concentrated with his might and time slowed until it almost stopped. He could see that Unty and his cohorts had built great strength, that they could kill everyone in that field given time. In the well of his heart and the height of his mind Chief Reddy allowed his own energy to grow. He called upon the flames inside him using the alien song of Ma'at and then prepared to let time flow again. "Separate them!" he shouted and then led the headlong race toward the onetime gangbanger and his crew. Henry Minter was with him. Nosome Blane was with him. A few of the men and women within earshot followed quickly. Minter hit the man on the left and Uncle 'Some took the one on the right. Chief leaped upon Luther Unty pitting his lean boy-limbs against the huge muscles of the killer while three of his acolytes did their best to restrain the fourth man. The congregation moved out into a wide circle obeying an unspoken command. They watched as Nosome climbed on his quarry's back deftly avoiding his blows. They watched Henry and his equally powerful opponent trade blows that would have laid low a heavyweight boxer in his prime. They watched as pitifully small Chief Reddy tried to hold back the arms of his brawny adversary. * * * CHIEF STRAINED AGAINST Luther Unty. He felt the impossible physical strength of the man, but, at the same time, he quailed under the assault of hatred and vituperation that flowed freely from Unty's heart. The young leader could see that Unty had been feeding on hatred since his childhood in the streets. Dozens of other souls were tortured in the angry man's memories. Raped women and children, men shot down as they walked out of their homes. There were beatings and one night where he had tortured a young man just for being weak. The malevolence in Unty's heart paralyzed Chief just as the unutterable evil of Harold Timmons had. He felt himself lose balance and fell. "Hah!" Unty yelled as he prepared to crush Chief's head. The young leader, at that moment, gave up the struggle thrust upon him by Destiny. He was once again a weak boy unable to rise or wipe his own ass, unable to call out and be heard. Above Unty he could see the towering semitransparent and golden image of Prometheus. The beautiful Olympian's features were somber and pensive. Once again time had slowed in Chief's mind. The respite could not save him, but only show in excruciating detail the hard-soled foot and the coming of death. It was then that he saw Rhonda coming through the image of the Titan. She had a big rock in her hands and was intent on the head of his slayer. Seeing Rhonda he thought about the kiss; he felt how it was to stand next to her, for her to hold him. Everything depends on timing, he thought clearly, without fear or sense of hopelessness. This simple phrase of detached observation filled Chief with glee. He laid there no longer prey to the pain and suffering that Unty had lived through and parsed out. It was just a matter of space and time and the flame that burned in his heart and his mind. Rhonda slammed the big rock against the right side of Luther Unty's skull. At that moment time, for Chief, began to move quickly again. The blow to Unty's head was powerful and true but the young killer was more than proof to this attack. He swayed to the left a few inches and he lost his balance because he was only standing on one foot. He had to right himself, prepare once more to snuff out the weak child who enraged him simply by existing. This momentary reprieve was more than enough for Chief's salvation. While Unty teetered on his feet the boy rose effortlessly, no longer paralyzed or appalled. While his friends fought doggedly, while the congregation of light watched, Unty righted himself and prepared to knock Chief down and to stomp his head and throat, balls and diaphragm. Then the killer would help his friends and they would kill as many of the followers as possible. But before any of this could happen Chief Reddy reached out and touched Luther Unty in the center of his chest—all of the flame from his heart and mind channeling through his fingers. The concussion threw Unty backward ten feet or more, leaving him unconscious with smoke rising from his bulky dark clothes. Luther's cohorts fell to their knees, their strength gone without Luther's hate to guide them. Everything was still in the field. There was magic in the air but no laughter or mirth. SIXTEEN "EACH ONE OF YOU come forward and touch him," Chief Reddy said to his ninety-three surviving followers. "You will feel the chill of evil and desperation. You will see in him those who you cannot help to convert. Understand this feeling, hold it close, and never give the cold fire to anyone with it." It was nighttime in the woods a few hundred yards from the first battle of Chief Reddy's life. The moon was nearly full and the young god's eyes burned brightly. Luther Unty, stripped to the waist and gagged, was lashed to a small pine. He struggled against the leather bonds made from belts donated by Redd's Revolutionaries. The followers went up one by one enduring the threats and curses that sang from the killer's muffled throat. They suffered his sickening touch and the waves of hatred that rolled off of him. The dead had been buried and Luther's followers were tied up in a ravine not far away. "It feel's like disease," Tana Chin said to Chief after touching Unty's forearm. Chief placed his hand upon her brow feeling the light flow from his center into hers. In the radiance of their connection he could see Tana's children and their children and her grandparents who were born in San Francisco but never learned English very well. Her life was a gray two-dimensional background and she was an illuminated, fully formed lark flying from that bleak canvas out into the world. They smiled at each other, the boy and the fifty-something grandmother. "I will go back to China," she said, "even though I've never been." There was a baker and dentist, a professional burglar and two prostitutes who had come to the park, witness to their own salvation. Chief touched each of them after they had tested the evil of Luther Unty. He imparted some of himself to each and in doing so became weaker and weaker. By the time he'd addressed everyone Chief Reddy could hardly stand. When the last of his followers had gone out to spread the word he fell to his knees, the light in his eyes barely a flicker. "I'ma kill this mothahfuckah now," Henry Minter said advancing on Unty. "No," Chief called out from the ground where his head was being cradled by his uncle and Rhonda held his hands. "Killing him would be worse. Leave him tied to the tree. Leave him." * * * WHEN HE WOKE up the next morning Chief found himself on the beloved, overgrown lawn of his backyard. Hundreds of starlings were lined up on the fences and telephone lines, on the edge of roofs and the backstairs of his mother's apartment and Rhonda's new place. "You almost died," Prometheus said. "It felt as if I wasn't a body anymore," the boy said to the Titan kneeling beside him. "You gave yourself to your followers," the immortal briefly named Foreman Prospect said. "They came looking for a leader but you took each one by the hand and spoke your names together. In the future you must remember never to meet with more than seventeen of your teachers at a time. Remember, Death stalks you from all sides—from above and below, and even from the people you serve." "Did you send Unty against me?" Chief asked. "Yes." "Why?" "I am dead," the Titan said simply. "I have no choice in where I go or what is seen through me. Luther Unty's hatred dragged me from the earth. I could not deny him answers to his prayers." "And he prayed to find me?" "Yes. But it had to happen. You will find much evil in this sundered world. Malice has grown where hope and faith once thrived. The gods have done this to your race." "But we created the gods," Chief argued with the shimmering image. "You have also created the poisons in the air and the ocean, the cities where no one can feel the earth or see the real sky. You have made Commerce a god that weighs on humanity like a twenty-four-pound boil on a man's back. That which is created is not necessarily good..." Prometheus faded with these last words and Rhonda McKinney came out of her house wearing a sheer coral robe and no shoes. "Come on in the house with me, baby," she said. * * * INSIDE SHE LET the robe fall to the floor. Chief's child's heart quailed. He was about to thank her for saving his life, maybe saving the world, but the shock of her young and voluptuous body threw all gratitude from his mind. "Come on up in the bed with me," the older child said to the younger. She led him into the Rodriguezes' bedroom. The family had moved in with the Joneses across the street and left their furniture for the god's girlfriend. There were religious icons everywhere: crucifixes, paintings, holy candles, and small sculptures of Mary and Jesus and the three kings. "Take off those old clothes," Rhonda said. When he balked she got down on one knee and began unbuttoning, then unzipping his pants. She pulled these down along with his underwear and didn't even seem to notice the straining erection. "Get in the bed, Junior," she said as if she hadn't seen him throw a man more than twice his size ten feet with only a touch. Chief did what she requested. He got under the blanket but she laid out on top, her breasts and legs leaning toward him. Chief, for his part, was silenced. The beauty of Rhonda, her courage and refusal to worship him or seek his gift sparked something in him that was unbearable and inescapable. "I ain't gonna do nuthin' wit' you until you a big man wit' big thing for me. You hear that, Junior?" "Aren't you afraid after what happened yesterday?" the boy asked. "What for? I seen worse than that in my own livin' room. One time my daddy beat Mama till one'a her eyes was hangin' outta her head. You know I seen all kindsa bad shit." "But you must have seen the power unleashed last night." "Yeah?" she said. "So?" "I don't know... can't you tell that it's different? That this is a very special moment in the history of the human race?" "Uh-huh," she said making a sexy sneer with her left nostril and upper lip. "I always known I was special. Why else God wanna test me like that? My grandmamma told me I was somebody an' no mattah what happened I was gonna be important an' not just 'cause'a my ass neither." Chief was momentarily aware that these times spent with Rhonda were most of the personal life he would ever have, that and a couple of stolen seconds with Uncle 'Some. He would always have these moments of clairvoyance; like when he was fighting Luther Unty and realized when he would be slain by those that feared and hated him. "Will you become my wife?" he asked again. "When you get bigger'n me an' got a hard on like a man, then I'll think about," she said affecting a hard tone and a somber visage. "I love you," Chief said and Rhonda's face softened. "The minute you got down next to me in the museum park I wanted to hold you. I've never felt like that before and won't ever again." "That's what you say now," Rhonda said, trying to get back her hard shell. "No," Chief said, placing his hand on her cheek. "I'm not like the others you've known or seen. My life, like yours, is fated. I have a journey and you are part of that. I love you, will never love anyone else like this." Rhonda McKinney's breath came faster and her mouth opened. And while she wasn't phased by demons and people getting on their knees to praise Chief, she was moved by the depth of his words. Even while she couldn't have complete faith in his promises, neither could she avoid his call. "What, what if I still say you cain't have no pussy till you the man I done wrote about in my diary?" "Tell Uncle 'Some to buy some steaks at the Ralphs," he replied. "I have to eat a lot of meat to build back my strength and to meet your demands." SEVENTEEN FOR SIX DAYS Chief remained in Rhonda's half of the family home. For three of those days she made him scrambled eggs in the morning, hamburgers for lunch, and steaks or chops at night. He ate and slept and looked upon her body feeling the blood flow and a distant call in his soul. "You hot, baby," Rhonda said to him on the third night. "Maybe you got a fever." "A fever for you." It wasn't the words but the tone of voice that made Rhonda's heart skip. She reached out to touch his straining manhood but he pushed her hand away. "Let's wait until I'm the man you need," he said. "But you ain't grown at all. And, and I love you, Junior. I want you." "Wait." * * * "HE OKAY," NOSOME BLANE said two days later when Rhonda called him over. "How can he be okay?" the young beauty asked angrily. "Here he hot enough to fry eggs on and I cain't wake him up. What's okay about that?" The old man's face took on an expression that was both confused and certain. "I don't know what to tell ya, Ron," he said. "All I know is that Foreman give me sumpin', sumpin' that let me know what's going on in Chief, in his life. I see him there. I feel his fever and I know that it's somethin' he gotta get through. It's almost ovah. He be fine in twenty-four hours." * * * THAT NIGHT RHONDA curled around Chief and held him despite the heat coming off his skin. She thought he was out of his head, rambling meaningless words, but in reality he was speaking in the ancient tongue, chanting the song of manhood. Though meaningless to her, the words had a lulling effect on Rhonda's feminine character. The boy's song entered her dreams.... She was standing next to a river that was very, very deep with water so clear that she could see all the way to the bottom. Huge fish with intelligent eyes moved gracefully under the surface. They were colored in reds and emerald, peach and snow white. Thousands of these behemoths traveled in the icy river. They shimmered and shone in the fast-flowing river that was filled with other life, too. Yellow crabs scuttled along the bottom while red-and-black razor fish flashed back and forth above them. Bright green birds dove for smaller fishes and a huge bear stood on the other side watching Rhonda watch the river. She wasn't afraid of the bear. She wanted to swim in the water but was worried that the beautiful fish would devour her. Seeming to sense her dilemma, one of the regal beasts swam close to shore, near her. She was drawn to him but still nervous. "I won't hurt you." These words came into her mind. "But you're so big," she said. "Come to the water's edge," the fish's words boomed in her mind. She hesitated and then did as he said. She got down on her knees and placed one hand into the clear waters. Suddenly the fish rose up out of the river halfway onto the bank. It dwarfed the girl. It was the size of her mother's house, larger. She fell back and was covered with a red-flecked emerald fin. "Rise up and stroke my whiskers, girl," the fish commanded. The white hairs flowing from the snout of the beast were soft and moist, surprisingly warm. As she rubbed his long mustache the fish vibrated and purred. He was like a kitten, she thought. Or maybe a lion tamed by a gentle touch. "Will you ride on my back downriver?" the voice spoke in her mind... * * * ... AND, JUST WHEN she was about to agree, a feeling came into her and she groaned with satisfaction. She opened her eyes and Chief was there caressing her, kissing her body. She reached out to touch him but the passion overwhelmed her again and she fell back in the bed crying out. He kissed her and caressed her, pinched her thighs and pressed his fingers into her mouth. "Come inside me, baby," she moaned. She'd never call him Junior again. "Will you marry me?" he asked. "What?" she cried. "Feel me," he said. "That's what I want, Daddy," she whimpered. "But that will mean we'll have a child," he said. "And I cannot create life without knowing we are together." Rhonda fell back from her lover. She reached for the lamp that was a ceramic statuette in the form of white Joseph. The bulb showed her that Chief had grown into a tall, powerfully built man who was dark-skinned perfection. His erection stood out and gave no sign of waning. "I don't even know you, baby." "Do you want me?" "Yes, I do." "Then give me your hand and I will be yours and only yours now and unto forever." It was as if Rhonda could see the walls built around her heart being torn down by the force of his promise. She wanted to say no. She wanted to have ignored the sad boy she'd met in the park. She wanted so much and now she was at the doorway. Finally she nodded and he made love to her the way she had always wanted. She felt that he was with her, traveling back through the pain she'd known as a child living a broken life in a broken home; as a young girl in hard streets and prison-like schools making her way among rough boys and girls who would become broken men and women like her father and the lovers that used her and then were gone. * * * WHEN THE SUN CAME UP Chief was ready to make love again because, even though he was now in a man's body, he was still a boy having sex for the first time. "Hold up, baby," she said. "I need to rest." "How long?" the boy inside the man asked. "At least a few hours. I ain't nevah had nobody make me feel like that. I got to wait a little while or I'ma go crazy." Chief put one hand on her outer thigh and the other against her cheek. He kissed her breasts lightly and then looked into her eyes. He knew that it was Prometheus who informed his lovemaking but he didn't care. He was also that man. * * * "SO WHEN YOU WANNA get married?" Rhonda asked the man of her dreams, the man who had created himself to be with her. "We already are." "What you mean?" "The moment you agreed we were joined," he said. "And now you're pregnant and we will be together with each other and through the child of our love." "So you don't wanna get married with a judge or nuthin'?" "Of course we can if that's what you want. I am yours in any ceremony you feel we need." Rhonda wasn't impressed with powers of magicks, she didn't care about what other people thought or wealth, but she was moved once again by the submission of the man who was only a boy a few days before. She was going to tell him that she was his, too; that there would never be anything between them and that she would die for him no matter what he said or did. She was about to make this declaration when the door to their bedroom broke down and six fully armed SWAT team members hurtled into the room. PART THREE EIGHTEEN "ON THE FLOOR!" yelled a man dressed in black battle gear, including a bulletproof vest and face mask. He was toting an automatic rifle, pointing it in Chief's face. The newly formed young man stared at the weapon unphased but curious. Rhonda screamed and grabbed him from behind. The centurion-like police surrounded the bed, leveling their weapons. "I said on the floor!" The policeman grabbed Chief by the arm and attempted to pull him from the bed, but it was like tugging on a stone statue. "What's the problem, Officer?" the boy/man asked. "On the floor or I will shoot!" "Why you wanna shoot me?" Chief asked, speaking through the voice of his people. "I ain't done nuthin'. Here I am up in the bed wit' my woman an' you break down my door." "Where's the boy!" another cop shouted. "What boy? Me an' Rhonda's the only one's up in here." This declaration caused three of the invaders to fan out through the bedroom and then the rest of the house. While they banged around searching for a child named Chief Reddy the man of the same name studied the ones who remained, training their rifles on him and Rhonda. He quickly discerned their names and potentials but he kept this information to himself. The serendipity of his transformation had saved him from their intentions, and he could not reveal himself. One of the policemen was named Thornton Mead. He came from a house lined with books and parents who spent their lives dreaming about ideas, ideals, and the transition of knowledge slowly creeping across the ages like a half-frozen serpent writhing through hoarfrost toward the warmth of day. The jittering energy thrumming through Thornton's arms and legs drove him from his wistful parents and made him first a thief and then a cop. He rarely visited them, tried to keep them out of his mind. It was this attempt that made them so prominent in his thoughts. Chief Redd closed his eyes and tried something new. Without speaking he imagined the fire in Thornton's soul. It was weak but well formed. There was the fuel of love and hope and curiosity deep inside this man. Breathing in through his nostrils Chief imagined himself inside the brutal young man's heart. There he planted the second flame upon the first. Thornton shifted his shoulders and Chief knew that he had discovered a new power. "Nobody here," a returning SWAT cop reported to the man that had wanted them on the floor. "Out of the bed," the leader said then. Naked both Chief and Rhonda rose and allowed themselves to be pushed and prodded into the backyard. There they were herded together with Nosome, Henry, and Mary, his mother. Mary mother of God, Chief mused thinking of all the religious iconography in Rhonda's borrowed house. No... mother of Man. Looking at his family Chief knew that his uncle 'Some had kept Henry from fighting back. The three were on their knees and surrounded by a dozen cops. "Where's Chief?" Mary cried looking at Rhonda. "He left last night," the Lover replied. "He said he needed to go out an' see the people." Chief and Rhonda were pushed toward their little broken family. They joined them on their knees. For a while then Chief was lost to the world of human struggle and fear. When his knees touched the overgrown, unnaturally healthy lawn he felt as if he were somehow returning home. The feel of the earth rose up through him and he groaned with the pleasure of living and having lived. His moon eyes gazed upward at the sky and were momentarily lost in their heights. Then he noticed the one thousand three hundred and seventy-four starlings that had come around to witness the trial of their master. Chief closed his eyes and concentrated, dousing the moonlight that wanted so desperately to come out from him. In the darkness he was weightless in a cocoon made from starling feathers and a blue so insistent that it hurt. "He off like the girl said, Officer," Nosome was saying. "We tried to make him heed but he's a willful boy, a pain in his poor mother's heart." When Chief opened his eyes again he saw that the SWAT team had been somewhat mollified by the older man's words. "And who are you?" The man who had tried to make him kneel had taken off his mask. He was asking questions in a milder tone though still he spoke like a master. "Foreman Prospect," the boy-now-a-man replied. "I'm from Kansas. I just met Ronnie last night." The other cop, the one whose fire Chief had kindled, was staring at him... wondering. * * * THERE WERE LOTS of questions and threats from the police. At first they were looking to arrest Chief and then they threatened to arrest Mary for being an unfit mother. But finally they agreed that she would return to the station with her son or call them if she found out where he was. Nosome assured them that nobody wanted wild Chief Reddy in their house. He was crazy and would be better off with the state. * * * "SO WHAT WE GONNA do now, Junior," Nosome asked late that afternoon in Mary Reddy's living room. The family was there: Nosome, Rhonda, Mary, Henry Minter, and Chief. Mary had already yelled and screamed at Chief demanding that he tell her where her son was. "It's me, Mama," he said over and over. But it wasn't until he allowed the spirit to fill his eyes again that she was partly convinced. "But how can you be a man if you ain't grown inta one?" she asked her son. "He's more of a man than the whole Marine Corps, Mrs. Reddy," Rhonda said with an emphasis that was deep and undeniable. * * * "WE HAVE TO LEAVE, Uncle 'Some," Chief said. "They comin' at us from all sides now. When I was lookin' at the sky today I saw a ripple." "What kinda ripple?" "A disturbance that means someone has come from above to destroy me. And Luther Unty is still alive and the man Harold Timmons has been sharpening his knives and perverting my flame. I can feel him searching, wanting to cut open my chest. He believes that if he can eat my heart that he will have all my power." "And then there's the cops," Henry said. "Yes," agreed Chief. "They want to lock me away to protect their power." The phone ringing caused Rhonda to jump. She grabbed Chief's arm. "It's for me," he said. "Chief Reddy?" Thornton Mead said in the god-boy's ear. "Yes." "They're watching your house front and back," the soon-to-be-ex-policeman said. "Go out your side door at five tomorrow mornin'. Go into the Danby's house next door an' I will come with my truck at six-thirty. I'll get you away from them." NINETEEN HENRY LAY IN THE BED with Mary that night, his unnaturally strong arms around her, her eyes staring into the luminescent blue night-light plugged into the socket. "I used to dream that he'd come out of that room on his own just like he did," she whispered. "And now he's a man," Henry said. "Better." "But I don't know what to do about him. I don't know how to help him." "Junior told me that you spent all day every day feedin' him an' washin' him an' tellin' him stories. He said that he seen you fightin' the doctors and makin' sure the nurses did what the doctors said." "That was only a mother fightin' for her child," she said miserably. "But now he's like magic and Nosome is too and so are you. All I am is just a woman. And I'm tired, Hank. So tired I cain't even sleep." "Don't mattah if you tired," Henry Minter, the madman, said. "'Cause, baby, if it wasn't for you Foreman Prospect would'a nevah been able to find Junior. He wouldn't have seen that you made a boy with a soul big enough to hold the hope of the world. All three of us, Rhonda an' me an' Nosome, got our own job to do, but you the only one already done it. You the onlyest one done proved herself. Reddy's alive an' he wakin' people up from the nightmare man done made. He woun't'a been there if you hadn't worked all them years to keep him alive." * * * RHONDA AND CHIEF made love through the night. For hours he massaged her with oils they found in the Rodriguezes' bathroom. She wanted to ask about how they would live, where they would go, but then she'd look at him and see that he was her ideal—a man that was strong enough to hold up the roof but who would never hit her or the child growing inside. * * * NOSOME BLANE LEFT the houses of his family and new friends. Since he'd been forced to give up drink he spent hours thinking about things. And thinking was a problem in a house where everybody was in love but him. He walked a mile or so, followed at a distance by an unmarked patrol car, until he reached the house of Tonya Poundman, his sister. He knocked and Rutherford answered the door. "Hey, man," Nosome said to the squat and powerful carpenter. "What you want?" Rutherford had rust-colored skin and sharp features. He looked like a man from another age who worked under a hot sun building pyramids in the jungles of South America. "I'm glad you come back, man," Nosome said. "Nearly broke my sister's heart when you wanted her to choose." "I said, what do you want?" "Nosome?" Tonya called from somewhere beyond the door. "I come to pay ya back for some'a what you done for me, Rutherford," Nosome said. He pulled out a wrinkled, brown paper lunch bag from his jacket pocket. It was filled with something. Nosome handed the bag to Rutherford who was staring suspiciously at his brother-in-law. Tonya came to the doorway then. She smiled seeing her brother. He smiled for her. "What's this?" Rutherford asked. "Seventeen thousand four hunnert ninety-two dollars an' few pennies," Nosome replied. "Where you get it at?" "Street ministry," Nosome said. He could feel the police watching him but that didn't matter. "Foreman used his healin' touch on Chief an' the boy been goin' around preachin' the good word." "No healer could help that boy," Rutherford said. "I'm just tellin' you what I know, brother. This is some'a the money the boy done made. He said he wanted his auntie to have it, for savin' me an' then for Foreman savin' him." "You wanna come in for a drink?" Tonya asked her brother. "I done give up the sauce, pumpkin, but I could use some water." * * * TWO HOURS LATER and sixteen blocks away Nosome Blane went into a bar named Tookie's Tavern. It was a place he'd frequented for a couple of dozen years back when he hadn't known more than a few minutes of sobriety a day; back when he woke up every morning and vomited before taking his first drink. "'Some!" hailed Anita Lanan, the owner and bartender of her deceased husband's pub. "Where you been? And look at you, dressed all nice. You want a beer?" "Naw, Nita. I done give up the sauce for a while." "How long?" she asked with a knowing leer on her brown face. "Not long. Just till the end of days," he said as lightly as he could. "I'll take some fizzy water though. That an' some'a them pretzels if you got 'em." * * * "HI," A DARK-SKINNED and handsome woman said as she pulled up a chair to Nosome's small table in the corner. He gauged her age at forty-five but there was still the playfulness of youth in her eyes and face. "Hello yourself," Nosome replied. "What you drinkin'?" "Water." "Water an' what?" "Just water tonight. I got some miles to travel in the mornin'," Nosome said. "What's your name?" "Cassandra." "An' why a pretty young thing like you wanna sit across from a old man don't have two nickels to rub together?" The woman opened her mouth, but for some reason the words remained unspoken. Nosome smiled. She tried to speak four times before the words made it out. By that time she had transformed her lies into plain language. "High yellah niggah outside give me fifty dollahs to get you drunk an' find out what you know about a boy named Chief Reddy." "An' here I'm only drinkin' water." "What's your name?" Cassandra asked as if starting the conversation over. "Nosome, Nosome Blane." "That's a funny name." The old man shrugged his shoulders. "What you do, Cassandra?" "I was a ho'," she said easily, "but now I got the HIV an' you know Jesus might forgive me sellin' my body but he ain't gonna look kindly on me killin' my clientele. An' seein' that I probably be meetin' him soon I figure that I should change my ways in a hurry." "Workin' for the cops?" Cassandra shrugged. Nosome ordered her a pitcher of Sangria. * * * WHEN THE PUNCH was almost gone and Cassandra and Nosome had become friends they began to talk philosophy. "You believe in somebody live on a higher plane?" Nosome asked the ex-prostitute. "You mean like God?" "Exactly," the old man proclaimed. "Like God but not him the way we always known. More like our better selves, the part of our hearts that's too good for this world." "Why the cops after you, Nosome?" she asked. "Because I believe in that higher plane an' they worried that I might be right." Cassandra, under the effects of the wine, squinted, trying to make sense out of the old man's words. "They scared'a your creed?" she asked. "You went to school huh, girl?" "High school. Why?" "You got to be able to read to use a word like creed when nobody already used it." "So answer my question," she said. "I knew a man named Foreman Prospect," Nosome spoke as if half in a trance. "He come from on top of a mountain somewhere and he had the power to touch somebody an' cure whatever it was made 'em sick—either in their soul or their body." "Where is this man?" "He passed on." "Too bad. I could use a man like that." "But he taught another man what he knew or at least some'a what he knew before he left us." "Could this man lay hands on me," Cassandra asked, "an' cure me?" "I haven't seen him cure nobody, but maybe he could." "I won't tell the cops nuthin' if you ask him." "You wouldn't tell 'em no way," Nosome said. "But you got to come wit' me tonight if you wanna see my friend. 'Cause you know by tomorrah he be gone." * * * CHIEF WASN'T ASLEEP when Nosome knocked on Rhonda McKinney's door. It was after one in the morning. The god-boy came out wearing a pair of Henry's pants with no shirt. "He sure look like somebody good," Cassandra said. "He is," Rhonda said with both hunger and satisfaction in her tone. She was wearing a yellow kimono and looking more beautiful with each passing moment. They sat in the ornate living room that had been torn up and upended by the police search. Nosome explained how he met Cassandra and what she needed from him. As Cassandra stared at the beautiful young man she began to make out licks of flame in his eyes. After a long time thinking Chief said, "I think I can cure you, Miss Harlow." "How you know my last name?" Nosome put a hand on Cassandra's wrist and she didn't pursue the question. "... but," Chief continued, "the cure will be painful in your heart. It will be like taking out all your secrets and all you ever did wrong and putting them in front of your mother and father in pictures, sounds, and smells that reveal everything. They will hear how you felt and what you said to the pimps and Johns and women you worked with. And you won't be able to turn away. Because if you stop I will not be able to cure you and you will live on with grief in your heart. "That's because when I heal I heal the whole person and you aren't only sick in your body. You have betrayed yourself and that needs fixing, too." Cassandra shivered and Nosome put his arm around her. She buried her face in his shoulder and nodded. "She ready," the old man said and Chief reached out to touch her... * * * THE WAILS THAT CAME from Cassandra Harlow were heartrending, filled with grief. Rhonda had to leave the apartment and go next door where Henry and Mary asked her what was happening. "He doin' sumpin' to this woman Nosome knows," Rhonda said. "She done clawed off her clothes an' now she beggin'. An' if you look at her you see things... terrible things that she done an' seen. It makes you feel like you her mama an' you got to see her let men do awful things but you cain't stop it an' she cain't neither." * * * IN THE MORNING, Chief was comatose, as was Cassandra. Henry carried the heir to Prometheus through the side of the house next door and Nosome took Cassandra, limp and unwieldy as a newly deceased corpse, in his arms. Thornton Mead came by in a locksmith's van and pulled into the neighbor's driveway where the god-boy's family loaded in and were driven away while the police sentries watched the house. TWENTY "YES," THE BEAUTIFUL dark-skinned minister said to the tent full of poor men and women of all colors and ages and stages of hopelessness. "There is a light inside you that grows when you work together with your neighbors, when you feed each other and heed each other and answer when your friend calls out in pain. "Pay no attention to the men with guns in uniforms. Pay no heed when the president tells you that you must kill. Don't pay taxes but help your neighbors. Build a monument in your hearts..." The Redds, as they came to be known in some places, had made it down to the outskirts of El Paso a little more than a week after escaping L.A. This was Chief's fourth sermon in six days. His practice was to walk through the town in the morning and afternoon addressing people by their names and mentioning the troubles on their minds. Those who leaned toward evil ran away but the ones who seemed like deer in oncoming headlights were handed a flyer by Uncle 'Some or Madman asking them to come to the revival meeting. Almost all who were called made the sermon if they could. After the lecture Nosome moved through the crowd with his porkpie hat upturned for donations. Afterward refreshments were served and those who had come from curiosity spoke to each other and bonded then and there. They would leave as a group planning a new life that wasn't based on buying and selling, living in isolation or accepting hatred of the unknown as a way of life. One or two whose light burned brighter were invited to spend a few minutes with Chief alone. He spoke to them as he had to the people in Will Rogers Park. He asked them to go out and talk to people who wanted to listen. "There are those whose fire burns darkly," he'd always say at the end. "They will resist you and you must let them go. There is evil in the world. Many have been infected by this malady and cannot know the gift without destroying it." * * * "I DON'T UNDERSTAND," Cassandra said to him that night at the Crossroads Hotel at the intersection of two lonely roads that were called highways. Everyone else was asleep when Cassandra came upon the beautiful young man sitting on a wooden crate in the half-empty asphalt parking lot. The moon was shining and Chief's eyes were on fire. "What, Cassie?" the man-boy-god asked. "What did you do to those people? What have you done to me?" Chief had come outside to see the millions of stars and look for a clue to reveal his celestial pursuers. He hoped that maybe Prometheus would come to him, but after three hours of waiting Cassandra was the only one to appear. "All my life I was confined to a bed," he said. "And then one day I was touched by something divine—" "Like you touched me," Cassandra said. "No, not really. I'm like a bug at the foot of the power that changed me. One day my flames might hope to reach his height. That is, if I'm not killed first." "What's gonna happen to those people you talk to almost every night?" Cassandra asked. "They'll go back to their homes and open their doors to each other. They'll move closer together and only one out of three will continue to work. They'll shop at the market and lay the food out on the lawn for their neighbors to come out and pick and choose what they need. They will not practice war or hatred and they will be amazed by spiders' webs and the kiss of death." "That just sounds crazy." "And the special ones will teach others to live outside the rancid lives they have been bunged into by meaningless labor and the codification of fear. They will have long talks in coffee shops with strangers and cousins that they've lost touch with." "But the cops will come after them like they come after us," Cassandra argued. "Then they will talk to their cell mates and jailors, their enemies and wardens. The prisoners will form into unions that will resist the will of retribution. After all, a man can only forgive himself, he can only truly be punished if he accepts the sin in his heart." "But what about me, Junior?" Cassandra asked. "You did something else to me. Nosome said that you almost died when you cured my disease." "The HIV was a little thing," he said. "It was like a wet spot that dried up instantly under the heat that you manufacture. The hard thing was the perversion of love and survival, the lies you told and the ways you made yourself ugly, hideous in the mirror. Your soul was paralyzed before you met Uncle 'Some. He awakened you and I cured you. I did this at the risk of my own life because the world needs minor deities like us. I am the Word and you, Cassandra, you are Healing. In time you can make the blind see and cancers wither. And every time you do it your life will be in the balance." For a while the two sat in silence; the lame boy in a demigod's body with a woman who had been intimate with ten thousand men, under more stars than anyone other than Chief could count. "What about you?" she asked after the long silence had run its course. "Me? I like TV. I used to dream about being able to hold a glass in my own hand without dropping it. And I remember being chained to a boulder and every day an eagle would come and rip the liver from my gut." "That's not you. You're not a TV show or a dream or a nightmare." Chief felt the healing hand of the woman fate had brought him. She touched his face. "I am the heir to the gods," he said without haughtiness or conceit. "They gave me powers to see and know and sometimes to offer the very beginnings of change. I have visions and now and then these revelations are not me but that from which I arise." An electric shock ran up Cassandra's fingers but she didn't pull her hand away. "Three hundred years ago there was a boy named Tumi captured in a raid near where men now call Ivory Coast. The boy lost his sister and mother to the king that defeated his people and he was sold to a Portuguese slave ship and chained in the hold with four hundred and twenty-seven others from different lands with different gods." As he spoke Cassandra could see these images, smell the foul odors and also the despair. "Tumi lost heart and after six weeks he died in his chains. He lay there open-eyed, still seeing even though he no longer breathed. "Desire rose up in that dead soul, the desire to undo all that had been done not only to him and his people but to that king who had sold his own soul, and even to the sailors who also condemned their descendants with the mark of evil. "I am Tumi," Chief Reddy said. "It does not matter what I ate for breakfast or why I travel from place to place setting fires in human hearts. It's not because my father abandoned me or because I broke my mother's spirit with my needs. I am nothing. Like any worker in a factory who puts the right front tire on the new car coming down the line. I'm just doing my job and nothing else matters at all." TWENTY-ONE A WEEK LATER Harold Timmons arrived in El Paso; drawn by the faint scent of god-spoor. He would murder one woman and devour her heart before the Community of Light, as the Redd Revolutionaries dubbed themselves in that town, became aware of him and hunted him until he had to run away. But Harold didn't care. He was after Chief Reddy's heart. He would have left anyway. Maybe he would have liked to stay long enough to eviscerate a child, but he was a practical killer. There was logic even in insanity. * * * WHILE HAROLD RODE buses and slaughtered hopefuls, Luther Unty was organizing youth and prison gangs around the nation. He gave wild bacchanals in empty warehouses from Spokane to Cicero, Illinois; spoke arcane verses over drugs that became more potent than anything any gang member had known before. Through his web of followers he searched for a boy-minister who could read peoples' minds and hearts. * * * ON JULY 16 of that year, the god named Mercury stood on line in a light cotton suit to shake hands with the smirking president. When the beady-eyed plebian president looked into the dark and bottomless orbs of the messenger of the gods he said, "Don't I know you?" "Ron Messenger, Mr. President," Mercury said. "I bring you an important piece of information." The president stalled because long ago—before electricity and psychoanalysis, before antibiotics and Christianity—a law had been set down in Man's genes that he must listen when the messenger of the gods spoke. The presidential handlers and the Secret Service guards tried to block the meeting set up that evening at the Wilton Hotel, but the president was adamant and Mr. Messenger was implacable. The two sat at a table of the forty-fourth-floor bar overlooking the city of Houston. Messenger wore a black suit and a black hat with twin yellow feathers on either side of the brim. "So where do we know each other from, Ron?" the president asked. "I don't remember exactly," the god said, "but it was somewhere in school." The president frowned and nodded. He wasn't sure either. So much of his early life was a blur, especially right then, when he was waging war in two countries and planning attacks on two more. "So what was so important?" the president asked, remembering that he had a meeting with the pansy general from the Pentagon early the next morning. "There's a young black man traveling around the country perverting the good book and turning people into enemies of democracy," Mercury said, reading key words of the shifty leader's mind. "His name is Reddy but they call him Chief Redd. Look into him and what he's done and I'm sure you'll agree that he needs to be destroyed." "Destroyed?" the president said, finding the claim incredulous. "Don't trust me, Mr. President. Just have the FBI or the Secret Service check him out. I'm sure you'll find that his brand of terrorism is worse than the whole Middle East combined." The most powerful man on earth turned his head to look out on the southern city. It was night outside but he knew that the temperature was over a hundred degrees. He turned back to ask the man named Messenger about the nature of the black man's sermons, but his old friend, who he didn't really remember, was gone.... * * * MEANWHILE THE REDDY FAMILY traveled through Louisiana and Mississippi, Arkansas and Florida. When federal agents tried to infiltrate one of his meetings five miles north of Tampa, Chief and his family borrowed a mobile home from one of his advanced acolytes and drove nonstop to Oregon. There they began the sermons again. They worked their way up into Washington State and out onto the islands of the Puget Sound. On their fourth night in the wild of the Sound Chief once again found himself sitting outside at night hoping for guidance from his creator. Before the Transition, Chief saw himself as a thing, something created and formed from the experiences of another. He had been a dreaming worm tended by his mother, too weak even to fall out of bed on his own. Now he was tall and powerful, women said that he looked like an onyx statue if the ancient Greeks had worked in that stone. He could read the light from any being and see into the dark dissipated hearts of men, but there was little to his own personality. There weren't the scars and blemishes of experience in him like there were in others. Madman, 'Some, Mary, Rhonda, and Cassandra all had deep marks on their souls, scoring that made them unique for their purpose. But to Chief his soul seemed like that of an infant, smooth and without consequence except for the events of the past few months. He had the worries of a young child before the Titan came into his room, his world. And now he, the least among men, was their guidepost, their one hope. Having these thoughts Chief Reddy didn't notice the two-hundred-pound timber wolf that approached from the nearby stand of pine. That morning the huge feral creature had scented the god-boy from fifty miles away. He left his pack and loped through the woods without stopping until he got to the edge of the trailer park camping grounds. He advanced on the musing demigod both fearful and ecstatic; for wolves still remembered the gods. Wolves had their own fires and those flames burned brightly and hot. That was why animals were drawn to Chief. They came to offer their fealty. When the boy lifted his head he was looking into the eyes of the brown and gray wolf. Before then Chief had seen the animals that followed him as curious beasts attracted to the spoor of Prometheus. But in this wild thing from the deep wood the heir of the flame felt a kinship and an answer to his innocence. The wolf gave to Chief his feral nature and his wild love of wind and scent, fresh blood and the clear notion of an eternal present. He held out a hand and the timber wolf batted it with his long snout. "What shall I call you, friend?" Chief asked. Timberman. The word came into Chief's mind. He tore off his clothes then and ran with his new comrade into the woods. They ran together all night feasting on rabbits and howling at the sky. TWENTY-TWO "NOW WE GOT TO BE TRAVELIN' wit' a wild animal?" Mary Reddy said the next morning. Rhonda had already made friends with the wolf. They were sitting on the ground together at the back of the mobile home. She was scratching his chest vigorously with both hands and he was licking her neck hungrily. Now and again he'd growl and leap to his feet as if realizing that he was in an alien and dangerous situation, but then he'd sniff the air and look toward Chief. The sight of the Heir calmed the wolfish fears and Timberman, as Chief dubbed him, would settle down with Rhonda again. "He's my heart, Mama," Chief said simply. "What's that mean? Your heart?" "I don't know what it means, Mama. But I need him to carry something for me." "Like a mule?" Mary, the daughter of country folk, said. "Not something like that," the god-boy answered, trying to find an answer within his own words. He failed and turned to his Titan-appointed guardian. "Where we goin' today, Uncle 'Some?" "Salt Lake City, Junior," Nosome Blane said. He and Cassandra had spent the night in a motel down the road. "Got some real religion up that way. Lotta people already half the way there." * * * ON THE RIDE Chief sat at the back of the mobile home with his arm around Rhonda and Timberman at his feet. Mary and Henry sat together further up talking about their disparate experiences; him fatherless and on the street picking fights and scrambling to survive and her waiting hand and foot on a boy she loved more than anything—a boy she now felt was lost to her, a boy whose heart was carried by a wild animal. Nosome drove while Cassandra read to him from all kinds of different books. Cassandra, it turned out, was well educated up until the age of eighteen. She'd used books as way to escape the pain of her life. And now that she was healthy and on the road with the god-boy, her greatest pleasure was reading biographies and novels, how-to books and religious tomes to her old, old man. Nosome for his part would drive hour after hour without getting tired or bored. He listened to every word Cassandra read, increasing his knowledge by leaps and bounds. Another thing that Foreman Prospect had done to him was to make his mind a sponge for knowledge and simple detail. He remembered every acolyte in every town in every state they drove through. Faces, street signs, passing comments by strangers in the street, Nosome even remembered errant sounds and noises. He himself was like some great tome recording all the aspects of the world. While the old man drove and listened he'd look up into a mirror above his head now and again to see Chief in the far back on the padded bench with his wolf and his girl. Every hour or so he'd catch the alien eye of his grandnephew and spiritual guide; thus entering into the experience of the vastly expanded, and yet inexperienced, mind. For the most part Nosome could tell that his nephew was just a boy with few experiences and little notion of all the little things in the world. The elder Blane realized that Foreman had empowered him to help guide the boy in the indispensable trivialities of life, the trace elements that bound the soul to the world. They often talked about everyday things like stoplights and jazz clubs, convenience stores and how people greeted each other on the street. "Junior," Nosome would say, "you are the servant to man not the master." "I know that, Uncle 'Some." "Yeah. And I knew for forty-sumpin' years that I should get sober, but you know that didn't make me put down the bottle." * * * ON THE DAYS while Nosome drove and Cassandra read, when the driver would look up and see into his nephew's eyes, there often came a moment of realization that was neither person but a temporary amalgam of their altered spirits. Nosome would ponder this experience in the dark after making love to Cassandra even though he thought that sex in his life was dead and gone. In his thoughts Nosome would be stopping at Salt Lake City, Boise, Chicago for two weeks, and on to Minneapolis, Cleveland, and Cincinnati. They traveled through dozens of towns and rode in the mobile home using the donations of newly turned Redd Revolutionaries to buy their gas. Nosome would remember everything and Chief would see these memories in their brief connections. Life for both men was growing like a fire in the dry woods of late summer. They traveled for only two months but it seemed, especially in those daily meetings of the mind, that they had been wandering for years in the spiritual desert of America. Through Chief's mind Nosome could see the sadness and desolation that filled the hearts of almost everyone they met. "How can you take it, Junior?"'Some asked his nephew at Howlin' Wolf Trailer Park, twenty-six miles outside of Memphis. "Take what, Uncle 'Some?" the boy asked. "All these poor people draggin' 'round they souls like hundred-pound sacks of dead fish stinkin' up everything and hardly able to take a step wit'out groanin' out loud from the strain." It was nighttime and Chief's eyes were ablaze. He and Rhonda had just made love in a little abandoned shack in the woods. He was feeling relaxed and unconcerned about the world. Where Nosome remembered everything, Chief's mind, when he wasn't preaching, was filled by sensual pleasures brought to him by Rhonda and Timberman. The only times he was forced to think were when Nosome would look in his eyes or have a late-night talk. "I don't like to think about the way they are, Uncle," Chief said. "I try to see 'em the way they will be when the fire takes hold and they turn their backs on the foolishness of their lives. "In twenty years we'll come back the way we've been and whole cities will be filled with people who live for each other opening their doors and their hearts to the world before them. They won't need leaders or guards or policemen to do right. They will make a heaven right here on earth and, and when they die they'll be their own angels in a place that even I can't imagine." Listening to Chief's answer, Nosome Blane learned something both terrible and exultant; something that was hidden between the lines of the boy's vision. Nosome was afraid because he didn't feel equal to the task that Foreman Prospect had set out for him. "Junior," the spry old man said then. "Yeah, Uncle 'Some?" "I wanna ask you sumpin' 'bout what we doin'." "What's that?" the boy said. He was thinking about Rhonda's body, how she could make him go crazy by just letting a shoulder-strap fall or a lazy finger graze his neck. Then he remembered her hitting Luther Unty with that rock even though she could have stayed hidden and run from him. "What if Foreman wasn't arrested and put in that cell wit' me?" Chief snapped out of his reverie and felt a slight pang of fear. His face clouded over for a few moments and then he smiled. "Either it happens or it don't," the boy uttered. "Come again,"'Some said. "Either it happens or it don't, Uncle 'Some. The world don't have no take backs. You sleepin' in a house an' the stove catch on fire. Later on you wake up in a hospital an' there's burns all ovah your body. You think, 'what if I was out when the spark flashed an' the fire grew,' but that's ovah. You in the hospital dyin' an' they ain't no way back to when you was whole." Chief realized that he was speaking the language of his uncle not of Prometheus the Titan. This seemed appropriate. He wasn't a single being like other men. He was Transcendent, made of the impossible combination of mud and sunlight. "But we not wounded, Junior," Nosome said. "We livin' in bliss." "Happiness has a cost, Uncle," Chief said sadly. In his heart he was saying good-bye to his childhood. He realized for reasons he did not yet fully understand that his uncle's question marked the end of innocence. "What kinda cost?" "By now there are seventeen thousand three hundred and twenty-four souls that have had their light reignited and the second flame placed upon that. They are forming into groups that will resist the darkness of the gods. There are eight hundred and sixteen lamplighters that have been sent out on the path to inflame many thousands more. And while the flames of the Titan grow so do our enemies. They are, even now, planning to destroy us." "Then let's make us a army," Nosome said. "Let's settle down in New York or L.A. and convert us a army to hold back thems that wants to kill you." "I could do that but then the Word I'm giving would become like Luther Unty and worse—Harold Timmons. The army I'd raise would never seek peace and civility. No, Uncle, I can only do what I'm doing just like we cannot escape the fate pressed upon us by the Titan." "Why you talk one way sometimes and then another way a few minutes later?"'Some asked as a way of accepting Chief's decree. "Because I'm just the vessel, the glass that holds the wine." "I cain't drink wine no mo'," Nosome said. The boy laughed as his uncle shook his head ruefully. The talk, Nosome knew, was over... but the trouble was just beginning. TWENTY-THREE THEY ARRIVED ON BEALE STREET that noon after having parked their vehicle on an empty lot five miles away. Nosome wanted to do the sermon on the outskirts of town but Henry "Madman" Minter wanted to do their work on the street of musicians because Chief had predicted that his father, Terrence, would be there. "He's been freed from prison," the young cult leader had said. "But he is crippled and near death." * * * CHIEF WAS AMAZED at the lights that blazed in the musicians' hearts and souls on that street. The spirit of Prometheus was emboldened by the possibilities he saw among the people there. Chief called out their names and they followed him down the sidewalks playing instruments and humming tunes that had been sung in men's hearts since the days before the printing press and even the loom. * * * RON MESSENGER ARRIVED at the Memphis Airport at 6:15 that morning with a friend of his named Bill Archer. Archer had come from the same place as Messenger and was now in the employ of a covert branch of special services. On Olympus Archer was known as Phoebus, a member of Zeus's elite guard. He was armed with a long-range hunting rifle that had no telescopic site or any other kind of aiming mechanism. "I could shoot a gnat off of a fly's ass from two miles away," Archer had said to the shifty-eyed president. "I was formed from the ideal of the hunt." "Huh?" the president, who looked short but was deceptively tall, said. "He's a born hunter," Ron Messenger said. "That's all he means, Mr. President." "Does he understand that we need this done clean and neat with no strings, no trail to follah?" "He's the best, sir," the god said. "Once the job is done he will be gone and not the whole of the FBI will find even a footprint behind him." "I'm not sure," the president said, hesitating. They were standing in an empty hangar at the farthest end of an abandoned airport twenty miles north of Baltimore. "You checked this Chief Reddy out, haven't you?" Messenger asked. The shifty-eyed president stared at this man he remembered so well and yet hardly knew. He wondered, not for the first time, why he felt compelled to believe his words. "People are cutting themselves off from their governments and their peoples," Messenger said. "They aren't paying taxes and most of them aren't even going to their jobs anymore. They're working against your policies, sir, and there are more of them every day." "But he has so many followers," the president argued. "What difference would killing him make now?" "He is the heart and soul of the movement, sir," Messenger said. "His demise will break the heart in them." Mercury neglected to say that a special team was, even at that moment, forming on Olympus; a team that would come to earth and kill or pervert all of Chief's followers. It's what they did with Prometheus's first gift of flame. "No trail?" the president asked. "Not one clue, sir." * * * TIMBERMAN WAS LEFT in the van when Chief and his friends departed for Beale Street. But the feral beast tore through the flooring of the mobile home and stealthily made his way after a scent only he could discern. * * * BEALE STREET WAS ALIVE with music and dancing, free-flowing libations and the words of the god-boy. "You are my people," Chief Reddy said aloud. Everyone everywhere in a two-block radius could hear the words in their minds. "There is no reason in war. There is not satisfaction in revenge. There is no god sitting on any throne without you tilling the earth and carving from stone and giving birth again and again. There is no reason to lie or elevate yourselves above your station because you are all a part of divinity. There is no thing that will make you better—no gold or jewel or piece of paper saying you own anything but your own body and your own joy. There is no you without the person standing next to you. There is no other way to live...." While he preached tubas boomed and trumpets blared, women cried and men did, too. While he exhorted them with words they already suspected they sang and danced and absorbed wisdom that had lain dormant in Man's soul for uncounted generations. * * * AT THE EDGE of the huge crowd of revelers Henry Minter saw an old man in a wheelchair gazing blissfully in his direction. Henry was supposed to be standing next to Chief but he sensed something and slowly drifted away. "Are you Terry Minter?" Henry asked the old man. "Yes, I am. Do I know you?" "I'm Henry... your son." * * * TWILIGHT WAS COMING on and Bill Archer moved across the roofs above Beale Street with inhuman agility and extreme focus. No one saw him. No one was looking for him. The people in the streets were having the celebration that they had been waiting for for well over a thousand years. The prophet had come and they were him. His words were their hearts. His life was their hope. * * * MARY SCANNED THE CROWD for Madman. Nosome was worried but not exactly sure why. Chief stood on the roof of a baby blue Ford Explorer exhorting the crowd and feeling for the first time his soul rising up out of his body. When he looked down upon the thousands that filled the streets, dancing and making music, he saw in a third-floor window the glittering image of Prometheus. The Titan waved—good-bye? * * * AT THAT MOMENT Timberman leaped through the air while Bill Archer pulled the trigger of his gun. Nosome Blane looked up to see the wolf tearing out a man's throat on a rooftop and then he heard the men shout and the women scream. He turned to see Chief hanging off the side of the SUV, his head oozing blood. Somewhere Henry Minter was running. A wolf howled its lament over the crowd's cries. Cassandra ran to help but life was gone from the man who had ripped contagion from her—body and soul. On the rooftop Phoebus looked up at the darkening sky, experiencing real death and cursing his fate. In a small nearby store Nosome found five cans of lighter fluid. And while the Redd Family held back the crowd he drenched the dead god-boy and set him afire. The flames leaped high into the night and expressed images that everyone could see. The music was over. The celebration was done. Somewhere a wolf howled and Hope had died... but not without leaving its legacy. The flames were beautiful and anyone who saw them could not grieve for long because the inferno that was Chief Reddy's body lit up the night and the hearts of Memphis. Blacks and whites and browns and every other color and persuasion of men and women and children came out of their houses to be inspired to live, for that same flame now blazed within them. Mary and Rhonda mourned the son and the lover. Nosome fell to his knees and prayed. Henry Minter ripped at his breast and Cassandra held him, filling his anguish with the restorative spirit that Chief had given her. * * * RON MESSENGER, UPON seeing the flames rise up above the buildings on Beale Street, was deeply wounded by the power and content of the fire. He ran away through the oncoming crowds cursing Prometheus, who had given his only friend, Nosome Blane, the knowledge to keep the flame of the gods alive even as their herald perished. Nor was the flame soon over. Fueled by the divinity of Chief Reddy's body the blaze grew hotter and hotter. Nearby buildings began to burn and the crowd moved into wider and wider circles away from harm, but not so far as to distance themselves from the Light and the Heat of their deliverance. Fifty-seven city blocks were leveled by the heat before the dawn had come. Stone buildings collapsed. Cars melted into slag where they stood. Sidewalks and asphalt streets turned molten tar and stone. People from hundreds of miles around came to bask in the curative balm of light. Reds and blues, ochres and violet shimmering lights cascaded down upon the citizens of Chief Reddy's new world. Bill Archer's body was reduced to ash anointing the new age. Mercury returned to heaven scorched and bleeding from the god-child's fire. Luther Unty awoke in a Gary, Indiana, crack house choking on smoke that he couldn't locate. Harold Timmons was preparing to cut the heart out of a child he had taken out from her bed when the room started spinning and he fell unconscious. When he awoke the police had arrested him for a dozen major crimes. * * * NO ONE NOTICED the three women, two men, and one sad wolf that fled the flames they knew too well. Behind them they left the devastation of enlightenment. And though fully a million citizens were transformed and elevated by the light, many dark hearts failed that night. Men and women who were formed by evil were sundered and their souls drifted down to Hades. And hope was born the way Chief had imagined it. EPILOGUE A YEAR LATER and a million citizens of Memphis had gone out into the greater world: ragged flames of hope and inspiration spread around the globe intent on igniting a movement for unity. Luther Unty worked to undermine the greater cause. His powers were increased by Vulcan and Dionysus, but the gods feared that their rule was over. Harold Timmons mastered his fellow prisoners and sent them out to do his evil bidding. From his cell he ordered crimes that would decimate many thousands of the Redd Revolutionaries. The shifty-eyed president ordered every person who wore red and preached on street corners to be surveiled and arrested under the second Patriot Act. Everywhere people were coming out of their houses to hear the ministries of those who saw the fire that leveled so much of Memphis. Bands played all over the world and even though hundreds were slaughtered by frightened presidents and dictators, holy men and criminals, thousands more arose to carry on the Word. * * * ON THE ANNIVERSARY of the death of Chief Reddy, Nosome Blane called together Mary, Henry, Rhonda, Cassandra, and the wolf Timberman. They met at the rock in Will Rogers Park. They were a sad lot. Mary had left Henry as, she believed, he had abandoned Chief. Madman fell back into his old ways wandering the streets and drinking. Cassandra also went homeless and Rhonda bore a child, a son she named Truth. They lived in the old mobile home, and though she loved her son Rhonda was rarely known to smile. * * * "WE HAVE COME here to remember, Junior," Nosome said. "Now I know that you all hurtin' and sore and maybe even you blame each other and yourselves for the boy's death. But you know he wouldn't want that. He would want you to get together again and remember him for what he did to this world... for what he's doin' every day. And so I want you all to gather 'round this stone where he taught his first teachers and here we will remember him." Mary moved away from Henry, who was drunk or high, but Cassandra drew him into the circle. Rhonda, her baby in her arms, smiled briefly and took her place at the memorial stone. Timberman leaped up onto the flat rock and curled down, a wolven way to show respect. Nosome Blane lifted his arms to the sky and exhorted the spirit of mankind to witness the ceremony of remembrance. "We want to remember Junior here today in the place where he proved that he was a man," Nosome said. "We want to stand here as a family at least one more time and deny the powers that would keep us as their slaves. We want to call up the light and the fire that that boy brought to us without ever questioning his fate or his own needs. There's a revolution goin' on out there. There's a war for peace bein' waged in Asia and Europe, Africa and right here in the streets of Los Angeles. There's a battle ragin' an' because Junior never shirked or flinched the war is being won. I call now on the fire of my friend Foreman Prospect. I call on it to show itself to us here so that we can see where Junior wanted us to go...." "Look!" Mary shouted, pointing at the midsection of the couching wolf. Smoke and then flame rose from the thick and shaggy brown and gray pelt of the huge canine, but he didn't whimper or even seem to notice. The fire jumped high in the air, covering every color seen and not seen by the naked eye. When the fire rose above them all the Redd Clan gazed upon it each in their own commune with the flame. The wolf's body dissipated, feeding the memorial fires. Mary crossed over to Henry and embraced him. A broad grin showed itself on Mary's visage. Again, Nosome Blane fell to his knees. Cassandra took a step toward the fire. If anyone had witnessed this movement they would have seen that the woman left an image of herself, the whore and drug addict, behind like the empty husk of a butterfly cocoon or snake. She reached into the fire and drew out the form of a beautiful naked man with a wolf cub in his arms. One of his eyes became the flame that had engulfed him while the other shone like a half-moon at midnight. "Thank you," Chief Reddy said. "I have traveled a long, long way to get here. Now it's time for us to move." ALSO BY WALTER MOSLEY LEONID McGILL MYSTERIES All I Did Was Shoot My Man When the Thrill Is Gone Known to Evil The Long Fall EASY RAWLINS MYSTERIES Blonde Faith Cinnamon Kiss Little Scarlet Six Easy Pieces Bad Boy Brawly Brown A Little Yellow Dog Black Betty Gone Fishin' White Butterfly A Red Death Devil in a Blue Dress OTHER FICTION The Tempest Tales Diablerie Killing Johnny Fry The Man in My Basement Fear of the Dark Fortunate Son The Wave Fear Itself Futureland Fearless Jones Walkin' the Dog Blue Light Always Outnumbered, Always Outgunned RL's Dream 47 The Right Mistake NONFICTION Twelve Steps Toward Political Revelation This Year You Write Your Novel What Next: A Memoir Toward World Peace Life Out of Context Workin' on the Chain Gang ABOUT THE AUTHOR WALTER MOSLEY is one of the most versatile and admired writers in America today. He is the author of more than thirty-four critically acclaimed books, including the major bestselling mystery series featuring Easy Rawlins. His work has been translated into twenty-one languages and includes literary fiction, science fiction, political monographs, and a young-adult novel. His short fiction has been widely published, and his nonfiction has appeared in The New York Times Magazine and The Nation. He is the winner of numerous awards, including an O. Henry Award, a Grammy, and PEN America's Lifetime Achievement Award. He lives in New York City. Visit his website at www.waltermosley.com. This is a work of fiction. All of the characters, organizations, and events portrayed in this novel are either products of the author's imagination or are used fictitiously. THE GIFT OF FIRE Copyright © 2012 by Walter Mosley Artist Credit: Greg Ruth All rights reserved. A Tor Book Published by Tom Doherty Associates, LLC 175 Fifth Avenue New York, NY 10010 www.tor-forge.com Tor® is a registered trademark of Tom Doherty Associates, LLC. e-ISBN 9781466816138 First Edition: May 2012
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Australian Hard Rock Legends Rose Tattoo Unleash "Outlaws" Album March 6, 2020! by Unqualified-Kritic | Feb 8, 2020 | News, Tour Rose Tattoo are extraordinarily proud to announce the release of "Outlaws", the album, on March 6. A re-recording of the famed first and iconic "Rock N Roll Outlaw", spawning classics like "Nice Boys" (Don't Play Rock 'n' Roll), "Rock N' Roll Outlaw", "One Of The Boys" and, "Bad Boy For Love" these songs have etched their way into the music landscape over the last 40+ years. The magik with a 'k' is that they've included three bonus tracks from the very early days, songs that were demoed and never made the final cut, being a 'love song' "Rosetta" written back in 1975 by Anderson and Rilens, "Snow Queen" and "Sweet Love Rock and Roll" penned by Ian Rilens. Loveable rogue, Angry Anderson recruited the best in the business to head up the assault on this one comprising legendary bass player, Mark Evans of AC/DC fame, iconic guitarist Bob Spencer – Finch, Skyhooks, The Angels, and, Pete Well's endorsed, "unbelievably talented maestro of rock mayhem", Dai Pritchard on slide. Keeping the drum stool warm on this outing is 'Genghis', the keeper of time, Jackie Barnes. "For me, that's critically important, cosmically important, because this album is all about history, heritage, honour and respect. Just as significant is the forging, the embracing, and the official signing on if you like, to that history by THIS band and having them make their own reverent mark. This really is about honouring the past and respecting the present." – Angry Anderson "Rock N' Roll Outlaw", was originally recorded at the famed Alberts studios, produced by the legendary team, Vanda & Young and, released through Albert, Repertoire Records in late 1978. The bands debut album, reviewed as "A dangerous, unpredictable, monster of a record whose power has hardly diminished an ounce in the decades since this album cemented the band's foundation for the years to come." "The universe owes a deep unpayable debt to this band's founders Peter Wells and Ian Rilen as well as Mick Cocks, Dallas 'Digger' Royal, Geordie Leach, Rockin' Robin Riley and Lobby Loyde. Paul DeMarco, Dai Pritchard and Steve King have of course honourably kept the temple flame burning brightly more recently, but the album also rightfully acknowledges the foresight and support of Ted Albert, Fifa Riccobono, George Young and Harry Vanda who saw something in this band back when most other record companies were calling the cops." – Angry Anderson "Outlaws" will be released by Cleopatra Records on all digital platforms as well as on CD and a special limited edition vinyl pressing in your choice of RED or SILVER! To purchase: https://orcd.co/rosetattoooutlaws Rose Tattoo will undertake their first U.S. tour since 1982 this May. Rose Tattoo US tour dates: 05.07.20 – W. HOLLYWOOD, CA – WHISKY A GOGO 05.10.20 – GARDEN GROVE, CA – GARDEN AMPHITHEATRE 05.12.20 – DENVER, CO – THE VENUE 05.14.20 – HOUSTON, TX – WAREHOUSE LIVE! 05.15.20 – SAN ANTONIO, TX – THE ROCK BOX 05.16.20 – DALLAS, TX – TREES 05.17.20 – AUSTIN, TX – THE LOST WELL 05.20.20 – CHICAGO, IL – REGGIE'S ROCK CLUB 05.21.20 – FLINT, MI – THE MACHINE SHOP 05.22.20 – COLUMBUS, OH – ACE OF CUPS 05.23.20 – DAYTON, OH – ODDBODY'S 05.25.20 – BROOKLYN, NY – MARKET HOTEL (2nd show added) 05.27.20 – BROOKLYN, NY – MARKET HOTEL (original show still on) 05.28.20 – MANCHESTER, NH – THE JEWEL 05.29.20 – PHILADELPHIA, PA – VOLTAGE LOUNGE 05.30.20 – NEW BEDFORD, MA – THE VAULT MUSIC HALL http://www.rosetattoo.com.au/ https://www.facebook.com/RoseTattoo/ New 5CD Box Set Celebrates Australian Hard Rock Legends ROSE TATTOO With Unreleased Vintage Concert Performances! by Nick | Oct 16, 2018 | News Los Angeles, CA – "There were two bands in Australia that everyone called 'the boys'," says Rose Tattoo frontman Angry Anderson. "There was AC/DC, and there was the Tatts." Homegrown heroes of the highest order, Rose Tattoo formed in Sydney, Australia in 1976 and quickly became one of the country's most lauded and celebrated acts. They released album after album of pure, unadulterated riff rock madness scoring hits with songs such as "Bad Boy For Love," "Rock N' Roll Outlaw," "Scarred For Life" and lots more! The group gained international fame when Stateside up-and-comers Guns N' Roses cited Rose Tattoo as a major influence and covered the band's "Nice Boys (Don't Play Rock 'N' Roll)" on the 1986 EP Live ?!*@ Like A Suicide, which was later re-released on the multi-platinum selling 1988 album, GN'R Lies. Now on the eve of another major international tour comes this special box set collection, a monster 5CD set that captures these Australian gods at the very peak of their powers from 1980 to 1982. Scarred For Live features all of the band's best loved hits and many deep cuts as well, all packaged in an attractive box with rare photos and detailed liner notes written by music journalist Dave Thompson based on a new interview with the beloved Angry Anderson. DISC 1: Mount Druitt, Sydney, Australia – January 1, 1980 1. Snow Queen 2. Tramp 3. Astra Wally 4. Sweet Love 5. Nice Boys (Don't Play Rock 'N' Roll) 6. The Butcher And Fast Eddy 7. Rock 'N' Roll Outlaw 8. Bad Boy For Love 9. One Of The Boys DISC 2: Bondi Lifesaver, Sydney, Australia – August 31, 1980 2. Astral Wally 4. Movin' On 5. Remedy 8. She's Gone 10. Rock 'N' Roll Outlaw 11. Oxford St. Nick 12. Going Down 13. Sweet Love 14. Bad Boy For Love DISC 3: Reading Rock Festival, United Kingdom – August 29, 1981 DISC 4: Hordern Pavillion, Sydney, Australia – April 30, 1982 1. Out Of This Place 3. Assault & Battery 6. Rock 'N' Roll Is King 7. Manzil Madness 8. One Of Boys 9. Chinese Dunkirk 11. Astra Wally 12. All The Lessons 13. Nice Boys (Don't Play Rock 'N' Roll) 14. Money (That's What I Want) 16. Suicide City DISC 5: Wax Museum, Washington DC, USA – December 12, 1982 3. We Can't Be Beaten 6. Juice On The Loose 8. Branded 9. Scarred For Life Catch Rose Tattoo On Tour: Oct 5 – Marrickville, NSW Australia – Marrickville Bowling Club Oct 6 – St. Clair, NSW – Australia Oct 12 – Vermont South, Australia – The Burvale Hotel Oct 13 – Melton, VIC – Macs Hotel Oct 19 – Adelaide, SA, Australia – Norwood Hotel Oct 20 – Wallaroo, SA, Australia – Coopers Alehouse Oct 26 – Wollongong, NSW, Australia – Waves Oct 27 – Pittwater, NSW, Australia – Pittwater RSL Nov 2 – Ravenswood, WA, Australia – Ravenswood Hotel Nov 3 – Perth, WA, Australia – Charles Hotel Nov 10 – Central Coast, NSW, Australia – Ettalong Diggers Buy the 5CD box set: https://cleorecs.com/store/shop/rose-tattoo-scarred-for-live-1980-1982-5-cd/
{ "redpajama_set_name": "RedPajamaCommonCrawl" }
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\section{Introduction} Solar flares are events of rapid energy release in the solar atmosphere that are observed as sudden brightenings particularly in chromospheric and transition region (TR) spectral lines. It is suggested that the flare energy is initially released in the corona, which heats the local plasma and accelerates charged particles. Then the energy is transported downward through either thermal conduction or non-thermal particle beams, and is mostly deposited in the chromosphere. The chromosphere is heated to generate an enhanced radiation that is seen as flare ribbons. During the heating process, hard X-ray (HXR) emission can be produced by non-thermal electrons impacting the ambient atoms. In particular, heating of the chromosphere results in an excessive pressure, which can drive the plasma flowing upward, a phenomenon known as chromospheric evaporation \citep{neup68, hira74, acto82}. There have been suggested two types of chromospheric evaporation, gentle and explosive ones \citep{fish85b, mill06a, mill06b}. For the explosive evaporation, the upward momentum is large enough that at the same time induces a noticeable downward compression front, known as chromospheric condensation \citep{fish87, canf90}. In the gentle evaporation, however, obvious signatures of condensation have not been observed.\\ Chromospheric evaporation and condensation are mostly manifested in the Doppler shifts of spectral lines of relatively high temperatures and those of relatively low temperatures, respectively. Extensive efforts have been devoted to the spectral observations and spectral diagnostics on the chromospheric evaporation and condensation. Generally speaking, the most prominent spectral feature of the hot emission lines, such as Ca {\sc xix}, Fe {\sc xix}, Fe {\sc xxi}, Fe {\sc xxiii}, and Fe {\sc xxiv} lines, is a blueshift or blueshifted component with a Doppler velocity of tens to hundreds of km s$^{-1}$, which corresponds to chromospheric evaporation \citep[e.g.,][]{anto82, bros04, bros09b, bros09a, mill09, raft09, wata10, trip12, dosc13, youn15}. For the relatively cool lines formed in the chromosphere and TR, such as H$\alpha$, Ca {\sc ii}, He {\sc i}, and O {\sc v} lines, they usually exhibit a redshift or red asymmetry with a Doppler velocity of a few tens of km s$^{-1}$, which is caused by chromospheric condensation \citep[e.g.,][]{dela92, falc92, neid93, ganw93, wuls94, ding95, ding96, kuri17}. In fact, simultaneous appearance of both blueshifts and redshifts in lines of different temperatures, indicative of an explosive evaporation, has been observed at flare ribbons by different instruments. Based on the data from the Coronal Diagnostic Spectrometer \citep[CDS;][]{harr95} on board the {\em Solar and Heliospheric Observatory}, \citet{teri06} reported strong blueshifts in the Fe {\sc xix} line and redshifts in the O {\sc v} and He {\sc i} lines. With a higher spatial resolution and cadence as well as a wider temperature coverage, the Extreme-ultraviolet Imaging Spectrometer \citep[EIS;][]{culh07} on board {\em Hinode} \citep{kosu07} also detected blueshifts and redshifts simultaneously in different lines that can be explained in terms of explosive evaporation \citep{wata10, bros16, liyd11, dosc13}. In particular, it has been found that some emission lines formed at relatively high temperatures also show redshifts \citep[e.g.,][]{mill09, chen10, youn13}. \\ In addition, the {\em Interface Region Imaging Spectrograph} \citep[{\em IRIS};][]{depo14} has provided plenty of spectral data with a high spatial and temporal resolution since its launch in 2013. It offers a good opportunity to study chromospheric evaporation and condensation in more details. For example, chromospheric evaporation has been detected as a whole blueshift in the hot Fe {\sc xxi} 1354.08 {\AA} line \citep[e.g.,][]{poli15, poli16, sady15, dudi16}. On the other hand, chromospheric condensation has been observed in a few important cool lines. \citet{tian14} found that the cool O {\sc iv}, Si {\sc iv}, C {\sc ii} and Mg {\sc ii} lines display evident redshifts at the flare loop footpoints. \citet{bros15} studied the Si {\sc iv}~1402.77 {\AA} line and reported downward velocities that are consistent with chromospheric condensation. \citet{grah15} revealed sudden and strong condensation downflows from the red asymmetry of the Mg {\sc ii} 2791.59 {\AA} line. Moreover, through a comparison between the blueshifted Fe {\sc xxi} 1354.08 {\AA} line and redshifted C {\sc i} 1354.29 {\AA} line, \citet{lidn15} studied explosive evaporation in two X1.6 flares, which is mainly driven by electron beam heating. By checking the temporal evolution of the line features, \citet{sady16} found a delay of the maximum blueshift of the Fe {\sc xxi} 1354.08 {\AA} line relative to the maximum redshift of the C {\sc ii} 1334.53 {\AA} line. \\ The two resonance Si {\sc iv}~lines at 1393.75 and 1402.77 {\AA} observed routinely by {\em IRIS} have been used in a number of studies on chromospheric condensation \citep[reviewed by][]{liyd19}. \citet{tian14} reported Si {\sc iv} 1402.77 {\AA} redshifts with a velocity of about 50 km s$^{-1}$ at the loop footpoint produced by chromospheric condensation. \citet{liyd15} displayed red-asymmetric Si {\sc iv}~1402.77 {\AA} line profiles at the flare ribbons caused by condensation plasma. \citet{tian15} also reported asymmetric Si {\sc iv}~1402.77 {\AA} line profiles with an enhanced red wing but not entirely redshifted, which indicate a larger speed (about 100 km s$^{-1}$) of chromospheric condensation than previously thought. In addition, \citet{bran15} showed the Si {\sc iv}~line profiles with a red asymmetry or even two separate peaks within the ribbon of an M-class flare. \citet{warr16} found that the Si {\sc iv}~1402.77 {\AA} line exhibits a stronger redshifted component but a weaker stationary component. Considering the line profile shape, \citet{leek17} used a multiple-component Gaussian function to fit the Si {\sc iv}~1402.77 {\AA} line and obtained the intensity, Doppler velocity and line width for each component. Furthermore, \citet{liyk17} studied an X-shaped flare and found that the Si {\sc iv}~1402.77 {\AA} line changes significantly in shape at different locations. They considered that the Si {\sc iv}~line is either wholly shifted or asymmetric, in relation to different kinds of energy deposition. \citet{zhan16} discussed periodic chromospheric condensation through analyzing the Si {\sc iv}~1402.77 {\AA} line at different locations in a C3.1 circular ribbon flare. \citet{tian18} discussed repeated chromospheric condensation and energy injection in the form of nonthermal electrons by analyzing the redshifts of the Si {\sc iv}~1402.77 {\AA}, Mg {\sc ii} 2791.59 {\AA}, and Mg {\sc ii} k 2796.35 {\AA} lines. \\ In this paper, we focus on the {\em IRIS} Si {\sc iv}~1402.77 {\AA} line and study its temporal variations in three flares. In particular, we perform different methods to the line profiles with different shapes to obtain the physical parameters. By comparing the line features with the HXR emission, we find that the whole redshift and red asymmetry of the Si {\sc iv}~line may correspond to different flare heating modes. In Section \ref{sec-instr}, we describe the instruments and data reduction. Then we describe the methods of moment analysis and Gaussian fitting in Section \ref{thi-met}. Section \ref{fou-res} presents the results for the three flares, and Sections \ref{fiv-dis} and \ref{six-sum} provide discussions and conclusions, respectively.\\ \section{Instruments and Data Reduction} \label{sec-instr} The data used in this study come from a few instruments that are described in the following. We select three flares (listed in Table \ref{tab-table}) for study that possess the following observational characteristics: a variety of flare magnitude (both large and small flares), high enough time cadence (say, $<$10 s), full spatial coverage (flare ribbons) and temporal coverage (rise phase of the flare), and HXR observations. \\ {\em IRIS} provides high-resolution slit-jaw images (SJIs) as well as spectral data via a slit. The slit has a width of $0.^{''}33$ and a maximum length of 175$^{''}$. For SJIs, the maximum field of view is $175^{''}\times175^{''}$ and the pixel size is $0.^{''}166$. There are two observation modes to acquire the slit spectra, either raster scan or sit-and-stare, for the latter of which the time cadence can be as high as 1--2 s. Note that all the three flares under study are observed with a sit-and-stare mode. The relevant observational parameters including the time cadence and the pixel size along the slit are given in Table \ref{tab-table}. The level 2 data ready for scientific use are analyzed here. \\ \begin{deluxetable*}{lcccccccccl}[b!] \tablecaption{List of the flares analyzed in the paper \label{tab-table}} \tablenum{1} \tablehead{ \colhead{Event} & \colhead{Date of} & \colhead{{\em GOES}} & \colhead{Start} & \colhead{Peak} & \colhead{End} & \colhead{Cadence\tablenotemark{1}} & \colhead{Pixel size\tablenotemark{2}} & \colhead{HXR emission\tablenotemark{3}} \\ \colhead{Number} & \colhead{Observation} & \colhead{Class} & \colhead{Time} & \colhead{Time} & \colhead{Time} & \colhead{(s)} & \colhead{($^{''}$)} & \colhead{(keV)} } \startdata Flare 1 & 2014-09-06 & M1.1 & 16:50 & 17:09 & 17:22 & 9.5 & 0.166 & {\em RHESSI} 25--50 \\ Flare 2 & 2014-09-10 & X1.6 & 17:21 & 17:45 & 17:45 & 9.5 & 0.166 & {\em Fermi} 51--102 \\ Flare 3 & 2016-12-02 & B1.8\tablenotemark{4} & 16:14 & 16:24 & 16:36 & 1.7 & 0.333 & {\em RHESSI} 6--12 \\ \enddata \tablenotetext{1}{ This is the time cadence of the {\em IRIS} spectra that were obtained with a sit-and-stare mode.} \tablenotetext{2}{ The pixel size is for {\em IRIS} SJIs. Note that there is a spatial binning for flare 3.} \tablenotetext{3}{ It represents the highest energy band from {\em RHESSI} or {\em Fermi}, at which evident HXR emission was detected.} \tablenotetext{4}{ There is no record of the flare in the {\em GOES} list. The flare class is determined based on the peak of 1--8 {\AA} flux.} \end{deluxetable*} {\em IRIS}\ spectral windows include lines formed over a wide temperature range from about 4500 K to 10 MK, thus reflecting physical properties from the lower atmosphere to the corona. We focus on the TR Si {\sc iv}~1402.77 {\AA} line \footnote{This line may suffer from a blending with some relatively weak lines \citep[e.g.,][]{doyl92}. However, the effects of the blending on the Si {\sc iv}~line intensity as well as Doppler velocity are trivial and can be safely neglected based on theoretical calculations.} that has a formation temperature of about 10$^{4.9}$ K. Generally, the Si {\sc iv}~line can be considered as optically thin; thus we can easily derive the Doppler velocity from the line profiles. In order to determine the reference wavelength of the Si {\sc iv}~line, we first apply the single Gaussian fitting to the line profiles in a quiet region before the flare and then make an average of the line centers. After the reference line center is determined like that, the Doppler shifts calculated from the lines in the flare region are purely flare-induced. The reference line centers determined here vary marginally for different events. For example, the reference wavelength is about 1402.7852 {\AA} for flare 1 and 1402.7910 {\AA} for flare 2. Note that in flare 3 (a microflare), the Si {\sc iv}\ line is too weak outside the flare region; thus we just use the theoretical value of 1402.7700 {\AA} as the reference wavelength. The uncertainty in Doppler velocity is estimated to be $\sim$5 km s$^{-1}$. \\ For HXR observations, flares 1 and 3 were captured by the {\em Reuven Ramaty High Energy Solar Spectroscopic Imager} \citep[{\em RHESSI};][]{linr02} and flare 2 by {\em Fermi} Gamma-ray Burst Monitor \citep[GBM;][]{meeg09}. {\em RHESSI} can image and observe the Sun in X-ray and $\gamma$-ray bands (3 keV--17 MeV) with an energy resolution of about 1 keV to 5 keV. The spatial resolution can be as high as $2.^{''}3$ and the temporal resolution is about 2 s or better. For flare 1, we reconstruct the 25--50 keV image using detectors 2--8 from 16:55:00 UT to 16:55:32 UT. For flare 3, however, the emission above 12 keV is very low, and thus we do not reconstruct the image here. {\em Fermi} GBM has an energy band ranging from 8 keV to 40 MeV. It has 12 detectors, whose viewing angles towards the Sun vary with time during their operation. During the period of flare 2, the viewing angles of detectors n2 and n4 towards the Sun are almost unchanged (about $60^{\circ}$). Here we choose the data from detector n2 for analysis. Note that in this work, we only plot the HXR light curve at the highest energy band that has evident emission for each of the flares (see Table \ref{tab-table}). \\ We also use the data from some other instruments including the Atmospheric Imaging Assembly \citep[AIA;][]{leme12} on board the {\em Solar Dynamics Observatory} \citep[{\em SDO};][]{pesn12} and also the {\em Geostationary Operational Environmental Satellites} ({\em GOES}). AIA is designed to make full-disk imaging observations from multiple bands, which include seven EUV (94 {\AA}, 131 {\AA}, 171 {\AA}, 193 {\AA}, 211 {\AA}, 304 {\AA}, and 335 {\AA}), two UV (1600 {\AA} and 1700 {\AA}) and one white-light (4500 {\AA}) channels. The data from AIA are of high spatial resolution ($\sim$$1.^{''}2$) and high cadence (12 or 24 s). {\em GOES} has two soft X-ray bands, 0.5--4 {\AA} and 1--8 {\AA}. The 1--8 {\AA} flux is usually used to monitor the response of a flare. \\ \section{Methods} \label{thi-met} The Si {\sc iv}~line formed in the TR is usually treated as optically thin, although it may have some opacity effects especially during flares \citep[e.g.,][]{math99, kerr19}. Considering that this line does not show any reversal or absorption feature in the line core and that the line profiles can be well fitted by a Gaussian function (with either a single or double components) in the three flares generally, we assume this line to be optically thin here. If there are no obvious dynamics in the atmosphere, this line can be well represented by a single Gaussian shape. However, this is not always the case in flares, especially when multi-velocity flows exist along the line of sight. Therefore, we need to perform different methods in different cases when quantitatively deriving the parameters from the line profiles, which include the moment analysis, a single-Gaussian fitting, and a multiple-Gaussian fitting. \\ \subsection{Moment Analysis} The zeroth, first, and second moments, denoted by $I$, $\lambda_c$, and $\sigma$, respectively, are defined as following: \begin{equation} I=\int(f(\lambda)-f_0)\,d\lambda, \end{equation} \begin{equation} \lambda_c=\left[ \int\lambda(f(\lambda)-f_0)\,d\lambda \right]/I, \end{equation} and \begin{equation} \sigma^2=\left[ \int(\lambda-\lambda_c)^2(f(\lambda)-f_0)\,d\lambda \right]/I, \end{equation} where $f(\lambda)$ is the observed line intensity at wavelength $\lambda$ and $f_{0}$ is the background intensity. \\ The zeroth moment $I$ corresponds to the integrated intensity over wavelength. The first moment $\lambda_c$ yields the centroid position of the line profile, from which the Doppler shift velocity is calculated by \begin{equation} v=c(\lambda_c-\lambda_r)/\lambda_r, \end{equation} where $c$ is the speed of light and $\lambda_r$ is the reference wavelength of the line. The second moment $\sigma$ is a measure of the line width reflecting the velocity dispersion of the plasma at the line formation layer. The moment analysis is robust and can be performed to all kinds of line profiles. However, it can only yield parameters on average (like the line shift and line width) that should be used with caution in particular when dealing with very asymmetric line profiles. \\ \subsection{Gaussian Fitting} If the line profile shows a Gaussian shape, it can be fitted by the following function: \begin{equation} f(\lambda)=I_p\exp \left[ -\frac{(\lambda-\lambda_c)^2}{(\Delta\lambda_D)^2} \right]+f_0, \end{equation} where $I_p$ is the peak intensity above the background and $\lambda_c$ is the central wavelength of the observed line profile. The parameter $\Delta\lambda_D$ refers to the Doppler width of the line profile. \\ For some asymmetric line profiles, we also employ a double-Gaussian fitting. This assumes that the profile consists of two Gaussian-shaped components and a constant background \citep{hong16}, so that \begin{equation} f(\lambda)=I_1\exp\left[ -\frac{(\lambda-\lambda_1)^2}{(\Delta\lambda_1)^2} \right]+I_2\exp\left[ -\frac{(\lambda-\lambda_2)^2}{(\Delta\lambda_2)^2} \right]+f_0, \end{equation} where $I_1$ and $I_2$ are the peak intensities, and $\Delta\lambda_1$ and $\Delta\lambda_2$ are the Doppler widths of the two components, respectively. The parameters $\lambda_1$ and $\lambda_2$ refer to the central wavelengths of the two components, which can be used to calculate the Doppler velocity via Equation (4). \\ For the three flare events under study, we first make a moment analysis on the observed Si {\sc iv}~line profiles to get the average parameters, and then adopt a single or double Gaussian fitting to further extract the line parameters more accurately. Two typical examples of the observed line profiles and their fitting curves are shown in Figure \ref{fig-met}. In the first case (Figure \ref{fig-met}(a)), the observed Si {\sc iv}~line profile presents a good single Gaussian shape. Accordingly, we can see that the Doppler velocities derived from the moment analysis and the single Gaussian fitting are almost the same. Note that there still exists a tiny difference between the two velocities, which is supposed to be caused by a deviation from a purely symmetric line profile. In the second case (Figure \ref{fig-met}(b)), the observed line profile displays an evident enhancement at the red wing. It is obvious that the single Gaussian fitting is not appropriate for such a line profile with an evident asymmetry. Hence we adopt a double Gaussian function to fit the line profile, which yields a better and more precise result. It is seen that the first Gaussian component is relatively stationary, while the second one is redshifted, which contributes to the red-wing enhancement. The Doppler velocity of the second component is of course larger than the velocities derived from the moment analysis and single Gaussian fitting. This suggests that adopting a single Gaussian fitting may underestimate the redshift velocity of the Si {\sc iv}~line profiles that show a significant red asymmetry. Quantitatively, we perform a double Gaussian fitting to those line profiles if the velocities derived from a single Gaussian fitting and the moment analysis differ by larger than $\sim$3 km s$^{-1}$. Note that there are very few line profiles that could be better fitted by a three-Gaussian function. However, considering that the third Gaussian component is quite weak in intensity, we still perform a double Gaussian fitting to those line profiles. \section{Observations and Results} \label{fou-res} \subsection{Flare 1: the M1.1 flare on 2014 September 6} \label{fou-res1} \subsubsection{Observation Overview} The M1.1 flare on 2014 September 6 occurred in NOAA AR 12157. The flare started at 16:50 UT and ended at 17:22 UT. There exist two peaks in the {\em GOES} SXR light curve (about 16:56 UT and 17:09 UT), which imply two episodes of energy release in this flare \citep{tian15}. Here we mainly focus on the first SXR peak during the time period of 16:45--17:01 UT. This flare was well observed by {\em IRIS}, {\em SDO}/AIA, and {\em RHESSI}. The {\em IRIS} SJIs were taken at 1330 and 1400 {\AA} with a cadence of $\sim$19 s. From $\sim$16:55 UT, the exposure time of the SJIs changed from 8 s to 2.4 s. Figures \ref{fig-rhe}(a)--(c) show the AIA 131 {\AA} image, the SJI at 1330 {\AA}, and the spectra of Si {\sc iv}~during the flare, with the {\em RHESSI} HXR sources at 25--50 keV overplotted on the former two panels. Note that the {\em RHESSI} HXR and AIA 131 {\AA} images have been rotated counterclockwise by 45$^{\circ}$ to fit the orientation of the {\em IRIS} SJIs. One can clearly see some flare loops in the AIA 131 \AA\ image as well as their footpoints (or flare ribbon) in the SJI at 1330 \AA\ that well match the HXR sources. The {\em IRIS} slit crossed some of the flare ribbons where prominent emission shows up in the Si {\sc iv}\ spectra. Figure \ref{fig-rhe}(d) shows the {\em GOES} SXR, {\em RHESSI} HXR, and SJI 1400 {\AA} light curves. For comparison, we also plot the time derivative of the SXR flux. It is seen that the HXR emission and the SXR time derivative show a similar temporal evolution with peaks at $\sim$16:55 UT, implying the validity of the Neupert effect for this flare. The SJI 1400 \AA\ emission also peaks at nearly the same time within the time resolution of the observations. \subsubsection{Results} Figure \ref{fig-mo1} shows the space-time diagrams of the total intensity, Doppler velocity, and line width of Si {\sc iv}~derived from the moment method. Note that for the saturation region (marked by a black contour), we still provide the results of the moment analysis for reference \footnote{In fact, we did some tests by truncating the unsaturated line profiles to mimic the saturated cases. It is found that applying the moment analysis to the saturated Si {\sc iv}~line profiles at flare ribbons seems to be OK especially for just tracing the evolving trend of the physical parameters.} (similarly for flare 2 as described in Section \ref{fou-res2}). As seen from Figure \ref{fig-mo1}(a), the Si {\sc iv}~intensity increases gradually with the occurrence of the flare, and the flare ribbon undergoes an apparent motion toward northwest from $\sim$16:53 UT. Mostly, the Si {\sc iv}~line shows notable redshifts as revealed from the velocity map. The redshift velocity can reach about 40 km s$^{-1}$, which is within the typical velocity range of chromospheric condensation. It is also found that the redshift velocity keeps over tens of km s$^{-1}$ for some minutes after the intensity peak time. In addition, the Si {\sc iv}~line width shows a similar trend to the Doppler velocity, which is also a typical feature for the chromospheric condensation plasma \citep{liyd15} \\ In Figure \ref{fig-sp1} we plot the temporal evolution of Si {\sc iv}~spectra as well as some example profiles at a selected ribbon location marked in Figure \ref{fig-mo1}. It is seen that, before the Si {\sc iv}~intensity peak time, the profiles are symmetric and show a good Gaussian shape; then we apply a single Gaussian function to fit them (see Figures \ref{fig-sp1}(b) and (c)). However, after the Si {\sc iv}~intensity peak, the profiles appear to be asymmetric, or the red wing is enhanced compared with the blue wing (Figures \ref{fig-sp1}(d)--(f)); then we make a double Gaussian fitting for such profiles, which are decomposed into a relatively static component and a redshifted one. The fitting results show that the redshifted component is stronger and broader than the static one. \\ Figure \ref{fig-f1} plots the Si {\sc iv}~line parameters deduced from the single or double Gaussian fitting at the selected ribbon location, together with the {\em GOES} SXR and {\em RHESSI} HXR light curves for comparison. It is clearly seen that, before the Si {\sc iv}~intensity peak time (also around the HXR peak time), the single Gaussian fitting and the moment method yield almost the same results. During this period, the Si {\sc iv}~intensity increases with time and the redshift velocity also increases somewhat from 4 to 13 km s$^{-1}$. We notice that the line width increases gradually as well. After that, the line profiles become asymmetric rather than wholly shifted and the Doppler velocity for the redshifted component is about 20 km s$^{-1}$. By comparison, the velocity of the relatively static component is almost zero in this case. The wholly redshifted profile and the subsequent redshifted component are supposed to be caused by a downflow in the line formation layer, which is known as chromospheric condensation. Note that if we still apply the single Gaussian fitting to the asymmetric line profiles, then the downflow velocity would be underestimated, as mentioned in Section \ref{thi-met}. \\ \subsection{Flare 2: the X1.6 flare on 2014 September 10} \label{fou-res2} \subsubsection{Observation Overview} The X1.6 flare on 2014 September 10 is from a sigmoid region in NOAA AR 12158. The flare started at 17:21 UT and peaked at 17:45 UT. Here we mainly focus on the time period before the flare peak, i.e., the rise phase of the flare. The {{\em IRIS}} SJIs were taken at a cadence of $\sim$19 s at 1400 and 2796 {\AA} and the exposure time changed from $\sim$8 s to 2.4 s after $\sim$17:27 UT. Since the {\em RHESSI} data are not available for this flare, we use the observation of {\em Fermi} GBM instead. Figures \ref{fig-fer}(a)--(c) show the AIA 131 {\AA} image, the SJI at 1400 {\AA}, and the slit spectra of Si {\sc iv}, respectively. We can see that the {\em IRIS} slit crossed the eastern flare ribbon at two locations. Note that this flare is so energetic that many pixels are saturated. The {\em GOES} SXR light curve and its time derivative, as well as the {\em Fermi} GBM 51--102 keV and SJI 1400 {\AA} light curves, are shown in Figure \ref{fig-fer}(d). It is seen that the Neupert effect seems also valid for this flare. In particular, when the HXR flux keeps growing, the SJI 1400 {\AA} emission rises and then reaches its maximum.\\ \subsubsection{Results} The space-time diagrams of the total intensity, Doppler velocity, and line width of Si {\sc iv}~for flare 2 are shown in Figure \ref{fig-mo2}. The intensity map reveals an apparent motion toward south of two brightening features that correspond to the two locations at the flare ribbon crossed by the {\em IRIS} slit. In the following study, we focus on the upper (north) ribbon location. It is seen that when the flare begins, the intensity, velocity, and width of the line increase rapidly at the same time. The region that we are interested in displays significant redshifts in the Si {\sc iv}~line. \\ In this flare, an evident line asymmetry appears before the Si {\sc iv}~intensity peak time. Figure \ref{fig-sp2} plots some typical Si {\sc iv}~line profiles at a selected ribbon location. It is seen that a red-wing enhancement appears very soon after the flare begins. In spite of the saturation effect, one can still notice that the red-wing enhancement increases with time, as revealed from the double Gaussian fitting. After the Si {\sc iv}~intensity peak, the strength of the redshifted component decreases but the velocity still increases for a while. The line profiles gradually become to be wholly redshifted with a small asymmetry during this period. \\ Figure \ref{fig-f2} shows the temporal evolution of the Si {\sc iv}\ line parameters obtained from either the single or double Gaussian fitting at the selected ribbon location, along with the {\em GOES} SXR and {\em Fermi} GBM 51--102 keV light curves. It is seen that the Si {\sc iv}~intensity has a rise-and-fall evolution during the rise of the SXR emission, while the Doppler velocity and the line width keep increasing with time. It is worth noting that the velocity of the relatively static component reaches almost 30 km s$^{-1}$ at a late time ($\sim$17:41 UT). The Doppler velocity of the redshifted component can be up to about 70 km s$^{-1}$. We also note that the Si {\sc iv}~redshifts appear around the same time as the rising of the HXR emission. \\ \subsection{Flare 3: the B1.8 flare on 2016 December 2} \label{fou-res3} \subsubsection{Observation Overview} The B1.8 microflare on 2016 December 2 started at $\sim$16:14 UT and peaked at $\sim$16:24 UT. {\em IRIS}\ observed this flare from 16:05 UT to 16:22 UT with a very high cadence of 1.7 s. The SJIs were only taken at 1400 {\AA} with a cadence of 2 s. Figures \ref{fig-rh3}(a)--(c) show the AIA 131 {\AA} image, the SJI at 1400 {\AA}, and the slit spectra of Si {\sc iv}. One can see that the {\em IRIS} slit crossed the narrow flare ribbon where evident redshifts appear in the Si {\sc iv}\ spectra. We show the {\em GOES} SXR, {\em RHESSI} 6--12 keV, and SJI 1400 {\AA} light curves in Figure \ref{fig-rh3}(d). It is seen that the SJI 1400 \AA\ emission starts to rise at about the same time as the {\em RHESSI} 6--12 keV emission, both of which reach their maxima before the {\em GOES} SXR peak time. Note that the intensity of the Si {\sc iv}~line in this microflare is much weaker compared with the former two large flares; therefore, we only choose a part of the flare region with an intensity above a certain threshold for analysis here. \\ \subsubsection{Results} Figure \ref{fig-mo3} shows the space-time diagrams of the intensity, velocity, and width of the Si {\sc iv}~line derived from the moment analysis for the region brighter than the intensity threshold. It is seen that, although the increase of the intensity is small, there still shows up an apparent motion of the flare ribbon with time. Note that the redshift of the Si {\sc iv}~line just appears in a very narrow region of less than 5$^{''}$ along the slit. However, the Doppler velocity can reach up to 60 km s$^{-1}$. In addition, the line width changes consistently with the Doppler velocity as in the former two flares. \\ Compared with the former two large flares, the Si {\sc iv}~line profiles in this microflare are quite simple, which only display a single Gaussian shape throughout the flare evolution. Figure \ref{fig-sp3} plots two example line profiles at the selected ribbon location. It is seen that the profiles are wholly redshifted during the flare. Thus, the redshift velocity derived from the single Gaussian fitting is almost the same as that obtained by the moment analysis. \\ Figure \ref{fig-f3} plots the temporal evolutions of the line parameters derived from the moment analysis and single Gaussian fitting, as well as the {\em GOES} SXR and the {\em RHESSI} 6--12 keV light curves. One can see that the intensity, velocity, and width of the Si {\sc iv} line rise and decline almost in phase, also in rough coincidence with the {\em RHESSI} 6--12 keV flux. \\ \section{Discussions} \label{fiv-dis} There have been published a number of studies on flares 1 and 2 from different aspects \citep{grah15, kuri15, lidn15, tian15}. For example, \citet{tian15} studied the chromospheric evaporation in these two flares by mainly focusing on the Fe {\sc xxi}\ line that is shown to be completely blushifted at the flare ribbons. In particular, the blueshift velocity of the Fe {\sc xxi}~line shows a good temporal relationship to the HXR 25--50 keV emission, suggesting a chromospheric evaporation caused by nonthermal electron heating. \citet{tian15} also reported enhanced emission in the red wings of the Si {\sc iv}\ line profiles. The Doppler velocity of the redshifts is found to be larger than the previously reported values and lasts for a relatively long time in the decay phase. They ascribed the redshifts in the cool lines to chromospheric condensation and also downward cooling plasma from the corona. In this work, we mainly focus on the rise phase of the flare. The redshifts detected in the Si {\sc iv}\ line at the flare ribbons are supposed to be a signature of chromospheric condensation. In addition, as pointed out by \citet{liyk17}, some of the Si {\sc iv}~line profiles at the flare ribbons can be well fitted by a Gaussian shape, but others cannot. Thus, we have distinguished two types of Si {\sc iv}~line profiles here and apply different methods to fit them accordingly. We also notice that flare 1 has been studied with both observations and radiative hydrodynamic simulations by \citet{kuri15}. The authors found red and blue asymmetries in the H$\alpha$ line but only a weak red asymmetry in the Ca {\sc ii} 8542 {\AA} line at the flare kernel. Resorting to the simulations, they interpreted the blue asymmetry in H$\alpha$ in terms of downflows owing to plasma condensation. This is consistent with our explanations although we detect red asymmetry in the Si {\sc iv}\ line. Note that the H$\alpha$ line is optically thick, which can show different spectral features from optically thin lines. \\ From the HXR emission and its relationship to the Si {\sc iv}\ line profiles, we could explore the heating mechanisms for the three flares under study. We find that in flares 1 and 2, the HXR emissions can be visible up to 50 keV or 100 keV, respectively, and they roughly show a relationship to the appearance of the asymmetry of the Si {\sc iv}~line. In flare 3, however, there is no detectable HXR emission above 12 keV or no obvious asymmetry in the Si {\sc iv}\ line profile. Therefore, we consider that nonthermal electron beam plays a major role in heating the chromosphere in flares 1 and 2, while thermal conduction may be the dominant heating way in flare 3. Both of the heating mechanisms are thought to be reflected in the different behaviors of the Si {\sc iv}\ line profiles. \\ The fact that the shape of the Si {\sc iv}\ line profile at the flare ribbon depends on the heating mechanisms has been investigated via numerical simulations. \citet{poli18} performed nanoflare simulations with nonthermal electron beam heating and in situ thermal heating. They found that in the case of thermal heating, the Si {\sc iv}~line more likely tends to be wholly redshifted, just like flare 3 in our study. In addition, \citet{kerr19} studied two resonance Si {\sc iv}~lines in flare models with electron beam heating. They reported that the energy flux, the low-energy cutoff, and the spectral index of the nonthermal electron beam could all influence the Si {\sc iv}~line profiles. For instance, with a higher injected energy flux, the Si {\sc iv}~line could more likely show a red asymmetry. Such a case seems to be consistent with our observations in flares 1 and 2 that have a relatively high magnitude of HXR flux. Furthermore, with the condensation propagating downward, there possibly exist two regions, a condensation front and a relatively stationary region, both of which contribute to the line emission. This might explain the two components of the line profile with different Doppler velocities. We notice that in the simulations of chromospheric condensation driven by electron beams, \citet{kowa18} revealed two flaring layers in the chromosphere: a chromospheric condensation region with downflowing, hot, and dense plasma as well as a stationary layer below it. Note that the stationary layer can also be heated by the high-energy electrons. The two layers can reproduce the red-wing asymmetry as shown in some chromospheric emission lines during flares. Finally, although the simulations mentioned above could reproduce the Si {\sc iv}\ spectral features as shown in our work, there still exist some discrepancies between the observations and simulations. For example, blueshifts of the Si {\sc iv}~line have been displayed in simulations, especially in the cases with a lower injected flux. However, we do not find any blueshifts in the Si {\sc iv}~line at the flare ribbon in our study. In addition, \citet{kerr19} emphasized the opacity effects in their study. However, our analysis of the Si {\sc iv}\ line for the three flares is based on the assumption that the line is formed in an optically thin condition. In the future, we will study more flare events as well as carry out radiative hydrodynamic simulations to further investigate the relationship between the Si {\sc iv}~line profiles and the heating mechanisms. \section{Conclusions} \label{six-sum} By using the high-resolution spectral observations from {\em IRIS}, we have tracked the temporal evolution of the Si {\sc iv}~1402.77 {\AA} line profiles at the flare ribbons in three flare events. In all of the events, we detect evident redshifts in this line, indicative of downward mass motions in the TR layer. The line intensity, Doppler velocity, and line width increase and reach their peaks at nearly the same time. It is interesting that in the two large flares, the Si {\sc iv}~profiles can transit from wholly redshifted to red-wing enhanced ones in the rise phase of the flare; however, in the third microflare, the Si {\sc iv}~profiles are wholly redshifted throughout the flare process. We have performed either a single or double Gaussian fitting to the line profiles. Generally speaking, the single Gaussian fitting is appropriate for dealing with the wholly shifted line profiles while it would underestimate the Doppler velocities in the case of red-asymmetric profiles. In the latter case, the double Gaussian fitting should be adopted, which can reveal two emission components, a relatively static one and a redshifted one for most cases. Specifically, the redshifted component is thought to originate from the region with a significant downward velocity. The downflow velocities corresponding to the redshifted components or the redshifts as a whole are measured to be a few tens of km s$^{-1}$. These values are among the typical speed of chromospheric condensation reported in previous studies. Existence of an observationally prominent chromospheric condensation implies that these flares belong to the events of explosive evaporation. \\ In order to explore the physical mechanism behind the Si {\sc iv}\ line profiles, we analyze the HXR emission for the three flares. We find that in the two large flares, the HXR emissions are prominent, which can be visible up to 50 keV or even 100 keV. In particular, these HXR emissions match in time with the development of the Si {\sc iv}\ redshifts and even the red asymmetries to some extent. In the microflare, however, the HXR emission is very weak and only visible up to 12 keV, and the Si {\sc iv}\ line appears to be wholly redshifted during the flare. Thus, we consider that nonthermal electron beam plays a major role in heating the plasma in the two large flares, while thermal conduction may be the dominant heating mechanism in the microflare. \\ \acknowledgments The authors would like to thank the anonymous referee for constructive comments. We also thank Dr. Hui Tian for helpful discussions. {\em IRIS} is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by the Norwegian Space Center (NSC, Norway) through an ESA PRODEX contract. {\em SDO} is a mission of NASA's Living With a Star Program. The project is supported by NSFC under grants 11733003, 11873095, 11903020, 11533005, 11961131002, and U1731241. Y.L. is also supported by CAS Pioneer Talents Program for Young Scientists and XDA15052200, XDA15320103 and XDA15320301. \bibliographystyle{apj}
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package co.cask.tephra.hbase98; import org.apache.hadoop.hbase.filter.Filter; import org.apache.hadoop.hbase.filter.FilterList; /** * Utility methods for working with HBase filters. */ public final class Filters { /** * Adds {@code overrideFilter} on to {@code baseFilter}, if it exists, otherwise replaces it. */ public static Filter combine(Filter overrideFilter, Filter baseFilter) { if (baseFilter != null) { FilterList filterList = new FilterList(FilterList.Operator.MUST_PASS_ALL); filterList.addFilter(baseFilter); filterList.addFilter(overrideFilter); return filterList; } return overrideFilter; } }
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AB InBev buys remaining stake in Craft Brew Alliance for $321M Published Nov. 12, 2019 • Updated June 16, 2020 Christopher Doering Senior Reporter UPDATE: June 16, 2020: Anheuser-Busch said it would sell the Kona Brewing operations in Hawaii, including the new brewery and two brewpubs, as part of its purchase of Craft Brew Alliance announced last November. The arrangement does not include CBA's Kona business outside of Hawaii. The companies said the divestiture was done to "expedite the regulatory review process and alleviate potential regulatory concerns regarding the proposed expanded partnership." AB InBev will purchase the 68.8% stake it doesn't own in Craft Brew Alliance for about $321 million, or $16.50 a share in cash, the companies said in a statement. Craft Brew owns brands including Kona Brewing, Widmer Brothers and Redhook Brewery. AB InBev had initially decided not to exercise an option to purchase these shares in Craft Brew for at least $24.50 each. This purchase option expired in August, and the brewing giant instead opted to pay a one-time $20 million fee to the company. As part of a long-standing agreement between AB InBev and Craft Brew, most of Craft Brew's brands are already distributed by the beer giant. The companies said the deal is expected to close in 2020. As consumers gravitate away from household names such as Budweiser, Coors Light and Miller Lite toward trendier craft beers, spirits and nonalcoholic or low-calorie beverages, it makes sense for AB InBev to expand its presence in these areas. One of the fastest growing segments during the last decade has come in craft beers, which have rolled out creative names and flavor concoctions. Last year, there were an estimated 7,450 craft breweries, roughly double the number in existence in 2014, according to the Brewers Association. Craft production rose 3.9% in 2018, with the segment responsible for about a quarter of all sales in $114 billion beer market, the group said. In recent years, AB InBev has purchased craft players such as Goose Island Beer, Devils Backbone, Wicked Weed Brewing and Karbach Brewing. In August, AB InBev acquired Platform Beer, a fast-growing regional brewery founded in Cleveland in 2014. The latest deal comes after AB InBev passed on a chance to purchase Craft Brew for $24.50 a share in August. Craft Brew had the chance to explore other options after August 23, but a transaction with AB InBev made the most sense for a number of reasons — most notably the sizable stake the beer giant retained that would have given it a signicant say in any other offer from a third party. Considering the lower share price in the actual acquisition, Craft Brew saw deeper value in solidifying the relationship with AB InBev. AB InBev, through its Anheuser-Busch division, has worked with Craft Brew and its brands in some capacity for more than 25 years. AB InBev​ is no doubt intimately familiar with Craft Brew's portfolio and growth prospects. It also already distributes Craft Brew's beers though its expansive network. With this acquisition, AB InBev will most likely not have to do much to incorporate the business into existing operations. "By combining our resources, our talented teammates, and dynamic brands, we will look to nurture the growth of CBA's existing portfolio as we continue investing in innovation to meet the changing needs of today's beverage consumers," Andy Thomas, CEO of Craft Brew, said in a statement. Brewbound, a website covering the craft beer space, said last year AB InBev's two largest craft brands — Goose Island (550,000 barrels) and Shock Top (430,000 barrels) — declined 7% and 23%, respectively. Citing data from the Brewers Association, Brewbound estimated AB InBev's 11 craft brands combined grew 1% in 2018. By adding Craft Brew to the fold, the publication said AB InBev will add roughly 756,959 barrels in production based on 2018 output. The Kona brand alone in 2019 is pacing at 500,000 barrels. "Kona is a scarce asset whose value is partially obscured by declines with Widmer and Red Hook brands," Bill Kirk, an analyst with MKM Partners, said in a note a few months ago cited by Seeking Alpha. "We estimate that Kona alone is worth [roughly] $350 [million] to equity holders." The deal also allows AB InBev to keep pace with other competitors in the beer space. Earlier this year, the Boston Beer Company, the manufacturer of Sam Adams, announced it would buy craft-beer maker Dogfish Head Brewery for $300 million in an effort to keep pace with "an intense amount of consolidation among many craft breweries in the U.S.," most notably the sale of many of them to large international beer giants. And last month, Molson Coors announced a sweeping restructuring that included job cuts, streamlining of its corporate structure and the addition of the word "beverage" to its name to "better reflect its strategic intent to expand beyond beer." While AB InBev has made other changes to its portfolio — including the introduction of Natural Light Seltzer in a bid to attract college-age fans and partnering in 2018 with Jim Bean to make a collaborative beer called Budweiser Reserve Copper Lager — buying Craft Brew is a low-risk and low-cost move for a company with a $154 billion market cap. As the beer industry continues to rapidly evolve, companies such as AB InBev can't afford to sit still, even if not all their product launches or acquistions end up succeeding. Craft Brew Alliance and Anheuser-Busch Announce Expanded Partnership Anheuser-Busch Filed Under: Corporate Operations, Beverages Latest in Corporate Operations New Tindle chef-curated meal kits elevate the plant-based meat experience Oatly debuts carbon footprint labeling on U.S. products
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\displaywidth\bgroup \global\@eqcnt\@ne\hfil $\@lign\displaystyle{##}$\tabskip\z@skip&\global\@eqcnt\tw@ $\@lign\displaystyle{{}##}$\hfil\tabskip\@centering& \llap{\@lign##}\tabskip\z@skip\crcr} \def\endeqalignno{\@@eqncr\egroup \global\advance\c@equation\m@ne$$\global\@ignoretrue} \@namedef{eqalignno*}{\@defeqnswfalse\eqalignno} \@namedef{endeqalignno*}{\endeqalignno} \def\eqaligntwo{\stepcounter{equation}\let\@currentlabel=\oldtheequation\alph{equation}} \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi \let\\=\@eqncr $$\displ@y \tabskip\@centering \halign to \displaywidth\bgroup \global\@eqcnt\m@ne\hfil $\@lign\displaystyle{##}$\tabskip\z@skip&\global\@eqcnt\z@ $\@lign\displaystyle{{}##}$\hfil\qquad&\global\@eqcnt\@ne \hfil$\@lign\displaystyle{##}$&\global\@eqcnt\tw@ $\@lign\displaystyle{{}##}$\hfil\tabskip\@centering& \llap{\@lign##}\tabskip\z@skip\crcr} \def\endeqaligntwo{\@@eqncr\egroup \global\advance\c@equation\m@ne$$\global\@ignoretrue} \@namedef{eqaligntwo*}{\@defeqnswfalse\eqaligntwo} \@namedef{endeqaligntwo*}{\endeqaligntwo} \newtoks\@stequation \def\subequations{\refstepcounter{equation}% \edef\@savedequation{\the\c@equation}% \@stequation=\expandafter{\oldtheequation\alph{equation}} \edef\@savedtheequation{\the\@stequation \edef\oldtheequation{\oldtheequation\alph{equation}}}% \setcounter{equation}{0}% \def\oldtheequation\alph{equation}}{\oldtheequation\alph{equation}}} \def\endsubequations{% \setcounter{equation}{\@savedequation}% \@stequation=\expandafter{\@savedtheequation}% \edef\oldtheequation\alph{equation}}{\the\@stequation}% \global\@ignoretrue} \def\big#1{{\hbox{$\left#1\vcenter to1.428\ht\strutbox{}\right.\n@space$}}} \def\Big#1{{\hbox{$\left#1\vcenter to2.142\ht\strutbox{}\right.\n@space$}}} \def\bigg#1{{\hbox{$\left#1\vcenter to2.857\ht\strutbox{}\right.\n@space$}}} \def\Bigg#1{{\hbox{$\left#1\vcenter to3.571\ht\strutbox{}\right.\n@space$}}} \catcode`@=12 \begin{document} \begin{titlepage} \maketitle \shorttitle{Metal--Insulator Transition} \pacs{71.30.+h, 71.55Jv, 72.15Rn} \jnl{Journal of Physics: Condensed Matter} \begin{abstract} The critical exponents of the metal--insulator transition in disordered systems have been the subject of much published work containing often contradictory results. Values ranging between $\half$ and $2$ can be found even in the recent literature. In this paper the results of a long term study of the transition are presented. The data have been calculated with sufficient accuracy (0.2\%) that the calculated exponent can be quoted as $s=\nu=1.54 \pm 0.08$ with confidence. The reasons for the previous scatter of results is discussed. \end{abstract} \thispagestyle{empty} \vfill \end{titlepage} \section{Introduction} The metal--insulator transition in disordered systems has been the subject of theoretical and experimental work at least since Anderson \def\citename##1{}\@internalcite{And58}. The similarities with thermodynamic phase transitions had been noted by several authors\def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{Tho74,Weg76} but it was not until 1979 that a usable formulation of the renormalisation group or scaling theory became available \def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{AALR79,Weg79c,Efe83}. The basic assumption of these theories, that the behaviour could be described by a single parameter scaling theory, was confirmed in numerical calculations by the present author \def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{mKK81,mKK83a}. For a recent review of the area see Kramer and MacKinnon \def\citename##1{}\@internalcite{KmK94}. In spite of the progress made the exponents, $s$ and $\nu$, describing the behaviour of the conductivity and the localisation length respectively have proven difficult to calculate reliably. For some time there appeared to be a consensus between theory and experiment that both exponents were equal to unity, but more recently this has been called into question from both the theoretical (e.g. \def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{KL84,Ler91a} ) and from the experimental \def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{SHLMvH93} side. Numerical results have been scattered at least between $0.5$ and $2$ with numerous attempts at developing alternative methods of calculation. A good example of the difficulties is given by the contrast between calculations for the Anderson model with rectangular or Gaussian disorder\def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{KBmKS90}. Using identical methods the exponents obtained were about $1.5$ and $1.0$ for the rectangular and Gaussian distributions respectively. It is clearly unreasonable for the exponents for these two cases to be different. In fact if they were different then it would call into question the justification of the use of any simple model Hamiltonian to describe the transition and so undermine the whole foundation of the subject. In this paper the results of calculations carried out over several years are presented. All the basic results have an accuracy of at least $0.2\%$ which enables the critical exponents to be calculated much more accurately than when the conventional $1\%$ is used. \section{Transfer Matrix Calculations} The transfer matrix method has been discussed in numerous papers \def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{mKK83a,PS81a} so only the briefest outline will be attempted here. The starting point is the usual Anderson\def\citename##1{}\@internalcite{And58} Hamiltonian \begin{equation} H = \sum_i \epsilon_i |i><i| + \sum_{i\not=j} V_{ij}|i><j| \ee{eq:1} where $V_{ij} = V_0$ between nearest neighbours on a simple cubic lattice and zero otherwise. In this work $V_0 =1$ is chosen and will therefore not be mentioned explicitly. The diagonal elements $\epsilon_i$ are independent random numbers chosen either from a uniform rectangular distribution with $-\half W < \epsilon_i < +\half W$ or from a Gaussian distribution of standard deviation $\sigma$. For purposes of comparison between the two cases an effective $W$ for the Gaussian case may be defined by equating the variances as $W^2 = 12\sigma^2$ . In terms of the coefficients $a_i$ of the wavefunctions on each site the Schr\"odinger equation may be written in the form \begin{equation} E a_i = \epsilon_i a_i + \sum_{j\not= i} a_j. \ee{eq:2} Consider now a long bar composed of $L$ slices of cross--section $M\times M$. By combining the $a_i$s from each slice into a vector $\bi A_i$ (\ref{eq:2}) can be written in the concise form \begin{equation} E \bi A_n = \bss H_n \bi A_n + \bi A_{n+1} + \bi A_{n-1} \ee{eq:3} where the subscripts $n$ now refer to slices and matrix $\bss H_n$ is the Hamiltonian for slice $n$. By rearranging (\ref{eq:3}) the transfer matrix is obtained \begin{subequations}\begin{eqnarray} \left(\begin{array}{l}\bi A_{n+1}\\ \bi A_n\end{array}\right) &=& \left(\begin{array}{ll}E - \bss H_n&-\bss I\\ \bss I & 0\end{array}\right) \left(\begin{array}{l} \bi A_n\\ \bi A_{n-1}\end{array}\right)\\ &=& \prod_{m=1}^{n} \left(\begin{array}{ll}E - \bss H_m&-\bss I\\ \bss I & 0\end{array}\right) \left(\begin{array}{l} \bi A_1\\ \bi A_0\end{array}\right)\\ &=& \bss T_n \left(\begin{array}{l} \bi A_1\\ \bi A_0\end{array}\right). \ees{eq:4} A theorem attributed to Oseledec\def\citename##1{}\@internalcite{Osc68} states that \begin{equation} \lim_{n\to\infty} \left(\bss T_n^\dagger \bss T_n\right)^{1/n} = \bss M \ee{eq:5} where $\bss M$ is a well defined matrix and $\bss T_n$ are products of random matrices. The logarithms of the eigenvalues of $\bss M$ are referred to as Lyapunov exponents and occur in pairs which are reciprocals of one another. By comparison with (\ref{eq:4}) the Lyapunov exponents may be identified with the rate of exponential rise (or fall) of the wave functions. In fact the smallest exponent corresponds to the longest decay length and hence to the localisation length of the system. In principle then it is necessary to calculate $\bss T_n$ for large $n$, and diagonalise $\bss T^\dagger \bss T$. Unfortunately the calculation is not quite so simple: the different eigenvalues of $\bss T^\dagger\bss T$ rise at different rates so that the smallest, which we seek, rapidly becomes insignificant compared to the largest and is lost in the numerical rounding error. Typically this happens after about 10 steps. \subsection{Orthogonalisation} In order to obtain the smallest Lyapunov exponent it is necessary to overcome this loss of numerical significance. This can be achieved in more than one way of which the orthogonalisation method is employed here. After about 10 matrices have been multiplied together the columns of the product matrix are orthogonalised to each other and normalised. This is equivalent to multiplying the product from the right by an appropriate matrix. This orthonormalisation process automatically separates the different exponentially growing contributions. The process is repeated every 10 or so steps and the logarithm of the length of the vector closest to unity is stored. The Lyapunov exponent is given by the mean value of these logarithms divided by the number of steps between orthonormalisations. In practice it is necessary to use only 50\% or $M\times M$ vectors rather than the full $2\times M\times M$ as the required vector is invariably the $M\times M$th. The error in the Lyapunov exponent can be estimated from the variance corresponding to the mean exponent. Although this estimate could be biased by correlations between the different contributions this is not found to be a serious problem in practice, at least when the localisation length is short compared with the distance between orthogonalisation steps. The optimum frequency of orthogonalisation steps can be estimated by comparing the length of the $M\times M$th vector before and after orthogonalisation. The ratio should not be allowed to get close to the machine accuracy. \section{Scaling Theory} The inverse of the smallest Lyapunov exponent is the localisation length $\lambda_M$. The renormalised length $\Lambda = \lambda_M/M$ is found to obey a scaling theory\def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{mKK81,mKK83a} such that \begin{equation} {\d\ln\Lambda\over\d\ln M} = {\mathop\chi\nolimits}\left(\ln\Lambda\right) \ee{eq:6} which has solutions of the form \begin{equation} \Lambda = {\mathop{\rm f}\nolimits}\left(M/\xi\right) \ee{eq:7} where $\xi$ is a characteristic length scale which can be identified with the localisation length of the insulator and which scales as the reciprocal of the resistivity of the metallic phase\def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{mKK83a}. In 3D (\ref{eq:6}) always has a fixed point $\chi = 0$ which corresponds to the metal--insulator transition. The behaviour close to the transition can be found by linearising (\ref{eq:6}) and solving to obtain \begin{equation} \ln\Lambda = \ln\Lambda_c + A(\tau - \tau_c)M^\alpha \ee{eq:8} where $\tau$ is the disorder $W$ or $\sigma$, $\Lambda_c$ and $\tau_c$ represent the critical $\Lambda$ and disorder respectively, and $A$ and $\alpha$ are constants. By comparing (\ref{eq:7}) and (\ref{eq:8}) an expression for $\xi$ can be obtained in the form \begin{equation} \xi \sim \left|\tau - \tau_c\right|^{1/\alpha} \ee{eq:9} so that the localisation length exponent $\nu$ is given by $\nu = 1/\alpha$. Since it is well known\def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{Weg76,AALR79} that the conductivity exponent $s$ is related to $\nu$ by $s = (d-2)\nu$ then by fitting (\ref{eq:8}) to the data and calculating $\alpha$ both exponents can be obtained. \subsection{Deviations from Scaling} One simple feature of (\ref{eq:8}) is that, when $\ln\Lambda$ is plotted against $\tau$, the curves for different $M$ intersect at a common point $(\ln\Lambda_c, \tau_c)$. In practice the data do not behave in exactly this way. There is a small deviation from scaling. This deviation could be taken into account by adding an extra term to (\ref{eq:8}) which depends on $M$ but not on $\tau$. Consider, however, the form \begin{equation} \ln\Lambda = A\tau M^\alpha + B(M) \ee{eq:10} which represents the most general form of such a correction. If a specific form for the correction were assumed it would require at least 4 independent fitting parameters to represent $B(M)$, including $\Lambda_c$ and $\tau_c$, and may still not represent the true deviation from scaling. It seems better therefore to fit an independent $B(M)$ for each value of $M$ and therefore to make no assumption about the nature of the deviation from scaling, other than that it is non--critical, and therefore independent of $\tau$, in the region of interest. By fitting the data to (\ref{eq:10}) in this way the exponent $\alpha$ is derived solely from the gradient of $\ln\Lambda$ \mbox{vs.} $\tau$ and the intercept is allowed to float. The results of such fits are shown in figure~\ref{fig:1}. \subsection{Data Fitting} The data can be fitted to (\ref{eq:10}) by iteratively using a standard least squares procedure. Care is required with the non--linear parameter $\alpha$. The quality of the fit can be tested by computing $\chi^2$ defined as \begin{equation} \chi^2 = \sum_i{\left(A\tau_i M_i^\alpha + B(M_i) - \ln\Lambda_i\right)^2 \over\sigma^2_i} \ee{eq:11} where $i$ runs over all data points and $\sigma_i$ is the error in point $i$. After fitting $\chi^2$ should be approximately equal to the number of data points less the number of fitted parameters. Hence the value of $\chi^2$ provides a measure of the quality of the fit. In the results presented here the range of values of disorder round the critical value was chosen such that $\chi^2$ conforms to this condition. Then a large number of additional points was calculated inside this range. An important side effect of this procedure is that the apparently acceptable range of disorder around the fixed point gets narrower as the calculations become more accurate. It is therefore important to test whether any apparent change in the fitted exponent is due to this narrowing. The values of the ideal and the fitted $\chi^2$ as well as the range considered are shown in table~\ref{tab:1}. Using $4\le M\le 12$ and the widest range of disorder $s=\nu=1.53\pm 0.04$ and $s=\nu=1.48\pm 0.05$ for rectangular and Gaussian cases respectively. \subsection{Statistical and Systematic Errors} The statistical error in the fitted critical exponent is easily estimated from the least squares fitting procedure. Systematic errors are more difficult to take into account. In this work an attempt is made to consider 3 sources of systematic error: \begin{itemize} \item Limited range of system sizes: $4\le M\le 12$ has been considered and the effect of ignoring the smaller system sizes tested. \item Width of the critical region: the maximum range of disorder is imposed by $\chi^2$ but may still be too large. The effect of narrowing this range still further has been tested. \item The choice of distribution of random numbers: this has been tested by comparing the rectangular and Gaussian cases. \end{itemize} These tests are represented in figure~\ref{fig:2}. Unfortunately the general increase in the error bars due to ignoring data tends to mask any systematic changes. There does however appear to be a general increase in the exponents when the $M=4$ data is eliminated and a tendency for the Gaussian data to lie below the rectangular. From this data $s=\nu\approx 1.54\pm0.08$ has been estimated, where the error bar may be somewhat wider than necessary. \section{Results and Conclusions} The results are summarised in table~\ref{tab:1}. All these results have been calculated in the middle of the band (i.e. $E=0$), but there is ample evidence that for the models considered here, this point is not special and is truly representative of the whole band, at least in the range $-6<E<6$. \begin{table}[htbp] \begin{tabular*}{\textwidth}{|l@{\extracolsep{\fill}}c|cc|cc|} \hline &&\bf Rectangular &&\bf Gaussian &\\ \hline Exponent &&$1.515\pm 0.033$ &&$1.484\pm0.048$&\\ Disorder Range &&$16.2\le W\le 16.8$ &&$21.0\le W\le21.5$&\\ System Sizes &&$4\le M\le 12$ &&$4\le M\le 12$&\\ $\chi^2$(expected) &&$142$ &&$97$ &\\ $\chi^2$(fitted) &&$126$ &&$75$ &\\ $W_c$ &&$16.50\pm 0.05$ &&$21.20\pm0.06$&\\ $\sigma_c$ &&$4.763\pm 0.015$ &&$6.120\pm0.018$&\\ $\Lambda_c$ &&$0.580\pm0.005$ &&$0.580\pm 0.005$&\\ \hline \end{tabular*} \caption{\label{tab:1} N.B: The estimates of $W_c$ and $\Lambda_c$ are based on the values given by several different fitting procedures.} \end{table} Unlike previous calculations \def\citename##1{\citen@me{##1}}\def\@ldcite{\relax}\@internalcite{KBmKS90} the exponents calculated for the two distributions now overlap well and are therefore consistent with the common assumption that simply changing the distribution does not change the universality class and hence the critical exponent. The discrepancy reported previously is presumably due to insufficient accuracy in the raw data and consequent assumption of a critical range of disorder which was too wide. This may have consequences for experiment as it seems to suggest that it is possible to obtain an exponent of unity simply by using too wide a range of data around the critical disorder, energy, pressure, etc. It should also be borne in mind that the influence of interactions may also account for differences between experimental results and those based on a model of non--interacting electrons. For this reason it may be more realistic to compare the present results with photonic or acoustic rather than electronic experiments. In summary, the critical exponent of the Anderson model of the metal--insulator transition is $s=\nu=1.54\pm 0.08$. \section*{Acknowledgements} This work has profited from many useful discussions with B.Kramer, M.Schreiber, J.B.Pendry, P.M.Bell, R.B.S.Oakeshott, E.A.Johnson, and P.J.Roberts. The financial support of the UK SERC and the European Union, through SCIENCE grant $\mbox{SCC}^*$--CT90--0020, is gratefully acknowledged. \section*{Figure Captions} \begin{enumerate} \item\label{fig:1} $\Lambda$ \mbox{vs.} $W$, for (a) rectangular and (b) Gaussian distributions. The data are represented by dots with differing symbols for different system sizes with $4\le M\le 12$ increasing in the direction of the arrow. Each point is accurate to $0.2\%$. The lines are fitted using (\ref{eq:10}). \item\label{fig:2} Fitted critical exponents for rectangular (Diamonds) and Gaussian (Squares) distributions. The absciss{\ae} represent the smallest system size taken into account (with small offsets for clarity). In each group the width of the fitted region is (from left to right) $(16.2 \le W \le 16.8)\to(16.3\le W\le 16.7)\to(16.4\le W\le 16.6)$ and $(21.0\le W\le 21.5)\to(21.05\le W\le 21.45)\to(21.1\le W\le 21.4)$ for rectangular and Gaussian cases respectively. The dotted lines represent the range $s=\nu=1.54\pm 0.8$. \end{enumerate}
{ "redpajama_set_name": "RedPajamaArXiv" }
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\section{Introduction} The Pierre Auger Observatory is the largest cosmic ray detector in the world ~\cite{Abraham:2004dt}. Currently under construction at Malarg\"ue (Argentina), it has begun taking data and already accumulated an exposure comparable to previous experiments such as AGASA and HiRes~\cite{auger}. In addition to studying the highest energy cosmic rays, Auger is also capable of observing ultra-high energy cosmic neutrinos~\cite{augersim}. At present, the AMANDA telescope at the South Pole holds the record for the most energetic neutrino interactions observed~\cite{amanda}; these events have energies up to $\sim 10^{5}$~GeV and are consistent with the predicted spectrum of atmospheric neutrinos. Auger, by contrast, is expected to detect neutrinos with energies above $\sim 10^{8}$~GeV. The ability to study neutrino interactions at such high energies will open a unique window on possible physics beyond the Standard Model (SM) of strong and electroweak interactions. A variety of models have been proposed in which neutrino interactions become substantially modified at very high energies, the most interesting being models of low scale quantum gravity (involving the exchange of Kaluza-Klein (KK) gravitons, production of microscopic black holes, and excitation of TeV-scale string resonances), and models featuring non-perturbative electro-weak instanton induced interactions. Moreover the neutrino flavor ratios predicted by the standard oscillation phenomenology can be modified in propagation over cosmological distances if processes such as neutrino decay occur. Auger is expected to detect the `cosmogenic' neutrino flux from interactions of extragalactic ultra-high energy cosmic rays with the cosmic microwave background. Thus, it will be sensitive to such effects, being capable of measuring the flux of ultra-high energy tau neutrinos in addition to the overall neutrino flux. In this article, we explore the relevant phenomenology and quantify the sensitivity of Auger to such new physics.\footnote{For a review of exotic neutrino interactions and their signatures in high-energy cosmic neutrino telescopes such as IceCube, see Ref.~\cite{Han:2004kq}.} In Sec.~\ref{detector}, we describe the Auger experiment and its ability to detect quasi-horizontal neutrino-induced air showers, as well as up-going showers induced by Earth-skimming tau neutrinos. In Sec.~\ref{fluxes} we discuss possible sources of cosmic ultra-high energy neutrinos. In section~\ref{general}, we infer the sensitivity of Auger to the neutrino-nucleon interaction cross-section and to the flavour content of the ultra-high energy cosmic neutrino flux. In Sec.~\ref{exotic} we consider specific models of physics beyond the SM and their signatures in Auger. We present our conclusions in Sec.~\ref{conclusions}. \section{Ultra-High Energy Neutrinos at Auger} \label{detector} \subsection{The Pierre Auger Observatory} Auger is a hybrid ultra-high energy cosmic ray detector, with a ground array of water Cerenkov detectors sampling air shower particles, overlooked by air fluorescence detector telescopes which observe the longitudinal development of the showers~\cite{Abraham:2004dt}. When completed in 2005--06, the Southern hemisphere site will have 1600 detectors on the ground covering 3000 km$^2$, and 4 fluorescence telescopes having 6 detectors each. A similar facility has been proposed for a Northern hemisphere site in Colorado (USA). In our calculations we consider only a single site and focus on the ground array. \subsection{Quasi-Horizontal, Deeply Penetrating Showers} \label{QH} At sufficiently high energies cosmic neutrinos can trigger atmospheric air showers similar to those due to high energy cosmic rays (hadrons or photons). However, unlike ordinary cosmic ray showers which are initiated near the top of the atmosphere, those generated by neutrinos can be initiated at any depth since the interaction cross-section is much smaller, hence the probability of interaction per unit length is approximately constant. Neutrino induced showers can thus be distinguished from cosmic ray showers by requiring that they be {\em deeply penetrating}. This is most useful for distinguishing the two kinds of showers, because within $\sim20^{\circ}$ of the horizon the electromagnetic component of hadron-induced showers is completely absorbed before reaching the detector. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{augerzenithqh.ps} } \caption{Angular distribution of neutrino-induced showers expected to be observed by the Auger ground array with different selection criteria. The upper solid line indicates all showers generated by cosmic neutrinos over the range of zenith angles shown, while the upper and lower dashed lines correspond to the cases when the shower is initiated at a depth exceeding 1000 and 2000 g/cm$^2$, respectively. The lower solid line corresponds to the case when the shower is initiated at a depth exceeding 2000 g/cm$^2$ {\it and} within 2000 g/cm$^2$ from the detector --- this last set of cuts is adopted in our calculations. For the four cases shown, the fraction of events which survive the various cuts are 1.0, 0.80, 0.60 and 0.33, respectively. We have adopted a neutrino spectrum $\propto E^{-2},$ saturating the Waxman-Bahcall flux bound, see Eq.~(\ref{WB}).} \label{zenith} \end{figure} The rate of neutrino-induced showers expected to be observed in an experiment such as Auger can be written as \begin{eqnarray} \frac{N_{\rm events}}{\Delta T_{\rm obs}} & = & 2 \pi N_{\rm A}\,\int{\rm d}E_\nu\,\int^1_0 {\rm d}y \frac{{\rm d}\sigma_{\nu N}}{{\rm d}y}(E_\nu) \,\int{\rm d}\cos\theta_{\rm z}\,A_{\perp}(\cos\theta_{\rm z}) \nonumber \\ & \times & \int^{X_{\rm ground}}_{X_{\rm min}} \, {\rm d}X \, P[E_{\rm sh},\cos\theta_{\rm z}, X] \, \frac{{\rm d}N_\nu}{{\rm d}E_\nu}(E_\nu) \,\,, \label{qh} \end{eqnarray} where $\Delta T_{\rm{obs}}$ is the observation time, $N_{\rm A}$ is Avogadro's number, ${\rm d}\sigma_{\nu N}/{\rm d}y$ is the differential neutrino-nucleon cross-section, $y$ is the inelasticity, $\theta_{\rm z}$ is the zenith angle, $A_{\perp}$ is the cross sectional area of the experiment as seen from a given zenith angle, $X$ is the atmospheric depth of the interaction (the atmospheric mass per unit area), $E_{\rm sh}$ is the total energy dissipated in the shower, and ${\rm d}N_{\nu}/{\rm d}E_{\nu}$ is the incoming neutrino flux. The function $P[E_{\rm{sh}},\theta_{\rm z}, X]$ is the probability of the experiment detecting a shower created at an atmospheric depth $X$ of energy $E_{\rm{sh}}$, at a zenith angle $\theta_{\rm z}$. In order to ensure that such a shower can be distinguished from one initiated by a hadron or photon primary, we require that for the Auger ground array: \begin{itemize} \item The zenith angle, $\theta_{\rm z} > 70^{\circ}$, \item The neutrino penetrates at least 2000 g/cm$^2$ into the atmosphere before interacting, \item The interaction takes place within 2000 g/cm$^2$ of the detector. \end{itemize} This third requirement is included due to the difficulty in reconstructing events beyond this range at Auger. At $\theta_{\rm z}=70^{\circ}$, the total path length traversed before the shower hits the Earth's surface is $X_{\rm{ground}} \approx 3000$ g/cm$^2$, hence relatively little atmosphere is present in which a neutrino primary can interact and be distinguished from an ordinary cosmic ray shower. At zenith angles larger than 85$^{\circ}$ however, the slant depth exceeds 10000 g/cm$^2$. In Fig.~\ref{zenith}, we show how these selection criteria will affect the observed angular distribution of quasi-horizontal showers in Auger. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{augercossm.ps} } \caption{Spectra of quasi-horizontal, deeply penetrating, neutrino induced showers as would be seen by Auger with the cuts shown in Fig.\ref{zenith}. The solid and dashed lines correspond to the cosmogenic flux and the Waxman-Bahcall flux (see section~\ref{fluxes} for details) which yield, respectively, 0.07 and and 0.22 events per year. For the latter case, the `bumps' at $\sim10^{7.5}$ GeV and $\sim10^{8.5}$ GeV correspond to NC and CC interactions respectively, while the $W^-$ resonance can be seen at $6.3 \times 10^{6}$~GeV.} \label{cossmqh} \end{figure} If the neutrino-nucleon cross-section is enhanced well above the SM prediction at ultra-high energies, it is possible that this depth of atmosphere will significantly attenuate the cosmic neutrino flux; to account for this, an additional factor of ${\rm e}^{-N_{\rm A} \, (X_{\rm{ground}}-2000\, \rm{g/cm}^2)\, \sigma_{\nu N}}$ should be included~\cite{atten} in Eq.~(\ref{qh}). For SM interactions, this factor is very nearly unity and can safely be neglected but it will be important for some of the exotic models we will consider. The neutrino-nucleon cross-section in Eq.~(\ref{qh}) describes both charged current (CC) neutrino-quark scattering and neutral current (NC) neutrino-quark scattering for which we adopt the cross-sections given in Ref.~\cite{cross}. The energy of the shower produced depends on the neutrino flavor and the type of interaction~\cite{Anchordoqui:2002vb}. Electron neutrinos undergoing CC interactions produce a shower with both an electromagnetic and hadronic component: $E_{\rm{sh, em}} = (1-y) E_{\nu}$, $E_{\rm{sh, had}} = y E_{\nu}$. Muon neutrinos undergoing CC interactions, as well as all neutrino flavors undergoing NC interactions, produce a hadronic shower with an energy, $E_{\rm{sh, had}} = y E_{\nu}$. Charged current interactions of tau neutrinos are somewhat more complicated but more interesting. The tau lepton produced in the initial CC neutrino interaction has a decay length of $L_{\tau} \approx$ 50~m $\times \, (E_{\tau}/10^6~{\rm GeV})$. Thus, at sufficiently high energies, the second hadronic shower from the tau decay will be spatially separated and be identifiable as a ``double bang'' event~\cite{doublebang}. Lacking a full-blown simulation, we estimate that a separation of the two bangs by 10~km would be adequate for definitive identification by the Auger ground array --- this requires that the primary neutrino energy exceed $\sim 3\times 10^{9}$~GeV and that the first interaction occurs 50 km or more above the ground (which is easily satisfied for neutrino induced showers inclined over $80^\circ$). However, if the energy exceeds $\sim10^{10}$~GeV, the tau lepton will hit the ground before decaying. A more careful analysis of shower profiles as seen by the Auger fluorescence detectors may allow significant acceptance for such events over a somewhat broader energy range. The scattering of electron flavor anti-neutrinos with electrons can occur efficiently via the resonant exchange of a $W^-$ boson~\cite{Glashow:W} at a neutrino energy of $6.3 \times 10^{6}$~GeV. Although we include this process in our calculations, the detector acceptance for showers at this energy is expected to be rather low, thus this process is of only marginal importance. To accurately determine the probability $P[E_{\rm{sh}},\cos \theta_{\rm z}, X]$ of the Auger ground array observing a shower with a given energy, zenith angle and initiated at a given depth, a detailed detector simulation is required, which is beyond the scope of this study. To make a reasonable estimate we have modelled the energy dependence such that we reproduce the acceptances found through the simulations performed in Ref.~\cite{augersim}. The probability function we arrive at is of order unity for shower energies of ${\cal O}(10^{12})$ GeV, decreases slowly down to energies of ${\cal O}(10^9)$~GeV, and then falls rapidly at lower energies. We treat hadronic and electromagnetic showers separately as in Ref.~\cite{augersim}; in the case of a mixed electromagnetic-hadronic shower, we treat it as two separate showers for the purpose of estimating the probability of detection. In Fig.~\ref{cossmqh} we plot the spectrum of quasi-horizontal, deeply penetrating, neutrino induced showers as would be seen by the Auger ground array, for two choices of the ultra-high energy cosmic neutrino spectrum. \subsection{Earth Skimming Tau Neutrinos} A second class of neutrino events potentially observable at Auger is generated by tau neutrinos which interact while skimming the Earth's surface \cite{tauevents}. Such interactions can generate tau leptons which escape the Earth and produce a slightly upgoing hadronic shower when they decay in the atmosphere.\footnote{Even when the tau lepton decays in the Earth, regeneration effects~\cite{regeneration} can extend the effective range, such that another tau lepton emerges and decay in the atmosphere.} This does not happen for electron neutrinos since the electrons produced in CC interactions are invariably absorbed in the Earth. For muon neutrinos, the produced muon can escape the Earth, but will not decay in the atmosphere since the decay length is $\agt 10^{8}$ times longer than for a tau. When a tau lepton is generated in the Earth, it loses energy via electromagnetic processes at a rate per unit length of~\cite{tauevents} \begin{equation} \frac{dE_{\tau}}{dx} \approx -\alpha -\beta \, E_{\tau}, \label{tauloss} \end{equation} where $\alpha$= 0.002 GeV cm$^2$/g and $\beta=6 \times 10^{-7}$ cm$^2$/g. At very high energies, this will often dramatically reduce the energy before the tau is able to decay. At moderate energies, this has little effect over a single decay length. In Fig.~\ref{tauprop} we show the effect of interactions for tau neutrino `Earth skimmers'. For incoming zenith angles only slightly below the horizon, the spectrum is not suppressed until above $\sim 10^9$~GeV, while there is a noticeable pile-up near $10^{7}$~GeV. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{augertauprop.ps} } \caption{Effect of interactions in the Earth on cosmic tau neutrinos with a spectrum $\propto E^{-2}_{\nu}$ extending up to $10^{12}$~GeV. The horizontal line is the unmodified spectrum and the other lines are for incoming angles 0.1$^{\circ}$, 1$^{\circ}$ and 5$^{\circ}$ degrees below the horizon.} \label{tauprop} \end{figure} Tau leptons produced in CC interactions near the Earth's surface can occasionally escape the Earth before decaying, and thus produce a hadronic shower which is potentially observable by Auger. We have calculated the spectrum of tau leptons escaping the Earth's surface by Monte Carlo using the energy loss rate of Eq.~(\ref{tauloss}). To calculate the probability of given tau neutrino induced shower being detected by Auger, we have used the same probabilities as employed in the case of quasi-horizontal showers. In addition to this function, however, we require that the shower be initiated at a height such that the shower is still able to be detected (for the details of this aspect of the calculation, see Ref.~\cite{zas}). This is particularly important for very high energy tau leptons which can escape the Earth's atmosphere before decaying~\cite{tauevents}. The Andes mountains near Auger's southern site are also a possible target for tau neutrinos; however, we have not included this effect in our calculations as the overall correction is less than 10\%~\cite{andes}. In Fig.~\ref{cossmtau} we show the spectrum of Earth skimmers as would be seen by Auger. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{augercossmtau.ps} } \caption{The spectrum of Earth skimming, tau neutrino induced showers as would be seen by Auger. The solid and dashed lines are, respectively, for the cosmogenic neutrino flux and the Waxman-Bahcall flux, which would yield 1.3 and 4.8 events per year in Auger.} \label{cossmtau} \end{figure} \section{Ultra-High Energy Cosmic Neutrino Fluxes} \label{fluxes} Ultra-high energy neutrinos may be produced in a wide range of astrophysical sources. In this section, we briefly discuss some of these possibilities. Interactions of ultra-high energy cosmic ray protons propagating over cosmological distances with the cosmic microwave background generates a cosmogenic flux of neutrinos~\cite{Berezinsky:1969} through the decay of charged pions produced in $p \gamma$ interactions~\cite{gzk}, which should also result in a suppression of the cosmic ray spectrum above the `GZK cutoff': $E_{\rm GZK} \sim 5 \times 10^{10}$ GeV. The intermediate state of the reaction $p \gamma_{\rm CMB} \to N \pi$ is dominated by the $\Delta^+$ resonance, because the $n$ decay length is smaller than the nucleon mean free path on the relic photons. Hence, there is roughly an equal number of $\pi^+$ and $\pi^0$. Gamma rays, produced via $\pi^0$ decay, subsequently cascade electromagnetically on the cosmic radiation fields through $e^+ e^-$ pair production followed by inverse Compton scattering. The net result is a pile up of $\gamma$ rays at GeV energies, just below the threshold for further pair production. On the other hand, each $\pi^+$ decays to 3 neutrinos and a positron. The $e^+$ readily loses its energy through synchrotron radiation in the cosmic magnetic fields. The neutrinos carry away about 3/4 of the $\pi^+$ energy, and therefore the energy in cosmogenic neutrinos is about 3/4 of the one produced in $\gamma$-rays. The normalisation of the neutrino flux depends critically on the cosmological evolution of the cosmic ray sources and on their proton injection spectra~\cite{Yoshida:pt,engel}. It also depends on the assumed spatial distribution of sources; for example, relatively local objects, such as sources in the Virgo cluster~\cite{Hill:1985mk}, would dominate the high energy tail of the neutrino spectrum. Another source of uncertainty in the cosmogenic neutrino flux is the energy at which there is a transition from galactic to extragalactic cosmic rays as inferred from a change in the spectral slope. While Fly's Eye data~\cite{Bird:1993yi} seem to favour a transition at $10^{10}$~GeV, a recent analysis of the HiRes data~\cite{Bergman:2004bk} points to a lower value of $\sim 10^{9}$~GeV. This translates into rather different proton luminosities at the sources~\cite{lowcrossover} and consequently different predictions for the expected flux of neutrinos~\cite{Fodor:2003ph}. A fourth source of uncertainty in the cosmogenic flux is the chemical composition --- if ultra-high energy cosmic rays are heavy nuclei rather than protons the corresponding cosmogenic neutrino flux may be somewhat reduced~\cite{heavycosmogenic}. Throughout this paper, we will adopt the cosmogenic neutrino spectrum as calculated in Ref.~\cite{engel}. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{fluxes.ps} } \caption{Different possibilities for the ultra-high energy cosmic neutrino spectrum. The solid horizontal line (WB) corresponds to the Waxman and Bahcall bound \cite{wb} assuming proton-proton interactions. The more rapidly falling solid line (WB (low crossover)) is obtained by the same argument but assuming a lower energy transition between galactic and extragalactic cosmic rays \cite{lowcrossover}. The dotted line (Top Down) is the predicted flux in models where the highest energy cosmic rays arise from the decays of superheavy dark matter particles to many body states \cite{dreesneutrinos}. Finally, the dashed line (Cosmogenic) is the spectrum of neutrinos produced in the intergalactic propagation of ultra-high energy protons \cite{engel}. In all cases, the curves show the sum of neutrinos and anti-neutrinos of all flavors.} \label{fluxfig} \end{figure} In addition to being produced in the propagation of ultra-high energy cosmic rays, neutrinos are also expected to be generated in their sources, such as gamma-ray bursts or active galactic nuclei \cite{review}. Although the details of the relationship between the cosmic ray spectrum and the cosmic neutrino spectrum are model dependent, some rather general arguments can be applied. In particular, Waxman and Bahcall~\cite{wb} have shown that for compact, cosmological sources, which are optically thin to proton-proton and proton-photon interactions, an upper limit can be placed on the flux of neutrinos. We will follow them in adopting a neutrino spectrum arising from proton-proton collisions (with an inelasticity of 60\%): \begin{equation} E_\nu^2 {\rm d}N_\nu/{\rm d}E_\nu \lesssim 4 \times 10^{-8}~{\rm GeV} \, {\rm cm}^{-2} \, {\rm s}^{-1}\, {\rm sr}^{-1} \,, \label{WB} \end{equation} summed over all flavors. After oscillations during propagation, one finds at Earth a nearly identical flux of the three neutrino flavors \cite{doublebang} with equal number of neutrinos and antineutrinos \cite{equal}. If the shape of the neutrino spectrum is not an $E^{-2}$ power law, or if the other assumptions of the Waxman-Bahcall argument are modified, this bound can be exceeded~\cite{mpr}. For example, if their bound is evaluated under the assumption of a low galactic to extragalactic crossover energy ($\sim 4 \times 10^8$ GeV rather than the $\sim10^{10}$ GeV used by Waxman and Bahcall) a larger flux with a steeper spectrum ($E^{-2.54}$) is obtained \cite{lowcrossover}. Furthermore, sources which are optically thick such that only neutrinos can escape (`hidden sources'), can easily exceed this bound~\cite{hidden}). Finally, if the highest energy cosmic rays are not accelerated in distant astrophysical sources but are instead produced relatively locally in the galactic halo in the decays of supermassive dark matter particles, then significantly higher fluxes of ultra-high energy photons and neutrinos will also be generated~\cite{topdown}. This model was motivated by the AGASA observation that the cosmic ray spectrum continues apparently without attenuation beyond $E_{\rm GZK}$ but at the same time the events are isotropically distributed on the sky. Such events have not yet been seen by Auger, which has moreover set a restrictive upper limit on the fraction of photons in the cosmic ray flux \cite{augerphoton}. We have normalized the theoretical expectations for the neutrino flux from QCD and electroweak fragmentation in heavy particle decay as calculated in Ref.~\cite{dreesneutrinos} by matching the flux of nucleons observed by Auger, and checked that the photon limits are not violated. In Fig.~\ref{fluxfig}, we plot the expected spectrum of ultra-high energy cosmic neutrinos in the models discussed above and give the corresponding event rates for Auger with standard QCD parton model calculations in Table~\ref{fluxestable}. In this calculation we have truncated the cosmic neutrino spectra above $10^{12}$ GeV --- this choice has only a mild effect on the estimated rates. \begin{table}[!ht] \begin{tabular} {| c || c | c |} \hline & Quasi-horizontal & Earth-skimming $\nu_{\tau}$ \\ \hline \hline Cosmogenic & 0.067 & 1.3 \\ \hline Waxman-Bahcall & 0.22 & 4.8 \\ \hline Waxman-Bahcall (low crossover) & 2.1 & 35 \\ \hline Top-Down & 0.16 & 4.1 \\ \hline \end{tabular} \caption{The number of neutrino induced events per year expected in Auger for various choices of the ultra-high energy neutrino spectrum, as shown in Figure~\ref{fluxfig}, calculated using the standard QCD parton model cross-section.} \label{fluxestable} \end{table} \section{Neutrino Physics with Auger} \label{general} \subsection{Prospects for Cross-Section Measurements} \label{sigmameasure} Deviations of the neutrino-nucleon cross-sections from the prediction of the simple parton model~\cite{cross} can signal new physics beyond the SM, but might alternatively be just due to saturation effects which can substantially modify the parton density at small $x$ (i.e. small energy fractions)~\cite{Gribov:1984tu}. These effects can significantly reduce the total cross-section at high energies, softening the power law behavior predicted by the simple `unscreened' parton model toward compliance with the Froissart bound~\cite{Kwiecinski:1990tb}. By contrast, new physics such as TeV-scale quantum gravity~\cite{Arkani-Hamed:1998rs,Randall:1999ee} can enhance the neutrino interaction cross-sections.\footnote{It is noteworthy that the neutrino-nucleon cross section can also be enhanced in some supersymmetric models through direct channel production of superpartner resonances~\cite{Carena:1998gd}.} This has been calculated in various different frameworks, e.g., arising from exchange of Kaluza-Klein (KK) gravitons~\cite{Nussinov:1998jt,Jain:2000pu}, black hole production~\cite{Feng:2001ib}, and TeV-scale string excitations~\cite{Domokos:1998ry}. In this section, we will discuss the ability of Auger to measure deviations in the neutrino-nucleon scattering cross-section from the SM prediction, without assuming any particular interaction model. The event rates for quasi-horizontal and Earth-skimming neutrinos have different responses to the inelastic cross-section~\cite{measureair}. The rate of quasi-horizontal showers {\em rises} proportional to the cross-section (although if this exceeds $\sim 10^{-28}$ cm$^2$, attenuation of the neutrino flux in the upper atmosphere becomes significant~\cite{atten}). By contrast, the rate of Earth skimming tau events is always {\em depleted} by an enhanced neutrino-nucleon cross-section because of absorption in the Earth. In order to probe deviations from the (unscreened) parton model calculation of the cross-section, it is necessary to note that the screening corrections affect CC and NC equally. To assess the experimental sensitivity to such effects, we assume a uniform suppression of the cross-section by a factor of 2 or 5 and show in Fig.~\ref{ox} the effect on the spectrum of Earth skimmers, as a function of the incoming angle. For a cross section reduced by a factor of 2, the total event rate of Earth skimmers is 1.4 yr$^{-1}$, which is slightly {\em larger} than for the unscreened parton model. The reduction in cross-section due to screening will be energy dependent in general, but as shown in Table~\ref{measuret}, the effect is mainly manifest at intermediate energies of $\sim 10^8-10^{10}$~GeV, corresponding to center-of-mass energies $\sqrt{s} \simeq 10^4 - 10^5$~GeV; at these energies the ratio of quasi-horizontal to Earth-skimming events is a useful diagnostic of any suppression in the cross-section. This is in fact primarily because the cosmogenic neutrino flux peaks at these energies, nevertheless since this represents a reasonable lower limit to the expected flux, this sensitivity is likely to be achieved and even surpassed. A factor of 2 reduction in the cross-section may appear extreme, even so it is clear that Auger can probe the behavior of parton distribution functions (pdfs) in a kinematic region out of reach of forseeable accelerators. This will be particularly beneficial for callibrating different hadronic interaction models of air shower development, which presently differ significantly in their predictions~\cite{Anchordoqui:1998nq}. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=1.7in,angle=90]{tauboostreduce1.ps} \,\, \includegraphics[width=1.7in,angle=90]{tauboostreduce2.ps} \,\, \includegraphics[width=1.7in,angle=90]{tauboostreduce5.ps} } \caption{The effect of interactions in the Earth on the tau neutrino spectrum when the (CC + NC) interaction cross-section is {\em suppressed}. As in Fig.~\ref{tauprop}, we adopt a spectrum $\propto E^{-2}_{\nu},$ which extends to $10^{12}$ GeV. In each frame, results are shown assuming the SM cross-section (solid-line) and a cross-section smaller by a factor of 2 (dotted-line) and a factor of 5 (dashed-line). The three frames are for incoming angles of 1$^{\circ}$, 2$^{\circ}$ and 5$^{\circ}$ degrees below the horizon.} \label{ox} \end{figure} With regard to enhancements of the cross-section by new physics, in general this will be different for CC and for NC interactions. To assess the sensitivity of Auger, we consider a toy model in which only the NC cross-section is enhanced by a factor ranging between 3 and 100, while assuming the inelasticity to be the same as in the SM.\footnote{This resembles the KK graviton exchange, as we discuss in the next section.} In Fig.~\ref{absorb} we show that this results in a suppression of Earth-skimming tau spectrum. By contrast the quasi-horizontal showers are enhanced, resulting in a steady increase of the ratio of quasi-horizontals to Earth skimmers, as the NC cross-section is increased (see Table~\ref{measurenc}). Clearly Auger would be sensitive to substantial increases of the NC cross-section. Thus both an increase and a decrease of the neutrino-nucleon cross-section from the na\"{\i}ve SM value will have distinctive observational signatures. To quantitatively assess the sensitivity of Auger to such effects, the uncertainty in the cosmic neutrino fluxes must also be taken into account~\cite{Anchordoqui:2005pn}. Moreover, to determine the acceptances to different types of events, a full detector simulation is clearly required to improve over the approximate estimates~\cite{augersim} adopted here. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=1.7in,angle=90]{tauboost01nc.ps} \,\, \includegraphics[width=1.7in,angle=90]{tauboost1nc.ps} \,\, \includegraphics[width=1.7in,angle=90]{tauboost5nc.ps} } \caption{The effect of interactions in the Earth on the tau neutrino spectrum when the (NC) interaction cross-section is {\em enhanced}. As in Fig.~\ref{tauprop}, we adopt a spectrum $\propto E^{-2}_{\nu},$ which extends to $10^{12}$ GeV. In each frame, results are shown for the cases of the SM cross-section (solid-line), a cross-section 10 times larger (dashed-line) and 100 times larger (dotted-line). The three frames are for incoming angles of 0.1$^{\circ}$, 1$^{\circ}$ and 5$^{\circ}$ degrees below the horizon.} \label{absorb} \end{figure} \begin{table}[!ht] \begin{tabular} {|c| c| c| c| c| c| c|} \hline $\sigma_{\nu N}$ & $10^6 - 10^7$ & $10^7- 10^8$ & $10^8-10^9$ & $10^9-10^{10}$ & $10^{10}-10^{11}$ & $10^{11}-10^{12}$ \\ \hline\hline $\,\,\,$ \,\,\,\,\,SM $\,\,\,$& $3.6 \times 10^{-5}$ & 0.056 & 0.85 & 0.41 & 0.020 & $1.1 \times 10^{-4}$ \\ \hline $\,\,\,\textstyle{\rm SM}\, \times \frac{1}{2}\,\,$ & $2.1 \times 10^{-5}$ & 0.057 & 0.86 & 0.45 & 0.026 & $1.8 \times 10^{-4}$\\ \hline \end{tabular} \caption{Variation of the rate (in yr$^{-1}$) of Earth-skimming tau neutrino induced events in various energy intervals (in GeV), for the SM (unscreened parton) cross-section, and for a cross section 2 times smaller (for {\em both} CC and NC). The cosmogenic neutrino flux has been assumed.} \label{measuret} \end{table} \begin{table}[!ht] \begin{tabular} {|c |c c| c c |c c|} \hline $\sigma_{\nu N}$ & Quasi-horizontal & \,\, & Earth-skimming $\nu_{\tau}$ &\,\,& Ratio & \\ \hline\hline Standard Model & 0.067 && 1.3 && 0.05 & \\ \hline SM $\times$ 3 & 0.096 && 1.1 && 0.09 & \\ \hline SM $\times$ 10 & 0.20 && 0.68 && 0.29 & \\ \hline SM $\times$ 100 & 1.5 && 0.081 && 19 & \\ \hline \end{tabular} \caption{ The energy integrated rate (in yr$^{-1}$) of quasi-horizontal and Earth-skimmers, as well as their ratio, for the SM cross-section and for different enhancements of the NC component alone. The cosmogenic neutrino flux has been assumed. } \label{measurenc} \end{table} \subsection{Prospects for Flavor Ratio Measurements} \label{flavormeasure} In most models of astrophysical neutrino sources, neutrinos are generated through the decay of charged pions: $\pi^+ \rightarrow \mu^+ \nu_{\mu} \rightarrow e^+ \nu_{e} \bar{\nu}_{\mu} \nu_{\mu}$ or $\pi^- \rightarrow \mu^- \bar{\nu}_{\mu} \rightarrow e^- \bar{\nu}_{e} \nu_{\mu} \bar{\nu}_{\mu}$, thus the flavor ratio at source is $\nu_e:\nu_{\mu}:\nu_{\tau} = 1/3:2/3:0$. However, oscillations modify this ratio as neutrinos propagate to Earth. Given the observed near maximal mixings~\cite{Eidelman:2004wy} and the long baselines involved, the predicted flavor ratio at Earth is $\nu_e:\nu_{\mu}:\nu_{\tau} \approx 0.36:0.33:0.31$ following Ref.~\cite{doublebang}. However, cosmic (anti-)neutrinos may also be generated in the decay of neutrons: $n \rightarrow p^+ e^- \bar{\nu}_e$. In this case, the initial flavor ratio of $\nu_e:\nu_{\mu}:\nu_{\tau} = 1:0:0$ becomes $\nu_e:\nu_{\mu}:\nu_{\tau} \approx 0.56:0.26:0.18$ at Earth~\cite{Anchordoqui:2003vc}. In either case a measured deviation from these predictions could indicate new physics if the neutrino production mechanism is well understood. To study the sensitivity of Auger to the flavor content, we plot in Fig.~\ref{ratiomeasure} the ratio of quasi-horizontal showers to Earth-skimming events as the $\nu_e$ flux is varied in ratio to the other flavors. As in Table~\ref{measurenc}, this ratio is 0.05 when the flux is equally spread among flavors. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{ratiomeasure.ps} } \caption{The ratio of neutrino induced quasi-horizontal showers to Earth-skimming tau neutrino induced upgoing showers as a function of the flavor content of the cosmic neutrino flux. We have assumed equal numbers of muon and tau neutrinos and adopted the cosmogenic neutrino flux.} \label{ratiomeasure} \end{figure} \section{Models of New Physics} \label{exotic} \subsection{Low Scale Quantum Gravity} Two of the most important scales in physics are the Planck scale ($M_{\rm Pl} = G_{\rm N}^{-1/2} \simeq 10^{19}$~GeV) and the weak scale ($M_W = G_{\rm F}^{-1/2} \simeq 300$~GeV), and a long standing problem is explaining the hierarchy between these scales. The traditional view is to adopt $M_{\rm Pl}$ as {\em the} fundamental scale and attempt to derive $M_W$ through some dynamical mechanism (e.g. renormalization group evolution). However recently several models~\cite{Arkani-Hamed:1998rs,Randall:1999ee} have been proposed where $M_W$ is instead the fundamental scale of nature. In the simplest construction of these models, the SM fields are confined to a 3+1-dimensional `brane-world' (corresponding to our observed universe), while gravity propagates in a higher dimensional `bulk' space. If space-time is assumed to be a direct product of a 3+1-dimensional manifold and a flat spatial $n$-dimensional torus $T^{n}$ (of common linear size $2\pi r_{\rm c}$), one obtains a definite representation of this picture in which the effective 4-dimensional Planck scale is related to the fundamental scale of gravity, $M_D$, according to~\cite{Arkani-Hamed:1998rs} \begin{equation} M_{\rm Pl}^2 = 8 \pi \,r_{\rm c}^n M_D^{n+2} \,\,, \end{equation} where $D = 4 + n$. If $M_D$ is to be not much higher than the electroweak scale, then this requires $r_{\rm c}$ to be large in Planck units and thus reformulates the hierarchy problem. For illustrative purposes in what follows we will consider only the case of flat extra-dimensions. The consequences of more exoteric scenarios (such as warped extra-dimensions~\cite{Randall:1999ee}) have been studied in detail by various authors~\cite{Davoudiasl:1999jd}. \subsubsection{Sub-Planckian Regime} From our 4-dimensional point of view, the higher dimensional massless gravitons then appear as an infinite tower of KK modes, of which the lowest is the massless graviton itself, while its excitations are massive. The mass-squared of each KK graviton mode reads, $m^2 = \sum_{i=1}^n \,\ell_i^2 / r_{\rm c}^{2},$ where the mode numbers $\ell_i$ are integers. Note that the weakness of the gravitational interaction is compensated by the large number of KK modes that are exchanged: the coupling $M^{-2}_{\rm Pl}$ of the graviton vertex is cancelled exactly by the large multiplicity of KK excitations $\sim \hat s^{n/2} \,r_{\rm c}^n$, so that the final product is $\sim \hat s^{n/2}/M_D^{2+n}$~\cite{Giudice:1998ck}. Here $\sqrt{\hat s}$ is the center-of-mass energy available for graviton-KK emission. Taking brane fluctuations into account, a form factor $\sim {\rm e}^{-m^2/M_D^2}$ is introduced at each graviton vertex~\cite{Bando:1999di}. This exponential suppression, which parametrizes the effects of a finite brane tension, provides a dynamical cutoff in the (otherwise divergent) sum over all KK contributions to a given scattering amplitude. Altogether, one may wonder whether the rapid growth of the cross-section with energy in neutrino-nucleon reactions mediated by spin 2 particles carries with it observable deviations from SM predictions. A simple Born approximation to the elastic neutrino-parton cross section (which underlies the total neutrino-proton cross-section) leads, without modification, to $\hat \sigma_{\rm el} \sim \hat s^2$~\cite{Nussinov:1998jt,Jain:2000pu}. Unmodified, this behavior by itself eventually violates unitarity. This may be seen either by examining the partial waves of this amplitude, or by studying the high energy Regge behavior of an amplitude $A_R (\hat s,\hat t) \propto \,\hat s^{\alpha(\hat t)}$ with spin-2 Regge pole, {\it viz.,} intercept $\alpha(0)=2$. For the latter, the elastic cross-section is given by \begin{equation} \frac{{\rm d}\hat\sigma_{\rm el}}{{\rm d}\hat t}\, \sim\, \frac{|A_R(\hat s, \hat t)|^2}{\hat s^2}\, \sim \hat s^{2\alpha(0)-2}\,\sim \hat s^2, \end{equation} whereas the total cross-section reads \begin{equation} \hat \sigma_{\rm tot}\, \sim \frac{\Im {\rm m} [A_R(\hat s,0)]}{\hat s}\,\sim \hat s^{\alpha(0)-1}\,\sim \hat s, \end{equation} so that eventually, $\hat \sigma_{\rm el}> \hat\sigma_{\rm tot}$~\cite{Anchordoqui:2000uh}. Eikonal unitarization schemes modify this behaviour. Specifically, for large impact parameter, a single Regge pole exchange amplitude yields $\hat \sigma_{\rm tot} \sim \ln^2(\hat s/s_0)$~\cite{Kachelriess:2000cb}, an estimate which is insensitive to the underlying theory at short distances (UV completion). Recently, the differential cross-section for such gravity-mediated interaction at large distances has been calculated~\cite{Emparan:2001kf}. Because of the large cross-sections, albeit with low inelasticity, there would be distinctive double and/or multiple bang events~\cite{Illana:2005pu} similar to those discussed in Sec.~\ref{QH}~\cite{Anchordoqui:2004bd}. For small impact parameters, it becomes difficult to respect partial wave unitarity as corrections to the eikonal amplitude are expected to become important. Note that graviton self interactions carry factors of $\hat t$ associated to the vertices, and thus as $\hat t$ increases, so does the attraction among the scattered particles. Eventually it is expected that gravitational collapse to a black hole (BH) will take place, absorbing the initial state in such a way that short distance effects are screened by the appearance of a horizon~\cite{Banks:1999gd,Feng:2001ib}. \subsubsection{Trans-Planckian Regime} According to Thorne's hoop conjecture~\cite{Thorne:ji}, a BH forms in a two-particle collision when and only when the impact parameter is smaller than the radius of a Schwarzschild BH of mass equal to $\sqrt{\hat s} \equiv \sqrt{xs}$. The total cross-section for BH production is then, \begin{equation} \hat\sigma_{\rm BH} = F(n)\,\pi r_s^2(\sqrt{\hat{s}}) \,, \label{hoopsigma} \end{equation} proportional to the area subtended by a ``hoop'' of radius~\cite{Myers:un} \begin{equation} \label{schwarz} r_s(\sqrt{\hat s}) = \frac{1}{M_D} \left[ \frac{\sqrt{\hat{s}}}{M_D} \right]^{\frac{1}{1+n}} \left[ \frac{2^n \pi^{\frac{n-3}{2}}\Gamma(\frac{n+3}{2})}{n+2} \right]^{\frac{1}{1+n}}\,, \end{equation} where $F(n)$ is a form factor of order unity. Recent work has confirmed the validity of Eq.~(\ref{hoopsigma}) and evaluated the dimension-dependent constant $F(n)$, analytically in four dimensions~\cite{Eardley:2002re} and numerically in higher dimensions~\cite{Yoshino:2002tx}. In the course of collapse, a certain amount of energy is radiated in gravitational waves by the multipole moments of the incoming shock waves~\cite{D'Eath:hb}, leaving a fraction $y \equiv M_{\rm BH}/\sqrt{\hat s}$ available to be emitted through Hawking evaporation~\cite{Hawking:1975sw}. Here, $M_{\rm BH}$ is a {\it lower bound} on the final mass of the BH and $\sqrt{\hat s}$ is the center-of-mass energy of the colliding particles, taken to be partons. This ratio depends on the impact parameter of the collision, as well as on the dimensionality of space-time~\cite{Yoshino:2002br}. Of course, this calculation is purely in the framework of classical general relativity, and is expected to be valid only for energies far above the fundamental Planck scale $M_D$, for which curvature is small outside the horizon and strong quantum effects are hidden behind the horizon. Extending this formalism to center-of-mass energies close to $M_D$ requires a better understanding of quantum gravity. String theory provides the best hope for understanding the regime of strong quantum gravity, and in particular for computing cross-sections at energies close to the Planck scale~\cite{Dimopoulos:2001qe}. In principle embedding TeV-scale gravity models in realistic string models might facilitate the calculation of cross-sections for BHs (and string excitations) having masses comparable to $M_D$. To be specific we will consider embedding of a 10-dimensional low-energy scale gravity scenario within the context of SO(32) Type I superstring theory, where gauge and charged SM fields can be identified with open strings localized on a 3-brane and the gravitational sector consists of closed strings that propagate freely in the internal dimensions of the universe~\cite{Antoniadis:1998ig}. After compactification on $T^6$ down to four dimensions, $M_{\rm Pl}$ is related to the string scale, $M_{\rm s}$, and the string coupling constant, $g_{\rm s}$, by $M_{\rm Pl}^2 = (2 \pi \,r_{\rm c})^6 \,M_{\rm s}^8/g_{\rm s}^2$ (hereafter, $D=10$, i.e. $n=6$). Subsequent to formation, the BH proceeds to decay~\cite{Chamblin:2003wg}. The decay of an excited spinning BH state proceeds through several stages. The initial configuration looses hair associated with multipole moments in a balding phase by emission of classical gravitational and gauge radiation. Gauge charges inherited from the initial state partons are discharged by Schwinger emission. After this transient phase, the subsequent spinning BH evaporates by semi-classical Hawking radiation in two phases: a brief spin-down phase in which angular momentum is shed~\cite{Frolov:2002xf}, and a longer Schwarzschild phase. In the latter the emission rate per degree of particle freedom $i$ of particles of spin $s$ with initial total energy between $(Q, Q + {\rm d}Q)$ is found to be~\cite{Han:2002yy} \begin{equation} \frac{{\rm d}\dot{N}_i}{{\rm d}Q} = \frac{\sigma_s (Q, r_s)\,\, \Omega_{d-3}}{(d-2)\,(2\pi)^{d-1}}\,\,Q^{d-2} \left[ \exp \left( \frac{Q}{T_{\rm BH}} \right) - (-1)^{2s} \right]^{-1} \,\,, \label{rate} \end{equation} where $T_{\rm BH} = 7/(4\,\pi\,r_s)$ is the BH temperature, \begin{equation} \Omega_{d-3} = \frac{2\,\pi^{(d-2)/2}}{\Gamma[(d-2)/2]} \end{equation} is the volume of a unit $(d-3)$-sphere, and $\sigma_s (Q, r_s)$ is the absorption coefficient (a.k.a. the greybody factor). Recall that SM fields live on a 3-brane ($d=4$), while gravitons inhabit the entire spacetime ($d=10$). The prevalent energies of the decay quanta are of ${\cal O}(T_{\rm BH}) \sim 1/r_s$, resulting in s-wave dominance of the final state. Indeed, as the total angular momentum number of the emitted field increases, $\sigma_s (Q,r_s)$ rapidly gets suppressed~\cite{Kanti:2002nr}. In the low energy limit, $Q \, r_s \ll 1,$ higher-order terms are suppressed by a factor of $3 (Q\,r_s)^{-2}$ for fermions and by a factor of $25 (Q\,r_s)^{-2}$ for gauge bosons. For an average particle energy $\langle Q \rangle$ of ${\cal O}(r_s^{-1})$, higher partial waves also get suppressed, although by a smaller factor. This strongly suggests that the BH is sensitive only to the radial coordinate and does not make use of the extra angular modes available in the internal space~\cite{Emparan:2000rs}. A recent numerical study~\cite{Harris:2003eg} has explicitly shown that the emission of scalar modes into the bulk is largely suppressed with respect to the brane emission. In order to contravene the argument of Emparan--Horowitz--Myers~\cite{Emparan:2000rs}, the bulk emission of gravitons would need to exhibit the opposite behavior -- a substantial enhancement into bulk modes. There is no {\it a priori} reason to suspect this qualitative difference between $s=0$ and $s=2$, and hence no reason to support arguments~\cite{Cavaglia:2003hg} favoring deviation from the dominance of visible decay. With this in mind, we assume the evaporation process to be dominated by the large number of SM brane modes. The lower bound on the mass radiated in the Schwarzschild phase could be somewhat reduced at large $b$~\cite{Yoshino:2005hi} compared to the estimate in Ref.~\cite{Yoshino:2002br} used here. On the other hand, the effective range of $b$ at which there is trapping is somewhat increase~\cite{Yoshino:2005hi}, with the result that there is not any significant change. The total number of particles emitted is approximately equal to the BH entropy, \begin{equation} S_{\rm BH} = \frac{\pi}{2}\,M_{\rm BH}\,r_s. \end{equation} At a given time, the rate of decrease in the BH mass is just the total power radiated \begin{equation} \frac{{\rm d}\dot{M}_{\rm BH}}{{\rm d}Q} = - \sum_{i} c_i\, \frac{\sigma_s(Q, r_s)}{8 \,\pi^2}\,\,Q^3 \left[ \exp \left( \frac{Q}{T_{\rm BH}} \right) - (-1)^{2s} \right]^{-1}\,\, , \label{rate2} \end{equation} where $c_i$ is the number of internal degrees of freedom of particle species $i$. Integration of Eq.~(\ref{rate2}) leads to \begin{equation} \dot{M}_{\rm BH} = - \sum_i c_i \,\,f\,\, \frac{\Gamma_s}{8\,\pi^2} \,\, \, \Gamma(4) \,\, \zeta(4)\, \,T^4_{\rm BH}\,A_4, \label{m} \end{equation} where $f=1$ (7/8) for bosons (fermions), and the greybody factor was conveniently written as a dimensionless constant, $\Gamma_s = \sigma_s(\langle Q \rangle,r_s)/A_4$, normalized to the BH surface area~\cite{Emparan:2000rs} \begin{equation} A_4 = \frac{36}{7}\,\pi\,\left( \frac{9}{2} \right)^{2/7}\ \, r_s^2 \label{area} \end{equation} seen by the SM fields ($\Gamma_{s=1/2} \approx 0.33$ and $\Gamma_{s=1} \approx 0.34$~\cite{greybody}). Now, since the ratio of degrees of freedom for gauge bosons, quarks and leptons is 29:72:18 (excluding the Higgs boson), from Eq.~(\ref{m}) one obtains a rough estimate of the mean lifetime, \begin{equation} \tau_{_{\rm BH}} \approx 1.67 \times 10^{-27}\,{\rm s}\, \left(\frac{M_{\rm BH}}{M_{10}}\right)^{9/7} \left(\frac{\rm TeV}{M_{10}}\right) \,, \label{lifetime} \end{equation} which indicates that BHs evaporate near-instantaneously into visible quanta. The semi-classical description outlined above is reliable only when the energy of the emitted particle is small compared to the BH mass, i.e. \begin{equation} T_{\rm BH} \ll M_{\rm BH}\,, \,\,\, {\rm or}\,\,\, {\rm equivalently,} \,\,\, M_{\rm BH} \gg M_{10} \,, \label{condition} \end{equation} because it is only under this condition that both the gravitational field of the brane and the back reaction of the metric during the emission process can safely be neglected~\cite{Preskill:1991tb}. For BHs with initial masses well above $M_{10}$, most of the decay process can be well described within the semi-classical approximation. However, the condition stated in Eq.~(\ref{condition}) inevitably breaks down during the last stages of evaporation. At this point it becomes necessary to introduce quantum considerations. To this end we turn to a quantum statistical description of highly excited strings. It is well-known that the density of string states with mass between $M$ and $M+{\rm d}M$ cannot increase any faster than $\rho (M) = {\rm e}^{\beta_{\rm H} M}/M,$ because the partition function, \begin{equation} Z (\beta) = \int_0^\infty {\rm d}M\, \rho(M) \,\,{\rm e}^{-M\, \beta} \,\,, \end{equation} would fail to converge~\cite{Hagedorn:st}. Indeed, the partition function converges only if the temperature is less than the Hagedorn temperature, $\beta_{\rm H}^{-1}$, which is expected to be $\sim M_{\rm{s}}$. As $\beta$ decreases to the transition point $\beta_{\rm H}$, the heat capacity rises to infinity because the energy goes into the many new available modes rather than into raising the kinetic energy of the existing particles~\cite{Frautschi:1971ij}. In the limit, the total probability diverges, indicating that the canonical ensemble is inadequate for the treatment of the system. However, one can still employ a microcanonical ensemble of a large number of similar isolated systems, each with a given fixed energy $E$. With the center-of-mass at rest, $E = M$ so the density of states is just $\rho(M)$ and the entropy $S = \ln \rho(M)$. In this picture, equilibrium among systems is determined by the equality of the temperatures, defined for each system as \begin{equation} T \equiv \left(\frac{\partial S}{ \partial M} \right)^{-1} = \frac{M}{\beta_{\rm H} M -1} \ . \end{equation} Equilibrium is achieved at maximum entropy when the total system heat capacity, $C$, is positive. Ordinary systems (on which our intuition is founded) have $C>0$. However, for a gas of massive superstring excitations the heat capacity, \begin{equation} C \equiv -\frac{1}{T^2} \left(\frac{\partial^2 S}{\partial M^2}\right)^{-1} = - \left(\frac{M}{T}\right)^2\, , \end{equation} is {\em negative}, as is the case for BHs~\cite{Hawking:de}. The positivity requirement on the total specific heat implies that strings and BHs cannot coexist in thermal equilibrium, because any subsystem of this system has negative specific heat, and thus the system as a whole is thermodynamically unstable. This observation suggests that BHs may end their Hawking evaporation process by making a transition to an excited string state with higher entropy, avoiding the singular zero-mass limit~\cite{Bowick:1985af}. The suggestion of a string $\rightleftharpoons$ BH transition is further strengthened by three other facts: (i) in string theory, the fundamental string length should set the minimum value for the Schwarzschild radius of any BH~\cite{Veneziano:1986zf}; (ii) $T_{\rm BH} \sim \beta_{\rm H}^{-1}$ for $r_s \sim M_{\rm s}^{-1}$~\cite{Susskind:ws}; (iii) there is an apparent correlation between the greybody factors in BH decay and the level structure of excited strings~\cite{Das:1996wn}. The string $\rightleftharpoons$ BH ``correspondence principle''~\cite{Horowitz:1996nw} unifies these concepts: When the size of the BH horizon drops below the size of the fundamental string length $\ell_{\rm s} \gg \ell_{10},$ where $\ell_{10}$ is the fundamental Planck length, an adiabatic transition occurs to an excited string state. Subsequently, the string will slowly lose mass by radiating massless particles with a nearly thermal spectrum at the unchanging Hagedorn temperature~\cite{Amati:1999fv}.(Note that the probability of a BH radiating a large string, or of a large string undergoing a fluctuation to become a BH is negligibly small~\cite{Horowitz:1997jc}.) The continuity of the cross-section at the correspondence point, at least parametrically in energy and string coupling, provides an independent supporting argument for this picture~\cite{Dimopoulos:2001qe}. Specifically, in the perturbative regime, the Virasoro-Shapiro amplitude leads to a ``string ball'' (SB) production cross-section $\propto g_{\rm s}^2 \hat{s}/M_{\rm s}^4$. This cross-section saturates the unitarity bounds at $g_{\rm s}^2 \hat{s}/M_{\rm s}^2 \sim 1$~\cite{Amati:1987wq}, so before matching the geometric BH cross-section $\propto r_s^2$, there is a transition region at which $\hat{\sigma} \sim M_{\rm s}^{-2}$. All in all, the rise with energy of the parton-parton $\rightarrow$ SB/BH cross-section can be parametrized as~\cite{Dimopoulos:2001qe} \begin{eqnarray} \hat \sigma (\sqrt{\hat{s}})\sim\left\{ \begin{array}{ll} \displaystyle \frac{g_{\rm s}^2\,\hat{s}}{M_{\rm s}^4} &\qquad M_{\rm s} \ll \sqrt{\hat{s}} \leq M_{\rm s}/ g_{\rm s}\,,\\ \displaystyle \frac{1}{M_{\rm s}^2}&\qquad M_{\rm s}/ g_{\rm s} < \sqrt{\hat{s}} \leq M_{\rm s}/ g_{\rm s}^2\,,\\ \displaystyle \frac{1}{M_{10}^2}\,\left[\frac{\sqrt{\hat{s}}}{M_{10}}\right]^{2/7} &\qquad M_{\rm s}/ g_{\rm s}^2<\sqrt{\hat{s}}\,, \end{array} \right. \end{eqnarray} where $M_{10} = (8\pi^5)^{1/8}\,M_{\rm s}/g_{\rm s}^{1/4}$ \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=0]{xminplot.eps} } \caption{Quantitative measures of the validity of the semi-classical analysis of BH production for $n=6$ extra dimensions, where $x_{\rm min} \equiv M_{\rm BH, min}/ M_{10}$.} \label{fig:xminplot} \end{figure} \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=0]{auger_nu_bh.eps} } \caption{The cross-section for BH production in neutrino nucleon collisions, for $n = 6$ extra dimensions, assuming $M_{10} = 1$~TeV and $M_{\rm BH, min} = M_{10}.$ Energy losses by gravitational radiation have been included. The SM $\nu N$ cross-section is indicated by the dotted line. For comparison the typical $pp$ cross-section is shown, as well as the cross-section required for triggering vertical and horizontal atmospheric showers. The cross-section for absorption by the Earth is also shown~\cite{Anchordoqui:2004xb}.} \label{bhcross} \end{figure} The inclusive production of BHs proceeds through different final states for different classical impact parameters $b$~\cite{Yoshino:2002br}. These final states are characterized by the fraction $y(z)$ of the initial parton center-of-mass energy, $\sqrt{\hat s}=\sqrt{xs}$, which is trapped within the horizon. Here, $z= b/b_{\rm max},$ where $b_{\rm max}= 1.3 \, r_s(\sqrt{\hat s})$~\cite{Yoshino:2002br}. With a lower cutoff $M_{\rm BH,min}$ on the BH mass required for the validity of the semi-classical description, this implies the joint constraint \begin{equation} y(z)\,\,\sqrt{x s} \ge M_{\rm BH,min} \label{constraint} \end{equation} on the parameters $x$ and $z$. Because of the monotonically decreasing nature of $y(z)$, Eq.~(\ref{constraint}) sets an {\it upper} bound $\bar z(x)$ on the impact parameter for fixed $x.$ The corresponding parton-parton BH cross-section is $\hat \sigma_{_{\rm BH}} (x) = \pi \bar b^2(x),$ where $\bar b=\bar z b_{\rm max}.$ The total BH production cross-section is then~\cite{Anchordoqui:2003jr} \begin{equation} \sigma_{_{\rm BH}}(E_\nu,M_{\rm BH,min},M_{10}) \equiv \int_{\frac{M_{\rm BH,min}^2}{ y^2(0) s}}^1 \, dx \,\sum_i f_i(x,Q) \ \hat \sigma_{_{\rm BH}}(x) \,\,, \label{sigma} \end{equation} where $i$ labels parton species and the $f_i(x,Q)$ are pdfs~\cite{Pumplin:2002vw}. The momentum scale $Q$ is taken as $r_s^{-1},$ which is a typical momentum transfer during the gravitational collapse process. The parameter $M_{\rm BH, min}$ plays in important role n interpreting the results derived below. The validity of the semi-classical calculation requires at least three criteria to be satisfied. First, $S_0$, the initial entropy of the produced BH should be large enough to ensure a well-defined thermodynamic description~\cite{Preskill:1991tb}. Second, the BH lifetime $\tau_{\rm BH}$ should be large compared to its inverse mass so that the black hole behaves like a well-defined resonance. Third, the BH mass must be large compared to the scale of the 3-brane tension $T_3$ so that the brane does not significantly perturb the BH metric~\cite{Frolov:2002as}. Quantitative measures of these three criteria are given in Fig.~\ref{fig:xminplot} for $n=6$, assuming $T_3 = \sqrt{8\pi}/(2\pi)^6 \ M_{10}^4$ for 6 toroidally-compactified dimensions~\cite{Polchinski:1996na}. We see that all three criteria are adequately satisfied for $M_{\rm BH,\ min} \agt 3 M_{10}$~\cite{Anchordoqui:2003ug}. The resulting $\nu N \to$ BH production cross-section is shown in Fig.~\ref{bhcross}. In the perturbative string regime, i.e. $M_{\rm SB,min} < \sqrt{\hat{s}} \leq M_s/g_s$, the SB production cross-section is taken to be \begin{equation} \sigma_{_{\rm SB}}(E_\nu,M_{\rm SB,min}, M_{10}) = \int_{\frac{M_{\rm SB,min}^2}{s}}^1 {\rm d}x \,\sum_i f_i(x,Q) \, \hat\sigma_{_{\rm SB}}(\hat{s}) \,, \end{equation} where $\hat\sigma_{_{\rm SB}} (\hat{s})$ contains the Chan-Paton factors which control the projection of the initial state onto the string spectrum. In general, this projection is not uniquely determined by the low-lying particle spectrum, so there are one or more arbitrary constants. The analysis in the $\nu q \rightarrow \nu q$ channel illustrates this point~\cite{Cornet:2001gy}. The $\nu g$ scattering, relevant for $\nu N$ interactions at ultra-high energies, introduces additional ambiguities. In our calculations we adopt the estimates given in Ref.~\cite{stringy2} considering the saturation limit and including both neutrino-quark and neutrino-gluon scattering. The resulting $\nu N \to$ SB cross-section is shown in Fig.~\ref{stringsigma}, setting the Chan-Paton factors equal to 1/2. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=90]{stringsigma.ps} } \caption{The neutrino-nucleon cross-section including the effects of TeV scale string resonances. The solid and dashed lines correspond to models with string tension $M_{\rm s}$ = 1 and 2 TeV, respectively. The Standard Model cross-section is shown as a dotted line for comparison.} \label{stringsigma} \end{figure} \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=2.6in,angle=90]{augerbhqh.ps} \includegraphics[width=2.6in,angle=90]{augerbhqhwb.ps}} \caption{The spectrum of quasi-horizontal, deeply penetrating, black hole induced showers as would be seen by Auger for the cosmogenic flux (left) and the Waxman-Bahcall flux (right). The dashed lines indicates different values of the fundamental Planck scale (from below $M_{10} = \,10,\, 7, \,5, \,4, \,3, \,2, \,1$~TeV; in all cases $M_{\rm BH,min} = 3 M_{10}$) while the solid line is the SM prediction.} \label{qhbh} \end{figure} \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=2.6in,angle=90]{augerbhtau.ps} \includegraphics[width=2.6in,angle=90]{augerbhtauwb.ps} } \caption{The spectrum of Earth skimming, tau neutrino black hole induced showers as would be seen by Auger for the cosmogenic flux (left) and the Waxman-Bahcall flux (right). The dashed lines indicates different values of the fundamental Planck scale (from below $M_{10} = \,1,\, 2, \,3, \,4, \,5, \,7, \,10$~TeV; in all cases $M_{\rm BH, min} = 3 M_{10}$), while the solid line is the SM prediction.} \label{taubh} \end{figure} As can be seen in Figs.~\ref{bhcross} and \ref{stringsigma}, although the neutrino interaction length is reduced below the SM value due to BH/SB production, it is still far larger than the Earth's atmospheric depth. Neutrinos therefore would produce BH/SBs with roughly equal probability at any point in the atmosphere. As a result, the light descendants of the BH/SB may initiate low-altitude, quasi-horizontal showers at rates significantly higher than SM predictions.\footnote{Additionally, neutrinos that traverse the atmosphere unscathed may produce black holes via interactions in the ice or water and be detected by neutrino telescopes~\cite{Kowalski:2002gb}.} Because of this the atmosphere provides a buffer against contamination by hadronic showers (for which the electromagnetic component is completely attenuated at such large zenith angles) allowing a good characterization of BH-induced showers when $S \gg 1$~\cite{Feng:2001ib,Dutta:2002ca}. If the quasi-horizontal deep shower rate is found to be anomalously large, it can be ascribed either to an enhancement of the incoming neutrino flux, or to an enhancement in the neutrino-nucleon cross-section. However, these possibilities may be distinguished by focusing on events which arrive at very small angles to the horizon. An enhanced flux will increase {\em both} the quasi-horizontal and Earth-skimming event rates, whereas a large BH cross-section {\em suppresses} the latter, because the hadronic decay products of BH evaporation do not escape the Earth's crust~\cite{Anchordoqui:2001cg}. To quantify the potential of Auger in discriminating BH/SB induced showers, we show separately the BH production event rates for quasi-horizontal and Earth skimming neutrinos in Figs.~\ref{qhbh} and \ref{taubh}. The SB production rates are similarly given in Figs.~\ref{qhstring} and \ref{taustring} and a summary of these event rates is provided in Tables~\ref{bhtable} and \ref{stringtable} respectively. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=2.6in,angle=90]{augerstqh.ps} \includegraphics[width=2.6in,angle=90]{augerstqhwb.ps} } \caption{The spectrum of quasi-horizontal, deeply penetrating, neutrino string-ball induced showers as would be seen by Auger for the cosmogenic flux (left) and the Waxman-Bahcall flux (right). The dashed lines refer to the string scale $M_{\rm s} = 1$ TeV (upper) and $M_{\rm s} = 2$ TeV (lower), while the solid line is the SM prediction.} \label{qhstring} \end{figure} \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=2.6in,angle=90]{augersttau.ps} \includegraphics[width=2.6in,angle=90]{augersttauwb.ps} } \caption{The spectrum of Earth skimming, tau neutrino string ball induced showers as would be seen by Auger for the cosmogenic flux (left) and the Waxman-Bahcall flux (right). The dashed lines refer to the string scale $M_{\rm s} = 1$ TeV (upper) and $M_{\rm s} = 2$ TeV (lower), while the solid line is the SM prediction.} \label{taustring} \end{figure} \begin{table}[!ht] \begin{tabular}{|c|| c| c|| c| c|| c| c|} \hline \multicolumn{1}{|c||}{$\sigma_{\nu N}$} & \multicolumn{2}{c||}{Quasi-horizontal} & \multicolumn{2}{c||}{Earth-skimming $\nu_{\tau}$} & \multicolumn{2}{c|}{Ratio}\\ \hline & Cosmogenic & Waxman-Bahcall & Cosmogenic & Waxman-Bahcall & Cosmo & WB\\ \hline\hline Standard Model & 0.067 & 0.22 & 1.3 & 5.0 & 0.050 & 0.044 \\ \hline $M_{10}=$ 1 TeV & 4.4 & 10.6 & 0.13 & 1.0 & 36 & 10.2 \\ \hline $M_{10}=$ 2 TeV & 0.95 & 2.4 & 0.48 & 2.6 & 2.0 & 0.91 \\ \hline $M_{10}=$ 3 TeV & 0.42 & 1.1 & 0.77 & 3.5 & 0.54 & 0.3 \\ \hline $M_{10}=$ 4 TeV & 0.25 & 0.66 & 0.96 & 4.1 & 0.26 & 0.16 \\ \hline $M_{10}=$ 5 TeV & 0.18 & 0.48 & 1.1 & 4.4 & 0.16 & 0.11 \\ \hline $M_{10}=$ 7 TeV & 0.12 & 0.34 & 1.2 & 4.7 & 0.1 & 0.073 \\ \hline $M_{10}=$ 10 TeV& 0.089 & 0.27 & 1.3 & 4.8 & 0.08 & 0.056 \\ \hline \end{tabular} \caption{Black hole producing event rates of quasi-horizontal showers and Earth-skimming tau neutrino induced showers expected to be observed per year by Auger for both the cosmogenic neutrino flux and the Waxman-Bahcall flux. In all cases $M_{\rm BH, min} = 3 M_{10}$.} \label{bhtable} \end{table} \begin{table}[!ht] \begin{tabular}{|c|| c| c|| c| c|| c| c|} \hline \multicolumn{1}{|c||}{$\sigma_{\nu N}$} & \multicolumn{2}{c||}{Quasi-horizontal} & \multicolumn{2}{c||}{Earth-skimming $\nu_{\tau}$} & \multicolumn{2}{c|}{Ratio}\\ \hline & Cosmogenic & Waxman-Bahcall & Cosmogenic & Waxman-Bahcall & Cosmo & WB\\ \hline \hline Standard Model & 0.067 & 0.22 & 1.3 & 5.0 & 0.05 & 0.044 \\ \hline $M_s=$ 1 TeV & 0.86 & 2.5 & 0.4 & 2.0 & 2.1 & 1.3 \\ \hline $M_s=$ 2 TeV & 0.17 & 0.48 & 1.5 & 5.7 & 0.11 & 0.084 \\ \hline \end{tabular} \caption{String ball producing event rates of quasi-horizontal showers and Earth-skimming tau neutrino induced showers expected to be observed per year by Auger for both the cosmogenic neutrino flux and the Waxman-Bahcall flux.} \label{stringtable} \end{table} \subsection{Non-perturbative Electroweak Interactions} The transition probability between two flat space vacua can be calculated in a Minkowski framework in analogy with WKB tunneling through non-vacuum fluctuations, or by evaluating the minimal action appropriate to a classical solution of Euclidean space in a given topological sector~\cite{Belavin:1975fg}. As is well known, in Yang-Mills theories the inclusion of massless fermions fundamentally alters the picture~\cite{'tHooft:1976fv}: transitions between vacua (separated by energy barriers whose minimum height is set by the sphaleron energy $E_{\rm sp}$~\cite{Klinkhamer:1984di}) will be totally suppressed unless accompanied by the simultaneous emission or absorption of {\em all} fermions coupled to the gauge field. In the Minkowski description, these fermions emerge during level-shifting in the strong ${\cal O}(1/g)$ gauge fields interpolating between vacua ($g=$ coupling constant). In the Euclidean description, the presence of a zero mode $\omega$ for each light fermion coupled to the gauge field will, because of the rules of Grassman integration, generate a 't Hooft vertex~\cite{'tHooft:1976fv} with all the different fermions appearing as legs, \begin{equation} {\cal L}_{\rm eff} \propto \prod_{i=1\dots N} \overline \omega F_i + {\rm h.c.} \; , \end{equation} where $F_i$ is a chiral fermion field. Whether these exotic processes occur with sizeable rates in high energy particle collisions is a long-standing open question~\cite{Aoyama:1986ej}. At center-of-mass energies $\sqrt{\hat s} < E_{\rm sp} \approx \pi M_W/\alpha_{W} \approx 7.5~{\rm TeV}$, the cross-section for electroweak instanton mediated processes is known to have an exponential form~\cite{McLerran:1989ab}. Here, $m_W = 80.423$~GeV is the W$^\pm$ boson mass and $\alpha_W (m_W) = 0.0338$ is the $SU(2)$ fine structure constant~\cite{Eidelman:2004wy}. Including essential pre-exponential factors~\cite{Khoze:1990bm}, one has, for the phenomenologically interesting case of fermion-fermion scattering ${\rm f+f}\stackrel{I}{\to}{\rm all}$, \begin{eqnarray} \nonumber \hat\sigma_{\rm ff}^{(I)} &\approx & \frac{1}{m_W^2} \, \left( \frac{2\pi}{\alpha_W}\right)^{7/2} \, \exp\left[ -\frac{4\pi}{\alpha_W}\, F_{\rm hg} \left( \frac{\sqrt{\hat s}}{4\pi m_W/\alpha_W} \right)\right] \\[1.5ex] \label{cross-qfd} & \simeq & 5.3\times 10^3\ {\rm mb}\ \exp\left[ -\frac{4\pi}{\alpha_W}\, F_{\rm hg} \left( \frac{\sqrt{\hat s}}{4\pi m_W/\alpha_W} \right)\right] \,. \end{eqnarray} where $F_{\rm hg}$ is the ``holy-grail'' function~\cite{Mattis:1991bj}. By means of perturbative calculations of the relevant exclusive amplitudes about the instanton ($I$), squaring them and summing over the final states, or, alternatively, by means of a perturbative calculation of the forward elastic scattering amplitude about the widely separated instanton anti-instanton ($I\overline{I}$) pair and determining the imaginary part to get the total cross-section via the optical theorem, one may calculate the decisive tunneling suppression exponent $F_{\rm hg}$, as a series in fractional powers of $\epsilon \equiv \sqrt{\hat s}/(4\pi m_W/\alpha_W) \simeq \sqrt{\hat s}/ (30\ {\rm TeV})$~\cite{Khoze:1990bm}, \begin{equation} \label{FW-pert} F_{\rm hg} (\epsilon ) = 1 - \frac{3^{4/3}}{2}\, \epsilon^{4/3} + \frac{3}{2}\,\epsilon^2 + {\mathcal O}(\epsilon^{8/3}) \,. \end{equation} Therefore, the total cross-section given in Eq.~(\ref{cross-qfd}) is exponentially growing for $\epsilon\ll 1$. At $\epsilon$ of ${\mathcal O}(1)$, however, the perturbative expression in Eq.~(\ref{FW-pert}) no longer applies. In this energy regime, only extrapolations of, and lower bounds on, the tunneling suppression exponent are available~\cite{Ringwald:2002sw}. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=3.5in,angle=0]{andreas.eps} } \caption{The allowed 90\%, 95\%, and 99\% CL regions for interpolation between the electroweak and QCD-like neutrino-nucleon cross-section consistent with existing data. Also shown with a dashed line is the predicted enhancement of the cross-section by electroweak sphalerons. For details see Ref.~\cite{Ahlers:2005zy,Han:2003ru}.} \label{andreas} \end{figure} Interestingly, at $\sqrt{\hat s} \sim 100$~TeV, the cross-section can rise to values characteristic of QCD interactions. Since the electroweak instanton-induced interaction applies equally to all fermions, neutrinos can thus acquire hadron-like cross sections at high energies. Moreover, the inelasticity of the process is {\em high}. Together, these facts imply that neutrino interactions mediated by instantons would induce air showers in the upper atmosphere with characteristics similar to those of proton-induced showers~\cite{Fodor:2004tr}. Conversely, the non-observation to date of deeply penetrating air showers constrains any sudden rise of the neutrino-nucleon cross-section~\cite{Morris:1993wg,Ahlers:2005zy}. Figure~\ref{andreas} shows the allowed region for transition from electroweak to QCD-like neutrino-nucleon cross-section, consistent with existing data. The dashed line indicates the neutrino-nucleon cross-section taken from Ref.~\cite{Han:2003ru} obtained taking $\hat \sigma_{ff} \agt 1~{\rm mb}$~\cite{Ringwald:2002sw}. As can be seen in Fig.~\ref{andreas}, this prediction is marginally consistent with the region allowed by current data. For this cross-section, the expected event rate at Auger would be 4.3 quasi-horizontal showers per year assuming the cosmogenic neutrino flux, and 14 quasi-horizontal showers per year assuming the Waxman-Bahcall neutrino flux; the rate of Earth-skimmers is 1.3 per year in both cases. As shown in Fig.~\ref{augerinstau}, the suppression of Earth-skimmers due to absorption in the Earth is negligible. However, the rate of quasi-horizontal showers is increased by about 2 orders of magnitude, and such events would be concentrated in a small energy range, as indicated in Fig.~\ref{augerinsqh}. This would provide a {\em clean} signal for electroweak instanton-induced interactions. Thus, if no deeply developing showers are observed, tighter constraints can be placed on this model, and more generally on any sudden rise in the neutrino-nucleon cross-section. \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=2.6in,angle=90]{augerinstau.ps} \includegraphics[width=2.6in,angle=90]{augerinstauwb.ps} } \caption{Suppression of Earth skimming events due to electroweak sphalerons as would be seen by Auger for the cosmogenic neutrino flux (left), and for the Waxman-Bahcall flux (right). The solid line is the SM prediction.} \label{augerinstau} \end{figure} \begin{figure}[tbh] \centering\leavevmode \mbox{ \includegraphics[width=2.6in,angle=90]{augerinsqh.ps} \includegraphics[width=2.6in,angle=90]{augerinsqhwb.ps} } \caption{The spectrum of quasi-horizontal showers mediated by electroweak sphalerons as would be seen by Auger for the cosmogenic neutrino flux (left), and for the Waxman-Bahcall flux (right). The solid line is the SM prediction.} \label{augerinsqh} \end{figure} \subsection{Neutrino Decay} Neutrinos are known to be sufficiently light that they are stable against tree-level electroweak decays. Moreover, decays of the form $\nu_i \to \nu_j \gamma$ or $\nu \to \nu \nu \overline \nu$ are severely constrained by experiment~\cite{Eidelman:2004wy}. However, some models of lepton number violation postulate the existence of a massless Goldstone boson, the Majoron, $X$. Consequently, decays such as $\nu_i \rightarrow \nu_j X$ or $\nu_i \rightarrow \overline{\nu}_j X$, are then possible, where $\nu_{i,j}$ denote mass eigenstates \cite{Schechter:1981cv}. Presently such possibilities are only weakly constrained by Solar neutrino data, which set the bound $\tau/m \gtrsim 10^{-4}$ s/eV~\cite{Bahcall:1986gq}. However, by studying cosmic neutrinos which have travelled over far longer baselines Auger can be more sensitive to their instability by a factor of $\sim10^2 - 10^4$, if an effective flavor ratio measurement can be made. Because of the extremely large energies probed by Auger, it will be complementary in this regard to the IceCube neutrino telescope~\cite{bdecay,Anchordoqui:2005gj}. The ratio of flavors observed in the cosmic neutrino spectrum depends on whether any species of neutrinos have decayed and on the decay channel. In the simple situation where all heavy neutrino species decay into the lightest mass eigenstate (or into non-interacting states, such as a sterile neutrino), we would expect to observe at Earth the flavor ratio \begin{equation} \phi_{\nu_e}:\phi_{\nu_{\mu}}:\phi_{\nu_{\tau}} = \cos^2\theta_{\odot} : \frac{1}{2} \sin^2 \theta_{\odot} :\frac{1}{2} \sin^2 \theta_{\odot} \approx 6:1:1, \end{equation} where $\theta_{\odot}$ is the solar neutrino mixing angle and we have assumed the normal hierarchy as well as $U_{e3} =0$. This result is independent of the flavor ratio at source. However, for the case of an inverted hierarchy, the predicted flavor ratio at Earth is \begin{equation} \phi_{\nu_e}:\phi_{\nu_{\mu}}:\phi_{\nu_{\tau}} = U^2_{e3}:U^2_{\mu 3}:U^2_{\tau 3} \approx 0:1:1, \end{equation} where $U_{\alpha_i}$ is the neutrino mixing matrix and we have taken the atmospheric mixing angle to be maximal. These results are in striking contrast to the expectation for stable neutrinos discussed earlier in Sec.~\ref{flavormeasure}. These two cases represent the most extreme deviations from the usual phenomenology and are robust in that they do not depend on the flavor composition at source. A variety of other (more baroque) possibilities have been considered, e.g. only the heaviest neutrino eigenstate decays, but the predicted flavor ratios after propagation then depend on the assumed flavor ratio at source~\cite{bdecay}. In Table~\ref{decaytable} we list some of these possibilities assuming the usual mass hierarchy and source flavor ratios as for pion decay. We cannot measure the flavor ratios directly at Auger. However, as discussed earlier, Earth-skimming events are generated uniquely by tau neutrinos, while quasi-horizontal showers can be generated by all neutrino flavors. Furthermore, because of maximal mixing of $\nu_\mu$ and $\nu_\tau$ we expect their fluxes to be always comparable. Therefore, by combining these two measurements Auger can potentially determine the flavor ratios of ultra-high energy neutrinos. \begin{table}[!ht] \begin{tabular} {|c|c|c|} \hline Decaying Mass Eigenstates & Decay Products & $\phi_{\nu_e}:\phi_{\nu_{\mu}}:\phi_{\nu_{\tau}}$ \\ \hline\hline $\nu_3$, $\nu_3$ & Irrelevant & 6:1:1 \\ \hline $\nu_3$ & Invisible & 2:1:1 \\ \hline $\nu_3$ & $\nu_2$ & 1.4--1.6:1:1 \\ \hline $\nu_3$ & $\nu_1$ & 2.4--2.8:1:1 \\ \hline $\nu_3$ & 50\%\,$\nu_1$, 50\%\,$\nu_2$ & 2:1:1 \\ \hline \end{tabular} \caption{The neutrino flavor ratios predicted for a variety of neutrino decay models with decay mode as indicated~\cite{bdecay}.} \label{decaytable} \end{table} Of course this requires a substantial event rate. As shown in Table~\ref{fluxestable} the standard cosmogenic neutrino flux is expected to generate only about 0.7 quasi-horizontal shower events over 10 years. This is certainly insufficient for making the precision measurements needed to identify the effects of neutrino decay. For the nominal Waxman-Bahcall flux, Auger is expected to detect about 2.2 quasi-horizontal events and about 48 Earth-skimming events in 10 yr. However, if the cosmic ray galactic--extragalactic transition happens at around $10^9$~GeV~\cite{Berezinsky:2005cq}, then the required proton luminosity in the extragalactic sources increases significantly. Then Auger would detect as many as 21 quasi-horizontal and 350 Earth-skimming events over 10 years. This corresponds to a 2$\sigma$ measurement of their ratio of $0.06 \pm 0.026$, which would exclude anomalous flavor composition with a $\nu_e$ content greater than $\phi_{\nu_e}:\phi_{\nu_{\mu}}:\phi_{\nu_{\tau}} \simeq 2.5:1:1$. Other possibilities for altering neutrino flavor ratios have been explored~\cite{bmeasure,lorentzcpt}. If Lorentz invariance is violated through modification of the usual dispersion relation for neutrinos by non-renormalizable operators induced by quantum gravity effects, then the fraction of tau neutrinos may be suppressed~\cite{lorentzcpt}. Auger would then observed the ratio of Earth-skimmers to quasi-horizontal events to decrease from about 20 to close to zero. \section{Conclusions} \label{conclusions} Our knowledge of fundamental interactions has largely been limited to the energies up to which collider experiments have been able to probe. The Tevatron, currently operating at Fermilab, produces collisions with a center-of-mass energy slightly below 2 TeV, while the Large Hadron Collider, under construction at CERN, will reach 14 TeV. By contrast, a typical neutrino observed at the Pierre Auger Observatory will have an energy of ${\cal O}(10^9)$ GeV, corresponding to a neutrino-nucleon center-of-mass energy exceeding 40 TeV. Although the number of collisions which will be observed (i.e. the beam luminosity) is far below that of collider experiments, Auger and other experiments sensitive to ultra-high energy cosmic neutrinos have in principle the ability to provide unique information on new physics beyond the reach of any planned accelerator. In addition to this advantage, cosmic neutrinos have traveled over very great distances before reaching Earth, thus their detection also constitutes an extremely long-baseline oscillation experiment. Instead of being limited to phenomena which occur over minuscule fractions of a second, cosmic neutrinos provide an exceptional window into phenomena only evident over cosmological scales of length or time. The Pierre Auger Observatory is capable of detecting two primary classes of neutrino induced events --- quasi-horizontal, deeply penetrating showers and (slightly) upgoing showers induced by Earth-skimming tau neutrinos. Used separately, the rates of such events are of limited use in probing new physics; since the spectrum of cosmic neutrinos is currently unknown, an event rate cannot by itself be used to determine the neutrino-nucleon interaction cross-section. However by combining these two classes of neutrino-induced events, it becomes possible to make a crude cross-section measurement. As this cross-section is increased (decreased), the rate of quasi-horizontal showers increases (decreases) accordingly, while by contrast, the rate of slightly upgoing showers is reduced (enhanced) since Earth-skimming tau neutrinos become absorbed more (less). Thus the ratio of quasi-horizontal, deeply penetrating showers to slightly upgoing showers provides a check of the behaviour of the neutrino-nucleon cross-section at ultra-high energies. The details of such a measurement, of course, depend on the energy dependence of such interactions, as well as their inelasticity and other characteristics. These two types of neutrino-induced events also provide the opportunity to constrain the ratios of flavors present in the ultra-high energy cosmic neutrino spectrum. If these neutrinos are generated through the decay of charged pions (as they are in most models), they will reach Earth in nearly equal quantities of each flavor after oscillations are taken into account. A larger than expected rate of quasi-horizontal, deeply penetrating showers, in comparison to the slightly upgoing shower rate, would thus indicate a suppression of the tau neutrino component in the ultra-high energy cosmic neutrino spectrum, due, for example, to neutrino decay. In this study, we have considered several specific models in which either the neutrino-nucleon cross-section, or the ratio of cosmic neutrino flavors, deviates substantially from the expectation of the perturbative Standard Model. We have studied enhancements in the neutrino-nucleon cross-section in models with low-scale gravity, variously described as due to the exchange of Kaluza-Klein gravitons, the production of microscopic black holes, and/or string resonances. We have also considered increases in the cross-section due to non-perturbative Standard Model electroweak instanton induced processes, which in contrast do not lead to a decrease in the inelasticity. Regarding flavor ratio measurements, we have discussed several models of decaying neutrinos. It is difficult to precisely delineate the reach of these techniques as this depends on the unknown flux of cosmic neutrinos at the energies to which Auger is sensitive. We have considered both the ``guaranteed'' cosmogenic flux which sets a lower bound and the Waxman-Bahcall flux which sets an upper bound. Further observations of ultra-high energy cosmic rays by Auger itself will help to pin down the expected neutrino flux. Over the next few years, the Pierre Auger Observatory may well identify the world's first ultra-high energy neutrino event. We have attempted to illustrate the exciting new possibilities for probing new physics that will be opened up by such a detection. \acknowledgements{We wish to thank Andreas Ringwald and Tom Weiler for a critical reading of the manuscript and helpful comments. LAA is partially supported by the US NSF grant PHY-0457004. TH is supported by the US DoE grant DE-FG02-95ER40896 and by the Wisconsin Alumni Research Foundation; he would also like to thank the Aspen Center for Physics for hospitality. SS acknowledges a PPARC Senior Fellowship (PP/C506205/1).} \newpage
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Indicators of Welfare Dependence: Annual Report to Congress, 2004 Contributors to this report include Gil Crouse, Sarah Douglas, Susan Hauan, Julia Isaacs, and Kendall Swenson of the Office of Human Services Policy under the direction of Don Winstead, Deputy Assistant Secretary for Human Services Policy, Office of the Assistant Secretary for Planning and Evaluation. For on-line versions of this report (and previous annual reports) see the Office of Human Services Policy's web page on the Indicators Reports:http://aspe.hhs.gov/hsp/indicators-rtc/index.shtml The Welfare Indicators Act of 1994 requires the Department of Health and Human Services to prepare annual reports to Congress on indicators and predictors of welfare dependence. The 2004 Indicators of Welfare Dependence, the seventh annual report, provides welfare dependence indicators through 2001, reflecting changes that have taken place since enactment of the Personal Responsibility and Work Opportunity Reconciliation Act (PRWORA) in August 1996. As directed by the Welfare Indicators Act, the report focuses on benefits under the Aid to Families with Dependent Children (AFDC) program, now the Temporary Assistance for Needy Families (TANF) program; the Food Stamp Program; and the Supplemental Security Income (SSI) program. Welfare dependence, like poverty, is a continuum, with variations in degree and in duration. Families may be more or less dependent if larger or smaller shares of their total resources are derived from welfare programs. The amount of time over which families depend on welfare might also be considered in assessing their degree of dependence. Although recognizing the difficulties inherent in defining and measuring dependence, a bipartisan Advisory Board on Welfare Indicators proposed the following definition, as one measure to examine in concert with other key indicators of dependence and well-being: A family is dependent on welfare if more than 50 percent of its total income in a one-year period comes from AFDC/TANF, food stamps and/or SSI, and this welfare income is not associated with work activities. Welfare dependence is the proportion of all families who are dependent on welfare. This 2004 report uses data from the Current Population Survey (CPS) and administrative data to provide updated measures through 2001 for several dependence indicators. Other measures are based on the Survey of Income and Program Participation (SIPP), the Panel Study of Income Dynamics (PSID), and other data sources. Drawing on these various data sources, this report provides a number of key indicators of welfare recipiency, dependence, and labor force attachment. Selected highlights from the report include the following: In 2001, 3.1 percent of the total population was dependent in that they received more than half of their total family income from TANF, food stamps, and/or SSI (see Indicator 1). While marginally higher than the 3.0 percent dependency rate measured in 2000, the 2001 rate is much lower than the 5.2 percent rate measured in 1996. Overall, 4.9 million fewer Americans were dependent on welfare in 2001 compared with 1996. Although the 2002 dependency rate cannot yet be calculated, preliminary data suggest it may increase slightly to 3.2 percent. The overall drop in dependence since 1996 parallels the more well-known drop in AFDC/TANF and food stamp caseloads. For example, the percentage of individuals receiving AFDC/TANF fell from 4.6 percent to 1.9 percent between 1996 and 2002 (see Indicator 3). Food stamp recipiency rates dropped from 9.5 percent to 6.6 percent over the same time period. In an average month in 2001, more than half (61 percent) of TANF recipients lived in families with at least one family member in the labor force. Comparable figures for food stamp and SSI recipients were 56 and 37 percent, respectively (see Indicator 2). Labor force participation, particularly full-time employment, increased considerably among TANF families in the last several years. Spells of AFDC/TANF receipt in the second half of the 1990s were shorter than spells of AFDC receipt in the early 1990s. Nearly half (47 percent) of AFDC/TANF spells for individuals entering the program between 1996 and 1999 lasted 4 months or less, compared to 30 percent of AFDC spells beginning between 1992 and 1994 (See Indicator 8). Longer-term welfare receipt was much less common during the 1990s compared to earlier decades. Less than 4 percent of those with some AFDC/TANF assistance between 1991 and 2000 received assistance in nine or ten years of the period, compared to 12 percent and 13 percent of AFDC recipients in the earlier two time periods (See Indicator 9). Since the causes of welfare receipt and dependence are not clearly known, the report also includes a larger set of risk factors associated with welfare receipt. The risk factors are loosely organized into three categories: economic security measures, measures related to employment and barriers to employment, and measures of nonmarital childbearing. The economic security risk factors include measures of poverty and well-being that are important not only as potential predictors of dependence, but also as a supplement to the dependence indicators, ensuring that dependence measures are not assessed in isolation. As such, the report includes data on the official poverty rate, one of the most common measures of economic well-being: As the dependency rate fell between 1996 and 2001, the poverty rate for all individuals fell also, from 13.7 percent in 1996 to 11.7 percent in 2001. In 2002, the poverty rate was slightly higher than in 2001 (12.1 percent), but was still lower than any year between 1980 and 1998 (see Economic Security Risk Factor 1, Figure ECON 1a). Finally, the report has four appendices that provide additional data on major welfare programs, alternative measures of dependence and non-marital births, as well as background information on several data and technical issues. Chapter I. Introduction and Overview The Welfare Indicators Act of 1994 (Pub. L. 103-432) directed the Secretary of Health and Human Services (HHS) to publish an annual report on welfare dependency. This 2004 report, the seventh annual indicators report, gives updated data on the measures of welfare recipiency, dependency, and predictors of welfare dependence developed for previous reports. It reflects changes that have taken place since enactment of the Personal Responsibility and Work Opportunity Reconciliation Act (PRWORA) in August 1996. The purpose of this report is to address questions concerning the extent to which American families depend on income from welfare programs. Under the Welfare Indicators Act, HHS was directed to address the rate of welfare dependency, the degree and duration of welfare recipiency and dependence, and predictors of welfare dependence. The Act further specified that analyses of means-tested assistance should include benefits under the Aid to Families with Dependent Children (AFDC) program, now the Temporary Assistance for Needy Families (TANF) program; the Food Stamp Program; and the Supplemental Security Income (SSI) program. This 2004 report provides updated measures through 2001 for dependency measures based on the Current Population Survey (CPS), with one preliminary estimate for 2002. Although more recent administrative data provide some information on recipiency through 2003, the survey data needed to examine overall welfare recipiency are not available past 2001 for the CPS-based measures, and are even less current for measures based on the Survey of Income and Program Participation (SIPP) and the Panel Study of Income Dynamics (PSID). This report presents analysis of PSID data through 2000 (IND 9 in Chapter II), an improvement over data through 1996 published in the previous two annual reports. These newly available PSID data allow for the examination of long-term recipiency since the enactment of welfare reform in 1996. As in the 2003 report, measures updated annually are presented at the front of each chapter, followed by the figures that are derived from data sources that are updated less frequently. Organization of Report This introductory chapter provides an overview of the specific summary measure of welfare dependence proposed by a bipartisan Advisory Board1 and adopted for use in this annual report series. It also discusses summary measures of poverty, following the Advisory Board's recommendation that dependence measures not be assessed in isolation from other measures of economic well-being. The introduction concludes with a discussion of data sources used for the report. Chapter II of the report, Indicators of Dependence, presents nine indicators of welfare dependence and recipiency. These indicators include dependence measures based on total income from all three programs – AFDC/TANF, SSI, and food stamps – as well as measures of recipiency for each of the three programs considered separately. Labor force participation among families receiving welfare and benefit receipt across multiple programs are also shown. The second half of the chapter includes longitudinal data on transitions on and off welfare programs and spells of dependence and recipiency. Newly updated for the current report, this section includes a measure of long-term program receipt of up to 10 years. Chapter III, Predictors and Risk Factors Associated with Welfare Receipt, focuses on predictors of welfare dependence – risk factors believed to be associated with welfare receipt. These predictors are shown in three different groups: (1) Economic security – including various measures of poverty, receipt of child support, food insecurity, and health insurance coverage – is important in predicting dependence because families with fewer economic resources are more likely to rely on welfare programs for their support. (2) Measures of the work status and potential barriers to employment of adult family members also are critical, because families must generally receive an adequate income from employment in order to avoid dependence without severe deprivation. (3) Finally, data on non-marital births are important since a high proportion of longterm welfare recipients first became parents outside of marriage, frequently as teenagers. Additional data and technical notes are presented in four appendices. Appendix A provides basic program data on each of the main welfare programs and their recipients; Appendix B shows how dependence is affected by the inclusion of benefits from the SSI program; Appendix C includes additional data on non-marital childbearing; and Appendix D provides background information on several data and technical issues. The main welfare programs included in Appendix A are: The Aid to Families with Dependent Children (AFDC) program, the largest cash assistance program, provided monthly cash benefits to families with children, until its replacement by the Temporary Assistance for Needy Families (TANF) program, which is run directly by the states. Data on the AFDC and TANF programs are provided in Appendix A, with AFDC data provided from 1977 through June 1997, and TANF data from July 1997 through 2002. The Food Stamp Program provides monthly food stamp benefits to individuals living in families or alone, provided their income and assets are below limits set in Federal law. It reaches more poor people over the course of a year than any other means-tested public assistance program. Appendix A provides historical data from 1970 to 2002. The Supplemental Security Income (SSI) program provides monthly cash payments to elderly, blind, or disabled individuals or couples whose income and assets are below levels set in Federal law. Though the majority of recipients are adults, disabled children also are eligible. Historical data from 1974 through 2002 are provided in Appendix A. 1 The first annual report was produced under the oversight of a bipartisan Advisory Board on Welfare Indicators, which assisted the Secretary in defining welfare dependence, developing indicators of welfare dependence, and choosing appropriate data. Under the terms of the original authorizing legislation, the Advisory Board was terminated in October 1997, prior to the submission of the first annual report. Measuring Welfare Dependence As suggested by its title, this report focuses on welfare "dependence" as well as welfare "recipiency." While recipiency can be defined fairly easily, based on the presence of benefits from AFDC/TANF, SSI or food stamps, dependence is a more complex concept. Welfare dependence, like poverty, is a continuum, with variations in degree and in duration. Families may be more or less dependent if larger or smaller shares of their total resources are derived from welfare programs. The amount of time over which a family depends on welfare might also be considered in assessing its degree of dependence. Nevertheless, a summary measure of dependence to be used as an indicator for policy purposes must have some fixed parameters that allow one to determine which families should be counted as dependent, just as the poverty line defines who is poor under the official standard. The definition of dependence proposed by the Advisory Board for this purpose is as follows: A family is dependent on welfare if more than 50 percent of its total income in a one-year period comes from AFDC, food stamps and/or SSI, and this welfare income is not associated with work activities. Welfare dependence is the proportion of all families who are dependent on welfare. This measure is not without its limitations. The Advisory Board recognized that no single measure could capture fully all aspects of dependence and that the proposed measure should be examined in concert with other indicators of well-being. In addition, while the proposed definition would count unsubsidized and subsidized employment and work required to obtain benefits as work activities, existing data sources do not permit distinguishing between welfare income associated with work activities and non-work-related welfare benefits. As a result, the data shown in this report overstate the incidence of dependence (as defined above) because welfare income associated with work required to obtain benefits is classified as welfare and not as income from work. This issue may be growing in importance under the increased work requirements of the TANF program. In 2002, over 33 percent of welfare recipients were working (including employment, work experience, and community service), compared to only 7 percent in 1992.2 This proposed definition also represents an essentially arbitrary choice of a percentage (50 percent) of income from welfare beyond which families will be considered dependent. However, it is relatively easy to measure and to track over time, and is likely to be associated with any very large changes in total dependence, however defined. For example, dependence under this definition declined as policy changes under welfare reform moved more recipients into employment. As shown in Figure SUM 1, 3.1 percent of the population would be considered "dependent" on welfare in 2001 under the above definition. This is about one-quarter of the percentage (12.6 percent) that lived in a family receiving at least some TANF, food stamp or SSI benefits during the year. Preliminary data from 2002 suggest that the dependency rate may increase slightly between 2001 and 2002.3 Figure SUM 1. Recipiency and Dependency Rates: 1996-2002 Note: Recipiency is defined as living in a family with receipt of any amount of AFDC/TANF, SSI, or food stamps during year. Dependency is defined as having more than 50 percent of annual income from AFDC/TANF, SSI and/or food stamps. Dependency rates would be lower if adjusted to exclude welfare assistance associated with working. The estimate for 2002 is preliminary Source: March CPS data, analyzed using the TRIM3 microsimulation model. While dependency and recipiency rates increased slightly to 3.1 and 12.6 percent, respectively, the 2001 dependency and recipiency rates remain significantly lower than the 1996 rates of 5.2 and 16.0, respectively. The drop in recipiency rates is consistent with administrative data showing declining TANF caseloads from 1996 to 2001. What is not apparent from administrative records, but is shown in these national survey data, is that the dependency rate also declined sharply after 1996. While 13.74 million individuals were dependent in 1996, only 8.86 million were dependent in 2001 – representing a decline of 4.88 million people. Recipiency and dependency rates are higher for non-Hispanic blacks and Hispanics than for nonHispanic whites, as shown in Table SUM 1. Recipiency and dependence also are higher for young children than for adults, and for individuals in female-headed families than for those in married-couple families. However, both recipiency and dependency rates are much lower for non-Hispanic blacks, Hispanics, children and individuals in female-headed families in 2001 compared to 1996. Measures of welfare dependency also vary based upon which programs are counted as "welfare programs." Dependency would be much lower – 1.4 percent – if only AFDC/TANF and food stamp benefits were counted (as shown in Appendix B and as is done in some measures in this report). Whereas the inclusion or exclusion of individuals receiving only SSI benefits had a relatively small effect on dependence indicators several years ago, in 2001 over two-fifths of dependent individuals are dependent on SSI income only. Another factor affecting dependence is the time period observed. The summary measures shown in Figure and Table SUM 1 focus on recipiency and dependency rates measured on an annual basis. Note that this report no longer provides ten-year measures of long-term dependency (as distinct from long-term recipiency) due to a cutback in PSID data collection that precludes further update of these measures. Longitudinal measures of program receipt, however, show that program spells are typically short and long-term recipiency is more rare (see Chapter II). Indicator 9, for example, shows that among individuals receiving AFDC/TANF at some point over a ten-year period ending in 2000, 18 percent received some welfare during six or more years. Another 31 percent were recipients in three to five years, and more than half (51 percent) received welfare in only one or two years. 2 The earnings of those in unsubsidized employment would be correctly captured as income from work in national surveys. Any welfare benefits associated with work experience, community service programs or other work activities, however, would be counted as income from welfare in most national surveys, a classification incompatible with the proposed definition. 3 While TRIM-adjusted CPS data for 2002 are not yet available, non-adjusted estimates from the Annual March Demographic Supplement to the CPS indicate a slight increase in the level of dependence between 2001 and 2002. Measuring Economic Well-Being To assess the social impacts of any change in dependence, changes in the level of poverty should be considered. This chapter focuses on the official poverty rate, the most common poverty measure; additional measures of poverty and need are also included under the Economic Risk Factors found in Chapter III. Poverty in 2002 remains much lower than in 1996, the year of passage of the Personal Responsibility and Work Opportunity Reconciliation Act. The official poverty rate for 2002 was 12.1 percent, compared to 13.7 percent in 1996. This difference in the poverty rate indicates that 1.96 million fewer people are in poverty and 2.33 million fewer children are in families with incomes below poverty than in 1996. There was a small increase in the overall poverty rate between 2001 and 2002, but the poverty rate for children was essentially unchanged (see Table ECON 1 in Chapter III). Table SUM 1. Recipiency and Dependency Rates: 1996-2001 Recipiency Rates (Rates of Any Amount of AFDC/TANF, Food Stamps, or SSI) All Persons 16.0 14.8 13.5 13.3 12.5 12.6 Racial/Ethnic Categories Non-Hispanic White 9.9 9.7 8.6 8.4 8.2 8.2 Non-Hispanic Black 35.6 30.2 29.6 29.8 27.0 26.3 Hispanic 32.0 28.0 24.5 23.4 21.0 21.6 Children Ages 0-15 24.7 22.1 20.0 19.7 18.1 20.8 Women Ages 16-64 16.0 14.7 13.6 13.6 12.5 12.5 Men Ages 16-64 11.7 11.1 10.0 9.6 9.2 9.6 Adults Age 65 and over 10.3 10.2 9.9 10.0 10.4 9.6 Family Categories Individuals in Married Couple Families 9.6 8.7 8.3 7.9 7.2 7.4 Individuals in Female-Headed Families 46.0 41.6 37.5 39.9 37.1 36.4 Individuals in Male-Headed Families 25.3 24.3 19.7 19.3 21.8 21.2 Unrelated Individuals 11.5 11.9 10.9 10.0 10.1 10.0 Dependency Rates (More than 50 Percent of Income from AFDC/TANF, Food Stamps or SSI) All Persons 5.2 4.5 3.8 3.3 3.0 3.1 Non-Hispanic Black 13.8 11.4 10.5 9.1 7.7 8.8 Hispanic 10.9 9.1 6.6 5.4 4.5 4.5 Children Ages 0-15 9.7 8.4 6.8 5.6 5.1 5.9 Women Ages 16-64 5.2 4.6 3.9 3.5 3.0 3.3 Men Ages 16-64 2.7 2.5 2.1 1.9 1.9 2.0 Adults Age 65 and over 2.4 2.1 2.1 2.0 2.1 1.9 Individuals in Male-Headed Families 5.4 5.6 4.2 3.0 4.4 4.0 Unrelated Individuals 4.2 4.2 4.2 3.4 3.8 3.8 Note: Recipiency is defined as living in a family with receipt of any amount of AFDC/TANF, SSI, or food stamps during the year. Dependency is defined as having more than 50 percent of annual family income from AFDC/TANF, SSI and/or food stamps. Dependency rates would be lower if adjusted to exclude welfare assistance associated with working. Spouses are not present in the Male-Headed and Female-Headed family categories. Persons of Hispanic ethnicity may be of any race. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders are included in the total for all persons but are not shown separately. Figure SUM 2. Percentage of Total Population in Poverty with Various Means-Tested Benefits Added to Total Cash Income: 1979-2002 Source: Congressional Budget Office tabulations of March CPS data. Additional calculations by DHHS. See ECON 4 in Chapter III for underlying table and further notes. Figure SUM 2 shows poverty estimates under both the official poverty rate and two other measures that adjust income to take into account cash benefits, non-cash benefits and taxes. The three measures in the graph are based on analyzing three different concepts of income against the poverty threshold: The solid line with filled squares shows the official poverty rate, based on total cash income, including earned and unearned income. The official poverty rate was 12.1 percent in 2002. The dotted line shows what poverty would be if means-tested cash assistance (primarily AFDC/TANF and SSI) were excluded from cash income. Income in this measure includes earnings and other private cash income, plus social security, workers' compensation, and other social insurance programs, as income. Poverty under this measure would be higher than the official measure, or 12.8 percent in 2002. The lowest line shows that poverty would be lower if the cash value of selected non-cash benefits (food and housing) and taxes, including refunds under the Earned Income Tax Credit (EITC), were counted as income.4 Under this definition, poverty rates in 2002 would be at least two percentage points lower than the official measure, or 10.0 percent. 4 The effects of selected non-cash benefits (food and housing) are shown separately from the effect of taxes in Figure ECON 4 in Chapter III. Prior to 1993, taxes increased poverty. Since 1993, taxes, including the refunds through the Earned Income Tax Credit, have caused reductions in poverty. The primary data sources for this report are the Current Population Survey (CPS), the Survey of Income and Program Participation (SIPP), the Panel Study of Income Dynamics (PSID), and administrative data for the AFDC/TANF, Food Stamp, and SSI programs. Beginning with the 2001 report, there was a shift to using CPS rather than SIPP data for several indicators and predictors of welfare recipiency and dependence. This change was necessary because CPS data are updated annually, while SIPP updates are available much less frequently. If it were not for the lags in data availability, the SIPP would be considered the most useful national survey for measuring welfare dependency. It was used most extensively in the first three annual dependence reports. Its longitudinal design, system of monthly accounting, and detail concerning employment, income and participation in federal income-support and related programs, make the SIPP particularly effective for capturing the complexities of program dynamics. It continues to be an important source of data in this report, particularly for measures related to AFDC/TANF spell duration and transitions in and out of AFDC/TANF recipiency, dependency, and poverty. For measures of receipt, dependency, and poverty at a single point in time, however, the report primarily uses the Annual March Demographic Supplement to the CPS, which measures income and poverty over an annual accounting period. As stated above, the CPS data are available on a timelier basis than the SIPP, and have been widely used to measure trends since the welfare reform legislation of 1996. However, because the CPS does not collect income in the same detail as the SIPP, it has been subject to criticism for underreporting of income, particularly welfare income. To address this concern, some of the indicators in this report are based on CPS data that have been analyzed by the Transfer Income Model (TRIM3), a microsimulation model developed by the Urban Institute under contract to the Office of the Assistant Secretary for Planning and Evaluation. Although its primary purpose is to simulate program eligibility and the impact of policy proposals, the TRIM model has also been used to correct for underreporting of welfare receipt and benefits. Welfare caseloads in TRIM3 are based on CPS data, adjusted upward to ensure that total estimates of recipients equal the total counts from administrative data. As shown in Figure SUM 3, the overall measures of dependency and recipiency have not been greatly affected by the change in data sources. Both data sources show a decline in dependence between 1996 and 1999, from 4.7 to 2.8 percent under the SIPP data, and from 5.2 to 3.3 percent under the TRIM-adjusted CPS data. Still, readers are cautioned against comparing measures for 1987-1995 from the SIPP data in the first three annual reports with the measures for 1996-2001 from the TRIM-adjusted CPS data. Figure SUM 3. Recipiency and Dependency Rates from Two Data Sources: 1987-2001 Note: Recipiency is defined as receipt of any amount of AFDC/TANF, SSI, or food stamps during year. Dependency is defined as having more than 50 percent of annual family income from AFDC/TANF, SSI and/or food stamps. Dependency rates would be lower if adjusted to exclude welfare assistance associated with working. The Panel Study of Income Dynamics (PSID) is another source of data used in this report. Like the SIPP it provides longitudinal data, but over a much longer time period than the three- to fouryear time period of the SIPP. With annual data on program receipt since 1968, the PSID provides vital data for measuring longer-term welfare use over periods of up to 10 years. Because the PSID indicators cover time spans as long as a decade, they are updated less frequently than the CPS-based and SIPP-based measures. This 2004 report provides the first updated analysis of PSID data beyond 1996, allowing examination of longer-term welfare receipt under the TANF program. However, the updated analysis of PSID data is only provided for Indicator 9 (Indicator 10 in last year's report). Reductions in the frequency and detail of data collection under the PSID have made it difficult to update indicators of long-term dependence (Indicator 9 in last year's report) and of reasons for entrance and exit from first spells of AFDC receipt (Indicator 11 in last year's report). Therefore, these indicators have been dropped from the report. A new measure of reasons for entrance and exit from AFDC/TANF is under development and will hopefully be published in next year's report. Finally, the report also draws upon administrative data for the AFDC/TANF, Food Stamp and SSI programs. These data are largely reported in Appendix A. Like the CPS data, administrative data are generally available with little time lags; these data are generally available through fiscal year 2002. To the extent possible, TANF administrative data are reported in a consistent manner with data from the earlier AFDC program, as noted in the footnotes to the tables in Appendix A. The fact remains that assistance under locally designed TANF programs encompasses a diverse set of cash and non-cash benefits designed to support families in making a transition to work, and so direct comparisons between AFDC receipt and TANF receipt must be made with caution. This issue also affects reported data on TANF receipt in national data sets such as the CPS and SIPP. For further technical information about the data presented in the report, specifically for information on race and ethnicity, unit of analysis, and annual versus monthly measures, please see Appendix D. Chapter II. Indicators of Dependence Following the format of the previous annual reports to Congress, Chapter II presents summary data related to indicators of dependence. These indicators differ from other welfare statistics because of their emphasis on welfare dependence, rather than simple welfare receipt. As discussed in Chapter I, the Advisory Board on Welfare Indicators suggested measuring dependence as the proportion of families with more than 50 percent of their total income in a one-year period coming from cash assistance through the AFDC (now TANF) program, food stamps and SSI benefits. Furthermore, this welfare income was not to be associated with work activities. The indicators in Chapter II were selected to provide information about the range and depth of dependence as defined by the Advisory Board. Existing data from administrative records and national surveys, however, do not generally distinguish welfare benefits received in conjunction with work from benefits received without work. Thus, it was not possible to construct one single indicator of dependence; that is, one indicator that measures both percentage of income from means-tested assistance and presence of work activities. This chapter focuses on recipients of three major means-tested cash and nutritional assistance programs: cash assistance through the Aid to Families with Dependent Children (AFDC) and the Temporary Assistance for Needy Families (TANF) programs, benefits under the Food Stamp Program, and Supplemental Security Income (SSI) benefits for elderly and disabled recipients. For some indicators, summary data and characteristics are provided for all recipients, not just those defined as welfare dependent. While a number of indicators focus on the percentage of recipients' income from means-tested assistance, other indicators focus on presence of work activities at the same time as welfare receipt. Here is a brief summary of each of the nine indicators: Indicator 1: Degree of Dependence. This indicator focuses most closely on those individuals who meet the Advisory Board's proposed definition of "dependence." In addition to examining individuals with more than 50 percent of their annual family income from AFDC/TANF cash assistance, food stamps and/or SSI benefits, it shows various levels of dependence by examining those with more than 0 percent, 25 percent, and 75 percent of their income from these sources (Indicators 1a and 1b). This indicator also shows the average percentage of income from meanstested assistance and earnings received by families with various levels of income relative to the poverty level (Indicators 1c and 1d). Indicator 2: Receipt of Means-Tested Assistance and Labor Force Attachment. This indicator looks further at the relationship between receipt of means-tested assistance and participation in the labor force. This is an important issue because of the significant number of low-income individuals that use a combination of means-tested assistance and earnings from the labor force. Indicator 3: Rates of Receipt of Means-Tested Assistance. This indicator paints yet another picture of dependence by measuring recipiency rates, that is, the percentage of the population that receives AFDC/TANF, food stamps, or SSI in an average month. Program administrative data make these figures readily available over time, allowing a better sense of historical trends than is available from the more specialized indicators of dependence. Indicator 4: Rates of Participation in Means-Tested Assistance Programs. While means-tested public assistance programs are open to all that meet their requirements, not all eligible households participate in the programs. This indicator uses administrative data and microsimulation models to reflect "take up rates" – the number of families that actually participate in the programs as a percentage of those who are legally eligible. Indicator 5: Multiple Program Receipt. Depending on their circumstances, individuals may choose a variety of different means-tested assistance "packages." This indicator looks at the percentage of individuals receiving AFDC/TANF, food stamps, and SSI in a month, examining how many rely on just one of these programs, and how many rely on a combination of two programs. Indicator 6: Dependence Transitions. This indicator uses data from the Survey of Income and Program Participation (SIPP) to look at the ability of individuals who are dependent on welfare in one year to make the transition out of dependence in the following year. Indicator 7: Dependence Spell Duration. Like Indicator 6, this indicator is concerned with dynamics of welfare receipt and welfare dependence. It shows the proportion of individuals with short, medium, and long spells, or episodes, of AFDC or TANF receipt. The focus is on individuals in AFDC/TANF families with no labor force participants. Indicator 8: Program Spell Duration. One critical aspect of dependence is how long individuals receive means-tested assistance. Like Indicator 7, this indicator provides information on short, medium, and long spells of welfare receipt. It differs from Indicator 7 in looking at all recipients, regardless of attachment to the labor force, and in analyzing recipients of each of the three major means-tested programs – AFDC/TANF, the Food Stamp Program, and SSI. Indicator 9: Long-Term Receipt. Many individuals who leave welfare programs cycle back on after an absence of several months. Thus it is important to look beyond individual program spells, measured in Indicator 8, to examine the cumulative amount of time individuals receive assistance over a period of several years. Indicator 1. Degree of Dependence Figure IND 1a. Percentage of Total Income from Means-Tested Assistance Programs: 2001 Only 3.1 percent of the total population in 2001 received more than half of their total family income from TANF, food stamps and SSI. As shown in Table IND 1b, the percentage of families dependent on public assistance has dropped by almost 50 percent since 1993, with most of the decline occurring since 1996. As noted in Chapter I, preliminary data suggest dependency may increase slightly but will still be near 3 percent in 2002. Under 13 percent of the overall population received at least one dollar in means-tested assistance in 2001. However, for over half of these individuals (7 percent of the total population), such assistance represented 25 percent or less of annual family income. The vast majority (87 percent) of the population received no means-tested assistance in 2001. As shown in Table IND 1a, individuals living in female-headed families were much more likely to be dependent on assistance from means-tested programs compared to individuals in married-couple or male-headed families (11.9 percent compared to 1.0 and 4.0 percent respectively). In 2001, one in four individuals receiving some public assistance reported that TANF, food stamps, and SSI accounted for more than half of their total family income. This number reflected a decline in dependence since 1996, when nearly one in three individuals receiving public assistance were dependent on it. Table IND 1a. Percentage of Total Annual Family Income from Means-Tested Assistance Programs, by Race/Ethnicity and Age: 2001 0% and <= 25% 25% and <= 50% 75% and <= 100% Total 50% All Persons 87.4 7.3 2.2 1.0 2.1 3.1 Non-Hispanic White 91.8 5.1 1.3 0.5 1.3 1.8 Non-Hispanic Black 73.7 12.2 5.3 3.0 5.8 8.8 Hispanic 78.4 13.0 4.1 1.5 3.0 4.5 Children Ages 0-15 81.9 9.5 3.5 1.9 3.2 5.1 Women Ages 16-64 87.5 7.2 2.1 1.0 2.3 3.3 Men Ages 16-64 90.4 6.2 1.4 0.5 1.5 2.0 Adults Age 65 and over 90.4 5.8 1.9 0.6 1.3 1.9 Individuals in Married-Couple Families 92.6 5.4 1.0 0.4 0.6 1.0 Individuals in Female-Headed Families 63.6 16.4 8.0 4.4 7.5 11.9 Individuals in Male-Headed Families 78.8 13.3 3.9 1.4 2.6 4.0 Unrelated Individuals 90.0 5.0 1.2 0.4 3.4 3.8 Note: Means-tested assistance includes AFDC/TANF, SSI, and food stamps. Total 50% includes all persons with more than 50 percent of their total annual family income from these means-tested programs. Income includes cash income and the value of food stamps. Spouses are not present in the Female-Headed and Male-Headed family categories. Persons of Hispanic ethnicity may be of any race. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders are included in the total for all persons but are not shown separately. Table IND 1b. Percentage of Total Annual Family Income from Means-Tested Assistance Programs: 1993-2001 >0% and <= 25% 1993 83.4 7.8 3.0 1.8 4.1 5.9 See above for note and source. Figure IND 1b. Percentage of Total Annual Income from Various Sources, by Poverty Status: 2001 Those in families with income below the poverty level received half (50 percent) of their total family income from earnings and 29 percent of their total family income from meanstested assistance programs (TANF, SSI, and food stamps) in 2001. In contrast, those with family income over 200 percent of the poverty level received the majority (87 percent) of their income from earnings and less than one percent of their income from means-tested assistance (a percentage so small, it is not visible in Figure IND 1b). The percentage of family income received from earnings is inversely proportional to overall family income relative to the poverty line. For example, the percentage of income received from earnings for those living in deep poverty (below 50 percent of poverty) was only 32 percent, compared to 50 percent for all poor individuals in 2001. On average, children were more likely than the elderly to live in families receiving a higher percentage of their income from means-tested assistance programs, as shown by Table IND 1c. The elderly received more income from other income sources, such as Social Security benefits and private pensions. The percentage of income received from earnings for families with incomes below the poverty level has increased over time, as shown in Table IND 1d. In 1995, poor families received 40 percent of their income from earnings; this percentage rose to 50 percent in 2001. Over the same time period, there was a decline in the percentage of income from means-tested programs among poor families from 41 percent to 29 percent. Table IND 1c. Percentage of Total Annual Family Income from Various Sources, by Poverty Status, Race/Ethnicity, and Age: 2001 < 50% Poverty <100% of Poverty 200% + of Poverty TANF, SSI, and Food Stamps 53.0 28.6 9.1 0.2 1.0 Earnings 31.6 49.9 69.3 87.1 85.5 Other Income 15.4 21.5 21.6 12.7 13.5 Non-Hispanic Black TANF, SSI, and Food Stamps 61.1 38.9 15.6 0.5 3.3 Other Income 10.4 12.7 10.0 7.1 7.8 Other Income 11.1 10.6 8.3 5.2 5.6 Children Ages 11-15 Adults Age 65 and over Earnings 10.2 5.9 9.4 36.1 32.6 Note: Total income is total annual family income, including the value of food stamps. Other income is non means-tested, nonearnings income such as child support, alimony, pensions, Social Security benefits, interest, and dividends. Poverty status categories are not mutually exclusive. Table IND 1d. Percentage of Total Income from Various Sources: Selected Years 200%+ of Poverty TANF, SSI, and Food Stamps 65.9 41.3 14.2 0.3 Earnings 22.5 40.4 64.8 85.4 Other Income 11.6 18.3 21.0 14.3 TANF, SSI, and Food Stamps 53.1 29.8 9.7 0.2 Indicator 2. Receipt of Means-tested Assistance and Labor Force Attachment Figure IND 2. Percentage of Recipients in Families with Labor Force Participants in that Month, by Program: 2001 In 2001, 61 percent of individuals who received TANF, 56 percent of individuals who received food stamps, and 37 percent of individuals who received SSI were in families with at least one person in the labor force, either part-time or full-time. About one-third of TANF and food stamp recipients lived in families with at least one fulltime worker in 2001, while approximately one-quarter had only a part-time labor force participant. In contrast, SSI recipients were more likely to live in families with no labor force participant. As shown in Table IND 2a, young children (under age six) in households receiving TANF, food stamps, and SSI were more likely to live with at least one full-time worker than were older children (ages 11-15) in such recipient households. The percentage of AFDC/TANF recipients living in families with at least one full-time worker increased from 19 percent in 1993 to 35 percent in 2001, as shown in Table IND 2b. Table IND 2a. Percentage of Recipients in Families with Labor Force Participants, by Program, Race/Ethnicity, and Age: 2001 No One in LF At Least One in LF, No One FT At Least One FT Worker TANF All Persons 38.7 26.0 35.3 Non-Hispanic White 35.9 25.8 38.3 Non-Hispanic Black 44.1 27.3 28.6 Hispanic 37.7 24.0 38.3 Children Ages 0-5 38.1 23.2 38.7 Children Ages 6-10 41.0 26.4 32.6 Children Ages 11-15 39.6 27.5 32.9 Women Ages 16-64 40.1 26.1 33.8 Men Ages 16-64 30.1 30.4 39.5 Adults Age 65 and over 66.3 16.5 17.2 FOOD STAMPS All Persons Adults Age 65 and over 88.9 6.4 4.7 SSI All Persons 63.5 8.7 27.8 Non-Hispanic White 69.7 8.1 22.3 Hispanic 52.2 8.8 39.0 Women Ages 16-64 72.8 8.9 18.2 Men Ages 16-64 64.1 8.0 27.9 Adults Age 65 and over 65.9 6.6 27.5 Note: Recipients are limited to those individuals or family members directly receiving benefits in a month. Full-time workers are those who usually work 35 hours or more per week. Part-time labor force participation includes part-time workers and those who are unemployed, laid off, and/or looking for work. This indicator measures, on an average monthly basis, the combination of individual benefit receipt and labor force participation by any family member in the same month. Table IND 2b. Percentage of AFDC/TANF Recipients in Families with Labor Force Participants 1993-2001 1993 57.0 24.2 18.8 Note: Recipients are limited to those individuals or family members directly receiving benefits in a month. Full-time workers are those who usually work 35 hours or more per week. Part-time labor force participation includes those who are unemployed, laid off, and/or looking for work. This indicator measures, on an average monthly basis, the combination of individual benefit receipt and labor force participation by any family member in the same month. Indicator 3. Rates of Receipt of Means-tested Assistance Figure IND 3a. Percentage of the Total Population Receiving AFDC/TANF, by Age: 1970-2002 Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, and U.S. Bureau of the Census (available online at http://www.census.gov). Although the survey data needed to examine overall welfare receipt and dependency are not yet available past 2001, administrative data for recipiency measures of AFDC/TANF, food stamps, and SSI are available through 2002, as shown in Figures IND 3a, IND 3b, and IND 3c. Additional administrative data are shown in Appendix A. Just under 2 percent of the population received TANF in 2002. This is the lowest rate of AFDC/TANF receipt in the past 30 years, as shown in Table IND 3a. The percentage of the total population receiving AFDC/TANF has dropped significantly since 1994, when it was at a 25-year high of over 5 percent. AFDC/TANF recipiency rates have been much higher over time for children than for adults, with the child recipiency rates also showing more pronounced changes over time. Between 1993 and 2002, AFDC/TANF receipt among children was cut by more than half (from 14 to under 6 percent), the most rapid decline in a generation. Table IND 3a. Number and Percentage of the Total Population Receiving AFDC/TANF, by Age 1970-2002 Total Recipients Adult Recipients Child Recipients Number (thousands) 1970 7,188 3.5 1,863 1.4 5,325 7.6 1972 10,345 4.9 2,848 2.0 7,497 10.8 Notes: See Appendix A, Tables TANF 2, TANF 12, and TANF 14, for more detailed data on recipiency rates, including recipiency rates by calendar year. Recipients are expressed as the fiscal year average of monthly caseloads from administrative data, excluding recipients in the territories. Child recipients include a small number of dependents ages 18 and older who are students. The average number of adult and child recipients in 1998 and 1999 are estimated using data from the National Emergency TANF Data Files and thereafter using the National TANF Data Files. Figure IND 3b. Percentage of the Total Population Receiving Food Stamps, by Age: 1975-2002 Source: USDA, Food and Nutrition Service, Office of Analysis, Nutrition, and Evaluation, Characteristics of Food Stamp Households, Fiscal Year 2001, and earlier reports, and U.S. Bureau of the Census (available online at http://www.census.gov). The food stamp recipiency rate increased to 6.6 percent in 2002, above the two previous years' rate of 6.1 percent – the lowest rate since the Food Stamp program became available nationwide. The 2002 recipiency rate is still significantly below the peak of 10.4 percent experienced in 1993 and 1994. As with AFDC/TANF, food stamp recipiency rates have been much higher over time for children than for adults. Between 1980 and 2002, the percentage of all children who received food stamps was between two and one-half to three times that for all adults ages 18 to 59. Similar trends in food stamps recipiency – largely reflecting changes in the rate of unemployment and programmatic changes – existed across all age groups over time, as shown in Table IND 3b. The percentages of individuals receiving food stamps within all age groups declined from 1984 through 1988, rose in the early 1990s until reaching a peak in 1994, and then declined through 2000 followed by a slight increase in 2002. Table IND 3b. Number and Percentage of the Total Population Receiving Food Stamps, by Age 1975-2002 Adult Recipients Age 60 and over Adult Recipients Ages 18-59 Child Recipients Ages 0-18 1975 16,320 7.6 – – – – – – 1976 17,033 7.8 – – – – 9,126 13.8 1980 19,253 8.5 1,741 4.9 7,186 5.6 9,876 15.5 1983 21,668 9.3 1,654 4.4 8,960 6.7 10,910 17.4 1992 25,369 9.9 1,687 3.9 10,550 7.2 13,349 20.1 1993 26,952 10.4 1,876 4.3 11,214 7.5 14,196 21.0 Note: See Appendix A, Tables FSP 1 and FSP 6 for more detailed data on recipiency rates. Recipients are expressed as the fiscal year average of monthly caseloads from administrative data, excluding recipients in the territories. From 1975 to 1983 the number of participants includes the Family Food Assistance Program (FFAP) that was largely replaced by the Food Stamp program in 1975. From 1975 to 1983 the number of FFAP participants averaged only 88 thousand. Source: USDA, Food and Nutrition Service, Office of Analysis, Nutrition, and Evaluation, Characteristics of Food Stamp Households, Fiscal Year 2001, and earlier reports and U.S. Bureau of the Census (available online at http://www.census.gov). Figure IND 3c. Percentage of the Total Population Receiving SSI, by Age: 1975-2002 Source: Social Security Administration, Office of Research, Evaluation, and Statistics, Social Security Bulletin, Annual Statistical Supplement 2003 (available online at http://www.ssa.gov/statistics) and U.S. Bureau of the Census (available online at http://www.census.gov). Unlike the recipiency rates for AFDC/TANF and food stamps, which have been influenced by outside factors such as the economy and welfare reform, overall recipiency rates for SSI show less variation over time. After trending downward slightly from 1975 to the early 1980s, the proportion of the total population that receives SSI has risen from 1.7 percent in 1985 to 2.5 percent in 1996 and subsequently declined slightly to 2.3 percent. As shown in Table IND 3c, the total number of recipients has grown by 66 percent over the same period, from 4.1 million in 1985 to 6.8 million people in 2002. Elderly adults (aged 65 and older) have much higher recipiency rates than any other age group. The gap has narrowed, however, as the percentage of adults aged 65 and older receiving SSI has been cut nearly in half, declining from 10.9 percent in 1975 to 5.6 percent in 2002. The proportion of children receiving SSI increased gradually between 1975 and 1990, and grew more rapidly in the early-to-mid 1990s, reaching a high of 1.4 percent in 1996. The rate has since fallen, with 1.2 percent of children receiving SSI in 2002. Table IND 3c. Number and Percentage of the Total Population Receiving SSI, by Age: 1975-2002 Adult Recipients Age 65 & over Dec 1975 4,314 2.0 2,508 10.9 1,699 1.3 107 0.2 Dec 1977 4,238 1.9 2,353 9.7 1,738 1.3 147 0.2 Note: December population figures used as the denominators are obtained by averaging the Census Bureau's July 1 population estimates for the current and the following year. See Appendix A, Tables SSI 2, SSI 8, and SSI 9 for more detailed data on SSI recipiency rates. In this report the categories of children under 18 and adults 18-64 differ from those in previous editions where the category of children included a small number of dependents 18 and older who were students. Source: Social Security Administration, Office of Research, Evaluation, and Statistics, Social Security Bulletin, Annual Statistical Supplement 2003 (available online at http://www.ssa.gov/statistics), and U.S. Bureau of the Census (available online at http://www.census.gov). Indicator 4. Rates of Participation in Means-tested Assistance Programs Figure IND 4. Participation Rates in the AFDC/TANF, Food Stamp and SSI Programs Selected Years Source: AFDC and SSI participation rates are tabulated using TRIM3 microsimulation model, while food stamp participation rates are from a Mathematica Policy Research, Inc. model. See Tables IND 4a, IND 4b, and IND 4c for details. Whereas Indicator 3 examined participants as a percentage of the total population (recipiency rates), this indicator examines participating families or households as a percentage of the estimated eligible population (participation rates, also known as "take up" rates). Only 48 percent of the families estimated as eligible for TANF cash assistance actually enrolled and received benefits in an average month in 2001. This is significantly lower than AFDC participation rates, which ranged from 77 percent to 86 percent between 1981 and 1996. See Table IND 4a for further information. The food stamp participation rate edged up slightly between 2000 and 2001, from 53 to 54 percent. The participation rate is still much lower than the 1994 rate of 70 percent. See Table IND 4b for further discussion. After rising steadily over the past several years, the SSI participation rate appears to have dropped nearly 6 percentage points between 2000 and 2001. At 70 percent it still is much higher than recent TANF and Food Stamp participation rates. See Table IND 4c for details by age and disability status. Table IND 4a. Number and Percentage of Eligible Families Participating in AFDC/TANF Selected Years Eligible Families(in millions) Participating Families(in millions) Participation Rate(percent) 1981 4.78 3.84 80.2 1994 (revised) 6.13 5.03 82.1 1997 (adjusted) 5.41 3.74 69.2 Notes: Participation rates are estimated by an Urban Institute model (TRIM3) which uses CPS data to simulate AFDC/TANF eligibility and participation for an average month, by calendar year. There have been small changes in estimating methodology over time, due to model improvements and revisions to the CPS. Most notably, since 1994, the model has been revised to more accurately estimate SSI participation among children, and in 1997 and 1998 the model was adjusted to more accurately exclude ineligible immigrants. In contrast to past editions, this table now includes families receiving assistance under Separate State Programs. Note that families subject to full-family sanctions are counted as nonparticipating eligible families due to modeling limitations. Also, the numbers of eligible and participating families include the territories and pregnant women without children, even though these two small groups are excluded from the TRIM model. The numbers shown here implicitly assume that participation rates for the territories and for pregnant women with no other children are the same as for all other eligibles. Source: U.S. Department of Health and Human Services, Administration for Children and Families, caseload tabulations and unpublished data from the TRIM3 microsimulation model. Between 2000 and 2001, eligibility for the TANF program increased slightly from 4.44 to 4.56 million families. This eligibility increase is primarily due to changes in the economy and/or population rather than changes in TANF eligibility rules. Despite the small increase in TANF eligibility in 2001, caseloads continued to fall, resulting in a drop in the participation rate for the sixth consecutive year. Participating families includes families receiving TANF cash assistance only. Families who receive services and benefits other than cash assistance are not included in the participation rate. Table IND 4b. Number and Percentage of Eligible Households Participating in the Food Stamp Program: Selected Years Eligible Households (in millions) Participating Households (in millions) Participation Rate (percent) September 76 16.3 5.3 32.6 February 78 14.0 5.3 37.8 August 80 14.0 7.4 52.5 August 92 16.7 10.2 61.6 September 94 (revised) 15.3 10.7 69.6 September 95 15.0 10.4 69.2 Note: Eligible households estimated from a Mathematica Policy Research, Inc. model that uses CPS data to simulate the Food Stamp Program. Caseload data are from USDA, FNS program operations caseload data. There have been small changes in estimating methodology over time, due to model improvements and revisions to the CPS. Most notably, the model was revised in 1994 to produce more accurate (and lower) estimates of eligible households. The original 1994 estimate and estimates for previous years show higher estimates of eligibles and lower participation rates relative to the revised estimate for 1994 and estimates for subsequent years. Source: U.S. Department of Agriculture, Food and Nutrition Service, Trends in Food Stamp Program Participation Rates: 1999 to 2001, July 2003. Between September 2000 and September 2001, there was a small increase in households eligible for the Food Stamp Program (from 13.5 to 13.9 million households). Caseloads grew at a slightly higher rate over the same year. The net effect was a small increase in the measured participation rate, from 53 to 54 percent. Over the longer run, there has been a significant drop in food stamp caseloads, from over 10 million in 1992 through 1995, to 7.5 million in 2001. This decline in caseloads occurred during a time when both the eligible population and the program participation rates were generally decreasing. These longer-term decreases are considerably larger than the small increases experienced in 2001. Table IND 4c. Percentage of Eligible Adult Units Participating in the SSI Program, by Type 1993-2001 All Adult Units One-Person Units Married-Couple Units 1993 62.0 57.0 71.0 37.0 Notes: Participation rates estimated using the TRIM3 microsimulation model, which uses CPS data to simulate SSI eligibility for an average month, by calendar year. There have been small changes in estimating methodology over time, due to model improvements and revisions to the CPS. In particular, the model was revised in 1997 to more accurately exclude ineligible immigrants. Thus the increased participation rate in 1997 is partly due to a revision in estimating methodology. Also note that the figure for married-couple units is based on very small sample sizes–for example, married-couple units were only about 7.5 percent of the eligible adults units and 5.1 percent of the units receiving SSI in the average month of 1998. Source: Unpublished data from the TRIM3 microsimulation model. There was an apparent drop in the SSI participation rate among adult units between 2000 and 2001, from 76 to 70 percent. This decline occurred across aged one-person units, disabled one-person units, and married-couple units that are either aged or disabled and it is due to a significant increase in the estimated eligible population of these groups. There have not been similar increases in the participating populations, perhaps due to lags between application and enrollment. The increase in the eligible population reflects a rise in the number of aged individuals, an increase in disabilities as reported on labor-market surveys (which may partially reflect tougher economic times), and a higher percentage of aged and disabled persons falling below the SSI eligibility limits. Some of the increase in the eligible population may be due to changes in the Current Population Survey (i.e., reweighting to reflect 2000 Census-based weights). In 2001, as in past years, disabled adults in one-person units had a higher participation rate (76 percent) than both aged adults in one-person units (64 percent) and adults in married-couple units (46 percent). Indicator 5. Multiple Program Receipt Figure IND 5. Percentage of Population Receiving Assistance from Multiple Programs (TANF, Food Stamps, SSI), Among Those Receiving Assistance: 2001 Of the 8 percent of the population in families receiving TANF, food stamps, or SSI benefits in an average month in 2001, about two-thirds (68 percent) received assistance from only one program. Most of these families received food stamps or SSI benefits only. However, other common patterns include food stamp and TANF receipt (19 percent) and food stamp and SSI receipt (12 percent). Children are more likely than other age groups to live in families receiving TANF and/or food stamps. For example, 16 percent of children under six lived in families receiving any public assistance in an average month in 2001, and 5 percent of children under six, lived in families receiving both TANF and food stamps, as shown in Table IND 5a. The percentage of individuals receiving assistance from at least one program among AFDC/TANF, food stamps, and SSI in an average month decreased during the mid-to-late 1990s (from 13 percent in 1994 to 8 percent in 2001), as shown in Table IND 5b. Table IND 5a. Percentage of Population Receiving Assistance from Multiple Programs (TANF, Food Stamps, SSI), by Race/Ethnicity and Age: 2001 Any Receipt One Program Only Two Programs TANF & FS FS & SSI Non-Hispanic Black 19.4 0.3 10.7 2.5 3.6 2.3 Children Ages 0-5 15.7 0.8 8.7 0.7 5.1 0.5 Children Ages 11-15 11.3 0.6 6.3 0.8 3.0 0.7 Note: Categories are mutually exclusive. SSI receipt based on individual receipt; AFDC/TANF and food stamp receipt based on full recipient unit. In practice, individuals do not tend to receive both AFDC/TANF and SSI; hence, no individual receives benefits from all three programs. The percentage of individuals receiving assistance from any one program in an average month (shown here) is lower than the percentage residing in families receiving assistance over the course of a year (shown in Table SUM 1 in Chapter I and Table IND 1a in Chapter II). Table IND 5b. Percentage of Population Receiving Assistance from Multiple Programs (AFDC/TANF, Food Stamps, SSI): 1993-2001 AFDC/ TANF AFDC/TANF & FS 1998 9.0 0.4 3.9 1.4 2.4 0.9 Indicator 6. Dependence Transitions Figure IND 6. Dependency Status in 1999 of Persons Who Received More than 50 Percent of Income from Means-Tested Assistance in 1998, by Race/Ethnicity Source: Unpublished data from the SIPP, 1996 panel. Recipients of means-tested assistance were more likely to move out of dependency in the late 1990s than in the early 1990s. Three-tenths (30 percent) of recipients who received more than 50 percent of their total income from means-tested assistance programs in 1998 transitioned out of this dependency status in 1999. The comparable transition rate was only 20 percent between 1993 and 1994, as shown in Table IND 6b. Of recipients who received more than 50 percent of their total income from AFDC/TANF, food stamps, and/or SSI in 1998, there was little difference among racial and ethnic categories in dependency transitions between 1998 and 1999. Past SIPP panels (data not shown) had found more movement among non-Hispanic whites than among non-Hispanic blacks. As shown in Table IND 6a, a slightly larger percentage of women who received more than half of their total income from means-tested assistance programs in 1998 remained "dependent" in 1999 compared to the same group of men (71 percent compared to 66 percent). Table IND 6a. Dependency Status in 1999 of Persons Who Received More than 50 Percent of Income from Means Tested Assistance in 1998, by Race/Ethnicity and Age Individuals Receiving more than 50% of Income from Assistance in 1998 Total (000's) Percentage of Persons Receiving No Aid in 1999 Up to 50% in 1999 Over 50% in 1999 All Persons 8,163 2.9 27.1 70.0 Non-Hispanic White 2,657 4.3 25.8 70.0 Non-Hispanic Black 2,925 2.0 27.8 70.1 Hispanic 1,895 2.0 26.3 71.7 Children Ages 0-5 1,271 3.6 29.7 66.6 Children Ages 6-10 1,056 2.1 27.4 70.6 Children Ages 11-15 998 2.9 29.0 68.1 Women Ages 16-64 2,847 3.7 25.5 70.8 Men Ages 16-64 1,337 2.7 31.6 65.7 Adults Age 65 and over 654 0.0 16.4 83.6 Note: Means-tested assistance is defined as AFDC/TANF, food stamps, and SSI. While only affecting a small number of cases, general assistance income is included within AFDC/TANF income. Individuals are defined as dependent if they reside in families with more than 50 percent of total annual family income from these means-tested programs. Because full calendar year data for 1997-1998 were not available for all SIPP respondents, some transitions were based on twelve-month periods that did not correspond exactly to calendar years. Table IND 6b. Dependency Status of All Persons Who Received More than 50 Percent of Income from Means Tested Assistance in Previous Year Total (000's) No Aid in Second Year Up to 50% in Second Year Over 50% in Second Year Transitions from: 1993 to 1994 14,810 1.6 18.6 79.8 1997 to 1998 9,672 3.1 28.8 68.1 Source: Unpublished data from the SIPP, 1993 and 1996 panels. Indicator 7. Dependence Spell Duration Figure IND 7. Percentage of AFDC/TANF Spells of Individuals in Families with No Labor Force Participants for Individuals Entering Programs During the 1993 and 1996 SIPP Panels, by Length of Spell In the late 1990s over two-fifths (41 percent) of AFDC/TANF spells for individuals in families with no one in the labor force ended within four months and over two-thirds (68 percent) ended within a year. These spells are measured for individuals entering AFDC/TANF between 1996 and 1999, during early implementation of the TANF program. Spells were much longer for families entering AFDC between 1993 and 1995, as shown in Figure IND 7 and Table IND 7b. Half (50 percent) of AFDC/TANF spells for individuals in families where no one participated in the labor force lasted more than 20 months in the 1993 SIPP panel, compared with only 19 percent of that length in the 1996 SIPP panel. As shown in Table IND 7a, the percentage of AFDC/TANF spells ending in four months or less was similar across racial/ethnic categories, ranging from 38 percent among non-Hispanic whites to 44 percent among non-Hispanic blacks. Spells shown in Figure IND 7 are limited to spells of recipients in families without any labor force participation. Spell lengths are slightly shorter in Figure IND 8, which shows spells for all recipients, including those in families with labor force participants. For example, whereas 81 percent of spells between 1996 and 1999 shown in Figure IND 7 end in 20 months or less, 87 percent of all AFDC/TANF spells during this same time period last 20 months or less, as shown in Figure IND 8. Table IND 7a. Percentage of AFDC/TANF Spells of Individuals in Families with No Labor Force Participants for Individuals Entering Programs During the 1996 SIPP Panel, by Length of Spell, Race/Ethnicity, and Age Spells <=4 Months Spells 5-12 Months Spells 13-20 Months Spells >20 Months All Persons 40.5 27.5 13.3 18.7 Non-Hispanic White 38.4 35.8 NA NA Non-Hispanic Black 44.1 22.4 11.5 21.9 Hispanic 39.6 23.2 NA NA Ages 0-15 Years 38.9 25.0 12.9 23.2 Ages 16-64 Years 42.2 31.4 NA NA Note: Spell length categories are not mutually exclusive. Spells separated by only 1 month are not considered separate spells. Due to the length of the observation period, actual spell lengths for spells that lasted more than 20 months cannot be observed. AFDC spells are defined as those spells starting during the 1996 SIPP panel for individuals in families with no labor force participants. For certain racial/ethnic and age categories, data are not available (N/A) due to insufficient sample size. Table Ind 7b. Percentage of AFDC/TANF Spells of Individuals in Families with No Labor Force Participants for Individuals Entering Programs During the 1993 and 1996 SIPP Panels 1993 Panel All Persons 27.2 16.2 6.9 49.7 1996 Panel All Persons 40.5 27.5 13.3 18.7 Indicator 8. Program Spell Duration Figure IND 8. Percentage of AFDC/TANF, Food Stamp, and SSI Spells for Individuals Entering Programs During the 1996 SIPP Panel, by Length of Spell Between the years 1996 and 1999, short spells lasting 4 months or less accounted for about 47 percent of AFDC/TANF spells, 43 percent of food stamp spells, and 34 percent of SSI spells. Approximately three-fourths of all AFDC/TANF and food stamp spells lasted one year or less (76 percent and 71 percent, respectively). In contrast, only 53 percent of SSI spells ended within one year. As shown in Table IND 8a, for TANF/AFDC spells, a smaller percentage of long spells (lasting more than 20 months) occurred among non-Hispanic whites compared to non-Hispanic blacks and Hispanics. Spells of welfare receipt were shorter in the second half of the 1990s than in the early 1990s, as shown in Table IND 8b. For example, only 13 percent of AFDC/TANF spells for individuals entering AFDC/TANF between 1996 and 1999 lasted 20 months or longer, compared with 34 percent of AFDC spells beginning between 1992 and 1994. Short spells are less common among recipients in families without labor force participants, as shown previously in Figure and Table IND 7. Length of TANF receipt varies across states, as shown in Appendix Table TANF 17, which shows an alternative measure of length of TANF receipt, using state administrative data. Table IND 8a. Percentage of AFDC/TANF, Food Stamp and SSI Spells for Individuals Entering Programs During the 1996 SIPP Panel, by Length of Spell, Race/Ethnicity, and Age AFDC/TANF All Recipients 46.6 29.2 11.5 12.7 Non-Hispanic White 47.4 33.0 10.7 8.9 Hispanic 46.3 25.4 10.5 17.9 Ages 0-5 Years 41.8 33.2 10.8 14.2 Ages 6 to 10 Years 49.4 24.6 9.0 17.0 Ages 11 to 15 Years 42.5 25.6 N/A N/A Ages 16 to 64 Years 48.6 30.7 12.0 8.7 65 Years and Older N/A N/A N/A N/A FOOD STAMPS All Recipients 43.1 27.7 9.3 19.8 Non-Hispanic White 46.5 27.5 9.4 16.7 Non-Hispanic Black 38.6 28.5 9.1 23.9 Hispanic 41.7 28.5 8.1 21.8 Ages 0 to 5 years 36.5 31.4 8.6 23.5 Ages 11-15 40.4 30.3 10.0 19.3 Ages 16-64 46.2 26.7 9.6 17.6 65 Years and Older 31.7 26.8 6.9 34.7 SSI All Recipients 34.1 19.2 9.1 37.6 Ages 0-10 N/A N/A N/A N/A Ages 11-15 30.9 N/A N/A N/A 65 Years and Older 22.1 16.7 11.9 49.3 Note: Spell length categories are not mutually exclusive. Spells separated by only 1 month are not considered separate spells. Due to the length of the observation period, actual spell lengths for spells that lasted more than 20 months cannot be observed. AFDC/TANF spells are defined as those starting during the 1996 SIPP Panel. For certain age categories, data are not available (N/A) because of insufficient sample size. Table IND 8b. Percentage of AFDC/TANF, Food Stamp and SSI Spells for Individuals Entering Programs During the 1992, 1993, and 1996 SIPP Panels 1992 Panel AFDC 30.4 24.7 10.5 34.4 Food Stamps 33.4 24.9 10.2 31.5 SSI 25.7 8.9 4.8 60.6 AFDC/TANF 46.6 29.2 11.5 12.7 Food Stamps 43.1 27.7 9.3 19.8 SSI 34.1 19.2 9.1 37.6 Source: Unpublished data from the SIPP, 1992, 1993, and 1996 Panels. Indicator 9. Long-term Receipt Figure IND 9. Percentage of AFDC/TANF Recipients, by Years of Receipt Between 1991 and 2000 Source: Unpublished data from the PSID public release data files, 1992-2001. Among all persons receiving AFDC/TANF at some point in the ten-year period ending in 2000, about half (51 percent) received assistance in only one or two of these years. Less than one third (31 percent) received AFDC/TANF in three to five years, and less than one fifth (18 percent) received AFDC/TANF during more than five of the ten years. A larger percentage of child recipients experienced long-term receipt (some receipt in at least six of the ten years) and a smaller percentage experienced short-term receipt in all three time periods relative to the percentages for all recipients, as shown in Table IND 9. Longer-term welfare receipt was much less common during the 1990s compared to earlier decades. Less than 4 percent of those with some AFDC/TANF assistance between 1991 and 2000 received at least one assistance payment in nine or ten years of the period, compared to 12 percent and 13 percent of AFDC recipients in the earlier two time periods. In the two ten-year time periods between 1971-1990, there was a large percentage difference in short-term AFDC receipt between all black and non-black recipients. In the ten-year period ending in 2000, this percentage difference was much smaller, with 49 percent of blacks and 53 percent of non-blacks receiving AFDC/TANF in only one or two years. Table IND 9: Percentage of AFDC/TANF Recipients Across Three Ten-Year Time Periods by Years of Receipt, Race, and Age All Races: All Recipients Child Recipients 0-5 Years received AFDC/TANF: 1-2 Years 44.0 44.8 50.9 36.3 36.1 37.9 9-10 Years 13.3 12.2 3.8 17.7 19.4 4.9 Black: 6-8 Years 18.6 17.5 NA 24.7 18.7 NA 9-10 Years 18.7 18.4 NA 22.8 28.7 NA Non-Black: 6-8 Years 9.4 15.7 NA 13.1 21.8 NA 9-10 Years 10.5 7.9 NA 14.1 12.3 NA Note: The base for the percentages consists of individuals receiving at least $1 of AFDC/TANF in any year in the ten-year period. Child recipients are defined by age in the first year of the 10-year period. This indicator measures years of recipiency over the specified ten-year time periods and does not take into account years of recipiency that may have occurred before or after each ten-year period. Race categories include those of Hispanic ethnicity. Due to small sample size, American Indians/Alaska Natives, Asians, and Native Hawaiians/Other Pacific Islanders are included in the estimates for non-black persons but are not shown separately. Data are not available (NA) separately by race for longer periods of cumulative receipt (6 or more years) in the most recent 10-year period. Chapter III. Predictors and Risk Factors Associated with Welfare Receipt The Welfare Indicators Act challenges the U.S. Department of Health and Human Services to identify and set forth not only indicators of welfare dependence and welfare duration but also predictors and causes of welfare receipt. However, welfare research has not established clear and definitive causes of welfare dependence. Instead, it has identified a number of risk factors associated with welfare use. For the purposes of this report, the terms "predictors" and "risk factors" are used somewhat interchangeably. Following the recommendation of the Advisory Board, this chapter includes a wide range of possible predictors and risk factors. As research advances, some of the "predictors" included in this chapter may turn out to be simply correlates of welfare receipt, some may have a causal relationship, some may be consequences, and some may have predictive value. The predictors/risk factors included in this chapter are grouped into three categories: economic security risk factors, employment-related risk factors, and risk factors associated with non-marital childbearing. Economic Security Risk Factors (ECON). The first group includes eight measures associated with economic security. This group encompasses five measures of poverty, as well as measures of child support receipt, food insecurity, and lack of health insurance. The tables and figures illustrating measures of economic security are labeled with the prefix ECON throughout this chapter. Poverty measures are important predictors of dependence, because families with fewer economic resources are more likely to be dependent on means-tested assistance. In addition, poverty and other measures of deprivation, such as food insecurity, are important to assess in conjunction with the measures of dependence outlined in Chapter II. Reductions in caseloads and dependence can reduce poverty, to the extent that such reductions are associated with greater work activity and higher economic resources for former welfare families. However, reductions in welfare caseloads can increase poverty and other deprivation measures, to the extent that former welfare families are left with fewer economic resources. Several aspects of poverty are examined in this chapter. Those that can be updated annually using the Current Population Survey include: overall poverty rates (ECON 1); the percentage of individuals in deep poverty (ECON 2), and poverty rates using alternative definitions of income (ECON 3 and 4). The chapter also includes data on the length of poverty episodes or spells (ECON 5). A ten-year measure of poverty (ECON 6 in last year's report) has been dropped due to reductions in the frequency and detail of data collection under the PSID. This chapter also includes data on child support collections (ECON 6), which can play an important role in reducing dependence on government assistance and thus serve as a predictor of dependence. Household food insecurity (ECON 7) is an important measure of deprivation that, although correlated with general income poverty, provides an alternative measure of tracking the incidence of material hardship and need, and how it may change over time. Finally, health insurance (ECON 8) is tied to the income level of the family, and may be a precursor to future health problems among adults and children. Employment and Work-Related Risk Factors (WORK). The second grouping, labeled with the WORK prefix, includes seven factors related to employment and barriers to employment. These measures include data on overall labor force attachment and the employment and earnings for low-skilled workers, as well as data on barriers to work. The latter category includes incidence of adult and child disabilities, adult substance abuse, and levels of educational attainment and school drop-out rates. Employment and earnings provide many families with an escape from dependence. It is important, therefore, to look both at overall labor force attachment (WORK 1), and at employment and earnings levels for those with low education levels (WORK 2 and WORK 3). The economic condition of the low-skill labor market is a key predictor of the ability of young adult men and women to support families without receiving means-tested assistance. The next two measures in this group (WORK 4 and WORK 5) focus on educational attainment. Individuals with less than a high school education have the lowest amount of human capital and are at the greatest risk of becoming poor, despite their work effort. Measures of barriers to employment provide indicators of potential work limitations, which may be predictors of greater dependence. Substance abuse (WORK 6) and disabling conditions among children and adults (WORK 7) all have the potential of limiting the ability of the adults in the household to work. In addition, debilitating health conditions and high medical expenditures can place a strain on a family's economic resources. Non-Marital Birth Risk Factors (BIRTH). The final group of risk factors addresses out-of-wedlock childbearing. The tables and figures in this subsection are labeled with the BIRTH prefix. This category includes long-term time trends in births to unmarried women (BIRTH 1), births to unmarried teens (BIRTH 2 and BIRTH 3), and children living in families with never-married parents (BIRTH 4). Children living in families with never-married mothers are at high risk of dependence, and it is therefore important to track changes in the size of this vulnerable population. As noted above, the predictors/risk factors included in this chapter do not represent an exhaustive list of measures. They are merely a sampling of available data that address in some way the question of how a family is faring on the scale of deprivation and well-being. Such questions are a necessary part of the dependence discussion as researchers assess the effects of welfare reform. Economic Security Risk Factors Economic Security Risk Factor 1. Poverty Rates Figure ECON 1. Percentage of Persons in Poverty, by Age: 1959-2002 Source: U.S. Bureau of the Census, "Poverty in the United States: 2002," Current Population Reports, Series P60-222 and data published online at http://www.census.gov/hhes/www/poverty.html. The official poverty rate was 12.1 percent in 2002, an increase over the rate of 11.7 percent in 2001. Even so, the percentage of persons living in poverty in 2002 was below the poverty rates experienced in most of the 1980s and 1990s. Children under 18 had a poverty rate of 16.7 percent in 2002, statistically unchanged from 2001. As in past years, the child poverty rate is considerably higher than the overall poverty rate. The poverty rate for the elderly (persons ages 65 and over) was 10.4 percent in 2002, an increase over the 2001 rate. This was a lower poverty rate than the rate for children under 18 (16.7 percent) and statistically indistinguishable from that of adults ages 18-64. Poverty rates by race are affected by a change in the questionnaire that allows individuals to report one or more races. The poverty rate for individuals reporting black race alone was 24.1 percent, as shown in Table ECON 1; the rate for those reporting black alone or in combination with other races was 23.9 percent (data not shown). Under either measurement, the gap between black and white poverty rates was close to 14 percentage points, slightly higher than the historic low of 13 percentage points in 2000 and 2001; but significantly lower than the early 1990s, when it exceeded 21 percentage points. Table ECON 1. Percentage of Persons in Poverty, by Race/Ethnicity and Age: Selected Years Related Children Hispanic Origin 65 & over 1959 NA NA 22.4 27.3 17.0 35.2 18.1 55.1 NA 1963 NA NA 19.5 23.1 NA NA 15.3 NA NA 1969 15.3 13.1 12.1 14.0 8.7 25.3 9.5 32.2 NA 1973 15.7 13.6 11.1 14.4 8.3 16.3 8.4 31.4 21.9 1980 20.3 16.8 13.0 18.3 10.1 15.7 10.2 32.5 25.7 1999 18.0 15.5 11.9 17.1 10.1 9.7 9.8 23.6 22.7 2000 17.8 14.7 11.3 16.2 9.6 9.9 9.5 22.5 21.5 2001 18.2 14.6 11.7 16.3 10.1 10.1 9.9 22.7 21.4 Notes: All persons under 18 include related children (own children, including stepchildren and adopted children, plus all other children in the household who are related to the householder by birth, marriage, or adoption), unrelated individuals under 18 (persons who are not living with any relatives), and householders or spouses under age 18. In this table, race categories include those of Hispanic ethnicity. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Whites and Blacks are for persons reporting a single-race only. Persons who reported more than one race are included in the total for all persons but are not shown under any race category. For example, the poverty rate of 10.2 percent shown for Whites in 2002 is for "White Alone including Hispanic." Though not shown, the rate for "White Alone or in Combination with other races" was 10.3 percent and for "White Alone, Non-Hispanic" the rate was 8 percent. American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders also are included in the total for all persons but are not shown separately, due to small sample size. Economic Security Risk Factor 2. Deep Poverty Rates Figure ECON 2. Percentage of Total Population Below 50 and 100 Percent of Poverty Level 1975-2002 Source: U.S. Bureau of the Census, "Poverty in the United States: 2002" Current Population Reports, Series P60-222 and unpublished tables available online at http://www.census.gov/hhes/www/poverty.html. The percentage of the population in "deep poverty" (with incomes below 50 percent of the federal poverty level) was 4.9 percent in 2002, compared to an overall poverty rate of 12.1 percent. In general, the percentage of the population with incomes below 50 percent of the poverty threshold has followed a pattern that reflects the trend in the overall poverty rate, as shown in Figure ECON 2. The percentage of people below 50 percent of poverty rose in the late 1970s and early 1980s, but then, after falling slightly, rose to a second peak in 1993. The overall poverty rate followed a somewhat similar pattern with more pronounced peaks and valleys. Over the past two decades, there has been an overall increase in the proportion of the poverty population in deep poverty. From a low of 28 percent of the poverty population in 1976, this population rose to nearly 41 percent in 2002. The total number of poor people in 2002 was 34.6 million, as shown in Table ECON 2. While higher than the previous year, this number was 4.7 million lower than the peak of 39.3 million in 1993. Table ECON 2. Number and Percentage of Total Population Below 50, 75, 100, and 125 Percent of Poverty Level: Selected Years Total Population (thousands) Below 50 percent Below 100 percent 1959 176,600 NA NA NA NA 39,500 22.4 54,900 31.1 1969 199,500 9,600 4.8 16,400 8.2 24,100 12.1 34,700 17.4 1981 227,200 11,200 4.9 20,700 9.1 31,800 14.0 43,800 19.3 1982 229,400 12,800 5.6 23,200 10.1 34,400 15.0 46,600 20.3 Note: The number of persons below 50 percent and 75 percent of poverty for 1969 are estimated based on the distribution of persons below 50 percent and 75 percent for 1969 taken from the 1970 decennial census. Source: U.S. Bureau of the Census, "Poverty in the United States: 2002," Current Population Reports, Series P60-222, unpublished tables available online at http://www.census.gov/hhes/www/poverty.html, and 1970 Census of Population, Volume 1, Social and Economic Characteristics, Table 259. Economic Security Risk Factor 3. Experimental Poverty Measures Figure ECON 3. Percentage of Persons in Poverty Using Various Experimental Poverty Measures, by Age: 2002 Source: U.S. Bureau of the Census, "Poverty in the United States: 2002," Current Population Reports, Series P60-222, available online at http://www.census.gov/prod/2003pubs/p60-222.pdf, and unpublished CPS data from the U.S. Census Bureau. Three experimental measures of poverty (developed by the Census Bureau in response to the recommendation of a 1995 panel of the National Academy of Sciences) yield poverty rates that are similar to the official poverty measure overall, but differ by age and other characteristics. Experimental measures generally show lower poverty rates among children than the official measure, partly because they take into account non-cash benefits that many children receive. Conversely, experimental measures show higher rates of poverty among the elderly than the official measure, in part due to the inclusion of certain out-of-pocket health costs in these measures. All three alternative measures shown in Figure Econ 3 take into account geographic adjustments (GA) in housing costs; the measures can also be calculated with no geographic adjustment (NGA), as shown in Tables ECON 3a and 3b. See note to Table ECON 3a. Table ECON 3a. Percentage of Persons in Poverty Using Various Experimental Poverty Measures, by Race/Ethnicity and Age: 2002 Alt1 MSI-NGA Alt2 MIT-NGA Alt3 CMB-NGA Alt1 MSI-GA Alt2 MIT-GA Alt3 CMB-GA All Persons 12.1 12.4 13.0 13.0 12.3 12.8 12.9 Non-Hispanic White 8.0 8.9 9.2 9.4 8.4 8.5 8.8 Non-Hispanic Black 24.1 21.2 22.2 22.3 20.6 21.1 21.3 Hispanic 21.8 21.09 22.7 22.2 23.3 25.4 24.8 Children Ages 0-17 16.7 13.8 15.3 14.7 13.9 15.2 14.6 Adults Ages 18-64 10.6 10.8 11.6 11.3 10.8 11.5 11.3 Adults Age 65 and over 10.4 16.7 14.4 17.6 16.0 13.4 16.9 Note: These experimental poverty measures implement changes recommended by a 1995 NAS panel, including: counting non-cash income as benefits; subtracting from income certain work-related, health, and child care expenses; and adjusting poverty thresholds for family size and geographic differences in housing costs. The three alternative measures are similar, except that each account for out-of-pocket medical expenses differently. For the first alternative ("MOOP subtracted from income" or MSI), medical out-of-pocket expenses (MOOP) are subtracted from income. The second alternative, ("MOOP in the threshold" or MIT) increases the poverty thresholds to take MOOP expenses into account. The third measure, CMB for combined methods, combines attributes of the previous two measures. Each of the three measures is calculated with and without accounting for geographic adjustments (GA and NGA). These experimental measures are different from those reported in last year's report because the Census Bureau changed its methodology based on research conducted to refine the NAS panel's experimental methods. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Non-Hispanic Whites and Non-Hispanic Blacks are for persons reporting a single-race only. Persons who reported more than one race, such as "White and Asian," are included in the total for all persons but are not shown under any race category. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders also are included in the total for all persons but are Source: U.S. Census Bureau, "Poverty in the United States: 2002," Current Population Reports, Series P60-222, available at http://www.census.gov/prod/2003pubs/p60-222.pdf, and unpublished CPS data from the U.S. Census Bureau. Table ECON 3b. Percentage of Persons in Poverty Using Various Experimental Poverty Measures 1999-2002 Official Measure 11.9 11.3 11.7 12.1 No Geographic Adjustment of Thresholds Medical costs alternative 1 (MSI-NGA) 12.2 12.1 12.4 12.4 Medical costs alternative 2 (MIT-NGA) 12.8 12.7 12.8 13.0 Medical costs alternative 3 (CMB-NGA) 12.9 12.8 13.0 13.0 Geographic Adjustment of Thresholds Medical costs alternative 1 (MSI-GA) 12.1 12.0 12.3 12.3 Medical costs alternative 2 (MIT-GA) 12.7 12.5 12.7 12.8 Medical costs alternative 3 (CMB-GA) 12.8 12.6 12.9 12.9 Economic Security Risk Factor 4. Poverty Rates with Various Means-tested Benefits Included Figure ECON 4. Percentage of Total Population in Poverty with Various Means-Tested Benefits Added to Total Cash Income: 1979-2002 Source: Congressional Budget Office tabulations of March CPS data. Additional calculations by U.S. Department of Health and Human Services. The official definition of poverty – which includes means-tested cash assistance (primarily TANF and SSI) in addition to pre-tax cash income and social insurance – was 12.1 percent in 2002, as shown in the bold line with empty boxes in Figure ECON 4. Without cash welfare, the 2002 poverty rate would be 12.8 percent, as shown by the top line in the figure above. Adding other non-cash, public assistance benefits to this definition has the effect of lowering the percentage of people who have incomes below the official poverty rate. Adding in the value of food and housing benefits reduces the poverty rate to 10.9 percent in 2002. When income is defined as including benefits from the Earned Income Tax Credit (EITC) and federal taxes, the percentage of the total population in poverty decreases to 10.0 percent in 2002. Taxes have had a net effect of reducing poverty rates since the significant increases in the size of the EITC in 1993 and 1995. The combined effect of means-tested cash assistance, food and housing benefits, EITC and taxes was to reduce the poverty rate in 2002 by 2.8 percentage points, as shown in Table ECON 4. Net reductions in poverty rates were somewhat lower during the recession of the early 1980s, and somewhat higher in the mid-1990s, largely due to expansions in the EITC. Table ECON 4. Percentage of Total Population in Poverty with Various Means-Tested Benefits Added to Total Cash Income: Selected Years Cash Income Plus All Social Insurance 12.8 16.0 14.5 13.8 15.6 14.9 13.5 12.0 12.8 Plus Means-Tested Cash Assistance 11.6 15.2 13.6 12.8 14.5 13.8 12.7 11.3 12.1 Plus Food and Housing Benefits 9.7 13.7 12.2 11.2 12.9 12.0 11.3 10.1 10.9 Plus EITC and Federal Taxes 10.0 14.7 13.1 11.8 13.0 11.5 10.4 9.5 10.0 Reduction in Poverty Rate 2.8 1.3 1.4 2.0 2.6 3.4 3.1 2.5 2.8 Note: The four measures of income are as follows: 1) "Cash Income plus All Social Insurance" is earnings and other private cash income, plus social security, workers' compensation, and other social insurance programs. It does not include means-tested cash transfers; (2) "Plus Means-Tested Assistance" shows the official poverty rate, which takes into account means-tested assistance, primarily AFDC/TANF and SSI; (3) "Plus Food and Housing Benefits" shows how poverty would be lower if the cash value of food and housing benefits were counted as income; and (4); "Plus EITC and Federal Taxes" is the most comprehensive poverty rate shown. EITC refers to the refundable Earned Income Tax Credit, which is always a positive adjustment to income whereas Federal payroll and income taxes are a negative adjustment. The fungible value of Medicare and Medicaid is not included. Economic Security Risk Factor 5. Poverty Spells Figure ECON 5. Percentage of Poverty Spells for Individuals Entering Poverty During the 1993 and 1996 SIPP Panels, by Length of Spell About half of all poverty spells that began during the 1996 SIPP panel ended within four months, and 80 percent ended within one year. Only 11 percent of all such spells were longer than 20 months. Spells of poverty that began between 1993 and 1995 were slightly longer; 47 percent ended within four months and 16 percent were longer than 20 months. Poverty spells among adults age 65 and older were more likely to last longer than 20 months (17 percent) than spells among other age groups, as shown in Table ECON 5a. Table ECON 5a. Percentage of Poverty Spells for Individuals Entering Poverty During the 1996 SIPP Panel, by Length of Spell, Race/Ethnicity, and Age All Persons 51.3 29.0 8.3 11.4 Racial/Ethnic Categories 51.3 29.0 8.3 11.4 Non-Hispanic White 54.6 28.1 7.6 9.7 Ages 0 to 5 Years 46.8 29.6 10.8 12.9 Ages 11 to 15 Years 49.5 30.9 7.9 11.7 Women Ages 16-64 years 50.7 29.3 8.5 11.5 Men Ages 16-64 Years 55.7 28.9 7.0 8.4 Adults Age 65 Years and Older 51.1 23.8 7.7 17.4 Note: Spell length categories are not mutually exclusive. Spells separated by only 1 month are not considered separate spells. Due to the length of the observation period, actual spell lengths for spells that lasted more than 20 months cannot be observed. Table ECON 5b Percentage of Poverty Spells for Individuals Entering Poverty During the 1993 and 1996 SIPP Panels, by Length of Spell and Year Economic Security Risk Factor 6. Child SUPPORT Figure ECON 6. Total, Non-AFDC/TANF, and AFDC/TANF Title IV-D Child Support Collections: 1978-2002 Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Child Support Enforcement, Child Support Collections: 2003 TANF Report to Congress (and earlier years), Washington, DC. Collections paid through the Child Support Enforcement system (Title IV-D of the Social Security Act) totaled $20.1 billion in 2002, over $1 billion more than in 2001. Since 1990, child support collections grew rapidly, at an average rate of almost $1.1 billion a year. In recent years, non-TANF collections have generally increased as a percentage of overall collections by the IV-D program. (Non-TANF collections include collections paid to former TANF families and families with no contact with the welfare system.) However, between 2001 and 2002, the $878 million growth in non-TANF collections was smaller in percentage terms than $302 million growth in TANF collections (5 percent compared to over 11 percent). A number of states have opted to pass through some or all of collections to the custodial TANF family, even though the 1996 welfare reform repealed the former requirement for a $50 "pass-through" to families. In recent years, the amount of TANF collections paid to TANF families has been difficult to track because of changes in data reporting forms. Available data suggest these payments declined in fiscal years 1997-2000, with a 100 percent increase shown in fiscal year 2001 and a 122 percent increase in 2002, as shown in Table ECON 6. Almost 75 percent of TANF collections (collections on behalf of TANF recipients and for past due support assigned to the state by former TANF recipients) were retained in 2002 to reimburse the state and federal governments for the cost of welfare benefits. Table ECON 6. Total, Non-AFDC/TANF, and AFDC/TANF Title IV-D Child Support Collections: 1978-2002 Total Collections (in millions) Total IV-D Administrative Expenditures AFDC/TANF Collections Non-AFDC/TANF Collections Current Dollars Constant '02 Dollars Payments to AFDC/TANF Families Federal & State Share of Collections 1978 $1,047 $2,829 $472 $13 $459 $575 $312 1979 1,333 3,307 597 12 584 736 383 1983 2,024 3,676 880 15 865 1,144 691 1984 2,378 4,138 1,000 17 983 1,378 723 1985 2,694 4,520 1,090 189 901 1,604 814 1987 3,917 6,235 1,349 278 1,070 2,569 1,066 1992 7,964 10,228 2,259 435 1,824 5,705 1,995 1995 10,827 12,794 2,689 474 2,215 8,138 3,012 1997 13,364 14,961 2,843 157 2,685 10,521 3,428 Note: Not all states report current child support collections in all years. Constant dollar adjustments to the 2000 level were made using a CPI-U-X1 fiscal year average price index. Due to changes in data reporting forms, data for fiscal years 1999 and thereafter relating to the Federal and State Share of TANF collections include assistance reimbursement for former TANF families and may not be exactly comparable to that of previous years. The total collection of payments to AFDC/TANF families can also include payments made to Medicaid only recipients. Economic Security Risk Factor 7. Food Insecurity Figure ECON 7. Percentage of Households Classified by Food Security Status: 2002 Source: U.S. Department of Agriculture, Economic Research Service, Household Food Security in the United States, 2002. A large majority (89 percent) of American households was food secure in 2002 – that is, showed little or no evidence of concern about food supply or reduction in food intake. The prevalence of food insecurity with hunger in 2002 was estimated to be 3.5 percent. During the twelve months ending in December 2002, one or more members of these households experienced reduced food intake and hunger as a result of financial constraints. Food insecurity would be lower measured over a monthly basis. An additional 7.6 percent of households experienced food insecurity, but were without hunger, during the twelve months ending in December 2002. Although these households showed signs of food insecurity in their concerns and in adjustments to household food management, little or no reduction in food intake was reported. Poor households have a higher rate of food insecurity with hunger (14.3 percent) than the 3.5 percent rate among the general population, as shown in Table ECON 7a. Only 1.5 percent of families with incomes at or above 185 percent of the poverty level showed evidence of food insecurity with hunger. Table ECON 7a. Percentage of Households Classified by Food Security Status and Selected Characteristics: 2002 Food Secure Food Insecure Total Food Insecure Without Hunger With Hunger All Households 88.9 11.1 7.6 3.5 Non-Hispanic White 92.0 8.0 5.3 2.6 Non-Hispanic Black 78.0 22.0 14.8 7.2 Hispanic 78.3 21.7 16.0 5.7 Households, by Age Households with Children Under 6 82.2 17.8 14.4 3.4 Households with Children Under 18 83.5 16.5 12.7 3.8 Households with Elderly 93.7 6.3 4.4 1.9 Household Income-to-Poverty Ratio Under 1.00 61.9 38.1 23.8 14.3 Under 1.85 70.8 29.2 19.5 9.7 1.85 and over 94.9 5.1 3.6 1.5 Note: Food secure households show little or no evidence of concern about food supply or reduction in food intake. Households classified as food insecure without hunger report food-related concerns, adjustments to household food management, and reduced variety and desirability of diet, but report little or no reduction in food intake. Households classified as food insecure with hunger report recurring reductions in food intake or hunger by one or more persons in the household Table ECON 7b. Percentage of Households Classified by Food Security Status: 1998-2002 Food Insecure Without Hunger Food Insecure With Hunger 1998 88.2 11.8 8.1 3.7 Economic Security Risk Factor 8. Lack of Health Insurance Figure ECON 8. Percentage of Persons without Health Insurance, by Income: 2002 Source: U.S. Bureau of the Census, "Health Insurance Coverage in the United States: 2002," Current Population Reports, Series P60-223 (March 2003 Current Population Survey). Online: Available at http://www.census.gov/prod/2003pubs/p60-223.pdf Poor persons were twice as likely as all persons to be without health insurance in 2002 (30 percent compared to 15 percent). While the ratio varied across categories, persons with family income at or below the poverty line were more likely to be without health insurance regardless of race/ethnicity, gender, educational attainment, or age. Hispanics were the ethnic group least likely to have health insurance in 2002, among both the general population and those with incomes below the poverty line. While white individuals in general were more likely to have insurance than black individuals, poor black individuals were more likely to have insurance than poor white individuals. Among all persons, the amount of education was inversely related to health insurance coverage. However, among poor persons, educational attainment made little difference as to whether individuals had health insurance. As shown in Table ECON 8, nearly half of poor people ages 25 to 34 are without health insurance. Among the general population, individuals ages 18 to 24 are the most likely to be without health insurance. Table ECON 8. Percentage of Persons without Health Insurance, by Income and Selected Characteristics: 2002 All Persons 15.2 30.4 Male 16.7 33.3 Female 13.9 28.1 White 14.2 31.4 Black 20.2 26.4 Hispanic 32.4 42.8 No High School Diploma 28.0 37.9 High School Graduate, No College 18.8 36.4 College Graduate 8.4 32.3 Age 18 and under 11.6 20.1 Ages 18-24 29.6 43.9 Age 65 and over 0.8 1.9 Note: "Poor persons" are defined as those with total family incomes at or below the poverty rate. Race categories include those of Hispanic ethnicity. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Whites and Blacks are for persons reporting a single-race only. Persons who reported more than one race, such as "White and Asian," are included in the total for all persons but are not shown under any race category. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders also are included in the total for all persons but are not shown separately. Employment and Work-Related Risk Factors Employment and Work-related Risk Factor 1. Labor Force Attachment Figure WORK 1. Percentage of Individuals in Families with Labor Force Participants, by Race/Ethnicity: 2002 Source: Unpublished tabulations of March CPS data. In 2002, 71 percent of the total population lived in families with at least one person working on a full-time, full-year basis, as shown in Table WORK 1a. The percent of full-time full-year workers was slightly lower than in 2001, although still higher than during most of the 1990s, as shown in Table WORK 1b. Overall, 14 percent of the population lived in families with no labor force participants and 15 percent lived in families with part-time and/or part-year labor force participants in 2002. Persons of Hispanic origin were less likely than non-Hispanic whites or non-Hispanic blacks to live in families with no one in the labor force in 2002 (10 percent compared to 15 and 17 percent, respectively). Working-age women in 2002 were more likely than working-age men to live in families with no one in the labor force (9 percent compared to 7 percent), as shown in Table Work 1a. Men were more likely than women to live in families with at least one full-time, full-year worker (80 percent compared to 76 percent). Table WORK 1a. Percentage of Individuals in Families with Labor Force Participants, by Race/Ethnicity and Age: 2002 During Year At Least One in LF No One FT/FY At Least One FT/FY Worker All Persons 14.2 14.7 71.1 Hispanic 9.7 15.4 74.9 Children Ages 0-5 5.5 16.2 78.3 Children Ages 6-10 5.9 14.8 79.3 Children Ages 11-15 5.9 13.6 80.5 Women Ages 16-64 8.8 15.5 75.7 Men Ages 16-64 6.8 13.7 79.5 Note: Full-time, full-year workers are defined as those who usually worked for 35 or more hours per week, for at least 50 weeks in a given year. Part-time and part-year labor force participation includes part-time workers and individuals who are unemployed, laid off, and/or looking for work for part or all of the year. This indicator represents annual measures of labor force participation, and thus cannot be compared to monthly measures of labor force participation in Indicator 2. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Whites and Blacks are for persons reporting a single-race only. Persons who reported more than one race, such as "White and Asian," are included in the total for all persons but are not shown under any race category. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders also are included in the total for all persons but are not shown separately. Table WORK 1b. Percentage of Individuals in Families with Labor Force Participants: 1990-2002 Employment and Work-related Risk Factor 2. Employment Among the Low-skilled Figure WORK 2. Percentage of All Persons Ages 18 to 65 with No More than a High School Education Who Were Employed: 1969-2002 Source: ASPE tabulations of March CPS data. Employment rates for women with a high school education or less continued to drop in 2002, following several years of rising employment, particularly among non-Hispanic black and Hispanic women. Low-skilled non-Hispanic white women continued to have the highest employment level (70 percent in 2002) among the three racial/ethnic groups. Employment levels for non-Hispanic white and Hispanic men with no more than a high school education have remained close to 85 percent for nearly to two decades. In contrast, employment levels for low-skilled non-Hispanic black men have varied over the same period. Between 1968 and 1983, employment rates for non-Hispanic black men with no more than high school education fell by 20 percentage points. Since 2000, these rates have fallen by more than 5 percentage points. As shown in Figure and Table WORK 2, employment levels for non-Hispanic black men with a high school education or less were 3 percentage points higher than those of similarly educated non-Hispanic black women in 2002. In contrast, there was a 13 percentage point difference in employment levels of non-Hispanic white men and women with a high school education or less, and a 28 percentage point difference between similarly educated Hispanic men and women. Table WORK 2. Percentage of All Persons Ages 18 to 65 with No More than a High School Education Who Were Employed: 1969-2002 1968 92.8 89.9 N/A 55.8 65.8 N/A 1975 88.2 78.8 86.2 58.3 57.2 49.7 Note: All data include both full and partial year employment for the given calendar year. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Whites and Blacks are for persons reporting a single-race only. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders are included in the total for all persons but are not shown separately. Hispanic origin was not available until 1975. Employment and Work-related Risk Factor 3. Earnings of Low-skilled Workers Figure WORK 3. Mean Weekly Wages of Women and Men Working Full-Time, Full-Year with No More than a High School Education, by Race (2002 Dollars): Selected Years Women's average weekly wages were lower than those of low-skilled men, across all race groups. In 2002, non-Hispanic white women had the highest average weekly wages among low-skilled women working full-time, full-year ($529). This level is a 15 percent increase over non-Hispanic white women's 1980 average weekly wages ($459 inflation adjusted). Non-Hispanic black women and Hispanic women's weekly wages increased at slower rate than non-Hispanic white women since 1980 (12 percent and 3 percent, respectively). For men, the gap between mean weekly wages for non-Hispanic white and non-Hispanic black men with low education levels has narrowed over time. In 1980, the mean weekly wage for low-skilled non-Hispanic black men working full-time was $564 (in 2002 dollars), or 74 percent of the $758 average for non-Hispanic white men. However, full-time working non-Hispanic black men with no more than a high school education received 78 percent of the mean weekly wages of non-Hispanic white men in 2002 ($578 compared to $745). Over the past fifteen years, both Hispanic women and men's wages have lagged behind non-Hispanic whites and blacks among low-skilled full-time workers. In 2002, Hispanic women's wages were 24 percent lower than non-Hispanic white women and 14 percent lower than non-Hispanic black women. Hispanic men had higher weekly wages than women but still trailed non-Hispanic white men by 29 percent and non-Hispanic black men by 8 percent. Table WORK 3. Mean Weekly Wages of Women and Men Working Full-Time, Full-Year with No More than a High School Education, by Race (2002 Dollars): Selected Years 1980 459 419 392 758 564 572 Note: Full-time, full-year workers work at least 48 weeks per year and 35 hours per week. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Whites and Blacks are for persons reporting a single-race only. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders are included in the total for all persons but are not shown separately. Employment and Work-related Risk Factor 4. Educational Attainment Figure WORK 4. Percentage of Adults Age 25 and Over, by Level of Educational Attainment: 1960-2002 Source: U.S. Bureau of the Census, "Educational Attainment in the United States: March 2002," Current Population Reports, Series PPL-169, March 2003, and earlier reports. There has been a marked decline over the past 40 years in the percentage of the population that has not received a high school education. This percentage fell from 59 percent in 1960 to 16 percent in 2002. The percentage of the population receiving a high school education only (with no subsequent college) was 25 percent in 1960 and rose to 39 percent in 1988. Since then this figure has fallen to 32 percent in 2002, although some of this decline is a result of a change in the survey methodology in 1992 (see note to Table WORK 4). Between 1960 and 1990, the percentage of the population with some college (one to three years) doubled, from 9 percent to 18 percent. The apparent jump in 1992 is a result of a change in the survey methodology (see note to Table WORK 4), but the trend continued upward, reaching 25 percent in 2002. The percentage of the population completing four or more years of college has more than tripled from 1960 to 2002, rising steadily from 8 percent to 27 percent. Table WORK 4. Percentage of Adults Age 25 and Over, by Level of Educational Attainment Selected Years Not a High School Graduate Finished High School, No College One to Three Years of College Four or More Years of College 1940 76 14 5 5 Note: Completing the GED is not considered completing high school for this table. Beginning with data for 1992, a new survey question results in different categories than for prior years. Data shown as Finished High School, No College were previously from the category "High School, 4 Years" and are now from the category "High School Graduate." Data shown as One to Three Years of College were previously from the category "College 1 to 3 Years" and are now the sum of the categories: "Some College" and two separate "Associate Degree" categories. Data shown as Four or More Years of College were previously from the category "College 4 Years or More," and are now the sum of the categories: "Bachelor's Degree," "Master's Degree," "Doctorate Degree," and "Professional Degree." Employment and Work-related Risk Factor 5. High-school Dropout Rates Figure WORK 5. Percentage of Students Enrolled in Grades 10 to 12 in the Previous Year Who Were Not Enrolled and Had Not Graduated in the Survey Year, by Race/Ethnicity: Selected Years Source: U.S. Department of Education, National Center for Education Statistics, Dropout Rates in the United States: 2000 and earlier years (based on Current Population Survey data from the October supplement). With the exception of a small upward movement in 1988, the dropout rates for teens in grades 10 to 12 declined steadily from 1979 to 1991. From a low of 4.0 percent, the rate began rising to a peak of 5.7 percent in 1995. Following this upturn, the overall rate again declined to 4.6 percent in 1997; since then it has fluctuated, moving up to 5.0 percent in 1999 and then back down again to 4.8 percent in 2000. Dropout rates among Hispanic and non-Hispanic black teens have fluctuated considerably over this period. Still, dropout rates are generally highest for Hispanic teens and lowest for non-Hispanic white teens. In 2000, the dropout rate was 7.4 percent for Hispanic teens, compared to 6.1 percent for non-Hispanic black teens and 4.1 percent for non-Hispanic white teens. Table WORK 5. Percentage of Students Enrolled in Grades 10 to 12 in the Previous Year Who Were Not Enrolled and Had Not Graduated in the Survey Year, by Race/Ethnicity: Selected Years 1974 6.7 5.8 11.6 9.9 1976 5.9 5.6 7.4 7.3 1978 6.7 5.8 10.2 12.3 Note: Beginning in 1987, the Bureau of the Census instituted new editing procedures for cases with missing data on school enrollment. Beginning in 1992, the data reflect new wording of the educational attainment item in the CPS. Persons of Hispanic ethnicity may be of any race. Due to small sample size, American Indians/Alaska Natives and Asian/Pacific Islanders are included in the total but are not shown separately. Employment and Work-related Risk Factor 6. Adult Alcohol and Substance Abuse Figure WORK 6. Percentage of Adults Who Used Cocaine or Marijuana or Abused Alcohol, by Age: 2002 Source: U.S. Department of Health and Human Services, Substance Abuse and Mental Health Services Administration, 2002 National Survey on Drug Use and Health. In 2002, young adults (ages 18 to 25) were more likely than older adults to report alcohol abuse, marijuana use, or cocaine use in the past month. More than one in six (17 percent) of adults 18 to 25 reported using marijuana in the past month during 2002, compared with 8 percent of adults 26 to 34 and 3 percent of adults 35 and older. Young adults were also significantly more likely to abuse alcohol than older adults. The percentage of persons reporting binge alcohol use was significantly larger than the percentages for all other reported behaviors across all age groups, as shown in Table WORK 6. Among all adult age categories, the use of cocaine, marijuana and alcohol abuse increased in 2002 to the highest level in 4 years, as shown in Table Work 6. Table WORK 6. Percentage of Adults Who Used Cocaine or Marijuana or Abused Alcohol, by Age: 1999 - 2002 Ages 18-25 1.7 1.4 1.9 2.0 Age 35 and Over 0.4 0.3 0.5 0.6 Binge Alcohol Use Age 35 and Over 16.0 16.4 16.2 18.6 Heavy Alcohol Use Note: Cocaine and marijuana use is defined as use during the past month. "Binge Alcohol Use" is defined as drinking five or more drinks on the same occasion on at least one day in the past 30 days. "Occasion" means at the same time or within a couple hours of each other. "Heavy Alcohol Use" is defined as drinking five or more drinks on the same occasion on each of five or more days in the past 30 days; all Heavy Alcohol Users are also Binge Alcohol Users. Employment and Work-related Risk Factor 7. Adult and Child Disability Figure WORK 7. Percentage of the Non-Elderly Population Reporting a Disability, by Age and Race/Ethnicity: 2002 Source: Centers for Disease Control and Prevention, National Center for Health Statistics, National Health Interview Survey. In 2002, non-elderly adults were more likely than children to have an activity limitation, 11.4 percent compared to 7.5 percent. While non-elderly adults were more likely than children to report an activity limitation, a higher percentage of children than adults were actually recipients of disability program benefits in 2002 (6.2 percent compared to 4.6 percent), as shown in Table WORK 7. Among both non-elderly adults and children, rates of activity limitation were somewhat similar for non-Hispanic whites and non-Hispanic blacks in 2002, but lower for Hispanics, as shown in Table WORK 7. Table WORK 7. Percentage of the Non-Elderly Population Reporting a Disability, by Race/Ethnicity and Age: 2002 Activity Limitation Long-Term Care Needs Disability Program Recipient Adults Ages 18-64 11.4 8.5 2.1 4.6 Children Ages 0-17 7.5 NA NA 6.2 Racial/Ethnic Categories (Adults Ages 18-64) Non-Hispanic Black 13.7 10.2 2.9 7.7 Hispanic 7.9 5.8 1.6 3.8 Racial/Ethnic Categories (Children Ages 0-17) Non-Hispanic White 7.7 NA NA 6.4 Non-Hispanic Black 9.4 NA NA 7.9 Hispanic 5.9 NA NA 5.0 Note: Respondents were defined as having an activity limitation if they answered positively to any of the questions regarding: (1) work disability (see definition below); (2) long-term care needs (see definition below); (3) difficulty walking; (4) difficulty remembering; (5) for children under 5, limitations in the amount of play activities they can participate in because of physical, mental, or emotional problems; (6) for children 3 and over, receipt of Special Educational or Early Intervention Services; and, (7) any other limitations due to physical, mental, or emotional problems. Work disability is defined as limitations in or the inability to work as a result of a physical, mental or emotional health condition. Individuals are identified as having long-term care needs if they need the help of others in handling either personal care needs (eating, bathing, dressing, getting around the home) or routine needs (household chores, shopping, getting around for business or other purposes). Disability program recipients include persons covered by Supplemental Security Income (SSI), Social Security Disability Insurance (SSDI), Special Education Services, Early Intervention Services, and/or disability pensions. Non-Marital Birth Risk Factors Non-marital Birth Risk Factor 1. Births to Unmarried Women Figure BIRTH 1. Percentage of Births to Unmarried Women, by Age Group: 1940-2002 Source: National Center for Health Statistics, "Nonmarital Childbearing in the United States, 1940 - 1999," National Vital Health Statistics Reports, Vol. 48 (16), 2000; "Births: Final Data for 2002," National Vital Statistics Reports, Vol. 52 (10), December 2003. The percentage of children born outside of marriage to women of all ages has increased over the past six decades, from 3.8 percent in 1940 to 34.0 percent in 2002. This increase reflects changes in several factors: the rate at which unmarried women have children, the rate at which married women have children, and the rate at which women marry. The percentage of children born outside of marriage is especially high among teen women and women ages 20-24. Four-fifths (80 percent) of all births to teens and a little over half (52 percent) to women ages 20-24 took place outside of marriage in 2002. Since 1994, the upward growth in percentage of unmarried births to all women has begun to level off. The growth in percentage of unmarried births to teen mothers also has slowed since 1994, although it is still rising (from 76 percent in 1994 to 80 percent in 2002). The steepest growth since 1994 is among the 20 to 24 year old age group, where the percentage of births to unmarried women has increased from 45 to 52 percent. Recently, the percentage of out-of-wedlock births has leveled off among black teens and all black women. Among white teens and all white women, the trend continues upward (see Table C-1 in Appendix C for non-marital birth data by age and race). Table BIRTH 1. Percentage of Births to Unmarried Women, by Age Group: Selected Years All Teens 1940 64.5 N/A N/A 14.0 3.4 3.8 1950 63.7 22.6 9.4 13.9 3.7 4.0 1955 66.3 23.2 10.3 14.9 4.3 4.5 1970 80.8 43.0 22.4 30.5 8.9 10.7 Note: Trends in non-marital births may be affected by changes in the reporting of marital status on birth certificates and in procedures for inferring non-marital births when marital status is not reported. Non-marital Birth Risk Factor 2. Births to Unmarried Teens Figure BIRTH 2. Percentage of All Births to Unmarried Teens Ages 15 to 19, by Race and Ethnicity 1940-2002 In contrast to the earlier Figure BIRTH 1, which showed births to unmarried teens as a percentage of all teen births, Figure BIRTH 2 shows births to unmarried teens as a percentage of births to all women. This percentage fell in the last four years, from 9.7 to 8.5 percent, reversing a long upward trend since 1940. This rate may be affected by several factors: the age distribution of women, the marriage rate among teens, the birth rate among unmarried teens, and the birth rate among all other women. The percentage of all births that were to unmarried teens has also dropped among white women over the past four years, declining to 7.2 percent in 2002. This drop is in contrast to the long upward trend, from less than 1 percent in 1960 to nearly 8 percent in 1998. Among black women, the percentage of all births that were to unmarried teens fell to 16.7 percent in 2002, the lowest percentage since 1969. This rate has varied greatly since 1940, rising sharply to a peak of 24 percent in 1975, and showing a gradual decline in most years since then. The sharp increase in the late 1960s and early 1970s reflects a 30 percent rise in non-marital teen births among black women concurrent with a 6 percent decline in total black births from 1969 to 1975. Table BIRTH 2. Percentage of All Births to Unmarried Teens Ages 15 to 19, by Race and Ethnicity: Selected Years 1940 1.7 0.8 N/A N/A 1970 5.1 2.6 18.8 N/A Note: Trends in non-marital births may be affected by changes in the reporting of marital status on birth certificates and in procedures for inferring non-marital births when marital status is not reported. Beginning in 1980, data are tabulated by the race of the mother. Prior to 1980, data are tabulated by the race of the child. Race categories include those of Hispanic ethnicity. Persons of Hispanic ethnicity may be of any race. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders are included in the total for all persons but are not shown separately. Non-marital Birth Risk Factor 3. Unmarried Teen Birth Rates Within Age Groups Figure BIRTH 3a. Births per 1,000 Unmarried Teens Ages 15 to 17, by Race: 1960-2002 Figure BIRTH 3b. Births per 1,000 Unmarried Teens Ages 18 and 19, by Race: 1960-2002 Source: National Center for Health Statistics, "Nonmarital Childbearing in the United States, 1940 - 1999," National Vital Statistics Reports, Vol. 48 (16), 2000; "Births: Final Data for 2002," National Vital Statistics Reports, Vol. 52 (10), December 2003. The birth rate per 1,000 unmarried teens fell again in 2002 for both black and white teens and for both younger (15 to 17 years) and older age groups (18 and 19 years). The rate for black teens ages 18 and 19, for example, fell from 140 per thousand in 1994 to 104 per thousand in 2002. Declines were larger among black teens than among white teens. Prior to 1994, birth rates among unmarried white teens in both age groups rose steadily for nearly three decades (from 4 to 24 percent among 15 to 17 year-olds and from 11 to 56 percent among 18 and 19 year-olds). The birth rate among unmarried black teens in both age groups was lower in 2002 than it has been in over four decades. While birth rates among unmarried black teens remain high compared to rates for unmarried white teens, the gap been black and white teens narrowed considerably during the 1990s. Table BIRTH 3. Births per 1,000 Unmarried Teen Women within Age Groups, by Race: 1950-2002 Ages 15 to17 Ages 18 and 19 1950 9.9 3.4 N/A 18.3 8.5 N/A 1955 11.1 3.9 N/A 23.6 10.3 N/A 1969 15.2 6.6 72.0 30.8 16.6 128.4 1977 19.8 10.5 73.0 34.6 18.7 121.7 Note: Rates are per 1,000 unmarried women in specified group. Trends in non-marital births may be affected by changes in the reporting of marital status on birth certificates and in procedures for inferring non-marital births when marital status is not reported. Beginning in 1980, data are tabulated by the race of the mother. Prior to 1980, data are tabulated by the race of the child. Rates for 1990-1999 have been revised on the basis of intercensal population estimates benchmarked to the 2000 decennial census and differ from earlier editions of this report. Race categories include those of Hispanic ethnicity. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders are included in the total for all persons but are not shown separately. Non-marital Birth Risk Factor 4. Never-married Family Status Figure BIRTH 4. Percentage of All Children Living in Families with a Never-Married Female Head, by Race/Ethnicity: 1982-2003 Source of CPS data: U.S. Bureau of the Census, "Marital Status and Living Arrangements," Current Population Reports, Series P20-212, 287, 365, 380, 399, 418, 423, 433, 445, 450, 461, 468, 478, 484, 491, 496, 506, 514, 537 various years, and ASPE tabulations of the CPS for 2003. Source of 1960 data: U.S. Bureau of the Census, 1960 Census of Population, PC(2)-4B, "Persons by Family Characteristics," Tables 1 and 19. The percentage of children living in families with never-married female heads increased from under 5 percent in 1982 to 10 percent in 2003. The percentage of white children living in families headed by never-married women has continued to rise over the past twenty years, from less than 2 percent in 1982 to 5.6 percent in 2003. Among Hispanics, the percentage of children living with never-married female heads more than doubled over the past twenty years, going from less than 6 percent in 1982 to 12 percent in 1996. Since then it has fluctuated up and down by about one-half a percentage point. The percentage of black children living in families headed by never-married women was much higher than the percentages for other groups throughout the time period. However, at 33 percent in 2003 it is two percentage points below its peak in 1999. Table BIRTH 4. Number and Percentage of All Children Living in Families with a Never-Married Female Head, by Race/Ethnicity: Selected Years Number of Children (in thousands) 1960 221 49 173 – 0.4 0.1 2.2 – 1970 527 110 442 – 0.8 0.2 5.2 – 1975 1,166 296 864 – 1.8 0.5 9.9 – 1980 1,745 501 1,193 210 2.9 1.0 14.5 4.0 1986 3,606 1,174 2,375 451 5.9 2.3 26.6 7.2 1992 5,410 2,016 3,192 757 8.4 3.9 33.1 10.3 1994 6,000 2,412 3,321 1,083 9.0 4.5 32.9 12.0 2003 7,008 3,028 3,454 1,497 10.0 5.6 33.3 11.9 Note: Data are for all children under 18 who are not family heads (excludes householders, subfamily reference persons, and their spouses). Also excludes inmates of institutions; children who are living with neither of their parents are excluded from the denominator. Based on Current Population Survey (CPS) except 1960, 1970, and 1980, which are based on decennial census data. In 1982, improved data collection and processing procedures helped to identify parent-child subfamilies. (See Current Population Reports, P-20, 399, Marital Status and Living Arrangements: March 1984.) Race categories include those of Hispanic ethnicity. Persons of Hispanic ethnicity may be of any race. Beginning in 2002, estimates for Whites and Blacks are for persons reporting a single-race only. Persons who reported more than one race, such as "White and Asian," are included in the total for all persons but are not shown under any race category. Due to small sample size, American Indians/Alaska Natives, Asians and Native Hawaiians/Other Pacific Islanders also are included in the total for all persons but are not shown separately. Nonwhite data are shown for Black in 1960. Source of CPS data: U.S. Bureau of the Census, "Marital Status and Living Arrangements," Current Population Reports, Series P20-212, 287, 365, 380, 399, 418, 423, 433, 445, 450, 461, 468, 478, 484, 491, 496, 506, 514, 537, various years, and ASPE tabulations of the CPS for 2003. The Welfare Indicators Act of 1994 specifies that the annual welfare indicators reports shall include analyses of families and individuals receiving assistance under three means-tested benefit programs: the Aid to Families with Dependent Children (AFDC) program authorized under part A of title IV of the Social Security Act (replaced with the Temporary Assistance for Needy Families (TANF) program by the Personal Responsibility and Work Opportunity Reconciliation Act of 1996), the Food Stamp Program under the Food Stamp Act of 1977, as amended, and the Supplemental Security Income (SSI) program under title XVI of the Social Security Act. This chapter includes information on these three programs, derived primarily from administrative data reported by state and federal agencies instead of the national survey data presented in previous chapters. National caseloads and expenditure trend information on each of the three programs is included, as well as state-by-state trend tables and information on the characteristics of program participants. Aid to Families with Dependent Children (AFDC) and Temporary Assistance for Needy Families (TANF) Aid to Families with Dependent Children (AFDC) was established by the Social Security Act of 1935 as a grant program to enable states to provide cash welfare payments for needy children who had been deprived of parental support or care because their father or mother was absent from the home, incapacitated, deceased, or unemployed. All 50 states, the District of Columbia, Guam, Puerto Rico, and the Virgin Islands operated an AFDC program. States defined "need," set their own benefit levels, established (within federal limitations) income and resource limits, and administered the program or supervised its administration. States were entitled to unlimited federal funds for reimbursement of benefit payments, at "matching" rates that were inversely related to state per capita income. States were required to provide aid to all persons who were in classes eligible under federal law and whose income and resources were within state-set limits. During the 1990s, the federal government increasingly used its authority under section 1115 of the Social Security Act to waive portions of the federal requirements under AFDC. This allowed states to test such changes as expanded earned income disregards, increased work requirements and stronger sanctions for failure to comply with them, time limits on benefits, and expanded access to transitional benefits such as child care and medical assistance. As a condition of receiving waivers, states were required to conduct rigorous evaluations of the impacts of these changes on the welfare receipt, employment, and earnings of participants. The Personal Responsibility and Work Opportunity Reconciliation Act of 1996 (PRWORA) replaced AFDC, AFDC administration, the Job Opportunities and Basic Skills Training (JOBS) program and the Emergency Assistance (EA) program with a block grant called the Temporary Assistance for Needy Families (TANF) program. Key elements of TANF include a lifetime limit of five years (60 months) on the amount of time a family with an adult can receive assistance funded with federal funds, increasing work participation rate requirements which states must meet, and broad state flexibility on program design. Spending through the TANF block grant is capped and funded at $16.5 billion per year, slightly above fiscal year 1995 federal expenditures for the four component programs. States must also meet a "maintenance of effort (MOE) requirement" by spending on needy families at least 75 percent of the amount of state funds used in FY 1994 on these programs (80 percent if they fail work participation rate requirements). TANF gives states wide latitude in spending both Federal TANF funds and state MOE funds. Subject to a few restrictions, TANF funds may be used in any way that supports one of the four statutory purposes of TANF: to provide assistance to needy families so that children can be cared for at home; to end the dependence of needy parents on government benefits by promoting job preparation, work and marriage; to prevent and reduce the incidence of out-of-wedlock pregnancies; and to encourage the formation and maintenance of two-parent families. Recent Legislative Action Legislative authority for the TANF block grant program expired September 2002. Since then, the program has been operated under a series of short-term extensions. In February 2002, President Bush proposed a plan, Working Toward Independence, to strengthen welfare reform, in order to help families remaining on welfare and other low-income families move toward self-sufficiency. The House of Representatives passed bills incorporating the key elements of the President's plan in both the 107th Congress (H.R. 4737) and the 108th Congress (H.R. 4). As of the end of 2003, a Senate version of TANF reauthorization was reported out of committee, but not yet taken up on the floor of the Senate. Final enactment of TANF reauthorization is expected in 2004. Data Issues Relating to the AFDC-TANF Transition States had the option of beginning their TANF programs as soon as PRWORA was enacted in August 1996, and a few states began TANF programs as early as September 1996. All states were required to implement TANF by July 1, 1997. Because states implemented TANF at different times, the FY 1997 data reflect a combination of the AFDC and TANF programs. In some states, limited data are available for FY 1997 because states were given a transition period of six months after they implemented TANF before they were required to report data on the characteristics and work activities of TANF participants. Because of the greatly expanded range of activities allowed under TANF, a substantial portion of TANF funds are being spent on activities other than cash payments to families. When tracking overall expenditure trends, the tables in this Appendix (e.g., Table TANF 4) include only those TANF funds spent on "cash and work-based assistance" and "administrative costs," not on work activities, supportive services, or other allowable uses of funds. Spending on these other activities is detailed in Table TANF 5. Note that TANF administrative costs include funds spent administering all activities, not just cash and work-based assistance. (Administrative costs under AFDC had included a small amount of funds for administering AFDC child care programs; such programs, and the costs of administering them, were transferred to the Child Care and Development Fund as part of PRWORA). There also is potential for discontinuity between the AFDC and the TANF caseload figures. For example, under TANF there is no longer a separate "Unemployed Parent" (UP) program, as there was under AFDC. While a separate work participation rate is calculated for two-parent families, this population is not identical to the UP caseload under AFDC. It is also possible that a limited number of families will be considered recipients of TANF assistance, even if they do not receive a monthly cash benefit. At present, the vast majority of families receiving "assistance"1 are, in fact, receiving cash payments; however, this may change over time. Once source of discontinuity has been removed in this edition of the Indicators report. Under TANF some states provide cash and other forms of assistance to specific categories of families (e.g., two-parent families) under Separate State Programs (SSPs), funded out of MOE dollars rather than federal TANF funds. This allows the states additional flexibility with regard to the time limits and work requirement. The official TANF caseload figures do not include these families. Starting with this edition, we have added recipients in SSPs into the caseload totals (the split between TANF and SSP caseloads is shown in Table TANF 3, nationally, and in Table TANF 15, by state). Expenditures for Separate State Programs are shown in Table TANF 5. AFDC/TANF Program Data The following tables and figures present data on caseloads, expenditures, and recipient characteristics of the AFDC and TANF programs. Trends in national caseloads and expenditures are shown in Figure TANF 1 and the first set of tables (Tables TANF 1-6). These are followed by information on characteristics of AFDC/TANF families (Table TANF 7) and a series of tables presenting state-by-state data on trends in the AFDC/TANF program (Tables TANF 8-13). These data complement the data on trends in AFDC recipiency and participation rates shown in Tables IND 4a and IND 5a in Chapter II. AFDC/TANF Caseload Trends (Figure TANF 1, Tables TANF 1-3). Welfare caseloads have stabilized over the past few years after declining dramatically during the 1990s. In fiscal year 2002, the average monthly number of TANF recipients was 5.65 million persons, down 1.9 percent from FY 2001. Moreover, this was 55 percent lower than the average monthly AFDC caseload in fiscal year 1996 and the smallest number of people on welfare since 1968. From the peak of 14.4 million in March 1994, the number of AFDC/TANF recipients dropped by 61.6 percent to 5.5 million in March 2003.2 Over three-fourths of the reduction in the caseload since March 1994 has occurred following the implementation of TANF. These are the largest welfare caseload declines in the history of U.S. welfare programs. As shown in Figure TANF 1, AFDC caseloads generally tended to increase in times of economic recession and decline in times of economic growth. The recent decline, however, has far outstripped that experienced in any previous period. Several studies have attempted to explain the unprecedented decline in caseloads and, specifically, to disentangle the effects of PRWORA and welfare reform from the simultaneous growth in the U.S. economy. Separating these effects is difficult, however, because PRWORA was enacted at a time when the economy was expanding dramatically, offering a uniquely conducive environment within which to move many recipients off the welfare rolls and into the labor market. Other policy changes, most notably expansions in the Earned Income Tax Credit, add further complexity. In general, studies have found that both economic conditions and welfare reform policies have played important roles in the recent caseload decline. A review of a dozen studies concluded that roughly 15 to 30 percent of the caseload decline prior to 1996 was attributed by most studies to welfare policies under waivers to the AFDC rules with approximately 30 to 45 percent of the decline explained by economic conditions (Schoeni and Blank, 2000). A study by the Council of Economic Advisers (1999) of the post-PRWORA period finds that just over one-third of caseload decline can be explained by welfare reform policy, while 8 to 10 percent is due to the economy. A more recent study estimates that over half the decline in caseloads after enactment of PRWORA were attributable to welfare reform (O'Neill and Hill, 2001). The relative stability of the caseload during the recent recession further supports the argument that the economy was only one of several factors driving caseloads down. AFDC/TANF Expenditures (Tables TANF 4-6 and Figure TANF 2). Tables TANF 4 and 5 show trends in expenditures on AFDC and TANF. Table TANF 4 tracks both programs, breaking out the costs of benefits and administrative expenses. It also shows the division between federal and state spending. Table TANF 5 shows the variety of activities funded under the TANF program. Figure TANF 2 and Table TANF 6 show that inflation has had a significant effect in eroding the value of the average monthly AFDC/TANF benefit. In real dollars, by 2001 the average monthly benefit per recipient had declined to 64 percent of what it was at its peak in the late 1970s. AFDC/TANF Recipient Characteristics (Table TANF 7). With the dramatic declines in the welfare rolls since the implementation of TANF, there has been a great deal of speculation regarding how the composition of the caseload has changed. Two striking trends are the increases in the proportion of families with no adult in the assistance unit and in employment among adult recipients. One of the most dramatic trends is the recent jump in the proportion of adult recipients who are working. In FY 2002, 25 percent of TANF adult recipients were employed, up from 11 percent in FY 1996 and 7 percent in FY 1992, as shown in Table TANF 7. Adding in those in work experience and community service positions, the percentage working was at an all-time high of over 33 percent in FY 2002 (data not shown). Similar upward trends are shown in data on income from earnings. These trends likely reflect positive effects of welfare-to-work programs, the strong economy, and the fact that, with larger earnings disregards, families with earnings do not exit welfare as rapidly. In addition, the increased employment of welfare recipients is consistent with broader trends in labor force participation. (For example, see Table Work 2 in Chapter III for trends in employment rates for women with no more than a high school education). Another dramatic change in the caseload is the increasing fraction of cases without an adult recipient. Such cases occur when the adults are ineligible (because they are a caretaker relative, SSI parent, immigrant parent, or sanctioned parent). Families with no adults in the assistance unit have climbed from 11.6 percent of the caseload in FY 1990 to 39.0 percent in FY 2002. Not counting cases with a sanctioned parent, 36.6 percent of the caseload was child-only in 2002. This dramatic growth has been due to an increase in the number of child-only cases during the early 1990s, followed by a decline in the number of adult-present cases. Even though child-only cases are generally not subject to the work requirements or time limits under TANF, the number of cases without an adult in the assistance unit has fallen by about 180,000 since 1996. In other areas, the administrative data show fewer changes in composition than might have been expected. There has been widespread anecdotal evidence that the most job ready recipients B those with the fewest barriers to employment B have already exited the welfare caseload and have stopped coming onto the welfare rolls, leaving a more disadvantaged population remaining. However, as the expectations for welfare recipients have increased, and fewer recipients are totally exempted from work requirements, others have speculated that the most disadvantaged recipients may also have been sanctioned off the rolls or terminated for failure to comply with administrative requirements. In fact, analyses of program data have not found much evidence of an increase or decline in readily observed barriers to employment in the current caseload. The question of whether the caseload has become more disadvantaged cannot be answered simply through administrative data provided by the states, which do not contain detailed information on such barriers to employment as lack of basic skills, alcohol and drug abuse, domestic violence, and disabilities. A few recent studies have found very high levels of these barriers among the TANF population. These studies also have found that the effects of these barriers are interactive; while any one barrier to employment can often be overcome, the more barriers a recipient faces, the less likely she is to find a job and maintain consistent employment over a period of time. AFDC/TANF State-by-State Trends (Tables TANF 8-17). There is a great deal of state-to-state variation in the trends discussed above. For example, as shown in Table TANF 10, while every state has experienced a caseload decline since 1993, the percentage change between the state's caseload peak and March 2003 ranges from 94 percent (Wyoming) to 26 percent (Indiana). Six states have experienced caseload declines of 75 percent or more. Table TANF 10 also shows that states reached their peak caseloads as early as May 1990 (Louisiana) and as late as June 1997 (Hawaii). Three new tables have been added to the state-by-state trends in this edition. Table TANF 15 shows TANF and Separate State Program (SSP) families and recipients, by state. Tables TANF 16 and 17 use a newly available data source, the High Performance Bonus data, which links TANF administrative records with quarterly earnings records, and allows examination of patterns of TANF receipt and employment. For example, Table TANF 16 shows the range across states in employment rates among TANF recipients (where employment is measured by presence of quarterly earnings in the same calendar quarter as one or more months of TANF recipient or in the immediately subsequent quarter). Table 17 complements the data on program spell duration provided in Table IND 8 in Chapter II, by examining state-by-state variation in the percentage of TANF recipients that receive benefits over the course of one year (four quarters) after a selected calendar quarter. Figure TANF 1. AFDC/TANF Families Receiving Income Assistance Note: "Basic families" are single-parent families and "UP families" are two-parent cases receiving benefits under AFDC Unemployed Parent programs that operated in certain states before FY 1991 and in all states after October 1, 1990. The AFDC Basic and UP programs were replaced by TANF as of July 1, 1997 under the Personal Responsibility and Work Opportunity Reconciliation Act of 1996. Shaded areas indicate NBER designated periods of recession from peak to trough. The decrease in number of families receiving assistance during the 1981-82 recession stems from changes in eligibility requirements and other policy changes mandated by OBRA 1981. Beginning in 2000, Total families includes TANF and SSP families. Last data point plotted is March 2003. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Planning, Research, and Evaluation. Figure TANF 2. Average Monthly AFDC/TANF Benefit per Recipient in Constant Dollars Note: See Table TANF 6 for underlying data. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, Quarterly Public Assistance Statistics, 1992 & 1993 plus unpublished data and Sixth TANF Annual Report to Congress, 2004. Table TANF 1. Trends in AFDC/TANF Caseloads, 1962 – 2002 Average Monthly Number (In thousands) Children as a Percent of Total Recipients Average1 Number of Children per Family Total Familie1 Unemployed Parent Families Unemployed Parent Recipients Total Children 1962.......…. 924 3,593 48 224 2,778 77.3 3.0 1963........... 950 3,834 54 291 2,896 75.5 3.0 1965........... 1,037 4,323 69 400 3,242 75.0 3.1 1971........... 2,531 9,557 143 726 6,963 72.9 2.8 1972........... 2,918 10,632 134 639 7,698 72.4 2.6 1974........... 3,170 10,845 93 434 7,825 72.2 2.5 1983........... 3,651 10,659 272 1,144 7,051 66.1 1.9 19972......... 3,937 10,935 2753 1,1583 7,7813 71.23 2.03 1998........... 3,200 8,790 179 7544 6,273 71.4 2.0 1999........... 2,674 7,188 NA NA 5,319 74.0 2.0 Note: Beginning in 2000, all caseload numbers include SSP families. 1 Includes unemployed parent families and child-only cases. 2 The Personal Responsibility and Work Opportunity Reconciliation Act of 1996 repealed the AFDC program as of July 1, 1997 and replaced it with the Temporary Assistance to Needy Families (TANF) program. 3 Based on data from the old AFDC reporting system which was available only for the first 9 months of the fiscal year. 4 Estimated based on the ratio of Unemployed Parent recipients to Unemployed Parent families in 1997. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, (Available online at http://www.acf.dhhs.gov/). Table TANF 2. Number of AFDC/TANF Recipients, and Recipients as a Percentage of Various Population Groups, 1970 – 2002 Total Recipients in the States & DC(in thousands) Child Recipients in the States & DC(in thousands) Recipients as a Percent of Total Populatio2 Recipients as a Percent of Poverty Populatio3 Recipients as a Percent of Pretransfer Poverty Population4 Child Recipients as a Percent of Total Child Population2 Child Recipients as a Percent of Children in Poverty3 1970 8,303 6,104 4.1 32.7 NA 8.8 58.5 1971 10,043 7,303 4.9 39.3 NA 10.5 69.2 1979 10,140 7,057 4.5 38.9 53.1 11.0 68.0 1997 10,224 7,077 5 3.7 28.7 40.7 10.0 50.1 1998 8,215 5,781 3.0 23.8 34.7 8.1 42.9 1 Total recipients are calculated here as the monthly average for the calendar year in order to compare with the calendar year counts of the poverty populations used to compute the recipiency rates. From 2000 onward, total recipients includes SSP recipients as well as TANF recipients. See Table IND 3a for fiscal year recipiency rates. 2 Population numbers used as denominators are resident population. See Current Population Reports, Series P25-1106 3 For poverty population data see Current Population Reports, Series P60-222 (Available online at http://www.census.gov/hhes/www/poverty.html). 4 The pretransfer poverty population used as denominator is the number of all persons in families with related children under 18 years of age whose income (cash income plus social insurance plus Social Security but before taxes and means-tested transfers) falls below the appropriate poverty threshold. See Appendix J, Table 20, 1992 Green Book; data for subsequent years are unpublished Congressional Budget Office tabulations. 5 Estimated based on the ratio of children recipients to total recipients for January through June of 1997. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance and U.S. Bureau of the Census, "Poverty in the United States: 2002," Current Population Reports, Series P60-222, and earlier years, (Available online at http://www.census.gov/hhes/www/poverty.html). Table TANF 3. TANF and Separate State Program (SSP) Families and Recipients, 2000 – 2002 2000 2,265 91 2,355 2002 2,065 128 2,194 Note: Some states provide cash and other forms of assistance to specific categories of families (e.g., two-parent families) under Separate State Programs (SSPs) which are funded out of Maintenance of Effort (MOE) dollars rather than federal TANF funds. See Table TANF 15 for SSPs by state. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, (available online at http://www.acf.dhhs.gov/) Table TANF 4. Total AFDC/TANF Expenditures on Cash Benefits and Administration, 1970 – 2002 [In millions of dollars] Federal Funds (Current Dollars) State Funds (Constant 2002 Dollars1) 1970 $2,187 $572 2 $1,895 $309 $4,082 $881 2 18,076 3,901 1971 3,008 271 2,469 254 5,477 525 23,219 2,226 1972 3,612 240 3 2,942 241 6,554 481 3 26,831 NA 1975 4,625 552 3,787 529 8,412 1,082 27,766 3,571 1977 5,626 595 4,762 583 10,388 1,177 29,878 3,385 1987 8,914 1,081 7,409 1,052 16,323 2,133 25,980 3,395 1990 10,149 1,358 8,390 1,303 18,539 2,661 25,770 3,699 1993 12,270 1,518 10,016 1,438 22,286 2,956 27,784 3,685 19974 9,748 1,273 7,799 1,098 17,547 2,371 19,644 2,654 1999 6,475 1,407 6,975 884 13,449 2,291 14,538 2,476 2002 4,554 1,633 4,854 983 9,408 2,617 9,408 2,617 Note: Benefits do not include emergency assistance payments and have not been reduced by child support collections. Foster care payments are included from 1971 to 1980. State funds for benefits include benefits under Separate State Programs. Beginning in fiscal year 1984, the cost of certifying AFDC households for food stamps is shown in the food stamp program's appropriation under the U.S. Department of Agriculture. Administrative costs include: Work Program, ADP, FAMIS, Fraud Control, Child Care administration (through 1996), SAVE and other State and local administrative expenditures. 1 Constant dollar adjustments to 2002 level were made using a CPI-U-X1 fiscal year price index. 2 Includes expenditures for services. 3 Administrative expenditures only. 4 The Personal Responsibility and Work Opportunity Reconciliation Act of 1996 repealed the AFDC program as of July 1, 1997 and replaced it with the Temporary Assistance to Needy Families (TANF) program. Under PRWORA, spending categories are not entirely equivalent to those under AFDC: for example administrative expenses under TANF do not include IV-A child care administration (which accounted for 4 percent of 1996 administrative expense). Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Financial Systems. Table TANF 5. Federal and State TANF Program and Other Related Spending Fiscal Years 1997 to 2002 (Millions) Cash & Work-Based Assistance Other Expenditures Total Expenditures Federal TANF Grants 1997 7,708 467 14 – 872 109 0 862 10,032 1998 7,168 763 252 – 938 224 6 1,136 10,487 1999 6,475 1,225 604 – 1,070 337 17 1,595 11,323 2000 5,444 1,606 1,553 496 1,328 242 – 2,715 13,384 State Maintenance of Effort Expenditures in the TANF Program 1997 5,955 311 752 – 704 101 9 926 8,758 1998 6,879 520 890 – 883 138 11 1,301 10,623 1999 6,541 503 1,135 – 743 118 23 1,334 10,397 2000 5,432 884 1,893 150 921 92 – 1,170 10,541 2001 4,887 685 1,730 113 920 83 – 1,195 9,613 State Maintenance of Effort Expenditures in Separate State Programs 1997 69 12 111 – 0 0 – 18 210 1998 216 3 137 – 6 1 – 28 391 1999 434 26 257 – 22 0 0 126 865 2000 305 11 73 17 19 0 – 431 856 2001 503 28 34 20 38 1 – 499 1,125 2002 860 24 72 24 41 -.5 – 652 1,673 1997 13,731 790 877 – 1,577 211 9 1,805 19,000 1998 14,264 1,286 1,280 – 1,828 362 17 2,465 21,502 2000 11,180 2,501 3,519 663 2,267 335 – 4,316 24,781 Note: Administration and Systems, shown separately here in Table TANF 5, can be combined to show total administrative costs, as in Table TANF 3. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Financial Services. Table TANF 6. Trends in AFDC/TANF Average Monthly Payments, 1962 – 2002 Monthly Benefit per Recipient Average Number of Persons per Family Monthly Benefit per Family (not reduced by Child Support) Weighted Average1 Maximum Benefit (per 3-person Family) 1962 $31 $171 3.9 $121 $664 NA NA 1963 31 169 4.0 126 681 NA NA 1969 43 201 4.0 173 803 $186 2 $867 1970 46 203 3.9 178 789 194 2 861 1975 63 209 3.3 209 689 243 802 1982 103 195 2.9 300 569 331 626 19973 130 146 2.8 362 405 420 470 Note: AFDC benefit amounts have not been reduced by child support collections. Constant dollar adjustments to 2002 level were made using a CPI-U-X1 fiscal-year price index. 1 The maximum benefit for a 3-person family in each state is weighted by that state's share of total AFDC families. 2 Estimated based on the weighted average benefit for a 4-person family. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, Quarterly Public Assistance Statistics, 1992 & 1993 and earlier years along with unpublished data. Table TANF 7. Characteristics of AFDC/TANF Families, Selected Years 1969 – 2002 Fiscal year1 Avg. Family Size (persons) 4.0 3.2 3.0 3.0 3.0 2.9 2.8 2.6 2.6 2.5 Number of Child Recipients One 26.6 37.9 42.3 43.4 42.5 42.5 43.9 44.2 44.8 47.0 Two 23.0 26.0 28.1 29.8 30.2 30.2 29.9 28.4 28.5 28.0 Three 17.7 16.1 15.6 15.2 15.8 15.5 15.0 15.3 14.8 14.2 Four or More 32.5 20.0 13.9 10.1 9.9 10.1 9.2 10.1 9.9 8.9 Unknown NA NA NA 1.5 1.7 0.7 1.3 2.0 2.0 1.9 Families with No Adult in Asst. Unit 10.1 12.5 14.6 8.3 9.6 14.8 21.5 34.5 37.1 39.0 Child-Only Families2 – – – – – – – 32.7 35.3 36.6 Families with Non-Recipients 33.1 34.8 NA 36.9 36.8 38.9 49.9 – – – Median Months on AFDC/TANF Since Most Recent Opening 23.0 31.0 29.0 26.0 26.3 22.5 23.6 – – – Presence of Assistance Living in Public Housing 12.8 14.6 NA 10.0 9.6 9.2 8.8 17.7 20.0 19.2 Participating in Food Stamp or Donated Food Program 52.9 75.1 75.1 83.0 84.6 87.3 89.3 79.9 80.9 80.1 Presence of Income With Earnings NA 14.6 12.8 5.7 8.4 7.4 11.1 23.63 24.33 21.83 No Non-AFDC/TANF Income 56.0 71.1 80.6 86.8 79.6 78.9 76.0 71.63 70.33 72.83 Adult Employment Status (percent of adults) Employed – – – – – 6.6 11.3 26.4 26.7 25.3 Unemployed – – – – – – – 49.2 47.5 47.2 Not in Labor Force – – – – – – – 24.3 25.8 27.5 Adult Women's employment status (percent of adult female recipients):4 Full-time job 8.2 10.4 8.7 1.5 2.2 2.2 4.7 – – – Part-time job 6.3 5.7 5.4 3.4 4.2 4.2 5.4 – – – Marital Status (percent of adults) Single – – – – – – – 65.3 66.9 66.6 Married – – – – – – – 12.4 11.7 11.5 Separated – – – – – – – 13.1 12.5 13.0 Widowed – – – – – – – 0.7 0.8 0.7 Divorced – – – – – – – 8.5 8.2 8.2 Basis for Child's Eligibility (percent children): Incapacitated 11.75 7.7 5.3 3.4 3.7 4.1 4.3 – – – Unemployed 4.65 3.7 4.1 8.7 6.5 8.2 8.3 – – – Death 5.55 3.7 2.2 1.8 1.8 1.6 1.6 – – – Divorce or Separation 43.35 48.3 44.7 38.5 34.6 30.0 24.3 – – – Absent, No Marriage Tie 27.95 31.0 37.8 44.3 51.9 53.1 58.6 – – – Absent, Other Reason 3.55 4.0 5.9 1.4 1.6 2.0 2.4 – – – Unknown – – – 1.7 – 0.9 0.6 – – – Note: Figures are percentages of families/cases unless noted otherwise. 1 Percentages are based on the average monthly caseload during the year. Hawaii and the territories are not included in 1983. Data after 1986 include the territories and Hawaii. 2 In this report, child-only families are those families with no adult in the assistance unit excluding those where there is no adult in the assistance unit as a result of the parent being sanctioned for non-compliance. 3 Presence of income is measured as a percentage of adult recipients, not families, in 1998 and subsequent years. 4 For years prior to 1983, data are for mothers only. 5 Calculated on the basis of total number of families. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, Characteristics and Financial Circumstances of TANF Recipients: 2003 TANF Annual Report to Congress and earlier years. Table TANF 8. AFDC/TANF Benefits by State, Selected Fiscal Years 1978 – 2002 [Millions of dollars] Alabama $78 $74 $68 $62 $62 $92 $75 $44 $36 $33 Alaska 17 37 46 54 60 113 107 77 55 55 Arizona 30 67 79 103 138 266 228 145 107 130 Arkansas 51 39 48 53 57 57 52 26 34 26 California 1,813 3,207 3,574 4,091 4,955 6,088 5,908 4,128 3,643 2,608 Colorado 74 107 107 125 137 158 129 80 48 53 Connecticut 168 226 223 218 295 397 323 305 166 128 Delaware 28 28 25 24 29 40 35 24 20 19 Dist. of Columbia 91 75 77 76 84 126 121 97 72 67 Florida 145 251 261 318 418 806 680 357 234 256 Georgia 103 149 223 266 321 428 385 313 180 109 Guam 3 5 4 3 5 12 14 NA NA NA Hawaii 83 83 73 77 99 163 173 153 141 85 Idaho 21 21 19 19 20 30 30 6 3 5 Illinois 699 845 886 815 839 914 833 771 269 146 Indiana 118 153 148 167 170 228 153 104 87 146 Iowa 107 159 170 155 152 169 131 104 79 76 Kansas 73 87 91 97 105 123 98 41 43 50 Kentucky 122 135 104 143 179 198 191 147 104 101 Louisiana 97 145 162 182 188 168 130 103 58 67 Maine 51 69 84 80 101 108 99 80 73 66 Maryland 166 229 250 250 296 314 285 192 196 227 Massachusetts 476 406 471 558 630 730 560 442 336 279 Michigan 780 1,214 1,248 1,231 1,211 1,132 779 589 386 326 Minnesota 164 287 322 338 355 379 333 276 193 184 Mississippi 33 58 74 85 86 82 68 60 18 37 Missouri 152 196 209 215 228 287 254 180 139 148 Montana 15 27 37 41 40 49 45 30 21 31 Nebraska 38 56 62 56 59 62 54 41 41 52 Nevada 8 10 16 20 27 48 48 39 28 48 New Hampshire 21 16 20 21 32 62 50 39 32 29 New Jersey 489 485 509 459 451 531 462 372 222 194 New Mexico 32 49 51 56 61 144 153 104 113 82 New York 1,689 1,916 2,099 2,140 2,259 2,913 2,929 2,149 1,554 1,465 North Carolina 138 149 138 206 247 353 300 211 140 139 North Dakota 14 16 20 22 24 26 21 22 12 10 Ohio 441 725 804 805 877 1,016 763 546 368 336 Oklahoma 74 85 100 119 132 165 122 72 78 45 Oregon 148 101 120 128 145 197 155 141 34 69 Pennsylvania 726 724 389 747 798 935 822 523 573 338 Puerto Rico 25 38 33 67 72 74 63 NA NA NA Rhode Island 59 71 79 82 99 136 125 117 105 89 South Carolina 52 75 103 91 96 115 101 52 91 35 South Dakota 18 17 15 21 22 25 22 14 10 11 Tennessee 77 83 100 125 168 215 190 108 146 132 Texas 122 229 281 344 416 544 496 315 248 203 Utah 41 52 55 61 64 77 64 50 40 41 Vermont 21 40 40 40 48 65 56 47 39 38 Virgin Islands 2 2 2 2 3 4 4 NA NA NA Virginia 136 165 179 169 177 253 199 123 186 101 Washington 175 294 375 401 438 610 585 450 312 295 West Virginia 53 75 109 107 110 126 101 52 49 71 Wisconsin 260 519 444 506 440 425 291 145 7 126 Wyoming 6 13 16 19 19 21 17 7 9 2 United States $10,621 $14,371 $15,236 $16,663 $18,543 $22,798 $20,411 $14,614 $11,180 $9,408 Note: Benefits refers to total cash benefits paid, (see Table TANF 4) but does not include emergency assistance payments. NA denotes data not available. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Program Support, Office of Management Services, data from the ACF-196 TANF Report and ACF-231 AFDC Line by Line Report. Table TANF 9. Comparison of Federal Funding for AFDC and Related Programs And 2002 Family Assistance Grants Awarded Under PRWORA [In millions] FY 1996 Grants for AFDC, EA & JOBS1 FY 2002 State Family Assistance Grant2 Increase from FY 1996 Level Percent Increase from FY 1996 Level Alabama $79.0 $124.2 $45.2 57 Alaska 60.7 60.3 -0.4 -1 Arizona 200.6 228.7 28.0 14 Arkansas 54.3 62.9 8.6 16 California 3,545.6 3,739.8 194.3 5 Colorado 138.9 169.4 30.5 22 Connecticut 221.1 280.1 59.1 27 Delaware 30.2 32.3 2.1 7 Dist of Columbia 77.1 117.1 39.9 52 Florida 504.7 622.7 118.0 23 Georgia 301.2 368.0 66.8 22 Hawaii 98.4 103.9 5.5 6 Idaho 31.3 35.0 3.7 12 Illinois 593.8 585.1 -8.8 -1 Indiana 121.4 217.1 95.8 79 Iowa 129.3 138.1 8.8 7 Kansas 86.9 101.9 15.0 17 Kentucky 171.6 190.4 18.7 11 Louisiana 122.4 189.2 66.8 55 Maine 73.2 78.1 4.9 7 Maryland 207.6 229.1 21.5 10 Massachusetts 372.0 459.4 87.3 23 Michigan 581.5 795.2 213.7 37 Minnesota 239.3 270.2 30.8 13 Mississippi 68.6 95.8 27.2 40 Missouri 207.9 227.9 20.0 10 Montana 39.2 46.4 7.2 18 Nebraska 56.2 58.4 2.2 4 Nevada 41.2 49.9 8.7 21 New Hampshire 36.0 39.0 2.9 8 New Jersey 353.4 404.0 50.7 14 New Mexico 129.9 121.9 -8.0 -6 New York 2,332.7 2,442.9 110.2 5 North Carolina 311.9 338.3 26.5 8 North Dakota 24.5 27.7 3.2 13 Ohio 564.5 728.0 163.5 29 Oklahoma 125.1 147.6 22.5 18 Oregon 146.4 166.8 20.4 14 Pennsylvania 780.1 719.5 -60.6 -8 Rhode Island 82.9 99.8 16.9 20 South Carolina 99.4 100.0 0.5 1 South Dakota 19.7 22.0 2.3 12 Tennessee 178.9 213.1 34.2 19 Texas 437.1 583.1 146.0 33 Utah 68.0 88.2 20.2 30 Vermont 42.4 49.7 7.4 17 Virginia 134.6 158.3 23.6 18 Washington 393.2 411.4 18.3 5 West Virginia 95.1 115.7 20.5 22 Wisconsin 241.6 331.0 89.4 37 Wyoming 14.4 19.6 5.2 36 United States $15,067 $17,004 $1,937 13 1 Includes Administration and FAMIS but excludes IV-A child care. AFDC benefits include the Federal share of child support collections to be comparable to the Family Assistance Grant. The 1996 figures have been revised since earlier versions of this report, to reflect upward revisions in states' reports of expenditures on the JOBS program. 2 The FY 2002 awards include State Family Assistance Grants, Supplemental Grants for Population Increases, Out of Wedlock Bonus and High Performance Bonus. Source: U.S. Department of Health & Human Services, Administration for Children and Families, Office of Financial Services. Table TANF 10. AFDC/TANF Caseload by State, October 1989 to March 2003 Peak [In thousands] Peak Caseload Oct '89 to Date Peak Occurred Oct '89 to Mar '03 Sept '96 Caseload Mar '03 TANF & SSP Caseload Percent Decline 1 Sept '96 to Mar '03 Percent Decline Peak to Mar '03 Alabama 52.3 Mar-93 40.7 19.5 52 63 Alaska 13.4 Apr-94 12.3 5.6 55 58 Arizona 72.8 Dec-93 61.8 48.3 22 34 Arkansas 27.1 Mar-92 22.1 10.9 51 60 California 933.1 Mar-95 870.3 496.6 43 47 Colorado 43.7 Dec-93 33.6 14.2 58 67 Connecticut 61.9 Mar-95 57.1 24.7 57 60 Delaware 11.8 Apr-94 10.5 5.7 45 51 Dist. of Columbia 27.5 Apr-94 25.1 16.8 33 39 Florida 259.9 Nov-92 200.3 58.5 71 77 Georgia 142.8 Nov-93 120.9 56.0 54 61 Guam 3.1 Oct-01 2.3 3.1 -36 0 Hawaii 23.4 Jun-97 21.9 13.5 39 42 Idaho 9.5 Mar-95 8.4 1.8 79 81 Illinois 243.1 Aug-94 217.8 37.4 83 85 Indiana 76.1 Sep-93 49.7 56.3 -13 26 Iowa 40.7 Apr-94 31.1 22.7 27 44 Kansas 30.8 Aug-93 23.4 15.4 34 50 Kentucky 84.0 Mar-93 70.4 34.8 51 59 Louisiana 94.7 May-90 66.5 22.2 67 77 Maine 24.4 Aug-93 19.7 10.3 48 58 Maryland 81.8 May-95 68.9 28.9 58 65 Massachusetts 115.7 Aug-93 84.3 49.1 42 58 Michigan 233.6 Apr-91 167.5 76.5 54 67 Minnesota 66.2 Jun-92 57.2 42.1 26 36 Mississippi 61.8 Nov-91 45.2 19.5 57 68 Missouri 93.7 Mar-94 79.1 44.4 44 53 Montana 12.3 Mar-94 9.8 6.4 34 48 Nebraska 17.2 Mar-93 14.4 11.9 17 31 Nevada 16.3 Mar-95 13.2 11.3 14 31 New Hampshire 11.8 Apr-94 8.9 6.2 31 48 New Jersey 132.6 Nov-92 100.8 44.2 56 67 New Mexico 34.9 Nov-94 33.0 16.3 51 53 New York 463.7 Dec-94 412.7 196.2 52 58 North Carolina 134.1 Mar-94 107.5 40.3 62 70 North Dakota 6.6 Apr-93 4.7 3.4 27 48 Ohio 269.8 Mar-92 201.9 84.0 58 69 Oklahoma 51.3 Mar-93 35.3 14.7 58 71 Oregon 43.8 Apr-93 28.5 19.1 33 56 Pennsylvania 212.5 Sep-94 180.1 80.3 55 62 Puerto Rico 61.7 Jan-92 49.5 19.0 62 69 Rhode Island 22.9 Apr-94 20.5 14.7 28 36 South Carolina 54.6 Jan-93 42.9 19.4 55 64 South Dakota 7.4 Apr-93 5.7 2.8 51 62 Tennessee 112.6 Nov-93 96.2 70.4 27 38 Texas 287.5 Dec-93 238.8 140.3 41 51 Utah 18.7 Mar-93 14.0 8.7 38 53 Vermont 10.3 Apr-92 8.7 5.3 39 48 Virgin Islands 1.4 Dec-95 1.3 0.5 66 68 Virginia 76.0 Apr-94 60.5 32.3 47 57 Washington 104.8 Feb-95 96.8 61.2 37 42 West Virginia 41.9 Apr-93 37.6 15.9 58 62 Wisconsin 82.9 Jan-92 49.9 21.0 58 75 Wyoming 7.1 Aug-92 4.3 0.4 91 94 United States 5,098 Mar-94 4,346 2,181 50 57 1Negative values denote percent increase. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, Division of Data Collection and Analysis. Table TANF 11. Average Monthly AFDC/TANF Recipients by State, Selected Fiscal Years Alabama 78 123 180 151 130 132 105 44 -19 -59 Alaska 5 8 15 16 20 38 36 18 79 -51 Arizona 40 51 51 72 124 201 172 94 38 -45 Arkansas 30 45 85 64 71 69 58 28 -19 -52 California 528 1,148 1,387 1,619 1,902 2,639 2,626 1,382 38 -47 Colorado 42 66 77 79 102 119 99 31 -4 -68 Connecticut 59 83 139 122 120 166 162 57 35 -65 Delaware 12 20 32 24 21 27 23 13 10 -45 Dist. of Columbia 20 40 85 58 49 74 70 43 44 -39 Florida 106 204 256 271 370 669 561 132 52 -77 Georgia 71 198 221 239 293 393 353 131 20 -63 Guam 1 2 5 6 4 7 8 11 91 37 Hawaii 14 25 60 51 44 62 67 50 52 -25 Idaho 10 16 21 17 17 23 23 2 38 -90 Illinois 262 368 672 735 636 712 655 135 3 -79 Indiana 48 73 157 165 154 216 148 150 -4 1 Iowa 44 64 104 123 98 110 89 55 -9 -38 Kansas 36 53 68 67 77 87 68 36 -11 -48 Kentucky 81 129 167 160 175 208 175 78 -0 -56 Louisiana 104 202 213 230 282 248 236 61 -16 -74 Maine 19 36 60 57 56 64 56 31 -0 -45 Maryland 80 131 212 195 186 222 204 71 10 -65 Massachusetts 94 208 350 235 263 307 237 108 -10 -54 Michigan 162 253 685 691 655 666 527 202 -20 -62 Minnesota 51 76 135 152 171 187 171 113 0 -34 Mississippi 83 115 173 155 179 159 129 40 -28 -69 Missouri 107 140 199 197 211 263 232 130 10 -44 Montana 7 13 19 22 29 35 31 16 8 -47 Nebraska 16 30 35 44 43 45 40 30 -7 -24 Nevada 5 12 12 14 23 38 38 32 66 -15 New Hampshire 4 9 22 14 16 30 24 14 48 -40 New Jersey 104 286 459 367 309 335 288 110 -7 -62 New Mexico 30 51 53 51 57 102 101 47 77 -53 New York 517 1,052 1,100 1,112 981 1,255 1,184 530 21 -55 North Carolina 111 124 198 166 223 333 278 91 24 -67 North Dakota 8 11 13 12 16 16 13 8 -14 -38 Ohio 183 266 513 673 632 685 546 191 -14 -65 Oklahoma 73 95 89 82 112 131 105 37 -6 -65 Oregon 31 75 102 74 89 114 87 41 -2 -53 Pennsylvania 303 426 629 561 521 620 544 211 4 -61 Puerto Rico 202 223 168 173 190 183 155 67 -18 -56 Rhode Island 24 38 52 44 46 63 58 44 27 -25 South Carolina 30 52 153 120 111 140 119 53 7 -55 South Dakota 11 16 20 16 19 19 16 7 -14 -59 Tennessee 76 129 162 155 211 300 260 168 23 -35 Texas 91 214 308 363 611 788 684 360 12 -47 Utah 22 33 37 38 45 50 40 20 -11 -50 Vermont 5 12 23 22 22 28 25 14 15 -44 Virgin Islands 1 2 3 4 3 4 5 2 55 -53 Virginia 46 87 166 154 151 195 162 71 7 -56 Washington 71 109 154 178 228 292 274 156 20 -43 West Virginia 116 93 77 106 111 114 95 42 -14 -56 Wisconsin 45 79 213 288 237 226 170 47 -28 -73 Wyoming 4 5 7 10 14 16 13 1 -9 -93 United States 4,323 7,415 10,597 10,813 11,460 14,226 12,645 5,654 10 -55 Note: Recipients in 2002 include SSP recipients. Source: U.S. Department of Health and Human Services, Administration for Children and Families, Office of Family Assistance, 2003 TANF Report to Congress. Table TANF 12. AFDC/TANF Recipiency Rates for Total Population by State: Selected Fiscal Years [In percent] Alabama 2.2 3.6 4.6 3.8 3.2 3.1 2.4 1.0 -24 -60 Alaska 1.8 2.6 3.7 3.0 3.7 6.3 5.9 2.7 63 -54 Arizona 2.6 2.9 1.9 2.3 3.4 4.7 3.7 1.7 11 -54 Arkansas 1.5 2.3 3.7 2.8 3.0 2.8 2.3 1.0 -25 -55 California 2.9 5.7 5.8 6.1 6.3 8.4 8.2 3.9 29 -52 Colorado 2.2 3.0 2.6 2.5 3.1 3.2 2.5 0.7 -19 -72 Connecticut 2.1 2.7 4.5 3.8 3.6 5.0 4.8 1.6 33 -66 Delaware 2.4 3.6 5.4 3.9 3.2 3.8 3.2 1.6 -0 -49 Dist. of Columbia 2.5 5.3 13.3 9.2 8.1 12.6 12.3 7.6 52 -38 Florida 1.8 3.0 2.6 2.4 2.8 4.7 3.8 0.8 33 -79 Georgia 1.6 4.3 4.0 4.0 4.5 5.5 4.7 1.5 4 -68 Hawaii 1.9 3.2 6.2 4.9 3.9 5.2 5.5 4.0 40 -27 Idaho 1.4 2.2 2.2 1.7 1.6 2.0 1.9 0.2 16 -91 Illinois 2.5 3.3 5.9 6.4 5.6 6.0 5.4 1.1 -2 -80 Indiana 1.0 1.4 2.9 3.0 2.8 3.7 2.5 2.4 -9 -3 Iowa 1.6 2.3 3.6 4.3 3.5 3.9 3.1 1.9 -12 -40 Kansas 1.6 2.4 2.9 2.8 3.1 3.4 2.6 1.3 -16 -50 Kentucky 2.5 4.0 4.6 4.3 4.7 5.4 4.5 1.9 -6 -57 Louisiana 2.9 5.6 5.0 5.2 6.7 5.7 5.4 1.4 -20 -75 Maine 1.9 3.6 5.4 4.9 4.5 5.2 4.5 2.4 -2 -47 Maryland 2.2 3.3 5.0 4.4 3.9 4.4 4.0 1.3 3 -67 Massachusetts 1.8 3.7 6.1 4.0 4.4 5.0 3.8 1.7 -12 -56 Michigan 2.0 2.9 7.4 7.6 7.0 6.9 5.4 2.0 -23 -63 Minnesota 1.4 2.0 3.3 3.6 3.9 4.1 3.6 2.2 -7 -38 Mississippi 3.6 5.2 6.9 6.0 6.9 5.9 4.7 1.4 -32 -70 Missouri 2.4 3.0 4.0 3.9 4.1 4.9 4.3 2.3 4 -46 Montana 1.0 1.9 2.4 2.7 3.6 4.0 3.5 1.8 -3 -49 Nebraska 1.1 2.0 2.2 2.8 2.7 2.8 2.4 1.7 -12 -27 Nevada 1.2 2.4 1.5 1.4 1.9 2.5 2.3 1.5 22 -34 New Hampshire 0.7 1.2 2.4 1.4 1.5 2.7 2.1 1.1 40 -45 New Jersey 1.5 4.0 6.2 4.9 4.0 4.2 3.5 1.3 -11 -64 New Mexico 3.0 5.0 4.1 3.5 3.8 6.1 5.8 2.6 53 -56 New York 2.9 5.8 6.3 6.2 5.4 6.8 6.4 2.8 17 -57 North Carolina 2.2 2.4 3.4 2.6 3.4 4.6 3.7 1.1 10 -70 North Dakota 1.2 1.7 2.0 1.8 2.4 2.6 2.1 1.3 -15 -36 Ohio 1.8 2.5 4.8 6.3 5.8 6.1 4.9 1.7 -17 -66 Oklahoma 3.0 3.7 2.9 2.5 3.6 4.0 3.1 1.1 -12 -66 Oregon 1.6 3.6 3.9 2.8 3.1 3.7 2.7 1.2 -14 -57 Pennsylvania 2.6 3.6 5.3 4.8 4.4 5.1 4.4 1.7 2 -62 Rhode Island 2.7 4.0 5.5 4.5 4.6 6.2 5.7 4.1 25 -29 South Carolina 1.2 2.0 4.9 3.6 3.2 3.8 3.1 1.2 -1 -61 South Dakota 1.6 2.4 2.9 2.3 2.7 2.6 2.2 0.9 -19 -60 Tennessee 2.0 3.3 3.5 3.3 4.3 5.7 4.8 2.9 11 -39 Texas 0.9 1.9 2.1 2.2 3.6 4.2 3.5 1.7 -1 -53 Utah 2.2 3.1 2.5 2.3 2.6 2.5 2.0 0.9 -25 -55 Vermont 1.4 2.6 4.4 4.2 3.9 4.8 4.3 2.3 10 -46 Virginia 1.0 1.9 3.1 2.7 2.4 3.0 2.4 1.0 -1 -60 Washington 2.4 3.2 3.7 4.0 4.7 5.4 4.9 2.6 6 -48 West Virginia 6.4 5.3 4.0 5.5 6.2 6.3 5.2 2.3 -16 -56 Wisconsin 1.1 1.8 4.5 6.1 4.8 4.4 3.3 0.9 -33 -74 Wyoming 1.1 1.5 1.4 2.0 3.1 3.4 2.6 0.2 -16 -94 United States 2.1 3.5 4.6 4.5 4.5 5.3 4.6 1.9 3 -58 Note: Recipiency rate refers to the average monthly number of AFDC recipients in each state during the given fiscal year expressed as a percent of the total resident population as of July 1 of that year. The numerators are from Table TANF 11. Sources: U. S. Department of Health and Human Services and U.S. Bureau of the Census, (Resident population by state available on line at http://www.census.gov/population/estimates/state/). Table TANF 13. Average Number of AFDC/TANF Child Recipients By State, Selected Fiscal Years Alabama 62 96 129 105 93 96 79 34 -14 -57 Arizona 31 39 38 50 87 136 118 70 36 -40 California 391 816 932 1,070 1,294 1,804 1,805 1,043 39 -42 Colorado 33 50 53 53 69 80 68 23 -2 -66 Connecticut 43 62 97 82 81 111 108 42 33 -62 Delaware 9 15 22 16 14 19 16 10 9 -38 Florida 85 160 184 191 264 463 395 104 49 -74 Guam 1 1 4 4 3 5 6 0 87 -100 Idaho 7 11 14 11 11 16 16 2 41 -88 Iowa 32 46 69 77 64 72 59 36 -7 -39 Kansas 28 41 49 45 52 59 48 25 -8 -47 Kentucky 58 93 118 107 117 137 120 57 3 -52 Louisiana 79 157 156 163 199 180 162 48 -19 -70 Maine 14 26 40 36 35 40 35 21 0 -41 Massachusetts 71 153 228 152 168 197 153 77 -9 -50 Minnesota 39 58 91 95 110 124 116 78 5 -32 Mississippi 66 93 128 112 129 116 96 30 -25 -68 Missouri 82 106 135 129 139 176 162 92 16 -43 Montana 6 10 13 15 19 23 21 11 10 -47 Nevada 4 9 8 9 16 27 27 23 71 -16 New Hampshire 3 7 15 9 11 19 16 10 48 -37 New Jersey 79 209 318 247 213 228 195 82 -8 -58 New Mexico 23 39 35 34 37 66 65 34 75 -48 New York 380 759 759 729 658 813 771 371 17 -52 North Carolina 83 94 141 113 152 223 191 70 26 -63 North Dakota 6 8 9 8 10 11 9 6 -12 -34 Ohio 136 198 348 424 414 455 382 142 -8 -63 Oklahoma 55 71 65 57 77 90 74 28 -4 -62 Oregon 23 52 65 49 60 76 60 30 0 -50 South Carolina 24 40 109 84 80 102 89 39 12 -57 South Dakota 8 12 15 11 13 14 12 5 -11 -55 Tennessee 58 99 115 105 144 203 181 121 26 -33 Vermont 4 8 14 14 14 17 16 9 15 -42 Virginia 35 66 116 103 104 134 114 51 10 -55 Washington 50 76 97 113 148 187 177 108 20 -39 West Virginia 80 65 58 64 68 72 62 28 -10 -54 Wyoming 3 4 5 7 9 11 9 1 -4 -92 United States 3,242 5,483 7,320 7,165 7,755 9,611 8,672 4,149 12 -52 Note: From FY 2000 onward, TANF child recipients include SSP child recipients. Table TANF 14. AFDC/TANF Recipiency Rates for Children by State, Selected Fiscal Years 1965 – 2002 Alabama 4.6 7.7 11.1 9.7 8.8 8.9 7.3 3.3 -17 -55 Alaska 3.1 5.0 8.0 5.9 7.4 12.8 12.4 6.2 67 -50 Arizona 4.8 6.0 4.8 5.9 8.6 12.1 9.7 4.7 12 -51 California 6.0 12.3 14.6 15.6 16.2 20.8 20.3 11.0 25 -46 Connecticut 4.4 6.1 11.8 10.8 10.8 14.2 13.7 4.8 27 -65 Delaware 4.7 7.5 13.4 10.2 8.7 10.5 8.9 5.1 2 -43 Dist. of Columbia 6.0 13.8 40.9 33.9 30.7 44.5 44.1 28.1 44 -36 Florida 4.3 7.6 7.8 7.6 8.8 14.1 11.6 2.6 31 -77 Georgia 3.2 9.1 9.8 10.1 11.8 14.6 12.8 4.4 9 -66 Hawaii 3.6 6.5 14.5 11.6 10.5 13.6 14.5 11.0 39 -24 Illinois 5.3 7.5 14.6 16.1 14.8 15.7 14.4 3.3 -3 -77 Iowa 3.2 4.7 8.4 10.2 8.8 9.9 8.2 5.1 -8 -37 Kentucky 4.9 8.3 10.9 10.5 12.4 14.1 12.4 6.1 -0 -51 Louisiana 5.5 11.3 11.8 12.2 16.5 14.6 13.3 4.0 -20 -70 Maine 3.9 7.7 12.5 11.7 11.5 13.1 11.8 7.4 3 -37 Maryland 4.6 7.3 12.4 11.4 10.6 12.0 11.1 3.8 5 -66 Massachusetts 3.8 8.1 15.3 11.2 12.4 13.9 10.6 5.2 -15 -51 Michigan 3.7 5.8 16.7 17.7 17.4 17.4 13.9 5.8 -20 -59 Minnesota 2.9 4.2 7.7 8.5 9.4 10.1 9.3 6.1 -0 -35 Mississippi 7.0 11.1 15.7 14.0 17.6 15.3 12.7 4.0 -28 -68 Missouri 5.2 6.9 9.9 9.8 10.6 12.9 11.6 6.4 10 -45 Montana 2.0 4.0 5.7 6.1 8.4 9.7 8.9 5.0 6 -44 New Jersey 3.4 8.8 16.0 13.5 11.7 11.7 9.9 3.8 -16 -61 New Mexico 5.2 9.5 8.5 7.8 8.3 13.5 13.1 6.7 59 -49 New York 6.3 13.0 16.2 16.7 15.4 18.0 17.0 8.0 11 -53 North Carolina 4.4 5.3 8.5 7.1 9.3 12.6 10.4 3.4 12 -68 Ohio 3.6 5.3 11.2 14.7 14.9 16.0 13.4 4.9 -10 -63 Oklahoma 6.4 8.5 7.6 6.3 9.1 10.4 8.5 3.1 -7 -63 Oregon 3.3 7.4 9.0 6.9 8.1 9.7 7.4 3.5 -8 -53 Pennsylvania 5.5 8.0 13.8 12.9 12.3 14.4 12.8 5.4 4 -58 Rhode Island 5.9 9.1 14.7 12.6 13.4 17.5 16.5 12.5 23 -24 South Carolina 2.3 4.2 11.6 9.1 8.7 10.8 9.4 3.8 8 -59 Tennessee 4.2 7.5 8.9 8.6 11.8 15.7 13.7 8.7 16 -37 Texas 1.7 4.1 5.2 5.4 8.7 10.4 8.8 4.4 1 -51 Vermont 2.7 5.4 9.9 9.9 9.5 11.7 10.8 6.5 13 -40 Virginia 2.2 4.1 7.9 7.1 6.8 8.4 7.0 2.9 3 -59 Washington 4.7 6.5 8.5 9.7 11.3 13.3 12.4 7.1 9 -43 West Virginia 12.2 11.2 10.4 12.6 15.7 16.8 14.6 7.2 -7 -51 Wisconsin 2.2 3.8 10.5 14.2 12.1 11.4 9.1 2.8 -25 -69 Wyoming 2.1 3.2 3.4 4.1 7.0 8.1 6.8 0.6 -2 -92 United States 4.4 7.6 11.3 11.2 11.9 14.0 12.4 5.6 4 -55 Note: Recipiency rate refers to the average monthly number of AFDC child recipients in each State during the given fiscal year as a percent of the resident population under 18 years of age as of July 1 of that year. The numerators are from Table TANF 13. Table TANF 15. TANF and Separate State Program (SSP) Families and Recipients, 2002 Alabama 18.0 0.2 18.2 42.8 0.9 43.6 34.0 0.5 34.5 Alaska 6.0 — 6.0 17.6 — 17.6 11.9 — 11.9 Arizona 40.1 — 40.1 94.3 — 94.3 70.3 — 70.3 Arkansas 12.0 — 12.0 27.7 — 27.7 20.6 — 20.6 California 462.3 50.6 512.9 1,160.9 220.6 1,381.5 911.5 131.0 1,042.5 Colorado 12.1 — 12.1 31.5 — 31.5 23.3 — 23.3 Connecticut 23.7 0.9 24.7 53.2 3.4 56.6 37.8 3.7 41.5 Delaware 5.5 0.1 5.6 12.4 0.5 12.9 9.4 0.3 9.7 Dist. of 16.2 0.3 16.5 42.2 0.9 43.0 31.4 0.6 32.0 Florida 59.0 2.1 61.1 123.2 8.4 131.7 99.5 4.3 103.8 Georgia 53.7 0.6 54.2 128.2 2.3 130.5 99.5 1.2 100.8 Guam 3.1 — 3.1 10.8 — 10.8 — — 0.0 Hawaii 11.1 4.7 15.9 30.5 19.4 49.9 21.3 11.2 32.5 Idaho 1.4 — 1.4 2.4 — 2.4 2.0 — 2.0 Illinois 48.1 0.7 48.8 133.7 1.3 135.0 106.9 0.5 107.3 Indiana 49.3 2.5 51.8 138.9 11.2 150.1 99.1 6.4 105.4 Iowa 20.2 1.5 21.7 53.4 1.5 55.0 35.9 — 35.9 Kansas 14.0 — 14.0 35.8 — 35.8 25.3 — 25.3 Kentucky 34.9 — 34.9 77.7 — 77.7 57.4 — 57.4 Louisiana 23.7 — 23.7 60.7 — 60.7 48.5 — 48.5 Maine 9.7 1.7 11.4 26.0 4.5 30.5 17.8 3.0 20.8 Maryland 27.1 2.1 29.3 64.9 6.5 71.4 48.1 4.2 52.3 Massachusetts 47.3 0.1 47.4 108.1 0.3 108.4 76.5 0.2 76.7 Michigan 74.3 — 74.3 201.7 — 201.7 148.8 — 148.8 Minnesota 35.9 3.9 39.7 94.6 18.0 112.6 68.1 10.3 78.4 Mississippi 17.6 — 17.6 40.4 — 40.4 30.5 — 30.5 Missouri 45.0 4.1 49.1 118.8 11.0 129.7 84.4 8.1 92.5 Montana 5.8 — 5.8 16.4 — 16.4 10.8 — 10.8 Nebraska 10.3 1.0 11.3 25.5 4.4 29.9 18.5 2.4 20.9 Nevada 11.0 1.0 12.0 27.6 4.4 32.1 20.5 2.5 23.0 New Hampshire 6.0 — 6.0 14.5 — 14.5 9.9 — 9.9 New Jersey 42.0 1.7 43.7 103.1 7.2 110.3 77.6 4.0 81.6 New Mexico 17.0 — 17.0 47.3 — 47.3 33.7 — 33.7 New York 170.4 33.9 204.4 412.5 117.0 529.5 292.8 77.9 370.7 North Carolina 42.9 0.0 42.9 91.1 0.1 91.2 70.2 0.1 70.3 North Dakota 3.2 — 3.2 8.3 — 8.3 6.0 — 6.0 Ohio 84.0 — 84.0 191.0 — 191.0 142.0 — 142.0 Oklahoma 14.8 — 14.8 36.9 — 36.9 28.3 — 28.3 Oregon 17.9 — 17.9 40.9 — 40.9 30.2 — 30.2 Pennsylvania 80.6 — 80.6 210.5 — 210.5 155.0 — 155.0 Puerto Rico 23.4 — 23.4 67.4 — 67.4 47.4 — 47.4 Rhode Island 14.4 1.2 15.6 39.0 4.6 43.5 27.1 2.6 29.7 South Carolina 21.5 — 21.5 53.3 — 53.3 38.5 — 38.5 South Dakota 2.9 — 2.9 6.6 — 6.6 5.4 — 5.4 Tennessee 63.0 1.0 64.0 164.6 3.7 168.3 118.8 2.2 121.0 Texas 129.9 6.6 136.5 331.4 28.5 359.9 253.1 15.5 268.6 Utah 7.8 0.1 7.8 19.9 0.2 20.1 14.3 0.1 14.4 Vermont 5.1 0.3 5.4 13.4 0.8 14.2 8.6 0.5 9.1 Virgin Islands 0.6 — 0.6 2.3 — 2.3 1.7 — 1.7 Virginia 30.1 0.9 30.9 67.3 3.5 70.8 49.1 1.9 50.9 Washington 54.2 4.2 58.4 137.8 18.4 156.1 95.7 11.8 107.6 West Virginia 15.9 — 15.9 41.6 — 41.6 28.2 — 28.2 Wisconsin 19.0 0.4 19.4 45.2 1.5 46.8 36.7 1.0 37.7 Wyoming 0.5 0.0 0.5 0.8 0.0 0.8 0.7 0.0 0.7 U.S. Total 2,065 128 2,194 5,149 505 5,654 3,841 308 4,149 Note: Some states provide cash and other forms of assistance to specific categories of families (e.g., two-parent families) under Separate State Programs (SSPs) funded out of Maintenance of Effort (MOE) dollars rather than federal TANF funds. Table TANF 16. Recipients with Earnings in Current and Following Quarters, Fiscal Year 2001 Adult TANF Recipients (Thousands) Percentage with Earnings Percentage without Earnings With Earnings inFollowing Quarter Alabama 10.7 39 74 61 22 Alaska 6.7 36 59 64 27 Arizona 24.5 40 73 60 20 Arkansas 8.8 45 76 55 27 California 287.9 43 83 57 14 Colorado 8.4 38 69 62 23 Connecticut 19.9 46 78 54 21 Delaware 3.6 46 74 54 24 Dist. of Columbia 12.6 37 73 63 17 Florida 36.0 42 79 58 23 Georgia 31.3 33 62 67 19 Hawaii 12.6 43 85 57 13 Idaho 0.6 47 79 53 30 Illinois 46.4 44 81 56 19 Indiana 37.5 50 80 50 22 Kansas 12.2 52 77 48 27 Kentucky 25.7 26 69 74 25 Louisiana 17.1 36 63 64 23 Maine 9.8 46 79 54 20 Maryland 20.8 37 71 63 20 Massachusetts 34.6 28 68 72 16 Michigan 61.2 36 68 64 19 Minnesota 41.0 48 77 52 21 Mississippi 10.2 36 70 64 21 Missouri 40.0 51 79 49 25 Montana 6.0 40 71 60 23 Nebraska 8.3 53 78 47 26 Nevada 6.2 48 76 52 21 New Hampshire 5.1 40 75 60 20 New Jersey 34.0 32 76 68 19 New Mexico 20.8 44 76 56 21 New York NA NA NA NA NA North Carolina 28.0 43 72 57 25 North Dakota 2.8 46 80 54 21 Ohio 65.5 44 76 56 22 Oklahoma 9.9 48 75 52 26 Oregon 11.7 30 71 70 16 Pennsylvania 69.7 29 71 71 19 Rhode Island 14.2 39 79 61 16 South Carolina 14.5 47 74 53 24 South Dakota 1.6 30 74 70 19 Tennessee 48.9 49 77 51 21 Texas 109.0 41 77 59 20 Utah 6.6 41 75 59 21 Vermont 6.2 42 77 58 19 Virginia 20.7 47 79 53 24 Washington 54.2 41 75 59 20 West Virginia 15.3 35 74 65 17 Wisconsin 8.3 39 72 61 23 Wyoming 0.3 39 65 61 27 All Reporting States 1,410 41 77 59 19 Note: "TANF adult recipients" is unduplicated roster of adults who received TANF benefits at any time during a quarter, averaged over four quarters in fiscal year. Data are not available for New York, which did not participate in the High Performance Bonus. Note also that TANF receipt and the presence of earnings may occur at different months within the quarter. Source: Unpublished ACF calculations of High Performance Bonus data. Table TANF 17. Patterns of TANF Receipt, Fiscal Year 2001 Adult TANF Recipients in Qtr(t) (Thousands) Percentage of Adult TANF Recipients Also Receiving Benefits in Following Quarters Qtr(t+1) Idaho 0.6 46 19 11 8 Note: "Adult TANF Recipients in Qtr(t)" is unduplicated roster of adults who received TANF benefits at any time during a quarter, averaged over four quarters in fiscal year. Data are not available for New York, which did not participate in the High Performance Bonus. This table examines length of receipt for all recipients receiving TANF in the selected quarter, in contrast to Table IND 8 in Chapter II, which looked at new entrants to AFDC/TANF. Another difference is that in this table, a recipient is counted as a recipient each quarter in which there is at least one month of receipt, even if the recipient has a gap of non-receipt for several months. 1 States are allowed to use TANF funds on a variety of services, including employment and training services, domestic violence services, child care, transportation, and other support services. Families receiving such services, however, generally should not be counted as recipients of TANF "assistance." Under the final regulations for TANF, "assistance" primarily includes payments directed at ongoing basic needs. It includes payments when individuals are participating in community service and work experience (or other work activities) as a condition of receiving payments (e.g., workfare). In addition to cash assistance, the definition also includes certain child care and transportation benefits (provided the families are not employed). It excludes, however, such things as: nonrecurrent, short-term benefits; services without a cash value, such as education and training, case management, job search, and counseling; and benefits such as child care and transportation when provided to employed families. 2 These values are slightly smaller than the usually cited figures on caseload decline, because these figures include recipients in SSPs, who are usually omitted from TANF caseload statistics. Food Stamp Program The Food Stamp Program (FSP), administered by the U.S. Department of Agriculture's (USDA) Food and Nutrition Service, is the largest food assistance program in the country, reaching more poor individuals over the course of a year than any other public assistance program. Unlike many other public assistance programs, FSP has few categorical requirements for eligibility, such as the presence of children, elderly, or disabled individuals in a household. As a result, the program offers assistance to a large and diverse population of needy persons, many of whom are not eligible for other forms of assistance. The Food Stamp Program was designed primarily to increase the food purchasing power of eligible low-income households to the point where they can buy a nutritionally adequate low-cost diet. Participating households are expected to be able to devote 30 percent of their counted monthly cash income (after adjusting for various deductions) to food purchases. Food stamp benefits then make up the difference between the household's expected contribution to its food costs and an amount judged to be sufficient to buy an adequate low-cost diet. This amount, the maximum food stamp benefit level, is derived from USDA's lowest-cost food plan, the Thrifty Food Plan (TFP). The federal government is responsible for virtually all of the rules that govern the program, and, with limited variations, these rules are nationally uniform, as are the benefit levels. Nonetheless, states, the District of Columbia, Guam, and the Virgin Islands, through their local welfare offices, have primary responsibility for the day-to-day administration of the program. They determine eligibility, calculate benefits, and issue food stamp allotments. The Food Stamp Act provides 100 percent federal funding of food stamp benefits. States and other jurisdictions have responsibility for about half the cost of state and local food stamp agency administration. In addition to the regular Food Stamp Program, the Food Stamp Act authorizes alternative programs in Puerto Rico, the Northern Mariana Islands, and American Samoa. The largest of these, the Nutrition Assistance Program in Puerto Rico, was funded under a federal block grant of over $1.3 billion in 2002. Unless noted otherwise, the food stamp caseload and expenditure data in this Appendix exclude costs for the Nutrition Assistance Program (NAP) in Puerto Rico. (Prior editions of this Appendix included NAP, but caseload and expenditure data in this Appendix are now limited to the Food Stamp Program, to be consistent with FSP data published by the USDA.) The Food Stamp Program offers assistance to nearly all financially needy households. To be eligible for food stamps, a household must meet eligibility criteria for gross and net income, asset holdings, work requirements, and citizenship or immigration status. The FSP benefit unit is the household. Generally, individuals living together constitute a household if they customarily purchase and prepare meals together. The income, expenses and assets of the household members are combined to determine program eligibility and benefit allotment. Monthly income is the most important determinant of household eligibility. Except for households composed entirely of TANF, SSI, or General Assistance recipients, gross income cannot exceed 130 percent of poverty. After certain amounts are deducted for living expenses, working expenses, dependent care expenses, excess shelter expenses, child support payment, and - for elderly/disabled households - medical expenses, net income cannot exceed 100 percent of poverty. Households also must not have more than $2,000 in cash, savings, stocks and bonds, and certain vehicles (households with an elderly or disabled member can have up to $3,000 in countable assets). All nonexempt adult applicants for food stamps must register for work. To maintain eligibility, they must accept a suitable job, if offered one, and fulfill any work, job search, or training requirements established by the FSP office. Nondisabled adults living in households with children can receive benefits for three months only, unless they work or participate in work-related activities. Participation is restricted for certain groups, including students, strikers, and people who are institutionalized. Legal immigrants who are disabled, under age 18, or have five years of legal US residency are eligible; all other noncitizens are not. Food stamp benefits are a function of a household's size, its net monthly income, its assets, and maximum monthly benefit levels. Allotments are not taxable and food stamp purchases may not be charged sales taxes. Receipt of food stamps does not affect eligibility for or benefits provided by other welfare programs, although some programs use food stamp participation as a "trigger" for eligibility and others take into account the general availability of food stamps in deciding what level of benefits to provide. Recent Legislative and Regulatory Changes Title IV and subtitle A of title VIII of the Personal Responsibility and Work Opportunity Reconciliation Act of 1996 (PRWORA) contain major and extensive revisions to the Food Stamp Program, including strong work requirements on able-bodied adults without dependent children, restricted eligibility of legal immigrants, and a reduction in maximum benefits. These three provisions, and subsequent amendments, are discussed below; their impact on program participation and expenditures begins to appear in food stamp administrative data for 1997, with the fuller impact shown in data for 1998 and beyond. First, a work requirement was added for able-bodied adult food stamp recipients without dependents (ABAWDs). Unless exempt, ABAWDs between the ages of 18 and 59 are not eligible for benefits for more than 3 months in every 36-month period unless they are (1) working at least 20 hours a week; (2) participating in and complying with a work program for at least 20 hours a week; or (3) participating in and complying with a workfare program. Under the original legislation, the Department of Agriculture was authorized to waive application of the work requirement to any group of individuals at the request of the state agency, if a determination is made that the area where they reside has an unemployment rate over 10 percent or does not have a sufficient number of jobs to provide them employment. The provision was further moderated under the Balanced Budget Act of 1997 (Public Law 105-33), which allowed states to exempt up to 15 percent of the ABAWD caseload (beyond those subject to waivers) and which increased funds for the food stamp employment and training program for the creation of job slots for able-bodied adults subject to time limits. Separately, title IV of PRWORA made significant changes in the eligibility of noncitizens for food stamp benefits. As first enacted, most qualified aliens, including legal immigrants (illegal aliens were already ineligible) were barred from receiving food stamps until citizenship. Subsequently, the Agriculture Research, Extension and Education Reform Act of 1998 (Public Law 105-185) restored food stamp eligibility to certain groups of qualified aliens who were legally residing in the United States before passage of PRWORA on August 22, 1996 and were over 65 years of age on that date or were under age 18 or disabled. Finally, the 1996 legislation restrained growth in future program expenditures by making changes in the benefit structure for eligible participants, including a reduction in the maximum food stamp allotment. Other provisions of the 1996 act disqualified from eligibility those convicted of drug-related felonies and gave states the option to disqualify individuals, both custodial and noncustodial parents, from food stamps when they do not cooperate with child support agencies or are in arrears in their child support. Recent regulatory and legislative changes have been made to increase access to food stamps among working poor families. Regulatory changes announced in July 1999 and expanded in November 2000 allow states to reduce reporting requirements and make it easier for working families to report income changes on a semiannual basis. Under the November 2000 regulations, states also have the option of providing a three-month transitional food stamp benefit to most families leaving TANF. In addition, the Agriculture Appropriations Bill for 2001 (P.L. 106-387) provides states with the option of liberalizing the treatment of vehicle assets to align with the states' TANF rules on vehicle eligibility. These changes were intended to address concerns that some of the decline in food stamp caseloads may be leaving poor families without nutritional assistance as they make the transition from welfare dependence to full self-sufficiency. The Farm Security and Rural Investment Act of 2002 - also known as the Farm Bill - reauthorized the Food Stamp Program through fiscal year 2007. This law brought a number of significant changes to the program, including some which supercede earlier changes made through PRWORA and subsequent FSP legislation and regulations. Specifically, the Farm Bill restores food stamp eligibility to legal immigrants who have lived in the country five years and to legal immigrants receiving disability benefits, regardless of entry date. Children of legal immigrants are also eligible for food stamps regardless of entry date. Effective in fiscal year 2004, the requirement that income and resources of an immigrant's sponsor be counted in determining the eligibility and benefit amounts for immigrant children is eliminated. Each provision became effective at different times, but all restorations were in effect by October 1, 2003. The Farm Bill also increased the asset limit from $2,000 to $3,000 for households with a disabled member, making it consistent with the limit for households with elderly, and replaced the fixed standard deduction with a deduction that varies according to household size and is indexed to cost-of-living increases, in recognition of the higher expenses larger households incur. For households in the 48 contiguous states and DC, Alaska, Hawaii and the Virgin Islands, the deduction is set at 8.31 percent of the applicable net income limit based on household size. (Households in Guam will receive a slightly higher deduction.) No household receives an amount less than the previous fixed standard deduction or more than the standard deduction for a household of six. Other Farm Bill changes include the authorization of $5 million per year for education and outreach grants to help inform the low-income public of their eligibility for food stamps, and increased flexibility for states in spending Employment and Training program funds to promote work. States also are now allowed to extend from three months to up to five months the period of time households may receive transitional food stamp benefits when they lose TANF cash assistance. Benefits are equal to the amount the household received prior to termination of TANF with adjustments in income for the loss of TANF. This change helps individuals moving off cash assistance to make the transition from welfare to work. The Farm Bill also implemented a number of administrative reforms and program simplifications, including: changing the quality control system so that only those states with persistently high error rates will face liabilities; awarding bonuses to states that improve the quality and accuracy of their service; allowing states to exclude certain types of income and resources not counted under TANF or Medicaid, such as educational assistance, when determining food stamp eligibility; allowing states to deem child support payments as income exclusions rather than deductions as an incentive for parents to pay child support; allowing states to simplify the standard utility allowance (SUA) if the state elects to use the SUA rather than actual utility costs for all households, thus reducing administrative burden, costs and errors; permitting states to use a standard deduction from income of $143 per month for homeless households with some shelter expenses; allowing states to extend simplified reporting procedures to all households, not just households with earnings; eliminating the requirement that the Electronic Benefit Transfer (EBT) system be cost-neutral to the federal government to help support the EBT conversion process; allowing USDA to use alternative methods for issuing food stamp benefits during times of disaster when use of EBT is impractical; requiring food stamp applications be made available through the Internet; and combining Puerto Rico and American Samoa's block grants into one grant and indexing both with inflation. Food Stamp Program Data The following six tables and accompanying figure provide information about the Food Stamp Program: Tables FSP 1-2 and Figure FSP 1 present national caseload and expenditure trend data on the Food Stamp Program as discussed below; Table FSP 3 presents some demogaphic characteristics of the food stamp caseload; and Tables FSP 4-6 present some state-by-state trend data for the FSP through fiscal year 2002. Food Stamp Caseload Trends (Table FSP 1). Average monthly food stamp participation was 19.1 million persons in fiscal year 2002, excluding the participants in Puerto Rico's block grant. This represents a significant increase over the fiscal year 2000 record-low average of 17.1 million participants. It is, however, far below the peak of 27.5 million recipients in fiscal year 1994. Both in absolute numbers and as a percentage of the population, food stamp recipiency in 2000 was lower than at any point in the previous twenty years. See also Table IND 3b and Table IND 4b in Chapter II for further data trends in food stamp caseload, specifically, food stamp recipiency and participation rates. Considerable research has demonstrated that the Food Stamp Program is responsive to economic changes, with participation increasing in times of economic downturns and decreasing in times of economic growth (see Figure FSP 1). Economic conditions alone did not explain the caseload growth in the late 1980s and early 1990s, however. Studies suggest that a variety of factors contributed to this caseload growth, including a weak economy and higher rates of unemployment, expansions in Medicaid eligibility, the legalization of 3 million undocumented immigrants, and longer participation spells (McConnell, 1991; Gleason, 1998). The decline in participation from 1994 to 2000 was caused by several factors, according to studies of this period. Part of the decline is associated with the strong economy in the second half of the 1990s. However, participation fell more sharply than expected during this period of sustained economic growth. Some of the decline reflected restrictions on the eligibility of noncitizens and time limits for unemployed nondisabled childless adults. The three groups where participation fell most rapidly included noncitizens and their US-born children, unemployed nondisabled childless adults, and persons receiving cash welfare benefits. As people left the welfare rolls, many also stopped participating in food stamps, even while remaining eligible (Genser, 1999; Wilde et al., 2000; Gleason et al., 2001; Kornfeld, 2002). The increase in FSP participation from 2000 to 2002 occurred during a period when unemployment increased from four percent to six percent, states took advantage of opportunities to expand categorical eligibility to those receiving in-kind TANF benefits and liberalize the treatment of vehicles, and the Food and Nutrition Service was encouraging states to conduct outreach efforts. Food Stamp Expenditures. Total program costs, shown in Table FSP 2, were considerably higher in 2002 than 2001, reflecting the increase in participation during that period as well as an increase in average benefits. Total federal program costs were $20.7 billion in 2002; the comparable 2001 cost was $18.1 billion (after adjusting for inflation). Average monthly benefits per person, also shown in Table FSP 2, were $79.60 per person in fiscal year 2002, up from $74.80 in 2001. This increase in benefits reverses a six-year decline in average monthly benefits adjusted to 2002 dollars. Food Stamp Household Characteristics. As shown in Table FSP 3, the proportion of food stamp households with earnings has increased, from about 20 percent for most of the 1980s and early 1990s, to 28 percent in 2002. At the same time, the proportion of households with income from AFDC/TANF has declined, from 43 percent in 1990 to 21 percent in 2002, following the dramatic decline in AFDC/TANF caseloads. Over half of all food stamp households have children, although the proportion has declined somewhat from over 60 percent in most of the 1980s and early 1990s to 54 percent in 2002. The vast majority (88 percent) of households have incomes below the federal poverty guidelines. Figure FSP 1. Persons Receiving Food Stamps: 1962 – 2002 Note: Shaded areas are periods of recession as defined by the National Bureau of Economic Research. Source: U.S. Department of Agriculture, Food and Nutrition Service, National Data Bank. Table FSP 1. Trends in Food Stamp Caseloads, Selected Years: 1962 – 2002 Food Stamp Participants Participants as a Percent of: Child Participants as a Percent of: Including Territories1 Excluding Territories Children Excld. Terr. Total Population2 All Poor Persons2 Pre-transfer Poverty Population3 Total Child Population2 Children in Poverty2 1962 6,554 6,554 NA 3.5 17.0 NA NA NA 1971 13,010 13,010 NA 6.3 50.9 NA NA NA 1975 4 17,152 16,320 NA 7.6 63.1 NA NA NA 1976 18,628 17,033 9,126 7.8 68.2 NA 13.8 88.8 1979 5 17,758 15,942 NA 7.1 61.1 57.1 NA NA 1980 21,173 19,253 9,876 8.5 65.8 60.7 15.5 85.6 1983 21,727 20,095 10,910 8.6 61.4 58.5 17.4 78.4 1993 26,982 26,952 14,196 10.4 68.6 63.8 21.0 90.3 1 Total participants includes all participating states, the District of Columbia, and the territories (including Puerto Rico from 1975 to 1982 —a separate Nutrition Assistance Grant for Puerto Rico was begun in July 1982). From 1962 to 1983 the number of participants includes the Family Food Assistance Program (FFAP) that was largely replaced by the FSP in 1975. The FFAP participants (as of December) for the seven years shown during the period from 1962 to 1974 were respectively: 6,411; 4,742; 3,977; 3,642; 3,002; 2,441; and 1,406 (all in thousands). From 1975 to 1983 the number of FFAP participants averaged only 88 thousand. 2 Includes all participating states and the District of Columbia only — the territories are excluded from both numerator and denominator. Population numbers used as denominators are the resident population — see Current Population Reports, Series P25-1106. For the persons living in poverty used as denominators, see Current Population Reports, Series P60-210. 3 The pretransfer poverty population used as denominator is the number of all persons in families or living alone whose income (cash income plus social insurance plus Social Security but before taxes and means-tested transfers) falls below the appropriate poverty threshold. See Appendix J, Table 20, 1992 Green Book; data for subsequent years are unpublished Congressional Budget Office tabulations. 4 The first fiscal year in which food stamps were available nationwide. 5 The fiscal year in which the food stamp purchase requirement was eliminated, on a phased- in basis. Source: U.S. Department of Agriculture, Food and Nutrition Service, National Data Bank, the 1996 Green Book, and U.S. Bureau of the Census, "Poverty in the United States: 2002," Current Population Reports, Series P60-222 and earlier years. Table FSP 2. Trends in Food Stamp Expenditures, Selected Years: 1975 – 2002 Total Federal Cost (Benefits + Administration) Benefitsh (Federal) Total Program Cost Average Monthly Benefit per Person 2002 Dollars2 1975 $4,619 $15,245 $4,386 $233 $175 $4,794 $21.30 $70.30 1976 5,685 17,567 5,326 359 270 5,955 23.90 73.80 19793 6,940 17,219 6,480 460 388 7,328 30.50 75.70 1981 11,225 22,769 10,630 595 504 11,729 39.50 80.10 19844 11,579 20,131 10,696 8835 805 12,384 42.70 74.20 1986 11,638 19,049 10,605 1,033 935 12,573 45.50 74.50 1988 12,316 18,832 11,149 1,168 1,080 13,396 49.80 76.10 1 Amounts include the federal share of state administrative and employment and training costs and certain direct federal administrative costs. They do not generally include approximately $60 million in food stamp- related federal administrative costs budgeted under a separate appropriation account (although estimates prior to 1989 do include estimates of food stamp related federal administrative expenses paid out of other Agriculture Department accounts). State and local costs are estimated based on the known federal shares and represent an estimate of all administrative expenses of participating states. 2 Constant dollar adjustments to 2002 level were made using a CPI-U-X1 fiscal year average price index. 3 The fiscal year in which the food stamp purchase requirement was eliminated, on a phased - in basis. 4 Beginning 1984 USDA took over from DHHS the administrative cost of certifying public assistance households for food stamps. Note: Total federal cost includes food stamps in Puerto Rico (1975-1982). This table differs from the versions published in previous years in that it does not include the costs of the Family Food Assistance Program in the period from 1975 to 1983. The cost of benefits does include food stamps in Puerto Rico from 1975 to 1982 but (for consistency with the reporting of the Food and Nutrition Service) the total expenditures for benefits does not include the funding for the Puerto Rico nutrition assistance grant from the last quarter of FY 1982 when it replaced Puerto Rico's food stamp program to the present (Puerto Rico's nutrition assistance grant was $778 million in 1983 and rose to over $1.3 billion in 2002. ) Source:USDA, Food and Nutrition Service unpublished data from the National Data Bank; and the 2000 Green Book. Table FSP 3. Characteristics of Food Stamp Households, 1980 – 2002 Year1 With Gross Monthly Income: Below the Federal Poverty Levels 87 93 92 92 92 90 91 90 89 88 Between the Poverty Levels and 130 Percent of the Poverty Levels 10 6 8 8 8 9 8 9 10 11 Above 130 Percent of Poverty 2 1 * * * 1 1 1 1 1 With Earnings 19 19 20 19 21 21 23 26 27 28 With Public Assistance Income2 65 71 72 73 66 69 67 65 63 56 With AFDC/TANF Income NA 42 42 43 40 38 37 31 26 21 With SSI Income 18 18 20 19 19 23 24 28 32 29 With Children 60 61 61 61 62 61 60 58 54 54 And Female Heads of Household NA 47 50 51 51 51 50 47 44 44 With No Spouse Present NA NA 39 37 44 43 43 41 38 37 With Elderly Members3 23 22 19 18 15 16 16 18 21 19 With Elderly Female Heads of Household3 NA 16 14 11 9 11 NA NA NA NA Average Household Size 2.8 2.8 2.8 2.7 2.6 2.6 2.5 2.4 2.3 2.3 1 Data were gathered in August in the years 1980-84 and during the summer in the years from 1986 to 1994. Reports from 1995 to the present are based on fiscal year averages. 2 Public assistance income includes AFDC/TANF, SSI, and general assistance. 3 Elderly members and heads of household include those of age 60 or older. * Less than 0.5 percent. Source: U.S. Department of Agriculture, Food and Nutrition Service, Office of Analysis, Nutrition, and Evaluation, Characteristics of Food Stamp Households, Fiscal Year 2002 and earlier years. Table FSP 4. Value of Food Stamps Issued by State, Selected Fiscal Years 1975 – 2002 Alabama $103 $246 $318 $328 $441 $357 $344 $417 Alaska 6 27 25 25 50 50 46 59 Arizona 41 97 121 239 414 253 240 386 Arkansas 78 122 126 155 212 206 206 265 California 361 530 639 968 2,473 2,020 1,639 1,707 Colorado 44 71 94 156 217 157 127 165 Connecticut 36 59 62 72 169 161 138 146 Delaware 6 21 22 25 47 34 31 39 Dist. of Columbia 31 41 40 43 92 85 77 76 Florida 207 421 368 609 1,307 845 771 878 Georgia 129 264 290 382 700 538 489 621 Guam 2 15 18 15 24 34 36 52 Hawaii 23 60 93 81 177 178 166 152 Idaho 11 29 36 40 59 47 46 62 Illinois 238 394 713 835 1,056 844 746 923 Indiana 58 154 242 226 382 263 268 408 Iowa 28 54 107 109 142 109 100 129 Kansas 12 38 64 96 144 83 83 113 Kentucky 135 211 332 334 413 345 337 410 Louisiana 148 243 365 549 629 467 448 587 Maine 31 60 62 63 112 100 81 97 Maryland 76 140 171 203 365 282 199 215 Massachusetts 75 171 173 207 315 222 182 209 Michigan 124 263 541 663 806 588 457 645 Minnesota 40 62 105 165 240 181 165 201 Mississippi 110 199 264 352 383 254 226 298 Missouri 82 142 212 312 488 345 358 477 Montana 11 18 31 41 57 52 51 58 Nebraska 11 25 44 59 77 68 61 74 Nevada 10 15 22 41 91 63 57 96 New Hampshire 11 22 15 20 44 30 28 35 New Jersey 125 226 260 289 506 384 304 314 New Mexico 48 81 88 117 196 144 140 154 New York 209 726 938 1,086 2,065 1,505 1,361 1,479 North Carolina 122 234 237 282 495 421 403 536 North Dakota 5 9 16 25 32 25 25 31 Ohio 253 382 697 861 1,017 613 520 726 Oklahoma 38 73 134 186 315 231 208 288 Oregon 56 80 142 168 254 198 198 319 Pennsylvania 175 373 547 661 1,006 764 656 700 Rhode Island 18 31 35 42 82 57 59 64 South Carolina 121 181 194 240 297 264 249 352 South Dakota 8 18 26 35 40 37 37 45 Tennessee 115 282 280 372 554 437 415 552 Texas 314 514 701 1,429 2,246 1,425 1,215 1,522 Utah 12 22 40 71 90 75 68 80 Vermont 9 18 20 22 46 34 32 34 Virgin Islands 6 19 23 18 28 22 21 17 Virginia 63 158 189 247 450 307 263 305 Washington 70 90 140 229 417 308 241 318 West Virginia 56 87 159 192 253 224 185 198 Wisconsin 29 68 148 180 220 130 129 197 Wyoming 3 6 15 21 28 21 19 22 United States $4,386 $8,721 $10,744 $14,186 $22,764 $16,889 $14,952 $18,257 Note: The totals for 1975 and 1980 include amounts for Puerto Rico of $366 and $828 million respectively. Source: U.S. Department of Agriculture, Food and Nutrition Service, unpublished data from the Food Stamp National Data Bank. Table FSP 5. Average Number of Food Stamp Recipients by State, Selected Fiscal Years Alabama 365 583 588 454 509 396 411 444 12 -13 Alaska 15 29 22 25 46 38 38 46 84 -0 Arizona 143 196 206 317 427 259 291 379 35 -11 Arkansas 267 301 253 235 274 247 256 284 17 4 California 1,455 1,493 1,615 1,955 3,143 1,832 1,668 1,710 61 -46 Colorado 150 163 170 221 244 156 154 178 10 -27 Connecticut 155 170 145 133 223 165 157 169 67 -24 Dist. of Columbia 122 103 72 62 93 81 73 74 49 -20 Florida 647 912 630 781 1,371 882 887 985 75 -28 Georgia 498 627 567 536 793 559 574 646 48 -19 Guam 6 22 20 12 18 22 23 24 50 39 Hawaii 75 102 99 77 130 118 108 106 69 -18 Idaho 39 61 59 59 80 58 60 70 36 -12 Illinois 926 903 1,110 1,013 1,105 760 825 886 9 -20 Indiana 392 353 406 311 390 300 347 411 25 5 Iowa 115 141 203 170 177 123 126 141 4 -21 Kansas 58 90 119 142 172 117 124 140 21 -18 Kentucky 472 468 560 458 486 403 413 450 6 -7 Louisiana 510 569 644 727 670 500 518 588 -8 -12 Maine 126 139 114 94 131 102 104 111 39 -15 Maryland 261 324 287 255 375 219 208 228 47 -39 Massachusetts 365 453 337 347 374 232 219 243 8 -35 Michigan 619 813 985 917 935 603 641 750 2 -20 Minnesota 167 171 228 263 295 196 198 217 12 -26 Mississippi 376 496 495 499 457 276 298 325 -8 -29 Missouri 300 335 362 431 554 423 454 515 28 -7 Montana 38 43 58 57 71 59 62 63 25 -10 Nebraska 49 66 94 95 102 82 81 88 7 -13 Nevada 32 32 32 50 97 61 69 97 94 0 New Hampshire 44 50 28 31 53 36 36 41 73 -22 New Jersey 490 605 464 382 540 345 318 320 42 -41 New Mexico 157 185 157 157 235 169 163 170 49 -27 New York 1,291 1,759 1,834 1,548 2,099 1,439 1,354 1,347 36 -36 North Carolina 466 582 474 419 631 488 494 574 51 -9 North Dakota 19 25 33 39 40 32 38 37 2 -8 Ohio 854 865 1,133 1,089 1,045 610 641 735 -4 -30 Oklahoma 171 209 263 267 354 253 271 317 33 -10 Pennsylvania 848 980 1,032 952 1,124 777 748 767 18 -32 South Carolina 410 426 373 299 358 295 316 379 20 6 South Dakota 33 43 48 50 49 43 45 48 -3 -2 Tennessee 397 624 518 527 638 496 522 598 21 -6 Texas 1,133 1,167 1,263 1,880 2,372 1,333 1,361 1,554 26 -34 Utah 46 54 75 99 110 82 80 90 11 -18 Vermont 44 46 44 38 56 41 39 40 47 -29 Virgin Islands 16 34 32 18 31 16 13 12 75 -59 Virginia 257 384 360 346 538 336 332 352 55 -34 Washington 253 248 281 340 478 295 309 350 41 -27 West Virginia 242 209 278 262 300 227 221 236 14 -21 Wisconsin 148 215 363 286 283 193 216 262 -1 -7 Wyoming 10 14 27 28 33 22 23 24 17 -29 United States 17,192 21,082 19,899 20,067 25,542 17,139 17,313 19,094 27 -25 Note: The totals for 1975 and 1980 include recipients in Puerto Rico of 810 thousand and 1.86 million respectively. Source: U.S. Department of Agriculture, Food and Nutrition Service, unpublished data from the National Data Bank. Table FSP 6. Food Stamp Recipiency Rates by State, Selected Fiscal Years Alabama 9.9 14.9 14.8 11.2 11.8 8.9 9.2 9.9 5 -16 Alaska 4.0 7.1 4.1 4.5 7.6 6.0 6.0 7.2 67 -6 Arizona 6.3 7.1 6.5 8.6 9.3 5.0 5.5 6.9 8 -26 Arkansas 12.4 13.1 10.9 10.0 10.6 9.2 9.5 10.5 7 -2 Colorado 5.8 5.6 5.3 6.7 6.2 3.6 3.5 4.0 -7 -36 Delaware 4.5 8.7 6.5 5.0 7.8 4.1 4.0 4.9 57 -37 Dist. of Columbia 17.2 16.1 11.4 10.3 16.2 14.1 12.8 13.0 58 -20 Georgia 9.8 11.4 9.5 8.2 10.6 6.8 6.8 7.5 28 -29 Hawaii 8.4 10.6 9.5 6.9 10.8 9.7 8.8 8.5 57 -21 Illinois 8.2 7.9 9.7 8.8 9.1 6.1 6.6 7.0 3 -23 Indiana 7.3 6.4 7.4 5.6 6.6 4.9 5.7 6.7 18 1 Iowa 4.0 4.8 7.2 6.1 6.2 4.2 4.3 4.8 0 -22 Kansas 2.5 3.8 4.9 5.7 6.6 4.3 4.6 5.2 15 -21 Kentucky 13.6 12.8 15.2 12.4 12.4 10.0 10.1 11.0 -0 -11 Louisiana 13.1 13.5 14.6 17.2 15.2 11.2 11.6 13.1 -12 -14 Maine 11.8 12.3 9.8 7.6 10.5 8.0 8.1 8.6 38 -18 Maryland 6.3 7.7 6.5 5.3 7.3 4.1 3.9 4.2 38 -43 Massachusetts 6.3 7.9 5.7 5.8 6.0 3.6 3.4 3.8 5 -38 Michigan 6.8 8.8 10.8 9.8 9.6 6.1 6.4 7.5 -3 -22 Minnesota 4.2 4.2 5.5 6.0 6.3 4.0 4.0 4.3 4 -31 Mississippi 15.7 19.6 19.1 19.4 16.6 9.7 10.4 11.3 -14 -32 Missouri 6.2 6.8 7.2 8.4 10.2 7.6 8.1 9.1 21 -11 Montana 5.1 5.5 7.1 7.1 8.0 6.6 6.8 7.0 13 -13 Nebraska 3.2 4.2 5.9 6.0 6.1 4.8 4.7 5.1 2 -16 New Jersey 6.7 8.2 6.1 4.9 6.6 4.1 3.7 3.7 35 -44 New Mexico 13.5 14.1 10.9 10.3 13.4 9.3 8.9 9.2 30 -31 New York 7.2 10.0 10.3 8.6 11.3 7.6 7.1 7.0 31 -38 North Dakota 2.9 3.9 4.9 6.1 6.1 5.0 5.9 5.8 -0 -5 Ohio 7.9 8.0 10.6 10.0 9.3 5.4 5.6 6.4 -7 -31 Oklahoma 6.2 6.9 8.0 8.5 10.6 7.3 7.8 9.1 25 -14 Oregon 8.6 7.5 8.5 7.6 8.9 6.8 8.2 10.2 17 15 Pennsylvania 7.1 8.3 8.8 8.0 9.2 6.3 6.1 6.2 15 -32 South Carolina 14.1 13.6 11.3 8.5 9.4 7.3 7.8 9.2 10 -2 South Dakota 4.8 6.2 6.9 7.2 6.6 5.7 5.9 6.3 -9 -5 Tennessee 9.3 13.6 11.0 10.8 11.8 8.7 9.1 10.3 9 -12 Texas 9.0 8.1 7.8 11.0 12.3 6.4 6.4 7.1 11 -42 Utah 3.7 3.7 4.6 5.7 5.3 3.7 3.5 3.9 -7 -27 Virginia 5.1 7.2 6.3 5.6 8.0 4.7 4.6 4.8 43 -39 Washington 7.0 6.0 6.4 6.9 8.6 5.0 5.1 5.8 24 -33 West Virginia 13.1 10.7 14.6 14.6 16.4 12.6 12.3 13.1 13 -20 Wisconsin 3.2 4.6 7.6 5.8 5.4 3.6 4.0 4.8 -7 -11 Wyoming 2.7 3.0 5.4 6.2 6.8 4.5 4.6 4.7 8 -30 United States 7.6 8.5 8.3 8.0 9.5 6.1 6.1 6.6 18 -30 Note: Recipiency rate refers to the average monthly number of food stamp recipients in each state during the particular fiscal year expressed as a percent of the total resident population as of July 1 of that year. The numerator is from Table FSP 5. Source: U.S. Department of Agriculture, Food and Nutrition Service, unpublished data from the National Data Bank and U.S. Bureau of the Census, (Resident population by state available online at http://www.census.gov). The Supplemental Security Income (SSI) Program is a means-tested, federally administered income assistance program authorized by title XVI of the Social Security Act. Established in 1972 (Public Law 92-603) and begun in 1974, SSI provides monthly cash payments in accordance with uniform, nationwide eligibility requirements to needy aged, blind and disabled persons. To qualify for SSI payments, a person must satisfy the program criteria for age, blindness or disability. Children may qualify for SSI if they are under age 18 and meet the applicable SSI disability or blindness, income and resource requirements. Individuals and married couples are eligible for SSI if their countable incomes fall below the Federal maximum monthly SSI benefit levels of $552 for an individual and $829 for a married couple in fiscal year 2003. SSI eligibility is restricted to qualified persons who have countable resources/assets of not more than $2,000, or $3,000 for a couple. The Social Security Administration (SSA) administers the SSI program. Since its inception, SSI has been viewed as the "program of last resort." Therefore, SSA helps recipients obtain any other public assistance that they are eligible to receive before providing SSI benefits. After evaluating all other income, SSI pays what is necessary to bring an individual to the statutorily prescribed income "floor." As of December 2001, 36 percent of all SSI recipients also received Social Security retirement or survivor benefits, which are the single greatest source of income for SSI recipients. Prior to the Personal Responsibility and Work Opportunity Reconciliation Act of 1996 (PRWORA), no individual could receive both SSI payments and Aid to Families with Dependent Children (AFDC) benefits. If eligible for both, the individual had to choose which benefit to receive. Generally, the AFDC agency encouraged individuals to file for SSI and, once the SSI payments had started, the individual was removed from the AFDC filing unit. Since states have the authority to set TANF eligibility standards and benefit levels under PRWORA, individuals are not prohibited from receiving both TANF benefits and SSI. With the exception of California, which converted food stamp benefits to cash payments that are included in the State supplementary payment, SSI recipients may be eligible to receive food stamps. If all household members receive SSI, the household is categorically eligible for food stamps and does not need to meet the Food Stamp Program's financial eligibility standards. If SSI beneficiaries live in households in which other household members do not receive SSI benefits, the household must meet the net income eligibility standard of the Food Stamp Program to be eligible for food stamp benefits. Several legislative changes made in the 104th Congress affected SSI participation and expenditures. Public Law 104-121, the Contract with America Advancement Act of 1996, prohibited SSI eligibility to individuals whose drug addiction and/or alcoholism (DAA) is a contributing factor material to the finding of disability. This provision applied to individuals who filed for benefits on or after the date of enactment (March 29, 1996) and to individuals whose claims were finally adjudicated on or after the date of enactment. It applied to current beneficiaries on January 1, 1997. PRWORA made several changes designed to maintain the SSI program's goal of limiting benefits to severely disabled children. First, the act replaced the former "comparable severity" test with a new definition of disability specifically for children, based on a medically determinable physical or mental impairment that result in "marked and severe functional limitations." Second, SSA discontinued use of the Individualized Functional Assessment (IFA) which it had implemented in 1991 following the Supreme Court's decision inSullivanv.Zebley, 493 U.S. 521(1990)(3).1Third references to "maladaptive behaviors" in certain sections of the Listing of Impairments (among medical criteria for evaluation of mental and emotional disorders in the domain of personal/behavioral function) were eliminated. The latter two provisions were effective for all new and pending applications upon enactment (August 22, 1996). Beneficiaries who were receiving benefits due to an IFA or under the Listings because of limitations resulting from maladaptive behaviors received notice no later than January 1, 1997, that their benefits might end when their case was redetermined. Additional provisions of the PRWORA with impact on enrollment are the requirement that eligibility be redetermined when beneficiaries reach age 18, using the adult disability standard; that "continuing disability reviews" be done for children; and that children who were eligible due to low birth weight have their eligibility redetermined at age one. Title IV of PRWORA also made significant changes in the eligibility of noncitizens for SSI benefits. Some of the restrictions were subsequently moderated, most notably by the Balanced Budget Act of 1997 (Public Law 105-33), which "grandfathered" immigrants who were receiving SSI at the time of enactment of the PRWORA. Those immigrants who entered the U.S. after August 22, 1996, may be eligible to receive SSI after having been "lawfully admitted for permanent residence." Several provisions aimed at reducing SSI fraud and improving recovery of overpayments were enacted in 1999 as part of the Foster Care Independence Act of 1999 (P. L. 106-169). Other legislation enacted in 1999 provides additional work incentives for disabled beneficiaries of SSI. SSI Program Data The following tables and figures provide SSI program data: Tables SSI 1 through SSI 5 present national caseload and expenditure trend data on the SSI program. Table SSI 6 presents demographic characteristics of the SSI caseload. Tables SSI 7 and SSI 8 present state-by-state trend data on the SSI program through fiscal year 2002. SSI Caseload Trends(Tables SSI 1-2 and Figure SSI 1). From 1990 to 1995, the number of SSI beneficiaries increased from 4.8 million to 6.5 million, an average growth rate of over 6 percent per year. Between 1995 and 2000, the number of beneficiaries fluctuated between 6.5 and 6.6 million persons. In December 2002, there were 6.8 million beneficiaries. Table SSI 1 presents information on the total number of persons receiving SSI payments in December of each year from 1974 through 2002, and also presents recipients by eligibility category (aged, blind and disabled) and by type of recipient (child, adult age 18-64, and adult age 65 or older). See also Table IND 4c in Chapter II for further data on trends in recipiency and participation rates. The composition of the SSI caseload has been shifting over time, as shown in Table SSI 2. The number of beneficiaries eligible because of age has been declining steadily, from a high of 2.3 million persons in December 1975 to less than 1.3 million persons in December 2002. At the same time, there has been strong growth in blind and disabled beneficiaries, from 1.7 million in December 1974 to 5.5 million in December 2002. Moreover, the number of disabled children has increased dramatically, particularly during the 1990s, when the number of disabled children receiving SSI increased from 309,000 in December 1990 to 955,000 in December 1996. The number of disabled children fell in the next three years, stabilized at 847,000 in 1999 and 2000, and rose to 915,000 in 2002. Several factors have contributed to the growth of the Supplemental Security Income program. Expansions in disability eligibility (particularly for mentally impaired adults and for children), increased outreach, overall growth in immigration, and transfers from state programs were among the key factors identified in a 1995 study by the General Accounting Office (GAO). GAO concluded that three groups— adults with mental impairments, children, and non-citizens— accounted for nearly 90 percent of the SSI program's growth in the early 1990s. The growth in disabled children beneficiaries is generally believed to be due to outreach activities, the Supreme Court decision in theZebleycase, expansion of the medical impairment category, and reduction in reviews of continuing eligibility.2 SSI Expenditures.While administrative costs increased by about 1 percent, the total amount paid out in SSI benefits increased from $33.6 billion (inflation adjusted) in 2001 to $34.6 billion in 2002, as shown in Table SSI 3. Average monthly benefits per person were $415 in 2002, up slightly from 2001 inflation adjusted benefit level of $413. For more details see Table SSI 4. SSI Recipient Characteristics.Over the last 20 years, the percentage of aged SSI recipients has dramatically decreased, while the percentage of disabled recipients has increased substantially. As shown in Table SSI 6, the proportion of SSI recipients aged 65 or older has decreased dramatically, from 54 percent in 1980 to 29 percent in 2002. Figure SSI 1.SSI Recipients by Age, 1974– 2002 Source:Social Security Administration, Office of Research, Evaluation, and Statistics,Social Security Bulletin· Annual Statistical Supplement· 2003(Data available online athttp://www.ssa.gov/statistics). Table SSI 1. Number of Persons Receiving Federally Administered SSI Payments 1974– 2002 Eligibility Category Type of Recipient Blind and Disabled Dec 1974 3,996 2,286 1,710 75 1,636 711 1,503 2,422 1Includes students 18-21 in 1974 only. Table SSI 2. SSI Recipiency Rates, 1974– 2002 All Recipients as a Percent of Total Population1 Adults 18-64 as a Percent of 18-64 Population1 Child Recipients as a Percent of All Children1 Elderly Recipients (Persons 65 & Older) as a Percent of All Persons 65 & Older1 All Elderly Poor2 Pretransfer Elderly Poor3 Dec 1974 1.9 1.2 0.1 10.8 78.5 NA Dec 1977 1.9 1.3 0.2 9.7 74.1 NA Dec 1979 1.8 1.3 0.3 8.8 61.3 66.8 1Population numbers used for the denominators are Census Bureau resident population estimates adjusted to the December date by averaging the July 1 population of the current year with the July 1 population of the following year (resident population estimates by age are available online athttp://www.census.gov). 2For the number of persons (65 years of age and older living in poverty) used as the denominator, seeCurrent Population Reports, Series P60-222. 3The pretransfer poverty population used as the denominator is the number of all elderly persons living in elderly-only units whose income (cash income plus social insurance plus Social Security but before taxes and means-tested transfers) falls below the appropriate poverty threshold. See Appendix J, Table 20,1992 Green Book;data for subsequent years are unpublished Congressional Budget Office tabulations. Notes:Numerators for these ratios are from Table SSI 1. Rates computed by DHHS. Source:1994 Green Bookand U.S. Bureau of the Census, "Poverty in the United States: 2002"Current Population Reports, Series P60-222 and earlier years, (Available online athttp://www.census.gov/hhes/www/poverty.html). Table SSI 3. Total, Federal, and State SSI Benefits and Administration, 1974– 20021 Total Benefits State Supplementation Administrative Costs (fiscal year) 2002 2 Dollars Federally Administered 1974 $8,183 $5,246 $3,833 $1,413 $1,264 $149 $285 1975 18,817 5,878 4,314 1,565 1,403 162 399 1984 17,958 10,372 8,281 2,091 1,792 299 864 1986 19,830 12,081 9,498 2,583 2,243 340 1,022 1987 20,510 12,951 10,029 2,922 2,563 359 976 1989 21,733 14,980 11,606 3,374 2,955 419 1,051 1Payments and adjustments during the respective year but not necessarily accrued for that year 2Data adjusted for inflation by ASPE using the CPI-U-X1 for calendar years Source:Social Security Administration, Office of Research, Evaluation, and Statistics,Social Security Bulletin· Annual Statistical Supplement· 2003, (Data available online at http://wwwssagov/statistics). Table SSI 4. Average Monthly SSI Benefit Payments, 1974– 2002 1974 $466 $135 $108 $64 $71 $35 1975 360 112 92 66 69 45 1977 349 123 104 69 72 53 1985 367 219 193 99 99 102 1994 411 338 310 105 99 152 1997 413 369 342 99 102 86 1Total is a weighted average of the Federal plus State average benefit, the Federal-only average benefit, and State-only average benefit. Note:The numerators for these averages are given in Table SSI 3 and the denominators are given in Table SSI 5. Averages were computed by DHHS. Data adjusted for inflation using a calendar-year average CPI-U-X1 index. Source:Number of persons receiving payments obtained from Social Security Administration, Office of Research, Evaluation, and Statistics,Social Security Bulletin· Annual Statistical Supplement· 2003. Table SSI 5. Number of Persons Receiving SSI Payments by Type of Payment, 1974– 2002 Federally Administered State Administered Jan 1974 3,249 2,956 1,839 1,480 358 Dec 1975 4,360 3,893 1,987 1,684 303 Table SSI 6. Characteristics of SSI Recipients, by Age, Sex, Earnings/Income,and Citizenship: Selected Years, 1980-2002 Ages 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 under 18 5.5 5.5 6.4 10.0 13.4 13.5 12.8 13.5 18-64 40.9 45.4 50.9 52.3 53.0 54.8 56.7 57.2 65 or older 53.6 49.1 42.7 37.7 33.7 31.6 30.5 29.3 Male 34.4 35.2 37.2 39.0 41.3 41.3 41.5 42.0 Female 65.5 64.8 62.8 61.0 58.7 58.7 58.5 58.0 Selected Sources of Income Earnings 3.2 3.8 4.7 4.4 4.2 4.5 4.4 4.1 Social Security 51.0 49.4 45.9 42.1 39.1 37.1 36.1 35.5 No other income 34.8 34.5 36.4 38.7 43.6 46.5 54.4 55.1 Noncitizens NA 5.1 9.0 10.8 11.7 10.0 10.5 10.4 Aged 43.6 36.4 30.2 26.4 23.3 21.0 19.5 18.4 Blind 1.9 2.0 1.7 1.5 1.4 1.2 1.2 1.1 Disabled 54.5 61.7 68.1 72.0 75.4 77.8 79.3 80.4 Noncitizens NA 9.7 19.4 25.4 30.0 27.0 28.5 29.2 Noncitizens NA 2.4 4.6 5.6 6.2 5.5 6.1 7.2 Under 5 11.7 NA NA 16.0 15.8 15.8 15.5 16.1 5-9 20.9 NA NA 26.9 28.5 30.2 28.5 26.8 10-14 28.8 NA NA 30.6 32.7 34.6 36.2 36.9 18-212 16.8 14.3 9.3 10.8 5.7 – – – Male NA NA NA 62.0 63.0 62.9 63.8 64.3 Female NA NA NA 38.0 37.0 37.1 36.2 35.7 1For 1980-1992 male-female classification reflects all blind and disabled, both children and adults; thereafter, it is based on adults only. 2In this table, students 18-21 are classified as children prior to 1998. Note:Data are for December of the year. Source:Social Security Administration,Social Security Bulletin· Annual Statistical Supplement· 2003and prior years. Table SSI 7. Total SSI Payments, Federal SSI Payments and State Supplementary Payments,Calendar Year 2002 Total Federal Federal SSI Total $34,566,844 $33,720,491 $29,898,765 $3,820,234 $847,845 Alabama 730,105 729,691 729,691 – 414 Alaska 99,520 43,872 43,872 – 55,648 Arizona 406,848 406,474 406,474 – 374 Arkansas 354,418 354,418 354,412 6 – California 7,230,494 7,230,494 4,460,666 2,769,828 – Colorado 331,543 243,234 243,234 – 88,309 Connecticut 319,446 236,055 236,055 – 83,391 Delaware 56,374 56,374 55,350 1,024 – District of Columbia 102,083 102,082 98,518 3,564 – Florida 1,824,115 1,814,407 1,814,392 15 9,707 Georgia 854,414 854,414 854,411 3 – Hawaii 110,657 110,658 98,495 12,163 – Idaho 94,024 86,514 86,514 – 7,510 Illinois 1,275,885 1,246,787 1,246,787 – 29,098 Indiana 427,283 423,503 423,503 – 3,780 Iowa 191,069 175,290 172,391 2,899 15,779 Kansas 164,412 164,412 164,412 – – Kentucky 821,504 802,898 802,898 – 18,606 Louisiana 761,420 760,944 760,944 – 476 Maine 138,104 130,762 130,762 – 7,342 Maryland 442,285 434,761 434,752 9 7,524 Massachusetts 849,101 849,101 683,294 165,807 – Michigan 1,097,109 1,065,066 1,039,390 25,676 32,043 Minnesota 389,321 303,434 303,424 10 85,888 Mississippi 542,847 542,847 542,845 2 – Missouri 541,328 515,040 515,040 – 26,288 Montana 62,959 62,959 62,136 823 – Nebraska 99,177 92,870 92,870 – 6,307 Nevada 132,907 132,908 127,780 5,128 – New Hampshire 67,382 55,785 55,785 – 11,597 New Jersey 721,272 721,272 640,486 80,786 – New Mexico 217,053 216,882 216,882 – 171 New York 3,407,767 3,407,767 2,849,925 557,842 – North Carolina 937,938 797,987 797,987 – 139,951 North Dakota 33,716 31,784 31,784 – 1,932 Ohio 1,189,946 1,189,946 1,189,936 10 – Oklahoma 365,295 327,859 327,859 – 37,436 Oregon 282,957 262,681 262,681 – 20,276 Pennsylvania 1,550,661 1,550,660 1,406,743 143,917 – Rhode Island 146,253 146,253 121,290 24,963 – South Carolina 465,841 454,062 454,062 – 11,779 South Dakota 54,751 52,251 52,246 5 2,501 Tennessee 705,106 705,106 705,106 0.286 – Texas 1,799,263 1,797,304 1,797,304 – 1,959 Utah 97,817 97,816 97,756 60 – Vermont 55,462 55,462 46,161 9,301 – Virginia 593,731 574,659 574,659 – 19,072 Washington 539,989 539,761 523,340 16,421 228 West Virginia 348,553 348,553 348,553 – – Wisconsin 508,120 386,334 386,334 – 121,786 Wyoming 25,353 24,680 24,680 – 673 Other: N. Mariana Islands 3,358 3,358 3,358 – – Table SSI 8. SSI Recipiency Rates by State And Program Type for 1979 and 2002 Total Recipiency Rate Rate for Adults 18-64 Rate for Adults 65 & Over Percent Change 1979-02 Alabama 3.6 3.6 1 1.8 3.5 91 21.0 6.9 -67 Alaska 0.8 1.5 95 0.5 1.5 178 14.0 6.0 -57 Arizona 1.1 1.6 44 0.9 1.6 80 5.0 3.2 -36 Arkansas 3.5 3.1 -11 1.9 3.0 60 17.1 5.8 -66 California 3.0 3.2 6 2.1 2.5 22 16.4 13.3 -19 Colorado 1.1 1.2 9 0.8 1.1 43 6.7 3.2 -52 Connecticut 0.8 1.5 100 0.6 1.5 138 2.7 2.6 -4 Delaware 1.2 1.6 34 0.9 1.4 49 5.4 2.3 -58 District of Columbia 2.3 3.5 54 1.9 3.1 61 8.6 6.7 -22 Florida 1.8 2.4 35 1.1 1.9 67 6.2 4.8 -23 Georgia 2.9 2.3 -20 1.9 2.1 11 17.7 6.8 -62 Hawaii 1.1 1.7 62 0.7 1.5 117 7.6 5.1 -33 Idaho 0.8 1.4 77 0.6 1.6 150 3.8 2.0 -47 Illinois 1.1 2.0 85 1.0 2.0 111 4.3 3.8 -11 Indiana 0.8 1.5 100 0.6 1.6 162 3.3 1.7 -49 Iowa 0.9 1.4 57 0.6 1.6 158 3.5 1.7 -51 Kansas 0.9 1.4 57 0.6 1.4 122 3.5 1.9 -45 Kentucky 2.5 4.3 69 1.8 4.4 146 12.5 7.0 -44 Louisiana 3.4 3.7 10 2.0 3.5 72 20.1 7.8 -61 Maine 2.0 2.4 23 1.4 2.7 94 8.6 3.1 -64 Maryland 1.2 1.6 39 0.9 1.5 60 5.4 4.0 -26 Massachusetts 2.2 2.6 16 1.3 2.5 95 10.8 5.6 -48 Michigan 1.3 2.1 67 1.1 2.3 115 5.9 3.0 -49 Minnesota 0.8 1.3 60 0.6 1.3 136 3.7 2.6 -30 Mississippi 4.5 4.4 -2 2.4 4.0 65 26.0 10.3 -60 Missouri 1.8 2.0 14 1.1 2.1 91 7.9 2.9 -63 Montana 0.9 1.6 80 0.7 1.7 136 3.8 2.0 -47 Nebraska 0.9 1.3 48 0.6 1.3 103 3.4 1.7 -50 Nevada 0.8 1.3 55 0.5 1.2 126 5.9 3.3 -44 New Hampshire 0.6 1.0 72 0.4 1.1 150 2.5 1.2 -53 New Jersey 1.1 1.7 49 0.9 1.4 63 4.7 4.5 -4 New Mexico 2.0 2.6 32 1.4 2.4 75 12.4 6.9 -44 New York 2.1 3.3 56 1.6 2.8 76 8.3 9.0 9 North Carolina 2.4 2.3 -4 1.6 2.0 27 13.6 5.4 -60 North Dakota 1.0 1.3 31 0.6 1.3 128 5.1 2.2 -56 Ohio 1.1 2.1 89 1.0 2.3 132 4.2 2.4 -42 Oklahoma 2.3 2.1 -9 1.3 2.1 58 11.6 3.8 -67 Oregon 0.9 1.6 86 0.7 1.7 143 3.3 2.8 -15 Pennsylvania 1.4 2.4 71 1.1 2.5 123 5.0 3.4 -31 Rhode Island 1.6 2.7 70 1.1 2.6 141 6.4 4.9 -24 South Carolina 2.7 2.6 -3 1.8 2.3 29 17.0 5.6 -67 South Dakota 1.1 1.7 49 0.7 1.6 122 5.0 3.0 -40 Tennessee 2.9 2.8 -2 1.9 2.7 44 14.8 5.5 -63 Texas 1.9 2.0 6 1.0 1.6 68 12.7 7.5 -41 Utah 0.6 0.9 64 0.5 1.0 96 3.0 1.9 -37 Vermont 1.8 2.1 19 1.3 2.1 60 8.1 3.6 -55 Virginia 1.5 1.8 20 1.0 1.6 57 8.5 4.6 -46 Washington 1.2 1.7 47 1.0 1.8 84 4.8 3.6 -25 West Virginia 2.1 4.1 92 1.9 4.7 153 8.0 4.6 -42 Wisconsin 1.4 1.6 11 1.0 1.6 67 6.5 2.3 -65 Wyoming 0.4 1.1 162 0.3 1.2 314 2.7 1.6 -42 Total 1.9 2.4 30 1.3 2.2 75 9.0 5.6 -38 Note:Recipiency rates for 2002 are the ratios of the number of SSI recipients (in the respective age groups) as of the month of December to the estimated population in the respective age group as of the month of July; calculations by DHHS. The 1979 rates are based on the average number of recipients during the year. Source:Social Security Administration,Social Security Bulletin· Annual Statistical Supplement· 2003and U.S. Bureau of the Census, (Resident population by state available online athttp://www.census.gov/population/estimates/state/). Table SSI 9. SSI Recipiency Rates by State, Selected Fiscal Years, 1975– 2002 Alabama 4.0 3.4 3.3 3.3 3.4 3.8 3.9 3.6 Alaska 0.8 0.8 0.7 0.8 0.9 1.1 1.2 1.5 Arizona 1.2 1.1 1.0 1.2 1.4 1.7 1.7 1.6 Arkansas 4.1 3.4 3.1 3.2 3.5 3.8 3.8 3.1 California 3.1 3.0 2.6 2.9 3.1 3.2 3.3 3.2 Colorado 1.4 1.0 0.9 1.1 1.3 1.5 1.5 1.2 Connecticut 0.8 0.8 0.8 1.0 1.1 1.3 1.4 1.5 Delaware 1.2 1.2 1.2 1.2 1.3 1.5 1.6 1.6 District of Columbia 2.2 2.4 2.5 2.7 3.0 3.5 3.7 3.5 Florida 1.9 1.8 1.6 1.7 1.9 2.3 2.4 2.4 Georgia 3.3 2.8 2.6 2.5 2.6 2.8 2.7 2.3 Hawaii 1.1 1.1 1.1 1.3 1.3 1.5 1.6 1.7 Idaho 1.1 0.8 0.8 1.0 1.2 1.4 1.5 1.4 Illinois 1.2 1.1 1.2 1.6 1.8 2.2 2.3 2.0 Indiana 0.8 0.8 0.9 1.1 1.3 1.5 1.6 1.5 Iowa 1.0 0.9 1.0 1.2 1.3 1.4 1.5 1.4 Kansas 1.1 0.9 0.9 1.0 1.1 1.4 1.5 1.4 Kentucky 2.8 2.6 2.7 3.1 3.4 4.1 4.4 4.3 Louisiana 3.9 3.2 2.9 3.2 3.5 4.1 4.2 3.7 Maine 2.3 1.9 1.9 1.9 2.0 2.4 2.2 2.4 Maryland 1.2 1.1 1.2 1.3 1.4 1.6 1.7 1.6 Massachusetts 2.3 2.2 1.9 2.0 2.2 2.6 2.7 2.6 Michigan 1.3 1.2 1.4 1.5 1.7 2.2 2.2 2.1 Minnesota 1.0 0.8 0.8 0.9 1.1 1.3 1.4 1.3 Mississippi 5.2 4.4 4.3 4.4 4.7 5.2 5.2 4.4 Missouri 2.1 1.7 1.6 1.7 1.8 2.1 2.2 2.0 Montana 1.1 0.9 0.9 1.3 1.4 1.6 1.6 1.6 Nebraska 1.1 0.9 0.9 1.0 1.1 1.3 1.3 1.3 Nevada 1.0 0.8 0.9 1.0 1.0 1.3 1.4 1.3 New Hampshire 0.7 0.6 0.6 0.6 0.7 0.8 0.9 1.0 New Jersey 1.1 1.2 1.2 1.4 1.5 1.8 1.8 1.7 New Mexico 2.3 1.9 1.8 2.1 2.3 2.6 2.7 2.6 New York 2.2 2.1 2.0 2.3 2.6 3.1 3.3 3.3 North Carolina 2.7 2.4 2.2 2.2 2.4 2.6 2.7 2.3 North Dakota 1.3 1.0 1.0 1.2 1.3 1.4 1.4 1.3 Ohio 1.2 1.1 1.2 1.4 1.6 2.1 2.3 2.1 Oklahoma 3.0 2.2 1.8 1.9 2.0 2.2 2.3 2.1 Oregon 1.1 0.8 1.0 1.1 1.2 1.5 1.5 1.6 Pennsylvania 1.2 1.4 1.4 1.6 1.8 2.1 2.2 2.4 Rhode Island 1.7 1.6 1.6 1.7 1.9 2.3 2.6 2.7 South Carolina 2.8 2.7 2.6 2.6 2.7 3.0 3.0 2.6 South Dakota 1.3 1.2 1.2 1.5 1.6 1.8 1.9 1.7 Tennessee 3.2 2.8 2.7 2.9 3.1 3.4 3.4 2.8 Texas 2.2 1.8 1.6 1.7 1.9 2.1 2.2 2.0 Utah 0.8 0.5 0.5 0.7 0.8 1.0 1.1 0.9 Vermont 1.9 1.7 1.8 1.8 2.0 2.2 2.2 2.1 Virginia 1.5 1.5 1.5 1.5 1.7 1.9 2.0 1.8 Washington 1.5 1.1 1.1 1.3 1.4 1.6 1.7 1.7 West Virginia 2.4 2.1 2.2 2.6 2.9 3.5 3.8 4.1 Wisconsin 1.4 1.4 1.5 1.8 1.9 2.2 1.8 1.6 Wyoming 0.7 0.4 0.5 0.8 0.9 1.2 1.2 1.1 Total1 2.0 1.8 1.7 1.9 2.1 2.4 2.5 2.4 1The number of SSI recipients used to calculate the total recipiency rate includes a certain number of recipients whose State is unknown. For 1975, 1985, and 1992, the numbers of unknown (in thousands) were 256, 14, and 71 respectively. 2For 1975-92 the percentages are calculated as the average number of monthly SSI recipients over the total population of each State in July of that year. For 1994-2002 the number of recipients is from the month of December; calculations by DHHS. Source:Social Security Administration,Social Security Bulletin· Annual Statistical Supplement· 2003, and Bureau of the Census, (Resident population by state available online athttp://www.census.gov/population/estimates/state/) 1In this case, the Supreme Court ruled that the IFA (or a residual functional capacity assessment) that applied to adults whose condition did not meet or equal a listing of medical impairments to determine eligibility should also be applied to children whose condition did not meet or equal the medical listing of impairments. 2The GAO study estimated that 87,000 children were added to the SSI caseload after the IFA for children was initiated. Alternative Definition of Dependence Based on Income from TANF and Food Stamps As directed by the Welfare Indicators Act of 1994 (Pub. L. 103-432), this annual report on Indicators of Welfare Dependence focuses on dependence on three programs: the Aid to Families with Dependent Children (AFDC) program, now Temporary Assistance for Needy Families (TANF); the Food Stamp Program; and the Supplemental Security Income (SSI) program. The summary measure of dependence proposed by the Advisory Board includes income from all three programs in its definition: A family is dependent on welfare if more than 50 percent of its total income in a one-year period comes from AFDC, food stamps and/or SSI, and this welfare income is not associated with work activities. This appendix examines an alternative definition of dependence that considers TANF and food stamps alone, excluding SSI. As shown in Table B-1, the rate of dependency would have been only 1.4 percent in 2001 if based on income from TANF and food stamps, as opposed to 3.1 percent when counting income from all three programs (TANF, food stamps, and SSI). In other words, less than half of individuals who are dependent under the standard definition also are dependent under the alternative definition that considers TANF and food stamps alone.1There is significant variation across the age groups, however. The elderly depend more on SSI than on TANF and food stamps; whereas 1.9 percent of elderly persons are dependent when counting the three major types of means-tested assistance, very few, 0.1 percent, are dependent when the definition is limited to TANF and food stamps. In contrast, children are primarily dependent on TANF and food stamps. Table B-1. Percentage of the Total Population with More than 50 Percent of Income from Various Means-Tested Assistance Programs, by Race and Age: 2001 TANF, SSI, & Food Stamps TANF & Food Stamps SSI Only All Persons 3.1 1.4 1.3 Non-Hispanic White 1.8 0.8 0.8 Non-Hispanic Black 8.8 4.3 3.3 Hispanic 4.5 2.1 1.8 Children Ages 0-5 5.9 4.1 1.2 Children Ages 6-10 5.4 3.2 1.2 Children Ages 11-15 4.4 2.3 1.2 Women Ages 16-64 3.3 1.4 1.5 Men Ages 16-64 2.0 0.7 1.1 Adults Age 65 and Over 1.9 0.1 1.6 Note: Income is measures as total family income. Hispanic may be of any race. 1In the early- to mid-1990s, 70 to 75 percent of individuals who were dependent under the standard definition were also dependent under the alternative definition. Additional Nonmarital Birth Data Table C-1. Percentage of Births that are to Unmarried Women Within Age Groups by Race, 1940-2002 1940 44.4 NA NA 7.2 1.9 NA NA NA NA NA 1945 50.7 NA NA 10.0 2.4 NA NA NA NA NA 1948 39.9 10.3 4.6 6.3 1.8 NA NA NA NA NA 1951 34.9 9.7 4.4 5.9 1.6 NA NA NA NA NA 1964 52.3 16.0 7.6 10.4 3.4 NA NA NA NA NA 1967 61.6 21.0 11.2 14.2 4.9 NA NA NA NA NA 1969 57.0 24.0 12.9 16.6 5.5 91.7 72.1 48.3 60.0 34.9 1980 75.4 45.4 27.1 33.6 11.2 98.6 93.1 79.9 86.2 56.1 Note:> Births to unmarried women in the United States for 1940 - 1979 are estimated from data for registration areas in which marital status of the mother was reported; see sources below. Beginning in 1980, births to unmarried women in the United States are based on data from states reporting marital status directly and data from non-reporting states for which marital status was inferred from other information on the birth certificate; see sources below. Source:> National Center for Health Statistics, "Nonmarital Childbearing in the United States, 1940 - 1999," National Vital Health Statistics Reports>, Vol. 48 (16), 2000; "Births: Final Data for 2002," National Vital Statistics Reports>, Vol. 52 (10), December 2003. Table C-2. Percentage of Births that are to Unmarried Women by State: Selected Years 1960-2002 Alabama 11 14 22 30 33 34 34 34 35 Alaska 5 9 16 26 27 29 31 33 34 Arizona NA 9 19 33 36 38 39 39 40 Arkansas NA 13 20 29 31 33 34 36 37 California NA NA 21 32 34 36 31 33 33 Colorado NA 9 13 21 24 25 25 25 27 Connecticut NA NA 18 27 29 30 31 29 29 Delaware 9 15 24 29 33 35 35 38 41 Dist of Columbia 20 38 56 65 67 69 66 60 57 Florida 9 14 23 32 34 36 36 38 39 Georgia NA NA 23 33 35 36 35 37 38 Hawaii 5 10 18 25 26 28 30 32 34 Idaho NA NA 8 17 18 19 21 22 22 Illinois 6 13 23 32 33 34 34 35 35 Indiana 4 8 16 26 29 32 32 35 36 Iowa 2 7 10 21 24 25 26 28 29 Kansas 3 7 12 22 24 26 27 29 31 Kentucky 5 8 15 24 26 28 30 31 33 Louisiana 9 15 23 37 40 43 43 46 47 Maine 3 7 14 23 25 28 29 31 33 Maryland NA NA 25 30 30 34 34 35 35 Massachusetts NA NA 16 25 26 27 25 27 27 Michigan 4 11 16 26 27 35 34 33 34 Minnesota 3 8 11 21 23 24 25 26 27 Mississippi 14 17 28 40 43 45 45 46 47 Missouri 6 11 18 29 32 33 33 35 35 Montana NA NA 13 24 26 26 28 31 33 Nebraska NA 8 12 21 23 25 25 27 29 Nevada 4 11 13 25 33 35 43 36 37 New Hampshire NA 6 11 17 19 22 23 25 25 New Jersey 4 10 21 24 26 28 28 29 29 New Mexico NA NA 16 35 39 42 42 46 47 New York NA NA 24 33 35 38 40 37 36 North Carolina 9 12 19 29 31 32 32 33 35 North Dakota 3 7 9 18 23 23 25 28 29 Ohio 4 NA 18 29 32 33 33 35 35 Oklahoma NA 8 14 25 28 30 31 34 36 Oregon 3 7 15 26 27 29 30 30 31 Pennsylvania 4 10 18 29 32 33 32 33 33 Rhode Island 3 7 16 26 30 32 33 35 36 South Carolina 12 15 23 33 35 37 37 40 40 South Dakota 3 7 13 23 27 28 30 33 35 Tennessee 9 12 20 30 33 33 33 35 36 Texas 5 9 13 18 17 29 30 31 32 Utah 2 4 6 14 15 16 16 17 17 Vermont NA NA 14 20 23 25 26 28 32 Virginia 8 11 19 26 28 29 29 30 30 Washington 3 9 14 24 25 26 27 28 29 West Virginia 6 6 13 25 28 30 31 32 33 Wisconsin 3 8 14 24 26 27 27 29 30 Wyoming 2 7 8 20 24 27 27 29 30 United States 5 11 18 28 30 33 32 33 34 Source:> National Center for Health Statistics, "Births: Final Data for 2002," National Vital Statistics Reports>, Vol. 52 (10), December 2003 and earlier reports available online at (http://www.cdc.gov/nchs/products/pubs/p ubd/vsus/1963/1963.htm>). Table C-3. Percentage of Births that are to Unmarried Women by Race/Ethnicity and State, 1994 – 2002 Non-Hispanic Alabama 35 35 16 20 16 19 71 68 19 25 Alaska 29 34 21 24 21 23 39 43 29 41 Arizona 38 40 35 38 25 25 65 62 51 52 California 36 33 36 34 23 20 63 63 46 42 Colorado 25 27 23 26 18 18 57 54 44 41 Connecticut 31 29 24 25 18 16 70 66 65 61 Dist. of Columbia 69 57 15 26 10 8 80 77 59 58 Florida 36 39 26 32 24 28 69 67 34 40 Georgia 36 38 18 25 18 21 68 66 23 43 Hawaii 28 34 16 17 15 17 20 19 44 44 Idaho 19 22 18 21 17 19 40 33 25 36 Illinois 34 35 23 27 18 21 79 77 38 43 Indiana 32 36 26 32 26 30 78 76 42 50 Iowa 25 29 23 28 23 27 75 74 37 41 Kansas 26 31 22 28 21 26 66 68 39 43 Kentucky 28 33 23 29 23 29 73 73 25 44 Louisiana 43 47 21 27 21 27 72 75 30 33 Maine 28 33 28 33 28 33 47 34 23 36 Maryland 34 35 19 24 18 21 64 59 39 45 Massachusetts 27 27 23 24 19 19 63 59 62 62 Michigan 35 34 24 26 23 25 79 74 42 42 Minnesota 24 27 21 24 20 21 73 58 46 51 Missouri 33 35 24 29 24 28 79 76 34 45 Montana 26 33 20 28 20 27 28 § 30 41 Nevada 35 37 31 35 27 28 70 70 44 44 New Jersey 28 29 19 24 13 14 67 64 48 53 New Mexico 42 47 37 44 23 27 61 57 49 54 New York 38 36 29 30 19 18 70 66 61 60 North Carolina 32 35 18 25 17 20 68 66 29 48 Ohio 33 35 25 29 25 28 78 75 50 50 Oklahoma 30 36 23 31 23 29 70 70 31 42 Oregon 29 31 28 31 27 28 71 61 35 42 Pennsylvania 33 33 25 27 23 24 79 75 63 61 Rhode Island 32 36 28 32 24 26 69 63 58 59 South Carolina 37 40 19 25 19 23 67 72 28 43 Tennessee 33 36 21 27 21 25 75 73 26 46 Texas 29 32 24 30 18 22 63 62 31 36 Vermont 25 32 25 32 25 32 33 59 34 § Virginia 29 30 19 22 18 20 64 62 38 40 Washington 26 29 24 28 23 25 55 53 35 42 West Virginia 30 33 29 32 29 32 76 72 22 35 Wisconsin 27 30 21 24 20 22 82 82 46 46 Wyoming 28 30 26 29 25 27 46 52 45 43 §> Figure does not meet standards of reliability or precision; based on fewer than 20 births in the numerator. Persons of Hispanic ethnicity may be of any race. Source:> National Center for Health Statistics, "Births: Final Data for 2002, " National Vital Statistics Reports>, Vol. 52 (10), December 2003 and earlier reports available online at (http://www.cdc.gov/nchs/products/pubs/pubd/vsus/1963/1963.htm.>). Table C-4. Birth Rates of Teens 15-19 Years, By State: Selected Years 1960-2002 [Births per 1,000 women in specified group] Alabama 104 90 78 68 64 71 69 61 55 Alaska 128 103 60 64 56 65 55 49 40 Arizona 112 79 67 65 67 76 74 68 61 Arkansas 116 93 84 75 73 80 72 66 60 California 103 69 52 53 53 71 67 47 41 Colorado 97 67 51 50 48 55 52 51 47 Connecticut 54 44 32 31 31 39 39 31 26 Delaware 100 73 49 51 51 55 55 48 46 Dist. of Columbia 132 116 73 62 72 93 85 53 69 Florida 117 86 64 59 58 69 60 51 45 Georgia 117 101 78 72 68 76 70 63 56 Hawaii 77 66 52 51 48 61 49 46 38 Idaho 102 66 59 59 47 51 49 43 39 Illinois 63 63 56 56 51 63 58 48 42 Indiana 100 75 64 57 52 59 57 49 45 Iowa 73 53 46 43 35 41 38 34 33 Kansas 94 65 57 57 52 56 52 46 43 Kentucky 108 86 78 72 63 68 62 55 51 Louisiana 113 84 79 76 72 74 70 62 58 Maine 93 65 55 47 42 43 34 29 25 Maryland 100 69 46 43 46 53 47 41 35 Massachusetts 51 40 31 28 29 35 33 26 23 Michigan 80 69 52 45 43 59 49 40 35 Minnesota 64 44 36 35 31 36 33 30 28 Mississippi 121 103 92 84 76 81 79 70 65 Missouri 99 72 59 58 54 63 55 49 44 Montana 97 62 54 48 44 48 42 37 36 Nebraska 82 54 45 45 40 42 38 38 37 Nevada 118 94 60 59 55 73 73 63 54 New Hampshire 76 55 41 34 32 33 30 23 20 New Jersey 58 50 37 35 34 41 38 32 27 New Mexico 127 79 67 72 73 78 74 66 62 New York 57 51 38 35 36 44 42 33 30 North Carolina 104 88 72 58 57 68 63 59 52 North Dakota 68 44 43 42 36 35 33 27 27 Ohio 84 65 56 52 50 58 53 46 40 Oklahoma 112 83 76 75 69 67 64 60 58 Oregon 88 58 48 51 43 55 50 43 37 Pennsylvania 67 53 44 41 40 45 41 34 32 Rhode Island 56 43 35 33 36 44 40 34 36 South Carolina 109 89 73 65 63 71 63 58 53 South Dakota 83 49 51 53 46 47 41 38 38 Tennessee 103 88 74 64 61 72 67 60 54 Texas 115 85 74 74 72 75 76 69 64 Utah 86 56 54 65 50 49 41 38 37 Vermont 74 54 43 39 36 34 28 23 24 Virginia 103 76 53 48 46 53 48 41 38 Washington 88 60 46 47 45 53 48 39 33 West Virginia 87 72 73 68 54 57 53 47 46 Wisconsin 64 46 41 40 39 43 38 35 32 Wyoming 112 71 68 79 59 56 48 42 40 Source:> National Center for Health Statistics, "Births: Final Data for 2002," National Vital Statistics Reports, Vol. 52 (10), December 2003 >and earlierreports >available online at (http://www.cdc.gov/nchs/products/pubs/pubd/vsus/1963/1963.htm>.) Table C-5. Birth Rates of Teens 15-19 Years, By Race, Ethnicity, and State: Selected Years 1994-1999 Alabama 72.2 62.8 55.1 52.5 54.8 50.8 108.1 83.2 71.8 136.2 Alaska 55.2 41.8 44.5 29.8 43.4 29.0 79.3 66.7 § 57.6 Arizona 78.7 69.6 77.3 69.7 49.2 39.6 99.7 74.9 136.3 125.4 Arkansas 76.3 68.1 64.1 59.9 63.1 57.4 120.2 96.5 118.4 121.2 California 71.3 50.7 76.6 55.2 38.1 25.2 89.2 58.4 118.4 83.4 Colorado 54.3 48.4 52.0 47.6 38.2 29.9 96.6 67.4 109.3 116.3 Connecticut 40.3 33.3 33.0 29.1 20.1 16.1 93.6 67.1 125.0 114.4 Delaware 60.2 54.3 43.0 40.4 38.4 35.7 115.4 99.8 § 116.6 Dist. of Columbia 114.7 83.5 16.9 23.2 15.3 § 138.5 127.8 96.7 § Florida 64.4 53.5 51.5 45.3 46.9 39.5 113.1 83.5 68.2 62.5 Georgia 71.7 65.1 54.1 55.8 51.5 49.3 106.9 84.4 133.8 154.5 Hawaii 53.5 43.8 33.0 16.9 29.8 14.7 § 31.0 107.7 98.2 Idaho 46.6 43.7 46.2 43.4 40.6 38.2 § § 117.8 91.9 Illinois 62.8 51.1 46.2 40.3 34.3 27.5 139.1 105.2 112.6 102.2 Indiana 57.9 51.6 51.8 46.7 50.8 44.6 115.3 97.2 82.2 99.6 Iowa 39.7 35.8 37.5 33.9 36.2 31.8 117.4 95.5 96.9 106.6 Kansas 53.5 47.4 48.7 44.0 44.9 38.3 116.4 97.6 106.9 108.3 Kentucky 64.5 56.4 60.4 53.8 60.3 53.2 113.5 85.3 § 112.1 Louisiana 74.7 62.8 49.2 45.3 49.6 45.6 115.3 89.7 49.3 33.8 Maine 35.5 29.8 35.0 29.5 35.0 29.4 § § § § Maryland 49.7 42.6 32.4 29.0 31.5 26.6 89.3 73.0 62.0 59.2 Massachusetts 37.2 28.7 32.6 25.4 23.5 17.9 90.5 68.0 132.9 101.9 Michigan 52.1 40.5 39.7 32.9 37.8 30.4 110.2 79.8 85.3 88.2 Minnesota 34.4 30.0 28.6 24.0 26.9 21.0 132.3 109.9 98.9 137.5 Mississippi 83.0 72.5 56.6 53.3 56.7 53.0 114.4 95.0 § 61.8 Missouri 59.0 49.6 49.1 43.1 48.7 42.0 123.1 92.0 65.4 87.7 Montana 41.2 35.1 34.7 29.7 34.0 28.8 § § § § Nebraska 42.8 37.0 37.9 32.9 34.5 28.6 119.3 97.5 110.6 97.2 Nevada 73.6 64.1 71.1 63.9 55.4 45.3 111.3 81.6 138.3 112.7 New Hampshire 30.1 24.0 30.1 24.3 29.6 23.5 § § § § New Jersey 39.3 32.8 27.2 25.3 16.5 13.5 99.7 72.6 81.1 76.6 New Mexico 77.4 67.4 76.2 68.6 43.7 37.7 66.4 50.4 102.4 91.8 New York 45.8 37.0 39.8 32.5 26.4 20.5 73.0 59.3 81.1 73.7 North Carolina 66.3 59.5 52.3 50.5 50.0 43.0 98.5 80.2 159.6 219.0 North Dakota 34.6 27.7 29.2 22.9 28.7 22.5 § § § § Ohio 55.0 46.0 46.1 39.6 45.2 38.6 116.1 88.6 83.6 76.0 Oklahoma 65.9 60.5 59.0 55.9 57.1 52.0 105.5 82.9 87.1 107.6 Oregon 50.7 46.5 49.8 46.2 43.8 38.7 101.6 64.5 136.8 119.3 Pennsylvania 43.8 36.2 34.0 29.2 30.5 25.5 118.1 93.6 129.3 114.0 Rhode Island 47.7 38.2 41.3 34.2 31.7 25.7 120.4 66.2 136.8 115.4 South Carolina 66.5 60.8 50.3 49.2 49.9 46.9 92.1 80.5 68.4 128.8 South Dakota 42.8 37.6 33.0 27.5 32.3 27.0 § § § § Tennessee 71.0 62.7 58.8 55.4 58.5 53.7 119.8 90.6 79.5 136.1 Texas 77.6 70.1 75.7 71.3 47.7 41.9 100.4 76.0 113.6 107.4 Utah 42.7 40.2 42.0 39.6 38.6 33.0 § § 96.9 118.8 Vermont 33.0 25.7 33.2 25.9 33.4 26.1 § § § § Virginia 50.7 42.7 40.7 33.7 38.8 31.1 87.9 73.8 79.4 73.6 Washington 48.2 40.1 47.2 39.3 40.5 32.6 80.9 60.7 125.8 98.0 West Virginia 54.3 47.9 53.7 47.2 53.8 47.1 80.7 71.7 § § Wisconsin 38.8 35.7 28.8 27.3 26.5 24.2 142.3 122.9 92.6 110.7 Wyoming 48.2 40.4 47.6 39.6 45.4 37.4 § § 74.9 65.0 §> Rates not calculated for states with less than 20 births to women in a given age and racial/ethnic group or if there were less than 1,000 women in the age and racial/ethnic group. Persons of Hispanic ethnicity may be of any race. Source:> National Center for Health Statistics, "Births to Teenagers in the United States, 1940-2000," National Vital Statistics Reports,> Vol. 49 (10), September 2001. Most of the indicators are shown by age categories, generally children ages 0-15, adults 16-64, and adults 65 and older. Youth 17 and 18 years of age are often classified with adults because they are considered potential members of the labor force in many labor force statistics. Many of the risk factors, however, use published data that define "children" to include all individuals less than 18 years of age. Annual and Monthly Measures There are differences between monthly and annual observation of benefit receipt. The measures of annual recipiency (that is, any receipt over the course of a year) shown in Figure and Table SUM 1 are higher than the more traditional measures of recipiency in an average month, as shown in several other indicators. Note that annual measures are for calendar years except where explicitly noted as fiscal years. Family Structure Categories For the primary measure of dependency in this 2004 report, estimates are provided for individual persons by family structure (see SUM1 and IND1). For these measures, the entire population is subdivided into the following four groups: individuals in married-couple families individuals in female-headed families, no spouse present individuals in male-headed families, no spouse present unrelated individuals. Most of the data sources allow analysis of the indicators and predictors of welfare dependence across several age and racial/ethnic categories. Where the data are available, statistics are shown for three racial/ethnic groups – Non-Hispanic white, Non-Hispanic Black, and Hispanic. Due to small sample size, American Indians/Alaska natives, Asians, and Native Hawaiians/Other Pacific Islanders are included in the totals for all persons but are not shown under separate race categories. In some instances, however, data are shown for "Whites" and "Blacks," rather than for "Non-Hispanic Whites" and "Non-Hispanic Blacks;" in such cases these racial categories include individuals of Hispanic Origin. Footnotes to the tables provide further documentation of issues related to race and ethnicity. Estimates based on 2002 CPS data are affected by a change in the CPS questionnaire that allows individuals to report one or more races (see ECON 1, ECON 9, WORK 1, WORK 2, and WORK 3). This change was implemented to comply with the 1997 Standards for Federal Data on Race and Ethnicity. In 2000, the Office of Management and Budget published guidelines for implementing these new standards. To accommodate the race categories under the new standards, CPS estimates for racial/ethnic categories beginning in 2002 are for persons who are non-Hispanic white (and no other race), non-Hispanic black (and no other race) and Hispanic (of any race). Persons who reported more than one race are included in the total for all persons but are not shown under any race category. Spells of dependency (Indicator 7) and recipiency (Indicator 8) are limited to those spells that begin during the SIPP panel of observation. Spells separated by only 1 month are not considered separate spells. If an individual has 2 or more spells of dependency or receipt, each is counted separately in the analysis. The individual, rather than the family or household, is the unit of analysis for most of the statistics in this report. The individual's dependency status, however, is generally based on total family income, taking into account means-tested assistance, earnings and other sources of income for all individuals in the family.1 This chapter, for example, has reported the percentage of individuals that are dependent (in SUM 1) or poor (in SUM 2) according to annual total family income. Recipiency status is also based on total annual family income in some instances; in SUM 1, for example, recipients are individuals in families receiving assistance at some point in the year. In most other indicators, recipiency is measured as the direct receipt of a benefit by an individual in a month. The difference between an individual and a family measure of recipiency is largest in the SSI program, which provides benefits to individuals and couples, not to families. 1 Family is generally defined as following the broad Census Bureau definition of family – all persons residing together that are related by birth, marriage, or adoption. ch2.pdf (pdf, 1.6 MB) Key Indicators | Welfare, Welfare Reform, & TANF | Poverty & Income Dynamics Report to Congress
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Q: Postgresql ERROR: syntax error at or near "PERFORM" when inserting a nested IF statement in a procedure I have the following procedure query that works fine: CREATE OR REPLACE FUNCTION table_update_notify() RETURNS trigger AS $$ DECLARE notification_channel text := TG_ARGV[0]; owner_id numeric := TG_ARGV[1]; owner_lat numeric := TG_ARGV[2]; owner_lng numeric := TG_ARGV[3]; trigger_radius numeric := TG_ARGV[4]; nearby_radius numeric := TG_ARGV[5]; changed_lat numeric; changed_lng numeric; user_id numeric; is_close boolean; name text; BEGIN IF TG_OP = 'INSERT' OR TG_OP = 'UPDATE' THEN changed_lat = NEW.lat; changed_lng = NEW.lng; user_id = NEW.user_id; name = NEW.name; ELSE changed_lat = OLD.lat; changed_lng = OLD.lng; user_id = OLD.user_id; name = OLD.name; END IF; -- If updated user's location is within the trigger radius of the trigger owner's location IF earth_box(ll_to_earth(owner_lat, owner_lng), trigger_radius) @> ll_to_earth(changed_lat, changed_lng) -- Don't notify owner if the owner's location changes AND user_id != owner_id THEN PERFORM pg_notify(notification_channel, json_build_object('user_id', user_id, 'name', name, 'is_close', is_close)::text); END IF; RETURN NEW; END; $$ LANGUAGE plpgsql; But if I insert another "IF" system after the "THEN", like so, I get an error: CREATE OR REPLACE FUNCTION table_update_notify() RETURNS trigger AS $$ DECLARE notification_channel text := TG_ARGV[0]; owner_id numeric := TG_ARGV[1]; owner_lat numeric := TG_ARGV[2]; owner_lng numeric := TG_ARGV[3]; trigger_radius numeric := TG_ARGV[4]; nearby_radius numeric := TG_ARGV[5]; changed_lat numeric; changed_lng numeric; user_id numeric; is_close boolean; name text; BEGIN IF TG_OP = 'INSERT' OR TG_OP = 'UPDATE' THEN changed_lat = NEW.lat; changed_lng = NEW.lng; user_id = NEW.user_id; name = NEW.name; ELSE changed_lat = OLD.lat; changed_lng = OLD.lng; user_id = OLD.user_id; name = OLD.name; END IF; -- If updated user's location is within the trigger radius of the trigger owner's location IF earth_box(ll_to_earth(owner_lat, owner_lng), trigger_radius) @> ll_to_earth(changed_lat, changed_lng) -- Don't notify owner if the owner's location changes AND user_id != owner_id THEN -- If the user is close enough to the user to be considered nearby IF earth_box(ll_to_earth(owner_lat, owner_lng), trigger_radius) @> ll_to_earth(changed_lat, changed_lng) THEN is_close = true; ELSE is_close = false; END IF PERFORM pg_notify(notification_channel, json_build_object('user_id', user_id, 'name', name, 'is_close', is_close)::text); END IF; RETURN NEW; END; $$ LANGUAGE plpgsql; And the error is: ERROR: syntax error at or near "PERFORM" LINE 39: PERFORM pg_notify(notification_channel, json_build_object(... According to my research, this happens when the language is not set to plpgsql, but I clearly am doing that. How can I execute this nested IF statement? A: You're missing a semi-colon after an END IF: END IF /* need semicolon here */ PERFORM pg_notify Best of luck.
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Applications for absentee ballots for the Jan. 19 U.S. senate election are available at the following locations: The Town Clerk�s office; the table located outside of the Town Clerk�s office; downloadable from the town�s Web site; or www.mass.gov. Absentee ballots have not arrived yet but will be available soon. The Historical Society�s 2010 calendar is available during regular hours at the Historical Society building, The Bigelow Tavern, 65 Worcester St., on Thursdays, from 9 a.m. to noon. The photos from this year�s calendar are from the society�s collection of glass negatives. Most of the early 20th century photos were taken by George F. Keyes, others were taken by William E. Parker. On Tuesday, Jan. 5, at the Senior Center, Anita Depatie, from the Auburn VNA will present a program on emergency preparedness. She will discuss how to plan and be prepared for an emergency and how to put an emergency kit together. She will be raffling off a pre-packaged emergency supply kit at the end of the program. Call to reserve a seat, (508) 835-6916. On Saturday, Jan. 9, Girl Authority, a local singing group, will be in concert at the West Boylston Middle-High School at 3:30 p.m. For more information on the group, visit girlauthority.com. Tickets are $12 per person. Seating is general admission. Advance ticket sales are available through the West Boylston Arts Foundation: wbaf.org. Snow date is Sunday, Jan. 10, at 3:30 p.m. All proceeds benefit the arts in the West Boylston public schools. The Boy Scouts of Troop 151 of West Boylston are offering curbside pickup of Christmas trees after the holiday. For a $10 fee, members of the troop will pickup used Christmas trees and bring them to the brush dump for chipping. The pickups will be held on the weekends of Jan. 9 to 10 and 16 to 17. This activity will help cover the cost of a trip to the 2010 National Boy Scout Jamboree. This service is for West Boylston residents only. To schedule a pickup, call Hillary at (508) 835-6966. The Rooke Chapel Ringers, a nationally-renowned handbell choir from Bucknell University, will be performing a holiday concert on Sunday, Jan. 10 at 3 p.m. at the First Congregational Church of West Boylston, 26 Central Street. Free admission, but donations will be gratefully accepted. Refreshments will be served. For more information, contact Anne Barnard at (508) 835-6779 or amb038@bucknell.edu. Debbie Seto, a member of the Golden Agers exercise class has invited the Golden Agers to her home on Jan. 21 at 11:30 a.m. for a potluck lunch and a showing of the movie, �Mama Mia� and a game of Wii Bowling. To sign up, call Debbie at (508) 854-8543. For directions and/or a ride, call Cathie Nickerson at (508) 835-2109. On Tuesday, Jan. 26, at the Senior Center, Susan Miller, from Memorial Hospital Diabetes Clinic will present a program on healthy eating with diabetes. This program is open to all and is especially good for those who have been diagnosed with pre-diabetes, as well as any other type of diabetes, even if it is not being treated with medicine at the moment. Miller will discuss a variety of meal planning tools that can assist in diabetes management. Call ahead to reserve a seat, (508) 835-6916. On Friday, Jan. 29, local brother-sister performers, Michael and Marisa, will be in concert at the West Boylston Middle-High School. Concert time is 6:30 p.m. The duo just opened for David Archeletta in Boston. Check out the group at michaelandmarisa.com. Tickets are $12 per person. Seating is general admission. Advance ticket sales are available through the West Boylston Arts Foundation, wbaf.org. Snow date is Saturday, Jan. 30, from 4 to 6:30 p.m. Buy tickets in advance to avoid being disappointed. All proceeds benefit the arts in the West Boylston public schools. The Trustees of the Beaman Memorial Public Library are seeking members of the community as well as library patrons to serve on a committee to evaluate the relative advantages of upgrading access to the collection of the Beaman Library. If someone is interested in shaping the future of library services in this way, contact Library Director, Louise Howland at the Beaman Memorial Public Library, 8 Newton Street, or call 508-835-3711. West Boylston Beneficiaries are reminded to make sure they do not lose their coverage. Open all mail from Social Security, Medicare, and other plans. Read all information. Save the paperwork and call with any questions. Call for a SHINE appointment at (508) 835-6916. The Beaman Memorial Library, 8 Newton St., will be open from 10 a.m. to 2 p.m. every Saturday through Memorial Day weekend. For a complete listing of the library hours or for more information about the services and programs the library offers, call (508) 835-3711. Take a trip to Italy with The Reverend Ken Cardinale May 15 through 27, 2010. Travel to Rome, Tuscany, Florence, Venice, Lake Como, Capri, Amalfi Coast, Pompeii and Switzerland. Cost is $3,990, airfare and all-included. Contact Erin McCarthy at Proximo Travel, 857 West Boylston St., Worcester. Or call (877)994-8259, or (508) 887-0556. Fax to (508) 854-8003. Email to erin@proximotravel.com, or visit www.proximotravel.com. Tower Hill�s annual �Holly Days� this year will highlight multicultural celebrations, now through Jan. 3. This year�s theme is �Holiday Celebrations Around the World.� Holly Days will be open on Mondays, from 10 a.m. to 5 p.m. and for extended evening hours on Wednesdays until 8 p.m. during Holly Days. Enjoy music and entertainment, as well as light refreshments on Wednesday evenings. The Holly Days exhibit and entertainment is included with garden admission; no admission is charged for shopping or caf� dining. Admission rates are $10 Adults, $7 Seniors (65 and over) and $5 Youth (6 to 18), children under 6 and members are admitted free. Discounted group rates are available. Hours daily through Jan. 3 are 10 a.m. to 5 p.m., Wednesdays until 8 p.m., closed Dec. 24, 25, 31 and Jan. 1. Tower Hill is located at 11 French Drive For more information, contact Tower Hill Botanic Garden at (508) 869-6111, or visit the website at www.towerhillbg.org. The Boylston Board of Health will hold an H1N1 vaccination clinic on Saturday, Jan. 9, from 9:30 a.m. to noon, at Tahanto Regional Middle-High School, located at 1001 Main St. This clinic is open to all residents of Boylston. Anyone planning to receive a vaccination should to go to www.boylston-ma.gov and in the Board of Health page download and fill out the screening questionnaire for injectable or intranasal vaccine and to bring the completed form with them. Support Boylston�s Habitat for Humanity Project by making a donation in honor of a friend or loved one. Those who do will receive a gift card to show that they have been honored with a donation. Money raised will support the construction on a duplex home in Boylston for two families. Every dollar will go to the Boylston Project. Make checks payable to Habitat for Humanity-Boylston Project and send them to Sue Olsen, P.O. Box 254, Boylston, MA 01505. Note addresses so acknowledgement and gift cards may be sent. Donations should include the contributors email address so an electronic card may be sent. For more information, contact Gary Quam at (508) 869-0049.
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© 2015 by Roseanna M. White Published by Bethany House Publishers 11400 Hampshire Avenue South Bloomington, Minnesota 55438 www.bethanyhouse.com Bethany House Publishers is a division of Baker Publishing Group, Grand Rapids, Michigan www.bakerpublishinggroup.com Ebook edition created 2015 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—for example, electronic, photocopy, recording—without the prior written permission of the publisher. The only exception is brief quotations in printed reviews. Library of Congress Cataloging-in-Publication Data is on file at the Library of Congress, Washington, DC. ISBN 978-1-4412-2881-9 Scripture quotations are from the King James Version of the Bible. This is a work of historical reconstruction; the appearances of certain historical figures are therefore inevitable. All other characters, however, are products of the author's imagination, and any resemblance to actual persons, living or dead, is coincidental. Cover design by Jennifer Parker Cover photography by Mike Habermann Photography, LLC Author represented by The Steve Laube Agency To Pappap was my dedication when I first penned this novel at age thirteen. After I finished my first rewrite at fourteen, it said, In loving memory of Pappap. Your life taught me to laugh in every possible moment; your death taught me to trust Him with all my might. You helped make me who I am, and I'll always love you. CWM # Contents Cover Title Page Copyright Page Dedication Character List One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Twenty-One Twenty-Two Twenty-Three Twenty-Four Twenty-Five Twenty-Six Twenty-Seven Twenty-Eight Twenty-Nine Thirty Thirty-One Thirty-Two Thirty-Three Epilogue Author's Note About the Author Back Ads Back Cover # Character List Brook's Family --- Brook Eden | The lost Eden heiress. Full name Elizabeth Brook Eden. Also called Baroness of Berkeley and Lady Berkeley (title inherited from mother). The Earl of Whitby | Brook's father. Given name of Ambrose Eden, but called Whitby or Lord Whitby, nicknamed Whit. The Countess of Whitby | Brook's mother, deceased; the previous Baroness of Berkeley. Given name of Elizabeth Brook, but called Lady Berkeley before her marriage and Lady Whitby afterward. Mary, Lady Ramsey | Brook's aunt, Whitby's sister. Brook calls her Aunt Mary but everyone else calls her Lady Ramsey. Lady Regan | Brook's first cousin, the elder of Lady Ramsey's daughters. Lady Melissa | Brook's first cousin, the younger of Lady Ramsey's daughters. The Marquess of Ramsey | Brook's step-cousin, Lady Ramsey's stepson and her daughters' half brother. Called Ram. Lady Catherine Rushworth | Brook's second cousin on her mother's side, called Lady Catherine formally, Kitty by her friends. Lord Rushworth | Brook's second cousin on her mother's side, Lady Catherine's brother and guardian. Given name of Crispin, called Lord Rushworth or Rush. Lord (John) Rushworth | Brook's mother's first cousin, deceased; the previous Lord Rushworth and father of Lady Catherine and Rush. Major Henry Rushworth | Brook's mother's first cousin, brother of John Rushworth, uncle of Lady Catherine and Rush. Justin's Family Justin Wildon | Brook's childhood friend in Monaco. Also called Lord Harlow or Harlow. Upon his father's death, becomes Marquess of Abingdon, sometimes called Bing. Upon his grandfather's death becomes Duke of Stafford. The Duke of Stafford (with subsidiary titles of Marquess of Abingdon and Earl of Harlow) | Justin's paternal grandfather. Given name of Samuel Wildon but called Stafford or Duke. William Wildon | Justin's father. Called Lord William. He inherited the courtesy title of Marquess of Abingdon after his older brother's death but refused to use it. Georgiana Wildon | Justin's mother, deceased. Edward Wildon | Justin's uncle, deceased; the oldest son of the Duke of Stafford, so the Marquess of Abingdon until his death. Caroline, Lady Abingdon | Justin's aunt, Edward's widow, Georgiana's sister. Justin calls her Aunt Caro but everyone else calls her Lady Abingdon. Susan, Lady Cayton | Justin's aunt, daughter of the Duke of Stafford, mother of Lord Cayton; Justin calls her Aunt Susan, everyone else calls her Lady Cayton. The Earl of Cayton | Justin's first cousin, Susan's son. Called Cayton, though Justin occasionally calls him James. Other Characters Deirdre O'Malley | Lady's maid to Brook. Earl Thate | Justin's best friend, called Lord Thate or Thate. Viscount Pratt | Neighbor to Brook and Whitby, very distant cousin to Whitby on his mother's side. Called Lord Pratt or Pratt. The Marquess of Worthing | Brook's friend. Given name of Brice Myerston, son and heir of the Duke of Nottingham. Called Lord Worthing, Worthing, and occasionally Brice. Lady Ella Myerston | Brook's friend, Worthing's younger sister. Called Lady Ella or Ella by her friends. # One MONTE CARLO, MONACO LATE AUGUST 1910 Temptation sat before her, compelling as the sea. Gleaming silver, green leather, the nearly silent rumble of engine . . . Brook trailed a gloved hand along the door, cast one glance over her shoulder, and let herself in. She couldn't stop the grin as she gripped the wheel of the Rolls-Royce. And why should she? Only a fool would leave such a car running right outside her door and not expect her to do something about it. "Don't even think it." His voice brought laughter to her lips, and she looked up to find her dearest friend at the opposite door—her first sight of him in five months. The warm Riviera wind had tousled his hair, making her wonder where his hat had gone today. "Teach me to drive it, Justin." He glared at her with an intensity to match the Mediterranean sun. All manner of men flooded Monaco in pursuit of its casino, and none could glower like the British. Well, perhaps the Russians, but theirs were more scowls than proper glowers. Though, if he expected her to be cowed by the look, he had taken leave of his senses. He leveled an accusatory finger at her nose. "I'm happy to take you for a drive in my new car, mon amie, but I will be behind the wheel." "Come, Justin." She said his name as it was meant to be said. In French. Soft J and long U, emphasis on the second syllable, the N silent—as she knew no one in his native country did. "Your gift will soon be back in England. We mustn't waste a moment of its time in Monaco. Get in and teach me." "A moment of its time?" But he laughed and slid into the left side of the car, shaking his head. The sun caught his hair and burnished it gold, caught the angles of his face and made it all the stronger. "The prince will have my head for this." Brook grinned at him. Once upon a time, she had dreamed that they would fall in love and live happily ever after—before she realized a future duke could never be more than friends with a nobody without a past. Before she came to understand Prince Albert wasn't really her grandfather. "He will be jealous, you mean. He must always have a chauffeur behind the wheel." Brook gripped the wheel tighter, until she could feel the thrum of the 40/50 engine in every cell. "Perhaps I will borrow one of the chauffeur's jackets and surprise him one day—after you've taught me." Justin pressed a hand to his brow, dark blond hair falling over his fingers. "Heaven help me. I'll be executed. My poor grandfather will expire from the shock of it, the dukedom will go extinct, and it will be all your fault. All because you grin at me and I can't say no." She grinned all the brighter now. "I don't intend to race in Grand-père's road rally—I only want to learn the basics." She made herself comfortable on the seat, positioning her feet on the pedals on either side of the steering column. She had read books and articles about the advances of the automobile, but the pages hadn't come close to conveying the power that came coursing through the floorboard. It was almost as heady a feeling as having a spirited horse under her. Almost. Justin slid closer, casting her a sideways look she couldn't read—making fear knot in her chest. She'd been waiting months for him to return, had begun to worry he never would, that his family would succeed in keeping him forever in the Cotswolds of England, and he would forget his promises to investigate the seal on the old, yellowed envelope she had pressed to his palm five months ago. She cleared her throat. "Did you learn anything? In England, I mean?" Justin adjusted the position of her hands on the wheel. "Of course I did. Literature and mathematics—" "Justin Wildon." "—philosophy and science." He ducked his head as if to make sure her feet were where they ought to be. Or to avoid her gaze. "I came across the papers of a German not long ago. Fellow by the name of Albert Einstein, a physics professor. Have you read him? He has interesting theories—" "Lord Harlow." She narrowed her eyes at him, but he still didn't look up. "—about Newtonian physics and something called special relativity, which I know you'd find interesting." He straightened, focus still on her feet. "There are pedals for clutch, brake, and accelerator. Throttle is on the steering column. You must press upon brake and clutch to begin." "I know." She pushed them without taking her eyes off his strong profile. "And you know well what I mean." He finally swung his face her way again, jaw set. "We can either talk about that or you can learn to drive. Choose one, for I don't intend to open such a conversation with you behind the wheel of my very new, very expensive automobile." "Bad as all that, is it?" She prayed again she could live with the answers she'd asked him to find. For eight years now she had known only who she wasn't—not the illegitimate daughter of opera star Collette Sabatini and Prince Louis Grimaldi, heir to the throne of Monaco. Not the petite-fille of the reigning Prince Albert, as his wife, Princess Alice, had shouted for all the palace to hear before she left him. So if not a daughter or granddaughter to the only family she knew . . . then who? "Release the hand brake, first of all. There by the wheel, on your right." Drawing in a long breath, she gripped the wooden handle and moved it as she had seen their drivers do, then checked for carriages or cars in the street. Seeing none, she mimicked the pedal work she had observed, moving her foot from the brake and aiming it at the accelerator. "Brook!" "Quoi?" She jammed her foot back on the brake. Justin ran a hand over his face. "Attends! Please—wait for my instruction." Another grin tickled her lips and pushed away the phantoms of the unknown. "When have I ever awaited instruction? But did I not let my first arrow fly with admirable accuracy? Am I not a better shot with a pistol than you? Can I not out-fence any young lord?" At last a breath of laughter relaxed his shoulders. Then he caught her gaze and held it, his eyes as deep as the ocean. "You think I don't know the thoughts rampaging through your mind? But I assure you, you've nothing to worry about. The news I bring is good." He gave her fingers a reassuring squeeze. "But it will change everything. You shouldn't try to digest it when behind the wheel of a car." She nodded and pushed the questions aside. For now. "Now I check the street again and transfer my foot from brake to accelerator while easing off the clutch." "A statement rather than a question, I see." His fingers left hers as he turned around to look at the street. "All clear. Angle the wheel hard to the left and gently—gently—press that foot to the accelerator." She obeyed, reveling in the increased thrum of the engine. Easing the car forward, a laugh slipped from her lips. She straightened the wheel and headed for the opera house. She could get the hang of this, given a bit more practice. Perhaps she could even convince Grand-père to let her drive one of theirs. Assuming she remained in Monaco. Risking a glance toward Justin, she barely kept from taking one hand off the wheel to play with the two pearls dangling from the gold filigree of her necklace. "You did verify I'm English, then?" He shot a look at the fingers she had nearly lifted. As if he knew exactly what habit she'd nearly indulged. "We already knew that." She sighed and let off the accelerator when they came upon a slow-moving barouche. "We knew Maman said so, but she was hardly in her right mind those last weeks." And for so many years, Brook had hoped and prayed that that had been the lie, as Grand-père so often assured her. "It was right enough. You are indeed English. Which, assuming you've looked in a mirror now and again, oughtn't to surprise you." Right on cue, the wind cast a tendril of her pale hair before her eyes. She certainly had nothing in common with the rest of the Grimaldis. How many times had she wished for their rich dark hair and fathomless brown eyes? The skin that the sun could kiss yet not burn? A delicate snort was all the response she could manage. Justin loosed a sigh nearly lost under the purr of the engine. "The story she told seems to be true—she was in York with the opera at the time but did not have a child of her own." Had Brook been anywhere else, she would have let her eyes slide closed so that she could summon the image of beautiful Maman, try to conjure the sound of her sterling soprano. But the memory had faded over the years, until now it was little more than a crystal echo. "So Prince Louis was right to keep me always at a distance—I am not his daughter." At least she wasn't another cause for scandal in the Grimaldi line. But it also meant Maman was not her mother. And Grand-père . . . He hadn't wanted her to ask these questions. She was, he had said, the only member of his family who acted like family, and what would he have if she left? But she had to. She couldn't live her life as a pretender. The people were already shouting against him, how much worse would it be if he continued to support her when she had no real claim to him, other than a bone-deep love? The barouche they followed turned down a side road, and Brook pressed on the accelerator. "What am I, then? A farmer's daughter? An abandoned waif?" His chuckle helped ease the band around her chest. "Mais non. It is as we imagined—you are a nymph from the fairy world." "A naiad you mean, ruling over a—" "—a brook. How could I have forgotten?" He captured the curl that obscured her vision and gave it a playful tug. "One of my favorites of our recent stories—'Brook of the Brook.' And where is my fairy princess taking us?" She smiled, but even the thought of the stories they created and picnics atop the ramparts overlooking Port Fontvieille couldn't erase the questions. "The theater. I have a ballet lesson. I keep threatening to join the Ballet Russes—Sergei says I am as talented as his Russian dancers." "An imp more than a naiad, surely." He tugged again on her curl and tucked it behind her ear. "I can only imagine how mad that drives the prince." "It hardly matters what I do." She slowed as her turn approached and prepared to wrestle the wheel around. Her heart thudded, but she drew in a deep breath. If she slipped, Justin would catch the wheel, would keep them from harm. "You will not take the stage." Justin sounded far harsher than Grand-père had. Perhaps her tone had been too blasé. Still, she could hardly resist teasing him—and fishing for more information. "Excuse me, your lordship, but why not? My mother was on the stage." "Collette would have been the first to tell you not to follow her example. And she was not your mother." "Quite right—I am an orphan, an unknown. Lizette Brook—a nobody." "You most certainly are not." "Who am I, then?" She glanced his way, brows arched. "Eyes on the road!" Hopefully he saw only that she turned her face square to the windscreen and not that she rolled those eyes in the process. "Was I right about the envelope? The seal?" Maman had left her with boxes upon boxes of correspondence, faded letters from faded loves. But one box of them had been different—they were in English. The tone was different too—not at all what amorous patrons had usually sent to Collette. And more, as she'd searched through the letters in the flat she'd shared with Maman before moving to the palace after her death, Brook had seen a variation of her own name on the ones on the top of the stack. Give Little Liz a kiss from her papa. But it had been signed only with Yours Forever, and the one envelope with the seal upon it had no address. Yet again she had to resist the urge to touch her necklace. The necklace Maman had confessed with her last breath had belonged to Brook's true mother. The woman killed in the carriage accident from which Collette had rescued Brook. The my love those English letters were written to? "The seal was helpful. Brook." He sighed again and rested a hand on her shoulder. "It led me to your mother. I saw a portrait of her, and it might as well have been you in a bustle. We found her. We found you." Her fingers curled around the wheel so tightly she feared she'd leave an impression in the wood. "Who, then? Who am I?" "We're nearly to the theater—pull over here. Foot off the gas, press the brake and then the clutch. Turn, turn." His fingers covered hers as he helped her guide the Rolls-Royce into an open spot nearer the casino than the theater. The moment the car halted, he reached over her to engage the hand brake and then switched off the magneto. The absence of the engine's noise barely made a difference with all the chatter from the street. But Brook didn't look at the gaily clad aristocrats making their way into the Casino Monte Carlo—she looked at the muscle gone tense in his jaw. "Justin." Her voice came out in a whisper so soft she couldn't be sure he heard her. "Tell me." He leaned against the green leather of the seat, elbow atop it, and rested his hand on her shoulder again. "You are a baroness." "A . . . what?" She knew the title—one couldn't be the friend of a duke's grandson without getting lessons in the British peerage. Which was why she knew she shouldn't have such a title unless by marriage. "How could I be a baroness?" The wind tried to toss that curl into her face again, but he caught it and tucked it away once more. "From your mother, who was a baroness in her own right. Passed from her mother, and her mother before her. You are Elizabeth Brook Eden, Baroness of Berkeley—one of only a handful of peeresses whose title is by right and not courtesy. And the heiress to a large estate." Little Liz. Maman had kept her name, just made it more French—Lizette Brook. Choosing to go by her middle name after Collette's death had been one of Brook's many small rebellions. Her eyes slid shut, her fingers found the warm pearls dangling from her necklace. Her mother's necklace. Her mother. "What was her name?" "Elizabeth as well, born with the surname Brook, which is where your middle name came from. Countess of Whitby." "Countess?" Her eyes flew open again. "My father was an earl?" Justin's free hand found hers, and he linked their fingers together. "Is an earl, Brooklet." Had she been standing, she would have had to sit. "My father . . ." "Is very eager to meet you." He squeezed her hand and ran his thumb over hers. "It's time to come home, Lady Berkeley." Brook drew in a long breath seasoned with fruit from the markets, the spice of Italian cooking, and the salty tang of the Mediterranean Sea. All her life, all her memory, this had been home. All the world she'd needed. "I . . . I must absorb all this." "Of course you must." He lifted her hand and kissed her knuckles as he had done ever since they played knight and damsel as children, back when she had dreamed it was real. But his eyes remained locked on hers now. "I know you have been praying about this as much as I have been. This is the answer to those prayers, mon amie. This is where the Lord wants you. And I will be with you every step of the way." No doubt he was right. And no doubt when her thoughts stopped crashing like waves in a tempest, the peace of the Lord would descend. But right this moment . . . "I must go. Au revoir, Justin." She leaned over, kissed him on either cheek, and let herself out of the car. A warm breeze gusted up the street. Brook touched her hat to make sure it was secure, then let her fingers fall to her necklace. A baroness, daughter of an earl. Of all the scenarios she had entertained, that had never been one of them. # Two Justin was probably the only man in all of Monaco who dreaded crossing the threshold of the famed Casino Monte Carlo. He'd done so enough that the opulence had no effect. The reliefs didn't turn his head, the paintings didn't draw his eye, and the crystal chandeliers were nothing but light for his feet. He could be thankful they had made their home here in Monte Carlo, because of Brook. But still he wished his father would find a different life. Perhaps if he lost more, he would. But no, Father had made a fortune at the tables over the years. It was hard to convince a successful gambler to turn over a new leaf when he could turn up a new card instead. Justin paused at the doorway of the baccarat room. Yes, there he was. A debonair smile upon his face, an impeccable suit on his lean figure, a pretty girl beside him. Drawing in a long breath, Justin closed his eyes for a moment. Prayed, for the millionth time that day alone, for the strength to have the needed conversation. Again. Prayed that this time Father would hear him. When he opened his eyes, he saw his father toss back the contents of his snifter—cognac, no doubt—and stand. He wobbled a bit as he straightened his jacket, but he was smiling. Blast. It may have been easier to convince him to return to England had he been fresh from a loss rather than a win. Father's smile grew when he spotted Justin, and he shook off the woman who had tried to tuck her hand into his arm. "There you are, Justin. Have you been out enjoying your birthday gift? Your Brook saw it the other week when it arrived and assured me it would suit you." How could he help but grin? Not just at the thought of his new Rolls-Royce, but at the man who had given it. Father had his faults—and twice the charm to offset them. "It is a magnificent car. Thank you." "Good, good." When near enough, his father clapped a hand to Justin's shoulder and steered him toward one of the washrooms. "I considered one designed by that Bugatti chap but knew you would appreciate the English touch." "Indeed." And it was as good an opening as any. "Speaking of things English—" "Save your breath, my son. I'm not going back." A footman bowed as he opened the washroom door for them. "Good evening, Lord William." "Pierre." While Father moved to the mirror, Justin sighed and sat on a plush chair. "Your refusal to come home doesn't change facts. Uncle Edward has been dead for twelve years—you are not Lord William anymore. You are the Marquess of Abingdon, the heir, and will be the next Duke of Stafford." "And facts don't change reality." Father undid his tie and started the knot afresh. "I have no interest in the duchy. When the old man kicks off, the title may come to me by law, but you'll manage the estate perfectly well without my help." Justin passed a hand over his hair—what had he done with his hat? He must have left it at the palace when he called on Prince Albert earlier. "He wants to see you. He isn't well, Father—it's time to make your peace." Father's reflected eyes met his in the mirror. One more tug on the tie and it was in a perfect bow. He turned, faced Justin. "There is no peace to be made." "But if you only—" "Don't ask it." Father sighed, and his face softened. "It's for the best. You will make the better duke, be the better overseer of the estate. If I tried to put my hand to it, I would foul it up." Justin stood again. "Nonsense. If you hadn't an innate sense of how to manage things, you wouldn't do so well here." Though he had to admit he was glad he took after his uncle Edward—and not so glad his cousin, Cayton, his father's sister's son, seemed to take after Father. Chuckling, the marquess headed toward the casino floor once again, then shifted tack and made for the front doors. "Knowing when to fold a hand and when to bet is a far cry from dealing with tenants and whatnot. As you ought to know, being excellent at the latter but an absolute dunce at cards." One corner of Justin's mouth tugged up, even as he fought down the desire to claim he could be good, if he tried. An experiment he had sworn to himself he would never perform. "Call it lack of interest." Though his smile remained bright, a shadow flitted through Father's eyes. "I suppose I ought to be glad I scared you onto the straight and narrow. But there are worse lives than the one I have chosen. You ought to toss responsibility to the wind and indulge yourself for once. Set up your singer's daughter as your mistress and—" "Oh, for—I will not make a mistress of Brook. Or anyone else." And why must they have this conversation? In public, no less? Another footman opened the main doors and handed Father's hat to him—obviously the employees knew the man's comings and goings far better than Justin did. With a shake of his head, he stepped out into the warmth of the evening. The sun was setting behind the mountains to the west, dusting the city with gold. Mischief had entered Father's eyes again. "Why must you always be so pious? I've seen how you've begun looking at her, and it's no wonder—she's grown into a beautiful young thing. But you can't wed her—one so set on doing right by his ducal grandfather would never disgrace the family name by marrying a performer's daughter." Justin tried clenching his jaw to keep from rising to Father's bait. But he couldn't stop himself. "She is more than that. Though—" "Well, she is no princess. She looks no more like Prince Louis than she did her mother." He followed when Father turned to the right. "Collette wasn't her mother. Brook is English. A baroness, as it happens." That brought Father's feet to a halt and his brows up. "Really. Well then, I suppose you can wed her. Do it here, will you, before you leave again? I don't fancy having to travel through the dratted English rain for your wedding." There was no reasoning with him. Why did he try? Justin shook his head as they started forward again. "Who said anything about marrying her?" "Well, you can't make a mistress of her—she's a baroness." A snort of laughter slipped out. "Where are you going? Shall we have a meal together?" "Not tonight, I'm afraid, though I wish you had made it home for your birthday—I had a regular gala planned out." Father tipped his hat to a couple strolling toward the casino. "Five and twenty now. Your grandfather must be hounding you to marry soon and be about the business of heirs, chanting nonsense about duty." "Mm." Regardless of his denial a moment ago, only one face ever came to mind when he considered a wife—but she never looked at him as anything but an old friend. Besides, Brook would have many changes to work through in the near future. "But he knows I have my hands full with learning the estates. And I will have to see Brook settled with the Earl of Whitby besides." "Here we are." Father halted before a gleaming roadster and rested his hand on the bonnet—apparently he had bought a car by "that Bugatti chap" after all. "The Earl of Whitby, you say?" Justin couldn't help but take a moment to admire the artistry of the lines. "Indeed. Do you know him?" "I used to." Father's voice went musing. "I heard he got rather eccentric after the death of his wife and the disappearance of his . . . Wait. Your Brook is his missing daughter? How the devil did she end up with Collette Sabatini, and here of all places?" "Collette was in Yorkshire at the time. She came upon the carriage accident. Though why she brought her here is a mystery." Justin frowned when Father got into the car. "Surely you don't mean to drive when you've been drinking. A car isn't a horse—it can't find its own way when you keel over in a stupor." Yet he tossed his hat to the seat and put on goggles and a cap. "You are my son, Justin, not my nursemaid. I am fine—and going to France to keep a dinner engagement. Your bed ought to be made up in the flat, and Fitzroy knows not to expect me." "Father—" "I will be home by luncheon tomorrow." He flashed a smile, all gleaming white teeth and charming irresponsibility. "Go and find yourself some trouble—it'll do you a world of good." Once more Justin had to shake his head. "One of these days we're going to finish a conversation without you riding off on some new lark." "Anything's possible, I suppose." The engine sprang to life with a roar and a rattle, and Father gave him a jaunty wave before backing out into the street without even looking behind him. Justin pressed a hand to his temple. He ought to go fetch his hat . . . after he walked off his hope and frustration. Grand-père found her on the ramparts. Brook's muscles were still warm and fluid from her ballet lesson, making her feel that if she stretched high enough, she could touch the clouds scuttling over the sky, or reach out and skim her hands through the warm waters. She grinned at him, but the prince's returning smile was small and tight. In his hands he clutched a worn leather book. Her chest went tight, her relaxation vanished. Her fingers pressed into the warm white stone. "You've spoken to Justin." "Before he found you." Grand-père didn't stop until he had pulled her tight to his chest and wrapped his arms around her. He smelled of security—ink and paper and a whiff of cologne. "I asked you to let it drop, ma fifille. To be content here, with me." She squeezed her eyes shut against the familiar worsted wool of his favorite jacket. "Grand-père . . . if it were only us, I would. You know that. I love you more than anyone else in the world. But with the people rioting—" "That had nothing to do with you. They want a constitution—that's all." It wasn't all. They all knew it wasn't all. She held him tighter. "Prince Louis was right all along. I'm not his. Charlotte clearly is—she is where your hope lies. Adopt her to keep the Grimaldi line going. Get to know her. Love her." Brook had never even met Charlotte—the illegitimate daughter with another performer, the daughter Prince Louis actually claimed as his own. But for a few years after the girl's birth—before Collette's deathbed confession—Brook had believed the child was her half sister. "You should never have taken me in after Maman—" "Hush." He pressed his lips to the top of her head. "You are my petite-fille. Whatever your blood, that will not change. And I wish you would stay." "Grand-père—" "Je sais. I know you will go, you are too headstrong to listen to your old grandfather when you have made up your mind." He pulled away, revealing a sad, proud smile. Touching a finger under her chin with one hand, he held up the book in his other. "You should have this, then. I promised Collette I would destroy it so you would never find it, but I couldn't. I think I always knew you would not be happy here forever—not when there were questions out there in need of answers. It is her journal." Brook's brows knit. "Whose? Maman's or . . . or my real mother's?" "Collette's." Though he pressed the journal to her hands, he held it still, held it shut. "Whatever answers it has, she thought they would hurt you. There must be a reason for that. Don't open this until you're ready to know what that reason is." Mutely, she nodded. Her fingers registered the worn leather, tried to feel what secrets might lie within. Part of her wanted to open it immediately, heedless of the warning, and learn what truth she could. But then she glanced up into Grand-père's troubled dark eyes and lowered the book to her side. She couldn't hurt him like that. It would be tantamount to shouting that all he'd given her, all he'd given up for her, meant nothing. "I will wait, Grand-père." Relief softened his eyes, and he nodded. "Come inside, ma fifille. Dress for dinner and then play for me. Let me hear you sing again before you leave me." Tucking her arm into the crook of his elbow, Brook let him lead her from the ramparts. Secrets could wait. An hour turned to two. Justin let the warm breeze soothe him, let the mixed scents of sweet and spice remind him of a childhood spent racing through these very streets. He had found trouble aplenty, adventure and happiness too. And Brook. He had found Brook on one of those unsupervised sprees. She had been but a sprite of a girl then, only five to his twelve, but the mischief in her eyes had intrigued him. Thate said it was strange that he had found such a steadfast friend in a girl seven years his junior. But it had never seemed so. At first she had simply amused him, and he had fancied her a sister to replace the one he'd barely known. Then it had been entertaining to teach her all the sport he shouldn't have. And now . . . now they had thirteen years of shared history. The guards let him pass with no more than a nod, and the footman merely pointed him toward the prince's private library. Once he reached the room, the sweet voice spilling out in an Italian aria brought him to a halt. Odd how much like Collette she sounded, though they shared no blood. Her maman had trained her well. He leaned into the doorway and saw Brook at the piano, accompanying herself as she sang, while Prince Albert lounged in his favorite chair. Her flaxen curls were twisted into some sort of chignon, an embellished band setting it off. As always, she wore the gold and pearl necklace Collette had said was her mother's. Its twin strands of links and pearls met at the filigree in the center, from which two dangling pearls drew attention downward. Justin forced a swallow. She had grown into a young lady too beautiful for his peace of mind. The notion of courting her had begun to niggle in the last few months. But he knew well she didn't look at him like he had begun to look at her. He would have to convince her. Win her. After she settled with her father at Whitby Park, after she had come to terms with being Baroness of Berkeley. After he was better grounded in his duties in Gloucestershire . . . then he would try to make her see that they could have so much more than friendship. When she finished the song with a flourish, Justin joined his applause to the prince's. She stood with one of those heart-stopping smiles of hers aimed his way. "Justin!" As always, the greeting made him smile. Only his closest family ever used his given name in England, and they never attempted the proper French pronunciation. "Bon soir, mon amie. Your Highness." The prince smiled, but Justin scarcely had time to note it, given that Brook came his way with her hands extended. He took them in his and leaned down to exchange the customary cheek kissing. And grinned at the thought of how her English family might react to the French ritual. "You look lovely tonight, Brook." An understatement, but it nevertheless brought a pretty blush to her ivory cheeks. "Merci. It is the new dress." She released his hands and did a pirouette worthy of the stage. "From Paris. Grand-père had it commissioned." "I told her she would be the envy of all the ladies in England." Prince Albert stood with an indulgent smile. Justin didn't miss the sorrow around its edges. "Indeed." Yet it wasn't the gown that would set her apart—it was her spirit. No other lady he'd met in England laughed with such abandon, moved with such grace, put such passion into her every pursuit. He prayed that spirit, and the faith beneath it, would be enough to sustain her through the transition ahead. As if the same thought had possessed her, her smile dimmed, as did the diamond gleam in her emerald eyes. "You'll join us for dinner, oui?" "I would be delighted." For now, he led her to the settee and took the cushion beside her. "Has it sunk in yet?" Her fingers toyed with the dual pearls dangling from her necklace. If there were a surer sign of her perplexity . . . "What if I am not this baroness? What if they turn me away?" The prince huffed. "That is simple. Then you will come home." He came to them and sat on the settee, resting a hand on Brook's shoulders. He had fought for her, fought to move her into the palace after Collette's death, though the rest of the family thought it a mistake. Because by then Brook had already been his fifille—his little girl. Prince Albert would always be her grandfather. Although even if she were not the baroness, Justin had no intentions of bringing her back to Monaco. He would convince her to stay, somehow or another. The thought of not seeing her for years wasn't to be borne. "We are not mistaken, Brooklet. Had I not been sure about this, I never would have said anything." "But—" "There is no reason to doubt, and every reason to believe this is who you are." He held out his hand until she put hers in it, then covered her slender fingers with his. "You have a father eager to love you. An aunt to usher you into society. Cousins near you in age waiting to become your friends. The Lord has prepared your place. There is no need to fear." He could see the trust returning to her eyes, the sparkle that brought light to the flecks of amber around her pupils, to the rings of sapphire around the emerald. His chest went tight. What would it be like to gaze into her eyes every day? To hear her laugh, her voice, to share stories whenever they pleased? To have the right to draw her into his arms and see if her lips were as soft as they looked? Maybe he wouldn't wait to declare himself. Maybe he could win her heart now and deliver her to Whitby as his fiancée—and use the wedding to lure Father home. Hurried footsteps intruded, startling enough to warrant the frown on the prince's face. When a footman charged into the room, the look of horror he wore brought Justin to his feet, Brook along with him. If some crisis of state were about to be announced—and with the revolt of a few months ago still fresh in their memories, he wouldn't discount it—he would take his leave so the prince could attend to business. But the servant looked to him. "Excusez-moi, Lord Harlow. Forgive me for bringing such news, my lord, but . . . your father. There has been an accident on the mountain road." His fingers went lax within Brook's tightened grip. Clouds gathered before his eyes. "What kind of accident?" # Three WHITBY PARK, NORTH YORKSHIRE, ENGLAND Deirdre O'Malley held the fresh sheets to her chest and sent an amused look toward the housekeeper. How much longer could his lordship's sister keep pacing the halls like a caged beast? Lady Ramsey had intercepted Deirdre nearly half an hour past to keep her from carrying out Lord Whitby's command to ready the Blue Room, but she had yet to decide which one ought to be prepared in its stead. Though Mrs. Doyle pressed her lips tight to suppress a smile, she sent Deirdre a wide-eyed, cautionary look. "The Rose Room, my lady? Is that one far enough from Lady Regan and Lady Melissa?" The marchioness sighed and pressed a hand to her brow. "It is too far. If we put the girl in there, my brother will know exactly what we're about. I don't want her near my daughters, but we can't put her at the opposite end of the wing." "It would show her plain as day what we think of her," Deirdre murmured into the sheets, though she knew she ought to keep the thought to herself. But her ladyship smiled and let her jet-clad wrist fall to her side again. "Ah, but my brother is convinced this one is real." "As he hoped the last three times." Mrs. Doyle started back toward the end of the hall nearer the stairs. "We all know how those ended." That they did—in each pretender being kicked to the drive. And with the earl becoming more a recluse than ever. "What about the Green Room?" Mrs. Doyle opened a door halfway down the hall. Lady Ramsey peered in. "It is awfully grand." The way the housekeeper's spine snapped even straighter than usual would have been more amusing had Deirdre not caught a glimpse of the clock on the chamber's mantel. Her half-day off duty would begin in another fifteen minutes, but she could hardly leave in the middle of a task without getting a scolding. Though, if she didn't make it into the village by two . . . "My lady, of course it is grand—they all are. This is Whitby Park, after all." "So I am aware." Her ladyship chuckled and touched a hand briefly to Mrs. Doyle's arm. "Very well, then—the Green Room it is. I will let my brother know I have changed his arrangements." Much as she liked Lady Ramsey, Deirdre breathed more easily once the lady had gone back down the stairs. She followed Mrs. Doyle into the bedchamber and set the sheets down. When she turned, the older woman was pulling off the coverlet. "Oh, you needn't trouble yourself, ma'am!" Mrs. Doyle didn't so much as pause. "Nonsense, Deirdre. Beatrix is putting the drawing room to rights, and making the bed yourself would take too long. With the earl's nieces here, you must be back from the village in time for the dressing gong." "Then I thank you." She unfolded the first of the sheets and handed one side to the housekeeper. "He swore after the last one that he wouldn't entertain any more pretenders." A long sigh accompanied her superior's brisk movements. "This one comes on the recommendation of Lord Harlow, a future duke. It is hard not to make an exception, given that." She tucked a corner with precision Deirdre had learned from her years ago. "Wish as we may that his lordship wouldn't have to go through this again, it is already set. The girl is coming. All we can do now is pray she leaves the earl's heart intact when she is dismissed." "Aye." They worked in silence for a moment, but Deirdre met the woman's eye again when they shook out the top sheet. "I have always wondered why his lordship didn't just remarry and hope for a son." A wistful smile settled on Mrs. Doyle's lips. "You would understand had you seen him with Lady Whitby. He'll mourn her for the rest of his life." "I suppose it's never easy, losing one's spouse." Mrs. Doyle fluffed a pillow and put it in place. "How is your mother faring these days?" "Getting on." As best as to be expected, anyway. Mum couldn't move past Da any more than the earl could his long-gone countess. She helped pull the coverlet back up, smooth it out, position the decorative pillows. "There we are." "And off you go. Remember—back by the dressing gong." Not wasting time on anything more than a curtsy and a smile, Deirdre hurried out and up the back stairs, untying her apron as she went. The sparse room she shared with Beatrix was silent and empty, so Deirdre laid the white apron carefully upon her bed and took up her coat, hat, and handbag. Inside the last she'd already tucked the letters she needed to post—one for Uncle Seamus in India and another for Mum and her siblings, including the pound notes. Half past one already. Heavens, but she had better hurry. Praying she didn't meet with Mrs. Doyle or Mr. Graham, the butler, to be scolded for her too-quick step, she flew belowstairs and headed for the back door. "Deirdre, wait! I'll walk with you to the village." She oughtn't to have to stifle a groan, not over Hiram. And any other day she would welcome the company of the second footman. Just not today. Still, she paused a step away from escape. Noise from the kitchen filled her ears, and its scents reminded her that she would miss tea—and she hadn't put aside any of her pay for frivolities like a biscuit from the baker in town, not this month. It would all head to Kilkeel. Little Molly would need a new coat for the coming winter, Mum had said. Hiram tugged a hat onto his head as he joined her. "Shall we, then?" "Aye." Though as soon as they were out in the cool air, she reached up to straighten his hat for him. "Much better." He laughed and skewed it again. "Stop your fussing, Dee. I'm not expected to look as polished as the silver when on my own time in the village." "Mr. Graham would disagree." A grin tugged at her lips. "I don't see him about, do you?" He checked over each shoulder to be sure, though, as they headed around the drive. "Safe and free. Have you any big plans this afternoon?" Her fingers tightened around the frayed strap of her handbag. "Letters to post, a bit of this and that by way of errands. You?" "As it happens, my cousin is on his way through the area, and we're grabbing a bite at the pub." Praise be to heaven—he'd be paying no mind to her, then. "Oh, won't that be a treat for you." "Aye." Hiram shot her a grin that faded to a comfortable silence. He took up a whistle as the long drive went round a bend. His ditty proved lighter than the sunshine flitting in and out of the clouds, warmer than the autumn air. She fussed with her jacket's buttons and tried not to sigh. How did he do it? Stay so bright and cheerful all the time, as if his parents were still alive, as if his brothers hadn't all been scattered, as if he hadn't been passed over for first footman when Mr. Graham's nephew arrived? As if life were fair? But she couldn't recall ever seeing Hiram frown for more than a minute, and they had both been working at Whitby Park for nigh onto seven years now. Made her wonder if there weren't a screw loose somewhere in that pleasant-looking head of his. His whistle came to a halt. "Hold up a moment, Dee. I've a lace untied here." She let her feet carry her a step farther while he bent down, let her eyes sweep across the moors that had never felt quite like home. Maybe one of these days she'd be able to return to Ireland. Settle down with a farmer or merchant who wouldn't mind that her best years had been spent in a lord's house in England, see that Mum passed her later years without working her fingers to the very bone. Assuming she could ever get ahead of the debt Da had taken on when the crops failed back in 1902. It wouldn't happen on a maid's salary, for sure and certain, though the extra she made as head housemaid certainly helped. "Dee!" The panic in Hiram's tone snapped her back to the present. Hooves thundered—and she had wandered into the crossroads. She hadn't any time to realize where the horses were coming from before she was yanked backward. Her feet tangled with Hiram's, and they both tumbled into the ditch. Pain shot through her bottom as she landed. At the loud whinny directly before her, she looked up to see that the two horses had reined in and one of the riders had dismounted. Hiram muttered something unintelligible and helped her to her feet as the rider strode their way. A mere glance showed her why her friend had been so quick to pull her up—Deirdre dropped into a wobbly curtsy. "Lord Cayton, my apologies." The young earl frowned and halted a few steps away. "We are the ones who must apologize for such a careless race. Are you injured?" "I am well, my lord." Deirdre smoothed her grey skirt and directed her gaze to the ground. No doubt Lord Cayton wouldn't recognize her from the times he'd come to Whitby Park, but it would take no great logic to realize from where they'd come. And his lordship may decide later it was their fault rather than his. "And you, man?" Hiram cleared his throat. "No worse for the wear, my lord." "Leave them to their outing, Cayton, and let's be on our way." The second voice brought Deirdre's gaze up, but only for a moment. A moment was sufficient to reveal the chiseled features and ebon hair that matched the smooth baritone. "Coming, Pratt. You're both certain you are well?" Deirdre nodded along with Hiram as Lord Cayton remounted his horse. They held their place until the riders had continued past and then stepped back onto the road toward Eden Dale. Hiram let out a whisper of breath and brushed something from Deirdre's shoulder. "Are you hurt, DeeDee?" "Nothing that hasn't passed already." She grinned to let him know she meant it. "And you?" "Fine." But he sent a rare frown after the gentlemen before he shook himself and smiled again. "We have an adventure to tell now. And some folks claim village life is too quiet." She had little choice but to laugh. The rest of the walk into town was uneventful, and they parted ways at the pub. Deirdre first posted her letters and then paused outside for a fortifying breath. A look around proved no one paid her any undue mind, so she headed for the church. Silence embraced her inside the sanctuary, and light slanted in with all the colors of the stained glass. It ought to have brought peace, reverence, but instead her pulse picked up as she slid into the next-to-last pew. Only then did she check her watch—two minutes to spare. No footsteps sounded, but she felt it when he came in, and she held her breath until he slipped into the pew behind her. Held it until, as always, he leaned forward and pressed a kiss to her jaw. "You nearly frightened me to death back there in the lane." Her eyes slid shut. "Nothing frightens you, Lord Pratt." Least of all the thought of her being harmed. "You think me such an ogre?" "I think you . . . too far above me to be disturbed by my stumbling." She slid away a few inches and turned to see his profile. The first time he had approached her, she had been struck dumb by his beauty. But it was the beauty of a dark angel—that she had learned quickly enough. His chuckle made no pretense of mirth. Much like the fingers he trailed down her neck never pretended they wouldn't as soon strangle as caress. "Tell me, my lovely Deirdre—how is it you know Lord Cayton?" Though she wanted to swallow, she didn't dare. Those fingers would note it and mark it against her. "He . . . he came to Whitby Park with his cousin last month. Lord Harlow. About the girl." "And that is the only time you've seen him? He hasn't come another time to call on Whitby's nieces?" He lifted a brow, his black gaze promising to know if she lied. "He came to dine once since. But he seemed more taken with Lady Melissa than Lady Regan." "Good. Good." Lord Pratt rested his arm on the back of the pew. "And Lady Regan—of whom has she been speaking lately?" Not him, though she wished she didn't have to admit that. "Her preference isn't clear, my lord. Though her sister teases her most about Lord Worthing." "Hmm." No one else she had ever met could pack so much displeasure into a hum. "You, of course, put in a word wherever you can." "Of course." "And Whitby—I heard he succeeded in breaking the entail on the estate." That, at least, should appease him. "Aye. With no possible heir through paternal lineage, they granted it. The estate will go wherever he wills it, and the title will go extinct when he passes on." She wasn't sure why so distant a maternal cousin as Pratt had any thought his lordship might name him heir—but then, he knew it was unlikely. That was why he was so determined to court Lady Regan. Lord Pratt leaned in until their noses all but touched. "And where will he will the estate?" "I . . . Mr. Graham thinks it certain Lady Regan will inherit, but Lord Whitby never speaks of such things in my presence." "Of course not." His smile did nothing to soften the steel in his eyes. "But he speaks of it to someone, and someone else overhears. Then that someone no doubt bandies it about in the kitchen later. I ask only that you keep your ears open, my sweet." Her nod was slight, lest it put her face any closer to his. "I do." "I know you do. After all, you realize my funds are not unlimited. I cannot keep supporting your family forever, not without—" "I know." She squeezed her eyes shut. "Unless, of course, you are willing to—" "Please. I understand." He laughed. "Very well, my lovely, cling to your so-called respectability a bit longer." The crinkling of paper drew her eyes open again, and she saw a banknote dangling before her. Eyes wide, she looked past the note and to him. "Why is it more than we agreed?" "Incentive." He reached over the pew back and slid it into the handbag she'd set at her side. There was nothing she could do but say thank-you. Even though she knew the devil never made a gift without demanding something in return. # Four Rain pelted the window, and the wind howled about the railway carriage. Brook pulled her coat tighter and wished for a blanket. Across from her, Justin pressed his lips together, but a smile still winked. "Cold, Brooklet?" Perhaps she ought not to have teased him so mercilessly over the years about his inability to adjust to the Mediterranean heat in the summers. Turnabout was fair play, after all. She crossed her arms and dug up a grin. "It is invigorating." As he laughed, Brook looked toward the door at the end of the car. Their companions would be back any moment—his valet, Peters, and her governess-turned-chaperone, Mademoiselle Ragusa. Perhaps Brook should have requested some coffee to warm her. She decided to settle for a body to block the chill from the window and so moved to Justin's side. Her book thudded to the floor, and he leaned down to pick it up. Then laughed again. "Dracula?" Lifting her chin, she snatched it away and set it beside her. "It has a portion that takes place in the town of Whitby. How was I to pass it up?" Though he shook his head, his eyes gleamed. A beautiful sight—for the week they remained in Monaco making arrangements for his father's funeral, he had been so silent she feared he would turn to marble. "Not exactly scientific research on your new hometown, mon amie." "Well, it was the best I could find in the meager ten minutes you afforded me in the book shop yesterday." And the thought of her "new home" made her every bit as anxious as the red-eyed stranger had made Harker in the first chapter. Justin studied her for a long moment, seeming as usual to divine her thoughts from her innocuous words. With a crooked half smile, he took her hand in his. And set the world to rights. "Look." He nodded toward the window. No new rain pattered the pane, though a few stubborn drops still clung and slipped along. Beyond them, sunshine broke through the clouds and painted the landscape with gold. Brook drew in a long breath. She had read of the English moors, and Justin had done his best to describe them to her. But nothing had prepared her for the sheer expanse. The land seemed to roll on forever, hardly touched by man. Heather blossomed purple and shone green as far as the eye could see. "It's beautiful. So . . . big. You could fit all of Monaco in that one valley." It made her itch to find a horse and let it have its head, to fly through the countryside until she lost herself in its grandeur. A new chill swept up her spine. Perhaps she didn't want to lose herself quite yet—not until she knew she had been found. "Another minute and you'll be able to see the North Sea. That should help you feel more at home." She kept her gaze fastened on the moors, not arguing when he slid closer to the window and pulled her along with him. She drew in a deep breath. "How do you survive in the Cotswolds without an ocean nearby? I don't know that I could." "Whenever it becomes unbearable, I simply go to Monaco." As he said that last word, the mirth faded from his eyes, and his tone went from cheerful to a low throb. His thumb stroked over her knuckle. "I suppose I have no reason to return there now." Her heart twisted at the pain in his voice. "It has only been a week. Give yourself time to take it in." Now he gripped her fingers so tightly they pulsed along with the memories. His face contorted for a fraction of a moment before he battled it back into a smooth, handsome mask over the agony. "I tried to warn him. He'd had too much to drink, he ought not to have—but he wouldn't listen. He would never listen, not about anything." Covering his hand with her other one, she prayed the gentle pressure she applied would steady him. "His choices were his own." "I know. But I . . ." He touched his head briefly to hers. "I'm sorry. I shouldn't prattle on about my loss when you've a reunion before you." "Please. Prattle." She tried to grin, though it felt unconvincing. For a moment he simply stared at her, the sapphire of his eyes going deeper with contemplation. Then he leaned over and kissed her forehead. "You have nothing to fear, mon amie. They will welcome you." She longed to believe him. But the heather outside stretched on and on, no civilization in sight. Everywhere she looked was green and soft purple instead of white and terra-cotta. Lovely, but not home. What if her family—assuming they were her family—were the same? The breath she drew in quavered. "And if not?" His fingers squeezed hers again. "Then you pay a visit to the Cotswolds. There is no ocean there, but there will always be a friend." Yes, better to focus on the unchanging. No matter what, Justin would always be there. Even if there was still too far away. Wishing she didn't feel like a lost child, she clung tight to his hand. "But you will stay in Yorkshire a little while, oui? At least until we are sure that I . . . that they . . ." "Until you are well and truly settled." His smile was his own now, not the shadow it had been the last week. "My cousin Cayton has a house an hour's drive from Whitby. I can stay with him as long as necessary." An hour's drive—in Monaco, that would take one into France, most of the way to Italy. Odd how it now kept one within the same neighborhood. She nodded and directed her gaze to the window again. Just in time. The train crested a little knoll, and there, out in the distance, beckoned the unmistakable sparkle of sun on a placid sea. Slate grey rather than emerald and azure, but that was no matter. It was the ocean, capable of raging and calm, of peace and war, of beauty and destruction. Her lips tugged up. Justin was right—wherever there was a sea, she could find her place. Mademoiselle Ragusa and Peters returned a moment later, the latter handing a cup of steaming coffee to Justin. The smell brought her to alert—though at the look on Justin's face when he sipped, she couldn't help but laugh. "Not to your liking, my lord?" "In some things I will always be Monegasque." Justin took another drink but then shook his head and handed the cup back to Peters. "Coffee, if not strong enough to wake a man from a coma, is not truly coffee." "Hear, hear." Brook raised an invisible cup of caffe espresso in salute. His valet chuckled and settled into his seat across the aisle. "Rest easy, my lord. Soon enough you'll be back at Ralin, where Mrs. Moore knows exactly how you like it—even though no visitor can stand the stuff." They could be sure at least one visitor would enjoy it, when Brook finally made her way to his home. All her life she had heard his stories of Ralin Castle, of its burgeoning flower gardens and centuries of lore, of the charming Cotswolds region with its thatched-roof stone cottages. A fairy-tale setting—with Justin as the brave prince atop his stately white horse. "There is Whitby." Justin nodded toward the window, where roofs and chimneys came into view abutting the sea. And atop a hill, a striking, crumbling old church. "And Whitby Abbey there. Your father's home is ten miles farther on. He'll have sent a carriage or a car, I should think, to meet us at the station." Brook clasped her hands together to keep them from shaking. The knots tied themselves tighter in her stomach as the train slowed. "We shall call this story 'The Beginning of the Baroness,'" Justin whispered into her ear as the locomotive screeched to a halt. "And it will be heartwarming—if dull for lack of conflict." Perhaps his jest didn't make the knots unravel as she stood, but it at least stilled the churning of her thoughts. Justin and his valet exited first and then reached around to help her and Mademoiselle Ragusa alight. The wind blustered around her the moment her foot touched the platform. Justin chuckled at her shiver. "Too brisk for you?" Brisk? It felt as though snow ought to be swirling—not that she'd ever experienced that phenomenon. "Not at all. I'm perfectly warm." "Liar." His laugh rang out warm and hearty, though. And when his gaze moved beyond the platform, his eyes lit still more. He raised an arm in greeting. "Thate! What are you doing here?" Brook followed his gaze toward a man leaning against the hood of a car. Having heard of Justin's closest English friend for years, she expected the unfashionably long hair, the laissez-faire that his folded arms shouted. And the grin that made him look more the piratical rogue than the respectable earl. She'd always thought she'd get on well with the notorious Alexander Thate. Here was a man who knew the benefits of tossing expectation to the wind and embracing one's dreams, who eschewed society's gossip. And who no doubt got away with it because of the good humor in his smile and his handsome face. Mademoiselle Ragusa leaned close. "If these two are an example of the gentlemen to be found here, you shall have a fine time, yes?" she whispered in Monegasque. Brook pressed her lips against a laugh as Thate pushed away from the automobile and jogged forward, hand extended. He and Justin met midway and clasped hands. She came close enough to hear the reply to Justin's question. "I headed this way when I got your wire, and Mother insisted we pay a visit to Lady Ramsey and her daughters at Whitby Park—so naturally when Whit said someone must meet your train, I volunteered." Then the handsome face went taut. "I'm so sorry about your father. Had there been time for me to come—" "I know." Again the leashed pain took hold of Justin's voice. "And with Grandfather too unwell to travel and my aunts afraid to leave him . . ." His shoulders coming up, he drew in a deep breath. "But enough of sad things." He beckoned Brook forward. Thate's eyes went wide as she approached. "Deuces, man, now it all becomes clear." Brook looked from Thate to Justin. Did he see a resemblance to the family at Whitby Park—as Justin had insisted? The roll of Justin's eyes made her think it something else. And that glower of his when his friend turned to her made her wonder what it might be. Thate executed a graceful bow and held out his hand to receive her fingers. Amusement winked in his eyes as he kissed them. "Enchanté, mademoiselle. Lord Thate at your service." "It is a pleasure, my lord. Justin has told me much about you." She curtsied in return and gave him a warm smile. Thate released her fingers, though the light in his eyes grew only more mischievous. "Likewise. And might I say, you have all the beauty for which the Eden family is famous." Yet, if she weren't mistaken, the interest gleaming wasn't in her so much as in making Justin's brow furrow still more. No doubt he knew how protective their friend could be and enjoyed seeing him riled. A grin stole over her lips. "Thank you, my lord." Justin all but stomped to her side and took her elbow, guiding her past his friend. "Thate, you promised." The earl tossed his head back in laughter, though Brook couldn't think what broken promise would be so funny. Then he hurried ahead of them and opened the rear door of the car. "And you, Bing, said she was 'pretty.'" Me? Brook tilted her head to look up at Justin. He halted a step away from the door, amusement now battling the temper in his eyes, brightening them from indigo back to sapphire. "Did you just call me Bing?" "Well I can hardly keep calling you Harry now, O illustrious Marquess of Abingdon." Brook gathered up the fabric of her skirt, ready to climb into the car. "Why did he ever call you Harry? I thought that a nickname for Henry." "It was his variation on Harlow. Thate has a remarkable knack for coming up with the most ridiculous nicknames for his friends. And," Justin added, pointing a finger at the makeshift chauffeur, "she is pretty." Thate lifted a single brow. "And fire is a bit warm." Again she got the impression he said it more to irritate Justin than to compliment her, which again made her grin. For a long moment, Justin made no reaction. Then he shook his head and gave in to a smile. "I ought to have known that having the two of you in the same country would give me nothing but headaches." "Your own fault for choosing us as friends." Thate offered Brook a hand to help her into the car. She settled upon the cushion that faced backward, directly behind the driver's seat, and slid over so that Justin might take the spot beside her. Mademoiselle Ragusa settled opposite, while Peters and their trunks moved to a separate carriage. "Are we ready, then?" Thate slid into the front, behind the wheel. Justin ran a hand over the trim. "This isn't one of yours, is it?" "Whitby's." "I thought so. Far too sensible for you." As Thate navigated out of the town, conversation lulled. But once countryside surrounded them, Justin angled himself on the seat. His smile was warm and clear. "One introduction made already. I daresay you'll meet Cayton soon, Brooklet. I have a feeling he will come with me often to visit—at least so long as your cousins are there." The engine begged for a shift and then protested when Thate ground the gears. Brook winced. Justin arched his brows—and grinned. "Yes indeed, you ought to have seen the look in his eyes when he met your cousin. He has probably been haunting Whitby this past month." Though she couldn't make out much of Thate's face, she saw the muscle tic in his jaw just before he said, "I daresay if he has tried it, Lady Regan sent the braggart packing." "I didn't say Lady Regan was the one who caught his eye." Justin's grin grew, teasing out creases in his cheeks that couldn't quite be called dimples. "Though I find it fascinating you would assume so." "Lady Melissa, then?" The edge left Thate's voice, and his next shift was smooth. "Hmm. Perhaps she could mold him into a more palatable human being." Justin's chuckle wove through the wind. "Do you hear that, Brooklet? For your younger cousin Cayton is worth saving, but for your elder he ought to be sent packing. Methinks Thate is smitten with Lady Regan." "Poppycock." Thate turned a bit too sharply around a corner, sending Brook sliding into Justin. "And even if I were, it would hardly matter. Everyone knows Lord Worthing will propose soon, and no young lady in her right mind would turn down a future duke." Justin made an impressed noise. "I shall keep that in mind, Alex old friend." "Except for you, of course. Ladies will turn you down by the dozen, what with that ugly mug of yours." He sent them bouncing over a rut in the road. Brook slapped a hand to her hat to keep it from flying off in search of a new mistress. "Will you stop teasing him before he jars us from the car?" With a laugh, Justin relented and turned the topic to automobiles. The ride smoother now, Brook settled in and let her gaze wander. The purple-sprigged countryside surrounded them, the North Sea in sight again with every crest of a knoll. Fingers at her necklace, she twisted the two dangling pearls together, apart, together the opposite way. The sun broke through the clouds more fully, and though its warmth was minimal, its gilding was unsurpassed. Brook drew in a long breath and watched the gold play over the heath, chasing the clouds' shadows. So beautiful. But could it ever be home? As the car overcame another small hill, the sea sparkled in the spreading sunshine. Justin leaned close. "There it is." Her breath fisted in her chest, her pulse hammered. She shifted, twisted, let her fingers fall from her necklace and grip Justin's hand. And looked. Whitby Park sprawled across the land, its central building a proud edifice of red-brown brick that seemed nearly as large as her home, the Palais Princier, in Monaco. The gardens were more expansive than anything Monaco-Ville could boast. The grandeur she had expected. The beauty she had anticipated. But this tugging in her chest . . . Of that she didn't know what to make. She couldn't possibly remember anything from her first months of life. Yet when Thate turned the car onto the long drive, she could have sworn something clicked inside her, like a piece of a puzzle finding its place. One part of the picture that might someday reveal who she really was. One small hint that made her wonder all the more at the blank spaces. # Five From the car, Justin surveyed the rows of people waiting before the grand front entrance to Whitby Park, the family aligned in front and the servants behind. All stood straight as arrows. All looked taut as bows. All wore cold cynicism under masks of welcome. His fingers wanted to fist, but he kept them relaxed so as not to alarm Brook. If any one of them dared to insult her, dared to upset her . . . If any one of them showed her anything but kindness . . . They wouldn't—not so long as he was there. He knew well she had only been granted this audience because Whitby wouldn't turn down the request of the Duke of Stafford's heir. But one audience was surely all it would take. Whitby would see in a glimpse what Justin had upon spotting the painting of the late Lady Whitby. The rest of the family, though? He scanned the faces, most of them vaguely familiar. Lady Ramsey and her elder daughter, Lady Regan, he had met a time or two during the Season, though the younger girl hadn't yet debuted. The matron stood close to her brother, the tilt of her chin giving away the steel behind her gracious smile. Lady Regan looked nearly bored, as if she had undergone this same scenario countless times. Which, likely, she had. How many pretenders had paraded through Whitby Park, hoping to charm their way into an inheritance? Thate switched off the magneto and slid out of the car, bringing Lady Regan's smile to the surface. Justin couldn't tell if she had any interest in his friend beyond that which every female seemed to have in society's most dedicated black sheep, but her demeanor shed some light on why Thate was taken with the raven-haired beauty. The true question, though, was whether her loveliness would be soured by hatred when her cousin was legitimized and named heiress in her place. If so, then hopefully Thate would forgive Justin for taking her to task. The younger daughter shifted beside her sister, curiosity coloring her expression as she tried to glimpse the visitors. Whitby himself looked the least at ease. He kept his hands clasped behind his back but rocked on his heels. No smile curved his lips—his jaw was clenched too tightly. Beside him, Brook toyed with her necklace. Twist, release, twist again. But it was the only indicator of her anxiety. Her face bore the mask learned first from Collette and then perfected under the Grimaldi tutelage. Did she realize how much the princess she looked? There was no pretending to that kind of bearing, the regal je ne sais quoi that made heads turn whenever she stepped onto the street. Or perhaps heads turned because she was, as Thate so helpfully pointed out, stunning. Not that it gave his friend any excuse to flirt with her. And not that, as he had implied, her beauty had anything to do with why Justin had so long been her friend. He gave her a little nudge toward Thate's outstretched hand. Brook let go of her necklace, drew in a long breath, and met his gaze. Her eyes had always told him more about her thoughts than her words. Right now they said she hoped—and she feared. And that something in her was beginning to believe. Justin smiled. When Brook believed in something, there was no stopping her. "Out you go, my lady." She leaned toward the door but then stopped and spun back around, brows creased. "Do I have to call you Lord Abingdon now?" He chuckled. "Try it, and I'll toss you in the drink. And if you think the air is cold, wait until you take a dip in the North Sea." She gave him that grin that nearly stopped his heart and let Thate help her out of the car. Justin hurried out behind her and urged her forward. Whitby took a step toward them and nodded at Justin. "My lord, welcome back to Whitby Park. I am so glad . . ." The man's face washed pale as he studied Brook's face. His hands fell to his sides, limp. "Lizzie." Lady Ramsey rushed to Whitby, though she didn't spare a glance toward Brook. The way her hands clamped onto his arm, Justin couldn't be sure if she meant to steady him or keep him still. "Now, Ambrose . . ." Before Justin could make the introductions, Brook eased forward, gaze tangled with Whitby's. She extended her hand and, when Whitby offered his, made a polite curtsy. "My maman called me Lizette—but since her death, I have been called by my middle name. Brook." The marchioness released Whitby's arm, her face going even paler than his. "That voice. Lizzie." She took a step and swayed. Whitby's face went from wonder to frustration in a heartbeat. "Mary, don't you dare—" "Mama! Don't!" Lady Melissa reached toward her mother. To no effect. Lady Ramsey's eyes rolled back, and she crumpled with surprising grace to the ground. Justin lurched forward, but none of the servants had budged, and Whitby waved him off. "Don't bother yourself, my lord, she is fine. Mary, do get up." He waited a moment, but her only response was a groan. To Brook he said, "She must always steal the show. We've grown accustomed to it. More or less." Lady Regan knelt with a sigh. "You're going to ruin your new frock, Mama. Lord Thate, would you be so good as to help me get her back on her feet?" Obviously a task his friend had no problem with, as it put him in immediate proximity to his would-be lady. Justin looked up from the flutter in time to see Lady Melissa shake her head and press her lips against a smile. Whitby rolled his eyes. "A cup of tea and she will be right as rain. Now." He drew in a breath and tugged his waistcoat into place. "Again, welcome to Whitby Park. My lord, allow me first to offer my sincere condolences. We heard about your father yesterday, with great distress." The mention of Father brought the clouds back again, rolling like thunder through Justin's being. Why must everyone mention it? Their condolences only made it worse. He pasted a smile into place and nodded. "He said you were an old acquaintance, when I mentioned you just before . . ." Blast! He ought to have known better than to let his thoughts take him so near their last moments together. He had to pause to clear any telltale emotion from his throat. "It was a great shock to us all." Whitby looked every bit as uncomfortable with the conversation. He nodded once and motioned toward the tall front doors. "You will stay here until Monday, of course. We will do our best to entertain you. A fox hunt or grouse hunting or billiards or . . . whatever it is young men do these days to fill their time." "Oh, Ambrose, really. Did you not look over the list I gave you?" They both turned to where Lady Ramsey had risen to her feet between her elder daughter and Thate. Aside from a curl out of place, she looked no worse for her faint. Whitby blinked at her and then pivoted back to Justin. "Have you need of a footman to assist you, my lord, or is your valet with you?" "He is, yes." Whitby paused a moment before looking once more at Brook, and he drew in a new breath as if to brace himself. Studied her as if she might vanish like the morning mist. Did he see any of himself in her, or just her mother? To Justin's eye, there was little resemblance. Where Brook was fair, Whitby had hair closer to his niece's raven, streaked through with silver. A broader face where hers was delicate and narrow. Brown eyes rather than green. But the lines around the earl's mouth relaxed, and he offered her his arm. "And have you a lady's maid, my dear?" The endearment must have loosed something inside him. His nostrils flared, and he promptly cleared his throat. Brook tucked her fingers into the crook of his elbow and smiled. "A temporary one—my former governess is chaperoning and assisting me during the journey." Lady Ramsey fell in on Brook's other side and motioned her daughters to follow. "Well, we shall see you have everything you need until you find a lady's maid. Deirdre has a fine hand with hair. Though yours doesn't need much work. It is lovely—exactly as Lizzie's used to be, with those curls." Some of the pressure eased from Justin's chest. Whitby and his sister both wanted to believe her their own, which meant they would look for proof of it instead of against it. They would accept her. She would find her place. He indulged in a quick sigh and cast a half smile Thate's way. "And where are you staying?" "Here." He looked around with a frown. "Mother ought to be about, unless she took tea elsewhere." Good. While the ladies fussed over Brook, he wouldn't be bored senseless. Nodding, he started after the others, shoes crunching over the carefully raked macadam. He halted when the butler stepped forward. The man bowed and held out a folded paper that flapped in a gust of salt-tinged wind. "Pardon me, Lord Abingdon. This was delivered for you not an hour ago." "Ah, thank you." Who would send him a telegram here? Not Cayton, certainly, being so near. Which left Grandfather—and which inspired him to open it now rather than wait for a bit of privacy. If there were more bad news, so soon after Father . . . "Everything all right?" Thate must have had the same thought. His voice was tight. But a quick scan elicited only minor concern. "It must be. Grandfather says he is coming to join us in Yorkshire." Thate's scowl mirrored Justin's thoughts. "He has hardly left Ralin for two years." "I know." He hadn't even felt able to attend the sessions this past spring—and Grandfather took his responsibilities in the House of Lords very seriously. "Maybe he is on the mend." "I hope so." Thate's sobriety fled in the face of a mischievous grin. "And if he isn't yet, he certainly will be after one look at your Brook's angelic face. Such beauty can surely produce miraculous—" "Oh, stop." Had they not been in company, he would have punctuated it with a shove, as they had done as boys at school. And Thate would have shoved him right back, even as they started after the others again. "And forgo seeing that unprecedented jealousy on your face? Unthinkable." "It is concern for her, not jealousy." If his friend so much as looked at her too long, he'd toss him in the drink. "And you had better watch yourself, or her cousin will overhear you singing her praises." Thate opened his mouth but just grunted. And gave him an elbow jab too discreet to be effective. "Adolescent." Had they not caught up with the group, Justin would have been honor bound to return the jab with increased force. He settled for swiping his hat from his head and handing it to a servant. Their hosts had led them through the tall front door and into the great hall with its intricately patterned floor tiling and plaster reliefs of classical scenes above the paneling. Brook, having spent much of her life in the prince's palace, would not be cowed by the display of wealth—but having also spent time amid performers, she would nonetheless appreciate it. Whitby directed them into the drawing room. Deep, rich colors took the place of the pastels that usually dominated such rooms, and reigning from the wall was the portrait that had convinced Justin this was Brook's home. She stood now in the center of the chamber, staring at the painting of the woman who could be no one but her mother. Justin slid to her side, ready to take whatever action she might need. To support her if her knees went weak, to assure her if the doubts rushed in. To protect her if the show of acceptance from the family turned to attack. All eyes were on her, but she seemed oblivious to that. The look on her face was the exact one she had worn when they first went to the Louvre—passionate awe. She gave a minuscule shake of her head. "She was so beautiful." Whitby took her other side, hands again clasped behind his back. "Indeed—the most beautiful woman I had ever seen. And the kindest, with the gentlest spirit." "You love her very much." Brook's voice was a soft echo in the room. His larynx bobbed. "She was everything. Everything. She and our daughter, for the few short months I knew her." He pivoted to face Brook, examined her countenance yet again. "You are Lizzie's very image." Brook's lips quirked, and a familiar light entered her eyes, the one that could keep an entire principality on its toes. "Not quite. Her nose was not so narrow, and her forehead higher. And our chins—we have very different chins. Mine is absent that crease there, below the mouth." Leave it to Brook to point out all the differences. Justin couldn't help but chuckle. "And you'll find her spirit about as gentle as a typhoon." Her laugh rang out like a chime. "I would warn you not to believe him, mais alors. He knows me too well." Whitby and his sister exchanged a glance, Lady Ramsey lifting a hand to her chest. "She sounds so very like her, Ambrose." "Mama." Lady Regan shook her head, though she looked more amused than anything. "After lecturing Uncle not an hour ago—" "You didn't know her, darling." Voice soft rather than harsh, Lady Ramsey gripped her daughter's hand. "Lizzie was my dearest friend. Eighteen years has not erased her memory. The voice, the face . . ." "The crest." Whitby's sharp gaze turned on Justin. "That was how you identified me, with an envelope with my crest on it. But you were not sure how the envelope came to be in her possession." Justin could only look to Brook, who uttered a quick "Oh!" and let her little lace bag fall from where it had been looped over her arm. After flipping it open, she pulled out a folded sheet of yellowed paper. She handed it to Whitby. "There is an entire box of correspondence in my trunk. Maman had it amid her collection, but something about them seemed different. I haven't read them all, not knowing what they were, but that one mentioned a baby." "What is it, Ambrose?" Lady Ramsey asked as her daughters claimed the settee. Justin watched the change come over Whitby's face. From unaffected to curious to certain. His eyes scanned the page, and then he lowered it to his side and focused on Brook. "Ambrose?" "It is only a letter, Mary. A letter I wrote to Lizzie." He spun away, raised a hand to his face, and pivoted back with a visage once again stoic. "Perhaps it is conceivable that these letters would have somehow ended up in a stranger's hands. And I have certainly seen many a young woman who bore a resemblance to my wife or my own family. But when one combines it all . . . well, there is only one thing I can say." He reached out and took Brook's hand. More, he let his lips quiver. "Welcome home." The unmistakable sound of breaking china came from the opposite side of the room, shattering the mood as surely as it had the plate. # Six Deirdre's face flamed hotter than a summer kitchen. She could only imagine how Beatrix felt, being the one to drop the saucer. Though it hardly mattered which of them had done it. All eyes were now on them, which was what they were to avoid at all costs. Bile burned Deirdre's throat, and her hands shook. Never in her seven years here had she done anything to earn a reprimand from the earl when he spoke to the staff after morning prayers, but it would surely come tomorrow. And what if he dismissed her? He usually wouldn't after one infraction. But then, never before had anyone drawn such attention to themselves during what he seemed to believe was the most important reunion of his life. "Oh, heaven above," Beatrix murmured under her breath. "I think I'll be sick." "Hush." Deirdre wrapped the broken shards of china into her apron and curtsied low. "Begging your pardon, my lords. My ladies. So very sorry." Oh, she should have been paying more attention to laying out the tea things and less to the conversation underway. Then she would have been able to catch the plate before it fell. But if his lordship honestly believed this girl was his daughter . . . well then, it would affect them all. And she dared not think how Lord Pratt might react. The marchioness sent her a scathing look, but Lord Whitby chuckled and waved a hand. "It is no matter, Deirdre. I always hated that tea service anyway." Lady Ramsey squeaked a protest. "That was Mother's favorite pattern." "It is hideous, Mary, and you obviously agree, since you wouldn't let me foist it on you when you married." Her ladyship narrowed her eyes at the earl. "That doesn't meant I want to see it dashed to pieces." "It is no tragedy." Lord Whitby moved a few steps nearer, pulling the girl along with him. He met first Beatrix's gaze, then Deirdre's. "Don't let it concern you. This is a day of celebration." Words that soured Deirdre's stomach more than any dressing down. She nodded, curtsied again, but then darted a glance at the newcomer. The earl had always seen through the others quickly enough, no matter how compelling their stories or appropriate their looks. What made this one different? "Tea is ready to be poured, Lady Ramsey." Deirdre was tempted to curtsy yet again but settled for a respectful nod toward the marchioness. "I shall fetch another saucer." The lady motioned for her daughters and started for the table. "How do you like your tea, Brook darling? Strong or weak?" Deirdre and Beatrix hurried out of the way of the encroaching family—and the newcomer, who for some reason laughed at the simple question. "Honestly, my lady, I have rarely drunk it. The prince always served coffee, as did my maman." Lord Harlow—or rather, Abingdon now—grinned. "I daresay you would like it best strong, Brook. Perhaps without the usual sugar and cream." Deirdre and Beatrix slipped from the room, and her friend frowned. "She doesn't drink tea? Who in the world doesn't drink tea?" "Not an Englishwoman, for sure and certain." Deirdre shook her head and wrapped the broken plate more tightly. With any of the other pretenders, that would have been proof enough that she lied. "Do you think she's really the baroness?" Beatrix looked over her shoulder, though they were too far away to see the family now. Deirdre led the way down the back hall and to the servants' stairs that would deliver them to the kitchen. Most days she didn't notice the abrupt change between the ornate and the plain, the decorative and the serviceable, but today it struck her soundly. For all they knew that nicely clad young woman would be more at home belowstairs. Like the last one, an orphan from the workhouse someone had decided to dress up. "I think," she said quietly enough that no one would be able to overhear, "that the babe died along with her mother, sad as that is. And that the earl will be doing a terrible disservice to his niece if he allows a charlatan into the family." "It does seem unfair to Lady Regan. Unless of course this girl is who she claims, in which case it would be unfair for her rightful inheritance to go to her cousin." Beatrix sighed and reached up to secure a lock of fair brown hair that threatened to escape her cap. "I'm glad I'm not making such decisions." They took the stairs as quickly as they dared and nearly ran into Mrs. Doyle at the bottom. She greeted them with a tight smile. "Are they settled with their tea?" Deirdre nodded. "Yes, ma'am." Mrs. Doyle frowned at her balled-up apron. "What have you there?" "Oh." Her cheeks flamed again as she revealed the broken plate. "I must hurry back up with a replacement." With a click of her tongue, Mrs. Doyle took the pieces. "Deirdre, how unlike you. This will have to be docked from your pay, you realize." The taste of bile returned. How much would a saucer edged in gold-leaf cost? Surely the price of a meal for the whole family in Kilkeel, if not a week's worth of meals. And she hadn't even been the one to . . . But Beatrix's family was no better off and relied as heavily on what she sent home. Deirdre dipped her head. "I understand, Mrs. Doyle." "Deirdre, no." Beatrix put a hand on her arm, then squared her shoulders. "It was me who broke it, ma'am. She made the earl think it her, but it wasn't." The woman's face softened, and a smile teased lines onto her face. "We shall worry with this later. Fetch another saucer, Deirdre, posthaste. Then go to the Green Room to unpack our guest's trunk." "Yes, ma'am." Deirdre patted Beatrix's hand and slipped away. "I daresay she won't be long considered a guest, Mrs. Doyle." Beatrix made no effort to speak softly now. "His lordship says she's his daughter." "What? So soon?" Mrs. Doyle sounded as dismayed as Deirdre felt. Perhaps even more so, having known the late Lady Whitby. Much as she would have liked to tarry to hear more, Deirdre didn't dare. She had only a few minutes to arrive with the new saucer while Lady Ramsey made each cup of tea. Her hands shook again by the time she reached the china cupboard. Though she hated to take even a single moment to herself, she had to, to calm down. The last thing she needed was to drop the one replacement saucer. And praise be to heaven that Lady Thate had taken tea in town, or they would be in a fine predicament. Plate in hand, she hurried back through the great hall and into the drawing room. Everyone sat round the tea table, those still without cups nibbling on biscuits. Lady Ramsey's brow was creased with thought as she tipped the pot over one of the last two teacups. "Collette Sabatini? Why does that sound familiar?" Deirdre skirted the edge of the room as the girl—what was she supposed to call her?—smiled. "You probably saw her perform at some point, my lady. She was a legendary opera star in her day." Not so much as a spoon clinking against china dented the silence. Deirdre paused a moment, then hurried to Lady Ramsey's side and slid the saucer into its place with a quick bob. The newcomer didn't look cowed by the riveted attention of the other ladies. She sighed and turned to Lord Abingdon. "Did you not mention that?" "He did." Lord Whitby took the cup from his sister and tested it. "I did not deem it worth mentioning to my sister until I knew whether you were my daughter. And, Mary, I'll thank you not to overreact." "Overreact?" The marchioness lifted her chin. "Certainly not. But of course this is information we must guard. It could ruin your reputation before you even have one, my dear." Deirdre started back around the perimeter of the room, but not before she saw the steel enter the blonde's eyes. "I will do my best not to offend anyone." The girl set her cup down with nary a clatter. "But I will not deny the woman who sacrificed so much to raise me when she had no obligation to do so." Deirdre could hardly resist peeking around to see how Lady Ramsey would respond to that. She found the woman's smile softening, but her eyes none too relenting. "Of course not, dear—in private. I only mean we need not bring up in society a relation so scandalous. We will simply emphasize your association with the Grimaldis." Lord Abingdon choked on a laugh. "You surely realize the royal family leads the way in scandal, my lady." The marchioness turned horrified eyes on the young woman as Deirdre nearly bumped into a chair. "Not me," the girl said on a laugh of her own. "I hadn't had the chance to scandalize anyone yet, other than by rehearsing with the Ballet Russes. And ignoring all opinions on the matter, which is to be expected of a Grimaldi." As Deirdre turned to the door, Lord Whitby snorted in amusement. Not surprising, since he had thumbed his nose at society for years. She slipped through the door, catching only one more glimpse of the family. Enough to see that Lady Regan had sat forward, desperation in her eyes. She never was one for conflict. "Your English is good, Brook. I detect only a hint of a French accent." Deirdre paused outside the door. If this was another case of an imposter having been schooled by someone who wanted a piece of Whitby's pie . . . "Justin has spoken it with me since I was five, and then I had formal lessons beginning at six. Prince Albert insisted I take my lessons at the palace even when I still lived in a flat with Maman." "Who is Justin?" Lady Ramsey's voice bespoke dread. It sounded like one of the young men who cleared his throat. "I am. Brook has long been like a sister to me, so I pray you indulge our familiarity." Deirdre stared at the wall, wishing she could see through the white panels. Not that watching the family would clear up any of the puzzlement. "Are you spying, DeeDee?" Hiram's whisper sent her a foot into the air. Barely holding back a scream of alarm, she clapped a hand over her chest and glared at him—then hurried away from the door. "I most certainly am not." He chuckled and kept pace, balancing a few hatboxes in one arm. "So you call standing there with your ear all but pressed against the door what, exactly?" "Curiosity." Had it been anyone else to catch her at it, she wouldn't have admitted that much. But Hiram wasn't to be fooled, and his exaggerated "Ahh" even made her grin. "You can hardly blame me. What do you think of her?" Hiram shrugged and opened the door that would give them quickest access to the servants' stairs. "What can I think? I only saw her for those moments outside. She's beautiful—that's all I can say with certainty." "I've a bad feeling about it all. I . . . Why are you carrying hatboxes?" "Hmm?" He glanced down as if surprised to find them in his arms, when he ought to have left it to the lower manservants. "Oh. Trying to be useful. Everyone's in a tizzy." He shifted his awkward burden to the other arm. "Now, why are you uneasy? This one isn't like the others—we've no reason to think a future duke would lie to us about who she is." "Don't we?" She frowned, though he wasn't likely to be able to see it in the dark hallway. Perhaps someday they would be able to flip a switch here for light, as in the master's part of the house. Today she would count the stairs as she always did. "Who's to say what shape the Stafford estate's in? Perhaps he fancies her but couldn't marry someone without a fortune." "Dee." Somehow his voice combined humor with disappointment. "You never used to be so cynical. All the other maids are tittering behind their hands at how handsome our gentlemen guests are, and all you can think of are dark motives?" His words were a fist, setting up an ache in her heart where they hit her. But she could hardly explain why handsome young lords all seemed little better than tyrants. She could hardly tell him it was easy to ascribe to one a motive she knew for a fact another had. A chill chased up her spine. Lord Pratt would find this news most interesting when they met next week. "Dee?" Luckily her feet paid better attention than her mind—she stopped on the landing by rote and opened the door so he could pass through with his burdens. "I don't want to see another imposter hurt the family. Strange as it seems to feel sorry for the masters, such wealth comes with too much deception." No one knew that better than she. Hiram waited for her to emerge into the hall and studied her with furrowed brow. "We'll have to trust that his lordship will know if she's really his daughter." A sigh found passage through her lips. "He thinks she is. He said as much in the drawing room." "Well then. Our part is to welcome her." "Oh, Hiram." Only he would try to make it so simple. But then, he would still answer to Mr. Graham and then Lord Whitby, while she and the other female servants would have to deal with the presumptuous girl when she tried to make herself mistress. With Hiram following behind, she hurried to the Green Room—and came to an abrupt halt when she saw the girl's chaperone within. "Beg pardon." The Frenchwoman looked up with acute relief. "Ah, bonjour. You can help here, oui? I can tie her corset and pin up her curls, but I am better with organizing books than the dresses of the princesse." Princess? Doubt compounded with doubt. If they were fabricating this story, would they have chosen such a difficult one to believe? Deirdre plastered on a smile and moved to take a heavily beaded gown from the woman's hands. "Of course. You're probably exhausted from your trip—why not head to the housekeeper's parlor? Or we've a chef who would delight in speaking French with someone." Hiram laughed and set the boxes upon the bed. "Monsieur Bisset—taking delight?" But the woman's eyes lit. "You have a chef de cuisine?" Much to the dismay of most of the servants. Temperamental as old Mrs. Wallis had been, she at least hadn't spat at them in a foreign language. "Aye, and I daresay, being French yourself, he would welcome you eagerly." The woman paused midstep, her dark brown eyes snapping. "I am not French. I am Monegasque." Deirdre shook out the gown, deemed it too heavy to hang, and pulled open a drawer of the armoire. "My apologies. I thought it French you were speaking." "Oui." The woman grinned. "Much like you speak English, but with an accent decidedly Irish. So if I were to call you an Englishwoman . . ." "I see your point." She stepped back over to the trunk and pulled out another gown, equally as exquisite. And gave the woman a smile. "Or you could rest until the dressing gong. I trust Mrs. Doyle showed you your room?" Understanding glinted in the woman's eyes. "Oui. Now I will remove myself from your way. Merci beaucoup for your help." "You're welcome." Deirdre watched her leave, glanced at Hiram lingering in the doorway, and turned back to the armoire. "Well. This girl has lovely things, I'll grant her that." "Would have taken a fortune to have all that commissioned. Too much a one to invest in a false story, eh?" Deirdre folded the dress around a square of tissue and placed it on top of the first. "Hadn't you better get back belowstairs, Hiram?" "I will. Should I move the trunk for you?" "I wouldn't object." She indicated a spot nearer the armoire and while he hauled the laden trunk, she moved to the smaller satchel sitting atop the bed. Inside she found the usual items a lady was wont to travel with, and a book that made her snort. "What?" She held the tome up for Hiram to see. "Dracula. Our so-called baroness apparently has a taste for gothic novels." "So do our marchioness's daughters." "True enough." After placing the book beside the bed, she moved to the dressing table to put the brush and pins and handkerchief in their drawer. "I can only imagine having time to spend on such nonsense." Hiram chuckled. "Can you imagine wearing all this fuss and bother day in and day out?" She spun and flew his way to snatch the pale-blue silk from his hands. "If you soil that—" "Easy, Dee, I wouldn't." Knowing him, he had indeed checked his hands for dirt before picking it up, but that was hardly the point. If so much as a bead were lost, she would be the one held accountable. She held it against herself, away from him, with exaggerated fervor, so it came off as a jest rather than testiness. Hiram's eyes went soft and teasing. "It's a good color for you. Do you ever wish you had such pretty things?" When the only way to get them would be to let Lord Pratt make a mistress of her? And then to know such a frock could have paid her family's way for a month or more? Nay. She would sooner wear burlap. "Given that you just accused me of spying, I dare not say yes, lest you also accuse me of conspiring to thievery." He chuckled, then took a long stride away. "Never. But, Dee . . . ?" "Hmm?" She folded the beautiful blue silk, careful not to make any hard creases. "Such lovely dresses would suit you. You've the face for them." She snapped upright, but he was already out the door. Still, the words echoed in the room, tangling in the emerald-green bed-curtains and sticking to the paler-papered walls. Her eyes slid closed, though it was her insides that felt heavy. Heaven help them all. She hoped he didn't mean anything by his words. Because nothing could lay down that road. Not so long as she was bound by debt to the farm. And worse, to Lord Pratt. Shaking the heaviness off, she turned back to the trunk and made quick work of storing the dresses. And then paused, fingers hovering over a leather-bound book. Its lack of words on the cover or spine made her think it must be some kind of journal. Should she put it out for the girl, with Dracula? Or store it with the other bandboxes that she'd discovered with a glance were full of correspondence? Lifting it out, she weighed it and glanced inside, at the last pages, to see if they were dated. If the girl wrote in it regularly, she would want it out. But the last dates were from 1902—yet the hand was too mature to have been the lady's when she was so young. The words looked like French. Slapping the cover closed again, Deirdre stood. It must be the journal of the opera singer. Which meant it might disclose who the girl actually was. If so, his lordship deserved to know. Not that Deirdre could read French to tell him anything she happened to see . . . but she knew someone who did. Checking over her shoulder out of habit, she slid the book into the large pocket beneath her apron. If the girl asked, she would say she had put it with the other letters. But with any luck she would have it back before it was missed. As soon as she knew whether the chit was a fraud or not. # Seven Brook jolted awake, a cry clawing at her throat, begging for release. Her chest still heaved, her pulse still galloped. It took all her might to keep from leaping from the bed and running, so fervent was the impression that she must escape. She tried to scrabble for the dream that had found her, but so little of it made sense. Thunder. Lightning. Darkness, consuming and pursuing. And that unmistakable impression that danger poised, ready to pounce. She squeezed her eyes shut and ran her hands over the unfamiliar blanket covering her. "Un rêve. C'était seulement un rêve." Only a dream. A dream could not chase her, could not hunt her. Could not hurt her. "Are you all right, my lady?" The soft question came from somewhere in the predawn shadows to her right. And the English words gave her pause. Whitby Park. Brook drew in a ragged breath and pushed her errant curls out of her face. "Oui. Je . . ." English, she must wrap her tongue around English. The servant stepped forward, away from the unlit fireplace. "My lady?" "I am well." She managed to speak in the correct language, though Brook heard the French in the words more than usual. She cleared her throat and concentrated on speaking as Justin would. "Only a bad dream. Apparently Dracula is not wise bedtime reading." But it hadn't been Transylvanian monsters hunting her through the darkness. A chill danced over her limbs and made her shiver. The maid must have seen it, as she hurried to the bedside and pulled the blankets up around Brook's chin. "There now, my lady. I shall light the fire for you, and you can go back to sleep. It is only half past six." Brook relented—for a moment, though she had no intention of succumbing to that dark dream again. Instead, she studied the face of the maid. She had seen her several times yesterday. Outside. Coming from her cousin's room before dinner. And in the drawing room at tea. "Deirdre, isn't it?" The young woman paused halfway back to the fireplace. "Aye." Brook nodded and nestled under the covers. Did every English morning have such a damp chill, or was it due to the mist tapping its fingers at her windowpane? "A fine Irish name—I have read some of the island's lore and remember the story of Deirdre." The maid turned, offered a tight smile, and went back to her task. "Hard to forget such a bloody tale, I imagine. I can't think why my parents gave me a name wrapped in violence." Brook noted the perfect profile, the creamy complexion, the rich dark hair peeking from the snow-white cap, and could well imagine why they would name her after the most beautiful woman in Irish history. But beauty had been a curse in the story, and the woman's manner wasn't one that invited compliments. The cold compounded. And lying abed certainly wouldn't hold it at bay. Brook tossed the covers aside and swung her legs over the edge of the mattress. Then, when Deirdre spun back to her, wondered if she had done something wrong. Though the maid's lips smiled, her eyes had narrowed. "Can I assist you in something, my lady?" Oh, how she missed her lady's maid. Odette knew her habits, her preferences, and had never once made her feel as if she'd committed a crime by standing up. She took a moment to stretch, wishing for a barre. Ballet was no doubt out of question this morning, but she could surely find some exercise somewhere. "If you would help me into my corset, I can otherwise manage for now, thank you. I think I'll dress and go outside." "At this hour?" Alarm saturated Deirdre's tone, though she cleared her throat as if to cover it. Brook poured hot water from the pitcher into the matching basin. "Is no one else up?" "Lord Whitby, perhaps, but the ladies never rise until after eight." "Ah." Brook would have to learn the way this house operated and change some habits accordingly, but on other things she couldn't compromise—and wasting so much time in bed was one of them. The early morning hours were her favorite. "I'm afraid I always rise with the sun. Or," she added, looking out at the grey morn, "with the fog, it would seem." "Of course, my lady." Perhaps most young ladies wouldn't have noticed the subtle disapproval in Deirdre's tone—but Brook had heard enough of it over the years from Prince Louis to pick it out of any voice. And had decided long ago not to waste her life trying to please those who did not want to approve of her. She chose a soft washcloth from the bottom shelf of the stand and wet it, wiped the residue of the nightmare from her face. "Shall I choose a walking dress for you?" A walking dress, not her riding habit. Brook turned and gave the girl, probably six or seven years her elder, her most endearing grin. "Is a ride out of the question?" The maid paused midreach into the armoire. "If you wish to ride, give me but a moment to rouse the grooms from their breakfasts and—" "Non. Never mind." Brook certainly didn't need the grooms to be put out with her. "A walk will be perfect." Finally, a smile absent the veiled frustration. Deirdre held out a clean chemise and drawers, and Brook took them with her behind the screen. A moment later she emerged ready to slip into her corset. Silence held as Deirdre pulled the stays tight, then helped her into a walking dress of fine grey silk satin as light as the mist, which had a matching kimono coat. "Would you like a tray of tea and toast before you venture out of doors, my lady?" Had she offered coffee, it may have been enough to tempt her. But tea? "No thank you." "Shall I assist with your hair, then?" "No need, just for a walk." To prove it, Brook ran her fingers through the curls and then twisted them to her head as she walked toward the dressing table. A few pins strategically jabbed, and it was as neat a chignon as one needed for a foggy morning promenade. She fastened her pearls around her neck and turned toward the door. Deirdre stood poker straight beside the unlit fire. Brook slid her coat on and then paused before the maid. "Thank you for your help—and I am sorry to have startled you this morning." "It was my pleasure to assist you." Brook let the lie slide and smiled. She then hurried from the room and toward the stairs that would lead her to the great hall and a garden exit. She passed a horde of housemaids busy polishing and dusting in the main rooms but otherwise saw no one—which suited her well. Stepping into the cool morning, she let the fog slide over her as she walked, until she felt like nothing more than a shadow in the obscured garden. At the moment, disappearing into the low-hanging cloud soothed her as nothing else could. All the previous evening, every single set of eyes about the place seemed trained on her. Watching, waiting for her to slip up, trying to discern who and what she was. If only she knew, so that she could show them. She passed the hulking forms of the shrubs, went into the flower garden. Other than the occasional birdsong, the fog dampened any noise and cocooned her in precious quiet. Then, after exiting the gardens and wandering across the lawn until she couldn't make out so much as an outline of the house behind her, after climbing a hill, she heard sweet music—the crash of waves on shore. Brook hurried up the remaining rise and sucked in a breath at the scene before her. Perhaps a storm raged somewhere out at sea, for the water rose and fell in a froth of whitecaps, choppy and savage. A blurry impression of white floated about the horizon, where the sun struggled to stake its claim on the day. A gull screeched and dove. This was beauty. This could be home. More than the high ceilings and masterful plasterwork, the gleaming chandeliers. Those had evoked something in her, yes. But they hadn't beckoned like the sea. The words she had read last night from Hosea echoed now in her mind. "Therefore they shall be as the morning cloud and as the early dew that passeth away, as the chaff that is driven with the whirlwind out of the floor, and as the smoke out of the chimney. Yet I am the Lord thy God. . . ." She drew in a deep breath, pulled her coat tighter around her. And listened for the Lord in the clap of surf, where she always heard Him best. Where He lurked from time eternal, no matter what else may change around her. Let me not be like the mist, mon Dieu, she prayed. Let me not vanish into it in this strange new place. A horse's pounding hooves broke through the stillness mere seconds before a startled whinny brought her around. The beast reared only a few feet away, sending a spray of sandy earth in her direction. It was a fine creature, one that spoke of wealth and a keen eye. She stepped to the side and murmured a soothing phrase in French while its master called out a harsh "Whoa!" Her focus traveled from horse to man, and she barely held in a gasp. Obviously a man of means, the rider bespoke masculine beauty in his every line. Muscled legs, tapered waist, broad shoulders, a perfect face. But it was the eyes, dark as jet, that made her stomach clench with the memory of the dream, that made her want to turn and run all the way to Monaco. "Good morning." His voice was all it should be. Smooth and cultured, a rich baritone. But it made her retreat a step. As did the way his gaze swept over her. "Are you lost, Miss . . . ?" She had the sudden urge to babble something fast and senseless in Monegasque. But it felt cowardly, so instead she lifted her chin in the way Maman had taught her. "I am not lost." Horse calm again, the man dismounted and held the reins in one hand. The smile he gave her made unease skitter over her neck. How far had she wandered from the house? Too far, certainly, for anyone to hear her if she screamed. But this was a gentleman. Surely it was only the nightmare, the mist, his unexpected appearance that made her uneasy. Surely she would laugh at herself once the sun broke through the clouds and she had a cup of strong coffee to bolster her. He bowed. "Forgive me if I frightened you. Lord Pratt—at your service. You must be a guest at Whitby Park." Brook inclined her head. "I am staying there, yes." "One of Lady Regan or Lady Melissa's friends, perhaps? I am Whitby's cousin." Lord Whitby hadn't mentioned any cousins in the area while they were on the topic of family during dinner. Brook lifted her brows. "Are you? I am his daughter." "Are you?" His smile turned to a smirk. "You must be the opera singer Harlow was accompanying from the Continent." "Abingdon. And though I was raised by a singer, I am not one myself." "Hmm." Again his gaze swept the length of her, making her hand itch to slap him. "My apologies. How long has the earl given you to convince him? Most receive two or three days of grace, though a few have been sent packing within an hour." Were she a cat, Brook's hackles would have risen. "I beg your pardon, my lord, but why is that any concern of yours?" His chuckle set her teeth on edge. "I would like a more formal introduction before you leave this place." "She isn't going anywhere." The voice came out of the fog like a lighthouse beam. Brook turned her head in its direction, but it was another moment before Lord Whitby became a silhouette and then a man. A man with a hard expression aimed solely at the young lord. "And you, Pratt, will speak with more respect to my daughter." Whitby stopped at her side, close enough to touch. And glowered with enough force to send the young man back to his horse. Brook pressed her lips against a smile. With such similar glowers, he and Justin ought to get on well. Pratt cleared his throat and bowed. "Morning, Whitby. And forgive me. There have been so many over the years." "And yet, were she a fraud, you would have been interested in an introduction?" Her father nodded toward the way from which Pratt had come. "Get on with you." Pratt's smile was as smooth as ice—and just as treacherous. "Of course, cousin. I know how you enjoy solitude on your morning walks. Good day." His gaze moved to Brook. It was too dark to be termed respect, but at least it was not so predatory. "And I look forward to meeting you again . . . my lady." She made no reply, other than to shift closer to Lord Whitby. Swinging back into the saddle with a grace that normally would have earned her appreciation, Pratt nodded, gave her another too-warm smile, and turned his mount around. Not until the horse's hoofbeats had faded away did her father let out a low sigh that sounded half like a growl. "Watch that one—he brings trouble wherever he goes. And I don't like the way he was looking at you." Yet, now that he was gone, the morning mist seemed to glow silver. Or perhaps that was thanks to Lord Whitby. She slipped her hand through his arm. When she looked up at him, there was no stirring of supposed memory, no thought of This is my father. Only the recognition of a man she could like well—kind, handsome, and of the sort of disposition she had always been drawn toward. And a lingering question that made her wonder why, in her dying moments, her mother hadn't asked Maman to see Brook safely into his arms. Whitby looked down at her, loosing a snort of laughter. "Listen to me. Twelve hours a father again, and already I'm threatening the young men to stay away from you." Brook smiled and let him lead her a few steps closer to the shore. "That is one man from whom I'm happy to steer clear—I didn't like the way he looked at me either. He is a cousin?" Her father sighed. "Unfortunately, though too distant for his tastes. I try to be patient with him, as it was through his father that I met your mother. But I have little use for those so blatantly trying to claim what is mine. He has been after your cousin Regan this past year. No doubt he'll now give his attention to you." Brook couldn't suppress a shiver, though she tried to tell herself it was from the frigid breeze off the water and not the thought of Pratt lingering too near, too often. She also couldn't quite get used to all those yours. Her mother, her aunt, her cousins . . . her father. Her gaze locked on the tossing waves, it took her a long moment to realize Whitby was studying her. She tilted her head and nodded toward the North Sea. "I have always been drawn to the ocean. Was . . . was my mother that way?" "No." His voice went soft, filled with yearning. "The house and gardens were her domain. This—" he swept a hand out toward the sea—"you apparently inherited from me." "Did I?" That helped—the thought that she was not just "the very image" of her mother, that she had some of him in her too. And yet. "Are you quite certain, beyond all doubt, that I am your daughter? Because if not, I do not want to prolong this, it will only make it harder. And with everyone so suspicious of my motives already . . ." He looked into her eyes long enough that she had to wonder what he saw. "I always believed . . ." He drew in a deep breath. "From the moment you were born, I adored you. Your mother and I, we doted on you ourselves when our friends entrusted their babes to nurses. I knew you—knew how to soothe your tears, knew what would make you smile. Knew, after the accident, that you were still alive, somewhere. And I always believed that when I found you, there would be no mistake." Something quivered inside. Not with unease. Non, more like a sprout unfurling its first leaf. "But there have been so many claiming they were your daughter." "Yes." He looked out over the sea as though it were a part of him. "Beginning as soon as your mother was buried. But I knew what my babe looked like, how she acted, though no one thought I would. And as the years passed, as I realized I would likely not know my daughter by sight . . . that was when my prayers grew more fervent. Something has always made it clear that the claims were false. Information did not match up." "Ought you not to look for that now?" He chuckled and turned them back toward the house. "We are more alike than you think, my dear. I already have—in the time since Lord Abingdon first came to me. I found nothing to make me doubt the truth of your story. Still, I knew the true test would be meeting you." Feet still moving steadily, he looked over at her. Lips unsmiling, his eyes gleamed with certainty. "I have no doubts. And the fact that my sister agrees—well, that is miraculous enough to speak for itself." Brook smiled and let the silence of the fog wrap around them as they crossed the wide expanse of lawn. Once in the garden again, he cleared his throat. "I called you Little Liz when you were a babe. Even then, you looked so much like her. Which pleased me to no end." Little Liz . . . like in that letter. His hand had penned those words to his love. She tried to picture this cynical man fawning over an infant and had to grin at the image. "That is very sweet." "Your mother didn't think so." A corner of his mouth quirked up. "She insisted you would be your own person. She . . . she called you Brook." "Truly?" The green life inside opened a little more. And its root shot down into the earth beneath her feet. "Truly." They said no more, traveling the garden path in a quiet uncannily comfortable. When they reached the house, a beam of sunshine arrowed through the mist and painted its gold upon the red brick. A warming sign to chase away the lingering chill of that terrible dream. "Shall we take breakfast with the others?" She hadn't realized they had been out so long. "Am I presentable?" A quick check of her dress proved it unsoiled by her walk, if damp, and he chuckled when she lifted a hand to her hair. "You look perfect." She grinned and let him guide her toward the dining room, from which welcoming voices spilled. The chandelier glowed above the polished cherry table, and a cheerful fire crackled in the hearth. Her aunt and cousins and Lady Thate sat already, plates before them. Thate pulled out a chair as they entered, and Justin was still at the sideboard, selecting a rather suspicious-looking piece of . . . meat? Her father let go her arm, and she smiled at him, then went to Justin's side. "What is that?" she asked in a whisper. He chuckled. "Kippers—smoked fish. If you ask anyone from Yorkshire, Whitby is the only place in the world where you can get them in their right proper form." She was saved the need to respond when Lord Thate made a noise like a wheezing animal. She looked over in time to see him lower his steaming mug and reach for a goblet of water. "Good heavens, Bing—how do you drink that stuff?" Brook arched a brow at Justin, who grinned and motioned toward the smaller of two carafes upon the sideboard. "Apparently the chef has an espresso machine." "Incroyable." She bypassed the plates and headed for the coffee cups. "Drink it at your own risk, my lady. Stiff enough to stand a spoon in." Thate coughed, widened his eyes, shook his head. "I shan't sleep for a week." An added benefit, if it fended off more of those dreams. Her aunt chuckled and then blinked in a way that Brook suspected was a warning. "You returned just in time, Ambrose. The girls and I were discussing the need for a house party." Whitby grunted. "The words need and house party should not be uttered in the same sentence." Brook grinned. Not so her aunt, who loosed a sigh bright with frustration. "Do be reasonable, Am. We must introduce Brook to the families of import, and it is far too long until next Season to wait until then. Though we must begin planning her debut now, along with Melissa's. With King George's coronation set for next summer, absolutely everyone will be in Town." "Debut?" Her father set the larger coffee carafe down with a bit more force than was necessary. "She isn't old enough to have society foisted upon her." "She is eighteen!" "Nonsense. Why, she is only, what . . . four months old?" Only the small twitch at the corner of his mouth betrayed his jest. The marchioness sighed again. "Why must you always be so absurd?" "Because the thought of sending her straight into a Season terrifies me." He spooned an egg onto his plate, added toast, and moved over to the table. "Being every bit as beautiful as Lizzie, she'll no doubt garner a dozen proposals by the end of summer, and then I shall be forced to give her away, after just getting her back. It doesn't bear thinking about." A rumble of thunder darkened Justin's eyes too. "Quite a valid fear, my lord. Might I suggest locking her in her chamber instead? You may have a small hope of keeping her out of trouble that way." Brook spared him only an obligatory scowl. Of more concern was picking up a plate and considering the offerings. Justin had often told her that English sausage didn't have nearly enough spice for a Mediterranean palate, and the kippers . . . non. Fish was to be served fresh, not like that. But eggs ought to be safe, and toast with some of that delectable-looking jam. And if all else failed, the coffee could be a meal in itself. "Back to the topic of the house party, if you please," her aunt said. Whitby sighed. "Why are you asking me, Mary? It is no concern of mine if you host a party when you return to London." "Oh, but it would have to be here, Uncle Whit." Brook turned to the table and found her younger cousin leaning forward. Taking the spot left open between Whitby and Justin, she smiled at Melissa, though the girl kept her gaze on Whitby. He forked a bite of egg and ignored his niece. Aunt Mary scooted forward on her chair. "She's quite right, Ambrose. The London house hasn't any ground for hunting or sport, and I don't want to impose upon Ram and his new wife so soon." Ram, she had learned last night, was Aunt Mary's stepson, and the Marquess of Ramsey these last two years—since his father's death. "And yet you feel no compunction in imposing upon me." The words may have sounded harsh, but the tone was light. Her aunt grinned. "I seem to recall a certain brother telling me, upon my marriage, that I ought always to consider Whitby Park my home." "Your brother was young and foolish at the time, and didn't realize you'd be forcing a house party upon him." "So it's settled, then." Her aunt clapped her hands together, though Brook couldn't think where she'd read the permission in Whitby's response. But he made no more objection—Mary obviously knew her brother far better than Brook did. "Two weeks ought to be enough for the planning." Her aunt proceeded to tick off names that meant nothing to Brook, all those she insisted they must invite, present company included. And then, "And I suppose we must invite Lord Pratt." Whitby, who had looked to be paying no attention until then, frowned. "Must we?" "Indeed. And the Rushworths—they are Brook's closest relatives on Lizzie's side. I wonder if their uncle is back from India yet. Major Rushworth was always fond of Lizzie." "Too fond," Whitby mumbled before taking a bite of toast. "I beg your pardon?" He gave his sister a closed-mouth grin, swallowed. "Nothing, Mary. Only I don't think he has left the subcontinent in a decade or two." "Never mind him, then. Ram and Phoebe, of course, and her siblings." "Oh heavens. We'll be overrun." Her father put down his cup and reached for the paper a footman held out on a silver salver. "Oh, and Ambrose, I've noticed the servants don't seem to know how to address Brook. After prayers this morning you must officially introduce her to them as your daughter." He sighed and unfolded the paper, amusement and frustration mingling in his expression. "Thank you, Mary. I never would have thought of that. It's a wonder the house continues to stand when you're not visiting." "Speaking of coming and going . . ." Justin put down his fork and smiled at the group at large. Brook's stomach knotted. "I have the honor of meeting my grandfather's train this afternoon. We'll go directly to Cayton's so he can rest." Her heart gave one thud. "You are leaving?" "I will not be able to continue my overnight stay, no. But we will call on you soon. Tomorrow, or the next day at the latest." "Which is it?" She asked the question in Monegasque—better to be rude than vulnerable—and tried to keep her tone cheerful. "Tomorrow or days from now? And when will you next be able to make the trip? For this party? In two weeks?" His smile looked normal, but his eyes sparked with concern. "Please, mon amie, you know my grandfather's health is fragile." Which made her feel like a selfish clod. "I know. But I . . . I need you." "This isn't how I planned it—I'm sorry." The conflict on his face made her feel even worse. "But you will be fine. They have accepted you." She could only swallow and reach for her coffee. He had promised. That soft light in his eyes was the one that usually accompanied his reaching for her hand. But he didn't. Not here, not now, not with this particular company around them. Which begged the question of when he ever would again. "If you need me," he said softly, still in Monegasque, "I will come in half a moment." She nodded. She understood that his grandfather relied on him. She understood that his responsibility was first and foremost to Stafford. But understanding didn't keep the mist from overwhelming the sun again. Didn't change the fact that she needed him in the coming days—and he would be far away. # Eight Justin spotted James Cayton near the train platform and told himself it would be good to see his cousin again. And more often, now that Justin had no reason to travel abroad. He told himself they could finally be friends. But when he saw the black-haired figure making Cayton laugh, his hand curled into a fist. What in thunder was Pratt doing here? He strode forward knowing well his face reflected the question. "James!" His cousin turned. Recognition lit his eyes—but no pleasure. "My lord." Were it not for Pratt's smirking gaze upon him, Justin would have winced. Instead, he forced a smile. "Are we so formal, cousin?" Cayton's smile looked every bit as strained. He motioned toward his companion. "Are you acquainted with Lord Pratt?" Justin bit back the unfortunately that threatened to spring from his tongue. Perhaps Pratt heard it anyway, given his bark of dry laughter. "We were at school together. You were, what, a year below me, Harlow? Two?" "Two, before you were expelled." Justin tried to convince his fingers to unclench, but in vain. Pratt had the gall to laugh again. "The headmaster had no sense of humor when it came to his daughter." At least Cayton's grin was short-lived. Though whether from lack of amusement or the glare Justin sent him . . . His cousin cleared his throat. "I am not certain if you've heard yet, Pratt, but my cousin is Lord Abingdon now. My uncle was recently killed in an automobile accident." Genuine grief lit his eyes. Justin drew in a deep breath. Pratt's smirk barely shifted. "Sorry to hear it. From what you tell me, Cay, your uncle was a man who knew how to enjoy himself. Gambling, women, and drink were his life, were they not?" Justin's fingers curled again, and his blood went hot. Yet how was he to argue? He swallowed back the irritation and made it a point to direct his gaze to the distance, where the rhythmic puff of steam marked a locomotive's approach. Cayton must have seen the flash of anger. He put a restraining hand on Pratt's shoulder and whispered something. Pratt snorted and shrugged away. Took a step nearer to Justin. "I came across Whitby this morning on my ride—and the girl you brought to him. He seems convinced she is his daughter." Brook had met Pratt already? And she had not told him? Justin pivoted slowly when he wanted to spin and lunge. The young lord's smirk had turned to an outright sneer. "Beautiful girl, isn't she? And the fire in her eyes—that one is passionate. Tell me, my lord, how well do you know her?" Insinuation hissed like a snake, and the answering outrage brought Justin a step closer, made his other hand clench into a matching fist. Cayton stepped between them, eyes wide with warning. He aimed his glare at Justin, though he said, "Pratt, have a care." Pratt's answering laugh slid over him like a shadow. "Oh, I do. I assure you. And I very much look forward to getting to know the baroness better myself. Very much indeed." He took a step backward, into the throng of people awaiting the train. "Good to see you again, Lord Abingdon. Cayton." His cousin gave Justin an angry glare as Pratt disappeared. "Must you be always such a prude, Justin? Why can you not laugh and wave things off like any normal man?" Were it not for the increased press of people, he may have given Cayton a shake. "And be more like him? No thank you. And you would do well to steer far clear of him too, James. That man is trouble." Cayton's face went hard, his chin lifted. Rebellion gleamed hot and sure in his eyes. "You may be the next duke, but you'll not dictate to me." Justin had to turn away, watch the approaching steam engine, and draw in a deep breath until his blood calmed. Was this what his father had been like as a young man, before he married and settled—somewhat—in Monaco? Had he chafed always against his family? Perhaps so. But Father, at least, had never lacked for charm, making it all too easy to overlook his failings. Not so Cayton. But they were family, and he had so little family left. Grandfather, whom they all knew was dying. Aunt Caro, his uncle Edward's widow—and also his mother's sister. Aunt Susan, Cayton's mother. Cayton himself. That was all. All the family he had left in the world. Four people, soon to be three unless Grandfather surprised all his physicians. He shifted closer to his cousin even as the noise of the train covered the babble of the people around them. "Can we not be friends, James? Please." Cayton kept his face toward where the engine would chuff to a halt. "Since when do you need me for a friend? You have Thate. And now your little would-be princess is here. They have always been enough for you." "But you are my cousin. I will be under your roof as long as Grandfather wants to stay in Yorkshire." Justin tried on a grin, though it felt strained. "And you can come with me to Whitby Park for visits. It sounded as though Lady Melissa will be in residence for some time to come." Cayton sent him a quirked brow. The hostility had faded from his eyes, and a smile finally teased the corner of his mouth. "Are you trying to buy my friendship with the promise of fair company?" "Will it work?" There, a genuine laugh. "Perhaps." The strain left Justin's smile as he turned to watch the train pull into the station, his gaze traveling its length until he found the duke's private carriage. A stride toward friendship, he hoped. The kind they had enjoyed as boys, those few times Mother had brought him to England before her death. Before James became an earl at the age of nine, when his father died. Before Uncle Edward's death meant Justin was heir apparent to the duchy, after his father. Before they grew into such different men. "Justin." Cayton shifted closer as the whoosh of the train's brakes sent steam billowing out around them. "I am sorry about your father. I didn't get the message in time, but I wanted to be there. Know that." It wouldn't have helped, not then. But knowing Cayton had wanted to come soothed now. "Thank you, James." They said no more, just made their way to the end of the platform and the door that one of Grandfather's servants opened. He would travel with a whole retinue. Not because he needed anyone but his aging valet, but to keep up appearances. To make this crowd of onlookers take note and realize someone of import had arrived. They looked. They whispered. When someone recognized the crest on the side of the car, they exclaimed. Perhaps Father had the right idea while he'd hidden in Monaco—ignore the station, ignore the title, ignore the expectations. Be whomever he wanted to be. Justin would never have that luxury. But neither did he intend to make such a fuss wherever he went. He drew in a long coal-dusted breath and clenched his teeth against another onslaught of emotion when Grandfather stepped down. He managed the two stairs with only the assistance of his silver-tipped cane, but the servant was there to make sure he didn't fall. His valet materialized behind him with concern etching his brows. Both of Justin's aunts soon rushed out to flank him. The duke's clothes hung on him, evidence of another bout of too-quick weight loss. His face had gone gaunt. His hair—brown two years ago, grey two months ago—was white as the chalk cliffs. Justin sucked in a breath to keep the pain of it from his face. "I was only gone a few weeks." His cousin jerked a nod, his face tight with worry too. Of course it would be. Having spent most of his growing-up years at Ralin Castle after his father died, Cayton was even closer to Grandfather than Justin was. The duke was more father to him than grandfather. Did he ever resent that Justin was heir to the duchy, just because Cayton was born to the duke's daughter rather than one of his sons? Grandfather looked up once his footing on the platform was sure, and he gave them a smile. "My boys. You both made it—good. I worried we had not given you enough notice." "Of course we made it." Cayton leaned over to kiss his mother's cheek and grip Grandfather's free hand. "Your favorite room is ready at Azerley Hall. We can—" "Soon." The duke's gaze went over Cayton's shoulder, to Justin. He lifted snowy brows. "We will stop at Whitby Park first, for tea. I will meet this princess of yours before I die, Justin." He knew not whether he should smile at the mention of Brook, sigh at how Cayton bristled at being dismissed, or shake his head at the mention of Grandfather's death. He settled for a nod. "Baroness—Whitby is certain she's his daughter. She is eager to meet you as well, sir." When the duke took a step forward, they all moved with him, a careful ballet set in time to his faltering stride. "What conveyance have we? Did you bring your new automobile, James?" "I did," his cousin replied with a smile in his voice, "as it is large enough for us all." Justin fell in beside Aunt Caro. With her silver-and-gold hair, her bluer-than-sapphire eyes, she was what he imagined his mother would have looked like now, had she lived. He smiled. "I accepted Whitby's offer of a carriage. Though wait until you see the Rolls-Royce that Fa—" His throat closed off. His nostrils flared. Grandfather sent a quick look to Cayton, then focused on Justin. "I will ride with you to Whitby Park, Justin. Then we will all proceed to Azerley Hall together after tea." Aunt Caro patted the duke's arm. "I will join you. It will give Susan time to question James on which young ladies he intends to keep in contact with now that the Season has ended." Justin forced the pain of his father's memory back, down, away, and dredged up a grin. "I believe he has set his sights on Lady Ramsey's younger daughter, Lady Melissa." Aunt Susan lifted her brows. "Is that so?" She tucked her hand into the crook of her son's elbow. "When did you meet her, dear? She is not out yet." Cayton sent Justin half a glare, though its force was negated by the amusement in it. And the flush in his cheeks. "When I went with Justin to Whitby Park last month. And I accepted Lord Whitby's invitation to dinner a fortnight later." "Well." Aunt Susan's smile was equal parts pleasure and . . . relief? "She would be an excellent match, to be sure." The servants had cut a path for them through the crush of other passengers coming and going, through friends and family greeting or sending off one another. Brook would be glad they were coming again so soon—or angry. A definite possibility, what with her passions reigning with Mediterranean abandon. He glanced down at his aunt Caro, over to his grandfather, to his other aunt, his cousin. All with pleasant masks over their thoughts. So unlike the families he knew in Monte Carlo, who greeted with a shout, with a kiss—who could roar in fury one moment and with laughter the next. His gaze drifted in Whitby Park's direction. And he found himself praying that the English rains wouldn't dim Brook's fire. "You seem quiet, Justin." Aunt Caro spoke in a volume to match her observation as the Whitby carriage came into view. "I hope you know I have been praying for you. Every hour, every day." "I know." Swallowing did little to relieve the lump in his throat. "I was trying to convince Father to come home. I thought the urgency was here, not there. Had I known it was our last conversation . . ." His aunt tipped her face up to study his. "Would you have done things differently? I daresay not. It is your nature to try to hold your family together." "And was it his to stubbornly cling to separation?" Something shifted in her eyes, went distant and cold. "He had his reasons. I pray you do not judge him without knowing them." "Caroline." Grandfather's tone was the one he had used on Justin and Cayton when they were children getting into mischief. Aunt Caro pressed her lips together, her eyes now flashing. Justin's chest went tight. He had thought that coming home, having Brook here, having nothing pulling him away anymore, would grant him a measure of peace in the wake of the turmoil. Apparently not. Cayton and Aunt Susan had wandered a few steps ahead, and their conversation sounded light and easy as they headed for their gleaming car. They climbed in with a wave to the rest of them as the servants loaded luggage into a carriage. Their little party climbed into Whitby's carriage in silence. Grandfather settled on one of the facing seats, determination etching the lines in his face deeper. Justin sat beside his aunt. The door shut behind him. Aunt Caro cleared her throat. "Susan will be pleased if James pursues Lady Melissa. I assume she is as lovely as her sister." "She is." Justin smiled, though less at the thought of Lady Melissa than at the way Cayton flushed over her. And at the memory of Thate's scowl when he thought it Lady Regan in whom Cayton was interested. How amusing it would be if his friend and cousin ended up married to sisters. The duke cleared his throat. "As a grandfather, I pray he chooses wisely. But as the duke, I am far more concerned with who you might wed, Justin. This princess turned baroness—do you intend to marry her?" Aunt Caro hissed out a breath. "Duke!" "Do not chide me, Caroline." Somehow, he managed to look both weary and authoritative. "The duchy has been in the Wildon family for nigh unto three hundred years. Am I an ogre for being concerned about the next generation, with ensuring an heir?" How could Grandfather say such a thing in Aunt Caro's presence, when he knew how sensitive she had always been about her childlessness? Justin could only see his aunt's profile, but he didn't miss the way her fingers dug into the plush seat beneath her as she said, "That depends, sir, entirely upon your methods of ensuring it." It felt as though a bare, live wire had been let loose in the carriage, sizzling and snapping. His grandfather and aunt's gazes clashed for a long moment, then Grandfather looked at Justin again. "This girl, Justin." As if she were just a girl. As if the question were so simple. Justin wanted to look away, but he knew his grandfather expected eye contact. "I don't know, sir. When I think of the future, I can imagine no other woman at my side through the years. But I . . . She loves me, but it has long been as a brother, a friend. Her feelings have not grown as mine have, and I fear if I push her, declare myself too soon, I would ruin any chances I have." The duke's faded brown eyes went soft. "I understand. But I would know, before I die, that you have chosen a worthy woman to assume the title of duchess. You have always spoken of this girl as you have none other, and now that she is here . . . Well, why do you think I dragged myself from the comforts of Ralin?" Must every conversation come back to death? "You could yet recover, Grandfather. There is no need to speak of—" "Hush, my boy." The duke leaned his head back, gripped his cane. "I am tired, and I have the peace that I leave Stafford in good hands. It is enough. I am ready, whenever the good Lord decides my time is complete." Justin could not say the same. He was not ready to let go of his grandfather. Not so close on the heels of losing his father. His gaze now sought the window, though he looked at it rather than through it. "I remember thinking perhaps I could lure Father home for my wedding, mere minutes before . . ." Grandfather snorted, drawing his focus back inside. "He would not have come. You ought to have realized that after all these years. If he did not return for his own brother's funeral, he—" "Why should he have?" Aunt Caro shifted, folded her arms over her middle. Her face looked as yielding as granite, and from this angle Justin could see her tension in the strained muscles of her neck. "Edward never gave a thought to William. Frankly, sir, nor did you, other than as a stopgap heir. You were far more concerned with molding Justin into your image." She reached over, patting Justin's hand as if the show of affection could soften the words. Then she turned eyes on him that were as scorching as blue flame. "Did William ever tell you?" Something sank into Justin's stomach. It was too numb to be called fear. "Tell me what?" "Caroline." The duke put a world of forbidding into her name. "He deserves to know how much William loved him. He needs to know—" "He already knows that, and it's all he needs to. You will keep your word. So long as there is breath left in my body, you will bite your tongue." To punctuate it, Grandfather lifted his cane and then drove it back to the floor. "Are we understood?" Now her fingers settled over Justin's and gripped. Hard. "Yes, sir." The rock in Justin's stomach doubled in size. "What? You cannot lead into a subject like that only to abandon it." But his aunt merely sniffled and averted her face. "Grandfather?" The duke's hard gaze turned on him, softening only the slightest degree. "It is nothing to worry over, Justin. A woman's nonsense. No more." Aunt Caro wasn't given to nonsense—she was given to faith, had been the one to teach him to pray, to seek the Father, to always trust in Him. And that place inside where his faith was born quivered now, warning him that whatever this truth was his aunt thought he needed to know, it was far from nonsense. But it seemed he wouldn't learn it while his grandfather ruled the house. # Nine Deirdre scurried behind Mrs. Doyle, her pulse quickening. "Will they stay the night, then?" Mrs. Doyle snatched a lamp from the stand near the passageway and lit it. "For tea, they said, but I'll not be caught unawares." She spun for the stairs that would take them up to the family levels. Deirdre tucked a stray wisp of hair into her cap, flying up the stairs after her superior. And pushing down her mounting concern. If they were going to prepare more rooms . . . if tea became an extravagant affair . . . "I am sorry you will miss your afternoon off." Mrs. Doyle must have read her mind. "You may take it tomorrow instead, but we can't spare you today." "Of course, ma'am." The words came out with nary a squeak. But tremors turned her stomach. Pratt would not be pleased if she missed their meeting. What was she to do though? She would just have to report next time they met that Whitby had a new heiress and ask him to translate the journal. The young lady hadn't missed it yet; surely she wouldn't in the next few days. Lady Berkeley—that was what the earl had told them to call her. A shudder overtook her that had little to do with the cool draft in the stairwell. "Mrs. Doyle . . . what do you think of her?" "It isn't my place to form an opinion." The housekeeper pushed open the needed door, and they stepped into a hallway filled with opulent tapestries and ancestral paintings. "Lady Berkeley looks much like the late Lady Whitby—God rest her soul." By rote, Deirdre crossed herself. "But . . ." "There is no but, Deirdre." Mrs. Doyle's voice sounded resigned, though, not chiding. "His lordship has decided. And if I might be so bold . . ." The older woman paused—and if she deliberately wasted time, it must be important indeed. Deirdre drew herself up, waited. Mrs. Doyle leaned close. "This could be your chance for advancement. That Frenchwoman will be leaving tomorrow, they said. Lady Ramsey offered to help her find a maid schooled in Paris, but there is a chance she could ask you to rise to the task instead." Deirdre's throat went dry. How was she to even try for that? And yet . . . yet if she could. She would get to take her meals in the housekeeper's parlor with the upper staff. They would all call her O'Malley instead of Deirdre. She would no longer have to polish all the silver and dust the furniture—her sole task would be seeing to the baroness and her things. Her wage would increase dramatically. Perhaps then she could get away from her ties to Pratt. As if he would release her so easily, especially if she served the girl he would no doubt set his sights on. That was too much to hope. She gathered a smile for Mrs. Doyle. "I shall do all I can, ma'am. But I daresay I oughtn't to get my hopes up, aye?" Mrs. Doyle acknowledged that with a movement of her brow, and then she spun toward the bachelor's wing. "There is little we can do. But I would rather welcome you to my parlor than some pretentious woman from the Continent." Well now. Even if she failed to convince her ladyship to take her on, that was something to treasure. Mrs. Doyle's respect was hard won and worth much. "I thank you for that, ma'am. Truly." When they reached the guest rooms, they had no more time for conversation. Lord Abingdon's room was still made up—thanks be to heaven—and they set to work on the one next to his for the duke. Beatrix and the other under-maids were seeing to rooms for the dowager ladies and Lord Cayton, but sure and Mrs. Doyle would not delegate the task of a chamber for the duke himself. They worked in efficient, precise silence and then hurried back downstairs. When she saw that the family lingered in the great hall, Deirdre slid into the shadows to wait for them to head to drawing room or parlor. The front door stood open, the line of footmen visible on the steps. His Grace must be having a difficult time exiting the carriage. The baroness stood by Lord Whitby, her hand resting lightly upon his arm. The pretenders had tried to cling—though his lordship never allowed it for more than a few seconds. They never carried themselves as this one did, either. Fluid grace, it seemed, but with an undertone of pride. Nothing would be the same again. Unless Lord Whitby changed his mind, this overconfident, self-assured princess would be the new mistress of Whitby Park. Deirdre's gaze slid over to Lady Regan. She didn't look upset, but she always kept her emotions in check, always strove for peace. Lady Melissa was the one more likely to shout her opinion for all to hear, and she at least watched the newcomer closely. Perhaps she would issue a warning to her uncle. Though just now both young ladies seemed far more concerned with the young men in attendance. Lady Melissa's gaze latched onto Lord Cayton when he entered. Lady Regan kept sending sidelong glances to Lord Thate. Deirdre sighed. At least Pratt wouldn't care anymore who Lady Regan fancied. Turning her eyes back to the blonde, Deirdre watched her loose his lordship's arm and step forward when Lord Abington entered with the duke. The chances of the lady wanting her for a maid were slim as waistlines in a famine. But she had to try. Even if she didn't like the girl, even if she hoped Lord Whitby soon saw reason and booted her out, she owed it to Mum to try. The family exchanged words with the visitors. There was bowing and curtsying, and they all paired off. And though she was more interested in the way Lady Regan flushed when she slid her hand into the crook of Lord Thate's elbow, Deirdre focused her gaze on Lady Berkeley and Lord Abingdon. The baroness didn't just rest her hand on his arm, she gripped it. And he didn't merely cover her fingers with his own, he clung to them. The look they exchanged—charged was the only word that came to mind. By sheer force of will, Deirdre kept her eyes from narrowing. They must be more than friends, those two. And oh, but she didn't want to be the one to tell Pratt that the new heiress was already in love with another. As they moved in the wake of the others, Deirdre caught the young lord's quiet, "Are you still angry with me?" Lady Berkeley's chuckle was low and taut. "Oui. But I will overlook it for now." He said something else, but he had shifted into the language her ladyship's maid had spoken last night. She again picked out a few words she recognized as French, but the cadence was wrong. Whatever he said, her ladyship's face went serious. She answered in English. "My aunt has arranged for some sport—archery and croquet. Assuming the rain holds off and Lord Whitby can convince your grandfather to stay after tea, we will all have a lovely, relaxing afternoon together." Lord Cayton and Lady Melissa partially turned around to share their enthusiasm with that plan, but Deirdre kept her gaze forward and her face clear as they passed by. What did it mean that this girl called Lord Whitby by his title, though she already greeted Lady Ramsey as Aunt Mary? Deirdre didn't know. But she would keep her mouth shut . . . and her eyes open. Brook shook her head, unable to think up what secrets Justin's family could be keeping from him. "The reason for the rift between your father and the rest of them, I should think. But as for what it is . . ." She had always wondered at what had caused it. It must have been more than William's penchant for gambling. But Justin had never known, and she hadn't been well enough acquainted with his father to ever ask him. Justin sighed. "Grandfather forbade her from telling me. And though I keep trying to convince myself it is likely just some old argument that makes little sense anymore, they both are—were, in Father's case—so adamant about the secret being kept." She held tight to his arm, studying his profile as he looked out to the breaking waves of the North Sea. Sometimes it mattered less what a secret was than that it was. And right now, Justin needed his family supporting him, not adding more to his burdens. "Ça va?" He sighed and squeezed her hand where it rested on his forearm, as he had done upon arriving three hours prior. "Je ne sais pas. It is all just . . . too much." Laughter from the gardens drew her gaze back toward the house, where the other young people were engrossed in a game of croquet. Brook had bowed out of that particular game. And after she'd bested everyone in their impromptu archery competition, they had all made a show of thanking her for excusing herself. Teasing . . . or were they glad to see her go, if only for a half-hour promenade? She turned to Justin again. His eyes had gone a darker blue, as they always did when he was troubled. "It seems we both have secrets in our families. Grand-père gave me Maman's journal before I left. He said she wanted it destroyed rather than letting me see it. He said . . . he said she thought coming back here would hurt me." Justin frowned and led her away from the frothing whitecaps, back down the hill toward the carefully structured shrubbery and the laughing couples on the lawn. "Have you read it? Does it explain why she took you away rather than delivering you to Whitby?" "I haven't." She wanted to tug on his arm, to slow him down, to stay away a little longer. But from the house came the sound of a gong, and the croquet game came to an immediate halt. They must all get inside to dress for dinner. She contented herself with pressing upon his arm. "I promised Grand-père I wouldn't read it until I was ready to face whatever it told me. And just now . . . I feel the coward for admitting it, but it is hard enough to accept the simple facts. That he is my father, that this is my home. I want to make my own impressions, not be colored by Maman's." "Wise." He looked down at her, a smile softening the corners of his mouth. "Curious as I know you always are, direct it now toward discovering this family of yours, not focusing on the past." To that, she could only draw in a deep breath and tilt her head in acknowledgment. Soon they reached the indoors and parted ways. Brook tried to catch whatever glimpses she could of the house as she passed through it—a tour had been on the schedule for this afternoon, but the duke's arrival had put a halt to that plan. She knew her way back up to the corridor that housed her and her cousins' rooms, though—and could have followed the sounds of giggling females had she not. Regan and Melissa were turning toward their doors as she topped the stairs. Regan smiled and waved at her. "After you dress, come in with us to have your hair arranged. It will give us a chance to talk about the gentlemen." Brook returned the smile. It had been so long since she could claim any true female friends. She had been in a strange position in Monaco—not quite a royal to rub elbows with the nobles paying court to Grand-père, but too much a one to be accepted by Maman's former ilk. Perhaps now things would be different. "Thank you. I shall." Her smile faded, though, when she caught Melissa's low, "Regan! How in the world can you be so accepting when . . ." Before she could hear the end of the question, their door shut. She told herself to shake it off—it was normal, after all, for them to have reservations. Slipping into her own chamber, she found Mademoiselle Ragusa holding up two of Brook's new evening dresses, tilting her head from side to side and humming a nonsense verse she had used to sing in their schoolroom. Brook closed the door behind her and gave the woman a grin. "Having trouble deciding which you will wear, mademoiselle?" The governess laughed and held them both up to her. "I could not tie my corset tight enough. But there are so many pretty things, and all will flatter you. Which tonight?" Brook joined her, running a hand down the pale green sleeve of one, along the violet beading of the other. She saw only Grand-père, that indulgent smile on his face as she tried to decide between the two silks, the way he had said, "Obtiens tous les deux." Get them both. Blinking against the burning in her eyes, she chose the green. It had been his favorite. Within minutes, she had changed, slid the matching slippers onto her feet, and bade her companion a good evening. Nerves fluttered in her stomach as she made her way to the door her cousins had gone into. She had to pull in a long breath. Pray for fortification. For peace. For . . . for a connection. At the first light rap of her knuckles upon the door, it opened inward. Melissa must have been standing there, waiting. Though her smile looked strained, she at least offered it. She motioned Brook inside, shutting the door behind her. Wariness gave way to pure feminine appreciation. "That gown! It's divine. Parisian?" Brook smiled. Just thinking of Paris brought the smell of baking baguettes to her memory, and the sound of a lazy concertina. "Oui." Regan stepped from behind a dressing screen, her smile calmer but far more welcoming. "Your Lord Abingdon's jaw will no doubt drop to the floor." Mischief lit her eyes. "Have you an understanding? You seem so close." "An under . . ." It took Brook a moment to process what that meant. Then she felt the heat scorch her cheeks. "Oh, no. We are only friends." "But?" Melissa raised her brows and spun toward the bed, where an evening gown lay waiting. "You cannot be happy with that only. He is nearly as handsome as his cousin." Cayton was not half so handsome as Justin. But Brook wasn't about to argue with a smitten girl. "He is like a brother to me." "The best way for a romance to begin." Regan chuckled and motioned Brook to the seat at the dressing table. "You first, cousin. Deirdre will return in a moment." Cousin. Brook sank onto the padded stool and smiled at Regan's reflection. They were both so beautiful, these cousins of hers, with the rich dark hair that felt like home. Melissa had disappeared behind the screen. "I wish they were staying more than a night." Regan shook her head and stepped to Brook's side, opening a traveling case that revealed rows of jewelry. "They will be back in a fortnight for Mama's house party." "An eternity. And don't pretend you didn't have to stifle a groan when Thate said he would leave tomorrow as well." Regan stifled a sigh now and pulled out a necklace dripping crystals. "It hardly matters. Thate never pays me any mind." Toying with one of the hairpins on the tabletop, Brook met her cousin's reflected gaze. It was not so unlike the backstage dressing room at the ballet. Girls were girls. They spoke of men. They fussed and dressed and yearned. She could grin. "Au contraire. He grew quite testy in the car yesterday when talk turned to one of your suitors—a duke's son. I cannot recall his name." "Lord Worthing." Melissa pronounced it with an exaggerated sigh as she reemerged. "He and your Lord Abingdon are the only two heirs to duchies between the ages of ten and fifty. Every young lady in London was in a dither when Worthing came to call on Regan." "He is a good man." Regan fastened her necklace, her voice so even, so calm that it was clear he was, to her, nothing more than that. "Of strong faith, which is rare. Handsome. Everything a lady could want." Her hands fell to her sides, and her gaze bore right through the looking glass. Melissa appeared at their side and slid her arm around her sister's waist. "But you have set your heart on an unfashionable young earl who will no doubt go careening into a ditch and get himself killed in one of those cars of his." Brook winced—she couldn't help it. And thanked the Lord her cousin hadn't said it when Justin could hear and be reminded of his father's death. Better to focus on Regan and Thate. "La vie est une fleur dont l'amour est le miel." "Hmm?" Regan looked down at her, her eyes still a bit distant. "Life is a flower?" ". . . of which love is the honey. Victor Hugo." Such pretty words. But maybe they were just the stuff of novels and poems. Never had Brook seen it play out in reality. Prince Louis refused to marry, and Prince Albert . . . Grand-père had been unlucky in matters of the heart. He had divorced his first wife well before Brook's day. Then came Princess Alice. Brook well remembered his argument with her, when he moved Brook to the palace. Such accusations had surfaced then—his paramour, her paramour, problems with his son, problems with her son. They had separated. Love, it seemed, had nothing to do with anything. A discreet knock signaled the entrance of the maid. Brook assumed the conversation would shift, but Melissa shook her head and looked at her sister. "But can you be sure you love him? I like Thate, but he is hardly a responsible, dependable man to choose as a husband. If Lord Worthing proposes, you can hardly say no." Regan loosed a long gust of breath. "If Lord Worthing hadn't come calling, everyone would consider Thate a fine catch. It isn't as though he's a pauper seeking my fortu—" She looked away, but Brook saw the flush in her cheeks. And why would she cut herself off at the mention of a fortune? Brook glanced at Melissa in the mirror, but the younger of her cousins pressed her lips together and looked from her sister to Brook and back again. She knew so little of this family. But they had that half-brother, Ram. He had inherited their father's titles, his estates. And from what Justin said, most money went with estates, to keep them up. So Regan would not have a fortune, aside from a dowry. But she would have . . . had Brook not come. Excluding their mother, these two were Whitby's closest relatives, and Regan was the elder. She would have been the heiress of all that was his, aside from the title itself. And simply by showing up, Brook had stripped her of that. "Regan. I . . ." She knew not what to say. The only words her tongue could find were Monegasque, and even they made little sense. Regan put a warm, steady hand on Brook's shoulder. "Think nothing of it, Brook. I never put my hopes in an inheritance—Uncle Whit only just succeeded in breaking the entail that required a male heir for it all. It is worth far more to see him finally find you. To gain a cousin and friend." It was no wonder Thate had been unable to keep his eyes off Regan all afternoon. She was far more than a lovely face. "I feel the same. When I asked Justin to help me find my family, I never imagined all this." Brook had thought that, at most, he would show her a photograph of her deceased parents. Perhaps a village in which she was born. Deirdre took up position behind her and gathered Brook's curls. Melissa pulled up a chair and eased to a seat upon it. "Justin. It sounds so very French. A friend of mine caught a glimpse of him during the Season, you know, and couldn't stop talking about him. So mysterious—gone most of his life on the Continent, scarcely ever making an appearance in Town." "And nearly as handsome as his cousin?" Regan winked and leaned against the wall. Melissa dimpled. "Nearly. Though you would probably put Thate or Worthing at the top of the list. What say you, Deirdre? Who is the handsomest of our gentlemen friends?" Brook expected the maid to demur, but instead she looked over at the sisters with a warm smile. "'Tis a hard matter to judge, sure enough. Though I must confess that one of the most striking men I've seen is that cousin of his lordship's—Lord Pratt." Unease skittered up Brook's spine at the mere mention of him. Melissa made a thoughtful hum. "He is striking. But there is something about him . . ." "He has no heart—that's what." Regan pushed up and stepped behind Brook. "How lovely. I do wish I had curls so you could do mine like that, Deirdre. But it would take hours to use the tongs on it first." "Your hair is beautiful, my lady. So glossy and thick. Though sure and Lady Berkeley's curls are a joy to work with too." Brook summoned a smile, though that title still felt so odd. How long would it take for her to get used to answering to it? To seeing all these new faces? To having a father who looked at her as if she were an answer to prayer? When Deirdre indicated she had finished, Brook rose. "Will Lord Whitby be ready, do you think? Could I find him before dinner?" Regan sat on the stool, elegant as a queen. "He will be in the library." "A library!" Of course a house this size would have one, but how could she have been here a complete day already without finding it? "Where is it?" Deirdre gave her a smile thirty degrees cooler than the one she had given Regan. "I can show you the way, my lady." "Thank you, but instruction will suffice. You are busy." Deirdre relented without a fuss, and told her where to find the library. With little more ado, Brook slipped out. Back to her chamber, into the attached dressing room. There, on the floor, she had already noted the bandbox that held Maman's letters . . . and theirs. Whitby and his Lizzie's. For a long moment she stared at it in the fading light. It had been years since she'd glanced through them. She'd never much wanted to read the love notes Maman had received over the years, especially the ones from the years before Brook was born, when they were more than pleas for a meeting, always refused. When they hinted at meetings enjoyed, instead. When Brook had started to wonder if the box of English ones were in fact to another woman altogether, when she had drawn out the one envelope with a seal to give to Justin, she had considered reading them all, more closely. Something had stayed her. She opened the bandbox and drew out the smaller wooden one. Held it to her middle. Now she knew the writer was not some long-gone, faceless name. He was here. Her father. And if anyone should go through these letters, it was he. She slid out of her room, padded down the stairs, took turn after turn until she heard the snapping of a fire in a grate and smelled the perfume of books and smoke and the magic of all those tales in one place. Whitby surged to his feet the moment she entered, as if he had been waiting for her. "Brook. Were you looking for me, or for a book?" "You." Though now she had to turn in a circle to take in the books. Shelves upon shelves of them, floor to vaulted ceiling. "If ever you cannot find me, look here first." Whitby chuckled and slid a step toward her. "I daresay I will already be here ahead of you. It is my favorite room in the house." Another thing they had in common. She put on a smile—though it felt slight and uncertain—and held out the box. "I mentioned these yesterday. The correspondence, from you to my mother. I wanted to return it." The look that crossed his face went beyond words. As if it were the echo of a million memories, the joyous and the bittersweet. He closed the space between them and reached for the box. His fingers were long and slender, like Brook's on a larger scale. They gripped it in one hand, traced its contours with the other. "My grandmother's box. I gave it to Lizzie thinking she would store her gloves in it, as Grandmama had." He flipped the latch, opened the lid. Shut it again and held it out to her with flaring nostrils. Knowing her confusion must be on her face, Brook reached for it slowly. Her father's larynx bobbed as he swallowed. "I have the ones she wrote to me. Her words, her script, her perfume once upon them. I don't need to read my own words. You may, if you like. If you think it would help you to know us. I can give you the others, too, that she wrote." Though she pressed the wood until it seemed her fingertips ought to dent it, reading their correspondence still seemed wrong somehow. "I do not want to pry." But his eyes went soft, and he skimmed his hand over her wrist. Fleeting, brief, the touch of a man unaccustomed to such gestures. All the more important because of it. "I want you to. It is a daughter's privilege." Was it also a daughter's privilege to ask questions? Because she wanted to know why her mother had had the box with her when she died. Why they were even in a carriage leaving Whitby Park in the middle of the night, on their way to York. Why, if this emotion that seemed to have drowned her father was love in its purest form, they were so disastrously separated. One hand went to the pearls dangling from her necklace. Twisted them, let them fall, twisted the opposite way. A knock sounded on the open door, and Brook jumped and spun. Whitby had introduced Mr. Graham and Mrs. Doyle earlier, and they both now stood in the doorway, faces pleasant but cool. The butler sketched a quick bow. "Mademoiselle Ragusa's train ticket has been purchased as you requested, my lord, and there will be one awaiting her at the port as well." The housekeeper directed a smile to Brook. "Deirdre will be happy to serve as your lady's maid until you can find one, my lady. Or indefinitely, if you find her work satisfactory." Though the features were still unfamiliar, she knew well the look on Mrs. Doyle's face. She had seen it often enough on the fearsome palace housekeeper's, the one who wore the keys to the Grimaldi home as if they could unlock heaven and hell themselves. It said: "My words suggest—my will commands." Brook told her lips to curve. She had gone head to head with her old housekeeper a time or two, but it would surely not be wise to do so with Mrs. Doyle on Brook's second day here. "I'm sure Deirdre will suit me quite well, Mrs. Doyle. I would be pleased to have her as my lady's maid." Whitby obviously knew the look as well as Brook did. His sigh blustered forth. "You needn't agree to anything so soon, my dear. We should speak with Mary about it, see what she recommends—and you can be sure she'll have an opinion." Mrs. Doyle stiffened. Brook offered a conciliatory smile, though she wasn't sure who she was trying to placate. "I would welcome a maid who is already a part of Whitby Park." "Brook—" "Je veux être comme l'un d'eux." I want to be one of you. She hoped her father spoke enough French to understand her meaning—and that the servants did not. Given their blinks, she was right at least on that score. Whitby she was not sure of. Not until his lips twitched. "You are the image of your mother, my dear. But it seems your disposition you inherited from me, stubbornness and all. My apologies." She repositioned the box of letters and returned his hint of a smile. "It serves me well." He dismissed the servants with a nod and then studied Brook for a long moment, obviously trying to see something beyond the visible. Perhaps he managed it. The lines around his eyes softened, as did the tension in his shoulders. "Tomorrow," he said quietly, "after our guests depart, I will show you your home. And your mother's room, her things. You can have whatever you wish. I want you to be one of us too, Brook. I want you to feel that this is where you should have been all along." She wanted it too, so very much. Wanted to walk these halls with all the abandon she had enjoyed in the prince's palace. Wanted to be welcomed and respected among the household as her cousins were. Wanted to make this man before her laugh and smile, and to know what to call him. But those desires were much like love—pretty on paper. So very hard won in reality. # Ten Never had the ride to Azerley Hall felt so interminable. Justin tried to watch the Yorkshire moors roll by beyond the rain-spattered windscreen, but it was no use. He kept darting glances to Grandfather, who kept ignoring him. No, that was unfair. Grandfather was exhausted. The creases in his face had deepened, and his cane had shaken nearly violently as he made his way to the car that morning. The trip had cost him precious energy. The fact that he had been unreadable yesterday through tea, supper, and Brook's performance on the piano afterward, meant only that he did not forget his breeding even in illness. The fact that he held his silence now meant only that he was too tired to converse, not that he had something bad to say. Still. Justin needed to know what the duke thought of her. Not that his opinion would change Justin's, but it mattered. It mattered a great deal. "Will you relax?" Cayton leaned close to issue the order. The others seemed not to hear him over the noise of the engine. "You are taut as a bow." "Sorry." But though he told his neck to ease, his shoulders to let go their bunching, they would not obey. His cousin rolled his eyes. "You can call again any time you please." Whitby had said as much when they departed, that they were always welcome. But Justin glanced again at Grandfather, and a knot tightened in his stomach. He daren't leave him. Not unless the duke improved drastically. Cayton's gaze followed Justin's. And he nodded. His shoulders went taut too. In this, they were of the same mind. Silence reigned again—other than the roar of the motor, the grumble of the tires over the pitted road, and the hiss of rain on glass. The air today was cool, to match the low grey clouds. And Brook, as she toured Whitby Park with her father, would be claiming she loved the rain, even as her fingers turned to ice. Another look at the duke proved as fruitless as the first hundred. He said nothing as they covered the last miles to Cayton's home. Nor as they parked and all climbed out. Nor, even, when Aunt Susan insisted on helping him up the stairs. Then, when they were all inside the towering great hall, he met Justin's gaze at last. "Join me in the study, if you would, Justin." With the command, years fell away. He was back at school, getting called into the headmaster's office for putting a toad in a tutor's drawer. At the palace, being called before Prince Albert after he'd taught Brook to use a pistol when she was twelve. At Ralin, about to get a dressing down for posing a suit of armor in a too-undignified manner. "Coming, Grandfather." "May God have mercy on your soul," Cayton muttered as he brushed past on his way to the stairs. Justin allowed himself only one fortifying glance at his aunts and then followed the duke's slow steps toward Cayton's study. To the room, behind the desk, to the massive leather chair that somehow made Grandfather's weakening frame seem bigger rather than smaller. Justin took the chair on the other side of the mahogany desk. And wished Father were there to make one of his jokes about tight laces and expectations. Grandfather gripped the arm of the chair with one hand, held on to his cane with the other, and drilled his gaze directly into Justin's. "I did not want to mention this with your aunts present. But we have a problem." Justin tried to straighten his spine, though it was already a ramrod. "Sir?" "I have tried to ease you into the running of the estates. Perhaps I shouldn't have. You know we have holdings in India, Africa, Canada, and the Caribbean." Of course he knew—and had been hinting for years that Grandfather needed to let him review the ledgers for them. Justin nodded. The duke sighed. "Apparently my stewards there were not as trustworthy as I thought. They have been lining their pockets with Stafford money. Money I was counting on to improve the tenements. The repairs on the cottages in the village cannot be put off another year, but I don't know now how we can afford them." The fear eased before it could slice. Those books he did know, better at this point than his grandfather did. "If we tighten up on our spending and cut loose a few holdings that we've no need of, we can find the money." And he had inherited a tidy sum from Father—though mentioning the fortune he had made at the tables wasn't a safe topic. Grandfather shook his head. "I am none too sure. You will have to travel to put these issues to rights, and doing so in the style you must to make a proper impression will necessitate spending yet more." He paused, focused his gaze on the far wall and its rows of scarcely populated bookshelves, and rocked his cane back and forth. "You should marry her, my boy. Sooner rather than later. Being out of society so long, Whitby has spent little and made much. He will give her an impressive dowry, and he said he will draw up the papers to name her his heir within the week." His throat went dry and tight. "You asked him?" The duke's gaze snapped back to him. Slapped at him. "We all know how these things work. You will make his daughter a duchess. No one would expect her to come to the union empty-handed. It would be a marriage of equals, beneficial to all." Justin clenched his teeth until he felt the tic of the muscle in his jaw. Grandfather's eyes went dark. "Why do you look at me like that? You are fond of the girl—it is obvious. You would likely have wed her even if she were a penniless nobody, nothing but the daughter of an opera singer. You ought to be praising the Lord she is more, so that it will bring good to Stafford rather than scandal." Justin could not convince his jaw to unclench. If he did, words would spill out that he could not in good conscience speak to the duke. Words about how he didn't want to marry for the good of Stafford. He didn't want to marry with even the thought of what money she could bring to him. If he married Brook it would be because they had always understood each other. They made each other better, stronger, more. Because he loved her. Had always loved her, would always love her. And he could not cheapen that by putting a pound sign on it. Grandfather, apparently assuming his word was, as always, law, leaned back in the chair. "Propose soon. Perhaps at this house party in a fortnight, before the rest of society meets her. She is too beautiful to remain unattached for long, and that aunt of hers is inviting all the leading families. Nottingham's son will be there, and he's likely to switch his affections to your baroness now that she is Whitby's heir. From what I hear, he has a silver tongue and a way with the young ladies. Claim her before he can." A stab, a twist in his gut. "Brook will not be so easily swept off her feet, Grandfather. She may sing the arias of love and romance, but she is practical and logical to a fault." When she wasn't flying off on some impulsive lark or another, anyway . . . but liberating cars and joining the ballet were a far cry from pledging her life to a stranger's. Weren't they? "Logic will tell her, then, to make the best match possible. And if word gets out that our estates are not in order, you will be second to Worthing. Move ahead of the rumors." Grandfather pushed himself up, slowly and obviously with pain. "You will be a good duke, Justin. A good husband. A good father. You will be all we prayed you would be. You have always made me proud." Justin had little choice but to nod and rise too. He knew the duke meant well, meant to assure his happiness and prosperity. But he had the wrong of it this time. Justin couldn't propose to Brook now. Not with her still settling with Whitby, not when he'd had no chance to convince her he could be more than a brother—and not with the shadow of debt over the house of Stafford. He could suffer it if the ton whispered that he had stationed her as Whitby's lost heiress so he could wed her and take the earl's estate. But if she ever thought it . . . no. He would never, never let her wonder that. Which left him only one choice. He must put Stafford in order first. And trust that if the Lord meant Brook for his wife, she would be waiting for him once he had. He trailed the duke out of Cayton's study, telling himself not to worry. She had always waited before. Always welcomed him to Monaco with sunshine and a kiss on each cheek. Had always made it clear he was her favorite person, aside from the prince. There was no reason to think she couldn't fall in love as he had. No reason to think another absence from her would change the bond between them, just because she was in England now. With her family. In a new home. With all the nation soon to be clamoring for a peek at the princess-turned-baroness, and sure to be enamored with what they would see. No, no reason at all to doubt. The room felt familiar. Brook trailed a finger along the edge of a shelf as her eyes drank in the honeyed woods, the polished metals, the touches of color and play of light. It smelled of faded flowers and crisp air, of comfort. Her mother's chamber felt familiar, but not as Whitby Park itself had when she first saw it. It stirred no imagined memories. What it brought to mind, rather, was the feel inside the sanctuary of Cathédrale Notre-Dame-Immaculée. Reverence. Sanctity. A heritage preserved with tireless care. The dressing table still sat in the corner, no doubt as Lady Whitby left it. A hairbrush beside a bottle of perfume, at an odd angle. A necklace glinting gold as it snaked around a pot of powder. A book still sat on the bedside table, a slip of paper marking a page halfway through. The Count of Monte Cristo. Brook smiled, though it faded fast. She had read the novel two years ago. It seemed her mother had never finished it. She stifled the urge to peek into the armoire. Were she to do so, she suspected she would see a rainbow of old-fashioned gowns. Time here had stood still. Whitby halted by her side, regarding the room with the solemnity of the sanctuary's priest. "Mary accuses me of making it a shrine. I've never known how to explain to her that I did not keep it just so for my own benefit." He moved to a chair with a length of wispy fabric draping the arm. Gathering it in one hand, he seemed to look into the past, perhaps to the ivory shoulders it had once graced. Then he let it slide back to its place. Just so. "But when I came in here after her funeral . . . or for months after . . . I felt that—that it was not finished. There were too many questions unanswered. And you, still missing. How was I to move on? It would have been wrong." Brook slid to the window and touched the fleur-de-lis pattern in the velvet drapes. She looked out but scarcely saw the maze cut into the shrubbery. Instead of the midday sun, she saw darkness. Heard thunder rumbling and felt the sizzle of lightning. That dream had plagued her again last night. "Are you all right, Brook?" "Hmm?" Her hand had found her pearls again. Her father's gaze focused upon her fingers, and a corner of his mouth turned up. "Your mother used to do that too, when she was lost in thought." A thought that brought the burning back to her eyes. "With this necklace?" He frowned. "That one?" "Maman said she was wearing it that night." She touched the pearls and then lowered her hand. Her father's face went taut. "What else did she say? Did she explain why . . . why Lizzie did not send you back to me?" "No." Perhaps it was in the journal. She should look, for his sake if not her own. Though—she frowned—she could not recall seeing it among her things since she arrived. Mademoiselle Ragusa must have slid it somewhere for safekeeping, but where? "She said only that we were in a carriage accident. That my mother took this necklace off and made her swear to keep it for me. I . . . I suppose I always assumed it was from you." He straightened his shoulders, forcing the torment from his face, and stepped closer. Narrowing his eyes upon it, he shook his head. "I never bought her pearls. They were, at the time, more for an unwed girl than a married woman. Perhaps her family gave it to her, though. For her debut, likely." And why would that make disappointment seep through Brook? "You do not recognize it?" Amusement glinted now in his gaze. "Lizzie had no shortage of pretty baubles. And I took great pleasure in showering her with more. Here." He motioned Brook to the left, toward a door he opened to reveal a dressing room bursting with those gowns in every shade and hue, the leg-o'-mutton sleeves the height of fashion eighteen years before. While Brook let her eyes feast on the fabrics and colors, her father headed straight for a cabinet built into the corner and pulled open the drawers. When he waved a hand at them, she noted the harder rainbow within. Rubies and topaz and emeralds and sapphires, garnets and jet and diamonds. More memories assaulted her. Not of this room, this mother. Non, now her mind went back to the little flat she had shared with Maman in Monaco-Ville before her death. She remembered playing with Collette's necklaces, earrings, and bracelets. Asking for the name of each jewel. Holding it up in chubby fingers to see how it would look on her. And Maman would laugh that crystalline laugh, would sing the names of the gems to her. Would refuse to answer her questions of where each piece came from. Now Brook was old enough to understand that Collette had once accepted such gifts from wealthy patrons like Prince Louis. But she had given up such a life to be Brook's mother. To raise her with a better example. Whitby lifted a collar necklace heavy with diamonds and emeralds. "I gave Lizzie this to celebrate our first anniversary. To match her eyes. The color of emeralds, with the light of diamonds." Brook's heart ached for him. His tone was still so full of love. Of the pain of loss. How could he have survived so long without his Lizzie? "It's lovely." His expression shifted, and his smile seemed lighter. "It is yours now. All of them are." "Non. Je ne peux pas." She stepped back too quickly and knocked her heels into the door, sending it into the wall with a bang. Her father looked at her as though she had spoken in Greek instead of French. "Why can you not? I certainly am not going to wear them." A breath of laughter escaped, despite herself. Still, she could not lay hold of English and spoke in French. "Tout le mond pensera . . ." He lifted a brow. "What does it matter what everyone thinks? Yes, plenty will declare that you have come back solely to inherit my fortune. But is that why you came home?" He asked the question with no doubt in his tone. But with discerning eyes. Eyes that had seen through imposters, eyes that had continually scanned the horizon for his lost daughter. Brook sighed and shook her head. "But I don't want to bring scandal and gossip down upon you." He snorted a laugh and put the necklace back, picking up a shorter string of diamonds in its stead. "I am an old favorite of the gossip-hounds. Eccentric Whitby, the recluse of North Yorkshire. According to your aunt, I have been seen haunting the abbey's ruins along with all the other ghosts, prowling the roads waiting for your mother's carriage to appear, and grabbing random blond children in the streets to see if they are my missing child." He held up the bracelet, indicated her wrist. She stretched it out and let him fasten on the clusters of diamonds. "Poppycock, of course. I only haunted the abbey once and couldn't tolerate the draft. I simply had to swear off it." He put on that crooked smile again and dropped his hands with a nod. "There. It suits you well. And she would be glad to know you have it. That piece has been around, I think, since the first Baroness of Berkeley." Brook let her wrist fall to her side, let the bracelet come to a glimmering rest against her hand. The prince had given her jewels before, but she had rarely worn anything more than the pearl necklace. All she had ever wanted was her own place. Her own things. Her own identity. She had never known what those were. "So long as you are certain. I have lived long enough on a borrowed name." He motioned her back out of the dressing room. "Then take the one that is yours—it has been waiting for you all this time." She stepped back into her mother's room, surrounded by her mother's things. And realized that he hadn't kept the room just so for himself—he had kept it for her. So that when she came home, she would find bits and pieces of the mother she had lost. And the father who loved her enough to preserve it for her. A nod was all she could manage. He must have spoken the language of nods well. He returned it with one of his own and led her back into the hallway, to the next door down. His room, she knew, and when he motioned her to follow, she stepped inside. In many ways it was like the prince's chambers. That same masculine presence, the lingering scent of shaving soap, the glass case of cufflinks bright against deep colors. But here, the windows weren't open to a warm, salt-tinged Mediterranean breeze, she couldn't look out to see terra-cotta roofs lining the streets. Couldn't hear the music of shouting, laughing tourists, street performers, and bustling city life. She saw only green through the glass, heard only the muted chirping of birds. She halted a step inside while Whitby strode directly to a chest of drawers against the far wall. Opening the third drawer, he moved aside some folded fabric and withdrew an ornate cigar box. He put the drawer to rights and was in front of her in the next moment, the box outstretched. She knew it must be the letters from her mother. And though she still felt a little odd at the thought of reading them, it was obviously important to him that she do so. That she know their story so she could understand her own. "Thank you." Such feeble words. But they were all she had, so she said them again as if to seal them. She offered him a smile and lifted the box. "I shall go and put them in my room. Then we can meet in the library?" His smile was warm, the long-borne pain hidden again under fresh joy. "Perfect." She hurried down the corridor, along another, along the maze of them until she reached the Green Room. Whitby had said they would move her to the family wing tomorrow, into the room that had always been meant to be hers. So she wasn't surprised to find Deirdre in her chamber, refolding and packing all the gowns that had only been out of her trunk for a few days. With a brief smile of acknowledgment, Brook bypassed her and went to the dressing room. The journal had been in the bottom of her trunk. But if the mademoiselle were putting it away, she would likely store it with the letters—she knew they were her maman's, and that the book was too. It would be the logical place for them. But no leather peeked out. She didn't see it on any shelf, or in any drawer in here. Perplexed, Brook set the new collection of missives down and headed back to her bedroom. With her regular reading, perhaps? Dracula or La Bible? Both of those tomes rested on her bedside table . . . but no journal. Deirdre cleared her throat. "Can I help you find something, my lady?" Brook sighed. "Yes, perhaps you've seen it. I had a leather journal in my trunk, an old one. It was my maman's." The maid's face remained blank. "A journal? I can't recall seeing it, my lady. But I shall keep an eye out for it as I repack everything." Brook couldn't have lost it. She knew she had packed it, she had put it in the trunk first thing, before Odette had added her gowns. Casting her gaze around the room again, she nodded. "Thank you. It must be here somewhere. I haven't even read it yet, I . . ." She shouldn't blabber about it to the staff. Summoning a smile, she nodded Deirdre back to her task. "I'm sure you'll find it as you pack, thank you. Will you let me know when you do? I'll be in the library." With Deirdre's quiet assurances following her out, Brook slipped into the hallway again. So much for being able to offer her father answers. Apparently they would have to wait for another day. # Eleven Deirdre had finally managed to escape the house, and without anyone making her wait so they could walk to the village together. Not that she would have minded Hiram's company, but she needed the time to clear her head. The rain poured down in earnest. Her half boots would be a muddy mess, and though she had donned her oiled cape and had her brolly opened above her, every time the wind gusted she got a face full of water. And naturally, when she reached the crossroads there was a carriage bearing down, ready to slosh by and send that entire puddle upon her. She backed up, hopefully out of splashing distance. The carriage pulled to a halt. For a moment she thought it must be someone in need of direction—then she saw the dark scowl on the man that swung open the door. Pratt. "Have you got your days confused, my lovely?" Not giving her time to answer, he jerked his head. "Get in. And make it quick." She told herself to be grateful for the escape from the rain. Though he would likely deduct the price of cleaning up her mud from her next payment. With a glance over her shoulder to be sure no one would see, she closed her umbrella and hoisted herself up. The interior was dim and smelled of spice and rain. Pratt tapped the ceiling to order the driver onward, never taking his eyes from her. "I expect you have an excuse for missing our rendezvous yesterday." Her umbrella was dripping a lake onto his floor. "The Duke of Stafford came unexpectedly. I could not be spared." "The Duke of Stafford." His glare chased away the light. "Why?" As if she dared to interpret the mind of a duke. "On his way to Azerley Hall, he said. Thought to stop in for tea so he could meet the new baroness and Lady Melissa, whom your friend Cayton could scarcely take his eyes from." "Is it official, then? Whitby has accepted this performer's daughter as his own?" She nodded, not bothering to ask how he knew that much, lurking around as he always did. "Although . . . you know French, don't you, my lord?" His answer was the arch of a dark brow. Perhaps this wasn't such a good idea. But she had already smuggled the book out in her handbag, she might as well see it through. She drew out the leather journal. "Her ladyship had this with her. I thought . . ." "Leave the thinking to me." He snatched it from her hands though, and flipped to the first page. The way his gaze darkened, she couldn't be sure if the words he found pleased or angered him. "This isn't the baroness's." "The singer's. The baroness hasn't even read it yet." There, his lips turned up. Because she figured it would only improve his mood, she added, "I am to be her lady's maid. His lordship will announce it after prayers tomorrow." "Moving up in the world, are we?" Yet his gaze said she was worth no more than ever. "Your instincts were good with this. And you'll be even more useful now. Earn her trust. And pay especial attention to her relations with Abingdon—I won't have him marrying her before I can so much as get a proper introduction." She hesitated, reached halfway out. "The journal, my lord. I need it back, to return to her things. She was looking for it yesterday. If you could just take a peek to see if it verifies the story she told . . ." That quickly, his mood turned. "My French is not so flawless that I can just glance at it. I'll read it at Delmore and return it when I am through." Unease clawed at her, but she knew better than to argue—it would only make him more determined. She cast around for something to distract him before he decided to keep it forever. "Lady Ramsey is throwing a house party in a fortnight's time, at Whitby Park. You are to be invited." His smile reemerged. "Good." Eden Dale was already coming into view, and Lord Pratt smacked the roof again, calling out, "Stop here!" Quiet and cold, he added to her, "Can't be seen together, can we?" "No, of course not." If only she had her old bin of brushes and cloths so she could wipe up the mess. "So sorry for the mud, your lordship." "I have servants to clean it." Quick as a snake, he grabbed her wrist and pulled her to his side of the carriage, pressed his mouth to hers. More poison than kiss, more shackles than embrace. She endured it—and promised herself a thorough scrubbing when she got home. He chuckled as he pulled away. "I saw her, you know, the other morning. She is nearly as beautiful as you." He dragged his finger down the side of her face, from temple to chin. "It will not be a hardship to marry her. And even less of one knowing you come with her." Deirdre prayed he wouldn't detect her shudder. She said nothing. But when he let her go, she lunged for the door and exited with more speed than grace. His laugh joined with rumbling wheels and pounding rain as the carriage rolled on again. She had left her umbrella inside. And deemed getting wet an even trade for escaping him. The horse was black as midnight and skittish as a phantom. Brook knew the moment she stepped into the stables and clapped her gaze upon the stallion that he would be her mount of choice. She had little use for a docile horse—when she rode, it was to give herself over to wind and earth and sky, to lay bare her soul to the Father who had crafted both beast and land across which it flew. When she rode, it was to push herself to the edge of reason and safety. When she rode, she lived. The stable master slurred some response to her question of the horse's name that she could scarcely understand, so thick was his accent. The groom interpreted with, "Him? Nay, milady, you don't be wanting Oscuro." "Oscuro." She whispered the name, but not as he had done, with the dreadful British enunciation. She accented it as the Italian dictated. Oscuro, the unknown darkness. Perhaps the horse knew his name had been said wrong all this time, for he tossed his black mane and nickered. Of course, he also reared up and pawed. She made for his end stall. "He ain't tame, milady!" The groom jogged to her side. "He was bred for the races, but he wouldna tolerate a rider. Broke his trainer's leg, he did. His lordship's only keeping him to stud." And she didn't intend to ride him today—she wasn't daft. But she would. Soon. And the start would be getting him used to her presence. She halted a bit away from his stall, out of range of hooves but close enough for him to catch her scent. "He is well groomed for being unbroken." The man grunted. "It takes two of us to get him secured, and then we draw lots to see who risks getting bit or kicked. Stay clear of him, milady, I beg you. He'd better to have been named after the devil than darkness." Oscuro pawed the air again, showing off his musculature and powerful frame. "He wants to run free." "Aye, and he can never be off the tether, or he'll be over the fence and gone. Leave him be, now. We have a mare, just as handsome, same coloring—share a sire, they do, but this one's trained for the sidesaddle. Her name's Tempesta, but she's got patience to match her spirit." Brook turned to face the groom—slowly, so as not to startle Oscuro. "I don't care for the sidesaddle. I will be riding astride." Temper flashed in his eyes. "The young ladies always ride sidesaddle, milady. Lady Melissa is a most excellent horsewoman too. I've accompanied her many a time. Let me saddle Tempesta for you and we can go. His lordship said we ought to show you all the estate." She tried on her sweetest smile. "I do appreciate the offer . . . what is your name?" He heaved a sigh, but the fight didn't leave his eyes. "Francis, milady." "Francis." He seemed immune to her grin, but she brightened it anyway. "I am happy to take the horse you recommend—though with a traditional saddle. But much as I appreciate your offer, I don't need an escort today. I'll not go far." Not too far. Francis's returning smile looked about as warm as last week's unrelenting rain. "I'll fetch the horses, milady." Horses, plural. She sighed as he strode away and then turned slowly back to Oscuro. He kicked at the stall. She nodded. "I know how you feel," she said in Monegasque. "I have not been alone for over a week, save for when I sleep, and I am about to kick something too." She was enjoying the time with her family. Aunt Mary was welcoming, if a bit aloof, Regan sweet as could be, and Melissa's offense on her sister's behalf seemed to be fading. But Brook had not been so surrounded by people . . . ever. The prince had given her the run of the palace, and more often than he liked, she slipped out without a chaperone and took herself to ballet lessons or for a spicy salsiccia. Or to find Justin, if he was in Monaco. Footsteps sounded behind her, along with a sigh she knew quite well already. "Naturally, you find the dangerous one." Her grin, she had discovered, worked quite well on her father. She flashed it at him now. "He is the handsomest. Are you the one who gave them Italian names?" Whitby hummed, nodded, and held out his palm. Oscuro ignored him, but given his behavior otherwise, it was surely the equivalent of a whinny of greeting from any other horse. "Not all of them, of course, but it seemed to suit him and his sister. Francis said he is saddling Tempesta for you." "Oui. F—" Father, she almost said, but stopped herself. She had not called him such yet, and she would not now, when she was trying to wheedle him into something. "Francis said he must come with me." Her father lifted his brow. "This is a problem?" She splayed her hands. "Do you always like company on your rides?" "I am a man." No doubt he tried to keep his expression clear—but she thought she detected amusement in it. Now Brook planted those hands on her hips. "And I inherited your disposition." "You'll never let me live that down." "You'd never want me to." Yes, definitely amusement. It made his lips twitch. "And you've only been home a week." She flashed her grin again. "Imagine when it's been a year." He clasped his hands behind his back, sent his gaze over her shoulder, and rocked on his heels. "An hour, and you must stay on Whitby land." "Two hours." His brows lifted. "But the boundary?" "Accepted." "Done." He held out a hand. She shook it, unable to stifle the laugh as she did. "You are an admirable negotiator." "Ha! You are an unabashed flatterer. Francis!" Brook scurried to keep pace as he strode down the open space between the stalls. "He seems to think I need a sidesaddle, too, if we could correct him on that at the same time." Her father came to an abrupt halt and turned to her with that expression of fond disbelief he had given her at least forty times in the last seven days. "You ride astride?" "Have you ever tried to ride sidesaddle?" A short laugh slipped out. "The prince taught you?" "Grand-père . . . allowed it." Whitby's eyes went to slits again. "Let me guess . . ." "Justin taught me." A phrase she had uttered a matching forty times. "It is all his fault, really, every bit of unconventionalness . . . Is that a word?" A snort was his only answer. He took two more steps, then halted again. Lifted a finger. "How, if you don't mind me asking, do you ride astride in a skirt?" Brook kicked a leg out a bit, revealing the split that was all but invisible when she stood still. He pressed a hand to his brow and moved onward. "My daughter is wearing trousers." "Oh, there's no need to sound so horrified. They are not trousers exactly." His grunt disagreed. "Your mother would kill me." "Nonsense. She wouldn't have let a little thing like a split skirt upset her. Although now that you mention it, trousers would be far more efficient." "Heaven help me. Next thing I know you'll be joining the suffragettes." She tucked her hand into the crook of his arm and grinned up at him. "I'd rather learn to drive. Justin taught me a bit in his Rolls-Royce in Monaco." "Of course he did." "But not nearly enough. Have you learned how? You could teach me." He sent her another look she had already learned—one that said her grin had reached the limits of its powers. For now. "I employ a chauffeur." "So I should learn from him?" They reached the stall where Francis worked on another midnight-black horse. This one greeted them properly, to the point of snuffling at her father's pockets. Apparently in search of sugar, since he produced a cube of it. "Try it and I'll lock you in your chamber as your Justin recommended. Francis—no sidesaddle for the baroness. And she has my permission to ride alone." He raised a finger and leveled it at her nose. "Two hours. On our land. Or I dig up the key—it is surely around somewhere." "Mrs. Doyle no doubt knows where it is." And, she suspected, would happily hand it over. Brook had yet to earn more than a polite turning of the lips from her. She held out a hand for Tempesta to sniff. Getting a damp snort of approval, she rubbed the mare's ebony nose. "But about the driving." "No." Francis exited the stall with the sidesaddle in hand, shooting her a look that said quite simply she was not what he thought a baroness should be. Brook focused on her father. "Another deal, then. If I can break Oscuro within two months' time, you let me learn to drive." He was a master at the arched eyebrow, this father of hers. "You expect to convince me to let you learn one dangerous task by promising to do another?" "He was born to race." Her tone went more serious than she'd intended. Her fingers curled into her skirt. "Some creatures have a harder time obeying the standards put before them. But if you can inspire them to, they will outdo all the rest. You must simply learn their language." She expected him to ask if she had ever broken a horse before. She was prepared to tell him all about her favorite mount in Monaco, how she had finally ridden him over the French hills after months of work. He didn't ask but studied her until Francis returned with a regular saddle and a stony countenance. Then he sighed. "Two months?" "Well. Assuming I'm not forbidden from stepping out in the rain." As she had been all last week. Granted, it had been abysmally chilly, but she would have suffered it for the sake of a horse. "And then the car." Whitby turned to face the end of the aisle again, where Oscuro still snorted and fumed. "This is certainly no life for him. If he can be trained—" "Your lordship!" Aghast, Francis paused in his reach for the girth strap. "You've the best trainers in all Yorkshire. If they canna break him, then he canna be broken." "Or . . ." Whitby faced her again, met her gaze. "They did not speak his language. Sometimes when we think someone should understand English, they really only know . . . French." Brook's heart swelled, warmed. Francis looked ready to snort along with Oscuro. "You want we should speak French to him?" Another twitch of his lips, but her father didn't turn to the groom. "If you can do it, my dear, then I will not only allow you to learn to drive, I will learn with you." She held out her hand as he had done before. "Done." Rather than shaking, he clasped her hand in both of his and squeezed it. "Be careful. If you're not back in precisely two hours, I'll send out the hounds to find you." "Thank you." She stretched up on her toes so she could kiss his left cheek, then his right. When she had said farewell to Justin that way a week ago, Aunt Mary had played her fainting trick again and had lectured her for a solid hour afterward. Her father half-smiled, as if remembering the same thing. "The sea abuts our property on the east, of course. To the south, you may go so far as the copse of trees beyond the duck pond. To the west, so far as the road leading to the village, and to the north, all the way to the hedge dividing our land from Delmore." "Delmore?" "Pratt's estate. It's a sprawling, mazelike monstrosity that has a strange charm I think you'll enjoy seeing." She wrinkled her nose at the name. "I'd just as soon not." He chuckled. "Enjoy yourself. Perhaps tomorrow, if the weather holds, we can take a morning ride together." "I can think of no better way of starting the day than with a ride." He moved off, greeting a few of the horses with the same muted affection he gave his family. Muted, but sure. Solid. Brook watched him step into the weak sunshine and turned to Tempesta. Francis led her out of the stall and handed Brook the reins with nary a word. He gave her the exact same flat stare her maid—whom she was apparently now to call O'Malley—had when she saw the split skirt. Silent, screaming disapproval. And they all wondered why she needed a solitary ride. She adjusted the stirrups and then swung up into the saddle. Its leather was supple, well worn and well cared for. She settled comfortably into it, gathered the reins, and patted the horse's neck. "Allons-y, ma fille." Go she did, at a high-stepping walk from the stables, into a trot southward with the barest of whispers, and to a full gallop when Brook gave her rein. Tempesta's hooves ate up the ground, raining clods of dirt down behind them. Before they left the lawn, Brook reached up and unpinned her hat so she could toss it to the ground. She needed the wind to whip through her hair and blow away all the frustrations. She needed to be free, to discover, to find her place. Find it she did, at the southeast corner, where the land rose before tumbling into the sea. The waves before her, a cliff under her, the moors rolling out behind . . . not exactly the seascape she had grown up with, but close enough. Beautiful enough. Enough. For a moment after reining Tempesta to a halt, she merely closed her eyes and breathed it in. Whispered a thank-you to the Lord, and then a please. An outpouring. An in-taking. Then she slid down so her own feet could test the earth. Were the wind not gusting off the ocean, she would have withdrawn from her pocket the two letters she had chosen to read today. One from her father, one from her mother. He had traveled a good deal, it seemed, in those days. And whenever they were apart, they would write. Of love. Of family. Of yearning to be together again. She had matched up the dates as best she could for the two stacks, which had taken most of one rainy afternoon . . . especially given how often she had to pause to laugh at something Regan or Melissa said as they all worked on their projects together in the upstairs salon. Reading them she was taking slowly as well. Familiarizing herself with each loop in her mother's hand, in the quick dash of her father's. Their favorite phrases, their nicknames for each other. According to the dates, she was drawing near to the time when they would mention her, as in that first letter of her father's she had spotted. Though she knew already there would not be many letters for her to read for that time—they had not been apart then. She had mentioned the gap in dates as she was correlating them, and Aunt Mary had given her an indulgent smile. "When Ambrose found out Lizzie was expecting, he could not be dragged from her side," she had said. "Not until necessity dictated it right before . . ." Before that night. The night the carriage careened off the road and everything changed. The wind shifted, and the warmth she had worked up on the ride went the way of the sunshine—swept away by the clouds. With a shiver, Brook pulled out her watch from her pocket. She still had time, but if the sun didn't reemerge, she would be half frozen before she reached home. After mounting again, she set a slower pace toward the house. By the time she had found her hat and gained the stables, she was shivering. She handed the reins back over to the brooding Francis. Coffee. She needed coffee. "Brook!" Regan waved to her from the terrace outside the library. She sat with her sister and mother, looking positively warm in her short-sleeved afternoon dress. "Tea?" Striding their way, Brook chafed her hands together and smiled for her cousin. "Aren't you cold out here?" Regan laughed. "Are you jesting? It's lovely." Aunt Mary reached for her teapot. "Strong or weak today, dear?" She had tried both. She cared for neither. Grinning, she said, "Caffe espresso. Can your pot produce that? If I beg?" Her aunt laughed and motioned toward the house. "No. But I daresay the chef's can. Ask him for some and join us." Funny—the two times she had dared request coffee since Justin left, she had been delivered a cup of pale, watery stuff unfit for consumption. "I've been warned away from the kitchen—how, then, do I put in this request?" "Oh, nonsense." Aunt Mary sipped at her tea, her stern gaze belying her pleasant smile. "Don't let the servants intimidate you, child, or you will never manage the house. You are mistress. Go where you will. Ask for what you want." Mistress. Not a role that seemed hers, with Aunt Mary presiding over teas and dinner and Whitby in control of all else. But her aunt was right. If she ever hoped to be accepted by the household, she had to earn their respect. And she wouldn't do that playing the mouse. With a smile, she nodded and made for the library door. "I shall return with caffe." The door opened noiselessly, shut with a click. A rustle of newspaper from the corner proved her presence had been noted though, and her father peered over the top of the page with smiling eyes. "With twenty minutes to spare, even. All in one piece, are you?" "So long as you are not counting hairpins. Although I would like to lodge a complaint—your air here is too cold. Might we import some Mediterranean breezes?" He chuckled and raised the paper again. "I'll have some shipped, posthaste." The rows of books were tempting, as was the fire in the grate. But the allure of coffee kept her feet moving through the room. She would settle into her leather chair after tea, before the dressing gong. It had become her favorite hour of the day. The halls grew less familiar as she neared the stairs down to the kitchen. Her mother must have walked this path countless times, on her way to plan the menu with the old cook. Brook tried to picture her here, the true mistress about her duties. She would have been comfortable, in her element. Humming, perhaps. She would have smiled as she descended and the sound of laughter drifted up to her. Brook felt like an interloper. "Aw, come now, DeeDee. Have a cup with your lowly friends." A male voice, though Brook couldn't place it. The answering laugh she knew, though Melissa and Regan had been the ones to draw it out before. "That's O'Malley to you, Hiram. And sure and if I do, her ladyship will return the self-same moment all covered in mud and needing my assistance." Her cue. Clearing her throat in warning, Brook descended the last steps and turned the corner into the kitchen. The servants all leaped to their feet or halted their work. Brook smiled at the group at large. "Don't mind me. I only need a word with Monsieur Bisset." Those about tasks resumed them. Those about their tea shifted from foot to foot without retaking their seats. She had learned that the English took their teatime quite seriously—so she would hurry. She turned to the rotund man frowning from his place at the stove. "Bon après-midi, monsieur. Ça va?" He turned back to the simmering pot. "I am busy," he answered in French. French . . . but not quite French. Hadn't they said he was from Paris? Or was it only that he was schooled in Paris? She stuck to français. "And it smells delicious. I will trouble you only for a moment." Her gaze went to the beautiful, miraculous, life-promising machine in the corner. How had he come by the exact model the prince had insisted on for the palace? They weren't cheap, and she couldn't think that Whitby had bought it, given that he never drank espresso. It must be the monsieur's, and he must have spent years of savings on it. Was he simply unwilling to share with her? But if so, then why had he produced a pot when Justin was here? Well, she would never know if she didn't ask. "I was hoping I could have a cup of espresso." The chef spun on her, his face red. "You would have me abandon my hollandaise, the most temperamental of sauces, the one I learned from my grandmother in Provence, which she had learned from hers, to make you coffee?" The kitchen went silent around them, but Brook merely folded her arms over her chest. The French bluster she knew well. His particular accent she did not. "Provence? I think not." More likely some corner of Quebec or another. He sputtered and muttered, though he used no words that she could make out. And the red in his cheeks faded to white. Blast. She hardly cared if he had lied about where he was from to secure a position as a French chef du cuisine. All she wanted was a cup of coffee. Why was that so much to ask? Deirdre held her breath with the others while the baroness and the monsieur all but spat at each other in French. Apparently her ladyship had no qualms about arguing with an employee. She answered him phrase for phrase, gesture for gesture. Proclaimed something emphatic with a sweep of her arms and then pointed at the odd machine that hissed and steamed whenever the chef used it, and spurted out a coffee black as night. Other than Bisset himself, no one but Lord Abingdon had ever suffered it. Well, and the baroness. Though when a cup had been requested for her last week, the chef had not set the thing to hissing, he'd merely tossed a few grounds into a kettle. Which her ladyship must have realized. Pure exasperation covered her face as she delivered another line of too-rapid French, ending with a s'il vous plaît that sounded more like command than request. Monsieur Bisset glared. Sighed. Asked something. "Non." The baroness motioned again at the machine. "Je veux seulement un espresso!" He huffed. But he nodded before he waved a hand at the stairs. The baroness echoed his huff and spun away. "Merci, monsieur." To the rest of them, she nodded. Then she stomped her way back upstairs. "Well, I never." Mrs. Doyle smoothed a hand over her shirtwaist and looked from the stairs to the chef. "What, pray tell, was that all about?" Monsieur Bisset barely glanced at the housekeeper—he lumbered to the machine with a kettle of water. "Coffee." "Coffee." The housekeeper's tone was cooler than February in Kilkeel. "Her ladyship raised her voice at you over coffee?" He didn't answer, not even in French. Odd—usually he greeted their questions with an unintelligible spout of nonsense. Now he cranked the coffee grinder. Mrs. Doyle looked to Deirdre with raised brows. "O'Malley, is she always like this?" Deirdre cleared her throat. "No, ma'am. Not that I've seen. Though she does lapse quite often into French, ma'am, and I can't be telling what she says." "DeeDee." Hiram breathed her name like a warning. But it was nothing but the truth, and she couldn't lie to the housekeeper. She lifted her chin. Mrs. Doyle lifted hers too. "I do detest anyone raising their voices at one of our own. Monsieur Bisset, give the coffee to me when it is ready. I will deliver it myself." Hiram raised his brows, but Deirdre could only shrug. She slid to her seat beside him as the chatter returned to the kitchen. But Hiram held silent, and she could think of nothing to say either. A few minutes later the monsieur slid a cup of inky coffee onto the table before Mrs. Doyle, and everyone else fell silent again too. Silent and somber as the housekeeper fetched a larger cup, poured half the espresso into it, and filled the rest with water. Their laughter followed her up the stairs. # Twelve Justin pulled into the drive of Whitby Park, lined with unfamiliar carriages and cars promising strangers he didn't feel up to meeting, and knew he shouldn't have come. Never mind that Grandfather had told him to—he couldn't erase from his mind the way the duke's hand had trembled when they parted yesterday. How short of breath he had been. Now Justin would have to paste on a smile and put aside his worry, though he would rather turn his Rolls-Royce around. Still, he followed Thate to the stables and parked. Peters hopped out the moment Justin switched off the magneto. "I'll see to your things, my lord." "Thank you." He slid the key into his trouser pocket as he got out and scanned the figures flocking the lawn. Given the direction of Thate's gaze, Lady Regan must be by the table. Perhaps Brook was near her. "There you are! I thought you would never arrive." Or perhaps she was in the stables. He pivoted, spotting her as she emerged into the sunshine, and grinned. Even though his usual reaction to her beauty made him remember Grandfather's parting remark. "You know what you must do." "Hiding, Brooklet?" She bypassed the hand he held out and greeted him as she always had, kissing him on each cheek. "Naturellement. She managed to get twenty people here, all to stay the week—and she has been going absolutely batty with the preparations." He could only assume the "she" was her aunt. "Whitby allowed it?" Her smile did his heart good. Even better was the gleam of contentment in her eyes. "I give him two days before he flees Yorkshire. And I will be there by his side—you're welcome to join us." "You are getting on, then." He took her hand, tucked it against his arm, and led her toward the lawn. Thate awaited them with lifted brows. "We are much alike. And Lord Thate, you can relax—for a few days at least." She flashed a grin that would likely have turned his friend to a puddle, had he not been one already over her cousin. "Lord Worthing and his sister will not be arriving until Tuesday. You have three whole days to win her, and I suggest you put them to use." Thate grinned, too, even as he said, "I don't know what you mean, my lady. But might I say, since my oaf of a friend failed to do so, that you are looking particularly lovely today?" It was drattedly true. She wore some blue thing that looked like a slice of the sky draping her too precisely. The nip in the air had put roses in her cheeks, and the sun—which must have shown up on order of Lady Ramsey—made her hair gleam purest gold. No doubt every male set of eyes would be glued to her all week, and Justin wouldn't be able to rid his mind of Grandfather's warnings. Grandfather's commands. Thate's low chuckle made Justin aware of his own scowl. "Predictable." He lifted his brows and glanced toward the table where Thate's gaze kept wandering. "Pot and kettle." Thate laughed, but the way Brook's brows knit made him wonder if she had not yet learned that particular idiom. She didn't seem to catch the meaning of their jest, praise be to heaven. With a tug on his arm, she spurred him onward. "Thank heavens you're here—now I can finally get a decent cup of espresso." He pulled her to a halt again, though Thate sighed when he did. "What do you mean? The chef obviously knows how to make it." She shrugged and looked out into the distance. When her gaze grazed the collection of people on her lawn, she leaned a bit closer to his side. He wasn't about to complain—though he wondered if she even realized she had done it. Or if she could possibly know how it made him want to catch the curl that the wind toyed with, give it the tug he always had . . . and then slide his hand to the back of her neck and lean down to touch his lips to hers. He forced his mind back to the issue of caffe. "What is the problem?" The light in her eyes dimmed. She shrugged again, a gesture so very Gallic that she might as well have broken into a rousing rendition of "La Marseillaise." "They don't like me." Now Thate faced them, frowning along with Justin. "Who? The kitchen staff?" "All of them." She smiled, but it was dim and forced. "The family has welcomed me, but the staff . . . It is their loyalty to my father, I think. They have seen so many pretenders over the years." But with loyalty should have come trust—and if they distrusted her, then they also distrusted Whitby's recognition of her. "Unacceptable. You are their mistress, and if they cannot serve you well, they ought to be replaced. Surely your father agrees . . . Except you've not told him, or you wouldn't look away with that"—oh so lovely—"flush in your cheeks." "I know I should. And I will." She forced a little smile. "After the house party." He wanted to press the issue, but it would do no good. She had that obstinate set to her chin. But even Thate looked concerned. He motioned them onward again but kept his focus on Brook. "Have you hired a lady's maid? Perhaps if you have someone loyal first to you . . ." "I chose to promote the head housemaid. She has a way with hair." Did she know how weak it sounded? She must, because she kept her gaze fastened on the ground ahead of them. "Brook." "I thought it would help." She looked up now, and her smile went cheeky. "You ought to have seen her horror the first time I pulled out my riding habit." He snorted a laugh at the thought. But given that she had first learned to ride astride in Justin's outgrown knee breeches, the split skirt ought to have been praised as a brilliant compromise. "I can well imagine. Have you chosen a horse yet? I hear Whitby has some of the best stock in the country." "I have been riding a black mare named Tempesta—she is a beautiful creature, with an admirable spirit. But . . ." Her eyes gleamed so bright, he knew trouble brewed. "It is her brother I want. They say he cannot be broken and keep him on a tether at all times. His name is Oscuro." Justin tightened his fingers around hers. Riding astride was one thing—toying with wild horses quite another. He had nearly had a fit when she'd told him last year of the prince's horse she had "helped train," and Prince Albert's quiet assurances that she had been well guarded had done little to allay the fears. "Whitby surely doesn't let you near him." Her impish grin said otherwise. "I've already got him tolerating me in the stall. Another week and I intend to put my weight on him. If I can ride him in two months' time, my father and I will learn to drive the car together. By next spring, Lord Thate, I may be racing you at Surrey." The woman needed to be locked in a tower somewhere. On a desert island. With no wild horses. Or racetracks. "Don't even think it." Thate laughed. "Our friend is quite right. You would never stand a chance in that touring car of your father's. You would need a proper racing car. Perhaps a Lancia. Or a Benz." "A Fiat," she countered. "They may not have won the Grand Prix in May, but they set the fastest lap times, n'est pas?" She was mad—stark, raving mad. "Before sliding off the road and killing one of their mechanics. You are not racing, Brook. And you." Justin spun on Thate, lifting his hand from hers to give his friend a helpful shove in the arm. "Stop encouraging her. In fact, if you hope to win Lady Regan before Nottingham's son shows up, perhaps you ought to quit talk of racing altogether." "I don't have—" He cut himself off with a huff, apparently realizing the absurdity of the claim. He pursed his lips and looked to Brook again. "I don't suppose she's mentioned me." Brook's silver laugh chimed, making Justin's stomach tighten. "Perhaps." "Hmm." Mouth still pursed, Thate drew to a halt a fair piece from the gathering. "My mother says no lady of quality will have me as I am." "Well, that's ridiculous. And I think my cousin would agree." She grinned as she looked toward Lady Regan. "For all her steady ways, she is a romantic." "She is . . . perfect." Squaring his shoulders, Thate sucked in a breath. "Excuse me, Bing. My lady. I have only three days, and I don't mean to waste another moment of them." Justin watched his friend stride off, smiled, and was content to hold Brook on the edge of the gardens for a while longer. "He has an honest chance with her?" Brook hummed and rested her cheek against his shoulder, making his pulse accelerate far too much. "She's in love with him. Melissa thinks it foolish, and Aunt Mary talks only of whether Lord Worthing will propose. But if Thate speaks up, she'll accept him in a heartbeat." For a moment they said nothing more, just watched the way Thate first greeted a gentleman, how he used the conversation to shift directions, and then just happened to find himself at Lady Regan's side. Deft. Justin hadn't known he had it in him. "We ought to fashion a story to commemorate this occasion. We can call it, 'The Day Thate Conformed to Normal Social Ritual.'" She tossed back her head in a laugh. "A bit unwieldy, that title. I prefer 'When Love Found Them.'" His smiled. It faded, though, when a dark-clad figure caught his attention. "Pratt came, I see." She looked his way, shuddered, and then tugged Justin toward the house. "Those are my other cousins he is talking with. The Rushworths—Lord Rushworth and Lady Catherine. My mother and their father and his brother, Henry, were first cousins. Both their parents have passed. They have only their uncle left, but he has been in India for most of their lives." From this distance, Lady Catherine could have been Brook. Blond hair, trim figure, fashionable. Though he certainly hoped Brook never clung to Pratt's arm like that one did. "Did you meet them yet?" "Briefly." Brook nodded toward where her father sat in a chair on the terrace, trying to disappear behind a newspaper. "Lord Rushworth said hardly a word, but his sister seemed nice enough—though Regan doesn't like her, and Regan is usually a sound judge of character." She yawned, though she tried to cover it. Justin eyed the bench adjacent to Whitby's chair. Perhaps they could find another newspaper and follow his example. "Tired already?" "I was up too late looking for that journal I mentioned—I've no idea where it got put, and poor O'Malley was obviously afraid I'd blame her for it. Though Odette must have moved it when she packed for me, or Mademoiselle Ragusa at some point." She shrugged, though her eyes did not lose their disturbed gleam. "And I have been having the strangest dream." Not a good one, if the set of her mouth were any indication. "Nightmare?" "Oui. The same one, over and again." Her words drifted into Monegasque. "It is very vague. A storm, fierce and frightening. Lightning, thunder, darkness . . . and always this feeling of some threat lurking." Her right fingers found her pearls, twisted. Justin frowned. The journal would turn up, and the dreams were likely nothing—the influence of an unfamiliar home, an unknown future, unanswered questions. Still. "Every night?" "Almost. But they will pass." She renewed her smile and removed her hand from his arm so she could sit. Justin sat beside her, trying to ignore how cold his arm now felt. Whitby looked up from his paper. Smiled at Brook—scowled at him. "You. You have some explaining to do, Lord Abingdon." Perhaps he would have worried, had Brook's laugh not been so carefree. He looked from the woman beside him to her father. Was it laughter in Whitby's eyes too, or irritation? "What have I done, my lord?" Whitby folded his paper and raised his hand, a finger up. "You taught my daughter to ride astride." He raised another finger. "To shoot a pistol." A third. "To drive an automobile." Four. "To swim." And his thumb. "To fence." Brook's next laugh interrupted him, and Justin felt his mouth tug upward into a grin too. Whitby narrowed his eyes. "What have you to say for yourself?" Looking at her, how she sat with such confidence, how she laughed with such abandon, how she faced the world with such brilliance, there was only one thing he could say. "You're welcome." Deirdre hated this time of morning. When all was still dark outside the many-paned windows of Whitby Park, when she should have had a peaceful hour to take her breakfast and go about her tasks. It had once been her favorite time of day. Now she dreaded it, knowing she had to rush if she hoped to have Lady Berkeley's room prepared before the baroness surged out of bed. Her ladyship was up before the sun most mornings. She seldom asked for anything, but that only made it worse. She knew Deirdre resented her presence, and by knowing made her ashamed of it with every apologetic smile. Sure and it was enough to spoil the whole day. She trod silently down the hall, pausing outside the baroness's new room. Granted, it had now been hers longer than the Green Room had been, but it still felt strange to Deirdre. This was a chamber she had once cleaned with a pervasive sense of pity for his lordship. One that Beatrix wouldn't even step foot in without crossing herself. The babe's room, they had used to call it. Lady Berkeley's now. Deirdre said a silent prayer that the lady would still be abed and turned the knob. No lamplight greeted her. No soft humming came from the window seat. The baroness's wrapper was still draped on the chair—a guarantee that she was yet beneath her covers, for the girl couldn't tolerate chill air. Deirdre loosed a breath of relief and headed for the dressing room. His lordship had been sending over jewels and hats, scarves and gloves. Anything belonging to the late Lady Whitby that had not gone absolutely out of fashion. Still, the girl only wore that pearl necklace she had arrived in. She would slip on a bracelet or ring of the late lady's now and then, but the heavily-jeweled items remained on their velvet trays. Deirdre flipped on the electric light once she'd closed herself in. The baroness had instructed her to have her riding habit ready this morning. A hunt was planned. Deirdre couldn't help the purse of her lips as she pulled it down. Had the woman no shame, to wear pants with all those guests around? At least the lady would have to dress for breakfast first. She had indicated no preference for that, so Deirdre selected an ivory morning dress with rose inlays, just because she fancied it. After gathering the necessary underthings, she turned the light back off and blinked against the darkness of the bedchamber as she stepped into it. She went stiff when she heard shifting on the bed. By now, she knew the sounds. The muttering in French, the thrashing of limbs. The non, non, non. Another nightmare. For a moment, she strained forward. Little Molly'd had the worst nightmares after Da died. Deirdre had always pulled her close, smoothed her hair, whispered until her sister woke up and stopped her trembling. She could still see the fear in the wee one's big brown eyes. The same fear she had glimpsed in the baroness's one morning when she sat bolt upright after such thrashing. But what could her ladyship have to haunt her? What had she lost to throw her into such turmoil? Nothing. All she had done was gain, gain, gain. Deirdre spun for the fireplace. The wood had already been set, and the scullery maid would be in soon to light it, but not soon enough. She would have to do it herself, or else when the lady snapped awake in a few minutes, she would be a-shiver. Have to pull her blankets close. Chafe her hands together. Silent condemnation of Deirdre's inability to see to her needs. Francis said Lords Abingdon and Thate had all but told her ladyship to find another lady's maid. No doubt she was waiting for an excuse to do so, and heaven help her if Deirdre would provide her with one. She already had the journal hanging over her head, though her ladyship seemed to think the Frenchwoman had misplaced it, praise be to the Lord. Still, she would ask Lord Pratt about it if she could find him alone. He had surely had time enough to translate it by now. The first flames chased away the sulfur's bite when a scratching came at the door. Satisfied that the tinder would catch, she rose and opened it. Beatrix stood wide-eyed in the hall. "I don't know what I'm to do, DeeDee!" "Shh." Glancing over her shoulder to make sure the lady still slept, Deirdre stepped into the hall and eased the door closed. "What is it?" Beatrix wrung her hands. "I was taking out the pots in the ladies' wing and went into Lady Catherine's room, but she wasn't alone. There was a man in her bed! Lord Whitby would—" "Shh." Deirdre held up a hand this time to illustrate her point and leaned over. "Hush, Bea. It's a house party, what do you think happens?" The younger girl's mouth fell open. "Well, not that. His lordship would be aghast—I know he would. And Lady Ramsey—" "Lady Ramsey knows the ways of the world, and so long as it isn't her daughters disgracing themselves, she is happy to turn a blind eye." Beatrix didn't look relieved. "But his lordship . . . You remember when he dismissed Bridey last year for getting caught with a village boy in the stables. He's no tolerance for such things under his roof, he said. And if not from us, then surely not from a lady." "Beatrix, listen." She gripped her friend's arm and steered her back toward the exit from the family hall. "We're not going to tell his lordship. We're going to mind our own and bite our tongues and not say a word. Do you understand?" "But—" "It isn't our business. And if you told Lord Whitby and he confronted her, she would only say you were lying and try to get you sacked. It isn't worth it. She'll be gone when the week is." At last, capitulation filled Beatrix's eyes. Her friend nodded. Deirdre did too, and released her. "Now back to your duties and I to mine." "Sorry to interrupt, Dee." "No matter." She produced a smile and made a shooing motion that had always sent her siblings on their way. "Off with you now." Once Beatrix had scurried along, Deirdre turned back to the baroness's door and slid inside. "Is everything all right? I thought I heard voices." Her ladyship's voice was thick with accent, as usual when she first awoke. Sometimes her first words weren't even English, and she didn't seem to realize it until Deirdre blinked at her. Now she smiled as warmly as she could manage. "Only Beatrix with a question, my lady. Let me get your wrapper, and then I'll fetch a cup of that coffee you like." It didn't produce the enthusiasm she had expected. Her ladyship pulled the blankets higher. Her thank-you was low and soft and mournful. She reached up and wiped at her cheek. Deirdre pulled the belted dressing gown from the chair and set it on the bed. The lady had closed her eyes again, but the fire's light caught on the moisture in her lashes. Hesitating, Deirdre almost reached out. But there was no use in that. So she slipped from the room again and hurried to the kitchen. Monsieur Bisset was in full steam, like a locomotive charging through the room, barking orders at the under cooks and assistants. Deirdre avoided whomever she could, sneaking a cup of the dark coffee and making her escape. Soon, the dumbwaiter would be coming up and down with platters of food bound for the breakfast room. The gentlemen would stir within the next hour, the ladies an hour after that. There would be dressing for breakfast, dressing for the hunt, dressing for tea, dressing for games out of doors, dressing for dinner. She, along with the other lady's maids and valets, would be brushing this garment, pressing that, cleaning shoes and more shoes and the next pair too when they came in muddy. The guests would laugh and gossip and flirt and relax. The staff would hustle and bustle and pray for the week to come to a quick end. Thank heavens his lordship didn't make a habit of this sort of thing. She moved cautiously through the halls, careful not to spill a drop of the scalding liquid. When she finally gained the baroness's room again, she found her in the window seat, her wrapper on and a blanket around her. The girl stared out the window into the thick fog. "Here we are, my lady." Perhaps the bright note felt false, but with any luck it wouldn't sound it. She held out the cup. Lady Berkeley took it with a smile every bit as feigned. "Thank you. Would you be so good as to hand me my Bible?" "Of course." It sat on the bedside table, as always. The gold-embossed letters read LA BIBLE: ANCIENT ET NOUVEAU TESTAMENT. She picked it up, handed it over. And said, for a reason she could scarcely fathom, "It looks old." The lady ran her fingers over the creased leather. "Justin gave it to me when I was ten—when Maman died. I understood so little of it then, but I made myself read, because he said it was important. So I read and grew and understood and believed and now . . . now these pages hold memories along with truth." Yet rather than open it, she set it on her knees and held the hot cup in both hands. Rested her head against the wall behind her. "Was it hard when you came here, O'Malley? From Ireland?" Her hands itched for a task. Her feet strained for the door. But she held her place. "It was a blessing—I daresay one I wouldn't have received had I not had my uncle's recommendation. I send my earnings home to my family, and I know it eases them to have it. It's been hard for Mum since my da died, with the farm mortgaged as it was." She hadn't meant to say so much, had only wanted to sound grateful. She cleared her throat. "Do you miss Monaco?" Lady Berkeley sighed and sipped at her coffee. "My grandfather. And the weather." A smile winked out, disappeared. "But I knew I couldn't stay there forever. I was not a Grimaldi, not by blood. And there was so much unrest—the people revolted in the spring, demanding a constitution. Even as Grand-père placated them, I kept wondering if I was more like them than him—if I belonged on the streets, protesting with the crowds, or if living behind the palace walls was my place. I wanted to know who I was. So I asked Justin to help me find out, and . . ." "And here you are. Home." How nice it must be, to go on a search for answers and find all this. Life didn't turn out so fine for most of them. "O'Malley." The baroness shifted, set the Bible on the seat beside her, and met Deirdre's gaze. "Please don't pretend. Your position is safe, I assure you. You needn't put on this front." Deirdre's back went stiff. "Sure and I don't know what you mean, my lady." "I know you don't like me—and you don't have to. I'm not . . . I'm not what you all want your baroness to be. That is mine to accept." She set the cup on its saucer, the saucer on the Bible. "But duplicity I will not." Deirdre knew not what to say to that, what she was meant to do. Any response she could make may well explode in her face. So she stood there, held the baroness's gaze until it felt disrespectful, and then lowered hers to the floor. "Do you wish to dress yet?" The lady stood, folded the blanket and put it back on the foot of the bed, and moved behind the screen. Deirdre handed her the shift and bloomers, the corset. Then came the dress. Her ladyship slipped it on and then stepped out, back to Deirdre, hair held up out of the way. She made quick work of the row of the buttons, and of putting up her ladyship's hair a minute later while she sipped her coffee. Then the baroness crossed back to her window, her Bible, and dismissed Deirdre with a few quiet words. The feeling of freedom she usually felt at the "That is all" didn't come as she stepped into the hall. She felt only the certainty that she should have woken the lady from her nightmare. Maybe the morning would have gone differently if she had. She paused at the break in the paneling that would open to the service staircase. And what would Mum say if she saw her now? Or Da, who had always called her his sunshine? She didn't feel so sunny anymore, hadn't since his death. But he would be pained to know it. He would be disappointed in seeing the resentment always a-boil inside her. Clouds came, he had always said. Sometimes they brought rain to nourish, sometimes hail to destroy. Some years were fat, others lean. She closed her eyes, heard his voice in her heart, so deep and sure, even as the fever consumed him. "Crops fail, DeeDee. People die. The bad comes, to one and all. What matters . . . sure and it's what we do with it. That's what makes a man strong or weak, good or bad. Not the outside—the in." The in. She pressed a hand to her ribs, where her heart beat a painful accusation. Aye, he would be disappointed in what she'd done with it. He'd look at the baroness and see a girl too long lost, not a pampered princess undeserving of all she'd been given. He'd see a hurting soul, not a pretender. But he wouldn't have made a fuss about it. He just would've said to Mum, "Bake an extra pie, Bonny-my-bonny. We've a neighbor who needs the smile." She straightened her shoulders and pivoted on her heel, knowing what peace offering she could give. A dash down the main stairs, a turn toward the library. A book. It couldn't make them friends, but they needn't be at odds. Stepping into the library, she moved to the right, where his lordship kept the novels. The young ladies had been talking last night about Jane Eyre, and the baroness had confessed she had never read it. Lady Melissa said Lord Whitby had a copy in his collection, though, and Lady Berkeley's eyes had danced. Deirdre would find it, deliver it to her room. The door clicked shut, and a hum as slick as darkness thrummed through the room. "Well, well. You have a taste for literature too? You are a woman of endless allure, Deirdre O'Malley." Though she wanted to jump, to spin, to face the devil so she could read his intent, she restrained herself. Continuing to the shelf, she took a deep breath to ensure her voice came out calm and even. "Good morning, Lord Pratt. I was unaware you passed much time with books." How could a laugh, quiet and short, sound so very menacing? "No. But the room I find intriguing. Has it always been Whitby's favorite spot in the house, do you think?" His voice stayed on the other side of the chamber, muffled as if he spoke toward the opposite shelves rather than her. Good. Whatever his intent, perhaps she could go about her business without ramming into it. "I should think so." Her eyes perused the titles, alphabetized by author. The D section was before her. She needed the Bs. To the side? No—drat. She craned her neck upward, to the row of shelves well above her head. "I'm glad you found me. I need the journal back, my lord. Her ladyship has turned her room upside down looking for it. She thinks it lost." "Let her think it so, lost in travel. I'm not finished with it." Was his French as bad as all that? "Could you make out none of it? Whether it supports the claim she's his or not?" "She's his daughter." His satisfied hum made her feel sick. "I hear she spends much time in here too." Deirdre shot a look over her shoulder at him. He stood with his gaze on a row of matching tomes. Should she press the point of the journal? Much as the thought of leaving it with him made panic nip, he wouldn't budge. She strode to the wheeled ladder and pulled it to the proper shelf. "Not this time of day, if that is your hope. My lord." "Not at all." He picked up a decorative book end, flipped it in his hands, put it back. "I seem to have gotten off on the wrong foot with her when we first met. But you are with her most of the day—tell me, what does she want in a man?" Deirdre climbed up the first few rungs, her eyes scanning for Brontë. "She's mum about such things, my lord, even with her cousins. But I can tell you she reads academic texts as often as novels, in assorted languages. I've heard them discussing scientific papers a time or two, even." No Brontë—neither Charlotte nor Emily nor Anne. She pursed her lips. They had all begun with pen names, hadn't they? His lordship must have early editions. Bell—that was it. She climbed up farther. The ladder shook beneath her. Gasping, she gripped the sides. "I am not interested in her reading material, my lovely. Tell me something useful." She glanced down only once at his stormy black eyes. "It is useful. She takes great interest in such things. And faith, she is all the time talking of her faith." He hissed out a breath and grabbed her right ankle. "You expect me to discuss religion with her?" The more he pulled on her leg, the tighter her throat went, so that she could barely croak out, "Horses. Automobiles. She wants to learn to drive." Her foot slipped off the rung. He chuckled and set it back on. "Better. And?" And his fingers went terrifyingly gentle on her ankle. She pulled it away under the guise of going up one more rung. "That is what she speaks of. Horses, cars, books." "How very dull she would be, were it not for that alluring face, figure, and fortune." Deirdre spotted Jane Eyre by Currer Bell and grabbed it. Though when she glanced down again, she saw him leaning against the ladder like a crocodile on the bank—or perhaps she had paid too much attention to Lady Melissa's reading of Peter Pan the other evening. He offered a patronizing smile. "Do you really think you can climb away from me?" Before she could form a response, he grabbed both her legs and pulled hard enough to yank her from the ladder. She tried to bite back the scream, tried to hold to the rungs with her free hand, but in vain. Before she could discern exactly how it happened, he had an arm clamped around her waist and pressed her to the bookshelf. Struggling was no use, but she averted her face—and caught a whiff of a distinctly floral perfume. The same too-strong scent that Lady Catherine wore. She had a feeling she knew with whom the young lady had been dallying last night. His lips found her jaw, his other hand turned her face. Try as she might, she couldn't hold back a whimper when he kissed her, when trying to twist away accomplished nothing but him pressing her harder to the shelves. She managed to turn her face again, at least. "Please, my lord. You promised. You promised if I gave you the information you wanted, you wouldn't—" "DeeDee?" The door slammed open, and Hiram charged in. "I heard a scream. Did you fall? Are you . . . ?" She squeezed her eyes shut against the horror on his face. Pratt had the gall to laugh again. "She did, as a matter of fact, but I was fortunately here to catch her." He backed away, tweaked her chin. "Tread carefully, old girl," he murmured. Then louder, "No harm done, I think." He whistled—whistled—his way out of the room. Deirdre didn't move, didn't open her eyes as the door clicked and footsteps hurried her way. Hiram's arms came about her. Gentle, warm. Comforting. "What happened, Dee? Did he hurt you?" His hand soothed her back where the shelf had bit, his lips settled on her hair. "Tell me." Too soothing. Too comforting. She let her forehead rest on his shoulder one moment more, and then she eased away. "It's nothing, Hi. I fell." Sorrow shone from the eyes usually bright with laughter. "I know you better than that, Deirdre O'Malley. He had his hands on you. He was—" "He kissed me—that's all." She spat it out in a gush, praying he would leave it at that. But he knew her too well. He stroked her cheek as Da had used to do, brushed his thumb over her lips as no man ever had. "'Tisn't all, Dee. It never is with men like that, who think they have the rights to whatever they want." A shudder overtook her. She knew it. It was why she'd been hoping and praying he wanted information more than he wanted her. "He didn't hurt me." "This time, praise the good Lord above." He leaned in, kissed her forehead. "You're too beautiful for this world. This place in it, anyway, where the fancy lords can treat you as naught but a plaything." He let his arms fall and took a step away. "You should have stayed in Ireland. Married a farmer—a big burly one that could fight off any what looked at you crossways." "Oh, Hiram." He made a muddle of her. And she couldn't even resent him for it. "It isn't so bad. It's a good house." It's why her uncle had recommended her here, and why she always filled her letters to him with naught but the good things about it. "It is." Determination lit Hiram's eyes—and lit panic in her stomach. She knew him too well, too, and grabbed at his arm. "No. You can't be telling his lordship. It'll only make Lord Pratt angry, and he'll no doubt find a way to take it out on us." She shook him, though it barely moved his arm and certainly didn't dim his gaze. "Promise me." "For now." He said it easily, without relenting at all. He cupped her cheek where Pratt had pushed it and made her forget the pain. Leaned down and brushed his lips over hers so softly she couldn't remember the bruising embrace. Then he stepped away again and held out his hand. "For now. But I'll not stand by and let him hurt you. I can't." Deirdre stifled a sigh and slid her fingers into his. Just for a moment, until they left the library. Then she'd pull away. Then she'd put the walls back up. Because Pratt would hurt her, before it was over. He would have his way, whatever that way was, and she would pay the price. But sure and she wouldn't let Hiram pay it with her. # Thirteen Tempesta thundered into the trees, leaving Brook little choice but to laugh. The horse wasn't after the hounds, nor the fox—she was after the run, which suited Brook fine. A tug on the reins brought her down to a walk, the better to draw in a breath and watch the sunlight shaft through the reddening leaves. In the distance, she could hear the shouts of the others. Her cousin Ram had led the way this morning, promising adventure with a wink aimed at Thate. If the whispers she had overheard in the hallway were correct, said adventure would involve sending Regan off with said young man, giving him a chance to propose. Brook heard Regan's laugh now, sweet and too near. Thate's murmur answered, a low thrum she couldn't make out. They must have broken away from the others. And it certainly wouldn't do for them to come upon her. She urged Tempesta behind a thicket, dismounted, and murmured French nothings into the mare's ear to keep her still. Crunching leaves, snapping twigs, snorting horses. "Are you certain you saw it come this way?" Regan's voice, breathless and excited. Who would have thought that staid Regan would be so eager on a hunt? "I am all but sure. Through here, I think. A little farther and we shall have it cornered." A little farther, and they would come out into the clearing by the duck pond—as perfect a spot for a proposal as any girl could dream. Brook shared her smile with her horse and rubbed Tempesta's nose. "I want them to be happy. Together," she whispered once they had moved beyond her hearing. "I want to believe it can be." And that it could last. That death would not snatch one of them too soon, that life would not tear them apart. It was surely possible. It had to be possible. She could think of no examples, but if she couldn't believe it then she might as well go back to her window seat and the grey, foggy mood that had enveloped her that morning. Non. She would not let the nightmare-induced ennui overtake her again. "Come, Tempesta. Viens." After mounting again, she wheeled the horse around and headed back the way she had come. Which way had Justin gone? Hard to say—he and Pratt had been insulting each other all morning and were no doubt now in a heated race to nab the wily fox. Her lips tugged up. Entertaining as it had been to listen to their repartee, she was not about to get in their way. She had joined the hunt for the riding, not the actual hunt. "Good morning, cousin." Tempesta took the last step into the clearing nearer the house, and Brook smiled a welcome for the cousin she didn't know so well. "Lady Catherine." "Kitty, please." She sat atop one of Whitby's milder horses, her knee up against the sidesaddle, her hat at a jaunty angle. Blond curls spilled over her shoulder. Did they look alike? They must, on the surface. Blond hair, green eyes. Her mother must have inherited it from the Rushworth side and passed it along to Brook. Though Brook never felt the same confidence in a crowd that this cousin exuded, nor did her wit lend itself to the clever-but-biting conversation Lady Catherine had apparently mastered. Brook couldn't quite decide if she found it entertaining or off-putting. Lady Catherine's brother sat the horse next to her. He greeted her with a nod and a quiet, "Lady Berkeley." "Lord Rushworth." Brook hadn't formed much of an opinion on him at all. Half the time she didn't even notice when he was in the room. Tempesta fell in alongside them. "I haven't seen the fox or the hounds this direction." Lady Catherine laughed, practiced and perfect. "Perhaps not, but Lords Pratt and Abingdon came tearing through, and that was enough to draw me." She grinned, making Brook lean toward liking her. "Though they seem to have vanished, so perhaps I ought to go back to the house and wait for Lord Worthing to arrive instead. I must say, cousin, your aunt has succeeded in gathering England's finest to come and meet you." Brook smiled again and looped the reins lazily through her fingers. "It has been a bit overwhelming, I confess. Though I'm glad to have met you two. You live near, do you not?" "An hour or so away, depending on the roads. It was during a visit to our parents that your mother met your father, you know." Catherine's green eyes looked sharp as flint as she scanned the area. "Mother used to tell stories of Lady Whitby's fame in London—she apparently came to our home for some quiet after her first Season. Well, I suppose she wasn't Lady Whitby then. Though even after she wed your father, the men still hounded her, it seems. Quite inspiring." Because Catherine laughed, Brook smiled. Though she remembered too well the trouble it caused Maman to be hounded by men, and she couldn't imagine finding it amusing. Lady Catherine leaned across the distance between their mounts, her eyes sparkling. "Can I tell you a secret? Uncle Henry never got over her—he has been hiding in India all these years, mourning first her marriage to Whitby and then her death." Rushworth shifted in his saddle, his brows pulling down half a degree. "Kitty. You oughtn't to gossip about Uncle." His sister waved that off with a laugh that sounded like silver bells. "If Uncle doesn't want to be gossiped about, then he should have taken more care. Everyone knew he was in love with Elizabeth. Mother said some even whispered that they . . . But of course that's nonsense. Your mother would never betray your father, even if he was away so often during their first years of marriage." Brook looked for evidence of cattiness in Catherine's tone but found none. Still, it pierced to think of people whispering so about her parents. Her father had lost enough. Thunder roaring, lightning sizzling. Darkness all around. Brook shook her head against the impressions of the dream. Called to mind the words she'd read in Thessalonians that morning. "Ye are all the children of light, and the children of the day: we are not of the night, nor of darkness." "I'm sorry, cousin." Catherine's horse shifted and pranced, ending up nose-to-nose with Tempesta. The horses greeted each other with friendly nickers. The lady offered a smile, soft and regretful. "Cris is right, I shouldn't have brought it up. I ought to know better than to repeat anything our mother told us, God rest her soul. She was a jealous creature, and she remembered Cousin Elizabeth through that lens." "Catherine," her brother said again. The lady huffed out a breath and sent her gaze heavenward. "Am I not allowed to say anything, Cris? It's no secret that Mother was jealous!" Rushworth pressed his lips together but said no more. Brook conjured up a smile. "My father told me it was through Pratt's family that he met my mother." Frustration with her brother apparently forgotten, Catherine beamed and turned her horse around again, motioning them all forward with a nod of her stylish top hat. "Oh, you'll find that the peerage is rather small, really. The late Lord Pratt was always close with both our father and uncle. The story goes that when Cousin Elizabeth came for a visit, she tired of our mother's less-than-warm company." Here she darted a look at her brother, though Rushworth made no response. "Lord Pratt came to visit Father one day and mentioned that he had a cousin about her age at Whitby Park—your aunt. So Elizabeth came here to call, she met your father, and Uncle Henry never forgave Lord Pratt for it." The last part she delivered on a laugh, tossing back her head. "Mother always said that had Henry not just got back to India when Pratt was killed, he would have been investigated for it." Brook's brows furrowed. "Pratt's father was killed? Accidentally, you mean?" "I'm afraid not." Rushworth's tone was several shades more somber than his sister's had been. "He was shot in a back alley of Whitby." Catherine nodded, her eyes alight despite the serious turn of her mouth. "I was no more than two at the time, but Cris says he remembers a bit of it—the whole region was in an uproar over two such high-profile losses so close together. It was only a fortnight or so after your mother." Brook directed Tempesta around a fallen log jutting out and then reined her in when the sound of pounding hooves and braying dogs reached her ears. A moment later the hounds tore by, Justin and Pratt hot on their heels, neither so much as noting the trio of horses still within the tree line. Brook had to smile. Catherine sighed, her gaze on the backs of the men. "Pratt was nine when it all happened. He still speaks, sometimes, of how he misses his father." Despite her dislike for him, pity stirred at that. "It is no easy thing, losing one's parent so young. What of his mother?" "She was ill for years—consumption—before passing away a year or so ago." Catherine sighed again and reached up to touch the hollow beneath her throat. "Sometimes I think he would rather let the grief drown him than be comforted by those who still love him." "That would be Kitty." Rushworth's tone was amused . . . or perhaps mocking. Brook wasn't quite sure. But Catherine sent him an easy, teasing glare. "I'll be Lady Pratt within the year, Crispin, mark my words." Then she grinned and turned her horse back toward the house. "Unless I toss him over for one of the future dukes available. Lady Regan seems to have thoroughly snared Worthing, but they aren't engaged yet, so there is still hope. Although I must say Lord Abingdon is every bit as handsome. Unless you've a claim to him, cousin?" "Only of friendship." It took a bit more effort than it should have to smile back at Catherine this time. Which made little sense. Brook had long ago banished the childhood dream of finding a happily-ever-after with him, but all these questions made her realize she wasn't sure what she would actually do when he declared his intentions for some young lady . . . perhaps even one of those here. Her cousin held out an arm through the space between them. "I am so glad you're home, Brook. It will be a delight to get to know you, to have another young lady nearby." Brook stretched out, too, to clasp the elegant fingers in her own. Regan and Melissa would soon return to London, after all. It would be good to make other friends. "Likewise, Kitty." Catherine smiled and nodded at Brook's wrist before releasing her. "What a lovely bracelet. Rubies, is it? And diamonds?" "Mm." She settled her hands on the pommel again and touched a gloved finger to the gems. "It was my mother's." "I thought it must be. Another contention of my mother." Catherine sent her eyes heavenward. "She was all the time claiming that Elizabeth inherited jewels that ought to have gone to Father. Though I daresay most of them are from the Brook side, not the Rushworth." She paused as if waiting confirmation, but Brook had to shrug. "I am afraid my father doesn't remember the history of many other than the ones he gave her." "Ah, it's no matter." Catherine pulled her mount to a halt and cast her gaze northward. "I think I'll have a look at Delmore while we're out here. Will either of you join me?" "I suppose I shall." Rushworth said it on a sigh, though. "My lady?" Brook shook her head. Though the boy-Pratt may have deserved pity, the man still made her uneasy, and she had no desire to go gawk at his home—Whitby had pointed it out once, and that was enough. "I think I'll go find my father. But I've enjoyed talking with you, Kitty. My lord." The siblings nodded and said their farewells, and Brook aimed Tempesta back toward the stables, the thought of their neighbor irritating her more with every hoof-fall. How could Catherine be in love with him? She surely saw beyond his handsome face, saw the way he looked at all the young ladies as if they were naught but pounds sterling and playthings. He was exactly the kind of man Maman had fumed about, the kind who thought women were good for nothing but satisfying men. The kind of person who valued nothing but himself. He was an arrogant, self-absorbed reprobate, and he didn't deserve the happiness Catherine would try to bring him. And how could a man who had a lady like her cousin waiting keep looking at Brook as he did? Irritation sizzling its way to anger, Brook dismounted at the stables and handed the reins to blank-faced Francis. It wasn't that Pratt liked her, that she knew. It was as Whitby had said that first morning—he wanted what was theirs. A piercing whinny at the end of the aisle drew her. She shouldn't approach Oscuro now, when her blood was high. Horses were too sensitive to mood. Still, she strode down the hay-strewn aisle until she stood before his stall. Oscuro snorted and kicked at the door. His leads were snapped on, anchoring him between the posts—they must be preparing to groom him. Brook stepped forward and opened the gate. He snorted again and tried to toss his head, whinnied low and pleading. "Je sais. I know." She offered her hand as she did every day. Sometimes he tried to nip. Sometimes he ignored her. Today his nostrils flared, and he turned as much as he could to look at her. She moved to his side to make it easier. "You will run soon, mon ami. Fast as the wind, free as the birds." Slowly, slowly she reached for his nose. The first stroke felt like victory. The second like fate. "You will see. There are boundaries—there always are. But you can find your place within them. Learn how to live within a fence but let your spirit soar." She rubbed up his nose, down, and then along his cheek. "Your sister has learned . . . but you're not like your sister, are you? I understand that too. You can look like another and be totally different. And do you know what?" She leaned closer, slid her other hand down his graceful neck. "That is as it should be. I don't want to make you Tempesta. I want to help you be the champion you were born to be. Fast as the wind. Free as the birds." The hand on his nose had stilled, and he nudged it. Sunshine scattered the last of the clouds the nightmare had gathered over her. She obliged Oscuro with another stroke. Her father eased up beside her. She hadn't heard him approach, but his presence didn't startle her, nor did it earn any acknowledgment from the horse. "Progress." Pride colored his tone. He reached, not to try to pet the horse, but to pat her shoulder. "I thought you were on the hunt." "I was. I spoke for a while with Lady Catherine and thought I'd come find you." "Ah, the Rushworths." His arms went to their usual position, hands clasped behind his back. "I hear Monaco is pleasant this time of year. We could plan a little trip—that left tomorrow." She grinned, gave Oscuro one more rub, and stepped aside for the grooms who approached with brushes and hoof pick. It must be Whitby's dislike of Henry Rushworth that made him wary of the whole family. "Perhaps in the spring. Once I have Oscuro ready for the races and have convinced you to buy a roadster." He touched a hand to her elbow to usher her toward the exit. "Hmm. That may aid us in escaping the dreaded Season, but it doesn't help with this infernal house party." The door came into view, and through it, the gleaming silver paint of Justin's Rolls-Royce. She grinned. "We could liberate Justin's car. Look at it, sitting there gloomy and ignored." Whitby chuckled and led the way out into the sunshine, his eyes on the trees in the distance and the horses emerging from it. "Perhaps later. I think we had better not miss this show—there are Regan and Lord Thate." Brook paused once the sunshine could envelope her and lifted her brows at her father. "You know." "Ram spoke to me this morning." He nodded toward the garden where the married ladies had congregated, Aunt Mary presiding. "He fully approves the match. Mary will not." "And you?" His eyes smiled, though his lips only hinted at it. "He makes her laugh, shakes her from the comfortable. She reminds him there is life beyond the racetrack. They will suit well." She linked her arm through his. "Well said." "And well done, it seems." He nearly grinned as he watched the goings-on in the distance. Thate all but leaping from his horse at the garden's edge, reaching for Regan, and swinging her down and around. Even from the distance, she could see Aunt Mary go still and could well imagine the wariness in her eyes as the new couple approached her. Thate gestured. Regan clasped his arm. Whitby chuckled. "One . . . two . . . three." Aunt Mary crumpled to the ground. Heaving a happy sigh, he nodded. "There. All is as it should be. She will come around, and your cousin will be thrown the most obnoxiously extravagant wedding this side of Buckingham. Let us pray she does the planning at her London house and doesn't drag all the nonsense here." Brook laughed and then turned to the driveway when a plume of dust appeared. "Our tardy guests?" Her father nodded when a car came around the bend. "It must be. Thate acted not a moment too soon." Though she half-expected him to lead her away in all haste, to let someone else greet the newcomers, Whitby instead lingered outside the stables amidst all the other parked cars and carriages. "You were talking with the Rushworths, then?" "Mm. Kitty was telling me of how my mother came to meet you. Well, that she came here to call on Aunt Mary." Brook would never tire of seeing the way his eyes went soft and warm at the mention of his Lizzie. Of the way his lips twitched. "Mary was out that day. I had seen your mother before, in London that Season—though only from a distance. She was always surrounded by crowds of adoring beaux, and I . . . I thought it all ridiculous, honestly. All that hubbub over one lovely face." She couldn't hold back the breath of laughter. "I am utterly shocked." Half a grin emerged. "I had to be in Town that year, Mary was just betrothed to Ramsey. But I had no intention of playing those games. Then when I walked into the great hall as she was leaving her card . . ." His gaze went distant, awe-filled. "I was stunned. Not just because of her beauty, but because up close, without the crowds, I could so easily see that she was a woman of heart." "And what did you do? Let her go, until the next time, when Aunt Mary was home?" "No." He chuckled and cast his gaze to the side of the house and the maze cut into the shrubbery. "I assured her my sister would return in but a few moments and asked her to walk the maze with me to pass the time. Then pretended to get lost." "Cunning." He tapped a finger to his temple and, when the arriving car pulled into an open space a fair distance away, led her that direction. "I had that to my advantage, if nothing else. We strolled, talked." "And fell in love?" It wasn't hard to picture it, not with that light in his eyes. "It didn't take long. She and Mary became fast friends, saw each other almost every day, and I . . . I think I knew within a week, though I couldn't fathom she would feel the same. Miraculously, though . . ." Brook patted her father's arm, even as her eyes tracked the two heads climbing from the car—one red as fire and the other dark as midnight—and the servants' carriage that followed behind, headed for the rear. "It is no miracle." "Love is always a miracle. Especially in this world." He waved at all that was his—the grand house, the grounds, the extravagance. "I pray you find it someday, Brook—though not," he added, spinning to her with a scowl, "anytime soon. I've only just got you back. Are we clear?" She was still laughing when the taller of the heads, the dark one, turned their way. It took only a glance to see why Thate had been worried over this future duke, and why Melissa always said his name on a wistful sigh. Lord Worthing could only be described as debonair—handsome, polished, and with a charm that all but knocked her over the moment he flashed his teeth in a grin. "Lord Whitby! Our apologies for our late arrival." He strode their way, hand outstretched. The other half of his our seemed to have disappeared, but Brook couldn't see where she'd gone. She let go her father's arm so he could shake the young man's hand. Unlike most of the other guests to greet them, he actually kept his gaze on Whitby rather than gawking at her. "No need to apologize, my lord." Her father didn't smile now, though he looked pleasant enough. And was probably adding a silent, The fewer the merrier. "We are only glad you could join us at all. I am acquainted with your father, you know." Lord Worthing's smile emerged again and nearly blinded her. He had dimples, even white teeth, and, what was more, seemed genuine in his enjoyment of life. "He speaks highly of you. He and Mother wanted to join us, but they had a few engagements yet in the Highlands they couldn't bow out of." Releasing Whitby's hand, Worthing turned the full force of his smile on her. "And this must be your daughter." "Lady Berkeley, yes." Her father touched a supportive hand to the small of her back. Brook held out a hand, acknowledging the skitter of pleasure that raced up her arm when Worthing took it in his and pressed his lips to her knuckles. "I've heard much about you, my lord." He laughed as he straightened, his fingers still clasping hers. "And despite that, I hope we will be friends. I am certain you and Ella . . ." Here he turned, and his smile gave way to a frown. "I seem to have misplaced my sister." He said it as one might say one had misplaced one's book . . . yet with obvious fondness. Brook could see her red hair over by Regan and Thate, but she decided that might not be the wisest place to direct his gaze just then. "She must have seen a friend." Brook glanced to her father for help, but Whitby was frowning down the driveway. He in fact patted her back and took a step away. "That appears to be a courier. Will you excuse me, my dear? My lord?" "Of course." In proof, Lord Worthing took the fingers he still held and tucked them into the crook of his elbow, beaming down at her. "It must be my lucky day. Only here for minutes, and already I have the lady of the hour on my arm. You are every bit as lovely as I had been warned to expect, my lady . . . and perhaps a bit more besides." The wool under her fingers was fine, worsted. The same texture as Grand-père's favorite jacket—for a moment the breeze felt warmer, the distant voices sounded Monegasque. For a moment, she was strolling through Monte Carlo, the scents of spice and salt in her nose. A princess again. No—a pretender again. Worthing drew her forward. "I've said something to upset you. Please, forgive me. Flattery is our language, but if it makes you uncomfortable . . . Though in my defense, it's hardly flattery when it's true." A taste of laughter tickled her throat, though she let only a small smile escape. "I am immune to flattery, my lord—I grew up in a prince's palace." When she glanced up, she saw his dark eyes had gone serious. And seemed to see far more of her than they ought. "You are permitted to miss it—your father will understand." She very nearly withdrew her hand and fled—a man she had known for all of a blink had no right to see what no one else ever seemed to. But then he glanced toward the side of the house where his sister had gone, where Regan still stood with her arm woven through Thate's, and he came to an abrupt halt. Brook sucked in a breath. Worthing looked down at her with an arched brow and eyes filled with . . . laughter? "I seem to have missed something." She could only stare at him. He must be upset at the woman he was courting attaching herself to another. And his quick stopping had shouted his surprise. Why, then, did his face reflect only amusement? Brook cleared her throat. "We just saw them come back from the hunt together. I haven't spoken to her yet . . ." Worthing put on a lopsided smile and faced forward again, his gaze fastened on the new couple. "I deserve the credit for that, I think. I can't tell you the number of times I caught him scowling at us in London." She could well imagine though, and had to fight back a chuckle. "Oh, Thate wouldn't scowl. Glowering, though—I have found the English to be masters of the glower." Lord Worthing's laugh rang out free and bright. "Bested only by the Russians, I daresay. Or are they ones with proper scowls?" Her very thought from that day in Monaco . . . which made her stomach knot up. "You've the right of it." She looked at her cousin, laughing and grinning in the distance, and then back to her companion. "You're not upset?" Lord Worthing sighed. "Your cousin is absolutely everything I could want in a wife, were I to make a list. And I think we both hoped we would fall in love. But . . ." He motioned toward Regan with his free hand. "We didn't. I've known for a while that the Lord had other plans for us." Brook tugged her hand free of his arm, so she could plant it on her hip. "Why, then, were you still courting her?" Perhaps she shouldn't get irritated with a near stranger. But it was her cousin he had been toying with. Sweet, selfless Regan. And he had the nerve to grin. "Oughtn't you to be chiding her, my lady, and demanding to know how she could dangle me while in love with Thate? It isn't as though I was courting anyone else at the same time. Surely I am the injured party here, not your cousin, who certainly looks happy with how things turned out." The fact that he had a point did nothing to defuse the anger so quick to burn today. It must be the fault of the dream, and the restless night's sleep it had caused yet again. "You certainly don't seem injured, my lord—you seem rather happy as well." Perhaps on another face, the arch of brow would have come off as a challenge. On him, it looked like a jest. "And now it is a crime to be glad that a young lady I care for has found the husband the Lord intended for her?" For a moment, the irritation still simmered. But the longer she held his gaze, the weaker the fire burned. And the more amusing it all seemed. Regan was happy, Thate was happy—and it was due in large part to Worthing inspiring Thate to jealousy. Who knew how long it would have taken him to act otherwise? It seemed no one was displeased with how it all turned out. With the exception of Aunt Mary, of course. Brook relented with a gusty sigh and nodded toward the redhead hurrying their way. "I believe your sister is coming to break the bad news to you." Lord Worthing chuckled and deftly tucked her fingers into the crook of his arm again. "Play along and I shall be forever in your debt—I can never get my fill of teasing Ella." "Play along with what exactly?" Rather than answer, he patted her fingers where they rested on his arm, as if she were a friend he'd known for years. "You'll get along well, I think. You'll find that she's annoyingly optimistic, but we love her anyway." Brook directed her gaze to the distraught girl—she looked to be about seventeen—and could well imagine liking her. There was no clever cunning in her eyes, no line of artistry in her carriage. She looked all brightness and innocence. Except for the concern in her cinnamon eyes as she rushed up. "Brice . . ." "I know, Ella-bell." Reaching out, he slung his other arm over her shoulders, so easily he must do it often. He loosed an exaggerated sigh. "And my heart has positively rent in two. But the Lord is good, and already He has provided me the most beautiful bandage a man could ask for. This is the Baroness of Berkeley, a succor to my crushed spirits." Ella stared at him a moment, agape, and then looked to Brook. This must be his game, though Brook wasn't sure how, exactly, she should play along. Was she to act lovestruck? Before she could decide, Lady Ella rolled her eyes. "Sometimes I can hardly tell when you're joking." Worthing winked at Brook. "It's a gift." "Did you warn her that you're an unabashed flirt, and that she had better not take a word you say seriously?" At that, Brook had to laugh. "I figured that much out for myself, my lady." "Call me Ella, please." Dimpling, she reached across her brother to clasp the hand Brook lifted. "I hope we'll be friends. I've been absolutely dying to hear about your life in Monaco. It sounds so romantic!" "Heaven help us—Ella, you don't need any more tales of romance in your life. Make it out to be a bore, my lady, I beg you, or she'll be running off to the casinos." Ella's eyes widened. "I would never! He's terrible, Lady Berkeley, ignore him. Don't believe anything he says. The only place I would ever run off to is Scotland—" "Hear that? She's threatening to elope, and she isn't even out yet." "Stop teasing, Brice." Ella slapped her brother in the arm, making Brook laugh. Then she looked around him again, to her. "Our mother's from Edinburgh, and we take our holiday every year at her family's lodge in the Highlands. Again, Lady Berkeley, just ignore him." "Don't worry." She found their banter refreshing—not unlike what she and Justin so often shared. "And please, call me Brook." Ella's smile was sunshine. Lord Worthing's was pure mischief. "Well, if you insist, but I suspect it will make your aunt faint dead away to hear me do so. And then you'll be obliged to call me Brice, and she might never recover." She would have laughed again, but Worthing's mirth faded as he looked at something beyond her. Her father, it seemed, though he wasn't coming their way. His jaw was clenched, his hand clutched around a piece of paper, and his course set for the house. No, not the house—the group of hunters just dismounting on the south lawn. She hadn't noticed them come up, though now she swore she could feel Justin's eyes shooting arrows into her. That Whitby was headed his way shouldn't have made alarm race up her spine. Not until he called out for Lord Cayton as he passed. "Oh no." Brook would have run forward, caught her father, passed him by. She would have run to Justin, gripped his hand, readied to hold him up again as she had those few short weeks ago. But her old friend turned his face away from her and strode forward to meet her father and his cousin. Rigidity in every line. Fingers curled into fists. Posture shouting that he needed, wanted no one. The fool of a man. Lord Worthing took her hand off his arm, let it go. But settled his fingers on her shoulder for a moment. "He's the one who brought you here, isn't he? You met in Monaco?" She could only nod, mute. "Everyone knows the Duke of Stafford is ill. Whatever news your father's carrying, it isn't good. Go to him. Even if he pushes you away, go. He needs you." It was all the impetus she needed to go tearing across the lawn. # Fourteen I knew I shouldn't have left." Justin pulled off his muddy boots and handed them to Peters. He needed to change. He needed to pack. He needed to leave, right now. No, yesterday. No, he shouldn't have come at all. His valet grasped the black leather too tightly, obviously as shaken as he and Cayton had been. "He told you to go, Your Grace. It was what he wanted." Your Grace. "Don't. Not yet, please. Please, just . . . let me be me until we get home." Peters turned away, toward his boot brushes. "I'm sorry, my lord." He dragged a hand through his hair. "Don't apologize. You've done nothing wrong." Justin had, though. God had tried to warn him, and he had listened to the duke instead. Yet not, because he hadn't come with any intention of proposing to Brook. And he'd been rewarded by seeing her laughing with who could only be Lord Worthing, her hand on his arm. Then this, moments later. It had been all he could do to escape the lawn before he fell to pieces, in front of her and the man he had no doubt would become her new beau. "You were going to change, my lord." "Right." Here he was standing in the middle of his room, shirtless, wasting precious time. He charged behind the screen and made quick work of peeling off mud-caked breeches. His trousers and shirt and waistcoat were already waiting, and the moment he stepped out in them, Peters was there, boots abandoned, to knot his tie. I shouldn't have left. Shouldn't have come. "Your aunts were there. He wasn't alone." Not like Father had been. And his aunts had each other—not like him. Still. "We should leave the car here and take the train. It'll be faster." To that, Peters nodded. "I can arrange it. You should find Lady Berkeley and your cousin to let them know." "Yes. Thank you." He spun for the door, yanked it open, and nearly collided with the fist Cayton had poised to knock. His cousin's face was pale, and he was still in his mud-spattered riding clothes. But at least he didn't knock on Justin's head in lieu of the door. "I was making arrangements," he said by way of greeting. "Your car will be taken to Azerley Hall, and we'll take the train. It'll be faster." Justin nodded and stepped into the hall. "I was thinking the same. When does the next one leave for Gloucestershire?" "Perhaps Whitby knows." They strode together down the bachelor wing, their strides matching. "Where is he?" "Library, I think." They traveled the distance in silence. Would likely travel all the way home in silence, and that was fine. He needed to think. Within a month, he had lost them both. Father and grandfather. Voices came from the library, soft and familiar. The moment he stepped inside, Brook was there. Her arms around him, her face pressed to his shoulder. Her aunt's lips thinned in obvious disapproval, but Justin closed his eyes against it and held Brook tight. When she was flying his way across the lawn, he only wanted escape. Maybe because he knew how much he needed her, needed this. He could crumble—she would piece him together again. He could refuse to let go, ask her to come with him—she would, despite the consequences. Which was why he knew he had to release her, though he couldn't convince his arms of it quite yet. He needed her, needed her warmth to chase away the chill inside. Her arms tightened around him. "I'm coming with you." "Elizabeth Brook! Ambrose, did you hear her?" "Easy, Mary. She meant we." Justin opened his eyes to find that Whitby had drawn near. He set a hand on Justin's shoulder. "I will bring her. We can be ready within the hour." He had a feeling the offer was spontaneous, and for Brook's sake. Clearing his throat, he set her a step away. Their gazes tangled. "Not today. Come tomorrow, or Wednesday." Temper snapped to life in her eyes. "Non." "You have guests." Whitby snorted. "Mary has guests. No one will even notice we have gone." Lady Ramsey huffed her disagreement. "Don't be absurd. We will end the party early, but we can hardly close the house on a minute's notice." Brook didn't glance at her aunt, just held Justin's gaze. "I want to come with you." Of course she would. Because he was her dearest friend, the closest thing she had to a brother. Swallowing did nothing to banish the lump in his throat. "I know. But this is what I need you to do." Confusion swirled through her eyes. "Why?" Because having her as a sister, a friend wasn't enough—and he couldn't ask her for more, not when she might grant it out of pity. He leaned down and kissed her left cheek, her right. "S'il vous plaît, mon amie. Crois-moi." Trust me. "Justin." She caught his hand and held it for a long moment, obviously debating whether to argue more. Then she let his fingers go. It was what he'd wanted—it shouldn't have broken him all the more. She nodded. "Tomorrow then. We'll be on the first train." Not more than a day behind him, which would give him precious little time to get hold of himself. He looked to Whitby. "And when is the next one, my lord? Do you know?" "Three o'clock." Then they had no time to waste. He lifted his hand, wanting to settle it on her cheek or in her hair or at her waist. To hold fast to hers, as he had a month ago. He let it drop back to his side. "I have to go. Thank you, my lord, for your hospitality. And you, Lady Ramsey." Animosity apparently forgotten, she dropped into a curtsy. "The pleasure is ours, Duke." A hand pressed upon him, heavy and unrelenting. Unable to utter another word, he turned and left, his silent cousin at his side. He wouldn't, apparently, have the journey home to come to grips with anything. He was the duke. And his every step had to take that into account from now on. Their voices a din in her ears, Brook stared at the empty doorway. He had left, just like that. Wrapped up in his own misery and unwilling to let her share it. He had deliberately pushed her away. He needed her—Worthing was right about that—but he wouldn't let her help this time. Her aunt's words came back into focus. "No, we must have the conversation, Ambrose." Aunt Mary grasped her wrist and tugged Brook around to face her. "A young lady of breeding does not offer to travel with a man. Surely you know that. You were raised in a palace, not a . . . a . . ." Words must have, thankfully, failed her. Brook sighed. "You don't understand, Aunt Mary. He has always been a brother to me, my dearest friend, and he is hurting." How could she not be with him when he was hurting? "I do understand, my dear." Her father snorted. "And well you show it." Aunt Mary shot him a glare. "But you are not children anymore, Brook. You must take your reputation into account." "Leave her alone, Mary." Whitby settled his hand on Brook's shoulder, his arm about her back. It was the closest thing to an embrace he had given her. "There is nothing wrong with traveling with one's father to a funeral." Lifting her hands in exasperation, Aunt Mary spun away. "You could not possibly have discussed it before she—" "We didn't need to." He squeezed her shoulder. "These things are understood." Not to all, apparently. "Ambrose—" "She is my daughter, Mary. Let me worry with her. You, I believe, have a wedding to plan." Brook leaned into him, savored the feel of his arm as it slid around her. Even so, it couldn't ease the place gone taut inside. First Justin had left her here, after promising to stay. Now he was pushing her away when he needed her most. She could fight him, fight for him, fight for what she had assumed would always be there between them. But what good would it do if he didn't fight alongside her? Justin stood at the window of his study, high in one of the turrets of Ralin Castle. His thumb kept rubbing at the heavy gold of the signet ring Aunt Caro had given him that morning. The seal of Stafford—the same ring that Wildon dukes had been using with their signature since the first of them, hundreds of years before. It didn't fit. Grandfather's knuckles had swollen with age, and he'd had the thing enlarged. Now it moved all about Justin's finger, up and down, round and round. Uncomfortable. Unfamiliar. Unsuited. What he wouldn't give to be outside on a ride through the familiar hills and dales. Instead, he stood in a somber black suit, trying to ignore the sea of people milling about below, all waiting to offer their condolences. In a matter of minutes, he would have to climb into the sedate coach, leading the procession to the chapel in town. Then another procession to the family cemetery on the far edge of the property, where they would all gather round him again. He turned the signet around. "Justin." Aunt Caro's voice came from the doorway, but he didn't turn around. He didn't want to see her draped in black. "We've only a few more minutes." His nod felt stiff, his body brittle. Like if he moved too much, he could snap in two. He turned, intending to move, to slip past her. But he made the mistake of looking up and saw her in her mourning, and it made a fist form in his gut. "What was it he wouldn't let you tell me two weeks ago? About Father?" Torment flickered over her face, the face so much like his mother's. "Now isn't the time, Justin." She'd said then he should know, he should know how much Father had loved him—something he could use right now, when the world felt so empty. But what did it really matter? He was gone, Grandfather was gone, everyone was gone . . . or maybe it was just Justin who was. Broken. Hollow. "Wait." Aunt Caro held up her hand, palm out. Face twisted. "I think now is the time, actually. I can't watch you do this, Justin. I can't let you turn into him." His brow furrowed as he shoved his hand into his trouser pocket. "Into Father? You needn't worry about that. I'm nothing like him." But Aunt Caro only looked all the sadder as she lowered her hands. "That's my fear. That you'll focus only on William's bad habits and not see his strengths. That you'll try to model Edward or your grandfather when . . . when you shouldn't. When you don't know what their single-mindedness did to this family." Justin lowered himself to the edge of his desk, not taking his gaze off his aunt's face. He'd long known her and Uncle Edward's marriage had been rocky at best. They'd married for love, she had said once, but when she failed to produce an heir, it had soured. But aside from the mistresses he then kept, his uncle had been a decent man. Always working for the good of Stafford—that's what he remembered of him. Aunt Caro sighed. "William . . . William wasn't your father." She might as well have taken the medieval sword from the wall and run him through. Justin couldn't breathe, couldn't move. Couldn't believe it. If he wasn't his father's son, then it meant he had no Wildon blood in his veins, that he wasn't the rightful heir to the duchy. Well, he was—it was a matter of legal name at birth and little else—but he shouldn't be. He was only . . . Aunt Caro's eyes slid shut. "Edward was." The sword pulled out, but it left a gaping wound in its place. Wildon blood then . . . but suddenly that didn't matter as his mind ground into gear. "Wait. You're telling me that my mother . . . No. She wouldn't have. She was—" "She was not to blame." A cynical laugh snorted from his aunt's lips, and she pressed a hand to her temple. "She was only seventeen, she had no idea, no defense—it was Edward. I knew then it was Edward, but still I was so furious, so hurt I couldn't see her pain. I couldn't see what it meant for my baby sister when she discovered she carried you. I . . ." She shook her head. Her lips quavered. "They wanted to keep her here through her term. Deliver the child and, if it was a boy, give him—you—to me to raise. Edward's heir. But I couldn't do it. I couldn't, not then." He felt as heavy as the stone walls around him. "Of course you couldn't. But Mother—" "Sweet Georgiana." Now the pain faded, and her eyes went soft. "It is just as well that I was too weak to save my sister—she never would have given you up. But my refusal forced the duke's hand. He ordered William to marry her. That way, you would be legitimate, a Wildon by name as well as blood. And assuming I never produced a son, the title would fall to your father and you after Edward. The line would be preserved." The line. Always about the line. Justin closed his eyes and shook his head, though it did nothing to make the awful truth go away. Never once had Father hinted to Justin that he had been forced into fatherhood. Even when Justin had all but accused him of being less a man than Uncle Edward—he winced now at the thought of those words—Father had merely grinned and said how glad he was that Justin had inherited all the best traits of the family. "But he loved her." That was a truth he couldn't question even now. "And she him." Aunt Caro folded her hands before her. "A blessing that happened quickly. It could not erase in your father's mind the injustice we had all forced upon him, but he was a good man, Justin. He held it against us, but never you. Never your mother. He loved you both more than anything in the world." Justin pushed to his feet and turned toward the window again, twisting the signet around his finger. In the silence that crept in, he sent his mind backward. Through the years, through the trips to England and home to Monaco. Trying, in retrospect, to find any flicker in Father's eyes. Any unexplained shadows. All he could see was the way Father had drawn Mother into his arms and danced with her when a band of street performers below their window had struck up a waltz. The way they had both drawn him close—he an awkward boy of ten—to cradle the tiny form of Amalie, how they had whispered in his ear that he would be the best brother in the world. How, when his mother and sister died, Father had pulled him closer instead of pushing him away. "We still have each other," he had said. "We at least have each other." He didn't know that he was that strong. Didn't think he could be like his father, not in the ways that mattered. Aunt Caro touched a hand to his arm. "Don't shut us all out, Justin. Don't be like Edward, please. It would break your mother's heart. Break your father's. You're better than that, better than him." Was he? He didn't feel it just now. He didn't feel anything, not even the promises he had stared at in his Bible last night, willing the black words to lighten his spirit. Maybe his mind knew the truth, but his heart was too raw. It had gone numb. He pivoted, shrugging off her hand, and headed for the door and the spiral stairs beyond it. Aunt Caro scurried behind. "Justin!" He ignored her, hurrying past one landing, down toward the next. Maybe Brook would still be inside somewhere. Maybe he could find her and . . . and what? When she had come upon him in the library last night and wrapped her arms around him, it had taken every ounce of strength he had not to press his lips to hers and beg her to love him. Beg her to make him feel alive again. She deserved better than that. She deserved to fall in love, not to be forced to marriage for fear of hurting him . . . and he suspected she loved him too much to say no if he asked. Just not for the right reasons. Not for the reasons he needed. Aunt Caro sighed behind him. "Do you intend to follow the duke's instructions on where and when to travel?" It would mean leaving almost immediately. Fixing things. Building things. Shoring up the holes inside. "I should be back by the start of the Season." His aunt slowed his step with a hand on his elbow. "What of your Brook? Have you considered how it will hurt her if you leave her now? I thought you meant to court her. But if you leave—you could lose her." "No." The oath whispered out, more prayer than denial. "Never. She is my very heart, Aunt Caro. Mon âme." His soul. Her smile softened, lost some of its sorrow. "You have more of William in you than you suppose." He prayed she was right. The silence resumed as they wound down the rest of the turret and joined Cayton and Aunt Susan in the foyer. Together, they stepped outside, into the masses. The sun was too hot. It seemed today, of all days, England's skies ought to have been grey and low and menacing. Instead, summer had pounced on them for one last hurrah, scorching all the mourners in their dull black frocks and coats. Justin's eyes scanned the crowds, looking for the gleaming golden head that would soak in the warmth so happily. She was there with her father, her eyes already on Justin. She didn't offer him a smile—she'd know he didn't want one. But she nodded. And it gave him strength enough to straighten his spine and head for the coach. The services passed in a blur. The church, the graveside. The mourners passed in an even hazier one. Faces he didn't know, names he wouldn't remember. He shook hands, nodded, and even managed a strained smile now and again. Even when they called him Duke or Stafford. When those without a title of their own called him Your Grace. Perspiration trickled down the back of his neck by the time the line had shrunk to a bare two dozen left to greet. That was when Brook appeared before him, on her father's arm. Whitby shook his hand and gripped his shoulder in one strong, quick move. He said nothing, just moved on to Justin's aunts and cousin. Brook's fingers somehow became tangled in his, though he couldn't be sure which of them had reached out. He held on and used them to pull her closer. Not as close as he would have liked. And then whispered, in Monegasque, "Say my name." She squeezed his fingers back. "Justin Wildon." Soft J. Long U. Silent N. As it was meant to be said. One knot of the pain loosened, and he felt his shoulders relax. "Brook. I will have to travel." The shadows in her eyes belied the understanding nod. "I thought you might. To where?" The names had been swirling around his head incessantly. "Canada and the Caribbean to start, so I can make it home again for Thate's wedding." His friend had looked almost apologetic as he shared his good news yesterday. "Then Africa, India." Her eyes emptied of emotion, the way they did when she fought for composure. Her shoulders seemed to have absorbed the tension that left his. "When will you be back for good?" His throat ached. "In time for your debut." "Seven months." She drew herself up taller, donned the invisible cloak of the Grimaldis. "It has never been so long." No, even when he was at school, he'd taken his holidays in Monaco. "I will be home for the wedding though. And I'll write. Tell you stories of my adventures. 'Justin Crusoe,' perhaps." "'Around the World in Two Hundred Days.'" Her smile was but a flutter, quickly gone. He lifted their hands and pressed his lips to her knuckles. "Pray for me?" "Every morning. Every night. Every noon." She raised up on her toes and kissed his cheeks as she always did. How he wished it were hello-again instead of good-bye. He gave her fingers one more squeeze. "Save your first dance for me." "C'est la tienne." He smiled and let her move to his aunt. Someday, God willing, she would be his, not just a dance. The smile faded when Pratt stepped up. Neither of them extended a hand. Pratt smirked. "No worries, old boy. I won't let her get too lonely in your absence." Justin bit his tongue. Someday, he would level in a fist in the reprobate's nose—and enjoy every bruised knuckle he earned. Darkness blanketed the house. It had been late when they got home from Ralin, later still by the time Brook bade her father good-night and retired. She had tried to sleep, tried to rest, tried to put aside the fear that nothing would ever be the same again between her and Justin, though she couldn't think what had caused the distance between them. Why he kept pushing her away. She wished her mother were here, to give her advice. Instead she had found only thunder behind her closed eyes. The lightning had flashed, the panic had nipped. The darkness had overwhelmed her. What was it about that infernal dream? A storm, but never any other details. Just impressions, fuzzy and vague and all the more frightful for it. She shivered and pulled her dressing gown tighter, holding the candle out before her so she wouldn't wake the house with the flip of electric switches. She had already tiptoed past Whitby's door. At her mother's, she paused. But no, if she went in there, her father might hear her. No reason to wake him. There were other places in the house to find her mother. Usually at the end of the corridor she turned for the stairs leading downward. Toward the outside, the dining rooms, the library. But according to her father, Mother's favorite room had always been her upstairs salon. And so Brook took the stairs going up, her candle providing scanty light in the dark stairwell. Shadows flitted to and fro in the room she let herself into. Tree limbs swaying before moonlight, night creatures in the skies. Despite herself, she shivered again and headed directly for the oil lamp, ornate and feminine, sitting upon the well-worn desk. Once she'd lit it and its cheery yellow glow illumined her corner of the room, her shoulders relaxed. The chair was small and dainty, woman-sized. Its padding had worn thin, evidence of how often her mother had sat just here, where she now did. Perhaps she had even brought Brook up with her when she was a babe, let her lie on the Turkish rug and coo while she attended her correspondence. Perhaps they had been together here, before it all went to pieces. She trailed her fingers over the embellishments carved into the edge of the desk. This, much like her Mother's bedroom, had been left unchanged aside from cleaning. She'd already poked around enough to know that the top center drawer contained pens and ink, wax, a seal. Paper was stored in the bottom right, correspondence she had saved in the left. Brook bent down and pulled open the deep right drawer—and realized she'd been wrong. What she had thought was a stack of paper was actually old letters. Well, she didn't know what she would have written to Justin yet anyway. She reached for the stack and pulled them out, thumbed through. The name Henry Rushworth was on enough of them to catch her eye, so she flipped one open at random. The handwriting was bold and bare. Well, Lizzie, I've arrived in India, and it's hot as blazes. You would hate it, I daresay . . . She scanned through descriptions of heat and insects, of the throngs in the marketplaces and the spice in the food. Of finding a bungalow to set up house in and locals to staff it. I'm fortunate to have O'Malley with me—I'd never trust the locals to fix my tea. At that, she looked up with a start. A different O'Malley, or was he related to Deirdre? She had said her uncle had recommended her here, had she not? Was this he? More scanning seemed to indicate he was Rushworth's valet . . . No, batman, he called him. The military equivalent. She read on. You should see the fabric they make here, Lizzie. Stunning, simply stunning, with beadwork that would put Paris to shame. I've got some to send home to Mother and Rush's wife . . . would send some to you, too, if I didn't think that husband of yours would dash around the world to put a fist to my nose in thanks. I hope you're happy there, Liz. I do. Though if ever the northern climes grow too harsh . . . Brook shook her head and flipped to another letter. Apparently her father had reason to remember this Henry as he did. The tone may not have been that of a man trying to lure a woman away from her husband, but it was certainly that of one with regrets, and whose affections were no secret. She glanced through a few more before she came to the last one by date. Just a few months before her mother's death. The locals have stories that would make the hair stand up on the back of your neck. Ancient curses, angry gods, marauding tigers . . . Perhaps when I'm home for leave, I'll tell you a few. I hope you'll see me, if only for tea. If he'll let you. I miss you, Liz. I know you're happy with your choice, that you'll have his babe any day now. But I miss you. Brook touched a finger to a telltale dried water drop that smeared the last word and wondered if her father had allowed a reunion. With a sigh, she gathered the letters to take with her to her room and stood. She had to rest, somehow or another. Tomorrow she intended to put her weight on Oscuro and see how he responded. Perhaps a wild horse could take her mind off all the things she couldn't change . . . and the ones she wished with all her might would stay the same. # Fifteen TWO MONTHS LATER NOVEMBER 1910 Thunder roared. Lightning sizzled. Brook loosed a laugh from the depths of her throat that felt a little bit mad. They should ride away from the storm, back to the safety of stables and hearth. Instead, she let Oscuro gallop toward the thunderheads coming in off the sea, bringing darkness hours too early. She braced her feet in the stirrups, rose off the saddle, and leaned into him. He soared over the fence and kept on flying. Fast as the wind. Free as a bird. The rain greeted them in another half mile, along with the property edge. Oscuro whinnied a protest when she reined him in, but he slowed, stopped. She checked her watch. "Faster even than yesterday." He would trounce all the other horses at the races in the spring—assuming she could convince him to let someone else onto his back. She could see Delmore from here, the sprawling maze of it. And, if her eyes and the rain didn't deceive her, the Rushworth carriage pulling away. Her lips tugged up. Would Kitty be there with her brother, trying for whatever unfathomable reason to convince Pratt to marry her? Perhaps they would stop at Whitby Park. Perhaps even stay the night. Another crack of lightning struck to the west, thunder tripping over it for its turn. She had hoped she and her father could take their drive into Eden Dale this afternoon so she could post her letter to Brice and Ella, and another to Grand-père, but it seemed that would have to wait for tomorrow. A visit would be worth the change of plans though, if the Rushworths decided not to chance the muddy roads. "Back we go, boy-o." She said it in her best—albeit poor—imitation of Deirdre's accent. The horse gave an obliging shake of his head, coiled his muscles, and prepared to fly homeward again. The storm raced them. Fat, cold raindrops struck her as the ground soaked up the torrent and turned to mud. For Oscuro's sake, she pulled up on the reins. If he slipped and fell—non, not on her account. Better to let the weather win and suffer the drenching. Lights were on in her father's study, and she saw his silhouette in the window when Oscuro trotted over the lawn. Waiting, as he always did, to make sure she came safely home. Most days, he did his waiting at the stables, so he could congratulate her on the day's progress. He wasn't quite so mad as she, though, when it came to the rain. Another glance, this time upward, and she saw the light on in her own room, and another silhouette. Deirdre stood at the window with hands on hips, and she gave a shake of her head before she turned away. No doubt muttering in her brogue about mud and wet and cold—but she'd be drawing a hot bath and laying out a warm change of clothes. Brook wouldn't claim her lady's maid as a friend, but they had reached a truce. Deirdre served her well, without pretense, often displaying consideration that took her by surprise. Other than a couple cups during the fateful house party, the maid hadn't managed to secure her a decent cup of coffee, but her quiet "It isn't me, my lady, nor is it the chef" had been all the conversation on the matter Brook had the heart for. She had been sending a few extra pound notes to the O'Malley farm every week—she hadn't told Deirdre she was doing it, nor had she mentioned it to anyone else on the staff, but her father had approved that use of her allowance. He had given her that proud look again and had patted her shoulder. One of these days, he would give her an actual embrace in those moments when he clearly wanted to. One of these days, she would form her lips around Father as she so often almost did . . . then couldn't. She cast another look at the closed-up carriage house where the new roadster hid, and then toward the village. One of these days, she'd be able to leave a letter on the table to be posted with her father's correspondence and trust that it wouldn't still be sitting there, alone, after his had been taken. The thunder laughed at her, mean and mocking. As it had in the dream last night. Oscuro slowed to a walk as they crossed the drive and gave his head a shake. No sign of the Rushworth carriage, which brought a twinge of disappointment—though she could hardly blame her cousins for seeking their own hearth on such an evening. Brook patted Oscuro's neck and dismounted, her boots squishing an inch into the muck. A disgusted noise slipped from her throat, and a shiver of cold skittered up her spine. "You should experience a Mediterranean rain sometime, boy. Warm even in November, by comparison." He nickered his agreement as she slid the reins over his head and led him toward the darkened stables. The nicker turned to a high whinny when she stepped inside, and he pranced backward rather than follow her in. "Shh. Calme toi, Oscuro. Allons-y." She frowned at the way he sidestepped. He never exactly liked going back to his stall, but he hadn't behaved like this in a month. She squinted into the darkness. Why were no lights on? "Francis? Russell?" The strike came without warning, a blow to her shoulder that forced her to her knees. She fumbled the reins, heard the horse's fearful scream. Or maybe it was her own. Up, she had to get up— Another blow, this one to the side of her head. Senses as muddy as the ground, she planted her hands, pulled her knees under her. Cruel hands seized her by the back of the jacket and whipped her upward only to slam her into the wall. A heavy, putrid form pinned her there, one rough-skinned palm pressing her cheek to the splintering wood. "Where are they, missy?" His voice rasped in her ear, and the smell of kippers and onions curdled her stomach. "Qui?" Her arms were trapped, one against the wall, one between their bodies at a strange angle, his meaty hand cuffed around her wrist. English. She needed English. "Who?" He growled and twisted her arm still higher, making her shoulder strain and pop. "Donnel be coil with me, girl, or I'll slit yer pretty throat when I'm done with ye. Where are the feral ice?" The pain must have addled her brain—his words were mere sounds strung together, no sense behind them. "Je ne sais quoi . . . I don't . . ." She couldn't clear the French from her whimpering mouth. "I don't understand." His next growl was more roar. There was a whisper of fabric, an unmistakable click, and a cold metal cylinder pressed to her temple. Her soul cried out. A wordless prayer for help, for strength, for clarity. Feet. It must have been the Lord, but He whispered into her ear with Justin's voice, and countless memories flooded her. Innocent tussles, fencing, boxing. So many lessons in how to move, to act, to spring. Your feet. The rest of her body was pinned, but her feet were free. She slid one until it found his foot. Lift, coil, slam. His scream set up a pounding in her ear, but he pulled away. Not much, but enough. It had to be enough. She jerked free of him and lunged for the doorway, back into the rain and thunder and sizzling lightning. Oscuro was still there, whinnying his warnings. She changed directions. If she could gain the saddle . . . The mud betrayed her, and the brute grabbed her shoulder, spinning her around. He had the gun up, pointing at her heart. Another frantic cry from Oscuro. Hooves flew, struck. The weapon flew, too, to the left. While the man cursed the horse, she dove for the gun. He caught her again when her fingers were only inches away, shoving her down into the mud, flipping her, pinning her legs with his knees. Lightning flashed against the evil in his eyes—and the wicked blade he had pulled out in lieu of the gun. "The feral ice, missy—ye must knowl where they are. Ye've all her things." "I don't . . ." She stretched, arched, writhed. Two more inches. One. ". . . know . . ." Her shoulder screamed, but she forced it farther. ". . . what you mean." There. Cold metal had never felt so beautiful. She gripped it and swung, striking him in the side of the head. It won her freedom, but at the price of his rage. Something struck her face, something bit her ribs before she could get the gun between them. He lunged away, so that her first shot found only air. Scrabbling to her knees, she cocked it again to load the next round in the chamber and took aim at his dark form in the gathering dusk. The next flash of lightning illuminated his raised arms. Knife still in hand, but the stance of surrender. She didn't trust it for a second. "Careful, missy. You donnel knowl how to use it." "Then it seems you ought to be careful, lest I mean to take a warning shot and send a round between your eyes by mistake." She could—her aim had always been better than Justin's, better than most of the palace guards'. If he so much as twitched the hand holding the knife . . . No. She gripped the gun tighter, fighting the rain and the mud for purchase of the handle. She couldn't kill him. She wanted answers, and dead men never offered enough of them. "Brook!" Her father's shout, half covered by a roll of thunder. The brute gripped his knife and came for her. She pulled the trigger, recocked, took aim again. But this bullet had found its mark, and he fell to the ground cradling his injured hand, screaming. "My lady! Are you all right?" Strange hands pulled at her, igniting pain in a thousand places. She pushed them away, elbowed and kicked. "Stop, my lady. It is only me. Pratt. I am trying to help you." She would sooner be left in the mud—but when his face appeared before her, he looked earnest and shaken. The rain washed the last of the fight out of her, and she let him help her to her feet and pry the revolver from her hands. "Brook!" Her father's cry was near now. Pain sliced through her side, her knees buckled. But the arms that caught her smelled of pipe tobacco and leather and ink, so she let them hold her. Let herself be crushed to her father's chest, even though the agony redoubled. It was worth it. She squeezed her eyes shut tight as heaven's tears streamed down her face. "Papa." He shuddered, wrapped his arms around her more securely. "I am here. Right here. Did he hurt you?" A pistol shot made her jump before she could form an answer. "Pratt!" Lord Pratt lowered the revolver as the man sagged to the ground—his hand around a second gun. Why had he not pulled it earlier? "My apologies, my lord. I meant only to disarm him but haven't the aim of your daughter, it seems." Brook eased away from her father, mainly so that she could press a hand to where fire ate at her side. Mud caked her everywhere, cold and slick, but this was warm. Sticky. Her father kept one arm anchored around her. "What are you doing here, Pratt?" Pratt wiped at the rain streaming down his face. "Some of your post was delivered to me by mistake. I thought to beat the rain—then was closer to your house than mine when it hit. Thank heavens." That meant he left before the Rushworths. It didn't seem right. It didn't . . . he . . . He turned to them, concern lining his face. "Were you hit, my lady?" "Only with his fists." Did the words come out in English or French? Or perhaps Monegasque? Another peal of thunder sent the sky spinning. "Perhaps, too, with his knife." The world tipped . . . but settled with her father's chest under her cheek and his chin in her line of vision. "Hold on, my dear. We'll get you help." "I'll be all right, Papa." That must have been why she could never call him Father—it wasn't his name. She let her eyes slide closed when pain crashed again. "Oscuro. He saved my life." "Oh, my Brook. Don't worry. Don't worry about a thing. Papa is here." The kitchen door crashed open, and Deirdre nearly dropped the new cake of scented soap she had fetched from the laundry. And when she saw his lordship straggle in, soaking wet and with the baroness limp in his arms, Pratt shadowing him, drop it she did. As the wind gusted the rain in with them, everyone in the kitchen leaped to their feet with a cacophony of questions. Deirdre's eyes remained fixed on Lady Berkeley. And on the red stain coloring the mud on her side. "Quiet, please!" His lordship's voice, so seldom raised, brought instant hush to the din. He wore the mask of barely held calm. "My daughter was attacked. I need O'Malley and Mrs. Doyle to come with me now. Mr. Graham, call for the physician and the constable." The butler dashed off even before he'd finished bowing. Deirdre stepped around Hiram to meet his lordship at the stairway. "The horse?" "A man. The horse saved her, she said." Mrs. Doyle pressed a hand to her chest and turned to the stairs. But before she did, Deirdre saw the look on her face. Regret . . . and determination. "Who would dare do such a thing?" "Those answers will have to await the constable. Jack?" The first footman hurried around the table. "My lord." "See Lord Pratt is shown to a room so he may dry out." "Yes, my lord." They were on the stairs then, hurrying up them without heed to the trail of water and mud they left behind. For a moment, Deirdre wondered who would have to scrub it all clean again. But it didn't matter. She would do it herself if necessary, and sure and the others would feel the same. So long as death didn't visit them tonight. So long as his lordship didn't fade away again into the man he had been before she came. There'd been laughter in the house, even with Lady Ramsey and her daughters gone back to London after the Duke of Stafford's funeral. Under her breath she whispered a prayer for perhaps the first time since Da died. "Save her, Lord Jesus. Save her." At the main floor, Jack led Pratt off in the direction of the bachelor's wing. Deirdre took the chance to slide around his lordship so she could hurry ahead to the baroness's room. She reached it half a minute ahead of him and Mrs. Doyle, giving her just enough time to snap open a spare sheet to lay across the coverlet. Lord Whitby lowered his daughter's muddied form onto it with agony on his face. "Look what he's done to her. The monster." Deirdre glanced only a moment at her face, scraped and bruised. It would hurt her, aye, but it wasn't what had knocked her into darkness. She undid the buttons on the lady's riding jacket and hissed out a breath at the bright red blood staining the side of her once-white shirt. "Step back now, your lordship," Mrs. Doyle said, her voice calm and soothing and brooking no argument. "Let us tend her as we can. Why don't you tell us what happened?" "I don't know." He sounded helpless. Looked it, as he sank into a chair and stared into the corner. "I saw her ride back in, dismount. The horse was skittish, but she tried to get him inside. Then . . . I don't know. The lights were out, but I thought perhaps the storm—then I heard a gunshot. And saw someone pushing her down when the lightning flashed. So I ran. Pratt reached her first and shot the man when he pulled out a second gun." Deirdre tried to ease the jacket off the lady's shoulders. She groaned and pulled away. "Shh, now, my lady. Sure and we have to get you out of these muddy clothes." The baroness blinked her eyes open, though they were glazed. "Deirdre?" "Aye." She smoothed the sodden locks from her ladyship's face. "Your jacket." The lady shifted but moaned again. "My shoulder." Mrs. Doyle came to her aid. "The jacket is ruined anyway, we'll cut it off. And don't you fret, my lady. You won't feel a thing." Lady Berkeley must have been clenching her teeth against the pain, given the pulse in her jaw. But she nodded and let them cut away the dark blue fabric. And, once free of it, said, "Papa?" Lord Whitby was on his feet again in half a blink, taking Mrs. Doyle's place when she turned to fetch the basin. "I am here." Deirdre had to give the lady credit—she nearly managed a smile. "I see that. And in quite a state. You should go and get dry." "Absolutely not." "They need to help me from the rest of my habit." She swallowed and pressed a hand to her oozing side. "It isn't so bad. I think the corset must have deflected the worst of the blade." Perhaps he believed her—or perhaps the mention of corsets did its work. Either way, Lord Whitby heaved a sigh but nodded and, after leaning down to kiss her forehead, headed for the door. "Ten minutes, and I'll be back. Is there anything I can get for you?" Mrs. Doyle stepped forward, setting the basin on the side table where La Bible usually rested. "She'll want coffee, my lord. That steam-pressed concoction the chef makes." His lordship chuckled and gripped his daughter's hand a moment. Her attempt at a smile faded. "Oscuro?" "Safe and well. The grooms had been knocked out and bound, but they were working themselves loose when you fainted. Francis is giving your horse an extra cup of oats for his heroics." She nodded, swallowed, and then fastened her eyes on her father. "Is he dead? The man?" Whitby hesitated a moment and then nodded. "I imagine the constable will want to speak with you. Tomorrow is soon enough for that though." Deirdre tucked away a wisp of hair that had slipped from her cap and turned to the baroness's feet. She would remove the muddy boots rather than stand idle. "No, don't put him off. I would as soon get it over with." "We shall see." They would see who was the more stubborn. Deirdre untied the riding boots and slipped them off as the earl finally left. Mrs. Doyle closed the door behind him. And they got to work. The scissors came out again to remove the ruined shirt. While Mrs. Doyle put it with the jacket pieces, Deirdre unhooked the corset and let it fall to the sides. From there, they could shift her chemise and get their first glimpse of the wound. The baroness sucked in a fast breath but made no complaints as Mrs. Doyle sponged away the blood. "It isn't as deep as I feared, and the bleeding is slow," the housekeeper said. "But it's long and will still require stitches." "And let's pray this eye doesn't blacken and the scrapes heal quickly." Deirdre picked up the wet rag that had already cooled and set it gently over the swollen side of the baroness's face. "Otherwise you'll be a fine sight for your cousin's wedding next week." Lady Berkeley lifted her uninjured arm to hold the cool cloth in place. "Aunt Mary will be furious with me." "She couldn't be, child. You were attacked." Mrs. Doyle pressed her lips together and shook her head. Still, Deirdre caught the glint of tears in her eyes, and sure and the baroness did as well. "I cannot think why anyone would do this to you." Deirdre's hands shook as they moved to assist her out of the split skirt. "Glad I am that Lord Pratt killed the monster." "No." The lady's eyes slid closed. "Now I'll never know what he wanted from me." "Leave it to the law and his lordship to figure that out, child." Mrs. Doyle held out a hand for the mud-caked skirt. "I agree with O'Malley. No one should be allowed to hurt one of our own. He got what he deserved." The baroness didn't open her eyes, but she sniffed, and her nostrils flared. "One of your own?" "Aye." Deirdre headed for the door when there was a knock upon it. She cracked it open, smiling when she saw Monsieur Bisset in the hall, a steaming cup in hand. His lordship couldn't have put in the order yet. But the chef had known. She took the espresso with a nod and could feel her da smiling down on her when she set it on the table. "And don't you be forgetting it, my lady." As soon as they had her dressed again and settled in to await the doctor, Deirdre gathered the ruined habit to take down to the laundress. The split skirt possibly could be saved—and she knew that was the important part for her ladyship. When she reached the bottom of the service stairs, those gathered in the kitchen all stood. Hiram stepped forward. "How is she?" Deirdre nodded. "Awake again, and the bleeding has stopped." A collective sigh filled the room, and chatter sprang up. She didn't try to make sense of all the mutters of outrage and sympathy. She headed for the laundry. Hiram fell in beside her. "Jack said Pratt will be staying the night—I wanted you to know. He's changed already and is in the library, so keep yourself above stairs with her ladyship, Dee." She paused in the empty, close hallway so she could look up at Hiram. "Don't be worrying for me, Hi. I know how to steer clear of the likes of him." "I can't help it." He shoved his hands into his trouser pockets and half turned toward the kitchen. "I know he comes sniffing around after the baroness whenever he can find the excuse, but usually his lordship boots him out as soon as is decent. Tonight he invited him to stay. It could make the lout bold." "But not so bold as to come to the baroness's room—and that's where I'll be, for sure and certain." She smiled, because she was glad he cared, even if she shouldn't be. Then she nodded toward the laundry. "I need to take care of these. I thank you for the warning, Hiram. It's good to know to mind my step." He gave her a thoughtful little smile that seemed to say I wonder and spun back for the kitchen. Deirdre sighed and shifted her muddied, bloodied burden. She would wonder too, if she dared to let herself. Laundry deposited for a scrubbing, she headed back up without speaking to anyone else. Not all the way to the family's floor though—no, she headed for the library, checking over her shoulder often to make sure no one saw her go that way. Ready to beard the lion, as they said, in his den. Feeling more certain with every step, she opened the door without hesitation, stepped inside, and clicked it shut behind her. Lord Pratt stood by the fire, an arm braced on the mantel. At her entrance, he glanced up but then back to the flames. "How is she?" "Well enough, I think." Squaring her shoulders, she marched over to the fireplace. "Are you behind this, my lord? Did you hire him? Because I swear if you did, I'm done helping you. She could have been killed!" "And you think I want that?" Temper flashing in his eyes, he straightened. "I want to marry her, you dolt, not attend her funeral. What possible good could she do me dead?" He came a menacing step closer, but she didn't retreat. Not today. "But your plan could have gone wrong. You could have hired him still, to scare her, then happened by at the right moment to rescue her. Play the hero, win Whitby's gratitude and her favor." He advanced another step, glared down at her. "You think me so low. So base. So willing to flirt with death for favor—yet you dare come in here and accuse me of it?" It might well be her undoing, but she lifted her chin. "Did you do it?" For a second, he held her gaze, and the familiar devil looked back at her. Then he looked away. "No." His voice had lost its edge. "I did not hire that sot to scare the baroness so I could rescue her. Satisfied?" She wasn't sure. She shouldn't be . . . Yet she believed him. Perhaps he had lied before, but this seemed different. She backed up a step. "I had to ask. I don't want to see her hurt again." "I assure you, Deirdre. Neither do I." He returned to his place by the grate, turning his face back to the flames. "Go tend your mistress." She eased toward the door, hesitant to turn her back on him. But he seemed lost in the dance of the fire. She spun and slipped out again. As she made her way back to the baroness's chamber, though, she could scarcely make sense of it. Was it possible he actually cared about her ladyship? No—he hadn't mentioned feeling, just that she wouldn't do him any good dead. She winced now, where she hadn't before. Something had to be dead inside him, to speak so. Voices came from the bedchamber when she arrived, and she found Lord Whitby inside with the doctor from Eden Dale. They were both smiling and making encouraging noises, so Deirdre slipped behind them and headed for the dressing room and its attached lavatory. Much as the baroness needed it, she wouldn't feel up for the bath Deirdre had drawn. She drained the water. Rising again, she set things to rights, taking her time. When she headed back through to the bedroom, the doctor was following Mrs. Doyle out. The baroness seemed to be asleep. "He gave her a bit of laudanum," his lordship said from the chair he had pulled up beside her bed. "Just enough to ease a bit of the pain so she can rest." Deirdre crossed to the other side of the bed and pulled up another chair. "I daresay she needs it." But it looked none too peaceful. Lady Berkeley turned her head from side to side, little restless noises coming from her lips. Then the "Non, non, non" Deirdre knew so well. Lord Whitby did not. He leaned forward, brow furrowed. "Perhaps I should have let her refuse it." "'Tisn't the laudanum, your lordship. It's the nightmare. She has it most every night." But she shouldn't have to suffer it this night. Deirdre sat on the bed, ran her fingers along her ladyship's face as she would have Molly's, and then caught up her hand. "Shh now, my lady. It's only a dream. Only a dream." "The same one? Every night?" She tilted her head toward Lord Whitby. "She never speaks of them—but they always look like this." "She's never said a thing to me." And the hurt of it made creases around his eyes. But still he took her other hand, cradled it in his. Murmured, "All is well, my little Brooklet. Hush now. Hush." For a second it seemed she would listen. Then she gasped, her eyes flew open, and her chest heaved. "My mother—it must be. 'You have all her things,' he said. All her things." Now his lordship looked to Deirdre, panicked question in his eyes. She could only shrug. "That must be the laudanum, my lord." He sighed and brushed the fair curls from his daughter's forehead. "Easy, precious. Go back to sleep." Her eyes unfocused, she shook her head. "Non. They always find me there. The lightning and the thunder and the night and . . ." "Shh. They'll not find you tonight. I'm here." "Papa." She blinked rapidly, and a measure of awareness lit her eyes. "What was I saying?" "Nothing." He smiled and kissed her forehead. "Rest. I'm here. Rest." Deirdre slipped from the mattress and went to the window. Arms folded across her middle, she fought back the burn of tears. Her da had done the same thing when one of them had the fever or woke up in a fright. He had looked at her and her siblings with that same light of love. Family, it seemed, crossed from abovestairs to below with few differences, at the heart of it. She sighed and looked past the pattering rain. The thunder had moved off. The lightning had ceased. But the night was full and dark and promised to be a long one. # Sixteen Whoever invented laudanum ought to be executed. Never in Brook's life had her head hurt so—though granted, it might not be all the fault of the drug. She had to take the stairs slowly, largely because of the dizziness. Her legs were sore, bruised where the ruffian's knees had pressed them, but not that sore. Her shoulder ached from the strained muscle, but she could have ignored it. And of course, her side was so tender and raw that a corset had been out of the question, necessitating Paul Poiret dresses that didn't require one. But it was the fuzzy head that was driving her batty. "Lady Berkeley, what are you doing? Where is O'Malley?" Brook gripped the banister tightly before trusting herself to look up. Mrs. Doyle was rushing up the stairs toward her, her frown not one to be ignored. Brook ignored it anyway. "I sent her on an errand. Papa said the constable will be here in an hour, Lady Catherine's note said she will be visiting not long after that, and I need to have my wits about me." The housekeeper pressed her lips together. And then looped her arm through Brook's. "You should have had O'Malley help you down, my lady. We can't have you falling and hurting yourself worse." A nearly valid point. She already looked a fright—bruised and scraped from face to foot—and they were to leave for London in three days. There was no way she could stand beside Regan at her wedding like this. Would Aunt Mary even allow visitors for her? Brice and Ella had promised to call as soon as she made Town. And it made her stomach hurt outright to think that Justin's first view of her in two months would be when she looked like the loser of a barroom brawl. Her hand shook against the railing as they continued down. A brawl it had been, but she hadn't been the loser. And she still couldn't think why the man had lain in wait for her. At least she would have another story to tell Justin. "Brook Tames the Darkness" for her victory with Oscuro . . . and "The Assailant in the Stables" for last night. "A hearty breakfast will bolster you, my lady. Chef made the eggs you like so well, a sausage so spicy it sent poor Jack running for water, and of course your coffee." She had to swallow before she could speak. Who knew breakfast and coffee could mean so much? "Thank you, Mrs. Doyle. I will thank Monsieur Bisset later." The grand staircase stretched on for miles, but at last her feet touched even floor, and they headed for the breakfast room at a normal pace. Or nearly normal. Almost, nearly normal. Her father's voice floated out to meet them. "I don't care if it takes a year, Constable, I want this man's identity found. If I have to pay an investigator to inquire in every village and hamlet in all the empire, I will." She halted outside the door, her brow taut. Papa had said the constable would be here to meet with her at nine o'clock. It was only eight. "And you may have to, your lordship—the folks in Eden Dale said they'd never seen him before, and he certainly isn't one of Whitby's usual drunks." She stepped into the room, extracting her arm from Mrs. Doyle's. "He wasn't drunk. He smelled of kippers and onions, not alcohol, and his reflexes were as quick as mine." The men came to a halt—all three of them. Her father with his tea halfway to his mouth, the man she presumed to be the constable with a click of his heels, and Pratt at the sideboard filling a plate with her eggs. No one had mentioned he was still here. Though she supposed after saving them the night before, her father could hardly begrudge him a change of clothes and a warm bed. Something niggled there, though. What, again, had he been doing here? Some bits were so muddled . . . "I don't know whether to scold or rejoice." Papa put down his cup and stood, motioning her in. He pulled out her usual chair. "I said we would bring him to the sitting room across from your chamber." "And I thought to breakfast with you first." She tried to give him her usual cheeky smile, but a nasty scrape forbade it. "Sit." He indicated her chair and then turned to the sideboard. "Eggs, sausage, and this stuff you so optimistically call coffee?" "Yes, please. And merci." She sat, though it was little relief to her side, and looked to the uniformed officer. "You've no idea who he was?" "Not yet, your ladyship. But the day is young, and we've only just started asking." A different song, it seemed, than the one he had sung for her father. She lifted a brow and kept her back straight, trying to keep all pressure off her side. "He had a strange accent, if that helps you. He put an L on the end of some words. Donnel for don't. Coil for coy." The constable sent a glance over her head. Papa put her plate and cup before her. His eyes, she saw when he retook his seat, had gone thoughtful. "Bristol." "Bristol?" Pratt echoed. He took a chair across from her with a shake of his head. "It's awfully far." For a man out for a random robbery, perhaps. For one on a mission . . . She took a sip of the coffee, nearly sighing in bliss. Her father ignored Pratt altogether. "So he said 'Don't be coy.' What else?" She took another sip to clear her head. "He asked me where they were. I at first thought he spoke of people, but he must have meant things. Something . . ." It had made so little sense. "It sounded like feral ice. And he said I must have it, I had all her things." She looked up, a blurry image surfacing of her father leaning over her, the dream still clouding her mind. Papa must have made the same connection. "Your mother. But what among her things could anyone be looking for? And why now, when she has been gone so long?" "I don't know." It made no more sense than it had last night, and trying to focus on it made her head hurt. "You are yet unwell, my lady." Pratt's voice sounded concerned—anxious even. "Pushing yourself will accomplish nothing. Rest, then send word to the constable if you think of anything else." "No. I am well enough." He ended his words with Ls. So perhaps it wasn't feral. Fear? But what was fear ice? More coffee—that was all she needed. Though her stomach disagreed with her tongue and her head, forcing her to test the food as well. She must have missed dinner last night. Kippers . . . so he had to have been in Whitby long enough for a meal at a pub. Perhaps he had rented a room. Maybe the constable's knocking on those doors would reveal something after all. Fire. Not fear, fire. Fire ice. Fire and ice. Ice . . . cold? Non. Jewels—diamonds. The British called them ice sometimes, did they not? Brook put down her fork, though the food was perfect. Diamonds . . . she had many of them, now, that had been her mother's. Bracelets, rings, necklaces. Papa leaned back to murmur something to the constable. What was it he had said when he offered that first necklace? "To match her eyes. The color of emeralds, with the light of diamonds." Eyes. Fire eyes . . . Written words flashed through her mind, though she couldn't be sure she remembered them correctly through the haze. She pushed away. Too slowly to be called abrupt, but still it brought the men to another halt. Brook forced a smile. "Excuse me, gentlemen. I'm afraid I'm not so well after all." Her father all but leaped from his chair. "I'll help you back to your room." Panic clawed at her throat. Yet it couldn't be. She would look at the letter again. Try to make sense of it. "No, Papa. You must finish your conversation here. I shall find . . ." Mrs. Doyle couldn't have gone too far. She looked to the door. No Mrs. Doyle. But Deirdre appeared as if summoned by her very thoughts. Or, given the exasperation upon her face, by Brook's disappearance from her bedroom. "There you are, my lady! You look pale as a ghoul. Let me see you back upstairs." "Thank you—I would appreciate it." Brook bent her knees—all the curtsy she could manage—and nodded at the men. "Pray continue, gentlemen." Deirdre slid a gentle arm around her waist, careful to avoid the injured side. "I'll have someone bring your plate and coffee. You need to rest, my lady. It's quite a trauma you received, and not so many hours ago." Brook's mind buzzed too much to argue. She gladly accepted the help up the stairs and into her room—though she declined the offer of bed in favor of a chair. And she only took the chair once she had first gone to her dressing room and tried to reach, not for the jewels, but for the box of her parents' letters. "Your ladyship!" Brook sighed . . . and winced. "You're right. I can't reach it. Would you be so kind?" Mumbling in Gaelic all the while, Deirdre pulled down the box from the shelf with ease and shooed Brook back to her chair. "I can't think what's so all-fired important . . ." Brook offered no explanation, just opened the box and pulled out the bundle of letters. She had finished reading through them all a month ago and had divided them again into his and hers, in their separate boxes. These were hers, from him. She flipped to the bottom of the stack. The very last one by date. It had been buried in the box when she first sorted them—though the rest had been in reverse order, newest on top. She'd thought it odd, but Regan and Melissa had distracted her from dwelling on it. Now she dwelled and unfolded the missive. Her eyes scanned over the first few paragraphs, but it wasn't there. She flipped it over. There, on the back. I know you have jewels enough already, my love, but when I saw this, I thought of you. Of how it would look against the cream of your skin, under the fire of your eyes. You have always been my Fire Eyes. Fire Eyes. But they weren't a thing, for a thief to demand. Yet he had tied them to a gift . . . "The letters again?" Deirdre was returning from the door with her breakfast tray. She slid it onto the table by Brook's side and raised her brows at the paper. "And who's that one from?" "My father to my mother." "Is it? Doesn't look like his lordship's hand." "No." It had been the first thing she had noted too, after sorting through so many of them. But the explanation for that lay in the first paragraph. "The letter says he'd hurt his hand—his valet wrote it for him." Though now that she knew him, she couldn't imagine her father sharing such intimate thoughts with any third party. Ever. Someone else had obviously penned it though. Another knock sent Deirdre back to the door, and Papa poked his head in the moment she opened it. "May I come in?" "Please." He could be trusted. She had known it all along, but now she was sure. "I would appreciate your help." Question in his eyes, he strode her way. She held out the letter. He took it, but without any change to that silent inquiry. "What's this?" "I wish I knew. It was with the letters you wrote my mother, signed with your name, but not in your hand. It says you dictated it to your valet." His gaze shot from the page to her. "I would never dictate a letter to my wife to my valet." "I know. So then . . ." "So then." His gaze fell to the sheet again, scanned, narrowed. "What is this gift?" She nearly smiled at the temper in his tone—jealous, nearly twenty years later, at the thought of someone else sending a gift to his Lizzie. Did Brice ever react so? Not that she'd seen, though he looked at her warmly. And Justin . . . he was too much her brother. He guarded her fiercely, but it wasn't the same, was it? "Some kind of jewelry, obviously." He had flipped the page, and she knew when he got to that last line by the quick breath he drew in. Knew, when he looked up, that his mind had made the same leap hers had. "Not feral ice. Fire Eyes." "Yes." She moistened her lips. "I first thought it might have been ice—like diamonds. Which is what got me thinking about this letter." "It must be one of the pieces I attributed to the Brooks or Rushworths. She— "Wait." Brook got slowly to her feet and walked into her dressing room, pulling out the card-paper bandbox where she'd put Mother's miscellaneous correspondence as she'd read them. Tossing it to her bed, she riffled through the contents. It didn't take long before she lifted a few folded sheaves. "I knew I recognized that script, try as he did to disguise it. I found these letters while reading through Mother's correspondence." Papa took the missives, and as he read, soon flushed. "That blighter." He threw the pages into the bandbox and turned abruptly. "O'Malley, find us fresh paper. We have a letter to write to one Major Henry Rushworth, in India." Justin hadn't attended many weddings, but this one seemed exceedingly long to his way of thinking. And dull. Much as he had enjoyed the few moments before the ceremony he'd had to poke fun at Thate, who had been grinning like a lunatic, this wasn't where Justin wanted to be. Not given the gaping absence of Brook. His ship had been days late to port, and he was convinced it was only prayer that had allowed him to make it into the city in time for the nuptials. He'd had no time to go to his townhouse, only to send Peters for his clothes while he headed for the church. Once there, of course, it had been straight into the room with Thate and their other friends from school who would stand with him. No one had mentioned that Brook would not be present—wasn't she to be one of the bridesmaids? He'd found Whitby in the crowd, had sent him a questioning look . . . but hadn't been able to decipher the mirroring one Whitby sent back. The moment the interminable ceremony finally ended and the impossible crowd made its way out to greet the Earl and new Countess Thate, Justin found Brook's father. "Where is she?" Whitby lifted a single brow. "And a cheerful hello to you too, Duke. She's at Mary's." But . . . "Why?" The other brow joined the first. "She wasn't well enough." "What? Is she ill?" It would have to be serious indeed to keep her away. Now Whitby sighed and pulled him back into the church, away from the milling nobility. "Did you not go home first, sir? She and I drove round yesterday and left a letter for you." At the shake of Justin's head, the earl nodded. "She is injured—a cut to her side that wouldn't, apparently, allow her to wear her bridesmaid's dress, and her face is a veritable rainbow of blues and greens that made my sister faint each morning for three days running." Panic vied with pity. "That horse?" It had to be. That stubborn girl— "No. She has Oscuro well in hand." He looked as if he were about to say more but then darted a worried look at the crowds. "I would keep the press out of it, so I'll say no more. It's all in the letter." Letter be hanged—he'd get the story from Brook herself, and he certainly wasn't going to lollygag here when she was but a few miles away. With a nod to Whitby, he exited again and gripped Thate's shoulder. His friend turned, that idiotic smile still in place. "There you are, Shep. I thought you'd run off to India already." His intentions paused, he blinked. "Shep?" "Stafford . . . Staff . . . Shepherd . . . surely you can follow the train of my thoughts by now." A grin stole Justin's lips. "Never—mine are too logical to take the twists and turns yours do." He gazed out over the sea of people, far too many of whom watched him. "I'm going to slip away for a bit, but I'll make my way to the ball when I can." Thate's smile went lopsided and knowing. "Any particular place you're slipping away to?" As if he didn't know perfectly well. Thate must have known exactly what kinds of injuries Brook had managed to sustain, even if he hadn't taken it upon himself to enlighten Justin before the wedding. Though he supposed he would have been a bit suspicious had the man's mind been on his bride's cousin rather than his bride. "I have to see her." Thate's smile was the exact one he'd given him in school when Justin had fallen into the pond after Thate had warned him not to trust that old log. Pure condescending glee. "Oh, I know you do. Go. Enjoy your freedom while it lasts—I daresay she'll have chains around you soon." And if he could be as happy in them as Thate seemed to be . . . Justin grinned. "Don't make me hurt you on your wedding day, Alex." Thate's laugh followed him down the sweeping stone steps outside the church before it got lost in the chatter of the crowd. He found the Rolls-Royce and gave the motor a crank. Slid into his seat, switched on the magneto, and turned the key. He had the direction for Lady Ramsey's home and knew it wasn't too far from his own townhouse on Grosvenor Square, so he set off in the general direction. The sun was already setting behind the buildings of London when he found the right street and then the right number. He parked, killed the magneto, and hopped out, sparing only a moment to smile at Whitby's words. "She and I drove over yesterday . . ." Which of them, he had to wonder, had been behind the wheel? The butler opened the door to his knock. Stepping inside, Justin handed over his card and received an immediate bow. "Good evening, Your Grace. But I am afraid the family is all out—" "At the wedding, I know." He took a step to the right, though, when the strains of a piano—and a soprano—reached him. "I was just there, where I learned of the baroness's injuries. I needed to see for myself she is well." The butler's eyes brightened. "Ah, you are that duke. Of course, Your Grace. I will let the baroness know—" "Please, don't interrupt her playing. It's been too long since I've heard it." And what other duke would come calling? The only possible answer made his palms go damp. He handed over his hat and overcoat and let his feet point him toward her siren's song. "This way?" "Yes, sir. Follow me." The butler led him a short way down the hall and indicated the double French doors to what must be the music room. He glimpsed a harp near the window, an old clavichord by the shelves. With a nod of thanks he stepped inside. And saw the piano. Her back was to him. Her hair was down—the chandelier's light shone on each spiraling strand of gold tumbling down her back, wild and free. A sight he hadn't seen in years. And which hadn't used to make him react like this. She played with the same abandon she applied to her every other pursuit, as if it might be the last song she ever sang, the last keys her fingers would touch. He recognized the song—it was from a Puccini opera, and Collette had earned her fame belting out this particular bittersweet refrain. Letting the music sweep through him, he eased into the room, careful to keep out of Brook's peripheral vision. The last time he had happened upon her like this had been that night in Monaco. When he had looked at her and thought how beautiful she had grown to be, how he would soon declare himself. Swallow as he might, the lump wouldn't ease from his throat. A few more months, a few more trips, a few more continents. Things had gone well in Barbados and Canada. Not well enough that he could avoid sinking the money Father left him into improvements for the Stafford tenants in Gloucestershire, but well. Promising. If he could put things to rights as efficiently in India and Africa . . . Brook lifted her voice in the final high, soaring note. Her fingers stilled for a measure, two, then flew over the keys in a heartrending finale. Once her voice had gone silent and her fingers still, he stepped forward, clapping. She spun around on the bench, her eyes going bright as she sprang up. "Justin!" Because he couldn't help it, he smiled—and because her face was mottled with bruises, that smile faded as she launched herself into his arms. He let her kiss his cheeks but knew he was scowling. It deepened when she flinched away from the hand he settled on her waist, pain flashing through her eyes. "What happened?" He didn't mean it to come out so harsh sounding. His hands slid to her back—until he realized he felt only cloth and flesh, no rigid boning. Far too alluring. He dropped them altogether. Her eyes flickered only briefly. "Did you not read the letter?" "I didn't get home. I had barely enough time to reach the church. What happened?" She sighed and rubbed a fingertip over a mostly healed scrape on her arm. "I was attacked one evening—we still don't know why, or who the man was. I took his gun and shot the knife from his hand—" "You what?" Images assaulted him: Brook held at gunpoint. Brook with a knife at her throat. Brook, one of the few people he had left in the world, nearly killed. The fear of it swallowed him, and he dragged her to his chest again and held her close. Let the solid feel of her, the proof that she had survived her ordeal, seep into every inch. "Justin—" "Hush. Give me a moment." He squeezed his eyes closed and buried his face in her golden, fragrant curls. Her arms were around him. Her breath on his neck. She even stroked a hand over the back of his head. Soothing, giving comfort, when she was the one who had been injured. He swallowed and forced himself to pull away, though he couldn't resist cupping her uninjured cheek as he met her gaze again. "Sorry. I can see you're all right, but the thought of it . . ." He shook his head. "You shot a knife from his hand?" Only Brook could nod about it with a hint of a smile. "And then Papa came, and Pratt. Pratt killed the brute when he drew out a second gun." He could only stare at her now, waiting for the words to clarify. She'd called Whitby Papa—that was a big step for her, and it must be a new one. But . . . "Pratt?" The arch of her brows looked amused. "Now you sound like Brice and Ella. Pratt finds any excuse he can to call, though that was certainly the first we welcomed him." His brain had hit another snag. "Who are Brice and Ella?" "Sorry—Lord Worthing and his sister, Lady Ella Myerston." Were there a seat handy, he would have sunk into it. As it was, his hand slipped from her cheek. "Lord Worthing." He was back at Whitby Park, on the day Grandfather died. Looking across the lawn at her on his arm. Seeing the way she laughed, the way he looked down at her. "You are on a first name basis with them?" With him? She spun away with a chuckle, toward a laden tea table that seemed to have everything but tea on it. A chuckle, as if it weren't paramount to claiming they were engaged. That, while he was an ocean away dreaming of declaring his love to her, she was forgetting he even existed. "You'd like them," she said, insensibly. "After the house party they had gone back to Scotland to finish their holiday with their mother's family, and they all stopped again at Whitby Park for a few days' rest on their way home to Sussex." She turned back toward him, cup of steaming black coffee in hand. "Have you seen your cousin yet? He has spent much of the fall in Town. Largely, it seems, because Melissa was here. She says she is certain he will propose soon, though Aunt Mary wants her to debut first." When Brook extended the coffee toward him, Justin took it without thinking. But he didn't feel the heat of it on his palm. He wasn't even certain the electrified chandelier still shone. So many times he had come home from months of school or travel, had sought her out at the palace—and she had spoken to him of academic papers or dignitaries or the latest advances of the automobile. Not weddings and debuts and cousins and friends known the empire over for their ability to make women fall at their feet. Friends she called by first name with a gleam in her eye. Friends who had looked at her as though seeing the sun for the first time. Justin downed half the cup of coffee in a single shot. It warmed him, but not in the way he'd hoped. "You've been busy." She paused with her hand on the gleaming silver coffeepot and looked right into his soul. "Would you have me stand around idle?" "No. Of course not." But he would have her not make him feel, the moment he stepped in the room, that he had become superfluous to her life. "It's good to know you've found your place. Made friends." He must not have sounded convincing. She planted her hands on her hips and narrowed her eyes. "What choice had I? To spend my every waking hour waiting for you to deign to write a letter?" "I wrote letters!" He held out a hand, palm up . . . though he had not written as much as he should have. Every time he put pen to paper, the only words that wanted to make their mark were I love you. I need you. She rolled her eyes and spun away again. "One. One letter." "More than one. Three, at the least. They must not have reached you." She sighed and put a pastry on a plate, handed that to him as well. Strawberry—his favorite. But he knew well he couldn't eat a bite. Not when she looked up at him like that. "Three letters, then. In over two months. You have never written so little, never. You abandon me here—" "I did not abandon you." He set the plate back down with a bit too much force. "I delivered you to your father!" "Yes, and then you left. And I know . . . I know you had to. I know that." She pressed a hand to her forehead, holding back the curtain of curls. "But you pushed me away, Justin. You wouldn't let me in—and then silence, except for that one brief letter that might as well have been to your solicitor, for all its personal tone. And now you have the gall to stand there and be upset that I have made other friends?" Deuces. She made him feel the fool, made Aunt Caro's warnings clang in his head. But what choice had he had? "You act as if you wrote me every day, as if your letters were about anything but your progress with that devil of a horse and the prattle of Lady Catherine. Where was the mention of our cousins being all but engaged? Of this . . . this Brice you apparently like so well?" She stared at him as if he'd grown another head. "I only heard about our cousins when I got to London three days ago, and why are you saying Brice's name like that? Surely Thate didn't poison you against him so thoroughly—" "By thunder, Brook, this has nothing to do with Thate and everything to do with you. You cannot go around calling a young man by his given name! Haven't you any idea how things are done here?" It was a mistake. He knew it the moment the words exited his mouth. The fury that snapped in her eyes only hammered it home. "So then I should call you Duke now, after all? Is that it?" What was wrong with her? He shook his head. "Don't be stupid. It's different for us—we were children together. But heaven help me, you better not be telling me that you feel as close to Worthing after two visits as you do to me after thirteen years, or I'll—" "You'll what?" Her shoulders had edged back, her chin had thrust out, making the bruises shout at him. Yet somehow, she didn't look like a petulant child ready for a brawl. She looked like a princess facing down an angry mob. "Run off to another continent and not bother to write?" "Brook, that's unfair!" He reached out, tempted to shake some sense into her—or perhaps to pull her tight and give in to the long-festering need to kiss her, to show her why he couldn't suffer another man be in her life like that. But when his hand gripped her shoulder, she hissed out a breath, her eyes went wide, and she pulled away, clutching the shoulder. He'd hurt her. Heaven help him. "What did I do?" She shook her head, though the denial was obviously a lie, given the way she squeezed her eyes shut. "It's nothing. It was just wrenched, is all, and bruised." He'd hurt her. A careless touch, and he made her wince away. Still, that was nothing. He'd done far worse in years past, he was sure, as he taught her his sports. But he'd hurt her. He saw it as she opened her eyes again and stared at him from too-dry eyes. He was, it seemed, his father's son. Not Father's—Father, who could grin and sweep his lady into his arms and make her forget all the agony that had come before. Not Father, who understood that when pain came, you clung to those who mattered most, you didn't push them away. But Edward's son. Stone-faced, coldhearted Edward's son. Nostrils flaring, he dragged in a shaky breath. "I'm sorry. I should . . . I should go. It seems I'm still not fit for company, so I'll just . . . I'll see you in the spring." He pivoted. Eyes unfocused, he made for the general direction of the door. "Wait!" Her long, delicate fingers caught his. Familiar. Warm. Perfect . . . But if he clung to them, it would hurt them both all the more. He wanted something she obviously didn't want to give. She hadn't been dreaming of him, hadn't been yearning for him. She couldn't have been, if she were so busy getting to know Brice. "Justin, you can't mean . . . When does your ship leave?" He slid his eyes closed to keep from looking at her. Told himself not to squeeze her fingers. "Tomorrow." "What?" Her fingers fell away. "No. You just got here, you can't possibly leave again so soon." "I thought to have a week in Town before the wedding. I cannot help that my ship was so late." His voice sounded hollow, empty. Just as he felt. He wasn't strong enough to keep from turning to see her. She shook her head, sending her curls swaying. "No. But you can help when you leave. Postpone a few days, Justin. Please. There are things . . ." She blinked and looked away, but one of the tears still overflowed and spilled onto her cheek. "There are things I cannot put in a letter." "Brook." There were things he couldn't either. The truth of his father. Of his heart. "I wish I could. But I risk missing an important rendezvous if I delay my departure. I have to go." She gripped his wrist. "I know you have responsibilities. But they cannot always take precedence over people." "They are about people—the hundreds upon hundreds of them who rely on the Stafford estates for their well-being." It was true. Why, then, did it sound like an excuse to his own ears? "I mean your family." "They understand. My friends understand. Everyone else—" "Everyone else?" She tossed his hand away from her and all but leaped back. "Now I am not family, not a friend?" Why could he say nothing right to her anymore? He lifted his hand, though then he let it fall again. "That is not what I meant." Fire snapped in her eyes. "Isn't it? It seems to me that it's exactly what you meant. That you can't bear the thought of not being able to do everything on your own, to control all you touch, O Mighty Duke of Stafford, and so you must push away those who make you feel and—" "You have no idea what I feel!" "Ça c'est sûr!" The French sent him reeling backward—not because of her claim that that was the point, but because she had stuck with English until then, which was unprecedented when her emotions ran so high. Proof that she had built a place for herself . . . and he had no part of it. He had ruined everything. And he didn't know what to do but turn, pray to God that He would help him mend it, and leave. # Seventeen Brook stood where he'd left her. The threatening tears made her nose ache, and pain from holding them back scorched her side. He'd left. He hadn't teased or cajoled or called himself a dunce. He hadn't shot back with an accusation of his own. He hadn't gathered her close again and told her why he had such shadows in his eyes. He'd left. A sob nearly escaped. So much she'd wanted to tell him—nightmares and jewels and lying letters in her mother's things—and now he was gone, and he was leaving tomorrow, and they would part for months with this between them, and then things would never be the same again. Things were already not the same. "My lady." Deirdre bustled into the room, concern in her eyes. "I heard shouting. Was that the duke?" "Yes." But no. Yes, it had been the Duke of Stafford talking about responsibilities and trips he couldn't postpone. The Duke of Stafford, with eyes so much older than his years. But beneath him, somewhere, was Justin. She darted around her maid. "I have to catch him." "But I heard his car start up." "Then I have to go after him. I need a horse. Or the car." "My lady." Deirdre caught her by the elbow, horror on her face. "No. You can't go out alone at night in London. Not looking as you do." Did she mean the bruises or the clothes? Either way. Shaking her arm free, Brook charged through the doorway. "You're right. I need livery. There should be something in the laundry." "My lady!" This time she was the one to halt. "You can help me or you can stay out of my way, O'Malley, but I am going after him." Indecision chased through the Irishwoman's eyes . . . then she crossed herself and flew down the corridor. "Heaven help me and may his lordship forgive me. I'll fetch the livery." Every step seemed to take an hour, every button an age, but the clock said it was not five minutes later that Brook flew from Aunt Mary's house in borrowed Ramsey livery, a chauffeur's cap hiding her hair and shadowing her face. Papa usually helped with the car's crank, but tonight she didn't have time to find other assistance. She did it herself, ignoring the strain to her side, and leaped behind the wheel. Thank heavens Papa had driven her to Justin's townhouse yesterday. She followed the same route now, praying he had gone home and not to the wedding ball. Surely, surely he was not so unmoved that he could feast and dance as if the world were still whole. She took the last two turns too fast, but her galloping heart would accept nothing less. When she squealed to a stop in the rear of his driveway and saw him just exiting the carriage house, she deemed it worth it. He spun at her ignominious entrance, light from the lamp outlining him in gold. He wouldn't know the car. She took it out of gear and pushed open the door, tossing her cap to the seat behind her. "Brook?" It was half disbelief, half relief in his tone. She wasted no time on words, not quite yet. Just ran for him and didn't stop until her face was buried in his chest, her arms wrapped around him. He hugged her back, so tightly she could feel the ache in his heart even above the one in her side. "Gently," she muttered into his ascot. "Sorry." His arms didn't loosen but shifted away from the sore spot. "I'm so sorry." "I couldn't let you leave like this. I didn't mean to fight with you." She squeezed him tighter, breathed in the scent of lemon and spice. "I'm sorry, Justin. I understand your duties. I do. But why must you push me away? I need you." He stroked a hand over her hair. Lingering . . . but sorrowful. Then he rested his head on hers. "No you don't. You've always been so strong. Independent. Look at you, flourishing in my absence." "No." He wouldn't say that if he knew what dreams haunted her. If he'd seen the evil glint in the eyes behind the gunman. "Don't leave like this. I know you must go, but not like this." Silence pulled her soul taut. London still made its noises, to be sure, but he made none. Made no move. He just stood there and held her and then loosed a breath that seemed to expel his every drop of energy. "Everything has changed." Her own thought—but hearing him say it made her shake her head and tip her face up more to look at him. "Non. Not everything. You are still my dearest friend." "Am I?" He put his hands on her shoulders and urged her away. "After acting as I have?" She gazed into his eyes and saw how dark the blues looked in the night, how darker still with what he kept pent up inside him. "Justin . . . it doesn't matter." It couldn't. His smile looked so sad. "Of course it matters. I've hurt you, and that's the last thing I meant to do. But I don't know how to remedy it, other than to promise you I'll try never to do it again." Now his hands dropped to his sides, and he backed up a step. "You need to go home, mon amie. The streets aren't safe." Neither was home. She slid closer again, found his hand. "Not yet. We haven't talked. I don't know what adventures you've found." Squeezing her eyes shut tight, she knotted her fingers around his. "Tell me a story." "All right." He squeezed her fingers . . . and then released them. "I believe this one is called 'The End of an Era.'" Her throat went so tight she could only whisper her reply. "What happens?" "I don't know yet." She shook—not with November's cold, but with his leaving. With the changing. It seemed all she could do was try to make the parting sweet. She lifted her hands, planted them on his shoulders, and strained up. "Go with God." She kissed his left cheek, soft and sorrowful. "Hurry back to me." She kissed his right. But when she tried to lower back to her heels, he pulled her against him. Tangling his hand in her hair, he tilted her face back, giving her a single glimpse of his eyes—deepest blue still, and flashing. Then he touched his lips to hers. Just a touch. But it lit a spark. Then a fire, a sweeping, a diving. She clung to his shoulders and parted her lips and was lost. Utterly, beautifully lost in a sea of sensation. He angled his head and took her deeper, making her want to dance, to sing, to fly. Then her arms were empty and only cold air kissed her. Her eyes flew open in time to see him shove a hand through his hair. What was that? Or rather, why was that? For a second, it all roiled through his eyes—question, regret, and . . . and something far warmer, far deeper. Then it was gone, locked away. Never in her life had she felt so very cold. "Justin." But the Duke of Stafford took another step backward, twirled the signet ring on his finger. "You need to go home. I'll have a groom follow." No. He couldn't just run away again after that, after changing everything on her. "Justin." He kept retreating. "Au revoir, mon amie." Until I see you again. Though only the Lord above knew when that would be . . . and if she would still be his amie. Her stomach clenched. "The End of an Era." She spun and ran for the roadster, closing herself in. The car was still running—she backed up, turned, and sped onto the square before he could rouse any of the grooms. Was back to Aunt Mary's likely before one could have saddled a horse. So very close. So very far. When she pulled back into the carriage house, she switched off the car and rested her head on the wheel for a minute. Tried to convince the breaths to come into her lungs in an orderly fashion, to exit one at a time. They seemed determined to trip and tangle. It was cold. Her hands stung. She needed a fire. A blanket. Justin's arms. No. Pocketing the key, she stumbled from the car and ran toward the servant's entrance. Reached for the handle. The door swung open before her, a man's figure looming against the lamplight within. "Elizabeth Brook Eden! Inside—now." She ought to have known her father wouldn't linger at the ball. Scarcely feeling the trudge of her feet, she slid by him. "Into the parlor, young lady." Of course. At home, the parlor was where he led prayers, where he doled out praise to the staff. Where, she heard, he would fire anyone to be dismissed. She'd yet to see him do it. Perhaps the fear of it won obedience. Or more likely their love of him. She made her way into Aunt Mary's green and gold parlor and stopped in the middle of the room, her head too heavy to hold high. Papa slammed shut the door behind them. "What in blazes were you thinking? Taking the car without permission is bad enough, but at night? Alone, in an unfamiliar city?" "I know. I'm sorry." "And running after a man? I don't care how good a man, how well you know him, some things are not done, Brook!" It seemed she was forever doing things that weren't done. "I know." She couldn't lift her gaze—it felt too heavy. So all she saw were his pacing feet. To the right, pivot, to the left. "You could have been accosted. Hurt even worse." She opened her mouth, but nothing came out. "You could have been killed." Her eyes slid shut. "And by thunder, Brook, why aren't you arguing with me?" The question broke her, made a strangled laugh escape her lips as tears wept from her eyes. "I'm sorry, Papa." His feet drew near and his arms came around her, fierce and tender at once. "No crying—it isn't fair." "I'm sorry." They were the only words she could find. She wiped at the tears and sagged against him. "He's leaving again tomorrow. And I'm a muddle. I think he's . . . in love with me." Papa's sigh gusted along with the wind outside the windows. "I know he is. And I strictly forbid it. You shouldn't be old enough for such things, not with all the years we've missed." A weak smile tugged at her lips. He led her to the sofa and sat beside her, her hands in his. His eyes searching hers. "What of you? Are you in love with him?" Was she? She stared into the dancing flames of the hearth, felt again the heat inside her when he'd kissed her. Felt again the cold when he'd backed away. "I don't know, Papa. When I was a girl, I would dream . . . but I knew it could never be. I resigned myself to that years ago. A duke cannot marry a singer's daughter." Papa pressed her fingers, holding them tight. "You are no longer that, though." Her brows pulled down, her heart squeezed. "But that shouldn't be enough, should it? That now that I'm suitable he would . . ." She closed her eyes against the firelight and shook her head. "I love him. I've always loved him. I don't know about the romance, but I know that." "And we all do foolish things for those we love." Papa cleared his throat, bringing her eyes open again, to latch on his pained face. "But I can't lose you again, Brook. You can't possibly know the fear that struck me when I realized you and the car were both gone." She could imagine it. "I was selfish. I didn't think." "Why? What happened?" She settled into the space at his side, where she could lean into him and rest her head on his shoulder, pretending she'd done so for years. "We fought—which is nothing new, but it was different this time. I don't know why. We said things, stupid things, and then he left. And was leaving Town in the morning, and I couldn't let him. Not like that." "Oh, Brook." His tone went weary. "Perhaps you are in love. We all say stupid things when we're in love. Argue over nothing." "It hurts." He snorted a laugh. "Love often hurts." "Then why would we do it?" He squeezed her hands, warming them. "Because it's worth it. Even when we lose them, it's worth it." She could only sigh. Papa planted a kiss on the top of her head. "Did you find him?" She nodded. "He's still reeling so from his losses and now feels alone, left behind. I try to understand that, but he won't let me in. And then he kissed me, and—" "He did what?" His shoulder jerked from under her. "Give me the key. I'm going over there—" "And what?" Seeing the ire, so purely paternal, sparked life into her heart. "You'll threaten him into marrying me?" "Hardly. I'll threaten his life if he dares to come home again." She nearly laughed. "Papa." "I've said it before—I'll not give you away so soon. I won't do it." He looked almost, nearly serious. And it almost, nearly made her wonder if that's what Justin wanted—to marry her. It sent an uneasy thrill through her middle. Did she want that? Did she want a lifetime in his arms? Maybe . . . possibly. The kiss had been beyond anything she had dreamed. But what if they couldn't be in love and still be friends? Was it worth the trade? She gripped her father's hand again. "Could we focus, please? If it isn't too much trouble?" He pursed his lips, one of those British glowers in place. "He hurt you." She dragged in a long breath. "Because everything's changed." "Everything does." The offense faded from his eyes again, and that hard-won, ready-to-be-amused peace replaced it. "That's no reason to scare a decade off your father's life and break his heart with your tears. Change can be so very good." She settled back against the couch and rested her gaze again on the crackling fire. Some change was good, yes. Coming home. Finding Papa. But what felt like a risk three months ago felt safe as a pony in contrast to this. "Sometimes. Sometimes it can tear us apart. How are we to know which is which?" "We can't. But we can pray." He cradled her fingers between both his hands, effectively pulling her gaze back to his. His eyes shone with certainty. "And know that whatever comes, we're not alone anymore. And that, my dear, certainly changes everything." Brook managed a smile, then looked again to the fire. Before, it had always been Justin beside her through the hard places. Her cold supposed-father ignoring her existence. Now she had Papa to work through the questions with her. Her fingers found her necklace and freed it from the collar of the livery jacket so that she could toy with the dangling pearls. Questions, so many questions plaguing her. And Justin still didn't even know what they were. # Eighteen FIVE MONTHS LATER LATE APRIL 1911 The sun shone through the window, the birds chorused their pleasure, and Brook dug her fingers into her palm. He would not leave the quicker if she shouted. Tempting as it was. "You cannot honestly have expected anything different, Lord Pratt." He prowled about her mother's drawing room, a stain of shadow against the jewel-toned fabrics. Though he smiled, it could shift to a snarl at any moment. "I beg you to reconsider, my lady. I can give you all you could ask for in a husband. Independence, respect, affection. And you could stay here, in the area you've come to love so well. When we combine our estates, we will be the single greatest landowner in Yorkshire." When? When they combined their estates? Un. Deux. Trois. She dragged in a seething breath. "We both know it's that property you want, not me." His gaze raked over her much as it had her first morning by the sea. At once hot and cold. Lingering and dismissive. "I assure you, my lady. I want both." Had he been close enough, she would have slapped him. "Watch yourself, Pratt." "I would rather watch you." Kitty would call the note in his voice charm—she must have been deaf to the conceit and greed. He slid around the wingback chair with the look of a panther readying to pounce. "Come, darling. Who else would overlook your eccentricities?" She bristled when he motioned toward her trousers. She only wore them riding, and only since the split skirt was ruined with blood and mud. "I don't much care if anyone 'overlooks my eccentricities.'" She planted her hands on her hips to prove it. "Let them think what they will. I will be who I am, and I will make no apologies. And if that means I eschew society and forgo the marriage mart . . . well, what a shame." Something flashed in his eyes, dark and impatient. "Do you think Stafford will come home and sweep you into his arms and make you a duchess?" Silence was the only answer she would give, along with a glare she hoped was stony and cool. But her fingers dug deeper into her palms. "But why would you want that?" The corners of his lips pulled up, though she wouldn't insult the word smile by calling it such. "You've the shared history, I realize. But you must have seen the man he's become. No room in his heart for anything but the duchy. He'll be like his uncle—cold, hard, unbending. A wife for the sole purpose of providing heirs, a mistress on the side whom he can dismiss at will. Safe and controlled and measurable." He prowled closer. "Does that sound like you, my dear? Safe and controlled and measurable?" What she wouldn't give for another six inches in height, so she could meet him eye to eye. A narrowing of them would have to suffice, and a tilt of her chin. "You know nothing of us." And given that she didn't know what she wanted when it came to Justin, Pratt certainly couldn't. "I know you write to him every week. I know he hasn't written back to you even once." A tempest crashed over her. More aimed at Justin than Pratt, but as he wasn't handy, she unleashed it where she might and slashed a hand through the air. "How could you possibly—" "Have you never actually spoken to the postmaster in Eden Dale? Friendly chap. Talkative." She drilled a hand into his shoulder, pushing him back a step. "To whom I write is none of your concern!" His dark eyes snapped, and he closed his hand around her wrist. "Now who had better watch herself?" Stupid. She should have retreated. Now when she tugged, his fingers tightened. "Release me." Instead he raised her wrist higher and placed a kiss on her palm. Her skin turned to ice. Kitty was due any minute, and if she came in upon this, it would break her heart. "I said—" "I heard you." So calm, so mocking. He lowered her wrist but didn't let it go. "Or do you think to turn to Worthing? Don't put your hopes there, my darling. He may flirt with you as he does every other female, but he doesn't intend to marry you. His estates are still flush from his mother's dowry, and he enjoys the hunt far too much to settle with just one woman before he must." Her nerves snapped. Without question, Brice flirted too much, with everyone. But she and Papa had stayed two weeks in Sussex with the Duke of Nottingham's family last month, and she had spent countless hours talking with Brice. There were moments when it wasn't just flirtation. Moments when he seemed to gaze into her very soul. Moments when she wondered if his lips would ignite the same fire Justin's had . . . and moments when she was sure they wouldn't. "You know nothing about my thoughts. Don't hazard to guess." "I know more than you think." He finally unfurled his fingers, letting her go. Stepped to the window. "I'm not a bad option for you, Brook." She had never given him permission to call her that—but pointing it out felt weak. "I don't need an option, Lord Pratt." His eyes narrowed at whatever he saw out the window. "I daresay you will when Kitty is through and your reputation is slashed to ribbons." He nodded in the direction of the drive. "You think to frighten me with that threat? Kitty is one of my dearest friends." Brook moved to a different window and spotted the familiar Rushworth carriage. An open one today, displaying Catherine in all her splendor. No Rush beside her, which meant no leash on her tongue. It always made for a more entertaining visit. Though it did occasionally make Brook wonder what her cousin said about her when she wasn't in the room. "My cue to disappear, I think." Pratt spun and reached for the hat he had tossed to a table when he barged in fifteen minutes prior. "And if you would deny having seen me . . ." Brook sent a pointed look toward the stables, in front of which his Benz was parked. "Say I've been with your father the whole time." "And why should I?" "I saved your life—now I'm calling in the favor." Justin would have said it with irony. Brice with mirth. Pratt delivered it with nothing but harsh sobriety as he reached the door in a full-length stride. She shook her head and sent a glance to the painting from which her mother reigned. Forever captured in the Frederick Worth gown Brook had discovered in her wardrobe, still beautiful with its deep green fabric shot through with gold. In the painting she wore the emerald and diamond necklace Papa had first shown Brook. And the bracelet she had worn to the hunt. The one Lady Catherine had admired. Rubies and diamonds. Actually, Kitty always took note of whatever jewelry she wore. In part it seemed polite interest, but Brook had begun to wonder if her cousin believed those tales her mother told . . . or if, perhaps, Henry Rushworth—who had never replied to their letter—had taken something from his brother and sister-in-law and sent it to Brook's mother. It would explain the letter—and Catherine's veiled interest. Fire eyes. Rubies? Diamonds? It seemed it ought to be one or the other. Unfortunately, that barely narrowed down her mother's collection. Mr. Graham cleared his throat from the drawing room door. "Lady Catherine Rushworth, my lady." Brook glanced down at her trousers. She could change first, but she still hoped to have time for a ride this afternoon. "Show her in, Mr. Graham. Thank you." The butler bowed and disappeared. It was scarcely half a minute later that Catherine stormed in. "Where is he?" Brook sighed. "And a sunny good-day to you, cousin." Lady Catherine narrowed her eyes. "I saw his car." Brook nodded, pressing her lips together. When would Catherine see that Pratt wasn't worth her affection? That he would do nothing but hurt her? "I don't know where he might have gone." For all she knew he was cataloguing the silverware he intended to add to his estate. Though if he tried it, Mr. Graham might personally give him the heave-ho. Worth seeing, that. Catherine advanced with startling speed. No amusement sparked in her green eyes today, no promise of biting jests or shared laughter. Just fierceness. Desperation. "You'll not have him." The him must still be Pratt. Though why Catherine thought Brook wanted him, she couldn't say. "On that we agree. You know I would never—" "Don't try to placate me. I know very well he was going to propose before you leave for London, but I am the one he will be marrying. Make no mistake about that." Brook almost put tongue to a flippant answer, but that glint in her cousin's eyes stilled it, made her opt for seriousness instead. "Catherine, I assure you I have no intention of marrying Pratt—or any man who is out only to get Whitby Park." Catherine lifted her chin. "At least you have brains enough to know that's all he wants—all any man will want, once the gossips in London realize you're cut from the same cloth as Whitby." Brook took an abrupt step back. "Why are you acting this way? I thought—" "We were friends?" The glint in her eyes was ice, hard and deadly. "For a girl raised by a prince's mistress, you can be charmingly naïve, cousin." Brook staggered another step back. She had spent more time with Catherine than with Regan or Melissa, had thought . . . All these months, she had ignored Papa's mutters about the Rushworths, had chalked it up to a lingering animosity toward his would-be rival for Mother's affections. "What are you saying?" Catherine shadowed her, no light in her eyes to speak of life. No curve to her lips to say she was joking now. "I've suffered your company long enough. Listening to you go on and on about that stupid beast of yours, your ridiculous cars, your precious duke—and now Worthing to boot. But I've had enough. Your family has taken enough from mine. First the Fire Eyes, and now Pratt." Though the glare hadn't cooled Brook's blood, the words did. She felt sculpted from ice. "The Fire Eyes?" She couldn't move. It hurt too much. "You? You were the one who hired him?" Lady Catherine lifted her perfectly plucked brows. "Hired whom, darling? I can't think what in the world you're talking about." Brook's fingers curled into her palms, finding the marks they had left from Pratt's visit. The Rushworths had been in the area that night, hadn't they? Somewhere in the muddle of memory, she remembered spotting their carriage leaving Delmore. But how, how could her cousin, her friend have a part in it? "I could have been killed, and I don't even know what these Fire Eyes are!" Before she saw it coming, a hand connected with her cheek, and Catherine followed it with a push that sent Brook stumbling back into a chair. "How stupid do you think I am?" She stood again, though slowly, ready to defend herself this time. Her cousin spun away. "Did you honestly think it would look like a coincidence, sending your duke off as you did, to the very place they were found? You're just like your mother." She wheeled around again, looking as though she would lunge. Brook stood prepared. Perhaps that was why Catherine stopped and contented herself with another snarl. "You see how it ended for her. Don't make the same mistake, my lady." Now it was Brook who lunged, though Catherine charged for the door. She caught her by the elbow in the threshold. "What are you talking about? What happened to my mother?" Catherine jerked her arm free and produced a heartless smile. "How am I to know, cousin? I was not yet two when she suffered that unfortunate accident. But I will say this." She stepped into the hall and dragged a scathing glare down Brook's riding habit. "Your family seems to have bad luck around horses. Perhaps you ought to take more care." Oh, she would take care all right. She would take care to get to the bottom of whatever this Fire Eyes business was—and would assuredly not be intimidated by the likes of Catherine Rushworth. Tempted to slam every door she could find, Brook stormed for the stables. And told herself the tears burning her eyes were from anger and not hurt at the betrayal. Deirdre would have screamed, had the hand over her mouth not cut off all her air. It took her only a moment to recognize the hand, the arm, the familiar cologne. Pratt. Her panic increased when he pushed her into the empty parlor and clicked the door shut behind them. Drawing a steadying breath in through her nose, she reminded herself that he was like any other beast, able to sense her fear. Calm was her only hope. His fingers peeled off her mouth, and he spun her around. Eyes hard and dark as jet, he backed her into the wall and trapped her there with an arm on either side of her. "I'm done being kind." His voice came out low and deadly. "She refused me." Deirdre's whole body shuddered. "I tried. She is willful and—" "I know what she is." One of his hands closed around her throat. Not squeezing, but making it clear he could. His gaze burned into hers. "I have a man in your village, ready to light a torch and toss it to the O'Malley roof one night if I but give the word." He didn't need to tighten his fingers—his words choked her, and she had to shut her eyes against the sight of him. Though then the images of her mum and siblings swam before her, from strong, near-grown Killian toiling in the fields, all the way down to little Molly. "What do you want from me? I've done all you asked." Stolen things. Told him things her ladyship would hate her for telling. She would get sacked, possibly arrested, if ever the Whitbys discovered it. But she had risked it, because she had known his favor would turn to threat if she refused. That the wee ones would pay for it if she tried to do the noble thing. He eased away, dropped his hand. "Nothing yet. But when I ask, I want no questions. I want obedience. Are we understood?" Her stomach churned, and bile rose in her throat. A blank check for evil—that was what he demanded. And she had no choice but to nod. Justin pressed the brake longer than necessary. Waited, though the carriage had long since passed, to turn the wheel. And when turn it he did, it was with a sigh. Brook must be furious with him—no, worse than furious. Hot anger would have been banked, cooled. She would be ice. Eden Dale lay behind him, Whitby yet ahead, but he let the Rolls-Royce motor its way up the long, winding drive to Whitby Park. He had already done his homework. Phoned Thate . . . and Cayton . . . and Aunt Caro to be sure no one had heard conflicting information. To guarantee that, indeed, the Whitbys would be at home yet today, not already in London for the Season. That was part of the plan. Catch her here, where she was most comfortable. Where he could more easily get her alone. That was critical. Utterly critical to his plan. Given the beautiful spring day and the looming departure, he was hopeful he could find her out of doors. The gardens . . . the seafront . . . anywhere he could come upon her by herself. Where he could charge right up to her, turn her around, and kiss her. By his calculations, he may well end up with a fist to his gut or a palm slapping his cheek. But that would be fine—it would get her back to fury, take her from ice to fire. From there, it would be a matter of apology and confession. "Please, Lord." His chest had felt so tight for months. Too many times he had relived that kiss outside his townhouse, the way she had clung to him, met him measure for measure. He could win her yet. He could. There was a fire inside her for him, and he could fan it, turn one kind of love into another. He hoped. But then, all the letters he wrote, pouring out his heart . . . and she had never written him back. Not except that once—a letter that had made precious little sense. A collection of still and again and yet that appealed to a context he didn't have. It seemed she had written others that hadn't reached him. "Please, Lord." To think that she had instead chosen not to write, not to reply—no. He couldn't accept that. It would undo him. Even if that one letter had mentioned plans to go to Sussex to spend a fortnight with the Duke of Nottingham's family. He set his mouth, beat back the fear. They would bridge the gap. They must. Pick up where they'd left off, as they had always used to do. A kiss, a punch, some heated Monegasque shouting . . . then hopefully another kiss, softer words, and the months would melt away. He would— He slammed on his brakes as he came around a bend, and coal-black forelegs pawed at the air beside him. The hooves barely missed taking a layer of paint off his door as the horse's rider pulled the beast back. His heart wouldn't slow for an hour. "Where the devil did you come from?" He asked the question of the horse . . . then noted the hands pulling on the reins. Feminine, elegant, perfect. He took the car out of gear and leaned back in his seat. Brook focused first on calming the horse and then lifted sparking green eyes to him. "One might ask you the same question, Duke." Oh yes, she was angry. And he had to smile. She was hatless, and the wind had whipped many a curl free of its chignon. Her habit was a deep green, bringing out the emerald of her eyes. His smile turned to a grin. "You're wearing trousers." She patted the horse's midnight neck. "That's what you say to me after a five-month silence? 'You're wearing trousers'? Really?" He chuckled and turned sideways to better look at her. It had been six years since he'd last seen her in them. And his castoffs had never hugged her legs quite like these did. "They look good on you. Though I have a feeling your aunt disagrees." Usually such an observation would have won him a grin, a laugh. Apparently she was in no mood to be amused today. She gathered the reins as if ready to turn the horse back into the open land. "Brook." He reached out, though she was too far away. He needed to touch her, even if only to put her hand on his arm. But he was in his car, she on the horse. Obviously a kiss could not bridge the gap. Lord, give me the words, please. I beg you. Help me make this right. "I know you're angry with me." She breathed a mirthless laugh. "Oh. Oh yes. But don't flatter yourself—you're not the one who sent me out here in a rage today." "Who did?" At her glare, his hand fell to the door and rested on the sun-warmed metal. He sighed. And latched his gaze upon the one thing that might draw her out. "This is Oscuro?" "Oui." The French warmed him. Let him smile. "He's magnificent." Nearly as magnificent as his rider. "You broke him." "Never." She rubbed a hand up the stallion's neck again. "But we've reached an agreement. I let him taste freedom, so long as he does so with me on his back." He still thought it had been foolish of her to try—but he wasn't about to say so again now. "Whatever you want to call it, you succeeded. Just as you said you would." She lifted her chin and spun Oscuro to face the house. "Some of us believe in keeping our word." A dagger obviously aimed at him . . . though he wasn't quite sure what he had done now, or failed to do, to deserve it. He put the car back in gear so he could keep pace when she clicked the horse into a walk. "I suppose, then, I should have made you promise to actually answer my letters." "Your letters?" She drew Oscuro to a halt again and sent Justin the look she had always called the English glower. When had she picked that up? "How am I to answer what does not exist? Though I suppose I oughtn't to be surprised that you yet again chose silence when you said you would write. Even after you . . ." After he kissed her? He took the car from gear, set the hand brake, and let himself out. "I wrote to you every other day at the start. Every week at the end, though it was disheartening never to hear back from you." "What?" She shook her head, though her glower shifted to a frown. "I wrote to you every week. But never once got a letter in return." Unease went tight inside him. Much like his chest had felt these five months when thinking of her, but more. More urgent. "You sent them first to my solicitor, for him to forward?" "Yes! Like Thate and Cayton—I checked the direction against theirs." Her gaze went distant for a moment, then she dismounted. Justin shook his head. "They can't all have gone astray." "No." She shoved a few stray curls from her face and spun toward the house, then all the way around, toward the village. "Someone has been tampering with my post." The way she said it, so calm, so sure—with dread certainty instead of outrage—tied another knot inside him. "Are the servants still unwelcoming?" She shook her head, her eyes distant. "They have been fiercely loyal and protective since the attack." The attack. Most of the time, he avoided thinking of that, or remembering the bruising on her face in November. He could not dwell on it if he wanted to remain sane. He reached out again, this time able to rest a hand on her shoulder. The feel of it was familiar and sent warmth flowing through his veins. It turned to ice when she shrugged him off. He swallowed down the hurt. "Thate said the man's identity is still in question." "He was using the name Fitz Jenkins, but it wasn't his real one." She turned back to the horse. "We need to talk to my father about this." This was not how he had envisioned their reunion going. He couldn't exactly follow conversation about her attacker with a passionate embrace, but he also couldn't just follow her up to the house like this. He reached over her to put a hand on the saddle, effectively blocking her way. When she sent him an exasperated look, he met it with a smile. "Will he let anyone else ride him?" Her eyes glinted. Her brows lifted. "The right someone. My father has, and one of the jockeys. Though only one." Justin inclined his head toward the Rolls-Royce. "Trade?" She looked to the car, and the corners of her lips curled up—the exact smile she'd first worn when she spotted the car idling outside the palace in Monaco. A soothing reminder that despite new English glowers, she was still Brook. When she returned her gaze to his, challenge gleamed. "If Oscuro will allow it." Nearly fourteen years of friendship, and he had to prove himself. But then, if he had his way, friendship would be only part of what they would have from now on. He grinned and backed away, toward the horse's head. Holding out a hand for the beast to sniff, he stroked the other down his neck and whispered in French into his ear. "I need your help, boy. I don't want to disappoint her—you understand that, I think." Oscuro nickered and bumped Justin's hand with his nose. "He isn't biting you—congratulations." Brook handed him the reins and took a step toward the car. Justin reached an arm to halt her. His hand settled on her waist. "Brook." Rather than look at him, she stared at the car. "How did your trip go, Justin? Did you set everything in order?" "I did." Though it gave him no pride to say so. The entire time he was away, he kept hearing Aunt Caro and Brook in his head, telling him it wasn't enough, not when he had hurt his family for it. He kept seeing his father with grief-stricken eyes, pulling him close. He kept remembering Uncle Edward, who never once looked at him with any warmth, saying, "Focus first on Stafford, boy. People come and go, but the land stays forever." God had dealt with Justin while he dealt with business. Dealt with him for trying to strengthen the outward when he should have been giving the inward to Him. He'd spent so many hours on his knees these past two months, he had memorized every stitch of the quilts over which his hands had been clasped. All of which he'd told her already . . . none of which she knew. "Good." Her voice came out quiet, but by no means soft. "I prayed you would succeed. I prayed . . . I prayed you would find whatever it was you needed." "I did." He wanted to draw her close, but she stood so stiff, so immovable. "I found that it was here all along. Which I knew, but . . . perhaps I needed the time alone with the Lord. To fully understand who I needed to be, and who I must not be, at all costs." She looked up at him now, though not as she used to. No smile teased her lips, no sparkle lit her eyes. "Good." That was all she could say? His fingers pressed against her waist, though she wouldn't come any closer. "Brook . . ." "It is my turn now, Justin." She stepped away, and her eyes went from blank to snapping—but not with the love he'd hoped to find in them. With determination. "My turn to find some answers. They've been waiting far too long." She opened the door of the car and slid in, scarcely smiling as she ran her hands over the wheel. So very unlike her. All of it, so very unlike her. He turned back to Oscuro. The horse at least proved she was still his Brook. Chasing the dangerous when a sane person would have chosen a known quantity. Never settling for the mere exceptional when the magnificent was just out of reach. The beast let him mount, though he shifted, skittish, as Justin settled his weight in the saddle. He murmured a few French nothings, as he always did with Alabaster, and Oscuro tossed his head in seeming recognition of the words. Brook put the Rolls-Royce in gear and, finally, shot him a look he knew well. Challenge. "I bet I can beat you there." He lifted a brow. "You think my car is faster than your horse?" The familiar, blessed, impish smile possessed her lips. "I think its driver is faster than his rider. Ready?" He twisted the reins around his hands and crouched forward. "Allons-y." # Nineteen Brook had meant the ride to clarify. She had meant to come back inside with a mind cleared of Catherine's insinuations and Pratt's ill-placed proposal. She had meant to put the hurts and suspicions and outrage in their proper places before she spoke with Papa. To her dismay, she wanted to stomp and scream and cry as much now as she had two hours ago, though it was no longer only the fault of her neighbors. Or maybe it was. The postmaster was talkative, Pratt said. Was he bribable? And was Pratt low enough to steal her mail to keep her from communicating with Justin? The question made ice chase the fire in her veins. She handed the key to the car back to Justin—and plotted how to get it from him again when she was better able to enjoy it. When the questions weren't buzzing so loudly. He slid it absently in his pocket, his gaze still on Oscuro as Russell led him into the stables. "Magnificent. Is he ready for the races?" "He will be by summer." She couldn't help the lift of her head, the tilt of her chin. "Ready to admit I knew what I was doing?" When he shot that grin at her, her stomach flipped. Which unsettled her all the more. She had hoped that by the time he returned, she would know her own mind and heart concerning him. Instead, she was more confused than ever. She wanted to kiss him again, to test her reaction . . . and yet wanted to steer far clear and force their relationship back to what it had been before. "I never doubted you knew what you were doing," he said. "I just didn't want to see you get hurt in doing it." "You started it." The tease slipped out. And somehow, her heart went cold in its wake instead of warmer. No, not cold. Sad. So much had changed. And communication about it had been stolen from them. Now where were they to go? Justin chuckled and offered his arm. "I taught you to ride astride, not to break horses. To shoot at targets, not the weapon from a villain's hand." "You gave me a taste for risk-taking." "My lessons were hardly risks." "They felt like it at the time." She slid her hand into the crook of his elbow. He must have kept active while he was away—his arm was firmer than before, the muscle larger. His gaze went to her wool-clad legs. "Those I like." The way he said it, the way he looked at her . . . he was flirting. Justin Wildon, Duke of Stafford, her oldest friend, was flirting with her. And she could think of nothing clever to say. Brice's words she could parry with skill, but it was different with Justin. She could think only of the inane. "I have your hat—you left it at Aunt Mary's." That night. "The End of an Era." He reached up as if surprised to find it not on his head and then grinned again. "Keep it. I've given up on the thing." Then he cast a glance over his shoulder again. "Finest stallion I've seen in years. Are you studding him out? I'm considering breeding Alabaster this year—" "And now we're going to talk about horse breeding?" She shook her head and pulled him toward the house. "Some would say a lady shouldn't discuss such things." "Some are idiots. Will your father make all the decisions for Oscuro while you sit quietly by?" He still knew how to make her smile, even if it faded quickly. "We'll agree to no fees until after the races, when he proves himself a champion." "So you can gouge me? Does a lifetime of friendship not gain me a discount?" The feigned outrage warmed some of the cold spots inside, though it didn't last. Not when her eyes fell to the bracelet on her wrist. She had bigger matters to attend than Oscuro. "We'll see. In the autumn." Her father would be in the library. She aimed them toward that door. Justin stepped into her path. "Brook, might we . . . have a moment first?" His eyes were a bright sapphire today, and the April sun caught his hair and set it alight. His jaw had gotten more chiseled in his absence, his shoulders broader. Her chest went tight. "What?" He sighed and glanced over his shoulder. Took her hands. "I was hoping . . . for some time alone with you before we join your father." Yes, she had dreamed of this, of him, while he was away. But the other dream always overshadowed it. Thunder and lightning and darkness. "You had time." He gave her half a smile, crooked and so very charming. "It didn't go as I planned." "And what did you plan?" His gaze dropped to her lips, warmed her. He eased closer, their clasped hands still between them. "Shall I show you?" Yes! "No." Saying it made the pressure compound behind her nose and eyes, but she held his gaze. Let go his hands. "There is too much you don't know." "You'll tell me. I'll tell you. We'll do as we've always done and—" "No." She had to look away. Otherwise she'd forget how to speak, gazing into those familiar eyes, forget the heavy truth that had settled in her heart during her ride. A truth too-long forgotten already. "It isn't like it always was." "No. It will be better." His voice thrummed over her nerves, and they caught fire when he feathered a hand over her cheek, into her hair. "We can make it better." "Justin—" His lips silenced hers, held them captive. A soft touch that promised so much more—that took her back those months to London, then forward again through the many nights she had lain awake agonizing over whether they could make each other happy or would destroy each other in trying. Perhaps if she could relax, give herself over to the sensation again, she would know. But her mind wouldn't still. Why hadn't he listened to her? She couldn't think about this now, not clearly, not given Catherine's hissing words and the threats in Pratt's eyes. Justin pulled away. His eyes were dark, his brow questioning. "Brooklet?" She could only shake her head and step around him. He should have listened. Or come back yesterday, or tomorrow, or even an hour from now. "Brook." "You don't understand." But he would in a moment, if he followed her. Which he did, as she strode for the library door and opened it. "Brook!" He reached for her arm. She all but leaped into the library to avoid his fingers. Her father looked up from behind his paper and took to his feet with a smile. "Duke! You made it home when planned, I see." Justin sighed and pasted on a smile. "I did, yes. Forgive me for arriving without warning." "You are always welcome here. You know that." But rather than striding forward with a hand outstretched, Papa frowned and moved toward Brook. "What is the matter, my dear?" "It was Catherine." Neither of them could know what she referenced, so she drew in a breath and shoved the flyaway curls from her face. "She's the one who hired my attacker—she must be. She said we took the Fire Eyes from her family." "Fire Eyes?" Justin looked from her to her father. Papa came to an abrupt halt, thoughts whipping through his eyes. "She said that?" Justin shifted, putting himself halfway between them. "What are the Fire Eyes?" Her father looked to him. "We are not entirely certain—jewels, but that is all we could determine. Did she say what they were when she mentioned them?" "No." Brook's hands curled into fists that did nothing to squeeze out the hurt in her heart. "But it links her to that man. You need to call the constable—" "She would deny it." Sighing, Papa set aside the paper still in his hand and came over to clasp her shoulder. "And I had him looking for a connection to them when we first suspected it was Rushworth jewels." She snapped upright, her mouth agape. But Papa had never trusted them. Not like she had. He shook his head. "There was nothing." "He nearly killed me, and there's nothing we can do?" Her father looked deep into her eyes, his brow still drawn. "Tell me everything she said to you." She did so, word for word, including the slap—at which point Justin lurched a step forward, outrage snapping in his eyes. "She struck you? And you let her?" Leave it to him to make it sound like her fault—and make her want to smile about it. "I assure you, had she advanced again, she would have taken a fist to her upturned nose." Papa sighed. "Then what?" "Then . . ." Swallowing did nothing to make the lump in her throat go away. "Then she said she was not so stupid as to think Justin's travels a coincidence, that he was in the place they originated. That I was just like my mother, and look where it led her. She intimated . . . Papa, I don't think the carriage accident was an accident. Not entirely, anyway." There. She'd said it, that truth that had pounded her brain with every thunder of Oscuro's hooves. Her father spun away, muttering a word she couldn't quite make out but that she suspected was a curse, given the way he seemed at a loss as to what to do with his hands. After a moment he clasped them behind his back in that way of his. "Catherine was trying to upset you. It was a fierce storm. The rain had wrought havoc on the roads. The carriage overturned. A tragic accident, nothing more." Storm? Thunder and lightning and darkness. Brook, hands shaking, sank to the edge of a chair. "No one ever mentioned that. Is that what I've been dreaming of all these months? The storm that killed her?" Papa looked at her as if the very question would make him unravel. Justin, when he stepped into view, instead looked at her like her sanity already had unraveled. "You were far too young to remember anything from that night." "I know that." And she didn't need him to make her feel ridiculous. She pivoted, strode to a shelf, though all the titles upon it blurred together. "It has always been so vague. So frightening. Impressions, nothing more. But you cannot know how it has tormented me." Justin held up his hands. "I can imagine. But focus on the facts for now. This is a serious accusation you're lobbing Lady Catherine's way. And linking it to your mother's death, which she could not possibly be responsible for, will do nothing to gain you believers among the constabulary." She wasn't trying to get the constable to believe her, though—just them, the two men who mattered most. Dragging in a long breath, she fixed her gaze on her father. "What about Catherine's parents? Her father—Mother's cousin?" He shook his head. "They were never close but never seemed at odds. His wife was jealous and contentious, but she would never have taken it so far." "But how far would she have taken it? Perhaps the accident was an accident, but what sent her on that journey?" Brook splayed her hands, begging them to understand. To believe. "Why would she leave here, with me, with the letters from you? Why, when Collette arrived, did my mother tell her to take me away and not to find you? Why?" Papa shook his head, the muscle in his jaw ticking. "Questions I have asked myself too long." Justin eased forward. "The more immediate question is what Lady Catherine wants, and how far she will go to get it. The hint about my having traveled to these Fire Eyes' origin is little help. I was in Africa and India both, and both are rich in mines of all kinds." Brook folded her arms over her middle. "Whatever they are, it seems my mother had them, perhaps unwittingly. It is all linked. That is certainly no coincidence." For a long moment, neither man made any response. Then Justin's eyes went dark. "You didn't write to me about any of these concerns, did you?" She shook her head, though his meaning still made her stomach churn. "I told you in November there were things I could not put in a letter." "Good. I think we need to operate on the assumption that someone has stolen your correspondence purposefully." "Stolen your—" Papa cursed again, louder this time. "Why have you said nothing of this to me, Brook?" "We just realized it." Justin shoved his hands into his pockets. His shoulders had edged back. His spine had gone straight. He looked, standing there in a casual suit of clothes, perfect confidence in his every line, like a duke. "I wrote her dozens of letters, she says she got none. She sent me dozens, I received only one." He hadn't mentioned that. "Which one?" His eyes flashed. "It was dated the twenty-third of February. A week before you were set to go to Sussex." He said Sussex as if it were the birthplace of all annoyance. "When we were still in London for Mary's birthday." Papa's eyes went calculating as he thought through it. "The one you posted yourself, that day we went out for a drive." She could see the suspicions mounting in his eyes, as they had in her mind. The implications were unmistakable—she had sent other letters from London and Sussex. But they had not reached him, either. The postmaster in Eden Dale could hardly be blamed. She sank onto the edge of Papa's favorite chair. Justin paced to the unlit fireplace. "Which servants travel with you?" "My valet, Lewis. Her maid, O'Malley. Clark, who drives the carriage with them and our luggage. That's all." Justin had turned back toward them but did not approach. "Does the maid still dislike you, Brook?" Her father sucked in a breath. "She . . . ? Brook! What else have you not told me?" A headache was gathering behind her eyes. "It was nothing to burden you with, Papa. The servants are all so loyal to you, it took them a while to accept that I was not out to steal all that is yours. That is all. Je promets." Her promise didn't seem to ease him any. "How long is 'a while'? How long did they not accept you after I specifically instructed them to welcome you as their mistress?" Given the paternal fire in his eyes, he might call the servants in and dismiss each and every one of them, even though at this point they all doted on her. Or so she thought. "Focus, Papa. We have only three suspects right now, and I daresay, whichever of them did it, it wasn't a matter of dislike. Pratt said something today about how I'd never received any letters from Justin—intimating he got the information from the postmaster." Her father narrowed his eyes. "And what was Pratt doing here?" She waved a hand. "Proposing. But the point is that he may have bribed—" "Proposing?" The twin responses from Justin and her father made Brook roll her eyes. "Oui, and I, of course, fell at his feet in adoration and said yes. Because we all know how much I like him. Again, could we please focus, gentlemen? On the possible bribery?" Papa tugged on his waistcoat. "What kind of man proposes to a young lady without first speaking with her father?" "The kind who knows well her father would refuse his blessing." She managed a smile for him and resisted the urge to glance at Justin. "Bribery, Papa." "Hmph." He stalked to the window, glaring in the direction of Pratt's land. "Lewis has been with me for twenty-five years. I cannot think he would do this—he has no family to support, and I have set aside a living for him when it is time for him to retire. But . . . those years have established a friendship, and if he believed you a pretender, as those who came before . . ." "O'Malley's family is struggling." She didn't want to say it, to admit it. Didn't want to think it could be Deirdre, with whom she'd finally established a rapport. "I've been sending extra funds, but she doesn't know that. I know little of Clark." "I know little more—he only joined us last year. O'Malley has been here nigh unto eight." Her father nodded, staring into space. "We will look into all of them. We cannot afford to assume." Justin was still glowering. "Have we two issues here, or one? Are Pratt and Lady Catherine working toward separate goals—he, you and she, the Fire Eyes—or are they somehow working together?" Brook drew in a breath and leaned back into the chair. "Pratt would have no claim on any Rushworth jewels. And Kitty—Catherine." She wouldn't use the familiar name, not anymore. "She's in love with him, so she certainly would not aid him in his pursuit of Whitby Park. They must be separate." "I agree." Justin nodded once, then shook his head. "You always have had a knack for finding trouble, Brooklet, but this . . . Pratt is obviously not opposed to stooping low to get his way. And if Lady Catherine would really hire a man to threaten you over jewels, what would she do because of Pratt's affection for you?" "It isn't affection—it's greed. But your point is valid." She raised a hand to rub at the muscles gone taut in her neck. So many hours spent laughing together. So many times she had listened while Catherine pined for Pratt. How could her cousin think Brook low enough to pose a threat to her relationship with him? "They may be unrelated at the core, but that does not mean that one will not exacerbate the other. Pratt thoughtfully warned me that Catherine will try to rip apart my reputation in London. I didn't believe him then, but . . ." Papa's face finally relaxed. "We can only hope. If you complement her gossip with that horrible pink thing your aunt commissioned for your debut, we might have reason to come home again by June." No doubt she would be ready well before then. Brook grinned. "I plan to wear the gown Grand-père sent. But have no fear, Papa—I'll not force you to too many balls." "Your aunt will try to have us at something every night of the week." "United, we can stand against her." Justin had lifted a brow and seemed to squelch a grin. "Pink? You look terrible in pink." "Thank you ever so much for noticing." His chuckle sounded like memories, indulgent and carefree. "You've always been quick to proclaim it—I don't know why your aunt would ever dare try to put it on you. What did the prince send?" "Oh, the loveliest gown." It seemed trivial, in light of all else they needed to talk about. And yet not, because it was a gift from her grandfather, one that proved he still thought of her, still loved her. "Pale green, with a blue overlay of beading. Wait until you see it." Justin smirked. "Green? For a debut? Only you would dare wear something other than white or pale pink, Brooklet." Then his eyes shifted. They went softer, and that flirtatious gleam entered them again. "Don't forget you've promised me your first dance—after you open the floor with your father, of course." "I haven't forgotten." Her smile, though, would only stretch halfway before it felt too heavy. Too false. Sighing, she met her father's gaze again. How was she supposed to worry with filling up her dance card when her mother's death still loomed over her, when mysterious jewels taunted her, when friends declared themselves enemies, when threats seemed to lurk everywhere? Papa moved to the chair and rested a hand on her shoulder. "She has been gone this long, my dear. Much as we both need the answers, there is no urgency." Because she must, she nodded. But she couldn't shake the feeling that in fact there was. # Twenty Twilight possessed the heath by the time Justin rolled to a halt at the carriage house of Azerley Hall. He had dined with the Edens, but when Whitby issued an invitation to stay, the pressing upon his spirit said he shouldn't. He still wasn't sure if Brook had looked disappointed or relieved. He still wasn't sure if he was disappointed or relieved. He parked his car, let himself out, and trudged his way toward the front door of his cousin's house. The drive had not, as he hoped, helped him collect his thoughts. They were still awhirl with it all. Proposals from Pratt. Threats from Lady Catherine. Something called the Fire Eyes. And she hadn't kissed him back. "Justin." He started at Cayton's voice. Looking up, he could barely make out his cousin's form at the edge of the garden. "James?" "Mm. Join me? I just ordered some wine." Out here? The evening had turned cool, but the moon held court in the heavens, and it was rare enough that his cousin actually asked for his company. Justin altered his course, thankful he had shrugged into his great coat for the drive. "Of course." He passed through the opening in the hedge as Cayton sat at a small table, in one of two chairs. His cousin motioned toward the second. "I need to talk to you." Justin's stomach went tight as he pulled out the cold metal seat and lowered himself into it. "Why does that not sound pleasant?" Cayton sighed and folded his arms, shirtsleeves gleaming white over his chest in the moonlight. "You are going to London tomorrow?" "Yes. You?" "Soon. But I . . . I need to go to Gloucestershire first." At that, Justin frowned. Aunt Susan was there with Aunt Caro. But they said they were traveling tomorrow too. "To Ralin? What do you need? We can phone the castle and have it sent with your mother." A servant emerged from the house, bottle of wine and goblets on a tray. After depositing it on the table, she scurried away. Cayton said nothing while he poured. Justin waited. Accepted a goblet, took a sip. His cousin's next sigh gusted forth. "I'm betrothed." That brought Justin's spine straighter, though he had been ready to try to recline against the wrought-iron back. He smiled—halfway, until he realized that Cayton didn't. "When? I was not aware you'd seen Lady Melissa lately." Cayton held his glass but didn't drink. Apparently he would rather stare into its burgundy depths. "I haven't seen her since last month, when I was in Town." The frown pulled at Justin's brows again. "You have been engaged a full month and have said nothing? Someone would have mentioned—" "No." No . . . what? That Cayton hadn't been engaged a full month, or that he hadn't said nothing? It must be the first. "You asked her by letter?" "No." Sounding exasperated now, Cayton looked up. The moonlight caught on the whites of his eyes. "It's not Melissa." "It's not . . ." The words made little sense. Justin gave up on the wine. "You told me you were in love with her." And the saying of such a thing had been striking, when he read his cousin's letter over the winter—he had not thought them close enough to warrant such a confession. "I know. I am. Or was. Or . . ." Cayton set his goblet down with a clatter of crystal upon marble—leaned forward and rested his forehead in his hands. "I'm strapped, Justin. And a second daughter's dowry isn't going to help." "James—" "Don't lecture me. I know you put your estate to rights, so you no doubt think I can do the same. But I can't. It's been languishing too long, and I had no idea. I thought the steward had it well in hand—he's been taking care of everything since before I was born. But when he passed away in January and I looked over everything . . ." Now it was Justin's turn to sigh. "I was not going to lecture. I certainly cannot judge. But are you sure marriage is the answer?" Cayton snorted. "I have no other alternatives. It seems I don't have the luck of your father." "James—you've been gambling?" His cousin winced. "The horse races." A breath of laughter slipped out before he could stop it. "Perhaps you should have tried baccarat—that was Father's game." Not that Justin was actually advising . . . but his cousin knew that. Cayton sent him a lopsided, sad smile. "Too late. I've already sworn off it all." For a long moment, the only sound was the chirping of the frogs from the pond. Justin took another sip of the wine. "Who, then, if not Lady Melissa?" Cayton picked his glass up again too. "Miss Adelaide Rosten." "Rosten." Justin held his burgundy halfway to his lips. "The name sounds familiar." "It should—she is your neighbor in Gloucestershire. Her grandfather made his fortune in the mills." "And she is the heiress." Cayton nodded. "She . . . she is a sweet girl. Unobtrusive. I knew her as a child, though I scarcely paid any mind to her. She has no family left." Try as he might, Justin could not put a face to the name. "So it is official?" "Yes. We haven't made the announcement yet, but yes. I wanted . . . Before anyone else knows, I wanted to speak with you. Mother isn't happy with me, nor is Aunt Caro. And of course, if we're all in London, the gossips will soon pick up on it all, and Miss Rosten . . . She doesn't deserve to be lambasted. If you stood with us, it would go a long way toward smoothing things over." For Cayton, yes. No doubt it would. But for Justin? He ran a hand over his face. Brook would no doubt be furious on behalf of her cousin. One more thing between them, if he stood beside Cayton. But what choice did he have? "Have you told Lady Melissa?" "Not yet. I will as soon as I get to London. I realize this will put you in a tight spot with your baroness. If you . . ." "You know I will support you, James." Cayton's shoulders sagged. "I couldn't be sure. I know you hoped it would be neat and tidy for you. Thate married to Regan, me to Melissa, you to Brook." It would have. But he should have known better than to expect it. "Reality is rarely so tidy though, hmm?" "Indeed. Let us pray it is simpler for you and you can win her back." Justin had been reaching for his glass again, but that brought his arm to a halt. "Win her back?" Cayton motioned in the direction of Whitby. "Melissa told me she and Worthing are always exchanging letters, that she visited him in Sussex and had nothing but happy tales to tell." He took a drink, set his glass down again. "Don't underestimate your competition, cousin. When you didn't write to her, she had to turn somewhere." "I did write her. More frequently than I ever had before, but—it seems someone intercepted the letters." His cousin stared at him for a long moment, brow creased. "Are you quite serious? Why the devil would anyone do that?" Justin shook his head. "I don't know. But someone did, and caught hers to me too, before they could be posted." "I suppose that helps, at least—that she now knows you did write." "Yes. Maybe." He, too, looked off toward Whitby. Only darkness met him. "But knowing it cannot undo the damage. Cannot tell us all the thoughts shared and not received. Knowing there is treachery does not bridge the gap." It just gave Brook another focus. Cayton trailed a finger along the crystal's edge, making it sing. "You should have won her before you left. Secured an engagement, if not married her then and there." Justin picked his glass up again, though he didn't drink. It wouldn't warm the places Brook's reception had left so cold and hopeless. "I know." "And what about me, I ask you? No one gives any thought to my reputation, and the fact that it will be left in absolute tatters if I don't get the first dance from either of you." Brook pressed her lips together against a grin as Brice splayed a hand over his heart, his face the archetype of a tragic hero. "No doubt you'll perish from the neglect, my lord." "I shall indeed. Cruel creatures." He turned to include Melissa in his sad-eyed gaze. "First your sister dashes my heart to pieces, and now the two of you show no regard for my tender feelings." "Careful, Worthing." Melissa angled her sweetest smile his way, though her fingers didn't pause in their embroidery. "Keep it up, and I may decide to toss Cayton over for you, out of pity. Then where would you be?" "Blessed beyond measure, to have the attention of a lady so fair." He grinned as he said it . . . then sank to a seat on the couch well away from Brook's cousin. "But let no one ever accuse me of being the means of another's heartbreak. You must resist my charm, my lady, for the sake of Lord Cayton." Brook chuckled and set aside the book she'd been reading before Brice arrived. Aunt Mary had already taken her and Melissa to the shops, spending obscene amounts of money on hats and gloves and wraps and who knew what else. Never in her life had Brook more longed for a horse, an open stretch of land, and the sea by her side. It had been nothing like shopping in Paris with Grand-père. Especially given Aunt Mary's stony silence when Brook insisted that—no, she would not wear the horrid pink thing to her debut—she would wear the green gown. Brook stood and moved to the window overlooking the street, telling herself she was not waiting to see a Rolls-Royce hum up the drive. Her fingers found the dangling pearls. Twisted, released, twisted again. She dropped her hand when Brice leaned into the wall beside her window. Though it took effort, she mustered a smile. "Did Ella pout at being left behind in Sussex?" He grinned. "She put up an admirable fuss, though of course it didn't budge our mother. She's got that stubborn Scotch blood, after all." His gaze went to the window, to the road she'd been not watching. "Have you seen him yet? Rumor says he's been back for a few days." Nothing ever slipped by him. It could get annoying. Nodding, Brook glanced to her cousin—and was surprised to see Regan sitting beside Melissa, though Brook hadn't heard her come in. They were talking, laughing, Regan's hand resting on the barely visible bump of the child she would deliver at the end of summer. "He came to Whitby Park before we left." "And?" Brice lifted a dark brow. "I hope you socked him right in the nose." A laugh slipped out. "You, who wouldn't even step on that spider at Midwynd?" "I didn't say I would have socked him. But I would have cheered for you, if you chose to." Despite his grin, his eyes were serious and warm. "He deserves it, after ignoring you as he's done." "He didn't, though." She cast another glance at her cousins, who knew nothing about mail-tampering or Fire Eyes or threats. And whom she would happily keep in the dark, since their knowing would only make their mother faint. "He wrote to me, apparently. But I never got his letters, nor did he receive mine." Brice straightened and faced the window, putting his back to her cousins. No doubt so they wouldn't see his frown. "On both ends—that is no quirk of the post." "No." "Brook." He reached for her hand and held it between both of his. "I've a bad feeling. I have had ever since you told me of that man in the stables, and it's only grown worse. Whatever this Fire Eyes business is about, it's dangerous." "I don't think this had anything to do with that. More likely it was Pratt." Brice shook his head and held her fingers tighter. "One explanation is always more likely than two. And I don't believe in coincidences—you know that." "I know." His faith often put hers to shame. But then, he could see things so much more clearly—it was hardly fair. "Have you any insight that could actually prove helpful, instead of worrying me more?" He held his tongue, held her gaze as thoughts marched through his eyes. His thumb stroked over her knuckles in an absent gesture—she'd seen him do the same to his mother or sister. Still, it sent a warm little tingle up her arm. Not exactly fire, not exactly hope. But perhaps it could be fanned into something. He, at least, wouldn't shove her away at the first possible moment. At length, Brice nodded. "This time next week, you will be the darling of London. Use it to your advantage." Frustration knotted in her chest, and she looked back to the window. "You are always so sure of how I will be received, but I am not. I am still so very Monegasque, and—" "And that is still so very intriguing. You were raised by a performer, Brook, and as a princess. You don't act quite like all the other girls. You carry yourself like a ballerina. And I am in no way trying to flatter you when I say yours is the loveliest face in Town." His tone was serious, quiet, a bid to look at him again. She did, and found his eyes dark and intent, as they had been that first day at the house party, when he'd told her to go to Justin, whether he wanted to let her or not. "Use it," he whispered. "Enchant them. Leave them wondering, seeking more—it will mean the press will show up wherever you are." She tried, in vain, to tug her fingers free as she loosed an exasperated sigh. "And why in the world would I—" "Because"—all teasing left his expression, and he gripped her hand tighter, held her arm straight down to keep her still—"where the press is, there is safety. Where reporters and photographers lurk, no one will dare make a move against you." A chill skittered up her spine. "You make the danger sound so real." "And the knife in your side didn't? The fists that pounded your face? The gun to your head?" Another shiver chased the first. "Point taken." "Then take the advice as well. I don't want to see you bruised and battered again." She was still trying to work the pent-up breath from her lungs when movement in the doorway caught her attention. Aunt Mary's butler—and behind him, Justin, whose gaze had already found her . . . and whose eyes had already narrowed. "The Duke of Stafford, my ladies. My lord." "Heaven help me." Brice dropped Brook's hand. "Promise you'll attend my funeral?" She shouldn't have laughed. When she did, Justin's narrowed eyes turned to his glower. Justin charged toward the carriage house, telling himself he was overreacting. That he ought to be glad Brook had made such good connections. Found someone to hold her hand and whisper in her ear when she thought Justin chose not to. He wanted to rip Worthing to shreds. Feed the pieces to hungry wolves. And then, if he were feeling spiteful, burn their waste. "Stafford!" He stopped within a few feet of the Rolls-Royce, his hand fisted around the key. The muscle in his jaw pulsed, but he could do nothing to calm it. He turned to see the man in question coming up behind him—and it didn't escape him that the lighthearted grin that had animated his face through the entire, interminable hour they had spent in the same room was now conspicuously absent. "Can I help you, my lord?" Worthing flashed a smile, fleeting as lightning, and motioned toward the house. "I think you misunderstood things." Like the way he had been holding Brook's hand so tightly in his? The way their heads were bent together? The fervor in both their expressions? A striking contrast to the way Brook had greeted him two days ago. "Oh, I think I understood perfectly." "I doubt it." Worthing had the gall to smile again, longer this time. "One of us may be in love with her, but it isn't me." Justin gripped the key until it hurt. "So you're toying with her—is that it? Flirting with her, courting her, inviting a familiarity you have no intention of seeing through?" He took a step forward. Worthing took one back, raised his hands in exaggerated surrender—but amusement had rekindled in his eyes. "You're spoiling for a fight, aren't you? You'll not have one from me. She means the world to me, but we are only friends." Justin snorted. "I know all about being only friends with Brook." "You used to." Lowering his hands again, Worthing's face went from mirthful to serious. Condemning. "I wonder if you've forgotten all you once knew. You've hurt her. That's unforgivable, and I won't stand by and watch you break her heart." Of all the arrogant, presumptuous . . . He stepped closer, close enough to realize they were of a height, close enough to think that Worthing's fine, straight nose could do with a knot from Justin's fist. "You have no idea—" "I'm not talking about the missing letters." Justin stepped back, sucked in a breath. She had told him of that? Already? Worthing didn't so much as flinch. "You pushed her away before you ever left—effectively tossing her heart to the ground. Then you come back and act as though she is to blame for not falling at your feet." "I did not—" "Shut up." Worthing eased half a step closer. "You weren't here. You didn't see it. You didn't see how it hurt her not to have your friendship to rely on. Yet you show up now intent on romance, as if you can charge across that half-burnt bridge and not cause even more damage. Well, I hate to tell you, but she has bigger concerns at the moment." Justin's lip curled. "You?" "Don't be an idiot. I'm not the one who attacked her in November—and I'm not the one set to tread on her heart." Worthing put his hands in his trouser pockets. Such a casual move, but it didn't make him look at ease. It made him look determined. "Break it, and—I warn you—you will have a fight on your hands. But not the kind you want." Justin lifted his brows and folded his arms across his chest. "You're threatening me?" The man smiled again. "Someone has to. And I suspect no one else would dare cross the mighty Duke of Stafford." Expelling an incredulous breath, Justin shook his head. "I don't need to be warned." "Good. Then we can be friends." Worthing withdrew one of his hands and held it out, as if actually expecting Justin to shake it. He glared at him. "If you're finished, I have somewhere I need to be." Worthing looked at his empty hand. With a shrug, he stepped back. "I know you have. What I don't know is why you're leaving it." Arrogant, presumptuous . . . Justin turned and climbed into his car. Friends? No. Pieces. Wolves. Waste. # Twenty-One Deirdre pinned the baroness's last curl into place and then stood back, unable to keep from smiling. "There. What do you think?" Her ladyship stood, ran gloved hands over her gown to smooth it, and looked in the mirror. Deirdre couldn't think why she sighed as she did. The gown the prince had sent fit her to perfection, the colors set off her skin and eyes, and the style was daring enough to steal the attention of everyone who would catch a glimpse of her. "Is something the matter, my lady?" "No." The baroness smiled, but she touched a hand to her pearl necklace, a sure tell of inner turmoil. "But I would rather be in Yorkshire. At home." Deirdre would be too, though Beatrix hadn't been able to fathom that she would rather stay at Whitby Park than come to Town. Perhaps if she didn't know Pratt was here too . . . if she wasn't looking over her shoulder every time she stepped out of doors, wondering when he might pounce and ask something terrible of her. . . . Then she'd have to confess that her ladyship hadn't been herself since they'd arrived—and especially not since she'd had the duke and Lord Worthing in the parlor two days ago, then hadn't seen hide nor hair of His Grace since. Combine the baroness's melancholy with Lady Melissa's increasing rancor that Lord Cayton had yet to pay a call, and the house was in a veritable tempest. Shaking it off, Deirdre smiled and unbuttoned the train of the gown. "You'll be the belle of every ball, my lady. No doubt you'll have all the gentlemen in love with you, and you'll have your pick of them. Though I can't imagine a better choice than the ones you already have." The lady muttered something in French and pressed a hand to her stomach. "Nothing feels right." At that, Deirdre's hands stilled. She rose, met her mistress's gaze. And prayed she spoke the truth when she said, "It will be." The light in her ladyship's eyes seemed to Deirdre desperate, anxious. No doubt due to the coming evening. "O'Malley . . ." She looked away, sighed. "How is your family?" If it was a distraction the baroness needed, Deirdre could provide. She chuckled. "Doing well, I hear. Mum said Uncle Seamus sent her a package of silk and spices last month—he's tried to take care of us since Da died, in addition to his mum. Though my stories are sure to match his when I take my holiday—rubbing elbows as I've been with dukes, handling gifts sent from princes . . ." The baroness smiled. It wasn't as bright as usual, had none of her characteristic abandon. But somehow, she thought it would serve the lady well in the ballrooms and drawing rooms of London. Though sure and she knew little enough about it. At the knock on the door, she stepped aside. "That'll be his lordship. It's time." "Where are they?" Brook's eyes scanned the room as surely as her cousin's did. It was crowded with people she had never met, names her aunt insisted were important ones, faces that all seemed to turn her way. But not Justin's. And not Cayton's. In response to Melissa's hushed, furious question, Brook could only shake her head. Papa patted her hand, which rested on his arm. "As I taught you, my dear. Trip. Run into the most ostentatiously dressed women. Step on toes, and snub anyone you can. Perhaps sneeze in a cup or two of punch, and Mary will be begging us to leave." "Ambrose, please." Aunt Mary slid behind them, tugging here and there on Brook's gown. Then she paused, clasped Brook's shoulders, and gave them a squeeze. "You look stunning." With that peace offering, she moved to Melissa. Brook grinned up at her father, then looked over to her cousins. Melissa looked beautiful, if a bit stormy, on her brother's arm. She spotted Brice near at hand, his usual grin in place . . . but with a shadow in his eyes. The musicians raised their bows and, of one accord, launched into the opening set. Her father sighed. "And so it begins. We have missed our chance to run away." He turned to her, his hand extended. She placed her fingers on his palm and smiled up at the grin hidden away in his eyes. "I'm glad to be here with you, too, Papa." He chuckled and led her onto the dance floor. Even above the music, she could hear bits and snips of the conversations they spun past. Whitby . . . all these years . . . carriage accident . . . missing . . . imposter . . . princess. They needled, but she shrugged them off and raised her chin. She was not an imposter, but she remembered how to be a princess. And they would have the answers to the other soon. She knew they would. "That's my girl." Her father's eyes gleamed as he spun with her to a different corner of the floor. All too soon, the music changed, and he delivered her back to the edge of the ballroom. Brice waited, apology in his eyes. He nodded to her father and held out a hand. "Stafford hasn't arrived yet. Just late, no doubt, but we can't have you without a partner so soon in the night. If I might step in?" Her lips tugged up, and she transferred her hand to his. "Selfless of you, with so many lovely young ladies about to flirt with." "They will swoon as well later as they would now." He led her out, his smile never faltering. "You look resplendent. That gown cannot be from London." "Paris." "Of course." He spun her with a flourish into his arms for the waltz. "It suits you. Shall we set the tongues to wagging?" A laugh tickled its way from her throat. "Is there any choice with you?" "Never." She wasn't surprised to find that he danced without flaw. Nor that he could keep up a steady stream of banter as they sidestepped the other couples. But there was no tingle tonight where his hand grazed her back or clasped hers. Instead, her gaze went to the door each time she spun to face it. "Keep that up, and I'll never live it down—that the beautiful baroness kept searching for another when in my arms. Cruel creature." Laughing, she turned her face back to him and returned his smile. "Perhaps you could keep my attention if you were ready to confess what you said to him the other day—" "I'm telling you, I threatened to pulverize him. Fisticuffs, bloody noses, the whole lot." Brice with fists raised—the picture wouldn't form. "Mm-hmm." "Challenged him to a duel. Sabers at dawn." "Right." Though it made her chuckle again. "Though mind your volume, mon ami, or that will appear in tomorrow's Times." Merriment danced in his eyes. "That would be a laugh. Though I daresay your duke would not agree. And speaking of said devil." He nodded toward the door. "I had better deliver you to him before he lops off my head." He timed it so that they reached that edge of the dance floor as the song drew to an end. Justin had worked his way to the edge of the crowd—the thunder in his brows no doubt clearing the way for him—and Brice greeted him with a bow and transferred her hand directly to his. Naturally, he grinned. "No need to thank me." Naturally, Justin glowered. "Didn't plan to." "You were late." "And you were quite happy, it seems, to take my place." She still couldn't wrap her head around Justin being jealous over her. "Gentlemen." Making sure her smile remained bright and her words quiet, Brook curtsied to Brice and tucked her hand into the crook of Justin's elbow. "We are far from alone, n'est pas?" Justin grunted and took his turn leading her onto the floor. "Five minutes late." "And the music did not wait for you." She didn't want to be irritated, not tonight, but it shivered over her skin. Or perhaps that was his touch. She wasn't sure. Having reached a bit of open space, Justin turned her toward him and slid a hand onto her waist. He held her closer than the two-step demanded, and her pulse sped—with that irritation, or with something better? His gaze dipped down to take in her dress, and his lips tugged up. "You look . . . nice." "Nice?" She laughed as he spun them into the dance. "As many hours as I spent getting ready, I had better look more than nice." "Pretty, then?" His eyes gleamed. She lifted her brows and gripped his hand. Maybe she could do this. Maybe she could slide easily from friendship to flirtation. "Your nemesis over there chose resplendent. Surely you can outdo that." He smiled, the challenge turning to a simmer. "How about this." He pulled her a little closer and leaned his head toward her ear. "I have traveled the world over these last months, but nowhere, on no continent, in no country, have I ever seen anyone half so beautiful as you." Her fingers gripped his shoulder. "Better." Wasn't it? The trip of her heart said so. But still that voice whispered in the back of her mind that he had never used to say such things, when she was just Brook Sabatini, illegitimate daughter of an opera star. He didn't used to think he could hold her close with his arms and push her away with his words. He didn't used to try to pair fire and ice. She didn't mean to sigh. But it built up inside and pushed its way past the music and glitter and seeped out when she spotted Melissa a few couples away, dancing with some young gentleman Brook had never seen before. Justin followed her gaze and winced. "Was she upset?" He must have known that Cayton had requested the first set of dances months ago. "More like furious." "Deservedly." "Mm. Where is he?" He shrugged, his muscles bunching under her hand. "I haven't seen him in a few days. We are to go riding in Hyde Park tomorrow though." Looking back to her, his eyes were deep and serious. "I have to support him. I hope . . . I hope you don't blame me for it." She glanced again at Melissa, whose laughter looked sincere rather than feigned. Missing a ball may have spoken to Cayton's character, but it need not add to the tension between them. "That is between our cousins, not us." "Good." He squeezed her fingers and then looked out over the crowd. "They're all watching you. And there's a veritable sea of young men around your father and aunt, no doubt begging to be introduced. I'll have to cut a swath to claim another dance later." Would he even bother? "Thate is in the back room, I believe. Regan said something about an airplane pilot who was coming, and now all Thate can talk about is the air race this summer." She couldn't blame Justin if he retreated that direction. Frankly, she would rather be back there too—talking of automobiles and aeronautics—than in the ballroom. Justin grinned. "Will he take to the skies next, do you think?" "Regan made him swear he would keep his feet firmly planted on the ground until the baby arrives. Then . . . who knows." His eyes went wide, and his smile crooked. "Baby? I hadn't heard. I'll have to find him and torment him about his settled and predictable life." They slowed when the music hit its cadence, and the fingers against her back splayed out. "May I pay you a visit tomorrow, my lady?" The low warmth of his tone belied the formality of his words. Not completely unfamiliar, that. And the gleam in his eyes . . . It had changed, yes, but he had always looked at her with more warmth than anyone else. Maybe it wasn't such a change. Maybe, if she gave him a chance to share his heart, he would put her fears to rest. Maybe he would kiss her again, and the sensations would swell, and she would know that no matter what had happened in her life, he would still have wanted her. Because looking up into his sapphire eyes, she knew without a doubt that she would have come here at some point. She would have left Monaco, and where else in the world would she have gone but to him? She had always loved him. Maybe . . . maybe she had always been in love with him. What, then, would he have done had Brook Sabatini come knocking upon Ralin Castle's door? She would have to find out. And it might as well be tomorrow. Pulling out a smile, she said, "I would be delighted, Duke." # Twenty-Two Brook poured steaming black life into a cup and prayed with the first sip that it would produce miracles. Her feet were sore. Her eyes were gritty. And so many names and faces buzzed in her head that she wanted to crawl back under her covers and shut out the world. Papa slid up next to her at the sideboard and began filling a plate—with her preferences, not his. "Now you've done it." She took another sip of the coffee—not espresso, but at least strong—and lifted her eyes to his. His lips were twitching, so she went ahead and grinned. "What have I done this time?" In answer, he handed her his newspaper, folded open, and indicated the table. She sat with cup and news, let him slide the plate in front of her . . . but wasn't sure how she would eat breakfast. It may have been well past noon already, but the headline made her stomach knot. THE LOST HEIRESS OF WHITBY "Eat." Papa dropped a kiss onto the top of her head and sat beside her. "Much as I detest being in the news, the article is not a bad one." It felt it, though, as she read and nibbled at her toast. A reminder of the carriage accident, an explanation of how Brook had ended up in the care of Collette Sabatini in Monaco, where the opera star passed her off as the child of Prince Louis. From there it shifted, touching on the countless girls paraded through Whitby Park over the last eighteen years trying to claim her inheritance. Then it sped back to the present, summarizing her arrival home in early September, her acceptance by her family, and how the reclusive Whitby was in London for the Season with her now. At least it didn't mention the attack in November. Nor—which, frankly, surprised her—did it mention Justin anywhere. No, instead it reported that after opening the floor with her father last night, she was seen in the arms of Lord Worthing, with whom she danced thrice more—which was not true. She had danced only once more with Brice, once more with Justin. Her aunt had told her she could not, under any circumstances, dance more than twice with any one man unless she intended to be the subject of every gossiping tongue in London. Apparently even obeying such rules did not guarantee avoidance of that fate. Her eyes finally moved to the last paragraph. In a Parisian gown of pale green silk with an exquisite overlay of blue beading, the baroness debuted in glory. As onlookers gazed upon her, many remembered the fame her mother had attained twenty years prior, and it seems only fitting that they gave to her the same name with which they had dubbed the late Elizabeth Brook and welcomed a new Baroness Beauty into their midst. Brook lowered the paper and looked over at her father. "They called her that?" Papa's smile was small and wistful. "They did." Brook grinned and might have replied had the butler not cleared his throat from the doorway. "Excuse me, but the Marquess of Worthing has arrived. Shall I show him in here or . . . ?" Brook stood even as her father did, coffee in her hand and paper in his. The food she would happily abandon. Aunt Mary employed an English cook, not a French—or French Canadian, as the case may be—chef, and her palate had not adjusted to the fare. "Drawing room," they answered in unison. Melissa was dragging herself down the stairs as they went by, dressed and coiffed but with eyes at only half-mast. They exchanged a grin, and her cousin fell in with them instead of heading for the breakfast room. Brice preceded them into the room by only a few feet and spun to face them the moment they were all inside. Brook expected him to be grinning, teasing. Instead, his eyes were serious. "Have you seen it?" "The article about our Baroness Beauty?" Papa patted her shoulder. "We did." "No. Well, yes, that too, but did you read the rest of the paper yet, Whit?" Her father shook his head. Brice indicated the folded newspaper, brows arched. "May I?" "Certainly." Brook strained onto her toes to try to see what section he was flipping toward. Though she couldn't tell—not until he said, "Here," and handed it back. "Engagement announcements?" Brow furrowed, Papa accepted the paper. Brook and Melissa leaned in on either side of him. Brook sucked in a breath when her gaze snagged on familiar names. "Pratt and Lady Catherine?" On the one hand—her cousin's hand—no surprise. But he . . . Did that mean he had given up his hope of joining their estates? Brice nodded. "One of the two surprises." "What el—" Melissa's shriek of outrage cut off her question, and she snatched the paper from her uncle's hand. "Who in blazes is Adelaide Rosten?" Frowning, Brook looked to Brice. "Cayton," he murmured. Cayton—in the engagement section? Her cousin looked ready to tear the paper to shreds. "That lying, swindling, misleading, snake-tongued, blackhearted . . ." She had seen Melissa in quite a few storms of temper since September, but never like this. "'Hell hath no fury . . .'" Brice grinned. "Chaucer, isn't it?" Brook rolled her eyes. Melissa had finished her list of adjectives, it seemed. "I'm going to kill him! I'm going to march over to his townhouse and pluck every hair from his head!" "He's not at home," Brice helpfully supplied, hands in his pockets and half a grin still on his mouth. "I passed him on the way here—he looked as if he were going to Hyde Park." Melissa shoved the paper back into Papa's chest. "Then so am I. And you"—she grabbed Brice by the arm—"are coming with me." Amusement gave way to panic on Brice's face. "Ah . . ." "You're going to look at me with that adoration you feign so well, and I'm going to laugh at your every ridiculous joke in sight of all London." "Oh. Um." He looked to Brook, eyes wide, and mouthed Help. Brook smiled and tucked her hand into the crook of her father's arm. "I'm sure you'd be blessed beyond measure to keep company with a lady so fair, Lord Worthing. Isn't that what you said the other day?" Brice narrowed his eyes at her while Melissa tugged him toward the door with the strength of a bull. "You're going to let her kidnap me?" Chuckling, she waved her fingers. "My cousin needs you." "Your cousin's terrifying." Melissa spun and must have given him quite the look, though Brook couldn't see her face. Brice pasted on a smile. "Terrifying . . . ly beautiful?" Melissa yanked him out the door, giving him time for only one more pleading look. Papa sighed. "When the anger fades, she will be heartbroken." It was true. And though Brook had never really liked Cayton, nor his readiness to arrange trysts with Melissa behind her mother's back, she had been ready to be happy for her cousin when he proposed. Had he only been toying with her all these months? She didn't know—but she knew who would. Perhaps he was summoned by her thoughts, for the moment she spun around, Justin stood in the doorway, question in his brows. "Where was Lady Melissa pulling Lord Fastidious?" Brook reclaimed her hand from Papa's arm so she could fold hers over her chest. "Is that what you were talking about last night? Your cousin is engaged?" He stared at her blankly for a moment and then sighed. "He said he told her." "He lied." "Coward." Justin pivoted, as if ready to chase after Melissa . . . then must have thought better of it. "She saw it in the paper? And that was the first she knew of it?" Brook wanted to ask him about his cousin's motives. She wanted to ask him how he could support him. She wanted to ask him if his affections could be trusted. She moved her arms down, over her stomach, and bit it all back. "I thought you were riding with him today." Justin looked her way again, conflict in his eyes. "I am. James was going to fetch Miss Rosten first, though, and as I've no desire to be a third wheel . . . I thought you might join us. But that was when I thought you knew already. I understand if you would rather not. Your cousin—" "Would not thank me if I passed up the chance to meet this Miss Rosten." Brook looked to her father, who nodded his permission. "Are you on Alabaster?" "In a landau." No need for her to change into her riding habit, then—which was good, since her aunt had insisted on one with a skirt, which would necessitate the dreaded sidesaddle. "I'll fetch my hat and wrap." Justin was vaguely aware of the sun shining. Of birds flitting from tree to tree. Of the scads of people walking, riding, driving along the paths through London's largest park. He wanted to focus on the woman beside him, on the sweet smell of lilacs that drifted from her hair. But Brook was focused on Cayton's landau. She had been the epitome of polite during the introductions, but now her lips were pressed together, and her fingers gripped the edge of her kimono coat. She didn't even mention the suffragettes shouting from their soapboxes as she turned a hard gaze on Justin. "Is it catching?" He expelled a bitter breath. Miss Adelaide Rosten was not what he had expected, to say the least. "I know little about her, except that she is my neighbor in Gloucestershire. They knew each other as children." "Tell me he met her again and fell in love and doesn't see the obvious. Tell me that is why he tossed over my cousin for her." If only he could. Ahead of them, Miss Rosten presented her profile as she looked to Cayton. She smiled, and it looked so sincere. So sweet. So . . . hopeful. But could do nothing to fill the hollow cheeks or lighten the shadows under her eyes. "She is an heiress. He is strapped." Brook shook her head. "She is ill. She looks . . . she looks like Maman did at the end." His hands tightened on the reins. "I know." "Mon ami." Her fingers landed on his arm, though they didn't stay there. "Tell me your cousin is not so low as to marry a dying woman for her money, knowing well she hasn't long to live, knowing well he can soon move on." "I . . . don't know." He didn't want to think so. Cayton, as he confessed his engagement at Azerley Hall, had seemed honest about his reasoning, and he certainly hadn't mentioned any illness. "Perhaps it is a childhood malady that she still bears the marks of. But perhaps she is well now." She didn't look well. But Brook didn't point it out again. "Look at how she watches him." "She cares for him." Which raised more questions in Justin's mind. Did Miss Rosten know Cayton's reasons for proposing, or had he spoken words of love to her? Had he misled her? "Perhaps he knew of her feelings. Perhaps . . . perhaps he wanted to give her some happiness." A delicate snort slipped from Brook's lips. "Forgive me for doubting Cayton's pure heart. Perhaps they'll be happy, though—it seems unions built on love always end miserably, so perhaps one arranged for pragmatic reasons will have better luck." Surely she jested. "Let us hope, for Regan and Thate's sake, that you're mistaken." The fleeting smile she managed didn't make it to her eyes. "If anyone can defy statistics, it is they." He studied her profile, shaking his head. "When did you get so cynical on the subject of love, Baroness Beauty?" Brook winced. "Saw that, did you?" "It wasn't so bad." Even if it had exaggerated her relationship with Worthing—and even if she did dodge his question about love. "The Lost Heiress. That's what they'll all know me as now." Her eyes went distant, and the fingers of one hand had abandoned her kimono's hem in favor of twisting the pearls on her necklace. He bumped his shoulder into hers. "You are an heiress, Brooklet. You can't expect society not to notice." "But for most of my life I was just . . . lost." She drew in a breath and twisted the pearls the other direction. "If you hadn't found Papa for me . . ." "Let us praise the Lord that I did, that the crest was enough." She looked up at him, dropping her hand back to her lap. "And what if I had been a lost nobody, instead of a lost heiress?" The question turned her eyes to flame. "I would have come, Justin. I would have shown up at Ralin one day and demanded that tour you always promised. Then what would you have done?" "I would have given you the tour." And likely drawn her into his arms and kissed her and . . . what? Even Father, who had eschewed all ducal responsibilities, claimed Justin couldn't marry her so long as she was only the illegitimate child of an opera singer. Though Grandfather had accused him of wanting to marry her even if it brought disgrace to Stafford. Which of them knew him better? Brook shook her head and looked away. "You asked me at the funeral to say your name. Say mine." The demand was unfair—his name hadn't changed, only his title. Hers . . . "Elizabeth Brook. Sabatini or Eden, it doesn't matter. You are my Brooklet." "I am your friend." "You are . . ." My heart. My soul. "So much more." Now anger sparked in the eyes she turned on him. "If I were still Brook Sabatini?" "You're not. Why are you dwelling on hypotheticals?" He motioned to the Ramsey barouche that crossed their path, to Melissa with her chin held high and Worthing with a laugh on his lips. "Do you think he would be your friend if you were still Brook Sabatini?" Her words changed to Monegasque as they rose in volume. "I think I never would have known him! You . . . you are the only one I could carry from one life to another. The only one who ought to know me and love me for my past, not just my present!" "I do." He swallowed, held her sparking gaze. "That doesn't mean I'm not grateful for the way things are." "It isn't enough." Looking away again, she pulled her kimono tighter, even though the sun was gaining in warmth. "I need to know, Justin. You are trying to change everything—I need to know why. I need to know you would have pursued me in the same way even had you discovered my father was a penniless nobody instead of the Earl of Whitby." "Well, of course it wouldn't have been in the same way!" How could it have been? He would have had to fight his family every step of the way, would have exchanged one set of difficulties for another. He certainly wouldn't have rejected the idea of an engagement months ago in order to prove to her he wasn't after her fortune. Brook slid to the opposite side of the bench. "Take me home. Now." Blast. That probably hadn't sounded the way he'd meant. "Brook—I didn't mean I wouldn't have pursued you, just that it would have been different." How could she look so dratted beautiful even as she snorted and folded her arms over her chest. "Oh, I'm sure, Duke. You would have found some suitable girl to court, and I would have been . . . What? Dismissed from your life? Or would you have tried to make a mistress of me?" His blood ignited, and he gripped the reins tight. "How could you say that? You know me better than—" "I know it's how things are done in your family! Even your sainted Uncle Edward—" "Don't compare me to him." His words sounded, oddly, cold rather than hot, despite the roar in his veins. Turning her face toward him again, she lifted a brow. "And why not? You always idolized him. 'If the shoe fits . . .' as the saying goes. . . ." He all but jerked the horses toward the nearest exit from the park. "I am not like him." "You are exactly like him!" "He raped my mother!" He didn't, couldn't look at her as the words, still in Monegasque, pulsed around them. His nostrils flared. "Got her with child on purpose, thinking to make Aunt Caro raise me. I am not like him." "Justin." Her voice went soft, filled with sympathy that did nothing to make his fists relax around the reins. "I'm sorry. I didn't know." "Of course you didn't know." He directed the horses back toward her aunt's, the fire only building. His words slipped back into English. "How could you? That would have required granting me ten whole minutes to speak of something other than you, wouldn't it? Something other than your problems, your mysteries. Oh, but this is your turn. Your time. My apologies." Her fingers landed on his arm, though the touch was brief, quickly gone. "Justin . . ." "Don't." For an eternity, he said nothing. He couldn't work any words past his clenched teeth. Couldn't dislodge those months of doubt, of wondering if she even cared or if she'd fallen for Worthing. And now here she was, saying his feelings didn't even matter. That what he may have done if she weren't who she was outweighed what he had actually done to protect her. He turned onto her aunt's street and forced a swallow. "I love you." The words, so long unsaid, nearly choked him. "Take your time. Decide if that's enough. And let me know when you've figured it out." He pulled to a halt in front of Lady Ramsey's and glanced her way. She stared at him, mouth agape, incredulity shifting to irritation before his eyes. "That is how you choose to tell me you love me? In the middle of an argument, followed by a statement that yet again you'll retreat behind your wall?" "When better? But if it's charm and smooth words you want, then I guess we know where your heart inclines." "You're an imbecile." Gathering her skirt into hand, she leaped down from the landau. Stomped toward the door, but then halted at the base of the stairs and spun back to him, fury flashing in her eyes. "I'm not in love with Brice." For a moment, hope sprouted. But she didn't follow it with anything, didn't say she was in love with him. He breathed a laugh and lifted the reins. "At the risk of sounding like an echo, my lady—that isn't enough." He was halfway down the street by the time he heard the door's slam. # Twenty-Three Deirdre handed the baroness the book she had fetched from her bedchamber, smiling at the yawn the lady tried to cover with a hand. "Perhaps you would adjust easier to the late nights, my lady, if you were consistent about them." Lady Berkeley sent her a tired scowl. "You sound like Aunt Mary, O'Malley. I have been to three balls and a soiree. That is surely enough for two weeks' time." Lady Ramsey didn't seem to think so—she and Lady Melissa had been out each and every night to something or another. Not that Deirdre could blame Lord Whitby and the baroness for staying in whenever they could finagle it. And if the papers were any indication, her absences only increased her fame. Deirdre made no attempt to keep track of the flood of young ladies and gentlemen who swarmed the parlor and drawing room for Ladies Berkeley and Melissa. Which would be why Lord Whitby and his daughter were now hidden away here in the upstairs salon. His lordship's paper rustled as he turned another page. "We can go home whenever you're ready, my dear. I have verified that the House of Lords cares no more for my opinion now than they ever did, so I've nothing to keep me here." Deirdre took a seat near to the baroness's, to be at hand when next she needed something, and picked up last night's ball gown. Some clumsy oaf had stepped on the train and caused a tear, and it would take all Deirdre's skill with a needle to mend it without it being noticeable. She opened her case of thread and selected the closest match to the lavender silk. A knock upon the open door earned a groan from the lady and brought Deirdre's gaze up. The butler stood there, silver salver in hand. "Not more callers, Mr. Vander. I'm not at home. I've run off on safari." The butler smiled and bowed. "A letter, my lady. Addressed to both you and his lordship." The baroness grinned, though sure and her smiles had none of them been very bright since she returned in a huff after her drive with the duke following her debut. "In that case, thank you very much." "And shall I tell your next callers you're on safari, Lady Berkeley?" She chuckled as her father stood to accept the thin envelope on the tray. "I leave that to your discretion." Deirdre threaded her needle and tied the end while his lordship picked up the letter opener from the salver and made a neat slit in the envelope. Putting it down again, he nodded his thanks and dismissal of the butler. And frowned at the letter. "This looks suspiciously like . . . Brook, it is from Major Rushworth!" Deirdre's hands went still even as the baroness leaped to her feet. "What does he say?" His lordship looked up from the page with wide eyes. "That he's back in Town and will call tomorrow morning at nine o'clock. He requests a private audience with the two of us." "Back in Town?" Deirdre realized she had spoken only when the two looked at her. She drew in a quick breath. "Pardon me. I . . . my uncle usually travels with the major." Lord Whitby frowned for a moment, though it quickly cleared. "Of course, I'd forgotten his batman is the one who recommended you to us. How long has it been since you've seen him, O'Malley?" She turned her gaze back to the gown. Sure and she hadn't meant to steal the floor. "Many years, my lord. Not since I was a girl, though he is always most faithful in writing. He and my da were close, and he's done his best to see to the family since . . ." Lord Whitby's warm smile reminded her of why Uncle Seamus had recommended his house to her. "The major is staying at the Hendon Hall Hotel, it seems. Why not take your afternoon off and see if your uncle is with him?" "Oh." She hadn't felt such a swell of joy since Da yet lived—because seeing Uncle Seamus would be a bit like seeing her father again. Her gaze flew to the baroness. "May I, my lady?" "Of course. Go." Her ladyship made a little shooing motion with her hands. She didn't need to be told again. Smiling her thanks, Deirdre put needle and dress aside and dashed from the room. She changed quickly into a matching skirt and jacket, grabbed her handbag, and fastened a hat over her chignon. Then it was down the stairs with her, and to the kitchen, where she found Lady Ramsey's housekeeper. "Pardon me, ma'am. Do you know how to get to the Hendon Hall Hotel?" The woman pursed her lips. "They've turned Hendon Hall to a hotel, have they? Pity. But yes, I know it—it's in the north part of the city. You'll want to take the tube." New excitement joined the flutter in her stomach. She had yet to have cause to use the underground railway. "How much?" "Two pence is all." "Thank you." Her grin felt as though it would split her cheeks. "Have you need of anything while I'm out, ma'am?" The old woman returned her smile. "No. Go on with you." Letting herself out the back door, Deirdre circled around to the street and all but skipped toward the heart of London. And screamed when a hand closed around her mouth and tugged her into an alley, though her cry was muffled behind the fingers. "Quiet." Pratt. Shuddering, she nodded. He let go her mouth and spun her around. His eyes were two black slits. "Where are you off to so merrily, my lovely?" Why was he always there to spoil everything? She backed into the brick wall behind her. "To see my uncle is all, my lord." "Uncle." Pratt lifted a single brow. She swallowed and pressed her hand to the cool bricks. "Aye. My da's brother. If you'll excuse me—" "Not so fast." He shifted when she did, to block her from making an escape back to the street. "I have missed her at every turn." Deirdre lifted her chin. "And why should you care? You're betrothed." "And will be married within a fortnight by special license, if Rush has anything to say about it." He put on a cold, unfeeling smile. "Which is why I must act now." "Special license?" Deirdre felt her eyes widen. "Is Lady Catherine—" "A liar? Most likely, but her brother believes whatever she tells him. I've a task for you, Deirdre." For a moment she could only stare. He had gotten Lady Catherine with child and still he meant to pursue Lady Berkeley? Deirdre's breath shook when she released it. "What?" Pratt withdrew a bundle of envelopes from his inner pocket, secured with a feminine-looking ribbon. "It's very simple. You aren't to open them; you aren't to glance at them. You're just to put them in the bottom of Lady Berkeley's trunk, where she'll not see them. Do you understand? Under something, hidden away. And whenever you return to Yorkshire, put them away with all the correspondence she'll have collected in London." Her hands shook as she took them and slipped them into her handbag. She pressed against the wall again when he loomed nearer. "What are they?" He backed away a step. "No questions—or your family goes up in flames. I'm watching you, my lovely." She shivered, closed her handbag, and said no more as he turned and strode away. It took her a long moment to push the fear down and convince her feet to move. Forward, she must go forward. She must push down the question of what he meant to do. Soon she handed over her two pennies at the tube station and climbed aboard the electric train with all the other passengers. Pratt's black eyes kept flashing before her, sapping the joy from the experience. When she finally climbed off in north London, she had only a blurred memory of the stops and starts, the small windows, the tunnel walls hurtling by outside them. The sunlight near to blinded her when she stepped back outside and asked a tube worker for directions to the hotel. It took her ten minutes of striding, then wandering, to find the columned exterior of what had so recently been a family's mansion. She stood on the street and stared up at it. Once a grand home—now open to strangers to sleep and dine in for a price. Heaven help her, she hoped such a fate never befell Whitby Park. Shaking it off, she followed the walk toward the back entrance and knocked on the door. A harried woman in a white cap and apron answered. "Yes?" "Good day, ma'am. I've come inquiring as to whether Major Rushworth has an O'Malley with him as batman." "And who's asking?" The voice boomed from behind her, deep and displeased. Deirdre spun, hand splayed over her heart, and spotted who could only be the major striding her way from the garden. He was in uniform, but for the missing hat. His head gleamed bald in the sunlight, his drooping moustache accentuating his frown. She dipped a curtsy. "Major. I'm Seamus O'Malley's niece, Deirdre. Please, did he come with you? I haven't seen him in ages." "I should think you haven't." His scowl didn't lessen. "No one was to know we were here now. How did you learn of it?" He looked as though he would as soon toss her into the shrubs as listen to her answer. She let her gaze fall to his boots. "I'm in service to the Baroness of Berkeley. I was there when she and Lord Whitby got your letter, and they—knowing as they do that my uncle is your batman—said I might come looking for him." "Whitby." The major spat it out like a curse. "Naturally he would ignore the part that said to tell no one where I was staying, or that I was even in Town." Her shoulders went tight. "Forgive me, sir. I didn't mean to step into a family quarrel. I only want to see my uncle." The moustache twitched. "Not at the moment, you don't. Old boy is ill—he's resting now." "Ill?" All her hope sagged within her. "Mightn't I see him, Major? Make sure he's comfortable?" "I said he's resting." His nostrils flared, but then his eyes softened. A mite. "Come back around tomorrow, girl. Or the next day. We'll be in Town for the week—then it's home to India." He turned back for the garden. "Too dratted cold and rainy on this godforsaken isle." With no other recourse, Deirdre gripped her bag in both hands and dragged her feet back the way she'd come. She'd go home. She'd put Pratt's envelopes in the baroness's trunk. And she'd wish she'd never stepped foot outside today. Brook wished, as she paced to the far corner of the music room, that they were at home. In their library. Books surrounding her instead of instruments. She had tried to play to soothe her nerves, but soft music wouldn't come—and she couldn't very well play thunderous songs while Aunt Mary and Melissa were still abed. She paused beside the window and looked out at the rain-soaked city. As always, her gaze sought a familiar form, a familiar car, and she chided herself for it. Justin wouldn't come any more today than he had the last fortnight. Apparently when he said she could let him know when she'd made up her mind, he meant he wouldn't grace her with his presence until she did so. How, then? How was she to apologize for comparing him to his uncle? How was she to tell him how miserable she'd been without him? How was she to tell him that she was sure, so very sure now, that she loved him? "Major Rushworth, my lord." Brook turned but didn't advance. Better to stay where she was, half-hidden behind the harp, and put Justin from her thoughts before she focused on the major. He strode in. Dressed in uniform, his skin was tan and leathery. His bald head gleamed in the chandelier's light, his moustache framed his mouth, and his brows were furrowed. He halted a few steps inside the door and glared at her father. "Whitby." "Major." Papa had stood to greet him, though he didn't move forward to offer a hand. "Kind of you to finally reply—though a letter would have sufficed." Major Rushworth snorted. "I think not. If I learned anything eighteen years ago, it is that letters are not secure." Papa darted a glance her way. They had learned that truth as well. Otherwise she'd have sent one to Justin. "Did you have a safe trip from India?" "I arrived, didn't I? And I'm eager to get back, so if we might dispense with the pleasantries—you said you found a letter I sent Lizzie in your name. What else did you find?" Papa's expression barely flickered, but Brook could read the frustration in his stance, and in the way his hands curled. "Mysteries." Brook edged out from behind the harp. "What are the Fire Eyes?" They both looked at her when she spoke, but it was the major she watched. He washed pale, his eyes bulged, and his larynx bobbed as he swallowed. "Lizzie." A corner of Papa's mouth tugged up. "We call her Brook." "Your daughter." He shook his head, though his gaze didn't shift. He still looked at her as though she were a phantom. "She is the exact image of . . ." Papa motioned her forward. "Not quite. Her nose is narrower, her forehead not so high. And their chins—they have very different chins." Brook grinned at her father and stopped at his side. He would remember her exact words from their first meeting. The major's nostrils flared. "But her smile. Her eyes." It took all her will to keep from stepping half-behind her father. She could not imagine her mother ever being close to this man before her. "The Fire Eyes, Major. Not mine. What are they? And why was I nearly killed over them?" Rushworth spun away, spitting out an expletive. "It has found you. I thought . . . with ignorance would come safety—that the curse could not strike those who didn't know about it." "Curse." Papa's incredulity saturated his tone. The major turned back to them with a glare worthy of the Russians. "Don't patronize me, Whitby, as if I am the fool. You think you can believe in your precious Lord in heaven without admitting there is another side? You think there is no power in darkness? I have felt it—I have heard it howling in the jungles while you've been safe in your mansion." Lightning and thunder and darkness. Brook suppressed a shudder and made no objection when her father rested a hand on her back. Children of the light. Children of the day. "The Fire Eyes—whatever they are—carry a curse?" His eyes found hers, and they were a roiling brown. "So goes the legend—that hatred and eventually death will follow whomever holds them. I dismissed it when I heard it. And when my every relationship crumbled to pieces, I called it man's greed, not a devil's curse. But perhaps the two are not so different." He nodded, and his gaze fell to her throat. "Your mother's pearls?" Her fingers sought and found the familiar dangling globes. "She was wearing this necklace when she died—it was all I had of her until I came home last autumn. I always wear it." Though the major's lips turned up, it scarcely resembled a smile. Though he laughed, it carried no mirth. "Then you have always been wearing the Fire Eyes." She pulled her hand away, the pearls scalding her. "What?" Papa turned a bit so they could see each other's face, his eyes wide. "That is what you sent her with that letter? The Fire Eyes are pearls?" "I sent it to her. But they are not pearls." He held a hand, motioned with his fingers. "If I might see it?" Papa seemed struck mute. So with a deep breath, Brook reached up and unhooked the necklace—stretched out her arm and let the gold and pearls drip into the major's palm. He reached into his pocket, drew out a pen knife, and sank into a chair. Her fingers rested on her bare neck as he fiddled with the longer of the dangling pearls. Each pulse thundered, fluttered. Try as she might, she couldn't make sense of it. Rushworth sent a hard glare to her father and then looked down again. "I meant only to hide the Fire Eyes—that's the only reason I sent them to Lizzie. It was not . . . I knew she would never accept a gift from my hand—she had made that clear. So I wrote a letter and signed your name to it, Whitby. Sent the necklace with it. I thought she would stash it in her safe with her other jewelry, wear it once or twice, but otherwise forget about it, not even thinking to mention it to you. I knew you sent her endless gifts. I thought they would, for all intents and purposes, vanish." Papa's nostrils flared. "What are they?" The major grunted, focused on his task. "They, Whitby . . ." The tip of his knife seemed to have found purchase. A moment later, the pearl split into two perfectly even half spheres. He shook a red gem out into his palm. "Here." He held the jewel on his outstretched palm, though it was a long moment before Papa reached for it. He frowned as he examined it. "Ruby?" "Ruby!" Rushworth barked a laugh. "Guess again." "Well, it certainly isn't a garnet." "No." Brook picked it up from her father's hand and held it up to catch the sunlight shafting through the window. Fire leaped to life within it, sending a scarlet-toned rainbow dancing on the opposite wall. "It's . . . it's a diamond. A red diamond." "Two red diamonds." Rushworth held out a second one, which must have come from the other dangling pearl. "Identical. Flawless. Two carats each. Worth a fortune." Brook accepted the second, held them up beside each other. The beauty of the stones, so pure a red, so bright with internal life, left her speechless. Her father shook his head. "They are lovely, but rather small, aren't they? They can't be worth more than a few thousand pounds." "Are you daft?" Rushworth motioned toward the jewels. "They are the rarest diamond in the world—only a few have ever been discovered, and the largest is only five carats. To have two of them—flawless, identical, and that large . . . Kings have killed for these jewels. Wise men have abandoned faith to search for them. According to Indian legend, entire villages were wiped out, burned to the ground, in the pursuit of them." Brook lowered them, let them fall back into her palm, and then held them out. "How did you come by them, then?" The major put his hands behind his back. "A stroke of luck—bad luck, I now firmly believe. They were being sold as rubies by some chap from the jungles desperate for enough money to buy food. I bought them more out of pity than anything. But when I examined them, I soon realized what I had. I asked questions—my second mistake." Since he wouldn't take them, she closed her fingers around the diamonds. "Because then word spread that you had them." "And everyone wanted them. The natives say the Fire Eyes were forged by the gods and given to Dakshin Ray, the tiger god. But humans stole them, and so Dakshin Ray put a curse upon them. To turn brother against brother, father against son, until chaos reigned—and then the tiger would come." "You can't believe that." But Papa's voice was not so firm, not so strong. Rushworth sank back down into the chair. "I didn't. But what I did believe were the three attempts on my life after I bought them. I thought it the craze of the locals, so I took my leave and came home. In England—sensible, staid, cool England—I knew logic would prevail." And yet it was here, in sensible, staid, cool England, that someone had nearly killed her for them. Where this man's niece had declared her an enemy because she thought Brook had them. Thought they must have been among her mother's things. And she had been right. Brother against brother . . . apparently cousin against cousin too. "You must have told your brother, and he his wife." She leaned into Papa's side. "That must be how Lady Catherine knew of them. They told her." The major sneered. "No doubt raised her to think they were by rights hers, and I stole them. The moment John set eyes on the things, he wanted them. The Indians would have said the curse's fangs sank into him. I never should have brought him in. I shouldn't have told anyone about them. But I had been too long out of the country. I needed help finding a reputable jewel dealer. But a finder's fee was all I offered, not the equal partnership they claimed. Next thing I knew . . ." He shook his head and averted his face. "They pitted us against one another. They would have torn us all apart. I did the only thing I could think to do and had this necklace made to hide them. Got rid of them." "Hidden is not rid of." Papa took a step away from her, putting himself back in the major's line of sight. "Why not actually do it? Toss them into the sea?" "You might as well ask why we don't destroy Rome, since people once fought over it. The history of those jewels, Whitby . . ." "It isn't the history you wanted to preserve." Brook strode over to the side table on which he'd set the necklace. She put the diamonds back inside the shells and found they snapped together without a visible seam. Such amazing craftsmanship, all to deceive. "If that were the case, you would have donated them to a museum. Instead, you sent them to my mother knowing well you could get them back someday. And hoped that in the meantime, your brother would forget about them." Rushworth ran a hand over his moustache but wouldn't meet her gaze. "I underestimated their potency then—I won't now. I gave them to your mother, young lady, so they are now yours. Donate them if you want, toss them in the drink if you'd prefer. But I want nothing to do with them again. I've lost enough thanks to those accursed diamonds. My brother, my best friend." "My mother." She traced a finger along the necklace with its vicious secrets. "Now, see here." The major rose, too fast, too close. "You can't lay the blame for that accident upon the Fire Eyes. I told John I'd sold them without his help, and I left the country. No one knew I'd sent the necklace to her, and it was so long afterward that she died . . ." "What?" Papa stepped to her side again. "You couldn't have even boarded the boat yet when she was killed." "Has your memory left you, Whitby? I went back in August. It was October she died." "No, it was August. The nineteenth. The day you left York for Bristol." "No." Rushworth's eyes went foggy. "It can't be. I got the telegram after I'd been back several weeks, and it said it had only just happened, after Pratt's murder . . . ." Brooke remembered Lady Catherine's mention of Pratt's father being shot in a back alley soon after her mother's accident. Might the two be connected? "I would not have . . ." The major's eyes widened. "You mean to tell me I could have gone to her funeral? Why did no one send a message while I was yet in Bristol?" Her father drew in a long breath and heaved it back out. "I cannot answer for your brother. For my part, I could think of nothing but the loss. Lizzie dead, Brook missing. Nothing else mattered. I can scarcely even remember the trip home from London after Mr. Graham wired the news to me." "John." Now it was his brother's name that Rushworth spat out like a curse. He passed a hand over his gleaming head. "Were he not dead, I would throttle him." "Do you think he . . . ?" Papa's breath came too fast and then seemed to bunch up. "Could he have found out somehow that you sent them to her? Could he have threatened her—could that be why she was on the road that night?" The major shook his head, but it didn't seem to be in answer. "I dare not say it's impossible, not at this point. Not if my niece has been asking your daughter for the diamonds." Silence pulsed through the room for a long minute. At last Rushworth stood, tugged his jacket down, and met her father's gaze. "I am sorry, Whitby. Much as I never liked you, I never meant to bring tragedy upon your house. I certainly never meant to hurt Lizzie." "I know that," her father said, voice hushed. The major's gaze shifted to Brook. "And it's then because of me that you were lost for so long. I am sorry for that too. I never wished anything but joy for Lizzie's girl." Brook could only nod. "I'll do what I can to set things right, though heaven knows I cannot undo the things that really matter. But I'll pay a visit to Crispin and Catherine. I documented everything, knowing I'd someday need proof the Fire Eyes were mine." They wouldn't believe him—there was no doubt of that. Documents could be so easily forged. But she appreciated that he wanted to try. "Thank you." He jerked his head once in a single nod and made for the door. She thought he would charge through it without another word, but instead he paused in the threshold and turned back. "My first stop when I leave here will be my solicitor, drawing up a new document verifying they have been legally given to you. But if my niece and nephew do not cease their pestering, you may want to consider that donation, my lady. And make it very public, with cameras flashing at every turn. Where the press is—" "There is safety. A friend of mine recently said as much." "You have wise friends. You'll need them." His shoulders rising with his breath, he nodded once more and disappeared. Brook turned to her father. "What now?" Papa reached for the necklace on the table and held it up. "I don't believe in curses." Thunder and lightning and darkness. Brook shuddered. "I do. There is a reason the Bible warns us not to dabble in such things—and it cannot be because they are fables." He granted that with a tilt of his head. "Allow me to rephrase—I do not believe curses can have more power than our Lord. We will pray for guidance. We will trust in Him. And . . ." He reached around her to fasten the necklace in its usual place. "Until we receive guidance from Him, we do nothing out of the ordinary. They don't know they're in this necklace, so it is, for the moment, the best place to keep them." She touched the pearls. So many times over the years she had done so, never guessing at what lay within. They would never feel the same to her again. "When she fled, she thought this from you. Whatever sent her to the road that night, she was wearing this because it was the most recent gift you'd given her." But it had all been a lie, and it could well have killed her. Brook shook her head. "We could give them to Catherine. Make the madness stop." Papa's nostrils flared, and he blinked. "No. If the Rushworths are somehow responsible for her death—no. I'll not let them profit from it. It isn't right." No, it wasn't. But then . . . so little seemed to be. # Twenty-Four Deirdre got past the door of the Hendon Hall Hotel this time. No Major Rushworth waited in the garden to halt her, and the frazzled maid who greeted her at the door waved her in and up the back stairs. Room six, the maid had said. She found it on the second floor easily enough and knocked. Lightly at first, though she'd been told the major was out, so she had little fear of disturbing him. When no reply came, she knocked louder. Was that a groan from within? Her pulse increased. "Uncle Seamus? Is that you?" She pounded harder but could hear nothing else from within. "Uncle, it's DeeDee." He was in there, and the major wasn't . . . so she put her hand to the knob and turned. It gave under her hand when she pushed. "I'm coming in." No objection sounded, so she pushed the door open the rest of the way. And screamed. 'Twasn't her uncle sprawled on the floor in a pool of red, that she saw in a moment. But it did nothing to keep her from screaming again, from pressing to the wall. Her knees went weak. The khaki pants, the scuffed boots, the gleam of sun on his head . . . the major. The maid had been wrong—he was here, and someone had . . . had . . . Did he live? She couldn't imagine how, with all the blood soaking into the wooden floor beneath him. But she had heard a groan, hadn't she? She heard it again, seconds before the pounding of feet sounded on the stairs. Not from the major—from the room to the right. Deirdre stumbled that direction and through the open doorway. "Uncle Seamus." He lay on a cot in a stained undershirt, a sheet pulled up to his waist. Despite the years since she had last seen him, she knew him immediately—he looked like Da, but for the silver hair. "Uncle Seamus." Her knees gave out before she could lower herself gracefully to the floor, but she made it to him and gripped his hand. No blood, praise be to the Lord. But plenty of shouting now, from the outer room, and a flurry in her uncle's room as well. It all blurred together. A cacophony, when all she wanted was a whisper from her uncle's lips. Figures darting around her periphery, when all she wanted to see was the lifting of his eyes. "Uncle Seamus. Uncle, it's me. It's DeeDee. Are you all right? Speak to me, I beg you." She said it over and over again. Finally he blinked. And then hands closed over her shoulders and pulled her away. She tried to shrug them off, to slap away the arms that turned her. Her struggle stopped cold when she caught sight of the Scotland Yard uniform on the man who held her. His eyes were icy and hard. "Are you the one who found him?" How had the police arrived so soon? Or was it soon? How long had she been on her knees, gripping her uncle's hand, ignoring the buzz around her? "I . . . I came to see my uncle is all. The maid said the major was out. I . . . He's sick. Uncle Seamus is sick." The officer's hands gripped her tighter. "Were you the one to find the major?" She wanted home. She wanted Hiram to put an arm around her. She wanted Lord Whitby to stalk up behind her and command this man to let her go. She wanted . . . she wanted to wake up and find this nothing but a nightmare to make her thrash about on her bed the way the baroness always did. "Miss!" "Yes." Her eyes slid shut, and her knees felt weak again. "Yes, I found him." "You forced the lock?" "What?" Her eyes opened again, though they refused to focus on him. "No, sir. It was unlocked. I turned the knob, is all. I . . . I wanted to see my uncle. The major was out, she said, and I heard him groan." His hands left her shoulders, and for a second she knew relief. Then he gripped her by the elbow instead and propelled her forward, out, into the buzz and cacophony. When he aimed her for the exit, she dug in her heels. "No! My uncle!" "We'll see to him. You need to come with me to the station." She nearly fell on the way down the stairs, earning her a curse from the officer—detective? He all but shoved her into the police carriage waiting outside, though then he left her in there alone for half of forever. The other half was saved for the agonizing ride through the city. Though when he hauled her into Scotland Yard as if she were a criminal, she began to wish the ride had lasted longer. He sat her down on a hard chair and positioned himself behind a desk. "Your name?" "Deirdre O'Malley." Oh, how her mum would be appalled to see her here now. She twisted her fingers around each other. "Your uncle is the batman of Major Rushworth?" The detective—there was a little board on his desk that labeled him as Detective Cole—scribbled something onto a page. "Aye. Seamus O'Malley." "You're unmarried?" He glanced at her with those beady eyes again. "Aye. I'm in domestic service." "To whom?" Oh, heaven help me. Would they call Lord Whitby in? But then, the major was a relation of the baroness. They would be calling on them with the news at any rate. She sucked in a breath. "Lord Whitby and his daughter, the Baroness of Berkeley." Cole added that to his notes. "The Baroness of . . . wait." Here he paused and looked up at her with, if it were possible, even less warmth. "Baroness Beauty?" "So she's been dubbed." She leaned forward. "Please, sir. Will they take my uncle to the hospital, do you think? Which one?" "Liller." The detective flagged another fellow walking by in identical dress. "Ring up Lord Whitby. At . . . ?" He lifted a brow at Deirdre. Her stomach knotted. She stuttered out Lady Ramsey's direction. Once the second chap bustled off, Cole shot question after question at her. Did she know Major Rushworth? Had she met him before? When was the first time? How well did she know him? What did she think of him? How long since she had seen her uncle? Did she honestly expect him to believe that his lordship had granted her two afternoons off to visit an ill relative? "He is a kind and fair employer, sir, who understands the importance of family. Yes, he let me off again after I was turned away yesterday! If you don't believe me—" "Then ask me yourself." Deirdre spun on her hard wooden chair, never so grateful to see his lordship. And the baroness had come, too, and now came to her chair and rested her hands on Deirdre's shoulders. A show of support. A touch of comfort. Tears stung the backs of her eyes. The detective rose, but slowly. "Lord Whitby, I presume?" His lordship didn't stretch out a hand to shake. Rather, he folded his arms over his chest and narrowed his eyes. "I would like to know why you're interrogating my employee for visiting her uncle. Unless familial concern has been made illegal in my absence from Town, and no one thought to inform me of it." Cole's lips pulled up in a hint of smile that dared to look mocking. "Your employee was at the scene of a murder, my lord. First at the scene, which more often than not denotes some involvement beyond happenstance." The baroness's hands went lax on her shoulders. "Murder?" The detective's eyes flicked to Lady Berkeley and swept her up and down. Judging, though Deirdre couldn't tell what verdict he came to. "Major Henry Rushworth was slain in his hotel room this afternoon." "No!" "It can't be." Lord Whitby stepped closer to them. He kept his gaze on the policeman. "We just saw him this morning." "Did you now." The detective sank back into his chair, that cynical little smile back in place. "Then have a seat, my lord. I have a few questions for you as well." Her father had lied to Scotland Yard—and Brook fully approved. He'd told them everything . . . except the small detail of the Fire Eyes. She stepped out into the sunshine and nearly stumbled back inside when a man with a pencil and pad sprang forward, another with a camera close behind. Lovely. She tucked a hand into her father's arm and let emotion wash over her face. A ballerina on a stage. A princess before an angry mob. A baroness sitting across from a detective who quite obviously had a bone to pick with the gentry. He had all but salivated at the prospect of linking her and her father to a murder. Never mind that Papa had gone to the House of Lords again today directly after Major Rushworth left them—saying he needed time to think where no real concerns would distract him. Never mind that Brook had been surrounded from ten o'clock onward by no fewer than a dozen young ladies and gentlemen. Those facts wouldn't have, Detective Cole had all but said, stopped them from hiring someone. The reporter licked his pencil. "My lady! What are you doing at Scotland Yard? Is it true someone tried to attack you this morning after you were out riding?" They had finally heard about that, had they? If months late . . . and a bit confused. She forced a sad, small smile to her lips when she would have preferred to storm by. Where the press was, there was safety. Please, Lord, help me. Help me not to crumble. Keep us safe. "No, I was not the victim of the crime today. My cousin, whom I met for the first time this morning, was murdered in his hotel room a few hours ago." She blinked several times and touched a fingertip to the corner of her eye, though no tears had gathered. They may have, had the anger not been so strong. Another person dead. And for what? Diamonds? Papa slipped his arm around her. Deirdre remained hidden behind them. The reporter scratched furiously at his pad. "Your cousin?" "My mother's cousin. Major Henry Rushworth." She looked over her shoulder at Scotland Yard and heaved a sigh she hoped was worthy of the stage. "I dare not say more. I don't want to hinder the detective's investigation. Justice must be done." She had her doubts that it would be. The camera flashed. Brook leaned into her father's side before it could flash again. A unified front, sorrow in the slope of their shoulders. Were it a dance, she would have pointed her toe, arched her back, brought her arms into a low circle to complete the picture. "Were you brought in for . . . for questioning?" The reporter's eyes were wide. Brook breathed a little laugh and tucked a stray curl under her hat. "No, no. We came in on our own the moment they called us. We must do anything we can to aid in the capture of my cousin's killer. We wanted to make sure the police had all the information we did, scant as it is." Not that they had even known about the murder when they were told to come collect Deirdre . . . but her maid didn't need the attention of the press. "Rushworth." The reporter tapped that line in his notes and looked up at her with raised brows. "He must be related to Lord Rushworth and Lady Catherine." "Their uncle." She turned her face up toward her father. "We should pay them a visit, Papa. They will surely be even more distressed than we are." "We will, my dear." His eyes applauded her. Then he nodded at the reporter. "If you'll excuse us." They didn't await an answer, just continued down the stairs with a measured step. Deirdre, Brook noted when she looked up, had slipped around them while they were distracting the reporters and waited at the car. She looked awful. Her face was pale, her eyes haunted, and she clasped her hands so tightly her knuckles were white. "I have to see my uncle," was her greeting when they joined her. Brook reached for her hands. "Of course you do." "We'll take you." Papa opened the door and ushered them both inside, shielding them from the camera until the door had shut. Brook settled beside Deirdre and kept ahold of her hands, which were cold and trembling. "We can go in with you too, if you want company. I wouldn't want to be alone so soon after seeing what you did." Deirdre's chin shook too. "Thank you. I would appreciate that." She sniffed and lowered her head. "I wish Hiram were here." Brook squeezed her hands. She had seen them together a few times, knew they were close. "I'm sure he would want to be too." Papa cranked the engine to life and slid into the driver's seat. Within the minute, they were pulling onto the busy streets, headed for a part of the city she had yet to see. No one seemed inclined to talk, so Brook let her gaze drift to the window. Let the truth drift into her heart. She wanted Justin. With no front, no walls between them. She wanted to be able to rush into his arms, to kiss his cheeks, to cry on his shoulder if the tears chose to come. To tell him what the major had said that morning, what the Fire Eyes were . . . how everyone connected with them seemed to end up dead before their time. She wanted to forget the anger, forget the questions, and just be Brook and Justin again. Her fingers found the faux pearls and twisted them together. The irony of the long habit hit her anew, and she let her fingers fall. She needed the things gone—but Papa was right. They had to tread carefully. Too many people had already died, and if the Rushworths were responsible, they had to bring them to justice, not give them what they wanted. For her mother. For the major. Eventually her father pulled up in front of a large, dreary-looking building stained with soot and time. Brook reined in her thoughts and gave Deirdre's hand an encouraging squeeze. They all exited in silence, traversed the walk without a word, and only spoke once inside to learn which ward Seamus O'Malley had been taken to. The hospital was utilitarian, the starkness unrelieved by color. Their shoes clicked loud against the tile floors. Brook and her father flanked Deirdre, and the maid darted a look her way. "I'm so sorry for bringing this upon you." "It isn't your doing." Brook's voice came out a whisper in the white corridor. Deirdre shook her head. "It's because of me you were called down there. Because of me the reporters saw you leaving." Papa sent encouragement from his gaze without the need to smile. "Circumstances that were outside your control. The only thing you did, O'Malley, was try to care for your uncle. There is no blame to be found in that." "The detective—" "Will keep an open, unbiased mind about it all or will find himself out of favor with his superiors." Her father's face went hard. "I have never much cared for those who use their influence amiss—but there is no guilt for this in my house, and if Cole tries to find any, I will use whatever force I must to see justice done. And if my influence alone doesn't suffice, we've two dukes in our corner." "At least one of whom would be eager for an excuse to let loose his temper." Brook's lips tugged up. Justin, with his Duke of Stafford glower, would be furious indeed when he learned how Cole had interrogated her. Even with all between them, she knew that. Papa nodded toward the door they'd been instructed to take. Inside were a row of cots filled with blanket-covered figures. A few sat up with book or newspaper in hand, others seemed to be sleeping. Brook touched a hand to Deirdre's back to indicate she should lead the way. Deirdre peered at each figure they passed, until finally she sucked in a breath and came to a halt. "Uncle." The man on the bed was pale as the moon with deep circles under his eyes, his skin wrinkled and cracked. His eyes fluttered open, though they stared up without recognition. "Who . . . ?" His voice sounded faint, scratched. Deirdre reached for a cup of water and lifted his head to help him sip. "It's Deirdre, Uncle Seamus. I'm here in London with Lord Whitby and heard you were here as well." "DeeDee." His eyes focused upon Deirdre's face. "All grown." "Aye. 'Tis been too long." She settled a hand on his forehead. "You're hot as blazes. How do you feel?" His eyes went cloudy again, and his face screwed up. "The major. Is he . . . ?" Brook reached for her father's hand. Deirdre swallowed audibly. "Dead." Seamus turned his face away. "I was too weak to help. All I . . . all I could do was lie there. Pretend to be dead myself." "You're ill. Better to pretend to death than meet it in fact." Deirdre dashed at her eyes and sniffed. "Have the police been to talk to you?" The man shook his head. "I heard . . . when they were taking me . . . something about being too far gone to have seen anything." He turned his face back to Deirdre, then beyond her. Recognition sparked when he spotted Papa. "But I did, milord. I saw him." Papa eased forward. "Saw who, O'Malley?" "Don't know. Young fellow. Spry—climbed . . . out window. Wore a hat. Long coat. Couldn't . . . couldn't see face, but . . ." "Easy, uncle." Deirdre trailed her fingers over his face. "Don't tax yourself, now." He reached up, though it looked like it took all his strength, and caught Deirdre's hand. "He took . . . papers. Solicitor." Brook's breath tangled in her throat, and she looked up at her father. The major had said he was having papers drawn up—and who but Lord Rushworth and Lady Catherine would have a reason to take them? Who else would stand to inherit anything that was his in light of his death? The flare of Papa's nostrils said he was thinking the same. "You heard his voice, then. Was he educated? Had he any accent?" "I . . . Educated. He was educated." Seamus closed his eyes for a long moment, then dragged in a deep breath. "Major seemed . . . to struggle to place him." Brook's brows pulled down. He wouldn't have struggled to place Rush—he looked just like his father's portraits. But any number of other people could have been vaguely familiar, she supposed. Her father nodded and gave the man a tight smile. "That's very helpful, O'Malley. We'll find his solicitor and get a copy of whatever was stolen. Justice will be done. You rest now." He patted Deirdre's shoulder. "Stay with him as long as you like. But if darkness falls before you leave, don't try to take the tube—hire a hack. Here's enough for the fare." Deirdre opened her mouth, obviously set on refusing the money Papa held out, but Brook shook her head. "Take it, O'Malley." There'd been tragedy enough for one day. They didn't need the too-lovely maid finding more in the tube tunnels. She obeyed, slowly, and sank down onto the edge of her uncle's cot. "I don't deserve your kindness, my lord." "Nonsense." He turned, ushering Brook along with him. "Family is the most important thing, always. You focus on yours right now. We'll give him some peace so he can rest." Brook cast one last look over her shoulder at the shriveled man, the broken, beautiful girl. Both with a pall of death over them. The whole world, it seemed, had one to match. # Twenty-Five The weight pressed upon Deirdre's shoulders until she thought she wouldn't be able to trudge her way down the hospital corridor. Last night when she finally left her uncle's side, it had been bad enough. Today, with the sun shining bright through the windows and catching on the baroness's hair, it was worse. Perhaps, had her ladyship merely granted her more time off, it wouldn't have weighed so heavily. But she had driven her. Cheerfully so, even though Lady Ramsey had apparently insisted Lady Berkeley go out to a dinner party with them last night, and it had left her exhausted today. Perhaps, had her uncle been as bad as yesterday, she could have shoved guilt aside and focused solely on him. But he was, praise God, much improved—and had looked at her with Da's eyes, with wise eyes, as if knowing exactly how she had treated this family that would do so much for her. She darted a glance at the young woman beside her. There were ladies aplenty in the hospital, most of them part of some aid group or another, out to do their good deeds for nameless faces. They came in flocks, in wide-brimmed hats overflowing with lace and silk flowers, in their best morning suits and dresses. Lady Berkeley had come in her simplest, her hat modest—her worth coming through all the louder. A nurse passed them, and Deirdre drew in a breath and tried to smile. "Did you have a nice time last night, my lady? Was His Grace there? Or Lord Worthing, perhaps?" Her ladyship sighed. "The duke was, surprisingly. And his cousin with Miss Rosten, which meant that my cousin spent the night flirting outrageously with some poor chap who's likely half blind with love now." Deirdre smiled. "It is hard to feel sorry for her when she goes about revenge with so much energy." Her ladyship chuckled. "It is, at that." "And His Grace? Did you speak with him?" Though her ladyship hadn't said a word about it, she'd watched the disappointment grow each day he hadn't come. She knew that whatever they had argued about this time, the baroness regretted it. Now all emotion drained from her countenance, the mask left in its place perfect but empty. "I did. Long enough to request he come by this morning at nine. Which, of course, he didn't." They opened the massive front door and stepped out into a fine mist caught halfway between fog and rain. Deirdre stopped her ladyship with a hand on her arm. "My lady . . . life can be so short. You mustn't let misunderstandings get in the way of happiness. You charged through the city at night last year to keep things right between you—why do you now wait around for him to come to you?" "Because I . . ." She looked away, but not before Deirdre saw the pain in her eyes. "Because everything has changed." A month ago—a week ago, a day ago—she wouldn't have dared to loop her arm through the lady's. Today, she couldn't imagine doing otherwise. "He's in love with you, my lady. And you with him." Lady Berkeley sighed. "What if it isn't enough?" And Deirdre knew, as she gazed on this hurting girl, that she could have been any hurting girl—baroness or not. She knew that if Mum realized how she'd come by the money she sent, she'd toss it into the pond. Knew that she couldn't keep serving these good people knowing how she'd betrayed them. Knew she had to throw herself on their mercy and let come what may. "My lady." A step away from the car, she drew them both to a halt. But she couldn't look into the familiar eyes or the inquisitive face. She drew in a breath that wasn't deep enough and locked her gaze on the embroidery at her ladyship's shoulder. "I need to confess. You've been so good, you and your father, especially about my uncle. But . . . but I really don't deserve it. I've done something terrible." The shoulder sagged. "Pratt. All my post." Of course she'd suspected, once His Grace got home and they talked. Deirdre's arm slipped from her ladyship's, down to her side. "I was only a housemaid when it began, and the money he gave me . . . they needed it, my mum and family. And it seemed harmless at first—he wanted to know which suitor Lady Regan favored, before you came home. Who was to be named Whitby's heir." "But stealing?" Her ladyship stepped away. Perhaps she'd hop in her car and leave Deirdre to find her own way home—heaven knew it would serve her right. As would finding all her things tossed to the curb when she got there. "Did that seem harmless too? Did he pay you more for that?" Deirdre winced at the bitter tone. "I couldn't get out. He turned to threats, if I tried. First that he would force me to his bed and then . . . then he threatened my family. Said he had a man in my village ready to burn the house to the ground." "So you come to us!" The baroness spun to face her again, her face a combination of anger and pity. Her accent deepened, the French curling around her vowels and consonants as it did in those first moments when she awoke from the nightmare. "Did you not pause to think that we could have helped? That we could have protected them? Protected you?" Had she? No. Never. Perhaps because she couldn't imagine they would go so far out of their way to help her—though they had just proven they would. Perhaps because she had never really believed that their good could win out over his evil. "I'm sorry, my lady. I know you have to dismiss me, at the least, perhaps even have me arrested for tampering with the mail. But I couldn't keep lying to you." If he was merciful, his lordship would take action now and not wait until they got back to Whitby Park so he could make an example of her before the rest of the staff. If she were beyond lucky, he would not involve the law, in order to keep his name from the press again. The baroness pressed the heel of her hand to her forehead, under the sloped brim of her hat. "Well, well. Are the conspirators squabbling?" Deirdre jolted at the voice, her gaze flying about the area until it clapped upon Detective Cole. Without allowing herself to think of the audacity of it, she stepped in front of her ladyship. "Detective. Have you come to talk to my uncle? He is awake, and he saw much of what happened yesterday." The man tilted his lips into a patronizing smile. "Oh, I already know what happened." "Good." She lifted her chin, even if she had to clutch her hands together to keep them from shaking. "Then you know it was an educated man what stabbed him, one he didn't know well." A condescending chuckle joined the smile. "That doesn't much narrow it down, does it? Given that the major has been on the subcontinent for almost two decades. Which is why—" he took a step nearer, and Deirdre could see the hard light gleaming in his eyes—"I find it so very odd that you, niece to his batman, end up working for them, the house of the major's archrival." Her back stiffened. "My uncle recommended me there—he said it was the finest house he'd seen." The baroness stepped to her side. "And you are better versed in ancient gossip than I supposed, Detective, if you know of that old rivalry. But let me guess—my cousins told you." He inclined his head. "Did they also tell you of the argument between the major and his brother—their father?" Such darkness . . . so like that always in Pratt's eyes. Deirdre shuddered. The detective's eyes narrowed. "Over the diamonds. Which are by rights theirs, but which they believe you have. Their theory . . . Lady Berkeley . . . is that when the major tried to reclaim them, you had him killed." Her ladyship drew herself up—but Deirdre's gaze was snagged by a new figure striding their way, fury in His Grace's every movement. She reached for the baroness's hand and gave it a little tug to get her attention. Lady Berkeley shifted and made a quick half curtsy. "Good morning, Duke." "My lady." The duke packed a world of feeling into the greeting, though it was the detective he speared with his glare. "Detective." "Your Grace." Cole's face went harder, a shutter coming over the gleam in his eyes. "Excuse us, but I'm engaged in official business with the baroness." "No you're not. You're engaged upon harassing a young lady whom your superiors have verified had absolutely no motive for arranging the death of her cousin." He jerked his head, a clear dismissal with an undertone of threat. "I suggest you return to Scotland Yard and take a look at the papers sent over by the major's solicitor." The detective held the duke's gaze for a long moment, then glanced back to the baroness. The muscle in his jaw ticked. His Grace moved nearer, looming over Cole. Deirdre hadn't thought the detective short, but in that moment he looked it. "And I suggest you tread carefully." "I always do." Cole narrowed his eyes. "What exactly is your interest in all this, Your Grace?" The duke lifted his brows. "You're a detective. Figure it out." "Oh, I will. Rest assured." His Grace stepped aside and made a flourishing gesture indicating the detective ought to leave. "It oughtn't to take you too long, if you know how to do your job. And do have a lovely day." Cole stalked off toward a horse hitched at the far corner of the hospital. The duke watched him for a moment, then spun back to them. His face had gone hard as granite, and fury blazed brighter than ever in his eyes as he locked them on the baroness. "O'Malley, excuse us for a moment." He took the lady's hand and pulled her the opposite direction. Were it anyone else looking at the baroness with such anger, Deirdre may have refused. But she wasn't about to get in the way of a man in love. Justin's blood was a roar in his ears, his heart a thundering tempest. It had begun that morning, when he'd opened the paper to see her plastered on the front cover, with the headline of MURDER HAUNTS BARONESS BEAUTY nearly sending him into a stroke. Had her father not shown up within minutes, he would have been pounding on her door long before the nine o'clock hour she'd asked him to come. As it was, he'd spent his morning pounding on doors with Whitby instead, trying to find the solicitor that Rushworth used. It had done little to cool his temper. Justin pulled Brook into a poor excuse for a garden at the side of the hospital and, for lack of privacy, turned to Monegasque as he spun her to face him. "Are you insane or just stupid?" Not, perhaps, the best greeting if his aim were to keep her calm. But at the moment he had no desire for calm. He wanted a fight, and no one else in the world would give him the one he needed. She pulled her hand free and looked as though she wanted to slap him with it. "Excuse me?" Her words were in Monegasque too. Justin waved a hand at the world at large. "You have detectives chasing you with murder charges, a killer on the loose slaying people connected to these stupid Fire Eyes, and what do you do? You head out into the city, alone but for a maid, without ever pausing to consider for even one second that you could be next!" He expected her to shout. Instead, she went calm—but seething. "What do you know of it? You didn't even bother to come this morning when I asked you to." "Because your father came to my house at eight. I assumed you knew that and would wait for me—that while I was off pounding on solicitors' doors with him, you wouldn't be darting off on your own, trying to get yourself killed." "I didn't know." Still, frustration overtook the realization in her eyes, and she pivoted away. "But how could you possibly expect me to sit idly by? It's fine and good for you to put yourself into the path of all this, but if I so much as take my maid to visit her uncle, I'm either stupid or insane?" "You don't think. Not about consequences. You never have." He turned, too, and took a step to put himself in front of her again. "You chase whatever impulse seizes you, valuing your blasted independence above common sense." "And what if I do?" Her eyes were ablaze, green fire spitting at him. "If it's a fault, it's mine, and one you've long known about. If you loved me like you claimed—" "If? You doubt me because I don't applaud when you run headlong into danger?" Now the seething gave way to fuming, and she sliced a hand through the air. "For once in your life, why can't you accept the fact that perhaps a person isn't wrong just because they don't agree with you?" He took a step back. "When have I—" "When have you not? 'You'll not take the stage.' 'You'll not race.' 'You'll not get near that horse.' You always have to be giving orders, the one in control, and it drives you mad when you're not!" She surged forward, poking a finger into his shoulder. "Well, Duke, you're not my father. You don't get to dictate to me." "You're my son, Justin, not my nursemaid." His father's words rang in his head. Yet again, being blamed for caring. For wanting someone to take two minutes to think about consequences, about how a decision might affect someone else. Might affect him. How he might feel if someone drove off the road or ran pell-mell into the clutches of a murderer. He held his arms wide. "I guess that's who I am. Who I've always been. If it's a fault, it's one you've long known about. What, then?" She breathed a laugh as dry as the withered flower stalk by her foot. "That would be the question, wouldn't it?" The temper in his eyes went darker, calmer, more treacherous. Turned to ice. No. He had already lost his father—he wasn't going to lose Brook. He couldn't lose Brook. Not to this Fire Eyes insanity, and not because of his own mistakes. He swallowed, breathed, sent heavenward a silent prayer. "Just tell me. Tell me what you need me to be." "Here." She thrust her hand downward, pointing at the ground by her side. "I need you to be here, but you never are." "I'm here." He stepped forward, clasping her elbows. She wrenched free. "You're not. Even when you are, you're not, you're behind that dashed wall you've built." She shook her head and wrapped her arms around her stomach. "You won't . . . ever since I came here, you . . ." When she averted her face, he caught the glistening of tears in her eyes. He reached out again, but she retreated and shook her head. "I thought I loved you. That we could make it work, but . . . but we don't. We don't work anymore. You can't just kiss me again and set the world to rights. Maybe . . . maybe God only meant you to bring me here. Maybe friends is all we were meant to be." The earth beneath him crumbled, opened, swallowed him into its yawning darkness. "I can't just be your friend anymore." "I know." She held herself tighter. "I guess that means we're . . . nothing." He couldn't move. Couldn't speak. Couldn't think. It was unfathomable. Because he needed her so much—how was it possible she could bid him farewell so easily? Yet she did. She stood there for a moment, no tears spilling over their rims, no uncertainty shaking her. And then she turned and walked away, her arms still clutched around her stomach. Justin could only stand there in the pathetic little garden and let his eyes slide closed. He tried to pray, but he had no words. Just a cry that came from his gut but couldn't find purchase on his tongue. And so it echoed through him, clanging and pounding. An accusation. A desperate plea. # Twenty-Six My lady—" "Don't." Brook didn't even look at Deirdre as she slid into the driver's seat of the roadster. She had already cranked it and had the key in her hands. Steady, those hands. As steady as her voice. Because inside, she'd ground to a halt. Still, if not peaceful. Too still for shaking. Too still for words. Deirdre said nothing more. Brook didn't let herself wonder what she had meant to say—no doubt it was some question about what she intended to do with the knowledge that she had acted as Pratt's spy. But Brook couldn't think about that right now either. She could only think of pressing the clutch, the accelerator, the brake. Where to turn, when to signal. How to park, and then to put one foot in front of the other to lead her inside. She paused at the door but still couldn't look at her maid. "O'Malley, when we get inside, I want you to pack—" "My things. I understand." "No. Well, yes. But mine too. We're going home." "We . . ." Wisely, she said no more. Not in the mood to wait for a bell to be answered, Brook pushed open the door. She bypassed the drawing room with its laughter and crowds of near-strangers and headed straight for the study, where Papa was most likely to be. Aunt Mary was there too, leaning over his shoulder and pointing at some paper or another on the desk. They both looked up when she entered. Her aunt smiled. Her father, when he saw her face, stood. "What is it?" Words. The only ones she could find were French. "Can we go home, Papa? Please?" "What?" Her aunt had obviously understood, given the outrage in her eyes, though she answered in English. "Absolutely not! You are the darling of Town, you cannot possibly leave before the king's coronation—" "Of course we can." Papa's voice was low and soft, his eyes seeing far beyond hers. "Did your Justin find you?" He tried, and failed, to pronounce it correctly. But his name still made a sob well up in that empty place, lodge in her throat. "He is not my Justin. He will never be. I . . . I want to go home." "Of course." He came around the desk and pulled her to his chest. "My darling girl." He said no more, because he was Papa, and he understood when silence was all that could soothe. Aunt Mary, to her credit, held her tongue, too, and didn't even faint. She just whisked by them. No doubt to go somewhere private to bemoan her niece's utter ignorance of society. Or perhaps to get reinforcements. A minute later, when Papa drew away, Melissa was there with wide eyes. "You're leaving?" Brook held out a hand for her cousin to grip, though she couldn't manage a smile. "I have to. I don't suppose you want to come?" She could use a friend to laugh with, to mourn with—one who may have been reserved at first but who loved her now. Who never feigned feeling just to turn on her. But Melissa sighed. "I can't. Mama would have a fit—and I need to stay here and snag myself a husband." "Oh, Lissa." She tugged her in for a tight embrace. "Not out of spite. Don't marry out of spite. You'll be stuck with him for all your life." "I know." Melissa pulled away, her face somber. "I promise. But I will stay. You need your open spaces and ocean to cope, I need my crowds and laughter." To that she could only nod. Papa, it seemed, was the only one who related to her need. So it would be just them again, and the staff who knew how she liked her coffee and sausage and to stir the fire earlier than usual in her grate. And a maid who would sell her secrets to a land-grubbing neighbor—but she would ignore that for now. She would get home, get settled. Then talk to Papa about Deirdre. If she were empty inside, should it not have made her feel lighter? But her legs, as she turned for the steps, felt heavy as despair. Justin exited the House of Lords and paused a moment to look up at the grand, towering facade of the palace. For years, anytime he saw Westminster's pointed spires and gothic styling, he had dreamed of being inside its cavernous chamber, taking the seat reserved for him. Facing the throne. A lot of good he was doing, finally there but his mind a few crucial miles away. He wanted to focus on the laws and debates—but he couldn't, not when Brook was still in danger . . . and had dismissed him so summarily. His feet hitched when he caught sight of the figure leaning against a shining new Austin parked a spot away from the Rolls-Royce. Maybe Worthing was waiting for his father—Justin had noted the Duke of Nottingham chatting with a few other lords of his generation after the session ended. With any luck, the son wouldn't even notice Justin walking by. He could hope. He had, after all, spent half the night on his knees in prayer before exhaustion had claimed him. And then the other half sleeping on his hard floor. Surely that was penance enough. Apparently not. Worthing straightened as Justin neared, that annoying grin on his face and his hands in his trouser pockets. "Stafford! Good day." A sigh fisted in his chest. He had no fight left in him. But little patience either. "What do you want, Worthing?" The idiot man's grin only grew. "To earn your eternal gratitude. She left Town this morning." "What?" Justin's feet planted themselves a few feet from Worthing, refusing to go a step farther. "For Yorkshire?" Worthing nodded. "Would have left yesterday afternoon, had it not taken so long to ready. But at first light . . ." He pulled one hand out of his pocket to illustrate his point, imitating a car driving away—complete with muted engine noises. Had it been Thate, and news of someone else's leaving, Justin would have laughed. "She told you she was going though." The grin turned patronizing. "Yes, you see, we take part in this bizarre social ritual called conversation. You should give it a try sometime. It's when you exchange words—at a normal volume—for the purpose of sharing information, rather than for accusation or inflicting emotional pain." Justin's shoulders slumped. Even at that, he could muster no anger. He was too weary. "It wasn't all me. I started it, I grant that, but—" "I know." Worthing clapped a hand to his shoulders, as if they were the best of friends. "She told me what was said, and I told her she was being an idiot, that you had a perfectly valid point and that you wouldn't have been so very fearful if you didn't love her so much—and had you not suffered enough losses this year. But you know Brook." He rolled his eyes and dropped his hand. "A mite stubborn, that girl." He . . . he had defended him? To Brook? Justin stared at him for a long moment. "Why?" "Is she stubborn? That is a question only the Almighty can answer. But if you mean why did I say such things to her, the answer ought to be obvious." Worthing met Justin's gaze, held it. "She's wrong. I don't know why she's so set on denying what she feels for you when it's obvious to anyone who sees her watching you, but she's wrong. You are meant for more than just getting her to England. God isn't finished with the two of you yet." "Know that, do you?" But the words didn't come out mocking—they emerged . . . hopeful. No smile touched Worthing's expression now. Peace, however, saturated it. "Yes. I do." Again, Justin was reduced to staring. What stared back at him made him feel the dunce—though, granted, a relieved one. "You're really not in love with her." Worthing chuckled and leaned into the side of his car again—at least, Justin assumed it was his. "Are you daft? I'd never survive it. If she isn't trying to bore me to death with some obscure academic work, she's trying to give me a heart attack, flying around on that wild stallion of hers." Though he'd never expected to experience such a thing, a grin tugged at Justin's lips. In the presence of Worthing. "She's magnificent, isn't she?" Worthing laughed outright this time. "That she is, and I adore her—in much the same way I adore my sister, who drives me nearly as mad." He paused and then gave a sideways nod in the direction his hand had motored. "Go after her, you imbecile. And don't relent until you have an actual conversation and have convinced her you can't live without her. Address whatever's keeping her from declaring her love for you and move on to all the happily-ever-after nonsense." For the first time in weeks, hope sparked to life. Justin took a step toward the Rolls-Royce but then paused. "Worthing . . . I'm in your debt." The grin reemerged. "Excellent. No doubt I'll need a favor one of these days, when I'm the one gone stupid over some young lady." Justin smiled again and hurried to his car. Worthing followed, saying nothing while Justin cranked it and slid inside, but then he leaned toward the window. "Listen." His voice was serious again, and as low as it could be and still be heard over the engine. "My first thought, when she said she was leaving, was that it was good—she'll be away from the Rushworths, Pratt, whoever killed her cousin. But I can't shake the feeling that the danger will follow her home." Cold dread overtook Justin's heart. Of course it would. Anyone who would kill so easily wouldn't let a few hundred miles get in his way. He nodded. So did Worthing. "My advice would be to resolve this thing between you as quickly as her stubborn will allows—and then get ready. The tempest, I think, has only just begun." Because the words felt like truth, Justin nodded again. And because they were a terrible omen, he sighed. "I trust you'll be in prayer." "Without ceasing. For the both of you." He stuck a hand in, and Justin clasped it without hesitation. "Keep in touch. And if you need me, give the word." Funny how, in that moment, this man he had thought for sure was an enemy seemed like a certain friend. "Let's pray I don't have to." Without further ado, he backed out and joined the stream of cars and carriages. A quick stop at his townhouse to collect Peters and their things, and he'd be on his way. He'd rent rooms somewhere in Whitby, to be close by. And he'd simply wear her down with his presence. He would be there. Every hour, every day, knocking upon her door. Praying, without ceasing. Until she let him in again. Darkness cloaked the familiar heath by the time Deirdre found a moment to step outside. Still, it was earlier than it should have been. She hadn't finished unpacking for the baroness yet, but she'd been dismissed. No doubt the lady chafed at her presence. The air had a nip to it, but it still smelled of spring in the country—a scent she had missed acutely in London. But she hadn't counted on being back so soon. And knew, now, she wouldn't be here long. The baroness would talk to his lordship soon. Then Deirdre would find herself called forward after prayers, denounced in front of them all. Mrs. Doyle would gasp and press a hand to her mouth. Mr. Graham would rumble out a cough of outrage. Beatrix's eyes would go wide with shock. And Hiram . . . Hiram would look at her with that profound disappointment that would shatter her heart into a million pieces. "Escaped finally, did you, Dee?" Her eyes slid shut against the warm, cheerful voice. She buttoned the jacket she had slipped on and sank onto the stone garden bench. "How have you been, Hi?" He chuckled as he took the seat next to her. "It was quiet while you were gone, as expected. Though I can't say as anyone was surprised at the wire saying you were on your way back. Murder though—didn't expect that." The image kept gnashing at her, popping up whenever she closed her eyes. The major, in a pool of his own blood, his limbs at odd angles. She shuddered. "I'm the one who found him. When I went to see Uncle Seamus." "Oh, Dee." His arm came around her shoulders, and he pulled her to his side. She sagged against him and wished she could stay there forever. But what was the point? She would soon be gone. Back to Kilkeel in disgrace. Then what would Mum do? "I've ruined everything, Hiram. Lord Whitby and the baroness were so kind, so supportive—but I'd tossed it all away long before that. They'll sack me soon." Hiram's hand stroked over her hair. "What do you mean, sweetheart? You've done nothing wrong." Sweetheart. She savored it for a moment, let it turn over in her mind. It clashed against the guilt. "I have, though. I already confessed it to the baroness. I . . . it was Pratt. He approached me in the village a year ago." Hiram went stiff, but he held her all the tighter. "Approached you how?" Her stomach hurt in the remembering. How she had turned down a side street to make it the quicker to the post office and had all but run into him. How, at first, she had been struck dumb by his beauty—up until then, she had only glimpsed him from afar when he prowled around Whitby Park. But he must have seen her. He knew her name, her position, her salary . . . her family's situation. "He . . . he said he knew how my family was struggling, and he wanted to help. That I had two choices—I could either become his mistress or . . . or feed him information on who Lord Whitby would name heir." "DeeDee." He turned a bit and wrapped his other arm around her too. Sorrow laced his tone. "Why'd you say nothing? You could have told me. Told his lordship." She should have. That was so clear now, but at the time . . . "It seemed so silly. I had little information to give, but he paid me well for it. But then the baroness came, and he'd grown so impatient. Threatening—which was always lurking under the surface; I knew that all along—that if I hadn't agreed, it would be trouble to find my family, not pound notes." She fisted her hands in his shirt and pressed her forehead to his shoulder. "Now what am I to do? I'll be dismissed, possibly arrested, and my mum . . ." "Your mum'll be fine." He pressed a kiss to her hair. "You'll be fine. His lordship won't want the attention of pressing charges, and we'll find other positions. I'll take up farming, if I must." "Hiram." She wanted to cling to that we, but it wasn't right. "No. It's my trouble, my wrong. You can't be the one to pay for it." "And you think it won't be punishment for me if you leave, if I must do without you?" He touched a hand to her face to turn it and then feathered his lips over hers. "I love you, Dee. Where you go, I go. We'll marry, and I'll help you take care of your family. I promise you." She should refuse. But she was too selfish. Sliding an arm around his neck, she kissed him soundly, letting the joy of it scrub at the bitterness and regret. It couldn't obliterate them, but it eased their harshness. "I love you, Hiram. I'd be honored to be your wife. Though sure and I'm sorry to come to you with such trouble at my heels." "We'll face it together." He brushed at the hair coming loose from its pins, and the moonlight gilded his smile. "Two are stronger than one, aye? We'll start looking for other positions. Together." She nodded and rested against him again. But her mind went back inside, up the stairs, to the chamber where, if the baroness had found sleep, she was no doubt thrashing about in the throes of her nightmare. Her ladyship couldn't escape her troubles, and heaven help her but Deirdre felt responsible for them. Bound to her through them, obligated to help. And she would, if she were given the chance. But that seemed a very big if. # Twenty-Seven From her seat at her window, Brook could hear the rumble of the Rolls-Royce as it made its way down the drive. She wouldn't look up from her book. She wouldn't. She had no need to see the silver paint, the golden head—though today the top would be up, as the rain was coming down in torrents. She had thought it would keep Justin at home, or wherever he'd been staying the past fortnight. No such luck. Of course, if her father wouldn't keep entertaining him . . . Her fingers curled around the edges of her book—Kant, and the German was nearly impossible. Especially when she was not watching the Rolls-Royce disappear over the knoll. With an exasperated breath, she tossed it to the window seat and took to her feet. Deirdre stepped out from the dressing room. "Do you need something, my lady?" Her words were quiet and eager, as they had been each of the interminable fifteen days since they'd left the hospital in London. Brook knew Deirdre was waiting for the proverbial shoe to drop. Waiting for Brook to tell her father, and for her father to dismiss her. And several times, she had nearly confided what Deirdre had confessed. But then she would stop. Dismissing her wouldn't get the letters back or erase the secrets told. Dismissing her would mean needing to find a replacement, and that meant someone new who could be bought and bribed. Deirdre would make no new betrayal. She might be, right now, the most trustworthy employee to be found. Brook forced half a smile. "Nothing. Thank you. I'm going to find my father." Not meeting her gaze, Brook kept on for the door. She didn't want to dismiss her . . . but she hadn't quite forgiven. She had tried. Had prayed the words. But she was still so empty inside. Papa was, as expected, in the library. When she entered, it wasn't just the scent of pipe tobacco and paper and leather that greeted her, though—there, too, lingered the scent of lemon and spice. Justin. She very nearly retreated, but then she'd be left with only her own company, and she had days ago grown annoyed with herself. "Have a pleasant chat, Papa?" Justin had been here hours today. Hours. And her father had the gall to smile over his newspaper. "I did. We were discussing the latest advancements in aeronautics. You should have joined us, my dear. You would have enjoyed it." "Papa." She sank into her usual chair, at right angles to his. "Why will you not turn him away?" "Because I enjoy his company." He reached for his pipe and put it between his teeth, though he didn't light it. He never did while she was in the room, after she'd once coughed. "Clever young man. I can see why you like him so well." "Liked." "Come now, my dear, we both know you're only so miserable because you're in love. One of these days you'll relent long enough to talk to him, and it will take but a single honest, earnest conversation for you to put aside your differences." He took the pipe out again and used it to point at her. "When that day comes, I would prefer the pleasure of saying 'I knew it all along' to the regret of saying 'I'm sorry for treating him poorly while you were at odds.'" "Papa." "You cannot avoid him forever." Why not? Why would he not go away? Back to London or Gloucestershire or India or Africa or anywhere—so long as it wasn't Whitby Park? She rested her elbow on the arm of the chair and then her head in her hand. "I don't want to see him. There's nothing left to say." "I think there is." He put newspaper and pipe aside and leaned forward, resting his hand on her knee. "Brook, whenever I walk into your room, I see the same book sitting on your bedside table. What does it say to do?" "That isn't fair." She had tried looking for comfort in La Bible. She had tried to find answers. But it had just been words these past weeks, never sinking deeper than her mind. "I know we are to forgive. And I will. But that doesn't mean that we can go back to the way things used to be." "Who ever said you should?" He sat back up, shaking his head. "But God does not just instruct us to forgive—He instructs us to trust. To trust that, even though life hurts us, He will take care of us. That even if we lose the ones we love, He will sustain us through it." Her brows knit. "Trust is not my problem." "Isn't it?" He gave her knee a squeeze. "You are afraid to love, my dear. Afraid that if you do, it will only come to a miserable end. And it may—life comes with no promises. But it's worth it. It's worth the risk." She shook her head, intending it to be a denial that she was afraid. But with each movement, her resolve shifted. "No. No, it's not worth it. How can I possibly love him when it means arguing like we have been? When it means he doesn't want me to spread my wings lest I get hurt in the flight—" "Brook, he is the one who taught you how to fly! But is it so unreasonable that he asks you to look before you leap?" How was it that Justin could make her feel the fool even when he wasn't in the room? "What is the point, though? We will only hurt each other. Or . . . or lose each other later." He gripped her hand, resting their clasped fingers on the book she'd left on the side table last night. "Must I quote Shakespeare at you, my dear? 'It is better to have loved and lost—'" "No! It isn't!" Silence greeted her outburst, and it reigned long enough to make her glance at her father's countenance. To see the patience there . . . and the sheen in his eyes. Of all the people for her to have said such a thing to . . . His fingers tightened around hers. "Should I not have loved your mother, then? Is that what you're saying?" "Papa . . ." "I lost her. I lost you. And it brought me to my knees. It tormented me for years and made me shut myself off from society. But it brought me to my knees—and the Lord was there, through it all, supporting me. The Lord was there, shaping me through my loss into the man He wanted me to be." She lowered her head, her gaze. "I didn't mean . . ." "You did. But you don't understand, Brook. Had I run away in fear from the things she made me feel, I would not have mourned any less when she died. I would have mourned more. Mourned the loss of the happiness we could have had and didn't. I would have mourned what could have been and wasn't. I would have been even more miserable, I would have turned bitter, I would have been hounded not just by questions but by crippling regrets." "But—" "If Justin were killed today, and you had all this between you, what would it do to you?" Her breath balled up in her chest, choked her. He patted her hand and then leaned back. "Love is much like Oscuro, my dear. Yes, it is dangerous. You may get hurt. But the victory of the ride . . . Would you be willing to miss out on that, just because at any time he might shy at something and send you to the ground?" And now her lips tugged up. He knew her language all too well. She sat up straighter—and then started when running steps burst into the room. Deirdre halted halfway in, her eyes wide and her hands shaking violently as they clutched at a slip of paper. "Beg pardon. But I—it's my mum." Brook pushed herself up even as her father did. He stepped forward, the pipe in his hand again. "What has happened?" She was glad he had asked. Her tongue was knotted. Pratt had made good on his threats. Deirdre must have known her thoughts. She looked her way, shook her head. "Sickness, it says. Bad. My brother, he says I need to come home. I know I oughtn't to ask—" "Of course you ought." Brook slid up beside her father, knowing he would have said the same. "You need to go to her." "You can make the afternoon train west if you hurry. I will send ahead to procure a steamer ticket for you." Deirdre blinked rapidly and clutched the paper to her chest. "I'm indebted to you, your lordship." "We've been through this, O'Malley. Family first. And give your uncle our regards—he's still there convalescing, is he not?" "Aye. And thank you. And again, thank you." With watery eyes, Deirdre flew from the room. Brook turned to her father. "May I drive her? She cannot walk in this weather." "Of course." He leaned over and kissed her forehead. "But be careful of all the mud. And consider what I've said. I hate seeing you like this, Brook. You are meant to be sunshine and tempests, not dreary fog and rain." Unable to think of any response, unable to think why it sounded like such a compliment, she could only wrap her arms around him and hold on for a long, fortifying moment. Then she ran from the room in search of Deirdre. "But I want to come. I want to meet her, DeeDee, and if it's bad enough that they're calling you home . . ." Tears stung at the implication, but Deirdre couldn't give them purchase yet. Couldn't let them overtake her. She swallowed the fear down and paused at the end of the servants' hall to put a hand to Hiram's face. "I know. And if I get there, and it's that bad, I'll send you word. I promise it, I will. But I couldn't take the time to explain to his lordship why you should come with me. We've said nothing, and now—" "Now the train leaves so soon, and you must be on it." Because he was Hiram, he brightened. Nodded. Leaned down to press a quick kiss to her lips. "Send me word no matter what. Let me know you've got there safe, or I'll worry all the week long." "I promise. I'll wire you before I board the ship, again when I dock, and then from Kilkeel. I promise." "Good." He kissed her again and then jumped away from her when hurried footfalls reached them. Deirdre recognized the step, though she heard it rarely in this part of the house, and straightened as the baroness came running down the last few steps. Her ladyship looked relieved to have caught her. "There you are. I'll drive you, but we must hurry. The roads will be a mess." She hadn't the time to argue, though she felt she should. Instead, she flew into her room and tossed what she hoped were suitable items into her bag, then dashed back out. The baroness no longer stood in the hall, though Hiram still did. "She went to fetch her hat and a wrap, said to meet her at the car." Nodding, Deirdre ran down the hall, figuring even Mrs. Doyle wouldn't chastise her for it in this case. Hiram kept pace, though she halted him at the door that would lead them out into the rain and muck. "You mustn't muddy your livery." He looked about to argue but must've decided not to waste the time. With a heave of breath, he kissed her again and drew her in for a quick embrace. "I love you, Dee. Go with God." "Pray for Mum. And I love you too." She held tightly to him as long as she dared and then darted out into the rain. The baroness had beaten her out and was already driving the car from the carriage house. Deirdre climbed in, barely getting the door shut before they were off. "Thank you. Though sure and I'm sorry to leave you without warning." Her ladyship didn't look over at her. She had scarcely touched her with a gaze these two weeks, except when it was unavoidable. "It is no great thing. You must be with your family. I know what they mean to you." "Aye." Her throat went tight. The baroness knew the lengths Deirdre would go to in order to provide and protect. And surely hated her for it. "Ought I to bother coming back?" The baroness sighed and shifted the gear lever. "O'Malley—Deirdre. What you did . . . were it not for the letters, it would be nothing." "Letters . . ." There were some yet she hadn't answered for. Hadn't even thought of them in the wake of the major's death. "Forgive me, my lady. He had me plant more in your trunk, in London. I forgot about it when all . . . And when we got home, you'd finished the unpacking on your own." She clutched the handles of her bag until her fingers hurt. "I'm so sorry. I can't say it enough. I never let myself think how it would hurt you. I never thought it would so ruin things with the duke." The lady's fingers tightened, too, on the wheel. "Can I trust you now? If Pratt comes to you again—" "I'd go straight to you and his lordship. I swear it." Her heart thudded in her chest. Was it possible her ladyship would grant her another chance? She shouldn't. But, oh, how Deirdre prayed she would. "Then . . ." She eased to a halt at the base of the drive, glanced both ways—and then at Deirdre. "Then come back when you're satisfied your mother is better. I'll handle my father." "Thank you." The tears pressed again, but she blinked and cleared her throat. "That sounds so feeble. But I've no other words." A corner of the baroness's mouth tipped up as she pulled out onto the road for Whitby. As her father's so often did. "If it's more words you're looking for, you could start with 'Hiram and I . . .' That wasn't the first I've seen you with a flush in your cheeks in his company." They heated now, though not from embarrassment. The one joy since they returned to Yorkshire had been those moments by his side. Knowing that he knew the worst of her and loved her anyway, that he would give up all he'd worked for to be with her, and to help her family. "He's asked me to marry him. I've said yes. Though we haven't said anything to anyone yet, not knowing if . . . if we'd have to leave." "Well." Her ladyship glanced her away again, her smile full and bright. "I'll be sure and alleviate that concern for him when I get back. Congratulations." "Thank you, my lady." There would be new problems to figure, now that she knew they could stay at Whitby Park. God willing, children would come, and rare it was that any of the domestic help had a child underfoot. "Don't fuss over the details yet." And when had her ladyship learned to read her so well? "We can worry over where you'll stay after you get back. For now, focus on your mum, knowing you've a good man awaiting your return." "Aye." The worry seized her mind again. How ill must Mum be for Killian to send for her? Had they money enough for a physician? She let her eyes slide closed so she could better pray. By the time they pulled up outside the railway station, the rain had gone from downpour to drizzle. Still, puddles splashed around the roadster's tires. And already a train puffed its steam from the tracks. She'd better hurry, in case it was hers. "Thank you again, my lady." "Let us know how she is—we'll be praying." Deirdre didn't know how much longer she'd be able to fend off the tears. Perhaps on the train, in the overcrowded anonymity of third class, she would indulge in a few of them. "I will. Drive safely home." Without wasting another moment, she let herself out and dashed up to the rain-soaked platform. Brook had waited a few minutes, studying the lightening clouds and the crowds of people, to make sure Deirdre did not come back, having missed her train. But when no raven hair reappeared after a while, she checked traffic and backed carefully out to the cobbled street. The abbey on the hill stole her attention as she headed out of town—she'd yet to explore the ruins. Papa had never shown any interest in that particular attraction, hounded as it was by tourists. But there would be no tourists flocking there today, she would guess. Perhaps she would walk up the hill and wander its once-hallowed chambers for a few minutes. She wouldn't linger long—Papa would be expecting her back, and she was more than a little curious to see what letters Pratt had had Deirdre put in her trunk—but she needed to pray. Earnestly, openly. Not like she'd been doing since they'd returned home. Decision made, she found a place to park on Church Street and let herself out. The famed one hundred ninety-nine stairs loomed, and the wind blew sprinkling rain into her face. But it was warm and raw and felt like heaven's way of washing away some of the dust inside her. She climbed quickly, having the stairs all to herself. Her legs felt a pleasant burn once she reached the top. The grass was green and bright, close cut. And the three remaining walls of the abbey towered huge and gold-grey. She squished her way through what must have once been the main doors . . . but rather than stepping into a room, she stepped instead into an unhindered view of the sea. Yes. This was worth seeing. A skeleton of a wall, graceful arches, pointed spires, and God's creation, all together. The wind whipped the water of the harbor and tried to snatch away her hat. She closed her eyes and considered letting it. Letting the fingers of air soothe and caress. Father God, mon Dieu. Please, I . . . Forgive me. I have been focusing only on my hurt. On my . . . my fears. Papa was right. I'm afraid of giving myself over to this. But what is it you say? Perfect love casts out fear. Cast it out of me, Lord, please. Of all the things I want to be, that is not one of them. I do not want to be a coward. I don't want . . . I don't want to miss the joys you have for me because I'm too frightened to grasp them. Purge me of the shadows, Lord, of the darkness. Fill me, please, and show me what I should do. Warmth touched her, intense enough that her eyes flew open, expecting to see summer's sun breaking through the clouds. But no, rain still misted over her upturned face. The heat came from within. It started in that cold, aching place surrounding her heart and seeped its way outward. Her breath shuddered. Her knees shook. And a warm gust of wind beckoned her to look to her left. There, perched on the base of what had once been a column, half-hidden behind the remaining column between them, he sat. With his eyes closed, his face turned to the sea, no hat to keep the misting rain from his face. And given how wet his clothes looked, he must have been there even when the rain had been torrent instead of drizzle. Justin. Her breath whispered out. For weeks she had avoided him—but the moment she obeyed the Lord's urging to stop and pray, there he was. And love for him nearly felled her. She moved toward him, though he must not have heard the squishing of her shoes above the whistling of the wind. He didn't open his eyes, showed no signs of awareness. Stopping in front of him, she let the smile come. This disheveled man with the dripping hair would never be mistaken for the Duke of Stafford, even if he sat there twirling the signet round and round his finger. He was Justin. That was all. "Haven't you the sense to go in out of the rain?" She said it lightly, still smiling. His eyelids rose slowly, his lips parted. Disbelief filled his gaze when he looked at her, and he surged to his feet. "Brook." His arms came around her, crushing her to his drenched, cool chest before she could protest. She didn't want to protest. She wanted to wrap her arms around his neck and hold on until all the foolish things they'd said were washed away by the rain. "Justin. I'm sorry." Those words tasted like honey and felt like balm. She said them again. "I'm sorry, so sorry." "But you were right. I do try to control everyone." His hands moved up her back, over her shoulders, and put enough space between them that he could frame her face. Bright as sapphires, his eyes gleamed. She put her fingers over his. "No, you were right. I'm impulsive. I always have been." He tossed her hat to the ground and rested his forehead on hers. "And much as that drives me mad with fear, I love it about you—that where I am cautious, you are bold; that where I think and never act, you charge ahead." "But I can be careless. And you're right to think of consequences. Had your father listened to you . . . You've always been the one to take care. And I love that about you—that you consider so far ahead of where I look." She gripped his fingers as much as she could without dislodging them. Strong and familiar, long and lean, with the bold circle of gold there to proclaim what he had become. He stroked his thumbs over her cheekbones. "Je t'aime. Tu es mon âme. Mon cœur." I love you. You are my soul. My heart. "Justin . . . I love you." She kept her words English, though she could hear the French in them. "And I was so very wrong. I do need you. I'm so much better with you than without you." He kissed her then, his chilled lips warming against hers as his hands slid to her back again and pulled her against him. Rain from his jacket seeped through hers, but it warmed instead of cooling. This was the kiss she had dreamed of all those months he was gone—gentle but demanding, deep and slow. The kind that made her want to savor, want to strain forward, want to never leave his arms. She slid her fingers into his hair, slick with water, and pressed close when he tried to pull away. Smiling against her mouth, he kissed her once more but then set her back. "We've still much that needs saying." She gripped his waterlogged lapel lest he get some foolish idea about putting more than a few inches of space between them. "It can wait. You can come home with me, and we can talk into the night. I daresay this will be harder to achieve, though, in my father's presence." He chuckled, his mouth hovering an infuriating breath away. "I was told that I couldn't expect to kiss you again and set the world to rights." Oh, how she loved the way his eyes flashed darker when feeling crashed through them. "I am, on occasion, happy to be proven wrong." "Really." Mirth sparkled in his eyes. "Not the Brook I know." She couldn't help but chuckle. "All right, just on this one occasion. So you had best take advantage of it and kiss me again." His lips brushed hers. "If I must." He pulled her closer and kissed her until her mind went muddled and her legs weak. She let her fingers trail down his neck, settling a moment at that place beneath his jaw, where she could feel how his pulse raced in time to hers. When next his lips broke away, he still held her flush against him. "I would have married you." Her mind must still be hazy. Had he said would have? "Hmm?" "Had you shown up at Ralin Castle one day, if we hadn't found your father. I would have married you." His lips trailed over her cheek, her jaw, and paused on her pounding pulse. "I would have agonized over it—I'll admit that—but at first I would have found some excuse to keep you close and told myself it was enough to have you near, to have your friendship. Expectation would have kept me up at night—all those centuries of dukes' voices telling me I must marry a noblewoman, and preferably a monied one, or landed." A delicious chill raced through her. It had been too long since he'd told her a story. And never one like this. "Then what?" He tilted her head back, kissed her throat. "At some point, I would have been unable to deny how my feelings for you had changed. And I would have begged you to marry me. You would have put up a fuss about it though, because you distrust unions based solely on love. You would have tried to argue that a future duke couldn't marry the daughter of an opera singer. Of course, I would have pointed out the many times the Grimaldis ignored such logic." The chuckle in her throat felt so different with his lips still resting there against her skin. "But I would have had to point out how rarely those unions ended well." "We would be different though, you and I. We have our faith to bind us, not just our love. But that love—it's too strong to stay silent forever. You probably would have tried to do something impulsive, like leave without telling me. But I'd have been there. I'd have galloped after you on Alabaster, though she'd have a hard time keeping pace with Oscuro." "I wouldn't have had Oscuro." "Shh." He laughed, trailed his nose back up her neck. "Fine, then. I would have had no trouble overtaking you on whatever pathetic mount you'd found for yourself." "Well, I wouldn't say I'd ever choose a pathetic—" He pressed his lips to hers. It was, she decided, the best way to be silenced. "The important thing," he said against her mouth, "being that I caught you." A happy sigh built in her chest. "Yes, you have." "And you would have given up your argument. We would have been married at Ralin, setting the press and gossips abuzz, but we wouldn't have cared." She hooked her wrists together behind his head. "And why should we? They are nothing to us." His smile went from simmering to warm. "And then, at some point, we would have traveled to visit my cousin. On the train, no doubt, coming through Whitby. And your father would have been here, seeing off his sister and nieces, and he would have seen you and thought you his Lizzie. He would have come up to us, apologized for staring, explaining how much you looked like someone he once knew. From there, it would have been easy to piece it all together. And so, the tale called 'If Brook Were Not Eden' would still have ended with the realization that she is." She loosed that happy sigh and rested her head against her arm and his shoulder. "The best yarn you've ever spun." One of his hands moved to her head, and he wrapped a loose curl around his finger in that way he'd always done. "Brook . . . I wanted this before we were sure you were Eden. I was ready to declare myself while we were still in Monaco, but then Father's death . . . And then again, months ago, before I left for our holdings. But Grandfather told me to . . . told me to use your money to put Stafford to rights. And I couldn't do that. I never wanted you to think that it had anything to do with your fortune." And she, foolish creature that she was, had believed just that. She stroked a hand over the back of his neck and smiled a little at the way he shivered. "That explains a lot." "I'm sorry. I meant to protect you, but I was protecting my own pride too, by pushing you away. I should have trusted the Lord and not tried to solve it all myself. Had I listened, I wouldn't have hurt you so." Her hand slid over his shoulder and rested against his heart. "It was my fear that hurt me, not you. But I'll not let it rule me, Justin. I want to see what the Lord has in store for us, together." He kissed her again, featherlight. And then grinned. "I would ask you a rather important question right now, but I had better speak with your father first. I don't want to be considered the kind of man who would propose to a young lady without seeking his approval." She laughed and shoved at him. "Justin Wildon—all those conversations this past fortnight, and you haven't already spoken of that?" "Are you daft? Had you happened by and overheard me asking such a thing, when you'd made it clear you never wanted to speak to me again, you would have challenged me to a duel—and I happen to know how good a shot you are." Laughing again, she went up on her tiptoes to kiss him. "I suppose now I must invite you back to Whitby Park, so you can request an audience with my father . . . and then one with me." His grin winked again. "Give me an hour to change, and I shall be there. I daresay he would have an opinion about a man showing up looking like he'd taken a plunge in the ocean too." "Deal." She pulled away and held out a hand, to shake on it. He took her hand, but then he raised it to his lips instead. "May I walk you back to your car, my lady?" "I would be honored, Duke." They traveled the steps together, quickly as they dared, and she let him steal one more kiss as he closed her into the roadster, even though other tourists were emerging from their hotels and inns now. Let them be scandalized, if they saw. She didn't care. Joy had filled the hollow inside, and she would gladly suffer hearing her father say he'd told her so. The rain stopped as she drove out of Whitby, and she hummed a happy refrain from Mozart's "Le Nozze di Figaro," tapping out the beat on the wheel. For a day that began so poorly and was marred with worry for Deirdre's family, the Lord had certainly surprised her. She would pray for her maid's family—and sing praises. Nothing could ruin the afternoon. Not the way the mud sucked at her tires with every revolution, not the clouds still rolling in off the North Sea, and not even the herd of sheep crossing the road amidst much bleating, which forced her to a halt two miles from her turn to Whitby Park. She might be unable to start again in this mud, but what did it matter? If she had to sit here until Justin came by, then it would give them something else to laugh about. She leaned back, waiting for the animals to clear the road and— Her door was wrenched open, and a rough hand pulled her out before she could think to react. She tried to scream—surely there was a shepherd with all those sheep—but glove-covered fingers clamped down over her mouth. "Not a sound, darling. Not unless you want to tell me here and now where the Fire Eyes are." Pratt? She wanted to kick, scream, something—but a sweet smell filled her nostrils, and the edges of her vision went black. # Twenty-Eight Justin knew well he was grinning like a fool, and he didn't much mind it. Even when Mr. Graham greeted him with a raised brow. "Back so soon, Your Grace?" He had changed into dry clothes as quickly as he could—though granted, it may have been quicker had he not kept trying to rush poor Peters, who had finally declared him hopeless and sent him out, laughing, with his tie askew. Still, by the time the Rolls-Royce chugged through the mud and ruts, an hour had indeed passed. "They're expecting me this time, Mr. Graham. Or Lady Berkeley is, anyway. I don't know about Whitby." "Don't know what about Whitby?" Brook's father emerged from his study, his focus on a stack of post that he flipped through as he walked. Justin's smile didn't dim. He was glad he'd had the time to get to know the earl better. Not that he would have chosen that particular reason for it, had a choice been given. "Whether Brook had told you yet that she asked me to call. We ran into each other at the abbey and finally talked." That brought Whitby's gaze up from the letters and lit a gleam in his eye. "Did you? Good—though she certainly hasn't found me to tell me so. I actually didn't think her back yet. Did I miss her, Mr. Graham?" The butler's brows drew together. "I am unaware of her return, my lord—though she has been known to sneak past us all before. Shall I send Beatrix up to check?" "Yes. Please." But Whitby's brows had pulled down too, and he moved toward the door. "I should have been able to hear the car. I was listening for it. I expected her back well before now." Justin's heart skipped a beat, though he told himself not to worry. "She must be here. She left well ahead of me—I watched her off. And had she got stuck along the road, I would have come across her." "I'm sure she is here . . . somewhere." But the earl's step quickened as he pushed open the door and stepped outside. Justin followed him down the front steps, along the macadam of the drive. And silently echoed the curse that Whitby muttered when they saw the empty stall in the carriage house. Her father spun, his eyes bordering on wild. "Horses. We need horses. Now. Horses!" Justin had to jog to keep up as the earl flew toward the stables, shouting for Oscuro and Tempesta to be saddled posthaste. His heart, he was fairly certain, had stopped. She had to be here. She had to be, because she had been nowhere between. He had been watching for her once he saw the state of the roads, half expecting to see her up to her wheel wells in mud. Mr. Graham came huffing into the stables as the harried grooms brought the horses out. "My lord. Your Grace. Lady Berkeley is not in the house. No one has seen her since she left with O'Malley." "I know. We're going to look for her." Whitby swung up onto Oscuro. "Mr. Graham?" "My lord?" "Gather the staff. Lead them in prayer. I want all work halted until my daughter is found." Justin put his foot in the stirrup and mounted Tempesta, watching how the butler's face paled. "Found, my lord?" But the earl wasn't looking at Mr. Graham anymore. His focus had gone to the slate-grey clouds. "It's those blasted Fire Eyes—it has to be. I shouldn't have let her out of my sight until it was all resolved." Justin nudged Tempesta forward. "We'll find her, Whit." Whitby pressed his lips together and his heels to Oscuro's flanks. At the crossroads they turned, without the need for discussion, toward Whitby. The horses ate up the first mile, Tempesta doing her best to keep up with Oscuro. As they closed on the second mile, Justin shouted, "Wait! I noticed ruts near here on my way over. Sheep prints, too." They reined in to a trot until the obvious place of crossing came into view. Oscuro pranced about as Whitby studied the road. "Someone must have had to stop for them. A car, not a carriage, given the width of the ruts." But the only tire tracks going through them were those of his Rolls-Royce, along with one set from a carriage. He nudged Tempesta forward, across the sheep prints. "Whit." Whitby came up beside him, his gaze following Justin's. Off the road, following the two muddy tracks through the grass and to one of the copses of trees that marked the edge of pastureland. They both urged the horses to follow them, Justin's gut going tighter with every hoof fall. He knew, even before he caught sight of the bumper gleaming in the weak sunlight. Even before he saw the familiar black paint of the Eden roadster. "No." The word tore from Whitby's throat with even more panic than had saturated the curse. The eyes he turned on Justin were tortured. "They've taken her." The no beat an echo in Justin's head, in his chest. He clenched his hands around the reins. "We'll find her. They've less than an hour on us. She is well. She's a fighter, she's bright." But she was a fighter—and sometimes fighting could get a body killed. Whitby turned Oscuro back to the road. "Constable. Hounds to pick up her scent. And while they're doing that, we're going to the Rushworths'. If they are back from London, then we have our answer." They went first to see the constable, then back onto their horses and through Eden Dale, heading southward toward Azerley Hall for about half an hour before Whitby motioned Justin left at a fork, rather than right. It took another fifteen minutes before the villages and farms parted to reveal an old manor house situated well off the road. Not all that grand compared to Whitby Park or Ralin Castle, though it looked well maintained and had a stunning profusion of flower gardens. Their approach didn't go unnoticed. They had no sooner dismounted before the front doors than Lord Rushworth emerged from the garden to the right, confusion in his brows. "Lord Whitby. Duke. What an unexpected pleasure. I was about to have tea—would you care to join me?" Whitby looked more inclined to throttle him. "Where is she?" The question in Rushworth's eyes only deepened. "I'm sorry—who? My sister?" "My daughter." Now the man's eyes went blank. "My lord, I haven't seen the baroness since I was in London. Why would I know her whereabouts now?" Whitby's fingers had curled into a fist—a feeling Justin knew well, though now a strange calm possessed him. He put a hand on Whitby's shoulder and stepped forward. "Forgive us, my lord. We came here on a whim. We were not even expecting you to be at home. I would have thought you and your sister would stay in London throughout the Season." Now the man's face went tight. "Kitty wanted the wedding to be here." "Wedding." Whitby said the word as if it were actually a funeral. And given that she had married Pratt, that comparison wasn't far off, by Justin's estimation. A bit of color stole into Rushworth's face. "It was Sunday. There was . . . a bit of a rush." A picture formed in Justin's mind's eye . . . and he didn't much like it. "So Pratt is back in Yorkshire too? Or did they go to the Continent for a honeymoon?" Please, Lord . . . "Kitty wanted to settle at Delmore." He exchanged a glance with Whitby. Lady Catherine—or rather, Lady Pratt—was the one blatantly pursuing the Fire Eyes. Had she filled Pratt's ears with the tales of them as well? Greedy, base, selfish, cruel-minded Pratt on the trail of priceless red diamonds? "Wait." Rushworth raised a hand and backed up a step. "Is the baroness missing? And you think we have something to do with it?" Whitby pointed a finger at the man's chest. "Your sister came to my house and demanded the Fire Eyes. She threatened my daughter. And then your uncle was murdered in his room the very day he came to tell us about them. Will you try and tell me you have nothing to do with it?" "I swear to you, my lord—you're mistaken." Rushworth backed up another step. "Yes, Kitty was enamored with the tales our mother told of the diamonds. But we would never hurt anyone over such trifles." Whitby advanced, seeming to tower over the younger man though he couldn't be more than an inch taller. "And your father? Will you tell me he did not threaten my wife, that fear of him did not send her into the night with our daughter when he learned Henry sent the jewels to her?" At that, Rushworth froze. "I cannot speak to my father's actions." Slowly, his raised hands sank. "Though heaven knows he was not a gentle man. I would not put such things past him." Justin didn't want to feel any compassion for this man, not when all fingers still pointed to his sister and Pratt as being behind their trouble. But digging up the feud from a generation past wouldn't help them now. What they needed to do was get to Delmore. "Whit." "Right." Whitby pivoted, his face granite. "Shall we give our congratulations to the happy couple, Duke?" Justin nodded, though he held Rushworth's gaze for a long moment. The man had to know what his sister was, had to know the kind of man she had married. He had to—yet he looked back at him evenly, without a flinch, without any indication that he considered the whole story of the Fire Eyes to be more than a fairy tale. Spinning back to Tempesta, Justin let it churn around in his mind. And spoke only once they were outside the gates. "Do you believe him?" "Not for a moment." "Do you think him involved?" Whitby hissed out a breath. "I don't know. He has always struck me as more a shadow than a man. But at the least, I don't think Brook is here. I know this house, these grounds, and there would be no good place to hide her." Justin shifted in his saddle. "And Delmore? How well do you know it?" Whitby's silence lasted three beats too long. "Not well enough." Deirdre woke to darkness and a pounding head. A groan slipped out as she tried to sit up. Her wrists hurt, her shoulder was sore, and her mouth was parched. "Deirdre—are you awake?" "Lady Berkeley?" No, no, that wasn't right. Deirdre should be on her way to Kilkeel, and her ladyship should be in Yorkshire. But this dark space didn't rock as a train should. And it smelled of damp earth and mold. "Yes. Here, I have some water." She heard rustling, shifting, and then a hand groped at her shoulder. Deirdre reached up, and her fingers closed around a canteen. Eagerly she raised it to her lips. The water was fresh and cool, and with its touch came a few snippets of memory. Running to the ticket counter. Being pulled to the other side of it, a gun barrel pressed to her back. Pratt. She handed the canteen back before her shudder could spill it. "He got you too. Oh, my lady, I'm so sorry. I had no idea he—" "This isn't your fault. He set it all up. We couldn't have known. He sent the telegram, he was lying in wait, he had his flocks ready to block the road whenever I came back." Deirdre squeezed her eyes shut, though doing so didn't change the darkness a whit. "You shouldn't have driven me." But at least Mum wasn't ill—the one spot of good in it all. He had said so when he pressed the gun to her back. "He said he would have taken me on my next ride, if I hadn't—and would have shot Oscuro to do so." The baroness's hands found hers and gripped them. "We are in this together, Deirdre. There is no room for regrets." Deirdre clung to those strong fingers. "Where are we?" "He put a hood over my head a few minutes after I roused from the chloroform, but I think we're at Delmore. Some sort of cellar?" It made no sense. His interest had been in Whitby Park, in marrying the baroness—how would kidnapping them help him attain that? She shook her head—and immediately regretted it when the ache turned to a slicing pain. A whimper escaped, and then the baroness's arm came around her shoulders. "I'm the one who owes you an apology, Deirdre." Her hand rubbed over Deirdre's shoulder. "He wants the diamonds." She didn't know what diamonds her ladyship meant—and it didn't much matter. "Well, if you know where these diamonds are, you mustn't tell him. Sure and he'll kill us once you do. He can't let us go, not without bringing the law upon himself. He's too smart not to know that." "I know. I know." "He's heartless. A devil. Put nothing past him." "I—" Noise from the right silenced them. A clanging, a scraping, and then sudden light blinded her and made the pain slice again. Wincing, Deirdre turned her face into the baroness's shoulder and blinked until the brightness wasn't so harsh to her eyes. "Ah, good. We're all awake." The door slammed shut, and a lamp came to a rest on a table across the room. No—an old desk. And the room didn't have the earthen walls she had expected, but stone ones. There was even a space that must have once been a window, now filled with bricks. Not a cellar, then. Pratt pulled the chair away from the desk. It, as opposed to everything else in here, looked solid and somewhat new. He sat and hooked an ankle over the opposite knee. The easy pose bore a marked contrast to the gun he kept pointed at them. "Now then. Ready to chat, my lady?" Her ladyship lifted her chin and somehow managed to look regal even here, on the floor. "Oh, quite. This ought to be interesting. Do tell me, my lord, why you think you have any claim to the Fire Eyes." The Fire Eyes—those she had certainly heard the baroness and Whitby discussing, though she hadn't ever heard they were diamonds. Pratt's nasty little smile curved his lips. "I forget how little you know of family history. My father was Henry Rushworth's dearest friend." The baroness's face shifted, though only slightly. "He is the one who introduced my mother to Aunt Mary." "And by extension, your father—for which ol' Hank never forgave him. Leastways, not until he came home from India in need of help in peddling a few jewels. Then he was all gracious words and generous offers to whomever would help him get rid of the things." He motioned with the gun. "Even shares, he said. A third to my father, a third to his brother, a third for himself." Deirdre rubbed at her wrists. They were chafed and red and had obviously been bound. "But that makes no sense. Why would he promise away so much of his profit, when they were in his possession?" Pratt narrowed his eyes on her. "Desperation, my lovely, can make one do stupid things." "And I suppose you have proof of this. Documentation. Evidence of a legal, binding agreement." The baroness folded her hands in her lap. Mud marred the walking dress Deirdre had chosen for her that morning. Pratt put his second foot down and leaned forward. "I have my father's word." "Is it worth more than his son's?" At the fury that snapped through his eyes, Deirdre tried to squeak out a warning. When he lunged for the baroness, she tried to scrabble before her to provide a barrier. All she achieved for her efforts was another blow to her head that sent her reeling. The baroness still ended up trapped between Pratt and an old trunk. Her ladyship was bent backward at an angle that looked painful, his gun pressed to the hollow beneath her jaw. "My father died for those gems! When Henry ran back to India like the coward he was, when he sent them to your mother, when he forced my father to renege on the deal he had struck with his buyer, he was killed. Murdered! If anyone has a right to them, it's me." He pushed her harder against the trunk. "I tried to do this the friendly way. All you had to do was marry me—then I could have searched for the jewels at Whitby Park at my leisure. So simple. But you're as stubborn and haughty as the rest of your family." The baroness didn't shake, didn't quake, didn't waver. She smiled. "You never would have found them. Not in a million years." "Oh, but you would have. You with your mother's face—the major would have told you where he'd hidden them. And he did, didn't he? He told you how he sent them to her . . . though I suspect he left out the part of how his own greed made him betray his brother and his oldest friend." "Greed and betrayal played a crucial role in his tale, actually." Deirdre pushed herself back up, cursing the weakness in her limbs, the pain in her skull. She needed to help—but what could she do? If she tried to knock him away, he could very well shoot the baroness. Indeed, he pressed the gun harder into her throat. "And now he's given them to you. Tried to sign them over to you, ignoring the first deal he'd struck. Forgetting his own brother, his friend, and the legacy their children ought to be receiving." Now the baroness's eyes slid shut. "You're the one. You're the one who killed him." "Blood for blood—his for my father's." Deirdre's stomach twisted so hard she had to pull her knees to her chest to try to ease the pain. If she needed any more proof that he'd never let them out of this alive . . . The baroness strained against him. "They are just diamonds, Pratt! I am sorry your father lost his life over them, but why would you keep the cycle of violence turning? Why?" "Why?" He laughed, and the room seemed to grow darker again. "Have you any idea how much those 'just diamonds' are worth, you idiot woman? My father didn't die for the jewels, he died for what they would mean to us. Never again, in my lifetime or my grandchildren's, would I have to worry about whether the rents will cover the expenses. If I can afford the necessary improvements. If I need to let a footman go. And that was with a third of their price. Now that Kitty and I are wed, we'll have two-thirds between us—even if we give Rush his share." "No one in his right mind would spend that much on a couple of pieces of red carbon." Red? Deirdre eased her knees back down. Red diamonds? She'd never even heard of such things. Pratt laughed again and pushed the baroness back harder against the trunk when she tried to twist away. "We can debate their sanity all you want, but I've a buyer already waiting, and I don't intend to share my father's fate by disappointing him. Your pieces of red carbon are destined to grace the throat of a Russian princess, my darling." He gave the baroness another push into the trunk but then stood up. Perhaps it was the new bit of freedom that allowed her ladyship to breathe a laugh. "No. You'll never find them unless I tell you where they are, which I will never do. That I promise you." "Oh. My darling. I think you will. Because it's very simple. Talk, and you live. Don't, and you die." "No matter what, I die. How stupid do you think I am, Pratt? You can't let me go after this." His lips turned up into that evil little grin Deirdre so hated. "I didn't say I'd let you go. I said I'd let you live." He sent his gaze down her in a way that surely made her ladyship's skin crawl. "More incentive to keep my lips sealed." Deirdre winced. She was all for standing against him—but didn't her ladyship realize that antagonizing him would only make things harder? Pratt chuckled. "It's going to be so pleasant, hearing you sing a different tune by the time we're through. Deirdre." He motioned for her to get up. With the gun. On shaking legs, she obeyed. She tried to promise the baroness with her eyes that she would do nothing to compromise her. Prayed she understood, and that she herself would have the strength to keep that promise. Pratt closed his fingers around her arm. "Now, as a gesture of good faith, I'm going to take your lovely little maid here for some refreshment for you. I'll let her bring in a cot, a pillow, a blanket. You're going to get a good night's sleep and consider all you have to lose by withholding from me. And then in the morning, my darling lady, you're going to talk. Are we understood?" Given the pulsing in her ladyship's jaw, she was clenching her teeth against whatever response she wanted to make. Deirdre loosed half a relieved exhale before Pratt jerked her toward the door. Perhaps she could get away somehow. Find help. He tossed her through the door and pulled it shut as she fell into the wall opposite. Then, before her addled mind could recover from the jarring, he pressed her to the damp stone. "Don't get any heroic ideas, my lovely, if you even have such things in you." The barrel of the gun touched her head, directly upon the wound. She whimpered before she could stop herself, though it only made him chuckle. "This is why I took you along with her. She will refuse me—I know that. But you—you're in there with her, a fellow victim of my cruelty. Get her to confide. Open up. Tell you where the diamonds are." Deirdre pressed her lips shut against the no that threatened to spew out. Better he think she was still on his side, however reluctantly. "Do that," he murmured into her ear, "and I'll see that your family is set up for all their miserable lives, and you'll be free to enjoy it with them. Knowing, of course, that if you ever breathe a word of this to anyone, it all disappears." She squeezed her eyes shut. He thought her so low . . . and why wouldn't he? She had proven herself to be little more than a worm, happy to sell her own soul for a few pound notes. Not anymore—and maybe this was how she could redeem herself. Earn his trust, fully, so that she could help the baroness escape with her life. It could very well cost her her own if she were caught in it . . . but it was a risk she had to take. If she were killed, the earl would see her family was cared for. And Hiram—Hiram would be proud, knowing she had done what was right. She swallowed and bent her mind into a silent prayer. "How much?" "Hmm?" "How much will you give me if I help you with this?" He chuckled and eased off her. "I thought you'd come around. Let's say . . . ten thousand pounds. That'll be enough to see your family through, won't it?" Undoubtedly. But if he thought greed her sole motive . . . "No. I went ten percent. Of whatever it is you get from the Russians. Ten percent." "Five." "Fifteen." Laughing again in his throat, he spun her around and pressed a kiss to her lips. It took all her willpower not to wipe it away. His eyes looked almost . . . affectionate as he tweaked her chin. "I knew I liked you. Pity you didn't accept my first offer—we would have suited well." She lifted her chin. "Do I have my ten or don't I?" "Fine." He took her hand and tugged her down the dim hallway. "But you're going to have to make it quick. Whitby and Stafford will be out looking for her by now." Please, God, lead them here! Help them find us. He stopped her at the end of the hall and motioned to a room on the right. Its windows were also bricked over, except for the transoms. But through them she could see only sky. "You come no farther than this. I'll leave the lamp in there for now, and you can take that tray of food and water. But warn her that this is the last of my generosity. If she doesn't talk by morning, she'll have nothing." He motioned to a folded metal cot that looked as if it belonged in a military barracks. "Drag that back for her, if you want. Or if you'd rather watch her suffer through a night on the floor, tell her I changed my mind." Deirdre nodded, kept her face neutral. And prayed she could keep up the deception until Lord Whitby came pounding upon the door. # Twenty-Nine Justin kept his hands in his pockets to hide how they'd fisted. His feet itched, his chest ached. He needed to be doing, not standing here in the drawing room with Brook's mother looking down on him, all but asking with her painted eyes why they weren't out there tracking down her baby. They'd come back only to exchange the horses and get some water for themselves. But the constable was waiting for them and insisted on a search of Brook's room before they went accusing another lord of kidnapping. Justin paced the library while they went about it. He had wanted to follow them up, but it hadn't seemed right. He almost wished he had, though, when Whitby returned, his face a thunderhead and eyes flashing lightning. The constable followed, flipping through a stack of what looked like letters. Justin's brows lifted. "Did you find something?" "Lies," Whitby all but spat. The constable sent their host a hard look. "Close as you've grown, she's still a young woman, my lord. And they all keep secrets from their fathers." Justin watched doubt flicker through Whitby's eyes—probably remembering all those things Brook hadn't told him. But then he straightened his shoulders and lifted his chin. "Not this, though. She would not have hidden a romance from me—especially given that she has been in love with him this whole time." He motioned toward Justin. Justin's throat went dry. "Would someone please enlighten me?" The constable motioned with the stack of folded papers. "Love letters, it seems. Dated from the time she arrived through a couple weeks ago. My French is rusty, but they seem to be from an actor. Someone she knew in Monaco. They speak of running off together." "Nonsense." Justin strode forward and held out a hand until the constable put one of the letters into it. "I know all her friends from Monaco, and there were precious few. No young men." None, other than him. He would have known it if there had been. He would have known if she'd been in communication with anyone other than Prince Albert. And he was shaking his head within moments of reading through the letter. "No. Whitby is right, this is a lie. Aside from the fact that I've never heard of the fellow, the writing is all wrong. This was most assuredly not written by a native French speaker." A knock came upon the open door before the others could respond. Mr. Graham stood there, a salver in hand. "Telegram, my lord." Whitby stepped forward to take it, trepidation in his eyes. It darkened to hurt but then blazed into anger as he read it. "No." The constable and Justin both flanked the earl to read over his shoulder. Forgive me, Papa STOP I do not mean to hurt you but must follow my heart STOP It is all too much STOP J is too cold and W not serious STOP Need someone who understands me STOP Met the son of a friend of Maman at train station STOP Left with him STOP Will wire when we get to Continent The constable sighed. "No doubt the same man these letters are from. Someone must have pinched her car from the station and then dumped it." "No." Justin balled up the paper in his hand. "No, this isn't from her. She didn't leave from the train station—she met me at the abbey after she dropped O'Malley off, and I watched her drive out of town." Whitby's mouth went firm. "Whoever sent this obviously didn't know that. Didn't know the two of you had made up." He was obviously the J in the note—and Worthing must be W. But she never called him Worthing. She called him Brice. He would have been a B. The constable didn't look entirely convinced. He held out a hand toward Whitby. "May I take it with me, my lord, and the letters? I'll see what we can discover about where it originated. And in the meantime, I'll thank you two not to go off half-cocked, accusing the neighbors of anything." The request ate him up inside like acid, and Whitby looked every bit as unwilling to agree. His jaw ticked for a moment before he gave a curt nod. "For tonight, Constable. But a father knows. A father knows when something bad has happened to his daughter, and I'll not sit here while she is hurt or worse. Not again. If you've no leads by the morning, the duke and I are paying a visit to Delmore." To his credit, the constable didn't dismiss it as an idle threat or get in a bluster over it. He merely nodded, considering that as he had the paper in his hands. "I've a cousin who's a groundsman at Delmore. I'll pay him a call, quietly. See if anything's amiss on the estate. But you know as well as I that the place is a maze—if by chance she is there, our barging in won't help us find her. We must go about this with thought and care. And with prayer." Praying—Justin had been praying constantly as they rode through Yorkshire. Mr. Graham had assured them the moment they stepped inside that the staff had spent the last hours on their knees. Still, he couldn't shake the feeling that they needed even more people beseeching heaven on Brook's behalf. The constable took his leave, promising to trace the telegram posthaste and to call first thing in the morning. As Justin watched him go, a hand settled on his spirit. And a name filtered into his mind, making him sigh. He turned to Whitby. "We need to let Worthing know. He seems to have an uncanny knack for knowing what to pray." Justin had wired him when he got to Whitby, and in the two weeks since, he'd received two letters from the man, both so very to the point that Justin had to wonder if the Lord whispered directly into his ear. Brook's father nodded—then shook his head. "We'd have to send a letter rather than a wire, and we certainly can't use the phone. The operators could well leak it to the press. But a letter is too slow." "No . . . wait." Ideas swirled. Motioning for Whitby to follow, he charged from the drawing room, down the hall, and into the library. Flicking on the electric lights as he entered, he headed straight for the chair Whitby had been in earlier. His newspaper still sat on the table beside it. Justin scooped it up and turned it face out. The earl lifted a brow at the picture, weeks old, of Brook that graced the cover. "My point exactly, Duke. The merest mention of my daughter makes the front page. This insipid article is about nothing but the fact that she hadn't been to a ball in two weeks, and they wondered if she'd left Town." "Exactly. Can you imagine if they learned she was kidnapped?" Pressure mounted in his chest, too desperate to be called excitement—but right. It had to be. "It would be in every newspaper in England. Front page. Every single person in this county and the next would see it and be on the lookout for her." Whitby's eyes sparked. "If the article made it clear there was a sizable reward to anyone who offered solid information as to her whereabouts . . ." Justin lifted a brow. "How well do you think Pratt can trust his servants?" This time, a hint of a smile touched Whitby's lips as he said, "Not well enough." Tossing the paper back to the table, Justin nodded. "Exactly. But we can't tip our hand until the constable is ready to intercept anyone coming or going from Delmore." More waiting—but waiting with purpose. "Worthing can help us with the press. He's as much their darling as Brook—but that again leaves us with how to reach him without tipping our hand too soon." Justin shook his head. "Let's not forget how uncanny he is. Ring him up. Say you need him to come. I daresay he'll be here by morning, with no other words needed." Whitby's features eased a bit, and then he spun for the door. "I'll be back as soon as I reach him—you had better stay here tonight, Duke. I'll send someone for your valet." "Thank you." Though he felt too antsy to sit, he sank down anyway, onto the seat he knew Brook favored. He ran his hands over the arms of the chair, knowing hers were the last to touch the upholstery. He reached over and rested his fingers on the book left on the side table. La Chartreuse de Parme. She'd read it before—he remembered her talking about how a Frenchman had captured the Italian spirit. So very Brook, this book. His eyes slid closed. "Help us find her, Lord. Please. Keep her safe until we do. Drape your protection over her, keep any harm from finding her. Please. Please." Nothing whispered into his ear. But peace seeped into his chest, and it spread warmth into places he hadn't realized were chilled. The lamp's oil ran out while she slept. Brook awoke to that cavernous darkness again, and with the sinking certainty that Pratt had meant the words Deirdre had relayed the day before. If she didn't cooperate, he would bring no more oil. No more water. No more food. She sat up, the rusty metal cot squeaking underneath her. Reaching up, she touched the pearls around her neck. If he knew they were here even now . . . that yesterday, as he held a gun to her head, he had been but inches from the things he desired most . . . What was she to do? She couldn't give them to him. He might, might let Deirdre go, which would mean she could fetch help, but that was a big if. And even if he did . . . she had a feeling that Pratt would not waste any time in teaching Brook a lesson. She would pay dearly for her impudence the moment he had the diamonds in hand. She couldn't turn them over. That was all there was to it. She needed some other way of escape, and it would have to come from the Lord—He would have to clear the way for her. "A fire goeth before him, and burneth up his enemies round about." The Scripture filtered into her mind—in English. Odd, given that her Bible reading was still entirely in French. Perhaps it had been in a recent sermon at the church in Eden Dale or from one of Papa's daily selections. That must be it—she could hear it in her father's voice. Deep and strong. Authoritative. Promising. Papa. Tears burned her eyes at the thought of him. He would be so worried. So afraid of losing her all over again, and over the same thing. And Justin, faced with losing yet another loved one in so short a time. . . . For their sakes, Lord, have mercy. You are my champion. You are my hope. Send out that fire before us to clear the way, mon Dieu. The rattle, the clang, and then the influx of light as Pratt came into the room. Perhaps one of these times she could be ready to dart around him, to leap out the door . . . though Deirdre had said the door at the end of the hall was locked too. She wouldn't get far enough to make it worth whatever punishment he'd dole out. When the light shone on her, she forced a smile. "Good morning, Lord Pratt." His smile was as dark as ever. "Good evening, Brook." Evening? No, it couldn't be. Deirdre had seen late afternoon sunshine yesterday, she said. They couldn't have slept that long . . . or that little. It was a ploy. "Is it? And you've not brought us any tea." "You wouldn't have drunk it if I had." He nodded toward where Deirdre was stirring on her pallet on the floor. Since Brook had the cot, she'd insisted Deirdre take the pillow and blanket. "Though perhaps your maid would have. I am willing to be civil, my darling. But civility must go both ways. You give me what I want, and I'll give you what you want." She would appeal to Pratt's humanity, if he had any. Deirdre's warning rang clear in her memory though. He was a heartless devil, capable of anything. Perhaps a slight exaggeration, but . . . she had to try something, didn't she? Drawing in a deep breath, she smoothed her wrinkled walking dress. If she couldn't appeal to his heart, perhaps she could appeal to his greed. "I will make you a deal. Make me one of your partners, divide it evenly with me when you sell, and I'll get them for you. You can let me go, and I'll say my car got stuck and I went out for help but got lost. No harm done." Not that Papa or Justin would ever believe that even if she did want to try it—she never got lost. But anything that would get her out. Pratt chuckled. "I'm afraid I'm not quite so stupid, darling, but good try. Let's try this instead though—you tell me where you've stashed them, and we send Deirdre in to get them. She can claim another message was waiting for her, saying her mother had recovered. Then everyone lives." "Except that I'll be trapped here. I cannot make that deal, Pratt. You need to offer better incentive." She stood, knowing she had better move her legs while she had the light. He would no doubt take it out with him again. She made it all of three steps before he'd come up behind her and clamped an arm around her waist. "Now, darling, it wouldn't be so bad. I'd give you a fine room. Lovely clothes. Books. You like those, don't you?" He trailed his nose along the side of her face, from temple to cheek. "And you'd come to like me. I'd show you what a . . . generous man I can be. You wouldn't want to leave. I've already sent your father a note saying you've run off with a performer you knew in Monaco, and I'll let you write him from time to time. I'd have to approve the letters, of course, but he needn't think you dead." A quaver formed in her stomach. Not so much at his words as at the way his hand slid down her hip. "You'll have to do better than that." She'd meant it to come out strong, daring. It hadn't. His chuckle mocked her attempt. "You seem to be under the delusion that you have room to bargain. But I'm afraid that ship has long since sailed. Now, had you told us where you'd hidden them when my rough-edged compatriot asked you in the stables—" "You said it wasn't you!" Deirdre had apparently sat up at some point. She looked on now with horrified eyes. Pratt's hand pressed harder against Brook's hip. "No, my lovely. I said I didn't hire him to scare her so that I could play the hero. I assure you, that was never my intent. Though I also gave the bloke strict instructions not to kill her, and he seemed to have forgotten that one. What I get for hiring riffraff, I suppose." Brook tried to swallow, though her throat didn't want to work. She had thought it Lady Catherine . . . but they were in this together all the time, it seemed. "I didn't even know what the Fire Eyes were. How was I to—" "You found out quickly enough, though, didn't you?" He drew away, but before she could take advantage of it, he shoved. She landed with a crash of rusty springs back onto the cot. He stood before her, a dark blot against the lamplight. "Sent your duke to India after the information. Though you must have been clever about how he was to get it back to you. I couldn't figure out any of his letters. Did you set up some kind of code?" "Quoi?" He was mad. Stark, raving mad. "No doubt you were furious when he got back—thinking he hadn't written." A chuckle rumbled out, cruel and low. "Did you think he wanted to keep it all for himself? But no—he wanted you too. That's why he brought you to England in the first place, wasn't it? Set you up as Whitby's lost heiress so he could marry you and make a fortune in the process." Her fingers curled over the rough, rusted edge of the cot's frame. Pratt stepped close and then closer, plunging a hand into her hair, which had long ago come loose from its chignon. "And oh, how angry he must have been to come back and find you all but engaged to Worthing, while he was away digging up your secrets for you. Is that why the two of you could do nothing but fight after his return? Did you not want him anymore, my darling? I can't say as I blame you. He was always a self-righteous, condescending—" "He is not!" "Ah." His hand wrapped around her hair, too tightly, and pulled her head back. "So you do still have feelings for him. Well, that makes this next part even more fun. Give me the diamonds, Brook, or he'll be the first one I kill." No. Her blood froze, her fingers released their hold on the cot. He couldn't. He wouldn't. He daren't. "Jenkins was so easy—and I was even applauded for it." He twisted her hair even tighter. "Henry—he was necessary. But Stafford . . . Stafford would be a genuine pleasure. I haven't decided yet if I'll put a bullet through his skull or a knife through his heart." "No." Her voice, blast it, came out weak and desperate. "You can't. If you keep killing everyone connected with this, you'll get caught." His laugh said otherwise. "Oh, but no one would know. He would get a wire saying he's needed abroad, and off he'd go. No one would think anything of it for a year or more, and by then, who would link it to me? Everyone knows the duchy comes first for him. No one would question it if he disappeared to tend it." Releasing her abruptly, he straightened. "Your father, on the other hand—he's too much a fixture in these parts. His death will have to look like an accident. Simple enough, really. The brakes could fail in his car. Or he could be tossed from that wild horse of yours. But he will certainly be next, after the duke." He turned, pacing toward Deirdre. "And then, if you still refuse to talk, you've an aunt. A pregnant cousin. And that fiery one that Cayton tossed over—though he's a friend, and he still loves her even though his wedding is only a fortnight away, so I had better save her for last. But then . . ." He turned back to face her. "I don't think it will take that long. Do you?" Though she refused to shut her eyes, she wanted to. She wanted to shake, to cry, to scream—or to lunge for him and wring his neck. She wanted, needed to think him bluffing. But Henry Rushworth lay in a fresh grave beside his brother. And Jenkins in a pauper's one, not far off. Her life for theirs—that was what he was proposing. She must give him the diamonds or everyone she loved would be killed. Lord! She wanted to believe He had some better alternative. Where, though? How? She thought—perhaps, maybe—she heard His quiet Trust me in the recesses of her spirit. But the fear clanged so much louder. A weight settled beside her on the cot, and Deirdre's arm slid around her. "Don't give up, my lady," she whispered into her ear. Then, louder, "Give her time to consider, my lord." Pratt's chuckle moved toward the door. "A few hours, and I'll leave the full lamp. But no food. No water. Not until you sing for me, my little chanteuse." He withdrew a leather-bound book from his jacket pocket, dropped it onto the worn surface of the desk. "Incentive—and a reminder of what we're capable of." The click of the door a moment later sounded like canon fire to her ears. Deirdre smoothed back Brook's hair. "He's bluffing." "He's not. He's already killed." She stood and slid over to the desk, her eyes going wide. The journal—Maman's journal, the one she had bemoaned as lost all this time. "How did he . . . ?" "My fault. My first crime against you." Deirdre appeared at her side, that familiar apology in her eyes. "I thought . . . I thought it would disprove your story, but I couldn't read the French." It hardly mattered now. "My father and Justin will be working with the constable—they won't believe that note he said he sent." Deirdre nodded. "They'll be surrounded by people, searching for you, everyone will know what they're doing. He can't make His Grace disappear." She wanted to believe that. Wanted to hope. But the lamp he left couldn't fend off the darkness. She clutched the journal to her chest and squeezed her eyes closed. # Thirty Whitby Park had never been exactly boisterous whenever Justin had visited, aside from at the house party. But that morning as he made his way down the stairs, it seemed downright melancholy—which suited his mood well. Sleep hadn't come, or not for long. He had lain there praying most of the night. Eventually he had given up and had risen, switched on a lamp, and pulled out the Bible that Peters had packed for him. He'd left a marker in the Psalms at some point or another, and that was where he'd turned. His eyes had found the ninety-seventh one: The Lord reigneth; let the earth rejoice; let the multitude of isles be glad thereof. Clouds and darkness are round about him: righteousness and judgment are the habitation of his throne. A fire goeth before him, and burneth up his enemies round about. His lightnings enlightened the world: the earth saw, and trembled. The hills melted like wax at the presence of the Lord, at the presence of the Lord of the whole earth. The heavens declare his righteousness, and all the people see his glory. Testimony of the Lord's greatness, His power. Assurance that the God of the universe was Lord of this too. Justin's part was to trust, to tremble. To cling to the promise that they were not of the darkness but children of light. He paused at the base of the stairs. Breakfast room? He nearly headed that way, but he suspected Whitby wouldn't be there. He'd taken no dinner last night, though Mrs. Doyle and Mr. Graham had both cajoled him. The chef claimed he couldn't pray without cooking, and so food had been prepared. Perhaps the staff had eaten it; Justin hadn't either and couldn't now. He angled his feet instead for the hall that would take him to the library. Whitby stood by the glass doors, looking out at the early morning sunshine. At Justin's entrance, the older man acknowledged him with a partial turn of his head. "She said, the first time she came into this room, that if ever she went missing, I should look for her here." A smile bade for leave to touch Justin's lips. He let it, though it no doubt looked as sorrowful as it felt. He moved to Whitby's side and shoved his hands into his pockets. "We'll find her." "We must. I already lost Lizzie to the greed for these diamonds, though she never even knew she had them. I'll not lose Brook to them. Not again." "We'll find her." If he said it often enough, perhaps the doubts and fears would flee in the face of the must. "Then your biggest concern will be whether or not to grant us your blessing." Whitby's chuckle had little mirth in it. "And yours will be learning to tolerate your father-in-law spending months of every year at your home." "I've rooms enough, I suppose." And it warmed him, to think that Whitby would be willing to spend part of his time in Gloucestershire. The earl drew a deep breath in through his nose, his hands clasped behind his back. "Looking back . . . I cannot fathom how I spent all those years without her. How the hole of her absence didn't swallow me up. Finding her has made my life so full." "I know." Justin didn't know what else to say. And needed to say no more. Whitby breathed a shaky laugh and nodded toward the door. Or rather, toward the disheveled man striding toward it. Any other day, seeing Worthing with his tie askew, his clothes rumpled, and his hair mussed would have inspired serious jesting. Today he settled for opening the French doors. Worthing charged through, his eyes absent any amusement this morning. "No one opened the front door, so I made a guess. What's wrong? Something's wrong. I've had the worst feeling the whole way here—and frankly, several hours before I got your cryptic call." Justin nodded to Whitby. "Told you he was uncanny." Whitby sighed. "She's missing. Kidnapped, it must be. We suspect Pratt." Justin stepped to his side. "But we have a plan." They explained it, along with the telegram and letters that had thrown the constable, and as they did, Worthing's expression went from outrage to determination. By the time they finished their ideas concerning the press, he was nodding. "I can help with that." "That was our hope." Justin would have said more, but Mr. Graham chose that moment to enter with the constable. The official looked none too happy, though he made an effort to smile when introductions were made to Worthing. Still, he turned without any more small talk to Whitby. "The magistrate wouldn't order a search warrant for Delmore, my lord. I dispatched an officer to the telegraph station at the next town, and he was able to verify that a young blond woman, well dressed, sent the telegraph yesterday afternoon." Justin folded his arms over his chest. "It couldn't have been Brook." But Pratt's new wife looked much like her—where had she been yesterday? The constable inclined his head. "I don't disbelieve you, Your Grace, given what you told me of your conversation with her at the abbey. But without a warrant, we cannot do anything but pay Pratt a friendly visit—which I suggest we do. Let's go as we would to any other neighbor and ask them all to be on the lookout for her. My cousin at Delmore has promised to keep watch for anything abnormal and report it to me." Whitby drew the constable toward the table, where a slew of paper and fountain pens had been set up. "We've another plan as well, involving the press." The constable nodded as Whitby laid it out for him. Worthing passed a hand through his hair—the cause of the mussing, it seemed—and stepped nearer to Justin. "You spoke to her yesterday?" His chest tightened and he nodded. "Just before. She found me at the abbey, and we . . . It was raining. I changed into dry clothes and headed here immediately, so I could speak to Whit. I wasn't that far behind her. If only I had gone with her . . ." "Don't." Worthing's hand gripped his shoulder. "He wouldn't have been able to take her if I had been there." "Or else he would have shot you and taken her anyway, and it would have been hours before Whitby knew what had happened." Worthing shook his head, his eyes intent. "Or even if it discouraged him from acting then, he would have found her another time. When she was out for one of her rides or on another drive or . . . She would have insisted on being alone at some point—you know it as well as I. And he would have been ready to pounce, whenever that was." At least this way, they realized it almost immediately. Perhaps Worthing was right, that it was better than the alternative. Worthing removed his hand, sighed, and focused his gaze on nothing. He had circles under his eyes and lines of weariness around them. "Evil men flourish. The righteous suffer. The Lord never promises we won't—only that He'll sustain us when the tribulation comes." Justin shook his head. "You are uncanny. You know that, right?" Worthing's grin made a showing—brief and muted. "She'll be glad to see you and I are friends." "When we find her." "We'll find her." But the nagging fear wouldn't be banished. "I pray it's in time. He can't mean to let her go. If it's Pratt, if she knows it's him . . . he must plan to kill her once she's told him where the diamonds are." Worthing inclined his head. "Then we pray the Lord stops her lips." "Gentlemen." Whitby, standing by the library table, motioned them over. "We need to get this drafted for the press with all speed. And then to Delmore." The writing went quickly, with Worthing acting as scribe. No doubt some of the papers would alter it here and there, but this would be what they sent over the wire, and this would rouse every able body to search for her. Including, he prayed, the able bodies at Delmore. They bolstered themselves with tea and caffe espresso and then headed, all of them somber-faced, outside. Because the constable and Worthing both warned that the roads were yet all but unpassable to anything wheeled, they opted for horses. When the grooms brought them out, Whitby handed Justin the reins for Oscuro. It nearly choked him up. He patted the quivering midnight shoulder and stroked the beast's strong neck. "Let's go find Brook, boy. Allons-y." The horse tossed his head in what Justin chose to interpret as eager agreement. Little conversation was exchanged along the way. But he could take some comfort in the size of their entourage, what with the constable's men and the mass of Whitby Park servants who followed behind on foot to fan out and search for any sign of their baroness. They would find her. They must. Unease crawled over his skin like spiders when Whitby led them down the lane marked with posts reading DELMORE. Heath gave way to pastures full of fluffy sheep only weeks away from shearing and horses grazing in their paddocks. Copses of trees, rising hills, and on the horizon a bluff that would tumble into the sea. Salt tinged the air when the breeze whistled by. The land, being so close to Whitby Park, ought to have seemed familiar. But it didn't feel right. Didn't feel peaceful. Didn't feel lovely and welcoming, as Brook's home had from the moment he first rode up the drive to see if perhaps she belonged there. Oscuro either sensed his discomfort or felt the oppression himself—he shied, whinnied, nearly sidestepped into Tempesta. Justin brought him back under control with a firm rein and quick French. The constable, as they neared the carriage house, nodded toward the copse of trees behind the building and the rickety old carriage that sat in high, dried grass. "I went right round the back last night to find Antony, so I didn't notice that. But look. Ruts leading to it, and the grass is flattened. Not to mention that it looks entirely too clean for what must be an unused antique. And what cause, do you think, would he have for taking it out?" A most excellent question—one that made those invisible spiders race over Justin's skin again. She was here, somewhere. She had to be. Lord, let her know we're coming. We're close. We're going to find her. They must have made a fairly impressive picture as they all dismounted and climbed the steps up to the door—the three gentlemen and five officers. The mighty wooden slab opened before they could even ring, and a perplexed butler stood before them. His gaze locked on Whitby's face, which he no doubt recognized. "My lord. Do come in. Is something the matter?" "Something is very much the matter." Whitby strode past the butler, the rest of them following in his wake. "My daughter is missing. Please fetch Lord Pratt at once, and assemble the staff. We need all available men out looking for her." The butler's alarm seemed genuine, and he certainly wasted no time in showing them into a parlor and going to fetch Pratt. Justin exchanged a glance with Worthing. If Pratt had her at Delmore, surely someone on his staff knew it. But if he were any judge, it wasn't that one. The purse of Worthing's mouth bespoke a similar thought. Silence held until Pratt strode in a moment later, a pale-faced Lady Catherine—Lady Pratt—behind him. "Whitby." His expression turned to half a sneer when he spotted Justin. "And Stafford and Worthing. I never expected to welcome the two of you into my home." "We haven't time for youth's rivalries just now, Pratt." Whitby's spine had gone straight as Stonehenge, and his face as hard. "Brook is missing." The lady gripped her husband's arm, horror on her face. Pratt frowned. "Missing how?" "Missing missing. She drove her maid to the train station yesterday and never returned. We found her car pushed off the road, into a copse of trees." The constable stepped forward. "My men scoured the area thoroughly. We found no trace of her, precisely, but there was a set of carriage tracks leading from the area in question and heading here." If Catherine pressed any closer to Pratt's side, it would require a tool to separate them. "You must be mistaken, sir." The constable blinked at her. "Mud doesn't lie, my lady." "Mud." She blinked too, with an innocence that they surely all knew was feigned. "Oh, you know, I do believe I heard the rain, now that you mention it. Although—" here she rested her head against her husband's shoulder—"I confess we've paid very little attention to the outside world. We were married on Sunday, you know." When Pratt smiled down at her, Justin could almost believe, for half a second, that love existed there. But if it did, Pratt wouldn't have been pursuing Brook so relentlessly, so recently. His expression looked more pragmatic when he looked over to the constable. "A carriage, you say? I haven't even used one in months. I've a new car, and when the roads are impassable for it, I ride." The constable folded his hands before him. "I noticed an old one behind the carriage house that has been out recently." Something flashed in Pratt's eyes—a flare, quickly gone. But there. "That thing . . . I've given the servants use of it—you'll have to ask them." Lifting his chin, the constable strode forward. "I'll go and see if they've gathered then, shall I?" The lady pried herself off Pratt's side. "I'll accompany you, sir. I hate to think of my poor cousin being missing!" Pratt watched her go, his gaze lingering on her hips. "You know, I wasn't certain how I would take to it, but I'm finding married life to be most enjoyable." "Our felicitations." Somehow Worthing managed to say it with a smile, yet in a tone that contained only irony. "But I'm afraid we've come to ask you to interrupt your honeymoon for a few hours. We need everyone we can muster out looking for her." Pratt lifted a brow. "Apparently, if they've called you in from London. Can the Season continue without you, Worthing? Or did you come to Yorkshire with amorous intentions?" Justin had never had cause to see Worthing bristle quite so much. "I came," he said with cold deliberation, "because my friends needed me. Will you join us or not?" "This was a bad idea." Justin stepped forward, unable to stand inactive anymore. "Whitby, I'll wait outside." Justin pushed past and made for the exit. His host followed. "Stafford, wait." He would rather get out of the house. Every moment he spent inside made him more ill at ease. So he didn't turn. He figured Whitby and Worthing weren't far behind, but he didn't verify that either. He charged for the sunshine, for fresh air. And made it down the front steps before a hand on his arm stopped him. He shook it off even as he spun. Maybe Pratt thought his expression was one of concern—but it was too dark, too hate-filled. Justin's fingers curled into his palm. "What?" Pratt's eyes narrowed. "I'll help in the search. I must make sure Kitty is well first—she has been ill every morning this week—but then I'll join you." He'd chased him down to say that? "Fine." Justin turned again. "Duke!" A growl formed in his throat as he slowly pivoted back. "What?" Pratt had a hand extended. "I know we've never liked each other. But we can put it aside for this, can't we? A truce." The last thing he wanted to do was put his hand in Pratt's. Those hands could well have hurt Brook. But he could hear the constable in his head, telling him not to tip their hand too much, too soon. With monumental effort, he uncurled his fingers and put his palm to Pratt's. "I will find her." Perhaps it came out more as a threat than a declaration . . . but if so, so be it. Pratt held too hard to Justin's fingers. He had to tug to free them, and then they curled of their own will back into a fist. Pratt smirked. "I know you'll not want to hear this from me, but have you considered the possibility that she left of her own volition?" His fingers dug into his palm. "Excuse me?" A lifted brow joined the smirk as Pratt shoved his hands into his pockets. "It's no secret the two of you have been at odds. What did you think she would do when you followed her here, hounding her steps? She probably ran away just to escape you." Before Justin was even aware of giving his arm the command, it had pulled back, flown forward, and his fist connected with the reprobate's nose. A satisfying crunch met his ears, and a pleasant pain scourged his knuckles. "Stafford!" Worthing cried, and his tone was a cross between warning, outrage, and a laugh. Pratt staggered back, his eyes glazed. He touched a hand to the blood dripping from his nose. Then his eyes flashed hot fury, and he lunged. # Thirty-One The sound of a gun's report brought Brook to her feet, sending the open journal to the floor. "What was that?" Deirdre, sitting at the desk, stood more slowly. "A shot?" "A shot." And it struck her right in the heart, bringing to life the fears Maman's words, and those she had written about Mother, had already ignited. Fire raced through her, and her legs insisted on moving. She went to the door, tried the latch. Flew to the bricked-over windows. Surely one, somewhere, was loose. "Likely someone hunting." "No." Her fingers bit into brick and crumbling mortar. Gripped, pushed, but they wouldn't give. "It was a pistol, not a rifle." "You can tell that?" Of course she could—though the sound had been distant. Still, her heart hammered, pressure seizing her head. A cloud of panic swirled around her. She slapped a hand to the brick. "Papa! Are you out there? Help!" "My lady, if Pratt hears you screaming—" "I don't care. Papa! Justin!" They must be out there. Why else would someone be firing a pistol? They had found her. Or trace of her. They were there. They were there—and a shot had been fired. By whom? She flew back to the door, pounded upon it. "Help! Let me out! Someone help!" She had to get to them. She must. They were there, so near, and bullets were flying—or one, anyway. Why had there not been a second? Had they killed Pratt? Or . . . "Help!" She had to get to Papa. She had to tell him what that journal said, the truth of what sent Mother into the night. She had to. He needed, finally, those answers. "My lady!" Deirdre tried to pull her away from the door—Brook shrugged off her hands. They landed again, and gripped her more firmly. "Stop. Please, I beg you." "Someone will hear. Someone will come and help." "Someone may hear, yes." Fear drenched Deirdre's tone. "And when they try to come, Pratt will kill them. And then be so furious with us . . ." No. No. She had to get out. She had to, that certainty gripped her far more strongly than Deirdre ever could. She broke free and went back to pounding and screaming. She wouldn't give up . . . though her hand stung. Her throat burned. Evidence that time was passing, though it all seemed frozen to her. Were they still there? Did they know she was? The door pushed inward, suddenly and forcefully enough to knock her down. For one glorious second she hoped—then she looked up and saw Pratt towering over her. Blood soaked his shirt, stained his chin. He had a laceration on his cheek. And such bright hatred in his eyes that she recoiled, scrabbling back along the floor until she bumped into the cot. Was that the look that had been in his father's eyes as he and John Rushworth chased down her mother? He whipped something at her head. She raised her arm to deflect it but gasped in pain when it hit her arm—though small, it was solid and heavy and clanged when it skidded across the floor. "You want your precious duke? That's all you'll ever get of him!" Justin? Resisting the urge to rub at what would surely become a welt, she pulled herself to her knees. There, glinting in the lamplight—gold. "No." Shaking too hard to stand, she crawled to it. It couldn't be—no. Justin would never, never take off his signet. He hadn't since his grandfather's death. It was there, always there on his right ring finger, where he could twirl it around. The familiar lion and cross of Stafford rose from the gold. The recessed places were dark and, when she picked it up with shaking fingers, sticky. Blood. She closed her fist around it. "What have you done?" Did the words even make it past her dry lips? They must have, because he laughed. "Exactly what I said I'd do. Except I didn't have to worry with hiding the body—he attacked me. I was defending myself, and the constable was there to see it." He grabbed her by the hair and pulled her to her feet. "Now you'll believe me, hmm? Your father's next, my lady." "Non!" She kicked him in the shins, slammed her ring-encasing fist into the laceration on his cheek. He cursed her, but his hand loosed its hold on her curls. The moment it did, she took off for the door. He hadn't locked it behind him, hadn't even closed it all the way. She need only reach it, get through it, and then— He slammed into her, slammed her into the door, slammed it closed. "Going somewhere, darling?" Held there, pinned between the damp wooden door and him, she smelled mold and blood. Justin's blood? She squeezed her eyes shut tight. It couldn't be. He couldn't be dead. He couldn't. Wouldn't her soul know it if he were? But hadn't she felt unaccountable fear at that gunshot? A sob balled up in her throat, surging upward but getting caught before it could do more than make her shudder. So much darkness. So much violence, and for what? A couple of diamonds stained red from it all? Had her arms been free, she would have reached up to rip the necklace from her throat. "You fool! You terrible, cruel fool. They're right here, you can have them. I don't care anymore! Just let me go to him. Maybe he's not dead, maybe he can be saved, maybe—" "What is she saying?" Pratt's words came out harsh, and he pushed her harder to the door. "It's Monegasque, my lord." Another sob started in her stomach and convulsed its way upward, this one making it all the way past her lips. She couldn't even speak the right language. Couldn't act, couldn't escape, couldn't help Justin—and that was assuming he wasn't beyond help. She couldn't give her father the truth, couldn't keep her maid safe, couldn't break the curse that greed had wrought. The necklace felt like hands around her throat. Pratt's hands, stained with blood. So much blood. "Justin." "Should have learned long ago to control that temper of his. Now talk. Where are the Fire Eyes?" "In my necklace." He pulled her back a few inches just to slam her to the door again. "English!" She was trying! But when she opened her mouth again, no words emerged at all, only a cry that snatched her breath away and made her every muscle shudder. Once open, the floodgates wouldn't be stopped. Her knees buckled, and she would have slid toward the floor if he hadn't still been holding her there. Pratt made a disgusted noise, gripped her shoulders, and tossed her aside. Landing on the floor, she drew her knees to her chest and shut her eyes against the light from the lamp. It had no place here, with all the darkness. With the thunder of his anger. With the lightning of his hatred. She wanted Justin. To hear his voice, whispering assurances. To feel his arms about her, promising a tomorrow worth fighting for. She wanted her father, with his dry sense of humor and fathomless understanding. She wanted home. All she had was a bloodied ring and a tongue that wouldn't speak English long enough to make it all stop. Hands soothed over her hair, so gentle that they must be Deirdre's. "She needs time to calm down." As if time could reverse the damage done. Could heal him, bring him to her door. "She has an hour—or Whitby's next. It would be easy enough for him to meet with an accident while out looking for her." "Non!" She forced her limbs to uncurl, forced herself up, away from Deirdre. To her knees and then her feet. "Non!" The door shut with a pistol's bang. The key in the lock ground like a bullet sliding into the chamber. She fell onto the door again, pounding. Screaming. Even she didn't know now what words she shouted, whether they were plea or command or denial. She didn't know what she meant to do when he reentered. She should have thought. Should have found something to use as a weapon. Should have . . . When the door pushed back, he had his gun in his hand and fury in his eyes. "Shut up!" Never. Bellowing at the top of her lungs, she threw herself at him. If he shot her, she'd at least draw some blood first. Her nails bit his cheek, raked down. A sickening thud echoed in her ears . . . in her skull. All other sound faded. The world went fuzzy and seemed to freeze, then shift. Slowly, as if she were viewing it all through morning fog, the room went sideways and the floor embraced her. Then the lamp went out. "Wake up, my lady. Please." Deirdre had said the words so often, they had begun to sound nonsensical. The burning of the lamp was the only measure she had of passing time. She had filled it while Pratt cursed and lifted the baroness onto the cot, the only thing she could think to do to look unconcerned, when she'd wanted to rush over and try to rouse her. She'd refilled it again since. That meant that at least sixteen hours had passed. More, now. A day must be done, a new one beginning. And still the baroness hadn't stirred. Hadn't wakened. He'd come back once, when the lamp was still half full. The tempest on his face when he saw the lady was still unconscious . . . To her utter surprise, he hadn't taken it out on Deirdre. He had, instead, left her with a key to this door, though that would only give her access to the hall. She had tried every door along it, tried the key in every lock, but the only one that would open was the one she'd seen before. He'd left food there, and water. She'd tried dribbling some onto her ladyship's lips, but that earned her no response either. She'd thought to try reading to her, but the journal was all they had, and it was in French. There had been a letter tucked into the last page though. That had been in English, and she'd read it aloud . . . then almost wished she hadn't. It had been from them, the elder Pratt and Rushworth. To the late Lady Whitby. Claiming they'd killed her husband, saying that the body found in York the night before—a newspaper clipping was included, about a body so badly mutilated as to be unidentifiable—was him. Warning that if she didn't hand over the Fire Eyes, the baby would be next. Deirdre's fingers went knotted as the words swam before her again. How horrified must the lady have been? A young mother getting such news, convinced, it seemed, by the horror. No wonder she had fled, thinking it the only way to save her babe. The floor was cold and hard under Deirdre's knees, and the lamp did little to make the shadows flee. "Lord God." She had prayed more these hours than at any time in her life—other than when it was Da who had lain unresponsive on a lumpy mattress. She picked up His Grace's ring from where it had skidded under the cot and put it in the baroness's hand, curled her fingers around it. "Lord above, I beg you. Restore her. Deliver her. Give her back to her father and . . ." She'd nearly said "His Grace." But that wasn't possible now, was it? She pressed her lips together. Pratt had said there would be no questions about killing him, that the constable had witnessed it. But no one could kill a duke without consequences, for sure and certain. Even if Pratt saw no prison term for it, there would be questions. He had to know that. It had to be what had put him in such a rage. And what if he were taken away to answer for it? What would become of them then, with neither water nor food? "Wake up, my lady. Come now." Deirdre rested her head against the side of the tick. Had she slept at all this night? If so, not for more than a minute here or there. "I've the key to the door. Not the outer one, only this one, but it's something. Wake up, and we can make a plan together. Lie in wait in the room by the outer door. You'll think of something, fearless as you are. But sure and you have to wake up first." Not a whimper. Not a flinch. Deirdre closed her eyes—jolted when her head slipped, and sure and that made her eyes fly to the lamp. Was the oil lower? She couldn't remember, now, what level it had been at. But enough remained that she could get up, walk to the end of the hall and see if new water awaited, or breakfast. Perhaps the aroma of food would stir her ladyship. Deirdre's joints creaked when she arose, her muscles screamed. And as she walked, her feet dragged. It took all her focus to get the key into the lock and turn it. She shuffled her way down the hall. A scratching reached her ears halfway down. She paused, the sound bringing her awake a bit more. Mice? A rat? Her pulse hammered at the thought. She lifted the lamp, though she saw no evidence of the rodent. But sure and it was the sound of claw on wood at the end of the hall. It stopped. Then came again, louder. She squealed, though quick as a flash she clamped a hand over her mouth. The scratching stopped. And in its place came . . . a hiss? Did rats hiss? No, wait—that was words! Praise be to the Almighty. Someone was at the door! "I'm coming." Her voice came out the barest whisper, but she hoped whoever it was could hear her. Tremors possessed her by the time she reached the door. What if it was a trick? Did Pratt doubt her? Was he testing her? It was a risk she had to take. "Is someone there?" "Bless my soul!" came the muted reply. "I didn't hear awry, then. Who is in there? Is this the baroness?" Someone knew! Deirdre pressed close against the door, her mouth at the crack. "Aye! I mean, not I, but I'm her maid, and she's in here too. Do you work for Pratt?" "Much to my dismay—but I'm cousin to the constable, and he told me to be on the lookout. I was seeing to repairs outside this wing and heard the screaming. Took me all night to find the hall what corresponded. Are ye well in there?" She splayed her fingers against the wood. "Nay. I'm well enough, but the baroness is hurt. He struck her in the head, and she's not woken for so very long. I don't know what to do." A shuffling sound reached her, one that went away from the door and then back. "He's coming. We haven't much time. But he's joining the search this morning, ordered his horse to be ready at eight. He'll be away. Two hours' time. I'll get you out, somehow or another. Aye?" "Aye! Aye." Two hours. She didn't know how she'd gauge it, but she knew answered prayer when it scratched. The constable's own cousin—praise be to heaven. He could help her carry the baroness out, help them sneak from the house. Then it would only be a matter of getting her the miles back to Whitby Park. Heaven help her—how was she to do that, if her ladyship didn't awaken? She would worry with that later. For now she rushed back to the cell, where the baroness lay as she'd left her. Golden curls tucked beside her, soiled gown half covered by the ratty blanket. Hands limp and useless at her side, with Pratt's blood still staining her nails. The distant creak of the door echoed down the hall. Footsteps. And then a curse. "Blast it, Deirdre—why the devil is the door open?" She spun, her fists at her sides. And took what was likely a sinful amount of pleasure in seeing the angry welts on his face, the bruises and cuts. "And what harm can it possibly do, when she hasn't so much as twitched a finger since you struck her? She needs a doctor." "A doctor would do nothing but wave smelling salts under her nose." Apparently doubting her word, he strode to the cot. Cursed again when the truth spoke for itself. "Idiot woman, forcing my hand." Never in her life had she been so tempted to strike a gentleman and add another mark to his once-beautiful face. She would do it, too, if the baroness didn't need her to keep his trust. But the words . . . the words came forth of their own volition. "You call her stupid? How did you expect her to react when you come barreling in here and tell her you've killed the man she loves?" He jerked toward her, looking ready to bite. Then, with a low mutter she couldn't discern, he knelt down and pressed a finger to her ladyship's neck. "Her pulse is still strong—she cannot be too hurt. She will wake up soon, and when she does, the door had better be locked. And you had better be ready to get answers from her." He stood straight again and strode for the door. Deirdre followed him out—closing the door behind her. "Sure and I will be, if she awakens. And what if she doesn't? Or what if you're arrested for killing the duke? Will you let us die of thirst?" He'd left a lamp at the end of the hall. Its light outlined the hard angle of his brows. "I won't be. I did nothing but defend myself." "But—" "Shut up, Deirdre, or I swear I'll lay you out along with her." Never, in the year she'd known him, had she seen his nerves so frayed, his temper so close to the surface. Perhaps, devil though he was, he hadn't been prepared for the effects of his own actions. Perhaps he staggered under the weight of his sins. Perhaps . . . perhaps he realized he'd dug himself too deep a pit. She followed him into the other open chamber, where a new tray had taken up residence on the table. He motioned to the bread, the cheese, the ham, the pitcher of water. "That ought to keep you alive, don't you think?" She folded her arms over her chest. It was more than he usually brought—which meant he intended not to be back by the midday meal, she would guess. And also, praise God, that there would be plenty for both of them when the baroness awoke. "Well then." He turned to the door. "Wait." She didn't know what she meant to say, only that she wanted to prick at him. Needle him in whatever way she could. She lifted her chin. "If thirst doesn't kill me, boredom might, while I wait for her ladyship to flutter her lashes. Have you a book in this house of yours? One written in English?" One she could actually read to her ladyship, that didn't speak of the horrors that had brought them here? Pratt snorted, though not with amusement. He stood stock-still for a moment and then reached into his jacket pocket. Pulled out a newspaper, still crisply folded and bound with twine, and threw it to the floor. "Don't get the pages out of order—I'll want to read it later." Not awaiting her response, he hurried out. Though he did toss over his shoulder, "And lock the blasted door!" # Thirty-Two Thunder roared, lightning sizzled, and darkness consumed her. Fear nipped, making a cry want to tear from her throat. But her throat wouldn't work. Brook couldn't make her body obey the command to run, flee, get away from the danger behind her. Then the words began. Some in French—Maman's words, but in Brook's own voice as she read the pages of the journal, softly. Some in English, filling in the gaps. Pratt and Rushworth had told Mother that Papa was dead—and that Brook was next. That's what had sent her out into the night, into the storm. Why she had the letters from Papa with her . . . and why she was wearing the pearls and gold she had thought were the last gift she would ever receive from him. When the storm raged, when the carriage tipped . . . That was where Maman's journal had begun. With watching the accident from the distance and rushing up. Hearing the wail of a baby. The groans of a dying woman—the driver was already dead. She recorded Mother's words, her pleas to take the babe, her Elizabeth Brook, and see her to safety. Somewhere far away, she said. She had no one left in England. Her husband was dead, and her family . . . How was she to know whom of her family she could trust, when it was a cousin who had done this to her? In the darkness, Brook felt tears gather. How alone Mother must have felt in those last moments. Giving away her child, mourning the husband she didn't realize would soon be mourning her. Thinking her whole family turned against her. Collette recorded her own fears too—suddenly having a child she didn't know how to care for. Fearing that whatever had sent the woman to her death would chase after her if she took the child . . . but being unable to leave the babe to the elements. She'd found nothing on the lady to offer identification—no doubt purposeful on Mother's part, if she were running away. But she took what she could for the baby. The box of letters. The necklace. And she had devised the best plan she could come up with for seeing to the girl's future. She went to the man she'd been involved in an affair with a year before—Prince Louis of Monaco. Prince Louis, who had never wanted to be Brook's father. Who had never loved her, never accepted her. But Grand-père had. Grand-père, always at odds with his son, had believed Maman's story. Had arranged for their care. Their flat. Had promised to provide for Brook's education. No wonder Maman had made him promise never to tell her. To destroy the journal with the story written inside it. She no doubt feared that if Brook ever returned to England, the violence would find her as it had her mother. And so it had. Her fingers curled into the damp mattress, closing around something warm and hard. Metallic. Her fingertips ran over it, tracing its contours . . . slipping into it. A convulsion rippled through her. Not just Maman and Mother. Justin, too, was gone. She squeezed her eyes shut against the darkness. They were all gone. Brook tried to sit, but her head pounded too hard, and her limbs all felt so heavy. How could she feel so tired, and yet as if she hadn't moved in an eternity? She used her fingertips to turn the large ring of gold around her finger. If Justin were here, he would prod her. Poke her if necessary, but he wouldn't let her lie about. He wouldn't let her weep away her life. He wouldn't let Pratt win. He'd tell her to get up and fight. She didn't want to fight. It hurt. And what was the point? Pratt had already won, had avenged his father's death, had taken what mattered most. Why fight anymore over the diamonds? Why should anyone else lose their lives over the Fire Eyes? "A fire goeth before him, and burneth up his enemies round about . . ." "Mon Dieu." She opened her eyes again, and the lamp seemed brighter than it had before. "Are you here in this? You must be, because you promise you are. But I can't feel you now. I can't see you." "His lightnings enlightened the world." She shuddered. The lightning had always been there, hand in hand with the darkness. They had seemed, somehow, of the enemy, not of God. But He was the author of that story. It was from His treasury that the winds came. By His hand that night overtook day. By His command that they died? No. "Ye are all the children of light, and the children of the day." Men made their own choices. And as some of them chose life, others chose death, chose evil. God could stop all the evil, all the violence, but if He did, He'd be rendering their choices for Him meaningless. But God did have a hand in this world. He was the one who had brought Brook home. Back to Papa. He was the one who had led her that day to Justin, in the abbey. He had led them to reconciliation before Pratt found her. She must praise Him for that. Papa was right. The hurt was unfathomable, the hole gaping. But it would have been even worse if they had still been at odds. And she knew, with every fiber of her being, that Justin would tell her to buck up. To mourn later. To focus, now, on beating Pratt. Getting free, somehow. Finding justice for him . . . and gathering close what family she had left. As William had taught him. Gritting her teeth with every contraction of muscle, she pushed herself up. "My lady!" The door's squeak must have blended with the cot's—but Deirdre flew through it and was on her in a moment, scarcely taking time to put down the tray in her hands before pulling Brook close in a hug so exuberant it made her head throb. "You're awake! Praise be to God, you're awake!" She pushed aside the pain and squeezed Deirdre back. "What day is it?" "You've been out almost an entire day, and sure and you scared a decade off my life." "Sorry." Brook pulled away and managed what she hoped was a smile. She gripped her friend's hands. "We need to get out of here. Somehow, some way. We'll lie in wait at the end of the hall if we must, and spring on him when next he comes, but—" Deirdre's laugh, light and a bit incredulous, cut her off. She shook her head. "I knew you would come up with something like that, once you roused. But the Lord has provided. There's a groundsman what heard your shouting yesterday. He's coming back in two hours to help us." Brook sagged in relief. "Two hours." "Aye. Enough time to eat and for it to revive us. Here, sit at the desk. You need water right off, and then some food." "You, too, from the looks of you. Have you slept at all?" Brook took slowly, carefully to her feet. Deirdre steadied her and then bent for the tray. "You were sleeping enough for the both of us." "You'll eat and then must rest. You'll need your strength." "Aye." Deirdre slid the tray onto the desk and gave her a smile. "It's good to have you back, my lady." Brook returned the smile and took the chair before her legs gave out. The bread smelled of heaven, and the water that Deirdre poured into a dented tin cup tasted of ambrosia. She took a slow sip, let it settle, and picked up the newspaper. "Really?" Deirdre held out her hands, palms up. "I asked him for reading material. Mostly to irritate him, but he tossed that at me." Tugging at the string with one hand, she reached with the other for a slice of cheese. Then she unfolded the paper. Her own picture stared back at her. This one was from the night of her debut, but the camera had caught her in an odd moment. She was looking over her shoulder at something, no smile on her lips. Rather, concern etched her brow—had she been wondering, in that moment, where Justin was? Not a picture they would have run then, but now it suited the headline. BARONESS BEAUTY KIDNAPPED! A startled breath escaped and brought Deirdre to her side. She quickly read through the paragraphs. Her lungs closed off when she reached the fifth one, and she jabbed a finger at it. In an interview given last evening, the Duke of Stafford stood with Lord Whitby and Lord Worthing and pronounced that he would match the reward . . . "What time?" Was it hope that fluttered, or new fear of it being dashed? "What time did Pratt come in with the ring?" "Morning." Deirdre's fingers dug into her shoulder, but she scarcely felt it. "This had to have been after. He's alive!" A sound came from Brook's throat that was half laugh, half cry. She pressed a hand to her mouth—the one that still had his ring slung loosely around one finger. "Pratt was lying." Justin was alive—which meant all she had to do was get to him. She read the rest of the article as she ate, her heart pounding with every word. The reward her father offered was substantial—and the fact that Justin had offered to match it would make it mighty tempting for anyone who had caught a glimpse of her. They'd done what they could to swing the tide. To win her allies. She would use them. When they finished eating, she banished a protesting Deirdre to the cot and let the words run through her mind time and again. The Duke of Stafford. Alive and giving quotes to the press. She stood, stretched, paced until her legs didn't feel so wooden and the tension in her neck eased a bit. She prayed and she praised and she plotted. They would have a considerable trek ahead of them, when they got free. They had to be at Delmore, and once the groundsman got them out of the house, she could find her way home easily enough. Find the sun, find the south, and go. Pratt land would lead straight to Eden. She had only to avoid him, and she would be home. Then she had to read the article again . . . and shake her head. He had used the same trick on her that his father had used on her mother—and she, too, had fallen for it. Had been mourning one who hadn't been lost at all . . . but who would be concerned about losing her. Well, it was time for the pattern to reach its end—and for Papa to finally have the answers he'd needed for nearly nineteen years. She retrieved the journal from the floor and made a makeshift sack for it and the canteen. Had it been up to Justin alone, they would have been out again the moment dawn streaked the sky. But they had waited for the paper, and he was glad of that too. As he finally strode out into the cool morning air, certainty settled in his chest. They had done right. They had given her what she needed. Even if the magistrate wouldn't budge, wouldn't let them search Delmore, they would find her. She would find a way out, find someone to help her, and they would be there when she did. "Stafford, Whitby! Wait!" Justin paused with one foot on the macadam and the other on the stair. Whitby was ten paces ahead of him, but he turned too. Worthing stood at the door, motioning to the footman who had been assigned as his valet. "Tell them, Hiram." Hiram seemed to be clinging to composure by no more than a thread as he waited for Whitby to join them. "Forgive me for not speaking sooner, my lord, but I tried to tell myself it was unrelated." Whitby shook his head. "Speak, Hiram." "It's Deirdre. She swore she'd wire at every stop, and she hasn't. I was worried, so yesterday afternoon when the search took me to town, I telegrammed her family. She never arrived in Kilkeel—and what's more, her mother isn't sick, they never sent her a message. Pratt must have taken her too." Justin felt his brows pull together. His thumb moved to his ring finger to twist the signet around, but its empty state made him want to utter a few choice words. He'd worry with that later, though. "Why would Pratt take her too?" Hiram glanced at Worthing, who gave him a helpful prod forward, his face stern. "Tell them." The footman swallowed. "She'd been giving him information. He'd threatened her family." Whitby pivoted away, muttered something unintelligible, and spun back to him. "Why did she not come to me?" Hiram spread his hands. "She sees the mistake now, my lord, which is what's to the point. She won't help him in this, though he might think she will. She must be with her ladyship. She'll help her. I know my DeeDee, and she'll help her get free." Justin wasn't so sure about that, but he didn't know the woman. Whitby, after a long moment of clenched fists and ticking jaw, nodded. So then. Justin headed for the stables once more and nearly drew his pistol when he caught sight of the rider who trotted their way. "Easy." Worthing stayed him with a hand on his arm. "No doubt he's keeping up appearances. Let him, for now. We'll have the noose around his neck soon enough." There was no "soon enough" when it came to bringing down Pratt. Justin planted his feet outside the stable door, folded his arms over his chest, glared. And took no small amount of satisfaction from the bruises and gashes on Pratt's face. And scratches—Justin hadn't scratched him. Pratt nodded at Whitby. "I'll head toward Eden Dale, Whitby. Are we meeting back here at noon?" Whitby shook his head. "We need you on the road toward the town, not the village. Those on foot will cover that area." Pratt worked his jaw, no doubt hating the idea of being sent farther from his house. But he nodded. "Very well. See you in a few hours, then." As he turned his horse and rode off, Worthing stepped close. "You didn't scratch his face." "No." "His wife?" Though the idea made him want to grin, he shook his head. "I think not." Worthing nodded. Not in general but toward his hand. "And why do you keep doing that with your thumb? Not that this is a bad time to develop a nervous tic, but . . ." Must the man notice everything? Justin loosed his arms, stretched his fingers. "I can't find my signet. Perhaps I left it in my room in town." But he hadn't. He couldn't remember when he'd last had it, but he'd have noticed its absence sooner if he'd left it at the hotel. Worthing's eyes went wide. "You lost your signet? Are you mad? You'll have centuries of dukes haunting you—" "It doesn't matter. Not today." He set his gaze to the north, toward where Brook had to be. "I can always have a new ring made. Just now, all that matters is finding her." A high-pitched whinny from inside the stables underscored his point. The grooms were bringing out the horses, but they were all skittish. No doubt because Oscuro reared and bucked and pulled at his lead, his eyes flashing whites. Whitby backed up to stand beside him and Worthing. The groom tried to get the stallion calm with a few words that did nothing. "Sorry, milord! He let us saddle him, but then he started acting like this the moment he got free of his stall. Not fit for riding today, it seems. We'll get him put away." "No." Whitby's voice was calm, deliberate. His eyes flashed certainty. "Don't put him away. Let him go." The groom looked at him like he was mad. "Pardon, milord?" "Secure the reins and let him go. Maybe he can find her where we can't." Justin's lips tugged up. "You're using a horse as a bloodhound?" "Have you a better suggestion? The bloodhounds couldn't find the right trail." Too much rain, their masters had wagered, and too many trails she'd set on her many rides through the country. The dogs had chased to and fro and to again. Justin strode to Oscuro, took the reins from the groom, and whispered to the beast in French. He calmed. Not enough that any sane man would try to ride him, but enough that he could slip the lead back over his head and pat his neck. "Go find her, boy. Va." The horse didn't even need a slap to the rump to send him on his way. The moment Justin let go the reins, he took off like a sleek black bullet. Worthing shook his head. "I do hope the idea isn't for us to keep up with him." Brook's father motioned for the other horses. "We'd never stand a chance." "And if he doesn't come back?" Then Yorkshire would have a wild stallion jumping its fences and scaring its sheep. But Justin chose to believe. "He'll come back. With Brook." He accepted the reins for Tempesta, mounted, and pointed her in the direction Oscuro had gone. # Thirty-Three The sound of hammer on brick brought Deirdre out of sleep with a start. She flew off the cot, heart pounding as pieces of mortar crumbled and spewed out from the once-windows, hitting the floor. The baroness was at her side in a beat, linking their arms. "Your groundsman?" "I assumed he meant he'd return to the hall." But surely someone out to harm them, someone on Pratt's side, would use the door. She edged a bit closer, though off to the side, where the shower of brick-pieces was at a minimum. "Hello? Is that you?" She didn't know who you even was, but the hammering stopped for a moment, and the same voice she'd heard earlier said, "Aye. It's me, Antony—the constable's cousin. Stand clear, it'll take only a minute." They clutched each other the tighter, both going more and more tense as sunlight found the cracks and shone down through. Blessed, beautiful summer sunlight. First just a few spots shafting in, then a whole beam. And finally, a rough-worn but friendly face peered down at them. He grinned. "Both awake, I see. Ready?" The baroness's smile looked exhausted with relief. "So very." "I've a ladder. You'll have to climb up." A moment later the face disappeared, and an old wooden ladder appeared through the space he'd made. Deirdre rushed forward to grab it and steady it. Then she waved the baroness over. "You first, my lady. I'll hold it steady for you." Her ladyship wobbled a few times on the way up, muttering in French as she did. No doubt frustrated at her own unsteadiness. Once her shoes disappeared through the gap, Deirdre followed. The ladder slipped when she reached the third rung, but she bit back a scream. And hands steadied it at the top. "Quickly now. I hammer often enough out here, but not usually upon the brick. I can't be sure no one heard, but it was that or an axe to the door, and there are fewer servants out here." Deirdre breathed a prayer and rushed up the last few rungs, exhaling in relief when strong hands grabbed her arms and pulled her out. Their window, it seemed, had its top at ground level—the bottom being within a paved moat that seemed to line this whole side of the house. For what purpose, she couldn't discern, but the stones and bricks looked old as the hills. Part of the original structure, perhaps, from an age long-since past. "Hurry, milady. Here." Antony grabbed up a patched jacket that had seen better days and handed it to the baroness, along with an equally battered skirt. "My wife's—she helps me in the gardens sometimes, is about your height, praise be to heaven. Put it on. And her hat, here. One for you too, miss." "Deirdre." She took the hat and prayed its broad rim would hide her. Her ladyship wasted no time. She slid the disguise directly over her muddied, bloodied dress—though even with the layers underneath and her sack concealed beneath it too, still she swam in it. Jamming the hat over the hair the sun was determined to catch and alight, she nodded. "Let's be off." "Aye." Antony guided them both with a hand to each of their elbows, shooting a look over his shoulder. "There are stairs at the end there. Keep your heads down, but act like you're talking. Hopefully, if anyone sees, they'll think you're my daughter, miss, coming from town to visit." They no sooner gained level ground than the first shout went up from the distance. It took all Deirdre's willpower not to spin to see who it belonged to, not to run for the trees. Antony gripped her elbow the tighter. "Easy. It's Roger—he's a friend. 'Tis the chauffeur we need to be wary of, methinks, and perhaps the footmen." He half turned in the direction of the shout and let go Deirdre's arm long enough to wave. "Morning, Rog! I'll come by later, aye?" The figure in the distance had a shovel in hand and didn't make for them. He merely raised a hand in salute and kept to his path. Deirdre couldn't bring her breath to even out though. Antony's fingers took her elbow again, and he guided them southward, toward a copse of trees. Five feet strode across, ten, twenty. Halfway. "Ho, Antony!" This voice sounded harsh. "Where are you going? His lordship said no one is to leave the immediate grounds today." Antony swallowed hard, his larynx bobbing under his kerchief. He let go their elbows again. "Stay here. Act disinterested. Don't turn around." He strode a few paces back toward the house and called out, "Just hunting truffles is all, Mr. Michaels." Deirdre clutched at the baroness's hands, bending close. As if they were talking, laughing. Or at least, she hoped it would look like she shook from laughter and not from fear. "His lordship ordered no truffles." Now suspicion edged the voice. "Lord, help us," Deirdre muttered. "What do we do?" The baroness gripped her fingers. "Hold still. A minute more." Antony loosed a guffaw of a laugh. "He might not have, but the new lady did, and I for one don't aim to get on her bad side so soon!" That earned him a snort from the man behind them. "Where are the hounds, then?" "Already in the trees—though they must not have found any yet, given how quiet they are. If they don't in half an hour, we'll turn right back, sir. Better the lady's disappointment than the lord's ire, aye?" Now a grunt. "Half an hour. Not a minute more." Deirdre's breath whooshed out. Could they reach Lord Whitby's land in the allotted time? Antony returned, on her ladyship's other side this time. He motioned them forward at an even, unhurried pace. "Easy now, until we reach the cover of the trees. Natural-like. And pray he don't go and see that the truffle hounds are still in their bays." Pray she did, for that and more. All was quiet as they tromped into the tree line, at which point their guide darted a glance over his shoulder. He nodded. "He's gone. We'd best run for it now." They did, though within a minute her ladyship called them to a halt. "Sorry. My corset." Her breath came in short, hard gasps. She tossed the hat down and shrugged from the jacket and skirt. "You'll have to loosen it for me, Deirdre, or I'll pass out before we get more than a quarter mile." Deirdre didn't need to be told twice. She went at the row of buttons while Antony made a show of turning his back to them. The baroness dragged in another half breath. "What if he finds you were lying, sir? You could get in trouble for helping us." A gnarled hand waved that away. "I've already sent my wife to our daughter in Eden Dale, first thing when my cousin spoke to me. I mean to join them there after I see you home, milady. I'll go to my cousin and tell him all I know. When Lord Pratt is behind bars, we'll decide our next step." Deirdre tugged at the stays, loosening them until her ladyship could breathe normally and then tying a cursory bow to keep the corset up. Her fingers knew the buttons well enough to make quick work of them. "My father's offered a reward to any who help me." Antony nodded. "Aye, I know. It's how I talked the ladder from George." Someone else knew of them? Deirdre's hands shook, making the last three buttons impossible. "There. Good enough." Antony turned back toward them. "I tried to manage it on my own, but . . ." "No matter. We'll be happy to compensate this George. And you, of course—" "Nay, milady." Antony's shoulders straightened even as he motioned them onward again. "I'll not have it said I did right just for a bit of quid. Though if you've a position open on Whitby grounds . . ." Her ladyship smiled as she broke into a run. "I'm certain we have, Antony. Absolutely certain of it." They hadn't the breath for any more conversation. Sticking to the trees as much as they were able, they concentrated on covering ground. Soon they'd be home. She'd be back in Hiram's arms, and her ladyship would be in her duke's, and her father's. Assuming Pratt didn't find them first. She whispered another prayer, one the baroness echoed in French. Thundering hooves interrupted before she could say her amen. Antony halted, cursed, motioned them deeper into the line of trees. The baroness stepped out instead, into the open. "It's Oscuro! How did he—he must have jumped the fence when they put him out or . . . No." A laugh broke free of her lips. "He's saddled! Oscuro!" The beast bore down on them, and though her ladyship hadn't the sense of get out of his way, sure and Deirdre did. The baroness laughed again as he shifted his course, looking as though he'd barrel right into her. Instead he circled her, nickered, and shoved his head into her side with enough force to send her back a few steps. Still laughing, she wrapped her arms about his neck and murmured something in French. Then she looked to Deirdre. "Our chariot, milady. I recognize this last stretch—on Oscuro, we'll be back on Whitby property within a minute." "Then it'll take no more than five on foot." Deirdre stepped closer to Antony's side, though she smiled weakly. "Horses and I don't get along. But you go ahead. He'll see you safe to your da's arms. I'll take the sensible path." "You don't know what you're missing." "Aye, but I do. Broken limbs and a heart that gives way from fear." Her ladyship laughed again and left the beast for a moment, long enough to come and wrap her arms about Deirdre. And who'd have thought they'd become friends, the grand baroness with her Parisian gowns, and her, little DeeDee from belowstairs? She held her back, tight as she would little Molly. "Be careful, my lady. He's out here somewhere." "And he'll be answering for his sins." Her ladyship pulled away and looked to Antony. "You'll see her safely to Whitby Park?" "Upon my honor, milady." He swept his cap from his head and held it over his heart. The baroness nodded, a breeze toying with her loose curls. "See you soon, then." Deirdre held her ground while the baroness swung up into the saddle. When they thundered off, Deirdre accepted the arm Antony held out. "We'd best hurry," he said as he pulled her along toward Whitby Park with a glance over his shoulder. "Our half hour was up some time ago." Today the wind raced them, and the sun urged them on. Brook made little use of the stirrups, which had been set for longer legs than hers, and held on with her knees. If Pratt were out supposedly helping look for her, he would, she hoped, stay near the roads. So she headed for the sea. It crashed its greeting. Waves on shore, clouds skidding overhead as accents to the sun. After days in darkness, she soaked up the warm light with joy. "Ye are all the children of light, and the children of the day." A promise too easily forgotten in the darkness. The moment they crossed over to that familiar mark where they would always turn around, her heart leaped. And Oscuro put on a renewed burst of speed. He, too, knew home. Figures appeared, mounted. For a moment she worried that it might be Pratt—but the lead horse was black. He had no black horse that she had ever seen. And given the way Oscuro shifted direction to angle for them, she let her heart leap. It must be Tempesta. No matter who rode her, it meant a friend, and a laugh tickled her throat. The laugh turned to a shout when she caught the gleam of sunlight on a blond head. Justin. It was Justin. An echo of a shout came her way too, woven into the rush of water and the cry of gulls overhead. He waved, but she didn't let go the reins to wave back. Better to hold on and let the stallion run faster than he ever had before. Justin pulled ahead of whoever rode alongside him on one of the bays—Papa? Brice? Soon she could make out her beloved's form, his face. His beautiful, strong, unmarked face. If he had indeed attacked Pratt yesterday, there was no question who the victor had been. She pulled Oscuro up to a halt and leaped from his back to cover the last few feet. "Brook!" Justin's feet hit the ground too, and seconds later his arms were around her. "Brook. Mon amour. Are you hurt? If he hurt you, I'll kill him." The laughter bubbled up again. Right then, the ache in her head meant nothing. She wrapped her arms tight around him and pulled his head down for a kiss. Quick but hard, exuberant. "I'm fine," she breathed against his mouth. "I'm home, and I'm fine. And you're fine. Pratt told me he'd killed you." "What?" His arms tightened around her, and he buried his face in her hair. "Never." "He had your ring." "He has my ring?" "Had. He threw it at me. It's in my pocket." "He must have slipped it off when I shook his hand—right before I socked him in the nose." He squeezed her, and then he set her back a few inches and traced her face with his gaze. His eyes darkened. "You're bruised. He did hurt you. Tell me that at least it was you who put the scratches on his face." "I would have gouged out his eyes had he not knocked me out with his pistol. What was the shot I heard? It set me to screaming, which got the attention of one of the groundsmen. He just led us out." Justin's grin was boyish, unrepentant. "We had a bit of a scuffle. The constable fired a shot into the air." "A bit of a scuffle?" Brice's voice brought Brook's gaze up. He dismounted, more leisurely. The usual mirth in his grin couldn't disguise the relief in it—or the circles under his eyes. "Stafford would have pounded him to a pulp." He stepped nearer and put a hand on her shoulder. "If I may, Duke." Brook wasn't sure what exactly had changed between these two, but Justin let her go with naught but a lifted brow. Brice gave her a quick embrace. "I knew something bad was going to happen." Stretching to her toes, she kissed his cheek. "And you came from London to help. You're a true friend, Brice." "They couldn't have handled the press without me." With a wink, he propelled her back into Justin's arms. "Did you see our article yet?" "It's what let me know Justin wasn't dead." And being tucked to his side was pure bliss. Nearly as much as hearing the rumble of Justin's chuckle. "You should have seen Worthing yesterday when he arrived. Clothes wrinkled. Hair out of place." "Extenuating circumstances." A metallic sound cut through her laugh, one she recognized only vaguely. A shotgun being pumped. "Non." They all spun at the same time, even as Pratt stepped out of the trees. He held the weapon at the ready, pointed at the three of them. "Well, well. Look at this lovely target. I bet I could fell all three of you with the scatter shot." Before she could even mutter a prayer, Justin and Brice had both put themselves between her and Pratt. Justin kept his hand clamped on her arm—he knew her too well, knew how she readied to elbow her way back up. "Are you too stupid to know when you've been beaten, Pratt?" Justin's fingers squeezed a warning into her arm. Begging, that pressure, begging her to stay put. "There's no winning now." How could a face look so shadowed in full sunlight? His eyes spewed hatred at them. "You think I didn't know this was a possibility? I'm about to disappear—and one of you is coming with me until I do, to assure my safety. Worthing? You look like you're in the mood for self-sacrifice. Spare the lovebirds another separation, hmm?" "Don't even think about it, Brice." Brook kept her voice too quiet for Pratt to hear over the pounding surf behind them and knotted a hand in the back of his jacket to make sure he took her advice. "He could well kill you—and even if not, he'll only come back. He'll not give up on the diamonds so easily." Brice shook his head. "You would abandon your wife, Pratt? And she with child?" Brook's hand nearly went lax. Pratt edged closer. "Kitty's resourceful. And she would fare better with an absent husband than an imprisoned one." No doubt they already had a plan to rendezvous. No doubt Kitty knew every facet of his plan, had helped him devise it. The betrayal still pierced. Justin turned his head a fraction toward her. "I have a pistol at my back," he said in Monegasque. "Pull it out, Brooklet—you're the better shot. I'll get Worthing out of the way." "What did you say?" Pratt stomped closer, his eyes wild and his finger twitching. "Don't try anything. A hostage would be handy, but if I have to kill you all and make a run for it, I'll do it." Father, help us. When Justin's fingers loosened, she moved her arm to his back, slid her hand under his jacket, doing her best not to move the fabric. The pistol was at the small of his back, the grip warm under her hand. Pratt's gaze arrowed into hers. "Step away from the baroness, gentlemen. Now." "Dive," she whispered. "Both of you. On the count of trois. Un." Pratt brought the butt of the shotgun to his shoulder, his lips compressed. "Deux." Brook pulled the pistol free. Pratt's finger moved to the trigger. She brought the weapon up, shouting, "Trois!" The men lunged to the side, but a shot ripped the air before her finger touched the trigger. Pratt jerked. The shotgun fell. Eyes glazed, he staggered to his knees and then collapsed. The constable stood behind him, pistol still smoking. Papa was at his side, looking ready to empty his revolver into Pratt's still form, but the constable put a hand on his arm. "I'll take care of him, my lord." A cloud cleared from her father's eyes. He passed his gun to the constable and ran forward. Brook handed Justin's back too and met Papa in a fierce embrace. The moment his arms came about her, a cry took hold of her throat. "Papa. I'm sorry. I never wanted you to go through that again." He held her tight, sucked in a deep breath. "You're safe. That's all that matters, my precious girl." "She thought you were dead." She pulled away enough to look into his face. "He had Maman's journal, and that's what Mother told her. She thought you were dead, thought they would kill me next. That's why she sent me away." Papa rested a hand on her cheek. "I would have gladly gone the rest of my life without knowing why, if it had spared you this." She covered his hand with hers. "But I'm safe. And now we know." "We do. And praise be to God, you are." He kissed her forehead. Hiram ran their way, panic on his face. "Lady Berkeley! Is Deirdre with you?" A smile tugged. "Following on foot, led by the constable's cousin. They both deserve a hero's welcome. She will be glad to see you, Hiram." Hiram needed no more urging—he took off at a run in the direction she indicated. The constable removed his hand from Pratt's neck and shook his head. "He's dead, which was not my goal. But Antony helped you?" "We never would have escaped without him." With a satisfied nod, the constable stood. "Good. Now—go home, have a meal, rest. When you're ready, I've questions." "And I've the answers." "When my men get here, I'll leave them to see to the body. I've a conversation to have with Lady Pratt—and no doubt a few servants to arrest." Brook's back went stiff at mention of Catherine. She had to have been involved—but Pratt hadn't once mentioned her. Brook had never seen her. Other than the one time she'd demanded the Fire Eyes, she had, it seemed, kept her hands clean. It had been Pratt who hired Jenkins to attack her, Pratt who killed the major. Pratt who kidnapped her and Deirdre. A sick knot twisted in her stomach. They would have nothing to accuse Catherine of. No proof of her involvement. She would walk free. Papa rubbed a hand over Brook's back, no doubt feeling the tension. "Dust yourselves off, gentlemen, and let's go home. I daresay the chef has cooked enough for an army as he prayed." Brook wouldn't let Catherine ruin her homecoming. She made herself grin at the exaggerated look on Brice's face as he brushed the sandy soil from his trousers, and then she turned to Justin, her hand in her pocket again. The gold of his ring was warm and smooth—she'd cleaned it off with some of the water earlier. As he straightened his jacket, she stepped away from her father and held it out to him. His grin bloomed, lopsided and mischievous, to match the gleam in his eyes. "Are you proposing, my lady, with that ring?" She grinned right back and dropped to one knee. "Will you marry me, Duke?" Laughter rang out all around her. Justin's loudest of all as he gripped her by the arm and pulled her back to her feet. "Get up, you fool woman. And yes." He planted a kiss soundly on her lips and snatched the ring from her hand. "I most assuredly will." The gold back where it belonged, he slid an arm around her and came back for a second, slower kiss. "I'll ask you properly once we're back to Whitby Park. I've a ring in my room there too. It's a bit smaller. Has more sparkle. Was my mother's." She nestled into his side as her father gathered the horses' reins. "Your yes was binding, sir—asking again would be redundant. But I'll be proud to wear your mother's ring." Justin leaned down again, fire in his eyes. Brice's hands appeared between them, forcing their faces apart. "I've had trauma enough for one day." He shoved his way between them, grinning all the while as he slung an arm over each of their shoulders. "Am I best man, Stafford? Or will I have to fight Thate for the honor?" "You'll have to fight me, if you don't get out of my way." "Touchy, touchy." With a wink, Brice slid his arms free and moved ahead of them as the constable called out a greeting for Antony and Deirdre, safely out of the trees. "Brook will defend me if you try to pummel me. Isn't that right, my lady?" "Not this time." She slid her arm around Justin's waist and tilted her face up toward his. She knew it, knew every feature and expression. And loved none so well as the way he looked at her now. As if she were his yesterday, his today. His tomorrow. "Je t'aime." His smile spoke as much as his words. "And I love you. Always." # Epilogue LATE AUGUST 1911 The summer sun beat down hot and glorious upon them. The North Sea wind whipped and refreshed. Justin let go of the hand he held so that he could slide his arm around her waist instead, content to stand in the sand with Brook and do nothing but watch the waves roll in. She rested her head against his shoulder. "I'm still not sure how I shall survive for months on end without the sea at hand." Chuckling, Justin pressed a kiss to the top of her head. "I'll keep you well distracted, Duchess. I promise. And whenever you can't suffer it anymore, we'll come back here." "If my father will have you, after you stole his footman." She gave him a cheeky grin and walked her fingers up his chest. It was nearly enough to ruin a man's concentration. Justin chuckled and indulged in a long, slow kiss. When he had mentioned before the honeymoon that Peters wanted to move on, out of domestic service, Whitby had been the one to suggest he take on Hiram, so that he and Deirdre could travel together whenever Justin and Brook did. A fine solution. Justin and Hiram didn't know each other well yet, but he could appreciate a man who went through each day with such good cheer. At least when such a man didn't constantly interrupt when he wanted to kiss his wife, with cleared throats and loud ahems. He pulled away with a scowl for Worthing, who stood a few feet away, his feet in the grass rather than the sand. "Have you made it your life's work to harass us, Worthing?" His friend grinned. "You would think so, but no. It only seems that way because there's never a moment when you're not sneaking off with your wife for a kiss." "I didn't know you were here—or coming." Brook left Justin's side long enough to greet Worthing with a kiss on his cheek. "On your way to Scotland?" "Aye, that we are," he said in a fine imitation of the Highland burr. He nodded back toward Whitby Park as Brook returned to Justin's side. "Ella and my parents are having tea with Whit, who has already convinced them to tarry here until tomorrow. He said you were greeted with a visit from Catherine upon your return yesterday." Justin settled a hand on Brook's back in time to feel her shudder. "A lovely homecoming from our honeymoon." They had envisioned a quiet evening telling Whitby all about their trip through the Mediterranean, their visit with Prince Albert. A quiet evening at home before their planned trip this evening to Azerley Hall, to get to know the new Lady Cayton. But then Lady Pratt had glided in, all sugary smiles over the venom they knew hovered beneath. "Put a pall on the whole evening." Worthing pressed his lips together. "Did she try to make friends again?" "She tried. As if I'm stupid enough to fall for her tricks a second time." Brook's fingers went to her necklace, to the pearl-hidden diamonds she still wore. They needed to decide what to do with them—but had all agreed to focus first on the wedding, on getting settled in at Ralin Castle. The long, cold winter would give them time enough to discuss red diamonds and Indian curses with her father. "As if I couldn't see the hate in her eyes. She loved Pratt, unfathomable as it seems. In her eyes, we killed him. Yet another person dead because of the Fire Eyes—yet another reason for her to think they should be hers." She was playing it smart, though, Justin had to grant her that. Gathering a horde of supporters, making herself into a celebrity. Hand-in-hand with every article about Brook had been one about Catherine—the poor, deceived fiancée and then pregnant wife, who had been used by her husband because of her connection with the jewels. The telegraph clerk hadn't been able—or willing—to identify Catherine as the one to send that false note. But Justin knew it. He knew it. Brook wrapped her arms around her middle. "This isn't over. She'll bide her time, she'll let us get comfortable and perhaps focus for now on her coming child. But she'll strike again." Justin drew in a slow breath. "Pratt waited nineteen years to avenge his father's death—I daresay Catherine won't be quite so patient to avenge his. We can't afford to relax, to let our guards down." Worthing shoved his hands in his pockets and stared past them, to the glimmering sea. "You should just get rid of the things. Donate them to a museum." "Even if we did, she would still seek revenge for him." Brook's fingers fell away from the pearls. "And she would still seek the diamonds. I know she would, and probably others besides her. If we donate them, then we pass along the curse to some unwitting museum staff. Guards would end up dead in attempted thefts. Other property destroyed. Other lives ruined because of these stupid things. I can't do that. I can't make someone else pay for them." With a sigh, Worthing looked at Brook, then at Justin. "I see your point. The poor chaps at a museum wouldn't know how to defend against this. Wouldn't know that the best way to hold the evil at bay is through prayer." A chill possessed Justin, despite the hot summer sun. He nodded. "We know, though. We know how to fight it." "And yet . . . you've lost so much already. Both of you. You've had so much sorrow this past year." Worthing's brow had a furrow as deep as the sea. "You deserve peace as you start your life together." "Brice—no." Brook shook her head wildly, sending curls into the clutches of the wind. "This isn't your fight. We appreciate all the prayers you've prayed for us, all the support you have given. But your involvement ends there. Don't try to take any of this upon yourself. I won't let you." Worthing's grin reemerged, bright if a touch sad. "But I've gotten a taste for adventure. Let me help here or I'll have to go find a mountain to scale. A horde of pirates to fight off. Maybe a sheik to challenge." "No." The mirth fell away. "I have to, though. The Lord has made that very clear—and I'll have no peace if I don't obey Him." Justin's fingers curled over his wife's shoulder. "Worthing—" "She wouldn't have forgotten that I was there, too, when Pratt was killed. If she blames you, she blames me. If she's made a target of you, she's made one of me." He shrugged. "Might as well make it count and tell her I have the diamonds too. Get her to focus more on me than you for a while." Brook shook her head. "She'd never believe it. She wants them too badly to think we'd ever give them up." Justin shook his head, too, looked off into the distance. Narrowed his eyes at the glint of sun on blond hair. "Don't look now, but I believe she's watching us as we speak. No doubt thinks we're plotting how to keep the things from her." "Then let's make it count." Worthing swallowed and pasted on a smile. "She'll believe it if she sees it. If you give them to me now." "Brice." No laughter laced Brook's voice. Worthing's grin faded again. "This is what we're supposed to do." Justin felt the breath she drew in and sucked in one to match. "You can't be sure of that." His breath of laughter sounded more cynical than amused. "You think not? If you have an argument with it, take it up with the Almighty. Perhaps you'll convince Him where I've failed." Only Worthing could talk so calmly about arguing with God. "You can't actually want them. If you try to sell them, if word gets out, you'll be hunted down just like Rushworth was." "What I want is for my friends to be safe!" He shoved a hand through his hair—his tell, Justin had learned, of the deepest unrest. "She could already be carrying your child, Stafford, or if not now, then soon. What then? Why would you not take whatever safety for them I can offer, meager as it is?" While Justin tried not to let the hope and fear of a possible coming child overwhelm him, Brook gripped the dangling pearls, the diamonds within. "It won't help. She'll still come after us." "Yes." Worthing held out a hand. "She'll come after all of us. But if I can get her to come after me first, then you two can focus on your marriage for now. On your baby—whenever one joins you." Brook's eyes went narrow. "Why do you keep speaking of—?" "Call it a hunch." A wink of a grin, quickly gone. Worthing wiggled his fingers. "Let me help you. I promise I'll tread with the utmost care. With constant prayer. I'll find a way to expose her for what she is, to see she meets justice. And then I'll return the diamonds. You have my word." Brook took her bottom lip between her teeth and then looked up into Justin's eyes. Hers were damp. "He could be right. We could . . . I could be . . ." She splayed a hand over her stomach. "I don't want to bring a child into the middle of this." Was she saying . . . ? She couldn't be sure, it was too soon. But if she thought it possible . . . Justin exhaled shakily. "All right. All right. But we'll help you plan. We'll help you catch her." Brook was already working at the pearls. A diamond dropped into her palm, and then, a moment later, its twin. Justin swallowed. All the times they'd spoken of them, but this was the first he'd seen them. She held out her palm, and the sun angled down and set the jewels aflame. Could Catherine see it, from where she stood on her bluff on Delmore land? Probably not—but she would guess. She would assume. Despite his words, Worthing stood there a long moment staring at them. He lifted his hand slowly and scooped them from hers. Held them up to catch the light . . . and perhaps the attention of their distant observer. "Hello, trouble." Lowering his hand again, he slid it and the gems into his pocket. "I had better at least be named the child's godfather for this." Brook breathed a strained laugh and leaned into Justin's side. "Be careful, Brice." He nodded, waved a hand at them, and turned back toward the house. "I'm going to go and tell my parents you've invited us to spend Christmas with you at Ralin Castle. It's the least you can do, after all." "We've rooms enough, I suppose." Justin chuckled as their friend stomped back down the hill. Brook rested her head on his shoulder. "Sometimes it still feels so unreal. One of your stories." He thought so nearly every morning, when he awoke with her in his arms. "This one must be called 'The Life of the Duchess.' And there are many adventures yet to be spun in it—all of which have the happiest of endings." She smiled up at him, then glanced back toward the house for which Worthing strode. "And no doubt quite a lot of excitement we would all rather do without." "You wouldn't know what to do with a boring life. If no danger found you, you'd create some." At that she laughed, tossing back her head so it could blend with the music of wind and surf. Then she sighed. "I suppose it's time we leave for Azerley Hall." "Mm." He smoothed back the curl that had blown into his face, tucked it behind her ear. "Cayton's note said Adelaide is excited to get to know you better." "And I her. Though I don't know if I have it in me to be anything but polite to Cayton. Not seeing how Melissa still mourns the loss of him." "He at least recognizes that he made a thorough mull of everything. Perhaps there's hope for him." He pressed a kiss to the top of her head. "Shall we, then?" He expected another sigh, another grumble about Cayton. Instead, she grinned in that way only Brook could, the way that nearly stopped his heart. And she held up the key to the Rolls-Royce that had been, a minute ago, in his pocket. "I'll drive." There was nothing for it but to laugh and chase her down the hill. # Author's Note I often say a book has been with me for a long time . . . but no book has been with me as long as this one. When I was twelve, Brook's story began in what I was determined would be my first completed novel, entitled Golden Sunset, Silver Tear. I finished it a year and a half later. After nine other published books, nineteen years, four titles, and countless rewrites, I'm beyond ecstatic to see Brook and Justin's story in print. And with Bethany House, the first publisher I queried about it at the age of fourteen! One of the first revisions I made to the story as a teen was to change the opening setting from a fictional kingdom to Monaco, after learning of the Grimaldis' longest monarchy in history. Though there was obviously never a Brook in that rich family, she fits well with the actual history. Prince Louis, who I billed as her father, was always at odds with his father, Prince Albert—largely because he refused to marry and instead kept an actress as his mistress. Their one daughter, Charlotte, was adopted into the Grimaldi family legally so she could be named the crown princess, and the principality could be kept from the hands of the next nearest relative—one Kaiser Wilhelm of Germany. The biggest change Brook and Justin underwent, though, was when I decided to change the setting of the story from the 1860s to the 1910s. The credit there belongs to my fabulous agent, Karen Ball—and the change was one of those that, once I'd thought of how to do it, earned an "Of course! How could I have missed this all these years? This is when Brook was supposed to have lived!" The changing times and ideas perfectly fit the spirit Brook had always had, and though it required a complete overhaul of the story, it was one I took joy in. For those wondering about the red diamonds, let me assure you that, though the Fire Eyes are fictional, the information shared about such jewels in general is true. They really are the rarest jewel in the world, and the largest red diamond is only five carats. On a similar note, while I set my heroine's home in a real area and descriptions of Whitby and Yorkshire are taken from research, Whitby Park and Eden Dale are fictional locations, as are the other homes mentioned. Like any story, The Lost Heiress couldn't have been written without help and input. Thanks to Patrick Collins of the National Motor Museum in Brockenhurst, UK, for taking the time to answer my questions about the Rolls-Royce that later became known as the Silver Ghost—and going above and beyond by scanning pages of its manual for me! And I'm also so grateful to my English reader, Elisabeth Allen, for volunteering to read over the manuscript and make sure no Americanisms worked their way into the story. You were a real godsend, Elisabeth! I also had to tap the immense knowledge of the British Raj of my Irish-born friend, Christine Lindsay—thanks so much for patiently answering my questions about what rank Henry should have, and what Deirdre would call her parents. And of course, Wendy Chorot for reviewing all my French for me—thanks, flower! I'm so blessed to be surrounded by encouraging family and friends, from my parents (who told me my thirteen-year-old version of this book was great) to my husband, David (who has read so many versions of it, it's amazing he hasn't gone cross-eyed). I've had priceless input on these characters over the years from critique partners (Stephanie!) and agents and editors, all of whom contributed to the story I ended up telling. And I can't begin to say how grateful I am to Charlene and the team at Bethany House for believing it was Brook's time to be published. Karen S., I still can't get over coming full circle on this after so many years since that pitch at my first conference! The Lost Heiress has become, in my mind, a symbol of determination—a lesson in how, when we're chasing our dreams, we should never stubbornly cling to the way we think our goals need to play out . . . but we should never give up on those loves the Lord has given us. I hope you enjoyed getting to know my first heroine. Over the years her name has changed, along with her station, her home, and her family. But her spirit is still the one I envisioned when I first sat down as a preteen with a pencil, a stack of loose-leaf notebook paper, and the determination to write a book. May her story be to you just a portion of what it has been to me all these years. Roseanna M. White pens her novels beneath her Betsy Ross flag, with her Jane Austen action figure watching over her. When not writing fiction, she's homeschooling her two small children, editing and designing, and pretending her house will clean itself. Roseanna is the author of ten historical novels and novellas, ranging from biblical fiction to American-set romances to her new British series. She makes her home in the breathtaking mountains of West Virginia. You can learn more about her and her stories at www.RoseannaMWhite.com. Resources: bethanyhouse.com/AnOpenBook Website: www.bethanyhouse.com Facebook: Bethany House
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The Guns of Navarone or Navarone may refer to: The Guns of Navarone (novel), 1957 World War II-set novel by Alistair MacLean Navarone Island, fictional Greek isle in MacLean's novel The Guns of Navarone (film), 1961 film based on MacLean's novel The Guns of Navarone (song), film's theme song, covered by Jamaican ska group The Skatalites in 1965 and later covered by The Specials "Guns of Navarone" (Sean Paul song), 2021 song by Sean Paul from his album Live N Livin! featuring Jesse Royal, Stonebwoy & Mutabaruka Force 10 From Navarone, 1968 novel by Alistair MacLean, sequel to The Guns of Navarone Force 10 from Navarone (film), released in 1978 and loosely based on MacLean's 1968 novel Navarone (video game), Japanese arcade video game released by Namco in 1980, loosely inspired by 1961 and 1978 films Navarone (band), Dutch rock band formed in 2008 See also Navarrone, American historical novel by Helen R. Myers; winner of 1993 RITA Award#Contemporary Romance
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In this blog post I simply wanted to share an introduction video lecture of my LifeBuff Pro course. The subject of Basic Psychology is so very important to understand at least 90% of your emotional struggles! I should also mention that in general there are already 59 lectures available! Since LifeBuff Pro is currently still under development by aiming for the total of 100 lectures it is also 50% OFF! Hi, my name is Alex. ... please be aware that music production is just 1/3 of the game. The other 2/3 are self promotion and a sane mind! Can't hurt to try, no? Don't worry, you can unsubscribe anytime! Where would you like us to send your freebies ? Don't worry, we don't like spam either, and you email adress will be safe.
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\section{Introduction} The interaction of \emph{recognition} and \emph{quantification} involves the analysis of spans of topological spaces of the form \begin{equation} \begin{aligned} \begin{tikzcd}[node distance = 20mm] \node (S) at (0,0) {\beta S}; \node (T) [below left of = S] {\beta T}; \node (X) [below right of = S] {X}; \draw[->] (S) to node[right, yshift = 2mm, xshift = -1mm] {g} (X); \draw[->] (S) to node[left, yshift = 2mm, xshift = 2mm] {\beta f} (T); \end{tikzcd} \end{aligned} \label{eq:7} \end{equation} where $T$ is the set of structures for the logic at hand, $S$ is the set of all models over these structure (i.e. a set of free variables is fixed and elements of $S$ consist of a structure from $T$ equipped with an interpretation of the free variables). In the case of the existential quantifier, the map $f: S \to T$ is just the map taking a model to the underlying structure, and $\beta$ is the Stone-\v Cech compactification (or, equivalently, $\beta f$ is the Stone dual of the Boolean algebra homomorphism $f^{-1}: \mathcal P} \newcommand{\cB}{\mathcal B(T) \to \mathcal P} \newcommand{\cB}{\mathcal B(S)$). The idea of recognition is that, instead of specifying a subalgebra of $\mathcal P} \newcommand{\cB}{\mathcal B(S)$ corresponding to the formulas with free variables we want to study, $g$ is a continuous map to a Boolean space dual to the subalgebra. If $\cB$ is the Boolean algebra of clopens of $X$, then the Boolean algebra $g^{-1}[\cB] = \{g^{-1}(U) \mid U \in \cB\}$ consists of the model classes of the formulas with free variables that we want to study. Also, for $L = g^{-1}(U) \cap S$, $L_\exists = f[L]$, is the set of structures satisfying the corresponding (existentially) quantified formula, and we are interested in building a recognizer for the Boolean subalgebra of $\mathcal P} \newcommand{\cB}{\mathcal B(T)$ generated by the sets $f[g^{-1}(U)]$, for $U \in \cB$. In~\cite{GehrkePetrisanReggio16}, it is shown that the desired recognizer is of the form \begin{equation} \label{eq:6} h: \beta T \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X) \end{equation} and is given dually by $\Diamond V \mapsto f[g^{-1}(V)]$, invoking Vietoris for Boolean spaces as the dual of the monad for modal logic. A categorical approach for a similar construction, also having in mind applications in formal language theory, may be found in~\cite{ChenUrbat16}. A much simpler analysis would ensue if we can obtain $h$ by lifting the pair~\eqref{eq:7} as follows \begin{equation} \label{eq:8} \beta T \hookrightarrow {\mathcal V}} \newcommand{\im}{{\sf Im}(\beta T) \xrightarrow{(\beta f)^*} {\mathcal V}} \newcommand{\im}{{\sf Im} (\beta S) \xrightarrow{{\mathcal V}} \newcommand{\im}{{\sf Im}(g)} {\mathcal V}} \newcommand{\im}{{\sf Im}(X). \end{equation} Here ${\mathcal V}} \newcommand{\im}{{\sf Im}(g)$ is the \emph{forward image} under $g$, while $(\beta f)^*$ is the \emph{inverse image} under~$f$. It is well known that forward image is always continuous and ${\mathcal V}} \newcommand{\im}{{\sf Im}$ is indeed viewed as a covariant endofunctor with ${\mathcal V}} \newcommand{\im}{{\sf Im}(g)$ given by forward image. On the other hand, inverse image under a continuous map is not in general continuous on the Vietoris spaces. Our short and simple analysis here consists in observing that for certain continuous maps (those whose duals have lower adjoints, and for the $\beta f$'s in particular) inverse image \emph{does} give a continuous map. Indeed, we show that all continuous maps of the form~\eqref{eq:6} come about from a span composing inverse and direct image. In the setting of recognition, we have to deal with a superposition of semigroup structure and topology. We show that a similar phenomenon is at play for semigroups and that these combine correctly to give a simple description of the topo-algebraic recognizing maps of the form~\eqref{eq:6} by means of a composition as in~\eqref{eq:8}. The work in~\cite{GehrkePetrisanReggio16} and further papers in this direction for quantification in \emph{first-order logic}~\cite{GehrkePetrisanReggio17, BorlidoCzarnetzkiGehrkeKrebs17} deal with a second complication which is much deeper, and which stems from the combined fact that the set $S$ of models does not carry an appropriate semigroup structure and that $f$ is not a homomorphism of semigroups. We also mention that, although they do not refer to logic on words, the corresponding treatment for the fragments of logic defining \emph{regular languages} may be inferred from~\cite{AlmeidaWeil1995}. Here, we treat a much simpler case, namely \emph{monadic second-order logic}, for which $T$ and $S$ are semigroups and $f$ is a length-preserving semigroup homomorphism (i.e., one of those for which inverse images lift homomorphically to the powerset semigroups). This note is organized as follows. In Section~\ref{sec:prelim}, we set up the notation and define the main concepts used later. In Section~\ref{sec:powers} we study power constructions both for compact Hausdorff and Boolean spaces and for semigroups. Finally, in Section~\ref{sec:power-bis} we show that forward images under length-preserving homomorphisms of languages recognized by a Boolean space with an internal semigroup are precisely those that are recognized by its Vietoris. This generalizes a known result for regular languages~\cite{Reutenauer79,Straubing79}. Interpreting the result in the context of logic on words leads to a topo-algebraic description of second-order existential quantification. \section{Preliminaries}\label{sec:prelim} The reader is assumed to have some acquaintance with Stone duality and with semigroups and the theory of formal languages, including logic on words. Nevertheless, we briefly recall the main concepts and results that we will need. For further reading on topology, we refer to~\cite{Willard70}, on Stone duality to~\cite{DaveyPriestley02}, on semigroup and formal language theory to~\cite{Almeida94}, and on logic on words to~\cite{Straubing94}. \paragraph{Stone duality.} \emph{Stone duality} establishes a correspondence between Boolean algebras and Boolean spaces.\footnote{A Boolean space is a compact Hausdorff topological space with a basis of clopen sets.} On the level of objects it goes as follows. Given a Boolean space~$X$, the set of its clopen (i.e., both open and closed) subsets, ${\rm \it{Clop}}(X)$, forms a Boolean algebra (actually, this is true for every topological space). Conversely, if~$\cB$ is a Boolean algebra, then the set of ultrafilters\footnote{An ultrafilter is a proper maximal nonempty upset $x \subseteq \cB$ closed under binary meets.} of $\cB$, $X_\cB$, is a Boolean space when equipped with the topology generated by the sets of the form \[\widehat {b} = \{x \in X_\cB \mid b \in x\},\] for $b \in \cB$. We have isomorphisms $\cB \cong {\rm \it{Clop}}(X_\cB)$ and $X \cong X_{{\rm \it{Clop}}(X)}$. For the morphisms, we have the following correspondence. If $f: X \to Y$ is a continuous function between Boolean spaces, then taking preimages of clopens defines a homomorphism $f^{-1}:{\rm \it{Clop}}(Y) \to {\rm \it{Clop}}(X)$ of Boolean algebras, and if $\alpha: \cB \to \mathcal C$ is a homomorphism of Boolean algebras, for every ultrafilter $y \in X_\mathcal C$, the set $\{b \in \cB \mid \alpha (b) \in y\}$ is an ultrafilter of $\cB$, and this correspondence yields a continuous function $X_\mathcal C \to X_\cB$. Dual spaces of powerset Boolean algebras will play a role in the sequel. For any set~$S$, the Stone dual of $\mathcal P} \newcommand{\cB}{\mathcal B(S)$ is the \emph{Stone-\v Cech compactification of $S$}, which we will denote by~$\beta S$. The set~$S$ embeds densely in~$\beta S$ via the map $s \mapsto \{x \in \beta S \mid s \in x\}$. We say that a subset $L \subseteq S$ is \emph{set-theoretically recognized} by a Boolean space~$X$ provided there is a continuous map $f: \beta S \to X$ and a clopen subset $C \subseteq X$ such that $L = f^{-1}(C) \cap S$. Any function of sets $f: S \to T$ yields a homomorphism of Boolean algebras $f^{-1}: \mathcal P} \newcommand{\cB}{\mathcal B(T) \to \mathcal P} \newcommand{\cB}{\mathcal B(S)$ and the corresponding dual map will be denoted $\beta f: \beta S \to \beta T$. \paragraph{Formal languages and recognition.} An \emph{alphabet} is a finite set of symbols $A$, also called \emph{letters}, a (non-empty) \emph{word} over $A$ is an element of the free semigroup $A^+$ and a \emph{(formal) language} $L$ is a set of words over some alphabet. The Boolean algebra of all languages over $A$, $\mathcal P} \newcommand{\cB}{\mathcal B(A^+)$, is naturally equipped with a biaction of $A^+$ given by \begin{equation} u^{-1}Lv^{-1} = \{w \in A^+\mid uwv \in L\},\label{eq:1} \end{equation} for every $u,v \in A^+$ and $L \subseteq A^+$. Languages of the form~\eqref{eq:1} are called \emph{quotients of $L$}. We say that a language $L$ is \emph{recognized} by a semigroup $S$ provided there is a homomorphism $f:A^+ \to S$ and a subset $Q \subseteq S$ satisfying $L = f^{-1}(Q)$, or equivalently, provided $L = f^{-1}(f[L])$. Notice that the set of all languages recognized by a given homomorphism forms a Boolean algebra closed under quotients. Given a homomorphism $h: B^+ \to A^+$ between finitely generated free semigroups and a language $L \subseteq A^+$ recognized by~$S$, we have that $h^{-1}(L)$ is also recognized by~$S$. The homomorphism~$h$ is said to be \emph{length-preserving}, or an \emph{lp-morphism}, if it maps letters to letters. Languages recognized by finite semigroups are called \emph{regular}. To handle non-regular languages topo-algebraically, one needs to consider richer structures, which we introduce below. Notice that an attempt to make the notion of recognition as uniform as possible was first made in~\cite{Bojanczyk15} and followed up by~\cite{ChenAdamekMiliusUrbat2016}. But in both cases the main concern is to capture several computational models within the same framework, and not going beyond \emph{regularity}. \paragraph{Boolean spaces with internal semigroups.} This concept, for monoids, was introduced in~\cite{GehrkePetrisanReggio16}, based on ideas of~\cite{GehrkeGrigorieffPin10}. We give a short introduction. For more details see~\cite{GehrkePetrisanReggio16} or~\cite{GehrkePetrisanReggio17-arxiv}. If $\cB \subseteq \mathcal P} \newcommand{\cB}{\mathcal B(A^+)$ is a Boolean algebra of languages, then dually, we have a continuous quotient $q: \beta (A^+) \twoheadrightarrow X_\cB$. If the Boolean algebra~$\cB$ is closed under quotients, then~$q[A^+]$ has a natural semigroup structure, which is inherited from the duals of the homomorphisms $u^{-1}(\_)v^{-1}: \cB \to \cB$. Moreover, the restriction of~$q$ to~$A^+$ induces a homomorphism $A^+ \twoheadrightarrow q[A^+]$ onto a dense subset of~$X_\cB$. The pair $(q[A^+], X_\cB)$ encodes the essential information about the Boolean algebra closed under quotients~$\cB$. This is the reasoning motivating the definition of a \emph{Boolean space with an internal semigroup}. A \emph{Boolean space with an internal semigroup}, or \emph{\bis{}} for short, is a pair $(S,X)$ where $S$ is a semigroup densely contained in a Boolean space~$X$, and such that the natural actions of $S$ on itself extend to continuous endomorphisms of~$X$, that is, for each $s \in S$, there are continuous functions $\lambda_s, \rho_s: X \to X$ such that the following diagrams commute: \begin{center} \begin{tikzpicture}[node distance = 15mm] \node (S) at (0,0) {$S$}; \node[below of = S] (SS) {$S$}; \node[right of = S] (X) {$X$}; \node[below of = X] (XX) {$X$}; % \node (Sr) at (6,0) {$S$}; \node[below of = Sr] (SSr) {$S$}; \node[right of = Sr] (Xr) {$X$}; \node[below of = Xr] (XXr) {$X$}; % \draw[->] (S) to node[left] {\footnotesize $s\cdot (\_)$} (SS); \draw[>->] (S) to (X); \draw[>->] (SS) to (XX); \draw[->] (X) to node[right] {\footnotesize $\lambda_s$} (XX); % \draw[->] (Sr) to node[left] {\footnotesize $(\_)\cdot s$} (SSr); \draw[>->] (Sr) to (Xr); \draw[>->] (SSr) to (XXr); \draw[->] (Xr) to node[right] {\footnotesize $\rho_s$} (XXr); \end{tikzpicture} \end{center} The prime example of a \bis{} is the pair $(A^+, \beta(A^+))$, which corresponds to taking $\cB = \mathcal P} \newcommand{\cB}{\mathcal B(A^+)$. A morphism of {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{}, $f: (S, X) \to (T, Y)$, is a continuous map $f:X \to Y$ so that $f$ restricts to a semigroup homomorphism $f{\upharpoonright} : S \to T$. A language $L \subseteq A^+$ is recognized by the \bis{} $(S, X)$ if there exists a morphism $f: (A^+, \beta(A^+)) \to (S, X)$ and a clopen subset $C \subseteq X$ such that $L = f^{-1}(C) \cap A^+$. Notice that each semigroup homomorphism $A^+ \to S$ completely determines a morphism of {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{} $(A^+, \beta(A^+)) \to (S, X)$. It is not hard to verify that the set of all languages recognized by a \bis{} via a fixed homomorphism is a Boolean algebra closed under quotients. \paragraph{Logic on words.} As the name suggests, \emph{logic on words} is meant to express properties of words. We consider two kinds of variables: first-order and second-order variables. Intuitively, first-order variables provide information about positions in a word, while the second-order variables stand for sets of positions. First-order variables are denoted by $x, x_1, x_2, \ldots$, and the second-order ones by $X, X_1, X_2, \ldots$. There are the following three types of atomic formulas: \begin{itemize} \item if $R \subseteq \mathbb N^k$, then $R(x_1, \ldots, x_k)$ is a (uniform) \emph{$k$-ary numerical predicate} (expressing that ``the tuple of positions $(x_1, \ldots, x_k)$ belongs to $R$''); \item if $a \in A$, then ${\bf a}(x)$ is a \emph{letter predicate} (expressing that in the ``position $x$ there is an $a$''); \item if $x$ is a first-order variable and $X$ is a second-order variable, then $X(x)$ is an atomic formula (expressing that ``$x$ belongs to $X$''). \end{itemize} Then, Boolean combinations of formulas are formulas and if $\phi$ is a formula, then both $\exists x \ \phi$ and $\exists X \ \phi$ are formulas. It is well known that every monadic second-order sentence is equivalent, over words, to a formula of the form $\exists X_1 \cdots \exists X_N \ \phi(X_1, \ldots, X_N)$, for some first-order formula $\phi$ with free variables in $\{X_1, \ldots, X_N\}$. To interpret formulas with $N$ second-order free variables, one usually considers words over the extended alphabet $A \times 2^N$. For instance, if $N = 1$, then the word $(a,1)(b,0)(b,1)(b,0)(a,1) \in (A \times 2)^+$ encodes the word $w = abbba$ with the only second-order variable interpreted in the set of odd positions of~$w$. Thus, every formula $\phi = \phi (X_1, \ldots, X_N)$ with free variables in $\{X_1, \ldots, X_N\}$ defines a language $L_{\phi(X_1, \ldots, X_N)} \subseteq (A \times 2^N)^+$, and the language over $A^+$ definable by $\exists X_1 \cdots \exists X_N \ \phi(X_1, \ldots, X_N)$ consists of all the words $w$ for which there is an interpretation of the free variables satisfying $\phi$. In other words, $T = A^+$ is the set of all structures for logic on words in~$A$, $S = (A \times 2^N)^+$ is the set of all MSO models in $N$ free variables on words in~$A$, and the lp-morphism $\pi_N:(A \times 2^N)^+ \twoheadrightarrow A^+$, given by the projection $A\times 2^N \twoheadrightarrow A$, gives rise to existential quantification in the sense that $L_{\exists X_1 \cdots \exists X_N \ \phi(X_1, \ldots, X_N)} = \pi_N[L_{\phi (X_1, \ldots, X_N)}]$. More generally, given a language $L \subseteq (A \times 2^N)^+$, we denote $L_{\exists_N} = \pi_N[L]$. \section{Some power constructions}\label{sec:powers} \paragraph{Compact Hausdorff and Boolean spaces.} The power of a compact Hausdorff spaces is the so-called \emph{Vietoris space}. \emph{Vietoris} is a covariant endofunctor on compact Hausdorff spaces which restricts to the category of Boolean spaces. At the level of objects it assigns to a space~$X$ the set of all its closed subsets, denoted ${\mathcal V}} \newcommand{\im}{{\sf Im}(X)$ and called the \emph{Vietoris space of $X$}, equipped with the topology generated by the sets of the form \[\Diamond U = \{C \in {\mathcal V}} \newcommand{\im}{{\sf Im} (X) \mid C \cap U \neq \emptyset\} \qquad \text{ and }\qquad \Box U = \{C \in {\mathcal V}} \newcommand{\im}{{\sf Im}(X) \mid C \subseteq U\},\] where $U \subseteq X$ ranges over all open subsets of~$X$. In the case where $X$ is a Boolean space, taking $\Diamond U$ and $\Box U$ for $U$ clopen gives a subbasis of clopen subsets for~${\mathcal V}} \newcommand{\im}{{\sf Im}(X)$. For more details see~\cite{Michael51}. Note that, since $X$ is Hausdorff, each singleton is closed and thus the map $i_X\colon X\to{\mathcal V}} \newcommand{\im}{{\sf Im}(X)$ sending each $x\in X$ to $\{x\}$ is well defined. Further note that, for any open $U\subseteq X$, we have \[ \Diamond U\cap \im(i_X)=i_X[U] \qquad \text{and} \qquad \Box U \cap \im(i_X)=i_X[U] \] so that $X$ is homeomorphically embedded in ${\mathcal V}} \newcommand{\im}{{\sf Im}(X)$ via the map $i_X$. When the space $X$ is clear from the context, we denote this embedding simply by~$i$. On morphisms, the Vietoris functor acts as follows: if $g: Z \to X$ is a continuous function, then so is \[{\mathcal V}} \newcommand{\im}{{\sf Im}(g): {\mathcal V}} \newcommand{\im}{{\sf Im}(Z) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X), \qquad C \mapsto g[C].\] Indeed, routine computations show that, for an open subset $U \subseteq X$, we have \begin{equation} {\mathcal V}} \newcommand{\im}{{\sf Im}(g)^{-1}(\Diamond U) = \Diamond(g^{-1}(U)) \quad \text{ and }\quad {\mathcal V}} \newcommand{\im}{{\sf Im}(g)^{-1}(\Box U) = \Box(g^{-1}(U)).\label{eq:2} \end{equation} On the other hand, given a continuous map $f: Z \to Y$, taking preimages also defines a function between the corresponding Vietoris spaces, but in a contravariant way. A natural question is then under which conditions the function $f^* := f^{-1}: {\mathcal V}} \newcommand{\im}{{\sf Im}(Y) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(Z)$ is continuous. \begin{proposition}\label{p:3} Let $f: Z \to Y$ be a continuous function between compact Hausdorff spaces. Then, the following are equivalent: \begin{enumerate}[label = (\alph*)] \item\label{item:1} $f^*: {\mathcal V}} \newcommand{\im}{{\sf Im}(Y) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(Z)$ is continuous, \item\label{item:2} $f^* \circ i: Y \to {\mathcal V}} \newcommand{\im}{{\sf Im}(Z)$ is continuous, \item\label{item:3} $f$ is open. \end{enumerate} \end{proposition} \begin{proof} Since $Y$ is homeomorphically embedded in $Y$ via the map $i$, it is clear that $\ref{item:1}$ implies $\ref{item:2}$. % The remainder of the proposition is essentially a consequence of the fact that, for any set map $f$, the forward image under $f$ is lower adjoint to the inverse image under $f$. That is, \[ \forall\ S\subseteq Y,\ T\subseteq Z\qquad \qquad(\ f[T]\subseteq S \ \iff\ T\subseteq f^{-1}(S)\ ). \] Using this, we see first of all that, for compact Hausdorff spaces, the map $f^*$ is always continuous with respect to the `box-part' of the topology. That is, for any open $U\subseteq Z$ and closed $K\subseteq Y$, we have \begin{align*} K\in (f^*)^{-1}(\Box U) \iff f^{-1}(K)\subseteq U &\iff U^c\subseteq(f^{-1}(K))^c=f^{-1}(K^c)\\ &\iff f[U^c]\subseteq K^c\iff K\subseteq(f[U^c])^c. \end{align*} Now, since $f$ is continuous, $Y$ is compact, and $Z$ is Hausdorff, it follows that $f$ is a closed mapping and thus $(f[U^c])^c$ is open. That is, we have shown that \[ (f^*)^{-1}(\Box U) = \Box (f[U^c])^c. \] Similarly, we have \begin{align*} K\in (f^*)^{-1}(\Diamond U) \iff f^{-1}(K)\cap U\neq\emptyset &\iff U\not\subseteq(f^{-1}(K))^c=f^{-1}(K^c)\\ &\iff f[U]\not\subseteq K^c\iff K\cap f[U]\neq\emptyset. \end{align*} Now, if $f$ is open, then the above calculation shows $(f^*)^{-1}(\Diamond U) = \Diamond f[U]$ and thus that $\ref{item:1}$ and $\ref{item:2}$ hold. Conversely, if $f^*\circ i$ is continuous, then, for any open $U\subseteq Z$, the set \begin{equation} (f^*\circ i)^{-1}(\Diamond U)=\{y\in Y\mid \{y\}\cap f[U]\neq\emptyset\}=f[U]\label{eq:4} \end{equation} must be open. \end{proof} In particular we have the following: \begin{corollary}\label{c:2} Let $X$, $Y$ and $Z$ be compact Hausdorff spaces and $f: Z \to Y$ and $g: Z \to X$ be continuous functions. If $f$ is an open map, then \[h = {\mathcal V}} \newcommand{\im}{{\sf Im}(g) \circ f^* \circ i: Y \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X), \qquad y \mapsto h(y) = g[f^{-1}(\{y\})]\] is continuous. Moreover, for every open subset $U \subseteq X$, the equality $h^{-1}(\Diamond U) = f[g^{-1}(U)]$ holds. \end{corollary} \begin{proof} The fact that $h$ is continuous follows immediately from Proposition~\ref{p:3}. Moreover, for an open subset $U \subseteq X$, we have: \[h^{-1}(\Diamond U) = (f^*\circ i)^{-1}({\mathcal V}} \newcommand{\im}{{\sf Im}(g)^{-1}(\Diamond U)) \just = {\eqref{eq:2}} (f^* \circ i)^{-1}(\Diamond g^{-1}(U)) \just = {\eqref{eq:4}} f[g^{-1}(U)]. \popQED\] \end{proof} In the next proposition we show that every continuous map $Y \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X)$ arises from a composition as in Corollary~\ref{c:2}. \begin{proposition}\label{p:5} Let $X$ and $Y$ be compact Hausdorff spaces and $h: Y \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X)$ be a continuous function. Then, the subspace \[Z = \{(y, x) \in Y \times X \mid x \in h(y)\}\] of $Y \times X$ is compact and Hausdorff. In particular, $h$ is of the form ${\mathcal V}} \newcommand{\im}{{\sf Im}(g)\circ f^* \circ i$, where $f$ and $g$ are the restrictions to $Z$ of the projections to~$Y$ and~$X$, respectively. \end{proposition} \begin{proof} The space $Z$ is compact Hausdorff if $Z$ is a closed subspace of~$Y \times X$. % We show that $Z^c$ is open. Let $(y,x) \notin Z$. Then, $x \notin h(y)$ and since $h(y) \subseteq X$ is closed and $X$ is compact and Hausdorff (and thus, regular), there exist disjoint open subsets $U, V \subseteq X$ so that $x \in U$ and $h(y) \subseteq V$. Since $h$ is continuous, it follows that $h^{-1}(\Box V) \times U$ is an open neighborhood of $(y,x)$ contained in $Z^c$. \end{proof} We finish this section with a characterization of the open maps between Boolean spaces. Recall that, in general, a \emph{lower adjoint} of a map $\alpha: P \to Q$ of posets is a map $\alpha_*: Q \to P$ satisfying \[ \forall\ p \in P,\ q \in Q \qquad \qquad(\ \alpha_*(q)\le p \ \iff\ q \le \alpha(p)\ ). \] \begin{proposition} Let $\cB$ and $\mathcal C$ be Boolean algebras, with duals $Y$ and $Z$, respectively. Then, a continuous function $f: Z \to Y$ is open if and only if the dual map $\alpha: \cB \to \mathcal C$ has a lower adjoint. \end{proposition} \begin{proof} First observe that, since taking forward images preserves arbitrary unions, $f$ is open if and only if $f[\,\widehat c\, ]$ is open for every $c \in \mathcal C$. Since $f[\, \widehat c\ ]$ is closed, we have \[ f[\, \widehat c \ ] = \bigcap \{\widehat b \mid b \in \cB, \ f[\,\widehat c \ ] \subseteq \widehat b\}.\] Thus, $f$ is open exactly when the set \begin{equation} \{\widehat b \mid b \in \cB, \ f[\,\widehat c \ ] \subseteq \widehat b\}\label{eq:5} \end{equation} has a minimum for inclusion. On the other hand, the fact that $\alpha$ and $f$ are dual to each other, translates to the fact that, for every $b \in \cB$ and $c \in \mathcal C$, we have \[ f[\, {\widehat c}\ ] \subseteq \widehat b \iff \widehat c \subseteq f^{-1}(\widehat b) = \widehat{\alpha(b)} \iff c \le \alpha(b).\] Therefore, \eqref{eq:5} has a minimum if and only if $\{b \in \cB \mid c \le \alpha(b)\}$ does. But this amounts to saying that $\alpha$ admits a lower adjoint. \end{proof} In particular, since every function $f: S \to T$ between sets $S$ and $T$ induces a complete homomorphism $f^{-1}: \mathcal P} \newcommand{\cB}{\mathcal B(T) \to \mathcal P} \newcommand{\cB}{\mathcal B(S)$ between complete Boolean algebras, thus having a lower adjoint, we have the following: \begin{corollary}\label{c:1} For every map of sets $f: S \to T$, the map $\beta f: \beta S \to \beta T$ is open. \end{corollary} \begin{remark}\label{r:2} We remark that, so far, we proved that, given a set map $f: S \to T$ and a continuous function $g: \beta T \to X$ the Boolean algebra generated by the subsets of the form $f[g^{-1}(U)]$, for $U \in {\rm \it{Clop}}(X)$, is precisely $h^{-1}[{\rm \it{Clop}}({\mathcal V}} \newcommand{\im}{{\sf Im}(X))] = \{h^{-1}(V) \mid V \in {\rm \it{Clop}}({\mathcal V}} \newcommand{\im}{{\sf Im}(X))\}$, where $h = {\mathcal V}} \newcommand{\im}{{\sf Im}(g) \circ f^* \circ i$. \end{remark} \paragraph{Semigroups.} We start by recalling that, given a semigroup $S$, the set $\mathcal P} \newcommand{\cB}{\mathcal B(S)$ of its subsets is equipped with a semigroup structure given by pointwise multiplication: $$Q_1 \cdot Q_2 = \{s_1s_2 \mid s_1\in Q_1, s_2 \in Q_2\},$$ for every subsets $Q_1,Q_2 \subseteq S$. In particular, there is an embedding of semigroups $i_S: S \hookrightarrow \mathcal P} \newcommand{\cB}{\mathcal B(S)$ given by $i_S(s) = \{s\}$. When $S$ is clear from the context, we just write~$i$. Notice that $\mathcal P} \newcommand{\cB}{\mathcal B(S)$ also admits a monoid structure, with the neutral element being the empty set, but we are only concerned with the semigroup structure of~$\mathcal P} \newcommand{\cB}{\mathcal B(S)$. Taking powers defines an endofunctor on semigroups. Indeed, for a homomorphism $g: S \to T$, taking forward images defines a homomorphism \[\mathcal P} \newcommand{\cB}{\mathcal B(g): \mathcal P} \newcommand{\cB}{\mathcal B(S) \to \mathcal P} \newcommand{\cB}{\mathcal B(T), \qquad Q \mapsto g[Q].\] Of course, the set $\mathcal P} \newcommand{\cB}{\mathcal B_{fin}(S)$ consisting of the finite subsets of $S$ forms a subsemigroup of $\mathcal P} \newcommand{\cB}{\mathcal B(S)$, and for a homomorphism $g: S \to T$, $\mathcal P} \newcommand{\cB}{\mathcal B(g)$ restricts to a homomorphism $\mathcal P} \newcommand{\cB}{\mathcal B_{fin}(g): \mathcal P} \newcommand{\cB}{\mathcal B_{fin}(S) \to \mathcal P} \newcommand{\cB}{\mathcal B_{fin}(T)$. This observation will be useful in Section~\ref{sec:powers}. On the other hand, if $f: S \to T$ is a semigroup homomorphism, then taking preimages defines a map between the corresponding powersets, which in general is not a homomorphism. However, we do have the next result. \begin{lemma}\label{l:1} Let $f: B^+ \to A^+$ be a homomorphism between free semigroups. Then, the following are equivalent: \begin{enumerate}[label = (\alph*)] \item\label{item:4} $f^{*}: \mathcal P} \newcommand{\cB}{\mathcal B(A^+) \to \mathcal P} \newcommand{\cB}{\mathcal B(B^+)$ is a homomorphism, \item\label{item:5} $f^*\circ i: A^+ \to \mathcal P} \newcommand{\cB}{\mathcal B(B^+)$ is a homomorphism, \item\label{item:6} $f$ is an lp-morphism. \end{enumerate} \end{lemma} \begin{proof} The equivalence between~\ref{item:4} and~\ref{item:5} is trivial. Suppose that $f$ is length-preserving. Then, given $w_1, w_2 \in A^+$ and $u \in B^+$, we have that $u \in f^*\circ i(w_1w_2)$ if and only if $f(u) = w_1w_2$. Since $f$ is length-preserving, this happens if and only if $u$ admits a factorization $u = u_1u_2$ satisfying $f(u_1) = w_1$ and $f(u_2) = w_2$, that is, $u \in f^*\circ i(w_1)\cdot f^*\circ i(w_2)$. This proves that $f^*\circ i$ is a homomorphism. Conversely, if $f$ is not length-preserving, then there exists a letter $b \in B$ so that $f(b)$ may be written as $aw$ for some $a \in A$ and $w \in A^+$. Then, $b$ belongs to $f^*\circ i(aw)$ but not to $f^*\circ i(a)\cdot f^*\circ i(w)$ and so, $f^*\circ i$ is not a homomorphism. \end{proof} \begin{remark}\label{r:1} Notice that for an lp-morphism $f: B^+ \to A^+$ and a word $w \in A^+$, the set $f^{-1}(w)$ is finite. Thus, by Lemma~\ref{l:1}, every such~$f$ defines a homomorphism of semigroups $f^*: A^+ \to \mathcal P} \newcommand{\cB}{\mathcal B_{fin}(B^+)$. \end{remark} The following is a particular case of a well known result in semigroup theory (see e.g.~\cite[Chapter~XVI, Proposition~1.1]{Pin-notes}). \begin{proposition}\label{p:4} Let $h: T \to \mathcal P} \newcommand{\cB}{\mathcal B(S)$ be a homomorphism. Then, the set \[R = \{(t, s) \mid t \in T, \ s \in h(t)\}\] is a subsemigroup of $T \times S$. In particular, $h$ is of the form $\mathcal P} \newcommand{\cB}{\mathcal B(g) \circ f^* \circ i$, where $f$ and $g$ are the restrictions to~$R$ of the projections to~$S$ and~$T$, respectively. \end{proposition} \section{The power construction for {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{} and MSO}\label{sec:power-bis} \paragraph{The power construction for {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{}.} Combining the power constructions of Section~\ref{sec:powers} provides a power construction that applies to {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{}: \begin{definition}[{\cite[Theorem III.1]{GehrkePetrisanReggio17}}] Let $(S, X)$ be a \bis{}. We define the \emph{Vietoris of $(S,X)$} to be the~\bis{} \[{\mathcal V}} \newcommand{\im}{{\sf Im}(S,X) = (\mathcal P} \newcommand{\cB}{\mathcal B_{fin}(S), {\mathcal V}} \newcommand{\im}{{\sf Im}(X))\] equipped with the actions \[\lambda_Q: {\mathcal V}} \newcommand{\im}{{\sf Im}(X) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X), \quad C \mapsto \bigcup_{s \in Q} \lambda_s[C] \qquad\text{ and }\qquad \rho_Q: {\mathcal V}} \newcommand{\im}{{\sf Im}(X) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(X), \quad C \mapsto \bigcup_{s \in Q} \rho_s[C],\] for each $Q \in \mathcal P} \newcommand{\cB}{\mathcal B_{fin}(S)$. \end{definition} We remark that, although the fact that ${\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$ is a \bis{} only appears explicitly in~\cite{GehrkePetrisanReggio17}, this is implicitly present already in~\cite{GehrkePetrisanReggio16}. We will say that a morphism $h: (B^+, \beta(B^+)) \to (A^+, \beta(A^+))$ of {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{} is \emph{length-preserving} provided its restriction to $B^+$ is length-preserving. Using Corollary~\ref{c:2} and Lemma~\ref{l:1}, and taking into account Remark~\ref{r:1}, we obtain: \begin{proposition} Let $A$ and $B$ be alphabets and $(S, X)$ a \bis{}. Then, for every span \begin{equation*} \begin{aligned} \begin{tikzcd}[node distance = 25mm] \node (S) at (0,0) {(B^+, \beta (B^+))}; \node (T) [below left of = S] {(A^+, \beta(A^+))}; \node (X) [below right of = S] {(S, X)}; \draw[->] (S) to node[right, yshift = 2mm, xshift = -1mm] {g} (X); \draw[->] (S) to node[left, yshift = 2mm, xshift = 2mm] {f} (T); \end{tikzcd} \end{aligned} \end{equation*} with $f$ length preserving, the map $h = {\mathcal V}} \newcommand{\im}{{\sf Im}(g) \circ f^* \circ i$ is a morphism of {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{}. \end{proposition} \begin{proposition}\label{p:9} Let $(S, X)$ and $(T, Y)$ be {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{} and $h: (T, Y) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$ a morphism. Then, there is a \bis{} $(R, Z)$ and morphisms $f:(R, Z) \to (T, Y)$ and $g: (R, Z) \to (S, X)$ so that $h = {\mathcal V}} \newcommand{\im}{{\sf Im}(g) \circ f^* \circ i$. \end{proposition} \begin{proof} We take $Z = \{(y,x) \mid y \in Y, \ x \in h(y)\}$ and $R = \{(t,s) \mid t \in T, \ s \in h(s)\}$. By Propositions~\ref{p:5} and~\ref{p:4} we already know that $Z$ and $R$ are, respectively, a Boolean space and a semigroup that do the job. Thus, it remains to show that $(R, Z)$ is a \bis{}. Since the pair $(T \times S, Y \times X)$ has a \bis{} structure, we only need to prove that $R$ is dense in $Z$. Let $V\subseteq Y$ and $U \subseteq X$ be open subsets and $(y, x) \in (V \times U) \cap Z$. We need to show that $(V \times U) \cap R$ is nonempty. Since $h$ is continuous, $h^{-1}(\Diamond U) \cap V$ is an open subset of~$Y$, and it is nonempty as it contains~$(y,x)$. Since $T$ is dense in~$Y$, there exists an element $t \in h^{-1}(\Diamond U) \cap V \cap T$. In particular, $h(t) \cap U \neq \emptyset$. Since $h$ restricts to a semigroup homomorphism $T \to \mathcal P} \newcommand{\cB}{\mathcal B_{fin}(S)$, this yields the existence of $s \in h(t) \cap U \cap S$ as required. \end{proof} As a consequence we obtain the desired result on recognition. \begin{corollary}\label{c:3} Let $(S, X)$ be a \bis{}. Then, a language $L \subseteq A^+$ is recognized by ${\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$ if and only if it is a Boolean combination of forward images under lp-morphisms of languages recognized by~$(S, X)$. \end{corollary} \begin{proof} The backwards implication is a trivial consequence of Remark~\ref{r:2} and Proposition~\ref{p:9}. % Conversely, let $h: (A^+, \beta(A^+)) \to {\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$ be a morphism recognizing $L \subseteq A^+$. Again by Proposition~\ref{p:9}, there exists a \bis{} $(R,Z)$ and morphisms $f:(R, Z) \to (A^+, \beta(A^+))$ and $g: (R, Z) \to (S, X)$ so that $h = {\mathcal V}} \newcommand{\im}{{\sf Im}(g) \circ f^* \circ i$. The only thing to notice is that $R = \{(u, s) \mid u \in A^+, s \in h(u)\}$ is the subsemigroup of $A^+ \times S$ generated by the finite alphabet $B = \{(a, s) \mid a \in A, \ s \in h(a)\}$. Therefore, we have a unique morphism of {\sf BiS}s} \newcommand{\bis}{{\sf BiS}{} $\pi: (B^+, \beta(B^+)) \to (R, Z)$ mapping $b \in B$ to $b \in R$, and this morphism is such that $f \circ \pi$ is length-preserving. The intended conclusion follows then from Remark~\ref{r:2}. \end{proof} We remark that this result takes care of the first stage of set-theoretic recognition in the first-order setting~\cite{GehrkePetrisanReggio16, BorlidoCzarnetzkiGehrkeKrebs17}. The complication in the first-order setting stems from the fact that the first-order models do not form a semigroup. Here we treat the considerable easier case of monadic second-order quantifiers. \paragraph{Monadic second-order existential quantification.} We address the following questions: Given a \bis{} $(S, X)$, which \bis{} recognizes the Boolean algebra generated by the languages of the form $L_{\exists N}$ where $L$ is recognized by $(S, X)$? Does it recognize much more? Unlike in the first-order case where an iterative construction is absolutely needed, we will see that, for second-order quantification, taking once the power of $(S, X)$ is enough to recognize every language $L_{\exists N}$ for every $N \in \mathbb N$, where $L$ is recognized by~$(S, X)$. In fact, since the projection $\pi_N: (A\times 2^N)^+ \twoheadrightarrow A^+$ modeling the quantifier $\exists_N$ is length-preserving, this essentially follows from the results above. As already mentioned, the corresponding problem for first-order quantifiers was considered in~\cite{GehrkePetrisanReggio16, BorlidoCzarnetzkiGehrkeKrebs17} and it is much more delicate, as the universe of models of formulas with free first-order variables does not admit a semigroup structure. \begin{proposition}\label{p:8} Let $(S, X)$ be a \bis{}. If $(S, X)$ recognizes the language $L \subseteq (A \times 2^N)^+$, then ${\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$ recognizes the language $L_{\exists_N} \subseteq A^+$. Conversely, if $K \subseteq A^+$ if recognized by ${\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$, then there exists a positive integer $N$, an alphabet $A' \subseteq A$ and a language $L \subseteq (A' \times 2^N)^+$ recognized by $(S, X)$ such that $K = L_{\exists_N}$. \end{proposition} \begin{proof} Since $L_{\exists_N} = \pi_N[L]$ and $\pi_N$ is an lp-morphism, the first part is a consequence of Corollary~\ref{c:3}. % Conversely, let $K \subseteq A^+$ be a language recognized by ${\mathcal V}} \newcommand{\im}{{\sf Im}(S, X)$. By the proof of Corollary~\ref{c:3}, there is a finite alphabet $B \subseteq A\times S$ such that $K = f[L]$ for a language $L \subseteq B^+$ recognized by $(S, X)$, where $f: B \to A$ is the restriction of the projection $A \times S \to A$. Take $A' = f[B]$ and choose a big enough~$N$ so that each of the sets $B \cap (\{a\} \times S)$, with $a \in A'$, has at most~$N$ elements. Then, there exists an onto homomorphism $\pi: (A' \times 2^N)^+ \twoheadrightarrow B^+$ satisfying $\pi_N = f \circ \pi$, and so, there is a language $L' = \pi^{-1}(L) \subseteq (A' \times 2^N)^+$ recognized by $(S, X)$ and such that $K = L'_{\exists_N}$. \end{proof} \begin{remark} Observe that, if $B' \subseteq B$ is an inclusion of alphabets, then a language $L \subseteq (B')^+$ can always be seen as a language over $B$ via the inclusion $(B')^+ \subseteq B^+$. Nevertheless, the fact that $L$ is recognized by $(S, X)$ as a language over $B'$ does not necessarily implies that it is recognized by $(S, X)$ as a language over $B$. On the other hand, the \bis{} $(S \times {\bf 2}, X \times \{0,1\})$, where ${\bf 2}$ denotes the two-element semilattice, does recognize $L$ as a language over~$B$. Since one is usually interested in studying fragments of logic and not a single formula, we can always express the property ``every letter belongs to $B'$'', and so, this is not really a constraint. \end{remark} Notice that, Proposition~\ref{p:8} implies that a language is recognized by the power of an aperiodic semigroup if and only if it is obtained by second-order quantification of a language recognized by aperiodic semigroups. On the other hand, the languages recognizable by an aperiodic semigroup are precisely those definable in ${\bf FO}[<]$~\cite{Schutzenberger65,McNaughtonPapert71}, and in turn, the second-order existential quantification of those yields ${\bf MSO}[<]$, a fragment of logic defining precisely the regular languages~\cite{Buchi60,Elgot61} (i.e., those recognized by a finite semigroup). We may thus derive the following: \begin{corollary} Every finite semigroup divides a power of an aperiodic one. \end{corollary} \footnotesize \bibliographystyle{plain}
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Die Stiftung Wald und Wild in Mecklenburg-Vorpommern ist eine durch Claus Robert Agte 1998 gegründete gemeinnützige Stiftung mit Sitz in Schildfeld. Im ehemaligen Pferdestall des Forsthofes Schildfeld betreibt die Stiftung ein Schulungs- und Begegnungsstätte. Der Gutswald Rodenwalde mit 650 ha Fläche, davon 470 ha Wald und 180 ha Offenland, im Landkreis Ludwigslust-Parchim, den Agte 1999 erwarb, wurde nach dem Tod Agtes an die Stiftung übertragen. Aufgaben Die Stiftung fördert wildbiologische und forstwissenschaftliche Projekte, engagiert sich für Biotopentwicklung, Landschaftsgestaltung und Artenschutz. Ein weiterer Tätigkeitsschwerpunkt ist die Öffentlichkeitsarbeit, um Verständnis für Wald und Wild, aber auch die Akzeptanz der Jägerschaft zu sichern. Schwerpunkte der Arbeit Waldbauliche Entwicklungs- und Schutzmaßnahmen Jagdliche Hege- und Schutzmaßnahmen im Sinne eines wildverträglichen Waldes und ausgewogener, wildfreundlicher Kulturlandschaften Weiterbildung von Förstern und Jägern in Mecklenburg-Vorpommern im Sinne der Erhaltung eines artenreichen Waldbiotops Erforschung der natürlichen Zusammenhänge zwischen Wald und Wild Erforschung und Förderung der Beziehung des Menschen zu Wald und Wild, insbesondere durch die Unterstützung von Bildungseinrichtungen Pflege und Erhaltung der Lehr- und Informationszentren der Landesforstverwaltung Mecklenburg-Vorpommern Projekte Seit 1998 wurden 50 Projekte in MV gefördert. Die Stiftung unterstützt das Rebhuhnprojekt in der Lewitz. Beim Rebhuhnprojekt wird auf 12.000 ha unter aktiver Mitarbeit von 42 Revierinhabern, 7 Landwirtschaftsbetrieben sowie des Amtes Parchim an einer Stabilisierung und Steigerung der Rebhuhnbestandes gearbeitet. Dabei werden Biotopgestaltungsmaßnahmen, verstärkte Prädatorenbejagung und Auswilderung gezüchteter Rebhühner durchgeführt. Das Rebhuhnprojekt betreibt eine Rebhuhnaufzuchtstation um Rebhuhnketten (Rebhuhn-Familien) auszuwildern. Seit 2019 läuft ein Forschungsprojekt, das auf Basis umfangreicher Lebensraumanalysen an den Schwarzstorch-Brutplätzen in MV Handlungsempfehlungen zur Verbesserung der Lebensraumsituation des Schwarzstorches in MV zum Ziel hat. Bereits 2011 lief ein Vorläuferprojekt. Die Stiftung initiierte 1998 ein Telemetrie-Untersuchungsprojekt am Damwild. Die Ergebnisse von 1999 bis 2010 wurden 2010 im Buch Untersuchungen zur Raumnutzung des Damwildes publiziert. Publikationen Die Ethik in der Jagd. Nordwest Media Verlag, Grevesmühlen 2005 Unser Damwild in Mecklenburg-Vorpommern. Nordwest Media Verlag, Grevesmühlen 2006 Weblinks Offizielle Website Einzelnachweise Wald und Wild in Mecklenburg-Vorpommern Naturschutzorganisation (Deutschland) Artenschutz Gegründet 1998
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<?php namespace Translate\Test\TestCase\Model\Table; use Cake\ORM\TableRegistry; use Cake\TestSuite\TestCase; use Translate\Model\Table\TranslateDomainsTable; /** * Translate\Model\Table\TranslateDomainsTable Test Case */ class TranslateDomainsTableTest extends TestCase { /** * Test subject * * @var \Translate\Model\Table\TranslateDomainsTable */ public $TranslateDomains; /** * Fixtures * * @var array */ protected $fixtures = [ 'plugin.Translate.TranslateDomains', 'plugin.Translate.TranslateProjects', 'plugin.Translate.TranslateStrings', ]; /** * setUp method * * @return void */ public function setUp(): void { parent::setUp(); $config = TableRegistry::exists('TranslateDomains') ? [] : ['className' => 'Translate\Model\Table\TranslateDomainsTable']; $this->TranslateDomains = TableRegistry::getTableLocator()->get('TranslateDomains', $config); } /** * tearDown method * * @return void */ public function tearDown(): void { unset($this->TranslateDomains); parent::tearDown(); } /** * @return void */ public function testInstance() { $this->assertInstanceOf(TranslateDomainsTable::class, $this->TranslateDomains); } /** * @return void */ public function testSave() { $data = [ 'name' => 'default', 'translate_project_id' => 1, ]; $entity = $this->TranslateDomains->newEntity($data); $result = $this->TranslateDomains->save($entity); $this->assertTrue((bool)$result); } /** * Test statistics method * * @return void */ public function testStatistics() { $this->markTestIncomplete('Not implemented yet.'); } }
{ "redpajama_set_name": "RedPajamaGithub" }
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package cn.sdgundam.comicatsdgo.gd_api.async_task; import android.os.AsyncTask; import java.util.HashMap; import java.util.Map; import cn.sdgundam.comicatsdgo.api_model.ApiResultWrapper; import cn.sdgundam.comicatsdgo.api_model.CheckOriginUpdateResult; import cn.sdgundam.comicatsdgo.gd_api.Communicator; import cn.sdgundam.comicatsdgo.gd_api.GDInfoBuilder; /** * Created by xhguo on 12/9/2014. */ public class CheckOriginUpdateAsyncTask extends AsyncTask<Integer, Void, ApiResultWrapper<CheckOriginUpdateResult>> { @Override protected ApiResultWrapper<CheckOriginUpdateResult> doInBackground(Integer... integers) { Map<String, String> parameters = new HashMap<String, String>(); parameters.put("origin-count", integers[0] + ""); // stub params parameters.put("p", "1"); parameters.put("s", "2"); ApiResultWrapper<CheckOriginUpdateResult> resultWrapper; try { String json = Communicator.requestApi("has-new-origin", parameters); CheckOriginUpdateResult result = GDInfoBuilder.buildOriginInfoList(json); return new ApiResultWrapper<CheckOriginUpdateResult>(result); } catch(Exception e) { resultWrapper = new ApiResultWrapper<CheckOriginUpdateResult>(e); return resultWrapper; } } }
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<?php class Twig_Tests_TokenStreamTest extends PHPUnit_Framework_TestCase { static protected $tokens; public function setUp() { self::$tokens = array( new Twig_Token(Twig_Token::TEXT_TYPE, 1, 0), new Twig_Token(Twig_Token::TEXT_TYPE, 2, 0), new Twig_Token(Twig_Token::TEXT_TYPE, 3, 0), new Twig_Token(Twig_Token::TEXT_TYPE, 4, 0), new Twig_Token(Twig_Token::TEXT_TYPE, 5, 0), new Twig_Token(Twig_Token::TEXT_TYPE, 6, 0), new Twig_Token(Twig_Token::TEXT_TYPE, 7, 0), new Twig_Token(Twig_Token::EOF_TYPE, 0, 0), ); } public function testNext() { $stream = new Twig_TokenStream(self::$tokens); $repr = array(); while (!$stream->isEOF()) { $token = $stream->next(); $repr[] = $token->getValue(); } $this->assertEquals('1, 2, 3, 4, 5, 6, 7', implode(', ', $repr), '->next() advances the pointer and returns the current token'); } }
{ "redpajama_set_name": "RedPajamaGithub" }
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// Copyright 2016 The Chromium Authors. All rights reserved. // Use of this source code is governed by a BSD-style license that can be // found in the LICENSE file. 'use strict'; /** * A class that listens for touch events and produces events when these * touches form gestures (e.g. pinching). */ class GestureDetector { /** * Constructs a GestureDetector. * @param {!Element} element The element to monitor for touch gestures. */ constructor(element) { this.element_ = element; this.element_.addEventListener('touchstart', this.onTouchStart_.bind(this)); this.element_.addEventListener('touchmove', this.onTouch_.bind(this)); this.element_.addEventListener('touchend', this.onTouch_.bind(this)); this.element_.addEventListener('touchcancel', this.onTouch_.bind(this)); this.pinchStartEvent_ = null; this.lastEvent_ = null; this.listeners_ = new Map([ ['pinchstart', []], ['pinchupdate', []], ['pinchend', []] ]); } /** * Add a |listener| to be notified of |type| events. * @param {string} type The event type to be notified for. * @param {Function} listener The callback. */ addEventListener(type, listener) { if (this.listeners_.has(type)) { this.listeners_.get(type).push(listener); } } /** * Call the relevant listeners with the given |pinchEvent|. * @private * @param {!Object} pinchEvent The event to notify the listeners of. */ notify_(pinchEvent) { let listeners = this.listeners_.get(pinchEvent.type); for (let l of listeners) l(pinchEvent); } /** * The callback for touchstart events on the element. * @private * @param {!TouchEvent} event Touch event on the element. */ onTouchStart_(event) { // We must preventDefault if there is a two finger touch. By doing so // native pinch-zoom does not interfere with our way of handling the event. if (event.touches.length == 2) { event.preventDefault(); this.pinchStartEvent_ = event; this.lastEvent_ = event; this.notify_({ type: 'pinchstart', center: GestureDetector.center_(event) }); } } /** * The callback for touch move, end, and cancel events on the element. * @private * @param {!TouchEvent} event Touch event on the element. */ onTouch_(event) { if (!this.pinchStartEvent_) return; // Check if the pinch ends with the current event. if (event.touches.length < 2 || this.lastEvent_.touches.length !== event.touches.length) { let startScaleRatio = GestureDetector.pinchScaleRatio_( this.lastEvent_, this.pinchStartEvent_); let center = GestureDetector.center_(this.lastEvent_); let endEvent = { type: 'pinchend', startScaleRatio: startScaleRatio, center: center }; this.pinchStartEvent_ = null; this.lastEvent_ = null; this.notify_(endEvent); return; } let scaleRatio = GestureDetector.pinchScaleRatio_(event, this.lastEvent_); let startScaleRatio = GestureDetector.pinchScaleRatio_( event, this.pinchStartEvent_); let center = GestureDetector.center_(event); this.notify_({ type: 'pinchupdate', scaleRatio: scaleRatio, direction: scaleRatio > 1.0 ? 'in' : 'out', startScaleRatio: startScaleRatio, center: center }); this.lastEvent_ = event; } /** * Computes the change in scale between this touch event * and a previous one. * @private * @param {!TouchEvent} event Latest touch event on the element. * @param {!TouchEvent} prevEvent A previous touch event on the element. * @return {?number} The ratio of the scale of this event and the * scale of the previous one. */ static pinchScaleRatio_(event, prevEvent) { let distance1 = GestureDetector.distance_(prevEvent); let distance2 = GestureDetector.distance_(event); return distance1 === 0 ? null : distance2 / distance1; } /** * Computes the distance between fingers. * @private * @param {!TouchEvent} event Touch event with at least 2 touch points. * @return {number} Distance between touch[0] and touch[1]. */ static distance_(event) { let touch1 = event.touches[0]; let touch2 = event.touches[1]; let dx = touch1.clientX - touch2.clientX; let dy = touch1.clientY - touch2.clientY; return Math.sqrt(dx * dx + dy * dy); } /** * Computes the midpoint between fingers. * @private * @param {!TouchEvent} event Touch event with at least 2 touch points. * @return {!Object} Midpoint between touch[0] and touch[1]. */ static center_(event) { let touch1 = event.touches[0]; let touch2 = event.touches[1]; return { x: (touch1.clientX + touch2.clientX) / 2, y: (touch1.clientY + touch2.clientY) / 2 }; } };
{ "redpajama_set_name": "RedPajamaGithub" }
335
//--------------------------------------------------------------------------- // For user: you can disable or enable it //#define MEDIAINFO_DEBUG //--------------------------------------------------------------------------- //--------------------------------------------------------------------------- // Pre-compilation #include "MediaInfo/PreComp.h" #ifdef __BORLANDC__ #pragma hdrstop #endif //--------------------------------------------------------------------------- //--------------------------------------------------------------------------- #include "MediaInfo/Setup.h" //--------------------------------------------------------------------------- //--------------------------------------------------------------------------- #include "MediaInfo/Reader/Reader_File.h" #include "MediaInfo/File__Analyze.h" #include "ZenLib/FileName.h" #ifdef WINDOWS #undef __TEXT #include "Windows.h" #endif //WINDOWS using namespace ZenLib; using namespace std; //--------------------------------------------------------------------------- // Debug stuff #ifdef MEDIAINFO_DEBUG int64u Reader_File_Offset=0; int64u Reader_File_BytesRead_Total=0; int64u Reader_File_BytesRead=0; int64u Reader_File_Count=1; #include <iostream> #endif // MEDIAINFO_DEBUG //--------------------------------------------------------------------------- namespace MediaInfoLib { #if MEDIAINFO_READTHREAD void Reader_File_Thread::Entry() { ReadSize_Max=Base->Buffer_Max>>3; for (;;) { Base->CS.Enter(); if (Base->Buffer_Begin==Base->Buffer_Max) { Base->IsLooping=false; Base->Buffer_End=Base->Buffer_End2; Base->Buffer_End2=0; Base->Buffer_Begin=0; } size_t ToRead; size_t Buffer_ToReadOffset; if (Base->IsLooping) { ToRead=Base->Buffer_Begin-Base->Buffer_End2; Buffer_ToReadOffset=Base->Buffer_End2; } else { ToRead=Base->Buffer_Max-Base->Buffer_End; Buffer_ToReadOffset=Base->Buffer_End; } Base->CS.Leave(); if (ToRead) { if (ToRead>ReadSize_Max) ToRead=ReadSize_Max; size_t BytesRead=Base->F.Read(Base->Buffer+Buffer_ToReadOffset, ToRead); if (!BytesRead) break; Base->CS.Enter(); if (Base->IsLooping) { Base->Buffer_End2+=BytesRead; } else { Base->Buffer_End+=BytesRead; if (Base->Buffer_End==Base->Buffer_Max) { Base->IsLooping=true; } } Base->CS.Leave(); #ifdef WINDOWS SetEvent(Base->Condition_WaitingForMoreData); #endif //WINDOWS } #ifdef WINDOWS else WaitForSingleObject(Base->Condition_WaitingForMorePlace, INFINITE); #endif //WINDOWS if (IsTerminating()) break; Yield(); } #ifdef WINDOWS SetEvent(Base->Condition_WaitingForMoreData); //Sending the last event in case the main threading is waiting for more data #endif //WINDOWS } #endif //MEDIAINFO_READTHREAD const size_t Buffer_NoJump=128*1024; //--------------------------------------------------------------------------- Reader_File::~Reader_File() { #if MEDIAINFO_READTHREAD if (ThreadInstance) { ThreadInstance->RequestTerminate(); SetEvent(Condition_WaitingForMorePlace); while (!ThreadInstance->IsExited()) Sleep(0); #ifdef WINDOWS CloseHandle(Condition_WaitingForMorePlace); CloseHandle(Condition_WaitingForMoreData); #endif //WINDOWS delete ThreadInstance; MI_Internal->Config.File_Buffer=NULL; MI_Internal->Config.File_Buffer_Size=0; MI_Internal->Config.File_Buffer_Size_Max=0; delete[] Buffer; } #endif //MEDIAINFO_READTHREAD } //--------------------------------------------------------------------------- size_t Reader_File::Format_Test(MediaInfo_Internal* MI, String File_Name) { //std::cout<<Ztring(File_Name).To_Local().c_str()<<std::endl; #if MEDIAINFO_EVENTS { string File_Name_Local=Ztring(File_Name).To_Local(); wstring File_Name_Unicode=Ztring(File_Name).To_Unicode(); struct MediaInfo_Event_General_Start_0 Event; memset(&Event, 0xFF, sizeof(struct MediaInfo_Event_Generic)); Event.EventCode=MediaInfo_EventCode_Create(MediaInfo_Parser_None, MediaInfo_Event_General_Start, 0); Event.EventSize=sizeof(struct MediaInfo_Event_General_Start_0); Event.StreamIDs_Size=0; Event.Stream_Size=File::Size_Get(File_Name); Event.FileName=File_Name_Local.c_str(); Event.FileName_Unicode=File_Name_Unicode.c_str(); MI->Config.Event_Send(NULL, (const int8u*)&Event, sizeof(MediaInfo_Event_General_Start_0)); } #endif //MEDIAINFO_EVENTS //With Parser MultipleParsing /* MI->Open_Buffer_Init((int64u)-1, File_Name); if (Format_Test_PerParser(MI, File_Name)) return 1; return 0; //There is a problem */ //Get the Extension Ztring Extension=FileName::Extension_Get(File_Name); Extension.MakeLowerCase(); //Search the theorical format from extension InfoMap &FormatList=MediaInfoLib::Config.Format_Get(); InfoMap::iterator Format=FormatList.end(); if (!MI->Config.File_ForceParser_Get().empty()) Format=FormatList.find(MI->Config.File_ForceParser_Get()); if (Format==FormatList.end()) { Format=FormatList.begin(); while (Format!=FormatList.end()) { const Ztring &Extensions=FormatList.Get(Format->first, InfoFormat_Extensions); if (Extensions.find(Extension)!=Error) { if(Extension.size()==Extensions.size()) break; //Only one extenion in the list if(Extensions.find(Extension+__T(" "))!=Error || Extensions.find(__T(" ")+Extension)!=Error) break; } ++Format; } } if (Format!=FormatList.end()) { const Ztring &Parser=Format->second(InfoFormat_Parser); if (MI->SelectFromExtension(Parser)) { //Test the theorical format if (Format_Test_PerParser(MI, File_Name)>0) return 1; } } size_t ToReturn=MI->ListFormats(File_Name); return ToReturn; } //--------------------------------------------------------------------------- size_t Reader_File::Format_Test_PerParser(MediaInfo_Internal* MI, const String &File_Name) { //Init MI_Internal=MI; #if MEDIAINFO_READTHREAD ThreadInstance=NULL; Buffer_End2=0; //Is also used for counting bytes before activating the thread #endif //MEDIAINFO_READTHREAD //Opening the file F.Open(File_Name); if (!F.Opened_Get()) return 0; //Info Status=0; MI->Config.File_Size=F.Size_Get(); MI->Config.File_Current_Offset=0; MI->Config.File_Current_Size=MI->Config.File_Size; MI->Config.File_Sizes.clear(); MI->Config.File_Sizes.push_back(MI->Config.File_Size); if (MI->Config.File_Names.size()>1) { #if MEDIAINFO_ADVANCED if (MI->Config.File_IgnoreSequenceFileSize_Get()) { MI->Config.File_Size=(int64u)-1; } else #endif //MEDIAINFO_ADVANCED { for (size_t Pos=1; Pos<MI->Config.File_Names.size(); Pos++) { int64u Size=File::Size_Get(MI->Config.File_Names[Pos]); MI->Config.File_Sizes.push_back(Size); MI->Config.File_Size+=Size; } } } //Partial file handling Ztring Config_Partial_Begin=MI->Config.File_Partial_Begin_Get(); if (!Config_Partial_Begin.empty() && Config_Partial_Begin[0]>=__T('0') && Config_Partial_Begin[0]<=__T('9')) { if (Config_Partial_Begin.find(__T('%'))==Config_Partial_Begin.size()-1) Partial_Begin=float64_int64s(MI->Config.File_Size*Config_Partial_Begin.To_float64()/100); else Partial_Begin=Config_Partial_Begin.To_int64u(); if (Partial_Begin) F.GoTo(Partial_Begin); } else Partial_Begin=0; Ztring Config_Partial_End=MI->Config.File_Partial_End_Get(); if (!Config_Partial_End.empty() && Config_Partial_End[0]>=__T('0') && Config_Partial_End[0]<=__T('9')) { if (Config_Partial_End.find(__T('%'))==Config_Partial_End.size()-1) Partial_End=float64_int64s(MI->Config.File_Size*Config_Partial_End.To_float64()/100); else Partial_End=Config_Partial_End.To_int64u(); } else Partial_End=(int64u)-1; if (Partial_Begin>MI->Config.File_Size) Partial_Begin=0; //Wrong value if (Partial_Begin>Partial_End) Partial_Begin=0; //Wrong value //Parser MI->Open_Buffer_Init((Partial_End<=MI->Config.File_Size?Partial_End:MI->Config.File_Size)-Partial_Begin, File_Name); //Buffer MI->Option(__T("File_Buffer_Size_Hint_Pointer"), Ztring::ToZtring((size_t)(&MI->Config.File_Buffer_Size_ToRead))); MI->Config.File_Buffer_Repeat_IsSupported=true; //Test the format with buffer return Format_Test_PerParser_Continue(MI); } //--------------------------------------------------------------------------- size_t Reader_File::Format_Test_PerParser_Continue (MediaInfo_Internal* MI) { if (MI == NULL) return 0; bool StopAfterFilled=MI->Config.File_StopAfterFilled_Get(); bool ShouldContinue=true; if (MI->Info) Status=MI->Info->Status; //Previous data if (MI->Config.File_Buffer_Repeat) { MI->Config.File_Buffer_Repeat=false; #if MEDIAINFO_DEMUX MI->Config.Demux_EventWasSent=false; #endif //MEDIAINFO_DEMUX Status=MI->Open_Buffer_Continue(MI->Config.File_Buffer, MI->Config.File_Buffer_Size); #if MEDIAINFO_READTHREAD if (ThreadInstance && !MI->Config.File_Buffer_Repeat) { CS.Enter(); Buffer_Begin+=MI->Config.File_Buffer_Size; #ifdef WINDOWS if (Buffer_Begin==Buffer_Max) { CS.Leave(); SetEvent(Condition_WaitingForMorePlace); } else #endif //WINDOWS CS.Leave(); } #endif //MEDIAINFO_READTHREAD #if MEDIAINFO_DEMUX //Demux if (MI->Config.Demux_EventWasSent) return 2; //Must return immediately #endif //MEDIAINFO_DEMUX //Threading if (MI->IsTerminating()) return 1; //Termination is requested if (Status[File__Analyze::IsFinished] || (StopAfterFilled && Status[File__Analyze::IsFilled])) ShouldContinue=false; } #if MEDIAINFO_DEMUX //PerPacket if (ShouldContinue && MI->Config.Demux_EventWasSent) { MI->Config.Demux_EventWasSent=false; Status=MI->Open_Buffer_Continue(NULL, 0); //Demux if (MI->Config.Demux_EventWasSent) return 2; //Must return immediately //Threading if (MI->IsTerminating()) return 1; //Termination is requested if (Status[File__Analyze::IsFinished] || (StopAfterFilled && Status[File__Analyze::IsFilled])) ShouldContinue=false; } #endif //MEDIAINFO_DEMUX if (ShouldContinue) { //Test the format with buffer while (!(Status[File__Analyze::IsFinished] || (StopAfterFilled && Status[File__Analyze::IsFilled]))) { //Seek (if needed) if (MI->Open_Buffer_Continue_GoTo_Get()!=(int64u)-1) { #ifdef MEDIAINFO_DEBUG std::cout<<std::hex<<Reader_File_Offset<<" - "<<Reader_File_Offset+Reader_File_BytesRead<<" : "<<std::dec<<Reader_File_BytesRead<<" bytes"<<std::endl; Reader_File_Offset=MI->Open_Buffer_Continue_GoTo_Get(); Reader_File_BytesRead=0; Reader_File_Count++; #endif //MEDIAINFO_DEBUG #if MEDIAINFO_READTHREAD if (ThreadInstance) { ThreadInstance->RequestTerminate(); SetEvent(Condition_WaitingForMorePlace); while (!ThreadInstance->IsExited()) Sleep(0); #ifdef WINDOWS CloseHandle(Condition_WaitingForMorePlace); CloseHandle(Condition_WaitingForMoreData); #endif //WINDOWS delete ThreadInstance; ThreadInstance=NULL; MI->Config.File_Buffer=NULL; MI->Config.File_Buffer_Size=0; MI->Config.File_Buffer_Size_Max=0; Buffer_Max=0; delete[] Buffer; Buffer=NULL; Buffer_Begin=0; Buffer_End=0; Buffer_End2=0; IsLooping=false; } if (Buffer_End2!=(size_t)-1) Buffer_End2=0; #endif //MEDIAINFO_READTHREAD int64u GoTo=Partial_Begin+MI->Open_Buffer_Continue_GoTo_Get(); MI->Config.File_Current_Offset=0; int64u Buffer_NoJump_Temp=Buffer_NoJump; if (MI->Config.File_Names.size()>1) { size_t Pos; #if MEDIAINFO_SEEK if (MI->Config.File_GoTo_IsFrameOffset) { Pos=(size_t)MI->Open_Buffer_Continue_GoTo_Get(); //File_GoTo is the frame offset in that case MI->Info->File_GoTo=(int64u)-1; MI->Config.File_GoTo_IsFrameOffset=false; GoTo=0; } else #endif //MEDIAINFO_SEEK { for (Pos=0; Pos<MI->Config.File_Names.size(); Pos++) { if (Pos==MI->Config.File_Sizes.size()) MI->Config.File_Sizes.push_back(F.Size_Get()); else if (MI->Config.File_Sizes[Pos]==(int64u)-1) MI->Config.File_Sizes[Pos]=F.Size_Get(); if (Pos>=MI->Config.File_Sizes.size() || MI->Config.File_Sizes[Pos]==(int64u)-1) break; if (GoTo<MI->Config.File_Sizes[Pos]) break; GoTo-=MI->Config.File_Sizes[Pos]; MI->Config.File_Current_Offset+=MI->Config.File_Sizes[Pos]; } if (Pos>=MI->Config.File_Sizes.size()) break; } if (Pos!=MI->Config.File_Names_Pos-1) { F.Close(); F.Open(MI->Config.File_Names[Pos]); if (Pos>=MI->Config.File_Sizes.size()) { MI->Config.File_Sizes.resize(Pos, (int64u)-1); MI->Config.File_Sizes.push_back(F.Size_Get()); } MI->Config.File_Names_Pos=Pos+1; MI->Config.File_Current_Size=MI->Config.File_Current_Offset+F.Size_Get(); Buffer_NoJump_Temp=0; } } if (GoTo>=F.Size_Get()) break; //Seek requested, but on a file bigger in theory than what is in the real file, we can't do this if (!(GoTo>F.Position_Get() && GoTo<F.Position_Get()+Buffer_NoJump_Temp)) //No smal jumps { if (!F.GoTo(GoTo)) break; //File is not seekable MI->Open_Buffer_Init((int64u)-1, MI->Config.File_Current_Offset+F.Position_Get()-Partial_Begin); } } #if MEDIAINFO_READTHREAD if (ThreadInstance==NULL && Buffer_End2!=(size_t)-1 && Buffer_End2>=16*1024*1024) { if (!MI->Config.File_IsGrowing && MI->Config.File_Names.size()==1) { delete[] MI->Config.File_Buffer; MI->Config.File_Buffer=NULL; MI->Config.File_Buffer_Size_Max=0; Buffer_Max=MI->Config.File_Buffer_Read_Size_Get(); Buffer=new int8u[Buffer_Max]; Buffer_Begin=0; Buffer_End=0; Buffer_End2=0; IsLooping=false; #ifdef WINDOWS Condition_WaitingForMorePlace=CreateEvent(NULL, FALSE, FALSE, NULL); Condition_WaitingForMoreData=CreateEvent(NULL, FALSE, FALSE, NULL); #endif //WINDOWS ThreadInstance=new Reader_File_Thread(); ThreadInstance->Base=this; ThreadInstance->Run(); } else Buffer_End2=(size_t)-1; } #endif //MEDIAINFO_READTHREAD //Handling of hints if (MI->Config.File_Buffer_Size_ToRead==0) break; //Problem while config if ( #if MEDIAINFO_READTHREAD ThreadInstance==NULL && #endif //MEDIAINFO_READTHREAD MI->Config.File_Buffer_Size_ToRead>MI->Config.File_Buffer_Size_Max) { delete[] MI->Config.File_Buffer; if (MI->Config.File_Buffer_Size_Max==0) MI->Config.File_Buffer_Size_Max=1; while (MI->Config.File_Buffer_Size_ToRead>MI->Config.File_Buffer_Size_Max) MI->Config.File_Buffer_Size_Max*=2; if (MI->Config.File_Buffer_Size_Max>=64*1024*1024) MI->Config.File_Buffer_Size_Max=64*1024*1024; //limitation of the buffer in order to avoid to big memory usage MI->Config.File_Buffer=new int8u[MI->Config.File_Buffer_Size_Max]; } //Testing multiple file per stream if ( #if MEDIAINFO_READTHREAD ThreadInstance==NULL && #endif //MEDIAINFO_READTHREAD F.Position_Get()>=F.Size_Get()) { #if MEDIAINFO_ADVANCED2 MI->Open_Buffer_SegmentChange(); #endif //MEDIAINFO_ADVANCED2 if (MI->Config.File_Names_Pos && MI->Config.File_Names_Pos<MI->Config.File_Names.size()) { MI->Config.File_Current_Offset+=MI->Config.File_Names_Pos<=MI->Config.File_Sizes.size()?MI->Config.File_Sizes[MI->Config.File_Names_Pos-1]:F.Size_Get(); F.Close(); #if MEDIAINFO_EVENTS MI->Config.Event_SubFile_Start(MI->Config.File_Names[MI->Config.File_Names_Pos]); #endif //MEDIAINFO_EVENTS F.Open(MI->Config.File_Names[MI->Config.File_Names_Pos]); while (!F.Opened_Get()) { #if MEDIAINFO_EVENTS MI->Config.Event_SubFile_Missing_Absolute(MI->Config.File_Names[MI->Config.File_Names_Pos]); #endif //MEDIAINFO_EVENTS if (MI->Config.File_Names_Pos+1<MI->Config.File_Names.size()) { MI->Config.File_Names_Pos++; F.Open(MI->Config.File_Names[MI->Config.File_Names_Pos]); } else //break the otherwise infinite loop { break; } } if (MI->Config.File_Names_Pos>=MI->Config.File_Sizes.size()) { MI->Config.File_Sizes.resize(MI->Config.File_Names_Pos, 0); MI->Config.File_Sizes.push_back(F.Size_Get()); } MI->Config.File_Names_Pos++; MI->Config.File_Current_Size+=F.Size_Get(); } } #if MEDIAINFO_READTHREAD if (ThreadInstance) { CS.Enter(); #ifdef WINDOWS if (Buffer_End2+Buffer_End-Buffer_Begin<Buffer_Max/8*7) { CS.Leave(); SetEvent(Condition_WaitingForMorePlace); CS.Enter(); } #endif //WINDOWS for (;;) { MI->Config.File_Buffer_Size=Buffer_End-Buffer_Begin; if (MI->Config.File_Buffer_Size) break; if (!ThreadInstance->IsExited()) { CS.Leave(); #ifdef WINDOWS WaitForSingleObject(Condition_WaitingForMoreData, INFINITE); #else //WINDOWS Sleep(0); #endif //WINDOWS CS.Enter(); } else { if (IsLooping) { IsLooping=false; Buffer_End=Buffer_End2; Buffer_End2=0; Buffer_Begin=0; } MI->Config.File_Buffer_Size=Buffer_End-Buffer_Begin; break; } } MI->Config.File_Buffer=Buffer+Buffer_Begin; CS.Leave(); if (MI->Config.File_Buffer_Size>MI->Config.File_Buffer_Size_ToRead) MI->Config.File_Buffer_Size=MI->Config.File_Buffer_Size_ToRead; } else #endif //MEDIAINFO_READTHREAD { MI->Config.File_Buffer_Size=F.Read(MI->Config.File_Buffer, (F.Position_Get()+MI->Config.File_Buffer_Size_ToRead<(Partial_End<=MI->Config.File_Size?Partial_End:MI->Config.File_Size))?MI->Config.File_Buffer_Size_ToRead:((size_t)((Partial_End<=MI->Config.File_Size?Partial_End:MI->Config.File_Size)-F.Position_Get()))); #if MEDIAINFO_READTHREAD if (ThreadInstance==NULL && Buffer_End2!=(size_t)-1) Buffer_End2+=MI->Config.File_Buffer_Size; #endif //MEDIAINFO_READTHREAD } /* High CPU usage #if MEDIAINFO_EVENTS if (MI->Config.File_Buffer_Size) { struct MediaInfo_Event_Global_BytesRead_0 Event; memset(&Event, 0xFF, sizeof(struct MediaInfo_Event_Generic)); Event.EventCode=MediaInfo_EventCode_Create(MediaInfo_Parser_None, MediaInfo_Event_Global_BytesRead, 0); Event.EventSize=sizeof(struct MediaInfo_Event_Global_BytesRead_0); Event.StreamIDs_Size=0; Event.StreamOffset=F.Position_Get()-MI->Config.File_Buffer_Size; Event.Content_Size=MI->Config.File_Buffer_Size; Event.Content=MI->Config.File_Buffer; MI->Config.Event_Send(NULL, (const int8u*)&Event, sizeof(MediaInfo_Event_Global_BytesRead_0)); } #endif //MEDIAINFO_EVENTS */ //Testing growing files int64u Growing_Temp=(int64u)-1; if (MI->Config.ParseSpeed>=1.0 && !MI->Config.File_IsGrowing && MI->Config.File_Current_Offset+F.Position_Get()>=MI->Config.File_Size) { if (MI->Config.File_Names.size()==1) { Growing_Temp=F.Size_Get(); if (MI->Config.File_Size!=Growing_Temp) MI->Config.File_IsGrowing=true; } else if (MI->Config.File_TestContinuousFileNames_Get()) { Growing_Temp=MI->Config.File_Names.size(); MI->TestContinuousFileNames(); if (MI->Config.File_Names.size()!=Growing_Temp) MI->Config.File_IsGrowing=true; } } if (MI->Config.File_IsNotGrowingAnymore) { MI->Config.File_Current_Size=MI->Config.File_Size=F.Size_Get(); MI->Open_Buffer_Init(MI->Config.File_Size, F.Position_Get()-MI->Config.File_Buffer_Size); MI->Config.File_IsGrowing=false; MI->Config.File_IsNotGrowingAnymore=false; } if (MI->Config.File_IsGrowing && (Growing_Temp!=(int64u)-1 || MI->Config.File_Current_Offset+F.Position_Get()>=MI->Config.File_Size)) { for (size_t CountOfSeconds=0; CountOfSeconds<(size_t)MI->Config.File_GrowingFile_Delay_Get(); CountOfSeconds++) { int64u LastFile_Size_Old=MI->Config.File_Sizes[MI->Config.File_Sizes.size()-1]; size_t Files_Count_Old=MI->Config.File_Names.size(); MI->TestContinuousFileNames(); int64u LastFile_Size_New=F.Size_Get(); size_t Files_Count_New=MI->Config.File_Names.size(); if (LastFile_Size_New!=LastFile_Size_Old || Files_Count_New!=Files_Count_Old) { if (MI->Config.File_Names.size()==1) //if more than 1 file, file size config is already done in TestContinuousFileNames() { MI->Config.File_Current_Size=MI->Config.File_Size=LastFile_Size_New; MI->Open_Buffer_Init(MI->Config.File_Size, MI->Config.File_Current_Offset+F.Position_Get()-MI->Config.File_Buffer_Size); } break; } #ifdef WINDOWS Sleep(1000); #endif //WINDOWS } } #ifdef MEDIAINFO_DEBUG Reader_File_BytesRead_Total+=MI->Config.File_Buffer_Size; Reader_File_BytesRead+=MI->Config.File_Buffer_Size; #endif //MEDIAINFO_DEBUG //Parser Status=MI->Open_Buffer_Continue(MI->Config.File_Buffer, MI->Config.File_Buffer_Size); #if MEDIAINFO_READTHREAD if (ThreadInstance && !MI->Config.File_Buffer_Repeat) { CS.Enter(); Buffer_Begin+=MI->Config.File_Buffer_Size; #ifdef WINDOWS if (Buffer_Begin==Buffer_Max) { CS.Leave(); SetEvent(Condition_WaitingForMorePlace); } else #endif //WINDOWS CS.Leave(); } #endif //MEDIAINFO_READTHREAD if (MI->Config.File_Buffer_Size==0) { #if MEDIAINFO_EVENTS MediaInfoLib::Config.Log_Send(0xC0, 0xFF, 0xF0F00101, "File read error"); #endif //MEDIAINFO_EVENTS break; } #if MEDIAINFO_DEMUX if (MI->Config.Demux_EventWasSent) return 2; //Must return immediately #endif //MEDIAINFO_DEMUX //Threading if (MI->IsTerminating()) break; //Termination is requested } } //Deleting buffer #if MEDIAINFO_READTHREAD if (ThreadInstance) { ThreadInstance->RequestTerminate(); SetEvent(Condition_WaitingForMorePlace); while (!ThreadInstance->IsExited()) Sleep(0); #ifdef WINDOWS CloseHandle(Condition_WaitingForMorePlace); CloseHandle(Condition_WaitingForMoreData); #endif //WINDOWS delete ThreadInstance; ThreadInstance=NULL; MI->Config.File_Buffer=NULL; MI->Config.File_Buffer_Size=0; MI->Config.File_Buffer_Size_Max=0; Buffer_Max=0; delete[] Buffer; Buffer=NULL; Buffer_Begin=0; Buffer_End=0; Buffer_End2=0; IsLooping=false; } else #endif //MEDIAINFO_READTHREAD { delete[] MI->Config.File_Buffer; MI->Config.File_Buffer=NULL; MI->Config.File_Buffer_Size_Max=0; } #ifdef MEDIAINFO_DEBUG std::cout<<std::hex<<Reader_File_Offset<<" - "<<Reader_File_Offset+Reader_File_BytesRead<<" : "<<std::dec<<Reader_File_BytesRead<<" bytes"<<std::endl; std::cout<<"Total: "<<std::dec<<Reader_File_BytesRead_Total<<" bytes in "<<Reader_File_Count<<" blocks"<<std::endl; #endif //MEDIAINFO_DEBUG if (!MI->Config.File_KeepInfo_Get()) { //File F.Close(); } //Is this file detected? if (!Status[File__Analyze::IsAccepted]) return 0; MI->Open_Buffer_Finalize(); #if MEDIAINFO_DEMUX if (MI->Config.Demux_EventWasSent) return 2; //Must return immediately #endif //MEDIAINFO_DEMUX return 1; } //--------------------------------------------------------------------------- #if MEDIAINFO_SEEK size_t Reader_File::Format_Test_PerParser_Seek (MediaInfo_Internal* MI, size_t Method, int64u Value, int64u ID) { size_t ToReturn=MI->Open_Buffer_Seek(Method, Value, ID); if (ToReturn==0 || ToReturn==1) { //Reset Status=0; } return ToReturn; } #endif //MEDIAINFO_SEEK } //NameSpace
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Last Minute! Stories! Ukulele! Hazard Tape! Too late to be messing around with paid ticketing so just let me know you're coming here: http://plateshow02.eventbrite.com I'll be passing the hat for pecuniary contributions.
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Georg Marschalk von Ebnet (died 1505) was the Prince-Bishop of Bamberg from 1503 to 1505. Biography Georg Marschalk von Ebnet was a member of the Marschalk von Ebnet family, which derived its name from being hereditary Marshal of Ebnet, now a district of Burgkunstadt. The cathedral chapter of Bamberg Cathedral elected Marschalk von Ebnet to be Prince-Bishop of Bamberg on 19 September 1503. Pope Julius II confirmed his appointment on 11 December 1503. He died on 30 January 1505 without ever having been consecrated as a bishop. References 1505 deaths Prince-Bishops of Bamberg Year of birth unknown
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{"url":"https:\/\/answers.opencv.org\/question\/197156\/loading-tensorflow-model-without-pbtxt-file\/","text":"Hi, complete DNN newbie here. I'm loading opencv_face_detector_uint8.pb with its associated opencv_face_detector.pbtxt file. When I omit the second argument to the readNet function, the unconnected output layer vector is different. Why?\n\nNet net = dnn::experimental_dnn_v5::readNetFromTensorflow(\"opencv_face_detector_uint8.pb\", \"opencv_face_detector.pbtxt\");\n\nvector<int> vi = net.getUnconnectedOutLayers();\nvector<string> node_names;\nfor(int i : vi)\nnode_names.push_back(net.getLayer(i)->name);\nfor(auto& s : node_names)\ncout << s << \" \";\n\/* outputs \"detection_out\"*\/\n\n...\n\/*outputs: mbox_loc mbox_conf_flatten, i.e. two nodes*\/\n\n\nA side question is: why can't I call\n\nnet.forward(node_name); \/*node_name is a string containing an output node name*\/\n\n\non the two nodes I get in the second example?\n\nI hope someone can clarify, thanks.\n\nedit retag close merge delete","date":"2020-09-26 05:56:29","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\": 0.19028812646865845, \"perplexity\": 14331.29870233902}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400234232.50\/warc\/CC-MAIN-20200926040104-20200926070104-00540.warc.gz\"}"}
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{"url":"https:\/\/iwaponline.com\/view-large\/1089554","text":"Based on Table\u00a010, the calculated values of WQI in this study were compared with the prescribed standards to show the water quality condition for agriculture purpose as represented in Figure\u00a010.\nTable\u00a010\n\nWQI categories and their classifications (Rown et al. 1972)\n\nWQIClassification\n<50\u00a0Excellent\n50\u2013100\u00a0Good\n100\u2013200\u00a0Poor\n>300\u00a0Unfit\nWQIClassification\n<50\u00a0Excellent\n50\u2013100\u00a0Good\n100\u2013200\u00a0Poor","date":"2021-12-09 10:45:50","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\": 0.8120922446250916, \"perplexity\": 616.6175146293725}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363791.16\/warc\/CC-MAIN-20211209091917-20211209121917-00500.warc.gz\"}"}
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title: Policy Language kind: documentation weight: 2 toc: true --- ```live:eg:module:hidden package example ``` OPA is purpose built for reasoning about information represented in structured documents. The data that your service and its users publish can be inspected and transformed using OPA's native query language Rego. ## What is Rego? Rego was inspired by [Datalog](https://en.wikipedia.org/wiki/Datalog), which is a well understood, decades old query language. Rego extends Datalog to support structured document models such as JSON. Rego queries are assertions on data stored in OPA. These queries can be used to define policies that enumerate instances of data that violate the expected state of the system. ## Why use Rego? Use Rego for defining policy that is easy to read and write. Rego focuses on providing powerful support for referencing nested documents and ensuring that queries are correct and unambiguous. Rego is declarative so policy authors can focus on what queries should return rather than how queries should be executed. These queries are simpler and more concise than the equivalent in an imperative language. Like other applications which support declarative query languages, OPA is able to optimize queries to improve performance. ## The Basics This section introduces the main aspects of Rego. The simplest rule is a single expression and is defined in terms of a [Scalar Value](#scalar-values): ```live:eg/pi:module pi := 3.14159 ``` Rules define the content of documents. We can query for the content of the `pi` document generated by the rule above: ```live:eg/pi:query:read_only,merge_down pi ``` ```live:eg/pi:output ``` Rules can also be defined in terms of [Composite Values](#composite-values): ```live:eg/rect:module rect := {"width": 2, "height": 4} ``` The result: ```live:eg/rect:query:read_only,merge_down rect ``` ```live:eg/rect:output ``` You can compare two scalar or composite values, and when you do so you are checking if the two values are the same JSON value. ```live:eg/rect/compare:query:merge_down rect == {"height": 4, "width": 2} ``` ```live:eg/rect/compare:output ``` You can define a new concept using a rule. For example, `v` below is true if the equality expression is true. ```live:eg/undefined:module v { "hello" == "world" } ``` If we evaluate `v`, the result is `undefined` because the body of the rule never evaluates to `true`. As a result, the document generated by the rule is not defined. ```live:eg/undefined:query:hidden v ``` ```live:eg/undefined:output:expect_undefined ``` Expressions that refer to undefined values are also undefined. This includes comparisons such as `!=`. ```live:eg/undefined/expression:query:merge_down v == true ``` ```live:eg/undefined/expression:output:expect_undefined,merge_down ``` ```live:eg/undefined/other_expression:query:merge_down v != true ``` ```live:eg/undefined/other_expression:output:expect_undefined ``` We can define rules in terms of [Variables](#variables) as well: ```live:eg/rules:module t { x := 42; y := 41; x > y } ``` The formal syntax uses the semicolon character `;` to separate expressions. Rule bodies can separate expressions with newlines and omit the semicolon: ```live:eg/rules/newlines:module:read_only t2 { x := 42 y := 41 x > y } ``` When evaluating rule bodies, OPA searches for variable bindings that make all of the expressions true. There may be multiple sets of bindings that make the rule body true. The rule body can be understood intuitively as: ``` expression-1 AND expression-2 AND ... AND expression-N ``` The rule itself can be understood intuitively as: ``` rule-name IS value IF body ``` If the **value** is omitted, it defaults to **true**. When we query for the value of `t` we see the obvious result: ```live:eg/rules:query:hidden t ``` ```live:eg/rules:output ``` The order of expressions in a rule does not affect the document's content. ```live:eg/expression_order:module s { x > y y = 41 x = 42 } ``` The query result is the same: ```live:eg/expression_order:query:hidden s ``` ```live:eg/expression_order:output ``` There's one exception: if you use assignment `:=` the compiler will check that the variable you are assigning has not already been used. ```live:eg/assignment_check:module:merge_down z { y := 41 y := 42 43 > y } ``` ```live:eg/assignment_check:output:expect_assigned_above ``` Rego [References](#references) help you refer to nested documents. For example, with: ```live:eg/references:module sites = [{"name": "prod"}, {"name": "smoke1"}, {"name": "dev"}] ``` And ```live:eg/references/basic:module r { sites[_].name == "prod" } ``` The rule `r` above asserts that there exists (at least) one document within `sites` where the `name` attribute equals `"prod"`. The result: ```live:eg/references/basic:query:hidden r ``` ```live:eg/references/basic:output ``` We can generalize the example above with a rule that defines a set document instead of a boolean document: ```live:eg/references/helper:module q[name] { name := sites[_].name } ``` The value of `q` is a set of names ```live:eg/references/helper:query:hidden q ``` ```live:eg/references/helper:output ``` We can re-write the rule `r` from above to make use of `q`. We will call the new rule `p`: ```live:eg/references/helper/composed:module p { q["prod"] } ``` Querying `p` will have the same result: ```live:eg/references/helper/composed:query:hidden p ``` ```live:eg/references/helper/composed:output ``` As you can see, rules which have arguments can be queried with input values: ```live:eg/references/helper/argument:query:merge_down q["smoke2"] ``` ```live:eg/references/helper/argument:output:expect_undefined ``` If you made it this far, congratulations! This section introduced the main aspects of Rego. The rest of this document walks through each part of the language in more detail. For a concise reference, see the [Policy Reference](../policy-reference) document. ## Scalar Values Scalar values are the simplest type of term in Rego. Scalar values can be [Strings](#strings), numbers, booleans, or null. Documents can be defined solely in terms of scalar values. This is useful for defining constants that are referenced in multiple places. For example: ```live:eg/scalars:module greeting := "Hello" max_height := 42 pi := 3.14159 allowed := true location := null ``` These documents can be queried like any other: ```live:eg/scalars:query:merge_down [greeting, max_height, pi, allowed, location] ``` ```live:eg/scalars/str:output ``` ## Strings Rego supports two different types of syntax for declaring strings. The first is likely to be the most familiar: characters surrounded by double quotes. In such strings, certain characters must be escaped to appear in the string, such as double quotes themselves, backslashes, etc. See the [Policy Reference](../policy-reference/#grammar) for a formal definition. The other type of string declaration is a raw string declaration. These are made of characters surrounded by backticks (`` ` ``), with the exception that raw strings may not contain backticks themselves. Raw strings are what they sound like: escape sequences are not interpreted, but instead taken as the literal text inside the backticks. For example, the raw string `` `hello\there` `` will be the text "hello\there", not "hello" and "here" separated by a tab. Raw strings are particularly useful when constructing regular expressions for matching, as it eliminates the need to double escape special characters. A simple example is a regex to match a valid Rego variable. With a regular string, the regex is `"[a-zA-Z_]\\w*"`, but with raw strings, it becomes `` `[a-zA-Z_]\w*` ``. ## Composite Values Composite values define collections. In simple cases, composite values can be treated as constants like [Scalar Values](#scalar-values): ```live:eg/cube:module cube := {"width": 3, "height": 4, "depth": 5} ``` The result: ```live:eg/cube:query:merge_down cube.width ``` ```live:eg/cube:output ``` Composite values can also be defined in terms of [Variables](#variables) or [References](#references). For example: ```live:eg/composite_variables:query:merge_down a := 42 b := false c := null d := {"a": a, "x": [b, c]} ``` ```live:eg/composite_variables:output ``` By defining composite values in terms of variables and references, rules can define abstractions over raw data and other rules. ### Objects Objects are unordered key-value collections. In Rego, any value type can be used as an object key. For example, the following assignment maps port **numbers** to a list of IP addresses (represented as strings). ```live:eg/objects:module:merge_down ips_by_port := { 80: ["1.1.1.1", "1.1.1.2"], 443: ["2.2.2.1"], } ``` ```live:eg/objects/lookup:query:merge_down ips_by_port[80] ``` ```live:eg/objects/lookup:output:merge_down ``` ```live:eg/objects/iteration:query:merge_down some port; ips_by_port[port][_] == "2.2.2.1" ``` ```live:eg/objects/iteration:output ``` When Rego values are converted to JSON non-string object keys are marshalled as strings (because JSON does not support non-string object keys). ```live:eg/objects/marshal:query:merge_down ips_by_port ``` ```live:eg/objects/marshal:output ``` ### Sets In addition to arrays and objects, Rego supports set values. Sets are unordered collections of unique values. Just like other composite values, sets can be defined in terms of scalars, variables, references, and other composite values. For example: ```live:eg/cube/sets:query:merge_down s := {cube.width, cube.height, cube.depth} ``` ```live:eg/cube/sets:output ``` > Set documents are collections of values without keys. OPA represents set documents as arrays when serializing to JSON or other formats that do not support a set data type. The important distinction between sets and arrays or objects is that sets are unkeyed while arrays and objects are keyed, i.e., you cannot refer to the index of an element within a set. When comparing sets, the order of elements does not matter: ```live:eg/set_equality:query:merge_down {1,2,3} == {3,1,2} ``` ```live:eg/set_equality:output ``` Because sets are unordered, variables inside sets must be unified with a ground value outside of the set. If the variable is not unified with a ground value outside the set, OPA will complain: ```live:eg/set_unification:query:merge_down {1,2,3} == {3,x,2} ``` ```live:eg/set_unification:output:expect_unsafe_var ``` Because sets share curly-brace syntax with objects, and an empty object is defined with `{}`, an empty set has to be constructed with a different syntax: ```live:eg/set_construction:query:merge_down count(set()) ``` ```live:eg/set_construction:output ``` ## Variables Variables are another kind of term in Rego. They appear in both the head and body of rules. Variables appearing in the head of a rule can be thought of as input and output of the rule. Unlike many programming languages, where a variable is either an input or an output, in Rego a variable is simultaneously an input and an output. If a query supplies a value for a variable, that variable is an input, and if the query does not supply a value for a variable, that variable is an output. For example: ```live:eg/variables:module sites := [ {"name": "prod"}, {"name": "smoke1"}, {"name": "dev"} ] q[name] { name := sites[_].name } ``` In this case, we evaluate `q` with a variable `x` (which is not bound to a value). As a result, the query returns all of the values for `x` and all of the values for `q[x]`, which are always the same because `q` is a set. ```live:eg/variables:query:merge_down q[x] ``` ```live:eg/variables:output ``` On the other hand, if we evaluate `q` with an input value for `name` we can determine whether `name` exists in the document defined by `q`: ```live:eg/variables/value:query:merge_down q["dev"] ``` ```live:eg/variables/value:output ``` Variables appearing in the head of a rule must also appear in a non-negated equality expression within the same rule. This property ensures that if the rule is evaluated and all of the expressions evaluate to true for some set of variable bindings, the variable in the head of the rule will be defined. ## References References are used to access nested documents. The examples in this section use the data defined in the [Examples](#example-data) section. The simplest reference contains no variables. For example, the following reference returns the hostname of the second server in the first site document from our example data: ```live:eg/data/ref1:query:merge_down sites[0].servers[1].hostname ``` ```live:eg/data/ref1:output ``` References are typically written using the "dot-access" style. The canonical form does away with `.` and closely resembles dictionary lookup in a language such as Python: ```live:eg/data/ref2:query:merge_down sites[0]["servers"][1]["hostname"] ``` ```live:eg/data/ref2:output ``` Both forms are valid, however, the dot-access style is typically more readable. Note that there are four cases where brackets must be used: 1. String keys containing characters other than `[a-z]`, `[A-Z]`, `[0-9]`, or `_` (underscore). 2. Non-string keys such as numbers, booleans, and null. 3. Variable keys which are described later. 4. Composite keys which are described later. The prefix of a reference identifies the root document for that reference. In the example above this is `sites`. The root document may be: * a local variable inside a rule. * a rule inside the same package. * a document stored in OPA. * a documented temporarily provided to OPA as part of a transaction. * an array, object or set, e.g. `[1, 2, 3][0]`. * a function call, e.g. `split("a.b.c", ".")[1]`. * a [comprehension](#comprehensions). ### Variable Keys References can include variables as keys. References written this way are used to select a value from every element in a collection. The following reference will select the hostnames of all the servers in our example data: ```live:eg/data/var_key:query:merge_down sites[i].servers[j].hostname ``` ```live:eg/data/var_key:output ``` Conceptually, this is the same as the following imperative (Python) code: ```python def hostnames(sites): result = [] for site in sites: for server in site.servers: result.append(server.hostname) return result ``` In the reference above, we effectively used variables named `i` and `j` to iterate the collections. If the variables are unused outside the reference, we prefer to replace them with an underscore (`_`) character. The reference above can be rewritten as: ```live:eg/data/meh_key:query:merge_down sites[_].servers[_].hostname ``` ```live:eg/data/meh_key:output ``` The underscore is special because it cannot be referred to by other parts of the rule, e.g., the other side of the expression, another expression, etc. The underscore can be thought of as a special iterator. Each time an underscore is specified, a new iterator is instantiated. > Under the hood, OPA translates the `_` character to a unique variable name that does not conflict with variables and rules that are in scope. ### Composite Keys References can include [Composite Values](#composite-values) as keys if the key is being used to refer into a set. Composite keys may not be used in refs for base data documents, they are only valid for references into virtual documents. This is useful for checking for the presence of composite values within a set, or extracting all values within a set matching some pattern. For example: ```live:eg/composite_key:module s := {[1, 2], [1, 4], [2, 6]} ``` ```live:eg/composite_key/1:query:merge_down s[[1, 2]] ``` ```live:eg/composite_key/1:output ``` ```live:eg/composite_key/2:query:merge_down s[[1, x]] ``` ```live:eg/composite_key/2:output ``` ### Multiple Expressions Rules are often written in terms of multiple expressions that contain references to documents. In the following example, the rule defines a set of arrays where each array contains an application name and a hostname of a server where the application is deployed. ```live:eg/data/multi:module apps_and_hostnames[[name, hostname]] { some i, j, k name := apps[i].name server := apps[i].servers[_] sites[j].servers[k].name == server hostname := sites[j].servers[k].hostname } ``` The result: ```live:eg/data/multi:query:merge_down apps_and_hostnames[x] ``` ```live:eg/data/multi:output ``` Don't worry about understanding everything in this example right now. There are just two important points: 1. Several variables appear more than once in the body. When a variable is used in multiple locations, OPA will only produce documents for the rule with the variable bound to the same value in all expressions. 2. The rule is joining the `apps` and `sites` documents implicitly. In Rego (and other languages based on Datalog), joins are implicit. ### Self-Joins Using a different key on the same array or object provides the equivalent of self-join in SQL. For example, the following rule defines a document containing apps deployed on the same site as `"mysql"`: ```live:eg/data/self_join:module same_site[apps[k].name] { some i, j, k apps[i].name == "mysql" server := apps[i].servers[_] server == sites[j].servers[_].name other_server := sites[j].servers[_].name server != other_server other_server == apps[k].servers[_] } ``` The result: ```live:eg/data/self_join:query:merge_down same_site[x] ``` ```live:eg/data/self_join:output ``` ## Comprehensions Comprehensions provide a concise way of building [Composite Values](#composite-values) from sub-queries. Like [Rules](#rules), comprehensions consist of a head and a body. The body of a comprehension can be understood in exactly the same way as the body of a rule, that is, one or more expressions that must all be true in order for the overall body to be true. When the body evaluates to true, the head of the comprehension is evaluated to produce an element in the result. The body of a comprehension is able to refer to variables defined in the outer body. For example: ```live:eg/data/comprehension_intro:query:merge_down region := "west" names := [name | sites[i].region == region; name := sites[i].name] ``` ```live:eg/data/comprehension_intro:output ``` In the above query, the second expression contains an [Array Comprehension](#array-comprehensions) that refers to the `region` variable. The region variable will be bound in the outer body. > When a comprehension refers to a variable in an outer body, OPA will reorder expressions in the outer body so that variables referred to in the comprehension are bound by the time the comprehension is evaluated. Comprehensions are similar to the same constructs found in other languages like Python. For example, we could write the above comprehension in Python as follows: ```python # Python equivalent of Rego comprehension shown above. names = [site.name for site in sites if site.region == "west"] ``` Comprehensions are often used to group elements by some key. A common use case for comprehensions is to assist in computing aggregate values (e.g., the number of containers running on a host). ### Array Comprehensions Array Comprehensions build array values out of sub-queries. Array Comprehensions have the form: ``` [ <term> | <body> ] ``` For example, the following rule defines an object where the keys are application names and the values are hostnames of servers where the application is deployed. The hostnames of servers are represented as an array. ```live:eg/data/array_comprehension:module app_to_hostnames[app_name] := hostnames { app := apps[_] app_name := app.name hostnames := [hostname | name := app.servers[_] s := sites[_].servers[_] s.name == name hostname := s.hostname] } ``` The result: ```live:eg/data/array_comprehension:query:read_only,merge_down app_to_hostnames[app] ``` ```live:eg/data/array_comprehension:output ``` ### Object Comprehensions Object Comprehensions build object values out of sub-queries. Object Comprehensions have the form: ``` { <key>: <term> | <body> } ``` We can use Object Comprehensions to write the rule from above as a comprehension instead: ```live:eg/data/object_comprehension:module app_to_hostnames := {app.name: hostnames | app := apps[_] hostnames := [hostname | name := app.servers[_] s := sites[_].servers[_] s.name == name hostname := s.hostname] } ``` The result is the same: ```live:eg/data/object_comprehension:query:read_only,merge_down app_to_hostnames[app] ``` ```live:eg/data/object_comprehension:output ``` Object comprehensions are not allowed to have conflicting entries, similar to rules: ```live:eg/data/object_comprehension_conflicting:query:merge_down {"foo": y | z := [1, 2, 3]; y := z[_] } ``` ```live:eg/data/object_comprehension_conflicting:output:expect_conflict ``` ### Set Comprehensions Set Comprehensions build set values out of sub-queries. Set Comprehensions have the form: ``` { <term> | <body> } ``` For example, to construct a set from an array: ```live:eg/data/set_comprehension:module:merge_down a := [1, 2, 3, 4, 3, 4, 3, 4, 5] b := {x | x = a[_]} ``` ```live:eg/data/set_comprehension:query:hidden a b ``` ```live:eg/data/set_comprehension:output ``` ## Rules Rules define the content of [Virtual Documents](../philosophy#how-does-opa-work) in OPA. When OPA evaluates a rule, we say OPA *generates* the content of the document that is defined by the rule. The sample code in this section make use of the data defined in [Examples](#example-data). ### Generating Sets The following rule defines a set containing the hostnames of all servers: ```live:eg/data/rules:module hostnames[name] { name := sites[_].servers[_].hostname } ``` When we query for the content of `hostnames` we see the same data as we would if we queried using the `sites[_].servers[_].hostname` reference directly: ```live:eg/data/rules:query:read_only,merge_down hostnames[name] ``` ```live:eg/data/rules:output ``` This example introduces a few important aspects of Rego. First, the rule defines a set document where the contents are defined by the variable `name`. We know this rule defines a set document because the head only includes a key. All rules have the following form (where key, value, and body are all optional): ``` <name> <key>? <value>? <body>? ``` For a more formal definition of the rule syntax, see the [Policy Reference](../policy-reference/#grammar) document. Second, the `sites[_].servers[_].hostname` fragment selects the `hostname` attribute from all of the objects in the `servers` collection. From reading the fragment in isolation we cannot tell whether the fragment refers to arrays or objects. We only know that it refers to a collections of values. Third, the `name := sites[_].servers[_].hostname` expression binds the value of the `hostname` attribute to the variable `name`, which is also declared in the head of the rule. ### Generating Objects Rules that define objects are very similar to rules that define sets. ```live:eg/data/rule_objects:module apps_by_hostname[hostname] := app { some i server := sites[_].servers[_] hostname := server.hostname apps[i].servers[_] == server.name app := apps[i].name } ``` The rule above defines an object that maps hostnames to app names. The main difference between this rule and one which defines a set is the rule head: in addition to declaring a key, the rule head also declares a value for the document. The result: ```live:eg/data/rule_objects:query:merge_down apps_by_hostname["helium"] ``` ```live:eg/data/rule_objects:output ``` ### Incremental Definitions A rule may be defined multiple times with the same name. When a rule is defined this way, we refer to the rule definition as *incremental* because each definition is additive. The document produced by incrementally defined rules is the union of the documents produced by each individual rule. For example, we can write a rule that abstracts over our `servers` and `containers` data as `instances`: ```live:eg/data/incremental_rule:module instances[instance] { server := sites[_].servers[_] instance := {"address": server.hostname, "name": server.name} } instances[instance] { container := containers[_] instance := {"address": container.ipaddress, "name": container.name} } ``` If the head of the rule is same, we can chain multiple rule bodies together to obtain the same result. We don't recommend using this form anymore. ```live:eg/data/incremental_rule_nr:module:read_only instances[instance] { server := sites[_].servers[_] instance := {"address": server.hostname, "name": server.name} } { container := containers[_] instance := {"address": container.ipaddress, "name": container.name} } ``` An incrementally defined rule can be intuitively understood as `<rule-1> OR <rule-2> OR ... OR <rule-N>`. The result: ```live:eg/data/incremental_rule:query:read_only,merge_down instances[x] ``` ```live:eg/data/incremental_rule:output ``` ### Complete Definitions In addition to rules that *partially* define sets and objects, Rego also supports so-called *complete* definitions of any type of document. Rules provide a complete definition by omitting the key in the head. Complete definitions are commonly used for constants: ```live:complete:module:read_only pi := 3.14159 ``` > Rego allows authors to omit the body of rules. If the body is omitted, it > defaults to true. Documents produced by rules with complete definitions can only have one value at a time. If evaluation produces multiple values for the same document, an error will be returned. For example: ```live:eg/conflicting_rules:module # Define user "bob" for test input. user := "bob" # Define two sets of users: power users and restricted users. Accidentally # include "bob" in both. power_users := {"alice", "bob", "fred"} restricted_users := {"bob", "kim"} # Power users get 32GB memory. max_memory := 32 { power_users[user] } # Restricted users get 4GB memory. max_memory := 4 { restricted_users[user] } ``` Error: ```live:eg/conflicting_rules:output:expect_conflict ``` OPA returns an error in this case because the rule definitions are in *conflict*. The value produced by max_memory cannot be 32 and 4 **at the same time**. The documents produced by rules with complete definitions may still be undefined: ```live:eg/conflicting_rules/undefined:query:merge_down max_memory with user as "johnson" ``` ```live:eg/conflicting_rules/undefined:output:expect_undefined ``` In some cases, having an undefined result for a document is not desirable. In those cases, policies can use the [Default Keyword](#default-keyword) to provide a fallback value. ### Functions Rego supports user-defined functions that can be called with the same semantics as [Built-in Functions](#built-in-functions). They have access to both the [the data Document](../philosophy/#the-opa-document-model) and [the input Document](../philosophy/#the-opa-document-model). For example, the following function will return the result of trimming the spaces from a string and then splitting it by periods. ```live:eg/basic_function:module:merge_down trim_and_split(s) := x { t := trim(s, " ") x := split(t, ".") } ``` ```live:eg/basic_function:query:merge_down trim_and_split(" foo.bar ") ``` ```live:eg/basic_function:output ``` Functions may have an arbitrary number of inputs, but exactly one output. Function arguments may be any kind of term. For example, suppose we have the following function: ```live:eg/function_input:module:read_only foo([x, {"bar": y}]) := z { z := {x: y} } ``` The following calls would produce the logical mappings given: | Call | ``x`` | ``y`` | | ------------------------------------------------------- | ---------- | ----------------------------- | | ``z := foo(a) `` | ``a[0]`` | ``a[1].bar`` | | ``z := foo(["5", {"bar": "hello"}])`` | ``"5"`` | ``"hello"`` | | ``z := foo(["5", {"bar": [1, 2, 3, ["foo", "bar"]]}])`` | ``"5"`` | ``[1, 2, 3, ["foo", "bar"]]`` | If you need multiple outputs, write your functions so that the output is an array, object or set containing your results. If the output term is omitted, it is equivalent to having the output term be the literal `true`. That is, the function declarations below are equivalent: ```live:eg/function_output_unset:module:read_only f(x) { x == "foo" } f(x) = true { x == "foo" } ``` The outputs of user functions have some additional limitations, namely that they must resolve to a single value. If you write a function that has multiple possible bindings for an output variable, you will get a conflict error: ```live:eg/function_single_output:module:merge_down p(x) := y { y := x[_] } ``` ```live:eg/function_single_output:query:merge_down p([1, 2, 3]) ``` ```live:eg/function_single_output:output:expect_conflict ``` It is possible in Rego to define a function more than once, to achieve a conditional selection of which function to execute: Functions can be defined incrementally. ```live:eg/double_function_define:module q(1, x) := y { y := x } q(2, x) := y { y := x*4 } ``` ```live:eg/double_function_define/1:query:merge_down q(1, 2) ``` ```live:eg/double_function_define/1:output ``` ```live:eg/double_function_define/2:query:merge_down q(2, 2) ``` ```live:eg/double_function_define/2:output ``` A given function call will execute all functions that match the signature given. If a call matches multiple functions, they must produce the same output, or else a conflict error will occur: ```live:eg/double_function_define_diff_out:module r(1, x) := y { y := x } r(x, 2) := y { y := x*4 } ``` ```live:eg/double_function_define_diff_out:query:merge_down r(1, 2) ``` ```live:eg/double_function_define_diff_out:output:expect_conflict ``` On the other hand, if a call matches no functions, then the result is undefined. ```live:eg/double_function_define_undefined:module s(x, 2) := y { y := x * 4 } ``` ```live:eg/double_function_define_undefined/1:query:merge_down s(5, 2) ``` ```live:eg/double_function_define_undefined/1:output ``` ```live:eg/double_function_define_undefined/2:query:merge_down s(5, 3) ``` ```live:eg/double_function_define_undefined/2:output:expect_undefined ``` ## Negation To generate the content of a [Virtual Document](../philosophy#how-does-opa-work), OPA attempts to bind variables in the body of the rule such that all expressions in the rule evaluate to True. This generates the correct result when the expressions represent assertions about what states should exist in the data stored in OPA. In some cases, you want to express that certain states *should not* exist in the data stored in OPA. In these cases, negation must be used. For safety, a variable appearing in a negated expression must also appear in another non-negated equality expression in the rule. > OPA will reorder expressions to ensure that negated expressions are evaluated after other non-negated expressions with the same variables. OPA will reject rules containing negated expressions that do not meet the safety criteria described above. The simplest use of negation involves only scalar values or variables and is equivalent to complementing the operator: ```live:eg/simple_negation:module t { greeting := "hello" not greeting == "goodbye" } ``` The result: ```live:eg/simple_negation:query:read_only,merge_down t ``` ```live:eg/simple_negation:output ``` Negation is required to check whether some value *does not* exist in a collection. That is, complementing the operator in an expression such as `p[_] == "foo"` yields `p[_] != "foo"`. However, this is not equivalent to `not p["foo"]`. For example, we can write a rule that defines a document containing names of apps not deployed on the `"prod"` site: ```live:eg/data/negation:module prod_servers[name] { site := sites[_] site.name == "prod" name := site.servers[_].name } apps_in_prod[name] { app := apps[_] server := app.servers[_] prod_servers[server] name := app.name } apps_not_in_prod[name] { name := apps[_].name not apps_in_prod[name] } ``` The result: ```live:eg/data/negation:query:read_only,merge_down apps_not_in_prod[name] ``` ```live:eg/data/negation:output ``` ## Universal Quantification (FOR ALL) Like SQL, Rego does not have a direct way to express _universal quantification_ ("FOR ALL"). However, like SQL, you can use other language primitives (e.g., [Negation](#negation)) to express FOR ALL. For example, imagine you want to express a policy that says (in English): ``` There must be no apps named "bitcoin-miner". ``` A common mistake is to try encoding the policy with a rule named `no_bitcoin_miners` like so: ```live:eg/data/incorrect_no_bitcoin:module:read_only no_bitcoin_miners { app := apps[_] app.name != "bitcoin-miner" # THIS IS NOT CORRECT. } ``` It becomes clear that this is incorrect when you use the [`some`](#some-keyword) keyword, because the rule is true whenever there is SOME app that is not a bitcoin-miner: ```live:eg/data/incorrect_no_bitcoin_some:module import future.keywords.in no_bitcoin_miners { some app in apps app.name != "bitcoin-miner" } ``` You can confirm this by querying the rule: ```live:eg/data/incorrect_no_bitcoin_some:query:merge_down no_bitcoin_miners with apps as [{"name": "bitcoin-miner"}, {"name": "web"}] ``` ```live:eg/data/incorrect_no_bitcoin_some:output ``` The reason the rule is incorrect is that variables in Rego are _existentially quantified_. This means that rule bodies and queries express FOR ANY and not FOR ALL. To express FOR ALL in Rego complement the logic in the rule body (e.g., `!=` becomes `==`) and then complement the check using negation (e.g., `no_bitcoin_miners` becomes `not any_bitcoin_miners`). For this policy, you define a rule that finds if there exists a bitcoin-mining app (which is easy using the `some` keyword). And then you use negation to check that there is NO bitcoin-mining app. Technically, you're using 2 negations and an existential quantifier, which is logically the same as a universal quantifier. For example: ```live:eg/data/correct_negation:module import future.keywords.in no_bitcoin_miners_using_negation { not any_bitcoin_miners } any_bitcoin_miners { some app in apps app.name == "bitcoin-miner" } ``` ```live:eg/data/correct_negation/1:query:merge_down no_bitcoin_miners_using_negation with apps as [{"name": "web"}] ``` ```live:eg/data/correct_negation/1:output ``` ```live:eg/data/correct_negation/2:query:merge_down no_bitcoin_miners_using_negation with apps as [{"name": "bitcoin-miner"}, {"name": "web"}] ``` ```live:eg/data/correct_negation/2:output:expect_undefined ``` {{< info >}} The `undefined` result above is expected because we did not define a default value for `no_bitcoin_miners_using_negation`. Since the body of the rule fails to match, there is no value generated. {{< /info >}} Alternatively, we can implement the same kind of logic inside a single rule using [Comprehensions](#comprehensions). ```live:eg/data/comprehesion_alternative:module:read_only no_bitcoin_miners_using_comprehension { bitcoin_miners := {app | some app in apps; app.name == "bitcoin-miner"} count(bitcoin_miners) == 0 } ``` By importing the future keyword "every", you get another option to express universal quantification: ```live:eg/data/every_alternative:module:read_only import future.keywords.every no_bitcoin_miners_using_every { every app in apps { app.name != "bitcoin-miner" } } ``` {{< info >}} Whether you use negation, comprehensions, or `every` to express FOR ALL is up to you. The `every` keyword should lend itself nicely to a rule formulation that closely follows how requirements are stated, and thus enhances your policy's readability. The comprehension version is more concise than the negation variant, and does not require a helper rule while the negation version is more verbose but a bit simpler and allows for more complex ORs. {{< /info >}} ## Modules In Rego, policies are defined inside *modules*. Modules consist of: * Exactly one [Package](#packages) declaration. * Zero or more [Import](#imports) statements. * Zero or more [Rule](#rules) definitions. Modules are typically represented in Unicode text and encoded in UTF-8. ### Comments Comments begin with the `#` character and continue until the end of the line. ### Packages Packages group the rules defined in one or more modules into a particular namespace. Because rules are namespaced they can be safely shared across projects. Modules contributing to the same package do not have to be located in the same directory. The rules defined in a module are automatically exported. That is, they can be queried under OPA's [Data API](../rest-api#data-api) provided the appropriate package is given. For example, given the following module: ```live:package_declaration:module:read_only package opa.examples pi := 3.14159 ``` The `pi` document can be queried via the Data API: ```http GET https://example.com/v1/data/opa/examples/pi HTTP/1.1 ``` Valid package names are variables or references that only contain string operands. For example, these are all valid package names: ``` package foo package foo.bar package foo.bar.baz package foo["bar.baz"].qux ``` These are invalid package names: ``` package 1foo # not a variable package foo[1].bar # contains non-string operand ``` For more details see the language [Grammar](../policy-reference/#grammar). ### Imports Import statements declare dependencies that modules have on documents defined outside the package. By importing a document, the identifiers exported by that document can be referenced within the current module. All modules contain implicit statements which import the `data` and `input` documents. Modules use the same syntax to declare dependencies on [Base and Virtual Documents](../philosophy#how-does-opa-work). ```live:import_data:module:read_only package opa.examples import data.servers http_servers[server] { server := servers[_] server.protocols[_] == "http" } ``` Similarly, modules can declare dependencies on query arguments by specifying an import path that starts with `input`. ```live:import_input:module:read_only package opa.examples import input.user import input.method # allow alice to perform any operation. allow { user == "alice" } # allow bob to perform read-only operations. allow { user == "bob" method == "GET" } # allows users assigned a "dev" role to perform read-only operations. allow { method == "GET" data.roles["dev"][_] == input.user } ``` Imports can include an optional `as` keyword to handle namespacing issues: ```live:import_namespacing:module:read_only package opa.examples import data.servers as my_servers http_servers[server] { server := my_servers[_] server.protocols[_] == "http" } ``` ## Some Keyword The `some` keyword allows queries to explicitly declare local variables. Use the `some` keyword in rules that contain unification statements or references with variable operands **if** variables contained in those statements are not declared using `:=` . | Statement | Example | Variables | | --- | --- | --- | | Unification | `input.a = [["b", x], [y, "c"]]` | `x` and `y` | | Reference with variable operands | `data.foo[i].bar[j]` | `i` and `j` | For example, the following rule generates tuples of array indices for servers in the "west" region that contain "db" in their name. The first element in the tuple is the site index and the second element is the server index. ```live:eg/data/some:module tuples[[i, j]] { some i, j sites[i].region == "west" server := sites[i].servers[j] # note: 'server' is local because it's declared with := contains(server.name, "db") } ``` If we query for the tuples we get two results: ```live:eg/data/some/i:query:hidden tuples ``` ```live:eg/data/some/i:output ``` Since we have declared `i`, `j`, and `server` to be local, we can introduce rules in the same package without affecting the result above: ```live:eg/data/some/i:module # Define a rule called 'i' i := 1 ``` If we had not declared `i` with the `some` keyword, introducing the `i` rule above would have changed the result of `tuples` because the `i` symbol in the body would capture the global value. Try removing `some i, j` and see what happens! The `some` keyword is not required but it's recommended to avoid situations like the one above where introduction of a rule inside a package could change behaviour of other rules. For using the `some` keyword with iteration, see [the documentation of the `in` operator](#membership-and-iteration-in). ## Every Keyword {{< info >}} To ensure backwards-compatibility, new keywords (like `every`) are introduced slowly. In the first stage, users can opt-in to using the new keywords via a special import: `import future.keywords` introduces _all_ future keywords, and `import future.keywords.every` introduces the `every` keyword described here. There is no need to also import `future.keywords.in`, that is **implied** by importing `future.keywords.every`. At some point in the future, the keyword will become _standard_, and the import will become a no-op that can safely be removed. This should give all users ample time to update their policies, so that the new keyword will not cause clashes with existing variable names. {{< /info >}} ```live:eg/data/every0:module:merge_down import future.keywords.every names_with_dev { some site in sites site.name == "dev" every server in site.servers { endswith(server.name, "-dev") } } ``` ```live:eg/data/every0:query:merge_down names_with_dev ``` ```live:eg/data/every0:output ``` The `every` keyword takes an (optional) key argument, a value argument, a domain, and a block of further queries, its "body". The keyword is used to explicity assert that its body is true for *any element in the domain*. It will iterate over the domain, bind its variables, and check that the body holds for those bindings. If one of the bindings does not yield a successful evaluation of the body, the overall statement is undefined. If the domain is empty, the overall statement is true. Evaluating `every` does **not** introduce new bindings into the rule evaluation. Used with a key argument, the index, or property name (for objects), comes into the scope of the body evaluation: ```live:eg/every1:module:merge_down import future.keywords.every p { every i, x in [1, 2, 3] { x-i == 1 } # array domain } q { every k, v in {"foo": "bar", "fox": "baz" } { # object domain startswith(k, "f") startswith(v, "b") } } r { every x in {1, 2, 3} { x != 4 } # set domain } ``` ```live:eg/every1:output ``` Semantically, `every x in xs { p(x) }` is equivalent to, but shorter than, a "not-some-not" construct using a helper rule: ```live:eg/every2:module:merge_down import future.keywords.every xs := [2, 2, 4, 8] p(x) := x > 1 r { every x in xs { p(x) } } s { not lte_one } lte_one { some x in xs not p(x) } ``` ```live:eg/every2:output ``` Negating `every` is forbidden. If you desire to express `not every x in xs { p(x) }`, please use `some x in xs; not p(x)` instead. ## With Keyword The `with` keyword allows queries to programmatically specify values nested under the [input Document](../philosophy/#the-opa-document-model) and the [data Document](../philosophy/#the-opa-document-model). For example, given the simple authorization policy in the [Imports](#imports) section, we can write a query that checks whether a particular request would be allowed: ```live:import_input/1:query:merge_down allow with input as {"user": "alice", "method": "POST"} ``` ```live:import_input/1:output ``` ```live:import_input/2:query:merge_down allow with input as {"user": "bob", "method": "GET"} ``` ```live:import_input/2:output ``` ```live:import_input/3:query:merge_down not allow with input as {"user": "bob", "method": "DELETE"} ``` ```live:import_input/3:output ``` ```live:import_input/4:query:merge_down allow with input as {"user": "charlie", "method": "GET"} with data.roles as {"dev": ["charlie"]} ``` ```live:import_input/4:output ``` ```live:import_input/5:query:merge_down not allow with input as {"user": "charlie", "method": "GET"} with data.roles as {"dev": ["bob"]} ``` ```live:import_input/5:output ``` The `with` keyword acts as a modifier on expressions. A single expression is allowed to have zero or more `with` modifiers. The `with` keyword has the following syntax: ``` <expr> with <target-1> as <value-1> [with <target-2> as <value-2> [...]] ``` The `<target>`s must be references to values in the input document (or the input document itself) or data document. > When applied to the `data` document, the `<target>` must not attempt to > partially define virtual documents. For example, given a virtual document at > path `data.foo.bar`, the compiler will generate an error if the policy > attempts to replace `data.foo.bar.baz`. The `with` keyword only affects the attached expression. Subsequent expressions will see the unmodified value. The exception to this rule is when multiple `with` keywords are in-scope like below: ```live:multiple_with:module:read_only inner := [x, y] { x := input.foo y := input.bar } middle := [a, b] { a := inner with input.foo as 100 b := input } outer := result { result := middle with input as {"foo": 200, "bar": 300} } ``` ## Default Keyword The `default` keyword allows policies to define a default value for documents produced by rules with [Complete Definitions](#complete-definitions). The default value is used when all of the rules sharing the same name are undefined. For example: ```live:eg/default:module default allow := false allow { input.user == "bob" input.method == "GET" } allow { input.user == "alice" } ``` When the `allow` document is queried, the return value will be either `true` or `false`. ```live:eg/default:query:hidden allow ``` ```live:eg/default:input:merge_down { "user": "bob", "method": "POST" } ``` ```live:eg/default:output ``` Without the default definition, the `allow` document would simply be undefined for the same input. When the `default` keyword is used, the rule syntax is restricted to: ``` default <name> := <term> ``` The term may be any scalar, composite, or comprehension value but it may not be a variable or reference. If the value is a composite then it may not contain variables or references. ## Else Keyword The ``else`` keyword is a basic control flow construct that gives you control over rule evaluation order. Rules grouped together with the ``else`` keyword are evaluated until a match is found. Once a match is found, rule evaluation does not proceed to rules further in the chain. The ``else`` keyword is useful if you are porting policies into Rego from an order-sensitive system like IPTables. ```live:eg/else:module authorize := "allow" { input.user == "superuser" # allow 'superuser' to perform any operation. } else := "deny" { input.path[0] == "admin" # disallow 'admin' operations... input.source_network == "external" # from external networks. } # ... more rules ``` ```live:eg/else:query:hidden authorize ``` In the example below, evaluation stops immediately after the first rule even though the input matches the second rule as well. ```live:eg/else/1:input:merge_down { "path": [ "admin", "exec_shell" ], "source_network": "external", "user": "superuser" } ``` ```live:eg/else/1:output ``` In the next example, the input matches the second rule (but not the first) so evaluation continues to the second rule before stopping. ```live:eg/else/2:input:merge_down { "path": [ "admin", "exec_shell" ], "source_network": "external", "user": "alice" } ``` ```live:eg/else/2:output ``` The `else` keyword may be used repeatedly on the same rule and there is no limit imposed on the number of `else` clauses on a rule. ## Operators ### Membership and iteration: `in` {{< info >}} To ensure backwards-compatibility, new keywords (like `in`) are introduced slowly. In the first stage, users can opt-in to using the new keywords via a special import: `import future.keywords` introduces _all_ future keywords, and `import future.keywords.in` introduces the `in` keyword described here. At some point in the future, the keyword will become _standard_, and the import will become a no-op that can safely be removed. This should give all users ample time to update their policies, so that the new keyword will not cause clashes with existing variable names. {{< /info >}} The membership operator `in` lets you check if an element is part of a collection (array, set, or object). It always evaluates to `true` or `false`: ```live:eg/member1:module:merge_down import future.keywords.in p := [x, y, z] { x := 3 in [1, 2, 3] # array y := 3 in {1, 2, 3} # set z := 3 in {"foo": 1, "bar": 3} # object } ``` ```live:eg/member1:output ``` When providing two arguments on the left-hand side of the `in` operator, and an object or an array on the right-hand side, the first argument is taken to be the key (object) or index (array), respectively: ```live:eg/member1c:module:merge_down import future.keywords.in p := [x, y] { x := "foo", "bar" in {"foo": "bar"} # key, val with object y := 2, "baz" in ["foo", "bar", "baz"] # key, val with array } ``` ```live:eg/member1c:output ``` **Note** that in list contexts, like set or array definitions and function arguments, parentheses are required to use the form with two left-hand side arguments -- compare: ```live:eg/member1d:module:merge_down import future.keywords.in p := x { x := { 0, 2 in [2] } } q := x { x := { (0, 2 in [2]) } } w := x { x := g((0, 2 in [2])) } z := x { x := f(0, 2 in [2]) } f(x, y) := sprintf("two function arguments: %v, %v", [x, y]) g(x) := sprintf("one function argument: %v", [x]) ``` ```live:eg/member1d:output ``` Combined with `not`, the operator can be handy when asserting that an element is _not_ member of an array: ```live:eg/member1a:module:merge_down import future.keywords.in deny { not "admin" in input.user.roles } test_deny { deny with input.user.roles as ["operator", "user"] } ``` ```live:eg/member1a:output ``` **Note** that expressions using the `in` operator _always return `true` or `false`_, even when called in non-collection arguments: ```live:eg/member1b:module:merge_down import future.keywords.in q := x { x := 3 in "three" } ``` ```live:eg/member1b:output ``` Using the `some` variant, it can be used to introduce new variables based on a collections' items: ```live:eg/member2:module:merge_down import future.keywords.in p[x] { some x in ["a", "r", "r", "a", "y"] } q[x] { some x in {"s", "e", "t"} } r[x] { some x in {"foo": "bar", "baz": "quz"} } ``` ```live:eg/member2:output ``` Furthermore, passing a second argument allows you to work with _object keys_ and _array indices_: ```live:eg/member3:module:merge_down import future.keywords.in p[x] { some x, "r" in ["a", "r", "r", "a", "y"] # key variable, value constant } q[x] = y { some x, y in ["a", "r", "r", "a", "y"] # both variables } r[y] = x { some x, y in {"foo": "bar", "baz": "quz"} } ``` ```live:eg/member3:output ``` Any argument to the `some` variant can be a composite, non-ground value: ```live:eg/member4:module:merge_down import future.keywords.in p[x] = y { some x, {"foo": y} in [{"foo": 100}, {"bar": 200}] } p[x] = y { some {"bar": x}, {"foo": y} in {{"bar": "b"}: {"foo": "f"}} } ``` ```live:eg/member4:output ``` ### Equality: Assignment, Comparison, and Unification Rego supports three kinds of equality: assignment (`:=`), comparison (`==`), and unification `=`. We recommend using assignment (`:=`) and comparison (`==`) whenever possible for policies that are easier to read and write. #### Assignment `:=` The assignment operator (`:=`) is used to assign values to variables. Variables assigned inside a rule are locally scoped to that rule and shadow global variables. ```live:eg/assignment1:module:read_only x := 100 p { x := 1 # declare local variable 'x' and assign value 1 x != 100 # true because 'x' refers to local variable } ``` Assigned variables are not allowed to appear before the assignment in the query. For example, the following policy will not compile: ```live:eg/assignment2:module:merge_down p { x != 100 x := 1 # error because x appears earlier in the query. } q { x := 1 x := 2 # error because x is assigned twice. } ``` ```live:eg/assignment2:output:expect_assigned_above,expect_referenced_above ``` A simple form of destructuring can be used to unpack values from arrays and assign them to variables: ```live:eg/assignment3:module:read_only address := ["3 Abbey Road", "NW8 9AY", "London", "England"] in_london { [_, _, city, country] := address city == "London" country == "England" } ``` ```live:eg/assignment3:output ``` #### Comparison `==` Comparison checks if two values are equal within a rule. If the left or right hand side contains a variable that has not been assigned a value, the compiler throws an error. ```live:eg/comparison1:module:merge_down p { x := 100 x == 100 # true because x refers to the local variable } ``` ```live:eg/comparison1:output ``` ```live:eg/comparison2:module:merge_down y := 100 q { y == 100 # true because y refers to the global variable } ``` ```live:eg/comparison2:output ``` ```live:eg/comparison3:module:merge_down r { z == 100 # compiler error because z has not been assigned a value } ``` ```live:eg/comparison3:output:expect_unsafe_var ``` #### Unification `=` Unification (`=`) combines assignment and comparison. Rego will assign variables to values that make the comparison true. Unification lets you ask for values for variables that make an expression true. ```live:eg/unification1:query:merge_down # Find values for x and y that make the equality true [x, "world"] = ["hello", y] ``` ```live:eg/unification1:output ``` ```live:eg/data/unification2:query:merge_down sites[i].servers[j].name = apps[k].servers[m] ``` ```live:eg/data/unification2:output ``` #### Best Practices for Equality Here is a comparison of the three forms of equality. ``` Equality Applicable Compiler Errors Use Case -------- ----------- ------------------------- ---------------------- := Everywhere Var already assigned Assign variable == Everywhere Var not assigned Compare values = Everywhere Values cannot be computed Express query ``` Best practice is to use assignment `:=` and comparison `==` wherever possible. The additional compiler checks help avoid errors when writing policy, and the additional syntax helps make the intent clearer when reading policy. Under the hood `:=` and `==` are syntactic sugar for `=`, local variable creation, and additional compiler checks. ### Comparison Operators The following comparison operators are supported: ```live:comparison_operators:module:read_only a == b # `a` is equal to `b`. a != b # `a` is not equal to `b`. a < b # `a` is less than `b`. a <= b # `a` is less than or equal to `b`. a > b # `a` is greater than `b`. a >= b # `a` is greater than or equal to `b`. ``` None of these operators bind variables contained in the expression. As a result, if either operand is a variable, the variable must appear in another expression in the same rule that would cause the variable to be bound, i.e., an equality expression or the target position of a built-in function. ## Built-in Functions In some cases, rules must perform simple arithmetic, aggregation, and so on. Rego provides a number of built-in functions (or "built-ins") for performing these tasks. Built-ins can be easily recognized by their syntax. All built-ins have the following form: ``` <name>(<arg-1>, <arg-2>, ..., <arg-n>) ``` Built-ins usually take one or more input values and produce one output value. Unless stated otherwise, all built-ins accept values or variables as output arguments. If a built-in function is invoked with a variable as input, the variable must be _safe_, i.e., it must be assigned elsewhere in the query. Built-ins can include "." characters in the name. This allows them to be namespaced. If you are adding custom built-ins to OPA, consider namespacing them to avoid naming conflicts, e.g., `org.example.special_func`. See the [Policy Reference](../policy-reference#built-in-functions) document for details on each built-in function. ### Errors By default, built-in function calls that encounter runtime errors evaluate to undefined (which can usually be treated as `false`) and do not halt policy evaluation. This ensures that built-in functions can be called with invalid inputs without causing the entire policy to stop evaluating. In most cases, policies do not have to implement any kind of error handling logic. If error handling is required, the built-in function call can be negated to test for undefined. For example: ```live:eg/errors:module:merge_down allow { io.jwt.verify_hs256(input.token, "secret") [_, payload, _] := io.jwt.decode(input.token) payload.role == "admin" } reason["invalid JWT supplied as input"] { not io.jwt.decode(input.token) } ``` ```live:eg/errors:input:merge_down { "token": "a poorly formatted token" } ``` ```live:eg/errors:output ``` If you wish to disable this behaviour and instead have built-in function call errors treated as exceptions that halt policy evaluation enable "strict built-in errors" in the caller: API | Flag --- | --- `POST v1/data` (HTTP) | `strict-builtin-errors` query parameter `GET v1/data` (HTTP) | `strict-builtin-errors` query parameter `opa eval` (CLI) | `--strict-builtin-errors` `opa run` (REPL) | `> strict-builtin-errors` `rego` Go module | `rego.StrictBuiltinErrors(true)` option Wasm | Not Available ## Example Data The rules below define the content of documents describing a simplistic deployment environment. These documents are referenced in other sections above. ```live:eg/data:module sites := [ { "region": "east", "name": "prod", "servers": [ { "name": "web-0", "hostname": "hydrogen" }, { "name": "web-1", "hostname": "helium" }, { "name": "db-0", "hostname": "lithium" } ] }, { "region": "west", "name": "smoke", "servers": [ { "name": "web-1000", "hostname": "beryllium" }, { "name": "web-1001", "hostname": "boron" }, { "name": "db-1000", "hostname": "carbon" } ] }, { "region": "west", "name": "dev", "servers": [ { "name": "web-dev", "hostname": "nitrogen" }, { "name": "db-dev", "hostname": "oxygen" } ] } ] apps := [ { "name": "web", "servers": ["web-0", "web-1", "web-1000", "web-1001", "web-dev"] }, { "name": "mysql", "servers": ["db-0", "db-1000"] }, { "name": "mongodb", "servers": ["db-dev"] } ] containers := [ { "image": "redis", "ipaddress": "10.0.0.1", "name": "big_stallman" }, { "image": "nginx", "ipaddress": "10.0.0.2", "name": "cranky_euclid" } ] ```
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package dmutex_test import ( "fmt" "testing" "time" "github.com/gigawattio/testlib" "github.com/gigawattio/zklib/dmutex" zktestutil "github.com/gigawattio/zklib/testutil" zkutil "github.com/gigawattio/zklib/util" "github.com/samuel/go-zookeeper/zk" ) func Test_DistributedMutexService(t *testing.T) { zkPath := fmt.Sprintf("/%v", testlib.CurrentRunningTest()) zktestutil.WithZk(t, 1, "127.0.0.1:2181", func(zkServers []string) { defer func() { if err := zkutil.ResetZk(zkServers, zkPath); err != nil { t.Error(err) } }() service := dmutex.NewDistributedMutexService(zkServers, 5*time.Second, zkPath) objectId1 := "my-app-1" timeout := 3 * time.Second if err := service.Lock(objectId1, "1", timeout); err != nil { t.Fatal(err) } if err := service.Lock(objectId1, "2", timeout); !dmutex.IsAcquisitionFailedError(err) { t.Fatalf("Locked object failure error did not match expected `dmutex.DistributedMutexAcquisitionFailed': %s", err) } objectId2 := "my-app-2" if err := service.Lock(objectId2, "3", timeout); err != nil { t.Fatal(err) } if err := service.Unlock(objectId1); err != nil { t.Fatal(err) } if err := service.Lock(objectId1, "4", timeout); err != nil { t.Fatal(err) } if err := service.Unlock(objectId1); err != nil { t.Fatal(err) } }) } func Test_DistributedMutexServiceCleaner(t *testing.T) { zktestutil.WithZk(t, 1, "127.0.0.1:2181", func(zkServers []string) { var ( timeout = 3 * time.Second zkPath = fmt.Sprintf("/%v/", testlib.CurrentRunningTest()) objectId = func(id interface{}) string { return fmt.Sprintf("my-app-%v", id) } service = dmutex.NewDistributedMutexService(zkServers, 5*time.Second, zkPath) ) defer func() { if err := zkutil.ResetZk(zkServers, strings.TrimRight(zkPath, "/")); err != nil { t.Errorf("Unexpected error while resetting zkPath=%q: %s", zkPath, err) } }() cleanAndVerifyNumChildren := func(expected int) error { if err := service.Clean(); err != nil { return fmt.Errorf("Unexpected error from Clean(): %s", err) } err := zkutil.WithZkSession(zkServers, timeout, func(conn *zk.Conn) error { children, _, err := conn.Children(zkutil.NormalizePath(zkPath)) if err != nil && err != zk.ErrNoNode { t.Logf("children=%v err=%s\n", children, err) return err } if actual := len(children); actual != expected { return fmt.Errorf("Expected num children under %q == %v but actual count is %v", zkPath, expected, actual) } return nil }) if err != nil { return err } return nil } if err := cleanAndVerifyNumChildren(0); err != nil { t.Fatal(err) } if err := service.Lock(objectId(1), "1", timeout); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(1); err != nil { t.Fatal(err) } if err := service.Lock(objectId(1), "2", timeout); !dmutex.IsAcquisitionFailedError(err) { t.Fatalf("Locked object failure error did not match expected `dmutex.DistributedMutexAcquisitionFailed': %s", err) } if err := cleanAndVerifyNumChildren(1); err != nil { t.Fatal(err) } if err := service.Lock(objectId(2), "3", timeout); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(2); err != nil { t.Fatal(err) } if err := service.Lock(objectId(3), "4", timeout); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(3); err != nil { t.Fatal(err) } if err := service.Unlock(objectId(1)); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(2); err != nil { t.Fatal(err) } if err := service.Unlock(objectId(3)); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(1); err != nil { t.Fatal(err) } if err := service.Lock(objectId(1), "5", timeout); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(2); err != nil { t.Fatal(err) } if err := service.Unlock(objectId(2)); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(1); err != nil { t.Fatal(err) } if err := service.Unlock(objectId(1)); err != nil { t.Fatal(err) } if err := cleanAndVerifyNumChildren(0); err != nil { t.Fatal(err) } }) }
{ "redpajama_set_name": "RedPajamaGithub" }
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Danone je nadnárodní potravinářská firma. Historie Firmu založil Isaac Carasso roku 1919 ve Španělsku, když ho inspiroval propagátor jogurtů Ilja Iljič Mečnikov. Jogurty se ale tehdy prodávaly v lékárně. Poté se přesunula i do Francie. Tam se sloučila s firmou Gervais a později s dalšími firmami. Firmě patřila například Opavia. Firma otevírá Danone Instituty po celém světě. V roce 2013 prodala svoji mlékárnu v Benešově americké společnosti Schreiber Foods. Odkazy Reference Externí odkazy Nadnárodní korporace Firmy založené roku 1919 Potravinářské firmy Francouzské firmy
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{"url":"http:\/\/www.journaltocs.ac.uk\/index.php?action=browse&subAction=subjects&publisherID=8&journalID=2768&pageb=1&userQueryID=&sort=&local_page=1&sorType=&sorCol=","text":"Subjects -> MATHEMATICS (Total: 1061 journals) \u00a0 \u00a0 - APPLIED MATHEMATICS (86 journals)\u00a0 \u00a0 - GEOMETRY AND TOPOLOGY (23 journals)\u00a0 \u00a0 - MATHEMATICS (783 journals)\u00a0 \u00a0 - MATHEMATICS (GENERAL) (43 journals)\u00a0 \u00a0 - NUMERICAL ANALYSIS (23 journals)\u00a0 \u00a0 - PROBABILITIES AND MATH STATISTICS (103 journals) MATHEMATICS (783 journals) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 1 2 3 4 | Last\n\n1 2 3 4 | Last\n\nSimilar Journals\n Computational ComplexityJournal Prestige (SJR): 0.381 Citation Impact (citeScore): 1Number of Followers: 4 \u00a0 \u00a0\u00a0 Hybrid journal (It can contain Open Access articles) ISSN (Print) 1420-8954 - ISSN (Online) 1016-3328 Published by Springer-Verlag \u00a0[2570 journals]\n\u2022 Approximate Nonnegative Rank is Equivalent to the Smooth Rectangle Bound\n\u2022 Abstract: Abstract We consider two known lower bounds on randomized communication complexity: the smooth rectangle bound and the logarithm of the approximate nonnegative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term. The logarithm of the nonnegative rank is known to be a nearly tight lower bound on the deterministic communication complexity. Our result indicates that proving an analogous result for the randomized case, namely that the log approximate nonnegative rank is a nearly tight bound on randomized communication complexity, would imply the tightness of the information complexity bound. Another corollary of our result is the existence of a Boolean function with a quasipolynomial gap between its approximate rank and approximate nonnegative rank. We also show that our method yields an alternative simple proof of the equivalence between the approximate rank and the approximate \u03bc norm, first shown by Lee and Shraibman.\nPubDate: 2019-03-01\n\n\u2022 Asymptotic tensor rank of graph tensors: beyond matrix multiplication\n\u2022 Abstract: Abstract We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For $${k \\geq 4}$$ , we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the exponent per edge can be below 2\/3, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalize to higher-order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.\nPubDate: 2019-03-01\n\n\u2022 Query-to-Communication Lifting for P NP\n\u2022 Abstract: Abstract We prove that the PNP-type query complexity (alternatively, decision list width) of any Boolean function f is quadratically related to the PNP-type communication complexity of a lifted version of f. As an application, we show that a certain \u201cproduct\u201d lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture PNP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).\nPubDate: 2019-03-01\n\n\u2022 The Average Sensitivity of Bounded-Depth Formulas\n\u2022 Authors: Benjamin Rossman\nPages: 209 - 223\nAbstract: Abstract We show that unbounded fan-in Boolean formulas of depth d\u00a0+\u00a01 and size s have average sensitivity $${O(\\frac{1}{d} \\log s)^d}$$ . In particular, this gives a tight $${2^{\\Omega(d(n^{1\/d}-1))}}$$ lower bound on the size of depth d\u00a0+\u00a01 formulas computing the parity function. These results strengthen the corresponding $${2^{\\Omega(n^{1\/d})}}$$ and $${O(\\log s)^d}$$ bounds for circuits due to H\u00e5stad (Proceedings of the 18th annual ACM symposium on theory of computing, ACM, New York, 1986) and Boppana (Inf Process Lett 63(5): 257\u2013261, 1997). Our proof technique studies a random process where the switching lemma is applied to formulas in an efficient manner.\nPubDate: 2018-06-01\nDOI: 10.1007\/s00037-017-0156-0\nIssue No: Vol. 27, No. 2 (2018)\n\n\u2022 Local Expanders\n\u2022 Authors: Emanuele Viola; Avi Wigderson\nPages: 225 - 244\nAbstract: Abstract A map $${f : \\{0,1\\}^{n} \\to \\{0,1\\}^{n}}$$ has locality t if every output bit of f depends only on t input bits. Arora et\u00a0al. (Colloquium on automata, languages and programming, ICALP, 2009) asked if there exist bounded-degree expander graphs on 2 n nodes such that the neighbors of a node $${x\\in\\{0,1\\}^{n}}$$ can be computed by maps of constant locality. We give an explicit construction of such graphs with locality one. We then give three applications of this construction: (1) lossless expanders with constant locality, (2) more efficient error reduction for randomized algorithms, and (3) more efficient hardness amplification of one-way permutations. We also give, for n of the form $${n=4\\cdot3^{t}}$$ , an explicit construction of bipartite Ramanujan graphs of degree 3 with 2 n \u22121 nodes in each side such that the neighbors of a node $${x\\in \\{0,1\\}^{n}{\\setminus} \\{0^{n}\\}}$$ can be computed either (1) in constant locality or (2) in constant time using standard operations on words of length $${\\Omega(n)}$$ . Our results use in black-box fashion deep explicit constructions of Cayley expander graphs, by Kassabov (Invent Math 170(2):327\u2013354, 2007) for the symmetric group $${S_{n}}$$ and by Morgenstern (J Comb Theory Ser B 62(1):44\u201362, 1994) for the special linear group SL $${(2,F_{2^{n}})}$$ .\nPubDate: 2018-06-01\nDOI: 10.1007\/s00037-017-0155-1\nIssue No: Vol. 27, No. 2 (2018)\n\n\u2022 Matrix rigidity of random Toeplitz matrices\n\u2022 Authors: Oded Goldreich; Avishay Tal\nPages: 305 - 350\nAbstract: Abstract A matrix A is said to have rigidity s for rank r if A differs from any matrix of rank r on more than s entries. We prove that random n-by-n Toeplitz matrices over $${\\mathbb{F}_{2}}$$ (i.e., matrices of the form $${A_{i,j} = a_{i-j}}$$ for random bits $${a_{-(n-1)}, \\ldots, a_{n-1}}$$ ) have rigidity $${\\Omega(n^3\/(r^2\\log n))}$$ for rank $${r \\ge \\sqrt{n}}$$ , with high probability. This improves, for $${r = o(n\/\\log n \\log\\log n)}$$ , over the $${\\Omega(\\frac{n^2}{r} \\cdot\\log(\\frac{n}{r}))}$$ bound that is known for many explicit matrices. Our result implies that the explicit trilinear $${[n]\\times [n] \\times [2n]}$$ function defined by $${F(x,y,z) = \\sum_{i,j}{x_i y_j z_{i+j}}}$$ has complexity $${\\Omega(n^{3\/5})}$$ in the multilinear circuit model suggested by Goldreich and Wigderson (Electron Colloq Comput Complex 20:43, 2013), which yields an $${\\exp(n^{3\/5})}$$ lower bound on the size of the so-called canonical depth-three circuits for F. We also prove that F has complexity $${\\tilde{\\Omega}(n^{2\/3})}$$ if the multilinear circuits are further restricted to be of depth 2. In addition, we show that a matrix whose entries are sampled from a $${2^{-n}}$$ -biased distribution has complexity $${\\tilde{\\Omega}(n^{2\/3})}$$ , regardless of depth restrictions, almost matching the known $${O(n^{2\/3})}$$ upper bound for any matrix. We turn this randomized construction into an explicit 4-linear construction with similar lower bounds, using the quadratic small-biased construction of Mossel et\u00a0al. (Random Struct Algorithms 29(1):56\u201381, 2006).\nPubDate: 2018-06-01\nDOI: 10.1007\/s00037-016-0144-9\nIssue No: Vol. 27, No. 2 (2018)\n\n\u2022 On the hardness of the noncommutative determinant\n\u2022 Authors: V. Arvind; Srikanth Srinivasan\nPages: 1 - 29\nAbstract: Abstract In this paper, we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits. For any field $${\\mathbb{F}}$$ we show that computing the $${n\\times n}$$ permanent over $${\\mathbb{F}}$$ is polynomial-time reducible to computing the $${2n\\times 2n}$$ (noncommutative) determinant whose entries are $${O(n^2)\\times O(n^2)}$$ matrices over the field $${\\mathbb{F}}$$ . We also derive as a consequence that computing the $${n\\times n}$$ permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of $${n^{O(1)}}$$ dimension. Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials.\nPubDate: 2018-03-01\nDOI: 10.1007\/s00037-016-0148-5\nIssue No: Vol. 27, No. 1 (2018)\n\n\u2022 Interactive proofs and a Shamir-like result for real number computations\n\u2022 Abstract: Abstract We introduce and study interactive proofs in the framework of real number computations as introduced by Blum, Shub, and Smale. Ivanov and de Rougemont started this line of research showing that an analogue of Shamir\u2019s result holds in the real additive Blum\u2013Shub\u2013Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model in which also multiplications can be performed and reals can be exchanged. The ultimate goal is to give a Shamir-like characterization of the real counterpart $${{\\rm IP}_\\mathbb{R}}$$ of classical IP. Whereas classically Shamir\u2019s result implies IP \u00a0=\u00a0 PSPACE \u00a0=\u00a0 PAT \u00a0=\u00a0 PAR, in our framework a major difficulty arises: In contrast to Turing complexity theory, the real number classes $${{\\rm PAR}_\\mathbb{R}}$$ and $${{\\rm PAT}_\\mathbb{R}}$$ differ and space resources considered separately are not meaningful. It is not obvious how to figure out whether at all $${{\\rm IP}_\\mathbb{R}}$$ is characterized by one of the above classes\u2014and if so by which. We obtain two main results, an upper and a lower bound for the new class $${{\\rm IP}_\\mathbb{R}.}$$ As upper bound we establish $${{{\\rm IP}_\\mathbb{R}} \\subseteq {\\rm MA\\exists}_\\mathbb{R}}$$ , where $${{\\rm MA} \\exists_\\mathbb{R}}$$ is a real complexity class introduced by Cucker and Briquel satisfying $${{\\rm PAR}_\\mathbb{R} \\subsetneq {\\rm MA}\\exists_{\\mathbb{R}} \\subseteq {\\rm PAT}_\\mathbb{R}}$$ and conjectured to be different from $${{\\rm PAT}_\\mathbb{R}}$$ . We then complement this result and prove a non-trivial lower bound for $${{\\rm IP}_\\mathbb{R}}$$ . More precisely, we design interactive real protocols verifying function values for a large class of functions introduced by Koiran and Perifel and denoted by UniformVPSPACE $${^{0}.}$$ As a consequence, we show $${{\\rm PAR}_\\mathbb{R} \\subseteq {\\rm IP}_\\mathbb{R}}$$ , which in particular implies co- $${{\\rm NP}_\\mathbb{R} \\subseteq {\\rm IP}_\\mathbb{R}}$$ , and $${{\\rm P}_\\mathbb{R}^{Res} \\subseteq {\\rm IP}_\\mathbb{R}}$$ , where Res denotes certain multivariate Resultant polynomials. Our proof techniques are guided by the question in how far Shamir\u2019s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVPSPACE $${^{0}}$$ are extremely helpful.\nPubDate: 2018-11-07\n\n\u2022 Lower Bounds and PIT for Non-commutative Arithmetic Circuits with\nRestricted Parse Trees\n\u2022 Abstract: Abstract We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $${\\mathbb{F}\\langle{x_1,\\ldots,x_N\\rangle}}$$ , where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits. We show exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde et\u00a0al. [Electronic Colloquium on Comput Complexity (ECCC) vol 23, no 94, 2016], who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree. The polynomial we prove a lower bound for is in fact computable by a polynomial-sized non-commutative circuit. We show exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye et\u00a0al. (Theory Comput 12(1):1\u201338, 2016) and the above lower bounds of Lagarde et\u00a0al. (2016), which are known to be incomparable. Here too, the hard polynomial is computable by a polynomial-sized non-commutative circuit. We make progress on a question of Nisan (STOC, pp 410\u2013418, 1991) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of $${n^{\\Omega(\\log d)}}$$ for any UPT formula computing the product of d $${n \\times n}$$ matrices. When $${d \\leq \\log n}$$ , we can also prove superpolynomial lower bounds for formulas with up to $${2^{o(d)}}$$ many parse trees (for computing the same polynomial). Improving this bound to allow for $${2^{o(d)}}$$ trees would give an unconditional separation between ABPs and Formulas. We give deterministic whitebox PIT algorithms for UPT circuits over any field, strengthening a result of Lagarde et\u00a0al. (2016), and also for sums of a constant number of UPT circuits with different parse trees.\nPubDate: 2018-09-29\n\n\u2022 Some observations on holographic algorithms\n\u2022 Abstract: Abstract We define the notion of diversity for families of finite functions and express the limitations of a simple class of holographic algorithms, called elementary algorithms, in terms of limitations on diversity. We show that this class of elementary algorithms is too weak to solve the Boolean circuit value problem, or Boolean satisfiability, or the permanent. The lower bound argument is a natural but apparently novel combination of counting and algebraic dependence arguments that is viable in the holographic framework. We go on to describe polynomial time holographic algorithms that go beyond the elementarity restriction in the two respects that they use exponential size fields, and multiple oracle calls in the form of polynomial interpolation. These new algorithms, which use bases of three components, compute the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case, the parity can also be computed for any one slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes.\nPubDate: 2018-09-01\n\n\u2022 Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via\nCommunication Complexity\n\u2022 Abstract: Abstract One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de Morgan formulas. Karchmer et\u00a0al. (Comput Complex 5(3\/4):191\u2013204, 1995b) suggested to approach this problem by proving that formula complexity behaves \u201cas expected\u201d with respect to the composition of functions $${f\\diamond g}$$ . They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds. The first step toward proving the KRW conjecture was made by Edmonds et\u00a0al. (Comput Complex 10(3):210\u2013246, 2001), who proved an analogue of the conjecture for the composition of \u201cuniversal relations.\u201d In this work, we extend the argument of Edmonds et\u00a0al. (2001) further to $${f\\diamond g}$$ where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in H\u00e5stad\u2019s work on random restrictions (H\u00e5stad in SIAM J Comput 27(1):48\u201364, 1998), our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer & Wigderson in (SIAM J Discrete Math 3(2):255\u2013265, 1990). In addition, our proof gives a new structural result, which roughly says that the naive way for computing $${f\\diamond g}$$ is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n 3-o(1) due to H\u00e5stad (1998).\nPubDate: 2018-09-01\n\n\u2022 Communication with Contextual Uncertainty\n\u2022 Abstract: Abstract We introduce a simple model illustrating the utility of context in compressing communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information $${X \\in \\{0,1\\}^n}$$ and Bob gets $${Y \\in \\{0,1\\}^n}$$ , where (X, Y) is drawn from a known distribution, and Bob wishes to compute some function g(X, Y) or some close approximation to it (i.e., the output is g(X, Y) with high probability over (X, Y)). In our variant, Alice does not know g, but only knows some function f which is a very close approximation to g. Thus, the function being computed forms the context for the communication. It is an enormous implicit input, potentially described by a truth table of size 2 n . Imprecise knowledge of this function models the (mild) uncertainty in this context. We show that uncertainty can lead to a huge cost in communication. Specifically, we construct a distribution $${\\mu}$$ over $${(X,Y)\\in \\{0,1\\}^n \\times \\{0,1\\}^n}$$ and a class of function pairs (f, g) which are very close (i.e., disagree with o(1) probability when (X, Y) are sampled according to $${\\mu}$$ ), for which the communication complexity of f or g in the standard setting is one bit, whereas the (two-way) communication complexity in the uncertain setting is at least $${\\Omega(\\sqrt{n})}$$ bits even when allowing a constant probability of error. It turns out that this blow-up in communication complexity can be attributed in part to the mutual information between X and Y. In particular, we give an efficient protocol for communication under contextual uncertainty that incurs only a small blow-up in communication if this mutual information is small. Namely, we show that if g has a communication protocol with complexity k in the standard setting and the mutual information between X and Y is I, then g has a one-way communication protocol with complexity $${O((1+I)\\cdot 2^k)}$$ in the uncertain setting. This result is an immediate corollary of an even stronger result which shows that if g has one-way communication complexity k, then it has one-way uncertain-communication complexity at most $${O((1+I)\\cdot k)}$$ . In the particular case where the input distribution is a product distribution (and so I\u00a0=\u00a00), the protocol in the uncertain setting only incurs a constant factor blow-up in one-way communication and error.\nPubDate: 2018-09-01\n\n\u2022 On the Relationship Between Statistical Zero-Knowledge and Statistical\nRandomized Encodings\n\u2022 Abstract: Abstract Statistical Zero-knowledge proofs (Goldwasser et\u00a0al. in SICOMP: SIAM J Comput, 1989) allow a computationally unbounded server to convince a computationally limited client that an input x is in a language $${\\Pi}$$ without revealing any additional information about x that the client cannot compute by herself. Randomized encoding (RE) of functions (Ishai & Kushilevitz in FOCS 2000) allows a computationally limited client to publish a single (randomized) message, $${{\\rm Enc}(x)}$$ , from which the server learns whether x is in $${\\Pi}$$ and nothing else. It is known that $${\\mathcal{SRE}}$$ , the class of problems that admit statistically private randomized encoding with polynomial-time client and computationally unbounded server, is contained in the class $${\\mathcal{SZK}}$$ of problems that have statistical zero-knowledge proof. However, the exact relation between these two classes, and, in particular, the possibility of equivalence was left as an open problem. In this paper, we explore the relationship between $${\\mathcal{SRE}}$$ and $${\\mathcal{SZK}}$$ , and derive the following results: In a non-uniform setting, statistical randomized encoding with one-side privacy ( $${\\mathcal{1RE}}$$ ) is equivalent to non-interactive statistical zero-knowledge ( $${\\mathcal{NISZK}}$$ ). These variants were studied in the past as natural relaxation\/strengthening of the original notions. Our theorem shows that proving $$\\mathcal{SRE}=\\mathcal{SZK}$$ is equivalent to showing that $${\\mathcal{1RE} = \\mathcal{SRE}}$$ and $${\\mathcal{SZK} = \\mathcal{NISZK}}$$ . The latter is a well-known open problem (Goldreich et\u00a0al. in CCC 1999). If $${\\mathcal{SRE}}$$ is non-trivial (not in $${\\mathcal{BPP}}$$ ), then infinitely often one-way functions exist. The analog hypothesis for $${\\mathcal{SZK}}$$ yields only auxiliary-input one-way functions (Ostrovsky in Sixth Annual Structure in Complexity Theory Conference 1991), which is believed to be a significantly weaker notion. If there exists an average-case hard language with perfect randomized encoding, then collision-resistance hash functions (CRH) exist. Again, a similar assumption for $${\\mathcal{SZK}}$$ implies only constant-round statistically hiding commitments, a primitive which seems weaker than CRH. We believe that our results sharpen the relationship between $${\\mathcal{SRE}}$$ and $${\\mathcal{SZK}}$$ ...\nPubDate: 2018-08-20\n\n\u2022 On semiring complexity of Schur polynomials\n\u2022 Authors: Sergey Fomin; Dima Grigoriev; Dorian Nogneng; \u00c9ric Schost\nAbstract: Abstract Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial $${s_\\lambda(x_1,\\dots,x_k)}$$ labeled by a partition $${\\lambda=(\\lambda_1\\ge\\lambda_2\\ge\\cdots)}$$ is bounded by $${O(\\log(\\lambda_1))}$$ provided the number of variables k is fixed.\nPubDate: 2018-06-04\nDOI: 10.1007\/s00037-018-0169-3\n\n\u2022 An adaptivity hierarchy theorem for property testing\n\u2022 Authors: Cl\u00e9ment L. Canonne; Tom Gur\nAbstract: Abstract Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of \u201crounds of adaptivity\u201d it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in $${k+1}$$ rounds, where the queries in the $${i}$$ \u2019th round may depend on the answers obtained in the previous $${i-1}$$ rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity' We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every $${n \\in \\mathbb{N}}$$ and $${0 \\le k \\le n^{0.33}}$$ there exists a property $${\\mathcal{P}_{n,k}}$$ of functions for which (1) there exists a $${k}$$ -adaptive tester for $${\\mathcal{P}_{n,k}}$$ with query complexity $${{\\tilde O}{(k)}}$$ , yet (2) any $${(k-1)}$$ -adaptive tester for $${\\mathcal{P}_{n,k}}$$ must make $${{\\tilde \\Omega}{(n\/k^2)}}$$ queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.\nPubDate: 2018-05-24\nDOI: 10.1007\/s00037-018-0168-4\n\n\u2022 Algebraic independence over positive characteristic: New criterion and\napplications to locally low-algebraic-rank circuits\n\u2022 Authors: Anurag Pandey; Nitin Saxena; Amit Sinhababu\nAbstract: Abstract The motivation for this work (Pandey et\u00a0al. 2016) comes from two problems: testing algebraic independence of arithmetic circuits over a field of small characteristic and generalizing the structural property of algebraic dependence used by Kumar, Saraf, CCC\u201916 to arbitrary fields. It is known that in the case of zero, or large characteristic, using a classical criterion based on the Jacobian, we get a randomized poly-time algorithm to test algebraic independence. Over small characteristic, the Jacobian criterion fails and there is no subexponential time algorithm known. This problem could well be conjectured to be in RP, but the current best algorithm puts it in NP $${^{\\#{\\rm P}}}$$ (Mittmann, Saxena, Scheiblechner, Trans.AMS\u201914). Currently, even the case of two bivariate circuits over $${\\mathbb{F}_2}$$ is open. We come up with a natural generalization of Jacobian criterion that works over all characteristics. The new criterion is efficient if the underlying inseparable degree is promised to be a constant. This is a modest step toward the open question of fast independence testing, over finite fields, posed in (Dvir, Gabizon, Wigderson, FOCS\u201907). In a set of linearly dependent polynomials, any polynomial can be written as a linear combination of the polynomials forming a basis. The analogous property for algebraic dependence is false, but a property approximately in that spirit is named as \u201cfunctional dependence\u201d in Kumar, Saraf, CCC\u201916 and proved for zero or large characteristics. We show that functional dependence holds for arbitrary fields, thereby answering the open questions in Kumar, Saraf, CCC\u201916. Following them, we use the functional dependence lemma to prove the first exponential lower bound for locally low algebraic rank circuits for arbitrary fields (a model that strongly generalizes homogeneous depth-4 circuits). We also recover their quasipoly-time hitting-set for such models, for fields of characteristic smaller than the ones known before. Our results show that approximate functional dependence is indeed a more fundamental concept than the Jacobian as it is field independent. We achieve the former by first picking a \u201cgood\u201d transcendence basis, then translating the circuits by new variables, and finally approximating them by truncating higher degree monomials. We give a tight analysis of the \u201cdegree\u201d of approximation needed in the criterion. To get the locally low-algebraic-rank circuit applications, we follow the known shifted partial derivative-based methods.\nPubDate: 2018-05-14\nDOI: 10.1007\/s00037-018-0167-5\n\n\u2022 Tensor surgery and tensor rank\n\u2022 Authors: Matthias Christandl; Jeroen Zuiddam\nAbstract: Abstract We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and asymptotic rank of the final tensors. We illustrate our method with a number of examples. Tensor surgery on the triangle graph, which corresponds to the matrix multiplication tensor, leads to nontrivial rank upper bounds for all odd cycle graphs, which correspond to the tensors of iterated matrix multiplication. In the asymptotic setting we obtain upper bounds in terms of the matrix multiplication exponent \u03c9 and the rectangular matrix multiplication parameter \u03b1. These bounds are optimal if \u03c9 equals two. We also give examples that illustrate that tensor surgery on general graphs might involve the absorption of virtual hyperedges and we provide an example of tensor surgery on a hypergraph. Besides its relevance in algebraic complexity theory, our work has applications in quantum information theory and communication complexity.\nPubDate: 2018-03-22\nDOI: 10.1007\/s00037-018-0164-8\n\n\u2022 Constructive non-commutative rank computation is in deterministic\npolynomial time\n\u2022 Authors: G\u00e1bor Ivanyos; Youming Qiao; K. V. Subrahmanyam\nAbstract: Abstract We extend the techniques developed in Ivanyos et\u00a0al. (Comput Complex 26(3):717\u2013763, 2017) to obtain a deterministic polynomial-time algorithm for computing the non-commutative rank of linear spaces of matrices over any field. The key new idea that causes a reduction in the time complexity of the algorithm in Ivanyos et\u00a0al. (2017) from exponential time to polynomial time is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen & Makam (Adv Math 310:44\u201363, 2017b), who were the first to observe that the blow-up parameter can be controlled. Both methods rely crucially on the regularity lemma from Ivanyos et\u00a0al. (2017). In this note, we improve that lemma by removing a coprime condition there.\nPubDate: 2018-03-22\nDOI: 10.1007\/s00037-018-0165-7\n\n\u2022 The Landscape of Communication Complexity Classes\n\u2022 Authors: Mika G\u00f6\u00f6s; Toniann Pitassi; Thomas Watson\nAbstract: Abstract We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between $${\\mathsf{P}}$$ and $${\\mathsf{PSPACE}}$$ , short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity. Among our new results we show that $${\\mathsf{MA} \\not\\subseteq \\mathsf{ZPP}^{\\mathsf{NP}[1]}}$$ , that is, Merlin\u2013Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one $${\\mathsf{NP}}$$ query. Here the class $$\\mathsf{ZPP}^{\\mathsf{NP}[1]}$$ has the property that generalizing it in the slightest ways would make it contain $${\\mathsf{AM} \\cap \\mathsf{coAM}}$$ , for which it is notoriously open to prove any explicit lower bounds. We also prove that $${\\mathsf{US} \\not\\subseteq \\mathsf{ZPP}^{\\mathsf{NP}[1]}}$$ , where $${\\mathsf{US}}$$ is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that $${\\mathsf{US} \\not\\subseteq \\mathsf{coDP}}$$ , where $${\\mathsf{DP}}$$ is the class of differences of two $${\\mathsf{NP}}$$ sets. Finally, we explore an intriguing open issue: Are rank-1 matrices inherently more powerful than rectangles in communication complexity' We prove a new separation concerning $${\\mathsf{PP}}$$ that sheds light on this issue and strengthens some previously known separations.\nPubDate: 2018-03-22\nDOI: 10.1007\/s00037-018-0166-6\n\n\u2022 On Space and Depth in Resolution\n\u2022 Authors: Alexander Razborov\nAbstract: Abstract We show that the total space in resolution, as well as in any other reasonable proof system, is equal (up to a polynomial and $${(\\log n)^{O(1)}}$$ factors) to the minimum refutation depth. In particular, all these variants of total space are equivalent in this sense. The same conclusion holds for variable space as long as we penalize for excessively (that is, super-exponential) long proofs, which makes the question about equivalence of variable space and depth about the same as the question of (non)-existence of \u201csupercritical\u201d tradeoffs between the variable space and the proof length. We provide a partial negative answer to this question: for all $${s(n) \\leq n^{1\/2}}$$ there exist CNF contradictions $${\\tau_n}$$ that possess refutations with variable space s(n) but such that every refutation of $${\\tau_n}$$ with variable space $${o(s^2)}$$ must have double exponential length $${2^{2^{\\Omega(s)}}}$$ . We also include a much weaker tradeoff result between variable space and depth in the opposite range $${s(n) \\ll \\log n}$$ and show that no supercritical tradeoff is possible in this range.\nPubDate: 2017-10-17\nDOI: 10.1007\/s00037-017-0163-1\n\nJournalTOCs\nSchool of Mathematical and Computer Sciences\nHeriot-Watt University\nEdinburgh, EH14 4AS, UK\nEmail: journaltocs@hw.ac.uk\nTel: +00 44 (0)131 4513762","date":"2020-01-18 01:35:27","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\": 0.8258237242698669, \"perplexity\": 739.6416410119799}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250591431.4\/warc\/CC-MAIN-20200117234621-20200118022621-00449.warc.gz\"}"}
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package org.apache.kafka.clients.admin; import org.apache.kafka.common.annotation.InterfaceStability; import java.util.Collection; /** * Options for {@link Admin#describeLogDirs(Collection)} * * The API of this class is evolving, see {@link Admin} for details. */ @InterfaceStability.Evolving public class DescribeLogDirsOptions extends AbstractOptions<DescribeLogDirsOptions> { }
{ "redpajama_set_name": "RedPajamaGithub" }
9,475
{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-3x-2-10x-3-0","text":"# How do you solve 3x^2-10x+3=0?\n\nJun 19, 2018\n\nSee a solution process below:\n\n#### Explanation:\n\nWe can factor the left side of the equation as:\n\n$\\left(3 x - 1\\right) \\left(x - 3\\right) = 0$\n\nNow, solve each term on the left side of the equation for $0$\n\nSolution 1:\n\n$3 x - 1 = 0$\n\n$3 x - 1 + \\textcolor{red}{1} = 0 + \\textcolor{red}{1}$\n\n$3 x - 0 = 1$\n\n$3 x = 1$\n\n$\\frac{3 x}{\\textcolor{red}{3}} = \\frac{1}{\\textcolor{red}{3}}$\n\n$\\frac{\\textcolor{red}{\\cancel{\\textcolor{b l a c k}{3}}} x}{\\cancel{\\textcolor{red}{3}}} = \\frac{1}{3}$\n\n$x = \\frac{1}{3}$\n\nSolution 3:\n\n$x - 3 = 0$\n\n$x - 1 + \\textcolor{red}{3} = 0 + \\textcolor{red}{3}$\n\n$x - 0 = 3$\n\n$x = 3$\n\nThe Solution Set Is:\n\n$x = \\left\\{\\frac{1}{3} , 3\\right\\}$","date":"2020-06-05 20:07:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 14, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9237533211708069, \"perplexity\": 1373.2205892828028}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590348502204.93\/warc\/CC-MAIN-20200605174158-20200605204158-00197.warc.gz\"}"}
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{"url":"http:\/\/openstudy.com\/updates\/508e735ce4b078e4677b7810","text":"## anonymous 4 years ago Given a geometric sequence whose sum of the first 10 terms is 4 and whose sum from the 11th to the 30th term is 48, find the sum from the 31st to the 60th term.\n\n1. cwrw238\n\nSum of 10 = a * (r^4 - 1) -------- = 4 r- 1\n\n2. anonymous\n\n@cwrw238 why is it r^4? I thought it's r^10.\n\n3. anonymous\n\nn is 10 not 4\n\n4. amistre64\n\nhmm, given is: $S_n=\\frac{1-r^n}{1-r}$ $S_{10}=4=\\frac{1-r^{10}}{1-r}$ $S_{30-10}=48=\\frac{1-r^{20}}{1-r}$\n\n5. anonymous\n\n@hitten101 yes yes :)\n\n6. shubhamsrg\n\nyou have been given the sum upto first 10 terms =4 you have also been given the sum upto first 30 terms = 4 + 48 =52 and you have 2 eqns with 2 variables ->solve for a and r now calculate sum for first 60 terms from that subtract sum of first 30 terms.. this should help..\n\n7. anonymous\n\n|dw:1351514867224:dw|\n\n8. anonymous\n\na(r^10 - 1) \/ r - 1 = 4 a(r^30 - 1) \/ r - 1 = 52 ? @shubhamsrg like this?\n\n9. anonymous\n\nsolve for a and r.. then find the sum of 60 terms subtract sum of 30 terms from sum of 60 terms\n\n10. shubhamsrg\n\n@kmeds16 yep @hitten101 mistake in your formulla in the denominator..\n\n11. anonymous\n\nI got, r^10 = 3. this is confusing :\/ 10th root of 3?!\n\n12. cwrw238\n\nmy mistake r^10 not r^4\n\n13. shubhamsrg\n\nhow'd you get that? o.O\n\n14. shubhamsrg\n\nahh k..got it\n\n15. anonymous\n\n@kmeds16 @shubhamsrg yes no exponent in the denominator.. you are right\n\n16. anonymous\n\nsecond equation divided by first equation. hehehehehe\n\n17. shubhamsrg\n\nyour main aim is not to find r,, your main aim is to find sum.. leave it as r^10 = 3\n\n18. anonymous\n\nfind the sum of S60 and subtract 52, right?\n\n19. shubhamsrg\n\nseems likely..\n\n20. shubhamsrg\n\ndo this,,this might simplify.. substitue r^10 =3 whereever you can leave r-1 as it is.. you can see a\/(r-1) = 4\/(r^10 -1) in calculation for sum of 60 terms ,make use of this eqn,, no need to find a.. :)\n\n21. shubhamsrg\n\nr^60 we all can find.. hmm.. hope that helped..\n\n22. anonymous\n\nsolving...hehehe ahm, thanks for the idea..","date":"2017-01-24 11:21:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8013215065002441, \"perplexity\": 4554.4693955833245}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560284405.58\/warc\/CC-MAIN-20170116095124-00136-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
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HASSIA PIACENZA is a unique loafer that is made of high-quality stretch material with embossed velvet. The soft leather lining and cushioned footbed provide comfort that you look for in a footwear. The footbed is also removable to provide extra depth to accomodate your own insoles or orthotics. The slightly-elevated rubber sole has a height of 2 cm.
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{"url":"https:\/\/ncatlab.org\/nlab\/show\/Bayesian+reasoning","text":"# nLab Bayesian reasoning\n\nBayesian reasoning\n\n### Context\n\n#### Measure and probability theory\n\nmeasure theory\n\nprobability theory\n\n# Bayesian reasoning\n\n## Idea\n\nBayesian reasoning is an application of probability theory to inductive reasoning (and abductive reasoning). It relies on an interpretation of probabilities as expressions of an agent\u2019s uncertainty about the world, rather than as concerning some notion of objective chance in the world. The perspective here is that, when done correctly, inductive reasoning is simply a generalisation of deductive reasoning, where knowledge of the truth or falsity of a proposition corresponds to adopting the extreme probabilities $1$ and $0$.\n\nSeveral advantages to this interpretation have been proposed. For one thing, it is possible to have (rational) degrees of belief about a wide range of propositions, including matters which are settled, yet unknown. For example, it is possible to speak of the probability of the value of a constant of nature falling within a given interval on the basis of various pieces of evidence, or of the outcome of a coin that has already been tossed. This interpretation of probability beyond limiting frequencies leads, so advocates such as Edwin Jaynes claim, to a more integrated treatment of probability theory and statistics.\n\n## Dutch Book justification\n\nIt can be shown by so-called \u201cDutch Book\u201d arguments, that a rational agent must set their degrees of belief in the outcomes of events in such a way that they satisfy the axioms of probability theory. The idea here is that to believe a proposition to degree $p$ is equivalent to being prepared to accept a wager at the corresponding odds. For instance, if I believe there is a 0.75 chance of rain today, then I should be prepared to accept a wager so that I receive $S$ units of currency if it does not rain, and pay out $S\/3$ units if it does rain. Note that $S$ may be chosen to be negative by the bettor.\n\nIt can be shown then that such betting odds must satisfy the probability axioms, otherwise it will be possible for someone to place multiple bets which will cause the bookmaker to suffer a certain loss whatever the outcome. For example, in the case above, my degree of belief that it will not rain today should be 0.25. Were I to offer, say, 0.5, someone could stake $(-3)$ units on the first bet and $(-2)$ units on the second bet, and will gain $1$ unit whether or not it rains. Of course, real bookmakers have odds which sum to more than 1, but they suffer no guaranteed loss since clients are only allowed positive stakes.\n\n## Cox\u2019s axioms\n\nSome consider the reliance on the idea of the undesirability of certain financial loss to be unbefitting for a justification of what is supposed to be an extension of ordinary deductive logic (Jaynes 2003). Axiomatisations in terms of the properties one should expect of degrees of plausibility have been given, and it can be shown from such axioms that these degrees satisfy the axioms of probability. Richard Cox is responsible for one such axiomatisation (for the moment see Wikipedia: Cox\u2019s theorem).\n\n## Conditionalizing\n\nUsing Bayes' Rule, degrees of belief can be updated on receipt of new evidence.\n\n$P(h|e) = P(e|h) \\cdot \\frac{P(h)}{P(e)},$\n\nwhere $h$ is a hypothesis and $e$ is evidence.\n\nThe idea here is that when $e$ is observed, your degree of belief in $h$ should be changed from $P(h)$ to $P(h|e)$. This is known as conditionalizing. If $P(h|e) \\gt P(h)$, we say that $e$ has provided confirmation for $h$.\n\nTypically, situations will involve a range of possible hypotheses, $h_1, h_2, \\ldots$, and applying Bayes\u2019 Rule will allow us to compare how these fare as new observations are made. For example, comparing the fate of two hypotheses,\n\n$\\frac{P(h_1|e)}{P(h_2|e)} = \\frac{P(e|h_1)}{P(e|h_2)}\\cdot \\frac{P(h_1)}{P(h_2)}.$\n\nHow to assign prior probabilities to hypotheses when you don\u2019t think you have an exhaustive set of rivals is not obvious. When astronomers in the nineteenth century tried to account for the anomalies in the position of Mercury\u2019s perihelion, they tried out all manner of explanations: maybe there was a planet inside Mercury\u2019s orbit, maybe there was a cloud of dust surrounding the sun, maybe the power in the inverse square law ought to be (2 - $\\epsilon$),\u2026 Assigning priors and changing these as evidence comes in is one thing, but it would have been wise to have reserved some of the prior for \u2018none of the above\u2019.\n\nInterestingly, one of the first people to give a qualitative sketch of how such an approach would work was George Polya in \u2018Mathematics and Plausible Reasoning\u2019 (Polya), where examples from mathematics are widely used. The idea of a Bayesian account of plausible reasoning in mathematics surprises many, it being assumed that mathematicians rely solely on deduction. (See also Chap. 4 of Corfield03.)\n\n## Objective Bayesianism\n\nFor some Bayesians, degrees of belief must satisfy further restrictions. One extreme form of this view holds that given a particular state of knowledge, there is a single best set of degrees of belief that should be adopted for any proposition.\n\nSome such restrictions are generally accepted. If, for example, all I know of an event is that it has $n$ possible outcomes, the objective Bayesian will apply the principle of indifference to set their degrees of belief to $1\/n$ for each outcome. On the other hand, if there is background knowledge concerning differences between the outcomes, indifference need not hold. This principle of indifference can be generalized to other kinds of invariance, such as the Jeffreys prior (wiki).\n\nOther objective Bayesian principles include maximum entropy (see Jaynes 2003). For instance, Jaynes argues that if all that is known of a die is that the mean value of throws is equal to, say, 4, then a prior distribution over $\\{1, 2, 3, 4, 5, 6\\}$ should be chosen which maximizes entropy, subject to the constraint that the mean is 4. Many familiar distributions are maximum entropy distributions, subject to moment constraints. For instance, the Normal distribution, $N(\\mu, \\sigma^2)$, is the distribution over the reals which maximises entropy subject to having mean $\\mu$ and variance $\\sigma^2$.\n\n## Exchangeability\n\nFrequentist statistics makes much use of independent and identically distributed (iid) random variables, for example in sampling situations. If, say, we were to toss a coin repeatedly and record the outcomes, the frequentist would typically understand this as sampling from a Bernoulli distribution for some fixed value $p$ of the coin showing heads. From the sample one could then calculate an estimate and confidence interval for the true value of $p$.\n\nMany Bayesians, in particular Bruno de Finetti, argue that this makes no sense since probability is not in the world, but rather it represents the strengths of our beliefs in different outcomes. Their formulation in such repeated sampling cases is to say that if our degrees of belief are such that, for all $n$, the probability we assign to any sequence of $n$ tosses is invariant under any permutation of $n$ elements, then we can represent our degrees of belief for all sequences as arising from a mixture of Bernoulli distributions for some prior distribution over the value of $p$.\n\nMore formally, given a sequence of random variables $\\{X_i\\}^\\infty_{i = 1}$ each taking the values $0$ and $1$, de Finetti\u2019s Representation Theorem says that the sequence is exchangeable if and only if there is a random variable $\\Theta: \\Omega \\to [0, 1]$, with distribution $\\mu_{\\Theta}$, such that\n\n$P\\{X_1=x_1,\\ldots,X_n=x_n\\}=\\int_{[0,1]} \\theta^s (1 - \\theta)^{n - s} d \\mu_{\\Theta}(\\theta),$\n\nin which $s = \\sum^n_{i=1} x_i$.\n\nOften, for ease of calculation, Beta distributions are chosen as priors on $p$, which can be taken as representing one\u2019s confidence as though one had already seen a certain number of heads and tails. Bayesians are sometimes criticized for the subjectivity inherent in the choice of a prior, but in many cases, such as this one, prior distributions will eventually be \u2018washed out\u2019 by the weight of the evidence. The de Finetti theorem has a generalization for multivariate distributions (BBF).\n\nOf course, exchangeability may not represent the strength of one\u2019s prior beliefs accurately. For example, in the case of coin tossing, I may have a suspicion of there being something in the tossing mechanism which would make the result of one toss depend on its predecessor. Then it would be quite reasonable for me, say, to have accorded a higher prior probability to the sequence of one hundred heads followed by one hundred tails than to some of its permutations. There are, however, generalizations of de Finetti\u2019s representation theorem to Markov chain situations (Diaconis and Freedman).\n\n## References\n\n### General\n\n\u2022 Edwin Jaynes, Probability Theory: The Logic of Science, Cambridge University Press, 2003.\n\n\u2022 George Polya, Mathematics and Plausible Reasoning: Vol. II: Patterns of Plausible Inference, Princeton University Press, 1954.\n\n\u2022 David Corfield, Towards a Philosophy of Real Mathematics, Cambridge University Press, 2003, Chap. 4.\n\n\u2022 Persi Diaconis and David Freedman, \u201cDe Finetti\u2019s theorem for Markov chains.\u201d Annals of Probability, 8(1), 115-130, 1980.\n\n\u2022 A. Bach, H. Blank, H. Francke, Bose-Einstein statistics derived from the statistics of classical particles, Lettere Al Nuovo Cimento Series 2, Volume 43, Issue 4, pp 195-198.\n\n### Category-theoretic treatment\n\nPlacing Bayesian inference in a category theoretic setting occurs in\n\n\u2022 Kirk Sturtz?, Bayesian Inference using the Symmetric Monoidal Closed Category Structure, (arXiv:1601.02593)\n\n\u2022 Jared Culbertson?, Kirk Sturtz? Bayesian machine learning via category theory, (arXiv:1312.1445)\n\n### Bayesian inference in physics\n\nDiscussion of applications in astronomy?, cosmology and particle physics includes\n\nLast revised on January 19, 2019 at 07:24:31. See the history of this page for a list of all contributions to it.","date":"2019-10-14 02:10:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 43, \"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\": 0.8205231428146362, \"perplexity\": 477.01599498987457}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986648481.7\/warc\/CC-MAIN-20191014003258-20191014030258-00263.warc.gz\"}"}
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package cloud.strategies; import cloud.CloudService; import com.google.inject.Inject; import com.google.inject.Singleton; import de.uniulm.omi.cloudiator.sword.api.extensions.KeyPairService; import models.KeyPair; import models.VirtualMachine; import models.service.KeyPairModelService; import java.util.Optional; import static com.google.common.base.Preconditions.checkState; /** * Created by daniel on 17.12.15. */ @Singleton public class KeyPairPerCredentialStrategy extends AbstractKeyPairStrategy { private final KeyPairModelService keyPairModelService; @Inject public KeyPairPerCredentialStrategy(CloudService cloudService, KeyPairModelService keyPairModelService) { super(cloudService); this.keyPairModelService = keyPairModelService; } @Override protected Optional<KeyPair> existsFor(VirtualMachine virtualMachine) { return keyPairModelService.getKeyPair(virtualMachine); } @Override protected KeyPair createKeyPairFor(VirtualMachine virtualMachine, KeyPairService keyPairService) { synchronized (KeyPairPerCredentialStrategy.class) { checkState(virtualMachine.owner().isPresent()); checkState(virtualMachine.location().isPresent()); de.uniulm.omi.cloudiator.sword.api.domain.KeyPair remoteKeyPair = keyPairService .create(virtualMachine.owner().get().getUuid(), virtualMachine.location().get().swordId().get()); checkState(remoteKeyPair.privateKey().isPresent(), "Expected remote keypair to have a private key, but it has none."); KeyPair keyPair = new KeyPair(remoteKeyPair.id(), remoteKeyPair.providerId(), remoteKeyPair.id(), virtualMachine.cloud(), virtualMachine.owner().get(), remoteKeyPair.privateKey().get(), remoteKeyPair.publicKey(), null); this.keyPairModelService.save(keyPair); return keyPair; } } }
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Since September I have been writing reviews for businesses in the Bendigo and surrounding areas. Here is a link to one if you'd like to read it, just copy and paste it into your browser. This entry was posted on November 20, 2008 by sharongreenaway in Uncategorized and tagged Uncategorized.
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Despite that previous statement, I do believe that the travel photography community is stronger than it has ever been. Yes, traditional forms of income-generation are dying, but the evolving world has opened up a thousand different doors for those willing to take the risks to make it in the industry. And, yes, there is a lot of competition out there in the travel photography world, but the community is strong and the amount of camaraderie rather than jealousy in the business absolutely floors me nearly every day. Tip from a pro: Instead of trying to work with a large media organization like a magazine or newspaper, become a small media icon yourself. If you have a large and influential presence on social media, such as Facebook and Twitter, you might be more appealing to these organizations than old school media. So, start a blog, gain followers, and who knows, you could be their next photographer. Hi Claire, thanks for your message. That's correct, it's not possible to upload photos from your computer to Instagram however it's quite popular to get around this by emailing the photos to yourself, then opening the email on your phone and storing the attached image in your phone's library. This then allows you to post to Instagram. Alternatively there are a number of apps or plugins that allow you to upload to Instagram, most however will require payment. This is one I suggest looking at 'LR/Instagram' but I can't promise anything as I don't personally use this method. In contrast with most of my peers, I seldom use Photoshop and have never used Lightroom. However, I rely on three post-processing/editing apps as my tools of choice...these are Color Efex Pro (originally of Google and now part of DxO Software, Iridient Developer (the raw image format processing software for macOS, and well known for its ability to process Fujifilm X-Trans raw files), and lastly ON1 Photo Raw ( a raw processor, photo editor and plug-in collection all in one). Tip from a pro: To work with the big brands, you need to market yourself in a way that will appeal to these types of clients. The kind of architectural or food photography a hotel chain needs is very different from what a tour company that specializes in extreme travel. Don't try to work in all genres and styles. That's a good path to becoming an inadequate photographer. Focus only on the genre and style you love and put all of your heart and effort into it. © 2018 Meredith Corporation Travel & Leisure Group. All rights reserved. TravelandLeisure.com is part of the Travel & Leisure Group. Travel + Leisure is a trademark of Meredith Corporation Travel & Leisure Group, registered in the United States and other countries. Travel + Leisure may receive compensation for some links to products and services on this website. Offers may be subject to change without notice. Another place to capture expressions are the subways; either on the platforms or in the cars themselves. My favorite images are the one of a young woman avidly watching a movie on her smartphone, while wearing a single hair roller to tame her fringe....and of the young girl who appears to be viewing a smart phone screen on an ad on a subway platform while her mother is busy texting on her real phone. If you are a top notch, worldwide known photographer, it is very likely that customers from all over the world will want to use your services (for fashion, events, sports, architecture, products etc.). But as we are focusing on travel photography, commercial organizations that deal with traveling and tourism are more likely to hire you. Notable examples are hotels, tour companies, airlines and so on. Approaching a new client can be a lot easier if you happen to be visiting that region, or if it's where you are based. Start local and contact businesses who you regularly use or that have less than desirable images on their website…put together a proposal and they'll more than likely say yes if it benefits them! If they're just starting out on social media you can offer to create a library of social media images they can use over a 3-6month period to generate interest in their product/region. Aga Szydlik is a professional culture photographer and a doctoral candidate based in South Africa. She tells us that her journey with photography started with Muay Thai (the famous Thai fight style) which she documented extensively. Based in Thailand, she able to explore South East Asia, onwards to Indonesia and South Africa. She is enthusiastic about alternative processes, analogue photography, Lomography and salt/albumin prints as well as mixed media. Some of these 'singalong' parlors still exist, faded and tired but otherwise unchanged, offering a taste of popular and cheap entertainment from a past era. How these survive in anyone's guess. The parlors usually have an organist (who can also play a guitar) and a handful of habitual customers who sing Cantonese songs...and occasionally Western oldies such as "Sealed With A Kiss" by the Canton Singing House organist. The Lower Omo River in south west Ethiopia is home to eight different tribes whose population is about 200,000 and it is there that they've lived there for many centuries. The tribes such as the Daasanach, Kara (or Karo), and the Mursi live along the Omo river and depend on it for their livelihood. The annual flooding of the Omo River feeds the biodiversity of the region and guarantees the food security of the tribes especially as rainfall is low and erratic. Chinese opera has a long, rich history that dates back to 200 A.D. Over the centuries, a handful of styles of opera emerged — each with its own distinct makeup, music, and acting traditions — reflecting the eras and tastes of the changing dynasties. Sichuan opera is the youngest style, emerging around 1700 in Chengdu, Sichuan province, where it is still performed today by a dwindling roster of troupes. If a bucket list is about doing extreme things, hydro Zorbing is a perfect fit. The activity, which was developed in New Zealand, involves rolling down a hill inside a transparent, water-filled orb. "From everything I've read, it's an enormous amount of fun. You're sort of in a controlled bounce down a hillside when you're half tumbling and half flying," Zackham says. newzealand.com Love your site! How do you go about not needing work visas to do photography for tourism boards, hotels etc? I see a lot of travel photography in foreign places, but in most countries its illegal to work there and next to impossible to get a work visa as a photographer. Any advice on reaching out to brands/hotels/tourism boards etc overseas without finding myself being deported for working in their country? My still-embryonic idea is to enlist the help of a local acquaintance who would wear a cheongsam (aka qi pao), and take the role of a sing-song girl. The photo shoot would take place in the streets of Yau Ma Tei, and in the parlor itself. Whether the parlor would allow it or not is an open question that will be answered when I'm there. The owners and clients seemed very laid back when I made these photographs. You have provided a great deal of information on a subject I am really interested in. I will be researching the websites on this list. I have started my own website at http://www.davidhintzphotography.com, I have sold some of my photos on microstock websites and now looking to sell directly from my own website. Thanks for all your work on this topic. I would be interested in your comments on my site if you had the time to look at it. Besides the travel publications like National Geographic Traveler, Conde Nast Traveler, etc., the demand for this genre exists in industries like Travel, Photo Education, etc. Many travel photographers are today leading photo-tours through companies such as Intrepid Exposures, utilising their knowledge of unique travel locations, experience of working as professional photographers and using this to help travel enthusiasts take great travel images during their trips. Many others are doubling up as educators in the field of ambient light photography. Some of them are doing assignments which intrinsically use their strengths, e.g. shooting exteriors or interiors of buildings for architects and interior designers. Photographers like Steve McCurry are often commissioned to shoot commercial advertising work using their skills from travel and documentary photography to produce powerful advertising images. Zackham got fascinated with manatees while he was writing a screenplay about dolphins. He learned it's quite easy to swim with the so-called sea cows, and took his family to this spot north of Tampa, where manatees congregate during winter. "To be able to jump in the water with something that big with your 5-year-old? It's amazing," he says. discovercrystalriverfl.com My personal opinion -after having met many such characters- in India; either in Varanasi, Rishikesh, Vrindavan et al, as well as at the Kumbh Mela, is that the majority of them are fake in the sense that they're not dedicated ascetics, but individuals who are adopted a vagabondage lifestyle, begging for alms and food...under the guise of being holy and religious. As an approved photographer on stock libraries, you can possibly get access to client briefs where you can submit your work direct to the client, meaning they'll consider you for the project and see your profile. Otherwise there's usually a marketplace type system for you to upload your images and have them added to collections based on themes, destinations and seasons. For the Aztec and Toltec pre-Hispanic cultures, death was a natural phase of life. The dead were still members of the community, kept alive in memory and spirit, and during Día de los Muertos, they temporarily returned to Earth. Nowadays, people flock to cemeteries to be with the souls of the dead, and build private altars with photographs of the dead, and their favorite foods and beverages. The gatherings are often joyous in tone, and the families remember the lives of the departed. As travel has become more accessible, more and more, the genre is opening up to amateurs and professionals alike. Amateur Travel photography is often shared through sites like Flickr, 500px and 1x. Travel photography, unlike other genres like fashion, product, or food photography, is still an underestimated and relatively less monetized genre, though the challenges faced by travel photographers are lot greater than some of the genres where the light and other shooting conditions may be controllable. Traditionally travel photographers earned money through Stock photography, magazine assignments and commercial projects. Nowadays, the stock photography market has collapsed and more and more photographers are using more innovative methods of earning a living such as through blogging, public speaking, commercial projects and teaching.
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Hi I have the daughters 156 jts 2.0 that needs a Cam and balance shaft belt. I was going to do the water pump at the same time but what else should I do? Can I just put the spring kit in the Variator or should I replace it. And in doing the spring do I need to do the variator bearing as well? Doing something with the variator is a good idea, the refurb kit is appreciably cheaper than a whole unit. A new seal is good practice. You'll still need the variator socket, the TDC tool set, all the usual kit plus I recommend a counterhold tool for the exhaust cam sprocket. Out of four TS motors I've been intimate with, one had slight weeping from the balance shaft oil seals. All had weeping from the balance shaft end caps at the opposite side but only one is accessible with the gearbox still attached. ALWAYS replace the Water Pump when doing belt change, 1/2 of belt failures is due to water pump bearing failure. You must remember the water pump is also a pulley. What is the deal with the TDC tool why can't Alfa just put in timings marks like every other car? There are some indexing marks but they're not to be considered reliable. With any older Euro car, purchase price is just the entry ticket. May I suggest just buying a new Variator, the last thing you want is to find out that a rebuild didn't work and having to the whole lot over again.
{ "redpajama_set_name": "RedPajamaC4" }
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\section{Introduction} Hereafter, we are interested in the explicit rate at which a system of $N$-interacting stochastic particle $(X^{1,N},X^{2,N},\dots,X^{N,N})$ satisfying \begin{equation} \label{eq:particle_intro} \left\{ \begin{aligned} &X^{i,N}_t=X^i_0+\int_{0}^{t}c(s,(X_r)_{0\leq r\leq s})\,ds\\ &\hspace{1cm}+\int_0^t A(s,(X^{i,N}_r)_{0\leq r\leq s})\Big(B(s,(X^{i,N}_r)_{0\leq r\leq s};\overline{\mu}^{N,N}_s)\,ds+\,dW^i_s\Big),\,i=1,\cdots,N,\,0\leq t\leq T,\\ &\overline{\mu}^{N,N}_t=\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,N}_r)_{0\leq r\leq t}\}},\,X^i_0\sim \mu^0,\,(X^1_0,X^2_0,\dots,X^N_0)\,\text{independent}, \end{aligned} \right. \end{equation} propagates chaos. The particle system is defined up to some finite time horizon $0<T<\infty$, with a given initial distribution on $\er^d$ and $W^1,\dots,W^N$ a sequence of independent $m$-dimensional standard Brownian motions ($m\geq 1$). The system of SDEs \eqref{eq:particle_intro} mainly endows an non-anticipative diffusion component $A$ and two non-anticipative drift components $c$ and $A B$ (resulting from the product of $A$ and $B$) and issued from some given progressively measurable mappings: \[ c:(t,x)\in[0,T]\times\Cc([0,T];\er^d) \mapsto c(t,x)=c\big(t,(\omega_{\theta\wedge t}(x))_{0\leq \theta\leq T}\big)\in \er^{d}, \] \[ A:(t,x)\in[0,T]\times\Cc([0,T];\er^d) \mapsto A(t,x)=A\big(t,(\omega_{\theta\wedge t}(x))_{0\leq \theta\leq T}\big)\in\er^{d\times m}, \] \[ B:(t,x,P)\in[0,T]\times\Cc([0,T];\er^d)\times \Pp(\Cc([0,T];\er^d))\mapsto B(t,x;P)=B\big(t,(\omega_{\theta\wedge t}(x))_{0\leq \theta\leq T},P\circ((\omega_{\theta\wedge t})_{0\leq \theta\leq T})^{-1}\big)\in\er^m, \] for $(\omega_t)_{0\leq t\leq T}$ the canonical process on $\Cc([0,T];\er^d)$. In particular, the interaction between particles are described by the component $B$ whose values range in the same dimension as of the Brownian diffusion driving each elements of \eqref{eq:particle_intro}. The propagation of chaos property will be here mainly understood for the law of the paths of \eqref{eq:particle_intro}; namely in the sense where, for a fixed number of particles $X^{1,N},\dots,X^{k,N}$ as the overall number $N$ of interacting particles increases, the chaos (independency) of the initial $X^1_0,\dots,X^N_0$ and diffusive $(W^1_t)_{t\geq 0},\,\dots,(W^N_t)_{t\geq 0}$ inputs of the system is restored in the particle dynamics of the group of particles yielding to the generic dynamic: \begin{equation} \label{eq:McKeanVlasov_intro} \left\{ \begin{aligned} &X^{\infty}_t=X_0+ \int_{0}^{t}c(s,(X^{\infty}_r)_{0\leq r\leq s})\,ds\\ &\hspace{1cm}+\int_0^t A(s,(X^{\infty}_r)_{0\leq r\leq s})\Big(B(s,(X^{\infty}_r)_{0\leq r\leq s};\Ll((X^{\infty}_r)_{0\leq r\leq s}))\,ds+\,dW_s\Big),\,0\leq t\leq T,\\ &\Ll((X^{\infty}_r)_{0\leq r\leq t}))=\text{Law of }((X^\infty_r)_{0\leq r\leq t}),\,X_0\sim \mu^0, \end{aligned} \right. \end{equation} and the weak limit behaviour: \[ \Ll((X^{1,N}_t,\dots,X^{k,N}_t)_{0\leq t\leq T})\underset{N\rightarrow \infty}{\longrightarrow} \Ll((X^{\infty}_t)_{0\leq t\leq T})\otimes\dots\otimes \Ll((X^{\infty}_t)_{0\leq t\leq T}). \] Due to the exchangeability of the particle system, this property is further equivalent to \[ \Ll\Big(\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{k,N}_t)_{0\leq t\leq T} \}}\Big)\underset{N\longrightarrow \infty}{\longrightarrow}\Ll((X_t)_{0\leq t\leq T})\,\text{in the weak sense on }\Pp(\Cc([0,T];\er^d)), \] whenever $k\geq 2$, [Sznitman \cite{Sznitman-89}, Proposition 2.2]. Particular cases of interest for \eqref{eq:McKeanVlasov_intro} that will be discussed later are the situations where the interaction kernel is of the form \[ \int b(t,x,\tilde{x})\,\nu(d\tilde{x}),\,t\geq 0,\,x\in\er^d,\,\nu\in\Pp(\er^d),\,b:[0,\infty)\times\er^d\times\er^d\rightarrow \er^m\,\text{bounded}, \] and where the diffusion component $A$ is either a $d\times d$-valued ($m=d$) bounded and uniformly elliptic matrix or, for $d=2m$, is of the form: \begin{equation}\label{eq:LangevinMcKean} A=\begin{pmatrix} 0 & 0\\ 0 & \sigma\\ \end{pmatrix} \end{equation} More precisely, the former case corresponds to the prototypical McKean-Vlasov dynamic: \begin{equation}\label{eq:ProtoMcKeanVlasovintro} dX_t=\Big(\int b(t,(Y_t,V_t),(y,v))\,\mu(t,dx)\Big)\,dt+\sigma(t,X_t)\,dW_t,\,\mu(t)=\Ll(X_t),\,X_0\sim \mu_0, \end{equation} while the later case can be further particularized into a Langevin dynamic $X_t=(Y_t,V_t)\in\er^m\times\er^m$ satisfying: \begin{equation}\label{eq:LangevinMcKeanintro} \left\{ \begin{aligned} &dY_t=V_t\,dt,\,\,(Y_0,V_0)\sim\mu_0\\ &dV_t=\Big(\int b(t,(Y_t,V_t),(y,v))\,\mu(t,dy,dv)\Big)\,dt+\sigma(t,X_t)\,dW_t,\,\mu(t)=\Ll(Y_t,V_t). \end{aligned} \right. \end{equation} The propagation of chaos property of stochastic interacting particle systems has received over of the years a tremendous amount of attention since its initial introduction in statistical physics (Kac \cite{Kac-56}) for its applications for the probabilistic interpretation of nonlinear pdes (McKean \cite{McKean-66}, \cite{McKean-67}; see the surveys Bossy \cite{Bossy-03}, Jabin and Wang \cite{JabinWang-17} for two global overviews on the theoretical and practical aspects related to McKean-Vlasov or McKean SDEs and related particles approximations) and in its modern utilization for the description of interacting economical agents models and game theory (see e.g. Kolokolstov \cite{Kolokolstov-10}, Carmona and Delarue \cite{CarDel-18a}, \cite{CarDel-18b} and references therein). The central result of the present paper (Theorem \ref{mainthm1:LinearCaseTV}) establishes an explicit (and optimal) rate of convergence for the propagation of chaos property between \eqref{eq:particle_intro} and \eqref{eq:McKeanVlasov_intro} in terms of the total variation distance: \[ \Vert\mu-\nu\Vert_{TV}=\sup_{A\in\Bb(\Cc([0,T];\er^d))}\left|\int \1_{\{x\in A\}}\mu(dx)-\int \1_{\{x\in A\}}\nu(dx)\right|,\,\mu,\nu\in\Pp(\Cc([0,T];\er^d)), \] Mainly this result rests on a generic criterion (see the condition $(\mathbf{C})$ below) which does not directly relies on some regularity properties of $B$ but rather ensure the control of some moments of the Doleans-Dale exponential martingale related to the Girsanov transformation which maps the $N$-system of McKean SDEs \eqref{eq:McKeanVlasovParticle} into the $N$-interacting particle system \eqref{eq:Nparticles}. The core idea of the main result of the present paper is based on a probabilistic interpretation of the proof techniques introduced in Jabin and Wang \cite{JabinWang-16} for the propagation of chaos in entropy (and by extension in total variation) of the one time-marginal distributions of McKean-Vlasov dynamics of the form \eqref{eq:LangevinMcKeanintro} with bounded interaction. More generally, the authors designed a guideline for establishing a sharp quantitative estimate of the propagation of chaos, in terms of a vanishing initial chaos (the particle being initially correlated) and (possibly) vanishing diffusion, through a powerful combination of pde analysis, entropy estimate and combinatorics. This guideline, combined with large deviations principles, was extended to the instance of McKean-Vlasov dynamics \eqref{eq:McKeanVlasov_intro} endowed with singular interaction kernels of the form $b\in W^{-1,\infty}$ (i.e. $b^{(k)}(x)=\sum_{l}\partial_{x_l} G^{k,l}(x),\,G\in L^\infty$). Linked to the probabilistic interpretation of the proof techniques of \cite{JabinWang-16}, let us mention that a (non-explicit) propagation of chaos property in entropy and in total variation distance was recently considered in Lacker \cite{Lacker-18} for the McKean SDE: \begin{equation} \label{eq:McKeanVlasovPastDepend} z_t=Z_0+\int_0^t B\big(s,(z_r)_{0\leq r\leq s},\Ll((z_r)_{0\leq r\leq s})\big)\,ds+\int_{0}^{t}\sigma(s,(z_r)_{0\leq r\leq s})\,dW_s,\,0\leq t\leq T. \end{equation} and its related particle approximation: \begin{equation} \label{eq:ParticlePastDepend} z^{i,N}_t=Z^i_0+\int_0^t B\big(s,(z^{i,N}_r)_{0\leq r\leq s},\frac{1}{N}\sum_{j=1}^N\delta_{\{(z^{j,N}_r)_{0\leq r\leq s}\}}\big)\,ds+\int_{0}^{t}\sigma(s,(z^{i,N}_r)_{0\leq r\leq s})\,dW^i_s,\,0\leq t\leq T, \end{equation} assuming the uniform ellipticity of $\sigma$, the boundedness and Lipschitz continuity (in terms of the total variation distance) of $\sigma^{-1}B$ and the continuity of \[ \nu\in\Pp(\Cc([0,T];\er^d))\mapsto \int_{\Cc([0,T];\er^d)}\int_{0}^{T} \left|\sigma^{-1}(t,z)\left( B(t,z,\mu)- B(t,z,\nu)\right)\right|^2\,dt\,\nu(dz), \] The core idea of \cite{Lacker-18} is closely connected to the original idea introduced in Mishura and Veretennikov \cite{MisVer-16} (from which the present paper owns also its initial step) linking the measurement of the total variation distance between two It\^o's diffusion processes in terms of the Girsanov transformation between the two processes and its applications for the weak uniqueness problems of the McKean SDEs \eqref{eq:ProtoMcKeanVlasovintro}. (It should also be noticed that the idea of establishing propagation of chaos through the Girsanov transformation was already hinted in the preprint Veretennikov \cite{Veretennikov-18} almost at the same time as \cite{Lacker-18}.) The dynamics \eqref{eq:particle_intro} and \eqref{eq:McKeanVlasov_intro} considered hereafter present a extended version of \eqref{eq:McKeanVlasovPastDepend} and \eqref{eq:ParticlePastDepend} which enable to relax elliptic assumption on the diffusion coefficients and embed the case \eqref{eq:LangevinMcKeanintro}. Let also mention that, compared to \cite{Lacker-18}, the wellposed problems related to \eqref{eq:McKeanVlasov_intro} and \eqref{eq:particle_intro} will not be addressed hereafter (assumptions \hypi and \hypii) to rather focus on quantifying explicitly the related propagation of chaos property. The main result of this paper (Theorem \ref{mainthm1:LinearCaseTV}) is stated in Section \ref{sec:MainResults} and proved in Section \ref{sec:Proof}. Section \ref{sec:SufficientConditions} is dedicated to applications of this main result in the particular cases \eqref{eq:ProtoMcKeanVlasovintro} and \eqref{eq:LangevinMcKeanintro} (see corollaries \ref{coro:BoundedCase} and \ref{coro:KineticBoundedCase} respectively) and to exhibit a sufficient condition for the condition $(\mathbf{C})$ in terms of the second order differentiability of $\nu\mapsto B(t,x,\nu)$ (Proposition \ref{prop:DifferentiabilityCondition}). Although \eqref{eq:ProtoMcKeanVlasovintro} and \eqref{eq:LangevinMcKeanintro} only presents applications of Theorem \ref{mainthm1:LinearCaseTV} where the interaction is bounded, more singular situations should be handled by cut-smoothing techniques. The particular case of conditional McKean Lagrangian models (see Bossy, Jabir and Talay \cite{jabir-11a}), which initially motivated the present work, will be discussed in \cite{JabMen-19}. \textbf{Assumptions}: (As before, $(A B)$ denotes the functional on $[0,\infty)\times\Cc([0,\infty);\er^d)\times \Pp(\Cc([0,\infty);\er^d))$ resulting from the product between the diffusion $A$ and drift component $B$ in \eqref{eq:McKeanVlasovParticle} and \eqref{eq:McKeanVlasov_intro}.) \noindent \hypo For any $\mu_0$ on $\er^d$, $0\leq T<\infty$, there exists a unique weak solution $(\Xx_t)_{0\leq t\leq T}$ satisfying the SDE: \begin{equation}\label{eq:IntermediateSDE} \left\{ \begin{aligned} &\Xx_t=X_0+ \int_{0}^{t}c(s,(\Xx_r)_{0\leq r\leq s})\,ds+\int_0^t A(s,(\Xx_r)_{0\leq r\leq s})\,dW_s,\,0\leq t\leq T,\\ &\Xx_0\sim \mu^0. \end{aligned} \right. \end{equation} and for $(\Xx^{1}_t)_{0\leq t\leq T},\,\dots,(\Xx^{1}_t)_{0\leq t\leq T}$ a family of $N$ independent copies of $(\Xx_t)_{0\leq t\leq T}$, it holds that $1\leq i\leq N$, \[ \int_0^T\left| \big (A B\big)\big(s,(\Xx^{i}_r)_{0\leq r\leq s};\nu^N_s\big)\right|^2\,ds<\infty, \] where $\nu^{N}_t=\frac{1}{N}\sum_{j=1}^{N}\delta_{\{(\Xx^j_r)_{0\leq r\leq t}\}}$. \noindent \hypi For any $\mu_0$, $0<T<\infty$, the SDE \eqref{eq:McKeanVlasov_intro} admits a unique weak solution $(X_t)_{t\geq 0}$ such that, almost surely, \[ \int_0^T\left|\big(A B\big)(s,(X_r)_{0\leq r\leq s};\Ll((X_r)_{0\leq r\leq s}))\right|^2\,ds<\infty. \] \noindent \hypii For any $\mu_0$, $0<T<\infty$, $N\geq 1$, the system of SDEs \eqref{eq:particle_intro} admits a unique weak solution $\{(X^{i,N}_t)_{t\geq 0};\,1\leq i\leq N\}$ such that, a.s. \[ \forall\,1\leq i\leq N,\,\int_0^T\left|\big(A B\big)(s,(X^{i,N}_r)_{0\leq r\leq s};\overline{\mu}^{N,N}_s)\right|^2\,ds<\infty, \] where $(\overline{\mu}^{N,N}_t)_{0\leq t\leq T}$ is the flow of (random) empirical measures given as in \eqref{eq:McKeanVlasov_intro}. \begin{remark} With the assumptions \hypi and \hypii, we deliberately leave aside the wellposedness problems of a weak solution to the $N$-interacting particle system \eqref{eq:particle_intro} and to the McKean SDE \eqref{eq:McKeanVlasov_intro} to rather focus on quantifying explicitly the related propagation of chaos property. Although not necessary, the assumption \hypo is used to ensure, in a simple way, the equivalency in law between \eqref{eq:particle_intro} and \eqref{eq:McKeanVlasov_intro}. Let us also mention that the assumptions on the weak uniqueness of \eqref{eq:particle_intro} and \eqref{eq:McKeanVlasov_intro} can be relaxed as long as there exist a solution to \eqref{eq:particle_intro} and a solution to \eqref{eq:McKeanVlasov_intro} for which \eqref{proofstp:i} hold. \end{remark} \textbf{Notation: For any integer $m\geq 1$, and any finite positive time horizon $T$, $\Cc([0,T];\er^{m})$ (respectively $\Cc([0,\infty);\er^{m})$) will denote the space of continuous functions defined on $[0,T]$ (resp. $[0,\infty)$) with values in $\er^m$ equipped with the uniform norm $\Vert x\Vert_{\Cc([0,T];\er^m)}=\max_{0\leq t\leq T}|x(t)|$ (resp. $\Vert x\Vert_{\Cc([0,\infty);\er^m)}=\max_{t\geq 0}|x(t)|\wedge 1)$. $\Pp(\Cc([0,T];\er^{m}))$ and $\Pp(\Cc([0,\infty);\er^{m}))$ will denote respectively the space of probability measures defined on $\Cc([0,T];\er^m)$ and on $\Cc([0,\infty);\er^m)$. Finally, $\Vert~\Vert_{TV,(0,T)}$ will denote the total variation norm on $\Pp(\Cc([0,T];\er^{m}))$, that is (see e.g. Equation $(3.2.13)$ in Rachev \cite{Rachev-91}): for all $P_1,P_2$ on $\Pp(\Cc([0,T];\er^{m}))$ \[ \Vert P_1-P_2\Vert_{TV,(0,T)}=\sup_{A \in \Bb(\Cc([0,T];\er^m))}\left|P_1(A)-P_2(A)\right|, \] where $\Bb(\Cc([0,T];\er^m))$ denotes the Borel $\sigma$-algebra of $\Cc([0,T];\er^m)$. Whenever $P_1, P_2\in \Pp(\Cc([0,\infty);\er^{m}))$ and $0<T<\infty$ is a finite time horizon, $\Vert P_1-P_2\Vert_{TV,(0,T)}$ will simply correspond to the total variation distance between the probability measures restrained to the sample space $(\Cc([0,T];\er^d),\Bb(\Cc([0,T];\er^d)))$. \section{Main result}\label{sec:MainResults} Let $(\Omega,\Ff,(\Ff_t;\,0\leq t\leq T),\PP)$ and $(\widetilde{\Omega},\widetilde{\Ff},(\widetilde{\Ff}_t;\,0\leq t\leq T),\widetilde{\PP})$ be two (possibly different) filtered probability spaces under each of which are defined a collection of $(X^i_0,(W^i_t)_{0\leq t\leq T})$ and $(\widetilde{X}^i_0,(\widetilde{W}^i_t)_{0\leq t\leq T})$ of independent copies of $(X_0,(W_t)_{0\leq t\leq T})$. Then, under \hypi and \hypii, consider a version of the particle system \eqref{eq:McKeanVlasovParticle} defined on $(\widetilde{\Omega},\widetilde{\Ff},(\widetilde{\Ff}_t;\,0\leq t\leq T),\widetilde{\PP})$ as \begin{equation} \label{eq:Nparticles} \left\{ \begin{aligned} &X^{i,N}_t=\widetilde{X}^i_0+\int_{0}^{t}c(s,(X_r)_{r\leq s})\,ds\\ &\hspace{1cm}+\int_0^t A(s,(X^{i,N}_r)_{0\leq r\leq s})\Big(B(s,(X^{i,N}_r)_{0\leq r\leq s};\overline{\mu}^{N,N}_s)\,ds+d\widetilde{W}^i_s\Big),\,0\leq t\leq T,\,i=1,\cdots,N,\\ &\overline{\mu}^{N,N}_t=\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,N}_r)_{0\leq r\leq t}\}},\,\widetilde{X}^i_0\sim \mu^0, \end{aligned} \right. \end{equation} and a system of $N$-independent copies of \eqref{eq:McKeanVlasov_intro} defined on $(\Omega,\Ff,(\Ff_t;\,0\leq t\leq T),\PP)$ as \begin{equation} \label{eq:McKeanVlasovParticle} \left\{ \begin{aligned} &X^{i,\infty}_t=X^i_0+\int_{0}^{t}c(s,(X^{i,\infty}_r)_{r\leq s})\,ds\\ &\hspace{1cm}+\int_0^t A(s,(X^{i,\infty}_r)_{0\leq r\leq s})\Big(B(s,(X^{i,\infty}_r)_{0\leq r\leq s};\Ll((X^{i,\infty}_r)_{0\leq r\leq s}))\,ds+\,dW^i_s\Big),\\ &\mu^{i,\infty}(t)=\Ll((X^{i,\infty}_r)_{0\leq r\leq t}),\,X^i_0\sim \mu^0. \end{aligned} \right. \end{equation} As the assumption \hypii ensures the uniqueness of each component of the system \eqref{eq:McKeanVlasovParticle}, the distribution $\Ll((X^{i,\infty}_t)_{0\leq t\leq T})$ is the common for all component and equal to the one of \eqref{eq:McKeanVlasov_intro}; the index $i$ may be dropped. The superscript $\infty$ in \eqref{eq:McKeanVlasovParticle} will be used as a pointer to remind that \eqref{eq:McKeanVlasovParticle} is (at least heuristically) the suitable limit system of \eqref{eq:Nparticles}. Our main result is given by the following theorem: \begin{theorem}\label{mainthm1:LinearCaseTV} Assume that \hypi and \hypii hold. Assume also that the following condition $\mathbf{(C)}$ holds: \begin{equation*} \mathbf{(C)}\,\,\left\|\,\, \begin{aligned} &\text{There exists a constant }0<\beta<\infty\text{ such that for any }0<T_0<T<\infty,\,0<\delta<\infty,\text{and, for all integer }p\geq 1,\\ &\hspace{4cm}\EE_{\PP}\left[\left(\int_{T_0}^{(T_0+\delta)\wedge T}\left| \triangle B^{i,N,\infty}_t\right|^{2}\,dt\right)^p\right]\leq \frac{p! \beta^{p}\delta^p}{N^p},\\ &\text{where}\,\triangle B^{N,\infty}_t=B(t,(X^{i,\infty}_r)_{0\leq r\leq t};\overline{\mu}^{N,\infty}_t)- B(t,(X^{i,\infty}_r)_{0\leq r\leq t};\Ll((X^{i,\infty}_r)_{0\leq r\leq t}),\\ &\overline{\mu}^{N,\infty}_t=\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,\infty}_r)_{0\leq r\leq t}\}}. \end{aligned} \right. \end{equation*} Then \[ \Vert \Ll\big((X^{1,N}_t,X^{2,N}_t,\dots,X^{N,N}_t)_{0\leq t\leq T}\big)- \Ll\big((X^{1,\infty}_t,X^{2,\infty}_t,\dots,X^{N,\infty}_t)_{0\leq t\leq T}\big)\Vert_{TV,(0,T)}\leq C(1+\beta T)\sqrt{\frac{k}{N}}, \] where $C$ is a constant only depending on $T$, $m$ and $\beta$. \end{theorem} The condition $\mathbf{(C)}$ can be understood as a local Novikov condition in the spirit the one key argument for the proof of Khasm'inskii's lemma (see e.g. [Simon \cite{Simon-82}, Lemma B.1.2.]). Alternatively the condition $\mathbf{(C)}$ in Theorem \ref{mainthm1:LinearCaseTV} can be viewed as a (non-asymptotic) large deviation principle or a sub-gaussian concentration property for the deviation between the "empirical" drift of \eqref{eq:Nparticles} evaluated along the $N$-system of McKean SDEs \eqref{eq:McKeanVlasovParticle}: \[ B\big(t,((X^{i,\infty}_r)_{0\leq r\leq t});\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,\infty_r})_{0\leq r\leq t}\}}\big), \] and its mean-field limit: \[ B\big(t,((X^{i,\infty}_r)_{0\leq r\leq t});\Ll((X^{i,\infty_r})_{0\leq r\leq t})\big). \] In the situations \eqref{eq:ProtoMcKeanVlasovintro} and \eqref{eq:LangevinMcKeanintro}, $\mathbf{(C)}$ is a direct consequence of the boundedness of the interaction kernel $b$. In more general situation the condition may result from a Lipschitz property of $\nu\in\Pp(\Cc([0,T];\er^d))\mapsto B\big(t,x;\nu\big)$ and a centering property (see Lemma \ref{lem:RatePathDependent}) or from a higher regularity property in terms of the variational- linear functional derivative of $\nu\in\Pp(\Cc([0,T];\er^d))\mapsto B\big(t,x;\nu\big)$ (see Definition \ref{def:FlatDerivative} and Proposition \ref{prop:DifferentiabilityCondition}). \section{Proof of Theorem \ref{mainthm1:LinearCaseTV}}\label{sec:Proof} \subsection{Preliminary on propagation of chaos for the total variation distance and control of the Girsanov transformation between $\Ll(X^{1,\infty},\dots,X^{N,\infty})$ and $\Ll(X^{1,N},\dots,X^{N,N})$} For notation convenience, define \[ P^{k,N}=\Ll((X^{1,N}_t,X^{2,N}_t,\cdots,X^{k,N}_t)_{0\leq t\leq T})\in \Pp(\Cc([0,T];\er^{dk})), \] the joint law of the first $k$ particles of \eqref{eq:McKeanVlasovParticle} and by \[ P^{k,\infty}=\Ll((X^{1,\infty}_t,X^{2,\infty}_t,\cdots,X^{k,\infty}_t)_{0\leq t\leq T})\in \Pp(\Cc([0,T];\er^{dk})), \] the joint law of the first $k$ independents copies of \eqref{eq:McKeanVlasov_intro}. The later reduces to \[ P^{k,\infty}=\underbrace{P^{\infty}\otimes P^{\infty}\otimes \cdots \otimes P^{\infty}}_{\text{k times}},\,\,\, P^{\infty}=\Ll\big((X^{\infty}_t)_{0\leq t\leq T}\big), \] as the assumption \hypi ensures the weak uniqueness of \eqref{eq:McKeanVlasov_intro}. The combination of the assumptions \hypo, \hypi and \hypii ensure that for all $1\leq k\leq N<\infty$, the measures $P^{k,N}$ and $P^{k,\infty}$ are equivalent and the Radon-Nikodym derivative formulates\footnote{The proof of \eqref{proofstp:i} under the sole assumptions \hypo, \hypi and \hypii is detailed in the appendix section.} is given by the Doleans-Dale exponential martingale: \begin{equation} \label{proofstp:i} \begin{aligned} &Z^{N}_T:=\frac{dP^{N,N}}{dP^{N,\infty}}\\ &=\exp\left\{-\sum_{i=1}^N\int_0^T \left(B\big(t,(X^{i,\infty}_r)_{0\leq r\leq t},\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,\infty}_r)_{0\leq r\leq t}\} }\big)-\int B\big(t,(X^{i,\infty}_r)_{0\leq r\leq t},\Ll((X^{i,\infty}_r)_{0\leq r\leq t})\big)\right)\cdot \,dW^{i}_t\right.\\ &\quad \left.-\frac{1}{2}\int_0^T \sum_{i=1}^N \left|B\big(t,(X^{i,\infty}_r)_{0\leq r\leq t},\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,\infty}_r)_{0\leq r\leq t}\}}\big)-\int B\big(t,(X^{i,\infty}_r)_{0\leq r\leq t},\Ll((X^{i,\infty}_r)_{0\leq r\leq t})\big) \right|^2\,dt\right\}\\ &=\exp\left\{-\sum_{i=1}^N\int_0^T \triangle B^{i,N}_t\cdot \,dW^{i}_t-\frac{1}{2}\sum_{i=1}^N\int_0^T \left|\triangle B^{i,N}_t\right|^2\,dt\right\}, \end{aligned} \end{equation} where $(\triangle B^{i,N}_t)_{0\leq t\leq T},\,i=1\dots,N$ are given as in $(\mathbf{C})$. By Csisz\'ar-Pinsker-Kullback's inequality, \begin{equation}\label{proofstp:g} \Vert P^{k,N}-P^{k,\infty}\Vert_{TV,\Pp((\Cc([0,T];\er^{kd})))} \leq \sqrt{2 H(P^{k,N}\,|\,P^{k,\infty})}, \end{equation} where $H(P^{k,N}\,|\,P^{k,\infty})$ is the relative entropy between $P^{k,N}$ and $P^{k,\infty}$ is given by \[ H(P^{k,N}\,|\,P^{k,\infty})=\int_{\mathbf{\omega}^k\in\Cc([0,T];\er^{dk})} \log(dP^{k,\infty}/dP^{k,\infty})(\mathbf{\omega}^{k})P^{k,N}(d\mathbf{\omega}^k) \] with $dP^{k,N}/dP^{k,\infty}$ being explicitly given by the conditional expectation $\EE_{\PP}\left[Z^N_T \,|\,(X^{1,\infty},\dots,X^{k,\infty})\right]$ valuing the average value of $Z^{N}_T$ given the path on $[0,T]$ of the $k$-first components of \eqref{eq:McKeanVlasovParticle}, $(X^{1,\infty}_t,\dots,X^{k,\infty}_t)_{0\leq t\leq T}$. At this stage, for $1\leq k<N$, decomposing the empirical measure $\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,\infty}_r)_{0\leq r\leq t}\}}$ into \[ \frac{1}{N}\sum_{j=1}^k\delta_{\{(X^{j,\infty}_r)_{0\leq r\leq t}\}}+\frac{N-(k+1)}{N}\left(\frac{1}{N-(k+1)}\sum_{j=k+1}^N\delta_{\{(X^{j,\infty}_r)_{0\leq r\leq t}\}}\right), \] and owing to the l.s.c. property of $H$ and as $(X^{k+1,\infty},\dots,X^{N,\infty})$ are i.i.d., a natural propagation of chaos property can be derived providing some boundedness and continuity properties on $\nu\mapsto B(t,x;\nu)$. (In \cite{Lacker-18}, an alternative route was proposed proving that $\lim_{N\rightarrow\infty}H(P^{k,\infty}\,|\,P^{k,N})=0$. This results was derived succeeding from a preliminary propagation of chaos results $\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{j,N}_r)_{0\leq r\leq T}\}}\rightarrow \Ll((X^\infty_r)_{0\leq r\leq T})$ derived from a large deviation principle.) An explicit estimate of the propagation of chaos can further be deduced from the super-additive property of the renormalized relative entropy (see e.g. [Hauray and Mischler 2014, Lemma 3.3-iv]), \[ \frac{1}{k}H\big(P^{k,N}|\,P^{k,\infty}\big)\leq \frac{1}{N}H\big(P^{N,N}\,|\,P^{N,\infty}\big). \] Plugged into \eqref{proofstp:g}, \begin{equation*} \Vert P^{k,N}-P^{k,\infty}\Vert_{TV,\Pp((\Cc([0,T];\er^{kd})))} \leq \sqrt{\frac{2k}{N} H(P^{N,N}\,|\,P^{N,\infty})}=\sqrt{\frac{2k}{N} \EE_{\PP}\left[Z^{N}_T\log(Z^{N}_T)\right]}. \end{equation*} from which emerges the optimal rate $1/\sqrt{N}$ provided $\sup_N\EE[(Z^N_T)^{1+\delta}]<\infty$, for some $\delta>0$. The necessity of the uniform control for a moment greater than $1$ of $(Z^N_t)_{0\leq t\leq T}$ can be observed more directly in the case of $P^{k,N}$ and $P^{k,\infty}$: Under \hypi and \hypii, the total variation distance between $P^{k,N}$ and $P^{k,\infty}$ can be expressed as:\eqref{proofstp:i}, for all $A\in\Bb(\Cc([0,T];\er^{kd}))$, we have \begin{align*} \widetilde{\PP}((X^{1,N},\dots,X^{k,N})\in A)=P^{k,N}(A)=\EE_\PP\left[Z^{N}_T\1_{\{(X^{1,\infty},\dots,X^{k,\infty})\in A\}}\right] \end{align*} from which we deduce that \begin{align*} \Vert P^{k,N}-P^{k,\infty} \Vert_{TV,(0,T)} &=\sup_{A \in\Bb(\Cc([0,T];\er^{2d}))} \left|\EE_\PP\left[\left(Z^{N}_T-1\right)\1_{\{(X^{1,\infty},\dots,X^{k,\infty})\in A\}}\right]\right|\\ &=\EE_\PP\left[\left|\EE_\PP\left[\left(Z^{N}_T-1\right)\1_{\{(X^{1,\infty},\dots,X^{k,\infty})\in A\}}\,|\,(X^{1,\infty},\dots,X^{k,\infty})\right]\right|\right]. \end{align*} Since \[ Z^N_T=1+\sum_{i=1}^{N}\int_0^T Z^N_t\triangle B^{i,N}_t\,\cdot dW^{i}_t=\sum_{i=1}^{N} \sum_{l=1}^m \int_0^T Z^N_t\triangle B^{i,N,(l)}_t\,\cdot dW^{i,(l)}_t, \] for \[ Z^N_t=\frac{dP^{N,N}}{dP^{N,\infty}}\Big{|}_{\Bb(\Cc([0,T];\er^d))}=\exp\left\{-\sum_{i=1}^N\int_0^t \triangle B^{i,N}_r\cdot \,dW^{i}_r-\frac{1}{2}\sum_{i=1}^N\int_0^t \left|\triangle B^{i,N}_r\right|^2\,dr\right\}, \] and since $(W^{k+1},\dots,W^{N})$ are independent from $(X^{1,\infty},\dots,X^{N,\infty})$, the conditional expectation \[ \EE_\PP\left[\left(Z^{N}_T-1\right)\1_{\{(X^{1,\infty},\dots,X^{k,\infty})\in A\}}\,|\,(X^{1,\infty},\dots,X^{k,\infty})\right], \] reduces into \[ \EE_\PP\left[\left(\sum_{i=1}^k\int_0^T Z^N_t\triangle B^{i,N}_t\,\cdot dW^{i}_t\right)\,|\,(X^{1,\infty},\dots,X^{k,\infty})\right]. \] This gives: \begin{equation}\label{eq:BoundTVa} \Vert P^{k,N}-P^{k,\infty} \Vert_{TV,(0,T)}= \EE_{\PP}\left[\left|\EE_{\PP}\left[\sum_{i=1}^k\int_0^T Z^{N}_t \triangle B^{i,N}_t\cdot \,dW^{i}_t\,\Big{|}\,(X^{1,\infty}_r,\dots,X^{k,\infty}_r)_{0\leq r\leq T}\right]\right|\right]. \end{equation} Using successively Burkh\"older-Davis-Gundy's inequality, Jensen's inequality, the exchangeability of $(X^{1,\infty},\dots,X^{N,\infty})$ and H\"older's inequality for an arbitrary $1<p< \infty$, it follows: \begin{align*} \Vert P^{k,N}-P^{k,\infty} \Vert_{TV,(0,T)}&\leq \EE_{\PP}\left[\left(\int_0^T (Z^{N}_t)^2 \sum_{i=1}^k\left|\triangle B^{i,N}_t\right|^2\,dt\right)^{1/2}\right]\leq \sqrt{k}\EE_{\PP}\left[\left(\int_0^T (Z^{N}_t)^2\left|\triangle B^{i,N}_t\right|^2\,dt\right)^{1/2}\right]\\ &\leq \sqrt{k}\EE_{\PP}\left[\max_{0\leq t\leq T}(Z^{N}_t)\left(\int_0^T\left|\triangle B^{i,N}_t\right|^2\,dt\right)^{1/2}\right]\\ &\leq \sqrt{k}\left(\EE_{\PP}\left[\max_{0\leq t\leq T}(Z^{N}_t)^p\right]\right)^{1/p}\left(\EE_{\PP}\left[\left(\int_0^T\left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p/(2(p-1))}\right]\right)^{(p-1)/p}. \end{align*} Applying Doob's inequality, we get \begin{equation}\label{eq:BoundTVb} \Vert P^{k,N}-P^{k,\infty} \Vert_{TV,(0,T)}\leq \sqrt{k}\frac{p}{p-1}\left(\EE_{\PP}\left[(Z^{N}_T)^p\right]\right)^{1/p}\left(\EE_{\PP}\left[\left(\int_0^T\left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p/(2(p-1))}\right]\right)^{(p-1)/p}. \end{equation} The display of the rate $1/\sqrt{N}$ is then directly related to the technical difficulty of controlling uniformly a $1+\delta$-moment of $Z^N_T$ as such uniform control would imply that the finiteness of the moments , $\EE_{\PP}[(\sum_{i=1}^N\int_0^T|\triangle B^{i,N}_t|^2\,dt)^{k}]$, which, owing to the exchangeability of $(X^{1,\infty},\dots,X^{N,\infty})$ amounts to establishing $\EE_{\PP}[(\int_0^T|\triangle B^{i,N}_t|^2\,dt)^{k}]$ is of order $1/N^k$. The proof of Theorem of \ref{mainthm1:LinearCaseTV} below is set by first establishing a local-in-time control of an arbitrary moment of $(Z^N_t)_{0\leq t\leq T}$, which combined with \eqref{eq:BoundTVb} and a careful split of the transformation from $(X^{1,\infty},\dots,X^{N,\infty})$ to $(X^{1,N},\dots,X^{N,N})$ to small time intervals enable to conclude the claim. \subsection{Proof of Theorem \ref{mainthm1:LinearCaseTV}} \begin{prop}\label{prop:ControlExpMart} Let $\{(X^{i,\infty}_t)_{0\leq t\leq T};\,1\leq i\leq N\}$ be given as in \eqref{eq:McKeanVlasovParticle} and assume that $\mathbf{(C)}$ hold true. Then, for all $0<T_0<T<\infty$, $0<\kappa<\infty$, \[ \sup_N \EE_\PP\left[(Z^N_{T_0+\delta}/Z^N_{T_0})^\kappa\right]=\sup_N\EE_\PP\left[\exp\left\{\kappa\sum_{i=1}^N \int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t-\frac{\kappa}{2}\int_{T_0}^{T_0+\delta} \left|\triangle B^{i,N}_t\right|^2 \,dt\right\}\right], \] is bounded from above by $1+\exp{\kappa^2}+\frac{2}{1-8 \kappa\delta \beta}$ provided that $\delta< (8\kappa \beta)^{-1}$. \end{prop} \begin{proof}[Proof of Proposition \ref{prop:ControlExpMart}] For the moment, let $\delta$ be an arbitrary positive real number and let us show that \[ \sup_N\EE_\PP\left[\exp\left\{\kappa\sum_{i=1}^N \int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t\right\}\right]<\infty. \] Using the Taylor expansion for the exponential function, \begin{align*} \EE_\PP\left[ \exp\left\{\kappa\sum_{i=1}^N\int_{T_0}^{T_0+\delta}\triangle B^{i,N}_t\cdot \,dW^{i}_t\right\}\right] \leq \sum_{k\geq 0}\frac{\kappa^k}{k!}\EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t\right)^k\right]. \end{align*} Splitting this sum into its even and odd components, and since, for all $r\in\er$, $r^{2p+1}\leq 1+r^{2p+2}$, we have \begin{equation} \label{proofstp:a} \begin{aligned} &\EE_\PP\left[ \exp\left\{\kappa\sum_{i=1}^N\int_{T_0}^{T_0+\delta}\triangle B^{i,N}_t\cdot \,dW^{i}_t\right\}\right]\\ &\leq \sum_{p\geq 0}\frac{\kappa^{2p+1}}{(2p+1)!}\EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t\right)^{2p+1}\right] +\sum_{p\geq 0}\frac{\kappa^{2p}}{(2p)!}\EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t\right)^{2p}\right]\\ &\leq 1+\sum_{p\geq 0}\frac{\kappa^{2p+1}}{(2p+1)!}+2\sum_{p\geq 0}\frac{|\kappa|^{2p}}{(2p)!}\EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t\right)^{2p}\right]. \end{aligned} \end{equation} Applying the martingale moment control of Carlen-Kr\'ee \cite{CarKre-91} (see Theorem \ref{thm:CarlenKree}, Appendix section, for a reminder), we have \begin{align*} \EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t\right)^{2p}\right] \leq 2^{2p} (2p)^{p} \EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p}\right]. \end{align*} Then, by Jensen's inequality and the exchangeability of the $N$-system of McKean-Vlasov dynamics, we get that \begin{align*} \EE_\PP\left[\left(\sum_{i=1}^N\int_{T_0}^{T_0+\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p}\right] & \leq N^{p}\EE_\PP\left[\left( \int_{T_0}^{T_0+\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p}\right]. \end{align*} Plugin the estimate of the condition $\mathbf{(C)}$ then ensures the upper bound \begin{equation} \label{proofstp:c} \begin{aligned} \EE_\PP\left[ \exp\left\{\kappa\sum_{i=1}^N\int_{T_0}^{T_0+\delta}\triangle B^{i,N}_t\cdot \,dW^{i}_t\right\}\right]\leq 1+\exp{\kappa^2}+2\sum_{p\geq 0}\frac{p!p^p2^{3p}\delta^p\beta^{p}\kappa^p}{(2p)!}. \end{aligned} \end{equation} Since $C:=\sup_{p}\big(p!p^p/((2p)!) \big)<\infty$, the sum is essentially geometric and the condition $\delta/(8\beta\kappa)<1$ ensures its finiteness with \[ \sup_N \EE_\PP\left[(Z^N_{T_0+\delta}/Z^N_{T_0})^\kappa\right]\leq 1+\exp{\kappa^2}+\frac{2}{1-8 \kappa\delta \beta} . \] \end{proof} Coming back to the proof of Theorem \ref{mainthm1:LinearCaseTV}], for an arbitrary integer $1<p<\infty$, and for $\overline{\delta}:=(8\beta p)^{-1}$, choose an arbitrary real number $\delta$ in $(0,\overline{\delta}(p))$ (this number will be specified at the end of the proof). For $M:=\llcorner T/\delta\lrcorner$, we define the partition $[0,T]=\cup_{m=0}^M[t_m,t_{m+1})$ with \[ t_0=0,\,t_{M+1}=T,\,t_{m+1}-t_{m}=\delta\,\mbox{for}\,0\leq m< M. \] Next, for each $m$, define the family of $N$-processes $(Y^{1,N,m,\infty}_t)_{0\leq t\leq T},\dots,(Y^{N,N,m,\infty}_t)_{0\leq t\leq T}$ as: for each $1\leq i\leq N$, \noindent $\bullet$ Whenever $0\leq t\leq m\delta$, the path $Y^{i,N,m,\infty}_t$ is given as a weak solution to \begin{align*} Y^{i,N,m,\infty}_t&=Y^{i,N,m,\infty}_0+\int_{0}^{t}c(s,(Y^{i,N,m,\infty}_r)_{r\leq s})\,ds\\ &\quad+\int_0^t A(s,(Y^{i,N,m,\infty}_r)_{0\leq r\leq s})\big(B(s,(Y^{i,N,m,\infty}_r)_{0\leq r\leq s};\Ll((Y^{i,N,m,\infty}_r)_{0\leq r\leq s}))\,ds+\,dW^i_s\big); \end{align*} $\bullet$ Whenever $m\delta <t\leq T$, \begin{align*} Y^{i,N,m,\infty}_t&=Y^{i,N,m,\infty}_{m\delta}+\int_{m\delta}^{t}c(s,(Y^{i,N,m,\infty}_r)_{r\leq s})\,ds\\ &\quad +\int_{m\delta}^t A(s,(Y^{i,N,m,\infty}_r)_{0\leq r\leq s})\big(B(s,(Y^{i,N,m,\infty}_r)_{0\leq r\leq s};\overline{\nu}^{N,N}_s)\,ds+\,dW^i_s\big), \end{align*} for \[ \overline{\nu}^{N,N}_t=\Ll((Y^{i,N,m,\infty}_r)_{0\leq r\leq m\delta})+\frac{1}{N}\sum_{j=1}^N\delta_{\{(Y^{j,N,m,\infty}_r)_{m\delta< r\leq t} \}}. \] By construction, the sequence $\{(Y^{i,N,1,\infty}_t)_{0\leq t\leq T};\,i=1,\dots,N\}$, ..., $\{(Y^{i,N,1,\infty}_t)_{0\leq t\leq T};\,i=1,\dots,N\}$ corresponds to a partially interacting particle corresponding, for any fixed $m$, to the McKean SDEs system \eqref{eq:McKeanVlasov_intro} up to the time $m\delta$, and integrate a mean-field interaction from $t=m\delta$ to $t=T$. Owing the uniqueness properties following \hypii and \hypiii, for $m=0$, $(Y^{1,N,0,\infty}_t,\dots,Y^{N,N,0,\infty}_t)_{0\leq t\leq T}$ corresponds to the McKean-Vlasov system \eqref{eq:ParticlePastDepend} and, for $m=M+1$, $(Y^{1,N,M+1,\infty}_t,\dots,Y^{N,N,M+1,\infty}_t)_{0\leq t\leq T}$ to the interacting particle system \eqref{eq:McKeanVlasovParticle}. Denoting by $P^{k,m,N}$ the probability measure generated by $(Y^{1,N,m,\infty}_t)_{0\leq t\leq T},\dots,(Y^{k,N,m,\infty}_t)_{0\leq t\leq T}$ on $(\Cc([0,T];\er^d),\Bb(\Cc([0,T];\er^d)))$, by the triangular inequality, \begin{equation} \label{proofstp:d} \Vert P^{k,\infty}-P^{k,N}\Vert_{TV,(0,T)}=\Vert P^{1,M+1,N}-P^{1,0,N}\Vert_{TV,(0,T)}\leq \sum_{m=0}^{M}\Vert P^{k,m+1,N}-P^{k,m,N}\Vert_{TV,(0,T)}. \end{equation} By definition, the cost in term of an exponential martingale reduces is given by the following: for some $0\leq m\leq M+1$, $1\leq i\leq N<\infty$, and and, for $0\leq m\leq M-1$, using Corollary \ref{coro:DensityTwoDiff}, \begin{align*} \frac{dP^{N,m,N}}{dP^{N,m+1,N}} &=\exp\left\{-\sum_{i=1}^N\int_{m\delta}^{(m+1)\delta} \triangle B^{i,N}_t\cdot \,dW^{i}_t-\frac{1}{2}\int_{m\delta}^{(m+1)\delta} \sum_{i=1}^N \left| \triangle B^{i,N}_t \right|^2\,dt\right\}=Z^N_{(m+1)\delta}/Z^N_{m\delta}, \end{align*} and \begin{equation}\label{proofstp:e} \begin{aligned} \frac{dP^{N,M,N}}{dP^{N,M+1,N}} &=\exp\left\{-\sum_{i=1}^N\int_{M\delta}^{T} \triangle B^{i,N}_t\cdot \,dW^{i}_t-\frac{1}{2}\int_{M\delta}^{T} \sum_{i=1}^N \left| \triangle B^{i,N}_t \right|^2\,dt\right\}=Z^{N}_T/Z^{N}_{M\delta}. \end{aligned} \end{equation} Replicating the preceding calculations from \eqref{eq:BoundTVa} to \eqref{eq:BoundTVb}, we immediately get, for any $0\leq m\leq M-1$, and $p^*=p/(p-1)$ the conjugate of $p$, \begin{equation}\label{proofstp:f} \begin{aligned} &\Vert P^{1,m+1,N}-P^{1,m,N}\Vert_{TV,(0,T)}\\ &\leq\sqrt{k} p^*\left(\EE_{\PP}\left[\left(Z^N_{(m+1)\delta}/Z^N_{m\delta}\right)^p\right]\right)^{1/p}\left(\EE_{\PP}\left[ \left(\int_{m\delta}^{(m+1)\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p^*}\right]\right)^{1/p^*}. \end{aligned} \end{equation} Using Jensen's inequality and $(\mathbf{C})$, for $\lfloor p^*\rfloor$ the (least) integer part of $p^*/2$, \begin{align*} \EE_{\PP}\left[\left(\int_{m\delta}^{(m+1)\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{p^*/2}\right] &=\EE_{\PP}\left[\left(\left(\int_{m\delta}^{(m+1)\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{\lfloor p^*/2\rfloor +1}\right)^{p^*/(2(\lfloor p^*/2\rfloor+1))}\right]\\ &\leq \left(\EE_{\PP}\left[\left(\int_{m\delta}^{(m+1)\delta} \left|\triangle B^{i,N}_t\right|^2\,dt\right)^{\lfloor p^*/2\rfloor +1}\right]\right)^{p^*/(2(\lfloor p^*/2\rfloor+1))}\\ &\leq \frac{((\lfloor p^*/2\rfloor+1)!)^{p^*/(2(\lfloor p^*\rfloor+1))}(\delta\beta)^{p^*/2}}{N^{p^*/2}}. \end{align*} Finally, coming back to \eqref{proofstp:f}, Proposition \ref{prop:ControlExpMart} gives: \begin{align*} &\Vert P^{k,m+1,N}-P^{k,m,N}\Vert_{TV,(0,T)}\leq\sqrt{k} \left(1+\exp{p^2}+\frac{2}{1-8 p\delta \beta}\right)\left(\frac{\overline{C}(p)\sqrt{\delta\beta}}{\sqrt{N}}\right),\,m=0,\dots,M-1,\\ &\overline{C}(p):=\frac{p}{p-1}((\lfloor p/(2(p-1))\rfloor+1)!)^{1/(\lfloor p/(2(p-1))\rfloor+1)}. \end{align*} In the same way, we get \begin{equation}\label{proofstp:h} \begin{aligned} \Vert P^{k,M+1,N}-P^{k,M,N}\Vert_{TV,(0,T)}&\leq \sqrt{k} \left(1+\exp{p^2}+\frac{2}{1-8 p(T-M\delta) \beta}\right)\left(\frac{\overline{C}(p)\sqrt{(T-\delta M)\beta}}{\sqrt{N}}\right). \end{aligned} \end{equation} Coming back to $\Vert P^{k,N}-P^{k,\infty}\Vert_{TV,(0,T)}$, we get \begin{align*} &\Vert P^{k,N}-P^{k,\infty}\Vert_{TV,(0,T)}\\ &\leq \frac{\sqrt{k}}{\sqrt{N}}\overline{C}(p)\times\left(\left(1+\exp{p^2}+\frac{2}{1-8 p\delta \beta}\right)\sqrt{\delta\beta}M +\left(1+\exp{p^2}+\frac{2}{1-8 p(T-M\delta) \beta}\right)\sqrt{(T-M\delta)\beta}\right)\\ &\leq \frac{\sqrt{k}}{\sqrt{N}}\overline{C}(p)\left(1+\exp{p^2}+\frac{2}{1-8 p\delta \beta}\right)\times\left(\sqrt{\frac{\beta}{\delta}}T+\sqrt{\delta\beta}\right). \end{align*} Then, choosing for instance $\delta=1/((8+\epsilon)p\beta)$ for some $\epsilon>0$, we conclude \begin{align*} &\Vert P^{k,N}-P^{k,N}\Vert_{TV,(0,T)}\leq C\frac{\sqrt{k}}{\sqrt{N}}(1+T\beta),\\ &C:=\inf_{p>1,\epsilon>0}\left\{ \frac{p}{p-1}\left(1+\exp{p^2}+\frac{8+\epsilon}{\epsilon}\right) \left(\frac{((\lfloor p/(p-1)\rfloor+1)!)^{p/(p-1)\times (\lfloor p/(p-1)\rfloor+1)^{-1}}}{\sqrt{(8+\epsilon)}}\sqrt{8+\epsilon}\right)\right\}. \end{align*} \section{Some applications and a sufficient condition for Theorem \ref{mainthm1:LinearCaseTV}}\label{sec:SufficientConditions} \subsection{Applications to McKean-Vlasov dynamics with bounded interaction kernel} As an immediate consequence of Theorem \ref{mainthm1:LinearCaseTV}, we have the following propagation of chaos result for McKean's toy model: \begin{equation*} dX_t=\int b(t,X_t,y)\mu(t,dy)\,dt+\sigma(t,X_t) dW_t,\,\mu(t,dy)=\Ll(X_t) \end{equation*} \begin{corollary}\label{coro:BoundedCase} Given $b:(0,\infty)\times\er^d\times\er^d\rightarrow \er^d$ a Borel bounded function, $\sigma=\sigma(t,x)$ is a uniformly bounded and continuous, positive definite matrix-valued function in the sense that there exist $0<\lambda<\Lambda<\infty$ such that \[ \lambda|\xi|^2\leq \xi\cdot \sigma\sigma^*(t,x)\xi\leq\Lambda|\xi|^2,\,\forall\,t\geq 0,x\in\er^d,\xi\in\er^d, \] let $(X^{1,N}_t,X^{2,N}_t,\dots,X^{N,N}_t)_{t\geq 0}$ and $(X^{1,\infty}_t,X^{2,\infty}_t,\dots,X^{N,\infty}_t)_{t\geq 0}$ satisfy \begin{align} &dX^{i,N}_t=\frac{1}{N}\sum_{j=1}^Nb(t,X^{i,N}_t,X^{j,N}_t)\,dt+\sigma(t,X^{i,N}_t) d\widetilde{W}^i_t,\label{eq:ProtoMcParticleSys}\\ &dX^{i,\infty}_t=\int b(t,X^{i,\infty}_t,y)\mu(t,dy)\,dt+\sigma(t,X^{i,\infty}_t) dW^i_t,\,\mu(t,dy)=\Ll(X^{i,\infty}_t),\label{eq:ProtoMcKeanVlasov}\\ \end{align} where $(X^1_0,\,(W^{1}_t)_{t\geq 0}),\,\dots,(X^N_0,\,(W^{N}_t)_{t\geq 0})$ and $(\widetilde{X}^{1,N}_0,\,(\widetilde{W}^{1}_t)_{t\geq 0}),\,\dots,(\widetilde{X}^{N,N}_0,\,(\widetilde{W}^{N}_t)_{t\geq 0})$ independent copies of $(X_0,(W_t)_{t\geq 0}),\,X_0\sim\mu_0$. Then, for any arbitrary $0<T<\infty$, we have \[ \Vert \Ll\big((X^{1,N}_t,X^{2,N}_t,\dots,X^{N,N}_t)_{0\leq t\leq T}\big)- \Ll\big((X^{1,\infty}_t,X^{2,\infty}_t,\dots,X^{N,\infty}_t)_{0\leq t\leq T}\big)\Vert_{TV,(0,T)}\leq C(1+2\Vert \sigma^{-1} b\Vert_{L^{\infty}}T)\sqrt{\frac{k}{N}}, \] where $C$ is given as in Theorem \ref{mainthm1:LinearCaseTV} and $\Vert \sigma^{-1} b\Vert_{L^{\infty}}:=\text{supess}_{0\leq t\leq T,\,x,y\in\er^d}\big(\sum_{l=1}^{d}|(\sigma^{-1}b)^{(l)}(t,x,y)|^2\big)^{1/2}$. \end{corollary} (Owing to the boundedness of the interaction kernel $b$, the wellposedness of the SDEs \eqref{eq:ProtoMcParticleSys} is immediately granted by a Girsanov transformation. For \eqref{eq:McKeanVlasov_intro}, the weak uniqueness property is immediately granted by [Jourdain \cite{Jourdain-97}, Theorem 3.2].) As a preliminary step for the proof, let us remind the following moment inequality for the sum of i.i.d. real random variables which is a simple consequence of the moment estimates for Sub-Gaussian r.v.s' (see e.g. Bougeron, Lugosi and Massart \cite{BoLuMa-16}, Theorem 2.1) and of Hoeffding's inequality (see e.g. \cite{BoLuMa-16}, Theorem 2.8): \begin{proposition}\label{prop:SubGaussianMoment} Let $X_1,X_2,\cdots,X_n$ be a sequence of i.i.d. random variables such that a.s. $|X_1|\leq \overline{m}<\infty$. Then, for all integer $q\geq 1$, \[ \EE[\left(\sum_{i=1}^n\left(X_i-\EE[X_i]\right)\right)^{2q}]\leq q!(2n\overline{m}^2)^q. \] \end{proposition} \begin{proof}[Proof of Corollary \ref{coro:BoundedCase}] The uniform ellipticity of $\sigma$ allowing to rewrite \eqref{eq:ProtoMcParticleSys} and \eqref{eq:ProtoMcKeanVlasov} can be rewritten into \begin{align*} &d\tilde{X}^{i,N}_t=\sigma(t,\tilde{X}^{i,N}_t) \big(\frac{1}{N}\sum_{j=1}^Nb(t,\tilde{X}^{i,N}_t,\tilde{X}^{j,N}_t)\,dt+d\tilde{W}^i_t\big),\\ &dX^{i,\infty}_t=\sigma(t,X^{i,N}_t) \big(\int \sigma^{-1}(t,X^{i,\infty}_t)b(t,X^{i,\infty}_t,y)\mu(t,dy)\,dt+ dW^i_t\big),\,\mu(t,dy)=\Ll(X^{i,\infty}_t). \end{align*} Owing to the boundedness of $(t,x,y)\mapsto (\sigma^{-1}b)(t,x,y)$, applying Proposition \ref{prop:SubGaussianMoment} yields, for all $1\leq l\leq d$, \begin{align*} \EE_\PP\left[\left|\sum_{j=2}^N\left( \sigma^{-1}(t,X^{1,\infty}_t)\left(b(t,X^{1,\infty}_t,X^{j,\infty}_t)-\int b(t,X^{1,\infty}_t,y)\,\mu(t,dy)\right)\right) \right|^{2p}\right]\leq p!\left(2(N-1)\Vert \sigma^{-1}b\Vert^2_{L^{\infty}}\right)^p. \end{align*} Setting \[ \triangle (\sigma^{-1}b)^{i,j,N}_t:= \sigma^{-1}(t,X^{i,\infty}_t)\left(b(t,X^{i,\infty}_t,X^{j,\infty}_t)-\int b(t,X^{i,\infty}_t,y)\,\mu(t,dy)\right). \] Jensen's inequality yields \begin{align*} &\EE_\PP\left[\left(\int_{T_0}^{T_0+\delta}\left| \frac{1}{N}\sum_{j=1}^N\triangle (\sigma^{-1}b)^{i,j,N}_t\right|^{2}\,dt\right)^p\right] \leq \frac{\delta^{p-1}}{N^{2p}}\int_{T_0}^{T_0+\delta} \EE_\PP\left[\left|\sum_{j=1}\triangle (\sigma^{-1}b)^{i,j,N}_t\right|^{2p}\right]\,dt\\ &\leq \frac{\delta^{p-1}}{N^{2p}}\int_{T_0}^{T_0+\delta} \EE_\PP\left[\left|\sum_{j=1,j\neq i}\triangle (\sigma^{-1}b)^{i,j,N}_t\right|^{2p}\right]\,dt +\frac{\delta^{p-1}}{N^{2p}}\int_{T_0}^{T_0+\delta} \EE_\PP\left[\left|\triangle (\sigma^{-1}b)^{i,i,N}_t\right|^{2p}\right]\,dt\\ &\leq \frac{\delta^{p}p!(N-1)^p}{N^{p}}\Vert \sigma^{-1}b\Vert^{2p}_{L^\infty} +\frac{\delta^{p}}{N^{2p}}\Vert (\sigma^{-1}b)\Vert^{2p}_{L^\infty}. \end{align*} The condition $(\mathbf{C})$ is then satisfy for $\beta=2\Vert \sigma^{-1}b\Vert^2_{L^{\infty}}$ and the estimate on the total variation distance then follows from Theorem \ref{mainthm1:LinearCaseTV}. \end{proof} The demonstration of Corollary \ref{coro:BoundedCase} can be easily extended to the case of Langevin dynamic yielding to the following propagation of chaos result: \begin{corollary}\label{coro:KineticBoundedCase} Given $b:(0,\infty)\times\er^d\times\er^d\rightarrow \er^d$ a Borel bounded function and $\sigma:(0,\infty)\times\er^d\rightarrow \er^{d\times d}$, a uniformly bounded positive definite matrix-valued function, let $((Y^{1,N}_t,V^{1,N}_t),\dots,(Y^{N,N}_t,V^{N,N}_t)_{t\geq 0}$ and $((Y^{1,\infty}_t,V^{1,\infty}_t),\dots,(Y^{N,\infty}_t,V^{N,\infty}_t)_{t\geq 0}$ satisfy \begin{equation*} \left\{ \begin{aligned} &dY^{i,N}_t=V^{i,N}_t\,dt,\,\,(Y^{i,N}_0,V^{i,N})=(\widetilde{Y}^i_0,\widetilde{V}^i_0),\label{eq:ProtoLangevinParticle}\\ &dV^{i,N}_t=\frac{1}{N}\sum_{j=1}^N b(t,(Y^{i,N}_t,V^{i,N}_t),(Y^{j,N}_t,V^{j,N}_t))\,dt+\sigma(t,Y^{i,N}_t,V^{i,N}_t)d\widetilde{W}^i_t, \end{aligned} \right. \end{equation*} \begin{equation*} \left\{ \begin{aligned} &dY^{i,\infty}_t=V^{i,\infty}_t\,dt,\,\,(Y^{i,\infty}_0,V^{i,\infty})=(Y^i_0,V^i_0),\label{eq:ProtoLangevinMcKean}\\ &dV^{i,\infty}_t=\Big(\int b(t,(Y^{i,\infty}_t,V^{i,\infty}_t),(y,v))\,\mu(t,dy,dv)\Big)\,dt+\sigma(t,Y^{i,\infty}_t,V^{i,\infty}_t)dW^i_t,\,\mu(t)=\Ll(Y^{i,\infty}_t,V^{i,\infty}_t). \end{aligned} \right. \end{equation*} where $((Y^1_0,V^1_0),\,(W^{1}_t)_{t\geq 0}),\dots,((Y^N_0,V^N_0),\,(W^{N}_t)_{t\geq 0})$ and $((\tilde{Y}^1_0,\tilde{V}^1_0),\,(\tilde{W}^{1}_t)_{t\geq 0}),\dots,((\tilde{Y}^N_0,\tilde{V}^N_0)),\,(\tilde{W}^{N}_t)_{t\geq 0})$ are two collections of independent copies of $(Y_0,V_0)\sim\mu_0$ and $(W_t)_{t\geq 0}$. Then, for any arbitrary $0<T<\infty$, we have \[ \Vert \Ll\big((Y^{1,N}_t,V^{1,N}_t),\dots,(Y^{k,N}_t,V^{k,N}_t))_{0\leq t\leq T}\big)- \Ll\big((Y^{1,\infty}_t,V^{1,\infty}_t),\dots,(Y^{k,\infty}_t,V^{k,\infty}_t)_{0\leq t\leq T}\big)\Vert_{TV,(0,T)}\leq C(1+2\beta T)\sqrt{\frac{k}{N}}, \] where $\beta=2\text{supess}_{0\leq t\leq T,\,x,y\in\er^d}\big(\sum_{l=1}^{d}|(\sigma^{-1}b)^{(l)}(t,x,y)|^2\big)^{1/2}$. \end{corollary} (We refer to \cite{JabMen-19} for a detailed discussion on the wellposedness, in the weak and strong sense, of \eqref{eq:ProtoLangevinMcKean}.) \subsection{A sufficient condition for Theorem \ref{mainthm1:LinearCaseTV}} In this section, we present a sufficient condition for the application of Theorem \ref{mainthm1:LinearCaseTV} which cover the corollaries \ref{coro:BoundedCase} and \ref{coro:KineticBoundedCase} as particular cases. As a warm-up, let us consider the following lemma: \begin{lemma}\label{lem:RatePathDependent} Assume that \hypi and \hypii hold. Assume also that, for all $0\leq t<\infty$, $x\in\Cc([0,\infty);\er^d)$ $\nu\in\Pp(\Cc([0,\infty);\er^d))\mapsto B(t,x;P)$ is Lipschitz continuous w.r.t. the total variation distance; that is there exists $0<K<\infty$ such that $P,Q\in\Pp(\Cc([0,\infty);\er^d))$, $0\leq t<\infty$, $x\in\Cc([0,\infty);\er^d)$, \begin{equation}\label{cond:TVLip} \left|B(t,x;P)-B(t,x;Q)\right|\leq K\Vert P-Q\Vert_{TV,(0,t)}. \end{equation} Assume finally that the following centering (conditional) property holds: \[ \EE_{\PP}\left[B\big(t,X^{i,\infty};\frac{1}{N-1}\sum_{j=1,j\neq i}^{N}\delta_{\{X^{i,\infty}\}}\big)\,\Big{|}\,X^{i,\infty}\right]=B(t,X^{i,\infty};\Ll(X^{i,\infty})) \] Then the condition $(\mathbf{C})$ is satisfied for $\beta= 4K$. \end{lemma} Prior to the proof let us recall the notion of functions with bounded difference and an annex concentration property: \begin{definition} Let $E$ be some measurable space. A function $f:E^n\rightarrow \er$ is said to have the bounded difference property if, there exists $c_1,c_2,\cdots,c_n>0$ such that for all $(x_1,x_2,\cdots,x_n),(y_1,y_2,\cdots,y_n)\in E^n$, we have for all $1\leq i\leq n$, \[ \left|f(x_1,\cdots,x_{i-1},x_i,x_{i+1},\cdots,x_n)-f(x_1,\cdots,x_{i-1},y_i,x_{i+1},\cdots,x_n)\right|\leq c_i. \] \end{definition} \begin{theorem}[Bounded Difference Inequality, \cite{BoLuMa-16}, Theorem $6.2$]\label{thm:BoundedDifferenceIneq} Let $E$ be some measurable space, $(Y_1,\cdots,Y_n)$ be a family of $E$-valued i.i.d. random variables and let $f:E^n\rightarrow \er$ be some function satisfying the bounded difference property. Then \[ \mathbf{Y}=f(Y_1,\cdots,Y_n) \] satisfies: for all $t\geq 0$, \[ \max\left(\PP\left(\mathbf{Y}-\EE[\mathbf{Y}]\geq t\right),\PP\left(\mathbf{Y}-\EE[\mathbf{Y}]\leq -t\right)\right)\leq \exp\{-\frac{t^2}{2\nu}\}, \] for $\nu=\sum_{i=1}^n (c_i)^2/4$. \end{theorem} In particular, the above ensure the following moment estimates: For all integer $k\geq 1$, \begin{equation}\label{proofstp:j} \EE\left[\left(\mathbf{Y}-\EE[\mathbf{Y}]\right)^{2k}\right]\leq k!(4\nu)^k. \end{equation} \begin{proof}[Proof of Lemma \ref{lem:RatePathDependent}] Fix $t\geq 0$ and $\nu$ an arbitrary probability measure on $\Cc([0,\infty);\er^d)$ and define the family of mappings \[ f^{(l)}_i:\mathbf{x}^N=(x_1,x_2,\cdots,x_n)\in \Cc([0,\infty);\er^d)\mapsto f_i(\mathbf{x}^N)= \left(B^{(l)}(t,x_i,\mu^{-i,N}(\mathbf{x}^N))-B^{(l)}(t,\mathbf{x},\nu)\right)\in\er,\,1\leq l\leq m, \] for $1\leq i\leq N$, $\mu^{-i,N}(\mathbf{x}^N)=\frac{1}{N}\sum_{j=1,j\neq i}^N\delta_{\{x_j\}}$ the empirical measure related to $\mathbf{x}^N$ deprived of $x_i$. For any $i$, $l$, observe that the Lipschitz condition \eqref{cond:TVLip} implies that: \begin{align*} &\left|f^{(l)}_i(x_1,\cdots,x_{k-1},x,x_{k+1},\cdots,x_n)-f^{(l)}_i(x_1,\cdots,x_{k-1},y,x_{k+1},\cdots,x_n)\right|\\ &=\left|B^{(l)}\big(t,x_i,\frac{1}{N}\sum_{j=1,j\neq i,k}^N\delta_{\{x_j\}}+ \frac{1}{N}\delta_{\{x\}}\big)- B^{(l)}\big(t,x_i,\frac{1}{N}\sum_{j=1,j\neq i,k}^N\delta_{\{x_j\}}+ \frac{1}{N}\delta_{\{y\}}\big) \right|\\ &\leq K\Vert \frac{1}{N}\delta_{\{x\}}-\frac{1}{N}\delta_{\{y\}}\Vert_{TV,(0,T)}\leq\frac{K}{N}. \end{align*} so that each of the $f_k$'s satisfies a bounded difference property with coefficients $c_i:=K/N$ for all $1\leq i\leq N$. Applying \eqref{proofstp:j} with $\nu=\sum_{i=1}^N(c_i)^2/4=K^2/4N$, it follows that \begin{align*} \EE_\PP\left[\left|\left(B^{(l)}\Big(t,X^{i,\infty},\frac{1}{N}\sum_{j=1,j\neq i}^N\delta_{\{(X^{k,N}_t)_{0\leq t\leq T} \}}\Big)-B^{(l)}\Big(t,X^{i,\infty},\Ll(X^{i,\infty})\Big)\right)\right|^{2p}\right]\leq p! \frac{K^{2p}}{N^p}, \end{align*} from which we deduce that \begin{align*} &\EE_\PP\left[\left|B^{(l)}\Big(t,X^{i,\infty},\frac{1}{N}\sum_{j=1}^N\delta_{\{(X^{k,N}_t)_{0\leq t\leq T} \}}\Big)-B^{(l)}\Big(t,X^{i,\infty},\Ll(X^{i,\infty})\Big)\right|^{2p}\right]\\ &\leq 2^{2p-1}\EE_\PP\left[\left|B^{(l)}\Big(t,X^{i,\infty},\frac{1}{N}\sum_{j=1,j\neq i}^N\delta_{\{X^{k,N}_.\}}\Big) -B^{(l)}\Big(t,X^{i,\infty},\Ll(X^{i,\infty})\Big)\right|^{2p}\right]\\ &\quad +2^{2p-1} \EE_\PP\left[\left|B^{(l)}\Big(t,X^{i,\infty},\frac{1}{N}\sum_{j=1}^N\delta_{\{X^{k,N}_.\}}\Big)-B^{(l)}\Big(t,X^{i,\infty},\frac{1}{N}\sum_{j=1,j\neq i}^N\delta_{\{X^{k,N}_.\}}\Big)\right|^{2p}\right]\\ &\leq p! \frac{2^{2p-1}K^{2p}}{N^p}+\frac{2^{2p-1}K^p}{N^{2p}}\leq p! \frac{4^{p}K^p}{N^p}. \end{align*} Therefore, \begin{equation*} \EE_\PP\left[\left(\int_{T_0}^{T_0+\delta}\left|B(t,X^{i,\infty},\mu^{N,\infty})-B(t,X^{i,\infty},\Ll(X^{i,\infty}))\right|^{2}\,dt\right)^p\right]\leq \frac{(4\delta m K^2)^p}{N^p}. \end{equation*} \end{proof} The core argument of Lemma \ref{lem:RatePathDependent} relies mostly on the centering property regularity of the drift component $(A B)$ in its measure argument formulated in terms of an analog of the linear derivative functional linear (see e.g. [\cite{Kolokolstov-10}, Appendix $F$], [\cite{CarDel-18a}, Section 5.4]) here below set on the sample space $\Cc([0,T];\er^d)$ : \begin{definition}\label{def:FlatDerivative} The $\er^m$-valued functional $B=\left(B^{(1)},B^{(2)},\dots,B^{(m)}\right)$ is said to admit a bounded second order flat derivative if, for all $1\leq l\leq m$ there exist two measurable bounded functionals: \[ \frac{d B^{(l)}}{dm}=:\in [0,T]\times\Cc([0,T];\er^d)\times \Pp(\Cc([0,T];\er^d))\times \Cc([0,T];\er^d)\rightarrow \er, \] \[ \frac{d^2 B^{(l)}}{dm^2}:(t,x,m;\omega_1,\omega_2)\in [0,T]\times\Cc([0,T];\er^d)\times \Pp(\Cc([0,T];\er^d))\times \Cc([0,T];\er^d)\times \Cc([0,T];\er^d)\rightarrow \er, \] such that, for all $0<T<\infty$, $0\leq t\leq T$, $x\in\Cc([0,T];\er^d)$, $P,Q\in\Pp(\Cc([0,T];\er^d)$, \[ B^{(l)}(t,x,Q)-B^{(l)}(t,x,P) =\int_{0}^{1}\int_{\omega\in\Cc([0,T];\er^d)}\frac{d B^{(l)}}{dm}(t,x,(1-\alpha)P+\alpha Q;\omega)\left(Q(d\omega)-P(d\omega)\right)\,d\alpha, \] and, for all $0<T<\infty$, $0\leq t\leq T$, $x\in\Cc([0,T];\er^d)$, $P,Q\in\Pp(\Cc([0,T];\er^d)$, $\omega\in\Cc([0,T];\er^d)$, \begin{align*} &\frac{d B^{(l)}}{dm}(t,x,Q;\omega)-\frac{d B}{dm}(t,x,P;\omega)\\ &=\int_{0}^{1} \int_{\tilde{\omega}\in\Cc([0,T];\er^d)}\frac{d^2 B^{(l)}}{d m^2}(t,x,(1-\alpha)P+\alpha Q;\omega,\tilde{\omega})\left(Q(d\tilde{\omega})-P(d\tilde{\omega})\right)\,d\alpha. \end{align*} where $(1-\alpha)P+\alpha Q,\,0\leq \alpha\leq 1$ is the set of probability measures given by the convex interpolations between $P$ and $Q$. \end{definition} \begin{proposition}\label{prop:DifferentiabilityCondition} Assume that \hypi and \hypii hold and that for all $0\leq t\leq T$, $x\in\Cc([0,T];\er^d)$, $\mu\in\Pp(\Cc[0,T];\er^d)\mapsto B(t,x,\mu)$ admits a uniformly bounded second order derivative in the sense of Definition \ref{def:FlatDerivative}. Then the condition $\mathbf{(C)}$ in Theorem \ref{mainthm1:LinearCaseTV} holds. \end{proposition} \begin{proof} For any $1\leq l\leq m$, using $\frac{d B^{(l)}}{d m}$, we have \begin{align*} &\triangle B^{i,N,(l)}_t:=B^{(l)}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^N_t)-B^{(l)}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\Ll((X^{i,\infty}_r)_{0\leq r\leq t}))\\ \end{align*} for $\overline{\nu}^{\alpha,N}_t=(1-\alpha)\overline{\nu}^{N}_t+\alpha\Ll((X^{i,\infty}_r)_{0\leq r\leq t})$. In the first sum, for fixed $j$, define the (partial) empirical measure $\overline{\nu}^{-j,N}_t=\frac{1}{N-1}\sum_{l=1,l\neq j}^{N}\delta_{\{(X^{l,\infty}_r)_{0\leq r\leq t}\}}$. Adding and subtracting to the above, \begin{align*} &\frac{1}{N}\sum_{j=1}^N\int_{0}^{1} \frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{-j,\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t})\\ &\quad-\int_{0}^{1}\int_{\omega\in\Cc([0,T];\er^d)}\frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{-j,\alpha,N}_t )(\omega)\Ll((X^{i,\infty}_r)_{0\leq r\leq t})(d\omega)\,d\alpha, \end{align*} for \[ \overline{\nu}^{-j,\alpha,N}_t=(1-\alpha)\overline{\nu}^{-j,N}_t+\alpha\Ll((X^{i,\infty}_r)_{0\leq r\leq t}), \] we have the decomposition: \[ \triangle B^{i,N,(l)}_t=I^{i,N,(l)}_t+J^{i,N,(l)}_t+K^{i,N,(l)}_t, \] where \begin{align*} I^{i,N,(l)}_t&:=\frac{1}{N}\sum_{j=1}^N\int_{0}^{1}\left( \frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t})-\frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{-j,\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t})\right)\,d\alpha, \end{align*} \begin{align*} J^{i,N,(l)}_t&:=\frac{1}{N}\sum_{j=1}^N\int_{0}^{1} \frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{-j,\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t})\\ &\quad-\int_{0}^{1}\int_{\omega\in\Cc([0,T];\er^d)}\frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{-j,\alpha,N}_t )(\omega)\Ll((X^{i,\infty}_r)_{0\leq r\leq t})(d\omega)\,d\alpha, \end{align*} \begin{align*} &K^{i,N,(l)}_t\\ &:=\int_{0}^{1}\int_{\omega\in\Cc([0,T];\er^d)}\left(\frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{-j,\alpha,N}_t ;\omega)-\frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},\overline{\nu}^{\alpha,N}_t ;\omega)\right)\Ll((X^{i,\infty}_r)_{0\leq r\leq t})(d\omega)\,d\alpha, \end{align*} Using the second order derivative $d^2B/dm^2$ and since \[ \overline{\nu}^{\alpha,N}_t(d\omega)-\overline{\nu}^{-j,\alpha,N}_t(d\omega)=\frac{1}{N}\delta_{\{(X^{j,N}_r)_{0\leq r\leq t}\}}+\frac{1}{N(N-1)} \sum_{l=1,l\neq j}^N\delta_{\{(X^{l,N}_r)_{0\leq r\leq t}\in d\omega\}} \] we immediately get for $I^{i,N}_t$: \begin{align*} I^{i,N,(l)}_t&=\frac{1}{N}\sum_{j=1}^N\int_{0}^{1}\int_{0}^1 \frac{d^2B^{(l)}}{dm^2}(t,(X^{i,\infty}_r)_{0\leq r\leq t},(1-r)\overline{\nu}^{\alpha,N}_t+r\overline{\nu}^{\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t};\tilde{\omega}) \left(\overline{\nu}^{\alpha,N}_t(d\tilde{\omega})-\overline{\nu}^{-j,\alpha,N}_t(d\tilde{\omega})\right)\,d\alpha\,dr\\ &=\frac{1}{N^2}\sum_{j=1}^N\int_{0}^{1}\int_{0}^1 \frac{d^2B^{(l)}}{dm^2}(t,(X^{i,\infty}_r)_{0\leq r\leq t},(1-r)\overline{\nu}^{\alpha,N}_t+r\overline{\nu}^{\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t},(X^{j,\infty}_r)_{0\leq r\leq t})\,d\alpha\,dr\\ &\quad + \frac{1}{N^2(N-1)}\sum_{j=1}^N\sum_{l=1,l\neq j}^N\int_{0}^{1}\int_{0}^1 \frac{d^2B^{(l)}}{dm^2}(t,(X^{i,\infty}_r)_{0\leq r\leq t},(1-r)\overline{\nu}^{\alpha,N}_t+r\overline{\nu}^{\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t},(X^{l,\infty}_r)_{0\leq r\leq t})\,d\alpha\,dr. \end{align*} In the same way, \begin{align*} &K^{i,N,(l)}_t\\ &=\frac{1}{N}\sum_{j=1}^N\int_{0}^{1}\int_{0}^1 \frac{dB^{(l)}}{dm}(t,(X^{i,\infty}_r)_{0\leq r\leq t},(1-r)\overline{\nu}^{\alpha,N}_t+r\overline{\nu}^{\alpha,N}_t ;(X^{j,\infty}_r)_{0\leq r\leq t};\tilde{\omega}) \left(\overline{\nu}^{\alpha,N}_t(d\tilde{\omega})-\overline{\nu}^{-j,\alpha,N}_t(d\tilde{\omega})\right)\,d\alpha\,dr\\ &=\frac{1}{N^2}\sum_{j=1}^N\int_{0}^{1}\int_{0}^1 \frac{d^2B^{(l)}}{dm^2}(t,(X^{i,\infty}_r)_{0\leq r\leq t},(1-r)\overline{\nu}^{\alpha,N}_t+r\overline{\nu}^{\alpha,N}_t ;\omega,(X^{j,\infty}_r)_{0\leq r\leq t})\Ll((X^{j,\infty}_r)_{0\leq r\leq t})\,d\alpha\,dr\\ &\quad + \frac{1}{N^2(N-1)}\sum_{j=1}^N\sum_{l=1,l\neq j}^N\int_{0}^{1}\int_{0}^1 \frac{d^2B^{(l)}}{dm^2}(t,(X^{i,\infty}_r)_{0\leq r\leq t},(1-r)\overline{\nu}^{\alpha,N}_t+r\overline{\nu}^{\alpha,N}_t ;\omega,(X^{l,\infty}_r)_{0\leq r\leq t})\Ll((X^{j,\infty}_r)_{0\leq r\leq t})\,d\alpha\,dr. \end{align*} These estimates ensure directly that \begin{align*} \EE\left[\left(\int_{T_0}^{T_0+\delta}\left|I^{i,N,(l)}_t\right|^2\,dt\right)^p\right]\leq \frac{2^p\delta^p}{N^p}\Vert \frac{d^2 B}{dm ^2}\Vert^{2p}_{L^\infty}, \end{align*} and \begin{align*} \EE\left[\left(\int_{T_0}^{T_0+\delta}\left|K^{i,N,(l)}_t\right|^2\,dt\right)^p\right]\leq \frac{2^p\delta^p}{N^p}\Vert \frac{d^2 B}{dm ^2}\Vert^{2p}_{L^\infty}. \end{align*} The final component $J^{i,N,(l)}$ can be estimated in the same way as in the proof of Lemma \ref{lem:RatePathDependent}. \end{proof} \paragraph{Acknowledgement:} This article was prepared within the framework of the Russian Academic Excellence Project '5-100'. The author is thankful to Lukasz Szpruch and Paul-Eric Chaudru de Raynal for having pointed out the use of linear functional derivative to derive the sufficient condition in Proposition \ref{prop:DifferentiabilityCondition}, and to Alexander Veretennikov for very fruitful discussions over the past year. \section{Appendix} \textbf{Carlen and Kr\'ee's optimal martingale moment control}: \begin{theorem}[Carlen and Kr\'ee \cite{CarKre-91}, Theorem $A$]\label{thm:CarlenKree} For $p\geq 1$, define \[ b_p=\sup_{(M_t)_{t\geq 0}}\left\{\frac{\EE\left[(M_t)^p\right]^{1/p}}{\EE\left[(\sqrt{\langle M\rangle_t})^p\right]^{1/p}}\right\}, \] where the supremum is taken over the set of real valued bounded and continuous martingales $(M_t)_{t\geq 0}$. Then \[ \sup_{p\geq 1}\frac{b_p}{\sqrt{p}}=2. \] \end{theorem} The boundedness condition, assumed in Carlen and Kr\'ee \cite{CarKre-91}, can be easily dropped, thanks to a truncation argument, to state the generic inequality: \begin{equation}\label{proofst:h} \EE\left[(M_t)^p\right]^{1/p}\leq 2\sqrt{p}\EE\left[(\sqrt{\langle M\rangle_t})^p\right]^{1/p}\,\text{whenever}\,\EE\left[(\sqrt{\langle M\rangle_t})^p\right]<\infty. \end{equation} Indeed, given $(M_t)_{t\geq 0}$ a continuous $L^p$-finite martingale and introducing the stopping time $\tau_\lambda=\inf\{t>0\,:\,|M_t|\geq \lambda\}$, the truncated process $(M_{t\wedge\tau_\lambda};\,t\geq 0)$ is bounded, so that \[ \EE\left[(M_{t\wedge \tau_\lambda})^p\right]^{1/p}\leq 2\sqrt{p}\EE\left[(\sqrt{\langle M\rangle_{t\wedge \tau_\lambda}})^p\right]^{1/p}. \] Taking the limit $\lambda\rightarrow \infty$, we conclude \eqref{proofst:h} \noindent \textbf{Proof of \eqref{proofstp:i}:} From this proposition, we deduce the following corollary that can be simply deduced from [Theorem $7.7$, Lipster and Shiryaev \cite{LipShi-01}]: \begin{corollary}\label{coro:DensityTwoDiff} Let $(\zeta^1_t)_{0\leq t\leq T}$ and $(\zeta^2_t)_{0\leq t\leq T}$ be two It\^o diffusion processes defined a filtered probability space $(\Omega,\Ff,(\Ff_t)_{t\geq 0},\PP)$, satisfying \[ d\zeta^i_t=\alpha_i(t,\zeta^i)\,dt+dW^i_t,\,\zeta_0=0,\,0\leq t\leq T,,\,i=1,2, \] Then assuming that \[ \PP\left(\int_0^T\left|\alpha_1(t,\zeta^1)\right|^2\,dt +\int_0^T\left|\alpha_2(t,\zeta^2)\right|^2\,dt<\infty\right)=1, \] and \[ \PP\left(\int_0^T\left|\alpha_1(t,W^1)\right|^2\,dt+\int_0^T\left|\alpha_2(t,W^2)\right|^2\,dt<\infty\right)=1, \] the probability measures $P_{\zeta_1}$ and $P_{\zeta_2}$ are equivalent and \[ \frac{dP_{\zeta^1}}{dP_{\zeta^2}}(T,\zeta^2)=\exp\left\{-\int_0^T \left(\alpha_1(t,\zeta_2)-\alpha_2(t,\zeta_2)\right)\cdot \,d\zeta^2_t-\frac{1}{2}\int_0^T \left|\alpha_1(t,\zeta^2)-\alpha_2(t,\zeta^2)\right|^2\,dt \right\}, \] \[ \frac{dP_{\zeta^2}}{dP_{\zeta^1}}(T,\zeta^1)=\exp\left\{-\int_0^T \left(\alpha_2(t,\zeta_1)-\alpha_2(t,\zeta_1)\right)\cdot \,d\zeta^1_t-\frac{1}{2}\int_0^T \left|\alpha_2(t,\zeta^1)-\alpha_1(t,\zeta^1)\right|^2\,dt\right\}. \] \end{corollary} Applying the preceding corollary to \eqref{eq:McKeanVlasovParticle} and \eqref{eq:Nparticles}, we deduce \eqref{proofstp:i} by applying two successive Girsanov transformations, first mapping the $\er^{dN}$-valued process: \begin{equation*} (X^{1,\infty}_t,\dots,X^{N,\infty}_t)_{0\leq t\leq T}, \end{equation*} into a system of $N$ (independent) copies of the solution to \eqref{eq:IntermediateSDE}. The interaction between the component is then introduced by a second Girsanov transformation yielding to \eqref{eq:Nparticles}.
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{-# LANGUAGE TypeFamilies, KindSignatures, ConstraintKinds #-} module BayesStack.Types ( Probability , HasLikelihood(..) , FullConditionable(..) ) where import GHC.Prim (Constraint) import Numeric.Log type Probability = Log Double class HasLikelihood p where type LContext p a :: Constraint type LContext p a = () likelihood :: LContext p a => p a -> Probability -- | A distribution for which a full conditional factor can be produced class FullConditionable p where type FCContext p a :: Constraint type FCContext p a = () sampleProb :: FCContext p a => p a -> a -> Double
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/intermediate-algebra-for-college-students-7th-edition\/chapter-6-section-6-5-synthetic-division-and-the-remainder-theorem-exercise-set-page-455\/59","text":"Intermediate Algebra for College Students (7th Edition)\n\n$x^2-9$.\n$x^2-9=(x+3)(x-3)$, thus the LCD of $x+3$ and $x-3$ is $x^2-9$.","date":"2019-11-18 09:25:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.944776713848114, \"perplexity\": 1047.2009358729372}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496669730.38\/warc\/CC-MAIN-20191118080848-20191118104848-00488.warc.gz\"}"}
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package skinny.micro.data import scala.language.reflectiveCalls import skinny.micro.SkinnyMicroException import skinny.micro.implicits.TypeConverter import scala.collection.JavaConverters._ import scala.collection.mutable.Map /** * Adapts attributes from servlet objects (e.g., ServletRequest, HttpSession, * ServletContext) to a mutable map. */ trait AttributesMap extends Map[String, Any] with MutableMapWithIndifferentAccess[Any] { protected def attributes: Attributes /** * Optionally returns the attribute associated with the key * * @return an option value containing the attribute associated with the key * in the underlying servlet object, or None if none exists. */ def get(key: String): Option[Any] = { if (attributes == null) None else { attributes.getAttribute(key) match { case null => None case v => Some(v) } } } /** * Optionally return and type cast the attribute associated with the key * * @param key The key to find * @tparam T The type of the value * @return an option value containing the attributed associated with the key in the underlying servlet object, * or None if none exists */ def getAs[T](key: String)(implicit mf: Manifest[T], converter: TypeConverter[Any, T]): Option[T] = { get(key) flatMap (converter(_)) } /** * Return the attribute associated with the key or throw an exception when nothing found * * @param key The key to find * @tparam T The type of the value * @return an value for the attributed associated with the key in the underlying servlet object, * or throw an exception if the key doesn't exist */ def as[T](key: String)(implicit mf: Manifest[T], converter: TypeConverter[Any, T]): T = { getAs[T](key) getOrElse (throw new SkinnyMicroException("Key " + key + " not found")) } /** * Return the attribute associated with the key or throw an exception when nothing found * * @param key The key to find * @tparam T The type of the value * @return an value for the attributed associated with the key in the underlying servlet object, * or throw an exception if the key doesn't exist */ def getAsOrElse[T](key: String, default: => T)( implicit mf: Manifest[T], converter: TypeConverter[Any, T]): T = { getAs[T](key) getOrElse default } /** * Creates a new iterator over all attributes in the underlying servlet object. * * @return the new iterator */ def iterator: Iterator[(String, Any)] = { attributes.getAttributeNames().asScala map { key => (key, attributes.getAttribute(key)) } } /** * Sets an attribute on the underlying servlet object. * * @param kv the key/value pair. If the value is null, has the same effect * as calling `-=(kv._1)`. * * @return the map itself */ def +=(kv: (String, Any)): AttributesMap.this.type = { attributes.setAttribute(kv._1, kv._2.asInstanceOf[AnyRef]) this } /** * Removes an attribute from the underlying servlet object. * * @param key the key to remove * * @return the map itself */ def -=(key: String): AttributesMap.this.type = { attributes.removeAttribute(key) this } }
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{"url":"http:\/\/brkmnd.com\/pages\/math\/Default.aspx?id=1","text":"Linear regression using python\n\nstatistics\/python :: 09-12-2017\n\nOK. So I have started an introductionary course in machine learning. Machine learning is, somewhat, making models using regression. And the first example involves linear regression. I have long pondered on effective ways to do this kind of things on a computer. Finally I have found one. So lets delve. Regression is fitting a functions values as close to a set of observed values as possible. Hence taking the mean of the distance of all function\/model values to the appropriate observed value. Let say we have a bunch of observed values: $$obsXs = [1.2,2.0,3.3,4.0,4.9]$$ $$obsYs = [2.5,3.5,4.0,4.5,5.0]$$\n\nLinear regression easily can be done using maple, simply by typing the commands:\n\nwith(Statistics)\nobsXs := Vector([1.2, 2.0, 3.3, 4.0, 4.9])\nobsYs := Vector([2.5, 3.5, 4.0, 4.5, 5.0]) Fit(a*x+b, obsXs, obsYs, x, summarize = embed)\n\nAnd we get the result $$0.633138751683879x+1.94993264481365$$ with the rather good R-squarred $0.965109$\n\nBut what is interresting is of course what is going on under the hood. So lets have a look. First define the squarred error as $$sqrErr = (y_n - f(x_n))^2$$ That is the squarred distance from a value from our model, f, and to the observed value. Second lets define a prototype for our model, that is a basic linear function $$f(x) = w_1 x + w_0$$ where $w_1,w_0$ are the unknowns we are interested in finding optimally: we want the mean squarred error to be as small as possible: $$argmin_{w_0,w_1} 1\/n \\sum_{n = 1} sqrErr(w_0,w_1)$$\n\nThese kind of problems are solved by differentiating and solving the result equal 0. First rewrite: $$1\/n \\sum (y_n - f(x_n))^2 = 1\/n \\sum (y_n^2 - f(x_n))^2 = 1\/n \\sum y_n^2 + f(x_n)^2 - 2y_n f(x_n)$$\n\nSubstitute: $$1\/n \\sum y_n^2 + f(x_n)^2 - 2y_n f(x_n) =\\\\ 1\/n \\sum y_n^2 + (w_1 x_n)^2 + w_0^2 + 2 w_1 w_0 x_n - 2y_n (w_1 x_n + w_0)$$\n\nNow differentiate: $$\\frac{\\partial sqrErr}{\\partial w_1} = 1\/n \\sum 2 w_1 x_n^2 + 2w_0 x_n - 2y_n x_n$$ And $$\\frac{\\partial sqrErr}{\\partial w_0} = 1\/n \\sum 2w_0 + 2w_1 x_n - 2y_n$$\n\nNow solve first the last expression equal 0: $$0 = 1\/n \\sum 2w_0 + 2w_1 x_n - 2y_n \\Rightarrow w_0 = - 1\/n \\sum w_1 x_n + y_n$$ This can be seen as: $$\\hat{w_0} = \\bar{y} - w_1 \\bar{x}$$ Where the bar denotes the mean value, and the hat denotes argmin. The first derivative equal 0: $$0 = 1\/n \\sum 2w_1 x_n^2 + 2w_0 x_n - 2y_n x_n \\Rightarrow \\\\ 0 = 1\/n \\sum w_1 x_n^2 + w_o x_n - y_n x_n \\Rightarrow \\\\ 0 = 1\/n \\sum (w_1 x_n^2 + \\bar{y} x_n - w_1 \\bar{x} x_n) - \\bar{yx} \\Rightarrow \\\\ 0 = w_1 (\\bar{x^2} - (\\bar{x})^2) + \\bar{y} \\bar{x} - \\bar{yx} \\Rightarrow \\\\ \\hat{w_1} = \\frac{\\bar{yx} - \\bar{y} \\bar{x}}{ \\bar{x^2} - \\bar{x}^2 }$$\n\nWhat we need is to check is that the second derivative is bigger than 0. But it is. Now we have a formula for doing linear regression. Lets apply with pyhton to the two vectors from above. First we define some aux functions:\n\ndef initA(f,l): retval = [] for i in range(0,l): retval.append(f(i)) return retval def foldA(f,a,acc): l = len(a) for i in range(0,l): acc = f(a[i],acc) return acc def zipA(f,a,b,acc): l = len(a) for i in range(0,l): acc = f(a[i],b[i],acc) return acc\n\nThen the two vectors\n\nobsXs = [1.2,2.0,3.3,4.0,4.9] obsYs = [2.5,3.5,4.0,4.5,5.0]\n\nSo we have to find the mean of the following values $$x, x^2, y, yx$$\n\nLets go:\n\nobsXs = [1.2,2.0,3.3,4.0,4.9] obsYs = [2.5,3.5,4.0,4.5,5.0] len_xs = len(obsXs) len_ys = len(obsYs) mean_x = foldA(lambda x,acc: x \/ len_xs + acc,obsXs,0) mean_y = foldA(lambda y,acc: y \/ len_ys + acc,obsYs,0) mean_x2 = foldA(lambda x,acc: x**2 \/ len_xs + acc,obsXs,0) mean_xy = zipA(lambda x,y,acc: (x * y) \/ len_xs + acc,obsXs,obsYs,0) w1 = (mean_xy - mean_x * mean_y) \/ (mean_x2 - mean_x ** 2) w0 = mean_y - w1 * mean_x print \"f_model1(x) = \" + str(w1) + \"x + \" + str(w0)\n\nAlthough this alone is pretty cool, lets find that squarred r thing. According to this WikiPage we can see that we have to calculate the following numbers:\n\ndef f_model1(x): return w1 * x + w0 ss_tot = foldA(lambda y,acc: (y - mean_y) ** 2 + acc,obsYs,0) ss_res = zipA(lambda x,y,acc: (y - f_model1(x)) ** 2 + acc,obsXs,obsYs,0) print \"r^2 = \" + str(1 - ss_res \/ ss_tot)\n\nDon't mind the ss-thing. 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eMedTV Home » Women Channel » Lybrel Lybrel is a birth control pill that can be taken continuously, as there are no inactive pills. It comes in the form of a tablet that must be taken at the same time every day. Possible side effects can include nausea, vomiting, and headaches. Even though you will not have a regular monthly period while taking Lybrel, you will probably have some bleeding from time to time. What Is Lybrel? Lybrel™ (levonorgestrel/ethinyl estradiol) is an oral contraceptive (birth control pill). It is the first combination oral contraceptive that is approved to be taken continuously, with no inactive pills at all. While taking Lybrel, women will not have regular monthly periods, although they will probably experience some unpredictable bleeding and spotting from time to time. (Click Lybrel Uses for more information on what it is used for, including possible off-label uses.) Who Makes Lybrel? Lybrel was made by Wyeth Pharmaceuticals, Inc., although it is no longer being manufactured. It is still available in generic form. Lybrel is a combined oral contraceptive, which means that it is a birth control pill that contains both an estrogen (ethinyl estradiol) and a progestin (levonorgestrel). It primarily works to prevent pregnancy by stopping ovulation (the maturation and release of eggs from the ovaries). However, it also prevents pregnancy in two other, minor ways. Lybrel alters the cervical mucus (the fluid of the cervix, which is the lower, narrow part of the uterus that is connected to the vagina), making it more difficult for sperm to enter the uterus. Lybrel also alters the lining of the uterus (called the endometrium), making it less receptive to an embryo. There is no reason for women to have a monthly period while taking birth control pills. In fact, the "period" you experience while taking birth control pills isn't really a period at all. Because ovulation does not occur, the body does not prepare for a possible pregnancy by building up the lining of the uterus, so there is no need to shed the lining (as with a regular period). Instead, the "period" that occurs due to birth control pills is actually caused by a withdrawal of the hormones in the pills, which causes bleeding. Even though you will not have a regular monthly period while taking Lybrel, you will probably have some bleeding from time to time. This bleeding may resemble a menstrual period, but it will not occur at regular intervals. More Headlines in Lybrel ‣ Effects of Lybrel ‣ When and How to Use Lybrel ‣ Side Effects of Lybrel ‣ Drug Interactions With Lybrel ‣ What If I Take a Lybrel Overdose? ‣ Storage Methods ‣ Strengths ‣ Is There a Generic Version of Lybrel? Lybrel Article Continues on Next Page » Lybrel Guide ‣ Lybrel ‣ Lybrel Side Effects ‣ Lybrel Uses ‣ Lybrel Dosage ‣ Lybrel Drug Interactions 5 Common Relationship Mistakes for Adults With ADHD Lybrel Side Effects Lybrel Uses Lybrel Dosage Lybrel Drug Interactions Lybrel Warnings and Precautions Lybrel Overdose Lybrel and Pregnancy Lybrel and Breastfeeding Generic Lybrel Lybrel [package insert]. Philadelphia, PA: Wyeth Pharmaceuticals, Inc.;2007 May. Food and Drug Administration, Center for Drug Evaluation and Research. Electronic orange book: approved drug products with therapeutic equivalence evaluations. FDA Web site. Available at: http://www.fda.gov/cder/ob/. Accessed July 21, 2013.
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\section{Introduction and Definitions} Thoroughly this study, we assume that the open unit disc$\left\{ z:z\in \mathbb{C} \text{ and }\left\vert z\right\vert <1\right\} $\ is shown by $\mathbb{E}$ , all normalized analytic functions of the form \begin{equation} f(z)=z+\overset{\infty }{\underset{n=2}{\sum }}a_{n}z^{n}. \label{eq1} \end{equation satisfying the conditions \begin{equation*} f(0)=0\text{ \ and \ }f^{\prime }(0)=0 \end{equation* in $\mathbb{E}$ is demonstrated by the symbol $\mathcal{A}$\ , and the subclass of all functions in $\mathcal{A}$ which are univalent in $\mathbb{E} $ is shown by $\mathcal{S}$. Due to the fact that the univalent functions are one to one, these functions are possessed of inverse .While the inverse of univalent functions are invertible they don't need to be defined on the entire unit disc $\mathbb{E}$ . Absolutely, according the Koebe one-quarter theorem , a disc of radius $\frac{1}{4}$ \ is in the image of $\mathbb{E}$ under every function $f\in \mathcal{S}$ $.$ Thus, every function $f\in \mathcal{S}$ own an inverse function and this inverse function can be defined on a disc of radius $\frac{1}{4}$. The inverse function of $f$ can be expressed by \begin{equation} g(w)=f^{-1}\left( w\right) =w~-a_{2}w^{2}+\left( 2a_{2}^{2}-a_{3}\right) w^{3}-\left( 5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right) w^{4}+\cdots . \label{eq1aa} \end{equation} If $f$ and $f^{-1}$ are univalent in $\mathbb{E},$then one can call the function $\ f\in \mathcal{A}$ is bi-univalent in $\mathbb{E}$. $~$We symbolize the class of bi-univalent functions defined in $\mathbb{E\ }$\ with $\Sigma $ . One can see the basic definitions of the analytic and bi univalent function class and their properties and intersting of functions in the class $\Sigma ,$ in the study of Srivastava \textit{et al.} \cit {Srivastava 2010}. Lewin (\cite{Lewin 67}) is the first mathematician working this subject. \cite{Lewin 67}) obtained the bound 1.51 for the modulus of the second coefficient $\left\vert a_{2}\right\vert .$ Later, Brannan and Clunie \cit {Brannan and Clunie 80} conjectured that $\left\vert a_{2}\right\vert \leqq \sqrt{2}$ for $f\in \Sigma .$ Later , if $f\left( z\right) \in \Sigma $ then $\max \left\vert a_{2}\right\vert =\frac{4}{3}$ is proven by Netanyahu \cit {Netanyahu 69} $.$ Then, a certain subclasses of class $\Sigma $ analogous subclasses $\mathcal{S}^{\star }\left( \beta \right) $ of starlike functions and $\mathcal{K}\left( \beta \right) $ convex functions of order $\beta $ \left( 0\leqq \beta <1\right) $ in $\mathbb{U}$ was expressed by Brannan and Taha \cite{Brannan and Taha 86}$,$ in turn (see \cite{Netanyahu 69}). The classes $\mathcal{S}_{\Sigma }^{\star }\left( \beta \right) $ and $\mathcal{ }_{\Sigma }\left( \beta \right) $ of bi-starlike functions of order $\beta $ in $\mathbb{U}$ and bi-convex functions of order $\beta $ in $\mathbb{U},$ corresponding to the function classes $\mathcal{S}^{\star }\left( \beta \right) $ and $\mathcal{K}\left( \beta \right) ,$ were also introduced congruently. For each of the function classes $\mathcal{S}_{\Sigma }^{\star }\left( \beta \right) $ and $\mathcal{K}_{\Sigma }\left( \beta \right) ,$ these mathematicians obtained some estimates for the initial coefficients but these estimatese were not sharp. Recently, motivated substantially by the following work on this area Srivastava \textit{et al.} \cite{Srivastava 2010}, many authors searched the coefficient bounds for diversified subclasses of bi-univalent functions (see, for instance, \cite{Frasin 2011}, \cite{Srivastava 2016}). Dealing with the bounds on the general coefficient \left\vert a_{n}\right\vert $ for $n\geqq 4,$ there isn't enough knowledge$. $ In the literature, only a few works has been made to identyfy the general coefficient bounds for $\left\vert a_{n}\right\vert $ for the analytic bi-univalent functions (see, for instance, \cite{Hamidi and Jahangiri 2014}, \cite{Jahangiri and Hamidi 2013}). Today, the problem of identyfying the coefficient for each of the coefficients $\left\vert a_{n}\right\vert $ \left( n\in \mathbb{N}\setminus \left\{ 1,2\right\} ;\;\mathbb{N}=\left\{ 1,2,3,\cdots \right\} \right) $ is an unsoluble problem . For two analytic functions $f$ and $F,$ as long as there is an analytic function $w$ defined on $\mathbb{E}$ by $\omega (0)=0,\left\vert \omega (z)\right\vert <1$ satisfying $f(z)=\varphi (\omega (z))$ , then the function $f$ \ is named to be subordinate to $F$ and demonstrated by $f\prec \varphi .$ The class of Ma-Minda starlike as well as convex functions \cit {MaMinda1992} are expreesed as follows: \begin{equation*} \mathcal{S}^{\star }\left( \varphi \right) =\left\{ f\in \Sigma :\frac zf^{\prime }(z)}{f(z)}\prec \varphi (z)\right\} \end{equation* and \begin{equation*} \mathcal{K}\left( \varphi \right) =\left\{ f\in \Sigma :\left( 1+\frac zf^{\prime ^{\prime }}(z)}{f^{\prime }(z)}\right) \prec \varphi (z)\right\} \end{equation* where $\varphi $ be analytic function having positive real part in $\mathbb{ },\varphi (0)=1$ and $\varphi ^{\prime }(0)>0$ also $\varphi (\mathbb{E)}$ is a region which is starlke with respect to 1 and symmetric with respect to the real axis. In 2013, the coefficient bounds for biunivalent Ma-Minda starlike and convex functions were described in the study of Ali at all. \cite{Ali2013} .\bigskip The classes $\mathcal{S}^{\star }\left( \varphi \right) $ and $\mathcal{K}\left( \varphi \right) $ includes several famous subclasses starlike functions like special case. \bigskip In 1970, the concept of quasi subordination was first defined by \cite{Robertson1970} . For the functions $f$ and $F$, if there exists analytic functions $F$ and $w,$ with $\left\vert F(z)\right\vert \leq 1,w(0)=0$ and $\left\vert w(z)\right\vert <1$ such that the equality \begin{equation*} f(z)=F(z)\varphi (\omega (z)) \end{equation* holds, then the function $f$ \ is said to be quasi subordinate to $\varphi ,$ demonstrated by \begin{equation} f(z)\prec _{q}\varphi (z),z\in \mathbb{E}\text{.} \label{eq1a} \end{equation} Prefering $F(z)\equiv 1$ , $\ $the quasi subordination given in (\ref{eq1a}) turns into the subordination $f(z)\prec \varphi (z).$ Thus, the quasi subordination is a universality of the well known subordination and majorization (see \cite{Robertson1970} ). \bigskip The work on quasi subordination is very wide and it includes some recent investigations (\cite{Lee1975}, \cite{Ren1991}, \cite{Robertson1970}, \cite{Mohoddarus2012}). From beginning to the end this study, it is assumed that\bigskip \begin{equation} \begin{array}{cc} F(z)=A_{0}+A_{1}z+A_{2}z^{2}+\cdots , & \left( \left\vert F(z)\right\vert \leq 1,z\in \mathbb{E}\right \end{array} \label{eq1b} \end{equation $\ $ \bigskip an \begin{equation} \begin{array}{cc} \varphi (z)=1+B_{1}z+B_{2}z^{2}+\cdots & \left( B_{1}>0\right \end{array \ \label{eq1c} \end{equation} \bigskip where $\varphi (z)$ is analytic function in $\mathbb{E}$. Guided by the above mentioned studies, we define a subclass of function class $\mathcal{S}$ in such a way. \begin{definition} The class $\mathcal{M}_{\Sigma }^{q,\varphi }(\gamma ,\lambda ,\delta )$ \left( 0\neq \gamma \in \mathbb{C} ,\lambda \geq 1,\delta \geq 0\text{ and }z,w\in \mathbb{E}\right) $\ consits of the function $f\in \Sigma $\ , expressed in the relation (\ref{eq1}), \ if the quasi subordination conditions \end{definition} \begin{equation} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)+\delta zf^{\prime \prime }(z)-1\right] \prec _{q}\left( \phi (z)-1\right) , \label{eq1d} \end{equation} \begin{equation} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)+\delta wg^{\prime \prime }(w)-1\right] \prec _{q}\left( \phi (w)-1\right) \end{equation} are satisfied, wehere the function $f^{-1}$ is the restruction of $g$ described in the relation (\ref{eq1aa}) , $\varphi $ is the function given in (\ref{eq1c}). By choosing the special values for $\delta ,\gamma ,\lambda $ and the class \mathcal{M}_{\Sigma }^{q,\varphi }(\gamma ,\lambda ,\delta )$ reduces to several earlier known classes of analytic and biunivalent functions studied in the literature. \begin{remark} Taking $\ \gamma =1$ and $\delta =0$ in the above class, so we obtain \end{remark} \begin{equation*} \mathcal{M}_{\Sigma }^{q,\varphi }(1,\lambda ,0)=\mathcal{R}_{\Sigma }^{q}(\lambda ,\varphi ) \end{equation* This study was firstly introduced by Patil and Naik (\cite{Patil2017}). The class $\mathcal{R}_{\Sigma }^{q}(\lambda ,\varphi )$ containes functions f\in \mathcal{S}$ satisfying \begin{equation*} \left[ \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)- \right] \prec _{q}\left( \phi (z)-1\right) \end{equation* and \begin{equation*} \left[ \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)- \right] \prec _{q}\left( \phi (w)-1\right) . \end{equation* where $z,w\in \mathbb{E}$ and $\lambda \geq 1.$ \begin{remark} Taking $F(z)\equiv 1,$ so the quasi subordination \ reduces the subordination and we get \end{remark} \begin{equation*} \mathcal{M}_{\Sigma }^{q,\phi }(\gamma ,\lambda ,\delta )=\mathcal{M _{\Sigma }^{\phi }(\gamma ,\lambda ,\delta ) \end{equation* This new class containes functions $f\in \Sigma $ satisfying \begin{equation*} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)+\delta zf^{\prime \prime }(z)\right] \prec \varphi (z) \end{equation* and \begin{equation*} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)+\delta wg^{\prime \prime }(w)\right] \prec \varphi (w). \end{equation*} The class $\mathcal{M}_{\Sigma }^{\varphi }(\gamma ,\lambda ,\delta )$ involves many well known classes, which are given following: \begin{enumerate} \item Taking $\ \gamma =1$ and $\delta =0$ , so we obtain \end{enumerate} \begin{equation*} \mathcal{M}_{\Sigma }^{q,\phi }(1,\lambda ,0)=\mathcal{M}_{\Sigma _{1,\lambda }}^{0}(\phi )=\mathcal{R}_{\Sigma }(\lambda ,\phi ). \end{equation* The class $\mathcal{R}_{\Sigma }(\lambda ,\phi )$ was first introduced by Kumar et all. (\cite{Kumar2013}). This class involves functions $f\in \Sigma $ satisfying \begin{equation*} \left[ \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)\right] \prec \varphi (z) \end{equation* and \begin{equation*} \left[ \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)\right] \prec \varphi (w) \end{equation* for the function $\phi $ analytic \bigskip and $\lambda \geq 1.$ The class $\mathcal{M}_{\Sigma }^{\varphi }(1,\lambda ,0)$ involves many earlier classes. These classes are given following: \begin{enumerate} \item[(i)] Taking $\varphi (z)=\frac{1+(1-2\beta )z}{1-z},0\leq \beta <1,$ we obtain the class $\mathcal{B}_{\Sigma }(\beta ,\lambda )$ defined by Frasin and Aouf ((\cite{Frasin 2011}), see Definition 3.1). \item[(ii)] Taking $\varphi (z)=\left( \frac{1+z}{1-z}\right) ^{\alpha },0<\alpha \leq 1,\lambda \geq 1,$ then we have the class $\mathcal{B _{\Sigma }(\alpha ,\lambda )$ defined by Frasin and Aouf ((\cite{Frasin 2011 ), see Definition 2.1). \item[(iii)] Taking $\lambda =1$ then the class reduces $\mathcal{H}_{\Sigma }(\varphi )$ investigated and defined by Ali et all. \cite{Ali2013}. \item[(\i v)] Taking $\lambda =1$ and $\varphi (z)=\frac{1+(1-2\beta )z}{1-z ,0\leq \beta <1,$ then the class reduces $\mathcal{H}_{\Sigma }(\beta )$ defined and investigated by Srivastava et al. \cite{Srivastava 2010}(see Definition 2). \item[(v)] Taking $\lambda =1$ and $\varphi (z)=\left( \frac{1+z}{1-z \right) ^{\alpha },0<\alpha \leq 1,$ then the class reduces $\mathcal{H _{\Sigma }^{\alpha },$ defined and investigated by Srivastava et al. \cit {Srivastava 2010}(see Definition 1). \item[2.] Taking $\ \gamma =1,\lambda =1$ and $\delta =0$ \ then we get \end{enumerate} \begin{equation*} \mathcal{M}_{\Sigma }^{\varphi }(1,1,0)=\mathcal{R}_{\Sigma }^{{}}(\varphi ). \end{equation* The class $\mathcal{R}_{\Sigma }^{{}}(\varphi )$ was introduced by Ali et al. \cite{Ali2013}. This class involves of functions $f\in \Sigma $ satisfying \begin{equation*} f^{\prime }(z)\prec \varphi (z) \end{equation* and \begin{equation*} g^{\prime }(w)\prec \varphi (w). \end{equation*} \begin{enumerate} \item[3.] Taking \bigskip $\lambda =1$ and $0\leq \delta <1,$ so we have \end{enumerate} \begin{equation*} \mathcal{M}_{\Sigma }^{\varphi }(\gamma ,1,\delta )=\mathcal{R}_{\Sigma }(\eta ,\lambda ,\varphi ) \end{equation* The class $\mathcal{R}_{\Sigma }(\eta ,\lambda ,\varphi )$ was studied by Deniz (\cite{Deniz2013}). This class involves of the functions $f\in \Sigma $ satisfying \begin{equation*} 1+\frac{1}{\gamma }\left[ f^{\prime }(z)+\eta zf^{\prime ^{\prime }}(z)- \right] \prec \varphi (z) \end{equation* and\bigskip \begin{equation*} 1+\frac{1}{\gamma }\left[ g^{\prime }(w)+\eta wg^{\prime ^{\prime }}(w)- \right] \prec \varphi (w). \end{equation*} \begin{enumerate} \item[4.] Taking $\ \gamma =1,\lambda =1$ and $\varphi (z)=\frac{1+(1-2\beta )z}{1-z},0\leq \beta <1$ , then we have \end{enumerate} \begin{equation*} \mathcal{M}_{\Sigma }^{q,\varphi }(\gamma ,\lambda ,1)=\mathcal{M}_{\Sigma }^{\alpha }(\lambda ,\delta ). \end{equation* The class $\mathcal{M}_{\Sigma }^{\alpha }(\lambda ,\delta )$ was defined and ivestigated by Bulut (\cite{Bulut2016}). This class involves of the functions $f\in \Sigma $ satisfying \begin{equation*} \func{Re}\left( \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)+\delta zf^{\prime \prime }(z)\right) >\alpha \end{equation* and \begin{equation*} \func{Re}\left( \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)+\delta wg^{\prime \prime }(w)\right) >\alpha \end{equation* where $0\leq \alpha <1.$ \section{Coefficient Estimates for the Function Class $\mathcal{M}_{\Sigma }^{q,\protect\varphi }(\protect\gamma ,\protect\lambda ,\protect\delta )$} Firstly, we will state the Lemma 4 to obtain our result. \begin{lemma} (\cite{Pommerenke1975}) If $p\in \mathcal{P}$, then $\left\vert p_{i}\right\vert \leq 1$ for each $i,$ where $\mathcal{P}$ is the family all functions $p$, analytic in $\mathbb{E}$, for which \end{lemma} \begin{equation*} \begin{array}{cc} \func{Re}\left\{ p(z)\right\} >0 & ,\left( z\in \mathbb{E}\right \end{array \end{equation* where \begin{equation*} \begin{array}{cc} p(z)=1+p_{1}z+p_{2}z^{2}+\cdots & \left( z\in \mathbb{E}\right) \end{array \end{equation*} We begin this section by finding the estimates on the coefficients \left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ for functions in the class $\mathcal{M}_{\Sigma }^{q,\varphi }(\gamma ,\lambda ,\delta )$ proposed by Definition 1. \begin{theorem} If $f\in \Sigma $ expressed by (\ref{eq1}) belongs the class $\mathcal{M _{\Sigma }^{q,\varphi }(\gamma ,\lambda ,\delta ),0\neq \gamma \in \mathbb{C} ,\lambda \geq 1,\delta \geq 0$ and $z,w\in \mathbb{E},$ then \bigskip \begin{equation} \left\vert a_{2}\right\vert \leq \min \left\{ \tfrac{\left\vert A_{0}\right\vert B_{1}}{1+\lambda +2\delta }\left\vert \gamma \right\vert \sqrt{\tfrac{\left\vert A_{0}\right\vert \left( B_{1}+\left\vert B_{2}-B_{1}\right\vert \right) }{1+2\lambda +6\delta }\gamma }\right\} \label{eq8} \end{equation an \begin{equation} \left\vert a_{3}\right\vert \leq \min \left\{ \begin{array}{l} \left[ \frac{A_{0}^{2}B_{1}^{2}}{\left( 1+\lambda +2\delta \right) ^{2} \left\vert \gamma \right\vert +\tfrac{\left( \left\vert A_{0}\right\vert +\left\vert A_{1}\right\vert \right) B_{1}}{1+2\lambda +6\delta }\right] \left\vert \gamma \right\vert , \\ \\ \tfrac{\left\vert A_{0}\right\vert \left( B_{1}+\left\vert B_{2}-B_{1}\right\vert \right) +\left( \left\vert A_{0}\right\vert +\left\vert A_{1}\right\vert \right) B_{1}}{1+2\lambda +6\delta }\left\vert \gamma \right\vert \end{array \right. . \label{eq9} \end{equation} \end{theorem} \begin{proof} \bigskip If $f\in $ $\mathcal{M}_{\Sigma _{\gamma ,\lambda }}^{q,\delta }(\varphi )$ then, there are analytic functions $u,v:\mathbb{E\rightarrow E}$ with $u(0)=v(0)=0$, $\left\vert u(z)\right\vert <1,\left\vert v(w)\right\vert <1$ and a function $F$ given by\ (\ref{eq1a}) , such that \begin{equation} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)+\delta zf^{\prime ^{\prime }}(z)-1\right] =F(z)\left[ \varphi \left( u(z)\right) -1\right] \label{eq10} \end{equation and \end{proof} \begin{equation} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)+\delta wg^{\prime ^{\prime }}(w)-1\right] =F(w)\left[ \varphi \left( u(w)\right) -1\right] \label{eq11} \end{equation Determine the functions $p_{1}$ and $p_{2}$ in $\mathcal{P}$ given by \begin{equation*} p_{1}(z)=\frac{1+u(z)}{1-u(z)}=1+c_{1}z+c_{2}z^{2}+\cdots \end{equation* and \begin{equation*} p_{2}(w)=\frac{1+v(w)}{1-v(w)}=1+d_{1}w+d_{2}w^{2}+\cdots . \end{equation* Thus, \begin{equation} u(z)=\frac{p_{1}(z)-1}{p_{1}(z)+1}=\frac{1}{2}\left[ c_{1}z+\left( c_{2} \frac{c_{1}^{2}}{2}\right) z^{2}+\cdots \right] \label{eq12} \end{equation and \begin{equation} v(w)=\frac{p_{2}(w)-1}{p_{2}(w)+1}=\frac{1}{2}\left[ d_{1}w+\left( d_{2} \frac{d_{1}^{2}}{2}\right) w^{2}+\cdots \right] . \label{eq13} \end{equation The fact that $p_{1}$ and $p_{2}$ are analytic in $\mathbb{E}$ with p_{1}(0)=p_{2}(0)=1$ and and have their real part in $\mathbb{E}$ is obvius. Due to the fact that all of the functions $u,v:\mathbb{E\rightarrow E}$ and p_{1}$, $p_{2}$ have their real part in $\mathbb{E}$ , the relations \left\vert c_{i}\right\vert \leq 2$ and $\left\vert d_{i}\right\vert \leq 2$ are true (\cite{Pommerenke1975}). Using (\ref{eq10}) and (\ref{eq14}) together with (\ref{eq1b}) \ and (\ref{eq1c}) in the right hands of the relations (\ref{eq10}) and (\ref{eq11}) , we obtain \bigskip \begin{equation} F(z)\left[ \varphi \left( u(z)\right) -1\right] =\frac{1}{2 A_{0}B_{1}c_{1}z+\left\{ \frac{1}{2}A_{1}B_{1}c_{1}+\frac{1}{2 A_{0}B_{1}\left( c_{2}-\frac{c_{1}^{2}}{2}\right) \right\} z^{2}+\cdots \bigskip \label{eq14} \end{equation and \begin{equation} F(w)\left[ \varphi \left( v(w)\right) -1\right] =\frac{1}{2 A_{0}B_{1}d_{1}w+\left\{ \frac{1}{2}A_{1}B_{1}d_{1}+\frac{1}{2 A_{0}B_{1}\left( d_{2}-\frac{d_{1}^{2}}{2}\right) \right\} w^{2}+\cdots \label{eq15} \end{equation By using the form of the functions $f$ and $g$ ,which are giben by (\ref{eq1 ) and (\ref{eq1aa}) $,g=f^{-1},$ we have \begin{equation} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{f(z)}{z}+\lambda f^{\prime }(z)+\delta zf^{\prime \prime }(z)\right] =\frac{1}{\gamma }\left[ \overset{\infty }{\underset{n=2}{\sum }}\left[ 1+(n-1)\lambda +n(n-1)\delta \right] a_{n}z^{n-1}\right] \label{eq16} \end{equation and \begin{equation} \frac{1}{\gamma }\left[ \left( 1-\lambda \right) \frac{g(w)}{w}+\lambda g^{\prime }(w)+\delta wg^{\prime \prime }(w)\right] =\frac{1}{\gamma }\left[ \overset{\infty }{\underset{n=2}{\sum }}\left[ 1+(n-1)\lambda +n(n-1)\delta \right] A_{n}w^{n-1}\right] . \label{eq17} \end{equation Comparing the coefficients of (\ref{eq14}) with (\ref{eq16}) and (\ref{eq15 ) with (\ref{eq17}), then we have \begin{equation} \frac{1}{\gamma }\left( 1+\lambda +2\delta \right) a_{2}=\frac{1}{2 A_{0}B_{1}c_{1} \label{eq18} \end{equation \bigskip \begin{equation} \frac{1}{\gamma }\left( 1+2\lambda +6\delta \right) a_{3}=\frac{1}{2 A_{1}B_{1}c_{1}+\frac{1}{2}A_{0}B_{1}\left( c_{2}-\frac{c_{1}^{2}}{2}\right) +\frac{A_{0}B_{2}}{4}c_{1}^{2} \label{eq19} \end{equation \bigskip \begin{equation} -\frac{1}{\gamma }\left( 1+\lambda +2\delta \right) a_{2}=\frac{1}{2 A_{0}B_{1}d_{1} \label{eq20} \end{equation \bigskip \begin{equation} \frac{1}{\gamma }\left( 1+2\lambda +6\delta \right) \left( 2a_{2}^{2}-a_{3}\right) =\frac{1}{2}A_{1}B_{1}d_{1}+\frac{1}{2 A_{0}B_{1}\left( d_{2}-\frac{d_{1}^{2}}{2}\right) +\frac{A_{0}B_{2}}{4 d_{1}^{2} \label{eq21} \end{equation From (\ref{eq18}) and (\ref{eq20}), we have \begin{equation} c_{1}=-d_{1} \label{eq22} \end{equation \bigskip and \begin{equation} \frac{8}{\gamma ^{2}}\left( 1+\lambda +2\delta \right) ^{2}a_{2}^{2}=A_{0}^{2}B_{1}^{2}\left( c_{1}^{2}+d_{1}^{2}\right) . \label{eq23} \end{equation Adding (\ref{eq19}) and (\ref{eq21}) \ , we get \begin{equation} \frac{1}{\gamma }\left( 1+2\lambda +6\delta \right) a_{2}^{2}=\frac 2A_{0}B_{1}\left( c_{2}+d_{2}\right) +A_{0}\left( B_{2}-B_{1}\right) \left( c_{1}^{2}+d_{1}^{2}\right) }{8}. \label{eq24} \end{equation Using (\ref{eq22}) end Lemma 4 in equalities (\ref{eq23}) and (\ref{eq24}), we obtain desired result given by th e inequality (\ref{eq8}). Now, to find the bound on $\left\vert a_{3}\right\vert ,$ by using the relations (\ref{eq21}) and (\ref{eq19}), then we have\bigskip \begin{equation} \frac{2}{\gamma }\left( 1+2\lambda +6\delta \right) \left( a_{3}-a_{2}^{2}\right) =\frac{2A_{1}B_{1}c_{1}+A_{0}B_{1}\left( c_{2}-d_{2}\right) }{2\left( 1+2\lambda +6\delta \right) }\gamma \label{eq25} \end{equation Substituting $a_{2}^{2}$ from (\ref{eq23}) and (\ref{eq24}) and putting (\re {eq25}) respectively, we obtain \begin{equation} a_{3}=\frac{A_{0}^{2}B_{1}^{2}\left( c_{1}^{2}+d_{1}^{2}\right) }{8\left( 1+\lambda +2\delta \right) ^{2}}\gamma ^{2}+\frac{\left( 2A_{1}B_{1}c_{1}+A_{0}B_{1}\left( c_{2}-d_{2}\right) \right) }{4\left( 1+2\lambda +6\delta \right) }\gamma \label{eq26} \end{equation an \begin{equation} a_{3}=\frac{2A_{0}B_{1}\left( c_{2}+d_{2}\right) +A_{0}\left( B_{2}-B_{1}\right) \left( c_{1}^{2}+d_{1}^{2}\right) }{8\left( 1+\lambda +2\delta \right) ^{2}}\gamma +\frac{2A_{1}B_{1}c_{1}+A_{0}B_{1}\left( c_{2}-d_{2}\right) }{4\left( 1+2\lambda +6\delta \right) }\gamma \label{eq27} \end{equation Using the Lemma in (\ref{eq26}) and (\ref{eq27}), we complete the proof of theorem. \section{Corollaries and Consequences} Choosing $\gamma =1$ and $\delta =0$ in Theorem 5, we get the consequences below \begin{corollary} (\cite{Patil2017}). Let the function $f\in \mathcal{R}_{\Sigma }^{q}(\lambda ,\phi )$ . Then \begin{equation*} \left\vert a_{2}\right\vert \leq \min \left\{ \tfrac{\left\vert A_{0}\right\vert B_{1}}{1+\lambda },\sqrt{\tfrac{\left\vert A_{0}\right\vert \left( B_{1}+\left\vert B_{2}-B_{1}\right\vert \right) }{1+2\lambda } \right\} \end{equation* an \begin{equation*} \left\vert a_{3}\right\vert \leq \min \left\{ \begin{array}{l} \frac{A_{0}^{2}B_{1}^{2}}{\left( 1+\lambda \right) ^{2}}+\tfrac{\left( \left\vert A_{0}\right\vert +\left\vert A_{1}\right\vert \right) B_{1}} 1+2\lambda }, \\ \\ \tfrac{\left\vert A_{0}\right\vert \left( B_{1}+\left\vert B_{2}-B_{1}\right\vert \right) +\left( \left\vert A_{0}\right\vert +\left\vert A_{1}\right\vert \right) B_{1}}{1+2\lambda \end{array \right. . \end{equation*} \end{corollary} \begin{remark} The estimates for $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ of Corollary 6 show that Theorem \ 5 (\ref{Thm5}) is an improvement of the estimates obtained by Patil nd Naik (7, (\cite{Patil2017 ), Theorem 2.2). \end{remark} \bigskip Choosing $F(z)\equiv 1$ in Theorem 5 , we obtain the following \begin{corollary} Let the function $f\in \mathcal{M}_{\Sigma }^{\phi }(\gamma ,\lambda ,\delta ).$ Then, \begin{equation} \left\vert a_{2}\right\vert \leq \sqrt{\left( \tfrac{B_{1}+\left\vert B_{2}-B_{1}\right\vert }{1+2\lambda +2\delta }\right) \gamma } \end{equation and \end{corollary} \bigskip \begin{equation} \left\vert a_{3}\right\vert \leq \left( \tfrac{B_{1}+2\left\vert B_{2}-B_{1}\right\vert }{1+2\lambda +2\delta }\right) \gamma . \end{equation} If we let $F(z)\equiv 1,$ $\gamma =1$ and $\delta =0$ in Theorem 5 , we get the consequences below: \begin{corollary} \bigskip (\cite{Kumar2013}) Let the function $f\in \mathcal{R}_{\Sigma }(\lambda ,\phi ).$Then, \end{corollary} \begin{equation*} \left\vert a_{2}\right\vert \leq \min \left\{ \tfrac{B_{1}}{1+\lambda } \sqrt{\tfrac{B_{1}+\left\vert B_{2}-B_{1}\right\vert }{1+2\lambda }}\right\} \end{equation* an \begin{equation*} \left\vert a_{3}\right\vert \leq \min \left\{ \frac{B_{1}^{2}}{1+2\lambda } \frac{B_{1}^{2}}{^{\left( 1+\lambda \right) ^{2}}},\tfrac{B_{1}+\left\vert B_{2}-B_{1}\right\vert }{1+2\lambda }\right\} . \end{equation*} \begin{remark} In light of the Corollary 9, we can state the following remarks: \begin{enumerate} \item The estimates for $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ of Corollary 9 (\ref{Cor8}) show that Theorem 5 (\re {Thm5}) is an improvement of the estimates obtained by Kumar et all. \bigskip (\cite{Kumar2013}). \end{enumerate} \end{remark} \begin{enumerate} \item[2] Further, if we let $\varphi (z)=\frac{1+\left( 1-2\beta \right) z} 1-z}\ \ \ \left( 0\leq \beta <1\right) ,$and $\lambda =1$, then Theorem 5 gives the bounds on $\left\vert a_{2}\right\vert \leq \sqrt{\tfrac{2(1-\beta )}{3}}$ for functions $f\in $ $R_{\sigma }(\beta )$ which coincides with the consequence of Xu et all (\cite{Xu2012}). \ Also if we let $\beta =1$, then estimates of Xu et all (\cite{Xu2012}) becomes $\left\vert a_{2}\right\vert \leq \sqrt{\tfrac{2}{3}}$ for functions in the class $R_{\sigma }(0).$Since the estimate on $\left\vert a_{2}\right\vert $ for $f\in R_{\sigma }(0)$ is improved over the assumed estimate $\left\vert a_{2}\right\vert \leq \sqrt{2} $ for $f\in \sigma ,$ the functions in $R_{\sigma }(0)$ can not be the nominee for the sharpness of the estimate in the class s $\ \sigma .$ \item[3.] Taking\bigskip\ $\lambda =1$ , then we get the consequences below \end{enumerate} \begin{corollary} Let the function $f\in \mathcal{R}_{\Sigma }(\phi ).$Then, \end{corollary} \begin{equation*} \left\vert a_{2}\right\vert \leq \min \left\{ \tfrac{B_{1}}{2},\sqrt{\tfrac B_{1}+\left\vert B_{2}-B_{1}\right\vert }{3}}\right\} \end{equation* an \begin{equation*} \left\vert a_{3}\right\vert \leq \min \left\{ \frac{B_{1}^{{}}}{3}+\frac B_{1}^{2}}{4},\tfrac{B_{1}+\left\vert B_{2}-B_{1}\right\vert }{3}\right\} . \end{equation*} \begin{remark} Corollary 11 is the improvement of the estimates given by Ali et all. (Theorem2.1., (\cite{Ali2013})) \end{remark} \textbf{Acknowledgements} The authors are extremely grateful to the reviewers for a careful reading of the manuscript and making valuable suggestions leading to a better presentation of the paper.
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Bruce McEver | President & Co-Founder Berkshire Capital Securities LLC, USA & UK Case Study Outline → Bruce McEver Video (above) → Learning Objectives → Main Category of Action → Bruce's Story → Summary of Case → Interview with Bruce McEver → Introduction to the United States – Demographics and Economy – Religious Demographics – Conflict and Violence related to Religion → More About Bruce McEver → Discussion Questions → Media and Added Resources ♦ Return to Templeton Religion Trust "Business Case Studies" Home Bruce McEver, co-founder and president of Berkshire Capital Securities LLC in New York and London, set up a foundation which works to cultivate inter-religious understanding through the promotion of religious literacy especially among business leaders. The learning objectives for this case study include: 1. Business and ethics can be and are enriched by spirituality and religious ethics. 2. Business people, like many people, draw on inner spiritual resources in their work. 3. Business can benefit from religious literacy. 4. Philanthropy usually follows a business leader's passions. Main Category of Action Social investment and philanthropy Financial and in-kind contributions, and strategic social investment support for NGOs, UN and/or multilateral agencies or directly to affected communities and/or contribution of functional expertise through volunteering efforts. Bruce's Story Divinity school is not the most likely place to find a venture capitalist, an investment banker or a published poet. Well, maybe a poet. Yet H. Bruce McEver, 72, is all of those things, and more. The man who founded Berkshire Capital Securities LLC, a global merger and acquisitions investment firm, in 1983 went on to complete a master's in theology at Harvard Divinity School in 2011. At Harvard, McEver noticed a lack of religion and culture courses for students at its prominent business school. With his friend Ron Thiemann, a theologian at Harvard, McEver founded a program called Business Across Religious Traditions — BART, for short — that brought the foundational ethics of the world's religious traditions to the business school classroom. "Jesus, the Buddha, Muhammad, Joseph Smith, these were religious entrepreneurs," McEver said recently from his New York City office. "Almost all of our ethics spring from the religious traditions these entrepreneurs founded. If you understand the religious background of these ethics it makes them much more full-fleshed, more powerful for businesspeople." McEver and Thiemann went on to start the Foundation for Religious Literacy, which oversees the BART program and several others. The foundation's Faith, Ethics and Leadership seminars bring together business leaders with religious thinkers, and its Religious Liberty Roundtable promotes tolerance and religious understanding as good global business practices. The foundation also funds a free online Harvard course dedicated to promoting religious literacy, and it recently launched a Massive Open Online Course, or MOOC, expected to reach 50,000 people globally. The foundation sponsors a curatorship at the Smithsonian dedicated to religious literacy and fosters a relationship between Georgia Tech University — McEver's alma mater — and Emory University's Candler School of Theology with its Leadership and Multifaith Program. The program, known by its acronym LAMP, offers a series of religious literacy courses now reaching thousands of students. "Business leaders don't exist in a vacuum," said Ben Marcus, a Harvard Divinity master's candidate and an adviser to the foundation. "They need to understand why we create wealth and to what end. They need to know how religion motivates their employees and clients. Religion affects their business calculus on any number of issues, so to be literate in it is smart." McEver was raised in the United Methodist Church and is now a member of a Congregationalist church. He had a kind of spiritual awakening after the death of his wife, something he describes in his poem "Many Paths," published in 2012 by The Cortland Review. Its final lines might be a foundational statement about the Foundation for Religious Literacy: I look up into a blinding, cold sun and feel a release — an energy courses the length of my body, and says again, then again: There are many paths. Nothing has ever been so clear. Charles Haynes, founding director of the Religious Freedom Center at the Newseum Institute, knows McEver through a program the Foundation for Religious Literacy is funding at the Newseum for journalists. "Bruce has worked hard to put religious literacy on the national agenda because he recognizes that ignorance is a root cause of division, hate and intolerance," Haynes said. "By investing in religious literacy education at Harvard Divinity School and elsewhere, Bruce is helping to ensure that educators, thought leaders and others have access to resources that are academically and constitutionally sound. This will be his legacy." Today, McEver is turning his energy toward ensuring the Foundation for Religious Literacy survives its founders. "I am always asking how do we have a broader reach," he said. "I am still searching. But I think, given our political climate, there has never been a bigger need than now for people to have an understanding of other people's religious traditions." Summary of Case Business leaders often underestimate the positive influence of faith in the workplace and society in general, especially in the way that religious freedom promotes peace and stability. H. Bruce McEver, Chairman and Founder of Berkshire Capital Securities LLC, created with the late Prof. Ron Thiemann of Harvard Divinity School, The Foundation for Religious Literacy. The foundation promotes religious understanding by bringing together business professionals along with outstanding academics and practitioners. The Foundation cultivates inter-religious understanding and practical skills through collaboration with partners such as the regional Harvard Business School Clubs via its Business Across Religious Traditions seminars. Through its Faith Ethics and Leadership seminars, leaders in business and other professions, the foundation also advocates values and ethics derived from religious and secular traditions, fostering a healthy pluralist and peaceful democracy through respect for differences. Interview with Bruce McEver The following interview by the Religious Freedom & Business Foundation was done during the inaugural Global Business & Interfaith Peace Awards, which were held in Rio de Janeiro on Tuesday, Sept. 6, a day before the Opening Ceremony of the 2016 Paralympic Games. The awards recognize business leaders – current or past CEOs – who have demonstrated leadership in championing interfaith understanding and peace. The Awards are a partnership initiative of the Religious Freedom & Business Foundation (RFBF), and the United Nations Global Compact Business for Peace (B4P) platform, with collaboration from the United Nations Alliance of Civilizations. The next awards will be given in Seoul, Korea, ahead of the 2018 PyeongChang Winter Paralympics. Note: Interview begins at 0:14 marker. Introduction to the United States Demographics and Economy* A 2015 estimate puts the U.S. population at 321,368,864. Of this, nearly 80% of the population is white, approximately 13% are African American, the Asian population is at 4%, and Amerindian and Alaskan natives represent close to 1%. The US has the most technologically powerful economy in the world, with a per capita GDP of $54,800. US firms are at or near the forefront in technological advances, especially in computers, pharmaceuticals, and medical, aerospace, and military equipment; however, their advantage has narrowed since the end of World War II. Based on a comparison of GDP measured at Purchasing Power Parity conversion rates, the US economy in 2014, having stood as the largest in the world for more than a century, slipped into second place behind China, which has more than tripled the US growth rate for each year of the past four decades. In the US, private individuals and business firms make most of the decisions, and the federal and state governments buy needed goods and services predominantly in the private marketplace. US business firms enjoy greater flexibility than their counterparts in Western Europe and Japan in decisions to expand capital plant, to lay off surplus workers, and to develop new products. At the same time, businesses face higher barriers to enter their rivals' home markets than foreign firms face entering US markets. Long-term problems for the US include stagnation of wages for lower-income families, inadequate investment in deteriorating infrastructure, rapidly rising medical and pension costs of an aging population, energy shortages, and sizable current account and budget deficits. The onrush of technology has been a driving factor in the gradual development of a "two-tier" labor market in which those at the bottom lack the education and the professional/technical skills of those at the top and, more and more, fail to get comparable pay raises, health insurance coverage, and other benefits. But the globalization of trade, and especially the rise of low-wage producers such as China, has put additional downward pressure on wages and upward pressure on the return to capital. Since 1975, practically all the gains in household income have gone to the top 20% of households. Since 1996, dividends and capital gains have grown faster than wages or any other category of after-tax income. Imported oil accounts for nearly 55% of US consumption and oil has a major impact on the overall health of the economy. Crude oil prices doubled between 2001 and 2006, the year home prices peaked; higher gasoline prices ate into consumers' budgets and many individuals fell behind in their mortgage payments. Oil prices climbed another 50% between 2006 and 2008, and bank foreclosures more than doubled in the same period. Besides dampening the housing market, soaring oil prices caused a drop in the value of the dollar and a deterioration in the US merchandise trade deficit, which peaked at $840 billion in 2008. Because the US economy is energy-intensive, falling oil prices since 2013 have alleviated many of the problems the earlier increases had created. In March 2010, President OBAMA signed into law the Patient Protection and Affordable Care Act, a health insurance reform that was designed to extend coverage to an additional 32 million Americans by 2016, through private health insurance for the general population and Medicaid for the impoverished. Total spending on healthcare – public plus private – rose from 9.0% of GDP in 1980 to 17.9% in 2010. Wars in Iraq and Afghanistan required major shifts in national resources from civilian to military purposes and contributed to the growth of the budget deficit and public debt. Through 2014, the direct costs of the wars totaled more than $1.5 trillion, according to US Government figures. * CIA Factbook Religious Demographics As shown in the Pew Research chart below, 78.3% of the U.S. population identify as Christian, while 16.4% identify as unaffiliated. Jews represent 1.8% of the population and Buddhists represent 1.2%. Muslims, Hindus and folk religions each respectively represent less than 1% of the population. In 2010, according to Pew Research, out of a population of 310,380,000, Christians had 243,060,000 adherents, with an expected growth to 258,410,00 in 2030. Unaffiliated U.S. citizens had 50,980,000 numbers in 2010, and are expected to expand to 75,740,000 in 2030. The median age for all religions is 37, with the Jewish population representing the oldest at 41, and Muslims representing the youngest at 24. The Muslim U.S. population has the highest fertility rate at 2.8, followed by Christians at 2.1, with the unaffiliated fertility rate at 1.6. Conflict and Violence Related to Religion The global war on terrorism and the backlash in the United States against religious minorities, including calls during the 2016 presidential election to limit Muslim immigration, add to social tensions involving religion. In the 2015 FBI's Uniform Crime Reporting (UCR) Program, the nation's law enforcement agencies reported that there were 7,173 victims of hate crimes. Of these victims, 1,402 were victims of anti-religious hate crimes: – 52.1 percent were victims of crimes motivated by their offenders' anti-Jewish bias. – 21.9 percent were victims of anti-Islamic (Muslim) bias. – 4.3 percent were victims of anti-Catholic bias. – 4.1 percent were victims of bias against groups of individuals of varying religions (anti-multiple religions, group). – 3.6 percent were victims of anti-Eastern Orthodox (Russian, Greek, Other) bias. – 3.4 percent were victims of anti-Protestant bias. – 1.3 percent were victims of anti-Other Christian bias. – 0.6 percent were victims of anti-Mormon bias. – 0.4 percent were victims of anti-Hindu bias. – 0.4 percent were victims of anti-Sikh bias. – 0.1 percent were victims of anti-Jehovah's Witness bias. – 0.1 percent were victims of anti-Buddhist bias. – 0.1 percent were victims of anti-Atheist/Agnostic bias. -7.6 percent were victims of bias against other religions (anti-other religion). As shown in the Pew Research chart below, the global median score for social hostilities involving religion is 2.4 on a 10-point scale, where 10 is high. The United States' rating is 3.1, meaning it has moderate social hostilities involving religion. The global median score for governmental restrictions on religious freedom is 3.1. The U.S. has low moderate restrictions on religious freedom with a score of 3.0. More About Bruce McEver Bruce McEver founded Berkshire Capital in 1983, pioneering the concept of providing independent merger, acquisition, and strategic advisory services for investment managers and securities firms. He directs long-term strategy and business development efforts. Previously, Bruce served as the Assistant to the Chairman of Paine Webber Group Inc. for mergers and acquisitions. He was formerly a Vice President with Blyth Eastman Dillon Inc. and a venture capital analyst at Bessemer Securities, Inc. Bruce earned a BIE from Georgia Institute of Technology, an MBA from Harvard Business School, and an MTS in Religion and Literature at Harvard Divinity School. Earlier in life, he studied at the Technische Hochschule in Germany. And as a Lieutenant, USN, he was on the staff of the Assistant Secretary of Defense (Systems Analysis). On the philanthropic and social engagement side, Bruce is President and Co-Founder of the Foundation for Religious Literacy through which he pursues his passion for interfaith understanding becoming the norm for corporate leaders. For instance, Mr. McEver funded Harvard University's Business Across Religious Traditions (BART) program, through which he and Ron Thiemann (former Dean of Harvard Divinity School) created educational modules on economic ethics in Buddhism, Confucianism, Christianity, Daoism, Hinduism, Islam, and Judaism. Programs have been offered in Boston, London, New York City, and San Francisco, often in collaboration with local Harvard Business School Clubs. Mr. McEver donated funds to create the Religious Literacy Project, which is housed at Harvard's Center for the Study of World Religions. The project is a collaboration with the Harvard Extension School. Bruce also created the Foundation for Religious Literacy (TFRL) to promote inter-religious understanding by bringing business leaders and other professionals together with outstanding academics and practitioners. The Foundation cultivates skills for considering values and ethics derived from religious and secular traditions, and acknowledges that a healthy pluralist democracy requires respect for difference. Through this seminar-style executive education series, business leaders with demonstrated interest in continuing opportunities for learning come together under the auspices of TFRL to learn about and discuss ways their professional work can draw on resources from religion across traditions. Each seminar hosts a leading academic or practitioner and addresses how the resources of religious traditions can increase professional effectiveness, ethical leadership, and personal conduct. Seminars are organized, not by shared religious affiliation, but by a common interest in how religion can inform action in the workplace. TFRL has also sponsored a Curatorship of American Religion at the Smithsonian's National Museum of American History to highlight religious diversity and the importance of religious freedom in America; the Leadership and Multifaith (LAMP) Program, a collaboration between the Georgia Institute of Technology and the Candler School of Theology at Emory University; and the Religious Literacy Project at Harvard Divinity School. Bruce is a member of the Dean's Council at Harvard Divinity School and the Candler School of Theology at Emory University. Bruce founded the Business across Religious Traditions (BART) Program between Harvard Business School and Harvard Divinity School and co-founded The Foundation for Religious Literacy. Bruce is a Professor of Practice at the Georgia Institute of Technology. He participates actively in environmental conservation efforts and is a hiker, biker, and author of three books of poetry. 1. How can business and ethics be enriched by spirituality and religious ethics? 2. In what ways can business people draw on inner spiritual resources in their work? 3. How can business benefit from religious literacy? 4. In what ways can philanthropy follow a business leader's passions? Media and Added Resources – Thomas Lux & Bruce McEver: The Poetry of Work (The Paula Gordon Show) – Bruce McEver Tribute Video (Ivan Allen College of Liberal Arts) – The Foundation for Religious Literacy This case study was prepared by Melissa Grim, J.D., M.T.S., a senior research fellow with the Religious Freedom & Business Foundation, and Brian Grim, Ph.D., president of the foundation. It is made possible by a generous grant from the Templeton Religion Trust.
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Q: ubuntu18.04 is stuck on black screen after installing nvidia driver and reboot Problem I want to install nvidia driver in my laptop. My laptop is always stuck on blackscreen after installing nvidia driver and executing reboot. I have tried tried different approaches and many nvidia driver versions. However, the problem is always here. My ubuntu18.04 has two kernel 5.4.0-122 and 5.4.0-117. I installed nvidia driver by software update. Then the output is normal after executing nvidia-smi. But when I reboot my laptop, a probelm appearred. If I choose 5.4.0-122 kernel, the laptop will be stuck in black screen. If I choose 5.4.0-117 kernel, NVIDIA-SMI has failed because it couldn't communicate with the NVIDIA driver. Make sure that the latest NVIDIA driver is installed and running will appear in shell after executing nvidia-smi. I will describe below some key information about my laptop. Hope someone could help me with this perplexing problem. Thanks! My laptop information * *sudo lshw -class video *-display UNCLAIMED description: 3D controller product: GP107M [GeForce GTX 1050 Mobile] vendor: NVIDIA Corporation physical id: 0 bus info: pci@0000:01:00.0 version: a1 width: 64 bits clock: 33MHz capabilities: pm msi pciexpress bus_master cap_list configuration: latency=0 resources: memory:de000000-deffffff memory:c0000000-cfffffff memory:d0000000-d1ffffff ioport:e000(size=128) memory:df000000-df07ffff *-display description: VGA compatible controller product: Intel Corporation vendor: Intel Corporation physical id: 2 bus info: pci@0000:00:02.0 version: 04 width: 64 bits clock: 33MHz capabilities: pciexpress msi pm vga_controller bus_master cap_list rom configuration: driver=i915 latency=0 resources: irq:131 memory:dd000000-ddffffff memory:b0000000-bfffffff ioport:f000(size=64) memory:c0000-dffff Installation approaches Preparation before installation Disable secure boot and disable the system's own Nouveau graphics card driver Installation Add ppa source Open Software Update --> Additional Programs --> Click on one of the drivers to install it --> Finish the installation --> reboot In addition, I tried nvidia-driver-515, nvidia-driver-510, nvidia-driver-470, nvidia-driver-418 and nvidia-driver-390. Problem solution I have tried Fixing Ubuntu Freezing at Boot Time, but it does not work. A: I reinsatll my OS from ubuntu18.04 to ubuntu20.04. The thing become simple. I install nvidia driver just from the "software and updates/additional drivers" and select the latest "tested" one. It works!
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Botanically speaking, both the redwoods and the "big trees" are species of the genus Sequoia - a pretty name given to them in honor of Sequoyah, the Cherokee Indian who invented letters for his people. They are both natives of California, the redwoods being confined to the coast ranges and the "big trees" to the western slopes of the Sierra Nevada Mountains. They are dis­tinguished by their peculiar fibrous bark and their rich color of cinnamon brown. The redwood grows in such large quantities that it is a fit material for commerce, and the red­wood industry of Humboldt County, California, where the trees abound, is enormous. The "big trees," on the other hand, are carefully guarded by the Government. The Mariposa Grove, which contains over seven hundred majestic trees, has been set apart by Con­gress as a National park, and the Government commissioners are able to resist the encroach­ments of every­thing except forest fires, which, at times, have sadly decimated and des­troyed the trees. Many of the trees are known throughout the world by charac­teristic names given to them in honor of popu­lar heroes and favorites of the hour. A section of one of the fallen kings of the Mariposa group is called "Chip of the Old Block," and our illustration gives but a faint idea of what the "old block" must have been in the day of its towering grandeur. Another tree, shown on the following page, has been called "Uncle Tom's Cabin," on account of the tent like opening at its base - an opening in which the stalwart figure of a man is dwarfed. The most famous tree is called "Grizzly Giant," which is over 93ft. in circum­ference at the ground, and over 64ft. at a distance of 11ft. from the base. It reaches a height of 200ft before throwing out a branch, and its first branch is 8ft. in diameter. "Grizzly Giant" is the largest living tree in the world, and stands over 275ft. high. These figures can, however, but barely suggest the mammoth girth of this celebrated sequoia, which people travel from al1 parts of the world to see. Nor can the visitor realize that it and its neighbors have been stand­ing for 2,500 years. Yet such is the estimated age of these forest giants. They were but bushes when Nero fiddled before burning Rome. The" big trees" were discovered in 1852 by a white hunter named Dowd, who in that year found himself in the neighborhood of Calaveras Grove. The date 1850 is carved on one of the trees, and this has led many people to think that the big trees had been visited previously to 1852. Since that time, the trees have been one of the remarkable natural "sights" of the United States. Botanists have quarreled over the proper name to give them, and have esti­mated their age from the rings in the fallen logs. Cross sections have been cut and forwarded to different parts of the country, in order that people might see for themselves that the stories of the "big trees" were true. In Boston, several years ago, one of these cross sections was erected, in a public square, and dances were held on its polished surface. The idea of using a tree for such a purpose originated in California, where a stump of one of the trees has had a house built upon it, to serve as a ball-room. A glance at our illustration of the prime­val redwood forest in Humboldt County, California, the first of several redwood photographs lent to us by the Hum­boldt Cham­ber of Com­merce, will give an idea of the massive trunks of these valuable trees, which stretch out for 100 miles along the coast, "not in sentinel groves," according to a poetical writer, "but in one continuous belt - dense, stately, dark, and forbidding." The forests are apparently imperishable, except through the axe of the woodsman, and this is wielded with care. The trees are never injured by fire. The wood resists combustion, and is hard to burn even when dry. The redwood is the only lumber that can take the place of the white pine, answer as a satisfactory substitute mahogany and black walnut, displace oak for redwood ties, cypress and cedar for shingles, and surpass all other woods for dura­bility when in contact with earth, or when exposed to moisture. These qualities make the redwood industry important to the builders of cities and homes, of rail­roads, flumes, and conduits, to those engaged in mining, manufacturing, and agri­culture all over the country. It is important to the con­sumers, and they should feel as gratified as do the people of Humboldt, that there is still a reserve forest containing 5o,ooo,ooo,ooo ft. of timber, which can be utilized for so many purposes. Redwood will make an enduring foundation, solid walls, and an imperish­able roof. Thus it provides the substantial equipment for any structure. But it may be made to embellish and adorn the home, as well as shelter the inmates. As a finishing wood it is un­equalled, and for cabinet material some qualities of it are superior. Even the stumps, it is said, refuse to perish or even to die, but send forth shoots and sprouts which, if left undisturbed, would renew the forests in course of centuries. With such superb natural resources at hand, it is not strange that the redwood forests should resound with the cry of the lumberman and the crash of the falling tree. With these may be heard the grating of the saw as it cuts its swath through the heart of the tree, the steady c1ick of the axe, the buzzing of fifty saw­mills in the neighborhood, and the puff of powerful locomotives engaged in pulling the heavy logs out of the woods, It is a scene of eternal hurry in the very heart of Nature. In order to cut down the trees, the chop­pers stand on platforms raised around the tree at some little distance from the base. The steady movement of the axe makes a quick impression on the massive timber, but it sometimes takes two weeks for two men to start the tree on its crashing fall to the ground. Most of the unskilled laborers of the county are employed in felling, although even this class of work requires a special amount of skill. The great bulk and weight of redwood logs, and the fact that operations in the logging regions are in progress only during summer months and the absence of snow, make lum­bering in Humboldt differ from the methods used elsewhere. The character of the country, mostly rugged, also in­troduces a distinct element into logging operations. Ingenuity combined with capital has intervened, and almost every extensive red­wood mill plant in Hum­boldt includes several miles of railroad, with locomo­tives, cars, and other equip­ments for transporting logs and lumber, numerous donkey engines for hauling logs out into the road, several miles of electric wire with instruments to supply telephone service to the remotest camps and connect them with the mill and yard, and, in many cases, a system of wire cable on the endless chain prin­ciple, with stationary engine to "snake" the logs to the railway landing. Oxen are still used in some camps, and it is an interesting sight to watch a long string of tugging oxen toiling down through the hills amid a cloud of dust, the logs after them like a gigantic snake. A redwood is ready for the donkey engine as soon as it has been sawed into sections. Chains and ropes are then attached. to the log, and it is drawn through the forest towards the platform cars or trolleys, upon which it is deposited. In the illustration at the top of this page we may see one of these mammoth sections in position on the car. When all the cars are loaded in this manner, they are made up into a train and attached to a power­ful locomotive. A not uncommon sight in the redwood region, but one which, to strangers, would appear remarkable, is illustrated at the bottom of this page, where we get a full view of a train load of twenty-four redwood logs wind­ing slowly from among the hills. By many lumbermen in California the rivers are used with great effect in the trans­port of logs. In the summer the logs are dumped into the bed of the stream to await a winter freshet, which carries the mass along with great speed to the mills, where they lie until they are ready to be sawed. The greater part of the logs, however, are transported by the railways direct to the side of the river or pond, and there shot into the water by means of in­clined ways made of other logs. The logs dash down with great swiftness, and enter the water with a huge splash, casting the spray high into the air with the force almost of a tor­pedo explosion. The illustrations on this page show a log - chute and the magnificent column of water sent up by the diving log. From the log­ pond to the sawmill is usually but a step. In some mills the logs are pulled up on small cars; in others, they are drawn up on greased ways by means of long cables. In the forest of the Bridal Veil Lumbering Company, at Bridal Veil, in Oregon, the logs are transferred to the mill by means of a curious railway, illustrated on the following page. The train, so-called, is made up of an ordinary locomotive and a string of logs, each one as large in diameter and some even larger than the boiler of the engine. Boards are nailed to the sleepers between the rails, and on these the logs slide. Except on descending grades, the boards are greased, and the train moves at good speed. Where the road is level or slightly ascend­ing the engine pulls the logs, and where it is descending it holds them back. At the mills of the company the manufactured lumber, regard­less of size, is run into a flume, and this is carried about two miles to the planing mill and shipping yard, the flume descending about 1,200ft. in that distance. One of the great items of expense in the lumber business is the cost of transportation from the forests to the consumer. Huge sums which might otherwise have been left in the pockets of householders have been placed in the coffers of railway and steamship companies. It was in order to lessen the cost of transport that the cigar shaped log ­raft was designed. These extraordinary rafts, of which we give five excellent illustrations, are the invention of Mr. Hugh R. Robertson, of St. John, New Brunswick. The first raft was built at Joggins, in Nova Scotia, and on account of its novelty quickly gained the nickname of the "Joggins raft." It was built in 1887, and its dimensions, of which a fair idea is given in the illustration below, were: length, 560ft. ; depth, 35ft. It took several months to build, and was composed of several hundred thousand logs, closely bound together in a cradle of logs, which rested upon timber foundations. The raft was pointed at one end, and lay on the shore slant-wise in order that it might be quickly and easily launched. During the process of construc­tion the inventor was much laughed at, but, nothing daunted in his scheme, he launched the raft and dispatched it to New York in tow. The first Joggins raft, however, quickly bore out the prophecies of Mr. Robertson's opponents, and came to grief in the wild and wintry Atlantic. The hawser which attached it to the tug was snapped by the force of the waves, the raft burst in pieces, and the huge logs, which represented many thousand pounds in gold, were rapidly distributed over the surface of the Atlantic, to the deep chagrin of the inventor, and the danger of mariners. Notwithstanding this accident, the Joggins raft had really come to stay. A second raft was quickly built in co-operation with Mr. J. D. Leary, and sent to New York, a distance of 700 miles, in ten days, where the lumber of which it was made was sold at a profit enormous in itself, and yet at a price remark­able for cheapness. The much derided inventor made a fortune, and, selling his idea to Mr. Leary, left for the Pacific Coast, where he is engaged to this day in transporting lumber by means of cigar­ shaped rafts with wonderful success. Our last two illustrations show the side and top view of one of these rafts lately built on the Columbia River. Its value was £9,000., and its length 528ft., with the width of 52ft. and a draught of 24ft. The heavy chains, which are so plainly seen in the illustration, enclose 560,000 lineal feet of timber. Originally published in The Strand Magazine. March 1898.
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4,622
Q: Click event for the wrong button being fired in jQuery? This happens very rarely but it still happens sometimes. I have two buttons next to each others with a jQuery click event on each: JS: $("#accepttrade").click(function(){ if(document.getElementById("agreeterms").checked ){ //accept process $("#acceptdeposit").slideUp(200); } }); $("#declinetrade").click(function(){ //decline $("#acceptdeposit").slideUp(200); }); HTML: <div id="acceptdeposit"> <button id="declinetrade" >Decline</button>&emsp; <button id="accepttrade" >Accept</button><input type="checkbox" id="agreeterms"> </div> But sometimes when someone click on decline, it occurs the click of accept button, and go through even if the checkbox is unchecked. I have never experienced it myself, but is it possible that this could happend? How can I be sure that "accept process" is never reached unless the user checks the box and click on accept? A: Try this, cancel your click event: $("#accepttrade").click(function(e){ if($("#agreeterms").is(':checked') ){ //accept process $("#acceptdeposit").slideUp(200); } e.preventDefault(); });
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DUBLIN--(Business Wire)--The "Certificate in Employee Relations Law Seminar" conference has been added to ResearchAndMarkets.com's offering. The Certificate in Employee Relations Law Seminar provides the most comprehensive, practical, up-to-date employment law training available. This is a 4 day seminar geared to the real-world needs of human resource professionals, attorneys, and managers. The seminar provides "best practices" insights and information on the full range of employee relations law issues.
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Котабато () град је на Филипинима и регионални центар Административног региона у Муслиманском Минданау. Град се налази на западној обали острва Минданао и има 271.786 становника према проценама из 2010. Географија Кроз град протиче река Минданао. Клима Становништво Партнерски градови Naga Референце Спољашње везе Sangguniang Panlungsod of Cotabato City Website NSCB details for cotabato city geographic code Philippine Census Information Department of Tourism Градови на Филипинима Википројект географија/Насеља на Филипинима
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{"url":"http:\/\/popflock.com\/learn?s=Metre_per_second_squared","text":"Metre Per Second Squared\nGet Metre Per Second Squared essential facts below. View Videos or join the Metre Per Second Squared discussion. Add Metre Per Second Squared to your PopFlock.com topic list for future reference or share this resource on social media.\nMetre Per Second Squared\nMetre per second squared\nUnit systemSI\nUnit\u00a0ofacceleration\nSymbolm\/s^2\u2002or\u2002m\/s\u00b2\n\nThe metre per second squared is the unit of acceleration in the International System of Units (SI). As a derived unit, it is composed from the SI base units of length, the metre, and time, the second. Its symbol is written in several forms as m\/s2, m\u00b7s-2 or m\u00a0s-2, ${\\displaystyle {\\tfrac {\\operatorname {m} }{\\operatorname {s} ^{2}}}}$, or less commonly, as m\/s\/s.[1]\n\nAs acceleration, the unit is interpreted physically as change in velocity or speed per time interval, i.e. metre per second per second and is treated as a vector quantity.\n\n## Example\n\nAn object experiences a constant acceleration of one metre per second squared (1\u00a0m\/s2) from a state of rest, when it achieves the speed of 5\u00a0m\/s after 5 seconds and 10\u00a0m\/s after 10 seconds. The average acceleration a can be calculated by dividing the speed v (m\/s) by the time t (s), so the average acceleration in the first example would be calculated: ${\\displaystyle a={\\frac {\\Delta v}{\\Delta t}}={\\frac {5{\\text{ m\/s}}}{5{\\text{ s}}}}=1{\\text{ (m\/s)\/s}}=1{\\text{ m\/s}}^{2}}$.\n\n## Related units\n\nNewton's second law states that force equals mass multiplied by acceleration. The unit of force is the newton (N), and mass has the SI unit kilogram (kg). One newton equals one kilogram metre per second squared. Therefore, the unit metre per second squared is equivalent to newton per kilogram, N\u00b7kg-1, or N\/kg.[2]\n\nThus, the Earth's gravitational field (near ground level) can be quoted as 9.8 metres per second squared, or the equivalent 9.8\u00a0N\/kg.\n\nAcceleration can be measured in ratios to gravity, such as g-force, and peak ground acceleration in earthquakes.\n\n## Unicode character\n\nThe \"metre per second squared\" symbol is encoded by Unicode at code point SQUARE M OVER S SQUARED ? m\/s^2 ?.[3]\n\n## Conversions\n\nBase value (Gal, or cm\/s2) (ft\/s2) (m\/s2) (Standard gravity, g0)\n1 Gal, or cm\/s2 1\n1 ft\/s2 1\n1 m\/s2 1\n1 g0 1","date":"2020-11-29 15:09:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"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\": 0.7485772967338562, \"perplexity\": 1667.8879294990281}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141198409.43\/warc\/CC-MAIN-20201129123729-20201129153729-00206.warc.gz\"}"}
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package org.ros2.rcljava.service; public class RMWRequestId { public byte[] writerGUID = new byte[16]; public long sequenceNumber; }
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\section{Introduction And Literature Review} Anomaly detection is an area of considerable importance and has been subject to increasing attention in recent years. Comprehensive reviews of the area can be found in \cite{chandola2009anomaly, pimentel2014review}. The field's growing importance arises from the increasing range of applications to which anomaly detection lends itself: from fraud prevention \citep{chandola2009anomaly,pimentel2014review}, to fault detection \citep{chandola2009anomaly,pimentel2014review}, and even the detection of exoplanets \citep{fisch2018linear}. More recently, the emergence of internet of things and the ubiquity of sensors has led to emergence of the online detection of anomalies as an important statistical challenge. Kalman filters \cite{Kalman1960} provide a convenient framework to detect anomalies within a streaming data context. In particular, they can be updated in a fully online fashion at a fixed computational cost. At each time point, Kalman filters also provide an estimate both for the expectation and variance of the next observation. These can be used to determine whether that observation is anomalous or not. However, the major drawback of Kalman filters is their lack of robustness to outliers: once the filter has encountered an outlier, it will often produce inaccurate predictions for many future time points. The anomaly detection literature distinguishes between two types of outliers. The first are additive outliers, sometimes referred to as observational outliers \citep{gandhi2009robust}, which affect the observational noise only. The other type of outliers are the innovative, or process \citep{huang2017novel}, outliers. These affect the updates of the hidden states. In practice, both have a similar effect on the next observation, but quite different effects on subsequent observations. Moreover, some innovative outliers cannot be detected immediately as their influence on the observations is only noticeable after, or over, a period of time. A range of robust Kalman filters has been proposed to date. Many side-step the problem of distinguishing between the two outlier types. By far the largest class of filters aims to be robust against heavy tailed additive outliers. Examples of such filters include \cite{ting2007learning, agamennoni2011outlier}, which assume $t$-distributed additive noise and perform inference using variational Bayes, \cite{ruckdeschel2014robust}, who use Huberised residuals, and \cite{chang2014robust} inflate the noise covariance matrix whenever an outlier is encountered. A few filters have also been developed with the aim of achieving robustness against innovative outliers \citep{ruckdeschel2014robust}. The problem with such filters is that they exacerbate the shortcomings of the Kalman filter when they encounter the other type of anomaly: additive outlier robust Kalman filters, for example, update their hidden states even less than the classical Kalman filter when encountering innovative outliers. In principle, it seems straightforward to combine the ideas of these two types of robust Kalman filter. One body of literature proposes to use Huberisation of both innovative and additive residuals \citep{gandhi2009robust,chang2014robust}. Others \citep{huang2017novel,huang2019novel} have modelled both additive and innovative outliers using $t$-distributions, by imposing Wishart priors on the precision matrix of both the innovations and additions and maintaining the posterior by using variational Bayes approaches. The issue with these filters comes from how they approximate the filtering distribution of the state. Both return uni-modal posteriors after encountering an anomaly. This is a shortcoming given that the posterior after an anomaly is likely to be multi-modal: if the outlying observation was caused by an additive anomaly, the state will be close to the prior, whereas if it was caused by an innovative anomaly, the state would be far from it. The ideal approach to constructing a robust filter would be to model the possibility of outliers in both the observation and system noise, and then use a filter algorithm that attempts to calculate, or approximate, the true filtering distribution for the model. An early attempt to do this was the spline based approach \cite{kitagawa1987non}, but the computational complexity increases very quickly with the number of dimensions and such a filter becomes impracticable when the state dimension is greater than 3. As a result we consider using particle filters \citep{gordon1993novel,fearnhead2018particle}. These are able to produce Monte Carlo approximations to the filtering distribution for an appropriate model that allows for outliers, and, in principle, can work even if the filtering distribution is multi-modal. However the Monte Carlo error of standard implementations of the particle can be prohibitively large \citep{chang2014robust}. In this paper, we develop an efficient particle filter by using a combination of Rao-Blackwellisation and well-designed proposal distributions. The idea of Rao-Blackwellisation is to integrate out part of the state so that the particle filter approximates the filtering distribution of a lower-dimensional projection of the state. In our application this projection is whether each component of the additive and innovative noise is an outlier, and if it is how much the variance of the noise has been inflated. Conditional on this information, the state space model becomes linear-Gaussian and we can implement a Kalman Filter to calculate exactly the conditional filtering distribution, while being able to fully capture multi modal posteriors. This idea is similar to that which underpins the Mixture Kalman Filter \citep{chen2000mixture}. Whilst Rao-Blackwellisation improves the Monte Carlo accuracy of the filter, such a filter can still have the shortcomings noted by \cite{chang2014robust} and perform poorly without good proposal distributions for the information we condition on. One of the main contributions of this work is a proposal distribution that accurately approximates the conditional distribution of the variance inflation for each component of the noise, and hence approximates the optimal proposal distribution \citep{pitt1999filtering}. As a result of this proposal, we find that accurate results can be obtained even with only a few particles. Another important challenge addressed by this paper is that certain innovative outliers can not immediately be detected. An innovative outlier in a latent trend component for instance can cause a trend changes which may only become apparent -- i.e.\ produce a visible outlier in the observations -- many observations after the innovative outlier in the trend occurred. It is nevertheless important to capture such outliers as they can affect a potentially unlimited number of observations to come. The proposed particle filter includes the possibility to back-sample the variance inflation particles in light of more recent observations, which enables it to capture these important anomalies. The remainder of this paper is organised as follows: We discuss our robust noise model, consisting of a mixture distribution of Gaussian noise, representing typical behaviour, and heavy tailed noise, representing atypical behaviour, for both the additive (observational) and innovative (system) noise process in Section \ref{sec:Model}. The model is shown to be very similar to that considered by \cite{huang2019novel}. We then introduce the proposal distribution for the scale of the noise in Section \ref{sec:Particle_Filter}, before extending it to anomalies which are not immediately identifiable in Section \ref{sec:Backsampling}. The proposed filter is compared to others in Section \ref{sec:Simulation} and applied to router data and a benchmark machine temperature data-set in Section \ref{sec:Application}. The proposed methodology, which we call Computationally Efficient Bayesian Anomaly detection by Sequential Sampling (CE-BASS) has been implemented in the the \texttt{R} package \texttt{RobKF} available from \texttt{https://github.com/Fisch-Alex/Robkf}. Derivations of theoretical results and complete pseudocode are available in the appendix. \section{Model And Examples}\label{sec:Model} Throughout this paper, we will consider inference about a latent state, $\textbf{X}_t$, through partial observations, $\textbf{Y}_t$, modelled as \begin{align}\label{eq:Main} \begin{split} \textbf{Y}_t &= \textbf{C} \textbf{X}_t + \textbf{V}_t^{\frac{1}{2}}\bm{\Sigma}_A^{\frac{1}{2}}\bm{\epsilon}_t, \\ \textbf{X}_t &= \textbf{A} \textbf{X}_{t-1} + \textbf{W}_t^{\frac{1}{2}}\bm{\Sigma}_I^{\frac{1}{2}}\bm{\nu}_t. \end{split} \end{align} Here the additive noise, $\bm{\epsilon}_t \in \mathbb{R}^p$, and the innovations $\bm{\nu}_t \in \mathbb{R}^q$ are both i.i.d.\ standard multivariate Gaussian. The diagonal matrices $\bm{\Sigma}_A$ and $\bm{\Sigma}_I$ denote the covariance of the additive and innovation noise respectively. The diagonal matrices $\textbf{V}_t$ and $\textbf{W}_t$ are used to capture additive and innovative outliers respectively, with large diagonal entries of $\textbf{V}_t$ corresponding to additive outliers and large diagonal entries of $\textbf{W}_t$ corresponding to innovative outliers. The classical Kalman model is recovered by setting $\textbf{W}_t=\textbf{I}$ and $\textbf{V}_t=\textbf{I}$ for all times $t$. \begin{figure} \begin{subfigure}[b]{0.49\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/example_rw.jpg} \caption{Random walk}\label{fig:rw_ex} \end{subfigure} \begin{subfigure}[b]{0.49\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/example_rwandt} \caption{Random walk with trends} \label{fig:rwandt_ex} \end{subfigure} \caption{Two examples of time series which are realisations of outlier infested Kalman models. (a) was simulated using the setup defined in Equation \eqref{eq:RW}, with $\sigma_A = 1$, $\sigma_I = 0.1$, and outliers defined by $W_{100} = 3600$, $V_{400} = 100$, and $W_{700} = 10000$. Conversely (b) second example was simulated using the model defined in Equation \eqref{eq:RWand_T} using $\sigma_A = 1$, $\sigma_I^{(1)} = 0.1$, $\sigma_I^{(2)} = 0.01$ and outliers defined by $W^{(1)}_{100} = 3600$, $V_{400} = 100$, and $W^{(2)}_{700} = 40000$.} \end{figure} The model in Equation \eqref{eq:Main} can be used to model a range of time series behaviours. We will use the following two examples throughout the paper: \textbf{Example 1}: The random walk model with both changepoints and outliers, similar to the problem considered by \cite{fearnhead2019changepoint}. It can be formulated as \begin{align}\label{eq:RW} Y_t = X_t + V_t^{\frac{1}{2}}\sigma_A\epsilon_t, \;\;\;\;\;\;\;\;\;\;\;\;\; X_t = X_{t-1} + W_t^{\frac{1}{2}}\sigma_I\nu_t. \end{align} Here atypically large values of $V_t$ correspond to outliers, whilst atypically large values of $W_t$ correspond to changes. A realisation of this model can be found in Figure \ref{fig:rw_ex}. \textbf{Example 2}: A time series with changes in trend, level shifts, as well as outliers, similar to the model considered by \cite{maeng2019detecting}. It can be formulated as \small \begin{equation}\label{eq:RWand_T} \begin{split} Y_t = X_t^{(1)} + V_t^{\frac{1}{2}}\sigma_A\epsilon_t \;\;\;\;\;\;\;\;\;\;\;\;\; X_t^{(1)} &= X_{t-1}^{(1)} + X_{t-1}^{(2)} + \left(W_t^{(1)}\right) ^{\frac{1}{2}}\sigma_I^{(1)}\nu_t^{(1)},\\ X_t^{(2)} &= X_{t-1}^{(2)} + \left(W_t^{(2)}\right) ^{\frac{1}{2}}\sigma_I^{(2)}\nu_t^{(2)}, \end{split} \end{equation} \normalsize with the first component of the hidden state denoting the current position and the second indicating the trend. Here, outliers are modelled by large values of $V_t$ whilst level shift and changes in trend are modelled by atypically large values of $W_t^{(1)}$ and $W_t^{(2)}$ respectively. A realisation of this model can be found in Figure \ref{fig:rwandt_ex}. A key feature of this second model is that an outlier in the trend component, $X_t^{(2)}$, may only become detectable many observations after the outlier -- this challenging issue mentioned in the introduction is addressed via the methods in Section \ref{sec:Backsampling}. A wide rage of other commonly used time series features, such as auto-correlation, moving averages, etc.\ can be incorporated in the model. To infer the locations of anomalies we use the model \begin{equation}\label{eq:Noise} \textbf{V}_t^{(i,i)} = 1 + \lambda_t^{(i)} \frac{1}{\tilde{\textbf{V}}_t^{(i,i)}} \;\;\;\;\;\;\;\;\;\textbf{W}_t^{(j,j)} = 1 + \gamma_t^{(j)} \frac{1}{\tilde{\textbf{W}}_t^{(j,j)}} \end{equation} for $1 \leq i \leq p$ and $1 \leq j \leq q$. The random variables $\lambda_t^{(i)} \sim Ber(r_i)$ and $\gamma_t^{(j)} \sim Ber(s_j)$ are indicators that determine whether an anomaly is present or not for $1 \leq i \leq p$ and $1 \leq j \leq q$ respectively. For additional interpretability, we impose that at most one anomaly is present at any given time $t$, and define $r_i$ and $s_j$ to be the probabilities that $\lambda_t^{(i)}=1$ and $\gamma_t^{(j)}=1$ respectively. The inverse scale, or precision, of an anomaly (if present) is given by the random variables $\tilde{\textbf{V}}_t^{(i,i)} \sim \tilde{\sigma_i}\Gamma(a_i,a_i)$ and $\tilde{\textbf{W}}_t^{(j,j)} \sim \hat{\sigma_j}\Gamma(b_j,b_j)$ for $1 \leq i \leq p$ and $1 \leq j \leq q$ respectively. The proposed model bears similarities to the model used by \cite{huang2019novel}. Both use a mixture of Gaussian and heavy tailed noise. The main difference is that the anomalous behaviour is characterised by noise which is the sum of a Gaussian and a $t$-distribution in our model as opposed to just a $t$-distribution in the model used by \cite{huang2019novel}. This ensures that anomalies coincide with strictly greater noise and makes the result more interpretable. In practice, however, the noise distribution considered in this paper and in \cite{huang2019novel} are likely to be of very similar shape. \section{Particle Filter}\label{sec:Particle_Filter} We now turn to filtering the model defined by Equations \eqref{eq:Main} and \eqref{eq:Noise}. The main feature we exploit is the fact that if we knew the value of $(\textbf{V}_t,\textbf{W}_t)$ at all times $t$, we could just run the classical Kalman filter over the data. Consequently, our approach will consist of sampling particles for $(\textbf{V}_t,\textbf{W}_t)$, conditional on which the classical Kalman update equations for the hidden state $\textbf{x}_t$ can be used. This approach, very similar to the mixture Kalman filter \citep{chen2000mixture,fearnhead2003line} is summarised by the pseudocode in Algorithm \ref{alg:Basic}. For each time, $t$, the code loops over the existing particles, $(\textbf{V}_{t},\textbf{W}_{t})$, and simulates $M'$ descendants for each of them in step 4. They are stored in a set of candidate particles. If we have $N$ particles at time $t$, keeping all candidates would produce $NM'$ particles at time $t+1$. To avoid growing the number of particles exponentially with $t$, Step 7 resamples the candidates to keep just $N$ particles. The filtering distribution for each of these particles is then calculated using the Kalman Filter updates in step 10. \begin{algorithm} \caption{Basic Particle Filter (No Back-sampling)} \label{alg:Basic} \begin{footnotesize} \begin{tabular}[h]{ll} {\bf Input:} & An initial state estimate $(\bm{\mu}_0,\bm{\Sigma}_0)$ \\ & A number of descendants, $M'=M(p+q)+1$ \\ & A number of particles to be maintained, $N$. \\ & A stream of observations $\textbf{Y}_1,\textbf{Y}_2,...$ \\ {\bf Initialise:} & Set $Particles(0) = \{(\bm{\mu}_0,\bm{\Sigma}_0)\}$ \end{tabular} \begin{algorithmic}[1] \For{$t \in \mathbb{N}^+ $} \State $Candidates \gets \{\}$ \For{$(\bm{\mu},\bm{\Sigma}) \in Particles(t-1)$} \State $(\textbf{V},\textbf{W},prob) \gets \text{Sample\_Particles}(M',\bm{\mu},\bm{\Sigma},\textbf{Y}_t,\textbf{A},\textbf{C},\bm{\Sigma}_A,\bm{\Sigma}_I)$ \State $Candidates \gets Candidates \cup \{(\bm{\mu},\bm{\Sigma},\textbf{V},\textbf{W},prob)\}$ \EndFor \State $Descendants \gets \text{Subsample}(N,Candidates)$ \State $Particles(t) \gets \{\}$ \For{$(\bm{\mu},\bm{\Sigma},\textbf{V},\textbf{W},prob) \in Descendants $} \State $(\bm{\mu}_{new},\bm{\Sigma}_{new}) \gets \text{KF\_Upd}(\textbf{Y}_t,\bm{\mu},\bm{\Sigma},\textbf{C},\textbf{A},\textbf{V}^{1/2}\bm{\Sigma}_A,\textbf{W}^{1/2}\bm{\Sigma}_I)$ \State $Particles(t) \gets Particles(t) \cup \{(\bm{\mu}_{new},\bm{\Sigma}_{new}) \}$ \EndFor \EndFor \end{algorithmic} \end{footnotesize} \end{algorithm} The main challenge in the above approach consists of selecting a good sampling procedure for the particles. Whilst it may be a natural choice to sample particles $(\textbf{V}_{t+1},\textbf{W}_{t+1})$ from their prior distribution, this is not suitable for the problem considered in this paper. In particular, this sampling procedure would not be robust to outliers: the stronger an anomaly was, the less likely we would be to sample a particle with an appropriate value of $(\textbf{V}_{t+1},\textbf{W}_{t+1})$, as discussed by \cite{chang2014robust}. \normalsize Adopting ideas from \cite{pitt1999filtering} and \cite{arulampalam2002tutorial}, we overcome the above challenge by sampling particles from an approximation to the conditional distribution of $(\textbf{V}_{t+1},\textbf{W}_{t+1})$ given observation $\textbf{Y}_{t+1}$. Denote the model's prior distribution for $(\textbf{V}_{t+1},\textbf{W}_{t+1})$ in \eqref{eq:Noise} by $\pi_0(\cdot)$. The conditional distribution $\pi(\textbf{W}_{t+1},\textbf{V}_{t+1}|\textbf{Y}_{t+1})$ for the descendants of a particle whose filtering distribution for $\textbf{x}_{t}$ is $N(\bm{\mu},\bm{\Sigma})$ is then proportional to \begin{equation*} \pi_0(\textbf{W},\textbf{V}) \mathcal{L}\left(\textbf{Y},\textbf{C}\textbf{A}, \textbf{C}\textbf{A}\bm{\Sigma}\textbf{A}^T\textbf{C}^T + \bm{\Sigma}_A \textbf{V}+ \textbf{C} \bm{\Sigma}_I\textbf{W}\textbf{C}^T\right). \end{equation*} Here we have dropped time indices for convenience, and $\mathcal{L}\left(\textbf{x}, \bm{\mu}, \bm{\Sigma}\right)$ denotes the likelihood of an observation $\textbf{x}$ under a $N(\bm{\mu}, \bm{\Sigma})$-model. Since at most one component is anomalous, we can re-write this as a sum over which, if any, component is anomalous \small \begin{align*} \mathbb{I}_{\left\{ \textbf{W}=\textbf{I},\textbf{V}=\textbf{I} \right\}}\pi(\textbf{I},\textbf{I}|\textbf{Y}) + \sum_{j=1}^q \mathbb{I}_{\left\{ \textbf{W}=\textbf{I} + \frac{\textbf{I}^{(j)}}{\tilde{\textbf{W}}^{(j,j)}},\textbf{V}=\textbf{I} \right\}}\hat{\pi}_j\left(\tilde{\textbf{W}}^{(j,j)}\right) + \sum_{i=1}^p \mathbb{I}_{\left\{ \textbf{W}=\textbf{I},\textbf{V}=\textbf{I} + \frac{\textbf{I}^{(i)}}{\tilde{\textbf{V}}^{(i,i)}} \right\}}\tilde{\pi}_i\left(\tilde{\textbf{V}}^{(i,i)}\right). \end{align*} \normalsize Here, we use the shorthand \footnotesize \begin{equation*} \tilde{\pi}_i\left(\tilde{\textbf{V}}^{(i,i)}\right) = \pi\left( \textbf{I},\textbf{I} + \frac{\textbf{I}^{(i)}}{\tilde{\textbf{V}}^{(i,i)}}|\textbf{Y} \right) \end{equation*} \normalsize and \footnotesize \begin{equation*} \hat{\pi}_j\left(\tilde{\textbf{W}}^{(j,j)}\right) = \pi\left( \textbf{I} + \frac{\textbf{I}^{(j)}}{\tilde{\textbf{W}}^{(j,j)}} ,\textbf{I}|\textbf{Y} \right). \end{equation*} \normalsize Since the target distribution $\pi(\textbf{W},\textbf{V}|\textbf{Y})$ is intractable, we construct an approximation to it, which we denote $q(\textbf{W},\textbf{V}|\textbf{Y})$, and use this as our proposal distribution. This proposal is proportional to \small \begin{align*} \mathbb{I}_{\left\{ \textbf{W}=\textbf{I},\textbf{V}=\textbf{I} \right\}}\beta_0 + \sum_{j=1}^q \mathbb{I}_{\left\{ \textbf{W}=\textbf{I} + \frac{\textbf{I}^{(j)}}{\tilde{\textbf{W}}^{(j,j)}},\textbf{V}=\textbf{I} \right\}} \hat{\beta}_j \hat{q}_j\left(\tilde{\textbf{W}}^{(j,j)}\right) + \sum_{i=1}^p \mathbb{I}_{\left\{ \textbf{W}=\textbf{I},\textbf{V}=\textbf{I} + \frac{\textbf{I}^{(i)}}{\tilde{\textbf{V}}^{(i,i)}} \right\}}\tilde{\beta}_i\tilde{q}_i\left(\tilde{\textbf{V}}^{(i,i)}\right). \end{align*} \normalsize Clearly, there is no benefit in simulating multiple identical descendants, so we wish to sample precisely one dependent that corresponds to no outliers. To do this, and also to have the same number of descendant particles for each possible type of outlier, we set $\beta_0 = \frac{1}{1+M(p+q)}$, $\tilde{\beta}_i = \frac{M}{1+M(p+q)}$, and $\hat{\beta}_j = \frac{M}{1+M(p+q)}$, and use stratified subsampling as in \cite{fearnhead2003line}. This leads to $M'=M(p+q)+1$ total descendants per particle, $M$ for each of the $p$ additive and $q$ innovative outliers, and one for no outlier. Each of these particles is then given a weight proportional to \begin{equation*} \frac{\pi(\textbf{W}_{t+1},\textbf{V}_{t+1}|\textbf{Y}_{t+1})}{q(\textbf{W}_{t+1},\textbf{V}_{t+1}|\textbf{Y}_{t+1})}. \end{equation*} The main challenge now consists of obtaining proposal distributions $\tilde{q}_i(\cdot)$ for $1 \leq i \leq p$ and $\hat{q}_j(\cdot)$ for $1 \leq j \leq q$ that provide good approximations to the conditional posteriors which are proportional to $\tilde{\pi}_i(\cdot)$ and $\hat{\pi}_j(\cdot)$ respectively. In the next subsection, we therefore derive proposal distributions that provide leading order approximations to the conditional posteriors. To simplify notation, we define the predictive variance $\hat{\bm{\Sigma}} = \textbf{C}\textbf{A}\bm{\Sigma}\textbf{A}^T\textbf{C}^T + \bm{\Sigma}_A + \textbf{C} \bm{\Sigma}_I\textbf{C}^T$ and use it throughout the remainder of this paper. We also begin by assuming that $\textbf{C}$ contains no $0$-columns. The proposal introduced in the following subsection also forms the basis of back-sampling introduced in Section \ref{sec:Backsampling}, which allows to relax this on $\textbf{C}$. \subsection{Proposal Distributions}\label{sec:Props} For $1 \leq i \leq p$, we would like the proposal distribution $\tilde{q}_i\left(\tilde{\textbf{V}}^{(i,i)}\right)$ for the precision, $\tilde{\textbf{V}}^{(i,i)}$, to be as close as possible to $\tilde{\pi}_i\left(\tilde{\textbf{V}}^{(i,i)}\right)$ or, equivalently, proportional to \footnotesize \begin{equation*} f_i\left(\tilde{\textbf{V}}^{(i,i)}\right) \frac{ \exp \left( -\frac{1}{2} \left(\textbf{Y}-\textbf{C}\textbf{A}\bm{\mu}\right)^T \left( \hat{\bm{\Sigma}} + \frac{\bm{\Sigma}_A^{(i,i)}}{\tilde{\textbf{V}}^{(i,i)}} \textbf{I}^{(i)}\right)^{-1} \left(\textbf{Y}-\textbf{C}\textbf{A}\bm{\mu}\right) \right) }{\sqrt{\left|\hat{\bm{\Sigma}} + \frac{\bm{\Sigma}_A^{(i,i)}}{\tilde{\textbf{V}}^{(i,i)}} \textbf{I}^{(i)} \right|}}, \end{equation*} \normalsize where $f_i()$ denotes the PDF of the $\tilde{\sigma}_i\Gamma(a_i,a_i)$-distributed prior of $\tilde{\textbf{V}}^{(i,i)}$. It should be noted that the intractable terms, \begin{equation}\label{eq:annoyingterms} \left|\hat{\bm{\Sigma}} + \frac{\bm{\Sigma}_A^{(i,i)}}{\tilde{\textbf{V}}^{(i,i)}} \textbf{I}^{(i)} \right| \;\;\;\;\;\;\; \text{and} \;\;\;\;\;\;\; \left( \hat{\bm{\Sigma}} + \frac{\bm{\Sigma}_A^{(i,i)}}{\tilde{\textbf{V}}^{(i,i)}} \textbf{I}^{(i)}\right)^{-1} \end{equation} can both be expanded using the matrix determinant lemma and the Sherman Morrison formula respectively, as they are rank 1 updates of a determinant and inverse respectively. Indeed, by the matrix determinant lemma, \small \begin{equation*} \left|\hat{\bm{\Sigma}} + \frac{\bm{\Sigma}_A^{(i,i)}}{\tilde{\textbf{V}}^{(i,i)}} \textbf{I}^{(i)} \right| = \frac{\left|\hat{\bm{\Sigma}}\right| }{\tilde{\textbf{V}}^{(i,i)}}\left( 1 + \bm{\Sigma}_A^{(i,i)}\left(\hat{\bm{\Sigma}}^{-1} \right)^{(i,i)} + O\left(\tilde{\textbf{V}}^{(i,i)} \right) \right), \end{equation*} \normalsize the leading order term is conjugate to the prior of $\tilde{\textbf{V}}^{(i,i)}$. Moreover, by the Sherman Morrison formula the second term in Equation \eqref{eq:annoyingterms} is equal to \small \begin{equation*} \hat{\bm{\Sigma}}^{-1} - \hat{\bm{\Sigma}}^{-1} \textbf{I}^{(i)} \hat{\bm{\Sigma}}^{-1} \left[\frac{1}{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}} - \left(\frac{1}{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}}\right)^2\frac{\tilde{\textbf{V}}^{(i,i)}}{\bm{\Sigma}_A^{(i,i)}}\right], \end{equation*} \normalsize up to $ O \left( \left(\tilde{\textbf{V}}^{(i,i)} \right)^2 \right) $. Crucially, the first two terms are constant in $\tilde{\textbf{V}}^{(i,i)}$, while the third is linear in $\tilde{\textbf{V}}^{(i,i)}$ and therefore returns a term which is conjugate to the prior of $\tilde{\textbf{V}}^{(i,i)} $. Furthermore, we are most concerned about accurately sampling the particle when an anomaly occurs in the $i$th component, which happens when the precision, $\tilde{\textbf{V}}^{(i,i)}$, and the higher order terms, become small. Keeping only the leading order terms in the determinant and the exponential term results in the proposal distribution \small \begin{equation*} \tilde{\textbf{V}}^{(i,i)}\sim \tilde{\sigma}_i\Gamma\left(a_i + \frac{1}{2},a_i + \frac{\tilde{\sigma}_i}{2\bm{\Sigma}_A^{(i,i)}}\left( \frac{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,:)} \left(\textbf{Y}-\textbf{C}\textbf{A}\bm{\mu}\right) }{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}}\right)^2\right) \end{equation*} \normalsize for $\tilde{\textbf{V}}^{(i,i)}$. More detailed derivations, including the associated weight are given by Theorem 1 in the appendix. This proposal has the property that as the observed anomaly in the $i$th component becomes larger, i.e.\ as \begin{equation*} \frac{1}{\bm{\Sigma}_A^{(i,i)}}\left( \frac{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,:)} \left(\textbf{Y}-\textbf{C}\textbf{A}\bm{\mu}\right) }{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}}\right)^2 \end{equation*} increases, the mean of the proposal for $\tilde{\textbf{V}}^{(i,i)}$ diverges from the prior mean and behaves asymptotically like \begin{equation*} (2a_i+1)\bm{\Sigma}_A^{(i,i)} \left( \frac{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}}{\left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,:)} \left(\textbf{Y}-\textbf{C}\textbf{A}\bm{\mu}\right) }\right)^2. \end{equation*} Consequently, the variance and the squared residual will be on the same scale, thus achieving computational robustness. A very similar approach can be used to obtain a proposal distribution $\hat{q}_j \left( \tilde{\textbf{W}}^{(j,j)} \right)$ which provides a leading order approximation for the distribution proportional to $\pi \left( \textbf{I} + \frac{1}{\tilde{\textbf{W}}^{(j,j)}} \textbf{I}^{(j)} ,\textbf{I} |\textbf{Y} \right)$. The proposal consists of sampling \footnotesize \begin{equation*} \tilde{\textbf{W}}^{(j,j)} \sim \hat{\sigma_j}\Gamma\left(b_j + \frac{1}{2},b_j + \frac{\hat{\sigma_i}}{2\bm{\Sigma}_I^{(j,j)}}\left( \frac{\left(\textbf{C}^T\right)^{(j,:)}\hat{\bm{\Sigma}}^{-1} \left(\textbf{Y}-\textbf{C}\textbf{A}\bm{\mu}\right) }{\left(\textbf{C}^T\hat{\bm{\Sigma}}^{-1}\textbf{C}\right)^{(j,j)}}\right)^2\right) \end{equation*} \normalsize and is of very similar form to the proposal distribution for particles with an additive outlier and well defined if $\textbf{C}$ has no $\textbf{0}$-columns. Further details, including the associated weight, are given in Theorem 2 in the appendix. Like the proposal distribution for particles with an additive anomaly this proposal is computationally robust: it ensures that the squared residual and the variance will be on the same scale as the anomaly in the $j$th innovative component becomes stronger. Finally, the ``proposal" for particles without anomalies consists of deterministically setting $\textbf{V} = \textbf{I}$ and $\textbf{W} = \textbf{I}$. The weight associated with this particle is proportional to the likelihood, the closed form of which is given in Theorem 3 in the appendix. \subsection{Choices of Parameters}\label{sec:weights} The choice of hyper-parameters, particularly $\hat{\sigma_i}$ and $\tilde{\sigma_i}$, has a significant effect of the performance of the proposed filter. One reason for this is that an outlier observation could be the result of either an additive or an innovative outlier. It may be that the root cause can only be determined after further observations are made. Thus, we wish to choose hyper-parameters in such a way as to ensure that observed anomalies, which are equally well explained by different classes of anomalies, are given similar importance weights. The following result describes such a choice: \setcounter{Thm}{3} \begin{Thm}\label{Thm:Weights} Let the prior for the hidden state $\textbf{X}_{t}$ be $N(\bm{\mu},\bm{\Sigma})$ and an observation $\bm{Y}_{t+1} := \bm{Y}$ be available. When \begin{equation*} \tilde{\sigma}_i = \Sigma_A^{(i,i)} \left(\hat{\bm{\Sigma}}^{-1}\right)^{(i,i)} \;\; \text{and} \;\; \hat{\sigma}_j =\Sigma_I^{(j,j)} \left(\textbf{C}^T\hat{\bm{\Sigma}}^{-1}\textbf{C}\right)^{(j,j)}, \end{equation*} and $a_1 = ... = a_p = b_1 = ... = b_q = c$, the weights of additive and innovative anomalies are asymptotically proportional to \begin{equation*} \frac{c^{c}\frac{1}{M}r_i\frac{\Gamma(c+\frac{1}{2})}{\Gamma(c)} \exp \left( \frac{1}{2} \delta ^ 2 \right) }{\left( \frac{\delta^2}{2} \right)^{c} } \;\; \text{and} \;\; \frac{c^{c}\frac{1}{M}s_j\frac{\Gamma(c+\frac{1}{2})}{\Gamma(c)} \exp \left( \frac{1}{2} \delta ^ 2 \right) }{\left( \frac{\delta^2}{2} \right)^{c} } \end{equation*} when \small \begin{equation*} \textbf{Y}-\textbf{CA}\bm{\mu} = \frac{\delta \textbf{e}_i}{ \sqrt{\left( \hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}} } \;\; \text{and} \;\; \textbf{Y}-\textbf{C}\textbf{A}\bm{\mu} = \frac{\delta\textbf{C}^{(:,j)}}{\sqrt{\left(\textbf{C}^T\hat{\bm{\Sigma}}^{-1}\textbf{C}\right)^{(j,j)}}}, \end{equation*} \normalsize respectively, as $\delta \rightarrow \infty$ \end{Thm} The above choice of hyper-parameters therefore leads to all components being given equal asymptotic importance weight under an anomaly they are able to account for. I.e.\ one which satisfies $\frac{\textbf{C}^{(:,j)}}{\sqrt{\left(\textbf{C}^T\hat{\bm{\Sigma}}^{-1}\textbf{C}\right)^{(j,j)}}}\delta = \textbf{Y}-\textbf{CA}\bm{\mu} = \frac{\delta \textbf{e}_i}{ \sqrt{\left( \hat{\bm{\Sigma}}^{-1}\right)^{(i,i)}} } $. Setting all the $a_i$s and $b_j$s to the same constant is advisable due to the fact that the convolution of two $t$-distributions whose means drift further and further apart yields two stable, i.e.\ non-vanishing modes if and only if they have the same scale parameter. While, $\hat{\bm{\Sigma}}^{-1}$ is not fixed but time dependent, it nevertheless converges to a limit under an observable Kalman filter model. In practice, we therefore use this limit to set $\tilde{\sigma}_i$ and $\hat{\sigma}_j$. \subsection{Example 1 - revisited} \begin{figure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/analysed_rw_100.jpg} \caption{t=100} \label{fig:rw_ex_100} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/analysed_rw_101.jpg} \caption{t=101} \label{fig:rw_ex_101} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/analysed_rw.jpg} \caption{Full data} \label{fig:rw_ex_1000} \end{subfigure} \caption{Robust particle filter output at various times. Additive anomalies are denoted by red points, innovative anomalies by blue lines. Grey observations are yet to be observed.} \label{fig:rw_ex_solved} \end{figure} The proposed filter can be applied to the data displayed in Figure \ref{fig:rw_ex} to detect anomalies in an online fashion. It is worth pointing out that the filter re-evaluates past anomalies as more data becomes available. This can be seen in Figure \ref{fig:rw_ex_solved}: When initially encountering the anomaly at time $t=100$ the filter gives approximately equal weight to the possibility of it being an additive outlier and to it being an innovative one. It is only when the next observation becomes available, that the filter (correctly) classifies it as an innovative anomaly. Note that only $N=20$ particles were used and only $M=1$ descendent of each anomaly type was sampled per particle. \section{Particle Filter With Back-Sampling -- CE-BASS}\label{sec:Backsampling} As mentioned in the introduction, it is possible that innovative outliers may not immediately be observed. One such example are innovative outliers in the trend component of the model described in \eqref{eq:RWand_T}. The filter as described in Algorithm \ref{alg:Basic} can not deal with such anomalies as it only inflates the variance of the innovative process at time $t$ when there is evidence in the observation at the same time $t$ that an outlier occurred. This can be remedied by back-sampling particles representing innovative outliers at a later time, $t+k$, once more observations and therefore evidence for an anomaly are available. This can be done using nearly identical approximation strategies as used in the previous section and allows to relax the assumptions made in the previous section from $\textbf{C}$ not having any $\textbf{0}$-columns to requiring that the system be observable. \subsection{Back-Sampling Particles Using the Last $k+1$ Observations} The proposed back-sampling strategy at time $t$ consists of sampling particles for $(\textbf{V}_{t+1-k},...\textbf{V}_{t+1},\textbf{W}_{t+1-k},...,\textbf{W}_{t+1})$ given a $N(\bm{\mu}_{t-k},\bm{\Sigma}_{t-k} )$ filtering distribution for $\textbf{x}_{t-k}$ and observations $\textbf{Y}_{t-k+1},...,\textbf{Y}_{t-k}$. Specifically, we sample particles with a innovative single anomaly in $\textbf{W}_{t+1-k}$ assuming no other innovative anomalies or additive anomalies. Conditional on these augmented particles classical Kalman updates can once more be used as shown in Algorithm \ref{alg:Back-sample}. It should be noted that Algorithm \ref{alg:Basic} is a special case of Algorithm \ref{alg:Back-sample} which arises from setting $\mathcal{B}_1 = ... = \mathcal{B}_q = \{1\}$. \begin{algorithm} \caption{Particle Filter (With Back Sampling) -- CE-BASS} \label{alg:Back-sample} \begin{footnotesize} \begin{tabular}[h]{ll} {\bf Input:} & An initial state estimate $(\bm{\mu}_0,\bm{\Sigma}_0)$. \\ & A number of descendants, $M'=M(p+q)+1$. \\ & A number of particles to be maintained, $N$. \\ & A stream of observations $\textbf{Y}_1,\textbf{Y}_2,...$ \\ {\bf Initialise:} & Set $Particles(0) = \{(\bm{\mu}_0,\bm{\Sigma}_0,1)\}$ \\ & Set $max\_horizon = \max \left(\cup_{i=1}^q \mathcal{B}_i\right)$ \end{tabular} \begin{algorithmic}[1] \For{$t \in \mathbb{N}^+ $} \State $Cand \gets \{\}$ \Comment{To Store Candidates} \For{$(\bm{\mu},\bm{\Sigma},prob_{prev}) \in Particles(t-1)$} \State $(\textbf{V},\textbf{W},prob) \gets \text{Sample\_typical}(\bm{\mu},\bm{\Sigma},\textbf{Y}_t,\textbf{A},\textbf{C},\bm{\Sigma}_A,\bm{\Sigma}_I)$ \State $Cand \gets Cand \cup \{(\bm{\mu},\bm{\Sigma},\textbf{V},\textbf{W},prob\cdot prob_{prev},1)\}$ \State $Add\_Des \gets \text{Sample\_additive}(\bm{\mu},\bm{\Sigma},\textbf{Y}_t,\textbf{A},\textbf{C},\bm{\Sigma}_A,\bm{\Sigma}_I,M)$ \For {$(\textbf{V},\textbf{W},prob) \in Add\_Des$} \State $Cand \gets Cand \cup \{(\bm{\mu},\bm{\Sigma},\textbf{V},\textbf{W},prob\cdot prob_{prev},1)\}$ \EndFor \EndFor \For {$hor \in \{1,...,max\_horizon\}$} \For{$(\bm{\mu},\bm{\Sigma},prob_{prev}) \in Particles(t-hor)$} \State $\tilde{\textbf{Y}} \gets \left[\textbf{Y}_{t-hor+1}^T,...,\textbf{Y}_{t}^T\right]^T$ \State $Inn\_Des \gets \text{BS\_inn}(\bm{\mu},\bm{\Sigma},\tilde{\textbf{Y}},\textbf{A},\textbf{C},\bm{\Sigma}_A,\bm{\Sigma}_I,M,hor)$ \For {$(\textbf{V},\textbf{W},prob) \in Inn\_Des$} \State $Cand \gets Cand \cup \{(\bm{\mu},\bm{\Sigma},\textbf{V},\textbf{W},prob\cdot prob_{prev},hor)\}$ \EndFor \EndFor \EndFor \State $Desc \gets \text{Subsample}(N,Cand)$ \Comment{Sampling proportional to $prob$} \State $Particles(t) \gets \{\}$ \For{$(\bm{\mu},\bm{\Sigma},\textbf{V},\textbf{W},prob,hor) \in Descendants $} \State $(\bm{\mu},\bm{\Sigma}) \gets \text{KF\_Upd}(\textbf{Y}_{t+1-hor},\bm{\mu},\bm{\Sigma},\textbf{C},\textbf{A},\textbf{V}^{1/2}\bm{\Sigma}_A,\textbf{W}^{1/2}\bm{\Sigma}_I)$ \If{$hor > 1$} \For {$i \in \{2,...,hor\}$} \State $(\bm{\mu},\bm{\Sigma}) \gets \text{KF\_Upd}(\textbf{Y}_{t+i-hor},\bm{\mu},\bm{\Sigma},\textbf{C},\textbf{A},\bm{\Sigma}_A,\bm{\Sigma}_I)$ \EndFor \EndIf \State $Particles(t) \gets Particles(t) \cup \{ (\bm{\mu},\bm{\Sigma},prob \cdot \frac{|Cand|}{|Desc|}) \}$ \EndFor \EndFor \end{algorithmic} \end{footnotesize} \end{algorithm} To sample a particle with an innovative anomaly in the $j$th component of $\textbf{W}_{t+1-k}$, we define an augmented observation vector $\tilde{\textbf{Y}}_{t+1-k}^{(k)} = (\textbf{Y}_{t+1-k}^T,...,\textbf{Y}_{t+1}^T)^T$. This is normally distributed with mean $\tilde{\textbf{C}}^{(k)} \textbf{A}\bm{\mu}_{t-k}$ and variance \footnotesize \begin{equation*} \tilde{\textbf{C}}^{(k)} \left( \textbf{A} \bm{\Sigma}_{t-k} \textbf{A}^T + \tilde{\textbf{Q}}^{(k)} \right) \left(\tilde{\textbf{C}}^{(k)}\right)^T + \tilde{\textbf{R}}^{(k)}, \end{equation*} \normalsize where $\tilde{\textbf{C}}^{(k)} = \textbf{C} \left(\left(\textbf{A}^0\right)^T,...,\left(\textbf{A}^k\right)^T\right)^T$ denotes the augmented matrix mapping the hidden states to the observations, \footnotesize \begin{equation*} \tilde{\textbf{R}}^{(k)} = \begin{bmatrix} \textbf{V}_{t+1-k}^{-1} \bm{\Sigma}_A & 0 & \ddots \\ 0 & \ddots & 0 \\ \ddots & 0 & \textbf{V}_{t+1}^{-1}\bm{\Sigma}_A \end{bmatrix} \end{equation*} \normalsize and \footnotesize \begin{equation*} \tilde{\textbf{Q}}^{(k)} = \begin{bmatrix} \textbf{W}_{t+1-k}^{-1} \bm{\Sigma}_I & 0 & \ddots \\ 0 & \ddots & 0 \\ \ddots & 0 & \textbf{W}_{t+1}^{-1}\bm{\Sigma}_I \end{bmatrix} \end{equation*} \normalsize In a similar spirit, we define the augmented predictive variance to be \begin{equation*} \hat{\bm{\Sigma}}^{(k)} = \tilde{\textbf{C}}^{(k)} \left( \textbf{A} \bm{\Sigma}_{t-k} \textbf{A}^T + \textbf{I}_{k+1} \otimes \bm{\Sigma}_I \right) \left(\tilde{\textbf{C}}^{(k)}\right)^T + \textbf{I}_{k+1} \otimes \bm{\Sigma}_A . \end{equation*} As a result of this reformulation, we retrieve update equations consisting of a single Kalman step, albeit with slightly different dimensions of the observation, $(k+1)p$ instead of $p$. It is therefore possible to use the sampling procedure for innovative outliers introduced in Section \ref{sec:Props}. This consists of sampling particles for $\tilde{\textbf{W}}_{t+1-k}^{(j,j)}$ from \footnotesize \begin{equation*} \hat{\sigma_j}\Gamma\left(b_j + \frac{1}{2},b_j + \frac{\hat{\sigma_j}}{2\bm{\Sigma}_I^{(j,j)}}\left( \frac{\left(\left(\tilde{\textbf{C}}^{(k)}\right)^T\right)^{(j,:)}\left(\hat{\bm{\Sigma}}^{(k)}\right)^{-1} \tilde{\textbf{z}}_{t+1-k}^{(k)} }{\left(\left(\tilde{\textbf{C}}^{(k)}\right)^T\left(\hat{\bm{\Sigma}}^{(k)}\right)^{-1}\tilde{\textbf{C}}^{(k)}\right)^{(j,j)}}\right)^2\right). \end{equation*} \normalsize for the residual $ \tilde{\textbf{z}}_{t+1-k}^{(k)} \tilde{\textbf{Y}}_{t+1-k}^{(k)}-\tilde{\textbf{C}}^{(k)} \textbf{A}\bm{\mu}_{t-k}$. The associated weight is given in Theorem 5 in the appendix. As in Section \ref{sec:weights}, we want to give different particles equal weights if they explain anomalies equally well. In particular, we therefore want to balance out the weights given to the back-sampled particles and the descendants of particles with an anomaly sampled at time $t-k+1$ using just $\textbf{Y}_{t+1-k}$. In order to do so, consider observations $\textbf{Y}_{t+1},...,\textbf{Y}_{t+1-k}$ which are such that they perfectly fit an innovative outlier in the $i$th innovative component at time $t-k+1$, i.e. \small \begin{equation*} \tilde{\textbf{Y}}_{t+1-k}^{(k)}-\left(\tilde{\textbf{C}}^{(k)}\right)\textbf{A}\bm{\mu}_{t-k} = \frac{\left(\tilde{\textbf{C}}^{(k)}\right)^{(:,j)}}{\sqrt{\left(\left(\tilde{\textbf{C}}^{(k)}\right)^T\left(\hat{\bm{\Sigma}}^{(k)}\right)^{-1}\left(\tilde{\textbf{C}}^{(k)}\right)\right)^{(j,j)}}}\delta. \end{equation*} \normalsize As $\delta$ grows, the importance weight behaves as \begin{equation*} \frac{b_j^{b_j}\frac{1}{M}s_j\frac{\Gamma(b_j+\frac{1}{2})}{\Gamma(b_j)} \exp\left(-\delta^2\right) }{\left( \frac{\hat{\sigma}_j}{2\bm{\Sigma}_I^{(j,j)} \left(\left(\tilde{\textbf{C}}^{(k)}\right)^T\left(\hat{\bm{\Sigma}}^{(k)}\right)^{-1}\left(\tilde{\textbf{C}}^{(k)}\right)\right)^{(j,j)} }\delta^2 \right)^{b_j} }, \end{equation*} up to the likelihood term and the $\left(1-\sum_{i=1}^{p}r_i - \sum_{j=1}^{q}s_j\right)^{k}$ factor. However, these terms are also present in the weights of the descendants of the particles sampled at $t+1-k$ if no further anomaly was sampled at times $t+2-k,...,t+1$. Therefore, setting \begin{equation*} \hat{\sigma}_j = \bm{\Sigma}_I^{(j,j)} \left(\left(\tilde{\textbf{C}}^{(k)}\right)^T\left(\hat{\bm{\Sigma}}^{(k)}\right)^{-1}\left(\tilde{\textbf{C}}^{(k)}\right)\right)^{(j,j)} \end{equation*} results in the same asymptotic probabilities as the one obtained in Section \ref{sec:weights}. Given $\hat{\sigma}_j$ can only take a single value we set \begin{equation*} \hat{\sigma}_j = \max_{k \in \mathcal{B}_j} \left( \bm{\Sigma}_I^{(j,j)} \left(\left(\tilde{\textbf{C}}^{(k)}\right)^T\left(\hat{\bm{\Sigma}}^{(k)}\right)^{-1}\left(\tilde{\textbf{C}}^{(k)}\right)\right)^{(j,j)} \right), \end{equation*} where $\mathcal{B}_j \subset \mathbb{N}$ denotes the set of horizons used to back-sample the $j$th component of the $\textbf{W}_t$. A range of observations guide the choice of the sets $\mathcal{B}_j$ for $1 \leq j \leq q$. We assume that the Kalman model is observable, i.e.\ that there exists a $k$ such that the matrix $ \left[ \left(\textbf{C}\right)^T , \left(\textbf{CA}\right)^T, ... ,\left(\textbf{CA}^k\right)^T\right] $ has full column rank. Let $k^*$ denote the lowest such $k$. It is advisable to choose the set $\mathcal{B}_j$ such that it contains at least one element greater or equal to $k^*$. The reason for this being that any innovative anomaly capable of eventually influencing the observations must do so within $k^*$ observations from occurring. It should also be noted that a horizon $h$ can only be in the set $\mathcal{B}_j$ if the $j$th column of the augmented mapping from the hidden states to the observations, $\tilde{\textbf{C}}^{(h)}$, is non-zero as this is required by the proposal. Consequently, setting $\mathcal{B}_j = \left\{k \in \{1,...,k^*\} : \left(\tilde{\textbf{C}}^{(k)}\right)^{(:,j)} \neq \textbf{0}\right\}$ is a natural choice. \subsection{Example} With back-sampling, we are now able to tackle the example from Figure \ref{fig:rwandt_ex}. We used $\mathcal{B}_1 = \{1,...,40\}$, $\mathcal{B}_2 = \{1,...,40\}$, to sample back up to 40 observations. We maintained $N=40$ particles and sampled $M=1$ descendants of each type. The output of the particle filter can be seen in Figure \ref{fig:rwandt_ex_solved}. As before, the filter updates its output as new observations become available. Whilst the trend innovation occurs at time $t=800$, the anomaly is first detected around time $t=820$. Even then, there is a large amount of uncertainty regarding the precise location of the anomaly which only gets resolved at a later time. \begin{figure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/analysed_rwandt_820.jpg} \caption{t=820} \label{fig:rwandt_ex_100} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/analysed_rwandt_821.jpg} \caption{t=821} \label{fig:rwandt_ex_101} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.9\linewidth]{Plots/analysed_rwandt.jpg} \caption{Full data} \label{fig:rwandt_ex_1000} \end{subfigure} \caption{Robust particle filter output at various times. Additive anomalies are denoted by red points, innovative anomalies by blue lines. Grey observations are yet to be observed.} \label{fig:rwandt_ex_solved} \end{figure} \section{Simulations}\label{sec:Simulation} \begin{figure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RW_Kalman.jpg} \caption{Case 1} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RW_IO} \caption{Case 1, IOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RW_AO} \caption{Case 1, AOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RW_AOIO} \caption{Case 1, Both} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWMult_Kalman.jpg} \caption{Case 2} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWMult_IO} \caption{Case 2, IOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWMult_AO} \caption{Case 2, AOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWMult_AOIO} \caption{Case 2, Both} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWT_Kalman.jpg} \caption{Case 3} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWT_IO} \caption{Case 3, IOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWT_AO} \caption{Case 3, AOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/RWT_AOIO} \caption{Case 3, Both} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/SV_Kalman.jpg} \caption{Case 4} \label{fig:meanchange_graph} \end{subfigure} \begin{subfigure}[b]{0.245\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/SV_IO} \caption{Case 4, IOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/SV_AO} \caption{Case 4, AOs} \label{fig:meanANOMchange_graph} \end{subfigure} \begin{subfigure}[b]{0.24\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Simulation_Plots/SV_AOIO} \caption{Case 4, Both} \label{fig:meanchange_graph} \end{subfigure} \caption{Violin plots for the average predictive log-likelihood of the five filters (IOAO: CE-BASS, KF: The classical Kalman Filter, AO T: \cite{agamennoni2011outlier}, AO H: \cite{ruckdeschel2014robust}, IO H: \cite{ruckdeschel2014robust}) over the four different scenarios under a range of models. Higher values correspond to better performance. Methods are omitted on the graphs if they can not be applied to the setting or if their performance is too poor.} \label{fig:LOG-LIK} \end{figure} We now turn to comparing CE-BASS against other methods. In particular, we compare against the $t$-distribution based additive outlier robust filter by \cite{agamennoni2011outlier}, the Huberisation based additive outlier robust filter by \cite{ruckdeschel2014robust}, the Huberisation based innovative outlier robust filter by \cite{ruckdeschel2014robust}, and the classical Kalman Filter \citep{Kalman1960}. All these algorithms are implemented in the accompanying package. We consider four different models and generate 1000 observations for each. For each of the four models, we consider a case in which no anomalies are present, a case in which only additive anomalies are present, a case in which only innovative anomalies are present, and a case in which both additive and innovative anomalies are present. When anomalies are added, they are added at times $t=100$, $t=300$, $t=600$, and $t=900$. Specifically we considered the following three models: \begin{enumerate} \item The model of Example 1 with $\sigma_A=1$ and $\sigma_I=0.1$. We consider a case with only additive outliers, a case with only innovative outliers, and a case where an additive outlier at $t=100$, is followed by two innovative outliers at times $t=300$ and $t=600$, which were then followed by an additive outlier at time $t=900$. To simulate additive anomalies, we set $V_t^{\frac{1}{2}}\sigma_A\epsilon_t = 10$ and to simulate the innovative outliers we set $W_t^{\frac{1}{2}}\sigma_I\nu_t = 10$. \item The random walk model with two measurements \small \begin{align*} Y_t^{(1)} &= X_t + \left( V_t^{(1)} \right)^{\frac{1}{2}}\sigma_A^{(1)}\epsilon_t^{(1)}, \;\; & \;\; X_t = X_{t-1} + W_t^{\frac{1}{2}}\sigma_I\nu_t \\ Y_t^{(2)} &= X_t + \left( V_t^{(2)} \right)^{\frac{1}{2}}\sigma_A^{(2)}\epsilon_t^{(2)}, \;\; & \end{align*} \normalsize where $\sigma_A^{(1)} = \sigma_A^{(2)} = 1$ for $i=1,2$ and $\sigma_I=0.1$. We consider a case with only additive outliers (one in the first component, then two in the second, then one in the first), a case with only innovative outliers, and a case where an additive outlier in the first component at time $t=100$ is followed by two innovative outliers at times $t=300$ and $t=600$, which are then followed by an additive outlier in the second component at time $t=900$. For additive anomalies, we set $\left( V_t^{(1)} \right)^{\frac{1}{2}}\sigma_A^{(1)}\epsilon_t^{(1)} = 10$ or $\left( V_t^{(2)} \right)^{\frac{1}{2}}\sigma_A^{(2)}\epsilon_t^{(2)} = 10$ and for innovative outliers, we set $W_t^{\frac{1}{2}}\sigma_I\nu_t = 10$. \item The model of Example 2 with $\sigma_A=1$, $\sigma_I^{(1)}=0.1$ and $\sigma_I^{(2)}=0.01$. We consider a case with only additive outliers, a case with only innovative outliers (one in the second component, then one in the first, then one in the second, then one in the first), and a case with an additive outlier at $t=100$, followed by an innovative outlier affecting the first component of the hidden state at times $t=300$, followed by an innovative outlier affecting the second component of the hidden state at times $t=600$, followed by an additive outlier at time $t=900$. The additive anomalies were instances where we set $V_t^{\frac{1}{2}}\epsilon_t = 30$ and the innovative outliers were instances where we set $\left( W_t^{(1)} \right)^{\frac{1}{2}}\eta_t^{(1)} = 100$ or $\left( W_t^{(2)} \right)^{\frac{1}{2}}\eta_t^{(2)} = 500$. \item An extension of Example 2 where the position is also observed. The equations governing the hidden state are as before whilst the equations governing the observations are \footnotesize \begin{align*} Y_t^{(1)} &= X_t^{(1)} + \left( V_t^{(1)} \right)^{\frac{1}{2}}\sigma_A^{(1)}\epsilon_t^{(1)}, \\ Y_t^{(2)} &= X_t^{(2)} + \left( V_t^{(2)} \right)^{\frac{1}{2}}\sigma_A^{(2)}\epsilon_t^{(2)}, \end{align*} \normalsize where $\sigma_A^{(1)} = \sigma_A^{(2)} = 1$. We consider a case with only additive outliers (in the first component only), a case with only innovative outliers (one in the second component, then one in the first, then one in the second, then one in the first), and a case with an additive outlier at time $t=100$, followed by an innovative outlier affecting the first component of the hidden state at time $t=300$, followed by an innovative outlier affecting the second component of the hidden state at time $t=600$, followed by an additive outlier at time $t=900$. For additive anomalies, we set $\left( V_t^{(1)} \right)^{\frac{1}{2}}\sigma_A^{(1)}\epsilon_t^{(1)} = 30$ and for innovative outliers, we set $\left( W_t^{(1)} \right)^{\frac{1}{2}}\sigma_I^{(1)}\eta_t^{(1)} = 100$ or $\left( W_t^{(2)} \right)^{\frac{1}{2}}\sigma_I^{(2)}\eta_t^{(2)} = 500$. \end{enumerate} We evaluate the different methods based on average predictive log-likelihood and average predictive mean squared error. We exclude all observations corresponding to anomalies from the calculation of these averages since the filters can not be expected to predict them. When calculating the average mean squared error we additionally remove one observation after the anomaly in the first setting and two observations in the third setting from the performance metric. This is to give the filter enough information to determine which type of anomaly the outlier corresponds to and return to a unimodal posterior, as the MSE is only an appropriate metric for unimodal posteriors. The average log-likelihoods across all models can be found in Figure \ref{fig:LOG-LIK}, while the qualitatively very similar results for the mean squared error can be found in the appendix. We see that the performance of CE-BASS compares favourably with that of the competing methods. In particular it is as accurate as the Kalman filter in the absence of anomalies and is more accurate than the additive outlier and innovative outlier robust filters even when only additive or innovative outliers are present, i.e.\ the settings for which these algorithms were designed. \section{Application}\label{sec:Application} In this section, we apply CE-BASS to two real datasets. We will use different types of models for the two applications to illustrate the way in which CE-BASS can be used. The first dataset is a labelled benchmark dataset which consists of temperature readings on a large industrial machine. Here, we will use a model which considerably restricts the movements of the hidden states when no anomalies are present, and thus emulates a changepoint model. The second is an unlabelled dataset which consist of repeated throughput measurements on a router. For that application we will use a model which has a considerable amount of flexibility and where the hidden states tend to follow the observations and therefore detect localised anomalies. \subsection{Machine Temperature Data} \begin{figure} \begin{subfigure}[b]{0.495\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/Raw} \caption{Raw data with labels} \label{fig:Rawdata} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.495\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/Analysed.jpg} \caption{CE-BASS output} \label{fig:Analyseddata} \vspace{20pt} \end{subfigure} \caption{Machine temperature dataset. The labelled anomalies are: a planned shutdown, an early warning sign of a problem, and the catastrophic system failure caused by the problem.} \label{fig:Machine_Temp} \end{figure} We now apply CE-BASS to the machine temperature data taken from the Numenta Anomaly Benchmark (NAB, \cite{lavin2015evaluating}) which can be accessed at \textit{https://github.com/numenta/NAB}. The data consists of over 20000 readings from a temperature sensor on a large industrial machine and is displayed in Figure \ref{fig:Rawdata} along the three periods of anomalous behaviour labelled by an engineer. The first corresponds to a planned shutdown and the second to an early warning sign of the third anomaly -- a catastrophic failure. In order to do so, we use the random walk model from Example 1 with the aim of detecting persistent changes in mean. We therefore use a maximum backsampling horizon of 250 by setting $\mathcal{B}_1=\{1,5,10,20,40,80,150,250\}$ and fix $\sigma_I = 1/10000\sigma_A$ to ensure that long and weak anomalies will not be interpreted as a persistent shift in the typical state. We use the first 15\% of the data, marked by \cite{lavin2015evaluating} as train data, to estimate the standard deviation $\sigma_A$ as well as the initial mean $\mu_0$ using the median absolute deviation and the median respectively. Using robust covariance methods we also detect very strong auto-correlation ($\rho=0.99$) and therefore took the default probabilities for anomalies to the power of $\frac{1}{1-\rho}$. The results of this analysis can be seen in Figure \ref{fig:Analyseddata}. We note that all anomalies flagged by the engineer are also being detected by CE-BASS. Two additional innovative anomalies around a prolonged drop which preceded the planned shutdown are also detected. They could be a false positive or an early warning sign of an anomaly prevented by the shutdown which has not been noticed by the engineer. \subsection{Router Data} \begin{figure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat11} \caption{Day 11} \label{fig:Day11} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat12} \caption{Day 12} \label{fig:Day12} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat13} \caption{Day 13} \label{fig:Day13} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat14} \caption{Day 14} \label{fig:Day14} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat15} \caption{Day 15} \label{fig:Day15} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat16} \caption{Day 16} \label{fig:Day16} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat17} \caption{Day 17} \label{fig:Day17} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat18} \caption{Day 18} \label{fig:Day18} \vspace{20pt} \end{subfigure} \begin{subfigure}[b]{0.32\linewidth} \centering \includegraphics[width=0.95\linewidth]{Plots/Paper/No_trend_analysed_day_no_cheat19} \caption{Day 19} \label{fig:Day19} \vspace{20pt} \end{subfigure} \caption{CE-BASS applied to 9 days of de-seasonalised router data. Lines correspond to innovative anomalies, i.e.\ spikes or level shifts.} \label{fig:Router} \end{figure} The online analysis of aggregated traffic data on servers is an important challenge in both predictive maintenance and cyber security. This is because anomalies in throughput can point towards problems in the network such as malfunctions or malicious behaviour. Detecting anomalies as soon as possible therefore means that the root cause can be addressed more quickly -- potentially even before user experience is affected or harm caused. In this section, we consider 19 days worth of data from a network IP router which has been gathered at a frequency of one observation every 30 seconds. To preserve confidentiality, we de-seasonalised the data for days 11 to 19 using a seasonality model trained on days 1 to 10 and, for the purpose of this paper, consider only the de-seasonalised data for days 11 to 19 which can be found in Figures \ref{fig:Day11} to \ref{fig:Day19}. The main features apparent in the daily series are spikes, outliers, and changepoints. In order to capture these, we use an AR(1) model with slowly changing mean to model the observations $Y_t$. Formally, we used the model \begin{align*} Y_t &= X_t^{(1)} + X_t^{(2)} + V_t\sigma_A \epsilon_t, \;\; & \;\; X_t^{(1)} &= X_{t-1}^{(1)} + W_t^{(1)}\sigma_I^{(1)} \eta_t^{(1)}, \\ & \;\;& \;\; X_t^{(2)} &= \rho X_{t-1}^{(2)} + W_t^{(2)}\sigma_I^{(2)} \eta_t^{(2)}. \end{align*} Here, anomalies in $\epsilon_t$ correspond to isolated outliers, anomalies in $\eta_t^{(1)}$ correspond to level shifts and outliers in $\eta_t^{(2)}$ correspond to spikes. We use the first 1000 observations of the first day, to obtain the estimates $\sigma_A = 0.0516$, $\sigma_I^{(1)} = 0.0157$, $\sigma_I^{(2)} = 0.516$, and $\rho = 0.815$. The result obtained from running CE-BASS with these parameters on the daily router data is displayed in Figures \ref{fig:Day11} to \ref{fig:Day19}. We note that very few of the anomalies returned can be classed as false positives. At the same time, a large number of anomalies are flagged, including a large number of outliers and spikes, but also some level shifts (Day 14). Discussion with engineers highlighted that the anomalies detected matched well with their knowledge of the data. This shows CE-BASS's ability to return a large number of diverse features which can be used as inputs to a supervised algorithm should labels become available. \section{Acknowledgements} This work was supported by EPSRC grant numbers EP/N031938/1 (StatScale) and EP/L015692/1 (STOR-i). The authors also acknowledge British Telecommunications plc (BT) for financial support, David Yearling and Trevor Burbridge in BT Research for discussions. \bibliographystyle{unsrt}
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(2684) Douglas est un astéroïde de la ceinture principale. Description (2684) Douglas est un astéroïde de la ceinture principale. Il fut découvert le à la station Anderson Mesa par Norman G. Thomas. Il présente une orbite caractérisée par un demi-grand axe de 3,05 UA, une excentricité de 0,04 et une inclinaison de 9,9° par rapport à l'écliptique. Compléments Articles connexes Liste des planètes mineures (2001-3000) Ceinture d'astéroïdes Références Planète mineure découverte en 1981 Astéroïde de la ceinture principale Objet céleste découvert par Norman G. Thomas Objet céleste découvert à la station Anderson Mesa
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Das eindimensionale Zuschnittproblem (englisch one-dimensional cutting stock problem) ist ein NP-schweres ganzzahliges lineares Optimierungsproblem mit dem Ziel, eindimensionale Teile in vorgegebenen Bedarfszahlen aus möglichst wenig Stücken Material gegebener Länge zuzuschneiden. Dieses Problem verdankt seine große Bedeutung auch dem Umstand, dass es als Relaxation für kompliziertere mehrdimensionale Pack- und Zuschnittprobleme verwendet wird, zum Beispiel beim Containerbeladeproblem mit Quadern, wenn man sich alle Teile in Streifen zerlegt denkt. Problemstellung und grundlegende Definitionen Gegeben ist eine unbegrenzte Zahl von Stücken eindimensionalen Rohmaterials vorgegebener Länge . Daraus sollen Teile der Länge zugeschnitten werden, mit insgesamt also Teile. Dafür sollen möglichst wenige Stücke des Rohmaterials verbraucht werden. Reststücke können nicht miteinander verbunden werden und zählen als Abfall. Sind die Bedarfszahlen sehr klein, spricht man auch vom (eindimensionalen) Behälterproblem (Bin-Pack-Problem). Unmittelbare Anwendungen sind zum Beispiel das Zuschneiden von Rohren oder das Abspeichern von nicht teilbaren und nicht weiter komprimierbaren Dateien auf möglichst wenig Datenträgern einheitlicher Kapazität. Die Verallgemeinerung auf mehrere verschiedene Längen an Rohmaterial wird später behandelt. Soll eine bestimmte Schnittbreite berücksichtigt werden, ist dies möglich, indem alle Längen, also und () um vergrößert werden. Damit wird das Problem auf eines mit Schnittbreite 0 zurückgeführt. Zunächst werden die Längen und die Bedarfszahlen (für ) zu Vektoren und zusammengefasst. Die Zusammenfassung aller Daten zu einer Instanz erfolgt als Quadrupel . Hierbei bedeutet der Begriff Problem immer eine Problemklasse, während erst mit konkreten Daten eine Instanz vorliegt. Eine Zuschnittvariante ist ein Vektor nichtnegativer ganzer Zahlen, der angibt, wie oft jedes Teil in dieser Variante vorkommt. Die Variante ist genau dann zulässig, wenn gilt. Die Indexmenge aller zulässigen Varianten   sei mit bezeichnet. Damit lautet das ganzzahlige lineare Optimierungsproblem: bei Dieses Problem ist stets lösbar, wenn für alle Teilelängen gilt, da die Zielfunktion (2) nach unten beschränkt ist und nur ganzzahlige Werte annimmt und eine zulässige Lösung existiert, etwa aus jedem Stück Ausgangsmaterial nur ein Teil zu fertigen. Der Bedarf an zuzuschneidenden Teilen aller Sorten wird aufgrund der Bedingung (3) gedeckt. Ersetzt man in ihr das Gleichheitszeichen durch ≥, entsteht ein zum Modell (2)–(4) äquivalentes, denn man kann aus Zuschnittvarianten, die zu Überproduktion führen, einzelne Teile herauslassen. Auf diese Weise erhält man aus einer Optimallösung des abgeänderten Problems eine für das Problem (2)–(4) mit gleichem Zielfunktionswert. Da die Anzahl der zulässigen Varianten und damit der Variablen in der Aufgabe (2)–(4) oft sehr groß ist, wurde auch nach alternativen Modellen gesucht. Ein solches besteht u. a. in der Formulierung als Optimierung eines Flusses in einem Netzwerk. Jenes Fluss-Modell stellt sich als äquivalent zur obigen Problemformulierung heraus. Wegen Einzelheiten sei z. B. auf verwiesen. Eine zulässige Zuschnittvariante heißt eigentlich, wenn gilt, also die Variante, für sich alleine einmal verwendet, keine Überproduktion liefert. Offensichtlich reichen eigentliche Varianten zur Lösung des Problems (2)–(4) aus. Das Problem ist -schwer, denn schon die Frage, ob alle Teile aus nur zwei Stück Ausgangsmaterial geschnitten werden können, führt auf das Rucksackproblem, und dieses ist -vollständig. Gemäß ist das eindimensionale Bin-Pack-Problem sogar stark -vollständig. Trotz dieser ungünstigen Komplexität können für viele Instanzen in akzeptabler Zeit Optimallösungen bestimmt werden, zum Beispiel mittels geeigneter Heuristiken. Um die Güte einer zulässigen Lösung zu bewerten, benötigt man möglichst scharfe untere Schranken. Eine einfache untere Schranke für den optimalen Zielfunktionswert der ganzzahligen Optimierungsaufgabe stellt die Materialschranke dar. Allerdings ist diese Schranke meist zu ungenau, denn die Differenz wächst im Allgemeinen unbeschränkt mit den Bedarfszahlen. Durch Abmilderung der Bedingung (4) gewinnt man aus dem Problem (2)–(4) Relaxationen mit oft deutlich schärferen Schranken, nämlich die stetige Relaxation: Einschränkung auf eigentliche Varianten: Die optimalen Zielfunktionswerte beider Relaxationen seien mit und bezeichnet. Dann zeigt man leicht . Von besonderem theoretischem Interesse ist die Lücke . Es stellt sich heraus, dass es schwer ist, für irgendeine Instanz zu prüfen. Eine Verschärfung der Relaxation (6) erhält man mit der Einführung oberer Schranken für die Variantenhäufigkeiten. So darf etwa eine Schnittvariante höchstens -mal verwendet werden. Doch auch für diese kompliziertere Relaxation konstruiert man leicht Instanzen, bei denen die Schrankenverschärfung versagt. Beispiel Für die Instanz sind die Daten in nebenstehender Abbildung nochmals angegeben (rosa eingefärbt), dazu (grün gefärbt) die eindeutige Optimallösung der stetigen Relaxation (5). Nur die zweite Zuschnittvariante, nämlich , weist Verschnitt auf. Außer dieser Variante sind alle in der Relaxationslösung in positiver Häufigkeit vorkommenden Varianten uneigentlich. Wie man die Relaxationen löst, erklärt ein späterer Abschnitt. Es gilt . Die Instanz ergibt die für den Fall, dass kein Teil mehr als die Hälfte des Ausgangsmaterials ausmacht, größte bisher bekannte Lücke (Stand 2007). Vervielfacht man die Bedarfszahlen mit elf oder ersetzt man sie durch den neuen Vektor , ergibt sich wieder die Lücke , jedoch wird für die abgeänderte Instanz . Nur noch die Verschärfung der Relaxation (6) verrät . Doch eine geringfügige Verkürzung einzelner der gemäß zu schneidenden 25 Teile ermöglicht, auch die Verschärfung der Relaxation (6) wirkungslos zu machen. Äquivalente Instanzen Zwei Instanzen und heißen äquivalent, wenn und gilt und jede für eine der beiden Instanzen zulässige Variante auch für die andere Instanz zulässig ist. Äquivalente Instanzen erhält man aus einer gegebenen zum Beispiel durch Multiplikation aller Längen mit einer positiven Konstanten oder indem man Teilelängen um bis zu ε verkleinert oder das Ausgangsmaterial um ε verlängert, falls hinreichend klein ist, weil keine neue Variante hinzu kommt. Somit kann man stets zu rationalen und nach Multiplikation mit dem Hauptnenner zu ganzzahligen Daten übergehen. Eine gemäß (1) zulässige Variante heißt maximal, wenn gilt, also der Verschnitt kleiner als das kleinste zuzuschneidende Teil ist. Um zu prüfen, ob die Instanzen und bei äquivalent sind, genügt es offensichtlich zu untersuchen, ob jede für eine Instanz maximale Variante auch für die andere Instanz zulässig ist und umgekehrt. Dagegen darf nicht schon auf Äquivalenz geschlossen werden, wenn jede für maximale Variante auch für maximal ist, wie das Gegenbeispiel , zeigt. Beispiel zur Äquivalenz: Thomas Gau fand bei Testrechnungen die Instanz mit Lücke . Ersetzt man die 3001 durch 3125, ergibt sich eine äquivalente Instanz, da alle anderen Längen durch 250 teilbar sind und die Variante maximal ist. Deshalb geht keine zulässige Variante verloren. Dividiert man nun alle Längen durch 125, ergibt sich wieder eine äquivalente Instanz, nämlich . Eine weitere äquivalente Instanz entsteht hieraus durch Multiplikation aller Längen mit und geeignetes Runden, nämlich . Um nachzuweisen, dass keine äquivalente Instanz mit durchgängig ganzzahligen Daten und kleinerer Länge des Ausgangsmaterials existiert, kann das duale Simplex-Verfahren ohne Zielfunktion eingesetzt werden. Drei Typen von Ungleichungen sind von den äquivalenten Instanzen zu erfüllen: für aufgrund der Sortierung für jede maximale Variante für jede unzulässige Variante wegen der Ganzzahligkeit. Die meisten dieser Ungleichungen sind überflüssig, d. h., sie folgen aus anderen. Im Simplexschema können derartige Zeilen gestrichen werden, wenn sie keine negativen Einträge enthalten und die zugehörige Basisvariable nicht zu den oder gehört. Zuletzt bleiben für die gesuchten Längen nur Zeilen übrig. Allerdings existieren Beispiele, die zur Beschreibung aller äquivalenten Instanzen mit durchgängig ganzzahligen Daten mehr Ungleichungen benötigen, wo also zusätzliche Zeilen mit mindestens einem negativen Eintrag im Endschema verbleiben. Beispiel: Gesucht werden alle zur Instanz äquivalenten Instanzen mit durchgängig ganzzahligen Längen, wobei für alle gilt. Offensichtlich ist zu fordern. Neben den Unzulässigkeitsbedingungen und könnten noch viele überflüssige Ungleichungen notiert werden, darunter zu nicht aufgeführten weiteren maximalen Varianten. Die zu diesen sechs Ungleichungen gehörenden nichtnegativen Schlupfvariablen seien mit , bezeichnet. Sie sind als Differenz ganzzahliger Größen ebenfalls ganzzahlig. Folgendes Schema entsteht: Demzufolge darf die Schlupfvariable nicht beliebig unabhängig von den anderen erhöht werden. Tauscht man gegen , geht die Ganzzahligkeit von verloren, falls alle Schlupfvariablen der Nichtbasis danach auf 0 gesetzt werden. Insgesamt ergibt sich, dass und , mit den vier Parametern beschrieben werden können, während gilt, also die ganzzahligen Punkte eines Intervalls zu nehmen sind. In anderen Beispielen können derartige Besonderheiten noch komplizierter aussehen. Eine weitere Schwierigkeit besteht jeweils darin, die Äquivalenz vollständig zu prüfen, ob also keine notwendige Ungleichung, insbesondere zu unzulässigen Varianten, fehlt. Eine andere Art der Gleichwertigkeit ergibt sich, indem man Teile mit Bedarfszahlen größer als 1 als mehrere verschiedene Teile, die jeweils genau einmal gefordert werden, auffasst. Wenn zum Beispiel dreimal ein Teil der Länge 5 gewünscht wird, kann man ebenso etwa und anstelle des einen Teils mit der Bedarfszahl 3 schreiben. Folglich ist das eindimensionale Bin-Pack-Problem, bei dem jedes Teil genau einmal in Behälter der Größe zu packen ist, gleichwertig zum oben eingeführten eindimensionalen Zuschnittproblem (2)–(4). Der Teilbarkeitsfall; modifizierte Ganzzahl-Aufrundungseigenschaft Eine zulässige Variante heißt elementar, wenn sie nur eine Teilesorte enthält, also von der Gestalt mit ist, wobei den -ten Basis-Einheitsvektor des bezeichnet, . Der Teilbarkeitsfall liegt vor, wenn ganzzahliges Vielfaches jeder Teilelänge ist. Dann ergibt sich sofort , indem in der stetigen Relaxation (5) nur maximale elementare Varianten verwendet werden. Beispiel: Die Instanz besitzt für die Relaxation (5) wegen den optimalen Zielfunktionswert . Hier gilt aber , d. h., es ist unmöglich, mit nur zwei Stück Ausgangsmaterial alle Teile zu fertigen. Diese Instanz ergibt die größte bisher im Teilbarkeitsfall bekannte Differenz , nämlich (Stand 2007). Für obige Instanz gilt die Ganzzahl-Aufrundungseigenschaft nicht. Da alle bisherigen Erfahrungen darauf schließen ließen, dass die Lücke für beliebige Instanzen des eindimensionalen Zuschnittproblems (2)–(4) stets klein ist, wurde der Begriff der modifizierten Ganzzahl-Aufrundungseigenschaft (englisch modified integer round-up property, MIRUP) geprägt. Eine Instanz weist diese Eigenschaft auf, wenn gilt. Die Vermutung, jede Instanz des eindimensionalen Zuschnittproblems (2)–(4) besitze MIRUP, konnte bisher (Stand 2007) nur in Spezialfällen nachgewiesen werden, zum Beispiel für den Teilbarkeitsfall. Ein einfacherer Beweis von Guntram Scheithauer und Johannes Terno wurde in der Dissertation noch verschärft. Es gilt folgender Satz: Für jede Instanz des Teilbarkeitsfalls gilt . Sind sämtliche Teile größer als des Ausgangsmaterials, gilt sogar . Das Vorhandensein unendlich vieler, paarweise nicht äquivalenter Instanzen des Teilbarkeitsfalls mit folgt unter anderem aus diesen Aussagen: Seien paarweise teilerfremde ganze Zahlen mit und . Für alle sei , und es gebe keine Lösung der ganzzahligen Aufgabe (2)–(4), in der diese Teile (mit den Bedarfszahlen ) in höchstens Varianten untergebracht werden. Ferner seien und für . Die so festgelegte Instanz besitzt eine Lücke . Für beliebiges sei das kleinste gemeinsame Vielfache von (oder ein Mehrfaches davon). Dann besitzt die Instanz eine Lücke . Sei beliebig und . Dann gilt für die Instanz . Lösung der stetigen Relaxation Schon für relativ kleine Parameter ist die Mächtigkeit der Menge oft so groß, dass eine vollständige Aufzählung aller zulässigen Zuschnittvarianten nicht in Frage kommt. Daran ändert sich auch nichts, wenn nur verschnittarme Varianten betrachtet werden. Da aber auch in einer Optimallösung gelegentlich verschnittreiche Varianten vorkommen, wäre dieser Lösungsansatz falsch. Aus dieser Not machten Gilmore und Gomory eine Tugend, indem sie die Relaxation mit dem revidierten Simplexverfahren lösten, als Start mit den einfachsten Varianten begannen und bessere bei Bedarf im Laufe der Optimierung suchten. In der revidierten Simplexmethode werden die zur Zielfunktion gehörenden Koeffizienten in Basis- und Nichtbasisanteil bzw. aufgeteilt, ebenso die Nebenbedingungen in der Weise , wobei die Basismatrix regulär ist. Löst man nach auf und setzt dies in die Zielfunktion ein, so ergibt sich . Da in unserem Zuschnittproblem alle Zielfunktionskoeffizienten 1 sind, ist eine Verbesserung des Zielfunktionswertes der stetigen Relaxation (5) folglich nur möglich, wenn eine gemäß (1) zulässige Variante mit existiert. Für diese Spaltengenerierung ist somit jeweils ein Rucksackproblem zu lösen, wobei gilt. Eine einfache Rechenkontrolle besteht in . Damit im Simplexverfahren Zyklen vermieden werden, empfiehlt sich die Regel von Bland (vgl. Simplex-Verfahren#Zeilenauswahl). Um diese Regel umzusetzen, hebt man jede gefundene Variante auf und prüft, bevor das Spaltengenerierungsproblem (7) gelöst wird, ob früher eine Variante abgespeichert wurde, für die gilt. In diesem Falle wird nicht das Rucksackproblem (7) bearbeitet, sondern von den abgespeicherten Varianten eine in die Basis getauscht, die den größten Wert für das Skalarprodukt ergibt. Ansonsten muss die Spaltengenerierungsaufgabe gelöst werden. Den Aufwand für eine exakte Lösung des Problems sollte man nicht scheuen, da sonst in der Regel wesentlich mehr Simplexschritte gebraucht werden. Ein einfaches Beispiel: Von eindimensionalem Ausgangsmaterial der Länge 11 sind in besonders hoher Stückzahl Teile der Längen 6, 4 und 1 zu schneiden, und zwar im Verhältnis . Die Materialausnutzung ist zu maximieren. Das bedeutet, hier ist eine Optimallösung der stetigen Relaxation (5) von der Instanz gesucht. Für die erste Basis werden maximale elementare Varianten gewählt, das sind , und , so dass anfangs eine Diagonalmatrix ist. Es ergeben sich die nachfolgenden revidierten Simplexschemata, unter denen die neue Variante angegeben ist. Die Pivotelemente sind jeweils mit einem Stern gekennzeichnet. Aus Gründen der einfacheren Programmierung wurden die rechten Seiten und der Vektor in der ersten Spalte bzw. Zeile untergebracht. Ganz rechts steht jeweils die transformierte neue Spalte, bestehend aus und . optimal Die Varianten und sind folglich im Verhältnis zu schneiden. Beim letzten Austausch war (und nicht ) aus der Basis zu tauschen, um der Regel von Bland zu gehorchen, nämlich bei mehreren wählbaren Zeilen immer diejenige auszuwählen, die zum Austausch der Variable mit dem kleinsten Index aus der Basis führt. Obwohl dieses Beispiel akademisch aussehen mag, zeigt es, dass auch negative Werte im Laufe der Rechnung auftreten können, wenn die Bedarfszahlen geeignet vorgegeben waren. Soll die Relaxation (6) mit Einschränkung auf eigentliche Varianten gelöst werden, kann dieser Effekt noch wesentlich stärker auftreten. Im Spaltengenerierungsproblem ist die Verwendung jener Teile verboten, so dass sich die Spaltengenerierungsaufgabe vereinfacht. Residuale Instanzen Um das ganzzahlige Problem (2)–(4) zumindest nahezu optimal zu lösen, kann zunächst die Relaxation (5) oder (6) herangezogen werden. Durch einfaches Aufrunden ergibt sich eine zulässige Lösung, wenn Überproduktion erlaubt wird. Bei einzelnen Zuschnittvarianten kann vielleicht sogar abgerundet werden. Doch selbst bei optimaler Rundung erhielte man im Allgemeinen einen Zielfunktionswert deutlich über dem optimalen Wert . Dieses Vorgehen erscheint deshalb nur sinnvoll, wenn die Anzahl verschiedener Varianten minimiert werden soll, weil die Umstellung der Fertigungsanlage auf andere Schnittpläne sehr aufwendig ist. Ansonsten empfiehlt es sich, durchgängig abzurunden und für die verbliebenen Teile entweder mit einer neuen Heuristik fortzufahren oder noch einmal die Relaxation (6) zu benutzen. Sei eine mittels Simplexverfahren ermittelte optimale Basislösung der stetigen Relaxation (5). Ersetzt man in der Instanz den Vektor der Bedarfszahlen durch , entsteht eine sogenannte residuale Instanz . Dabei dürfen durchaus auch Nullen im Bedarfsvektor auftreten. Bei vielen Abschätzungen der Lücke hilft folgendes Lemma: Seien beliebig. Dann gilt für residuale Instanzen die Implikation . Beweis: Nach Voraussetzung gibt es eine optimale Basislösung, in der jede Variante eine Häufigkeit hat. Einen ganzzahligen Zuschnittplan mit Überproduktionen erhält man aus der Optimallösung der Relaxation (5) durch einfaches Aufrunden. Das ergibt und somit . Zum -fachen dieser Ungleichung addieren wir die gemäß Voraussetzung gültige Ungleichung und erhalten . Division durch liefert die Behauptung. Stets gilt , so dass für theoretische Untersuchungen die Betrachtung residualer Instanzen ausreicht. Insbesondere kann man aus einer Optimallösung des ganzzahligen Problems (2)–(4) für eine für konstruieren, falls die Ganzzahl-Aufrundungseigenschaft erfüllt. Kennt man eine gute ganzzahlige Lösung für , dann auch für . Leider gilt dies nicht stets auch für die Optimalität, wie folgendes Gegenbeispiel zeigt: hat eine eindeutige Optimallösung der Relaxation (5), nämlich für , wobei die (in der Relaxation) verschnittfreien Varianten , , und zugrunde gelegt wurden. Es gilt und , wobei es mehrere verschiedene Optimallösungen für das ganzzahlige Problem (2)–(4) gibt. Die residuale Instanz ist hier eindeutig (mit ), und es gilt , so dass das Abtrennen des ganzzahligen Anteils von der Optimallösung der stetigen Relaxation zu einer Erhöhung der Lücke führte. Bei Verwendung der Relaxation (6) wäre dieser Effekt nicht eingetreten. Doch bei Vorgabe des Bedarfsvektors vermag auch die Einschränkung auf eigentliche Varianten diese Unannehmlichkeit nicht zu verhindern. Ein Näherungsalgorithmus mittels Relaxation Soll das ganzzahlige Problem (2)–(4) exakt gelöst werden, erweist es sich oft als vorteilhaft, gelegentlich innerhalb der Optimierung mit Hilfe eines Näherungsverfahrens nach guten zulässigen Lösungen zu suchen. Eine entsprechende Verfahrensskizze ist diese: Löse die Relaxation (6) und trenne den ganzzahligen Anteil ab. Es verbleibe eine residuale Instanz . Wähle von den verbliebenen Varianten eine mit maximaler Häufigkeit. Diese Variante sei . Bei streiche überzählige Teile aus . Ergänze, wenn möglich, weitere noch zuzuschneidende Teile in . Falls der Nullvektor ist, ist nichts mehr zu schneiden, halt. Solange es möglich ist, ersetze in jeweils ein Teil durch ein größeres noch zu schneidendes. Schneide die Variante möglichst oft zu, passe die Bedarfszahlen an und gehe zum Punkt 2. Der Aufwand für Punkt 6 ist mit abzuschätzen. Die in Punkt 1 ausgewählten Varianten können vor dem Abtrennen mitunter auch wie in den Punkten 2–7 nachbearbeitet werden. Die so verbesserte Heuristik fand bei pseudozufällig erzeugten Testinstanzen oft Optimallösungen. Falls der erzielte Zielfunktionswert für das ganzzahlige Problem (2)–(4) doch größer als wird, kann in Punkt 2 auch noch einmal die Relaxation (6) gelöst werden. Die Farley-Schranke Um eine gute zulässige Lösung des ganzzahligen Zuschnittproblems (2)–(4) zu ermitteln, genügt manchmal eine zulässige, nicht optimale Lösung der Relaxation (5) oder (6). Falls man aus diesem Grunde die Optimierung vorzeitig abbrechen möchte, zum Beispiel weil die erzielten Zielfunktionswerte zu stagnieren beginnen, benötigt man dennoch Gewähr, dass man tatsächlich nahe am Optimum ist. Dies leistet eine untere Schranke nach A. A. Farley, die auf der Dualitätstheorie der linearen Optimierung beruht. Das zur stetigen Relaxation (5) gehörige duale Optimierungsproblem lautet bei für jede zulässige Variante . Jedes , welches zum zulässigen Bereich des dualen Problems gehört, liefert somit eine untere Schranke für den optimalen Zielfunktionswert. Ein geeignetes findet man leicht, sobald das Rucksackproblem (7) gelöst wurde. Die ermittelte Variante sei . Gilt , liegt eine Optimallösung der Relaxation vor. Andernfalls setzen wir und erhalten für jede beliebige zulässige Zuschnittvariante die Abschätzung laut Annahme über . Folglich ist das gewählte für die duale Aufgabe zulässig. Daraus ergibt sich die gewünschte untere Schranke , die sich bei Fortsetzung der Optimierung dem optimalen Zielfunktionswert der Relaxation nähert. Analog sieht die Schranke für (6) aus. Beispiel: Für die Instanz wurden oben die berechneten Simplexschemata angegeben. Vor dem ersten Austausch galten und sowie , also . Nach dem Eintauschen der Variante in die Basis wurden , und ermittelt. Folglich sinkt die untere Schranke vorübergehend auf und wächst nicht monoton. Für mehrdimensionale Zuschnittprobleme kann eine entsprechende untere Schranke ebenso aufgestellt werden. Gemäß A. A. Farley ermöglicht diese Schranke die Einsparung vieler Spaltengenerierungs- und Austauschzyklen, ohne Gefahr zu laufen, weit vom Optimum entfernt zu sein. (k,k+1)-Instanzen Sei eine positive ganze Zahl. Die Instanz heiße -Instanz, falls für alle zutrifft, also jedes Teil genau - oder -mal (zuzüglich Verschnitt) in das Ausgangsmaterial passt. Das Studium dieser Instanzen ermöglicht einerseits den Bau besonders bösartiger Beispiele, andererseits Abschätzungen – auch im allgemeinen Fall – für die Lücke . Umfangreiche Untersuchungen enthält, während die diesbezüglichen Beiträge in und Vorläufer darstellen. Unter einer -Teile-Variante verstehen wir eine beliebige zulässige Zuschnittvariante , mit der genau Teile zugeschnitten werden (), d. h. , wobei für den aus lauter Einsen bestehenden Einsvektor steht. (k,k+1)-Instanzen mit Δ=1 Offensichtlich gilt bei die Ganzzahl-Aufrundungseigenschaft, weil nur möglich ist. Bei ändert sich die Situation jedoch grundlegend. Ist , ergibt sich und für die Instanz , wobei die Teile nicht nach der Länge geordnet wurden. Verwendet man in der stetigen Relaxation (5) die verschnittfreien Varianten , und je -mal, die Varianten , und je -mal so bestätigt man schnell . Für folgende Instanzen gilt jeweils und : : : : Es gibt keine (2,3)-Instanz mit . Dagegen gilt und für die Instanzen bei bei . Komplementäre Instanzen Zuerst betrachten wir (2,3)-Instanzen mit . Mit der Festlegung für alle erhalten wir eine (2,3)-Instanz mit , die wir komplementäre Instanz zu nennen. Die zu komplementäre Instanz ist wieder . Ist eine (für ) verschnittfreie Variante mit genau drei Teilen, ist auch für zulässig und verschnittfrei. Dagegen entsprechen verschnittbehafteten 3-Teile-Varianten in unzulässige in und umgekehrt. Folglich kann man nicht von Optimallösungen der Aufgaben (2)–(4) oder (5) oder (6) für auf die entsprechenden Optimallösungen für schließen. Dennoch fällt die Verallgemeinerung auf -Instanzen nicht schwer. Gegeben sei nun eine -Instanz mit , für die zutrifft, in der also in jeder Optimallösung der stetigen Relaxation nur verschnittfreie Varianten mit je Teilen in positiver Häufigkeit verwendet werden. Dann ist für hinreichend großes die Instanz eine zu äquivalente Instanz, denn in -Instanzen sind alle Varianten mit höchstens Teilen zulässig und alle Varianten mit mindestens Teilen unzulässig, während der Verschnitt einer -Teile-Variante in ebenso groß wie in ist. Für unsere Zwecke reicht die Forderung aus. Die zu komplementäre Instanz definieren wir als Dabei braucht nicht notwendig -Instanz zu sein. Hat die Instanz eine Lücke , gilt das auch für fast alle in der Instanz , da es nur endlich viele Kombinationen von höchstens Teilen gibt, die eine verschnittfreie Variante ergeben. Folglich erfüllt auch die komplementäre Instanz , für fast alle , die der oben angegebenen Bedingung genügen, die Ganzzahl-Aufrundungseigenschaft nicht. Beispiel: weist für alle eine Lücke 1 auf, und es gilt . Für alle ist eine (2,3)-Instanz. Die zugehörige komplementäre Instanz lautet und besitzt für alle ebenfalls die Lücke 1. Für ist auch eine (2,3)-Instanz. (2,3)-Instanzen mit auffälligen Nennern Bevor (2,3)-Instanzen mit Lücke angegeben werden, demonstrieren wir, wie vielfältig der Nenner von werden kann. Dazu sei beliebig gewählt. Die Festlegungen , , und , für liefern eine (2,3)-Instanz mit . Der Bruch kann nur durch 9 gekürzt werden. Dagegen ergeben sich bei der (2,3)-Instanz für die optimalen Zielfunktionswerte und für die stetige Relaxation (5), was zur Vermutung führt. Der Beweis steht noch aus. Für sind Zähler und Nenner teilerfremde natürliche Zahlen. Variiert man die Längen geringfügig oder streicht man einzelne Teile aus der Instanz, können sich noch ganz andere Nenner ergeben, zum Beispiel: , , , , , (2,3)-Instanzen mit Δ>1 Verknüpft man die Instanzen mit den Zweierpotenznennern und diejenigen mit Lücke 1 in geeigneter Weise, entstehen nach wenigen weiteren Umgestaltungen (2,3)-Instanzen mit Lücke , nämlich , , , , . Die Instanzen mit den ungeraden Nennern eignen sich für diese Konstruktion nur bedingt, weil sie verhältnismäßig viel Verschnitt verursachen. Rückführung auf einfachere Instanzen Um obere Schranken für die Lücke nachzuweisen, erscheint es aufgrund vorstehender Ergebnisse notwendig, die Instanzen zu vereinfachen. Dies ist leicht möglich. Um nicht den Rahmen zu sprengen, verzichten wir hier auf Beweise. Sei eine beliebige -Instanz. Bezeichnen und die Häufigkeiten aller -Teile-Varianten in Optimallösungen der Relaxation (5) bzw. des ganzzahligen Problems (2)–(4), gilt . Gilt , können die größten Teile in -Teile-Varianten zugeschnitten werden, und nach deren Abtrennung verbleibt eine Instanz mit und . Im Gegensatz zu den residualen Instanzen wächst hier die Lücke nicht. Mit jedem weiteren Abtrennen des jeweils größten verbliebenen Teils nimmt der optimale Zielfunktionswert der stetigen Relaxation (5) nochmals um ab. Daraus ergibt sich eine wichtige Folgerung: Jede -Instanz mit kann auf eine Instanz mit , und reduziert werden. Das heißt insbesondere: Ist so, dass für alle -Instanzen mit die modifizierte Ganzzahl-Aufrundungseigenschaft (MIRUP) gilt, dann auch für alle -Instanzen mit . Beispiel: In der (2,3)-Instanz gilt und , also . In Optimallösungen des ganzzahligen Problems (2)–(4) kommen folglich mindestens zwei Varianten mit je höchstens zwei Teilen vor. Vier Teile der Länge 4097 werden in zwei 2-Teile-Varianten zugeschnitten. In der entstehenden Instanz gilt und . Schneidet man nun die beiden größten Teile, also mit Längen 4097 und 3073, in einer gesonderten Variante, sinkt der optimale Zielfunktionswert der Relaxation (5) auf . Trennt man nun noch zwei Teile der Länge 3073 ab, ergibt sich für die Restinstanz . Dennoch gibt es viele nicht ganzzahlige Optimallösungen der Relaxation (5). Gemäß der Folgerung bedeutet es keine Einschränkung, wenn ab hier nur noch -Instanzen mit und betrachtet werden. Ohne Beschränkung der Allgemeinheit seien die Teile nach fallenden Längen geordnet, also . Die Anzahl aller zuzuschneidenden Teile mit Länge oberhalb sei . Für bezeichne jeweils die Gesamthäufigkeit aller in einer vorliegenden Optimallösung der stetigen Relaxation (5) vorkommenden Varianten, die genau Teile aus (und größere Teile) enthalten. Aufgrund dieser Einteilung der Teile in größere und kleinere ergibt sich und , also auch . Gilt , können die größten Teile aus in einer abgesonderten Variante geschnitten werden, und die Abtrennung dieser Variante ändert die Lücke nicht. Unter Umständen findet man anstelle dieser eine zulässige -Teile-Variante, die ein größeres Teil enthält, aber keine kleineren. Dann kann die Reduktion eventuell häufiger vorgenommen werden. Beispiel: In der (2,3)-Instanz gilt , so dass gewiss siebenmal je drei Teile der Länge 2731 in einer Variante abgetrennt werden können. Diese Variante wird aber durch und dominiert. Diese beiden verschnittfreien Varianten können zwei- bzw. fünfmal zugeschnitten werden, ohne Überproduktion zu erzielen. In der erhaltenen Instanz gilt wieder , und nun kann die Variante einmal abgetrennt werden. (Teil 5 war schon verbraucht.) Das führt auf die Instanz mit und , so dass noch neunmal die Variante abzutrennen ist, ohne die Lücke zu ändern. Auf der Grundlage obiger Aussagen ist keine weitere Reduktion mehr möglich. Mittels Heuristiken findet man nun leicht eine ganzzahlige Optimallösung, und . A-priori-Zulässigkeit von Varianten Seien , , und . In der -Instanz mit seien und für wieder wie im vorigen Abschnitt definiert. Die Zuschnittvariante ist unabhängig von den Längen und zulässig, kurz geschrieben, wenn eine der folgenden Bedingungen erfüllt ist: Bei gilt auch . Gemäß den Reduktionen im vorigen Abschnitt kann dies erzwungen werden, auch und . Dies wird im Folgenden vorausgesetzt, wenn obige sehr technische Aussagen auf (2,3)-Instanzen angewandt werden. Dann gilt auch . Unter den oben angegebenen Voraussetzungen ist in einer (2,3)-Instanz die Variante zulässig, kurz geschrieben, wenn eine der folgenden Bedingungen zutrifft: (a) oder (b) oder (c) (erfordert Fallunterscheidung bezüglich ) (d) Die gemäß diesen hinreichenden Kriterien zulässigen Zuschnittvarianten sollen (a)-, (b)-, (c)- bzw. (d)-Variante heißen. Von diesen Kriterien kann keins durch die anderen ausgedrückt werden, obwohl jede (d)-Variante mit auch (a)-Variante ist und die (a)-Varianten die (b)-Varianten dominieren. Bedingung (a) ist aber bei nicht anwendbar. (a)-Varianten sind bei am günstigsten, (c)-Varianten bei , weil wird. Beispiel: Seien und . Bei findet man z. B. die (a)-Variante (5,16,26), denn . (d) hätte wegen kein Ergebnis gebracht. Eine (b)-Variante ist (2,18,27), denn . Wegen durfte (a) nicht benutzt werden. Bei findet man wegen keine (a)- und (b)-Varianten. Eine (d)-Variante ist (9,10,27), denn . Wegen (gemäß Reduktionen) und ist und daher (9,19,21) eine (c)-Variante, denn und . Hier hätte das Kriterium (d) wegen versagt. Nun seien , und gegeben. Dann gilt und , so dass es die (a)-Varianten (1,4,6), (2,3,6) und (3,4,5) gibt. Wäre , könnte man schnell einen Widerspruch konstruieren oder zeigen, dass (2,4,5) die einzige verwendete Variante für Teil 2 in der Relaxation (5) wäre. Beide Fälle ergeben . Aufstellung ganzzahliger Schnittpläne Es kommt nicht nur darauf an, die Zulässigkeit einzelner Zuschnittvarianten nachzuweisen, sondern dass kein nur einmal vorhandenes Teil in mehreren Varianten verwendet wird. Wieder betrachten wir nur -Instanzen mit , in denen die obigen Reduktionen vorgenommen wurden, wo also auch und gilt. Alleine von , und abhängige Abschätzungen für werden angegeben. Zwei Zuschnittvarianten sollen unkorreliert heißen, wenn sie eigentlich sind und kein Teil gleichzeitig in beiden Varianten vorkommt, also gilt. Sei die Anzahl derjenigen Teile aus , die nicht gleichzeitig in paarweise unkorrelierten -Teile-Varianten untergebracht werden können, einerlei welcher Plan für das Problem (2)–(4) zugrunde gelegt wird. Dann gilt: mit , speziell bei . Daraus folgt für (2,3)-Instanzen mit sofort MIRUP, also . Bei kann man geeignete (a)- und/oder (d)-Varianten auswählen, so dass höchstens sechs große Teile übrig bleiben, was auch MIRUP ergibt. Zum Beispiel gibt es bei , stets die (d)-Varianten (7,17,30), (8,18,29), (9,19,28), (10,20,27), (11,21,26), (12,22,25) und (13,23,24). Bei ist damit alles gezeigt, ansonsten ersetzt man die erste Variante durch die (a)-Variante (7,14,30) und bei noch weitere Varianten entsprechend. Dagegen kann alleine mittels obiger Aussagen MIRUP im Fall nicht mehr bewiesen werden. Aussagen zum allgemeinen Fall Inwieweit die Lücke für eine beliebige Instanz beschränkt ist, konnte noch nicht abschließend erforscht werden. So blieb beispielsweise noch offen, ob durch eine Konstante beschränkt ist oder linear mit wachsen kann oder etwa eine Abschätzung der Art mit einer Konstante für alle Instanzen des eindimensionalen Zuschnittproblems zutrifft. Aus der Betrachtung residualer Instanzen folgt die fast triviale Aussage . Deutlich komplizierter ist der Beweis für die Abschätzungen und sowie der Nachweis der Ganzzahl-Aufrundungseigenschaft für alle Instanzen mit . Dagegen ist es eine leichte Übung, nachzuweisen, falls ganzzahlig ist. Ferner konnte MIRUP, also , für folgende Fälle bewiesen werden: , wenn außerdem alle Teile größer als ein Viertel des Ausgangsmaterials sind. Dazu diente auch folgende Aussage, für die eine gute Vorlage enthielt: Lemma: Über die Instanz werden für alle , ferner sowie und vorausgesetzt. Gilt , so sei , andernfalls und . Dann besitzen die Instanzen und die gleiche Lücke , d. h., beim einmaligen Zuschneiden des größten Teils und des ggf. größten dazu passenden Teils ändert sich unter den angegebenen Voraussetzungen die Lücke nicht. Keine der Voraussetzungen kann weggelassen werden, wie die Gegenbeispiele und mit und zeigen. Das MAXGAP-Problem; Konstruktionssätze für große Δ Das MAXGAP-Problem (gap ‹engl.› Lücke, Schlucht, Aussparung, …) lautet: Man finde Instanzen des eindimensionalen Zuschnittproblems (2)–(4) mit möglichst großer Lücke . Die größte bisher (Stand 2007) erreichte Lücke beträgt . Eine Instanz mit und ist diese: , Der Nachweis, dass die Ganzzahl-Aufrundungseigenschaft nicht gilt, erfolgte mit Schnittebenenverfahren. Sollte für unbeschränkt sein, kann das MAXGAP-Problem sinnvollerweise wie folgt abgeändert werden: Man bestimme Instanzen mit möglichst großer Lücke zu vorgegebener oberer Schranke für die Anzahl verschiedener Teile. Inzwischen konnten mit Hilfe zweier Konstruktionssätze unter anderem Instanzen mit folgenden Werten für gefunden werden: Erster Konstruktionssatz: Die Instanz mit rationalen Längen weise die Lücke auf. Sei . Ferner sei so gewählt, dass für jeden Vektor mit und auch gilt. Dann besitzt die Instanz eine Lücke größer oder gleich . Sind alle Daten ganzzahlig, erfüllt stets die Voraussetzungen, da jede unzulässige Variante mindestens eine Einheit zu viel Material benötigt. Die Anwendung dieses Konstruktionssatzes mit auf die Instanzen , und das am Anfang angegebene illustrierte Beispiel, die die Lücken , bzw. aufweisen, ergibt die Instanzen , und mit den Lücken , und . Vor weiteren Konstruktionen für Instanzen mit großer Lücke erklären wir zusammengesetzte Instanzen. Für zwei Instanzen und gelte zunächst . In diesem Fall bedeutet die zusammengesetzte Instanz den Auftrag, alle Teile aus und in den jeweils geforderten Stückzahlen aus dem einheitlichen Material der Länge zuzuschneiden. In der Situation werden alle Längen in einer Instanz mit einer geeigneten Konstante multipliziert. Man kann auch beide Instanzen so anpassen. Bis auf Äquivalenz ergibt sich damit dasselbe. Beispiel: ist genau die obige mit der Abbildung illustrierte Beispielinstanz . Hier addieren sich die Lücken 0,1 und 1,0 zu 1,1. Spezielle parametrisierte Instanzen mit Lücke sind die folgenden: Diese Instanzen erweisen sich als nützlich zur Konstruktion großer Lücken. Die Teilelängen der beiden ersten Instanzen nähern sich für , während in der dritten Instanz das vierte bis neunte Teil relativ klein sind. Für beliebige mit gibt es natürliche Zahlen , , so dass für zutrifft. Analoges gilt für . Zweiter Konstruktionssatz: In der Instanz mit und ist ausdrücklich auch erlaubt. Erhöht man um 1, so nehme der optimale Zielfunktionswert der ganzzahligen Aufgabe (2)–(4) zu. Dann gibt es natürliche Zahlen , mit , so dass die zusammengesetzte Instanzen und die Lücke aufweisen, vgl. auch Beispiel: Die Instanz erfüllt die Voraussetzungen, denn in keiner Optimallösung des Problems (2)–(4) findet sich eine Zuschnittvariante mit mindestens zwölf Einheiten Verschnitt. Mit entsteht die Instanz mit Lücke . Die Annahme bedeutet wegen , dass der größtmögliche Verschnitt in einer Variante, die in einer Optimallösung des ganzzahligen Problems (2)–(4) in positiver Häufigkeit vorkommen kann, kleiner ist als die Hälfte des kleinsten Teils in der Instanz. Diese Bedingung zu erfüllen, stellt die Hauptschwierigkeit bei der Anwendung des zweiten Konstruktionssatzes dar. So genügt es nicht, wenn der größtmögliche Verschnitt genau halb so groß wie das kleinste Teil ist. Zum Beispiel ist die Lücke nicht mit konstruierbar, sondern erfordert einige Teile mehr. Die Idee für obige Instanz mit 34 verschiedenen Teilen und einige verwandte Instanzen mit gleicher Lücke, aber einer geringeren Anzahl unterschiedlicher Teile, besteht in der Zusammensetzung mehrerer der oben angegebenen parametrisierten Baustein-Instanzen, zu denen noch vier Teile der Länge hinzugefügt werden. Die Längen der Teile Nr. 4–28 verhalten sich zur Länge des Ausgangsmaterials ungefähr wie , , , und . Betrachtet man annähernd gleich lange Teile als identisch, entsteht eine Modellinstanz, die die Voraussetzungen des zweiten Konstruktionssatzes erfüllt, nämlich mit der Zusatzbedingung, dass die gemäß verschnittfreien Varianten je einmal nicht benutzt werden dürfen. In Optimallösungen des ganzzahligen Problems (2)–(4) für die Modellinstanz wird der Gesamtverschnitt von neun Längeneinheiten in oder aufgeteilt, und wegen ist die Verschnittbedingung erfüllt. Anstelle dieser Plausibilitätserklärung müssen für die fertig konstruierten Instanzen exakte Verfahren eingesetzt werden, um zu beweisen. Im Gegensatz zum ersten Konstruktionssatz ist der zweite kaum wiederholt auf eine Instanz anzuwenden, da ein kleines Teil, welches in eine andere Variante verlegt werden kann, zu großen Verschnitt hinterlässt. Dafür konnten mit dem zweiten Konstruktionssatz Lücken bis aufgebaut werden; mit dem ersten wurden es . Um die Anzahl notwendiger verschiedener Teilelängen zu reduzieren und dennoch dieselbe Lücke zu erhalten, könnte man versuchen, die entsprechenden Baustein-Instanzen geeignet zu verändern. So entstand die Instanz aus im Wesentlichen dadurch, dass gleiche Teilelängen erzwungen wurden. Dies stößt jedoch auf Hindernisse, wie folgendes Lemma aus zeigt: Lemma: In der Instanz mit seien die Zuschnittvarianten , , , , und zulässig. Dann gilt , , , , . Gilt außerdem , so folgt , , , und . Minimale Instanzen ohne Ganzzahl-Aufrundungseigenschaft Wir setzen ohne Beschränkung der Allgemeinheit voraus, dass sämtliche Daten ganzzahlig sind. Dann kann die Minimalität einer Instanz mit immer noch verschieden verstanden werden: minimale Anzahl verschiedener Teilelängen: mit ist so eine Instanz. Wegen musste sein. minimale Länge des Ausgangsmaterials: Bisher (Stand 2007) ist hier nur mit bekannt; bei der Zusatzforderung wäre zum Beispiel zu nennen. Ob auch kleinere als 16 bzw. 18 möglich sind, ist noch offen. Im Teilbarkeitsfall wird . minimale Anzahl insgesamt zuzuschneidender Teile oder anders ausgedrückt, minimale Teilezahl im Bin-Pack-Problem: Man kann mit einer längeren Fallunterscheidung zeigen, dass mindestens fünf Teile benötigt werden und in diesem Falle wird. Entsprechende Instanzen sind und , jeweils mit . Online-Optimierung Bei Online-Optimierungsproblemen werden die Daten erst im Laufe der Optimierung bekanntgegeben, zum Beispiel wenn kontinuierlich ein Prozess optimal gesteuert werden soll. In so einem Falle wird nicht nur verlangt, eine möglichst gute zulässige Lösung in Echtzeit zu liefern, sondern die Teile können noch nicht einmal vorher nach ihrer Größe sortiert werden. Mitunter dürfen wenige Teile zurückgestellt werden, so dass zumindest für diese Teile einige Freiheit besteht. Es versteht sich von selbst, dass bei Online-Optimierungsproblemen eine exakte Optimierung ausgeschlossen ist und man sich mit schnellen Heuristiken begnügen muss. Einen Überblick über Näherungsalgorithmen und ihre Güte gibt zum Beispiel. Einige Ergebnisse aus diesem Übersichtsartikel werden im Folgenden vorgestellt. Das Problem (2)–(4) werde mit dem Approximationsalgorithmus bearbeitet. Für eine beliebige Instanz sei der ermittelte Zielfunktionswert . Ein absolutes Güteverhältnis im ungünstigsten Fall ergibt sich zu . Für eine asymptotische Güteeinschätzung werden Folgen von Instanzen mit betrachtet. Der entsprechende Limes superior des Verhältnisses werde mit bezeichnet. Die Heuristik Next Fit (NF), stets nur eine Zuschnitt- oder Packvariante offen zu halten und, falls das nächste Teil nicht mehr hineinpasst, mit der Eröffnung einer neuen Packvariante die letzte zu schließen, also für kein weiteres Teil mehr zu verwenden, ergibt im ungünstigsten Fall für fast als erzielten Zielfunktionswert, etwa wenn immer abwechselnd Teile der Größen und mit sehr kleinem gepackt werden sollen. Folglich ist . Die Heuristiken First Fit (FF) und Best Fit (BF) sind ungefähr gleich gut; sowohl als auch können sich für verschiedene Instanzen ergeben. Beiden Heuristiken ist gemeinsam, dass alle bereits angefangenen Varianten für noch zu packende Teile verwendet werden dürfen, sofern das Teil passt. Bei First Fit wird die erste passende Packvariante gewählt, bei Best Fit eine mit minimalem verbleibendem freien Platz. Eine besonders ungünstige Folge zu packender Teile für diese Heuristiken liegt beispielsweise vor, wenn wiederholt Teile mit den Längen mit sehr kleinem positivem in der Reihenfolge monoton wachsender Größe gepackt werden sollen. Ein noch ungünstigeres Beispiel enthält mit (beliebig) und . Da gleichzeitig gemäß jenem Artikel und für jede Instanz gilt, folgt . Any Fit (AF) heiße eine Heuristik, die neue Packvarianten erst beginnt, wenn das nächste zu packende Teil in keine vorherige Variante passt. Für beliebige solche Online-Heuristiken gilt . Falls immer höchstens Packvarianten gleichzeitig offen sein dürfen, wobei eine endliche Konstante ist, schränkt das die möglichen Heuristiken ein. Unter dieser Bedingung wird für Online-Algorithmen stets . Um ein besseres asymptotisches Verhalten zu gewährleisten, braucht man folglich Heuristiken, die beliebig viele Varianten offenhalten dürfen und in Abhängigkeit von der Teilegröße bereit sind, eine neue Packvariante zu beginnen, obwohl das nächste zu packende Teil noch in eine bereits vorhandene Variante hineinpasst. Auch ohne Sortierung kann auf diese Weise ein asymptotisches Verhältnis erzielt werden, jedoch existiert kein Online-Algorithmus mit . Bessere Verhältnisse können erzielt werden, wenn die zu packenden Teile relativ klein gegenüber sind. Werden die Teile vor Optimierungsbeginn nach monoton fallender Größe sortiert, kann nur noch von Offline-Optimierung gesprochen werden. Für die zu First Fit und Best Fit gehörenden entsprechenden Heuristiken gilt . Die absolute Güteeinschätzung ist hier . Mit verfeinerten Algorithmen kann nach der Sortierung das asymptotische Verhältnis beliebig dem Wert 1 genähert werden, wobei weiterhin lineare Zeit erforderlich ist, d. h., für alle gibt es Approximationsalgorithmen mit . Darüber hinaus existieren Approximationsalgorithmen mit polynomialer Komplexität und . Allerdings wächst die absolute Differenz bei solchen Algorithmen stark an. Die vorstehenden Aussagen beleuchten jeweils das Verhalten im ungünstigsten Fall. Soll der durchschnittliche Fall untersucht werden, sind zusätzliche Annahmen über die eingehenden Zufallsgrößen erforderlich. Unter geeigneten Voraussetzungen sind Aussagen über gewisse Erwartungswerte möglich. Verallgemeinerungen und Erweiterungen Falls mehrere verschiedene Materialien mit Längen und Preisen gegeben sind, sind anstelle des Spaltengenerierungsproblems (7) mehrere Rucksackprobleme zu lösen, nämlich mit Rucksackkapazität für . Eine Verbesserung des Zielfunktionswertes der stetigen Relaxation ist möglich, wenn wird und keine Entartung vorliegt. Auch Vorratsbeschränkungen für gewisse Materialien können leicht in das lineare Optimierungsmodell eingearbeitet werden. Um eine zulässige Startlösung zu erhalten, borgt man sich gegebenenfalls fiktives Ausgangsmaterial ausreichender Länge und mit sehr hohem Preis. Wenn trotz Vorratsbeschränkungen das Problem lösbar ist, werden alle fiktiven Materialien durch tatsächlich vorhandene ersetzt. Diese Idee ist auch für Plattenzuschnittprobleme und ähnliche anwendbar. Nun kann umgekehrt gefragt werden, welche Vorräte verschieden langen Ausgangsmaterials gekauft werden müssen, um mehrere Zuschnittaufträge möglichst billig erledigen zu können. Dabei wird angenommen, dass längere Stücke Material stets teurer sind als kürzere und aus organisatorischen Gründen die Zuschnittaufträge nicht vermischt werden dürfen. Das heißt, aus einem Stück Material dürfen Teile immer nur für einen Auftrag geschnitten werden. Dieses Sortimentsproblem wird einschließlich numerischer Experimente in der Habilitationsschrift behandelt. Eine andere Verallgemeinerung des eindimensionalen Zuschnittproblems ist das mehrdimensionale Vektorpackproblem, das aus Planungsaufgaben entstehen kann. Anstelle der Zulässigkeitsbedingung (1) werden hier mehrere derartige Bedingungen gleichzeitig gestellt, zum Beispiel dass eine gewisse geometrische Länge und zugleich ein bestimmtes Gesamtgewicht der gepackten Teile nicht überschritten werden dürfen. Ebenso könnte auch die Zeit zu einer derartigen Nebenbedingung führen. Das Streifenpackproblem stellt eine weitere aus Planungsproblemen entstehende Verallgemeinerung des eindimensionalen Zuschnittproblems dar. Die Aufgabe besteht darin, in ein Rechteck der Abmessungen mit fester Breite und möglichst kleiner Höhe nicht drehbare kleinere Rechtecke mit gegebenen Ausdehnungen und Bedarfszahlen überlappungsfrei zu packen. Wären die Höhen aller anzuordnenden Rechtecke gleich, handelte es sich um das eindimensionale Zuschnittproblem. Eine unmittelbare Anwendung des Streifenpackproblems besteht in der Planung mit einer begrenzt verfügbaren Ressource, so dass nach möglichst kurzer Zeit eine Liste von Aufträgen, die die Ressource benötigen, vollständig abgearbeitet ist. Eine völlig andere Erweiterung des eindimensionalen Zuschnittproblems liegt vor, wenn wegen begrenzten Platzes neben der Zuschnittanlage bei einem Großauftrag gefordert wird, dass stets höchstens verschiedene Teilesorten in Bearbeitung sind, wobei die positive ganze Zahl fest vorgegeben ist. Bevor also die -te Sorte begonnen werden kann, muss eine andere Teilesorte abgeschlossen worden sein. Gesucht wird zusätzlich zu den Zuschnittvarianten eine Reihenfolge, die dafür sorgt, dass möglichst wenig Material verbraucht und die Zusatzbedingung eingehalten wird. Ein Überblick über eine Vielzahl weiterer Pack- und Zuschnittprobleme, für die das eindimensionale Problem (2)–(4) auch als Relaxation dienen kann, wird unter anderem in gegeben. Das "Zuschnittproblem mit Muster-Minimierung" ("cutting stock problem with pattern minimization") ist eine Erweiterung des klassischen Zuschnittproblems. Die Erweiterung "Muster-Minimierung" bedeutet, dass neben dem Rohmaterialverbrauch zusätzlich die Anzahl der verschiedenen verwendeten Zuschnittsvarianten minimiert werden soll. Dieses Problem tritt auf, wenn für jede Zuschnittsvariante einmalige Produktionskosten (Rüstkosten) anfallen, unabhängig davon, wie oft die Variante produziert wird, z. B. wenn für jede Variante, ein Rohr in mehrere Teilstücke zu zersägen, eine eigene Zuschnittschablone benötigt wird. Bezeichnet die Kosten für ein Stück Rohmaterial sowie die einmaligen Kosten, die für jede Zuschnittsvariante anfallen, muss hier an Stelle der Gleichung (2) folgender Wert minimiert werden: Dieses Problem gilt seit den 1960er Jahren als ungelöst. Die Artikel befassen sich mit dem Problem, indem sie die nicht stetige Funktion (8) durch stetige Funktionen approximieren und mit nicht-linearer Optimierung arbeiten. Zur Geschichte Bereits 1939 gab Leonid Witaljewitsch Kantorowitsch ein ganzzahliges Modell für das eindimensionale Zuschnittproblem an, aber die zu seinem Modell gehörende stetige Relaxation ist sehr schwach; sie liefert nur die Materialschranke. Nachdem bis 1960 die Grundlagen der linearen Optimierung, darunter das revidierte Simplexverfahren, bereitgestellt worden waren, veröffentlichten Gilmore und Gomory bereits 1961/63 das Lösungsverfahren für die stetige Relaxation, nämlich die Simplexmethode mit der Spaltengenerierung zu kombinieren. Damit konnten, ausreichend Rechenzeit vorausgesetzt, für nicht zu große Instanzen (fast) optimale Lösungen ermittelt werden. Da aber die Rechenzeit für größere Instanzen inakzeptabel hoch wird, weil viele Simplexschritte notwendig sind und viele Rucksackprobleme gelöst werden müssen, interessierte man sich auch für schnelle Heuristiken und ihre Qualitäten. Dies geschah in den 1970er Jahren. Da immer wieder , also Ganzzahl-Aufrundungseigenschaft beobachtet wurde, drängte sich eine entsprechende Vermutung auf, bis Odile Marcotte 1985 anhand des 3-Matching-Problems nachweisen konnte, dass es Instanzen des eindimensionalen Zuschnittproblems mit Lücke geben müsse. 1986 veröffentlichte Frau Marcotte ein konkretes Beispiel. Da dieses Beispiel aber (ganze) Zahlen zwischen einer und zehn Millionen enthielt, wurde gesagt, so etwas käme in der Praxis nicht vor. Als Fieldhouse 1990 die wesentlich einfachere Instanz angab, sah man zwar ein, dass Derartiges durchaus in der Praxis vorkommt, aber es handele sich um singuläre Einzelfälle. Mit der Angabe unendlich vieler, paarweise nichtäquivalenter Instanzen mit Differenz fiel die falsche These über die Ganzzahl-Aufrundungseigenschaft im eindimensionalen Zuschnittproblem endgültig. Noch Ende des 20. Jahrhunderts gelang es, für die exakte Lösung des ganzzahligen Problems (2)–(4) als Alternative zum Verzweigungsverfahren das Schnittebenenverfahren I von Gomory erfolgreich anzupassen. Inzwischen wurde das Verfahren weiterentwickelt zu Branch and price and cut. Einige noch offene Fragen Neben den Fragestellungen aus der Komplexitätstheorie besteht zum eindimensionalen Zuschnittproblem noch einiger Forschungsbedarf, zum Beispiel: Bezeichnet die Menge aller Instanzen des eindimensionalen Zuschnittproblems mit höchstens verschiedenen Teilelängen, fragen wir nach . Wie kann möglichst scharf abgeschätzt werden? Die gleichen Fragen stellen wir für Spezialfälle wie den Teilbarkeitsfall oder wenn eine ganze Zahl mit für alle existiert oder wenn kein Teil länger als die Hälfte des Ausgangsmaterials ist. Wenn insgesamt genau fünf Teile zuzuschneiden sind, d. h. , kann man ausrechnen, dass von allen diesen Instanzen der Anteil der Instanzen mit Lücke ungefähr ein Zwanzigtausendstel beträgt. Wie groß ist der Anteil, wenn mehr Teile zuzuschneiden sind? Wodurch sind diejenigen Instanzen gekennzeichnet, die eine kleinere Differenz aufweisen als die zugehörigen residualen Instanzen? Für welche minimalen bei durchgängig ganzzahligen Daten kann werden? Sind für kleinere als 30, 18 und 16 möglich? Quellenangaben Weblinks Auf der Seite der Interessengruppe Cutting and Packing an der TU Dresden finden Interessenten unter anderem 53 schwierige Testbeispiele ohne Ganzzahl-Aufrundungseigenschaft zum eindimensionalen Zuschnittproblem. winvedaga: Lösung des Eindimensionalen Zuschnittproblems mit Hilfe des Approximationsalgorithmus Lineare Optimierung
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Ing. Vladimír Nedoma (* 1888, Praha) byl autobusový dopravce ze Zbraslavi. Provozoval za první republiky autobusovou dopravu a je známý jako výrazný konkurent Elektrických podniků hl. m. Prahy, který měl kvůli nedodržování předpisů regulujících provozování autobusové dopravy opakované konflikty s úřady. Je označován za jednoho z předních a zároveň nejkontroverznějších soukromých autobusových dopravců v Praze v období před druhou světovou válkou. Než začal podnikat v autobusové dopravě, byl dílovedoucím autosprávkárny na Zbraslavi II č. 79, kde pak sídlil i jeho autobusový podnik. Historie autobusové dopravy 4. března 1927 požádal o koncesi pro "periodickou dopravu osob automobilovými omnibusy" v trase Praha (Reprezentační dům) – Smíchov – Zlíchov – Chuchle – Zbraslav – Štěchovice – Slapy – Nový Knín. Ještě před rozhodnutím o žádosti začal 25. června 1927 bez koncese provozovat dopravu v trase Praha – Zbraslav. O koncesi na trasu Praha – Zbraslav žádaly v březnu 1927 též Elektrické podniky hl. m. Prahy, a proti neoprávněnému provozování dopravy pak protestovaly. Zemská správa politická v Praze však v září 1927 žádost o koncesi projednala a 29. listopadu 1927 Nedomovi požadovanou koncesi udělila. Elektrickým podnikům zemská správa žádost o koncesi zamítla s odůvodněním, že již je o dopravu na Zbraslav dostatečně postaráno. Koncem roku 1930 jezdily dva autobusy v trase Karlovo náměstí – Štěchovice, jeden v trase Karlovo náměstí – Nový Knín, dva v trase Jungmannovo náměstí – Zbraslav a jeden údajně v trase Zlíchov – Velká Chuchle. Vzhledem k tomu, že v koncesi byla trasa linky popsaná jen letmo, nedokázaly úřady posoudit, zda linka zajíždějící ve Velké Chuchli až do obce koncesi porušuje. V září 1930 (podle jiného zdroje od roku 1931) byla všeobecně upravena stanoviště dálkových linek v Praze a Nedomovy linky byly zkráceny ke Štefánikovým kasárnám na Smíchově (dnešní náměstí Kinských). Pražský magistrát v roce 1931 vydal vyhlášku, kterou se snažil usměrnit umístění zastávek v Praze, avšak protože vyhláška byla v rozporu s koncesemi vydanými zemskými úřady, Nejvyšší správní soud ji v roce 1934 zrušil. V květnu 1928 začal Nedoma bez koncese provozovat jedním autobusem Škoda autobusovou dopravu z Jungmannova náměstí do Modřan, s jízdným 4 Kč. Elektrické dráhy si opakovaně stěžovaly, například u policejního ředitelství. Nedoma na policejním ředitelství vypověděl, že o koncesi již magistrát dávno požádal. Magistrát se Nedomovou žádostí zabýval v září 1928. O výsledku vyřízení Nedomovy žádosti o koncesi se doklady nedochovaly, ale z pozdějších zpráv se zdá, že byla zamítnuta a Nedoma dopravu pravděpodobně na čas zastavil. V roce 1928 zemský úřad zamítl žádost Elektrických podniků o koncesi na autobusovou dopravu Praha – Zbraslav s odůvodněním, že "jest o dopravu na Zbraslav dostatečně postaráno soukromými podnikateli a autobusy ČSD". Mezitím 18. října 1929 zavedly Elektrické podniky svou vlastní linku "L" z Braníka do Hodkoviček a od 1. června 1930 ji prodloužily do Modřan k čáře potravní daně. Jak popisují Elektrické dráhy ve své další stížnosti u živnostenského referátu z 25. srpna 1930, od 16. srpna 1930 Nedoma začal dvěma autobusy provozovat dopravu v úseku Braník – Modřany, přičemž jeho autobusy zastavují ve všech zastávkách Elektrických podniků včetně konečné v Braníku, ležící na pozemku EP, a jezdí těsně před autobusy Elektrických podniků, čímž mu přebírají cestující. Podobně si Elektrické podniky stěžovaly opakovaně a Nedoma byl živnstenským úřadem pokutován. Jízdní řád platný od 1. září 1930 dokládá, že na Nedomově lince jezdilo ve všední dny 29 párů spojů z Čechovy čtvrti do zastávky U křížku a zpět, a v sobotu odpoledne a v neděli jezdily v patnáctiminutových intervalech v trase z Tylova náměstí do zastávky U křížku. Jízdné bylo sníženo na 2 Kč. Od 30. prosince 1930 na 1,50 Kč, což už bylo obecně považováno za dumpingovou cenu, linka v té době jezdila až do centra Prahy na Jungmannovo náměstí a u Elektrických podniků stála v té době jízda s přestupem 1,70 Kč. V červenci 1931, kdy již Nedomova linka nejezdila, magistrátní živnostenský referát udělil Nedomovi za porušení živnostenského řádu spočívajícího v provozování dopravy bez koncese pokutu 500 Kč. Poté, co Nedoma svou dopravu na modřanské lince zastavil, začal do Modřan, rovněž bez koncese, jezdit čtyřmi autobusy Jan Holub, švagr Vladimíra Nedomy přičemž zachoval i Nedomovy ceníky. Když se úřady začaly jeho případem zabývat, 16. listopadu 1931 Holub provoz této linky zastavil a další své aktivity přesunul do Břevnova (1933 do Ruzyně), kde postupoval podobně. Podle jiného zdroje těchže autorů jezdil Holub v této trase v letech 1928–1930 a po něm v roce 1931 opět a bez koncese Vladimír Nedoma. Koncem roku 1930 nabídl Nedoma svou firmu, zahrnující 8 autobusů, garáže na Zbraslavi, Elektrickým podnikům, a to za 1,43 milionu Kč, a dalších 15 tisíc Kč měsíčně za koncesi na štěchovickou linku a údajné výlučné stanovištní místo ve Velké Chuchli. Elektrické podniky hl. m. Prahy nabídku odmítly. Elektrické podniky kritizovaly Nedomu také za to, že nedodržuje osmihodinovou pracovní dobu řidičů, ale řidiči pracují na témž voze po celý provozní den, a údajně i údržba autobusů byla téměř nulová. Vozy musely mezi linkami různě přejíždět. Po roce 1935 je obecně o autobusové dopravě málo dochovaných dokumentů, protože byla liberalizována a směla být provozována bez koncese. V roce 1936 již linku na Zbraslav provozovaly i Elektrické podniky, přičemž v této relaci si konkurovalo mnoho firem včetně Nedomovy. Další a poslední dochovaný doklad o existenci Nedomovy linky a podniku se vztahuje k datu 10. října 1939, kdy byl z důvodů válečných opatření omezen provoz na dva páry spojů na lince Praha – Nový Knín, přičemž navíc byly zřízeny spoje Nový Knín – Davle, Nový Knín – Měchenice, Štěchovice – Měchenice, Slapy – Měchenice a Čím – Měchenice s návazností na železnici. Nadále byla v provozu linka Praha – Zbraslav, na níž některé spoje končily na náměstí, 5 spojů jezdilo až do Lipan (dnes Lipence) a některé ke Kostrounkům. V Praze končily spoje mimo špičku v Hlubočepích a ve špičce jezdily až na Smíchov na Štefánikovo náměstí. Krom toho 10 párů spojů jezdilo ze Štefánikova náměstí do Velké Chuchle. Podnik a nucená správa Zázemí podniku tvořila nemovitost s adresu Zbraslav II, č. 79 (dnes Žitavského čp. 501), k níž patřila budova s šesti obytnými místnostmi s příslušenstvím, garáž pro 7 autobusů o užitné ploše 24×10 metrů a větší pozemek. Vozový park v roce 1930, kdy Nedoma nabízel podnik k odkoupení, představovalo 8 autobusů: 2 autobusy Škoda 550, 1 autobus Škoda 125, 1 autobus Škoda 505, 1 autobus Praga NO a 1 autobus Fiat 510 karosovaný jako hotelový vůz pro 10 osob. Nedoma dlužil necelých 125 tisíc Kč za čtyři autobusy značky Škoda. Proto byla 27. srpna 1931 na jeho podnik s názvem "Koncesovaná autobusová doprava osob na trati Praha – Zbraslav – Štěchovice – Nový Knín Vladimíra Nedomy" uvalena na základě žaloby výrobce nucená správa. Správcem koncese se stal Otto Pick. Ten navrhl Elektrickým podnikům, aby svými vozy převzaly provoz na lince Praha – Nový Knín na účet vnucené správy a pod dozorem vnuceného správce koncese, přičemž by měsíčně platily 2500 Kč ve prospěch věřitelů Nedomovy firmy. Elektrické podniky návrh odmítly, v té době ještě nebylo vyřízeno jejich odvolání proti zamítnutí koncese na jejich vlastní linku na Zbraslav. Vnucenými správci byli od roku 1931 do roku 1937 (jiný zdroj uvádí jen doklad z roku 1936) Václav Dudek ze Zbraslavi (který v roce 1932 provozoval autobusovou dopravu ze Smíchova do Velké Chuchle), a od roku 1937 Václav Kryml. Zdá se, že kolem roku 1937 podnik pod nucenou správou úspěšně fungoval. Poslední doložená zpráva o podniku je z roku 1939, další jeho osud není znám. Reference Zaniklí autobusoví dopravci v Česku Autobusová doprava v Praze Autobusová doprava ve Středočeském kraji Čeští podnikatelé Lidé ze Zbraslavi Narození v Praze Narození v roce 1888 Osoby s nejistým datem úmrtí Muži
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Way back in January 2011, Peter Rose, editor of the Australian Book Review, described what he looks for in new reviewers. Of course, he's writing about reviewers for a serious journal, but it got me thinking about what blog readers look for when they visit litblogs. And so, I thought I'd ask. I know what I like, but what do you? First, do you look for something different in a litblog review from one in a newspaper? And, whether yay or nay, what do you want in a litblog review? A summary of the plot or a longer description of the story? How important is it to avoid spoilers? Information about the author and other background material to the book? In the review, or as a link to another site such as Wikipedia? An objective analysis of the book in terms of its literary merit or a more conversational chat about what the reviewer likes? Lots of quotes/excerpts, few or none? Essay-style or headings and dot-points? A discussion of well-known easy-to-get books or lesser-known and perhaps harder-to-find ones? Tags/Categories/Labels? If so, what sort of categorising do you find most useful for delving into a blog? Information about where to buy the book? And, for fun, the Oxford comma or not! I know most of these are not necessarily either/or propositions, and you don't have to answer them all, but I'm sure you have preferences. Hmm, Oxford comma, now that's a knotty problem. Why do they call it an Oxford comma when the Brits don't use it, I wonder? Anyway, I think I'll stick with what Mrs Sheedy taught us in Grade 6. No comma before 'and' in a series. That suits my British heritage too LOL. LOL Lisa. I'm like Karen Lee. I use it occasionally to avoid ambiguity … I have noticed overall that commas are disappearing, sometimes to the detriment of sense and rhythm. Fashion in prose! As I was clicking on the link to comment, I was thinking 'Hmm, Oxford comma, now…' so I am still chuckling to see my opening thoughts mirroring Lisa's. However, from there our opinions differ. I often use the Oxford comma to avoid ambiguity. If there is no ambiguity, I leave it out. Ambiguous enough for you? For what it's worth, I'll tell you some of the things I look for (although I don't always do my reviews in the same way). 1) I do like a bit of background on the author, particularly if it relates in any way to the book they have written. 2) I definitely love quotes because that's really how I can tell if it is a book I want to read. 3) A good bit of white space for blogs so it's easy on the eye and, yes, I love a pic sometimes. More than anything, I like to be surprised, just as I so often am by good writers. Instead of reading the latest from an author that is similar to their last book, I like to be blown out of the park by something really NEW. In the same way, I like reviewers who 'mix it up a bit' so I never really know what to expect. Oh, love your answer, Karen Lee. Surprise … That's a good thing to aim for though not so easy to achieve! Agree re comma. And white space is a good thing … Particularly in the electronic environment I think. There are two ways I use reviews: to get a handle on a reading experience of my own, and to get ideas for new reads. I generally google for the first and follow litblogs for the second, but I like it when a blogger's literary journey takes me back to books I've already read. A litblog reviewer can use the perpendicular pronoun, adopt a chatty style, and review a book without necessarily having read anything else by the author; but they still need to draw out what makes a book good or bad in terms of literary merit, as well as infering what happens without spoilers. I like a pithy essay style with short summaries, links to more information about the book and author, and quotes to illustrate a point. Publication details add context, and links to other reviews are useful, but I have my own places to go to buy books. Thanks Judith, that all makes perfect sense to me. Having for so long avoided using the perpendicular pronoun I must say I enjoy being free to be more personal in the blog and be a bit chatty, say why I chose the read the book, why it relates to me, etc. Sometimes I'd like to free myself up a bit more … work in progress! A thought-provoking post, helpful too as we think about what it is that we as book bloggers do offer. I'd like to be informed in a succinct way about the story, a summary, but not spoilers if they would affect a reader's enjoyment of the work. Author's background and relevant info of the work such as awards are important info too I think. While the literary analysis of structure, plot, characterization are helpful, I tend to look for a more personal perspective and the reviewer's own insights, something like a ripple effect. 😉 After all, as Roger Ebert said, 'all reviews are subjective.' And yes, stylish writing can make a review more convincing and enjoyable as well. Thanks for an interesting post, WG. I'm curious to see what other bloggers think. Thanks Arti … when I started blogging I resented avoiding spoilers because I like to explore the "meaning" of a work rather than just "review" it, and it is hard to do that without giving away the ending but I'm gradually working out ways around it! I do like to do some literary analysis because I love to think about "how" writers go about engaging us and achieving their goals. Which is a bit funny really because I have no aims to be a fictional writer myself (that is, a creative writer). But, I like to think about why they chose to tell their story the way they did, if that makes sense. I'm interested too to see what others think … and am glad you've added your voice to the mix. I did like Peter Rose's points – sweetness and grammar – and just had a laugh at the Oxford comma quandary (To my parents, Ayn Rand and God!). I do like your style of reviews as they are I must say. Focussed but flowing, honest and clear, neither harsh nor too laudatory. I also like some quotes, some past links and a sort of quiet consideration of the author's intentions and his or her ability to carry these out. I like to read about language, no spoilers please and at least a couple of characters who might draw me in. A little about the author, just enough. And sorry to harp but having read a range of reviews lately while researching, a simple Sorry, not for me! instead of unbuckled slamming should the reviewer dislike the book. Being a writer always makes me consider the hours and effort involved in producing and finding a publisher for 200-odd pages, so I do prefer fair, not mean. Last week there was a good article in the Guardian about the incestuous rapport between publishers and reviewers in the newspapers VS the validity of Amazon reviews with their full range of dad's praise or utter slaughtering. It's incredible how venomous and cursory some reviewers can be, and ridiculous how some books are praised to the heavens. Thank goodness there are so many great book review blogs for something more measured. Thanks Catherine … I liked the way Peter Rose expressed himself too! Thanks for the compliment (though I wasn't asking for that.) As you know already I'm not one to slam a book. The only books I really disike are those that are cliched and wooden, and I'm pretty good at avoiding those. A book might be overwritten, a little predictable, murky in theme, convoluted in plot or antithetical to my views, but if the work has some sense of originality then I'll give it the time of day. It's partly about respect I think. Thanks Hannah. Yes, the Warning … thanks for bringing that up. In my early bogging days, I did do some spoilers and would give warnings. I think that is a valid way to go as long as the warning is clear and not too close the the actual spoiler. No Oxford comma thanks! I work on loads of different magazines (here in UK) as a freelancer these days and I can happily confirm that not one of them uses it — unless, of course, it is to avoid ambiguity. As for book reviews, can I tell you what I don't want? There seems to be a real fashion among book bloggers for simply copying the blurb verbatim and then writing "my thoughts" underneath. I no longer follow blogs that do this. A bit of info on the author is OK, but it's not of great interest to me (which is probably why I just work it into the body of my reviews). I often wonder where people source this info anyway — most of it is copied word for word from the author's own website or wikipedia. If you're going to do that, then simply provide a link. And does anyone bother to respect the copyright of author photographs? I never see pic credits on them. But this is a great post, Sue. Lately there's been much controversy in press and on Twitter over whether bloggers actually write reviews. I've avoided getting involved in the argument. Many think reviews MUST put the book into context with the author's oeuvre and the literary world as a whole. But I personally don't think this matters: how does knowing that kind of information help you decide whether you want to read the book or not? Thanks kimbofo. Yes, I agree with the Oxford comma … just to avoid ambiguity. I have seen a couple of blogs as you describe and was a little astonished. I didn't go back! I agree with what you say about author info … I tend to not include info but just provide a link, usually to Wikipedia because it usually provides other links and it's not likely to break. As I'm sure you know, I always check copyright and attribute my photos. It's an article of faith for me. I don't really keep up with Twitter and only follow a few anyhow (including you), so I hadn't seen the current controversy, though I have seen discussions in the past. I agree that putting a work in literary context isn't a "must". It might be relevant sometimes depending on a work but blog reviews aren't literary essays OR they are only if the blogger wants them to be. I've never seen that Updike article before but I can see why you like it. It's from the author's perspective, but it's pretty reasonable it seems to me. Thanks for sharing it. I *so* agree with Kim about blogs that copy a plot summary from the press release and then relate their 'thoughts'. It's structurally lazy. Thanks Angela. The responses have all been great … I will do a follow up. – some info about the author but this is not a must. If I've read a book by the same author I will mention that and also if there's some interesting detail as well – when I read "The Yellow Wallpaper and Other Writings" by Charlotte Perkins Gilman I found out that she was the niece of Harriett Beecher Stowe, who wrote "Uncle Tom's Cabin"; I thought that was interesting to mention. – I don't want an academic review, what I want to see is the impression that particular book has left in the mind of the reviewer. Did they like it? If yes, why, if not, why? I'm not fond of very long posts either. If it's something that goes on and on, I might just skip it. – some quotes are fine. It's interesting to see the passages people like in a book, as I think they tell something about the reader. – not really interested in the awards won but there's nothing wrong if they are mentioned. Thanks for joining in Delia. I'm loving everyone's comments … and pretty much agree with all you say here. I think how the blogger came to read the book is often interesting and personalises the review nicely. I think I sometimes don't make the impression the book left on my as clear as I could. In trying to avoid cliches I sometimes say nothing! Must work on that one! This is a great, thought provoking post. I have to confess that I do the dreaded back of the book synopsis, followed by my thoughts, and after having read kimbofo's remarks, I need to give some thoughts on restyling. * A basic summary of the plot, either above review or incorporated into review. I do not like the whole book regurgitated at the beginning of the review. It's boring (to me), and it serves no purpose (IMO) in convincing me to read or not read. Frankly, if I'm reading the book review, it usually because I'm familiar with the author or it's a title I have heard or had recommended. * I like both objective analysis of literary merit and a conversational chat about what the reviewer likes. I tend to lean toward more informal, non-academic reviews, although I avoid reviewers that are just gushing (or complaining) about content. Obviously content is important, but reviewing (again, my opinion) is not about pointing out all the ways I would have done it differently, but rather evaluating whether or not the finished product works or does not. * Some quotes, but they shouldn't be the overwhelming bulk of the review. * Essay vs. Bullets depends on the reviewer I think, and (to some extent) the book reviewed. I have employed that technique when I have several specific points to address, but most of the time I write in essay format. * Keep it short. It is rare that I stick around for a long review, unless the reviewer's writing style grabs me immediately. I think it's important to make your points and be succinct. That is what I gravitate toward reading, and that's the aim when I write my own reviews. * Images? Only if they are illustrative of a point. * Awards? Not an important point for me personally. * Discussion of other works, etc.? Again, if pertinent or illustrative of a particular point. * Tags / Categories / Labels? Yes, yes, yes! * Reading challenges? Not important to me. * I think publication details are important, but they should be brief. Title, author, format, narrator (if audio), isbn, copyright date. This is especially important to me on audiobooks that have several versions, becuase some narrators are vastly better than others. * I hardly ever follow links to other reviews unless I follow the blogger and know them to recommend things that I will like. Otherwise I read reviews on Goodreads, Amazon.com, or wherever. * Unless the book is extremely hard to find, purchase info is unnecessary IMO. Thanks for this post. Loved it. Lots to think about! LOL Laura, thanks for commenting and 'fessing up! I don't think there's necessarily anything wrong with that style…different readers like different types of reviews. Also, it's a matter of why you are writing your blog, what you want to achieve, and how much time you've got to write them. The one great thing about blogging is that none is paying us so we can write exactly (as long as it's legal of course) what we want, the way we want to write it. Thanks for all your reflections on my questions … I think we're building up quite a picture and I might do a follow up post summarising them in a few days (or a week or so!). * avoidance of spoilers or hints of them although a warning beforehand is ok. * Brief publishing info is good but not essential. I really like to know the reviewers thoughts on the book, how they came to choose it, what they liked and honesty about what they didn't. So I guess I'm more into the personal aspect of blogging but I do apprecicate a well researched and thought out review. No matter how well written a review is, I'll only want to read it if I'm likely to want to read the book so I like to read bloggers who have a similar taste in books to me. Now with all those requirements I wonder if I do any of that myself! Thanks Tracey … so glad you decided to join the discussion. It's interesting that a few people would like to know "why" the blogger decided to read the book. I know I like that sort of personal info too … and you're right, in the end we're most likely to read reviews 9and accept a variety of styles and approaches) from bloggers with similar tastes. None or all of the things you listed. Difficult to say for sure but some little phrase or quote or aside that makes me think, I gotta read that! Now that was short and sweet Nicola, but said it all really! Thanks for joining in. Super blog post! Funny how these things work, isn't it? I look for different things from different reviews. If I trust the reviewer then all I look for is a recommendation. Your list certainly makes me think about my own reviews – do you present what you like to see in the reviews of others or do you present what you think other people want to read?? Food for thought! Thanks Justine. I think you make a good point about looking for different things from different reviews. I knew as I was writing it that nothing is cut and dried but I thought it was worth asking the questions and seeing if there were some preferences. I'm rather relieved to find that few people have preferences set in concrete. I'll add a third question to your two: Do you write what you want to record so that you'll remember the book in the future! That was one of the reasons I started … and I felt that the discipline of writing in on a blog than a private journal might better keep me on task! I think it has! I love your added question: 'Do you write what you want to record so that you'll remember the book in the future?' Truthfully, no. I write about the book so that I will remember it because my brain is rapidly turning into a sieve (either because I am getting older or because I have simply read way too many books!). My real motivation for reviewing is that I think I've read enough to be able to recommend (or not) books to other people. I always try to include information on how I came across the book … For example, there was a time when I was Jeanette Winterson mad and any book she wrote was one I had to have … my opinion, clearly biased! I think it's a good indication of the calibre of the reviewer if you can connect with their reading history. As we all know, there are different strokes for different folks! Oh thanks Justine for considering the third question, which I expressed pretty poorly I now see. I like your idea about the bloggers' "reading history". I love that you refer people to your blog for recommendations. I hope they come back and comment later on what they thought! Thanks Stu … I like your point about comparison with like writers. That's sometimes easier said than done but I like the way it can built up a literary picture. Fair point about not finding reviews for lesser known works, such as translated works and those published by smaller publishers. Thanks for asking and thanks for all of you who have already responded. I agree with much of what you have already said. My own additions follow. Not a plot summary or longer description of the story, please. I dislike spoilers greatly. I like to get information in the order the author intends. For non-fiction, I definitely want a summary of main points. I do like some indication of what any book is about—plot, characters, mood, setting, style. I don't think analysis of literary merit is ever "objective." I don't want a review to be judgmental, but some comments on literary aspects of a book are very welcome since I don't always catch them myself. I like a more personal style and response. I like reviews that link the book being discussed to other larger issues or other books—like our discussion of Cather did. I appreciate suggestions for related books. Some well-chosen, generally short quotes are welcome. Essay-style only. Headings and dot-points are not reviews. Long or short, depends on the book. More innovative, creative books and those not reviewed elsewhere need longer reviews. A discussion of lesser-known books. What is easy to find in Australia may not be in the USA. MINOR FEATURES, nice but should not predominate. Ah thanks Marilyn … I think I'm with you about plot. I tend, I think, to give a broad sense but that's all. In my first ever blog review (it was on a group blog), a commenter said it was very interesting but she'd like to know what it was about! I realised I'd said pretty much nothing about that. I edited that post! And "depends on the book" just about sums it all up doesn't it. I like your good point about longer reviews for those not reviewed elsewhere. And sometimes for those well (as in frequently) reviewed books, it can be good to do something different from the traditional review – pick an angle and work on that. Our Cather discussion was great, I agree. A book can have so many different angles to relate it too – the form/genre, the voice, the setting, the themes, etc. Depending on which of these you look at, a book can re related to quite widely divergent works, which I find fascinating. When was it that Oxford claimed the comma? Thirty-five years ago, or thereabouts, I heard it referred to as the "serial comma". I find that I almost never read reviews of fiction. For non-fiction, I want to know what the book is about. If it is a subject I know something about, I want to learn what the book has to say about the subject that I may not have heard before. If it is a subject I know little or nothing of, tell me why I should repair my ignorance. Also, I had a look at the paper that compares Amazon reviews to professional ones, and it appeared to me to make much more modest claims than that they were generally comparable. I have no idea George. There could be a PhD in that! I'm probably a bit like you re reviews – I tend to mostly read reviews AFTER I've read books and written my own. Helps me be fresher … I must read that Amazon review comparison article. Hi Sue, I thought I'd just add some thoughts discussed at the Sydney Writers' Festival. In a session entitled 'Friends Reviewing Friends', Gideon Haigh commented that he believed we have a review culture in Australia but not a critical one. Borrowing a thought from one of our best critics, Geordie Williamson, from another session ('Classic!'), he said that culture is the links between all our books. So my take on things is that to move toward a critical culture one should try to describe not just the book itself but its links to other books, its place in our culture. That means being very well read, of course, but it's a nice aim I think. John. Thanks John … and that's using "critical" in its wider more analytical sense isn't it. I like that sense of "culture" being the link. My problem is that as well as being well-read it also requires a good memory, something I fear I'm losing with time! You ask such good questions! What I like is for the review to be appropriate to the book being reviewed. If the book is fluffy and all plot then I don't expect in-depth character analysis. But if the book has depth, I expect the review to have a bit more depth too. A brief plot summary is nice and I don't really care about spoilers (except for whodunits) becasue I very likely will have forgetten them when I get around to reading the book. I like background information that is relevant to the book. A few quotes are nice especially if the blogger mentions how gorgeous the language is, but I don't like quotes that are really long unless it is somehow unavoidable. Definitely essay-style, mid-length to long-ish. If it starts getting too long I generally think the post maybe should be divided into two. What I like to hear is about the blogger's reading experience and engagement with the book and some analysis of what made the book good or bad for them. I really dislike it if the blogger declares the book horrible or stupid but doesn't bother to examine it any further. Likewise on the gush side. I also like it when a blogger picks out a theme or character or idea and mulls it over a bit. I don't want New York Review of Books-type posts. I like them personal and quirky. As for the Oxford comma, only to solve issues of ambiguity. Another great response, Stefanie. I'm thrilled that people have taken time to express their views on this. Your spoiler comment made me laugh – I can related to that. I take your point on long quotes – it's hard sometimes but I agree that too many too long ones can make me glaze. But, like anything, it depends! I like your point about splitting a review into two. I sometimes do that with Delicious Descriptions but making a real second post is a good way to go I think (in some circumstances!). I read litblogs to see books discussed, rather than reviewed, the difference being that a discussion can concentrate on one tiny thing, it doesn't have to give you a fair overview, and it isn't trying to let you know whether you should buy the book or not — a discussion wants to let you know that the blogger is interested enough to choose this book, out of all the other books in the world, to talk about, and then the reason, and then more reasoning beyond that; it gives you an idea of the blogger's tastes, and then you might be interested without feeling reviewed-at; you know the writer hasn't been working under the pressure of a target. But if you're reviewing then you might as well make it as dense as possible — get it thick and nice, throw in the author's background for good grist, make it "essay-style" not "dot-points" because "dot-points" make me wonder if you think I'm too much of a boofhead to follow an argument without regular signposts (reading, YOU ARE HERE, with big arrows, NOW WE DISCUSS THEME B, A FAMILY TORN APART). Quotes to back up opinions are useful, but, speaking anecdotally, quotes that're meant to prove that the author "writes lyrically" or "beautifully" are usually a waste of time; lyrical writing works best when it's cumulative (which you cannot quote), and beauty is subjective. Question: "An objective analysis of the book in terms of its literary merit or a more conversational chat about what the reviewer likes?" Answer: a subjective analysis of the book in terms of its literary merit, revealing the reviewer's area of preference. Short or long: whatever works best. If long, have plenty to say. Images: there or not, it doesn't matter. Entirely up to the blog-stylist. A picture of the cover or author is not necessary. Would actually rather not know that book written by nicely-smiling blonde lady in attractive garden deckchair with Tudor decor in background. Awards: nice to know but not essential, can be reviewerishly handy if they tell you what kind of book it is. "All right, so it's the kind of book that wins that prize." Shorthand. There are people who avoid Booker winners. But the reviewer can use the mention of the prize to discuss the prize. "It's an atypical Miles Franklin winner, in that it's wildly avant-garde and set in a Patagonian diamond mine." Labels and categories: if I'm searching through a blog then Author and Country are helpful. Oxford comma: do not mind. Let it appear or disappear, whatever makes it happy, like a sea turtle. Entertaining … and thoughtful … reply as usual DKS. I like your point about "discussed" rather than "reviewed". It's worth thinking about approaching litblogs more in that way. I've realised recently that I tend not to produce neat little quotable bites (like, "heartfelt writing that soars" or somesuch) … I've seen people struggling to get them from my blog but I don't think I'm going to change my style because it's not really me. Like Lizzie Bennet I'm a "rational creature" rather than an emotive one. And I take your point about quotes. Sometimes I think that a book is "beautifully written" so want to give some examples, but then discover it's hard to find something that works out of the context of the whole. I still try though. Usually for my own benefit to provide a hook for my memory later, if that makes sense. You made me laugh about not wanting to know the author was "a nicely-smiling blonde lady…". As for dot-points, I do use them sometimes – not because I think my readers are boofheads but because I'm either lazy or, more often, want to create some white space to give the readers a breather. I promise not to overdo it but!! I've seen someone promoting the idea of dot points for that reason. White space, not laziness. Less intimidating than large chunk of text. Increases confidence in audience members. "I can handle this." Individual launches self on sea of prose like indomitable dinghy. Reviews of "lesser-known" books are the best justification for the whole practice of blog-reviewing, your review of Fergus Hume and his Hansom Cab, for example, a bit of literary history I might never have heard of if you hadn't mentioned it, or the review I saw on the Aboriginal Art & Culture blog, of Minoru Hokari's Gurindji History, a book I would never have imagined. You come across the idea of a book like this and it's Andre Breton finding startling objects in the Paris junk-marketplace all over again. You do have way with words! I can see people launched on a prose sea in their indomitable dinghies. I agree that giving air to lesser known works is great justification. It's also very satisfying isn't it? I like blogs to be brief and to the point. I want to know if the blogger enjoyed the book, and if not, why not. I appreciate a few lines from the book to give me an idea of the writing style. I also like to know if it is a charactor or plot driven story, the time period and the setting of the story. If it is non fiction, I do expect more detail and the relevance of the book. Thanks Meg for contributing. I'm really appreciating hearing everyone's ideas. I like your comments about what you particularly like to know – character or plot driven, time period and place setting. I think a round-up post is coming in a week or so because some patterns are emerging.
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using System; using Newtonsoft.Json; using VkNet.Utils; namespace VkNet.Model.GroupUpdate; /// <summary> /// Объект, который содержит сообщение и информацию о доступных пользователю функциях. /// </summary> [Serializable] public class MessageNew : IGroupUpdate { /// <summary> /// Сообщение. /// </summary> [JsonProperty("message")] public Message Message { get; set; } /// <summary> /// Информация о доступных пользователю функциях. /// </summary> [JsonProperty("client_info")] public ClientInfo ClientInfo { get; set; } #region Методы /// <summary> /// </summary> /// <param name="response"> </param> /// <returns> </returns> public static MessageNew FromJson(VkResponse response) => new() { Message = response["message"], ClientInfo = response["client_info"] }; /// <summary> /// Преобразование класса <see cref="MessageNew" /> в <see cref="VkParameters" /> /// </summary> /// <param name="response"> Ответ сервера. </param> /// <returns>Результат преобразования в <see cref="MessageNew" /></returns> public static implicit operator MessageNew(VkResponse response) { if (response == null) { return null; } return response.HasToken() ? FromJson(response) : null; } #endregion }
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Biografia Di origine tedesca, nato Arnold Kaiser a Rochester nello stato di New York il 16 giugno 1894, cambiò il suo nome in Norman Kerry nel corso della prima guerra mondiale. Kerry iniziò a lavorare a New York come commesso per una famiglia dedita al commercio di pellicce ma, ritenendo di non aver attitudine per questo tipo di lavoro, lo lasciò e divenne un agente teatrale. Nel 1916 incontrò Rodolfo Valentino, di cui divenne amico e al quale consigliò di provare a cimentarsi nella giovane industria del cinema. Kerry interpretò il suo primo ruolo nel film L'allegra favola di Black Burke (1916) di Allan Dwan, con Douglas Fairbanks, ma divenne popolare nel 1923 interpretando una parte nel film Il gobbo di Notre Dame. In seguito apparve in diversi film e al fianco delle principali attrici del momento, tra cui Lillian Gish in Annie Laurie la fanciulla scozzese (1927) e Joan Crawford. Sposatosi tre volte, con l'avvento del cinema sonoro Kerry non mantenne il successo sperato e, dalla metà degli anni '30, le sue apparizioni cinematografiche furono scarse. Raggiunta la maturità, dimostrò la sua propensione all'avventura arruolandosi nella Legione straniera francese e ritornò negli Stati Uniti solamente nel 1940, quando la Francia fu invasa dall'esercito nazista di Hitler. Norman Kerry morì a Los Angeles il 12 gennaio 1956, all'età di 61 anni, vittima d'una malattia del fegato. Filmografia Vanity, regia di John B. O'Brien (1916) L'allegra favola di Black Burke (Manhattan Madness), regia di Allan Dwan (1916) The Little American, regia di Cecil B. DeMille e Joseph Levering (1917) The Little Princess, regia di Marshall Neilan (1917) Amore d'artista (Amarilly of Clothes-Line Alley), regia di Marshall Neilan (1918) Up the Road with Sallie, regia di William Desmond Taylor (1918) Rose o' Paradise, regia di James Young (1918) Good Night, Paul, regia di Walter Edwards (1918) The Talk of the Town, regia di Allen Holubar (1918) Toton the Apache, regia di Frank Borzage (1919) Getting Mary Married, regia di Allan Dwan (1919) Virtuous Sinners, regia di Emmett J. Flynn (1919) The Dark Star, regia di Allan Dwan (1919) Soldiers of Fortune, regia di Allan Dwan (1919) Passion's Playground, regia di J.A. Barry (1920) A Splendid Hazard, regia di Arthur Rosson (1920) Buried Treasure, regia di George D. Baker (1921) Proxies, regia di George D. Baker (1921) The Wild Goose, regia di Albert Capellani (1921) Little Italy, regia di George Terwilliger (1921) Three Live Ghosts, regia di George Fitzmaurice (1922) Find the Woman, regia di Tom Terriss (1922) The Man from Home, regia di George Fitzmaurice (1922) Till We Meet Again, regia di William Christy Cabanne (1922) Brothers Under the Skin, regia di E. Mason Hopper (1922) Is Money Everything?, regia di Glen Lyons (1923) Donne viennesi (Merry-Go-Round), regia di Rupert Julian e, non accreditato, Erich von Stroheim (1923) The Hunchback of Notre Dame, regia di Wallace Worsley (1923) The Acquittal, regia di Clarence Brown (1923) The Thrill Chaser, regia di Edward Sedgwick (1923) The Satin Girl, regia di Arthur Rosson (1923) The Shadow of the East, regia di George Archainbaud (1924) Daring Youth, regia di William Beaudine (1924) True as Steel, regia di Rupert Hughes (1924) Cytherea, regia di George Fitzmaurice (1924) Scadenza tragica (Between Friends), regia di J. Stuart Blackton (1924) Butterfly, regia di Clarence Brown (1924) So This Is Marriage?, regia di Hobart Henley (1924) Il prezzo del potere (The Price of Pleasure), regia di Edward Sloman (1925) Fifth Avenue Models, regia di Svend Gade (1925) The Phantom of the Opera, regia di Rupert Julian (1925) Lorraine of the Lions, regia di Edward Sedgwick (1925) Under Western Skies, regia di Edward Sedgwick (1926) The Barrier, regia di George W. Hill (1926) Mademoiselle Modiste, regia di Robert Z. Leonard (1926) The Love Thief, regia di John McDermott (1926) The Claw, regia di Sidney Olcott (1927) Annie Laurie la fanciulla scozzese (Annie Laurie), regia di John S. Robertson (1927) Lo sconosciuto (The Unknown), regia di Tod Browning (1927) Body and Soul, regia di Reginald Barker (1927) The Irresistible Lover, regia di William Beaudine (1927) Love Me and the World Is Mine o The Affairs of Hannerl, regia di Ewald André Dupont (1927) Legione straniera (The Foreign Legion), regia di Edward Sloman (1928) La dama di Mosca (The Woman from Moscow), regia di Ludwig Berger (1928) Man, Woman and Wife, regia di Edward Laemmle (1929) Trial Marriage, regia di Erle C. Kenton (1929) The Woman I Love, regia di George Melford (1929) The Bondman, regia di Herbert Wilcox (1929) The Prince of Hearts, regia di Cliff Wheeler (1929) Ex-Flame, regia di Victor Halperin (1930) Bachelor Apartment, regia di Lowell Sherman (1931) Air Eagles, regia di Phil Whitman (1931) Phantom of Santa Fe aka The Hawk , regia di Jacques Jaccard (1931) Tanks a Million, regia di Fred Guiol (1941) Film o documentari dove appare Norman Kerry The City of Stars, regia di H. Bruce Humberstone - sé stesso (1924) Hello, 'Frisco, regia di Slim Summerville - sé stesso (1924) The Voice of Hollywood No. 6, cortometraggio - sé stesso (1924) The Cohens and Kellys in Hollywood, regia di John Francis Dillon - filmato d'archivio (1932) The Legend of Rudolph Valentino, regia di Graeme Ferguson - filmato d'archivio (1961) Altri progetti Collegamenti esterni Cinema muto statunitense Persone legate alla Legione straniera francese
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{"url":"http:\/\/oftankonyv.reak.bme.hu\/tiki-index.php?page=T1+relaxation&structure=Tank%C3%B6nyv+Fizikusoknak","text":"# T1 relaxation\n\nIt is known by many experimental results that if a bunch of magnetic moments are placed in external magnetic field, a certain part of these moments will align to the direction of the field. If we are talking about quantum mechanical objects, for example spin-half particles, this means that in the equilibrium state of the system a bit more than half of the spins will be parallel to the external field, and a bit less than half of them will be antiparallel to it. The ratio of these two depends on their energy level in the magnetic field, which is of course determined by the projection of the angular momentum to the field:\n\n(1)\n\nAnd the equilibrium ratio of the parallel () and antiparallel () spins is the well-known Boltzmann factor with the energy level difference:\n\n(2)\n\nIf we alternate the ratio for example by applying an RF excitation the system will somehow return to the equilibrium state described in (2) after the excitation effect ceases. In a simple model where we assume the spins to be completely independent we can describe the relaxation method by two transition probabilities per unit time: the probability of a parallel spin becoming antiparallel denoted by , and the probability of the opposite, i. e. an antiparallel spin becoming parallel denoted by . Note that these two are generally not equal because of the environment of the spins end the existence of the external magnetic field. With these probabilities, the time derivatives of the spin population can be written as:\n\n(3)\n(4)\n\nThe measureable quantity is the difference between these populations, so we will use the followings:\n\n(5)\n(6)\n\nUsing (3) and (4) the time derivative of the population difference will be:\n\n(7)\n\nWhereof the equilibrium value of the population difference is\n\n(8)\n\nFrom this we define a time-dimension quantity as\n\n(9)\n\nUsing this the time derivative of the population difference is as follows:\n\n(10)\n\nWith the solution of\n\n(11)\n\nAs can be seen, the time dependence of the spin population difference - with the latter proportional to the longitudinal component of the magnetization pointing to the direction of the external field - shows exponential decay to the equilibrium state with a time constant . This effect is called longitudinal or relaxation.","date":"2022-09-26 00:53:18","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\": 0.8642673492431641, \"perplexity\": 263.16249730551306}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030334620.49\/warc\/CC-MAIN-20220925225000-20220926015000-00030.warc.gz\"}"}
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Template Tester October 23, 2010 at 2:09 am · Filed under Uncategorized Yet . . . July 16, 2009 at 7:29 pm · Filed under Uncategorized There are significant differences between the earliest manuscripts of the story, dating from the last few centuries before Christ, and those of later centuries, with Goliath's height increasing from an original six and a half feet (200cm) to nine and a half feet (290cm), and David changing from a young man to a boy. The story may have originated in oral traditions about David – the name "Goliath" appears to be an authentic 10th century Philistine one – but Goliath's famous battle may originally have been with the obscure Elhanan, rather than with the great king of Jewish folklore, and the version preserved in the Book of Samuel shows clear parallels with G October 29, 2008 at 7:58 pm · Filed under Uncategorized Google Reader: Coding/link Test 1800 Glorious First of June was the first and largest fleet action of the naval conflict between the Kingdom of Great Britain and the First French Republic during the French Revolutionary Wars. Mann is a surname of Germanic and also separately of Punjab origin. The Germanic name translates roughly as "person" or "man". The first uses of the name date to approximately the 9th century. The name was often taken by common persons, and not nobility. The Punjab surname is found in the northern state of India. It literally means 'honour'. The surname is common among the Sikh people. In fiction, Glass Horses, by author James R. Mittag, chronicles the lives of the brothers John and Mark Mann. The Memba are a tribal population of 3,500 is centered around Tuting and Geling, near the Siang river in the West Siang and Upper Siang district of Arunachal Pradesh in India not very far from the Tibetan border. A sizeable population can be found in the nearby Yargab-Chu valley in Mechuka (West Siang), where the population is around 4000 to 5000 and as well as Medog county in Tibet. Dr. Lalmani Misra (August 11, 1924 – July 17, 1979), M.A., Ph.D., D. Mus. (Veena), M.Mus. (Vocal), B.Mus. (Sitar, Tabla), (Sahitya Ratna) Dean & Head, Facutly of Music and Fine Arts, Banaras Hindu University, Varanasi, was an eminent Indian classical musician known as much for his art as for his scholarship. Link Test / Coding Test. July 28, 2008 at 1:24 am · Filed under Uncategorized From The Wiki: The Sindhia state of Gwalior became a major regional power in the latter half of the eighteenth century and figured prominently in the three Anglo-Maratha Wars. They held sway over many of the Rajput states, and conquered the state of Ajmer. The genus Saxicola[1], the stonechats or chats, is a genus of 14 species of small passerine birds restricted to the Old World. They are insectivores of open scrubland and grassland with scattered small shrubs. New Haven is one of the areas of Enugu that was mapped out in the 1960s and grown from a residential suburb to a major commercial area especially along Chime Avenue, the main high street. Plato (Greek: Πλάτων, Plátōn, "broad")[1] (428/427 BC[a] – 348/347 BC), was a Classical Greek philosopher, who, together with his mentor, Socrates, and his student, Aristotle, helped to lay the foundations of Western philosophy.[2] Plato was also a mathematician, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the western world. Plato was originally a student of Socrates, and was as much influenced by his thinking as by what he saw as his teacher's unjust death. Meet the Profs Sample Feeds The Good News of the Gospel NEW FEED Playlist Test http://www.musicplaylist.us/mc/mp3player_new.swf
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Switching on this years Christmas lights means that part of Bryngwyn Road and Maescanner Road in Dafen will be closed temporarily next month. Part of Bryngwyn Road, from its junction with Havard Road for a total distance of 225 metres in an easterly direction through to the entrance to Dafen Park at Maescanner Road will be closed between 5.45pm and 6.15pm on Friday, November 27. This is to ensure public safety during the festive lights switch-on and a walking parade into Dafen Park. Drivers are advised to use alternative routes during this brief period, and if possible to avoid travelling in this direction for up to 30 minutes before hand because of the high number of pedestrians expected to gather in the area outside the Church Hall. Visitors to Dafen Park for the festive celebrations are advised wherever possible to approach on foot; there will be no parking facilities near the Park. Non-residential parking in nearby streets, especially on pavements, is also discouraged to avoid congestion and potential risks to pedestrians. The closest public car park, with limited spaces, is next to Dyfed Steels.
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{"url":"http:\/\/www.scientificlib.com\/en\/Mathematics\/LX\/GilmanGriessTheorem.html","text":"# .\n\nIn finite group theory, a mathematical discipline, the Gilman\u2013Griess theorem, proved by (Gilman & Griess 1983), classifies the finite simple groups of characteristic 2 type with e(G) \u2265 4 that have a \"standard component\", which covers one of the three cases of the trichotomy theorem.\n\nReferences\n\nGilman, Robert H.; Griess, Robert L. (1983), \"Finite groups with standard components of Lie type over fields of characteristic two\", Journal of Algebra 80 (2): 383\u2013516, doi:10.1016\/0021-8693(83)90007-8, ISSN 0021-8693, MR 691810\n\nMathematics Encyclopedia","date":"2021-09-26 00:58:40","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\": 0.8400393128395081, \"perplexity\": 1394.114373462059}, \"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-2021-39\/segments\/1631780057787.63\/warc\/CC-MAIN-20210925232725-20210926022725-00576.warc.gz\"}"}
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