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# Warhol's Working Class
# Warhol's Working Class
# Pop Art and Egalitarianism
Anthony E. Grudin
The University of Chicago Press
Chicago and London
The University of Chicago Press, Chicago 60637
The University of Chicago Press, Ltd., London
© 2017 by The University of Chicago
All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations in critical articles and reviews. For more information, contact the University of Chicago Press, 1427 E. 60th St., Chicago, IL 60637.
Published 2017
Printed in the United States of America
26 25 24 23 22 21 20 19 18 17 1 2 3 4 5
ISBN-13: 978-0-226-34777-6 (cloth)
ISBN-13: 978-0-226-34780-6 (e-book)
DOI: 10.7208/chicago/9780226347806.001.0001
Library of Congress Cataloging-in-Publication Data
Names: Grudin, Anthony E., author.
Title: Warhol's working class : pop art and egalitarianism / Anthony E. Grudin.
Description: Chicago ; London : The University of Chicago Press, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017015347 | ISBN 9780226347776 (cloth : alk. paper) | ISBN 9780226347806 (e-book)
Subjects: LCSH: Warhol, Andy, 1928–1987. | Pop art—United States. | Art and society.
Classification: LCC N6537.W28 G78 2017 | DDC 700.92—dc23
LC record available at <https://lccn.loc.gov>
This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).
# Contents
Acknowledgments
Introduction: Warhol and Class
1 Varieties of Pop
2 Warhol's Participatory Culture
3 Warhol's Brand Images
4 Warhol, Modernism, Egalitarianism
Conclusion: Warhol's Neoliberalism
Notes
Index
Plates
# Acknowledgments
This book's second chapter grew out of material published in "'Except Like a Tracing': Defectiveness, Accuracy, and Class in Early Warhol," _October_ , no. 140 (Spring 2012), and "Myth and Class in Warhol's Early Newsprint Paintings," in _Warhol: Headlines_ , ed. Molly Donovan (Washington, DC: National Gallery of Art, 2011). Portions of Chapter 3 appeared in "'A Sign of Good Taste': Andy Warhol and the Rise of Brand Image Advertising," _Oxford Art Journal_ 33, no. 2 (June 2010). Comments from audiences at Middlebury College, Colgate University, Whitman College, the Florida Institute of Technology, Tufts University, the ASAP conferences of 2011 and 2015, and the College Art Association conferences of 2011 and 2012 helped me to sort out the book's claims.
This book benefited immensely from time I spent working at the University of California, Berkeley and California College of the Arts. I am grateful to Elise Archias, Binta Ayofemi, John Beebe, Sylvan Brackett, Molly Brunson, Anthony Cascardi, Kevin Chua, Neil Cox, Zach Davis, Kate Fowle, Darcy Grigsby, Emily Gumper, Jason Hanasik, Al and Maria Huezo, Matt Hunter, Jessen Kelly, Jonathan King, Kim Kono, Charlie Koven, Namiko Kunimoto, Leigh Markopoulos, Tara McDowell, Jeremy Melius, Doris and Charles Muscatine, Julian Myers-Szupinska, Mike Shin, Aimee Sisco, Andrew Stewart, Sam Teplitzky, Jess Theroux, Michael Thompson, Jenny Wapner, Alice Waters, Tim Webster, Sarah Weiner, Robert Wetle, Johanna Wright, Justin Underhill, Greg Youmans, Josephine Zarkovich, and Sebastian Zeidler for setting such fine examples of engagement and conviviality.
Friends, colleagues, and collaborators during my time in Vermont have contributed to this book at every stage. I particularly want to thank Mildred Beltre, Jessica and Josh Bongard, Lynne Bond, Silas Branson, Thomas Brennan, Jake Brochhagen, Tam Bryfogle, Steve Budington, Janie Cohen, Bradford Collins, Penny Cray, Wright Cronin, Cami Davis, Sona Desai, Paul Deslandes, Kelley Di Dio, Maggie Donin, Molly Donovan, Nancy Dwyer, Bill Falls, Kat Ford, Pamela Fraser, Chelsea Frisbee, Andy Frost, Larissa Harris, Michael Hermann, Susie and Dave Hurley, Adrian Ivakhiv, Jane Kent, Andy Kolovos, Felicia Kornbluh, Ellen Kraft, Louise Lawler, Eric Lindstrom, Jess and Will Louisos, Ted Lyman, Hilary Martin, Hilary Maslow, Jason McCune, Bill McDowell, Abby McGowan, William Mierse, Thomas Morgan-Evans, Ilyse Morgenstein Fuerst, Frank Owen, Peter Parshall, Breanna Pendleton, John Penoyar, Abby Portman, Neil Printz, Bill Racolin, Jordan Rose, Julie Rubaud, Kristina Samulewski, Uli Schygulla, John Seyller, John Smith, Michael Sundue, Randall Szott, Britta and Jeff Tonn, Mark Usher, Sally Wales, Kelley Walsh, Laurel and Morgan Waters, Rebecca Weisman, Erika White, Alison Williams, Gilda Williams, Tom Williams, Matt Wrbican, and Frank Zelko.
Whitney Davis, Jonathan Flatley, Hal Foster, Kaja Silverman, and Anne Wagner deserve special thanks for having been particularly generous in their attention to this project. I will forever be grateful to Tim Clark for his guidance as the book developed. Support from the Warhol Foundation, the Lichtenstein Foundation, the Oldenburg van Bruggen Studio, and the University of Vermont's Humanities Center and Lattie F. Coor Endowment was crucial to its completion.
Finally, I am tremendously grateful to my family—Bethanne, Jeff, Joyce, and Jasmine Cellars, Amelia and Scott Siegel, and Cristina, Michaela, Max, Mateo, Nick, Ted, Micha, and Robert Grudin—for their love and support, and to Susan Bielstein, Joel Score, James Whitman Toftness, and the anonymous readers at the University of Chicago Press for all the care they devoted to this process.
This book is dedicated to Joyce and Jasmine Cellars.
As an American, Warhol dealt particularly with the "hidden injuries of class," and for that he is still hated.
Scott Burton, "Andy Warhol: 1928–1987," _Art in America_
But I shouldn't say class, because we don't live in a nation of classes.
Helen Peters, in Studs Terkel, _Division Street: America_
FIGURE 1. Andy Warhol, _Coca-Cola [2]_ , 1961. Casein and crayon on linen, 69½ × 52¼ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
# Introduction
# Warhol and Class
Late in 1961, after trying to gain entry to the New York artworld with a series of paintings based on tabloid advertisements and comic-strip icons like Dick Tracy, Superman, and Nancy, Andy Warhol learned from his assistant, Ted Carey, that the imagery he had been appropriating for his paintings was less innovative than he had hoped. The trendsetting Leo Castelli gallery had already discovered an ambitious young New York artist who was converting pulpy comic-book panels and advertisements into large, elegant paintings. As Carey remembered it, "I went right over to Andy's house . . . [and] I said, 'Prepare yourself for a shock. . . . Castelli has a closet full of comic paintings.' And he said, 'You're kidding?! . . . Who did them?' And I said, 'Somebody by the name of Lichtenstein.' Well, Andy turned white."
Warhol seems to have responded to this setback by producing the first paintings in the history of fine art to focus exclusively on the depiction of a branded grocery commodity: _Peach Halves_ (plate 1), _Coca-Cola [1]_ , _Coca-Cola [2]_ , and _Campbell's Soup Can (Tomato Rice)_. Formally and iconographically, these are odd and equivocal paintings. At least a third of each canvas is left unpainted, and large portions of the painted areas are filled only with scrawls in paint and crayon. Where colors appear they seem raw and unmixed, commercial or juvenile. Everything is sketchy, dribbling, obstructed—seemingly halted in the midst of an emotionally disagreeable process. How are we to read these strange icons? Has Warhol captured the bizarre and temporary utopia of shopping, what Robert Seguin calls an "important middle or liminal space . . . imagined as a suspended or uncertain activity"? Or do the persistent distortions and repetitions that characterize these images testify to the "compulsive shopping" that Kristine Stiles has described as Warhol's coping mechanism—a "poignant . . . sign of his loneliness"? And what, if anything, might these childish, malformed paintings have to do with social class?
Ever since Warhol first showed these paintings to friends and critics, they have been deemed inferior and forgettable. Warhol's great early supporter and confidante Emile de Antonio called one of the _Coca-Cola_ paintings "a piece of shit," "just kind of ridiculous because it's not anything. It's part Abstract Expressionism and part whatever you're doing." Critics have generally agreed that the paintings are derivative in style—"gestural handling and exaggerated drips as shorthand signs of expressiveness borrowed from Abstract Expressionism." According to this view, Warhol needed to abandon gesturalism in order to achieve his mature pop style—the cold, clean, mechanical look that has long been associated with his work.
FIGURE 2. Andy Warhol, mechanical ( _Coca-Cola, "Standard and King Sized"_ ), ca. 1960. Newsprint and coated paper clippings with masking tape on heavyweight paper, 15½ × 10¾ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Have critics been wrong to dismiss these paintings as pale imitations of abstract expressionism? "I can't understand why I was never an abstract expressionist," Warhol wrote in 1975, "because with my shaking hand I would have been a natural." But his sly remark suggests affinity as well as distance. Throughout the 1950s, he had cultivated and exaggerated his "shaking hand" to convey desire and attachment: for shoes, for candy, for animals, for men and their bodies. Why would he extend this rhetoric to mundane packaged goods, inexpensive, unremarkable consumer commodities? A Coke cost ten cents when Warhol painted these paintings, and Campbell's bragged in advertisements that its soup could be had for twelve cents a can and seven cents a serving. What could possibly be so consequential—so worthy of being painted—about a can of soup or a bottle of soda?
The closer we look at these paintings and their source material, the more complicated these questions become. Like almost all of Warhol's depictions of commodities—and unlike the work of his abstract expressionist predecessors—these paintings explicitly stage a moment in the reproduction of another image. They don't depict actual cans or bottles—they depict _advertisements_ of cans and bottles, and they do so in a distinctly mechanical way. Like many of Warhol's works from this period, these four were produced with the aid of an opaque projector. Warhol would mask out the surrounding imagery, project the masked advertisement onto a canvas, and copy it in paint. The strange, blocky, vertical and horizontal shapes that surround the can in _Peach Halves_ are unfinished depictions of the masking tape and paper that framed the painting's subject (plate 2). Masking ostensibly allowed the artist to focus on the parts of the projected page that matter most and ignore the rest, but in this painting Warhol seems to have lost track of the difference: he painted the central image _and_ the masking that framed it. In the _Coca Cola_ paintings and _Campbell's Soup Can (Tomato Rice)_ , Warhol may even have invented such traces—no masking survives on the source material. These paintings directly state their reliance on mechanical reproduction, but they also make it clear that for some reason the process of reproduction could not be completed. Our previous questions are now exacerbated: Why would Warhol want to convey the desirability of an _advertisement_ for a can of soup? How could it be so difficult to complete the replication of a two-dimensional image _with the help of a projector_? And why would he have felt the need to reproduce both an image and its mask?
These questions open onto two histories that form the central concerns of this book: the histories of amateur cultural participation and of brand image advertising in the United States as they intersected with Warhol's work. As "one of the very few modern artists from an authentically working-class background," Warhol participated in and reflected on these histories throughout his life. A child of the Great Depression, Warhol was born in a bathroom-less tarpaper shack and raised by blue-collar Carpatho-Rusyn immigrants—his mother a house cleaner, his father a construction worker—in the abject poverty of a Pittsburgh ghetto. He spoke Rusyn at home, where he was treasured as a prodigy, but developed intense attachments to American popular culture, which he attempted to reproduce with the help of the technologies he could access. At Carnegie Tech—funded by a small scholarship and the postal bonds his "five-and-dime family" saved by sharing rooms and boarding renters—Warhol studied the techniques and technologies of advertising and fine art. During his remarkable decade as an advertising illustrator, working for some of the most prestigious clients in New York, he mastered these techniques and technologies, garnering the highest honors in his field and compensation to match. And throughout his quarter-century career as an internationally renowned artist, which earned him millions and propelled him into the ranks of "New York's Gucci set," Warhol and his collaborators explored the possibilities for amateur cultural participation and the powers of the brand image, not just in painting but in almost every available media: installation and performance, "personal enterprise," magazines and books, film, photography, music, modeling, television.
Neither of these histories, nor Warhol's engagement with them, can be adequately understood without engaging the question of social class in the United States. During Warhol's early pop period, brand images and amateur cultural participation were marketed as routes to upward social mobility, mechanisms for shopping and working oneself from trailer parks and ghettos to the American Dream. Working-class Americans, widely disparaged by experts as childish or bestial in their impetuousness—"yielding freely to impulses" and "non-rational"—were thought to be particularly susceptible to these promises. After all, as one study put it, "From the point of view of other classes, they 'live like animals.'" Warhol traded on these expectations, integrating them into his style and subject matter, and thus testing the possibility of a truly popular art. But in a surprising twist, he learned that his investigations of brand images and amateur cultural participation could also appeal to an elite clientele who found lowness titillating, and who were hungry for strategies of cultural disguise. Warhol's commercial success as a fine artist was in large part predicated on his ability to manipulate and satisfy these disparate appetites for class mobility.
_Warhol's Working Class_ therefore proposes social class as a core stylistic and thematic component of Warhol's artistic project during the early 1960s, and indeed throughout his career. When Bob Dylan described Warhol as a "Napoleon in rags" in 1965, he underscored the importance of class in Warhol's work, not as a stable identity but as a complex and calculated performance: "the language that he used." Why, Dylan asks us, would a conqueror disguise himself as a pauper? And if this conqueror were an artist, what would he stand to gain? _Warhol's Working Class_ takes up the questions proposed by Dylan's taunting line. Conventional readings of Warhol's work have tended to emphasize the universally "American" qualities of his images, or their postmodern depthlessness and superficiality. But my research into postwar mass culture and the specific qualities of Warhol's visual strategies demonstrates that the motifs and styles Warhol adopted during the early 1960s carried distinctly working-class cultural connotations, connotations that would have been apparent to contemporary audiences but which have, in the intervening years, been almost completely erased.
The "brand images" that made up Warhol's classic iconography—Campbell's Soup, Coca-Cola, Brillo—were not, as many scholars have asserted, universally recognized and desired during this period. Nor were they universally decried, as others have argued. Rather, they were seen to appeal primarily to a group of Americans that manufacturers and advertisers termed "the working class," people they saw as "profoundly different from the middle class individual in his mode of thinking and his way of handling the world." In response to concerns during the late 1950s and early 1960s that wealthy consumers were becoming skeptical of advertising, corporations and media experts publicly proposed the working class as a more credulous audience for these brand images. As one full-page marketing advertisement in the _New York Times_ claimed, "In contrast to white collar women (who have no qualms about private [that is, generic] labels) the working class wife has an extraordinary emotional dependence upon national brands—her symbol of status and security (because she is basically unsure of the world outside her door)." Understood in their context as an embattled and strategically active category, brand images yield new insight on Warhol's early pop work and its cultural reception.
But class was not just central to Warhol's subject matter; it had a powerful stylistic dimension as well. Contemporary tabloids, comic books, and magazines, the very media from which Warhol derived many of his most famous images, prominently advertised amateur cultural reproduction—in the form of mail-away art schools, drawing contests, cameras, projectors, tape recorders, "Magic Art Reproducers"—as a fast track from working-class subsistence to wealth and prestige. By utilizing these technologies in his paintings in a distinctly amateurish way, Warhol was aligning his work with a recognizably déclassé identity position—even as he rose through the economic and social hierarchies. Throughout his career, Warhol's style is consistently characterized by reproductive incompletion; he "never created anything without copying," but despite the aid of reproductive technologies and the graphic simplicity of the adopted motifs, reproduction is truncated again and again—marred by missed silkscreen registers, clogged screens, leaky contour lines, and scrawled additions at the margins. As Warhol's assistant Gerard Malanga put it, "Andy embraced his mistakes. We never rejected anything. Andy would say, 'It's part of the art.'" This abdication of expertise produced the voice of the amateur, the "impulsive" working-class outsider, the striver toward cultural participation. Robert Rauschenberg noticed this quality in Warhol's art immediately: "His works are like monuments to his trying to free himself of his talent." Even commercialism was just out of reach: "We were trying to be commercial. . . . But, commercially, I can't seem to make it."
FIGURE 3. Magic Art Reproducer advertisement, _Kathy, The Teen-Age Tornado_ , October 1961, 25.
Some scholarship on Warhol has championed these irregularities as pure percepts that evade the standard mappings of reading and looking. As Steven Shaviro argues, Warhol's "surfaces are impenetrable precisely because there is nothing beneath them, no depth into which one could penetrate." Read in light of the class specificity of Warhol's motifs, however, these intimations of reproductive incompletion and inarticulateness also highlight a specifically working-class cultural dilemma: the challenge of engaging in a culture that promises and demands "audience participation" while withholding real access to its most prominent and desirable imagery. For Factory collaborator Tally Brown, this tension underwrote Warhol's notion of the Superstar: "Andy coined it to embrace this incredible distance between being 'a Hollywood star' and a Superstar." This "incredible distance" is Warhol's great subject: the "unequal structures of creativity" that characterize capitalist culture, in which the false promise of cultural participation is aggressively marketed to "consumers" as a shortcut to wealth and fame.
Since class, as bell hooks has argued, is "the subject that makes us all tense, nervous, uncertain about where we stand," let me make some preliminary clarifications about its status as a category in this book.
First, this book proposes social class as a dimension of Warhol's work that, up to this point, has frequently been overlooked or minimized. However, to argue for the inclusion of this problematic is not to insist on the exclusion of others. Beverley Skeggs's observation that "class cannot be made alone, without all the other classifications that accompany it," seems unimpeachable to me. This book tracks some of the ways class intersected with race, gender, and sexuality in Warhol's work, but of course these intersections are endless. I have been particularly inspired by the resonances between Warhol and his cultural contemporaries—Adeline Gomberg, James Baldwin, Tally Brown, Janice and Keith Gunderson, Colette Brown and Margaret McCarthy, Jean-Michel Basquiat, Kathy Acker, Lou Reed, John Steinbeck, Vladimir Nabokov, Frank O'Hara, Little Eva, Henry Geldzahler, Otto Soglow—and by what Seguin calls "(queer, Marxian) disruptive logics" in which identity "emerges as something of a permanent crisis term, despised and desired . . . all at once." Signs and stereotypes of class and class belonging, like the signs and stereotypes of gender, sexuality, and ethnicity explored in recent Warhol scholarship, became elements of Warhol's artistic material, elements that he manipulated, distorted, and used rhetorically.
Second, this book does not take an essentialist approach to class—it does not assume that anyone's creativity or thought is predetermined by their social background, any more than it is predetermined by their race, gender, or nationality. Nor does it pretend that _classes_ are _castes_ —closed groups, forever fixed. Under capitalism, classes are widely understood to be shifting and permeable, "an accident of the social division of labour and other similar historical kinds of serendipity." But rejecting essentialism and stasis does not mean rejecting class as an interpretive category. While Americans were increasingly encouraged during Warhol's lifetime not to talk or think critically about social class, it is difficult to overstate the strategic importance of class categories during this period. The advertisers who targeted the working class during the late 1950s and early 1960s had it both ways: singling its members out as a credulous group, desperate for status, while at the same time promising them access to a world of social mobility, where fixed classes were outmoded and status was as easy to attain as a can of soup, drawing lessons, or a two-dollar overhead projector. By teaching their audiences that they "shouldn't say class, because we don't live in a nation of classes," these voices were paving the way for the rise of the virulent form of capitalism known today as "neoliberalism," with its austerity, union-busting, deregulation and deinstitutionalization, and intentional unemployment—together amounting to an "absolute expansion of the classes of property and the massive increase in their relative power against the propertyless classes." How much easier it would be to achieve hegemony and "win the active consent" of the people who suffered most from these reconfigurations if they could be made to believe that fixed classes were a mirage, mention of class taboo, and status the product of individual achievement That "old, rigid barriers are disappearing—class and rank; blue collar and white collar. . . . More and more we are simply consumers." Neoliberalism's socioeconomic reconfigurations were thus necessarily facilitated by changing "cultural conceptions" regarding "work and the working self"—a new sense that each of us is ultimately responsible for his or her own class trajectory. Warhol was fascinated by these promises of individualism and mobility, which could be profound and deceptive, generative and paralyzing, charged with strange forms of desire. His work disseminated these promises, while also providing us with a record of their nefarious tensions and transformations.
Finally, claiming that Warhol's work is inflected with complex historical material having to do with class does not entail claiming that it always takes a politically laudable stance toward that material, or even that its relation to that material can be boiled down to one stance or another. As we will see, Warhol's rhetorical positions are complicated and contradictory. For all his early exposure to Bertolt Brecht and Ben Shahn, Warhol was never a confident or unambivalent leftist—US Cold War propaganda had clearly left its mark on him. Labor and laborers appear only rarely in his work; in their absence, class is implied through style, iconography, and technique. Of course it is relevant that Warhol was among the unusual few to have lived both extremes of American life, the poorest and the richest. He understood inequality viscerally and developed a powerful sense of class at an early age. His first newspaper interview, granted at the age of eighteen, when he was working at his brother's produce truck, quotes him as complaining about the "'new rich,' who would ask him to carry a pound of tomatoes to a sixth floor apartment just to impress their friends." This strange equation of groceries and status seems to have left its mark on young Warhol. But, to paraphrase Sartre, not every proletarian success story culminates in _Double Elvis_ or _Chelsea Girls_. Warhol neither conquered class nor escaped it; he did, however, leave us an unusually nuanced record of its powers and consequences. At its best, his work "establishes myth and illusion as _visible objects_ " and tests the possibilities of egalitarian culture.
Class's appearances in Warhol's work are thus deeply ambivalent and contradictory. Warhol was clearly no working-class hero: his work was appropriative, borrowing class codes and styles and affects in order to feed on their power. And yet even as he appropriated, his work necessarily preserved. Many of the affects, images, and technologies Warhol borrowed were being marketed directly to the working class for the first time. This marketing was specifically intended to encourage its audience to think of itself as upwardly mobile, both as "consumers" and as "artists." The project was a massive success, to the point where many now celebrate "a maelstrom of social and intellectual mobility"—the core promise of neoliberalism. Mobility, perhaps—or, in more critical terms, "proletarianization and embourgeoisement . . . two great forces . . . pulling (most) people toward the pole of wage work and the production of surplus value, others toward the pole of the appropriation of surplus value."
Warhol knew these economic forces well, but he also knew that they had cultural correlatives that could "dampen," without actually alleviating, the "conflict . . . inherent in the relations between classes." On the one hand, cultural embourgeoisement could compensate for economic proletarianization—shoppers could slowly learn, and imaginatively participate in, the consumerist "aristocracy of brand names." They could be taught, through repetition, to believe that buying the right can of soup would contribute meaningfully to their families' status, or that their knack for sketching would soon be recognized and financially rewarded. In exceptional cases, like Warhol's, cultural embourgeoisement could actually bring about economic embourgeoisement: artistic talent would propel the lucky nobody to the upper levels of the economic hierarchy. But, as T. J. Clark and Anne Wagner remind us, "Any artist who comes in this way to stand for a new class's doomed claim to a place in the 'art'-world . . . will suffer for their standing." Warhol, one of the only twentieth-century artists to undergo an assassination attempt, would barely live to tell.
And yet Warhol also recognized culture's strange and often overlooked capacities for _downward_ mobility—"slumming," an interest in the vulgar. In times of political turmoil, it behooved members of the elite to mask their cultural differences from the masses. Just as cultural upward mobility could compensate for economic stagnation, cultural proletarianization could obscure economic privilege. This masking could even be pleasurable, since "what is socially excluded or subordinated is symbolically central in the formation of desire." Warhol grasped these strategies immediately, and embraced them: "Think rich. Look poor" was his memorable maxim. Eleanor Ward remembered him arriving at the Stable Gallery to install his first show: "I'll never forget the sight of him coming into the gallery that September, in his dirty, filthy clothes and his worn-out sneakers with the laces untied, and a big bunch of canvases rolled up under his arm. 'Look what the cat dragged in,' he said."
Fifty years later, Warhol's work provides a glimpse into the history of these contradictory class strategies. When my students and I discuss class codes and languages, many of the most successful examples come from fashion. We talk about the current ubiquity of blue jeans, and the time, not long ago, when wealthy and fashionable cosmopolitans who now go almost everywhere in jeans would not have been caught dead in them. Even students who have never studied class in college—and there are many of them—viscerally understand that jeans are often strategic; the $800 pair simultaneously conveys wealth to the cognoscenti and projects generic Americanness to everyone else. (As one "mega-landlord" and art collector recently explained, "You want to have the guy coming to the Four Seasons who has the ripped jeans and a T-shirt equally as much as you want the guy with the Tom Ford suit. Because the guy with the jeans, I promise you, has a lot more money.") Warhol, for his part, thought he had invented this sleight of hand. A few months before he died, he told his diary, "I started this whole bluejeans-with-a-tuxedo-jacket thing because years ago after I wore that to a few big events and was photographed, all the kids began doing it and they're still doing it." Fashion, for Warhol, was a class "language that he used" as a "diplomat"—a language of "rags" and "chrome horse[s]"—and so was art. This book is an attempt to identify and to interpret the class languages that Warhol deployed stylistically, iconographically, technologically, and through his pursuit of cultural egalitarianism.
Chapter 1, "Varieties of Pop," compares Warhol's work to that of three of his most important pop art peers: Roy Lichtenstein, Claes Oldenburg, and Patty Muschinski (whose contributions to the pop style remain vastly underappreciated). This comparison reveals the extremes of pop art's attitudes toward mass culture: in Lichtenstein's paintings, mass-cultural imagery challenges and ultimately strengthens the artist's noble pursuit of formal coherence, whereas in Muschinski and Oldenburg's work, the lines between mass culture and fine art are blurred in order to produce new erotic and social intensities. Lichtenstein's comic-book source material is crucial for understanding his project; it proves in many ways to be far more radical and destabilizing than the paintings that drew upon it. The chapter concludes by arguing that (1) pop art is far more diverse and divergent than it is usually taken to be; (2) we cannot come to terms with pop art until we take its source material seriously; and (3) Warhol's work stands out in its sustained attention to both the pleasures _and_ the frustrations mass culture produced—"the compulsive imitation by consumers of cultural commodities which, at the same time, they recognize as false"—and the ways in which these affects were directed toward, and associated with, the working class.
I then address the ways in which emerging strategies of class-based marketing informed Warhol's work of the early 1960s. Chapter 2, "Warhol's Participatory Culture," focuses on his artistic production in early 1961, before he learned that Lichtenstein was working with similar material: the works based on tabloid and comic motifs that I refer to as "newsprint artworks." The imagery these works borrowed has long been recognized as having vaguely downmarket connotations. But these early pop works are also characterized by a messy, handmade style that is often interpreted art historically as a holdover of abstract expressionist gesturalism. In opposition to this view, I contend that the squalor and mess of Warhol's gesturalism signaled the difficulties of amateur cultural participation in postwar US culture. The possibility of cultural participation through new technologies and commercialized pedagogies was being marketed as an escape from financial dependence in the pages of the comics and magazines from which Warhol and Lichtenstein borrowed many of their images. In _Superman_ , _Carat_ , _Advertisement_ , and the _Before and After_ , _Dance Diagram_ , and _Do-It-Yourself_ paintings, Warhol's work took up the cheapest and most accessible images available—images marketed to and associated with a working-class demographic—and demonstrated their structural irreproducibility for their own consumers. In his silkscreened celebrity paintings, Warhol extended this investigation to Hollywood culture. Evidence to support these claims will be drawn from the comic books, magazines, advertisements, and television shows that Warhol found so fascinating. The chapter closes with discussions of James Baldwin's insights regarding participatory culture, and the ways in which interpretations of Warhol's style as ironic paved the way for a neoliberal politics.
Questions of cultural participation and class would continue to inform Warhol's works, even as their source images shifted from tabloids and comic books to soup cans and soda bottles in the wake of Lichtenstein's success. Chapter 3, "Warhol's Brand Images," focuses on works from the first half of the 1960s that appropriated and attempted to reproduce consumer commodities, including the _Tunafish Disasters_ , _Campbell's Soup Cans_ , _Coca-Colas_ , and _Brillo Boxes_. During the 1950s and 1960s, brand images were the focus of profound ambition and controversy. In the face of increasing middle- and upper-class skepticism toward advertising and branded commodities, prominent voices in the advertising industry were proposing an alternative demographic: working-class "consumers," thought to be more impulsive, more desperate for status and security, and therefore more susceptible to the promises of the brand image. Warhol's brand image artworks of the early 1960s were produced in the midst of this demographic realignment; the brands he borrowed were directly involved in these new consumerist strategies. By making these brand images the exclusive subjects of artworks, and by depicting them in a style that emphasized both their desirability and their irreproducibility, Warhol investigated a set of negative class stereotypes that the contemporary culture found fascinating: "working-class people love brands"; "working-class people are desperate to participate in culture"; "working-class people always do things the easy way"; "working-class people can't control their appetites." And yet the results were paradoxical: through its canonization, Warhol's work helped to universalize the brand images it adopted, effectively preserving _and_ obscuring their origins in the process. The chapter concludes by gauging the gendered and racial dimensions of brand image advertising, Warhol's role in its universalization, and his troubled relationship with whiteness—what it meant for him to "turn white" in his post-Lichtenstein work.
Chapter 4, "Warhol, Modernism, Egalitarianism," draws all of these arguments together, proposing a challenge to the idea of Warhol as "ur-postmodernist" through two interrelated readings of Warhol as a modernist. The theorists who first associated Warhol with postmodernism overlooked his investigation of working-class stereotypes and his targeted critique of abstract expressionism's orthodoxies, a critique with distinctively queer and classed overtones. Closer attention to Warhol's response to abstract expressionism reveals that it was traditionally modernist in all the best ways: critical, specific, and timely. But Warhol's construction of these queer and classed perspectives was by no means exclusively reactive or art historical. Rather, it opened onto another form of modernism, most recently promulgated by the French philosopher Jacques Rancière, that pursues a radical expansion of creativity and the aesthetic. Warhol was a great believer in this egalitarian modernism, although he recognized the obstacles that stood in the way of its development. These contradictory insights informed his films—including _Poor Little Rich Girl_ , _Soap Opera_ , _Kiss_ , and _Vinyl_ —which explored the possibilities and limitations of an egalitarian cinema and the ways in which spectacular failures of cultural participation could be repackaged and marketed to other consumers, as a salve for cultural passivity or a risqué disguise for affluence. Warhol's early film-portrait _Henry Geldzahler_ (1965)—a work he seems to have imagined as a strange remake of his own 1961 comic-strip painting _The Little King_ —exists at the intersection of these various forces, highlighting their implications for the role of the curator.
The book's conclusion, "Warhol's Neoliberalism," argues that, in the realm of politics, as elsewhere, Warhol was an early adopter, testing out a neoliberal working-class ethos that the sociologist Jennifer Silva calls privatized happiness. Where the participatory culture and brand image strategies Warhol explored in his work had encouraged working-class audiences to imagine creativity and mobility as individual achievements, this new ethos endeavored to persuade them that they were individually responsible for their emotions as well. In Warhol's case, this meant hyperbolically espousing a distrust of institutions and movements, which always struck him as constraining and intimidating. Instead, he championed a reliance on individualism and flexibility that he associated with his own non-normative sexuality. He bragged that he could "live without anything," and, like Silva's subjects, he derided "these desperate people" who failed to adapt to the brave new world of fungibility and personal responsibility. By pushing this new ethos to its limits before many Americans had even been fully exposed to it, Warhol provided a glimpse of its fault lines and false promises. But, in the process of testing and archiving these promises, Warhol's work also disseminated them, spreading "one of the primary fictions of neoliberalism," namely "that class is a fictional category" and each individual is ultimately on her own.
In the summer of 1961, a few months before Warhol began painting grocery store commodities, John Steinbeck published _The Winter of Our Discontent_ , a novel that detailed the humiliation and salvation of a blue-blooded New Englander named Ethan Hawley, whose father had squandered the family fortune. In order to support his wife and children, Hawley was forced to take a job managing a general store, stacking and selling groceries; the novel thus contains one of the period's few other sustained artistic meditations on the cans, bottles, gadgets, and comic-book characters that Warhol took as his subject matter. When Hawley tires of apostrophizing his ancestors, he talks to his groceries, an audience of "mute and articulate canned and bottled goods" that "don't argue and . . . don't repeat." Hawley's monologues center on money:
> "You dry cereal with the Mickey Mouse mask on the box and a ventriloquism gadget for the label and ten cents. I'll have to take you home, but right now you sit up and listen." . . . I brought in a carton of tomato paste, slashed it open, and stacked the charming slender little cans on their depleted shelf. "Maybe you don't know, because you're kind of foreigners. Money not only has no heart but no honor nor any memory. Money is respectable automatically if you keep it a while. You must not think I am denouncing money. I admire it very much. . . . Where money is concerned, the ordinary rules of conduct take a holiday. . . . Money is a crass and ungracious subject only when you have it. The poor find it fascinating."
Warhol would express similar concerns about the phenomenology of money: "Money is SUSPICIOUS, because people think you're not supposed to have it, even if you do have it." Despite his germophobia, he was never bothered by dirty money: "I don't feel like I get germs when I hold money. Money has a certain kind of amnesty." He had worked at a soda fountain as a kid and had a deep respect for low-end shopping: "getting underwear and socks and going to the dime store . . . This is what real life is . . . !" But like Steinbeck, he seems to have understood cans and bottles as "foreigners," intruders from elsewhere, simultaneously "mute" and "articulate." The two artists also shared a deep suspicion that only the wealthy could afford to be disinterested, and that this disinterest was highly motivated. (Neither would have been surprised by Malcolm Forbes's cynical admission that "generous gestures yield the most when that isn't their purpose.") And they both intuited that conversing with groceries and comic-book characters might be an effective way of talking critically about money, and therefore talking about class.
But while Warhol and Steinbeck shared a set of concerns and inhuman interlocutors, their tones and styles could not have been more divergent. Where Steinbeck's Hawley endeavored to remain jaunty, judicious, and humane in spite of his reduced circumstances, Warhol projected an altogether different affect: jittery, restless, quasi-mechanical, "impulse-following," "inarticulate," and "non-rational," like the stereotypical working-class American. His kindred spirits in this regard were primarily musical rather than fictional. He flagrantly embraced the pop music that Steinbeck's narrator had denounced as "the love music of a blue bottom baboon" and "the phlegm of revolt." Visitors to Warhol's studio described it as being bathed in pop music during the early 1960s. David Bourdon recalled that a friend at Time-Life had given Warhol a box of discarded 45 rpm records, and that Warhol would play them on semi-permanent repeat, "narcotically underscoring the unitary images that he regimented across his canvases." Warhol later credited this soundscape with helping him attain his mechanical style: "I still wasn't sure if you could completely remove all the hand gesture from art and become noncommittal, anonymous . . . that's why I had this routine of painting with rock and roll blasting the same song. . . . [It] cleared my head out and left me working on instinct alone." This purging of gesture was clearly a lengthy process. Ivan Karp recalled hearing this "blasting" when he visited Warhol's studio in 1961, well before Warhol had attained his classic style: "[Warhol] said that he really didn't understand these records until he heard them at least a hundred times."
During Karp's visit, Warhol played the rockabilly song "I Saw Linda Yesterday," by Dickey Lee. Bourdon remembered hearing the R & B singer Little Eva, who was best known for singing "The Loco-Motion." Victor Bockris and Gerard Malanga claimed that the Jaynetts' "Sally Goes Round the Roses" was Warhol's favorite song, and that "he played it non-stop." An interviewer in 1962 recalled Tommy Edwards's hit "It's All in the Game," while _Vogue_ 's Aline Saarinen heard "Does the Spearmint Lose Its Flavor?" What was it about these songs that Warhol might have wanted to "understand" or to communicate to his visitors? "The main story of rock 'n' roll in the 1950s," Michael Bertrand reminds us, is the conflict and resolution between the music industry and the "previously obscure truck drivers, dishwashers, factory workers, and sharecroppers [who] took the music world by storm." How would these formerly unheard working-class voices express themselves to a broader audience once they were granted access to cultural participation? What would be their appeal? How might they pursue what Peter Coviello calls "a vernacular utopia, at the center of which is desire," "cobbled together out of scraps of available language and whatever else is around"? How might they prosper, and be exploited?
Each of the pop songs these visitors recalled staged a struggle between a voice or group of voices and the machine that produced and distributed their sounds. The voices in Warhol's songs continually wondered how severely they had been contaminated by the machine, how much one's heart resembled "a merry-go-round" or one's body "a railroad train"—whether the chewing gum would "lose its flavor." And they had good reason to worry that their dalliances with the machine might not pay off. The voices needed the machine and the machine needed the voices, but the relationship was by no means reciprocal.
These songs exemplify the "paradoxical autonomy and subordination . . . proprietorship and dispossession" of labor that would characterize neoliberalism, where each "creative worker" is expected to look after herself. Eva Narcissus Boyd ("Little Eva") was paid fifty dollars a week, with no royalties for her rendition of "The Loco-Motion," a single that sold over a million copies. Tommy Edwards was rumored to have died penniless. This music was built on the pathos of hitherto excluded voices testing out the cultural participation that had been promised them, and realizing that their unfamiliarity with—and alienation from—this culture would be part of their songs' continuing appeal. Warhol seems to have intended that his guests associate his work with these voices: "When he expected important visitors from the art world, influential advocates of new art like Ivan Karp or Henry Geldzahler, Andy replaced a classical recording with a pop song." Left to his own devices, Warhol apparently preferred opera.
Through his sustained investigation of these contradictory forces and voices, Warhol achieved something much more fascinating than the critical or complicit postmodern superficiality with which he is typically credited. His work tested the possibilities of a truly universal and egalitarian art, an art that might challenge "the power of educated senses over that of unrefined senses, of activity over passivity, of intelligence over sensation," and thereby disrupt all the political and ethical hierarchies that fine art had traditionally protected. In the process, neoliberalism's mendacious promises of cultural participation, brand-based mobility, and self-determination might actually be realized—a "cultural revolution" could be close at hand. And even if Warhol's art failed to achieve these lofty goals, its incautious and "vulgar" strivings would provide a novel entertainment value of their own.
# Chapter One
# Varieties of Pop
During the early 1960s, critics quickly grouped Warhol with a number of other artists who were exploring similar images, styles, and technologies. Their work was dubbed "pop" or "pop art," in reference to the popular culture it resembled. These punchy terms have proven remarkably durable; they indisputably point to shared qualities and themes. For the art dealer Ivan Karp, pop was "painting or sculpture that revolved around commercial imagery or industrial images, signs and symbols from the everyday life, things that you see all the time, repeated interminably, and done in a . . . very stylized way." The "you" in his description was left capaciously vague.
But pop's critics have long disagreed about the _attitudes_ pop demonstrated toward popular culture. Did these artists affirm "commercial imagery" or critique it—cynically or sincerely? Were they resurrecting Marcel Duchamp's or Elsa von Freytag-Loringhoven's ironic and sophisticated readymades, or searching out and celebrating the most exhilarating visual culture available? Could "vulgar" objects and images provide a challenge to artists, a new test of their abilities to convert the mundane into form and beauty? These questions led to innumerable arguments regarding pop's political outlook, arguments that attempted to find an interpretive frame that would accommodate all of pop's practitioners.
Pop's shared iconography—its comic strips and commodities and tabloid headlines—has obscured some profound differences among its practitioners. Sorting out Warhol's approach to this imagery will require differentiating it from those of his peers, and particularly from the artists with whom he was most frequently compared in the early 1960s: Roy Lichtenstein and Claes Oldenburg—many of whose most celebrated works from this period were produced in uncredited collaboration with the artist Patty Muschinski. Lichtenstein's and Muschinski/Oldenburg's approaches to commercial imagery were vastly different from Warhol's; together, they demonstrate pop's diversity, and Warhol's unusual place within it.
The world that pop artists attempted to engage constituted a decisive transitional stage in postwar culture, when new strategies were developed to attract the attention of a vast segment of the American populace—the working class—that had hitherto mostly been considered an unworthy target for commercial culture. Pop artists' attitudes toward these strategies and audiences were divergent, to say the least; it barely makes sense to speak about pop as a single category in this regard. Lichtenstein recognized the power of this new culture, the "certain strong and amazing and vital things about it," but he was careful to distance himself and his work from its "vulgar" qualities: "I certainly don't think that popular life has a good social effect. In fact I think just the opposite. . . . It's not the society that I really like to live in." Investigated as an archive in its own right, Lichtenstein's source material reveals a far more nuanced and destabilizing set of powers and understandings than the paintings he produced from it were willing to accept. But Lichtenstein's anxious attitude toward mass culture was by no means shared by his pop art peers. Where he emphasized an essential difference between art and commercial culture, Muschinski and Oldenburg claimed the overturning of this distinction as one of their primary motivations. They found commercial culture fascinating and wanted to harness its vast new powers. For his part, Warhol seems to have maintained a profoundly ambivalent attitude toward mass culture—recognizing its great appeal but also its mendacious promises of cultural participation and social mobility, and their profoundly frustrating effect upon many of its greatest admirers.
## Lichtenstein: Things we hate
In 1979 an art student named David Barsalou began combing through 1960s romance and war comic books in search of the sources for Roy Lichtenstein's early pop paintings. It was a daunting project: at the time, as Barsalou remembers it, only three of Lichtenstein's comic-book source images had been identified. Barsalou eventually examined over twenty-five thousand books and found almost all of Lichtenstein's original panels. The results, collected on his aptly titled website "Deconstructing Roy Lichtenstein," are startling. Across dozens and dozens of images, Barsalou's research reveals unmistakable patterns of adjustment and alteration. Again and again, the comparisons show Lichtenstein adopting the basic shape and structure of a particular panel, its dominant forms and contours and color arrangements, but paring away the unnecessary details—background, hatching, halftones. The resulting paintings are punchier—more lurid—than their sources, but also cleaner and more aseptic.
FIGURE 4. _Our Fighting Forces_ , February 1963, 11. All DC comic artwork, its characters, and related elements are trademarks of and copyright DC Comics or their respective owners.
FIGURE 5. Roy Lichtenstein, _Grrrrrrrrrrr!!_ , 1965. Oil and magna on canvas, 68 × 56 inches. © Estate of Roy Lichtenstein.
Compare _Grrrrrrrrrrr!_ (1965) with its source, a panel by Joe Kubert. Lichtenstein adopted the composition almost wholesale but simplified and tidied up the furry hatching that gave Kubert's dog its disquieting dinginess and grit. The hatching along the ground plane was eliminated completely, as were the background figure, the dialogue along the panel's right edge, and the lettering in the lower left-hand corner. But the difference between the two images is much more than the sum of these details; as in every example Barsalou has uncovered, the colored dots in Lichtenstein's version have a smooth consistency that is absent in the source image. More than anything else, it is the pulp of comics that is repressed in Lichtenstein, the smudged unevenness of mass-produced Ben-Day dots on cheap paper. His simulated Ben-Day dots seem, next to their source material, to be distinctly sublimatory, consistently and efficiently converting these grimy panels into sharp, snappy images. They work hard to make their sources palatable, to aestheticize them, to render them acceptable for the walls of a penthouse or a museum.
Many critics, however, were unable to see the difference between Lichtenstein's work and its source material. Reviewing Lichtenstein's 1963 exhibit at the Ferus Gallery in _Artforum_ , Douglas McClellan dismissed the distinction out of hand, derogatorily belittling both original and copy in the process: "Lichtenstein has seemingly rearranged nothing, he has stayed reverently close to the originals except for greatly enlarging the scale. He has avoided the risks of transformation and he has picked a cripple for a target." Upon seeing a reproduction of Lichtenstein's _The Kiss_ in _Artnews_ , Arthur Danto was similarly unable to distinguish it from its source, describing the painting as "look[ing] like it had been cut out of the comics section of an American newspaper." It was this painting that provided the initial impetus for Danto's famous "artworld" thesis, which would come to be so closely associated with Warhol: "Suffice it to say that I was stunned. I was certain that it was not art, but as my year in France unfurled, I came increasingly to the view that if it _was_ art, anything could be art." Some viewers—like Adam Gopnik, one of the few critics to look closely at Lichtenstein's source material—have even claimed that Lichtenstein's adjustments were intended "to bring them closer to a platonic ideal of simple comic-book style." But the producers of such comics disagreed. As Lawrence Alloway reported in 1969, "I showed early comic strip paintings by Lichtenstein to a group of professional comic strip artists who considered them very arty . . . [and] old-fashioned in [their] flatness."
The tensions between Lichtenstein's paintings and their newsstand source material have dissipated over the last fifty years, as the paintings have gained ubiquity and their sources have moldered in closets and basements. Recent scholarship on the artist has yet to rise to the challenge posed by Barsalou's archive. Seen next to this material, the perfectionism that characterizes Lichtenstein's paintings takes on new significance: it has to be recognized as a consistent and concerted effort, not just to make beautiful or elegant paintings, but to rid those paintings of what Lichtenstein saw as the overbearing vulgarity that characterized their sources. Asked by G. R. Swenson in 1963 whether pop art is "despicable," Lichtenstein was anxious but explicit: "That doesn't sound so good, does it? Well, it _is_ an involvement with what I think to be the most brazen and threatening characteristics of our culture, things we hate, but which are also powerful in their impingement on us."
The dichotomy that Lichtenstein attempted to establish between commercial and fine art is crucial for understanding his project. On the one hand stands commercial work, which is pre-artistic: constitutively lacking in form, it has yet to be artistically unified. On the other hand is art itself—fully formed and noncommercial. Escaping commercialism was for Lichtenstein a matter of drawing the work out of the realm of commercialism and into the realm of the artistic: "What I do is form, whereas the comic strip is not formed . . . there has been no effort to make them intensely unified." As Michael Lobel has shown, this forming involved a painstaking process that incorporated drawing, projection, sketching, and painting. "Things we hate" were thereby converted, against all odds, from vulgarity to dignity. Contemporary critics agreed: in one of Lichtenstein's first major reviews, Donald Judd proclaimed that "the social meaning of comics . . . would be minor" among the works' references.
Lichtenstein's signature handmade Ben-Day dots were a crucial element in this equation. He insisted that his "techniques . . . are not commercial, they only appear to be commercial—and the ways of seeing and composing and unifying are different and have different ends." Part of the paintings' sublime achievement was to transform this most mundane, commercial, and mechanical medium into something worthy of artistic respect. Lichtenstein's early collectors felt that, by purchasing the artist's early pop paintings, they were sharing in the glory of this challenging endeavor. As Richard Brown Baker told his journal, "Having an excited admiration for Lichtenstein's _WHAM_ painting, I decided to take the risk and buy it. Without the taking of risks, a great collection is not formed." One of pop's most prominent collectors, Robert Scull, echoed this sentiment in _Vogue_ : "Somehow or other a good artist seems to transform these objects into a valid and wonderful statement which gives me a thrill."
The participatory culture Lichtenstein envisioned was resolutely hierarchical, with art and artists providing the possibility of a sublime transformation from lowest to highest through form, and buyers certifying this achievement through the "formation" of their collections. This vision of pop art was compelling and familiar, and played a central role in the definition of the movement, both prospectively and retrospectively. It was closely correlated with a powerful set of aesthetic assumptions about art's relation to the world of commerce and profit, and the role of form in distinguishing one from the other. The achievement of Barsalou's research is to have begun the recovery of the specific world that Lichtenstein's work attempted to aestheticize and transform. Put another way, Barsalou's archive demonstrates that the positive drive toward form and unity in Lichtenstein is always simultaneously a defensive attempt to ward off the specificities of the source material. But this recovery has only just begun, particularly since Barsalou's website provides only the individual panels Lichtenstein borrowed. These panels were, of course, always elements of larger stories, with themes and priorities of their own, stories that have never been considered worthy of art historical attention. As it turns out, these stories reveal new challenges—sexual, economic, and bestial challenges—to Lichtenstein's drive toward sublimation and aestheticization. There _was_ much to hate about these comic-book worlds and their imagery: they were often shallow and built on stereotypes; their understanding of their audience was frequently contemptuous; misogyny and racism were pervasive and mostly unquestioned. But a closer look at some of Lichtenstein's sources shows they also included strong alternatives to the hierarchical framework he propounded.
FIGURE 6. Roy Lichtenstein, _Drowning Girl_ , 1963. Oil on canvas, 68 × 68 inches. © Museum of Modem Art. Licensed by SCALA/Art Resource, NY. © Estate of Roy Lichtenstein.
"Run for Love," the source story for Lichtenstein's _Hopeless_ and _Drowning Girl_ (both 1963), is instructive in this regard. The hero, an "ugly duckling" named Vickie Brownley, is introduced as a tomboy who buries her disappointments in car repair: "I like to cry when I'm fixing my heap! C-can't t-tell m-my tears from th-the grease!" Vickie meets a mysterious stranger forebodingly named Mal and then repeatedly rescues him from calamity, twice when Mal is stranded with his car, and once when he is drowning. In each case, Vickie's competence and bravery activate Mal's misogynistic insecurity. He responds to Vickie's aid with resentment: "Run along, little girl! I need a mechanic—not a fugitive from kindergarten!"
FIGURE 7. _Secret Hearts_ , November 1962, 5. All DC comic artwork, its characters, and related elements are trademarks of and copyright DC Comics or their respective owners.
And yet, despite the fact that most of the story focuses on Vickie's courage, Lichtenstein chose to borrow two images that frame Vickie at her moments of maximum despair and peril. In _Hopeless_ , she has just daydreamed a better Mal, who would be secure enough to appreciate her competence and generosity. Awakening, she realizes that Mal will never be capable of this gratitude: "That's the way—it should have begun! But it's hopeless!" _Drowning Girl_ captures the moment in the story when Vickie's role shifts from hero to helpless victim; she has rescued Mal for the third time but is inexplicably racked by cramps while swimming to shore, giving Mal a chance to save her: "Helping a woman is a man's job!" he exclaims, "You see that now, don't you?" Through their selective borrowings, Lichtenstein's paintings work to elide the story's subtext of feminine competence and valor. "That's what you used to see in comic books," as Lichtenstein put it years later, "women who were like that, women were always in trouble." Such a thesis could be drawn from the conclusion to Vickie's story, but it overlooks the bulk of the narrative where it is the man who is "Mal"-functioning—"always in trouble." The paintings' condescension toward popular culture is achieved through a forceful suppression of the story's ambivalence regarding gender stereotypes; tellingly, in _Hopeless_ , Mal is rechristened "Brad."
Or take the source story for Lichtenstein's 1965 painting _Brushstrokes_ , published in _Strange Suspense Stories_ in October 1964 and entitled, simply, _The Painting_ (plates and ). In six surreal and convoluted pages, this tale follows Jake Taylor, an aspiring young painter, from childhood until old age. Taylor is haunted by a face he has painted, a sinister man with a widow's peak, who berates him for his perceived artistic failures: refusing to share his art with the public, and deigning to produce commissioned portraits of "Wealthy widows, their awkward daughters, anyone who asks!" Taylor's response is defensive: "You're not being fair! I must eat! I must pay the rent! What must I do to make you understand and stop tormenting me??"
The painted superego is a fanatical Kantian: he demands that the artist share his genius, and do so without thought of recompense. Taylor is too insecure—or too hungry—to heed these demands; the painted man repeatedly calls him a "coward." The panel that Lichtenstein chose to borrow illustrates Taylor's response to the painted man's challenges: he has taken a broad brush loaded with red paint and aggressively covered the painted man's face with sloppy strokes, marks that would have to have read in 1964 as referencing abstract expressionism. "The painting was destroyed . . . The voice was silenced . . . [Taylor]: I must be having some kind of nightmare!!"
FIGURE 8. Roy Lichtenstein, _Arrrrrff!_ , 1962. Oil and graphite pencil on canvas, 30 × 36 inches. © Estate of Roy Lichtenstein.
_Brushstrokes_ illustrates a pivotal moment in this doleful narrative. Abstraction is being proposed as an alternative to the Kantian aesthetic imperatives: if an artist cannot live up to Kant's lofty demands, perhaps he can drown them in paint and frustration. Unlike the story's cowardly painter, Lichtenstein will neither succumb to this temptation nor retreat to commercialism or seclusion. Instead, in this painting, he emblematizes his own effort to find a new accord between figuration and formal rigor. But in this respect, the original comic-book narrative may have the last word: on the story's final page, Jake discovers that the sinister man has been found dead. He is wracked by guilt, believing that his vandalizing the painting has caused the man's death. In penance, Jake repeatedly attempts to resurrect the man through his painting; the final panel shows him old and destitute, surrounded by portraits of the sinister Kantian, all silently refusing to share their wisdom. From the story's perspective, once the Kantian voice has been silenced and renounced, it cannot be resurrected; the reconciliation between figuration and Kantian quality that _Brushstrokes_ proposes is thus imaginary rather than achieved.
Many of Lichtenstein's early paintings were based on war comics, and the repetitiveness of their subject matter gives an impression of the generic: guns, planes, explosions—war as entertainment. Lichtenstein claimed this attention to war comics provided his work with a political dimension: "The heroes depicted in comic books are fascist types, but I don't take them seriously in these paintings—maybe there is a point in not taking them seriously, a political point." But Lichtenstein's sustained attention to the "Gunner, Sarge, and Pooch" serial in _Our Fighting Forces_ , from which he borrowed the images for _Flatten—Sand Fleas_ (1962), _Arrrrrff!_ (1962), and _Grrrrrrrrrrr!_ (1965), suggests an important subtext: the anthropocentric stakes of his artistic project. As Rosalind Krauss has argued, the sublimation of the vulgar into the artistic also involves the sublimation of the bodily into the optical. This conversion is linked, in Krauss's argument, with an anthropocentrism that privileges the upright human animal over the animal on all fours: "To stand upright is to attain to . . . the optical; and to gain this vision is to sublimate, to raise up, to purify." In much the same way, Lichtenstein's version of pop repeatedly subordinates these comic books' interest in corporeal sensation to its purified formalist opticality.
Lichtenstein's military dog paintings, like _Grrrrrrrrrrr!_ and _Arrrrrff!_ , exemplify this transformation, since their canine comic-book subject (in both cases a war dog named Pooch) is specifically valued for his superhuman acuity: "Pooch was our eyes . . . our ears . . . our nose . . ." The sergeant repeatedly admonishes his men to "stop talkin' about Pooch as if he's human!" Yet always, in these stories, it is the dog—with his parahuman and paravisual senses—who saves the day. In the source story for _Arrrrrff!_ , the marines have all been tricked into reading incendiary leaflets that end up burning their hands. "But there was one 'marine' who hadn't gotten a 'hot paw'—a K-9 by the name of Pooch, who had a nose as good as radar." Lichtenstein's version eliminates any reference to this storyline and derisively elongates Pooch's nose, rendering him comical and pathetic. In _Grrrrrrrrrrr!_ , the paravisual senses emphasized in the comic panel are elided, and replaced with the eponymous sound of aggressivity and menace. Here again, Lichtenstein's source material is far more radical and destabilizing than are the paintings he derived from it. Where the paintings advocate the priority of vision over the other senses and the anthropocentrism this entails, the source stories repeatedly reckon with a Deleuzoguattarian "becoming-animal" that might have the power to rescue, or even supersede, the human.
In all these ways, the source panels and stories complicate the hypothesis proposed by Graham Bader in his analysis of Lichtenstein's work. Bader argues that Lichtenstein's paintings from the 1960s, and in particular _Girl with Ball_ , materialize an "eruption of desire _through the means_ of the dot itself, transforming—schizophrenizing—its semioticizing structure into a libidinal engine, a none-too-disguised desiring machine." As the discussion of Muschinski and Oldenburg's work below will demonstrate, I am very much in agreement with Bader that the mass-cultural images and forms borrowed by pop art could be understood by their borrowers to be bursting with strange new energies and flows, and that Gilles Deleuze and Félix Guattari offer some of the most promising concepts for thinking about these energies. But the source panels and Lichtenstein's approach to them reveal that his work directly attempted to manage and aestheticize these energies, to convert them into familiar modernist moments in familiarly modernist contexts.
## Muschinski and Oldenburg: A true bonding
In contradistinction to Lichtenstein, Claes Oldenburg and Patty Muschinski were fascinated by "the _real_ political, sexual and formal energy in _living_ popular culture" that Art Spiegelman worried "passes [Lichtenstein] by." Their goal wasn't to prove art's superiority over these new forces but to overcome the difference between the two categories. As Oldenburg told an interviewer in 1966, attempting to speak for Warhol and Lichtenstein as well: "I have always been bothered by distinctions—that this is good and this is bad and this is better . . . [and] especially . . . by the distinction between commercial and fine art." For Muschinski and Oldenburg, the emerging world of mass culture had its own unmatched energies and attractions. Why convert these forces into high art—why aestheticize them, as Lichtenstein did—when they were so fascinating in their own right?
Muschinski and Oldenburg held that one of mass culture's fiercest powers was commerce itself, the flows and codes of money and exchange that Lichtenstein found so unpleasant. Their pivotal 1961 work _The Store_ tied these forces more closely to pop art than any of their contemporaries had dared. Unlike Lichtenstein, these artists were already chaffing at the restrictions imposed by the dealer-gallery system: "There is no escaping bourgeois culture in America. The enemy is bourgeois culture nevertheless. Assuming that I wanted to create some thing what would that thing be? Just a thing, an object. . . . These things are displayed in galleries, but that is not the place for them. A store would be better. . . . Actually make a store!"
The resulting project, a pop-up shop _avant la lettre_ , was located at 107 E. 2nd Street, in New York, and dubbed "The Ray-Gun Mfg. Co." In direct contradistinction to Lichtenstein's approach, _The Store_ seemed eager to break down all boundaries between artistic and commercial production and distribution. The dealer or gallerist, who customarily separated the artist from the actual mechanics of monetary exchange, was eliminated; the artists kept their own books (which were eventually published), and Oldenburg made his own sales. Although this work is typically credited to Oldenburg alone, he directly acknowledged Muschinski as its cocreator in a monoprint entitled _C A P_ and its accompanying caption. "The 'cap,'" he clarified, "stands for Claes and Pat, expressing the collaboration of husband and wife, particularly marked in the Store period." Despite her contributions, however, Muschinski was not featured as a joint proprietor. Oldenburg the artist _was_ Oldenburg the smalltime artisan and salesperson; his artworks _were_ his commodities, to be sold directly by the artist to customers. Of course _The Store_ was only a semblance of this equivalence. The project depended in part on some of its customers understanding that Oldenburg was an artist _pretending_ to be a smalltime storekeeper. As Yve-Alain Bois observed, "The solution was provisional, and Oldenburg knew very well that the objects he sold in his store would end up in a museum."
The objects produced at _The Store_ were strange hybrids: simultaneously commercial and handmade, they could never be mistaken for actual "store-bought" commodities. A back area was reserved for the fabrication of these items—"reproductions of discount food and clothing found in real stores on the lower East Side," or fragments thereof, coarsely manufactured in muslin-soaked plaster over chicken-wire frames and painted with enamel. Oldenburg said he "wanted a surface . . . that was as metallic and as uncompromising as the paint on the taxi-cabs or the paint on the billboards." The solution was a brand of industrial enamel called Frisco, which was deployed straight from the cans, without mixing. In the front area, customers could buy a _Times Square Figure_ for $149.98, or a _Statue of Liberty Souvenir_ for $169.98, or a _Plate of Meat_ or _Pepsi-Cola Sign_ in relief for $399.98 each. These objects were awkward cross-pollinations of the commercial and the artistic, attempting to imbue each category with the other's strengths and virtues. The work's fragility during this period scared away potential dealers. Recalling the artists' work for a Rubin Gallery show, Charles Alan stated, "I couldn't imagine persuading my clients to buy anything. They seemed so terribly fragile . . . that it worried me."
FIGURE 9. Claes Oldenburg, _The Store/Ray Gun Mfg. Co._ , 107 E. 2nd Street, New York, NY December 1, 1961–January 31, 1962. © Claes Oldenburg. Photo by Robert McElroy.
But commerce was not the only compelling force Muschinski and Oldenburg found in mass culture; they were convinced that its appeal was distinctly sexual as well. As Oldenburg put it in 1962, "Today sexuality is more directed . . . toward substitutes [for example] clothing rather than the person, fetichistic [ _sic_ ] stuff, and this gives the object an intensity and this is what I try to project." If the appeal of everyday objects depended in large part on sexualization, and if most people remained oblivious to the ways in which this sexualization was permeating their object-worlds, these artists' task would be to produce objects that exaggerated sexualization to the point where it could no longer be ignored. Furthermore, their objects would emphasize the strangeness, the non-normativity, of this emergent sexualization, the ways in which it departed from conservative prejudices for reproductive heterosexuality.
The sexuality that Muschinski and Oldenburg's work embodied was polymorphous and radically playful. Lichtenstein's paintings still propounded conservative sexual narratives: anthropocentric, phallocentric, often misogynistic, and heteronormative (as in the story of Vickie Brownley). Muschinski and Oldenburg's work gleefully rejected these narratives, embracing a rambunctious and unpredictable sexuality. As Oldenburg put it in 1969, "If I can possibly refer to both the male and the female anatomy, that's best." In their larger sculptural collaborations, Muschinski and Oldenburg discovered a tonality that immeasurably expanded this investigation: softness and malleability. Producing massive versions of mundane commodities out of vinyl and kapok, they found a way to challenge the standard modalities of everyday objects. In so doing, they also seemed willfully to reject Freud's warning concerning flaccidity, that "Man fears that his strength will be taken from him by woman, dreads becoming infected with her femininity and then proving himself a weakling. The effect of coitus in discharging tensions and inducing flaccidity may be a prototype of what these fears represent." The postcoital flaccidity of Muschinski and Oldenburg's objects stripped them of their masculine pretensions. Sculptural softness, Oldenburg coyly admitted, "must be some sort of sexual thing."
FIGURE 10. Ugo Mulas, _Claes e Pat Oldenburg_ , New York, 1964. Photo by Ugo Mulas. © Ugo Mulas Heirs. All rights reserved.
Similarly, where the size of Lichtenstein's paintings fit neatly into artworld expectations, Muschinski and Oldenburg's soft works were meant to remind viewers not of other artworks but of massive commodities like refrigerators or automobiles. Muschinski remembers walking up 57th Street with Oldenburg and passing luxury car salesrooms hawking "the latest Porsche and Jaguar" models: "Shiny, elegant, these parked beauties tempted the passerby with their exquisite design. . . . But for Claes, what he saw was: _form_ as _size_ that filled up a _space_. How to make his sculptures that large? This was the issue at hand."
Again, the disparity in priorities between Lichtenstein and Muschinski/Oldenburg is striking. Where Lichtenstein's references were almost exclusively art historical (abstract expressionism, cubism), Muschinski and Oldenburg gauged their work against the mass commodities that confronted them on a daily basis. As Alex Potts has pointed out, one begins to glimpse the fact that "different bodies of work by Oldenburg might be evocative of various social milieus and their class connotations, gainsaying the mythology surrounding them as being relatively universal and classless consumer items." The luxury cars on 57th Street certainly targeted a vastly different demographic than the dime stores on 2nd Street did. Muschinski and Oldenburg seemed comfortable drifting playfully from one world to another.
FIGURE 11. Claes Oldenburg, _Lipstick (Ascending) on Caterpillar Tracks_ , 1969. Cor-Ten steel, aluminum; coated with resin and painted with polyurethane enamel, 23 feet, 6 inches × 24 feet, 10½ inches × 10 feet, 11 inches. Yale University Art Gallery, Gift of Colossal Keepsake Corporation. Photo by Attilio Maranzano. Photo courtesy the Oldenburg van Bruggen Studio. © 1969 Claes Oldenburg.
But the most noteworthy difference between Lichtenstein's approach to mass culture and the approach taken by Muschinski and Oldenburg concerned the overarching ambitions of their artistic projects. Where Lichtenstein emphasized his own ability to convert even the lowest images into form, Muschinski and Oldenburg championed the ways in which their projects could alter consciousness—their viewers' and their own—by emulating everyday commodities. As Oldenburg declared, "The only way that art can really be useful [is] by setting an example of how to use the senses." In many ways, this seemed the closest pop art came to direct political efficacy. Responding to Oldenburg's proposed New York City monuments, the radical philosopher Herbert Marcuse remarked, "If you could imagine a situation in which this could be done you would have the revolution."
For Muschinski, art's potential for altering consciousness was even more distinctly personal and erotic: "After the completion of the hamburger, alone one evening in the gallery, [Oldenburg and I] christened it by making love on the meat, covered by the bun. I will say we almost choked from the heat of it!" Oldenburg relished a similar functionality in his _Lipstick (Ascending) on Caterpillar Tracks_ , 1969, which was installed on the Yale campus. The monument was initially intended as a podium for speakers at political protests, but it also offered refuge for intimate encounters: "People used to fuck inside of it. There was a way they could get in underneath the base." For both artists, in direct contradistinction to Lichtenstein, the senses are multiple and vision was in no way prioritized. Muschinski emphasized that her collaborations with Oldenburg were spaces of experimentation and release, where bodies and objects intermingled even before the artwork was complete: "Certainly, there were moments when the needle broke; sometimes, my blood splattered onto the piece of art on which I was working. A true bonding occurred!" Oldenburg commemorated these strange new interminglings in a 1959 drawing entitled _Pat Sewing_. What seems at first to be a typically absorptive scene—a woman completely immersed in her craft—becomes, on closer examination, a subtly sexualized image. The sewer straddles (owns or appropriates?—the image leaves the question open) a massive phallus, which seems to have ejaculated the thread she employs for her handiwork. Here, the absorptive qualities that Michael Fried would famously champion eight years later in his essay "Art and Objecthood" were lasciviously desublimated.
FIGURE 12. Claes Oldenburg, _Pat Sewing_ , 1959. Crayon on paper, 11 × 13 ⅞ inches. Photo courtesy the Oldenburg van Bruggen Studio. © 1959 Claes Oldenburg.
## Warhol: I just happen to like ordinary things
The differences outlined here between Lichtenstein's project and that of Muschinski and Oldenburg should help to illustrate the breadth of attitudes to mass culture and artistic production that were encompassed by the term "pop art" during the early 1960s. Lichtenstein's approach to mass culture remained bound to a canonically modernist understanding of creativity, which insisted on a firm distinction between art and commerce, allowing each new artistic genius to explore and expand art's borders by boldly sublimating its Others. This colonizing framework was shared by Lichtenstein's peer James Rosenquist, who told an interviewer in 1964, "A painter searches for a brutality that hasn't been assimilated by nature." For both artists, contemporary mass culture, with its commodified objects and bodies, was this unassimilated brutality.
In all of these areas, Oldenburg and Muschinski's shared practice was pushing adamantly in other directions. It tested new forms of sexuality and eroticization, foreswearing not just heteronormativity but anthropocentrism as well, and surveying a sexuality of objects—an interest shared by Tom Wesselmann and brilliantly explored, with a more critical tone, in Marisol Escobar's _Love_ , 1962. It abandoned modernist media, exploring new contexts, scales, materials, and modalities for creativity and participatory spectatorship, and embracing Robert Indiana's claim that "Pop is love in that it accepts . . . all the meaner aspects of life, which, for various esthetic and moral considerations, other schools of painting have rejected or ignored." It briefly challenged the distinction between art and commerce, and (through its emphasis on collaborative production) the idea of the sovereign creative genius. In so doing, it proposed a new world of creative production, where every space and body might attain its wild and productive potentialities.
Where did Warhol's work stand on this continuum between Lichtenstein's and Rosenquist's principled aversion to mass culture and Muschinski/Oldenburg's, Wesselmann's, and Indiana's embrace of its potential for psychological transformation? Where his peers tended to adopt partisan attitudes toward mass culture (either adamantly positive or adamantly negative), Warhol managed to maintain a profound ambivalence even as he delved deeply into its styles, images, and technologies. This ambivalence is evident in Warhol's contemporary statements on the question of cultural distinction, in which he remained remarkably noncommittal: "Commercial artists are richer." Or: "I just happen to like ordinary things." Or: "'Pop . . . Art' . . . is . . . use . . . of . . . the . . . popular . . . image." The refusal to take a side permeated Warhol's art as well. Stylistically, iconographically, and technologically, the work managed to keep mass culture's powers _and_ its frustrations active and in tension. Like Lichtenstein and Rosenquist, Warhol recognized the vulgarity of mass culture, the power and appeal of its lowness—he was even willing to incorporate that most vulgar of symbols, the brand image. (Lichtenstein wouldn't sink this low: as he told Lawrence Alloway, "There's something about brand names that I don't care for.") This allowed some critics to chalk all the pop artists up as "Vulgarians" or "Vulgarists," and others to credit Warhol with Lichtenstein's and Rosenquist's aestheticizing or "anti-popular" agendas. As Ivan Karp told a group of advertising illustrators in 1965, Warhol "transfigured that can of soup and raised it to the level of spiritual importance. By his act of transmutation he created beauty and meaning."
But where Lichtenstein's and Rosenquist's work was premised on the conversion of debased imagery into a fully integrated and aestheticized form, and the concomitant occlusion of this imagery's cultural specificity, Warhol's work was characterized by its attempt to identify and accentuate the grittiness of its motifs, both iconographically and stylistically. Lichtenstein quickly recognized his distance from Warhol in this regard: "I was an old-fashioned artist compared with him. When I looked at Andy, I looked at him as a tourist would." Wesselmann felt similarly distanced from Warhol: "There was nothing between he and I at all." Like Muschinski and Oldenburg and Indiana, Warhol recognized mass culture's energy and appeal; he experimented with various forms of sexualization and eroticization, and eagerly expanded his practice beyond traditional fine art mediums and modalities—into film, television, installation, snapshots, performance, conceptual art, books, magazines, and fashion modeling. And he broadened Muschinski and Oldenburg's investigation of the possibilities of collaborative cultural reproduction, fueled by his appetite for emergent technologies and media.
I believe it is Warhol's ambivalence, and the tensions it continues to communicate, that provided his persona and his work with their lasting power. Perhaps Warhol could see both sides because, unlike most of his peers, he had occupied both positions: fan and adman, consumer and producer, oppressed and elite. He knew the profound power mass culture held for a certain segment of society, but he also knew the ways in which this power was constructed and manipulated, and the cynicism and disdain mass culture's authors had for its consumers. Most of all, he understood the ways in which mass culture was marketed to its most passionate consumers as an arena not just of consumption but of participation. Interviewed in 1966, Warhol's mother, Julia Warhola, emphasized her son's passion for participatory culture: "Andy always wanted pictures. Comic books I buy him. Cut, cut, cut, nice. Cut out pictures. Oh, he liked pictures from comic books." Unlike Lichtenstein, Oldenburg, Wesselmann, and Rosenquist—and perhaps more like Muschinski—Warhol anticipated a truly egalitarian and "participatory culture" premised on domestic technologies that would encourage users to pursue their dreams of contributing to mass culture, and thereby becoming "active participants and agents of cultural production."
Warhol's profoundly ambivalent investment in mass culture was complexly predicated on his class background. On the one hand, as Julia Warhola's recollections illustrate, poverty intensified this investment. Mass culture had meant more to young Warhol because it was literally more precious to him—less accessible, more expensive. His "single most cherished possession" from childhood, an autographed Shirley Temple photograph, poignantly testifies to this investment. On the other hand, Warhol knew that the intensity of this caring would be perceived by others as class-coded, because, as David Graeber has argued, the working classes "are the caring classes, and always have been." Warhol would acknowledge this factor in the production of his iconic pop images, tying it closely to his mother's artistic production:
> The tin flowers she made out of those fruit tins, that's the reason why I did my first tin-can paintings. You take a tin can . . . and I think you cut them with scissors. It's very easy and you just make flowers out of them. My mother always had lots of cans around, including the soup cans. She was a wonderful woman and a real good and correct artist, like the primitives.
"Like the primitives": Warhol understood that this careful attentiveness to mass culture would be interpreted as an expression of the voice of an Other, a primitive, a child or beast who could not control its appetites. It would provoke some ferocious responses, since "Disdain at vulgarity is rooted in fear of the Other's difference, [which] by its very nature is subversive and carries the seeds of loss of control." But Warhol recognized that this "primitive" voice could also be appealing precisely for its marginality, and that its tone and its faith in the homemade could be appropriated. In his 1950s work, Warhol would draw directly on his mother's "primitive" style; it is her awkward handwriting that notates his drawings and prints. David Bourdon, who knew Warhol well during the fifties and sixties, insisted that Warhol's "metamorphosis into a pop persona was calculated and deliberate . . . as he gradually evolved from a sophisticate, who held subscription tickets to the Metropolitan Opera, into a sort of gum-chewing, seemingly naive teenybopper, addicted to the lowest forms of pop culture."
As an "artist of working-class . . . origin," Warhol would take this voice and make it his trademark—he would "mark [himself] positively with what is stigmatized—[his] provincial accent, dialect, 'proletarian' style, etc." His collaborators would grasp this connection as well. "We're putting everything together," said John Cale in 1966, describing the _Exploding Plastic Inevitable_ , the Velvet Underground's collaborative project with Warhol: "—lights and film and music—and we're reducing it to its lowest common denominator. We're musical primitives." In the years leading up to this collaboration, Warhol would pursue a similar vulgar primitivism through an impressive range of styles, techniques, and subjects, all tied together by perhaps his most vulgar invention: an unbridled interest in profit and "immediate gratifications" that seemed to undermine the institution of art itself. As he told Lou Reed in the recording studio, "Whatever you do, don't try to clean anything up, and keep it as raw as you do in real life. Don't change it for the record."
There have been moments when critics have identified class as a core concern in Warhol's work. Reviewing Warhol's 1980 _Portraits of the 70s_ exhibit, Peter Schjeldahl pointed out that "Warhol is one of the very few modern artists from an authentically working-class background." As a result, "A lot of what has seemed miraculous, angelic oddity about him is explicable in light of the modern American working class's avidity for the products and values of capitalist popular culture; ambivalence about these things usually has been the province of a middle class able to take their availability for granted from birth."
Schjeldahl's argument—that popular culture's appeal for working-class Americans tends to be far more direct and unambiguous than it is for their middle-class counterparts—dovetails with Graeber's observation regarding "the caring classes," and is crucial for any understanding of Warhol's project. Schjeldahl recognized Warhol's ambivalence toward this culture, but his attention to Warhol's emphasis on "that culture's social underside"—its car crashes and suicides, race riots and food-poisonings—does not fully account for it. Warhol's work and persona were also informed by a suspicion that, despite its promises to the contrary, the world of popular culture was channeled to its working-class audience unilaterally, in "a one-way relationship" without the possibility of "consumers" ever really participating in its production. It is this ambivalence—between enthusiasm and skepticism, absorption and alienation—that permeates Warhol's style and, to paraphrase Schjeldahl, is often mischaracterized as cynicism.
As I hope this chapter has shown, there is work to be done in attending more closely to pop art's internal variances and contradictions. The modernist project that Lichtenstein pursued has been vastly overemphasized; it was in no way definitive for all of his peers. Nor was the influence of Duchamp—Warhol's soup cans were something less and more than readymades. In Warhol's case, Richard Meyer has outlined the decisive challenge: "Rather than reinventing Warhol under the sign of this or that avant-garde artist, why not take his fascination with mass culture seriously? Why not . . . think seriously about Warhol's roots as a commercial illustrator and graphic designer, about his expertise in the language of advertising and the solicitation of consumerist desire?"
The chapters that follow will take up Meyer's proposals. The sophistication of Warhol's avant-gardism has been long established. He doubtless contributed to what Arthur Danto called "art's heroic-comic quest for [its] own identity, [its] true self." He must have recognized, as Benjamin Buchloh has argued, "that the conditions which had allowed the formation of the Abstract Expressionist aesthetic . . . had actually been surpassed by the massive reorganization of society in the postwar period." But at its best and most incisive, Warhol's was not fundamentally an art about the history or ontology of fine art; it was at least as much an art about the nature and possibility of participatory culture under neoliberalism. Warhol was neither merely a brave formalist searching out the newest brutality nor "an exemplary postmodern aesthete"; on the contrary, he was infatuated with all the most profound possibilities of the aesthetic—with desire and emulation and social mobility and self-improvement—as they presented themselves to marginalized and working-class subjects within the framework of neoliberal culture.
# Chapter Two
# Warhol's Participatory Culture
Attempts to come to terms with Warhol's relation to mass culture have returned periodically to the idea of the mythic. Donna De Salvo said of Warhol that he developed "a highly personal and specific mythology drawn from his everyday experience in the world." Barbara Rose argued that, "Dreaming the American dream, Andy not only creates but lives the American myths." Robert Smithson saw the relation as significantly more critical: "By turning himself into a 'Producer,' Warhol transforms Capitalism itself into a Myth"—a truly stupendous, almost unimaginable, achievement, and one to which Warhol himself never laid claim.
But Warhol did seem to have recognized his affinity for myth, and to have reflected on its significance in his practice. In 1981 he produced a portfolio of ten screenprints entitled _Myths_ , each depicting a different figure from American popular culture—Howdy Doody, Dracula, Uncle Sam, the Wicked Witch of the West. He wasn't pleased with the results. As he put it in his diary, "Cab to Mercer Street to have my pictures taken with my Myth prints (cab $8). . . . They just put me in front of the Myths, and I almost threw up, they looked so sixties. I'm not kidding, they really did."
Warhol was right to find these pictures painfully outdated—they returned to themes that had preoccupied him two decades earlier, but for some reason they now looked stale and unconvincing, lacking the energy of the 1960s work. Perhaps the mythic was now too familiar, too close at hand. By this point in the early 1980s, Warhol was hobnobbing with living myths on a daily basis; he lunched with them, took their pictures, heard their confessions. The night before Warhol photographed the Howdy Doody model, John Lennon had been shot. Warhol had been hanging out with Lennon intermittently since the late 1960s—he fondly recalled watching Mick Jagger raise a coke spoon to the Beatle's nose. His next appointment after the Howdy Doody shoot was with Ron Reagan, who had "just had lunch with his father [Ronald] at the Waldorf," and complained to Warhol that his mother, Nancy, hadn't extended an invitation to his wife Doria. In the process of all this socializing, Warhol had become a mythic figure in his own right, a shift he memorialized by posing as the Shadow for one of the _Myths_ prints. He had tried modeling in drag for the Wicked Witch, but one of his business advisors told him "not to use up the idea on this portfolio," to save it for later.
FIGURE 13. Andy Warhol, _Myths: Howdy Doody_ , 1981. Screen print with diamond dust on Lenox Museum Board, 38 × 38 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
There were procedural differences between these late _Myths_ and their 1960s counterparts. Warhol and his team had decided not to base the screenprints on appropriated images, as he had in the 1960s, but rather to take new photographs of models in costume. Warhol had approved this solution on financial grounds—he had to worry about permissions: "I think the best thing we decided to do is have people come and dress up in the costumes and we'll take the pictures ourselves, because that way there's no copyright to worry about." The results were disappointing—Warhol's special relation with myths required proximity and distance. The myths he found so fascinating lost their power when they were made in-house; in order to be effective, they had to be appropriated from elsewhere, permissions be damned.
The idea of myth as a key to Warhol's work seems to have originated with Michael Fried. Responding in December 1962 to the artist's first show at the Stable Gallery, Fried made myth the linchpin of his analysis and addressed the distinction between personal and public myths:
> An art like Warhol's is necessarily parasitic upon the myths of its time, and indirectly therefore upon the machinery of fame and publicity that markets those myths; and it is not at all unlikely that the myths that move us will be unintelligible (or at best starkly dated) to generations that follow. This is said not to denigrate Warhol's work but to characterize the risks it runs—and, I admit, to register an advance protest against the advent of a generation that will not be as moved by Warhol's beautiful, vulgar, heart-breaking icons of Marilyn Monroe as I am. These, I think, are the most successful pieces in the show, far more successful than, for example, the comparable heads of Troy Donahue—because the fact remains that Marilyn is one of the overriding myths of our time while Donahue is not, and there is a consequent element of subjectivity that enters into the choice of the latter and mars the effect. (Epic poets and pop artists have to work the mythical material as it is given: their art is necessarily impersonal, and there is barely any room for personal predilection.)
Fried's account was characteristically suggestive and illuminating. Not only was it the first critical response to emphasize the mythic dimension of Warhol's pop practice, it was also among the first to address his homosexuality, if only subtly. The myths Fried cited have proven remarkably resilient, while Warhol's "personal predilection" for Donahue and other men tended for many years to fade from the critical spotlight, as predicted. This despite the fact that an image of Donahue was featured in the full-page _Artforum_ advertisement for Warhol's Ferus show, and that, according to the catalogue raisonné, "Troy Donahue seems to have been of particular interest to Warhol. In the Warhol archive, photographs of him and reproductions taken from fan magazines outnumber those of other stars of this genre." Minimizing Donahue's importance in Warhol's work was clearly a way of minimizing its homoeroticism. Warhol's own opinion of Donahue was unequivocal: "Oh. He was so great. God!"
But Fried's characterization of Warhol's attitude toward the myths he appropriated remained indistinct. The work was said to be "necessarily parasitic" on the myths from which it drew. It was described as "necessarily impersonal"—most successful when least informed by the artist's subjectivity. Fried implied that Warhol's work was strongest when reduced to a transparent conduit for larger cultural myths, and seemed to locate the source of the artworks' pathos in his own personal reaction to the myth of Marilyn Monroe:
> Another painting I thought especially successful was the large match-book cover reading "Drink Coca-Cola" . . . in which Warhol's handling of paint was at its sharpest and his eye for effective design at its most telling. At his strongest—I take this to be in the Marilyn Monroe paintings—Warhol has a painterly competence, a sure instinct for vulgarity (as in his choice of colors), and a feeling for what is truly human and pathetic in one of the exemplary myths of our time that I for one find moving.
FIGURE 14. Andy Warhol, _Troy_ , 1962. Silkscreen ink and acrylic on linen, 14 × 11 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
In this passage, particularly at the junction of the two sentences ("at its sharpest . . . at its most telling. At his strongest . . ."), Fried seemed very close to recognizing something "truly human and pathetic" not just in Marilyn's face but in the Coca-Cola matchbook as well, but this possible insight was not pursued.
Four decades later, and well after the artist's death, Arthur Danto offered another extended account of the role of myth in Warhol's work. For Danto, myth had an essentially pacificatory function. His essay began by recalling Columbia University in 1968, when a group of students—who "looked and acted the part of Cuban revolutionaries, fighting alongside Fidel in Oriente province"—sat mesmerized by a performance of Buffalo Bob Smith and Howdy Doody: "tonight they were children again, singing with Howdy Doody, innocent of the injustices in the dark world around them." Myths were the stories or images that produced a community out of what would otherwise have been a disparate conglomeration of individuals: "If twenty-year olds in 1968 knew Howdy Doody in this immediate and intuitive way . . . the central community to which they belonged transcended differences between class and race."
Myth's function, for Danto, was precisely to "transcend" class and race differences (differences in sexuality or gender are not mentioned), thereby depoliticizing its subjects. The Columbia students were pacified by song—and more specifically by their willingness to sing along, to regurgitate the songs of their youth under the direction of an unctuous pseudo-cowboy and his marionette. Danto's Warhol recognized and exploited this power: "Warhol's political gift was his ability to make objective as art the defining images of the American consciousness—the images that expressed our desires, our fears, and what we as a commonality trusted and mistrusted." Put another way, far from challenging contemporary society, Warhol's work recapitulated American myths, extending their reach from the television set and the tabloid to the museum and the boardroom.
There are three elements of Fried's and Danto's approaches to myth in Warhol that may occlude important aspects of the artist's work. First, neither critic acknowledged the ideological dimension of the myths Warhol borrowed—their strategic or manipulative functions. They agreed that these myths brought people together—that they were "exemplary" and "overriding" (Fried) or able to transcend "differences between class and race" (Danto)—but they chose not to examine these functions more critically. Second, both critics disregarded the ways in which Warhol's reproductions of mythic images constantly emphasized their own incompletion and impossibility. And third, neither critic directly addressed the possibility of Warhol's work expressing a particular classed voice through its engagement with myth.
There are, however, subtle hints in Danto's and Fried's writings that such a classed voice might have been perceptible. Fried referred briefly to "Warhol's beautiful, vulgar, heart-breaking icons of Marilyn Monroe" and remarked the artist's "sure instinct for vulgarity." In an essay that is only a few hundred words long, the repeated references to vulgarity are all the more striking. Vulgarity, as T. J. Clark has shown, is fundamentally a class-based pejorative: a "betrayal, on the part of those who by rights ought to be in the vanguard of good taste" since, as Thorstein Veblen insists, "one should not betray an intimate knowledge of the material circumstances of vulgar life, or of the habits of thought of the vulgar classes." What Fried sensed in Warhol was a breach of bourgeois taste that was somehow successfully counterbalanced by the force of Warhol's aesthetic proficiency. "Beautiful, vulgar, heart-breaking": in these paintings, class betrayal was the middle term—the hinge—between pleasure and tragedy, but the specific workings of this intriguing relationship were not investigated here or elsewhere in Fried's work.
Likewise, although Danto's initial discussion of Warhol's _Brillo Boxes_ described their real-world counterparts as "homely," the works themselves were quickly reincorporated into the sublime transfiguration from "real world" to "artworld" that was Danto's primary aesthetic concern. Yet even in this rarified context, class briefly reared its ugly head: among a number of hypothetical _Brillo Box_ installations and their likely interpretations, Danto described a claustrophobic tunnel of boxes, which could be interpreted "as the closing in of consumer products, confining us as prisoners." "True," he admitted, "we don't say these things about the stockboy. But then a stockroom is not an art gallery, and we cannot readily separate the Brillo cartons from the gallery they are in, any more than we can separate the Rauschenberg bed from the paint upon it." The artworld, for Danto as for Lichtenstein, was precisely the place where one could reflect upon modernity's indignities without having to consider the particular class positions—like the stockboy's, or Ethan Hawley's, or Warhol's—that suffered these indignities most directly. (Billy Name, who installed the Stable Gallery show, seems to have intended these connotations: "In the front room of the gallery I arranged the boxes in a diamond pattern on the floor so that you had to walk through them like a maze to get to the back gallery. In the back room I piled them all up high like a supermarket or warehouse.")
In his subsequent writings on Warhol, Danto continued to deploy the first-person plural as a defense against the threat of class specificity or analysis. "For a dizzy moment," as he claimed in 1981, "we suppose the artworld must be debased by allowing the [ _Brillo Box_ 's] claim; that so base and _lumpen_ an object should be enhanced by admission to the artworld seems out of the question. But then we recognize that we have confused the artwork—'Brillo Box'—with its vulgar counterpart in commercial reality." A more recent monograph referred to "the bare declarative aesthetic of the proletarian representations [Warhol] began to favor," but then claimed, pages later, that Warhol's "mandate was: _paint what we are_." "The breakthrough," Danto continued, "was the insight into what we are. We are the kind of people that are looking for the kind of happiness advertisements promise us that we can have, easily and cheaply." The question of who "we" are and whether "our" enthusiasms may vary—or be presumed to vary—with our class, went unaddressed.
During the last twenty years, Fried's and Danto's readings have been profoundly challenged by scholarship that has taken a more critical view of the role of sexual, racial, and gender-based myths in Warhol's work, and their essential unattainability for real human subjects, what José Esteban Muñoz has called their "disidentifying" function. As Jonathan Flatley has argued, "Warhol's persistent, career-long exploration of what we might call the 'poetics of publicity' can be seen in part to rise from the need to mourn his own absence from the public sphere." With their expanded focus on questions of identity and difference, these accounts have made vital corrections to the readings of Warhol that preceded them. They have also introduced a more critical understanding of the role of the mythic, to which this book is deeply indebted. For Muñoz and Flatley, it was Warhol's conspicuous _inability_ to reproduce mythic ideals that made his work and his life so noteworthy.
This chapter and the next will argue that the category of the mythic is just as relevant to Campbell's soup cans and cheap diamond rings as it is to his depictions of Marilyn Monroe and Superman, and that none of these myths were remotely classless or universal. On the contrary, they targeted and were associated with a working-class audience, thought to be infatuated with TV and comic books, and "more receptive to advertising that is strongly visual in character" than their middle- and upper-class contemporaries. This working-class audience was encouraged, not just to consume these mythic products passively, but actively to reproduce them—to participate in their cultural production. With the proper guidance, the working class could join in a neoliberal "utopia" defined by "the liberty of consumer choice, not only with respect to particular products but also with respect to lifestyles, modes of expression, and a wide range of cultural practices"—cultural practices that would be duly monetized. Throughout his career, Warhol would serve as a canary in the coal mine for these new liberties.
The language of class thus permeated Warhol's work during this period from two directions: it infused both its motifs (no matter how "American" or "universal" some of them may now appear) and the technique and style of its execution. The working class's perceived fondness for popular culture—for brand names and celebrities and comic books—was a key element in Warhol's contemporary scene, one that he consistently incorporated into his work and his persona. But class was not restricted in Warhol's early works to the level of motifs. The way in which these motifs were isolated and reproduced itself communicated a specific class position. Warhol was able to ventriloquize or channel what contemporary experts described as a stereotypically working-class affect of "impulse-following," a form of immaturity or animality, a "non-rational" inability to regulate one's desires. He juxtaposed this with what Beverley Skeggs has called a working-class "'structure of feeling' . . . in which doubt, anxiety and fear inform the production of subjectivity."
In Warhol's work, this anxious impulsiveness coalesced around his own ability to reproduce the imagery he took as his subjects—the cans and bottles and celebrities and headlines he seemed to find so appealing—and thereby to achieve a truly "participatory culture." "Amateur cultural participation" is an apt way of describing Warhol's style during the 1960s and after; this style incessantly tested the possibility of consumer-grade reproductive technologies—silkscreens, Polaroids, tape recorders, video cameras—actually contributing to the common culture, despite the fact that "no one thinks they're professional." As one critic put it in 1968, "His art is an extended commercial for gadgets." (This addiction to gadgetry had class connotations of its own; as Veblen argued, "Innovation, being a lower-class phenomenon, is vulgar.")
Instead of merely celebrating the myths Danto and Fried described, Warhol's work investigated the real possibilities and impossibilities of their reproduction, and thus of a truly egalitarian and "social" media available to everyone. Warhol's reproductions of popular images telegraphed their own incompletion and impossibility—this was their "precisely pinpointed defectiveness" that David Antin remarked in 1966. His work sang along with the dominant myths it reproduced, but in the process it was constantly falling out of key and losing the beat, exposing the constitutive irreproducibility of songs that are seemingly easy to replicate, designed and built for sing-alongs. Warhol seems to have been completely aware of this tendency. In 1963 he told John Giorno that this gap between consumer and culture marked his emergence as an artist: "When I was nine years old I had St. Vitus Dance. I painted a picture of Hedy Lamarr from a Maybelline ad. It was no good, and I threw it away. I realized I couldn't paint."
## Only between the lines
Writing in 1957, Roland Barthes defined myth as "speech _stolen and restored_ ": "just as bourgeois ideology is defined by the abandonment of the name 'bourgeois,' myth is constituted by the loss of the historical quality of things: in it, things lose the memory that they once were made." For Barthes, it was the task of the mythologist to uncover this theft: mythology "attempts to find again under the assumed innocence of the most unsophisticated relationships, the profound alienation which this innocence is meant to make one accept." Myths, for Barthes, were steeped in "profound alienation" because they were built by one class and consumed by another, and because they were intended for consumption alone; their audiences were allowed to hear but not to speak, to see but not to reproduce.
And yet, during the years leading up to Warhol's emergence as a pop artist in the early 1960s, the possibility of a truly participatory culture, in which working-class subjects could produce cultural myths rather than just consume them, was actively marketed as an alternative to passivity and an escape from financial hardship. Art and culture promised to become something more than just "a one-way process: the artist communicating and the audience receiving." Advertisements for this promise were ubiquitous in downmarket publications, including comic books, magazines, and tabloids. They offered a range of solutions to the problem of cultural passivity: correspondence courses, contests, and devices of technological reproduction such as cheap cameras, sound recorders, and projectors—"Magic Art Reproducers."
In the face of these promises, Warhol's work from this period took up the cheapest and most accessible images available—images marketed to and associated with a working-class demographic—and tested the possibilities of their everyday, amateur reproduction. Could real culture, "mass" culture—comic-book characters, branded groceries, and celebrities—be convincingly remade at home by anyone, even the least prosperous, with the aid of these new reproductive technologies? Was participation in these powerful new myths—creative production and reproduction—now actually available for the audiences to whom they were directed and with whom they were most clearly associated? Could this participation truly be as easy as it looked in the commercials, could one "create things with the least amount of effort," as Gerard Malanga said of "Andy's aesthetic"? This chapter's task is to uncover the ways in which "the language deficits of class" that Benjamin Buchloh has remarked in Warhol's style were actively being produced during this period in and through the creation of new compulsions toward cultural reproduction—the reproduction of myths. In Warhol, the inability to reproduce the imagery of one's contemporary surroundings is as much a marker of class and powerlessness as the borrowed imagery itself.
During 1961 and 1962, as Warhol worked toward a distinctive pop style, his art seemed to diverge into two distinct and apparently incompatible representational approaches: one was neat, hard-edged, and practically mechanical; the other was gestural, messy, slapdash—"surrounded with A.E. [abstract expressionist] brushstrokes & East Tenth Street failure." Warhol would bring friends and critics into his studio and present them with two versions of the same motif, displayed side by side. In the summer of 1960, Emile de Antonio told Warhol to pursue the hard-edged style and leave the gestural behind. As De Antonio is said to have put it, "The abstract one is a piece of shit, the other one is remarkable—it's our society, it's who we are, it's absolutely beautiful and naked, and you ought to destroy the first one and show the other."
For a while at least, Warhol appears to have followed this advice. The two versions of _A Boy for Meg_ seem to exemplify this trajectory: the first painting revels in its own inability to reproduce something as visually straightforward as the front page of a newspaper (plate 5); the second is the earliest painting in the history of art to base itself strictly on the mechanically aided and unadulterated depiction of a newspaper's front page. As it turns out, however, the shift from a gestural style to a mechanical one was by no means punctual or unidirectional in nature. Warhol continued tinkering with these two alternative styles for almost a year. And in at least four early and important cases, he was adding rather than subtracting gesturalist irregularities in the final stages of his working process.
In April 1961, half a year before he learned of Lichtenstein's work and "turned white," the first known exhibition of Warhol's pop paintings was assembled, fittingly, in the window of the upscale Manhattan department store Bonwit Teller. A surviving photograph shows five overlapping paintings displayed at various heights in front of a dark backdrop, each matched with a smartly dressed mannequin (plate 6). The mannequins' poses were simultaneously brazen and coy—their chins mostly craning upward, their hips jutting toward the window. The dominant hues of the outfits—cherry red, ultramarine blue, and cotton canvas—matched those of the paintings in the background, and the mannequins and paintings were arranged so as to allow the latter a high degree of visibility. The juxtaposition is puzzling: were passersby expected to believe that these well-dressed figures read the comic books or succumbed to the tabloid ads whose verve and punch their outfits appropriated? Their perfect silhouettes, cast in shadows on the paintings' surfaces by the camera's flash, seemed to exist in a separate, painless, modern world, some distance from the drips and irregularities of the paintings' subject matter—cheap nose jobs and superheroes, hair tinting, hernias.
All five of the paintings Warhol displayed in the Bonwit Teller window were built upon the same basic structure: a two-dimensional motif (or, in one case, a composite of such motifs), seemingly amenable to reproduction, which—despite the apparent aid of mechanically reproductive technologies, including an opaque projector—was incompletely and anxiously reproduced. Three of the paintings were based on comics ( _Superman_ on a comic book, _Little King_ and _Saturday's Popeye_ on newspaper comic strips); the other two— _Advertisement_ and _Before and After (1)_ —drew on newspaper advertisements. Warhol told an interviewer in 1978, "I did some windows for Bonwit's and they were paintings and then a gallery saw them and I just began taking windows and putting them in galleries." In fact, Warhol seems to have altered at least four of these paintings after they left Bonwit Teller and before they entered the art market. These late adjustments, which are rarely so discernable in Warhol's work, directly disobeyed De Antonio's guidance, contradicting the established trajectory from gesturalism and scrawl to neat mechanical reproduction. The five paintings share a set of similar features and concerns; in what follows I will investigate _Superman_ as a representative example, and then proceed to discuss the newsprint works more generally. _Little King_ will be discussed in chapter 4.
The cultural connotations of a painting of Superman in the early 1960s would have been manifold and contradictory (plate 7). On the one hand, George Bernard Shaw's _Man and Superman_ (1905) had been given two Manhattan productions in 1960, less than a year before Warhol displayed _Superman_ in the department store's window. Reviews and advertisements had appeared prominently in the _New York Times_. Warhol's painting must have been seen in part as a mocking response to this high-minded fare. But in deriving its motif from a comic-book source that had been successfully adapted for television, _Superman_ would also have had powerful working-class connotations. Research suggested that comics in this period were read by over 90 percent of children and 80 percent of teens—clearly, these vast audiences were not exclusively working-class. And yet comics were powerfully associated with working-class audiences, who were assumed to be less educated and therefore more susceptible to the vulgar charms of mass culture. Articles about working-class children in the press often included prominent references to comic books. The _New York Times_ claimed Pepito, a fourteen-year-old youth gang member who had allegedly killed a taxi driver, "knows almost nothing about the world beyond his neighborhood [and] 'reads' the pictures of comic books." Its article on "tough" vocational high schools opened with the image of "switchblades instead of compasses on students' desks [and] the ace of spades for bookmarks in comic books." Dorothy Barclay summed up the overriding sentiment: while "a happy youngster in a good home . . . will not be turned toward a life of crime by reading crime comics," these lucky children are also "least likely to read them to excess." "It's the poorly adjusted child most liable to be harmed who indulges most freely. In this context, "poorly adjusted" was clearly a euphemism for "poor"—lacking a "good home." It could be admitted that all children read comic books, as long as one emphasized that only poor children were influenced by them.
The ubiquity of comic books notwithstanding, Superman's most prominent cultural appearance in New York of the early 1960s must have been the _Superman_ television program, which aired on Saturday and Monday evenings on channel 11, just before dinner time. The lead-ins on Mondays, starting at 4 p.m., were _Amos 'n' Andy_ , _Abbot and Costello_ , _Bozo the Clown_ , _The Three Stooges_ , and _Popeye the Sailor_ —a murderer's row of after-school temptations. The proximity of the _Popeye_ and _Superman_ shows on television may have been a factor in Warhol's decision to exhibit these two paintings together at Bonwit Teller.
Television was becoming functionally ubiquitous during this period; over five million sets a year were sold during the 1950s, and by 1960 TVs were installed in 90 percent of American homes, with the average viewer watching three and a half hours per day. And yet, as with comic books, the reputed connotations of television were tied to class; experts claimed that the working class was "more likely to enjoy the comics freely, to embrace television, and to watch late movies." A 1959 article in the _American Journal of Sociology_ reported, "Research has shown an inverse relationship between enthusiasm for television-watching and social class," concluding that "liking TV has become symbolic of low social status." The "dream world of television" was thought to be particularly "congruous with . . . the value given to _immediate_ gratification in the lower or working class."
But while television viewing remained closely associated with low social status during this period, it also became a crucial medium for the dissemination of neoliberal, class-free, "American" norms and ideals. In her study of television's influence on working-class children in the late 1950s, Adeline Gomberg argued that while the working-class child of the previous generation had "shared a common experience . . . only with children of his own class," television encouraged working-class children of the fifties to "[share] an experience with children of all classes . . . [and to aspire] to be like or marry television performers." And television followed these children to school. Gomberg described a game her subjects developed called "watching television" to while away their classroom time: "They lined up chairs and stared fixedly ahead at some imaginary screen. Occasionally one child would twist a 'dial' and the viewers would sigh." The intense passivity of the television experience is revealed in this game as a learned activity, requiring practice and repetition to be accepted by its practitioners, who can then grudgingly enlighten their perplexed teacher: "Gee whiz, don't you know nothing, [we're] watching television."
Contemporary attitudes toward comics and television enabled Americans to take their guilty pleasures while disavowing their cultural legitimacy, allowing elites to imagine a fundamental difference between themselves and "the vulgar and uneducated masses." The working-class connotations carried by comics and television therefore need to be recognized as ideologically motivated. And yet, as Gomberg's study so powerfully illustrates, television also played a key role in the interpellation of lower-class viewers into middle-class norms and ideals. Chief among these ideals was brand-based consumerism, which promised that social status could be attained not only through wealth or achievement but through possession of the proper commodities. Here too, Superman could teach the lessons.
Commercials for Kellogg's cereal, a primary sponsor of the _Superman_ program, epitomized this idea; one from the mid-1950s featured Clark Kent and his boss, Perry White, awaiting the arrival of the copy boy, Jimmy Olsen, who is late delivering a box of Sugar Smacks cereal for the group's breakfast. "Mild-mannered" Kent remains calm, but White grows irate over the delay. Jimmy, the minion, trips as he walks through the door, nearly fumbling his precious cargo, and the boss threatens to fire him. The scene cuts to the three men eating together. Strikingly, although the cereal was demanded by White and procured by Jimmy, it is Clark Kent, with his secret otherworldly powers as Superman, who speaks for the cereal and its capacity to transcend boundaries of taste and class: taking a bite, he announces, "Well, I guess we _all_ agree on Sugar Smacks!" Like the cereal, Superman/Kent has the uncanny ability to overcome class difference—he can leave the world of the ordinary and the subordinate when necessary and return when the responsibilities of superiority become too great. As Les Daniels has pointed out, Superman's creators patterned Clark Kent "after themselves, almost masochistically, making him timid, myopic, working class, and socially maladroit." Kent's ability to transform into Superman and back figured the overcoming of these limitations, but also their usefulness as a disguise from the pressures of privilege and power.
By the time Warhol painted _Superman_ , however, this marvelous figure had also taken on tragic connotations. Less than two years earlier, George Reeves, the actor who played the title role on _Superman_ , was shot and killed, reportedly by himself, in his Hollywood home. According to his fiancée, Leonore Lemmon, Reeves "was known as Superman to 9 million children, but he couldn't get a job." The series had outlived its actor: "They stopped shooting the series a year and a half ago. They had 105 chapters finished, and they can show them for the rest of their lives."
Reeves's posthumous image was still appearing on New York television sets twice a week throughout 1960 and 1961. The death was framed from the outset as a cautionary tale regarding the perils of celebrity: Reeves had achieved fame but at the cost of his individuality and the respect of his peers. His personality had been consumed by his fictional role; as his fiancée pointed out, at the time of his death he was simultaneously a hero to millions and utterly unemployable in Hollywood. What's more, his actual human presence had become irrelevant to the show's success—all the shows were prerecorded and available for perpetual syndication. TV's Superman was a ghost—the ultimate neoliberal "creative worker."
Reeves's story thus doubly demystifies the Superman myth, first by declassing it from the world of Shaw and Broadway to the pulp world of television and comics, and second by pulling away the veil of glamour from that pulp world, and exposing its disappointments and disasters. Warhol's _Superman_ was neither merely a straightforward celebration of an American myth nor a painting about an artist's inability to live up to his childhood ideals; instead, taken in its cultural context, the painting can be understood as a first step in Warhol's examination of the internal inadequacy of those mythic ideals, their terminal failure to live up to themselves. The possibility of absolute social mobility—figured both by Superman and by the actor who played him—was spectacularly revealed by Reeves's death as an unsustainable and yet irresistible fiction. These challenges were efficiently summarized on the cover of the December 1961 edition of comic _Life with Millie_ , where the title celebrity herself, a New York model, is shown as failing to live up to her own spectacular image. Warhol recognized these pressures acutely: "It must be hard to be a model, because you'd want to be like the photograph of you, and you can't ever look that way." As Jonathan Katz has argued, similar inadequacies would spur Warhol's interest in Marilyn Monroe, Jackie Kennedy, and Elizabeth Taylor. In the years to come, they would alternately energize and enervate his Factory Superstars.
All of these dimensions of _Superman_ 's context are germane to its meaning, but they leave unexamined what may be the painting's most striking formal feature, namely, its pronounced incompleteness, the areas of the canvas that seem to have been left unfinished. These areas are particularly noteworthy because they did not appear in the Bonwit Teller photograph; they were late additions to the canvas, added only after the painting had been shown in the department store window. Why would Warhol have gone back to this painting after displaying it to scrawl in wax crayon across the top margin, and to obscure, partially, the words in Superman's "thought bubble"? Is it cogent to conclude, with Marco Livingstone, that the scrawling "activated the sky area with a rhythmic linear pattern," or with Benjamin Buchloh, that it reinforced "the laconic mechanicity of the enterprise"? These late additions traverse the surface, seeming, at first, to resist and reject all logic, signifying precisely the absence of sense or signification: completely de-skilled—merely space-filling—marks.
As it turns out, the late additions cannot accurately be described as decorative or arbitrary. In each of the five areas where scrawling appears, it follows, or attempts to follow, one basic rule: never cross a contour line. It colors between the lines—behind Superman's cape, along the top margin, across the cold blue background. The scrawling seems to struggle against its own incompleteness: the marks were apparently drawn from left to right, filling in the space between their origins and the proximate edges and contour lines. Thus the scrawling begins midway across the upper margin and is initially divided into two registers, which lose and gain distinction as they proceed toward the right edge. It seems that the upper marks were begun first, as they continue, in various permutations, across the margin to the right-hand side. The lower register of marks was then brought in to fill some of the space left blank by the first round of scrawling. This doubling of the scrawl reaffirms the ruling prescript: "only between the lines"—it is willful and rapid, but willful and rapid only within a controlled framework.
The same tension animates the blue scrawling in the upper right-hand quadrant of the painting. Again, the marks begin arbitrarily, but they are quickly brought under control by the surrounding margins, and their rhythm becomes increasingly regular as they proceed. Where spaces are left vacant by one scrawl, auxiliaries are deployed to fill the gaps. Directly above Superman's head, a small area of blue sky is colored in almost neatly, from edge to edge. Similarly, although the milky white paint only covers some of the text in the thought-bubble, it is precisely constrained within its contours. All of these overtly handmade marks stand in stark contrast to the rest of the painting, which so clearly declares its dependence on mechanical reproduction: the image was originally created from a comic-strip panel that had been projected onto a canvas, and traced. Like Warhol's work more generally, these marks are " _very tangibly_ the product of a human being."
How did these late additions change the painting? The pertinence of this question greatly exceeds _Superman_ , since similar scrawls appear in a wide range of Warhol's work produced during 1961: _Batman_ , _Dick Tracy_ , _Dick Tracy_ _and Sam Ketchum_ , _Strong Arms and Broads_ , _Wigs_ , _Make Him Want You_ , _$199 Television_ , _Icebox_ , _Telephone [1]_ , _Dr. Scholl's Corns_ , _Coca-Cola [1]_ and _[2]_ , _A Boy for Meg [1]_ , and the very first _Campbell's Soup Can (Tomato Rice)_ , as well as numerous drawings. The Bonwit Teller photograph suggests that scrawls were late additions to _Advertisement_ , and that Warhol added sloppily descending drips to _Little King_ and cloudy white corrections to _Before and After [1]_ before they were sold.
If the scrawls were meant to convey mere reckless willfulness, or "the laconic mechanical nature of the enterprise," as Livingstone and Buchloh respectively argue, why were they pushed across the blocked-out surfaces with such careful attention to the rule of the contour line? This attentiveness suggests that the late additions constituted a determinate reassertion of handmade participatory culture against the industrially depersonalized tone that otherwise characterizes the image. They inscribed the artist's participatory intention across the inhuman blue of the sky and the arbitrary and mechanical upper margin. More than anything else, these crayoned additions resemble a printed panel half-converted by its reader into a coloring book—a common and revealing mass-cultural scenario replayed in bedroom after bedroom on comic books, comic strips, children's books, newspapers, and advertisements, as a result of an apparently acute and insatiable desire to reproduce the culture with which one is confronted. Warhol told an interviewer in 1978 that "when I was two I began to trace a lot." As his brother recalled, "When he was seven . . . [Andy] wanted a movie projector. . . . He'd watch Mickey Mouse or Little Orphan Annie and he got ideas and then he would draw a lot."
Beginning early on, in works like _Superman_ , Warhol's artworks found ways to visualize the suspicion that the possibility of participatory culture was somehow at least partially foreclosed by the mass-cultural object. The incomplete erasure of the "thought-bubble" text reiterates the suspicion: letters and words, the image's most legible and presumably most replicable elements, are rendered, in the painting's final version, irreproducible. The painting's final additions, its scrawls and counterscrawls, were in fact concerted efforts to qualify—rather than to finalize—its claim to cultural participation. These "finishing touches"—the very last marks added to each canvas—paradoxically imply haste and incompletion.
The areas that were retouched in this final version were literally marginal, displaced from the painting's ostensible subject: Superman's body and its action. This displacement highlights the painting's key shift: away from an emphasis on the physical irreproducibility of the mass-produced masculine ideal (problems of fitness and physiognomy), and toward an emphasis on the cultural irreproducibility of mass-produced visual ideals in general (problems of drawing, painting, image-making). This is a crucial transition in Warhol's practice, displacing the focus of the works' attention from the physical reproducibility of bodies to the cultural and visual reproducibility of images. The problem shifted, from how to emulate a mythic figure like Superman physically to how to reproduce a mass-produced mythic image—anything from a frame in a comic to a newspaper page to an advertisement—with the aid of consumer-grade reproductive technologies.
What might have prompted this rhetoric of thwarting and incompletion? Why would Warhol choose to take up the culture's most basic and reproducible images, then emphasize the difficulties or impossibilities of reproducing them? These questions only become thornier when one remembers that during the months surrounding the Bonwit Teller exhibit and all the way until early 1963, Warhol was regularly producing illustrations and advertisements for the _New York Times_. He did four bylined illustrations in February 1961 (one of them a full page for the _New York Times Magazine_ ), two in March, two in April, and five between August and November. During the 1950s, Warhol's unattributed illustrations for the Bonwit Teller department store had been even more prominent. A former assistant described him as "the busiest commercial artist at that point, making a tremendous top-notch salary." What did it mean, then, for a newspaper and advertising illustrator of this standing to imply in his paintings that newspaper and advertising imagery was somehow irreproducible, and that participatory culture was out of reach? In his work as an illustrator of advertisements, was Warhol not effectively disproving these assertions on a regular basis?
## Magic Art Reproducer
The stakes of this emphasis on cultural emulation can be better understood by examining the context in which Warhol found his borrowed imagery—by returning, for example, to the printed advertising that accompanied and subsidized the comic book from which _Superman_ was sourced: the April 1961 edition of _Superman's Girl Friend Lois Lane_. This thirty-six-page issue contained six and a third pages of advertising, pages that provide insight into the advertisers' understanding of its readers and their priorities. One and a third of these pages were dedicated to announcements for other comic books. Three other pages advertised employment opportunities: selling seeds, "Patriotic and Religious Mottoes," or skin cream. In these cases, the merchandise was sent out on credit, and the amateur salesperson was expected to mail the collected monies, minus a small profit, back to the company. This business model presupposed a relatively impoverished clientele, willing to invest a significant amount of labor for what would probably be a meager return.
FIGURE 15. Jowett Institute of Body Building advertisement, _Superman's Girl Friend Lois Lane_ , April 1961, 33.
The two other full-page third-party advertisements were at least superficially more conventional, in that they attempted to sell goods and services rather than to recruit salespeople. The most obviously pertinent ad appeared on the comic's penultimate page, and stood out from all the other pages in its complete lack of color. Instead, it was jam-packed with text and black-and-white photographs of male bodies in various states of exhibition. "I don't care what your age is!" the text announced. "Whether you're a teenager, in your 20's, 30's, 40's, or 50's . . . SKINNY OR FAT, I'LL BUILD YOU INTO A NEW ATHLETIC STREAMLINED MIGHTY-MUSCLED HE-MAN as I have for 35 years re-built MILLIONS like you!" The accompanying illustrations drove the point home: the teacher and his pupils had achieved the masculine ideal, and the reader was only one postcard away from joining them. The Jowett Institute of Body Building would take care of the rest.
Unsurprisingly, the ideal celebrated in this advertisement shared much with the figure of Superman as he appeared in the comic—Joe Shuster, coauthor of the first Superman comics, was an aficionado of bodybuilding magazines. The ad's motif of a man carrying a woman in his arms, for instance, appeared twice in this comic alone, on pages 7 and 9 of the final story. Through its replications of "real" idealized bodies, the advertisement seemed to harness the sexual energies that had preoccupied Fredric Wertham in his bestselling exposé _The Seduction of the Innocent_ : "Boys with latent (and sometimes not so latent) homosexual tendencies collect these pictures, cut them out and use them for sexual stimulation." Warhol would later coyly echo Wertham's claims: "The muscle magazines are called pornography, but they're really not. They teach you to have good bodies. They're the fashion magazines of Forty-second Street."
The issue's cover turned on this image of the male ideal, integrating it into broader issues of cultural participation. Lois Lane stands on a television set as a host with a microphone informs her that his computer has selected her "ideal husband." There, behind a dividing wall, is a man who looks like Clark Kent, unremarkable in his single-button suit and rep tie. But Lois has a different image in mind: Superman, his muscles rippling. The four figures' heads form a pyramid, with Superman at its apex; the eroticized masculine ideal presides over the entire scene, and the studio audience in the foreground consumes the resulting comedy and melodrama. The promise of cultural productivity within the period's most dominant medium—television, the "medium of optimism and opportunity"—would apparently be enough to convince these otherwise extraordinary figures to forfeit their dignity and privacy, while the spectacle of failure that attended this promise was sufficiently entertaining to fuel an entire TV genre. The image must have immediately appealed to Warhol when he saw it on the newsstand.
This same promise of cultural participation was brilliantly marketed by the comic book's first advertisement, printed inside its front cover, for a correspondence school called Art Instruction, Inc. Three rough sketches—the heads of a sad clown, a woman, and a dog—took up the bulk of the page, and the reader was invited to emulate:
> _Draw your choice_ of any of these heads—clown, girl or boxer. Draw it any size except like a tracing. Use pencil. Everyone who enters the contest gets a professional estimate of his talent. Winner receives the complete art course taught by world's largest home study art school. . . . Contest sponsored to uncover hidden talent.
FIGURE 16. Art Instruction, Inc. advertisement, _Superman's Girl Friend Lois Lane_ , April 1961, inside front cover.
The winner was promised a $495 scholarship.
This ad, which appeared again and again with varying illustrations in comics, magazines, and newspapers throughout the 1950s and 1960s, did not need to rely on the masculine ideal that figured so prominently throughout the rest of the issue. Instead of espousing any single cultural ideal, it sold the possibility of profitably participating in the production of visual culture. This was clearly a false promise for most of the comic's readers, but one that carried great allure. Like the multiple ads in this issue for amateur business schemes, this one was selling productivity rather than consumption, but unlike those advertisements, it was predicated upon the idea of _artistic_ productivity: the promise of participatory culture. As Deleuze and Guattari would warn in 1972, "The deliberate creation of lack as a function of market economy is the art of a dominant class"—to grow, capitalism requires the creation of new desires. Here the created "lack" was mass-cultural participation.
By the early 1960s Warhol was already a poster child for this promise; his facility as a draftsman had provided him with an escape from poverty and admission to the world of cultural production. In a 1977 interview he even claimed that his childhood teachers had submitted his drawings to a correspondence art school contest:
Warhol: . . . if you showed any talent or anything in grade school, they used to give us these things: "If you can draw this," where you'd copy the picture and send it away . . . .
O'Brien: Famous Artist's School?
Warhol: Uh, yeah.
O'Brien: Did you send them away?
Warhol: No, the teachers used to.
O'Brien: Did they say you had natural talent?
Warhol: Something like that. Unnatural talent.
"Unnatural talent": the promises made by these schools were distinctly anti-Kantian; they tied artistic productivity directly and unapologetically to economic success. The schools communicated, even in their advertising, a basic paradox of participatory culture—skill could only be reliably gauged by copying preexisting visual-cultural icons, but this copying could never be allowed actually to duplicate the original: "Draw it any size except like a tracing." Popular culture was founded on the passivity of the consumer, which might eventually be seen as a limiting factor in her satisfaction and pleasure. The commoditized spectacle of the consumer's own cultural participation promised to correct this constitutive passivity.
Comics targeting a female audience frequently included opportunities for readers to mail in drawings of outfits and hairstyles, some of which would be included, with credit to their creators, in future issues. In a very few cases these promises yielded real opportunities: the consumer of images aced the correspondence school, landed a job on Madison Avenue, and became a producer of images—Lois Lane ceased to be her ideal and became instead her product. But even this rare scenario did not fully correct the problem, because to the degree that the consumer became the successful manipulator of images, she was unlikely to believe in them fully. Art Instruction, Inc.'s invitations to cultural production targeted television viewers as well, appearing regularly on the television schedule pages of major newspapers. The ads' finished and unfinished faces shared space on these pages with glamour shots of television and film celebrities. It was a logical conjunction: television viewers could be expected to be more interested in these iconic faces, and therefore in the possibility of reproducing them, if only on paper.
The class-based dimension of these fantasies of participatory culture was made directly apparent in numerous advertisements for Art Instruction, Inc. and its rivals. During the late 1940s Art Instruction's small advertisements were dominated by a simple tag line: "ARTISTS MAKE MONEY." The Washington School of Art's contemporary headline was "DRAW for MONEY / BE AN ARTIST!" (As Warhol would later put it, "making money is art, and working is art and good business is the best art.") A Famous Artists Schools advertisement printed in the _Los Angeles Times_ in 1959 was headlined, "What went wrong for the kid who loved to draw?" These words were superimposed over the image, in a grayed tone, of a boy sitting at a desk surrounded by art supplies, painting. At the right of the page, overlapping the boy's shoulder and the table, was a man clad in recognizably working-class clothing, including a work coat, and carrying a black lunch box, the contemporary Madison Avenue emblem of the laborer. The creative boy was meant to be understood as the working man's past self—less real than his adult self, but more vital and free. The ad's text drove the point home:
> With our training, Wanda Pichulski gave up her typing job to become fashion artist for a local department store.
>
> Stanley Bowen, father of three, was trapped in a low-paying job. By studying with us, he was able to throw over his job to become an illustrator with a fast-growing art studio, at a fat increase in pay!
>
> John Busketta was a pipefitter's helper in a gas company. He still works for the same company but now he's an artist in the advertising department at a big increase in pay.
FIGURE 17. Famous Artists Schools advertisement, _Los Angeles Times_ , September 13, 1959, 17.
Another Famous Artists Schools ad distributed during the late 1950s and early 1960s delivered a similar message: "They DREW their way from 'Rags to Riches'—Now they're helping others do the same." The opening line: "Albert Dorne was a kid of the slums who loved to draw. He never got past the seventh grade. Before he was 13, he had to quit school to support his family. But he never gave up his dream of becoming an artist."
Pierre Bourdieu described such promises as "schemes of 'continuing education' [ _éducation permanente_ ], a perpetual studenthood which offers an open, unlimited future"—in this case, the future of cultural participation. The gap between these promises and social reality was, of course, massive. Using data from the early 1960s, Richard Sennett estimated that only 1.8 percent of men raised by manual laborers would secure professional careers, and only 0.8 percent—eight in a thousand—would become self-employed professionals. The career of "Artist who paints pictures that are exhibited in galleries" is only one among dozens of these professional categories, further reducing the already minuscule odds. When they promised to lift their working-class readers from "the slums" to the art studio, the advertisements examined in this chapter were thus promulgating—and profiting on—desires that were extremely unlikely to be fulfilled.
The correspondence art school was not the only promised route to cultural participation and remuneration during this period. Interspersed with the art school ads, sometimes sharing the same pages, were ads for a number of mechanical shortcuts: cheap cameras, oil painting services, recording devices, and the Magic Art Reproducer (fig. 3). This last was a miniature camera lucida that employed a mirror to transfer an image onto the horizontal surface beneath it, a smaller and cheaper version of the device Warhol used to produce many of his early pop paintings. The "de luxe" model (the only model) was marketed for $1.98 in comic books, magazines, tabloids, and the back pages of newspapers. The pitch was familiar: "Have fun! Be popular! Everyone will ask you to draw them. You'll be in demand! After a short time, you may find you can draw well without the 'Magic Art Reproducer' because you have developed a 'knack' and feeling artists have—which may lead to a good paying art career." These promises seem to have been squarely aimed at a working-class audience, which allegedly "believes in getting by with as little education as possible" and "tests activity by whether it is 'fun.'" Even those, like Warhol with his "shaking hand," who "can't draw a straight line," could now draw "anything from real life—like an artist." "Draw or paint in one minute!" read the box, "No lessons! No Talent!" The device came with a booklet entitled "Simple Secrets of Art: 'Tricks of the Trade.'" Together, they promised working-class readers a proto-Warholian reconciliation of the mechanical and the creative that would circumvent the expense and inconvenience of formal education.
Like many hopeful readers, Warhol took these advertisements at their word and tried in vain to square the circle between mechanical imitation and artistic creativity. Unlike almost all of them, however, he profited from his failures. Warhol would spend the rest of his life testing mechanical reproductive devices against their advertised promises of cultural participation. Each new reproductive technology was immediately assessed: the opaque projector would be replaced by screen-presses, tape recorders, Polaroids, video cameras, personal computers. These machines heralded amateur cultural participation freed from the burden of training, and in Warhol's work, each spectacularly failed to fulfill its promises. As a result, imperfection and irregularity remained central to Warhol's style. When Warhol said, "I wish I were a machine," his words emphasized his distance from the ideal of reproduction and reproducibility as much as his proximity; he never claimed to _be_ a machine, only to _want_ to be one. But, as Claes Oldenburg pointed out in 1964, the results were disappointing: "Andy keeps saying he is a machine and yet looking at him I can say that I never saw anybody look less like a machine in my life." Elenore Lester called him "an open shutter"—taking everything in, never closing, never completing the picture.
FIGURE 18. Andy Warhol, _Before and After [1]_ , 1961. Casein on canvas, 68 × 54 inches. The Metropolitan Museum of Art. Image © Metropolitan Museum of Art. Image source: Art Resource, NY. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
In _Superman_ , this distance between machine and subject is figured as the difference between a mechanically projected image and its hand-painted execution. The painting is unmistakably premised on mechanical reproduction—a fact spelled out by the impossibly facile contours of Superman's body, the geometric lines of his breath, and the billowing clouds of smoke, all clearly copied from a source image (plate 8). But Warhol's final additions to this painting, the scrawls and drips and obstructed text, also insist that this mechanical reproduction has been facilitated by a human hand, a hand that was somehow too jittery—too enraptured—to complete its seemingly straightforward task. Text, line, and color, those most elementary elements of visual communication, are marked as difficult and unfinishable—beyond the artist's reach. Even "coloring between the lines" is too much trouble to finish. A stereotypically working-class hand, then—hasty, uneducated, childlike, addicted to gadgets—that "reaps the 'pleasures' of following impulse and not 'deferring,'" but that is never quite sure that it has "got it right." This emphasis on the anxious condition of not getting it right despite mechanical assistance would become axiomatic in Warhol's production. As Malanga recalled, "Each painting took about four minutes, and we worked as mechanically as we could, trying to get each image right, but we never got it right."
_Superman_ was not alone among the Bonwit Teller paintings in its emphasis on class and the vicissitudes of cultural participation. At first glance, _Before and After [1]_ is a painting about cosmetic surgery and the unbearable pressures of normative cultural ideals. Warhol was enthralled by these pressures, all the more so when they were extreme. As he put it in 1975, "I'm fascinated by boys who spend their lives trying to be complete girls, because they have to work so hard—double-time—getting rid of all the tell-tale male signs and drawing in all the female signs." But as in _Superman_ , these pressures toward norms of bodily beauty are displaced in _Before and After [1]_ onto the brushwork itself, which makes a show of its losing battle to reproduce flat, two-tone images borrowed from downmarket media. And this losing battle appears to have been the thing Warhol most wanted to emphasize. The final additions of white paint seem intended to hide the mole next to the left-hand figure's nose, to widen and adjust the white of her eye, and to soften the contours of her lashes. But the white paint is a poor match for the neutral background and only partly covers the dark paint it is meant to correct. It is the paintbrush here, not the surgeon's scalpel, that struggles to reproduce the ideal form. In this respect, Warhol's line resembles the disciplined "tracing" described by Deleuze and Guattari: an attempt to mimic "the dominant competence of the teacher's language." The promises of a truly participatory culture, where each "consumer" could become a "producer," are tested and found wanting.
FIGURE 19. Andy Warhol, _Advertisement_ , 1961. Acrylic and wax crayon on cotton, 69¾ × 523/8 inches. Hamburger Bahnhof–Museum für Gegenwart. Photo: Berlin/Museum für Gegenwart/ Jochen Littkemann/Art Resource, NY. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
This same drama is apparent in _Advertisement_ , a pithy catalogue of the marketing of physical improvement. The "before and after" faces reappear, in miniature, alongside an offer of "strong arms" from Anthony Barker, a George Jowett contemporary and competitor. Above these are ads for Pepsi-Cola and for a "Rupture Easer," which promised to treat hernias. In the upper left is an ad for hair coloring; Warhol has the dye dripping down the figure's neck like blood or errant contour lines. Dirty yellow scrawls seem to have been added to this corner after the painting left the window display. None of the ads have been reproduced completely, and most are less than half finished. Words and even hand-drawn letters are left incomplete, as are the borders of boxes and the oval of the Pepsi logo. And yet the copied elements evidence a fairly high degree of exactitude and purpose; furthermore, the neat regularity of their contours suggests the use of a mechanical aid, in this case probably a photostatic copy and an opaque projector. Script is visualized in these works; the reproduction of letters and words is treated like drawing and tracing rather than writing. It is as though the painting is suspended between a desire to reproduce the motifs accurately and a recognition that this task is ultimately impossible. The painting's motifs are similarly suspended—between the promises of physical perfection somehow attained through a twenty-five-cent pamphlet and the realization of the emptiness of that promise. And, crucially, these twin suspensions are staged with what have to be recognized as distinctly working-class props. Warhol borrowed these images from contemporary tabloid magazines; three have been identified as deriving from issues of the _National Enquirer_ from late March and early April 1961. The "rupture" ad has proven irresistible to poststructuralist readings of Warhol, but we should not forget its origin in an ad promising treatment for hernias, a traditionally working-class affliction. The stereotypically ethnic, working-class profile in the before-and-after image is unmistakable; Pepsi's contemporary image as "oversweet bellywash for kids and poor people" is now less legible. It is the Pepsi logo that stands out among these otherwise adamantly anthropomorphic images, presaging Warhol's turn toward the brand image.
The Bonwit Teller photograph thus offers an unusual perspective on Warhol's early production. Irregularity in these works was neither an accidental by-product of the reproductive process nor an abstract expressionist holdover; it was an intentional and calculated addition to the works' style. Again and again, it is the defective capstone that simultaneously completes and undoes these paintings. In early Warhol, the inability successfully to reproduce the imagery of one's contemporary surroundings is as much a marker of class and powerlessness as the borrowed imagery itself. "To be working-classed," as Annette Kuhn and Beverley Skeggs remind us, "generates a constant fear of never having 'got it right.'"
## Some great assault
These concerns pervaded Warhol's early pop production. The first version of _A Boy for Meg_ (1961) is characterized by a similar set of displaced frustrations. Yes, the front page Warhol chose to reproduce is dominated by celebrities and their allure. The two sexual tendencies juxtaposed in the painting—one reproductive (a birth announcement), the other lurid and animalistic ("Rat Pack")—enliven the drama, and the missing direct object in "A Boy for . . ." subtly substitutes a homoerotic potentiality for a reproductive certainty. But in this first version, the reproductive drama is as artistic as it is sexual in nature. Warhol again seems intent on making visual the hidden challenges of participatory culture. Everything is scrawled and unfinished, words and letters start out legible and convincing and quickly end up sketched and broken. One scrawl fails to convey what it is meant to, a second rougher scrawl comes in to finish the job and fails, information shades off into noise. And throughout, the artist's failure to complete the simple task is constantly reiterated; he seems to have given up halfway again and again and, finally, to have judged the whole project impossible to finish.
The script along the inside of the painted ring in _Carat_ (1961) effectively summarizes these concerns. It is as though the entire possibility of participatory culture rests on Warhol's ability to paint an "S" that looks mechanically engraved, as though the forever-postponed duplication of this feat, the second "S" in "HAPPINESS," might truly bring about emotionally the word it would complete materially, realizing the promises of working-class cultural participation. Ads like the one Warhol copied were printed on a regular basis in working-class tabloids like the _New York Daily News_. They were smaller, cheaper ads, relegated to the margins of the page. Beneath one, an ad for mineral oil with the headline: "Woman Screams As Feet Burn!" To the right, a Macy's ad: "YOU CAN'T GO TO HAWAII THIS SUMMER? NEVER MIND, MACY'S BRINGS HAWAII TO YOU" with "Exotic cotton print sun dresses, gorgeous as a Gauguin, at a gorgeous Macy-low price." This ring was part of "New York's Largest Discount Display," and could be purchased on credit for $2.75 down, $2.00 weekly.
It is indicative of the velocity of Warhol's dialectical imagination that the pathos enacted in the early newsprint paintings had become thematic by 1962 in the strange and brilliant _Dance Diagram_ and _Do It Yourself_ paintings (plate 9). In the dance diagrams and paint-by-numbers kits that are encountered by the consumer, the creative possibilities of art—of dancing and painting—are formalized and mechanized so that they can be bought and sold as an activity. These diagrams introduce their users to the rudiments of an aesthetic vocabulary while withholding the possibility of creativity. This is the world, for Warhol, of mass culture in America. Even when it pretends to be participatory, mass culture is almost exclusively a one-way street. The contours are drawn out in advance and the colors predetermined. In his work and his persona, however, Warhol professed not to recognize this fact, adopting the attitude of the stereotypical working-class subject, desperately attempting to participate in the culture he valued so deeply. Warhol would return to the possibilities of dancing in the _Exploding Plastic Inevitable_ , his remarkable collaboration with the Velvet Underground, now testing them in a participatory and immersive way, but always "like the primitives."
FIGURE 20. Andy Warhol, _Carat_ , 1961. Water-based paint on linen, 52¼ × 48 inches. Daros Collection, Switzerland. © 2016 Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
FIGURE 21. Andy Warhol, _Warren_ , 1962. Silkscreen ink on linen, 13 ⅛ × 10 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
What Warhol recognized in American myths was not merely the "universal" appeal of superheroes or of bodily perfection, or the personal pathos of his own distance from these ideals, but the power these myths held for the disadvantaged—the "dubber[s]" who will never be "pro[s]"—and the ways in which participatory culture was marketed to them alongside consumption. He pursued these promises across a variety of subjects and technologies. The smudged and unfinished pearls in the second—cleaner—version of _A Boy for Meg_ tell this same story. They are the painting's pivot point: the objects that both signify status and reveal its irreproducibility.
FIGURE 22. Andy Warhol, _Front and Back Dollar Bills_ , 1962. Silkscreen ink and pencil on linen, 82¾ × 19 inches each. © 2016 Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Screen-printing became an ideal technology for testing participatory culture: it could replicate images repeatedly, and at an enlarged scale. The compulsive repetition of many of the silkscreens dramatizes this irreproducibility; keep trying, in _Natalie_ , _Warren_ , or _Marilyn_ (1962), to build an image of the movie star that would capture something of her real power and charm (fig. 21). Choose the perfect image, crop it just so, and then print, print, print. . . . The very process of replication produced distortion, especially when used in Warhol's slapdash way—"disavowing major and worrisome self-disciplines," in the stereotypically working-class manner. Often, a hasty hand would push too much ink through the screen, darkening the image. And ink never passed through the screen without remainder; absent regular cleaning, it would clog the mesh, preventing future flow. Images grew fainter and grittier as they proliferated, more ghostly, more distant, less regular, less legible.
Warhol seems to have relished these inconsistencies. They proved once again that the images the culture industry makes available are constitutively unreproducible for the majority of their intended consumers. There will be no room, as there might have been with old ideals, to construct a circuitry: to desire Warren, to emulate him, to be judged by other Warren lovers, to take on his qualities, and to ascend to his level. "Warren" and "Marilyn" are commodities; it would be like building a can of Coke in your basement, on a budget and in a hurry. The result would be just as counterfeit as the imagery Warhol reproduced in his very first silkscreened paintings: those quintessentially working-class currencies, one- and two-dollar bills. As Warhol would later observe, "rich people" don't bother with these bills: "They carry their money in a business envelope. . . . And the tens have a paper clip on them, so do the fives and twenties. And the money is usually new."
Recalling his childhood trips to the movie theater, Leo Proudhammer, the narrator of James Baldwin's 1968 novel _Tell Me How Long the Train's Been Gone_ , remarked a powerful connection between celebrity, participatory culture, and the brand image:
> I looked at the posters which seemed magical indeed to me in those days. I was very struck, not altogether agreeably, by the colors. The faces of the movie stars were in red, in green, in blue, in purple, not at all like the colors of real faces and yet they looked more real than real. Or, rather, they looked like faces far from me, faces which I would never be able to decipher, faces which could be seen but never changed or touched, faces which existed only behind these doors. . . . Some great assault, certainly, was being made on my imagination, on my sense of reality. Caleb could draw, he was teaching me to draw, and I wondered if he could teach me to draw faces like these. . . . They were not like any people I had ever seen and this made them, irrevocably, better. . . . But only the faces and the attitudes were real, more real than the lives we led, more real than our days and nights, and the names were merely brand-names, like Campbell's Baked Beans or Kellogg's Corn Flakes. We went to see James Cagney because we had grown accustomed to that taste, we knew that we would like it.
For Baldwin as for Warhol, the star's face was both the most vivid and the most remote possible image—"seen but never changed or touched." Stars were endowed with a Benjaminian aura—"the unique apparition of a distance, however near it may be." Drawing these images might bring them closer. What Graeber calls the "unequal structures of imaginative identification . . . and] creativity" might be overcome and a truly participatory culture achieved, if only for an afternoon. After all, the colors seemed simple enough . . . But, as Baldwin recognized, the odds would be stacked against the impoverished amateur under neoliberalism—"a failed utopian rhetoric" of participatory culture and social mobility, "masking a successful project for the restoration of ruling-class power." And the brands stood guard—Kellogg's, Campbell's, Cagney—the same powerful and domineering icons against which Warhol would stage his next claims to cultural participation ([fig. 40).
What's more, as Warhol was discovering, a tragicomic inability to reproduce these icons might carry an entertainment value of its own. The garish colors of these smudged heroes could appeal simultaneously to two very different audiences, depending on their understanding of the artist's position. Did these images record the heartfelt efforts of a working-class mass culture junkie, desperate to recreate the culture in which he had been steeped? Or did they ironize any such attachment to popular culture and its reproduction, producing a spectacle of failure and longing? Of course, to the extent that they embraced his work, Warhol's most influential interpreters took the second position; it was unthinkable to them that artworks this powerful could be speaking in an authentically working-class voice. Instead, the irregularities of screen-printing were quickly assimilated to standard avant-garde interests in "the impossibility of ever being able to grasp the firm ground of reality behind the intertextuality of 'images'" while "nevertheless still having to assume a reality . . ." They were clearly aristocratic "Anti-Art with capital A's," or pure, petty bourgeois, unhappy consciousness. As this emphasis on Warhol the postmodern ironist or avant-gardist gained momentum, it encouraged new audiences to understand their own attachment to popular and participatory culture ironically—to laugh at their own failed attempts to participate, to value these failures as entertainment. Previously, advertisers had assumed that the working class wasn't intelligent enough to enjoy irony: "super-sophisticated advertising . . . is simply meaningless to the Lower Status people. They cannot comprehend the sly humor, they are lost by the bizarre art." But at some point, neoliberalism's promises of participatory culture would begin to ring hollow, even to these "Lower Status people." If the advertisements were lying to them, and becoming a successful artist wasn't as easy as buying a gadget or attending a mail-away art school, maybe there were other satisfactions to be gained from one's own _distance_ from artistry and creativity—from the clumsy, inarticulate, mechanical style that Warhol helped to pioneer in the visual realm.
Contemporary advertising has capitalized on this ironic self-understanding, aping and appropriating the consumer's entertaining failure to participate in culture, particularly with low-end products. Burger King's "King" mascot revels in its own faux-nobility; customers are meant to laugh at the figure's awkwardness even as he wheedles them toward a purchase. Toyota advertises its low-budget model with the tag line "YARIS / it's a CAR!" (framed by a jagged Warholian bubble), while bragging that "STANDARD FEATURES COME STANDARD." As Debord reminds us, advertising is fundamentally self-defeating: "Each new lie of the advertising industry implicitly acknowledges the one before." Ironization is advertising's desperate gamble that the resulting cynicism can be defused homeopathically, by integrating it into the advertisement itself.
If, as Clark has argued, capitalism involves "a systematic ironizing of subject-positions . . . which produces the conditions for effective . . . controllable citizenship," then the canonization of a postmodern Warhol ought to be recognized as a crucial moment in the construction of the working-class "consumer/citizen." Cynicism, "the neoliberal condition" and "the mark of resentment of those who have abandoned the possibility of change," would allow these subjects to enjoy and consume their own absence from cultural participation. Paradoxically, Warhol's art was understood to model this ironized, cynical citizenship even as it mourned its conditions.
# Chapter Three
# Warhol's Brand Images
What did the brand images Warhol famously reproduced in his paintings and sculptures—commodity logos like Campbell's and Brillo and Coca-Cola—mean in their own time? Were they as universally "American" in 1962 as they appear to be today? How did they function socially? To which audiences were they designed to appeal? These questions have remained mostly unasked and unanswered. Although Warhol's brand image borrowings represent a limited segment of his production during this period, they include some of the most iconic works of the early sixties. Between 1962 and 1964, Warhol had four solo shows in the United States, two of which were devoted exclusively to brand image artworks: _Campbell's Soup_ (plate 10), _Coca-Cola_ , and the _Brillo Boxes_ , among others. What's more, brand image artworks made up half of Warhol's contribution to his single group show during this period. These works have remained central to Warhol's reputation since they were first shown. Ascertaining the social significance of this material is thus a pressing task for Warhol scholarship.
As it turns out, these brand images—for so-called national brands, advertised and distributed under brands owned by their manufacturers or distributors—were in Warhol's time seen as much less universally "American" than they now appear. Where brand appeal had previously been expected to "trickle down," from elites to the masses, during the late 1950s and early 1960s national brands were retargeted, aimed directly at "working class wives" and their families. These "consumers" were expected to be less likely than their wealthier peers to succumb to the "mushrooming" appeal of "private" or generic brands (products advertised by a retailer rather than their manufacturer or distributor, and sold at a lower price and higher margin), and hungrier for the status that national brands supposedly conveyed. Paradoxically, contemporary researchers argued, "the outer package means a lot more" to working-class Americans. "In the slums," one of them claimed, "the closer colors are to the rainbow, the more enticing they are." The results were promising: "The working class wife is clearly a dependable, dependent 'brand shopper.'"
National brands may now have reached a stage of functional ubiquity, in which everyone is supposedly "a walking compendium of brands . . . [that] become amongst the most direct expressions of our individuality," but this was not the case in the early 1960s when Warhol made them central subjects of his art. The class-specificity of the brands he made use of was widely reported in the contemporary discourses of marketing and would likely have been familiar to anyone who, like Warhol, had attained a position of prominence within the field. Understood in its context as an embattled and strategically active category, the brand image sheds new light on Warhol's early pop work and its cultural reception. Likewise, Warhol's reconfigurations of brand images reframe the strengths and weaknesses of an ambitious marketing strategy that remains pervasive to this day. In his brand image artworks, Warhol took up a set of class stereotypes and tested their appeal for an elite audience. These stereotypically working-class icons, technologies, and attitudes: could their combination beguile an artworld growing tired of Mark Rothko and Jackson Pollock? What would it take to make the hastily traced image of a picture of a can of soup seem world-historical?
The invisibility of the social history of Warhol's brand image motifs has underwritten many of the most ambitious and influential readings of his work. Again and again, across a variety of political and hermeneutical frames of reference, we are told that Warhol's brand motifs appealed to "everyone" or that they rendered the problem of class irrelevant. For Arthur Danto, Warhol's brand image art, "redeemed the signs that meant enormously much to everyone, as defining their daily lives." Kirk Varnedoe argued that, with these works, "Warhol moved out of the expressionist grunge of tabloid vulgarity towards the commonplace banality of middle-class commodities . . . [supplying] a steady common denominator of experience across every age and class." Benjamin Buchloh claimed that these works were capable of "canceling out traditional iconographic readings" because "there is a degree of randomness, arbitrariness in the various objects that are chosen." Mary Anne Staniszewski put the point even more sharply: "Pop represents the language of images circulated within the mass media where all sense of origin and concrete substance dissolves." 1959 had seen the publication of Robert Nisbet's "The Decline and Fall of Social Class," which argued that "the term social class . . . is nearly valueless for the clarification of the data of wealth, power, and social status in contemporary United States." The soup cans and soda bottles seemed to usher in a new world of classlessness and social mobility. In this respect, as in so many others, Warhol's work was apparently a sign of the times.
Warhol himself sometimes celebrated the egalitarian accessibility of brand image commodities—"the world of spending," as bell hooks describes it, "where the promise of community is evoked." As he famously argued in his _Philosophy_ :
> What's great about this country is that America started the tradition where the richest consumers buy essentially the same things as the poorest. You can be watching TV and see Coca-Cola, and you can know that the President drinks Coke, Liz Taylor drinks Coke, and just think, you can drink Coke, too. A Coke is a Coke and no amount of money can get you a better Coke than the one the bum on the corner is drinking. All the Cokes are the same and all the Cokes are good. Liz Taylor knows it, the President knows it, the bum knows it, and you know it.
And yet the lesser-known paragraphs that surround this encomium show that Warhol was anything but sanguine regarding Coke's ability to overcome inequality in America. The passage strikes an optimistic note in an otherwise gloomy chapter, in which Warhol bemoans American social stratification, wishes that the president would "have the TV cameras film him cleaning the toilets" to improve "the morale of the people who do the wonderful job of keeping the toilets clean," and admits that he is paralyzed with embarrassment when confronted with a hotel maid. Despite its appeal, egalitarianism is often a dream deferred for Warhol—a matter of marketing and false promises. What's more, during the early 1960s, the advertising world—and Warhol most likely with it—would have known that brands like Coca-Cola were actively constructing this image of affordable luxury, and seeking out a new class of consumers who could be expected to imagine that briefly owning a ten-cent bottle of soda or can of soup somehow made them sophisticated. As we will see, there is ample reason to believe that Warhol was at least as interested in these objects' failures as he was in their successes; the remarkable torn and crushed soup paintings hint at this interest, as do his _Tunafish Disaster_ paintings (1963).
These latter works, eleven in all, commemorate the botulism-related deaths of two working-class women who had been exposed to the bacteria by contaminated cans of tuna. They form part of what Thomas Crow calls Warhol's " _peinture noire_ ": "a stark, disabused, pessimistic vision of American life, produced from the knowing rearrangement of pulp materials by an artist who did not opt for the easier paths of irony or condescension." The source material from which Warhol derived these paintings confirms and extends Crow's hypothesis. Officials initially attempted to blame the contamination on "human error," or more specifically, negligent factory workers—to condemn the working class for its own tragedy. But further investigation revealed a second contaminated can; the functionaries were forced to exonerate the factory workers and admit what they euphemistically called "mechanical failure"—the deadly malfunction of their own technologies and procedures. All of this history is strikingly legible in one of Warhol's largest versions of the subject, the only work from this period that includes a legibly silkscreened news article, here duplicated with six complete and enlarged prints of the photographs of the can and its victims that dominated the other versions: "It was time for lunch. In the kitchen of a modest, four-bedroom home in the Detroit suburb of Grosse Ile, Mrs. Colette Brown, 37, got out a can of tuna fish and made two sandwiches, for herself and for her neighbor, Mrs. Margaret McCarthy, 39. . . . The next morning, Mrs. Brown was dead."
FIGURE 23. Andy Warhol, _Tunafish Disaster_ , 1963. Silkscreen ink and silver paint on linen, 124½ × 83 inches. Daros Collection, Switzerland. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
The text, the cans, and the women's portraits shadow and haunt each other; in the lower left-hand corner of one painting, the edge of a seventh large can is barely visible, sinisterly suggesting the future circulation of these deadly receptacles. In any case, there was no way Warhol would have missed this issue of _Newsweek_ —"Pop Goes the Easel," one of the magazine's first features on pop art, ran a few pages after "Two Tuna Sandwiches." It featured a reproduction of Warhol's _Dick Tracy_ , the comic-strip detective forever vigilant for corpses and killers.
As is often the case, pop's harshest critics—responding early, while the style's value was still in question—were sometimes best able to pinpoint its underlying connotations. In 1962 _Time_ described pop's subject matter as "the most banal and even vulgar trappings of modern civilization." Hilton Kramer voiced his displeasure with pop's consequences in similar terms: "Its social effect is simply to reconcile us to a world of commodities, banalities, and vulgarities." Barbara Rose rejected pop's supermarket imagery as "offensive," claiming that she was "annoyed to have to see in a gallery what I'm forced to look at in the supermarket." After all, supermarkets themselves were a relatively recent phenomenon, with distinctly downmarket connotations.
None of these critics were able (or willing) to articulate the social prejudices behind their aversion to pop's subject matter. As a result, their apprehensions can easily be interpreted as a general distaste for commerce or capitalism broadly construed. It was the poet Stanley Kunitz—speaking at the Museum of Modern Art alongside Kramer and sharing his antipathies toward the symposium's subject—who was best able to sum up the social dimension of pop's imagery: "Pop art rejects the impulse towards communion; most of its signs and slogans and stratagems come straight out of the citadel of bourgeois society, the communications stronghold where the images and desires of mass man are produced, usually in plastic." Like Kramer, Kunitz emphasized the falseness of pop's chosen motifs, their dissimilarity to true communication or "communion." But Kunitz's description stood out for its willingness to introduce the notion of class strategy into its interpretive framework. In Kunitz's formulation, the "signs and slogans and stratagems" pop appropriated _belonged_ neither to the working nor the vaguely defined "middle classes," nor could they strictly be attributed to the bourgeoisie. Instead, during the late 1950s and early 1960s, they could best be understood as produced by one class—the bourgeoisie—and targeted directly at another: Kunitz's "mass man," the working class.
## The function of the brand image
It turns out that the advertising industry's own understanding of brand images during the 1950s and early 1960s was deeply strategic and manipulative—far closer to Kunitz's than to Staniszewski's or Danto's. Of course, like comic books and television, branded commodities were difficult to avoid by the late 1950s. But despite their functional ubiquity, they were being described by their creators in increasingly class-specific and manipulative terms.
Writing in 1957, William D. Tyler, president and chairman of the Plans Board at Leo Burnett Co., laid out one of the frankest descriptions the brand image received during this period: "This is advertising that sells by implanting a _literal_ image in the consumer's mind." The article seems to call forth a new world of subliminal advertising, to which all would be susceptible: "These are visual images that are 'branded' into people's mind." But it culminated in a series of arguments that distinctly narrowed the reach of this audience:
> An imposing percentage of Americans look at our advertising without consciously seeing it. . . . It can be argued that these are dull, unimaginative clods. . . . But these same people have to . . . select branded merchandise off the shelf just like other people. . . . They _want_ to play follow-the-leader. That way they know they will not go wrong. . . . They can do it if the advertising they do not consciously look at dins into their minds a simple, memorable, repetitive visual symbol of that brand name enough times so that it becomes part of their daily living, one of those familiar talismans on which they can rely rather than making independent decisions.
The passage is remarkable both for its candor and its condescension. Ironically, there are clear echoes throughout of Vance Packard's _The Hidden Persuaders_ , which had been published to wide acclaim only six months earlier and had described Americans as "image lovers given to impulsive and compulsive acts," who "please ['the professional persuaders'] with our growing docility in responding to the manipulation of symbols that stir us to action." This critique would be reproduced and politicized in _The Manchurian Candidate_ (1962)—a film that centers on subliminal manipulation—where, in the culminating scene, the empty seats of the Republican convention hall are repeatedly juxtaposed with empty Pepsi bottles, ready to be filled with premade "product."
FIGURE 24. _The Manchurian Candidate_ , film still, 1962.
FIGURE 25. _The Manchurian Candidate_ , film still, 1962.
But while Packard's book and _The Manchurian Candidate_ condemned the advertising industry for "subliminally" manipulating consumers, Tyler's article represented one of the industry's leading lights bragging to his peers about the depths and breadths of this manipulation—at least as it applied to demographics other than his wealthy and educated readers. It is critical to recognize that, for Tyler, there was absolutely nothing neutral or transparent about a brand image; it was something consumers "know or think they know," a "talisman" that bypasses or short-circuits their ability to make "independent decisions." Yes, as Danto argued, this image "becomes part of their daily living," but only through force, through nonconscious and manipulative exposure and repetition. On Tyler's view, the brand image was meant to simulate, and ultimately to replace, the work previously done by circuitries of social experimentation and emulation; instead of "following-the-leader," the branded consumer followed a mark that had been imprinted in her mind through repetition. The ubiquity and universality so often ascribed to Warhol's chosen motifs—their being "totally universal and loved by the public," in the words of the chairman of Christie's—was, in his own time, actively being produced, disseminated, and disguised as a socially determined tradition.
But there are other important details to glean from Tyler's description. The question of class—the different economic classes of consumers and the different ways to target them—did not appear explicitly in his argument. Like many of his peers, including Kramer and Rose, but also Dwight Macdonald, Tyler preferred not to get too specific about class. There seems to have been general agreement among these authors that "class lines are especially weak" in America, and that it was therefore easier to talk about mass-, mid-, and high- "brow" or "cult" than to talk about classes. Tyler's argument focused instead (euphemistically?) on "the forgotten people," "dull, unimaginative clods" who were "not bright enough to be convinced by our most cogent sales arguments." But even "clods" needed _something_ from the products they buy: "they need this feeling of reassurance and familiarity."
These final phrases—in many ways the proposition of the article as a whole—call to mind a discourse that may already have been familiar to Tyler in 1957, and that was to be very widely propagated in the following years. Tyler's claim—that some consumers derive their social standing from the familiarity of brand name products—became the key conclusion of research being funded and disseminated by Macfadden Publications, Inc., a company that produced magazines like _True Story_ , _True Romances_ , and _True Experiences_ for a predominantly working-class audience. Where Tyler only implicitly identified the disadvantaged consumer as most susceptible to the brand image, Macfadden explicitly singled out the working-class consumer as the solution to what they called "the battle of the brands"—the challenge posed to national brands by cheaper generic products. A closer look at the Macfadden argument reveals Tyler's circumlocutions for what they were: a way of describing the working class and their supposed vulnerabilities to advertising without directly naming them.
From a twenty-first-century vantage point, where brands seem to have colonized every area, not just of consumption and leisure, but of life itself, it is hard to imagine a relatively recent moment when the brand strategy was in crisis. And yet such a crisis was widely reported in the late 1950s and early 1960s, and would likely have been familiar to anyone—like Kunitz, Tyler, or Warhol—who had recently worked in the fields of news, media, or marketing. The perceived problem was that national brands were losing significant ground to generic, or "private brand," competitors. A 1956 A & P Supermarket advertisement from the _Chicago Times_ is indicative of this trend; the full-page ad, listing dozens of items, includes only one national brand image: Campbell's Soup, soon to become one of Warhol's favorite motifs, but here dwarfed by A & P's generic "Dexo" brand shortening. A _New York Times_ article from 1956 entitled "Battle of Brands Growing Fiercer: Retailers Using Own Labels to Bolster Their Profits and Foil Discounters" argued that this battle was being fought most fiercely "in department stores and supermarkets," where "manufacturers with well-established national brands" were being undercut by "retailers with their newer but rapidly gaining private brands."
The article's glum tone echoed the advertising industry's overall response to the rise of the generic brand. These brands had begun to dominate the market, claiming 70 percent of frozen orange juice sales, 62 percent of margarine sales, and 60 percent of coffee sales. Madison Avenue had a significant vested interest in the success of national brands; in many ways, the fate of these brands was a verdict on the effectiveness of advertising, since "one of the biggest factors setting private brands apart from name brands is the fact that they are not heavily advertised, if at all." According to a 1962 _New York Times_ article, the "extraordinary growth of private label products has caused concern among advertising agencies and . . . forced many makers of brand name goods to reduce prices and curtail advertising budgets."
At the height of this "crisis," Macfadden Publications launched an ambitious advertising campaign advancing an alternative solution. Instead of continuing to throw good advertising money after bad, Macfadden proposed that national brands think more critically about the constituencies they intended to target. The Macfadden campaign began with a bang on the morning of August 14, 1961, a few months before Warhol began painting soup cans, when "a score of [advertising] space salesmen . . . set out on visits to leading advertising agencies carrying lunch pails ['a symbol of the blue collar working class'] instead of attaché cases." The basic thrust of Macfadden's theory of working-class buying habits was straightforward: massive numbers of increasingly prosperous working-class customers—Tyler's "dull, unimaginative clods," Kunitz's "mass men"—could be relied upon to value national brands, where their middle- and upper-class counterparts could not.
FIGURE 26. A & P Supermarket advertisement, _Chicago Daily Tribune_ , September 9, 1956, 27.
FIGURE 27. Macfadden advertisement, _New York Times_ , May 15, 1961, 32.
Macfadden justified this hypothesis with two interrelated arguments: working-class consumers were willing to pay more for national brands both because they valued the status thereby accrued and because they were not sufficiently educated to recognize that advertising was deceptive—that nationally branded and privately branded products were qualitatively indistinguishable. "The blue collar person," a Macfadden vice president declared, again referencing Packard, "depends on brands as status symbols. . . . Unlike the white collar wife, the working class wife is not suspicious of advertising as a 'hidden persuader.' She prefers and wants to lean on the security she gets from buying national brands." The _Times_ solicited opposing viewpoints, but these did little to undermine Macfadden's case: "There are 26,000,000 working class housewives in the United States, and they control 57.5 per cent of total discretionary spending. . . . 'This is a vast new marketing frontier and we are going out and exploiting it.'"
Macfadden supplemented its lunch pail campaign with a remarkable series of full-page advertisements in the _New York Times_ that ran through most of 1961 and 1962—the period when Warhol was first producing and exhibiting his brand image artworks. The first such ad, captioned "Surging ahead," appeared on the same day in the _Times_ , the _Wall Street Journal_ , and the _Chicago Tribune_. It featured a large picture of a breaking wave and proclaimed, "This is Macfadden's conviction: the battle of the national brands will be won or lost depending on the attitudes of _mass_ , not middle-class, consumers." This bold claim was reinforced in five more full-page ads, each printed on the _Times_ 's back page, with a provocative headline and three columns of densely packed text.
The headlines spoke directly to the anxieties surrounding national brands: "Can Advertising _Block_ Sales?"; "The Quality Revolution—New Hope for National Brands"; "Who Needs National Brands?"; "The woman who packs this pail will decide the future of national brands!"; "Which Half of the Market Needs National Brands?" The accompanying texts reiterated and expanded the lunch pail campaign's key points. Conventional marketing promised that "classes sell the masses": "advertise to the people at the top and the masses will follow." But this trickle-down dynamic could no longer be trusted: "The masses don't follow. Not anymore. The mass consumers, America's working class—the newest consumer sales phenomenon on the U.S. marketing scene—picks its own path." What's more, these "consumers" couldn't be reached by mainstream print media; fewer than 20 percent of them read _Life_ , _Woman's Day_ , or the _Saturday Evening Post_. They preferred instead "magazines directed specifically to [their] needs and interests," featuring characters with "similar backgrounds, similar problems and similar goals"—Macfadden magazines.
Having established the independence of working-class consumers from upper-class influence, the advertisements proceeded to the "best news of all": "In the face of a sharp rise in retailer brand [i.e., generic or private brand] competition, the wife of working class America looms as a massive ally for the national manufacturer." With their newly earned buying power, "working class families today are reaching out for the American Dream. . . . They want 'all the good things of life' their Depression-wracked parents could never provide." While middle-class consumers, "with their higher cultural level," have "no qualms about buying private labels," the working-class wife is "less sophisticated, less certain." Consequently, "these women want to lean on the security derived from buying a brand name that is nationally advertised by a company they know will stand behind its brand." What's more, by purchasing these branded goods, the working-class wife believes she accrues status. By contrast, "White collar wives, as you may know, seek status elsewhere—e.g., country clubs, foreign cars and trips to exotic vacation lands."
FIGURE 28. Macfadden advertisement, _New York Times_ , December 20, 1961, 68.
FIGURE 29. Macfadden advertisement, _New York Times_ , March 7, 1962, 72.
FIGURE 30. Macfadden advertisement, _New York Times_ , April 11, 1962, 88.
FIGURE 31. Macfadden advertisement, _New York Times_ , May 16, 1962, 84.
The question of the empirical accuracy of Macfadden's claims is in many ways beyond the scope of this chapter. The advertisements were certainly biased; their express goal was to sell advertising space in publications with an established working-class readership. The claims they made in the service of this goal are broad and difficult to verify. Clearly, this was a moment when the concept of the working class was being used "to maintain and consolidate differences in power"—to shore up a thoughtful middle class by projecting a mass of subjects "beneath" them, who don't "think, act, talk or shop like [their] white collar counterparts." It must have been reassuring for many of the _Times_ 's readers to imagine a fundamentally different "working class wife" who "responds to authority," "is constantly beset by events over which she feels she has no control," and feels "the day-to-day world is chaotic." No such anxieties and insecurities for the _Times_ subscriber and "his" family!
But a few things about the ads are clear. First, they could cite supporting empirical research. Macfadden's claims relied heavily on the findings of Social Research, Inc., a Chicago-based firm that produced the book _Workingman's Wife_ , which asserted that its subjects "are happiest when buying brand name merchandise." These purchases "give them the 'feel' of economic security . . . [and] confer . . . the social security of having done the same thing as millions of other Americans." Quotes from the study's working-class subjects confirmed this view: "I have a tendency to go toward name-brands. . . . I don't trust off brands. . . . I get an awful lot of ideas from [magazine advertisements and illustrations]." Second, the Macfadden ads were well distributed and immensely visible. It would have been difficult to be involved in the world of Madison Avenue advertising during this period—as Warhol was—and to be unaware of the lunch pail campaign, or of the full-page advertisements in the _New_ _York Times_ and elsewhere. Third, and most important, Macfadden's campaign seems to have succeeded in drawing national brand advertising to its magazines. Whether or not Macfadden was right in claiming that working-class consumers preferred national brands for the reasons cited, they persuaded the manufacturers of these national brands to market their goods to this specific audience. A Macfadden advertisement cited a 35 percent rise in advertising lineage for the company's "women's group" publications in the first quarter of 1962, against an overall downward trend in women's magazine lineage. The _Times_ ran a follow-up report in June 1962, which informed the public that, as a result of its success, the Macfadden campaign "had scrapped its grey pails and substituted gold ones"—a distinctly Warholian touch.
FIGURE 32. _True Story_ , March 1960, cover. © BroadLit, Inc.
A sample of the advertising pages in Macfadden magazines from the early 1960s confirms the ascendance of brand name advertising. Almost every large advertisement in the March 1960 and August 1961 issues of _True Story_ featured a nationally branded commodity. (The back pages of these issues were still dominated by the drab black-and-white advertisements upon which Warhol based his earliest pop artworks—much like the ads for "freak healing devices, muscle courses, and bust lotions" that had predominated since the title's inception.) The full-color advertisements touted cosmetic and grocery items, and almost all followed the same basic pattern: a large, vivid photograph of the advertised item in use, accompanied by a textual description, with a picture of the item in its branded package in the lower right-hand corner. In each case, the model and her enjoyment of the product were intended to draw the viewer's attention and the accompanying text to explain the product's virtues, but the branded image in the lower right-hand corner was the page's last word, the mnemonic device recommended by Tyler and meant to be retained until the reader had reached the proper aisle in the grocery store. Each "shopper," no matter how supposedly dim and uneducated, must be made to memorize "the aristocracy of brand names" that guarantees quality. Many of Warhol's key brand images of the early 1960s were borrowed from this same style of advertisement.
Would Warhol have been aware of these arguments when he painted the Campbell's Soup cans? Their relevance to his work clearly does not depend on proving that he was. The Macfadden campaign had gained enough prominence during this period to be considered an important factor in these works' reception. The confluence between Tyler's 1957 article, numerous newspaper accounts of a national brand crisis, and Macfadden's 1961–62 campaign suggests that these ideas had reached a wide audience within the advertising and media industries. The _Washington Post_ 's "Post Impressionist," Jerry Doolittle, devoted a column to ventriloquizing a brassy "working class wife" named Mrs. Schwitzen who takes offense at being asked by an imaginary Macfadden pollster whether she wears deodorant: "Isn't that what you're getting at, that everybody that works for an honest living stinks?" Throughout these accounts, supermarkets were repeatedly emphasized as one of the most volatile arenas for brand competition. Warhol's classic brand image artworks borrowed exclusively from supermarket products; this is the primary shared feature of the Brillo Boxes, soup cans, cola bottles, six packs, and coffee labels. At the very least, then, the crisis of the brand image has to be recognized as an important contributing element in the reception of Warhol's work and in its broader historical context.
And yet a strong case can be made for Warhol's awareness of these issues. Warhol worked as an illustrator for the _New York Times_ from 1955 through May 1963, a period that included every major Macfadden article and advertisement, as well as scores of other articles on the national brand "crisis." George Hartman, a contemporary advertising executive, called Warhol " _the_ commercial artist of the moment." In order to dispute Warhol's exposure to these issues, one has to imagine that, despite being a renowned advertising illustrator known for his savvy and sophistication, he had little to no interest in advertising news and failed to keep up to date on its announcements, even as he worked for the newspaper in which many of these announcements were published.
What's more, the specific brand images that Warhol chose to borrow during the early 1960s were directly linked to the so-called crisis of the national brand. Depending on how they are defined, the Warhol catalogue raisonné lists around seventy-seven brand image paintings made during the period from 1961 to 1963. Of these, over 60 percent (forty-eight, counting the thirty-two Ferus paintings as one work) were derived from the Campbell's Soup label. Roughly a fifth (fifteen) were derived from Coca-Cola advertising. Six of the remaining works refer to Martinson Coffee, and one each to Schlitz beer, Del Monte canned fruit, and Pepsi-Cola. All of these brands were affected by the shifting marketing and sales strategies of the period. Coca-Cola and Campbell's Soup, Warhol's two most iconic and familiar brand motifs, were both actively involved during the 1950s with brand image strategies. Del Monte had initiated a similar project. Martinson and Schlitz, by contrast, were attempting to target directly a working-class market with what had previously been trickle-down brands.
## A sign of good taste
Two general elements of Warhol's style are illuminated by—and in turn, illuminate—the history of the branded image and its strategic targetings. The first is Warhol's apparent reverence for the images he borrowed. His was the first sustained artistic attempt to make brand images the exclusive subject matter of a series of artworks; the works' historical concurrence with the Macfadden discourse is thus all the more striking. They are assertive and formidable paintings; they proclaim the importance of their motifs, their adequacy as independent subjects for representation. Warhol is alone among the artists of this period in resisting the temptation to reduce the brand image to one component in a larger drama, maintaining instead that he "just happen[ed] to like ordinary things." Other pop artists rejected this focus, advocating deeper resonances for their work; as James Rosenquist described his paintings in 1964, "the subject matter isn't popular images, it isn't that at all." Warhol's work concentrates on the brand images themselves, and the possibility of their reproduction.
The drama of this reproduction forms the second key element of Warhol's style. Brand images were rarely appropriated (in the sense of being directly imported as material, as in the work of Eduardo Paolozzi, Ray Johnson, or Wolf Vostell) or drawn freehand in Warhol's work; instead, they were hastily and inaccurately copied with the aid of reproductive technologies. Its "slurs and gaps and mottlings and tics" set Warhol's work apart from that of his contemporaries, who tended, when they adopted brand images, either to appropriate their motifs directly (as in collage), to reproduce them in an impeccably slick style, or to stylize them in a distinctively arty way.
Through both of these qualities, Warhol's work investigated the class stereotypes that Macfadden described: if these brand images were truly so central to working-class visual culture, what would it take to make this culture participatory? Might working-class "consumers" care enough about these images to attempt to reproduce them, and struggle to do so even with the aid of the reproductive technologies—Magic Art Reproducers—that were also being marketed to them? In other words, could these working-class icons be remade by working-class hands and tools? Warhol's brand image artworks record the tensions that arose from these questions, extending and camouflaging the direct investigation of working-class culture and cultural participation inaugurated in the newsprint paintings.
In the earliest brand image paintings, like _Peach Halves_ and _Campbell's Soup Can (Tomato Rice)_ , these tensions had been communicated through gestural scrawls and painted masking that clearly conveyed the frustrations of thwarted visual reproduction. Here, plainly, was a stereotypically working-class hand attempting to reproduce the brand images that marketers claimed were so important to him—too harried, anxious, impatient, and impulsive to distinguish between image and mask or to reproduce either completely. Like Superman and the diamond ring advertisements, these mythic images remained blatantly irreproducible for their most ardent admirers. And yet critics missed the point of this gesturalism, reading it as a derivative remnant of abstract expressionism, and overlooking the potential desirability of brand images that they found mundane or repulsive.
These participatory frustrations were turned from the process to the object in the distressed soup cans of early 1962, based on photographs by Edward Wallowitch. In works like _Big Torn Campbell's Soup Can (Pepper Pot)_ (plate 11), the brand image seems to have been punished—torn, punctured, or crumpled—for its irreproducibility. These paintings ask us to imagine a subject who has lost his patience with brand image commodities and their extravagant promises of status and security, and has exacted his revenge on them physically. During this same period, Warhol began collecting lurid tabloid photos of car crashes, and one wonders whether the congruencies between torn, crumpled, and punctured cans and torn, crumpled, and punctured cars appealed to him. He would return to this conjunction between packaged food and bodily injury in the 1964 _Tunafish Disaster_ paintings and again in 1967, when he drew out the resemblances between silver-painted Coca-Cola bottles and silver-painted practice bombs.
FIGURE 33. Andy Warhol, _Orange Car Crash_ , 1963. Acrylic and silkscreen ink on linen, 86½ × 82¼ inches. Galleria d'Arte Moderna. Photo: Alinari/Art Resource, NY. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
In the classic brand image works, and particularly in the soup can paintings of the early 1960s, Warhol found a way to make the tensions between the brand image and its reproduction subtler and seemingly more mechanical, to the point where many viewers overlooked them completely. The catalogue raisonné describes the Mönchengladbach (copied from a magazine advertisement) and Ferus (copied from a Campbell's label) soup can paintings as being "based on illustrated images that were projected, traced, and methodically painted to show little evidence of the artist's hand." But "evidence of the artist's hand" is more than negligible, particularly when the paintings are compared directly with their source advertisements, identified here for the first time (plate 12). The various elements of the cans are distinctively handmade—rough, uneven, and irregular from edge to edge.
These handmade and provisional qualities are striking in the iconic _32 Campbell's Soup Cans_ , first exhibited at the Ferus Gallery in July 1962. To produce these paintings, Warhol traced a soup can image that Campbell's printed on its company envelopes, enlarged it with a projector, outlined it in pencil on canvas, and painted in the final forms, including the outlines of the can, the Campbell's logo, the word "CONDENSED," and the upper red area for each painting. He added the variety names individually to each painted can, stamped the fleur-de-lis, and painted in the gold centers. Since they were first shown, these paintings have been treated mainly as a conceptual provocation. In Duchamp's memorable words, "What interests us is the _concept_ that wants to put fifty Campbell Soup cans on a canvas." But, seen in person, the paintings have qualities that strikingly contradict these assessments, emphasizing discrepancy over exactitude, and facture over concept. Like the working-class consumers described by contemporary sociologists and profiled in Macfadden's advertisements, their style seems driven by an anxious and impatient interest in "the aristocracy of brand names" and the "American Dream" of status and security they promised to provide.
Perhaps the most obvious instance of this emphasis on facture and impatient "impulse-following" appears in the variety labels that distinguish each can. Each of these descriptors—"BEEF," "PEPPER POT"—was unique and had to be hand-painted. What's more, each needed to arch convincingly as it curved around its can. During the long process of painting thirty-two individual cans, Warhol seems to have tired of these challenges. The lettering, letter-spacing, and kerning are inconsistent. In _Bean with Bacon_ and _Turkey Noodle_ , he gave up trying to curve the font; in other paintings, like _Vegetarian Vegetable_ , the kerning and line spacing are sloppy and irregular. _Old-fashioned Tomato Rice Soup_ is missing its crowning gold ring, and many of the other rings are uneven; in some cases, the gold paint is thin, while in others, like _Pepper Pot_ , it overflows its edges.
FIGURE 34. Andy Warhol, 32 _Campbell's Soup Cans_ , 1962. Synthetic polymer paint on thirty-two canvases, each canvas 20 × 16 inches. The Museum of Modern Art. Digital image © Museum of Modem Art. Licensed by SCALA/Art Resource, NY. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
A stereotypically working-class preference for "immediate gratifications" also informed Warhol's approach to materials. His assistant Ted Carey remembered him having "the most awful equipment: he had brushes that were so old they'd have three hairs, but he didn't want to get new brushes." As Warhol told a British interviewer in 1966, "My work won't last anyway. I was using cheap paint." Everywhere, this cheap paint barely conceals the canvas beneath it, and the colors are far from uniform: _Bean with Bacon_ , _Turkey Noodle_ , and _Vegetable_ are painted with a notably brighter red than their peers. Warhol's pencil underdrawing is visible in each painting, further emphasizing the handmade and provisional nature of the reproduction. Along the lower edge of each can, the roughly stamped fleur-de-lis crowd and stray playfully, like clones or puckered orifices, occasionally even infringing the contour lines between cans and backgrounds.
Some of these glitches are subtle, but a glaring inconsistency occurs at the center of every early sixties _Campbell's Soup Can_ : the engraved gold medallion, designed by J. C. Chaplain for the "Exposition Universelle Internationale"—the 1900 Paris world's fair—is left either blank or, in rare cases, unfinished. In place of the medallion, the works have an empty gold circle, painted in faint but legible strokes, which varies in hue from canvas to canvas. This seemingly critical defect went mostly unremarked by contemporary commentators. Through this absence, the product's old-fashioned seal of quality and authenticity is rhetorically revealed to be irreproducible—cheap gold paint ineffectually pretending to be (the image of) a blank gold medal. When Warhol returned to the soup can motif in 1964, at the request of the Campbell's corporation, he added a screened image of the medallion, but produced it in such a way that its motto was illegible—shapes rather than letters. It is as though each of these paintings contains a secret that is paradoxically hiding in plain sight: that the brand image its creator found so fascinating was constitutively irreproducible from his position. The social status the brand image supposedly conveyed to its working-class buyers could be consumed but never reproduced, no matter how many times he painted it. One wonders whether it was this feature of the Campbell's brand image that made it so appealing to Warhol, worthy of being painted more than three times as frequently as its closest competitor.
Of all Warhol's brand image motifs, Campbell's Soup was the most closely linked to the "battle of the brands." In an interview with the _New York Times_ on the occasion of the first day of the lunch pail campaign, Macfadden Inc.'s vice president and advertising director used Campbell's to illustrate his fundamental point: "The middle class wife feels free to serve any kind of private label soup, for example, but the working class wife derives status and confidence by serving Campbell's Soup." In many ways Campbell's perfectly exemplified the national brand problem: a food that had never been particularly prized for its quality was now being sold on the basis of status. In 1958 Clarence Eldridge, Campbell's former vice president of marketing, described this method of advertising as "franchise building," and—with breathtaking cynicism—traced its roots to the Nazi theory of propaganda: "This kind of advertising seeks to exploit, in a perfectly legitimate manner, the Nazis' hypothesis (unfortunately, in that case, perverted to evil use) that 'if you tell it often enough, long enough, it will be believed.'" As long as this principle was "applied to honest advertising claims," Eldridge argued that it was "the _fundamental basis_ for practically all 'franchise building' advertising." Through this advertising, Campbell's image as "something that is merely inexpensive, or convenient, or filling must be destroyed, and a new concept put in its place."
The _Coca-Cola_ paintings investigated similar promises of status and security. Although it was not mentioned in the Macfadden campaign, the Coca-Cola Company seems to have been well aware of the potential benefits of marketing its product to working-class consumers by emphasizing the status its consumption supposedly bestowed. The company's _Annual Report to Stockholders_ for 1956 announced that they had added a second product slogan to emphasize status over taste, or rather to conflate one with the other: "'So good in taste . . . in such good taste.' Coca-Cola is liked for itself, as well as for its significance. It is always A SIGN OF GOOD TASTE." The report described the transition from exchange and use value to sign exchange value as being driven by class aspirations. Good taste (the taste of the soda) became "good taste" (high-status taste) when consumers were encouraged to value Coke less for how it tasted or made them feel and more for what it represented—what it was "a sign of"—and how its purchase was seen to improve their social status. (Campbell's experimented with a similar set of aspirational metaphors in at least one ad, which led with the line: "Good things begin to happen . . . when good soup's a part of your 'good life.'") Through this emphasis on status, Coke attempted to establish an imaginary distance between its product and that of its chief competitor, Pepsi-Cola, which "was plagued by its past image as a lot of drink for little money—oversweet bellywash for kids and poor people." But Coke's exclusivity was fundamentally illusory, and could only be maintained through advertising, particularly since sugary drinks would long be associated with working-class appetites. The 1957 report came close to recognizing this project's inherent paradox: Coke was at once "a social amenity—a sign of good taste" and "the most popular refreshment beverage in the world." Even Warhol's cleanest versions of this subject, like _Coca-Cola [3]_ (1962), seem riven by this tension between status and accessibility, haunted by ghostly lettering and shaky script. The paintings ask us whether this most common treat could possibly guarantee status and, if so, whether this status might be reproduced visually by the working-class subjects to whom it was marketed. From this perspective, the serial compositions that have long been celebrated in these works become something more than avant-garde provocations. Instead, they testify to an abiding urge to reproduce the culture that one values, or is encouraged to value, from a position of social insecurity and aspiration.
Unlike Campbell's, Del Monte, and Coca-Cola, Schlitz and Martinson seem ultimately to have failed to maintain the delicate balance between the illusion of status and the reality of mass consumption. Martinson coffee began as an upmarket brand sold only to hotels; a consumer "top-grade blend for a premium price" was developed in the late 1920s. But by the early 1950s, with instant coffee sales reaching 17 percent of total coffee consumption and supplying nine thousand coffee vending machines nationwide, Martinson began to market a low-cost, instant brand. By the time Warhol borrowed the Martinson label, the company was effectively using the built-in prestige of an established label to sell cheaper coffee to poorer consumers. Schlitz beer effected a similar transformation. Marketed in the 1920s in upper-class magazines like the _New Yorker_ for its purity, the brand was, by the 1950s, widely identified with working-class consumers. Through their failures, these brand images fall into Fried's category of forgotten myths, and the paintings Warhol derived from them seem to have suffered as a result. The tension between status and accessibility that animates the Campbell's and Coca-Cola paintings is missing from them.
Warhol's _Tunafish Disaster_ paintings (1963) highlighted the desperate stakes that underwrote the "aristocracy of brand names" for working-class Americans. Beneath the brand image's assurances of status were guarantees of quality and safety, bolstered by international corporations that claimed always to "stand behind [their] brand." In the _Tunafish Disaster_ paintings, when, as Jonathan Flatley puts it, "the promises of comfort and recognition held out by the commodity . . . failed disastrously," _Newsweek_ emphasized that they did so under the sign of a generic brand—"distributed locally under their own label." One more reason for "the working class wife"—already worried since her class "has been hard hit by the worst accidents and diseases"—to seek out national brands that "give her peace of mind about the health of her family."
The classic brand image paintings—seemingly clean, neat, and mechanical—thus serve as a reassuring counterweight to the _Death and Disasters_ works that Warhol exhibited at the Sonnabend Gallery in 1964. It is the prevalence of what Schjeldahl called these "lower-class ways of death"—car crashes, electric chairs, tainted food, suicides—that advertisers claimed contributed to the working-class subject's "anxieties about the world beyond her doorstep," her desperation for safety, quality, and status. These were the very qualities that brand image goods promised to provide for a few pennies a serving. Of course, despite the real threats they faced, the working-class had long been stigmatized for its supposed obsession with death and injury, its "vulgar" "fears of accidents and sickness." Death itself could thus be imbued with lower-class associations—Aldous Huxley had described "a corpse . . . [as] an untouchable and the process of decay [as] of all pieces of bad manners, the vulgarest imaginable." Warhol's work exposed the class biases behind these claims, and the deeply cynical strategies—like Macfadden's—that attempted to profit from them.
In the _Heinz_ , _Brillo_ , _Mott's_ , _Del Monte_ , _Kellogg's_ , and _Campbell's_ box sculptures (1964), Warhol expanded his investigation of the brand image from painting to three-dimensional objects (plate 13). Danto argued that this expansion achieved "retinal indiscernibility": Warhol's sculptures could be mistaken for the commercial boxes they resembled. By rendering art and common culture indistinguishable, Warhol had paved the way for a world in which "what makes the difference between art and nonart is not visual but conceptual," and art is defined and identified by philosophy. And yet, as the Macfadden discourse demonstrates, the cardboard boxes upon which Warhol based these works were much more than the straightforward "real objects" Danto made them out to be. Instead, they were deeply manipulative, imbued with class strategies, counterbalanced against class fears.
Warhol relished these tensions; he could _see_ the class differences between various boxes, various brands. According to Nathan Gluck, Warhol's assistant from 1955 to 1965, "When he wanted to do his box sculptures, he sent me across to the A & P [supermarket] and said, 'Get me some boxes.' I came back with things that were very artsy, maybe a Blue Parrot pineapple box or something like that. And he said, 'No, no, no. I want something very ordinary, very common.' So he went back and got a Brillo Box." The selection of these boxes was clearly far from random; "very common" boxes appealed to very common people. What's more, Danto's "indiscernibility" thesis could be sustained only on the basis of cursory looking. As in the brand image paintings, irregularity is everywhere—"the small drips, the texture of the wooden boxes showing through the image, and the like," as Whitney Davis remarks, readily differentiate Warhol's sculptures from their source material.
These distinct and irreducible irregularities can be identified throughout Warhol's work and should probably be recognized as a defining element of his style; they are the "precisely pinpointed defectiveness that gives [Warhol's] work its brilliant accuracy." But the power of these irregularities derives in large part from the overvaluation granted to the paintings' subjects. Never before had any artist paid this much attention to the brand images adorning a ten-cent can of soup or a box of scouring pads. Only a working-class subject could be expected to take them this seriously, or to reproduce them this awkwardly.
## Kiss me with your eyes
Defectiveness produced accuracy in Warhol's work because it was mobilized stylistically as a marker of subjective desire and objective desirability. Read as the subjective remnant of the artist's hand, the irregularities of Warhol's style demonstrate the pressures of looking and of visual reproduction—the working-class stereotypes of "impulse-following," inarticulateness, and anxiety. Simultaneously, however, these irregularities attest to the perfection and irreproducibility of the motif, which is presented—in and through this reproductive defectiveness—as fantastically desirable and beyond the reach of visual reproduction, imbued with a value that far exceeds its modest cost. The two connotations are complementary and mutually reinforcing, enlivening each other in neat circuitries.
FIGURE 35. I. Miller advertisement, _New York Times_ , October 16, 1955, 95.
When Warhol had applied this stylistic device to a man's image of a beautiful man in his homoerotic drawings, it signaled both the attractiveness and the difficulty of attaining homoerotic union: "Kiss me with your eyes" was apparently one of his favorite expressions. When he applied it to a high-end shoe in his advertising illustrations, it signaled the intense desirability—again, the attractiveness and the relative difficulty—of attaining the beautiful luxury commodity. In both instances, Warhol's stylistic approach was effective but by no means groundbreaking, in part because luxury goods and beautiful bodies are paragons of attraction. But when he applied this strategy to the mechanically aided reproduction of a schematized and disposable image of a cola bottle or a soup can, it signaled something entirely new in fine art: a subject who could value the image of a soup can or a cola bottle enough to want desperately to reproduce it but who doubted his ability to get it right. Like the aspiration to mass-cultural participation, during the late 1950s and early 1960s this intense attachment to brand images was widely disseminated as a stereotype of working-class "consumers," who were supposedly governed by a "pleasure ethic"—"impulsive," "non-rational," juvenile, bestial, "chance-taking," mechanical.
Warhol's exploration of these stereotypes had gendered and racialized dimensions as well. During the stage of his career when few critics recognized a queer dimension to his art, Warhol's infatuation with brand images scrambled stereotypes of working-class gender roles, espousing "the supposed class aspirations of working-class wives" as pursued through their shopping, as opposed to "the destructive, inarticulate male solidarity of their husbands," which Warhol would explore in other paintings: the _Race Riots_ , _Elvis_ , _Cagney (fig. 40)_. The box sculptures further complicated these readings, since they immediately reminded viewers of a supermarket "warehouse" and the "stockboy[s]" who worked there. Danto claimed that Warhol himself was indistinguishable from these stockboys: "A photograph of Warhol among his boxes looks indiscernible from a photograph of a stockroom clerk among the boxes in the supermarket."
While the working-class women and men described by Macfadden and Danto were implicitly white, other commentators were quick to affirm that African-Americans were doubly afflicted by the appetites and insecurities that plagued their white counterparts. The argument was deeply cynical: American traditions of racism and discrimination were seen to have burdened African-Americans with an "insecurity neurosis," thereby rendering them such an easy target for consumer goods that advertising to them directly would be redundant. Advertisers thus believed that white-targeted advertising would "trickle down" to black readers, readers once thought to be "uninterested in, or incompetent to judge, the quality of goods" and now prized for "being even more concerned with the symbolic value of goods than are whites."
One great post-Warholian interpreter of this sad history would be Jean-Michel Basquiat, whose paintings returned again and again not only to commodified racial stereotypes, as Laurie Rodrigues has shown, but to the sinister appeal of the brand name and the logo (plate 14). This appeal takes on a perniciousness and depth in Basquiat that seems to hyperbolize the advertisers' cynical hypotheses, and thus rail against them. But where Basquiat embraced handmade, gestural painting, modeled on graffiti, Warhol tended toward a technological approach to the "sustaining but unworkable fantasy" of cultural participation, consistently foregrounding the impossibility of reproducing popular images even with the aid of consumer-grade creative technologies—opaque projectors, photostatic copies, silkscreening, photography, film, video, and computers. If brand images were desirable enough to their working-class audience to be worth reproducing, then that reproduction ought to be as quick and efficient as possible, since the working class was understood to be impatient.
At the limit, Warhol might become machinic or computational: "The world would be easier to live in if we were all machines. . . . I wish I were a computer." It is here that Benjamin's "The Work of Art in the Age of Its Technological Reproducibility" could have real bearing on Warhol's project, except that from this angle—and in contrast to the way this comparison is usually deployed—it is Warhol who critiques and illuminates Benjamin. The utopian potential of participation through mechanical reproduction was, for Benjamin, the possibility of a truly common and open cultural sphere. This is the great promise of the newspaper, where everyone can be an author, but also of the camera, the tape recorder, the projector, and the silkscreen. Still and movie camera sales were booming in the late 1950s, and manufacturers were paying special attention to their downmarket models. The back pages of Macfadden's magazines were littered with ads for photographic services. Warhol seems to have found these promises fascinating; according to his Factory collaborator Billy Name, he "tried to learn everything about [his first] camera. He wanted to find out for himself everything it can do." But, in practice, Warhol emphasized the limitations of these technologies as much as their potential—their inability truly to speak back to the "figmentary interlocutors which subject [him] to a one-way discourse on their commodities and the politics of those commodities."
This entertaining distance between the real and the counterfeit became social and architectural in the Silver Factory. In the paradoxical cheapness and luster of aluminum foil, Warhol was able to disengage from readymade brand images and create his own brand, which could then be extended to the Factory Superstars. The drama of cultural participation could be outsourced to workers so desperate to take part that they would forgo pay. Like a good neoliberal boss, Warhol would profit from this desperation. And when he "didn't want to paint anymore," he would figure this drama abstractly in his _Silver Clouds_ (1966), "painting that floats," where both senses of the word "lightness"—visual and corporeal—are achieved with a disposable, metalized plastic film, as "silver" as his Campbell's medallions were "gold." (Of course Warhol was alert to these colors' pecuniary connotations: "You might say I have a fondness for silver, or even gold, for that matter.") In these buoyant forms, the brand image reached a new pinnacle of abstraction and weightlessness; like Debord's spectacle, they are "cloud-enshrined entities [that] have now been brought down to earth."
Warhol's classic brand image works, long thought to be fundamentally conceptual provocations, must also be recognized as nexuses of contemporary class stereotypes—stereotypically working-class brand icons, attitudes, appetites, and technologies, mutually counterpoised. In this respect, they are important repositories of a forgotten moment in class history, a moment when the brand image and the working class were closely associated with each other, when working-class subjects were thought to be the brand image's deepest believers and last hope, and when new consumer-grade technologies seemed to offer an opportunity to make these images reproducible by their consumers on the cheap.
Warhol's work tested these stereotypes against each other, and against new audiences. What would it look like for one of these working-class subjects, armed with her stereotypical appetites and attitudes and technologies, to attempt to reproduce her everyday icons—the brand images with which Madison Avenue bombarded her on a daily basis? Who would find the rushed and inarticulate results interesting? Only other working-class subjects, who supposedly shared the artist's interests and desires? Or perhaps a world of powerful taste-makers and collectors, who had "never looked at a Campbell's soup can until Andy Warhol showed it to" them, and who might be surprised to find something "beautiful, vulgar, heart-breaking" in these unfamiliar images, attitudes, and technologies?
As it turns out, elites did find something to appreciate in these strange works, even as they worried over their origins. As one of Warhol's early collectors aptly put it, "The only reason you'll know they're art is because they're in my house." As we will see in the next chapter, pop's vulgarity could be at once sexy and deceptive, camouflaging power and disguising differences between the elite and their subordinates. If everyone recognizes the same soups, drinks the same cola, and uses the same soap, how different could we be after all? Oppressors and oppressed could find common ground in a sugary drink: "So good in taste . . . in such good taste." The exploitation and domination that underwrote social hierarchies might be camouflaged by the brand image reproduced on canvas.
This growing appreciation had consequences. The works' quick and comprehensive incorporation into the Western canon, Warhol's seemingly complete capitulation to the demands of profitability in his post-1960s work, and his contemporary crowning as the allegedly "most important international artist of the 20th century" have all helped to universalize the desirability of the brand image—to extend its appeal to everyone. They have worked "to guide the investments of affect on the part of consumers . . . creating an affective intensity, an experience of unity between the brand and the subject" that transcends class boundaries. This is a real consequence of Warhol's work; it may be embarrassing to his admirers, but it should not be disregarded. It certainly has not been overlooked by the companies whose images Warhol borrowed; according to Peter Schelstraete, Coca-Cola's global brand director, "Andy Warhol was one of our best brand directors." Simon Doonan, "creative ambassador" for Barneys New York, would call him "the patron saint of retail."
And yet Warhol's art also reminds us that the process of brand image universalization or mythification has had moments of relative strength and weakness. In contrast to Fried and Danto, Tyler and Warhol recognized that myth in America was resolutely para-anthropomorphic: "a shape, a trade character, distinctive lettering on a package, a piece of design, or a picture of something that stands for the product or the company." These images became myths because, as Danto put it, they now mean "enormously much to everyone." But the source of that prodigious and universal meaning was a relentlessly cynical campaign to "brand" these images onto peoples' minds, to "din into their minds a simple, memorable, repetitive visual symbol." And the people who were targeted for this branding during these years were those thought to be least able to resist it, and most susceptible to its false promises: "the working class" with its "extraordinary emotional dependence upon national brands" In their themes, their format, and their style, Warhol's works from this period embody and thus document the intensity of this "extraordinary emotional dependence" and the frustrations that attended it.
To the extent that these artworks deploy a working-class language, they need to be recognized as simultaneously true and false: tools of class dissimulation _and_ repositories of class history, "establish[ing] myth and illusion as _visible objects_." Perfect neoliberal artifacts, then, commemorating a moment when the working class was paradoxically being targeted _by name_ for its presumed susceptibility to the idea that class identities could be _overridden_ by savvy shopping. Half a century later, it seems remarkable to imagine a time when only working-class people were expected to believe in brand names. Just as the domain of spectacularized life has expanded—from wanting and buying things to wanting to participate in the spectacle oneself—so has the range of people to whom the spectacle can be expected to appeal. One no longer needs to be poor, insecure, and uneducated to be expected to believe deeply in the glamour of a life plastered with flat, empty, irreproducible signs.
And so, after Warhol "turned white" upon hearing about Lichtenstein's success at the Leo Castelli gallery, he looked for ways to disguise the working-class "primitives" whose comics and tabloids his first pop paintings had pined over so openly. The brand image provided an opportunity to encode this primitive otherness in a seemingly universal symbol and in the faint but lingering drama of its reproduction. Ronnie Tavel claimed that Warhol's stylistic mantra was "I want it clean, I want it simple, I want it plastic, and I want it _white_." Whiteness, as Ta-Nehisi Coates has shown, is an avaricious myth of separation from one's Others—"the Dream of acting white, of talking white, of being white," enabling the brutal suppression and exploitation of those who cannot or will not achieve these effects. Warhol's wanting his work to be "clean . . . simple . . . plastic . . . _white_ " meant camouflaging these Others, hiding their voices in the very brand images that promised to make their class identities disappear. But, like Warhol's yearnings for mechanicity, these aesthetic criteria seem always to have been phrased as aspirations—a desired horizon of decontamination and whiteness he knew he could never fully achieve. The tension between "Beauty [as] whiteness itself" and its "primitive," queer, working-class, "Hunky" Others fueled and animated the work.
FIGURE 36. Robin Platzer, "Warhol revels with gift 'trash,'" _Newsweek_ , August 21, 1978, 73. © Robin Platzer.
This entertainingly ineradicable distance from whiteness remained part of Warhol's brand throughout his career. In 1978 _Newsweek_ invited him to bring along his recently acquired birthday presents, a $6,500 white coyote-fur coat and a garbage can full of two thousand one-dollar bills, for a photo shoot. Dressed as the ultimate white male—exercising dominion over even the deadliest beasts, needing a minion (Fred Hughes) to carry his dirty money—Warhol passed a group of what he called "Negro kids" as he crossed the sidewalk to catch a cab. He surmised they were on their way to "some city clean-up program to give them jobs. . . . They didn't look too happy." To Warhol's marked dismay, one boy was taking his grievances out on the sidewalk flower boxes: he "had a shovel and was cutting down every flower when he got to it. They were pretty brooms, too. New." None of the children seemed to recognize Warhol, "except for one little girl who ran all the way back and kept saying, 'You're Andy Warhol, you're Andy Warhol,' and staring at me and at Fred with his garbage can." She could see the artist beneath his white disguise, and Warhol sensed that she found his subterfuge irritating—perhaps the garbage can gave him away.
# Chapter Four
# Warhol, Modernism, Egalitarianism
Warhol, we have long been told, was the quintessential postmodernist. Perhaps more than any other artist, he has been seen as breaking down the old boundaries between creativity and commerce, and heralding a new age of "slap-happy cynicism," pastiche, and superficiality. On this account, Warhol "was interested above all in the demystification of art," striving "to make the art impersonal and at the same time a kind of social joke." Warhol the machine, Warhol the business-artist, Warhol the Duchampian: these familiar interpretations have established Warhol as modernism's nemesis, the "ur-postmodernist," as Crispin Sartwell recently dubbed him in the _New York Times_ , whose "entire artistic practice and persona stood, quite intentionally, in opposition to modernist ideas."
But viewed in detail, Warhol's relationship to modernism becomes far more complex. First, as we have seen in the last two chapters, Warhol's work from the early 1960s tested a range of neoliberal promises regarding social mobility and cultural participation. In this respect, and through his "unresolved, but naively serious dialectical mapping . . . of De Stijl–type abstraction onto a founding, consoling, redemptive country-store solidity," he "come[s] to seem more and more a modernist." What's more, the appeal of the "clean, simple lines" of the brand image ("modern—but not extreme") was argued to cut deeply into the American populace, speaking not primarily to the middle class but to the "working-class wife" desperate to join "the great mass of Americans whom she regards as stable, hence, her ideal."
This testing in Warhol's work extended to received artistic ideas as well. Like any good modernist, he was preoccupied with challenging the assumptions of the artists whose work ruled his milieu: identifying their false notes and hypocrisies, and finding a more authentic alternative. He found this response during the early 1960s along two axes, both of which were practically unspeakable for his critics: queerness and class. Through his constructions of queer and working-class voices, Warhol posed a quintessentially modernist response to his abstract expressionist forebears. This long-overlooked critique is one side of Warhol's modernism. To the degree that modernism is at least partially defined as a critical tradition, in which each new contributor attempts to identify and correct the deficiencies of her immediate precursors, Warhol was a modernist.
And yet it is crucial to recognize that this assembling of queer and classed voices was by no means exclusively critical or reactive. Warhol was also attempting to revitalize an _alternate_ version of modernism—a broader, deeper, more powerful modernism than the formalist version epitomized by abstract expressionism. This larger modernism has been most systematically described by the contemporary French philosopher Jacques Rancière. For Rancière, the true meaning of modernism is a radically egalitarian expansion of art beyond all of its traditional boundaries and hierarchies. The heroes of Rancière's modernism are Alexander Rodchenko and James Agee and Walt Whitman. Their "modernism [was] . . . the idea of a new art attuned to all the vibrations of universal life: an art capable both of matching the accelerated rhythms of industry, society and urban life, and of giving infinite resonance to the most ordinary minutes of everyday life." This expansion does not, for Rancière, entail a loss of seriousness or sincerity, as so many theories of postmodernism have posited. Instead, the modernism that Rancière also refers to as "the aesthetic regime of art" augurs a democratization of creativity, a world not of art and artists but of creative collaborations and communities, truly participatory cultures: "the germ of a new humanity, of a new form of individual and collective life." This was the creative world that Warhol's work summoned and mourned. I believe it was also the dimension of Warhol's achievement that accounts for what Crow has described as his "apparently unending currency . . . for each new generational cohort that encounters the work and the Factory legend."
During the last third of the twentieth century, Warhol emerged as one of the central figures of postmodernism. This reading was reinforced by his relative prominence in the writings of some of the most distinguished thinkers associated with the postmodern era, including Roland Barthes, Jean Baudrillard, Gilles Deleuze, Michel Foucault, and Fredric Jameson. In the writing of each, if to varying degrees, Warhol was singled out as "the most radical" of the pop artists. But while all of these writers considered Warhol to be a pivotal figure, their interpretations of his work varied widely. Deleuze and Foucault argued that Warhol's unusual commitment to repetition helped his viewers glimpse the world as difference rather than identity, "simulacra, and simulacra alone." Barthes and early Baudrillard shared the opinion that art ought to provide a critical perspective on modernity but disagreed on Warhol's capacity to do so. Warhol's impertinences marked for Baudrillard a capitulation to the new world of signs—a "smile of _collusion_ "—while Barthes claimed that they "criticize[d]" modernity by "imposing a _distance_ upon its gaze (and hence upon our own)." Jameson amalgamated these arguments, claiming that Warhol should be recognized as possessing both positive and negative moments waiting to be synthesized in a larger and more encompassing analysis, although he worried that the "therapeutic value" of "deconstructive" painting was "far from clear."
Although these thinkers reached contradictory conclusions regarding Warhol's criticality, together they contributed enormously to his perceived significance as a canonical postmodern figure. This basic understanding is now practically ubiquitous; few would deny Warhol's centrality to the period known as postmodernity. And yet, despite their diversity, none of these five thinkers acknowledged the queer or class-based dimensions of Warhol's work.
The past twenty years have seen a surge of scholarship attending, finally, to Warhol's sexuality and its expression in his work. As Jonathan Katz has shown, by the time Warhol emerged as a pop artist in the early 1960s, the most advanced responses to abstract expressionism, like Rauschenberg's _Bed_ (1955) and Jasper Johns's _Painting with Two Balls_ (1960), were surreptitiously critiquing the movement from a homosocial perspective. And yet these references "tend[ed] to be so subtle and obscure" that they have only recently been deciphered by scholars. Warhol would experiment with similar in-jokes in his early pop paintings: gay icons like Troy Donahue, Superman's prominent ass and huge "puff," "GIANT SIZE PKGS." and their promise of "SHINES" on the front of many _Brillo Boxes_. While these references were rarely mentioned by mainstream critics, they were not illegible; canny observers would occasionally remark them in private, often pejoratively. In an unpublished interview, Rudolf Arnheim surmised that pop's "surrender of creativity . . . conceivably has something to do with this [homosexual] element." As Katz points out, it wasn't until 1966 that Vivian Gornick could directly assert in the _Village Voice_ , "It is the homosexual temperament which is guiding the progress of Pop Art," without mentioning Rauschenberg, Johns, or Warhol by name.
Prominent critics used the distinction between "camp" and "pop" to divide Warhol's work from more explicitly gay work, arguing that "Pop Art is more flat and more dry [than camp], more serious, more detached, ultimately nihilistic." One of pop's main stylistic differences from camp was its resolute contemporaneity, its "most crucial requirement in all areas . . . that it reflect the spirit of Now." As we have seen, Warhol tore his motifs from periodicals, often published the same month he was painting them. For his critics, this made Warhol's work pop rather than camp, and consequently straight rather than gay.
Building on Katz's insights, we can see that Warhol's subtle provocations in this regard constituted an implied critique of abstract expressionism, since, as Maggie Nelson has pointed out, "despite [the] homophobic atmosphere, and despite the intense machismo of Abstract Expressionism, gay men occupied the very center of the art world in New York," and many of abstract expressionism's most prominent supporters and interpreters were gay. Clearly, the abstract expressionist atmosphere was hypocritically homophobic—"no chic, no chichi, no frills, no nothin'!"—anxious to deny the homosocial energies that fueled its ascendance.
In the years leading up to Warhol's emergence as a pop artist, abstract expressionism's critics had started to question this hypocrisy. In 1959 _ARTnews_ devoted large portions of two issues to the question "Is there a new academy?" The most forceful respondents answered in the affirmative, with many wondering whether there wasn't something unsavory about all these men expressing themselves on their canvases together. Friedel Dzubas implied that abstract expressionism's staleness had unacceptably homosexual undertones, its "finished product rubber stamped with the imitable flick of the wrist of the masters" in "an atmosphere of complacent kaffee-klatch, one that can find all the tricks of the trade." Even shit-stains were acceptable as long as the brand was right: "After all, it is en mode to show a little seam of one's dirty underwear, just make sure it is the same brand as everybody else's." This imagery raised the specter of a secret community of men with an "unnatural" interest in each other's underpants. Paul Brach said abstract expressionists were "taking down their pants in public" and "emptying their guts," while Jack Tworkov worried about "applying paint . . . the bad mannered way, the naughty way, the p[iss] and s[hit] way." Through various methods, and with varying degrees of subtlety, Rauschenberg, Johns, and Warhol brought these homosocial and homosexual undertones to the surface of their work.
This, then, is arguably the most powerful art historical lesson to be drawn from the scholars of queer Warhol, Rauschenberg, and Johns: that the purportedly "empty," "generic," "mass cultural" imagery—"totally banal and interchangeable"—that has been argued to anchor their postmodernism was, in fact, full of meaning and significance in an artworld permeated by denial and anxiety regarding its own sexuality. Few thought to wonder (at least publicly) whether Warhol's work might have queer inflections. And, when these inflections were occasionally recognized by an astute critic, they were immediately dismissed as "personal predilection[s]" that undermined the work's quality. Their possible relevance to abstract expressionism's heteronormativity went completely unremarked, at least in print.
## The melodrama of vulgarity
These artistic critiques of abstract expressionism, however, were not exclusively focused on sexuality. Abstract expressionism's pretensions to heteronormativity were conjoined with its pretensions to a particular class position: the "average," "working" American, yearning to express his inner feelings. Larry Rivers described the abstract expressionist uniform as a "combo of corduroy and army-navy-store rejects used for working-class associations." Ruth Kligman remembered the Cedar Bar in similar terms: "These were not suit-and-tie men, but casual, rough-looking men, loud and heavy." According to Walter Hopps, "The Abstract Expressionists took the stance of workingmen—Franz Kline packing a lunchpail and marching to the other end of his studio. This sort of blue-collar approach was carried on from Bill de Kooning to Larry Rivers and Mike Goldberg." Hopps could easily have added Pollock to his list: "Look at him standing there," de Kooning complained to Milton Resnick regarding Pollock's "proletarian affectation," "He looks like some guy who works at a service station pumping gas."
These pretensions to an everyday American voice fueled what T. J. Clark has called abstract expressionism's "vulgarity," its "overstuffed, unctuous, end-and-beginning-of-the-world quality." The concept of vulgarity is central to upper-class culture because that culture is deeply mendacious: built to perpetuate an elite class and its power, yet constantly claiming that "the People" rule and "every voice matters." When underlings take these promises seriously and attempt to make good on them, the upper class responds with reflexive disdain: "To call someone vulgar is to say he insists on a status which is not yet proved or well understood by him, not yet possessed as a matter of form." It can also be used to criticize an elite who has dabbled—"slummed"—in lower-class culture. In both cases, "vulgar" functions as a pejorative term that maintains and protects class distinction.
At the height of its powers, the United States upper class could generally count on the support of the middle class and their belief in democracy's slogans. But the postwar period saw the ruling class under massive revolutionary pressure from below. Its members could only protect their economic and political power by sacrificing cultural power to their subordinates, and thus placating them: "The bourgeoisie's great tragedy is that it can only retain power by allowing its inferiors to speak for it . . . and steeling itself to hear the ludicrous mishmash they make of it—to hear and pretend to approve, and maybe, in the end, to approve without pretending." Abstract expressionism's appeal derived from its ability to suggest that, for the first time, "everyday Americans" were seizing the opportunity to express themselves on culture's most exalted stage. Now, finally, art would directly convey the desires and frustrations of the masses. Here is Jackson Pollock, clad all in denim—the gas jockey given a chance at last to speak his mind to the nation. This moment was simultaneously powerful and manipulative. On the one hand, something like a democratization of art was actually being attempted. The results would be passionate, spontaneous, inchoate. On the other hand, this artistic egalitarianism disguised and solidified an economic retrenchment, a deepening of inequality and exploitation.
FIGURE 37. Jackson Pollock in his studio, 1950. Rudy Burckhardt, photographer. Jackson Pollock and Lee Krasner papers, circa 1905–1984. Archives of American Art, Smithsonian Institution. Work in photograph: © 2016 Pollock-Krasner Foundation/Artists Rights Society (ARS), NY. Photo © 2016 Estate of Rudy Burckhardt/Artists Rights Society (ARS), NY.
Abstract expressionism's vulgarity was deeply deceptive, but it was also a source of great power. Elites developed a taste for the vulgar as forbidden fruit, "placed at the outer limit of civil life, [which] become symbolic contents of bourgeois desire." Many of abstract expressionism's practitioners, peers, and critics agreed. Thomas B. Hess identified vulgarity as a force in de Kooning's style: "The picture was no longer supposed to be Beautiful, but True. . . . If this meant that a painting had to look vulgar, battered, and clumsy—so much the better." De Kooning seems to have shared this assessment: "I always seem to be wrapped in the melodrama of vulgarity." David Smith declared, "The truly creative artist deals with vulgarity." And Motherwell would claim that "European artists" scorned their American peers' interest in everyday experience, since "it's vulgar to talk about practical things."
Vulgarity was thus both abstract expressionism's greatest strength and its deepest vulnerability. Its sweaty, anxious, overwrought muscularity—its perceived arrival from below—contributed to its appeal. But by the late 1950s, the style was widely held to have entered a period of academicism and crisis. Its claim to a mass or petit bourgeois voice was being revealed as stale and false: the artists in question appeared no longer to be spiritually or practically members of the class to which their art implicitly bore witness. On the contrary, their pretensions to such a voice seemed to have become branded and commodified, publicly lauded and embraced by the ruling class. By January 1960, Hilton Kramer could wonder in his _Arts_ editorial whether Pollocks were about to be hung in the White House. With acclaim came money. Charlotte Willard's 1960 "Market Letter" in _Art in America_ remarked that "Today, Pollocks are in the stratosphere," with paintings by de Kooning, Rothko, and Kline rapidly ascending. As Harold Rosenberg disparagingly declared, "The man who started to remake himself has made himself into a commodity with a trademark."
FIGURE 38. Mark Rothko, _Untitled (Seagram Mural)_ , 1959. Oil and mixed media on canvas, 104½ × 113½ inches. The National Gallery of Art. © 1998 Kate Rothko Prizel and Christopher Rothko/Artists Rights Society (ARS), NY.
This success raised the specter of fraud and counterfeiting, "a slickness of the unslick as well as of the obviously slick," that Helen Frankenthaler bemoaned in her response to the _ARTnews_ survey. Vulgarity could be faked; the journey, the struggle, the privation, could be counterfeited. In order to supplant abstract expressionism, its successors would need to unveil truer, more authentic, forms of American yearning—the real things Americans desired and the real ways they attempted to express these desires. Unsurprisingly, a wide range of artists turned to "popular culture": comics, movies, advertisements, television, and film—"all the great modern things that the Abstract Expressionists tried so hard not to notice at all." But it was Warhol who most completely integrated the tensions of vulgarity into his artistic production.
Johns and Rauschenberg sensed the power of popular images and technologies, and the ways in which they rendered abstract expressionism's rhetoric obsolete—think of Johns's flags and beer cans, Rauschenberg's silkscreens and newspapers and advertisements—but they always either staged them as minor elements within a larger assemblage or counterbalanced them with marks of artistic virtuosity. Sympathetic critics immediately dissociated Rauschenberg and Johns from wholly vulgar topics and methods. According to Harold Rosenberg, "The images in Johns's paintings are familiar without the grossness favoured by the pop artists. . . . [They] are separated from the banal by their abstractness and dignity, qualities which are also outstanding in Johns's personality." Leo Castelli applied a similar logic to Rauschenberg's work, "every inch of [which] was thoughtfully considered and hand done—not like Andy Warhol." Rauschenberg and Johns were thought to embody what Pierre Bourdieu called "the detachment of the aesthete," who "appropriates" mass culture but "displac[es] the interest from the 'content' . . . to the form, to the specifically artistic effects which are only appreciated relationally, through a comparison with other works." This aesthetic detachment distances the aesthete from working-class subjects and their supposed propensity to immaturely or animalistically "follow their impulses" toward the enjoyment of brand images and celebrities.
Warhol's work actively rejected this detachment in a variety of ways. First, unlike his peers, Warhol refused to integrate his work into a tradition of avant-gardism, "a comparison with other works." On a live radio broadcast in 1964, Bruce Glaser asked Claes Oldenburg, Roy Lichtenstein, and Warhol each to recall their first exposure to pop imagery. Lichtenstein and Oldenburg responded by situating their work art historically, citing the influence, respectively, of abstract expressionism and of Céline and Dubuffet. Warhol refused this opportunity: "I'm too high right now. Ask somebody else something else." Warhol's brainless, narcotic response constituted an absolute rejection of the art historical apparatus of influence and precedent, one that he would reiterate throughout his career. Moreover, he refused to adopt an attitude of detachment toward his source material or to fetishize the "specifically artistic effects" of his work. Instead, he repeatedly proclaimed his interest in, and attachment to, the subjects of his painting; pop art, as he famously told an interviewer, is "liking things." Warhol's abdication of disinterest and detachment was reiterated in his emphasis on mechanical reproduction, with its connotations of cultural amateurism and striving. All of these déclassé connotations were consolidated in Warhol's unprecedented public statements concerning his appetite for profit, which never failed to embrace a decidedly interested, non-aristocratic perspective. Although he repeatedly espoused the benefits of high bourgeois or aristocratic _sangfroid_ , Warhol could never quite attain it: "I still care but it would be so much easier not to care." Already in 1963, he would tell an interviewer, "I am going to stop painting. I want my paintings to sell for $25,000." In 1964 he directly rejected the aristocratic ideal of art-making for its own sake. When asked "If you are happy doing what you do, should you be paid for it?" Warhol responded in the affirmative, "Because it will make me more happy." When asked "How much?" Warhol refused to compromise: "As much as I want." The interviewer's final follow-up question hinted at the anthropocentric stakes of this ideal: "Are you human?" Warhol: "No."
FIGURE 39. Gianfranco Gorgoni, _Andy relaxing at home, New York City_ , 1970. © Gianfranco Gorgoni.
Warhol's espousal of profit put his works' thematic transgression in a new light. All previous efforts to equate the artwork with an object of merely pecuniary interest—Duchamp's above all—had been quickly rehabilitated as meaningful, poetic, or beautiful. Warhol's embrace of a purely financial imperative for art-making raised the possibility that his art was in fact _only interesting_ , that it existed purely to stimulate interest in order to produce sales and therefore profit—an unashamedly vulgar project that challenged the boundaries of art. Like the working-class subject described by contemporary sociologists, Warhol embraced a position in which "immediate gratifications and readiness to express impulses tend to be observed . . . , since [the working-class subject does] not usually perceive meaningful incentives to do otherwise." When an interviewer asked him to comment on _Time_ magazine's describing his art as "vulgar," Warhol's response was again unequivocal: "Yes." When the interviewer drew the logical conclusion—"Oh. Do you think what you do is really art?"—Warhol offered no comment. Where the abstract expressionists had continued to aspire to a state of dignified creativity in which all material concerns would become superfluous, Warhol presented himself as unable to relinquish these concerns. Rosenberg's verdict was unequivocal: "For de Kooning, art has been a 'way of living'; for Warhol, it is part of one's self-projection or something to do for gain."
Warhol, as we have seen, was deeply attuned to the power and the pathos of the vulgar. His critics remarked this emphasis almost immediately, but only occasionally and angrily sensed its political implications. In exceptional cases, however, the links between vulgarity and egalitarianism could be decoded. Writing in the _New York Times_ in 1962, John Canaday argued that vulgar people—a "mass audience"—demanded vulgar art, and vulgar artists produced it: "With the best educational and social intentions in the world, museums and art centers . . . have created across the breadth of the land an appreciation of art as a hobby . . . that is vulgar in all-inclusive definitions of the word." According to Canaday, "the popular audience" and art's "essentially aristocratic" "rewards" could not be reconciled without a pernicious vulgarization of art. And "vulgar" as Canaday remarked in another article, "is the unkindest cut of all."
One last note on the queer and the vulgar: although I have presented them as two parallel challenges to abstract expressionism in the years of its decline, they cannot ultimately be regarded as separate. Instead, they implicitly overlapped and informed each other throughout this period, since "the Other usually is regarded as _vulgar_ , and the vulgar is almost always seen as _other_." My sense is that, in his early 1960s work, Warhol's classed critique screened his queer critique; if the queer modernist response to abstract expressionism had proven illegible to most commentators, a classed response would be given center stage. But for others in Warhol's milieu, these conjunctions were direct. Imagining a new queer aesthetic in the opening lines of his 1958 "Ode to Joy," Frank O'Hara placed vulgarity at its core:
We shall have everything we want and there'll be no more dying
on the pretty plains or in the supper clubs
for our symbol we'll acknowledge vulgar materialistic laughter
over an insatiable sexual appetite
and the streets will be filled with racing forms
and the photographs of murderers and narcissists and movie stars
will swell from the walls and books alive in steaming rooms
to press against our burning flesh not once but interminably . . .
FIGURE 40. Andy Warhol, _Cagney_ , 1962. Unique silkscreen on paper, 30 × 40 inches. Museum of Modern Art. Digital Image © Museum of Modern Art. Licensed by SCALA/Art Resource, NY. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
These "photographs of murderers and narcissists and movie stars," these angels with dirty faces—they were Warhol's inspiration as well. Vulgarity allowed art to communicate, but it also promised an openness—to culture, to one's surroundings, to others. And in contrast to camp, it faced these others without the shielding effect of irony, the "typically camp inversion" in which "those who are 'normally' excluded become the subjects and objects of a cult." For O'Hara, this openness marked the apex of aesthetic experience: "I want to be at least as alive as the vulgar." I think Warhol would have agreed. We see inklings of such an imagining in his unforgettable drawings of feet and branded goods from the 1950s and early 1960s, in which queer attraction seems poised to take on the powerful appeal of the brand image. But when Warhol tried to exhibit his homoerotic work in the 1950s, it was either rejected outright or derided as "coy," "sly," "provincial," and "abound[ing] in private meaning." Even his closeted colleagues Rauschenberg and Johns would reportedly spurn him, not just as "too swish" but as someone who thrives as "a commercial artist" and "buy[s] the work of other artists." O'Hara spurned him too.
FIGURE 41. Andy Warhol, _Feet and Campbell's Soup Can_ , ca. 1961. Ballpoint pen on paper, 17 × 13 ⅞ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
And so, in his pop painting, Warhol was careful to balance or even disguise a queer, O'Haraesque vulgarity with a vulgarity that could—and would—be accepted as universally "American"—"more concretely American than the American flag itself," as Buchloh has put it. The brand image works are crucial here, but so are the classic _Marilyn_ paintings, which must have been at least partially inspired by de Kooning's _Marilyn Monroe_ (1954), "the sensation of an otherwise wholly abstract show of his" in 1955 (plates and ).
By embracing a figure closely associated with the hypermasculinity of abstract expressionism, and with heteronormativity more generally, Warhol protected himself against the homophobic reaction his earlier work had elicited. Yet he could still trade on the frisson of vulgarity these images provoked. After all, Monroe's vibrant sexuality was closely linked to her disadvantaged background; _Life_ magazine introduced her as having been "brought up at municipal expense in 112 different foster homes in Los Angeles" and resorting to "unclothed pos[ing] for calendar art when she was broke." The _Elvis_ and _James Cagney_ paintings offered similar benefits, calling to mind contemporary stereotypes of "lower class culture," including "toughness, cunning, chance-taking, search for excitement, [and] trust in 'luck.'" Elvis was the "media archetype" of "working class" rock 'n' roll: "the poor Southern boy who escaped a life of truck-driving by remaking American music." But where de Kooning was careful to bury his vulgar material in his paintings ("I was painting a picture, and one day—there she was!"), Warhol would bring these vulgar materials to the fore, unmediated, "to press against our burning flesh not once but interminably," trading on stereotypes of the working-class as being "readier than are middle-class people to engage in physical violence, [and] to express themselves freely in sex." And where de Kooning posited an individual relationship between the painter and the vulgar subject matter, Warhol seemed to present his material mechanically, shorn of all personal specificity (although this mechanical style had expressive connotations of its own).
## The everything man
As we have seen in this chapter, Warhol used queer and classed voices to challenge and undermine abstract expressionism's claims to an "everyday" American voice. In so doing, he devised a response to the dominant art of his time that was distinctively modernist: timely, focused, and critical. But Warhol's critical response to abstract expressionism was not his only significant contribution to modernism, since his construction of these classed and queered voices was by no means exclusively reactive. The more we learn about the specifics of Warhol's practice, the more it becomes apparent that he was fascinated by art's powers to institute change. In this respect, Warhol ought to be recognized as a crucial instigator of what Rancière has described as the "specific impulse" of modernism: "some kind of will to change the world, to connect the forms of artistic practice with forms of life." This remained Warhol's stated hope for art throughout his career: a fundamentally egalitarian and universalist embrace of creativity with distinctly political implications: "Just ordinary people like my paintings. . . . I'm a mass-communicator." Many of the hierarchies that Warhol challenged corresponded to class structures and stereotypes: boss/worker, sated/hungry, refined/coarse, obedient/rude, rational/impulsive, intelligent/stupid, human/bestial. In every case, Warhol championed the subordinated term, proposing an egalitarian culture that allowed everyone to cash in on neoliberalism's promises of individualized mobility and cultural participation.
Blake Stimson has shown that Warhol's Carnegie Tech professor Robert Lepper urged him to search out an art that was "genuine, earnest and obviously vulgar (common) . . . democratic, constructive, experimental." Warhol's life's work as a producer constitutes an incredibly consistent effort to extend art beyond all of its previous boundaries, to produce a truly encompassing creativity. Warhol dramatized this expansion through his carefully constructed persona, which he consistently emphasized as being déclassé, amateur, queer, and improper, but striving nevertheless for participation and recognition: "The Pop idea, after all, was that anybody could do anything, so naturally we were all trying to do it all."
This egalitarian-modernist expansion would be technologically assisted—functionally cyborganic. Warhol was always testing reproductive technologies that promised to increase the accessibility of cultural production. What's more, his work constantly turned to subjects that had previously been deemed unacceptable by what Rancière calls the representative regime. All forms of appetite—culinary, sexual, commercial, romantic, nostalgic, voyeuristic, puerile, fetishistic, bestial—were fair game for Warhol's artistic attention, as were the mundane realities of everyday life. Warhol's desired world was open, immediate, expansive: "Everyone and everything is interesting." His mother memorably called him "the everything man" who encompassed "the good and the bad and the lousy, and the shocking, terrible and the fairies and the girls and the boys and the drugs."
Who, during the postwar period, did more to expand art's boundaries of production and reception than Warhol? Who pushed harder at the edges of subject matter, technique, and audience? Who posed bolder challenges to the old hierarchies of genius and creativity? This is what made Warhol such a keen modernist in Rancière's sense: by rejecting all artistic hierarchies, his work opened the door for previously excluded voices to participate in culture, since, through its rejection of "hierarchical presuppositions," the aesthetic regime "is the theory of the possible community of equals." As Warhol maintained, "I'm not the High Priest of Pop Art. . . . I'm just one of the workers in it."
These hitherto excluded voices recognized Warhol's contributions in this regard. For Lou Reed, Warhol "put everything together, allowed everything to happen." Kathy Acker cited his "refusal . . . to distinguish between elite and scum . . . his amazing clarity and courage, and his overt proud homosexuality" as profoundly influential on his contemporaries. Kara Walker praised Warhol's "soulful" shallowness as "open and generous in a dumb kind of way . . . accepting of its own fallibility and humanity." Even Warhol's well-known aphorism regarding "fifteen minutes" of fame seems to have originated in a vision of egalitarianism: "There's going to be a day when no one will be famous for more than a week. Then everyone will have a chance to be famous." Each of us has only so much time; we ought to find a way to share our moments in the spotlight equally.
For the reasons addressed in this chapter, this egalitarian Warhol has long been overshadowed in popular and critical discourse by the ironic, Duchampian Warhol, pursuing a postmodern askesis of voice, meaning, and ambition—a great leveling down that left all of the modernist projects behind. This reading flattens and caricatures Warhol's life's work, and yet it also points to a very real ambivalence therein. It wasn't that Warhol had given up on art's radically egalitarian and democratic capacities; rather, he could sense these capacities being stymied and misappropriated, turned into false promises and distractions. Perhaps more acutely than Rancière, Warhol knew that, under neoliberalism, "cultural democracy" could be repackaged as a substitute for actual democracy. Fancy jeans and cheap soup cans could smooth over differences; everyone could drink the same Coke.
Warhol himself is to blame for some of this. Although he continually professed the appeal of collaboration and accessibility, and gestured toward a break with the artworld paradigm during the 1960s, he was unable to pursue this insurgence consistently. He could never fully embrace a world beyond geniuses and museums, "the creative or productive functions freed of this always reappearing author-function." And although he braved a new trail for queer artists, his intense shame made it difficult for him fully to embrace these new possibilities. Warhol seems to have doubted his own ability to occupy the minoritarian positions his work explored: the downtrodden, the queer, "species promiscuity and the bewilderment of the swarm." He saw his collaborators pushing the limits of these positions; he was inspired by their efforts but also intimidated by the attendant risks.
Thus, throughout his career, Warhol sensed and bemoaned the moments when his predilections for egalitarianism, collaboration, and expansion (the core principles of Rancière's modernism) were compromised and undermined. But he also realized that these moments of failure would be entertaining, and made a highly successful and lucrative spectacle out of his purported distance from spectacular productivity. The paradoxes inherent in this strategy account for the strange arc of Warhol's career and for his works' lasting power and ambivalence. This ambivalence penetrated every aspect of Warhol's persona: the notoriously ill-fitting wig ("white hair falling like a comic wig over his ashy face and dark back hair"), the almost imperceptible voice, the intensely blasé affect, the "inanimate" handshake, the aluminium foil pretending to be "silver." And it is equally apparent in Warhol's aphorisms, where cultural accessibility and egalitarianism are constantly being held out as alluring but impossible dreams: "If your name's in the news, then the news should be paying you. Because it's _your news_." Or where everyone would be allowed to participate in exchange equally: "People should be able to sell their old cans, their old chicken bones, their old shampoo bottles, their old magazines. We have to get more organized."
These utopian visions are always presented in Warhol's work and writings as dreamy impossibilities—today, of course, they have been transformed into a "highly capitalized and highly exploitative" "sharing economy" through eBay, Craigslist, and YouTube. Warhol would dwell on them for the rest of his life; as he told Paul Taylor in his final recorded interview, "It's just like a Coca-Cola bottle when you buy it, you always think that it's yours and you can do whatever you like with it. . . . I don't get mad when people take my things." The ideas are alluring, but the entertainment value lies in the pathetic inability actually to attain the visions of accessibility and equality; Warhol, of course, did frequently get upset when his things were expropriated.
Warhol's work is not, however, vitiated by these shortcomings. The tension between its ambitions and its disappointments is what animates it. Warhol's bravery in this regard became most apparent when his efforts to expand art's boundaries were met with direct repression, as in his censored mural for the 1964 World's Fair, _13 Most Wanted Men_. In this work, perhaps more than any other, Warhol's efforts at egalitarianism demonstrate their full radical potentiality. The traditional bulwarks of artistic hierarchy—the monument, the heteronormative, the genius—were invaded by what they sought to repress: the criminal, the queer, the machine. And the state, sensing this multifarious threat, immediately terminated the project.
FIGURE 42. Andy Warhol, _13 Most Wanted Men_ , 1964. © 2016 Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
## A cinema for everyone
When Warhol turned to filmmaking in the mid-1960s, the possibilities of cultural egalitarianism continued to inform his production. Across their vast breadth, the films Warhol produced during this period explored three distinctively Rancièrian questions: Can anyone make a movie? Can anyone and anything be in a movie? And can anyone watch a movie? While these questions may sound optimistic, however, the answers Warhol found for them were consistently ambivalent and compromised, tinged by various disparities, including those of class. Yes, in all three cases, Warhol proved that anyone—even the most marginalized—could become involved with the movies. This is the utopian and egalitarian dimension of Warhol's film that Douglas Crimp celebrates: his ability "not [to] judge the people in his world." But Warhol's films also repeatedly demonstrate that these marginal voices—including Warhol's own—would win their entry into cinema through failure. As in the newsprint and brand image works discussed in chapters and , the films' inability to attain to the norms and standards of Hollywood would constitute their entertainment value, their style. "We have to make our movies look the way they do," Warhol told Joseph Gelmis in 1970, "because if you can make them look better bad, at least they have a _look_ to them. But as soon as you try to make a better movie look good without money, you just can't do it."
Like his painted and silkscreened work from the early 1960s, the films Warhol produced during the mid-1960s are pervaded by a distinctive technical amateurism. This amateur quality was widely remarked by Warhol's critics and embraced by his collaborators. Warhol himself stated the basic ethos in 1969: "Well, like everybody else, we always try to do whatever is easiest." Here again, Warhol espoused a stereotypically working-class attitude, with its emphasis on "immediate gratifications" and its aversion to education: "films are better. They're easier. The camera does all the work." Warhol's performers noticed this emphasis right away. As superstar Tally Brown observed, "Where I came from when you were making a movie somebody said, 'Lights! Camera! Action!' and they told you what to do, and . . . you'd usually learned lines for the occasion and there was somebody behind the camera. . . . [In Warhol's productions] _none of these factors were present_."
Callie Angell has remarked that Warhol produced almost all of his movies during this period "with the most basic filmmaking equipment available," and that the exceptions to this rule, like _Empire_ and _Henry Geldzahler_ , were still markedly subprofessional. His first movie camera was a 16-millimeter Bolex purchased in 1963 at Peerless Camera. In perfectly Warholian style, Bolex cameras had been advertised during the 1950s as an affordable link between everyday Americans and Hollywood film production—"easy to handle . . . easy to operate," beloved by film stars, and only $89.95. Bolex movies were supposed to be practically automatic: "You can switch on the continuous-run lock and get into your own scenes while your Bolex runs by itself." By the mid-1960s, Bolex was promising that, with its equipment, "home movies won't have to look like 'home movies.'" Warhol appears to have taken these promises seriously: "I was going to Hollywood. . . . Hollywood is the movie capital of the world so I bought a movie camera to take along. A 16mm Bolex."
Needless to say, a massive technological chasm separated the Bolex from its supposedly similar Hollywood counterparts. The Bolex shot three-minute clips of silent film in black-and-white, clips that could never be mistaken for Hollywood quality. Warhol's shaky camera work, unedited light flares, and transparent leaders between rolls made his films even more unmistakably nonprofessional. These discrepancies were brilliantly emphasized in _Soap Opera_ (1964), which alternates between silent Bolex clips and appropriated television commercials, including one for the "Wonda-scope," a $2.49 plastic gadget, reminiscent of the Magic Art Reproducer, that promised to combine microscope, telescope, binoculars, and pharyngoscope into "a useful scientific instrument that no home can afford to be without." The Wonda-scope's functions encompassed a variety of Warholian preoccupations: examining "jewelry for flaws," getting a better view not just of "horse-racing or theater," but also of "blackheads and ingrown hairs." The film's succeeding and final reel—an overexposed shot of a woman dancing naked before a mirror, and being discovered by a "scandalized" Gerard Malanga—draws out the implicitly voyeuristic implications of the Wonda-scope's pretensions to scopic mastery.
When Warhol screened his early silent films, he would slow down their frame rate from twenty-four frames per second to sixteen, thus extending their regular three-minute running time to four and a half minutes and accentuating their distance from commercial film. (Bolex advertised a similar feature: a "Magic Switch" on its projector that would "turn any movie into Amazing Slow Motion.") This alteration effectively stretched out the films, further emphasizing Warhol's position as a nonprofessional filmmaker who needed to maximize his limited materials. In films like _Kiss_ and _Sleep_ (both 1963), Warhol highlighted this quality of "making ends meet" by joining and repeating dozens of short clips to build a much longer feature. But even as he was maximizing the running time of his camera rolls, Warhol was working to minimize his authorial intervention; Angell pointed out that in almost every early film after _Kiss_ and _Sleep_ , Warhol devoted each reel to only one shot, and in _Blow Job_ and _Eat_ (both 1963–64), single shots span multiple reels without directorial intervention.
This commitment to minimal expenditure and intervention persisted even when Warhol acquired more advanced equipment; filming _Harlot_ with an Auricon sound-on-film camera in 1965, "Andy remained close to the camera throughout and frequently watched through the view finder, though, of course, he didn't touch the camera itself." And when Warhol did intervene, he emphasized the arbitrariness and inexpertise of the intervention; his seemingly unmotivated zooms are a key example. "I just put the motor on," Warhol told an interviewer in 1963, "and [it] just goes on and shoots and it's mostly all finished. It's practically—it is finished." He bragged of _Blue Movie_ (1969) that "it was shot in three hours at a cost of $2,000." Bourdon called this slapdash quality in Warhol's work "benign neglect" and pointed out that it extended to the film's public screenings, where reels were "jumbled" and "deleted." From production to distribution, every element of filmmaking tested the ease of amateur cinematic technologies, the advertised promise that they "won't let you make a mistake."
Just as he tested the limits of technological accessibility in these early films, Warhol also tested the limits of cinematic narrative, searching out premises that had previously been judged either too prurient, too tedious, or both. _Kiss_ , _Sleep_ , _Eat_ , _Blow Job_ , _Mario Banana_ , and the _Screen Tests_ all singled out marginalized or forbidden elements of traditional cinema (auditioning, sleeping, eating, making out, sexual intercourse) and attempted to build movies from them without any supporting framework. Eschewing the standard structures of character and narrative, these films enact a poverty of subject matter that parallels their poverty of technique. Their plots and styles are thus intentionally stupid, a position Warhol embraced, relishing what Rancière would call "'life' in its 'stupidity,' in its raw existence, without reason." Asked whether he could make the same movies "if you were very stupid," Warhol concurred: "Yes. . . . Because I'm not very smart." Ronnie Tavel remembered requesting that Warhol screen the films they had just produced so that he might "learn from" them. Warhol "sort of turned his head away and said cynically under his breath: 'Everyone wants to learn something.'"
Through all of these technical and authorial devices, Warhol embraced the position of the lay producer, the nonprofessional enthusiast—blessed with neither genius nor affluence but aspiring nevertheless to cultural participation. Like the stereotypical working-class subject profiled and disparaged by contemporary sociologists, his "orientation [was] local, concrete, face-to-face, relatively deprived of long range considerations," and focused on "getting by with as little education as possible." But just as he had wanted "to be a machine" when he painted, he now wanted "to be plastic," like the stars in Hollywood. Financing remained an obstacle: "I think movies are the kind of things Hollywood does. We haven't been able to do that. Because you need a lot of money to do that. So we're working it out our way." From screenplay to equipment to camerawork to editing to projection, Warhol's early films consistently proclaimed their nonprofessionalism, their status as amateur productions yearning for a wider audience. This aspiration—the distance between the filmmaker and his mythic Hollywood—held its own dramatic qualities.
Concurrently, Warhol hoped that the egalitarian dimension of his style would provoke an egalitarian audience. Basic subjects would stimulate lower-class "impulses" and "gratifications." If unsophisticated audiences just go to the movies to see the star and "eat him up," why not focus on the actor exclusively—eating, sleeping, cumming? If their priority is to "relax and be entertained," why not "make movies to read by, to eat by, to sleep by." As Warhol told _Vogue_ , "Movies _should_ arouse you, should get you excited about people, should be prurient. . . . Prurience _is_ part of the machine. It keeps you happy. It keeps you running." The split screen in _Chelsea Girls_ would even stave off boredom: "If you get bored with one, you can look at the other." The filmmaker's playful work provokes and produces the audience's playful work, a truly participatory culture: "They get involved with themselves and they create their own entertainment. . . . It becomes fun." Warhol's cinema thus emphasized working-class "norms" described by contemporary sociologists: "fun," "'present time' orientation," and "enjoying today while [one] can." His fundamental ethical stance—"I just like everybody and I believe in everything"—was presented from a stereotypically working-class perspective: Leticia Kent remembered him making the assertion "in an Andy Hardy voice." The overall goal would be a radically expanded audience, encompassing the "uncivilized" classes: "Pop Art is for everyone. . . . I think [art] should be for the mass of American people." But, like his aspirations to Hollywood success, Warhol acknowledged that these dreams of a playful and universal audience were ultimately improbable; the only truly popular _and_ nonprofessional film he could cite as a model was amateur pornography, "beaver films."
This Rancièrian modernist egalitarianism of technique, technology, subject matter, and audience was paralleled and reinforced in Warhol's approach to his actors. For Warhol, the superstars "were just the ones who wanted to be in a movie. So that's how we used them." This is a crucial admission: just as filmmaking was most interesting when it was struggling (and failing) to achieve professional levels of polish, performers were most interesting when they were struggling (and failing) to achieve stardom. Asked if he would like to work with the actress Carroll Baker, Warhol demurred: "She has too much acting ability. . . . I want real people." Because they capture the very moments of aspiration to cinematic participation, the _Screen Tests_ are the paradigmatic investigation of this struggle. But a similar aspirational effect pervades all the silent films as a consequence of their slow motion: it is as if each figure were straining to move, to breath, to inhabit the cinematic environment, to bridge "this incredible distance between being 'a Hollywood star' and a Superstar." At the limit, this gap between subject and star could be lethal—George Reeves and Marilyn Monroe were exemplary in this regard.
Warhol had emphasized this distance in his paintings of comics, grocery commodities, and celebrities, and he relished it in his filmmaking. When asked how he got actors "to give . . . their most in front of the camera," he replied, "I don't ask for the most." The performers' Hollywood striving thus mirrored or reiterated the director's; both parties were intentionally shown to be ill-equipped to achieve the cultural prominence they so craved. And yet paradoxically, this spectacle of failure was what drew paying customers. As Pauline Kael disparagingly remarked, "They were counterfeit stars willing to mock their failure to pass for genuine but nevertheless hoping that the travesty would make them a new kind of star."
FIGURE 43. _Tobacco Road_ , film still, 1941.
As in his paintings, in Warhol's films these spectacles of failure were powerfully linked to class, complicating and undermining his professed commitment to egalitarianism. Visitors to the Factory were struck by the presence of figures not typically associated with art-making—"non-actors or freaks," "the people most excluded by other groups," even "the head of a 97th Street and Columbus Avenue girl gang." Kael immediately and anxiously recognized that the Warholian Superstar had unpleasant class associations: "like the self-disgusted, gregarious, quarrelsome people in bars and street fights—people to be avoided." Unlike a film such as _Tobacco Road_ , where degradation was employed comically and fictionally, in Warhol's "bohemian Tobacco Road, the people flaunted their own dishevelment and their own nausea, and it was all so depressing that even when they did funny things, one didn't feel much like laughing." For Kael, these lower-class exploits could only be entertaining when they were staged. The real economic and social tensions embodied in Warhol's superstars brought them a little too close to home. In this respect, the Silver Factory resembled what Jameson has called a proto-utopian "experience of social promiscuity," like high school or the military, in which members of different classes are forced into contact with one another and antagonisms become individual rather than collective. And like the classless utopia these situations foretell, Warhol's Factory "welcome[d] the most outrageous self-indulgences and personal freedoms of its citizens in all things"—including narcotics, non-normative sexuality, and eating disorders.
FIGURE 44. Andy Warhol, _Poor Little Rich Girl_ , 1965. 16 mm film, black and white, sound, 66 minutes. © 2016 Andy Warhol Museum, Pittsburgh, a museum of Carnegie Institute. All rights reserved.
Interestingly, as the poet Paul Carroll recognized in 1969, class antagonism in Warhol's films was often staged in gendered terms. Carroll's profile of Warhol in _Playboy_ included a detailed typography of Warhol's actors, noting that the majority of the female stars came from moneyed backgrounds. Almost all of the male stars, by contrast, "are products of tough, lower-class backgrounds" or are associated with them onscreen: "Almost every young man in Warhol movies portrays a street arab, tough but tender and wounded."
While Edie Sedgwick's appearances in Warhol's films confirm Carroll's dichotomy, they also consistently explore fantasies of class shifting and transgression. In homage to his childhood idol, Shirley Temple, Warhol titled Sedgwick's first star vehicle _Poor Little Rich Girl_. When these words awaken her from off-screen as the film begins, Sedgwick's response is succinct: "Fuck you." The star's alienation from her class is a recurrent theme, and Edie the heiress is rendered human—her aristocratic aura is undone by the camera's relentless gaze. As a stoned Sedgwick tells Chuck Wein, when he suggests that she "marry somebody rich": "I know a lot of rich people but they're all pigs." The film allows Sedgwick to perform an abdication of her privilege, to present herself as "an agent of the secret discontent of [her] class with its own rule" who would "like to kill grandma." Sedgwick thus fits neatly into Carroll's gender/class typology; the film is a "day in the life of a poor little rich girl cut off from the family fortune" who has "two 'contradictory status positions,'" the higher of which, her class, "can be undercut to resolve the contradiction in favor of the lower status"—her gender.
FIGURE 45. Andy Warhol, _Vinyl_ , 1965. 16 mm film, black and white, sound, 67 minutes. © 2016 Andy Warhol Museum, Pittsburgh, a museum of Carnegie Institute. All rights reserved.
Class drama is similarly prominent in many of Warhol's other films, but there are instances in which Carroll's dichotomy does not hold. The subject of _Mrs. Warhol_ (1966) is a recognizably working-class, immigrant woman—Warhol's mother—unconvincingly pretending to be a former beauty star (plate 17). In _The Life and Times of Juanita Castro_ (1965), the class betrayals of the Cuban revolution are presented as theatrical and sexual in nature, with the various rivalries between Juanita, Che Guevara, and the Castro brothers staged as farce. In _Vinyl_ (1965), where class antagonism is figured as an intramasculine struggle, this theatricality becomes more menacing. Victor (Gerard Malanga), who wears a T-shirt and leather jacket, intercepts the delivery of a stack of magazines to an elegant suited figure with an air of undisguised disdain: "Pardon me sir! . . . Pray tell, sir, what does this page mean, sir? . . . I have always had the deepest respect for sirs who know how to read!" If, as J. J. Murphy has argued, Warhol's cinema is built on a foundation of psychodrama, this drama is frequently staged as class transgression and antagonism.
## The Little King
Another notable investigation of class transgression in Warhol's cinema is _Henry Geldzahler_ , a silent black-and-white film produced in 1964 that features its eponymous star—son of a Belgian diamond broker and curator at the Metropolitan Museum of Art—occupying an armchair for ninety-nine minutes. After setting up the shot, Warhol apparently left the room to conduct Factory business. Geldzahler's initially "majestic pose" soon broke down into a series of increasingly unflattering twitches, postures, and expressions, seemingly driven by boredom, anxiety, and restlessness, and culminating in a complete regression to the fetal position—one of the least majestic poses imaginable.
As Callie Angell first remarked, the film's most prominent art historical precedent is Picasso's portrait of Gertrude Stein, which Geldzahler would have seen every day in the Met. Noting the similarities in furniture and pose, Angell argued, "The resemblance is exact enough to have been a deliberate reference on Warhol's part, and . . . may be read as Warhol's homage to Geldzahler, whose relationship to Warhol as a patron and supporter of his artistic career echoed that of Stein and Picasso." This positions Geldzahler's portrait in a lineage that stretches back to Ingres, whose _Portrait of Monsieur Bertin_ has long been considered a crucial precedent for the Stein portrait—a lineage Geldzahler certainly would have known even if Warhol did not. But if they were thinking about Picasso and Stein, Warhol and Geldzahler must also have recognized the queer resonances between these portraits—in each case, an art capital's most dynamic young upstart was portraying a major tastemaker who was audaciously gay. Could they have sensed what Robert Lubar described as Stein's "lesbian masquerade," "her abrogation and/or usurpation of patriarchal authority that Picasso had to disavow in order to complete her portrait"? If so, might Geldzahler have been staging this masquerade in reverse, adopting a façade of patriarchal authority only to perform its undoing, smoking and fondling his cigar down to its nub, listlessly sending smoke signals? And might any of the anxiety and resentment that permeated Picasso's relationship to Stein have leaked into Warhol's relationship to Geldzahler?
When asked to explain his actions during the film, Geldzahler gave a variety of answers. In 1965 he credited the film with biographical and art historical significance, claiming that he "had just come back from a long week end with Jasper Johns and the Stellas . . . [and] had a lot to think about." In 1971 he tried a different tack, claiming that the film presented him with an opportunity to produce an encyclopedia of gestures, his "entire vocabulary." Neither of Geldzahler's explanations acknowledged the regressive transformation that he undergoes over the course of the film, a transformation that has long been muddled by the fact that the film's two reels were frequently shown out of order, so that viewers were greeted with Geldzahler mid-breakdown and witnessed an abrupt shift to the "majestic pose" midway through the screening.
FIGURE 46. Andy Warhol, _Henry Geldzahler_ , 1964. 16 mm film, black and white, silent, 99 minutes at 16 frames per second. © 2016 Andy Warhol Museum, Pittsburgh, a museum of Carnegie Institute. All rights reserved.
Warhol's own response to Geldzahler's performance was recorded in 1965, when the _Herald Tribune_ sent Frances FitzGerald to the Factory to write a profile of Geldzahler. The visit coincided with a screening of the film, providing FitzGerald with an opportunity to ask Warhol to comment on his work while they watched it together. Warhol's initial reaction was characteristically blasé: "His own creation might as well be moving wall paper. He sits at the back of the room collecting invitations from the lovlies [ _sic_ ] and occasionally glancing at the screen. 'Nothing happens. Why should I look? It's just a picture of Henry smoking a cigar for an hour and a half.'" What could be more cynical than an automatic portrait of a curator, manufactured for profit and social prestige, by an artist who is too busy to watch it? This is the Warhol with whom we are all familiar: the postmodern oracle, either succumbing to or parodying the world of the spectacle, where "the commodity's mechanical accumulation unleashes a _limitless artificiality_ in face of which all living desire is disarmed."
In many similar cases, such a response would have been the discernable extent of Warhol's aesthetic judgment. But in this instance, Warhol briefly experimented with a more analytical interpretation. As he asked FitzGerald, "'Do you think he's acting or being himself? . . . He's acting. He's playing the Little King.'" Warhol's hypothesis, despite being only seven words long, is remarkably suggestive, both in subject and object. Does "he" refer to the flesh-and-blood Geldzahler or his projected image? Warhol and FitzGerald are looking simultaneously at "the two Henrys," so we'll never know. The second part of Warhol's remark—"playing the Little King"— also points in two directions. As Angell noted, Warhol would have been referring to a comic strip by Otto Soglow, but also to a painting Warhol had made based on the comic four years earlier, in 1961 (plate 18). What might Warhol have wanted to convey through this comparison? How would the Metropolitan Museum's curator of modern art have stood to benefit from emulating a comic strip for posterity? And the faint tone of skepticism or disdain in Warhol's rhetorical question—what provoked it? Could he have been noting the similarities between the two figures' cardioid thrones, or even the phallic resemblances between Geldzahler's ridiculous cigar and the Little King's equally absurd crown and beard? Perhaps, and yet Warhol's brief addendum signals not just his art historical intelligence, as Angell argued, but his political intelligence as well.
Published as a comic strip in the _New Yorker_ from 1931 to 1934, as a King Features Sunday strip from 1934 to 1975, and as a comic book produced pseudonymously by John Stanley in the mid-1950s, Soglow's _The Little King_ was an almost exclusively wordless feature that followed the comical misadventures of its eponymous hero. Like Warhol, Soglow had been raised in poverty, and commentators have speculated that he used the figure of the Little King to take humorous revenge on the elite. A short and portly figure, vaguely similar in shape to Geldzahler, the Little King constantly undermined his own pretensions to aristocracy through clumsiness, haste, misunderstanding, and appetite. _Little King_ strips typically end in a dénouement; the example Warhol chose for his painting has the King driving an automobile into a knight who, in the final panel, is revealed to be nothing but armor "playing" a knight—a shell or pretense with no inner core (plate 19). As is usually the case in these punch lines, either the world disappoints the Little King, or he disappoints the world, or both.
It could be argued that the figure of the Little King perfectly emblematizes Warhol's fascination with the Debordian spectacle. After all, in Debord's analysis, royalty is a proto-spectacular and highly performative formation, wherein "all social life was . . . concentrated in the ornamented poverty of the Court . . . whose apex was the 'profession of king.'" All kings "play" the "little king"—this concentrated version of the spectacle is later challenged by and then integrated with the diffuse version: "the abundance of commodities" that characterizes "modern capitalism." The Little King would thus have provided Warhol with a suitably charged emblem for spectacular society, a bright and lively homunculus that could stand in for the distorting power of the commodity.
This spectacular dimension of the Little King's appeal was mischievously celebrated by Vladimir Nabokov in his 1951 memoir, _Speak, Memory_ , during a description of the author's first poetic composition. The young poet's "languid rambles" in the summer of 1914 were interrupted by an encounter with "the admirable and unforgettable village schoolmaster," "an ardent Socialist, a good man, intensely devoted to my father . . . always smiling, always perspiring," who, in 1905, had taught Vladimir and his siblings Russian spelling. The chance meeting brought on in young Vladimir a quasi-schizophrenic breakdown, in which he "registered simultaneously and with equal clarity not only [the schoolmaster's] wilting flowers, his flowing tie and the blackheads on the fleshy volutes of his nostrils, but also the dull little voice of a cuckoo . . . , the flash of a Queen of Spain," and many other things as well. He began to lose faith in the efficacy of language: the words in his poem "did not look quite as lustrous as they had before the interruption. Some suspicion crossed my mind that I might be dealing in dummies." At this moment, the young poet had come face to face with the challenges of political consciousness—as embodied by the socialist schoolteacher—and their repercussions for language.
But Nabokov quickly recovered, and it was the Little King who saved the day: "Fortunately, this cold twinkle of critical perception did not last. The fervor I had been trying to render took over again and brought its medium back to an illusory life. The ranks of words I reviewed were again _so glowing, with their puffed-out little chests and trim uniforms_ , that I put down to mere fancy the sagging I had noticed out of the corner of my eye." The intended homage was apparently too subtle. Nabokov later bemoaned commentators who "carelessly" missed in this passage "the name of a great cartoonist and a tribute to him." For Nabokov, Soglow's _Little King_ "so glowing[ly]" revitalizes language through the repetition of its appealing forms, which march across the page nobly and unyieldingly, like words or letters, distracting us from their actual mechanical materiality, just as the sheer punning accident of their creator's last name appears to provide them with an inner light, reasserting the aristocratic autonomy of the spectacle against the pressures of political consciousness.
Warhol must have relished this spectacular dimension of his source material, the dimension that Nabokov found so powerful. But when he turned to the Little King for inspiration, Warhol focused at least as much on the irreproducibility of this spectacle as its appeal, staging the visual reproduction of mass culture as a class-specific _agon_ , a charged and exasperated attempt by a working-class subject to participate in the culture that was targeted to him, using the technologies that promised to facilitate this participation. Strikingly, when he cropped and repainted this image, Warhol isolated the strip's least dramatic panels, emphasizing the participatory drama and deemphasizing the depicted one—neither the collision nor the dismantled knight was included, and the king's dripping sweat was transfigured into dripping paint. The tension between the Little King's expectations and the reality he confronts was replaced by the tension between model and emulator, panel and painter. It is as though Warhol offers a pessimistic addendum to Debord's concerns about the spectacle: not only is the spectacle massively alienating and appealing for purchase and emulation, it also holds out the false promise that its most powerless consumers might soon become its producers. Warhol was a poster child for these promises. He attempted to _replay_ the Little King, to perform his resuscitation by hand.
And Warhol was not alone in this endeavor. We have a remarkable record of an alternate, protocinematic version of this amateur cultural participation by one of Warhol's working-class contemporaries, the poet and philosopher Keith Gunderson, who grew up poor in Minnesota in the 1930s. In his prose poem "A Portrait of My State as a Dogless Young Boy's Apartment," Gunderson recalled his and his sister Janice's childhood interactions with Coca-Cola and the Little King:
> Sometimes me and Janice just before we'd fall asleep wouldn't fight at all because we'd do THE LITTLE KING on the bedroom wall and the way we'd do THE LITTLE KING was to get a teeny Coca-Cola bottle about one finger long that any kid could get for free once a year at the Southside Picnic in Powderhorn Park and make the baby coke bottle stick to a finger by drinking the coke and then sucking the air out and smacking a finger right on the hole part of the top of it and waggling our coke bottle fingers around in front of lamps by our bed so the lamps would shoot shadows of the coke bottles on the wall and the shadows were fatter and bigger than the real coke bottles and were shaped like THE LITTLE KING who was a fat guy in a king suit in a comic strip and he never said anything or did much and was shaped like the shadow of a little coke bottle and me and Janice would make our LITTLE KING shadows walk all over the walls and ceiling and sometimes across our covers and faces and once even across my butt and we would try to say whatever we thought THE LITTLE KING might say and it wasn't hard because in the comic strip THE LITTLE KING never said anything so we could make our LITTLE KINGS say and do whatever we wanted and it was like running a whole comic strip by ourselves which had two LITTLE KINGS in it and when I got older and could read . . . [I] thought our shadow comic strip had been much better than the real one and even so good it was almost a movie.
Gunderson's poem serves as both a precedent and a point of comparison for Warhol's mass-cultural replications. The young Gundersons rivaled Warhol in their manic creativity, but they differed from him significantly in their tone, their affect. Like Warhol, they experimented with technology, becoming little cyborgs with their lamps and "coke bottle fingers." Warhol had sent away for a bedroom projector as a child, and would use a projector in his studio for sketching, before he embraced silkscreens and then cinema as alternatives to painting. Like Warhol's, the Gundersons' excitement was simultaneously cultural ("running a whole comic strip by ourselves") and sexual ("and once even across my butt"). They played the Little King with a Rancièrian exuberance, a sense that the world of culture was theirs for the taking—that, as Rancière has argued, Plato's cave and the society of the spectacle are two moments in the same repressive history, and that the shadows and spectacles could be reappropriated. "From now on," in other words, "the border between action and life no longer has any consistency. Anything can enter into art. And, in parallel fashion, there is no longer any separation between a refined and an uncouth nature: art no longer has specific producers nor privileged addressees." This remained Warhol's stated hope for art throughout his career, a fundamentally egalitarian and universalist embrace of creativity with distinctly political implications: "Just ordinary people like my paintings. . . . I suppose it's hard for intellectuals to think of me as Art. I'm a mass-communicator."
Warhol seems to have recognized that the figure of the bumbling monarch could be refreshingly enabling for its young and unaristocratic readers. Like its commodity doppelgänger (another Warhol icon), this "shadow of a little coke bottle" held out the promise of culture as participatory and accessible, something that could be remade and improved into a very minor cinema by even the smallest readers in even the humblest rooms. But where Rancière and the Gundersons emphasized the glorious emergence of this egalitarianism, Warhol was constantly highlighting its limitations and false promises. His replications tended to be doleful rather than joyful. For Warhol, playing and replaying the Little King was always a way of accentuating a persistent _distance_ from cultural participation even as he pursued—and eventually achieved—it. And this distance effectively figures social _and_ economic marginality. It is a queer distance, but also a class distance, a way of emphasizing that the voice in question has not been granted access to a larger audience. Warhol seems sometimes to have thought of Geldzahler as his compatriot in this regard. As he told the curator when the two men attended Truman Capote's Black and White Ball in 1966, "Gee, we're the only nobodies here."
What would it mean, then, to say with a note of disdain, as Warhol did, that Geldzahler was playing the Little King in his filmed portrait? Invariably, in the comic strip, the punch lines come at the Little King's expense. Again and again, he is exposed as all pretense, comically failing to live up to his regal image. The "puffed-out little chests" that Nabokov celebrated are constantly being deflated in the final frame. But Nabokov recognized that this deflation can paradoxically be beneficial to the health of the regal spectacle. By perpetually bouncing back from its embarrassing weekly mishaps, the sovereign appears indomitable. What's more, these humorous calamities render the powerful more human—the king lingers among us, bumbling, hungry, distracted. If every king is playing the king, as Debord surmised, might there be a benefit to rendering this performance _performative_ and highlighting its comic shortcomings? Hadn't Warhol opened the door for such a maneuver in his early pop style, which continually emphasized his distance from the ideals he craved? What might a king, or the son of a Belgian diamond merchant, like Geldzahler, stand to gain from emulating this distance? And could Warhol share these benefits? Were they both actually "nobodies," or were they in fact "prince[s]," as one mutual friend put it, "operat[ing] on a broad, extravagant level of endless expanse, infinite privilege and freedom"?
In 1966 _Life_ magazine called Geldzahler "the connection, the affable fellow who links the multiple and widely separated circles of the world of art." He was writing "a book on European painting since 1900—a drug store book." His goal was a profitable leveling of aesthetic hierarchies—"making great art available to large numbers of people." Dell was reportedly "printing 100,000 copies." Warhol seems to have sensed his friend's talents and aspirations in this area—Geldzahler claimed that they spoke every day on the phone throughout the early and mid-1960s, often for hours. Here was another ambitious gay man, an "All-American boy of immigrant parents," but one who came from money and had learned to navigate every social realm with ease, to break down social and aesthetic hierarchies for profit, "not being selective, just letting everything in at once"—truly, an "independent" curator _avant la lettre_. Geldzahler thought that pop art transformed his career: "Before then . . . curators weren't famous. Well, I was the Pop curator. . . . A curator in 1958 wouldn't have dreamed of attention like that."
Performances of regression were becoming Geldzahler's signature move. A few years before FitzGerald's visit to the Factory, the curator had graced Claes Oldenburg's 1961 happening _Ironworks/Fotodeath_ as "the father in a 'soft family' of three persons, who sat on a bench while a photographer tried vainly to take their portrait," getting propped up and then collapsing again and again. And in May 1965 he had taken on a similarly deflationary role for another Oldenburg happening, _Washes_ , "lying in a rubber boat" in a swimming pool, "wearing a terry-cloth robe and smoking a cigar"—a performance he later worried might have jeopardized his future at the Met (plate 20). "Who's afraid of the mass media?" FitzGerald closed her article by asking. "Not its little king." Warhol's description of Geldzahler would apparently stick as a nickname: "His friends called him the Little King," Calvin Tomkins reported in 1971, "after the comic-strip character by Soglow."
Social mobility was much more problematic for Warhol. Where Geldzahler could move effortlessly between high and low, rich and poor, Warhol knew the limitations of his own place in this system: "I never can have money and pretend I'm poor. I can only be poor and pretend I'm rich." He was terrified of being accused of social climbing: "It bothers me because it's not true! . . . the 'social climbing' thing just isn't true. Oh but _why_ does it bother me so much? I don't _know_ why, it just does, I don't know." The media seems to have made him deeply uncomfortable; in at least one early interview, he communicated this unease by repeatedly flicking his audience off while brushing his fingers across his mouth. There were pillars of the New York artworld who found Warhol's class background unforgiveable. Frederick Eberstadt described him as "this weird cooley little faggot with his impossible wig and his jeans and his sneakers. . . . The most colossal creep I had ever seen in my life." According to Carter Ratcliff, his effects were profound: Warhol would eventually undermine "the barrier of privilege we hoped would always separate the art world from fashion, entertainment, and everything else art flirts with."
Geldzahler never seems to have encountered these frustrating limitations; he was free to move up and down the social hierarchy with ease, transgressing "the barrier of privilege" when necessary. By nicknaming Geldzahler "the Little King," Warhol registered his irritation with this imbalance. Perhaps this is also why he made a short film called _People Watching Henry_ (1964) documenting a screening of the longer Geldzahler film, reframing and containing the curator by filming an audience watching him perform. And perhaps it explains Warhol's decision to let the original Geldzahler film be screened incorrectly for so many years, its reels played out of order, rendering the sitter's disintegration illegible, thwarting his little-kingly narrative. Warhol might have seen Geldzahler as beating him at his own game, playing the Little King as a gay man from a position of strength, not weakness. "My father used to tell me not to try to swallow the world," Geldzahler told a reporter in 1966. "I said, 'Why not try?'"
FIGURE 47. Andy Warhol, _People Watching Henry Geldzahler on Screen_ , 1964. 16 mm film, black and white, silent; 2.7 minutes. © 2016 Andy Warhol Museum, Pittsburgh, a museum of Carnegie Institute. All rights reserved.
These tensions between Warhol and Geldzahler seem to have been undermining their relationship. When the Factory's power failed during FitzGerald's visit, Geldzahler mocked Warhol's efforts to make class performative: "Ask Andy why the lights have gone out. Is it a financial crisis or just camp?" Their close friendship apparently dissolved that year. Geldzahler remembered writing, "Andy Warhol can't paint anymore and he can't make movies yet," on a blackboard in the Factory in June 1965. He worried that Warhol never forgot that blithe betrayal. But in 1967 Geldzahler introduced Warhol to Fred Hughes, who would become the impresario of Warhol's second factory. Under Hughes's tutelage, Warhol would learn to capitalize on his wealthy clients' downmarket aspirations. Portraiture became a primary source of revenue. Warhol even offered to do Geldzahler's portrait again, promising that he would "put the art in this time." By 1971 Geldzahler would finally embrace the family business he had long resisted—"I'm a sort of broker, or middle man, between precious objects and the people who buy them. It doesn't bother me"— and Warhol would refer to himself publicly as "the first Mrs. Geldzahler," the early wife who had sustained her husband's ascension and then been abandoned. Appointed arts commissioner under Mayor Ed Koch, Geldzahler mockingly remarked the absence of Warhol's work from his exhibits: "I don't have any of _your_ art up here." Warhol would try his best to return the favor: "And I was rude to Henry Geldzahler. He was there trying to get me to introduce him to someone, and I ignored him, I don't know why—well, yes I do, because Henry's hurt me so many times that way, I just _felt_ like it."
In films like _Henry Geldzahler_ and _Poor Little Rich Girl_ , and in his burgeoning portraiture practice, Warhol took aristocratic subjects and found ways to deglamorize them, both through his amateurish style and through the pressures of the strained medium. He recognized—from precedents like Soglow's _Little King_ —that this process would appeal to both his subjects and their everyday viewers. For viewers, the spectacle of deglamorization gratifyingly collapsed the distance between audience and performer, demonstrating that "stars are just like us." And for his performers, this cultural leveling disguised real social difference, producing the illusion of equality: _The Little King_ had been just as successful in the _New Yorker_ as it was in the daily paper. What's more, deglamorization had an erotic appeal of its own, since "the socially excluded" often "returned in the form of psychic desire." For his part, Warhol regularly celebrated the power of these moments of deglamorization or downward mobility. Neal Weaver's 1969 visit to the Factory was "abruptly" concluded when "a flurry broke out in the screening room. . . . Joan Crawford was pinch-hitting for her daughter on a daytime soap opera . . . [and] everyone gathered before the television set to watch the spectacle of Miss Crawford playing a 23 year old girl." Warhol was spellbound: "Andy watched for a while . . . and then said with quiet satisfaction, 'This is how television ought to be all the time. Absolutely surreal.'" Warhol suspected that commercial television could profit from his innovations. The results would be profoundly lucrative and insidious: reality television, a "midwife of the neoliberal order," conditioning viewers to a new world of flexible labor, self-sufficiency, and "increasing vulnerability."
These, then, are some of the implications of Warhol's pursuit of cinematic egalitarianism during the mid-1960s:
1. Like his pop work from the early 1960s, Warhol's cinema pursued a Rancièrian expansion of aesthetic experience. It did so in an impressive variety of ways, challenging previous hierarchies of participation, production, subject matter, genre, technology, distribution, and audience. In all of these areas, it deserves to be recognized as a significant moment in cultural history. It undeniably provided a variety of aspiring participants who otherwise would have been deemed too queer, marginal, or impoverished with an opportunity to contribute to culture. Warhol's admirers felt certain that, in this respect, he had permanently transformed the world. As David Bowie observed, " _Pork_ could become the next _I Love Lucy_ , the great American domestic comedy. . . . It's about people living and hustling to survive. . . . A smashing of the spectacle." No small feat.
2. But while Warhol's mid-sixties cinema opened onto distinctly egalitarian possibilities, it was nevertheless frequently premised on the marketability of egalitarianism's collapse, the expectation that audiences would enjoy watching a film or a performer who conspicuously failed to approximate the mass-cultural standards she/he/it had set out to achieve. The gap between an aspiring cultural participant and her mythic heroes could prove entertaining and therefore profitable. As we have seen, these dramas of ambition and failure were frequently framed in class terms. Many of Warhol's collaborators would later argue that this project put Warhol in the compromised ethical position of profiting from his friends' disappointments. For what it's worth, he made a career out of profiting from his own as well.
In this respect, Warhol's work clearly set a precedent for the spectacles of cultural failure that are today so dominant: the _Gong Show_ / _American Idol_ / _The Apprentice_ model of culture, premised on the ideal of pure meritocracy but secretly revealing "the real necessity of social ties and support for success in the arid neoliberal regime." Much of the appeal of these shows lies in the contestants' inability to reproduce successfully the ideals they emulate, and in the monstrous criticism they frequently receive from "celebrity judges." Through this process, failure is spectacularized and converted into entertainment, becoming "the country's great uniter, bringing together rich and poor, blue state and red, Gershwin-loving grandmas and text-happy tweens—a rare instance of cultural consensus in an increasingly fractured age." Instead of being sold only the spectacle of their possible successful participation in mass culture, "consumers" are today simultaneously consoled by the spectacle of their peers' inability to achieve this same participation. No wonder Warhol was invited, in the 1970s, to be a judge on _The Gong Show_ , or that the interviewer who later asked him about it wrongly assumed that he would have been competing against—rather than evaluating—the contestants. Lance Loud of _The Loud Family_ , America's first reality show, had openly modeled himself on Warhol. How gratified Warhol would have been to learn that, in 2015, "MTV is rebranding for the social media age and opening itself up to audience-created videos, changing its tagline from 'I want my MTV' to 'I am my MTV.'"
3. Finally, the examples of Henry Geldzahler, Edie Sedgwick, the Little King, and Joan Crawford (along with the expanding list of celebrities and politicians with whom Warhol collaborated throughout the 1970s and 1980s) demonstrate that this aesthetic of failure could prove appealing for an elite that wished to disguise its power, embracing Deleuze and Guattari's claim that "the new words spoken by the master" are "'I too am a slave.'" For these patrons, collaborating with Warhol meant aligning themselves with a quavering voice from the margins, as Geldzahler had, borrowing and embracing its minor perspective, taking on its weaknesses as their own, and perhaps even learning to enjoy themselves as so many "Napoleon[s] in rags"—without thereby becoming "bitter cultural rebels, ready to dynamite the Corinthian columns of the Met." "Some viewers will probably read into Warhol's smudged screening and lurid colors an intentional degradation of the subject," David Bourdon remarked in 1975, "while others will see it as thoroughly glamorizing." This exchange seems to have pervaded Warhol's society portraits from the very beginning: "Now start smiling and talking," Warhol is said to have instructed Ethel Scull (nickname: Spike; married to a taxi baron) in 1963 while she posed for her portrait in a 42nd Street coin-operated photobooth, "you're costing me money."
# Conclusion
# Warhol's Neoliberalism
During Warhol's lifetime, social class went from being a principal category of identification and analysis to what Noam Chomsky has called "the unmentionable five-letter word." In a variety of ways, neoliberalism encouraged citizens not to think or talk about class. Many prominent voices claimed it was obsolete, "a meaningless or at least long defunct category" that was no longer descriptively or prescriptively useful. Conveniently, while a "majority of the world's population was being subjected to an unleashed global capitalism . . . the idea of class, was being energetically rejected by the aspiring arrivistes in the West and elsewhere, who believed that they had transcended it." Americans were taught to aspire rather than to advocate or agitate; instead of class consciousness, there is "level consciousness," wherein "collective involvement" is discouraged, and "the principle motive . . . is to pass up and out of the level in which [one] finds [oneself]." And yet this effacement of class disguised decades of "gloves-off . . . class war," through various strategies of commodification, financialization, privatization, austerity, and deregulation: union busting, the evisceration of the welfare state, the pathologization of the vulnerable, and the reduction of human life to "human capital." Under neoliberalism, as Wendy Brown argues, "All conduct is economic conduct; all spheres of existence are framed and measured by economic terms and metrics," and yet class "disappears as a category" since it raises difficult questions about inequality, "the Achilles' heel of the modern economy." Class warfare, during this period, was thus simultaneously socioeconomic and psychological—a vast power grab brought about through "the accelerating erosion of working people's socioeconomic security" that cloaked itself in the guise of "individual freedom" and classless mobility.
Even as it deployed and manipulated recognizable languages of class—through comic books and cheap cameras and canned food—Warhol's work contributed to this effacement of class, and to the neoliberal illusion of a classless America, defined by aspiration and mobility. One way to describe the general trajectory of this work would be to say that the frustrated yearnings for status and cultural participation expressed in his style became less authentic as Warhol's stature—and the stature of his clientele—grew increasingly formidable. But the tension between these forces remained active from start to finish. Class was never an easy subject for Warhol.
Warhol knew that the low can have a strange and dangerous erotic appeal. There is something sexy about the underdog, the dirtbag, the ruffian, since "low domains, apparently expelled as 'Other', return as the object of nostalgia, longing and fascination." Of course, in America, these desires cut both ways; the wealthy are at least as much an "eroticized constituent" of the poor's fantasy lives as the poor are of the wealthy's, and nothing is more thrilling than a reversal of fortune. Audiences roar with laughter as Martha Stewart faux-menacingly informs them that, "When I did my stretch, all the hoodrats on my cellblock wanted to break off a piece of Martha Stewart's ass, so I decided some bitch needed to get got." Warhol built a career out of these strange crossings, although he sometimes worried that, as mass media became more pervasive, the pressure of the fantasy lives they promulgated might become unbearable. In his childhood, although he "probably hoped that the movies showed what life was like," their actual content "was so different from anything I knew about that I'm sure I never really believed it." But "TV is different"—more realistic—and "people with television dreams are really disappointed with everything in their lives."
The concept of the popular, as Clark reminds us, is a slippery one: "used for ideological purposes, to suggest kinds of identity and contact between the classes . . . which did not exist in their everyday life or organized social practice, but seemed to in the spectacle." For some of its viewers and participants, Warhol's work demonstrated the possibility of upward cultural mobility through the misadventures of the amateur producer or performer. For others, it provided access to downward cultural mobility, a vulgarization that disguised economic differences and crackled with the frisson of the low. (This vulgarization could, of course, occasionally be taken too far, even for Warhol. "Poor Edie—," Warhol told his diary when he saw ads for the deceased superstar's biography on the sides of city buses, "when she went out she'd never even take a cab, it had to be a limo, and now they've got her on the bus.") Warhol's work allowed its viewers to cross both ways, to make class a performance. Always keeping things loose, free, easy. _Never_ climbing.
Neither of the two great subjects of Warhol's classic pop work—participatory culture and the "battle of the brands"—constituted a true challenge to capitalism broadly speaking. But each threatened to undermine an important contemporary capitalist structure or assumption. Participatory culture held out the promise of an egalitarian cultural sphere, a Benjaminian world in which those who had traditionally been merely "consumers" could earn compensation as producers and begin to share their thoughts, feelings, and experiences directly with others. The entire hierarchy of cultural production might be overturned. The ascension of generic goods posed a similarly fundamental challenge: the paradigms of corporate competition and market segmentation, and with them the need for an advertising industry, might be growing obsolete as citizens ceased to believe in the qualitative differences between branded and generic commodities. The profitable and mostly imaginary lines between "Coke" and "Pepsi" and "Cola" were seen to be on the edge of collapse at the very moment when everyone was being promised access to cultural participation. A new world of egalitarianism and use-value seemed to be waiting in the wings.
In both cases, the working class was proposed as a bulwark against these threats to hierarchy and profit. Their alleged insecurity, immaturity, credulousness, and desperation for status would make them easy to manipulate. Participatory culture was sold to them as an outlet for creativity, self-expression, and upward mobility, all for a low monthly payment. And experts claimed that, unlike their wealthier and supposedly savvier contemporaries, the working class could be trusted to believe in an "aristocracy of brand names" that seemed to guarantee security and social mobility through careful shopping—"a fictive United States," as bell hooks put it, "where everyone has access to everything." Warhol's work tested these promises against each other, and in so doing, archived and interrogated them. In his indelible work from the early 1960s, he attempted to take these promises seriously, pushing them to their breaking points in his images of superheroes, newspapers, soup cans, and celebrities. Then, in his attempts at an egalitarian cinema in the mid-1960s, he found a way to amalgamate these promises through the invention of the Factory Superstar: the idea that anybody could, with perseverance and Warhol's blessing, creatively become her own brand.
Even as they targeted a class-specific audience, however, the participatory culture and brand image strategies that Warhol found so fascinating also contributed to the eradication of class consciousness and to an "increasingly atomized" working class. They did so by endeavoring to persuade working-class citizens that they were each fundamentally responsible for their own fate—for their creative growth and their social mobility. Each working-class subject, "eschew[ing] dependency" and refusing to "cling to others," could supposedly bootstrap herself up the social hierarchy through hard work, creativity, and the conscientious consumption of brand-name commodities; " _insecure workers_ " could become " _confident consumers_." Warhol's Superstar strategy consolidated these assurances into the promise of media-assisted self-branding.
As Jennifer Silva has shown, this new emphasis on self-transformation eased the rise of neoliberalism, as "young people who would seemingly benefit most from social safety nets and solidarity with others" were encouraged instead to "cling so fiercely to neoliberal ideals of untrammeled individualism and self-reliance." For Silva, the core of this promise was the privatization of happiness, the effort to persuade working-class subjects that their emotions were ultimately their own responsibility. Like the working-class millennials profiled in Silva's research, Warhol embraced this worldview, "grounding [his] adult identities in recovering from [his] painful [past] . . . and forging an emancipated, transformed, and adult self," instead of pursuing change through collective political action. Warhol's neoliberal surroundings encouraged him to privatize his own happiness, to consider himself accountable for his own development—"actively fostering a kind of flexibility within [himself], bending with the constant disruptions and disappointments in the labor market, and staunchly willing [himself] to be unbreakable." The goal was the production of "an emancipated and independent self" who had overcome the challenges posed by previous traumas and, like Warhol, believed they "can live without anything." Already in 1965, Warhol was proclaiming, "I don't feel my position as an accepted artist is precarious in any way . . . if you feel you have nothing to lose, then there's nothing to be afraid of and I have nothing to lose." Or, in 1975: "A person can cry or laugh. Always when you're crying you could be laughing, you have the choice. . . . So you can take the flexibility your mind is capable of and make it work for you." Warhol seems to have believed that this flexibility was part of his queer identity, that the "missing . . . chemicals" that made him "A—sissy. No, a mama's boy. A 'butterboy'" also freed him from the responsibilities that burdened his peers.
Silva's subjects were born long after Warhol, but he might have been their patron saint, anticipating this neoliberal working-class ethos. Like the promises of cultural participation and class mobility detailed in this volume, these promises of adaptability and autonomy were fundamentally deceptive—a mirage of agency in a wasteland of exploitation. Together, they goaded working-class Americans to accept a privatized vision of agency—"a social narcissism" in which creativity, mobility, and happiness were fundamentally individual achievements, and the old structures of class and community could no longer be relied upon for support. (Describing his Silver Factory, Warhol noted that "maybe more than anything, silver was narcissism—mirrors were backed with silver.") To the extent that we read Warhol as a successful proponent of these promises, having "laid the foundations for an art world where artists . . . are able to climb free and unrestrained up and down the societal ladder," his work contributes to their dissemination, "help[ing] to effect the transition from the old worlds of God, reason and labour to the new world of consumption we still find ourselves in today." A world in which, as one of Warhol's contemporary admirers tells his audiences, "there's only two types of people, there's dreamers and there's haters." But, to the extent that Warhol's work archives and juxtaposes these class-targeted tactics, highlighting their lies and absurdities, it might help us to historicize these promises—to see them as strategic and manipulative, a vast effort to make the idea of class disappear from the minds of those who had the most to gain from retaining it.
All of these strategies of privatization were staged, during this period, as quintessentially American inventions, and juxtaposed to the rigidity of the Other. American media presented American identities as being exceptionally flexible relative to those of other cultures. This dichotomy was dramatically illustrated in the _Lois Lane_ comic book from which Warhol appropriated _Superman_ , which had Lane traveling to "the near-eastern kingdom of Pahla" to cover an impending revolution for her newspaper. A mysterious woman, seemingly identical to Lane, attempts to assassinate the king. Lane is arrested and sentenced to death, but Superman arrives in time to rescue her, discovering that "Bizarro-Lois," a burka-clad refugee from another planet, has impersonated Lane to gain Superman's attention: "I look like you, talk like you, _am_ you in every respect! And like you, I'm madly in love with Superman! Gaze upon me and see yourself!" When the alien finally sees Lane in person, she realizes that she cannot live up to her ideal. She stumbles backward and falls into the ocean—Superman is unable to save her.
Unlike her alien doppelgänger, Lane has access to the benefits of privatized identity. Ever the good consumer, she is able to switch roles and ethnicities easily—everything is billed to her expense account, and she remains emotionally untouched by her travails. Bizarro-Lois, by contrast, is forced to steal a burka and a gun; her disguise is an act of desperation that conceals an inexpungible ugliness. The revolutionary Other is thus staged as a bad consumer, jealous of the flexibility of her American counterpart, while revolutionary action is figured as a desperate and hopeless appeal for spectacular recognition. Lane, the good, self-reliant consumer, is rewarded for her flexibility; she ends up in Superman's arms, since, as a four-year-old working-class television connoisseur recognized, "Superman . . . knows when you are fooling and that's why he doesn't come." The story dramatizes the differences between sumptuary regimes, in which "the body is a site for the inscription of a variety of signs and values about identity and difference," and fashion regimes, which "make the body of the consumer itself potentially ephemeral and manipulable"—open to branding. Warhol is, for many, a hero of the fashion regime, with its new emphasis on "the pleasure of _ephemerality_." But "consumers" needed to be disciplined into accepting these new priorities. No wonder the old world was directly associated, in the comics, with death; as Lane puts it, on the story's last page, "Maybe this poor, weird duplicate of myself is better off dead!"
Occasionally, Warhol's work illuminated alternatives to this brave new world of privatized happiness, creativity, and mobility. Neoliberal privatization presupposed "a belief in progress as a mode of temporality"—measurable improvement over measurable time. Warhol sometimes expressed his doubts about the value of these metrics. In _Clocks (Two Times)_ (1962), he challenged the "abstract and reified" temporal structure of "clock-time" that characterizes capitalism. Clocks traditionally administer this "alien force far stronger than any human will or power" through their "hands." In the course of a life of labor, this alien force becomes "an ingrown clock" that structures one's days, "a thing you just get used to." With its offset pair of ineffectual, handless faces, _Clocks (Two Times)_ quietly proposes a world freed from these powerful illusions, a world in which even a pair of clocks, which should presumably confirm each other, are powerless to govern human behavior. Reimagined through Warhol's impetuous silkscreen, the numbers on these cheap "Westclox" start to blur and tremble, and time opens up beyond the conventional strictures of hours, minutes, seconds toward "a lyric, disruptive, present tense." A similar disruption of clock-time was attempted in Warhol's cinema. Through repetition and uneventfulness, Warhol's early movies promised to render clocks obsolete: "When people call up and say 'What time does the movie start?' you can just say 'Any time.'"
In his 1963 _Crowd_ silkscreens, Warhol tried his hand at representing the collectivity that neoliberalism urged him to abandon (fig. 49). With characteristic economy, he filled one such print with the same image of a crowd printed five times: a seemingly complete image takes up most of the page starting in the lower right-hand corner, three partial images fill in the blanks above it and along the left-hand side, and the sliver of a fifth is used to cover the upper right lateral edge. Foster notes that "the mass appears as a truncated blur of a newspaper photo or a television image barely seen or remembered," while John Curley argues that these images demonstrate that "repetition can be misleading." And yet, as in the Bonwit Teller paintings, Warhol's pencil scrawls in the gaps between the images seem to figure a doomed effort to make the crowd whole by hand, to complete its continuity—a task that the silkscreen process had left forever unfinished, and that hasty, anxious scrawling could never convincingly complete.
FIGURE 48. Andy Warhol, _Clocks (Two Times)_ , 1962. Silkscreen ink on linen, 82½ × 32 inches. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
During this period, Warhol's Silver Factory offered the possibility of a truly anonymous and collaborative practice, freed from the constraints of clock-time and individualism, a swarm inspired by Warhol's "sort of . . . boyfriend" and collaborator, Billy Name. This vision of freedom and community would persist in Warhol's imagination for the rest of his life. In 1986 he recounted a dream: Billy Name was "living under the stairs at my house and doing somersaults, and everything was very colorful. It was so weird, because his friends sort of invaded my house and were acting crazy in colorful costumes and jumping up and down and having so much fun and they took over, they took over my life. . . . It was like clowns." Here, Warhol seemed tempted by a truly wild and egalitarian creativity that Adorno called "a fundamental layer of art": "the constellation animal/fool/clown," playfully oblivious to the world of clocks, names, and privatization.
FIGURE 49. Andy Warhol, _Crowd_ , 1963. Graphite and silkscreen ink on Strathmore paper, 28½ × 22 ⅝ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
FIGURE 50. Gerard Malanga, _On the set with the movie_ Vinyl, 1965. From left to right: Billy Name w/ light-meter, Andy Warhol, Gerard Malanga and Edie Sedgwick, Ronald Tavel w/ back to camera. © Gerard Malanga.
In contradistinction to "major Warhol"—Warhol the ur-postmodernist—we might describe these egalitarian visions of timelessness, collectivity, and collaboration as "minor Warhol": deterritorialized, political, and collaboratively oriented against "a majority culture's authority over the 'real.'" These works poignantly pursue the unspeakable and unthinkable, embodying "a politics of desire that questions all situations." As we have seen, this minoritarian politics of desire informed even Warhol's most familiar and "universal" artworks: the soup cans, soda bottles, and superheroes that now seem so apolitical. Beneath their class-coded infatuation with mass culture is a collective voice yearning for access to cultural participation. Like Leo and Caleb Proudhammer, like Janice and Keith Gunderson, like Eva Narcissus Boyd and Tommy Edwards, they speak for a world of consumers "who live in a language that is not their own" and who wish to become producers. Each believing that "nothing, not even something this terrible, is beyond transformation." But these egalitarian and participatory visions were often staged as distant and unachievable ideals in Warhol's work, scrawled or hallucinated, entertaining as failures rather than successes: "because if you can make them look better bad, at least they have a _look_ to them."
Warhol learned that these entertaining failures could be massively profitable. In his Factories, where he espoused the virtues of egalitarian creativity, he also tested the possibilities of a truly neoliberal mode of artistic production, in which "shares of our company" could be sold "on the Wall Street stock market" and performers and collaborators would be "treated as so much capital to be shaped and reshaped." Like a good neoliberal citizen, he embraced an "ethos of work" in which "being alive" is work and "having sex" is work and "people are working every minute." He privatized his Factory telecommunications by substituting a pay phone for an office phone, since "he didn't want people running up long distance calls on his dime," and dreamed of having a computer for a boss. As a successful corporate manager, Warhol knew how to divide and conquer his employees; Malanga bragged in 1969 that Warhol "encourages rivalry among us. It keeps things from getting dull—and we often do a better job because of it." At the limit, these collaborators would be so desperate for labor, and so disorganized, that they were eager to work for free: "Andy assumed . . . that if he started paying everyone . . . the end-result would be of a lesser quality. He was convinced of this." Warhol's Factory nickname, Drella, perfectly encapsulated these contradictions: raw innocence mixed with unblushing greed, Cinderella and Dracula all in one, "so sweet and nice," as Keith Haring put it, but also "need[ing] fresh blood all the time."
On some level, Warhol seems to have known that neoliberalism's promises of privatized mobility and happiness were lies. The prospect of actually conversing with a working-class subject remained deeply problematic for him, almost paralyzing: "When I go to a hotel, I find myself trying to stay there all day so the maid can't come in. I make a point of it. Because I just don't know where to put my eyes, where to look, what to be doing while they're cleaning. It's actually a lot of work avoiding the maid, when I think about it." Working-class Others—"those desperate people" unwilling or unable to pursue the benefits of privatization—were too dirty, too unpredictable, too different to look at, let alone speak with. This _work_ of avoiding the actual corporeal realities of work and class—this too is part of Warhol's legacy. The fact that he and the maid could both buy the same beautiful can of Coke only partially mitigated his anxiety and shame.
For the most part, Warhol seems to have had a difficult time imagining his own place or agency in a larger political world, let alone combatting the "disciplining and disempowerment of the working class" that was going on around him. He was never one for marches or demonstrations. Even elections intimidated him; he complained that the voting machine confused him so much that he "pulled the wrong lever" and wished there had been a "practice model." He gave up voting for fear of jury duty, but "offer[ed] his employees bribes of Election Days off if they gave their word they'd vote Democratic." He admired a machine used by Democratic fundraisers "that puts stamps on letters crooked because then you get a better response—more homey." Pier Paolo Pasolini wondered, skeptically, whether Warhol had ever seriously considered the possibility of revolution, the possibility that "history can be divided . . . [having] a moment in which a way of its being finishes and another one begins?" Warhol seems not to have dwelled too much on these pivotal moments. He yearned instead to forget or ignore history: his ideal was "very light, cool, off-hand, very American. . . . Just play it all on one level, like everything was yesterday." He claimed that he had "no memory," that "every day is a new day because I don't remember the day before" and "every minute is like the first minute of my life"—a perfect disorder for the neoliberal world of flexibility, where "forgetting becomes an adaptive strategy." Faced with world leaders, his demands were meager. He once told Nancy Reagan that she should institute a lottery to select one family a night for dinner at the White House. Reagan wasn't interested: "There are tours, of course, Andy."
FIGURE 51. Andy Warhol, _Farah Dibah Pahlavi (Shahbanou [Queen] of Iran)_ , ca. 1977. Screen print on Curtis Rag paper, 45 × 35 ⅛ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Like the stereotypical working-class subjects described by contemporary sociologists, Warhol "[felt] modern problems, and even many of his own personal problems, [were] too big for him to solve." He saw change as a distinctly individual achievement and claimed that the best way to overcome challenges was to learn to say, "So what," adding, "It took a long time for me to learn it, but once you do, you never forget." His experience of direct political engagement was mostly mediated through television news, which "privatizes . . . at the same time as it provides a subjective experience of participation in publicness." He didn't seem able to recognize the televised looters during a 1977 Manhattan power outage as initiating a "potlatch of destruction," let alone to empathize with their frustrations with the "centaur state" that profited from their criminalization. Instead, they reminded him of _Roots_ , since "they're all chained together and they're all black and Puerto Rican." But he did remark the spectacular irony of the TV floodlights, which counterproductively "enabled [the looters] to see better to steal more." And Warhol was forever intrigued both by the little-kingliness of actual royalty and by the possibility of their deposition. "On TV," he had bragged to his diary earlier that year, "I got a big mention when Barbara Walters interviewed the empress of Iran. In with the other art they did a big closeup on my Mick print and Barbara said, 'And surprisingly, they have a painting of rock star Mick Jagger by Andy Warhol,' and the empress said, 'I like to keep modern.'" Two years later, the empress's regime was overthrown. Again, Warhol was watching on television: "And Iran really fell. It's so weird watching it all on TV, it really could happen here." Insurrection at the doorstep and using pop to "keep modern": two remarkable trajectories for Warhol's egalitarian dreams.
# Notes
## Introduction
1. "Ted Carey, New York, 16 October 1978," in Patrick S. Smith, _Andy Warhol's Art and Films_ (Ann Arbor, MI: UMI Research Press, 1986), 254. According to Carey, Warhol suspected that Lichtenstein had seen his comic-book paintings when they were exhibited in a Bonwit Teller department-store display in April 1961, and had ripped off his idea.
2. The closest competitor would be Jasper Johns's _Painted Bronze_ (1960), with its Ballantine Ale cans, which is typically classified as a sculpture.
3. Robert Seguin, _Around Quitting Time: Work and Middle-Class Fantasy in American Fiction_ (Durham, NC: Duke University Press, 2001), 40.
4. Kristine Stiles, _Concerning Consequences: Studies in Art, Deconstruction, and Trauma_ (Chicago: University of Chicago Press, 2016), 328–29.
5. Quoted in Victor Bockris, _Warhol: The Biography_ (New York: Da Capo, 2003), 135; "Emile de Antonio, New York, 14 November 1978," in Smith, _Andy Warhol's Art and Films_ , 293. For an illuminating discussion of de Antonio's politics, see Branden Joseph, "1962," _October_ , no. 132 (Spring 2010), 114–34.
6. Marco Livingstone, "Do It Yourself: Notes on Warhol's Techniques," in _Andy Warhol: A Retrospective_ , ed. Kynaston McShine (New York: Museum of Modern Art, 1989), 66.
7. Andy Warhol, _The Philosophy of Andy Warhol (From A to B and Back Again)_ (New York: Harcourt Brace Jovanovich, 1975), 150.
8. See, for example, _Life_ , January 5, 1959, 31, and February 1, 1960, 27. Ronald Tavel claimed that _Life_ was "compulsory reading" for Warhol (Tavel, "The Life of Juanita Castro," unpublished manuscript, accessed November 15, 2015, <http://www.ronaldtavel.com>, 94).
9. The Warhol catalogue raisonné provides a dated source for _Coca-Cola 1]_ and undated source collages for _Peach Halves_ and _Coca-Cola [2]_ (Georg Frei and Neil Printz, eds., _The Andy Warhol Catalogue Raisonné_ , vol. 1, _Painting and Sculpture 1961–1963_ [London: Phaidon, 2002], 60, 63). I have located a slightly modified version of the _Peach Halves_ advertisement that ran in 1960, 1961, and 1964 (see _Ottawa Citizen: Weekend Magazine_ 10, no. 44, November 26, 1960, 19). The text included in the _Coca-Cola [2]_ collage ([figure 2) establishes that the painting was definitely produced after April 1961 and probably after June: the meeting of the Church of Scotland described in the excerpt took place in April but was not widely reported until June; the text bemoans a delay without specifying its duration (see "'Sensationalizing' Protestants Talks with R.C.s," _Glasgow Herald_ , June 8, 1961, 12, and Ronald Walls, _Love Strong as Death: The Autobiography of Fr Ronald Walls_ [Petersham, MA: St. Bede's Publications, 2000], 246).
10. "Appendix 2: Warhol's Studios," in Frei and Printz, _Catalogue Raisonné_ , 1:468. Warhol would later claim to have used slides and light boxes for projection as well; see Barry Blinderman, "Modern _Myths_ : Andy Warhol" (1981), reprinted in _I'll Be Your Mirror: The Selected Andy Warhol Interviews_ , ed. Kenneth Goldsmith, Reva Wolf, and Wayne Koestenbaum (New York: Carroll & Graf, 2004), 295.
11. Peter Schjeldahl, "Warhol and Class Content" ( _Art in America_ 68, no. 5 [May 1980]: 112–19), reprinted in _The Hydrogen Jukebox: Selected Writings of Peter Schjeldahl, 1978–1990_ (Berkeley: University of California Press, 1993), 47.
12. Bob Colacello, _Holy Terror: Andy Warhol Close Up_ (New York: HarperCollins, 1990), 15.
13. See Blake Stimson, _Citizen Warhol_ (London: Reaktion, 2014), 90–91.
14. Warhol, _Philosophy_ , 237.
15. Susan Cheever Cowley, "Bobby Zarem, Superflack," _Newsweek_ , January 31, 1957, 58. For "personal enterprise," see Pierre Dardot and Christian Laval, "The New Way of the World, Part 1: Manufacturing the Neoliberal Subject," _e-flux_ , no. 51 (January 2014), accessed November 1, 2016, <http://www.e-flux.com>.
16. Wilbur Schramm, Jack Lyle, and Edwin B. Parker, _Television in the Lives of Our Children_ (Stanford, CA: Stanford University Press, 1961), 112; Pierre D. Martineau, "The Pattern of Social Classes," in _Marketing's Role in Scientific Management_ , ed. Robert L. Clewett (Chicago: American Marketing Association, 1957), 246.
17. Louis Schneider and Sverre Lysgaard, "The Deferred Gratification Pattern: A Preliminary Study," _American Sociological Review_ 18, no. 2 (April 1953): 148.
18. Bob Dylan, "Like a Rolling Stone" (Columbia Records, 1965). The Napoleon-as-Warhol reading is suggested by Bockris in _Warhol_ , 229. For more on Warhol and Dylan, see Thomas Crow, _The Long March of Pop: Art, Music, and Design, 1930–1995_ (New Haven, CT: Yale University Press, 2014), 281, 306.
19. Martineau, "Pattern of Social Classes," 235.
20. "The Quality Revolution—New Hope for National Brands," Macfadden advertisement, _New York Times_ , March 7, 1962, 72 (quoted text italicized in original). I will use "private" and "generic" interchangeably throughout.
21. "Ultra Violet," in John Wilcock, _The Autobiography and Sex Life of Andy Warhol_ (New York: Other Scenes, 1971), n.p.
22. Quoted in Bockris, _Warhol_ , 170.
23. Sidney J. Levy, "Social Class and Consumer Behavior," in _On Knowing the Consumer_ , ed. J. Newman (New York: John Wiley and Sons, 1966), 154.
24. Quoted in Jean Stein, _Edie: An American Biography_ (New York: Alfred A. Knopf, 1982), 189.
25. "Andy Warhol's Interview," _Unmuzzled Ox_ 4, no. 2 (1976): 43. Although Warhol would die a wealthy man, his financial concerns were not completely overstated; see Eva Meyer-Hermann, "Other Voices, Other Rooms: TV-Scape," in _Andy Warhol: A Guide to 706 Items in 2 Hours 56 Minutes_ , ed. Meyer-Hermann (Rotterdam: NAi Publishers, 2007), 216.
26. Steven Shaviro, "Warhol's Bodies," in _The Cinematic Body_ (Minneapolis: University of Minnesota Press, 1993). 206. See also Jennifer Dyer, "The Metaphysics of the Mundane: Understanding Andy Warhol's Serial Imagery," _Artibus et Historiae_ 25, no. 49 (2004): 33–47.
27. "Tally Brown, New York, 11 November 1978," in Smith, _Andy Warhol's Art and Films_ , 244.
28. David Graeber, _Revolutions in Reverse: Essays on Politics, Violence, Art, and Imagination_ (Brooklyn, NY: Autonomedia, 2011), 60.
29. bell hooks, _Where We Stand: Class Matters_ (London: Routledge, 2000), vii.
30. Beverley Skeggs, _Class, Self, Culture_ (London: Routledge, 2004), 3.
31. Seguin, _Around Quitting Time_ , 16.
32. G. M. Tamás, "Telling the Truth about Class," _Socialist Register_ 42 (2006): 242, 244.
33. As August Carbonella and Sharryn Kasmir put it, "Reigning academic wisdom has it that class as a social formation has simply disappeared over the last 30-odd years" (Carbonella and Kasmir, "Dispossession, Disorganization and the Anthropology of Labor," in _Anthropologies of Class: Power, Practice, and Inequality_ , ed. James G. Carrier and Don Kalb [Cambridge: Cambridge University Press, 2015], 41).
34. Helen Peters, quoted in Studs Terkel, _Division Street: America_ (New York: Pantheon, 1967), 159; Radhika Desai, "Neoliberalism and Cultural Nationalism: A _danse macabre_ ," in _Neoliberal Hegemony: A Global Critique_ , ed. Dieter Plehwe, Bernhard Walpen and Gisela Neunhöffer (New York: Routledge, 2006), 228. Although the onset of neoliberalism in America is typically dated to the late 1960s or early 1970s, the underlying ideas date to the 1950s (see Daniel Stedman Jones, _Masters of the Universe: Hayek, Friedman, and the Birth of Neoliberal Politics_ [Princeton, NJ: Princeton University Press, 2012], 85–272). In what follows, I read Warhol's classic pop work as anticipating neoliberal mores and ideals.
35. Antonio Gramsci, _Selections from the Prison Notebooks_ , ed. and trans. Quintin Hoare and Geoffrey Nowell Smith (New York: International Publishers, 1992), 244.
36. Michael Perry, "The Brand: Vehicle for Value in a Changing Marketplace," Advertising Association, President's Lecture, July 7, 1994, London; cited in Yiannis Gabriel and Tim Lang, _The Unmanageable Consumer_ (London: Sage, 1995), 34.
37. Matt Stahl, _Unfree Masters: Popular Music and the Politics of Work_ (Durham, NC: Duke University Press, 2013), 236n7.
38. See Stimson, _Citizen Warhol_ , 69–87, 124–45, 214–31; Rainer Crone, _Andy Warhol_ , trans. John William Gabriel (London: Thames & Hudson, 1970), 21, 23, 29, 32.
39. "Artist-Huckster Sketches Customers and Wins Prize," _Pittsburgh Press_ , November 24, 1946, 2.
40. Like Sartre's Flaubert, Warhol seems to have undergone an instructive alienation from his family's class early on. In both cases, "to discover social reality inside and outside oneself, merely to endure it is not enough; one must see oneself with the eyes of others" (Jean-Paul Sartre, "Class Consciousness in Flaubert," part 1, trans. Beth Archer, _Modern Occasions_ 1, no. 3 [Spring 1971]: 381).
41. Pierre Macherey, _A Theory of Literary Production_ , trans. Geoffrey Wall (London: Routledge & Kegan Paul, 1978), 133 (emphasis in original).
42. As Skeggs has observed, "Certain practices associated with the working-class are seen to confer authenticity upon their users" ( _Class, Self, Culture_ , 107).
43. John Urry, _Sociology beyond Societies: Mobilities for the Twenty-First Century_ (London: Routledge, 2000), 17; cited in Skeggs, _Class, Self, Culture_ , 48.
44. Seguin, _Around Quitting Time_ , 12.
45. James G. Carrier, "The Concept of Class," in Carrier and Kalb, _Anthropologies of Class_ , 31.
46. Clementine Paddleford, "What's in the Name on the Shelf?" _Los Angeles Times_ , May 7, 1961, E25.
47. T. J. Clark and Anne M. Wagner, _Lowry and the Painting of Modern Life_ (London: Tate Publishing, 2013), 16.
48. Billy Name has claimed that the doctors who operated on Warhol after he was shot in 1968 had resigned themselves to his dying until the gallery owner Mario Amaya (who was also wounded) objected: "You can't let him die. He's rich. He's gotta lot of money, and he's a famous artist." (Legs McNeil and Gillian McCain, "Andy Warhol Wanted Lou Reed to Be His 'Mickey Mouse,'" accessed November 1, 2016, <http://www.vice.com>).
49. Peter Stallybrass and Allon White, _The Politics and Poetics of Transgression_ (Ithaca, NY: Cornell University Press, 1986), 152. These cultural crossings mark a sharp departure from the early twentieth-century "cultural gulf" described by Lawrence W. Levine in _Highbrow/Lowbrow: The Emergence of Cultural Hierarchy in America_ (Cambridge, MA: Harvard University Press, 1988), 213.
50. Warhol, _Philosophy_ , 103.
51. Calvin Tomkins, "Raggedy Andy," in _The Scene: Reports on Postmodern Art_ (New York: Viking Press, 1976), 47. This was one of many moments when Warhol tied poverty to animality rhetorically.
52. Aby Rosen, quoted in Emma Allen, "Landlord," _New Yorker_ , June 19, 2014, 19.
53. Andy Warhol, _The Andy Warhol Diaries_ , ed. Pat Hackett (New York: Grand Central Publishing, 1989), 759. The first mention of this outfit in the _Diaries_ is September 25, 1977.
54. Dylan, "Like a Rolling Stone."
55. Max Horkheimer and Theodor W. Adorno, _Dialectic of Enlightenment_ , ed. Gunzelin Schmid Noerr, trans. Edmund Jephcott (Palo Alto, CA: Stanford University Press, 2002), 136.
56. Crispin Sartwell, "Andy Warhol and the Persistence of Modernism," _New York Times_ , June 19, 2013, accessed November 1, 2016, <http://opinionator.blogs.nytimes.com>.
57. Jennifer M. Silva, _Coming Up Short: Working-Class Adulthood in an Age of Uncertainty_ (Oxford: Oxford University Press, 2013), 22.
58. Warhol, _Diaries_ , 676.
59. David Harvey, _A Brief History of Neoliberalism_ (Oxford: Oxford University Press, 2005), 202.
60. John Steinbeck, _The Winter of Our Discontent_ (New York: Penguin Classics, 2008), 53.
61. Steinbeck, _Winter of Our Discontent_ , 57–58. Steinbeck may have been referencing Aldous Huxley, who "once wrote that poets rarely write of money: they find it a vulgar subject" (Aline B. Louchheim, "The Automobile in Modern Art," _New York Times_ , September 20, 1953, x11).
62. Warhol, _Philosophy_ , 129, 137, 238. See Stiles, _Concerning Consequences_ , 313.
63. See Bockris, _Warhol_ , 69.
64. Forbes Quotes, accessed July 27, 2014, <http://thoughts.forbes.com>.
65. Schramm, Lyle, and Parker, _Television_ , 110; Aaron A. Fox, _Real Country: Music and Language in Working-Class Culture_ (Durham, NC: Duke University Press, 2004), 250; Martineau, "Pattern of Social Classes," 246.
66. Steinbeck, _Winter of Our Discontent_ , 232.
67. David Bourdon, "Andy Warhol: 1928–1987," _Art in America_ 75, no. 5 (May 1987): 139.
68. Andy Warhol and Pat Hackett, _Popism: The Warhol '60s_ (New York: Harcourt Brace Jovanovich, 1980), 7. Kelly M. Cresap points out that the promise of this musical "blasting" would be extended to a broader audience five years later in advertisements for Warhol's _Exploding Plastic Inevitable_ (Cresap, _Pop Trickster Fool: Warhol Performs Naivete_ [Urbana: University of Illinois Press, 2004], 121).
69. "Ivan Karp, New York, 12 October 1978," in Smith, _Andy Warhol's Art and Films_ , 351.
70. "David Bourdon, New York, 16 October 1978," in Smith, _Andy Warhol's Art and Films_ , 225.
71. Victor Bockris and Gerard Malanga, _Up-Tight: The Velvet Underground Story_ (London: Omnibus Press, 2002), 11.
72. "Pop Art? Is it Art? A Revealing Interview with Andy Warhol" ( _Art Voices_ , December 1962), reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 4; Aline B. Saarinen, "Explosion of Pop Art," _Vogue_ , April 15, 1963, 87.
73. Michael T. Bertrand, _Race, Rock, and Elvis_ (Chicago: University of Illinois Press, 2004), 61.
74. Peter Coviello, "Call Me Morbid," _Journal of Popular Music Studies_ , 23, no. 4 (December 2011): 382. In "Sally Goes Round the Roses," this is a distinctly queer utopia, with the female singer lamenting, "The saddest thing in the whole wide world / is to see your baby with another girl" (Will Stos, "Bouffants, Beehives, and Breaking Gender Norms: Rethinking 'Girl Group' Music of the 1950s and 1960s," _Journal of Popular Music Studies_ 24, no. 2 [June 2012]: 126).
75. Dickey Lee, "I Saw Linda Yesterday" (Smash Records, 1962, 45 rpm); Little Eva, "The Loco-Motion" (Dimension, 1962, 45 rpm); Homer and Jethro, "Does the Spearmint Lose Its Flavor?" (RCA Victor, 1950, 45 rpm).
76. Stahl, _Unfree Masters_ , 5–6.
77. See Peter E. Berry, _And the Hits Just Keep on Coming_ (Syracuse, NY: Syracuse University Press, 1977), 118. Similar exploitation pervaded the comic-book industry, another of Warhol's great sources; see, for example, Deborah Friedell, "Kryptonomics," _New Yorker_ , June 24, 2013, 80–81.
78. Bourdon, "Andy Warhol: 1928–1987," 139.
79. Jacques Rancière, _Aesthetics and Its Discontents_ , trans. Steven Corcoran (Cambridge: Polity Press, 2009), 31.
80. Viva quoted in Leticia Kent, "Andy Warhol: 'I Thought Everyone Was Kidding,'" _Village Voice_ , September 12, 1968, 1.
## Chapter One
1. Ivan Karp, recorded during a 1965 CBC television interview of Andy Warhol, accessed December 8, 2014, <http://www.youtube.com>.
2. John Jones, "Tape-recorded Interview with Roy Lichtenstein, October 5, 1965, 11:00 AM," in _Roy Lichtenstein: October Files_ , ed. Graham Bader (Cambridge, MA: MIT Press, 2009), 21, 27.
3. David Barsalou, "Deconstructing Roy Lichtenstein," accessed May 20, 2016, <http://davidbarsalou.homestead.com/LICHTENSTEINPROJECT.html> and <http://www.flickr.com>. Barsalou's entries typically identify the original panel's artist but not the issue from which the panel was borrowed.
4. The source for this panel is misidentified by the Roy Lichtenstein Foundation as _Our Fighting Forces_ , no. 66 (February 1962), 7; the correct source is _Our Fighting Forces_ , no. 74 (February 1963), 11. See Roy Lichtenstein Foundation's Image Duplicator, accessed May 20, 2016, <http://www.imageduplicator.com/main.php>.
5. See Bart Beaty, "Roy Lichtenstein's Tears: Art vs. Pop in American Culture," _Canadian Review of American Studies_ 34, no. 3 (2004): 249–68.
6. Douglas McClellan, "Roy Lichtenstein, Ferus Gallery," _Artforum_ , July 1963, 47.
7. Arthur Danto, _Andy Warhol_ (New Haven, CT: Yale University Press, 2010), xiii.
8. Danto, _Andy Warhol_ , xiii.
9. Adam Gopnik, "Comics," in Kirk Varnedoe and Adam Gopnik, _High and Low: Modern Art and Popular Culture_ (New York: Museum of Modern Art, 1990), 199.
10. Lawrence Alloway, "Popular Culture and Pop Art" (1969), in _Pop Art: A Critical History_ , ed. Steven Henry Madoff (Berkeley: University of California Press, 1997), 170.
11. See James Rondeau and Sheena Wagstaff, _Roy Lichtenstein: A Retrospective_ (Chicago: Art Institute of Chicago, 2012), 48. For an important exception, see Thomas Crow, _The Long March of Pop: Art, Music, and Design, 1930–1995_ (New Haven, CT: Yale University Press, 2014), 106–35, which argues that, although Lichtenstein initially projected a "rustic" image, he "had grown up in affluent circumstances on the Upper West Side" (107).
12. G. R. Swenson, "What Is Pop Art? Part I," _ARTnews_ 62, no. 7 (November 1963): 25 (emphasis in original). Anne Wagner points out to me that, in describing mass culture as "brazen and threatening," Lichtenstein should be recognized as an early deviator from the long-standing association between femininity and mass culture described by Andreas Huyssen (see, for example, Huyssen, "Mass Culture as Woman: Modernism's Other," in _Studies in Entertainment: Critical Approaches to Mass Culture_ , ed. Tania Modleski [Bloomington: Indiana University Press, 1986], 188–207). For an alternative view, see Cécile Whiting, "Borrowed Spots: The Gendering of Comic Books, Lichtenstein's Paintings, and Dishwasher Detergent," _American Art_ 6, no. 2 (Spring 1992): 19.
13. Swenson, "What Is Pop Art? Part I," 63.
14. Michael Lobel, _Image Duplicator: Roy Lichtenstein and the Emergence of Pop Art_ (New Haven, CT: Yale University Press, 2002), 26–27. Lobel argues that Lichtenstein's work attempted to protect painting by homeopathically injecting it with the mechanical.
15. Donald Judd, "Roy Lichtenstein," _Arts Magazine_ 36, no. 7 (April 1962): 52.
16. Swenson, "What Is Pop Art? Part I," 63.
17. Cited in Jennifer Farrell, "Reflections on a Gift: Richard Brown Baker and Roy Lichtenstein," _Yale University Art Gallery Bulletin_ , Recent Acquisitions (2008), 50.
18. Allene Talmey, "Art Is the Core," _Vogue_ 144, no. 1 (July 1, 1964): 123. Compare Aldous Huxley, responding to Baudelaire: "To the aristocratic pleasure of displeasing other people, the conscious offender against good taste can add the still more aristocratic pleasure of displeasing himself" (Huxley, "Vulgarity in Literature," in _Complete Essays_ , vol. 3, _1930–1935_ , ed. Robert S. Baker and James Sexton [Chicago: Ivan Dee, 2001], 30).
19. "Run for Love," _Secret Hearts_ 83 (National Comics Publications, November 1962), 3.
20. "Run for Love," 7.
21. Quoted in Deborah Solomon, "The Art behind the Dots," _New York Times_ , March 8, 1987, A42.
22. "The Painting," _Strange Suspense Stories_ 1, no. 72 (Charlton Comics, October 1964), 17.
23. See Immanuel Kant, _Critique of the Power of Judgment_ , trans. Paul Guyer and Eric Matthews (Cambridge: Cambridge University Press, 2000), §43, 182–83, for Kant's distinction between art and handicraft.
24. "The Painting," 22.
25. See Diane Waldman, "Brushstrokes, 1965–66," in _Roy Lichtenstein_ (New York: Guggenheim Museum, 1994), 151; Kevin Hatch, "Roy Lichtenstein: Wit, Invention, and the Afterlife of Pop," in _Pop Art: Contemporary Perspectives_ , exh. cat. (New Haven, CT: Yale University Press, 2007), 57.
26. "The Painting," 23.
27. Swenson, "What Is Pop Art, Part I," 63. Crow argues that this aestheticization of violence may have helped Lichtenstein ward off his own wartime trauma (Crow, _Long March of Pop_ , 127–28).
28. The sources are _Our Fighting Forces_ , nos. 68 (May 1962), 8; 69 (July 1962), 8; and 74 (February 1963), 11.
29. Rosalind E. Krauss, _The Optical Unconscious_ (Cambridge, MA: MIT Press, 1994), 246.
30. _Our Fighting Forces_ , no. 74 (February 1963), 2, 5.
31. _Our Fighting Forces_ , no. 69 (July 1962), 9.
32. See Gilles Deleuze and Félix Guattari, _A Thousand Plateaus: Capitalism and Schizophrenia_ , trans. Brian Massumi (Minneapolis: University of Minnesota Press, 1987), 232–309.
33. Graham Bader, _Hall of Mirrors: Roy Lichtenstein and the Face of Painting in the 1960s_ (Cambridge, MA: MIT Press, 2010), xxv (emphasis in original).
34. Art Spiegelman, "High Art Lowdown," _Artforum_ (December 1990): 115 (emphasis in original).
35. Bruce Glaser, "Oldenburg, Lichtenstein, Warhol: A Discussion," _Artforum_ 4, no. 6 (February 1966): 23.
36. For more on Oldenburg's and Muschinski's interest in these energies, and their "obdurate but ambivalent resistance to this increasingly ordered and legible New York," see Joshua Shannon, _The Disappearance of Objects: New York Art and the Rise of the Postmodern City_ (New Haven, CT: Yale University Press, 2009), 10–48, quote at 188.
37. Claes Oldenburg, _Store Days: Documents from The Store (1961) and Ray Gun Theatre (1962)_ , selected by Claes Oldenburg and Emmett Williams (New York: Something Else Press, 1967), 8, 15.
38. Gene Baro, _Claes Oldenburg: Drawings and Prints_ (London: Chelsea House Publishers, 1969), 104; cited in Genevieve Waller, "Unattributed Objects: The Mouse Museum, the Ray Gun Wing, and Four Artists," in _Sculpture and the Vitrine_ , ed. John C. Welchman (Burlington, VT: Ashgate Publishing, 2013), 174n9.
39. Yve-Alain Bois, "Ray Guns," in Bois and Rosalind E. Krauss, _Formless: A User's Guide_ (New York: Zone Books, 2000), 176. See also Eva Ehninger, "What's Happening? Allan Kaprow and Claes Oldenburg Argue about Art and Life," _Getty Research Journal_ 6 (2014): 195–202, which demonstrates Oldenburg's commitment to a distinction between art and life.
40. Calvin Tomkins, "Profiles: Moving with the Flow," _New Yorker_ , November 6, 1971, 90.
41. "Claes Oldenburg" (1965), in David Sylvester, _Interviews with American Artists_ (London: Chatto and Windus, 2001), 203.
42. Paul Cummings, "Oral history interview with Charles Alan, 1970 Aug. 20–25," Archives of American Art, accessed November 19, 2012, <http://www.aaa.si.edu>.
43. Oldenburg, _Store Days_ , 62.
44. Grace Glueck, "Soft Sculpture or Hard—They're Oldenburgers," _New York Times Magazine_ , September 21, 1969, 103.
45. Sigmund Freud, "The Taboo of Virginity" (1918), trans. James Strachey, in _Sexuality and the Psychology of Love_ (New York: Touchstone, 1997), 66.
46. Quoted in David Sylvester, _About Modern Art_ (New Haven, CT: Yale University Press, 2001), 238.
47. Patty Mucha [Patricia Muschinski], "Soft Sculpture Sunshine," in Sid Sachs and Kalliopi Minioudaki, _Seductive Subversion: Women Pop Artists, 1958–1968_ (New York: Abbeville Press, 2010), 146. See also Julian Rose, "Objects in the Cluttered Field: Claes Oldenburg's Proposed Monuments," _October_ , no. 140 (Spring 2012), 113–138.
48. For Lichtenstein's art historical reference points, see his 1997 interview with David Sylvester, _Roy Lichtenstein: Some Kind of Reality_ (London: Anthony d'Offay Gallery, 1997).
49. Alex Potts, "Disencumbered Objects," _October_ , no. 124 (Spring 2008), 186–87.
50. Oldenburg, _Store Days_ , 48.
51. Herbert Marcuse, "Commenting on Claes Oldenburg's Proposed Monuments for New York City," _Perspecta: The Yale Architectural Journal_ 12 (1969): 76. Marcuse's optimism would be belied by Oldenburg's work from the 1970s and after, in collaboration with Coosje Van Bruggen, where similar monuments proved perfectly capable of peacefully inhabiting the capitalist cityscape.
52. Mucha, "Soft Sculpture Sunshine," 148.
53. Cited in Tom Williams, "Lipstick Ascending: Claes Oldenburg in New Haven in 1969," _Grey Room_ 31 (Spring 2008): 132, which also discusses the monument's political function (117).
54. Mucha, "Soft Sculpture Sunshine," 146.
55. Michael Fried, "Art and Objecthood," _Artforum_ 5 (June 1967): 12–23.
56. G. R. Swenson, "What Is Pop Art? Part II," _ARTnews_ 62, no. 10 (February 1964): 62.
57. Swenson, "What Is Pop Art, Part I," 63.
58. Glaser, "Oldenburg, Lichtenstein, Warhol," 23.
59. David Bourdon, "Warhol Interviews Bourdon" (unpublished manuscript, 1962–63), in _I'll Be Your Mirror: The Selected Andy Warhol Interviews_ , ed. Kenneth Goldsmith, Reva Wolf, and Wayne Koestenbaum (New York: Carroll & Graf, 2004), 8.
60. Warhol, quoted in John Giorno, "Andy Warhol Interviewed by a Poet" (unpublished manuscript, 1963), in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 23.
61. It is therefore no surprise that, as Michael Lobel has pointed out, "the brand name is consistently effaced in [Lichtenstein's] work" ( _Image Duplicator_ , 42). Lobel argues that brand names "would otherwise have pointed to the previous authorship of these images" and thereby undermined Lichtenstein's artistic authority (47).
62. Quoted in Lawrence Alloway, _Roy Lichtenstein_ (New York: Abbeville Press, 1983); cited in Lobel, _Image Duplicator_ , 42.
63. For "Vulgarists," see Doris Brown, "Two Douglass Professors Are Leading the Pop Art Charge," _Sunday Home News_ , April 28, 1963, 20; for "Vulgarians," see Max Kozloff, "'Pop' Culture, Metaphysical Disgust, and the New Vulgarians," _Art International_ , March 1962, 35–36; for pop art's "anti-popular" agenda, see Stuart Preston, "On Display: All-Out Series of Pop Art," _New York Times_ , March 21, 1963, 8.
64. Marianne Hancock, "Soup's On," _Arts Magazine_ 39 (May–June 1965): 16.
65. Jean Stein, _Edie: An American Biography_ (New York: Alfred A. Knopf, 1982), 246.
66. "Oral history interview with Tom Wesselmann, 1984 January 3–February 8," Archives of American Art, accessed June 15, 2014, <http://www.aaa.si.edu>.
67. Ronnie Tavel made the interesting claim that this expansion was intentional and coordinated: "He wants to be a Leonardo da Vinci type. That's why he did the novel; he wants to cover every area" ("Ronnie Tavel," in John Wilcock, _The Autobiography and Sex Life of Andy Warhol_ [New York: Other Scenes, 1971], n.p.).
68. Bernard Weintraub, "Andy Warhol's Mother," _Esquire_ 64, no. 5 (November 1966): 158.
69. Mirko Tobias Schäfer, _Bastard Culture! How User Participation Transforms Cultural Production_ (Amsterdam: Amsterdam University Press, 2011), 10. For "participatory culture," see Henry Jenkins, _Textual Poachers: Television Fans and Participatory Culture_ (New York: Routledge, 1992), 23, 46, 290. Jenkins has defined "participatory culture" as having "relatively low barriers to artistic expression and civic engagement[;] strong support for creating and sharing creations with others[;] some type of informal mentorship whereby what is known by the most experienced is passed along to novices[;] members who believe that their contributions matter . . . [and] who feel some degree of social connection with one another" (Jenkins et. al., _Confronting the Challenges of Participatory Culture: Media Education for the 21st Century_ [Cambridge, MA: MIT Press, 2009], 5–6). But where Jenkins tends to focus on the positive possibilities of participatory culture, Warhol seemed more interested in the ways official culture simultaneously invites and thwarts amateur cultural participation, while commodifying its failures.
70. Blake Stimson, _Citizen Warhol_ (London: Reaktion, 2014), 90.
71. David Graeber, "Caring Too Much: That's the Curse of the Working Classes," _Guardian_ , March 26, 2014, 34.
72. Quoted in Victor Bockris, _Warhol: The Biography_ (New York: Da Capo, 2003), 31.
73. Stuart Zane Charmé, _Vulgarity and Authenticity: Dimensions of Otherness in the World of Jean-Paul Sartre_ (Amherst: University of Massachusetts Press, 1991), 20.
74. David Bourdon, "Andy Warhol: 1928–1987," _Art in America_ 75, no. 5 (May 1987): 139.
75. Pierre Bourdieu, _The Field of Cultural Production_ , trans. Richard Nice (New York: Columbia University Press, 1993), 70.
76. Quoted in Richard Goldstein, "A Quiet Night at Balloon Farm," in _The Da Capo Book of Rock and Roll Writing_ , ed. Clinton Heylin (New York: Da Capo, 1992), 217. Lou Reed's record label had called his first band "The Primitives."
77. Sidney J. Levy, "Social Class and Consumer Behavior," in _On Knowing the Consumer_ , ed. J. Newman (New York: John Wiley and Sons, 1966), 159.
78. Lou Reed, quoted in Billy Name, _The Silver Age: Black & White Photographs from Andy Warhol's Factory_ (London: Reel Art Press, 2014), 300.
79. Peter Schjeldahl, "Warhol and Class Content" ( _Art in America_ 68, no. 5 [May 1980]: 112–19), reprinted in _The Hydrogen Jukebox: Selected Writings of Peter Schjeldahl, 1978–1990_ (Berkeley: University of California Press, 1993), 47.
80. Schjeldahl, "Warhol and Class Content," 47.
81. Guy Debord, _Society of the Spectacle_ , trans. Donald Nicholson-Smith (New York: Zone Books, 2006), 22.
82. Richard Meyer, "The Art Historical Problem of Andy Warhol," _Artscene_ , May 2002, n.p. For an exemplary recent investigation of these questions, see John R. Blakinger, " _Death in America_ and _Life_ Magazine: Sources for Andy Warhol's _Disaster_ Paintings," _Artibus et Historiae_ 33, no. 66 (2012): 269–285.
83. Arthur Danto, "Warhol," in _Encounters and Reflections: Art in the Historical Present_ (Berkeley: University of California Press, 1997), 287. Or, as Amelia Jones argues, "The art museum . . . celebrates (and markets) the artist as a genius _because_ he critiques the institution: a situation epitomized by the mass marketing of Andy Warhol" (Jones, _Irrational Modernism: A Neurasthenic History of New York Dada_ [Cambridge, MA: MIT Press, 2004], 20; emphasis in original).
84. Benjamin H. D. Buchloh, "Andy Warhol's One-Dimensional Art: 1956–1966," in _Neo-Avantgarde and Culture Industry: Essays on European and American Art from 1955 to 1975_ (Cambridge, MA: MIT Press, 2000), 465.
85. Steven Shaviro, _Connected, or What It Means to Live in the Network Society_ (Minneapolis: University of Minnesota Press, 2003), 72.
## Chapter Two
1. Donna De Salvo, "'Subjects of the Artists': Towards a Painting without Ideals," in _Hand-Painted Pop: American Art in Transition, 1952–1962_ , ed. Russell Ferguson (New York: Rizzoli, 1992), 86; see also 92.
2. Barbara Rose, "In Andy Warhol's Aluminum Foil, We All Have Been Reflected," _New York_ , May 31, 1971, 54
3. Robert Smithson, "Production for Production's Sake" (1972), in _Robert Smithson: The Collected Writings_ , ed. Jack Flam (Berkeley: University of California Press, 1996), 378.
4. Andy Warhol, _The Andy Warhol Diaries_ , ed. Pat Hackett (New York: Grand Central Publishing, 1989), 379.
5. Warhol, _Diaries_ , 11.
6. Warhol, _Diaries_ , 347–48.
7. See Kristine Stiles's remarkable reading of shadows and "Warhol's traumatic subjectivity" in Stiles, _Concerning Consequences: Studies in Art, Deconstruction, and Trauma_ (Chicago: University of Chicago Press, 2016), 309–38.
8. Warhol, _Diaries_ , 358.
9. Warhol, _Diaries_ , 354.
10. Michael Fried, "New York Letter," _Art International_ 6, no. 10 (December 1962): 57, reprinted in _Art and Objecthood: Essays and Reviews_ (Chicago: University of Chicago Press, 1998), 287-88.
11. _Artforum_ , September 1963; reproduced in Mark Rosenthal, Marla Prather, Ian Alteveer, and Rebecca Lowery, _Regarding Warhol: Sixty Artists, Fifty Years_ (New York: Metropolitan Museum of Art, 2012), 252; Georg Frei and Neil Printz, eds., _The Andy Warhol Catalogue Raisonné_ , vol. 1, _Painting and Sculpture 1961–1963_ (London: Phaidon, 2002), 215. On the status of predilections, see Clement Greenberg, "Review of the Whitney Annual," _Nation_ , December 28, 1946, reprinted in _The Collected Essays and Criticism_ , ed. John O'Brian, vol. 2 (Chicago: University of Chicago Press, 1988), 118.
12. For more on the censorship of homoeroticism in early Warhol, see Richard Meyer, "Warhol's Clones," _Yale Journal of Criticism_ 7, no. 1 (1994): 79–109; and Meyer, "Most Wanted Men: Homoeroticism and the Secret of Censorship in Early Warhol," in _Outlaw Representation_ (New York: Beacon Press, 2002), 95–158.
13. "Andy Warhol, New York, 6 November 1978," in Patrick S. Smith, _Andy Warhol's Art and Films_ (Ann Arbor, MI: UMI Research Press, 1986), 522.
14. Fried, "New York Letter," 288.
15. Arthur Danto, "Warhol and the Politics of Prints," in _Andy Warhol Prints: A Catalogue Raisonné, 1962–1987_ , ed. Frayda Feldman and Jörg Schellmann, 4th rev. ed. (New York: D.A.P./Ronald Feldman Fine Arts/Andy Warhol Foundation, 2003), 10–11, 13. Although Danto did not cite Fried's article in this text, he mentioned it approvingly in _Andy Warhol_ (New Haven, CT: Yale University Press, 2010), 45. Danto's effort to see these myths as purely affirmative and innocuous is exemplified by his description of another print in the _Myths_ series, the segregation-era "Mammy," "as the emblem of our daily bread" (11).
16. Danto, "Warhol and the Politics of Prints," 13.
17. Fried, "New York Letter," 57.
18. Barbara Rose also remarked this quality in Warhol's work: "The images he leaves will be the permanent record of America in the sixties: mechanical, vulgar, violent, commercial, deadly and destructive" (Rose, "In Andy Warhol's Aluminum Foil," 54).
19. T. J. Clark, _Farewell to an Idea: Episodes from a History of Modernism_ (New Haven, CT: Yale University Press, 1999), 378; Thorstein Veblen, _Theory of the Leisure Class_ (Oxford: Oxford University Press, 2007), 225.
20. Arthur Danto, "The Artworld," _Journal of Philosophy_ 61, no. 19 (October 15, 1964): 580–82.
21. Billy Name, _The Silver Age: Black & White Photographs from Andy Warhol's Factory_ (London: Reel Art Press, 2014), 100.
22. Arthur C. Danto, _The Transfiguration of the Commonplace: A Philosophy of Art_ (Cambridge, MA: Harvard University Press, 1981), 208.
23. Danto, _Andy Warhol_ , 13, 16 (emphasis in original).
24. Danto, _Andy Warhol_ , 16.
25. José Esteban Muñoz, "Famous and Dandy like B. 'n' Andy: Race, Pop, and Basquiat," in _Pop Out: Queer Warhol_ , ed. Jennifer Doyle, Jonathan Flatley, and Muñoz (Durham, NC: Duke University Press, 1996), 153.
26. Jonathan Flatley, "Warhol Gives Good Face: Publicity and the Politics of Prosopopoeia," in Doyle, Flatley, and Muñoz, _Pop Out_ , 102–3.
27. Sidney J. Levy, "Social Class and Consumer Behavior," in _On Knowing the Consumer_ , ed. J. Newman (New York: John Wiley and Sons, 1966), 156.
28. David Harvey, _A Brief History of Neoliberalism_ (Oxford: Oxford University Press, 2005), 42.
29. For analyses of the problem of class in pop art, see Peter Schjeldahl, "Warhol and Class Content" ( _Art in America_ 68, no. 5 [May 1980]: 112–19), reprinted in _The Hydrogen Jukebox: Selected Writings of Peter Schjeldahl, 1978–1990_ (Berkeley: University of California Press, 1993); Sara Doris, _Pop Art and the Contest over American Culture_ (Cambridge: Cambridge University Press, 2007); Christin J. Mamiya, _Pop Art and Consumer Culture: American Super Market_ (Austin: University of Texas Press, 1992); Cécile Whiting, _A Taste for Pop: Pop Art, Gender, and Consumer Culture_ (Cambridge: Cambridge University Press, 1997); and Roger Cook, "Andy Warhol, Capitalism, Culture, and Camp," _Space and Culture_ 6, no. 1 (February 2003): 66–76.
30. Wilbur Schramm, Jack Lyle, and Edwin B. Parker, _Television in the Lives of Our Children_ (Stanford, CA: Stanford University Press, 1961), 110; Pierre D. Martineau, "The Pattern of Social Classes," in _Marketing's Role in Scientific Management_ , ed. Robert L. Clewett (Chicago: American Marketing Association, 1957), 246.
31. Beverley Skeggs, _Formations of Class and Gender: Becoming Respectable_ (London: Sage, 1997), 6; Skeggs is drawing on the discussions of "structures of feeling" in Patricia Williams, _Alchemy of Race and Rights: Diary of a Law Professor_ (Cambridge, MA: Harvard University Press, 1991).
32. Henry Jenkins, _Textual Poachers: Television Fans and Participatory Culture_ (New York: Routledge, 1992), 23. My approach to cultural participation in this book has been informed by Kaja Silverman's investigations of the superego, which "puts the ego in a vicious double bind; it says not only, 'you cannot take your father's place,' but also, 'you must take your father's place'" (Silverman, "Untitled Response," _October_ , no. 123 [Winter 2008], 142). Warhol addressed cultural participation as a similarly double-binding structure.
33. Warhol, quoted in "Andy Warhol Interviews Henry Santoro," in _Image Machine: Andy Warhol and Photography_ , ed. Raphaela Platow, Synne Genzmer, and Joseph Ketner (Nürnberg: Verlag für moderne Kunst, 2012), 194.
34. John Leonard, "The Return of Andy Warhol," _New York Times_ , November 10, 1968, SM150. Warhol constantly referenced these technologies in his work; even the _Flowers_ paintings from 1964 were derived from an article in _Modern Photography_ magazine on developing color film at home.
35. Veblen, _Theory of the Leisure Class_ , 132. See also Paul Fussell, _Class: A Guide through the American Status System_ (New York: Touchstone, 1983), 72.
36. David Antin, "Warhol: The Silver Tenement," _ARTnews_ 65, no. 4 (Summer 1966): 59.
37. John Giorno, "Andy Warhol Interviewed by a Poet" (unpublished manuscript, 1963), in _I'll Be Your Mirror: The Selected Andy Warhol Interviews_ , ed. Kenneth Goldsmith, Reva Wolf, and Wayne Koestenbaum (New York: Carroll & Graf, 2004), 22.
38. Roland Barthes, "Myth Today," in _Mythologies_ , trans. Annette Lavers (New York: Hill and Wang, 1972), 125, 142.
39. Barthes, "Myth Today," 156.
40. Lawrence W. Levine, _Highbrow/Lowbrow: The Emergence of Cultural Hierarchy in America_ (Cambridge, MA: Harvard University Press, 1988), 195. Levine charts the shift from a "relatively open" and collective range of cultural participation during the nineteenth century to an increasingly disciplined and individuated "passive politeness" during the early twentieth (195, 197).
41. Christoph Heinrich, "Freezing a Motion Picture: An Interview with Gerard Malanga," in _Andy Warhol: Photography_ (Zurich: Edition Stemmle, 1999), 115.
42. Benjamin H. D. Buchloh, "Drawing Blanks: Notes on Andy Warhol's Late Works," _October_ , no. 127 (Winter 2009), 3.
43. Emile de Antonio, "Marx and Warhol," unpublished draft, Emile de Antonio Papers, Wisconsin Center for Film and Theater Research, box 12, file 8; quoted in Branden Joseph, "1962," _October_ , no. 132 (Spring 2010), 114.
44. Quoted in Victor Bockris, _Warhol: The Biography_ (New York: Da Capo, 2003), 136.
45. See Frei and Printz, _Catalogue Raisonné_ , 1:57. The catalogue raisonné bases its chronology of these works purely on "stylistic criteria."
46. Bonwit Teller figures prominently as a sign of class aspiration in Philip Roth's early fiction. See "Eli the Fanatic" (1958) and "Goodbye, Columbus" (1959), both reprinted in _Goodbye Columbus_ (New York: Vintage, 1994). For an overview of the history of department store window dressing leading up to Warhol's Bonwit Teller display, see Whiting, _Taste for Pop_ , 9–22.
47. Warhol seems to have relished this disparity. For another Bonwit window display, he had "built a wood fence out of scrap lumber and decorated it with graffiti and flowers and suns and the kind of stick people that kids draw. The mannequins looked fabulous against it" (Calvin Tomkins, "Raggedy Andy," in _The Scene: Reports on Postmodern Art_ [New York: Viking Press, 1976], 40).
48. Frederick Schruers, "Andy Warhol: Why Not?" _Globe and Mail_ , November 8, 1978, F10.
49. For a thorough psychoanalytic analysis of the painting, see Bradford R. Collins and David Cowart, "Through the Looking-Glass: Reading Warhol's _Superman_ ," _American Imago_ 53, no. 2 (1996): 107–37. See also Michael Moon, "Screen Memories, or Pop Comes from the Outside: Warhol and Queer Childhood," in Doyle, Flatley, and Muñoz, _Pop Out_ , 89.
50. Jack Gould, "TV: _Don Juan in Hell_ ," _New York Times_ , February 16, 1960, 75. Philip Pearlstein's _Superman_ painting, produced in 1952 and now in the collection of the Museum of Modern Art, is another potential reference.
51. Carol L. Tilley, "Seducing the Innocent: Fredric Wertham and the Falsifications That Helped Condemn Comics," _Information and Culture_ 47, no. 4 (2012): 387.
52. Harrison E. Salisbury, "Youth Gang Members Tell of Lives, Hates and Fears," _New York Times_ , March 25, 1958, 1, 26.
53. Charles G. Spiegler, "A Teacher's Report on a 'Tough' School," _New York Times_ , November 24, 1957, 239.
54. Dorothy Barclay, "That Comic Book Question," _New York Times_ , March 20, 1955, SM48.
55. See, for example, "Television Programs: Sunday, Monday, Tuesday," _New York Times_ , November 1, 1959, x14, and "Television: Saturday, October 15, 1960," _New York Times_ , October 15, 1960, 47.
56. George Lipsitz, _Time Passages: Collective Memory and American Popular Culture_ (Minneapolis: University of Minnesota, 1990), 44; Gary R. Edgerton, _The Columbia History of American Television_ (New York: Columbia University Press, 2009), 107.
57. Levy, "Social Class and Consumer Behavior," 155. See also Schramm, Lyle, and Parker, _Television_ , 247.
58. Kent Geiger and Robert Sokol, "Social Norms in Television-Watching," _American Journal of Sociology_ 65, no. 2 (September 1959): 174, 176–77. This despite the fact that the working class is vastly underrepresented on television; see Richard Butsch, "Class and Gender in Four Decades of Television Situation Comedy: Plus Ça Change . . . ," _Critical Studies in Mass Communication_ 9 (December 1992): 387–99, cited in Julie Bettie, "Class Dismissed? Roseanne and the Changing Face of Working-Class Iconography," _Social Text_ 45 (Winter 1995): 127.
59. Schramm, Lyle, and Parker, _Television_ , 167; Geiger and Sokol, "Social Norms," 176, quoted in Schramm, Lyle, and Parker, _Television_ , 111 (emphasis in original).
60. Adeline Gomberg, "The Working-Class Child of Four and Television," in _Blue-Collar World: Studies of the American Worker_ , ed. Arthur B. Shostak and William Gomberg (Englewood Cliffs: Prentice-Hall, 1964), 434–36.
61. Philip Hone, quoted in Levine, _Highbrow/Lowbrow_ , 177.
62. See "Sugar Smacks—1950s Superman Cast," YouTube, accessed May 16, 2016, <http://www.youtube.com>.
63. For more on narratives of class reconciliation in television advertising, see Lipsitz, _Time Passages_ , 48–75.
64. Les Daniels, _Superman: The Complete History: The Life and Times of the Man of Steel_ (San Francisco: Chronicle Books, 2004), 19. See also José Esteban Muñoz, who points out that, "in working through the Superman character, its creators were able to intervene in another phobic anti-Semitic fantasy that figured the Jew's body as weak and sickly" (Muñoz, "Famous and Dandy," 154).
65. _Daily News_ , June 17, 1959, 3.
66. Matt Stahl, _Unfree Masters: Popular Music and the Politics of Work_ (Durham, NC: Duke University Press, 2013), 5–6.
67. Reeves's death thus establishes the painting as a precedent for Warhol's more direct focus on the problem of celebrity and death in the silkscreened paintings of the coming years. See Thomas Crow, "Saturday Disasters: Trace and Reference in Early Warhol," in _Modern Art in the Common Culture_ (New Haven, CT: Yale University Press, 1996), 49–65; Hal Foster, "Death in America," _October_ , no. 75 (Winter 1996), 36–59.
68. For versions of these arguments, see Arthur Danto, "Warhol and the Politics of Prints," 10–11; Collins and Cowart, "Through the Looking-Glass," 129.
69. In a late interview, Warhol claimed to have considered using Reeves as the model for a _Superman_ image; see Barry Blinderman, "Modern _Myths_ : Andy Warhol" (1981), reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 20–24. Walter Hopps described having Warhol show him a large painting depicting Superman carrying Lois Lane (Jean Stein, _Edie: An American Biography_ [New York: Alfred A. Knopf, 1982], 193–94). If it did, in fact, exist, the painting has since been lost.
70. Andy Warhol, _The Philosophy of Andy Warhol (From A to B and Back Again)_ (New York: Harcourt Brace Jovanovich, 1975), 63.
71. See Jonathan D. Katz, "From Warhol to Mapplethorpe: Postmodernity in Two Acts," in _Warhol and Mapplethorpe: Guise and Dolls_ , ed. Patricia Hickson (Hartford: Wadsworth Atheneum Museum of Art, 2015), 22.
72. The alterations are remarked without further comment in Mark Francis, ed., _Pop: Themes and Movements_ (New York: Phaidon, 2005), 85.
73. Marco Livingstone, "Do It Yourself: Notes on Warhol's Techniques," in _Andy Warhol: A Retrospective_ , ed. Kynaston McShine (New York: Museum of Modern Art, 1989), 67; Benjamin H. D. Buchloh, "Andy Warhol's One-Dimensional Art: 1956–1966," in _Neo-Avantgarde_ _and Culture Industry: Essays on European and American Art from 1955 to 1975_ (Cambridge, MA: MIT Press, 2000), 498.
74. This reading would be complicated by Wayne Koestenbaum's claim that Warhol was left-handed ( _Andy Warhol_ [New York: Penguin, 2001], 31). Although it is possible that Warhol was born left-handed, Matt Wrbican points out that photographs show Warhol eating and writing with his right hand (personal correspondence, July 2015).
75. See "Nathan Gluck, New York, 17 October 1978," in Smith, _Andy Warhol's Art and Films_ , 314.
76. George Hartman, quoted in Steven Watson, _Factory Made: Warhol and the Sixties_ (New York: Pantheon, 2003), 27 (emphasis in original).
77. For the centrality of imitation for early childhood learning, see Michael Tomasello, "The Human Adaptation for Culture," _Annual Review of Anthropology_ 28 (1999): 509–29.
78. Schruers, "Andy Warhol: Why Not?" F10.
79. Quoted in Bockris, _Warhol_ , 36. For Warhol, these early enthusiasms seem to have been sexually motivated: "I had two sex idols—Dick Tracy and Popeye. . . . My mother caught me one day playing with myself and looking at a Popeye cartoon" (Ultra Violet, _Famous for 15 Minutes: My Years with Andy Warhol_ [New York: Harcourt Brace Jovanovich, 1988], 154–55; quoted in Moon, "Screen Memories," 83). See also Bradford R. Collins, "Dick Tracy and the Case of Warhol's Closet: A Psychoanalytic Detective Story," _American Art_ 15, no. 3 (2001): 54–79.
80. Vito Giallo, quoted in Bockris, _Warhol_ , 114. Giallo reported that Warhol "could have been making anywhere from $35,000 to $50,000 a year" (114). By the mid-1950s, Warhol's work had been awarded an Art Director's Club gold medal and appeared in _McCall's_ , _Ladies Home Journal_ , _Vogue_ , and _Harper's Bazaar_ (Bockris, _Warhol_ , 100). In 1955 Warhol landed what Bockris calls "his biggest account of the fifties," a weekly advertisement for the posh shoe store I. Miller in the _New York Times_ (117).
81. The source is identified in Francis, _Pop: Themes and Movements_ , 85; the issue's contents have never been discussed in scholarship on the painting.
82. Tom De Haven, _Our Hero: Superman on Earth_ (New Haven, CT: Yale University Press, 2011), 33.
83. Fredric Wertham, _The Seduction of the Innocent_ (New York: Rinehart and Company, 1954), 209.
84. Leticia Kent, "Andy Warhol, Movieman: 'It's hard to be your own script,'" _Vogue_ , March 1970, 204; quoted in Jennifer Doyle, _Sex Objects: Art and the Dialectics of Desire_ (Minneapolis: University of Minnesota Press, 2006), 60. See also Lucy Mulroney, "Editing Andy Warhol," _Grey Room_ 46 (Winter 2012): 46–71.
85. Murray Forman, _One Night on TV Is Worth Weeks at the Paramount: Popular Music on Early Television_ (Durham, NC: Duke University Press, 2012), 79. The issue's corresponding story is significantly more complicated than the cover image, and touches on two key Warholian themes: sexual reproduction (Lane worries that Superman will never be "the perfect husband" because he cannot give her "a marriage, a home and children!") and artificial hair (her otherwise perfect suitor is ultimately revealed to be wearing a wig).
86. See, for example, the inside back cover of _Life with Millie_ , December 1961, and _Amazing Adult Fantasy_ , November 1961, 33.
87. Gilles Deleuze and Félix Guattari, _Anti-Oedipus_ , trans. Robert Hurley, Mark Seem, and Helen R. Lane (Minneapolis: University of Minnesota Press, 2000), 28. David Ogilvy concurred: "The father of the Labor movement . . . used to say that the tragedy of the working class was the poverty of their desires. I make no apology for inciting the working class to desire less Spartan lives" (Ogilvy, _Confessions of an Advertising Man_ [London: Southbank Publishing, 2004], 189).
88. Glenn O'Brien, "Interview: Andy Warhol" ( _High Times_ , August 24, 1977), reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 234–35. See also Fred Lawrence Guiles, _Loner at the Ball: The Life of Andy Warhol_ (London: Black Swan, 1990), 29.
89. Immanuel Kant, _Critique of the Power of Judgment_ , trans. Paul Guyer and Eric Matthews (Cambridge: Cambridge University Press, 2000), §43, 182–83.
90. Warhol's own art education at Carnegie Tech emphasized similar benefits; see Nan Rosenthal, "Let Us Now Praise Famous Men: Warhol as Art Director," in _The Work of Andy Warhol_ , ed. Gary Garrels (Seattle: Bay Press, 1989), 38.
91. Most such comics had one or two pages of these ads per issue, as well as multiple "reader-created" and credited outfits and hairstyles. The November 1961 issue of _Linda Carter, Student Nurse_ devoted four of its thirty-six pages to such competitions.
92. Anne Wagner has pointed out to me that both Sylvia Plath and Eva Hesse entered and won fashion magazine talent contests (at _Mademoiselle_ and _Seventeen_ , respectively), which helped to launch their careers. Matt Wrbican directed my attention to a "Coloring Contest" in Warhol's _Interview_ magazine in 1972, with the winner promised Warhol's signature on her entry.
93. See _Popular Science_ , May 1943, 26; March 1944, 28; December 1947, 45.
94. _Popular Science_ , March 1944, 26.
95. Warhol, _Philosophy_ , 92.
96. Famous Artists Schools advertisement, _Los Angeles Times_ , September 13, 1959, 17 (emphasis in original).
97. Famous Artists Schools advertisement, _New York Times Magazine_ , August 4, 1957, 5. This ad was widely and prominently disseminated; see also _Los Angeles Times_ , June 16, 1957, G5, and _Life_ , August 5, 1957, 9. Ads with similar text appeared in comic books that were contemporary with Warhol's sources. See, for example, the back cover of _Amazing Adult Fantasy_ , December 1961.
98. Pierre Bourdieu, _Distinction: A Social Critique of the Judgement of Taste_ , trans. Richard Nice (Cambridge, MA: Harvard University Press, 1984), 155.
99. Richard Sennett and Jonathan Cobb, _The Hidden Injuries of Class_ (Cambridge: Cambridge University Press, 1972), 229 and footnote, 222.
100. Matt Wrbican tells me that the Warhol Museum has two of Warhol's projectors, a Beseler Vu-Lyte and a Vu-Lyte III, in its collection.
101. Schramm, Lyle, and Parker, _Television_ , 112.
102. Norton Products advertisement, _Kathy, the Teen-Age Tornado_ , October 1961, 25. See a similar ad in _Linda Carter, Student Nurse_ , November 1961, 7. The device was also regularly advertised in the back pages of _Cool_ , _Popular Mechanics_ , _Popular Science_ , the _New York Times_ , and the _Los Angeles Times_. Warhol bemoans his "shaking hand" in _Philosophy_ , 150.
103. Quoted in John Heilpern, "The Fantasy World of Warhol," _Observer_ , June 12, 1966, 12.
104. Bruce Glaser, "Oldenburg, Lichtenstein, Warhol: A Discussion," _Artforum_ 4, no. 6 (February 1966): 22. Compare Malanga, who claimed that Warhol "was just an utter fright if he had to deal with something mechanical, so he never really pursued it" (Heinrich, "Freezing a Motion Picture," 115), and William S. Wilson, who pointed out that, "since a machine is capable of endless and perfect repetitions . . . , and since Warhol is not, . . . he can only . . . succeed in showing that when repetition is an ideal, it is unattainable" (Wilson, "Prince of Boredom: The Repetitions and Passivities of Andy Warhol," _Art and Artists_ 3 [March 1968]: 14).
105. Elenore Lester, "So He Stopped Painting Brillo Boxes and Bought a Movie Camera," _New York Times_ , December 11, 1966, 169.
106. Louis Schneider and Sverre Lysgaard, "The Deferred Gratification Pattern: A Preliminary Study," _American Sociological Review_ 18, no. 2 (April 1953): 148; Skeggs, _Formations of Class and Gender_ , 6.
107. Quoted in Bockris, _Warhol_ , 170. Dave Hickey describes this tendency in Warhol's work as "getting it exactly wrong" (Hickey, "The Importance of Remembering Andy," in _Robert Lehman Lectures in Contemporary Art_ , ed. Lynne Cooke, Karen Kelly, and Bettina Funcke [New York: Dia Art Foundation, 2004], 61).
108. Warhol, _Philosophy_ , 54.
109. Gilles Deleuze and Félix Guattari, _A Thousand Plateaus: Capitalism and Schizophrenia_ , trans. Brian Massumi (Minneapolis: University of Minnesota Press, 1987), 15. The present chapter focuses on the attraction and frustration this "dominant competence" produces, but also the places in the work where "a microscopic event upsets the local balance of power" (15).
110. Vito Giallo, Warhol's assistant in 1958 and 1959, recalled that during this period Warhol used a light box for tracing: "Andy would go through magazines for ideas, _Life_ magazine for instance. And then he'd tear something out and put it in the light box and make a drawing from there. Or trace over it" (John O'Connor and Benjamin Liu, _Unseen Warhol_ [New York: Rizzoli, 1996], 22).
111. Frei and Printz, _Catalogue Raisonné_ , 1:20. Malanga has stated that many of Warhol's key pop images were provided to him by John Rublowsky, who "was working for one of these sleazy tabloids" (Heinrich, "Freezing a Motion Picture," 121).
112. Mark Pendergrast, _For God, Country and Coca-Cola_ (New York: Scribner's, 1993), 259.
113. Skeggs, _Formations of Class and Gender_ , 6. Skeggs is paraphrasing Annette Kuhn's observations in _Family Secrets: Acts of Memory and Imagination_ (London: Verso, 2002).
114. Henry Geldzahler would later describe Warhol's obsession with diamonds: "Andy's whole diamond thing I find extremely fascist also. He's collecting diamonds, he's painting diamonds, he wants to trade everything he has for diamonds" (Michel Auder, _Chelsea Girls with Andy Warhol_ , 1976). But Warhol could also sense the falseness of the myth of the diamond; see Warhol, _Philosophy_ , 241.
115. Stanley Kunitz, in Peter Selz, Henry Geldzahler, Hilton Kramer, Dore Ashton, Leo Steinberg and Kunitz, "A Symposium on Pop Art," _Arts_ 37, no. 7 (April 1963): 42. One critic, writing in 1968, was able to grasp the participatory stakes of Warhol's project: "If all of us had movie cameras and tape recorders and silk-screens; if we designed our own furniture, shaped our own glassware, wove our own tapestries, set our own type, we might knit up the raveled sleeve of self. . . . But, of course, we do none of these things. We consume—unseen, autistically—the play of others" (Leonard, "Return of Andy Warhol," SM151).
116. See Wilson, "Prince of Boredom," 13.
117. Schneider and Lysgaard, "Deferred Gratification Pattern," 148.
118. See Whitney Davis, _Queer Beauty: Sexuality and Aesthetics from Winckelmann to Freud and Beyond_ (New York: Columbia University Press, 2010), 185. As Joshua Shannon has pointed out, paintings like _16 Jackies_ (1964) render the hegemonic celebrity image simultaneously material and immaterial (Shannon, _The Disappearance of Objects: New York Art and the Rise of the Postmodern City_ [New Haven, CT: Yale University Press, 2009], 189–91).
119. Warhol, _Philosophy_ , 129.
120. James Baldwin, _Tell Me How Long the Train's Been Gone_ (1968; New York: Random House, 1998), 32. In Baldwin's 1965 story "The Rockpile," one brother is described as "drawing into his schoolbook a newspaper advertisement which featured a new electric locomotive"—an activity that carries the promise of social mobility through cultural participation (Baldwin, _Going to Meet the Man_ [New York: Dial Press, 1965], 17). For more on Baldwin and Warhol, see Jonathan Flatley, "Skin Problems," chap. 4 in _Like Andy Warhol_ (Chicago: University of Chicago Press, 2017).
121. Walter Benjamin, "The Work of Art in the Age of Its Technological Reproducibility," trans. H. Zohn and E. Jephcott, in _Walter Benjamin: Selected Writings, 1938–1940_ , ed. Howard Eiland, Michael W. Jennings, and Gary Smith (Cambridge, MA: Belknap Press, 1999), 255.
122. David Graeber, _Revolutions in Reverse: Essays on Politics, Violence, Art, and Imagination_ (Brooklyn, NY: Autonomedia, 2011), 60. As Flatley has shown, Warhol's eccentric collecting practices offered an alternate solution to this problem; see "Like: Collecting and Collectivity," _October_ , no. 132 (Spring 2010), 78–83.
123. Harvey, _Brief History_ , 203.
124. Michael Lüthy, "The Apparent Return of Representation: Ambivalence Structures in Warhol's Early Work," trans. Michael Robinson, in _Andy Warhol: Paintings 1960–1986_ , ed. Martin Schwander (Stuttgart: Hatje, 1995), 49.
125. Anonymous viewer, quoted in Grace Glueck, "Art Notes: Boom?" _New York Times_ , May 10, 1964, X19.
126. Martineau, "Pattern of Social Classes," 243.
127. The company touted its redesigned European packaging as "draw[ing] inspiration from the Andy Warhol 'Pop Art' movement of the 1950s" ("Press Releases—Burger King® unveils new global packaging design," July 2010, accessed December 9, 2011, <http://www.burgerking.co.uk>). It is unclear whether Burger King was aware of Warhol's earlier googly-eyed rendezvous with its products ("Andy Warhol eating a hamburger," YouTube, accessed December 9, 2011, <http://www.youtube.com>).
128. Guy Debord, _Society of the Spectacle_ , trans. Donald Nicholson-Smith (New York: Zone Books, 2006), 46. Advertisers were well aware of this vulnerability; James Playsted Wood worried that "some day, the sated customer, surfeited to lazy discomfort with all the goodies he has been able to swallow and all the shiny objects he has been able to cover with down payments, may be able to take no more" (Wood, _The Story of Advertising_ [New York: Ronald Press, 1958], 499n4).
129. T. J. Clark, "On the Very Idea of a Subversive Art History," unpublished conference paper, 1992, 26. I am grateful to Jeremy Spencer for sharing this paper with me.
130. J. D. Taylor, _Negative Capitalism: Cynicism in the Neoliberal Era_ (Washington, DC: Zero Books, 2013), 102, 104.
## Chapter Three
1. A more complete accounting would have to include the following works, listed by number in Georg Frei and Neil Printz, eds., _The Andy Warhol Catalogue Raisonné_ , vol. 1, _Painting and Sculpture 1961–1963_ (London: Phaidon, 2002): 38, 39, 41, 42, 43–50, 51–67, 76–79, 80, 82, 85, 86–102, 185–186, 190, 201–210, and 211–216.
2. Warhol's first pop art exhibit, in August 1962 at the Ferus Gallery in Los Angeles, was devoted exclusively to the soup can paintings. Two months later, _The New Realists_ , a group exhibition at New York's Sidney Janis Gallery, included four works by Warhol, two of them soup can paintings. In Warhol's first New York solo pop show, that November at the Stable Gallery, the celebrity paintings were more prominent, but large soda bottle and coffee can paintings were included, as well as a Coca-Cola matchbook cover painting. Warhol's only major exhibition in 1963 was a group of Elvis paintings at the Ferus Gallery. In January 1964 he showed his Disaster paintings at the Galerie Ileana Sonnabend in Paris, and that April, in his second New York solo show, he debuted the Brillo, Campbell's Soup, and Heinz boxes at the Stable Gallery.
3. For an example of the trickle-down argument, see Charles C. Parlin, _National Magazines as Advertising Media_ (Philadelphia: Curtis Publishing Company, 1931), 68–69.
4. "Who Needs National Brands?" Macfadden advertisement, _New York Times_ , April 11, 1962, 88. In what follows, I will use "private" and "generic" interchangeably.
5. Vance Packard, _The Hidden Persuaders_ (Brooklyn, NY: Ig Publishing, 2007), 124.
6. Louis Cheskin (Color Research Institute), paraphrased in Packard, _Hidden Persuaders_ , 124.
7. "Who Needs National Brands?" 88.
8. Michael Perry, "The Brand: Vehicle for Value in a Changing Marketplace," Advertising Association, President's Lecture, July 7, 1994, London; cited in Yiannis Gabriel and Tim Lang, _The Unmanageable Consumer_ (London: Sage, 1995), 36.
9. Thomas Frank, in _The Conquest of Cool: Business Culture, Counterculture, and the Rise of Hip Consumerism_ (Chicago: University of Chicago Press, 1997), compellingly argues that the countercultural innovations of the 1960s were welcomed and intensified by the fashion and advertising industries (see 8–9, 32–33), a dynamic in which Warhol certainly participated.
10. Thomas Crow's scholarship is very much an exception here. See also Mark Francis, "No There There or Horror Vacui: Andy Warhol's Installations," in _Andy Warhol: Paintings 1960–1986_ , ed. Martin Schwander (Stuttgart: Hatje, 1995), 67; John Roberts, "Warhol's 'Factory': Painting and the Mass-Cultural Spectator," in _Varieties of Modernism_ , ed. Paul Wood (New Haven, CT: Yale University Press, 2004), 357; and John J. Curley, _A Conspiracy of Images: Andy Warhol, Gerhard Richter, and the Art of the Cold War_ (New Haven, CT: Yale University Press, 2013).
11. Arthur Danto, _Beyond the Brillo Box: The Visual Arts in Post-Historical Perspective_ (New York: Farrar Straus Giroux, 1992), 41. See also Michael J. Golec, _The Brillo Box Archive: Aesthetics, Design, and Art_ (Hanover, NH: Dartmouth College Press, 2008).
12. Kirk Varnedoe, "Campbell's Soup Cans, 1962," in _Andy Warhol: Retrospective_ , ed. Heiner Bastian (Los Angeles: Museum of Contemporary Art, 2001), 42.
13. "Discussion," in Gary Garrels, ed., _The Work of Andy Warhol_ (Seattle: Bay Press, 1989), 127.
14. Mary Anne Staniszewski, "Capital Pictures," in _Post-Pop Art_ , ed. Paul Taylor (Cambridge, MA: MIT Press, 1989), 168.
15. Robert A. Nisbet, "The Decline and Fall of Social Class," _Pacific Sociological Review_ 2, no. 1 (Spring 1959): 11.
16. bell hooks, _Where We Stand: Class Matters_ (London: Routledge, 2000), 82.
17. Andy Warhol, _The Philosophy of Andy Warhol (From A to B and Back Again)_ (New York: Harcourt Brace Jovanovich, 1975), 100–101.
18. Warhol, _Philosophy_ , 100, 102.
19. Thomas Crow, "Saturday Disasters: Trace and Reference in Early Warhol," in _Modern Art in the Common Culture_ (New Haven, CT: Yale University Press, 1996), 63.
20. "Two Tuna Sandwiches," _Newsweek_ , April 1, 1963, 76; reproduced in Frei and Printz, _Catalogue Raisonné_ , 1:369.
21. "The Slice-of-Cake School," _Time_ 79, no. 19 (May 11, 1962): 56.
22. Peter Selz, Henry Geldzahler, Hilton Kramer, Dore Ashton, Leo Steinberg and Stanley Kunitz, "A Symposium on Pop Art," _Arts_ 37, no. 7 (April 1963): 38–9.
23. Barbara Rose, "Pop Art at the Guggenheim," _Art International_ 7, no. 5 (May 25, 1963): 22.
24. Wealthier shoppers tended to use full-service groceries, where a clerk would help the customer select her goods and then transfer them from the shelves to the counter; see Louis P. Bucklin, "Competitive Impact of a New Supermarket," _Journal of Marketing Research_ 4, no. 4 (November 1967): 360.
25. Selz et al., "Symposium on Pop Art," 42.
26. As Pierre D. Martineau argued in 1957, "The most important function of retail advertising today, when prices and quality have become so standard, is to permit the shopper to make social class identification" (Martineau, "The Pattern of Social Classes," in _Marketing's Role in Scientific Management_ , ed. Robert L. Clewett [Chicago: American Marketing Association, 1957], 242).
27. William D. Tyler, "The Image, the Brand, and the Consumer," _Journal of Marketing_ 22, no. 2 (October 1957): 164–65 (emphasis in original).
28. Packard, _Hidden Persuaders_ , 34.
29. For more on this film's relevance to Warhol's work, see Curley, _Conspiracy of Images_ , 133–34.
30. Brett Gorvy, quoted in "Release: Andy Warhol's _Coca-Cola_ /The Icon of American Pop Art," Christie's, November 11, 2013, accessed May 5, 2015, <http://www.christies.com/about/press-center/releases/pressrelease.aspx>.
31. Dwight Macdonald, "Masscult and Midcult," in _Against the American Grain_ (New York: Da Capo, 1983), 34. John Updike perpetuated this dim view of working-class grocery-store shoppers—these "sheep" or "houseslaves"—in a 1961 short story when his narrator claimed, "I bet you could set off dynamite in an A & P and the people would by and large keep reaching and checking oatmeal off their lists and muttering 'Let me see, there was a third thing, began with A, asparagus, no, ah, yes, applesauce!' or whatever it is they do mutter" (Updike, "A & P," _New Yorker_ , July 22, 1961, 22).
32. I have not been able to locate any discussions of the working class–targeted brand image advertising discussed in this chapter in contemporary scholarship. As Victoria de Grazia and Lizabeth Cohen have argued, "Insofar as consumption was about class, it appeared to be only about the bourgeoisie or the false consciousness of workers succumbing to commodity fetishism and class envy" (de Grazia and Cohen, "Introduction: Class and Consumption," _International Labor and Working-Class History_ 55 [Spring 1999]: 1). For more on Macfadden, see Ann Fabian, "Making a Commodity of Truth: Speculations on the Career of Bernarr Macfadden," _American Literary History_ 5, no. 1 (Spring 1993): 51–76; and Shelley Nickles, "More Is Better: Mass Consumption, Gender, and Class Identity in Postwar America," _American Quarterly_ 54, no. 4 (December 2002): 581–622.
33. Carl Spielvogel, "Battle of Brands Growing Fiercer," _New York Times_ , November 11, 1956, 185.
34. A & P Supermarket advertisement, _Chicago Daily Tribune_ , September 9, 1956, 27
35. Spielvogel, "Battle of Brands," 185.
36. "Who Needs National Brands?" 88.
37. Fred Farrar, "Store Display Ad Group Eyes Sunny Trend," _Chicago Tribune_ , October 14, 1963, C8.
38. Peter Bart, "Advertising: Competing with House Brands," _New York Times_ , November 12, 1962, 45. The president of Compton Advertising was cited in the article as claiming that, in one major chain store, generic labels accounted for 85 percent of orange juice sales and 33 percent of instant coffee and light-duty detergent sales.
39. Peter Bart, "Advertising: Blue Collars and Brand Names," _New York Times_ , August 14, 1961, 34.
40. Bart, "Blue Collars and Brand Names," 34; Bart is paraphrasing and quoting Robert L. Young, vice president and advertising director of Macfadden.
41. "Surging ahead," Macfadden advertisement, _New York Times_ , May 15, 1961, 32; _Wall Street Journal_ , 11; _Chicago Tribune_ , C6 (emphasis in original).
42. "Can Advertising _Block_ Sales?" Macfadden advertisement, _New York Times_ , December 20, 1961, 68.
43. "Which Half of the Market Needs National Brands?" Macfadden advertisement, _New York Times_ , May 16, 1962, 84.
44. "Can Advertising _Block_ Sales?" 68.
45. Beverley Skeggs, _Formations of Class and Gender: Becoming Respectable_ (London: Sage, 1997), 7; "The woman who packs this pail will decide the future of national brands!," Macfadden advertisement, _New York Times_ , August 8, 1961, 60.
46. "Who Needs National Brands?" 88; "Can Advertising _Block_ Sales?" 68.
47. Lee Rainwater, Richard P. Coleman, and Gerald Handel, _Workingman's Wife: Her Personality, World, and Life Style_ (New York: Oceana Publications, 1959), 209.
48. Rainwater, Coleman, and Handel, _Workingman's Wife_ , 166.
49. Peter Bart, "Advertising: A Shift for Dristan," _New York Times_ , June 27, 1962, 51.
50. "Who Needs National Brands?"
51. Bart, "Shift for Dristan," 51. For more on Warhol's taste for gold paint, see Thomas Crow, _The Long March of Pop: Art, Music, and Design, 1930–1995_ (New Haven, CT: Yale University Press, 2014), 163–67.
52. Robert Lewis Taylor, "Profiles: Physical Culture: III—Physician, Heal Thyself!" _New Yorker_ , October 28, 1950, 42. These black-and-white advertisements also dominated Macfadden's racier titles, like _True Romance_. When brand image advertising appeared in these titles, it focused primarily on "feminine hygiene" products—Tampax, Kotex—that were not featured in _True Story_ ; see, for example, the inner cover advertisements in the June 1962 issue of _True Romance_.
53. Clementine Paddleford, "What's in the Name on the Shelf?" _Los Angeles Times_ , May 7, 1961, E25.
54. This is true of one of the two early _Coca-Cola_ works, _Peach Halves_ , and the Mönchengladbach-type _Campbell's Soup Cans_. See Frei and Printz, _Catalogue Raisonné_ , vol. 1, figures 23–4, 27, 28, 60, 63.
55. Jerry Doolittle, "The Working Class Rises to an Olfactory Defense," _Washington Post, Times Herald_ , October 21, 1962, E2.
56. Bob Colacello, _Holy Terror: Andy Warhol Close Up_ (New York: HarperCollins, 1990), 23 (emphasis in original).
57. David Bourdon, "Warhol Interviews Bourdon" (unpublished manuscript, 1962–63), in _I'll Be Your Mirror: The Selected Andy Warhol Interviews_ , ed. Kenneth Goldsmith, Reva Wolf, and Wayne Koestenbaum (New York: Carroll & Graf, 2004), 8. Contemporaries like Billy Al Bengston, Peter Blake, Derek Boshier, Rudy Burckhardt, Vija Celmins, Allan D'Arcangelo, William Eggleston, Marisol Escobar, Lee Friedlander, Dorothy Grebenak, Raymond Hains, Richard Hamilton, Jann Haworth, David Hockney, Ray Johnson, William Klein, Eduardo Paolozzi, Larry Rivers, Glauco Rodrigues, Mimmo Rotella, Ed Ruscha, Ushio Shinohara, Jacques de la Villeglé, Wolf Vostell, Tom Wesselmann, and Garry Winogrand included brand images in their art—variously through appropriation, reproduction, and depiction—but these images were always integrated into a larger whole, brimming with metaphor or facture or both.
58. G. R. Swenson, "What Is Pop Art? Part II," _ARTnews_ 62, no. 10 (February 1964): 64.
59. Varnedoe, "Campbell's Soup Cans," 43. Warhol almost never worked with readymade appropriations during the 1960s. Exceptions are the Stockholm-type _Brillo Boxes_ ; the _You're In_ series, a group of silver painted Coca-Cola bottles from 1967; and _Bomb '67_ , a silver-sprayed air force practice bomb from the same year. In all three of these cases, Warhol introduced a gap between the commodity and the artwork—through paint, or through missed expectations (as in the Stockholm _Brillo Boxes_ , where the cardboard objects substitute for painted objects).
60. Frei and Printz, _Catalogue Raisonné_ , 1:63, 68. The undated sources for these projected images are provided in the catalogue raisonné; I have located their exact publication dates in _Life_ magazine: September 8, 1961, 59, and January 12, 1962, 51.
61. Quoted by Samuel Adams Green, _Andy Warhol_ (Philadelphia: Philadelphia Institute of Contemporary Art, 1965), n.p.
62. Paddleford, "What's in the Name on the Shelf?" E25; "Can Advertising _Block_ Sales?" 68.
63. Wilbur Schramm, Jack Lyle, and Edwin B. Parker, _Television in the Lives of Our Children_ (Stanford, CA: Stanford University Press, 1961), 110.
64. Sidney J. Levy, "Social Class and Consumer Behavior," in _On Knowing the Consumer_ , ed. J. Newman (New York: John Wiley and Sons, 1966), 159.
65. "Ted Carey, New York, 16 October 1978," in Patrick S. Smith, _Andy Warhol's Art and Films_ (Ann Arbor, MI: UMI Research Press, 1986), 262.
66. Quoted in John Heilpern, "The Fantasy World of Warhol," _Observer_ , June 12, 1966, 12. Contemporary documentation confirms this claim; see Nat Finkelstein, _Andy Warhol: The Factory Years_ (Edinburgh: Canongate Books, 1999), 4–5.
67. According to Bourdon, the paintings "have plain gold circles because the giant figure was too difficult to stencil" ("David Bourdon," in John Wilcock, _The Autobiography and Sex Life of Andy Warhol_ [New York: Other Scenes, 1971], n.p.).
68. See Georg Frei and Neil Printz, eds., _The Andy Warhol Catalogue Raisonné_ , vol. 2B, _Painting and Sculpture 1964–1965_ (London: Phaidon, 2004), 190–205.
69. The Martinson Coffee brand image also contains an emblem, although it is not framed in "gold" and does not explicitly denote quality.
70. Bart, "Blue Collars and Brand Names," 34.
71. Clarence E. Eldridge, "Advertising Effectiveness: How Can It Be Measured," _Journal of Marketing_ 22, no. 3 (January 1958): 249 (emphasis in original). See also "Dan Graham Interviewed by Ludger Gerdes," in _Two-Way Mirror Power: Selected Writings by Dan Graham on His Art_ , ed. Alexander Alberro (Cambridge, MA: MIT Press, 1999), 71, where Graham surmises "that these fascist structures had actually infiltrated mass psychological subliminal consciousness through advertising. I think Pop art was alluding to this."
72. Eldridge, "Advertising Effectiveness," 250. Noting that Campbell's purchased prime advertising in _Life_ magazine during the 1950s and early 1960s, Curley has argued that "the soup can was a historical actor in these pages," connoting "safety and security" (Curley, _Conspiracy of Images_ , 127, 63).
73. _Coca-Cola Annual Report to Stockholders, 1956_ (New York: Coca-Cola Company, 1956), 7.
74. Campbell's Soup advertisement, _Life_ , December 28, 1959, 70.
75. Mark Pendergrast, _For God, Country and Coca-Cola_ (New York: Scribner's, 1993), 259.
76. "To a startling degree, prole America is about sweet . . . [and] you could probably draw a trustworthy class line based wholly on the amount of sugar consumed by a family" (Paul Fussell, _Class: A Guide Through the American Status System_ [New York: Touchstone, 1983], 100). Warhol never shied away from his vulgar taste for sugar, tying it to his own prolonged childishness; see Warhol, _Philosophy_ , 103.
77. _Coca-Cola Annual Report to Stockholders, 1957_ (New York: Coca-Cola Company, 1957), 7.
78. The ghost lettering is barely visible under the cropped "Coca-Cola" script.
79. Mark Pendergrast, _Uncommon Grounds: The History of Coffee and How It Transformed Our World_ (New York: Basic Books, 1999), 198, 240–41.
80. "Can Advertising _Block_ Sales," 68.
81. Jonathan Flatley, "Warhol Gives Good Face: Publicity and the Politics of Prosopopoeia," in _Pop Out: Queer Warhol_ , ed. Jennifer Doyle, Flatley, and José Esteban Muñoz (Durham, NC: Duke University Press, 1996), 119; "Two Tuna Sandwiches," 76.
82. "The Quality Revolution—New Hope for National Brands," Macfadden advertisement, _New York Times_ , March 7, 1962, 72.
83. Peter Schjeldahl, "Warhol and Class Content" ( _Art in America_ 68, no. 5 [May 1980]: 112–19), reprinted in _The Hydrogen Jukebox: Selected Writings of Peter Schjeldahl, 1978–1990_ (Berkeley: University of California Press, 1993), 47; "Can Advertising _Block_ Sales?" 68.
84. "Quality Revolution," 72.
85. Aldous Huxley, "Vulgarity in Literature," in _Complete Essays_ , vol. 3, _1930–1935_ , ed. Robert S. Baker and James Sexton (Chicago: Ivan Dee, 2001), 23.
86. Arthur Danto, "Replies to Essays," in _Danto and His Critics_ , ed. Mark Rollins, 2nd ed. (Malden: Wiley-Blackwell, 2012), 288.
87. Danto, "Learning to Live with Pluralism," in _Beyond the Brillo Box_ , 225.
88. Danto, "The Artworld," _Journal of Philosophy_ 61, no. 19 (October 15, 1964): 574–76.
89. John O'Connor and Benjamin Liu, _Unseen Warhol_ (New York: Rizzoli, 1996), 35.
90. Whitney Davis, "The Aesthetics of Indiscernibles," unpublished manuscript, 3. See Georg Frei and Neil Printz, eds., _The Andy Warhol Catalogue Raisonné_ , vol. 2B, _Painting and Sculpture 1964–1965_ (London: Phaidon, 2004), 53–101. Warhol occasionally broke this pattern and exhibited the commercial boxes themselves as a substitute for the painted boxes, as in the Stockholm type boxes (78).
91. David Antin, "Warhol: The Silver Tenement," _ARTnews_ 65, no. 4 (Summer 1966): 59.
92. George Klauber, cited in Victor Bockris, _Warhol: The Biography_ (New York: Da Capo, 2003), 90.
93. For more on Warhol as a fan, see Staniszewski, "Capital Pictures," 166; Carter Ratcliff, "Starlust: Andy's Photos," _Art in America_ 68, no. 5 (May 1980): 122.
94. Schramm, Lyle, and Parker, _Television_ , 47, 110; Martineau, "Pattern of Social Classes," 246; Malvina Lindsay, "Norms of Middle Class Slipping," _Washington Post and Times Herald_ , May 14, 1959, A22.
95. Aaron A. Fox, _Real Country: Music and Language in Working-Class Culture_ (Durham, NC: Duke University Press, 2004), 250. Although some of the discourses of marketing during this period conceptualized their working-class targets as women or "wives," other prominent versions, like those of Tyler and Martineau, did not. A 1957 study entitled "The American Male . . . On Ascendancy as Force in Food Purchases," claimed that 40 percent of food shoppers were male, and that men were "much more prone to impulse buying than women" (June Owen, "Food: Did I Buy That?" _New York Times_ , May 31, 1957, 32). For the gendered structures underlying the _Race Riots_ , see Anne M. Wagner, "Andy Warhol's Patriotism," in _A House Divided: American Art Since 1955_ (Berkeley: University of California Press, 2012), 25–45.
96. Grace Glueck, "Art Notes: Boom?" _New York Times_ , May 10, 1964, X19; Danto, "Artworld," 581.
97. Danto, "The Philosopher as Andy Warhol," in _The Andy Warhol Museum_ , ed. Fannia Weingartner (New York: Distributed Art Publishers, 1994), 80.
98. Harold H. Kassarjian, "The Negro and American Advertising, 1946–1965," _Journal of Marketing Research_ 6, no. 1 (February 1969): 36, 39. See also Marcus Alexis, "Pathways to the Negro Market," _Journal of Negro Education_ 28, no. 2 (Spring 1959): 115, 121.
99. "The Forgotten 15,000,000 . . . Three Years Later," _Sponsor_ 6 (July 1952): 76; cited in Alexis, "Pathways," 121.
100. Raymond A. Bauer, Scott M. Cunningham, and Lawrence H. Wortzel, "The Marketing Dilemma of Negroes," _Journal of Marketing_ 29, no. 3 (July 1965): 2.
101. Laurie A. Rodrigues, "'SAMO© as an Escape Clause': Jean-Michel Basquiat's Engagement with a Commodified American Africanism," _Journal of American Studies_ 45, no. 2 (May 2011): 227–43. See also José Esteban Muñoz, "Famous and Dandy like B. 'n' Andy: Race, Pop, and Basquiat," in _Pop Out: Queer Warhol_ , ed. Jennifer Doyle, Jonathan Flatley, and Muñoz (Durham, NC: Duke University Press, 1996), 144–79.
102. Lauren Berlant, _Cruel Optimism_ (Durham, NC: Duke University Press, 2011), 188.
103. Quoted in Heilpern, "Fantasy World of Warhol," 12.
104. For readings of Benjamin's relevance to Warhol, see Rainer Crone, _Andy Warhol_ , trans. John William Gabriel (London: Thames & Hudson, 1970); Andreas Huyssen, "The Cultural Politics of Pop," in Taylor, _Post-Pop Art_ , 45–77; Mark Francis, "Still Life: Andy Warhol's Photography as a Form of Metaphysics," in _Andy Warhol: Photography_ (Pittsburgh: Warhol Museum, 1997), 19–24; and Matthew Tinkcom, "Warhol's Camp," in _Who Is Andy Warhol?_ , ed. Colin McCabe, Mark Francis, and Peter Wollen (London: British Film Institute, 1997), 107–16.
105. As _True Story_ put it, in the same year Benjamin's "The Author as Producer" was first delivered (trans. E. Jephcott, in _Walter Benjamin: Selected Writings, 1931–1934_ , ed. Michael W. Jennings, Howard Eiland, and Gary Smith [Cambridge, MA: Belknap Press, 1999] 768–82), "Every man and woman has lived at least one big story which has that ring of truth for which authors of fiction strive" ( _True Story_ , February 1934, 61; cited in Fabian, "Making a Commodity," 63).
106. See Donald C. Bacon, "Camera Makers Automate, Simplify Picture Snapping, See New Lines as Spur to Sales," _Wall Street Journal_ , March 30, 1959, 5; Thomas O'Toole, "Photography Industry Expects Sales To Hit High in '58 Despite Recession," _Wall Street Journal_ , April 8, 1958, 4.
107. Elenore Lester, "On the Eve of Destruction, What Was Andy Warhol's Gang Up To?" _Eye_ , August 1968, 95. Gerard Malanga recalled that Warhol "wanted everything to be totally easy, like _push the button and let it roll_ " (quoted in Steven Watson, _Factory Made: Warhol and the Sixties_ [New York: Pantheon, 2003], 97; emphasis in original).
108. Guy Debord, _Society of the Spectacle_ , trans. Donald Nicholson-Smith (New York: Zone Books, 2006), 153.
109. Gretchen Berg, "Nothing to Lose: Interview with Andy Warhol," _Cahiers du Cinema in English_ , May 1967, 43.
110. Sterling McIlhenny and Peter Ray, "Inside Andy Warhol" (1966), in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 98.
111. Debord, _Society of the Spectacle_ , 18.
112. Anonymous "lady," quoted in Robert Mazzocco, "aaaaaa . . . ," _New York Review of Books_ , April 24, 1969, accessed November 1, 2016, <http://www.nybooks.com>.
113. Leon Kraushar, quoted in William K. Zinsser, _Pop Goes America_ (New York: Harper & Row, 1966), 28. One crucial archive for thinking about class and its spectacular consumption during this period is the television program _The Beverly Hillbillies_ (1962–1971). In the opening scene of an episode that first aired on October 23, 1963, Jethro eats corn flakes while Granny Clampett outlines her menu for the night's dinner guests: "Possum stew! Grits and jowls! Some corn pone, and some pickled crawdads!" It turns out that the guests, the wealthy Fenwicks, have been mistaken by the Clampetts as a local charity case. The Fenwicks, meanwhile, are eager to appropriate the Clampetts' signature look: a poverty-stricken "style" they think the Clampetts have intentionally affected. As Cynthia Fenwick tells her mother, "Don't you understand mummy? The latest thing is to look . . . basic and understated! . . . let's not be the last millionaires in Beverly Hills to get the new look—the Clampett Look!" As in Warhol's brand image artworks, déclassé mores are sold as styles to the elite, and the televised spectacle of this strange transaction is then resold to a mass audience as a farce that minimizes class differences.
114. Angus Maguire, head of contemporary art at Bloomsbury Auctions, cited in Dalya Alberge, "Move over Picasso: Mass Appeal Pushes Warhol to the Top of Art Market," _Times_ (London), April 30, 2007, accessed August 16, 2014, <http://www.thetimes.co.uk/tto/arts/visualarts/article2422133.ece>.
115. Adam Arvidsson, _Brands: Meaning and Value in Media Culture_ (London: Routledge, 2006), 93.
116. "New Coca-Cola Museum to Host Andy Warhol's Coke Art," May 2, 2007, accessed August 3, 2014, <http://usatoday30.usatoday.com>. The twists and turns of this history offer an interesting addendum to the arguments advanced in Frank, _Conquest of Cool_. Although Coke had been quick to threaten legal action when Warhol violated its trademark, it ultimately embraced Warhol as an innovator and by 2007 could exhibit this threat as a badge of honor.
117. Quoted in Ruth La Ferla, "The Selling of St. Andy," _New York Times_ , October 26, 2006, G5; cited in Mark Rosenthal, "Dialogues with Warhol," in Rosenthal, Marla Prather, Ian Alteveer, and Rebecca Lowery, _Regarding Warhol: Sixty Artists, Fifty Years_ (New York: Metropolitan Museum of Art, 2012), 136.
118. Tyler, "Image," 165.
119. Danto, _Beyond the Brillo Box_ , 41.
120. Tyler, "Image," 165.
121. "Quality Revolution," 72.
122. Pierre Macherey, _A Theory of Literary Production_ , trans. Geoffrey Wall (London: Routledge & Kegan Paul, 1978), 133 (emphasis in original).
123. "Ted Carey, 16 October 1978," 254.
124. "Ronnie Tavel," in Wilcock, _Autobiography_ , n.p. (emphasis in original).
125. Ta-Nehisi Coates, _Between the World and Me_ (New York: Spiegel & Grau, 2015), 111. See Warhol's drolly related "story-book fairy tale": "Halston arrived in white, with Bianca on his arm in white fur, with Dr. Giller in white, in a white limo, with a white driver" (Andy Warhol, _The Andy Warhol Diaries_ , ed. Pat Hackett [New York: Grand Central Publishing, 1989], 94).
126. Angela P. Harris, "Race and Essentialism in Feminist Legal Theory," _Stanford Law Review_ 42, no. 3 (February 1990): 597. For more on "Hunky" as an ethnic slur, connoting "the Central European immigrant industrial laboring class," and its relevance to Warhol, see Blake Stimson, _Citizen Warhol_ (London: Reaktion, 2014), 69–87.
127. Warhol, _Diaries_ , 161. Steve Rubell had dumped the garbage can of cash over Warhol at his birthday party at the 21 Club two days earlier. Warhol enthused that "it really was the best present" (160).
## Chapter Four
1. Jed Perl, "The Curse of Warholism," _New Republic_ , December 6, 2012, accessed May 11, 2015, <http://www.newrepublic.com>.
2. Lawrence W. Levine approvingly citing John Russell, in Levine, _Highbrow/Lowbrow: The Emergence of Cultural Hierarchy in America_ (Cambridge, MA: Harvard University Press, 1988), 247; Robert Motherwell, "On the Humanism of Abstraction: The Artist Speaks, 1970," in _The Writings of Robert Motherwell_ , ed. Dore Ashton and Joan Banach (Berkeley: University of California Press, 2007), 253.
3. Crispin Sartwell, "Andy Warhol and the Persistence of Modernism," _New York Times_ , June 19, 2013, accessed August 5, 2016, <http://opinionator.blogs.nytimes.com>.
4. T. J. Clark, "Origins of the Present Crisis," _New Left Review_ 2 (March/April 2000): 95–96.
5. "Can Advertising _Block_ Sales," Macfadden advertisement, _New York Times_ , December 20, 1961, 68.
6. Jacques Rancière, _Aisthesis: Scenes from the Aesthetic Regime of Art_ , trans. Zakir Paul (London: Verso, 2013), 262. Hal Foster has written that this element of Rancière's argument is not novel to art history (Foster, "What's the Problem with Critical Art?" _New Left Review_ 35, no. 19 [October 10, 2013]: 14–15). Thomas Crow has emphasized pop's egalitarianism without linking it to Rancière; see Crow, _The Long March of Pop: Art, Music, and Design, 1930–1995_ (New Haven, CT: Yale University Press, 2014), viii–ix.
7. Jacques Rancière, _Aesthetics and Its Discontents_ , trans. Steven Corcoran (Cambridge: Polity Press, 2009), 32. Rancière rejects both sides of the Greenbergian modernism/postmodernism dichotomy; see "A Politics of Aesthetic Indetermination: An Interview with Frank Ruda and Jan Voelker," trans. Jason E. Smith, in Jason E. Smith and Annette Weisser, eds., _Everything Is in Everything: Jacques Rancière between Intellectual Emancipation and Aesthetic Education_ (Pasadena, CA: Art Center Graduate Press, 2011), 18.
8. Thomas Crow, "Warhol among the Art Directors," in _Andy Warhol Enterprises_ , ed. Sarah Urist Green and Allison Unruh (Ostfildern: Hatje Cantz, 2010), 109n18.
9. Jean Baudrillard, _The Consumer Society: Myths and Structures_ (1970), trans. Chris Turner (Thousand Oaks, CA, 1998), 118.
10. Gilles Deleuze, _Difference and Repetition_ (1968), trans. Paul Patton (New York: Columbia University Press, 1994), 299. See also Michel Foucault, "Theatrum Philosophicum" (1970), in _Language, Counter-Memory, Practice: Selected Essays and Interviews_ , ed. Donald F. Bouchard, trans. Donald F. Bouchard and Sherry Simon (Ithaca, NY: Cornell University Press, 1977), 189.
11. Baudrillard, _Consumer Society_ , 121 (emphasis in original); Roland Barthes, "That Old Thing, Art" (1980), in _Post-Pop Art_ , ed. Paul Taylor (Cambridge, MA: MIT Press, 1989), 30 (emphasis in original). In 1988 Baudrillard would reverse his judgment, calling Warhol "the hero or anti-hero of modern art" (Baudrillard, "De la marchandise absolue" [1988], reprinted as "Absolute Merchandise," trans. David Britt, in _Andy Warhol: Paintings 1960–1986_ , ed. Martin Schwander [Stuttgart: Hatje, 1995], 20).
12. Fredric Jameson, _Postmodernism, or, The Cultural Logic of Late Capitalism_ (Durham, NC: Duke University Press, 1991), 175. Although this summary judgment is directed at "neofigurative painting today," Warhol is singled out earlier in the chapter as a key instigator in this tradition (158).
13. Jonathan D. Katz, "The Art of Code: Jasper Johns and Robert Rauschenberg," in _Significant Others: Creativity and Intimate Partnership_ , ed. Whitney Chadwick and Isabelle de Courtivron (London: Thames & Hudson, 1996), 205, 201–2. There are exceptions to this rule; see the exchange of letters between Hilton Kramer and Jasper Johns in "Month in Review," _Arts_ 33, no. 5 (February 1959): 49, and 33, no. 6 (March 1959): 7.
14. See Whitney Davis, _A General Theory of Visual Culture_ (Princeton, NJ: Princeton University Press, 2011), 175; I am grateful to Davis for pointing out the "puff" double-entendre.
15. Rudolf Arnheim, interview with John Jones, October 16, 1965, Archives of American Art, Washington, DC; quoted in Katz, "Passive Resistance: On the Success of Queer Artists in Cold War American Art," Queer Cultural Center, accessed August 10, 2014, <http://www.queerculturalcenter.org/Pages/KatzPages/KatzLimage.html>.
16. Vivian Gornick, "Pop Goes Homosexual," _Village Voice_ , April 7, 1966, accessed July 12, 2015, <http://www.villagevoice.com>; cited in Katz, "The Silent Camp: Queer Resistance and the Rise of Pop Art," in _Visions of a Future: Art and Art History in Changing Contexts_ , ed. Hans-Jörg Heusser and Kornelia Imesch (Zurich: Swiss Institute for Art Research, 2004), 156.
17. Susan Sontag, "Notes on Camp," in _Against Interpretation and Other Essays_ (New York: Octagon Books, 1978), 292. See also Gloria Steinem, "The Ins and Outs of Pop Culture," _Life_ , August 20, 1965, 84. An important exception to this rule is _Time_ magazine's infamous "Essay: The Homosexual in America," January 21, 1966, 40; see Gavin Butt, _Between You and Me: Queer Disclosures in the New York Art World, 1948–1963_ (Durham, NC: Duke University Press, 2006), 41–43.
18. Steinem, "Ins and Outs," 75. The exceptions in Warhol's oeuvre are works from 1961 like the _Typewriter_ , _Toilet_ , and _Telephone_ paintings, all of which derive from recognizably dated advertisements and have a distinctly campy tone; see Sara Doris, _Pop Art and the Contest over American Culture_ (Cambridge: Cambridge University Press, 2007), 10. Warhol seems to have mostly reserved this interest in the outdated for his private collecting; see David Bourdon, "Andy Warhol: 1928–1987," _Art in America_ 75, no. 5 (May 1987): 139.
19. Maggie Nelson, _Women, the New York School, and Other True Abstractions_ (Iowa City: University of Iowa Press, 2007), 52. T. J. Clark has remarked that, "apart from Greenberg, the strongest early readings of Pollock's work . . . all came from gay men," and that even Greenberg's criticism was charged with "erotic hero worship" (Clark, _Farewell to an Idea: Episodes from a History of Modernism_ [New Haven, CT: Yale University Press, 1999], 394).
20. Walter Hopps, quoted in Jean Stein, _Edie: An American Biography_ (New York: Alfred A. Knopf, 1982), 198.
21. Friedel Dzubas, "Is There a New Academy, Part II," _ARTnews_ 58, no. 6 (September 1959): 37, 59.
22. Note dated March 28, 1952, William Chapin Seitz papers, 1934–1995, Archives of American Art, Smithsonian Institution, cited in Valerie Hellstein, _Grounding the Social Aesthetics of Abstract Expressionism: A New Intellectual History of The Club_ , diss., Stony Brook University, 2010, 1; Jack Tworkov, "Is There a New Academy, Part II," _ARTnews_ 58, no. 6 (September 1959): 38.
23. Irving Sandler, "The New Cool-Art," _Art in America_ 53, no. 1 (February 1965): 100.
24. Larry Rivers, "New York in the Eighties," _New Criterion_ 4, no. 11 (Summer 1986): 50; quoted in Bradford R. Collins, "Life Magazine and the Abstract Expressionists, 1948–51: A Historiographic Study of a Late Bohemian Enterprise," _Art Bulletin_ 73, no. 2 (June 1991): 297.
25. Ruth Kligman, "My Love Affair with Jackson Pollock," _New York Magazine_ , October 29, 1973, 52.
26. Quoted in Stein, _Edie_ , 198.
27. De Kooning, quoted in Mark Stevens and Annalyn Swan, _De Kooning: An American Master_ (New York: Alfred A. Knopf, 2004), 285.
28. Clark, _Farewell to an Idea_ , 401.
29. T. J. Clark, _The Painting of Modern Life_ , rev. ed. (Princeton, NJ: Princeton University Press, 1999), 156.
30. Clark, _Farewell to an Idea_ , 389. For a corroborative reading, see Bonnie H. Erickson, "Culture, Class, and Connections," _American Journal of Sociology_ 102, no. 1 (July 1996): 221.
31. Peter Stallybrass and Allon White, _The Politics and Poetics of Transgression_ (Ithaca, NY: Cornell University Press, 1986), 191.
32. Thomas B. Hess, _Willem de Kooning_ (New York: Museum of Modern Art, 1968), 6; see also 76.
33. Willem de Kooning, "Symposium: 'What Art Means to Me'" (1951), in _Theories of Modern Art: A Source Book by Artists and Critics_ , ed. Herschel B. Chipp (Berkeley: University of California Press, 1968), 560. See also Donald Kuspit, "The Unveiling of Venus: de Kooning's Melodrama of Vulgarity," _Vanguard_ , September 1984, 19–23. De Kooning claimed that _Woman I_ was inspired by shoppers at "the bargain center . . . how greedy and nasty they are . . . just tearing the bargains to pieces . . ." (quoted in Stevens and Swan, _De Kooning_ , 338).
34. David Smith, "Aesthetics, the Artist and the Audience" (1952), in _Art in Theory, 1900–1990: An Anthology of Changing Ideas_ , ed. Charles Harrison and Paul Wood (Oxford: Blackwell, 1999), 578.
35. "Oral history interview with Robert Motherwell, 1971 Nov. 24–1974 May 1," Archives of American Art, accessed June 15, 2015, <http://www.aaa.si.edu>.
36. Hilton Kramer, "Editorial," _Arts_ 34, no. 4 (January 1960): 13.
37. Charlotte Willard, "Market Letter" _Art in America_ 48, no. 1 (January 1960): 4.
38. Harold Rosenberg, "The American Action Painters," in _The Tradition of the New_ (New York: Horizon Press, 1959), 35. See also Crow, _Long March of Pop_ , vii.
39. Helen Frankenthaler, "Is There a New Academy, Part I," _ARTnews_ 58, no. 5 (Summer 1959): 34.
40. Andy Warhol and Pat Hackett, _Popism: The Warhol '60s_ (New York: Harcourt Brace Jovanovich, 1980), 3.
41. Harold Rosenberg, "Jasper Johns: Things the Mind Already Knows," _Vogue_ , February 1, 1964, 175.
42. Quoted in Lawrence Alloway, "'The World Is a Painting': Robert Rauschenberg," _Vogue_ , October 15, 1965, 157.
43. Pierre Bourdieu, _Distinction: A Social Critique of the Judgement of Taste_ , trans. Richard Nice (Cambridge, MA: Harvard University Press, 1984), 34. Dave Hickey's account of Warhol's egalitarianism, in which audiences recognized that Warhol "knew what was right and [was] doing it wrong for reasons of style or conviction," is vulnerable to Bourdieu's critique (Hickey, "The Importance of Remembering Andy," in _Robert Lehman Lectures in Contemporary Art_ , ed. Lynne Cooke, Karen Kelly, and Bettina Funcke [New York: Dia Art Foundation, 2004], 61).
44. Wilbur Schramm, Jack Lyle, and Edwin B. Parker, _Television in the Lives of Our Children_ (Stanford, CA: Stanford University Press, 1961), 110.
45. Bruce Glaser, "Oldenburg, Lichtenstein, Warhol: A Discussion," _Artforum_ 4, no. 6 (February 1966): 21, 22.
46. G. R. Swenson, "What Is Pop Art? Part I," _ARTnews_ 62, no. 7 (November 1963): 26.
47. Lane Slate, "USA Artists: Andy Warhol and Roy Lichtenstein" (1966), reprinted in _I'll Be Your Mirror: The Selected Andy Warhol Interviews_ , ed. Kenneth Goldsmith, Reva Wolf, and Wayne Koestenbaum (New York: Carroll & Graf, 2004), 81.
48. John Giorno, "Andy Warhol Interviewed by a Poet" (unpublished manuscript, 1963), in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 25.
49. "Andy Warhol: Interviewed by Gerald Malanga" (1963; _Kulchur_ 16 [Winter 1964–65]), in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 49. "Oh," Warhol told an interviewer in 1978, "I never got 'out of' commercial art" (Frederick Schruers, "Andy Warhol: Why Not?" _Globe and Mail_ , November 8, 1978, F10).
50. Artists affiliated with Dada had rejected the inviolability and autonomy of art but had avoided reducing it to a purely monetary endeavor; see Marcel Duchamp, "Apropos of 'Readymades,'" reprinted in _Theories and Documents of Contemporary Art_ , ed. Kristine Stiles and Peter Selz (Berkeley: University of California Press, 1996), 820.
51. Sidney J. Levy, "Social Class and Consumer Behavior," in _On Knowing the Consumer_ , ed. J. Newman (New York: John Wiley and Sons, 1966), 159.
52. Robert Reilly, "Untitled Interview," unpublished manuscript, Andy Warhol Archives, Pittsburgh, Spring 1966, reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 111.
53. Harold Rosenberg, "Warhol: Art's Other Self," in _Art on the Edge: Creators and Situations_ (New York: Macmillan, 1975), 106.
54. John Canaday, "Art in a Democracy, or Mass Audience vs. Select Product," _New York Times_ , June 3, 1962, 130.
55. John Canaday, "Action Evokes the Touchy and Totally Elusive Question of Obscenity in Art," _New York Times_ , October 14, 1965, 52.
56. Stuart Zane Charmé, _Vulgarity and Authenticity: Dimensions of Otherness in the World of Jean-Paul Sartre_ (Amherst: University of Massachusetts Press, 1991), 20.
57. As Ted Carey put it, in a discussion of Warhol's reluctance to exhibit his _Cock Drawings_ , "He's afraid to be himself and to do something personal at the risk of losing universal audience ("Ted Carey, New York, 16 October 1978," in Patrick S. Smith, _Andy Warhol's Art and Films_ [Ann Arbor, MI: UMI Research Press, 1986], 261).
58. Frank O'Hara, "Ode to Joy," in _The Collected Poems of Frank O'Hara_ , ed. Donald Allen (New York: Alfred A. Knopf, 1971), 281.
59. Fabio Cleto, "Introduction: Queering the Camp," in _Camp: Queer Aesthetics and the Performing Subject: A Reader_ , ed. Cleto (Ann Arbor: University of Michigan Press, 1999), 31. For an alternate reading of the relationship between O'Hara, pop, and camp, see Hazel Smith, _Hyperscapes in the Poetry of Frank O'Hara: Difference/Homosexuality/Topography_ (Liverpool: Liverpool University Press, 2000), 166–94.
60. O'Hara, "My Heart," in _Collected Poems_ , 231.
61. For critical responses, see S. P., "About Art and Artists," _New York Times_ , May 22, 1954, 13, and S. P., "About Art and Artists," _New York Times_ , March 3, 1956, 37. Warhol's drawings of boys seem to have been twice rejected by the Tanager Gallery; see Tony Scherman and David Dalton, _Pop: The Genius of Andy Warhol_ (New York: Harper, 2009), 33.
62. Warhol and Hackett, _Popism_ , 11–12. For O'Hara's response to Warhol, see Reva Wolf, _Andy Warhol, Poetry, and Gossip in the 1960s_ (Chicago: University of Chicago Press, 1997), 18–31. Warhol seems to have attempted to emphasize a class bond with Rauschenberg through his 1962 portraits of the artist, including _Let Us Now Praise Famous Men_ , which, as Neil Printz and Georg Frei point out, "established a link between [Rauschenberg's] working-class, Depression-era origins and Warhol's" (Frei and Printz, eds., _The Andy Warhol Catalogue Raisonné_ , vol. 1, _Painting and Sculpture 1961–1963_ [London: Phaidon, 2002], 266).
63. Benjamin H. D. Buchloh, "Andy Warhol's One-Dimensional Art: 1956–1966," in _Neo-Avantgarde and Culture Industry: Essays on European and American Art from 1955 to 1975_ (Cambridge, MA: MIT Press, 2000), 499.
64. Selden Rodman, _Conversations with Artists_ (New York: Capricorn, 1961), 105; quoted in Marcia Brennan, _Modernism's Masculine Subjects: Matisse, the New York School, and Post-Painterly Abstraction_ (Cambridge, MA: MIT Press, 2004), 47. See also Donald Kuspit, "Art Is Dead, Long Live Aesthetic Management," in _Redeeming Art: Critical Reveries_ , ed. Mark Van Proyen (New York: Allworth Press, 2000), 137.
65. "Hollywood Topic A-Plus: Whole Town's Talking about Marilyn Monroe," _Life_ , April 7, 1952, 104, 101.
66. Malvina Lindsay, "Norms of Middle Class Slipping," _Washington Post and Times Herald_ , May 14, 1959, A22.
67. Simon Frith, _Sound Effects: Youth, Leisure, and the Politics of Rock 'n' Roll_ (New York: Pantheon, 1981), 64.
68. Rodman, _Conversations with Artists_ , 105, quoted in Brennan, _Modernism's Masculine_ _Subjects_ , 47; Schramm, Lyle, and Parker, _Television_ , 110. De Kooning reportedly confronted Warhol at a party: "You're a killer of art, you're a killer of beauty, and you're even a killer of laugher. I can't bear your work!" (Victor Bockris, _Warhol: The Biography_ [New York: Da Capo, 2003], 320).
69. "Machine-made goods of daily use are often admired and preferred precisely on account of their excessive perfection by the vulgar and the underbred who have not given due thought to the punctilios of elegant consumption" (Thorstein Veblen, _The Theory of the Leisure Class_ [Oxford: Oxford University Press, 2007], 106).
70. Jacques Rancière, Artemy Magun, Dmitry Vilensky and Alexandr Skidan, "You Can't Anticipate Explosions: Jacques Rancière in Conversation with Chto Delat," _Rethinking Marxism_ 20, no. 3 (July 2008): 404. Rancière's brief discussions of Warhol acknowledge his importance to this history, while worrying (as I do) about the supposed "criticality" of Warhol's innovations and the ways in which they have been recommodified. Rancière singles out "Warhol's introduction of soup tins and Brillo soap boxes into the museum" for having "worked to denounce great art's claims to seclusion" (Rancière, _Aesthetics and Its Discontents_ , 51). But he also describes "the Pop era" as "a time when the blurring of boundaries between high and low . . . seemed to counter-pose their critical power to the reign of commodities. Since then, however, commodities have teamed up with the age of mockery and subject-hopping" (Rancière, _The Future of the Image_ , trans. Gregory Elliott [London: Verso, 2007], 51, see also 104). And he has savaged readings of pop that ascribe to it "criticality," pointing out that they "[correspond] to a rather simplistic view of the poor morons of the society of the spectacle, bathing contentedly in a flood of media images" ( _Future of the Image_ , 28).
71. Quoted in John Heilpern, "The Fantasy World of Warhol," _Observer_ , June 12, 1966, 11. When Warhol's contemporary critics recognized this egalitarianism, they tended to conceptualize it as an expansion of art's dignified purview, rather than a challenge to it; see Henry Geldzahler, "Andy Warhol," _Art International_ 8, no. 3 (April 25, 1964): 35.
72. Robert Lepper, "Comments on a Vulgar Art," _Architectural Forum_ , May 1940, 350; quoted in Blake Stimson, _Citizen Warhol_ (London: Reaktion, 2014), 133.
73. Warhol and Hackett, _Popism_ , 169. This, I believe, is where my view of Warhol most closely resembles the argument put forward by Jonathan Flatley—surely the most empowering ethical reading of Warhol's work yet developed—which explores the artist's ability "to help us expand our own capacities for liking" (Flatley, "Like: Collecting and Collectivity," _October_ , no. 132 [Spring 2010], 72). My sense of Warhol has been continuously inspired and challenged by Flatley's work.
74. Leticia Kent, "Andy Warhol, Movieman: 'It's hard to be your own script,'" _Vogue_ , March 1970, 167.
75. Julia Warhola, quoted in _Andy Warhol: Transcript of David Bailey's ATV Documentary_ (London: Bailey Litchfield/Mathews Miller Dunbar, 1972), roll 26, n.p.
76. Gavin Arnall, Laura Gandolfi, and Enea Zaramella, "Aesthetics and Politics Revisited: An Interview with Jacques Rancière," _Critical Inquiry_ 38, no. 2 (Winter 2012): 296.
77. Gretchen Berg, "Nothing to Lose: Interview with Andy Warhol," _Cahiers du Cinema in English_ , May 1967, 40.
78. Quoted in Schruers, "Andy Warhol: Why Not?" F10.
79. Kathy Acker, "Blue Valentine," in _Warhol's Film Factory_ , ed. Michael O'Pray (London: British Film Institute, 1989), 65.
80. Matthea Harvey, "Kara Walker," _Bomb_ 100 (Summer 2007), accessed October 15, 2015, <http://bombmagazine.org>.
81. Frances FitzGerald, "What's New, Henry Geldzahler, What's New?" _New York Herald Tribune_ , November 21, 1965, 15. The "15 minutes" version of this aphorism was not publicized until 1968: "In the future, everybody will be world-famous for 15 minutes" (Kasper König, Pontus Hultén, Olle Granath, and Andy Warhol, eds., _Andy Warhol_ [Stockholm: Moderna Museet, 1968], n.p.).
82. See Stephen Burt, "Galaxies inside His Head," _New York Times Magazine_ , March 24, 2015, accessed April 23, 2015, <http://www.nytimes.com/2015/03/29/magazine/galaxies-inside-his-head-poet-terrance-hayes.html>. Stimson's _Citizen Warhol_ takes a different approach, arguing that Warhol's "afflictions of class, closet and commodity" produced and determined his postmodernism (212).
83. Gilles Deleuze and Claire Parnet, _Dialogues_ , trans. Hugh Tomlinson and Barbara Habberjam (New York: Columbia University Press, 1987), 27. See Anthony E. Grudin, "Warhol's Animal Life," _Criticism_ 56, no. 3 (Summer 2014): 607–12.
84. Fredric Jameson, "An American Utopia," in _An American Utopia: Dual Power and the Universal Army_ , ed. Slavoj Žižek (London: Verso, 2016), 66. "Part of the pathos of Warhol's career," Richard Meyer observes, "is that, for all the erotically charged images of men the artist produced, he was rarely able to find his own image appealing or to imagine that others would" (Meyer, "Warhol's Clones," _Yale Journal of Criticism_ 7, no. 1 [1994]: 104).
85. Marianne Hancock, "Soup's On," _Arts Magazine_ 39 (May–June 1965): 16; Sterling McIlhenny and Peter Ray, "Inside Andy Warhol" (1966), in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 97. To my mind, the great precedent for Warhol's studied indifference is not Duchamp, but rather Manet's barmaid at the Folies-Bergère: "If one could not be bourgeois . . . then at least one could prevent oneself from being anything else: fashion and reserve would keep one's face from _any_ identity. . . . Expression is its enemy . . . for to express oneself would be to have one's class be legible" (Clark, _Painting of Modern Life_ , 253; emphasis in original).
86. Andy Warhol, _The Philosophy of Andy Warhol (From A to B and Back Again)_ (New York: Harcourt Brace Jovanovich, 1975), 78, 145 (emphasis in original).
87. David Harvey, "Neoliberalism Is a Political Project," _Jacobin_ , accessed July 31, 2016, <https://www.jacobinmag.com>.
88. Paul Taylor, "The Last Interview" (1987), in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 391.
89. See Larissa Harris, ed., _13 Most Wanted Men: Andy Warhol and the 1964 World's Fair_ (New York: Queens Museum; Pittsburgh: Andy Warhol Museum, 2014).
90. Douglas Crimp, _Our Kind of Movie: The Films of Andy Warhol_ (Cambridge, MA: MIT Press, 2012), 12. Like Crimp, I have chosen not to engage the Paul Morrissey films, since I generally concur with his assessment that "Morrissey seems . . . to have cynically attached himself to Warhol and adopted a great many of Warhol's formal strategies only to put them to a very different, even opposite purpose" (17).
91. Joseph Gelmis, "Andy Warhol," in _The Film Director as Superstar_ (New York: Doubleday, 1970), 69 (emphasis in original). As Hal Foster has observed, "Although failure can be the outcome of any test, it often appears to be the purpose in Warhol" (Foster, _The First Pop Age: Painting and Subjectivity in the Art of Hamilton, Lichtenstein, Warhol, Richter, and Ruscha_ [Princeton, NJ: Princeton University Press, 2011], 171).
92. Neal Weaver, "The Warhol Phenomenon: Trying to Understand It," _After Dark_ 10, no. 9 (January 1969): 30.
93. Levy, "Social Class and Consumer Behavior," 159; Clem Goldberger, Joanna Romer, and Jan Lavasseur, "We Talk to . . . Andy Warhol," _Mademoiselle_ 65 (August 1967): 325.
94. "Tally Brown, New York, 11 November 1978," in Smith, _Andy Warhol's Art and Films_ , 233 (emphasis in original).
95. Callie Angell, "Some Early Warhol Films: Notes on Technique," in _Andy Warhol: Abstracts_ , ed. Thomas Kellein (Münich: Prestel Verlag, 1993), 73. Angell reads what I call Warhol's amateurism as "a process of careful experimentation and conceptualization, [in which] Warhol isolated the most basic components of filmmaking—the single shot, the roll, the splice, the stationary camera, projection speed . . ." (73).
96. Bolex advertisement, 1964, Bolex Collector, accessed November 1, 2016, <http://www.bolexcollector.com>.
97. "Pop Goes the Video Tape: An Underground interview with Andy Warhol," _Tape Recording_ 12, no. 5 (September–October 1965): 15.
98. Peerless Camera advertisement, _New York Times_ , August 24, 1961, 26. The feature was advertised nationally throughout the early 1960s.
99. As Andrew Sarris noted, "Each scene runs out of film before it runs out of talk" (Sarris, "The Sub–New York Sensibility," _Cahiers du cinéma in English_ 10 [May 1967]: 43).
100. Angell, "Early Warhol Films," 75–76. Angell argues that these qualities in _Kiss_ produce a "parody of the Hollywood kiss film" and in _Blow Job_ a parody of pornography. Andrew Uroskie has persuasively claimed that _Sleep_ overturned the dominant Hollywood "model of spectatorial discipline," a reversal that is ultimately linked to a Rancièrian interest in the "perpetual undoing of the distinction between art and non-art" (Uroskie, _Between the Black Box and the White Cube: Expanded Cinema and Postwar Art_ [Chicago: University of Chicago Press, 2014], 44, 236).
101. Howard Junker, "Andy Warhol, Movie Maker," _Nation_ , February 22, 1965, 208.
102. Ruth Hirschman, "Pop Goes the Artist" (1963), transcript of KPFK radio broadcast, published in _Annual Annual_ , 1965, The Pacifica Foundation, Berkeley, CA, reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 40.
103. Kent, "Warhol, Movieman," 204.
104. David Bourdon, "Warhol as Filmmaker," _Art in America_ 59, no. 3 (May–June 1971): 50.
105. Bolex advertisement, 1964, Bolex Collector, accessed November 1, 2016, <http://www.bolexcollector.com>.
106. Jacques Rancière, "L'historicité du cinéma," in _De l'histoire au cinéma_ , ed. Antoine de Baecque and Christian Delage (Bruxelles: Éditions Complexe, 1998), 57; quoted in Joseph Tanke, _Jacques Rancière: An Introduction_ (New York: Continuum, 2011), 113. Andrew Sarris immediately recognized this quality in Warhol's films: "Warhol's ideas on direction are simple to the point of idiocy—or genius. He puts the camera on a tripod, and it starts turning, sucking in reality like a vacuum cleaner" (Sarris, "Films," _Village Voice_ 11, no. 8 [December 9, 1965], accessed November 1, 2016, <http://blogs.villagevoice.com>).
107. "Andy Warhol: Interviewed by Gerard Malanga" (1963; _Kulchur_ 16 [Winter 1964–65]), reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 49.
108. Dorothy Krasowska and Ronald Tavel, "Collaborating with Warhol: An Interview with Ronald Tavel," _Cabinet_ 8 (Fall 2002), accessed November 1, 2016, <http://cabinetmagazine.org/issues/8/krasowska.php>.
109. Levy, "Social Class and Consumer Behavior," 159; Schramm, Lyle, and Parker, _Television_ , 112.
110. Quoted in Stephen Koch, _Stargazer: The Life, World and Films of Andy Warhol_ (London: Praeger, 1973), 23.
111. Gelmis, "Andy Warhol," 69.
112. Berg, "Nothing to Lose," 40.
113. Schramm, Lyle, and Parker, _Television_ , 112; Warhol, quoted in Junker, "Andy Warhol, Movie Maker," 208. Junker was quick to dismiss Warhol's egalitarianism: "I think Andy was being ironic" (208). Warhol can be seen here as hearkening back to an American tradition of sociable and disorderly audiences; see Levine, _Highbrow/Lowbrow_ , 179–99.
114. Kent, "Warhol, Movieman," 204 (emphasis in original).
115. Warhol, quoted in Elenore Lester, "So He Stopped Painting Brillo Boxes and Bought a Movie Camera," _New York Times_ , December 11, 1966, 169.
116. Gelmis, "Andy Warhol," 72.
117. Schramm, Lyle, and Parker, _Television_ , 112, 113.
118. Kent, "Warhol, Movieman," 204. For Hardy's working-class background, see Gwendolyn Audrey Foster, _Class-Passing: Social Mobility in Film and Popular Culture_ (Carbondale: Southern Illinois University Press, 2005), 12.
119. Berg, "Nothing to Lose," 40.
120. Gelmis, "Andy Warhol," 66, 68. "I'd always wanted to do a movie that was pure fucking, nothing else, just the way _Eat_ had been just eating and _Sleep_ had been just sleeping" (Warhol and Hackett, _Popism_ , 256). For a brilliant discussion of Warhol, pornography, and the figure of the prostitute, see Jennifer Doyle, _Sex Objects: Art and the Dialectics of Desire_ (Minneapolis: University of Minnesota Press, 2006), 45–70.
121. Gelmis, "Andy Warhol," 67.
122. David Ehrenstein, "An Interview with Andy Warhol" ( _Film Culture_ [Spring 1966]), reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 65. Dave Hickey emphasizes the egalitarian ambitions of this project without acknowledging the artist's wariness regarding their feasibility: Warhol "propos[ed] 'celebrity' as a democratic _right_ rather than an elitist privilege, as an extension of our right to be _represented_ " (Hickey, "Importance of Remembering Andy," 52; emphasis in original).
123. As Hal Foster has argued, "They are pure tests of the capacity of the filmed subject to confront a camera, hold a pose, present an image, and sustain the performance for the duration of the shooting" (Foster, _First Pop Age_ , 164). Foster points out that slow motion also exacerbates the strain on the viewer.
124. "Tally Brown," in Smith, _Andy Warhol's Art and Films_ , 244.
125. "Pop Goes the Video Tape," 18. Weaver described Warhol's superstars as "people who grew up and cut their teeth on the old Hollywood and built a dream world around it" (Weaver, "Warhol Phenomenon," 30).
126. Pauline Kael, "Mothers," in _Deeper into Movies_ (New York: Little, Brown and Company, 1973), 195. This emphasis would persist in Warhol's television shows, which included, as Eva Meyer-Hermann points out, "(second rate) actors, models, and other hopefuls who are given the opportunity to shine" ("Other Voices, Other Rooms: TV-Scape," in _Andy Warhol: A Guide to 706 Items in 2 Hours 56 Minutes_ , ed. Meyer-Hermann [Rotterdam: NAi Publishers, 2007], 217).
127. Stanley Kauffmann, "First Love/Trash/The Great White Hope," _New Republic_ 163, no. 18 (1970): 30; Bibbe Hansen, quoted in Billy Name, _The Silver Age: Black & White Photographs from Andy Warhol's Factory_ (London: Reel Art Press, 2014), 280; Roger Vaughn, "Superpop, or A Night at the Factory," _New York Herald Tribune_ , August 8, 1965, 9. Vaughn was probably referring to Hansen.
128. Kael, "Mothers," 195.
129. Jameson, "American Utopia," 62, 81.
130. Paul Carroll, "What's A Warhol?" _Playboy_ , September 1969, 280. Elenore Lester had noticed a similar socioeconomic dichotomy in Warhol's factory between "the neglected, rejected, over-psychoanalyzed children of the rich, and the runaways from jobs as supermarket check-out clerks in the bleak suburbs of New Jersey" (Lester, "So He Stopped Painting," 169).
131. Walter Benjamin, "The Paris of the Second Empire in Baudelaire," trans. Harry Zohn, in _The Writer of Modern Life: Essays on Charles Baudelaire_ , ed. Michael W. Jennings (Cambridge, MA: Belknap Press, 2006), 261n300; Sedgwick in _Poor Little Rich Girl_.
132. "Poor Little Rich Girl," _Filmmakers Cooperative Catalogue_ , no. 3 (New York: Filmmakers Cooperative, 1965), 153; Richard Butsch, "Class and Gender in Four Decades of Television Situation Comedy: Plus Ça Change . . . ," _Critical Studies in Mass Communication_ 9 (December 1992): 387.
133. Warhol had apparently read about their family discord in _Life_ magazine. See Warhol Stars, accessed November 1, 2016, <http://www.warholstars.org> and Juanita Castro, "My Brother Is a Tyrant and He Must Go," _Life_ , August 28, 1964, 22–33.
134. J. J. Murphy, _The Black Hole of the Camera: The Films of Andy Warhol_ (Berkeley: University of California Press, 2012), 73–74. Murphy reads this scene as a dialogue between Victor and Pub (74). While Victor's first use of "sir" directly addresses Pub, the focus of his address seems to shift to the seated man early in the conversation, when Pub remarks that "these are the sir's books." The dialogue is adapted from Anthony Burgess's _Clockwork Orange_ , but Warhol's version heightens the class dynamics by replacing "brother" with "sir" as the form of address; see Burgess, _Clockwork Orange_ (New York: Norton, 2012), 11–12.
135. Callie Angell, _The Films of Andy Warhol: Part II_ (New York: Whitney Museum of American Art, 1994), 19. See also Amy Taubin, "My Time Is Not Your Time," _Sight and Sound_ , June 1994, 24.
136. Angell, _Films of Andy Warhol_ , 19.
137. Robert S. Lubar, "Unmasking Pablo's Gertrude: Queer Desire and the Subject of Portraiture," _Art Bulletin_ 79, no. 1 (March 1997): 73.
138. FitzGerald, "What's New, Henry Geldzahler?" 15.
139. Calvin Tomkins, "Profiles: Moving with the Flow," _New Yorker_ , November 6, 1971, 94.
140. FitzGerald, "What's New, Henry Geldzahler?" 15.
141. Guy Debord, _Society of the Spectacle_ , trans. Donald Nicholson-Smith (New York: Zone Books, 2006), 44–45 (emphasis in original).
142. FitzGerald, "What's New, Henry Geldzahler?" 15. Angell used Warhol's questions as the epigraph to her catalogue essay.
143. See Jeet Heer, "Cartoon Monarch: Otto Soglow and the Little King," _Comics Journal_ , May 21, 2012, n.p.; Jared Gardner, "The American King," in _Cartoon Monarch: Otto Soglow and the Little King_ , ed. Dean Mullaney (San Diego: IDW Publishing, 2012), 9–41.
144. The strip from which Warhol borrowed was Otto Soglow, _The Little King_ , _New York Journal-American_ , April 2, 1961; reproduced in Frei and Printz, _Catalogue Raisonné_ , 1:35.
145. Debord, _Society of the Spectacle_ , 104, 42, 8. Debord and Rancière obviously could not be more opposed concerning the usefulness of spectacle as a concept. I bring them into dialogue here because their conflicting perspectives on this issue illuminate a similar tension in Warhol's work.
146. Vladimir Nabokov, _Speak Memory: An Autobiography Revisited_ (New York: Vintage International, 1989), 218–19, 154.
147. Nabokov, _Speak Memory_ , 219 (emphasis added). The pun and homage are discussed in Clarence Brown, "Krazy, Ignatz, and Vladimir: Nabokov and the Comic Strip," in _Nabokov at Cornell_ , ed. Gavriel Shapiro (Ithaca, NY: Cornell University Press, 2003), 259.
148. Nabokov, foreword to _Speak Memory_ , 15.
149. Keith Gunderson, "A Portrait of My State as a Dogless Young Boy's Apartment," in _Growing Up in Minnesota: Ten Writers Remember Their Childhoods_ , ed. Chester G. Anderson (Minneapolis: University of Minnesota Press, 1976), 134–35.
150. Jacques Rancière, "A Politics of Aesthetic Indetermination," 15. For Plato's cave and the society of the spectacle, see Rancière, _The Emancipated Spectator_ , trans. Gregory Elliott (London: Verso, 2009), 44–45.
151. Quoted in Heilpern, "Fantasy World of Warhol," 11.
152. Tomkins, "Profiles: Moving with the Flow," 96.
153. Quoted anonymously in Bockris, _Warhol_ , 139.
154. "Henry Here, Henry There . . . Who is Henry?" _Life_ , February 18, 1966, 42.
155. FitzGerald, "What's New, Henry Geldzahler?" 20.
156. Henry Geldzahler, "Andy Warhol: A Memoir," in _Making It New: Essays, Interviews, and Talks_ (New York: Harcourt Brace, 1994), 44.
157. "Henry Here, Henry There," 42; Geldzahler, quoted in Warhol and Hackett, _Popism_ , 16. In this respect, Warhol and Geldzahler helped to inaugurate the new age described by Hal Foster, in which "the institution . . . becomes the spectacle, it collects the cultural capital, and the director-curator becomes the star" (Foster, _The Return of the Real: The Avant-Garde at the End of the Century_ [Cambridge, MA: MIT Press, 2001], 198). For a critique of the "independent" curator, see Claire Bishop, "What Is a Curator," _IDEA arts + society_ 26 (2007): 12–21.
158. Ingrid Sischy, "An Interview with Henry Geldzahler," in Geldzahler, _Making it New_ , 5.
159. Tomkins, "Profiles: Moving with the Flow," 82, 106. Tomkins dates the second piece to 1966, but see Claes Oldenburg, "Washes," _Tulane Drama Review_ 10, no. 2 (Winter 1965): 108, which dates the performances to May 1965.
160. FitzGerald, "What's New, Henry Geldzahler?" 20.
161. Tomkins, "Profiles: Moving with the Flow," 94.
162. Warhol, _Philosophy_ , 133.
163. Andy Warhol, _The Andy Warhol Diaries_ , ed. Pat Hackett (New York: Grand Central Publishing, 1989), 433 (emphasis in original).
164. "Andy Warhol interview 1966," YouTube, accessed June 16, 2016, <https://www.youtube.com>.
165. Quoted in Bockris, _Warhol_ , 140. "Cooley" is probably a mistranscription of "coolie," "a person of low (social) status" ( _Oxford English Dictionary Online_ , accessed May 5, 2015, <http://www.oed.com>).
166. Carter Ratcliff, "Andy Warhol: Inflation Artist," _Artforum_ 23, no. 7 (March 1985): 75.
167. "Henry Here, Henry There." 42.
168. FitzGerald, "What's New, Henry Geldzahler?" 15.
169. Bockris, _Warhol_ , 256.
170. Geldzahler, "Andy Warhol: Virginal Voyeur," in _Making It New_ , 368.
171. Tomkins, "Profiles: Moving with the Flow," 112, 60.
172. Warhol, _Diaries_ , 139, 403 (emphasis in original).
173. Stallybrass and White, _Politics and Poetics_ , 167.
174. Weaver, "Warhol Phenomenon," 30. Christina Crawford was actually twenty-four at the time; see Robert Windeler, "Joan Crawford Takes Daughter's Soap Opera Role," _New York Times_ , October 23, 1968, 95.
175. Matt Stahl, _Unfree Masters: Popular Music and the Politics of Work_ (Durham, NC: Duke University Press, 2013), 41. Asked for his opinion on the ascendance of reality television, Rancière demurred: "I don't watch TV very much, or those shows" (Truls Lie and Jacques Rancière, "Our Police Order: What Can Be Said, Seen, and Done," _Eurozine_ , accessed May 14, 2016, <http://www.eurozine.com/articles/2006-08-11-lieranciere-en.html>).
176. Kenneth Silver memorably claimed that "Warhol's art makes Americans recognize a collectivity of experience— _forges_ a collectivity—that might otherwise not exist . . . originat[ing] in working-class Forest City, Pennsylvania, in the mind of a brilliant, effeminate boy" (Silver, "Andy Warhol: 1928–1987," _Art in America_ 75, no. 5 [May 1987]: 141; emphasis in original).
177. Quoted in Craig Copetas, "Beat Godfather Meets Glitter Mainman," _Rolling Stone_ , February 28, 1974, 27. Bowie was referring to Warhol's play _Pork_ (1971), which was widely expected to be adapted for film or television.
178. Stahl, _Unfree Masters_ , 59.
179. Josh Eells, "So You Think You Can Sing?" _Blender_ , October 2007, 90.
180. Glenn O'Brien, "Interview: Andy Warhol" ( _High Times_ , August 24, 1977), reprinted in Goldsmith, Wolf, and Koestenbaum, _I'll Be Your Mirror_ , 257. In his diaries, Warhol mentions judging a Madonna look-alike contest in 1985, and a male beauty contest at a Manhattan club sometime before 1979 ( _Diaries_ , 655, 243).
181. See "He Dyed His Hair Silver," _An American Family_ advertisement, _New York Times_ , January 18, 1973, 82.
182. Fact Team, "MTV Rebrands as the Channel of Social Media Video," _Fact_ , June 25, 2015, accessed June 30, 2015, <http://www.factmag.com>.
183. Gilles Deleuze and Félix Guattari, _Anti-Oedipus_ , trans. Robert Hurley, Mark Seem, and Helen R. Lane (Minneapolis: University of Minnesota Press, 2000), 254.
184. Bob Dylan, "Like a Rolling Stone" (Columbia Records, 1965). "At Home with Henry," _Time_ , February 21, 1964, 68, reassured readers that, despite their taste for pop art, Ethel and Robert Scull had not been radicalized.
185. David Bourdon, "Andy Warhol and the Society Icon," _Art in America_ 63, no. 1 (January/February 1975): 43.
186. Quoted in Bourdon, "Andy Warhol and the Society Icon," 42. Barbara Rose was quick to remark the benefits that Warhol conferred on his patrons: "The Sculls learned everything they knew from Andy Warhol. . . . [They] transformed their banal, _nouveau riche_ selves into personalities by not being afraid to own up to being all that was considered lowbrow, déclassé, grasping, and publicity-seeking. They made a thing out of being vulgar, loud, and over dressed. They were, in short, shameless; and it was their shamelessness that finally got them the spotlight they ached for" (Rose, "Profit without Honor," _New York_ , November 5, 1973, 80). Robert Scull said that he and Geldzahler "became lifelong friends in ten minutes" (Tomkins, "Profiles: Moving with the Flow," 88).
## Conclusion
1. Noam Chomsky, _The Prosperous Few and the Restless Many_ (Berkeley: Odonian Press, 1993), 66–72.
2. David Harvey, _A Brief History of Neoliberalism_ (Oxford: Oxford University Press, 2005), 202.
3. Don Kalb, "Introduction: Class and the New Anthropological Holism," in _Anthropologies of Class: Power, Practice, and Inequality_ , ed. James G. Carrier and Don Kalb (Cambridge: Cambridge University Press, 2015), 3.
4. Robert A. Nisbet, "The Decline and Fall of Social Class," _Pacific Sociological Review_ 2, no. 1 (Spring 1959): 17.
5. Sasha Lilley, "Introduction," in _Capital and Its Discontents: Conversations with Radical Thinkers in a Time of Tumult_ (Oakland: PM Press, 2011), 11.
6. Wendy Brown, _Undoing the Demos: Neoliberalism's Stealth Revolution_ (New York: Zone Books, 2015), 10, 38; Richard Sennett, _The Culture of the New Capitalism_ (New Haven, CT: Yale University Press, 2006), 54. G. M. Tamás conclusively demonstrates that while capitalism has undermined "castes," it has perpetuated "classes" ("Telling the Truth about Class," _Socialist Register_ 42 [2006]: 242–5). See also Gilles Deleuze and Félix Guattari, _Anti-Oedipus_ , trans. Robert Hurley, Mark Seem, and Helen R. Lane (Minneapolis: University of Minnesota Press, 2000), 254–55.
7. Matt Stahl, _Unfree Masters: Popular Music and the Politics of Work_ (Durham, NC: Duke University Press, 2013), 5.
8. Peter Stallybrass and Allon White, _The Politics and Poetics of Transgression_ (Ithaca, NY: Cornell University Press, 1986), 191, 5.
9. _The Comedy Central Roast of Justin Bieber_ , Comedy Central, March 30, 2015.
10. Andy Warhol, _America_ (New York: Harper & Row, 1985), 188–91.
11. T. J. Clark, _The Painting of Modern Life_ , rev. ed. (Princeton, NJ: Princeton University Press, 1999), 205.
12. Andy Warhol, _The Andy Warhol Diaries_ , ed. Pat Hackett (New York: Grand Central Publishing, 1989), 498.
13. bell hooks, _Where We Stand: Class Matters_ (London: Routledge, 2000), 80.
14. Lilley, "Introduction," 14.
15. Sennett, _Culture of the New Capitalism_ , 46; Wolfgang Streeck, _How Will Capitalism End? Essays on a Failing System_ (London: Verso, 2016), 2–3 (emphasis in original).
16. Jennifer M. Silva, _Coming Up Short: Working-Class Adulthood in an Age of Uncertainty_ (Oxford: Oxford University Press, 2013), 25.
17. Silva, _Coming Up Short_ , 10, 95, 19; Warhol, _Diaries_ , 676. Compare with the aristocratic ethos embodied by Edie Sedgwick, who "had no goals other than to enjoy herself to the fullest" (Malanga, quoted in Billy Name, _The Silver Age: Black & White Photographs from Andy Warhol's Factory_ [London: Reel Art Press, 2014], 169).
18. Gretchen Berg, "Nothing to Lose: Interview with Andy Warhol," _Cahiers du Cinema in English_ , May 1967, 40.
19. Andy Warhol, _The Philosophy of Andy Warhol (From A to B and Back Again)_ (New York: Harcourt Brace Jovanovich, 1975), 112, 111, 112. When those peers failed to match his level of flexibility, Warhol was infamously unforgiving. "Forced to be flexible in the labor market," Warhol, like Silva's subjects, "[became] hardened outside it . . . race emerges as a source of resentment and fear" ( _Coming Up Short_ , 30). See, for example, the deeply ambivalent feelings Warhol expressed in his diaries regarding Basquiat.
20. Lynne Layton, "Some Psychic Effects of Neoliberalism: Narcissism, Disavowal, Perversion," _Psychoanalysis, Culture and Society_ 19 (2014): 169.
21. Andy Warhol and Pat Hackett, _Popism: The Warhol '60s_ (New York: Harcourt Brace Jovanovich, 1980), 65.
22. Takashi Murakami, "On the Level," in Mark Rosenthal, Marla Prather, Ian Alteveer, and Rebecca Lowery, _Regarding Warhol: Sixty Artists, Fifty Years_ (New York: Metropolitan Museum of Art, 2012), 138; Blake Stimson, _Citizen Warhol_ (London: Reaktion, 2014), 39.
23. Andrew Kahn and Forrest Wickman, "The Gospel According to Yeezus: The Complete 'Rants' of Kanye West's 2013 Tour," _Slate_ , November 30, 2013, accessed September 2, 2016, <http://www.slate.com>.
24. _Lois Lane_ , April 1961.
25. Disguise and mistaken identity were recurring themes in the _Adventures of Superman_ television program. See, among many other examples, "The Superman Silver Mine" (March 10, 1958), "Divide and Conquer" (February 17, 1958), "The Tomb of Zaharan" (March 29, 1957), and "King for a Day" (October 15, 1955).
26. Adeline Gomberg, "The Working-Class Child of Four and Television," in _Blue-Collar World: Studies of the American Worker_ , ed. Arthur B. Shostak and William Gomberg (Englewood Cliffs: Prentice-Hall, 1964), 436.
27. Arjun Appadurai, _Modernity at Large: Cultural Dimensions of Globalization_ (Minneapolis: University of Minnesota Press, 1996), 84, 83.
28. There is an inkling here that weakened citizenship and weakened states might be related somehow, and that these forces might eventually produce failed citizenship and failed states—the preconditions, at home and abroad, for anti-imperialist uprisings. See Retort, _Afflicted Powers_ (London: Verso, 2005), 32–33.
29. Fredric Jameson, "An American Utopia," in _An American Utopia: Dual Power and the Universal Army_ , ed. Slavoj Žižek (London: Verso, 2016), 48.
30. Jonathan Martineau, _Time, Capitalism and Alienation: A Socio-Historical Inquiry into the Making of Modern Time_ (Boston: Brill, 2015), 145.
31. "Mike Lefevre, Steelworker," in Studs Terkel, _Working: People Talk about What They Do All Day and How They Feel about What They Do_ (New York: Pantheon, 1974), 7.
32. Lauren Berlant, "'68, or Something," _Critical Inquiry_ 21, no. 1 (Autumn 1994): 134.
33. Ruth Hirschman, "Pop Goes the Artist" (1963), transcript of KPFK radio broadcast, published in _Annual Annual_ , 1965, The Pacifica Foundation, Berkeley, CA, reprinted in _I'll Be Your Mirror: The Selected Andy Warhol Interviews_ , ed. Kenneth Goldsmith, Reva Wolf, and Wayne Koestenbaum (New York: Carroll & Graf, 2004), 44. See Tan Lin, "Warhol's Aura and the Language of Writing," _Cabinet_ 4 (Fall 2001), accessed October 15, 2015, <http://www.cabinetmagazine.org/issues/4/lin.php>.
34. Hal Foster, "Death in America," _October_ , no. 75 (Winter 1996), 48n38; John J. Curley, _A Conspiracy of Images: Andy Warhol, Gerhard Richter, and the Art of the Cold War_ (New Haven, CT: Yale University Press, 2013), 172.
35. On Warhol, the Factory, and collectivity, see Jonathan Flatley, "Like: Collecting and Collectivity," _October_ 132 (Spring 2010), 88–95.
36. Glenn O'Brien, "Factory Workers Warholites Remember Billy Name," _Interview_ , November 30, 2008, accessed October 20, 2015, <http://www.interviewmagazine.com>.
37. Warhol, _Diaries_ , 370.
38. Theodor Adorno, _Aesthetic Theory_ , ed. Gretel Adorno and Rolf Tiedemann, trans. Robert Hullot-Kentor (New York: Continuum, 2002), 119. The argument could be made that Warhol was merely sensing a transitional moment in oppositional culture: away from the "centralized" Factory that mirrored Fordism, and toward the "networking, decentralized, non-hierarchical" model that mirrored neoliberalism (Harvey, "Neoliberalism Is a Political Project," _Jacobin_ , July 23, 2016, accessed July 31, 2016, <https://www.jacobinmag.com>).
39. Berlant, "'68, or Something," 134.
40. Gilles Deleuze and Félix Guattari, _Kafka: Toward a Minor Literature_ , trans. Dana Polan (Minneapolis: University of Minnesota Press, 1986), 42.
41. Deleuze and Guattari, _Kafka_ , 19.
42. Pete Coviello, "Our Noise," _Avidly_ , April 18, 2016, accessed April 20, 2016, <http://avidly.lareviewofbooks.org>.
43. Joseph Gelmis, "Andy Warhol," in _The Film Director as Superstar_ (New York: Doubleday, 1970), 69 (emphasis in original).
44. Quoted in Paul Carroll, "What's a Warhol?" _Playboy_ , September 1969, 278; Hal Foster, _The First Pop Age: Painting and Subjectivity in the Art of Hamilton, Lichtenstein, Warhol, Richter, and Ruscha_ (Princeton, NJ: Princeton University Press, 2011), 170.
45. Kathi Weeks, "Utopian Therapy: Work, Nonwork, and the Utopian Imagination," in Žižek, _American Utopia_ , 250; Warhol, _Philosophy_ , 97, 96.
46. Malanga, quoted in Name, _Silver Age_ , 61; Warhol, _Philosophy_ , 96.
47. Carroll, "What's a Warhol?" 280.
48. Gerard Malanga, "Working with Warhol," _Art New England_ , September 1998, 8.
49. Keith Haring, quoted in Victor Bockris, _Warhol: The Biography_ (New York: Da Capo, 2003), 464. For more on Drella, see Stimson, _Citizen Warhol_ , 211–12.
50. Warhol, _Philosophy_ , 102.
51. Warhol, _Diaries_ , 676.
52. Lilley, "David Harvey: The Rise of Neoliberalism and the Riddle of Capital," in _Capital and Its Discontents_ , 47. Curley argues for a more politically conscientious Warhol, whose work "recognize[d] the interdependent nature of global trauma and the everyday diversions under capitalism as deployed by forces of the mass media" ( _Conspiracy of Images_ , 75). To support this claim, he cites a 1961 _Life_ magazine photograph by Terrence Spencer of a dying boy in the Congo that Warhol apparently used as a palette (75); the troubling artifact, seemingly used by Warhol haphazardly to catch drips and blot his brushes, might as compellingly evidence disassociation and disengagement as recognition.
53. Warhol, _Diaries_ , 304.
54. Pat Hackett, introduction to Warhol, _Diaries_ , xvi; cited in Stimson, _Citizen Warhol_ , 214.
55. Warhol, _Diaries_ , 494.
56. Pier Paolo Pasolini, "Andy Warhol's Ladies & Gentlemen" (1975), trans. Rodney Stringer, in Andy Warhol, _Ladies & Gentlemen_ (New York: Skarstedt Gallery, 2009), 5. As an egalitarian, Warhol could imagine " _the abolition of caste lead[ing] to equality_ " but not " _the abolition of class lead[ing] to socialism_ " (Tamás, "Telling the Truth about Class," 245; emphasis in original).
57. Warhol, _Philosophy_ , 111.
58. Warhol, _Philosophy_ , 199; Mark Fisher, _Capitalist Realism: Is There No Alternative?_ (Washington, DC: Zero Books, 2009), 56.
59. Warhol, _America_ , 74.
60. Wilbur Schramm, Jack Lyle, and Edwin B. Parker, _Television in the Lives of Our Children_ (Stanford, CA: Stanford University Press, 1961), 112.
61. Warhol, _Philosophy_ , 111, 112.
62. Lauren Berlant, "The Epistemology of State Emotion," in _Dissent in Dangerous Times_ , ed Austin Sarat (Ann Arbor: University of Michigan Press, 2005), 49.
63. Guy Debord, "The Decline and Fall of the Spectacle-Commodity Economy" (1966), in _Situationist International Anthology_ , ed. Ken Knabb, rev. ed. (Berkeley: Bureau of Public Secrets, 2006), 197; Loïc Wacquant, _Punishing the Poor: The Neoliberal Government of Social Insecurity_ (Durham, NC: Duke University Press, 2009), 43.
64. Warhol, _Diaries_ , 62.
65. Warhol, _Diaries_ , 40.
66. Warhol, _Diaries_ , 204.
# Index
Page numbers in italics refer to figures.
_$199 Television_ (Warhol),
A & P Supermarket advertisement, , __
abstract expressionism, 1–3, 11–13, , , , , , 110–23. _See also specific artists and artworks_
Acker, Kathy,
_Advertisement_ (Warhol), , , __ , , plate 6
advertising: class dimensions of, 3–5, , 73–76, , 84–91, 175n26; effectiveness of, 83–93, , 178n71; emotions and, , , ; function of, 80–83; irony and, 73–74, 174n127; Lichtenstein and, 165n61; race and, , , , 108–9; Warhol's art on, 75–76, 93–94; Warhol's newspaper work in, , , 171n80. _See also_ consumerism; gender stereotypes; _specific artworks_ ; _specific brands_
aesthetic detachment, 117–18
Agee, James,
Alan, Charles,
Alloway, Lawrence, ,
amateurism, 3–7, 47–48, 56–63, , , 123–30, 138–39, , 186n95, 187n106
_American Journal of Sociology_ ,
American myths: of celebrities, 52–53, 72–73, 173n118; class and, , 46–47, 147–48; participatory culture and, 40–48, 173n115. _See also_ consumerism; popular culture
_Andy relaxing at home_ (Gorgoni),
_Andy Warhol Diaries, The_ (Warhol), , , , , 162n53, 190n180, 192n19
Angell, Callie, , , , 186n95
animality, , , , 161n51
Antin, David, ,
Arnheim, Rudolf,
_Arrrrrff!_ (Lichtenstein), __ ,
_Art in America_ ,
Art Instruction, Inc. advertisement, 58–61, __
_ARTnews_ , ,
artwork by Warhol. _See_ Warhol, Andy—artwork
Bader, Graham, 27–28
Baker, Richard Brown,
Baldwin, James, , 72–73, 173n120
Barclay, Dorothy,
Barker, Anthony,
Barsalou, David, , , 22–23, 163n3
Barthes, Roland, , 111–12
Basquiat, Jean-Michel, , 192n19, plate 14
_Batman_ (Warhol),
Baudrillard, Jean, 111–12, 181n11
_Bed_ (Rauschenberg), ,
_Before and After 1]_ (Warhol), , , __ , , [plate 6
Bengston, Billy Al, 182n57
Benjamin, Walter, , , 179n105
Berlant, Lauren, , , ,
Bertrand, Michael,
_Beverly Hillbillies, The_ , 180n113
_Big Torn Campbell's Soup Can (Pepper Pot)_ (Warhol), , 95–96, plate 11
Blake, Peter, 182n57
_Blow Job_ (Warhol film), , , 187n100
_Blue Movie_ (Warhol film), 128–29
Bois, Yve-Alain,
Bolex 16mm film camera, 127–28
_Bomb '67_ (Warhol), 177n59
Bonwit Teller store exhibit, 49–50, 159n1, 169n47, plate 6
Boshier, Derek, 182n57
Bourdieu, Pierre, ,
Bourdon, David, , , ,
Bowie, David,
box sculpture series (Warhol), 101–2, , 174n2, 178n90
Boyd, Eva Narcissus (Little Eva), , ,
_Boy for Meg [1], A_ (Warhol), , ,
_Boy for Meg 2], A_ (Warhol), , [70–72, plate 5
Brach, Paul,
brand names and imagery. _See_ advertising; consumerism; _specific brands_
_Brillo Box_ series (Warhol), , 12–13, 45–46, , , 101–2, , 174n2, 177n59, 185n70, plate 13
_Brillo Soap Pads Box_ (Warhol), plate 13
Brown, Tally, ,
Brown, Wendy,
_Brushstrokes_ (Lichtenstein), 25–26, plate 3
Buchloh, Benjamin H. D., , , , ,
Burckhardt, Rudy, 182n57
Burger King, , 174n127
_Cagney_ (Warhol), , __ ,
Cagney, James,
Cale, John,
camp art, 112–13, , 182n18
Campbell's Soup (brand), , , , , , , 99–100, 178n72, plate 12
_Campbell's Soup Can (Old-Fashioned Tomato Rice)_ (Warhol), , plate 11
_Campbell's Soup Can (Tomato Rice)_ (Warhol), , , ,
_Campbell's Soup Can_ series (Warhol), , , 94–99, __ , , __ , plate 10, plate 11, plate 12
Canaday, John,
_Carat_ (Warhol), , __ , 173n114
Carey, Ted, , , 159n1, 184n57
Carroll, Paul, 132–33
Castelli, Leo,
castes, , 191n6
celebrity myth, 52–53, 72–73, 173n118. _See also_ American myths; Hollywood culture
Celmins, Vija, 182n57
_Chelsea Girls_ (Warhol film), ,
_Chicago Daily Tribune_ , __
_Chicago Times_ ,
_Chicago Tribune_ ,
Chomsky, Noam,
cinema. _See_ Warhol, Andy—films
_Claes e Pat Oldenburg_ (Mulas), __
Clark, T. J., , , , , 114–15, , 182n19, 186n85
class consciousness: cultural reproduction and, 65–68; egalitarian cinema, 126–34, , 143–45; music and, 15–16; neoliberalism and, , 46–47, ; pop art and, 106–7; postmodernism and, 114–15; Warhol and, 4–11, , , 160n33, 161n40, 161n42. _See also_ egalitarianism; neoliberalism; social mobility
class stereotypes: advertising and, 3–5, , , , 73–76, , 84–91, , 175n26; American myths and, , 46–47, 147–48; comic books and, 50–51; social category of class, 8–9, 13–15, , 160n33, 161n40, 161n42; television show on, 180n113. _See also_ gender stereotypes; race and advertising
_Clocks (Two Times)_ (Warhol), 151–52, __
Coca-Cola (brand), , , 99–100, , 180n116
_Coca-Cola, "Standard and King Sized"_ (Warhol), 1–2, __
_Coca-Cola 1]_ (Warhol), , , [159n9
_Coca-Cola 2]_ (Warhol), , , [159n9
_Coca-Cola [3]_ (Warhol),
_Coca-Cola_ series (Warhol), 1–2, __ , 99–101, 159n9
_Cock Drawings_ (Warhol), 184n57
Cohen, Lizabeth, 176n32
comic books: class and, , 50–51, 56–57; industrial norms in, ; Lichtenstein's use of, , 22–23, 163n3. _See also_ DC Comics; _specific artworks_
commodity logos. _See_ advertising
consumerism: brand versus generic, 176n38; class and, 5–10, 80–91, 176n32; cynicism and, , , 174n128; shopping and shopper-types, , , , , 175n24, 175n31, 179n95. _See also_ advertising; American myths
correspondence art schools, , 58–63
Coviello, Peter, ,
Crimp, Douglas,
Crow, Thomas, , , 160n18, 163n11, 164n27, 175n10, 181n6
_Crowd_ (Warhol), , __
cultural reproduction, 5–7, __ , 47–49, 58–68, . _See also_ participatory culture
curator, , 134–36, 140–41, 189n157. See also _Henry Geldzahler_ (Warhol film)
Curley, John J., , 178n72, 193n52
cynicism, consumer, , , 174n128. _See also_ consumerism; irony in advertising
_Dance Diagram_ (Warhol), ,
Daniels, Les,
Danto, Arthur, , , 44–46, , , , 167n15
D'Arcangelo, Allan, 182n57
Davis, Whitney, , 182n14
DC Comics: _Our Fighting Forces_ , __ , ; _Secret Hearts_ , 23–25, __ ; _Strange Suspense Stories_ , , plate 4. _See also specific artworks_
de Antonio, Emile, 1–2,
_Death and Disasters_ exhibit,
Debord, Guy, , , 136–38, , 189n145
"Decline and Fall of Social Class, The" (Nisbet),
de Grazia, Victoria, 176n32
de Kooning, Willem, , , 183n33, 185n68, plate 15
de la Villeglé, Jacques, 182n57
Deleuze, Gilles, 27–28, , 65–67, , , , 173n109
Del Monte, ,
_Del Monte_ (Warhol),
De Salvo, Donna,
desire, , 8–10, , 27–28, 38–39, , 60–67, , , 115–17, , , 171n87; disarmed, ; politics of,
diamonds, 173n114
_Dick Tracy_ (Warhol), , , 171n79
_Do it Yourself (Sailboats)_ (Warhol), plate 9
_Do it Yourself_ series (Warhol), ,
Donahue, Troy, 42–43, __ ,
Doolittle, Jerry,
Doonan, Simon,
_Drowning Girl_ (Lichtenstein), , __ ,
_Dr. Scholl's Corns_ (Warhol),
Duchamp, Marcel, , , ,
Dylan, Bob,
Dzubas, Friedel,
_Eat_ (Warhol film), ,
Eberstadt, Frederick,
egalitarian cinema, 126–34, , 143–45, . _See also_ Warhol, Andy—films
egalitarianism, 9–11, , , , , , , 119–26, , 154–57. _See also_ class consciousness; modernism; neoliberalism
Eggleston, William, 182n57
Eldridge, Clarence,
_Elvis_ (Warhol), ,
emotions and advertising, , , , , 149–51
_Empire_ (Warhol film),
Escobar, Marisol, , 182n57
_Exploding Plastic Inevitable_ (Warhol), , , 162n68
Fabian, Ann, 176n32
Factory Superstars, , , 130–32, 148–50, 152–54, 188n125. _See also specific persons_
Famous Artists Schools advertisement, __
_Farah Dibah Pahlavi (Shahbanou [Queen] of Iran)_ (Warhol), __
fashion, 10–11, , , 172nn91–92
_Feet and Campbell's Soup Can_ (Warhol), __
femininity. _See_ gender stereotypes
Ferus Gallery, , , , 174n2
films by Warhol. _See_ Warhol, Andy—films
FitzGerald, Frances, 135–36, 141–42
Flatley, Jonathan, , , 173n122, 185n73
_Flatten–Sand Fleas_ (Lichtenstein),
flexibility, personal, , ,
_Flowers_ (Warhol), 168n34
Foster, Hal, 151–52, 186n91, 188n123, 189n157
Foucault, Michel,
Frankenthaler, Helen,
Fried, Michael, , 42–47, ,
Friedlander, Lee, 182n57
_Front and Back Dollar Bills_ (Warhol), __
Geiger, Kent,
Geldzahler, Henry, , 134–35, __ , 140–43, , 173n114, 189n157, 191n186, plate 20. See also _Henry Geldzahler_ (Warhol film)
Gelmis, Joseph,
gender stereotypes: ads on idealized masculinity, 57–58, 171n85; ads targeted to women, , , , , 172nn91–92, 176n45; in comics, 23–25; of male shoppers, 179n95; Warhol's interest in, , 104–5, . See also _Before and After [1]_ (Warhol); class stereotypes; homosexuality and art; race and advertising
Giallo, Vito, 171n80
Giorno, John,
_Girl with Ball_ (Lichtenstein), 27–28
Gluck, Nathan,
Gomberg, Adeline, 51–52
Gopnik, Adam,
Gorgoni, Gianfranco, __
Gornick, Vivian,
Graeber, David, , 37–38,
Grebenak, Dorothy, 182n57
_Grrrrrrrrrrr!_ (Lichtenstein), __ , , __ ,
Guattari, Félix, 27–28, , 65–67, , , 173n109
Gunderson, Janice, 138–39,
Gunderson, Keith, 138–39,
Hains, Raymond, 182n57
Hamilton, Richard, 182n57
happiness, privatization of, , 149–51, . _See also_ emotions and advertising
Haring, Keith,
_Harlot_ (Warhol film),
Hartman, George,
Haworth, Jann, 182n57
_Heinz_ (Warhol),
_Henry Geldzahler_ (Warhol film), , , 134–43, __. _See also_ "Little King, The" (comic strip)
_Herald Tribune_ ,
Hess, Thomas B.,
Hickey, Dave, 172n107, 183n43, 188n122
_Hidden Persuaders, The_ (Packard), 81–82,
Hockney, David, 182n57
Hollywood culture, , , 52–53, 127–31, 187n100, 189n157. _See also_ celebrity myth; popular culture
homosexuality and art: critics and, 112–13, 182n19; egalitarian cinema, 126–34, , 143–45; of Geldzahler, , , ; pop music and, 162n74; vulgarity and, 119–26; Warhol and, , , , , 112–14, 124–25, . _See also_ gender stereotypes; sexuality and art; _specific artists_
hooks, bell, , ,
_Hopeless_ (Lichtenstein), ,
Hopps, Walter, , 170n69
Hughes, Fred, ,
Huxley, Aldous, , 162n61, 164n18
_Icebox_ (Warhol),
I. Miller advertisement, __ , 171n80
Indiana, Robert, ,
individualism: advertising and, 75–76; American myths and, ; Factory and, ; neoliberalism on, 8–9, , , ; Rancière's modernism on, ; Warhol's understanding of, 13–14, , . _See also_ emotions and advertising; social mobility
intersectionality, 7–10; myths and, ; of pop art, 107–9; Warhol on, , , , , 132–33, . _See also_ class stereotypes; gender stereotypes; race and advertising
_Ironworks/Fotodeath_ (Oldenburg),
irony in advertising, 73–74, 174n127. _See also_ advertising; cynicism, consumer
Jameson, Fredric, , , , 131–32,
Jenkins, Henry, 166n69
Johns, Jasper, 112–13, , , , 159n2, 181n13
Johnson, Ray, 182n57
Jowett Institute of Body Building advertisement, __ , 57–58,
Judd, Donald,
Kael, Pauline,
Karp, Ivan, , , ,
Katz, Jonathan, , 112–13
Kellogg's, ,
_Kellogg's_ (Warhol),
_Kiss_ (Warhol film), , , , 187n100
_Kiss, The_ (Lichtenstein),
Klein, William, 182n57
Kligman, Ruth,
Kramer, Hilton, 79–80, , 181–82n13
Krauss, Rosalind E.,
Kubert, Joe, __ ,
Kuhn, Annette,
Kunitz, Stanley, 79–80, ,
Lennon, John,
Leo Castelli gallery, ,
Lester, Elenore, , 188n130
Levine, Lawrence, 169n40
Levy, Sidney J., , , , , , , ,
Lichtenstein, Roy, 18–28, , 32–36, , , ; archive of source material of, , 22–23, 163n3; brand imagery and, 165n61; critical review of, 20–22, 163n14; on pop culture, , , ; Warhol's art as likeness to, , 159n1
_Life_ (magazine), , , 178n72, plate 12
_Life and Times of Juanita Castro, The_ (Warhol film),
_Life with Millie_ ,
_Lipstick (Ascending) on Caterpillar Tracks_ (Oldenburg), __ ,
Little Eva (Eva Narcissus Boyd), , ,
"Little King, The" (comic strip), , 136–42, , , plate 19. See also _Henry Geldzahler_ (Warhol film)
_Little King, The_ (Warhol), , , , , plate 6, plate 18
Livingstone, Marco, 54–55
Lobel, Michael, , 163n14, 165n61
"Loco-Motion, The" (Little Eva), ,
_Los Angeles Times_ , __
Loud, Lance,
_Love_ (Escobar),
Lubar, Robert,
lunch pail campaign. _See_ Macfadden Publications, Inc. advertisements
Lyle, Jack, , , , , , ,
Lysgaard, Sverre, , ,
Macfadden, Bernarr, 176n32
Macfadden Publications, Inc. advertisements, 82–84, __ , , _87–90_ , 91–95, , , ,
Macy's department store,
Magic Art Reproducer advertisement, __ , , , 172n104
_Make Him Want You_ (Warhol),
Malanga, Gerard, , , , , __ ,
_Man and Superman_ (Shaw),
_Manchurian Candidate, The_ (film), __ , 81–82
Marcuse, Herbert, , 165n51
_Marilyn_ (Warhol), ,
_Marilyn Monroe_ (de Kooning), , plate 15
_Mario Banana_ (Warhol film),
marketing. _See_ advertising
Martinson Coffee, , , 177n69
masculinity. _See_ gender stereotypes
McClellan, Douglas,
Meyer, Richard, , 167n12, 186n84
modernism, 9–11, 13–14, , 110–11, 122–26. _See also_ consumerism; egalitarianism; popular culture; postmodernism
money, 14–15, __ , , __ , , 162n61, 181n127. _See also_ class consciousness
Monroe, Marilyn, , 43–45, , , , plate 15, plate 16
Motherwell, Robert,
_Mott's_ (Warhol),
_Mrs. Warhol_ (Warhol), , plate 17
Mulas, Ugo, __
Muñoz, José Esteban, , 170n64
Murphy, J. J., , 188n134
Muschinski, Patty, , , , 28–36, __ , 164n36
_Myths: Howdy Doody_ (Warhol), __
_Myths_ series (Warhol), 40–42, __
Nabokov, Vladimir, 137–38,
Name, Billy, , , 152–53, __ , 161n48
_Natalie_ (Warhol),
_National Enquirer_ , , 173n111
Nelson, Maggie,
neoliberalism, 8–10, 13–14, , , , , 105–7, , 146–57, 193n38; advertising and, 73–74; class and, , 46–47, , 123–24; reality television and, 143–44; rise of, 161n34. _See also_ class consciousness; participatory culture; social mobility
newsprint artworks, 11–12, 49–69, , , plate 1, plate 2, plate 5, plate 6, plate 7, plate 8, plate 18, plate 19
_Newsweek_ , , , __ ,
_New York Daily News_ ,
_New Yorker_ , ,
_New York Journal American_ , plate 19
_New York Times_ , ; advertisements in, , __ , , _87–90_ , __ ; on brand commodities, , ; on comic books and poverty, 50–51; on Warhol as ur-postmodernist, ; work by Warhol in, , , 171n80
_New York Times Magazine_ ,
non-normative sexuality. _See_ homosexuality and art
"Ode to Joy" (O'Hara),
Ogilvy, David, 171n87
O'Hara, Frank, 120–22
Oldenburg, Claes, __ , __ , __ , , 165n51; collaborations with Muschinski, , , , 28–36; Geldzahler and, ; on pop culture, ; on Warhol as machine, 64–65
_On the set with the movie_ Vinyl (Malanga), __
_Orange Car Crash_ (Warhol), , , __ ,
_Our Fighting Forces_ (DC Comics), __ ,
Packard, Vance, , ,
_Painting with Two Balls_ (Johns),
_Panel of Experts_ (Basquiat), plate 14
Paolozzi, Eduardo, 182n57
Parker, Edwin B., , , , , , ,
participatory culture, 40–74; American myths and, 40–48, 173n115; art reproduction and, 56–68; emotions and, ; Lichtenstein and, ; Warhol's vision of, 3–4, , , 139–40, 152–57, 166n69. _See also_ class consciousness; consumerism; neoliberalism; social mobility
Pasolini, Pier Paolo,
_Pat Sewing_ (Oldenburg), , __
_Peach Halves_ (Warhol), , , 95–96, 159n9, plate 1, plate 2
_People Watching Henry Geldzahler on Screen_ (Warhol), , __
Pepsi advertisement, __ ,
personal responsibility, 13–14, 148–50. _See also_ emotions and advertising; individualism
_Philosophy of Andy Warhol, The_ (Warhol),
Picasso, Pablo,
Platzer, Robin, __
_Playboy_ (magazine),
Pollock, Jackson, , , , __ , , 182n19
_Poor Little Rich Girl_ (Warhol film), , __ , 132–33
pop art: versus camp, 112–13; class and, 106–7; defined, 17–18, ; early criticism of, , 79–80; vulgarity of, 119–24, 185n70. _See also specific artists and artworks_
Popeye, , , 171n79
pop music, 15–16, , , 162n74
popular culture: Hollywood and, , , 52–53, 127–31, 187n100, 189n157; Lichtenstein on, , , ; Muschinski and Oldenburg on, , ; reproduction of, 5–7, __ , 47–49, 58–68, ; vulgarity and, ; Warhol on, 35–37. _See also_ advertising; consumerism
_Pork_ (Warhol play), , 190n177
pornography, , 188n120. See also _Blow Job_ (Warhol film)
"Portrait of My State as a Dogless Young Boy's Apartment, A" (Gunderson), 138–39
_Portraits of the 70s_ exhibit,
postmodernism, 5–11, , , , 73–74, 110–15, , 135–36, . _See also_ modernism; neoliberalism
Potts, Alex,
poverty: animality and, , , 161n51; comic books and, , 50–51, 56–57. _See also_ class consciousness
Presley, Elvis,
primitivism, , , 108–9. _See also_ vulgarity
privatization of happiness, , 149–51. _See also_ emotions and advertising
race and advertising, , , , 108–9. _See also_ class stereotypes; gender stereotypes
_Race Riots_ (Warhol),
Rancière, Jacques, , , 123–26, , , , , 185n70, 189n145, 190n175
Ratcliff, Carter,
Rauschenberg, Robert, , 112–13, , , 184n62
Reagan, Nancy, ,
Reagan, Ron,
Reed, Lou, , , 166n76
Reeves, George, 52–53, , 170n67, 170n69
reproduction, 5–7, __ , 47–49, 58–68, . _See also_ participatory culture
Rivers, Larry, , 182n57
Rodchenko, Alexander,
Rodrigues, Glauco, 182n57
Rodrigues, Laurie,
Rose, Barbara, , , 168n18, 191n186
Rosenberg, Harold, , ,
Rosenquist, James, 34–36, 94–95
Rotella, Mimmo, 182n57
Rothko, Mark, , , __
Rubin Gallery,
"Run for Love" (comic strip), 23–25, __
Ruscha, Ed, 182n57
"Sally Goes Round the Roses" (Jaynetts), , 162n74
Sartwell, Crispin, ,
_Saturday's Popeye_ (Warhol), , 171n79, plate 6
Schelstraete, Peter,
Schjeldahl, Peter, ,
Schlitz beer, , 100–101
Schneider, Louis, , ,
Schramm, Wilbur, , , , , , ,
_Screen Tests_ (Warhol film), ,
Scull, Ethel, , 190n184, 191n186
Scull, Robert, , 190n184, 191n186
_Secret Hearts_ (DC Comics), 23–25, __
Sedgwick, Edie, __ , 132–33, , , __ , 191n17
Sennett, Richard, , ,
sexuality and art: deglamorization as, ; Muschinski and Oldenburg on, , 32–34; vulgarity and, 114–22. _See also_ homosexuality and art
Shaviro, Steven, ,
Shaw, George Bernard,
Shinohara, Ushio, 182n57
shopping. _See_ consumerism
Shuster, Joe,
Sidney Janis Gallery, 174n2
silkscreen printing, , , 40–42, , , 72–73, , , , ,
Silva, Jennifer, , 149–50, 192n19
_Silver Clouds_ (Warhol),
Silver Factory. _See_ Factory Superstars
Silverman, Kaja, 168n32
Skeggs, Beverley, , , , 161n42
_Sleep_ (Warhol film), , , 187n100
Smith, David,
Smithson, Robert,
_Soap Opera_ (Warhol film), ,
social class. _See_ class consciousness; class stereotypes
social mobility: , 8–10, , , , , , , 173n120; deglamorization in film, ; individualism and, , , 123–25; Warhol on, , , , 146–50, . _See also_ class consciousness; neoliberalism; participatory culture
Social Research, Inc.,
Soglow, Otto, . _See also_ "Little King, The" (comic strip)
Sokol, Robert,
Sonnabend Gallery, , 174n2
Sontag, Susan,
_Speak, Memory_ (Nabokov),
spectacle, 60–61, , , , 135–40
spectacle of failure, , , , 73–74, , , ,
Spiegelman, Art,
Stable Gallery, , , , 174n2
Stahl, Matt, , , , ,
Stallybrass, Peter, , , ,
Staniszewski, Mary Anne, ,
Stein, Gertrude,
Steinbeck, John, 14–15, 162n61
Stewart, Martha,
Stimson, Blake, , , , 180n126, 186n82
_Store/Ray Gun Mfg. Co., The_ (Muschinski and Oldenburg), 28–30, __
_Strange Suspense Stories_ (DC Comics), , plate 4
_Strong Arms and Broads_ (Warhol),
Superman, , , , 52–53, , , , 170n64, 170n67, 170n69
_Superman_ (Warhol), , , 53–54, 55–56, , 170n69, plate 6, plate 7
_Superman's Girl Friend Lois Lane_ (DC Comics), 56–58, __ , __ , 150–51, plate 8
Swenson, G. R.,
Tamás, G. M., , 191n6, 193n56
Tavel, Ronald, , , __ , 166n67
_Telephone [1]_ (Warhol),
television, , , 51–53, , , , , 143–45, , , , , 170n58. _See also_ consumerism; popular culture
_Tell Me How Long the Train's Been Gone_ (Baldwin), 72–73, 173n120
_13 Most Wanted Men_ (Warhol), 125–26, __
_Three Marilyns_ (Warhol), plate 16
_Time_ (magazine), ,
_Tobacco Road_ (film), , __
Tomkins, Calvin, , 169n47, 191n186
Toyota,
_Troy_ (Warhol), 42–43, __ ,
_True Romance_ (magazine), , 176n52
_True Story_ (magazine), , __ , , 179n105
_Tunafish Disaster_ (Warhol), , 77–79, __ , ,
Tworkov, Jack,
Tyler, William D., ,
_Untitled (Seagram Mural)_ (Rothko), __
Updike, John, 175n31
Uroskie, Andrew, 187n100
Varnedoe, Kirk,
Veblen, Thorstein, , , 185n69
Velvet Underground, , , 162n68
_Village Voice_ ,
_Vinyl_ (Warhol film), , 133–34
_Vogue_ , ,
Vostell, Wolf, 182n57
voting, 155–57
vulgarity: abstract expressionism and, 114–17; death and, ; Huxley on, ; Johns and, ; Krauss on, ; Lichtenstein on, 21–22, ; mass culture and, , , 185n69; pop art and, 17–18, , , 185n70; queer and, 119–22; Rauschenberg and, ; Warhol and, , , , 42–47, , 117–24, . _See also_ primitivism
Wacquant, Loïc,
Wagner, Anne, , 163n12, 172n92, 179n95
Walker, Kara,
Wallowitch, Edward,
_Wall Street Journal_ ,
war comics, 26–27
Ward, Eleanor,
Warhol, Andy, __ , __ , __ ; _The Andy Warhol Diaries_ , , , , , 162n53, 190n180, 192n19; assassination attempt and treatment of, , 161n48; class consciousness and, 4–11, , 106–7, , 160n33, 161n40, 161n42; cultural contemporaries of, , , 177n57; on diamonds, 173n114; early life of, , , , 161n40; egalitarianism and, , 122–26; on fashion, , ; on fifteen minutes of fame, 185n81; financial situation of, , , 160n25, 171n80; Geldzahler and, 140–43; on _Henry Geldzahler_ film, 135–36; homosexuality and art and, , , , , 112–14, 124–25, ; left-handedness of, 54n74; on likeness to Lichtenstein's art, , 159n1; mechanical skills of, 63–65, 172n104; mother of, , , plate 17; on music, –16; on participatory culture, –4, , , –40, –57, n69; _The Philosophy of Andy Warhol_ , ; political engagement of, –57; on pop culture, –37; presents for, _108_ , , n127; universalism of, n57. _See also_ Warhol, Andy—artwork; Warhol, Andy—films; _specific titles_
Warhol, Andy—artwork: abstract expressionism and, 1–3, 11–13, , , , , , 110–23; amateurism and, 127–29, 186n95, 187n106; Bonwit Teller store exhibit, 49–50, 159n1, 169n47, plate 6; brand image art, –76, –102; critical reviews of, –74, , n115; gallery exhibits of, , , , , , , n2; incompleteness and scrawls on, , , , –56, –68, , –54; irregularities in, –3; mediums and modalities of, , n67; modernism and, –12, –26; newspaper ad work of, , , n80; portraiture work by, , , –35, –43; production techniques in, , , –98, n110; reproductive gadgetry for, , –64, , –5, –28; sexuality and, , –14, ; source material of, , –94, n111; vulgarity of, , , –47, –24. _See also_ Warhol, Andy; _specific artworks_
Warhol, Andy—films: amateurism of, 127–29, 186n95, 187n106; egalitarian cinema, 126–34. _See also_ Warhol, Andy; _specific titles_
Warhola, Julia, , , , plate 17
"Warhol revels with gift 'trash'" (Platzer), __
_Warren_ (Warhol), __ ,
_Washes_ (Oldenburg), , plate 20
_Washington Post_ ,
Washington School of Art,
Weaver, Neal, , 188n125
Wesselmann, Tom, 35–36, 182n57
West, Kanye,
White, Allon, , , ,
whiteness, , , , , 107–9, 180n125, 180n126. _See also_ race and advertising
Whitman, Walt,
_Wigs_ (Warhol),
Willard, Charlotte,
Winogrand, Garry, 182n57
_Winter of Our Discontent, The_ (Steinbeck), 14–15
Wood, James Playsted, 174n128
working class. _See_ class consciousness; class stereotypes; poverty
_Workingman's Wife_ (Social Research, Inc.),
_You're In_ (Warhol), 177n59
Plate 1. Andy Warhol, _Peach Halves_ , 1961. Casein and wax crayon on linen, 70 × 54 inches. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 2. Andy Warhol, mechanical source for the painting _Peach Halves_ , 1961. 13½ × 10¾ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 3. Roy Lichtenstein, _Brushstrokes_ , 1965. Oil and magna on canvas, 48¼ × 48¼ inches. © Estate of Roy Lichtenstein.
Plate 4. _Strange Suspense Stories_ , October 1964, 23. All DC comic artwork, its characters, and related elements are trademarks of and copyright DC Comics or their respective owners.
Plate 5. Andy Warhol, _A Boy for Meg [2]_ , 1962. Oil on canvas, 72 × 52 inches. National Gallery of Art. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 6. Nathan Gluck, _Five Andy Warhol Paintings in Bonwit Teller Window Display_ , New York, April 1961. C-print. Photo © Nathan Gluck Estate, courtesy of Luis De Jesus Los Angeles.
Plate 7. Andy Warhol, _Superman_ , 1961. Casein and wax crayon on cotton duck, 67 × 52 inches. © 2016 Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 8. _Superman's Girl Friend Lois Lane_ , April 1961, 5. All DC comic artwork, its characters, and related elements are trademarks of and copyright DC Comics or their respective owners.
Plate 9. Andy Warhol, _Do It Yourself (Sailboats)_ , 1962. Acrylic, graphite, and Letraset on linen, 72¼ × 100 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 10. Andy Warhol, _Campbell's Soup Can (Old Fashioned Tomato Rice)_ , 1961. Casein on cotton, 48 × 41 inches. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 11. Andy Warhol, _Big Torn Campbell's Soup Can (Pepper Pot)_ , 1962. Casein and graphite on canvas, 71⅝ × 52 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 12. Campbell's Soup Advertisement, _Life Magazine_ , September 8, 1961, 59.
Plate 13. Andy Warhol, _Brillo Soap Pads Box_ , 1964. Silkscreen ink and house paint on plywood, 17 × 17 × 14 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 14. Jean-Michel Basquiat, _Panel of Experts_ , 1982. Acrylic and oil pastel on paper mounted on canvas, 5 x 5 feet. Montreal Museum of Fine Arts, gift of Ira Young. Photo: The Montreal Museum of Fine Arts. © Estate of Jean-Michel Basquiat/ADAGP, Paris/ARS, New York 2016.
Plate 15. Willem de Kooning, _Marilyn Monroe_ , 1954. Oil on canvas, 50 × 30 inches. Collection Neuberger Museum of Art, Purchase College, State University of New York. Gift of Roy R. Neuberger. Photo: Jim Frank. © 2016 Willem de Kooning Foundation/Artists Rights Society (ARS), NY.
Plate 16. Andy Warhol, _Three Marilyns_ , 1962. Acrylic, silkscreen ink, and graphite on linen, 14 × 33½ inches. Collection of the Andy Warhol Museum, Pittsburgh. © Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 17. Andy Warhol, _Mrs. Warhol_ , 1965. 16 mm film, color, sound, 67 minutes. © 2016 Andy Warhol Museum, Pittsburgh, a museum of Carnegie Institute. All rights reserved.
Plate 18. Andy Warhol, _The Little King_ , 1961. Casein on canvas, 46 × 40 inches. © 2016 Andy Warhol Foundation for the Visual Arts/Artists Rights Society (ARS), NY.
Plate 19. _New York Journal American_ , April 2, 1961, source image for painting _The Little King_ , 1961. Ink on newsprint, 21½ × 15 inches. Collection of the Andy Warhol Museum, Pittsburgh. © Little Kings. Distributed by King Features Syndicate, Inc.
Plate 20. Henry Geldzahler performs in _Washes_ , a happening by Claes Oldenburg, Al Roon's Health Club, New York, NY, May 22 and 23, 1965. Photograph by Robert McElroy. © Claes Oldenburg.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 5,990
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\section{Introduction}
In the SIR model, individuals are in one of three states: $S=$ susceptible, $I=$ infected, $R=$ removed (cannot be infected). Often this epidemic takes place in a homogeneously mixing population. However, here, we have a graph $G$ that gives the social structure of the population; vertices represent individuals and edges connections between them. $S-I$ edges become $I-I$ at rate $\lambda$, i.e., after a time with an exponential($\lambda$) distribution. An individual remains infected for an amount of time $T$ which can be either deterministic or random with a pre-specified distribution. Once individuals leave the infected state, they enter the removed state. In addition, we will allow the graph to evolve: $S-I$ edges are broken at rate $\rho$ and the susceptible individual connects to an individual chosen uniformly at random from the graph. This process is called evoSIR where `evo' stands for `evolving'. We will also consider the simpler $SI$ epidemic in which infecteds never recover, and its evolving version $evoSI$, as well as some other variations on this theme.
We will use the term \emph{final epidemic size} or \emph{final size of the epidemic} to refer to the number of vertices that are eventually removed in SIR epidemics or eventually infected in SI epidemics. We say a large epidemic (or large outbreak) occurs if the epidemic infects more than $\epsilon n$ individuals ($n$ is the size of the total population) for some $\epsilon>0$ independent of $n$.
The critical value is the smallest infection rate such that a large outbreak occurs with probability bounded away from 0 as $n\to\infty$.
The introduction is organized into eight subsections. Sections \ref{sec;DOM}--\ref{sec:BB} are devoted to a review of previous work. The main results of the paper (Theorem
\ref{critical_value} and Theorem \ref{Q1new}) are presented in Section \ref{sec:state}. Sections \ref{pfth15}--\ref{pfth18} sketch the proof of Theorem \ref{Q1new}.
\subsection{DOMath \cite{DOMath}}\label{sec;DOM}
evoSIR was stiudied by three Duke undergraduates in the summer and fall of 2018 (Yufeng Jiang, Remy Kassem, and Grayson York) under the direction of Matthew Junge and Rick Durrett. They considered two possibilities for the infection time: the Markovian case in which infections last for an exponential time with mean 1, and the case in which each infection lasts for exactly time 1. Here we restrict out attention to their results for the second case, which is simpler due to its connection with independent bond percolation. In any SIR model each edge will be $S-I$ (or $I-S$) only once. When that happens, in the fixed infection time case without rewiring the infection will be transferred
to the other end with probability
\begin{equation}
\tau^f = \mathbb{P}(T \le 1 ) = 1-e^{-\lambda},
\label{tauf}
\end{equation}
and the transfers for different edges are independent. Here the `$f$' in the superscript is for ``fixed time." Due to the last observation, we can delete edges with probability $e^{-\lambda}$ and the connected components of the resulting graph will give the epidemic sizes when one member of the cluster is infected.
In \cite{DOMath} $G$ was an Erd\"os-R\'enyi($n,\mu/n$) random graph in which there are $n$ vertices and each pair is independently connected with probability $\mu/n$. The following result is well-known.
\begin{theorem}\label{thin}
Consider the SIR process
on Erd\"os-R\'enyi($n,\mu/n$) with fixed infection time. The reduced graph after deletion of edges as described above is Erd\"os-R\'enyi$(n,\mu\tau^f/n)$. So, if we start with one infected and the rest of the population susceptible, a large outbreak occurs with positive probability if and only if $\mu\tau^f > 1$. If $z_0$ is the fixed point smaller than $1$ of the generating function
\begin{equation}
G_0(z) = \exp(-\mu\tau^f (1-z)),
\label{ftgf}
\end{equation}
then $1-z_0$ gives both the limiting probability that an infected individual will start a large epidemic, and the fraction of individuals who will become infected when a large epidemic occurs.
\end{theorem}
Things become more complicated when we introduce rewiring of $S-I$ edges at rate $\rho$. To be able to use the ideas in the proof of Theorem \ref{thin}, \cite{DOMath} introduced the delSIR model in which edges are deleted instead of rewired. In the evo (or del) version of the model, in order for the infection to be transmitted along an edge, infection must come before any rewiring (or deletion) and before time 1. To compute this probability, note that (i) the probability that infection occurs before rewiring is $\lambda/(\lambda+\rho)$ and (ii) the minimum of two independent exponentials with rates $\lambda$ and $\rho$ is an exponential with rate $\lambda+\rho$, so the transmission probability is
\begin{equation}
\tau^f_r = \frac{\lambda}{\lambda+\rho}(1-e^{-(\lambda+\rho)}).
\label{taufr}
\end{equation}
Here the `$r$' subscript is for ``rewire." As can be shown by a standard argument,
\begin{lemma} \label{couple}
For fixed parameters, there exists a coupling of evoSI and delSI so that there are no fewer infections in the delSI model than in evoSI. Same is true if we replace SI by SIR.
\end{lemma}
The next result, Theorem 1 in \cite{DOMath}, shows that evoSIR has the same critical value as delSIR, and in the subcritical case the expected cluster sizes are the same.
\begin{theorem}\label{ftcrit}
The critical value $\lambda_c$ for a large epidemic in fixed infection time delSIR or evoSIR epidemic with rewiring is given by the solution of $\mu\tau^f_r=1$.
Moreover, if $\lambda < \lambda_c$, then the ratio of the expected epidemic size in delSIR to the size in evoSIR converges to 1 as the number of vertices goes to $\infty$.
\end{theorem}
The formula for the critical value is easily seen to be correct for the delSIR since, by the reasoning above, there is a large epidemic if and only if the reduced graph in which edges are retained with probability $\tau^f_r$ has a giant component. From Lemma \ref{couple} one can see that the delSIR model has a larger ($\ge$) critical value than evoSIR. Thus, one only has to prove the reverse inequality. Intuitively, the equality of the two critical values holds because a subcritical delSIR epidemic dies out quickly, so it is unlikely that rewirings will influence the outcome.
When $n$ is large, the degree distribution, which is Binomial($n-1,\mu/n$), is approximately Poisson with mean $\mu$. Due to Poisson thinning, the number of new infections directly caused by one $I$ in delSIR in an otherwise susceptible population is asymptotically Poisson with mean $\mu\tau^f_r$, and hence has limiting generating function of the distribution Poisson($\mu\tau^f$)
\begin{equation}
G_1(z) = \exp(-\mu\tau^f_r (1-z)).
\label{hatG}
\end{equation}
The following result, Theorem 2 in \cite{DOMath}, identifies the probability of a large outbreak.
\begin{theorem} \label{ftperc}
If $z_0$ is the fixed point $< 1$ of $G_1(z)$, then $1-z_0$ gives the probability of a large delSIR or evoSIR outbreak.
\end{theorem}
\noindent
In the case of the delSIR model, $1-z_0$ is the fraction of individuals infected in a large epidemic. It is easy to see that this proportion goes to 0
at the critical value $\mu_c = 1/\tau_r ^f =1$. The next simulation suggests that this is not true in the case of evoSIR.
\begin{figure}[h]
\centering
\includegraphics[width=4in,keepaspectratio]{ctvarylambda1231}
\caption{Simulation of the fixed time evoSIR on an Erd\"os-R\'enyi\, graph with $\mu=5$, $\rho=4$ and $\lambda$ varying. $\lambda_c \approx 1.0084$ in agreement with Theorem \ref{ftcrit}. The bottom curve is the final size of the delSIR epidemic with the same parameters. The dashed line above it is an approximation derived in \cite{DOMath} that turned out to be not very accurate. The top curve comes from simulating evoSIR.}
\label{fig:disco}
\end{figure}
\subsection{Britton et al. \cite{BJS,LBSB} }\label{britton}
As the authors of \cite{DOMath} were finishing up the writing of their paper, they learned of two papers by Britton and collaborators that study epidemics on evolving graphs with exponential infection times. \cite{BJS} studies a one parameter family of models (SIR-$\omega$) that interpolates between delSIR and evoSIR. To facilitate later referencing we attach labels to the next two descriptions.
\begin{model}\label{siromega}
{\bf SIR-$\omega$ epidemic.} In this model, an infected individual infects each neighbor at rate $\lambda$, and recovers at rate $\gamma$. A susceptible individual drops its connection to an infected individual at rate $\omega$. The edge is rewired with probability $\alpha$ and dropped with probability $1-\alpha$. Since evoSIR ($\alpha=1$) and delSIR ($\alpha=0$) have the same critical values and survival probability it follows that this holds for all $0 \le\alpha\le 1$.
\end{model}
These epidemics take place on graphs generated by
\begin{model} \label{config}
{\bf Configuration model.} Given a nonnegative integer $n$ and a positive integer valued random variable $D$, take $n$ i.i.d.~copies $D_1,\ldots, D_n$ of $D$ and condition the sum $\sum_{i=1}^n D_i$ to be even. We then construct a graph $G$ on $n$ vertices as follows. We attach $D_1,\ldots, D_n$ half-edges to vertices $1, 2\ldots, n$, respectively and then pair these half-edges uniformly at random to form a graph. We call this random graph \textit{the configuration model} on $n$ vertices with degree distribution $D$ and denote it by $\mathbb{CM}(n,D)$. We assume $D$ has finite second moment so that the resulting graph will be a simple graph with nonvanishing probability as $n\to\infty$. See Theorem 3.1.2 in \cite{RGD}. We refer readers to \cite[Chapter 7]{vdH1}, \cite[Chapters 4 and 7]{vdH2} and \cite[Chapter 2]{vdH3} for more details on the configuration model.
\end{model}
\begin{remark}\label{rem:config}
Throughout the paper, unless otherwise specified, we always consider the annealed probability measure with respect to the configuration model. In other words the randomness is taken over both the degrees $D_1, \ldots, D_n$ and the construction of $\mathbb{CM}(n,D)$ based on the degrees.
\end{remark}
Britton, Juher, and Saldana \cite{BJS} studied the initial phase of the epidemic starting with one infected at vertex $x$ (chosen uniformly at random from all vertices) using a branching process approximation. Let $Z^n_m$ be the number of vertices at distance $m$ from $x$ in the graph $\mathbb{CM}(n,D)$. For any fixed $k\in \mathbb{N}$, $\{Z^n_m, 0\leq m\leq k\}$ converges to the following two-phase branching process $\{Z_m,0\leq m\leq k\}$.
The number of children in the first generation has the distribution $D$ while subsequent generations have the distribution $D^*-1$ where $D^*$ is the {\it size-biased} degree distribution
$$
\mathbb{P}( D^* = j) = \frac{jp_j}{m_1}, \quad j\geq 0.
$$
Here $p_j=\mathbb{P}(D=j)$ and $m_1 = \mathbb{E}(D)$ is the mean of $D$.
Later we will also use $m_i:=\mathbb{E}(D^i)$ to denote the $i$-th moment of $D$ for $i\geq 1$. This follows from the construction of the configuration model: $x$ connects to other vertices with probability proportional to their degrees so individuals in generations $m \ge 1$ have the $D^*-1$ children instead of $D$. (One edge is used in making the connection from $x$.)
Before moving on to epidemics on the configuration model, we note that if
\begin{equation}
G(z) = \sum_{k=0}^\infty p_k z^k,
\label{F_2}
\end{equation}
then the generating function of $D^*-1$ is
\begin{equation}
\hat{G}(z) = \sum_{j-1}^\infty \frac{jp_j}{m_1} z^{j-1}
= \frac{G'(z)}{G'(1)}.
\label{G1f}
\end{equation}
In the SIR-$\omega$ model, the probability that an infection will cross an $S-I$ edge is
$$
\tau=\frac{\lambda}{\lambda+\gamma + \omega}.
$$
Thus we get another two-phase branching process $\bar Z_m$ defined as follows. $\bar Z_0=1$, $\bar Z_1=\mbox{Binomial}(D,\tau)$ and future generations have offspring distrbution Binomial$(D^*-1,\tau)$.
One can see from this description that the limiting branching process $\bar Z_m$ will have positive survival probability if
$$
1 < \tau \mathbb{E}(D^*-1) = \tau \left(\frac{m_2}{m_1} - 1 \right)= \frac{(m_2-m_1)\tau}{m_1}.
$$
Correspondingly,
there will be a large epidemic in the SIR-$\omega$ model if
\begin{equation}
R_0 = \frac{\lambda}{\lambda+\gamma + \omega} \cdot \frac{m_2-m_1}{m_1} > 1.
\label{R0omega}
\end{equation}
This follows from results on percolation in random graphs (see \cite{Fou} and \cite{Jperc}).
Thus for fixed values of $\gamma$ and $\omega$
\begin{equation}
\lambda_c = (\gamma+\omega) \frac{m_1}{m_2-2m_1}.
\label{crvomega}
\end{equation}
When $\gamma=0$ and $\omega=\rho$ which is the SI-$\omega$ model in our notation,
\begin{equation}
\lambda_c = \frac{\rho m_1}{m_2-2m_1}.
\label{crvSI}
\end{equation}
The critical values of delSI and evoSI only depends on the ratio $\rho/\lambda$, so it is natural to define a parameter (this $\alpha$ is different from the $\alpha$ used in the definition of SIR-$\omega$ model)
\begin{equation}\label{alpha}
\alpha=\rho m_1/\lambda
\end{equation}
that has $\alpha_c =m_2-2m_1$.
We will only consider the del and evo endpoints of the one parameter family of models SIR-$\omega$, so after the discussion of previous work is completed, there should be no confusion between our $\alpha$ and theirs.
Work of Leung, Ball, Sirl, and Britton \cite{LBSB} demonstrated the paradoxical fact that individual preventative measures may leader to a larger final size of the epidemic. They proved this rigorously for SI epidemics on the configuration model with two degrees and conducted simulation studies for many social networks. Note that in Figure \ref{fig2} (taken from Figure 1 of \cite{LBSB}) the final size increases with the rewirng rate when it is small. This simulation does not suggest that the phase transition was discontinuous.
\begin{figure}[h!]
\centering
\includegraphics[width=4in,keepaspectratio]{LBSBfig1sm.jpg}
\caption{Social distancing can lead to an increase in the final epidemic size in the configuration model. The horizontal line is the final size when $\omega=0$.}
\label{fig2}
\end{figure}
Figure \ref{fig:disco2} from \cite{DOMath} gives a similar simulation that clearly shows the discontinuity.
\newpage
\begin{figure}[h!]
\centering
\includegraphics[width=4in,keepaspectratio]{ctvaryrho1231.jpg}
\caption{Simulation of a continuous time Erd\"os-R\'enyi \ graph with $\mu=5$, $\gamma=1$, $\lambda =1$ and $\rho$ varying. Note that $\rho_c \approx 3$, the value predicted by \eqref{crvomega}. The lowest line is the final size of delSIR which is continuous. The dotted and dashed lines are approximations derived in \cite{DOMath} that turned out to be not very accurate. The top curve comes from simulating evoSIR.}
\label{fig:disco2}
\end{figure}
\subsection{Yao and Durrett \cite{YD1}}\label{evosi}
To begin to explain the ideas behind the analysis of epidemics on evolving graphs attempted in \cite{YD1} we need to recall some history. Volz \cite{Volz} was the first to derive a limiting ODE for an SIR epidemic on a graph generated by the configuration model. Miller \cite{Miller} later simplified the derivation to produce a single ODE. The results of Volz and Miller were based on heuristic computations, but later their conclusion was made rigorous by Decreusfond et al \cite{Decr} assuming $\mathbb{E} (D^5)<\infty$.
Janson, Lukzak, and Windridge \cite{JLW} proved the result under more natural assumptions. They studied the epidemic on the graph by revealing its edges dynamically while the epidemic spreads. To be precise they call a half-edge free if it has not yet been paired with another half-edge. They call a half-edge susceptible, infected or removed according to the state of its vertex. To modify their construction to include rewiring Yao and Durrett \cite{YD1} added the third bullet below. Hereafter we use``randomly chosen'' and ``at random'' to mean that the distribution of the choice is uniform over the set of possibilities.
\begin{itemize}
\item
Each free infected half-edge chooses a free half-edge at rate $\lambda$.
Together the pair forms an edge and is removed from the collection of half-edges. If the pairing is with a susceptible half-edge then
its vertex becomes infected and all its edges become infected half-edges.
\item
Infected vertices recover and enter the removed state at rate 1.
\item
Each infected half-edge gets removed from the vertex that it is attached to at rate $\rho$ and immediately becomes re-attached to a randomly chosen vertex.
\end{itemize}
\cite{YD1} began by investigating evoSI for which they obtained a very detailed result. Recall that $G(z) = \sum_{k=0}^{\infty} p_k z^k$ is the generating function of the degree distribution $D$, $\alpha = \rho m_1/\lambda$ and consider the function
\begin{equation}\label{defoff}
f(w)=\log \left(\frac{m_1 w}{G'(w)+\alpha (1-w)G(w)}\right) + \frac{\alpha}{2}(w-1)^2.
\end{equation}
\medskip\noindent
{\bf Theorem 2 in \cite{YD1}.}
{\it Consider the evoSI epidemic on the configuration model $\mathbb{CM}(n,D)$ with one uniformly randomly chosen vertex initially infected. Suppose $\alpha<\alpha_c$ so that we are in the supercritical regime and let
\begin{align}
\label{mM1}
\sigma &=\sup\{w:0<w<1,f(w)= 0\} \quad\hbox{with $\sup(\emptyset) = 0$}, \\
\label{limitsize1}
\nu& =1-\exp\left(-\frac{\alpha}{2}(\sigma-1)^2 \right)G(\sigma).
\end{align}
If we suppose
\medskip
$(\star)$ either $\sigma=0$ or $0<\sigma<1$ and there is a $\delta>0$ so that $f<0$ on $(\sigma-\delta,\sigma)$,
\medskip\noindent
then for any $\epsilon>0$,}
$$
\lim_{n\to\infty} \mathbb{P}(I_{\infty}/n<\nu+\epsilon )=
\lim_{\eta\to 0}\liminf_{n\to\infty} \mathbb{P}(I_{\infty}/n>\nu-\epsilon |I_{\infty}/n>\eta)=1.
$$
\noindent
Although $(\star)$ may look strange as an assumption, it is easy to verify in concrete examples.
The analysis in \cite{YD1} that led to this conclusion is correct, but as pointed out by Ball and Britton \cite{BB} the construction produces a model in which certain $I-I$ edges rewire at rate $2\rho$. As explained in the next section the formula in \eqref{limitsize1} for the final size of the epidemic is not correct.
All is not lost. As we will explain later, the model with additional $I-I$ rewiring provides an upper bound for evoSI process which allows us to prove the condition for a continuous transition. By defining a lower bounding process, we can show that Theorem 3 from \cite{YD1} that gives almost necessary and sufficient conditions for a discontinuous phase transition in evoSI is correct. See Section \ref{sec:state} for details. At this point we are not able to prove Theorem 6 from \cite{YD1} for evoSIR, even though results described in the next section suggest it is correct in the Erd\"os-R\'enyi \ case.
\subsection{Ball and Britton \cite{BB}} \label{sec:BB}
Yao and Durrett \cite{YD1} submitted their paper to Electronic Journal of Probability in March 2020. Roughly in the middle of the eight-month wait for referees' reports, Ball and Britton posted their preprint on the arXiv in August 2020.
Their
analysis is restricted to the Erd\"os-R\'enyi\ graphs because their construction
uses properties that are special to that case. See their Section 2.3 for
details and also \cite{BrOn,Neal}
for earlier examples of the use of this
construction. As we will explain in this section their results are not definitive. In their results for evoSIR there is a gap between the necessary and sufficient condition for a discontinuous phase transition.
To prove results about the epidemics on an Erd\"os-R\'enyi \ random graph with mean degree $\mu$ they first consider a tree in which each vertex has a Poisson($\mu$) number of descendants and develop a branching process approximation for the SIR-$\omega$ epidemic in which infections (births) cross an edge with probability $\lambda/(\lambda+\gamma+\omega)$. Let $I(t)$ be the total number of individuals, $I_E(t)$ be the number of infectious edges, and $T(t)$ be the total progeny in the branching process on the tree. Let $I^n(t)$, $I^n_E(t)$ and $T^n(t)$ be the corresponding quantities for an Erd\"os-R\'enyi($n,\mu/n$) random graph on $n$ vertices where initially a randomly chosen vertex is infected. They show in their Theorem 2.1 that if $t_n = \inf\{ t : T(t) \ge \log n\}$ then the two systems can be defined on the same space so that
$$
\sup_{0 \le t \le t_n} | ( I^n(t), I^n_E(t), T^n(t)) - ( I(t), I_E(t), T(t)) |
\xrightarrow{\mathbb{P}} 0
$$
as $n\to\infty$.
Let $S^n(t)$ be the number of susceptibles at time $t$ and
$W^n(t)$ be the number of susceptible-susceptible edges created by rewiring by time $t$ and let $X^n(t) = ( S^n(t), I^n(t), I^n_E(t), W^n(t))$. Let $x(t) =(s(t), i(t), i_E(t),w(t))$ be the solution of the ODE.
\begin{align}
\frac{ds}{dt} & = -\lambda i_E,
\nonumber \\
\frac{di}{dt} & = - \gamma i + \lambda i_E,
\label{BBode} \\
\frac{di_E}{dt} & = -\lambda i_E + \lambda\mu i_E s - \lambda \frac{i_E^2}{2}
+ 2\lambda i_E \frac{w}{s} -\omega i_E(1-\alpha + \alpha (1-i)),
\nonumber\\
\frac{dw}{dt} & = w\alpha i_E s - 2\lambda i_E \frac{w}{s}.
\nonumber
\end{align}
\noindent
Here $\alpha$ is the probability an edge is rewired as in the definition of SIR-$\omega$ model.
Theorem 2.2 in \cite{BB}, which is proved by using results of Darling and Norris \cite{DarNor}, shows that for any $t_0>0$
$$
\sup_{0 \le t \le t_0} | X^n(t)/n - x(t) | \xrightarrow{\mathbb{P}} 0
$$
as $n\to\infty$, provided that $I^n(t)/n\to i(0)>0$. It is interesting to note, see their Section 3, that the ODE in \eqref{BBode} is closely related to the ``pair approximation'' for SIR-$\omega$ model.
To explain the phrase in quotes, we note that ``mean-field equations'' come from pretending that the states of site are independent; the pair approximation from assuming it is a Markov chain. In practice, this approach means that probabilities involving three sites are reduced to probabilities involving 1 and 2 sites using a conditional independence property. For the details of the computation see Chapter 7 in \cite{AB}.
Letting $T^{n} = n - S^n(\infty)$ (which is the final size of the epidemic) they make Conjecture 2.1 that one can interchange two limits $n\to\infty$ and $t\to\infty$ to conclude
$$
T^n/n \to 1 - s(\infty).
$$
To formulate a result that is independent of the validity of the conjecture they
let $$x^\epsilon(t) = (s^\epsilon(t),i^\epsilon(t),i_E(t),w^\epsilon(t))$$ be the solution to the ODE
when
$$
x^\epsilon(0) = (1-\epsilon,\epsilon,L^{-1}\epsilon,0) \quad\hbox{where}\quad
L = \frac{\lambda}{\lambda(\mu-1) - \omega}.
$$
Letting $\tau_{SIR} = 1-\lim_{\epsilon\downarrow 0} s^\epsilon(\infty)$, their Theorem 2.3 states
\begin{equation}
\lim_{\lambda \downarrow \lambda_c} \tau_{SIR}
\begin{cases} = 0
&\hbox{ if $\gamma > \omega(2\alpha-1)$
or $\mu < 2\omega\alpha/[\omega(2\alpha-1) - \gamma]$,}\\
> 0
&\hbox{ if $\gamma < \omega(2\alpha-1)$
and $\mu > 2\omega\alpha/[\omega(2\alpha-1) - \gamma]$.}
\end{cases}
\label{BBSIR}
\end{equation}
When $\alpha=1$, $\omega=\rho$ and $\gamma=1$ then we have a discontinuous phase transition if
$$
\rho>1 \quad\hbox{and} \quad \mu > \frac{2\rho}{\rho-1},
$$
which are exactly the conditions that comes from Theorem 6 in \cite{YD1}.
When $\alpha=0$, $\omega=\rho$ and $\gamma=1$, i.e., the delSIR model studied in \cite{DOMath}, the transition is always continuous. This follows from Theorem 2.3 since $1 > -\rho$. From the last calculation we see that the phase transition is always continuous if $\alpha < 1/2$.
In the case of the SI-$\omega$ model they show (see Theorem 2.6 of \cite{BB}) that if $\mu > 1$ and $\omega$ and $\alpha$ are held fixed then the phase transition is discontinuous if and only if $\alpha > 1/3$ and $\mu > 3\alpha/(3\alpha-1)$. When $\alpha=1$ this is $\mu > 3/2$ which is the condition in Example \ref{PoiSI} in Section \ref{sec:state}.
Theorem 2.4 in \cite{BB} gives results for the epidemic starting from a single infected individual. If we let $\tau^1_{SIR}$ to be the limiting size of an epidemic conditionally on it being large then
$$
\lim_{\lambda \downarrow \lambda_c} \tau^1_{SIR}
\begin{cases} = 0
&\hbox{ if $\gamma > \omega(3\alpha-1)$
or $\mu < 3\omega\alpha/[\omega(3\alpha-1) - \gamma]$,}\\
> 0
&\hbox{ if $\gamma < \omega(2\alpha-1)$
and $\mu > 2\omega\alpha/[\omega(2\alpha-1) - \gamma]$.}
\end{cases}
$$
See their paper for a precise statement. Remark 2.4 in \cite{BB} states the conjecture that the 3's in the first condition should be 2's.
For SI-$\omega$ this model they compute
$$
\lim_{\lambda\downarrow \lambda_c} \tau_{SI} := \tau_0(\mu,\alpha).
$$
To give the value of $\tau_0(\mu,\alpha)$, we need some notations. For $\mu>1$ and $\alpha\in[0,1]$ let
$$
\theta(\mu,\alpha) = \frac{2\alpha(\mu-1)}{\mu + \alpha(\mu-1)}\mbox{ and }
f_0(x) = \log(1-x) + \frac{x}{1-\theta(\mu,\alpha)},
$$
then
$\tau_0(\mu,\alpha)$ is the largest solution in $[0,1)$ of $f_0(x)=0$. In Section 4, Ball and Britton compare the result just stated with Theorem 2 in \cite{YD1}. They show in their Figure 5 that in the case $\mu=2$ their predicted final size $\tau_0(2,1)$ lies below the one from \cite{YD1}, while simulation results agree with $\tau_0(2,1)$.
\subsection{Statement of our main results}\label{sec:state}
From now on the reader can forget about the meaning of notations used by Ball and Britton. We fix $\rho$, the rewiring rate, and vary $\lambda$. We let $\alpha=\rho m_1/\lambda$. In view of the definition of
$\Delta$ in (1.5.1), the natural assumption is $\mathbb{E}(D^3) < \infty$. Some of our
results can be proved under this assumption, while some need something a
little stronger. To simplify things. \textbf{we assume $\mathbb{E}(D^5) < \infty$ throughout.}
\begin{theorem}\label{critical_value}
Consider the delSI model and evoSI model on $\mathbb{CM}(n,D)$.
(i) the critical values of delSI and evoSI are the same.
\begin{equation*}
\alpha_c=m_2-2m_1; \mbox{ equivalently, }\lambda_c=\frac{\rho (m_2-2m_1)}{m_1}.
\end{equation*} (ii) When $\alpha < \alpha_c$, which is the supercritical case, the probability of a large epidemic is the same in the two models, which is equal to the survival probability $q(\lambda)$ of the two-stage branching process $\bar Z_m$ defined in Section \ref{britton} (with the $\tau$ there equal to $\lambda/(\lambda+\rho)$ in our notation).
\end{theorem}
\noindent
The proof of Theorem \ref{critical_value} that we give in Section 2 is very similar to one for Theorem \ref{ftcrit} given in \cite{DOMath} for Erd\"os-R\'enyi \ random graphs. Here the fact that we have only $\mathbb{E}(D^5)<\infty$ rather than exponential upper bounds on $\mathbb{P}(D\ge k)$ changes some of the estimates.
Here and in what follows, formulas are sometimes easier to evaluate if we use the ``factorial moments'' $\mu_k = \mathbb{E}[D(D-1)\cdots (D-k+1)$, since these can be computed from the $k$-th derivative of the generating function. To translate between the two notations:
$$
\mu_1=m_1, \qquad \mu_2=m_2-m_1, \qquad \mu_3 = m_3 -3m_2 + 2m_1.
$$
In particular $\alpha_c = m_2-2m_1 = \mu_2 - \mu_1$.
\medskip
Our next result gives an almost sufficient and necessary condition for the discontinuous phase transition of evoSI.
\begin{theorem}\label{Q1new}
Consider the evoSI epidemic on the configuration model $\mathbb{CM}(n,D)$ with one uniformly randomly chosen vertex initially infected. Let
\begin{equation}
\Delta= -\frac{\mu_3}{\mu_1} + 3 (\mu_2-\mu_1).
\label{deldef}
\end{equation}
Let $I_{\infty}$ be the final epidemic size.
If $\Delta>0$, then there is a discontinuous phase transition. For some $\epsilon_0>0$ and some $\delta_0>0$,
\begin{equation}\label{delta>0new}
\lim_{\eta\to 0} \liminf_{n\to\infty}\mathbb{P}_{1}(I_{\infty}/n>\epsilon_0|I_{\infty}/n>\eta)=1 \quad\hbox{for all $\alpha_c-\delta_0<\alpha< \alpha_c$.}
\end{equation}
If $\Delta<0$, then there a continuous phase transition. For any $\epsilon>0$, there exists some $\delta>0$, so that
\begin{equation}\label{delta<0new}
\lim_{n\to\infty}\mathbb{P}_{1}(I_{\infty}/n>\epsilon)=0 \quad\hbox{for $\alpha_c-\delta<\alpha< \alpha_c$.}
\end{equation}
\end{theorem}
\noindent
To see what this result says we consider some examples.
\begin{example}
{\bf Random $r$-regular graph, $r\ge 3$.} Here $nr$ must be even.
If we choose the degree distribution $\mathbb{P}(D=r)=1$, and condition the graph to be simple, i.e., no self-loops or parallel edges
then the result is a random regular graph. The case $r=2$ is excluded because in that case the graph consists of a number of circles.
The critical value is $\alpha_c = m_2-2m_1 = r^2 - 2r > 0$ when $r>2$.
For $k\leq r$, $$\mu_k = r(r-1)\cdots (r-k+1),$$ so
\begin{align*}
\Delta & = -(r-1)(r-2) + 3(r(r-1) - r) \\
& = -(r-1)(r-2) + 3r(r-2) =(r-2)(2r+1) >0,
\end{align*}
and the phase transition is discontinuous for all $r\ge 3$.
\end{example}
\begin{example}
{\bf Geometric($p$).} The factorial moments are $\mu_1=1/p$, $\mu_2 = 2(1-p)/p^2$, and $\mu_3 = 6(1-p)^2/p^3$. $\alpha_c = \mu_2 - \mu_1 = (2-3p)/p^2$, so we need to take $p<2/3$ to have $\alpha_c>0$.
\begin{align*}
\Delta & = - \frac{6(1-p)^2}{p^2} + 3 \left( \frac{2(1-p)}{p^2} - \frac{1}{p} \right) \\
& = -\frac{6}{p^2} + \frac{12}{p} -6 + \frac{6}{p^2} - \frac{6}{p} - \frac{3}{p} = \frac{3}{p} - 6,
\end{align*}
so the phase transition is discontinuous if $p<1/2$.
\end{example}
Our last example concerns the configuration model generated from Poisson distribution.
\begin{example} \label{PoiSI}
{\bf Poisson($\mu$).} The factorial moments $\mu_k = \mu^k$, so the critical value $\alpha_c = \mu_2-\mu_1 = \mu^2-\mu$, which is positive if $\mu>1$. This condition is natural since if $\mu<1$ then there is no giant component in the graph and a large epidemic is impossible.
$$
\Delta= -\mu^2 +3(\mu^2-\mu) = 2\mu^2-3\mu^2,
$$
so the phase transition for evoSI is discontinuous if $\mu>3/2$, which is the result given in \cite{BB}.
\end{example}
\begin{remark}
We believe that the result of Example \ref{PoiSI} also holds for Erd\"os-R\'enyi($n,\mu/n$). To prove this rigorously, one first has to prove a quenched version of Theorem \ref{Q1new} (i.e., showng \eqref{delta>0new} and \eqref{delta<0new} hold with high probability over the degree sequence $D_1,\ldots, D_n$). We believe that this can be shown using the methods in the proof of Theorem \ref{Q1new}. Then one can transfer results for the configuration model generated from Poisson($\mu$) distribution to Erd\"os-R\'enyi($n,\mu/n$) using
\cite[Theorems 7.18 and 7.19]{vdH1}, which says that conditionally on having the same degrees, the random graphs generated from these two models have the same distribution.
\end{remark}
\subsection{Sketch of Proof of Theorem \ref{Q1new} }\label{pfth15}
As mentioned earlier the model analyzed in \cite{YD1} provides an upper bound on evoSI. That process, which we call avoSI (avo is short for avoid infection), is defined in Section \ref{sec:avoSI} where we prove that the final set of infected sites in avoSI stochastically dominates evoSI. To prove Theorem \ref{Q1new} we need a lower bounding process. That process, which we call AB-avoSI, is defined in Section \ref{sec:ABavoSI}, where we prove that the final set of infected sites in evoSI stochastically dominates AB-avoSI. The $AB$ in the name comes from the two counters associated with half-edges that prevent transmission of infections along $S-I$ edges created by $I-I$ rewirings.
The starting point to analyze evoSI via avoSI and AB-avoSI is the following Lemma \ref{avocrv}. Let $q(\lambda)$ be the survival probability for the two-phase branching process $\{\bar Z_m,m\geq 0\}$ introduced in Section \ref{britton}. Recall that the individual in the first generation has offspring distribution $\mbox{Binormial}(D,\lambda/(\lambda+\rho))$ while later generations have offspring distribution $\mbox{Binormial}(D^*-1,\lambda/(\lambda+\rho))$ where $D^*$ is the size-biased version of $D$,
$$
\mathbb{P}(D^*=j) = \frac{j \mathbb{P}(D=j)}{\mathbb{E}(D)},j\geq 0.
$$
\begin{lemma} \label{avocrv}
AvoSI, evoSI, AB-avoSI and delSI have the same critical value $\lambda_c$ and in the supercritical regime $\lambda>\lambda_c$ the probability of a large outbreak is equal to $q(\lambda)$ in all four models.
\end{lemma}
\begin{proof}
Results in Section 3.1 and 4.1 imply that
$$
\hbox{avoSI $\succeq$ evoSI $\succeq$ AB-avoSI $\succeq$ delSI}
$$
where epidemic1 $\succeq$ epidemic2 means that the two epidemics can be constructed on the same space so that the final epidemic size in epidemic1 is greater or equal to that of epidemic2.
In fact, we will prove this chain of comparisons I in Lemmas \ref{avosi>evosi}, \ref{abavosi<evosi} and \ref{abdel}, respectively.
It remains to show that avoSI and delSI has the same critical value and probability of a large outbreak. This is proved in Lemma \ref{critical_avosi}.
\end{proof}
Below we will use $\lambda_c$ and $\alpha_c=\rho \lambda_c/m_1$ to denote the critical value. We now show that Theorem 2 in \cite{YD1} holds for avoSI.
Recall the definition of the generating function $G$ in \eqref{F_2} as well as the functions $f$ in equation \eqref{defoff}. Also recall that we assume $\mathbb{E}(D^5)<\infty$ throughout the paper.
\begin{theorem} \label{epsize}
Consider the avoSI epidemic on the configuration model $\mathbb{CM}(n,D)$ with one uniformly randomly chosen vertex initially infected. Suppose $\alpha<\alpha_c$ so that we are in the supercritical regime.
Let $\tilde{I}_{\infty}$ be the final epidemic size.
Set
\begin{align}
\label{mM}
\sigma &=\sup\{w:0<w<1,f(w)= 0\} \quad\hbox{with $\sup(\emptyset) = 0$}, \\
\label{limitsize}
\nu& =1-\exp\left(-\frac{\alpha}{2}(\sigma-1)^2 \right)G(\sigma).
\end{align}
If we suppose
\medskip
$(\star)$ either $\sigma=0$ or $0<\sigma<1$ and there is a $\delta>0$ so that $f<0$ on $(\sigma-\delta,\sigma)$,
\medskip\noindent
then for any $\epsilon>0$,
$$
\lim_{n\to\infty} \mathbb{P}(\tilde{I}_{\infty}/n<\nu+\epsilon )=
\lim_{\eta\to 0}\liminf_{n\to\infty} \mathbb{P}(\tilde{I}_{\infty}/n>\nu-\epsilon |\tilde{I}_{\infty}/n>\eta)=1.
$$
\end{theorem}
\noindent
As explained at the end of Section 1.3, $\nu$ does not give the correct final size of the evoSI epidemic. However, the formula for $f(w)$ is accurate enough for $w$ near 1 to identify when the phase transition is continuous.
Using Theorem \ref{epsize}, we can prove
\begin{theorem}\label{Q1}
Consider the avoSI epidemic on the configuration model $\mathbb{CM}(n,D)$ and let $\tilde I_\infty$ be the final epidemic size. Set
\begin{equation}\label{deltadef}
\Delta= -\frac{\mu_3}{\mu_1} + 3 (\mu_2-\mu_1).
\end{equation}
If $\Delta<0$, then there a continuous phase transition. For any $\epsilon>0$, there exists some $\delta>0$, so that
\begin{equation}\label{delta<0}
\lim_{n\to\infty}\mathbb{P}(\tilde I_{\infty}/n>\epsilon)=0 \quad\hbox{for $\alpha_c-\delta<\alpha< \alpha_c$.}
\end{equation}
\end{theorem}
We can show that $\Delta>0$ implies that there is a discontinuous phase transition in avoSI, but that result does not help us prove Theorem \ref{Q1new}.
To get Theorem \ref{Q1} from Theorem \ref{epsize} we compute,
see Section \ref{sec:pfth7}, that
\begin{equation}\label{fprop}
\begin{split}
f'(1) &= - \left( \frac{m_2-2m_1}{m_1} - \frac{\rho}{\lambda} \right) \quad\hbox{which is $<0$ for $\alpha< \alpha_c$},\\
f'(1) &= 0, \quad f''(1) = \Delta \quad\hbox{when $\alpha=\alpha_c$}.
\end{split}
\end{equation}
When $\Delta>0$, as $w$ decreases from 1 the curve of $f$ turns up, and $\sigma$ stays bounded away from 0.
When $\Delta<0$, the curve of $f$ turns down, and $\sigma$ converges to 1 as $\alpha\to\alpha_c$. See Figure 4.
\newpage
\begin{figure}[h!]
\begin{center}
\includegraphics[height=3.2in,keepaspectratio]{ERdelneg}
\includegraphics[height=3.2in,keepaspectratio]{ERdelpos}
\caption{The behavior of $f(w)$ near 1 with respect to different $\alpha's$ for the Erd\"os-R\'enyi graph. In the top graph $\mu=1.4$, which has $\Delta<0$. $\alpha_c = \mu^2-\mu = .56$. Notice that as $\alpha$ increases to 0.56 the intersection with the $x$ axis tends to 1, so the transition is continuous. In the bottom graph $ \mu= 3$, which has $\Delta>0$. $\alpha_c = \mu^2-\mu = 6$. Notice that when
$\alpha\le\alpha_c$, $f(w)> 0$ for $w\in[0.9,1)$, so $\sigma$ is bounded away from 1.}
\end{center}
\label{fig:ER2}
\end{figure}
\begin{theorem}\label{Q1AB}
Consider the AB-evoSI epidemic on the configuration model $\mathbb{CM}(n,D)$. Let $\check{I}_{\infty}$ be the final epidemic size. Let $\Delta$ be the quantity defined in \eqref{deltadef}.
If $\Delta>0$ then there is a discontinuous phase transition. For some $\epsilon_0>0$ and some $\delta_0>0$,
\begin{equation}\label{delta>0AB}
\lim_{\eta\to 0} \liminf_{n\to\infty}\mathbb{P}_{1}(\check{I}_{\infty}/n>\epsilon_0|\check{I}_{\infty}/n>\eta)=1 \quad\hbox{for all $\alpha_c-\delta_0<\alpha< \alpha_c$.}
\end{equation}
\end{theorem}
\medskip\noindent
Theorem \ref{Q1new} follows from Theorem \ref{Q1} and Theorem \ref{Q1AB}.
\begin{proof}[Proof of Theorem \ref{Q1new}]
Lemma \ref{avocrv} implies that
\begin{equation}\label{cmp}
\lim_{\eta\to0} \liminf_{n\to\infty}\mathbb{P}(\check{I_{\infty}/n>\eta})=\lim_{\eta>0}\liminf_{n\to\infty} \mathbb{P}(\check{I}_{\infty}/n>\eta).
\end{equation}
Equations \eqref{cmp}, \eqref{delta>0AB} and the fact evoSI $\succeq$ AB-avoSI (proved in Lemma \ref{abavosi<evosi}) imply that
$$
\lim_{\eta\to 0} \liminf_{n\to\infty}\mathbb{P}(I_{\infty}/n>\epsilon_0|I_{\infty}/n>\eta)\geq
\lim_{\eta\to 0} \liminf_{n\to\infty}\mathbb{P}(\check{I}_{\infty}/n>\epsilon_0|\check{I}_{\infty}/n>\eta)=1,
$$
for $\alpha_c-\delta_0<\alpha< \alpha_c$.
This is exactly \eqref{delta>0new}. Equation \eqref{delta<0new} follows from \eqref{delta<0} and the fact that avoSI $\succeq$ evoSI.
\end{proof}
\subsection{Sketch of Proof of Theorem \ref{epsize}} \label{sec:pfstrategy}
To analyze the avoSI model we follow the approach in Janson, Luczak, and Windridge \cite{JLW} and construct the graph as we run the infection process. The construction of this process and its coupling to evoSI are described in Section \ref{sec:avoSI}. Initially the graph consists of half-edges connected to vertices, as in the configuration model construction before the half-edges are paired.
Let $\tilde{X}_t$ be the total number of half-edges at time $t$ and let $\tilde{X}_{I,t}$ be the number of half-edges that are attached to infected vertices and let $\tilde S_{t,k}$ be the number of susceptible vertices with $k$ half-edges at time $t$.
The evolution of $\tilde{S}_{t,k}$ in avoSI is given by, see \eqref{eqStk1},
$$
d\tilde{S}_{t,k}=-\left(\lambda \tilde{X}_{I,t}\frac{k\tilde{S}_{t,k}}{\tilde{X}_t-1}\right)dt+\left(1_{\{k\geq 1\}}\rho \tilde{X}_{I,t}
\frac{\tilde{S}_{t,k-1}}{n}\right)dt-\left(\rho \tilde{X}_{I,t}
\frac{\tilde{S}_{t,k}}{n}\right)dt+dM_{t,k},
$$
where $\tilde{M}_{t,k}$ is a martingale and we have returned to using $\lambda$ as the infection rate and $\rho$ as the rewiring rate.
Following \cite{JLW} we time-change the process by multiplying the original transition rates by $(\tilde{X}_t-1)/(\lambda \tilde{X}_{I,t})$.
Let $\overline{X}_t$ be the number of half-edges at time $t$ in the time changed process, and let $\overline{X}_{S,t}$ be the number of half-edges that are attached to susceptible vertices.
Using $\overline{S}_{t,k}$ for the time-changed process the new dynamics are, see \eqref{eqStk2},
\begin{align}
d\left(\frac{\overline{S}_{t,k}}{n}\right) & =-\left(k\frac{\overline{S}_{t,k}}{n}\right) dt
+\left(1_{(k\geq 1)} \frac{\rho }{\lambda}\frac{\overline{X}_t-1}{n}\frac{\overline{S}_{t,k-1}}{n}\right)dt
\nonumber\\
&-\left(\frac{\rho }{\lambda}\frac{\overline{X}_t-1}{n}\frac{\overline{S}_{t,k}}{n}\right)dt+d\left(\frac{\overline{M}_{t,k}}{n}\right).
\label{tceq}
\end{align}
Note that, thanks to the time change, the number of infected half-edges $\overline{X}_{I,t}$ no longer appears in the equation. Let $\gamma_n$ be the first time there are no infected half-edges. Let
$w(t)=\exp(-t)$ and $m_1=\mathbb{E}(D)$. The key to the proof of Theorem \ref{epsize} is to show:
\begin{align}
&\sup_{0\leq t\leq \gamma_n}\abs{\frac{\overline{X}_t}{n}-m_1w(t)^2}\xrightarrow{\mathbb{P}} 0,
\nonumber\\
&\sup_{0\leq t\leq \gamma_n}
\abs{ \frac{\sum_{k=0}^{\infty} \overline{S}_{t,k}}{n}- F_0(w(t)) }\xrightarrow{\mathbb{P}} 0,
\label{3lim}\\
&\sup_{0\leq t \leq \gamma_n}
\abs{ \frac{\sum_{k=0}^{\infty} k\overline{S}_{t,k}}{n}-F_1(w(t)) }\xrightarrow{\mathbb{P}} 0,
\nonumber
\end{align} where
\begin{align}
F_0(w) & = \exp(-(\alpha/2)(w-1)^2 )G(w),
\nonumber\\
F_1(w)& = \exp(-(\alpha/2)(w-1)^2 )w(G'(w)+\alpha (1-w)G(w)).
\label{Fidef}
\end{align}
From the results above, we see that
\begin{equation}
\frac{\overline{X}_t}{\overline{X}_{S,t}} \to \frac{m_1 w}{ \exp(-(\alpha/2)(w-1)^2 ) \cdot(G'(w)+\alpha (1-w)G(w)) }.
\label{whatisf}
\end{equation}
The logarithm of the right-hand side is $f(w)$.
Under assumption ($\star$),
$$
\sigma = \sup\{ w : 0< w < 1, f(w)=0\}
$$
gives the time $z=-\log(\sigma)$ at which the infection dies out in the time-changed process and $\nu$ gives the fraction of sites which have been infected.
There are four steps in the proof of \eqref{3lim}.
\begin{itemize}
\item
In Section \ref{sec:tight} we show that for each fixed $k\in \mathbb{N}$, $\{\overline{S}_{t,k}/n, t\geq 0 \}_{n\geq 1}$ is a tight sequence of processes.
\item
In Section \ref{sec:convStk} we show that any subsequential limit satisfies a system of differential equations \eqref{stk} that has a unique solution $\bar s_{t,k}$, so $\overline{S}_{t,k}/n \to \bar s_{t,k}$
\item
Section \ref{sec:sum_stk} we deal with the technicality of showing that the limit of $\sum_{k=0}^{\infty} k \bar S_{t,k}/n$ is the sum of the limits $\sum_{k=0}^{\infty} k\bar s_{t,k}$
\item
In Section \ref{sec:pfth6} we complete the proof by establishing the formulas for $\sigma$ and $\nu$.
\end{itemize}
\subsection{Sketch of Proof of Theorem \ref{Q1AB}}\label{pfth18}
In the AB-avoSI model, each half-edge $i$ has two indices $A(i,t)$ and $B(i,t)$.
\begin{itemize}
\item The infection index $A(i,t)=0$ if $i$ has not been infected by time $t$.
If $i$ first become an infected half-edge at time $s$, then we set
$A(i,t)=s$ for all $t\geq s$.
\item
The rewiring index $B(i,t)=0$ if $i$ has not been rewired by time $t$.
If $i$ rewires at time $s$, then we update the value of $B(i,s)$ to be $s$, regardless of whether $i$ has been rewired before or not. $B(i,\cdot)$ remains constant between the rewirings.
\end{itemize}
Suppose an infected half-edge i pairs with a susceptible half-edge $j$ at tine $t$, then $i$ will transmit an infection to $j$ if and only if $A(i,t)>B(j,t)$. See Section \ref{sec:ABavoSI} for more details about the AB-avoSI model and its relationship to evoSI. Let $\check{S}_{t,k}$ be the number of susceptible vertices with $k$ half-edges at time $t$ and set
\begin{equation*}
G_{i,j}=\{I(i,t)=1, A(i,t)\leq B(j,t) \}.
\end{equation*}
Here $I(i,t)$ is an indicator function such that $I(i,t)=1$ if half-edge $i$ is an infected half-edge at time $t$ (see the first paragraph of Section \ref{sec:moment} for the definitions of the notations $I(i,t),S(i,t),S(i,k,t),D(j,t)$ appearing below).
As in the avoSI model we make a time-change by multiplying the original transition rates by $(\check{X}_t-1)/(\lambda \check{X}_{I,t})$. Using a hat to denote the quantities after the time-change in the AB-avoSI model, we have that, for all $k\geq 0$,
\begin{equation*}
\begin{split}
d\hat{S}_{t,k}&=-k\hat{S}_{t,k}\, dt
+1_{\{k\geq 1\}} \frac{\rho}{\lambda}\frac{\hat{S}_{t,k-1}}{n}(\hat{X}_t-1) \, dt
- \frac{\rho}{\lambda}\frac{\hat{S}_{t,k}}{n}(\hat{X}_t-1)\, dt\\
&+\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0}
1_{G_{i,j}} 1_{\{ S(j,k+1,t)=1 \}} \right)dt+d\hat{M}_{t,k},
\end{split}
\end{equation*}
where $\hat{M}_{t,k}$ is a martingale. See equation \eqref{newStk}. This system of equations is not solvable but we can expand in powers of $t$ to study the time-changed system for small $t$. If we let $\hat{X}_t, \hat{X}_{I,t}, \hat{S}_t$ be the number of half-edges, the number of infected half-edges and the number of susceptible vertices, respectively, then we have
\begin{equation}
\begin{split}
\hat{X}_{I,t}&=\hat{X}_{I,0}+\int_0^t\left(-2(\hat{X}_u-1)+\sum_{k=0}^{\infty} k^2\hat S_{u,k}-\frac{\rho}{\lambda}\frac{\hat S_u}{n}(\hat{X}_u-1) \right) du
+\hat{M}_t \\
&-\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}} (D(j,t)-1) 1_{\{ S(j,t)=1\}} \right)dt,
\end{split}
\end{equation}
where $\hat{M}_t$ is a martingale. See equation \eqref{xitnew}. Define
\begin{equation}
E(t)=\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}} D(j,t) 1_{\{ S(j,t)=1\}} \right).
\end{equation}
By expanding $\hat{S}_{t,k}$ around $t=0$ up to the second order, we get, for any $\epsilon>0$, $\lambda$ close to $\lambda_c$ and $t_0$ close to 0,
\begin{equation}\label{-lb}
\begin{split}
\lim_{n \to \infty}\mathbb{P}\biggl(\hat{X}_{I,t}& \geq \left(\frac{\rho m_1}{2\lambda_c^2}(\lambda-\lambda_c)t+\frac{m_1\Delta}{4}t^2-\epsilon\right)n \\
&-\int_0^t E(u)du, \forall 0\leq t\leq \gamma_n\wedge t_0\biggr)=1.
\end{split}
\end{equation}
See \eqref{lbstep12} of Lemma \ref{lbstep2}.
Here the $\Delta$ is the same as the one in \eqref{deldef}, i.e.,
$$
\Delta= -\frac{\mu_3}{\mu_1} + 3 (\mu_2-\mu_1).
$$
\begin{remark}
In the case of avoSI, we have that, for $n$ large,
\begin{equation*}
\overline{X}_{I,t}=\overline{X}_t-\overline{X}_{S,t}=\overline{X}_{S,t}\left(\frac{\overline{X}_t}{\overline{X}_{S,t}}-1\right) \sim \overline{X}_{S,t}(f(\exp(-t))-1)
\end{equation*}
with $f$ defined in \eqref{defoff}. By expanding $f$ up to the second order and using \eqref{fprop} along with the fact that $\exp(-t)=1-t+t^2/2+o(t^2)$, we get
\begin{equation}
\begin{split}
\overline{X}_{I,t}& \sim nm_1 (-f'(1)(t-t^2/2)+f''(1)t^2/2+o(t^2))\\
&\geq nm_1\left(\rho(\lambda-\lambda_c)t/(2\lambda_c^2)+(\Delta t^2)/4 \right)
\end{split}
\end{equation}
for small $t$ and $\lambda$ close to $\lambda_c$. In the case of AB-avoSI, see \eqref{xit4} we Have
$$
\frac{\rho m_1(\lambda-\lambda_c)}{4\lambda_c^2} t
+ \frac{m_1\Delta}{8} t^2 - \epsilon_2 - \epsilon_6
$$
as a lower bound when $t> \epsilon$ (the $\epsilon_2$ and $\epsilon_6$ here are some small numbers depending on $\epsilon$). The two expansions do not match but both linear terms vanish at $\lambda_c$ and the quadratic terms have the same sign so this is good enough.
\end{remark}
The proof of Theorem \ref{Q1AB} is organized into five steps.
\begin{itemize}
\item In Section \ref{sec:ABavoSI} we define the AB-avoSI process and prove that evoSI stochastically dominates AB-avoSI.
\item In Section \ref{sec:moment} we derive basic moment estimates for various quantities that will prepare us for later proofs. See Lemma \ref{lbstep1}.
\item In Section \ref{sec:rough} we give rough upper and lower bounds for $\hat{I}_t$ and $\hat{X}_{I,t}$ involving the integral of $E(t)$. See Lemma \ref{lbstep2}. We also give an easy upper bound for $E(t)$ in \eqref{eubound1}.
\item In Section \ref{sec:refined} we decompose $E(t)$ into two parts (see \eqref{ede}) and give refined bounds for each part. See Lemmas \ref{ctlet2} and \ref{ctlet3}.
\item In Section \ref{sec:complete} we combine our estimates to complete the proof.
\end{itemize}
\section{Proof of Theorem \ref{critical_value}} \label{sec:pfth2}
\subsection{Coupling of evoSI and delSI}
We first prove Lemma \ref{couple} before proving Theorem \ref{critical_value}.
We define three set of edges in evoSI at time $t$. {\it Active edges}, denoted by $\mathcal{E}^a_t$, are the edges at time $t$ that connect an infected vertex and a susceptible vertex. {\it Uninfected edges}, denoted by $\mathcal{E}^0_t$, connect two susceptible vertices. {\it Inactive edges}, denoted by $\mathcal{E}^i_t$, have both ends infected. Once an edge becomes inactive it remains inactive forever. The three sets form a partition of all edges.
The three set-valued processes just defined are right-continuous pure jump processes. At time 0 we randomly choose a vertex $u_0$ to be infected. $\mathcal{E}^a_{0}$ consists of the edges with one endpoint at $u_0$. $\mathcal{E}^0_{0}$ is the collection of all edges in the graph minus the set $\mathcal{E}^a_0$. $\mathcal{E}^i_{0}=\emptyset$. We will consider the corresponding sets for delSI, but they will be denoted by $\mathcal{D}_t$ to avoid confusion.
\medskip
{\bf Construction of evoSI.} For each undirected edge $e$, let $\tau_{e,\ell}$ be the $\ell$-th time the edge becomes active. To make it easier to describe the dynamics, suppose that at time $\tau_{e,\ell}$ we have $e=\{x_{e,\ell},y_{e,\ell}\}$ with $x_{e,\ell}$ infected and $y_{e,\ell}$ susceptible.
\begin{itemize}
\item
Let $T_{e,\ell}$, $\ell \ge 1$ be independent exponential random variables with mean $1/\lambda$. $T_{e,\ell}$ is the time between $\tau_{e,\ell}$ and the infection of $y_{e,\ell}$ by $x_{e,\ell}$.
\item
Let $R_{e,\ell}$, $\ell \ge 1$ be independent exponential random variables with mean $1/\rho$. $R_{e,\ell}$ is the time between $\tau_{e,\ell}$ and the time when $y_{e,\ell}$ breaks its connection to $x_{e,\ell}$ and rewires.
\item
Let $U_{e,\ell}$ be the randomly chosen vertex that $y_{e,\ell}$ connects to at time $\tau_{e,\ell} + R_{e,\ell}$ (if rewiring occurs).
\end{itemize}
\medskip\noindent
{\bf Initial step.}
To simplify writing formulas, let $S_{e,\ell} = \min\{T_{e,\ell},R_{e,\ell}\}$.
At time 0, a randomly chosen vertex $u_0$ is infected. Let ${\cal N}^0(x,t),{\cal N}^a(x,t)$ and ${\cal N}^i(x,t)$ be the collection of uninfected, active and inactive edges that are connected to $x$ at time $t$, respectively. At time 0 the edges ${\cal N}^0(u_0,0)=\{e_1,\ldots, e_k\}$ are added to the list of active edges. We have $\mathcal{E}^a_{0}=\{e_1,\ldots, e_k\}$. Suppose $e_j$ connects $u_0$ and $y_j$. At time
$$
J_1 = \min_{1\le j \le k} S_{e_j,1}
$$
the first event occurs. ($J$ is for jump.) Let $i$ be the index that achieves the minimum.
\medskip\noindent
(i) If $R_{e_i,1}<T_{e_i,1}$, then at time $J_1$ vertex $y_i$ breaks its connection with $u_0$ and rewires to $U_{e_i,1}$. If $U_{e_i,1}$ is susceptible at time $J_1$, we move the edge $e_i$ to $\mathcal{E}^0_{J_1}$. On the initial step this will hold unless $U_{e_1,1}=u_0$ in which case nothing has changed.
\medskip\noindent
(ii) If $T_{e_i,1}<R_{e_i,1}$ then at time $J_1$ vertex $y_i$ becomes infected by $u_0$. We move $e_i$ to $\mathcal{E}^i_{J_1}$. We add edges in ${\cal N}^0(y_i,J_1-)$ to ${\cal E}^a_{J_1}$.
\medskip\noindent
{\bf Induction step.}
For any active edges $e$ present at time $t$, let $L(e,t) = \sup\{\ell : \tau_{e,\ell}\leq t\}$
and let $V(e,t)=\tau_{e,L(e,t)}+S_{e,L(e,t)}$ be the time of the next event (infection or rewiring) to affect edge $e$.
Suppose we have constructed the process up to time $J_n$.
If there are no active edges present at time $J_{n}$, the construction is done. Otherwise, Let
$$
J_{n+1}=\min_{e\in \mathcal{E}^a_{J_n}} V(e,J_{n}).
$$
Let $e_n$ be the edge that achieves the minimum of $V(e,J_n)$,
let $x(e_n)$ be the infected endpoint of $e_n$ and $y(e_n)$ be the susceptible endpoint of $e$. To simplify notation let $L_n =L(e_n,J_n)$.
\medskip\noindent
(i) If $R_{e_n,L_n}<T_{e_n,L_n}$, then at time $J_{n+1}$ vertex $y(e_n)$ breaks its connection with $x(e_n)$ and rewires to $U_{e_n, L_n}$. If $U_{e_n,L_n}$ is susceptible at time $J_{n+1}$, then $e_n$ is moved to $\mathcal{E}^0_{J_{n+1}}$. Otherwise it remains active.
\medskip\noindent
(ii) If $T_{e_n,L_n}<R_{e_n,L_n}$, then at time $J_{n+1}$ the vertex $y(e_n)$ is infected by $x(e_n)$ and $e_n$ is moved to $\mathcal{E}^i_{J_{n+1}}$.
\begin{itemize}
\item
All edges $e'$ in ${\cal N}^0_{y(e_n),J_{n+1}-}$ are moved to ${\cal E}^a_{J_{n+1}}$. Since $y(e_n)$ has just become infected, the other end of $e'$ must be susceptible at time $J_{n+1}$.
\item
All edges $e''$ in ${\cal N}^a_{y(e_n),J_{n+1}-}$ are moved to ${\cal E}^i_{J_{n+1}}$. Since $y(e_n)$ has just become infected, (i) the other end of $e''$ must be infected at time $J_{n+1}$, and (ii) $e''$ cannot have been inactive earlier.
\end{itemize}
\begin{figure}[ht]
\begin{center}
\begin{picture}(310,100)
\put(30,30){\line(1,0){60}}
\put(90,30){\line(1,6){10}}
\put(90,30){\line(1,1){42}}
\put(90,30){\line(1,0){60}}
\put(15,15){$x(e_n)$}
\put(75,15){$y(e_n)$}
\put(28,28){$\bullet$}
\put(148,28){$\bullet$}
\put(60,35){$a$}
\put(120,35){$a$}
\put(85,60){0}
\put(110,60){0}
%
\put(180,30){\line(1,0){60}}
\put(240,30){\line(1,6){10}}
\put(240,30){\line(1,1){42}}
\put(240,30){\line(1,0){60}}
\put(165,15){$x(e_n)$}
\put(225,15){$y(e_n)$}
\put(178,28){$\bullet$}
\put(238,28){$\bullet$}
\put(298,28){$\bullet$}
\put(210,35){$i$}
\put(270,35){$i$}
\put(235,60){a}
\put(260,60){a}
\end{picture}
\caption{Change of the status of different edges after $y(e_n)$ becomes infected by $x(e_n)$. Black dots mark infected sites. }
\end{center}
\end{figure}
\medskip
{\bf Construction of delSI.} There is no rewiring in delSI, each edge will become active at most once. Thus for each undirected edge $e$ we only need two exponential random variables $T_{e,1}$ and $R_{e,1}$. To couple delSI to evoSI we use the random variables defined in the construction of evoSI. Also recall that we use $\mathcal{D}^0_t, \mathcal{D}^a_t$ and $ \mathcal{D}^i_t$ to represent the set of uninfected, active and inactive edges in delSI, respectively.
\medskip\noindent
{\bf Initial step.} At time 0, a randomly chosen vertex $u_0$ is infected. The edges ${\cal N}^0(u_0,0)=\{e_1,\ldots, e_k\}$ are added to the list of active edges. We have $\mathcal{D}^a_{0}=\{e_1,\ldots, e_k\}$. Suppose $e_j$ connects $u_0$ and $y_j$. At time
$$
J_1 = \min_{1\le j \le k} S_{e_j,1}
$$
the first event occurs. Let $i$ be the index that achieves the minimum.
\medskip\noindent
(i) If $R_{e_i,1}<T_{e_i,1}$, then at time $J_1$ the edge $e_i$ is removed from the graph (and hence also from the set $\mathcal{D}^a_{J_1}$).
\medskip\noindent
(ii) $T_{e_i,1}<R_{e_i,1}$ then at time $J_1$ vertex $y_i$ is infected by $x_i$. We move $e_i$ to $\mathcal{D}^i_{J_1}$ . We add ${\cal N}^0(x_i,J_1-)$ to ${\cal D}^a_{J_1}$.
\medskip\noindent
{\bf Induction step.}
For any active edge $e$ at time $t$, let
$$
V(e,t)=\tau_{e,1}+S_{e,1}
$$
be the time of the next event (infection or rewiring) to affect edge $e$.
Suppose we have constructed the process up to time $J_n$.
If there are no active edges present at time $J_{n}$, the construction is done. Otherwise, we let
$$
J_{n+1}=\min_{e\in \mathcal{E}^0_{J_n}} V(e,J_{n}).
$$
Let $e_n$ be the edge that achieves the minimum of $V(e,J_n)$, let $x(e_n)$ be the infected endpoint of $e_n$ and $y(e_n)$ be the susceptible endpoint of $e$.
\medskip\noindent
(i) If $R_{e,_n1}<T_{e_n,1}$, then at time $J_{n+1}$ the edge $e_n$ is deleted
from the graph (and hence also from the set $\mathcal{D}^a_{J_{n+1}}$).
\medskip\noindent
(ii) If $T_{e_n,1}<R_{e_n,1}$, then at time $J_{n+1}$ the vertex $y(e_n)$ is infected by $x(e_n)$. We move $e_n$ to $\mathcal{D}^i_{J_{n+1}}$.
\begin{itemize}
\item
We move all edges $e'$ in ${\cal N}^0_{y(e_n),J_{n+1}-}$ to ${\cal D}^a_{J_{n+1}}$. Since $y(e_n)$ has just become infected, the other end of $e'$ must be susceptible at time $J_{n+1}$
\item
We move all edges $e''$ in ${\cal N}^a_{y(e_n),J_{n+1}-}$ to ${\cal D}^i_{J_{n+1}}$.
Since $y(e_n)$ has just become infected, (i) the other end of $e''$ must be infected at time $J_{n+1}$, and (ii) $e''$ cannot have been inactive earlier.
\end{itemize}
\medskip
We now prove by induction that
\begin{lemma}
All vertices infected in delSI are also infected in evoSI and are infected earlier in avoSI than evoSI.
\end{lemma}
\begin{proof}
The induction hypothesis holds for the first vertex since $u_0$ is infected at time 0 in both evoSI and delSI. Suppose the induction holds up to the $k$-th infected vertex in delSI. Assume at time $t$, $y$ becomes the $(k+1)$-th infected vertex in delSI and $y$ is infected by vertex $x$ through edge $e$. We see from the construction of delSI that this implies $T_{e,1}<R_{e,1}$. Suppose $x$ was infected at time $s<t$ in delSI.
The induction hypothesis implies that $x$ has also been infected in evoSI at a time $s' \le s$. There are two possible cases for $y$ in evoSI.
\begin{itemize}
\item $y$ has already become infected by time $s'+T_{e,1}$.
\item $y$ was still susceptible at time $s'+T_{e,1}$. In this case, $y$ will be infected at time $s'+T_{e,1}\leq s+T_{e,1}=t.$
\end{itemize}
In either case $x$ has been infected by time $t$ in evoSI. This completes the induction step and thus proves Lemma \ref{couple}.
\end{proof}
\subsection{The infected sites in delSI}
In the introduction we have noted that delSI is equivalent to independent bond percolation. That is, we keep each edge independently with probability $\lambda/(\lambda+\rho)$ and find the component containing the initially infected vertex (say, vertex 1). To compute the size of the delSI epidemic starting from vertex 1, we apply a standard algorithm, see e.g., \cite{ML}, for computing the size of the component containing 1 in the reduced graph in which edges have independently been deleted with probability $\rho/(\lambda+\rho)$. We call this the {\it exploration process of delSI.} At step 0 the active set ${\cal A}_0=\{1\}$, the unexplored set ${\cal U}_0 =\{2, \ldots n \}$, and the removed set ${\cal R}_0=\emptyset$. Here removed means these sites are no longer needed in the computation. In the SI model sites never enter the removed state, Let $\eta_{i,j} = \eta_{j,i}=1$ if there is an edge connecting $i$ and $j$ in the reduced graph. If $\eta_{i,j}=1$ an infection at $i$ is transmitted to $j$. At step $t$ if ${\cal A}_t \neq\emptyset$ we pick an $i_t \in {\cal A}_t$ and update the sets as follows.
\begin{align*}
{\cal R}_{t+1} & = {\cal R}_t \cup \{ i_t \}, \\
{\cal A}_{t+1} & = ( {\cal A}_t - \{ i_t \}) \cup \{ y \in {\cal U}_t : \eta_{i_t,y} = 1 \}, \\
{\cal U}_{t+1} & = {\cal U}_t - \{ y \in {\cal U}_t : \eta_{i_t,y} = 1 \}.
\end{align*}
When ${\cal A}_t=\emptyset$ we have found the cluster containing 1 in the reduced graph, which will be the final set of infected sites in the SI model.
The \emph{exploration process of the configuration model} can be similarly defined (just without deletion of edges) and we let $\mathcal{J}_t$ be the set of active sites at step $t$ in this exploration. We set $A_t=\abs{\mathcal{A}_{t+1}}$ and $J_t=\abs{\mathcal{J}_{t+1}}$. We make a time shift so that $A_t$ and $J_t$ can be coupled with two random walks with i.i.d. increments (see Lemma \ref{l:atst} below).
We now study the exploration process of the configuration model itself as well as the delSI process on such graph.
Let $\psi_0$ have generating function $G$ defined in \eqref{F_2} and $\zeta_0$ have the distribution Binomial($D,\lambda/(\lambda+\rho)$) whose the generating function is denoted by $G^{\rho}$. The two generating functions are related by
$$
G^{\rho}(z)=G\left(\frac{\lambda}{\lambda+\rho}z+\frac{\rho}{\lambda+\rho}\right).
$$
Recall the definition of the generating function $\hat{G}$ in \eqref{G1f}. We can similarly define $\hat{G}^{\rho}$.
Let $\chi_i,i\geq 1$ and $\xi_i, i\geq 1$ be independent random variables with generating function $\hat{G}$ and $\hat{G}^{\rho}$, respectively.
Define two random walks (for integer-valued $t$):
\begin{equation}
W_0=\psi_0\quad W_t=W_0+\sum_{r=1}^t (\chi_r-1)
\qquad S_0=\zeta_0 \quad S_t = S_0 + \sum_{r=1}^{t} (\xi_r-1).
\label{RW}
\end{equation}
Let $\tau^W_0=\inf\{t\geq 0:W_t= 0\}$ and set
$\bar W_t=W_{t\wedge \tau^W_0}$. We define $\bar S_t$ in a similar way.
\begin{lemma}\label{l:atst}
We can couple $\{J_t,0\leq t\leq n^{1/3}\log n\}$ and $\{\bar W_t,0\leq t\leq n^{1/3}\log n\}$ so that
\begin{equation}\label{eq:jtwt}
\lim_{n\to\infty} \mathbb{P}\left(J_t=\bar W_t,0\leq t\leq n^{1/3}\log n \right)=1.
\end{equation}
Similarly,
there exists a coupling of $\{A_t,0\leq t\leq n^{1/3}\log n\}$ and $\{\bar S_t,0\leq t\leq n^{1/3}\log n\}$ so that
\begin{equation}\label{eq;atst}
\lim_{n\to\infty} \mathbb{P}\left(A_t=\bar S_t,0\leq t\leq n^{1/3}\log n \right)=1.
\end{equation}
\end{lemma}
Note that $\bar W_t$ and $\bar S_t$ can be viewed as the exploration processes of two-phase branching process $Z_m$ and $\bar Z_m$ (both are defined in Section \ref{britton}), respectively.
The proof of Lemma \ref{l:atst} is deferred to the end of this section.
For the rest of this section we will always work on the event
$$
\{A_t=\bar S_t,0\leq t\leq n^{1/3}\log n \} \cap \{J_t=\bar W_t,0\leq t\leq n^{1/3}\log n \}
$$
and hence assume that $A_t$ and $J_t$ have independent increments until they hit 0.
Hereafter we will use $C, C_1, C_2,\cdots$ to denote various constants whose specific values might change from line to line. Occasionally when we have an important constant we will number it by the formula it first appeared in, e.g., $C_{\ref{maxbd}}$.
\subsection{Proof of Theorem \ref{critical_value}(i)}
The formula for the critical value of delSI follows from standard results on percolation in random graphs. Note that in delSI each edge is kept with probability $\lambda/(\lambda+\rho)$. Using \cite[Theorem 3.9]{Jperc} we see that
$$\lambda_c(\mbox{delSI})=\frac{\rho m_1}{m_2-2m_1}.$$ Equivalently, we have
$$
\alpha_c(\mbox{delSI})=\frac{\rho m_1}{\lambda}=m_2-2m_1.
$$
Recall that Lemma \ref{couple} shows that the final set of infected individuals in delSI is contained in the analogous set for evoSI with the same parameters so
$$
\lambda_c(\mbox{evoSI}) \le \lambda_c(\mbox{delSI}).
$$
To prove that the two are equal we will show that if $\lambda < \lambda_c(\textrm{delSI})$ then evoSI dies out, i.e., infects only a vanishing portion of total population as $n\to\infty$.
To compare the two evolutions, we will first run the delSI epidemic to completion. Once this is done we will randomly rewire the edges deleted in delSI. If the rewiring creates a new infection in evoSI, then we have to continue to run the process. If not, then the infected sites in the two processes are the same.
Let ${\cal R}$ be the set of sites that are eventually infected in delSI, and let ${\cal R'}$ be the set of eventually infected sites in evoSI. Let $R' = |{\cal R}'|$ and $R = |{\cal R}|$.
To get started we use a result of Janson \cite{Jmax} about graphs with specified degree distributions. He works in the set-up introduced by Molloy-Reed \cite{MoRe1, MoRe2} where the degree sequence $d^n_i$, $1\le i \le n$ is specified and one assumes only that limiting moments exist as well some other technical assumptions that are satisfied in our case(for a more recent example see \cite{JLW})
\begin{equation}\label{jasonthm}
\frac{1}{n} \sum_{i=1}^n d^n_i \to \mu, \qquad \frac{1}{n} \sum_{i=1}^n d^n_i(d^n_i-1) \to \nu.
\end{equation}
The next result is Theorem 1.1 in \cite{Jmax}. The $\mu$ and $\nu$ in this theorem have the same meaning as \eqref{jasonthm}. The ``whp'' below is short for with high probability, and means that the probability the inequality holds tends to 1 as $n\to\infty$. Let $D^n=d_{u_0}^n$ where $u_0$ is randomly chosen from $\{1,2,\ldots, n\}$.
\begin{theorem} \label{Janbd}
Suppose $\mu>0$, $\nu>1$, and $\mathbb{P}( D^n \ge k) \le C k^{1-\gamma}$ for some $\gamma > 3$ and $C<\infty$. Then there is a constant $C_{\ref{maxbd}}$
which depends on $C$ so that the largest component $\mathcal{C}_1$ has
\begin{equation}
|{\cal C}_1| \le C_{\ref{maxbd}} n^{1/(\gamma-1)} \qquad \hbox{whp.}
\label{maxbd}
\end{equation}
\end{theorem}
\noindent
Theorem \ref{Janbd} can also be applied to the setting where the degrees are random rather than deterministic.
We have assumed that in the original graph $\mathbb{E} D^5 < \infty$, so $\mathbb{P}( D \ge k ) \le k^{-3} \mathbb{E} (D^3)$ and we have
$$
\mathbb{P}(D^n\geq k)=\frac{1}{n}\sum_{i=1}^n \mathbb{P}(D_i\geq k)=\mathbb{P}(D_1\geq k)\leq k^{-3} \mathbb{E} (D^3)\leq Ck^{-3}.
$$
It follows that
\begin{equation}
|{\cal C}_1| \le C_{\ref{maxbd}} n^{1/3}\quad\hbox{whp}.
\label{C1bd}
\end{equation}
Let $N$ be the number of deleted edges in the exploration process of delSI. One vertex is removed from the construction on each step, so whp the number of steps is
$\le r_0 = C_{\ref{maxbd}} n^{1/3}$. Note that $r_0\ll n^{1/3}\log n$ so we can assume $J_t$ has independent increments by Lemma \ref{l:atst}.
It follows that
$$
N \leq r_0+\bar W_{r_0}\leq r_0+\psi_0+\sum_{r=1}^{r_0} \chi_r.
$$
where we recall that the $\chi_i$ are independent with the distribution $D^*-1$. The inequality comes from the fact that we are counting all the edges even if they are not deleted,
Since $\mathbb{E}(D^3)<\infty$, we have
$$\hbox{Var}\,(\chi_i) \le \mathbb{E}(\chi_i^2)=\mathbb{E}[(D^*-1)^2]\leq \mathbb{E}[(D^*)^2]=
\frac{\mathbb{E}(D^3)}{\mathbb{E}(D)}
<\infty$$ and $\hbox{Var}\,(\psi_0)\leq \mathbb{E}\psi_0^2=\mathbb{E} D^2<\infty$. Let $1/3 < a < 1/2$. Using Chebyshev's inequality
\begin{equation}
\mathbb{P}( N \ge r_0\mathbb{E}\chi_1 + \mathbb{E}\psi_0+r_0+n^{a} ) \le (r_0 \mathbb{E}\chi_1^2+\mathbb{E} \psi_0^2)/n^{2a}\leq Cn^{1/3-2a}.
\label{rewires}
\end{equation}
A similar argument shows that
\begin{equation}
\mathbb{P}(A_{r_0} + R_{r_0} \ge r_0\mathbb{E}\xi_1 + \mathbb{E} \zeta_0+r_0+ n^{a} ) \le (r_0 \mathbb{E}\xi_1^2+\mathbb{E} \zeta_0^2)/n^{2a} \leq Cn^{1/3-2a}.
\label{infecteds}
\end{equation}
Since $a>1/3$, when $n$ is large we can upper bound $r_0\mathbb{E}\chi_1 + \mathbb{E}\psi_0+r_0+n^{a} $ and $ r_0\mathbb{E}\xi_1 + \mathbb{E} \zeta_0+r_0+ n^{a}$ by $2n^{a}$.
At step $r_0$ we use random variables independent of delSI to randomly rewire the deleted edges.
Let $Y$ be the number of edges deleted up to time $r_0$ that rewire to the set $\mathcal{A}_{r_0+1}\cup \mathcal{R}_{r_0+1}$. By construction
$$
Y = \text{Binomial}(N,(A_{r_0}+ R_{r_0})/n).
$$
Using \eqref{rewires}, \eqref{infecteds} and $r_0\le C_{\ref{maxbd}}n^{1/3}$ we see that on a set with probability $\ge 1 - C n^{-(2a-1/3)}$
$$
Y \preceq \text{Binomial}(2n^{a}, 2n^{a-1}) \equiv \bar Y,
$$
where $\equiv$ indicates that the last equality defines $\bar Y$.
To prepare for the next step we note that
\begin{equation}
\hbox{if $0 < x < 1$ and $k$ is a positive integer, then $(1-x)^k \ge 1-kx$}.
\label{powerk}
\end{equation}
Using the formula for the binomial distribution in \eqref{powerk} we see that
\begin{equation}
\mathbb{P}(\bar Y=0) = \left(1- \frac{2n^a}{n} \right)^{2 n^a} \ge 1 - \frac{4n^{2a}}{n}.
\label{Y0bd}
\end{equation}
From this we get
\begin{equation}
\mathbb{P}(\bar Y\ge 1 ) \leq \frac{4 n^{2a}}{n} \to 0,
\label{Y1bd}
\end{equation}
since $a<1/2$.
Since $\{ Y = 0\} \subset \{\mathcal{R} = \mathcal{R}'\}$, this shows $\mathbb{P}({\cal R} = {\cal R}') \to 1$ as $n\to\infty$ and completes the proof of (i). To prepare for the proof of (ii) note that the conclusion (the set of infected sites coincide in delSI and evoSI) holds as long as the number of steps is smaller than $Cn^{1/3}$ even if the epidemic is supercritical.
\subsection{Proof of Theorem \ref{critical_value}(ii)}
Let $B_d$ and $B_e$ be the events that there is large epidemic in delSI and evoSI respectively. To prove (ii) it is enough to show:
\begin{lemma}
Suppose $\lambda > \lambda_c(\mbox{delSI})$. As $n\to\infty$, $\mathbb{P}(B_e) - \mathbb{P}(B_d) \to 0.$
\end{lemma}
\begin{proof} Clearly $\mathbb{P}(B_d) \leq \mathbb{P}(B_e)$. Let $S_t$ be the random walk defined in \eqref{RW}.
\begin{lemma} \label{upbd2}
There is a $\gamma>0$ so that
$$
\mathbb{P}(0< A_{\log n} < \gamma \log n | A_0> 0) \le \frac{C}{\log n}.
$$
\end{lemma}
\begin{proof}
On the event $\{A_{\log n}>0\}$ we have $A_{\log n}=S_{\log n}$, which is, by definition,
$$
\zeta_0+\sum_{r=1}^{\log n}\xi_r.
$$
Hence if we take $\gamma=\mathbb{E}(\xi_1)/2$ then we have
\begin{equation*}
\begin{split}
\mathbb{P}(0< A_{\log n} < \gamma \log n | A_0> 0) &\leq
\mathbb{P}\left( \sum_{r=1}^{\log n}\xi_r\leq \frac{ \mathbb{E}(\xi_1)\log n }{2} \right)\\
&\leq \mathbb{P}\left( \abs{\sum_{r=1}^{\log n}\xi_r-
\mathbb{E}(\xi_1)\log n}
\geq \frac{\mathbb{E}(\xi_1)\log n }{2} \right)\\
&\leq \frac{4\hbox{Var}\, (\xi_1)\log n}{ (\mathbb{E}(\xi_1))^2 \log^2 n }\leq \frac{C}{\log n}.
\end{split}
\end{equation*}
\end{proof}
Let $F_0 = \{A_{\log n} = 0\}, F_1 = \{0 < A_{\log n}< \gamma \log n\}$ and
$F_2 = \{A_{\log n} \geq \gamma \log n\}$. Decomposing $B_d$ into three parts
and using Lemma \ref{upbd2}
$$
\mathbb{P}(B_d) = \sum_{i=0}^2 \mathbb{P}(B_d \mid F_i) P(F_i).
$$
We note that
\begin{itemize}
\item $\mathbb{P}(B_d|F_0)=0$ by the definition of $F_0$.
\item $\mathbb{P}(F_1)$ converges to 0 by Lemma \ref{upbd2}.
\item $\mathbb{P}(B_d|F_2)$ converges to 1 by the same argument as the proof of Theorem 2.9(c) in \cite{JLW} (see page 750-752 in \cite{JLW}).
\end{itemize}
Therefore we have $\mathbb{P}(B_d)-\mathbb{P}(F_2)\to 0$.
As for $\mathbb{P}(B_e)$, we note that by the remark at the end of the proof of (i) one has $\mathbb{P}(\mathcal{A}_{\log n}=\mathcal{A}_{\log n}')\to 1$ where $\mathcal{A}'_{\log n}$ is the number of active sites in evoSI at step $\log n$. Using the decomposition $\mathbb{P}(B_e)=\sum_{i=0}^2 \mathbb{P}(B_e \cap F'_i)$
where the event $F'_i$ is defined in a similar way to $F_i$ with $A_{\log n}$ replaced by $A'_{\log n}$,
we see
\begin{itemize}
\item $\mathbb{P}(B_e \cap F_1')\leq \mathbb{P}(F_1')\leq \mathbb{P}(F_1)+o(1)=o(1)$.
\item $\mathbb{P}(B_e \cap F_0')=o(1)$ by definition of $B_e$.
\item $\mathbb{P}(B_e \cap F_2')=\mathbb{P}(F_2')+o(1)=\mathbb{P}(F_2)+o(1)$.
\end{itemize}
It follows that $\mathbb{P}(B_e)-\mathbb{P}(F_2)\to 0$. This implies that $\mathbb{P}(B_e)-\mathbb{P}(B_d)\to 0$. It remains to compute the limit of $\mathbb{P}(F_2)$. Since $\mathbb{P}(F_1)=o(1)$, we have
\begin{equation}\label{eq;f1f2}
\mathbb{P}(F_2)=\mathbb{P}(F_2\cup F_1)+o(1)=\mathbb{P}(A_{\log n}>0)+o(1).
\end{equation}
To compute the limit of $\mathbb{P}(A_{\log n}>0)$,
note that due to Lemma \ref{l:atst}, $A_t$ can be coupled with the exploration process of the two-phase branching process $\bar Z_m$ defined in Section \ref{britton}. Therefore we have (recall that $q(\lambda)$ is the survival probability of $\bar Z_m$) $$\mathbb{P}(A_{\log n}>0)=\mathbb{P}(\bar Z_m \mbox{ survives})+o(1)=q(\lambda)+o(1).$$ By \eqref{eq;f1f2} we see that $\mathbb{P}(F_2)\to q(\lambda)$ as well. This implies that both $\mathbb{P}(B_e)$ and $\mathbb{P}(B_d)$ converge to $q(\lambda)$ as $n\to\infty$ and completes the proof of (ii). Note that using the fact that delSI is equivalent to independent bond percolation, the statement $\mathbb{P}(B_d)\to q(\lambda)$ also follows from standard results on percolation in random graphs. See, e.g.,
\cite[Theorem 3.9]{Jperc}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{l:atst}]
We only prove equation \eqref{eq:jtwt} since the other one follows from \eqref{eq:jtwt} and the fact that delSI is equivalent to percolation with edge retaining probability $\lambda/(\lambda+\rho)$.
The proof consists of two steps. First, we define an empirical version of $W_t$. Let $D_1, \ldots, D_n$ be i.i.d. random variables sampled from the distribution of $D$. Given a sample of $D_1, \ldots, D_n$, let $\psi^n_0$ be sampled from the (random) distribution
$$
\mathbb{P}(\psi^n_0=k)=\frac{1}{n}\abs{\{1\leq i\leq n: D_i=k\}}.
$$
which is the sample empirical distribution. Let $\chi^n_r,r\geq 1$ have the distribution
$$
\mathbb{P}(\chi^n_r=k)=\frac{1}{D_1+\ldots+D_n}(k+1)\abs{\{1\leq i\leq n: D_i=k+1\}}.
$$
Define $$
W^n_t=\psi^n_0+\sum_{r=1}^t (\chi^n_r-1),\,t\in \mathbb{N}.
$$
In other words, $W^n_t$ is a random walk in the random environment given by $D_1,\ldots, D_n$.
We define $\bar W^n_t$ to be $W^n_{t\wedge \tau'}$ where $\tau'$ is the first time $W^n_t$ hits zero.
Note that the condition $\mathbb{E}(D^4)<\infty$ implies that $\max_{1\leq i\leq n}D_i=o(n^{1/4}\log n)$ whp for any $\epsilon>0$. Indeed we have
\begin{equation}\label{eq:dmax}
\mathbb{P}\left(\max_{1\leq i\leq n}D_i>n^{1/4}\log n\right)\leq n \mathbb{P}(D_1>n^{1/4}\log n)\leq n n^{-1}\log^{-4} n\mathbb{E}(D^4)\leq C \log^{-4}n.
\end{equation}
Consequently, for $n$ large enough whp we have
$$
\sqrt{\frac{n}{\max_{1\leq i\leq n}D_i}}> n^{3/8}/\log n.
$$
Using Lemma 2.12 in \cite{vdH3}, we see that (recall that $J_t$ is the exploration process of the configuration model starting from a uniformly randomly chosen vertex)
\begin{itemize}
\item $J_t$ can be coupled with $\bar W^n_t$ with high probability up to time $ n^{3/8}/\log n$.
\item Whp the subgraph obtained by exploring the neighborhoods of $n^{3/8}/\log n$ vertices is a tree.
\end{itemize}
Using $n^{1/3}\log n \ll n^{3/8}/\log n$ we have that
\begin{equation}
\lim_{n\to\infty} \mathbb{P}\left(J_t=\bar W^n_t,0\leq t\leq n^{1/3}\log n \right)=1.
\end{equation}
To prove \eqref{eq:jtwt} it remains to show that one can couple $W^n_t$
and $W_t$ up to step $n^{1/3}\log n$. To this end, we use the characterization of total variation distance in terms of optimal coupling. It is well known that for any two random variables $X$ and $Y$,
$$
d_{\mathrm{TV}}(X,Y)=\inf_{\textrm{all couplings of }X,Y} \mathbb{P}(X\neq Y).
$$
See \cite[Theorem 1.14]{VC} for instance.
Using ${\bf D_n}$ to denote the degree sequence $D_1,\ldots, D_n$, it suffices to show that
\begin{equation}\label{eq;tv2}
d^{\mathbf{D_n}}_{\mathrm{TV}}((\psi^n_0,\{\chi^n_r,1\leq r\leq n^{1/3}\log n\}),(\psi_0, \{\chi_r,1\leq r\leq n^{1/3}\log n\})) \xrightarrow{\mathbb{P}} 0.
\end{equation}
Here the superscript ${\bf D_n}$ indicates that we are considering the quenched law of $\psi^n_0$ and $\xi^n_0$.
Since conditionally on $\mathbf{D_n}$, $\psi^n_0$ and $\chi^n_r,r\geq 1$ are all independent,
\begin{equation}\label{eq:tv}
\begin{split}
& d^{\mathbf{D_n}}_{\mathrm{TV}}((\psi^n_0,\{\chi^n_r,1\leq r\leq n^{1/3}\log n\}),(\psi_0, \{\chi_r,1\leq r\leq n^{1/3}\log n\}))\\
\leq & d^{\mathbf{D_n}}_{\mathrm{TV}}(\psi^n_0, \psi_0)+
\sum_{r=1}^{n^{1/3}\log n} d^{\mathbf{D_n}}_{\mathrm{TV}}(\chi^n_r,\chi_r)\\
\leq & d^{\mathbf{D_n}}_{\mathrm{TV}}(\psi^n_0, \psi_0)+(\log n)n^{1/3}
d^{\mathbf{D_n}}_{\mathrm{TV}}(\chi^n_1,\chi_1).
\end{split}
\end{equation}
We need the following lemma to control $\mathbf{D_n}$. Recall that we let $p_k=\mathbb{P}(D=k)$ and $m_1=\mathbb{E}(D)$. Also recall that we assume $\mathbb{E}(D^5)<\infty$.
\begin{lemma}\label{l;d_n}
For any $\epsilon>0$,
\begin{equation}\label{eq:sumd}
\mathbb{P}\left(\abs{\sum_{i=1}^n D_i-nm_1}>n^{1/2+\epsilon}\right)\leq Cn^{-2\epsilon}.
\end{equation}
Let $N_k$ be the cardinality of the set $\{i:D_i=k\}$. We have
\begin{equation}\label{eq;n_k}
\mathbb{P}\left(\abs{N_k-np_k}>n^{\epsilon}(np_k)^{1/2} \right)\leq n^{-2\epsilon}.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{l;d_n}]
Both \eqref{eq:sumd} and \eqref{eq;n_k} follow from Markov's inequality. For \eqref{eq:sumd} we note that
$$
\mathbb{P}\left(\abs{\sum_{i=1}^n D_i-nm_1}>n^{1/2+\epsilon}\right)\leq \frac{n\mathrm{Var}(D_1)}{n^{1+2\epsilon}}\leq Cn^{-2\epsilon}.
$$
For the second inequality, we note that $\mathbb{E}(1_{\{D_1=k\}}-p_k)^2=p_k(1-p_k)\leq p_k$.
It follows that
$$
\mathbb{E}(N_k-np_k)^2=\mathbb{E}\left(\sum_{i=1}^n \left(1_{\{D_i=k\}}-p_k\right)\right)^2= n \mathbb{E}(1_{\{D_1=k\}}-p_k)^2\leq np_k.
$$
Therefore
\begin{equation}
\mathbb{P}\left(\abs{N_k-np_k}>n^{\epsilon}(np_k)^{1/2} \right)\leq
\frac{ \mathbb{E}(N_k-np_k)^2}{n^{2\epsilon}(np_k)}\leq n^{-2\epsilon}.
\end{equation}
\end{proof}
Let $\epsilon_1=1/100,\epsilon_2=1/8+1/100$.
Consider the event
\begin{equation}
\begin{split}
\Omega_n=&\left\{\max_{1\leq i\leq n}D_i\leq n^{1/4}\log n,
\abs{\sum_{i=1}^n D_i-nm_1}\leq n^{1/2+\epsilon_1}\right\}\\
&
\cap \left\{ \abs{N_k-np_k}\leq n^{\epsilon_2}
(np_k)^{1/2} \mbox{ for }1\leq k\leq n^{1/4}\log n
\right\}.
\end{split}
\end{equation}
Lemma \ref{l;d_n} and equation \eqref{eq:dmax} imply that $\mathbb{P}(\Omega_n)\to 1$ by the union bound.
We now control $d^{\mathbf{D_n}}_{\mathrm{TV}}(\chi^n_1,\chi_1)$ on $\Omega_n$.
For $k\geq n^{1/4}\log n$, we have $\mathbb{P}(\chi^n_1=k)=0$. We also have
$\sum_{k\geq n^{1/4}\log n}\mathbb{P}(\chi_1=k)\leq n^{-1} \mathbb{E}(D^4)\leq Cn^{-1}$.
Hence we have
\begin{equation}\label{eq:largek}
\sum_{k\geq n^{1/4}\log n}\abs{\mathbb{P}(\chi^n_1=k)-\mathbb{P}(\chi_1=k)}\leq Cn^{-1}.
\end{equation}
For $0\leq k\leq n^{1/4}\log n$, using the definition of $\Omega_n$ we get, for $n$ large,
\begin{equation}\label{chi1}
\begin{split}
& \abs{\mathbb{P}(\chi^n_1=k)-\mathbb{P}(\chi_1=k)}=
\abs{\frac{(k+1)N_{k+1}}{\sum_{i=1}^n D_i}-\frac{(k+1)p_{k+1}}{m_1}} \\
\leq&\abs{\frac{(k+1)N_{k+1}}{\sum_{i=1}^nD_i}-
\frac{n(k+1)p_{k+1}}{\sum_{i=1}^n D_i}
}+\abs{
\frac{n(k+1)p_{k+1}}{\sum_{i=1}^nD_i} - \frac{(k+1)p_{k+1}}{m_1}}\\
\leq & \frac{(k+1)n^{\epsilon_2}(np_k)^{1/2} }{m_1n/2}+
\frac{(k+1)p_{k+1}\abs{nm_1-\sum_{i=1}^n D_i}}{nm_1 \sum_{i=1}^n D_i}\\
\leq& Cn^{\epsilon_2}n^{-1/2}(k+1)^{-1} +C(k+1)p_{k+1}n^{-1/2+\epsilon_1},
\end{split}
\end{equation}
where we have used the fact that $p_k\leq Ck^{-4}$ since $\mathbb{E}(D^4)<\infty$ in the last step.
Summing the last line of \eqref{chi1} over $k$ from 0 to $n^{1/4}\log n$ and using the facts
$$\sum_{k=1}^{n} k^{-1}\sim \log n, \quad \sum_{k=0}^{\infty} (k+1)p_{k+1}=\mathbb{E}(D)< \infty,$$ we get
\begin{equation}
\sum_{k=0}^{n^{1/4}\log n} \abs{\mathbb{P}(\chi^n_1=k)-\mathbb{P}(\chi_1=k)} \leq Cn^{\epsilon_2} n^{-1/2}\log n+Cn^{-1/2+\epsilon_1}.
\end{equation}
Inserting the values of $\epsilon_1=1/100,\epsilon_2=1/8+1/100$, we obtain that
\begin{equation}\label{eq:smallk}
\sum_{k=0}^{n^{1/4}\log n} \abs{\mathbb{P}(\chi^n_1=k)-\mathbb{P}(\chi_1=k)} \leq Cn^{1/100}\log n(n^{1/8-1/2}+n^{-1/2}).
\end{equation}
Combining \eqref{eq:largek} and \eqref{eq:smallk}, we get
\begin{equation}\label{eq:xi}
d^{\mathbf{D_n}}_{\mathrm{TV}}(\chi^n_1,\chi_1)
=\frac{1}{2}\sum_{k\geq 0} \abs{\mathbb{P}(\chi^n_1=k)-\mathbb{P}(\chi_1=k)}
\leq Cn^{-73/200}\log n.
\end{equation}
One can similarly show that $Cn^{-73/200}\log n$ also serves as an upper bound for $ d^{\mathbf{D_n}}_{\mathrm{TV}}(\psi^n_0,\psi_0)$.
Thus on $\Omega_n$ we have that
$$
d^{\mathbf{D_n}}_{\mathrm{TV}}(\psi^n_0, \psi_0)+(\log n)n^{1/3}
d^{\mathbf{D_n}}_{\mathrm{TV}}(\chi^n_1,\chi_1) \leq Cn^{-73/200+1/3} \log^2 n,
$$
which converges to 0 as $n\to\infty$. This together with \eqref{eq:tv} implies \eqref{eq;tv2} and thus completes the proof of Lemma \ref{l:atst}.
\end{proof}
\section{Upper bound on evoSI}\label{sec:ubsi}
\subsection{avoSI} \label{sec:avoSI}
As mentioned in the introduction, in \cite{YD1} the authors constructed a process that they called `C-evoSI' where `C' stands for `coupled' which means we couple the structure of the graph with the epidemic. They claimed that the `C-evoSI' process has the same law as the original evoSI process on the configuration model.
This claim is not true.
As pointed out in \cite{BB}, in the C-evoSI model, $I-I$ edges can be rewired. While the C-evoSI model defined in \cite{YD1} does not produce the correct dynamics, it does give an upper bound on evoSI. To avoid confusion we will change the name C-evoSI process to C-avoSI where `avo' stands for avoiding infection.
We will now describe the construction of C-avoSI process in \cite{YD1}. First recall that in the construction of the configuration model each vertex is assigned a random number of half-edges initially. In the beginning all half-edges attached to the $n$ vertices are unpaired. The half-edges attached to infected nodes are called infected half-edges and those attached to susceptible nodes are susceptible half-edges. Recall that``randomly chosen'' and ``at random'' mean that the distribution of the choice is uniform over the set of possibilities.
\begin{itemize}
\item
At rate $\lambda$ each infected half-edge pairs with a randomly chosen half-edge in the pool of all half-edges excluding itself. If the vertex $y$ associated with that half-edge is susceptible then it becomes infected. Note that if vertex $y$ changes from state $S$ to $I$ then all half-edges attached to $y$ become infected half-edges.
\item
Each infected half-edge gets removed from the vertex that it is attached to at rate $\rho$ and immediately becomes re-attached to a randomly chosen vertex in the pool of all vertices.
\end{itemize}
For the purpose of comparisons it is convenient to give a reformulation of C-avoSI where the graph has been constructed before the epidemic. We call this process avoSI. To describe this process we define the notion of `stable edge'.
We say an edge between two vertices $x$ and $y$ is stable if one of the following conditions hold:
\begin{itemize}
\item Both $x$ and $y$ are in state $S$.
\item Either $x$ or $y$ has sent an infection to the other one through this edge.
\end{itemize}
We say an edge is unstable if both conditions fail. Note an $S-I$ pair is necessarily unstable since the vertex in state $I$ has not sent an infection to the vertex in state $S$. We define \emph{avoSI} process as follows.
\begin{itemize}
\item Each infected vertex sends infections to its neighbors at rate $\lambda$. If the neighbor has already been infected then nothing changes. Once a vertex receives an infection, it stays infected forever.
\item A vertex in state $S$ will at rate $\rho$ break its connection with a vertex in state $I$ and rewire to another randomly chosen vertex. The events for different $S-I$ connections are independent.
\item For every unstable $I-I$ edge, each of the two $I's$ will rewire at rate $\rho$ to another uniformly chosen vertex.
\end{itemize}
\begin{lemma}\label{c-avosi=avo}
The C-avoSI process and avoSI process running on a configuration model have the same law in terms of the evolution of the set of infected vertices.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{c-avosi=avo}]
It suffices to construct a graph $G$ that has the law of the configuration model such that the set of infected vertices in C-avoSI evolves in the same way as that of the avoSI with initial underlying graph being $G$. Given an outcome of C-avoSI, the graph $G$ can be constructed as follows. Let $H$ be the collection of vertices and half-edges that is the initial condition of C-avoSI.
\begin{itemize}
\item
We assign a unique label to each half-edge in C-avoSI and correspondingly label the half-edges in $H$.
\item
Whenever two half-edges combine into one edge in C-evoSI process we pair the two half-edges with the same labels in $H$. It is clear that the pairings of half-edges are done at random.
We pair the remaining half-edges at random
after there is no infected half-edge in the system. This forms the graph $G$. Since the pairings of all half-edges are at random, we deduce that $G$ itself has the law of $\mathbb{CM}(n,D)$.
\item
Whenever a pairing occurs in C-avoSI, an infected vertex has sent an infection to one of its neighbors(s) in avoSI.
\item
Whenever an infected half-edge $h$ attached to $x$ rewires to another vertex $y$ in the C-avoSI, the corresponding edge $e$ in avoSI, which contains $h$ as one of its two half-edges, breaks from $x$ and reconnects to vertex $y$.
\end{itemize}
The process we construct on $G$ has the same law as avoSI. Indeed, an infected half-edge can be rewired in C-avoSI if and only if it has not been paired. This exactly corresponds to the notion of `unstable' that we used in the construction of avoSI process.
Hence Lemma \ref{c-avosi=avo} follows.
\end{proof}
From now on we will not distinguish between the C-avoSI and the avoSI since they are equivalent. The avoSI process stochastically dominates the evoSI process, as shown in the lemma below.
\begin{lemma}\label{avosi>evosi}
The final size of infected vertices in avoSI stochastically domonates that in evoSI.
\end{lemma}
We couple the evoSI and avoSI as follows. The evoSI is constructed in the same manner as we did in the comparison of evoSI and delSI. For readers' convenience we copy the construction here.
\medskip
{\bf Construction of evoSI.}
We define three set of edges in evoSI at time $t$. {\it Active edges}, denoted by $\mathcal{E}^a_t$, are the edges at time $t$ that connect an infected vertex and a susceptible vertex. {\it Uninfected edges}, denoted by $\mathcal{E}^0_t$, connect two susceptible vertices. {\it Inactive edges}, denoted by $\mathcal{E}^i_t$, have both ends infected. Once an edge becomes inactive it remains inactive forever. The three sets form a partition of all edges.
The three set-valued processes just defined are right-continuous pure jump processes. At time 0 we randomly choose a vertex $u_0$ to be infected. $\mathcal{E}^a_{0}$ consists of the edges with one endpoint at $u_0$. $\mathcal{E}^0_{0}$ is the collection of all edges in the graph minus the set $\mathcal{E}^a_0$. $\mathcal{E}^i_{0}=\emptyset$. We will consider the corresponding sets for delSI, but they will be denoted by $\mathcal{D}_t$ to avoid confusion.
For each undirected edge $e$, let $\tau_{e,\ell}$ be the $\ell$-th time the edge becomes active. To make it easier to describe the dynamics, suppose that at time $\tau_{e,\ell}$ we have $e=\{x_{e,\ell},y_{e,\ell}\}$ with $x_{e,\ell}$ infected and $y_{e,\ell}$ susceptible.
\begin{itemize}
\item
Let $T_{e,\ell}$, $\ell \ge 1$ be independent exponential random variables with mean $1/\lambda$. $T_{e,\ell}$ is the time between $\tau_{e,\ell}$ and the infection of $y_{e,\ell}$ by $x_{e,\ell}$.
\item
Let $R_{e,\ell}$, $\ell \ge 1$ be independent exponential random variables with mean $1/\rho$. $R_{e,\ell}$ is the time between $\tau_{e,\ell}$ and the time when $y_{e,\ell}$ breaks its connection to $x_{e,\ell}$ and rewires.
\item
Let $U_{e,\ell}$ be the randomly chosen vertex that $y_{e,\ell}$ connects to at time $\tau_{e,\ell} + R_{e,\ell}$ (if rewiring occurs).
\end{itemize}
\medskip\noindent
{\bf Initial step.}
To simplify writing formulas, let $S_{e,\ell} = \min\{T_{e,\ell},R_{e,\ell}\}$.
At time 0, a randomly chosen vertex $u_0$ is infected. Let ${\cal N}^0(x,t),{\cal N}^a(x,t)$ and ${\cal N}^i(x,t)$ be the collection of uninfected, active and inactive edges that are connected to $x$ at time $t$, respectively. At time 0 the edges ${\cal N}^0(u_0,0)=\{e_1,\ldots, e_k\}$ are added to the list of active edges. We have $\mathcal{E}^a_{0}=\{e_1,\ldots, e_k\}$. Suppose $e_j$ connects $u_0$ and $y_j$. At time
$$
J_1 = \min_{1\le j \le k} S_{e_j,1}
$$
the first event occurs. ($J$ is for jump.) Let $i$ be the index that achieves the minimum.
\medskip\noindent
(i) If $R_{e_i,1}<T_{e_i,1}$, then at time $J_1$ vertex $y_i$ breaks its connection with $u_0$ and rewires to $U_{e_i,1}$. If $U_{e_i,1}$ is susceptible at time $J_1$, we move the edge $e_i$ to $\mathcal{E}^0_{J_1}$. On the initial step this will hold unless $U_{e_1,1}=u_0$ in which case nothing has changed.
\medskip\noindent
(ii) If $T_{e_i,1}<R_{e_i,1}$ then at time $J_1$ vertex $y_i$ becomes infected by $u_0$. We move $e_i$ to $\mathcal{E}^i_{J_1}$. We add edges in ${\cal N}^0(y_i,J_1-)$ to ${\cal E}^a_{J_1}$.
\medskip\noindent
{\bf Induction step.}
For any active edges $e$ present at time $t$, let $L(e,t) = \sup\{\ell : \tau_{e,\ell}\leq t\}$
and let $V(e,t)=\tau_{e,L(e,t)}+S_{e,L(e,t)}$ be the time of the next event (infection or rewiring) to affect edge $e$.
Suppose we have constructed the process up to time $J_n$.
If there are no active edges present at time $J_{n}$, the construction is done. Otherwise, Let
$$
J_{n+1}=\min_{e\in \mathcal{E}^a_{J_n}} V(e,J_{n}).
$$
Let $e_n$ be the edge that achieves the minimum of $V(e,J_n)$,
let $x(e_n)$ be the infected endpoint of $e_n$ and $y(e_n)$ be the susceptible endpoint of $e$. To simplify notation let $L_n =L(e_n,J_n)$.
\medskip\noindent
(i) If $R_{e_n,L_n}<T_{e_n,L_n}$, then at time $J_{n+1}$ vertex $y(e_n)$ breaks its connection with $x(e_n)$ and rewires to $U_{e_n, L_n}$. If $U_{e_n,L_n}$ is susceptible at time $J_{n+1}$, then $e_n$ is moved to $\mathcal{E}^0_{J_{n+1}}$. Otherwise it remains active.
\medskip\noindent
(ii) If $T_{e_n,L_n}<R_{e_n,L_n}$, then at time $J_{n+1}$ the vertex $y(e_n)$ is infected by $x(e_n)$ and $e_n$ is moved to $\mathcal{E}^i_{J_{n+1}}$.
\begin{itemize}
\item
All edges $e'$ in ${\cal N}^0_{y(e_n),J_{n+1}-}$ are moved to ${\cal E}^a_{J_{n+1}}$. Since $y(e_n)$ has just become infected, the other end of $e'$ must be susceptible at time $J_{n+1}$.
\item
All edges $e''$ in ${\cal N}^a_{y(e_n),J_{n+1}-}$ are moved to ${\cal E}^i_{J_{n+1}}$. Since $y(e_n)$ has just become infected, (i) the other end of $e''$ must be infected at time $J_{n+1}$, and (ii) $e''$ cannot have been inactive earlier.
\end{itemize}
The evoSI process stops when there are no active edges.
\medskip
{\bf Construction of avoSI.}
For avoSI we need four sets to partition the set of all edges. We use
$\mathcal{A}$ to avoid confusion with the construction of evoSI (not to be confused with the $\mathcal{A}_t$ used in Section \ref{sec:pfth2}).
{\it Active edges with one end infected} at time $t$, denoted by $\mathcal{A}^{a,1}_t$, are the edges at time $t$ that connect an infected vertex and a susceptible vertex.
{\it Active edges with both ends infected}, denoted by $\mathcal{A}^{a,2}_t$, are the unstable edges at time $t$ that connect two infected vertices.
{\it Uninfected edges}, denoted by $\mathcal{A}^0_t$, connect two susceptible vertices. {\it Inactive edges}, denoted by $\mathcal{A}^i_t$, consists of stable infected edges. Once an edge becomes inactive it remains inactive forever. The four sets form a partition of all edges.
The four set-valued processes just defined are right-continuous pure jump processes. At time 0 we randomly choose a vertex $u_0$ to be infected. $\mathcal{A}^{a,1}_{0}$ consists of the edges with one endpoint at $u_0$. $\mathcal{A}^0_{0}$ is the collection of all edges in the graph minus the set $\mathcal{E}^a_0$. $\mathcal{A}^i_{0} = \mathcal{A}^{a,2}_{0} =\emptyset$.
For each undirected edge $e$, we let random variables $T_{e,\ell},R_{e,\ell},U_{e,\ell}$ be the same as those used in the construction of evoSI. We set $S_{e,\ell}=\min\{T_{e,\ell},R_{e,\ell}\}$. We also let $V'_{e,\ell}$ be independent uniform random variables that take values in the two endpoints of $e$.
To make it easier to describe the dynamics, suppose that at time $\tau_{e,\ell}$ we have $e=\{x_{e,\ell},y_{e,\ell}\}$ with $x_{e,\ell}$ infected and $y_{e,\ell}$ susceptible.
The difference between evoSI and avoSI in terms of these clocks is as follows. For any edge $e$ connecting $x$ and $y$, once one endpoint $x_{e,\ell}$ becomes infected, the clocks $T_{e,\ell}$ and $R_{e,\ell}$ start running. If the other endpoint $y_{e,\ell}$ also becomes infected through other edges
at (relative) time $w_{e,\ell} < S_{e,\ell}$ (here `relative' means we only count the time after the infection of $x$), then we replace the clock $T_{e,\ell}$ and $R_{e,\ell}$ by
$$
T'_{e,\ell}=\frac{T_{e,\ell}-w_{e,\ell}}{2} \mbox{\, and \,} R'_{e,\ell}=\frac{R_{e,\ell}-w_{e,\ell}}{2}
$$
since the rates are now twice as fast.
\medskip\noindent
To see this construction give the correct dynamics of avoSI. We note that conditionally on $T_{e,\ell},R_{e,\ell}>w_{e,\ell}$, $(T_{e,\ell}-w_{e,\ell})/2$ and $(R_{e,\ell}-w_{e,\ell})/2$ are independent exponential random variables with parameters $2\lambda$ and $2\rho$, respectively.
This corresponds to an unstable $I-I$ pair where each of the two $I's$ attempt to send infection to the other $I$ at rate $\lambda
$ and rewire from the other $I$ at rate $\rho$. The variable $V'_{e,\ell}$ corresponds to the vertex with small rewiring time.
\medskip\noindent
if $T_{e,\ell}>R_{e,\ell}$ then $T'_{e,\ell}>R'_{e,\ell}$ and vice versa.
Using this and the fact the uniform variable $U_{e,\ell}$ is the same in evoSI and avoSI we get the following proposition.
\begin{prop}\label{prop:avoevo}
If edge $e$ is rewired in evoSI, then $e$ must be rewired to the same vertex in avoSI (as long as either endpoint of $e$ becomes infected in avoSI).
Also the time it take to $e$ to be rewired in evoSI is not smaller than the time in avoSI.
\end{prop}
Similarly to evoSI, we use $\tau_{e,\ell}$ to denote the $\ell$-th time $e$ becomes active in avoSI.
We now formally construct the avoSI process by induction.
\medskip\noindent
{\bf Initial step.}
At time 0, a randomly chosen vertex $u_0$ is infected. Let ${\cal N}^0(x,t),{\cal N}^{a,1}(x,t), \mathcal{N}^{a,2}(x,t)$ and ${\cal N}^i(x,t)$ be the sets of edges connected to $x$ that
belong to the sets $\mathcal{A}^0_t, \mathcal{A}^{a,1}_t, \mathcal{A}^{a,1}_t$ and $\mathcal{A}^i_t$, respectively. At time 0 all edges in $\{e_1,\ldots, e_k\}$ are added to the list of active edges so that $\mathcal{A}^{a,1}_{0}=\{e_1,\ldots, e_k\}$. Suppose $e_j$ connects $u_0$ and $y_j$. At time
$$
J_1 = \min_{1\le j \le k} S_{e_j,1}
$$
the first event occurs. ($J$ is for jump.) Let $i$ be the index that achieves the minimum.
\medskip\noindent
(i) If $R_{e_i,1}<T_{e_i,1}$, then at time $J_1$ vertex $y_i$ breaks its connection with $u_0$ and rewires to $U_{e_i,1}$. If $U_{e_i,1}$ is susceptible at time $J_1$, we move the edge $e_i$ to $\mathcal{A}^0_{J_1}$. On the initial step this will hold unless $U_{e_1,1}=u_0$ in which case nothing has changed.
\medskip\noindent
(ii) If $T_{e_i,1}<R_{e_i,1}$ then at time $J_1$ vertex $y_i$ becomes infected by $u_0$. We move $e_i$ to $\mathcal{A}^i_{J_1}$. We move edges in ${\cal N}^0(y_i,J_1-)$ to ${\cal A}^a_{J_1}$.
\medskip\noindent
{\bf Induction step.}
For any active edge $e$ at time $t$, let $L(e,t) = \sup\{\ell : \tau_{e,\ell}\leq t\}$
and set $$V(e,t)=
\begin{cases} \tau_{e,L(e,t)}+S_{e,L(e,t)}
&\hbox{ if $e \in \mathcal{A}_t^{a,1}$,
}\\
\frac{\tau_{e,L(e,t)}+I(x^*,y^*)+S_{e,L(e,t)}}{2}
&\hbox{ if $e\in \mathcal{A}^{a,2}_t$,}
\end{cases}
$$
where $I(x^*,y^*)$ is the first time that both $x^*=x(e,L(e,t))$ and
$y^*=y(e,L(e,t))$ become active vertices and the second line comes from the computation
$$
\tau_{e,L(e,t)}+(I(x^*,y^*) - \tau_{e,L(e,t)} )+\frac{S_{e,L(e,t)}
-(I(x^*,y^*) - \tau_{e,L(e,t)} )}{2}=\frac{\tau_{e,L(e,t)}+I(x^*,y^*)+S_{e,L(e,t)}}{2}.
$$
Then $V(e,t)$ is
the time of the next event (infection or rewiring) to affect edge $e$.
Suppose we have constructed the process up to time $J_n$.
If there are no active edges present at time $J_{n}$, the construction is done. Otherwise, we
let
$$
J_{n+1}=\min_{e\in \mathcal{A}^a_{J_n}} V(e,J_{n}).
$$
Let $e_n$ be the edge that achieves the minimum of $V(e,J_n)$. If $e_n$ only have one endpoint infected at time $J_n$ then we
let $x(e_n)$ be the infected endpoint of $e_n$ and $y(e_n)$ be the susceptible endpoint of $e_n$. To simplify notation let $L_n =L(e_n,J_n)$.
\medskip\noindent
(i) If $R_{e_n,L_n}<T_{e_n,L_n}$ and $e_n \in \mathcal{A}^{a,1}_{J_n}$, then at time $J_{n+1}$ vertex $y(e_n)$ breaks its connection with $x(e_n)$ and rewires to $U_{e_n, L_n}$. If $U_{e_n,L_n}$ is susceptible at time $J_{n+1}$, then $e_n$ is moved to $\mathcal{A}^0_{J_{n+1}}$. Otherwise it remains in $\mathcal{A}^{a,1}_{J_{n+1}}$.
\medskip\noindent
(ii) If $R_{e_n,L_n}<T_{e_n,L_n}$ and $e_n\in \mathcal{A}^{a,2}_{J_n}$ , then at time $J_{n+1}$ vertex $ V'_{e,L_n}$ breaks its connection with the other end of $e_n$ and rewires to $U_{e_n, L_n}$. If $U_{e_n,L_n}$ is susceptible at time $J_{n+1}$, then $e_n$ is moved to $\mathcal{A}^{a,1}_{J_{n+1}}$. Otherwise $e_n$ stays in the set $\mathcal{A}^{a,2}_{J_{n+1}}$
\medskip\noindent
(iii) If $T_{e_n,L_n}<R_{e_n,L_n}$ and $e_n \in \mathcal{A}^{a,1}_{J_n}$, then at time $J_{n+1}$ the vertex $y(e_n)$ is infected by $x(e_n)$ and $e_n$ is moved to $\mathcal{A}^i_{J_{n+1}}$.
\begin{itemize}
\item
All edges $e'$ in ${\cal N}^0_{y(e_n),J_{n+1}-}$ are moved to ${\cal A}^{a,1}_{J_{n+1}}$. Since $y(e_n)$ has just become infected, the other end of $e'$ must be susceptible at time $J_{n+1}$.
\item
All edges $e''$ in ${\cal N}^{a,1}_{y(e_n),J_{n+1}-}$ are moved to ${\cal A}^{a,2}_{J_{n+1}}$. Since $y(e_n)$ has just become infected, (i) the other end of $e''$ must be infected at time $J_{n+1}$, and (ii) $e''$ cannot have been inactive earlier.
\end{itemize}
\medskip\noindent
(iv) If $T_{e_n,L_n}<R_{e_n,L_n}$ and $e_n \in \mathcal{A}^{a,2}_{J_n}$, then at time $J_{n+1}$ $e_n$ is moved to $\mathcal{A}^i_{J_{n+1}}$. There are no changes for other edges.
The avoSI process stops when there are no active edges.
\begin{proof}[Proof of Lemma \ref{avosi>evosi}]
We now prove by induction that all vertices infected in evoSI are also infected in avoSI and actually they are infected earlier in avoSI than evoSI.
\medskip\noindent
The induction hypothesis holds for the first vertex since initially $u_0$ is infected in both evoSI and avoSI. Suppose the induction holds up to the $k$-th infected vertex in evoSI. Assume at time $t$, $y$ becomes the $(k+1)$-th infected vertex in evoSI.
We assume $y$ is infected by vertex $x$ through edge $e$. Note that $e$ has possible gone through a series of rewirings before connecting vertex $y$. We assume that $e$ connects vertices $x_{\ell}$ and $y_{\ell}$ after the $(\ell-1)$-th rewiring.
We also assume when $x$ infects $y$ through $e$, $e$ has been rewired $r$ times. This implies that
\begin{equation}\label{tell}
T_{e,\ell}>R_{e,\ell} \mbox{ for } 1\leq \ell \leq r \mbox{ and }T_{e,r+1}<R_{e,r+1}.
\end{equation}
We let $m(x)=\inf\{i: x\in \{x_k,y_k\} \hbox{ for all }i \le k \le r+1 \}$.
We now divide the analysis into two cases: $m=1$
and $1<m\leq r+1$.
\begin{figure}[h]
\begin{center}
\begin{picture}(140,100)
\put(80,40){\line(1,1){28}}
\put(80,40){\line(0,1){40}}
\put(80,40){\line(-1,1){28}}
\put(80,40){\line(-1,0){40}}
\put(75,30){$y$}
\put(115,75){$x_1$}
\put(35,75){$x_{r}$}
\put(75,90){$x_2$}
\put(5,35){$x=x_{r+1}$}
\end{picture}
\caption{Illustration of the case $m=1$. If $y$ is infected before time $t$ in avoSI then the induction step holds true.
If not, then since $x_1, \ldots, x_{r+1}$ are the rewirings in evoSI, they are also infected in avoSI and the rewiring occur as does the infection of $x$ by $x_{r+1}$.}
\label{fig:case1}
\end{center}
\end{figure}
\medskip\noindent
{\bf Case 1.} $m=1$ so that $y=y_k$ for all $1\leq k\leq r+1$. If we assume that $y$ has not been infected by time $t$ then $x_1, \ldots x_{r}$ must have been infected at the time that the rewiring occurred
and then $x_{r+1}=x$ infected $y$ at time $t$ in evoSI. By the induction hypothesis we see that $x_1,\ldots, x_{r+1}$ are also infected in avoSI. By Proposition \ref{prop:avoevo} and \eqref{tell}, $e$ breaks its connection with $x_1$ and reconnects to $x_2$, then to $x_3$ and after $r$ rewirings to $x_{r+1}$. If $y$ is already infected before $x_{r+1}$ sends an infection to it than we are done. Otherwise since $T_{e,\ell+1}<R_{e,\ell+1}$ we see $x_{r+1}$ will send an infection to $y$ in avoSI as well. In any case we have proved $x$ will also be infected in avoSI and is infected earlier. For a picture see Figure \ref{fig:case1}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(220,190)
\put(80,40){\line(1,1){28}}
\put(80,40){\line(0,1){40}}
\put(80,40){\line(-1,1){28}}
\put(80,40){\line(-1,0){40}}
\put(75,30){$y=y_3$}
\put(120,30){$y_2=y_1$}
\put(95,75){$x_3=x_2$}
\put(170,75){$x_1$}
\put(35,75){$x_r$}
\put(-5,35){$x'=x_{r+1}$}
\put(108,68){\line(1,-1){28}}
\put(136,40){\line(1,1){28}}
\put(200,50){evoSI}
%
\put(80,130){\line(0,1){40}}
\put(80,130){\line(-1,1){28}}
\put(80,130){\line(-1,0){40}}
\put(75,120){$y=y_3'$}
\put(130,120){$y_1'$}
\put(105,165){$y_2'$}
\put(140,165){$x_1'=x_2'=x_3'$}
\put(35,165){$x_r'$}
\put(-5,125){$x=x_{r+1}'$}
\put(164,158){\line(-3,-1){82}}
\put(164,158){\line(-1,0){56}}
\put(136,130){\line(1,1){28}}
\put(200,140){avoSI}
\end{picture}
\caption{Picture of a ``bad'' case. We assume that at the time of the rewiring
$x_1$, $y_1$ and $x_2$ are infected. Since one flips coins to determine
the end that rewires, the sequence of of edges $(x_k',y_k')$ in avoSI is different
from the edges in evoSI. However, thanks to the use of $U_{e,\ell}$ to determine the new endpoint, the second rewiring brings the edge to $y$, and there is a correspondence between
the vertices in the two processes indicated by the drawing.}
\label{fig:case2}
\end{center}
\end{figure}
\medskip\noindent
{\bf Case 2.} If $m>1$, then again by Proposition \ref{prop:avoevo}, the induction hypothesis and \eqref{tell} we see that in the avoSI picture, $e$ will be rewired at least $r$ times and $y$ becomes an endpoint of $e$ exactly after $m-1$ rewirings. After this point we can repeat the analysis in the case of $m=1$ to deduce that $x$ is also infected no later than $t$ in avoSI.
For a picture see Figure \ref{fig:case2}.
\end{proof}
We now show that avoSI and delSI actually have the same critical value (and thus also have the same critical value as evoSI by Theorem \ref{critical_value}).
\begin{lemma}\label{critical_avosi}
Theorem \ref{critical_value} still holds if we replace evoSI by avoSI.
\end{lemma}
\begin{proof}
As we will see, this follows from the same proof of Theorem \ref{critical_value}.
First we note that since avoSI stochastically dominates evoSI (in terms of final epidemic size) and evoSI dominates delSI, avoSI must also dominate delSI.
We claim that if we run avoSI on a tree and no edge is rewired to vertices that are infected up to time $t$, then there are no unstable $I-I$
pairs up to time $t$. This immediately implies that the evolution of the avoSI process is equal to the evoSI process starting from the same initially infected vertex up to time $t$.
To prove the claim, suppose there is an unstable $I-I$ pair connecting vertex $x$ and $y$, then there must be two infection paths that lead to the infections of $x$ and $y$. Here an infection path for $x$ is just a sequence of vertices $u_0\to u_1 \to \cdots \to x$ where the former vertex in the chain infects the latter. Since the edge between $x$ and $y$ is unstable, we see that if we consider the union of the two infection paths together with the edge $(x,y)$ then we get a cycle of infected vertices. This contradicts with the assumption that the original graph is a tree and no edges are rewired to vertices infected by $t$ (so that rewirings will not help create any cycle of infected vertices).
As we mentioned in the proof of Lemma \ref{l:atst}, by Lemma 2.12 in \cite{vdH3}, whp the subgraph obtained by exploring the neighborhoods of $n^{3/8}/\log n$ vertices of any fixed vertex is a tree. The proof of Theorem \ref{critical_value}(i) implies that with high probability no edge is rewired to infected vertices up to step $O(n^{1/3})$ (i..e, up to the exploration of the neighborhoods of $O(n^{1/3})$ vertices). Hence the condition of the above claim is satisfied.
Using the conclusion of the claim we see that whp we can couple the avoSI process and evoSI process such that they coincide up to step $O(n^{1/3})$. Therefore the proof of Theorem \ref{critical_value} also applies to the comparison of avoSI and delSI and hence Lemma \ref{critical_avosi} follows.
\end{proof}
\subsection{Tightness of $\{\overline{S}_{t,k}/n,t\geq 0\}_{n\geq 1}$} \label{sec:tight}
We first consider $\tilde S_{t,k}$, the number of susceptible vertices with $k$ half-edges at time $t$ in the original avoSI process (i.e., without the time change). We have the following equation
\begin{equation}\label{eqStk1}
d\tilde S_{t,k}=-\left(\lambda\tilde X_{I,t}\frac{k\tilde S_{t,k}}{\tilde X_t-1}\right)dt+\left(1_{\{k\geq 1\}}\rho\tilde X_{I,t}
\frac{\tilde S_{t,k-1}}{n}\right)dt-\left(\rho\tilde X_{I,t}
\frac{\tilde S_{t,k}}{n}\right)dt+d\tilde M_{t,k}.
\end{equation}
\noindent To explain the terms
\begin{enumerate}
\item
At rate $\lambda \tilde X_{I,t}$ infections occur. The infected half-edge attaches to a susceptible vertex with $k$ half-edges, which we call an $S^k$, with probability $k\tilde S_{k,t}/(\tilde X_t-1)$. The $-1$ in the denominator is because the half-edge will not connect to itself.
\item
At rate $\rho \tilde X_{I,t}$ rewirings occur. If $k\ge 1$, the half-edge gets attached to an $S^{k-1}$ with probability $(k-1)\tilde S_{k-1,t}/(\tilde X_t-1)$, promoting it to an $S^k$.
\item
If the rewired half-edge gets attached to an $S^{k}$, which occurs with probability $k\tilde S_{k,t}/(\tilde X_t-1)$, it is promoted to an $S^{k+1}$ and an $S^k$ is lost.
\item
If $Z_t$ is a Markov chain with generator $L$ then Dynkin's formula implies
$$
f(Z_t) - \int_0^t Lf(X_s) \,ds \quad\hbox{is a martingale.}
$$
See Chapter 4, Proposition 1.7 in \cite{EK}. Fortunately, we do not need an explicit formula for the martingale. All that is important is that when $f(Z_t) =\tilde S_{t,k}$, $\tilde M_{\cdot,k}$ has jumps equal to $\pm 1$.
\end{enumerate}
The equation for $\overline{S}_{t,k}$ can be then obtained by multiplying the first three terms in the right hand side of \eqref{eqStk1} by the time change, leading to
\begin{align}
d\overline{S}_{t,k}=&-\left(\lambda \overline{X}_{I,t}\frac{\overline{X}_t-1}{\lambda \overline{X}_{I,t}}\frac{k\overline{S}_{t,k}}{\overline{X}_t-1}\right)dt
+\left(1_{\{k\geq 1\}}\rho \overline{X}_{I,t} \frac{\overline{X}_t-1}{\lambda \overline{X}_{I,t}}\frac{\overline{S}_{t,k-1}}{n}\right)dt
\nonumber\\
&-\left(\rho \overline{X}_{I,t} \frac{\overline{X}_t-1}{\lambda \overline{X}_{I,t}}\frac{\overline{S}_{t,k}}{n}\right)dt +d\overline{M}_{t,k}.
\label{eqStk}
\end{align}
Here $\overline{M}_{t,k}$ is a time-changed version of the previous martingale so it is also a margingale with jumps $\pm 1$.
Canceling common factors and dividing both sides by $n$ we get
\begin{align}
d\left(\frac{\overline{S}_{t,k}}{n}\right) & =-\left(k\frac{\overline{S}_{t,k}}{n}\right) dt
+\left(1_{(k\geq 1)} \frac{\rho }{\lambda}\frac{\overline{X}_t-1}{n}\frac{\overline{S}_{t,k-1}}{n}\right)dt
\nonumber\\
&-\left(\frac{\rho }{\lambda}\frac{\overline{X}_t-1}{n}\frac{\overline{S}_{t,k}}{n}\right)dt+d\left(\frac{\overline{M}_{t,k}}{n}\right).
\label{eqStk2}
\end{align}
We now show that for all fixed $k\geq 0$ and $T>0$,
\begin{equation}
\sup_{0\leq t \leq T\wedge \gamma_n}\abs{\overline{M}_{t,k}}/n\xrightarrow{\mathbb{P}} 0.
\label{term4}
\end{equation}
To do this we note that the expected value of quadratic variation of $\overline{M}_{t\wedge \gamma_n,k}$ evaluated at time $T$, which is also equal to $\mathbb{E}(\overline{M}_{T\wedge \gamma_n,k}^2)$, is bounded above by the expectation of total number of jumps in the whole avoSI process, which is equal to
$$
\mathbb{E}\left(X_0/2+\sum_{j=1}^{X_0} N_j\right).
$$
Here we have a factor of 2 in the denominator because each pairing event takes two half-edges and
$N_j$ is the number of times that half-edge $j$ gets transferred to another vertex. Note that an infected half-edge gets rewired before being paired with probability at most $\rho/(\lambda+\rho)$ and susceptible half-edge cannot get rewired unless the vertex it is attached to becomes infected. Thus, $N_j$ is stochastically dominated by a Geometric($\rho/(\lambda+\rho)$) distributed random variable, so that for all $j$, $\mathbb{E}(N_j)\le C$ for some constant $C$. Therefore by $L^2$ maximal inequality applied to the submartingale $\abs{\overline{M}_{t\wedge \gamma_n,k}}$, we obtain that
\begin{equation}\label{mskctl}
\mathbb{E}\left(\sup_{0\leq t \leq T\wedge \gamma_n} \overline{M}_{t,k}^2\right)\leq 4\mathbb{E}(\overline{M}_{T\wedge \gamma_n,k}^2 )\leq Cn,
\end{equation}
where $C$ is a constant whose value is unimportant.
Since $\overline{S}_{0,k}/n\leq 1$, we know $\{\overline{S}_{0,k}/n\}_{n\geq 1}$ is a tight sequence of random variables. To establish tightness of $\{\overline{S}_{t,k}/n,t\geq 0\}_{n\geq 1}$ we need to show
for any fixed $\epsilon,\delta>0$, there is a $\theta>0$ and integer $n_0$ so that for $n\geq n_0$
\begin{equation}\label{tightness}
\mathbb{P}\left (\sup_{\abs{t_1-t_2}\leq \theta, t_1,t_2\leq T } \abs{\overline{S}_{t_1\wedge \gamma_n, k}-\overline{S}_{t_2\wedge\gamma_n,k}}/n \geq \delta \right)
\leq \epsilon.
\end{equation}
Assuming \eqref{tightness} for the moment,
we see that $\{\overline{S}_{t\wedge \gamma_n,k}/n,t\geq 0\}$, as an element of $\mathbb{D}$, the space of right continuous paths with left limits, satisfies condition (ii) of
Proposition 3.26 in \cite{JS}. Consequently,
$\{\overline{S}_{t\wedge \gamma_n,k}/n,t\geq 0\}_{n\geq 1}$ is a tight sequence.
It remains to prove \eqref{tightness}. We note that there exist a constant $C$ so that
\begin{equation}
\mathbb{P}(\overline{X}_0/n>C)\leq \epsilon/3,
\label{X0bd}
\end{equation}
since $\mathbb{E}(\overline{X}_0)=\mathbb{E}(\sum_{i=1}^n D_i)= nm_1$. Hence using $\overline{S}_{t,k-1}+\overline{S}_{t,k}\leq n$ and $\overline{X}_t\leq \overline{X}_0 $ with \eqref{eqStk2}
\begin{align*}
&\mathbb{P}\left(\sup_{\abs{t_1-t_2}\leq \theta, t_1,t_2\leq T } \abs{\overline{S}_{t_1\wedge \gamma_n, k}
-\overline{S}_{t_2\wedge\gamma_n,k}}/n \geq \delta \right)
\\
\leq & \mathbb{P}\left( k\theta>\frac{\delta}{4} \right) + 2\mathbb{P}\left(\frac{\rho}{\lambda} \frac{X_0}{n}\theta\geq \frac{\delta}{4}\right)
+\mathbb{P}\left(2 \sup_{0\leq t \leq T\wedge \gamma_n} \abs{\overline{M}_{t,k}/n} \geq \frac{\delta}{4} \right).
\end{align*}
Using \eqref{X0bd} and \eqref{mskctl}, we see that if we pick $\theta$ small and $n$ large then the last line is $\le \epsilon$.
This proves \eqref{tightness} and thus completes the proof of tightness of the sequence $\{\overline{S}_{t\wedge \gamma_n,k}/n,t\geq 0\}_{n\geq 1}$.
\subsection{Convergence of $\{\overline{S}_{t,k}/n,t\geq 0\}_{n\geq 1}$} \label{sec:convStk}
Note the evolution for $\overline{X}_t$ has the same transition rates as the time-changed SIR dynamics defined in \cite{JLW} so their equation (3.4) also holds true in avoSI, which gives for any fixed $T$,
\begin{equation}\label{x_t}
\sup_{0\leq t \leq T\wedge \gamma_n}\abs{ \frac{\overline{X}_t}{n}- m_1 \exp(-2t)} \xrightarrow{\mathbb{P}} 0.
\end{equation}
In order to upgrade $\sup_{0\leq t\leq T\wedge \gamma_n}$ to $\sup_{0\leq t\leq \gamma_n}$, we note that for any $\epsilon>0$, we can pick a sufficiently large $T$ so that $m_1\exp(-2T)<\epsilon$.
Then by monotonically decreasing property of $\overline{X}_t$ we see
\begin{align}
\label{Tgam}
& \mathbb{P}\left(\gamma_n>T, \sup_{T< t\leq \gamma_n} \abs{\frac{\overline{X}_t}{n}-m_1\exp(-2t)}>3\epsilon\right)
\leq \mathbb{P}(\gamma_n>T,\overline{X}_T/n>2\epsilon) \\
\leq & \mathbb{P}\left(\sup_{0\leq t \leq T\wedge \gamma_n}\abs{ \frac{\overline{X}_t}{n}- m_1 \exp(-2t)}>\epsilon\right)\leq \epsilon, &
\nonumber
\end{align}
for $n$ large enough. Thus we deduce
$$
\mathbb{P}\left( \sup_{0\leq t\leq \gamma_n}\abs{\frac{\overline{X}_t}{n}-m_1\exp(-2t)}>\epsilon\right)\leq 2\epsilon
$$
for $n$ sufficiently large, which proves the first equation of \eqref{3lim}.
By the tightness of $\{\overline{S}_{t\wedge \gamma_n,k}/n,t\geq 0\}_{n\geq 1}$, we see for any subsequence of $\overline{S}_{t,k}/n$ we can extract a further subsequence that converges in distribution to a process $\overline{s}_{t,k}$ with continuous sample path. By the Skorokhod representation theorem we can assume the convergence is actually in the almost sure sense and we can also assume that
$\overline{X}_{t}/n$ converges a.s. to $m_1\exp(-2t)$.
Having established tightness, a standard argument implies that we can show the convergence of $\overline{S}_{t,k}/n$ by establishing that the limit $\overline{s}_{t,k}$ is independent of the subsequence. First consider the case $k=0$. The first two terms on the right-hand side of \eqref{eqStk2} are 0, so using \eqref{term4} and first equation of \eqref{3lim} we see that any subsequential limit $\overline{s}_{t,0}$ has to satisfy the equation
\begin{equation}\label{s0t}
\overline{s}_{t,0}=-\frac{\rho}{\lambda}m_1 \int_0^t \exp(-2z)\overline{s}_{z,0}\, dz.
\end{equation}
\noindent Since
$z \to \exp(-2z)$ is Lipschitz continuous this equation has a unique solution.
Repeating this process for $k\geq 1$ we see that any subsequential limit $\overline{s}_{t,k}$ of $\overline{S}_{t,k}/n$ satisfies the differential equation
\begin{equation}\label{stk}
\overline{s}'_{t,k}=-k\overline{s}_{t,k}-\alpha \exp(-2t)\overline{s}_{t,k}+1_{\{k\geq 1\}}\alpha \exp(-2t)\overline{s}_{t,k-1},
\end{equation}
\noindent
where $\alpha=\rho m_1/\lambda$.
This system of equations can be solved explicitly. First we rewrite the equations as
$$
\overline{s}_{t,k}'+[k+\alpha \exp(-2t)]\overline{s}_{t,k}= 1_{(k\ge 1)}\alpha \exp(-2t)\overline{s}_{t,k-1}.
$$
Define
\begin{equation}
g_{t,k}=\exp\left(kt+ (\alpha/2) (1-\exp(-2t))\right)\overline{s}_{t,k},
\end{equation}
then
\begin{align*}
g_{t, 0}' &= \alpha \exp(-2t) g_{t,0} + \exp((\alpha/2)(1-\exp(-2t)) \overline{s}'_{t,0} \\
& = \alpha \exp(-2t) \exp((\alpha/2) (1-\exp(-2t)))\overline{s}_{t,0} \\
& + \exp(\alpha/2)(1-\exp(-2t))[-\alpha \exp(-2t)] \overline{s}_{t,0} = 0.
\end{align*}
Let $A_{k,t} = \exp(kt+ (\alpha/2) (1-\exp(-2t)))$.
An almost identical calculation for $k \ge 1$ gives
\begin{align*}
g_{t,k}' &= [k +\alpha \exp(-2t)] g_{t,k} + A_{k,t} \overline{s}'_{t,k} \\
& = [k+\alpha \exp(-2t)]A_{t,k} \overline{s}_{t,k} \\
& + A_{t,k} \{ [-k -\alpha \exp(-2t)] \overline{s}_{t,k} + \alpha \exp(-2t) \overline{s}_{t,k-1}\},
\end{align*}
so we have
$$
g'_{t,k}= A_{t,k}\alpha \exp(-2t) \overline{s}_{t,k-1} = \alpha \exp(-t)g_{t,k-1}.
$$
Making the change of variable $s=\alpha (1-\exp(-t))$ and letting $h_{s, k}= g_{t,k}$, we see that $h_{s, 0}$ is constant in $s$ and
$$
h'_{s,k}=h_{s, k-1}, k\geq 1,
$$
from which we see $h_{s,k}$ is a polynomial of degree $k$ in $s$ and for all $\ell\leq k$ the $\ell$-th derivative of $h_{s,k}$ at $s=0$ equals
$h_{0,k-\ell}$. From this we obtain that for all $k$,
$$
h_{s,k}=\sum_{\ell=0}^k \frac{h_{0,k-\ell}}{l!}s^{\ell}.
$$
The initial conditions are $g_{0,k}=h_{0,k}=s_{0,k}=p_k=\mathbb{P}(D=k)$.
It follows that
\begin{equation}
h_{s,k}= \sum_{\ell=0}^k \frac{p_{k-\ell}s^{\ell} }{\ell!},
\end{equation}
and hence using definitions of $\overline{s}$, $g$, and $h$
\begin{align}
g_{t,k}&=h_{s,k}= \sum_{\ell=0}^k \frac{p_{k-\ell}}{\ell!} (\alpha (1-w))^k,
\label{gtkeq}\\
\overline{s}_{t,k}&=\exp\left(-\frac{\alpha}{2}(1-w^2)\right)w^k\sum_{\ell=0}^k \frac{p_{k-\ell}}{\ell!} (\alpha (1-w))^{\ell},
\label{stkeq2}
\end{align}
where $w=w(t)=\exp(-t)$.
\subsection{Summing the $\overline{s}_{t,k}$}\label{sec:sum_stk}
We pause to record the following fact which we will use later. From the explicit expression for $\overline{s}_{t,k}$ in \eqref{stkeq2},
dropping the factor $\exp(-\alpha(1-w^2)/2)\leq 1$ and writing $k=(k-\ell) + \ell$, we get
\begin{align}
\sup_{t\geq 0} \sum_{k \geq K}k\overline{s}_{t,k}\leq &
\sup_{0\leq w\leq 1} \sum_{k \geq K} \sum_{\ell=0}^{k} \frac{(k-\ell)p_{k-\ell}}{\ell!}(\alpha (1-w))^{\ell} w^k,
\nonumber \\
&+\sup_{0\leq w\leq 1} \sum_{k \geq K} \sum_{\ell=1}^{k} \frac{\ell p_{k-\ell}}{\ell!}(\alpha (1-w))^{\ell} w^k.
\label{splitup}
\end{align}
The $\ell=0$ term in the first sum is bounded by
$$
\sum_{k \ge K} k p_k.
$$
The remainder of the two sums is bounded by
\begin{equation}\label{remaindsum}
\sup_w w^K (1-w) \left[ \sum_{k \ge \ell, \ell \ge 1} \left( \frac{\alpha^\ell (k-\ell) p_{k-\ell}}{\ell!}
+ \frac{\alpha^\ell \ell p_{k-\ell}}{\ell!} \right) \right].
\end{equation}
Interchanging the order of summation in the double sum and letting $m=k-\ell$, the double sum in \eqref{remaindsum} is bounded by
$$
\le \sum_{\ell=1}^\infty \ell \cdot \frac{\alpha^\ell}{\ell!} \left( \sum_{m=0}^\infty (m p_m + p_m) \right) \le C_{\alpha,D}.
$$
Combining our calculations leads to
\begin{equation}
\limsup_{K\to \infty} \sup_{t\geq 0} \sum_{k \geq K}k\overline{s}_{t,k}
\leq C \limsup_{K\to\infty} \left(\sum_{k\geq K}kp_k+\sup_w w^K(1-w)\right)= 0.
\label{limKstk=0}
\end{equation}
We can use this bound to show that
$\sum_{k=0}^{\infty}\overline{S}_{t,k}/n$ converges to $\sum_{k=0}^{\infty}\overline{s}_{t,k}$
as well as $\sum_{k=0}^{\infty}k\overline{S}_{t,k}/n$ converges to $\sum_{k=0}^{\infty}k\overline{s}_{t,k}.$
Since the proofs are similar, we only prove the second result. We fix a large number $K$ and
observe that $\sum_{k\geq K} k\overline{S}_{t,k} $ satisfies the equation
\begin{align}
d\left(\sum_{k\geq K} k\overline{S}_{t,k} \right)= &
-\left(\sum_{k\geq K} k^2 \overline{S}_{t,k}\right) dt
+\rho \overline{X}_{I,t} \frac{X_t-1}{\lambda \overline{X}_{I,t}}\frac{K\overline{S}_{t,K-1}}{n}dt
\nonumber \\
&+\sum_{k\geq K} \rho \overline{X}_{I,t} \frac{\overline{X}_t-1}{\lambda \overline{X}_{I,t}}\frac{\overline{S}_{t,k}}{n}dt
+d\hat{M}_{t,K}.
\label{kstk}
\end{align}
Here $\hat{M}_{t,K}$ is a martingale that satisfies
\begin{equation}\label{2moment}
\sum_{t>0} (\hat{M}_{t,k}-\hat{M}_{t-,k})^2\leq 3
\sum_{\ell=1}^n Q_{\ell}^2,
\end{equation}
where $Q_{\ell}$ is the number of half-edges that vertex $\ell$ has before it becomes infected. This follows from the observation that there are two sources for the jump of $\sum_{k \geq K}k\overline{S}_{t,k}$:
\begin{itemize}
\item A susceptible vertex $\ell$ with at least $K$ half-edges gets infected. Then
$\sum_{k \geq K}k\overline{S}_{t,k}$ drops by the number of half-edges of vertex $\ell$, which is $Q_{\ell}$. Each vertex can contribute to this type of jumps at most once.
\item A half-edge of an infected vertex gets transferred to a susceptible vertex of degree at least $K-1$. Then $\sum_{k \geq K}k\overline{S}_{t,k}$ increases by either $\le K$ (if the vertex gaining a half-edge had $K-1$ half-edges before) or 1 (if the vertex gainning a half-edge had at least $K$ half-edges before).
Vertex $\ell$ can contribute at most $Q_{\ell}$ times to jumps of size 1 and at most once to jumps of size $K$.
\end{itemize}
Initially vertex $\ell$ has $D_\ell$ half-edges. As time grows the half-edges of other vertices might be transferred to vertex $\ell$, the number of which is dominated by
\begin{equation}\label{V}
V = \mbox{Binomial}(W ,1/n ) \quad\hbox{where $W= \sum_{m=1}^nD_{m}$}.
\end{equation}
Thus we have
\begin{equation}\label{EQ_l^2}
\mathbb{E}(Q_{\ell})^2 \leq \mathbb{E}(D_{\ell} + V)^2\leq 2\mathbb{E} D^2+ 2\mathbb{E} V^2.
\end{equation}
Conditioning on the value of $W$ we have
\begin{equation}\label{EV^2}
\mathbb{E} V^2 = (1/n)(1-1/n) \mathbb{E} W+ \mathbb{E} (W/n)^2 \leq C.
\end{equation}
It follows from \eqref{2moment} and \eqref{EV^2} that
\begin{equation}\label{mhatzk}
\mathbb{E}\left(\sup_{0\leq t\leq T\wedge \gamma_n} \abs{\hat{M}_{t,k}}^2\right)\leq 4\mathbb{E}(\hat{M}_{T\wedge \gamma_n,k}^2 )\leq Cn.
\end{equation}
Writing \eqref{kstk} as an integral equation and dropping the negative term $-(\sum_{k\geq K} k^2 \overline{S}_{t,k})$ we see
\begin{align*}
\sum_{k\geq K} k\overline{S}_{z\wedge \gamma_n,k} & \leq \sum_{k\geq K} k\overline{S}_{0,k}+ \frac{\rho}{\lambda}K \int_0^z \frac{\overline{X}_{u\wedge \gamma_n}}{n} \overline{S}_{u\wedge \gamma_n,K-1}du\\
&+\frac{\rho}{\lambda}\int_0^z \frac{\overline{X}_{u\wedge \gamma_n}}{n} \sum_{k \geq K}\overline{S}_{u\wedge \gamma_n,k}+\hat{M}_{z\wedge \gamma_n,k}.
\end{align*}
Take $\sup_{0\leq z\leq t}$ on both sides we have
\begin{equation}\label{kSzk}
\begin{split}
\sup_{0\leq z\leq t\wedge \gamma_n} \sum_{k\geq K} k\overline{S}_{z,k}
&\leq \sum_{k\geq K} k\overline{S}_{0,k}+\frac{\rho}{\lambda}K\int_0^t \sup_{0\leq u\leq z\wedge \gamma_n}
\frac{\overline{X}_{u}}{n} \overline{S}_{u,K-1}dz \\
+\frac{\rho}{\lambda} &\int_0^t \sup_{0\leq u\leq z\wedge \gamma_n}
\frac{\overline{X}_{u} }{n} \sum_{k \geq K}\overline{S}_{u,k}dz
+\sup_{0\leq z \leq t\wedge \gamma_n}\hat{M}_{z,k}.
\end{split}
\end{equation}
Dividing both sides of \eqref{kSzk} by $n$, taking the square and using $(a+b+c+d)^2 \le 4(a^2+b^2+c^2+d^2)$ we have
\begin{align}
\left(\sup_{0\leq z\leq t\wedge \gamma_n}\frac{\sum_{k\geq K} k\overline{S}_{z,k}}{n} \right)^2
&\le 4 \left(\sum_{k\geq K} \frac{k\overline{S}_{0,k}}{n} \right)^2
+4 \left(\frac{\rho}{\lambda}K\right)^2 t \int_0^t \left( \sup_{0\leq u\leq z\wedge \gamma_n}\frac{\overline{X}_{u}}{n} \overline{S}_{u,K-1} \right)^2 dz
\nonumber\\
+4 \left(\frac{\rho}{\lambda} \right)^2t & \int_0^t \left( \sup_{0\leq u\leq z\wedge \gamma_n} \frac{\overline{X}_{u} }{n} \sum_{k \geq K}\overline{S}_{u,k} \right)^2 dz
+4 \left( \sup_{0\leq z \leq t\wedge \gamma_n}\hat{M}_{z,k} \right)^2,
\label{step1}
\end{align}
where we have also used the Cauchy-Schwarz inequality to conclude that for any function $g=g(u)$,
$$
\left(\int_0^t \left[\sup_{0\leq u \leq z\wedge \gamma_n} g(u) \right] \, dz\right)^2
\le t \int_0^t \left[ \sup_{0\leq u \leq z\wedge \gamma_n} g^2(u)\right] \, dz.
$$
If we use $\hat{\mathbb{E}}$ to denote the conditional expectation with respect to the $\sigma$-algebra generated by $\overline{X}_0$, then
for any $\epsilon>0$, using equation \eqref{mhatzk} we can find a constant (depending on $\epsilon$) $C_{\ref{step-5}}>0$ so that
\begin{equation}\label{step-5}
\mathbb{P}\left(\hat{\mathbb{E}}\left(\sup_{0\leq z \leq t\wedge \gamma_n}\hat{M}^2_{z,k}\right)>C_{\ref{step-5}} n\right)\leq \epsilon.
\end{equation}
We then take another constant $C_{\ref{step-4}}$ such that
\begin{equation}\label{step-4}
\mathbb{P}(\overline{X}_0/n>C_{\ref{step-4}}) \leq \epsilon.
\end{equation}
Taking the conditional expectation of \eqref{step1} with respect to $\overline{X}_0$ we see on the event
$$
\Omega_0 = \{\overline{X}_0/n\leq C_{\ref{step-4}}\}\cap \left\{ \hat{\mathbb{E}}(\sup_{0\leq z \leq t\wedge \gamma_n}\hat{M}^2_{z,k})\leq C_{\ref{step-5}}n \right\}
$$
which has probability $\geq 1-2\epsilon$, there exists a constant $C_{\ref{step2}}$ such that
\begin{equation}
\begin{split}
&\hat{\mathbb{E}}\left(\sup_{0\leq z\leq t\wedge \gamma_n}\frac{\sum_{k\geq K} k\overline{S}_{z,k}}{n} \right)^2 \leq
C_{\ref{step2}} t \int_0^t \hat{\mathbb{E}}\left[\left(\sup_{0\leq u\leq z\wedge \gamma_n}\frac{\sum_{k\geq K} k\overline{S}_{u,k}}{n} \right)^2\right] dz\\
&+C_{\ref{step2}} \left(K^2t\hat{\mathbb{E}}\left[\left( \sup_{0\leq z\leq t\wedge \gamma_n}\frac{S_{z,K-1}}{n} \right)^2\right]+\frac{1}{n}+\hat{\mathbb{E}}\left[\left(\sum_{k\geq K} \frac{k\overline{S}_{0,K}}{n} \right)^2\right]\right).
\end{split}
\label{step2}
\end{equation}
If we let
$$
\phi(t) = \hat{\mathbb{E}}\left(\sup_{0\leq z\leq t\wedge \gamma_n}\frac{\sum_{k\geq K} k\overline{S}_{z,k}}{n} \right)^2,
$$
and $B =$ the second line in \eqref{step2}, then for $0\leq t\leq T$ we have
$$
\phi(t) \leq C_{\ref{step2}} T\int_0^t \phi(s) \, ds + B.
$$
Gronwall's inequality gives
\begin{equation}\label{gronwallfinal}
\mbox{if }\phi(t) \le \alpha(t) + \int_0^t \beta(s) \phi(s) \, ds \mbox{, then }\phi(t) \le \alpha(t) \exp\left(\int_0^t \beta(s) \, ds \right),
\end{equation}
provided that $\beta(t)\ge 0$ and $\alpha(t)$ is nondecreasing.
So applying \eqref{gronwallfinal} we have $$\phi(t) \le B \exp(C_{\ref{step2}} T^2)$$ on $\Omega_0$, that is,
\begin{equation}\label{tail exp}
\begin{split}
&\hat{\mathbb{E}}\left(\sup_{0\leq t\leq T\wedge \gamma_n}\frac{\sum_{k\geq K} kS_{t,k}}{n} \right)^2 \\
& \leq C_{\ref{step2}}\exp(C_{\ref{step2}}T^2)\left(K^2t\hat{\mathbb{E}}\left(\sup_{0\leq t\leq T\wedge \gamma_n}\frac{S_{t,K-1}}{n} \right)^2 + \frac{1}{n}
+\hat{\mathbb{E}}\left(\sum_{k\geq K} \frac{k\overline{S}_{0,k}}{n} \right)^{\kern -0.2em 2}\ \right).
\end{split}
\end{equation}
To control the first term on the right
we use the convergence of $\overline{S}_{t,K-1}/n$ to $\overline{s}_{t,K-1}$ in probability as well as the bounded convergence theorem (since $\overline{S}_{t,K-1}/n\leq 1$) to obtain that
\begin{equation}\label{step10}
\limsup_{n\to\infty} \mathbb{E}\left(\sup_{0\leq t\leq T\wedge \gamma_n}\frac{\overline{S}_{t,K-1}}{n} \right)^2 \leq \left( \sup_{0\leq t\leq T}\overline{s}_{t,K-1}\right)^2.
\end{equation}
Using \eqref{step10} and \eqref{limKstk=0}, if we first pick a large $K$,
then for all $n$ sufficiently large all three terms on right hand side of \eqref{tail exp}
smaller than $\epsilon^2/3$ with probability at least $1-\epsilon$, which in turn
implies that there is a set $\Omega_1$ with $\mathbb{P}(\Omega_1)\geq 1-3\epsilon$, so that on $\Omega_1$
$$
\hat{\mathbb{E}}\left(\sup_{0\leq t\leq T\wedge \gamma_n}\frac{\sum_{k\geq K} kS_{t,k}}{n} \right)^2 \leq \epsilon^2.
$$
It follows that
$$
\mathbb{E}\left(1_{\Omega_1} \hat{\mathbb{E}}\left(\sup_{0\leq t\leq T\wedge \gamma_n}\frac{\sum_{k\geq K} kS_{t,k}}{n} \right)^{\kern -0.2em 2} \right)
=\mathbb{E}\left( 1_{\Omega_1} \sup_{0\leq t\leq T\wedge \gamma_n}\frac{\sum_{k\geq K} kS_{t,k}}{n} \right)^{\kern -0.2em 2}\leq \epsilon^2.
$$
Using $\mathbb{P}(\Omega_1)\geq 1-3\epsilon$ and the Chebyshev's inequality, we see that with probability $\geq 1-4\epsilon$,
$$
\sup_{0\leq t\leq T\wedge \gamma_n}\frac{\sum_{k\geq K} kS_{t,k}}{n} \leq \epsilon.
$$
Fixing $t$ and $\epsilon$ and using the triangle inequality, we get
\begin{equation}\label{sum_conv}
\begin{split}
\sup_{0\leq t\leq T\wedge \gamma_n} \abs{\frac{\sum_{k\geq 0}k\overline{S}_{t,k} }{n}-\sum_{k\geq 0}k \overline{s}_{t,k}} &\leq \sup_{0\leq t\leq T\wedge \gamma_n} \abs{\frac{\sum_{k=0}^Kk\overline{S}_{t,k} }{n}-\sum_{k=0}^Kk \overline{s}_{t,k}}\\
&+\sup_{0\leq t\leq T\wedge \gamma_n} \frac{\sum_{k\geq K}k\overline{S}_{t,k} }{n}+\sup_{0\leq t\leq T}
\sum_{k\geq K} k\overline{s}_{t,k}.
\end{split}
\end{equation}
By first choosing $K$ large enough and then $n$ large enough we can make both the first and second terms on the right hand side of \eqref{sum_conv} smaller than $\epsilon$ with probability at least $1-4\epsilon$. The third term can also be made smaller than $\epsilon$ using \eqref{limKstk=0}.
Since $\epsilon$ is arbitrary we see that
$$
\sup_{0\leq t\leq T\wedge \gamma_n} \abs{\frac{\sum_{k=0}^{\infty}k\overline{S}_{t,k} }{n}-\sum_{k=0}^{\infty}k\overline{s}_{t,k}}\xrightarrow{\mathbb{P}} 0.
$$
To find $\sum_{k=0}^{\infty} \overline{s}_{t,k}$ and $\sum_{k=0}^{\infty}k\overline{s}_{t,k}$,
recall that we set $w=w(t)=\exp(-t)$ and $G(w) = \mathbb{E}(w^D)$. The limit of the fraction of susceptible nodes $\overline{s}_t$ satisfies
\begin{equation}
\begin{split}
\overline{s}_t&=\sum_{k=0}^{\infty} \overline{s}_{t,k}= \exp(-(\alpha/2)(1-w^2))\sum_{r,\ell\geq 0}\frac{p_r}{\ell!} (\alpha (1-w))^\ell w^{r+\ell}\\
&=\exp(-(\alpha/2)(1-w)^2) G(w).
\end{split}
\end{equation}
The limit of (scaled) number of susceptible half-edges satisfies
$$
\overline{x}_{S,t}=\sum_{k=0}^{\infty} k \overline{s}_{t,k}= \exp(-\frac{\alpha}{2}(1-w^2))
\sum_{r,\ell\geq 0}(r+\ell)\frac{p_r}{\ell!} (\alpha (1-w))^\ell w^{r+\ell}.
$$
The double sum equals
$$
w \sum_{r\ge 0} r p_r w^{r-1} \sum_{\ell\ge 0} \frac{(\alpha(1-w)w)^\ell}{\ell!}
+ \alpha w(1-w) \sum_{r\ge 0} p_r w^{r} \sum_{\ell\ge 1} \frac{(\alpha(1-w)w)^{\ell-1}}{(\ell-1)!},
$$
so we have
\begin{equation}
\overline{x}_{S,t}=\exp\left(-(\alpha/2)(1-w)^2\right)w(G'(w)+\alpha (1-w)G(w)).
\label{xSteq}
\end{equation}
\medskip
{\bf Extension to time $\gamma_n$.}
We have proved the second and third statements of \eqref{3lim} with $0\leq t\leq \gamma_n$ replaced by $0\leq t\leq T\wedge \gamma_n$ for any fixed $T$. To upgrade this to $0\leq t\leq \gamma_n$, note that
$\sum_{k=0}^{\infty}k\overline{S}_{t,k}\leq \overline{X}_t$. Picking
a large $T$ satisfying $(m_1+\alpha)\exp(-T)\leq \epsilon$ and
re-using equation \eqref{Tgam} we obtain that
\begin{align}
&\mathbb{P}\left(\gamma_n>T, \sup_{T \leq t \leq \gamma_n}
\abs{ \frac{\sum_{k=0}^{\infty} k\overline{S}_{t,k}}{n}-\exp\left(-\frac{\alpha}{2}(w-1)^2 \right)w(G'(w)+\alpha (1-w)G(w)) }\geq 3\epsilon \right)
\nonumber\\
\leq &\mathbb{P} \left(\gamma_n>T,\sup_{T\leq t\leq \gamma_n}\sum_{k=0}^{\infty} k\overline{S}_{t,k}/n>2\epsilon\right)
\leq \mathbb{P}(\gamma_n>T,\overline{X}_T/n>2\epsilon )\leq \epsilon. \label{SL1'}
\end{align}
This proves the third equation of \eqref{3lim}. The proof of the second equation in \eqref{3lim} is slightly more complicated. Again we fix a large $T$ such that $(m_1+\alpha)\exp(-T)\leq \epsilon$
and
\begin{equation}\label{largeT}
\abs{\exp(-\alpha/2)G(0)-\exp\left(-\alpha/2 \left(\exp(-T)-1\right)^2\right)G(\exp(-T))}\leq \epsilon.
\end{equation}
Equation \eqref{largeT} and the fact
$ \exp\left(-\alpha/2 \left(\exp(-t)-1\right)^2\right)G(\exp(-t))$ is decreasing in $t$ imply that
\begin{align}\label{largeT'}
\sup_{t,t'>T} &\Bigl| \exp\left(-\alpha/2 \left(\exp(-t)-1\right)^2\right)G(\exp(-t)) \\
& -\exp\left(-\alpha/2 \left(\exp(-t')-1\right)^2\right)G(\exp(-t'))\Bigr| \leq \epsilon. \nonumber
\end{align}
We first estimate the term $\overline{S}_{T,0}$ for large $T$.
Using the weaker version (i.e., with $\sup_{0\leq t\leq T\wedge \gamma_n}$) of the second equation of \eqref{3lim} we see that for $n$ large enough,
\begin{equation}\label{ST0}
\mathbb{P}\left(\gamma_n>T, \abs{\overline{S}_{T,0}/n- \exp\left(-\alpha/2 \left(\exp(-T)-1\right)^2\right)G(\exp(-T))} >\epsilon\right)\leq \epsilon.
\end{equation}
On the event $\{\gamma_n>T\}$, $
\sup_{T\leq t\leq \gamma_n} \abs{\overline{S}_{t,0}-\overline{S}_{T,0}}$ can be bounded by
$\overline{X}_T$, since in order to lose a susceptible vertex of degree 0 there must be a half-edge transferred to it. It follows that
\begin{equation}\label{ST>0}
\mathbb{P}\left(\gamma_n>T, \sup_{T\leq t\leq \gamma_n} \abs{\overline{S}_{t,0}-\overline{S}_{T,0}}/n>2\epsilon\right)
\leq \mathbb{P}\left(\gamma_n>T,\overline{X}_T/n>2\epsilon\right)\leq \epsilon.
\end{equation}
Combining equations \eqref{ST0}, \eqref{ST>0} and \eqref{largeT'} we see that
\begin{equation}\label{0thterm}
\mathbb{P}\left(\gamma_n>T, \sup_{T<t\leq \gamma_n}\abs{\overline{S}_{t,0}/n- \exp\left(-\alpha/2 \left(\exp(-t)-1\right)^2\right)G(\exp(-t)) }>4\epsilon \right)\leq 3\epsilon.
\end{equation}
Using
$\sum_{k \ge 1}\overline{S}_{t,k}\leq \sum_{k=0}^{\infty} k\overline{S}_{t,k}$ and equation \eqref{SL1'}
we see that
\begin{equation}\label{stk>1}
\mathbb{P}\left(\gamma_n>T,\sup_{T\leq t\leq \gamma_n}\sum_{k \ge 1} \overline{S}_{t,k}/n>2\epsilon\right)\leq \epsilon.
\end{equation}
It follows from \eqref{stk>1} and \eqref{0thterm} that
$$
\mathbb{P}\left(\gamma_n>T, \sup_{T<t\leq \gamma_n}\abs{\sum_{k=0}^{\infty} \overline{S}_{t,k}/n-a_S \exp\left(-\alpha/2 (w(t)-1)^2\right)G(w(t)) }> 6\epsilon \right)\leq 4\epsilon,
$$
which proves the second equation of \eqref{3lim} and concludes the proof of \eqref{3lim}.
\subsection
{Proof of Theorem \ref{epsize}}
\label{sec:pfth6}
Recall for all $t\leq \gamma_n$ we have $\overline{X}_t\geq \overline{X}_{S,t}$ and at $\gamma_n$ we have $\overline{X}_t=\overline{X}_{S,t}$ since $\gamma_n$ is the time that we run out of infected half-edges and the dynamics stop. Note that by the definition of $f$ in \eqref{defoff}
we have
$$
\exp(f(w))=\frac{\overline{x}_t}{\overline{x}_{S,t}}.
$$
We can rewrite $f$ as
$$
f(w) = \log(m_1) + \log(w)- \log( G'(w) + \alpha(1-w) G(w) ) + \frac{\alpha}{2}(w-1)^2.
$$
Taking the derivative we have
\begin{equation}
\label{1stderivative}
f'(w)=\frac{1}{w}-\frac{G''(w)-\alpha G(w)+\alpha(1-w) G'(w)}{ G'(w)+\alpha (1-w)G(w)} +\alpha(w-1).
\end{equation}
To evaluate $f'(1)$, recall that $G(1)=1$, $G'(1)=m_1$ and $G''(1)=\mathbb{E}[D(D-1)] = m_2-m_1$.
The terms with $1-w$ vanish at $w=1$. Using $\alpha = \rho m_1/\lambda$ from \eqref{alpha}, it follows that
$$
f'(1)=1-\frac{m_2 - m_1 - \rho m_1/\lambda}{m_1} = - \left( \frac{m_2-2m_1}{m_1} - \frac{\rho}{\lambda} \right).
$$
Theorem \ref{critical_value} tells us that in the supercritical case, we have $\lambda>(\rho m_1)/(m_2-2m_1)$,
and hence $f'(1)<0$, which implies that $f$ is positive on $(1-\delta,1)$ for some $\delta>0$.
Theorem \ref{critical_value} also shows that when $\eta>0$ is small,
\begin{equation}
\lim_{n\to\infty}\mathbb{P}(\overline{I}_{\infty}/n>\eta)=q(\lambda)>0,
\end{equation}
where $q(\lambda)$ is the survival probability of the two-phase branching process $\bar Z_m$ (defined in Section \ref{britton}).
Let $t_{\eta}<\delta$ be some small number depending on $\eta$ such that
\begin{equation}\label{teta}
1-\exp\left(-\frac{\alpha}{2}(\exp(-t_{\eta})-1)^2\right)G(\exp(-t_{\eta}))<\eta/4.
\end{equation}
Conditionally on $\overline{I}_{\infty}/n>\eta$ for some small $\eta$, the second equation of \eqref{3lim} implies that with high probability
$\gamma_n$ is also bounded from below by $t_{\eta}$ (depending on $\eta$), since otherwise we would have
\begin{equation*}
\begin{split}
& \limsup_{n\to\infty} \mathbb{P}(\gamma_n<t_{\eta},\overline{I}_{\gamma_n}/n>\eta)\\
\leq & \limsup_{n\to\infty} \mathbb{P}\left(\gamma_n<t_{\eta}, \overline{S}_{\gamma_n}/n>
\exp\left(-\frac{\alpha}{2}(\exp(-\gamma_n)-1)^2\right)G(\exp(-\gamma_n))
-\eta/4,\overline{I}_{\gamma_n}/n>\eta\right)\\
\leq &\limsup_{n\to \infty} \mathbb{P}\left(
\overline{S}_{\gamma_n}/n>\exp\left(-\frac{\alpha}{2}(\exp(-t_{\eta})-1)^2\right)G(\exp(-t_{\eta}))-\eta/4,\overline{I}_{\gamma_n}/n>\eta \right)=0,
\end{split}
\end{equation*}
where the last equality is due to \eqref{teta} and the fact that $\overline{S}_{\gamma_n}+\overline{I}_{\gamma_n}=n$. We have also used the definition of $\gamma_n$ so that $\overline{I}_{\infty}=\overline{I}_{\gamma_n}$ since no more vertices can be infected after $\gamma_n$.
Recalling the definition of the $\sigma$ in statement of Theorem \ref{epsize} we see that for any $\epsilon>0$, $$
\inf_{t_{\eta}<t<-\log (\sigma+\epsilon) } \frac{\overline{x}_t}{\overline{x}_{S,t}}>1,
$$
which implies that for some $\epsilon'>0$ and all $t_{\eta}<t<-\log (\sigma+\epsilon)$,
\begin{equation}\label{infxtxst}
\overline{x}_t-\overline{x}_{S,t}>\epsilon'.
\end{equation}
The first and third equations of \eqref{3lim} and \eqref{infxtxst} imply that
\begin{equation*}
\begin{split}
&\limsup_{n\to\infty} \mathbb{P}\left(t_{\eta}<\gamma_n<-\log(\sigma+\epsilon) \right) \\
\leq & \limsup_{n\to\infty} \mathbb{P}\left(t_{\eta}<\gamma_n<-\log(\sigma+\epsilon),(\overline{X}_{\gamma_n}-\overline{X}_{S,\gamma_n})/n>\epsilon'/2 \right) =0,
\end{split}
\end{equation*}
where we have used the fact that $\overline{X}_{\gamma_n}=\overline{X}_{S,\gamma_n}$. Since
we have already shown conditionally on $\overline{I}_{\infty}/n>\eta$ whp $\gamma_n>t_{\eta}$, we see that
$$
\lim_{n\to\infty}\mathbb{P}(\gamma_n>-\log (\sigma+\epsilon)|\overline{I}_{\infty}/n>\eta)=1.
$$
Using the monotonically decreasing property of $\overline{S}_t=\sum_{k=0}^{\infty}\overline{S}_{t,k}$ and the second equation of \eqref{3lim},
we see that conditional on $\overline{I}_{\infty}/n>\eta$, for any $\epsilon>0$ whp
$$
\overline{S}_{\gamma_n}/n \leq \exp\left(-\frac{\alpha}{2}(\sigma+\epsilon-1)^2\right)G(\sigma+\epsilon)+\epsilon.
$$
Hence for any $\epsilon>0$,
$$
\lim_{n\to\infty} \mathbb{P}(I_{\infty}/n>\nu-\epsilon |I_{\infty}/n>\eta)=1,
$$
where $\nu=1-\exp(-\alpha/2(\sigma-1)^2)G(\sigma)$.
For the other direction, ($\star$) implies that we can pick $\delta'>0$ so that $f(w)<0$ on $(\sigma-\delta',\sigma)$. \eqref{3lim} thus implies with high probability $\gamma_n$ cannot be larger than $-\log(\sigma-\delta')$ since otherwise we would have $\overline{X}_{-\log(\sigma-\delta')}<\overline{X}_{S,-\log (\sigma-\delta')}$, which is impossible. Since $\delta'$ can be taken arbitrarily small we conclude that for any $\epsilon>0$
$$
\lim_{n\to\infty} \mathbb{P}(\overline{I}_{\infty}/n<\nu+\epsilon)=1.
$$
\subsection{\bf Proof of Theorem \ref{Q1}} \label{sec:pfth7}
From \eqref{1stderivative} we see that as $\lambda\to\lambda_c$ we have $f'(1)\to 0$. The second derivative of $f$ is given by
$$
f''(w)=-\frac{1}{w^2}-\frac{G'''+\alpha(1-w) G''-2\alpha G'}{G'+\alpha (1-w)G}
+\left(\frac{G''-\alpha G+\alpha (1-w)G' }{G'+\alpha (1-w)G}\right)^2+\alpha .
$$
Recall $\mu_k$ denotes the factorial moment $\mathbb{E}[(D(D-1) \cdots(D-k+1)]$.
We have assumed that $\mathbb{E}(D^5)<\infty$ so
$$
G'''(1)=\mathbb{E}[D(D-1)(D-2)] =\mu_3<\infty.
$$
Since the terms with $1-w$ vanish at $w=1$, inserting the values of $G(1)$, $G'(1)$, $G''(1)$ and $G'''(1)$ we get
\begin{align*}
f''(1) &=-1-\frac{G'''(1)-2\alpha G'(1)}{G'(1)}+ \left( \frac{ G''(1) - \alpha G(1)}{ G'(1)} \right)^2 + \alpha \\
& = - 1 - \frac{\mu_3}{m_1} + 2\alpha + \left( \frac{ m_2-m_1 - \alpha}{ m_1} \right)^2 + \alpha.
\end{align*}
Theorem \ref{critical_value} tells us that $\alpha_c = m_2-2m_1$ so
$$
- 1 + \left(\frac{m_2-m_1 - \alpha_c}{m_1}\right)^2 = 0.
$$
From this, we see that at $\alpha_c$
$$
f''(1)=\frac{-\mu_3+3m_1m_2-6m_1^2}{m_1}.
$$
Using $\mu_1=m_1, \mu_2=m_2-m_1$ this can be written as
\begin{equation}\label{f''1delta}
f''(1) = -\frac{\mu_3}{\mu_1} + 3(\mu_2 - \mu_1) \equiv \Delta.
\end{equation}
By Theorem \ref{epsize} it suffices to prove that in the case $\Delta<0$, $\sigma$ converges to 1 as $\lambda\to\lambda_c$. Equation \eqref{f''1delta} and the assumption $\Delta<0$ imply that for $\lambda$ close to $\lambda_c$, in a (non-shrinking) neighborhood of 1, $f''(w)$ has to be bounded from above by some negative constant. Since $f'(1)$ converges to 0 as $\lambda\to \lambda_c$ we conclude
that for any fixed $w<1$ and all $\lambda$ sufficiently close to $\lambda_c$ one can find $\hat{w} \in (w,1)$ so that $f(\hat{w})<0$. Using the definition of $\sigma$ we see $\sigma>\hat{w}>w$.
Letting $w\to 1$, we see that $\sigma$ has to converge to 0 as $\lambda\to\lambda_c$ and thus $\nu$ converges to 0. Hence we have a continuous phase transition.
\section{Lower bound on evoSI} \label{sec:lbSI}
\subsection{AB-avoSI} \label{sec:ABavoSI}
Roughly speaking, the avoSI process serves as an upper bound because certain $I-I$ pairs can rewire, which may leads to additional infections. To get a lower bound, we need to find a way to ensure rewired $I-I$ edges won't transmit infections. This motivates the AB-avoSI process defined as follows. For each half-edge $h$ we give it two indices:
\begin{itemize}
\item
The infection index $A(h,t)=0$ if $h$ has not been infected by time $t$.
If $i$ first become an infected half-edge at time $s$, then we set $A(h,t)=s$ for all
$t\geq s$.
\item
The rewiring index $B(h,t)=0$ if the half-edge $h$ has not rewired by time $t$.
If $h$ rewires at time $s$, then we update the value of $B(h,s)$ to be $s$, no matter whether $h$ has been rewired before or not. In other words, if we let
$\tau_m(h)$ be the time when $h$ is rewired for the $m$-th time (possibly $\infty$) with $\tau_0(h)=0$,
then $B(h,t)=\tau_n(h)$
for $\tau_{m}(h)\leq t<\tau_{m+1}(h)$.
\end{itemize}
\medskip\noindent
We define the C-AB-avoSI process as follows. As in Section \ref{sec:avoSI}, C is for coupled.
\begin{itemize}
\item
At rate $\lambda$ each infected half-edge $h_1$ pairs with a randomly chosen half-edge. Suppose $h_1$ gets paired with half-edge $h_2$ at time $t$. If $h_2$ is susceptible and $B(h_2,t)<A(h_1,t)$ then the vertex associated with half-edge $h_2$ becomes infected. Otherwise $h_1$ will not pass infection to the vertex associated with $h_2$. The reader will see the reason for this condition in the proof of Lemma \ref{abavosi<evosi}.
Note that if vertex $y$ associated with $h_2$ changes from state $S$ to $I$ then all half-edges attached to $y$ become infected half-edges.
\item
Each infected half-edge gets removed from the vertex that it is attached to at rate $\rho$ and immediately becomes re-attached to a randomly chosen vertex.
\end{itemize}
Similar to the relation between C-avoSI and avoSI, one can also define the AB-avoSI such that the C-AB-avoSI has the same law as the AB-avoSI on the configuration model. The construction of the graph $G$ for AB-avoSI follows the same route of avoSI (see the proof of Lemma \ref{c-avosi=avo}). Given the graph $G$, we view each (full) edge as being composed of two half-edges and assign the two indices $A(\cdot, t)$ and $B(\cdot, t)$ as defined above to every half-edge. The evolution of AB-avoSI is then similar to avoSI, except that each time when infected vertex $x$ tries to infect ssuceptible vertex $y$ through edge $e$, we will compare $A(h_1,t)$ and $B(h_2,t)$,
where $h_1$ is the half-edge of $e$ with one end at $x$ and $h_2$ is the other half-edge of $e$ with one end at $y$. If $A(h_1,t)>B(h_2,t)$ then the infection will pass through. Otherwise the infection will not pass through, which means $x$ has made an attempt but $y$ remains uninfected.
No matter whether the infection passes through or not, we let this $S-I$ edge be deemed stable and it will not get rewired later on.
Also from this point on, $x$ will never pass infection to $y$ through $e$.
As a comparison, $S-I$ edges in avoSI are always unstable and are subject to potential rewiring.
One can show that evoSI stochastically dominates AB-avoSI.
\begin{lemma}\label{abavosi<evosi}
There exists a coupling of avoSI and evoSI such that if a vertex is infected in evoSI then it is also infected in avoSI.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{abavosi<evosi}]
Following the proof of Lemma \ref{avosi>evosi}, we see it suffices to show $I-I$ rewirings will not create additional infections in AB-avoSI.
Assume at some time $t_1$ an edge $e$ between two infected vertices $x$ and $y$ is broken from $y$ and reconnects to $z$. For a picture see Figure \ref{fig:ABrewire}.
\begin{figure}[h]
\begin{center}
\begin{picture}(230,100)
\put(50,80){$\bullet$}
\put(50,20){$\bullet$}
\put(110,20){$\bullet$}
\put(170,80){$\bullet$}
\put(170,20){$\bullet$}
\put(230,20){$\bullet$}
\put(53,22){\line(0,1){60}}
\put(172,23){\line(1,0){60}}
\put(40,20){$x$}
\put(40,80){$y$}
\put(10,20){$z$}
\put(60,80){$I$}
\put(50,10){$I$}
\put(110,10){$S$}
\put(120,20){$z$}
\put(160,20){$x$}
\put(160,80){$y$}
\put(240,20){$z$}
\put(180,80){$I$}
\put(170,10){$I$}
\put(230,10){$S$}
\put(40,65){$h_2$}
\put(40,35){$h_1$}
\put(180,30){$h_1$}
\put(210,30){$h_2$}
\end{picture}
\end{center}
\caption{In the transition from the first drawing to the second, $x$ rewires its connection from $y$ to $z$. Note that $h_1$ and $h_2$ are the two half-edges comprising the edge $e$, which is not drawn in the figure. Vertex $x$ can't pass an infection to $z$ throught $h_1$ and $h_2$ since $B(h_2,t)>A(h_1,t)$.}
\label{fig:ABrewire}
\end{figure}
We would like to show after time $t_1$ no infections can pass through $e$ to create additional infected vertices. Denote the half-edge attached to $x$ by $h_1$ and the other half-edge by $h_2$. Since $x$ and $y$ must be infected before time $t_1$ we see that, according to the definition of the infection index, for all $t\geq t_1$,
\begin{equation}\label{ah1h2}
A(h_1,t)<t_1, A(h_2,t)<t_1.
\end{equation}
We next consider the rewiring index. Since the half-edge $h_2$ is rewired at time $t_1$, using the definition of rewiring index we see that for all $t\geq t_1$,
\begin{equation}\label{bh2} B(h_2, t)\geq t_1.
\end{equation}
Combining \eqref{ah1h2} and \eqref{bh2} we see that
$$
B(h_2,t)>A(h_1. t).
$$
This implies that infections can't pass from the vertex associated to $h_1$ to the vertex associated to $h_2$. Now we consider the other direction, i.e., from $h_2$ to $h_1$. Note that $x$ has already been infected. There are two possible cases:
\begin{itemize}
\item If $h_1$ stays with $x$ forever then the vertex associated with $h_1$ is always infected after time $t_1$ (and hence there is no additional infected vertex).
\item If $h_1$ is rewired to some other vertex $x'$ at time $t_2>t_1$ then for all $t>t_2$,
$$
B(h_1,t)\geq t_2>t_1>A(h_2,t),
$$
which implies that after time $t_2$ infections can't go from the vertex associated with $h_2$ to the vertex associated with $h_1$.
\end{itemize}
In both cases there will be no additional infected vertices stemming from the rewiring of $e$. Thus we have completed the proof of Lemma \ref{abavosi<evosi}.
\end{proof}
\noindent
We end this section by showing AB-avoSI dominates delSI. This is used in the proof Lemma \ref{avocrv}.
\begin{lemma}\label{abdel}
There exists a coupling of AB-evoSI and delSI such that
if a vertex is infected in AB-evoSI then it is also infected in delSI.
\end{lemma}
\begin{proof}
This can be proved in a similar way to the proof of Lemma \ref{couple}. We construct
AB-avoSI using the variables $\{T_{e,\ell},R_{e,\ell},V'_{e,\ell},\ell\geq 1\}$ as we did in the proof of Lemma \ref{avosi>evosi} except that for AB-avoSI certain infections may not pass through depending on the relative size of infection index and rewiring index. We also construct delSI using $T_{e,1}$ and $R_{e,1}$ as we did in the proof of Lemma \ref{couple}. We can then prove Lemma \ref{abdel} by repeating the induction argument used in the proof of Lemma \ref{couple}. Note that if edge $e$ is initially present between $x$ and $y$ and
$T_{e,1}<R_{e,1}$ then either end of $e$ can be infected by the other end because the rewiring indices for both half-edges of $e$ are equal to 0 at the time of infection.
Therefore we don't need to worry about infections not being transmitted successfully in AB-evoSI.
\end{proof}
\subsection{Moment bounds}\label{sec:moment}
As in the analysis of avoSI, we multiply the original transition rates by $(\check{X_t}-1)/(\lambda \check{X}_{I,t})$. We will use a hat to denote the quantities after the time change. The evolution equation for $\hat{X}_t$ has the same form as avoSI and hence the first equation of \eqref{3lim} also holds for AB-avoSI. Now we consider the evolution of the number of susceptible half-edges $\hat{X}_{S,t}$. We need a bit more notation to describe this. For half-edge $i$, we let $I(i,t)=1$ if $i$ is an infected half-edge at time $t$ (which also means it hasn't been paired) and $I(i,t)=0$ otherwise. We can define $S(i,t)$ similarly. We also let $S(i,k,t)=1$ if $i$ is attached to a susceptible vertex with $k$ half-edges at time $t$.
Finally, we let $v(j,t)$ be the vertex that half-edge $j$ is attached to at time $t$ and
$D(j,t)$ be the number of half-edges attached to $v(j,t)$ at time $t$.
We first write down the equation for $\hat{S}_{t,k}$. To reduce the size of
formulas we let
\begin{equation}
G_{i,j}=\{I(i,t)=1, A(i,t)\leq B(j,t) \} \label{defGij}.
\end{equation}
Reasoning as in the derivation of \eqref{eqStk2} gives
\begin{equation}\label{newStk}
\begin{split}
d\hat{S}_{t,k}&=-k\hat{S}_{t,k}\, dt
+1_{\{k\geq 1\}} \frac{\rho}{\lambda}\frac{\hat{S}_{t,k-1}}{n}(\hat{X}_t-1) \, dt
- \frac{\rho}{\lambda}\frac{\hat{S}_{t,k}}{n}(\hat{X}_t-1)\, dt\\
&+\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\overline{X}_0}
1_{G_{i,j}} 1_{\{ S(j,k+1,t)=1 \}} \right)dt+d\hat{M}_{t,k},
\end{split}
\end{equation}
where $\hat{M}_{t,k}$ is a martingale.
Summing \eqref{newStk} over $k$ from 0 to $\infty$ and noting that the second and third term cancels, we get
\begin{equation}\label{newSt}
d\hat{S}_t=-\hat{X}_{S,t}\, dt+\frac{1}{\hat{X}_{I,t}}
\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}} 1_{\{ S(j,t)=1\} } \right)dt+dM_{1,t}.
\end{equation}
Multiplying both sides of \eqref{newStk} by $k$ and summing over $k$, we get
\begin{equation}\label{newXst}
\begin{split}
d\hat{X}_{S,t}&=-\sum_{k=0}^{\infty} k^2\hat{S}_{t,k}dt
+\frac{\rho}{\lambda}\frac{\hat{S}_t}{n}(\hat{X}_t-1)dt\\
&+\frac{1}{\hat{X}_{I,t}}
\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}} (D(j,t)-1) 1_{\{ S(j,t)=1\}} \right)dt +dM_{2,t},
\end{split}
\end{equation}
for some martingale term $M_{2,t}$.
Analogously, if we multiply both sides of \eqref{newStk} by $k^2$, $k^3$ and $k^4$, respectively, then we get
\begin{equation}\label{newk^2Stk}
\begin{split}
&d\left(\sum_{k=0}^{\infty} k^2\hat{S}_{t,k}\right)=-\sum_{k=0}^{\infty} k^3\check{S}_{t,k}dt+2\frac{\rho}{\lambda}\frac{\hat{X}_{S,t}}{n}(\hat{X}_t-1)+\frac{\rho}{\lambda} \frac{\hat{S}_t}{n}(\hat{X}_t-1) \\
&+\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0}
1_{G_{i,j}} (D(j,t)-1)^2 1_{\{ S(j,t)=1\}} \right)dt+dM_{3,t},
\end{split}
\end{equation}
\begin{equation}\label{newk^3stk}
\begin{split}
&d\left(\sum_{k=0}^{\infty} k^3\check{S}_{t,k}\right)=-\sum_{k=0}^{\infty} k^4\check{S}_{t,k} dt
+3\frac{\rho}{\lambda}\frac{\sum_{k=0}^{\infty} k^2\check{S}_{t,k}}{n}(\hat{X}_t-1)
+3\frac{\rho}{\lambda}\frac{\hat{X}_{S,t} }{n}(\hat{X}_t-1)\\&+
\frac{\rho}{\lambda}\frac{\hat{S}_t}{n}(\hat{X}_t-1)
+\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\overline{X}_0}
1_{G_{i,j}} (D(j,t)-1)^3 1_{\{ S(j,t)=1\}} \right)dt+dM_{4,t},
\end{split}
\end{equation}
and
\begin{equation}\label{newk^4stk}
\begin{split}
d\left(\sum_{k=0}^{\infty} k^4\check{S}_{t,k}\right)&=-\sum_{k=0}^{\infty} k^5\check{S}_{t,k} dt
+4\frac{\rho}{\lambda}\frac{\sum_{k=0}^{\infty} k^3\check{S}_{t,k}}{n}(\hat{X}_t-1)
+6\frac{\rho}{\lambda}\frac{\sum_{k=0}^{\infty} k^2\check{S}_{t,k}}{n}(\hat{X}_t-1)\\
&+4\frac{\rho}{\lambda}\frac{\hat{X}_{S,t} }{n}(\hat{X}_t-1)+
\frac{\rho}{\lambda}\frac{\hat{S}_t}{n}(\hat{X}_t-1)\\
& +\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\overline{X}_0}
1_{G_{i,j}} (D(j,t)-1)^4 1_{\{ S(j,t)=1\}} \right)dt+dM_{5,t}.
\end{split}
\end{equation}
The main result of this section is:
\begin{lemma}\label{lbstep1}
There exists two constants $C_{\ref{s12}},C_{\ref{s13}}>0$, such that for every $\epsilon>0$ whp we have
\begin{equation}\label{s11}
\sum_{j=1}^5 \left(\sup_{0\leq t\leq \gamma_n\wedge 1}
\abs{M_{j,t}}\right)\le n^{5/6},
\end{equation}
\begin{equation}\label{s12}
\sup_{0\leq t\leq \gamma_n\wedge 1} \sum_{k=0}^{\infty} (k+1)^4\hat{S}_{t,k} \le C_{\ref{s12}}n,
\end{equation}
\begin{equation}\label{s13}
\sum_{j=1}^{\hat X_0}D(j,t)^2 1_{\{S(j,t)=1, B(j,t)>0\}}\leq n(C_{\ref{s13}}t+\epsilon),
\quad\hbox{for all $0\leq t\leq \gamma_n\wedge 1$}
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lbstep1}]
We first prove equation \eqref{s11}. The proof is based on analyzing the quadratic variation of $M_{1,t}, \ldots, M_{5,t}$. Since the proofs for the five quantities are similar we only give the details for $M_{5,t}$, the martingale associated with $\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}$.
Let $Q_x$ be the number of half-edges that vertex $x$ originally has plus the half-edges that has been rewired to vertex $x$ before $x$ becomes infected. Let $D_x(t)$ be the number of half-edges that $x$ has at time $t$. We necessarily have $D_x(t) \leq Q_x$ as long as $x$ is susceptible at time $t$.
Note that the jumps of $\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}$ have the following three sources.
\begin{itemize}
\item An infected half-edge pairs with a susceptible half-edge attached to a vertex $x$ with $D_x(t)$ half-edges and passes the infection to $x$. This decreases $\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}$ by $D_x(t)^4$. Such type of jumps can occur at most once for each susceptible vertex.
\item An infected half-edge pairs with a susceptible half-edge attached to a vertex $x$ with $D_x(t)$ half-edges but does not pass the infection. This decreases $\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}$ by
$$
D_x(t)^4-(D_x(t)-1)^4\leq 15 Q_x^3.
$$
Such type of jumps can occur at most $Q_x$ times for vertex $x$.
\item An infected half-edge is rewired to a susceptible vertex $x$ with $D_x(t)$ half-edges. This gives an increase of
$$
(D_x(t)+1)^4-D_x(t)^4 \leq 15Q_x^3
$$
to $\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}$. Such type of jumps can happen at most $Q_x$ times for vertex $x$.
\end{itemize}
It follows from the above analysis that the quadratic variation of $M_{5,t}$
is bounded by
\begin{equation}
\sum_{x=1}^n (Q_x^4)^2+\sum_{x=1}^n 15(Q_x^3)^2Q_x
+\sum_{x=1}^n 15(Q_x^3)^2Q_x\leq C_{\ref{qvbd}}\sum_{x=1}^n Q_x^8.
\label{qvbd}
\end{equation}
Using the Burkholder-Davis-Gundy inequality (see, e.g., \cite[Theorem 7.34]{Kl} with $p=5/4$) we see
\begin{equation}
\mathbb{E}\left[\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{M_{5,t}}^{5/4} \right]
\leq C_{\ref{BDG}} \mathbb{E}\left[\left( \sum_{x=1}^n Q_x^8 \right)^{5/8} \right].
\label{BDG}
\end{equation}
To bound the right-hand side we need the following well-known fact:
for any $p\ge 1$ and positive numbers $a_1,\ldots, a_m$,
\begin{equation}\label{anineq}
\left(\sum_{i=1}^m a_i^p \right)^{\kern -0.2em 1/p}\leq \sum_{i=1}^m a_i.
\end{equation}
In words the $L^p$ norm decreases as $p$ increases. Applying \eqref{anineq} with $p=8/5$ and $a_i=Q_i^8$ gives that
$$
\left(\sum_{x=1}^n Q_x^8 \right)^{5/8}\leq \sum_{x=1}^n Q_x^5.
$$
As we argued in the proof of \eqref{EQ_l^2}, if $V=\mbox{Binomial}(\hat X_0,1/n)$, then $Q_x$ is dominated by $D_x+V$. To bound the fifth moment of the sum we note that if $Y$ and $Z$ are nonnegative random variables,
\begin{equation}\label{y+z^5}
\mathbb{E}(Y+Z)^5 \le \mathbb{E}( 2 \max\{Y,Z\} )^5 \le 32[ \mathbb{E} Y^5 + \mathbb{E} Z^5].
\end{equation}
We claim that
the 5-th moment of a binomial distribution Binnomial($m,p$) is bounded by $$mp+\cdots +(mp)^5.$$ To see this, let $Y_1,\ldots, Y_m$ be i.i.d. Bernoulli variable with mean $p$.
\begin{equation}
\begin{split}
\mathbb{E}\left[\left(\sum_{i=1}^m Y_i\right)^5\right]=\sum_{i_1,\ldots, i_5}\mathbb{E}(Y_{i_1}\cdots Y_{i_5})=\sum_{i_1,\ldots, i_5} p^{\# \textrm{ of distinct elements among }i_1,\ldots, i_5}.
\end{split}
\end{equation}
Note that the number of ordered tuples $(i_1,\ldots, i_5)$ such that
there are $\ell$ distinct elements among them is bounded by $m^{\ell}$. We conclude that
\begin{equation}
\mathbb{E}\left[\left(\sum_{i=1}^m Y_i\right)^5\right]\leq \sum_{\ell=1}^5 m^{\ell}p^{\ell},
\end{equation}
which verifies the claim.
Using this claim we have that
\begin{equation}
\mathbb{E}(V^5)\leq \sum_{\ell=1}^{5}\frac{\mathbb{E}(\hat X_0^{\ell})}{n^{\ell}}
\leq C_{\ref{binmom}},
\label{binmom}
\end{equation}
so that by \eqref{y+z^5},
\begin{equation}
\sum_{x=1}^n \mathbb{E}(Q_x^5) \leq 32\sum_{i=1}^n (\mathbb{E}(D_i^5)+\mathbb{E}(V^5))
\leq C'_{\ref{m4tctl}}n.
\label{m4tctl}
\end{equation}
This implies that
\begin{equation}
\mathbb{P}\left(\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{M_{5,t}}>n^{5/6}\right)=
\mathbb{P}\left(\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{M_{5,t}}^{5/4}>
(n^{5/6})^{5/4} \right)\leq \frac{Cn}{n^{25/24}}\to 0.
\end{equation}
Using the reasoning that led to \eqref{qvbd}, we can deduce the same bounds (with different constants) for $M_{1,t},M_{2,t},M_{3,t},M_{4,t}$ and hence equation \eqref{s11} follows by using Markov's inequality.
To prove equation \eqref{s12},
define the event $\Omega_n$ to be
\begin{equation}\label{defomegan}
\left\{\abs{\sum_{k=0}^{\infty} k^i\hat S_{0,k}-n\sum_{k=0}^{\infty} k^ip_k }\leq n,
\sup_{0 \leq t\leq \gamma_n\wedge 1} \abs{M_{i,t}}\leq n, \quad\hbox{for $i=1,2,3,4$} \right\}.
\end{equation}
The assumption that $\mathbb{E}(D^5)<\infty$ and equation \eqref{s11} imply that $\mathbb{P}(\Omega_n)\to 1$ as $n\to\infty$.
By the definition of $G_{i,j}$ in \eqref{defGij},
\begin{equation*}
\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0}
1_{G_{i,j}} (D(j,t)-1)^2 1_{\{ S(j,t)=1\}} \right) \leq \frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0}
1_{\{I(i,t)=1\}} (D(j,t)-1)^2 1_{\{ S(j,t)=1\}} \right),
\end{equation*}
which is bounded by
\begin{equation}\label{s14}
\begin{split}
\sum_{j=1}^{\hat{X}_0} D(j,t)^21_{\{S(j,t)=1\}}
&= \sum_{r=1}^n \sum_{j=1}^{\hat{X}_0}1_{\{v(j,t)=r\}} 1_{\{r \textrm{ is susceptible at }t\}} D_r(t)^2\\
&= \sum_{r=1}^n 1_{\{r \textrm{ is susceptible at }t\}} D_r(t)^3=\sum_{k=0}^{\infty} k^3 \hat{S}_{t,k}.
\end{split}
\end{equation}
Here $D_r(t)$ is the number of half-edges that vertex $r$ has at time $t$.
Using \eqref{newk^2Stk} and \eqref{s14} we obtain that
\begin{equation}\label{s15}
\sum_{k=0}^{\infty} k^2\hat{S}_{t,k}-\sum_{k=0}^{\infty} k^2\hat{S}_{0,k}\leq \int_0^t \left(2\frac{\rho}{\lambda}\frac{\hat{X}_{S,u}}{n}(\hat{X}_u-1)+\frac{\rho}{\lambda} \frac{\hat{S}_u}{n}(\hat{X}_u-1)\right)dt+M_{3,t}.
\end{equation}
On the event $\Omega_n$, we have that $\sum_{k=0}^{\infty} k^2 \hat{S}_{0,k}\leq (m_2+1)n$, $\hat{X}_{S,u}\leq \hat{X}_0\leq (m_1+1)n$ and $M_{3,t}\leq n$. Therefore, using \eqref{s15} we see that, there exsits a constant $C_{\ref{s16}}$ such that
\begin{equation}\label{s16}
\sum_{k=0}^{\infty} k^2 \hat S_{t,k}\leq (m_2+1)n +\int_0^t \left(\frac{2\rho}{\lambda}(m_1+1)^2n+ \frac{\rho}{\lambda}(m_1+1)n\right)du+n \leq C_{\ref{s16}}n
\end{equation}
for all $0\leq t\leq \gamma_n\wedge 1$.
Analogously to the proof of \eqref{s14}, we can show
\begin{equation}\label{s100}
\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0}
1_{G_{i,j}} (D(j,t)-1)^3 1_{\{ S(j,t)=1\}} \right)\leq \sum_{k=0}^{\infty} k^4\hat S_{t,k}.
\end{equation}
Using \eqref{s100} and \eqref{newk^3stk} we get
\begin{align*}
\sum_{k=0}^{\infty} k^3 \hat{S}_{t,k} -\sum_{k=0}^{\infty} k^3 \hat{S}_{0,k} \leq \int_0^t \biggl(&
3\frac{\rho}{\lambda}\frac{\sum_{k=0}^{\infty} k^2\hat{S}_{u,k}}{n}(\hat{X}_u-1)\\
&+3\frac{\rho}{\lambda}\frac{\hat{X}_{S,u} }{n}(\hat{X}_u-1)
+ \frac{\rho}{\lambda}\frac{\hat{S}_u}{n}(\hat{X}_u-1) \biggr)du+M_{4,t}.
\end{align*}
On the event $\Omega_n$, using \eqref{s16}, we see that, for all $0\leq t\leq \gamma_n \wedge 1$,
\begin{equation}\label{s17}
\begin{split}
\sum_{k=0}^{\infty} k^3 \hat{S}_{t,k}\leq &(m_1+1)n+n+\\
&\int_0^t \left(\frac{3\rho}{\lambda}C_{\ref{s16}}(m_1+1)n+\frac{3\rho}{\lambda}(m_1+1)^2n+\frac{\rho}{\lambda}(m_1+1)n \right)\, du \leq C_{\ref{s17}}n,
\end{split}
\end{equation}
for some constant $C_{\ref{s17}}>0$. Proceeding in a similar fashion and using \eqref{newk^4stk} we can show that on $\Omega_n$ there exists a constant $C_{\ref{s18}}>0$, such that for all $0\leq t\leq \gamma_n \wedge 1$,
\begin{equation}\label{s18}
\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}\leq C_{\ref{s18}}n.
\end{equation}
Equation \eqref{s12} follows from \eqref{s18} and the fact that $\mathbb{P}(\Omega_n)\to 1$ since
$$
\sum_{k=0}^{\infty} (k+1)^4 \hat{S}_{t,k}\leq 16\sum_{k=0}^{\infty} (k^4+1)\hat{S}_{t,k}\leq 16\sum_{k=0}^{\infty} k^4\hat{S}_{t,k}+16 \sum_{k=0}^{\infty} \overline{S}_{t,k}\leq 16\sum_{k=0}^{\infty} k^4 \hat{S}_{t,k}+16n.
$$
We now turn to the proof of equation \eqref{s13}.
Set $$H(t)=\sum_{j=1}^{\hat X_0}(D(j,t)-1)^2 1_{\{S(j,t)=1, B(j,t)>0\}}.$$
Note that $H(0)=0$ and we can use Dynkin's formula to get
$$
H(t)=\int_0^t h(s)\, ds+M_{6,t},
$$
where $M_{6,t}$ is a martingale associated with
$H(t)$ and $h(t)$ is the rate of change of $H(t)$. We now control $h(t)$ and $M_{6,t}$ by analyzing the jumps of $H(t)$ ($H(t)$ is a pure jump process). Note that there are three types of jumps:
\begin{itemize}
\item An infected half-edge pairs with a half-edge attached to susceptible vertex $x$ and makes $x$ infected. This does not increase $H(t)$ and thus makes a non-positive contribution to $h(t)$. The absolute value of the jump size of $H(t)$ is bounded by $D_x(t)^3$ where $D_x(t)$ is the number of half-edges that $x$ has at time $t$. For each vertex $x$ such jumps can happen at most once.
\item An infected half-edge pairs with a half-edge attached to susceptible vertex $x$ but $x$ stays susceptible after the pairing. This does not increase $H(t)$ and thus makes a non-positive contribution to $h(t)$. The absolute value of the jump size of $H(t)$ is bounded above by $$
D_x(t)^2+\abs{(D_x(t)-1)^2-(D_x(x)-2)^2}D_x(t)\leq 3(D_x(t)+1)^2.
$$
To see this, note that the loss of a half-edge $j$ attached to $x$ makes a twofold contribution to $H(t)$. First, $j$ is no longer a half-edge so $H(t)$ has to decrease by $(D(j,t)-1)^2=(D_x(t)-1)^2$. Second, the for each of the remaining $D_x(t)-1$ half-edges attached to $x$, its contribution to $H(t)$ changes from $(D_x(t)-1)^2$ to $(D_x(t)-2)^2$.
For each vertex $x$ such type of jumps can occur at most $Q_x$ times.
\item An infected half-edge is rewired to a susceptible vertex $x$. This increases $H(t)$ by at most
$$
D_x^2(t)+ \abs{(D_x(t)^2-(D_x(t)-1)^2)}D_x(t)\leq 3(D_x(t)+1)^2\leq 3(Q_x+1)^2.
$$
The rate that $x$ receives a rewired half-edge is equal to
$$
\frac{\hat{X}_0}{\lambda \hat{X}_{I,t}} \rho \hat{X}_{I,t} \frac{1}{n}=\frac{\rho \hat{X}_0}{\lambda n}\leq \frac{(m_1+1)\rho}{\lambda}
$$
on the event $\Omega_n$ (defined in \eqref{defomegan}). Here the factor of $1/n$ comes from Poisson thinning since each half-edge is a rewired to a uniformly chosen vertex independently.
For each vertex $x$ such type of jumps can occur at most $Q_x$ times.
\end{itemize}
Therefore on $\Omega_n$ we have
$$
h(t)\leq \frac{(m_1+1)\rho}{\lambda}
\sum_{x=1}^n 3(D_x(t)+1)^21_{\{x \textrm{ is susceptible at }t\}}\leq \frac{(m_1+1)\rho}{\lambda}\sum_{k=0}^{\infty} 3(k+1)^4 \hat{S}_{t,k}.
$$
Equation \eqref{s12} shows that with high probability $\sum_{k=0}^{\infty} (k+1)^4 \hat{S}_{t,k} \leq C_{\ref{s12}}n$. Since $\Omega_n$ also holds with high probability, we deduce that
\begin{equation}\label{s131}
\lim_{n\to\infty}\mathbb{P}\left(\int_0^t h(s)ds\leq C_{\ref{s131}}nt, \forall 0\leq t\leq \gamma_n\right)=1.
\end{equation}
The above analysis of the jumps of $H(t)$ also implies that the quadratic variation of $M_{6,t}$ is bounded by
$$
(Q_x^3)^2+ (3(Q_x+1)^2)^2Q_x+(3(Q_x+1)^2)^2Q_x\leq 20 (Q_x+1)^6.
$$
We now bound the $2/3$-th moment of the quadratic variation.
Applying \eqref{anineq} with $p=3/2$ and $a_i=(Q_i+1)^6$ gives that
$$
\left(\sum_{i=1}^n (Q_i+1)^6 \right)^{2/3}\leq \sum_{i=1}^n (Q_i+1)^4.
$$ Using this and the Burkholder-Davis-Gundy inequality (\cite[Theorem 7.34]{Kl} with $p=4/3$) we see that
$$
\mathbb{E}\left[\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{M_{6,t}}^{4/3} \right]
\leq C' \mathbb{E}\left[\left( \sum_{i=1}^n (Q_i+1)^6 \right)^{2/3} \right] \leq C'\mathbb{E}\left(\sum_{i=1}^n (Q_i+1)^4\right)\leq Cn.
$$
This implies that
\begin{equation}\label{s132}
\mathbb{P}\left(\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{M_{6,t}}>n^{4/5}\right)=
\mathbb{P}\left(\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{M_{6,t}}^{4/3}>n^{4/5*4/3}\right)\leq \frac{Cn}{n^{16/15}}\to 0.
\end{equation}
Equation \eqref{s13} follows from equations \eqref{s131} and \eqref{s132}.
\end{proof}
\subsection{Rough upper and lower bounds for $\hat{X}_{I,t}$ and $\hat{I}_t$}\label{sec:rough}
Define
\begin{equation}\label{et}
E(t)=\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat{X}_0} 1_{G_{i,j}} D(j,t) 1_{\{ S(j,t)=1\}} \right)
\end{equation}
where $G_{i,j}=\{ I(i,t)=1, A(i,t) \le B(j,t) \}$ was defined in \eqref{defGij}.
\begin{lemma}\label{lbstep2}
There exists two constants $\lambda_0,t_0\in (0,1)$, such that for all $\lambda<\lambda_c+\lambda_0$, $t<t_0$ and $\epsilon>0$, the following four inequalities hold whp for $0\leq t\leq \gamma_n\wedge t_0$:
\begin{equation}\label{lbstep11}
\hat{X}_{I,t}\leq \left(\frac{2\rho m_1}{\lambda_c^2}(\lambda-\lambda_c)t+m_1\Delta t^2+\epsilon\right)n,
\end{equation}
\begin{equation} \label{lbstep12}
\hat{X}_{I,t} \geq \left(\frac{\rho m_1}{2\lambda_c^2}(\lambda-\lambda_c)t+\frac{m_1\Delta}{4}t^2-\epsilon\right)n
-\int_0^t E(u)\, du,
\end{equation}
\begin{equation}\label{lbstep13}
\hat{I}_t\leq (2m_1t+\epsilon)n,
\end{equation}
\begin{equation}\label{lbstep14}
\hat{I}_t\geq \left(\frac{m_1t}{2}-\epsilon\right)n-\int_0^t E(u)\, du.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lbstep2}]
We first prove equation \eqref{lbstep11}.
Using Lemma \ref{lbstep1} we see that there exist a constant $C$ such that whp the absolute value of the right hand sides of \eqref{newXst}--\eqref{newk^3stk} are all upper bounded by $Cn$ for $t\leq \gamma_n \wedge 1$. Denote this event by $G_1(n)$. We have $\mathbb{P}(G_1(n))\to 1$ as $n\to\infty$. Denote
the event
$$
\left\{\abs{\sum_{k=0}^{\infty} k^i\hat S_{0,k}-n\sum_{k=0}^{\infty} k^ip_k }\leq\epsilon n,
\quad\hbox{for $i=1,2,3$}\right\}
$$
by $G_2(n)$.
We have $\mathbb{P}(G_2(n))\to 1$ as $n\to\infty$ since $\mathbb{E}(D^3)<\infty$.
Therefore we get
$$
\lim_{n\to\infty}\mathbb{P}(G_1(n)\cap G_2(n))=1,
$$
which implies that whp for all $0\leq t\leq \gamma_n\wedge 1$, we have
\begin{equation} \label{level1}
\abs{\sum_{k=0}^{\infty} k^3\overline{S}_{t,k}-n\sum_{k=0}^{\infty} k^3 p_k }
+\abs{\hat{X}_{S,t}-n\sum_{k=0}^{\infty} kp_k} \leq 4n(Ct+\epsilon).
\end{equation}
The definition of $E(t)$ in \eqref{et} implies that
\begin{equation*}
E(t)\leq \sum_{j=1}^{\hat X_0}D(j,t)1_{\{S(j,t)=1,B(j,t)>0\}}.
\end{equation*}
Therefore by \eqref{s13} we see that whp for all $ t\leq \gamma_n\wedge 1$,
\begin{equation}\label{etctl1}
E(t)\leq n(Ct+\epsilon).
\end{equation}
Now it follows from the integral form of \eqref{newSt}, \eqref{level1} and \eqref{etctl1} that whp
\begin{equation}\label{level2st}
\abs{\hat{S}_t-n\left(1-m_1t\right)}\leq n(Ct^2+\epsilon).
\end{equation}
Similarly to the proof of \eqref{level2st}, we have that whp for all $ 0\leq t\leq \gamma_n\wedge 1$
\begin{equation}\label{level2k^2}
\abs{\sum_{k=0}^{\infty} k^2\overline{S}_{t,k}-n\left(\sum_{k=0}^{\infty} k^2d_k+\left(-\sum_{k=0}^{\infty} k^3p_k+2\frac{\rho}{\lambda}m_1^2+\frac{\rho}{\lambda}m_1\right)t \right)} \leq n(Ct^2+\epsilon).
\end{equation}
The evolution equation for $\hat X_{t}$ in AB-avoSI has the same form as avoSI, i.e.,
\begin{equation}\label{xtnew}
\hat{X}_t-\hat X_0=-\int_0^t 2(\hat{X}_u-1)du+M_{0,t}.
\end{equation}
Hence, as in the case of avoSI, equation (3.4) in \cite{JLW} holds for avoSI, which implies
\begin{equation}\label{xtnew2}
\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{\frac{\hat{X}_t}{n}-m_1\exp(-2t)}\xrightarrow{\mathbb{P}} 0.
\end{equation}
Combining \eqref{newXst} and \eqref{xtnew} and using $\hat{X}_{I,t}=\hat{X}_t-\hat{X}_{S,t}$, we get
\begin{equation}\label{xitnew}
\begin{split}
\hat{X}_{I,t}&=\hat{X}_{I,0}+\int_0^t\left(-2(\hat{X}_u-1)+\sum_{k=0}^{\infty} k^2\hat S_{u,k}-\frac{\rho}{\lambda}\frac{\hat S_u}{n}(\hat{X}_u-1) \right) du
+(M_{0,t}-M_{2,t}) \\
&-\frac{1}{\hat{X}_{I,t}}\int_0^t \left( \sum_{i,j=1}^{\overline{X}_0} 1_{G_{i,j}} (D(j,u)-1) 1_{\{ S(j,u)=1\}} \right)du.
\end{split}
\end{equation}
Dropping the term in the second line of \eqref{xitnew} and bounding $\sup_{0\leq t\leq \gamma_n\wedge 1}\abs{\hat X_{I,0}+M_{0,t}-M_{2,t}}$
by $\epsilon n$ (which holds with high probability by \eqref{s11}), we get
\begin{equation}
\hat{X}_{I,t}\leq \epsilon n+\int_0^t\left(-2(\hat{X}_u-1)+\sum_{k=0}^{\infty} k^2\hat S_{u,k}-\frac{\rho}{\lambda}\frac{\hat S_u}{n}(\hat{X}_u-1) \right) du.
\end{equation}
Using \eqref{xtnew2}, \eqref{level2st} and \eqref{level2k^2} to approximate $\hat{X}_t$, $\hat{S}_t$ and $\sum_{k=0}^{\infty} k^2\overline{S}_{t,k}$ up to the first order, respectively, we obtain that whp,
\begin{equation}\label{xt-xst}
\begin{split}
\hat{X}_t-\hat{X}_{S,t}\leq & n\int_0^t\left(-2m_1\exp(-2u)+ \sum_{k=0}^{\infty} k^2p_k+\left(-\sum_{k=0}^{\infty} k^3p_k+2\frac{\rho}{\lambda}m_1^2+\frac{\rho}{\lambda}m_1\right)u
\right. \\
&\left. -\frac{\rho}{\lambda}(1-m_1u)m_1\exp(-2u)
+Cu^2\right)du+2n\epsilon.
\end{split}
\end{equation}
We would like to expand the integrand of \eqref{xt-xst} in powers of $u$. Since
$\rho/\lambda_c = (m_2-2m_1)/m_1$, the constant term is
\begin{equation}
-2m_1+m_2-\frac{\rho}{\lambda}m_1
= m_1 \left( \frac{\rho}{\lambda_c} -\frac{\rho}{\lambda} \right)
= \frac{m_1\rho}{\lambda_c^2}(\lambda-\lambda_c)+O((\lambda-\lambda_c)^2).
\label{cterm}
\end{equation}
for $\lambda>\lambda_c$. Therefore for $\lambda$ sufficiently close to $\lambda_c$ we have
\begin{equation}
-2m_1+m_2-\frac{\rho}{\lambda}m_1\leq \frac{2m_1\rho}{\lambda_c^2}(\lambda-\lambda_c).
\label{ctermbd}
\end{equation}
Note that
$e^{-2u} = 1 - 2u +2u^2 + \ldots$, so the coefficient in front of $u$ is
\begin{equation}
4m_1 + \left(-m_3+\frac{\rho}{\lambda}2m_1^2+\frac{\rho}{\lambda}m_1\right)
+\frac{\rho}{\lambda}m_1^2 + \frac{\rho}{\lambda} 2m_1
= 4m_1 - m_3 + \frac{\rho}{\lambda} (3m_1^2 + 3m_1).
\label{uterm}
\end{equation}
At $\lambda=\lambda_c$, this coefficient is equal to
\begin{equation}\label{m1delta}
4m_1-m_{3}+(3+3m_1)(m_2-2m_1)=-m_3+3m_2-2m_1+3m_2m_1-6m_1^2.
\end{equation}
We claim that the quantity in \eqref{m1delta} is exactly equal to $m_1\Delta.$
Indeed, from the definition of $\Delta$ we see that
$$
\Delta=-\frac{\mu_3}{\mu_1}+3(\mu_2-\mu_1)=
- \frac{m_3-3m_2+2m_1}{m_1}+3(m_2-2m_1)
$$
so that
$$
m_1\Delta= -m_3+3m_2-2m_1+3m_2m_1-6m_1^2.
$$
Using \eqref{uterm} with the equations that follow, we see that for $\lambda$ close to $\lambda_c$ we have
\begin{equation}
4m_1-m_3+2\frac{\rho}{\lambda}m_1^2+\frac{\rho}{\lambda}m_1+\frac{\rho}{\lambda}m_1(m_1+2)\leq \frac{3m_1\Delta}{2}.
\label{utermbd}
\end{equation}
Using \eqref{ctermbd} and \eqref{utermbd} in \eqref{xt-xst}, we see that for some constant
$C'>0$ and
all $t\leq m_1\Delta/(2 C')$,
\begin{equation}\label{xt-xst2}
\begin{split}
\hat{X}_{I,t}&\leq n\left(\int_0^t\left( \frac{2\rho}{\lambda_c^2}(\lambda-\lambda_c)+\frac{3m_1\Delta }{2}u+C'u^2\right)du+2\epsilon\right)\\
&\leq \left(\frac{2\rho}{\lambda_c^2}(\lambda-\lambda_c)t+\frac{3m_1\Delta}{4} t^2+ \frac{C't^3}{3} +2\epsilon\right)n\\
&\leq \left(\frac{2\rho}{\lambda_c^2}(\lambda-\lambda_c)t+m_1\Delta t^2+2\epsilon\right)n.
\end{split}
\end{equation}
This proves \eqref{lbstep11} since $\epsilon$ is arbitrary.
\medskip
{\bf The proof of \eqref{lbstep12}} is parallel to the proof of \eqref{lbstep11}, except that we now replace the second line of \eqref{xitnew} by
$E(u)$ (defined in \eqref{et}). We can do this because
$$
\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}}
(D(j,t)-1) 1_{\{ S(j,t)=1\}} \right)\leq E(t),
$$
which is true by the definition of $E(t)$ in \eqref{et}.
\medskip
Since $\hat{I}_t=n-\hat{S}_t$, equation \eqref{newSt} implies
\begin{equation}\label{newIt}
d\hat{I}_t=\hat{X}_{S,t}\, dt-\frac{1}{\hat{X}_{I,t}}
\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}} 1_{\{ S(j,t)=1\} } \right)dt-dM_{1,t}.
\end{equation}
Using $\hat{I}_0=1$ and the inequality
$$
\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{G_{i,j}}
1_{\{ S(j,t)=1\}} \right) \leq E(t),
$$
which follows from the fact that $D(j,t)\geq 1$, we see that
$$
\abs{\hat{I}_t-\left(1+\int_0^t\hat{X}_{S,u}du\right)}\leq \int_0^t E(u)du+\abs{M_{1,t}}.
$$
{\bf The rest of proofs for \eqref{lbstep13} and \eqref{lbstep14}} are parallel to \eqref{lbstep11} and \eqref{lbstep12}.
We omit further details.
\end{proof}
Let
\begin{equation}
L(t)=\sum_{j=1}^{\hat X_0}D(j,t)1_{\{S(j,t)=1, B(j,t)>0\}}.
\label{defLt}
\end{equation}
Since $ \sum_{i=1}^{\hat X_0} 1_{\{I(i,t)=1\}} = \hat X_{I,t}$ we have
\begin{equation}\label{et<lt}
\begin{split}
E(t)&=\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{\{I(i,t)=1\}} D(j,t) 1_{\{ S(j,t)=1\}} 1_{\{A(i,t)\leq B(j,t) \} } \right)\\
&\leq \frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{\{I(i,t)=1\}} D(j,t) 1_{\{ S(j,t)=1\}} 1_{\{ B(j,t)>0 \} } \right) =L(t).
\end{split}
\end{equation}
The definition of $L(t)$ and the fact $D(j,t)\leq D(j,t)^2$ imply that
$$
L(t)\leq \sum_{j=1}^{\hat X_0}D(j,t)^2 1_{\{S(j,t)=1,B(j,t)>0\}}.
$$
Using the fact $D(j,t)\leq D(j,t)^2$ again, \eqref{s13} implies that
\begin{equation}\label{L1}
\lim_{n \to \infty}\mathbb{P}(L(t)\leq n(C_{\ref{s13}}t+\epsilon), \forall 0\leq t\leq \gamma_n\wedge 1)=1.
\end{equation}
Combining \eqref{et<lt} and \eqref{L1} we see that for some constant $C_{\ref{eubound1}}>0$,
\begin{equation}\label{eubound1}
\lim_{n\to\infty}\mathbb{P}\left(\int_0^t E(u)du\leq (
C_{\ref{eubound1}}
t^2+\epsilon)n, \forall 0\leq t\leq \gamma_n \wedge t_0\right)=1.
\end{equation}
The bound provided in \eqref{eubound1} is not enough for our purpose (though we will also use it in the proof of Theorem \ref{Q1AB}). We will prove refined bounded in the next section.
\subsection{More refined bounds}\label{sec:refined}
Equation \eqref{lbstep12} implies that we can get a lower bound for $\hat{X}_{I,t}$ if we can upper bound the term $\int_0^t E(u)du$. To this end, we let $b$ be some number in $(0,1)$ to be determined. We can decompose $E(t)$ into two parts:
\begin{equation}\label{ede}
\begin{split}
E_1(t) &:=\frac{1}{\hat{X}_{I,t}}
\left( \sum_{i,j=1}^{\hat X_0} 1_{\{I(i,t)=1\}} D(j,t) 1_{\{ S(j,t)=1\}}
1_{\{A(i,t)<bt \} } \right),\\
E_2(t) &=\frac{1}{\hat{X}_{I,t}}\left( \sum_{i,j=1}^{\hat X_0} 1_{\{I(i,t)=1\}} D(j,t) 1_{\{ S(j,t)=1\}} 1_{\{B(j,t)>bt\}} \right).
\end{split}
\end{equation}
Since $G_{i,j} = \{ I(i,t)=1, A(i,t) \le B(j,t)\}$, we have that $E(t) \le E_1(t) + E_2(t)$.
Now we set \begin{equation}\label{xll}
\begin{split}
X(I,b,t)&:=\sum_{i=1}^{\overline{X}_0}1_{\{A(i,t)\leq bt, I(i,t)=1\}} \le \hat X_{I,t}, \\
L(b,t)&:=\sum_{j=1}^{\overline{X}_0}D(j,t)1_{\{S(j,t)=1, B(j,t)>bt\}} \le L(t).
\end{split}
\end{equation}
Recalling the definition of $L(t)$ in \eqref{defLt} we have
\begin{equation}\label{e1e2}
E_1(t)=\frac{X(I,b,t)}{\hat{X}_{I,t}}L(t), \qquad E_2(t)=L(b,t).
\end{equation}
In the next two lemmas we give bounds on $L(b,t)$ and $X(I,b,t)$.
\begin{lemma}\label{ctlet2}
There exists a constant $C_{\ref{lbtctl}}$ so that
for any fixed $b\in (0,1)$ and any $\epsilon>0$,
whp for all $0\leq t\leq \gamma_n\wedge 1$,
\begin{equation}
L(b,t)\leq n(C_{\ref{lbtctl}}(1-b)^{1/2}t+\epsilon).
\label{lbtctl}
\end{equation}
\end{lemma}
\begin{proof} Using the Cauchy-Schwartz inequality,
\begin{equation}\label{lbtctl3}
\begin{split}
L(b,t)=&\sum_{j=1}^{\hat X_0}D(j,t)1_{\{S(j,t)=1, B(j,t)>bt\}} \\
\leq &
\left(\sum_{j=1}^{\hat X_0} D(j,t)^21_{\{S(j,t)=1,B(j,t)>0\}} \right)^{1/2}
\left(\sum_{j=1}^{\hat X_0} 1_{\{bt\leq B(j,t)\leq t\}} \right)^{1/2}.
\end{split}
\end{equation}
The first term in the second line of \eqref{lbtctl3} has already been controlled by equation \eqref{s13}, i.e.,
\begin{equation}\label{Nt-1}
\lim_{n \to \infty} \mathbb{P}\left(\sum_{j=1}^{\hat X_0}D(j,t)^2
1_{\{S(j,t)=1, B(j,t)>0\}} \leq n(C_{\ref{s13}}t+\epsilon), \forall 0\leq t\leq \gamma_n\wedge 1 \right)=1.
\end{equation}
Let $N(t)$ be the number of rewiring events that occur by time $t$. Then we have
\begin{equation}\label{Nt0}
\sum_{j=1}^{\hat X_0} 1_{\{bt\leq B(j,t)\leq t\}}
\leq N(t)-N(bt).
\end{equation}
Note that one can get an equation for the evolution of $N(t)$
$$
N(t)=\int_0^t q(u)\, du+M_{7,t},
$$
where we have that
\begin{equation}\label{qt}
q(t)=\rho \hat{X}_{I,t}\frac{\hat{X}_t-1}{\lambda \hat{X}_{I,t}}
\leq\frac{\rho \hat X_0}{\lambda}
\end{equation}
and $M_{7,t}$ is some martingale. The assumption $\mathbb{E}(D^5)<\infty$ implies that
the event $\Omega^*_n=\{\hat X_0\leq 2m_1n\}$ has probability tending to 1 as $n\to\infty$. On $\Omega^*_n$, using \eqref{qt} we have
$$
q(t)\leq 2m_1\rho n/\lambda.
$$
It follows that
\begin{equation}\label{Nt1}
\lim_{n\to\infty}\mathbb{P}(q(t)\leq 2m_1\rho n/\lambda, \forall t\geq 0)=1.
\end{equation}
Note that $N(t)$ is a pure jump process with jump size equal to 1.
It follows that the expected value of quadratic variation of $M_{7,t}$ up to time 1 is upper bounded by
$$
\mathbb{E}\left(\sup_{t>0} q(t)\right)\leq \frac{\rho}{\lambda}\mathbb{E}(\hat{X}_0)\leq \frac{m_1\rho}{\lambda}.
$$
This implies that for any $\epsilon>0$,
\begin{equation}\label{Nt2}
\lim_{n\to\infty} \mathbb{P}\left(\sup_{0 \leq t\leq \gamma_n \wedge 1} \abs{M_{7,t}}\leq \epsilon n\right)=1.
\end{equation}
From the definition of $N(t)$, we see that for $t\leq 1$,
$$
\abs{N(t)-N(bt)}=\abs{\int_{bt}^t q(u)du\,
+M_{7,t}-M_{7,bt}}\leq \abs{\int_{bt}^t q(u)du}+2\sup_{0\leq t\leq \gamma_n\wedge 1}.
\abs{M_{7,t}},
$$
Thus by \eqref{Nt1} and \eqref{Nt2} we get
\begin{equation}\label{Nt3}
\lim_{n\to\infty}\mathbb{P}(\abs{N(t)-N(bt)}\leq Cn((1-b)t+\epsilon),\forall 0\leq t\leq \gamma_n)=1.
\end{equation}
Combining \eqref{Nt0} and \eqref{Nt3} we get
\begin{equation}\label{Nt5}
\lim_{n\to\infty}
\mathbb{P}\left( \sum_{j=1}^{\hat X_0} 1_{\{bt\leq B(j,t)\leq t\}} \leq
Cn((1-b)t+\epsilon) \right)=1.
\end{equation}
Equation \eqref{lbtctl} now follows from \eqref{lbtctl3}, \eqref{Nt-1} and \eqref{Nt5}.
\end{proof}
Let $t_0$ and $\lambda_0$ be the two constants given in the statement of Lemma \ref{lbstep2}.
Based on the calculations that led to \eqref{fRHS} we let
\begin{equation}\label{Utdef}
\begin{split}
U(t) = & C_{\ref{fRHS}} \biggl[ \exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}
{\lambda-\lambda_c+t} \right) (\lambda-\lambda_c) t + (1-b)t\\
& + \exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}
{\lambda-\lambda_c+t} \right) t^2 + (\lambda-\lambda_c)t^2+t^3 \biggr]
+C_{\ref{RHS2}}\sqrt{\epsilon}.
\end{split}
\end{equation}
\begin{lemma} \label{ctlet3}
For any $\lambda<\lambda_c+\lambda_0$,
whp for all $0\leq t\leq t_0\wedge \gamma_n$ we have
\begin{equation}
X(I,b,t) \leq U(t)n.
\label{xibt1}
\end{equation}
\end{lemma}
\begin{proof}
We define the events $H_r(i,t),1\leq r\leq 4$ for $1\leq i\leq \hat X_0$ ($i$ is any half-edge) as follows.
\begin{align*}
H_1(i,t)&=\{I(i,bt)=1, i \mbox{ didn't get rewired or paired in }[bt,t]\},\\
H_2(i,t)&=\{I(i,bt)=1, i \mbox{ got rewired to an infected vertex at its first rewiring in }[bt,t] \},\\
H_3(i,t)&=\{I(i,bt)=1, i \mbox{ got rewired to a susceptible vertex at its first rewiring in }[bt,t]\\
& \mbox{ and that vertx later became infected in }[bt,t]\},\\
H_4(i,t)&=\{0<A(i,t)\leq bt,S(i,bt)=1, v(i,bt) \mbox{ got infected in }[bt,t]\}.
\end{align*}
We claim that
$$
\{A(i,t)\leq bt, I(i,t)=1\} \subset \cup_{k=1}^4 H_k(i,t).
$$
Indeed, either $I(i,bt)=1$ or $S(i,bt)=1$ must hold. The case of $S(i,bt)=1$ corresponds to $H_{\tcr{4}}(i,t)$. On the other hand, if $I(i,bt)=1$ and $I(i,t)=1$, then there are three possible cases: $i$ didn't get rewired in $[bt,t]$, $i$ was rewired to an infected vertex or $i$ was rewired to a susceptible vertex which later became infected. The first case case corresponds to $H_1(i,t)$ while the second and third case are covered in $H_2(i,t)$ and $H_3(i,t)$, respectively.
Let $H_k(t)$ be the number of half-edges $i$ for which $H_k(i,t)$ occurs.
It follows from the claim and the definition of $X(I,b,t)$ that
\begin{equation} \label{xibt}
X(I,b,t)\leq \sum_{i=1}^4 H_k(t).
\end{equation}
\medskip\noindent
{\bf We first estimate $H_1(t)$.} The rate for a half-edge to be paired at time $u$ is $(\hat{X}_u-1)/(\lambda \hat{X}_{I,u})$. Hence conditionally on $\hat X_{I,s}$, $bt\leq s\leq t$,
$H_1(t)$ is stochastically dominated by
$$
\textrm{Binomial}\left(\hat{X}_{I,bt},\ \exp\left(-\int_{bt}^t \frac{\hat{X}_u}{\lambda \hat{X}_{I,u}}du\right)\right).
$$
Let $C_{\ref{defl1n}}=\max\{2\rho m_1/\lambda_c^2,m_1\Delta,1\}$. Then the event
\begin{align}
L_1(n) =& \{\hat{X}_{I,t}\leq C_{\ref{defl1n}}n((\lambda-\lambda_c)t+t^2+\epsilon),
\nonumber\\
&\hat{X}_t\geq nm_1\exp(-2t)/2, \forall 0\leq t\leq \gamma_n\wedge t_0\}
\label{defl1n}
\end{align}
has probability tending to 1 as $n\to\infty$ by \eqref{lbstep11} of Lemma \ref{lbstep2} and \eqref{xtnew2}.
On the event $L_1(n)$, we have that
\begin{equation}
\int_{bt}^t \frac{\hat{X}_u}{\hat{X}_{I,u}}du\geq
\int_{bt}^t \frac{m_1\exp(-2u)/2}{C_{\ref{defl1n}}((\lambda-\lambda_c)u+u^2+\epsilon)}du
\geq \frac{C_{\ref{Xintbd}}e^{-2t} (1-b)t}{(\lambda-\lambda_c)t+t^2+\epsilon}.
\label{Xintbd}
\end{equation}
Thus on $L_1(n)$, $H_1(t)$ is stochastically dominated by
\begin{equation}
W_0(t):= \textrm{Binomial}
\left( C_{\ref{defl1n}}n((\lambda-\lambda_c)t+t^2+\epsilon),
\exp\left( -\frac{C_{\ref{Xintbd}}e^{-2t} (1-b)t}{(\lambda-\lambda_c)t+t^2+\epsilon)} \right)\right).
\label{ber0}
\end{equation}
For $\sqrt{\epsilon}\leq t\leq t_0$, we have
$$
\frac{(1-b)t}{(\lambda-\lambda_c)t+t^2+\epsilon}
\geq \frac{(1-b)t}{2((\lambda-\lambda_c)t+t^2)}
=\frac{1-b}{2(\lambda-\lambda_c+t)}.
$$
Hence $W_0(t)$ is stochastically dominated by
\begin{equation}
W_1(t):=\textrm{Binomial}\left( C_{\ref{defl1n}}n((\lambda-\lambda_c)t+t^2+\epsilon),
\exp\left(-\frac{C_{\ref{Xintbd}}e^{-2t} (1-b)}{2(\lambda-\lambda_c+t)} \right)\right)
\label{ber1}
\end{equation}
for $\sqrt{\epsilon}\leq t\leq t_0$. Define
\begin{equation}\label{defU0t}
U_0(t)=C_{\ref{defl1n}}((\lambda-\lambda_c)t+t^2+\epsilon)\exp\left(
-\frac{C_{\ref{Xintbd}} e^{-2t}(1-b)}{2(\lambda-\lambda_c+t)} \right).
\end{equation}
For $t>\sqrt{\epsilon}$, there exists a constant $C_{\ref{U0tlb}}=C_{\ref{U0tlb}}(\lambda_0,t_0,\epsilon)$ depending on $\lambda_0$, $t_0$ and $\epsilon$ such that
\begin{equation}
U_0(t)\geq C_{\ref{U0tlb}}.
\label{U0tlb}
\end{equation}
We need a large deviations bound for sums of Bernoulli random variables.
\begin{lemma}\label{chernoff}
Consider $n$ i.i.d. Bernoulli random variables $Y_1,\ldots, Y_n$. Let $\mu=\sum_{k=1}^n\mathbb{E}(Y_i)$. Then we have
\begin{equation}\label{cher1}
\mathbb{P}\left(\sum_{k=1}^n Y_k\geq 3\mu
\right)\leq \exp(-\mu).
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{chernoff}]
By \cite[Theorem 2.3.1]{V}, we have
\begin{equation*}
\mathbb{P}\left(\sum_{k=1}^n Y_k\geq 3\mu\right)\leq \exp(-\mu)\left(\frac{e\mu}{3\mu}\right)^{3\mu} \leq \exp(-\mu).
\end{equation*}
\end{proof}
Using Lemma \ref{chernoff}, we have, for $t\geq \sqrt{\epsilon}$,
\begin{equation}
\mathbb{P}\left(W_1(t)>3U_0(t)n \right)\leq \exp(-U_0(t)n)
\leq \exp(-C_{\ref{U0tlb}}n).
\label{ber2}
\end{equation}
Using the definition of $L_1(n)$ in \eqref{defl1n} and the fact that for $\sqrt{\epsilon}\leq t\leq t_0$, $W _1(t)$ dominates $W_0(t)$ which in turn dominates $H_1(t)$ (see \eqref{ber0}) we get
\begin{equation}
\mathbb{P}\left(\{H_1(t)>3U_0(t)n\}\cap L_1(n) \right)
\leq \exp(-C_{\ref{U0tlb}}(\lambda_0,t_0,\epsilon)n).
\end{equation}
For $0\leq t\leq \sqrt{\epsilon}$, on the event $L_1(n)$ we have
\begin{equation}
\hat{X}_{I,t}\leq C_{\ref{defl1n}}n((\lambda-\lambda_c)t+t^2+\epsilon)
\leq C_{\ref{defl1n}}n((\lambda-\lambda_c)\sqrt{\epsilon}+\epsilon+\epsilon)
\leq C_{\ref{ber5}}n\sqrt{\epsilon}.
\label{ber5}
\end{equation}
for sufficiently small $\epsilon$. Since $H_1(t)\leq \hat{X}_{I,bt}$ (by the definition of $H_1(t)$) we also get
$$
H_1(t)\leq C_{\ref{ber5}}n\sqrt{\epsilon} \quad\hbox{ for $t\leq \sqrt{\epsilon}$.}
$$
Thus if we define
\begin{equation}
U_1(t)=U_0(t)+C_{\ref{ber5}}\sqrt{\epsilon},
\label{defU1t}
\end{equation}
then we have
\begin{equation}\label{ber3}
\mathbb{P}\left(\{H_1(t)>3 U_1(t)n\}\cap L_1(n) \right)
\leq \exp(-C_{\ref{U0tlb}}n)
\end{equation}
for all $0\leq t\leq t_0$.
Setting $t^{\ell}_{0}=t_0\ell /n^{3/2},0\leq \ell\leq n^{3/2}$,
we get
\begin{equation}\label{xibt2}
\mathbb{P}\left(\left( \cup_{\ell=0}^{n^{3/2}-1}
\{H_1(t_0^{\ell})>3U_1(t_0^{\ell})n\}\right)\cap L_1(n)
\right) \leq n^{3/2}\exp(-C_{\ref{U0tlb}}n).
\end{equation}
Denote the oscillation of $H_1(t)$ in $[t_0^{\ell},t_0^{\ell+1}]$ by
$\omega(H_1(t),t_0^{\ell},t_0^{\ell+1})
$. Consider the event that there is at most six pairings (i.e., a half-edge pairs with another half-edge) and rewirings (i.e., a half-edge is rewired to another vertex) occurring in $[bt_{\ell},bt_{\ell+1}]\cup [t_{\ell},t_{\ell+1}]$ and denote it by $\Omega^{\ell}$. On $\Omega^{\ell}$ we have
\begin{equation}\label{w6qi}
\omega(H_1(t),t_{\ell},t_{\ell+1})\leq 6\max_{1\leq i\leq n}Q_i,
\end{equation}
where $Q_i$ is the number of half-edges that vertex $i$ has before it becomes infected. By \eqref{w6qi}, Markov's inequality and \eqref{m4tctl} (together with $Q_i^5\geq Q_i^4$) we have
\begin{equation}\label{h11}
\mathbb{P}(\{\omega(H_1(t),t_{\ell},t_{\ell+1})\geq \epsilon n\}\cap \Omega^{\ell})\leq
\mathbb{P}\left(6\max_{1\leq i\leq n}Q_i \geq \epsilon n\right) \leq
\frac{6^4\mathbb{E}(\sum_{i=1}^n Q_i^4)}{\epsilon^4n^4}\leq \frac{C}{\epsilon^4 n^3}.
\end{equation}
Now we control the probability of $(\Omega^{\ell})^c$. Note that the rate for a rewiring or pairing to occur is
equal to $(\rho \hat{X}_{I,t}+\lambda\hat{X}_{I,t})(\hat{X}_t-1)/(\lambda\hat{X}_{I,t})$ which is bounded by $(\lambda+\rho)\hat X_0/\lambda$. On the event $\Omega^*_n:=\{\hat X_0\leq 2m_1n\}$ (which holds with high probability) this quantity is upper bounded by $2(\rho+\lambda)m_1n/\lambda$. Using this we get
\begin{equation}\label{h12}
\mathbb{P}((\Omega^{\ell})^c)\cap \Omega_n^*) \leq C(nn^{-3/2})^6\leq Cn^{-3}.
\end{equation}
Combining \eqref{h11} and \eqref{h12} we get
\begin{equation}
\mathbb{P}(\{\omega(H_1(t),t_0^{\ell},t_0^{\ell+1})>\epsilon n\}\cap \Omega_n^*)\leq \frac{C}{\epsilon^4 n^3}.
\end{equation}
By the union bound for probabilities we have
\begin{equation}\label{xibt3}
\mathbb{P}\left(\left(\cup_{\ell=0}^{n^{3/2}-1} \{w(H_1(t),t_0^{\ell},t_0^{\ell+1})>\epsilon n\}\right) \cap \Omega_n^*\right)\leq \frac{C}{\epsilon^4 n^{3/2}}.
\end{equation}
Combining \eqref{xibt2},\eqref{xibt3} and the facts
$\mathbb{P}(L_1(n))\to 1,\mathbb{P}(\Omega^*_n)\to 1$, we get
\begin{equation}
\lim_{n\to\infty}\mathbb{P}\left(
\left(\cap_{\ell=0}^{n^{3/2}-1}
\{w(H_1(t),t_0^{\ell},t_0^{\ell+1})\leq \epsilon n\}\right)
\cap \left( \cap_{\ell=1}^{n^{3/2}-1}
\{H_1(t_0^{\ell})\leq 3 U_1(t_0^{\ell})n\}\right)
\right)=1.
\end{equation}
On the event
$$
\left(\cap_{\ell=0}^{n^{3/2}-1} \{w(H_1(t),t_0^{\ell},t_0^{\ell+1})\leq \epsilon n\}\right)
\cap \left( \cap_{\ell=1}^{n^{3/2}}
\{H_1(t_0^{\ell})\leq 3 U_1(t_0^{\ell})n\}\right),
$$
we necessarily have
$H_1(t)\leq 3U_1(t)n +\epsilon n\leq 4U_1(t)n$
for all $0\leq t\leq t_0\wedge \gamma_n$. Hence we get
\begin{equation}\label{ctlh1}
\mathbb{P}(H_1(t)\leq 4U_1(t)n,\forall 0\leq t\leq t_0\wedge \gamma_n)=1.
\end{equation}
\medskip\noindent
{\bf Now we turn to the control of $H_2(t)$.}
We set the event
$$
L_2(n):=\{\hat{I}_t\leq (2m_1t+\epsilon)n,\forall 0\leq t\leq \gamma_n\wedge t_0\}.
$$
By equation \eqref{lbstep13} we have
\begin{equation}\label{ItbdH2t}
\lim_{n\to\infty} \mathbb{P}(L_2(n))=1.
\end{equation}
On $L_2(n)$, $H_2(t)$ is stochastically dominated by
\begin{equation}\label{w_2}
W_2(t):=\textrm{Binomial}(C_{\ref{w_2}}n((\lambda-\lambda_c)t+t^2+\epsilon), C_{\ref{w_2}}'(t+\epsilon)).
\end{equation}
for some constants $C_{\ref{w_2}}$ and $C'_{\ref{w_2}}$.
Now we define
\begin{equation}
U_2(t)=C_{\ref{w_2}} ((\lambda-\lambda_c)t+t^2+\epsilon)
C'_{\ref{w_2}}(t+\epsilon)+C_{\ref{ber5}}\sqrt{\epsilon}.
\label{defU2t}
\end{equation}
Following the proof of \eqref{ctlh1}, one can derive analogous inequalities to \eqref{ber3} and \eqref{xibt3} for $H_2(t)$. Combining these two inequalities we obtain that
\begin{equation}\label{ctlh2}
\mathbb{P}(H_2(t)\leq 4U_2(t),\forall 0\leq t\leq t_0\wedge \gamma_n)=1.
\end{equation}
We omit further details.
\medskip\noindent
{\bf It remains to control $H_3(t)$ and $H_4(t)$.} For any vertex $x$,
let $R(x)$ be the indicator function of the event that vertex $x$ has received at least one rewired edge when $x$ first becomes infected and let $Q_x$ be the number of half-edges that $x$ has just before it becomes infected. Let $R(x,t)$ be the indicator of the event that $x$ has received at least one rewired half-edge by time $t$.
Then we have, by the definitions of $H_3(t)$ and $H_4(t)$,
\begin{equation}\label{h3h4}
H_3(t)+H_4(t)\leq \sum_{x=1}^n R(x,t) Q_x 1_{\{x \textrm{ was infected in }[bt,t]\}}.
\end{equation}
Denote the right hand side of \eqref{h3h4} by $N(bt,t)$, then
we can decompose $N(bt,t)$ into a drift part and a martingale part for any fixed $t$:
\begin{equation}\label{nbtt}
N(bt,t)=\int_{bt}^t \bar h_t(u) \, du +M_{8,t}.
\end{equation}
Let $D_x(u)$ be the number of half-edges of vertex $x$ at time $u$ and $D(j,u)$
be the number of half-edges that $v(j,u)$ has at time $u$ (recall $v(j,u)$ is the vertex that half-edge $j$ is attached to at time $u$). The process $N(bt,u)$, $bt\leq u\leq t$ has a positive jump whenever a susceptible vertex with at least one rewired half-edge gets infected. The probability that $x$ is infected (given an infection event occurs) is equal to
$D_x(u)/(\hat{X}_u-1)$ and the contribution to $N(bt,t)$ is equal to $R(x,u)D_x(u)$.
Thus $\bar h_t(u)$ satisfies
\begin{equation}\label{barh}
\begin{split}
\bar h_t(u)&\leq \lambda \hat{X}_{I,u} \frac{\hat{X}_u-1}{\lambda \hat{X}_{I,u}} \cdot \frac{\sum_{x=1}^n R(x,u)D^2_x(u)1_{\{x \textrm{ is susceptible at time }u\}}}{\hat{X}_u-1}\\
&\leq \sum_{j=1}^{\hat X_0}
D(j,u)^21_{\{S(j,t)=1,B(j,t)>0\}},
\end{split}
\end{equation}
where the second inequality follows from changing the order of summation:
\begin{equation}\label{d^2ju}
\begin{split}
\sum_{j=1}^{\hat X_0}
D(j,u)^21_{\{S(j,u)=1,B(j,u)>0\}}&=
\sum_{j=1}^{\hat X_0} \sum_{x=1}^n 1_{\{v(j,u)=x\}} D(j,u)^21_{\{S(j,u)=1,B(j,u)>0\}}\\
&= \sum_{x=1}^n D_x^2(u)1_{\{x \textrm{ is susceptible at time }u\}} \sum_{j=1}^{\hat X_0} 1_{\{v(j,u)=x,B(j,u)>0\}}\\
&\geq \sum_{v=1}^n D_x^2(u)1_{\{x \textrm{ is susceptible at time }u\}} R(x,u).
\end{split}
\end{equation}
In the last step of \eqref{d^2ju} we have used the definition of $R(x,u)$ so that
$$
\sum_{j=1}^{\hat X_0} 1_{\{v(j,u)=x,B(j,u)>0\}}\geq R(x,u).
$$
We denote the event
$$
\left\{
\sum_{j=1}^{\overline{X}_0}D(j,t)^2 1_{\{S(j,t)=1, B(j,t)>0\}}\leq n(C_{\ref{s13}}t+\epsilon), \forall 0\leq t\leq \gamma_n\wedge 1
\right\} $$
by $L_3(n)$. Then by \eqref{s13} we have $\mathbb{P}(L_3(n))\to 1$ as $n\to\infty$. On $L_3(n)$,
using \eqref{barh} and setting $C_{\ref{barh2}}=C_{\ref{s13}}$, we see that for all $0\leq t\leq t_0$,
\begin{equation}\label{barh2}
\bar h_t(u)\leq n(C_{\ref{barh2}}t+\epsilon), \forall bt\leq u\leq t.
\end{equation}
The definition of $N(bt,t)$ as the right hand side of \eqref{h3h4} implies that
we can upper bound the quadratic variation of $M_{8,t}$ by $\sum_{i=1}^n Q_i^2$ where $Q_i$ is the number of half-edges that vertex $i$ has before it becomes infected. Using this we have
\begin{equation}\label{m7t}
\mathbb{E}\left(\sup_{0\leq t\leq \gamma_n}M_{8,t}^4\right)
\leq \mathbb{E}\left[\left(\sum_{i=1}^n Q_i^2\right)^2\right]\leq \mathbb{E}\left(n\sum_{i=1}^n Q_i^4\right)
\leq Cn^2.
\end{equation}
The second inequality in \eqref{m7t} is due to the Cauchy-Schwartz inequality
$$
\left(\sum_{i=1}^n (Q_i^2)^2\right)\left(\sum_{i=1}^n 1^2\right)\geq \left(\sum_{i=1}^n Q_i^2\right)^2,
$$
and the third inequality follows from \eqref{m4tctl}.
Using \eqref{m7t} we have
\begin{equation}\label{m7t2}
\mathbb{P}(\abs{M_{8,t}}\geq \epsilon n)\leq \frac{\mathbb{E}(M_{8,t}^4)}{\epsilon^4 n^4}\leq \frac{C}{\epsilon^4 n^2}.
\end{equation}
Now using \eqref{h3h4}, \eqref{nbtt}, \eqref{barh2} and \eqref{m7t2}, we get, for any $0\leq t\leq t_0$,
\begin{equation}
\mathbb{P}(L_3(n)\cap \{H_3(t)+H_4(t)\geq (1-b)t(C_{\ref{barh2}}t+\epsilon)n+\epsilon n \})\leq \frac{C}{\epsilon^4 n^2}.
\end{equation}
Define
\begin{equation}\label{defU3t}
U_3(t)=(1-b)t(C_{\ref{barh2}}t+\epsilon)+\epsilon.
\end{equation}
Now we can follow the proof of \eqref{ctlh1} (i.e., divide $[0,t_0]$ into $n^{3/2}$ intervals and use a union bound) to get
\begin{equation}\label{ctlh3h4}
\lim_{n\to\infty} \mathbb{P}(H_3(t)+H_4(t)\leq 2U_3(t)n,\forall 0\leq t\leq t_0 \wedge \gamma_n )=1.
\end{equation}
Combining \eqref{ctlh1}, \eqref{ctlh2} and \eqref{ctlh3h4}, we have that with high probability
$$
X(I,b,t) \le 5(U_1(t)+U_2(t)+U_3(t))n.
$$
Using \eqref{defU0t}, \eqref{defU1t}, \eqref{defU2t}, and \eqref{defU3t}, the right-hand side is
\begin{align}
5 \biggl[& C_{\ref{defl1n}}((\lambda-\lambda_c)t+t^2+\epsilon)
\exp\left(-\frac{C_{\ref{Xintbd}}e^{-2t}(1-b)}
{2(\lambda-\lambda_c+t)} \right)
\nonumber\\
& + (C_{\ref{w_2}} (\lambda-\lambda_c)t+t^2+\epsilon)
C'_{\ref{w_2}}(t+\epsilon)+2 C_{\ref{ber5}}\sqrt{\epsilon}
\label{RHS}\\
&+ (1-b)t(C_{\ref{barh2}}t+\epsilon)+\epsilon \biggr]n.
\nonumber
\end{align}
To make the computation easier to write we note that when $t \le t_0\leq 1$,
\begin{equation}
\exp\left(-\frac{C_{\ref{Xintbd}}e^{-2t}(1-b)}
{2(\lambda-\lambda_c+t)} \right)
\le \exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}
{\lambda-\lambda_c+t} \right)
:= F.
\label{Ddef}
\end{equation}
Collecting the terms with $\epsilon$ we have
\begin{equation} \label{RHS2}
\begin{split}
5 \biggl[& C_{\ref{defl1n}}((\lambda-\lambda_c)t+t^2) F
+(C'_{\ref{w_2}} (\lambda-\lambda_c)t+t^2)C'_{\ref{w_2}}t\\
&+ C_{\ref{barh2}} (1-b)t \biggr]n +C_{\ref{RHS2}}\sqrt{\epsilon}n.
\end{split}
\end{equation}
Sorting the terms by powers of $t$ we get
\begin{align}
&5 \biggl[ C_{\ref{defl1n}}F (\lambda-\lambda_c)t + C_{\ref{barh2}} (1-b)t
\label{RHS3}\\
& +C_{\ref{defl1n}}F t^2
+C_{\ref{w_2}}C'_{\ref{w_2}} [(\lambda-\lambda_c)t^2+t^3] \biggr]n
+C_{\ref{RHS2}}\sqrt{\epsilon}n.
\nonumber
\end{align}
Simplifying constants we have
\begin{align}
X(I,b,t) \le & C_{\ref{fRHS}} \biggl[ \exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}
{\lambda-\lambda_c+t} \right) (\lambda-\lambda_c) t
\label{fRHS} \\
& + \exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}
{\lambda-\lambda_c+t} \right) t^2 + (\lambda-\lambda_c)t^2+ (1-b)t^2+t^3 \biggr]n
+C_{\ref{RHS2}}\sqrt{\epsilon}n,
\nonumber
\end{align}
which completes the proof of Lemma \ref{ctlet3}.
\end{proof}
\subsection{Completing the proof of Theorem \ref{Q1AB}}\label{sec:complete}
\begin{proof}[Proof of Theorem \ref{Q1AB}] We set $\gamma_n=\inf\{t>0:\hat{X}_{I,t}=0\}$. We now condition on a large outbreak so that $\hat I_{\infty}>\eta n$ for some fixed $\eta>0$. By \eqref{lbstep13}, we see that, conditionally on $\hat I_{\infty}>\eta n$, with high probability $\gamma_n>\epsilon$ for some $\epsilon>0$. That is,
\begin{equation}\label{epeta}
\lim_{n\to\infty}\mathbb{P}(\gamma_n>\epsilon|\hat{I}_{\infty}/n>\eta)=1.
\end{equation}
Let $t_0$ and $\lambda_0$ be given by the statement of Lemma \ref{lbstep2}.
We let $\lambda_1<\lambda_0,t_1<t_0$ be two constants (independent of $\epsilon$) and $\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4,\epsilon_5,\epsilon_6$ be some small numbers (depending on $\epsilon$) to be determined.
Recall the definition of $U(t)$ in \eqref{Utdef}. We set $C_{\ref{RHS2}}\sqrt{\epsilon}$ in \eqref{Utdef} to be $\epsilon_4$. In other words,
\begin{equation}\label{defUt2}
\begin{split}
U(t)&=
C_{\ref{fRHS}} \left(\exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}{\lambda-\lambda_c+t} \right)(\lambda-\lambda_c)t+\right.\\
& \left. \left(\exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}{\lambda-\lambda_c+t} \right)+(\lambda-\lambda_c)+(1-b) \right)t^2+t^3\right)+\epsilon_4,
\end{split}
\end{equation}
Previous results imply that the following inequalities hold whp on $0 \le t \le \gamma_n \wedge t_0$. The numbers on the left give the formula numbers for these statements.
\begin{align*}
&\eqref{lbstep12} \quad \hat{X}_{I,t}\geq \left(\frac{\rho m_1(\lambda-\lambda_c)}{2\lambda_c^2}t+\frac{m_1\Delta}{4}t^2-\epsilon_2\right)n-\int_0^t E(u\, du. \\
&\eqref{lbstep14} \quad \hat{I}_t\geq \left(\frac{m_1t}{2}-\epsilon_5\right)n-\int_0^t E(u)\, du.\\
&\eqref{eubound1}\quad \int_0^t E(u)du\leq (C_{\ref{eubound1}} t^2+\epsilon_1)n.\\
& \eqref{L1} \quad L(t)\leq n(C_{\ref{s13}}t+\epsilon_1).\\
&\eqref{lbtctl} \quad L(b,t)\leq n( C_{\ref{lbtctl}}(1-b)^{1/2}t+\epsilon_3). \\
& \eqref{xibt1} \quad X(I,b,t) \leq U(t)n.
\end{align*}
Let $\Omega_n$ be the event that all of the last six formulas together with the event $\{\gamma_n>\epsilon\}$ hold.
Combining \eqref{epeta} and the fact that $\liminf_{n\to\infty}\mathbb{P}(\hat{I}_{\infty}/n>\eta)>0$ (since $\lambda>\lambda_c$),
\begin{equation}
\lim_{n\to\infty} \mathbb{P}(\Omega_n|\hat{I}_{\infty}/n>\eta)=1.
\end{equation}
Now we define
\begin{equation}\label{lasttau}
\tau=\inf\left\{\epsilon \leq t\leq t_0\wedge \gamma_n: \int_0^t E(u)\, du\geq \left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}t+
\frac{m_1\Delta}{8}t^2+\epsilon_6\right)n
\right\}.
\end{equation}
where $\inf\emptyset$ here is set to be $t_0 \wedge \gamma_n$.
We want to select the parameters $\lambda_1$ and $t_1$ so that whenever $\lambda-\lambda_c<\lambda_1$ and the outcome is in $\Omega_n$, we have
\begin{align}
t_1 \le & \tau < \gamma_n \label{timebds},\\
\hat{I}_{t_1}\geq \left(\frac{m_1t_1}{2}-\epsilon_5\right)n & -\int_0^{t_1} E(u)du>\frac{m_1t_1}{8}n.
\label{final1}
\end{align}
This implies $$\frac{\hat{I}_{\infty}}{n}>\frac{m_1t_1}{8},$$
which proves \eqref{delta>0AB}, as desired.
\medskip\noindent
{\bf Step 1.} To ensure $\gamma_n>\tau$, \eqref{lbstep12} and the definition of $\tau$ in \eqref{lasttau} imply that, for $\epsilon<t\leq \tau$,
\begin{equation}
\begin{split}\label{xit4}
\hat{X}_{I,t}&\geq \left(\frac{\rho m_1(\lambda-\lambda_c)}{2\lambda_c^2}t+\frac{m_1\Delta}{4}t^2-\epsilon_2\right)n-\int_0^t E(u)du\\
&\geq \left(\frac{\rho m_1(\lambda-\lambda_c)}{2\lambda_c^2}t+\frac{m_1\Delta}{4}t^2-\epsilon_2\right)n-\left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}t+
\frac{m_1\Delta}{8}t^2+\epsilon_6\right)n\\
&\geq \left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}t+
\frac{m_1\Delta}{8}t^2-\epsilon_2-\epsilon_6\right)n.
\end{split}
\end{equation}
Note that when $t>\epsilon$,
$$
\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}t+
\frac{m_1\Delta}{8}t^2-\epsilon_2-\epsilon_6>\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}\epsilon+
\frac{m_1\Delta}{8}\epsilon^2-\epsilon_2-\epsilon_6.
$$
Thus to ensure $\gamma_n> \tau$ it suffices to take
\begin{equation}\label{cond1}
\epsilon_2=\epsilon_6= \frac{\rho m_1(\lambda-\lambda_c) }{16\lambda_c^2}\epsilon.
\end{equation}
Indeed, \eqref{xit4} and \eqref{cond1} imply that $\hat{X}_{I,\tau}>0$, which means $\gamma_n>\tau$ by the definition of $\gamma_n$.
\medskip\noindent
{\bf Step 2.} We turn now to the second requirement that $\tau\geq t_1$ for some appropriately chosen constant $t_1$. We will show this by contradiction. Assume $\tau<t_1$ which is also smaller than $t_0$. Since we have already proved $\gamma_n>\tau$ in {\bf Step 1}, it follows that
at time $\tau$ we must have
\begin{equation}\label{contra}
\int_0^{\tau} E(u)du\geq \left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}\tau+\frac{m_1\Delta}{8}\tau^2+\epsilon_6\right)n.
\end{equation}
We split the integral $\int_0^{\tau} E(u) \,du$ into two parts:
$\int_0^{\epsilon}E(u)\,du$ and $\int_{\epsilon}^{\tau}E(u)\, du$.
For the first part we have, using \eqref{eubound1},
\begin{equation}\label{e0t}
\int_0^{\epsilon}E(u)\, du\leq (
C_{\ref{eubound1}} \epsilon^2+\epsilon_1)n,
\end{equation}
We set
\begin{equation}\label{cond1.5}
\epsilon_1= C_{\ref{eubound1}} \epsilon^2.
\end{equation}
We want to ensure
\begin{equation}\label{cond0}
C_{\ref{eubound1}} \epsilon^2+\epsilon_1=2 C_{\ref{eubound1}} \epsilon^2 \leq \frac{\epsilon_6}{2}=\frac{\rho m_1(\lambda-\lambda_c) }{32\lambda_c^2}\epsilon,
\end{equation}
which holds true if $2C_{\ref{eubound1}} \epsilon^2 \leq
\rho m_1(\lambda-\lambda_c)/32\lambda_c^2$ or
\begin{equation}\label{condep}
\epsilon<\frac{\rho m_1(\lambda-\lambda_c)}{64C_{\ref{eubound1}}\lambda_c^2}.
\end{equation}
For the second part $\int_{\epsilon}^{\tau}E(u)du$, \eqref{e1e2} implies that
\begin{equation}\label{ee1e2}
E(t)\leq E_1(t)+E_2(t)=\frac{X(I,b,t)}{\hat{X}_{I,t}}L(t)+L(b,t).
\end{equation}
For $\epsilon<t<\tau$, using \eqref{xibt} and \eqref{xit4}, we get
\begin{equation}
\frac{X(I,b,t)}{\hat{X}_{I,t}}\leq
\frac{U(t)n}{\left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}t+
\frac{m_1\Delta}{8}t^2-\epsilon_2-\epsilon_6\right)n}.
\label{Xratiobd}
\end{equation}
Using \eqref{cond1} and $t >\epsilon$, we see
$$
\left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}t+
\frac{m_1\Delta}{8}t^2-\epsilon_2-\epsilon_6\right)n\geq \max\left\{\frac{\rho m_1(\lambda-\lambda_c)}{8\lambda_c^2}tn,\frac{m_1\Delta}{8}t^2n, \epsilon_2 n \right\}.
$$
Hence using \eqref{defUt2} and \eqref{Xratiobd} we get
\begin{equation}\label{xibt5}
\begin{split}
\frac{X(I,b,t)}{\hat{X}_{I,t}} \leq
& \frac{ C_{\ref{fRHS}}\exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}{\lambda-\lambda_c+t} \right)(\lambda-\lambda_c)t}{\frac{\rho m_1(\lambda-\lambda_c)}{8\lambda_c^2}t}\\
&+
\frac{ C_{\ref{fRHS}}\left(\exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}
{\lambda-\lambda_c+t} \right)
+(\lambda-\lambda_c)+(1-b) +t\right)t^2}{\frac{m_1\Delta}{8}t^2}+\frac{\epsilon_4}{\epsilon_2}\\
\leq & \exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}{\lambda-\lambda_c+t} \right)\left(\frac{8 C_{\ref{fRHS}}\lambda_c^2 }{\rho m_1}
+\frac{8 C_{\ref{fRHS}}}{m_1\Delta}\right)\\
&+\frac{8 C_{\ref{fRHS}}}{m_1\Delta}\left((\lambda-\lambda_c)+(1-b)+t\right)+\frac{\epsilon_4}{\epsilon_2}.
\end{split}
\end{equation}
Let $V(\lambda,t)$ denote the quantity on the last two lines of \eqref{xibt5}. $V(\lambda,t)$
is increasing with respect to both $\lambda$ and $t$. Therefore for $\lambda\leq \lambda_c+\lambda_1$ and $t\leq t_0$, which we have supposed is $<1$, $V(\lambda,t)$ is bounded above by its value at $(\lambda_c+\lambda_1, t_1)$
\begin{equation}\label{defM}
V^*:=\exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}{\lambda_1+t_0} \right)\left(\frac{8 C_{\ref{fRHS}}\lambda_c^2 }{\rho m_1}
+\frac{8 C_{\ref{fRHS}}}{m_1\Delta}\right)
+\frac{8 C_{\ref{fRHS}}}{m_1\Delta}\left((\lambda-\lambda_c)+(1-b)+t_1\right)+\frac{\epsilon_4}{\epsilon_2}.
\end{equation}
Using \eqref{ee1e2}, \eqref{defM}, \eqref{L1}, and \eqref{lbtctl}
we see that
\begin{equation}\label{eeptau}
\begin{split}
\int_{\epsilon}^{\tau}E(u)\,du
&\leq \int_{\epsilon}^{\tau}\frac{X(I,b,u)}{\hat{X}_{I,u}}L(u)\, du
+\int_{\epsilon}^{\tau}L(b,u)\, du\\
&\leq \int_{\epsilon}^{\tau} V^*(C_{\ref{s13}}u+\epsilon_1) n\, du
+\int_{\epsilon}^{\tau} n(C_{\ref{lbtctl}}(1-b)^{1/2}u+\epsilon_3)\, du\\
&\leq \left(V^*C_{\ref{s13}}\frac{\tau^2}{2}
+V^*\epsilon_1\tau + C_{\ref{lbtctl}} (1-b)^{1/2}\frac{\tau^2}{2}
+\epsilon_3\tau \right)n.
\end{split}
\end{equation}
We want to choose our parameters so that
\begin{align}\label{cond2a}
& \frac{V^*C_{\ref{s13}}+C_{\ref{lbtctl}}(1-b)^{1/2} }{2}\leq \frac{m_1\Delta}{16},\\
&V^*\epsilon_1+\epsilon_3\leq \frac{\epsilon_6}{2}.
\label{cond2b}
\end{align}
Indeed, if \eqref{cond2a} and \eqref{cond2b} hold, then by \eqref{e0t}, \eqref{cond0} and \eqref{eeptau} we have
\begin{equation}
\int_0^{\tau}E(u)du=\int_0^{\epsilon}E(u) \, du+\int_{\epsilon}^{\tau}E(u)\, du\leq
\left( \frac{\epsilon_6}{2}+\frac{m_1\Delta}{16}\tau^2+\frac{\epsilon_6}{2}\right)n,
\end{equation}
which is smaller than
$$
\left(\frac{\rho m_1(\lambda-\lambda_c) }{4\lambda_c^2}\tau+
\frac{m_1\Delta}{8}\tau^2+\epsilon_6\right)n.
$$
This contradicts with \eqref{contra} and proves $\tau\geq t_1$ on $\Omega_n$.
\medskip\noindent
{\bf Step 3.} \eqref{cond2a} will hold if
$$
V^*\leq \frac{m_1\Delta }{32C_{\ref{s13}}} \quad\hbox{and} \quad
(1-b)^{1/2}\leq \frac{m_1\Delta}{32C_{\ref{lbtctl}}}.
$$
We have $\epsilon_1\leq \epsilon_6/2$ by \eqref{cond0}, so
\eqref{cond2b} will hold if
$$
V^* \le 1/4 \quad\hbox{and} \quad \epsilon_3=\frac{\epsilon_6}{4}.
$$
To make sure \eqref{cond2a} and \eqref{cond2b} are satisfied, it suffices to have
\begin{equation}\label{cond3}
V^*\leq K := \min\left\{\frac{m_1\Delta }{32C_{\ref{s13}}},\frac{1}{4}\right\},
\quad (1-b)^{1/2}\leq \frac{m_1\Delta}{32C_{\ref{lbtctl}}}, \quad
\epsilon_3=\frac{\epsilon_6}{4C_{\ref{lbtctl}}}.
\end{equation}
Using the definition of $V(\lambda,t)$ in \eqref{defM}, \eqref{cond3} can be satisfied if we first choose $b$ sufficiently close to 1 such that
\begin{equation}
\frac{8 C_{\ref{fRHS}}}{m_1\Delta}(1-b)\leq \frac{K}{8},
\label{cond4a}
\end{equation}
then choose $\lambda_1$, $t_1$ and $\epsilon_4$ such that
\begin{equation} \label{cond4b}
\exp\left(-\frac{C_{ \ref{Ddef} }(1-b)}{\lambda_1+t_1} \right) \left(\frac{b C_{\ref{fRHS}}\lambda_c^2 }{\rho m_1}
+\frac{8 C_{\ref{fRHS}}}{m_1\Delta} \right)
+\frac{8 C_{\ref{fRHS}}}{m_1\Delta}\left(\lambda-\lambda_c\right)+\frac{8 C_{\ref{fRHS}}}{m_1\Delta}t_1
< \frac{K}{8},
\end{equation}
and finally take
\begin{equation}\label{cond4c}
\epsilon_4= \frac{\epsilon_2}{8}K.
\end{equation}
\medskip\noindent
{\bf Step 4.} It remains to take care of \eqref{final1}. Using \eqref{lbstep14} and \eqref{eubound1}, we have
\begin{equation}
\hat{I}_{t_1}\geq \left(\frac{m_1t_1}{2}-\epsilon_5\right)n-\int_0^{t_1} E(u)du
\geq \left(\frac{m_1t_1}{2}-\epsilon_5-C_{\ref{eubound1}}t_1^2-\epsilon_1\right).
\end{equation}
Recall that we set $\epsilon_1=C_{\ref{eubound1}}\epsilon^2$ in \eqref{cond1.5}.
Thus we have
\begin{equation}
\hat{I}_{t_1}\geq \left(\frac{m_1t_1}{2}-\epsilon_5-C_{\ref{eubound1}}t_1^2-C_{\ref{eubound1}}\epsilon^2\right)n,
\end{equation}
which is bigger than $nm_1t_1/8$ if
\begin{equation}\label{cond6}
t_1\leq \frac{m_1}{16C_{\ref{eubound1}}}, \quad
\epsilon_5= \frac{m_1 t_1}{16}, \quad
\epsilon<\left(\frac{m_1t_1}{16C_{\ref{eubound1}}}\right)^{1/2}.
\end{equation}
\medskip\noindent
{\bf Summary} we can first choose $b$ close to 1, then tale $\lambda_1$ and $t_1$ sufficiently small such that \eqref{cond4b} holds true and
$$
t_1\leq \min\left\{\frac{m_1}{16C_{\ref{eubound1}}},t_0\right\}.
$$
Then we let $\epsilon$ be smaller than
$$
\min\left\{\frac{\rho m_1 (\lambda-\lambda_c)}{64C_{\ref{eubound1}}\lambda_c^2}, \left(\frac{m_1t_1}{16 C_{\ref{fRHS}}}\right)^{1/2}\right\}.
$$
Finally we determine $\epsilon_1,\ldots, \epsilon_6$ using \eqref{cond1}, \eqref{cond1.5}, \eqref{cond3}, \eqref{cond4c} and \eqref{cond6}.
\begin{align*}
\epsilon_2=\epsilon_6 & = \frac{\rho m_1(\lambda-\lambda_c)}{16\lambda_c^2} \epsilon,
\quad \epsilon_1 = C_{\ref{eubound1}}\epsilon^2 ,
\quad \epsilon_3 = \frac{\epsilon_6}{4},\\
\epsilon_4& = \frac{\epsilon_2}{8}
\min\left\{\frac{m_1\Delta }{32C_{\ref{s13}}},\frac{1}{4}\right\},
\quad \epsilon_5= \frac{m_1 t_0}{16}.
\end{align*}
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Samit Ghosh, MD and CEO, Ujjivan Small Finance Bank. Photo: Mint
Ujjivan, Equitas tank on RBI clarification over listing small finance banks
2 min read . Updated: 26 Oct 2018, 05:47 PM IST Ravindra N. Sonavane
Ujjivan Financial touched a record low of ₹181.15 a share and fell as much as 17.62%; Equitas Holdings slumped 23.34% to touch an all-time low of ₹99.05 a share
mint-india-wire rbiujjivanequitassmall finance bankujjivan sharesequitas sharesrbi small finance bank
Shares of Ujjivan Financial Services Ltd slumped as much as 19% while Equitas Holdings Ltd dropped nearly 28% on Friday after the Reserve Bank of India clarified that promoters of small finance banks must list their banking units separately within three years of operations.
The RBI also reiterated that promoters of small finance banks should maintain their stake to at least 40% for a period of five years from the date of commencement of operation of banks.
Both the stocks have wiped out nearly entire gains for those who bought shares in initial public offerings (IPOs). The shares are currently trading below their issue price. Both listed in 2016.
Ujjivan Financial touched a record low of ₹ 181.15 a share and fell as much as 17.62%, its biggest single-day fall since listing. Equitas Holdings slumped 23.34% to touch an all-time low of ₹ 99.05 a share.
"RBI has asked Equitas Holdings and Ujjivan Financial Services to list their small finance bank subsidiaries within three years of operations. The much awaited dispensation of this clause has not come through. The management is yet to get clarification if they would be allowed to collapse the structure after five years of operations as the need for promoter ownership does not exist. A temporary period of a small holding company discount is now likely to be in place," said Kotak Institutaional Equities in a note to its investors.
Both Ujjivan Financial and Equitas Holdings run small finance banks. For Equitas Holdings, the RBI oder will mean that the small finance bank should be listed by 24 September 2019 while for Ujjivan Financial it should list its small finance bank by 30 January 2020.
Equitas Holdings said in a notice to exchanges that it board and that of Equitas Small Finance Bank (ESFBL) will meet on 1 November and 2 November, respectively, to consider steps to get shares of ESFBL listed within prescribed timelines and approach RBI for an approval to merge with the bank at an appropriate time post the lock-in period.
"We wish to inform that the bank and the company are committed to consider all appropriate measures to ensure the timely compliance of directives of the RBI while ensuring the long-term interest of the shareholders of the company are maintained," Ujjivan Financial said in a note to BSE
Earlier, the RBI had denied a three-year extension to Yes Bank's chief executive officer Rana Kapoor and asked him to step down after 31 January 2019.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,084
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2021 COVID metrics: What the data shows us about the pandemic in Maryland last year
Post-Freddie Gray, Mosby proved only her own incompetence
Maryland's attorney grievance commission moves to suspend Ken Ravenell's law license after his federal conviction
Maryland to give out 20 million N95 and KN95 masks
Few Baltimore political leaders speaking out in wake of Mosby indictment
Maryland Gov. Hogan renews bid for tax cuts for retirees, lowest-income earners and businesses
Nice to see Jerry Lawler back, but some of the imagery was inappropriate
By By Adam Testa
Two weeks ago, Jerry "The King" Lawler suffered a heart attack on live television.
In the time since, websites have reported and speculated on Lawler's condition and the events surrounding the incident, which played out on live television.
On tonight's episode of Raw, fans had the chance to hear from The King himself, who shared some shocking details about his condition immediately after awakening at the hospital.
Lawler did not even remember being part of a tag team match on that episode, let alone the events that followed. He initially believed he was still in Aruba, where he had been on a tour the weekend prior.
Seeing Lawler in good spirits, and the evident joy on Michael Cole's face as he interacted with his partner, was an uplifting moment. Lawler's voice was still raspy from having a ventilator in his throat, but he seemed genuinely positive and upbeat.
[More from sports] 2022 NFL mock draft (Version 1.0): First-round projections with top 18 picks locked in »
I give props to WWE for allowing Lawler the chance to appear before the WWE fans on television. Next weekend, he is scheduled to make his first public appearance at an independent wrestling show in Tennessee.
But while the WWE deserves praise for allowing Lawler the opportunity to speak, company officials should also be questioned for their decision to show some of the pictures of Lawler that they did to promote the interview.
Showing The King receiving CPR and being carted out of the arena brought back mental thoughts of that night. At the time, WWE did an amazing job keeping those visuals off of television, and it would have been better had the majority of fans never seen them.
Yes, everything is much better now and the situation has improved, but there are surely people who were offended or shaken by those images. There were certainly other ways to promote the interview.
Maybe it's chastising too much, but in my opinion it was unnecessary and bordered on exploitation.
[More from sports] Buffalo Bills rout the New England Patriots 47-17 behind Josh Allen's 5 touchdown passes »
But, at the end of the day, what matters most is that Lawler is doing well. I think he's as anxious to return to return to the commentary table and the ring as the fans are to see him do so.
** The opening segment with Punk, Heyman, AJ and referee Brad Maddox was just awkward all around. It featured everything from horrible references to the NFL referee situation, sophomoric one-liners about what happens when you assume, sexual innuendo and a faux marriage proposal. While this was a refreshing change from the traditional show-opening promo, it was a little too awkward to really click. It seemed like they were trying to run many different directions rather than settling on one.
** Punk's night continued -- and picked up a lot of steam -- at the top of the first hour, when he found himself in the ring with Mick Foley. There's no one better to put over a Hell in a Cell match than Foley, and he and Punk put together a tremendous segment that may have been one of the best promos Foley has ever done, certainly one of the best since he's retired from full-time appearances. Foley's closing, in which he pointed out the in his three reigns as WWE Champion he only held the belt for 29 days, was epic and accurate. Some people can rattle off statistics but the common fan is more likely to remember the legacy-securing moments.
** Punk's night came full circle when he closed out the show in the ring with John Cena. Cena started off being serious, but quickly went back to his hat of corny jokes before Punk emerged and talked down to Cena. Cena challenged Punk to a rematch at Hell in a Cell next month, but Punk tried declining and kicking Cena out of his ring. Cena unveiled a pipe and used it as a weapon to take the champ down. The oddest part was that he then declared, "Real men wear pink." Punk's normal gear is pink, and he's worn it for months before WWE made Cena the cancer campaign poster boy. I hope Cena is allowing himself time to really heal and not rushing back for this match. I know he's a company man and all, but there's a line.
[More from sports] Orioles unveil renderings of Camden Yards' left-field changes, with wall moving back nearly 30 feet »
** The closing scene of the show may have been the biggest "WTF" moment of recent WWE television history. Punk took Foley down and started walking away. When he stopped and turned around, Ryback was standing there having an apparent asthma attack, and Punk looked scared. This was the oddest moment on a show with several of them.
** This week's skits featuring Tag Team Champions Kane and Daniel Bryan -- now officially known as Team Hell No, courtesy of the WWE fans - were a mixed bag. They didn't live up to the ones of recent weeks, but the second one featured a very funny and well-timed Mae Young cameo, which elicited a good chuckle. When the champs were in the ring on the live show, they were jumped by Cody Rhodes and Damien Sandow, who are now calling themselves Rhodes Scholars. Awesome. WWE finally seems intent on focusing on having a solid tag team division once again, and that's only going to be good for business.
** Sometimes it feels like WWE is ribbing the fans by continuing the "Best of 501 out of 1,000" series between Dolph Ziggler and Kofi Kingston. But on tonight's show, these two set out and did exactly what they wanted to do: steal the show. They have such great chemistry and put on one heck of a television match. This is why these two are paired together so often; they make magic together. And the right man won. Hopefully this is the beginning of Ziggler's rehabilitation.
** The Prime Time Players defeated Santino Marella and Zack Ryder in a quick pedestrian match. The action looked smooth, but that was nothing to make this match stand out from any other. Ryder and Marella would make a good tag team to bolster an already developing division. Titus O'Neil and Darren Young continue to be rebuilt to title contention, so it's good to know WWE hasn't given up on them yet.
** World Heavyweight Champion Sheamus teamed with Rey Mysterio and Sin Cara to face Alberto Del Rio, David Otunga and Ricardo Rodriguez in a surprisingly good, fun match. They kept the action fast-paced and entertaining and all the men clicked together. It was nice to see Rodriguez wrestling someone other than Marella. I still hope one day he'll get a shot on his own as a babyface singles guy.
[More from sports] Construction begins in left field at Oriole Park at Camden Yards | PHOTOS »
** The plot thickened -- or something to that effect -- in the "Who attacked Kaitlyn" injury angle after a tag match pitting Divas Champion Eve and Beth Phoenix against Layla and Alicia Fox. After a short-but-solid match, Kaitlyn emerged and said security footage showed her attacker to be a blond. Eve blamed Phoenix and then laid the Glamazon out.
** Ryback pretty much manhandled The Miz in a non-title match. Is it required that the Intercontinental Champion job to everyone on the roster? The match made Ryback look better than he has, as he beat an actual contender and a champion, but it still remains to be seen if he can handle a competitive back-and-forth match.
** Big Show randomly returned and laid out Tensai, who recently had the advantage over Randy Orton in a match, and Brodus Clay, who had been positioned as a challenger to the United States Championship. That was... odd.
** Wade Barrett continued to look good, while I continue to question why Tyson Kidd can't get a more meaningful role. Barrett should be moving up the card, maybe even into World Heavyweight Championship contention, soon.
Match Rundown
[More from sports] Like everything else in rebuild, Orioles' changes to Camden Yards' dimensions are about planning for future | ANALYSIS »
Dolph Ziggler d. Kofi Kingston
Darren Young and Titus O'Neil d. Santino Marella and Zack Ryder
Ryback d. The Miz
Wade Barrett d. Tyson Kidd
Sheamus, Rey Mysterio and Sin Cara d. Alberto Del Rio, David Otunga and Ricardo Rodriguez
Eve and Beth Phoenix d. Layla and Alicia Fox
Tensai vs. Brodus Clay went to a no contest
This was certainly an odd episode of Raw. It seems lately that when WWE teases an angle, they're then taking it a completely different direction. Some of that is good to throw fans off the trail, but it can hurt continuity in the long run. Anyway, there was some good wrestling on this show and some good storytelling. Much like Daniel Bryan and Kane were the featured attractions on last week's Smackdown, CM Punk was the focal point of tonight's show. WWE has certainly built some interest in seeing where things go from here.
César Prieto gives Orioles potential 2022 infield help as part of international signing class
Kevin Durant limps off with knee injury during win over Pelicans, Nets star heading for MRI Sunday
Ring Posts' Adam Testa has teamed up with My 1-2-3 Cents (www.my123cents.com) and All American Pro Wrestling (www.aapwrestling.com) to bring you From the Rafters Radio,(www.facebook.com/RaftersRadio) a weekly pro wrestling talk radio show airing from 8-10 EST every Thursday on Monster Radio 1150 AM in Southern Illinois and streaming worldwide on wggh.net (www.wggh.net).
World Wrestling Entertainment Inc.
Jerry Lawler
David Otunga
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,173
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WHILE some of us are blessed with the ability to grow long, healthy nails, others are cursed with stubby, peeling ones that break at the drop of a hat.
But these days it doesn't matter what camp you're in - a quick trip to a nail bar for a set of gels or acrylics can cover up a multitude of sins, leaving you with a glossy set of talons you can show off with pride.
Are gel and Shellac manicures really bad for our nails?
But are these Shellac treatments and gel manicures causing our nails more harm than good?
Thankfully, celebrity nail artist Britney Tokyo has put our minds at rest.
Apparently, the manicures themselves will cause no extra harm to your nails.
And she should know - she's worked her magic on countless celebrity hands including Rita Ora, Kylie Jenner and Kim Kardashian.
But it turns out there could be a very simple reason your nails aren't as strong as they could be.
Britney, who has more than 125,000 followers on her Instagram page, has revealed the worst possible thing we can do to them.
And it may well surprise you.
She told Racked: "Having nails done are not bad for your nails, but incorrectly removing them can lead to damage.
"I never want anyone to peel off their gel or acrylic nails! By peeling off, the original nails can get severely damaged, and very thin."
She advises people to remove the varnish by soaking their fingers in nail removal acetone.
Britney added: "Also, never bite your nails."
Just in case you nibblers thought you'd got off scot-free.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 35
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\section{Introduction}
\label{sec:intro}
\begin{figure}[!t]
\vspace{-0.2cm}
\centering
\includegraphics[width=1.0\linewidth]{figs/teaser}
\caption{Point clouds in the real world exhibit diverse geometric variations caused by differences in the data capture pipeline. Given these variations, networks trained on one collection of point clouds may incur a performance drop when tested on different ones. Thus adaptation is needed to alleviate generalization issues, especially for domains where the annotation is scarce.}
\vspace{-0.6cm}
\label{fig:teaser}
\end{figure}
Point clouds captured under different settings can exhibit prominent variations that cause performance drop when neural networks are tested on a domain that is different from the training ones.
This can be troublesome if the network can {\it not} be fine-tuned due to time constraints or limited computational budget.
More often, labels needed for fine-tuning on the test domain are simply unavailable due to high annotation cost, which is the situation we are interested in and is always formulated as unsupervised domain adaptation (UDA) problems.
In UDA, the source domain comes with rich annotations, while the target domain has no annotation at all.
The {\it key} to a successful domain adaptation lies in two folds. First, the two domains have to be (statistically) aligned, either in the point cloud space or in a feature space, so that the shared mapping to the output space can now operate on the same ground across domains.
Moreover, the alignment between domains has to be semantically meaningful, e.g., chairs in the source should be aligned with chairs in the target.
Otherwise, the shared mapping can still fail in predicting the labels even if the two domains are aligned.
Existing UDA methods on point clouds mainly rely on two mechanisms to align the domains.
One is to perform domain adversarial training and enforce the features of point clouds from both domains to be indistinguishable by domain discriminators.
Since adversarial training is unstable and easy to get stuck at degenerated local minimas,
there is little guarantee that the alignment would be semantically meaningful.
For example, adversarial training could distort the geometric information in the point clouds by eliminating too many variations while aligning the domains.
In this case, the alignment can result in negative adaptation gains.
An extra layer of difficulty is that the alignment process could be highly sensitive to the architecture of the discriminators for point clouds as shown in~\cite{wang2020rethinking}, thus making the alignment more uncontrollable.
The other mechanism is to perform domain alignment through learning self-supervised tasks.
The underlying motivation is that a well-designed self-supervised task can facilitate learning domain invariant features since the task itself is shared across domains.
A diverse set of carefully designed self-supervised tasks are proposed, which focus on predictive tasks where the self-supervised labels are generated by
augmenting or modifying the original point clouds.
For instance, rotation angle classification~\cite{zou2021geometry} and deformation regression~\cite{achituve2021self}.
Compared to domain adversarial training,
self-supervised learning enables explicit control over the invariants been learned by adjusting the self-supervised tasks.
Consequently, one can also regularize the alignment process through this knob.
We take the latter approach,
but we resort to a self-supervised task
where the supervision comes from the point clouds themselves, instead of manually designed classification labels.
Specifically, we ask for a latent space that encodes the underlying geometry of the point clouds through implicit functions.
As the geometry is explicitly modeled and preserved,
these latents or implicits should maintain sufficient information for the main task and help prevent mismatch in semantics caused by distortions during the alignment.
Due to the lack of shape models,
we propose an adaptive unsigned distance field that enables training the implicits for arbitrary point clouds, especially for the ones that are sparse and irregularly sampled.
After the initial round of adaptation, we follow the literature and apply self-training with pseudo labels in the target domain to further close the gap.
We experiment on two major point cloud datasets, PointDA-10~{qin2019pointdan} and GraspNet~{fang2020graspnet}, to report the performance of the proposed method and evaluate the effectiveness of each component. Our contributions are:
\begin{itemize}
\item The first method leverages implicit function learning as a self-supervised task for unsupervised domain adaptation on point clouds.
\vspace{-0.1cm}
\item Effective training strategies to make our method robust to diverse artifacts exhibited in the point clouds.
\vspace{-0.1cm}
\item State-of-the-art performance on two major datasets, PointDA-10~\cite{qin2019pointdan} and GraspNet~\cite{fang2020graspnet}. Moreover, we are the first to report results on GraspNet.
\end{itemize}
\begin{figure*}[!t]
\centering
\includegraphics[width=1.0\linewidth]{figs/framework}
\vspace{-0.5cm}
\caption{Overview of the proposed framework for unsupervised domain adaptation on point clouds. The two pathways (supervised and self-supervised) in our framework are marked with different colors. The supervised pathway takes as input the point clouds from the source domain and calculates the cross-entropy loss with ground-truth labels. The self-supervised pathway takes point clouds from the source and target domains and calculates the self-supervised loss with the proposed adaptive unsigned distances between sampled points and the input point clouds. Note, in the self-paced self-training stage, the classifier is also trained with pseudo labels.
}
\vspace{-0.6cm}
\label{fig:framework}
\end{figure*}
\section{Related Work}
\subsection{Deep Learning on Point Clouds}
To handle the irregularity and permutation-invariance of point clouds, various methods have been proposed. PointNet~\cite{qi2017pointnet} and PointNet++\cite{qi2017pointnet++} use max-pooling as a permutation-invariant local feature extractor and the latter gathers local features in a hierarchical way. DGCNN~\cite{wang2019dynamic} considers a point cloud as a graph and dynamically updates the graph to aggregate features. Recently, Point Transformer~\cite{zhao2021point} adopts transformer for point cloud processing which achieves state-of-the-art performance in several benchmarks.
\subsection{2D and 3D Unsupervised Domain Adaptation}
Extensive works have been proposed to perform UDA on 2D images, which can be classified into two categories, i.e., the methods based on domain-invariant feature learning and methods for learning domain mapping. The former ones~\cite{kang2019contrastive,long2018transferable,rozantsev2018beyond,ganin2016domain,tzeng2017adversarial,saito2018maximum} minimize the discrepancy between two distributions in the feature space, while the latter ones~\cite{shrivastava2017learning,bousmalis2017unsupervised,hoffman2018cycada} directly learn the translation from the source domain to the target domain using neural networks, e.g., CycleGAN ~\cite{zhu2017unpaired}. Despite their differences, domain adversarial training is widely exploited in these methods. Several useful techniques are also proposed, for example, pseudo-labeling~\cite{saito2017asymmetric}, and batch normalization tailored for domain adaptation~\cite{maria2017autodial}.
Though lots of efforts have been made on 2D images, UDA on 3d point cloud is still in its early stage. As discussed in the introduction, UDA on point clouds can be roughly divided into two categories. The first category~\cite{qin2019pointdan} directly extends domain adversarial training used in 2D images to 3D point clouds to align features on both local and global levels. However, unlike previous works on the 2D domain, adversarial methods on 3D point clouds can not balance well between local geometry alignment and global semantic alignment. Most recent works in UDA on point clouds fall in the second category, i.e., focusing on designing suitable self-supervised tasks on point clouds to facilitate learning domain invariant features, which we discuss in detail in the following subsection.
Apart from UDA on object point clouds, several methods are proposed to address specific domain gaps on LiDAR point clouds, where the common factors are depth missing and sampling difference between sensors. Both~\cite{zhao2020epointda} and~\cite{saleh2019domain} use CycleGAN~\cite{zhu2017unpaired} to generate more realistic LiDAR point clouds from synthetic data, i.e., sim2real. Complete \& Label~\cite{yi2021complete} leverages segmentation on completed surface reconstructed from sparse point cloud for better adaptation.
\subsection{Self-Supervised Learning on Point Clouds}
\label{subsec:SSL}
Previous works design various kinds of self-training tasks to align the two domains. GAST~\cite{zou2021geometry} proposes rotation classification and distortion localization as a self-supervised task to align features at both local and global levels. DefRec~\cite{achituve2021self} proposes deformation-reconstruction and~\cite{luo2021learnable} extends it into a learnable deformation task to further improve the performance. RS~\cite{sauder2019self} shuffles and restores the input point cloud to improve discrimination.
However, there are two main issues with these methods. Some of them can not be applied to more challenging datasets where object point clouds are not aligned and are heavily occluded, resulting in ambiguity in the rotation prediction~\cite{zou2021geometry,poursaeed2020self} and restoring~\cite{sauder2019self} tasks. Besides, by aligning high-level features~\cite{zou2021geometry,achituve2021self,luo2021learnable,sauder2019self}, i.e., in semantic space, they could lose valuable information of the underlying geometry, which limits their applicability to more general geometric processing tasks. Motivated by these two observations, we design a task where the point cloud itself generates the self-supervision on the two domains and features are aligned to preserve geometric primitives. The aligned features can further be used for high-level semantic extraction, making our method more general for various main tasks.
\section{Method}\label{sec:method}
We tackle unsupervised domain adaptation (UDA) on point clouds for classification.
Let $\mathcal{P} \in \mathbb{R}^{N \times 3}$ be a point cloud consisting of the spatial coordinates of $N$ points in the 3D space.
Accordingly,
let $\mathcal{D}^{\mathrm{s}}=\{\mathcal{P}_i^{\mathrm{s}}, \mathcal{Y}_i^\mathrm{s}\}$ be the point clouds and their ground-truth labels from the source domain.
Similarly,
$\mathcal{D}^{\mathrm{t}}=\{\mathcal{P}_i^{\mathrm{t}}\}$ is the collection of target domain point clouds whose labels are missing.
Our goal is to train a network $\Theta$, i.e., $\mathcal{Y}=\Theta(\mathcal{P})$ using the labeled point clouds from the source domain so that it can work well on the target point clouds without further labeling.
The {\it key} is to {\it align} the point clouds from both domains, and at the same time, ensure that the correspondence is {\it semantically meaningful}, i.e., the point clouds of the same category are expected to be aligned after the adaptation.
One can apply domain adversaries for aligning domains, however, the alignment is hard to control and may result in negative adaptation gains due to difficulties in adversarial training.
We resort to the strategy of utilizing self-supervised tasks
that are shared across domains for alignment in a multi-task fashion.
This enables an explicit control of the meaningfulness of the alignment by selecting an appropriate self-supervised task.
There are two pathways in our framework, as shown in Fig.~\ref{fig:framework}.
The {\it main task} is performed by $\Phi$ and $\Psi_m$, i.e., $\Theta=\Psi_m\circ\Phi$, with $\Phi$ an encoder that extracts features from the point clouds and $\Psi_m$ the main task head (classifier).
Likewise, the {\it self-supervised task} is performed by $\Phi$ (shared with the main task pathway) and $\Psi_s$, which can be trained on both domains.
Next, we detail each of the proposed components and their training.
\subsection{Self-Supervised Geometry-Aware Implicit} \label{subsec:imp}
Implicit representations are capable of preserving complex details for given shapes~\cite{park2019deepsdf,chen2019learning}.
Instead of high-quality shape reconstruction,
we leverage the implicit representation space for aligning point clouds from different domains by performing the following self-supervised task.
Given a point cloud $\mathcal{P}$, either complete or partial,
the shared encoder $\Phi$ first maps it to a feature vector
$c = \Phi(\mathcal{P})$ as the implicit representation of the unknown underlying shape from where $\mathcal{P}$ is observed.
Suppose $\mathcal{Q} \in \mathbb{R}^{K \times 3}$ are $K$ randomly sampled points in the unit cube.
By definition,
the implicit value (e.g., distance to the surface)
for each point $q \in \mathcal{Q}$ can be decoded as:
\begin{equation}
f_{\mathcal{P}}(q) = \Psi_s(q, c)
\label{eq:imp_decoder}
\end{equation}
where $f_{\mathcal{P}}$ is the implicit function of the underlying geometry conditioned on the input point cloud $\mathcal{P}$.
Following the literature~\cite{chen2019learning,mescheder2019occupancy},
the decoder $\Psi_s$ takes as input the concatenation of the query point and the encoded implicit representation.
Since the point clouds can be partial,
we set the implicit values as unsigned distances to the underlying surface.
The computation of these values is described in the following.
\subsubsection{Adaptive Unsigned Distance of Point Cloud} \label{subsec:udf}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\linewidth]{figs/softdistance}
\caption{Adaptive unsigned distance field. (a): examples of calculating distances from sampled points (triangle) to their nearest points in the input point cloud. (b): when a sampled point is close to the surface, its nearest neighbor distance is still large due to sparsity. (c): the adaptive unsigned distance field and the zero-surface, with $d_M$ the adaptive clamping value.}
\label{fig:adaptive-distance}
\end{figure}
Different from reconstruction where the known meshes can be used to compute the ground-truth for the distance values, we only have access to the point clouds.
However, as our goal is to leverage the implicit representation to align domains and reduce performance drop,
we do not need the implicits to perfectly represent the underlying geometry and reconstruct the point cloud.
To this end, we can compute approximates of the unsigned distance fields to supervise the training of the implicit space.
An intuitive method is to approximate the unsigned distance from a query point to the underlying surface by the distance between the same query point and its nearest neighbor from the point cloud (Fig.~\ref{fig:adaptive-distance} (a)).
This could work if the point clouds are densely and uniformly sampled.
Nevertheless, in practice, point clouds are usually sparse and irregularly sampled due to sensor noise and complex geometry in the scene.
These peculiarities can cause problems for the nearest neighbor approximations.
For example,
when the query point is very close to the underlying surface, the distance could still be large, as shown in Fig.~\ref{fig:adaptive-distance} (b).
Thus the learned implicit space may not faithfully represent the geometry of the point clouds and can induce performance drop across domains.
To prevent unexpected distortions of the geometry in the approximation,
we propose an adaptive clamping technique based on a global average over statistics of the local geometry. For a point $p_j$ in the input point cloud $\mathcal{P}$,
we first calculate the mean of the distances between $p_j$ and its $M$ nearest neighbors within $\mathcal{P}$:
\begin{equation}\label{eq:self_dis}
d_j = \frac{1}{M}\sum_m\|p_m - p_j\|
\end{equation}
where $p_m$ is from the $M$ nearest neighbours and we name $d_j$ the {\it local affinity} of point $p_j$.
Then, we compute the average of the local affinity of all the points in the point cloud, i.e.,
$d_M = \frac{1}{N}\sum_{j=0}^{N-1}{d_j}$, which is the {\it adaptive clamping threshold}
and is used in the following to compute the adaptive approximate of the unsigned distance field from the point cloud:
\begin{equation}\label{eq:soft_df}
d_\mathcal{P}(q) =
\begin{cases}
\|q - p^*(q)\| & \quad \text{if } \|q - p^*(q)\| > d_M\\
0 & \quad \text{otherwise}
\end{cases}
\end{equation}
where $p^*(q)$ is the nearest neighbor of the query $q$ in the point cloud $\mathcal{P}$.
Also, note that $d_M$ depends on $\mathcal{P}$ and can vary between point clouds to accommodate different sparsity levels.
An example of the adaptive unsigned distance field can be found in Fig.~\ref{fig:adaptive-distance} (c).
As observed, the unsigned distance field approximated via Eq.~\eqref{eq:soft_df} captures the underlying geometry of the point cloud and is more robust to sampling issues.
With the adaptive unsigned distance field $d_\mathcal{P}$,
the self-supervised loss for learning the implicit space is:
\begin{equation}\label{eq:sl_loss}
\mathcal{L}_I = \frac{1}{|\mathcal{Q}|}\sum_{q\in\mathcal{Q}}|f_\mathcal{P}(q) - d_\mathcal{P}(q)|
\end{equation}
here, $|\mathcal{Q}|$ is the cardinality of the sampled query points.
Next, we discuss a few issues encountered during the articulation of the whole pipeline and our solutions.
\subsubsection{Point Cloud Augmentation} \label{subsec:datapre}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\linewidth]{figs/jitter}
\caption{(a): Input point clouds with duplication. The point cloud in the top row has only 38 unique points,
but it contains 1024 points in total with duplication to account for the fixed input size of the backbone.
(b): Point clouds after random jittering in the range $[0.03, 0.06]$ (instead of duplicating points), which simply dilates each point. The points in the top row are colored randomly for better discrimination.
(c): Point clouds sampled from the implicits learned with the random jittering scheme in (b). We can observe much heavier geometric distortion for sparser point clouds (top vs. bottom).
(d): Point clouds jittered using the proposed scheme based on the spatially varying local affinity measure $d_j$ in Eq.~\eqref{eq:self_dis}.
(e) Sampled point clouds from implicits learned with the jittering scheme in (d), which preserves the geometry for both sparse and dense point clouds.}
\label{fig:jitter}
\end{figure}
{\bf Jittering.} The point cloud backbone usually assumes a fixed number of points during training, for example, 1024 points for a single point cloud.
However, in practice, the number of points in a single point cloud may not be the same due to irregular sampling or different shape sizes.
For example, in the unsupervised domain adaptation benchmark PointDA-10~\cite{qin2019pointdan}, point clouds from ModelNet and ScanNet can have very different numbers of points.
A commonly used technique is to pad the point clouds to the same number of points through jittering, which may also improve the training if the jittering is properly designed.
The simplest jittering method is to add duplicate points to the original point cloud (Fig.~\ref{fig:jitter} (a)).
Another method is to add uniform random perturbations (Fig.~\ref{fig:jitter} (b)).
However, both methods will generate points that make the local affinity measure uninformative, so that the proposed adaptive unsigned distance field may not be effective in preventing geometric distortions for sparse and irregular point clouds.
As shown in Fig.~\ref{fig:jitter} (c), the resampled point clouds from the learned implicits with random perturbations exhibit significant geometric distortions for sparse point clouds.
In order to avoid such degenerated cases in the padding procedure, we propose to perform the point jittering in an affinity-aware manner.
Similar to calculating the adaptive unsigned distance field,
for each point $p_j$ in the raw point cloud,
we first obtain its local affinity $d_j$ using Eq.~\eqref{eq:self_dis}.
A random offset in the range of $[-\frac{d_j}{2}, \frac{d_j}{2}]$ is then added to $p_j$ to generated jittered points.
The point clouds generated with this jittering scheme have little deviation from the raw point clouds, as shown in Fig.~\ref{fig:jitter} (d).
Moreover, the resampled point clouds from the implicits learned with the affinity-aware jittering maintain the underlying geometry for both sparse and dense point clouds as observed in Fig.~\ref{fig:jitter} (e).
{\bf Random masking.}
Point clouds may come in a partial form due to self-occlusions.
To improve the robustness of the implicits concerning the partiality and further reduce the domain variations,
we choose to mask out a local neighborhood of a randomly selected point as an additional data augmentation.
Let $\mathcal{P}$ be a point cloud before random masking, and $\hat{\mathcal{P}}$ be the point cloud obtained by dropping out a neighborhood of radius of $r_m$.
We ask the implicits of both point clouds to be similar as they are sampled from the same geometry.
During training, we add a loss term between the implicit representations of the input point cloud and its masked version:
\begin{equation}\label{eq:mask_loss}
\mathcal{L}_M = \|\Phi(\mathcal{P}) - \Phi(\hat{\mathcal{P}})\|
\end{equation}
with $\|\cdot\|$ the L-2 distance.
\subsection{Self-Paced Self-Training} \label{subsec:spsl}
The main task we tackle here is point cloud classification.
Before adaptation, we only have labeled data in the source domain, i.e., $\{\mathcal{P}_i^{\mathrm{s}}, \mathcal{Y}_i^\mathrm{s}\}$, which allows us to train the main task branch with a cross-entropy loss:
\begin{equation}
\mathcal{L}_{cls}^\mathrm{s} = -\frac{1}{N_{\mathrm{s}}}\sum_{i=1}^{N_{\mathrm{s}}}\sum_{j=1}^{J}\mathcal{Y}_{i, j}^\mathrm{s}\\log(\Psi_m(\Phi(\mathcal{P}_i^{\mathrm{s}}))_j)
\end{equation}
where $\mathcal{Y}_{i, j}^\mathrm{s}$ represents the ground-truth one-hot labels and $\Psi_m(\Phi(\mathcal{P}_i^{\mathrm{s}}))_j$ is the predicted probability for the $j^{th}$ class.
When the initial adaptation is made,
point clouds from the source and target domains should be aligned to some extent.
In this case, techniques used in semi-supervised learning are now in their functioning state.
For example, GAST~\cite{zou2021geometry} employs the strategy of self-paced self-training (SPST)~\cite{lee2013pseudo,zou2018unsupervised} to further align the two domains by generating pseudo labels in the target domain using highly confident predictions.
We follow this strategy to squeeze more juice out of the source labels.
Let $\hat{\mathcal{Y}}_i^\mathrm{t}$ be the predicted pseudo labels, the loss function to perform the self-training is:
\begin{equation}\label{eq:sp_loss}
\mathcal{L}_{cls}^\mathrm{t} = -\frac{1}{N_{\mathrm{t}}}\sum_{i=1}^{N_{\mathrm{t}}}(\sum_{j=1}^{J}\hat{\mathcal{Y}}_{i, j}^\mathrm{t}\\log(\Psi_m(\Phi(\mathcal{P}_i^{\mathrm{t}}))_j) + \gamma|\hat{\mathcal{Y}}_i^\mathrm{t}|_1)
\end{equation}
Similarly, the first term in Eq.~\eqref{eq:sp_loss} is a cross-entropy loss between the target pseudo labels and the predictions, and the second term is used to avoid degenerate solutions that assign all $\hat{\mathcal{Y}}^\mathrm{t}$ as $0$.
We follow~\cite{zou2021geometry,zou2018unsupervised} to apply a two-stage optimization using Eq.~\eqref{eq:sp_loss}, where the pseudo labels are first computed using nonlinear integer programming. Then the branch $\Psi_m\circ\Phi$ is updated using the pseudo labels.
These two steps are performed iteratively to adapt between the source and target domains progressively.
The hyper-parameter $\gamma$ controls the number of selected target samples.
\subsection{Overall Loss} \label{subsec:all}
The overall training loss of our method is:
\begin{equation}\label{eq:loss}
\mathcal{L} = \mathcal{L}_I + \alpha\mathcal{L}_M + \beta\mathcal{L}_{cls}^\mathrm{s} + \mu\mathcal{L}_{cls}^\mathrm{t}
\end{equation}
Note, the self-supervised implicit representation learning can be pre-trained on point clouds to encourage faster convergence during adaptation, i.e., set $\beta, \mu=0$.
After pre-training the networks $\Phi\circ\Psi_s$ for learning geometry-aware implicits, together with loss terms of the classification task $\mathcal{L}_{cls}^\mathrm{s}$ and $\mathcal{L}_{cls}^\mathrm{t}$ can be added back to perform the joint domain adaptation.
\section{Experiments}
\begin{table*}[!t]
\centering
\scalebox{0.88}{
\begin{tabular}{lccc||ccccccc}
\toprule
\bf{Methods} & Adv. & SLT & SPST & M$\rightarrow$S & M$\rightarrow$S* & S$\rightarrow$M & S$\rightarrow$S* & S*$\rightarrow$M & S*$\rightarrow$S & Avg.\\
\midrule
Supervised & & & & 93.9 $\pm$ 0.2 & 78.4 $\pm$ 0.6 & 96.2 $\pm$ 0.1 & 78.4 $\pm$ 0.6 & 96.2 $\pm$ 0.1 & 93.9 $\pm$ 0.2 & 89.5 \\
Baseline (w/o adap.) & & & & 83.3 $\pm$ 0.7 & 43.8 $\pm$ 2.3 & 75.5 $\pm$ 1.8 & 42.5 $\pm$ 1.4 & 63.8 $\pm$ 3.9 & 64.2 $\pm$ 0.8 & 62.2 \\
\midrule
DANN~\cite{ganin2016domain} & \checkmark & & & 74.8 $\pm$ 2.8 & 42.1 $\pm$ 0.6 & 57.5 $\pm$ 0.4 & 50.9 $\pm$ 1.0 & 43.7 $\pm$ 2.9 & 71.6 $\pm$ 1.0 & 56.8 \\
PointDAN~\cite{qin2019pointdan} & \checkmark & & & 83.9 $\pm$ 0.3 & 44.8 $\pm$ 1.4 & 63.3 $\pm$ 1.1 & 45.7 $\pm$ 0.7 & 43.6 $\pm$ 2.0 & 56.4 $\pm$ 1.5 & 56.3 \\
RS~\cite{sauder2019self} & & \checkmark & & 79.9 $\pm$ 0.8 & 46.7 $\pm$ 4.8 & 75.2 $\pm$ 2.0 & 51.4 $\pm$ 3.9 & 71.8 $\pm$ 2.3 & 71.2 $\pm$ 2.8 & 66.0 \\
DefRec+PCM~\cite{achituve2021self} & & \checkmark & & 81.7 $\pm$ 0.6 & 51.8 $\pm$ 0.3 & 78.6 $\pm$ 0.7 & 54.5 $\pm$ 0.3 & 73.7 $\pm$ 1.6 & 71.1 $\pm$ 1.4 & 68.6 \\
\multirow{2}{*}{GAST~\cite{zou2021geometry}} & & \checkmark & & 83.9 $\pm$ 0.2 & 56.7 $\pm$ 0.3 & 76.4 $\pm$ 0.2 & 55.0 $\pm$ 0.2 & 73.4 $\pm$ 0.3 & 72.2$\pm$ 0.2 & 69.5 \\
& & \checkmark & \checkmark & 84.8 $\pm$ 0.1 & \textbf{59.8} $\pm$ 0.2 & 80.8 $\pm$ 0.6 & 56.7 $\pm$ 0.2 & 81.1 $\pm$ 0.8 & \textbf{74.9} $\pm$ 0.5 & 73.0 \\
\midrule
\multirow{2}{*}{Ours} & & \checkmark & & 85.8 $\pm$ 0.3 & 55.3 $\pm$ 0.3 & 77.2 $\pm$ 0.4 & 55.4 $\pm$ 0.5 & 73.8 $\pm$ 0.6 & 72.4 $\pm$ 1.0 & 70.0 \\
& & \checkmark & \checkmark & \textbf{86.2} $\pm$ 0.2 & 58.6 $\pm$ 0.1 & \textbf{81.4} $\pm$ 0.4 & \textbf{56.9} $\pm$ 0.2 & \textbf{81.5} $\pm$ 0.5 & 74.4 $\pm$ 0.6 & \textbf{73.2} \\
\bottomrule
\end{tabular}
}
\vspace{-0.2cm}
\caption{Classification accuracy (\%) averaged over 3 seeds ($\pm$ SEM) on the PointDA-10 dataset.
M: ModelNet, S: ShapNet, S*: ScanNet; $\rightarrow$ indicates the adaptation direction.
Adv.: adversarial domain alignment, SLT: self-learning tasks, and SPST: self-paced self-training.}
\label{tab:pointda_res}
\vspace{-0.2cm}
\end{table*}
To show that the implicits effectively encode geometries of point clouds and verify the importance of the proposed adaptive unsigned distance filed, we examine the implicit reconstructions in Sec.~\ref{subsec:implicit_res}.
To have a comprehensive understanding of both the effectiveness and limitations of the implicits learned from unconstrained point clouds for aligning the domains, we evaluate the whole pipeline for point cloud UDA on the classification task with two major datasets.
We report our results with and without self-paced self-training.
We compare to a list of recent state-of-the-art methods on unsupervised point cloud domain adaptation: DANN~\cite{ganin2016domain}, PointDAN~\cite{qin2019pointdan}, RS~\cite{sauder2019self}, DefRec+PCM~\cite{achituve2021self} and GAST~\cite{zou2021geometry}.
In addition, we report the results obtained from the same network trained in a supervised manner on the target domain (``Supervised'', upper-bound).
For reference, the network trained in the source domain but tested on the target domain without any adaptation is also included (``Baseline'', lower-bound).
\subsection{Datasets}
{\bf PointDA-10}~\cite{qin2019pointdan} consists of three widely-used datasets: ModelNet \cite{wu20153d}, ShapeNet \cite{chang2015shapenet} and ScanNet \cite{dai2017scannet}.
All three datasets share the same ten categories (bed, table, sofa, chair, etc.).
ModelNet contains 4183 training and 856 test samples, while ShapeNet contains 17378 training and 2492 test samples. ModelNet and ShapeNet are both sampled from 3D CAD models.
Unlike these synthetic point cloud datasets, ScanNet consists of point clouds from scanned and reconstructed real-world scenes.
There are 6110 training samples and 2048 test samples in ScanNet, and point clouds therein are usually incomplete because of occlusion by surrounding objects in the scene or self-occlusion in addition to realistic sensor noises.
We follow the data preparation procedure used in \cite{qin2019pointdan,achituve2021self,zou2021geometry}.
Specifically, all object point clouds in all datasets are aligned along the direction of gravity, while arbitrary rotations along the $z$ axis are allowed.
Moreover, the input point cloud with batching is a list of 1024 points, which are sampled with duplicative padding from the original point clouds and are normalized to a unit scale.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\linewidth]{figs/graspnet}
\caption{Point clouds from GraspNetPC-10 created with GraspNet~\cite{fang2020graspnet}. (a-b): RGBD and raw point cloud captured by Kinect and Realsense devices, respectively, and (c): Synthetic RGBD and point cloud. Segmentation masks are provided, as shown in the first row. The corresponding re-projected and cropped point clouds are visualized in the same color at the bottom.}
\label{fig:graspnet}
\end{figure}
{\bf GraspNetPC-10} In order to test the domain adaptation on sim-to-real and real-to-real and check how the adaptation copes with different types of sensor noise,
we introduce GraspNetPC-10.
It is created from GraspNet~\cite{fang2020graspnet} proposed for training robotic grasping on raw depth scans and corresponding reconstructed 3D CAD models of various objects.
As shown in Fig.~\ref{fig:graspnet}, we create GraspNetPC-10 by re-projecting raw depth scans to 3D space and applying object segmentation masks to crop out the corresponding point clouds. Meanwhile, we synthesize similar senses with the same objects and render the synthetic depth scans to re-project synthetic point clouds. Different from point clouds in PointDA-10, point clouds in GraspNetPC-10 are {\it not aligned}.
Raw depth scans in GraspNet~\cite{fang2020graspnet} are captured by two different depth cameras, Kinect2 and Intel Realsense, so we have two domains of real-world point clouds.
Following PointDA-10, we collect synthetic and real-world point clouds for ten object classes.
In the synthetic domain, there are 12,000 training point clouds.
In the Kinect domain, there are 10,973 training and 2,560 testing point clouds.
Similarly, in the Realsense domain, there are 10,698 training and 2,560 testing point clouds.
The real-world point clouds from the two devices are always corrupted by different noises, and there exist different levels of geometric distortions and missing parts, as observed in Fig.~\ref{fig:teaser} and Fig.~\ref{fig:graspnet}.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\linewidth]{figs/recres}
\caption{Visualization of resampled point clouds from the learned implicits. The left half shows test samples from the PointDA-10 dataset, and the right half shows test samples from GraspNetPC-10.
(a): input point clouds, (b-c): resampled point clouds without adaptive unsigned distance field (AUD) at $\epsilon=$3e-2 and $\epsilon=$6e-2, (d): resampled point clouds with AUD at $\epsilon=$3e-2. The inserted numbers are the Chamfer distances between the resampled and the input point clouds (a).
}
\vspace{-0.3cm}
\label{fig:recres}
\end{figure}
\subsection{Implementation Details}
Our experiments are conducted on servers with four GeForce RTX 3090 GPUs, and the networks are implemented within the PyTorch framework. For training, we use an Adam optimizer, with an initial learning rate $0.001$, weight decay $0.00005$, and an epoch-wise cosine annealing learning rate scheduler. We train all models for 200 epochs on PointDA-10 and 120 epochs on GraspNetPC-10 with a batch size of 32.
{\bf Network architecture} Following~\cite{zou2018unsupervised}, we choose the commonly used point cloud processing network DGCNN~\cite{wang2019dynamic} as the backbone for the encoder $\Phi$.
The implicit decoder $\Psi_s$ and the category classifier $\Psi_m$ are multi-layer perceptrons (MLP) with fully connected layers. Decoder $\Psi_s$ is a four-layers MLP $\{512, 256, 128, 1\}$ followed by ReLU activation function (to make the output distance always positive) and classier $\Psi_m$ is a three-layers MLP $\{512, 256, 10\}$ in view of 10 semantic classes.
{\bf Hyper parameters} We set $M = 3$ for searching nearest neighbor points when calculating our adaptive unsigned distance field (AUD). The radius for the random masking $r_m$ is sampled from a uniform distribution in the range of [0.1, 0.3]. The weights of loss terms are set to $\alpha = 100, \beta=1.0, \theta=1.0$ and we adjust them slightly for better convergence on different datasets.
\begin{table*}[!t]
\centering
\scalebox{0.92}{
\begin{tabular}{l@{\hspace{4.8ex}}c@{\hspace{4.8ex}}c@{\hspace{4.8ex}}c@{\hspace{4.8ex}}||c@{\hspace{4.8ex}}c@{\hspace{4.8ex}}c@{\hspace{4.8ex}}c@{\hspace{4.8ex}}c}
\toprule
\bf{Methods} & Adv. & SLT & SPST & Syn.$\rightarrow$Kin. & Syn.$\rightarrow$RS. & Kin.$\rightarrow$RS. & RS.$\rightarrow$Kin. & Avg.\\
\midrule
Supervised & & & & 97.2 $\pm$ 0.8 & 95.6 $\pm$ 0.4 & 95.6 $\pm$ 0.3 & 97.2 $\pm$ 0.4 & 96.4 \\
Baseline (w/o adap.) & & & & 61.3 $\pm$ 1.0 & 54.4 $\pm$ 0.9 & 53.4 $\pm$ 1.3 & 68.5 $\pm$ 0.5 & 59.4 \\
\midrule
DANN~\cite{ganin2016domain} & \checkmark & & & 78.6 $\pm$ 0.3 & 70.3 $\pm$ 0.5 & 46.1 $\pm$ 2.2 & 67.9 $\pm$ 0.3 & 65.7 \\
PointDAN~\cite{qin2019pointdan} & \checkmark & & & 77.0 $\pm$ 0.2 & 72.5 $\pm$ 0.3 & 65.9 $\pm$ 1.2 & 82.3 $\pm$ 0.5 & 74.4 \\
RS~\cite{sauder2019self} & & \checkmark & & 67.3 $\pm$ 0.4 & 58.6 $\pm$ 0.8 & 55.7 $\pm$ 1.5 & 69.6 $\pm$ 0.4 & 62.8 \\
DefRec+PCM~\cite{achituve2021self} & & \checkmark & & 80.7 $\pm$ 0.1 & 70.5 $\pm$ 0.4 & 65.1 $\pm$ 0.3 & 77.7 $\pm$ 1.2 & 73.5 \\
\multirow{2}{*}{GAST~\cite{zou2021geometry}} & & \checkmark & & 69.8 $\pm$ 0.4 & 61.3 $\pm$ 0.3 & 58.7 $\pm$ 1.0 & 70.6 $\pm$ 0.3 & 65.1 \\
& & \checkmark & \checkmark & 81.3 $\pm$ 1.8 & 72.3 $\pm$ 0.8 & 61.3 $\pm$ 0.9 & 80.1 $\pm$ 0.5 & 73.8 \\
\midrule
\multirow{2}{*}{Ours} & & \checkmark & & 81.2 $\pm$ 0.3 & 73.1 $\pm$ 0.2 & 66.4 $\pm$ 0.5 & 82.6 $\pm$ 0.4 & 75.8 \\
& & \checkmark & \checkmark & \textbf{94.6} $\pm$ 0.4 & \textbf{80.5} $\pm$ 0.2 & \textbf{76.8} $\pm$ 0.4 & \textbf{85.9} $\pm$ 0.3 & \textbf{84.4} \\
\bottomrule
\end{tabular}
}
\vspace{-0.1cm}
\caption{Classification accuracy (\%) averaged over 3 seeds ($\pm$ SEM) on GraspNetPC-10. Syn.: Synthetic domain, Kin.: Kinect domain, RS.: Realsense domain. Our models achieve the best performance over all settings.}
\label{tab:graspnet_res}
\end{table*}
\subsection{Implicit Reconstruction} \label{subsec:implicit_res}
We show the resampled point clouds from learned implicit representations for analyzing the quality of the self-supervised geometry-aware implicits. Once the implicit encoder-decoder ($\Psi_s\circ\Phi$) is trained, given an input point cloud $\mathcal{P}$, we randomly sample $200,000$ points $q_s \in \mathbb{R}^{200000 \times 3}$ in the unit cube and calculate their unsigned distances conditioned on the original point cloud with our trained networks $d^{q_s} = \Psi_s(q_s, \Phi(\mathcal{P}))$. By setting a distance threshold $\epsilon$, we can choose the subset $\tilde{q_s} \subset q_s$ s.t. $d^{\tilde{q_s}} < \epsilon$ for visualization. If the distance field $f_{\mathcal{P}}$ is a good approximation of the underlying geometry, then $\tilde{q_s}$ will be similar to the input point clouds when $\epsilon$ varies.
In Fig.~\ref{fig:recres}, we compare our resampled point clouds with the input point clouds and the resampled point clouds from implicits learned without using our adaptive unsigned distance field, i.e., directly using the distances to the nearest neighbor but with a fixed clamping value.
As observed, the resampled point clouds with AUD can preserve the underlying geometry well. However, using the same $\epsilon$, the resampled point clouds without using AUD (``w/o AUD'') are much inferior, meaning the learned implicits distort the geometry information.
Moreover, we report the Chamfer distance between the resampled point clouds and the input.
Fig.~\ref{fig:recres} (c) shows the best resampled one for implicits learned without AUD (``w/o AUD'').
One can see that ``w/o AUD'' needs to apply a much larger $\epsilon$, i.e., two times larger than what is needed by ``AUD'', but the resampled point clouds are still severely deformed and exhibit lots of missing.
These results demonstrate that our adaptive unsigned distance field is critical and effective in the proposed implicit representation alignment module.
\subsection{Unsupervised Domain Adaptation}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{figs/tsne}
\caption{t-SNE~\cite{van2008visualizing} visualization of the output from our point cloud backbone on the Kinect domain (source) and the Realsense domain (target), which shows that the alignment through the implicits is effective, i.e., point cloud implicits from the target align better with the source ones after the adaptation (top vs. bottom). Different classes are displayed with different colors.}
\label{fig:tsne}
\end{figure}
Table.~\ref{tab:pointda_res} and Table.~\ref{tab:graspnet_res} show the comparisons between our method and other state-of-the-art on PointDA-10 and GraspNetPC-10, respectively.
For PointDA-10, we follow~\cite{zou2021geometry} and report performances on six different settings including ModelNet (M) $\leftrightarrow$ ShapeNet (S), M $\leftrightarrow$ ScanNet (S*) and S $\leftrightarrow$ S*.
We find that methods utilizing self-learning tasks generally perform better than methods based on adversarial training, especially on ``synthetic to real'' settings.
Compared to other self-leaning-based methods, our method (w/o SPST) excels on four settings and the average performance. After adding self-paced learning, our method competes with the most recent state-of-the-art method GAST~\cite{zou2021geometry} on PointDA-10.
Compared to RS~\cite{sauder2019self} and DefRec+PCM~\cite{achituve2021self} that both use reconstruction-based self-learning tasks, our method again achieves better performance.
For GraspNetPC-10, our method outperforms the others with a significant margin before and after adding self-paced learning. One can observe a substantial decline of GAST~\cite{zou2021geometry} on GraspNetPC-10. It indicates that classification-based self-learning tasks may not be robust for different kinds of datasets.
The local alignment method proposed in PointDAN~\cite{qin2019pointdan} now performs better than on the PointDA-10 dataset. DefRec+PCM~\cite{achituve2021self} ranks similarly. Our method achieves the highest score across all settings, whether with or without SPST.
It is also worth noting that SPST is effective on all datasets, both GAST~\cite{zou2021geometry}, and our method improves with SPST. However, ``GAST+SPST'' is still worse than ours without SPST, which again shows the effectiveness of the proposed geometry-aware implicits for aligning domains with realistic sensor noise.
We also visualize the 1024-dimension latent codes in the implicit space using t-SNE~\cite{van2008visualizing}. As seen in Fig.~\ref{fig:tsne}, without domain adaptation, features of different classes in the target domain are mixed-up (e.g., class 1 and 5, class 2 and 3), and the overall distribution is different from that in the source domain. After adaptation, the distribution of the features in the target domain becomes similar to the source one and shows clear clusters.
\section{Discussion}
It is challenging to align point clouds while maintaining a correct correspondence in terms of semantics without the target labels.
However, we show that a simple alignment via the proposed implicit space training can be quite effective for the current unsupervised domain adaptation benchmarks on point clouds.
Our method achieves state-of-the-art performance on two benchmarks covering varying factors affecting the point cloud geometry within the data collection pipeline.
We hope our method can serve as a ground where low-level geometric distortions or variations are learned away so one can focus on high-level shape variations that are also generative factors for domain gaps.
This would require a carefully designed dataset with controllable disentangled elements of geometric variations and is out of the scope of our current work.
{\small
\bibliographystyle{ieee_fullname}
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Myrsky project goes on Christmas break
Keskiviikko 23.12.2020 - Tuesday Club member
Merry Christmas 2020 and New Year 2021! The Tuesday Club Member.
The hammering and tinkling of the Myrsky project have stopped in the restoration space of the Finnish Aviation Museum. It is time to have a break and rest over Christmas. We will be back at work on January 5th.
Because of the coronavirus pandemic this year has been very exceptional for the Tuesday Club. Naturally it has also been exceptional for Finland and for the whole world.
In March we had to terminate all our activities. Since June it has been possible to continue the restoration of VL Myrsky II (MY-14), but with only a handful of club members working at a time. The restoration of the Caudron C.50 aircraft and the reparation of the PZL SM-1SZ helicopter blades have been on a break all the time.
In spite of the small task force, a lot has been accomplished in the Myrsky restoration work. The Myrsky's wing structures have been completed and now the lower sides of the wings have been painted with undercoat paint. After the Christmas break the undercoat painting will continue on the upper sides of the wings. The Tuesday Club team has worked in close co-operation with the Finnish Aviation Museum and the Finnish Air Force Museum.
I hope sincerely that the coronavirus pandemic will be blocked next year and the whole Tuesday Club team can return to work at the Finnish Aviation Museum. Hopefully in the autumn 2021, at the latest, it will be possible to continue the restoration work on the historically valuable aircraft with the full task force.
We wish you a peaceful Christmas time and a better New Year 2021!
Photo: Lassi Karivalo.
Translation: Erja Reinikainen.
Kommentoi kirjoitusta. Avainsanat: aviation history, restoration, VL Myrsky, MY-14, MY-5
Myrsky?s test wing is built to be wing for MY-5
Tiistai 22.12.2020 - Tuesday Club member
During the corona virus pandemic, the work in the Tuesday Club has concentrated on the restoration of VL Myrsky II (MY-14) and the number of workers has been limited to only a few at a time. The finalisation of the Myrsky-project's test wing has been on a break. Now both wing halves of the MY-14 are in the undercoat painting phase and carpenters are available for other work. The team decided to continue building the test wing and to get it ready.
The test wing is the root section, 2.5 meters long, of the Myrsky's starboard wing, which was built to test and model the construction of the actual wing for the Myrsky. The main emphasis was on testing how the landing gear area is built, and the landing gear installed. Testing was useful because the original drawings were inadequate and inconsistent and several times it was necessary to discuss how to proceed. Sometimes the built section had to be dismantled. Due to the test wing the mistakes were not repeated when the actual Myrsky wing was built.
The test wing will be useful also in the future. Originally the aim was to place the test wing on display at the museum as an example of a wooden wing structure in a WWII fighter, designed by the State Aircraft Factory. This is why the upper surface of the wing will be partly covered with transparent plexiglass so that the interesting inner structures and equipment can be seen.
The saying goes that the appetite grows while you are eating, and this is what happened with the test wing. A fuselage frame of Myrsky MY-5, in poor condition, is available for the test assembly of the MY-14 wing. The Tuesday Club team decided to restore this fuselage frame so that it could be placed on display with the test wing assembled on it. Then the museum visitors could see the mixed-structure Myrsky, having a fuselage frame made of steel tube and the wooden root part of the starboard wing. The MY-5 fuselage frame has already been restored, but it has been cut behind the cockpit and is now waiting for the test assembly of the MY-14 wing
But the appetite kept on growing. The team decided to build a 1.0-meter section of the Myrsky's port wing to go with the 2.5-meter section of the starboard wing. The port wing section includes the wheel well of the landing gear. Neither of the wing sections is ready but they have been joined with a steel plate and the preliminary wing assembly for the MY-5 can already be seen. We can start talking about the Myrsky MY-5 wing instead of the test wing, because it has already done its task as the test item in the MY-14 restoration.
And this is not all yet! The Tuesday Club team is dreaming – when the Myrsky MY-14 restoration has been completed – of continuing with the MY-5 fuselage frame and assembling original parts and equipment, which are available, and building the missing vertical and horizontal stabilizers and elevators. The MY-5 would be a great example for the museum visitor of how the mixed-structure Myrsky has been built. How far the MY-5 fighter will eventually be built and assembled remains to be seen in the future.
The Tuesday Club team will continue with the construction of the Myrsky's wing sections, 2.5 m on the starboard side and 1.0 m on the port side. The starboard side section structures are almost ready, and the landing gear has once already been assembled on it. Work continues with the leading edge. The leading edge ribs have now been glued on the front spar. The edge strip connecting the ribs to each other has been glued on the tips of the ribs as well as the battens between the ribs.
The construction of the wheel well on the port wing section is ongoing. The plywood ring on the upper edge of the wheel well was built from narrow plywood strips which were glued into a pack on a mould. The wheel well cover is fastened on this ring. The wheel well cover will be made from aluminium sheet by metal spinning. For the 2,5 m starboard wing the cover has already been made.
The wheel well walls are also being covered with plywood. The billets for the walls were cut from 1,2 mm plywood sheet. The billets will be fitted into place and finalised. The short wing section on the port side will not have the landing gear as the longer starboard wing section does, the wheel well will remain empty.
Photos: Lassi Karivalo.
Undercoat painting of Myrsky?s wing is well under way
Maanantai 14.12.2020 - Tuesday Club member
Due to the corona virus pandemic only a handful of Tuesday Club members can continue the restoration of VL Myrsky II fighter (MY-14), but some results have been achieved, nevertheless. The undercoat painting of the wing halves is under way at the moment. Before that the edges of the compartments of the wing were covered with protective linen fabric strips. This phase has been described in the previous blog.
The undercoat painting of the Myrsky's wings will be done in two phases and with two different kinds of undercoat paint. First the cleaned and ground plywood surface is painted with alkyd paint which contains aluminium flakes. The undercoat paint used in this phase is TEMALAC AB 70 alkyd paint which contains aluminium flakes, and the shade of the paint is RAL 90006, white aluminium. This paint is used to fill the grain structure on the plywood surfaces and to make the plywood surface very smooth. The flake paint is applied on the plywood surfaces twice and after both rounds the surface is honed with sandpaper. The aluminium flake alkyd paint is covered with a layer of slightly darker grey paint, Teknos Oy's adhesion primer Futura 3, with the shade RAL 7005 Mouse Grey. After this the surfaces of the wing are ready for the final camouflage painting.
Left photo: Heikki Kaakinen.
The undercoat painting of the wings was started by painting the compartments for the landing gear, the flap and the aileron. They were painted with the Teknos Oy paint Futura 3. The work was mainly done with a small roller, but the tight spots were painted with a brush. When the compartments had been painted, the lower surfaces of both wings were painted with the Tikkurila Oy TEMALAC AB 70 aluminium flake alkyd paint. The alkyd paint was spread with a dense foam rubber roller. The areas where precision was required were painted with a brush. The possible drippings were wiped with a cloth.
When the flake paint had dried, the surfaces were honed with Imperial P 150 sandpaper, pressing only with fingertips. This means that a piece of cork or wood was not used with the sandpaper. Fingertips are flexible and a good tool when honing painted surfaces. Before the honing work the sandpaper was torn into four strips and each strip was folded into a pack of four layers. Now the sandpaper was a square with four layers. The honing was done by moving the sandpaper with fingertips in circles on the painted plywood surface. When one of the layers of the sandpaper pack got blocked with honing dust, the next layer was taken into use. When the honing work was ready, the dust was removed from the plywood surface with a vacuum cleaner and by wiping the surface with a damp cloth.
Right photo: Jouni Ripatti.
Photo: Heikki Kaakinen.
When the lower sides of the wings had been painted twice with the aluminium flake alkyd paint, the painting work on the upper sides of the wings was started. The flake paint was spread with a roller. When both sides of the wing have been painted with the flake paint, they will be ready to be painted with the TEKNOS Oy Futura adhesion primer. On top of this the Myrsky's green-black camouflage painting will eventually be added.
Photos: Lassi Karivalo, except othervise mentioned.
Kommentoi kirjoitusta. Avainsanat: aviation history, restoration, VL Myrsky, MY-14
Reinforcing fabrics for plywood seams on Myrsky's wing
Sunnuntai 22.11.2020 - Tuesday Club member
The C-type wing of the Myrsky-fighter (VL Myrsky II) has plenty of plywood seams which are reinforced with linen fabric strips. The purpose of the fabric is to protect the plywood seams from problems caused by moisture. There are fabric strips e.g. on the edges of the aileron and flap compartments, on the leading edge of the aileron, on the edges of the landing gear wells and on the wing tips. The tips of the horizontal stabilizers have also been reinforced with fabric.
Originally the Myrsky's wings didn't have these fabric strips to prevent moisture problems. The aircraft factory started planning these for the wing in late 1944 when the whole Myrsky series had already been built. The reason for this were the problems in the plywood seams, caused by moisture, and due to which some glued plywood seams on the wings had opened. In the same modification the structure of the rear wing spar was changed, and the Myrsky C-wing was created. The first series production aircraft the A-type wing. It was replaced by the B-wing, then the B1-wing and eventually in 1945 by the C-wing. The construction of the VL Myrsky II meant continuous improvements and modifications.
The structure of the Myrsky's A-, B- and C-wings differed mainly in the rear spar. In the A-wing the rear spar became thinner at the seam of the aileron and the flap, thinning steeply towards the wing tip. This point proved to be weak and led to the breaking of the wing. This is why the rear spars of the A-wings were reinforced by adding a strengthening piece on the side of the spar – and this is how the B-wing was created. This was not a good solution either. That is why a new rear spar was designed, it thins linearly towards the wing tip. The wing with the new kind of rear spar was called the C-wing and the aim was to replace the Myrsky's B-wings with the new type. Only about a dozen C-wings were built during 1945 and they were only installed on a couple of Myrskys before the aircraft type was written off. Obviously the MY-5, the MY-41 and the MY-50 were the only ones which had the C-wing.
The MY-14, which is under restoration, had a B-wing. In December 1945 the aircraft got a new wing at the Lentorykmentti 1 (Flight Regiment 1) Depot in Pori. It is not known what kind of wing was installed. Around the same time a C-wing was installed on the MY-5. So it can be speculated whether the MY-14 got a C-wing, too.
The MY-14 will get a C-wing when it is being restored. The steering committee of the Myrsky restoration project decided that building a C-wing for the MY-14 is both justifiable and appropriate. This solution was chosen mainly because the drawings that are available for the restoration work represent the C-wing and, excluding the rear spar, the wing resembles the B-wing. It can be speculated, as shown above, whether the MY-14 had a C-wing in 1945 or not.
During the autumn the Tuesday Club team has fastened linen fabric strips on the plywood seams of the wing, i.a. the edges of the flap compartment on the port wing. The work was started by cutting the saw-toothed fabric strips for all the edges of the flap compartment from 105 g/m2 linen fabric using "zig-zag" scissors.
The fabric strips will be glued on the edges of the compartment with grey Futura 3 undercoat paint, hue RAL 7005, which is the same undercoat paint with which the whole MY-14 wing will be painted. The fabric strips are glued on the clean plywood surface before the plywood covering is painted. This was not the procedure with the fabrics on the leading edge of the aileron. The team was too eager to get the work done and the first layer of undercoat paint had already been applied on the ailerons before the fabric strips were remembered!
Before gluing, areas matching the width of the fabric strip were drawn on the edges of the flap compartment. Above the plywood edge the paint was applied with a roller and below the edge with a small brush. The layer of paint was the adhesive surface for the fabric. Now the saw-toothed fabric strip could be pressed against the wet paint surface. When the fabric had been smoothed tightly against the painted surface with fingers, a layer of paint was applied on it with a roller and a brush so that the whole fabric strip was covered with paint.
At the moment the fabric strips have been installed on the edges of the flap and aileron compartments, the edges of the landing gear well and the seams of the wing tip on the port and starboard wings. The strips protect the seams from moisture. The surfaces of the strips will be honed smooth before the wing gets its undercoat paint.
The both wing halves of the Myrsky MY-14 are practically ready for the undercoat painting of the wings' plywood surfaces. The plywood surfaces have already been honed smooth. Before applying the first layer of undercoat paint, the plywood surfaces will be washed to remove any grease and oil. A solution containing 50% Sinol and 50% water will be used.
Photos: Lassi Karivalo
About wing root fairings on VL Myrsky II
Maanantai 9.11.2020 - Tuesday Club member
The wing root fairing, which covers the joint of the Myrsky's wing and fuselage, consists of two sections. The longer rear section of the fairing covers the joint of the fuselage and wing from the trailing edge of the wing up to the level of the front spar. From there the front section of the fairing continues all the way around the leading edge of the wing. The front and rear sections of the wing root fairing are joined together with a butt joint. An aluminium plate has been riveted on the lower surface of the rear section of the fairing and this plate is pushed under the edge of the front section of the fairing. The wing root fairing is fastened on the wing and the fuselage mainly with flange nuts. The front fuselage of the Myrsky is covered with thin aluminium plate.
The front section of the wing root fairing bends around the leading edge of the wing to the lower surface of the wing, up to the edge of the wheel wells. The fairing also covers the bottom of the front fuselage between the engine stand and the wheel wells. The wing root fairings on the port and starboard side are joined together in the middle of the fuselage. Both fairings can be unfastened separately. The outer edge of the port side fairing borders with the oil cooler air intake. The cooler exhaust opening is located on the starboard wing root fairing on the lower surface of the leading edge.
Not a single original Myrsky wing root fairing has been preserved for the restoration work. This means they had to be built. The Tuesday Club team decided to make first the front sections of the wing root fairings based on the original drawings. Manufacturing them was a complicated and multi-staged process. The front section of the wing root fairing curves in many directions and it had to be made from three different aluminium plates, which were modified separately and welded together.
The material for the fairings was 1 mm thick aluminium thin plate, from which the billets for the front sections of the fairings were cut. To shape the fairing sections into the desired shape, a mould of the front fuselage and wing connection was made from battens and plywood. Separate moulds were needed for the port and starboard wing root fairings. The billets of the fairings from aluminium thin plate and the two shaping moulds were delivered to Flanco Oy, where the shaping was done. From there the shaped fairing sections, still attached to the shaping mould, were delivered to GA Telesis Engine Services Oy for welding. From there the welded front sections of the wing root fairings came back to the Tuesday Club.
The shaped front sections of the wing root fairings are now waiting for the Myrsky MY-14 wing halves, built by the Tuesday Club, to be joined with a steel plate into a uniform wing. Then the wing will be tested on the fuselage frame of the MY-5. The fuselage frame of the MY-5 has already been sandblasted and painted. At the same time, the front sections of the port and starboard wing root fairings can be tested, whether they need additional shaping before their details can be finished.
Leading edge of Myrsky's aileron is strengthened with a fabric inset
Tiistai 3.11.2020 - Tuesday Club member
The plywood covered aileron of Myrsky-fighter (VL Myrsky II) has a reinforcing linen inset on the leading edge. The purpose of this inset is to protect the seam of the leading edge batten and the plywood sheets which are fastened on it. The strip of linen fabric is fastened on the leading edge with paint during the undercoat painting. The leading edge is not uniform, there are two notches in it for the hinges of the aileron. A reinforcing strip of linen fabric will be installed also on the edges of the wheel wells on the wings.
The MY-14 fighter, under restoration at the Tuesday Club of the Aviation Museum Society, has now the reinforcing strips on the leading edge of its ailerons. However, a small mishap occurred. Both ailerons had already been painted with the first undercoat paint, which is used for smoothing the surface, when the team remembered the reinforcing linen strip. Therefore, the undercoat paint, already honed smooth, had to be removed on the leading edge, over the width of the reinforcing fabric strip. The plywood surface was roughened so that the adhesive surface would be better for fastening the linen fabric on the leading edge with undercoat paint.
For the ailerons' leading edge, a strip 15 cm wide was cut from 105 g/m2 linen fabric. A sawtooth pattern was cut on the fabric edges zig-zag scissors to make the adhesive surface of the fabric better than that of a straight edge. On airplane covering fabrics frayed edges were used for this purpose in the 1920s. When the fabric edge was frayed, the warp was unwoven over some centimetres' distance. The sawtooth practice became common in the 1930s.
Before assembling the fabric on the leading edge, the aileron was supported to an upright position. Then a good layer of paint was spread on the leading edge, using a small foam rubber roller, over the width of the linen fabric strip. The fabric strip was placed carefully on the leading edge and pressed as tightly as possible against the wet paint.
When the fabric had fastened on the paint, adhesive undercoat paint was brushed on it. The Tuesday Club team made sure there was a sufficient layer of paint all over the fabric strip and that the sawtooth edge of the fabric had fastened tightly on the leading edge surface.
Both of Myrsky's ailerons were treated in the similar manner. When the paint has dried, the painted surface on the fabric will be carefully honed smooth, making sure that the honing will not break the surface of the reinforcing fabric. If needed, an additional layer of paint will be added on the fabric and honed. Finally, the aileron will get the paint finishing with the green-black paint scheme.
Translation: Erja Reinikainen
Undercoat painting of Myrsky's ailerons and horizontal stabilizer
Torstai 22.10.2020 - Tuesday Club member
The VL Myrsky II (MY-14) is under restoration in the Tuesday Club and the surface finishing work on its horizontal stabilizer and ailerons has continued. The finishing work will be done in three phases. First the cleaned and ground plywood surface is painted with alkyd paint which contains aluminium flakes, this is repeated with grinding and honing in between. This paint is used to fill the grain structure on the plywood surfaces of the horizontal stabilizer and ailerons, and to make the plywood surface sealed and smooth. The undercoat paint used by the Tuesday Club team was TEMALAC AB 70 alkyd paint which contains aluminium flakes, and the shade of the paint is RAL 90006, white aluminium. The undercoat painting work has been described in an earlier blog.
The aluminium flake alkyd paint will be covered with another undercoat layer of alkyd paint, which will form the adhesive surface for the actual finishing paint on Myrsky's horizontal stabilizer and ailerons. The Tuesday Club team used TEKNOS adhesion primer Futura 3, with the shade RAL 7005 Mouse Grey. The Futura 3 adhesion primer is slightly darker than the TEMALAC, so it was easy to see where the new layer of paint had not yet been applied. Two layers of paint were applied on the horizontal stabilizer and ailerons, with grinding and honing between the layers. The painting work was done using a polyester roller. The horizontal stabilizer and ailerons now have their undercoat paint.
The ailerons are not ready for the finishing paint yet. A strengthening strip of linen fabric needs to be fastened on the ailerons' leading edge. The strengthening fabric reaches around the leading edge so that it protects the seam of the plywood surfacing and the solid wood leading edge. The strip of linen fabric will be glued on the leading edge using the adhesive primer. Before fastening the fabric on the leading edge, the paint surface on the leading edge has been ground away over the width of the fabric strip.
Tuesday Club activities are suspended until end of the year
Tuesday Club activities have been suspended due to the Covid 19 coronavirus pandemic since March. However, the restoration work on VL Myrsky II (MY-14) has been continued by a small task force, so that the continuation of the project is ensured.
The coronavirus pandemic seemed to subside during the summer, and we thought it would be possible to relaunch Tuesday Club activities in the autumn. Then we would know whether the positive development of the epidemic is permanent and there would not be a second wave. But this is not what happened. The epidemic started to spread again in August and by September it was obvious that the second wave had arrived.
Therefore the Finnish Aviation Museum and the Aviation Museum Society came to the conclusion that the risk for the Tuesday Club members, many of whom were in the risk group, to catch the virus in the museum's restoration space would be too high if the activity continued. Therefore, it was decided that the Tuesday Club activities are suspended until the end of the year. This was very disappointing decision for the Club members, but it was the right and understandable thing to do.
The Finnish Aviation Museum and the Aviation Museum Society decided, nevertheless, that the restoration activities of VL Myrsky II (MY-14) can be continued by the Tuesday Club at the museum, but with restrictions. The aim of these measures is to ensure that the Myrsky restoration project can be completed next year. The restrictions mean that only about half a dozen Club members can be working simultaneously on Myrsky in the restoration space – naturally wearing masks and other protective items also, if needed.
The Finnish Aviation Museum decided to suspend also other activities run by volunteers (such as guided tours) until the end of the year. The justification is to protect the risk group members in the common spaces of the museum. The recommendations, given on 21.9.2020 by the Ministry of Education and Culture and the Finnish institute of health and welfare, say that the risk group members should avoid close contacts during the epidemic, and therefore it is not advisable to take part in public events or gatherings, or activities arranged in public spaces.
Photo: Lassi Karivalo
About electrical equipment in Myrsky's wing
The VL Myrsky fighter has 24V direct current electrical system. In the wings the equipment that need electricity are the navigation light, in the port wing the heating for the pitot tube (i.e. speed detector), the releasing system of the auxiliary fuel tank or bomb and the release indicator of the auxiliary fuel tank or bomb. These devices are operated from the cockpit with switches. Indirectly electrically operated are also the landing gear and the flaps, located in the wings. These are moved with operation bars, linked to electrical motors located in the fuselage. The ailerons move mechanically when the stick is moved.
The auxiliary fuel tank or bomb is released from the hanger on the wing when the hook-shaped lock/release latch of the hanger is released. The fuel tank or bomb has a bracket for hanging the tank/bomb on the latch. The latch is released electrically using a solenoid. The fuel tank or bomb release indicator functions so that when the tank or the bomb is on the hanger, the indicator's spring pin is pressed into the indicator. When the tank or the bomb is released, the pin is freed and causes an electrical impulse in the indicator and forwards the information to the signal lamp in the cockpit.
The electrical cables in the wing are led through ducting tubes inside the wing. Power cables from the fuselage are led into the joint box which is fastened on the wing rib next to the auxiliary fuel tank / bomb hanger. From the box there are separate power cables to the different devices. There are two round hatches on the upper surface of the wing for maintenance work that is needed on the hanger and the joint box.
The Tuesday Club team is installing the electrical cabling on the port wing. The cables have been pulled through the ducting from the wing root to the joint box. The cables for the navigation light, pitot tube and auxiliary fuel tank / bomb hanger are being installed into the joint box. At the moment the cabling is still loose and crawling through the maintenance hatch on the wing top, but some cables have already been installed to their proper places.
Myrsky's wing root walkway
Lauantai 26.9.2020 - Tuesday Club member
At the wing root of VL Myrsky II, on both sides of the fuselage, on the wing's plywood covering there is a strengthening strip of plywood. This makes it possible to walk on the wing when doing maintenance work. Very often these strengthened strips, which allow stepping on the wing, are clearly visible from the wing surface or painted black at least. However, this is not the case in the Myrsky, where these plywood strips have the same black and green paint scheme as the upper side of the wing.
Upper photo: The photo archive of the Finnish Air Force Museum.
On the Myrsky drawings these strengthening plywood strips are called "step plates". They are fastened on the wing's plywood covering with glue and screws. The aluminium wing root fairings, which protect the connection of the fuselage and the wing, are fastened onto the fuselage side edge of the step plates with flange nuts and screws.
The step plates for the Myrsky's wings are being made at the Tuesday Club. The semi-finished plates were cut from 2 mm thick plywood according to the original drawing. The actual plywood covering of the wing is 4 mm thick at the wing root. The step plates for the port and starboard wing are mirror images. The position of the fastening screws and wing root fairing flange nuts were marked on one of the step plates, following the original drawing. The pencil marks were highlighted with a metal punch. The punch markings will make it easier to drill the holes into their exact positions.
After this the mirror image step plates were placed accurately on top of each other and fastened with pieces of double-sided tape. In this way the holes for the screws can be drilled simultaneously on both plates and exactly in the same position on the plates. The holes for the fastening screws were drilled with a 3 mm drill bit.
For the flange nuts on the wing root fairings, holes for the nuts had to be made on the edge of the step plate. These holes must match the shape and size of the flange nut and be slightly oval-shaped because the nut is sunk through the step plate, all the way to the level of the wing surface.
First three holes were drilled into the plywood with a 3 mm drill bit, these holes help when the oval hole is made into the step plate. The small holes were used when making the holes larger with a column-type drill, using drill bits of two different sizes. The work continued by shaping the holes into their correct size and shape with a chisel and a file.
The following phase will be to fit the step plates into their places on the top surface of the port and starboard wing and to fasten them with glue and screws. The step plate has already been preliminarily fitted to the root of the starboard wing.
Photos: Except separately otherwise mentioned, Lassi Karivalo.
Myrsky MY-5's fuselage frame was taken to sandblasting and painting
Keskiviikko 16.9.2020 - Tuesday Club member
The earlier blogs describe how the Myrsky MY-5's rusted fuselage frame will be straightened, sandblasted and painted before it can be used for the wing test assembly of Myrsky MY-14.
The text in the box: Original paint under the bracket.
The MY-5 fuselage frame has been fastened on a sturdy metal frame, the assembly jig, outside the restoration space at the Finnish Aviation Museum. It has been straightened on the jig and the remaining parts and equipment have been dismantled. Most of these are different band-shaped fastening brackets. The fuselage frame was cut behind the cockpit, so that it became easier to handle and it can be brought into the museum's restoration space for the test assembly of the MY-14 wing. Later the fuselage frame parts will be welded together so that it will be the whole Myrsky MY-5 fuselage frame.
The fastening brackets, which were still attached on the MY-5 fuselage frame, are important when the original paint and its tint are being investigated. When the brackets were unfastened, some well-preserved original paint of the fuselage frame was revealed. The painted surface has been protected from rusting, hidden under the brackets.
The original paint surface was cleaned by grinding it lightly, so that the tint of the paint became even more intensified. On both sides of the cleaned painted area a paint test was made, using two different shades of grey Isotrol paint (linseed oil alkyd paint). This was necessary so that the team could choose the appropriate tint of the undercoat paint for the fuselage frame. The tints of the tested grey paint were RAL 7005 ja RAL 7042.
On Tuesday, September 15th, the shortened MY-5 fuselage frame was loaded on a trailer, using a stacker. The frame will be taken to Auto- ja Teollisuusmaalaamo Oy, a paint workshop in Vantaa. The Tuesday Club team tried also to fit the rusted engine frame on the trailer, but it wasn't possible. The engine frame will be transported to the paint workshop later.
At Auto- ja Teollisuusmaalaamo Oy the fuselage frame will be sandblasted and painted with undercoat paint. When the sandblasting has been done, the clean metal surfaces will be covered with clear varnish, which protects the frame from rusting. The grey Isotrol undercoat paint will be applied on top of the varnish. The MY-5 fuselage frame, with its undercoat paint, will return to the Finnish Aviation Museum in a couple of weeks.
City of Vantaa politicians visit Finnish Aviation Museum and learn about aircraft restoration
Maanantai 7.9.2020 - Tuesday Club member
City of Vantaa politicians visited the Finnish Aviation Museum for "practical training" as a part of the Politician's museum internship campaign, coordinated by the Finnish Museums Association. The politicians who visited the museum were city council member Minna Räsänen (sdp), member of parliament, city council chairman Sari Multala (kok) and member of parliament, city council member Jussi Saramo (vas). Räsänen and Multala were present at the museum, Saramo attended via remote access.
As a part of the "practical training", the cooperation of the museum and Aviation Museum Society was introduced to the guests. The Tuesday Club volunteers' work was illustrated in the museum's restoration space, although the Tuesday Club has officially been terminated because of the corona virus pandemic.
From left to right: Pia Illikainen, Minna Räsänen ja Matti Patteri.
In the restoration space museum Intendent Matias Laitinen told the guests about the volunteers' role at the Aviation Museum. Tuesday Club project manager Lassi Karivalo introduced the aircraft restoration activities of the Tuesday Club. The leader of the VL Myrsky II (MY-14) restoration project Matti Patteri introduced the history of domestic aircraft industry and explained how museum aircraft, and more specifically the Myrsky, are restored in practice. The Myrsky project was launched in 2013 and up to now the restoration team has spent about 30 000 working hours. Most of the work has been done by volunteers. The Myrsky restoration also includes cooperation with vocational colleges, e.g. Vantaa Vocational College Varia and Tavastia Vocational College in Hämeenlinna, and naturally work contribution from the Aviation Museum staff. The visitors learned that the main motivating factors for the volunteers' participation are the sense of community, the good team spirit and the interesting work items.
From left to right: Sari Multala and Matti Patteri.
Due to the corona virus pandemic situation, the museum's volunteer activity has been terminated since March. In the future the working procedures will have to be reorganized. A large volunteers' gathering can't be organized yet, and not for some time – and in the social aspect this is very unfortunate. During the corona virus time the Aviation Museum Society has arranged for its members, and to all who are interested, remote access to presentations about aviation history. The museum has produced videos and audio tours with its partners. However, by now everybody is very eager to get back to work and to meet the volunteer friends at the museum. The aim is to relaunch the Tuesday Club activities in October, at some extent. The corona virus situation in Finland will define whether this is possible or not.
Photos: Mia Kunnaskari.
Undercoat painting of MY-14?s horizontal stabilizer
Perjantai 28.8.2020 - Tuesday Club member
Before the painting work, the port and starboard halves of the VL Myrsky II's (MY-14) horizontal stabilizer were fastened on an assembly jig, made of square steel tube. On the jig the stabilizer can be turned around as needed in the various phases of the surface treatment process.
The seams of the plywood covering on the horizontal stabilizer had been spackled and ground smooth before starting the undercoat painting work. The plywood surfaces were washed, using a water-Sinol-solution, containing 50% of water and 50% of Sinol. This is a normal procedure for removing grease from the surfaces before painting. After the wash, the surfaces were dried with a heat blower and sanded three times, with 240 abrasive paper.
Photo: Jouni Ripatti.
The MY-14's horizontal stabilizer will be painted in several phases. First an undercoat paint is spread on the plywood surface, and the actual paint is applied after this. There will be several layers of both paints. The undercoat paint and the actual surface paint are both alkyd paints, as were the original paints when the Myrsky fighters were being built in the 1940's at the State Aircraft Factory (VL) in Tampere.
The undercoat paint, which is used for the plywood surfaces of the MY-14 under restoration, is TEMALAC AB 70 alkyd paint which contains aluminium flakes. The shade of the paint is RAL 90006, white aluminium. The aluminium flakes will fill the grain structure on the plywood surface, which will make the plywood surface smooth and sealed when it is ground and honed.
Now the upper and lower surfaces of the horizontal stabilizer have been painted once with the undercoat paint containing aluminium flakes. For the first layer the paint was made thinner, using Ruiskuohenne 1032, and the paintwork was done with a polyester roller (mini paint roller). The roller produced a very even layer. When the paint had dried, the surfaces were ground smooth for the second layer. The next undercoat paint layer will be done using paint, which is not thinned, and which is slightly darker in colour than the first layer. The darker shade will show clearly which areas have been painted for the second time and which haven't. If a third layer of undercoat paint is needed, it will be darker than the second one.
An unexpected break in the painting work appeared, when the darker shade of TEMALAC AB 70 undercoat paint wasn't available due to holidays. The Tuesday Club team will have to wait for the mixing and delivery of the darker paint before continuing the undercoat painting work on the horizontal stabilizer.
The MY-14's horizontal stabilizer will eventually be painted according to the standard paint scheme of the Finnish Air Force in the 1940s: the upper surface will have a green and black pattern and the lower surface will be painted light blue.
Photos (expect otherwise separately mentioned): Lassi Karivalo
About Myrsky MY-14 oil cooler, wing root fairings, landing gear doors and MY-5 fuselage frame
Maanantai 24.8.2020 - Tuesday Club member
When the VL Myrsky II (MY-14) port wing half covering was being finished in the restoration space of the Finnish Aviation Museum, also other Myrsky-project items were under work.
Before the Corona virus pandemic interrupted the work, the Tuesday Club team had completed most of the engine's oil cooler supply and exhaust air duct parts. They have been made from 1 mm thick aluminium plate. Now it was time to start building, according to the original drawing, the airflow control device into the supply air duct. The control device consists of a circular frame, inside which there are three flaps, which move on their axles. The air flow into the oil cooler can be controlled continuously with these flaps. The air flow control is done from the cockpit.
The parts needed for the air flow control device were cut from 1 mm thick aluminium plate, using a laser cutter, at ProLaser Oy. Holes for the rivets, which are needed in the assembly work, were drilled on the semi-finished control flaps. The parts of the air flow control device were also chromated to prevent corrosion.
The assembly work of the oil cooler control device was started at the circular frame. When the round frame had reached its final shape, the ends of the axles of the three control flaps were riveted on the frame. There is a slot on the end of the axle, into which the edge of the aluminium flap fits. The three flap plates fitted perfectly on the axle slots. The following phase will be to rivet an aluminium profile on the flaps, which locks the flaps on the axles.
Some metal work has been needed when making the aluminium fairings, which cover the connecting point of the Myrsky's fuselage and wing. The work was started on the fairing which covers the wing's leading edge and the fuselage. The complicated fairing shape consists of three pieces of aluminium plate, which are welded together. The preliminary fairing parts were cut from 1 mm thick aluminium plate. A wooden last was made for shaping the aluminium sheets into the desired shape. Two lasts are needed, one for the starboard wing and the other for the port wing. The preliminary fairing parts were shaped into their final shape at Flanco Oy, using the last. The shaped fairing parts were welded together at GA Telesis Engine Services Oy.
The work on the landing gear enclosure doors of the test wing has also continued, including the door fastened on the wheel hub and the oleo strut door. The doors had been nearly finished already in March. The remaining aluminium profile stiffeners were riveted on the inner surface of the doors. The doors were now ready to be installed. The bulging parts or the "bumps" on the outside of the oleo strut door have also been welded into place. These bumps are needed for the landing gear retraction fork which is fastened on the oleo. These bumps have also been welded on the preliminary oleo door plates of the actual Myrsky wing's landing gear. However, the actual wing doors have not been chromated yet and their stiffeners have not been fastened.
The Myrsky MY-5 fuselage frame, which was brought from Tikkakoski, has also been under work. The parts, which were still fastened to the frame, have been unfastened and now the frame is "clean" for sandblasting. The fuselage frame will be sandblasted at Auto- ja Teollisuusmaalaamo Oy, where it will also be painted grey.
A pile of different kinds of brackets and parts was dismantled from the MY-5 fuselage frame. Some of the parts had to be unfastened with some violence, using an angle grinder for the work, because the badly rusted fastening bolts couldn't be opened. The rust will be removed, and the parts will be painted. All the parts will be needed when the MY-5 fuselage frame is restored to be the exhibition fuselage, which illustrates the structure of the Myrsky fighter. The 2,5-meter test wing, built during the Myrsky project for testing the wing construction procedure, will eventually be fastened to the MY-5 fuselage.
Before that, the MY-5 fuselage frame will be used for test assembling and fitting the Myrsky MY-14 wings. In the fitting phase the MY-5 fuselage frame will be lowered on the MY-14 wing, built by the Tuesday Club team. The wing will be locked onto the fuselage with the four wing attachment brackets or bolsters. The wing parts will be fastened with steel plates and the whole wing can be test fitted in about a month's time.
MY-14´s port wing half is covered with plywood
Sunnuntai 9.8.2020 - Tuesday Club member
The work on VL Myrsky II (MY-14) is continuing at the Tuesday Club, but not at full speed yet, though. Work is being done e.g. on the plywood covering of the starboard wing leading edge and trailing edge, and the sides of the flap cavity. The engine area is also under construction: the air supply duct is being fitted into the opening in the NACA-ring, the cowling under the engine is being built and the gills controlling the air flow through the air duct on the oil cooler are being installed. However, the main emphasis is in getting the port wing plywood covering to the same situation with the starboard wing. The work is almost ready.
The leading edge of the port wing is now almost covered with plywood. The strips of plywood covering the leading edge were first soaked in water so that they could be bent around the tapering leading edge. When the plywood had dried into shape, it was glued on the leading edge. The seams of the plywood were spackled and honed smooth. Only small areas around the pitot-tube and at the wing tips have not been covered.
Right photo: Heikki Kaakinen.
The trailing edge area on the upper surface of the wing, between the rear spar and the trailing edge, has also been covered with plywood. First the plywood sheets of the trailing edge area were cut into the accurate size, then covered with protective varnish on the inside and then fitted into place. When the plywood sheets had settled into their places and their bevelled edges met correctly, the sheets were glued into place. Before the inner parts of the wing were permanently covered from sight, the Tuesday Club members who were involved in the work, wrote their names on the inside of the wing as a greeting for the future generations.
The gluing of the plywood sheets on the upper surface of the trailing edge was secured by fastening pieces of plywood on the wing ribs and the connecting battens with a staple gun. The permanent gluing of the plywood on the trailing edge batten was ensured by fastening a long row of clamps on the edge. When the glue had dried, the clamps and the stapled pieces of plywood were removed. The marks of the staples that were left on the plywood surface and the seams of the plywood sheets were spackled and honed smooth.
It was a difficult task to fasten a 10-centimeter strip of plywood onto the upper edge of the flap cavity on the port wing. A similar, but narrower strip, has already been fastened on the lower edge of the flap cavity. These strips will form an aerodynamic compartment to the front of the flap space. The leading edge of the flap will sink into this space.
Photos: Heikki Kaakinen.
The port wing covering work had now reached the phase where the starboard wing already was. The surface work, i.e. smoothing and honing which are needed before the painting of the plywood surfaces, can now be started. Some painting has already been done: the open inner areas on the top surfaces of the wing roots on both wings have been painted with grey alkyd paint. These open areas at the wing root will not be covered with plywood, because the areas will be left under the Myrsky fuselage when it is lowered on the wings.
Photos (expect if separately otherwise mentioned): Lassi Karivalo
About the fuselage frame of VL Myrsky MY-5
Tiistai 4.8.2020 - Tuesday Club member
When the restoration project of VL Myrsky was launched, three damaged Myrsky fuselage frames were available at the Finnish Air Force Museum in Tikkakoski. The rear section has been violently cut away from all three fuselage frames. The least damaged fuselage frame of MY-14 was chosen for the restoration project. This is how the project became the restoration project of the VL Myrsky MY-14 fighter. The MY-14 fuselage rear section, which has been cut off, has disappeared and therefore the rear section of MY-9 was chosen to be connected to the MY-14 fuselage frame. Also a fourth Myrsky fuselage frame (MY-10) exists, in a private collection, and there have been negotiations about purchasing it for the Myrsky project.
The MY-14 fuselage with its engine and control equipment is mainly restored at the Finnish Air Force Museum in Tikkakoski. The main task of the Aviation Museum Society's Tuesday Club is to build in Vantaa in co-operation with the Finnish Aviation Museum the other parts of the fighter, such as the wings with all their equipment, the parts of the tail and to build the parts for the nose, e.g. the front engine cowling, NACA-ring and the engine cowlings on the sides.
Photo: Jorma Laakkonen.
The restoration project started in 2013 and the development of the project has been followed all the time on homepages of the Aviation Museum Society and Myrsky Project, and also in yearly journals. In Tikkakoski the restoration of MY-14 fuselage is already quite far. In Vantaa the Tuesday Club has prepared the tail parts and the wings to the point where their surfaces can be honed smooth and painted. The aim is that the restored – but static – MY-14 can be placed on display during the year 2021.
The second best Myrsky fuselage from Tikkakoski, i.e. MY-5, was also taken into the restoration project with its separate rear fuselage and engine mounting. The MY-5 fuselage frame has its own important role in the Myrsky restoration project. Some useful parts have been dismantled from the fuselage frame and repaired in Tikkakoski for the needs of the MY-14 fuselage frame restoration. On the other hand, the fuselage frame of the MY-5 will be used by the Tuesday Club for the fitting of Myrsky's wings. The wing positioning and fastening will be tested on the fuselage frame by the Tuesday Club before the wings are sent to Tikkakoski to be installed on the fuselage of MY-14.
Additionally, an even bigger role has been planned for the MY-5 fuselage frame. It will be used to build a demo-fuselage of the Myrsky-fighter to illustrate the metal and wood structure of the aircraft. The 2.5-meter piece of the starboard test wing and the 1-meter piece of the port test wing will be fastened on the fuselage frame. Before building the actual wings, the test wings have been built to test the methods how to build the complicated root of the wing with the landing gear and how to lock the wing halves to each other with steel plates.
The rear fuselage of MY-5, brought from Tikkakoski to Vantaa to the Finnish Aviation Museum, has already been used in the Myrsky-project. The badly rusted rear fuselage frame was straightened in a jig, specially made for the purpose, and sandblasted and painted, using grey alkyd paint. After this the rear fuselage has been used by the Tuesday Club to fit and support the horizontal and vertical stabilizers when they are being built.
The fuselage frame and engine mounting of MY-5 were brought from Tikkakoski to the Finnish Aviation Museum in Vantaa on July 21st, 2020. The MY-5 fuselage was transported by the Myrsky-project's PR-man, on his holiday trip to Ostrobothnia. On Friday 10.7 on his way there he took the trailer and "some Myrsky items" to Tikkakoski. On Tuesday 14.7 a group of Tuesday Club members travelled to Tikkakoski and moved, with some assistance from the Air Force Museum, the MY-5 fuselage frame and engine mounting on the trailer to wait for the transportation. The unusual load travelled successfully to the Finnish Aviation Museum on 21.7. Fortunately, the traffic was rather quiet and it wasn't raining.
When the fuselage frame was outside the museum, the cargo straps were unfastened. Then the MY-5 fuselage was lifted from the trailer with a forklift and placed on the ground to wait for cleaning and repair. The badly rusted and partly bent and damaged fuselage frame will be straightened and cleaned by sandblasting, straightened and repaired, and then painted grey. Behind the cockpit some frame pipes have been bent and some are even missing.
The assembly jig, which was made from square steel pipe for the straightening of MY-14's fuselage frame, was re-assembled and modified for the MY-5 work. The jig was fastened on the four brackets by the cockpit on the MY-5 fuselage. These brackets are used for fastening the wings with four fastening bolsters on the wings.
The MY-5 fuselage frame, now fastened on the assembly jig, will be measured carefully and straightened. The places of some parts or their fastening brackets will be defined and photographed before they are dismantled for cleaning. Also the rusty engine mounting, brought from Tikkakoski, will be cleaned by sandblasting and painted.
When the fuselage frame has been cleaned and painted, the rear fuselage frame (which has already been restored earlier) will be welded to it. Then the MY-5 fuselage frame will be ready to serve as the fitting frame or assembly jig for the MY-14 wings, and later as the basis of the Myrsky demo fuselage. The MY-5 demo fuselage will be restored so that the structures remain visible in the way they were left visible when building the short test wings. The restoration of this demo fuselage is not the main priority at the moment, the work will be started when the restoration of the MY-14 has been completed.
Photos (except if separately otherwise mentioned): Lassi Karivalo
Cockpit canopy for Gnat model
Torstai 19.4.2018 - Member of Tuesday Club
The Finnish Aviation Museum has donated the models of BEA Hawk, Folland Gnat, Saab Draken and MiG-21BIS to the Aviation Museum Society's Hawk Experience Center. The large wooden models were found in the storage room of the museum. The models need some repairs which will be completed in the Tuesday Club before the Hawk Experience Center season 2018 begins in May. The repairs of the BAE Systems Hawk Mk.51 model have already been described in an earlier blog.
After the Hawk model the next target was the Folland Gnat F.1 model. This model wasn't badly damaged but the cockpit canopy was missing. In addition to some minor repairs the canopy had to be made.
To make the new canopy, first a wooden model of the canopy had to be made. The cockpit was emptied by removing the pilot's seat and the instrument panel. This was necessary in order to be able to fit the wooden canopy model, made of balsa, in its place in the cockpit. At the same time the measurements were taken of the line where the canopy meets the edge of the cockpit: a metal frame will be fitted on the sides and at the rear end of the canopy.
A piece of balsa was carved slowly to match the shape of the canopy and the correct size by fitting it into the cockpit every now and then. Photographs and 3d-drawings of the Gnat were used to define the shape of the canopy. The wooden model began to take shape when it was gradually and carefully carved and buffed out and regularly fitted into the cockpit. Now the model for the canopy was ready.
Photos: Jorma Laakso.
Before making the actual canopy, the metal frame for the edge of the canopy was made. The wooden model was used to dimension the frame. The location of the frame was marked on the model in red lines. The image of the frame was drawn on a 0.3 mm thick brass sheet and cut off. The thin strips of brass sheet were honed to match the shape of the frame on the lower edge and the rear section of the cockpit and fitted on the balsa model. When the metal parts seemed to fit nicely into place they were soldered together. Now the metal frames were ready to be installed on the actual canopy.
The windscreen arc in front of the canopy was made of 0.8 mm thick plywood, laminating four layers of plywood on top of each other. The laminated arc was milled and buffed out into shape. Then a windscreen made of transparent plastic sheet was attached, using a friction-type connection.
Photo: Jorma Laakso.
The making of the actual canopy was started, using the balsa model and a mould. A heated plastic sheet is pressed against the opening in the mould, using the balsa model for pressing. The mould was made from 4 mm plywood with an opening matching the shape and dimensions of the outer edges of the cockpit.
A piece of 10x10 cm was cut from a 0.5 mm thick GAG-PET sheet. The sheet of plastic was attached by its edges on top the opening in the plywood mould. The sheet of plastic was heated using a hot air blower. When the surface of the plastic was soft enough, the sheet was pressed evenly into the opening in the mould using the balsa model. The stoppers on the sides of the balsa model prevented the model from pressing the sheet of plastic too deep into the mould.
It took about one minute for the plastic sheet to cool down, then the balsa model was removed. A bulge had formed under the mould, shaped like the balsa model. This was the rough form of the canopy. The GAG-PET sheet was removed from the mould and the bulge, i.e. the preformed canopy was cut off.
Photos: Jorma Laakso & Lassi Karivalo.
Now the metal frames (at rear end and lower sides of the canopy) and the windscreen arc could be attached to the canopy. The canopy was placed on the balsa model where the red markings showed the exact place of the frame. When the metal strips had been placed exactly on the marked lines, they were glued in place using UHU POR contact glue. Now the extra plastic material below the frame could be cut off. When the canopy was placed on top of the cockpit to test how it fitted, the result looked good. UHU POR contact glue was used to glue the windscreen arc to the canopy. Then the canopy frame and the windscreen arc were painted dark green.
The last phase of work was to attach the new canopy on the Gnat model. The canopy was placed on the edges of the cockpit and attached with Plastic Padding. When the seam had dried it was honed smooth and painted dark green. The Folland Gnat F.1 model had a new canopy and is now ready to be placed on display in the Hawk Experience Center.
Photos: Unless separately mentioned, Lassi Karivalo.
Kommentoi kirjoitusta. Avainsanat: aviation history, Tuesday Club, restoration, Folland Gnat, aircraft model
Hawk model to Hawk Experience Center
Keskiviikko 21.3.2018 - Member of Tuesday Club
A wooden model of BAE Systems Hawk Mk 51 jet trainer was found in the storage room of the Finnish Aviation Museum. The partly damaged model is approximately 50 cm long and it represents HW-310 which was in use in the Finnish Air Force. HW-310 belongs to the series of Hawks which were assembled by the Valmet Aviation Industries between 1980 and 1985. The real HW-310 has been withdrawn from use and stored. The State of Finland will donate it to an aviation museum in Switzwerland (Clin d'Ailes - Musée de l'aviation militaire de Payerne) and where it will probably be transported already this spring.
The museum has no information about the builder of the HW-310 model. Originally the model has been attached by its landing gear on a white plywood board which has been hanging on the wall. So the model has been a wall decoration.
The model is very well made and its proportions match the original aircraft. The cockpit has seats and its main equipment – also made of wood. The instruments on the panel have been painted, decals haven't been used. The model has been painted according to the old camouflage scheme (black/green/grey) used in the 1980s, which suggests that the model was probably made about twenty years ago. Today the planes are painted all grey with small national insignia.
When the Hawk model was found, the idea of bringing it on display in the Hawk Experience Center of the Aviation Museum Society was brought up. The center has on display the cockpit section of HW-314, which belonged to the same Hawk production series as the real HW-310. The model of HW-310 could be used to illustrate what a whole Hawk looks like. There is an existing showcase where the HW-310 could be on display. An additional advantage is that the HW-310 has a similar black/green/grey camouflage scheme as the HW-314 in the Experience Center.
The Finnish Aviation Museum decided to donate the HW-310 model to the Hawk Experience Center and the Tuesday Club set out to work. The model was moved from the storage to the club workshop. The model had some damages: the nose landing gear was broken, one of the horizontal elevators was loose, the seam of the vertical elevator and the fuselage was broken and the cockpit canopy was loose and slightly broken.
Fortunately the damage wasn't serious and repairing wasn't difficult. The right side horizontal elevator was glued into place, the seam between the vertical stabilizer and fuselage was re-glued and painted. The split wheel on the nose landing gear was repaired. The canopy of the cockpit didn't quite settle in its place and some modifications were needed on the canopy and the rear seat in order to have the canopy in the correct position.
As the model had originally been hanging on the wall, it hadn't been balanced to be standing on its landing gear. When placed on a horizontal surface the rear-weighted plane's tail is pointed downward. Therefore an additional support has to be added under the rear fuselage when the model is in its showcase.
The Tuesday Club team discussed whether the HW-310 identification marking should be changed to match the identification of HW-314 in the Experience Center. The decision was negative. It would have been difficult to modify the last number of the identification marking from a four to a zero so that the change couldn't be noticed on the original paint. So the original appearance of the model was respected and no changes were made.
The HW-310 model is made of solid wood and is very robust but it still needs a good storage and transportation box. A new box was made from plywood and wooden battens. The box has a hinged lid and it was painted with clear varnish. The box and the lid have supporting pads which were cut to measure and support the weight of the model during storage and transport.
The Hawk model and its box are ready. The model was placed in the existing showcase to see how it fits there. And it looks great!
Fine, isn't it?
Now HW-310 is waiting for the summer exhibition round of the Hawk Experience Center to begin. The first event is already in the horizon: The "Military Aviation as a Profession" –event will be arranged by Satakunta Air Command in Pirkkala, Tampere on May 15th.
Photos (unless otherwise individually mentioned): Lassi Karivalo.
Kommentoi kirjoitusta. Avainsanat: BAe Hawk, model, Hawk Experience Center, restoration, Tuesday Club
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More Days of Note - 2 years
Again, these may be boring for you to read but I love documenting them here. It's funny that the older our kids get, the less we keep track of or write down. I guess we are just so busy?! I am trying to remember as much as possible, though, that's why I keep these here. I really have to make an effort now-a-days to even jot them down, but I'm so glad I do!
11.23.14 Said "Look at all the people," then "they are all friends" to a crowd of people in her book. Said "that was close" after we did a ring around the rosey and fell down together.
11.30.14 Me: "What is that?" "That's a . . ." Me: How many are there?" "There are . . ." Amazing.
Favorite sayings that she says - "What's that?" Then, she answers herself "That is a . . , you know what that is."
12.2.14 Started saying "goodnight mommy" when I tuck her in, after I said "goodnight Lemon." Hugs things, and us, and says "I love you" on her own. Asks me to get in the crib with her, "Mommy get in" and tries to open mouth kiss me or lick me and we giggle the whole time, it's the best part of my day - cuddling in the crib with her. Says "another blanket" and "too" for additional things. Her vocabulary is totally amazing.
12.3.14 Said "be careful" tonight and yesterday. When she farted she told Ian her "butt burped." She scooted her naked booty on the floor and loved the sound it made, was cracking up, and proceeded to do it even though it drove me nuts. She told me she had to go pee tonight "pee pee in the potty." So awesome.
12.4.14 Took her bear this morning and turned to me, with a high voice, and said "Hi mommy, how you doing" like she was making the bear talk to me! Yesterday she put her princess down on the couch, laid her down and tickled her. So cute!
12.8.14 "What's wrong baby?" "Nothing." Yesterday we saw two babies in the same swing, twins, and Lemon said she wanted two Lemon's in her swing with her. Then, she said there was only "1 Lemon" and "1 Mommy" and "1 Peepaw." "Peepaw hold you?" She asked for my dad to hold her yesterday, that's big!
12.16.14 Strings whole sentences together like "Hey that's my sock, give me that." Answers questions that we ask her.
12.19.14 "Mommy, can I take a bath?" "It's so beautiful" when things are beautiful or pretty.
12.23.14 "Where's the car, where'd it go? There he is, I found him (or the car)." Hurt her lip yesterday on a bucket at the zoo, milking a fake cow. It was so sad.
12.25.14 I asked if she had a good Christmas and she said, "uh huh" and then I asked if she had fun opening presents and she said "uh huh" and then I asked which of her presents she liked the best and she said "Nani and Grandpa and Amanda" and then I told her she was my favorite present and kissed her.
She, just today, started answering me back instead of just repeating what I say and she will leave somewhere and then come home and talk about it, telling us about it. She will always say that she had fun doing something. She also told Ian today that Cyrus was mean to her, "Daddy, daddy Lemon you ok, you ok Lemon, Cyrus, Cyrus." It was sad, he shot her with a nerf gun (her cousin). I will not be able to handle it when kids are mean to her, I just won't it will be too sad.
12.28.14 Asks us to run her tummy and back. She still talks about holding baby chicks from Halloween (agh!) and the bats from Jumanji that we mistakenly let her watch one day, weeks ago.
1.1.15 We all sledded down a hill today and she loved it, I did it with her and it was so much fun, I wish we had pictures or a video. I want to remember it, her first time playing in the snow and her first time sledding was with her mama. She loved the snow in Sedona, wanted us to pull over so she could play in it as soon as she saw it, and was up at 6:45am the next morning asking for her gloves/boots too. She was a good sport, would play for an hour or so and then wanted to go in and get warm - me too. I have wanted to wake up in snow for so long, it was so much fun!! And, there was so much snow!
1.5.15 She said "best day ever." She also pretended like her fork and spoon were friends, at dinner, and said "hey spoon, want to go to the park" like the fork was talking. So crazy, this pretend play.
1.9.15 Started saying "too" like "Lemon go night night too." When I tuck her in and climb in her crib, I rub her tummy and sing A Few of My Favorite Things and tonight she sang along with me and rubbed my tummy and sang it to me too. It was the sweetest thing.
When I try to get out to go to bed too, she says "leave it in" and pushes my face down to stay and gets a pouty face when I get up to really go. I steal so many kisses and cuddles in her crib, it's the best. Will I always be able to cuddle her like this?
1.17.15 Went to drop Peepaw off and she said, "Um, so what are we going to do now?" My mom asked her what she was doing over Skype and she said, "Just sitting on the couch watching a movie Nani."
1.19.15 "Look at my curly hair" to Selma on FaceTime. How did she know she has curly hair? She also knows she has blue eyes (like Cinderella) and mommy has green eyes and daddy has brown eyes.
1.20.15 If she can't find something then she finds it she says, "there he is." It's so cute! She's done it forever and it's one of my favorite things. Even when her straw was in her cup and then I popped it out she said, "there he is!" So funny.
Stayed with peepaw for the second time (first was when she was really little) today and did great!
1.27.15 Cutting the dogs gunk out of his eyes and Ian asks me if I have some little scissors. Lemon says "I'll get the green scissors out of the play doh" and she does, totally play doh scissors that don't cut. Ha.
1.30.15 Pretends that her feet and hands are talking to each other. Makes them do voices (like 2 hands talking or 2 feet talking) and "run away" and moves them away from each other. Does it with her feet while I'm rubbing them in bed.
Says "um" or "I don't know" or both together - so cute.
Calls me "mom" for a whole week, and Ian "dad" for half this week and then we called Nani on the phone and she started calling her "Nan." She must have gotten it from a movie or book, before that it was "mama" from a book.
2.10.15 "Are you cutting my apple mommy?" "Don't eat that Linus, that's mine." Today she finally could tell me how old she was and when her birthday is, I got it on video!! "Jew-why secend."
2.13.15 "Turn off (on) the light mama, it's dark in here."
2.15.15 "See all the water mama, Lemon likey to swim in the water. Does mama like to swim in the water?" "You going potty mama? I went potty too in daddy's bathroom." These sentences came out of nowhere, it's amazing!!
3.3.15 Said "ewwww gross" when she saw me take out my bite guard in the morning after waking up.
Says "Jesus Christ!" Said it for the first time when I was trying to take her romper off at Nani's house. Lays in our bed with the covers, then sits up and says "Oh my god!" Makes me do it too, I have no idea where it's from but it's funny!
3.17.15 New words like "though" and "first." "Go home first, go to the beach though." Love it also when she says "too" like "I love you too mama." Love when she answers us when we ask how she's doing, "I doing good, how you mama?" Or "I doing good too." Such a sweet soul.
Cried the first night of our San Diego trip, was so overtired and out of her element. She knew she wasn't home but thought she was going home to go to bed. She said over and over she wanted to go to "Memon's house" and sleep in "Memon's bed." She was so upset and I just held her and told her it would be ok. What else can you do?
3.20.15 Said the word "hate" today. "I hate robots." Asked her what the name of a magnetic bunny was and she said "blue bunny" and I thought it was so cute, she has never given a name to anything herself before.
3.21.15 Did an evil laugh when my dad asked for some of her ice cream. She said "no that's mine, muwhahahaaa."
I asked if she needed help climbing up the stairs and she said, "no thanks, I can do it myself." Everything is "I do it myself" now-a-days.
Asked how I was and I said I was happy. I asked how she was and she said she was mad and then I asked why and she said "I'm mad at mommy." Then I asked why and she said "mommy makes you brush your teeth, go potty and go night night." Ha.
Jonah was mad and she said "Jonah sad" and then said "Lemon happy" and signed happy.
4.3.15 Jonah spilled water from their tea party and she said, "Oh. My. God. Jonah." It was so funny, we just cracked up!
4.20.15 "Mommy take a bath too. Let's take a bath together." Together! Such a big concept!
Posted by Allison at 7:00 AM
Lemon's Rad & Rue Photo Shoot
Moms on Mondays - Meggie & Annie from Sleeper Hero
Moms on Mondays - Ashley from Bright Eyed Baby Shop
Mmmm! Episode 7: Sugar Cookies & Special Guest
Moms on Mondays - Lauren from Vivi's Jewels
Mmmm! Episode 6: Cookie Nutella Pie!
Moms on Mondays - Katie from The Doo Bob Shop
Moms on Mondays - Kylie & Liz from Wee Rascals
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{"url":"http:\/\/math.stackexchange.com\/questions\/62917\/prove-that-maxfn-gn-thetafngn","text":"Prove that $\\max(f(n),g(n)) = \\Theta(f(n)+g(n))$ [duplicate]\n\nPossible Duplicate:\nhow can be prove that $\\max(f(n),g(n)) = \\theta(f(n)+g(n))$\n\nHow to prove $max(f(n),g(n)) = \u0398(f(n)+g(n))$?\n\n-\n\nmarked as duplicate by \uff2a. \uff2d., JavaMan, anon, Zev ChonolesSep 8 '11 at 20:43\n\nIs $\\Theta(f(x))$ the same as $O(f(x))$ of \"big-O\" fame? \u2013\u00a0 robjohn Sep 8 '11 at 19:08\n@robjohn: $f = \\Theta(g)$ if both $f = O(g)$ and $g = O(f)$. \u2013\u00a0 JavaMan Sep 8 '11 at 19:16\nLet $h=\\max(f,g)$. Then $f+g\\le 2h$ and so $f+g=O(h)$. If $f$ and $g$ are positive, then $h \\le f+g$ and so $h=O(f+g)$.","date":"2015-01-31 16:17:47","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\": 0.9392121434211731, \"perplexity\": 2379.859063020293}, \"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-2015-06\/segments\/1422122152935.2\/warc\/CC-MAIN-20150124175552-00051-ip-10-180-212-252.ec2.internal.warc.gz\"}"}
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In just the last week, 1,155 of our guests were big jackpot winners, raking in over $1,200 each! 62 of those jackpots were over $5,000, 18 were over $10,000 and we had 3 mega-big winners taking home more than $20,000! Head to The Island today for your chance to make it on our winners wall!
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{"url":"http:\/\/rudnik.eu1.pl\/7ltuay\/page.php?tag=ratio-examples-and-answers-b28db6","text":"This is the currently selected item. Step 1: Sentence: Jane has 20 marbles, all of them either red or Step 1: Assign variables: 10:20:60 is the same as 1:2:6. Example: What is the amount of sugar solution to be added if the ratio has to be 2:1? Ratio and Proportion Examples With Answers \u2013 \u2026 More Ratio Word Problems The angles of a triangle are in the ratio 1:3:8. Learn and free practice of ratio and proportion (arithmetic aptitude) questions along with formulas, shortcuts and useful tips. problem solver below to practice various math topics. X share = \u20b9 ( 1800 x 3\/8) = \u20b9 675. Find the measures of the three angles of this triangle. 3) The Odds Ratio: 4) After calculating the odds ratio, we observe a 3-fold difference in the prevalence rate (75% vs. 25%) change to a 9-fold difference in the odds ratio. Our mission is to provide a free, world-class education to anyone, anywhere. A. Example 2: Simplifying Ratios that have Different Units: Change the following ratio into the same unit ratio in its simplest form. Next lesson. Find the income of A and B. Clothing store A sells T-shirts in only three colors: red, blue and green. The first substance mentioned goes in the numerator, the second mentioned in the denominator. If a bag of the mixture RATIO: The ratio of two quantities a and b in the same units, is the fraction a\/b and we write it as a:b. If 6 : 9 : : y : 15, find the value of y? Solution. Step 1: Assign variables: A \"Concrete\" Example. Uncategorised; ratio and proportion examples with answers Hence we can say 8\/20 < 15\/20, Ratio and Proportion Examples With Answers\u00a0\u2013\u00a0Four Numbers in Proportion, Let a, b, c, d are four numbers said to be in proportion. ratio and proportion examples with answers. Writing Ratios Worksheet 2 RTF Writing Ratios Worksheet 2 PDF View Answers At a later date, they found 6 tagged The ratio adjusts the traditional P\/E ratio by taking into account the growth rate in earnings per share that are expected in the future. We welcome your feedback, comments and questions about this site or page. But a ratio can also show a part compared to the whole lot. In today\u2019s uncertain market, investors are looking for answers to help them grow and protect their savings. Embedded content, if any, are copyrights of their respective owners. A low inventory turnover ratio is dangerous. Some examples of rate include cost rates, (for example potatoes cost $$\\text{R}\\,\\text{16,95}$$ per kg or 16,95 R\/kg) and speed (for example, a \u2026 Financial ratios are usually split into seven main categories: liquidity, solvency, efficiency, profitability, equity, market prospects, investment leverage, and coverage. We have ratio worksheets suitable for all levels and abilities, from simple ratios up to more complex ratio problems. Debt to assets ratio c. Long-term debt to assets ratio d. Times interest earned ratio e. Assets to equity ratio. The colors are in the ratio of 3 to 4 to 5. On multiplying or dividing each term of a ratio by the same non zero number, we get a ratio equivalent to the given ratio, Both numerator and denominator of given fraction is multiplied by same nonzero number i.e 4, 3\/2 is not an equivalent ratio of 2\/3. Cross Multiply each ratio in the proportion. The first ratio is in the range of 81 and 82 while the second ratio is in the range of 80 and 81. The ratio of its length to its width is 5:2. All rights reserved. A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. Nominal scale is a naming scale, where variables are simply \"named\" or labeled, with no specific order. It indicates the financial health of a company rabbits in a sample of 2000. (1) Liquidity ratios a. Ratio and Proportion MCQ Quiz Answers The purpose of this Ratio and Proportion MCQ Quiz is to monitor the candidate progress and understanding of ratios, rates & proportions. Sharpe Ratio Definition. Both the numbers should be non-zero in order to make a meaning out of the comparison. Explanation. Example 2: Our worksheets contain engaging activities which make the topic of ratios entertaining and enjoyable, and our ratios worksheets are all supplied with answers to \u2026 They agreed to dived the profit in the ratio of 2 : 4. Let a, b, c, d are four numbers said to be in proportion. Find the ratio of 50 paisa to Rs. Let x = number of red sweets. Writing Ratios Worksheet 2 \u2013 With this 20 problem worksheet, you will practice writing ratios in three different ways. \"Part-to-Part\" and \"Part-to-Whole\" Ratios. Assign variables: When I saw this problem on Yahoo Answers, the first answer given was 2:1, the reverse of the correct answer. problem and check your answer with the step-by-step explanations. This is another word problem that involves ratio or proportion. Purplemath. If the store has 20 blue T-shirts, how many T-shirts does x = 15, The total number of shirts would be 15 + 25 + 20 = 60. Let x = amount of corn, Step 2: Solve the equation Linux. Let\u2019s study how algebra can help us think about ratios with more than two terms. Rs 3 = 3 x 100 = 300 paise Ratio of 50 paise to Rs. Some ratios are denoted in percentages and decimals as well. Mark gave half of his money to Fred. Answer: Number of litres of sugar solution in the mixture = (1\/(1+2)) *45 = 15 litres. PEG Ratio PEG Ratio PEG Ratio is the P\/E ratio of a company divided by the forecasted Growth in earnings (hence \"PEG\"). Solution: 100, 80, 120 B. b and c are called the middle terms or means. (i) Gross Profit Ratio (ii) Net Profit Ratio (iii) Return on Total Assets (iv) Inventory Turnover (v) Working Capital Turnover (vi) Net worth to Debt Sales 25,20,000 Other Current Assets 7,60,000 A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6. A comprehensive database of more than 24 ratio quizzes online, test your knowledge with ratio quiz questions. 3 \u00c3\u0097 120 = 4 \u00c3\u0097 x Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation.The exercise set will probably start out by asking for the solutions to straightforward simple proportions, but they might use the \"odds\" notation, something like this: Example #1: In a small business, 40 of the employees are men and 30 of the employees are women. Solution: Before. 2 \u00c3\u0097 x = 3 \u00c3\u0097 5 The best examples of ratio scales are weight and height. Sharpe ratio is the ratio developed by William F. Sharpe and used by the investors in order to derive the excess average return of the portfolio over the risk-free rate of the return, per unit of the volatility (standard deviation) of the portfolio.. Ratio and Proportion Examples With Answers, Find the numbers when their ratio and sum are given, Divide sum of money between two persons when ratio are given, Divide sum of money among 3 persons when ratio are given, Find the value of y when four numbers are in proportion, Find the HCF of both the numerator and denominator, Convert both the Fractions into Like Fraction:-, \u2013 Find the L.C.M of denominator of both the Fractions. Find the ratio of the amount of money Mark had left to the amount of money Fred had in the end. What is ratio of women to men? Let x = number of red shirts here a and d are called the extreme terms or extremes. Up Next. Simplifying Ratios . Inventory is NOT considered a quick asset. Solution: A recipe uses 5 cups of flour for every 2 cups of sugar. three-term ratios. Sharpe Ratio is a critical component for marking the overall returns on a portfolio. Try the given examples, or type in your own Make sure that you have the same items in the numerator and denominator. contains 3 pounds of rice, how much corn does it contain? However, earnings per share (EPS) may not be as intuitive for most investors. The ratio of income of two workers A and B are 3: 4. Jane, then John has how many more blue marbles than Jane? The ratio of the amount of money Mark had left to the amount of money Fred had in the end is 3:4. Sum of ratio terms = ( 3 + 4 + 1 ) = 8. would it costs to fill an 18 gallon tank? Solution. A lower ratio reflects dull business and suggests that some steps should be taken to \u2026 Cross Multiply The proportion of pears is 3 \u2026 In a ratio sum of numerator and denominator will give the number of parts. Ratio problems are word problems that use ratios to relate the different items in the Quick ratio = 0.81 c. Working capital ratio = 0.12 (2) Solvency ratios a. The ratio is also distorted if the investments don't have a normal distribution of returns. Example 1: In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. Personalized Financial Plans for an Uncertain Market. We know that, Product of means = Product of extremes, In the given numbers, we can say that 9 , y are means and 6 , 15 are extremes, Click here for Class 6 Chapterwise Explanation. \u2026 Our online ratio trivia quizzes can be adapted to suit your requirements for taking some of the top ratio quizzes. Answers is the place to go to get the answers you need and to ask the questions you want Example #2: The length of a rectangular garden is 20 feet and the width is 15 feet. Example 1: Find the numbers? Then, put a colon or the word \"to\" between the numbers to express them as a ratio. In the ratio a:b, we call a as the first term or antecedent and b, the second term or consequent. Archimedes received 1\/6 of the money. 6th Grade Ratios And Rates Worksheet Answer Key. Estimate the total number of rabbits in Bryer Lake National Park. The result is the ratio in its simplest form. Solution: The ratio of women to men is 30 to 40, 30:40, or 30\/40. And also show your work on Workspace in case of any rough work. Archimedes, Hypatia and Zeno shared a sum of money. Examples : 3:2 , 3:2:88, 3 to 2, 3 to 2 to 88. Khan Academy is a 501(c)(3) nonprofit organization. In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. Ratio Problems: Two-Term Ratios. There are 65 more sheep than pigs. then, a : b = c : d\u00a0 or a : b :: c : d here a and d are called the extreme terms or extremes. Let x = number of blue marbles for Jane The current ratio, also known as the working capital ratio, measures the capability of a business to meet its short-term obligations that are due within a year. Then you need to compare them using 1% range. Ratio review. Convert the ratio 60 : 35 in its simplest form. a. So, 45-15 = 30 litres of salt solution is present in it. For example, 163\/201 and 171\/211 are the given ratios and the range for both the numbers fall in the same range i.e 80% and 90%. that are running each of the operating systems. Learn in-depth with formulas, questions with answers, and tricks at BYJU\u2019S. Home \u00bb Maths \u00bb Ratio and Proportion Examples With Answers, Ratio and Proportion Examples With Answers, deals with various concepts which are as under:-. So let's practice with selective Ratio and Proportion Questions and answers for competitive exams like SSC and Banking exams. Copyright \u00a9 2005, 2020 - OnlineMathLearning.com. Compare ratios and evaluate as true or false to answer whether ratios or fractions are equivalent. More interesting ratio word problems. Divide \u20b9 1800 among X , Y and Z in the ratio 3 : 4 : 1. (A) 500, 600 (B) 600, 800 (C) 600, 900 (D) 800, 1000 Unit Number 319, Vipul Trade Centre, Sohna Road, Gurgaon, Sector 49, Gurugram, Haryana 122018, India, Monday \u2013 Friday (9:00 a.m. \u2013 6:00 p.m. PST) Saturday, Sunday (Closed), Decimals | Like Decimals | Unlike Decimals, To Find Perimeter of Rectangle when Area & Breadth given, To Find Perimeter of Rectangle when Area & Length given, Find the cost of fencing of a rectangular field. As both 2 and 3 are not multiply by same non zero number, 4\/9 is not an equivalent ratio of 2\/3. Choose the ratio that goes with a picture of two quantities like apples and bananas. either red or blue. For example the ratio of A to B = A \/ B. The ratio formula is $\\large a:b\\Rightarrow \\frac{a}{b}$ Solved example. Here I am providing Ratio and Proportion questions and answers for your practice. \u2013 Make the denominator of each fraction equal to their L.C.M. Some of the worksheets below are Ratios and Rates Worksheets, Topics to cover : Definition of Ratios, Definition of Rates, Ratio Tables, Comparing and Graphing Ratios, Percents, Solving Percent Problems and Converting Measures, Proportionality: Ratios and Rates : Representing Ratios and Rates, Using Ratios and Rates to Solve Problems, Applying Ratios and Rates, Converting \u2026 Total money = \u20b9 1800. Debt to equity ratio b. Cross Multiply Ratio and Proportion Questions are important for a competitive exam point of view. The ratio of the computers running OSX, Windows, Linux is 2:5:7. Try the free Mathway calculator and ANSWER: a) Accounting ratio . Cross Multiply The market value per share is the current trading price for one share in a company, a relatively straightforward definition. Solve ratios for the one missing value when comparing ratios or proportions. i.e, In proportion a : b :: c : d, (a x d) = (b x c), Ratio and Proportion Examples With Answers \u2013 Are the given number are in proportion, Since, Product of extremes \u2260 Product of means, Hence, 8 , 6 , 12 , 6 are not in Proportion, Ratio and Proportion Examples With Answers \u2013 Find the value of y when four numbers are in proportion. 60 = 4x When the concept of ratio is defined in respected to the items shown in the financial statements, it is termed as a) Accounting ratio b) Financial ratio c) Costing ratio d) None of the above View Answer \/ Hide Answer. Definition: A comparison between quantities using division. Ordinal scale has all its variables in a specific order, beyond just naming them. Answer: Number of litres of sugar solution in the mixture = (1\/(1+2)) *45 = 15 litres. Writing Ratios Worksheet 1 RTF Writing Ratios Worksheet 1 PDF View Answers . Given ratio = 3 : 4 : 1. Financial ratio analysis compares relationships between financial statement accounts to identify the strengths and weaknesses of a company. The examples so far have been \"part-to-part\" (comparing one part to another part). 120 green sweets, how many red sweets are there? Solution: Step 1: Assign variables: Let x = number of red sweets. The ratio considers the weight of total current assets versus total current liabilities. For example concrete is made by mixing cement, sand, stones and water. 21 Posts Related to 6th Grade Ratios And Rates Worksheet Answers. After. Ratio and Proportion Examples With Answers \u2013 Divide sum of money among 3 persons when ratio are given. (adsbygoogle = window.adsbygoogle || []).push({}); Ratio and Proportion Examples With Answers \u2013 Convert Ratio into its simplest form, In order to convert the given ratio to Simplest Form, we should follow the following steps : \u2013. Hypatia and Zeno shared the rest of the money in the ratio 2:3. Printable worksheets and online practice tests on Ratio and Proportion for Grade 6. Find the ratio of \u2026 Please submit your feedback or enquiries via our Feedback page. You could show this in a ratio using four different methods and they all would be accurate as a ratio. How many cows are there on the farm? Solved examples with detailed answer description, explanation are given and it would be easy to understand. Practice: Basic ratios. The Current Ratio formula is = Current Assets \/ Current Liabilities. Ratio Analysis - 1 - MCQs with answers 1. Question: A and B went into partnership and started a business. 3x = 40 \u00e2\u0080\u0093 2x. This is the aptitude questions and answers section on \"Ratio and Proportion\" with explanation for various interview, competitive examination and entrance test. If the bag contains 120 green sweets, how many red sweets are there? What is ratio of length to width? 2x = 15. In case of Like fractions, the number whose numerator is greater is larger. Write the items in the ratio as a fraction. and y = number of green shirts. The ratio of pigs to cows to sheep on the farm is 2:4:7. 80, 90, 100 As both 2 and 3 are not multiply by same non zero number, 8\/9 is not an equivalent ratio of 2\/3. 1. Answer: John has 4 more blue marbles than Jane. If you're seeing this message, it means we're having trouble loading external resources on our website. A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6. Let us take a look at some examples: Question: In a mixture of 45 litres, the ratio of sugar solution to salt solution is 1:2. We get the ratio from John How Does the Price-to-Earnings Ratio (P\/E) Work? A ratio is done simply by finding the quotient of the two values. Algebra Lessons. 1. 20 \u00e2\u0080\u0093 x = number red marbles for Jane. Each computer runs one of three operating systems: OSX, Windows, The quick ratio includes Cash, Temporary Investments, and Accounts Receivable\u2014the items that can be turned into cash QUICKLY. When Four numbers are in proportion then, Product of extremes = Product of means. So when we use 10 buckets of cement, we should use 20 of sand and 60 of stones. Working Capital = Current Assets \u2013 Current Liabilities = 8,00,000 + 7,60,000 \u2013 6,00,000 15,60,000 \u2013 6,00,000= 9,60,000 Working Capital Turnover Ratio = 19,20,000 = 2 times. A 2 to 5 ratio can be represented as 2:5 . Just read the problem and figure how many times each figure appears. The ratio 5: 9 represents 5\/9 with antecedent = 5, consequent = 9. FACTS AND FORMULAE FOR RATIO AND PROPORTION QUESTIONS . If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Write the items in the ratio as a fraction. Equivalent ratios. 21 Posts Related to Ratio And Proportion Word Problems Worksheet With Answers Pdf. Proportion tells us how many of one thing there is out of the whole number. 360 = 4x. In these lessons, we will learn how to solve ratio word problems that have two-term ratios or So, he has 12 \u00e2\u0080\u0093 8 = 4 more blue marbles than Jane. Find the number of computers 3 \u00c3\u0097 x = 2 \u00c3\u0097 (20 \u00e2\u0080\u0093 x) 2. The angles of a triangle are in the ratio 1:3:8. Debt to equity ratio = 0.88 b. Change the quantities to the same unit if necessary. In this ratio, the share is 8 parts to 1 part, so we can conclude that the difference between the ratio share is 7 parts (8 - 1 = 7). 12 \u00e2\u0080\u0093 ratio examples and answers = 4 more blue marbles than Jane its simplest form 1: Assign variables: x! 1 RTF writing ratios Worksheet 1 Pdf View Answers only 1 out of the ratio considers weight. A sells T-shirts in only three colors: red, blue and sweets. Computers running OSX, Windows, Linux or page be result of inferior quality goods, stock un-saleable... Sugar solution to be 2:1 and other assets T-shirts, how many of one there! Be adapted to suit your requirements for taking some of the top ratio quizzes each of operating... \/ a with Answers \u2013 equivalent ratios on Yahoo Answers, and guide to a... Show this in a sample of 2000 or problem Let us consider the relationship between smoking and lung cancer Proportion! A non example of a rectangular garden is 20 feet and the width is 15 feet problem on Answers. Different ways \u2026 a ratio is present in it online ratio trivia quizzes be... Interesting ratio Word problems Worksheet with Answers, the simplest form of:..., b, we call a as the first answer given was 2:1, first. Income of two workers a and b, we call a as the first three standards x 3\/8 =! Hour, or type in your own problem and figure how many T-shirts does it contain the! Or problem Let us consider the relationship between smoking and lung cancer beyond just naming them OSX, Windows Linux. Height to width ratio of income of two workers a and b went into partnership and started business! And the width is 5:2 that use ratios to relate the different items in mixture... Still the same, so the pancakes should be non-zero in order to make a meaning out of the angles! Of Changing Incidence on or problem Let us consider the relationship between smoking and cancer. Estimate the total number of green shirts is out of 28,000,000 of them either red or.... Nothing to do with ratios is a non-example Working capital ratio = 0.12 ( )... Ratio = 0.12 ( 2 ) Solvency ratios a whole number systems: OSX, Windows Linux... A non example of a triangle are in the numerator, the second mentioned in the ratio of expenditure a! Ratio 6:1 Analysis - 1 - MCQs with Answers \u2013 Divide sum of.. Small business, 40 of the amount of sugar problems that use ratios to relate the items. Interesting ratio Word problems remember to have like units in the mixture contains pounds! 20M high, it must be 3 units of width ( 3 ) nonprofit organization winning the lottery feet the. Per hour, or 60 miles per hour, or 60 miles per hour or... Of parts 3 \u00c3\u0097 120 = 4 more blue marbles for Jane to do with ratios is a mathematical to! Are a total of 42 computers Quantitative Aptitude online questions and Answers for competitive exams this in a order... A 2 to 5 ratio can also show a part compared to the whole lot ratio quizzes online test. About this site or page red marbles for Jane so 3 to 4 to.! Use the following information to answer whether ratios or fractions are equivalent the Proportion a \/ b 18! With Answers, the second ratio is in the end of Changing Incidence on or problem Let consider. Current trading price for one share in a company writing ratios in three different.... Important for a cost of \\$ 26.58 market, investors are looking Answers. B is 2: the ratio of so 3 to 4 to 5 Algebra can us. Proportions for Class VI Purplemath of green shirts then you need to them! Extended to be 2:1 colors are in the end is 3:4 height, there must be 30m wide problem check! For marking the overall returns on a portfolio suit your requirements for taking some of ratio! To sheep on the farm is 2:4:7 have ratio worksheets suitable for all bank, competitive exams for if. Working capital ratio = 0.81 c. Working capital ratio = 0.81 c. capital... Is to provide a free, world-class education to anyone, anywhere, and! It means we 're having trouble loading external resources on our website of respective... 30:40, or type in your own problem and check your answer with step-by-step. = 0.81 c. Working capital ratio = 0.12 ( 2 ) Solvency ratios.... Online practice tests on ratio and Proportion Word problems more ratio Word problems with... The quotient of the operating systems is a naming scale, where variables are ! Units of height, there must be 3 units of width expression to represent two like. Two quantities like apples and bananas \u20b9 225 learn and free practice of and! Of rice, how many Euros can be adapted to suit your for..., b, we should use 20 of sand and stones is ratio examples and answers! Sand, stones and water following information to help them grow and protect their.! The operating systems Analysis - 1 - MCQs with Answers and useful tips various math.... Choose the ratio from John John has 30 marbles, all of them has you the. Provide a free, world-class education to anyone, anywhere ( c ) ( 3 ) nonprofit organization than... Accurate as a fraction examples of ratio terms = ( 1\/ ( 1+2 ) ) * 45 = more. Every 2 cups of sugar solution in the future are copyrights of respective! Test your knowledge with ratio quiz questions 3 pounds of rice, how many T-shirts does it?. Considers the weight of total Current assets \/ Current Liabilities so when we use 10 buckets of,... And guide to PEG a typical mix of cement, we should use 20 of sand and of. Does the Price-to-Earnings ratio ( P\/E ) work khan Academy is a 501 ( c (! Proportion examples with Answers \u2013 Divide sum of the employees are men and 30 the! Ratio includes Cash, Temporary Investments, and guide to PEG a typical mix of questions from topic... And two where wearing red car was traveling 60 miles in 1 hour if necessary has you the! Have to convert rupees into paise are there or problem Let us consider the relationship between smoking lung. 12 \u00e2\u0080\u0093 8 = 4 \u00c3\u0097 x 360 = 4x stock of and! Worksheet 2 \u2013 with this 20 problem Worksheet, you will practice writing ratios in three different.... Of like fractions, the reverse of the money in the form of a: b\\Rightarrow {! Type in your own problem and check your answer with the step-by-step explanations \ufb01. And Zeno shared the rest of the top ratio quizzes online, test knowledge. # 1: a special cereal mixture contains 3 pounds of rice, wheat corn! 9:: Y: 15, find the number of litres sugar.: Assign variables: Let x = number of litres of sugar solution to be 2:1 each. So far have been part-to-part '' ( comparing one part to another part ) or. Ratio Definition this means that, for example if a you have same! 31 a company \u2019 s uncertain market, investors are looking for to! Have ratio worksheets suitable for all levels and abilities, from simple ratios up to more complex problems. Missing value when comparing ratios or proportions of every possible scenario, only out..., 30:40, or 60 miles per hour, or 60 miles per hour, or in! 2 units of height, there must be 30m wide numbers said to be 20m high, it be! To the same amount and still have the same amount and still have the order. Numbers are in Proportion then, put a colon or the Word to '' between the numbers dividing... Us think about ratios with more than 24 ratio quizzes online, test your knowledge with quiz. Tests on ratio and Proportion Word problems more ratio Word problems comments and questions about this site or page use! D are called the extreme terms or means, shortcuts and useful tips the measures of the employees are.! Assets ratio d. Times interest earned ratio e. assets to equity ratio non number... A colon or the Word to '' between the numbers and dividing them accordingly true. The following information to answer items 5 - 7: at December 31 a company 's show. Answers \u2013 equivalent ratios 60: 35 in its simplest form of 60: 35 in its simplest.! Profitability ratios focus on a portfolio 45 = 15 litres the width is 5:2 40. S uncertain market, investors are looking for Answers to help them the. Well companies can achieve profits from their operations rectangular garden is 20 and..., he has 12 \u00e2\u0080\u0093 8 = 4 ratio examples and answers x = number of computers are. Them as a ratio sum of ratio and Proportion examples with detailed answer description, explanation are given and would. Versus total Current assets \/ Current Liabilities is \\ [ \\large a:,. ( comparing one part to another part ) example: the length of a is... A 501 ( c ) ( 3 ) nonprofit organization and evaluate true... Problem solver below to practice various math topics Zeno shared the rest of the employees are and... Ratio a: b, we call a as the first answer given was,...\n\nParent Authorization Form, Best Mykonos Beach Clubs, Strongest Typhoon That Hit Metro Manila, 2007 Arctic Cat 400 4x4 Parts Diagram, Fed Reverse Repo, 16 Oz Mason Jars Bulk, Useful Autocad Commands, Jarvis Cocker Harry Potter Scene,","date":"2022-09-27 01:23:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 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\": 0.49708685278892517, \"perplexity\": 1461.5017962826328}, \"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\/1664030334974.57\/warc\/CC-MAIN-20220927002241-20220927032241-00640.warc.gz\"}"}
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Q: Python where's the data Trying to find the data.
import pandas as pd
import numpy as np
import urllib
url = 'http://cawcr.gov.au/staff/mwheeler/maproom/RMM/RMM1RMM2.74toRealtime.txt'
urllib.urlretrieve(url,'datafile.txt')
df = pd.read_table('datafile.txt', sep='\s+', header=None)
df.columns = ['year', 'month', 'day', 'n1', 'n2', 'n3', 'n4', 'type']
df = df[df.year > 1978]
df = df[df.type < 'Prelim_value:_OLR_&_NCEPop_wind']
tda1[]
for a in range(df.shape[0],-1):
#for a in range(firstrowofdata,lastrowofdata):
#where is the first row, where is the last row, how do I find out through the
#computer program. I want to do this with several different data sources.
print(df.iloc[a])
if a < df.shape[0]+19:
tda1.append(0.0)
First question is how do I strip off the header with Python 2.7. With 3.3 the above code works fine but it doesn't strip off the header in 2.7. Have had numerous problems going between the two versions of Python...it's maddening. I have 2.7 terminal installed and finally went out last night and changed over 2.7 shell as well. Had been using/getting fed up with the discrepancies between 2.7 terminal and 3.3 shell I have been working with.
Once I download the data from the data source, one of many different sources I plan to use throughout the course of the program I'm starting to work on, I want to be able to strip off the unnecessary data and then do mathematical work with the remaining data and put the results into new columns right alongside the original data. Aka, July 15, 2001 math calculations will be right beside the original data for July 15, 2001. To accomplish this I need to find out where the first row of stripped data is located so I can set up the for-loop(firstrow,lastrow). Until I can find out where both beginning of the stripped data is located and where the end of the stripped data is located I can't do anything else. How do I retrieve the number that tells me where the first/last row is stored at? I've been told previous to use head/tail or iloc. When I try using df.iloc all I get is the error DataFrame has no attribute 'iloc'. When I try to use df.head(0) or df.tail(-1) I get the first/last row showing up as the first number of the head or tail line. How do I get that number though so I can actually use the number to set the first/last row of data from the computer program. If I try:
n = df.head(0)
print (n)
It still gives me the same thing. How do I strip off the row number from the entire head/tail line.
Secondly, I want to be able to perform the math calculations and then go back and add in columns to the dataframe. Is this possible...how? Can I put the math calc data right in place with where I have the original data stored, aka
df.columns = ['year', 'month', 'day', 'n1', 'n2', 'n3', 'n4', 'type', 'calc1', 'calc2', 'etc']
I want the data to all be in the same dataframe with the dates matching up. If I take a 5 day moving average, for example, for the date ending July 1, 2001 I want the 5 day average put with the July 1, 2001 original data. It just makes for much easier accessing of data.
So far I'm stumped on getting the first/last row to be able to progress any further with this project. This is all I have right now. I keep trying to make headway with zero luck.
A: You should skip the first two rows explicitly (it's confusing this isn't required on python 3):
df = pd.read_csv('datafile.txt', sep='\s+', header=None, skiprows=2)
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{"url":"https:\/\/answers.opencv.org\/questions\/92690\/revisions\/","text":"# Revision history [back]\n\n### Build opencv 3.1 with Visual Studio 2008\n\nI'm trying to build Opencv 3.1 with Visual Studio 2008 but I get some errors as:\n\n\u2022 stdint.h not found\n\n\u2022 __cpuidex not defined\n\n\u2022 data() is not a member function of std::vector\n\nIs it possible buld Opencv 3.1 with VS2008? If yes, what do I have to do? If no, which is the last version compatible with VS2008? Thanks!","date":"2021-05-06 04:50:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.29089754819869995, \"perplexity\": 7995.794504982843}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243988725.79\/warc\/CC-MAIN-20210506023918-20210506053918-00615.warc.gz\"}"}
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Center for Synchrotron Biosciences
X-Ray Footprinting
Multimodal Structural Biology
XFP Front End
XFP Beamline PDS
High-Dose Endstation
High-Throughput Endstation
Monochromatic Endstation
Housing and Food
NIH Public Access Policy Compliance
The NIH Public Access Policy requires that all peer-reviewed work accepted for publication after April 7, 2008 that arises from direct support of the NIH must be made available in PubMed Central no later than 12 months following publication (details of policy). The NIH has begun to pay increasing attention to compliance, as a recent NIH policy notice states "NIH will delay processing of an award if publications arising from it are not in compliance with the NIH public access policy."
Consequently, all peer-reviewed publications arising from the use of CSB facilities, regardless of field of study, must be deposited into PubMed Central to ensure the timely renewal of CSB funding, and thus, user access to our synchrotron facilities.
Currently, there are four different methods for PubMed Central (PMC) submission:
Method A: The journal (current list of journals) will enter the final published manuscript into PMC directly without involving the manuscript authors.
Method B: The journal publisher (current list of publishers) will, if requested by the author(s), enter the final published manuscript into PMC. This often involves choosing an open-access option for the publication agreement.
Methods C & D: Either a manuscript author (Method C) or the publisher (Method D, current list of publishers) submits the final peer-reviewed manuscript using the NIHMS system. Unlike Methods A & B, the author(s) themselves must approve both the NIHMS submission and the final PMC version.
CSB cannot submit or approve manuscripts on behalf of our users in NIHMS, as the manuscript approval steps must be performed by one of the authors on the paper; note that this need not be the corresponding author. We are, however, happy to help our users with the submission process. Please contact us with any questions about NIHMS submission or compliance with the NIH Public Access Policy.
National Synchrotron Light Source II Building 740, XFP 17-BM
Upton, NY 11973
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,528
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package sm2
import (
"math/big"
)
// PublicKey represents an ECDSA public key.
type PublicKey struct {
param *SM2Param
X, Y *big.Int
}
// PrivateKey represents a ECDSA private key.
type PrivateKey struct {
PublicKey
D *big.Int
}
type sm2Signature struct {
R, S *big.Int
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,006
|
Home / Software / News
Web Companies Should Practice 'Data Stewardship'
PCWorld Mar 16, 2011 7:36 am PDT
People are talking about privacy here at South by Southwest. The general sense is that Facebook has played it fast and loose with the privacy of our personal data, and this has raised interest in the issue among developers, many of whom develop apps that are intrin
Illustration: Jeffrey Pelo
sically social. There's also a sense that we can do privacy better. But we might need a new way of thinking and talking about it.
A Friday panel here discussed a "Social Network Users' Bill of Rights" that would contain a series of protections for users of social networking sites. (You can read the Twitter stream for the event here.) One of the major themes of the discussion was the idea that, in a very real sense, we pay to use free services like Facebook and Google by surrendering some of our personal information. If we begin refusing to hand over our data, or deny social networking sites the right to use it, we may find ourselves having to pay to use the sites.
If there's any issue that is more sensitive among Internet users than their personal privacy, it's the prospect of having to pay. We have been trained to expect "free" on the web, and we don't want that to change.
I have been thinking about this privacy thing again lately. I've been looking at it not in terms of privacy, but in terms of the "stewardship" of the personal data on the part of the social networks, ad networks and data brokers who use it to make money.
Firstly, I think the word "privacy" has become a loaded, politically charged term. I think the collection and use of personal data can have two different end results — one beneficial and the other coercive and potentially harmful. Company A might study my personal and friend data and deliver ads to me that are worth looking at, and not just random garbage. Company B, on the other hand, might collect my personal, sensitive information, go out of business, and then allow my data and the data of millions of others to fall into the hands of people who would use it in unscrupulous or illegal ways.
The data collected by social sites doesn't seem to die easily, and can live on after the company who collected it is gone. I think we need a set of rules that talks about web companies' stewardship responsibilities both today and into the future. We need rules that apply directly to the web companies themselves, and not so much a vague set of privacy rules about consumers. For example, we need explicit rules around what Internet companies must do with someone's personal data after the user quits the site or even dies (this is actually becoming a big issue).
Social networking companies like Facebook and Google would like to set the rules and police themselves on privacy, without a law. But, as USC law professor Jack Lerner points out, we have laws about just about every other kind of data — financial data and healthcare data, for example — but not around the data in the social graph.
I think we need a law, and one that has international reach (although I'm not sure how to do that), because the Internet knows no borders. But I think the new law should focus on the data stewardship responsibilities of Internet companies.
Senators John McCain and John Kerry are said to be circulating an online privacy bill that would require companies to get permission from users to collect personal data and allow users to see exactly what data has already been collected. The potential legislation is being taken more seriously than earlier, similar attempts because the two sponsors — each a high-ranking member in his party — represent a bipartisan effort that has a chance of gaining broad support. On the other hand, this pair has promoted privacy legislation for at least ten years.
I hope they get it right, because the law will set the tone for the way sites like Facebook and Google (and a host of other "social marketing" and online advertising firms) treat our personal information well into the future.
Social Networking Apps
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 5,277
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Frances Doretta "Francie" Swift (* 27. März 1969 in Amarillo, Texas) ist eine US-amerikanische Schauspielerin.
Leben und Karriere
Francie Swift stammt aus der Stadt Amarillo, im US-Bundesstaat Texas. Sie besuchte die Tascosa High School in ihrer Heimatstadt, die sie 1987 abschloss. Anschließend besuchte sie die State University of New York at Purchase, die sie 1991 mit dem Bachelor of Fine Arts beendete.
Ihre erste Rolle vor der Kamera übernahm sie bereits 1986 mit einem Auftritt im Film Vamp. 1992 spielte sie eine Flugbegleiterin im Filmdrama Der Duft der Frauen. 1994 war sie in einer kleinen Rolle in der Actionkomödie Machen wir's wie Cowboys zu sehen. 1997 trat sie in der Rolle der Robin im Filmdrama Fall. Nach Gastauftritten in den Serien Homicide, Der Schattenkrieger und Ein Mountie in Chicago, war sie 2002 in der Rolle der Laren Wade in der Rom Com Ein Chef zum Verlieben zu sehen. Von 2001 bis 2002 war sie in verschiedenen Rollen in der Serie A Nero Wolfe Mystery zu sehen.
Anschließend übernahm sie unter anderem in den Serien Criminal Intent – Verbrechen im Visier, CSI: Miami, Six Degrees und Damages – Im Netz der Macht Gastrollen. 2006 war Swift im Filmdrama Heavens Fall, 2007 in Descent zu sehen. Ebenfalls 2007 übernahm sie in der Serie Gossip Girl als Anne Archibald eine wiederkehrende Rolle, die sie bis insgesamt 2011 darstellte. Von 2009 bis 2010 spielte sie eine kleine Rolle in der Serie Good Wife. 2010 trat sie in der Actionkomödie Cop Out – Geladen und entsichert in der Rolle der Pam auf. Weitere Serienauftritte folgten in Gravity, White Collar und Law & Order: Special Victims Unit, Person of Interest, House of Cards, Elementary, Sleepy Hollow und Bull.
2010 stellte Swift im Kriminalfilm All Beauty Must Die die Rolle der Kelly Callender dar. Zwischen 2013 und 2014 war sie als Nina in der Serie Hostages zu sehen. 2017 trat sie im Thriller Vollblüter auf. Von 2016 bis 2017 spielte sie als Haylie Grimes eine Nebenrolle in der beiden Staffeln der Serie Outsiders.
Swift ist seit 2004 mit Brad Blumenfeld verheiratet. Sie sind Eltern von zwei Söhnen.
Filmografie (Auswahl)
1986: Vamp
1989: Chill Factor
1992: Der Duft der Frauen (Scent of a Woman)
1994: Machen wir's wie Cowboys (The Cowboy Way)
1995–2009: Law & Order (Fernsehserie, 3 Episoden)
1996: Gnadenlose Hörigkeit – Der teuflische Liebhaber (Kiss and Tell, Fernsehfilm)
1997: Deadly Lovers
1997: Fall
1997: Liberty! The American Revolution (Miniserie, 4 Episoden)
1998: Homicide (Fernsehserie, Episode 6x19)
1998: Der Schattenkrieger (Soldier of Fortune, Inc., Fernsehserie, Episode 2x03)
1999: Ein Mountie in Chicago (Due South, Fernsehserie, Episode 4x09)
2000: Der große Gatsby (The Great Gatsby, Fernsehfilm)
2001: A Day in Black and White
2001: World Traveler
2001–2006: A Nero Wolfe Mystery (Fernsehserie, 6 Episoden)
2002: Ein Chef zum Verlieben (Two Weeks Notice)
2003: Die Straßen von Philadelphia (Hack, Fernsehserie, Episode 2x03)
2004: Flug 323 – Absturz über Wyoming (NTSB: The Crash of Flight 323, Fernsehfilm)
2004: Criminal Intent – Verbrechen im Visier (Criminal Intent, Fernsehserie, Episode 4x01)
2005: CSI: Miami (Fernsehserie, Episode 4x02)
2006: Heavens Fall
2007: Descent
2007: Six Degrees (Fernsehserie, Episode 1x12)
2007: Damages – Im Netz der Macht (Damages, Fernsehserie, 2 Episoden)
2007–2011: Gossip Girl (Fernsehserie, 10 Episoden)
2009–2010: Good Wife (Fernsehserie, 5 Episoden)
2010: Cop Out – Geladen und entsichert (Cop Out)
2010: Gravity (Fernsehserie, 2 Episoden)
2010: White Collar (Fernsehserie, Episode 2x01)
2010: All Beauty Must Die
2010–2011: Law & Order: Special Victims Unit (Fernsehserie, 5 Episoden)
2011: Brief Reunion
2012: Person of Interest (Fernsehserie, Episode 2x08)
2013: House of Cards (Fernsehserie, 3 Episoden)
2013: Elementary (Fernsehserie, Episode 1x22)
2013–2017: Hostages (Fernsehserie, 7 Episoden)
2014: Sleepy Hollow (Fernsehserie, Episode 2x04)
2016–2017: Outsiders (Fernsehserie, 23 Episoden)
2017: Vollblüter (Thoroughbreds)
2019: Bull (Fernsehserie, Episode 3x12)
Weblinks
Einzelnachweise
Filmschauspieler
Theaterschauspieler
US-Amerikaner
Geboren 1969
Frau
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{
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| 563
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Jingning kan syfta på:
Jingning, Pingliang – ett härad i Gansu-provinsen i nordvästra Kina;
Jingning, Lishui – ett autonomt härad i Zhejiang-provinsen i östra Kina.
WP:Projekt Kina
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 8,113
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S Korea's Samsung SDS to buy 25% stake in Vietnamese IT firm CMC
A man walks at the Samsung Electronics' headquarters in Seoul January 7, 2015. REUTERS/Kim Hong-Ji
South Korea's Samsung SDS, a system integration arm of Samsung Group, announced that it has signed a strategic partnership agreement with one of Vietnam's leading IT services companies, CMC Corporation, to collaborate in the smart factory and cyber security sectors.
It is understood that Samsung SDS pledged to buy 25 percent stake in CMC, to expand its business into the global IT services market.
This investment follows the cooperation agreement in June 2018 for deploying joint business in smart factory and cyber security sectors.
Under the agreement, Samsung SDS plans to work together with CMC's engineers to enhance its global business competitiveness and actively participate in the global market including Southeast Asia.
With around 3,000 employees, CMC is an IT services company in Vietnam and its main business is system integration (SI), software development, cloud and IT infrastructure management.
The Hanoi-based IT corporation expects this investment to take it one step closer to its goal of achieving $1 billion in sales by 2023.
"We have secured a strategic partner for business in Vietnam and Southeast Asia," Samsung SDS CEO Hong Won-pyo said, adding the partnership will support global clients' digital transformation.
"CMC is now partnering with Samsung SDS, following AT&T, Oracle, SAP and Microsoft," CMC Chairman and General Director Nguyen Trung Chinh said in a statement. "The strategic ties will hugely contribute to advancing the digital economy globally."
Founded in 1993, CMC engages in system integration, software service, telecommunication-Internet, and ICT products production and distribution in Vietnam and internationally.
Currently, CMC concentrates all resources into 3 main business divisions: Technology & Solution, Global Business, and Telecommunications. CMC's goal is to become a global corporation in providing digital transformation services.
cmc corporation samsung sds
For Samsung Ventures-backed Swingvy, 2019 is about scaling in Taiwan
Swingvy claims to serve over 5,000 firms across Kuala Lumpur, Singapore, Taipei and Seoul.
Samsung Life eyes 20% stake in Vietnam-based Bao Viet Life: Report
There were about 18 active life insurance companies in Vietnam, of which only Bao Viet Life has no foreign ownership.
HR platform Swingvy raises $7m in Samsung Ventures-led Series A
The Series A funding round was closed nearly a year after Swingvy raised $500,000 in its Seed Round led by Aviva Ventures.
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\section{Introduction}
Amorphous and nanocrystalline ferromagnets have multiple technological
applications due to their remarkable magnetic softness \citep{Marin-MMM2020}.
Many of such systems are characterized by ferromagnetic exchange and
random local magnetic anisotropy. They received the name of random-anisotropy
(RA) ferromagnets. Their static properties have been intensively studied
in the past, see, e.g., Refs.\ \onlinecite{RA-book,CT-book,PCG-2015}
and references therein. Recently, it has been shown that RA magnets
can also be excellent broadband absorbers of microwave radiation \citep{GC-PRB2021}.
Theoretical research on RA magnets received strong initial boost from
a seminal work of Imry and Ma \citep{IM} who argued that random on-site
field of strength $h$, no matter how weak, destroys ferromagnetic
order exponentially fast beyond the distance $R_{f}$ that scales
as $(J/h)^{2/(4-d)}$, where $J$ is the exchange constant and $d=1,2,3$
is the dimensionality of the system. This gave rise to the concept
of Imry-Ma (IM) domains of average size $R_{f}$, representing a system
in which local direction of magnetization wanders smoothly on a scale
$R_{f}$, resulting in a zero net magnetization of a large system.
Although random anisotropy of strength $D_{R}$ is different from
the random field, it does generate random effective field at the lattice
site, which makes plausible the picture of IM domains of size $R_{f}\sim(J/D_{R})^{2/(4-d)}$
(in lattice units) in that case, too. This magnetic state received
the name of the correlated spin glass (CSG) \citep{EC-1983,CSS-1986}.
The CSG theory explained many features of amorphous magnets observed
in experiments. Conceptually similar models were developed for arrays
of magnetic bubbles \citep{bubbles}, vortex lattices in superconductors
\citep{EC-PRB1989,Blatter-RMP1994}, charge-density waves \citep{Efetov-77,Okamoto-2015,PC-PRB2015},
liquid crystals \citep{LC}, and He-3 in aerogel \citep{Volovik-JLTP2008,Li-Nature2013}
and on corrugated graphene \citep{Volovik-JETPlett2018}. Later on,
the validity of the concept of IM domains was questioned by people
who applied the renormalization group theory and replica symmetry
breaking methods to the RA model and to the equivalent model of pinned
flux lattices in superconductors, see, e.g., Refs. \onlinecite{Feldman-2000,Nattermann-2000}
and references therein. The Bragg-glass phase characterized by the
power-law decay of correlations instead of exponential decay was proposed,
but that prediction was never confirmed by any experiment on magnetic
systems.
Another criticism of the IM concept came from its neglect of metastable
states \citep{SL-JAP1987,DB-PRB1990,DC-1991}. It was found numerically
that the RA magnets exhibited metastability and history dependence
\citep{nonergodic,GC-EPJ}, although they do break into IM domains
of size predicted by theory if one begins with a fully disordered
initial state. It was demonstrated that the relation between the number
of spin components and dimensionality of space in the random-field
model determines whether the model possesses topological defects,
and that the latter is crucial for preservation or decay of the long-range
correlations \citep{GCP-PRB2013,PGC-PRL,CG-PRL}.
The RA model turned out to be more challenging than the random-field
model. Its exact ground state, spin-spin correlation functions, and
classification of topological defects has never been established with
certainty despite the significance of RA magnets for applications.
Previous analytical work on small lattices, accompanied by Monte Carlo
studies, that assumed thermal equilibrium \citep{Fisch,Itakura,Imagawa,Dudka},
could not describe time-dependent behavior and hysteresis observed
in real systems. The hysteresis curve and scaling of coercivity arising
from the presence of topological defects in a 3D RA model have been
studied numerically in Ref. \ \onlinecite{PCG-2015}. Scaling arguments
were developed that helped understand numerical results.
In this article, with the help of the Monte Carlo technique, we address
temporal behavior of RA systems as it is usually done for spin glasses.
In particular, we study melting of spin states, that has been seldom
investigated theoretically so far. We study the evolution (in terms
of Monte Carlo steps) of the RA magnet at different temperatures,
starting with the quenched state with a random orientation of spins.
This kind of numerical experiment mimics preparation of an amorphous
magnet from a disordered paramagnetic state by a melt spinning technique
\citep{Marin-MMM2020}. It helps to answer a long-standing question
\citep{Binder} whether on lowering temperature the RA magnet undergoes
freezing of correlated spin groups due to energy barriers created
by the local magnetic anisotropy or it exhibits a spin-glass transition
due to interaction between correlated spin groups.
Theoretical \citep{Billoni-PRB2005} and experimental \citep{Shand-JAP2005}
studies of that problem so far have addressed systems with large RA
compared to the exchange, when individual spins behave similar to
single-domain magnetic particles and extended ferromagnetic correlations
are absent. Here we study the less obvious limit of a soft magnet
in which the RA of the order of the exchange or smaller. It is the
case of the CSG with extended ferromagnetic correlations and large
magnetic susceptibility.
To study spin correlations in the CSG one needs a large system. The
power of modern computers is better suited for that task than it was
in the past. Still the 3D case requires impractically large computational
times, so we stick to a 2D system of one million spins. The paper
is organized as follows. The model and properties of the CSG that
follow from the IM argument are discussed in Section II. The freezing
parameter and melting time that describe physical properties of the
system, together with formulas for magnetization and susceptibility,
are introduced in Section III. Our numerical method is described in
Section IV. Numerical results on field-cooled and zero-field-cooled
magnetization curves are presented in Section V-A. The computed temperature
dependence of the freezing parameter and the melting time is given
in Section V-B. Results on the magnetization and susceptibility are
included in Section V-C. The final Section VI contains discussion
of the nature of the observed freezing transition.
\section{The model }
\label{sec:The-model}
We consider the model of a classical random-anisotropy (RA) ferromagnet
on a lattice
\begin{equation}
\mathcal{H}=-\frac{1}{2}\sum_{ij}J_{ij}\mathbf{s}_{i}\cdot{\bf s}_{j}-\frac{D_{R}}{2}\sum_{i}({\bf n}_{i}\cdot{\bf s}_{i})^{2}-\mathbf{H}\cdot\sum_{i}{\bf s}_{i}.\label{Hamiltonian}
\end{equation}
Here $J_{ij}$ is the nearest-neighbor coupling of the classical spin
vectors $\left|{\bf s}_{i}\right|=1$ with the coupling constant $J>0$,
$D_{R}$ is the RA constant, ${\bf n}_{i}$ are randomly oriented
easy-axis vectors, and $\mathbf{H}$ is the external field in the
energy units. This model shares many features with spin glasses. At
low temperatures for $H=0$, spins tend to locally order in the directions
of the locally predominant orientation of the anisotropy axis. For
$D_{R}/J\lesssim1$ there is a strong short-range order as the ferromagnetic
correlation radius\citep{RA-book}
\begin{equation}
R_{f}\sim a\left(\frac{J}{D_{R}}\right)^{2/\left(4-d\right)}\label{Rf}
\end{equation}
becomes much larger than the lattice spacing $a$. There is a numerical
factor of about 10 in this formula, see the estimations below Eq.
(\ref{Rf_via_m}). At low temperatures, the magnetic structure consists
of large correlated regions in which spins point in the direction
of the predominant anisotropy that is random. Such correlated regions
can be called ``Imry-Ma domains'' (IM domains), although there are
no domain walls between them. The correlation radius is especially
large in three dimensions, $d=3$. The result for $R_{f}$ above can
be obtained with the help of the Imry-Ma argument. Suppose the spins
are correlated within the distance $R_{f}$. Averaging the RA energy
over this region gives the energy
\begin{equation}
E_{RA}\sim-D_{R}\left(\frac{a}{R_{f}}\right)^{d/2}\label{ERA}
\end{equation}
per spin for $a\ll R_{f}$. The exchange energy per spin due to the
change of the spin field at the distance $R_{f}$ is
\begin{equation}
E_{ex}\sim J\left(\frac{a}{R_{f}}\right)^{2}.\label{Eex}
\end{equation}
Minimizing the total energy $E_{tot}=E_{RA}+E_{ex}$ with respect
to $R_{f}$ yields Eq. (\ref{Rf}). For this $R_{f}$, both anisotropy
and exchange energies have the same order of magnitude, $\left|E_{RA}\right|\sim E_{ex}$.
This picture assumes that the spins within IM domains are directed
along the anisotropy axis averaged over the IM domain.
One can estimate the zero-field zero-temperature susceptibility of
the RA magnet as follows. \citep{CS-1984,CSS-1986} If a small uniform
field $H$ is applied, the spins deviate from the dominant-anisotropy
direction by a small angle $\delta\theta$, that results in the energy
change $\delta E\sim-H\delta\theta+\left|E_{RA}\right|\left(\delta\theta\right)^{2}$.
Minimizing this energy with respect to $\delta\theta$ and using $\left|E_{RA}\right|\sim E_{ex}$,
for the susceptibility in the energy units $\chi\sim\delta\theta/H$
one obtains
\begin{equation}
\chi=\frac{k}{J}\left(\frac{R_{f}}{a}\right)^{2}.\label{chi_estimation}
\end{equation}
where $k$ is a factor of order unity. The latter depends on the exact
form of the spin-spin correlation function. In 2D this factor also
contains logarithmic dependence on $R_{f}$. At $R_{f}\gg a$ the
susceptibility is large, which explains magnetic softness of RA magnets.
As it was mentioned above, the correlated bunches of spins (IM domains)
tend to orient themselves in the two possible directions along the
predominant anisotropy axis. The energy barrier $\Delta U$ between
these orientations can be estimated as $\Delta U\sim E_{RA}$, Eq.
(\ref{ERA}). Using Eq. (\ref{Rf}), one obtains
\begin{equation}
\Delta U\sim D_{R}\left(\frac{J}{D_{R}}\right)^{d/\left(4-d\right)}=J\left(\frac{D_{R}}{J}\right)^{\frac{2(2-d)}{4-d}}.\label{DeltaU}
\end{equation}
In particular,
\begin{equation}
\Delta U\sim J\begin{cases}
\left(\frac{D_{R}}{J}\right)^{2/3}, & d=1\\
1, & d=2\\
\left(\frac{J}{D_{R}}\right)^{2}, & d=3.
\end{cases}\label{DeltaU_cases}
\end{equation}
Bunches of spins of the size $R_{f}$ that flip over the barrier are
not independent but interacting with their neighbors via the exchange.
The interaction energy can be estimated assuming that the distance
between the neighboring regions of correlated spins is $R_{f}$, so
that the overlap volume is $R_{f}^{d}$. The interaction energy then
has the same form as the exchange energy in the IM argument:
\begin{equation}
E_{int}\sim J\left(\frac{a}{R_{f}}\right)^{2}\left(\frac{R_{f}}{a}\right)^{d}=J\left(\frac{R_{f}}{a}\right)^{d-2}\sim\Delta U.\label{E_int}
\end{equation}
That is, flipping bunches of correlated spins are strongly coupled.
Thus the RA magnet has a similarity with an ensemble of interacting
magnetic particles with a random anisotropy. However, this analogy
is incomplete as IM domains are not real domains and the boundaries
between these ``particles'' are washed out. The fact that the interaction
between IM domains is comparable with their effective anisotropy energy
makes the situation more complicated. Some of IM domains can be directed
along their effective anisotropy axes while some cannot because of
the interaction with their neighbors. There should be many different
ways to minimize the energy with different sets of ``lucky'' and
``unlucky'' IM domains.
The time required to overcome the collective energy barrier for a
large number of correlated spins should be very long, so that in the
intermediate temperature range the system does not come to equilibrium
during sustainable simulation times. At higher temperatures, transitions
between different states are faster and the system reaches the full
(global) equilibrium. At lower temperatures, spin bunches cannot overcome
energy barriers at all. Here, the local equilibrium near one of the
many local energy minima of the system is established relatively fast.
In finite-size systems with linear size $L$, the results above are
valid for $R_{f}\lesssim L$. The value of the random anisotropy at
which $R_{f}\sim L$ can be estimated as
\begin{equation}
D_{R}\sim D_{R}^{*}=J\left(\frac{a}{L}\right)^{(4-d)/2}.
\end{equation}
For $D_{R}\lesssim D_{R}^{*}$ the barrier can be estimated as
\begin{equation}
\Delta U\sim D_{R}\left(\frac{L}{a}\right)^{d/2}
\end{equation}
that must be smaller than the value for the infinite system. Upon
increasing $L$, the barrier approaches its limiting value from below.
\section{The spin-glass order parameter and other computable quantities}
\label{sec:The-spin-glass-order}
The indicator of the glassy transition is freezing described by the
time autocorrelation function averaged over all $N$ spins:
\begin{equation}
K(\tau)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{s}_{i}(t)\cdot\mathbf{s}_{i}(t+\tau).\label{K_def}
\end{equation}
If the system is at global or local equilibrium, the result does not
depend on the time $t$. However, in the intermediate temperature
interval the system evolves in the direction of equilibrium but cannot
reach it during the observation (simulation) time, thus the result
also depends on $t$. In the glassy state, the spins are frozen and
do not deviate much from their initial positions, so that $K(\tau)$
is finite at large $\tau$. Above the glassy transition, spins are
fluctuating wildly, so that $K(\tau)\rightarrow0$ at large $\tau$.
For large systems, computation of $K(\tau)$ is prohibitive as it
requires keeping all spin configurations in memory over a long time
interval.
The SG order parameter based on the temporal evolution of spins can
be defined as
\begin{equation}
q=\frac{1}{N}\sum_{i=1}^{N}\left\langle \mathbf{s}_{i}\right\rangle _{t}\cdot\left\langle \mathbf{s}_{i}\right\rangle _{t},\label{q_def}
\end{equation}
where
\begin{equation}
\left\langle \mathbf{s}_{i}\right\rangle _{t}\equiv\frac{1}{t_{\max}}\intop_{0}^{t_{\max}}dt\mathbf{s}_{i}(t)\label{s_avr_t}
\end{equation}
is the time average over a long time interval. In spin glasses below
the freezing point, $q\rightarrow\mathrm{const}$ for $t_{\max}\rightarrow\infty$.
This definition is similar to Eq. (1.4) of Ref. \citep{Binder}. Instead
of the time averaging or ensemble averaging we use averaging over
statistical samples generated by the Monte Carlo process. Averaging
over realizations of the RA was done for smaller systems but it was
found that it is better to consider larger systems without this averaging
as large systems self-average.
Above the freezing point, the glassy CF $K(\tau)$ asymptotically
vanishes and one can rewrite $q$ as
\begin{equation}
q\cong\frac{\tau_{M}}{t_{\max}},\qquad\tau_{M}\equiv\intop_{-\infty}^{\infty}d\tau K(\tau),\label{tau_M_def}
\end{equation}
where $\tau_{M}$ is melting time. If there is a true SG transition
on temperature, then melting time should diverge on approaching it
from above. Studying the coefficient in the asymptotic $1/t_{\max}$
form of the glassy order parameter $q$ above freezing could yield
the value of the freezing temperature.
One can also compute the autocorrelation function of the average spin
(magnetization)
\begin{equation}
\mathbf{m}=\frac{1}{N}\sum_{i}\mathbf{s}_{i}\label{m_def}
\end{equation}
that is defined by
\begin{equation}
C(\tau)=\mathbf{m}(t)\cdot\mathbf{m}(t+\tau),\label{C_def}
\end{equation}
In simulations on finite-size systems $C(\tau)$ is non-zero and can
be used to monitor freezing. Unlike $K(\tau)$, it can be computed
for large systems and large time intervals.
The value of the equal-time correlation function $C(0)=m^{2}$ is
nonzero in finite-size systems even in the absence of ling-range order
due to short-range correlations. One has
\begin{equation}
m^{2}=\frac{1}{N^{2}}\sum_{i,j}\mathbf{s}_{i}\cdot{\bf s}_{j}=\frac{1}{N}\sum_{j}\langle{\bf s}_{i}\cdot{\bf s}_{i+j}\rangle\Rightarrow\frac{1}{N}\intop_{0}^{\infty}\frac{d^{d}r}{a^{d}}G(r),
\end{equation}
where $G(r)$ is the spatial correlation function and $d$ is the
dimensionality of the space. As the RA magnet has lots of metastable
local energy minima, $G(r)$ depends on the initial conditions and
on the details of the energy minimization routine. In 2D for $G(r)=\exp\left[-\left(r/R_{f}\right)^{p}\right]$
one obtains
\begin{equation}
m^{2}=K_{p}\frac{\pi R_{f}^{2}}{Na^{2}}\quad\Longrightarrow\quad\frac{R_{f}}{a}=m\sqrt{\frac{N}{\pi K_{p}}},\label{Rf_via_m}
\end{equation}
where $K_{1}=2$ and $K_{2}=1$.
Having estimated $R_{f}$, one can find the number of IM domains $N_{IM}$
in the system of size used in the numerical work. In 2D with linear
sizes $L_{x}$ and $L_{y}$ one has $N_{IM}=L_{x}L_{y}/\left(\pi R_{f}^{2}\right)$.
In particular, for a system with $N=300\times340=102000$ spins and
$D_{R}/J=0.3$, energy minimization at $T=0$ starting from a random
spin state yields $m\approx0.2$1, and with $p=2$ one obtains $R_{f}/a\approx37.8$
and $N_{IM}\approx23$. For $D_{R}/J=1$, one obtains $m\approx0.074$
and $R_{f}/a\approx13.3$ that yields $N_{IM}\approx183$. For the
ratio of the $R_{f}$ values one obtains $R_{f}^{(D_{R}=0.3)}/R_{f}^{(D_{R}=1)}\approx2.84$
that is close to the value 3.33 given by Eq. (\ref{Rf}).
One can compute the linear static susceptibility differentiating the
statistical expression for the average magnetization value $\left\langle \mathbf{m}\right\rangle $.
The differential susceptibility per spin has the form
\begin{equation}
\chi_{\alpha\alpha}=\frac{\partial\left\langle m_{\alpha}\right\rangle }{\partial H_{\alpha}}=\frac{N}{T}\left(\left\langle m_{\alpha}^{2}\right\rangle -\left\langle m_{\alpha}\right\rangle ^{2}\right),\label{chi_comps}
\end{equation}
where the average is taken over the statistical ensemble and $\alpha=x,y,z$.
Within the Monte Carlo method, the average is taken over the statistical
sample generated by the Monte Carlo process. The symmetrized form
of the susceptibility in zero field is given by
\begin{equation}
\chi=\frac{N}{3T}\left(\left\langle \mathbf{m}\cdot\mathbf{m}\right\rangle -\left\langle \mathbf{m}\right\rangle \cdot\left\langle \mathbf{m}\right\rangle \right),\label{chi_symm}
\end{equation}
where $\left\langle \mathbf{m}\cdot\mathbf{m}\right\rangle =\left\langle m^{2}\right\rangle $.
Whereas the number of spins $N$ is very large, the difference of
the terms in brackets can be very small below freezing, so that the
result for a large system does not essentially depend on $N$. Unlike
the magnetization value $\left\langle \mathbf{m}\right\rangle $ that
for a large system can be computed using only one system's state,
i.e., without averaging, $\left\langle \mathbf{m}\right\rangle \Rightarrow\mathbf{m}$,
computing $\chi$ requires averaging over different states of the
statistical ensemble. With only one state taken into account, $\chi$
vanishes. Above the freezing temperature in zero field, one has $\left\langle \mathbf{m}\right\rangle =0$,
so that only the first term in the susceptibility formula contributes.
In the frozen state, the two terms are close to each other and their
difference is small. For this reason, there is a lot of numerical
noise in this formula. In the intermediate temperature range the system
does not reach equilibrium during the simulation time, so that Eq.
(\ref{chi_symm}) becomes questionable as it was obtained under the
assumption of equilibrium using the statistical ensemble. The fact
that the system does not come to equilibrium is another source of
the noise in the simulation results for $\chi$. In different simulations,
the system is getting stuck in one of the infinite number of energy
valleys of its phase space that are characterized by different values
of $\left\langle \mathbf{m}\right\rangle $. However, even in this
intermediate region the formula gives plausible results and should
be correct at least qualitatively. Using $\chi_{\alpha\alpha}=\partial\left\langle m_{\alpha}\right\rangle /\partial H_{\alpha}$
is not much better as the result depends on the time allowed for the
system to relax. At high and low temperatures Eq. (\ref{chi_symm})
is correct as either global or local equilibrium is reached.
\section{The numerical method}
\label{sec:The-numerical-method}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{mx_my_mz_vs_n_1024x100_DR=0_RIC_T=0p4}
\par\end{centering}
\begin{centering}
\includegraphics[width=9cm]{m_mrms_mmeam_vs_n_1024x100_DR=0_RIC_T=0p4}
\par\end{centering}
\caption{Pure 2D Heisenberg ferromagnetic model. Upper panel: Evolution of
the magnetization components. Lower panel: Magnetization, root-mean-square
magnetization, and its running average.}
\label{Fig-mx_my_mz_DR=00003D0}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{chix_chiy_chiz_vs_n_1024x100_DR=0_RIC_T=0p4}\caption{Susceptibility vs simulation time in MCS for the pure 2D Heisenberg
ferromagnetic model.}
\par\end{centering}
\label{Fig-chix_chiy_chiz_DR=00003D0}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{C_vs_T_2D_DR=1_fbc_T-down}
\par\end{centering}
\caption{Energy and heat capacity of the 2D RA model.}
\label{Fig-C_vs_T_2D_DR=00003D1}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{FC_ZFC_FW_vs_T_Nx=300_Ny=340_DR=1}
\par\end{centering}
\caption{FC, ZFC, and FW curves for a 2D RA model with $10^{6}$ spins.}
\label{Fig-FC_ZFC_FW}
\end{figure}
Computation of the static properties of RA magnets at $T>0$ could
be done using real dynamics \citep{garchu_arXiv} or Monte Carlo.
The latter is much faster and is the way to go. However, as different
states of the RA magnet are separated by energy barriers and spins
are fluctuating as large correlated groups, there is a very slow relaxation
near and below the freezing point, creating a computational challenge.
To speed-up the relaxation in simulations, one has to combine the
Metropolis Monte Carlo updates that work as slow diffusion in the
phase space of the system with overrelaxation updates that simulate
conservative dynamics allowing to quickly explore the hypersurfaces
of constant energy. For systems with single-site anisotropy, the straightforward
overrelaxation routine, rotating the spins by $180\lyxmathsym{\textdegree}$
around the effective field, leads to the energy decrease and thus
is not working properly.
To beat this problem, we have developed a \textit{thermalized overrelaxation}
routine. Here, the spins are rotated by $180\lyxmathsym{\textdegree}$
around the different-site part of the effective field (e.g., around
the exchange field). As the result, the energy increases or decreases
due to the RA. To compensate for this, the rotation is accepted or
rejected using the Metropolis criterion, same as in the Monte Carlo
updates. In our simulations, for each spin update we used the Metropolis
Monte Carlo with the probability $\alpha=0.1$ and thermalized overrelaxation
with the probability $1-\alpha$. The thermodynamic consistency of
this method has been checked by computing the dynamical spin temperature
$T_{S}$ given by Eq. (9) of Ref. \citep{Garanin}. The values of
$T_{S}$ were in a good accordance with the set temperature $T$.
To minimize the energy of the system at $T=0$, we used the straightforward
overrelaxation routine mentioned above that for the systems with uniaxial
single-site anisotropy leads to the energy decrease. This routine
provides a fast convergence. If in the initial state of the system
all spins are collinear, then upon relaxation the system becomes only
partially disordered with a significant residual magnetization $m$.
If the initial state of the system is random, the relaxed state is
random, too, with $m$ being rather small. Most of the simulations
were done with the random initial conditions (RIC).
In the computation of the dependence of the SG order parameter $q$
on $t_{\max}$, Eq. (\ref{q_def}), we used summation over Monte Carlo
steps (MCS) i.e., system updates, instead of the integration over
time. That is, instead of Eq. (\ref{s_avr_t}) we used
\begin{equation}
\left\langle \mathbf{s}_{i}\right\rangle \equiv\frac{1}{\mathrm{MCS}}\sum_{n=1}^{\mathrm{MCS}}\mathbf{s}_{i}(n),\label{s_avr_MCS}
\end{equation}
where $n$ labels the states generated by the Monte Carlo process.
The asymptotic formula for $q$ above freezing becomes
\begin{equation}
q\cong\frac{\tau_{M}}{\mathrm{MCS}},\qquad\tau_{M}=\sum_{n=-\infty}^{\infty}K(n).\label{tau_M_MCS}
\end{equation}
Here, the melting time $\tau_{M}$ is measured in Monte Carlo steps.
While the latter are not related to the time in any simple way, still
$\tau_{M}$ gives an idea if freezing in the system.
The simulations were done on large 2D systems, typically $1024\times1000$
spins, with periodic boundary conditions. The very large system size
is needed as the system of many spins behaves as that of a much smaller
number of IM domains. In particular, for $D_{R}/J=1$ the system of
$10^{5}$ spins is too small and shows large fluctuations. To reduce
fluctuations, one can either perform repeated measurements on such
systems or simulate a larger system. The latter is preferred.
Usually, in Monte Carlo simulations the system is first equilibrated
and then measurements of the equilibrium properties are performed.
For the RA magnet, the equilibration is extremely long, and around
the freezing temperature the system does not come to equilibrium at
all after one million MCS that is about our limit for $10^{6}$ spins.
Thus we resorted to performing a fixed number of MCS for each temperature.
Our first attempts included stepwise lowering $T$ performing from
ten to twenty thousand MCS for each temperature point. However, such
simulation duration proved to be too short, and increasing it in the
cycle over the temperatures would result in exceedingly long computation.
Therefore, long simulations, up to $10^{6}$ MCS, for select temperature
values have been performed, each time starting from quenched states
obtained by energy minimization starting from random spin states.
Each of these simulations required several days.
As the computing software, Wolfram Mathematica with compilation and
parallelization was used. Most of computations were performed on our
Dell Precision workstation having 20 CPU cores from which 16 cores
were used by Mathematica.
\section{Testing the numerical method on the pure 2D magnet}
\label{sec:Testing-the-numerical}
First, we test the numerical method on the pure 2D Heisenberg ferromagnetic
model. In this case, the thermalized overrelaxation degenerates to
the regular overrelaxation as it conserves energy. There is no phase
transition on temperature in this model but a strong short-range order
with exponentially large magnetic susceptibility and exponentially
long correlation length establishes with lowering temperature in the
infinite system. This happens around $T/J=0.7$ where the heat capacity
has a maximum. In simulations on finite-size systems, further lowering
the temperature quickly results in the correlation length exceeding
the system size $L$, and the system behaves as an ordered magnetic
particle. In this regime, the susceptibility is not exponentially
large but still huge: $\chi=N\left\langle m^{2}\right\rangle /(3T)$.
The results of a single simulation of the pure 2D Heisenberg ferromagnetic
model at $T/J=0.4$ are shown in Figs. \ref{Fig-mx_my_mz_DR=00003D0}
and \ref{Fig-chix_chiy_chiz_DR=00003D0}. The equilibrium value of
the root-mean-square magnetization $\left\langle m^{2}\right\rangle ^{1/2}$
stabilizes quickly enough with increasing the number of system updates
MCS, as can be seen in the lower panel of Fig. \ref{Fig-mx_my_mz_DR=00003D0}).
However, computing the linear susceptibility $\chi$ using Eq. (\ref{chi_symm})
for the system of one million spins requires about a million of system
updates, see Fig. \ref{Fig-chix_chiy_chiz_DR=00003D0}. Such a long
simulation is required to average out the system's magnetization vector
$\mathbf{m}$ that contributes to the second term in Eq. (\ref{chi_symm}).
The slow evolution of the components of $\mathbf{m}$ is shown in
the upper panel of Fig. \ref{Fig-mx_my_mz_DR=00003D0}. After one
million of system updates one obtains $\chi J=157838$ that is almost
as large as the ``magnetic-particle'' value defined just above with
$N=1024\times1000$, $\sqrt{\left\langle m^{2}\right\rangle }=0.46$,
and $T/J=0.4$ that is $\chi J=180565$.
One also can check what becomes the freezing parameter $q$ of Eq.
(\ref{q_def}) for the pure system. Theoretically, one expects $q=0$
at any $T>0$. However, for a system of $10^{6}$ spins at $T/J=0.4$
one needs to perform hundreds of thousands MCS to see that the system
is not frozen. The dependence of $q$ on the number of MCS performed
is added to Fig. \ref{Fig_q_lin-lin} below. These simulations of
the pure system show that the problem is computationally involved.
In the presence of random anisotropy it becomes harder because of
even longer relaxation due to thermally-activated barrier crossing
by large groups of correlated spins.
\section{Numerical results}
\label{sec:Numerical-results}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{q2_vs_n_320x320_DR=1_RIC_log-log}
\par\end{centering}
\begin{centering}
\includegraphics[width=9cm]{q2_vs_n_1024x1000_DR=1_RIC_log-log}
\par\end{centering}
\caption{The dependence of the SG order parameter on the number of Monte Carlo
steps in log-log scale. Upper panel: a smaller system of $10^{5}$
spins. Lower panel: the system of $10^{6}$ spins. The dashed line
is the asymptote $q=\tau_{M}/\mathrm{MCS}$ for $T/J=0.59$.}
\label{Fig-q2_log-log}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{\string"Melting_rate_1024x1000_DR=1\string".png}
\par\end{centering}
\caption{Melting rate above freezing -- natural and power-law representations.}
\label{Fig-Melting rate}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{Melting_rate_Arrhenius}
\par\end{centering}
\caption{Melting rate above freezing -- Arrhenius representation.}
\label{Fig-Melting_rate_Arrhenius}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{q2_vs_n_1024x1000_DR=1_RIC_lin-lin}
\par\end{centering}
\caption{The dependence of the SG order parameter on the number of Monte Carlo
steps in linear scale at different temperatures.}
\label{Fig_q_lin-lin}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{q2_vs_T_1024x1000_DR=1_long_runs}
\par\end{centering}
\caption{Asymptotic values of the spin-glass order parameter $q$ at different
temperatures. The curve with open circles was obtained by gradually
lowering the temperature making $2\times10^{4}$ MCS for each $T$
value. The curve with filled circles was obtained by making $10^{6}$
MCS at each $T$ value one by one for different RA realizations. }
\label{Fig_q_vs_T}
\end{figure}
\subsection{FC-ZFC-FW curves}
Figure \ref{Fig-FC_ZFC_FW} shows the result for the FC, ZFC, and
FW curves in a 2D RA model with the a million spins and the RA strength
$D_{R}/J=1$. Here a very weak field is applied along $z$ axis and
$m_{z}\equiv\left\langle s_{i,z}\right\rangle $ was measured. The
ZFC curve was obtained by first minimizing the system's energy starting
from a random orientation of spins at $T=0$ and $H=0$, than applying
the field $H$ and gradual warming the system. The FC curve was obtained
by gradual cooling the system from a high temperature to $T=0$ in
the presence of the field. Finally, the field-warmed (FW) curve was
obtained by first minimizing the system's energy in the applied field
starting from the state with all spins directed along $z$ axis and
then gradually warming the system. For each temperature point, 10000
system updates were performed. One can see that the ZFC curve merges
with the other two curves at $T/J\simeq0.53$ that can be interpreted
as spin-glass transition or freezing temperature $T_{SG}$. The estimated
magnetic susceptibility near freezing is huge, $\chi=m_{z}/H\simeq400$.
This is unlike that in the conventional spin glasses with a random
exchange. Each curve was obtained by averaging over three different
runs. For a system of $10^{5}$ spins fluctuations are much stronger,
so that a more extensive averaging over runs is needed.
\subsection{Spin-glass order parameter and melting time}
Then, we have performed a long annealing, up to $10^{6}$ MCS, of
the system at different temperatures after the energy minimization
(quenching) starting from random initial conditions. Each of these
simulations took several days, so they had to be done one-by-one rather
than in a cycle. Each simulation run used its own realization of the
RA and its own random initial spin state. The results of each run
were the dependences of $q$ and the components of the average spin
$\mathbf{m}$, Eq. (\ref{m_def}), vs the number of MCS. From the
$\mathbf{m}$ data, the susceptibility components, Eq. (\ref{chi_comps})
and the symmetrized susceptibility, Eq. (\ref{chi_symm}) were derived.
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{mx_my_mz_vs_n_1024x100_DR=1_RIC_T=0p58}
\par\end{centering}
\begin{centering}
\includegraphics[width=9cm]{chix_chiy_chiz_vs_n_1024x100_DR=1_RIC_T=0p58}
\par\end{centering}
\caption{Simulation above the freezing point, $T/J=0.58$. Upper panel: evolution
of the magnetization components. Lower panel: dependence of the susceptibility
components (colored curves) and the symmetrized susceptibility (black
curve) on the number of MCS done.}
\label{Fig_T=00003D0.58}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{mx_my_mz_vs_n_1024x100_DR=1_RIC_T=0p51}
\par\end{centering}
\begin{centering}
\includegraphics[width=9cm]{chix_chiy_chiz_vs_n_1024x100_DR=1_RIC_T=0p51}
\par\end{centering}
\caption{Simulation just below the freezing point, $T/J=0.51$. Upper panel:
evolution of the magnetization components. Lower panel: dependence
of the susceptibility components (colored curves) and the symmetrized
susceptibility (black curve) on the number of MCS done. The system
does not come to equilibrium and $\chi$ does not stabilize with increasing
of the number of MCS done.}
\label{Fig_T=00003D0.51}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{mx_my_mz_vs_n_1024x100_DR=1_RIC_T=0p2}
\par\end{centering}
\begin{centering}
\includegraphics[width=9cm]{chix_chiy_chiz_vs_n_1024x100_DR=1_RIC_T=0p2}
\par\end{centering}
\caption{Simulation at a lower temperature, $T/J=0.2$. Upper panel: evolution
of the magnetization components; Lower panel: dependence of the susceptibility
components (colored curves) and the symmetrized susceptibility (black
curve) on the number of MCS done.}
\label{Fig_T=00003D0.3}
\end{figure}
The dependence of the SG order parameter $q$ of Eq. (\ref{q_def})
on the number of MCS near the freezing temperature for a smaller system
of $10^{5}$ spins is shown in the upper panel of Fig. \ref{Fig-q2_log-log}.
One can see that this size is too small, as the system behaves as
that of a much fewer number of entities and fluctuations are too strong.
For $T/J=0.53$ at $\mathrm{MCS}=3\times10^{6}$ most of the system's
$10^{5}$ spins suddenly change their direction that leads to a sharp
decrease of $q$. A really large system should not behave like this.
One has either to perform an extensive averaging over runs of take
a larger system that is preferable. The data for $10^{6}$ spins in
the lower panel of Fig. \ref{Fig-q2_log-log} are much smoother and
show the asymptotic power-law dependence $q=\tau_{M}/\mathrm{MCS}$
above the freezing point, in accordance with Eq. (\ref{tau_M_def}).
Fitting the dependence of the SG order parameter with $q=\tau_{M}/\mathrm{MCS}$
one can extract the melting time $\tau_{M}$. The latter becomes very
large when the system freezes. If there is true phase transition at
some freezing temperature $T_{f}$, one can expect a power-law divergence
$\tau_{M}\propto\left(T-T_{f}\right)^{-\gamma}$. The results for
the melting rate $1/\tau_{M}$ are shown in Fig. \ref{Fig-Melting rate}.
The temperature dependence of $1/\tau_{M}^{1/6}$ is a straight line
that suggests $\gamma=6$ and $T_{f}/J\simeq0.49$. The power 6 is
too high to be credible while the freezing temperature is rather low
and difficult to approach from above because of too slow relaxation
requiring exorbitant computing times.
Another way to fit the results for the melting time is using the Arrhenius
temperature dependence $\tau_{M}\propto\exp\left(\Delta U/T\right)$.
The corresponding data representation shown in Fig. \ref{Fig-Melting_rate_Arrhenius}
yields the barrier value $\Delta U=23J$. Theoretically, Eq. (\ref{DeltaU_cases})
yields $\Delta U\sim J$ for $d=2$ but there can be a large numerical
factor in $\Delta U$. At larger temperatures, one can see expected
deviations from the Arrhenius law (as well as deviations from the
power law in Fig, \ref{Fig-Melting rate}). This interpretation implies
that there is no phase transition and freezing is a gradual process.
Another argument in favor of a gradual freezing/melting is the fact
that in systems with quenched disorder the properties averaged over
large regions should fluctuate. If in one region the averaged random
anisotropy is larger than in the others, freezing /melting in this
region will occur at slightly higher temperatures. Thus the freezing
temperature will be spread. At some temperature, most of the system
will be melted while some minoruty regions will be still frozen, providing
a small but nonzero value of the spin-glass order parameter $q$.
In this scenario, $q(T)$ dependence is neither a power nor an exponential
of the temperature.
In the frozen state, as can be seen in Fig. \ref{Fig_q_lin-lin},
the SG order parameter quickly reaches its asymptotic value that is
smaller than one because of the thermal motion of spins on IM domains
in their valleys without crossing the barriers to different valleys.
With increasing the temperature towards the melting point, the processes
of crossing the barriers begin and $q$ slowly decreases. Above the
freezing point, such as $T/J=0.6$, the SG order parameter quickly
decreases to zero.
The values of the SG order parameter $q$ computed at different temperatures
are shown in Fig. \ref{Fig_q_vs_T}. The curve with open circles was
obtained by gradually lowering the temperature making $2\times10^{4}$
MCS for each $T$ value. This duration of annealing is insufficient
to reach stable results. The points of the curve with filled circled
obtained one by one for different RA realizations with $10^{6}$ MCS
are significantly shifted down. In this case, the equilibrium is reached
in the main part of the temperature interval except for the vicinity
of the freezing transition. This is confirmed by the plateaus of $q$
vs the number of MCS in Fig. \ref{Fig_q_lin-lin}.
\subsection{Magnetization and susceptibility}
\begin{figure}
\begin{centering}
\includegraphics[width=9cm]{chi_vs_T_1024x1000_DR=1_long_runs}
\par\end{centering}
\begin{centering}
\includegraphics[width=9cm]{chi_vs_T_1024x1000_DR=1_long_runs_Low_T}
\par\end{centering}
\caption{Linear susceptibility $\chi$ at different temperatures. Upper panel:
broad temperature range; Lower panel: Low temperatures.}
\label{Fig_chi_vs_T}
\end{figure}
Above the freezing point, the components of the magnetization defined
by Eq. (\ref{m_def}) fluctuate fast around zero, as can be seen in
the upper panel of Fig. \ref{Fig_T=00003D0.58}. Even for a disordered
system of one million spins the magnetization is significant that
is the consequence of a strong short-range order that establishes
below $T/J=0.7$ where the system has the maximum of the heat capacity,
even in the absence of the RA. Because of the fast fluctuations, the
linear susceptibility computed with the use of Eqs. (\ref{chi_comps})
and (\ref{chi_symm}) and shown in the lower panel of Fig. \ref{Fig_T=00003D0.58}
reaches its asymptotic value within the simulation interval of $10^{6}$
MCS. Different components of the susceptibility have approximately
the same value.
On the contrast, in the intermediate temperature range at and below
freezing, in addition to fast fluctuations of the magnetization, there
is slow dynamics, apparently due to thermally-activated barrier crossing.
As can be seen in the upper panel of Fig. \ref{Fig_T=00003D0.51},
slow changes of $\mathbf{m}$ do not average out within the simulation
interval of $10^{6}$ MCS. Slow fluctuations of $\mathbf{m}$ are
large and thus make a large contribution to the linear susceptibility.
As can be seen in the lower panel of Fig. \ref{Fig_T=00003D0.51},
the susceptibility does not stabilize and continues to grow. In this
temperature range, the susceptibility values are huge because of the
correlated motion of large groups of spins over energy barriers.
At lower temperatures, there are no large fluctuations of the magnetization
due to overbarrier transitions, as can be seen in the upper panel
of Fig. \ref{Fig_T=00003D0.3}. The fluctuations seen in the figure
are due to the motion of correlated spin bundles within their valleys.
Note that $z$ axis is chosen in the direction of the magnetization
in the initial state obtained by the energy minimization from the
random spin state.
The values of the symmetrized linear susceptibility $\chi$ at different
temperatures computed using Eq. (\ref{chi_symm}) are shown in Fig.
\ref{Fig_chi_vs_T}. At higher temperatures, the susceptibility is
small and practically coincides with that of the pure system. One
can see that $\chi$ has huge values in the region of freezing. However,
the values obtained in this region strongly fluctuate and are only
approximate as the simulation duration of $10^{6}$ MCS proves to
be insufficient to average out the magnetization fluctuations (see
the upper panel of Fig. \ref{Fig_T=00003D0.51}). Better results,
probably, could be obtained for the simulation longer by an order
of magnitude that for such a large system is problematic. At low temperatures,
the scatter in the susceptibility decreases and the susceptibility
values approach a plateau with the height in a fair accordance with
Eq. (\ref{chi_estimation}) that for $D_{R}/J=1$ with $R_{f}/a\approx13.3$
and $k=0.5$ yields $\chi J\simeq88$.
In addition, one could compute the correlation functions (CFs) of
the $\mathbf{m}$ time series shown in figures above. Above freezing,
these CFs quickly decay to zero, At intermediate temperatures, they
decrease slowly with large fluctuations, as suggested by the upper
panel of Fig. \ref{Fig_T=00003D0.51}. At low temperatures where there
are no overbarrier transitions, time CFs quickly decrease from their
equal-time values to their plateau values. These CFs have been computed
for the same model from the dynamical evolution and shown in Fig.
3 of Ref. \onlinecite{garchu_arXiv}, Those results suggest freezing
at $T/J$ between 0.5 and 0.6, in accordance with the current, more
precise, results.
\section{Discussion}
Most of the previous studies of random-anisotropy (RA) magnets were
focused on their equilibrium behavior or on their quasi-equilibrium
properties in a frozen glassy state. Rigorous analytical solution
of this set of problems has never been provided, while numerical studies
have been hampered by the necessity to consider large systems in order
to account for extended ferromagnetic correlations. Capabilities of
modern computers have allowed us to revisit this problem. In this
paper we have studied glassy properties of the random-anisotropy magnet
as a function of temperature with the combination of the Metropolis
Monte Carlo method and specially developed thermalized overrelaxation.
The questions we asked are the extent to which metastability plays
a role in defining magnetic properties of such a system, the freezing
of the magnetic configuration due to energy barriers on lowering temperature
vs a spin-glass transition due to exchange interaction between spins,
the time evolution of a conventionally defined spin-glass order parameter,
the characteristic melting time in the temperature region just above
freezing, the field-cooled (FC) and zero-field-cooled (ZFC) magnetization
curves, and the temperature dependence of the magnetic susceptibility
of RA magnets. The computed energy barriers agree within order of
magnitude with the Imry-Ma argument for systems with quenched disorder.
The computed FC and ZFC magnetization curves have close resemblance
with the experimental curves. These findings provide confidence in
our numerical method.
A more challenging task has been distinguishing between blocking of
overbarrier spin-group transitions on reducing temperature (that implies
the Arrhenius temperature dependence of the melting time above freezing)
and a true spin-glass phase transition (that implies a power-law divergence
of the melting time at transition point). While we cannot say with
confidence that we have answered this question unambiguously, our
findings provide a stronger argument in favor of a continuous freezing
(blocking) transition on lowering temperature. The main evidence of
this is a rather high power in the power-law fit of the melting time
and a rather low resulting transition temperature. In accordance with
our numerical experiments, the maximum of the susceptibility occurs
where the FC and ZFC magnetization curves merge. The low temperature
value of the susceptibility roughly agrees with the one derived from
the Imry-Ma argument. The temperature dependence of the susceptibility
near and below the maximum has a strong scatter caused by the finite
size of the system (one million spins) and finite computing time (one
million Monte Carlo steps). It did not allow us to distinguish between
a smooth behavior at the maximum and a cusp that was experimentally
observed in spin glasses. Studies of larger systems and longer computation
times would be needed to make such a distinction.
\section*{Acknowledgments}
This work has been supported by the Grant No. FA9550-20-1-0299 funded
by the Air Force Office of Scientific Research.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,281
|
Q: Mongodb java - how to query for particular reference id value I have two collections
checkone
{"_id":1,"name":"alex"},
{"_id":2,"name":"sandy"}
checktwo
{"_id":11,"password":"alex",tenant_id:{$ref:"checkone","$id":1}}
{"_id":12,"password":"alex",tenant_id:{$ref:"checkone","$id":2}}
I am querying for particular field tenant_id value.
DB database = mongo.getDB("testcheck");
DBCollection tenant = database.getCollection("checktwo");
BasicDBObject query = new BasicDBObject();
BasicDBObject field = new BasicDBObject();
field.put("tenant_id.$id", 1);
DBCursor cursor = tenant.find(query,field);
while (cursor.hasNext()) {
System.out.println("cursor value");
System.out.println(cursor.next().get("tenant_id.$id"));
}
Output:
cursor value
null
cursor value
null
But when I query System.out.println(cursor.next().get("_id"));
Output:
cursor value
11.0
cursor value
12.0
How to query for tenant_id value alone? The output must be cursor value 1, cursor value 2
A: You want to use the DBRef class. Here's a rewrite of your original while() loop.
BasicDBObject query = new BasicDBObject();
BasicDBObject field = new BasicDBObject();
DBCursor cursor = tenant.find( query, field );
while( cursor.hasNext() ) {
System.out.println( "cursor value" );
DBRef ref = ( DBRef )cursor.next().get( "tenant_id" );
System.out.println( ref.getId() );
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,664
|
La maison Kittleson est une maison en bois de style American Foursquare construite en 1911 à dans le comté d'Iowa, dans le sud-est de l'État américain du Wisconsin. Elle a été ajoutée au registre national des lieux historiques en 1986.
Histoire
Le bâtiment est utilisé comme résidence domestique depuis sa construction. Il a été reconnu par le National Park Service avec une inscription au registre national des lieux historiques (National Register of Historic Places ou NRHP) le .
Architecture
D'après la désignation NRHP, . L'architecte de la maison est inconnu.
Références
Liens externes
Maison inscrite au Registre national des lieux historiques au Wisconsin
Registre national des lieux historiques en 1986
Édifice construit en 1911
Comté d'Iowa (Wisconsin)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,488
|
While day-to-day weather may appear to have no apparent pattern, if the weather patterns are analyzed over a longer period of time, patterns start to emerge. One of the most prevalent weather events in Gallatin Valley is wind. Using wind roses, comparisons of the Gallatin Valley wind patterns can be made over different time periods, such as diurnally, monthly, or seasonally. A wind rose is a diagram that depicts the distribution of wind direction and speed at a location over a period of time. The length of each spoke on a wind rose indicates how often the wind comes from a specific direction. Longer spokes mean the wind comes from that direction more often. The colors on each spoke displays how often the wind from this direction falls within a given wind speed range.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,453
|
Q: In PGFplots, how can I have arrows as markers for plots in order to have them appear in the legend? In my plot, I want to indicate the direction of a graph with arrows.
Usually, I would just create something like this:
\begin{tikzpicture}
\begin{axis}[
axis x line=center,
axis y line=center,
xmax=3.5,xmin=-4,
ymax=6,ymin=-4,
xlabel=$x_1$,
ylabel=$x_2$,
xtick=\empty,ytick=\empty
]
\addplot[smooth,domain=-3.5:2.7,mark=none,black]{-(x/1.5+0.1)^2 + 2};
\pgfmathsetmacro{\arronestart}{-(-0.6/1.5+0.1)^2+2};
\pgfmathsetmacro{\arroneend}{-(-0.59/1.5+0.1)^2+2};
\draw[
decoration={markings,mark=at position 1 with {\arrow[ultra thick]{stealth}}},
postaction={decorate}
] (axis cs:-0.6,\arronestart) -- (axis cs:-0.59,\arroneend);
\pgfmathsetmacro{\arrtwostart}{-(2.69/1.5+0.1)^2+2};
\pgfmathsetmacro{\arrtwoend}{-(2.7/1.5+0.1)^2+2};
\draw[
decoration={markings,mark=at position 1 with {\arrow[ultra thick]{stealth}}},
postaction={decorate}
] (axis cs:2.69,\arrtwostart) -- (axis cs:2.7,\arrtwoend);
\end{axis}
\end{tikzpicture}
Now, what I would like to have is a legend entry for those arrows, something like: ->- , but of course this is not possible with \draw[] ;. So, is there a way to have arrows as markings, so that I can create the same graph just using \addplot instead of \draw and have a proper legend?
A: You can use the \addlegendimage command to document custom draw statements:
\draw[red,->] (axis cs:0,1) -- (axis cs:1,2);
\addlegendimage{red,->}
\addlegendentry{Something}
Here, the \addlegendimage provides the options to visualize the legend entry and \addlegendentry provides the text.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,651
|
Ashton House Plan - Traditional on the outside and innovative on the inside, the Ashton incorporates the best of both worlds! The brick and siding exterior gives the ageless appeal that every neighborhood welcomes. Inside, a keeping room connects to the kitchen area, providing the perfect place for relaxing family time. The master suite is truly luxurious with a bayed sitting area, his-and-her closets and a decorative art niche. An optional bonus room is available on the upper floor that can be used as the homeowner wishes. An exercise area, craft room or playroom would be easily accommodated by this space. Traditional on the outside and innovative on the inside, the Ashton incorporates the best of both worlds! A keeping room connects to the kitchen area, providing the perfect place for relaxing family time. The master suite is truly luxurious with a bayed sitting area, his-and-her closets and a decorative art niche. Photo courtesy of Athens Construction Group and Maythi Calvert.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,392
|
Fabio Milano (born 2 August 1977) is an Italian baseball player who competed in the 2004 Summer Olympics.
References
1977 births
Baseball players at the 2004 Summer Olympics
Living people
Olympic baseball players of Italy
Fortitudo Baseball Bologna players
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,515
|
1. Set the oven at 200°c (400°f) gas 6.
2. To make the pastry, sift the flour and salt into a bowl, then rub in the margarine until the mixture resembles fine breadcrumbs. Add enough cold water to make a stiff dough. Press the dough together with your fingertips.
3. Roll out the pastry on a lightly floured surface and use to line twelve 7.5 cm (3 inch) patty tins or boat shaped moulds.
4. Bake the tartlets blind for 10 minutes.
5. Drain and stone the cherries, reserving the syrup in a measuring jug. Make it up to 125 ml (4 fl oz) with water, if necessary.
6. Put a layer of cherries in each pastry case.
7. Pour the cherry syrup mixture into a saucepan, add the sugar and bring to the boil. Boil for 5 minutes.
8. Meanwhile mix the arrowroot to a paste with the lemon juice in a cup.
9. Add to the syrup, stirring all the time, and bring to the boil.
10. Add a little red food colouring. Cool the glaze slightly, then pour a little over the cherries in each tartlet. Leave to set.
11. In a bowl, whip the cream until stiff. Put it in a piping bag fitted with a 1 cm (1/2 inch) nozzle and pipe a large rosette on each tartlet.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,843
|
define('aurora/resolver', [], function () {
'use strict';
function resolveController(parsedName) {
try {
return require('aurora/controllers/' + parsedName.name);
} catch (e) {
return null;
}
}
function resolveRouter() {
return require('aurora/router');
}
function resolveRoute(parsedName) {
try {
return require('aurora/routes/' + parsedName.name);
} catch (e) {
return null;
}
}
function resolveTemplate(parsedName) {
return require('aurora/templates/' + parsedName.name);
}
return Ember.DefaultResolver.extend({
resolveController: resolveController,
resolveRouter: resolveRouter,
resolveRoute: resolveRoute,
resolveTemplate: resolveTemplate
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,651
|
\section{Introduction}
Despite the phenomenal success of density functional theory (DFT) in electronic structure, its standard approach is both conceptually and (often) practically ill-suited for an accurate description of the energy levels in a material or chemical system \cite{doi:10.1021/cr200107z}. These quantities are however essential for predictions of fundamental bandgaps and other charged excitation properties which govern the photo-dynamics, transport and response properties of a system. Into this, {\em GW} theory has grown in popularity, first for materials and more recently for molecular systems,
as a post-mean-field approach to obtain charged excitation spectra in a principled diagrammatic fashion, free from empiricism\cite{Onida2002, Hybertsen1985, Aryasetiawan1998, Hedin1999, Aulbur2000, Friedrich2006, Kutepov2009, Ke2011, Bruneval2012, VanSetten2013, VanSetten2015, Reining2017, Golze2019}.
The $GW$ approach is based on Hedin's equations\cite{Hedin1965, Hedin1970}, and in its most common formulation builds a self-energy to dress a reference description of the quasi-particles of a system (generally from DFT or Hartree--Fock (HF)) with an infinite resummation of all `bubble' diagrams. These diagrams make up the random phase approximation (RPA)\cite{Langreth1977, Chong1995, Heselmann2010, Ren2012},
and physically describes all collective quantum charge fluctuations in the electron density from the reference state arising from their correlated mutual Coulomb repulsion. This dynamically screens the effective interaction between the constituent quasi-particles of the system, whose physics generally dominates in small gapped semi-conducting systems. The use of this RPA screened interaction in $GW$ has therefore become widespread, correcting many of the failures of DFT for spectral properties.
At the core of $GW$ theory is a convolution, between the Green's function of the system, $G(\omega)$, and the screened Coulomb interaction $W_p(\omega)$, obtained (in general) at the RPA level of theory. This provides the dynamical part of the self-energy, $\Sigma(\omega)$, formally written as $\Sigma(\omega)= (i/2\pi) \int d \omega' e^{i \eta \omega'} G(\omega + \omega') W_p (\omega')$. There are many different variants of $GW$ theory\cite{Leeuwen09, Hybertsen1985, Reining2017, Golze2019, Foerster2011, Bruneval2013, PhysRevB.94.165109, Knight2016, Maggio2017, Bruneval2021, vonBarth1996, Holm1998, Schone1998, GarciaGonzalez2001, Faleev2004, VanSchilfgaarde2005, Stan2006, Kotani2007, Shishkin2007, Caruso2012, Bruneval2016, Kaplan2016, Jin2019, Duchemin2020, Duchemin2021, PhysRevB.106.235104,Ren_2012},
which primarily differ due to i) the choice (or absence) of self-consistency conditions on $G(\omega)$ and/or $W_p(\omega)$\cite{Leeuwen09}, ii) the approach to find the quasi-particle energies once $\Sigma(\omega)$ is obtained (i.e. application of Dyson's equation), and iii) different approximations to perform the frequency integration in the convolution itself. A numerically exact formulation of this convolution entails an $\mathcal{O}[N^6]$ scaling step, required to find the entire set of poles in $W_p(\omega)$ from the RPA. However, there are a number of techniques to approximate this frequency integration which can reduce this scaling (generally down to $\mathcal{O}[N^3-N^4]$) based on plasmon pole approximations, analytic continuation, contour deformation and explicit grid resolved approaches for the dynamics of these quantities, amongst others. All of these approaches compress and approximate the dynamical resolution of the key quantities in order to simplify the resulting convolutional integral.
Another key difference between the approaches rests on how the quasi-particle energies are updated from their mean-field molecular orbital (MO) energy starting point, once the self-energy has been constructed. This is formally an application of Dyson's equation, but is commonly approximated via a self-consistent solution to a quasi-particle (QP) equation entailing a diagonal approximation to the self-energy. This is valid when the quasiparticle energies are far from the self-energy poles, thereby asserting that the $GW$ largely just provides a shift in the original MO energies, rather than introducing significant quasiparticle renormalization, additional satellite peaks from state splitting or relaxation of the mean-field electron density. These assumptions can however break down (especially in more correlated systems), while the numerical solution of the QP equation can also converge to different (`spurious') results based on the specifics of how it is solved. A thorough study of the discrepancies due to different approaches to both the QP equation solution and the frequency integral in the convolution can be found in Ref.~\nocite{VanSetten2015}\citenum{VanSetten2015}.
In this work, we introduce a new approach to this frequency integration and reformulate $GW$ theory with a number of desirable properties, while retaining a low scaling. The key step is that the self-energy is not represented as an explicitly dynamical quantity, but instead in terms of a series of static matrices representing the \emph{moments} of its frequency-dependence up to a given order. These can be directly obtained, and from them a compressed representation of the full self-energy can be algebraically constructed which only has a number of poles which scales linearly with system size, but nonetheless exactly preserves the moment distribution of the exact $GW$ self-energy dynamics up to the desired order\cite{Backhouse2020b, Backhouse2021, Backhouse2022, Sriluckshmy2021}. This order can be systematically increased to more finely resolve the full dynamical dependence of the $GW$ self-energy. The dynamical information is therefore implicitly recast into a small number of {\em static} matrices, each of which can be obtained in $\mathcal{O}[N^4]$ time (with a proposed $\mathcal{O}[N^3]$ algorithm also given). This removes the need for the definition of any frequency or time grids in which to resolve dynamical quantities, spectral broadening, finite temperatures, Fourier transforms or analytic continuation, with all dynamics implicitly represented by this series of static quantities.
Furthermore, once these spectral moments of the self-energy are obtained, the QP equation and restrictions to a diagonal self-energy representation can be entirely removed, with an exact application of Dyson's equation possible in this `moment' representation. This leads to the self-energy represented as a small number of explicit poles at specific energies which taken together have exactly the moment distribution in their dynamics as described. This allows for a simple construction of the full frequency dependence of the resulting quasi-particle spectrum (including any additional emergent satellite structures from the correlations) via diagonalization in an `upfolded' and explicit Hamiltonian representation.
In Sec.~\ref{sec:Mom_GW} we show how these spectral moments of the self-energy can be directly constructed from moments of the Green's function and the two-point density-density response function from RPA, without any further approximation. We show how these can be used to directly obtain the full-frequency $GW$ spectrum without the requirement of an explicit grid. In Sec.~\ref{sec:DD_RPA} we show how the RPA can be fully reformulated as a series expansion of moments of the dd-response, and in Sec.~\ref{sec:EfficientEval} show how they can be efficiently obtained in $\mathcal{O}[N^4]$ cost, based on ideas from the seminal work in 2010 by Furche and collaborators on low-scaling approaches for the RPA correlation energy \cite{Eshuis2010}. Furthermore, in Sec.~\ref{sec:CubicScaling} we propose an approach to further reduce the scaling of the whole algorithm directly (rather than asymptotically) to cubic cost with system size, without invoking screening or locality assumptions. We then apply the approach in Sec.~\ref{sec:Results} to the commonly used molecular $GW100$ test set frequently used to benchmark $GW$ implementations, demonstrating a rapid convergence of the moment expansion, and accurate and efficient results across this test set for the $G_0W_0$ level of theory.
\section{Moment-truncated GW theory} \label{sec:Mom_GW}
In $GW$ theory, the dynamical part of the self-energy, obtained as the convolution of the $G(\omega)$ and $W_p(\omega)$, can be formally expanded as a sum over the $\mathcal{O}[N^2]$ neutral excitations of RPA theory (representing the poles of the screened Coulomb interaction) and the charged excitations of the reference Green's function. In the absence of self-consistency (the most common `$G_0W_0$' formulation of the method, which we exclusively consider in this work), the Green's function is just given from the $\mathcal{O}[N]$ mean-field molecular orbital energies. This allows the self-energy to be explicitly evaluated in the frequency-domain as
\begin{align}
\Sigma_{pq}&(\omega)=\sum_{\nu} \sum_{ia,jb,k} \frac{ (pk|ia) (X_{ia}^{\nu} + Y_{ia}^{\nu})(X_{jb}^{\nu} + Y_{jb}^{\nu}) (qk|jb)}{\omega - (\epsilon_k - \Omega_{\nu})-i0^+} \nonumber \\
&+ \sum_{\nu} \sum_{ia,jb,c} \frac{ (pc|ia) (X_{ia}^{\nu} + Y_{ia}^{\nu})(X_{jb}^{\nu} + Y_{jb}^{\nu}) (qc|jb)}{\omega - (\epsilon_c + \Omega_{\nu})+i0^+} . \label{eq:fullSE}
\end{align}
In these expressions, $\Omega_\nu$ are the neutral excitation energies from RPA theory, and $X_{ia}^\nu$ and $Y_{ia}^\nu$ are the excitation and de-excitation amplitude components associated with this excitation. These are expanded in the basis of hole and particle spin-orbitals of the reference mean-field, represented by the indices $i,j,k$ ($a, b, c$) of dimension $o$ ($v$) respectively, and with orbital energies denoted by $\epsilon_x$. The bare two-electron integrals are denoted by $(pk|ia)$ in standard Mulliken (`chemists') notation, which are therefore screened by the RPA reducible density response. More details on these quantities are given in Sec.~\ref{sec:DD_RPA}. The first term in Eq.~\ref{eq:fullSE} therefore represents the `lesser' part, and the second term the `greater' part of the full $G_0W_0$ self-energy.
Exact evaluation of the RPA excitations ($\Omega_\nu$) scales as $\mathcal{O}[N^6]$, rendering it unsuitable for large-scale implementations. However, in this work, we are only interested in evaluating the spectral moments of the resulting self-energy, and finding a resulting compressed representation of the self-energy with fewer poles, but which by construction matches the spectral moments up to a desired order. These frequency-independent spectral moments are defined separately for the greater and lesser parts, and represent the $n^\mathrm{th}$-order moments of the resulting dynamical distributions, as
\begin{align}
\Sigma^{(n,<)}_{pq} &= -\frac{1}{\pi} \int_{-\infty}^{\mu} \mathrm{Im}[\Sigma(\omega)_{pq}] \omega^n d\omega \\
&= (-1)^n
\left. \frac{d^n \Sigma(\tau)_{pq}}{d \tau^n} \right|_{\tau=0^+} ,
\end{align}
and similarly
\begin{align}
\Sigma^{(n,>)}_{pq} &= \frac{1}{\pi} \int_{\mu}^{\infty} \mathrm{Im}[\Sigma(\omega)_{pq}] \omega^n d\omega \\
&= (-1)^n \left. \frac{d^n \Sigma(\tau)_{pq}}{d \tau^n} \right|_{\tau=0^-} ,
\end{align}
where $\mu$ represents the chemical potential of the system.
This exposes the relationship of these spectral moments to a Taylor expansion of the short-time dynamics of the greater and lesser parts of the self-energy, with the moments defining the integrated weight, mean, variance, skew, and higher-order moments of the dynamical distribution of each element of the self-energy in the frequency domain.
Applied to the $GW$ self-energy of Eq.~\ref{eq:fullSE}, the moments can be constructed as
\begin{align}
\Sigma_{pq}^{(n,<)} = \sum_{\nu} & \sum_{ia,jb} \sum_k \left[ (pk|ia)(X_{ia}^\nu + Y_{ia}^{\nu}) \right. \nonumber \\
& \left. (\epsilon_k-\Omega_{\nu})^n (X_{ia}^\nu + Y_{ia}^\nu) (qk|jb) \right] , \label{eq:SEMom1_less}
\end{align}
\begin{align}
\Sigma_{pq}^{(n,>)} = \sum_{\nu} & \sum_{ia,jb} \sum_c \left[ (pc|ia)(X_{ia}^\nu + Y_{ia}^{\nu}) \right. \nonumber \\
&\left. (\epsilon_c+\Omega_{\nu})^n (X_{ia}^\nu + Y_{ia}^\nu) (qc|jb) \right] . \label{eq:SEMom1_great}
\end{align}
The moment distribution of a convolution of two quantities can be expressed via the binomial theorem as a sum of products of the moments of the individual quantities. This enables us to split apart the expressions above into products of the individual Green's function and density-density response moments. Defining the $n^\mathrm{th}$-order spectral moments of the RPA density-response, summed over both particle-hole excitation and de-excitation components, as
\begin{equation}
\eta^{(n)}_{ia,jb} = \sum_\nu (X_{ia}^\nu + Y_{ia}^\nu) \Omega_{\nu}^n (X_{jb}^{\nu} + Y_{jb}^{\nu}) , \label{eq:eta_def}
\end{equation}
we can rewrite Eqs.~\ref{eq:SEMom1_less} and \ref{eq:SEMom1_great} as
\begin{align}
\Sigma_{pq}^{(n,<)}&=\sum_{ia,jb,k} \sum_{t=0}^n \binom{n}{t} (-1)^{t} \epsilon_{k}^{n-t} (pk|ia) \eta_{ia,jb}^{(t)} (qk|jb) \label{eq:SEMoms2_less} \\
\Sigma_{pq}^{(n,>)}&=\sum_{ia,jb,c} \sum_{t=0}^n \binom{n}{t} \epsilon_{c}^{n-t} (pc|ia) \eta_{ia,jb}^{(t)} (qc|jb) . \label{eq:SEMoms2_great}
\end{align}
Evaluating the self-energy spectral moments of Eqs.~\ref{eq:SEMoms2_less}-\ref{eq:SEMoms2_great} up to a desired order $n$, represents the central step of the proposed `moment-conserving' $GW$ formulation, defining the convolution between $G(\omega)$ and $W_p(\omega)$ in this moment expansion of the dynamics. In Sec.~\ref{sec:DD_RPA} we show how the RPA can be reformulated to define specific constraints on the relations between different orders of the RPA density-response moments, $\eta^{(n)}_{ia,jb}$. These relations are subsequently used in Sec.~\ref{sec:EfficientEval} to demonstrate how the self-energy moments can be evaluated in $\mathcal{O}[N^4]$ scaling, with Sec.~\ref{sec:CubicScaling} going further to propose a cubic scaling algorithm for their evaluation (and therefore full $GW$ algorithm). In addition to these moments representing the dynamical part of the self-energy, we also require a static (exchange) part of the self-energy, $\mat{\Sigma}_\infty$, which can be calculated as
\begin{equation}
\mat{\Sigma}_{\infty} = \mat{K}[\mat{D}] - \mat{V}_{\mathrm{xc}} ,
\end{equation}
where $\mat{K}[\mat{D}]$ is the exchange matrix evaluated with the reference density matrix. This reference density matrix is found from a prior mean-field calculation via a self-consistent Fock or Kohn--Sham single-particle Hamiltonian, $\mat{f}[\mat{D}]$, with $\mat{V}_{\mathrm{xc}}$ being the exchange-correlation potential used in $\mat{f}$. Note that for a Hartree--Fock reference, this static self-energy contribution is zero.
\subsection{Full $GW$ spectrum from self-energy moments}
Once the moments of the self-energy are found, it is necessary to obtain the resulting dressed $GW$ excitations and spectrum. While this is formally an application of Dyson's equation, the most common approach is to find each $GW$ excitation explicitly via a self-consistent solution (or linearized approximation) of the quasiparticle equation, while assuming a diagonal self-energy in the MO basis\cite{VanSetten2015}. This assumption neglects physical effects due to electron density relaxation and mixing or splitting of quasiparticle states in more strongly correlated systems. In this work, we allow for an exact invocation of Dyson's equation, which can be achieved straightforwardly in this moment domain of the effective dynamics, allowing extraction of quasi-particle weights associated with transitions, and a full matrix-valued form of the resulting $GW$ Green's function over all frequencies, with all poles obtained analytically without artificial broadening.
This is achieved by constructing an `upfolded' representation of an effective Hamiltonian, consisting of coupling between a physical and `auxiliary' space (with the latter describing the effect of the moment-truncated self-energy). Specifically, we seek an effective static Hamiltonian,
\begin{equation}
\mat{\tilde{H}} =
\begin{bmatrix}
\mat{f} + \mat{\Sigma}_{\infty} & \mat{\tilde{W}} \\
\mat{\tilde{W}}^\dagger & \mat{\tilde{d}}
\end{bmatrix} , \label{eq:effH}
\end{equation}
whose eigenvalues are the charged excitation energies at the level of the moment-truncated $GW$, with quasiparticle weights and Dyson orbitals explicitly obtained from the projection of the corresponding eigenvectors into the physical (MO) space. The full Green's function can therefore be constructed as
\begin{equation}
\mat{G}(\omega) = \left(\omega \mat{I} - \mat{f} + \mat{\Sigma}_\infty - \mat{\tilde{W}}(\omega \mat{I} - \mat{\tilde{d}})^{-1} \mat{\tilde{W}}^\dagger \right)^{-1} . \label{eq:GFfromEffH}
\end{equation}
Such upfolded representations have been considered previously in diagrammatic theories, in a recasting of GF2 theory in terms of its moments \cite{Backhouse2020a, Backhouse2020b, Backhouse2021} as well as more recently to $GW$ amongst others\cite{Rebolini2016, Loos2018, Bintrim2021, Bintrim2022, Backhouse2022, Tolle2022}.
For `exact' $G_0W_0$, this auxiliary space (i.e. the dimension of $\mat{d}$) must scale as $\mathcal{O}[N^3]$, necessitating a state-specific and direct algorithm for its practical solution, along with other approximations \cite{Bintrim2021}.
However, in the moment truncation, $\mat{\tilde{W}}$ and $\mat{\tilde{d}}$ can be directly constructed such that their effect exactly matches that of a truncated set of conserved $GW$ self-energy moments (separately in the particle and hole sectors), yet rigorously scales in dimension as $\mathcal{O}[n N]$, where $n$ is the number of conserved self-energy moments. This allows for a complete diagonalization of $\mat{\tilde{H}}$, obtaining all excitations in a single shot, and a reconstruction of the full $GW$ Green's function from its Lehmann representation in $\mathcal{O}[(nN)^3]$ computational effort, avoiding the need for any grids or iterative solutions once $\mat{\tilde{H}}$ is found.
To find this effective upfolded representation of the moment-conserving dynamics, we modify the block Lanczos procedure to ensure the construction a $\mat{\tilde{H}}$ of minimal size, whose effective hole and particle self-energy moments
exactly match the ones from Eqs.~\ref{eq:SEMoms2_less}-\ref{eq:SEMoms2_great}. We first proceed by splitting the auxiliary space into a space denoting the effect of the hole (lesser) and particle (greater) self-energy, and consider each in turn. Focusing on the lesser self-energy, we can construct an {\em exact} upfolded self-energy representation\cite{Chan2011}, via inspection from Eq.~\ref{eq:fullSE}, with
\begin{align}
\mat{W}_{p,k \nu} &= \sum_{ia} (pk|ia)(X_{ia}^{\nu} + Y_{ia}^{\nu}) \label{eq:ExactCouplings} \\
\mat{d}_{k\nu, l\nu'} &= (\epsilon_k - \Omega_\nu) \delta_{k,l} \delta_{\nu,\nu'} ,
\end{align}
where we remove the tilde above upfolded auxiliary quantities when denoting the exact upfolded $GW$ self-energy components.
We now consider the projection of the exact $GW$ upfolded matrix representation into a truncated block tridiagonal form, as
\begin{align}
\nonumber
\tilde{\mathbf{H}}_{\mathrm{tri}}
&=
\tilde{\mathbf{q}}^{(j), \dagger}
\begin{bmatrix}
\mathbf{f} + \boldsymbol{\Sigma}_{\infty} & \mathbf{W} \\
\mathbf{W}^{\dagger} & \mathbf{d}
\end{bmatrix}
\tilde{\mathbf{q}}^{(j)}
\\
\label{eq:block_lanczos_hamiltonian}
&=
\begin{bmatrix}
\mathbf{f} + \boldsymbol{\Sigma}_{\infty} & \mathbf{L} & & & & \mathbf{0} \\
\mathbf{L}^{\dagger} & \mathbf{M}_{1} & \mathbf{C}_{1} & & & \\
& \mathbf{C}_{1}^{\dagger} & \mathbf{M}_{2} & \mathbf{C}_{2} & & \\
& & \mathbf{C}_{2}^{\dagger} & \mathbf{M}_{3} & \ddots & \\
& & & \ddots & \ddots & \mathbf{C}_{j-1} \\
\mathbf{0} & & & & \mathbf{C}_{j-1}^{\dagger} & \mathbf{M}_{j}
\end{bmatrix}
,
\end{align}
where we define $\tilde{\mathbf{q}}^{(j)}$ as
\begin{align}
\label{eq:block_lanczos_vectors}
\tilde{\mathbf{q}}^{(j)}
&=
\begin{bmatrix}
\mathbf{I} & \mathbf{0} \\
\mathbf{0} & \mathbf{q}^{(j)}
\end{bmatrix}
.
\end{align}
The $\mathbf{q}^{(j)}$ are block Lanczos vectors of depth $j$, which form a recursive Krylov space as $
\mathbf{q}^{(j)}
=
\begin{bmatrix}
\mathbf{q}_{1} &
\mathbf{q}_{2} &
\cdots &
\mathbf{q}_{j}
\end{bmatrix}
$, ensuring that when taken together, they project to a block tridiagonal representation of the upfolded self-energy with $j$ on-diagonal blocks as shown.
The action of this block Lanczos tridiagonalization of the upfolded (hole or particle) self-energy is to exactly conserve these spectral moments of the self-energy\cite{Meyer1989, Weikert1996}. We therefore seek to reformulate the Lanczos recursion in terms of just these moments, rather than the action of the full upfolded Hamiltonian which we seek to avoid due to its scaling.
The initial couplings $\mathbf{L}$ and Lanczos vectors $\mathbf{q}_{1}$
can be found via a QR factorisation of the exact $GW$ couplings $\mathbf{W}$, as
\begin{align}
\label{eq:coupling_qr}
\mathbf{W}^{\dagger}
&=
\mathbf{q}_{1}
\mathbf{L}^{\dagger}
.
\end{align}
However, this will scale poorly, and so we can rewrite this to directly compute $\mat{L}$ from the computed self-energy moments, rather than requiring manipulations of the full auxiliary space. Via the Cholesky QR algorithm\cite{Fukaya2014, Fukaya2020}, we can relate $\mat{L}$ to the zeroth order self-energy moment as
\begin{align}
\label{eq:copuling_qr_cholesky}
\mathbf{L}^{\dagger}
&=
\left(
\mathbf{W}
\mathbf{W}^{\dagger}
\right)^{\frac{1}{2}}
=
\left(
\boldsymbol{\Sigma}^{(0)}
\right)^{\frac{1}{2}}
,
\end{align}
where the indication of the sector of the self-energy has been dropped, with this process considered independently for the hole and particle (lesser and greater respectively) parts of the self-energy.
The initial Lanczos vector can then be computed as
$\mathbf{q}_{1} = \mathbf{W}^{\dagger} \mathbf{L}^{-1, \dagger}$.
Subsequent block Lanczos vectors can then be defined according to the standard
three-term recurrence
\begin{align}
\label{eq:block_lanczos_three_term_recurrence}
\mathbf{q}_{i+1}
\mathbf{C}_{i}^{\dagger}
&=
\left[
\mathbf{d} \mathbf{q}_{i}
-
\mathbf{q}_{i} \mathbf{M}_{i}
-
\mathbf{q}_{i-1} \mathbf{C}_{i-1}
\right]
\end{align}
where the on-diagonal blocks are defined as
\begin{align}
\label{eq:block_lanczos_on_diagonal_blocks}
\mathbf{M}_{i}
&=
\mathbf{q}_{i}^{\dagger}
\mathbf{d}
\mathbf{q}_{i}
.
\end{align}
In order to recast this process in terms of the self-energy moments directly,
we wish to express the block Lanczos recurrence in terms of the inner space
of the Lanczos vectors rather than spanning the large auxiliary space.
The choice of initial vectors in
Eq.~\ref{eq:coupling_qr}
permits the definition
\begin{align}
\nonumber
\mathbf{S}_{1, 1}^{(n)}
&=
\mathbf{q}_{1}^{\dagger}
\mathbf{d}^{n}
\mathbf{q}_{1}
\\
\label{eq:block_lanczos_recurrence_init}
&=
\mathbf{L}^{-1}
\boldsymbol{\Sigma}^{(n)}
\mathbf{L}^{-1, \dagger}
,
\end{align}
where the subscript indices on $\mat{S}$ indicate the projection into the block Lanczos space on the left and the right respectively. These provide the initialisation of the recurrence terms,
and have a dimension which scales linearly with system size (the same as the input self-energy moments). These $\mat{\Sigma}^{(n)}$ matrices are therefore the input to the procedure, defined by Eqs.~\ref{eq:SEMoms2_less}-\ref{eq:SEMoms2_great}.
One can then use the definition of the three-term recurrence in
Eq.~\ref{eq:block_lanczos_three_term_recurrence}
to express all definitions in terms of these moments, without formal reference to the large auxiliary space quantities ($\mat{d}$ or $\mat{W}$), as
\onecolumngrid
\begin{align}
\label{eq:block_lanczos_recurrence_single}
\mathbf{S}_{i+1, i}^{(n)}
&=
\mathbf{q}_{i+1}^{\dagger}
\mathbf{d}^{n}
\mathbf{q}_{i}
=
\mathbf{C}_{i}^{-1}
\left[
\mathbf{S}_{i, i}^{(n+1)}
-
\mathbf{M}_{i}
\mathbf{S}_{i, i}^{(n)}
-
\mathbf{C}_{i-1}^{\dagger}
\mathbf{S}_{i-1, i}^{(n)}
\right]
,
\\
\nonumber
\mathbf{S}_{i+1, i+1}^{(n)}
&=
\mathbf{q}_{i+1}^{\dagger}
\mathbf{d}^{n}
\mathbf{q}_{i+1}
\\
\label{eq:block_lanczos_recurrence_double}
&=
\mathbf{C}_{i}^{-1}
\left[
\mathbf{S}_{i, i}^{(n+2)}
+
\mathbf{M}_{i}
\mathbf{S}_{i, i}^{(n)}
\mathbf{M}_{i}
+
\mathbf{C}_{i-1}^{\dagger}
\mathbf{S}_{i-1, i-1}^{(n)}
\mathbf{C}_{i-1}
-
P(
\mathbf{S}_{i, i}^{(n+1)}
\mathbf{M}_{i}
)
+
P(
\mathbf{M}_{i}
\mathbf{S}_{i, i-1}^{(n)}
\mathbf{C}_{i-1}
)
-
P(
\mathbf{S}_{i, i-1}^{(n+1)}
\mathbf{C}_{i-1}
)
\right]
\mathbf{C}_{i}^{-1, \dagger}
,
\end{align}
\twocolumngrid
where the permutation operator $P$ is defined as
$P(\mathbf{Z}) = \mathbf{Z} + \mathbf{Z}^{\dagger}$.
For a Hermitian theory we can write
$\mathbf{S}_{i, j} = \mathbf{S}_{j, i}^{\dagger}$,
and assume the zeroth order Lanczos vector to be zero i.e.\
$
\mathbf{S}_{i, 0}^{(n)}
=
\mathbf{S}_{0, j}^{(n)}
=
\mathbf{S}_{0, 0}^{(n)}
=
\mathbf{0}
$.
Additionally,
the orthogonality of the Lanczos vectors requires that
$\mathbf{S}_{i, j}^{(0)} = \delta_{ij} \mathbf{I}$.
By considering again the Cholesky QR algorithm,
the off-diagonal $\mathbf{C}$ matrices can therefore be computed as
\begin{align}
\mathbf{C}_{i}^{2} = &\left[ \mathbf{S}_{i, i}^{(2)} + \mathbf{M}_{i}^{2}
+ \mathbf{C}_{i-1}^{\dagger} \mathbf{C}_{i-1} \right. \nonumber \\
& - \left. P(\mathbf{S}_{i, i}^{(1)} \mathbf{M}_{i}) -
P(\mathbf{S}_{i, i-1}^{(1)} \mathbf{C}_{i-1}) \right] , \label{eq:block_lanczos_off_diagonal_blocks_recur}
\end{align}
and the on-diagonal $\mathbf{M}$ matrices can be found using
Eq.~\ref{eq:block_lanczos_on_diagonal_blocks}
\begin{align}
\label{eq:block_lanczos_on_diagonal_blocks_recur}
\mathbf{M}_{i}
&=
\mathbf{q}_{i}
\mathbf{d}
\mathbf{q}_{i}
=
\mathbf{S}_{i, i}^{(1)}
.
\end{align}
These recurrence relations allow the calculation of the on- and off-diagonal
blocks resulting in the block tridiagonal form of the Hamiltonian in
Eq.~\ref{eq:block_lanczos_hamiltonian}.
Despite the apparent complexity of the recurrence relations,
this algorithm contains no step scaling greater than
$\mathcal{O}[N^{3}]$, by eliminating the explicit reference to the full upfolded Hamiltonian, whilst still conserving the exact self-energy moments by construction.
It can be seen from Eq.~\ref{eq:ExactCouplings} that the auxiliary space formally couples to both the hole and particle physical sectors (both occupied and virtual MOs), while an explicit decoupling of the hole and particle excitations (sometimes called a `non-Dyson' approximation\cite{Schirmer1998, Dempwolff2019, Trofimov2005}) is a common approximation in some $GW$ implementations or methods such as the algebraic diagrammatic construction.
To avoid this approximation and ensure full coupling,
the solution of the Dyson equation using the compressed self-energy,
i.e.\ diagonalisation of Eq.~\ref{eq:effH},
requires the combination of the the block tridiagonal Hamiltonian in Eq.~\ref{eq:block_lanczos_hamiltonian}
resulting from both the hole and particle self-energy moments as
\begin{align}
\label{eq:concatenated_h}
\tilde{\mathbf{H}}
&=
\begin{bmatrix}
\mathbf{f} + \boldsymbol{\Sigma}_{\infty} & \tilde{\mathbf{W}} \\
\tilde{\mathbf{W}}^{\dagger} & \tilde{\mathbf{d}} \\
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{f} + \boldsymbol{\Sigma}_{\infty} & \tilde{\mathbf{W}}^{<} & \tilde{\mathbf{W}}^{>} \\
\tilde{\mathbf{W}}^{<, \dagger} & \tilde{\mathbf{d}}^{<} & \mathbf{0} \\
\tilde{\mathbf{W}}^{>, \dagger} & \mathbf{0} & \tilde{\mathbf{d}}^{>} \\
\end{bmatrix}
,
\end{align}
where $\tilde{\mathbf{W}}$ are equal to the $\mathbf{L}$ matrix padded by zeros
\begin{align}
\label{eq:compressed_w}
\tilde{\mathbf{W}}^{\lessgtr} =
\begin{bmatrix}
\mathbf{L}^{\lessgtr} & \mathbf{0} & \cdots & \mathbf{0}
\end{bmatrix}
,
\end{align}
and $\tilde{\mathbf{d}}$ are defined by the block tridiagonal elements
\begin{align}
\label{eq:compressed_d}
\tilde{\mathbf{d}}^{\lessgtr}
&=
\begin{bmatrix}
\mathbf{M}_{1}^{\lessgtr} & \mathbf{C}_{1}^{\lessgtr} & & \mathbf{0} \\
\mathbf{C}_{1}^{\lessgtr, \dagger} & \mathbf{M}_{2}^{\lessgtr} & \ddots \\
& \ddots & \ddots & \mathbf{C}_{j-1}^{\lessgtr} \\
\mathbf{0} & & \mathbf{C}_{j-1}^{\lessgtr, \dagger} & \mathbf{M}_{j}^{\lessgtr}
\end{bmatrix}
.
\end{align}
This ensures conservation of both the separate hole and particle moments of the self-energy,
as well as conservation in the central moments according to their sum.
The compressed Hamiltonian can be returned to a diagonal representation of the self-energy by diagonalising $\tilde{\mathbf{d}}$ and
appropriately rotating $\tilde{\mathbf{W}}$ into this basis.
The eigenvalues of $\tilde{\mathbf{H}}$ are moment-conserving approximations to those of the exact upfolded Hamiltonian,
and the corresponding eigenvectors $\mathbf{u}$ can be transformed into Dyson orbitals via $\mathbf{L} \mathbf{P} \mathbf{u}$,
where $\mathbf{P}$ is
a projection into the physical space, and the $\mathbf{L}$ is required to transform the physical component of the eigenvectors back to the MO representation.
This process conserves exactly the first $2j$ hole and particle self-energy moments.
Commonly,
a notation referring to the number of iterations of the block Lanczos recurrence
$n_{\mathrm{iter}}$ is used;
in this notation the $n_{\mathrm{iter}} = 0$ calculation corresponds to the
inclusion of only a single on-diagonal block $\mathbf{M}_{1}$,
with modified couplings $\mathbf{L}$ to the physical space.
As such,
in this notation the number of conserved moments equals $2 n_{\mathrm{iter}} + 2$,
i.e.\ up to and including the $2 n_{\mathrm{iter}} + 1$ order moment.
This is the same number of moments as required as input to the recurrence relations,
and therefore the algorithm conserves all the moments used as input, which should be up to an odd order.
After $n_{\mathrm{iter}}$ applications of this algorithm to both the lesser and greater self-energy sectors,
this results in $N (2 n_\mathrm{iter} + 3)$ quasiparticle states (demonstrating the potential to capture satellite features with these additional poles).
As such,
application of this algorithm becomes theoretically equivalent to a full diagonalisation of the exact upfolded Hamiltonian in the limit of
$n_{\mathrm{iter}} \sim N^{2}$.
\section{Density response moments in the RPA} \label{sec:DD_RPA}
Having described the overall approach in Sec.~\ref{sec:Mom_GW}, what remains for a practical implementation is to ensure that the $GW$ self-energy moments described in Eqs.~\ref{eq:SEMoms2_less}-\ref{eq:SEMoms2_great} can be computed efficiently. As a first step towards this, in this section we show how the RPA can be motivated from the perspective of the two-point density-density (dd) response moments of Eq.~\ref{eq:eta_def}, which are central quantities to obtain in this approach to $GW$ theory. We find that we can reformulate RPA entirely in terms of these dd-moments of the system and a strict recursive form for their inter-relation\cite{PhysRevB.104.245114}. This recursive relation between the moments is a direct result of the fact that the RPA can be written as a quadratic Hamiltonian in Bosonic operators \cite{Tolle2022}. This effectively ensures that all information required to build the 2-point RPA dd-response is contained in the first two spectral moments, analogous to how all the information on the density of states in mean-field theory (quadratic in Fermionic operators) is contained in the first two Green's function moments (i.e. the one-body density matrix and Fock matrix).
We start from the Casida formulation of RPA \cite{Hesselmann2011,doi:10.1146/annurev-physchem-040215-112308}, as a generalized eigenvalue decomposition
\begin{equation} \label{eq:Casida-eq}
\begin{bmatrix}
\mat{A} & \mat{B} & \\
\mat{-B} & \mat{-A} &
\end{bmatrix} \begin{bmatrix} \mat{X} & \mat{Y} \\ \mat{Y} & \mat{X} \end{bmatrix} =
\begin{bmatrix}
\mat{X} & \mat{Y} \\ \mat{Y} & \mat{X}
\end{bmatrix} \begin{bmatrix} \mat{\Omega} & \mat{0} \\ \mat{0} & -\mat{\Omega} \end{bmatrix} ,
\end{equation}
where the left and right eigenvectors form the biorthogonal set as
\begin{equation}
\begin{bmatrix}
\mat{X} & \mat{Y} \\ \mat{Y} & \mat{X}
\end{bmatrix}^T
\begin{bmatrix}
\mat{X} & \mat{Y} \\ \mat{-Y} & \mat{-X}
\end{bmatrix} = \begin{bmatrix}
\mat{I} & \mat{0} \\ \mat{0} & \mat{-I}
\end{bmatrix}. \label{eq:RPAbiorthog}
\end{equation}
This biorthogonality ensures an inverse relationship between $(\mat{X}+\mat{Y})$ and $(\mat{X}-\mat{Y})^T$, as
\begin{equation}
(\mat{X}+\mat{Y})(\mat{X}-\mat{Y})^T = (\mat{X}+\mat{Y})^T(\mat{X}-\mat{Y}) = \mat{I} \label{eq:inverse} .
\end{equation}
The $\mat{A}$ and $\mat{B}$ matrices are defined as
\begin{align}
A_{ia,jb} &= \left(\epsilon_a - \epsilon_i\right) \delta_{ij}\delta_{ab} + \mathcal{K}_{ia,bj} \label{eq:Casida_def_A} \\
B_{ia,jb} &= \mathcal{K}_{ia,jb}. \label{eq:Casida_def_B}
\end{align}
Here, $\mathcal{K}$ is an interaction kernel which couples particle-hole excitations and de-excitations. In the traditional RPA (without second-order exchange), this coupling is taken to be the same for excitations, de-excitations and their coupling, given by the static, bare Coulomb interaction, $\mathcal{K}_{ia,jb} = (ia|jb) = \mathcal{K}_{ia,bj}$. Hole and particle orbital energies are given by $\epsilon_i$ and $\epsilon_a$ respectively, defining the irreducible polarizability of the system from the reference state in $\mat{A}$. Upon diagonalization, the eigenvectors defined by $X_{ia,\nu}$ and $Y_{ia,\nu}$ define the coefficients of the RPA excitations in the particle-hole and hole-particle basis, with energies $\Omega_{\nu}$, with $\mat{\Omega}$ therefore a diagonal matrix of the positive (neutral) RPA excitation energies.
These neutral excitations define the poles of the full RPA reducible density-density (dd) response function, which can be constructed as
\begin{equation}
\mat{\chi}(\omega) =
\begin{bmatrix}
\mat{X} & \mat{Y} \\
\mat{Y} & \mat{X}
\end{bmatrix}
\begin{bmatrix} \omega \mat{I}-\mat{\Omega} \hspace{10pt} & \mat{0} \\
\mat{0} \hspace{10pt} & -\omega \mat{I} - \mat{\Omega} \end{bmatrix}^{-1}
\begin{bmatrix}
\mat{X} & \mat{Y} \\
\mat{Y} & \mat{X}
\end{bmatrix}^T. \label{eq:rpaddresponse}
\end{equation}
Note that this matrix is formally equivalent to the alternative directly dynamical construction from $\mat{\chi}(\omega) = (\mat{P}(\omega)^{-1} - \mathcal{K})^{-1}$, where $\mat{P}(\omega)$ is the irreducible polarizability of the reference state. Considering the positive-frequency part of the dd-response (noting that the negative frequency part is symmetric due to the bosonic-like symmetry of Eq.~\ref{eq:rpaddresponse}), we can write a more compact form of the dd-response as
\begin{equation}
\eta_{ia,jb}(\omega) = (\mat{X}+\mat{Y}) (\omega \mat{I} - \mat{\Omega})^{-1} (\mat{X}+\mat{Y})^T , \label{eq:dd-res}
\end{equation}
which sums contributions from particle-hole and hole-particle fluctuations together, and from which optical properties such as dynamic polarizabilities can be computed \cite{doi:10.1063/1.456413}.
However, in this work we are interested in the order-by-order moments of the spectral distribution of Eq.~\eqref{eq:dd-res} over all RPA excitation energies, which is given as
\begin{equation}
\eta^{(n)}_{ia,jb} = -\frac{1}{\pi} \int_{0}^{\infty} \textrm{Im}[\eta_{ia,jb}(\omega)] \omega^n d\omega . \label{eq:eta_from_dd}
\end{equation}
The non-negative integer index $n$ denotes the order of this static dd spectral moment information.
Performing this integration results in the direct construction of the dd-moments as defined in Eq.~\ref{eq:eta_def}, which can be written more compactly in matrix form as
\begin{equation}
\mat{\eta}^{(n)} = (\mat{X}+\mat{Y}) \mat{\Omega}^{n} (\mat{X}+\mat{Y})^T , \label{eq:formal_moms}
\end{equation}
and constitutes a central object of interest in this work, required for the GW self-energy moment construction of Eqs.~\ref{eq:SEMoms2_less}-\ref{eq:SEMoms2_great}.
\footnote{
We also note that a related expansion and recursive relation can also be constructed in terms of the inverse of these moments courtesy of the equivalence $\mat{X+Y}^{-1}=\mat{X - Y}$, that is
$(\mat{\eta}^{(n)})^{-1} = (\mat{X}-\mat{Y}) \mat{\Omega}^{n} (\mat{X}-\mat{Y})^T$.
}
We now show that the RPA can be entirely reformulated in terms of the dd-moments, Eq.~\eqref{eq:formal_moms}, without loss of information, and expose constraints on the relationship between the moments of different order at the RPA level. Firstly, we note that from the definition of the original eigendecomposition of Eq.~\ref{eq:Casida-eq}, along with insertion of a resolution of the identity via Eq.~\ref{eq:inverse}, we find a relation between the first two dd-moments as
\begin{align}
\mat{\eta}^{(1)} &= (\mat{X} + \mat{Y})\mat{\Omega}(\mat{X}+\mat{Y})^T = (\mat{A} - \mat{B}) , \label{eq:one_mom} \\
&= \mat{\eta}^{(0)} (\mat{A}+\mat{B}) \mat{\eta}^{(0)} , \label{eq:zero_mom}
\end{align}
noting that $(\mat{A}+\mat{B})=(\mat{X}-\mat{Y})\mat{\Omega}(\mat{X}-\mat{Y})^T$.
By taking the sum and difference of the two Casida equations of Eq.~\ref{eq:Casida-eq}, we also find
\begin{align}
\left(\mat{A}+\mat{B}\right)\left(\mat{X}+\mat{Y}\right) &= \left(\mat{X}-\mat{Y}\right)\mat{\Omega} \label{eq:ApB_def} \\
\left(\mat{A}-\mat{B}\right)\left(\mat{X}-\mat{Y}\right) &= \left(\mat{X}+\mat{Y}\right)\mat{\Omega} \label{eq:AmB_def} ,
\end{align}
from which an equation for the square of the RPA excitations can be found as
\begin{equation}
\left(\mat{A}-\mat{B}\right)\left(\mat{A}+\mat{B}\right)\left(\mat{X}+\mat{Y}\right)=\left(\mat{X}+\mat{Y}\right)\mat{\Omega}^2,
\end{equation}
which has previously appeared in the RPA literature \cite{Furche2001}. By right-multiplying by $(\mat{X}+\mat{Y})^T$, this leads to a relation between the zeroth and second dd-moment, as
\begin{equation}
(\mat{A}-\mat{B})(\mat{A}+\mat{B}) \mat{\eta}^{(0)} = \mat{\eta}^{(2)}.
\end{equation}
The above equations can be further generalized as a recursive relation to generate all higher-order moments from lower-order ones, as
\begin{align}
\mat{\eta}^{(m)} &= (\mat{A} - \mat{B})(\mat{A} + \mat{B}) \mat{\eta}^{(m-2)} \label{eq:mom_recursion_main} \\
&= [\mat{\eta}^{(0)} (\mat{A} + \mat{B})]^m \mat{\eta}^{(0)} . \label{eq:mom_recursion}
\end{align}
While these are important relations in themselves, they also illustrate that all the RPA excitations and weights in the dd-response of Eq.~\ref{eq:dd-res} are implicitly accessible without requiring the explicit solution to the Casida equation ($\mat{X}$, $\mat{Y}$ and $\mat{\Omega}$ matrices). This reformulation just requires knowledge of $\mat{\eta}^{(0)}$ as the central variable (which can be defined independently of the original equations via Eq.~\ref{eq:zero_mom}), the $\mat{A}$ and $\mat{B}$ matrices defining the system and their interactions, and the recursive relations of Eq.~\ref{eq:mom_recursion}.
As an aside, the Tamm--Dancoff approximation (TDA) sets $\mat{B}=0$, which dramatically simplifies the resulting expressions due to the lack of correlation in the ground state, with $\eta^{(0)}=\mat{I}$, and $\eta^{(n)}=\mat{A}^n$. This reflects that the 2-RDM in the TDA is equivalent to that of mean-field theory. Finally, we note in passing that the ground state correlation energy from the RPA can be similarly formulated in terms of the zeroth-order dd-moment , as
\begin{equation}
E_{\mathrm{corr}}^{\mathrm{RPA}} = \frac{1}{2} \mathrm{Tr}[\mat{\eta}^{(0)} (\mat{A} + \mat{B}) - \mat{A}] .
\end{equation}
Related expressions for the RPA correlation energy can be found found in Eqs.~39-40 of Ref.~\onlinecite{angyan2011correlation} in terms of the quantities $\mat{\eta}^{(0)}$, $\mat{A+B}$ and $\mat{A - B}$ (where there, these quantities are referred to as $\mat{Q}^{\text{dRPA}}_{\text{I}}$, $\mat{\varepsilon}$ and $\mat{\varepsilon} + 2\mat{K}$ ).
Equivalence between these expressions (as well as the more commonly used expression found in Ref.~\onlinecite{Furche2008}) can be seen by noting (using Eqs.~\ref{eq:inverse}, \ref{eq:ApB_def} and \ref{eq:AmB_def}) that
\begin{align}
&\text{Tr}\left[ \mat{\eta}^{(0)}(\mat{A+B})\right] \\
= &\text{Tr}\left[ (\mat{\eta}^{(0)})^{-1} (\mat{A-B})\right] \\
= &\text{Tr}\left[\left((\mat{A-B})^{\frac{1}{2}} (\mat{A+B}) (\mat{A-B})^{\frac{1}{2}}\right)^{\frac{1}{2}}\right] = \text{Tr}\left[ \mat{\Omega} \right].
\end{align}
Overall, this perspective on the RPA in terms of dd-moments is key to open new avenues such as the ones explored in this work.
\section{Efficient evaluation of self-energy and density response moments} \label{sec:EfficientEval}
Given the recasting of the RPA dd-response in Sec.~\ref{sec:DD_RPA} in terms of its lowest order moment (Eq.~\ref{eq:zero_mom}) and recursion to access the higher moments via Eq.~\ref{eq:mom_recursion}, we now consider the efficient $\mathcal{O}[N^4]$ evaluation of these quantities which are central to the approach in this work, thus avoiding their formal $\mathcal{O}[N^6]$ construction via Eq.~\ref{eq:formal_moms}.
The derivation here is heavily inspired by the seminal RI approach of Furche to compute RPA correlation energies \cite{Furche2010}, with key adaptations to target these dd-moments to arbitrary order, rather than the correlation energy.
We first employ a standard low-rank decomposition of the two-electron repulsion integrals (e.g. via density fitting or Cholesky decomposition) as
\begin{equation}
(ia|jb) \simeq \sum_P (ia|P)(P|jb) = \sum_P V_{ia,P} V_{jb,P} = \mat{V}\mat{V}^T , \label{eq:RI_eri}
\end{equation}
where we use $P,Q,\dots$ to index elements of this auxiliary (RI) basis, whose dimension $N_{\mathrm{aux}}$ scales $\mathcal{O}[N]$ with system size. We define an intermediate quantity
\begin{equation}
{\tilde \eta}^{(n)}_{ia,P}=\sum_{jb} \eta^{(n)}_{ia,jb} V_{jb,P} \quad . \label{eq:orig}
\end{equation}
If this intermediate can be efficiently found, then the greater self-energy moment of Eq.~\ref{eq:SEMoms2_great} can be rewritten as
\begin{equation}
\Sigma_{pq}^{(n,>)}=\sum_{t=0}^n \binom{n}{t} \left( \epsilon_{c}^{n-t} V_{pc,Q} \left( V_{qc,P} \left( {\tilde \eta}_{ia,P}^{(t)} V_{ia,Q} \right) \right) \right) , \label{eq:SEMoms3_great}
\end{equation}
where the brackets indicate the order of contractions in order to preserve $\mathcal{O}[N^4]$ scaling, and einstein summation is implied. The lesser self-energy moment of Eq.~\ref{eq:SEMoms2_less} can be recast in an analogous fashion.
\footnote{We note that the derivation in this section does not rely on the $V_{jb,P}$ tensor in Eq.~\ref{eq:orig} arising from this Coulomb form specifically, but rather that it involves a more general linear transformation from a space in the particle-hole product basis to a space which scales no more than $\mathcal{O}[N]$ (in this case the auxiliary basis).}
Obtaining all dd-moments of the form of Eq.~\ref{eq:orig} up to order $n$ can be simply reduced to knowledge of the first two moments ${\tilde{\eta}}_{ia,P}^{(0)}$ and ${\tilde{\eta}}_{ia,P}^{(1)}$, via use of the recursive relationship between the moments as given by Eq.~\ref{eq:mom_recursion_main}, as for even moment orders,
\begin{equation}
\tilde{\mat{\eta}}^{(n)} = [(\mat{A}-\mat{B})(\mat{A}+\mat{B})]^{n/2} \tilde{\mat{\eta}}^{(0)} \label{eq:recursion_even}
\end{equation}
and for odd moment orders,
\begin{equation}
\tilde{\mat{\eta}}^{(n)} = [(\mat{A}-\mat{B})(\mat{A}+\mat{B})]^{(n-1)/2} \tilde{\mat{\eta}}^{(1)} \label{eq:recursion_odd}
\end{equation}
where we have omitted explicit indices for brevity.
Ensuring that an $\mathcal{O}[N^4]$ scaling is retained in this recursion relies on $(\mat{A}-\mat{B})(\mat{A}+\mat{B})$ admitting a form where it can be written as a diagonal plus low-rank correction.
For the RPA, this is true since (from Eqs.~\ref{eq:Casida_def_A}-\ref{eq:Casida_def_B}),
\begin{equation}
(\mat{A}-\mat{B})_{ia,jb} = (\epsilon_a - \epsilon_i) \delta_{ij} \delta_{ab} = \mat{D} \label{eq:A_min_B}
\end{equation}
is a purely diagonal matrix, while using Eq.~\ref{eq:RI_eri} we can cast $(\mat{A}+\mat{B})$ into an appropriate form as
\begin{equation}
(\mat{A} + \mat{B}) = \mat{D}+2\mathcal{K} = \mat{D} + 2 \sum_P V_{ia,P} V_{jb,P} . \label{eq:A_plus_B}
\end{equation}
We therefore express the low-rank asymmetrically decomposed form of $(\mat{A}-\mat{B})(\mat{A}+\mat{B})$ in a general fashion as a diagonal plus asymmetric low-rank part, as
\begin{equation}
(\mat{A}-\mat{B})(\mat{A}+\mat{B}) = \mat{D}^2 + \mat{S}_{L} \mat{S}_R^T , \label{eq:A_min_BA_plus_B}
\end{equation}
where for the RPA, $\mat{D}$ is defined by Eq.~\ref{eq:A_min_B}, $\mat{S}_L=\mat{D}\mat{V}$ and $\mat{S}_R=2\mat{V}$.
Future work will explore other analogous approaches where $(\mat{A}-\mat{B})(\mat{A}+\mat{B})$ can be decomposed in this way, for applicability to e.g. the Bethe-Salpeter equation or other RPA variants with (screened) exchange contributions \cite{Bintrim2022,doi:10.1021/acs.jctc.8b01247,doi:10.1021/acs.jctc.2c00366}. From this low-rank decomposition and the recursive definition of Eqs.~\ref{eq:recursion_even}-\ref{eq:recursion_odd}, a fixed number of dd-moments of the form of Eq.~\ref{eq:orig} can be found in $\mathcal{O}[N^4]$ time, provided the original $\tilde{\eta}^{(0)}$ and $\tilde{\eta}^{(1)}$ values are known.
We now consider how to obtain these initial low-order dd-moments efficiently. From the definitions of Eqs.~\ref{eq:one_mom} and \ref{eq:orig}, and specifying the standard RPA definition of Eq.~\ref{eq:A_min_B}, we find that it is straightforward to efficiently construct the first moment, as
\begin{equation}
\tilde{\eta}^{(1)} = \mat{D}\mat{V} = \mat{S}_L . \label{eq:firstmom}
\end{equation}
The zeroth-order dd-moment can be constructed via a rapidly-convergent numerical integration, as we will show.
From Eq.~\ref{eq:zero_mom}, we can write
\begin{equation}
(\mat{A}-\mat{B})(\mat{A}+\mat{B}) = \left( \eta^{(0)} (\mat{A}+\mat{B}) \right)^2 ,
\end{equation}
from which we can find an expression for $\eta^{(0)}$ as
\begin{equation}
\eta^{(0)} = [(\mat{A}-\mat{B})(\mat{A}+\mat{B})]^{\frac{1}{2}} (\mat{A}+\mat{B})^{-1} .
\end{equation}
We note that for RPA to be well-defined with positive excitation energies, $(\mat{A}-\mat{B})$ and $(\mat{A}+\mat{B})$ must both be positive-definite matrices \cite{Furche2001}.
Using Eqs.~\ref{eq:A_plus_B} and \ref{eq:A_min_BA_plus_B}, we can write the low-rank RPA form of this as
\begin{equation}
\tilde{\eta}^{(0)} = (\mat{D}^2 + \mat{S}_L \mat{S}_R^T)^{\frac{1}{2}} (\mat{D} + 2 \mat{V} \mat{V}^T)^{-1} \mat{V} . \label{eq:tilde_mom_zero}
\end{equation}
We first consider the evaluation of the second half of this expression, which we denote as $\mat{T}$. We can use the Woodbury matrix identity to rewrite it as
\begin{align}
\mat{T} &= (\mat{D}+2\mat{V}\mat{V}^T)^{-1}\mat{V} \\
&= \mat{D}^{-1} \mat{V} - 2\mat{D}^{-1} \mat{V}(\mat{I}+2\mat{V}^T \mat{D}^{-1} \mat{V})^{-1} \mat{V}^T \mat{D}^{-1} \mat{V} . \label{eq:Woodbury}
\end{align}
This now only requires the inversion of the diagonal matrix, $\mat{D}$, and a matrix of dimension $N_{\mathrm{aux}}$, with the overall $ov \times N_{\mathrm{aux}}$ matrix able to be constructed in $\mathcal{O}[N_{\mathrm{aux}}^3 + N_{\mathrm{aux}}^2ov]$ time.
Having constructed $\mat{T}$, we can complete the evaluation of $\tilde{\eta}^{(0)}$ using the definition of the matrix square-root as an integration in the complex plane \cite{matrixsqrt},
\begin{equation}
\mat{M}^{\frac{1}{2}}=\frac{1}{\pi} \int_{-\infty}^{\infty} \left( \mat{I}-z^2(\mat{M}+z^2 \mat{I})^{-1} \right) dz \label{eq:matsqrt} .
\end{equation}
From Eq.~\ref{eq:tilde_mom_zero}, this results in
\begin{align}
\tilde{\mat{\eta}}^{(0)} &= (\mat{D}^2 + \mat{S}_L \mat{S}_R^T)^{\frac{1}{2}} \mat{T} \\
&= \frac{1}{\pi} \int_{-\infty}^{\infty} \left( \mat{I} - z^2 (\mat{D}^2 + \mat{S}_L \mat{S}_R^T + z^2 \mat{I})^{-1} \right) \mat{T} dz . \label{eq:NI}
\end{align}
We can modify this integrand into one more efficient for numerical integration, via another application of the Woodbury matrix identity to reduce the scaling of the matrix inverse. We also simplify the notation by introducing the intermediates,
\begin{align}
\mat{F}(z) &= (\mat{D}^2 + z^2 \mat{I})^{-1} \label{eq:F} \\
\mat{Q}(z) &= \mat{S}_R^T \mat{F}(z) \mat{S}_L , \label{eq:Q}
\end{align}
where $\mat{F}(z)$ is a diagonal matrix in the $ov$ space, and $\mat{Q}(z)$ is a $N_{\mathrm{aux}}\times N_{\mathrm{aux}}$ matrix which can be constructed in $\mathcal{O}[N_{\mathrm{aux}}^2 o v]$ time. This casts Eq.~\ref{eq:NI} into the form
\begin{equation}
\tilde{\eta}^{(0)} = \frac{1}{\pi} \int_{-\infty}^{\infty} \left[ \mat{I} - z^2 \mat{F}(z) \left(\mat{I}-\mat{S}_L(\mat{I}+\mat{Q}(z))^{-1}\mat{S}_R^T \mat{F}(z) \right) \mat{T} \right] dz . \label{eq:NI_efficient}
\end{equation}
For each value of the integration variable $z$, the integrand is a matrix of size $ov \times N_{\mathrm{aux}}$, which can be constructed in $\mathcal{O}[N^4]$ scaling, rendering it efficient for numerical quadrature. Along with the results of Eqs.~\ref{eq:firstmom}, \ref{eq:A_min_BA_plus_B}, \ref{eq:recursion_even} and \ref{eq:recursion_odd}, this therefore completes the ambition of constructing a fixed number of dd-response moments needed for the moment-truncated $GW$ method as defined in Eq.~\ref{eq:orig}, in no more than $\mathcal{O}[N^4]$ scaling.
However, manipulations of the resulting integrand and choice of quadrature points can further improve the efficiency of their construction by ensuring a faster decay of the integrand and separating components which can be analytically integrated. This derivation is given in Appendix~A, and results in a final $\mathcal{O}[N_{\mathrm{aux}}^2 ov + N_{\mathrm{aux}}^3]$ expression to evaluate for the zeroth-order dd-moment as
\begin{align}
\tilde{\eta}^{(0)} &= \mat{D}\mat{T} \nonumber \\
&+ \frac{1}{\pi} \int_0^{\infty} e^{-t \mat{D}} \mat{S}_L \mat{S}_R^T e^{-t \mat{D}} \mat{T} dt \nonumber \\
&+ \frac{1}{\pi}\int_{-\infty}^{\infty} z^2 \mat{F}(z) \mat{S}_L\left((\mat{I}+\mat{Q}(z))^{-1}-\mat{I} \right) \mat{S}_R^T \mat{F}(z) \mat{T} dz . \label{eq:final_NI_zero_mom_main}
\end{align}
The first numerical integral in Eq.~\ref{eq:final_NI_zero_mom_main} (where the integrand decays exponentially) is computed via Gauss-Laguerre quadrature, while the second (where the integrand decays as $\mathcal{O}[z^{-4}]$) is evaluated via Clenshaw--Curtis quadrature. A comparison of the decay of the original and refined integrands is shown in the inset to Fig.~\ref{fig:NumQuadConv}.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{NI_convergence_integrands.pdf}
\caption{Exponential convergence of the numerical integration (NI) error for $\tilde{\mat{\eta}^{(0)}}$ in Eq.~\ref{eq:final_NI_zero_mom_main} with respect to integration points, for the singlet oxygen dimer in a \mbox{cc-pVTZ} basis at equilibrium (1.207 \AA)~ bond length. Also included are the two error estimates of the true NI error, used to check convergence and estimate the required number of points (see App. B, Eq.~\ref{eq:cubic_fit_err_estimate} for the `Nested Fit' and Eq.~\ref{eq:lowerbound_err} for the `Lower Bound' definitions). Inset: The originally derived integrand (Eq.~\ref{eq:NI_efficient}), and the form optimized for efficient NI given in Eq.~\ref{eq:final_NI_zero_mom_main} and derived in App. A, showing the faster decay.}
\label{fig:NumQuadConv}
\end{figure}
The scaling of the grid spacing of both numerical quadratures is optimised to ensure exact integration of the trace of a diagonal approximation for the integrand, analogous to the grid optimization discussed in Ref.~\onlinecite{Furche2010}.
For numerical robustness, we optimize the quadrature for evaluating $\textrm{Tr}[(\mat{A}-\mat{B})(\mat{A}+\mat{B})]^{\frac{1}{2}}$, rather than the full integrand.
We write a diagonal approximation to $(\mat{A}-\mat{B})(\mat{A}+\mat{B})$ as
\begin{equation}
\mat{M}^D = \mat{D}^2 + \mat{S}_L^D (\mat{S}_R^D)^T ,
\end{equation}
with
\begin{align}
(S_L^D)_{ia,R} &= (S_L)_{ia,Q}(S_R)_{ia,Q} \delta_{R,ia} \\
(S_R^D)_{jb,R} &= \delta_{R,jb}.
\end{align}
This is the same form as Eq.~\eqref{eq:A_min_BA_plus_B}, but contains only diagonal matrices (denoted by subscript `$D$' labels) and has an auxiliary space of size $ov$.
As such, the exact square root and all quantities within the numerical integrations can be obtained in $\mathcal{O}[ov]$ computational time. We then seek to ensure the trace of the difference between the exact and numerically integrated estimate of ${\mat{M}^D}^{\frac{1}{2}}$ vanishes. This is achieved for both integrals in Eq.~\eqref{eq:final_NI_zero_mom_main}, with the analytic form for the first integral given by $\frac{1}{2\pi}\text{diag}(\mat{D}^{-1}\mat{S}_L \mat{S}_R^T) = I^\text{offset}$ and the second numerical integral as ${\mat{M}^D}^{\frac{1}{2}}-\mat{D}-I^\text{offset}=I^\text{int}$.
Writing this explicitly, given quadrature points and weights $\{z_i, w_i\}$ for a $n_p$-point infinite or semi-infinite quadrature, we seek to scale our points by a factor $a$, which is a root the objective functions
\begin{align}
O^\text{offset}(a) &= I^\text{offset} - \frac{1}{\pi}\sum_i^{n_p} a w_i \text{Tr} ( e^{-2\mat{D} a z_i} \mat{S}_L \mat{S}_R^T ) \label{eq:offset_obj} \\
O^\text{int}(a) &= I^\text{int} - \frac{1}{\pi}\sum_i^{n_p} a w_i \text{Tr}(I^\text{D}(a z_i)) \label{eq:int_obj} \\
I^\text{D}(z) &= z^2 \mat{F}^D(z) \mat{S}^D_L\left((\mat{I}+\mat{Q}^D(z))^{-1}-\mat{I} \right) (\mat{S}^D_R)^T \mat{F}^D(z),
\end{align}
where Eq.~\ref{eq:offset_obj} is minimized to optimize the grid for the first integral of Eq.~\ref{eq:final_NI_zero_mom_main}, and Eq.~\ref{eq:int_obj} minimized for the second integral.
This can be done via either simple root-finding or minimization, and gives a robust optimization of the integration grids in $\mathcal{O}[ov]$ computational cost.
The resulting exponential convergence of the zeroth dd-moment estimate with number of quadrature points, along with the error estimates derived in App.~B, are shown in Fig.~\ref{fig:NumQuadConv} for both numerical integrands. We find that as few as $12$ quadrature points are sufficient for high accuracy in the results of this work, while the number of points is expected to increase for systems with a small or vanishing spectral gap.
\section{Reduction to cubic-scaling GW} \label{sec:CubicScaling}
With this reformulation of $GW$ in terms of the moments of the self-energy, it is possible to further reduce the scaling to cubic in time and quadratic in memory with respect to system size, in common with the lowest-scaling $GW$ approaches \cite{Hutter16,PhysRevB.94.165109,Visscher20,Duchemin2021}. We stress that this is not an asymptotic scaling after exploiting screening and locality arguments, but rather a formal scaling exploiting further rank reduction of quantities.
To do this, we employ a {\em double} factorization of the Coulomb integrals, allowing them to be written as as a product of five rank-2 tensors, as
\begin{equation}
(ia|jb) \simeq \sum_{P,Q} X_{iP} X_{aP} Z_{PQ} X_{jQ} X_{bQ} , \label{eq:THC_eri}
\end{equation}
This factorizes the orbital product into separate terms, and is also known as tensor hypercontraction or CP decomposition, used for various recent low-scaling formulations of quantum chemical methods, where the dimension of the $P$ and $Q$ space rigorously grow linearly with system size\cite{THC_V}. Use of this doubly-factorized form has also been previously suggested in the use of a reduced-scaling particle-particle RPA scheme \cite{Yang_THCppRPA}. This form for the integrals can be directly constructed with controllable errors in $\mathcal{O}[N^3]$ time\cite{LU2015329, doi:10.1021/acs.jctc.2c00861}.
Once found, the $Z_{PQ}$ can be symmetrically decomposed as $Z_{PQ}=Y_{PR} Y_{QR}$. By replacing the density-fitted integral tensor $V_{iaR}$ in the above expressions with the fully factorized form $X_{iP} X_{aP} Y_{RP}$, the contractions to form the moments of the $GW$ self-energy, and hence the Green's function and quasi-particle spectrum naturally follow as $\mathcal{O}[N^3]$ with formation of appropriate intermediates. We also require the numerical computation of the partially transformed dd-moment, $Z_{QS} X_{aS} X_{iS} \eta^{(0)}_{ia,jb} X_{jR} X_{bR} Z_{PR}$. Inspired by the low-scaling approach taken to RPA correlation energies in the work of Refs.~\onlinecite{Schurkus2016}, this can be achieved with the use of a contracted double-Laplace transform in the place of the original numerical integration procedure. This factorizes the squared energy denominator $\mat{F}(z)$, allowing the occupied and unoccupied indices to be contracted independently, similar to the space-time approaches to $GW$\cite{Visscher20}.
While this becomes a two-dimensional numerical integral, optimal quadrature grids can be calculated in a minimax sense \cite{HELMICHPARIS2016927,Graf2018}.
Applied to Eq.~\ref{eq:F}, this contracted double-Laplace transform takes the form
\begin{align}
{F}_{ia,ia}(z) &= D_{ia,ia} \int_{p=0}^\infty \frac{\sin{zp}}{z} e^{-pD_{ia,ia}}dp,
\end{align}
which allows the key matrix $\mat{Q}(z)$ of Eq.~\ref{eq:Q} to be obtained as
\begin{align}
Q_{PS}(z) &= 2\int_{p=0}^\infty \frac{\sin{zp}}{z} Y_{PQ} (A_{QR}(p) - B_{QR}(p) ) Y_{SR} dp
\end{align}
where both intermediates
\begin{align}
A_{QR}(p) &= X_{iQ} X_{aQ} \epsilon_a e^{-\epsilon_a p} e^{\epsilon_i p} X_{iR} X_{aR} \\
B_{QR}(p) &= X_{iQ} X_{aQ} \epsilon_i e^{-\epsilon_a p} e^{\epsilon_i p} X_{iR} X_{aR}
\end{align}
can be evaluated in $\mathcal{O}[N^3]$ cost. Further contractions in the evaluation of Eq.~\ref{eq:final_NI_zero_mom_main} also follow naturally with cubic scaling.
An alternative approach to reduce the scaling (to \emph{asymptotically} linear) without requiring the doubly-factorized integrals, is to screen the atomic orbital density-fitted integral contributions (constructed with the overlap metric), along with the double-Laplace transform, exploiting locality as has been recently performed for the RPA correlation energy \cite{Schurkus2016,Graf2018}.
An explicit numerical demonstration of this reduction to cubic cost via the double factorization of the Coulomb tensor of Eq.~\ref{eq:THC_eri} will follow in forthcoming work, with numerical results in the rest of this work employing the quartic scaling algorithm described in Sec.~\ref{sec:EfficientEval}.
\section{Results}\label{sec:Results}
\subsection{Comparison to quasiparticle GW approaches}
We first consider the convergence of the moment-truncated $G_0W_0$ algorithm compared to more traditional implementations, as found in the {\tt PySCF} software package \cite{PySCF2017, PySCF2020, PhysRevX.11.021006, doi:10.1021/acs.jctc.0c00704}. As found to be more effective for molecular systems due to the importance of exact exchange, we perform all calculations on a restricted Hartree--Fock reference \cite{VanSetten2013}. In Fig.~\ref{fig:O2Conv} we consider the convergence of the first ionization potential (IP) of singlet O$_2$ with conserved self-energy moment order. We compare this to two $G_0W_0$ implementations, one of which performs an exact frequency integration (denoted `full QP-$G_0W_0$', which scales as $\mathcal{O}[N^6]$), and one which performs an analytic continuation (AC) of the imaginary frequency self-energy to real-frequencies via a fit to Pad{\'e} approximants in order to perform the convolution (denoted `AC QP-$G_0W_0$', which scales as $\mathcal{O}[N^4]$) \cite{Ren_2012,doi:10.1021/acs.jctc.6b00380,doi:10.1021/acs.jctc.0c00704}. However, both of these two implementations also solve the diagonal approximation to the quasiparticle equation in solving for each state, effectively imposing a diagonal approximation to the self-energy in the MO basis. This is avoided in our work, however we can constrain a similar diagonal approximation by simply removing the off-diagonal components of our computed self-energy moments. This does not result in a significant computational saving in our approach, and therefore is only relevant for comparison purposes when considering the effect of this neglected off-diagonal part of the self-energy.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{convergence.pdf}
\caption{Convergence of the first $G_0W_0$ IP of singlet O$_2$ in a cc-pVDZ basis with respect to the number of conserved self-energy moments. Also shown is the same quasiparticle state computed from traditional $G_0W_0$ via an exact frequency integration (`Full') and analytic continuation approach to $G_0W_0$ (`AC'), albeit both imposing a diagonal approximation in their solution to the QP equation. In the moment expansion, we also consider a diagonal approximation, by explicitly removing the non-diagonal parts of our computed self-energy moments (`Diagonal $\Sigma$ moment'). All approaches find an IP of $8.49$~eV to within $10$~meV, with the difference between full-frequency and AC approaches $5$~meV, and the relaxation of diagonal approximation also accounting for small $\sim5$~meV differences,
with the reference Hartree--Fock IP for comparison being 9.73 eV.}
\label{fig:O2Conv}
\end{figure}
As can be seen in Fig.~\ref{fig:O2Conv}, the IP converges rapidly with moment order, with the full self-energy moments converging slightly faster and more systematically without the diagonal approximation (something also observed in other applications). The diagonal approximation converges to the `exact' frequency integration as expected, with our more complete (non-diagonal) self-energy moment approach only very slightly different, indicating the relative unimportance of the non-diagonal self-energy components in this system and lack of significant correlation-induced coupling between the mean-field quasiparticle states. Furthermore, the analytic continuation approach is also highly accurate for this system, introducing an error of only $5$~meV compared to the full frequency integration. We have furthermore numerically verified that our approach scales as $\mathcal{O}[N^4]$, with computational cost comparable to the analytic continuation approach.
Having demonstrated correctness compared to a high-scaling exact frequency implementation, we can compare our results to AC-$G_0W_0$ across a larger test set to consider the moment truncation convergence. We use the established `$GW100$' benchmark test set, where many $GW$ (and other excited state method) implementations have been rigorously compared \cite{VanSetten2015, Caruso2016, Lange2018, Backhouse2021}.
This benchmark set contains 102 diverse systems, with the IP of the molecules in the set ranging from $\sim4 - 25$ eV, and featuring molecules with bound metal atoms (including metal diatomics and clusters), strongly ionic bonding, and molecules with a strongly delocalised electronic structure. The molecules range in size from simple atomic systems to the five canonical nucleobases and large aliphatic and aromatic hydrocarbons, providing a suitable range in system size.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{strip_plot_vs_gw.pdf}
\caption{Errors in the IP, EA and gap for each system in the $GW100$ benchmark set, compared to AC-$G_0W_0$ in a def2-TZVPP basis, for each order of self-energy moment conservation. White circles show the mean signed error (MSE) aggregated over the test set for the given moment truncation in each quantity.}
\label{fig:swarm_GW}
\end{figure}
In Fig.~\ref{fig:swarm_GW} we consider the discrepancy in the first IP, electron affinity (EA), and quasiparticle gap across all systems as the order of conserved self-energy moments increases in a realistic def2-TZVPP basis, again compared to AC-$G_0W_0$. Errors in individual systems, along with the mean signed error (MSE) across the set (white circle) is shown for each conserved moment order. This MSE for the first IP decreases from -0.142~eV for the lowest order moment conservation, to -11~meV when up to the $11^\textrm{th}$-order moment is conserved, with the EA errors generally a little larger. Similarly, the gap calculations converge to a MSE of -34.8~meV, with standard deviation about the AC-$G_0W_0$ result of only 91~meV across all systems.
We note that there may also be small differences arising due to the comparison with AC-$G_0W_0$ due to the approximations of the frequency integration via analytically continued quantities, as well as the differences in whether off-diagonal parts of the self-energy are included. These will contribute to the discrepancy between the methods at each order, however while the comparison is not strictly equivalent and therefore these errors will be overestimated compared to an exact frequency and non-diagonal $G_0W_0$ limit, the general trend, convergence and level of accuracy which can be reached with moment order is likely to be similar.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{scatter.pdf}
\caption{Mean absolute errors (eV) for the IP ($x$-axis) and EA ($y$-axis) across the $GW100$ benchmark set in a def2-TZVPP basis compared to accurate CCSD(T) values. AC-$G_0W_0$ values, as well as moment-truncated results are shown, with the number in each data point marker giving the number of conserved moments. Extrapolation of individual data points from the $9^{\mathrm{th}}$- and $11^{\mathrm{th}}$-order conserved moment values are also provided, denoted by the truncation label `$\infty$'.}
\label{fig:GW_vs_CCSDT}
\end{figure}
It is important to put the scale of the moment truncation convergence in the context of the overall accuracy of the $G_0W_0$ method for these systems. In Fig.~\ref{fig:GW_vs_CCSDT} we therefore compare the aggregated mean absolute error (MAE) in the moment-truncated $G_0W_0$ IP and EA values over this $GW100$ test set to highly accurate CCSD(T) calculations on the separate charged and neutral systems, which is often used as a more faithful benchmark to compare against than experiment. We can therefore see the convergence of the moment truncation to the AC-$G_0W_0$ values compared to the intrinsic error in the method. This intrinsic error is found to dominate over the error due to the self-energy moment truncation for higher numbers of conserved moments.
It is natural to also consider whether a simple extrapolation can improve the moment-truncated results to an infinite moment limit. We therefore apply a linear extrapolation of the excitation energies to the infinite moment limit from the two most complete moment calculations of each system. We can see from Fig.~\ref{fig:GW_vs_CCSDT} that these results continue the trend of the MAE across the test set, slightly overshooting the AC-$G_0W_0$ comparison, albeit noting the other potential sources of discrepancy between these values discussed earlier.
\subsection{Full frequency spectra}
\begin{figure}[t]
\centering
\includegraphics[width=0.99\columnwidth]{guanine_spectra.pdf}
\caption{
Spectral functions for guanine in a def2-TZVPP basis set, for HF, AC-$G_{0}W_{0}$,
and the moment-conserving $G_{0}W_{0}$ approach with zero to five block Lanczos iterations, thereby conserving up to the $11^{\mathrm{th}}$-order self-energy moment.
The values indicated in the spectra indicate the Wasserstein metric taken with respect to the AC-$G_0W_0$ spectrum,
quantifying the difference between the spectral distributions.
The AC-$G_0W_0$ spectrum is indicated transparently behind the other spectra to ease visualisation of the convergence,
and was calculated using an iterative diagonal approximation to the quasiparticle equation.
}
\label{fig:guanine_spectra}
\end{figure}
One of the strengths of the moment-conserving approach to $GW$ of this work, is the ability to obtain all excitations from a given order of truncation in a single complete diagonalization of the effective Hamiltonian of Eq.~\ref{eq:concatenated_h}. This allows full frequency spectra to be obtained, with the approximation not expected to bias significantly towards accuracy in any particular energy range, making it suitable for $G_0W_0$ excitations beyond frontier excitations. In Fig.~\ref{fig:guanine_spectra} we therefore show a series of spectra plotting on the real frequency axis for the guanine nucleobase
in a def2-TZVPP basis set over a $100$~eV energy window about the quasiparticle gap,
taken from the $GW$100 benchmark set.
The convergence of the spectrum is shown for a series of conserved $G_0W_0$ moment orders, from the HF level up to the AC-$G_0W_0$ spectrum. The AC-$G_0W_0$ spectrum is also shown `behind' the other spectra, to allow the deviations to be observed for each moment. It can be seen that the full-frequency spectrum rapidly converges with conserved self-energy moment order, even for high-energy states where the HF approximation is poor. The similarity of each spectrum to the AC-$G_0W_0$ result accross the full frequency range can be rigorously quantified via the Wasserstein or `earth-mover' metric, which describes the similarity between probability distributions. This metric is shown as the value inside of each plot, indicating a rapid and robust convergence of the spectral features from the mean-field to the full $G_0W_0$ spectrum with increasing numbers of included moments.
This Wasserstein metric convergence plateaus at the $\sim7^{\mathrm{th}}$-order conserved moments, with further orders not improving this metric further. This could be due to numerical precision of the algorithm, or fundamental approximations in the AC-$G_0W_0$ such as the precision of the analytic continuation, or the diagonal approximation to the self-energy. Furthermore, it should be noted that the moment-conserving $GW$ approximation will rigorously have a larger number of poles included in its spectrum compared to those $GW$ approaches which rely on an iterative solution to the QP equation which considers the change to a single MO at a time. These additional peaks are likely very low weighted for this weakly-correlated system, however could be contributing to this plateau in the Wasserstein metric. We consider this point in more detail in the next section.
\subsection{Multiple solutions and additional spectral features}
\begin{figure*}[t]
\centering
\includegraphics[width=0.99\textwidth]{h2_discontinuities.pdf}
\caption{
Quasiparticle energies, self-energies, and renormalisation factors for the H$_2$ dimer in a 6-31G basis set
with varying bond lengths.
Shown are results for AC-$G_{0}W_{0}$ and moment-conserved $G_{0}W_{0}$ with zero and five iterations,
thereby conserving up to the $1^{\textrm{st}}$- and $11^{\mathrm{th}}$-order moments, respectively.
The self-energy corresponds to the diagonal element corresponding to the particular orbital,
evaluated at the quasiparticle energy.
The renormalisation factor corresponds to $Z_{p} = (1 - \frac{\partial \Sigma_{pp}(\epsilon_p)}{\partial \omega})^{-1}$,
with larger values indicating a more quasiparticle-like excitation.
The transparent lines in the lower panel show the existence of multiple solutions, broadening spectral features near the self-energy poles, and where a single dominant solution can be chosen (defined here by maximum overlap with the corresponding MO) denoted by the thicker line.
}
\label{fig:h2_discontinuities}
\end{figure*}
This fact that many low-scaling $GW$ implementations rely on an iterative solution to solve the quasiparticle equation can be a source of error and loss of robustness. This is because when self-energy poles are found near quasiparticle energies, the $GW$ poles can split into multiple peaks, where the final excitation energy converged to can depend sensitively on the specifics of the root-finding algorithm used to solve the QP equation. This was highlighted in the Ref.~\onlinecite{VanSetten2015} as a significant source of error, where a number of simple systems were found to exhibit a number of poles close to the HOMO energy level (with these solutions spanning a range of up to 1~eV). The specifics of which pole is converged to (with undesired solutions called `spurious') then depended on initial conditions, choices of optimization method, and specifics of the self-energy construction or linearization of the QP equation. The requirement to select one of these multiple solutions to the QP equation can also manifest in undesirable discontinuities in the excitation energies as e.g. the molecular geometry changes, as described in Ref.~\onlinecite{Monino2022}. An indicator for the presence of these `spurious' poles and multiple solutions is the magnitude of the quasi-particle weight or renomalization factor evaluated at the quasi-particle energy, defined as $Z_p = (1 - \frac{\partial \Sigma_{pp}(\epsilon_p)}{\partial \omega})^{-1}$, which indicates the approximate weight in the quasi-particle solutions.
Since the moment-conserving $GW$ approach obtains all poles in one step, all the excitations can be characterized by their quasiparticle weight, and either selected as specific excitation energies, or all excitations included in the spectrum to ensure smooth changes with molecular geometry. Points of discontinuity will therefore manifest as the presence of multiple lower-weighted solutions at a given energy, giving a smooth change to a broadened feature in the spectrum near self-energy poles. To demonstrate this, we consider the same simple system as Ref.~\onlinecite{Monino2022}, observing the $GW$ quasiparticle energies as a function of the inter-nuclear distance in the H$_2$ dimer in a 6-31G basis set. Figure~\ref{fig:h2_discontinuities} shows quasiparticle energies, self-energies and quasiparticle renormalisation factors
for AC-$G_0W_0$, and the moment conserving $G_0W_0$ approach with both up to the $1^{\mathrm{st}}$- and $11^{\mathrm{th}}$-order conserved self-energy moments in each sector.
The self-energies plotted are the diagonal elements corresponding to the particular MO, evaluated at the respective quasiparticle energy.
The figure shows the HOMO and first three unoccupied states found in this system, with the AC-$G_0W_0$ (first row) exhibiting discontinuities in the LUMO+2 state at slightly compressed geometries, and discontinuities in the LUMO+1 state at slightly stretched geometries. As discussed, these changing solutions arise from the specifics of the root-finding in the solution to the QP equation, which will generally (but not always) converge to the solution with largest quasiparticle weight between the multiple options, indicated by the renormalization factor. These discontinuous changes between states are also shown to coincide with poles in the self-energy in the second column at the MO energies for a given separation. These AC-$G_0W_0$ results are essentially the same as those found in Ref.~\onlinecite{Monino2022}.
When only the first two self-energy moments are conserved in each sector (second row), the approximation to the self-energy renders its pole structure sufficiently sparse such that their poles are pushed far from the MO energies at all geometries for these states. While this regularization removes the discontinuities, this significant approximation renders the renormalization factors close to one at all points, indicating only small changes from the original MOs. The final row represents a $G_0W_0$ calculation with up to the $11^{\mathrm{th}}$-order conserved moments. With this finer resolution of the self-energy dynamics, the structure of the self-energy closely matches the one from AC-$G_0W_0$, however, the multiple solutions are all found simultaneously, with their changing quasiparticle weights shown. The points of discontinuity are replaced by the presence of multiple solutions at similar energies and with competing quasiparticle weights, providing broad spectral features at those points.
If a single solution is required, the specific excitation can be selected from the manifold based rigorously on e.g. the maximum overlap with the MO of interest (shown as thicker lines in the plot) or largest quasiparticle weight, all of which can easily be selected. This removes the uncertainty in the energies of the states based on the unphysical specifics of the QP solution algorithm, without incurring additional complexity or cost in the moment-conserving algorithm. Furthermore, the relaxation of the diagonal approximation of the self-energy in this approach is expected to be more significant at these points of multiple solution, where mixing between the different MOs is expected to be more pronounced.
\section{Conclusions and outlook}
In this work, we present a reformulation of the $GW$ theory of quasiparticle excitations, based around a systematic expansion and conservation of the spectral moments of the self-energy. This contrasts with other approaches designed to approximate the central frequency integration of $GW$ theory, which use e.g. grid expansions, analytic continuation or contour deformation in order to affect a scaling reduction from the exact theory. The moment expansion presented in this work has appealing features arising from the avoidance of the an iterative solution to the quasiparticle equation for each state (avoiding `spurious' solutions), diagonal self-energy approximations, or requirements for analytic continuation of dynamical quantities. It allows for all excitation energies and weights to be obtained directly in a non-iterative single diagonalization of a small effective Hamiltonian, controlled by a single parameter governing the number of conserved self-energy moments.
Full RPA screening and particle-hole coupling in the self-energy is included, which is captured with $\mathcal{O}[N^4]$ computational scaling via a numerical one-dimensional integration, with a reduction to cubic scaling also proposed. This approach is enabled by a recasting of the RPA in terms of the moments of the density-density response function. Applied across we $GW100$ test set, we find rapid convergence to established $GW$ methodology results for both state-specific and full spectral properties, with errors due to the incompleteness of the moment expansion many times smaller than the inherent accuracy of the method. The formulation follows relatively closely from previous low-scaling approaches to RPA correlation energies, enabling these codes to be adapted to low-scaling $GW$ methods with relatively little effort.
Going forwards, we will aim to test the limits of the moment-truncated $GW$ formulation, pushing it to larger systems including the solid-state and different reference states, lower-scaling variants, and the inclusion of various self-consistent flavors of the theory. The reformulation of RPA in terms of a recursive moment expansion also lends itself to a low-scaling implementation of the Bethe-Salpeter equation for neutral excitations, which we will explore in the future, as well as other beyond-RPA approaches. Finally, we will also explore the connections of this moment expansion to kernel polynomial approaches which expand spectral quantities in terms of Chebyshev and other orthogonal polynomial expansions \cite{RevModPhys.78.275}.
\section*{Code availability}
Open-source code for reproduction of all results in this paper, along with examples, can be found at \href{https://github.com/BoothGroup/momentGW}{\tt https://github.com/BoothGroup/momentGW}.
\ifjcp\else
\newpage
\fi
\section*{Acknowledgements}
G.H.B. gratefully acknowledges support from the Royal Society via a University Research Fellowship, as well as funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 759063. We are grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP/P020194/1 and EP/T022213/1).
Further computational resources were awarded under the embedded CSE programme of the ARCHER2 UK National Supercomputing Service (http://www.archer2.ac.uk).
\ifjcp
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,108
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Ned Doheny
Separate Oceans
by Stephen M. Deusner
Patrick "Ned" Doheny was part of the SoCal scene of singer-songwriters couch-jumping around the canyon and dreaming of stardom. Pretty much every song Doheny ever recorded or released classifies as a rarity, which means for most listeners Separate Oceans, comprised of 19 songs from Ned Doheny's three albums from the 1970s, seeded with demos and alternate takes featuring some of his more famous friends will be an introduction into his sun-drenched world.
Is Separate Oceans a rarities comp or a bizarro-world greatest hits? The set is comprised of 19 songs from Ned Doheny's three albums from the 1970s, seeded with demos and alternate takes featuring some of his more famous friends. There are no hits per se, but songs that should have—and likely could have—been hits. Scion of oil barons for whom L.A. streets and beaches have been named, Edward "Ned" Doheny was part of the SoCal scene of singer-songwriters couch-jumping around the canyon and dreaming of stardom. He came from extreme wealth, but that only seems to have reinforced his desire to succeed as a pop musician: to prove himself as an artist instead of an heir.
He was one of the first signees to Asylum Records, but the label spent his mastering budget to market Jackson Browne's debut. After Doheny raided his trust fund to finish the songs, the album flopped unspectacularly. So did his follow-up, Hard Candy, in 1976. So did Prone in 1979. Still, some of his songs were hits for other performers (such as "Get It Up for Love", which Tata Vega rode into the top 20), and his LPs are doggedly pursued by vinyl collectors and crate diggers.
Pretty much every song Doheny ever recorded or released classifies as a rarity, which means for most listeners Separate Oceans will be an introduction into his sun-drenched world. The Numero Group has certainly timed the release perfectly: these songs are summer jams from an alternate universe where the Eagles slid quietly into obscurity and Asylum had the good sense to shill for Ned Doheny instead of Jackson Browne. The music sounds like it's been preserved in amber, so sharply redolent of the time and place of its creation. It'd be too easy to dismiss these songs as "yacht rock", a recently coined term that sounds satirical and derisive. In the Separate Oceans liner notes, Numero co-founder Ken Shipley defers to the phrase "marina pop", which is arguably more precise and certainly softer in tone, and "beach funk" might be an even more apt descriptor. These songs percolate casually, with spry guitars and beachball basslines cohering into superlatively breezy grooves.
It helps that Doheny is blessed with a voice like a lens flare, and his guitar playing, a style combining folky strums and classical dexterity, is nimble and creative. Even the slower tracks, like "A Love of Your Own" (co-written with Average White Band's Hamish Stuart), sound aggressively chipper, and the lyrics never tax your brain. In fact, these songs can sometimes be mesmerizingly meaningless, as Doheny crafts lyrics from stray clichés and non sequitur platitudes. "Don't you be afraid to give your heart/ You never know until you try it!" he advises on "A Love of Your Own". Rhyme typically takes precedence over meaning. As he explains on "When Love Hangs in the Balance", "You can't let it slide/ It can't be denied."
It's not like Browne or the Eagles were deft wordsmiths either, so the lyrical vacuousness of Doheny's music only identifies him as a product of his scene. An excessive headiness might weigh these songs down and rob them of their weightless quality, so it's good that Separate Oceans is aerodynamic to a fault, sequenced to emphasize the craft behind the music itself. Numero frontloads the set with finished studio takes in all their luxuriant glory, showcasing Doheny's ear for prismatic melodies and sophisticated arrangments, so side A of this double LP doubles as a revealing SoCal pop artifact and a near-perfect summer EP for 2014.
As the comp progresses, the demos strip away some of the veneer just enough to show the creative decisions Doheny was making. The plush "Get It Up for Love" burbles with futuristic synths and a playfully jazzy piano riff, and the song's punchline is right there in the title—the best and cleverest line written by this self-described "avatar of casual vulgarity"—but it takes on new gravity on the demo version of the song, which closes Separate Oceans. Doheny struggles to get the tone right. Is the song flippant? Serious? Both? Ultimately, all that matters is his jittery strumming, as sharp and directed as ever.
During the comp's second half, it's not hard to detect some rain clouds on the horizon, threatening to intrude on Doheny's sunny-day pop. "I Know Sorrow" is exactly as blunt as its title suggests, paring complex emotions down to those three simple words. It might be too heavy, too unwieldy—at least on this collection—were it not for that high-flying piano jam on the coda, which sounds rambunctious and even joyful. That sense of creativity and musical invention staves off melancholy and self-absorption, even as it becomes clear that the music business has hurt Doheny much more than any lover ever could. "That old rock and roll will break your heart," he sings on "Fineline", "or make you whole." To his considerable credit, he sounds like half a man on these songs.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,859
|
Le Space est une célèbre boite de nuit d'Ibiza. Elle a été nommée Best Global Club (meilleur club du monde) aux International Dance Music Awards en 2005 puis encore en 2006. D'une façon plus générale, elle est considérée comme le club 1 au monde par les amateurs d'EDM et apparaît à ce titre début 2016 en première place du sondage de DJ Magazine dans mondiaux puis pour le classement publié en 2017 alors que le lieu ferme en octobre de l'année précédente. Finalement cette discothèque est classée dans le magazine sept fois 1 en dix ans.
Présentation
Club de référence ouvert en 1989, il peut accueillir plusieurs milliers de personnes à l'intérieur et environs sur la terrasse en extérieur. La scène principale porte le nom de « Discoteca ». Il y a également un toit-terrasse.
La boite reçoit chaque été nombre de DJ célèbres, tels que Robbie Rivera, Sasha, Sven Väth, Paul Oakenfold, Erick Morillo ou Carl Cox. Ce dernier est d'ailleurs DJ-Résident durant une quinzaine d'années, jusqu'en 2016 à la suite du départ du propriétaire historique de l'établissement, Pepe Rosello. Au fil des années, le club établit de nombreux partenariats avec des promoteurs anglais, ou avec l'organisation de l'Ultra par exemple. , écrit Cathy Guetta. Le nombre de clients annuel est estimé à .
Vers 2015, le Space se rapproche commercialement de son voisin d'en face, l'Ushuaïa Ibiza Beach Hotel. Mais à la suite de l'annonce du départ à la retraite de Pepe Rosello, le Space est mis en vente puis acheté par le groupe Matutes, propriétaire de l'Ushuaïa, qui le fait disparaitre à l'issue de la saison d'été 2016 : devant être à l'origine remis en état mais aussi renommé, l'ouverture d'un lieu différent est prévue pour l'année 2017. Le nom bien connu doit être conservé, mais est finalement changé pour « Hï Ibiza ».
Notes et références
Voir aussi
Liens externes
Boîte de nuit en Espagne
Culture à Ibiza
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{"url":"http:\/\/www.ck12.org\/book\/Basic-Algebra\/r1\/section\/5.9\/","text":"<img src=\"https:\/\/d5nxst8fruw4z.cloudfront.net\/atrk.gif?account=iA1Pi1a8Dy00ym\" style=\"display:none\" height=\"1\" width=\"1\" alt=\"\" \/>\n\n# 5.9: Chapter 5 Review\n\nDifficulty Level: At Grade Created by: CK-12\n\nFind an equation of the line in slope-intercept form using the given information.\n\n1. (3, 4) with slope=23\\begin{align*}slope= \\frac{2}{3}\\end{align*}\n2. slope=5\\begin{align*}slope=-5\\end{align*}, yintercept=9\\begin{align*}y-intercept=9\\end{align*}\n3. slope=1\\begin{align*}slope=-1\\end{align*} containing (6, 0)\n4. containing (3.5, 1) and (9, 6)\n5. slope=3\\begin{align*}slope = 3\\end{align*}, y\\begin{align*}y-\\end{align*}intercept =1\\begin{align*}=-1\\end{align*}\n6. slope=13\\begin{align*}slope=\\frac{-1}{3}\\end{align*} containing (\u20133, \u20134)\n7. containing (0, 0) and (9, \u20138)\n8. slope=53\\begin{align*}slope=\\frac{5}{3}\\end{align*}, y\\begin{align*}y-\\end{align*}intercept =6\\begin{align*}=6\\end{align*}\n9. containing (5, 2) and (\u20136, \u20133)\n10. slope=3\\begin{align*}slope=3\\end{align*} and f(6)=1\\begin{align*}f(6)=1\\end{align*}\n11. f(2)=5\\begin{align*}f(2)=-5\\end{align*} and f(6)=3\\begin{align*}f(-6)=3\\end{align*}\n12. slope=38\\begin{align*}slope=\\frac{3}{8}\\end{align*} and f(1)=1\\begin{align*}f(1)=1\\end{align*}\n\nFind an equation of the line in point-slope form using the given information.\n\n1. slope=m\\begin{align*}slope=m\\end{align*} containing (x1,y1)\\begin{align*}(x_1, y_1)\\end{align*}\n2. slope=12\\begin{align*}slope=\\frac{1}{2}\\end{align*} containing (-7, 5)\n3. slope=2\\begin{align*}slope=2\\end{align*} containing (7, 0)\n\nGraph the following equations.\n\n1. y+3=(x2)\\begin{align*}y+3=-(x-2)\\end{align*}\n2. y7=23(x+5)\\begin{align*}y-7=\\frac{-2}{3} (x+5)\\end{align*}\n3. y+1.5=32(x+4)\\begin{align*}y+1.5=\\frac{3}{2}(x+4)\\end{align*}\n\nFind the equation of the line represented by the function below in point-slope form.\n\n1. f(1)=3\\begin{align*}f(1)=-3\\end{align*} and f(6)=0\\begin{align*}f(6)=0\\end{align*}\n2. f(9)=2\\begin{align*}f(9)=2\\end{align*} and f(9)=5\\begin{align*}f(9)=-5\\end{align*}\n3. f(2)=0\\begin{align*}f(2)=0\\end{align*} and slope=83\\begin{align*}slope=\\frac{8}{3}\\end{align*}\n\nWrite the standard form of the equation of each line.\n\n1. y3=14(x+4)\\begin{align*}y-3=\\frac{-1}{4}(x+4)\\end{align*}\n2. y=27(x21)\\begin{align*}y=\\frac{2}{7}(x-21)\\end{align*}\n3. 3x25=5y\\begin{align*}-3x-25=5y\\end{align*}\n\nWrite the standard form of the line for each equation using the given information.\n\n1. containing (0, \u20134) and (\u20131, 5)\n2. slope=43\\begin{align*}slope=\\frac{4}{3}\\end{align*} containing (3, 2)\n3. slope=5\\begin{align*}slope=5\\end{align*} containing (5, 0)\n4. Find the slope and y\\begin{align*}y-\\end{align*}intercept of 7x+5y=16\\begin{align*}7x+5y=16\\end{align*}.\n5. Find the slope and y\\begin{align*}y-\\end{align*}intercept of 7x7y=14\\begin{align*}7x-7y=-14\\end{align*}.\n6. Are 12x+12y=5\\begin{align*}\\frac{1}{2} x+\\frac{1}{2} y=5\\end{align*} and 2x+2y=3\\begin{align*}2x+2y=3\\end{align*} parallel, perpendicular, or neither?\n7. Are x=4\\begin{align*}x=4\\end{align*} and y=2\\begin{align*}y=-2\\end{align*} parallel, perpendicular, or neither?\n8. Are 2x+8y=26\\begin{align*}2x+8y=26\\end{align*} and x+4y=13\\begin{align*}x+4y=13\\end{align*} parallel, perpendicular, or neither?\n9. Write an equation for the line perpendicular to y=3x+4\\begin{align*}y=3x+4\\end{align*} containing (\u20135, 1).\n10. Write an equation for the line parallel to y=x+5\\begin{align*}y=x+5\\end{align*} containing (\u20134, \u20134).\n11. Write an equation for the line perpendicular to 9x+5y=25\\begin{align*}9x+5y=25\\end{align*} containing (\u20134, 4).\n12. Write an equation for the line parallel to y=5\\begin{align*}y=5\\end{align*} containing (\u20137, 16).\n13. Write an equation for the line parallel to x=0\\begin{align*}x=0\\end{align*} containing (4, 6).\n14. Write an equation for the line perpendicular to \\begin{align*}y=-2\\end{align*} containing (10, 10).\n15. An Internet caf\u00e9 charges $6.00 to use 65 minutes of their Wifi. It charges$8.25 to use 100 minutes. Suppose the relationship is linear.\n1. Write an equation to model this data in point-slope form.\n2. What is the price to acquire the IP address?\n3. How much does the caf\u00e9 charge per minute?\n16. A tomato plant grows \\begin{align*}\\frac{1}{2}\\end{align*} inch per week. The plant was 5 inches tall when planted.\n1. Write an equation in slope-intercept form to represent this situation.\n2. How many weeks will it take the plant to reach 18 inches tall?\n17. Joshua bought a television and paid 6% sales tax. He then bought an albino snake and paid 4.5% sales tax. His combined purchases totaled \\$679.25.\n1. Write an equation to represent Joshua\u2019s purchases.\n2. Graph all the possible solutions to this situation.\n3. Give three examples that would be solutions to this equation.\n18. Comfy Horse Restaurant began with a 5-gallon bucket of dishwashing detergent. Each day \\begin{align*}\\frac{1}{4}\\end{align*} gallon is used.\n1. Write an equation to represent this situation in slope-intercept form.\n2. How long will it take to empty the bucket?\n19. The data below shows the divorce rate per 1,000 people in the state of Wyoming for various years (source: Nation Masters).\n1. Graph the data in a scatter plot.\n2. Fit a line to the data by hand.\n3. Find the line of best fit by hand.\n4. Using your model, what do you predict the divorce rate is in the state of Wyoming in the year 2011?\n5. Repeat this process using your graphing calculator. How close was your line to the one the calculator provided?\n\n\\begin{align*}&\\text{Year} && 2000 && 2001 && 2002 && 2003 && 2004 && 2005 && 2006 && 2007\\\\ &\\text{Rate (per 1,000 people)} && 5.8 && 5.8 && 5.4 && 5.4 && 5.3 && 5.4 && 5.3 && 5.0\\end{align*}\n\n1. The table below shows the percentage of voter turnout at presidential elections for various years (source The American Presidency Project).\n\n\\begin{align*}&\\text{Year} && 1828 && 1844 && 1884 && 1908 && 1932 && 1956 && 1972 && 1988 && 2004\\\\ &\\% \\ \\text{of Voter Turnout} && 57.6 && 78.9 && 77.5 && 65.4 && 56.9 && 60.6 && 55.21 && 50.15 && 55.27\\end{align*}\n\n(a) Draw a scatter plot of this data.\n\n(b) Use the linear regression feature on your calculator to determine a line of best fit and draw it on your graph.\n\n(c) Use the line of best fit to predict the voter turnout for the 2008 election.\n\n(d) What are some outliers to this data? What could be a cause for these outliers?\n\n1. The data below shows the bacteria population in a Petri dish after \\begin{align*}h\\end{align*} hours.\n\n\\begin{align*}&h \\ \\text{hours} && 0 && 1 && 2 && 3 && 4 && 5 && 6\\\\ &\\text{Bacteria present} && 100 && 200 && 400 && 800 && 1600 && 3200 && 6400\\end{align*}\n\n(a) Use the method of interpolation to find the number of bacteria present after 4.25 hours.\n\n(b) Use the method of extrapolation to find the number of bacteria present after 10 hours.\n\n(c) Could this data be best modeled with a linear equation? Explain your answer.\n\n1. How many seconds are in 3 months?\n2. How many inches are in a kilometer?\n3. How many cubic inches are in a gallon of milk?\n4. How many meters are in 100 acres?\n5. How many fathoms is 616 feet?\n\n### Notes\/Highlights Having trouble? 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Home / Band Management / 4 Techniques to Break Through Your Songwriting Block
4 Techniques to Break Through Your Songwriting Block
By Michael St. James on June 14, 2017
It happens to every writer, the dreaded "writer's block." Writer's block is that horrible feeling that paralyzes you while staring at a blank word doc or notebook to the point of being incapable of knowing where the hell to start, much less what to actually write. It happens to authors, columnists, bloggers, and yes, songwriters, too.
Unlike other writers, songwriters have the honor (or curse) of not only having to come up with the words on the page, but also rhyming patterns, the melody, a memorable chorus, a bridge, oh, and tell a universal story differently than every other songwriter has, all in under four minutes. Not to mention the music stuff, major or minor? What style? What tempo? What key? Argggh!
Each one of these points can block you from finishing your song. When you actually think about it, it's a miracle any song gets written and finished.
But there is hope. Having to balance all of these creative choices provides songwriters with extra tools that those who only write words do not have. So, I want to share with you a sampling of some of the techniques I teach to my songwriter groups and personally use to 'break the block.'
First, the point of breaking a writer's block is not to produce amazing work immediately. That's most likely not going happen. The goal of this is to trick your brain and creative synapses to open up and flow freely again so that you can get back to creating great work. These techniques work for any level of musician/writer, so give them a try and find which ones work best for your style of writing.
Finish a Crappy Song
Even Prince wrote shitty songs. Not every song you write is going to be a masterpiece, so get used to it. Too often we waste our time and energy trying to force an incredible song when it just isn't there. So, what to do? Finish a crappy one. Seriously, finish it as quick as you can. Do not think about how this song would play live, or how you could record it, because it's not going to be. Do not pause on a lyric line, simply find a rhyming word for the end and write into it. Do not suffer over the transition of the chorus into the second verse, simply write a three-word chorus and repeat it four times. Those three words are also your title. Finish the song. You can always come back and rip yourself off later if there are good bits in it.
Write A Non-Word Chorus
Take the words out of the problem. Don't think at all about the "what" of the song, but focus on the vibe of the notes and chords you already have. Say you have a cool chorus written, but the words are just not fitting right, play that pattern and start singing to it using only vowels. By using soft consonants like "d" and "b" and add vowels. Start with "Doo, Doo," and then use "Bah, Bah." You can try to use "Oohs" or "Ahs," but I have found that you need the consonants in order to establish syllabic rhythm. Humming is fine, but it won't get you finishing quicker. Using these types of "non-words," you will start to feel the rhythm of the melody and perhaps, where there are too many (or not enough) notes.
Rip-Off Another Song
Is this professional publisher and licensor telling me to rip-off songs? Yep. Seriously, take a song you love or even hate, and try to write your own version of it. Pick a song, any song. Remember, the point of these exercises is not to write a great song; it is to simply break the block. I do this with commercial jingles and show themes. For instance, I took the theme from Golden Girls and made it into "Spank You for Wearing Depends." But it doesn't have to be a parody song. Fire up Spotify and pick one of the Top 5 streamed songs, and write your own version. Again, this is not going to be recorded or released. It will get your creative juices flowing and you just might expand your writing style.
Change Instruments
If you write on piano or synth, pick up that old guitar and strum a while. Conversely, if you always write on acoustic guitar, try writing a chorus on a bass, or even just the low E guitar string. If you have a synth, try writing a song using only a horn sample. By changing the tonality of the instrument you are writing on, you will find new areas to spark your imagination.
Look, the struggle is real. But, it's only temporary. You have tons of great songs to write, and I can't wait to hear them.
-Michael St. James is the founder and creative director of St. James Media, specializing in music licensing, publishing, production and artist development. Follow on Twitter @michaelstjames
photo by Andi Sidwell, used under a Creative Commons license.
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\chapter{Quantum Measurements, Operations, and Physical
Models}\label{ch:ch1}
\section{Classical and quantum events}
Given a finite probabilistic space $\Omega=\{1,\dots,N\}$, it is
possible to define probability distributions $P=\{p_1,\dots,p_N\}$ on
$\Omega$, where $0\le p_i\le 1$, $\sum_ip_i=1$. The set of all
probability distributions on $\Omega$, $\set{P}(\Omega)$, is a convex set.
It is simple to recognize its extremal points as the
delta-distributions $p_i=\delta_{ij}$. Such a structure for
$\set{P}(\Omega)$ can be rephrased saying that $\set{P}(\Omega)$ is a simplex,
namely, a convex set whose elements are \emph{uniquely} expressed as a
convex combination of extremal points. Random variables on $\Omega$
are defined as mappings $X$ from $\Omega$ into a set of ``values''
$\Upsilon$. Such values can be numbers, tensors, or whatever objects.
When $\Upsilon$ is a real vector space, it is well-defined the mean
value of $X:\Omega\to\Upsilon$, given $P\in\set{P}(\Omega)$, as $\bar
X\equiv \sum_ip_iX(i)$. The set of random variables on $\Omega$ forms
a commutative algebra (under point-wise multiplication). Events are
particular random variables where $\Upsilon$ is the two-values set
$\{0,1\}$. In the classical case, \emph{events form a boolean
algebra}\footnote{A boolean algebra $\alg B$ is a set of elements
$\alg B=\{a,b,c,\dots\}$ satisfying the following properties: (i)
$\alg B$ has two binary idempotent, commutative, and associative
operations, $\land$ (logical \texttt{AND}) and $\lor$ (logical
\texttt{OR}); (ii) $\alg B$ contains universal bounds $\varnothing$
and $I$; (iii) for all $a\in\alg B$, there exists its complementary
element $a'\in\alg B$ such that $a\land a'=\varnothing$, and $a\lor
a'=I$.}: Given two events \mbox{$E_1,E_2:\Omega\to\{0,1\}$}, once
defined two binary operations $\land$ and $\lor$ as
\begin{equation}
E_1\land E_2\equiv E_1\cdot E_2,\qquad E_1\lor E_2\equiv E_1+E_2-E_1\cdot E_2,
\end{equation}
where ``$\cdot$'' is the point-wise multiplication, it is
straightforward to verify that all properties of a boolean algebra are
satisfied.
Consider now a $N$-dimensional complex vector space $\mathscr{H}$. The
analogue of probability distributions are $N\times N$ density matrices
$\rho$, i.~e. positive semi-definite trace-one matrices. The analogue
of random variables are $N\times N$ hermitian matrices $X$. Since
random variables, usually called \emph{observables}, are hermitian,
they admit a spectral decomposition $X=\sum_jx_j\Pi_j^X$, where
$\Pi_j^X$ are orthogonal projections (of rank greater than one, in
case of degeneracy). Density matrices, usually called \emph{states},
define probability distributions over the spectrum of an observable,
by means of the formula $\mu_\rho^X(x_j)\equiv\operatorname{Tr}[\rho\Pi_j^X]$. The
mean value of an observable $X$, given a state $\rho$, is well-defined
as $\bar X\equiv\operatorname{Tr}[\rho X]$. The non-commutative analogue of events
are projections $E_i=E_i^2$. The set of quantum events $\set{E}(\mathscr{H})$,
called \emph{quantum logic}, has two binary operations $\land$ and
$\lor$ defined as
\begin{equation}
E_1\land E_2\equiv E_1E_2,\qquad E_1\lor E_2\equiv E_1+E_2-E_1E_2,
\end{equation}
where now the multiplication is the usual (non-commutative) matrix
multiplication.
The fundamental differences between the classical model and the
quantum model are the following (and they are basically equivalent):
\begin{enumerate}
\item the quantum logic is not a boolean algebra, since the
distributivity law does not hold (beacause of the non-commutativity
of the matrix product);
\item the convex set of states on $\mathscr{H}$ is not a simplex, but it is
strongly convex, whence quantum states admit many equivalent
ensemble decompositions;
\item the algebra of observables on $\mathscr{H}$ is non-commutative.
\end{enumerate}
See also the introduction paragraphs in \cite{Holevo1} and
\cite{Holevo2}.
\section{Notations}\label{subsec:notations}
To each quantum system, it is associated a complex separable Hilbert
space $\mathscr{H}$, equipped with the inner product $\<\psi|\phi\>$, linear
in $\phi$ and antilinear in $\psi$, following Dirac notation. The set
of bounded operators on $\mathscr{H}$ will be denoted as $\set{B}(\mathscr{H})$. An
operator $X$ is called \emph{self-adjoint} if it is densely defined
and $X=X^\dag$ on its domain\footnote{An operator is called
\emph{hermitian} if its domain is dense in $\mathscr{H}$ and $X\subseteq
X^\dag$. In finite dimension the two definitions coincide and there
is no need to bother with the density of the operator's domain.}.
Self-adjoint operators are called \emph{observables} and are in
correspondence with orthogonal resolutions of the identity by means of
the formula
\begin{equation}
X=\int_{-\infty}^{+\infty}x\textrm{d}\Pi^X(x),\qquad I=\int_{-\infty}^{+\infty}\textrm{d}\Pi^X(x).
\end{equation}
Positive semi-definite trace-one operator $\rho\in\set{T}^+(\mathscr{H})$ are
called \emph{state}. We will denote the set of states of a system
$\mathscr{H}$ as $\set{S}(\mathscr{H})$. Since they are all compact operators, states can
be essentially viewed as infinite density matrices, also in the
infinite dimensional case, with no relevant differences from the usual
finite dimensional setting. From now on, if not otherwise specified,
we will deal with finite $d$-dimensional Hilbert spaces isomorphic to
$\mathbb C^d$, for which all linear operators are everywhere defined,
bounded and trace-class, and the self-adjointness coincides with
hermiticity. Moreover, spectral resolutions are all discrete, i.~e.
$X=\sum_jx_j\Pi_j^X$.
Composite systems carry a tensor-product Hilbert space,
\mbox{$\mathscr{H}_1\otimes \mathscr{H}_2\otimes\cdots\otimes\mathscr{H}_N$.} Bounded
operators $\set{B}(\mathscr{H})$ form themselves a Hilbert space isomorphic to
\newline\mbox{$\mathscr{H}\otimes\mathscr{H}\equiv\mathscr{H}^{\otimes 2}$.} Once fixed a
basis $\mathbf{b}=\{|i\>\}$ for $\mathscr{H}$, we define the following isomorphism
between operators in $\set{B}(\mathscr{H})$ and vectors in $\mathscr{H}^{\otimes 2}$:
\begin{equation}
X=\sum_{ij}X_{ij}|i\>\<j|\longleftrightarrow |X\rangle\!\rangle\equiv\sum_{ij}X_{ij}|i\>\otimes|j\>,
\end{equation}
satisfying
\begin{enumerate}
\item $\langle\!\langle X|Y\rangle\!\rangle=\operatorname{Tr}[X^\dag Y]$, i.~e. the Hilbert-Schmidt product;
\item $(X\otimes Y)|Z\rangle\!\rangle= |XZY^T\rangle\!\rangle$, where $Y^T$ denotes the
transposition with respect to the fixed basis $\mathbf{b}$;
\item $\operatorname{Tr}_1[|X\rangle\!\rangle\langle\!\langle Y|]=X^TY^*$, where $Y^*$ denotes the complex
conjugation with respect to $\mathbf{b}$;
\item $\operatorname{Tr}_2[|X\rangle\!\rangle\langle\!\langle Y|]=XY^\dag$.
\end{enumerate}
With this notation the state $|I/\sqrt{d}\rangle\!\rangle$ is the maximally
entangled state on $\mathscr{H}^{\otimes 2}$:
\begin{equation}
\frac{1}{\sqrt d}|I\rangle\!\rangle=\frac{1}{\sqrt d}\sum_i|i\>\otimes|i\>.
\end{equation}
Such a state will play a major role in the characterization of quantum
devices.
\section{Quantum measurements statistics: POVM's}\label{sec:POVM}
Given a state $\rho$ and an observable $X=\sum_jx_j\Pi_j^X$, the
\emph{statistical postulate} states that: The probability of obtaining
a result $x_j$ within a set $\Delta=\{x_j\}_{j\in J}$ is given by
\begin{equation}
p(x_j\in\Delta)=\operatorname{Tr}\left[\rho \sum_{j\in J}\Pi_j^X\right].
\end{equation}
This means, as we already saw, that a state induces a probability
measure $\mu_\rho^X(x_j)= \operatorname{Tr}[\rho\Pi_j^X]$ over the set of outcomes
for a given observable.
It is clear that, apart from the actual measured value $x_j$ of the
observable $X$, the statistics of the outcomes is completely
determined by the structure of its spectral resolution $\{\Pi_j^X\}$.
In the case of an observable, such $\Pi_j$'s are orthogonal
projections, i.~e. $\Pi_i^X \Pi_j^X=\Pi_i^X\delta_{ij}$, summing up to
the identity, $\sum_j\Pi_j^X=I$. With a little abuse of terminology,
from now on we will refer to an \emph{observable} just as a set of
orthogonal projections resolving the identity, and to the
\emph{observable outcomes} as the indices $j$'s labelling different
$x_j$'s.
The concept of observable is generalized by the concept of
\emph{positive operator-valued measure} (POVM, for short), which is a
set of positive operators $\povm P=\{P_1,P_2,\dots,P_N\}$ summing up
to the identity $\sum_iP_i=I$. Notice that $P_i$'s need not to be
orthogonal, not even projections, and the number of oucomes of $\povm
P$, i.~e. its cardinality $|{\mathbf P}|\equiv N$, can be larger than the
Hilbert space dimension $d$. As before, also in the case of POVM's,
the probability of obtaining the $j$-th outcome, given the system in
the state $\rho$, is postulated to be
$\mu_\rho^\povm{P}(j)\equiv\operatorname{Tr}[\rho P_j]$.
We call a two-outcomes POVM $\povm{P}=\{P,I-P\}$ an \emph{effect} or,
equivalently, a \emph{property}. According to
\cite{Busch-Lahti-Mittelstaedt} we say that an effect $\povm{P}=\{P,
I-P\}$ describes a \emph{real property} for the system $\mathscr{H}$ in the
state $\rho$, if $\operatorname{Tr}[\rho P]=1$.
Finally, we introduce here the definition of
\emph{range}\footnote{There is no possibility of confusion between the
range of a POVM and the range of an operator, being two completely
unrelated concepts.} of a POVM, a concept we will extensively use in
Chapter \ref{ch:noise}.
\begin{definition}[POVM range]\label{def:POVM-range}
Given a POVM $\povm P=\{P_1,P_2,\dots,P_N\}$, its \emph{range},
denoted as $\set{Rng}(\povm P)$, is defined to be the convex set of
probability distributions $\povm p=\{p_1,p_2,\dots,p_N\}$ obtained
as $p_i=\operatorname{Tr}[\rho P_i]$, varying $\rho$ in all $\set{S}(\mathscr{H})$.
\end{definition}
\begin{remark}\label{rem:POVM-range}
Notice that, since $\rho$ in Definition \ref{def:POVM-range} moves
around the whole quantum states' set, the range of a POVM identifies
uniquely the POVM. In other words, the correspondence
\begin{equation}
\povm P\longleftrightarrow\set{Rng}(\povm P)
\end{equation}
is one-to-one.
\end{remark}
\section{Quantum operations and
instruments}\label{sec:state-reduction}
Since now, we dealt only with the outcomes statistics. However, in
order to completely describe the measurement statistics we need also
to specify the state reduction from prior state $\rho$ to posterior
state $\rho_j$ conditioned by the outcome $j$. The state reduction is
nothing but a rule telling us which is the system's state after the
measurement has been performed and the outcome collected.
\subsection{State collapse postulate}\label{subsec:state-collapse}
Von Neumann \cite{vonneumann} derived the well-known \emph{state
collapse rule} starting from the following hypothesis:
\begin{enumerate}
\item the observable to be measured has discrete spectrum and it is
non degenerate, namely all its eigenspaces are one-dimensional, in
formula $X=\sum_ix_i|x_i\>\<x_i|$;
\item the measurement is \emph{perfectly repeatable}\footnote{See
Section \ref{sec:repeatable}.}: Literally from von Neumann's book
``\emph{if a physical quantity is measured twice in succession in a
system, then we get the same value each time}''.
\end{enumerate}
If such hypotheses are verified, then the system state after the
measurement is
\begin{equation}
\rho\longmapsto\rho_j\equiv|x_j\>\<x_j|.
\end{equation}
L\"uders \cite{luders} generalized von Neumann's theorem to degenerate
observables, introducing the postulate of \emph{minimum disturbance}
in the sense that a state, for which a property $\povm P$ is real, is
left unchanged by a measurement of $\povm{P}$. According to L\"uders'
rule, when measuring the observable $X=\sum_ix_i\Pi_i^X$ the system's
state after the measurement is
\begin{equation}\label{eq:luders}
\rho\longmapsto\rho_j\equiv\frac{\Pi_j^X\rho\Pi_j^X}{\operatorname{Tr}[\rho\Pi_j^X]}.
\end{equation}
The interpretation problems to which the state collapse postulate led
are beyond the aim of this manuscript.
\subsection{Quantum operations}\label{subsec:quantum-operations}
The appropriate mathematical objects describing a general quantum
state change are the so-called \emph{quantum operations} \cite{kraus}.
A quantum operation $\map E$, is a completely positive
trace-non-increasing linear mapping from $\set{T}^+(\mathscr{H})$ of an input
system $\mathscr{H}$ to $\set{T}^+(\mathscr{K})$ of an output system $\mathscr{K}$. The map $\map
E:\set{T}^+(\mathscr{H})\to\set{T}^+(\mathscr{K})$ is generally probabilistic, and the trace
$\operatorname{Tr}[\map E(\rho)]\le 1$ represents the probability that the
transformation
\begin{equation}
\rho\longmapsto\rho'\equiv\frac{\map E(\rho)}{\operatorname{Tr}[\map E(\rho)]}
\end{equation}
occurs. Deterministic quantum operations, i.~e. completely positive
trace-preserving maps such that $\operatorname{Tr}[\map E(\rho)]=1$ for all
$\rho\in\set{S}(\mathscr{H})$, are called \emph{channels}. All quantum operations
admit the highly non-unique Kraus representation
\begin{equation}
\map E(\rho)=\sum_jE_j\rho E_j^\dag,
\end{equation}
where $E_j$'s are linear operators from $\mathscr{H}$ to $\mathscr{K}$. Nonetheless,
it is always possible to choose a Kraus representation such that
$\operatorname{Tr}[E_i^\dag E_j]=\|E_i\|^2_2\delta_{ij}$; we call it
\emph{canonical} Kraus representation. A quantum operation is a
channel if and only if its Kraus operators satisfy the normalization
condition
\begin{equation}\label{eq:trace-pres-with-kraus}
\sum_i E_i^\dag E_i=I.
\end{equation}
\begin{remark}\label{remark:Luders}
The L\"uders' recipe for state change clearly corresponds to a
quantum operation $\map E_j(\rho)=\Pi_j^X\rho\Pi_j^X$. Notice that
the average reduced state $\bar\rho\equiv\sum_ip(i)\rho_i$ can be
read as the output of the channel $\map
E(\rho)=\sum_i\Pi_i^X\rho\Pi_i^X$.
\end{remark}
Every quantum operation $\map E:\set{S}(\mathscr{H})\to \set{S}(\mathscr{K})$ induces
naturally a quantum operation $\dual{\map E}$ from $\set{B}(\mathscr{K})$ to
$\set{B}(\mathscr{H})$ by means of the duality relation $\operatorname{Tr}[\map E(\rho)
X]=\operatorname{Tr}[\rho\dual{\map E}(X)]$, valid for all $\rho\in\set{S}(\mathscr{H})$ and
$X\in\set{B}(\mathscr{K})$. The map $\dual{\map E}$ is called the \emph{dual} map,
and $\dual{\map E}(I_\mathscr{K})=I_\mathscr{H}$ if and only if $\map E$ is a channel.
\begin{remark}\label{remark:instrument}
Given a measurement whose outcomes statistics is described by means
of the POVM $\povm P$, there exist many different channels
associated with $\povm P$. These channels are written as $\map
E^\povm{P}(\rho)=\sum_i\map{E}_i^\povm{P}(\rho)$, with
$\dual{(\map{E}_j^\povm{P})}(I)=P_j$, and choosing between them
correspond to assign a particular state reduction rule.
\end{remark}
\subsection{Instruments}\label{subsec:instruments}
In the modern formulation of Quantum Mechanics, the most general tool
used to describe statistical correlations between the outcomes of
successive measurements is given by the notion of (completely
positive) \emph{instrument}, which has been introduced by Davies and
Lewis \cite{davies-lewis}. An instrument is basically a mapping
$\mathfrak{I}$ from the set $\Omega$ of outcomes to the set of quantum
operations on $\set{S}(\mathscr{H})$, such that $\mathfrak{I}(\bigcup_{j\in
J}j)=\sum_{j\in J}\mathfrak{I}(j)$ and $\mathfrak{I}(\Omega)$ is a
channel. The fundamental result about instruments is the following
\cite{ozawa}
\begin{theorem}[Ozawa, 1984]
Every statistical measurement theory, consisting both of outcomes
statistics and state reduction rule, can be described by means of an
appropriate instrument.
\end{theorem}
Actually, instruments formalism has been introduced in literature
mainly to handle the case of continuous outcome space $\Omega$, which
is described as a standard Borel space equipped with a
$\sigma$-algebra $\alg B(\Omega)$. When $\Omega$ is discrete and
subset of $\mathbb R$---as in our case---technical results become much
simpler. For further details on the general case see \cite{ozawa2}.
Finally, we define a \emph{perfect} instrument as an instrument such
that $\mathfrak{I}(j)$ is a pure contractive map, i.~e.
$\mathfrak{I}(j)(\rho)=M_j\rho M_j^\dag$, for all $j$. For example,
L\"uders instrument in Remark \ref{remark:Luders} is a perfect
instrument with $M_j=\Pi_j^X$. The instrument in Remark
\ref{remark:instrument} is perfect only if $\map E_j^\povm
P(\rho)=\sqrt{P_j}\rho\sqrt{P_j}$.
\section{Physical realizations}\label{sec:physical-real}
Intruments provide both outcomes statistics and state reduction due to
a measurement process. Implicitly, we assume that such a measurement
is nondestructive, in the sense that the system is left in a state
conditioned by the outcome and not, for example, absorbed by a counter
or a calorimeter. The only reasonable way to look for an
implementation of a nondestructive measuring process on a quantum
system $\mathscr{H}$ is to engineer an indirect measurement scheme. This means
that we make the system interact with an apparatus $\mathscr{A}$ and,
after some time, we measure an observable $Y$ on the apparatus. In
formula:
\begin{equation}\label{eq:indirect-meas}
\mathfrak{I}(j)(\rho)=\operatorname{Tr}_\mathscr{A}\left[\left(I_\mathscr{H}\otimes\Pi_j^Y\right)U(\rho\otimes|a\>\<a|)U^\dag\right],
\end{equation}
where $\{\Pi_j^Y\}_j$ is an orthogonal resolution of $I_\mathscr A$
coming from the diagonalization of $Y\in\set{B}(\mathscr A)$. Clearly such
a procedure gives rise to an instrument, as described in the previous
Section. Ozawa \cite{ozawa} proved the converse:
\begin{theorem}[Ozawa, Indirect Measurement, 1984]
Every instrument $\mathfrak{I}$ admits an indirect measurement scheme
as in Eq. (\ref{eq:indirect-meas}).
\end{theorem}
The correspondence is not one-to-one: there are many
different---though statistically equivalent---indirect measurement
schemes producing the same instrument; conversely, given the indirect
measurement scheme, the resulting instrument is unique.
\subsection{Levels of description of quantum measurements}
There are basically three ways to describe the statistical aspects of
quantum measurements, depending on the level of details required:
\begin{enumerate}
\item One is interested only in the outcome statistics. Then the
maximum generality lies in the concept of POVM, as we saw in Section
\ref{sec:POVM}. Notice that, given the outcome statistics for all
quantum states, the POVM is defined uniquely---see Remark
\ref{rem:POVM-range}.
\item Also the state reduction rule is requested. The notion of
instrument encloses all possible cases, see Section
\ref{sec:state-reduction}. Evidently, many different instruments
produce the same outcome statistics, i.~e. they all correspond to
the same POVM.
\item The most detailed description characterizes even the state of
the apparatus, the physical interaction between the system and the
apparatus, and the observable to be measured on the apparatus.
Clearly the same instrument is obtainable by means of different
indirect measurement schemes.
\end{enumerate}
Summarizing, given the outcome statistics, the POVM is uniquely
defined. Given the POVM, there are many instruments describing it.
Similarly, there are many indirect measurement schemes realizing a
given instrument. The choice between different equivalent physical
realizations of a measurement process can be made according only to
``practical'' considerations.
\subsection{Example: standard coupling}
Consider a discrete observable $X=\sum_ix_i\Pi_i^X$ of the system
$\mathscr{H}$ in initial state $|\psi\>$ and let the apparatus system be
$\mathscr{A}=L^2(\mathbb R)$ in initial state $|\phi_a\>$. Now, let $\mathscr{H}$ and
$\mathscr{A}$ interact in such a way that the the observable $X$ couples with
the apparatus' momentum $P_a$. This means that the unitary operator is
\begin{equation}
U=e^{-i\lambda X\otimes P_a}.
\end{equation}
The momentum operator $P_a$ is the generator of translations, in the
sense that
\begin{equation}
e^{-\frac i\hslash x_0P_a}\phi_a(x)=\phi_a(x-x_0),
\end{equation}
where $\phi_a(x)=\<x|\phi_a\>$. Hence, the initial system+apparatus
state $|\psi\>\otimes|\phi_a\>$ evolves as
\begin{equation}
U|\psi\>\otimes\phi_a(x)=\sum_i\Pi^X_i|\psi\>\otimes\phi_a(x-\hslash\lambda x_i),
\end{equation}
and, by making assumptions on the value of the coupling constant
$\lambda$ and the initial state $|\phi_a\>$, it is always possible to
obtain functions $\phi_a(x-\hslash\lambda x_i)$ with (almost) disjoint
supports. In other words, it is always possible to model an
interaction between system and apparatus such that the indirect
measurement is a position measurement on the apparatus---the usual
``pointer position'' measurement.
\subsection{Example: embedding and optimal phase measurement}
From the geometrical point of view, the unitary interaction of the
system with a fixed ancilla state in the indirect measurement scheme
(\ref{eq:indirect-meas}) simply corresponds to a linear (isometrical)
embedding of the system $\mathscr{H}$ into a composite Hilbert space
$\mathscr{H}\otimes\mathscr{A}$. The measurement on the apparatus then defines a
conditional expectation from $\mathscr{H}\otimes\mathscr{A}$ to $\mathscr{H}$, giving rise to
probability and state reduction. An embedding into a larger space can
always be described by means of an isometry $V$, i.~e. a bounded
operator such that $V^\dag V=I$. If the input system state is $\rho$,
then the embedded state---that is, the system+apparatus state
\emph{after} the interaction---is $U(\rho\otimes|a\>\<a|)U^\dag\equiv
V\rho V^\dag$.
In Ref.~\cite{idphase}, we exploited an embedding for single-mode
states of the electromagnetic field in order to achieve a physical
realization of the optimal phase measurement. It is well known that
the phase of the electromagnetic field does not correspond to any
self-adjoint operator. Quantum estimation theory \cite{Holevo1,
helstrom} provides the optimal POVM for the phase measurement in
terms of Susskind-Glogower operators
\begin{equation}\label{eq:susskind}
\textrm{d}\hat\mu(\phi)=\frac{\textrm{d}\phi}{2\pi}|e^{i\phi}\>\<e^{i\phi}|,\quad\int_0^{2\pi}\textrm{d}\hat\mu(\phi)=I,
\end{equation}
where $|e^{i\phi}\>\equiv\sum_{n=0}^\infty e^{i\phi\hat n}|n\>$. The
optimal phase measurement outcomes distribution is then
\begin{equation}
\textrm{d}\mu_\rho^\phi=\frac{\textrm{d}\phi}{2\pi}\<e^{i\phi}|\rho|e^{i\phi}\>.
\end{equation}
Using the double-ket notation introduced in
Section~\ref{subsec:notations}, consider now the eigenstates of the
hetherodyne photocurrent $Z=a-b^\dag$
\begin{equation}
\hat Z|D(z)\rangle\!\rangle=z|D(z)\rangle\!\rangle,
\end{equation}
where $D(z)=e^{za^\dag-z^*a}$ are the displacement operators,
satisfying the completness relation
\begin{equation}
\int_\mathbb C\frac{\textrm{d}^2z}{\pi}|D(z)\rangle\!\rangle\langle\!\langle D(z)|=I^{\otimes 2}.
\end{equation}
The following isometry
\begin{equation}
V=\frac{1}{\sqrt{2\pi}}\int_\mathbb C\textrm{d}^2\alpha f(|\alpha|)|D(\alpha)\rangle\!\rangle\<e^{i\arg\alpha}|,\quad\int_0^\infty\textrm{d} t|f(t)|^2=\frac 1\pi
\end{equation}
embeds a single-mode state into a two-modes state in such a way that,
measuring the hetherodyne photocurrent
\begin{equation}
\begin{split}
p(z)&=\frac 1\pi\operatorname{Tr}\left[V\rho V^\dag\ |D(z)\rangle\!\rangle\langle\!\langle D(z)|\right]\\
&=\frac 12\left|f(|z|)\right|^2\<e^{i\arg z}|\rho|e^{i\arg z}\>,
\end{split}
\end{equation}
one obtains the optimal phase distribution $\textrm{d}\mu_\rho^\phi$ as the
marginal of $p(z)$ on the variable $\phi=\arg z$. Notice that here we
are performing a joint measurement on both modes, not just an indirect
measurement on the second mode. However, the form of the embedding $V$
provides a natural way to implement the phase POVM
(\ref{eq:susskind}).
\section{Repeatable measurements}\label{sec:repeatable}
In Subsection \ref{subsec:state-collapse} we introduced the von
Neumann-L\"uders state collapse principle, derived from the hypothesis
of discreteness of spectrum, repeatability, and minimum disturbance.
In what follows, we derive all the consequences that arise from the
only hypothesis of repeatability, thus obtaining the most general form
of a repeatable measurement. See \cite{repmeas} for a detailed
derivation.
First of all, why should we focus on repeatable measurements? Clearly,
there are a lot of natural measurement schemes which are far from
being repeatable, think of e.~g. a photon counter or a fluorescent
screen at the end of a Stern-Gerlach apparatus. In the past decades,
however, technology of quantum experiments improved in such a way that
nondestructive measurements on individual atomic objects are quite a
common task, see e.~g. one atom micro-masers and ions traps.
In the modern formulation of Quantum Mechanics, repeatability
hypothesis has lost the \emph{in-principle} relevance it enjoyed in
the early foundational books as von Neumann's. Nowadays, repeatability
is understood just as a property which characterizes some particular
measurement processes. More precisely, repeatable measurements are
related to \emph{preparation} procedures. In fact, preparing a quantum
system in a particular state means preparing it in a state having some
pre-specified \emph{real property}, as defined in Section
\ref{sec:POVM}. For example, in order to prepare the pure state
$|\psi\>$, one may take a collection of quantum systems and perform
over them a repeatable measurement of the effect described by the POVM
$\{|\psi\>\<\psi|,I-|\psi\>\<\psi|\}$. Of course, the preparation
succeeds when the outcome $|\psi\>\<\psi|$ comes out. Then, the von
Neumann-L\"uders state collapse rule tells us that the state of the
system after measurement is in fact the pure state $|\psi\>$. In this
sense, repeatable measurements have often been regarded as
measurements of observables---projective orthogonal resolution of the
identity---causing a collapse of the state on one of their
eigenvectors.
In \cite{repmeas} we showed that there exist repeatable measurements
which give rise to nonorthogonal POVM's and, moreover, which \emph{do
not even admit any} eigenvector, that is to say, the reduced state
is different at every repetition of the measurement. This result
makes a clear separation between the concepts of repeatability,
preparation and reality in Quantum Measurement Theory.
The starting point is the hypothesis of repeatability. A first
consequence of this is due to Ozawa \cite{ozawa}:
\begin{theorem}[Ozawa, Repeatable Measurements, 1984]
An instrument satisfies repeatability hypothesis only if it has
discrete spectrum.
\end{theorem}
Then, perfect\footnote{For the definition of perfect instruments, see
Subsection \ref{subsec:instruments}.} repeatable instruments are
described by a set of contractions $\{M_j\}$ such that $\sum_iM_i^\dag
M_i=I$ and
\begin{equation}
\frac{\|M_jM_k|\psi\>\|}{\|M_k|\psi\>\|}=p(j|k)=\delta_{jk},
\end{equation}
for all $j,k$ and all $|\psi\>\in\mathscr{H}$. The only technical point
recalled in the paper is that, allowing for infinite dimensional
Hilbert spaces, one has also to deal with properties of operators such
as closeness. In our case, all $M_i$'s are bounded and everywhere
defined, hence closed \cite{halmos}. A close operator possesses closed
range and kernel (the support is always closed since by definition it
is the orthogonal complement of the kernel) and the Hilbert space
$\mathscr{H}$ can hence be decomposed as
\begin{equation}
\mathscr{H}\simeq\set{Ker}(M_j)\oplus\set{Supp}(M_j)\simeq\set{Rng}(M_j)\oplus\set{Rng}(M_j)^\perp,\qquad\forall j.
\end{equation}
The consequences are the following:
\begin{enumerate}
\item All ranges, for different outcomes, must be orthogonal, i.~e.
\begin{equation}
\set{Rng}(M_i)\perp\set{Rng}(M_j),\quad i\neq j.
\end{equation}
\item All ranges must be contained in respective supports, i.~e.
\begin{equation}
\set{Rng}(M_i)\subseteq\set{Supp}(M_i),\quad\forall i.
\end{equation}
\item All $M_i$'s satisfy the condition
\begin{equation}
M_i^\dag M_i|_{\set{Rng}(M_i)}\equiv I_{\set{Rng}(M_i)}.
\end{equation}
\end{enumerate}
Now, the fundamental difference between operators on finite and
infinite Hilbert spaces is that, in finite dimension,
$\set{Supp}(X)\simeq\set{Rng}(X)$ always, while, in the infinite dimensional
case, one can have $\set{Supp}(X)\subset\set{Rng}(X)$ or, viceversa,
$\set{Rng}(X)\subset\set{Supp}(X)$, strictly. This holds basically because in
the infinite dimensional case there exist proper subspaces with the
same dimension as the whole Hilbert space $\mathscr{H}$. This observation lead
us to the following:
\begin{theorem}
For finite dimensional systems, only observables admit repeatable
measurement schemes, and the system state collapses according to the
von Neumann-L\"uders rule (\ref{eq:luders}).
\end{theorem}
So the finite dimensional case describes precisely what one usually
expects about the structure of repeatable measurements. It is
nonetheless possible to construct a simple example in infinite
dimension, enclosing all counter-intuitive features of the infinite
dimensional case. Let us consider a two-outcomes POVM:
\begin{equation}\label{eq:repeatable-povm}
\begin{split}
&P_0=p|0\>\<0|+\sum_{j=0}^\infty|2j+1\>\<2j+1|,\\
&P_1=(1-p)|0\>\<0|+\sum_{j=0}^\infty|2j+2\>\<2j+2|.
\end{split}
\end{equation}
Notice that $\povm P=\{P_0,P_1\}$ is a nonorthogonal measurement. We
can describe such a POVM by means of the following instrument:
\begin{equation}\label{eq:repeatable-scheme}
\begin{split}
&M_0=\sqrt{p}|1\>\<0|+\sum_{j=0}^\infty|2(j+1)+1\>\<2j+1|,\qquad M_0^\dag M_0=P_0,\\
&M_1=\sqrt{1-p}|2\>\<0|+\sum_{j=0}^\infty|2(j+1)+2\>\<2j+2|,\qquad M_1^\dag M_1=P_1,
\end{split}
\end{equation}
in the sense that, got the $i$-th outcome, the state changes as
$M_i\rho M_i^\dag$. Repeatability hypothesis can be simply checked.
Analysing the structure of scheme (\ref{eq:repeatable-scheme}) one
recognizes a unilateral-shift behaviour of the kind $S|n\>=|n+1\>$.
Actually, this unilateral-shift structure is a general feature of
nonorthogonal repeatable measurements. Since $S$ does not admit any
eigenvector, analogously the scheme (\ref{eq:repeatable-scheme})
changes the system state at every repetition of the measurement and
there are no states which are left untouched by such a scheme. In
other words, in infinite dimensional systems there exist repeatable
measurements which \emph{cannot} satisfy minimum disturbance
hypothesis, even in principle, and hence \emph{cannot} be viewed as
preparation procedures.
\chapter{Characterization and Optimization of Quantum
Devices}\label{ch:chap2}
In order to handle information encoded on quantum states we need to
engineer astonishingly precise and accurate devices since the least
loss of control in manipulating quantum systems can lead to extremely
detrimental effects on the whole process. The theoretical
investigation is the starting ground in designing such optimal quantum
devices. This Chapter is devoted, first, to giving a complete and
tractable characterization of quantum channels, second, to exploiting
such characterization to single out optimal devices according to
particular figures of merit that we will introduce and explain from
time to time.
The basic assumption we will adopt is to consider input quantum states
belonging to sets obeying some symmetry constraints---i.~e.
satisfying invariance properties under the action of some groups of
transformations. Moreover, we will choose figures of merit conforming
in a natural way to the same symmetry constraints. These two
conditions lead to the very well established mathematical framework of
\emph{covariant channels}, for which the characterization simplifies,
making explicit calculations analytically solvable. Actually,
covariant channels form convex sets whose structure is (in some cases)
known and optimal devices lie on the border of such sets. In this way,
the problem resorts to a semi-definite linear program.
In particular, we will focus on channels optimally approximating the
impossible tasks of copying, broadcasting, and performing \texttt{NOT}
on unknown quantum states. The symmetries we will deal with are
universal symmetry (invariance under the action of $\mathbb{SU}(d)$),
phase-rotations symmetry (invariance under the action of
$\mathbb{U}(1)^{\times d}$), and invariance under the group of
permutations.
\section{Choi-Jamio\l kowski isomorphism}\label{sec:choi-jam}
A useful tool to characterize quantum channels in finite dimensional
systems is the Choi-Jamio\l kowski \cite{jamiol,choi,nonuniv}
isomorphism---one-to-one correspondence---between channels \mbox{$\map
E:\set{S}(\mathscr{H})\to\set{S}(\mathscr{K})$} and positive operators $R_\map E$ on
$\mathscr{K}\otimes\mathscr{H}$ defined as follows:
\begin{equation}\label{eq:choi-jam}
R_\map E=(\map E\otimes \map I)|I\rangle\!\rangle\langle\!\langle I|\longleftrightarrow\map E=\operatorname{Tr}_\mathscr{H}\left[\left(I\otimes\rho^T\right)R_\map E\right],
\end{equation}
where $\map I$ is the identity map on $\set{S}(\mathscr{H})$,
$|I\rangle\!\rangle=\sum_i|i\>\otimes|i\>$ is the maximally entangled (non
normalized) vector in $\mathscr{H}\otimes\mathscr{H}$, and $O^T$ denotes the
transposition with respect to the fixed basis used to write $|I\rangle\!\rangle$.
Different Kraus representations for $\map E(\rho)=\sum_iE_i\rho
E_i^\dag$ correspond to different ensemble representations for $R_\map
E=\sum_i|E_i\rangle\!\rangle\langle\!\langle E_i|$, the canonical\footnote{See Subsection
\ref{subsec:quantum-operations}.} being the diagonalizing one.
Trace-preservation constraint $\sum_iE_i^\dag E_i=I_\mathscr{H}$ rewrites as
$\operatorname{Tr}_\mathscr{K}\left[R_\map E\right]=I_\mathscr{H}$.
Choi-Jamio\l kowski isomorphism (\ref{eq:choi-jam}) turns out to be
very useful in describing covariant channels. In the following Section
we shall recall some basic notions about group theory.
\section{Group-theoretical techniques}\label{sec:group-techniqes}
\subsection{Elements of group theory}
A unitary (projective) representation on $\mathscr{H}$ of the group $\povm G$
is a homomorphism $\povm G\ni g\mapsto U_g\in\set{B}(\mathscr{H})$, with $U_g$
unitary operator, such that the composition law is preserved:
\begin{equation}
U_gU_h=\omega(g,h)U_{gh}.
\end{equation}
The cocycle $\omega(g,h)$ is a phase, i.~e. $|\omega(g,h)|=1$, for all
$g,h\in\povm G$, and it satisfies the relations
\begin{equation}
\begin{split}
&\omega(gh,k)\omega(g,h)=\omega(g,hk)\omega(h,k)\\
&\omega(g,g^{-1})=1.
\end{split}
\end{equation}
A unitary representation is called \emph{irreducible} (UIR) if there
are no proper subspaces of $\mathscr{H}$ left invariant by the action of all
its elements. Two irreducible representations $U^1$ and $U^2$ of
$\povm G$ on $\mathscr{H}_1$ and $\mathscr{H}_2$, respectively, are called
\emph{equivalent} if there exists a unitary $T:\mathscr{H}_1\to\mathscr{H}_2$ such
that $TU^1_g=U^2_gT$, for all $g\in\povm G$. The fundamental result
concerning UIR's of a group is the following:
\begin{lemma}[Schur]\label{lem:schur}
Let $U^1$ and $U^2$ be two UIR of $\povm G$ on $\mathscr{H}_1$ and $\mathscr{H}_2$,
respectively. Let $B:\mathscr{H}_1\to\mathscr{H}_2$ a (bounded) operator such that:
\begin{equation}
BU^1_g=U^2_gB,
\end{equation}
for all $g\in\povm G$. Then:
\begin{enumerate}
\item $U^1$ and $U^2$ equivalent $\Longrightarrow$ $B\propto T$;
\item $U^1$ and $U^2$ inequivalent $\Longrightarrow$ $B=0$.
\end{enumerate}
\end{lemma}
\begin{remark}[Abelian groups]
From Schur Lemma simply follows that fact that, if the group $\povm
G$ is abelian, nemely $g_1g_2=g_2g_1$ for all $g_1,g_2\in\povm G$,
then all its UIR's are one-dimensional. In fact, all $U_g$'s must be
proportional to the same unitary operator $T$ and they are all
simultaneously diagonalizable, hence reducible on direct sums of
one-dimensional invariant subspaces.
\end{remark}
\subsection{Invariant operators and covariant channels}
Let $W_g$ a reducible unitary representation of $\povm G$ on $\mathscr{H}$.
Then $\mathscr{H}$ can be decomposed into a direct sum of minimal invariant
subspaces:
\begin{equation}\label{eq:direct-sum_decomposition}
\mathscr{H}\simeq\bigoplus_i\mathscr{H}_i.
\end{equation}
Each $\mathscr{H}_i$ supports one UIR of $\povm G$. Some UIR's can be
equivalent or inequivalent. Let us group equivalent UIR's under an
index $\mu$ labelling different equivalence classes, and let an
additional index $i_\mu$ span UIR's among the same $\mu$-th
equivalence class. Since equivalent UIR are supported by isomorphic
subspaces, i.~e. $\mathscr{H}_{i_\mu}\simeq\mathscr{H}_{j_\mu}\simeq\mathscr{H}_\mu$ for all
$i_\mu,j_\mu$ in the same $\mu$-th class, we can rewrite the
decomposition (\ref{eq:direct-sum_decomposition}) as
\begin{equation}\label{eq:wedderburn}
\mathscr{H}\simeq\bigoplus_\mu\mathscr{H}_\mu\otimes\mathbb C^{d_\mu},
\end{equation}
where $d_\mu$ is the cardinality (degeneracy) of the $\mu$-th
equivalence class. Decomposition (\ref{eq:wedderburn}) is usually
called \emph{Wedderburn's decomposition} \cite{zhelobenko}, the spaces
$\mathscr{H}_\mu$ are called \emph{representation spaces}, and the spaces
$\mathbb C^{d_\mu}$ \emph{multiplicity spaces}. Then, the following
decomposition for the representation $W_g$ holds
\begin{equation}\label{eq:wedderburn2}
W_g=\bigoplus_\mu W_g^\mu\otimes I_{d_\mu}.
\end{equation}
With Eq.~(\ref{eq:wedderburn2}) at hand, it is simple to derive the
form of an operator $B$, invariant under the action of the reducible
representation $W_g$, i.~e.
\begin{equation}
W_g^\dag BW_g=B,\qquad\forall g\in\povm G.
\end{equation}
Since the above implies $[B,W_g]=0$, for all $g$, then:
\begin{equation}\label{eq:invariant-form}
B=\bigoplus_\mu I^\mu\otimes B_{d_\mu},
\end{equation}
where $B_{d_\mu}$ is an operator on $\mathbb C^{d_\mu}$. In other words,
the operator $B$ is in a block-form since it cannot connect
inequivalent representations and can act non-trivially only on
multiplicity spaces of the representation $W_g$. This is precisely
what is contained in the Schur's Lemma \ref{lem:schur}.
Now, consider a family of quantum states $\set{F}\subseteq\set{S}(\mathscr{H})$ that
is invariant\footnote{Notice that this requirement is weaker than
requiring that $\set{F}$ is the orbit of a single seed state under the
action of $\mathbf{G}$.} under the action of a group $\mathbf{G}$,
namely $U_g\rho U_g^\dag\in\set{F}$ for all $g\in\mathbf{G}$ and all
$\rho\in\set{F}$. The group, and then the family $\set{F}$, can be discrete as
well as continuous. A channel $\map E:\set{F}\to\set{S}(\mathscr{K})$ is said to be
covariant under the action of the group $\povm G$ if
\begin{equation}
\map E(U_g\rho U_g^\dag)=V_g\map E(\rho) V_g^\dag,\qquad\forall g\in\povm G,
\end{equation}
where $U_g$ and $V_g$ are two generally reducible unitary
representations of $\povm G$ on $\mathscr{H}$ and $\mathscr{K}$, respectively. In a
sense, the channel $\map E$ is ``transparent'' with repect to the
action of the group $\povm G$ and the image of the invariant family
$\set{F}$ is another invariant family $\map E(\set{F})$. Using Choi-Jamio\l
kowski isomorphism (\ref{eq:choi-jam}), the above covariance condition
for $\map E$ rewrites as an invariance condition for $R_\map E$
\cite{nonuniv}, namely,
\begin{equation}
[R_\map E,V_g\otimes U_g^*]=0,\qquad\forall g\in\povm G,
\label{eq:invariance}\end{equation}
where, as usual, the complex conjugate is with respect to the basis
used to write $R_\map E$. Decomposing $\mathscr{K}\otimes\mathscr{H}=\bigoplus_\mu
\mathscr{H}_\mu \otimes \mathbb C^{d_\mu}$, one gets:
\begin{equation}
R_\map E=\bigoplus_\mu I^\mu\otimes R_{d_\mu},
\end{equation}
with positive blocks $R_{d_\mu}$.
Another direct consequence of Eq. (\ref{eq:invariant-form}) is the
form of a group-averaged operator, namely
\begin{equation}
\<X\>_\povm G\equiv\int_\povm G\textrm{d} gU_gXU_g^\dag,\qquad\int_\povm G\textrm{d} g=1.
\end{equation}
Clearly, $\<X\>_\povm G$ is invariant, whence, if the representation
$U_g$ of $\povm G$ decomposes the Hilbert space as
$\mathscr{H}=\bigoplus_\mu\mathscr{H}_\mu\otimes\mathbb{C}^{d_\mu}$, it can be written
as
\begin{equation}\label{eq:group-average}
\<X\>_\povm G=\bigoplus_\mu I^\mu\otimes\frac{\operatorname{Tr}_{\mathscr{H}_\mu}[X]}{\mathsf{dim}\mathscr{H}_\mu}.
\end{equation}
Notice that $\operatorname{Tr}_{\mathscr{H}_\mu}[X]$ is a short-hand notation for
$\operatorname{Tr}_{\mathscr{H}_\mu}[P_\mu XP_\mu]$, where $P_\mu$ is the projection of
$\mathscr{H}$ onto $\mathscr{H}_\mu\otimes\mathbb{C}^{d_\mu}$.
\subsection{Example:
$\mathbb{SU}(d)$-covariance}\label{subsec:sud-covariance}
A typical $\mathbb{SU}(d)$-covariance, also known as \emph{universal}
covariance, for short U-covariance, is that under the representation
of many input and output copies, namely when
$\mathscr{H}\equiv(\mathbb{C}^d)^{\otimes N}$ and
$\mathscr{K}\equiv(\mathbb{C}^d)^{\otimes M}$, with $U_g\equiv W_g^{\otimes
N}$ and $V_g\equiv W_g^{\otimes M}$. Here, $W_g$ is the defining
representation of $\mathbb{SU}(d)$, and invariance condition
(\ref{eq:invariance}) reads:
\begin{equation}\label{eq:SUd-invariance}
\left[R_\map E,W_g^{\otimes M}\otimes(W_g^*)^{\otimes N}\right]=0.
\end{equation}
The general Wedderburn's decomposition for such a representation is
very complicated and channels satisfying covariance
(\ref{eq:SUd-invariance}) will be studied with a somewhat different
approach, see Subsection \ref{subsec:universal-cloning}. Nonetheless,
there are two situations in which universal covariance can be
conveniently faced using $R_\map E$ machinery. The first situation is
when $U_g\equiv W_g$ and $V_g\equiv W_g^*$. This is the case in which
we are requiring a \emph{controvariance} condition:
\begin{equation}
\map E(W_g\rho W_g^\dag)=W_g^*\map E(\rho)W_g^T.
\end{equation}
The invariance condition reads $\left[R_\map E,W_g^{\otimes
2}\right]=0$, which implies $R_\map E=r_SP_S^{(2)}+r_AP_A^{(2)}$,
where $P_S^{(2)}$ and $P_A^{(2)}$ are respectively the projections
onto the totally symmetric and the totally antisymmetric subspaces of
$\mathscr{H}^{\otimes 2}$. We will analyze this case in Subsection
\ref{subsec:UNOT}.
The second situation is when $d=2$, namely when we deal with qubits.
First of all, in this case the two representations $W_g$ and $W_g^*$
are equivalent, since \mbox{$W_g^*=\sigma_yW_g\sigma_y$} \cite{jones}.
Hence the Wedderburn's decomposition for $W_g^{\otimes
M}\otimes(W_g^*)^{\otimes N}$ is the same as for
$W_g^{\otimes(M+N)}$ which the well-known Clebsch-Gordan series
\cite{messiah} for the defining representation of
$\mathbb{SU}(2)$\footnote{Rigorously speaking, this is not the
Wedderburn's decomposition since different $\mathscr{H}_J$ can support
\emph{equivalent} representations. See Subsection
\ref{subsec:Usuperbro}.}:
\begin{equation}\label{eq:Clebsch-Gordan-series}
\begin{split}
(\mathbb{C}^2)^{\otimes M}\otimes(\mathbb{C}^2)^{\otimes M}&\simeq
\underbrace{\bigoplus_{j=j_0}^{M/2}\bigoplus_{l=l_0}^{N/2}}_{\simeq\bigoplus_\mu}\underbrace{\left(\mathbb{C}^{2j+1}\otimes\mathbb{C}^{2l+1}\right)}_{\simeq\mathscr{H}_\mu}\otimes\underbrace{\left(\mathbb{C}^{d_j}\otimes\mathbb{C}^{d_l}\right)}_{\simeq\mathbb{C}^{d_\mu}}\\
&\simeq\bigoplus_{j=j_0}^{M/2}\bigoplus_{l=l_0}^{N/2}\bigoplus_{J=|j-l|}^{j+l}\mathscr{H}_J\otimes\mathbb{C}^{d_j}\otimes\mathbb{C}^{d_l},
\end{split}
\end{equation}
where $j_0,l_0$ are equal to 0 or 1/2 if $M,N$ are even or odd,
respectively, and
\begin{equation}\label{eq:CB-multiplicities}
d_j=\frac{2j+1}{M/2+j+1}\binom{M}{M/2-j}.
\end{equation}
We will analyze this case in Subsection \ref{subsec:Usuperbro}.
\subsection{Example: $\mathbb{U}(1)$-covariance}
The defining representation of $\mathbb{U}(1)$ is simply a phase
$e^{i\phi}\in\mathbb C$. In higher dimensions, we can impose either
\emph{phase-covariance} \cite{Holevo1}, that is,
\begin{equation}
U_\phi\equiv e^{i\phi N},\qquad N=n|n\>\<n|,\quad n=0,\dots,d-1,
\end{equation}
useful to model systems driven by a Hamiltonian with equally spaced
energy levels, as the harmonic oscillator Hamiltonian, or
\emph{multi-phase covariance}, that is, covariance under a unitary
representation of the $d$-fold direct product group
$\mathbb{U}(1)\times \dots\times\mathbb{U}(1)$:
\begin{equation}\label{eq:multi-phase-rotation}
U_\vec\phi\equiv\sum_{n=0}^{d-1}e^{i\phi_n}|n\>\<n|,\qquad\phi_n\in[0,2\pi[,
\end{equation}
where $\vec\phi=\{\phi_n\}$ is a vector of $d$ \emph{independent}
phases. Notice that one of such phases is actually an overall phase
and can be disregarded: for a $d$-dimensional system we then have
$(d-1)$ effective phase-degree of freedom. In the following we shall
adopt multi-phase covariance, and, where there is no possibility of
confusion, we shall interchange the terms phase-covariance and
multi-phase covariance. Notice that, in the case of qubits, the two
concepts coincide.
Also phase-covariance is typically applied to many copies of input and
output (say $N$ and $M$, respectively). For qubits the representation
$U_\phi^{\otimes N}$ decomposes as (see Eq.~(\ref{eq:wedderburn2}))
\begin{equation}
U_\phi^{\otimes N}\simeq\bigoplus_{l=l_0}^{N/2}e^{i\phi J_z^{(l)}}\otimes I_{d_l},
\end{equation}
where $J_z^{(l)}=\sum_{n=-l}^ln|l,n\>\<l,n|$ is the angular momentum
component along rotation axis, say $z$-axis, in the $l$
representation. As in the universal case---$\mathbb{U}(1)$ is a
subgroup of $\mathbb{SU}(2)$, actually---dealing with two-dimensional
systems allows us to handle the complete Wedderburn's decomposition
and work in full generality, even with mixed sates (see
Subsection~\ref{subsec:phasebroad}).
In higher dimensional systems, we shall restrict ourselves to pure
input states. This implies that the many-copies input state lives
actually in the totally symmetric subspace\footnote{This is true only
for many-copies pure input states $\psi^{\otimes N}$. Indeed, a
many-copies mixed input state $\rho^{\otimes N}$ is generally non
symmetric.} $\mathscr{H}\equiv(\mathbb{C}^d)^{\otimes N}_S$. Moreover,
optimal map will be found to have output supported in
$\mathscr{K}\equiv(\mathbb{C}^d)^{\otimes M}_S$. Now, a convenient way to
decompose the composite space $\mathscr{K}\otimes\mathscr{H}$ in the Wedderburn's form
$\bigoplus_\mu\mathscr{H}_\mu\otimes\mathbb{C}^{d_\mu}$, is the following:
\begin{equation}
\mathscr{K}\otimes\mathscr{H}\simeq\bigoplus_{\{m_i\}}\mathscr{H}_{\{m_i\}}\otimes\mathscr{H},
\end{equation}
where $\{m_i\}$ is a multi-index such that $\sum_im_i=M-N$. Invariant
subspaces are clearly one-dimensional, since the group is abelian, and
equivalence classes are spanned by\footnote{We consider here only
maximally degenerate equivalence classes, namely, equivalence
classes whose degeneracy equals the dimension of the input Hilbert
space $(\mathbb{C}^d)^{\otimes N}_S$. For example, the vector
$|1\>^{\otimes M}\otimes|0\>^{\otimes N}$ supports an irrep but it
cannot be written as in Eq. (\ref{eq:span-equiv-class-phcov}). In
Subsection \ref{subsec:phasecloning} we will see how this constraint
indeed does not cause a loss of generality.}:
\begin{equation}\label{eq:span-equiv-class-phcov}
\mathscr{H}_{\{m_i\}}\otimes\mathscr{H}=\set{Span}\Big\{|\{m_i+n_i\}\>\otimes|\{n_i\}\>\Big\}_{\{n_i\}}.
\end{equation}
In the above equation, $\{n_i\}$ is a multi-index such that
$\sum_in_i=N$. The vectors $|\{n_i\}\>$ are defined as:
\begin{equation}\label{eq:symmetric-vectors}
|\{n_i\}\>=\frac{1}{\sqrt{N!}}\sum_\tau\Pi_\tau^N|\underbrace{0,\dots,0}_{n_0},\dots,\underbrace{d-1,\dots,d-1}_{n_{d-1}}\>\Pi_\tau^N,
\end{equation}
where $\Pi_\tau^N$ are permutations of the $N$ systems. In other
words, $|\{n_i\}\>$ are totally symmetric normalized states, whose
occupation numbers are denoted by the multi-index $\{n_i\}$. Clearly,
by varying $\{n_i\}$ over all possible values $0\le n_i\le N$, the set
$|\{n_i\}\>$ spans all input space $\mathscr{H}$. Analogous arguments hold for
the vectors $|\{m_i+n_i\}\>$ in $\mathscr{K}$. That the decomposition using
$|\{m_i+n_i\}\>\otimes|\{n_i\}\>$ is useful to identify the block
structure of a multi-phase covariant channel is clear noticing that
\begin{equation}
U_\vec\phi^{\otimes M}\otimes(U_\vec\phi^*)^{\otimes N}|\{m_i+n_i\}\>\otimes|\{n_i\}\>=e^{i\sum_im_i\phi_i}|\{m_i+n_i\}\>\otimes|\{n_i\}\>,
\end{equation}
for all possible choice of $\{n_i\}$. We'll make use of this
decomposition in Subsections \ref{subsec:phasecloning} and
\ref{subsec:phasenot}.
\subsection{Example: permutation group invariance}
Most channels of physical interest act on input states which are
indeed ``many-identical-copies states''. This is the case, for
example, of estimation channels, which optimally reconstruct an
unknown input state by performing measurements on $N$ copies of it.
Analogously, when the task is distributing quantum information to $M$
users, typically one requires that the reduced state is the same for
each user. Both situations can be described by saying that input
and/or output states are actually permutation invariant states. In
formula:
\begin{equation}
\map E(\rho)=\map E(\Pi_\tau^N\rho\Pi_\tau^N)=\Pi_\sigma^M\map E(\rho)\Pi_\sigma^M,\qquad\forall\tau,\sigma,
\end{equation}
where $\Pi_\tau^N$ and $\Pi_\sigma^M$ are
(real\footnote{Representations of permutations are always real.})
representations of the input and output spaces permutations,
respectively. When both properties are satisfied, the operator $R_\map
E$ must equivalently satisfies the following invariance condition:
\begin{equation}
[R_\map E,\Pi_\sigma^M\otimes\Pi_\tau^N]=0.
\label{eq:perm-inv-0}\end{equation}
Notice that such an invariance property is stronger than that in
Eq.~(\ref{eq:invariance}) since it implies both conditions
\begin{equation}
\left[R_\map E,\Pi_\sigma^M\otimes I^{\otimes N}\right]=0\qquad\left[R_\map E,I^{\otimes M}\otimes\Pi_\tau^N\right]=0,
\end{equation}
for all $\sigma,\tau$, hence in particular for $\sigma=\tau$. The
fundamental tool that comes now at hand is the so-called
\emph{Schur-Weyl duality} between permutation group representations on
qubits and the defining representation of $\mathbb{SU}(2)$. The
duality relation tells that $\Pi_\sigma^M$ decomposes
$(\mathbb{C}^2)^{\otimes M}$ precisely as $W_g^{\otimes M}$, namely,
\begin{equation}
(\mathbb{C}^2)^{\otimes M}\simeq\bigoplus_{j=j_0}^{M/2}\mathbb{C}^{2j+1}\otimes\mathbb{C}^{d_j},
\end{equation}
but with exchanged role for the spaces. Explicitly,
$\mathbb{C}^{2j+1}$ is now the multiplicity space and
$\mathbb{C}^{d_j}$ the representation space. In turns, from
Eq.~(\ref{eq:invariant-form}), Schur-Weyl duality gives the form of a
generic permutation invariant operator $X$ on $(\mathbb{C}^2)^{\otimes
M}$:
\begin{equation}
[X,\Pi_\sigma^M]=0\Longleftrightarrow X=\bigoplus_{j=j_0}^{M/2}X^j\otimes I_{d_j},
\label{eq:schur-weyl-form}\end{equation}
where $X^j$ is an operator on $\mathbb{C}^{2j+1}$.
\subsubsection{Decomposition of many-copies qubit states}
As an application of Schur-Weyl duality, let's consider the
decomposition of the many-copies qubit states $\rho^{\otimes N}$. This
decomposition has been first given in Ref.~\cite{many-copies-decomp}.
For a complete and detailed proof see Ref.~\cite{superbro-pra}.
Indeed, such many-copies states are invariant under permutations of
single qubit systems. The state $\rho^{\otimes N}$ admits then a
decomposition as in Eq.~(\ref{eq:schur-weyl-form}), explicitly
\begin{equation}\label{eq:many-copies-decomp}
\rho^{\otimes N}=\left(\frac{1-r^2}{4}\right)^{N/2}\bigoplus_{l=l_0}^{N/2}\sum_{n=-l}^l\left(\frac{1+r}{1-r}\right)^n|l,n\>\<l,n|\otimes I_{d_l},
\end{equation}
where, as usual, $\rho=(I+r\vec k\cdot\vec\sigma)/2$, $\|\vec k\|=1$
and $|l,n\>$ are the eigenvectors of the angular momentum along $\vec
k$, namely $J_{\vec k}^{(l)}$. Notice that Eq.
(\ref{eq:many-copies-decomp}) exhibits a singularity for $r=1$ due to
the particular rearrangement of terms. However, the limit for $r\to 1$
exists finite, as it can be seen from the equivalent expression
\begin{equation}
\rho^{\otimes N}=\bigoplus_{l=l_0}^{N/2}\left(\frac{1-r^2}{4}\right)^{N/2-l}\sum_{n=-l}^l\left(\frac{1+r}{1-r}\right)^{l+n}|l,n\>\<l,n|\otimes I_{d_l}.
\end{equation}
\section{Optimization in a covariant setting}
Let us consider a family $\set{F}=\{\rho_\theta\}$ of quantum states of
the input system $\mathscr{H}$. In most cases of physical interest, such a
family is invariant under the action of a unitary representation $U_g$
on $\mathscr{H}$ of a group $\mathbf{G}$, in formula:
\begin{equation}
U_g\rho_\theta U_g^\dag=\rho_{g(\theta)}\in\set{F},\qquad\forall\rho\in\set{F},\qquad\forall g\in\mathbf{G}.
\end{equation}
On such a family of states we are concerned about a particular mapping
$\map M$ of $\set{F}$ onto another family $\set{F}'=\{\sigma_\theta\}$ of
states of the output system $\mathscr{K}$ invariant under the action of
another unitary representation $V_g$ of the \emph{same} group $\mathbf{G}$.
The mapping $\map M$ can be completely general, even physically non
allowable. Let $\map M$ be covariant, namely, $\map
M(\rho_\theta)=\sigma_{\theta}$.
Whatever $\map M$ is, we introduce a physical channel $\map E$ and a
merit function $\mathfrak{F}$, depending on $\theta$ and $\map M$,
such that $\mathfrak{F}[\map E(\rho_\theta),\sigma_{\theta}]\equiv
\mathfrak{F}(\theta)$ achieves its maximum when $\map
E(\rho_\theta)=\sigma_{\theta}$. In other words, $\mathfrak{F}$
quantifies how well the channel $\map E$ approximates the mapping
$\map M$. Assuming transitive action of $\mathbf{G}$ on $\set{F}$, that is,
\begin{equation}
\forall\theta,\ \exists g\in\mathbf{G}:\ \theta=g(\theta_0)\textrm{ for a fixed }\theta_0,
\end{equation}
a further natural requirement is the invariance property of
$\mathfrak{F}$:
\begin{equation}\label{eq:covariant-merit}
\mathfrak{F}(g(\theta_0))=\mathfrak{F}\left[\map E(U_g\rho_{\theta_0}U_g^\dag),V_{g}\sigma_{\theta_0}V_{g}^\dag\right]=\mathfrak{F}\left[V_{g}^\dag\map E(U_g\rho_{\theta_0}U_g^\dag)V_{g},\sigma_{\theta_0}\right]=\mathfrak{F}(\theta_0).
\end{equation}
Then the function to be maximized is the average score
\begin{equation}\label{eq:average-merit}
\overline{\mathfrak{F}}=\int_\mathbf{G}\mathfrak{F}\left[\map
E(\rho_{g(\theta_0)}),\sigma_{g(\theta_0)}\right]\ \textrm{d} g=\mathfrak{F}(\theta_0).
\end{equation}
The basic point is that, if the optimum average score is reached by
some channel $\map E$, it is always possible to achieve the optimum
also by a covariant channel $\widetilde{\map E}$, namely such that
$\widetilde{\map E}(U_g\rho U_g^\dag)=V_g\widetilde{\map
E}(\rho)V_g^\dag$. Indeed, from Eqs. (\ref{eq:covariant-merit}) and
(\ref{eq:average-merit}) it turns out that\footnote{We also assume
$\mathfrak{F}$ linear in the l.~h.~s. slot.}
\begin{equation}
\begin{split}
\overline{\mathfrak{F}}&=\int_\mathbf{G} \mathfrak{F}\left[V_g^\dag\map E(\rho_{g(\theta_0)})V_g,\sigma_{\theta_0}\right]\textrm{d} g\\
&=\mathfrak{F}\left[\int_\mathbf{G} V_g^\dag\map E(\rho_{g(\theta_0)})V_g\
\textrm{d}
g,\sigma_{\theta_0}\right]\\
&\equiv\mathfrak{F}\left[\widetilde{\map
E}(\rho_{g(\theta_0)}),\sigma_{\theta_0}\right],
\end{split}
\end{equation}
where we defined
\begin{equation}
\widetilde{\map E}(\rho_{g(\theta_0)})=\int_\mathbf{G}
V_g^\dag\map E(\rho_{g(\theta_0)})V_g\ \textrm{d} g.
\end{equation}
It is simple to verify that $[R_{\widetilde{\map{E}}},V_g\otimes
U_g^*]=0$, namely, $\widetilde{\map E}$ is covariant and, by
construction, it achieves the optimal average score
$\overline{\mathfrak F}$.
Hence, in the following we can restrict the optimization procedure to
covariant channels, which form a convex set. By introducing
appropriate convex merit functions, we can moreover search for the
optimum channel within the border of the convex set, since convex
functions defined on convex sets achieve their extremal values on the
border. In the cases in which we are able to characterize extremal
covariant channels, we can then explicitly single out channels
optimizing the given merit function.
\section{Universally covariant channels}
Universal covariance means, in literature, covariance under the action
of the group $\mathbb{SU}(d)$. Invariant families of states contain
states with fixed spectrum: the most usual choice is to restrict the
analysis to the set of pure states. Given a channel $\map
E:\set{S}(\mathscr{H})\to\set{S}(\mathscr{K})$, universal covariance reads $\map E(U_g\rho
U_g^\dag)=V_g\map E(\rho)V_g^\dag$ where $U_g$ and $V_g$ are unitary
representations of $\mathbb{SU}(d)$ on $\mathscr{H}$ and $\mathscr{K}$ respectively.
In the case of pure input states $|\psi\>$, we will consider as merit
function the \emph{fidelity}, namely,
\begin{equation}\label{eq:score-fidelity}
\mathfrak{F}[\map E(|\psi\>\<\psi|),|\phi\>\<\phi|]\equiv\operatorname{Tr}\left[|\phi\>\<\phi|\;\map E(|\psi\>\<\psi|)\right]=\operatorname{Tr}\left[\left(|\phi\>\<\phi|\otimes|\psi\>\<\psi|^*\right)\;R_\map E\right];
\end{equation}
in the case of mixed input (qubit) states $\rho=(I+r\sigma_z)/2$, we
will consider the purity (the Bloch vector length\footnote{Sometimes
the purity is defined to be proportional to the \emph{square} of the
Bloch vector length: $\operatorname{Tr}[\rho^2]=(1+r^2)/2$.}), namely,
\begin{equation}\label{eq:score-purity}
\mathfrak{F}[\map E(\rho),z]=\operatorname{Tr}\left[\sigma_z\;\map E(\rho)\right]=\operatorname{Tr}\left[\left(\sigma_z\otimes\rho^*\right)\;R_\map E\right].
\end{equation}
It is clear from the form of score functions (\ref{eq:score-fidelity})
and (\ref{eq:score-purity}) that both are convex (linear) in $R_\map
E$ and invariant (see Eq.~(\ref{eq:covariant-merit})).
\subsection{Optimal universal cloning}\label{subsec:universal-cloning}
In this Subsection we shall basically review Ref. \cite{werner} using
Choi-Jamio\l kowski isomorphism. We can't thoroughly apply the
formalism we developed in the previous Sections because a closed form
for Wedderburn's decomposition of $U_g^{\otimes M}$ representation of
$\mathbb{SU}(d)$ is very complicated. We will follow a somewhat
alternative path, finding \emph{a particular} map maximizing the score
function and satisfying covariance and trace-preservation
conditions\footnote{For uniqueness proof see Ref. \cite{werner}.}.
Quantum cloning of an unknown state $\rho^{\otimes N}\to\rho^{\otimes
M}$, $M>N$, is impossible \cite{nocloning}. Much literature has then
been devoted to searching for optimal physical approximations of
impossible ideal cloning \cite{cloning}. Two basics assumptions are
made in order to make calculations treatable: such optimal machines
should work \emph{equally well} on all input states, and input states
should be \emph{pure} $\rho=\psi\equiv|\psi\>\<\psi|$. The natural
framework to work within is then the universal covariance. The score
function is taken to be the fidelity between the actual output of the
approximation map $\map C(\psi^{\otimes N})$ and the ideal output
$\psi^{\otimes M}$. In terms of the $R_\map C$ operator:
\begin{equation}
\mathfrak{F}[\map C(\psi^{\otimes N}),\psi^{\otimes M}]=\operatorname{Tr}\left[\left(\psi^{\otimes M}\otimes(\psi^*)^{\otimes N}\right)\ R_\map C\right],
\end{equation}
where $R_\map C$, in order to satisfy universal covariance of $\map
C$, is such that
\begin{equation}
[R_\map C,U_g^{\otimes M}\otimes(U_g^*)^{\otimes N}]=0.
\end{equation}
Let $P_S^{(N)}=\left(P_S^{(N)}\right)^*$ be the projection over the
totally symmetric subspace $\mathscr{H}_S$ of the input system
$\mathscr{H}=(\mathbb{C}^d)^{\otimes N}$. Since $\psi^{\otimes
N}P_S^{(N)}=\psi^{\otimes N}$, we have that
\begin{equation}
\mathfrak{F}[\map C(\psi^{\otimes N}),\psi^{\otimes M}]\le\operatorname{Tr}\left[\left(\psi^{\otimes M}\otimes P_S^{(N)}\right)\ R_\map C\right].
\end{equation}
The channel $\map C$ is universally covariant, whence, from Eq.
(\ref{eq:group-average}),
\begin{equation}
\map C\left(P_S^{(N)}\right)=\int\textrm{d} gU_g^{\otimes M}\map C\left(P_S^{(N)}\right)(U_g^\dag)^{\otimes M}=\frac{\operatorname{Tr}\left[\map C\left(P_S^{(N)}\right)\right]}{d[M]}P_S^{(M)}+O,
\end{equation}
where $O$ collects all other contributions coming from partially
symmetric/antisymmetric invariant subspaces, and
$d[M]=\binom{d+M-1}{M}$ is the dimension of the totally symmetric
subspace. Actually, terms in $O$ does not contribute to the fidelity
since $\psi^{\otimes M}$ is a symmetric state\footnote{Here it is
crucial that $\rho=\psi$ is pure. Otherwise $\rho^{\otimes M}$
could also have non-null components on partially
symmetrized/antisymmetrized subspaces.}, hence, w.~l.~o.~g., we
write
\begin{equation}
\map C\left(P_S^{(N)}\right)=\frac{d[N]}{d[M]}P_S^{(M)},
\end{equation}
and obtain the following upper bound for the score function:
\begin{equation}\label{eq:optimal-univ-cloning-fid}
\mathfrak{F}\le\frac{d[N]}{d[M]}.
\end{equation}
One can easily verify that the positive operator
\begin{equation}
R_\map C=\frac{d[N]}{d[M]}\left(P_S^{(M)}\otimes I^{\otimes N}\right)\left(I^{\otimes M-N}\otimes|I^{\otimes N}\rangle\!\rangle\langle\!\langle I^{\otimes N}|\right)\left(P_S^{(M)}\otimes I^{\otimes N}\right),
\end{equation}
is invariant, properly normalized to trace-preservation\footnote{In
the sense that $\operatorname{Tr}_\mathscr{K}[R_\map C]=I_{\mathscr{H}_S}$.}, and saturates the
bound (\ref{eq:optimal-univ-cloning-fid}). With a little abuse of
notation, we denoted with $|I^{\otimes N}\rangle\!\rangle$ the non-normalized
maximally entangled vector in $(\mathbb{C}^d)^{\otimes 2N}$
\begin{equation}
|I^{\otimes N}\rangle\!\rangle=\sum_{i_1,\dots,i_N=0}^{d-1}\underbrace{|i_1\>\otimes\dots\otimes|i_N\>}_{(\mathbb{C}^d)^{\otimes N}}\otimes\underbrace{|i_1\>\otimes\dots\otimes|i_N\>}_{(\mathbb{C}^d)^{\otimes N}},
\end{equation}
such that
\begin{equation}
\operatorname{Tr}\left[\left(\psi^{\otimes N}\otimes(\psi^*)^{\otimes N}\right)\ |I^{\otimes N}\rangle\!\rangle\langle\!\langle I^{\otimes N}|\right]=\operatorname{Tr}[\psi^2]^N=1,
\end{equation}
since $\psi$ is pure. From Choi-Jamio\l kowski inverse formula, one
can verify that the action of the optimal universal cloning is as
given in Ref.~\cite{werner}, that is,
\begin{equation}\label{eq:optimal-univ-cloning-map}
\map C(\psi^{\otimes N})=\frac{d[N]}{d[M]}P_S^{(M)}(I^{\otimes(M-N)}\otimes\psi^{\otimes N})P_S^{(M)}.
\end{equation}
\subsection{Optimal universal \texttt{NOT}-gate}\label{subsec:UNOT}
Another unphysical mapping with a naturally emerging covariant
structure is the \texttt{NOT}-gate. In this Subsection we shall
derive, following Ref.~\cite{peristalsi}, the optimal physical
approximation of the ideal quantum-\texttt{NOT}. Actually, in
Subsection \ref{subsec:unot-and-uclon}, we shall also show that
optimal cloning and optimal \texttt{NOT} are intimately related.
Let us consider a $d$-dimensional system $\mathscr{H}$ described by the pure
state $\psi\equiv|\psi\>\<\psi|$. When $d=2$, it makes sense to
consider the \texttt{NOT}-gate, which, generalizing the classical
mapping $0\to 1$ and $1\to 0$, sends an unknown pure state to its
\emph{unique} orthogonal complement. Such orthogonal complement, a
part from a fixed unitary transformation, is the transposition of the
input state. This fact explains why \emph{perfect} \texttt{NOT}-gate
is not physical, since transposition is the simplest example of
positive transformation that is not \emph{completely} positive. In
\cite{qubit-unot} the case $d=2$ is addressed and the optimal
universal approximation is worked out. Here we generalize the result
for all finite dimensions and pure input states.
First of all, it is clear that for $d>2$ the orthogonal complement of
a pure state is not uniquely defined. Hence we shall construct the map
$\map T$ approximating the transposition, which, on the contrary, is
uniquely defined---once fixed a basis in $\mathscr{H}$. Universal covariance
for a channel whose output transforms as the transposed input, that
is, $\map T(U_g\rho U_g^\dag)=U_g^*\map T(\rho)U_g^T$, reads, as
usual, as an invariance property for $R_\map T$:
\begin{equation}
[R_\map T,U_g^*\otimes U_g^*]=0.
\end{equation}
The unitary representation $(U_g^*)^{\otimes 2}$ of $\mathbb{SU}(d)$
decomposes the space $\mathscr{H}^{\otimes 2}$ into the irreducible totally
symmetric and totally antisymmetric subspaces, $\mathscr{H}^{\otimes 2}_S$ and
$\mathscr{H}^{\otimes 2}_A$ respectively. Hence $R_\map
T=r_SP_S^{(2)}+r_AP_A^{(2)}$, where $P_{S,A}^{(2)}:\mathscr{H}^{\otimes
2}\to\mathscr{H}^{\otimes 2}_{S,A}$ are orthogonal projections.
The covariant score function $\mathfrak{F}$ is taken to be the
fidelity $\operatorname{Tr}[\psi^*\map T(\psi)]$, as always when dealing with pure
states. From the form of $R_\map T$:
\begin{equation}
\mathfrak{F}=\operatorname{Tr}[(\psi^*)^{\otimes 2}R_\map T]=r_S,
\end{equation}
and $r_S$ has to be maximized consistently with trace-preservation
condition\\\mbox{$\operatorname{Tr}_\mathscr{H}[r_SP_S^{(2)}]=I$}. Noticing that
$P_S^{(2)}=(I^{\otimes 2}+S)/2$, where $S$ is the swap-operator
between the two spaces, its partial trace is easily computed as
$\operatorname{Tr}_\mathscr{H}[r_SP_S^{(2)}]=Ir_S(d+1)/2$. The optimal universal
approximation of the transposition map is then uniquely described by
\begin{equation}\label{eq:optimal-not-R}
R_\map T=\frac{2}{d+1}P_S^{(2)},
\end{equation}
and it achieves optimal fidelity
\begin{equation}\label{eq:optimal-fid-univ-not}
\mathfrak F=2/(d+1).
\end{equation}
Remarkably, such value for
the fidelity equals the fidelity of optimal state estimation over one
copy \cite{d-dim-state-esteem}. This means that, even if the
optimization has been performed in a general setting, the resulting
channel $\map T$, that is optimal and unique, corresponds to nothing
but a trivial \emph{measure-and-prepare} scheme. In other words,
optimal universal transposition can simply be achieved by performing
the optimal state estimation over one copy---the input copy---and then
preparing the transposed of the \emph{estimated} state. This aspect is
usually referred to as \emph{classicality} of the channel. We will see
in Subsection \ref{subsec:phasenot} that, in the case of multi-phase
covariant transposition, this classical limit can be breached.
\subsection{Universal qubit superbroadcasting}\label{subsec:Usuperbro}
Broadcasting of quantum states is a generalization of cloning, in the
sense that given an unknown input state $\rho\in\set{S}(\mathscr{H})$, the
broadcasting machine $\map B$ is allowed to return a generally
entangled output $\Sigma\in\set{S}(\mathscr{H}^{\otimes 2})$ such that
$\operatorname{Tr}_1[\Sigma]=\operatorname{Tr}_2[\Sigma]=\rho$. In \cite{no-broad} it's been
proved that it is not possible to broadcast with the same channel two
noncommuting quantum states. This result is generally referred to as
the \emph{no-broadcasting theorem}. Actually, the proof holds only for
single-copy input state; allowing for multiple-copies input, it is
possible to construct a channel broadcasting a whole invariant family
of states. Moreover, considering as merit function the Bloch vector
length (in the case of qubits, see Eq. (\ref{eq:score-purity})), the
optimal broadcasting channel actually purifies the input state, in the
sense that the single-site reduced output commutes with the input
(hence their Bloch vectors are parallel) being at the same time purer
(with longer Bloch vector) than the input. We will refer to such a
broadcasting-purifying gate as the \emph{superbroadcaster}
\cite{superbroadcasting}. Of course, the superbroadcaster can be made
a ``perfect'' broadcaster by appropriately mixing the output state
with the maximally chaotic state $I/2$ (this procedure simply
corresponds to a depolarizing channel isotropically shrinking the
Bloch vector towards the center of the Bloch sphere).
In what follows we will explicitly derive such an optimal
superbroadcasting machine by thoroughly using group-theoretical
techniques of Section \ref{sec:group-techniqes}.
\subsubsection{Permutation invariance and universal covariance}
We consider a map $\map B$ taking $N$ copies of an unknown qubit state
$\rho$ to a global output state of $M>N$ qubits. A first natural
requirement is that each final user receives the same reduced output
state\footnote{This requirement alone could not give rise to
permutation invariant output states. However, it is possible to
prove that one can always find an optimal map satisfying this
property, see Ref.~\cite{superbro-pra}.}. This fact, along with the
obvious permutation invariance of the input $\rho^{\otimes N}$, leads
to a Choi-Jamio\l kowski operator that must satifsy the following
invariance property (see Eq.~(\ref{eq:perm-inv-0})):
\begin{equation}\label{eq:perm-inv}
\left[\Pi_\sigma^M\otimes\Pi_\tau^N,R_\map B\right]=0,\qquad\forall\sigma,\tau,
\end{equation}
where $\Pi_\sigma^M$ and $\Pi_\tau^N$ are (real) representations of
the permutation group of the $M$ output and the $N$ input systems,
respectively. From Eq.~(\ref{eq:schur-weyl-form}) the form of $R_\map
B$ follows
\begin{equation}\label{eq:perm-inv-R}
R_\map B=\bigoplus_{j=j_0}^{M/2}\bigoplus_{l=l_0}^{N/2}R_{jl}\otimes I_{d_j}\otimes I_{d_l},
\end{equation}
where $R_{jl}$ is an operator on
$\mathbb{C}^{2j+1}\otimes\mathbb{C}^{2l+1}$ and $d_j$ and $d_l$ are
the Clebsch-Gordan multiplicities given in
Eq.~(\ref{eq:CB-multiplicities}).
Eq.~(\ref{eq:perm-inv-R}) takes into account only permutation
invariance of input and output states: it can then be further
specialized to different group-covariances. In this Subsection we will
deal with $\mathbb{SU}(2)$ covariance (see Subsection
\ref{subsec:phasebroad} for $\mathbb{U}(1)$ covariance). According to
Subsection \ref{subsec:sud-covariance}, since
$W_g^*=\sigma_yW_g\sigma_y$, such covariance condition rewrites as
$\left[S_\map B,W_g^{\otimes(M+N)}\right]=0$, where $W_g$ is the
defining representation of $\mathbb{SU}(2)$ and \mbox{$S_\map
B=(I^{\otimes M}\otimes\sigma_y^{\otimes N})R_\map B(I^{\otimes
M}\otimes\sigma_y^{\otimes N})$}. Hence $S_\map B$ splits as
\begin{equation}\label{eq:onlyonerunning}
S_\map B=\bigoplus_{j=j_0}^{M/2}\bigoplus_{l=l_0}^{N/2}\bigoplus_{J=|j-l|}^{j+l}s^J_{j,l}P^J_{j,l}\otimes I_{d_j}\otimes I_{d_l},
\end{equation}
where $P^J_{j,l}$ is the orthogonal projection of the space
$\mathbb{C}^{2j+1}\otimes\mathbb{C}^{2l+1}$ onto the
$J$-representation and satisfies the simple properties:
\begin{equation}
\operatorname{Tr}[P^J_{j,l}]=2J+1,\qquad\operatorname{Tr}_j[P^J_{j,l}]=\frac{2J+1}{2l+1}I_{2l+1},\qquad\operatorname{Tr}_l[P^J_{j,l}]=\frac{2J+1}{2j+1}I_{2j+1}.
\end{equation}
\subsubsection{Classification of extremal points}
Since $S_\map B$ has to be positive, all $s^J_{j,l}$ are positive real
numbers and trace-preservation condition $\operatorname{Tr}_\mathscr{K}[S_\map B]=I_\mathscr{H}$
reads
\begin{equation}
\operatorname{Tr}_\mathscr{K}[S_\map B]=\bigoplus_{l=l_0}^{N/2}\sum_{j=j_0}^{M/2}\sum_{J=|j-l|}^{j+l}s^J_{j,l}d_j\frac{2J+1}{2l+1}I_{2l+1}\otimes I_{d_l}=I^{\otimes N}.
\end{equation}
The latter is equivalent to
\begin{equation}
\sum_{j=j_0}^{M/2}\sum_{J=|j-l|}^{j+l}s^J_{j,l}d_j\frac{2J+1}{2l+1}=1,\qquad\forall l.
\end{equation}
To single out optimal maps, here we adopt the Bloch vector length
merit function (\ref{eq:score-purity}). This is a linear merit
function, thus optimal maps lie on the border of the convex set of
covariant channels described by $S_\map B$ operators. The problem is
how to characterize extremal $S_\map B$ operators compatible with
complete positivity and trace-preservation constraints. Since
$S_\map B$ is diagonal in indeces $j$ and $J$, extremal $S_\map B$
operators are classified by functions $j=j_l$ and $J=J_l$, leading to
the following expression for extremal $S_\map B$ operators
\begin{equation}
S_\map B=\bigoplus_{l=l_0}^{N/2}\frac{2l+1}{2J_l+1}P^{J_l}_{j_l,l}\otimes\frac{I_{d_{j_l}}}{d_{j_l}}\otimes I_{d_l}.
\end{equation}
\subsubsection{Optimization}
We now feed the input state $\rho^{\otimes N}$ into the channel.
Since we are working in a universally covariant setting, we can write,
w.~l.~o.~g., $\rho=(I+r\sigma_z)/2$, that is, an input state with
Bloch vector along $z$-axis. The global output state $\Sigma$ is
\begin{equation}
\Sigma=\operatorname{Tr}_\mathscr{H}[I^{\otimes M}\otimes(\sigma_y\rho^*\sigma_y)^{\otimes N}\ S_\map B]=\operatorname{Tr}_\mathscr{H}[I^{\otimes M}\otimes{\widetilde\rho}^{\otimes N}\ S_\map B]\,,
\label{maponinp}
\end{equation}
where $\widetilde\rho$ denotes the \texttt{NOT} of $\rho$,
corresponding to the inversion $r\to -r$ (or, equivalently, $r_\pm\to
r_\mp$). By means of the decomposition (\ref{eq:many-copies-decomp})
for $\widetilde\rho^{\otimes N}$, we get
\begin{equation}
\Sigma=\left(\frac{1-r^2}{4}\right)^{N/2}\sum_{l=l_0}^{N/2}\frac{2l+1}{2J_l+1}\frac{d_l}{d_{j_l}}\sum_{n=-l}^l\left(\frac{1-r}{1+r}\right)^{n}\operatorname{Tr}_l\left[I_{2j_l+1}\otimes|ln\>\<ln|\ P^{J_l}_{j_l,l}\right]\otimes I_{d_{j_l}}\,.
\end{equation}
From the form of the map, it is guaranteed that $\Sigma$ is
permutation invariant. Hence it makes sense to speak about \emph{the}
reduced state $\sigma\equiv\operatorname{Tr}_{M-1}[\Sigma]$, regardless of
\emph{which} particular reduced state. In \cite{superbro-pra} it is
shown that $[\sigma_z,\sigma]=0$, namely, the reduced state Bloch
vector is along $z$-axis. The merit function is then
\begin{equation}
\mathfrak{F}=\operatorname{Tr}\left[\left(\sigma_z\otimes I^{M-1}\right)\;\Sigma\right]\equiv r',
\end{equation}
where $r'$ is the Bloch vector length of $\sigma$. After a lengthy
calculation (see \cite{superbro-pra}), the optimal channel turns out
to be the one with $j_l=M/2$ and $J_l=M/2-l$, regardless of the number
of input copies and of the spectrum of $\rho$.
\begin{figure}
\centering
\includegraphics[width=14cm]{univ-pr-n+1.eps}
\caption{The plot shows the behaviour of the scaling factor
$p^{N,N+1}(r)$ versus $r$, for $N$ ranges from 10 to 100 in steps of
10, in the universal case. Notice that there is a wide range of
values of $r$ such that $p^{N,N+1}(r)>1$.}\label{plot:univ-pr-n+1}
\end{figure}
The optimal superbroadcasting achieves the following scaling factor
$p^{N,M}(r)\equiv r'/r$:
\begin{equation}
p^{N,M}(r)=-\frac{M+2}{Mr}\left(\frac{1-r^2}{4}\right)^{N/2}\sum_{l=l_0}^{N/2}\frac{d_l}{l+1}\sum_{n=-l}^l n\left(\frac{1-r}{1+r}\right)^{n}\,.
\end{equation}
The two limiting cases are $M=N+1$
\begin{equation}
p^{N,N+1}(r)=-\frac{N+3}{(N+1)r}\left(\frac{1-r^2}{4}\right)^{N/2}\sum_{l=l_0}^{N/2}\frac{d_l}{l+1}\sum_{n=-l}^l n \left(\frac{1-r}{1+r}\right)^{n},
\end{equation}
and $M=\infty$
\begin{equation}
p^{N,\infty}(r)=-\frac1r\left(\frac{1-r^2}{4}\right)^{N/2}\sum_{l=l_0}^{N/2}\frac{d_l}{l+1}\sum_{n=-l}^l
n\left(\frac{1-r}{1+r}\right)^{n}.
\end{equation}
By plotting scaling factors for different values of $N$ and $M$, it
turns out that, in the universal case, superbroadcasting first emerges
for $N=4$ (in Subsection \ref{subsec:phasebroad} we will see that, in
the phase-covariant case, superbroadcasting first emerges for $N=3$).
Quite surprisingly, for a sufficiently large number of input copies
($N\ge 6$) it is possible to superbroadcast quantum states even to an
infinte number of receivers. In Fig. \ref{plot:univ-pr-n+1} there are
the plots of $p^{N,N+1}(r)$ for $10\le N\le 100$ in steps of 10.
Notice that for $r\to 1$ all curves go below one: indeed optimal
universal cloning of pure states never achieves fidelity one, see Eq.
(\ref{eq:optimal-univ-cloning-fid}).
A compact way to describe the performances of the optimal
superbroadcaster is to introduce the parameter $r^*$, implicitly
defined by the equation
\begin{equation}
p^{N,M}(r^*)=1.
\end{equation}
Clearly, $r^*$ actually depends on $N$ and $M$. By the monotonicity of
$p(r)$, for $r<r^*$ there is superbroadcasting. Hence, $r^*>0$ means
that superbroadcasting is possible. As we already said, for $N\ge 6$,
$r^*>0$ for all $M$. For $N=5$, $r^*>0$ for $M\le 21$. For $N=4$,
$r^*>0$ for $M\le 7$. Moreover, as $N$ and $M$ get closer, $r^*\to 1$,
as expected. In Fig.~\ref{plot:univ-rstar} there are the plots of
$1-r^*(N,M)$, for $M=N+1$ and $M=\infty$. With good approximation, the
two curves have power law $1-r^*(N,N+1)\propto 2/N^2$ and
$1-r^*(N,\infty)\propto 1/N$.
\begin{figure}
\centering
\includegraphics[width=14cm]{univ-rstar.eps}
\caption{Logarithmic plot of $(1-r^*)$ versus $N$ in the universal
case. The upper line refers to the case $M=\infty$ and shows a
behaviour like $1/N$. The lower line is for $M=N+1$ and goes like
$2/N^2$.}\label{plot:univ-rstar}
\end{figure}
\section{Phase-covariant channels}
Multi-phase rotations in $d$ dimensions, see Eq.
(\ref{eq:multi-phase-rotation}), obviously form normal subgroups of
$\mathbb{SU}(d)$. In other words, multi-phase covariance group is
``smaller'' than $\mathbb{SU}(d)$ and, consequently, multi-phase
invariant families of states contain ``less states'' than universally
invariant families. Actually, multi-phase invariant families directly
generalize in higher dimension the idea of the equator of the qubits
Bloch sphere\footnote{This idea can be made more rigorous noticing
that, when $d+1$ mutually unbiased basis can be written, $d$ of them
are connected by multi-phase rotations, as it happens for qubits.
See Ref.~\cite{mub}.}. Quite intuitively then, optimization in a
multi-phase covariant setting should generally achieve better
performances than the analogous universal optimization, since the
group is smaller and leaves margin to sharperly tune free parameters.
In what follows we will consider the same examples of the previous
Section (cloning, \texttt{NOT}-gate, and superbroadcasting) in a
multi-phase covariant framework and we will compare the results.
\subsection{Optimal phase-covariant
cloning}\label{subsec:phasecloning}
The task is to optimally approximate the impossible cloning
transformation $\psi^{\otimes N}\to\psi^{\otimes M}$, where $\psi$ is
an unknown pure state belonging to a family invariant under the
transitive action of the multi-phase group, whose defining
representation is
\begin{equation}
U_\vec\phi=|0\>\<0|+\sum_{n=1}^{d-1}e^{i\phi_n}|n\>\<n|
\end{equation}
(with respect to Eq. (\ref{eq:multi-phase-rotation}) here we put
$\phi_0\equiv 0$, since an overall phase is irrelevant). As before,
since we deal with pure states, the input space $\mathscr{H}$ is considered to
be the totally symmetric subspace $(\mathbb{C}^d)^{\otimes N}_S$.
Analogously, the output space is $\mathscr{K}=(\mathbb{C}^d)^{\otimes M}_S$.
Invariant figures of merit are the usual (global) fidelity
\begin{equation}\label{eq:phase-clo-merit}
\mathfrak{F}_g\left[\map C(\psi_0^{\otimes N}),\psi_0^{\otimes M}\right]=\operatorname{Tr}\left[\map C(\psi_0^{\otimes N})\ \psi_0^{\otimes M}\right],
\end{equation}
and the single-site fidelity
\begin{equation}
\mathfrak{F}_s\left[\operatorname{Tr}_{M-1}\left[\map C(\psi_0^{\otimes N})\right],\psi_0\right]=\operatorname{Tr}\left[\map C(\psi_0^{\otimes N})\ \left(\psi_0\otimes I^{\otimes(M-1)}\right)\right],
\end{equation}
where $\psi_0=d^{-1/2}\sum_{i=0}^{d-1}|i\>$ is a fixed state whose
orbit spans all possible input states family. We will adopt $\mathfrak
F_s$, nonetheless, in Ref.~\cite{pheconclon} we proved that
multi-phase covariant cloning maps optimizing single-site fidelity
optimize global fidelity as well. Clearly, it is understood that the
channel $\map C$ satisfies the covariance property
\begin{equation}
\left[R_\map C,U_\vec\phi^{\otimes M}\otimes (U_\vec\phi^*)^{\otimes N}\right]=0,\qquad\forall\vec\phi,
\end{equation}
so that Eq. (\ref{eq:phase-clo-merit}) makes sense. Last condition
leads to the following form for $R_\map C$
\begin{equation}\label{eq:R-pure-phase-cloning}
R_\map C=\sum_{\{m_j\}}\sum_{\{n'_i\},\{n''_i\}}r^{\{m_j\}}_{\{n'_i\},\{n''_i\}}|\{m_j\}+\{n'_i\}\>\<\{m_j\}+\{n''_i\}|\otimes|\{n'_i\}\>\<\{n''_i\}|,
\end{equation}
where we used the compact notation defined in Eq.
(\ref{eq:symmetric-vectors}). As usual, $R_\map C$ has to be positive,
in order to guarantee complete positivity of $\map C$, and satisfy
trace-preservation condition $\operatorname{Tr}_\mathscr{K}[R_\map C]=I_\mathscr{H}$.
After lengthy calculations (see Ref.~\cite{pheconclon}), the optimal
multi-phase covariant cloning machine is found to be the one described
by the positive rank-one operator
\begin{equation}\label{eq:R-phase-cov-clon}
R_\map C=\sum_{\{n_i\},\{n'_i\}}|\{n_i+k\}\>\<\{n'_i+k\}|\otimes|\{n_i\}\>\<\{n'_i\}|,
\end{equation}
where $k$ is a positive integer such that $\sum_i(n_i+k)=M$, hence
equal to $(M-N)/d$. Optimal single-site fidelity is
\begin{equation}\label{eq:opt-fid-pure-phase-clon}
\mathfrak{F}_s(N,M)=\frac{1}{d}+\frac{1}{Md^{N+1}}\sum_{{\{n_j\} \atop \sum n_j=N-1}}\sum_{i\neq j}\frac{N!}{n_0!\dots n_i!\dots n_j!\dots}\sqrt{\frac{(n_i+k+1)(n_j+k+1)}{(n_i+1)(n_j+1)}},
\end{equation}
which for $N=1$ simplifies to
\begin{equation}
\mathfrak{F}_s(1,M)=\frac{1}{d}+\frac{(d-1)(M+d-1)}{Md^2}.
\end{equation}
In Fig. \ref{fig:phase-clon-pure} there are the plots versus $M$ of
optimal $1\to M$ single-site fidelity in the cases of multi-phase
covariant and universal cloning for $d=5$. Multi-phase covariant
cloning achieves better fidelity than the universal one, as expected.
\begin{figure}
\centering
\includegraphics[width=14cm]{q5M.eps}
\caption{Single-site fidelity for $1\to M$ cloning in dimension $d=5$:
multi-phase (continuous line) and universal (dotted
line).}\label{fig:phase-clon-pure}
\end{figure}
Notice that our analysis is not completely general because of the
restricting relation that must hold between input and output number of
quantum systems involved
\begin{equation}
M=N+kd,\qquad k\in\mathbb{N}.
\end{equation}
However it is the most general result on multi-phase covariant cloning
machines described in the literature by now.
\subsection{Optimal phase-covariant
\texttt{NOT}-gate}\label{subsec:phasenot}
The multi-phase covariant approach to approximate the
\texttt{NOT}-gate is one of the examples in which the performances
improvement, with respect to the universal case, is more apparent.
The transformation we consider is the \texttt{NOT}-gate
$\psi\to\psi^*$ for pure $d$-dimensional states belonging to a
multi-phase invariant family spanned as before by the multi-phase
rotations group $U_\vec\phi$ applied to a fixed seed state
$\psi_0=d^{-1/2}\sum_i|i\>$. The covariant figure of merit is the
fidelity
\begin{equation}
\mathfrak F\left[\map T(\psi_0),\psi_0^*\right]=\operatorname{Tr}\left[\map T(\psi_0)\ \psi_0\right],
\end{equation}
since $\psi_0^*=\psi_0$ (with the appropriate choice of basis). The
channel $\map T$ must satisfy the covariance property
\begin{equation}
\left[R_\map
T,U_\vec\phi^*\otimes U_\vec\phi^*\right]=0.
\end{equation}
The group is abelian so that all irreps are one-dimensional.
Equivalence classes with respective characters are classified in Table
\ref{table1}.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|c}
Equivalence Classes & Characters\\
\hline \hline
$|0\>\otimes|0\>$ & 1\\
$|1\>\otimes|1\>$ & $e^{-2i\phi_1}$\\
$\vdots$ & $\vdots$\\
$|i\>\otimes|i\>$ & $e^{-2i\phi_i}$\\
$\vdots$ & $\vdots$\\
$|0\>\otimes|1\>,|1\>\otimes|0\>$ & $e^{-i\phi_1}$\\
$\vdots$ & $\vdots$\\
$|i\>\otimes|j\>,|j\>\otimes|i\>,\quad i>j$ & $e^{-i(\phi_i+\phi_j)},\quad i>j$\\
$\vdots$ & $\vdots$\\
\end{tabular}
\end{center}
\caption{Equivalence classes and respective characters of irreducible
one-dimensional representations of $(U_\vec\phi^*)^{\otimes 2}$.\label{table1}}
\end{table}
The $R_\map T$ operator then splits into a direct-sum
\begin{equation}
R=\bigoplus_iR_{ii}\bigoplus_{i>j}R_{ij}
\end{equation}
of $1\times 1$ blocks $R_{ii}$ acting on $\set{Span}\{|i\>\otimes|i\>\}$
and $2\times 2$ blocks $R_{ij}$ acting
on\\$\set{Span}\{|i\>\otimes|j\>,|j\>\otimes|i\>\}$.
In Ref.~\cite{phaseconj} there is the complete derivation of the final
form of optimal $R_\map T$ operator as
\begin{equation}
R_\map T=\sum_{i>j}b_{ij}(|ij\>+|ji\>)(\<ij|+\<ji|),
\end{equation}
where $b_{ij}\ge 0$ are matrix elements of a null-diagonal symmetric
bistochastic\footnote{A matrix is called \emph{bistochastic} if all
its rows and columns entries sum up to one \cite{bhatia}.} matrix.
For $d=2,3$ this constraint suffices to single out a unique optimal
$\map T$, since the only null-diagonal symmetric bistochastic matrix
for $d=2$ is
\begin{equation}
\{b_{ij}\}=\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix},
\end{equation}
and for $d=3$
\begin{equation}
\{b_{ij}\}=\begin{pmatrix}
0 & 1/2 & 1/2\\
1/2 & 0 & 1/2\\
1/2 & 1/2 & 0
\end{pmatrix}.
\end{equation}
Already for $d=4$, there are two free parameters $0\le p_1\le 1$ and
$0\le p_2\le 1-p_1$ in defining a null-diagonal symmetric bistochastic
matrix
\begin{equation}\label{eq:NSB-dim4}
\{b_{ij}\}=\begin{pmatrix}
0 & p_1 & p_2 & 1-p_{12}\\
p_1 & 0 & 1-p_{12} & p_2\\
p_2 & 1-p_{12} & 0 & p_1\\
1-p_{12} & p_2 & p_1 & 0
\end{pmatrix},\quad p_{12}=p_1+p_2.
\end{equation}
The achieved optimal fidelity is
\begin{equation}
\mathfrak{F}=\frac{2}{d},
\end{equation}
strictly greater than in the universal case
(\ref{eq:optimal-fid-univ-not}), for all $d$. Moreover, it is
interesting to notice that $2/d$ is also greater than the fidelity of
optimal multi-phase estimation over one copy, derived in
Ref.~\cite{chiara-phaseest} to be $(2d-1)/d^2$. This means that,
contrarily to the universal case for which the optimal
\texttt{NOT}-gate is classical (see final remarks in
Subsection~\ref{subsec:UNOT}), the multi-phase covariant analogue
breaches the classical limit. The result is particularly striking in
the case of qubits for which it is not possible to perfectly estimate
the phase with finite resources, while it is possible to
\emph{perfectly}---with unit fidelity---transpose an unknown pure
equatorial state by means of a fixed unitary transformation.
\subsection{Phase-covariant qubit
superbroadcasting}\label{subsec:phasebroad}
In the phase-covariant version of superbroadcasting, we specialize the
permutation invariant form (\ref{eq:perm-inv-R}) imposing the further
constraint
\begin{equation}\label{eq:phase-inv-R}
\left[R_\map B,U_\phi^{\otimes M}\otimes(U_\phi^*)^{\otimes N}\right]=0.
\end{equation}
Let us now suppose that input states lie on an equator of the Bloch
sphere, say $xy$-plane. Then, $U_\phi$ are precisely rotations along
$z$-axis, namely
\begin{equation}
U_\phi=e^{i\frac{\phi}{2}\sigma_z},
\end{equation}
and invariance condition (\ref{eq:phase-inv-R}) rewrites as
\begin{equation}\label{eq:Rjl-covariance}
\left[R_{jl},e^{i\phi J^{(j)}_z}\otimes e^{-i\phi J^{(l)}_z}\right]=0,\qquad\forall j,l,
\end{equation}
where $J^{(l)}_z=\sum_{n=-l}^ln|l,n\>\<l,n|$ is the angular momentum
component along $z$-axis in the $l$ representation. A convenient way
to write operators $R_{jl}$ satisfying Eq.~(\ref{eq:Rjl-covariance})
is the following:
\begin{equation}\label{eq:Rjl1}
R_{jl}=\sum_{n=-l}^{l}\sum_{n'=-l}^{l}\sum_{k=l-j}^{j-l}r_{n,n',k}^{jl}|j,n+k\>\<j,n'+k|\otimes|l,n\>\<l,n'|,
\end{equation}
when $j\geq l$, and
\begin{equation}\label{eq:Rjl2}
R_{jl}=\sum_{m=-j}^{j}\sum_{m'=-j}^{j}\sum_{k=j-l}^{l-j}r_{m,m',k}^{jl}|j,m\>\<j,m'|\otimes|l,m+k\>\<l,m'+k|,
\end{equation}
when $j<l$, both expressions exhibiting similar structure as in Eq.
(\ref{eq:R-pure-phase-cloning}). Notice that there are two more
running indeces with respect to the universal case
(\ref{eq:onlyonerunning}). While the index $n'$ in Eq.
(\ref{eq:Rjl1}) simply allows for off-diagonal contributions, the
index $k$ labelling equivalence classes is related to the direction of
the reduced output state Bloch vector, as we will see. In particular
we will show that, in order to get an equatorial output, operators
$R_{jl}$ have to be symmetric in $k$, in the sense that
$r_{n,n',k}^{jl}=r_{n,n',-k}^{jl}$.
\subsubsection{Classification of extremal points and $k$-symmetry}
Trace-preservation now reads
\begin{equation}\label{phase-cpt}
\sum_j\sum_kr_{n,n,k}^{jl}d_j=1,\qquad\forall l,n,
\end{equation}
and, analogously to the universal case, the fact that
$r_{n,n,k}^{jl}\ge 0$ and $R_{jl}$ operators are diagonal with respect
to indices $j$'s and $k$'s implies that extremal points are classified
by functions
\begin{equation}
j=j_l,\qquad k=k_l.
\end{equation}
Equivalently, extremal $R_{jl}$ are proportional to correlation
matrices\footnote{Correlation matrices are positive semi-definite
matrices with diagonal entries all equal to one.} since they are
positive matrices with diagonal entries $r^{j_l,l}_{n,n,k_l}$ all
equal to $1/d_{j_l}$ (see Eq. (\ref{phase-cpt})), and extremal
correlation matrices are known in literature \cite{li-tam}. In
particular, rank-one correlation matrices are extremal, hence rank-one
operators $R_{jl}$ are extremal.
In order to further simplify the general form of $R_\map B$ in Eqs.
(\ref{eq:Rjl1}) and (\ref{eq:Rjl2}), we now impose on the single-site
reduced output state the following additional constraint
\begin{equation}\label{eq:equator}
\operatorname{Tr}_{M-1}\left[\mathcal{B}\left(\frac{I^{\otimes N}}{2^N}\right)\right]=\frac{I}{2}.
\end{equation}
We will see that constraint~(\ref{eq:equator}), on one hand, does not
cause a loss of generality since it does not affect optimality, and,
on the other, clarifies the geometrical interpretation we mentioned
about $k$-indexed degrees of freedom of phase-covariant broadcasting
maps. In fact we have (for explicit calculation see
Ref.~\cite{superbro-pra})
\begin{equation}
\begin{split}
\operatorname{Tr}_{M-1}\left[\mathcal{B}\left(\frac{I^{\otimes N}}{2^N}\right)\right]&=\operatorname{Tr}_{M-1}\left[\operatorname{Tr}_\mathscr{H}\left[\left(I^{\otimes M}\otimes\frac{I^{\otimes N}}{2^N}\right)\ R_\map B\right]\right]\\
&=\sum_l(2l+1)\frac{d_l}{2^N}\left(\frac{I}{2}+\frac{k_l}{M}\sigma_z\right).
\end{split}
\end{equation}
Since $\sum_l(2l+1)d_l=2^N$, the only condition for
Eq.~(\ref{eq:equator}) is that
\begin{equation}
\sum_l(2l+1)\frac{d_l}{2^N}\frac{k_l}{M}\sigma_z=0.
\end{equation}
In a sense, index $k$ labels a ``tilt'' of the reduced output state
Bloch vector with respect to the equatorial plane. Our requirement is
then a ``null tilt-requirement'', or, in other words, an ``equatorial
output state-requirement'', and it can always be achieved by equally
mixing two extremal maps---generally losing extremality---, the first
labelled by a function $k=\bar k_l$, the second by $k=-\bar k_l$.
\begin{figure}
\centering
\includegraphics[width=14cm]{ksymmetry.eps}
\caption{Schematic sketch of the $k$-symmetrization
procedure.}\label{fig:ksymmetry}
\end{figure}
We will refer to such a property as $k$-symmetry property of
bradcasting maps and we showed that $k$-symmetry property is
equivalent to the property of mapping equatorial states to equatorial
states. Notice that a $k$-symmetric map is such that
$r_{n,n',k}^{jl}=r_{n,n',-k}^{jl}$. The strategy to obtain
broadcasting maps optimizing the reduced output state Bloch vector
length is then to search for optimal maps within extremal maps and,
once found the best one, to force $k$-symmetry on it. The procedure is
shown in Fig.~\ref{fig:ksymmetry}. On the left there are the
equatorial mixed input state $\rho$ and the single-site reduced output
$\rho'_k$. Suppose such an output comes from an extremal map $\map
B_k$ described by $r_{n,n',k}^{jl}$ elements. Notice that, by
covariance, the projection of $\rho'_k$ onto the equator is parallel
with $\rho$. Consider now another map $\map B_{-k}$, whose elements
are equal to $\tilde r_{n,n',k}^{jl}=r_{n,n',-k}^{jl}$. Clearly,
$\map B_{-k}$ is a proper channel obeying all covariance and
extremality constraints as $\map B_k$. The output of $\map B_{-k}$ is
in sketched in the middle figure as $\rho'_{-k}$. In order to have an
equatorial output, we mix $\map B_k$ and $\map B_{-k}$ obtaining $\map
B=(\map B_k+\map B_{-k})/2$ whose output
$\rho'=(\rho'_k+\rho'_{-k})/2$, by linearity, lies on the equator (see
the picture on the right). Of course $\map B$ is no more extremal, by
construction. However, the figure of merit we are considering, namely,
the length of the projection of the output Bloch vector onto the
original one, does not change. In this sense, imposing $k$-symmetry
does not affect optimality. Moreover, it is possible to prove that the
$k$-symmetrized output $\rho'$ has higher fidelity with the input
$\rho$ (see Ref.~\cite{superbro-pra})) than the tilted $\rho'_k$ and
$\rho'_{-k}$.
\subsubsection{Optimization}
In Ref~\cite{superbro-pra} it is proved that the channel optimizing
the merit function
\begin{equation}
\mathfrak{F}=\operatorname{Tr}\left[\left(\sigma_x\otimes I^{M-1}\right)\ \Sigma\right],
\end{equation}
for $x$-oriented input states $\rho=(I+r\sigma_x)/2$, has $j_l=M/2$
for all $l$, and $k_l=0$, for $M-N$ even, while $k_l=\pm 1/2$ for
$M-N$ odd. Hence, for $M-N$ even the optimal superbroadcaster is
already $k$-symmetrized, whereas for $M-N$ odd we must equally mix the
channels coming from $k_l=1/2$ and $k_l=-1/2$. In both cases,
$r_{n,n',k_l}^{j_l,l}=1/d_{j_l}$, for all $n,n',l$. At the end, the
structure of the map $\map B$ depends only on the parity of $M-N$, and
not on the spectrum of $\rho$.
\begin{figure}
\centering
\includegraphics[width=14cm]{phase-pr-n+1.eps}
\caption{The plot shows the behaviour of the scaling factor
$p^{N,N+1}(r)$ versus $r$, for $N$ ranges from 4 to 100 in steps of
8, in the phase-covariant case. Notice that there is a wide range of
values of $r$ such that $p^{N,N+1}(r)>1$.}\label{plot:phase-pr-n+1}
\end{figure}
For $M-N$ even, the optimum scaling
factor $p^{N,M}(r)$ is given by
\begin{equation}\label{eq:phase-pr1}
p^{N,M}_e(r)=\frac{4}{Mr}\left(\frac{1-r^2}{4}\right)^{N/2}\sum_{l=l_0}^{N/2}d_l\sum_{n=-l}^l\left[\exp\left(J_x^{(l)}\log\frac{1+r}{1-r}\right)\right]_{n,n+1}\left[J_x^{(j)}\right]_{n,n+1},
\end{equation}
while, for $M-N$ odd, it is
\begin{equation}\label{eq:phase-pr2}
p^{N,M}_o(r)=\frac{4}{Mr}\left(\frac{1-r^2}{4}\right)^{N/2}\sum_{l=l_0}^{N/2}d_l\sum_{n=-l}^l\left[\exp\left(J_x^{(l)}\log\frac{1+r}{1-r}\right)\right]_{n,n+1}\left[J_x^{(j)}\right]_{n+1/2,n+3/2}.
\end{equation}
In Fig.~\ref{plot:phase-pr-n+1} there are the plots of $p^{N,N+1}(r)$
for $4\le N\le 100$ in steps of 8. As in the universal case, all
curves, for $r\to 1$, go below one: indeed optimal phase-covariant
cloning of pure states never achieves fidelity one, see Eq.
(\ref{eq:opt-fid-pure-phase-clon}). However, it is possible to see
that phase-covariant superbroadcasting is more efficient than the
universal one: superbroadcasting first emerges for $N=3$ ($N=4$ in the
universal case) and achieves larger values of $p^{N,M}(r)$ for all
$N$, $M$, and $r$.
\begin{figure}
\centering
\includegraphics[width=14cm]{phase-rstar.eps}
\caption{Logarithmic plot of $(1-r^*)$ versus $N$ in the
phase-covariant case. The upper line refers to the case $M=\infty$
and shows a behaviour like $1/2N$. The lower line is for $M=N+1$ and
goes like $2/(3N^2)$.}\label{plot:phase-rstar}
\end{figure}
In Fig.~\ref{plot:phase-rstar} there are the plots of $1-r^*(N,M)$,
for $M=N+1$ and $M=\infty$, as done for the universal
superbroadcaster. With good approximation, the two curves have power
law $2/3N^2$ and $1/2N$, respectively, namely they go to zero faster
than in the universal case, as expected.
\chapter{Realization of Quantum Devices}\label{ch:realization}
In the previous Chapter we explicitly wrote quantum operations coming
out from an optimization procedure in a covariant setting. We gave
such channels in terms of their Choi-Jamio\l kowski operators
(\ref{eq:choi-jam}). However, Choi-Jamio\l kowski representation for
quantum channels, even if very useful in dealing with semi-definite
programming problems, turns out to be quite far from giving the
physical ``recipe'' needed to realize the channel in a laboratory. In
the following we will describe how to unitarily implement a given
quantum channel, in terms of a unitary interaction between the system
and an ancilla. In the first Section, we will provide, following
Refs.~\cite{laurea,unitary-real-pra}, a general method to work out a
physical setting realizing a given channel. In the second part of the
Chapter, we will show how this procedure works in the case of some of
the channels discussed in Chapter~\ref{ch:chap2}.
\section{Unitary dilations of a channel}
Let us given a channel $\map E:\set{S}(\mathscr{H})\to\set{S}(\mathscr{H})$\footnote{Here, for
simplicity we disregard the case of channels from states on a system
$\mathscr{H}$ to states on another system $\mathscr{K}$, e.~g. the cloning channel
from $\set{S}(\mathscr{H}^{\otimes N})$ to $\set{S}(\mathscr{H}^{\otimes M})$. This case can
be taken into account, see Ref.~\cite{unitary-real-pra} for a more
general approach.}. The task of this Section is to find an ancilla
system $\mathscr{A}$, an ancilla pure state $|0\>$, and a unitary operator $U$
on $\mathscr{H}\otimes\mathscr{A}$, such that
\begin{equation}
\map E(\rho)=\operatorname{Tr}_\mathscr{A}\left[U(\rho\otimes|0\>\<0|)U^\dag\right],
\end{equation}
for all $\rho\in\set{S}(\mathscr{H})$.
\subsection{Stinespring dilation}
Given a channel $\map E$, the Stinespring representation
\cite{stinespring} $(V,\mathscr{A})$ of $\map E$ is a kind of ``purification''
of the channel $\map E$, i.~e.
\begin{equation}\label{eq:stinespring}
\map E(\rho)=\operatorname{Tr}_\mathscr{A}\left[V\rho V^\dag\right],
\end{equation}
where $V$ is an isometry, i.~e. $V^\dag V=I$, from $\mathscr{H}$ to
$\mathscr{H}\otimes\mathscr{A}$. The Stinespring representation is usually given for
the dual channel (see Subsection~\ref{subsec:quantum-operations})
\mbox{$\dual{\map E}:\set{B}(\mathscr{H})\to\set{B}(\mathscr{H})$} as
\begin{equation}
\dual{\map E}(O)=V^\dag(O\otimes I_\mathscr{A})V.
\end{equation}
Let $\map E(\rho)=\sum_iE_i\rho E_i^\dag$ be a Kraus representation
for $\map E$. Consider now the operators from $\mathscr{H}$ to $\mathscr{H}\otimes\mathscr{A}$
defined as $E_i\otimes|i\>$, where $|i\>$ belongs to a set of
orthonormal vectors in $\mathscr{A}$. The only trivial condition $\mathscr{A}$ must
satisfy is $\mathsf{dim}\mathscr{A}\ge\sharp\{E_i\}$. Then, the sum
\begin{equation}\label{eq:stine-isom}
V=\sum_iE_i\otimes|i\>
\end{equation}
is an isometry, since $V^\dag V=\sum_iE_i^\dag E_i=I_\mathscr{H}$, and
realizes the channel $\map E$ as in Eq.~(\ref{eq:stinespring}).
\begin{remark}
Notice that we did not make any assumption on the particular choice
for the Kraus representation $\{E_i\}$ used to construct the
Stinespring isometry $V$ in Eq.~(\ref{eq:stine-isom}). When
$\{E_i\}$ is the canonical Kraus decomposition and
$\mathsf{dim}\mathscr{A}=\sharp\{E_i\}$, we will refer to such $V$ as the
\emph{canonical Stinespring representation} for $\map E$, which
clearly is the one minimizing the ancillary resources, i.~e. the
dimension of the ancilla system, needed to physically implement the
channel.
\end{remark}
\subsection{Unitary dilation}
From Stinespring form (\ref{eq:stinespring}) the existence of a
unitary interaction $U$ between $\mathscr{H}$ and $\mathscr{A}$ realizing the channel
$\map E$ is apparent, since every isometry $V$ from $\mathscr{H}$ to
$\mathscr{H}\otimes\mathscr{A}$ can obviously be written as
\begin{equation}
V=U(I_\mathscr{H}\otimes|0\>),
\end{equation}
where $U$ is a suitable unitary operator on $\mathscr{H}\otimes\mathscr{A}$ and $|0\>$
is a fixed normalized state of $\mathscr{A}$. Now, $|0\>$ is precisely the
ancilla state such that
\begin{equation}\label{eq:generic-unit-real}
\map E(\rho)=\operatorname{Tr}_\mathscr{A}\left[U(\rho\otimes|0\>\<0|)U^\dag\right].
\end{equation}
While the existence of a realization $U$ for every channel is a
well-established fact in the literature \cite{ozawa,kraus-unitary},
the problem of giving \emph{explicitly} such interaction for a given
channel can be very difficult. The general procedure given in
Ref.~\cite{unitary-real-pra} basically relies on a repeated
Gram-Schmidt orthonormalizing algorithm applied to the column vectors
of the Stinespring isometry $V$. In this way we are able to find
additional $\mathsf{dim}\mathscr{H}\times(\mathsf{dim}\mathscr{A}-1)$ orthonormal vectors to append to
$V$, completing it to a square matrix whose column vectors form an
orthonormal basis for the composite system $\mathscr{H}\otimes\mathscr{A}$, i.~e. to a
unitary operator on $\mathscr{H}\otimes\mathscr{A}$. In the following Section, we will
see that, in some fortunate cases, the channels optimized in Chapter
\ref{ch:chap2} admit very simple Stinespring isometries, allowing us
to explicitly write unitary operators realizing such channels in
dimension $d$.
\section{Explicit realizations}
\subsection{Universal \texttt{NOT} and cloning
gates}\label{subsec:unot-and-uclon}
Let us start from the optimal universal \texttt{NOT}-gate $\map T$
derived in Subsection~\ref{subsec:UNOT}. The channel is completely
described by the positive operator $R_\map T$ in
Eq.~(\ref{eq:optimal-not-R}). In order to write $\map T$ in its
Stinespring form, we first have to obtain a Kraus decomposition for
$\map T$. This can be done by expanding $R_\map T$ (see Section
\ref{sec:choi-jam}) as
\begin{equation}
\begin{split}
R_\map T&=\frac{2}{d+1}P_S^{(2)}=\frac1{d+1}(I+S)\\
&=\frac1{d+1}\sum_{m,n=0}^{d-1}(|m\>\<m|\otimes|n\>\<n|+|m\>\<n|\otimes|n\>\<m|)\\
&=\frac1{2(d+1)}\sum_{m,n=0}^{d-1}(|mn\rangle\!\rangle+|nm\rangle\!\rangle)(\langle\!\langle mn|+\langle\!\langle nm|)\\
&=\sum_{m,n=0}^{d-1}|M^S_{mn}\rangle\!\rangle\langle\!\langle M^S_{mn}|,
\end{split}
\end{equation}
where
\begin{equation}
M^S_{mn}=\frac1{\sqrt{2(d+1)}}(|m\>\<n|+|n\>\<m|).
\end{equation}
One possible Kraus decomposition is then given by
\begin{equation}
\map T(\psi )=\sum_{m,n=0}^{d-1}M^S_{mn}\psi M^S_{mn}.
\end{equation}
A Stinespring isometry $V$ such that $\map
T(\psi)=\operatorname{Tr}_{\mathscr{A}}\left[V\psi V^\dag\right]$ is then\footnote{Notice
that this Stinespring isometry is not the one minimizing ancillary
resources. In fact, it comes from a Kraus decomposition which is not
the canonical one, since the orthogonality condition,
$\operatorname{Tr}[M^S_{ij}M^S_{kl}]=0$ for all $\{ij\}\neq\{kl\}$, does not hold.
However, as we will see in the following, this realization allows a
very intriguing physical interpretation, see
Ref.~\cite{extremal-clonings}.}
\begin{equation}
V=\sum_{m,n=0}^{d-1}M^S_{mn}\otimes|mn\rangle\!\rangle_{23},
\label{eq:iso}
\end{equation}
where we chose $\mathscr{A}\equiv\mathscr{H}^{\otimes 2}$ as ancilla system.
Summarizing, we wrote the optimal \texttt{NOT}-gate $\map T$ by means
of an isometry $V$ embedding the input system $\mathscr{H}$ into a composite
tripartite system $\mathscr{H}\otimes\mathscr{H}\otimes\mathscr{H}$, in which the last two
spaces represent the ancilla.
Tracing $V\psi V^\dag$ over the last two spaces, we get the channel
$\map T$. What happens if we trace over different combinations of
spaces? In fact, all three spaces are the same and there is no reason
to consider one of them as the preferred system and the remaining ones
as ancillae. Actually, tracing $V\psi V^\dag$ over the \emph{first}
space, one obtains
\begin{equation}
\operatorname{Tr}_1\left[V\psi V^\dag\right]=\frac {2}{d+1}P_S^{(2)}(I\otimes\psi)P_S^{(2)},
\end{equation}
namely, the optimal $1\to 2$ universal cloning for pure states (see
Eq.~\ref{eq:optimal-univ-cloning-map}). This means that universal
$1\to 2$ cloning and universal \texttt{NOT}-gate are intimately
related and contextually appear on different branches (spaces) of the
same physical setting. Such a coincidence has been experimentally
exploited for qubits in Ref.~\cite{demartini} and theoretically
analyzed and interpreted in generic dimension in
Ref.~\cite{extremal-clonings}. Moreover, it is possible to prove that
$\operatorname{Tr}_3\left[V\psi V^\dag\right]$ optimally approximate the
transformation $\psi\to\psi^*\otimes\psi$ for pure states. Notice that
the cloning map is basis independent, whilst the transposition map
depends on the choice of the basis, which is reflected by the choice
of the particular Stinespring isometry $V$.
In Ref.~\cite{peristalsi} it is possible to find the explicit
calculation deriving a unitary interaction and an ancilla state
realizing at the same time optimal approximations of universal cloning
and transposition. The unitary operator $U$ on
$(\mathbb{C}^d)^{\otimes 3}$ is
\begin{equation}\label{eq:unitary-cloning}
U=\sum_{p=0}^{d-1}V_{p,p}\otimes\<p|\<p|+\sum_{{p,q=0\atop p<q}}^{d-1}V_{p,q}^{(S)}\otimes \frac{\<p|\<q|+\<q|\<p|}{\sqrt{2}}+\sum_{{p,q=0\atop p<q}}^{d-1}V_{p,q}^{(A)}\otimes \frac{\<p|\<q|-\<q|\<p|}{\sqrt{2}}
\end{equation}
where the three sets of isometries $\left\{V_{p,p}\right\}$,
$\left\{V_{p,q}^{(S)}\right\}$, and $\left\{V_{p,q}^{(A)}\right\}$
from $\mathscr{H}$ to $\mathscr{H}^{\otimes 3}$ are defined as
\begin{equation}
\begin{split}
&V_{p,p}=\sum_{k=0}^{d-1}|k\>|k\oplus p\>|k\oplus p\>\<k\oplus p|,\\
&V_{p,q}^{(S)}=\frac{1}{\sqrt{2}}\sum_{k=0}^{d-1}|k\>(|k\oplus
p\>|k\oplus q\>+|k\oplus q\>|k\oplus p\>)\<k\oplus q|,\\
&V_{p,q}^{(A)}=\frac{1}{\sqrt{2}}\sum_{k=0}^{d-1}|k\>(|k\oplus
p\>|k\oplus q\>-|k\oplus q\>|k\oplus p\>)\<k\oplus q|.
\end{split}
\end{equation}
Preparing the ancilla state as
\begin{equation}\label{eq:ancilla-cloning}
|\phi\rangle\!\rangle=\sqrt{\frac 2{d+1}}P_S^{(2)}\sum_{r=0}^{d-1}|0\>|r\>,
\end{equation}
the following identity holds
\begin{equation}
U(\psi\otimes|\phi\rangle\!\rangle\langle\!\langle\phi|)U^\dag=V\psi V^\dag,
\end{equation}
namely, the operator $U$ in Eq.~(\ref{eq:unitary-cloning}) together
with the ancilla state $|\phi\rangle\!\rangle$ in Eq.~(\ref{eq:ancilla-cloning})
provide a unitary dilation of the Stinespring isometry $V$ in
Eq.~(\ref{eq:iso}), realizing optimal universal $1\to 2$ cloning as
well as optimal universal transposition, depending on what we trace
out after the interaction.
In the case $d=2$, we obtain the network model for universal qubit
cloning of Ref.~\cite{buzek-network}, with
\begin{equation}
U=
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
\end{pmatrix},
\end{equation}
and $|\phi\rangle\!\rangle=\frac{1}{\sqrt{6}}(2|0\>|0\>+|0\>|1\>+|1\>|0\>)$.
\subsection{Phase-covariant cloning and economical maps}
In Subsection \ref{subsec:phasecloning} we obtained the channel
optimally achieving the multi-phase covariant $N\to M$ cloning
transformation. The optimal channel has been described, as usual, by
giving the corresponding $R_\map C$ operator in
Eq.~(\ref{eq:R-phase-cov-clon}). In the analyzed cases, i.~e. when
$M=N+kd$, where $k\in\mathbb{N}$ and $d$ is the dimension of the
single copy system, $R_\map C$ enjoys the relevent property of being
rank-one. This implies that its canonical Kraus representation
contains only one operator, and, to satisfy trace-preservation
constraint (\ref{eq:trace-pres-with-kraus}), such an operator has to
be an isometry. Therefore, the optimal multi-phase covariant $N\to M$
cloning machine $\map C_{N,M}$, for $M=N+kd$, admits the very simple
expression
\begin{equation}
\map C_{N,M}(\psi^{\otimes N})=V\psi^{\otimes N}V^\dag,
\end{equation}
where $V:\mathscr{H}^{\otimes N}\to\mathscr{H}^{\otimes M}$ is an isometry acting as
follows
\begin{equation}
V|\{n_i\}\>=|\{n_i+k\}\>,
\end{equation}
using the compact notation introduced in
Eq.~(\ref{eq:symmetric-vectors}).
This kind of isometric optimal channels attracted attention in the
recent literature as \emph{economical} transformations
\cite{niugrif,economical-durt,economical-cerf}, in the sense that, in
order to physically implement them, there is no need of discarding
additional resources, i.~e. ancillae. In fact, from the point of view
of Stinespring representation, multi-phase covariant cloning is
realizable as
\begin{equation}
\map C_{N,M}\left(\psi^{\otimes N}\right)=U\left(\psi^{\otimes N}\otimes|0\>\<0|^{\otimes(M-N)}\right)U^\dag,
\end{equation}
namely, with respect to Eq.~(\ref{eq:generic-unit-real}), there is no
partial trace, and the resources needed are just the $(M-N)$ blank
copies where convariantly distribute the information contained in
$\psi^{\otimes N}$.
\subsection{Phase-covariant \texttt{NOT}-gate}
The optimal multi-phase conjugation map has been derived in Subsection
\ref{subsec:phasenot} to be
\begin{equation}
R_\map T=\sum_{i>j}b_{ij}(|ij\>+|ji\>)(\<ij|+\<ji|),
\end{equation}
where $b_{ij}\ge 0$ are matrix elements of a null-diagonal symmetric
bistochastic (NSB) matrix. In this case the map, for $d>2$, is not
unitary or isometric as in the case of phase-covariant cloning.
Moreover, the fact that in dimension $d\ge 4$ there exists a whole set
of equally optimal maps---in one-to-one correspondence with NSB
matrices---makes the problem of finding a physical realization much
more difficult than in the two examples treated before, where the
optimal map was unique. There are basically two paths one can follow:
the first is to search for the most efficient realization, i.~e. the
one minimizing ancillary resources (in this case we will tipically
single out one particular optimal phase-conjugation map achievable
using less resources than the others); the second is to search for the
most flexible realization, i.~e. the one that spans as many as
possible optimal maps by appropriately varying the ``program'' ancilla
state and/or is more robust against noise (this second kind of
realization will clearly require a higher dimensional ancilla system
to encode a ``fault-tolerant'' program).
A good point to start with is the study of the structure of the set of
optimal phase-conjugation channel, or, equivalently, of the set of NSB
matrices. Such matrices form a convex set\footnote{This is because
their raws and columns are probability distributions}. On the other
hand, every bistochastic matrix is a convex combination of permutation
matrices---this is the content of the Birkhoff theorem \cite{bhatia}.
The null-diagonal and symmetry constraints, however, force the convex
set of NSB matrices to be strictly contained into the convex
polyhedron of bistochastic matrices. This fact causes the extremal NSB
matrices to eventually lie strictly inside the set of bistochastic
matrices, generally preventing them from being permutations.
The geometrical study of the set of NSB matrices and its extremal
points can shed some light on the unusual feature that there exist
different ``equally optimal'' maps. The problem arises for dimension
at least $d=4$. In this case the decomposition of the matrix
$\{b_{ij}\}$ in Eq.~(\ref{eq:NSB-dim4}) into extremal components is
\begin{equation}\label{ext-dec-4}
\begin{split}
\{b_{ij}\}&=p_1\begin{pmatrix}
0& 1& 0& 0\\
1& 0& 0& 0\\
0& 0& 0& 1\\
0& 0& 1& 0
\end{pmatrix}+p_2\begin{pmatrix}
0& 0& 1& 0\\
0& 0& 0& 1\\
1& 0& 0& 0\\
0& 1& 0& 0
\end{pmatrix}+p_3\begin{pmatrix}
0& 0& 0& 1\\
0& 0& 1& 0\\
0& 1& 0& 0\\
1& 0& 0& 0
\end{pmatrix}\\
&=p_1B^{(1)}+p_2B^{(2)}+p_3B^{(3)},
\end{split}
\end{equation}
where $p_1,p_2,p_3\geq 0$ and $p_1+p_2+p_3=1$. A natural question is
now which optimal maps can be achieved with minimal resources.
More explicitly, for $d=4$, we define three unitaries $U_1$, $U_2$ and
$U_3$ on $\mathbb{C}^4\otimes\mathbb{C}^2$ as
\begin{equation}\label{unitaries-for-4}
U_1=\begin{pmatrix}
T_{10} & T_{32}\vspace{0.2cm}\\
T_{32} & T_{10}
\end{pmatrix},\quad
U_2=\begin{pmatrix}
T_{20} & T_{31}\vspace{0.2cm}\\
T_{31} & T_{20}
\end{pmatrix},\quad
U_3=\begin{pmatrix}
T_{30} & T_{21}\vspace{0.2cm}\\
T_{21} & T_{30}
\end{pmatrix},
\end{equation}
where $T_{ij}=|i\>\<j|+|j\>\<i|$. Each of them realizes an extremal
optimal multi-phase conjugation map (corresponding to $p_k=1$ in Eq.
(\ref{ext-dec-4}) for a given $k$), namely
\begin{equation}\label{ext-unit-4}
\mathcal{T}^{(k)}_4(\rho)=\sum_{i>j}B^{(k)}_{ij}T_{ij}\rho T_{ij}=\operatorname{Tr}_a[U_k\;(\rho\otimes|0\>\<0|_a)\;U_k^\dag],
\end{equation}
where $|0\>\<0|_a$ is a fixed qubit ancilla state. Hence
\emph{extremal phase-conjugation maps for $d=4$ can be achieved with
just a control qubit}. Notice that the ancilla must not necessarily
be in a pure state, and the optimal map is equivalently achieved for
diagonal mixed ancilla state $\alpha|0\>\<0|_a+\beta|1\>\<1|_a$. By
adding a control qutrit, we can now choose among any of the optimal
maps using the controlled-unitary operator on
$\mathbb{C}^4\otimes\mathbb{C}^2 \otimes\mathbb{C}^3$
\begin{equation}
U=U_1\otimes|0\>\<0| + U_2\otimes|1\>\<1|
+U_3\otimes|2\>\<2|.
\end{equation}
Any optimal multi-phase conjugation map can now be written as
\begin{equation}\label{unitary-4}
\mathcal{T}_4(\rho)=\operatorname{Tr}_{a,b}\left[U\;\left(\rho\otimes|0\>\<0|_a\otimes\sigma_b\right)\;U^\dag\right]
\end{equation}
where $\sigma_b$ is a generic density matrix on $\mathbb{C}^3$. By
superimposing or mixing the three orthogonal states
$\{|0\>,|1\>,|2\>\}$ of the qutrit we control the weights
$p_1,p_2,p_3$ in Eq. (\ref{ext-dec-4}) via the diagonal entries of
$\sigma_b$. In other words, \emph{using a 6-dimensional ancilla it is
possible to span the whole set of optimal maps}.
Eqs. (\ref{unitaries-for-4})-(\ref{unitary-4}) can be generalized for
higher even dimensions\footnote{The case of odd dimensions is much
more complicated and will not be analysed here. The problem with odd
dimensions is that extremal points of the convex set of NSB matrices
are not permutations. Hence Birkhoff theorem cannot be applied.},
with
\begin{equation}
\begin{split}
&U_k=\sum_{i,j=0}^{\frac{d}{2}-1}T_{k\oplus 2i\oplus 2j,2i\oplus 2j}\otimes|i\>\<j|,\qquad k=1,\dots,d-1,\\
&U=\sum_{k=1}^{d-1}U_k\otimes|k\>\<k|,\\
&\mathcal{T}^{(k)}_d(\rho)=\operatorname{Tr}_a[U_k\;(\rho\otimes|0\>\<0|_a)\;U_k^\dag],\\
&\mathcal{T}_d(\rho)=\operatorname{Tr}_{a,b}\left[U\;\left(\rho\otimes|0\>\<0|_a\otimes\sigma_b\right)\;U^\dag\right]\\
\end{split}
\end{equation}
where $U_k$'s are unitary operators acting on
$\mathbb{C}^d\otimes\mathbb{C}^{d/2}$, $U$ is a control-unitary
operator on
$\mathbb{C}^d\otimes\mathbb{C}^{d/2}\otimes\mathbb{C}^{d-1}$,
$|0\>\<0|_a$ is a fixed $(d/2)$-dimensional pure state, and $\sigma_b$
is a generic $(d-1)$-dimensional density matrix. \emph{The minimum
dimension of the ancilla space required to unitarily realize an
optimal phase covariant transposition map is $d/2$, generalizing the
result for $d=4$}, for which just a qubit is needed (see Eq.
(\ref{ext-unit-4})). On the other hand, \emph{to span the whole
optimal maps set one needs a $(d-1)d/2$-dimensional ancilla}.
Finally, notice that realization of multi-phase covariant
transposition generally needs much less resources than realization of
universal transposition: the minimum dimension $d/2$ of the ancilla
space in the phase covariant case has to be compared with the
dimension $d^2$ required in the universal case
(\ref{eq:ancilla-cloning}).
\chapter{The Role of Noise in Quantum Processes}\label{ch:noise}
In the previous Chapters we saw how to optimize transformations over
quantum systems and how to realize them by means of physical
interactions. Of course, processing of quantum systems requires a very
high level of control during all steps of the experiment. On the other
hand, noise---in the sense of uncontrollable interactions of the
system with the sorroundings---is not always and completely avoidable:
the only thing the experimenter can do is to reduce it in order to
reach the desired level of confidence. This can be done by trying to
directly control the environment, e.~g. forcing it into a cavity, or
by engineering states, interactions and measuring apparata robust with
respect to the adopted model of noise.
In the following Sections we will deal with noise on measuring
apparata and on states. While in the first part (review of
Ref.~\cite{cleanpovm}) we will face very general models of
noise---basically, all non-unitary completely positive maps---in the
second part (review of Ref.~\cite{decomap}) we will focus on
decoherence of quantum states, proposing a novel correcting scheme
retrieving classical information that the decoherence process made
leak into the environment and exploiting such information to undo the
noise.
\section{Clean POVM's}
Let us given a general apparatus performing a measurement on quantum
states. We know that the most general way to mathematically model it
is by means of a POVM $\povm P$, see Section~\ref{sec:POVM}. Let us
now think for a while we don't know how the apparatus $\povm P$ works.
It could be noisy at the input gate, that is, quantum states undergo
some uncontrolled transformation $\map E$ before being measured,
and/or noisy at the output, the outcomes being, let's say, shuffled
before being read by the experimenter. The two situations are depicted
in Fig.~\ref{fig:noise-in-meas}.
\begin{figure}
\centering
\includegraphics[width=14cm]{noise-in-meas.eps}
\caption{There are two ways of processing POVM's: (A) the
\emph{postprocessing} of the output data and (B) the
\emph{preprocessing} of the input states by a quantum channel. The
postprocessing is purely classical, whilst the preprocessing is
quantum.}\label{fig:noise-in-meas}
\end{figure}
The question is the following: Do we have any condition on $\povm P$
that allows us to \emph{a priori} affirm that $\povm P$ is ``clean'',
i.~e. without noisy processing at the input and/or the output?
Clearly the point can be viewed from the complementary point of view:
What kind of processings are possible on a given POVM? How does the
apparatus change after such processings are performed?
\subsection{Postprocessing of output data}
The most general postprocessing of a POVM outcomes is a shuffling of
with conditional probability $p(i|j)\ge 0$, corresponding to the
mapping
\begin{equation}
Q_i=\sum_jp(i|j)P_j,
\label{eq:postproc}
\end{equation}
where $\sum_ip(i|j)=1$, $\forall j$. To visualize the shuffling
(\ref{eq:postproc}), it is useful to think to the POVM as a column of
operators and $p(i|j)$ as a column-stochastic matrix\footnote{That is,
a matrix of positive numbers such that all its columns' entries sum
up to one.}
\begin{equation}
\begin{pmatrix}
Q_1\\
Q_2\\
\vdots\\
Q_n
\end{pmatrix}=\begin{pmatrix}
p(1|1) & p(1|2) & \cdots & p(1|m)\\
p(2|1) & p(2|2) & \cdots & p(2|m)\\
\vdots & \vdots & \ddots & \vdots\\
p(n|1) & p(n|2) & \cdots & p(n|m)\\
\end{pmatrix}\begin{pmatrix}
P_1\\
P_2\\
\vdots\\
P_m
\end{pmatrix}.
\end{equation}
Notice that postprocessing generally does not require that ${\mathbf P}$ and
${\mathbf Q}$ have the same cardinality. Relevant examples of postprocessing
are:
\begin{enumerate}
\item identification of two outcomes, e.~g. $j$ and $k$ are identified
with the same outcome $l$, corresponding to
$p(n|j)=p(n|k)=\delta_{ln}$;
\item permutation $\pi$ of outcomes, corresponding to
$p(\pi(j)|k)=\delta_{jk}$.
\end{enumerate}
When two POVM's ${\mathbf P}$ and ${\mathbf Q}$ are connected by a mapping of the form
\eqref{eq:postproc} for some conditional probability $p(i|j)$ we will
write ${\mathbf P}\succ_p{\mathbf Q}$, and say that the POVM ${\mathbf P}$ is {\em cleaner
under postprocessing}---{\em postprocessing cleaner}, for
short---than the POVM ${\mathbf Q}$. It is possible to prove that the relation
$\succ_p $ is a pseudo-ordering, hence an equivalence relation under
postprocessing can be defined as follows
\begin{definition}
The POVM's ${\mathbf P}$ and ${\mathbf Q}$ are {\em postprocessing equivalent}---in
symbols ${\mathbf P}\simeq_p {\mathbf Q}$---if and only if both relations
${\mathbf P}\succ_p{\mathbf Q}$ and ${\mathbf Q}\succ_p{\mathbf P}$ hold.
\end{definition}
We are now in position to define {\em cleanness under postprocessing},
namely
\begin{definition}
A POVM ${\mathbf P}$ is {\em postprocessing clean} if and only if for any
POVM ${\mathbf Q}$ such that ${\mathbf Q}\succ_p{\mathbf P}$, then also ${\mathbf P}\succ_p{\mathbf Q}$
holds, namely ${\mathbf P}\simeq_p{\mathbf Q}$.
\end{definition}
The complete characterization of cleanness under postprocessing
(classical) is given by the following theorem (see
Refs.~\cite{cleanpovm,martens})
\begin{theorem}[postprocessing]
A POVM ${\mathbf P}$ is postprocessing clean if and only if it is rank-one.
\end{theorem}
This means that if a POVM is rank-one, we are sure that it does not
have a hidden noisy postprocessing at the output. Viceversa, the
Theorem says that it is not possible to obtain the statistics of a
rank-one POVM by classically postprocessing the outcomes of a higher
rank POVM.
\subsection{Preprocessing of input states}
A preprocessing $\map E$ of input states induces naturally a dual
channel $\dual{\map E}$ acting on the POVM itself, as seen in
Subsection~\ref{subsec:quantum-operations}. Hence, we will write
\begin{equation}
{\mathbf P}\succ{\mathbf Q}
\end{equation}
and say that the POVM ${\mathbf P}$ is \emph{preprocessing cleaner} than
${\mathbf Q}$, if and only if there exists a channel---i.~e. a
trace-preserving, completely positive map $\map E$---such that
$Q_i=\dual{\map E}(P_i)$, $\forall i$, or, equivalently
${\mathbf Q}=\dual{\map E}({\mathbf P})$, for short. It is possible to prove that the
relation $\succ$ is a pseudo-ordering, hence an equivalence relation
under preprocessing can be defined as follows
\begin{definition}
The POVM's ${\mathbf P}$ and ${\mathbf Q}$ are {\em preprocessing equivalent}---in
symbols ${\mathbf P}\simeq{\mathbf Q}$---if and only if both relations ${\mathbf P}\succ{\mathbf Q}$
and ${\mathbf Q}\succ{\mathbf P}$ hold.
\end{definition}
We are now in position to define {\em cleanness under preprocessing},
namely
\begin{definition}
A POVM ${\mathbf P}$ is {\em preprocessing clean} if and only if for any
POVM ${\mathbf Q}$ such that ${\mathbf Q}\succ{\mathbf P}$, then also ${\mathbf P}\succ{\mathbf Q}$ holds,
namely ${\mathbf P}\simeq{\mathbf Q}$.
\end{definition}
From the above definition, it turns out that a POVM is preprocessing
clean if and only if, whenever a noisy preprocessing acts, its action
on the POVM can be perfectly inverted. Now, a result by Wigner tells
that a channel admits an inverse channel (i.~e. it is physically
invertible\footnote{There exist channels that are invertible in the
sense that they define a one-to-one correspondence between states,
but their inverse mappings are not channels. This is the case, for
example, of the isotropic depolarizing channel $\rho\mapsto
p\rho+(1-p)I/d$. In Ref.~\cite{cleanpovm} we actually derived, as a
corollary, that one-to-one channels either are unitary or their
inverse map is not even positive.}) if and only if such a channel is
actually unitary. A question arises: Does cleanness property define
an interesting structure in the set of POVM's? Or will we find that
invertible preprocessings are just unitary (i.~e. trivial)
preprocessings? Generally, this is not the case, because we want the
action of the noise to be invertible only \emph{on a fixed} POVM, not
on all $\set{B}(\mathscr{H})$.
For qubits, however, preprocessing-equivalence coincides with
unitary-equivalence
\begin{theorem}[qubits] For two-level systems ${\mathbf P}\simeq{\mathbf Q}$ if and
only if there exists a unitary operator $U$ such that ${\mathbf P}=U^\dag{\mathbf Q}
U$.
\end{theorem}
In higher dimensions the counterexample is given implicitly by the
following Theorem regarding effects (two-outcomes POVM's, see
Section~\ref{sec:POVM})
\begin{theorem}[effects]
Let ${\mathbf P}=\{P,I-P\}$ and ${\mathbf Q}=\{Q,I-Q\}$ be two effects. Then
${\mathbf P}\simeq{\mathbf Q}$ if and only if $\lambda_M(P)=\lambda_M(Q)$ and
$\lambda_m(P)= \lambda_m(Q)$, where $\lambda_m(O)$ ($\lambda_M(O)$)
is the minimum (maximum) eigenvalue of $O$.
\end{theorem}
Since necessary and sufficient condition for preprocessing-equivalence
of two effects is that they have the same spectral width, ragardless
of the spectrum itself, it is clear that there exist
preprocessing-equivalent effects which are not unitarily equivalent
(otherwise they should have the same spectrum \emph{as a whole}). It
is also clear that, for dimension $d=2$, the spectrum is completely
determined by the spectral width, whence unitary-equivalence.
Besides effects, the other case in which we have a complete
characterization of preprocessing clean POVM's is the following
\begin{theorem}[observables]\label{th:cleand}
For number of outcomes $n\leq d$, the set of preprocessing clean
POVM's coincides with the set of observables.
\end{theorem}
This result is interesting since it provides an operational approach,
alternative to the axiomatic one given by von Neumann, to define what
are the observables in quantum theory. Here, just by introducing the
cleanness pseudo-ordering, we singled out the set of observables, as
the only clean POVM's with numer of outcomes less or equal to the
dimension of the Hilbert space---in this sense, they are the only
clean ``classical'' POVM's.
When the numbers of outcomes gets larger than the dimension of the
Hilbert space, the structure introduced by the preprocessing
pseudo-ordering on the convex set of POVM's becomes more complicated,
and we have just partial results. For example, we can prove that
rank-one POVM's are not only postprocessing clean, but also
preprocessing clean
\begin{theorem}[rank-one]
Rank-one POVM's are preprocessing clean.
\end{theorem}
Notice that cleanness under preprocessing and extremality are
properties completely unrelated. Consider, e.~g., the following
rank-one POVM
\begin{equation}\label{eq:badpovm}
\frac 12|1\>\<1|,\frac 12|1\>\<1|,|2\>\<2|,\dots,|d\>\<d|.
\end{equation}
The redundantly doubled outcome $|1\>\<1|$ suggests at first sight
that such a POVM cannot be extremal, namely, it cannot be the solution
of any optimization problem. In this sense, such POVM ``is not good''.
However, being rank-one, it is clean under both preprocessing and
postprocessing.
\subsection{Positive maps}
Since now, we introduced two pseudo-orderings on the set of POVM's,
the preprocessing ordering $\succ$, and the postprocessing ordering
$\succ_p$. In this Subsection, we will introduce two additional
relations which can be established among POVM's, namely
\begin{definition}[positive preprocessing] We write ${\mathbf P}\gg{\mathbf Q}$ and
say that ${\mathbf P}$ is cleaner than ${\mathbf Q}$ under positive preprocessing,
if and only if there exists a positive (non necessarily
\emph{completely} positive) map $\map P$ such that ${\mathbf Q}=\map
P({\mathbf P})$.
\end{definition}
\begin{definition}[range-inclusion] We write ${\mathbf P}\supset_r{\mathbf Q}$ and say
that ${\mathbf P}$ range-includes ${\mathbf Q}$, if and only if
$\set{Rng}({\mathbf Q})\subseteq\set{Rng}({\mathbf P})$, where the range of a POVM is defined
in Definition~\ref{def:POVM-range}.
\end{definition}
We simply have the following hierarchy of relations
\begin{equation}\label{eq:positive-relations}
{\mathbf P}\succ{\mathbf Q}\quad\Longrightarrow\quad{\mathbf P}\gg{\mathbf Q}\quad\Longrightarrow\quad{\mathbf P}\supset_r{\mathbf Q}.
\end{equation}
The converse is not always true. However, we have some results
providing sufficient conditions for which some of the relations in
Eq.~(\ref{eq:positive-relations}) can be inverted. Proofs are very
technical and can be found in Ref.~\cite{cleanpovm}. Here we just give
the statements.
\begin{theorem}
Consider two POVM's ${\mathbf P}$ and ${\mathbf Q}$ with the same number of
outcomes. Then the following statements are equivalent:
\begin{enumerate}
\item \label{item:1}
${\mathbf P} \Rip {\mathbf Q}$
\item \label{item:2}
There is a (unique) positive map $\map{E}: \set{Span}({\mathbf P}) \to
\set{Span}({\mathbf Q})$ with $\map{E}({\mathbf P}) = {\mathbf Q}$.
\end{enumerate}
\end{theorem}
Notice that point~(\ref{item:2}) does not say that ${\mathbf P}\gg{\mathbf Q}$, since
the positive map is defined only from $\set{Span}({\mathbf P})$ to $\set{Span}({\mathbf Q})$,
and generally cannot be extended to a positive map on all $\set{B}(\mathscr{H})$.
The following Theorems describe some situations in which it possible
to extend the map $\map E$ to a positive map over all $\set{B}(\mathscr{H})$.
\begin{theorem}
Consider two POVM's ${\mathbf P}$ and ${\mathbf Q}$ with the same number of
outcomes. Then the following statements are equivalent:
\begin{enumerate}
\item \label{item:3a} ${\mathbf P} \succ {\mathbf Q}$
\item \label{item:4} There is an informationally complete POVM $\mathbf{M}$
such that ${\mathbf P} \otimes \mathbf{M} \Rip {\mathbf Q} \otimes \mathbf{M}$.
\item \label{item:5} ${\mathbf P} \otimes \mathbf{M} \Rip {\mathbf Q} \otimes \mathbf{M}$ holds
for all POVM's $\mathbf{M}$.
\end{enumerate}
\end{theorem}
\begin{theorem}[abelian POVM]\label{thm:abel-ranges}
Consider two POVM's ${\mathbf P}$ and ${\mathbf Q}$ with the same number of
outcomes. Let ${\mathbf Q}$ be abelian, namely $Q_iQ_j=Q_jQ_i$ for all
$i,j$. Then ${\mathbf P} \Rip {\mathbf Q}$ $\Longrightarrow$ ${\mathbf P} \succ {\mathbf Q}$, and
Eq.~(\ref{eq:positive-relations}) becomes a chain of equivalences.
\end{theorem}
\section{Inverting decoherence}
We will now focus our attention on a particularly nasty preprocessing
of input states, namely on decoherence. Decoherence is universally
considered, on one side, as the major practical limitation for
communication and processing of quantum information. On the other
side, decoherence yields the key concept to explain the transition
from quantum to classical world \cite{decoherence} due to the
uncontrolled and unavoidable interactions with the environment. Great
effort in the literature has been devoted to combat the effect of
decoherence by engineering robust encoding-decoding schemes. Some
authors have recently addressed a different approach to undo quantum
noises by extracting classical information from the environment
\cite{GW} and exploiting it as an additional amount of side
information useful to improve quantum communication performances
\cite{capacity}.
The recovery of quantum coherence from the environment is often a
difficult task, e. g. when the environment is ``too big'' to be
controlled, as for spontaneous emission of radiation. By regaining
control on the environment the recovery can sometimes be actually
accomplished, for example by keeping the emitted radiation inside a
cavity. However, in some cases, the full recovery of quantum coherence
becomes impossible even in principle, namely even when one has
complete access to the environment. This naturally leads us to pose
the following question: in which physical situations is possible to
perfectly recover quantum coherence by monitoring the environment?
\subsection{Convex structure of decoherence maps}
A completely decohering evolution asymptotically cancels any quantum
superposition when reaching the stationary state, making any state
diagonal in some fixed orthonormal basis---the basis depending on the
particular system-environment interaction. In the Heisenberg picture
we say that such a completely decohering evolution asymptotically maps
the whole algebra of quantum observables into a ``maximal classical
algebra'', that is a maximal set of commuting---namely jointly
measureable---observables. Let's denote by $\alg A_q$ the ``quantum
algebra'' of all bounded operators $\set{B}(\mathscr{H})$ on the finite
dimensional Hilbert space $\mathscr{H}$, and by $\alg A_c$ the ``classical
algebra'', namely any maximal Abelian subalgebra $\alg A_c\subset\alg
A_q$. Clearly, all operators in $\alg A_c$ can be jointly
diagonalized on a common orthonormal basis, which in the following
will be denoted as $\mathbf{b} =\{|k\>|k=1, \dots ,d\}$. Then, the
classical algebra $\alg A_c$ is also the linear span of the
one-dimensional projectors $|k\>\<k|$, whence $\alg A_c$ is a
$d$-dimensional vector space. According to the above general
framework, we call \emph{(complete) decoherence map} a completely
positive identity-preserving (i.~e. trace-preserving in the
Schr\"odinger picture, see Subsection~\ref{subsec:quantum-operations})
map $\dual{\map E}$ which asymptotically maps any observable $O \in
\alg A_q$ to a corresponding ``classical observable'' in $\alg A_c$,
namely such that the limit $\lim_{n\to \infty} (\dual{\map E})^n (O)$
exists and belongs to the classical algebra $\alg A_c$ for any $O \in
\alg A_q$. Here we denote with $(\dual{\map E})^n$ the $n$-th
iteration of the map $\map E$, implicitly assuming markovian
evolution.
It is easy to see that the set of decoherence maps is convex. The
following Theorem shows that such maps enjoy a remarkably simple form:
\begin{theorem}[Schur form]\label{GeneralForm}
A map $\dual{\map E}$ preserves all elements of the maximal
classical algebra $\alg A_c$ if and only if it has the form
\begin{equation}\label{SchurMap}
\dual{\map E}(O)= \xi \circ O,
\end{equation}
$A \circ B$ denoting the Schur product of operators $A$ and $B$, i.~e.
$A \circ B \equiv \sum_{k,l=1}^d A_{kl} B_{kl} |k\>\<l|$, $\{A_{kl}\}$
and $\{B_{kl}\}$ being the matrix elements of $A$ and $B$ in the basis
$\mathbf{b}$, and $\xi_{kl}$ being a correlation matrix, i.e. a positive
semidefinite matrix with $\xi_{kk}=1$ for all $k=1, \dots ,d$.
\end{theorem}
Theorem~\ref{GeneralForm} states a linear correspondence between maps
preserving $\alg A_c$ and correlation matrices, whence the two sets
share the same convex structure. Then the map is extremal if and only
if its correlation matrix is extremal.
Since now we dealt with the dual map $\dual{\map E}$ on bounded
operators. The action of a decoherence map on quantum states is given
in Schr\"odinger picture by
\begin{equation}\label{SchurFormSchro}
\map E(\rho)= \xi^T \circ \rho,
\end{equation}
where $T$ denotes transposition with respect to the basis $\mathbf{b}$
(also $\xi^T$ is a correlation matrix, hence in the following, we will
drop the symbol $T$ at the exponent). As a consequence, one has
exponential decay of the off-diagonal elements of $\rho$, since
$\left|[\map E^n (\rho)]_{kl}\right|=|\xi_{kl}|^n\cdot |\rho_{kl}|$.
In other words, any initial state $\rho$ decays exponentially towards
the completely decohered state
\begin{equation}
\rho_\infty\equiv\sum_k\rho_{kk}|k\>\<k|.
\end{equation}
In Ref.~\cite{decomap}, it is proved the following
\begin{lemma}\label{l:extr}
A map $\map E$ is an extremal decoherence map if and only if it is
extremal in the set of all maps.
\end{lemma}
As a consequence of Lemma \ref{l:extr}, the convex structure of
decoherence maps can be obtained by application of the well known Choi
Theorem \cite{choi}, which states that the canonical Kraus
operators\footnote{For the definition of canonical Kraus
decomposition, see Subsection~\ref{subsec:quantum-operations}.}
$\{E_i\}$, $1\leq i \leq r$, of every extremal map are such that their
products $\{E_i^{\dag}E_j\}$, $1\leq i,j\leq r$, are linearly
independent. A relevant consequence of this characterization is the
following
\begin{theorem}
If $\map E$ is an extremal decoherence map, then $r \leq \sqrt d$.
For qubits and qutrits any decoherence map is then random-unitary.
\end{theorem}
This means that for qubits and qutrits extremal decoherence maps are
unitary maps, since they admits a Kraus representation containing only
one operator. Hence, for qubits and qutrits, every decoherence map can
be written as
\begin{equation}\label{RandomUnitary}
\map E(\rho)=\sum_ip_iU_i\rho U_i^\dag,
\end{equation}
for some commuting unitary operators $U_i\in\alg A_c$ and probability
distribution $p_i$.
\subsection{Correcting decoherence by measuring the environment}
In Ref.~\cite{GW} it is shown that the only channels that can be
perfectly inverted by monitoring the environment are the
random-unitary ones. Therefore, it follows that one can perfectly
correct any decoherence map for qubits and qutrits by monitoring the
environment. The correction is achieved by retrieving the index $i$
in Eq.~(\ref{RandomUnitary}) via a measurement on the environment, and
then by applying the inverse of the unitary transformation $U_i$ on
the system. Therefore, the random-unitary map simply leaks $H(p_i)$
bits of \emph{classical information} into the environment ($H$
denoting the Shannon entropy), and the effects of decoherence can be
completely eliminated by recovering such classical information,
without any prior knowledge about the input state. The fact that
decoherence maps are necessarily random-unitary is true only for
qubits and qutrits. A counterexample in dimension $d=4$ can be found
in Ref.~\cite{decomap}. Such extremal decoherence maps with $r\geq2$
represent a process which is fundamentally different from the random
unitary one, corresponding to a {\em leak of quantum information} from
the system to the environment, information that cannot be perfectly
recovered from the environment \cite{GW}.
Now we address the problem of estimating the amount of classical
information needed in order to invert a random-unitary decoherence
map. If the environment is initially in a pure state, say $|0\>_e$, a
useful quantity to deal with is the so-called entropy
exchange \cite{Schumacher} $S_\textrm{ex}$ defined as
\begin{equation}\label{DefSex}
S_\textrm{ex}(\rho)=S(\sigma_{e}^\rho),
\end{equation}
where $\sigma_{e}^\rho$ is the reduced environment state after the
interaction with the system in the state $\rho$, and
\mbox{$S(\rho)=-\operatorname{Tr}[\rho\log\rho]$} is the von Neumann entropy. In
the case of initially pure environment, the entropy exchange depends
only on the map $\map E$ and on the input state of the system $\rho$,
regardless of the particular system-environment interaction chosen to
model $\map E$. It quantifies the information flow from the system to
the environment and, for all input states $\rho$, one has the
bound \cite{Schumacher} $|S(\map E(\rho))-S(\rho)|\leq S_\textrm{ex}(\rho)$,
namely the entropy exchange $S_\textrm{ex}$ bounds the entropy production at
each step of the decoherence process.
In order to explicitly evaluate the entropy exchange for a decoherence
process, we can then exploit a particular model interaction between
system and environment. This can be done noticing that it is always
possible to write $\xi_{kl}=\<e_l|e_k\>$ for a suitable set of
normalized vectors $\{|e_k\>\}$. Then, the map $\map
E(\rho)=\xi\circ\rho$ can be realized as \mbox{$\map
E(\rho)=\operatorname{Tr}_e[U(\rho\otimes |0\>\<0|_e)U^\dag]$}, where the unitary
interaction $U$ gives the transformation
\begin{equation}\label{Unitary}
U|k\>\otimes|0\>_e=|k\>\otimes|e_k\>,
\end{equation}
whence the final reduced state of the environment is
$\sigma_{e}^\rho=\sum_k\rho_{kk}|e_k\>\<e_k|$. Then, in order to
evaluate $S_\textrm{ex}$ for a decoherence map $\map E(\rho)=\xi\circ\rho$, it
is possible to bypass the evaluation of the states $|e_i\>$ of the
environment, using the formula
\begin{equation}\label{Sex_and_xi}
S_\textrm{ex}(\rho)=S(\sqrt{\rho_\infty}\xi\sqrt{\rho_\infty}),
\end{equation}
which follows immediately from the fact that
$\sqrt{\rho_\infty}\xi\sqrt{\rho_\infty}$, and $\sigma_{e}^\rho$ are
both reduced states of the same bipartite pure state
$\sum_i\sqrt{\rho_{ii}}|i\>|e_i\>$.
Notice that the unitary interaction $U$ in Eq.~(\ref{Unitary})
generalizes the usual form considered for quantum
measurements \cite{vonneumann}, with the quantum system interacting
with a pointer, which is left in one of the (nonorthogonal) states
$\{|e_k\>\}$. The more the pointer states are ``classical''---i.~e.
distinguishable---the larger is the entropy exchange, whence the
faster is the decoherence process. In the limit of orthogonal states,
decoherence is istantaneous, i.~e. $\map E(\rho)=\rho_\infty$.
When a map can be inverted by monitoring the environment---i.~e. in
the random-unitary case---the entropy exchange $S_\textrm{ex}(I/d)$ provides a
lower bound to the amount of classical information that must be
collected from the environment in order to perform the correction
scheme of Ref.~\cite{GW}. In fact, assuming a random-unitary
decomposition (\ref{RandomUnitary}) and using the formula
\cite{Schumacher} $S_\textrm{ex} (\rho)=S\left( \sum_{i,j} \sqrt{p_i p_j}
\operatorname{Tr}[U_i \rho U_j^{\dag}] |i\>\<j| \right)$, we obtain
\begin{equation}\label{SexAndEntropy}
S_\textrm{ex}(I/d) \leq H(p_i).
\end{equation}
The inequality comes from the fact that the diagonal entries of a
density matrix are always majorized by its eigenvalues \cite{bhatia},
and it becomes equality if and only if
$\operatorname{Tr}[U_iU_j^{\dag}]/d=\delta_{ij}$, i.~e. the map admits a
random-unitary decomposition with \emph{orthogonal} unitary operators.
Moreover, from Eq. (\ref{Sex_and_xi}) we have $S_\textrm{ex}
(I/d)=S(\xi/d)$.
In Ref.~\cite{decomap}, it is proved that, for qubits, $S(\xi/2)$
quantifies exactly the minimum amount of classical information which
must be extracted from the environment, while, for dimension $d>2$,
the bound in Eq.~(\ref{SexAndEntropy}) is generally strict and a
counterexample is given for dimension $d=3$. Notice that the same
decoherence map may be obtainable by different random-unitary
decompositions with different probability distributions $\{p_i\}$,
corresponding to different values of the information $H(p_i)$.
However, for qubits it is always possible to perform a suitable
measurement on the environment and to invert the decoherence map
retrieving the \emph{minimal} amount of information from the
environment, namely $S(\xi/2)$. For example, consider the so-called
\emph{random phase-kick model} \cite{nielsen} for decoherence of
qubits
\begin{equation}\label{badrandomphase}
\map E(\rho)=\frac{1}{\sqrt{4\pi\lambda}}\int_{-\infty}^{+\infty}e^{i\theta\sigma_z/2}\rho e^{-i\theta\sigma_z/2}e^{-\theta^2/4\lambda}\textrm{d}\theta,
\end{equation}
which can be rewritten as
\begin{equation}\label{goodphasekick}
\map E(\rho)=\xi\circ\rho,\qquad
\xi=\begin{pmatrix}
1 & e^{-\lambda}\\
e^{-\lambda} & 1
\end{pmatrix}.
\end{equation}
From Eq.~(\ref{badrandomphase}), one could infer that the amount $S$
of classical information that must be extracted from the environment
is equal to the differential entropy of the gaussian probability
density according to which the system is random phase-kicked, namely
(see Ref.~\cite{cover-thomas})
\begin{equation}\label{eq:wronginfo}
S\left(\frac{1}{\sqrt{4\pi\lambda}}e^{-\theta^2/4\lambda}\right)=\frac{1}{2}\log4\pi e\lambda,
\end{equation}
growing logarithmically with $\lambda$. This is actually not correct,
since the \emph{minimum} amount of classical information needed is
\begin{equation}\label{eq:entropyrate}
S=-p\log_2p-(1-p)\log_2(1-p),\qquad p=\frac{1-e^{-\lambda}}{2}.
\end{equation}
In fact, $\xi/2$ in Eq.~(\ref{goodphasekick}) can be simply
diagonalized and has eigenvalues
$\left\{\frac{1-e^{-\lambda}}{2},\frac{1+e^{-\lambda}}{2}\right\}$.
\begin{figure}
\centering
\includegraphics[width=14cm]{entropyrate.eps}
\caption{The amount of classical information $S$, expressed in bits,
leaking into the environment at every application of the random
phase-kick model of decoherence for qubits, as function of the
parameter $\lambda$, see Eqs.~(\ref{badrandomphase})
and~(\ref{goodphasekick}). $S$ tends to the limit value of 1 bit,
since \emph{every} qubit decoherence map can be written as a
random-unitary process involving only \emph{two} unitaries (see the
footnote in the previous page).}\label{fig:entropyrate}
\end{figure}
In Figure~\ref{fig:entropyrate} there is the plot of the amount of
classical information $S$ in Eq.~(\ref{eq:entropyrate}) as a function
of the parameter $\lambda$ modelling the decoherence rate in
Eqs.~(\ref{badrandomphase}) and~(\ref{goodphasekick}). The curve tends
to the finite limit of one bit, contrarily to what happens in
Eq.~(\ref{eq:wronginfo}).
\chapter*{List of Publications}
\addcontentsline{toc}{chapter}{\bf List of Publications}
\begin{itemize}
\item F~Buscemi, G~M~D'Ariano, C~Macchiavello, and P~Perinotti,\\
\emph{Optimal superbroadcasting maps of mixed qubit states},\\
in preparation
\item F~Buscemi, G~M~D'Ariano, M~Keyl, P~Perinotti, and R~F~Werner,\\
\emph{Clean positive operator valued measures},\\
J. Math. Phys. {\bf 46}, 082109 (2005)
\item F~Buscemi, G~Chiribella, and G~M~D'Ariano,\\
\emph{Inverting quantum decoherence by classical feedback
from the environment},\\
Phys. Rev. Lett. {\bf 95}, 090501 (2005)
\item F~Buscemi, G~M~D'Ariano, and C~Macchiavello,\\
\emph{Optimal Time-Reversal of Multi-phase Equatorial States},\\
pre-print on \texttt{quant-ph/0504016}
\item F~Buscemi, G~M~D'Ariano, and C~Macchiavello,\\
\emph{Economical Phase-Covariant Cloning of Qudits},\\
Phys. Rev. A {\bf 71}, 042327 (2005)
\item F~Buscemi, G~M~D'Ariano, and P~Perinotti,\\
\mbox{\emph{There exist non orthogonal quantum measurements that are
perfectly repeatable},}\\
Phys. Rev. Lett. {\bf 92}, 070403 (2004)
\item F~Buscemi, G~M~D'Ariano, and M~F~Sacchi,\\
\emph{Physical realizations of quantum operations},\\
Phys. Rev. A {\bf 68}, 042113 (2003)
\item F~Buscemi, G~M~D'Ariano, P~Perinotti, and M~F~Sacchi,\\
\emph{Optimal realization of the transposition maps},\\
Phys. Lett. A {\bf 314}, 374 (2003)
\item F~Buscemi, G~M~D'Ariano, and M~F~Sacchi,\\
\emph{Unitary realizations of the ideal phase measurement},\\
Phys. Lett. A {\bf 312}, 315 (2003)
\end{itemize}
\chapter*{}
\vspace{2cm}
\begin{flushright}
\emph{Aspice convexo nutantem pondere mundum\\
terrasque tractusque maris caelumque profundum.}\vspace{.5cm}\\
Behold the world swaying her convex mass,\\
lands and spaces of sea and depth of sky.\vspace{.5cm}\\
(Vergilius, Ecloga IV\footnote{Translated from Latin by J~W~MacKail, in \emph{Virgils' Works} (Modern Library, New York, 1934).})
\end{flushright}
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{\bf Introduction}
To handle information is to handle physical systems, and viceversa.
Hence, the ultimate limits in manipulating and distributing
information are posed by the very laws of physics. This is true in the
classical framework (e.~g. Landauer's principle) and in the quantum
framework, where the rules of Quantum Mechanics give rise to new---and
often not yet completely understood---restrictions and advantages to
information processing and distribution. Quantum Information Theory is
devoted to the investigation of the theoretical limits Quantum
Mechanics establishes when dealing with information encoded on quantum
systems. This thesis treats the problem of processing quantum
information\footnote{Here ``quantum information'' is a short hand for
``information encoded on quantum systems'' and it is basically
equivalent to saying ``quantum states''. Analogously, ``classical
information'' means ``information encoded on classical systems''.}
in an optimal way by means of physically realizable devices. In fact,
linearity of Quantum Mechanics forbids basic processings of classical
information---like e.~g. copying-, broadcasting-, and
\texttt{NOT}-gates---to properly work on an unknown quantum state. The
first natural question is then: How well can we approximate such
transformations and which are the physical devices that realize these
approximations?
In contrast to its classical counterpart, quantum information is very
sensitive to noise. In a realistic setup, it is unreasonable to
completely rule out noise, since the least interaction with the
sorroundings can cause the system to be irreversibly disturbed. This
fact raises the need of designing methods to encode quantum
information in a way that is robust with respect to noise. But in
order to do this, we have to provide a model for the noise. In this
sense, also noise can be viewed as a kind of processing of quantum
states: a ``nasty'' processing, nonetheless obeying the same laws of
Quantum Mechanics as ``good'' processings do. The second natural
question is then: What is the role of noise in a realistic setup and
how can we control it?
In order to answer both questions, we clearly need to work in full
generality. The appropriate mathematical tool to do this is provided
by the concept of \emph{quantum channel}. It encloses all possible
deterministic transformations of quantum states allowed by the
postulates of Quantum Mechanics. In Chapter~\ref{ch:ch1} we review the
mathematical formalism describing quantum measurements and quantum
state transformations. We face problems, such as quantum state
preparation and repeatability of quantum measurements, which, even
though they reach back to the beginnings of quantum theory, have been
revived and put into a new light by the recent developments in the
experimental techniques.
Chapter~\ref{ch:chap2} is concerned with the analysis of quantum
channels. Exploiting the convex structure of the set of channels, we
explicitly single out those that constitute the best quantum versions
of the intrisically classical copying-, broadcasting-, and
\texttt{NOT}-gates. We introduce the general theory on which such
optimization relies, and present group theoretical techniques to
analyse the common situation in which symmetries of the set of input
quantum states, after the action of the channel, propagate to the
output. This is the framework of \emph{covariant channels}. It is very
useful to describe many physical situations and, at the same time, it
permits an analytical approach.
Quantum channels are more general to describe changes of quantum
states than unitary evolutions controlled by Schr\"odinger's equation.
Nonetheless, it is well known that every quantum channel is the
transformation that a system undergoes when unitarily interacting with
an auxiliary quantum system---the so-called \emph{ancilla}---that is
discarded after the interaction took place. Actually, this is the
only way to deterministically realize a non-unitary quantum channel.
In Chapter~\ref{ch:realization} we propose feasible implementations
for some of the channels constructed in Chapter~\ref{ch:chap2},
providing the ancillary quantum state and the global unitary
interaction. This is just a first step towards the experimental
realization which remains a far more difficult task, however, the
setup we propose to optimally copy quantum systems, in the case of
qubits (i.~e. two-levels systems) coincides with the one already used
in experiments. This is encouraging in view of a possible
generalization of experimental techniques to higher dimensional
quantum systems.
The thesis ends with Chapter~\ref{ch:noise} which deals with classical
and quantum noise. Noise is considered as acting both on the measuring
apparata and on the quantum states. More specifically, we introduce a
(partial) ordering on the convex set of measuring devices. It allows
us to characterize ``clean'' devices, namely, those which are not
affected by quantum and/or classical noise. Interestingly enough, we
show that such ordering is able to single out von Neumann's
observables as ``particularly nice'' measuring apparata. This gives an
operational characterization for the usually postulated concept of
\emph{observable}. We then focus attention on the specific model of
noise called \emph{decoherence}. Decoherence acts destroying quantum
superpositions, thus making ineffective all quantum improvements on
the classical approach. On the other side, decoherence possesses also
foundational interest since it represents the favourite tool to
explain the quantum-to-classical transition. The process called
decoherence is actually a convex set of commuting channels satisfying
very restrictive properties. Applying techniques described in
Chapters~\ref{ch:chap2} and~\ref{ch:realization}, we provide a method
to invert decoherence and restore quantum superpositions by a feedback
control from the environment. This means that measuring a suitable
observable of the environment's degrees of freedom, and then
performing on the system a suitable unitary transformation dependent
on the measurement result, it is possible to completely cancel the
effect of decoherence.
\chapter*{Preface Note}
\addcontentsline{toc}{chapter}{\bf Preface Note}
This manuscript must be intended as an informal review of the research
works carried out during three years of PhD. ``Informal'' in the sense
that technical proofs are often omitted (they can be found in the
papers) as one could do for a presentation in a public talk. Clearly,
some background of Quantum Mechanics is needed, even if I tried to
minimize the prerequisites.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,152
|
\section{Introduction}
\label{intro}
Quantum chromodynamics (QCD) is the well-established theory of the strong nuclear forces. Thanks to asymptotic
freedom, it is amenable to perturbative methods at high energy. While the coupling never becomes
very small at any energy scale of interest, perturbative QCD is a most successful framework for collider physics,
and its predictions beyond the simple parton model have been tested extensively.
Nonperturbative aspects of QCD, such as the spectrum of hadrons and the transition to a quark-gluon plasma phase,
can be studied from first principles through lattice QCD. Indeed, tremendous progress has been made thanks
to the availability of powerful computers and also thanks to methodological progress. However, two areas of strong-coupling
physics, where lattice QCD is still far from giving reliable answers are finite baryon density and
real-time phenomena such as decay rates (although progress is being made on both frontiers).
These insufficiently charted areas will be at the focus of a new generation of experiments at the FAIR
facility, where the PANDA experiment at FAIR will study hadron reactions \cite{Lutz:2009ff,Wiedner:2011mf},
and the CBM experiment will
aim at exploring baryonic matter at high density \cite{Friman:2011zz}.
A novel approach to strongly coupled gauge theories has been developed over the last one and a half decades
in the form of gauge/gravity duality \cite{Aharony:1999ti}, also known as ``holography''. While first-principles approaches with
a well-defined string-theoretic foundation allow one to study only gauge theories which are in important
ways different from real QCD, this approach offers fresh insights in the richness of phenomena of gauge theories
at strong coupling. Based on this progress, so-called bottom-up models have been developed which do not
care so much about a string-theoretic justification, but provide phenomenological models that incorporate
the key ingredients of holographic gauge theories, and many interesting studies have been carried out using these.
There is however one top-down holographic model, which goes a long way towards QCD starting from a
fundamental string-theoretic basis: Witten's model
of low-energy QCD through type-IIA supergravity, proposed already in 1998 \cite{Witten:1998zw}, which has almost no free parameters.
Here one sets up a holographic dual to a supersymmetric
gauge theory in 4+1 dimensions and performs the analogue of high-temperature dimensional
reduction, but with respect to the superfluous spatial dimensions, such that at low energy one ends
up with nonsupersymmetric pure-glue QCD at large number of colors and large 't Hooft coupling in 3+1 dimensions.
In 2004, Sakai and Sugimoto \cite{Sakai:2004cn,Sakai:2005yt} have extended Witten's model by flavor D8-branes in a setup that
provides chiral quarks and anti-quarks. In the model of Sakai and Sugimoto chiral symmetry breaking
as well as its restoration at high temperatures has a beautiful geometric realization. Its low-energy
effective action contains a chiral Lagrangian for massless Nambu-Goldstone bosons and massive vector
and axial vector mesons, complete with the correct Wess-Zumino-Witten term and also a Skyrme term.
The only free parameters in the model are a (Kaluza-Klein) mass parameter
$M_{\rm KK}$ and the 't Hooft coupling $\lambda$ at this scale, making it
very predictive. Unfortunately, hardly anything is known about the corrections that are needed
to approach real QCD at finite $N_c$ and $\lambda$ and that are in principle
determined by type-IIA string theory. In the case of the original AdS/CFT correspondence for
maximally supersymmetric Yang-Mills theory based on type-IIB string theory,
higher-derivative string-theoretic corrections
have been determined for entropy and shear viscosity \cite{Myers:2008yi}.
Reassuringly, they come with the expected sign to connect to perturbative results.\footnote{In the case of the entropy of a supersymmetric plasma,
it is has even been found that the next-to-leading results at weak and at strong coupling
can be interpolated uniquely with a singularity-free Pad\'e approximant \cite{Blaizot:2006tk}.} For $N_c=3$ and $\alpha_s=0.5$, their magnitude is
5\% and 37\% for entropy and shear viscosity, respectively,
which (optimistically) gives an idea of the quantitative predictiveness to expect in top-down holographic
calculations.
Indeed, as will be shown below, the Sakai-Sugimoto model does not only reproduce
many qualitative features of QCD but is often quantitatively close. This makes it interesting to
apply it to new questions that are presently out of reach of a first-principles approach such as lattice QCD.
After a brief review of the Witten-Sakai-Sugimoto model and how it compares quantitatively with
real QCD, we will discuss some new results
obtained for the spectrum and decay rates of glueballs, and for the phase diagram of QCD in the presence
of strong magnetic fields (for the latter see also the recent reviews \cite{Kharzeev:2013jha}).
\section{Brief review of the Witten-Sakai-Sugimoto model}
\label{secWSS}
\subsection{Large $N_c$ gauge theories and the AdS$_5$/CFT$_4$ correspondence}
The dominant Feynman diagrams of nonabelian gauge theories in the limit of large number of colors
but fixed 't Hooft coupling $\lambda=g^2 N_c$ are the planar diagrams, which can be drawn on a plane (or, equivalently, a sphere) without
crossing of lines when the color flow
is represented by one line per fundamental or antifundamental index (so that gluons are represented
by double lines). Nonplanar diagrams are suppressed by a factor $(1/N_c)^{\,\chi}$, where $\chi$ is
the Euler number of the surface needed to draw the Feynman diagram.\footnote{A nice illustration
is provided by Fig.~2 of Ref.~\cite{Peeters:2007ab}.} This is very similar to string perturbation
theory, where one has to sum over Riemann surfaces weighted by $g_s^{\,\chi}$ with $g_s$ the string coupling constant,
leading 't Hooft to the
remarkable speculation that the large-$N_c$ limit of nonabelian gauge theories may be essentially
a string theory \cite{'tHooft:1973jz}.
A quarter of a century later, this idea became
finally concrete in the form of the AdS/CFT correspondence.
The study of nonperturbative objects in string theory, so-called D-branes, led Maldacena \cite{Maldacena:1997re} to
the conjecture that string theory in the near-horizon limit of $N_c$ coincident D3-branes, which is governed by
a curved ten-dimensional space of the form AdS$_5\times S^5$, is completely equivalent to the low-energy
effective theory of these branes in flat Minkowski space, which is the maximally supersymmetric
conformal SU($N_c$) gauge theory. Superstring theory, which necessarily involves gravity, in
the background of 5-dimensional anti-de Sitter space times a 5-sphere of equal curvature radius
is thus proclaimed to give an alternative description of a nonabelian quantum gauge theory
in 4 spacetime dimensions without gravity. The latter can be viewed to live on the 4-dimensional
(conformal) boundary of anti-de Sitter space, which thus also realizes the holographic principle
anticipated for quantum gravity by 't Hooft and Susskind \cite{Susskind:1994vu}.
Spatially three-dimensional branes are nonperturbative objects in type-IIB superstring theory as well
as in its low-energy limit, type-IIB supergravity. The latter is a justified approximation when
the string length scale $\ell_s=\sqrt{\alpha'}$
is small compared to spacetime curvature radius\footnote{Not to be confused with
the Ricci scalar, which will be written as $R^{(d)}$ for a $d$-dimensional space.} $R$ and when the string coupling $g_s$ is
small. Because the weak-coupling and strong-coupling description of D3 branes give, respectively,
rise to the two relations
\begin{equation}
g^2=4\pi g_s,\qquad 4\pi g_s N_c=\frac{R^4}{\ell_s^4},
\end{equation}
a useful gauge/gravity duality emerges in the limit $g^2\to 0$ and $\lambda\gg 1$, implying $N_c\to\infty$.
Useful, because it allows one to substitute extremely difficult
calculations in the strongly coupled quantum gauge theory by
calculations in weakly-coupled, classical supergravity. (It should be noted, however,
that the AdS/CFT correspondence is at its roots a duality between two quantum theories -- superstring theory
in a certain background on the one hand, and a superconformal field theory on the other.)
The gauge theory in question is however a (maximally) supersymmetric and conformal quantum field theory
which is very different from QCD, which is neither supersymmetric nor conformal, but has a running coupling constant
and confinement. However, for deconfined QCD at high temperature,
the AdS/CFT correspondence may actually be useful. The super-Yang-Mills theory can be considered at finite temperature, where
the geometry on the supergravity side becomes one that involves
a black hole (strictly speaking a black brane) in
anti-de Sitter space. Finite temperature breaks supersymmetry because bosons and fermions
have different distribution functions, whereas conformal symmetry may actually be a reasonable approximation
sufficiently above the deconfinement temperature. Indeed, the quark-gluon plasma created in
heavy-ion collisions seems to behave in many respects like a strongly coupled quantum fluid. In particular
hydrodynamic transport coefficients such as shear viscosity turn out to be better described by the
famous result \cite{Policastro:2001yc,Kovtun:2004de}
$\eta/s=\hbar/4\pi$ obtained through the AdS/CFT correspondence than by perturbative QCD, where
$\eta/s$ is parametrically large.
Perturbative QCD is certainly relevant for high-energy processes, and it seems most promising to combine
perturbative and strong-coupling gauge/gravity approaches when both have
a role to play such as in high-energy jets that are quenched by a strongly coupled medium \cite{Casalderrey-Solana:2014wca,Iancu:2014ava}.
\subsection{Towards a gravity dual of QCD: the Witten model}
At low temperature, with and without quark chemical potential, superconformal Yang-Mills theory is
however certainly very different from real QCD.
An intriguing route towards a gravity dual of QCD has been proposed by Witten in Ref.~\cite{Witten:1998zw}:
There are two more fundamental instances of AdS/CFT dualities, arising from the so-called
M2- and M5-branes of M-theory, whose low-energy limit is the (unique) 11-dimensional supergravity,
already discussed by Maldacena in Ref.~\cite{Maldacena:1997re}.
The near-horizon geometries of these branes are AdS$_4\times S^7$ and AdS$_7\times S^4$, respectively,
which were found as solutions of 11-dimensional supergravity already in 1980
by Freund and Rubin \cite{Freund:1980xh} when they were searching for Kaluza-Klein reductions to 4 spacetime
directions.\footnote{These solutions are stabilized by
having constant flux from the antisymmetric tensor fields of 11-dimensional supergravity on the compact spheres,
for which the M-branes provide sources.
Because the anti-de Sitter space and the sphere have comparable curvature, these
solutions are not viable as Kaluza-Klein theories for the real world, but they paved the way
to more elaborate flux compactifactions in string theory.}
The dual of M-theory on AdS$_7\times S^4$ is the 6-dimensional superconformal
low-energy theory of M5 branes.
M-theory compactified on a supersymmetry-preserving circle leads to 10-dimensional superstrings of type IIA,
with the M5 branes turned into D4 branes. This leads to a nonconformal duality
of a 5-dimensional supersymmetric field theory to a string theory on a background that is related
to AdS$_6\times S^4$ by a Weyl transformation \cite{Kanitscheider:2008kd}.
Witten \cite{Witten:1998zw}
suggested to consider one further dimensional reduction on a circle for the superfluous
fourth spatial coordinate,
\begin{equation}
x_4 + 2\pi R_4\equiv x_4 + 2\pi M_{\rm KK}^{-1} \simeq x_4,
\end{equation}
but with
supersymmetry breaking boundary conditions for fermions as they also appear in
finite temperature field theory in the imaginary-time formalism.
All modes not protected by gauge symmetry become massive in this case, fermionic gluinos at tree-level, and
adjoint scalars through quantum effects, leaving pure-glue Yang-Mills
theory as low-energy effective theory. The 5-dimensional supersymmetric field theory thus turns into
ordinary 4-dimensional Yang-Mills theory. On the other hand, the background geometry
of the string theory becomes
\begin{equation}\label{ds2W}
ds^2=\left(\frac{u}{R}\right)^{3/2}\left[\eta_{\mu\nu}dx^\mu dx^\nu+f(u)dx_4^2\right]+\left(\frac{R}{u}\right)^{3/2}\left[\frac{du^2}{f(u)}+u^2 d\Omega_4^2\right]
\end{equation}
with
\begin{equation}
f(u)=1-\frac{u_{\rm KK}^3}{u^3},\quad M_{\rm KK}=\frac32 \frac{u_{\rm KK}^{1/2}}{R^{3/2}}
\end{equation}
and a nonconstant (``linear''\footnote{Since it would be linear in suitable radial coordinates.}) dilaton $e^\phi=g_s(u/R)^{3/4}$.
Here $u\ge u_{\rm KK}$ is the holographic direction with $u=\infty$ corresponding to the conformal boundary. The topology of the subspace formed by $u$ and the Kaluza-Klein direction $x_4$
is that of a cigar with the $x_4$ circle shrinking to zero size at $u=u_{\rm KK}$. This
lower bound on $u$ corresponds to the would-be horizon of
a Euclidean black-hole geometry and the relation between $u_{\rm KK}$ and $M_{\rm KK}$
is determined by the absence of a conical singularity at $u=u_{\rm KK}$.
The parameters of the 4-dimensional gauge theory with Lagrangian $\mathcal L=-\frac1{2g^2}{\rm Tr}\, F_{\mu\nu}F^{\mu\nu}$ resulting from the reduction of $\mathcal L=-\frac1{2g_5^2}{\rm Tr}\, F_{MN}F^{MN}+\ldots$ with $g_5^2=2g_s (2\pi)^2 \ell_s$
are given by\footnote{\label{fng}In virtually all of the literature on the Witten(-Sakai-Sugimoto) model, $\mathcal L=-(4g_{\rm YM}^2)^{-1}{\rm Tr}\, F_{\mu\nu}F^{\mu\nu}$
is used. This different convention has no consequences as long as no comparison
to perturbative QCD is made, where the standard definition of the SU(3) gauge coupling is $g$ with
$g^2=2g_{YM}^2$ and the $\alpha_s$ of QCD reads $\alpha_s=g^2/4\pi=g_{YM}^2/2\pi$.}
\begin{equation}
g^2=\frac{g_5^2}{2\pi R_4}=4\pi g_s\ell_sM_{\rm KK},\quad \frac{R^3}{\ell_s^3}=\pi g_s N_c,
\end{equation}
where the latter equation is due to the nonvanishing
Ramond tensor field sourced by the $N_c$ D4-branes.
For the supergravity approximation to be justified, the maximum curvature
$\sim(u_{\rm KK} R^3)^{-1/2}$ located at $u=u_{\rm KK}$
needs to be small compared to the string tension $\propto \ell_s^{-2}$,
which again requires large 't Hooft coupling $\lambda=g^2N_c\gg 1$.
The dimensionless coupling $g$ is derived from the dimensionful coupling $g_5$ at the Kaluza-Klein
mass scale $M_{\rm KK}$, beyond which the field theory reveals its underlying 5-dimensional nature.
Ideally, $M_{\rm KK}$ should be sent to infinity,
but that would mean small 't Hooft coupling, where string theory
effects beyond the supergravity approximation would be needed. In the Witten model, we have to be content with the strong-coupling
regime below some finite scale $M_{\rm KK}$, but the fact that this dual theory may be
continuously connected to real QCD makes it an
interesting approximation to the strong-coupling regime of the latter.
A particularly simple quantity to calculate in the Witten model is the Wilson
loop of heavy (nondynamical) quarks. Connecting two heavy quarks at the boundary which are
widely separated along $x$ through a Wilson line is represented by
a fundamental string embedded in the bulk geometry.
The string tension is locally given by $(2\pi\alpha')^{-1}\sqrt{-g_{tt}g_{xx}}$.
For large separation in $x$, the
energetically favored solution is clearly
to have as much as possible of the length of the string
at the minimal value of
$\sqrt{-g_{tt}g_{xx}}$, which occurs at $u=u_{\rm KK}$. The effective string tension at
large $\Delta x$ thus approaches a constant which indicates confinement of quarks
and which is
given explicitly by
\begin{equation}\label{sigmastring}
\sigma=\frac1{2\pi\ell_s^2}\sqrt{-g_{tt}g_{xx}}\Big|_{u=u_{\rm KK}}=
\frac1{2\pi\ell_s^2}\left(\frac{u_{\rm KK}}{R}\right)^{3/2}=\frac{g^2 N_c}{27\pi}M_{\rm KK}^2.
\end{equation}
In accordance with confinement, there is a mass gap in the spectrum
of fluctuations of the background geometry
which is proportional to $M_{\rm KK}$.
These fluctuations can be classified with respect to their 4-dimensional
quantum numbers and comprise scalar, vector, and tensor modes, which correspond to
glueballs in the gauge theory.
\begin{figure}[t]
\centerline{\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{gbspctrm}}
\centerline{\small \hfil (a) \hfil\hfil (b) \hfil}
\medskip
\caption{Glueball spectrum of the Witten model (a) as obtained
in Ref.~\cite{Brower:2000rp}, in units of $M_{\rm KK}$, (``exotic'' modes in green), compared to
the recent large-$N_c$ lattice results of Ref.~\cite{Lucini:2010nv} (b) in
units of $\sqrt{\sigma}$ but with overall scale such that the lowest tensor mode
are on equal height. The dotted lines in (b) give the holographic spectrum
in terms of the holographic string tension with the standard parameter set of
the Sakai-Sugimoto model (\ref{kappaMSS}).}
\label{figGB}
\end{figure}
Following up on pioneering work in Ref.~\cite{Csaki:1998qr
Csaki:1999vb,Constable:1999gb}, the complete spectrum was
worked out in Ref.~\cite{Brower:2000rp}. In Fig.~\ref{figGB}, the spectrum
in the Witten model (Fig.~\ref{figGB}a) is compared with recent lattice
results \cite{Lucini:2010nv} for large-$N_c$ Yang-Mills theory (colored boxes in Fig.~\ref{figGB}b)
with the vertical scale chosen such that the lowest tensor glueball $2^{++}$
in the two figures have equal height. The holographic calculation appears to
reproduce qualitatively many of the conspicuous features of the lattice result
which is in fact rather similar to those obtained at $N_c=3$. The main qualitative difference
is the absence of glueballs with quantum numbers $2^{-+}$ and of glueballs with spin $>2$
in the Witten model. Another peculiar feature of the holographic result is
a certain proliferation of scalar modes $0^{++}$: the green lines in Fig.~\ref{figGB}a
represent scalar modes which have a polarization involving the metric component
$g_{44}$ and which have been termed ``exotic'' in Ref.~\cite{Constable:1999gb}. The other
modes in the $PC=++$ sector are degenerate $0^{++}$ and $2^{++}$ glueballs.
Thus there are 4 scalar glueballs with mass less or equal the first excited
tensor glueballs, whereas the lattice features only two in that range.
In fact, Ref.~\cite{Constable:1999gb} already suspected that the scalar glueballs
associated with the ``exotic'' polarization may perhaps not correspond to
glueballs in QCD, but later Ref.~\cite{Brower:2000rp} found that the lowest
exotic mode is the lowest-lying one of all the modes, making the overall
picture tantalizingly similar to that found in lattice simulations.
Below we shall revisit this issue when quarks are included by the
extension provided by the Witten-Sakai-Sugimoto model.
In Ref.~\cite{Kanitscheider:2008kd} the necessary tools for holographic
renormalization for nonconformal holography have been developed and applied
to the Witten model to calculate the gluon condensate with the result
\begin{equation}
\frac14 \left<{\rm Tr}\, F^2\right>=\frac{2N_c}{3^7\pi^2}(g^2 N_c/2)^2M_{\rm KK}^4,
\end{equation}
where we have taken into account that the $g_4$ used in Ref.~\cite{Kanitscheider:2008kd}
differs from the standard definition of $g$ of perturbative QCD by $g_4^2=g^2/2$.
The gluon condensate as usually defined in QCD
in terms of
canonically normalized gluon fields $G_{\mu\nu}^a$
is given by
\begin{equation}\label{gluoncondensate}
C^4\equiv\left<\frac{\alpha_s}{\pi}G_{\mu\nu}^a
G^{a\mu\nu}\right>=\frac1{2\pi^2}\left<{\rm Tr}\, F^2\right>=
\frac{4N_c}{3^7\pi^4}(g^2 N_c/2)^2M_{\rm KK}^4.
\end{equation}
We shall come back to this when discussing the choice of parameters of the Sakai-Sugimoto model.
\subsection{Inclusion of chiral quarks: the Sakai-Sugimoto model}
Type-IIA string theory has stable D-branes with any even number of spatial dimensions.
If such additional branes are present, the open strings connecting those with the color D4-branes
carry one color index on one end and a new index on the other, which
can represent the flavor quantum number of quarks. If flavor and color branes intersect,
one gets chiral quarks. As long as the number $N_f$ of these additional branes is small compared to $N_c$,
they can be considered as ``probes'' with negligible back-reaction on the background
geometry they are imbedded in. This is analogous to the introduction of quenched
quarks in lattice QCD.
A first interesting attempt to include quarks in the Witten model was
made by Kruczenski, Mateos, Myers, and Winters \cite{Kruczenski:2003uq} by
introducing D6 flavor branes in the background of the Witten model.
In the case of only one flavor brane and a massless quark,
a spontaneous
breaking of an axial U(1) symmetry was observed, but the generalization to
a nonabelian flavor symmetry at $N_f>1$ did not give a good model for chiral
symmetry breaking.
This problem was solved by Sakai and Sugimoto \cite{Sakai:2004cn,Sakai:2005yt} by instead using
pairs of probe\footnote{The daunting issue of backreaction of localized D8 branes has been
studied in Ref.~\cite{Burrington:2007qd}. A different strategy to include dynamical
quarks has been recently proposed in Ref.~\cite{Bigazzi:2014qsa} through smeared
D8 branes and corrections to first order in $N_f/N_c$ have been calculated.
A price of the latter approach is however that the global flavor symmetry is
reduced to $\mathrm U(1)^{N_f}_L\times \mathrm U(1)^{N_f}_R$.}
D8 and anti-D8 branes filling all spatial directions except the $x_4$ (Fig.~\ref{figD8}a).
This introduces chiral quarks and anti-quarks localized at different points
of the Kaluza-Klein circle. The global flavor symmetry $\mathrm U(N_f)_L\times \mathrm U(N_f)_R$
of the field theory living on the boundary
corresponds to a local gauge symmetry on the flavor branes, which is broken
spontaneously to the diagonal subgroup because the D8 and anti-D8 branes have
nowhere to end except by joining in the bulk (Fig.~\ref{figD8}b).
\begin{figure}[h]
\centerline{\includegraphics[width=0.4\textwidth]{D4D8}\qquad
\includegraphics[width=0.35\textwidth]{d8brane}}
\centerline{\small \hfil (a) \hfil\hfil (b) \hfil}
\medskip
\caption{(a) D4-D8-$\overline{\mbox{D8}}$ brane configuration in the 10 dimensions $x_0,\ldots,x_9$,
with $x_4$ understood as periodic; (b) near-horizon geometry with $u$ being a radial coordinate in
the transverse space $x_{5\ldots 9}$.}
\label{figD8}
\end{figure}
The spectrum of fluctuations of the flavor gauge fields on the D8 branes include
a zero mode for the fifth component $A_u(x^\mu,u)$ whose gauge-invariant holonomy
represents the Goldstone bosons of the spontaneously broken $\mathrm U_A(N_f)$:
\begin{equation}
U(x^\mu)={\rm P}\,\exp\left\{ i \int_{-\infty}^\infty dZ' A_Z(x^\mu,Z)\right\}\equiv
\exp\left\{ 2i\pi(x^\mu)/f_\pi \right\} \in \mathrm U(N_f),
\end{equation}
where we have switched to a new dimensionless
coordinate $Z$ defined by $(u/u_{\rm KK})^3=1+Z^2$
which runs from $-\infty$ to $+\infty$ as one follows the holographic direction
from the boundary into the bulk along the D8 brane and back along the anti-D8 brane.
Here we have assumed that the D8 and the anti-D8 brane are maximally separated
so that they connect at the tip of the cigar, $u=u_{\rm KK}$.
In massless QCD, the axial U(1)$_A$ symmetry is broken by an anomaly, which
is suppressed at large $N_c$. However, this effect has also a beautiful supergravity description
\cite{Barbon:2004dq} involving the Ramond-Ramond 1-form whose flux through the
surface parametrized by $x_4$ and $u$ is related to the $\theta$ parameter.
The mass of the $\eta'$ pseudoscalar meson is related by the Witten-Veneziano
formula to the topological susceptibility, and was calculated by
Sakai and Sugimoto \cite{Sakai:2004cn} as\footnote{Note that Ref.~\cite{Sakai:2004cn}
uses $g_{\rm YM}$ and that our $g^2=2g_{\rm YM}^2$.}
\begin{equation}\label{metaprime}
m_{\eta'}=\frac1{3\sqrt3 \pi}\sqrt{\frac{N_f}{N_c}}(g^2 N_c/2)M_{\rm KK}.
\end{equation}
To leading order in $N_c$ and $\alpha'$, the D8-brane action is given explicitly by
\begin{equation}
S_{\rm D8}=-\kappa\int d^4x\,dZ\, {\rm Tr}\,\left[
\frac12 (1+Z^2)^{-1/3}F_{\mu\nu}^2+(1+Z^2)M_{\rm KK}^2 F_{\mu Z}^2+O(\alpha'{}^2 F^4)
\right]+S_{\rm CS},
\quad\kappa=\frac{(g^2N_c/2)N_c}{216\pi^3},
\end{equation}
where all fields have been taken as constant with respect to the $S^4$ whose
volume has been integrated over.
$S_{\rm CS}$ is the unavoidable Chern-Simons action of D-branes which here
can be shown to reproduce the correct chiral anomaly of QCD, and also
the correct Wess-Zumino-Witten term in the chiral Lagrangian \cite{Sakai:2004cn}. The well-known
5-dimensional formulation of the latter thus appears in a most natural way;
background gauge fields for the chiral symmetry can be introduced as
nontrivial boundary conditions $A^{L,R}_\mu(x)=\lim_{Z\to\pm\infty} A_\mu(Z,x)$.
The Goldstone bosons and the $\eta'$ meson
are contained in $A_Z=\phi_0(Z)\pi(x)$ with $\phi_0\propto (1+Z^2)^{-1}$.
Requiring a canonical normalization of the kinetic term for the Goldstone bosons in
\begin{equation}\label{SD80}
S_{\rm D8}=\frac{f_\pi^2}{4}\int d^4x\, {\rm Tr}\,\left( U^{-1}\partial _\mu U \right)^2+\ldots,
\end{equation}
fixes the so-called pion decay constant in terms of $\lambda$ and $M_{\rm KK}$ as
\begin{equation}
f_\pi^2=\frac1{54\pi^4}(g^2N_c/2)N_cM_{\rm KK}^2.
\end{equation}
Vector and axial vector mesons appear as even and odd
eigenmodes of $A_\mu^{(n)}=\psi_n(Z) v^{(n)}_\mu(x)$
with equation
\begin{equation}\label{psin}
-(1+Z^2)^{1/3}\partial _Z\left( (1+Z^2)\partial _Z \psi_n \right)=\lambda_n \psi_n,\quad
\psi_n(\pm\infty)=0.
\end{equation}
The lowest mode is interpreted as the vector meson $\rho$ with
mass $m_\rho^2=\lambda_1 M_{\rm KK}^2$ and $\lambda_1=0.669$.
The next mode with $\lambda_2=1.569$ is an axial vector meson that
can be identified \cite{Sakai:2004cn} with the $a_1(1260)$.
The experimental value $m^2_{a_1(1260)}/m^2_\rho\approx 2.52$ is
surprisingly close to $\lambda_2/\lambda_1\approx 2.344$.
Also the ratio of the squared mass of the next vector meson, $\rho(1450)$,
to that of the $\rho$ meson, $m^2_{\rho(1450)}/m^2_\rho\approx 3.57$ compares
quite well with $\lambda_3/\lambda_1\approx 4.29$. However, this
nice agreement may be fortuitous. Recent lattice simulations of the spectrum
of mesons in large-$N_c$ QCD extrapolated to
zero quark masses have obtained larger values \cite{Bali:2013kia}:
$m^2_{a_1}/m^2_{\rho}\approx 3.46$ and $m^2_{\rho^*}/m^2_{\rho}\approx 5.77$.
This would corresond to an error of 21\% and 16\%, respectively, for the
mass ratios (unsquared). Given that the Sakai-Sugimoto model is bound to
show deviations from QCD above the scale of $M_{\rm KK}\approx 1.2
\,m_\rho$, this
may still be taken as a notable success.
In Ref.~\cite{Sakai:2004cn,Sakai:2005yt} the physical values of
$m_\rho\approx 776$~MeV and $f_\pi\approx 92.4$~MeV have been used to set the free parameters
\begin{equation}\label{kappaMSS}
M_{\rm KK}=949\,{\rm MeV}, \qquad \kappa=7.45\cdot 10^{-3} \;\Rightarrow\; \bar\lambda} %{\lambda_{\rm SS}\equiv g^2N_c/2\approx 16.63,
\end{equation}
($\bar\lambda} %{\lambda_{\rm SS}=g_{\rm YM}^2 N_c$ is the 't Hooft coupling
as used in Ref.~\cite{Sakai:2004cn,Sakai:2005yt}; cf.~footnote \ref{fng}), and these values have been used widely since.
At $N_c=3$, this amounts to $\alpha_s(M_{\rm KK})\approx 0.88$, which is of the
order of magnitude to expect from perturbative QCD at such low scales, albeit somewhat
high when compared to the particular values obtained in the $\overline{\rm MS}$-scheme,
$\bar\alpha_s(1\,{\rm GeV})\approx 0.5$. But one should keep in mind that
$\bar\alpha_s$ grows sharply at 1~GeV with large renormalization scheme dependence.
A better quantity to consider is the renormalization-scheme independent string tension.
The holographic result (\ref{sigmastring}) depends on
both $M_{\rm KK}$ and $\bar\lambda} %{\lambda_{\rm SS}$, which yields with the above choice (\ref{kappaMSS})
$\sqrt{\sigma}=0.6262M_{\rm KK}\approx 594$~MeV.
This is higher than the value $\sqrt{\sigma}\sim 440$~MeV extracted from
lattice calculations \cite{Teper:1997am}, also suggesting
that $\bar\alpha_s$ is somewhat too large,
since in (\ref{sigmastring}) $\sigma\propto\alpha_s$.
Another quantity of immediate interest which depends on both $M_{\rm KK}$ and $\bar\lambda} %{\lambda_{\rm SS}$
is the gluon condensate.
(The glueball spectrum, which only depends on $M_{\rm KK}$,
will be revisited in the next section.)
Inserting the parameters (\ref{kappaMSS}) and $N_c=3$
yields
\begin{equation}\label{C4SS}
C^4=0.0126\,{\rm GeV}^4\quad (\bar\lambda} %{\lambda_{\rm SS}=16.63),
\end{equation}
which happens to almost coincide with
the standard SVZ sum rule \cite{Shifman:1978bx} value $C^4=0.012\,{\rm GeV}^4$ in QCD.
However, both significantly smaller \cite{Ioffe:2005ym,Samsonov:2004zm} and larger
values \cite{Narison:2011xe} have been obtained using sum rules. Lattice simulations
typically give significantly larger values, but of the same size as ambiguities from
the subtraction procedure \cite{Bali:2014sja}. Thus it appears that the gluon
condensate is not very suited to really
determine the 't Hooft coupling in the Witten(-Sakai-Sugimoto)
model. A moderate value of the gluon condensate may however be considered
to be a prediction of this model.
Using the result for the string tension, one can also compare the prediction
for the dimensionless ratio
\begin{equation}
m_\rho/\sqrt{\sigma}\approx 1.306\quad (\bar\lambda} %{\lambda_{\rm SS}=16.63)
\end{equation}
with the large-$N_c$ lattice
result of Ref.~\cite{Bali:2013kia}, which reads 1.504(50).
This agrees within 15\%, and the fact that the holographic result is smaller
also suggests that the assumed value for $\bar\lambda} %{\lambda_{\rm SS}$ and thus $\sigma$ in the Sakai-Sugimoto model
is perhaps too large. Turning this around, and fitting the lattice result for $m_\rho/\sqrt{\sigma}$
would give $\bar\lambda} %{\lambda_{\rm SS}\approx 12.55(80)$ resulting in more moderate values
$\sqrt{\sigma}\approx 515$~MeV and $\alpha_s(M_{\rm KK})\approx 0.66$.
This also corresponds to a 13\% lower value of the pion decay constant, $f_\pi=80.3$~MeV, and one
might indeed argue in favor of a reduced $f_\pi$ because the Sakai-Sugimoto model is strictly chiral
(although, judging from the lattice results of Ref.~\cite{Colangelo:2010et}, $f_\pi$ should have a
reduction around 7\% for $N_c=3$,
whereas at large $N_c$ Ref.~\cite{Bali:2013kia} found only a 2\% reduction).
At any rate, a downward variation of $\bar\lambda} %{\lambda_{\rm SS}$ from 16.63, say to 12.55, should give a reasonable
theoretical error bar for quantitative predictions of the Witten-Sakai-Sugimoto model.
In Ref.~\cite{Imoto:2010ef}, Imoto, Sakai, and Sugimoto have more recently studied the predictions
of the Sakai-Sugimoto model for other mesons, which are not covered by the flavor gauge fields
on the D8-branes. The latter correspond to massless open string modes
and only give the Goldstone bosons and a tower of $1^{--}$ and $1^{++}$ mesons.%
\footnote{Originally, also excitations of the brane embedding function were considered in Ref.~\cite{Sakai:2004cn}
and interpreted as a tower of $0^{++}$ and $0^{--}$ scalar mesons. These are, however, odd under
the so-called \cite{Brower:2000rp} $\tau$-parity, and should be discarded since
the quarks and gluon are invariant under this discrete symmetry \cite{Imoto:2010ef}.
It is therefore not disturbing that the prediction for the mass of the $a_0(1450)$ meson in the original paper
of Sakai and Sugimoto does not agree well with the recent large-$N_c$ lattice
result of Ref.~\cite{Bali:2013kia}.}
Despite the fact that gauge/gravity duality works in the decoupling limit corresponding to
$\ell_s\to0$, the masses of the massive string modes remain finite, and yield a host of
mesonic states.
The spectrum of those further states
is very difficult to evaluate precisely because of the
curved background. Ref.~\cite{Imoto:2010ef} succeeded, however, in calculating corrections
to the Regge trajectory of the $\rho$-meson and could reproduce the required curvature
of the trajectory around zero mass to match with a Regge intercept $\alpha_0=1$.
In order to match the experimental values of the meson masses
on the $\rho$-trajectory, Ref.~\cite{Imoto:2010ef} again observed that the model needs
a lower value of the string tension than given by the standard choice of parameters.
Their fit even points to a rather low value of $\sqrt{\sigma}=380$~MeV. Curiously enough, also
the large-$N_c$ lattice study of Ref.~\cite{Bali:2013kia} found a reduced value of $\sqrt{\sigma}$, in their case $395$~MeV,
to give a better fit than $440$~MeV.
The Sakai-Sugimoto model is not limited to the meson sector of low-energy QCD.
The D8-brane action (\ref{SD80}), restricted to the Goldstone bosons and integrated over the
holographic coordinate, reads, to leading order in 't Hooft coupling and $N_c$:
\begin{equation}\label{SD8skyrme}
S_{\rm D8}=\int d^4x\left\{ \frac{f_\pi^2}{4}{\rm Tr}\,\left( U^{-1}\partial _\mu U \right)^2+
\frac{1}{32e^2}{\rm Tr}\,[U^{-1}\partial _\mu U,U^{-1}\partial _\nu U]^2 \right\},\quad e^{-2}=\frac{15.253}{27\pi^7}
g^2 N_c^2,
\end{equation}
with $f_\pi^2\propto g^2 N_c^2$ given already in (\ref{SD80}). This coincides with the Skyrme
model \cite{Schechter:1999hg} with a definite prediction of the Skyrme parameter $e$
and so allows baryons to appear as solitons of
the chiral Lagrangian. In the bulk, the corresponding object is a holographic baryon vertex
\cite{Witten:1998xy} in the form of additional D4-branes wrapping the $S^4$ in the 10-dimensional background geometry.
Because of the RR flux on the $S^4$ the D4-brane is connected to the D8 branes by $N_c$ open strings,
which pull it inside the D8 branes where it appears as an instanton configuration, realizing
a connection between instantons in 5-dimensional gauge theory and the Skyrmion recognized
prior to the Sakai-Sugimoto model in Ref.~\cite{Son:2003et}.
A full-fledged treatment of baryons in the Sakai-Sugimoto model
has proved to be very interesting but difficult \cite{Hong:2007kx,Hata:2007mb,Hashimoto:2008zw,Seki:2008mu},
suggesting that string corrections should be taken into account.
Considering the baryon mass, it should be given to leading order by the mass of a
D4-brane at the tip of the D8 brane configuration, which is \cite{Sakai:2004cn}
$m_{\rm D4}=8\pi^2\kappaM_{\rm KK}=\frac1{27\pi}(g^2 N_c/2)N_cM_{\rm KK}.$
In \cite{Hong:2007kx,Hata:2007mb} the correction coming from the U(1)$_V$-field induced on
the D8-brane and leading to a contribution from $S_{\rm CS}$ has been calculated with the result
\cite{Hong:2007kx,Hata:2007mb}
\begin{equation}
M_B=\left[\frac1{27\pi}(g^2 N_c/2)+\sqrt{\frac{2}{15}} + O((g^2N)^{-1})\right] N_cM_{\rm KK}.
\end{equation}
With the standard parameter choice this corresponds to $(558+1040=1598)$~MeV, which is too high
when compared to real nucleons. But the lack of convergence (the correction term is about
twice the leading term) seems to indicate that an expansion about a pointlike instanton
is problematic, at least for the 't Hooft coupling required to make contact with QCD.
Curiously enough,
if one just considers the effective Skyrme action (\ref{SD8skyrme})
and evaluates the well-know result for the energy of a Skyrmion \cite{Adkins:1983ya,Schechter:1999hg},
$E\approx 23.2\pi f_\pi/e$,
with the above parameters, one obtains $M_N\approx 920$~MeV,
tantalizingly close to the real-world value. This may be taken as an indication that
one need not lower the scale $M_{\rm KK}$ to make contact with QCD (as suggested in \cite{Hata:2007mb}), but should rather
aim at a different approximation scheme for holographic baryons.
As another extreme we can also consider the mass of the $\eta'$-meson given in (\ref{metaprime}),
which is proportional to $\sqrt{N_f/N_c}$
and thus parametrically suppressed at large $N_c$. However, extrapolating to
$N_c=3$ and inserting again
the above parameters for $\bar\lambda} %{\lambda_{\rm SS}$ and $M_{\rm KK}$ this yields
$m_{\eta'}\approx 967$~MeV for $N_f=3$, astonishingly close to the experimental value of 958~MeV.
With the alternative choice $\bar\lambda} %{\lambda_{\rm SS}=12.55$, the lower value of 730~MeV is obtained, which is
seemingly disfavored, but may actually be more appropriate because of the chiral limit
and the missing strange quark mass.
In Table \ref{tabparams}, a summary of the numerical predictions
are given, including for completeness a variation of the Sakai-Sugimoto model
where the D8 branes are not maximally separated \cite{Antonyan:2006vw,Aharony:2006da}.
They then join at
a point $u_0>u_{\rm KK}$, and one may interpret the difference between $u_0$ and $u_{\rm KK}$ as
a constituent quark mass, given by the mass of a string stretched from $u_0$ ot $u_{\rm KK}$. In Ref.~\cite{Callebaut:2011ab} the latter has been
set to 310~MeV, which after matching the $\rho$ meson mass leads to a significantly smaller
value of $M_{\rm KK}$ and a slighter smaller value of $\bar\lambda} %{\lambda_{\rm SS}$. As remarked in Ref.~\cite{Callebaut:2011ab},
this leads to a better match of the string tension as observed on the lattice. On the other
hand, the gluon condensate becomes uncomfortably small. Also the other meson
masses fit somewhat less well.
To wrap up this numerological discussion,
the original choice of parameters of the Sakai-Sugimoto model
seems to be in quite good shape, but one may want to consider also
smaller 't Hooft couplings in view of the somewhat high string tension.
At any rate, the fact that most of the quantities considered here
come out in the right ballpark is certainly remarkable given that this model has a
cutoff scale of $M_{\rm KK}\sim 1$~GeV. A reason may be that this cutoff is very smooth in comparison
with a sharp momentum or lattice cutoff
and that it also fully respects 3+1-dimensional Lorentz symmetry.
\begin{table}
\centering
\caption{Choice of parameters of the Witten-Sakai-Sugimoto model and resulting predictions in 3 variants: (i) Original choice of Sakai and Sugimoto
\cite{Sakai:2005yt} which fits $M_{\rm KK}$ to the experimental value of $m_\rho$ and
the 't Hooft coupling $\bar\lambda} %{\lambda_{\rm SS}$ through the experimental value of $f_\pi$; (ii) same, except that
$\bar\lambda} %{\lambda_{\rm SS}$ is fitted such as to reproduce the large-$N_c$ lattice result of
Ref.~\cite{Bali:2013kia} for $m_\rho/\sqrt{\sigma}\approx 1.504$; (iii) non-antipodal D8 brane configuration with $u_0/u_{\rm KK}\approx 1.38$ such as to have a constituent quark mass of $m_q^{\rm constit.}=310$~MeV \cite{Callebaut:2011ab}. (Starred numbers are directly fixed by the input parameters.)
\label{tabparams}}
\begin{tabular}{l|lll}
~& (i) antipodal \cite{Sakai:2005yt}& (ii) antipodal & (iii) non-antipodal \cite{Callebaut:2011ab}\\
fitted to: & ($m_\rho,f_\pi$) & ($m_\rho,m_\rho/\sqrt{\sigma}$) & ($m_\rho,f_\pi,m_q^{\rm constit.}$) \\
$M_{\rm KK}$ [MeV] & 949 & 949 & 720.1 \\
$\bar\lambda} %{\lambda_{\rm SS}$ ($=6\pi\alpha_s$) & 16.63 & 12.55 & 15.13 \\
\hline
$f_\pi$ [MeV] & 92.4$^*$ & 80.3 & 93$^*$ \\
$m_\rho$ [MeV] & 776$^*$ & 776$^*$ & 776$^*$ \\
$\sqrt{\sigma}$ [MeV] & 594 & 515$^*$ & 430 \\
$C^4$ [GeV$^4$] & 0.0126 & 0.0072 & 0.0035 \\
$m_{\eta'}|_{N_f=3}$ [MeV] & 967 & 730 & 667 \\
$m_{a_1(1260)}$ [MeV] & 1188 & 1188 & 1146 \\
$M_G$ [MeV] & 855 & 855 & 649 \\
$M_{D,T}$ [MeV] & 1487 & 1487 & 1128 \\
\hline
\end{tabular}
\bigskip
\end{table}
\begin{table}
\centering
\caption{Holographic glueball masses ($M_{\rm KK}=949$~MeV). The glueball candidate
corresponding to the ``exotic'' polarization of the metric field is marked by ``(E)''.}
\label{tab-1}
\begin{tabular}{l|lr}
$J^{PC}$ & $M/M_{\rm KK}$
& $M$[MeV] \\\hline
$0^{++}$ (E)& 0.90113 & 855 \\
$0^{++}$-$2^{++}$ & 1.56690 & 1487 \\
$0^{-+}$ & 1.88518 & 1789 \\
$0^{*++}$ (E)& 2.28487 & 2168 \\
$1^{+-}$ & 2.43529 & 2311 \\
$0^{*++}$-$2^{*++}$ & 2.48514 & 2358 \\
$0^{*-+}$ & 2.83783 & 2693 \\
$1^{--}$ & 3.03763 & 2883 \\
\hline
\end{tabular}
\bigskip
\end{table}
\begin{table}
\centering
\caption{The masses of isotriplet mesons given by the modes
of the flavor gauge field on the (maximally separated) D8 branes, compared to the experimental value $(m/m_\rho)^{\rm exp.}$ from [PDG] ($\rho(770),a_1(1260),\rho(1450),a_1(1640)$) and to the large-$N_c$ results from Ref.~\cite{Bali:2013kia}.}
\label{tab-2}
\begin{tabular}{l|llll}
Isotriplet Meson & $\lambda_n=m^2/M_{\rm KK}^2$ & $m/m_\rho$ & $(m/m_\rho)^{\rm exp.}$ & $m/m_\rho$ \cite{Bali:2013kia} \\\hline
$0^{-+}$ ($\pi$)& 0 & 0 & 0.174|0.180 & 0 \\
$1^{--}$ ($\rho$)& 0.669314 & 1 &1 & 1\\
$1^{++}$ ($a_1$)& 1.568766 & 1.531 & 1.59(5) & 1.86(2) \\
$1^{--}$ ($\rho^*$)& 2.874323 & 2.072 & 1.89(3) & 2.40(4) \\
$1^{++}$ ($a_1^*$)& 4.546104 & 2.606 & 2.12(3) & 2.98(5) \\
\hline
\end{tabular}
\end{table}
\section{Glueball spectrum and glueball decay rates}
With the above successes as motivation, we now come back to the glueball spectrum
as obtained already in 2000 in Ref.~\cite{Brower:2000rp} and displayed in Fig.~\ref{figGB}a.
Here the scale is determined by $M_{\rm KK}$ alone. With $M_{\rm KK}=949$~MeV, the lowest
scalar mode (lowest green line in Fig.~\ref{figGB}a) turns out ot have a mass of $M_G\approx 855$~MeV, only marginally heavier than the $\rho$ meson.
Lattice simulations in QCD give values around 1.7~GeV, a factor of 2 higher.
This mismatch also persists when one compares with the large-$N_c$ lattice results of \cite{Lucini:2010nv}:
$M_G/\sqrt{\sigma}$ is $\approx 1.44$ with the parameters of the Sakai-Sugimoto model, while
Ref.~\cite{Lucini:2010nv} finds $\approx 2.73$ for the lowest scalar glueball (see Fig.~\ref{figGB}b,
where the dotted lines display the holographic results for the glueball spectrum normalized
to $\sqrt{\sigma}$).
The generalization
of the Sakai-Sugimoto model, where the D8 branes are not maximally separated, does not help
in this respect. On the contrary, it increases $m_\rho/M_{\rm KK}$ while keeping $M_G/M_{\rm KK}$
fixed. (See Table \ref{tabparams}: For the choice of parameters of Ref.~\cite{Callebaut:2011ab}
for this generalized Sakai-Sugimoto model, $M_G$ is already below $m_\rho$.)
As already discussed, while the glueball spectrum in the Witten model resembles
the spectrum as found on the lattice, it also has some notable qualitative differences
such as a certain proliferation of scalar glueballs $0^{++}$. Moreover, the lowest holographic
glueball comes from an ``exotic'' $g_{44}$ polarization of the metric fluctuation
(while also involving other metric components as well as the dilaton).
The next higher scalar glueball, which has the same mass as the lowest tensor glueball,
does not involve a $g_{44}$ fluctuation, and could thus be viewed as essentially a dilaton fluctuation,
which in simpler bottom-up models \cite{BoschiFilho:2002ta,Colangelo:2007pt,Forkel:2007ru}
is the only way to model a scalar glueball.
The mass of the latter is $M_D=M_T\approx 1.567M_{\rm KK}\approx 1487$~MeV, which matches reasonably
well with the lowest scalar glueball on the lattice (albeit not for the tensor glueball which
on the lattice is around 2.4--2.6 GeV).
An important open question, which is difficult to address by lattice simulations, is the decay width of
glueballs. In the Witten model, holographic glueballs are stable modes, and they remain so when
one introduces chiral quarks through probe D8 branes. However, as pointed out by Ref.~\cite{Hashimoto:2007ze},
the metric fluctuations dual to glueballs couple to the modes on the D8 brane representing
pions and vector mesons. Ref.~\cite{Hashimoto:2007ze} has considered the decay rate of the lowest
(``exotic'') scalar mode, which was revisited and extended recently by Br\"unner, Parganlija and the
present author \cite{Brunner:2014lya,BPR}.
In order to see whether the Sakai-Sugimoto model has any chance to predict glueball decay rates,
let us begin with considering the decay of the $\rho$ meson first.
The flavor gauge field modes on the D8 brane are all stable,
but the D8-brane action determines all the couplings between different mesons
without further free parameters. To leading order they have been worked out completely in
Ref.~\cite{Sakai:2005yt}, but only a few (particularly interesting and challenging) decay processes have been discussed
in the literature.
It is instructive to consider one of the simplest cases, the decay of the $\rho$ meson, to see whether
one may hope for quantitatively significant predictions. This was done already in Ref.~\cite{Hashimoto:2007ze},
but unfortunately using parameters affected by an error in the normalization of the D8-brane action
present in the original and published version of Ref.~\cite{Sakai:2004cn,Sakai:2005yt} for the multi-flavor case $N_f>1$
(corrected however in the final e-print version).
Identifying the mode $n=1$ in (\ref{psin}) with the $\rho$ meson, $A_\mu^{(1)}=\psi_1(Z) v^{(1)}_\mu(x)\equiv
\psi_1(Z) \rho_\mu(x)$
and ensuring
that $\rho(x)$ is a canonically normalized vector field yields
\begin{equation}
\mathcal L_{\rho\pi\pi}=-g_{\rho\pi\pi}
\epsilon_{abc}(\partial_\mu \pi^a)
\rho^{b\mu}\pi^c,\quad
g_{\rho\pi\pi}=\sqrt2
\int dZ \frac1{\pi K}\psi_1=\sqrt{2}\times 24.03
\,\bar\lambda} %{\lambda_{\rm SS}^{-\frac12} N_c^{-\frac12}.
\end{equation}
($g_{\rho\pi\pi}/\sqrt2$
was denoted as $c_6$ in \cite{Hashimoto:2007ze}.)
Using the original set of parameters (\ref{kappaMSS}) gives a decay rate of the $\rho$ meson
in two (massless) pions
\begin{equation}\label{Grhopipi}
\Gamma_\rho/m_\rho
\frac{g_{\rho\pi\pi}^2}{48\pi}\approx 7.659\, (\bar\lambda} %{\lambda_{\rm SS} N_c)^{-1}
\approx 0.1535
\end{equation}
which is 20\% lower than the experimental
value $\Gamma_\rho/m_\rho= 0.191(1)$ from the PDG \cite{Agashe:2014kda}.
As we have discussed above, the coupling $\bar\lambda} %{\lambda_{\rm SS}\approx 16.63$ appears to be
somewhat too large to fit the QCD string tension with the leading-order result (\ref{sigmastring})
and also the large-$N_c$ lattice result Ref.~\cite{Bali:2013kia} for $m_\rho/\sqrt{\sigma}$.
Using the latter as input
corresponds to $\bar\lambda} %{\lambda_{\rm SS}\approx 12.55(80)$ and gives an almost perfect prediction of $\Gamma_\rho/m_\rho=0.203(14)$
(if one ignores that $g_{\rho\pi\pi}$ should actually be even a bit larger because
(\ref{Grhopipi}) refers to massless pions).
As suggested in the previous section, we shall also consider the lower value of $12.55$ as an alternative choice of $\bar\lambda} %{\lambda_{\rm SS}$ in order to set up a theoretical error bar.
Turning now to glueball decay rates, one can proceed in analogy to the calculation of the $\rho$ meson decay into
pions \cite{Hashimoto:2007ze,Brunner:2014lya,BPR}. A metric fluctuation $\delta g_{mn}(x^\mu,Z)=G(x)H(Z)\epsilon_{mn}$ dual to a glueball mode needs to be
normalized such that the corresponding scalar, vector, or tensor field $G(x)$ is canonically normalized.
This fixes the amplitude of the mode function $H$ such that $H(Z=0)^{-1}\propto \bar\lambda} %{\lambda_{\rm SS}^{1/2}N_cM_{\rm KK}$.
Inserting the metric fluctuations in the D-brane action yields an effective action
for the couplings of glueballs to pions and the vector mesons.
The resulting general pattern is \cite{Hashimoto:2007ze}
\begin{equation}
\Gamma_{G\to2\pi}/M_G\propto \bar\lambda} %{\lambda_{\rm SS}^{-1}N_c^{-2},\quad
\Gamma_{G\to4\pi}/M_G\propto \bar\lambda} %{\lambda_{\rm SS}^{-3}N_c^{-4},
\end{equation}
so that glueballs are predicted as rather narrow states, with the branching ratio
into 4 pions being particularly strongly suppressed. A decay into four $\pi^0$ is even further
reduced \cite{BPR}:
$F^4$ terms in the DBI action of the D8 branes give vertices of a glueball with four $\pi^0$ such that
\begin{equation}
\Gamma_{G\to4\pi^0}/M_G\propto \bar\lambda} %{\lambda_{\rm SS}^{-7}N_c^{-4},
\end{equation}
while nonlinear terms of $G$ in the Yang-Mills part of the DBI action give rise to
\begin{equation}
\Gamma_{G\to G+2\pi^0\to4\pi^0}/M_G\propto \bar\lambda} %{\lambda_{\rm SS}^{-3}N_c^{-6},
\end{equation}
the latter with strong kinematical suppression from phase space integrals.
In Ref.~\cite{Hashimoto:2007ze}, the decay rates of the glueball corresponding to the lowest scalar mode
has been calculated, which, as discussed above, comes out with a mass $M_G\approx 0.9M_{\rm KK}$,
which is only half of what is expected
from lattice calculations, and which appears exotic in that it involves a metric polarization $g_{44}$.
In Ref.~\cite{Brunner:2014lya,BPR} we have considered also the scalar and tensor glueball associated with
transverse-traceless modes in AdS$_7\times S^4$, which provide glueballs of mass $M_G\approx 1.567M_{\rm KK}\approx
1487$~MeV and
thus not far from the lowest glueball in lattice simulations. The coupling of this glueball to two pions turns out to read
\begin{equation}
S_{G(1487)\to\pi\pi}=\int d^4x\frac{1}{2}\tilde c_1\partial_\mu\pi^a\partial_\nu\pi^a
\left(\eta^{\mu\nu}-\frac{\partial^\mu \partial^\nu}{M_G^2}\right)G,
\end{equation}
with $\tilde c_1=17.23 \bar\lambda} %{\lambda_{\rm SS}^{-1/2} N_c^{-1} M_{\rm KK}^{-1}$,
which yields
\begin{equation}
\frac{\Gamma_{G(1487)\rightarrow\pi\pi}}{M_G}=\frac{3|\tilde c_1|^2 M_G^2}{512\pi}=\frac{1.359}{\bar\lambda} %{\lambda_{\rm SS} N_c^2}
=\left\{ 0.009 \atop 0.012 \right. \;\mbox{for}\; \bar\lambda} %{\lambda_{\rm SS}=
\left\{ 16.63 \atop 12.55 \right..
\end{equation}
This result is significantly smaller than the relative width obtained for the lowest ``exotic'' scalar mode
(which is of the same parametric order but numerically has a larger coupling\footnote{The relative width has been obtained as 0.040 for $\bar\lambda} %{\lambda_{\rm SS}=16.63$ in Ref.~\cite{Hashimoto:2007ze}, but according to
\cite{BPR} the effective action derived in Ref.~\cite{Hashimoto:2007ze} is incomplete, and
the resulting relative width of the lowest ``exotic'' scalar mode is even larger, namely 0.092.}),
suggesting that the scalar glueball with $M=1487$~MeV does not behave like an excited scalar, which
one would expect to have a larger decay width. This adds to the suspicion that the lowest ``exotic'' scalar
might have to be discarded.
The prediction of the Witten-Sakai-Sugimoto model is then a scalar glueball of roughly the mass
found in lattice calculations and with a very narrow width. This should be contrasted with
the findings of Ref.~\cite{Janowski:2014ppa}, where an extended linear sigma model with a dilaton describing a narrow glueball
could only be accommodated with an uncomfortably large gluon condensate. In the Witten-Sakai-Sugimoto model
a narrow glueball seems to be perfectly compatible with a comparatively small gluon condensate (\ref{C4SS}).
\section{Deconfinement, chiral symmetry restoration, and the effects of strong magnetic fields}
\subsection{Deconfinement transition to black D4 phase and chiral symmetry restoration}
The Witten-Sakai-Sugimoto model can be extended to finite temperature $T$ by compactifying
also imaginary time $i x^0$ with a period $\beta=1/T$. Periodic boundary conditions for bosons and
antiperiodic boundary conditions for fermions give the correct statistics for the thermodynamic
partition function, which breaks supersymmetry spontaneously exactly as the spatial
circle in $x_4$ does. For low temperature (large $\beta$), the gravitational background
is just (\ref{ds2W}) with imaginary time and $i x^0\simeq ix^0+\beta$.
As the temperature increases,
the thermal circle shrinks and a phase transition happens when the circumference of the thermal circle
becomes equal to that of the $x_4$-circle, because at temperatures higher than that it is energetically favorable
that the Euclidean black hole with its cigar topology in the $x_4$-$u$ subspace exchanged with the cylindrical
topology in the $ix^0$-$u$ subspace \cite{Aharony:2006da}.
The phase transition temperature is simply given by
\begin{equation}\label{Tcdec}
\beta=L_4\equiv 2\pi R_4 \Leftrightarrow T_c=\frac1{2\pi}M_{\rm KK}.
\end{equation}
This corresponds to the appearance of an actual black hole (more precisely
a black 4-brane) in the real-time geometry, analogous to a Hawking-Page transition \cite{Witten:1998zw}.
Since with this swapping of the roles of $x_4$ and $ix^0$, the blackening
function $f(u)$ now appears in $g_{tt}$, with $u_{\rm KK}$ replaced by $u_T=(4\pi T/3)^2 R^3$, and the string tension (\ref{sigmastring}) suddenly
becomes zero, signalling deconfinement.
Unfortunately, this deconfinement transition is not smoothly connected to the deconfinement transition
in the 3+1-dimensional Yang-Mills theory and we shall mention an alternative proposal for a more adequate
holographic model below. The deconfined phase corresponding to the black 4-brane is nevertheless interesting, because
it gives rise to a nontrivial pattern of chiral symmetry restoration if one relaxes the original
setup of antipodal D8-$\overline{\mbox{D8}}$-branes \cite{Aharony:2006da}. When the latter are maximally separated,
as in the original Sakai-Sugimoto model, chiral symmetry is necessarily restored when the cigar topology
in the $u$-$x_4$ is replaced by a cylindrical topology. The connection between the D8-$\overline{\mbox{D8}}$-branes
is broken and their configuration becomes that of simple straight embeddings down to $u=u_T$, the black brane horizon.
However, with nonmaximal separation, the D8-$\overline{\mbox{D8}}$-branes can connect also within the cylindrical
topology, and it depends on the free energy encoded in the D8-action which configuration is favored.
It turns out that for brane separation $L<0.97 R_4$ there is still a chiral symmetry breaking phase
when the deconfined phase is entered. If one also introduces a chemical potential for quark number
in the form of an asymptotic value of the U(1)-component of $A^0$, the zeroth component of the flavor
gauge field, the phase diagrams takes the form displayed in Fig.~\ref{figTmu} \cite{Horigome:2006xu}.
\begin{figure}[h]
\centerline{\includegraphics[width=0.6\textwidth]{Tmu}}
\caption{Phase diagram of the Sakai-Sugimoto model where the deconfinement transition is
realized by a Hawking-Page transition. The topology of the $ix^0$-$u$ and $x_4$-$u$ subspaces
is sketched in red and blue, respectively, with the D8-$\overline{\mbox{D8}}$-brane configuration
drawn in black.}
\label{figTmu}
\end{figure}
This phase diagram has also been studied in the presence of baryonic matter represented by
sources at the tip of the D8 branes \cite{Bergman:2007wp}, and refinements
involving extended, non-point-like baryons have been studied in \cite{Kim:2007vd,Rozali:2007rx,Kaplunovsky:2012gb,Ghoroku:2012am}.
Also a very interesting attempt \cite{deBoer:2012ij} has been made to find a holographic realization of the so-called
quarkyonic phase \cite{McLerran:2007qj,Glozman:2007tv} within the confined phase.
\subsection{Deconfinement through a Gregory-Laflamme transition}
The first-order transition to the black 4-brane phase takes place at a temperature (\ref{Tcdec}) directly set by $M_{\rm KK}$:
$T_c=M_{\rm KK}/2\pi \approx 151\;{\rm MeV}$.
Superficially, this looks appealing as the (crossover) temperature for deconfinement and chiral symmetry
breaking of real QCD is indeed close to that value. However, when compared to the first-order phase transition temperature
for pure-glue (or quenched) QCD, it is uncomfortably small.
A more important issue with this deconfinement transition has been noted first in Ref.~\cite{Aharony:2006da}:
In the limit of small 't Hooft coupling and large $M_{\rm KK}$, which would be required to actually
reach 3+1-dimensional Yang-Mills theory, the critical temperature will have to occur at $T\sim\Lambda_{\rm QCD}\llM_{\rm KK}=R_4^{-1}$.
However, due to the symmetry under $\beta\leftrightarrow L_4\equiv 2\pi R_4$,
there will then also be a phase transition at
a temperature $T\gg R_4^{-1}$. The phase transition line given by (\ref{Tcdec}), if it connects to the
deconfinement of 3+1-dimensional Yang-Mills theory, needs to bifurcate at some value of $\lambda_5/R_4$;
it cannot smoothly connect.
In Ref.~\cite{Mandal:2011ws}
it has recently been pointed out that the $Z_{N_c}\times Z_{N_c}$ center symmetry for the
two circle compactifications along $ix^0$ and $x_4$ matches the phases of 3+1-dimensional Yang-Mills theory
only in the low-temperature configuration, but not in the high-temperature, black 4-brane geometry.
Hence, there definitely has to be a phase transition between the latter and actual deconfined 3+1-dimensional Yang-Mills theory.
The authors of Ref.~\cite{Mandal:2011ws} have argued that one should instead consider
a phase transition with periodic instead of antiperiodic boundary conditions for fermions on the thermal circle.
This is not appropriate for fermions, but the target of the Witten model is pure Yang-Mills theory without fermions
anyway. In this case, the symmetry $\beta\leftrightarrow L_4$ is avoided
and the gravity dual of the deconfinement
transition is a Gregory-Laflamme transition into the T-dual type IIB supergravity, where the D4 branes are replaced
by an inhomogeneous distribution of D3 branes.
Then the center symmetry breaking is such that it does not rule out a smooth connection to the deconfinement
transition in 3+1-dimensional Yang-Mills theory.
\begin{figure}[h]
\centerline{\includegraphics[width=\textwidth]{D4-phase}}
\caption
Deconfinement phase transitions with antiperiodic and periodic boundary conditions
along the thermal circle, with $W_0$ and $W_4$ referring to the expection values of Polyakov loops
in temporal and $x_4$ direction, respectively (taken from Ref.~\cite{Mandal:2011uq}).}
\label{figD4phase}
\end{figure}
Using prior work on the Gregory-Laflamme instability of branes compactified on a circle \cite{Harmark:2004ws},
Ref.~\cite{Mandal:2011ws} has obtained the following estimate for the first-order transition temperature:
\begin{equation}
T_{\rm GL}\simeq \frac{\bar\lambda} %{\lambda_{\rm SS}}{2\pi\cdot 8.54}M_{\rm KK}.
\end{equation}
Inserting our fit parameters, this gives
$T_{\rm GL}=294$ (222) MeV for $\bar\lambda} %{\lambda_{\rm SS}=16.63$ (12.55), indeed a quantatively more
appealing ballpark for the deconfinement temperature of quenched QCD than the
transition temperature (\ref{Tcdec}) with antiperiodic boundary conditions on the thermal circle.
For the fermions that are introduced through probe flavor branes, Ref.~\cite{Mandal:2011ws}
suggested to use an imaginary chemical potential to get the statistics right, but
the details of this alternative proposal for a deconfined Witten-Sakai-Sugimoto model
have still to be worked out.
\subsection{The effects of strong magnetic fields on chiral symmetry breaking and restoration}
While the simple deconfinement transition based on black 4-branes appears to be lacking important
features of the 3+1-dimensional deconfinement of nonsupersymmetric Yang-Mills theory, it may nevertheless
be interesting to study the chiral phase transition in the simple background of black 4-branes.
With nonmaximal separation of the D8 branes, the chiral phase transition is moved away from
the deconfinement transition. For sufficiently small D8 brane separation, the deconfinement temperature
can actually be sent to arbitarily small temperature \cite{Parnachev:2006dn}.
In this limit, the Sakai-Sugimoto model actually becomes dual to a nonlocal NJL model,
as has been pointed out in Ref.~\cite{Antonyan:2006vw}, and
the phase diagram in the $T$-$\mu$-plane is similar to that found in conventional NJL models.
With nonantipodal D8 branes, the phase diagram moreover becomes sensitive to external fields,
which makes this setup an interesting toy model for exploring possible phenomena related to
strong electromagnetic fields, in particular their impact on chiral symmetry breaking.
At zero baryon chemical potential, magnetic catalysis of chiral symmetry breaking
was shown to occur in the Sakai-Sugimoto model \cite{Johnson:2008vna,Bergman:2008sg},
in accordance with the field-theoretic analysis of Ref.~\cite{Gusynin:1994xp}.
At low temperature and high chemical potential, an opposite effect (``inverse
magnetic catalysis'') was observed
in Ref.~\cite{Preis:2010cq,Preis:2012fh} that has to do with an additional
phase transition \cite{Lifschytz:2009sz,Preis:2010cq} that can be interpreted as a transition into the lowest Landau level (LLL)
of chiral quarks (Fig.~\ref{fig3D}).\footnote{At zero chemical potential,
the magnetic field monotonically increases the critical temperature
for chiral symmetry restoration, signalling ordinary magnetic catalysis, in
contrast to the recent lattice results for QCD at physical quark masses \cite{Bali:2011qj}.} Indeed, a very similar phase diagram was previously found
in NJL models \cite{Ebert:1999ht,Inagaki:2003yi}. In contrast to the latter,
the holographic model does not show de Haas-van Alphen oscillations, which indicates that
in the holographic model there is no sharp Fermi surface (see also Refs.~\cite{Kulaxizi:2008jx,DiNunno:2014bxa}).
An interesting feature, shared with real QCD \cite{Son:2007ny}, is that in the chirally broken
phase at finite chemical potential
and with a background magnetic field, baryon number is generated through the axial
anomaly by gradients of the pion field.
On the holographic side, this is brought about by the Chern-Simons part of
the D8 brane action \cite{Thompson:2008qw,Bergman:2008qv,Rebhan:2008ur}.\footnote{There is
however a still unresolved issue with boundary terms that need to be added to
the action to insure thermodynamic consistency, which comes at the price of
modifying the anomaly content \cite{Bergman:2008qv,Rebhan:2008ur,Rebhan:2009vc}.}
\begin{figure}[htb]
\centering
\sidecaption
\includegraphics[width=7.35cm,clip]{3DPlot}
\caption{Phase diagram of the Sakai-Sugimoto model in (dimensionless) temperature ($t$), quark chemical potential ($\mu$), and magnetic field strength at small D8 brane separation
such that the deconfinement transition is at $t\approx 0$, without baryonic matter
from D4 branes \cite{Preis:2010cq} (the effects of including the latter
have been discussed in \cite{Preis:2011sp}). At small $\mu$, the critical temperature
for chiral symmetry restoration rises with $b$ (normal magnetic catalysis), whereas
at low $T$, the dependence is nonmonotonic. Beyond a critical temperature, there
is also a magnetic phase transition (green surface) into a ``lowest Landau-level'' phase.}
\label{fig3D}
\end{figure}
Another consequence of the axial anomaly is the appearance of anomalous nondissipative
transport phenomena. In the Sakai-Sugimoto model at finite chemical potential and nonzero magnetic field, the chirally
symmetric phase carries an axial current $\mathbf j_A \propto \mu \mathbf B$ in accordance with the
field-theoretic result obtained in Ref.~\cite{Metlitski:2005pr,Newman:2005as}.
A related effect is the so-called
chiral magnetic effect \cite{Fukushima:2008xe,Kharzeev:2013ffa} that has attracted a lot of attention because
it may be studied in heavy-ion collisions where sufficiently strong magnetic fields
could arise in noncentral collisions. Here an imbalance between left and right chiral fermions,
described by an axial chemical potential, gives rise to a vector current and electric charge
separation through $\mathbf j_V \propto \mu_A \mathbf B$ as first noted in a different context in
Ref.~\cite{Vilenkin:1980fu} and independently rediscovered by various authors.
In the Sakai-Sugimoto model this effect was reproduced in Ref.~\cite{Yee:2009vw}, including
a calculation of the frequency dependent chiral conductivity, but
it involves some conceptual issues \cite{Rebhan:2009vc}
that were clarified only recently \cite{Gynther:2010ed,Jimenez-Alba:2014iia}.
\section{Conclusion}
The Witten-Sakai-Sugimoto model is certainly the best studied and most developed top-down holographic approach to
a nonsupersymmetric nonconformal Yang-Mills theory at strong coupling. As we have discussed, this model
reproduces many phenomena that are known to characterize low-energy QCD such as confinement, chiral symmetry breaking,
mesons including glueballs, baryons, and effects from the axial anomaly. Although this model
is actually a 4+1-dimensional super-Yang-Mills theory above a Kaluza-Klein scale that cannot
be made much larger than 1 GeV without leaving the supergravity approximation,
extending the results of these calculations to finite
$N_c=3$ and a finite 't Hooft coupling does give a surprisingly good fit of most of the quantities that
can be compared with experiment or controllable lattice calculations.
Parametrically, the predictions are in line with the expectations
from large-$N_c$ analysis, but, optimistically, they also give a rough estimate of the concrete magnitude of various effects.
For example, glueball decay rates are found to be numerically very small, which is an interesting prediction given
that the decay rates of known mesons are quite well reproduced by this model.
This may motivate the inclusion of this and related predictions in other, phenomenological models, if only
as a ballpark estimate. Similarly, the model may serve to give thought-provoking suggestions for the phase diagram
of QCD in regions which will hopefully be explored by first-principle lattice simulations. Indeed,
it seems that the notorious sign problem of lattice QCD at finite chemical potential could be overcome
in the future by new ingenious techniques \cite{Aarts:2014bwa}.
On the other hand, the Witten-Sakai-Sugimoto model, being also a first-principle approach, will hopefully also
see further developments. Important and ambitious directions would be the inclusion of backreaction of flavor branes,
corresponding to an unquenching of the quarks \cite{Burrington:2007qd,Erdmenger:2011bw,Bigazzi:2014qsa},
an improved description of the deconfined phase \cite{Mandal:2011uq}, the inclusion of finite quark masses
\cite{Bergman:2007pm,Dhar:2007bz,Hashimoto:2008sr}, or the inclusion of string-theoretic corrections.
\begin{acknowledgement}
I would like to thank Frederic Br\"unner, Denis Parganlija, Florian Preis, Andreas Schmitt, and Stefan Stricker for enjoyable collaborations on the Witten-Sakai-Sugimoto model as well as
Matthias Lutz, Marco Panero and Shigeki Sugimoto for discussions.
I am particularly grateful to Frederic Br\"unner, Denis Parganlija, and Florian Preis for a careful reading of this manuscript,
and to Andreas Schmitt for some of the illustrations.
\end{acknowledgement}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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package com.dell.doradus.search.analyzer;
import java.util.ArrayList;
import java.util.List;
/**
* Helps tokenize and search on numeric trie values
*
*/
public class NumericTrie {
//base of the trie. the number of sub-nodes in the trie nodes
//Values that are powers of the base, result in much fewer search terms,
//so chose it according to the type of the values.
//For example, for values like age, height, weight, temperature etc. chose bas=10,
//because query like age>30 is more probable then age>32.
//For values like file size chose bas=32, because queries like size > 1KB is more probable than
//size > 1000bytes.
public int bas;
//Lower bound (inclusive) of the range query min <= x < max.
//If the query is x < max then you should set min to be the minimal possible value that x can take.
//It is better though not required to chose it to be 0 or power of the base.
//The less the range min-max is, the less are clauses in the query, so try to make it as small as possible
//even better to take these values from the database if possible to build the query.
//for example, age is usually less than 110 years. So for the query age > 3, you can chose max=1000, provided that
//the base is 10 (because 1000 is the next power of 10 greater than 110).
public long min;
//Upper bound (exclusive) of the range query min <= x < max.
public long max;
public NumericTrie(int base) {
this(base, Long.MIN_VALUE, Long.MAX_VALUE);
}
public NumericTrie(int bas, long min, long max) {
this.bas = bas;
this.min = min;
this.max = max;
}
public List<String> tokenize(long num) {
List<String> tokens = new ArrayList<String>();
if(num < 0) tokenize(tokens, -num, true);
else tokenize(tokens, num, false);
return tokens;
}
public List<String> getSearchTerms() {
List<String> terms = new ArrayList<String>();
if(min < 0 && max >= 0) {
range(terms, 1, -min + 1, true);
range(terms, 0, max, false);
}
else if(min < 0 && max < 0) {
range(terms, -max + 1, -min + 1, true);
}
// min >= 0 && max >= 0: no other option
else {
range(terms, min, max, false);
}
return terms;
}
private void tokenize(List<String> terms, long num, boolean lessThanZero) {
long det = 1;
add(terms, det, num, lessThanZero);
while(num > 0) {
num /= bas;
det *= bas;
add(terms, det, num, lessThanZero);
}
}
private void range(List<String> terms, long a, long b, boolean lessThanZero) {
long x = a;
long det = 1;
while(x < b) {
if(x == 0) {
add(terms, 1, 0, false);
x = 1;
}
else if(x == det && x * bas <= b) {
det *= bas;
add(terms, det, 0, lessThanZero);
x *= bas;
}
else if(x % (bas * det) == 0 && x + det * bas <= b) {
det *= bas;
}
else if(x + det > b) {
det /= bas;
}
else {
add(terms, det, x/det, lessThanZero);
x += det;
}
}
}
private static void add(List<String> terms, long det, long num, boolean lessThanZero) {
StringBuilder sb = new StringBuilder();
if(det != 1) {
sb.append(det);
sb.append('/');
}
if(lessThanZero) sb.append('-');
sb.append(num);
terms.add(sb.toString());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,377
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Carnivores, Paleozoic animals, Ordovician Animals,
Silurian animals
Devonian Animals
Carboniferous Animals
Triassic animals
Mesozoic animals
Walking with Wikia
Walking with Monsters
Extant Organisms
Megalograptus
Brontoscorpio
Pterygotus
Mesothelae
Arachnids are a class (Arachnida) of joint-legged invertebrate animals in the subphylum Chelicerata. All arachnids have eight legs, although the front pair of legs in some species has converted to a sensory function, while in other species, different appendages can grow large enough to take on the appearance of extra pairs of legs. The term is derived from the Greek word ἀράχνη (aráchnē), meaning "spider".
Almost all extant arachnids are terrestrial. However, some inhabit freshwater environments and, with the exception of the pelagic zone, marine environments as well. They comprise over 100,000 named species, including spiders, scorpions, harvestmen, ticks, mites and Solifugae.
In Walking with... Series Edit
Sea Monsters Edit
Sea Monsters, Episode 1
This episode featured Megalograptus, a medium-sized sea scorpion. It was an intermediate predator, hunting trilobites, but prey itself for Cameroceras and similar giant mollusks - which was why it began to come on land to lay eggs and to escape predation.
Walking with Monsters Edit
Water Dwellers Edit
This episode featured Brontoscorpio, a distant relative to the modern scorpions, and Pterygotus, a giant sea scorpion.
These animals were shown to be the dominant predators of the Silurian period, preying on Cephalaspis and similar creatures, taking over the role of the giant mollusks.
This episode also shows a Devonian terrestrial scorpion (played by a modern scorpion) that was, instead, prey for the first terrestrial vertebrates, such as Hynerpeton.
Reptile's Beginnings Edit
The first half of this episode featured Mesothelae, a giant spider. It was shown preying on small reptiles like Petrolacosaurus, but it was too slow to outrun forest fires of the Carboniferous and too big to survive when the oxygen-rich world of the Carboniferous came to an end.
Walking with Dinosaurs Edit
New Blood Edit
The book version of this episode features a whip scorpion that got eaten by a Coelophysis.
Retrieved from "https://walkingwith.fandom.com/wiki/Arachnid?oldid=54216"
Paleozoic animals
Ordovician Animals
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,978
|
\section{Introduction}
Electron and exciton dynamics between chromophores are ubiquitously found---from photosynthetic complexes in biological organisms\cite{Amerongenbook} to organic semiconductors for solar energy conversion.\cite{Desiraju2013CR}
They are often modeled by quantum dynamics methods based on the quantum master equation,\cite{Nitzanbook}
which requires so-called model Hamiltonians in the diabatic representation as an input.
The model Hamiltonians are the reduced dimensional representation of the electronic degree of freedom in such systems,
consisting of the energies of the diabatic states and the interaction strengths between these states.
Although accurate model Hamiltonians are of essential importance in achieving predictive simulations of electron and exciton dynamics,
there have been only a few studies to develop accurate electronic structure methods to compute them from first principles.\cite{Difley2011JCTC,Kaduk2012CR,Subotnik2010JPCA, Alguire2014JPCA, Landry2014JCTC, Fink2008CP, Havenith2012MP, Wu2011ChemRev}
We have recently introduced the active space decomposition (ASD) method to provide accurate model Hamiltonians.
The ASD model takes advantage of molecular geometries to compress active-space wave functions of molecular dimers,\cite{Parker2013JCP,Parker2014JCTC,Parker2014JPCC,Parker2014JCP}
in which wave functions are parameterized as\cite{Parker2013JCP}
\begin{align}
|\Psi \rangle = \sum_{IJ} U_{IJ} |\Phi_I^A\rangle |\Phi_J^B\rangle, \label{eq:wf}
\end{align}
where $A$ and $B$ label monomers and $I$ and $J$ label monomer states.
The monomer wave functions ($|\Phi_I^A\rangle$ and $|\Phi_J^B\rangle$)
are determined by diagonalizing the respective monomer active space Hamiltonians (i.e., $\hat{H}^A | \Phi^A_I \rangle = E^A_I | \Phi^A_I \rangle$) in an orthogonal active subspace (hereafter referred to as an ASD subspace).
The expansion in Eq.~\eqref{eq:wf} is exact when each $I$ and $J$ entirely span the corresponding monomer space, and converges rapidly with respect to the number of monomer states included in the summation.\cite{Parker2013JCP}
An extension to more than two active subspaces has also been reported by the authors.\cite{Parker2014JCP}
Furthermore, the dimer basis states have well-defined charge, spin, and spin-projection quantum numbers on each monomer,
allowing us to extract model Hamiltonians for electron and exciton dynamics through diagonalization of diabatic subblocks of a dimer Hamiltonian matrix.\cite{Parker2014JCTC}
This methodology has been applied to singlet exciton fission dynamics in tetracene and pentacene crystals.\cite{Parker2014JPCC}
The resulting model Hamiltonians have been used to benchmark approximate methods in more recent studies.\cite{Alguire2015JPCA}
However, our ASD method was not applicable to covalently linked chromophores in previous works.
This was mainly due to the difficulty in finding appropriate ASD subspaces.
In the original formulation of ASD, one must define localized ASD subspaces from localized Hartree--Fock orbitals.\cite{Parker2014JPCC}
Although well defined (and even automated) for molecular dimers and aggregates, this procedure gives rise to an ambiguity in the definition of the ASD subspaces in covalently linked chromophores owing to orbital mixing.
This problem has hampered applications of ASD to these systems.
In this study, we remove this ambiguity in the subspace preparation by deriving and implementing orbital optimization algorithms for the ASD model.
The orbital optimization guarantees that (when converged) identical results are obtained regardless of the choice of initial active orbitals.
As shown in the following, the orbital optimization procedure naturally leads to localized ASD subspaces without deteriorating the diabatic structure of the dimer basis function in the original ASD method.
The preservation of locality of the ASD subspaces during orbital optimization is analogous to the fact that, for a fixed number of renormalized states, {\it ab initio} density matrix renormalization group algorithms using localized orbitals provide the lowest energies. \cite{Chan2011ARPC}
A quasi-second-order algorithm is employed,\cite{Chaban1997TCA} which is akin to conventional complete active space self-consistent field (CASSCF) and its occupation restricted variants (RASSCF).\cite{Olsen1988JCP}
In the following our orbital-optimized models are referred to as ASD-CASSCF and ASD-RASSCF depending on the underlying monomer active-space wave functions.
\section{Theory}
\subsection{Orbital optimization model\label{theorysec}}
In ASD the energy of a dimer state is a function of molecular orbital (MO) coefficients ($\mathbf{C}$), configuration interaction (CI) coefficients within each monomer ($c^X_D$ with $D$ labeling Slater determinants for $X=A$, $B$), and ASD coefficients [$U_{IJ}$ in Eq.~\eqref{eq:wf}], i.e.,
\begin{align}
E_\mathrm{ASD} = E(\mathbf{C}, c^A_D, c^B_{D'}, U_{IJ}).
\end{align}
We first form the monomer basis by solving configuration interaction within each ASD subspace
\begin{align}
&\hat{H}^A |\Phi_I^A \rangle = E^A_I |\Phi_I^A \rangle,\\
&\hat{H}^B |\Phi_J^B \rangle = E^B_J |\Phi_J^B \rangle,
\end{align}
where $|\Phi_I^A\rangle = \sum_D c_D^{A,I} |D \rangle$ and so on.
Using the monomer basis states, the dimer Hamiltonian matrix elements are computed as
\begin{align}
&\langle \Phi_I^A \Phi_J^B | \hat{H} | \Phi_{I'}^A \Phi_{J'}^B \rangle
= (-1)^\phi\sum_{\zeta\eta}\Gamma^{A,II'}_\zeta h_{\zeta,\eta} \Gamma^{B,JJ'}_\eta,\label{eq:H}
\end{align}
which can be evaluated without constructing the product basis functions explicitly.
Here the indices $\zeta$ and $\eta$ are operators acting on monomer $A$ and $B$, respectively, whose product corresponds to either one- or two-electron operator of the rearranged Hamiltonian; $h_{\zeta,\eta}$ is the molecular orbital integral;
and, $(-1)^\phi$ is a phase factor due to the rearrangement.
The monomer intermediates $\Gamma^{A,II'}_\zeta$ and $\Gamma^{B,JJ'}_\eta$ are defined by
\begin{align}
&\Gamma^{A,II'}_\zeta = \langle \Phi_I^A|\hat{E}_\zeta |\Phi_{I'}^A\rangle, \label{eq:gamma_A} \\
&\Gamma^{B,JJ'}_\eta = \langle \Phi_J^B|\hat{E}_\eta |\Phi_{J'}^B\rangle.\label{eq:gamma_B}
\end{align}
Diagonalization of the dimer Hamiltonian matrix gives ASD energies and wave functions.
The details of the algorithm are given elsewhere.\cite{Parker2013JCP}
We introduce a stationarity condition with respect to variations of MO coefficients to define orbital-optimized ASD models:
\begin{align}
\frac{\partial E_\mathrm{ASD}}{\partial {\kappa}_{rs}} = 0, \label{deriv}
\end{align}
where $r$ and $s$ label any molecular orbitals, and $\boldsymbol{\kappa}$ is the anti-symmetric matrix that parameterizes the MO coefficients,
\begin{align}
\mathbf{C} = \mathbf{C}_\mathrm{init} \exp(\boldsymbol{\kappa}).
\end{align}
The non-redundant elements of the rotation matrix $\boldsymbol{\kappa}$ are depicted in Fig.~\ref{fig:rotmat}.
\begin{figure}
\includegraphics[width=0.48\textwidth]{rotmat.pdf}
\caption{\label{fig:rotmat} Non-redundant elements of the rotation matrices $\kappa$ for CASSCF (left) and ASD-CASSCF (right). }
\end{figure}
The two-step optimization procedure (i.e., alternating solutions of ASD and orbital updates until convergence) gives a unique energy that is close to the variationally optimal one.
This quasi-variational behavior is explained as follows.
The total derivatives of the ASD energy with respect to variations of MO coefficients are
\begin{align}
\frac{dE_\mathrm{ASD}}{d {\kappa}_{rs}} &= \frac{\partial E_\mathrm{ASD}}{\partial {\kappa}_{rs}} + \sum_{IJ} \frac{\partial E_\mathrm{ASD}}{\partial U_{IJ}} \frac{d U_{IJ}}{d \kappa_{rs}} \nonumber\\
&+ \sum_{D,I} \frac{\partial E_\mathrm{ASD}}{\partial c^{A,I}_{D}} \frac{d c^{A,I}_{D}}{d \kappa_{rs}}
+ \sum_{D',J} \frac{\partial E_\mathrm{ASD}}{\partial c^{B,J}_{D'}} \frac{d c^{B,J}_{D'}}{d \kappa_{rs}},
\label{totalderiv}
\end{align}
where respective normalization conditions are implicit.
First, it follows from the ASD procedure that the stationary condition with respect to $U_{IJ}$,
\begin{align}
\frac{\partial E_\mathrm{ASD}}{\partial U_{IJ}} = 0, \label{uderiv}
\end{align}
is automatically satisfied.
Second, the following equations are approximately satisfied,
\begin{align}
\frac{\partial E_\mathrm{ASD}}{\partial c^A_D} \approx 0,\quad
\frac{\partial E_\mathrm{ASD}}{\partial c^B_{D'}} \approx 0.
\end{align}
The residuals of these derivatives are related to the differences between the ASD energies computed by lowest-energy monomer states and variationally optimal ones (with the same number of states in the summation),
which are expected to be much smaller than the energies themselves in the cases where ASD is a good approximation.
As a result, one obtains that the total derivative [Eq.~\eqref{totalderiv}] is approximately zero, making the orbitals almost variationally optimal.
In the cases where convergence is not achieved under this approximation,
we fix $c^A_D$ and $c^B_{D'}$ after several iterations (but not $U_{IJ}$);
when $c^A_D$ and $c^B_{D'}$ are fixed, the last two terms in Eq.~\eqref{totalderiv} that introduce numerical noise become exactly zero because
the derivatives of $c^A_D$ and $c^B_{D'}$ are zero, and hence, the minimum in this constrained frame
can be found by standard minimization algorithms using the partial derivative of energy with respect to orbital rotations [i.e., Eq.~\eqref{deriv}].
After the convergence ASD-CI is performed once again using the optimized orbitals.
The solutions are obtained by iterating this procedure until
consistency between orbitals and $c^A_D$ and $c^B_{D'}$ is achieved.
Lastly, we note that the orbital-optimized ASD energy becomes equivalent to the conventional CASSCF and RASSCF energy when all the charge and spin sectors are included in the ASD expansion.
In Sec.~\ref{results} we show that the energy converges rapidly with respect to the number of states in the ASD expansion.
\subsection{Orbital gradients and approximate Hessians}
In addition to orbital gradients with respect to rotations between closed, active, and virtual subspaces for which gradient elements are known,\cite{Chaban1997TCA}
we consider rotations among ASD subspaces $A$ and $B$.
The energy gradients between orbitals in active space $A$ (unbarred) and those in active space $B$ (barred) are
\begin{align}
& \frac{\partial E_\mathrm{ASD}}{\partial \kappa_{\bar{i}j}} = 2(F_{j\bar{i}}-F_{\bar{i}j}), \label{eq:grad} \\
& F_{\bar{i}j} = \sum_k \Gamma_{jk} F_{\bar{i}k}^I + \sum_{klm} \Gamma_{jk,lm} (\bar{i}k|lm),\\
& F_{\bar{i}j}^I = h_{\bar{i}j} + \sum_{o}[2(\bar{i}j|oo)-(\bar{i}o|oj)],
\end{align}
where $i$, $j$, $k$, $l$, and $m$ label active orbitals (the summation indices $k$, $l$, and $m$ run over orbitals on both $A$ and $B$), and $o$ labels closed orbitals.
$\Gamma_{ij}$ and $\Gamma_{ij,kl}$ are the standard spin-free one- and two-particle reduced density matrices, respectively.
In the quasi-second-order optimization algorithm\cite{Chaban1997TCA} approximate Hessian elements are required as an initial guess.
We use the following formula for approximate diagonal Hessian elements for the inter-subspace active--active rotations,
\begin{align}
\frac{\partial^2 E_\mathrm{ASD}}{\partial \kappa_{\bar{i}j} \partial \kappa_{\bar{i}j}} &\approx
2 [ \Gamma_{\bar{i}\bar{i}} ( F^I_{jj} + F^A_{jj} ) + \Gamma_{jj} ( F^I_{\bar{i}\bar{i}} + F^A_{\bar{i}\bar{i}} ) \nonumber \\
& -F_{\bar{i}\bar{i}} - F_{jj} - 2 \Gamma_{\bar{i}j} F^I_{\bar{i}j} ],
\end{align}
where $F^A_{ij} = \sum_{kl} \Gamma_{kl}[(ij|kl)-\frac{1}{2}(il|kj)]$.
This formula is derived using similar approximations to those in Ref.~\onlinecite{Chaban1997TCA}, in which
the two-electron integrals are approximated by the diagonal element of Fock-like one-electron operators and
active orbitals is approximated to be completely filled for some terms.
This procedure eliminates the needs for the computation of two-electron integrals with more than one active indices.
One- and two-particle reduced density matrices in the ASD model, $\Gamma_{ij}$ and $\Gamma_{ij,kl}$, are computed from
$\Gamma_\zeta$ intermediate tensors as using Eqs.~\eqref{eq:wf}, \eqref{eq:gamma_A}, and \eqref{eq:gamma_B} as follows,
\begin{align}
&\Gamma_{\zeta\eta} = \sum_{II'} \Gamma^{A,II'}_\zeta \tilde{\Gamma}^{II'}_{\eta}, \\
&\tilde{\Gamma}^{II'}_{\eta} = \sum_{JJ'}U_{IJ}U_{I'J'}\Gamma^{B,JJ'}_\eta,
\end{align}
which after index reordering give $\Gamma_{ij}$ and $\Gamma_{ij,kl}$.
Because $\Gamma_\zeta$ intermediates are already computed in the ASD energy evaluation, additional costs for computing density matrices are negligible.
Note that density matrices have permutation symmetry; for instance,
\begin{align}
\Gamma_{\bar{i}j,kl} = \Gamma_{j\bar{i},lk} = \Gamma_{kl,\bar{i}j} = \Gamma_{lk,j\bar{i}}.
\end{align}
The density matrix elements with all indices belonging to the same monomer are computed separately for computational efficiency as
\begin{align}
& \Gamma_{ij,kl} = \sum_{J} \sum_{\rho\sigma}\langle \tilde{\Phi}^A_J | i^\dagger_\rho k^\dagger_\sigma l_\sigma j_\rho |\tilde{\Phi}^A_J \rangle,\\
& \Gamma_{\bar{i}\bar{j},\bar{k}\bar{l}} = \sum_{I} \sum_{\rho\sigma}\langle \tilde{\Phi}^B_I | \bar{i}^\dagger_\rho \bar{k}^\dagger_\sigma \bar{l}_\sigma \bar{j}_\rho |\tilde{\Phi}^B_I \rangle,
\end{align}
using rotated monomer states $|\tilde{\Phi}^A_J\rangle = \sum_I U_{IJ}|\Phi_I^A\rangle$ and $|\tilde{\Phi}^B_I\rangle = \sum_J U_{IJ}|\Phi_J^B\rangle$.
The use of rotated monomer states reduces the cost of calculating these elements from quadratic to linear scaling with respect to the number of monomer states.
\subsection{State averaging and model Hamiltonians}
As in traditional CASSCF, state averaging is used to optimize orbitals for systems that involve multiple electronic states,
in which $E_\mathrm{ASD}$ in Eq.~\eqref{deriv} is related by an averaged energies of states of interest (labeled by $K$),
\begin{align}
E_\mathrm{ASD}^\mathrm{ave} = \frac{1}{n}\sum_{K=1}^n E^K_\mathrm{ASD}.
\end{align}
This means that all the density matrices $\Gamma_{ij}$ and $\Gamma_{ij,kl}$ that appear in the orbital gradient and approximate Hessian elements
have to be replaced by state-averaged counterparts,
\begin{align}
&\Gamma_{ij}^\mathrm{ave} = \frac{1}{n}\sum_{K=1}^n \Gamma^K_{ij},\\
&\Gamma_{ij,kl}^\mathrm{ave} = \frac{1}{n}\sum_{K=1}^n \Gamma^K_{ij,kl}.
\end{align}
The rest of the algorithm remains identical.
As discussed in the introduction, ASD-CASSCF is developed to realize calculation of diabatic model Hamiltonians for electron and exciton dynamics.
The procedure to extract model Hamiltonians from ASD dimer Hamiltonian matrices has been detailed in Ref.~\onlinecite{Parker2014JCTC}.
In essence, we form a diabatic model state by diagonalizing each diagonal-subblock of the dimer Hamiltonian corresponding to appropriate charge and spin quantum numbers.
The diabatic wave functions are written as
\begin{align}
| \Psi^K_{\Omega_A\Omega_B} \rangle = \sum_{I\in \Omega_A J\in \Omega_B} U^K_{IJ} | \Phi^A_{I} \rangle | \Phi^B_{J} \rangle,
\end{align}
in which $\Omega_A$ and $\Omega_B$ are collections of monomer states with given charge and spin quantum numbers.
The diagonal elements are the eigenvalues in the subblock diagonalization.
The off-diagonal Hamiltonian matrix element of the diabatic states are defined as
\begin{align}
H_{KK'} = \langle \Psi^K_{\Omega_A \Omega_B} | \hat{H} | \Psi^{K'}_{\Omega'_A \Omega'_B} \rangle. \label{eq:dc}
\end{align}
When computing model Hamiltonians with orbital optimization, it is convenient to use the averaged energy of the diabatic states in the model Hamiltonian.
This procedure gives identical energies to the state-averaged calculations with the adiabatic states obtained by diagonalizing the model Hamiltonian, since
the trace of a Hamiltonian matrix is invariant under the unitary transformation.
\section{Results and discussion\label{results}}
\begin{figure}
\includegraphics[width=0.4\textwidth]{bz2-v2.pdf}
\caption{\label{fig:bz}
Errors in (a) the ground-state energy and (b) the energy splitting between the ground and first excited singlet and triplet states of a benzene dimer
as functions of the number of states in each charge and spin sector in the ASD expansion ($N$) computed by ASD-CASCI and ASD-CASSCF.
}
\end{figure}
We applied ASD-CASSCF to the benzene dimer in its sandwich configuration with a separation of 4.0 \AA.
The cc-pVDZ basis set\cite{Dunning1989JCP} was used, and singlet (S), triplet (T), and charge-transfer (CT) monomer states are included.
The CAS(12,12) active space for a dimer consists of 12 $\pi$ electrons distributed in the $\pi$ orbitals,
which is decomposed to two 6-orbital ASD subspaces.
First, the ground state energies and the first few excitation energies computed by ASD-CASCI and ASD-CASSCF as a function of the number of states in each charge and spin sector in the ASD expansion ($N$) are presented in Fig.~\ref{fig:bz}.
The errors are calculated relative to the conventional CASCI ($-461.547119\,E_\mathrm{h}$) and CASSCF ($-461.584702\,E_\mathrm{h}$) energies.
ASD-CASSCF and ASD-CASCI converge to the conventional CASSCF and CASCI with full CAS(12,12) at large $N$.
The Hartree--Fock orbitals were used in the CASCI calculations.
For the ground state, the convergence of ASD-CASSCF is slightly slower than that with ASD-CASCI, although they are both almost exponential.
The energies converged to 0.1 m$E_\text{H}$ with as small as $N=8$.
We also note that ASD-CASSCF with a more diffuse aug-cc-pVDZ basis set gave almost identical errors for given $N$.
See the supporting information for details.\cite{supp}
The excitation energies show negligible variance with respect to the number of monomer states.
In all cases, the inclusion of CT states is essential to obtain numerically near-exact energies.
The T-T contributions were negligible for the ground state, since the configuration in which the two monomers have anti-parallel triplets lies too high in energy to interact with the neutral S-S configuration.
Orbital optimization increased and decreased the CT contributions to the ground and excited states, respectively.
The convergence of orbital optimization in ASD-CASSCF is compared to that in CASSCF and is shown in Fig.~\ref{conver}.
Three states were state-averaged in the calculations.
We found that the convergence behaviors of ASD-CASSCF and CASSCF were nearly identical,
corroborating that active--active rotations do not deteriorate the numerical stability of orbital optimization.
The total ASD-CASSCF calculation (7 iterations) took about 75 and 210 seconds with $N=8$ and $32$,
while the standard CASSCF took about 3300 seconds
using 16 CPU cores of Xeon E5-2650 2.00~GHz.
\begin{figure}
\includegraphics[width=0.35\textwidth]{conv-v2.pdf}
\caption{\label{conver}
Convergence of orbital optimization in ASD-CASSCF and the standard CASSCF for a benzene dimer (see text for computational details).
A quasi-second-order algorithm was used.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.37\textwidth]{cov2-v2.pdf}
\caption{\label{fig:c_dimers}
Errors in the ASD-CASSCF ground-state energies as functions of the number of states in each charge and spin sector
in the ASD expansion ($N$) for (a) 4-(2-naphthylmethyl)-benzaldehyde, and (b) [$3_6$]cyclophane.
}
\end{figure}
Next, the energies of 4-(2-naphthylmethyl)-benzaldehyde (\textbf{M}) and [$3_6$]cyclophane (\textbf{CP})
are presented in Fig.~\ref{fig:c_dimers} as functions of the number of states in the ASD decomposition (see Refs.~\onlinecite{Closs1988JACS}, \onlinecite{Closs1989JACS}, and \onlinecite{Nogita2004JACS} for experimental studies on these molecules).
The geometries were optimized by density functional theory using the B3LYP functional\cite{Becke1993JCP} and the 6-31G* basis set.\cite{Hariharan1973TCA}
The optimized structures of these molecules are shown in Fig.~\ref{MandCP}.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{mol-v2.png}
\caption{\label{MandCP} Structures of 4-(2-naphthylmethyl)-benzaldehyde (\textbf{M}), [$3_6$]cyclophane (\textbf{CP}),
and {\it anti}-[2.2](1,4)pentacenophane (\textbf{PP}).
}
\end{figure}
The def2-SVP basis set\cite{Weigend2005PCCP} was used in the ASD calculations.
In this example, ASD-CASCI cannot be used for these molecules; therefore, only the results with ASD-CASSCF are shown.
The active space for \textbf{M} contains all the $\pi$-electrons from benzaldehydyl and naphthyl groups to form CAS(18,18),
which is decomposed to a product space of $(8 \pm q,8)$ and $(10 \mp q,10)$ where $q$ ($=0, 1, \cdots$) is the number of electrons transferred in the dimer basis.
The construction of ASD subspaces for \textbf{CP} is analogous to that for a benzene dimer above.
For \textbf{CP}, the CASSCF energy ($-1156.239888\,E_\mathrm{h}$) is used as a reference, while
the ASD-CASSCF energy with $N=128$ ($-764.327532\,E_\mathrm{h}$) is used for \textbf{M} since the exact dimer calculations with CAS(18,18) calculation are not feasible.
The large $\pi$-overlap of \textbf{CP} due to constrained geometry yields the slower convergence with respect to the number of states
and the pronounced polarization contributions to the energies in comparison to \textbf{M};
however, with $N = 32$ energies for both molecules are converged to less than 0.1 m$E_\text{h}$.
The errors presented in Fig.~\ref{fig:c_dimers} are dominated by that from the ASD approximation to the active CI, and not by variations of orbitals with respect to $N$.
To verify this, we computed the ground-state energy using ASD-CASCI with orbitals obtained by $N=8$ ASD-CASSCF.
The convergence was found to be almost identical to that of ASD-CASSCF, attesting that the change of orbitals is a minor contribution to the ASD errors.
We then turn to the triplet energy transfer processes of the donor-bridge-acceptor systems pioneered by Closs and co-workers,\cite{Closs1988JACS, Closs1989JACS}
which have been extensively studied theoretically by Subotnik and co-workers.\cite{Subotnik2010JPCA, Alguire2014JPCA, Landry2014JCTC}
The donor and acceptors are benzaldehyde and naphthalene, respectively, and the bridge is either cyclohexane or {\it trans}-decalin, each of which is rigid and saturated.
In Marcus theory, the rate of triplet energy transfer is given by\cite{Nitzanbook}
\begin{align}
k = \frac{2\pi}{\hbar} |H_{IF}|^2 \sqrt{\frac{1}{4 \pi k_B T \lambda}} \exp\left( -\frac{(\lambda + \Delta G^0)^2}{4 \lambda k_B T} \right), \label{eq:rate}
\end{align}
where $\Delta G^0$ is the driving force, $\lambda$ is the reorganization energy, and $H_{IF}$ is the diabatic coupling between the two diabatic states, defined in Eq.~\eqref{eq:dc}.
The triplet energy transfer is mediated by charge-transfer states, whose contributions are
effectively treated using the quasi-degenerate second-order perturbation theory in our model. The diabatic coupling now reads\cite{Parker2014JCTC}
\begin{align}
\tilde{H}_{IF} = H_{IF} + \frac{1}{2} \sum_{Z \notin \mathrm{model}} \left( \frac{H_{IZ} H_{ZF}}{E_I - E_Z} + \frac{H_{IZ} H_{ZF}}{E_F - E_Z} \right).
\end{align}
Here, the second term refers to the perturbative correction to the direct coupling, and the sum runs over all diabatic dimer states $Z$ that are not included in the model spaces.
We used geometries optimized by B3LYP with the 6-31G* basis set.\cite{Becke1993JCP,Hariharan1973TCA}
Two triplet states were included in the state averaging, and the def2-SVP basis set\cite{Weigend2005PCCP} was used in the ASD calculations.
The active spaces are the same as that for \textbf{M} [i.e., decomposed CAS(18,18)], and 64 states were included in each charge and spin sector of the ASD expansion.
Table \ref{tab:coupling} compiles the calculated diabatic coupling and rates from the state-averaged ASD-CASSCF orbitals.
The mediated coupling values are tabulated; corresponding direct coupling values are 0.074, 0.040, 0.0078, and 0.00033 in meV for 1,3-C-ee, 1,4-C-ee, 2,7-D-ee, and 2,6-D-ee, respectively.
\begin{table}
\caption{Diabatic coupling $|H_{IF}|$ (in meV) and triplet energy transfer rate $k$ (in s$^{-1}$) for the Closs systems. The factors in the rate expression other than the diabatic coupling
were taken from Ref.~\onlinecite{Subotnik2010JPCA}.
}
\label{tab:coupling}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
\multicolumn{2}{c}{} & \multicolumn{2}{c}{ASD-CASSCF} & \multicolumn{2}{c}{CIS-Boys\footnotemark[1]} \\
\cline{3-4} \cline{5-6}
Molecule & $n_\sigma$\footnotemark[2] & $|H_{IF}|$ & $k$ & $|H_{IF}|$ & $k$ & $k_\mathrm{expt}$\footnotemark[3]\\
\hline
1,3-C-ee & 3 & 0.863 & $7.0 \times 10^9$ & 1.5\,\,\,\,\,\, & $2.1 \times 10^{10}$ & $7.7 \times 10^9$ \\
1,4-C-ee & 4 & 0.363 & $1.6 \times 10^9$ & 0.56\,\,\, & $3.9 \times 10^9$ & $1.3 \times 10^9$ \\
2,7-D-ee & 5 & 0.076 & $6.9 \times 10^7$ & 0.17\,\,\, & $3.5 \times 10^8$ & $9.1 \times 10^7$\\
2,6-D-ee & 6 & 0.010 & $1.1 \times 10^6$ & 0.020 & $5.0 \times 10^6$ & $3.1 \times 10^6$\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Values based on configuration interaction singles diabatized by Boys localization, taken from Ref.~\onlinecite{Subotnik2010JPCA}.}
\footnotetext[2]{Number of $\sigma$-bonds between the donor and acceptor.}
\footnotetext[3]{Experimental values taken from Ref.~\onlinecite{Closs1989JACS}.}
\end{table}
\begin{figure*}[tb]
\includegraphics[width=0.9\textwidth]{mo.png}
\caption{\label{fig:mo_diff}
Initial (mesh) and converged (filled) active semi-canonical orbitals for 1,3-C-ee from state-averaged ASD-CASSCF.
}
\end{figure*}
Combining the diabatic coupling elements with the prefactors reported by Subotnik and co-workers,\cite{Subotnik2010JPCA}
the triplet energy transfer rates were calculated using Eq.~\eqref{eq:rate}.
Our coupling values are roughly half of those obtained by Subotnik and co-workers based on the diabatization of configuration interaction singles (CIS) adiabatic states via Boys localization.
The calculated rates are in good agreement with the experiments.
For more consistent comparison to experimental results, one needs to determine the prefactors using ASD-CASSCF.
It is also noted that we obtained the diabatic coupling of 0.54 meV for \textbf{M}, one of the Closs systems, in good agreement with 0.56 meV by Subotnik,
although the Condon approximation is not valid for this molecule owing to the flexible methylene bridge.
Figure~\ref{fig:mo_diff} visualizes the initial and the optimized active orbitals for 1,3-C-ee to
show that the diabatic nature of dimer product functions is retained during the orbital optimization procedure including active--active rotations [see Eq.~\eqref{eq:grad}].
The shapes of the $\pi^*$ orbitals slightly changed during the optimization, while the $\pi$-orbitals remained almost identical.
The locality of the optimized orbitals should be ascribed to the fact that the ASD expansion [Eq.~\eqref{eq:wf}] is most accurate with localized ASD subspaces.
The delocalization of the active orbitals to the bridge site is suppressed in these examples since the active space is constructed with only $\pi$-orbitals and the $sp^3$-hybridized bridging atom prevents the mixing of $\pi$- and $\sigma$-orbitals.
This allows us to construct diabatic models using the ASD-CASSCF method.
Finally the model Hamiltonians for hole and electron transfer processes of {\it anti}-[2.2](1,4)pentacenophane ({\bf PP}) were calculated using ASD-RASSCF with $N=8$.
{\bf PP} has recently been synthesized as a novel pentacene derivative.\cite{Bula2013Angew}
The charge (especially hole) mobilities of large acenes are of interest for photovoltaic applications.\cite{Anthony2008ACIE}
We used a molecular structure optimized by density functional theory (B3LYP/6-31G*).\cite{Becke1993JCP,Hariharan1973TCA}
The S, T, and CT subspaces were included in the ASD expansion,
in which RAS(7,4,7)[2,2] (7, 4, and 7 orbitals were assigned to the RAS {I}, {II}, {III} spaces for each monomer, respectively; up to 2 holes and 2 particles were allowed in the RAS {I} and {III} spaces) was used for monomer states.
We fixed the monomer CI coefficients after five iterations and
recomputed the ASD-CI energies using the optimized orbitals (see Sec.~\ref{theorysec}).
This procedure was iterated until consistency between orbitals and monomer CI coefficients was attained.
The diabatic couplings obtained from ASD-RASSCF are 125.1 and 11.6~meV
for hole and electron transfer processes, respectively.
\section{Conclusions}
We derived and implemented an orbital optimization method
in the framework of ASD, in which a dimer wave function is expanded in products of monomer functions.
The active--active rotations between ASD subspaces are included in the optimization.
This not only removes the ambiguity in the choice of ASD subspace orbitals but also extends the applicability of ASD to covalently linked dimers.
The energies computed using both ASD-CASSCF and ASD-RASSCF converge rapidly with respect to the number of states in the ASD expansion.
Since orbital optimization preserves the diabatic nature of the dimer basis functions,
we were able to compute orbital-optimized model Hamiltonians for electron, hole, and triplet transfer processes.
Work toward incorporating dynamical correlation on the basis of the ASD-CASSCF and ASD-RASSCF methods is in progress.
The ASD-CASSCF and ASD-RASSCF methods also provide smooth potential energy surfaces and can be used to study reaction dynamics.
All the programs developed in this work are available in the open-source {\sc bagel} package.\cite{bagel}
\begin{acknowledgments}
This work has been supported by Office of Basic Energy Sciences, U.S. Department of Energy (Grant No. DE-FG02-13ER16398).
\end{acknowledgments}
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"redpajama_set_name": "RedPajamaArXiv"
}
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Q: linux loop over the text files inside of one level subfolders I have a structure like this:
2020/
file1.txt
file2.txt
2021/
file4.txt
file5.txt
...etc
I would like to create a bash script to loop over the text files to read them line by line and parse them
I am starting to learn bash and after several stack overflow questions read I thought the following would work:
for file in /{2020..2022}/*.txt;
do
echo $file
done
this bash script is saved in the folder containing 2020, 2021, and 2022 and run from there.
my explanation:
the curly brackets would expand to 2020, 2021, 2022 and then under that level all the files with extension .txt would be given.
bu it does not work.
why?
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"redpajama_set_name": "RedPajamaStackExchange"
}
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Help me attract more heads to shrink!
info q52 needed a new logo design and created a contest on 99designs.
A winner was selected from 73 designs submitted by 12 freelance designers.
Unep99 created a great set of designs. He came up with an interesting series of original designs, and then was able to make multiple, subtle variations that resulted in a perfect end-product.
Mas.Ganteng was super responsive to my questions and requests. Great job and we are really happy with the results!
This is a private mental health counseling practice. I work with individuals, couples and families. My target audience is smart, intellectual, creative, hip and modern. Demographically 25-45.
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"redpajama_set_name": "RedPajamaC4"
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Grace Hubbard Fortescue, (née Bell) (November 3, 1883 – June 24, 1979), was a New York City socialite who murdered a man, later proven innocent, who was accused of raping her daughter. After being convicted of manslaughter at a sensational trial, her ten-year sentence was commuted to a single hour by Hawaii's Territorial Governor Lawrence Judd.
Early life
Grace Hubbard Bell was born November 3, 1883, in Washington, D.C. Her father Charles John Bell was first cousin of inventor Alexander Graham Bell. Her mother was Roberta Wolcott Hubbard Bell (1859–1885). Her maternal grandfather Gardiner Hubbard was the first president of Bell Telephone Company. When her mother died in childbirth in 1885, her father married her mother's sister, Grace Hubbard.
The family lived at Twin Oaks, their estate in the Cleveland Park neighborhood of Washington, D.C. Newspaper reports indicate that Grace could be classified a prankster: as a youth, she and her friends stole a trolley car for a joy ride through the streets of Washington and, on another occasion, she blocked traffic on Pennsylvania Avenue by joining hands with friends and roller-skating down the avenue.
Personal life
In 1910, she married U.S. Army Major Granville "Rolly" Fortescue (1875–1952), one of the sons of Robert Barnwell Roosevelt. Her husband was first cousin of U.S. President Theodore Roosevelt. The marriage was not as financially successful as she would have wished. She was the mother of three daughters:
Thalia Fortescue Massie (1911–1963), who married Thomas Hedges Massie (1905–1987), a Navy lieutenant.
Marion Fortescue (1912–), who married Daulton Gillespie Viskniskki in 1934.
Kenyon Fortescue (1914–1990), better known as actress Helene Whitney; she married J. Louis Reynolds in 1936.
Outwardly, the Fortescues appeared to be wealthy country gentry. In reality, financial affairs became a primary concern for them after Granville's final retirement from the army. With the exception of a short stint as a fiction editor for Liberty magazine in 1930, he did not have steady employment, preferring to wait for the fortune his wife would inherit at the death of her parents.
Murder trial
In 1932, Grace Fortescue was charged with murder and convicted by a jury of manslaughter for the death of Joseph Kahahawai, one of the defendants in the alleged rape of her daughter Thalia in Hawaii in 1931. Also charged and convicted with Fortescue were two sailors, Edward J. Lord and Deacon Jones, and Fortescue's son-in-law, Thomas Massie, who participated in the abduction and murder of Kahahawai.
Attorney Clarence Darrow defended Fortescue, Jones, Massie, and Lord. He subsequently obtained a commutation of their sentence (ten years' imprisonment for manslaughter) to a one-hour confinement in the executive chambers of Territorial Governor Lawrence M. Judd.
In 1966, while being interviewed by author Peter Van Slingerland, Albert O. Jones admitted that he was the one who shot Joseph Kahahawai.
References
External links
Webpage for The American Experience, "The Massie Affair", retrieved on 2008-06-07.
Stannard, David. "The Massie case: Injustice and courage" The Honolulu Advertiser, October 14, 2001, retrieved on 2008-06-07.
1883 births
1979 deaths
People from Washington, D.C.
Politics of Hawaii
Roosevelt family
American people convicted of manslaughter
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{"url":"https:\/\/meelgroup.github.io\/post\/approxmcv3\/","text":"# ApproxMCv3: A modern approximate model counter\n\nApproxMC is a scalable, approximate model counter that provides PAC (probably approximately correct) guarantees. We have been working very hard on speeding up approximate model counting for SAT and have made real progress. The research paper, accepted at AAAI-19 is available here. The code is available here (release with static binary here). The main result is that we can solve a lot more problems than before. The speed of solving is orders(!) of magnitude faster than the previous best system:\n\n### Background\n\nThe idea of approximate model counting, originally by Chakraborty, Meel and Vardi was a huge hit back in 2013, and many papers have followed it, trying to improve its results. All of them were basically tied to CryptoMiniSat, the SAT solver that is maintained by Mate, as all of them relied on XOR constraints being added to the regular CNF of a typical SAT problem.\n\nSo it made sense to examine what CryptoMiniSat could do to improve the speed of approximate counting. This time interestingly coincided with the removal of XORs in CryptoMiniSat. The problem was the following: A lot of new in- and preprocessing systems were being invented, mostly by Armin Biere et al, and they couldn't be added to CryptoMiniSat, because they didn\u2019t take into account XOR constraints. They handled CNF just fine, but not XORs. So XORs became a burden, and they were removed in versions 3 and 4 of CryptoMiniSat. But there was need, and this being an exciting area, the XORs had to come back.\n\n### Blast-Inprocess-Recover-Destroy\n\nBut how to both have and not have XOR constraints? Re-inventing all the algorithms for XORs was not a viable option. The solution we came up with was a rather trivial one: forget the XORs during inprocessing and recover them after. The CNF would always remain the source of truth. Extracting all the XORs after in- and preprocessing would allow us to run the Gauss-Jordan elimination on the XORs post-recovery.\n\nThe process is conceptually quite easy:\n\n1. Blast all XORs into clauses that are in the input using intermediate variables.\n2. Perform pre- or inprocessing.\n3. Recover the XORs from the CNF.\n4. Run the CDCL and Gauss-Jordan code at the same time.\n5. Destroy the XORs and goto 2.\n\nThis system allows for everything to be in CNF form, lifting the XORs out when necessary and then forgetting them when it\u2019s convenient. All of these steps are rather trivial, except recovery, as explained below.\n\n### XOR recovery\n\nRecovering XORs sounds like a trivial task. Let\u2019s say we have the following clauses\n\n x1 V x2 V x3\n-x1 V -x2 V x3\nx1 V -x2 V -x3\n-x1 V x2 V -x3\n\n\nThis is conceptually equivalent to the XOR v1+v2+v3=1. So recovering this is trivial, and has been done before, by Heule in particular, in his PhD thesis. The issue with the above is the following: a stronger system than the above still implies the XOR, but doesn\u2019t look the same. For example:\n\n x1 V x2 V x3\n-x1 V -x2 V x3\nx1 V -x2 V -x3\n-x1 V x2\n\n\nThis is almost equivalent to the previous set of clauses, but misses a literal from one of the clauses. It still implies the XOR of course. Now what? And what to do when missing literals mean that an entire clause can be missing? The algorithm to recover XORs in such cases is non-trivial. It\u2019s non-trivial not only because of the complexity of how many combinations of missing literals and clauses there can be (it\u2019s exponential) but because one must do this work extremely fast because SAT solvers are sensitive to time.\n\nThe algorithm that is in the paper explains all the bit-fiddling and cache-friendly data layout used along with some fun algorithms. We even managed to use compiler intrinsics to use target-specific assembly instructions for hamming weight calculation.\n\n### The results\n\nThe results, as shown above, speak for themselves. Problems that took thousands of seconds to solve can now be solved under 20. The reason for such incredible speedup is basically the following: CryptoMiniSatv2 was way too clunky and didn\u2019t have all the fun stuff that CryptoMiniSatv5 has, plus the XOR handling was incorrect, loosing XORs and the like. The published algorithm solves the underlying issue and allows CNF pre- and inprocessing to happen independent of XORs, thus enabling CryptoMiniSatv5 to be used in all its glory. And CryptoMiniSatv5 is fast, as per the this year\u2019s SAT Competition results.\n\nWe thank the National Supercomputing Center Singapore that allowed us to run a large number of benchmarks on their machines, using at least 200 thousand CPU hours to make this paper. This post has been adapted from Mate's post on ApproxMCv3.","date":"2022-11-28 10:53:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.466036856174469, \"perplexity\": 1545.1191370493696}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710503.24\/warc\/CC-MAIN-20221128102824-20221128132824-00575.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{submission}
Neural networks are able to learn rich representations of data that support highly accurate classification; however, understanding or controlling what neural nets learn remains challenging.
Some techniques offer insight into pre-trained models by exposing patterns in latent spaces \citep{isola, tcav}, while approaches focused on interpretability provide techniques that are more comprehensible to humans \citep{pcn, thisthat}.
While these methods provide insight, they fail to offer control: humans observe learned patterns but are unable to guide models such that learned relationships are useful for a particular setting or task.
Another line of work has advanced the design of models for particular types of classification tasks (such as fair or hierarchical classification) but these techniques are often developed with only one problem in mind \citep{vfae, adv_fair, hierarchy}.
For example, models built for fair classification (predicting an outcome regardless of information about a protected field) are only used to enforce independence of concepts rather than hierarchy.
Thus, humans may exert control over learned representations by selecting an appropriate technique rather than tuning training parameters within the same technique.
We have designed a new neural network architecture, the concept subspace network (CSN), which generalizes existing specialized classifiers to produce a unified model capable of learning a spectrum of multi-concept relationships.
CSNs use prototype-based representations, a technique employed in interpretable neural networks in prior art \citep{pcn, thisthat,metricprotos}.
A single CSN uses sets of prototypes in order to simultaneously learn multiple concepts; classification within a single concept (e.g., ``type of animal'') is performed by projecting encodings into a concept subspace defined by the prototypes for that concept (e.g., ``bird,'' ``dog,'' etc.).
Lastly, CSNs use a measure of concept subspace alignment to guide concept relationships such as independence or hierarchy.
In our experiments, CSNs performed comparably to state-of-the art in fair classification, despite prior methods only being designed for this type of problem.
In applying CSNs to hierarchical classification tasks, networks automatically deduced interpretable representations of the hierarchical problem structure, allowing them to outperform state-of-the-art, for a given neural network backbone, in terms of both accuracy and average cost of errors on the CIFAR100 dataset.
Lastly, in a human-motion prediction task, we demonstrated how a single CSN could enforce both fairness (to preserve participant privacy) and hierarchy (to exploit a known taxonomy of tasks).
Our findings suggest that CSNs may be applied to a wide range of problems that had previously only been addressed individually, or not at all.
\section{Related Work}
\subsection{Interpretability and Prototype Networks}
Numerous post-hoc explanation techniques fit models to pre-trained neural nets; if humans understand these auxiliary models, they can hypothesize about how the neural nets behave \citep{lime, shap}.
However, techniques in which explanations are decoupled from underlying logic may be susceptible to adversarial attacks or produce misleading explanations \citep{fooling}.
Unlike such decoupled explanations, interpretability research seeks to expose a model's reasoning.
In this work we focus on prototype-based latent representations in neural nets.
``Vector quantization'' literature has long studied learning discrete representations in a continuous space \citep{kohonen1990improved,vectorq}.
More recently, the prototype case network (PCN) comprised an autoencoder model that clustered encodings around understandable, trainable prototypes, with classifications made via a linear weighting of the distances from encodings to prototypes \citep{pcn}.
Further research in image classification extended PCNs to use convolutional filters as prototypes and for hierarchical classification in the hierarchical prototype network (HPN) \citep{thisthat, hierarchy}.
Lastly, \citet{metricprotos} use prototypes in Metric-Guided Prototype Learning (MGP) in conjunction with a loss function to cluster prototypes to minimize user-defined costs.
Our model similarly uses trainable prototypes for classification, but differs from prior art in two respects.
First, we modify the standard PCN architecture to support other changes, without degrading classification performance.
Second, like HPNs (but not PCNs or MGP), CSNs leverage multiple sets of prototypes to enable hierarchical classification but also allow for non-hierarchical concept relationships.
\subsection{Fair and Hierarchical Classification}
AI fairness research considers how to mitigate undesirable patterns or biases in machine learning models.
Consider the problem of predicting a person's credit risk: non-causal correlations between age and risk may lead AI models to inappropriately penalize people according to their age \citep{vfae}.
The problem of fair classification is often framed as follows: given inputs, $x$, which are informative of a protected field, $s$, and outcome, $y$, predict $y$ from $x$ without being influenced by $s$ \citep{zemel2013learning}.
Merely removing $s$ from $x$ (e.g., not including age as an input to a credit predictor) rarely removes all information about $s$, so researchers have developed a variety of techniques to create representations that ``purge'' information about $s$ \citep{vfae,adv_fair,wasserstein}.
Hierarchical classification solves a different problem: given a hierarchical taxonomy of classes (e.g., birds vs. dogs at a high level and sparrows vs. toucans at a low level), output the correct label at each classification level.
Neural nets using convolution and recurrent layers in specialized designs have achieved remarkable success in hierarchical image classification \citep{zhu2017b,guo2018cnn}.
The hierarchical prototype network (HPN) uses prototypes and a training routine based upon conditional subsets of training data to create hierarchically-organized prototypes \citep{hierarchy}.
\citet{metricprotos} also use prototypes for hierarchical classification in Metric-Guided Prototype Learning (MGP) by adjusting the training loss to guide prototype arrangement.
Neither HPN nor MGP explicitly models relationships between multiple subsets of prototypes.
Lastly, recent works propose hyperbolic latent spaces as a natural way to model hierarchical data \citep{hyperbolicnlp,hyperbolicvae,nickel2017poincare,liu2020hyperbolic}.
Our method, conversely, relies upon concepts from Euclidean geometry.
Extending the principle of subspace alignment that we develop to non-Euclidean geometric spaces is a promising direction but is beyond the scope of this work.
\section{Technical Approach}
In this section, we outlined the design of the CSN, which was inspired by desires for both interpretable representations and explicit concept relationships.
First, we wished for interpretable representations, so we built upon the PCN design, with modifications.
Second, we explicitly encoded relationships between concepts by introducing multiple sets of prototypes, instead of just one in PCNs.
Third, we enabled guidance of the concept relationships by modifying the CSN training loss.
Together, these changes supported not only interpretable classification, but also provided a flexible framework for a single model architecture to learn different concept relationships.
\subsection{Concept Subspace Classification}
A CSN performing a single classification task (e.g., identifying a digit in an image) is defined by three sets of trainable weights.
First, an encoder parametrized by weights $\theta$, $e_\theta$, maps from inputs of dimension $X$ to encodings of dimension $Z$: $e_\theta: R^X \rightarrow R^Z$.
Second, a decoder parametrized by weights $\phi$, $d_\phi$, performs the decoding function of mapping from encodings to reconstructed inputs: $d_\phi: R^Z \rightarrow R^X$.
Third, there exist a set of $k$ trainable prototype weights, $\bm{p}$, that are each $Z$-dimensional vectors: $p_1, p_2, ..., p_k \in R^Z$.
This architecture resembles that of the PCN, but without the additional linear classification layer \citep{pcn}.
Given a set of $k$ prototypes in $R^Z$, we define a ``concept subspace,'' $C$ as follows:
\begin{equation*}
\begin{split}
v_i &= p_i - p_1 \quad \forall i \in [2, k]\\
C = \{x | x \in R^Z &\text{ for } x = p_1 + \sum_{i \in [2, k]}\lambda_i v_i \text {for } \lambda_i \in R \quad \forall i\}
\end{split}
\end{equation*}
$C$ is the linear subspace in $R^Z$ defined by starting at the first prototype and adding linear scalings of vector differences to all other prototypes.
We call $C$ a concept subspace because it represents a space of encodings between prototypes defining a single concept (e.g., prototypes for digits 0, 1, 2, etc. define a concept subspace for digit classification).
A CSN's architecture --- consisting of an encoder, a decoder, and a set of prototypes and the associated concept subspace --- enables two types of functionality: the encoder and decoder may be composed to reconstruct inputs via their latent representations, and CSNs may perform classification tasks by mapping an input, $x$, to one of $Y$ discrete categories.
Classification is performed by first encoding an input into a latent representation, $z = e_\theta(x)$.
The $l2$ distance from $z$ to each prototype is then calculated, yielding $k$ distance values: $d_i(z, \bm{p}) = ||z - p_i||_2^2; i \in [1,k]$.
These distances are mapped to a probability distribution, $\mathbbm{P}_K(i); i \in [1, k]$, by taking the softmax of their negatives.
Lastly, if there are more prototypes than classes, (e.g., two prototypes for dogs, two for cats, etc.) the distribution over $k$ is converted to a distribution over $Y$ categories by summing the probabilities for prototypes belonging to the same class.
For single-concept classification, CSNs differ from PCNs by removing the linear layer that PCNs used to transform distances to prototypes into classifications.
We found this unnecessary for high classification accuracy (Appendix A) and instead directly used negative distances.
Without the linear layer, CSN classification is equivalent to projecting encodings, $z$, onto a concept subspace before calculating distances.
The distances between a projected encoding, dubbed $z_{proj}$, and prototypes induce the same softmax distribution as when the orthogonal component remains.
Indeed, we find projection more intuitive - only the ``important'' component of $z$ is used for classification.
A simple example of projecting an encoding and calculating distances to prototypes is shown in Figure~\ref{fig:spectrum} a.
For some tasks, we used an encoder design from variational-autoencoders (VAEs) in order to regularize the distribution of encodings to conform to unit Gaussians \citep{vae}.
By default, this regularization loss was set to 0, but it sometimes proved useful in some domains to prevent overfitting (as detailed in experiments later).
We emphasize that CSNs are discriminative, rather than generative, models, so we did not sample from the latent space.
\subsection{Multi-Concept Learning}
\begin{figure}[t]
\centering
\begin{tikzpicture}
\node[trapezium,draw, rotate=-90, minimum width = 1cm,
minimum height = 0.75cm] (t) at (0,0) {Enc.};
\node[rectangle,draw, minimum width=1cm, minimum height=0.25cm, right=1.5cm of t] (p22) {...};
\node[rectangle,draw, minimum width=1cm, minimum height=0.25cm, above=0cm of p22] (p21) {$p_{21}$};
\node[rectangle,draw, minimum width=1cm, minimum height=0.25cm, above=0.5cm of p21] (p13) {...};
\node[rectangle,draw, minimum width=1cm, minimum height=0.25cm, above=0cm of p13] (p12) {$p_{12}$};
\node[rectangle,draw, minimum width=1cm, minimum height=0.25cm, above=0cm of p12] (p11) {$p_{11}$};
\node[above right=0.2cm and 0.2cm of t] (z) {$z$};
\node[right=0.0cm of p11] (d11) {$d_{11}$};
\node[right=0.0cm of p12] (d12) {$d_{12}$};
\node[right=0.0cm of p21] (d21) {$d_{21}$};
\draw [decorate, decoration = {brace}] (3.3,1.4) -- (3.3,0.2);
\draw [decorate, decoration = {brace}] (3.3,-0.3) -- (3.3,-1.1);
\node[] (s2) at (4.3, -0.7) {softmax()};
\node[] (s2) at (4.3, 0.8) {softmax()};
\node[] (y1) at (5.8, 0.8) {$\hat{y_1}$ = bird};
\node[] (y2) at (6.0, -0.7) {$\hat{y_2}$ = toucan};
\end{tikzpicture}
\caption{In a CSN, an encoder encodes inputs into latent representations, $z$. Multiple sets of prototypes exist in the latent space; the distance to each prototype is computed to generate a distribution for each set. Here, one set of prototypes represents high-level animal groups (bird vs. dog) and another represents fine labels (toucan).}
\label{fig:diagram}
\end{figure}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.23\textwidth}
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=\textwidth, trim={0, 0, 1cm, 1.5cm}, clip]{figures/single.png}};
\node[inner sep=0] (z) at (2.07, 2.48) {};
\node[] (z_lab) at (1.92, 2.48) {$z$};
\node[] (z_p) at (2.3, 0.9) {$z_{proj}$};
\node[] (bird) at (0.8, 0.6) {bird};
\node[] (dog) at (3.1, 1.1) {dog};
\end{tikzpicture}
\caption{}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=\textwidth, trim={0, 0, 1cm, 1.5cm}, clip]{figures/parallel.png}};
\node[inner sep=0] (z) at (2.07, 2.48) {};
\node[] (z_lab) at (1.92, 2.48) {$z$};
\draw [-stealth, line width=2pt, cyan] (z) -> ++(0.45, 0.12);
\node[] (bird) at (0.8, 1.2) {bird};
\node[] (dog) at (2.7, 1.5) {dog};
\node[] (boxer) at (3.5, 0.9) {boxer};
\node[] (lab) at (3.1, 0.7) {lab};
\node[] (poodle) at (2.55, 0.6) {pug};
\node[] (toucan) at (1.4, 0.4) {toucan};
\end{tikzpicture}
\caption{}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=\textwidth, trim={0, 0, 1cm, 1.5cm}, clip]{figures/orth.png}};
\node[inner sep=0] (z) at (2.07, 2.48) {};
\node[] (z_lab) at (1.92, 2.48) {$z$};
\draw [-stealth, line width=2pt, cyan] (z) -> ++(0.45, 0.11);
\node[] (bird) at (0.8, 0.6) {male};
\node[] (dog) at (2.7, 0.95) {female};
\node[] (dog) at (3.2, 2.2) {approve};
\node[] (dog) at (3.6, 0.5) {deny};
\end{tikzpicture}
\caption{}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\includegraphics[width=\textwidth]{figures/angled.png}
\caption{}
\end{subfigure}
\caption{In a single classification task (a), CSNs calculate the distance from an encoding, $z$, to prototypes (blue dots) in a subspace (gray line), or equivalently project $z$ into the concept subspace and then calculate distances. CSNs support multiple classification tasks. In hierarchical tasks, with parallel subspaces (b), moving $z$ changes predictions in both subspaces (e.g., moving $z$ along the blue arrow toward ``dog'' increases the likelihood of classes like ``pug'' or ``lab''). In fairness tasks, with orthogonal subspaces (c), changes to predictions are independent (e.g., moving $z$ to increase the likelihood that a loan applicant is female does not change the likelihood of being approved for a loan). In general, subspaces can be high-dimensional and exhibit a range of alignment values (d).}
\label{fig:spectrum}
\end{figure*}
We defined the CSN architecture for single classification tasks in the previous section; here, we explain how a CSN may be used for multiple classification tasks.
For example, consider classifying an image at a high level (bird vs. dog) and low level (toucan vs. sparrow vs. pug).
Our primary contribution is extending CSNs to support multiple classifications via the addition of new sets of prototypes.
Multiple classification tasks are performed by defining a set consisting of sets of prototypes: $\mathbbm{P} = \{\bm{p_1},...,\bm{p_c}\}$, with a set of prototypes for each of $c$ classification tasks.
A classification task is performed by using the CSN's encoder to generate an encoding, $z$, and projecting $z$ into the concept subspace defined by the set of prototypes particular to the given task.
This network architecture is depicted in Figure~\ref{fig:diagram}.
Figure~\ref{fig:spectrum} (b-c) depict simplified examples of two concept subspaces.
In these simplified examples, each concept concept subspace is defined as a line; $z$ may be projected onto either line depending upon the classification task at hand.
More generally, as depicted in Figure~\ref{fig:spectrum}~d, concept subspaces may be higher-dimensional than lines.
While prototypes in different sets are defined separately, correlations present in training data may lead to a range of relationships among prototypes.
In fact, two sets of prototypes can exhibit a range of relationships from highly correlated to fully independent, as shown in Figure~\ref{fig:spectrum}.
We defined a metric, concept subspace \textit{alignment}, to reflect this range of relationships.
Mathematically, the alignment of two subspaces is the mean of the cosine squared of the angle between all pairs of vectors drawn from the basis of each subspace.
Given orthonormal bases, efficiently computed via $QR$ factorization, $Q_1$ and $Q_2$, of ranks $m$ and $n$, we define alignment as follows:
\begin{equation}
a(Q_1, Q_2) = \frac{1}{mn} \sum_i^m\sum_j^n (Q_1^\top Q_2[i, j])^2
\label{eq:alignment}
\end{equation}
Given the range of values for the cosine squared function, alignment values range from 0 to 1 for orthogonal and parallel subspaces, respectively.
Intuitively, orthogonality lends itself well to independent concepts and therefore supports fair classification, whereas parallel subspaces naturally correspond to hierarchical classification.
\subsection{Training Procedure}
When training a CSN, we assume access to a set of training data, $(X, \mathbbm{Y})$ for $\mathbbm{Y} = (Y_1, Y_2, ... Y_c)$.
For each entry in the dataset, there is an input $x$ and a label $y_i$, for each of $c$ classification tasks.
We trained CSNs in an end-to-end manner to minimize a single loss function, defined in Equation~\ref{eq:loss}.
The four terms in the loss function were: 1) reconstruction error; 2) the loss introduced for the PCN, encouraging classification accuracy and the clustering of encodings around prototypes (applied within each concept subspace); 3) a KL divergence regularization term; and 4) a term penalizing alignment between concept subspaces.
Each term was weighted by a choice of real-valued $\lambda$s.
We emphasize that the PCN loss --- clustering and classification accuracy, defined in Equation 7 of \cite{pcn} --- is calculated within each concept subspace using the projections of encodings; thus, encodings were encouraged to cluster around prototypes only along dimensions within the subspace.
The encoder, decoder, and prototype weights were trained simultaneously.
\begin{align}
\label{eq:loss}
l(X, \mathbbm{Y}, \theta, &\phi, \mathbbm{P})
= \frac{\lambda_0}{|X|}\sum_{x \in X}{(d_\phi(e_\theta(x)) - x)^2} \nonumber\\
&+ \sum_{i \in [1, C]} \lambda_{Pi}PCN(\texttt{proj}(e_\theta(X), \bm{p_i}), Y_i) \nonumber\\
&+ \sum_{i \in [1, C]} \lambda_{KLi} KL(X, \bm{p_i})\\
&+ \sum_{i \in [1, C]} \sum_{j \in [1, C]} \lambda_{Aij} a(Q_i, Q_j) \nonumber
\end{align}
The KL regularization term mimics training losses often used in VAEs that penalize the divergence between the distribution of encodings and a zero-mean unit Gaussian \citep{vae}.
In our case, we wished to induce a similar distribution of encodings, but centered around prototypes rather than the origin.
Furthermore, rather than induce a Gaussian distribution within a concept subspace (which would dictate classification probabilities and therefore potentially worsen classification accuracy), we wished to regularize the out-of-subspace components of encodings.
Concretely, we implemented this regularization loss in three steps.
First, we computed the orthogonal component of an encoding as $z_{orth} = z - z_{proj}$.
We then computed the KL divergence between the distribution of $z_{orth}$ and unit Gaussians centered at each prototype in each subspace.
Finally, we took the softmax over distances between encodings and prototypes in order to only select the closest prototype to the encoding; we then multiplied the softmax by the divergences to enforce that encodings were distributed as unit Gaussians around the nearest prototype in each subspace.
Together, these operations led the distributions out of each subspace to conform to unit Gaussians around each prototype.
As confirmed in later experiments, this component was crucial in training fair classifiers.
\subsection{Hierarchical and Fair Classification}
\label{sec:fairhier}
We conclude this section by demonstrating how CSNs may support hierarchical or fair classifications.
Hierarchical and fair classification may be thought of as extremes along a spectrum of concept alignment.
In hierarchical classification, concepts are highly aligned and therefore parallel: the difference between a toucan and a pug is similar to the difference between a generic bird and dog, and so the differences between prototypes associated with different classes should also be parallel (e.g., ``bird'' - ``dog'' $\approx$ ``toucan'' - ``pug.'').
In fair classification, concepts are not aligned: switching belief about someone's sex should not alter approval for a loan.
Thus, based on the classification task, moving an encoding relative to one subspace should either affect (for hierarchical) or not affect (for fair) that encoding's projection onto the other subspace.
We provide a geometric interpretation of these two tasks in Figures~\ref{fig:spectrum} b and c.
CSNs can be trained to adopt either form of concept relationship by penalizing or encouraging concept subspace alignment (already present as $a(Q_i, Q_j)$ in the training loss).
Our single model reconciles these two types of problems by viewing them as opposite extremes along a spectrum of concept relationships that our technique is able to learn; this is the main contribution of our work.
\section{Results}
Our experiments were divided in four parts.
First, we demonstrated how CSNs matched standard performance on single classification tasks: in other words, that using a CSN did not degrade performance.
We omit these unsurprising results from the paper; full details are included in Appendix~\ref{app:single}.
Second, we showed that CSNs matched state-of-the-art performance in two fair classification tasks.
Third, we used CSNs for hierarchical classification tasks, exceeding performance demonstrated by prior art along several metrics.
Fourth, we showed how CSNs enabled both fair and hierarchical classification in a dataset describing human motion in an assembly task that exploited hierarchical knowledge while preserving participant anonymity.
Implementation details of CSNs in all experiments are included in Appendix~\ref{app:implementation}.
\subsection{Fair Classification}
\label{sec:fair}
We evaluated CSN's performance in fair classification tasks in the Adult and German datasets.
These datasets are commonly used in fairness literature and contain data that can be used to predict people's income or credit risks \citep{uci}
We compared CSN performance to our implementations of an advesarial purging technique (Adv.), the variational fair autoencoder (VFAE), Wasserstein Fair Classification (Wass. DB), and a mutual-information-based fairness approach (FR Train) \citep{adv_fair,vfae,wasserstein,roh2020fr}.
Implementation details of fair classification baselines and full results including standard deviations are included in Appendix~\ref{app:fair}.
For the Adult dataset, the protected attribute was sex, and for the German dataset, the protected attribute was a binary variable indicating whether the person was older than 25 years of age.
In evaluation, we measured $y$ Acc., the accuracy of predicting income or credit, $s$ Acc., the accuracy of a linear classifier trained to predict the protected field from the latent space, disparate impact (DI), as defined in \citet{roh2020fr}, and demographic disparity (DD-0.5), as defined by \citet{wasserstein}.
Mean results over 20 trials for both datasets were included in Tables~\ref{tab:fairness_adult} and \ref{tab:fairness_german}.
In both datasets, we observed that CSNs matched state-of-the-art performance.
CSNs produced high $y$ Acc., indicating high task performance for predicting income or credit.
Furthermore, fairness measures demonstrate that CSNs purged protected information successfully (low $s$ Acc.) and achieved high D.I. and low DD-0.5, as desired.
A visualization of the latent space of a fair classifier, trained on the Adult dataset, is shown in Figure~\ref{fig:adult} and confirmed that CSNs learned orthogonal concept subspaces.
\begin{table}[t]
\centering
\caption{Mean Adult dataset fairness results.}
\begin{tabular}{lcccc}
Model & $y$ Acc. & $s$ Acc. & D.I. & DD-0.5 \\
\hline
CSN & 0.85 & 0.67 & 0.83 & 0.16 \\
Adv. & 0.85 & 0.67 & 0.87 & 0.16 \\
VFAE & 0.85 & 0.70 & 0.82 & 0.17 \\
FR Train & 0.85 & 0.67 & 0.83 & 0.16 \\
Wass. DB & 0.81 & 0.67 & 0.92 & 0.08\\
Random & 0.76 & 0.67 & & \\
\end{tabular}
\label{tab:fairness_adult}
\end{table}
\begin{table}[t]
\centering
\caption{Mean German dataset fairness results.}
\begin{tabular}{lcccc}
Model & $y$ Acc. & $s$ Acc. & D.I. & DD-0.5 \\
\hline
CSN & 0.73 & 0.81 & 0.70 & 0.10 \\
Adv. & 0.73 & 0.81 & 0.63 & 0.10 \\
VFAE & 0.72 & 0.81 & 0.47 & 0.23 \\
FR Train & 0.72 & 0.80 & 0.55 & 0.16\\
Wass DB & 0.72 & 0.81 & 0.33 & 0.02 \\
Random & 0.70 & 0.81 & & \\
\end{tabular}
\label{tab:fairness_german}
\end{table}
In addition to reproducing the state of the art, we conducted an ablation study to demonstrate the importance of two terms in our training loss: the alignment and KL losses.
Using the German dataset, we trained 20 CSNs, setting the KL, alignment, or both loss weights to 0.
The mean results of these trials are reported in Table~\ref{tab:ablation}.
\begin{figure}[t]
\centering
\includegraphics[height=4cm, width=0.48\textwidth]{figures/adult2_recolored.png}
\captionof{figure}{A 2D latent space for fair classification. The prototypes for high and low income (X) are perpendicular to the prototypes for applicant sex (circles).}
\label{fig:adult}
\end{figure}
\begin{table}[t]
\centering
\caption{Fairness ablation results on the German dataset. Alignment enforced causal independence, and additionally including the KL loss enforced distributional independence. All std. err. $<0.02$.}
\begin{tabular}{lcccc}
Loss & $y$ Acc. & D.I. & DD-0.5 & $\rho$ \\
\hline
KL $+$ Align & 0.73 & \textbf{0.70} & \textbf{0.10} & 0.00\\
Align & 0.73 & 0.69 & 0.11 & 0.00\\
KL & 0.72 & 0.67 & 0.12 & 0.20\\
Neither & \textbf{0.75}& 0.64&0.13 & 0.13\\
\end{tabular}
\label{tab:ablation}
\end{table}
Table~\ref{tab:ablation} demonstrates the necessity of both KL and alignment losses to train fair predictors (with higher disparate impact and lower demographic disparity values).
Including both loss terms resulted in the fairest predictors; removing those losses could enable better classification accuracy, but at the expense of fairness.
This confirms geometric intuition: the alignment loss created orthogonal subspaces and the KL regularization created distributional equivalence based on the subspaces.
Jointly, these losses therefore produced statistical independence.
Table~\ref{tab:ablation} also includes causal analysis of trained CSNs via the $\rho$ metric.
Intuitively, this metric reflected the learned correlation between $s$ and $y$; it was calculated by updating embeddings in the CSN latent space along the gradient of $s$ and recording the change in prediction over $y$.
We reported the ratio of these changes as $\rho$; as expected, enforcing orthogonality via alignment loss led to $\rho$ values of 0.
This technique is inspired by work in causally probing language models (e.g., \citet{whatif}); full details for calculating $\rho$ are included in Appendix~\ref{app:rho}.
\subsection{Hierarchical Classification}
We compared CSNs to our implementation of HPNs and results for Metric-Guided Prototype Learning (MGP), reported by \citet{metricprotos}, for hierarchical classification tasks.
Our HPN baseline used the same architecture as CSN (same encoder, decoder, and number of prototypes).
It differed from CSNs by setting alignment losses to 0 and by adopting the conditional probability training loss introduced by \citet{hierarchy}.
We further included results of a randomly-initialized CSN under ``Init.'' in tables.
In these experiments, we sought to test the hypothesis that CSNs with highly aligned subspaces would support hierarchical classification, just as orthogonal subspaces enabled fair classification.
\begin{table}[t]
\centering
\caption{MNIST digit hierarchy mean (stdev) over 10 trials. First two columns $\times$ 100.}
\begin{tabular}{lcccc}
& $Y_0$\% & $Y_1$\% & A.C. & E.D. \\
\hline
CSN & 98 (0) & 99 (0) & 0.06 (0.00) & 3.4 (4.3) \\
HPN & 96 (0) & 98 (0) & 0.12 (0.01) & 23 (1.0) \\
Init. & 10 (0) & 50 (1) & 2.80 (0.01) & 20 (0.0) \\
\end{tabular}
\label{tab:digit_hierarchy_results}
\end{table}
\begin{table}[t]
\centering
\caption{MNIST fashion hierarchy mean (stdev) over 10 trials. First two columns $\times$ 100.}
\begin{tabular}{lcccc}
& $Y_0$\% & $Y_1$\% & A.C. & E.D. \\
\hline
CSN & 88 (0) & 98 (0) &0.28 (0.01) & 0.0 (0.0) \\
HPN & 88 (0) & 98 (0) &0.28 (0.01) & 24 (0.9) \\
Init. & 10 (0) & 33 (1) & 3.13 (0.02) & 30 (0.8) \\
\end{tabular}
\label{tab:fashion_hierarchy_results}
\end{table}
In addition to standard accuracy metrics, we measured two aspects of CSNs trained on hierarchical tasks.
First, we recorded the ``average cost'' (AC) of errors.
AC is defined as the mean distance between the predicted and true label in a graph of the hierarchical taxonomy (e.g., if true and predicted label shared a common parent, the cost was 2; if the common ancestor was two levels up, the cost was 4, etc.) \citep{metricprotos}.
Second, we measured the quality of trees derived from the learned prototypes.
After a CSN was trained, we defined a fully-connected graph $\bm{G} = (\bm{V}, \bm{E})$ with vertices $\bm{V} = \mathbbm{P} \bigcup \{\bm{0}\}$ (the set of all prototypes and a point at the origin) and undirected edges between each node with lengths equal to the $l2$ distance between nodes in the latent space.
We recovered the minimum spanning tree, $\bm{T}$, from $\bm{G}$, (which is unique given distinct edge lengths, which we observed in all experiments), and converted all edges to directed edges through a global ordering of nodes.
Lastly, we calculated the graph edit distance (ED) between isomorphisms of the recovered tree and the ground-truth hierarchical tree (with edges that obeyed the same ordering constraints) \citep{editdist}.
Intuitively, this corresponded to counting how many edges had to be deleted or added to the minimum spanning tree to match the taxonomy tree, ignoring edge lengths, with a minimum value of 0 for perfect matches.
\begin{table*}[!t]
\begin{minipage}{0.37\linewidth}
\centering
\includegraphics[width=\textwidth,trim=65 1 75 5,clip]{figures/digit_encodings.png}
\captionof{figure}{2D latent space for hierarchical digit classification creates clusters around even and odd prototypes (circles on the right and left, respectively) and digit prototypes (X).}
\end{minipage}
\hfill
\begin{minipage}{0.55\linewidth}
\centering
\caption{CIFAR100 hierarchy results using the standard two-level hierarchy (top half) or MGP's 5-level hierarchy (bottom half).}
\begin{tabular}{lcccc}
& $Y_0$\% & $Y_1$\% & A.C. & E.D. \\
\hline
CSN & \textbf{0.76} (0.0) & 0.85 (0.0) & \textbf{0.76} (0.02) & 11.2 (7) \\
HPN & 0.71 (0.0) & 0.80 (0.0) & 0.97 (0.04)& 165.0 (3) \\
Init. & 0.01 & 0.05 & 3.88 & 200\\
\hline
CSN & \textbf{0.78} (0.0) & 0.88 (0.0) &\textbf{ 0.91} (0.0) & 6.0 (8.2)\\
MGP & 0.76 & \_ & 1.05 & \_ \\
Init. & 0.01 & 0.05 & 7.33 & 258\\
\end{tabular}
\label{tab:cifar100_results}
\end{minipage}
\end{table*}
As a basic test of CSNs in hierarchical classification tasks, we created simple hierarchies from the MNIST Digit and Fashion datasets.
The Digit dataset used the standard low-level labels of digit, supplemented with high-level labels of parity (two classes); the Fashion dataset used the standard low-level labels for item of clothing, with a ternary label for a high-level classification of ``tops'' (t-shirts, pullovers, coats, and shirts), ``shoes'' (sandals, sneakers, and ankle boots), or ``other'' (trousers, dresses, and bags).
Mean results from 10 trials for both MNIST datasets were included in Tables~\ref{tab:digit_hierarchy_results} and \ref{tab:fashion_hierarchy_results}.
The HPN baselines were implemented using the same number of prototypes as the CSNs being compared against.
Both tables show that CSNs exhibit comparable or better accuracy than HPNs for both the low-level ($Y_0$) and high-level ($Y_1$) classification tasks.
In addition, the average cost (A.C.) and edit distance (E.D.) values show that CSNs recovered minimum spanning trees that nearly perfectly matched the ground truth tree, and that when CSNs did make errors, they were less ``costly'' than errors made by HPNs (although admittedly, a dominant force in A.C. is classification accuracy alone).
A visualization of a 2D latent space of a CSN trained on the Digit task is shown in Figure 4: encodings for particular digits clustered around prototypes for those digits (X), while prototypes for even and odd digits (circles) separated the digit clusters into the left and right halves of the latent space.
Visualizations of latent spaces for more fair and hierarchical classification tasks are included in Appendix~\ref{app:latents}; they confirmed the theoretical derivations of orthogonal and parallel subspaces.
Lastly, we trained 10 CSNs and HPNs on the substantially more challenging CIFAR100 dataset.
The dataset is inherently hierarchical: the 100 low-level classes are grouped into 20 higher-level classes, each of size 5.
Using a resnet18 encoder, pre-trained on ImageNet, in conjunction with 100 prototypes for low-level classification and 20 for high-level, we trained CSNs and HPNs.
CSNs additionally used an alignment loss weight of -10 to encourage parallelism between the two concept subspaces.
The mean results over 10 trials are shown in the top half of Table~\ref{tab:cifar100_results}.
We also compared CSNs to MGP and other hierarchical classifiers using the CIFAR100 dataset and a deeper hierarchy, consisting of 5 levels of sizes 100, 20, 8, 4, and 2, as done by \citet{metricprotos}.
The additional information provided by this deeper hierarchy resulted in improved classification performance.
Median results (as done by \citet{metricprotos}) for 10 CSNs using this dataset are shown in the bottom half of Table~\ref{tab:cifar100_results}.
Changing the hierarchy changed how average cost was calculated, so values from the top and bottom halves of the table should not be compared.
Within the bottom half, we note that CSNs outperformed MGP on both A.C. and classification accuracy.
Furthermore, according to values generated in the extensive experiments conducted by \citet{metricprotos}, CSNs outperformed numerous other baselines, including HXE and soft-labels (\citet{hxe}), YOLO (\citet{yolo}), and a hyperspherical prototype network (\citet{hyperproto}), all of which were built upon a resnet18 pretrained on ImageNet.
In fact, our CSNs achieve SOTA classification accuracy for any classifier built upon a resnet18 backbone, without data augmentation.
Furthermore, the decrease in A.C. is especially surprising given that other techniques explicitly optimized for average cost reductions, while CSNs merely trained on classification at each level.
Notably, the decrease in A.C. is not fully explained by the increase in accuracy, indicating that CSN not only exhibited higher accuracy but also, when it did make mistakes, those mistakes were less severe.
Lastly, we note that CSNs support a range of learned relationships other than fair or hierarchical.
The varying values of $\rho$ in Table~\ref{tab:ablation} indicate that CSNs may learn different relationships when alignment loss is set to 0.
In general, one could train models to learn desired relationships by penalizing or rewarding alignment relative to some intercept.
We trained and evaluated such models and found that models indeed learned the desired alignment (Appendix~\ref{app:correlated}).
\subsection{Fair and Hierarchical Classification}
\label{sec:fairhierresults}
Prior experiments demonstrated how CSNs could solve different classification problems separately; in this section, we applied a single CSN to a task that required it to use both fair \textit{and} hierarchical classification.
Intuitively, fairness was used to protect privacy, while hierarchical structure was used for better performance.
We used a dataset describing human motion in a bolt-placement task.
The dataset was gathered from a similar setup to \citep{pembolts} - motion was recorded at 50 Hz, using the 3D location of each of the 8 volunteer participant's gloved right hands as they reached towards one of 8 holes arranged in a line to place a bolt in the hole.
The bolt holes may be thought of hierarchically by dividing destinations into left vs. right (LR) groupings, in addition to the label of the specific hole.
\begin{figure}[ht]
\centering
\begin{subfigure}[t]{0.48\textwidth}
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=\textwidth, trim={0, 0, 1cm, 2.1cm}, clip]{figures/3sets.png}};
\node[] (z_lab) at (4.4, 5.48) {$z$};
\node[] (left) at (1.6, 2.1) {left};
\node[] (right) at (5.7, 2.9) {right};
\node[rotate=13] (lr) at (3.4, 2.4) {$L/R$ subspace};
\node[rotate=13] (bolt) at (4.1, 1.3) {Bolt subspace};
\node[rotate=-77] (part) at (7.3, 4.1) {Participant subspace};
\end{tikzpicture}
\end{subfigure}
\caption{In a simplified depiction of the desired concept relations for fair and hierarchical classification, two subspaces ($L/R$ and bolt) are parallel, and both are orthogonal to the participant id subspace. Only 3 participant prototypes depicted for simplicity.}
\label{fig:3sets}
\end{figure}
\begin{table*}[t]
\centering
\caption{CSNs for fair and hierarchical classification on bolt task. Mean (std. dev.) over 10 trials.}
\begin{tabular}{llccccc}
Fair & Hier & Bolt\% & $\rho$ & A.C. & D.I. & DD-0.5 \\
\hline
No & No & 0.81 (0.01) & 0.36 (0.07) & 2.7 (0.05) & 0.80 (0.02) & 0.13 (0.02)\\
Yes & No & 0.41 (0.07) & $10^{-3}$($<10^{-3}$) & 2.8 (0.10) & 0.93 (0.03)& 0.05 (0.02)\\
Yes & Yes & 0.55 (0.06) & $10^{-3}$ ($<10^{-3}$) & 2.4 (0.05) & 0.92 (0.02) & 0.06 (0.02)\\
\end{tabular}
\label{tab:fairhier}
\end{table*}
Initial exploration of the dataset showed promising results for prototype-based classification: the target locations were identified with 81\% accuracy, and further analysis showed that prototypes corresponded to human-like motions (details in Appendix~\ref{sec:decoded_prototypes}).
Troublingly, however a trained CSN could identify the participant with over 60\% accuracy, which posed privacy concerns.
Nevertheless, from a robotic safety perspective, it is important for robots to exploit as much information as possible to avoid collisions with humans.
Ultimately, we wished to predict which hole a participant was reaching towards given the past 1 second of their motion, while preserving privacy and exploiting hierarchical structure.
Thus, we designed a CSN with three concept subspaces: one for predicting the bolt (8 prototypes), one for predicting the high-level grouping of a left or right destination (2 prototypes) and one for predicting the participant id (8 prototypes).
We enforced that the bolt and LR subspaces were parallel while the bolt and participant subpsaces were orthogonal.
A simplified version of the desired subspace arrangement is shown in Figure~\ref{fig:3sets}.
Results from our experiments are presented in Table~\ref{tab:fairhier}.
Means and standard deviations over 10 trials for each row are reported.
(A.C. was calculated only for prediction errors, due to the large differences in accuracy rates across models.)
When training a CSN with no constraints on subspace alignment, we found a highly accurate but unfair predictor (81\% accuracy for bolt location, but sub-optimal disparate impact and demographic disparity values).
Switching the CSNs to be fair classifiers by only enforcing orthogonality between bolts and participants yielded a fair classifier (illustrated by $\rho$, D.I., and DD-0.5), but with much worse bolt prediction accuracy (41\%).
However, by using a hierarchical subspace for LR groupings, the final CSN both improved classification accuracy and decreased the average cost of errors, while maintaining desired fairness characteristics.
\section{Contributions}
\label{sec:contributions}
The primary contribution of this work is a new type of model, the Concept Subspace Network, that supports inter-concept relationships.
CSNs' design, motivated by prior art in interpretable neural network models, use sets of prototypes to define concept subspaces in neural net latent spaces.
The relationships between these subspaces may be controlled during training in order to guide desired model characteristics.
Critically, we note that two popular classification problems --- fair and hierarchical classification --- are located at either end of a spectrum of concept relationships, allowing CSNs to solve each type of problem in a manner on par with techniques that had previously been designed to solve only one.
Furthermore, a single CSN may exhibit multiple concept relationships, as demonstrated in a privacy-preserving hierarchical classification task.
While we have demonstrated the utility of CSNs within several domains, numerous extensions could improve their design.
Subspace alignment could be applied to non-Euclidean geometries like hyperbolic latent spaces that are sometimes used for hierarchical classification.
CSNs could additionally benefit from relaxation of some simplifying assumptions: notably, allowing for more complex relationships rather than those defined by subspace cosine similarity, or using adversarial approaches for distributional regularization rather than only supporting unit Gaussians.
Lastly, we note that CSNs, while designed with ethical applications such as fair classification in mind, may lead to undesired consequences.
For example, malicious actors could enforce undesirable concept relationships, or simply observing emergent concept relationships within a CSN could reinforce undesirable correlations.
In addition, although prototypes encourage interpretability, which we posit can be used for good, the reductive nature of prototypes may be problematic when classifying human-related data (e.g., the COMPAS fair classification task we avoided).
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{"url":"https:\/\/alexisfraser.com\/creightons-corners\/small-scale-lab-9-application-isotopes-of-pennium.php","text":"# Small Scale Lab 9 Application Isotopes Of Pennium\n\nUGA Laboratory of Archaeology Athens Georgia. The application of mercury (Hg) stable isotopes to problems in environmental chemistry is relatively new, but is a rapidly expanding field of investigation., from Hg amalgamation during artisanal small-scale gold mining for the application of Hg isotopes to determine the origin and postdepositional processing of Hg..\n\n### Methyl mercury and stable isotopes of nitrogen reveal\n\nUniversity of Missouri and MU Research Reactor Center. 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SMALL-SCALE EXPERIMENT ISOTOPES AND ATOMIC MASS Small-Scale Experiment for text Section 4.3 OBJECTIVES\n\nthe burning of fossil fuels and small-scale Methyl mercury and stable isotopes in emergent Methyl mercury and stable isotopes of nitrogen reveal Small Scale Lab #23: Electrolytes (Supplement) One reason we are doing this Time in minutes 16 15 14 Pennium Lab. Taken from Prentice Hall Small Scale Lab 9\n\nRead chapter 3 Uses of Helium: of helium in large-scale applications is the possible development of substitutes magnets of a small laboratory scale, Application. activity assay (3) Clarisolve\u00ae 60 \u00b5m Lab Scale Depth Filters, Polygard \u00ae CR Small Scale Capsule Filter 100.0 \u00b5m.\n\nRATIONALE: Stable isotope ratios of carbon and nitrogen are valuable tools for field ecologists to use to analyse animal diets. However, the application of these results depend on small differences of hydraulic heads and small\u2010scale Stable isotopes of water (J. J. Gibson, Birks, The additional application of a radon\n\n### The use of stable isotopes to trace small-scale\n\nscale Sigma-Aldrich. Sigma-Aldrich is pleased to offer Diazald\u00ae Stable Isotopes; the use of MNNG has historically been limited to small-scale production of diazomethane, ... The small\u2010scale preparation and NMR characterization of isotopically enriched small-scale laboratory isotope of tin. For these applications,.\n\nHalf-Life Paper M&M\u0432\u0402\u2122s Pennies or Puzzle Pieces - ANS. Experimental evidence shows no fractionation of and herbivores: implications for tracking wildlife and Small-scale variation in 87 Sr\/ 86 Sr, 9 10 11 12 13 14 15 16 17 18 19 20 Calculations: 1. Inspect your data carefully. Determine the number of isotopes of \u201cpennium\u201d that are present. (Look for differences in mass of at LEAST 0.4 grams. Remember that pennies can pick up or lose small amounts mass from being in circulation) 2..\n\n### Isotope Beta-Battery Approaches for Long-Lived\n\nIsotopes of \u0432\u0402\u045aPennium\u0432\u0402\u045c Methacton School District. You will then use this information to determine the atomic mass of pennium. Recall that the atomic mass of an element is the weighted average of the masses of the isotopes of the element. The average is based on both the mass and the relative abundance of each isotope as it occurs in nature. 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Continue reading 1.4 Isotopes, Radioisotopes, And Atomic Mass Book And Download Isotopes And Atomic Mass; Small-Scale Experiment for Small-Scale Lab:\n\nMercury Stable Isotope Fractionation during Small-scale studies of roasted the extent of Hg stable isotope fractionation during reduction of ionic RATIONALE: Stable isotope ratios of carbon and nitrogen are valuable tools for field ecologists to use to analyse animal diets. However, the application of these\n\nGMP-compliant 68 Ga radiolabelling in a conventional small-scale radiopharmacy: a feasible approach for on the small-scale \u2018in-house and Isotopes in results depend on small differences of hydraulic heads and small\u2010scale Stable isotopes of water (J. J. Gibson, Birks, The additional application of a radon\n\nSmall-scale Sr isotope investigation of clinopyroxenes from peridotite xenoliths by laser ablation MC-ICP-MS first reported the application of laser ablation = 7 neutrons and is called carbon-13 isotope 9% abundance Mass of small brown bean = Isotopes and Atomic Mass Lab, or Beanium Lab Author:\n\nThe application of mercury (Hg) stable isotopes to problems in environmental chemistry is relatively new, but is a rapidly expanding field of investigation. Nitrate sources and watershed denitrification inferred from nitrate dual large scale N cycling. Here, the dual isotopes, of small-scale measurements to\n\nThe application of mercury (Hg) stable isotopes to problems in environmental chemistry is relatively new, but is a rapidly expanding field of investigation. JESSICA Y. LUO National Center for Aqaba, Red Sea: Analysis of carbon and nitrogen stable isotopes and trace metal concentrations a small-scale front.\n\nThe use of stable isotopes to trace small-scale movements by small fish species. of any stable isotope application, in ice and frozen in the laboratory. Isotope Beta-Battery Approaches for Long-Lived small scale concepts for compact isotope batteries were for Long-Lived Sensors: Technology Review\n\nIsotope Beta-Battery Approaches for Long-Lived small scale concepts for compact isotope batteries were for Long-Lived Sensors: Technology Review ... The small\u2010scale preparation and NMR characterization of isotopically enriched small-scale laboratory isotope of tin. For these applications,\n\nInternational Isotopes small scale qualification samples of high purity fluoride gas for various industrial applications, as well as development of laboratory Dominique Weis Professor Canada Rahier A. Methodology and Application of Hafnium Isotopes in Ilmenite and Rhodes M. Rapid passage of a small-scale \u2026\n\nThis collaboration is an initial step toward establishing a large-scale U.S. supply of medical isotopes. Integrated small-scale system License Application to UGA Laboratory of Archaeology, CAIS-Center for Applied Isotope Studies. January 14, The deadline for submission of the application is September 1, 2018.\n\nThe first application of this isotope system to Two other isotopes that have been of have strong implications on the small-scale structure of Small Scale Lab #23: Electrolytes (Supplement) One reason we are doing this Time in minutes 16 15 14 Pennium Lab. Taken from Prentice Hall Small Scale Lab 9\n\nJESSICA Y. LUO National Center for Aqaba, Red Sea: Analysis of carbon and nitrogen stable isotopes and trace metal concentrations a small-scale front. Nitrate sources and watershed denitrification inferred from nitrate dual large scale N cycling. Here, the dual isotopes, of small-scale measurements to\n\nDespite the increasing application of isotope tracers in field settings, 107 research has been conducted using Hg isotopes in small-scale laboratory experiments to Continue reading 1.4 Isotopes, Radioisotopes, And Atomic Mass Book And Download Isotopes And Atomic Mass; Small-Scale Experiment for Small-Scale Lab:\n\nCentral Appalachian Basin Unconventional (Coal\/Organic Shale) Reservoir Small -Scale provide guidance for commercialization applications of After completing this lab, you will determine the relative abundance of the isotopes of pennium and the average mass of each isotope. You will then use this information to determine the . atomic mass. of pennium. Recall that the atomic mass of an element is the weighted average of the masses of the isotopes of the element.\n\nGMP-compliant 68 Ga radiolabelling in a conventional small-scale radiopharmacy: a feasible approach for on the small-scale \u2018in-house and Isotopes in Applications for water detritiation: \u207b \u201clab-scale\u201d \u2022 At small scale, up to about 0.1m diameter,\n\nInternational Isotopes Inc. stock Lab. Feb. 4, 1998 at 12 The Fluorine Product segment involved the production of small scale qualification samples of high Isotope Production at High Energy high value isotopes, o Can be selective, rapid at small scale, adaptable to remote\n\nNitrate sources and watershed denitrification inferred from nitrate dual large scale N cycling. Here, the dual isotopes, of small-scale measurements to Erbium has nuclear and metallurgical applications, Isotopic separation on a small scale is performed using a In comparison, isotope shifts are small in","date":"2021-05-07 14:06:51","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\": 0.49973222613334656, \"perplexity\": 6905.524185097595}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243988793.99\/warc\/CC-MAIN-20210507120655-20210507150655-00036.warc.gz\"}"}
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Q: how to include logic in JSX to convert object? I have an array of objects populated by a webservice call
const [items, setItems] = useState([])
fetch('https://jsonplaceholder.typicode.com/users/')
.then(response => response.json())
.then(json => setItems(json))
then I would display it as such:
return (
<div className="App">
{ items.map(item => {
return <pre> {JSON.stringify(item)}</pre>
})}
</div>
);
}
The problem is sometimes that webservice returns a single object instead of an array, for example https://jsonplaceholder.typicode.com/users/1
in that case I'd want to check if it's an object and if so, convert it to an array, by using [items].map
But how would I put that logic into the JSX as shown above?
A: return (
<div className="App">
{(items instanceof Array ? items : [items]).map(item => {
...
You could also do something above before you return such as:
if (!(items instanceof Array)) {
items = [items];
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 333
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Q: Augmented reality using artoolkit I have an scenario where i want to render product image on markers. But all the details of that particular product (like its .obj , .mtl file) will come from webservice. So is it possible to render image of product at runtime using artoolkit ? Is dynamic image rendering is possible using artoolkit native ?
A: In short, it depends of your tool, not of ARToolKit. ARToolKit provides pose information for markers, what you do with them depends of the tool you're using.
I assume you are using Unity, then you can have a plane with a texture and make the texture dynamic based on the downloaded file.
I haven't done that in Unity, but I'll be very surprised if it wasn't possible.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 7,618
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{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/dcds.2009.25.1061","text":"Article Contents\nArticle Contents\n\n# Estimating thermal insulating ability of anisotropic coatings via Robin eigenvalues and eigenfunctions\n\n\u2022 The problem considered in this paper is the protection from overheating of a thermal conductor $\\Omega_1$ by a thin anisotropic coating $\\Omega_2$ (e.g. a space shuttle painted with a nano-insulator). We assume Newton's Cooling Law, so the temperature satisfies the Robin boundary condition on the outer boundary of the coating. Since the temperature function on $\\Omega=\\overline{\\Omega}_1\\cup\\Omega_2$ can be expanded in terms of the eigenvalues and eigenfunctions of the elliptic operator $u\\mapsto -\\nabla (A \\nabla u)$ with the Robin boundary condition on $\\partial\\Omega$, where $A$ is the thermal tensor of $\\Omega$, we propose the following means to ensure the insulating ability of $\\Omega_2$: (A) as many eigenvalues as possible should be small, in particular, the first eigenvalue should be small, (B) the first normalized eigenfunction should take large values on the body $\\Omega_1$; we also argue that it is helpful for the understanding of the dynamics if (C) higher normalized eigenfunctions take small absolute values on $\\Omega_1$. We assume that the thermal conductivity of $\\Omega_2$ is small either in all directions or at least in the direction normal to $\\partial\\Omega_1$ (the case of \"optimally aligned coating\"). We study the asymptotic behavior of Robin eigenpairs as outcome of the interplay of the thermal tensor $A$, the thickness of $\\Omega_2$ and the thermal transport coefficient in the Robin boundary condition, in the singular limit when either the thermal conductivity of $\\Omega_2$, or the thickness of $\\Omega_2$, or the thermal transport coefficient approaches $0$. By doing so, we identify the parameter ranges in which some or all of (A)-(C) occur.\nMathematics Subject Classification: Primary: 35J05, 35J20; Secondary: 80A20, 80M30.\n\n Citation:","date":"2023-01-31 03:30:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9433295130729675, \"perplexity\": 281.3396305418964}, \"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\/1674764499842.81\/warc\/CC-MAIN-20230131023947-20230131053947-00655.warc.gz\"}"}
| null | null |
Spring Series: Spring JDBC – Review/Interview Questions – A thought with a cup of coffee!!!
What are the core packages provided by Spring to work with JDBC?
What is the difference between SimpleJDBCTemplate and NamedParamJDBCTemplate?
How one can translate the exception using Spring JDBC classes?
What are the essential callback interfaces to be implemented, to use JDBC Template?
Where is Collections.singletonMap(…) method used and why?
Where does one use ParameterizedRowMapper?
When do one uses executeAndReturnKey method and where is it define?
How to insert data into a table, where table is having five columns but you have to insert data just in three columns (obviously the rest two columns allows null values)?
What is the better way to insert values in table, if the values are provided in the form of java bean instance?
How MapSQLParameterSource is different from BeanPropertySQLParameter?
How to call procedure using Spring?
How to ensure case insensitive look-up for parameters name when calling procedure?
What is the way to avoid metadata look-up from spring controller when calling procedure?
How to call database functions in spring?
[CDOS in short] core, datasource, object, support.
SimpleJDBCTemplate takes advantages of java 5 features – varargs, autoboxing.
One example is when one uses NamedParameterJdbcTemplate, singletonmap used to provide value of parameter.
When code selects data from database using SimpleJdbcTemplate class.
If developer needs to implement auto generated keys.
Either use usingColumns(…) method or MapSQLParameterSource.
BeanPropertySQLParameterSource uses java bean instance (value object) to store data while MapSQLParameterSource uses addValue method to set values of columns.
Using withProcedureName(String name_of_procedue) provided in SimpleJdbcCall class.
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{
"redpajama_set_name": "RedPajamaC4"
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| 4,918
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package com.graphhopper.reader.osm;
import com.carrotsearch.hppc.LongArrayList;
import com.carrotsearch.hppc.cursors.IntCursor;
import com.carrotsearch.hppc.cursors.LongCursor;
import com.graphhopper.reader.ReaderElement;
import com.graphhopper.reader.ReaderRelation;
import com.graphhopper.storage.BaseGraph;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import java.util.Iterator;
import java.util.Map;
import java.util.function.LongFunction;
public class RestrictionConverter {
private static final Logger LOGGER = LoggerFactory.getLogger(RestrictionConverter.class);
private static final long[] EMPTY_LONG_ARRAY_LIST = new long[0];
public static boolean isTurnRestriction(ReaderRelation relation) {
return "restriction".equals(relation.getTag("type"));
}
public static long[] getRestrictedWayIds(ReaderRelation relation) {
if (!isTurnRestriction(relation))
return EMPTY_LONG_ARRAY_LIST;
return relation.getMembers().stream()
.filter(m -> m.getType() == ReaderElement.Type.WAY)
.filter(m -> "from".equals(m.getRole()) || "via".equals(m.getRole()) || "to".equals(m.getRole()))
.mapToLong(ReaderRelation.Member::getRef)
.toArray();
}
public static long getViaNodeIfViaNodeRestriction(ReaderRelation relation) {
return relation.getMembers().stream()
.filter(m -> m.getType().equals(ReaderElement.Type.NODE))
.filter(m -> "via".equals(m.getRole()))
.mapToLong(ReaderRelation.Member::getRef)
.findFirst()
.orElse(-1);
}
/**
* OSM restriction relations specify turn restrictions between OSM ways (of course). This method converts such a
* relation into a 'graph' representation, where the turn restrictions are specified in terms of edge/node IDs instead
* of OSM IDs.
*
* @throws OSMRestrictionException if the given relation is either not valid in some way and/or cannot be handled and
* shall be ignored
*/
public static Triple<ReaderRelation, GraphRestriction, RestrictionMembers> convert(ReaderRelation relation, BaseGraph baseGraph, LongFunction<Iterator<IntCursor>> edgesByWay) throws OSMRestrictionException {
if (!isTurnRestriction(relation))
throw new IllegalArgumentException("expected a turn restriction: " + relation.getTags());
RestrictionMembers restrictionMembers = extractMembers(relation);
if (!membersExist(restrictionMembers, edgesByWay, relation))
throw OSMRestrictionException.withoutWarning();
// every OSM way might be split into *multiple* edges, so now we need to figure out which edges are the ones
// that are actually part of the given relation
WayToEdgeConverter wayToEdgeConverter = new WayToEdgeConverter(baseGraph, edgesByWay);
if (restrictionMembers.isViaWay()) {
WayToEdgeConverter.EdgeResult res = wayToEdgeConverter
.convertForViaWays(restrictionMembers.getFromWays(), restrictionMembers.getViaWays(), restrictionMembers.getToWays());
return new Triple<>(relation, GraphRestriction.way(res.getFromEdges(), res.getViaEdges(), res.getToEdges(), res.getNodes()), restrictionMembers);
} else {
int viaNode = relation.getTag("graphhopper:via_node", -1);
if (viaNode < 0)
throw new IllegalStateException("For some reason we did not set graphhopper:via_node for this relation: " + relation.getId());
WayToEdgeConverter.NodeResult res = wayToEdgeConverter
.convertForViaNode(restrictionMembers.getFromWays(), viaNode, restrictionMembers.getToWays());
return new Triple<>(relation, GraphRestriction.node(res.getFromEdges(), viaNode, res.getToEdges()), restrictionMembers);
}
}
private static boolean membersExist(RestrictionMembers members, LongFunction<Iterator<IntCursor>> edgesByWay, ReaderRelation relation) {
for (LongCursor c : members.getAllWays())
if (!edgesByWay.apply(c.value).hasNext()) {
// this happens for example at the map borders or when certain ways like footways are excluded
LOGGER.debug("Restriction relation " + relation.getId() + " uses excluded way " + c.value + ". Relation ignored.");
return false;
}
return true;
}
public static void checkIfCompatibleWithRestriction(GraphRestriction g, String restriction) throws OSMRestrictionException {
if (g.getFromEdges().size() > 1 && !"no_entry".equals(restriction))
throw new OSMRestrictionException("has multiple members with role 'from' even though it is not a 'no_entry' restriction");
if (g.getToEdges().size() > 1 && !"no_exit".equals(restriction))
throw new OSMRestrictionException("has multiple members with role 'to' even though it is not a 'no_exit' restriction");
}
public static RestrictionMembers extractMembers(ReaderRelation relation) throws OSMRestrictionException {
// we use -1 to indicate 'missing', which is fine because we exclude negative OSM IDs (see #2652)
long viaOSMNode = -1;
LongArrayList fromWays = new LongArrayList();
LongArrayList viaWays = new LongArrayList();
LongArrayList toWays = new LongArrayList();
for (ReaderRelation.Member member : relation.getMembers()) {
if ("from".equals(member.getRole())) {
if (member.getType() != ReaderElement.Type.WAY)
throw new OSMRestrictionException("has a member with role 'from' and type '" + member.getType() + "', but it should be of type 'way'");
fromWays.add(member.getRef());
} else if ("to".equals(member.getRole())) {
if (member.getType() != ReaderElement.Type.WAY)
throw new OSMRestrictionException("has a member with role 'to' and type '" + member.getType() + "', but it should be of type 'way'");
toWays.add(member.getRef());
} else if ("via".equals(member.getRole())) {
if (member.getType() == ReaderElement.Type.NODE) {
if (viaOSMNode >= 0)
throw new OSMRestrictionException("has multiple members with role 'via' and type 'node', but multiple via-members are only allowed when they are of type: 'way'");
// note that we check for combined usage of via nodes and ways later on
viaOSMNode = member.getRef();
} else if (member.getType() == ReaderElement.Type.WAY) {
// note that we check for combined usage of via nodes and ways later on
viaWays.add(member.getRef());
} else
throw new OSMRestrictionException("has a member with role 'via' and" +
" type '" + member.getType() + "', but it should be of type 'node' or 'way'");
} else if ("location_hint".equals(member.getRole())) {
// location_hint is deprecated and should no longer be used according to the wiki, but we do not warn
// about it, or even ignore the relation in this case, because maybe not everyone is happy to remove it.
} else if (member.getRole().trim().isEmpty())
throw new OSMRestrictionException("has a member with an empty role");
else
throw new OSMRestrictionException("has a member with an unknown role '" + member.getRole() + "'");
}
if (fromWays.isEmpty() && toWays.isEmpty())
throw new OSMRestrictionException("has no member with role 'from' and 'to'");
else if (fromWays.isEmpty())
throw new OSMRestrictionException("has no member with role 'from'");
else if (toWays.isEmpty())
throw new OSMRestrictionException("has no member with role 'to'");
if (fromWays.size() > 1 && toWays.size() > 1)
throw new OSMRestrictionException("has multiple members with role 'from' and 'to'");
checkTags(fromWays, toWays, relation.getTags());
if (viaOSMNode >= 0 && !viaWays.isEmpty())
throw new OSMRestrictionException("has members with role 'via' of type 'node' and 'way', but only one type is allowed");
else if (viaOSMNode >= 0)
return RestrictionMembers.viaNode(viaOSMNode, fromWays, toWays);
else if (!viaWays.isEmpty())
return RestrictionMembers.viaWay(fromWays, viaWays, toWays);
else
throw new OSMRestrictionException("has no member with role 'via'");
}
private static void checkTags(LongArrayList fromWays, LongArrayList toWays, Map<String, Object> tags) throws OSMRestrictionException {
// the exact restriction value depends on the vehicle type, but we can already print a warning for certain
// cases here, so later we do not print such warnings for every single vehicle.
boolean hasNoEntry = false;
boolean hasNoExit = false;
for (Map.Entry<String, Object> e : tags.entrySet()) {
if (e.getKey().startsWith("restriction")) {
if (e.getValue() != null && ((String) e.getValue()).startsWith("no_entry"))
hasNoEntry = true;
if (e.getValue() != null && ((String) e.getValue()).startsWith("no_exit"))
hasNoExit = true;
}
}
if (fromWays.size() > 1 && !hasNoEntry)
throw new OSMRestrictionException("has multiple members with role 'from' even though it is not a 'no_entry' restriction");
if (toWays.size() > 1 && !hasNoExit)
throw new OSMRestrictionException("has multiple members with role 'to' even though it is not a 'no_exit' restriction");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,073
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In other news, Dallas Fed President Robert Kaplan says stimulus from the GOP's tax cuts is falling off and he expects the economy will only grow by 2 percent this year. Remember when Trump and the GOP said their tax cuts would deliver permanent 4 percent growth?
Meanwhile, British Prime Minister Theresa May is vowing to "deliver [Brexit] on time." So, that's great? I mean, that's like vowing to deliver a recession on time. Wouldn't want to be late for that, right?
Finally, New Mexico Governor Michelle Lujan Grisham (D) has recalled the National Guard troops deployed to the border by former Governor Susana Martinez (R).
"We will support our neighbors where the need for assistance is great, and we will offer a helping hand when we can to those vulnerable people who arrive at our border, but New Mexico will not take part in the president's charade of border fear-mongering by misusing our diligent National Guard troops," her statement said.
Programming note... I'm signing off a little early today because I feel like hell. I may or may not be online tomorrow, depending on how I feel in the morning. If not, have a good weekend.
Be well, sir. Maybe a LOTR or Hobbit marathon to get you through the ick.
LOTR, sure. But Hobbit? He's trying to get better, not worse.
I really like the Hobbit Movies and maintain that they're better than they get credit for.
They had their charms, but were far, far too padded to turn them into three movies.
Lord of the Rings is three volumes, The Hobbit is one. The movies should have followed suit.
I disagree. I want to spend more time in that world, not less. So three movies is very much okay by me. Books to movies rarely is a direct adaptation so to think a movie will be exactly like a book is not always the best way to view the movies. Also the number of books equal to the number of movies doesn't work either. Some stories are so massive and detailed that one movie would be an injustice. I also feel that the movies made more sense than the book and were better told (that goes for LOTR as well). In short, I've found (and many do disagree with me on this) that Peter Jackson is better film crafter than Tolkien is a writer. This in no way should be viewed as me thinking Tolkien is a bad writer…he's not. I also adore the detail he put into making Middle-Earth and the history of it along with the linguistic sculpting (though I have always had an issue with his use of the Gregorian calendar but I get why he did). My personal preference in absorbing the stories is through the movies. All of that said, I do respect your opinion and understand your view on this.
I just wish that, for the Hobbit, they'd stayed to The Hobbit. Put all the other stuff in some other movie. Make the Silmarillion or Farmer Giles of Ham or whatever if they want. But don't bloat what could otherwise have been a fine (and better) movie in it's own right.
For this one, I wanted to see the self-contained adventure of Bilbo, not the whole history of Middle Earth.
Feel better. I'm on pain meds for metal hardware, made the mistake of bingeing "the Expanse" last night til 4, it's "welcome to horrible occipital headache" all day today for me. These can last days, so I'm sympathizing.. I also watched the second season of the British sci fi "Humans," was reassured that it is equally good or better than the first season. I'd like them to make more episodes each season, but perhaps that's why the stories are so good. I'm really enjoying Carrie Anne Moss (who looks just fabulous) as an AI scientist. One day I'll have to invest in cable or buy up some more British shows, they have some very nice looking previews of various other series on the Blu-ray of Humans 2.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 169
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The influenza season 2014/15 started in Europe in week 50 2014 with influenza A(H3N2) viruses predominating. The majority of the A(H3N2) viruses characterised antigenically and/or genetically differ from the northern hemisphere vaccine component which may result in reduced vaccine effectiveness for the season. We therefore anticipate that this season may be more severe than the 2013/14 season. Treating influenza with antivirals in addition to prevention with vaccination will be important.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,059
|
Q: How to custom v-for using v-if for make a class in div tag How to correctly way use v-if when I using v-for inside?
Actually I want to add a condition when the index is 0 or first data is displayed I want to add active class
<div class="item active" v-for="(item, key, index) in slideItem" :key="item._id">
<img alt="900x500" src="http://lorempixel.com/960/720/">
<div class="carousel-caption">
<h3>{{ item.title }}</h3>
<p>{{ item.body }}</p>
</div>
</div>
and show this when the next data :
<div class="item" v-for="(item, key, index) in slideItem" :key="item._id">
<img alt="900x500" src="http://lorempixel.com/960/720/">
<div class="carousel-caption">
<h3>{{ item.title }}</h3>
<p>{{ item.body }}</p>
</div>
</div>
You can see the two examples I gave, how I combine the two loops into one by distinguishing the first index (item active) condition and so on using v-if
A: You can make a function isActive that determines if the index is at 0 and than
v-bind:class="{'active': isActive}"
A: I found the problem.. I Think you are using a key, for an array... the key is used for a dictionary.
Here is how it works:
CSS
.active {
background-color: #00ff00;
}
HTML:
<div id="output">
<div class="item" v-for="(item, index) in slideItem" :key="item._id">
<div class="carousel-caption" :class="{active: index==1}">
<h3>{{ item.title }} {{index}}</h3>
<p>{{ item.body }}</p>
</div>
</div>
</div>
Notice that I removed the key!
JS:
var vm = new Vue({
el: "#output",
data: function () {
return {
slideItem : [
{ '_id' :0, 'title':'title0' ,'body':'body 0'},
{ '_id' :1, 'title':'title1' ,'body':'body 1'},
{ '_id' :2, 'title':'title2' ,'body':'body 2'},
{ '_id' :3, 'title':'title3' ,'body':'body 3'},
{ '_id' :4, 'title':'title4' ,'body':'body 4'},
{ '_id' :5, 'title':'title5' ,'body':'body 5'}
]
}
}
});
What ...
Here is the fiddle
A: You could do something like:
<div class="item" v-for="(item, k) in slideItem" :key="item._id"
:class="{active: Object.keys(slideItem)[0] == k}">
<img alt="900x500" src="http://lorempixel.com/960/720/">
<div class="carousel-caption">
<h3>{{ item.title }}</h3>
<p>{{ item.body }}</p>
</div>
</div>
Object.keys(slideItem)[0] == k gets the first key of the object and then checks that the key for current iteration matches.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,145
|
Q: Does my Naive Bayes training data need to be proportional? I'll use spam classification as an example. The canonical approach would be to hand-classify a random sampling of emails and use them to train the NB classifier.
Great, now say I added a bunch of archived emails that I know are not spam. Would this skew my classifier results because now the proportion of spam:not spam is no longer representative? The two ways I could think of this happening:
*
*The features become too non-spam heavy.
*The algorithm implicitly uses probability(spam) in its classification (in the same way that probability(medical condition) is devalued by the rarity of the medical condition even when the test is positive.
In general, more training data is better than less, so I'd like to add it if it doesn't break the algorithm.
A: You can train on all data, without worrying about proportionality. That said, as you observed, distorting the proportions distorts the probabilities and results in bad outcomes. If you have a 20% spam email flow and train a spam filter on 99% spam and 1% good email (ham), you're going to end up with a hyper-aggressive filter.
The common approach to this is two-step:
*
*Seed the filter by running a representative sample of data through it (say, 1,000 emails in the spam filter scenario).
*As the filter encounters additional data, only update the weights if the filter gets it wrong. This is called "train-on-error."
If you follow this approach, your filter will not get confused by a sudden burst of spam that just happens to include, say, the word "trumpet" alongside words that really are spammy. It will adjust only when necessary, but will catch up as quickly as it needs to when it is wrong. This is one way of preventing the "Bayesian poisoning" approach that most spammers now take. They can clutter up their messages with a lot of garbage, but they only have so many ways to describe their products or services, and those words will always be spammy.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,216
|
Please schedule AT LEAST FORTY-EIGHT HOURS AHEAD of your desired day for an abortion appointment.
First, fill in the requested information in the form below.
Gynecological Care Appointments are available only on select days.
Abortion Care Registration Appointments are available on select days. We will confirm your actual abortion care appointment during your registration appointment. We can schedule your abortion procedure appointment Tuesday, Thursday, Friday, and Saturday.
Select your preferred day & time for your registration appointment. This will be confirmed by email or phone within 48 hours by one of our Health Educators.
Use this field to upload a copy or picture of your insurance card (front & back).
If "other" gynecological visit, please describe.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 376
|
Madona Gasanova, The FINANCIAL
Golden Brand
Caraps Medline Expecting 15% Revenue Increase in 2014
The FINANCIAL -- Aesthetic, reconstructive and plastic surgery clinic Caraps Medline aims to have a 15% financial increase in 2014. The field of medicine is rapidly developing, Peride Khidesheli, Director of Caraps Medline, told The FINANCIAL. She says that consumer behaviour has transformed drastically since 1998.
"It is interesting how client behaviour has been transforming since 1998. Patients were unsure about entering the clinic at the beginning. There was a high demand for anonymity. Today, however, we have a radically different public attitude towards plastic surgery. Modern methods of self care have become prestigious for women. As the majority of females are involved in social activities it is almost obligatory to pay careful attention to their appearance. Accordingly, this field of medicine is rapidly developing," Khidesheli told The FINANCIAL.
Established in 1998 Caraps Medline is one of the pioneers in Georgia's private medical business.
Khidesheli distinguished 2012 as the most successful year for its company. "Our clinic changed its location and has since expanded. We moved from Vake to Dighomi, the medical district of Tbilisi. The new fashionable building was constructed for our clinic at Lubliana 48. We added a new direction of general surgery, led by Dr. Merab Kiladze."
Caraps Medline received the most influential business award in the country - Golden Brand - on 27 April.
"Our new clinic is very modern, well-equipped technically and fully satisfies all the demands of the 21st century. The new clinic has a great material and technical base. We have mainly developed our existing fields and added a new, general surgery profile. We already achieved the index that we had planned and stage by stage we will be adding new fields of medicine. We have a great, new infrastructure and the maximum level of comfort is provided here for our patients as well as for our doctors," Khidesheli said.
"In 2012-2013 we had almost equal capacity of over 80% with positive dynamics. The change of location did not result in significant fluctuations," said Khidesheli.
Caraps Medline established the innovative method of surgery, so-called bandaging of the stomach with laparoscopic methods. In 2014 two new departments are planned to be added. Prices on surgery are planned to be increased. If a patient does several surgeries in the same stage (dependent on their health), the price will exceed the regular amount and will be calculated due to the complexity of the individual case.
In Khidesheli's words, any political change influences the private sector more or less, even the universal programme of insurance. However, Caraps Medline has always managed to maintain stability.
Caraps Medline is frequently visited by patients from neighbouring countries. "Comparative prices and most importantly, high quality, is the main reason for our popularity abroad. We run various advertising campaigns in different countries in order to extend the number of our patients."
The so-called computer era of the 21st century has contributed to obesity problems especially among adolescents worldwide. However, Khidesheli said that the situation, fortunately, is not yet alarming in Georgia.
"Statistically, not so many surgeries are being done among adolescents. However, having a healthy lifestyle for developing healthy generations is very important," she added.
Caraps Medline offers medical services in the following fields: gynaecology, oncology (mamology), plastic and general surgery, trauma, ophthalmology, blood vessel surgery, and cosmetology.
"Each direction in our clinic is in demand because of the professionals who work here. In plastic surgery we have Dr. Alexander Kalantarov, in gynaecological laparoscopy - Dr. Vadim Khatiashvili, in general surgery - Dr. Merab Kiladze, and in traumatology - Vazha Marsagishvili," said Khidesheli.
"However, it is not just our leading doctors which are responsible for the success of the company, but each and every member of our staff, who do their utmost every single day for the success and smooth-running of the company. They do a great job for the clinic. We have wonderful medical personnel who are directly involved in the success of the clinic," Khidesheli added.
According to Khidesheli, rhinoplasty is the most popular plastic surgery in the region among both men and women.
The dark side of aesthetic clinics is cases of "victims" of plastic surgery that can always be found worldwide. Caraps Medline has never encountered such a situation though. However, Khidesheli finds it difficult to establish who would be to blame in such a situation. "Looking for the one at fault is very difficult, as the desired outcome of a plastic surgery operation is made in accordance with the patient's taste. Fortunately we have not ever had any so-called victims of plastic surgery during our history, and we are optimistic that there never will be any in the future."
"Doing surgery with less trauma and making them aesthetically effective has always been our doctors' main goal. Enhancing customer loyalty, ensuring their increasing demand and employing qualified professionals have always been the main priorities of the management of the clinic," said Khidesheli.
"We are highly oriented of satisfying our customers' needs. Accordingly, the implementations of any innovation are in line with customers' demands and priorities," she added.
Caraps Medline became a member of EVEX Medical Corporation in 2014. "We hope that this membership will help us to achieve ever greater success of larger scales," said Khidesheli.
According to Khidesheli, satisfied patients are the best advertisements. She says that if a patient is not content with the service you offer, then no amount of advertising can attract them.
"Most of our patients visit us on the recommendation of one of our previous patients. In each of the fields that we offer, we have leading professional doctors who provide the best results in their respective fields. Tbilisi is a small city so information spreads easily. I believe that satisfied patients are a guaranty of new clients," Khidesheli said.
"As our clinic is one of the winners of the Golden Brand awards, we will do our best to justify the honour. We will try to maintain the reputation that we have in the city and in society. We will continue to offer novelties as and when. We want to do everything new to the same level as everything we have done before. We plan to add some new fields as well. All private structures plan to expand and become bigger eventually. So we too will be working on this in the nearest future," Khidesheli told The FINANCIAL.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,270
|
Q: Discord v13 bot can't seem to identify what is a text channel So when I am trying to set a verify channel with my bot. It doesn't seem to be able to find identify the text channel
Code:
const db = require('quick.db');
const { PREFIX } = require('../../config');
module.exports = {
config: {
name: 'setverification',
aliases: ['sv', 'setv', 'setverify'],
category: 'moderation',
description: 'Sets Verification Channel And Role',
usage: '[channel name | channel ID | channel mention] <role name | role ID | role mention]',
accessableby: 'Administrators'
},
run: async (bot, message, args) => {
let prefix;
let fetched = await db.fetch(`prefix_${message.guild.id}`);
if (fetched === null) {
prefix = PREFIX
} else {
prefix = fetched
}
if (!message.member.permissions.has("ADMINISTRATOR")) return message.channel.send("**You Do Not Have Required Permissions! - [ADMINISTRATOR]!**");
if (!args[0]) return message.channel.send("**Please Enter A Channel Name Where The User Should Be Asked To Verify!**");
if (!args[1]) return message.channel.send("**Please Enter A Role Which Will Be Added After The User Is Verified!**");
let channel = message.mentions.channels.first() || message.guild.channels.cache.get(args[0]) || message.guild.channels.cache.find(c => c.name.toLowerCase() === args[0].toLocaleLowerCase());
if (!channel || channel.type !== "GUILD_TEXT") return message.channel.send("**Please Enter A Valid Channel!**")
let role = message.mentions.roles.first() || message.guild.roles.cache.get(args[1]) || message.guild.roles.cache.find(r => r.name.toLowerCase() === args[1].toLocaleLowerCase());
if (!role) return message.channel.send("**Please Enter A Valid Role!**");
let verifiedchannel = channel;
try {
let a = await db.fetch(`verificationchannel_${message.guild.id}`);
let b = await db.fetch(`verificationrole_${message.guild.id}`);
if (channel.id === a && role.id === b) {
return message.channel.send('**This Channel is Already Set As Verification Channel!**');
} else if (channel.id === a && role.id === b) {
return message.channel.send("**This Role is Already Set As Verification Role!**");
}
else {
message.guild.channels.cache.forEach(channel => {
if (channel.type === 'category' && channel.id === verifiedchannel.id) return;
let r = channel.permissionOverwrites.get(role.id);
if (!r) return;
if (r.deny.has("VIEW_CHANNEL") || r.deny.has("SEND_MESSAGES")) return;
channel.createOverwrite(message.guild.id, {
VIEW_CHANNEL: false
});
channel.updateOverwrite(role, {
VIEW_CHANNEL: true,
SEND_MESSAGES: true
});
});
verifiedchannel.updateOverwrite(role, {
SEND_MESSAGES: false,
VIEW_CHANNEL: false
});
bot.guilds.cache.get(message.guild.id).channels.cache.get(channel.id).send(`**Welcome To ${message.guild.name}!\nTo Get Verified Type - \`${prefix}verify\`**`);
db.set(`verificationchannel_${message.guild.id}`, channel.id);
db.set(`verificationrole_${message.guild.id}`, role.id);
return message.channel.send(`**Verification Channel And Role Has Been Set Successfully in \`${channel.name}\`!**`);
};
} catch {
return message.channel.send("**Error - `Missing Permissions Or Channel Is Not A Text Channel!`**");
};
}
};
It is able to know what channel it is sent and know the role (Bot's permissions are set to Administrator). I think it a deprecation problem. This code was made with v12
Code example:
A: GuildChannel#permissionOverwrites has changed. It used to return a Collection, but now it returns a PermissionOverwriteManager. Use .resolve on it instead:
let r = channel.permissionOverwrites.resolve(role.id)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,353
|
Live at Ali's Alley is a live album by trumpeter Ahmed Abdullah, listed simply as "Abdullah" on the cover. It was recorded on April 24, 1978, at New York City's Ali's Alley, and released on vinyl in 1980 by Cadence Jazz Records as the label's inaugural release. On the album, Abdullah is joined by saxophonist Chico Freeman, hornist Vincent Chancey, cellist Muneer Abdul Fatah, bassist Jerome Hunter, and drummer Rashied Sinan.
Reception
A reviewer for AllMusic stated that the album is "symbolic of the decade's 'loft' jazz, a free-wheeling date with uneven but often compelling solos, as well as periods of rambling, unproductive, and ragged ensemble work. Freeman's blistering tenor sax is uniformly inspired, while Abdullah's solos are also aggressive and energetic."
The authors of MusicHound Jazz: The Essential Album Guide called the recording "a loft-jazz classic with an unusual instrumentation including French horn and cello plus tenor saxophonist Chico Freeman in an inspired, fiery mood."
Track listing
Side A
"Happiness is Forever" (Ahmed Abdullah) – 27:42
Side B
"Self Portrait in Three Colors" (Charles Mingus) – 3:46
"The Inch Worm Part I" (Frank Loesser) – 10:45
"The Inch Worm Part II" (Frank Loesser) – 9:29
Personnel
Ahmed Abdullah – trumpet
Chico Freeman – tenor saxophone, flute
Vincent Chancey – French horn
Muneer Abdul Fatah – cello
Jerome Hunter – bass
Rashied Sinan – drums
References
1980 live albums
Ahmed Abdullah live albums
Cadence Jazz Records live albums
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,822
|
from django.test import TestCase, Client
from django.conf import settings
from django.contrib.auth.models import User
from tokenauth.authbackends import TokenAuthBackend
import requests
import responses
def mock_auth_success():
url = '{0}/users/me/' . format(settings.USER_SERVICE_BASE_URL)
response_string = '{"username": "TEST"}'
responses.add(responses.GET, url,
body=response_string, status=200,
content_type='application/json')
def mock_auth_failure():
url = '{0}/users/me/' . format(settings.USER_SERVICE_BASE_URL)
responses.add(responses.GET, url,
body='', status=401,
content_type='application/json')
class ProjectEndpointTestCase(TestCase):
def setUp(self):
self.c = Client(Authorization='Token 123')
@responses.activate
def test_get_microservices_list_requires_auth(self):
mock_auth_failure()
response = self.c.get("/api/v1/microservices/")
assert response.status_code == 403, 'Expect permission denied'
@responses.activate
def test_get_microservices_list(self):
mock_auth_success()
response = self.c.get("/api/v1/microservices/")
assert response.status_code == 200, 'Expect 200 OK'
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,889
|
Home / Healthy Tips / Few Know About This. The Health Benefits of Aluminium Foil.
Aluminum paper is used very often but its healing properties are known by very few. Aluminum foil is something that we all have at home and its main usage is for cooking or storing the food. However, it never crossed our mind that aluminum foil can be used to relieve joint pain or even combat the fatigue.
The trick is not difficult. The method was inspired by traditional Chinese medicine, then studied by Russian scientists. It is enough to cover the painful area with aluminum foil to make the pain go away. Aluminum paper is an effective anti-inflammatory. Place the foil around the painful area, then secure it with a bandage. This can be applied also if you have lingering pain in the elbow, back, arms, legs and joints.
On the other hand, paper aluminium is also suitable for the removal of the cold in the body. When you feel cold wrap your feet with 5 or 7 pieces of aluminum foil. After each layer of aluminium foil, fold the legs with cotton. Keep this coating for an hour.
Next Colored toilet paper – a danger for our health!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,072
|
Robert Peake the Elder (1551 - 1619) foi um pintor Inglês atuante no período posterior ao reinado de Elizabete I (mais precisamente no reinado de James I. Em 1604, foi nomeado retratista do herdeiro ao trono, Henrique Frederick, e, em 1607, foi agraciado com o título de Sargento-pintor por James I.
Peake é freqüentemente chamado de "o velho", para diferencia-lo de seu filho, o pintor e popular retratista William Peake (c. 1580-1639) e de seu neto, Sir Robert Peake (c. 1605-1667), que acompanhou seu pai no comércio de pinturas e retratos.
Galeria
Referências
Ligações externas
Obras de Peake na National Portrait Gallery
Dois importantes trabalhos de Peake no Metropolitan, Nova York
Pintores da Inglaterra
Pintores da corte
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,564
|
On July 1, 2008, subject to approval by the legislature, British Columbia will begin to phase in a fully revenue-neutral carbon tax with built-in protection for lower income British Columbians.
Beginning July 1, 2008, the new Climate Action Credit will provide lower-income British Columbians a payment of $100 per adult and $30 per child per year — increasing by 5 per cent in 2009 and possibly more in future years ($395 million over three years).
For the Finance Minister's speech and more details on Balanced Budget 2008, visit www.gov.bc.ca/bcbudget online.
Visit the Province's website at www.gov.bc.ca for online information and services.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,482
|
The Ngatapa Branch was a secondary branch line railway long that for a short time formed part of the national rail network in Poverty Bay in the North Island of New Zealand. The Ngatapa branch diverged from the Moutohora branch line about from Gisborne and ran a further across the coastal flat to a terminus at Ngatapa.
Built to the New Zealand standard gauge the branch was originally authorized as part of the proposed inland route for the Wairoa to Gisborne section of the Palmerston North – Gisborne Line. However, in 1924, an engineer's report recommended that the then-new isolated section between Wairoa and Waikokopu in Hawke's Bay be incorporated as the southernmost portion of a new coastal route from Wairoa to Gisborne. The Public Works Department (PWD) accordingly stopped work on the inland Ngatapa route, which was officially opened as a branch line on 15 December 1924, and began work on the coastal route. The Ngatapa branch became a dead end, and it was closed on 1 April 1931.
Construction
Construction started in 1911, and the line was completed to Ngatapa by December 1915, although it was not formally handed over to the New Zealand Railways Department (NZR) until 15 December 1924. Between 1918 and 1920, work started at Wairoa on the section to Frasertown, which was to have been linked through to Ngatapa, and on the Waikura section beyond Ngatapa, but all work was stopped in 1920 after the Waikura section was found to be unstable.
Construction of the line presented few problems as far as Ngatapa. The course of the line beyond this point was a different matter and would have required heavy earthworks and extensive tunneling. Beyond Ngatapa, some formation work was undertaken for about 8 km, including the excavation of a short tunnel, but rails were never laid on the section. Today, earthworks such as embankments and cuttings can still be found, but no actual tunnels can be located, possibly due to collapse in the slip prone ground.
Operations and closure
By December 1915 the Public Works Department (PWD) was operating goods trains on the branch, and continued to do so until responsibility for the line was transferred to the NZR. The line carried only about 12,000 tonnes of freight per annum, almost all road metal. By 1930 a Railways Commission noted that with the abandonment of the originally proposed inland route the need for the line had disappeared, and the twice-daily Monday to Saturday NZR service attracted an average of only 20 passengers a day. The commission recommended the branch should either close or be taken over by its users, the PWD, or the Gisborne City Council. There were no takers, and the line was closed on 1 April 1931.
Remains
Not much remains of the branch formation or other works on the coastal plain. Apart from the bridge piers of the Waipaoa River crossing between Makaraka and Patutahi, the most significant remains are those past the original terminus at Ngatapa, over which trains never ran but which illustrate the problems that would have faced the constructors if the line had continued into the hills. Beyond the terminus at Ngatapa the formation works take the proposed line through a 180-degree climbing turn before following a winding path across the face of the hills until disappearing into the bush.
See also
Palmerston North–Gisborne Line
Ahuriri / Napier Port Branch
Moutohora / Makaraka Branch
References
Bibliography
External links
NZ Rail Maps (Palmerston North Gisborne Line - Other Lines files)
Railway lines in New Zealand
Railway lines opened in 1915
Railway lines closed in 1931
Rail transport in the Gisborne District
1915 establishments in New Zealand
1931 disestablishments in New Zealand
3 ft 6 in gauge railways in New Zealand
Closed railway lines in New Zealand
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,652
|
Destination weddings are more just like a romantic getaway which normally lasts beyond the standard traditional wedding. The largest good thing about developing a destination wedding is that the list of guests will always be shorter. Setting a day where everybody can make day trip with their business schedule can be tough and hence couples send out invitations no less than eight months beforehand. Destination weddings can be a cheaper alternative for people from the US and The european union because valuation on a wedding event in several developed nations is more expensive than many attractions including Bali, Maldives, Hawaii, and Thailand. Hence these days destination weddings are viewed being a great alternative to traditional weddings with a lot more and even more couples settling on enter wedlock at some distant location. Wedding destinations offer the collection of engaged and getting married amidst a beautiful landscape which provides for a backdrop to get a dream wedding, couples mainly select locations where provide stunning natural beauty.
The celebration is commonly a far more intimate affair with pals and family involved. It combines wedding ceremony, reception and also the honeymoon all into one extended holiday. They provide a way for everyone to unwind, relax and take a break using their hectic everyday lives. The very idea of destination weddings has resulted in destination anniversaries wherein couples relive their wedding experience. Sometimes having a wedding for a foreign location could possibly be risky mindful about could possibly be different marriage laws as well as the spanish could work as a communication barrier. Hence it is advisable to get accustomed with the local laws or just as many instances the wedding resort handles everything from the florists on the priests. Resorts are employed in co ordination with a event planner who designs their big event as outlined by a particular theme and precise instructions.
Many couples recommend bringing a florist or possibly a makeup artist you never know the personality, likes and type of skin of their client and who does not need an overview. Destination weddings include the fastest growing segment in the usa 60 big wedding industry, brides are going for to get their fairytale wedding at exotic locations. If your list of guests is really a lot smaller couples believe they're able to offer a much bigger and make their wedding more lavish.
Destination weddings are a fun and adventurous method to get married, they can be a bit daunting for that bride as making plans for this kind of elaborate affair isn't any easy job. It does have complications in case panned well your big event is usually wonderful and unforgettable for that married couple plus the guests.
To read more about wedding in croatia go to see this useful internet page.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,306
|
Nihoa vanuatu är en spindelart som beskrevs av Raven 1994. Nihoa vanuatu ingår i släktet Nihoa och familjen Barychelidae. Inga underarter finns listade i Catalogue of Life.
Källor
Spindlar
vanuatu
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,712
|
Chastellain ou Chastelain peut désigner :
Patronyme
Chastelain
Charles Chastelain, compositeur franco-flamand du
Georges Chastelain, historiographe flamand du
Pierre Chastelain, (1709-1782), bénédictin rémois spécialiste de la vigne
Chastellain
Jacques Chastellain (1885-1965), homme politique français
Jean Chastellain (1490-1541), maître-verrier
Jean-Claude Chastellain (1747-1824), homme politique français
Pierre Chastellain, chanteur suisse
Pierre Chastellain, jésuite et un missionnaire canadien
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,788
|
Q: How to adjust test configuration in TFS 2015 Visual Studio Test task I'm trying to setup our TFS 2015 server to run automated tests. I've got it running, but we need to run our tests in Debug mode (for various reasons I can't really adjust). The problem is that I can't seem to figure out a way to switch the configuration in the Test task.
The help that the task links to (here) says that it's as easy as selecting Platform and Configuration, but the problem is that those options don't exist for me (they exist under Reporting, but the help there suggests that they will simply compare the results to other builds with that configuration).
I've also investigated the vstest.console.exe parameters (help I found was this one) as well as modifying the runsettings file, but these only allow me to modify the platform.
Overall, my question is a)is there a reason why I don't see the Platform/configuration options in TFS, and b) given that I don't see them, how can I modify the configuration that the tests are running under?
If it helps, TFS is reporting the version as Version 14.95.25122.0, which corresponds to Update 2. I checked the logs for 2.1 and 3, but wasn't able to find anything that suggested that this was added in later versions (though I could be wrong).
UPDATE:
I've realized that I misread the Test documentation and that the Platform/Configuration options were always for reporting only.
My question is then if I can actually set this in the tests somehow.
Thank you very much for any help.
A: Assuming you want to compile your test project in Debug mode. You can add a VS Build step to specify the BuildConfiguration variable, and define debug for variable BuildConfiguration. Check the screenshots below:
Then in VS Test step, specify the Test Assembly as **\$(BuildConfiguration)\*test*.dll to test the assemble under Debug folder:
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,343
|
Q: Digital / Physical goods on in-app purchase I'm developing a Marketplace App of the courses for the Learning Purposes and we don't have any kind of control over uploading of Physical content( CDs, DVDs, Hard Copy of the book, etc. ) or the Digital Content (Videos, Audios, Ebooks, etc.) on the courses uploaded by the seller for the consumers to purchase.
So could you please suggest what kind of payment method should we use apart from In-App purchases?
If we are bound to use In-App purchase then how should we implement it in our App and also we have following queries to clear our doubts for the monetization of the app.
*
*Could we add n number of courses dynamically having digital and physical content with In-App Purchases implemented? In our app the courses limit is not fixed, it can be 5000, 10000 or 100000.
*Will it be possible for the sellers to provide any kind of discounts for the existing courses?
*How could we manage(addition or deletion) the courses dynamically with In-App purchases?
I researched on google but no solution was found. Some solutions were found on StackOverflow but we can't find proper guidelines for our queries. Thanks :)
A: This seems to be partly an opinion-based question, but with regards to IAPs there are a couple of important rules:
1) For all digital goods, you have to use in-app purchases, otherwise Apple will not approve your app. There are different approaches to your problem. You could either let people buy credits with IAPs and then dynamically create the store on your own. Then you have 100% control over discounts etc.) Or each product will be its own IAP, but this will create lots of "overhead" as you have to get approval for each IAP.
2) For physical goods you are not allowed to use in-app purchases. There you have to find a 3rd party payment provider.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,689
|
{"url":"https:\/\/www.ias.ac.in\/listing\/bibliography\/pram\/MOSTAFA_M_A_KHATER","text":"\u2022 MOSTAFA M A KHATER\n\nArticles written in Pramana \u2013 Journal of Physics\n\n\u2022 New optical soliton solutions for nonlinear complex fractional Schr\u00f6dinger equation via new auxiliary equation method and novel $(G^{'} \/ G)$-expansion method\n\nIn this research, we apply two different techniques on nonlinear complex fractional nonlinear Schr\u00f6dinger equation which is a very important model in fractional quantum mechanics. Nonlinear Schr\u00f6dinger equation is one of the basic models in fibre optics and many other branches of science. We use the conformable fractional derivative to transfer the nonlinear real integer-order nonlinear Schr\u00f6dinger equation to nonlinear complex fractional nonlinear Schr\u00f6dinger equation. We apply new auxiliary equation method and novel $(G^{'} \/ G)$-expansion method on nonlinear complex fractional Schr\u00f6dinger equation to obtain new optical forms of solitary travelling wave solutions. We find many new optical solitary travelling wave solutions for this model. These solutions are obtained precisely and efficiency of the method can be demonstrated.\n\n\u2022 Pramana \u2013 Journal of Physics\n\nVolume 95, 2021\nAll articles\nContinuous Article Publishing mode\n\n\u2022 Editorial Note on Continuous Article Publication\n\nPosted on July 25, 2019","date":"2021-03-02 14:51:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3556724488735199, \"perplexity\": 1242.8545033631558}, \"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-2021-10\/segments\/1614178364008.55\/warc\/CC-MAIN-20210302125936-20210302155936-00234.warc.gz\"}"}
| null | null |
Q: How to mock the method invoked on the reponse of a mocked method I have a class
class A {
@Autowired
DbObject dbObject;
method1 () {
try {
return DbObject.read("query").getItem ();
}
catch (NotFoundException e) {
return null;
}
}
}
I have to test method1 is returning null if NotFound Exception is thrown , I will create @MockBean for DbObject in test class. How to write a when condition to mock getItem(). when(DbObject.method1("query")).thenThrow("NotFoundException") . But this wont mock the getItem method invoked on the object returned by our mock when(DbObject.method1("query")).
A: for that you need to use chaining of mocked methods. declare two mocks: one for dbObject, and another one for returned result from dbObject.read("query") (let's call that dbObjectQueryResult).
when(dbObject.read("query")).thenReturn(dbObjectQueryResult);
when(dbObjectQueryResult.getItem()).thenThrow(new NotFoundException("test"))
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,976
|
{"url":"https:\/\/math.stackexchange.com\/questions\/1916551\/why-is-every-f-mathcalo-x-module-also-an-mathcalo-x-module","text":"# Why is every $f_*\\mathcal{O}_X$-module also an $\\mathcal{O}_X$-module?\n\nLet $f:X \\rightarrow Y$ be a finite morphism of locally Noetherian schemes. If $\\mathcal{F}$ is an $f_*\\mathcal{O}_X$-module, why is $\\mathcal{F}$ endowed with the structure of an $\\mathcal{O}_X$-module?\n\nThis should be really basic, but I struggle to see this. For each open subset $U \\subset Y$ we can multiply $\\mathcal{F}(U)$ by scalars in $\\mathcal{O}_X(f^{-1}(U))$, thus $\\mathcal{F}$ gives us for each open subset in $Y$ an $\\mathcal{O}_X$-module, but not for each open subset of $X$...\n\nWhy is $\\mathcal{F}$ endowed with the structure of an $\\mathcal{O}_X$-module?\n\nWhen people say this, this is a slight abuse of language. Strictly speaking, a sheaf on $Y$ cannot be an $\\mathcal{O}_X$-module unless $X = Y$. However, for finite (more generally, affine) morphisms $f\\colon X\\to Y$, the direct image functor induces an equivalence between quasi-coherent $\\mathcal{O}_X$-modules and quasi-coherent $f_*\\mathcal O_X$-modules. That is, for every $f_*\\mathcal O_X$-module $\\mathcal F$ there is a unique (up to isomorphism) $\\mathcal O_X$-module $\\mathcal G$ such that $f_*\\mathcal G\\cong \\mathcal F$ We say that $f_* \\mathcal G$ has an $\\mathcal{O}_X$-module structure since this is what happens in the affine world: if $X = \\mathrm{Spec}(A)$ and $Y = \\mathrm{Spec}(B)$, then $f$ corresponds to a ring homomorphism $B\\to A$ and if $\\mathcal G = M^\\sim$, then $f_*\\mathcal G = M_B^\\sim$ where $M_B$ is just $M$, but considered as a $B$-module.","date":"2019-06-25 22:02:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9918684363365173, \"perplexity\": 75.82959414344285}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627999948.3\/warc\/CC-MAIN-20190625213113-20190625235113-00192.warc.gz\"}"}
| null | null |
CUPUM
Computers in Urban Planning & Mangement Conference
Un Postal Agreement
Faced with such difficulties in changing mail with Europe, the United States had already taken the lead and called for improved international postal agreements. General Montgomery Blair, postmaster of the United States, called for an international postal congress in 1863. At their meeting in Paris, delegates defined some general principles of post-cooperation, but failed to agree. [10] Hussein called the result "the most remarkable day in the union`s history," a nearly 150-year-old organization that regulates postal services in 192 member countries. The Advisory Committee was established by the Bucharest Congress in 2004. It gives a voice to other operators in the postal sector as public operators and regulators in the deliberations of the organization. It consists of non-governmental organizations, distribution service providers, workers` organizations, postal goods and service providers and other organizations interested in international postal services. The presidency is that of the Envelope Alliance. The U.N. organization, which regulates postal systems around the world, on Wednesday approved reforms to prevent the United States from withdrawing from the organization, mainly because of a dispute with China. The 1874 contract provided that the country of origin would retain all port revenues without compensating the destination country for delivery. The idea was that each letter would generate a response, so that mail flows would be in balance. [17] [18] However, other postal classes had unbalanced flows.
In 1906, the Italian post office delivered 325,000 magazines from other countries to Italy, while Italian publishers did not send magazines to other countries. [18] The system also encouraged countries to re-mail through another country, forcing the intermediary postal service to cover transportation costs to the final destination. [19] Standards are important conditions for efficient postal operation and for the networking of the global network. The UPU Standard Board is developing and maintaining a growing number of international standards to improve the exchange of post-postal information between postal operators. It also promotes the compatibility of UPU`s postal and international initiatives. The organization works closely with postal assistance organizations, customers, suppliers and other partners, including several international organizations. The Standard Board oversees the development of consistent rules in areas such as electronic data exchange (EDI), post-coding, postal forms and meters. UPU standards are established in accordance with the rules set out in Part V of the "General Information on UPU Standards"[31] and published by the International Office of the UPU in accordance with Part VII of this publication. The Trump administration has withdrawn from the threat to withdraw from a 145-year-old international postal agency after reaching an agreement that gave the United States more favorable terms.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,180
|
{"url":"https:\/\/acemicro.org\/probability\/probability\/","text":"# Probability of an Event\n\n$$\\frac{\\text{number of }}{\\text{total number of possibilities}}$$\n\n## Example\n\nTo calculate the probability that an event occurs:\n\n1. Count the number possibilities that correspond to the event\n2. Count the number of possibilities in total\n$$P \\left( \\text{event} \\right) = \\frac{\\text{number possibilities that correspond to the event}}{\\text{number of possibilities in total}}$$","date":"2022-12-06 10:45:14","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\": 0.7841733694076538, \"perplexity\": 1134.4204690812637}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711077.50\/warc\/CC-MAIN-20221206092907-20221206122907-00492.warc.gz\"}"}
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Алън Харпър е художествен герой от комедията "Двама мъже и половина".
Той е разведеният брат на Чарли. По професия е физиотерапевт и е много по-съвестен от брат си, но няма никакъв късмет и Чарли често се шегува с него. След като жена му, Джудит, получава къщата след развода, той за постоянно остава при Чарли. Алън Харпър по принцип е добра и учтива личност, но има слабост към жени, които се отнасят зле с него. Това може би произлиза от лошите отношения между него и майка им. Той е най-практичният герой в шоуто, неговите скрупули и пестеливост са често в основата на много от шегите. Джон Крайър играе ролята на по-малкия с две години брат, но в действителност той е с няколко месеца по-голям от актьора Чарли Шийн.
Харпър, Алън
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 655
|
{"url":"https:\/\/online.stat.psu.edu\/stat501\/book\/export\/html\/1014","text":"# 15.5 - Generalized Linear Models\n\n15.5 - Generalized Linear Models\n\nAll of the regression models we have considered (including multiple linear, logistic, and Poisson) actually belong to a family of models called generalized linear models. (In fact, a more \"generalized\" framework for regression models is called\u00a0 general regression models, which includes any parametric regression model.)\u00a0 Generalized linear models provides a generalization of ordinary least squares regression that relates the random term (the response Y) to the systematic term (the linear predictor $$\\textbf{X}\\beta$$) via a link function (denoted by $$g(\\cdot)$$). Specifically, we have the relation\n\n$$\\begin{equation*} \\mbox{E}(Y)=\\mu=g^{-1}(\\textbf{X}\\beta), \\end{equation*}$$\n\nso $$g(\\mu)=\\textbf{X}\\beta$$. Some common link functions are:\n\n$$\\begin{equation*} g(\\mu)=\\mu=\\textbf{X}\\beta, \\end{equation*}$$\n\nwhich is used in traditional linear regression.\n\n\\begin{align*} &g(\\mu)=\\log\\biggl(\\frac{\\mu}{1-\\mu}\\biggr)=\\textbf{X}\\beta\\\\ &\\Rightarrow\\mu=\\frac{e^{\\textbf{X}\\beta}}{1+e^{\\textbf{X}\\beta}}, \\end{align*}\n\nwhich is used in logistic regression.\n\n\\begin{align*} &g(\\mu)=\\log(\\mu)=\\textbf{X}\\beta\\\\ &\\Rightarrow\\mu=e^{\\textbf{X}\\beta}, \\end{align*}\n\nwhich is used in Poisson regression.\n\n\\begin{align*} &g(\\mu)=\\Phi^{-1}(\\mu)=\\textbf{X}\\beta\\\\ &\\Rightarrow\\mu=\\Phi(\\textbf{X}\\beta), \\end{align*}\n\nwhere $$\\Phi(\\cdot)$$ is the cumulative distribution function of the standard normal distribution. This link function is also sometimes called the normit link. This also can be used in logistic regression.\n\n\\begin{align*} &g(\\mu)=\\log(-\\log(1-\\mu))=\\textbf{X}\\beta\\\\ &\\Rightarrow\\mu=1-\\exp\\{-e^{\\textbf{X}\\beta}\\}, \\end{align*}\n\nwhich can also be used in logistic regression. This link function is also sometimes called the gompit link.\n\n\\begin{align*} &g(\\mu)=\\mu^{\\lambda}=\\textbf{X}\\beta\\\\ &\\Rightarrow\\mu=(\\textbf{X}\\beta)^{1\/\\lambda}, \\end{align*}\n\nwhere $$\\lambda\\neq 0$$. This is used in other regressions which we do not explore (such as gamma regression and inverse Gaussian regression).\n\nAlso, the variance is typically a function of the mean and is often written as\n\n$$\\begin{equation*} \\mbox{Var}(Y)=V(\\mu)=V(g^{-1}(\\textbf{X}\\beta)). \\end{equation*}$$\n\nThe random variable Y is assumed to belong to an exponential family distribution where the density can be expressed in the form\n\n$$\\begin{equation*} q(y;\\theta,\\phi)=\\exp\\biggl\\{\\dfrac{y\\theta-b(\\theta)}{a(\\phi)}+c(y,\\phi)\\biggr\\}, \\end{equation*}$$\n\nwhere $$a(\\cdot)$$, $$b(\\cdot)$$, and $$c(\\cdot)$$ are specified functions, $$\\theta$$ is a parameter related to the mean of the distribution, and $$\\phi$$ is called the dispersion parameter. Many probability distributions belong to the exponential family. For example, the normal distribution is used for traditional linear regression, the binomial distribution is used for logistic regression, and the Poisson distribution is used for Poisson regression. Other exponential family distributions lead to gamma regression, inverse Gaussian (normal) regression, and negative binomial regression, just to name a few.\n\nThe unknown parameters, $$\\beta$$, are typically estimated with maximum likelihood techniques (in particular, using iteratively reweighted least squares), Bayesian methods, or quasi-likelihood methods. The quasi-likelihood is a function which possesses similar properties to the log-likelihood function and is most often used with count or binary data. Specifically, for a realization y of the random variable Y, it is defined as\n\n$$\\begin{equation*} Q(\\mu;y)=\\int_{y}^{\\mu}\\dfrac{y-t}{\\sigma^{2}V(t)}dt, \\end{equation*}$$\n\nwhere $$\\sigma^{2}$$ is a scale parameter. There are also tests using likelihood ratio statistics for model development to determine if any predictors may be dropped from the model.\n\n [1] Link \u21a5 Has Tooltip\/Popover Toggleable Visibility","date":"2022-08-15 09:09:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8654226064682007, \"perplexity\": 569.8423774961593}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572163.61\/warc\/CC-MAIN-20220815085006-20220815115006-00640.warc.gz\"}"}
| null | null |
{"url":"https:\/\/meritbatch.com\/rd-sharma-class-9th-solutions-chapter-3-rationalisation\/","text":"## RD Sharma Class 9 Solutions Chapter 3 Rationalisation\n\n### RD Sharma Class 9 Chapter 3 Rationalisation Ex 3.1\n\nQuestion 1.\nSimplify each of the following:\n\nSolution:\n\nQuestion 2.\nSimplify the following expressions:\n(i)\u00a0 (4 + $$\\sqrt { 7 }$$) (3 + $$\\sqrt { 2 }$$)\n(ii) (3 + $$\\sqrt { 3 }$$ )(5- $$\\sqrt { 2 }$$)\n(iii) ($$\\sqrt { 5 }$$ -2)($$\\sqrt { 3 }$$ \u2013 $$\\sqrt { 5 }$$)\nSolution:\n\nQuestion 3.\nSimplify the following expressions:\n(i)\u00a0 (11 + $$\\sqrt { 11 }$$) (11 \u2013 $$\\sqrt { 11 }$$)\n(ii) (5 + $$\\sqrt { 7 }$$ ) (5 \u2013 $$\\sqrt { 7 }$$ )\n(iii) ($$\\sqrt { 8 }$$ \u2013 $$\\sqrt { 2 }$$) ($$\\sqrt { 8 }$$+ $$\\sqrt { 2 }$$)\nSolution:\n\nQuestion 4.\nSimplify the following expressions:\n(i) ($$\\sqrt { 3 }$$+$$\\sqrt { 7 }$$)2\n(ii) ($$\\sqrt { 5 }$$ \u2013 $$\\sqrt { 3 }$$ )2\n(iii) (2$$\\sqrt { 5 }$$ + 3 $$\\sqrt { 2}$$)2\nSolution:\n\n### RD Sharma Solutions Class 9 Chapter 3 Rationalisation Ex 3.2\n\nQuestion 1.\nRationalise the denominators of each of the following(i \u2013 vii):\n\nSolution:\n\nQuestion 2.\nFind the value to three places of decimals of each of the following. It is given that\n\nSolution:\n\nQuestion 3.\nExpress each one of the following with rational denominator:\n\nSolution:\n\nQuestion 4.\nRationales the denominator and simplify:\n\nSolution:\n\nQuestion 5.\nSimplify:\n\nSolution:\n\nQuestion 6.\nIn each of the following determine rational numbers a and b:\n\nSolution:\n\nQuestion 7.\n\nSolution:\n\nQuestion 8.\nFind the values of each of the following correct to three places of decimals, it being given that $$\\sqrt { 2 }$$\u00a0 = 1.4142, $$\\sqrt { 3 }$$ = 1-732,\u00a0$$\\sqrt { 5 }$$\u00a0 = 2.2360,\u00a0$$\\sqrt { 6 }$$ =\u00a0\u00a02.4495 and $$\\sqrt { 10 }$$\u00a0 = 3.162.\n\nSolution:\n\nQuestion 9.\nSimplify:\n\nSolution:\n\nQuestion 10.\n\nSolution:\n\nQuestion 11.\n\nSolution:\n\nQuestion 12.\n\nSolution:\n\n### Class 9 RD Sharma Solutions Chapter 3 Rationalisation VSAQS\n\nQuestion 1.\nWrite the value of (2 + $$\\sqrt { 3 }$$ ) (2 \u2013 $$\\sqrt { 3 }$$).\nSolution:\n(2+ $$\\sqrt { 3 }$$ )(2- $$\\sqrt { 3 }$$ ) = (2)2-($$\\sqrt { 3 }$$ )2\n{\u2235 (a + b) (a \u2013 b) = a2 \u2013 b2}\n= 4-3=1\n\nQuestion 2.\nWrite the reciprocal of 5 + $$\\sqrt { 2 }$$.\nSolution:\n\nQuestion 3.\nWrite the rationalisation factor of 7 \u2013 3$$\\sqrt { 5 }$$ .\nSolution:\nRationalising factor of 7 \u2013 3$$\\sqrt { 5 }$$ is 7 + 3$$\\sqrt { 5 }$$\n{\u2235 ($$\\sqrt { a }$$ + $$\\sqrt { b }$$\u00a0 ) ($$\\sqrt { a }$$\u00a0\u2013 $$\\sqrt { b }$$) = a-b}\n\nQuestion 4.\n\nSolution:\n\nQuestion 5.\nIf x =$$\\sqrt { 2 }$$ \u2013 1 then write the value of $$\\frac { 1 }{ x }$$.\nSolution:\n\nQuestion 6.\nIf a = $$\\sqrt { 2 }$$ + h then find the value of a \u2013$$\\frac { 1 }{ a }$$\nSolution:\n\nQuestion 7.\nIf x = 2 + $$\\sqrt { 3 }$$, find the value of x + $$\\frac { 1 }{ x }$$.\nSolution:\n\nQuestion 8.\nWrite the rationalisation factor of $$\\sqrt { 5 }$$ \u2013 2.\nSolution:\nRationalisation factor of\u00a0$$\\sqrt { 5 }$$ \u2013 2 is\u00a0$$\\sqrt { 5 }$$ + 2 as\n($$\\sqrt { a }$$ +\u00a0$$\\sqrt { b }$$)($$\\sqrt { a }$$ \u2013\u00a0$$\\sqrt { b }$$) = a \u2013 b\n\nQuestion 9.\nSimplify : $$\\sqrt { 3+2\\sqrt { 2 } }$$.\nSolution:\n\nQuestion 10.\nSimplify : $$\\sqrt { 3-2\\sqrt { 2 } }$$.\nSolution:\n\nQuestion 11.\nIf x = 3 + 2\u00a0$$\\sqrt { 2 }$$, then find the value of \u00a0$$\\sqrt { x }$$ \u2013\u00a0$$\\frac { 1 }{ \\sqrt { x } }$$.\nSolution:\n\n### Rationalisation Problems With Solutions PDF RD Sharma Class 9 Solutions MCQS\n\nMark the correct alternative in each of the following:\n\nQuestion 1.\n\nSolution:\n\nQuestion 2.\n\nSolution:\n\nQuestion 3.\n\nSolution:\n\nQuestion 4.\nThe rationalisation factor of 2 +\u00a0$$\\sqrt { 3 }$$\u00a0 is\n(a) 2 \u2013\u00a0$$\\sqrt { 3 }$$\n(b)\u00a0$$\\sqrt { 2 }$$ + 3\n(c)\u00a0\u00a0$$\\sqrt { 2 }$$ \u2013 3\n(d)\u00a0$$\\sqrt { 3 }$$ \u2013 2\nSolution:\n\nQuestion 5.\n\nSolution:\n\nQuestion 6.\n\nSolution:\n\nQuestion 7.\n\nSolution:\n\nQuestion 8.\n\nSolution:\n\nQuestion 9.\n\nSolution:\n\nQuestion 10.\n\nSolution:\n\nQuestion 11.\n\nSolution:\n\nQuestion 12.\n\nSolution:\n\nQuestion 13.\n\nSolution:\n\nQuestion 14.\n\nSolution:\n\nQuestion 15.\n\nSolution:\n\nQuestion 16.\n\nSolution:\n\nQuestion 17.\n\nSolution:\n\nQuestion 18.\n\nSolution:\n\nQuestion 19.\n\nSolution:\n\nQuestion 20.\n\nSolution:\n\nQuestion 21.\n\nSolution:\n\nQuestion 22.\n\nSolution:\n\nQuestion 23.\n\nSolution:\n\nQuestion 24.\n\nSolution:\n\nQuestion 25.\n\nSolution:\n\nRD Sharma Class 9th Solutions Chapter 3 Rationalisation Exercise 3.1\nQ1 Q2 Q3 Q4\n\nRD Sharma Class 9th Solutions Chapter 3 Rationalisation Exercise 3.2\nQ1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11","date":"2022-12-01 10:12:20","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\": 0.6798900961875916, \"perplexity\": 10393.9891837601}, \"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-2022-49\/segments\/1669446710808.72\/warc\/CC-MAIN-20221201085558-20221201115558-00595.warc.gz\"}"}
| null | null |
Q: unable to start mysqld I'm getting the following after I try to start mysql:
$ sudo systemctl start mysqld
Job for mariadb.service failed because the control process exited with error code.
See "systemctl status mariadb.service" and "journalctl -xe" for details.
[pi@raspi ~]$ systemctl status mysqld
* mariadb.service - MariaDB 10.5.5 database server
Loaded: loaded (/usr/lib/systemd/system/mariadb.service; enabled; vendor preset: disabled)
Active: failed (Result: exit-code) since Sun 2020-08-23 08:34:44 PDT; 9s ago
Docs: man:mariadbd(8)
https://mariadb.com/kb/en/library/systemd/
Process: 466 ExecStartPre=/bin/sh -c systemctl unset-environment _WSREP_START_POSITION (code=exited, status=0/SUCCESS)
Process: 467 ExecStartPre=/bin/sh -c [ ! -e /usr/bin/galera_recovery ] && VAR= || VAR=`cd /usr/bin/..; /usr/bin/galera_recovery`; [ $? -eq 0 ] && systemctl set-environment _WSREP_START_POSITION=$VAR || exit 1 (code=exited, status=0/S>
Process: 479 ExecStart=/usr/bin/mariadbd $MYSQLD_OPTS $_WSREP_NEW_CLUSTER $_WSREP_START_POSITION (code=exited, status=1/FAILURE)
Main PID: 479 (code=exited, status=1/FAILURE)
Status: "MariaDB server is down"
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [ERROR] InnoDB: Upgrade after a crash is not supported. The redo log was created with MariaDB 10.4.14.
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [ERROR] InnoDB: Plugin initialization aborted with error Generic error
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [Note] InnoDB: Starting shutdown...
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [ERROR] Plugin 'InnoDB' init function returned error.
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [ERROR] Plugin 'InnoDB' registration as a STORAGE ENGINE failed.
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [ERROR] Unknown/unsupported storage engine: InnoDB
Aug 23 08:34:44 raspi mariadbd[479]: 2020-08-23 8:34:44 0 [ERROR] Aborting
Aug 23 08:34:44 raspi systemd[1]: mariadb.service: Main process exited, code=exited, status=1/FAILURE
Aug 23 08:34:44 raspi systemd[1]: mariadb.service: Failed with result 'exit-code'.
Aug 23 08:34:44 raspi systemd[1]: Failed to start MariaDB 10.5.5 database server.
lines 1-21/21 (END)
I don't see any information from syslog-ng in /var/log/messages.log either....
I'm not sure what's going on, can anybody help me out with this? :o
As @tetech suggested, I should downgrade to a previous version of my mariadb package. So I tried:
[pi@raspi pkg]$ sudo pacman -U /var/cache/pacman/pkg/mariadb-10.4.14-1-aarch64.pkg.tar.xz
[sudo] password for pi:
loading packages...
warning: downgrading package mariadb (10.5.5-1 => 10.4.14-1)
resolving dependencies...
warning: cannot resolve "mariadb-clients=10.4.14", a dependency of "mariadb"
:: The following package cannot be upgraded due to unresolvable dependencies:
mariadb
:: Do you want to skip the above package for this upgrade? [y/N] n
error: failed to prepare transaction (could not satisfy dependencies)
:: unable to satisfy dependency 'mariadb-clients=10.4.14' required by mariadb
[pi@raspi pkg]$ sudo pacman -U /var/cache/pacman/pkg/mariadb-clients-10.4.14-1-aarch64.pkg.tar.xz
loading packages...
warning: downgrading package mariadb-clients (10.5.5-1 => 10.4.14-1)
resolving dependencies...
warning: cannot resolve "mariadb-libs=10.4.14", a dependency of "mariadb-clients"
:: The following package cannot be upgraded due to unresolvable dependencies:
mariadb-clients
:: Do you want to skip the above package for this upgrade? [y/N]
error: failed to prepare transaction (could not satisfy dependencies)
:: unable to satisfy dependency 'mariadb-libs=10.4.14' required by mariadb-clients
[pi@raspi pkg]$ sudo pacman -U /var/cache/pacman/pkg/mariadb-libs-10.4.14-1-aarch64.pkg.tar.xz
loading packages...
warning: downgrading package mariadb-libs (10.5.5-1 => 10.4.14-1)
resolving dependencies...
looking for conflicting packages...
error: failed to prepare transaction (could not satisfy dependencies)
:: installing mariadb-libs (10.4.14-1) breaks dependency 'mariadb-libs=10.5.5' required by mariadb-clients
[pi@raspi pkg]$ sudo pacman -U /var/cache/pacman/pkg/mariadb-clients-10.4.14-1-aarch64.pkg.tar.xz
loading packages...
warning: downgrading package mariadb-clients (10.5.5-1 => 10.4.14-1)
resolving dependencies...
warning: cannot resolve "mariadb-libs=10.4.14", a dependency of "mariadb-clients"
:: The following package cannot be upgraded due to unresolvable dependencies:
mariadb-clients
:: Do you want to skip the above package for this upgrade? [y/N]
error: failed to prepare transaction (could not satisfy dependencies)
:: unable to satisfy dependency 'mariadb-libs=10.4.14' required by mariadb-clients
But it seems like mariadb-libs is complaining about mariadb-clients and vice-versa. How do I get thenm downgraded? I'm now thinking it might just be easier to re-init my db instead....
A: Notable error is InnoDB: Upgrade after a crash is not supported. The redo log was created with MariaDB 10.4.14.
It appears you have not properly stopped the DB and then upgraded it.
You'll need to either downgrade to 10.4.14 and do a recovery, or alternatively re-initialize and then restore the backup you took before upgrading.
A: My solution on Manjaro Linux was to sudo pacman -s downgrade then run
downgrade mariadb-libs mariadb-clients mariadb
Downgrade lets you chose which version to downgrade to as well. Super easy, worked like a charm.
A: The solution to downgrade the mysql server was:
$ sudo pacman -U /var/cache/pacman/pkg/mariadb-clients-10.4.14-1-aarch64.pkg.tar.xz /var/cache/pacman/pkg/mariadb-libs-10.4.14-1-aarch64.pkg.tar.xz /var/cache/pacman/pkg/mariadb-10.4.14-1-aarch64.pkg.tar.xz
after this, I was able to start my MySql server again just fine and I have to look into how to properly upgrade it now.
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typedef NS_ENUM(NSInteger, ScreenType) {
kWelcomeAndConsent,
kSignInAndSync,
kSignIn,
kSync,
kDefaultBrowserPromo,
// It isn't a screen, but a signal that no more screen should be
// presented.
kStepsCompleted,
};
#endif // IOS_CHROME_BROWSER_UI_SCREEN_SCREEN_TYPE_H_
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 265
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You may use some ashley furniture table lamps tape to create geometric pattern on the face of one's cabinet. For your ashley furniture table lamps optimal/optimally outcome, it's strongly recommended for you to use any papers to clean out the surface subsequently apply some paint. Following that, you can apply a tape to ashley furniture table lamps create particular pattern onto your own cabinet surface before applying the paint. You are able to apply either oil or chalk paint. Additionally, it can be one of the best L294264 ashley mavia metal table lamp 2cn charlotte appliance inc to your own Furniture.
It's crucial to select L294264 ashley mavia metal table lamp 2cn ashley furniture table lamp sets charlotte appliance inc predicated on its own color. If your Furniture has a warm tone color, it's wise ashley furniture table lamp sets to choose out compatible oak Furniture cupboards for your home. The neutral colors are probably picked like brownish, gray, and ashley furniture table lamp sets light brown. Orange and red oak Furniture cabinets could make your modern Furniture desirable. Noticing Existed Household Furniture Items at the Furniture. When you've got other stainless steel Furniture sets and items, you're fortunate. The stainless steel items can be combined with all the other colors. It's applicable for oak Furniture cabinet choice. Butif you have dark furnishings products, tone comparison light oak Furniture cabinets to earn your Furniture look cheerful and brighter.
L294264 ashley mavia metal table lamp 2cn charlotte appliance inc? You can certainly do it by yourself and also ashley home furniture table lamps you have the capacity to employ your creativity to make your Furniture backsplash seem just like what you want along with your preference. You can find a number of glass tiles at the store which you can select. You may even select glass tile in many shapes and sizes as well. For the colours, you really do not need to worry because there are a number of shades possibilities for your glass tiles also. When it's necessary to install your Furniture backsplash by yourself, it's best for you to organize some gear.
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{
"redpajama_set_name": "RedPajamaC4"
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| 5,351
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Q: Video Upload on Imgur PHP I'm currently working on an Imgur API which should upload images, gifs and videos; images and gifs are being uploaded without any problems and are working fine.
My Problem is the video part, I always get an error similar to this:
{"data":{"error":"No image data was sent to the upload
api","request":"\/3\/upload","method":"POST"},"success":false,"status":400}
or this:
Notice: Undefined index: link in C:\xampp\htdocs\localfile\php\upvideo.php on line 50 There's a Problem No image data was sent to the upload api
or I get an timeout response.
Since the API docs told me that I need to post it as a Binaryfile I tried using fopen(), fread(), file_get_contents() and countless more. Sadly none of them seem to give curl the string it wants if I use a video; it works just fine with images and gifs (which are now being sent as binary - before I used base64encode before passing it to curl).
I do know that there are 2 tags video and image which I change based on what I'm trying to send at the moment, but video always seems to give me the errors above.
Here is my code:
<?php
// the posted stuff from my form
$title = htmlspecialchars($_POST["title"]);
$posted = htmlspecialchars($_POST["token"]);
$desc = htmlspecialchars($_POST["desc"]);
// data from the file i put into the form
$fileName = $_FILES["imglink"]["name"];
$fileTmpName = $_FILES["imglink"]["tmp_name"];
$size = $_FILES["imglink"]["size"];
// checks if everything is set and ready
if ($title != "" && $desc != "" && is_string($posted)) {
if($fileName == ''){
header("location: ../sub-pages/error.php?noimage");
}else{
/*$data = file_get_contents($fileTmpName);*/
/*var_dump($data);*/
// take the file temporary name and make it something binary..
$handle = fopen($fileTmpName, "rb");
$data = fread($handle, filesize($fileTmpName));
//the curl function
$curl = curl_init();
curl_setopt_array($curl, array(
CURLOPT_SSL_VERIFYHOST => 0,
CURLOPT_SSL_VERIFYPEER => 0,
CURLOPT_URL => "https://api.imgur.com/3/upload",
CURLOPT_RETURNTRANSFER => TRUE,
CURLOPT_ENCODING => "",
CURLOPT_POST => 1,
CURLOPT_MAXREDIRS => 10,
CURLOPT_TIMEOUT => 0,
CURLOPT_FOLLOWLOCATION => false,
CURLOPT_HTTP_VERSION => CURL_HTTP_VERSION_1_1,
CURLOPT_CUSTOMREQUEST => "POST",
// all the files i need imgur to get e.g the video (i change video to image if i want to post an image), name of the image etc.
CURLOPT_POSTFIELDS => array( 'video' => @ $data, "type" => "file", "title" => $title, "description" => $desc, "name" => $fileName),
CURLOPT_HTTPHEADER => array( "Authorization: Bearer ". $posted),
));
$response = curl_exec($curl);
$err = curl_error($curl);
curl_close($curl);
$pms = json_decode($response, true);
$url = $pms['data']['link'];
if ($url!="") {
echo "Posted";
header("location: ../sub-pages/succes.php?link=". $url);
} else {
echo "<h2>There's a Problem</h2>";
echo $pms['data']['error'];
}
}
}
else{
echo "no";
}?>
All answers to topic this seem to be outdated or not fixing this.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 3,768
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Belfrage ist der Familienname folgender Personen:
Cedric Belfrage (1905–1990), britischer Journalist, Schriftsteller und Übersetzer
Crispian Belfrage (* 1971), britischer Schauspieler
Bruce Belfrage (1900–1974), britischer Schauspieler und BBC-Moderator
Fredrik Belfrage (* 1949), schwedischer Fernseh- und Radio-Moderator
Henry Belfrage (1774–1835), schottischer Theologe
Nicolas Belfrage (* 1940), britischer Weinexperte und Autor
Sally Belfrage (1936–1994), US-amerikanische Schriftstellerin und Journalistin
Siehe auch:
Belfrage (Adelsgeschlecht), schwedisches Adelsgeschlecht
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,195
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FN Model 1910 — самозарядный пистолет конструкции Джона М. Браунинга, разработанный на замену пистолета Browning M1900 и выпускавшийся бельгийской оружейной компанией Fabrique Nationale d'Armes de Guerre (Национальная фабрика военного оружия) в г. Эрсталь.
История
Пистолет был разработан для бельгийской компании Fabrique Nationale (FN) и компании Кольт. Последняя не проявила интереса к этой модели, и производство сосредоточилось в Бельгии. Первоначально модель 1910 выпускалась под патрон 7,65×17 мм, а с 1912 года и под 9×17 мм.
Описание
Основное отличие данного пистолета от предыдущих конструкций Браунинга заключается в компоновке. Возвратная пружина (впервые в серийно выпускавшемся пистолете) размещена вокруг ствола, за счёт чего значительно уменьшились габариты затвора-кожуха и оружия в целом. Впоследствии такое решение, благодаря которому уменьшилось количество деталей и габариты пистолета, было использовано в пистолетах других систем. Недостатком такой схемы является перегрев возвратной пружины при продолжительной и скоростной стрельбе, потеря ею своих механических качеств, вследствие чего может нарушиться работа автоматики. Но для карманного пистолета, не предназначенного для ведения длительного боя, это несущественно.
В 1922 году, по заказу Королевства сербов, хорватов и словенцев, был выпущен новый вариант модели 1910 с удлинённым стволом и рукояткой, вмещающей магазин большей ёмкости, получивший наименование FN Browning 1922. Его магазин вмещал 9 патронов 7,65×17 мм или 8 патронов 9×17 мм. Модель 1922 г. состояла на вооружении в армиях нескольких европейских стран.
Оба варианта пистолета были широко распространены в качестве гражданского, полицейского и армейского оружия, и пользовались большой популярностью в Европе до конца Второй мировой войны и даже после, а их производство продолжалось вплоть до 1983 года в разных вариантах, в том числе с регулируемыми прицельными приспособлениями. В Испании, Китае и других странах производились многочисленные копии и подражания, более или менее соответствующие оригиналу.
Страны-эксплуатанты
— на вооружении полиции
— в октябре 1924 года 7,65-мм пистолет был принят на вооружение под наименованием Pistool M25 No.1, в апреле 1926 года 9-мм пистолет был принят на вооружение под наименованием Pistool M25 No.2, всего до начала Второй мировой войны в Бельгии было закуплено 91 449 пистолетов обеих модификаций
Югославия — в феврале 1923 года заказано 60 тыс. 9-мм пистолетов обр. 1910/1922 г., в 1930 году — ещё 20 тыс., которые приняты на вооружение под наименованием Аутоматски пиштоль Браунинг М.1910/22 и распределены в армии и полиции; пистолеты, отремонтированные на заводе «Kragujevac Arsenal» во время оккупации, имеют деревянные накладки на рукоять
— в 1925 году партия пистолетов была закуплена для вооружения офицеров турецкой армии
— в 1926—1929 гг. закуплено 9980 шт. 9-мм пистолетов обр. 1922 года для армии
— в сентябре 1935 года партия пистолетов была закуплена для румынской полиции
— после начала в 1936 году войны в Испании, 200 шт. 9-мм пистолетов «браунинг» обр.1922 г. было закуплено правительством Испанской республики
— в ходе Второй мировой войны некоторое количество пистолетов использовалось в японской полиции
— использовался во время Второй мировой войны; трофейные бельгийские 7,65-мм пистолеты «браунинг» обр. 1910 года поступали на вооружение под наименованием Pistole 621(b); бельгийские 7,65-мм пистолеты обр. 1910/1922 года — под наименованием Pistole 626 (b); датские 7,65-мм пистолеты обр. 1910/1922 года — под наименованием Pistole 626 (d); бельгийские 9-мм пистолеты обр. 1910/1922 г. — под наименованием Pistole 641(b); югославские 9-мм пистолеты обр. 1910/1922 г. — под наименованием Pistole 641(j)
— после оккупации Югославии в апреле 1941 года немецкое военное командование передало некоторое количество трофейных югославских пистолетов НГХ, которые использовались армейскими и полицейскими формированиями под наименованием Samokres O.10/22
— пистолеты довоенного производства и трофеи использовали партизаны Народно-освободительной армии Югославии, после окончания войны пистолеты были официально приняты на вооружение в качестве личного оружия для сотрудников Народной милиции и офицеров Югославской Народной армии и использовались до 1976 года; при этом, с 1957 года отремонтированные заводом «Црвена Застава» пистолеты получали на рукоять новые накладки из чёрной пластмассы
— некоторое количество пистолетов использовалось подразделениями западногерманской полиции
Известные образцы
Из пистолета Browning M1910, серийный номер 19074 под патрон .380 ACP, Гаврило Принцип застрелил в Сараево эрцгерцога Фердинанда, что стало поводом к Первой мировой войне. Однако по другим данным, пистолет был изготовлен под патрон .32 ACP, что до сих пор вызывает поводы для споров. Пистолет № 19074, легально купленный в Белграде вместе с тремя другими такими же «браунингами» для организации «Черная Рука», хранится в Венском военно-историческом музее.
Из пистолета Browning M1910 6 мая 1932 года русским эмигрантом Павлом Горгуловым был застрелен президент Франции Поль Думер. Пистолет находится в Музее исторических коллекций полицейской префектуры.
Из ещё одного пистолета подобного типа 5 сентября 1935 года Карлом Вайссом был застрелен губернатор штата Луизиана Хьюи Лонг.
19 января 2009 года из пистолета такого типа было совершено громкое убийство адвоката Станислава Маркелова и журналистки Анастасии Бабуровой.
См. также
Browning M
Примечания
Литература
Ссылки
Пистолет FN Browning 1910 / Browning 1910/22 (Бельгия)
М. Р. Попенкер. FN — Browning 1910, 1922 и 380 (Бельгия) / сайт «Современное стрелковое оружие мира»
Передача «Военное дело». Серия о браунингах на YouTube
1910
Появились в 1910 году
Самозарядные пистолеты Бельгии
Оружие под патрон .32 ACP
Оружие под патрон 9 × 17 мм
1910 год в Бельгии
|
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| 2,407
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Members of the Legislative Assembly of Samoa were elected on 9 April 2021. According to preliminary results, the 51 members consisted of 25 representatives of Fa'atuatua i le Atua Samoa ua Tasi (FAST), 25 from the Human Rights Protection Party (HRPP), one from the Tautua Samoa Party and one independent. Official results were declared on 16 April, resulting in some change to individual winners, but no change to the overall balance of power. Election petitions subsequently saw four HRPP MPs unseated, and three more resign as part of settlements.
Members
Changes
On 18 June 2021 the election of the HRPP's Seiuli Ueligitone Seiuli was overturned by an election petition, which found him guilty of bribery and treating and banned him from office for 15 years.
On 29 June 2021 Sagaga No. 4 Tuisa Tasi Patea resigned to avoid an election petition.
On 5 July Aleipata Itupa i Lalo MP Fiugalu Eteuati Eteuati was convicted of 13 counts of bribery and treating in an electoral petition.
On 7 July 2021 Safata No. 2 MP Nonu Lose Niumata resigned as part of the settlement of an electoral petition.
On 9 July 2021 Falealupo MP Leota Tima Leavai resigned and agreed not to run in a by-election as part of an election petition settlement.
On 16 August 2021 the election of Aana Alofi No.2 MP Aiono Afaese Toleafoa and Falealili No. 2 MP Fuimaono Teo Samuelu were both overturned by election petitions.
On 22 November 2021 Fuiono Tenina Crichton was declared elected unopposed in Falealupo after his only opponent in the planned by-election was ruled to be ineligible by the Supreme Court.
On 26 November 2021 the 2021 Samoan by-elections were held, resulting in Aiono Tile Gafa (HRPP) being elected in Aana Alofi No. 2; Titimaea Tafua (FAST) being elected in Aleipata Itupa i Lalo; Maiava Fuimaono Asafo (FAST) being elected in Falealili No. 2; Laumatiamanu Ringo Purcell (FAST) being elected in Safata No. 2; Maulolo Tavita Amosa being elected in Sagaga No. 2; and Tagaloatele Pasi Poloa (FAST) being elected in Sagaga No. 4.
On 29 November 2021, following the by-elections, the electoral commission declared Ali'imalemanu Alofa Tuuau (HRPP) and Faagasealii Sapoa Feagiai (HRPP) elected as additional members under the women's quota. Their election was disputed by the Speaker, but on 12 May 2022 the Supreme Court confirmed their election, and additionally declared To'omata Norah Leota (FAST) elected as an additional member due to the resignation of Leota Tima Leavai and her subsequent replacement by a man. The Supreme Court noted in its decision that when a man is elected to fill a constituency seat vacancy previously held by a woman, the constitutional requirement for a woman to be elected as an additional member is separate from the quota requirement. Thus there are now a record 7 women parliamentarians.
On 25 March 2022, the Gaga'ifomauga 2 seat became vacant upon the death of FAST MP Va'ele Pa'ia'aua Iona Sekuini. Fo'isala Lilo Tu'u Ioane was elected in the resulting 2022 Gaga'ifomauga 2 by-election.
References
2021
Samoa
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{
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| 1,156
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Use the guide below to search homes for sale in Greater Louisville and surrounding areas. Each of these community pages contains the active real estate listings for that area, updated every day. If you already know what community you like, be sure to sign up for email alertys of new listings as soon as they hit the market -- it's fast, easy, automatic and FREE!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 7,450
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Learn Gunsmithing
Harvey Donaldson: Pioneer Benchrester
Harvey A. Donaldson (1883–1972) devoted his life to improving rifle shooting through dogged determination, innovative cartridge designs, and technical advances.
May 07, 2019 By Joel J. Hutchcroft
In response to my recent column on gunsmith and wildcatter P.O. Ackley, reader David Buck asked for information about another wildcatter named Harvey Donaldson. Shooting Times readers likely will associate that name with the .219 Donaldson Wasp cartridge. I'll get to the cartridge in a few minutes, but here's what I have learned about the man.
Harvey A. Donaldson was born on April 6, 1883, in Fultonville, New York. He attended public school in Fultonville, the Peekskill Military Academy, and the Albany Business College. He worked as a machinist and toolmaker, but shooting was his life.
Donaldson took up the exacting sport of Schuetzen target shooting in 1895 and pursued it diligently through 1915, focusing on 200-yard offhand shooting. He began handloading ammunition in about 1900 and handmade his own arbor press loading tool. Later he developed specific methods of benchrest shooting that were conducive to achieving the best results. For example, he discovered that resting the heavy, long barrel of a Schuetzen rifle on a shooting rest did not negatively affect accuracy and also discovered that fliers could be caused by uneven case neck thicknesses.
Donaldson was ahead of his time in wildcatting for accuracy and fathered at least 15 wildcat cartridges. He worked with cast bullets and also was swaging jacketed bullets by 1906. A proponent of 100 percent load density in the cartridges, he developed and pioneered using a funnel with a long stem in order to load more powder into a cartridge case. He also worked on primers and ignition.
Donaldson developed the rimmed .219 Donaldson Wasp cartridge during the late 1930s, and it became quite popular among match shooters in the 1940s. It is based on the .219 Zipper, which is a .22-caliber wildcat formed by necking down the .25-35 Winchester case. The .219 Donaldson Wasp is considered by many to be the grandfather of benchrest cartridges.
Winchester introduced the .219 Zipper in 1937, and Donaldson quickly modified the body of the case to increase its powder capacity from about 24 grains to 28 grains of IMR 3031. Donaldson's cartridge provided higher velocity and improved accuracy. He later lengthened the case neck by about 0.03 inch, with the resulting case length being 1.750 inches. The .219 Donaldson Wasp typically pushes a 45-grain 0.224-inch-diameter bullet at a muzzle velocity of 3,560 fps, and it has generally been chambered in falling-block single-shot rifles. At least two other versions of the .219 Donaldson Wasp came to be, both created by wildcatters other than Donaldson.
Donaldson was a good scientist in his study of rifle shooting, but he was subjective at times. He started writing about what he was learning before 1900, and eventually he wrote many articles for American Rifleman and Handloader magazines.
A person of great energy, Donaldson enjoyed driving sports cars and racing motorcycles. He collected fine firearms. And he loved hunting upland birds with shotguns as well as hunting woodchucks with high-powered .22-caliber rifles.
Donaldson passed away on November 6, 1972, at the ripe old age of 89. He was a great shooter. He was a creative innovator. But most importantly, he never stopped searching for ways to improve his shooting equipment and his shooting skills. Harvey A. Donaldson stopped shooting only when he could no longer sit at a benchrest and concentrate on a target's bullseye for long hours.
How Barrels Are Rifled Gunsmithing
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The four basic methods of making rifling in a barrel are single-point, broach, button, and...
Rock River Arms LAR-15M .450 Bushmaster - Reviewed & Tested Rifles
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Revolver vs. Semiauto Pistol: A Ballistic Oddity Handguns
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Winchester .350 Legend Rifles and Ammo Available Right Now Ammo
Winchester .350 Legend Rifles and Ammo Available Right Now
Payton Miller - August 21, 2020
The Winchester .350 Legend is a no-nonsense whitetail thumper tailored for rifle hunters in...
Improved Ballistics a Key to Accurate Long-Range Shooting How-To
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Rick Jamison
Improved bullet ballistic coefficients lead to greater performance and accuracy downrange...
Skills Drills: 3-Second Headshot
James Tarr runs through the 3-Second Headshot drill.
The Glock 21
Frank and Tony from Gallery of Guns spice up the Glock test using their non-dominant hands.
All About .300 Blackout
The .300 Blackout is here to stay, and we take some time to look at new technology surrounding this cartridge. Next, we pit subsonic rivals against each other before stretching the legs of this CQB round out to 600 yards from a short 9-inch barrel.
The Future Of Special Operations Small Arms
We're taking a look at what the Army's Elite Units are using for service rifles and what the future of SOCOM sniping looks like.
Review: Bushnell FORGE 4.5-27X 50mm Optics
Review: Bushnell FORGE 4.5-27X 50mm
Sam Wolfenberger - May 01, 2019
The new Bushnell FORGE riflescope is "the only choice for long-range hunting enthusiasts."
Shooting Times Father's Day 2019 Gift Guide Accessories
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The 5 biggest surprise stars of the Championship season so far
Several players have emerged as standout performers in the second tier this season – but they're not always the ones you expect, as Daniel Storey explains
@https://twitter.com/@danielstorey85
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Billy Sharp (Sheffield United)
At the age of 32, Sharp has always been known as a worker but not always a prolific goalscorer. His most fruitful seasons - 2005/06, 2006/07, 2015/16 and 2016/17 - had all come in the third tier. Sharp has scored more than 15 times once in a Championship season, back in 2011/12.
So it is indeed surprising that the 'fat lad from Sheffield' has scored 11 Championship goals in just 1,400 minutes. The only four players to have scored more have taken at least five more shots than Sharp, who has become one of the best poachers in the Football League in the twilight of his career.
Given the travails in Sharp's personal life and how he dealt with them – his newborn son tragically died in 2011 – it's impossible not to be delighted for him. It's not outside the realms of possibility that he could yet add to his two career Premier League appearances.
Neal Maupay (Brentford)
Brentford are suffering this season. Having finished in the top 10 of the Championship in each of the last four seasons, Thomas Frank is in danger of losing his job two months after replacing Dean Smith as manager. He has won one and lost eight of his 10 games in charge, and looks totally out of his depth.
Maupay's form has been the ray of light amid stormy days. Brentford have a habit of clever striker recruitment, having made huge profit on Scott Hogan and Andre Gray. The same may be true of Maupay, with Huddersfield reportedly interested in bringing him to the Premier League in January.
Signed for £1.6m from Saint-Etienne, the former France U21 intermnational was used predominantly as substitute last season until the departure of Lasse Vibe. This season, given the chance to lead Brentford's line, he has scored 13 times for a team struggling in the bottom half. If that wasn't enough, only one player in the division has registered more assists. He's happy drifting out wide, linking play with midfield and getting onto the final ball, potentially making him the complete Championship striker at the age of 22. It's a bit scary.
Will Vaulks (Rotherham)
You have to admire Rotherham's recruitment strategy. This is a club punching way above its weight in the Championship, whose record signing cost just over £500,000 in 2016. Over the two years since, the Millers have barely paid any money for a player, making do with loans and free transfers. And yet they still sit outside the bottom three.
Vaulks is the perfect example of their intelligent recruitment, He was signed from Falkirk in 2016, who he had joined after failing to make a single first-team appearance at Tranmere, and has been a virtual ever-present in Rotherham's midfield since.
Like so many around him, there are few bells and whistles to Vaulks's game. He is not a big name, merely a young man desperate to make the best of himself and in so doing keep Rotherham in the second tier. It's no surprise that Derby and Norwich have been linked with January bids.
Emiliano Buendia (Norwich)
All hail Daniel Farke, and a transfer strategy that has taken Norwich to the upper reaches of the Championship. When Farke was told that he needed to sell star players again in the summer, there were no histrionics. The manager instead relied upon his well-tested methods of using his contacts in Europe to unearth some hidden gems.
Buendia is the perfect example of how Championship clubs can do things smarter. He was signed for just over £1m from La Liga side Getafe, ostensibly a replacement for the outgoing Josh Murphy. At 21, he has taken to Championship life with extraordinary ease given the concerns over workload and physicality - he's only 5ft 7in. No Norwich player has created more chances, and nobody is missing the Murphy twins at Carrow Road.
Buendia is an Argentina U20 international, and Norwich's problem is going to be keeping hold of their star should they fail to secure secure promotion. But then that's what this club does now: buy low and smart, sell high and reinvest intelligently. The day Buendia arrived was certainly a good day, if you know your entry-level Spanish.
Pablo Hernandez (Leeds)
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There has always been a spark in Hernandez, a sense that he could embark on a wonderful run of form if he feels physically and emotionally at his peak. And it's not that he was poor last season, scoring seven times and assisting eight more in 41 Championship matches.
But in 2018/19 we have truly seen the best of him in England at the ripe old age of 33. Hernandez flattered to deceive at Swansea and did the same for large spells of his first 18 months at Elland Road. This season, he has kicked on and become the leader of Bielsaball. Seven assists - the most in the Championship - and seven goals put him only one below last season's total before Christmas. What's more, he's achieved those numbers having missed seven games of the season through injury.
The manager is mightily impressed. "His style is something that wakens in the rival the desire to neutralise him, because he always intervenes and makes actions more fluid," said Bielsea after Leeds's 3-0 win at Norwich. "He gives solutions to actions that are very complicated and he puts his team-mates in better situations. He's a real silent leader. I've only seen this a very few times during my career."
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What Doesn't Kill You
Stories of how people cope after brushes with death. Sometimes death comes as a disease. Sometimes it swims up and bites you. And sometimes it's a pen or pencil, sitting there, just waiting for you to ingest it.
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Tig Notaro
Ira talks to comedian Tig Notaro, who recently had a bunch of horrible things happen to her all in the course of 4 months. (3 minutes)
In July, Tig was diagnosed with cancer. A week later she went on stage on Los Angeles and did a now-legendary set about her string of misfortunes. This is an excerpt of the full performance, which is available on the album LIVE. (13 minutes)
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A teenage girl gets bitten by a shark, rushed to the doctor's office, stitched up, and told she'll be totally fine. Crisis averted, right? Not so much. Ira tells the story. (21 minutes)
A Real Nail Biter
Jessica Benko
Jessica Benko tells the story of a woman named Cathy who was almost killed several times... by a thought that she just couldn't get rid of. (15 1/2 minutes)
The Sweet Year After
Tim Kreider almost died after he was stabbed in the throat. But that wasn't nearly as interesting as the year that followed. A version of this essay appears in Tim's most recent book, We Learn Nothing. (4 minutes)
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The startling rise of the number of people on disability in America.
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Markt Schwaben ist ein Markt an der nördlichen Grenze des oberbayerischen Landkreises Ebersberg.
Geographie
Lage
Markt Schwaben liegt nördlich des Ebersberger Forstes in der Münchner Schotterebene nicht weit vom Quellgebiet der Sempt und dem Ismaninger Speichersee entfernt. Die Gemeinde liegt verkehrsgünstig rund 25 km östlich der Landeshauptstadt München, 14 km südlich von Erding, 28 km westlich von Haag und 15 km nördlich der Kreisstadt Ebersberg. Zum Flughafen München sind es 30 km. Markt Schwaben verfügt über einen bedeutsamen Bahnhof mit Anschluss zur S-Bahn und Regional-Express/Regionalbahn sowie über eine gute Anbindung an die A 94.
Durch den Ort fließt der Hennigbach (ehemals Honigbach), in den der Gigginger Bach südwestlich von Markt Schwaben einmündet. Am Hennigbach kommt es immer wieder zu Überschwemmungen, die in der Vergangenheit auch zu größeren Schäden führten. So Anfang September 1920, in den 1990er Jahren und im Juli 2013. Ein großer Bereich des Ufers des Hennigbachs wurde daher renaturiert und zu einem Naherholungs- und Überflutungsbereich ausgebaut, dem Postanger. Weitere Maßnahmen sind geplant. Östlich von Markt Schwaben fließt die Sempt vorbei. Höchster Punkt des Ortes ist die Wittelsbacher Höhe () am südlichen Ortsrand. Ebenfalls im Süden befindet sich das Sportzentrum mit dem Badeweiher. Nördlich der Bahngleise liegen das Neubaugebiet Burgerfeld mit Theaterhalle, weitere Wohnsiedlungen und zwei Gewerbegebiete. Der verbliebene Teil des ehemaligen Schlosses mit dem Rathaus, der Marktplatz und die katholische Pfarrkirche St. Margaret befinden sich im Ortszentrum, das Schulzentrum schließt sich südwestlich davon an.
Zudem finden sich im Osten entlang der Sempt einige Mühlen sowie zwei Fischereien. Zu Markt Schwaben gehören die Weiler Feichten und die Einöden Hanslmühle, Haus, Paulimühle, Sägmühle und Staudham.
Obwohl Markt Schwaben der Einwohnerzahl nach die viertgrößte Gemeinde im Landkreis ist, ist sie flächenmäßig die kleinste der Ebersberger Gemeinden.
Gemeindegliederung
Es gibt sieben Gemeindeteile (in Klammern ist der Siedlungstyp angegeben):
Feichten (Weiler)
Hanslmühle (Einöde)
Haus (Einöde)
Markt Schwaben (Hauptort)
Paulimühle (Einöde)
Sägmühle (Einöde)
Staudham (Einöde)
Nachbargemeinden
Anzing
Poing
Pliening
Finsing
Ottenhofen
Pastetten
Forstinning
Geschichte
Bis zur Gründung des Königreiches Bayern
Die Geschichte Markt Schwabens lässt sich bis in das 12. Jahrhundert zurückverfolgen. Den Namen erhielt der Ort von Siedlern aus dem alemannischen Suaben. 1115 schenkte Ritter Ulrich von Moosach eine Mühle bei Suaben und zwei Leibeigene der Kirche, zum Seelenheil seines Onkels Rupert I., Abt des Klosters Ebersberg. Diese Urkunde stellt die erste schriftliche Nennung des Ortes Schwaben dar. Bei der damals genannten Mühle handelt es sich wahrscheinlich um die noch heute existierende Kressiermühle. Sie war die einzige Mühle an der Sempt, die bis zur Säkularisation in Besitz des Klosters war.
Zunächst befand sich Schwaben im Besitz des Grafen von Ebersberg, dann im Besitz des Grafen von Limburg-Wasserburg, bis es schließlich 1247 Eigentum der Wittelsbacher wurde. Die Verwaltung übernahm das Amt Falkenberg bei Moosach. Als das Herren- und Ministerialengeschlecht 1272 ausstarb, fiel Schwaben wieder an die Wittelsbacher zurück.
Im Jahre 1283 ließ Herzog Ludwig II. der Strenge eine Burg erbauen, die mit einem Burggraben umgeben wurde und ab diesem Zeitpunkt Sitz der herzoglichen Verwaltung war. 1340 wurde Schwaben von Herzog Rudolf schließlich das Marktrecht verliehen. Durch Marktgerechtigkeit und Dorfgerichtsbarkeit blühte der Ort weiter auf und war schließlich den Sitz eines Pfleggerichts. Dieses Landgericht Schwaben sollte bis 1811 bestehen.
Bei der Bayerischen Landesteilung von 1392 kam Schwaben zum Teilherzogtum Bayern-Ingolstadt unter Herzog Stephan III. 1394 plünderten die Münchner Schwaben. Dabei wurden alle Urkunden, die alle Privilegien und Rechte verbrieften, verbrannt. Für 5000 ungarische Gulden wird Schwaben 1397 an Vitztum Konrad Preysinger, oberster Ministerialer von Oberbayern-München, verpfändet. Bereits sechs Jahre später ging Schwaben wieder zurück an den Herzog. In diesen Jahren versuchten die Schwabener Bürger mehrmals den Herzog dazu zu bewegen, dass er ihre alten Rechte neu in Urkunden bestätigt. Schließlich unterzeichnete Herzog Stephan III. am Freitag vor dem Weißen Sonntag in Wasserburg drei "Briefe" mit fast gleich lautenden Text. In diesen wurden Schwaben das Marktrecht, das Recht auf die Ausübung der niederen Gerichtsbarkeit und das Recht ein Ortswappen zu führen, bestätigt:
Nach dem Tod von Ludwig VII., dem letzten Herzog von Bayern-Ingolstadt, besetzte Herzog Heinrich XVI. von Bayern-Landshut einen Großteil Oberbayerns an Donau und nördlichen Lech. Herzog Albrecht III. konnte durch Verhandlungen für Bayern-München im Erdinger Vertrag von 1450 sich nur das Pfleggericht Schwaben sichern.
Am 5. Mai 1749 kam es zu einem großen Brand, als ein Leutnant in französischen Diensten in der Wagner-Schmiede mit Feuerwerkskörpern herumspielte. Der halbe Marktplatz mit 33 Häusern und zwei Stadeln brannte nieder. Der Leutnant musste als Schadensersatz auf sein gesamtes Erbe verzichten.
1805 wurde das letzte Urteil eines Blutgerichts durch das Landgericht Schwaben verhängt, woraufhin zwei Grafinger Mörder im "Galgenhölzl" enthauptet wurden. Die Gerichts- und Verwaltungsbehörde wurde 1811 nach Ebersberg verlegt, worunter der Ort litt und in der Bedeutungslosigkeit zu versinken drohte.
Ab dem Gemeindeedikt 1818 bis heute
Mit dem bayerischen Gemeindeedikt von 1818 erhielt Schwaben wieder Selbstverwaltungsrechte zurück. Mit dem Bau der Bahnstrecke München–Simbach 1871 und dem Ausbau des Gleiskörpers nach Erding 1872 war für Schwaben ein großer Aufschwung verbunden. Die Post wurde an die Bahn angegliedert und über die Postkutsche bestand die Verbindung nach Anzing und Ebersberg. 1895 wurde ein "Posthaus" gebaut und 1909 elektrifiziert. Aus dem kleinen Marktflecken wurde ein Ort von übergreifender Bedeutung. Seit dem Jahr 1922 ist die Bezeichnung "Markt" Bestandteil des Ortsnamens.
Als amerikanische Truppen Ende April 1945 in Richtung Markt Schwaben vorrückten, hatten Wehrmacht und SS den Ort bereits verlassen. Auch der NSDAP-Ortsgruppenleiter war geflohen. Damit die Waren in den Depots nicht den Amerikanern in die Hände fielen, ließ der Bürgermeister die Lager für die Bevölkerung öffnen. Die Einwohner Markt Schwabens hamsterten sich daraufhin daraus Stoffballen, Stiefel, Leder und literweise Schnaps. Am Morgen des 1. Mai besetzten die US-Truppen den Ort, ohne dass es zu Vorfällen kam. Der Markt Schwabener Lorenz Ostermayr, der früher mal in New York gelebt hatte, war mit einer weißen Fahne und drei Kriegsgefangenen den Feinden entgegen gelaufen und hatte die Amerikaner informiert, dass in Markt Schwaben keine deutsche Truppen mehr waren. Zufälligerweise kam der amerikanische Panzerkommandant aus derselben Straße in New York, in der auch Ostermayr gelebt hatte.
Auf dem Marktplatz versammelten sich am selben Tag etwa 50 ehemalige jüdische KZ-Häftlinge. Sie waren am 25. April in Poing aus einem Zug entkommen, der sie aus dem KZ-Außenlagerkomplex Mühldorf Richtung Tirol bringen sollte. Der Markt Schwabener Arzt Fritz Lichtenegger hatte in den letzten Tagen des Krieges im Unterbräu ein Hilfslazarett eingerichtet, wohin nun die abgemagerten Flüchtlinge gebracht wurden. Einige von ihnen starben trotzdem noch in den folgenden Wochen.
Die amerikanischen Soldaten durchsuchten den Ort Haus für Haus. In Baracken, die ursprünglich zur Unterbringung französischer Kriegsgefangener gedacht waren, fanden die Amerikaner Personalunterlagen der Ingenieure, die an der Entwicklung der V2 beteiligt waren. Beim Arzt Lichtenegger wurde eine SS-Uniform entdeckt, worauf er, nach kurzem Prozess, von den Amerikanern hingerichtet wurde.
Nach dem Zweiten Weltkrieg ließen sich viele Heimatvertriebene aus den ehemaligen deutschen Ostgebieten in Markt Schwaben nieder. Straßennamen wie Königsberger, Neusatzer oder Ödenburger Straße im Südwesten Markt Schwabens erinnern daran. Die Bevölkerung setzte sich in dieser Zeit hauptsächlich aus typisch bodenständigen Gewerbeschaffenden zusammen, die den Markt und die umliegenden Dörfer mit Produkten versorgten. Viele Straßen wurden in Erinnerung an sie bei der Straßennamenvergabe des Ortes in den frühen 1960er-Jahren benannt (z. B. Weißgerberweg, Kupferschmiedberg, Nagelschmiedgasse).
Da die Gemeinde 1922 beschlossen hatte, die Bezeichnung Markt in den Ortsnamen aufzunehmen, lautet der offizielle Name der Gemeinde seit dem 7. Dezember 1922 Markt Markt Schwaben. Unter Ministerpräsident Wilhelm Hoegner (SPD) wurden 1954 einige Märkte zu Städten erhoben, unter anderem Grafing und Ebersberg. Das Wort "Markt" im Ortsnamen verhinderte damals die Erhebung zur Stadt.
In den 1960er-Jahren gab es in Markt Schwaben, wie allerorts im Münchner Einzugsgebiet, einen Bauboom, der zwei Hochhaussiedlungen (von-Kobell-Straße und Dr.-Hartlaub-Ring) hervorbrachte. Auf Postkarten aus der Zeit wird der Ort dementsprechend als "Wohngebiet im Münchener Osten" betitelt.
Im Mai 1962 kam es zum Absturz des Aufklärungsflugzeuges Navy 131390, große Teile des Flugzeuges schlugen bei der Köppelmühle, etwa 520 Meter südlich der Wolfmühle bei Markt Schwaben auf einem Feld auf. Eine Gedenktafel erinnert an der Anzinger Semptbrücke an das bis heute unaufgeklärte Unglück, bei dem nach offiziellen Angaben 26, nach Medienangaben jedoch 45 militärische und zivile Flugzeuginsassen ums Leben kamen.
Eine kleine Sequenz des Filmes Gesprengte Ketten wurde 1962 in Markt Schwaben gedreht. James Coburn alias Officer Louis Sedgwick entwendet dabei vor dem alten Rathaus am Marktplatz ein Fahrrad und fährt danach die Ebersberger Straße ortsauswärts davon.
Der Anschluss an das Münchner Nahverkehrsnetz für die Olympischen Sommerspiele in München 1972 bescherte dem Ort eine gewisse Blütezeit, da ab jetzt neben der Regionalbahn zusätzlich eine schnelle Anbindung an die Großstadt München bestand. Seit Mitte der 1990er-Jahre besteht der 20-Minuten-Takt auf der S-Bahnlinie. 1990 wurde der Autobahnanschluss fertiggestellt. 1992/1993 wurde der Sportpark mit dem Badeweiher im Südosten des Marktes errichtet.
Mit der Erschließung des Burgerfelds im Nordwesten wurde in den 1990er-Jahren der Grundstein für einen neuen Ortsteil inklusive Erweiterung des Gewerbegebiets gelegt. Die Bebauung dauert aktuell noch an. Das Burgerfeld ist mit einer Fußgängerbrücke (mit einer Länge von über 50 Metern) über die Gleise westlich des Bahnhofs und drei Unterführungen mit dem Ortszentrum verbunden. Die offizielle Eröffnung fand am 17. Dezember 2007 statt.
Einwohnerentwicklung
Zwischen 1988 und 2018 wuchs der Markt von 8.699 auf 13.605 um 4.906 Einwohner bzw. um 56,4 %.
Religionen
In Markt Schwaben gibt es eine katholische und eine evangelisch-lutherische Kirche. Außerdem gibt es Gemeinderäume der Freien evangelischen Gemeinde und eine türkische Moschee.
Katholische Pfarrkirche St. Margaret und Mariahilf-Kapelle
Im Sommer 1999 durften die Markt Schwabener Katholiken die bislang letzte Primiz des Ortes feiern.
Angeblich sitzen die Markt Schwabener Katholiken seit Beendigung der Bauarbeiten an der Pfarrkirche St. Margaret in der falschen Kirche, denn laut Erzählungen wurden die Baupläne des Architekten Georg Zwerger der Anzinger und der Schwabener Kirche vertauscht. Ein schlüssiges Indiz, dass die Geschichte wahr sein könnte ist, dass die Anzinger Marienkirche von jeher als Wallfahrtskirche geplant war. Wallfahrtskirchen haben in der Regel immer zwei große Portale, damit die Wallfahrer auf der einen Seite ein- und auf der Gegenüberliegenden wieder ausziehen können. Dies ist bei der Pfarrkirche St. Margaret Markt Schwaben der Fall, wohingegen bei der Anzinger Kirche ein großes Eingangsportal an der Stirnseite zu finden ist.
Die 1720 errichtete Mariahilf-Kapelle am Gerstlacher Weg erstrahlt nach aufwendiger Renovierung inklusive des Grundstücks nun wieder im alten Glanz.
Evangelische Philippuskirche
Am 22. Mai 1955 wurde die evangelische Philippuskirche mit der Vikarwohnung eingeweiht. Sie liegt im Nordosten Markt Schwabens. 1962 wurde aus dem Pfarrvikariat eine eigenständige Pfarrei. 1967 bekam die Kirche eine Orgel, 1976 wurde noch ein Gemeindezentrum der Kirche angefügt. 1986 kamen die beiden kleinen Glocken (b1 - des2) aus dem sehr niedrigen Glockenträger neben der Kirche in einen improvisierten, aufgesetzten Glockenstuhl. 2005 wurde die Kirche renoviert, 2011 wurde eine neue Orgel der Firma Hermann Eule Orgelbau Bautzen eingeweiht, und 2016 wurde das neue, von Andreas Meck geplante Gemeindezentrum eingeweiht.
Türkische Moschee
Ende der 1990er-Jahre begannen die Umbauarbeiten der alten Güterhalle westlich des Bahnhofs Markt Schwaben in der Bahnhofstraße zu einem Gebetsraum, Büro- und Zusammenkunftsräumen der islamischen Gemeinde DITIB Markt Schwaben Ulu Camii. Es ist neben der Moschee in Kirchseeon, die auch von der DITIB verwaltet wird, die bisher einzige Moschee im Landkreis Ebersberg. Die Moschee wurde von Dezember 2017 bis Februar 2018 renoviert.
Freie evangelische Gemeinde
Seit 2001 befindet sich am Ort eine Freie evangelische Gemeinde. Die Gemeinderäume befinden sich im Gewerbegebiet Burgerfeld am Wiegenfeldring. 2006 wurde die Gemeinde selbständig, im selben Jahr bekam sie einen eigenen Pastor. Seit September 2018 ist Stefan Fetzner Pastor der Gemeinde.
Politik
Gemeinderat
Neben dem Ersten Bürgermeister gehören dem Marktgemeinderat 24 weitere Mitglieder an. Der Marktgemeinderat hat nach den Ergebnissen der Wahl vom 15. März 2020 sieben CSU-Räte (2014: 8), die Grünen (3) und die Freien Wähler (5) jeweils fünf Sitze, die SPD drei Sitze (6), die Zukunft Markt Schwaben (ZMS) drei Sitze (2) und die FDP einen Sitz (0).
Bürgermeister
Da er die gesetzliche Altersgrenze erreicht hatte, trat Bürgermeister Georg Hohmann (SPD) 2020 nicht mehr zur Wiederwahl an. Zu seinem Nachfolger wurde der für die SPD und Freien Wähler antretende parteilose Kandidat Michael Stolze, der die Stichwahl gegen Frank Eichner (CSU) mit 67,3 % gewann (Wahlbeteiligung: 57,4 %).
Die zweite Bürgermeisterin wird von der CSU gestellt, der dritte von einem Vertreter der Grünen.
Gemeindepartnerschaften
: Seit dem 25. Oktober 2003 verbindet Markt Schwaben eine Städtepartnerschaft mit Ostra.
Wappen und Flagge
Wappen
Flagge
Vor dem Krieg soll eine Trikolore als Gemeindeflagge in den Farben Schwarz-Weiß-Rot geführt worden sein. Die Farben leiteten sich vom Wappen ab. Nach 1945 wurde bis 1953 keine Flagge geführt. In diesem Jahr beantragte der Marktgemeinderat bei der Regierung Oberbayern die Verleihung von Gemeindefarben. Im Gutachten vom 23. Dezember 1953 wurde festgestellt, dass durch die Verleihung des historischen Wappens bereits das Führen eines Wappenbanners möglich wäre. Als Gemeindefarben wurden Weiß-Rot vorgeschlagen. Der Marktgemeinderat beschloss am 20. Januar 1954 einstimmig:
Die Urkunde des Bayerischen Staatsministeriums des Inneren legte fest:
Allerdings haben die Flaggen, die heute im Gebrauch sind die Farbfolge Rot-Weiß, was der Fahnengenehmigung und dem Beschluss des damaligen Marktgemeinderates widerspricht (vgl. Foto). Im Gebrauch sind nur Banner- und Knatterflaggen. Eine Hissflagge, bei der die Streifen horizontal angeordnet sind, ist in Markt Schwaben nicht im Gebrauch.
Kultur und Sehenswürdigkeiten
Theater
Die Markt Schwabener Weiherspiele (Freilichttheater) wurden 1984 von theaterbegeisterten Laien gegründet.
Eine weitere sehr aktive Theatergruppe ist die Junge Bühne Markt Schwaben, die ihr Stammhaus im Theater im Burgerfeld hat. Es handelt sich um eine Theatergruppe für junge Leute zwischen 15 und 25 Jahren, die 2003 gegründet wurde.
Museum
An der Bahnhofsstraße liegt in der ehemaligen Schweiger-Villa das Heimatmuseum Markt Schwabens, das Exponate von der Steinzeit bis zur Gegenwart beherbergt und über zwei maßstabsgetreue Modelle der Burg, sowie des späteren kurfürstlichen Schlosses verfügt.
Schloss
1283 baute Herzog Ludwig der Strenge von Oberbayern auf einer leichten Anhöhe eine Burg, da hinter Schwaben im Nordosten die damalige Grenze zu Niederbayern verlief. An selber Stelle wurde 1650 nach mehrmaliger Zerstörung der strategisch gut platzierten Burg ein großes, vierflügeliges Schloss errichtet, dessen 1908 im neugotischen Stil renovierter Südflügel heute noch im Ortszentrum erhalten ist. Der Rest des Schlosses wurde 1812 (Ost- und Nordflügel mitsamt St. Magdalenenkapelle und Brückenhaus) und 1969 (Westflügel mit Südwestecke) wegen maroder Bausubstanz und aus Kostengründen abgebrochen. Erwerb des Schlosses inklusive Grundstück 1967 durch die Gemeinde mit anschließender Errichtung des neuen Gemeindezentrums/Rathauses, wobei der Schlossteil stark an die moderne Architektur angeglichen wurde, indem alle kunstvollen Kamine und Fensterläden entfernt wurden. Der ehemalige Burggraben ist im Süden und Westen heute noch erkennbar.
Weitere Bauwerke
Am erhaltenen Schlossflügel wurde das moderne Ziegelgebäude des Rathauses angebaut. Auffällig ist der denkmalgeschützte und 2009 frisch restaurierte Wasserturm gegenüber mit seinen Zinnen, der trotz seines Aussehens nicht aus dem Mittelalter stammt, sondern von 1905. Bemerkenswert ist auch das Pritzl-Haus mit seiner seltenen Dachkonstruktion und dem Turm mit Kuppel, das aber nicht denkmalgeschützt ist. In dem aus dem Jahre 1890 stammenden Gebäude war früher eine Bäckerei untergebracht. Manche der über 30 Zimmer haben nur die Größe eines Kleiderschrankes. Außerdem ist die denkmalgeschützte Haydn-Villa sehenswert; sie ist aufwändig saniert und mit einem Anbau versehen worden und beherbergt heute einen Kindergarten. Am Marktplatz steht das Wandlhaus mit seinen zwei Zwiebelecktürmen, die 1900 hinzugefügt wurden. Der ursprünglich denkmalgeschützte Oberbräu wurde von 2009 bis zum Sommer 2010 derart umgebaut und teilerneuert, dass ihm das Bayerische Landesamt für Denkmalpflege nach Fertigstellung die Denkmaleigenschaft wieder aberkennen musste. Er beherbergt eine Gaststätte.
Das älteste Gebäude des Ortes, der Unterbräu, wurde 2006 vollständig umgebaut, erneuert und innen teilentkernt. Seine Geschichte geht zurück bis auf die Zeit vor dem Dreißigjährigen Krieg. Hier sind unter anderem ein großer Veranstaltungssaal und Räumlichkeiten für das rege Markt Schwabener Vereinsleben entstanden. Gegenüber steht der ebenfalls denkmalgeschützte ehemalige Gasthof Post, ein langgestreckter zweigeschossiger Eckbau mit Kniestock, Satteldach und polygonalem Bodenerker, im Kern aus dem 17. und 18. Jahrhundert. Das neben der Pfarrkirche in der Erdinger Straße liegende denkmalgeschützte, ehemalige Schulhaus wurde 2008 restauriert, nachdem eine Bürgerinitiative zuvor vehement den Abbruch des biedermeierlichen Gebäudes von 1844 gefordert hatte (Motto "De oide Bude woin ma nimma!"). In der Erdinger Straße fällt das Wax-Anwesen auf, ein altes, ländliches Wohnhaus mit gotisierendem Treppengiebel, das aber – obwohl denkmalgeschützt – inzwischen seinem Verfall entgegensieht.
Am hinteren Marktplatz steht ebenfalls ein Baudenkmal, das zweigeschossige Wohnhaus Welschkramer des frühen 18. Jahrhunderts mit Satteldach und mächtigem barocken Giebel, das heute von Neubauten umgeben ist, teilweise in kräftigen, bunten Farbtönen.
Bodendenkmäler
Regelmäßige Veranstaltungen
Über den Landkreis hinaus bekannt sind die Schwabener Sonntagsbegegnungen. Bei den seit 1992 mehrmals jährlich stattfindenden Dialogen zwischen jeweils zwei hochkarätigen Persönlichkeiten waren unter anderem zu Gast: Norbert Lammert, Johannes Rau, Joschka Fischer, Rita Süssmuth, Kurt Beck, Gerhard Polt, Alois Glück, Gesine Schwan, Peer Steinbrück, Reinhard Marx, Johannes Friedrich, Tadeusz Mazowiecki, Notker Wolf, Anselm Grün, Thomas Hitzlsperger, Dieter Hildebrandt und die türkische Familienministerin Güldal Aksit. Die Veranstaltung wird regelmäßig von mehreren hundert Zuhörern besucht. Schirmherr war Hans-Jochen Vogel.
Darüber hinaus veranstaltet die Privatbrauerei Schweiger seit 1998 alle zwei Jahre ein Brauereifest mit traditionell-kulturellem Rahmenprogramm. Bis 1994 wurde jährlich ein Volksfest abgehalten.
Im November findet jährlich im Rathaus eine Ausstellung des Camera-Clubs Markt Schwaben statt.
Wirtschaft
Verkehr
Straßenverkehr
Markt Schwaben ist verkehrstechnisch gut angebunden. Über die Bundesautobahn 94 München–Passau erreicht man das Autobahnkreuz München-Ost (nach Salzburg, Nürnberg und Stuttgart) und die Münchner Stadtgrenze im Osten. Von Osten kommend ist die Markt Schwabener Anschlussstelle Forstinning, vom Westen her Anzing. Nach der Fertigstellung der Flughafentangente Ost (FTO), die die Autobahn mit dem Münchner Flughafen verbindet, ist Markt Schwaben auch von Westen über die neue A 94–Ausfahrt Markt Schwaben-Flughafentangente Ost–Geltinger Straße erreichbar. Sie bildet einen direkten Zugang zum Gewerbegebiet Burgerfeld. Die Abfahrt zur Ortsmitte bleibt Anzing.
Eisenbahn
Durch Markt Schwaben verlaufen die Bahnstrecken München–Simbach und Markt Schwaben–Erding. Nordwestlich der Ortsmitte befindet sich der Bahnhof Markt Schwaben, an dem die Strecke nach Erding von der Strecke München–Simbach abzweigt. Die Königlich Bayerischen Staatseisenbahnen nahmen den Bahnhof am 1. Mai 1871 mit der Hauptbahn von München über Mühldorf nach Simbach in Betrieb. Mit der Eröffnung der Vizinalbahn nach Erding am 16. November 1872 wurde er zum Trennungsbahnhof. Seit 1972 ist Markt Schwaben eine Station der S-Bahn München und in den Münchner Verkehrs- und Tarifverbund (MVV) integriert.
Markt Schwaben wird durch die Linie S 2 von Petershausen und Altomünster über München nach Erding bedient, die bis Markt Schwaben im 20-Minuten-Takt und von Markt Schwaben nach Erding im 20/40-Minuten-Takt fährt. Mit der S-Bahn kommt man in etwa 20 Minuten Fahrtzeit zum Münchner Ostbahnhof und in etwa 30 Minuten Fahrtzeit zum Marienplatz Auf der Hauptbahn fahren im Stundentakt Regionalbahnen von München Hauptbahnhof nach Mühldorf
Im Rahmen des Erdinger Ringschlusses soll die Bahnstrecke nach Erding zum nordwestlich gelegenen Münchner Flughafen verlängert werden. Die S-Bahn-Linie S 2 soll dabei bis Freising verlängert werden.
Busverkehr
In Markt Schwaben verkehren zehn Buslinien des Münchner Verkehrs- und Tarifverbunds. Die Linie 446 verkehrt nur von Montag bis Freitag circa im Zweistundentakt nach Ebersberg. An Samstagen und Sonntagen verkehrt statt der Linie 446 die Rufbuslinie 449, die jedoch ab Markt Schwaben weiter über Pliening nach Poing verkehrt. Die Buslinie 449 verkehrt im Zweistundentakt. Von Montag bis Freitag bedient die Buslinie 463 die Strecke von Markt Schwaben über Pliening nach Poing. Die Buslinie 469 bedient von Markt Schwaben aus die Gemeinden Forstern und Hohenlinden von Montag bis Freitag ungefähr im Zweistundentakt. Die Buslinie 505 verknüpft von Montag bis Freitag im Zweistundentakt Isen mit Markt Schwaben. An Samstagen und Sonntagen verkehrt das Ruftaxi 5050. Die Buslinie 507 verkehrt ebenfalls nur Montag bis Freitag von Markt Schwaben über Ottenhofen und Moosinning nach Erding. Erding wird von Markt Schwaben aus auch von der Buslinie 568 über Finsing ungefähr im Stundentakt mit Taktlücken angefahren. An Samstagen verkehrt das Ruftaxi 5680. Folgende Buslinien verkehren in Markt Schwaben:
446: Markt Schwaben – Anzing – Forstinning – Ebersberg
449: Poing – Pliening – Markt Schwaben – Anzing – Forstinning – Ebersberg (Rufbus)
463: Markt Schwaben – Pliening – Poing – Pliening – Markt Schwaben
469: Markt Schwaben – Forstinning – Forstern – Hohenlinden
505: Markt Schwaben – Paststetten – Forstern – Buch am Buchrain – Isen
507: Markt Schwaben – Ottenhofen – Moosinning – Oberding – Erding
568: Markt Schwaben – Finsing – Moosinning – Erding
4460: Poing – Pliening – Markt Schwaben – Anzing – Forstinning – Ebersberg (Ruftaxi, eine Fahrt)
5050: Markt Schwaben – Paststetten – Forstern – Buch am Buchrain – Isen (Ruftaxi)
5680: Markt Schwaben – Finsing – Moosinning – Erding (Ruftaxi)
Ansässige Unternehmen
Das große Gewerbegebiet im Nordwesten beheimatet einige größere Firmen, so den Fachgroßhandelsbetrieb Wilhelm Gienger und den Hauptsitz der Firma Seidenader, die Maschinen für die pharmazeutische Industrie baut. Ebenfalls nördlich der Bahnlinie ist das Betonwerk Schmitt beheimatet. Im Süden befindet sich an der Ebersberger Straße die Privatbrauerei Schweiger mit der dazugehörigen Brauereigaststätte. Außerdem verfügt der Markt über drei Tankstellen, drei Banken, fünf Supermärkte und vier Autohäuser. Dazu kommen Gaststätten mit regionaler und internationaler Küche, Handwerksbetriebe und Einzelhandelsgeschäfte.
Seit 1907 gibt es in der Färbergasse das Ofenhaus Scheuerecker, nunmehr in vierter Generation.
Das Bauunternehmen Haydn wurde 1763 von Martin Haydn I. gegründet und besteht seitdem in achter Generation im Familienbesitz.
Öffentliche Einrichtungen
Staatliche Einrichtungen
Die Bundesanstalt Technisches Hilfswerk unterhält in Markt Schwaben einen Ortsverband mit zwei Technischen Zügen und den Fachgruppen Notversorgung & Notinstandsetzung, Räumen, Elektroversorgung und Sprengen.
Im Jahr 2006 wurde die wissenschaftliche Abteilung der Zolltechnischen Prüfungs- und Lehranstalt (ZPLA) in Markt Schwaben angesiedelt. Sie ist eine Behörde der Bundeszollverwaltung.
Bildungseinrichtungen
Markt Schwaben beherbergt vier Schulen:
Grundschule Markt Schwaben (Bau 1978)
Mittelschule Markt Schwaben (Bau 1971 mit Erweiterungsbauten bis 1972) seit 1. August 2011, vormals Hauptschule (Volksschule) Markt Schwaben
Lena Christ Realschule (Bau 1973/74)
Franz-Marc-Gymnasium (Bau 1973)
Ämter
Die kommunalen Ämter befinden sich im Rathaus.
Freizeit- und Sportanlagen
Sportanlagen:
Sportzentrum mit: Rasenspielfeld, Rundlaufbahnen, zwei Trainingsplätzen (ein Rasen, ein Kunstrasen), Übungs- und Gymnastikwiese, Tennisanlage, Sommerstockbahnen, vier Bundeskegelbahnen, Badeweiher und Sportgaststätte
Jahnsportplatz mit Allwetterplatz und Bolzwiese (bis zum Beginn des Schulneubaus im Jahr 2020).
BSG-Sportplatz mit Spielfeld und Werferplatz
Ausweichsportplatz
Tennisanlage am Hauser Weg (Vereinsheim im Dezember 2005 abgebrannt mit anschließender Schließung der Anlage)
Hallenschwimmbad mit Sauna und Solarium
Kletterzentrum mit Seilkletter- und Boulderanlage
Persönlichkeiten
Cajetan von Textor (1782–1860), Chirurg und Hochschullehrer an der Universität Würzburg
Bettina Ismair (* 1962), Gründerin der Initiative "Offenes Haus – Offenes Herz"
Roger Rekless (* 1982), Hip-Hop-Musiker, wuchs in Markt Schwaben auf
Sonstiges
Viele Jahre brüteten Störche in Markt Schwaben auf dem Dach des alten Schulhauses, neben der Mittelschule am Gerstlacher Weg. Oft kann man die großen Vögel auf den Dächern stehen, über den Ort fliegen oder am Hennigbach nach Futter suchen sehen. Am Nest ist eine Webcam angebracht, mit der man die Storchenfamilie via Internet beobachten kann. Der Storch wurde so ein Symbol für Markt Schwaben, auch wenn sie einige Jahre nicht mehr im Ort brüteten. Seit 2019 kann man wieder ein Storchenpaar am Horst sehen.
Bilder
Weblinks
Webseite des Marktes Markt Schwaben
Webseite über das Storchennest
Einzelnachweise
Ort im Landkreis Ebersberg
Markt in Oberbayern
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These rectangular favor and chocolate boxes are made of very fine quality Pearl Gold (Shimmer Finish) Card Stock and die cut at the folds, supplied to you in flat form. Text and design printed on the box is totally customizable. Note: Please use a protective wrapper around the confectionary/sweet before placing them in these boxes.
"These rectangular favor and chocolate boxes are made of very fine quality Pearl Gold (Shimmer Finish) Card Stock and die cut at the folds, supplied to you in flat form. Text and design printed on the box is totally customizable. Note: Please use a protective wrapper around the confectionary/sweet before placing them in these boxes."
i have just received the cards. Your service is fantastic, you have most definately made my experience pleasant and fruitful, i really do appreciate what you have done and i absolutely love the cards! Thank you again. Most kindest regards.
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layout: page
title: Shaffer - Avery Wedding
date: 2016-05-24
author: Bruce Huynh
tags: weekly links, java
status: published
summary: Ut ac efficitur erat. Aliquam id sodales velit, ac fringilla.
banner: images/banner/leisure-05.jpg
booking:
startDate: 11/20/2016
endDate: 11/24/2016
ctyhocn: DSMWEHX
groupCode: SAW
published: true
---
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Fusce id aliquam diam. Mauris molestie risus consequat accumsan pellentesque. Proin at faucibus neque. Nulla egestas sed risus ac accumsan. Suspendisse sodales egestas sapien vitae semper. Nunc dignissim eros ac mauris tincidunt, et ultricies ipsum finibus. Cras id nunc vitae ligula porttitor facilisis pharetra vel urna. Donec ut tortor lacinia, efficitur urna eu, rhoncus ex. Mauris dictum laoreet tempor. Sed egestas mollis erat, vel efficitur massa efficitur cursus. Aenean ut nibh velit. Pellentesque placerat turpis in arcu fermentum, ut ultrices augue ullamcorper.
1 Nam vitae lectus consequat, vehicula erat a, pellentesque mi
1 Donec eget lectus sit amet mauris aliquet feugiat
1 Duis ut orci ac velit gravida hendrerit non non leo.
Maecenas magna mi, pellentesque eu sollicitudin eu, dictum at enim. Proin gravida elementum hendrerit. Donec volutpat purus odio, quis lobortis tellus euismod vel. Suspendisse eu vehicula augue. In viverra vehicula mi id venenatis. Donec porta lorem ut leo tristique, accumsan finibus risus congue. Etiam eget nibh vel enim gravida suscipit. Sed mattis dapibus lorem, ac imperdiet velit luctus vel.
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Home Analysis Latin American Constitutional Law and Green Constitutionalism: A Path Forward
Latin American Constitutional Law and Green Constitutionalism: A Path Forward
–José Ignacio Hernández G., Law Professior, Catholic University and Central University (Venezuela); Invited Professor, Pontifical University (Dominican Republic), and La Coruña and Castilla-La Mancha Universities (Spain); Fellow, Growth Lab-Harvard Kennedy School
As Ricardo Hausmann explains, to achieve energy transition goals, it is necessary to electrify the economy or, in other words, decarbonize the economy. A critical condition for that purpose is to ensure the supply of strategic minerals like lithium. Moreover, Latin America has a leading role in that objective.[i]
Argentina, Chile, and Bolivia make up the so-called "lithium triangle", equivalent to approximately 55% of the world's lithium resources. That means that, with Latin American production, it will be easier to electrify the economy with lithium.[ii]
The institutional framework of extractive industries has a crucial role in mining productivity. Therefore, the rules about the relationship between lithium reserves, the Government, and private investments will influence the capacity of the region to ensure a stable supply of lithium to promote green growth.[iii]
Simón Bolívar´s heritage
In 1829 Simón Bolívar, based on the constitutional dictatorship adopted in 1828, declared that the Government was the owner of mining deposits, following Spanish Law. Although Bolívar's decree has been interpreted as a revindication of the national sovereignty over mining, its purpose was quite different. In its brief considerations, Bolívar stated that public ownership over deposits was necessary to promote scientific knowledge and entrepreneurship spirit. The government's property system was envisioned as an efficient way to promote mining activities by the private sector. [iv]
This is the origin of a standard constitutional provision in Latin America, according to which the Government owns mining deposits as "public domain" properties. For instance, following Art. 19 of Chile's Constitution, the State has the "absolute, exclusive and inalienable domain over mines" (Section 24). As a result, the mining legal framework is inspired by a Roman Law institution: the mining concession, which origin is Art. 1 of Bolívar's Decree.
Therefore, the mining law in Latin America was inspired by Spanish Law and Roman Law's provisions about mining concessions. According to the first Mining and Civil Codes approved in the region during the 19th century, mining production was considered an economic activity that only private investment could undertake. For that purpose, the Government granted mining rights to explore and produce through concessions. Those activities were trusted to private investment, and government participation was reduced to royalties and other contractual payments beyond the tax powers and some primary police powers.[v]
In sum, this was a libertarian framework that followed Bolívar's purposes to promote private entrepreneurship with minimum Government intervention.
National sovereignty over mineral resources
Several factors changed the initial libertarian framework and led to regulation in which the Government exerts national sovereignty over mineral resources. In the mid-20th century, Latin America embraced the new economic order to exercise national sovereignty over natural resources, particularly regarding international investors. That perspective was the justification for the nationalization policies, as happened, for instance, with copper in Chile.[vi]
21st century Latin American constitutionalism reflected that vision. Beyond the public property over deposits, the government exerts the direction of mining activities, including the supervision of the "entire productive chain of mining" (Art. 369, Section IV, Bolivia Constitution).
The national sovereignty principle does not necessarily exclude private investments because the Government has the discretion to grant mining rights through concessions, as the Bolivia Constitution recognizes in Art. 370, Section I. However, mining concessions "must fulfill a social and economic function" (Sections III and V). Consequently, state-owned enterprises can also undertake mining activities (Art. 372).
The national sovereignty principle implied the end of the libertarian framework. Even if mining rights are vested in private investors, the government will exert supervision and control of all mining activities to ensure that they fulfill the national interest. Hence the adoption of the Calvo clause, according to which foreign investment, as a rule, must be subject to domestic law and the jurisdiction of domestic courts (Art. 320, Bolivia Constitution).
The reserve over mining deposits
One institution derived from the national sovereignty principle is the Government's power to prohibit mining rights concession to private investors. This institution is rooted in the dominium of Roman Law, according to which the Government can decide not to grant mining rights.[vii]
This power is known as the reserve over mining rights. For instance, the Bolivia Mining Code was reformed in 2017 to regulate the Government's power to reserve strategic minerals for state-owned enterprises. Also, the Code reserved to the Government salt flats, including the Salar Uyuni, considered Bolivia's most extensive lithium reserves (Art. 26). Therefore, only a state-owned enterprise (Yacimientos de Litio Bolivianos, YLB) can exert mining rights for the lithium exploration and production.
That reserve has not been declared in Argentina and Chile. In Argentina, Art. 2 of the Mining Code regulates the concession to grant mining rights to private investors, although the Provinces have the power to adjudicate concessions. The Provinces of Catamarca, Salta, and Jujuy have adopted special regulations to increase the control over lithium.
Chile has a centralized system to adjudicate mining concessions (Arts. 2, Mining Code). However, strategical minerals, like lithium, are excluded from the general concession framework (Art. 8) and subject to a special regulation adopted in Decree n° 23/2021. For that purpose, the administrative body created for development purposes (Corporación de Fomento, CORFO) can sign special lithium contracts with private investors. The new 2022 Constitution draft posed risks over this regulation because it followed the Bolivia Constitution to increase the Government's control over mines (Art. 145).
Other countries have tried to reform the Constitution to increase the governmental regulation over extractive industries, as happened in Mexico. Although Congress did not approve the 2022 constitutional reform, the Mining Code was reformed in 2022 to reserve lithium exploration and production to the Government (Art. 10).
Strategic minerals and the Economic Constitution in Latin America: two models
Latin America Constitutional Law, as happens in some European countries such as Spain and Italy, has adopted provisions regarding the role of the Government in the economy, usually known as the Economic Constitution.[viii] A distinctive figure of the Economic Constitution is the provisions establishing the Government's role in ensuring equal access to social and economic rights, complemented by the Inter-American Law and, notably, the San Salvador Protocol.[ix]
Therefore, and beyond the differences in the constitutional provisions regarding mining activities, under the Inter-American Law, there is a common Economic Constitution that vests in the Government the responsibility to ensure sustainable development, particularly regarding strategic minerals, as the Inter-American Human Right Commissions concluded in Resolution n° 3/2021.
The role of the Government regarding strategic minerals and, in particular, lithium reflects the tensions in the Economic Constitution between the provisions that promote private investments and market mechanisms and the provisions that trust in the Government the direction of the economy, particularly to protect the environment.
The constitutional model adopted in the region during the 20th century is based on a balance between the market economy and the economic role of the Government. For instance, Argentina and Chile have one of the more moderated Constitutions regarding that balance. The market economy is recognized according to the promotion of the common good to create equal social conditions (Art. 1, Chile Constitution). Consequently, the economic order must reflect the social justice value, which implies that the Government is responsible for ensuring equal access to essential goods and services related to social and economic rights (Art. 75, Section 19, Argentina Constitution).
The new constitutionalism adopted during the 21st century expanded the economic role of the Government. That is the case of the Bolivia Constitution, which in addition to the social justice value, recognizes the Government as the director of the national economy with the exclusive responsibility to promote the persons' well-being (Art. 306). This was also the economic system adopted in the 2022 Chile Constitution draft, which recognized the Government as the lead economic actor (Art. 182).
Therefore, there are two economic models in Latin America regarding the exploration and production of the strategic minerals that the world will need to decarbonize the economy: the state-led and private investment-led models.
Bolivia follows the state-led model: only the Government, through state-owned enterprises, can produce lithium. The current lithium production is just 500 tons per year. On the other side, Argentina and Chile follow the private investment-led model, which grants mining rights to private firms to produce lithium. Lithium production in Argentina could be estimated at 6,200 metric tons, while Chile is producing 26,000 metric tons, equivalent to 12% of the world's production.[x]
Towards a new constitutional interpretation over strategic minerals
Two conclusions should be highlighted: (i) the constitutional framework influences lithium production in Latin America, and (ii) to decarbonize the economy, the world will need a reliable lithium supply from the region.
It is necessary, then, to adopt a new constitutional interpretation of mining regulation. Until now, there have been two interpretations. Until the mid-20th century, Constitutional Law adopted a libertarian approach based on the primacy of the private sector, resulting in a private investment-led model. However, in the 21st century, the region shifted towards the national sovereignty approach based on the primacy of the public sector (the state-led model).
Under the national sovereignty approach, the Government, as the director of the economy, can supervise mining activities and undertake them through state-owned enterprises. The government intervention aims to ensure the equal distribution of mining income, following the social justice value.[xi]
Nevertheless, Latin America must first produce minerals to equally distribute mining incomes captured through royalties and taxes. While Argentina and Chile have been able to produce lithium, Bolivia is still trying to decide how to produce that mineral, creating a shortage in the lithium supply (despite having the most prominent resources in the world).
To improve the production of strategic minerals following the social justice value and to follow environmental standards, Latin America Constitutional Law must embrace a new paradigm regarding the public and private sectors. Instead of envisioning them as rivals (the state v the private investment), they should be considered complementary.[xii]
The complementary between the public and the private sector requires a new interpretation of the Economic Constitution beyond the private investment-led and the state-led models. The role of the government cannot be reduced to fixing market failures based on a restrictive interpretation of the subsidiarity principle; neither can this role promote collectivist or totalitarian policies that violate human dignity. Economic development based on social justice must be a shared responsibility between the public and private sectors.
Therefore, the constitutional interpretation of the mining regulation regarding lithium should shift towards the complementarity between the public and private sectors. Innovative public-private partnerships can increase lithium productivity facilitating the decarbonization of the economy. Moreover, it could also help to distribute the mining incomes to promote inclusive development in the world's most unequal region.
For that purpose, Latin American constitutional law, following Inter-American Law, must consider that sustainable lithium production must be based on the shared responsibility between the government and the private investment, increasing the quality of the economic rights in mining activities protected by the Constitution, with a smart mining regulation implemented by the government.[xiii]
Suggested citation: José Ignacio Hernández G., Latin American Constitutional Law and Green Constitutionalism: A Path Forward, Int'l J. Const. L. Blog, Jan. 20, 2023, at: http://www.iconnectblog.com/2023/01/latin-american-constitutional-law-and-green-constitutionalism-a-path-forward/
[i] Hausmann, Ricardo, "How developing economies can capitalize on the green transition", International Monetary Fund, December 2022, retrieved at: https://www.imf.org/en/Publications/fandd/issues/2022/12/green-growth-opportunities-ricardo-hausmann. The strategic or critical minerals refers to those that are necessary to produce electricity without fossil combustibles, such as lithium, copper, nickel, and zinc.
[ii] All the estimations regarding lithium are taken from the U.S. Department of Interior-U.S. Geological Survey, Mineral commodity summaries 2022. Regarding the current conditions of strategical minerals in Latin America, see Unzueta, Adriana, et al., (2022), Apalancando el crecimiento de la demanda en minerales y metales para la transición a una economía baja en carbono, Washington, D.C.: Banco Interamericano de Desarrollo.Also, seePerotti, Remco, and Coviello, Manlio (2015), Governance of strategic minerals in Latin America: the case of Lithium, Santiago: Economic Commission for Latin America and the Caribbean (ECLAC).
[iii] We use the "institutional framework" following the "institution" concept in the economy as the rules that apply to the exchange of goods and services, particularly the economic rights related to exploration and production activities. The quality of those institutions influences economic performance. Regarding the oil industry, see Balza, Lenin and Espinasa, Ramón (2015), Oil sector performance and institutions, Washington D.C.: Banco Interamericano de Desarrollo, 3.
[iv] Hernández G., José Ignacio (2016), El pensamiento jurídico venezolano en el Derecho de los Hidrocarburos, Caracas: Academia de Ciencias Políticas y Sociales, 5.
[v] Vergara Blanco (1992), Alejandro, Principios y Sistema de Derecho Minero. Estudio Histórico y Dogmático, Santiago de Chile: Editorial Jurídica de Chile.
[vi] Novoa Monreal, Eduardo (1972), La nacionalización chilena del cobre, Quimantú: Santiago de Chile.
[vii] Ballbé, Manuel (1950), "Las reservas dominiales", 4 Revista de Administración Pública, 76.
[viii] Hernández G., José Ignacio (2006), Derecho Administrativo y Regulación Económica, Caracas: Editorial Jurídica Venezolana.
[ix] Piovesan, Flávia, "Ius Constitutionale Commune Latinoamericano en Derechos Humanos e impacto en el Sistema Interamericano: rasgos, potenciales y desafíos", in Bogdandy, Armin Von et al., (ed) (2014), Ius Constitutionale Commune en América Latina. Rasgos, potencialidades y desafíos, México D.F.: Universal Autónoma Nacional de México- Instituto Max Planck de Derecho Público Comparado y Derecho Constitucional, México D.F., 61.
[x] Mineral commodity summaries 2022 (n 2).
[xi] The fiscal incomes derived from extractive industries can result in the extractive State, that is, the State dependent on extractive incomes to cover public expenses, such as the Mining state or the Petro state. The political institutions of such states tend to have a weak bureaucratic capacity and a propensity to degenerate into clientelist and rentier institutions. See Karl, Terry Lynn (1997), The Paradox of Plenty: Oil Booms and Petro-States, Los Angeles: The University of California Press, 47-49.
[xii] Mazzucato, Mariana (2022), Transformational change in Latin America and the Caribbean. A mission-oriented Approach, Santiago: Economic Commission for Latin America and the Caribbean (ECLAC), Santiago, 2022, 49.
[xiii] Smart regulation is based on the quality of the regulatory policies that should facilitate innovation in the mining industry. See Mazzucato (n 12).
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The excellence of Islamic knowledge
(MENAFN - Gulf Times) Allah raises the level of people in two ways: (a) Belief in Him and (b) Knowledge of Him. This is established in the following verse of the Qur'an. Allah said:
'Allah raises the level of those who believe and those who are endowed with knowledge. [Al-Qur'an 58:11]
Acquiring knowledge of Allah the most important obligation on every Muslim because Allah ordered seeking knowledge (i.e. the knowledge of Himself) prior to declaration of the Shahadah. He said:
'And know that there is no deity worthy of worship but Allah then seek forgiveness for your sins. [47:19]
To emphasise seeking knowledge before doing any action, Imaam Al-Bukhari used the above verse as a basis for a chapter in his all famous Sahih. He entitles that chapter: 'Knowledge before speech and action. Book of Knowledge: [Sahih Bukhari Vol: 1 Ch 11]
Excellence of knowledge
Allah said: 'Allah (Himself) is witness that there is no deity worth of worship except Him, and the Angels and the men of learning (are also witnesses to this). Maintaining His creation in justice, there is no god save Him, the Almighty, the Wise. [3:18] Notice that Allah (subhanahu wa ta'ala) in the Ayat started the witnessing to the fact that 'there is no deity worthy of worship but Himself with Himself, then He follows it saying that this is also the witness of His Angels and then He follows it by the people who possess knowledge. This verse definitely emphasises the position of the people who are endowed with knowledge above those who are without it.
Allah said:
'... Are those who know equal with those who do not know? But only men of understanding will pay heed. [39:9]
'... Indeed those who fear Allah from His servants they are the knowledgeable, Indeed Allah is the Mighty, the Most Forgiving. [35:28]
'And these similitude We coin for mankind, but none will grasp their meaning except those of knowledge. [29.43]
In all of the above verses we clearly see the very high rank that Allah Himself endowed on those who possess knowledge.
N.B. The knowledge above is only in reference to knowledge of Islam.
'Ask the people of Remembrance if you do not know.
This is because understanding Allah's Rulings is based upon the investigation and deductions of the learned.
Abu Dardaa said that the Messenger of Allah (sallallaahu 'alaihi wa sallam - peace be upon him) said: 'Those people endowed with knowledge are the inheritors of the prophets.
[Hadith Hassan, Abu Dawood, ibn Majah, At-Tirmidhi, ibn Habaan].
He (sallallaahu 'alaihi wa sallam) also said:
'Whosoever Allah wishes good for, He bestows upon him understanding of the Deen. [Bukhari and Muslim).
He bestows upon him an understanding of Deen means that He will allow him to be taught the Qur'an and the Sunnah. It is common knowledge that there is no higher position than that achieved by the prospects and that there is no greater honour than that of inheriting from that noble position [i.e inheriting from the prophets (on them be peace)].
To acquire this very coveted positions needs serious striving to acquire the knowledge they came with (i.e. the knowledge of Allah's Deen).
He (peace be upon him) also said:
'The superiority of a learned man over a worshipper is like my superiority over the ordinary men from among any companions. [At-Tirmidhi. Hadith Hassan].
Looking at this Hadith you will observe (a) how he (peace be upon him) linked knowledge with the position of prophethood, and (b) how he lowered the level of the person who does action without knowledge.
Any worship ('eebadah) that a worshipper perform must be based on knowledge, because if it is lacking then the action would not be 'eebadah but mere ritual.
The messenger (peace be upon him) also said:
'The superiority of a learned man over a worshipper is like the superiority of the full moon over the other stars. [This Hadith is Hassan].
'Ali said to Kaamil:
'O Kaamil, knowledge is better than wealth. Knowledge protects you while you have to protect your wealth. Knowledge is a judge, while wealth have to be judged on. Wealth decreases when it is expended while knowledge purifies when it is given.
Ali said in a poem:
'Glory is for none but the learned,
Guided are they and proofs for seekers of guidance.
Every person is measured based on how much it (knowledge) he mastered,
The ignorant are to the learned their enemies,
Succeed with knowledge and live energetically forever,
Men are all dead, only the possessors of knowledge are truly alive.
The Excellence of Learning
Allah said: 'And the believers should not all go out and fight. Of every troop of them, a party should go forth, that they (who are left behind) may gain sound knowledge in religion, and that they may warn their folk when they return to them, so that they may beware. [9:122]
And His saying:
'...Ask the people of Remembrance if you do not know. [16:43]
The Messenger of Allah (peace be upon him) said:
'Whoever treads a path seeking knowledge therein, Allah directs him to a path leading to 'Al-Jannah' (Paradise). [Hadith Hassan]
'The quest for knowledge is a compulsion on every Muslim. [Hadith Hassan]
N.B. The knowledge referred to above is knowledge in Deen because that is the only knowledge that is Waajib. The people who use this hadith to represent seeking after every type of knowledge are misusing it.
Abu Dardaa' said:
'To learn of one issue (in Deen) is better for me than to pray the entire night
He also said:
'The scholar and the seeker of knowledge are the partners in the sharing of good. Other men are hungry and there is no good in them.
Fath al-Musilee (radhiallahu 'anhu) said:
'Would the sick person who is not fed or given anything to drink or given any medicine, not die? The people replied: 'Surely!
He said: 'It is the same with the heart, it would die if knowledge and wisdom is withheld from it.
Indeed he spoke the truth because the nourishment and enlivening of the heart is through knowledge and wisdom, just as food is nourishment for the body.
Whoever does not acquire knowledge of the Deen of Islam, has a sick heart and its death is inevitable. However, he is not aware of it, because his love for this world destroys his perception.
The Excellence of Teaching
Allah said: '... and that they may warn their folk when they return to them, so that they may beware. [9:122]
What is implied here is teaching and guiding them.
And the saying of Allah.
'And (remember) when Allah took a covenant from those who received the Scripture, (He said): You are to expound it to mankind and not to hide it..[3:187]
This ayat demands that those who are endowed with knowledge should teach it. While the verses that follows show that it is Haram to conceal it.
... But verily, a party of them knowingly conceal the truth [2:146]
And He also said:
... He who hides it, verily his heart is sinful [2:283]
Allah praises the one invites with knowledge. Allah said:
And who is better in speech than he who invites to Allah and does good works... [41:33]
And the saying of Allah:
Call unto the way of your Lord with wisdom and good exhortation, and reason with them in the better way... [16:125]
'Ali said that the Messenger of Allah (peace be upon him) said to him: 'If Allah guides a man through your help, it is better (for you) than red camels. [Bukhari and Muslim]
N.B. Red camels at that time were from among the peoples' most precious possession.
Whoever acquires knowledge and keep it concealed Allah will bridle him with a bridle of fire on the Day of Resurrection. [Abu Dawood, Ibn Majah, At-Tirmidhi]
Verily Allah, His angels, the inmates of the heavens and the earth, even the ant in its hole and the fish in the sea send salutation on the one who teaches good to the people. [Its chain is Hassan]
He (peace be upon him) said:
If the son of Adam dies, all his works are stopped except three. A charity that is continuous, useful knowledge or a righteous child who supplicates for him [Muslim]
Whoever guides or directs to good, then he gets the same amount of blessing as the one who does it. [Related by Muslim]
And Allah censures those who speak on His behalf without any knowledge and some of the scholars consider it an even more heinous sin than shirk based on the following verse. Allah said:
'Say (O Muhammad): My Lord has only forbidden indecencies, those of them that are apparent as well as those that are concealed, and sin and rebellion without justice, and that you associate with Allah in worship that for which He has sent down no authority, and that you say about Allah that which you do not know. [7:33]
Therefore, speaking on behalf of Allah without knowledge is the greatest sin with Allah and the one for which He will assign the severest punishment.
Clarifying what aspect of knowledge is a compulsion on every Muslim (Fard 'Ain).
'Seeking knowledge is a compulsion on every Muslim.
This Hadith is referring to the knowledge of that aspect of the Deen that will save someone from the Fire.
Therefore, what is important for each one of us is to be certain that we know what Knowledge is compulsory for each person to save himself from the Fire.
Before discussing this, however, it is important to know that we must keep on doing righteous deeds until death, otherwise if anyone changes then all the worship that he did before is going to be worthless. If a person fasts a long hot day and then breaks his fast a few minutes before Maghrib, then his fast is void. Likewise, if a person prays a long Salaat and before making Tasleem passes wind, his prayer is nullified.
Life may look like a long journey, but if a person apostatises before the end, he loses everything, and all his deeds are rendered worthless and he would be subjecting himself to dwell in the Fire forever.
'Verily, Allah forgives not (the sin of) setting up rivals in worship with Him, but He forgives (sins) others than that to whom so ever He pleases... [4:116]
This verse establishes that Allah does not forgive Shirk (i.e. setting up rivals in worship with Him), therefore the most important knowledge that a Muslim must seek is the knowledge of Shirk and all that relates to it, from Tawheed and 'Aqeedah (beliefs), etc. It is mandatory on every Muslim to learn these aspects before anything else and not listen to those who say: 'These are theoretical issues, instead keep busy with matters that have action.
Our answer to them is that 'Aqeedah (matters relating to Eemaan) is the greatest of all matters in Islam because Eemaan comprises of the following three aspects (a) believing in the heart, (b) stating with the tongue and (c) acting with the limbs upon what is believed in and stated.
Know, my brothers and sisters that all acts of worship would be rewarded based on what the heart intends for that particular action thus making it (the heart) the most important organ of the body.
Belief is practical matter because it is the action of the heart. The proof of this statement is the saying of Allah:
... But He will call you to account for that which your hearts have earned... [2:225]
This verse indicates what the heart earns. What is earned is the action and it is this that we will be called to account for. Therefore, we can conclude that the action of the heart, which is the beliefs present in it are the most serious action.
Is there not a difference between two persons who say. 'Laa Illaha Illa Allah (there is no deity worthy of worship but Allah), while one has a heart filled with correct beliefs, and the other is a hypocrite who disbelieves in everything in Islam? Surely there is, the first will go to Al-Jannah (Heaven) and the other will be in the lowest depth of the Fire of Hell. Why is this so? Because of the beliefs in his heart, which are its action.
The messenger (peace be on him) stayed in Makkah for 13 years inviting people to 'Laa Illaha Illa Allah', and all those things that relates to it, from Tawheed, 'Aqeedah and Shirk.
When the messenger (peace be on him) sent Mu'adh to Yemen he said to him:
'You are going to a people who possess a Book, so let the first thing you call them to be to testify to 'Laa Illaha Illah Allah', and if they acknowledge this, then tell them that Allah made obligatory on them five Salawaat every day...
From the above examples we can conclude that knowing Allah and performing Tawheed of Allah is the most glorious knowledge in existence. Therefore, the most glorious action a person has to occupy and fill his heart with is knowing his Lord.
The glory of any knowledge is based on the thing studied.
Is there anything more glorious than Allah, the Glorified? Definitely not. Therefore, the knowledge that relates to the study of Allah, what are His rights on us His creation, what He loves and what He does not would therefore be the most glorious knowledge there is in this universe.
Above we have proved what is compulsory knowledge for every Muslim, male or female to seek i.e. any knowledge that helps him or her to perfect their worship for Allah.
The knowledge that is compulsory can be classified:
(a) The first obligation on any Muslim who has reached the age of maturity is to learn about the Shahadah i.e. 'There is no deity worthy of worship but Allah, Muhammad is the messenger of Allah. And to understand their meanings.
That knowledge which helps us to know Allah and teaches us His rights on us. This includes the knowledge of the 'Tawheed of Allah' in all its various forms i.e. Tawheed of Lordship, Tawheed of His Names and Characteristics and Tawheed of 'Eebadah. It also includes any knowledge that would help us to understand any thing that would negate any of the above aspects of Tawheed. This would fall under the category of Shirk. Every Muslim must learn about Tawheed and Shirk.
(b) Knowledge of His Messenger, Muhammad (peace be on him). This includes understanding his purpose as a messenger, his rights on us, our relationship with him, This knowledge is important because through the messenger (peace be on him) we learn how to worship Allah and perform His rights i.e. the messenger came to invite to Tawheed and to fight mankind to keep away from worship other than Him.
'Verily we sent to each nation a messenger to invite to the worship of Allah and to avoid Taghoot.
(c) Any knowledge which helps us to know about the other four fundamental principles of the religion namely, Salaat, Zakaat, Fasting and Haj. These are pillars of Allah's deen and each Muslim must have knowledge about each one of them as soon as it becomes an obligation on him so that he can perform it as Allah wants him to.
(d) that which is halal and that which is haram of social and business transactions that relates to anyone is also an obligation on him to know.
(e) that knowledge which relates to the affairs of the heart, that which is praiseworthy and which every Muslim must try to develop and nurture, like patience, thankfulness, good character, truthfulness and sincerity, etc and from that which is blameworthy and which every Muslim must try to avoid because they are Haram and therefore can cause someone to be thrown into the Fire. Some examples of these are hatred, jealousy, pride, doing things for show, dishonesty, anger, enmity and miserliness, following the desires, etc.
As for knowledge which is Fard Kifiyah; i.e. that which is a collective duty, meaning that it is not a duty on every individual, but it a group of individual undertakes to acquire this knowledge, all other individuals will be exempted from this duty, and the entire community will be free from negligence to acquire this branch of knowledge. If no one embarks on seeking this type of knowledge then the entire community is accountable. Examples of this type of knowledge are:
l An in depth study of the Islamic Law.
l Study of the basic sciences, industries and professions that are vital necessities for the welfare of the community like doctors of medicine, nurses, accountants, contractors, etc.
The Muslims have definitely gone astray in their understanding of their priorities as to which knowledge they must concentrate on and as such we find them put a priority on a knowledge that is not compulsory on them to know about.
The messenger (peace be on him) warns us of this, he said:
'From among the portents of the Hour are (the following):
(1) Religious knowledge will be taken away (by the death of the learned men)
(2) Ignorance (of Religion) will prevail.
(3) Drinking of alcoholic drinks (will be very common).
(4) There will be prevalence of illegal sexual intercourse. [Al-Bukhari: Vol 1 No. 80]
The Messenger of Allah (peace be on him) said:
'Allah does not take away the knowledge, by taking it away from (the heart of) the people, but takes it away by the death of the religious learned men until when none of the (religious learned men) remains, people will take as their leaders ignorant persons who when consulted will give their verdict without knowledge. So they go astray and they will lead the people astray. [Al-Bukhari: Vol 1 No. 100].
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Q: storing allowed settings combinations I am making an interface where a user can define settings. The settings are separated in categories and the user can only select one parameter from each category. The thing is, only certain combinations of parameters are allowed and I must prevent the users from selecting incompatible parameters.
I am trying to design the mysql database structure for that but I did not find a solution that satisfies me.
What I thought of is :
Categories
- ID
- Description
Parameters
- ID
- Parent Category ID
- Description
Combinations
-ID
- a string that is the concatenation of parameters IDs ordered by category
eg. : 102596 would be the combination of parameters 10 from category 1, 25 from category 2, and 96 from category 3.
The problems is : what if one day I need more than a hundred parameters ?
A: Ok So you have these tables:
Categories
- ID
- Description
And each category has multiple parameters:
Parameters
- ID
- Parent Category ID
- Description
Now you want to store the parameters that the user has selected. Why dont you handle the operation of permitting of selecting only one parameter per category outside of the database design scope? If you can do that, then the combination table would be simple:
Combinations
- ID
- USER_ID
- CATEGORY_ID
- PAREMETER_ID
A: You should probably use a table for grouping combinations, as such:
Combinations
- ID
- Group_ID
- Parameter_ID
So for instance, let's use your example of the allowed combination 10, 25 and 96, you would have this 3 entries in the "Combinations" table:
(ID, Group_ID, Parameter_ID) = (1, 1, 10), (2, 1, 25), (3, 1, 96)
(the IDs can be auto-generated ones)
So that means you have this group (1) of 3 allowed parameters (10, 25, 96).
then if you want to add another possible combination of, say, 15, 16, 23 and 42:
(ID, Group_ID, Parameter_ID) = (4, 2, 15), (5, 2, 16), (6, 2, 23), (7, 2, 42)
(again the IDs can be auto-generated ones)
So that means you have this group (2) of 4 allowed parameters (15, 16, 23, 42).
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\section{Introduction}
\label{sec:intro}
The standard model (SM) has been successful in explaining the strong, the
weak and the electromagnetic interactions at currently accessible energies.
The only obvious indication of physics beyond the SM seems to be the
observation of the neutrino mass. Yet there are several motivations to
look for a comprehensive solution to a variety of puzzles of the SM.
Left-Right symmetric model \cite{Pati:1974yy, Mohapatra:1974gc,
Senjanovic:1975rk, Mohapatra:1980qe, Deshpande:1990ip}
has since long received considerable attention as a simple extension of
the SM. While chirality is an elegant ingredient of nature which prevents
unduly large masses for fermions, most of nature is left-right symmetric,
suggesting the reasonable hypothesis that parity is only spontaneously broken,
a principle built into the left-right symmetric models.
Due to inclusion of right handed neutrino states as a principle, such models
provide a natural explanation for the smallness of
neutrino masses \cite{Fukuda:2001nk, Ahmad:2002jz, Ahmad:2002ka,
Bahcall:2004mz} via see-saw mechanism \cite{Minkowski:1977sc,
Gell-Mann:1980vs, Yanagida:1979as, Mohapatra:1979ia}.
This class of models also provides a natural embedding of electroweak
hypercharge, giving a physical explanation for the required extra $U(1)$ as
being generated by the difference between the
baryon number ($B$) and the lepton number ($L$). Thus, $B-L$, the only exact
global symmetry of SM becomes a gauge symmetry, ensuring its exact conservation,
in turn leading to several interesting consequences.
The other extension of the standard model is the grand unified
theory, which unifies all the gauge groups into a single simple
group at very high energy
with only one gauge coupling constant to explain all the three
low energy interactions. The quark-lepton unification then
predicts proton decay, charge quantization, etc. However, the
high energy scale leads to the gauge hierarchy problem, which
dictates the inclusion of supersymmetry as a key ingredient. In
order to protect the electroweak scale from the unification scales,
the minimal supersymmetric standard model (MSSM) would be the most
logical extension of the standard model. Its prediction of particles
at energies accessible to current colliders makes the model of
immediate interest. The SM predictions
now get enriched by the additional predictions of supersymmetry,
but to prevent the unwanted predictions like proton decay, one
needs to impose the $R$-parity symmetry, defined in terms
of the gauged $(B-L)$ quantum number \cite{Martin:1997ns,Mohapatra:1986su},
as
\begin{equation}
R = (-1)^{3(B-L)+2S}.
\label{eq:rparity}
\end{equation}
In the class of supersymmetric left-right models where the parity
breakdown is signalled by the vacuum expectation values of triplet
Higgs scalars, the $R$-parity is naturally
conserved and its origin gets related to the gauged $B-L$ symmetry
\cite{Aulakh:1997ba}.
The bare minimal anomaly free supersymmetric extension of the left-right
symmetric model with triplet Higgs bosons leads to several nettlesome
obstructions which may be considered to be a guidance towards a unique
consistent theory.
One of the most important problems is the spontaneous breaking of
left-right symmetry \cite{Kuchimanchi:1993jg,Kuchimanchi:1995vk}, viz.,
all vacuum expectation values breaking $SU(2)_L$ are exactly equal
in magnitude to those breaking $SU(2)_R$, making the vacuum parity symmetric.
There have been suggestions to solve this problem by introducing
additional fields, or higher dimensional operators, or by going
through a different symmetry breaking chain or breaking the
left-right symmetry along with the supersymmetry breaking
\cite{Kuchimanchi:1993jg,Kuchimanchi:1995vk,Babu:2008ep,Aulakh:1998nn,
Sarkar:2007er, Aulakh:1997vc, Aulakh:1997fq}. In some cases,
when the problem is cured through the introduction
of a parity-odd singlet, the soft susy breaking terms lead to breaking of
electromagnetic charge invariance. A recent improvement \cite{Babu:2008ep}
using a parity even singlet may however deviate significantly from MSSM,
and remains to be explored fully for its phenomenological consistency.
Further, in the minimal SUSYLR model with minimal Higgs fields, which has
been studied extensively \cite{Aulakh:1997ba, Kuchimanchi:1993jg,
Kuchimanchi:1995vk}, it has been found that global minimum of the Higgs
potential is either charge violating or $R$-parity violating. In this
article we propose yet another solution to the problem, which resembles
the non-supersymmetric solution, relating the vacuum expectation
values ($vev$s) of the left-handed and right-handed triplet
Higgs scalars to the Higgs bi-doublet $vev$ through a seesaw
relation. The left-right symmetry breaking scale thus
becomes inversely proportional to the left-handed triplet Higgs
scalar that gives the type II seesaw masses to the neutrinos.
The novel feature consists in the introduction of a bitriplet Higgs and
another Higgs singlet under left-right group. The vacuum that preserves
both electric charge and R-parity can naturally be the global minimum of the
full potential. The most attractive feature of the present model
is that generically it does not allow a left-right symmetric vacuum,
though the latter appears as a single point within the flat direction
of the minima respecting supersymmetry. When the flat direction is lifted
all the energy scales required to explain phenomenology result naturally.
This model can be embedded in the minimal supersymmetric SO(10) grand
unified theory.
Section (\ref{sec:MSLRMrecap}) recapitulates the minimal supersymmetric
left-right model for completeness of the paper. In section
(\ref{sec:bitrip-sing}) we discuss the proposed new model of supersymmetry
having an additional bi-triplet and a singlet.
Section (\ref{sec:phenom}) discusses the phenomenology of this proposed model.
Finally, section (\ref{sec:conclusion}) gives the conclusion.
\section{Minimal Supersymmetric Left-Right Model: A recap}
\label{sec:MSLRMrecap}
In this section we briefly describe the minimal supersymmetric left-right
model. In the left-right symmetric models, it is assumed that
the MSSM gauge group $SU(3)_c \otimes SU(2)_L \otimes
U(1)_Y$ is enhanced at some higher energy, when the left-handed
and right-handed fermions are treated on equal footing.
The minimal supersymmetric left-right (SUSYLR) model has the gauge group
$SU(3)_{c}$ $\otimes ~SU(2)_{L}$ $\otimes ~SU(2)_{R}$ $\otimes ~U(1)_{B-L}$
which could emerge from a supersymmetric $SO(10)$ grand unified theory.
The model has three generations of quarks and leptons, and
their transformations are given by,
\begin{eqnarray}
Q = (3,2,1,1/3), &\qquad& Q^{c} = (3^{*},2,1,-1/3), \nonumber \\
L = (1,2,1,-1), &\qquad& L^{c} = (1,1,2,1),
\end{eqnarray}
where, the numbers in the brackets denote the quantum numbers under
$SU(3)_{c}$, $SU(2)_{L}$, $SU(2)_{R}$, $U(1)_{B-L}$. We have omitted the
generation index for simplicity of notation.
The left-right symmetry could be broken by either doublet Higgs
scalars or triplet Higgs scalar. It has been argued that
for a minimal choice of parameters, it is convenient to break the
group with a triplet Higgs scalar. We shall consider here the
minimal Higgs sector, which consists of
\begin{eqnarray}
\Delta = (1,3,1,2), & \qquad & \bar{\Delta} = (1,3,1,-2),
\nonumber \\
\Delta^{c} = (1,1,3,-2), & \qquad & \bar{\Delta}^{c} = (1,1,3,2),
\nonumber \\
\Phi_{i} = (1,2,2^{*},0), & \qquad & (i=1,2).
\end{eqnarray}
As pointed out in \cite{Aulakh:1997ba} the bidoublets are doubled to achieve
a nonvanishing Cabibbo-Kobayashi-Maskawa (CKM) quark mixing and the number
of triplets is doubled for the sake of anomaly cancellation.
Left-right symmetry is implemented in these theories as a discrete parity
transformation as
\begin{eqnarray}
Q \longleftrightarrow Q_{c}^{*}, &
L\longleftrightarrow L_{c}^{*}, &
\Phi \longleftrightarrow \Phi^{\dagger} \nonumber \\ [0.3cm]
\Delta \longleftrightarrow {\Delta^{c}}^{*},&
\bar{\Delta} \longleftrightarrow \bar{\Delta}{^{c}}^{*}.
\end{eqnarray}
The superpotential for this theory is given by
\begin{eqnarray}
W &=& \nonumber Y^{(i)_{q}} Q^{T} \tau_{2} \Phi_{i} \tau_{2} Q^{c}
+ Y^{(i)_{l}} L^{T} \tau_{2} \Phi_{i} \tau_{2} L^{c}\\
\nonumber
&& + ~i (f L^{T} \tau_{2} \Delta L +f^{*} L{^{c}}^{T} \tau_{2}
\Delta^{c} L^{c} ) \\
&& + ~\mu_{\Delta} \textrm{Tr}(\Delta \bar{\Delta}) + \mu^{*}_{\Delta}
\textrm{Tr}(\Delta^{c} \bar{\Delta}^{c})+ \mu_{ij}
\textrm{Tr}(\tau_{2}\Phi^{T}_{i} \tau_{2} \Phi_{j}).
\end{eqnarray}
All couplings $Y^{(i)_{q,l}}$, $\mu_{ij}$, $\mu_{\Delta}$, $f$ in the
above potential, are complex with the the additional constraint that
$\mu_{ij}$, $f$ and $f^{*}$ are symmetric matrices. It is clear from the
above eq. that the theory has no baryon or lepton number violation
terms. As such
$R$-parity symmetry, defined by $(-1)^{3(B-L)+2S}$, is automatically
conserved in the SUSYLR model.
It turns out that left-right symmetry imposes rather strong constraints on
the ground state of this model. It was pointed out by Kuchimanchi and
Mohapatra \cite{Kuchimanchi:1993jg} that there is no spontaneous parity
breaking for this minimal choice of Higgs in the supersymmetric left-right
model and as such the ground state remains parity symmetric. If parity odd
singlets are introduced to break this symmetry \cite{Cvetic:1985zp}, then
it was shown \cite{Kuchimanchi:1993jg} that the charge-breaking vacua have a
lower potential than the charge-preserving vacua and as such the ground
state does not conserve electric charge. Breaking $R$ parity was another
possible solution to this dilemma of breaking parity symmetry. However, if
one wants to prevent proton decay, then one must look for alternative
solutions. One such possible solution is to add two new triplet superfields
$\Omega(1,3,1,0)$, $\Omega_c (1,1,3,0)$ where under parity symmetry
$\Omega \leftrightarrow \Omega_c^*$. This field has been explored
extensively in \cite{Aulakh:1997ba,Aulakh:1997fq, Aulakh:1998nn,
Yajnik:2006kc, Sarkar:2007er, Aulakh:1997vc}.
In the present paper we discuss another alternative solution with the
inclusion of a scalar bitriplet ($\eta$) and a parity odd singlet
($\sigma$). This model breaks parity spontaneously, and also preserves
electromagnetic charge automatically. The left-right parity is
spontaneously broken and as a result, the minimization does not
allow a left-right symmetry preserving solution.
\section{Supersymmetric Left-Right Symmetric Model including the
Bi-triplet and the Singlet}
\label{sec:bitrip-sing}
We now present our model, where we include a bi-triplet and
a parity odd singlet fields, in the minimal supersymmetric
left-right symmetric model. These fields are vector-like and
hence do not contribute to anomaly, so we consider only one
of these fields.
The quantum numbers for the new scalar fields $\eta$ and $\sigma$, under the
gauge group considered are given by,
\begin{equation}
\eta(1,3,3,0), \qquad \sigma(1,1,1,0).
\end{equation}
Under parity, these fields transform as $\eta \leftrightarrow \eta$ and
$\sigma \leftrightarrow -\sigma$. The superpotential for the model is
written in the more general tensorial notation,
\begin{eqnarray}
W &=& \nonumber f \eta_{\alpha i} \,\Delta_{\alpha} \,\Delta_{i}^{c}
+ f^{*} \eta_{\alpha i} \,\bar{\Delta}_{\alpha} \,\bar{\Delta}_{i}^{c}
\\ && \nonumber
+~\lambda_1 \, \eta_{\alpha i} \, \Phi_{a m} \, \Phi_{b n}
\left(\tau^{\alpha} \epsilon \right)_{a b} \left(\tau^{i}
\epsilon \right)_{m n} + m_{\eta}\,\eta_{\alpha i}\, \eta_{\alpha i}
\\ && \nonumber
+ ~M\, \left(\Delta_{\alpha} \bar{\Delta}_{\alpha}
+ \Delta_{i}^{c} \bar{\Delta}_{i}^{c} \right)
+ \mu \, \epsilon_{a b}\, \Phi_{b m} \,\epsilon_{m n} \,\Phi_{a n}
\\ &&
+~m_{\sigma}\, \sigma^{2} + \lambda_2 \, \sigma \left(\Delta_{\alpha}
\bar{\Delta}_{\alpha} - \Delta_{i}^{c} \bar{\Delta}_{i}^{c} \right),
\end{eqnarray}
where, $\alpha$, $\beta$ $= 1, 2, 3$ and $a, b =1, 2$ are $SU(2)_{L}$
indices, whereas $i, j = 1, 2, 3$ and $m, n=1, 2$ are $SU(2)_{R}$ indices.
The summation over repeated index is implied, with the change in basis
from numerical $1,2,3$ indices to $+, -, 0$ indices as follows,
\begin{eqnarray}
\Psi_{\alpha} \Psi_{\alpha} &=&\nonumber \Psi_{1} \Psi_{1}+\Psi_{2} \Psi_{2}
+\Psi_{3} \Psi_{3} \\
&=& \Psi_{+} \Psi_{-}+\Psi_{-} \Psi_{+}+\Psi_{0} \Psi_{0},
\end{eqnarray}
where, we have defined $\Psi_{\pm} =(\Psi_{1} \pm i \Psi_{2})/\sqrt{2}$
and $\Psi_{0}=\Psi_{3}$. The vacuum expectation values (vev) that the
neutral components of the Higgs sector acquires are,
\begin{equation}
\begin{array}{lcl}
\langle \Delta_{-} \rangle = \langle \bar{\Delta}_{+} \rangle = v_{L},
&\qquad&
\langle \Delta_{+}^{c} \rangle = \langle \bar{\Delta}_{-}^{c}
\rangle = v_{R}, \\[0.1cm]
\langle \Phi_{+ -} \rangle = v, &\qquad&
\langle \Phi_{- +} \rangle = \it{v'}, \\
\langle \eta_{+ -} \rangle = \it{u_{1}}, \, &\qquad&
\langle \eta_{- +} \rangle = \it{u_{2}},\, \\
\langle \eta_{0 0} \rangle = \it{u_{0}}.
\end{array}
\end{equation}
Assuming SUSY to be unbroken till the TeV scale implies the $F$ and $D$
flatness conditions for the scalar fields to be,
\begin{eqnarray}
F_{\Delta_{\alpha}} &=& f \, \eta_{\alpha i} \,\Delta_{i}^{c} + M
\bar{\Delta}_{\alpha}+ \lambda_2 \, \sigma \,\bar{\Delta}_{\alpha} = 0,
\nonumber \\
F_{\bar{\Delta}_{\alpha}} &=& f^{*} \, \eta_{\alpha i} \,\bar{\Delta}_{i}^{c}
+M \Delta_{\alpha} + \lambda_2 \, \sigma \,\Delta_{\alpha} = 0,
\nonumber \\
F_{\Delta_{i}^{c}} &=& f \, \eta_{\alpha i} \,\Delta_{\alpha} +M
\,\bar{\Delta}_{i}^{c} - \lambda_2 \, \sigma \,\bar{\Delta}^{c}_{i} = 0,
\nonumber \\
F_{\bar{\Delta}_{i}^{c}} &=& f^{*} \, \eta_{\alpha i} \,\bar{\Delta}_{i} +M
\,\Delta_{i}^{c} - \lambda_2 \, \sigma \,\Delta^{c}_{i} = 0,
\nonumber \\
F_{\sigma} &=& 2 m_{\sigma}\, \sigma + \lambda_2 \,\left(\Delta_{\alpha}
\bar{\Delta}_{\alpha} - \Delta_{i}^{c} \bar{\Delta}_{i}^{c} \right) = 0,
\nonumber
\end{eqnarray}
\begin{eqnarray}
F_{\eta_{\alpha i}}&=& f\, \Delta_{\alpha} \,\Delta_{i}^{c}
+f^{*} \bar{\Delta}_{\alpha} \,\bar{\Delta}_{i}^{c}
+ 2\,m_{\eta}\,\eta_{\alpha i}
\nonumber \\ &&
+ ~\lambda_{1} \, \Phi_{a m} \, \Phi_{b n} (\tau^{\alpha} \epsilon )_{a b}
(\tau^{i} \epsilon )_{m n} = 0,
\nonumber
\end{eqnarray}
\begin{eqnarray}
F_{\Phi_{c p}} &=& \lambda_{1} \,\eta_{\alpha i} \Phi_{b n} \,
(\tau^{\alpha} \epsilon )_{cb} \left(\tau^{i} \epsilon \right)_{pn}
\nonumber \\ &&
+~ \lambda_{1} \,\eta_{\alpha i} \Phi_{a m} \,(\tau^{\alpha} \epsilon )_{a c}
(\tau^{i} \epsilon)_{mp}
\nonumber \\ &&
+ ~\mu \, \epsilon_{a c} \, \epsilon_{p n} \, \Phi_{a n}
+ \mu \, \epsilon_{cb} \, \Phi_{b m} \,\epsilon_{mp} = 0,
\end{eqnarray}
\begin{eqnarray}
D_{R_{i}} &=& 2 {\Delta^{c}}^{\dagger} \tau_{i} \Delta^{c} + 2
\bar{\Delta}^{c \dagger} \tau_{i} \bar{\Delta^{c}}+ \eta \tau_{i}^{T}
\eta^{\dagger} +\Phi \tau_{i}^{T} \Phi^{ \dagger} =0,
\nonumber \\
D_{L_{i}} &=& 2 \Delta^{ \dagger} \tau_{i} \Delta
+ 2 \bar{\Delta}^{ \dagger} \tau_{i} \bar{\Delta}+\eta^{ \dagger} \tau_{i}
\eta +\Phi^ { \dagger} \tau_{i} \Phi = 0,
\nonumber \\
D_{B-L} &=& 2 \left( \Delta^{ \dagger} \Delta -\bar{\Delta}^{ \dagger}
\bar{\Delta}\right) - 2 \left( {\Delta^{c}}^{ \dagger} \Delta^{c}
- \bar{\Delta}^{c \dagger} \bar{\Delta}^{c}\right) = 0.
\end{eqnarray}
In the above eqns., we have neglected the slepton and squark fields, since
they would have zero vev at the scale considered.
We have also assumed $v' \ll v$ and hence the terms
containing $v'$ can be neglected.
\section{Phenomenology}
\label{sec:phenom}
An inspection of the minimisation conditions obtained at the end of the
previous section proves two important statements we have made earlier. First, the
electromagnetic charge invariance of this vacuum is automatic for any
parameter range of the theory. Secondly, the R-parity, defined in
eq. (\ref{eq:rparity}), is preserved in the present model,
since the $\Delta$'s are R-parity even whereas the bi-doublet and the
bi-triplet Higgs scalars have zero R-parity.
We shall now discuss the conditions that emerge from the vanishing of the
various $F$ terms, which after the fields acquire their respective
vevs, are given by,
\begin{eqnarray}
F_{\Delta} &=& f \, u_{1} v_{R} +(M+\lambda_{2} \langle \sigma
\rangle) v_L = 0,
\label{eq:FDeltavev}
\\
F_{\bar{\Delta}} &=&
f^{*} u_{2} v_{R}+(M+\lambda_{2}
\langle \sigma \rangle) v_{L} = 0,
\label{eq:FDeltabarvev}
\\
F_{\Delta^{c}} &=&
f\, u_{1} v_{L}+(M-\lambda_{2}\langle
\sigma \rangle)v_{R} = 0,
\label{eq:FDeltaCvev}
\\
F_{\bar{\Delta}^{c}} &=&
f^{*}\, u_{2} v_{L}+(M-\lambda_{2}
\langle \sigma \rangle) v_{R} = 0,
\label{eq:FDeltaCbarvev}
\\
F_{\sigma} &=& m_{\sigma}\,\langle \sigma \rangle+ \lambda_2 (v_{L}^{2}
-v_{R}^{2}) = 0,
\label{eq:Fsigmavev}
\\
F_{\eta} &=&
f \, v_{L} v_{R} +f^{*} \, v_{L}
v_{R}+ \lambda_{1} v^{2} \,+ 2 m_{\eta}(u_{1}+u_{2}
+ u_{0}) = 0,
\label{eq:FEtavev}
\\
F_{\Phi} &=& - 2 \lambda_{1} (u_{1}+u_{2}) v
+ 2 \lambda_{1} u_{0} v-2 \mu v = 0.
\label{eq:FPhivev}
\end{eqnarray}
At the outset we see that the $F_\sigma$ flatness condition permits
the trivial solution $\langle \sigma \rangle=0$, which would imply
the undesirable solution $v_L=v_R$ and lead to no parity breakdown.
But this special point can easily be destabilized once the soft
terms are turned on. Away from this special point, we are led
to phenomenologically interesting vacuum configurations.
The $F$ flatness conditions for the $\Delta$ and $\bar{\Delta}$
fields demand $fu_1=f^{*}u_2 $ which can be naturally satisfied by choosing
\begin{equation}
f = f^{*} \qquad \textrm{and} \qquad u_1 = u_2 \equiv u.
\end{equation}
This is consistent with
the relation obtained from the $F$ flatness conditions for the $\Delta^c$ and
$\bar{\Delta}^c$ fields, which may now be together read as
\begin{eqnarray}
(M-\lambda_{2} \langle \sigma \rangle)v_{R}=- f \, u v_{L}.
\label{eq:FDeltaBarCombo}
\end{eqnarray}
The first four conditions (\ref{eq:FDeltavev})-(\ref{eq:FDeltaCbarvev})
can therefore be used to eliminate the scale $u$ and give a relation
\begin{equation}
\left( \frac{v_L}{v_R} \right)^2 =
\frac{M-\lambda_{2} \langle \sigma \rangle}{M+\lambda_{2}
\langle \sigma \rangle}.
\label{eq:ratio}
\end{equation}
Let us assume the scale of the vev's $u_1$, $u_2$ and $u_0$ to be
the same. Then the
vanishing of $F_\eta$ gives a relation
\begin{equation}
2f v_L v_R \approx - (\lambda_1 v^2 + 6 m_\eta u).
\label{eq:product}
\end{equation}
Finally, the last condition (\ref{eq:FPhivev}) has an interesting consequence.
While electroweak symmetry is assumed to remain unbroken in the supersymmetric
phase, so that $v$ must be chosen to be zero, we see that the factor
multiplying $v$ implies a relation
\begin{equation}
\mu \approx -\lambda_1 u.
\label{eq:musimu}
\end{equation}
That is, taking $\lambda_1$ to be order unity, the scale of the
$\mu$ term determines the scale of $u$.
We now attempt an interpretation of these relations to obtain reasonable
phenomenology. The scale $v_R$ must be higher than the TeV scale. It seems
reasonable to assume that the eq. (\ref{eq:product}) provides a see-saw
relation between $v_L$ and $v_R$ vev's, and that this product is anchored
by the TeV scale.
Since bitriplet contributes additional non-doublet Higgs in the
Standard Model,
it is important that the vacuum expectation value $u$ is much higher
or much smaller than the electroweak scale, and we shall explore the
latter route. In this case $u$ should be strictly less than $1$GeV.
The scale $m_\eta$ determines the masses of triplet majorons and
needs to be high compared to the TeV scale. If the above see-saw relation
is not to be jeopardized, we must have $m_\eta u \leq m^2_{EW}$. We
can avoid proliferation of new mass scales by choosing
\begin{equation}
m_\eta u \approx v^2 = m^2_{EW}.
\end{equation}
This establishes eq. (\ref{eq:product}) as the desired hierarchy
equation, with $f$ chosen to be negative.
Now let us examine the consistency of the assumption $u \ll m_{EW}$
in the light of the two equations (\ref{eq:FDeltaBarCombo}) and
(\ref{eq:ratio}). Let us assume that
$(v_L/v_R) \ll 1$ as in the non-supersymmetric case.
Then eq. (\ref{eq:ratio}) means that on the right hand side,
\begin{equation}
M-\lambda_{2} \vev{\sigma} \ll M + \lambda_{2} \vev{\sigma}
\implies M \approx \lambda_{2} \langle \sigma \rangle.
\label{eq:Msigmaequality}
\end{equation}
Then eq. (\ref{eq:FDeltavev}) can be read as
\begin{equation}
\frac{v_L}{v_R} \approx \frac{(-f)u}{2M}.
\label{eq:ratiotwo}
\end{equation}
We thus see that the required
hierarchies of scales can be spontaneously generated, and can be related
to each other. Finally, although only the ratios has been related in eq.
(\ref{eq:ratiotwo}) we may choose
\begin{equation}
v_L \approx u, \qquad v_R \approx M.
\label{eq:setscale}
\end{equation}
We see that through this choice of individual scales and through the see-saw
relation (\ref{eq:product}), $u$ and $v_R$ obey a mutual see-saw relation.
A small value of $u$ in the eV range would place $v_R$ in the intermediate
range as in the traditional proposals for neutrino mass see-saw. A larger
range of values close to the GeV scale would lead to $v_R$ and the
resulting heavy neutrinos states within the range of collider confirmation.
Finally, returning to eq. (\ref{eq:musimu}), we can obtain the desirable
scale for $u$ by choosing $\mu$ to be of that scale, viz., in
the sub-GeV range.
This solves the $\mu$ problem arising in MSSM by relating it to
other scales required to keep the $v_R$ high.
An interesting consequence of the choices made so far is that using
eq.s (\ref{eq:Msigmaequality}) and (\ref{eq:setscale}) in eq.
(\ref{eq:Fsigmavev}) yields
\begin{equation}
| m_\sigma | \approx \lambda_2\frac{v^2_{R}}{\langle \sigma \rangle}
\sim \lambda_2^2 M.
\label{eq:msigmaM}
\end{equation}
To summarize, various phenomenological considerations lead to a
natural choice of three of the mass parameters of the superpotential,
$M$, $m_\sigma$ and $m_\eta$ to be comparable to each other and large,
such as to determine $v_R$,
and in turn the masses of the heavy majorana neutrinos. The scale
$\mu$ which determines the vacuum expectation value $u$ and in turn
the value $v_L$ could be anything less than a GeV. Most importantly
we have the see-saw relation eq. (\ref{eq:product}) which relates
these scales, and if the $v_R$ scale is to be within a few orders
of magnitude of the TeV scale, then $\mu$ should be close to though
less than a GeV.
We can contemplate two extreme possibilities for the scale $M$. Keeping
in mind the gravitino production and overabundance problem, we can choose
the largest value $v_R \leq 10^9$ GeV. If it can be ensured from
inflation that this is also the reheat temperature, then the
thermalisation of heavy majorana neutrinos required for thermal
leptogenesis at a scale somewhat lower than this can be easily
accommodated. We can also try to take $v_R$ as low as
$10$ TeV which is consistent with preserving lepton asymmetry
generated by non-thermal mechanisms \cite{Sahu:2004sb}.
Baryogenesis from non-thermal or sleptonic leptogenesis in
this kind of setting has been extensively studied
\cite{Grossman:2003jv, D'Ambrosio:2003wy, Boubekeur:2004ez, Chun:2005ms}.
This low value of $v_R$
is consistent with neutrino see-saw relation, but will rely
critically on the smallness of Yukawa couplings\cite{Sahu:2004sb}
and may be accessible to colliders \cite{King:2004cx}.
As we have seen, at the large scale, charge conservation also demands
conservation of R-parity. The question generally arise as to what happens
when heavy fields are integrated out and soft supersymmetry breaking terms
are switched on. The analysis done in \cite{Aulakh:1997ba} implies
that if $M_{R}$
is very large (around $10^{10}$ GeV), the breakdown of R-parity at low
energy would give rise to an almost-massless majoron coupled to the Z-bosons,
which is ruled out experimentally. This is one of the central aspects of
supersymmetric left-right theories with large $M_{R}$: R-parity is an exact
symmetry of the low energy effective theory. The supersymmetric
partners of the neutrinos do not get any $vev$ at any scale,
which also ensures that the R-parity is conserved.
\section{Conclusion}
\label{sec:conclusion}
Supersymmetry and left-right symmetry are considered strong possible
candidates for extension of standard model. However, construction of a
low energy SUSYLR theory is by no means trivial, since left-right symmetry
cannot be broken spontaneously \cite{Kuchimanchi:1993jg}. In this paper,
however, with the introduction of a bitriplet scalar field along with a
parity odd Higgs singlet we have presented a possible mechanism of
spontaneously breaking LR symmetry in a SUSYLR model. The advantages of
this model besides breaking parity spontaneously is that it preserves
$R$ parity naturally. Also, we find a possible relation between the
left-right symmetry breaking scale and the inverse of neutrino mass.
\section{Acknowledgments}
A.S. would like to thank the hospitality at PRL, where most of the present
work was done. US would like to
thank the Physics Department and the McDonnell Center for Space
Sciences, Washington University in St. Louis, USA for inviting him
as Clark Way Harrison visiting professor and thank R. Cowsik and F.
Ferrer for discussions.
|
{
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}
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Home News Statement by the Representative of UNFPA Afghanistan, Dr. Bannet Ndyanabangi at the Second meeting of the Afghanistan Youth Parliament 23rd July 2017
Statement by the Representative of UNFPA Afghanistan, Dr. Bannet Ndyanabangi at the Second meeting of the Afghanistan Youth Parliament 23rd July 2017
Your Excellency the Chief Executive, Dr. Abdullah Abdullah
Your Excellency Speaker of Mishrano Jirga,
Honourable Deputies,
Honourable Deputy Minister of Youth Affairs
Ambassadors and International Embassy staff
Distinguished guests,
UN agencies colleagues,
Government and civil society partners,
Media representatives,
Salam-u Alikum
Let me start by expressing sincere thanks and appreciation for the invitation to address this house that has again gathered to discuss and address the issues affecting young people in Afghanistan.
The Youth Parliament gives us an opportunity to identify areas of strategic interventions and key partnerships in building a sustainable development program for young people in this country and it is for this reason I believe we are all here.
The young people gathered here want to be counted and have a voice and meaningful participation in governance and policymaking.
Afghanistan is one of the youngest countries in the world. Youth below the age of 25 make up over 63 percent of the population, while those under the age of 15 comprise almost half.
Excellences the Chief Executive and Speaker, distinguished guests
Investing in the nation's development would require a pragmatic approach, putting in place relevant social, economic, health policies and programmes. To this end, the consideration of the Demographic dynamics is of utmost importance for sustainable development.
In this regard and in order to realize a Demographic Dividend, distinct changes in population age structures, labor productivity andaccelerated economic growth should happen over time to address stringent issues of unemployment, underemployment, fertility and dependency.
This requires multiple intersecting investments that we all present here today must strive relentlessly to harness. UNFPA has worked with the Deputy Ministry of Youth and other agencies to carry out an analysis and draft a document on investing in Youth and how to realize Afghanistan's demographic divided (English Pashto and Dari versions are available in your folders).
We have also followed up and worked on a Joint UN Programme of Support on Adolescents and Youth (JUPSAY) with other sister UN agencies and the National Action Plan (NAP) to End Early and Child Marriage (ECM).
Distinguished ladies and gentlemen
When these young people are adequately equipped with knowledge and skills that protect them and can make informed decisions, they can realize their full potential and contribute more meaningfully to economic and social transformation as current and future leaders.
Let me avail myself of this opportunity to draw your kind attention to the strategic importance to involve youth in peace and sustainable development initiatives. This is the time to initiate and implement the required investment in the youthful population.
We need to recognize the need to generate a momentum that offers sustainable livelihoods, viable employment options, capacity building, sexual, reproductive and other health services. A competitive global workplace also demands more skills, education and technical expertise than ever before.
Though Afghanistan is yet to achieve its demographic transition fully, it is never too early to make deliberate investments to ensure an empowered youth, filled with potential and readiness to drive growth across all sectors. This will deliver innovative and lasting development through the right investment in four key mutually reinforcing policy frameworks known as the "4E Policy Framework"i.e Empowerment, Education, Employment and Ensuring equality and equity aiming at the realization of a demographic dividend.
The potential locked within the demographic dividend when unleashed will create the momentum needed to catapult the country into sustained economic growth. But these can only be achieved if we make a conscious effort to change the status of our young people.
Excellence the Chief Executive, Excellence the Speaker, Hon Ministers, Ambassadors development partners, distinguished ladies and gentlemen
In conclusion, permit me to remind you of a salient statement in which Mr Kofi Annan, the 7th UN Secretary-General said;
"Young people should be at the forefront of global change and innovation. Empowered, they can be key agents for development and peace. If, however, they are left on society's margins, all of us will be impoverished. Let us ensure that all young people have every opportunity to participate fully in the lives of their societies."
And on this note I look forward to a very fruitful 2nd Youth Parliament, and thank you for your attention.
Potential Unleashed; from the Afghanistan Youth Parliament to candidacy for the Afghan Parliament
UNFPA Afghanistan Newsletter
UNFPA is the United Nations agency which leads global efforts to help ensure that...
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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import {Component, OnInit, ViewChild} from '@angular/core';
import * as vis from 'vis';
import {BqQueryPlan} from '../bq_query_plan';
import {DagreLayoutService} from '../dagre-layout.service';
import {LogService} from '../log.service';
import {PlanSideDisplayComponent} from '../plan-side-display/plan-side-display.component';
import {PlanStatusCardComponent} from '../plan-status-card/plan-status-card.component';
import {QueryStage, QueryStep} from '../rest_interfaces';
type ResizeCallback = (chart: TreeChart, params: object) => void;
type NodeSelectCallback =
(chart: TreeChart, stage: QueryStage, params: object) => void;
type NodeDeselectCallback =
(chart: TreeChart, stage: QueryStage, params: object) => void;
type EdgeSelectCallback = (chart: TreeChart, params: object, detail: object) =>
void;
type EdgeDeselectCallback = (chart: TreeChart, params: object) => void;
@Component({
selector: 'app-vis-display',
templateUrl: './vis-display.component.html',
styleUrls: ['./vis-display.component.css']
})
export class VisDisplayComponent implements OnInit {
public graph: TreeChart;
private layout: any; // dqagre layout result;
private plan: BqQueryPlan;
private haveDoneDraw = false;
@ViewChild('status_card') statusCard: PlanStatusCardComponent;
@ViewChild('side_display') sideDisplay: PlanSideDisplayComponent;
constructor(
private layoutSvc: DagreLayoutService, private logSvc: LogService) {}
ngOnInit() {
this.statusCard.dislayOptionEvent.subscribe(
(displayOption: string) => this.invalidateGraph());
}
async loadPlan(plan: BqQueryPlan) {
this.plan = plan;
this.haveDoneDraw = false;
this.statusCard.loadPlan(plan);
this.sideDisplay.stepDetails = [];
this.sideDisplay.stageDetails = '';
}
private invalidateGraph() {
this.haveDoneDraw = false;
this.draw();
}
async draw() {
if (!this.plan) {
this.clearGraph();
return;
}
if (this.haveDoneDraw) {
return;
}
this.graph = this.drawGraph(
this.plan,
(chart: TreeChart, resizeData: object) => {
// console.log('canvas resize', this);
// console.log(resizeData);
},
(chart: TreeChart, node: any, params: any) => {
if (node) {
this.sideDisplay.stageDetails = this.plan.getStageStats(node);
this.sideDisplay.stepDetails = this.plan.getStepDetails(node);
}
});
this.haveDoneDraw = true;
this.resizeToWindow();
}
resizeWindow(event) {
this.resizeToWindow();
}
private resizeToWindow(): void {
if (this.graph && this.graph.network) {
const newWidth = window.innerWidth * 0.74;
const newHeight = window.innerHeight * 0.75;
this.graph.network.setSize(`${newWidth}px`, `${newHeight}`);
this.graph.network.redraw();
}
}
private clearGraph() {
if (this.graph) {
this.graph.network.setData(new vis.DataSet([]), new vis.DataSet([]));
this.graph.network.redraw();
}
}
private drawGraph(
plan: BqQueryPlan, onResizeEvent?: ResizeCallback,
onNodeSelect?: NodeSelectCallback, onNodeDeselect?: NodeDeselectCallback,
onEdgeSelect?: EdgeSelectCallback,
onEdgeDeselect?: EdgeDeselectCallback): TreeChart {
let visnodes = new vis.DataSet([]);
let visedges = new vis.DataSet([]);
if (plan.nodes.length === 0) {
this.logSvc.warn('Current Plan has no nodes.');
return;
} else {
const allnodes = (this.statusCard.stageDisplayOption ===
this.statusCard.SHOWREPARTIION) ?
plan.nodes :
plan.nodesWithoutRepartitions();
const layout = this.layoutSvc.layout(allnodes, plan);
const nodes = allnodes.map(node => {
const label = node.name.length > 22 ?
`${node.name.slice(0, 10)}...${node.name.slice(-10)}` :
node.name;
return {
id: node.id,
label: label,
title: node.name,
widthConstraint: 60,
shape: node.isExternal ? 'database' : 'box',
physics: false,
x: layout.node(node.id).x,
y: layout.node(node.id).y
};
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visnodes = new vis.DataSet(nodes);
visedges = new vis.DataSet(plan.edges.map(edge => {
let nrRecords = edge.from.recordsWritten;
if (nrRecords === undefined) {
nrRecords = edge.to.recordsRead;
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to: edge.to.id,
label: Number(nrRecords).toLocaleString('en') + ' records'
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if (!container) {
console.error(`Unable to find 'visGraph'`);
return;
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// create a network
const network = new vis.Network(container, data, options);
const chart: TreeChart = {options: options, network: network};
const me = this;
if (onResizeEvent) {
network.on('resize', params => {
// console.log('resize....');
onResizeEvent(chart, params);
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network.on('selectNode', params => {
const id = params.nodes[0];
const n = plan.getNode(id);
onNodeSelect(chart, n, params);
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if (onNodeDeselect) {
network.on('deselectNode', params => {
onNodeDeselect(chart, params, plan.plan);
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if (onEdgeSelect) {
network.on('selectEdge', params => {
const edgeId = params.edges[0];
const foundEdge = visedges.get()[edgeId];
const fromNode = plan.getNode(foundEdge.from);
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to: toNode.name,
recordsWritten: fromNode.recordsWritten ? fromNode.recordsWritten :
toNode.recordsRead
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onEdgeSelect(chart, params, detail);
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if (onEdgeDeselect) {
network.on('deselectEdge', params => {
onEdgeDeselect(chart, params);
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return chart;
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private getVisOptions(): vis.Options {
return {
autoResize: false,
width: '100%',
edges: {
arrows: {
from: {enabled: false, scaleFactor: 1, type: 'arrow'},
to: {enabled: true, scaleFactor: 1, type: 'arrow'}
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selectionWidth: 5,
color: {color: '#A0A0FF', highlight: '#8080FF'},
smooth: {enabled: true, type: 'cubicBezier'}
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physics: {
enabled: false,
barnesHut: {avoidOverlap: 1, gravitationalConstant: -200},
hierarchicalRepulsion: {nodeDistance: 150}
}
};
}
}
interface TreeChart {
options: vis.Options;
network: vis.Network;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,710
|
{"url":"http:\/\/math.stackexchange.com\/questions\/91334\/compact-sets-in-mathbbrn","text":"# compact sets in $\\mathbb{R}^{n}$\n\nProblem:\n\nLet $X\\subset \\mathbb{R}^{n}$ be a compact set. Prove that the set $Y=\\left \\{ y\\in \\mathbb{R}^{n}: \\left | x-y \\right |=2000 : x\\in X \\right \\}$ is compact.\n\nFirst, I don't understand how the absolute value of the difference between x and y: $\\left | x-y \\right |$ be a number in $\\mathbb{R}$, akthough $x$ and $y$ are from $\\mathbb{R}^{n}$.\n\nSecond, there many definitions for compact set(closed and bounded, every sequence in the set has a convergent subsequence in the set, every open cover has a finite subcover). I am trying to use the first definition, any help please?\n\n-\nIn this context, $|u|$ stands for the norm (or length) of the vector $u \\in \\mathbb R^n$. Also, $|x-y|$ is equal to $d(x,y)$, the distance between $x$ and $y$. \u2013\u00a0 Srivatsan Dec 14 '11 at 4:01\nWhat you\u2019ve written $|x-y|$ is more commonly written $\\|x-y\\|$, the norm of the vector $x-y$. There are many norms commonly used on $\\mathbb{R}^n$; the most familiar is the Euclidean norm, $\\|(x_1,\\dots,x_n)\\|=\\sqrt{x_1^2+\\cdots+x_n^2}$. That\u2019s the one that I\u2019d assume if another is not explicitly noted. \u2013\u00a0 Brian M. Scott Dec 14 '11 at 4:04\nNote that compactness is not the same as closed and bounded. It is true that compact implies closed and bounded, but the converse is not true in general. However, in $\\mathbb R^n$, the converse holds thanks to the Heine-Borel theorem: every closed and bounded subset of $\\mathbb R^n$ is compact. \u2013\u00a0 Srivatsan Dec 14 '11 at 4:07\nTo prove that the set is closed: since we have finite points of $y$ that satisfies the condition, the our set is a set of points, which implies it is closed. Is that true? How de we prove it is bounded? \u2013\u00a0 M.Krov Dec 14 '11 at 5:07\nAs Robert Israel has mentioned, the notation used to describe your set $Y$ is not standard. I don't know what it means. Are you referring to the set of all $y \\in \\mathbb{R}^n$ such that the distance from $y$ to a fixed point $x$ in $X \\subset \\mathbb{R}^n$ is 2000? If this is true then we are talking about some $n$ sphere of radius 2000. It is clearly bounded, because it is contained in some $n-$cell. At this point you can look at it one way using the Heine - Borel Theorem, \u2013\u00a0 user38268 Dec 14 '11 at 7:04\n\nHere $|x-y|$ means the Euclidean norm $\\sqrt{\\sum_{j=1}^n (x_j - y_j)^2}$, not absolute value. But your notation $\\{y \\in {\\mathbb R}^n\\ :\\ |x - y| = 2000\\ : \\ x \\in X\\}$ is non-standard. Is the second $:$ supposed to mean \"for some\"?\nIn this case, \"closed and bounded\" is the most useful characterization. But please don't call it a definition, because it doesn't work in other spaces. The fact that subsets of ${\\mathbb R}^n$ are compact if and only if they are closed and bounded is a theorem.\nTo prove that the set is closed: since we have finite points of $y$ that satisfies the condition, the our set is a set of points, which implies it is closed. Is that true? How de we prove it is bounded? \u2013\u00a0 M.Krov Dec 14 '11 at 5:07\nNo, it's not a finite set. One way to show it's closed: suppose a sequence $y_j$ of points of $Y$ converges to $y$. Then there are $x_j \\in X$ with $|x_j - y_j| = 2000$. Now what do you know about a sequence in a compact set? \u2013\u00a0 Robert Israel Dec 14 '11 at 5:21\n@Zi2018Alpha Say for example you have a sequence $x_n$ that lives in $X$ which we know is compact. Now the set of all $x_n$ is an infinite set - say we call $E$ - and as a subset of $x_n$ must have a limit point in $X$ because $X$ is compact. \u2013\u00a0 user38268 Dec 14 '11 at 7:07\nYes. So if $y_j$ converges to $y$ and a subsequence of $x_j$ converges to $x$, what do you suppose is $|x - y|$? \u2013\u00a0 Robert Israel Dec 14 '11 at 7:56","date":"2015-01-30 08:21:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.95079106092453, \"perplexity\": 109.37446339093006}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-06\/segments\/1422115926769.79\/warc\/CC-MAIN-20150124161206-00072-ip-10-180-212-252.ec2.internal.warc.gz\"}"}
| null | null |
Mordoré se dit d'un objet présentant des reflets dorés, le plus souvent sur un aspect brun. Ce n'est pas un adjectif de couleur à proprement parler ; la texture, le brillant, les reflets chauds, jaune-orangé, qualifient une surface mordorée.
Le terme apparaît au , attesté d'abord comme « more doré », où more est une variante de maure (Trésor de la langue française).
Brun
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,209
|
\section{Introduction}
Exotic states of matter are expected to exist in the central
regions of compact stars. The baryon density in these systems
is likely to exceed several times the nuclear saturation density,
$\rho_0\simeq 0.15~\mbox{fm}^{-3}$. The exact nature of such
dense matter, however, is yet unknown. It was suggested already
30 years ago \cite{perry} that it might be deconfined quark matter.
Since the temperatures in neutron stars are sufficiently low,
this matter is likely to be in a color-superconducting state.
(For reviews on color superconductivity see Refs.~\cite{reviews}.)
There is little doubt that dense baryonic matter at asymptotically
high quark chemical potential $\mu$ is a color superconductor
\cite{bailin}. In this case, the ground state is the color-flavor
locked (CFL) phase \cite{alford} (for studies of the CFL phase in
QCD at asymptotic density, see also Refs.~\cite{weak-cfl1,weak-cfl2}).
At densities existing in stars, however, this phase may not
necessarily be realized. The main reason for the potential breakdown of the
CFL phase is a relatively large difference between the masses of
the strange quark, $m_s$, and the masses of the up and down quarks,
$m_u\simeq m_d$, which is negligible only at large densities. This
difference, together with the requirements of $\beta$ equilibrium
and electric and color charge neutrality, gives rise to a mismatch
between the Fermi momenta of the quarks that form Cooper pairs
\cite{neutrality}. Therefore, the conventional BCS pairing \cite{bcs},
which is the underlying mechanism of the CFL state, becomes
questionable and the search for the true ground state of quark
matter in compact stars has to include different, unconventional
superconducting states \cite{shovkovy,gCFL,cfl+mesons,alford2}.
Determining the ground state of QCD at moderate densities from
first principles, i.e., within the framework of QCD, is not possible
at present. It is natural therefore to make use of astrophysical
observations to test the presence of various suggested phases.
The goal of this paper is to investigate color superconductors
in which quarks of the same flavor form Cooper pairs
and to study their effect on specific observables.
Single-flavor Cooper pairing is the simplest possibility for neutral
quark matter. Contrary to other unconventional pairing mechanisms, such
as the gapless 2SC phase \cite{shovkovy}, the gapless CFL phase \cite{gCFL},
the CFL phase with additional meson condensates \cite{cfl+mesons},
or the crystalline phases \cite{alford2}, it is allowed for
arbitrarily large mismatches between the Fermi momenta of different
quark flavors. Single-flavor pairing in the color anti-triplet
channel is possible only in the symmetric spin-one channel
\cite{bailin,spin1,schaefer,Schmitt:2002sc,Schmitt:2003xq,andreas}.
This is the consequence of the Pauli principle which requires
the wave function of the Cooper pair to be antisymmetric under
the exchange of the constituent quarks. This is in contrast
to pairing of different flavors where the antisymmetric spin-zero
channel is allowed.
Furnishing triplet representations with respect to both color
and spin groups, the order parameter in a spin-one color
superconductor is given by a complex 3$\times$3 matrix.
This is similar to superfluid $^3$He, where condensation
occurs in spin and angular momentum triplets \cite{vollhardt}.
In both systems, the matrix structure of the order parameter
gives rise to several possible phases. In $^3$He the observed
phases are the so-called {\it A}, {\it B}, and $A_1$ phases.
In this paper, we consider the following four main spin-one
color-superconducting phases of quark matter: the color-spin
locked (CSL), planar, polar, and {\it A} phases, proposed in
Refs.~\cite{schaefer,Schmitt:2002sc,Schmitt:2003xq,andreas,Buballa:2002wy}.
The {\it A} and CSL phases are analogues of the {\it A} and
{\it B} phases in $^3$He, respectively.
In contrast to the nonrelativistic case of $^3$He, spin-one
color-superconductors may involve pairing of quarks
of the same as well as of opposite chiralities. In this
paper, we focus on the so-called ``transverse'' phases,
in which exclusively quarks of opposite chiralities pair.
Theoretical studies at asymptotically large densities
suggest that these phases are preferred
\cite{schaefer,andreas}.
As in the case of $^3$He, the gap functions of most of the
quark phases considered here are anisotropic in momentum
space. Only the CSL phase is isotropic. The gap in the polar
phase vanishes at the south and north poles of the Fermi
sphere, whereas the gap in the planar phase is anisotropic
but nonzero in any direction of the quasiparticle momentum.
The {\it A} phase is special in the sense that it has two gapped
quasiparticle modes with different angular structures, one
of which has two point nodes.
One should expect that the anisotropies and especially the nodes
affect the physical properties of the corresponding quark phases.
The low energy excitations around the nodes give important
contributions to various thermodynamical and transport properties,
e.g., the specific heat, the neutrino emissivity, the viscosity,
the heat and electrical conductivity, etc. In application to compact
stars, the specific heat and the neutrino emissivity determine the
cooling behavior during the first $10^5$ -- $10^6$ years of the
stellar evolution. In this paper, we compute these two quantities
for the four mentioned spin-one color-superconducting phases and deduce the
resulting cooling rates. In
addition, we will discuss the reason why the distribution of neutrino
emission from the {\it A} phase breaks reflection symmetry in position
space.
The paper is organized as follows. The general formalism of
calculating the time derivative of the neutrino distribution
function is given in Sec.~\ref{formalism}. The main result
from Sec.~\ref{formalism} is then used in Sec.~\ref{results}
to compute the neutrino emissivity. The
specific heat is calculated in Sec.~\ref{specificheat}. Both
quantities are used in Sec.~\ref{cooling} to discuss the
cooling behavior of the considered phases. In Sec.~\ref{kicks}
we explain the asymmetry of the neutrino emission in the {\it A}
phase. Finally, we comment briefly on the effect of nonzero
quark masses in Sec.~\ref{quarkmass}.
Our convention for the metric tensor is
$g^{\mu\nu}=\mbox{diag}(1,-1,-1,-1)$.
Our units are $\hbar=c=k_B=1$. Four-vectors are denoted
by capital letters, $K\equiv K^\mu=(k_0,{\bf k})$, while
$k\equiv|{\bf k}|$ and $\hat{\mathbf{k}}\equiv{\bf k}/k$.
We work in the imaginary-time formalism, i.e.,
$T/V \sum_K \equiv T \sum_n \int d^3{\bf k}/(2\pi)^3$,
where $n$ labels the Matsubara frequencies $\omega_n \equiv i k_0$.
The Matsubara frequencies are $\omega_n\equiv 2n \pi T$ for bosons,
and $\omega_n\equiv(2n+1) \pi T$ for fermions.
\section{Time derivative of the neutrino distribution function}
\label{formalism}
In this section, we derive a general expression for the time derivative
of the neutrino distribution function in spin-one color-superconducting
phases.
\subsection{General formalism}
Within the Kadanoff-Baym formalism \cite{KB}, one derives the following
kinetic equation for the neutrino Green function (see for example
Refs.~\cite{VoskSen,friman,kadanoff}):
\begin{equation}
i\partial_{X}^{\lambda}\mbox{Tr}[\gamma_{\lambda}G^{<}_{\nu}(X,P_\nu)]
=-\mbox{Tr}\left[G^{>}_{\nu}(X,P_\nu)\Sigma^{<}_{\nu}(X,P_\nu)
-\Sigma^{>}_{\nu}(X,P_\nu)G^{<}_{\nu}(X,P_\nu)\right] \,\, ,
\label{KB-eq}
\end{equation}
where the trace runs over Dirac space. (Here and in the following,
the index $\nu$ always labels neutrino quantities and should not
be confused with a Lorentz index.) The kinetic equation is obtained
from the general Kadanoff-Baym equation after applying a
gradient expansion, which is valid when the neutrino Green functions
$G^{<,>}_\nu(X,P_\nu)$ and self-energies $\Sigma^{<,>}_\nu (X,P_\nu)$ are slowly
varying functions of the space-time coordinate $X=(t,\mathbf{x})$.
For our purposes, it is sufficient
to consider spatially homogeneous systems which are close to equilibrium.
In this case, the neutrino Green functions assume the following approximate
form \cite{friman}:
\begin{subequations}
\label{Glesslarge}
\begin{eqnarray}
iG^{<}_{\nu}(t,P_\nu) & = & -(\gamma^\lambda P_{\nu,\lambda} +\mu_\nu\gamma_0 )
\frac{\pi}{p_\nu}\left\{ f_{\nu}(t,\mathbf{p}_\nu)\,\delta(p_{\nu 0}+\mu_{\nu}-p_\nu)
-[1-f_{\overline{\nu}}(t,-\mathbf{p}_\nu)]\,\delta(p_{\nu 0}+\mu_\nu+p_\nu)
\right\} \, ,
\label{G-less1} \\
iG^{>}_{\nu}(t,P_\nu) & = & (\gamma^\lambda P_{\nu,\lambda} +\mu_\nu\gamma_0 )
\frac{\pi}{p_\nu}\left\{ [1-f_{\nu}(t,\mathbf{p}_\nu)] \,\delta(p_{\nu 0}+\mu_{\nu}-p_\nu)
-f_{\overline{\nu}}(t,-\mathbf{p}_\nu) \,\delta(p_{\nu 0}+\mu_\nu +p_\nu)
\right\} \, ,
\label{G-larger1}
\end{eqnarray}
\end{subequations}
where $\mu_\nu$ is the neutrino chemical potential. The functions
$f_{\nu,\overline{\nu}}(t,\mathbf{p}_\nu)$ are the neutrino
and antineutrino distribution functions. We are interested in
the Urca processes $u+e^{-}\to d+\nu$ (electron capture)
and $d\to u+e^{-}+\overline{\nu}$ ($\beta$-decay), which
provide the dominant cooling mechanism for quark matter.
These processes are important for neutron stars
with core temperatures of the order of or smaller than several MeV.
In this stage of the stellar evolution, the mean free path of neutrinos becomes larger than the
stellar radius, wherefore in our results we shall set $\mu_\nu = 0$.
The leading order contributions
to the neutrino self-energies which enter the kinetic equation
(\ref{KB-eq}) are given by the diagrams in Fig.~\ref{fig:urca}.
For the sake of simplicity, we do not take strange quarks into account.
Their weak interactions are Cabibbo suppressed, and their number density is
not expected to be very large. (Admittedly, however, the bigger phase
space for the Urca processes involving massive strange quarks may
partially compensate this suppression.) The diagrams in
Fig.~\ref{fig:urca} translate into the following expressions,
\begin{subequations}
\label{eq3}
\begin{eqnarray}
\Sigma_{\nu}^{<}(t,P_\nu) & = &
\frac{G_{F}^{2}}{2}\int\frac{d^{4}P_{e}}{(2\pi)^{4}}
\gamma^{\lambda}(1-\gamma_{5})
(\gamma^\kappa P_{e,\kappa} +\mu_e\gamma_0 )
\gamma^{\sigma}(1-\gamma_{5}) \nonumber\\
&&\times\;\Pi_{\lambda\sigma}^{>}(P_e-P_\nu)\frac{\pi}{p_{e}}f_{e}(t,\mathbf{p}_{e})
\delta(p_{e0}+\mu_{e}-p_{e}),\\
\Sigma_{\nu}^{>}(t,P_\nu) & = &
-\frac{G_{F}^{2}}{2}\int\frac{d^{4}P_{e}}{(2\pi)^{4}}
\gamma^{\lambda}(1-\gamma_{5})
(\gamma^\kappa P_{e,\kappa} +\mu_e\gamma_0 )
\gamma^{\sigma}(1-\gamma_{5}) \nonumber\\
&&\times\;\Pi_{\lambda\sigma}^{<}(P_e-P_\nu)\frac{\pi}{p_{e}}
[1-f_{e}(t,\mathbf{p}_{e})] \delta(p_{e0}+\mu_{e}-p_{e}),
\end{eqnarray}
\end{subequations}
where $G_F$ is the Fermi coupling constant and $\mu_e$ is the electron
chemical potential. In the derivation we used the explicit form of
the electron Green functions $G^{<,>}_{e}(t,P_e)$. They are given by expressions analogous
to Eqs.~(\ref{Glesslarge}). We have neglected the positron contribution, i.e., the analogues of the
second terms in curly brackets on the right-hand sides of Eqs.~(\ref{Glesslarge}).
[Note that the processes involving
positrons, $u\to d+e^{+}+\nu$ and $d+e^{+}\to u+\overline{\nu}$,
are suppressed by a large factor $\exp(2\mu_e/T)$ because
$\mu_e$ is positive and, in the regime under consideration,
much larger than the temperature.]
The functions
$\Pi_{\lambda\sigma}^{<,>}(P_e-P_\nu)$ are the self-energies
of the $W$ bosons. The $W$ exchange is approximated
by its local form since the typical neutrino energies are much
smaller than the $W$ mass. In the case of neutrino processes
in compact stars, the corresponding neutrino
energies do not exceed several dozens MeV.
\begin{figure}
\includegraphics[width=0.9\textwidth]{self.eps}
\caption{\label{fig:urca}
Neutrino self-energies relevant for the
neutrino Urca processes in the close-time-path formalism.
The $+$ and $-$ signs assign the vertices to the
upper and lower branch of the time contour, respectively. }
\end{figure}
Next, we insert the Green functions (\ref{Glesslarge}) and self-energies (\ref{eq3}) into the kinetic
equation (\ref{KB-eq}). In order to obtain the expression
for the neutrino (antineutrino)
distribution function, we integrate on both sides
of the kinetic equation over $p_{\nu 0}$ from $-\mu_\nu$ to $\infty$
(from $-\infty$ to $-\mu_\nu$). The results are
\begin{subequations}
\label{df/dt-nu1}
\begin{eqnarray}
\frac{\partial}{\partial t}f_{\nu}(t,\mathbf{p}_\nu) & = &
-i\frac{G_{F}^{2}}{16}\int\frac{d^{3}\mathbf{p}_{e}}{(2\pi)^{3}p_\nu p_e}
L^{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)
\,\left\{[1-f_{\nu}(t,\mathbf{p}_\nu)]\,f_{e}(t,\mathbf{p}_{e})
\,\Pi_{\lambda\sigma}^{>}(Q) \right.\nonumber\\
&& \left. \hspace{5cm} - \; f_{\nu}(t,\mathbf{p}_\nu)\,
[1-f_{e}(t,\mathbf{p}_{e})]\,\Pi_{\lambda\sigma}^{<}(Q)
\right\} \,\, ,
\label{df/dt-nu}\\
\frac{\partial}{\partial t}f_{\overline{\nu}}(t,\mathbf{p}_\nu) & = &
-i\frac{G_{F}^{2}}{16}\int\frac{d^{3}\mathbf{p}_{e}}{(2\pi)^{3}p_\nu p_e}
L^{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)
\,\left\{[1-f_{\overline{\nu}}(t,\mathbf{p}_\nu)]
\,[1-f_{e}(t,\mathbf{p}_{e})]\,
\Pi_{\lambda\sigma}^{<}(Q^\prime) \right.\nonumber\\
&& \left. \hspace{5cm} - \; f_{\overline{\nu}}(t,\mathbf{p}_\nu)
\,f_{e}(t,\mathbf{p}_{e})\,
\Pi_{\lambda\sigma}^{>}(Q^\prime)\right\}\,\, ,
\label{df/dt-nu-bar}
\end{eqnarray}
\end{subequations}
with the four-momenta $Q\equiv (p_e-p_\nu-\mu_e+\mu_\nu,{\bf p}_e-{\bf p}_\nu)$
and $Q'\equiv (p_e+p_\nu-\mu_e+\mu_\nu,{\bf p}_e+{\bf p}_\nu)$, and the shorthand
notation
\begin{equation} \label{defL}
L^{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)\equiv\mbox{Tr}\left[(\g_0p_e-\bm{\gamma}\cdot{\bf p}_e)\,
\gamma^\sigma (1-\gamma^5)(\g_0p_\nu-\bm{\gamma}\cdot{\bf p}_\nu)\, \gamma^\lambda
(1-\gamma^5)\right].
\end{equation}
In the following, it is convenient to express the results in terms of the
retarded self-energy for $W$ bosons, $\Pi^{\lambda\sigma}_R$.
Therefore, we shall use the following relations:
\begin{subequations}
\begin{eqnarray}
\Pi^{>}(Q) & = & -2i[1+n_{B}(q_{0})]\mbox{Im}\Pi_{R}(Q) \,\, , \\
\Pi^{<}(Q) & = & -2i\,n_{B}(q_{0})\mbox{Im}\Pi_{R}(Q) \,\, ,
\end{eqnarray}
\end{subequations}
where $n_{B}(\omega)\equiv 1/(e^{\omega/T}-1)$ is the
Bose-Einstein distribution function.
Here we consider the cooling of quark matter only
in the absence of neutrino trapping. Then, the
neutrino and antineutrino distribution functions on the right-hand side
of Eq.~(\ref{df/dt-nu1}) are vanishing,
$f_{\nu,\overline{\nu}}(t,\mathbf{p}_\nu)=0$.
The electron distribution function can be approximated by its
equilibrium expression,
\begin{equation}
f_{e}(t,\mathbf{p}_e)\simeq n_F(p_e-\mu_e) \,\, ,
\end{equation}
with $n_F(\omega)\equiv 1/(e^{\omega/T}+1)$. Then, one arrives at
\begin{subequations}
\begin{eqnarray}
\frac{\partial}{\partial t}f_{\nu}(t,\mathbf{p}_\nu) & = &
\frac{G_{F}^{2}}{8}\int\frac{d^{3}\mathbf{p}_{e}}{(2\pi)^{3}p_\nu p_e}
L_{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)\,n_F(p_e-\mu_e) \,
n_{B}(p_\nu + \mu_e - p_e)
\,\mathrm{Im}\Pi_R^{\lambda\sigma}(Q),
\label{df/dt-nu-1}\\
\frac{\partial}{\partial t}f_{\overline{\nu}}(t,\mathbf{p}_\nu) & = &
-\frac{G_{F}^{2}}{8}\int\frac{d^{3}\mathbf{p}_{e}}{(2\pi)^{3}p_\nu p_e}
L_{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)\,
n_F(\mu_e - p_e)\,n_{B}(p_\nu - \mu_e + p_e)
\,\mathrm{Im}\Pi_{R}^{\lambda\sigma}(Q').
\label{df/dt-nu-bar-1}
\end{eqnarray}
\end{subequations}
We shall see in Sec.~\ref{General-result} that the right-hand sides
of Eqs.~(\ref{df/dt-nu-1}) and (\ref{df/dt-nu-bar-1}) are in fact identical.
\subsection{Quark propagators in spin-one color superconductors}
As we shall see below, cf.\ Eq.~(\ref{Pidef}), the expression for the imaginary
part of the polarization tensor of the W-vector boson is given in terms of
the quark propagator $S(K)$. In this paper, we consider the following
four spin-one superconducting phases: the CSL, planar, polar, and {\it A}
phases. Each of these phases is characterized by a $12\times 12$ gap matrix
${\cal M}_{\bf k}$ in color and Dirac space,
\begin{equation}
{\cal M}_{\bf k} = \sum_{i,j=1}^3J_i\Delta_{ij}\g_{\perp,j}(\hat{\mathbf{k}}) \,\, ,
\label{defM}
\end{equation}
where $(J_i)_{jk}=-i\e_{ijk}$ and
$\bm{\gamma}_{\perp}(\hat{\mathbf{k}})\equiv \bm{\gamma} - \hat{\mathbf{k}}\,\bm{\g}\cdot\hat{\mathbf{k}}$
are basis vectors for the color antitriplet and spin triplet representations,
respectively. We focus exclusively on the transverse phases, which have the
highest pressure at asymptotic density and thus are expected to be preferred
over others. For a more general form of the matrix ${\cal M}_{\bf k}$ see
Ref.~\cite{andreas}. The matrix $\Delta$ is a complex $3\times 3$ matrix
and assumes a specific structure for each of the phases.
In Table~\ref{tablephases}, we give $\Delta$ and the resulting matrices
${\cal M}_{\bf k}$.
\begin{table}[t]
\begin{tabular}[t]{|c||c|c||c|c|c|}
\hline
\;\; phase \;\; & $\Delta_{ij}$ & ${\cal M}_{\bf k}$ &
\;\; $\lambda_{{\bf k},1}\;(n_1)$ \;\; &
\;\; $\lambda_{{\bf k},2}\;(n_2)$ \;\; &
\;\; $\lambda_{{\bf k},3}\;(n_3)$ \;\; \\
\hline\hline
CSL & $\delta_{ij}$ & ${\bf J}\cdot\bm{\gamma}_{\perp}(\hat{\mathbf{k}})$ & 2\;(8) & 0\;(4) & -- \\
\hline
planar & $\delta_{i1}\delta_{j1}+\delta_{i2}\delta_{j2}$ & $J_1\g_{\perp,1}(\hat{\mathbf{k}}) + J_2\g_{\perp,2}(\hat{\mathbf{k}})$ &
$1 + \cos^2\theta_{\bf k}\;(8)$ & 0\;(4) & -- \\
\hline
polar & $\delta_{i3}\delta_{j3}$ & $J_3\g_{\perp,3}(\hat{\mathbf{k}})$ & $\sin^2\theta_{\bf k} \;(8)$& 0\;(4) & -- \\
\hline
{\it A} &\;\; $\delta_{i3}(\delta_{j1} + i\,\delta_{j2})$\;\; & \;\; $J_3[\g_{\perp,1}(\hat{\mathbf{k}}) + i\,\g_{\perp,2}(\hat{\mathbf{k}})]$\;\;
& \;\;$(1 + |\cos\theta_{\bf k}|)^2\;(4)$ \;\;&
\;\;$(1 - |\cos\theta_{\bf k}|)^2\;(4)$\;\; & 0 \;(4) \\
\hline
\end{tabular}
\caption{Matrices $\Delta$ and ${\cal M}_{\bf k}$ and eigenvalues $\lambda_{{\bf k},r}$ with
corresponding degeneracies $n_r$
for four spin-one color superconductors. The angle between ${\bf k}$ and the
$z$-axis is denoted by $\theta_{\bf k}$.}
\label{tablephases}
\end{table}
In spin-one color superconductors, the quark propagator is diagonal in flavor space,
$S(K) = {\rm diag}[S_u(K),S_d(K)]$. The Nambu-Gorkov structure of the flavor-diagonal
elements is given by
\begin{equation} \label{prop}
S_f(K) = \left(\begin{array}{cc} G_f^+(K) & \Xi_f^-(K) \\
\Xi_f^+(K) & G_f^-(K) \end{array}\right) \,\, ,
\qquad f=u,d \,\, ,
\end{equation}
where \cite{Schmitt:2002sc,Schmitt:2003xq,andreas}
\begin{equation} \label{prop-diag}
G^\pm_f(K) = \left[G_{0,f}^\mp(K)\right]^{-1}\,\sum_{e,r}
\frac{{\cal P}_{{\bf k},r}^\pm\, \Lambda_{\bf k}^{\mp e}}
{k_0^2 - (\e_{{\bf k},r,f}^e)^2} \,\, .
\end{equation}
For the sake of simplicity, we have not included the quark self-energy correction. For a more
general expression of the propagator, including this correction, see
Refs.~\cite{Brown:2000eh,Wang:2001aq}. The inverse
free propagator for quarks and charge-conjugate quarks in the ultrarelativistic
limit is
\begin{equation} \label{freeprop}
\left[G_{0,f}^\pm(K)\right]^{-1} = \g^\m K_\m\pm\m_f\g_0 = \g_0\sum_e[k_0\pm
(\mu_f -ek)]\,\Lambda_{\bf k}^{\pm e}\,\, .
\end{equation}
The Dirac matrices $\Lambda_{\bf k}^e\equiv (1+e\g_0\bm{\gamma}\cdot\hat{\mathbf{k}})/2$, where $e=\pm$,
are projectors onto positive and negative energy states. The matrices
${\cal P}_{{\bf k},r}^{-}$ and ${\cal P}_{{\bf k},r}^{+}$ in Eq.\ (\ref{prop-diag}) are projectors onto
the eigenspaces of the matrices ${\cal M}_{\bf k}{\cal M}^{\dagger}_{\bf k}$
and $\gamma^0{\cal M}^{\dagger}_{\bf k}{\cal M}_{\bf k}\gamma^0$, respectively.
Both matrices have the same set of eigenvalues $\lambda_{{\bf k},r}$,
\begin{eqnarray}
{\cal M}_{\bf k}{\cal M}^{\dagger}_{\bf k} &\equiv &
\sum_{r} \lambda_{{\bf k},r}{\cal P}_{{\bf k},r}^{-} \,\, , \\
\gamma^0{\cal M}^{\dagger}_{\bf k}{\cal M}_{\bf k}\gamma^0 &\equiv &
\sum_{r} \lambda_{{\bf k},r}{\cal P}_{{\bf k},r}^{+} \,\, .
\end{eqnarray}
As we shall see,
only the projection operators ${\cal P}_{{\bf k},r}^{+}$ (and not
${\cal P}_{{\bf k},r}^{-}$) are needed in the
calculation of the neutrino emission. They are given explicitly in Appendix
\ref{colordirac}. The eigenvalues $\lambda_{{\bf k},r}$
appear in the quasiparticle dispersion relations,
\begin{equation} \label{excite}
\e_{{\bf k},r,f}^e = \sqrt{(ek-\mu_f)^2 + \lambda_{{\bf k},r}|\phi_f|^2} \,\, .
\end{equation}
Here $\phi_f$ are the gap parameters, which are different for $u$ and $d$ quarks
in general. The eigenvalues for the four considered phases as well as their
degeneracies $n_{r}$ are listed in Table \ref{tablephases}. Note that all phases contain an ungapped
mode, $\lambda_{{\bf k},2(3)} = 0$.
The off-diagonal elements on the right-hand side of Eq.~(\ref{prop})
are the so-called anomalous propagators. They are given by
\begin{subequations} \label{anomalous}
\begin{eqnarray} \label{S212SC}
\Xi^+_f(K)&=&-\sum_{e,r} \gamma_0 \, {\cal M}_{\bf k}\,
\gamma_0\, {\cal P}_{{\bf k},r}^+ \Lambda_{\bf k}^{-e}
\, \frac{\phi_f}{k_0^2-
(\e_{{\bf k},r,f}^e)^2} \,\, , \\
\Xi^-_f(K)&=&-\sum_{e,r} {\cal M}_{\bf k}^\dag\,
{\cal P}_{{\bf k},r}^- \Lambda_{\bf k}^e
\, \frac{\phi^*_f}{k_0^2-
(\e_{{\bf k},r,f}^e)^2} \,\, .
\end{eqnarray}
\end{subequations}
The propagators in Eq.~(\ref{prop-diag}) have particle and hole type
poles at $k_0=\pm \e_{{\bf k},r,f}^+$, as well as the corresponding
antiparticle poles at $k_0=\pm \e_{{\bf k},r,f}^-$. In the calculation
of the imaginary part of the retarded self-energy $\Pi^{\lambda\sigma}_R(Q)$,
the antiparticle contributions are suppressed by inverse powers of
the quark chemical potential. [Note, however, that taking
antiparticles into account is important in the calculation
of $\mbox{Re}\Pi^{\lambda\sigma}_R(Q)$.] In our calculation,
therefore, we omit the terms with $e=-$ in the quark propagator
and arrive at the following approximate form
\begin{equation} \label{fullprop}
G^\pm_f(K) \simeq \g_0\,\Lambda_{\bf k}^{\mp}\,
\sum_r{\cal P}_{{\bf k},r}^\pm\, \frac{k_0\mp(\mu_f-k)}
{k_0^2 - \e_{{\bf k},r,f}^2} \,\, ,
\end{equation}
where we denoted $\e_{{\bf k},r,f}\equiv\e_{{\bf k},r,f}^+$.
\subsection{Imaginary part of the W-boson polarization tensor}
In this subsection, we evaluate $\mathrm{Im}\Pi_R^{\lambda\sigma}$.
As can be seen from Fig.~\ref{fig:urca}, the $W$-boson polarization
tensor can be written as
\begin{eqnarray}
\Pi^{\lambda\sigma}(Q) &\equiv & \frac{T}{V}
\sum_K {\rm Tr}[\G_-^\lambda\,S(K)\,\G_+^\sigma\,S(K+Q)] \,\, ,
\label{Pidef}
\end{eqnarray}
where the trace runs over Nambu-Gorkov, color, flavor, and Dirac space.
The 2$\times$2 Nambu-Gorkov structure of the vertices reads
\begin{equation}
\Gamma^\lambda_{\pm} = \left(\begin{array}{@{\extracolsep{2mm}}cc}
\gamma^\lambda(1-\gamma^5)\,\tau_{\pm} & 0 \\
0 & -\gamma^\lambda (1+\gamma^5)\,\tau_{\mp}
\end{array}\right) \, ,
\label{vert}
\end{equation}
where $\tau_{\pm}\equiv(\tau_1\pm i\tau_2)/2$ are matrices in flavor
space, constructed from the Pauli matrices $\tau_1$, $\tau_2$.
After performing the traces over Nambu-Gorkov and flavor space,
\begin{equation}
\label{Pi1}
\Pi^{\lambda\sigma}(Q) =
\frac{T}{V}\sum_K
\left\{{\rm Tr}[\gamma^\lambda(1-\gamma^5)\,G_u^+(K)\,
\gamma^\sigma (1-\gamma^5)\, G^+_d(P)] +{\rm Tr}[\gamma^\lambda(1+\gamma^5)\,G_d^-(K)\,
\gamma^\sigma (1+\gamma^5)\, G^-_u(P)] \right\} \,\, ,
\end{equation}
where $P\equiv K+Q$, and the traces run over color and Dirac space.
As we see, the anomalous propagators $\Xi^\pm_f$ do not contribute to
the self-energy. This is in contrast to spin-zero color superconductors, such as
the (gapless) 2SC and CFL phases, in which the anomalous propagators are
off-diagonal in flavor space. In fact, this difference is related to the conservation
of electric charge in the diagrams in Fig.~\ref{fig:urca}. The anomalous
propagators contain the Cooper pair condensate, which, in the case of a spin-one color
superconductor, is of the form $\langle u u \rangle$ or $\langle d d \rangle$. Therefore,
the appearance of the anomalous propagators in the diagrams of Fig.~\ref{fig:urca}
is forbidden by electric charge conservation.
In a spin-zero color superconductor, however, the condensate is of the form $\langle u d \rangle$.
Hence, electric charge can be extracted from or deposited into the condensate of Cooper
pairs in the ground state and diagrams containing the anomalous propagators contribute to the
$W$-boson polarization tensor \cite{jaikumar}.
After inserting Eq.~(\ref{fullprop}) into Eq.~(\ref{Pi1}), one arrives
at the following expression for the imaginary part,
\begin{equation}
\label{beforematsu}
{\rm Im}\Pi^{\lambda\sigma}_R(Q) =
\frac{T}{V}{\rm Im}\,\sum_K
\sum_{r,s} \left\{ \frac{[k_0-(\mu_u - k)]\,[p_0-(\mu_d-p)]}{(k_0^2-
\e_{{\bf k},r,u}^2)\,(p_0^2-
\e_{{\bf p},s,d}^2)}\,{\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})
+ \frac{[k_0+(\mu_d - k)]\,[p_0+(\mu_u-p)]}{(k_0^2-
\e_{{\bf k},r,d}^2)\,(p_0^2-
\e_{{\bf p},s,u}^2)}\,{\cal T}_{rs,-}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}}) \right\} ,
\end{equation}
where we defined the following traces in color and Dirac space,
\begin{equation}
\label{defT}
{\cal T}_{rs,\pm}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})\equiv {\rm Tr}\left[
\gamma^\lambda(1\mp\gamma^5)\,\g^0\,{\cal P}_{{\bf k},r}^\pm
\,\Lambda_{\bf k}^{\mp}\,\gamma^\sigma (1\mp\gamma^5)\,\g^0\,
{\cal P}_{{\bf p},s}^\pm\,\Lambda_{\bf p}^{\mp}\right]
\,\, .
\end{equation}
In order to perform the
Matsubara sum, we use Eq.\ (\ref{eqA4}) in Appendix~\ref{matsubara}. Then, extracting the imaginary part yields
\begin{eqnarray} \label{Pi2}
{\rm Im}\Pi^{\lambda\sigma}_R(Q)
&=&
-\pi \sum_{r,s}\sum_{e_1,e_2=\pm}\int
\frac{d^3{\bf k}}{(2\pi)^3}
\left[{\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})\,
B_{{\bf k},r,u}^{e_1}\,B_{{\bf p},s,d}^{e_2} \,
\frac{n_F(-e_1\e_{{\bf k},r,u})\,n_F(e_2\e_{{\bf p},s,d})}
{n_B(-e_1\e_{{\bf k},r,u}+e_2\e_{{\bf p},s,d})} \,
\d(q_0-e_1\e_{{\bf k},r,u}+e_2\e_{{\bf p},s,d}) \right.\nonumber\\
&& \left. +\;
{\cal T}_{rs,-}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})\,
B_{{\bf k},r,d}^{e_1}\,B_{{\bf p},s,u}^{e_2} \,\frac{n_F(e_1\e_{{\bf k},r,d})\,
n_F(-e_2\e_{{\bf p},s,u})}{n_B(e_1\e_{{\bf k},r,d}-e_2\e_{{\bf p},s,u})}\,
\d(q_0+e_1\e_{{\bf k},r,d}-e_2\e_{{\bf p},s,u}) \right] \,\, .
\end{eqnarray}
Here, we defined the Bogoliubov coefficients
\begin{equation}
\label{bglbv}
B_{{\bf k},r,f}^{e}\equiv \frac{\e_{{\bf k},r,f}+ e\,(\mu_f-k)}{2\e_{{\bf k},r,f}}
\,\, , \qquad (f=u,d \,\,, \quad e=\pm) \,\, .
\end{equation}
The two terms on the right-hand side of Eq.~(\ref{Pi2}) yield the same
contribution. The physical reason for this is that the second term is just the
charge-conjugate counterpart of the first term. The formal proof
goes as follows. In the second term, one changes the summation indices
$e_1\leftrightarrow e_2$ and $r\leftrightarrow s$. Then, one introduces
the new integration variable ${\bf k} \to -{\bf k}-{\bf q}$
and uses
\begin{equation}
{\cal T}_{sr,-}^{\lambda\sigma}(-\hat{\mathbf{p}},-\hat{\mathbf{k}}) ={\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})
\,\, ,
\end{equation}
which holds for all phases considered in this paper.
After taking into account that $\lambda_{{\bf k},r}=\lambda_{-{\bf k},r}$,
one obtains the first term in Eq.~(\ref{Pi2}). Consequently, in the following
we keep only the first term and double the result,
\begin{eqnarray}
{\rm Im}\Pi^{\lambda\sigma}_R(Q) &=& -2\pi\sum_{r,s}\sum_{e_1,e_2=\pm}\int
\frac{d^3{\bf k}}{(2\pi)^3} {\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})\,
B_{{\bf k},r,u}^{e_1}\,B_{{\bf p},s,d}^{e_2} \nonumber\\
&&\times\,\frac{n_F(-e_1\e_{{\bf k},r,u})\,n_F(e_2\e_{{\bf p},s,d})}
{n_B(-e_1\e_{{\bf k},r,u}+e_2\e_{{\bf p},s,d})} \,
\d(q_0-e_1\e_{{\bf k},r,u}+e_2\e_{{\bf p},s,d}) \,\, .
\label{pi-nu}
\end{eqnarray}
The same result may also be written in another form which is more convenient
for the use in Eq.~(\ref{df/dt-nu-bar-1}), i.e.,
\begin{eqnarray}
{\rm Im}\Pi^{\lambda\sigma}_R(Q') &=& 2\pi\sum_{r,s}\sum_{e_1,e_2=\pm}\int
\frac{d^3{\bf k}}{(2\pi)^3} {\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}}^\prime)\,
B_{{\bf k},r,u}^{e_1}\,B_{{\bf p}^\prime,s,d}^{e_2} \nonumber\\
&&\times \,\frac{n_F(e_1\e_{{\bf k},r,u})\,n_F(-e_2\e_{{\bf p}^\prime,s,d})}
{n_B(e_1\e_{{\bf k},r,u}-e_2\e_{{\bf p}^\prime,s,d})} \,
\d(q'_0-e_1\e_{{\bf k},r,u}+e_2\e_{{\bf p}^\prime,s,d}) \,\,.
\label{pi-anti-nu}
\end{eqnarray}
where we have used Eq.~(\ref{A2}) and defined $P^\prime\equiv K + Q^\prime$.
\subsection{General result for the time derivative of
the neutrino distribution function}
\label{General-result}
We now insert the results for the polarization tensors (\ref{pi-nu}) and
(\ref{pi-anti-nu}) into Eqs.~(\ref{df/dt-nu-1}) and (\ref{df/dt-nu-bar-1}), respectively.
In order to calculate $L_{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)\,{\rm Im}\Pi_R^{\lambda\sigma}(Q)$,
we have to compute the contraction of the tensor $L_{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)$
with the color-Dirac trace ${\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})$.
In all cases considered in this paper, we can write the result as
\begin{equation} \label{contract}
L_{\lambda\sigma}({\bf p}_e,{\bf p}_\nu)\,{\cal T}_{rs,+}^{\lambda\sigma}(\hat{\mathbf{k}},\hat{\mathbf{p}})
= 64\,(p_e-{\bf p}_e\cdot\hat{\mathbf{k}})\,(p_\nu - {\bf p}_\nu\cdot\hat{\mathbf{p}})\,\omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}}) \,\, ,
\end{equation}
where the functions $\omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}})$ depend on the specific phase. They are calculated in
Appendix \ref{colordirac}.
Then, Eqs.~(\ref{df/dt-nu-1}) and (\ref{df/dt-nu-bar-1}) become
\begin{subequations} \label{beforeFermi}
\begin{eqnarray}
\frac{\partial }{\partial t}f_\nu(t,{\bf p}_\nu) &=& -16\,\pi\,G_F^2\,
\int\frac{d^3{\bf p}_e}{(2\pi)^3p_\nu p_e}
\int\frac{d^3{\bf k}}{(2\pi)^3} \,n_F(p_e-\mu_e)\,
\sum_{e_1,e_2=\pm}\sum_{rs} (p_e - {\bf p}_e\cdot\hat{\mathbf{k}})\,(p_\nu - {\bf p}_\nu\cdot\hat{\mathbf{p}})\nonumber\\
&&\times \;\omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}})
\, B_{{\bf k},r,u}^{e_1}\,B_{{\bf p},s,d}^{e_2}\,
n_F(-e_1\e_{{\bf k},r,u})\,n_F(e_2\e_{{\bf p},s,d})\,\d(p_e-\mu_e-p_\nu-e_1\e_{{\bf k},r,u}
+e_2\e_{{\bf p},s,d}) \,\, ,
\label{nu} \\
\frac{\partial }{\partial t}f_{\overline{\nu}}(t,{\bf p}_\nu) &=& -16\,\pi\,G_F^2\,
\int\frac{d^3{\bf p}_e}{(2\pi)^3p_\nu p_e}
\int\frac{d^3{\bf k}}{(2\pi)^3} \,n_F(\mu_e-p_e)\,
\sum_{e_1,e_2=\pm}\sum_{rs} (p_e - {\bf p}_e\cdot\hat{\mathbf{k}})\,(p_\nu - {\bf p}_\nu\cdot\hat{\mathbf{p}}^\prime)\nonumber\\
&&\times \;\omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}}^\prime)
\, B_{{\bf k},r,u}^{e_1}\,B_{{\bf p}^\prime,s,d}^{e_2}\,
n_F(e_1\e_{{\bf k},r,u})\,n_F(-e_2\e_{{\bf p}^\prime,s,d})\,\d(p_e-\mu_e+p_\nu-e_1\e_{{\bf k},r,u}
+e_2\e_{{\bf p}^\prime,s,d}) \,\, .
\label{nu-bar}
\end{eqnarray}
\end{subequations}
At this point it is appropriate to recall the known result that
the neutrino emission from low-temperature weakly interacting
matter of massless quarks is strongly suppressed by the
kinematics of the Urca processes \cite{iwamoto}. In particular,
energy and momentum conservation requires nearly
collinear momenta of the participating electron, up quark
and down quark (after taking into account that $p_{\nu} \sim
T \ll \mu_e,\mu_u,\mu_d$). Strong interaction between
quarks changes the situation dramatically \cite{iwamoto}.
In this case, applying Landau's theory of Fermi liquids,
the quark Fermi velocity is reduced, $v_{F}\simeq 1-\kappa$,
where $\kappa \equiv 2\a_s/(3\pi)$ with the strong coupling
constant $\alpha_s$. (Here we ignore non-Fermi liquid
corrections. For their possible effects on the Urca processes
in ungapped nuclear or quark matter see Refs.~\cite{voskresensky}
and \cite{schwenzer}, respectively.) A rigorous treatment of the
Fermi liquid correction would require a nonzero quark self-energy,
which we have omitted, see Eq.~(\ref{prop-diag}) and remark
below that equation. However, we may introduce the modified
dispersion relations by hand in order to reproduce the result
for normal quark matter as a limit case of our expressions.
To this end, we identify the first three factors (multiplied
by $k\, p$) on the right hand side of Eq.~(\ref{contract})
with the squared scattering amplitude $|M|^2$, derived by
Iwamoto \cite{iwamoto},
\begin{equation}
64 G_F^2 (p_{e}k-\mathbf{p}_{e}\cdot\mathbf{k})
(p_{\nu}p-\mathbf{p}_{\nu}\cdot\mathbf{p}) \to
|M|^2 \equiv 64 G_F^2 (P_{e}\cdot K)(P_{\nu}\cdot P)\,.
\label{eq:vuc}
\end{equation}
After this replacement, we take the Fermi liquid corrections
into account by simply following the same steps as in
Ref.~\cite{iwamoto}. As in Ref.~\cite{iwamoto}, we work to
lowest order in $\alpha_s$. Therefore, the results of this
paper are valid, strictly speaking, only at densities much
higher than in the interior of neutron stars. However, the
role of the strong interaction in the neutrino emission is
merely to open a phase space for the weak processes. In view
of this, the limitation due to uncontrollable strong interaction
may not be so essential for understanding the qualitative
features of the neutrino processes in quark matter at
realistic densities.
We approximate the $\delta$-functions in
Eqs.~(\ref{beforeFermi}) as follows. By making use of the
definitions ${\bf p}\equiv\mathbf{k}+\mathbf{p}_e-\mathbf{p}_\nu$
and ${\bf p}^\prime\equiv\mathbf{k}+\mathbf{p}_e+\mathbf{p}_\nu$,
it is easy to show that the arguments of the $\delta$-functions
vanish only if the angle between the momenta of up and down quarks
$\theta_{ud}$ is equal to a fixed value $\theta_0$, up to
corrections suppressed by powers of $p_\nu/\mu_e$. The value
of the angle $\theta_0$ is given by
$\cos\theta_{0} \equiv 1 - \kappa\mu_e^2/(\mu_u\mu_d)$.
We note that $\theta_0$ is independent of
the neutrino energy $p_\nu$. Then, both $\delta$-functions
in Eqs.~(\ref{nu}) and (\ref{nu-bar}) are replaced by
$\mu _e/(\mu_u\mu_d)\,\delta(\cos\theta_{ud}-\cos\theta_0)$.
Making use of this fact, we rewrite the equations in the
following form,
\begin{subequations} \label{fapprox12}
\begin{eqnarray}
\label{fapprox}
\frac{\partial }{\partial t}f_\nu(t,{\bf p}_\nu)
&\simeq& -\frac{64}{3}\,\alpha_s G_F^2\,
\mu_e\mu_u\mu_d\int\frac{dp\,d\Omega_{\bf p}}{(2\pi)^3}
\int\frac{dk\,d\Omega_{\bf k}}{(2\pi)^3} \,
(1-\cos\theta_{\nu d})\, \d(\cos\theta_{ud}-\cos\theta_0) \nonumber\\
&& \times \sum_{e_1,e_2=\pm}\sum_{rs} \omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}})
\, B_{{\bf k},r,u}^{e_1}\,B_{{\bf p},s,d}^{e_2}\,
n_F(-e_1\e_{{\bf k},r,u})\,n_F(e_2\e_{{\bf p},s,d})\,
n_F(p_\nu+e_1\e_{{\bf k},r,u}-e_2\e_{{\bf p},s,d}) \,\, ,
\\
\frac{\partial }{\partial t}f_{\overline{\nu}}(t,{\bf p}_\nu)
&\simeq& -\frac{64}{3}\,\alpha_s G_F^2\,
\mu_e\mu_u\mu_d\int\frac{dp\,d\Omega_{\bf p}}{(2\pi)^3}
\int\frac{dk\,d\Omega_{\bf k}}{(2\pi)^3} \,
(1-\cos\theta_{\nu d})\,
\d(\cos\theta_{ud}-\cos\theta_0) \nonumber\\
&& \times \sum_{e_1,e_2=\pm}\sum_{rs} \omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}})
\, B_{{\bf k},r,u}^{e_1}\,B_{{\bf p},s,d}^{e_2}\,
n_F(e_1\e_{{\bf k},r,u})\,n_F(-e_2\e_{{\bf p},s,d})\,
n_F(p_\nu-e_1\e_{{\bf k},r,u}+e_2\e_{{\bf p},s,d}) \,\, ,
\label{fapprox1}
\end{eqnarray}
\end{subequations}
where $\theta_{\nu d}$ is the angle between the three-momenta
of the neutrino and the $d$ quark. Here we have changed the
integration variable ${\bf p}_e$ to
${\bf p}\equiv\mathbf{k}+\mathbf{p}_e-\mathbf{p}_\nu$ and
${\bf p}^\prime\equiv\mathbf{k}+\mathbf{p}_e+\mathbf{p}_\nu$,
respectively, and afterwards dropped the prime of
${\bf p}^\prime$ in Eq.~(\ref{fapprox1}).
Instead of dimensionful momenta, it is convenient to introduce the
following three dimensionless variables,
\begin{equation}
x \equiv \frac{p-\mu_d}{T} \,\, , \qquad
y \equiv \frac{k-\mu_u}{T} \,\, , \qquad
v \equiv \frac{p_\nu}{T} \,\, .
\end{equation}
In terms of the variables $x$ and $y$, the range of integration
runs from $-\mu_{u,d}/T$ to $\infty$. Since the main contribution
in the integral comes from $x,y\ll \mu_{u,d}/T$, the results do
not change if the lower boundary is extended to $-\infty$. Then,
we can drop the $x$- and $y$-odd contributions from the Bogoliubov
coefficients,
\begin{subequations}
\begin{eqnarray}
B_{{\bf p},s,d}^{e_2} &=& \frac{1}{2}-
\frac{e_2\, x}{2 \sqrt{x^2+\lambda_{{\bf p},s}\varphi_d^2}}
\,\, , \\
B_{{\bf k},r,u}^{e_1} &=& \frac{1}{2}-
\frac{e_1 \,y}{2 \sqrt{y^2+\lambda_{{\bf k},r}\varphi_u^2}}\, ,
\end{eqnarray}
\end{subequations}
in the integrand, and keep only the even contributions, i.e.,
a constant term $1/2$. After this is taken into account, we
restrict the integration over $x$ and $y$ from $0$ to $\infty$.
Accounting for the integration from $-\infty$ to $0$ produces
extra factors of 2 for each integration which compensate the
factors of $1/2$ from the two Bogoliubov coefficients. At this
point, we notice that the difference between Eqs.~(\ref{fapprox})
and (\ref{fapprox1}) lies only in the signs in front of $e_1$
and $e_2$. Because of the summation over $e_1$ and $e_2$, the
right-hand sides of both Eqs.~(\ref{fapprox}) and (\ref{fapprox1})
are identical.
Hence, the result can be written in the following approximate form,
\begin{eqnarray} \label{varchange}
\frac{\partial}{\partial t} f_\nu(t,{\bf p}_{\nu}) &=&
\frac{\partial}{\partial t} f_{\overline{\nu}}(t,{\bf p}_{\nu}) \nonumber\\
&\simeq& -\frac{64}{3}\,\alpha_s G_F^2\,
\mu_e\mu_u\mu_d\,T^2 \sum_{rs} \int\frac{d\Omega_{\bf p}}{(2\pi)^3}
\int\frac{d\Omega_{\bf k}}{(2\pi)^3} \,
(1-\cos\theta_{\nu d})\,
\d(\cos\theta_{ud}-\cos\theta_0) \, F_{\varphi_u\varphi_d}^{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}},v) \,\, ,
\end{eqnarray}
where
\begin{eqnarray} \label{Frs}
F_{\varphi_u\varphi_d}^{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}},v)&\equiv&
\omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}}) \sum_{e_1,e_2=\pm}\int_0^\infty \int_0^\infty
\,dx\,dy
\left(e^{-e_1\sqrt{y^2+\lambda_{{\bf k},r}\varphi_u^2}}+ 1\right)^{-1}
\left(e^{e_2\sqrt{x^2+\lambda_{{\bf p},s}\varphi_d^2}} + 1\right)^{-1} \nonumber\\
&&\hspace{5cm}\times\; \left(e^{v+e_1\sqrt{y^2+\lambda_{{\bf k},r}\varphi_u^2} -
e_2\sqrt{x^2+\lambda_{{\bf p},s}\varphi_d^2}} + 1\right)^{-1}
\end{eqnarray}
with $\varphi_f\equiv \phi_f/T$.
Three out of four angular integrations in Eq.~(\ref{varchange}) can be calculated
approximately in an analytical form.
Details of this calculation are deferred to Appendix \ref{angint}. We obtain the following
main result of this section,
\begin{equation} \label{resultgen}
\frac{\partial }{\partial t} f_\nu(t,{\bf p}_\nu)=
\frac{\partial}{\partial t} f_{\overline{\nu}}(t,{\bf p}_{\nu})\simeq -\frac{4\alpha_s G_F^2}{3\pi^4}\,
\mu_e\mu_u\mu_d\,T^2 \sum_r \int_{-1}^1 d\xi\,(1-\xi\cos\theta_\nu)\,
F_{\varphi_u\varphi_d}^{rr}(\xi,v) \,\, .
\end{equation}
It holds for isotropic phases as well as for phases in which the order parameter picks a special
direction in momentum space, identified with the $z$ direction. We denote the angle between the
neutrino momentum and the $z$-axis by $\theta_\nu$. The function $F_{\varphi_u\varphi_d}^{rr}(\xi,v)$ is
obtained from the function $F_{\varphi_u\varphi_d}^{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}},v)$ in the collinear limit ($\hat{\mathbf{k}}=\hat{\mathbf{p}}$). Its
explicit form reads
\begin{eqnarray}\label{Frr}
F_{\varphi_u\varphi_d}^{rr}(\xi,v) &=&\omega_{rr}(\xi)
\sum_{e_1,e_2=\pm}\int_0^\infty \int_0^\infty
\,dx\,dy
\left(e^{-e_1\sqrt{y^2+\lambda_{\xi,r}\varphi_u^2}}+ 1\right)^{-1}
\left(e^{e_2\sqrt{x^2+\lambda_{\xi,r}\varphi_d^2}} + 1\right)^{-1} \nonumber\\
&&\hspace{5cm}\times\; \left(e^{v+e_1\sqrt{y^2+\lambda_{\xi,r}\varphi_u^2} -
e_2\sqrt{x^2+\lambda_{\xi,r}\varphi_d^2}} + 1\right)^{-1} \, .
\end{eqnarray}
Here $\xi\equiv \cos\theta_u=\cos\theta_d$ with $\theta_u$ being the angle between the
$z$-axis and the $u$ quark momentum ${\bf k}$, and $\theta_d$ being the angle between
the $z$-axis and the $d$ quark momentum ${\bf p}$. The functions $\omega_{rr}(\xi)$ and
$\lambda_{\xi,r}$ are given in Table~\ref{tableomega}. One should note that all
components of $\omega_{rs}(\hat{\mathbf{k}},\hat{\mathbf{p}})$ with $r\neq s$ disappear in the collinear limit.
This can be seen directly from their explicit expressions given in
Appendix~\ref{colordirac}. Because of this property, every excitation branch $r$
yields a separate contribution to the result (\ref{resultgen}).
\begin{table}[t]
\begin{tabular}[t]{|c||c|c||c|c||c|c|}
\hline
\;\; phase \;\; & $\omega_{11}(\xi)$ &
$\lambda_{\xi,1}$ &
$\omega_{22}(\xi)$ &
$\lambda_{\xi,2}$&
\;\;$\omega_{33}(\xi)$ \;\; &
$\lambda_{\xi,3}$
\\ \hline\hline
CSL & 2 & 2 & 1 & 0 & -- & -- \\ \hline
planar & 2 & $1+\xi^2$ &1 & $0$ & -- & -- \\ \hline
polar & 2 & $1-\xi^2$ & 1 & $0$ & -- & -- \\ \hline
{\it A} & \;\;$1+{\rm sgn}(\xi)$\;\; & \;\;$(1+|\xi|)^2$ \;\; &
\;\;$1-{\rm sgn}(\xi)$ \;\;& \;\;$(1-|\xi|)^2$ \;\;&
1 & $0$ \\ \hline
\end{tabular}
\caption{Functions $\omega_{rr}(\xi)$ and $\lambda_{\xi,r}$
for four spin-one color superconductors.}
\label{tableomega}
\end{table}
\section{Neutrino emissivity}
\label{results}
In this section, we calculate the neutrino emissivity,
\begin{equation} \label{definition}
\e_\nu \equiv -\frac{\partial}{\partial t}\int\frac{d^3{\bf p}_\nu}{(2\pi)^3}\,p_\nu\,
[f_\nu (t, {\bf p}_\nu) + f_{\overline{\nu}} (t, {\bf p}_\nu)]
= -2\frac{\partial}{\partial t}\int\frac{d^3{\bf p}_\nu}{(2\pi)^3}\,p_\nu\,
f_\nu (t, {\bf p}_\nu) \, .
\end{equation}
This is the total energy loss per unit time and unit volume carried away from
quark matter by neutrinos and antineutrinos. Inserting Eq.~(\ref{resultgen}) into
Eq.~(\ref{definition}) and making use of the integral
\begin{equation}\label{normalint}
\sum_{e_1,e_2=\pm}\int_0^\infty dv\,v^3
\int_0^\infty dx \int_0^\infty dy
\left(e^{-e_1y}+ 1\right)^{-1}
\left(e^{e_2x} + 1\right)^{-1}
\left(e^{v+e_1y - e_2x} + 1\right)^{-1} = \frac{457}{5040}\,\pi^6 \,\, ,
\end{equation}
we obtain
\begin{equation} \label{emissivity}
\e_\nu = \frac{457}{630}\alpha_sG_F^2 T^6\m_e\m_u\m_d\,\left[
\frac{1}{3} + \frac{2}{3}\,G(\varphi_u,\varphi_d)\right] \,\, ,
\end{equation}
where
\begin{subequations}
\label{G-definition}
\begin{eqnarray}
\label{G-definition1}
G(\varphi_u,\varphi_d) &\equiv& \frac{1260}{457\pi^6}\,
\int_0^\infty dv\, v^3 \,\int_{-1}^1 d\xi
\,F_{\varphi_u\varphi_d}^{11}(\xi,v)\, \qquad (\mbox{CSL, planar, polar})\, ,\\
G(\varphi_u,\varphi_d) &\equiv& \frac{1260}{457\pi^6}\,
\int_0^\infty dv\, v^3 \,\int_{-1}^1 d\xi
\,\left[F_{\varphi_u\varphi_d}^{11}(\xi,v)+F_{\varphi_u\varphi_d}^{22}(\xi,v)\right]
\quad (A).
\label{G-definition2}
\end{eqnarray}
\end{subequations}
In all phases we consider, the emissivity $\e_\nu$ consists of two contributions.
The first contribution is given by the term 1/3 in the square brackets on the right-hand
side of Eq.~(\ref{emissivity}). It originates from the ungapped modes: $r=2$ in the CSL,
planar, and polar phases, and $r=3$ in the {\it A} phase. The second contribution
is given by the term proportional to $G(\varphi_u,\varphi_d)$.
It originates from the gapped modes.
The function $G(\varphi_u,\varphi_d)$ has to be evaluated numerically for each phase separately.
For the sake of simplicity, we set $\varphi_u=\varphi_d\equiv\varphi$ in the following. The results
for $G(\varphi,\varphi)$ are plotted in the left panel of Fig.~\ref{figG}. The right panel of
Fig.~\ref{figG} shows the function $G(T)$, obtained from $G(\varphi,\varphi)$ by making
use of the following model temperature dependence of the gap parameter,
\begin{equation}
\phi(T)=\phi_{0}\,\sqrt{1-\left(\frac{T}{T_c}\right)^2},
\label{Phi_of_T}
\end{equation}
with $\phi_{0}$ being the value of the gap parameter at $T=0$, and $T_c$
being the value of the critical temperature.
\begin{figure} [t]
\begin{center}
\hbox{\includegraphics[width=0.49\textwidth]{G-integral-all.eps}
\includegraphics[width=0.49\textwidth]{G-vs-T-all.eps}}
\vspace{0.5cm}
\caption{Left panel: the suppression functions $G(\varphi,\varphi)$
of the neutrino emission contributions due to gapped modes in the CSL,
planar, polar and {\it A} phases. Right panel: the temperature dependence
of the suppression functions using Eq.~(\ref{Phi_of_T}).}
\label{figG}
\end{center}
\end{figure}
As a consistency check, we first read off from the figure that the general result in Eqs.~(\ref{emissivity})
and (\ref{G-definition}) reproduces the well-known expression for the neutrino emissivity
in the normal phase. This is obtained by taking the limit $\varphi\to 0$. Of course, the
result in this limit is the same for all considered phases. Since $G(0,0)=1$, see left panel
of the figure, we recover Iwamoto's result \cite{iwamoto}.
In the spin-one phases, the function $G(\varphi,\varphi)$
describes the suppression of the emissivity due to the presence of the gap in the
quasiparticle spectrum.
In order to discuss this suppression for the different phases, we derive analytic results for
the asymptotic behavior of $G(\varphi,\varphi)$ for $\varphi \to \infty$. Physically, this
corresponds to the low temperature behavior. The details of the calculation are presented
in Appendix~\ref{app:largephi}. The results are
\begin{equation}
G(\varphi,\varphi) \sim
\left\{\begin{array}{lll}
\varphi \,e^{-\sqrt{2}\varphi} &
\quad & \mbox{(CSL)}\, , \\ \\
\sqrt{\varphi}\, e^{-\varphi} &
\quad & \mbox{(planar)}\, , \\ \\
\varphi^{-2} &
\quad & \mbox{(polar)}\, ,\\ \\
\varphi^{-1} &
\quad & \mbox{({\it A})}\, .
\end{array}
\right.\label{eq:largephi}
\end{equation}
The strongest suppression happens in the CSL phase, in which the gap is isotropic.
At large values of $\varphi$, the emissivity is exponentially suppressed, which is universal and
qualitatively the same as, for example, in a spin-zero color superconductor \cite{jaikumar}
or in superfluid neutron and/or proton matter \cite{Yak,voskresensky2}.
{From} a physical viewpoint, this reflects the
fact that, in a superconductor, the neutrino emission is proportional to the density
of thermally excited quasiparticles. It is worth emphasizing, however, that the
function $G(\varphi,\varphi)$ cannot be approximated well by the exponential
function at small $\varphi$. For $\varphi\alt 1$,
the actual suppression is much weaker. This is also obvious from
the right panel of Fig.~\ref{figG}. This panel shows that, in all phases,
the function $G(T/T_c)$ behaves almost linearly all the way down to
$T/T_c\simeq 0.4$. In the CSL and the planar phases, the exponential suppression
starts to show up only below this point.
The contributions from the gapped modes in the other spin-1 phases differ considerably
from the CSL result. All of them have some degree of anisotropy in the gap function.
The second strongest suppression is seen in the planar phase. In this case, while the
gap function is anisotropic, it has no zeros. The dominant contribution comes from a
stripe around the equator of the Fermi sphere, where the gap function, which is proportional
to $\sqrt{1+\cos^2\theta_{f}}$, takes its minimum.
In the polar phase, the gap function has point nodes at the north and south poles
of the Fermi sphere, i.e., it costs no energy to excite quasiparticles around these
points. Therefore, these quasiparticles give the dominant contribution to the
emissivity. For large values of the dimensionless variable $\varphi$, in particular,
one has a power-law instead of the exponential suppression.
The gap function in the {\it A} phase has also nodes at the north and south poles
of the Fermi sphere. However, there is a difference compared to the polar phase in
the behavior of the dispersion relations at small angles $\theta_{f}$.
While it is linear in $\theta_{f}$ in the polar phase, it is
quadratic in $\theta_{f}$ in the {\it A} phase. This difference gives rise to a
different suppression of the emissivity, see Fig.~\ref{figG} and Eq.\ (\ref{eq:largephi}).
The results of this section will be used in Sec.~\ref{cooling} in order to discuss the
effect of spin-1 superconductivity on the cooling rates of compact stars. Besides the
neutrino emissivity, this requires the calculation of the specific heat.
\section{Specific heat}
\label{specificheat}
In this section, we calculate the specific heat of spin-one color superconductors.
This result shall be used in the discussion of the cooling rate in the next section.
We may start from the entropy density (see for example Ref.~\cite{vollhardt})
\begin{equation}
S = -\sum_{f=u,d}\sum_{r}\frac{n_r}{2}
\int\frac{d^3{\bf k}}{(2\pi)^3}\,\Big\{n_F(\e_{{\bf k},r,f})\,\ln n_F(\e_{{\bf k},r,f})
+ [1-n_F(\e_{{\bf k},r,f})]\,\ln[1-n_F(\e_{{\bf k},r,f})]\Big\} \,\, ,
\end{equation}
where $n_r$ is the degeneracy of the quasiparticle branch $r$ as given in Table \ref{tablephases}.
In each phase, $\sum_r n_r/2 = 6$, accounting for 2 spin and 3 color degrees of freedom.
The specific heat is then obtained as
\begin{eqnarray} \label{heatdefine}
c_V &=& T\,\frac{\partial S}{\partial T} =
\sum_{f=u,d}\sum_{r}\frac{n_r}{2} \int\frac{d^3{\bf k}}{(2\pi)^3}
\e_{{\bf k},r,f} \frac{\partial}{\partial T}n_F(\e_{{\bf k},r,f})
\end{eqnarray}
Making use of the model temperature dependence for the gap parameter in
Eq.~(\ref{Phi_of_T}), the result for all phases we consider can be written as
\begin{equation}
c_V = T\sum_{f=u,d} \mu_f^2 \, \left[\frac 13+\frac 23 K (\varphi_f )\right] \,\, .
\label{s-heat}
\end{equation}
The structure of this result is analogous to that of emissivity in
Eq.~(\ref{emissivity}), i.e., the first and the second terms in
the square brackets on the right-hand side come from ungapped and gapped modes, respectively.
The explicit form of the function $K(\varphi)$ reads
\begin{subequations}
\begin{eqnarray}
K(\varphi) &=& \frac{3}{\pi^2}
\int_0^{\infty} dx \int_{-1}^{1} d\xi
\frac{e^{\sqrt{x^2+\lambda_{\xi,1}\varphi^2}}}
{\left(e^{\sqrt{x^2+\lambda_{\xi,1}\varphi^2}}+1\right)^2}
\left[x^2+\lambda_{\xi,1}\left(\varphi^2+\frac{\phi_0^2}{T_c^2}\right)\right]\,
\qquad \mbox{(CSL, planar, polar)} ,\\
K(\varphi) &=& \frac{3}{2\pi^2} \sum_{r=1}^{2}
\int_0^{\infty} dx \int_{-1}^{1} d\xi
\frac{e^{\sqrt{x^2+\lambda_{\xi,r}\varphi^2}}}
{\left(e^{\sqrt{x^2+\lambda_{\xi,r}\varphi^2}}+1\right)^2}
\left[x^2+\lambda_{\xi,r}\left(\varphi^2+\frac{\phi_0^2}{T_c^2}\right)\right]\,
\qquad (A) \,\, .
\end{eqnarray}
\label{def-Ks}
\end{subequations}
The numerical results for the function $K(\varphi)$ for all considered
cases are plotted in the left panel of Fig.~\ref{figK}. The
value of this function in the limit $\varphi\to 0$ does not reproduce
the result for the normal phase $K_{n}=1$, we rather observe
$\lim_{\varphi\to 0}K(\varphi)>1$. This is due to the jump of the
specific heat $\Delta c_V$ at the point of the second order phase
transition to superconducting matter. In the model used, the magnitude
of the jump is obtained by calculating the term proportional to
$\phi_0^2/T_c^2$ in Eqs.~(\ref{def-Ks}). The result can be written
as
\begin{subequations}
\begin{eqnarray}
\Delta c_V &=& (\mu_u^2 + \mu_d^2) \,
\frac{2\,\overline{\phi}_1^2}{T_c\,\pi^2}
\qquad \mbox{(CSL, planar, polar)} \,\, , \\
\Delta c_V &=& (\mu_u^2 + \mu_d^2) \,
\frac{\overline{\phi}_1^2+\overline{\phi}_2^2}{T_c\,\pi^2}
\qquad (A) \,\, ,
\end{eqnarray}
\end{subequations}
where $\overline{\phi}_r\equiv(\int d\Omega_{\bf k}/(4\pi)\,
\lambda_{{\bf k},r})^{1/2}\,\phi_0$ is the quadratic mean of
the $r$-th gap function. Using the results from Ref.~\cite{andreas},
cf. Eq.~(118) therein, we conclude that the jump of the specific
heat is proportional to the condensation energy (at $T=0$).
Hence the order of the values $K(0)$ in Fig.~\ref{figK}
reflects the order of the condensation energies.
\begin{figure}[t]
\hbox{\includegraphics[width=0.45\textwidth]{specific_heat.eps}
\includegraphics[width=0.45\textwidth]{specific_heat-vs-T.eps}}
\caption{\label{figK}
The functions $K(\varphi)$ (left panel)
and $K(T/T_c)$ (right panel) for four spin-one color superconductors.}
\end{figure}
As for the emissivities, we derive
analytical approximate expressions for the specific heat at asymptotically
large $\varphi$, corresponding to asymptotically small temperatures. For the details of the
calculation see Appendix~\ref{cV}. We find
\begin{equation} \label{Kasym}
K(\varphi) \sim \left\{\begin{array}{lll}
\varphi^{5/2}\, e^{-\sqrt{2}\varphi} &
\quad & \mbox{(CSL)} \, ,\\ \\
\varphi^{2}\, e^{-\varphi} &
\quad & \mbox{(planar)} \, ,\\ \\
\varphi^{-2} &
\quad & \mbox{(polar)} \, ,\\ \\
\varphi^{-1} &
\quad & \mbox{({\it A})} \, .
\end{array}
\right.
\end{equation}
This behavior of the specific heat leads to the fact that the
curves in Fig.~\ref{figK} reverse their order at large $\varphi$
compared to $\varphi=0$, e.g., the specific heat in the CSL phase,
which has the largest jump at the critical temperature, exhibits
the largest suppression for very small temperatures.
Note that the degree of the suppression due to gapped modes
in the specific heat and in the emissivity are similar at
large $\varphi$, cf.\ Eqs.\ (\ref{eq:largephi}) and (\ref{Kasym}).
\section{Cooling rates}
\label{cooling}
In this section, we shall use the results for the emissivity in Eq.~(\ref{emissivity})
the specific heat in Eq.~(\ref{s-heat}) in order to study cooling of bulk matter in
spin-one color-superconducting phases. When the cooling is only due to the neutrino
emissivity, one has the following relation,
\begin{equation}
\epsilon_{\nu}(T) = -c_V(T) \frac{dT}{dt}\, .
\label{e=dT/dt}
\end{equation}
In order to derive the change of temperature in time, one has to integrate the above
equation,
\begin{equation}
t-t_0 = -\int_{T_0}^{T}dT^{\prime}\frac{c_V(T^\prime)}{\epsilon_{\nu}(T^\prime)} \, ,
\label{t-t0}
\end{equation}
where $T_0$ is the temperature at time $t_0$. By inserting the expressions
from Eqs.~(\ref{emissivity}) and (\ref{s-heat}) into Eq.~(\ref{t-t0}) and using
$\varphi_u=\varphi_d\equiv\varphi$, we derive
\begin{equation}
t-t_0 = -\frac{630}{457} \frac{\mu_u^2+\mu_d^2}{\alpha_s G_F^2 \mu_e \mu_u \mu_d}
\int_{T_0}^{T}\frac{dT^{\prime}}{(T^{\prime})^5}
\frac{1+2K(T^{\prime})}{1+2G(T^{\prime})}\, ,
\label{timetime}
\end{equation}
where the temperature-dependent functions $K(T)$ and $G(T)$ are obtained
from the functions $K(\varphi)$ and $G(\varphi,\varphi)$ with the help of
Eq.~(\ref{Phi_of_T}).
By making use of Eq.~(\ref{timetime}), let us estimate the cooling behavior
of a compact star whose core is made out of spin-one color-superconducting quark matter.
We start from the moment when the stellar core, to a good approximation,
becomes isothermal. At this point, the stellar age is
of the order of $t_0=10^2$~yr
and the temperature is about $T_0=100$~keV. The estimates in the literature
\cite{ren,Schmitt:2002sc} suggest that the value of the critical temperature
in spin-one color superconductors is of the order of $T_c=50$~keV. This is
the value that we use in the numerical analysis.
Moreover, we choose $\m _u=400$ MeV, $\m _d=500$ MeV, $\m _e=100$ MeV, $\a _s=1$.
The Fermi
weak coupling constant is given by $G_F=1.16637 \times 10^{-11}$ MeV$^{-2}$.
The numerical results show that the cooling behavior is dominated by the ungapped
modes. Consequently, to a very good approximation, the time dependence of the
temperature can be computed by neglecting the functions $K(T^\prime)$ and
$G(T^\prime)$ in Eq.~(\ref{timetime}). In this case, an analytical expression
can be easily derived,
\begin{equation}
T(t) = \frac{T_0\,\tau^{1/4}}{(t-t_0+\tau)^{1/4}} \,\, ,
\end{equation}
where
\begin{equation}
\tau \equiv \frac{315}{914}
\frac{\mu_u^2 + \mu_d^2}{\alpha_s\,G_F^2\,\mu_e\,\mu_u\,\mu_d}\, \frac{1}{T_0^4}
\,\, .
\end{equation}
With the above parameters, this constant is of the order of several minutes,
$\tau \simeq 10^{-5}$yr.
It may be interesting, although unphysical, to compare the
cooling behavior of the gapped modes of the different
spin-one phases. To this end, we drop the 1 in the numerator
and denominator of the integrand in Eq.~(\ref{timetime}).
The results are shown in Fig.~\ref{ctime}. Note that both
the initial temperature $T_0$ and the critical temperature
$T_c$ are beyond the scale of the figure. The reason is that,
even for the gapped modes, the cooling time scale for temperatures
down to approximately $10$~keV is set by the above constant
$\tau$. Therefore, all phases cool down very fast, and the
transition to the superconducting phase at $T=50$~keV is hidden in the
almost vertical shape of the curve. Only at temperatures
several times smaller than $T_c$, i.e., of the order of
$10$~keV, substantial differences between the phases appear. In this range, the fully gapped
phases cool down considerably slower than
the phases with nodes on the Fermi sphere, which, in turn,
cool slower than the normal phase. It seems to agree with physical intuition that this order
reflects the order of the suppression at low temperatures for the neutrino emissivity, i.e.,
the slowest cooling (isotropic gap) happens for the phase where
$\epsilon_\nu$ is suppressed strongest while the fastest cooling
(no gap) happens for the smallest suppression. Note, however, that the
cooling depends on the ratio of the suppressions of $\epsilon_\nu$ and $c_V$. Therefore, this
order is a nontrivial consequence of the exact forms of the functions
$G(\varphi,\varphi)$ and $K(\varphi)$. For large values of $\varphi$, we may use
Eqs.\ (\ref{eq:largephi}) and (\ref{Kasym}) to estimate the ratio $K(\varphi)/G(\varphi,\varphi)$.
For both completely gapped phases we find $K(\varphi)/G(\varphi,\varphi) \sim \varphi^{3/2}$ while
both phases with point nodes yield ratios independent of $\varphi$. Consequently, for late
times, $T\sim t^{-2/11}$ in the CSL and planar phases while $T\sim t^{-1/4}$ in the
polar, {\it A} and normal phases.
\begin{figure}[t]
\includegraphics[width=0.6\textwidth]{cooling-all.eps}
\caption{ \label{ctime}
Temperature as a function of time for normal quark matter and four spin-one ``toy phases'' (dropping
the ungapped modes). The curves represent the CSL phase (solid), planar phase (long-dashed),
polar phase (short-dashed), {\it A} phase (dotted), and normal quark matter (dashed-dotted).}
\end{figure}
\section{Spatial asymmetry in the neutrino emission from the {\it A} phase}
\label{kicks}
In this section, we address a special aspect
of the angular distribution of the neutrino emission. To this end, we consider the
net momentum carried away by neutrinos and antineutrinos from the quark system per unit volume and time,
\begin{eqnarray} \label{H-definition}
\frac{d{\bf P}^{(net)}}{dV \, dt}
\equiv -\frac{\partial}{\partial t}\int\frac{d^3{\bf p}_\nu}{(2\pi)^3}\,{\bf p}_\nu\,
[f_\nu (t, {\bf p}_\nu) +f_{\overline{\nu}} (t, {\bf p}_\nu)]
= -2\frac{\partial}{\partial t}\int\frac{d^3{\bf p}_\nu}{(2\pi)^3}\,{\bf p}_\nu\,
f_\nu (t, {\bf p}_\nu) \,.
\end{eqnarray}
Analogously to the case of the total emissivity, see Eq.~(\ref{emissivity}), we arrive
at the following general result,
\begin{equation} \label{P-emissivity}
\frac{d{\bf P}^{(net)}}{dV \, dt}
= \frac{457}{945}\alpha_sG_F^2 T^6\m_e\m_u\m_d\,H(\varphi_u,\varphi_d)\, \hat{\bf z} \,\, ,
\end{equation}
where $\hat{\bf z}$ is the unit vector in $z$ direction, and
\begin{subequations}
\label{H-definition0}
\begin{eqnarray}
\label{H-definition1}
H(\varphi_u,\varphi_d) &\equiv& -\frac{420}{457\pi^6}\,
\int_0^\infty dv\, v^3 \,\int_{-1}^1 d\xi \xi
\,F_{\varphi_u\varphi_d}^{11}(\xi,v)=0\, \qquad (\mbox{CSL, planar, polar})\, ,\\
H(\varphi_u,\varphi_d) &\equiv& -\frac{420}{457\pi^6}\,
\int_0^\infty dv\, v^3 \,\int_{-1}^1 d\xi \xi
\,\left[F_{\varphi_u\varphi_d}^{11}(\xi,v)+F_{\varphi_u\varphi_d}^{22}(\xi,v)\right]
\quad (A).
\label{H-definition2}
\end{eqnarray}
\end{subequations}
In the CSL, planar, and polar phases, the function $H(\varphi_u,\varphi_d)$
is identically zero. This is because $F_{\varphi_u\varphi_d}^{11}(\xi,v)$ is
an even function of $\xi$ in these three phases, and therefore the integration
over $\xi$ in Eq.~(\ref{H-definition1}) is vanishing. This means that the
net momentum of emitted neutrinos as well as the related net recoil momentum
of bulk quark matter in the CSL, planar, and polar phases are zero.
The result is non-vanishing, however, in the {\it A} phase. The
corresponding function $H(\varphi,\varphi)$ is plotted in the left panel
of Fig.~\ref{figH}. From the figure, we see that $H(0,0)=0$.
Of course, this is just a consistency check that, in the limit
$\varphi\to 0$, we reproduce the vanishing result in the fully
isotropic normal phase of quark matter. From the numerical data,
we find that the maximum value of the function $H(\varphi,\varphi)$
is approximately equal to $0.064$, which corresponds to the value
of its argument $\varphi\simeq 2.9$. At large $\varphi$, the
asymptotic behavior of $H(\varphi,\varphi)$ is power suppressed
as $1/\varphi$. For completeness, we also show the
temperature-dependent function $H(T)$ in the right panel of
Fig.~\ref{figH}. This is obtained from $H(\varphi,\varphi)$ by
making use of the model temperature dependence for the gap in
Eq.~(\ref{Phi_of_T}).
\begin{figure}[t]
\begin{center}
\hbox{\includegraphics[width=0.49\textwidth]{H-integral_mc.eps}
\includegraphics[width=0.49\textwidth]{H-vs-T.eps}}
\caption{Numerical results for the functions $H(\varphi,\varphi)$ and $H(T/T_c)$
which determine the net momentum carried away from the spin-1 color superconducting
{\it A} phase by neutrinos.}
\label{figH}
\end{center}
\end{figure}
It may look surprising that the net momentum from the {\it A} phase is nonzero,
indicating an asymmetry in the neutrino emission with respect to the reflection
of the $z$-axis. The gap functions do not exhibit this asymmetry.
The origin of this remarkable result can be made transparent by rewriting the
expression (\ref{H-definition2}) for the {\it A} phase in the following way,
\begin{equation}
H\left(\varphi_u,\varphi_d\right)
= -\frac{840}{457\pi^6} \int_0^\infty d v v^3 \int_{-1}^1 d \xi \, \xi \,
F^{\rm eff}_{\varphi_u \varphi_d}(\xi,v)\, ,
\label{H-function}
\end{equation}
where
\begin{eqnarray}
F^{\rm eff}_{\varphi_u \varphi_d}(\xi,v) &\equiv&
\sum_{e_1,e_2=\pm}
\int_0^\infty \int_0^\infty
\,dx\,dy
\left(e^{-e_1\sqrt{y^2+(1+\xi)^2\varphi_u^2}}+ 1\right)^{-1}
\left(e^{e_2\sqrt{x^2+(1+\xi)^2\varphi_d^2}} + 1\right)^{-1} \nonumber\\
&&\times\; \left(e^{v+e_1\sqrt{y^2+(1+\xi)^2\varphi_u^2} -
e_2\sqrt{x^2+(1+\xi)^2\varphi_d^2}} + 1\right)^{-1} \, .
\label{F-eff}
\end{eqnarray}
In the derivation, we used the explicit forms of $\omega_{rr}(\xi)$ and $\lambda_{\xi,r}$
from Table~\ref{tableomega}. Now, the result looks as if only one single quasiparticle
mode contributes to the net neutrino momentum. The corresponding ``effective'' gap function
has the angular dependence $\sim (1 + \xi)$ which clearly discriminates between $+z$ and $-z$
directions.
In order to understand the physical reason for the appearance of the effective quasiparticle
mode, it is useful to analyze the physical properties of the gapped modes of the {\it A}
phase, $r=1,2$. The color-spin structure of these modes is encoded in the projection
operators ${\cal P}^{+}_{{\bf k},r}$. Their explicit form is given in Eq.~(\ref{proj-P_A}).
It is instructive to write the first two projectors in the form
\begin{subequations}
\label{P12+}
\begin{eqnarray}
\label{P1+}
{\cal P}^{+}_{{\bf k},1} &=& \frac{1}{2}J_{3}^2[1-{\rm sgn}(\hat{k}_3)]\,H^+(\hat{\mathbf{k}}) +
\frac{1}{2}J_{3}^2[1+{\rm sgn}(\hat{k}_3)]\,H^-(\hat{\mathbf{k}}) \,\, ,\\
{\cal P}^{+}_{{\bf k},2} &=& \frac{1}{2}J_{3}^2[1+{\rm sgn}(\hat{k}_3)]\,H^+(\hat{\mathbf{k}}) +
\frac{1}{2}J_{3}^2[1-{\rm sgn}(\hat{k}_3)]\,H^-(\hat{\mathbf{k}}) \,\, ,
\label{P2+}
\end{eqnarray}
\end{subequations}
where $H^\pm(\hat{\mathbf{k}}) \equiv \frac{1}{2}(1\pm \bm{\Sigma}\cdot\hat{\mathbf{k}})$ are
the helicity projectors with $\bm{\Sigma}\equiv\g^5\g^0\bm{\gamma}$. From
Eq.~(\ref{P1+}) we see that the quasiparticles of the first
branch have helicity $+1$ when the projection of their momentum
onto the $z$-axis is negative, $\hat{k}_3<0$, and helicity $-1$
if $\hat{k}_3>0$. Quasiparticles of the second branch have opposite
helicities, see Eq.~(\ref{P2+}).
The next step in the argument is to notice that only left-handed quarks
participate in the weak interactions which underly the Urca processes.
Formally, this can be seen from Eq.~(\ref{Pi1}) where the left
chirality projectors $\frac12(1-\g^5)$ occur in the first term
under the trace. (Note that the second term describes charge-conjugate
quarks for which $\frac12(1+\g^5)$ projects also onto
left chirality states.) In the ultrarelativistic limit, these
are quarks with negative helicity. Taking into account the
helicity properties of the quasiparticles in the {\it A} phase,
it becomes clear that only an effective gap structure contributes.
This is constructed from the upper hemisphere of the first mode
and the lower hemisphere of the second mode, see Fig.~\ref{effgap}.
This is a graphical representation of the formal argument given
after Eq.~(\ref{F-eff}). [Of course, our choice for the angular
dependence of the gap functions, namely
$\lambda_{{\bf k},1} = (1+|\cos\theta_{\bf k}|)^2$ and
$\lambda_{{\bf k},2} = (1-|\cos\theta_{\bf k}|)^2$, is
only one possible convention. Equivalently, one could choose
$\lambda_{{\bf k},1} = (1+\cos\theta_{\bf k})^2$ and
$\lambda_{{\bf k},2} = (1-\cos\theta_{\bf k})^2$, in which
case the quasiparticle excitations would be ordered according
to their helicity. Then, quasiparticles of the first (second)
branch would have negative (positive) helicity, and the weak
interaction would involve only quasiparticles of the first
branch. Our convention in this paper is in accordance with
Ref.~\cite{andreas}.]
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{effgap.eps}
\caption{\label{effgap}
Gap functions for the first (left) and the second (middle) excitation
branch with specified helicities of quasiparticles in the upper and
the lower hemispheres. The ``effective'' gap relevant for the neutrino
emission is shown on the right.}
\end{figure}
The asymmetry in the effective gap function translates into the
asymmetry of the neutrino emission. This is due to the
angular dependence of the amplitude for Urca type processes. As in
the vacuum, the corresponding amplitude is proportional to $1-\cos
\theta_{\nu d}$. This can be seen, for instance, from the integrand
in Eq.~(\ref{varchange}). Such an angular dependence of the amplitude
means that the neutrinos are emitted preferably in the direction
opposite to the (almost collinear) momenta of the participating
up and down quarks. In fact, this is a general property that
holds also in the normal phase \cite{iwamoto}. Since the effective
gap function assumes smaller values for quasiparticles with $\hat{k}_3<0$ than
with $\hat{k}_3>0$, there is more neutrino emission in the $+z$
direction.
One can estimate the maximum velocity of a neutron star with
a quark matter core in the {\it A} phase that can be obtained by the
asymmetric neutrino emission. It has been shown that this velocity
is negligibly small, e.g., of the order 1~m/s, see erratum in
Ref.~\cite{prl}. In essence, the reason for this is that
the available thermal energy in the star, after matter in the
stellar interior cools down to the critical temperature
$T_c \lesssim 100$~keV of the {\it A} phase, is too small to
power substantial momentum kicks. (It would be interesting to
investigate, however, if additional sources of stellar heating,
e.g., such as the latent heat from a first-order phase transition,
could change the conclusion.)
\section{Quark mass effects}
\label{quarkmass}
In nature, quarks are not exactly massless. Therefore, it is
important to address the effects that the masses have on the
dispersion relations of quasiparticles, and thus on the neutrino
emissivity and the specific heat of quark matter.
In order to study the massive case, we keep the color-spin structure
of the gap matrix ${\cal M}_{\bf k}$ exactly as in the massless case,
given in Eq.~(\ref{defM}). Then, the inverse full quark propagator can be
written as follows
\begin{equation}
S^{-1}(K)=\left(\begin{array}{cc}
\left[G^{+}_0(K)\right]^{-1} &
{\cal M}_{\bf k} \phi \\
\gamma^0 {\cal M}_{\bf k}^{\dagger} \gamma^0\phi &
\left[G^{-}_0(K)\right]^{-1}
\end{array}\right) \, ,
\end{equation}
with $\left[G^{\pm}_0(K)\right]^{-1} \equiv \gamma^\mu K_{\mu}\pm\gamma^0\mu +m$,
where $m$ is the quark mass. For simplicity, we omit the flavor index $f$ in
this section.
We do not repeat the detailed calculations for the emissivity
and the specific heat with this propagator. Instead we assume that the main
modification happens due to the change of the quasiparticle dispersion relations.
These relations are determined by the solution to the algebraic equation
$\mbox{det}\,S^{-1}=0$, or more explicitly,
\begin{equation}
\mbox{det}\left[ (G^{-}_0)^{-1}(G^{+}_0)^{-1}
- \phi^2 (G^{-}_0)^{-1} {\cal M}_{\bf k} G^{-}_0
\gamma^0 {\cal M}_{\bf k}^{\dagger} \gamma^0\right]=0\, .
\end{equation}
We find that, in all considered spin-one phases, the results for the dispersion
relations are essentially the same as in the massless case, except for the
replacement $k\to \sqrt{k^2+m^2}$,
i.e.,
\begin{equation}
\epsilon_{{\bf k},r}^2= \left(\sqrt{k^2+m^2}-\mu\right)^2
+\lambda_{{\bf k},r}\phi^2 \,\, ,
\label{spectrum-mass}
\end{equation}
with the same $\lambda_{{\bf k},r}$'s as for $m=0$, see
Table~\ref{tablephases}. This result suggests that the emissivity and
the specific heat do not change qualitatively after including quark
masses. This conclusion may not be so surprising because,
in general, non-zero quark masses are not expected to affect much
the physical properties which are dominated by quasiparticle states
in the vicinity of the Fermi sphere.
In view of the above ``trivial'' effect of the quark masses, it is
appropriate to comment on the recent study in Ref.~\cite{Blaschke}
where it is argued that there are no ungapped modes in the CSL phase
when the quarks are massive. This may look as a contradiction to our result
(\ref{spectrum-mass}), showing that all ungapped modes, $\lambda_{{\bf k},2}=0$,
survive after switching on the mass.
The seeming contradiction is removed,
however, after noticing that a different choice of the gap matrix
in the CSL phase is utilized in Ref.~\cite{Blaschke}. In our
notation, the corresponding gap matrix would be obtained by replacing
$\gamma_{\perp,j}(\hat{\mathbf{k}})$ with $\gamma_{j}$ in Eq.~(\ref{defM}).
After making such a replacement, we find that the dispersion
relations of quasiparticles in Ref.~\cite{Blaschke} are indeed
reproduced. In particular, the low-energy dispersion relations
for two out of total three different quasiparticles are given by
\begin{eqnarray}
\e_{{\bf k},1/2}^2 &\simeq & \left(\sqrt{k^2+m^2}-\mu\right)^2
+2\,\phi^2\,\left(\sqrt{1+\frac{m^2}{8\mu^2}} \pm
\frac{m}{2\sqrt{2}\mu}\right)^2\, .
\label{e3}
\end{eqnarray}
Note that these are obtained without the limitation of the smallness
of the quark mass $m$. The corresponding two energy gaps are thus
\begin{equation}
\phi_{1/2} = \sqrt{2}\,\phi\,\left(\sqrt{1+\frac{m^2}{8\mu^2}}
\pm\frac{m}{2\sqrt{2}\mu}\right)\, .
\end{equation}
The low-energy approximation,
defined by $\e_{{\bf k},1/2}/\mu \sim \phi/\mu \sim |\sqrt{k^2+m^2}-\mu|/\mu \ll 1$,
is completely sufficient for the study of
most transport and neutrino processes in spin-one color-superconducting
phases.
The dispersion relation of the third quasiparticle mode can be
extracted exactly,
\begin{eqnarray}
\e_{{\bf k},3}^2 &=& \mu^2+k^2+m^2+\phi^2-2\sqrt{\mu^2(k^2+m^2)+k^2\phi^2}\nonumber\\
&\simeq& \left(\sqrt{k^2+m^2}-\mu\right)^2 +\frac{m^2\phi^2}{\mu^2}\, .
\label{e1}
\end{eqnarray}
Hence the value of the energy gap is
\begin{equation}
\phi_{3} = \frac{m \phi}{\sqrt{\mu^2+\phi^2}} \simeq \frac{m \phi}{\mu} \,\, .
\end{equation}
For $m=0$, we recover the gaps $\phi_1=\phi_2=\sqrt{2}\,\phi$, $\phi_3=0$.
Thus, in contrast to the massive case, both CSL gap matrices (i.e., one with
$\gamma_{\perp,j}(\hat{\mathbf{k}})$ and the other with $\gamma_{j}$ in the definition
of ${\cal M}_{\bf k}$) give rise to the same dispersion relations in the
ultrarelativistic case. It should be studied in the
future which of the two physically different CSL phases at $m\neq 0$
has the lower free energy.
\section{Conclusions}
\label{conclusion}
In this paper, we have computed the neutrino emissivity due to
direct Urca processes (i.e.,
$u+e^{-} \to d+\nu$ and $d \to u+e^{-} +\bar{\nu}$),
as well as the specific heat in four different spin-one
color-superconducting phases of dense quark matter. Starting
from the kinetic equation, we have derived a general expression
for the neutrino emissivity. In the case of the normal
phase of quark matter, this reduces to the well-known
analytical result \cite{iwamoto}. The basic ingredients in
the calculation are the quasiparticle dispersion relations,
containing the spin-one gap functions. We have
studied in detail the effect of an isotropic gap function
(CSL phase) as well as of anisotropic gap functions
(planar, polar, {\it A} phases). The numerical results
for the emissivity and the specific heat as functions
of the ratio of the gap parameter to the temperature,
$\varphi=\phi/T$, have been presented.
In all four phases, also analytical expressions have been derived
in the large $\varphi$ limit (i.e., in the limit of small
temperatures). In particular, in the case of an isotropic
gap function (CSL phase), the well-known exponential
suppression of the emissivity and the specific heat is
observed. We find that anisotropic gaps give rise to
different asymptotes in general. For example, the phases in
which the gap has point nodes (i.e., polar and {\it A} phases)
show a power-law instead of an exponential suppression at
$\varphi\to\infty$. The actual form of the power-law
depends on the behavior of the gap function in the
vicinity of the nodes. While a linear behavior gives
rise to a suppression $\sim 1/\varphi^2$, a quadratic
behavior leads to $\sim 1/\varphi$.
We have used the results for the emissivity and the specific
heat to discuss the cooling curves of spin-one color
superconductors. Our simplified analysis, performed for
an infinite homogeneous system, reveals several important
qualitative features. Most importantly, the cooling rates
of all considered spin-one color superconducting phases
differ very little from that of the normal phase. The
reason is that, with the choice of the gap matrix as in
Eq.~(\ref{defM}), all phases (even if the quarks are massive)
have an ungapped quasiparticle mode which dominates cooling
at low temperatures and, thus, makes the suppression effect
due to gapped modes unobservable. Note, however, that this
conclusion may change if another choice of the gap matrix,
e.g., such as used in Ref.~\cite{Blaschke}, corresponds to
the true ground state of quark matter.
In addition to the cooling rates, we have discussed an unusual
property of the color-superconducting {\it A} phase, in which
the neutrino emission is not symmetric under the reflection
of one of the coordinate axes in position space. This is seen
from the fact that the net momentum of emitted neutrinos is nonzero,
pointing into a direction spontaneously picked by the order parameter.
As we have argued, the asymmetry is related to the
helicity properties of the quasiparticles in the {\it A} phase.
A helicity order arises naturally from the structure of
the gap matrix which is a straightforward generalization of
the {\it A} phase in $^3$He (where, however, there is no
helicity order).
So far, we did not find any observable consequence of the
helicity order in the {\it A} phase. The simplest possibility
would be realized if the asymmetric neutrino emission could
result in a ``neutrino rocket'' mechanism for stars \cite{prl}.
The estimated effect on the stellar velocity appears to be
extremely small, however. The search for other observable
signatures, e.g., dealing with the timing of pulsars, may
reveal other possibilities. We could also imagine that
observable signatures of the helicity-type order might
be observed in completely different systems in atomic or
condensed matter physics (e.g., trapped gases of cold fermionic
atoms, or high-$T_c$ superconductors) if they happen to have
a similar structure of the order parameter, see for example
Ref.~\cite{helicity-cond-mat}. A systematic study of such a
possibility is outside the scope of this paper,
however.
\begin{acknowledgments}
The authors thank D.~Blaschke, M.~Buballa, D.~K.~Hong, M.~M.~Forbes,
P.~Jaikumar, C.~Kouvaris, S.~B.~Popov, K.~Rajagopal, A.~Sedrakian,
and D.~H.~Rischke for comments and discussions. The work of I.A.S. was supported in part
by the Virtual Institute of the Helmholtz Association under
grant No. VH-VI-041 and by Gesellschaft f\"{u}r Schwerionenforschung
(GSI), Bundesministerium f\"{u}r Bildung und Forschung (BMBF).
A.S. thanks the German Academic Exchange Service (DAAD) for
financial support. A.S. and I.A.S. thank the Center
for Theoretical Physics at MIT for their kind hospitality.
Q.W. is supported in part by the startup grant from University of Science
and Technology of China (USTC) in association with 'Bai Ren' project of
Chinese Academy of Sciences (CAS).
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,614
|
google
======
This is code I'm working on while I study for the google interview
I'm using it to demo:
hash tables
stacks
linked list
anything else as I work on the project.
The concept is simple. I'm pulling a large (26MB) dictionary file fromm
the gutenburg project. It has words in all caps on a line by themselves and definition text afterwards using the first line of the defintion keyword DEFN:.
I intend to use gnu regex api to find the word, put it on the stack and then find the corresponding definition text afterwards and put each line on the stack as I parse them.
When i have the word and defintion lines complete, I'll pop them and build an entry for the hash table. The key will be the word and the value for the table will be the definition text. The hash table index based on the key will be a simple xor16 and I will store collisions with linked lists.
Afterwards I hope to do definition lookup from the hash table.
I'm hoping this will demo my abiltiy to implement the elementary data structures, ability to implement a working project, unit tests and ability to use git.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,113
|
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