Academic literature on the topic 'Multi-r-ic Depth Four Circuits'

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

IntroductionThe 2009 American film Trash Humpers, directed by Harmony Korine, was released at a time when the hipster had become a ubiquitous concept, entering into the common vernacular of numerous cultures throughout the world, and gaining significant press, social media and academic attention (see Žižek; Arsel and Thompson; Greif et al.; Stahl; Ouellette; Reeve; Schiermer; Maly and Varis). Trash Humpers emerged soon after the 2008 Global Financial Crisis triggered Occupy movements in numerous cities, aided by social media platforms, reported on by blogs such as Gawker, and stylized by multi-national youth-subculture brands such as Vice, American Apparel, Urban Outfitters and a plethora of localised variants.Korine’s film, which is made to resemble found VHS footage of old-aged vandals, epitomises the ironic, retro stylizations and “counterculture-meets-kitsch” aesthetics so familiar to hipster culture. As a creative stereotype from 1940s and ‘50s jazz and beatnik subcultures, the hipster re-emerged in the twenty-first century as a negative embodiment of alternative culture in the age of the Internet. As well as plumbing the recent past for things not yet incorporated into contemporary marketing mechanisms, the hipster also signifies the blurring of irony and authenticity. Such “outsiderness as insiderness” postures can be regarded as a continuation of the marginality-from-the-centre logic of cool capitalism that emerged after World War Two. Particularly between 2007 and 2015, the post-postmodern concept of the hipster was a resonant cultural trope in Western and non-Western cultures alike, coinciding with the normalisation of the new digital terrain and the establishment of mobile social media as an integral aspect of many people’s daily lives. While Korine’s 79-minute feature could be thought of as following in the schlocky footsteps of the likes of Rob Zombie’s The Devil’s Rejects (2006), it is decidedly more arthouse, and more attuned to the influence of contemporary alternative media brands and independent film history alike – as if the love child of Jack Smith’s Flaming Creatures (1963) and Vice Video, the latter having been labelled as “devil-may-care hipsterism” (Carr). Upon release, Trash Humpers was described by Gene McHugh as “a mildly hip take on Jackass”; by Mike D’Angelo as “an empty hipster pose”; and by Aaron Hillis as either “the work of an insincere hipster or an eccentric provocateur”. Lacking any semblance of a conventional plot, Trash Humpers essentially revolves around four elderly-looking protagonists – three men and a woman – who document themselves with a low-quality video camera as they go about behaving badly in the suburbs of Nashville, Tennessee, where Korine still lives. They cackle eerily to themselves as they try to stave off boredom, masturbating frantically on rubbish bins, defecating and drinking alcohol in public, fellating foliage, smashing televisions, playing ten-pin bowling, lighting firecrackers and telling gay “hate” jokes to camera with no punchlines. In one purposefully undramatic scene half-way through the film, the humpers are shown in the aftermath of an attack on a man wearing a French maid’s outfit; he lies dead in a pool of blood on their kitchen floor with a hammer at his feet. The humpers are consummate “bad” performers in every sense of the term, and they are joined by a range of other, apparently lower-class, misfits with whom they stage tap dance routines and repetitively sing nursery-rhyme-styled raps such as: “make it, make it, don’t break it; make it, make it, don’t fake it; make it, make it, don’t take it”, which acts as a surrogate theme song for the film. Korine sometimes depicts his main characters on crutches or in a wheelchair, and a baby doll is never too far away from the action, as a silent and Surrealist witness to their weird, sinister and sometimes very funny exploits. The film cuts from scene to scene as if edited on a video recorder, utilising in-house VHS titling sequences, audio glitches and video static to create the sense that one is engaging voyeuristically with a found video document rather than a scripted movie. Mainstream AlternativesAs a viewer of Trash Humpers, one has to try hard to suspend disbelief if one is to see the humpers as genuine geriatric peeping Toms rather than as hipsters in old-man masks trying to be rebellious. However, as Korine’s earlier films such as Gummo (1997) attest, he clearly delights in blurring the line between failure and transcendence, or, in this case, between pretentious art-school bravado and authentic redneck ennui. As noted in a review by Jeannette Catsoulis, writing for the New York Times: “Much of this is just so much juvenile posturing, but every so often the screen freezes into something approximating beauty: a blurry, spaced-out, yellow-green landscape, as alien as an ancient photograph”. Korine has made a career out of generating this wavering uncertainty in his work, polarising audiences with a mix of critical, cinema-verité styles and cynical exploitations. His work has consistently revelled in ethical ambiguities, creating environments where teenagers take Ritalin for kicks, kill cats, wage war with their families and engage in acts of sexual deviancy – all of which are depicted with a photographer’s eye for the uncanny.The elusive and contradictory aspects of Korine’s work – at once ugly and beautiful, abstract and commercial, pessimistic and nostalgic – are evident not just in films such as Gummo, Julien Donkey Boy (1999) and Mister Lonely (2007) but also in his screenplay for Kids (1995), his performance-like appearances on The Tonight Show with David Letterman (1993-2015) and in publications such as A Crackup at the Race Riots (1998) and Pass the Bitch Chicken (2001). As well as these outputs, Korine is also a painter who is represented by Gagosian Gallery – one of the world’s leading art galleries – and he has directed numerous music videos, documentaries and commercials throughout his career. More than just update of the traditional figure of the auteur, Korine, instead, resembles a contemporary media artist whose avant-garde and grotesque treatments of Americana permeate almost everything he does. Korine wrote the screenplay for Kids when he was just 19, and subsequently built his reputation on the paradoxical mainstreaming of alternative culture in the 1990s. This is exemplified by the establishment of music and film genres such “alternative” and “independent”; the popularity of the slacker ethos attributed to Generation X; the increased visibility of alternative press zines; the birth of grunge in fashion and music; and the coining of “cool hunting” – a bottom-up market research phenomenon that aimed to discover new trends in urban subcultures for the purpose of mass marketing. Key to “alternative culture”, and its related categories such as “indie” and “arthouse”, is the idea of evoking artistic authenticity while covertly maintaining a parasitic relationship with the mainstream. As Holly Kruse notes in her account of the indie music scenes of the 1990s, which gained tremendous popularity in the wake of grunge bands such as Nirvana: without dominant, mainstream musics against which to react, independent music cannot be independent. Its existence depends upon dominant music structures and practices against which to define itself. Indie music has therefore been continually engaged in an economic and ideological struggle in which its ‘outsider’ status is re-examined, re-defined, and re-articulated to sets of musical practices. (Kruse 149)Alternative culture follows a similar, highly contentious, logic, appearing as a nebulous, authentic and artistic “other” whose exponents risk being entirely defined by the mainstream markets they profess to oppose. Kids was directed by the artist cum indie-director Larry Clark, who discovered Korine riding his skateboard with a group of friends in New York’s Washington Square in the early 1990s, before commissioning him to write a script. The then subcultural community of skating – which gained prominence in the 1990s amidst the increased visibility of “alternative sports” – provides an important backdrop to the film, which documents a group of disaffected New York teenagers at a time of the Aids crisis in America. Korine has been active in promoting the DIY ethos, creativity and anti-authoritarian branding of skate culture since this time – an industry that, in its attempts to maintain a non-mainstream profile while also being highly branded, has become emblematic of the category of “alternative culture”. Korine has undertaken commercial projects with an array skate-wear brands, but he is particularly associated with Supreme, a so-called “guerrilla fashion” label originating in 1994 that credits Clark and other 1990s indie darlings, and Korine cohorts, Chloë Sevigny and Terry Richardson, as former models and collaborators (Williams). The company is well known for its designer skateboard decks, its collaborations with prominent contemporary visual artists, its hip-hop branding and “inscrutable” web videos. It is also well known for its limited runs of new clothing lines, which help to stoke demand through one-offs – blending street-wear accessibility with the restricted-market and anti-authoritarian sensibility of avant-garde art.Of course, “alternative culture” poses a notorious conundrum for analysis, involving highly subjective demarcations of “mainstream” from “subversive” culture, not to mention “genuine subversion” from mere “corporate alternatives”. As Pierre Bourdieu has argued, the roots of alternative culture lie in the Western tradition of the avant-garde and the “aesthetic gaze” that developed in the nineteenth century (Field 36). In analysing the modernist notion of advanced cultural practice – where art is presented as an alternative to bourgeois academic taste and to the common realm of cultural commodities – Bourdieu proposed a distinction between two types of “fields”, or logics of cultural production. Alternative culture follows what Bourdieu called “the field of restricted production”, which adheres to “art for art’s sake” ideals, where audiences are targeted as if like-minded peers (Field 50). In contrast, the “field of large-scale production” reflects the commercial imperatives of mainstream culture, in which goods are produced for the general public at large. The latter field of large-scale production tends to service pre-established markets, operating in response to public demand. Furthermore, whereas success in the field of restricted production is often indirect, and latent – involving artists who create niche markets without making any concessions to those markets – success in the field of large-scale production is typically more immediate and quantifiable (Field 39). Here we can see that central to the branding of “alternative culture” is the perceived refusal to conform to popular taste and the logic of capitalism more generally is. As Supreme founder James Jebbia stated about his brand in a rare interview: “The less known the better” (Williams). On this, Bourdieu states that, in the field of restricted production, the fundamental principles of all ordinary economies are inversed to create a “loser wins” scenario (Field 39). Profit and cultural esteem become detrimental attributes in this context, potentially tainting the integrity and marginalisation on which alternative products depend. As one ironic hipster t-shirt puts it: “Nothing is any good if other people like it” (Diesel Sweeties).Trash HipstersIn abandoning linear narrative for rough assemblages of vignettes – or “moments” – recorded with an unsteady handheld camera, Trash Humpers positions itself in ironic opposition to mainstream filmmaking, refusing the narrative arcs and unwritten rules of Hollywood film, save for its opening and closing credits. Given Korine’s much publicized appreciation of cinema pioneers, we can understand Trash Humpers as paying homage to independent and DIY film history, including Jack Smith’s Flaming Creatures, William Eggleston’s Stranded in Canton (1973), Andy Warhol’s and Paul Morrissey’s Lonesome Cowboys (1967) and Trash (1970), and John Waters’s Pink Flamingos (1972), all of which jubilantly embraced the “bad” aesthetic of home movies. Posed as fantasized substitutions for mainstream movie-making, such works were also underwritten by the legitimacy of camp as a form of counter-culture critique, blurring parody and documentary to give voice to an array of non-mainstream and counter-cultural identities. The employment of camp in postmodern culture became known not merely as an aesthetic subversion of cultural mores but also as “a gesture of self-legitimation” (Derrida 290), its “failed seriousness” regarded as a critical response to the specific historical problem of being a “culturally over-saturated” subject (Sontag 288).The significant difference between Korine’s film and those of his 1970s-era forbears is precisely the attention he pays to the formal aspects of his medium, revelling in analogue editing glitches to the point of fetishism, in some cases lasting as long as the scenes themselves. Consciously working out-of-step with the media of his day, Trash Humpers in imbued with nostalgia from its very beginning. Whereas Smith, Eggleston, Warhol, Morrissey and Waters blurred fantasy and documentary in ways that raised the social and political identities of their subjects, Korine seems much more interested in “trash” as an aesthetic trope. In following this interest, he rightfully pays homage to the tropes of queer cinema, however, he conveniently leaves behind their underlying commentaries about (hetero-) normative culture. A sequence where the trash humpers visit a whorehouse and amuse themselves by smoking cigars and slapping the ample bottoms of prostitutes in G-strings confirms the heterosexual tenor of the film, which is reiterated throughout by numerous deadpan gay jokes and slurs.Trash Humpers can be understood precisely in terms of Korine’s desire to maintain the aesthetic imperatives of alternative culture, where formal experimentation and the subverting of mainstream genres can provide a certain amount of freedom from explicated meaning, and, in particular, from socio-political commentary. Bourdieu rightly points out how the pleasures of the aesthetic gaze often manifest themselves curiously as form of “deferred pleasure” (353) or “pleasure without enjoyment” (495), which corresponds to Immanuel Kant’s notion of the disinterested nature of aesthetic judgement. Aesthetic dispositions posed in the negative – as in the avant-garde artists who mined primitive and ugly cultural stereotypes – typically use as reference points “facile” or “vulgar” (393) working-class tropes that refer negatively to sensuous pleasure as their major criterion of judgment. For Bourdieu, the pleasures provided by the aesthetic gaze in such instances are not sensual pleasures so much as the pleasures of social distinction – signifying the author’s distance from taste as a form of gratification. Here, it is easy to see how the orgiastic central characters in Trash Humpers might be employed by Korine for a similar end-result. As noted by Jeremiah Kipp in a review of the film: “You don't ‘like’ a movie like Trash Humpers, but I’m very happy such films exist”. Propelled by aesthetic, rather than by social, questions of value, those that “get” the obscure works of alternative culture have a tendency to legitimize them on the basis of the high-degree of formal analysis skills they require. For Bourdieu, this obscures the fact that one’s aesthetic “‘eye’ is a product of history reproduced by education” – a privileged mode of looking, estranged from those unfamiliar with the internal logic of decoding presupposed by the very notion of “aesthetic enjoyment” (2).The rhetorical priority of alternative culture is, in Bourdieu’s terms, the “autonomous” perfection of the form rather than the “heteronomous” attempt to monopolise on it (Field 40). However, such distinctions are, in actuality, more nuanced than Bourdieu sometimes assumed. This is especially true in the context of global digital culture, which makes explicit how the same cultural signs can have vastly different meanings and motivations across different social contexts. This has arguably resulted in the destabilisation of prescriptive analyses of cultural taste, and has contributed to recent “post-critical” advances, in which academics such as Bruno Latour and Rita Felski advocate for cultural analyses and practices that promote relationality and attachment rather than suspicious (critical) dispositions towards marginal and popular subjects alike. Latour’s call for a move away from the “sledge hammer” of critique applies as much to cultural practice as it does to written analysis. Rather than maintaining hierarchical oppositions between authentic versus inauthentic taste, Latour understands culture – and the material world more generally – as having agency alongside, and with, that of the social world.Hipsters with No AlternativeIf, as Karl Spracklen suggests, alternativism is thought of “as a political project of resistance to capitalism, with communicative oppositionality as its defining feature” (254), it is clear that there has been a progressive waning in relevance of the category of “alternative culture” in the age of the Internet, which coincides with the triumph of so-called “neoliberal individualism” (258). To this end, Korine has lost some of his artistic credibility over the course of the 2000s. If viewed negatively, icons of 1990s alternative culture such as Korine can be seen as merely exploiting Dada-like techniques of mimetic exacerbation and symbolic détournement for the purpose of alternative, “arty” branding rather than pertaining to a counter-hegemonic cultural movement (Foster 31). It is within this context of heightened scepticism surrounding alternative culture that the hipster stereotype emerged in cultures throughout the world, as if a contested symbol of the aesthetic gaze in an era of neoliberal identity politics. Whatever the psychological motivations underpinning one’s use of the term, to call someone a hipster is typically to point out that their distinctive alternative or “arty” status appears overstated; their creative decisions considered as if a type of bathos. For detractors of alternative cultural producers such as Korine, he is trying too hard to be different, using the stylised codes of “alternative” to conceal what is essentially his cultural and political immaturity. The hipster – who is rarely ever self-identified – re-emerged in the 2000s to operate as a scapegoat for inauthentic markers of alternative culture, associated with men and women who appear to embrace Realpolitik, sincerity and authentic expressions of identity while remaining tethered to irony, autonomous aesthetics and self-design. Perhaps the real irony of the hipster is the pervasiveness of irony in contemporary culture. R. J Magill Jnr. has argued that “a certain cultural bitterness legitimated through trenchant disbelief” (xi) has come to define the dominant mode of political engagement in many societies since the early 2000s, in response to mass digital information, twenty-four-hour news cycles, and the climate of suspicion produced by information about terrorism threats. He analyses the prominence of political irony in American TV shows including The Daily Show with Jon Stewart, The Simpsons, South Park, The Chappelle Show and The Colbert Report but he also notes its pervasiveness as a twenty-first-century worldview – a distancing that “paradoxically and secretly preserves the ideals of sincerity, honesty and authenticity by momentarily belying its own appearance” (x). Crucially, then, the utterance “hipster” has come to signify instances when irony and aesthetic distance are perceived to have been taken too far, generating the most disdain from those for whom irony, aesthetic discernment and cultural connoisseurship still provide much-needed moments of disconnection from capitalist cultures drowning in commercial hyperbole and grave news hype. Korine himself has acknowledged that Spring Breakers (2013) – his follow-up feature film to Trash Humpers – was created in response to the notion that “alternative culture”, once a legitimate challenge to mainstream taste, had lost its oppositional power with the decentralization of digital culture. He states that he made Spring Breakers at a moment “when there’s no such thing as high or low, it’s all been exploded. There is no underground or above-ground, there’s nothing that’s alternative. We’re at a point of post-everything, so it’s all about finding the spirit inside, and the logic, and making your own connections” (Hawker). In this context, we can understand Trash Humpers as the last of the Korine films to be branded with the authenticity of alternative culture. In Spring Breakers Korine moved from the gritty low-fi sensibility of his previous films and adopted a more digital, light-filled and pastel-coloured palette. Focussing more conventionally on plot than ever before, Spring Breakers follows four college girls who hold up a restaurant in order to fund their spring break vacation. Critic Michael Chaiken noted that the film marks a shift in Korine’s career, from the alternative stylings of the pre-Internet generation to “the cultural heirs [of] the doomed protagonists of Kids: nineties babies, who grew up with the Internet, whose sensibilities have been shaped by the sweeping technological changes that have taken place in the interval between the Clinton and Obama eras” (33).By the end of the 2000s, an entire generation came of age having not experienced a time when the obscure films, music or art of the past took more effort to track down. Having been a key participant in the branding of alternative culture, Korine is in a good position to recall a different, pre-YouTube time – when cultural discernment was still caught up in the authenticity of artistic identity, and when one’s cultural tastes could still operate with a certain amount of freedom from sociological scrutiny. Such ideas seem a long way away from today’s cultural environments, which have been shaped not only by digital media’s promotion of cultural interconnection and mass information, but also by social media’s emphasis on mobilization and ethical awareness. ConclusionI should reiterate here that is not Korine’s lack of seriousness, or irony, alone that marks Trash Humpers as a response to the scepticism surrounding alternative culture symbolised by the figure of the hipster. It is, rather, that Korine’s mock-documentary about juvenile geriatrics works too hard to obscure its implicit social commentary, appearing driven to condemn contemporary capitalism’s exploitations of youthfulness only to divert such “uncool” critical commentaries through unsubtle formal distractions, visual poetics and “bad boy” avant-garde signifiers of authenticity. Before being bludgeoned to death, the unnamed man in the French maid’s outfit recites a poem on a bridge amidst a barrage of fire crackers let off by a nearby humper in a wheelchair. Although easily overlooked, it could, in fact, be a pivotal scene in the film. Spoken with mock high-art pretentions, the final lines of the poem are: So what? Why, I ask, why? Why castigate these creatures whose angelic features are bumping and grinding on trash? Are they not spawned by our greed? Are they not our true seed? Are they not what we’ve bought for our cash? We’ve created this lot, of the ooze and the rot, deliberately and unabashed. Whose orgiastic elation and one mission in creation is to savagely fornicate TRASH!Here, the character’s warning of capitalist overabundance is drowned out by the (aesthetic) shocks of the fire crackers, just as the stereotypical hipster’s ethical ideals are drowned out by their aesthetic excess. The scene also functions as a metaphor for the humpers themselves, whose elderly masks – embodiments of nostalgia – temporarily suspend their real socio-political identities for the sake of role-play. It is in this sense that Trash Humpers is too enamoured with its own artifices – including its anonymous “boys club” mentality – to suggest anything other than the aesthetic distance that has come to mark the failings of the “alternative culture” category. In such instances, alternative taste appears as a rhetorical posture, with Korine asking us to gawk knowingly at the hedonistic and destructive pleasures pursued by the humpers while factoring in, and accepting, our likely disapproval.ReferencesArsel, Zeynep, and Craig J. Thompson. “Demythologizing Consumption Practices: How Consumers Protect Their Field-Dependent Identity Investments from Devaluing Marketplace Myths.” Journal of Consumer Research 37.5 (2011): 791-806.Bourdieu, Pierre. Distinction: A Social Critique of the Judgement of Taste. Trans. Richard Nice. Cambridge: Harvard University Press, 1984.Bourdieu, Pierre. The Field of Cultural Production Essays on Art and Literature. Edited by Randal Johnson. London: Polity Press, 1993.Carr, David. “Its Edge Intact, Vice Is Chasing Hard News.” New York Times 24 Aug. 2014. 12 Nov. 2016 <https://www.nytimes.com/2014/08/25/business/media/its-edge-intact-vice-is-chasing-hard-news-.html>.Catsoulis, Jeannette. “Geriatric Delinquents, Rampaging through Suburbia.” New York Times 6 May 2010. 1` Nov. 2016 <http://www.nytimes.com/2010/05/07/movies/07trash.html>.Chaiken, Michael. “The Dream Life.” Film Comment (Mar./Apr. 2013): 30-33.D’Angelo, Mike. “Trash Humpers.” Not Coming 18 Sep. 2009. 12 Nov. 2016 <http://www.notcoming.com/reviews/trashhumpers>.Derrida, Jacques. Positions. London: Athlone, 1981.Diesel Sweeties. 1 Nov. 2016 <https://store.dieselsweeties.com/products/nothing-is-any-good-if-other-people-like-it-shirt>.Felski, Rita. The Limits of Critique. Chicago: University of Chicago Press, 2015.Greif, Mark. What Was the Hipster? A Sociological Investigation. New York: n+1 Foundation, 2010.Hawker, Philippa. “Telling Tales Out of School.” Sydney Morning Herald 4 May 2013. 12 Nov. 2016 <http://www.smh.com.au/entertainment/movies/telling-tales-out-of-school-20130503-2ixc3.html>.Hillis, Aaron. “Harmony Korine on Trash Humpers.” IFC 6 May 2009. 12 Nov. 2016 <http://www.ifc.com/2010/05/harmony-korine-2>.Jay Magill Jr., R. Chic Ironic Bitterness. Ann Arbor: University of Michigan Press, 2007.Kipp, Jeremiah. “Clean Off the Dirt, Scrape Off the Blood: An Interview with Trash Humpers Director Harmony Korine.” Slant Magazine 18 Mar. 2011. 1 Nov. 2016 <http://www.slantmagazine.com/house/article/clean-off-the-dirt-scrape-off-the-blood-an-interview-with-trash-humpers-director-harmony-korine>.Latour, Bruno. “Why Has Critique Run Out of Steam? From Matters of Fact to Matters of Concern.” Critical Inquiry 30.2 (2004): 225-248.Maly, Ico, and Varis, Piia. “The 21st-Century Hipster: On Micro-Populations in Times of Superdiversity.” European Journal of Cultural Studies 19.6 (2016): 637–653.McHugh, Gene. “Monday May 10th 2010.” Post Internet. New York: Lulu Press, 2010.Ouellette, Marc. “‘I Know It When I See It’: Style, Simulation and the ‘Short-Circuit Sign’.” Semiotic Review 3 (2013): 1–15.Reeve, Michael. “The Hipster as the Postmodern Dandy: Towards an Extensive Study.” 2013. 12 Nov. 2016. <http://www.academia.edu/3589528/The_hipster_as_the_postmodern_dandy_towards_an_extensive_study>.Schiermer, Bjørn. “Late-Modern Hipsters: New Tendencies in Popular Culture.” Acta Sociologica 57.2 (2014): 167–181.Sontag, Susan. “Notes on Camp.” Against Interpretation. New York: Octagon, 1964/1982. 275-92. Stahl, Geoff. “Mile-End Hipsters and the Unmasking of Montreal’s Proletaroid Intelligentsia; Or How a Bohemia Becomes BOHO.” Adam Art Gallery, Apr. 2010. 12 May 2015 <http://www.adamartgallery.org.nz/wp-content/uploads/2010/04/adamartgallery_vuwsalecture_geoffstahl.pdf>.Williams, Alex. “Guerrilla Fashion: The Story of Supreme.” New York Times 21 Nov. 2012. 1 Nov. 2016 <http://www.nytimes.com/2012/11/22/fashion/guerrilla-fashion-the-story-of-supreme.html>.Žižek, Slavoj. “L’Etat d’Hipster.” Rhinocerotique. Trans. Henry Brulard. Sep. 2009. 3-10.

In this work we study the multi-r-ic formula model introduced by [KS15c] and improve upon the lower bound for multi-r-ic depth four circuits given in [KST16b], when viewed as a function of the number of input variables N. The improvement leads to superpolynomial lower bounds for values of r signi cantly higher than what is known from prior works. A (syntactically) multi-r-ic formula is an arithmetic formula in which the formal degree with respect to every variable is at most r at every gate. The formal degree of an input gate with respect to a variable x is de ned to be 1 if the gate is labelled with x and 0 if it is labelled with a eld element or a di erent variable. The formal degree of a sum (respectively, product) gate with respect to x is de ned as the maximum (respectively, sum) of the formal degrees of its children with respect to x. Naturally, a multi-r-ic formula computes a polynomial with individual degree of every variable bounded by r. Multi-r-ic formulas are a natural extension of the relatively well-studied multilinear formulas [Raz09, RY09]. In this work, we focus on multi-r-ic formulas that compute multilinear polynomials. They are interesting because they allow the formal degree of the root node of the formula to be as high as r times the number of underlying variables. This gives extra room for `clever' cancellations of the high degree components inside the formula thereby making this type of formulas harder to analyze (as formula homogenization is not known to be doable without blowing up the size superpolynomially unless degree is very small [Raz10]). Most lower bound proofs in the literature operate under the restriction of low formal degree or multilinearity [Raz09, RY09, KSS14, KLSS]. In this light, multi-r-ic formulas computing multilinear polynomials form a reasonable intermediate model to study in order to gain some insight on how to deal with high formal degree in general formulas. Another motivation for understanding the high formal degree case better (even at depth three) comes from the depth reduction result in [GKKS14]. With the aim of making progress on multi-r-ic formula lower bound, [KST16b] gave a ( N d r2 ) ( p d=r) lower bound for multi-r-ic depth four formulas computing the N-variate Iterated iii Matrix Multiplication (IMM) polynomial of degree d. As a function of N, the lower bound is at most 2 ( p N=r3) when d = (N=r2). In this thesis, our focus is on getting multi-r-ic depth four formulas with larger r into the arena of models that provenly admit a superpolynomial lower bound. In [KST16b], r can be at most N1=3 for the bound to remain superpolynomial. Our result (stated below) gives a superpolynomial lower bound for multi-r-ic depth four formulas where r can be as high as (N logN)0:9. Theorem. Let N; d; r be positive integers such that 0:51 N d 0:9 N and r (N logN)0:9. Then there is an explicit N-variate degree-d multilinear polynomial in VNP such that any multi-r-ic depth four circuit computing it has size 2 p N logN r . The theorem yields a better lower bound than that of [KST16b], when viewed as a function of N. Also, the bound matches the best known lower bound (as a function of N) for multilinear (r = 1) depth four circuits [RY09] which is 2 ( p N logN). The improvement is obtained by analyzing the shifted partials dimension (SPD) of an N- variate polynomial in VNP (as opposed to a VP polynomial in [KST16b]) of high degree range of (N), and comparing it with the SPD of a depth four multi-r-ic circuit. In [KST16b] a variant of shifted partials, called shifted skewed partials, is critically used to analyze the IMM polynomial (which is in VP) and obtain a lower bound as a function of N and d (particularly for low d). We observe that SPD (without `skew') is still e ective for the Nisan-Wigderson polynomial (which is in VNP), and yields a better lower bound as a function of only N when degree d is naturally chosen to be high. Our analysis gives a better range for r and a better lower bound in the high degree regime, not only for depth four multi-r-ic circuits but also for the weaker models: multi-r-ic depth three circuits and multi-r-ic depth four circuits with low bottom support. These (weaker) models are instrumental in gaining insight about general depth four multi-r-ic circuits, both in [KST16b] and our work.

Abstract The capability of the Scanning Electron Acoustic Microscopy (SEAM) technique for high resolution non-destructive subsurface imaging at different depths for a multi-level integrated circuit is assessed. Experimental results using a beveled DRAM IC sample are used to quantify the effect of the electron beam energy and modulation frequency on contrast, spatial resolution and depth of focus of SEAM amplitude and phase images.

Abstract The adoption of 3D packaging technology necessitates the development of new approaches to failure electronic device analysis. To that end, our team is developing a tool called the quantum diamond microscope (QDM) that leverages an ensemble of nitrogen vacancy (NV) centers in diamond, achieving vector magnetic imaging with a wide field-of-view and high spatial resolution under ambient conditions. Here, we present the QDM measurement of 2D current distributions in an 8-nm flip chip IC and 3D current distributions in a multi-layer PCB. Magnetic field emanations from the C4 bumps in the flip chip dominate the QDM measurements, but these prove to be useful for image registration and can be subtracted to resolve adjacent current traces in the die at the micron scale. Vias in 3D ICs display only Bx and By magnetic fields due to their vertical orientation and are difficult to detect with magnetometers that only measure the Bz component (orthogonal to the IC surface). Using the multi-layer PCB, we show that the QDM’s ability to simultaneously measure Bx, By, and Bz is advantageous for resolving magnetic fields from vias as current passes between layers. We also show how spacing between conducting layers is determined by magnetic field images and how it agrees with the design specifications of the PCB. In our initial efforts to provide further z-depth information for current sources in complex 3D circuits, we show how magnetic field images of individual layers can be subtracted from the magnetic field image of the total structure. This allows for isolation of signal layers and can be used to map embedded current paths via solution of the 2D magnetic inverse. In addition, the paper also discusses the use of neural networks to identify 2D current distributions and its potential for analyzing 3D structures.