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LLL

The Lenstra–Lenstra–Lovász (LLL) basis reduction is a -time method for transforming an arbitrary basis of a full-rank in into a reduced basis featuring short, nearly orthogonal vectors. Developed by Arjen Lenstra, , and , it was introduced in 1982 as part of a broader framework for solving hard problems in the , including relation detection and factoring over the rationals. The iteratively applies size and orthogonality reduction steps guided by Gram-Schmidt orthogonalization, ensuring the output basis satisfies the LLL reduction conditions—namely, that consecutive basis vectors are not too long relative to their projections and remain sufficiently independent—while running in time in the dimension and bit length of the input. Although it yields only an exponential approximation factor (roughly $2^{n/2} for dimension n) to the NP-hard shortest vector problem, rather than an exact solution, this efficiency has made LLL a foundational tool in algorithmic and . Key applications include breaking early knapsack cryptosystems by exposing hidden short vectors in associated lattices, enabling attacks on subset-sum instances underlying such schemes. It also underpins solvers by reducing high-dimensional problems to tractable cases and supports cryptanalytic techniques against systems like through partial key recovery via short vector approximation. Despite subsequent improvements like BKZ offering better approximations at higher computational cost, LLL remains the standard for initial basis preprocessing due to its speed and reliability across diverse fields, from error-correcting codes to simultaneous .

Computing and mathematics

Low-level programming language

A low-level programming language provides minimal from a computer's , enabling direct manipulation of resources such as registers, addresses, and instructions. These languages contrast with high-level languages by requiring programmers to manage low-level details like allocation and instruction sequencing explicitly, resulting in code that closely mirrors the binary machine instructions executed by the CPU. The primary types of low-level languages are and . consists of binary instructions (sequences of 0s and 1s) that the interprets and executes directly, without any translation step; for instance, on an x86 architecture, a simple might be represented as 00000001 00000010 in . , a symbolic representation of , uses mnemonics (e.g., "ADD" for ) and labels for operands, making it slightly more readable for humans while still being architecture-specific; it must be translated into via an assembler program before execution. This one-to-one correspondence between assembly instructions and machine operations ensures high efficiency but demands knowledge of the target hardware's specifics, such as and conventions. Characteristics of low-level languages include platform dependence, as code written for one processor family (e.g., versus x86) is generally incompatible with others without significant rewriting. They offer fine-grained control over system resources, leading to optimized performance in applications like operating system kernels, device drivers, and embedded systems, where predictability and minimal overhead are critical; for example, code can execute in fewer clock cycles than equivalent high-level constructs due to the absence of interpretation or overhead. However, this control comes at the cost of error-prone development, as programmers must handle issues like buffer overflows and pointer arithmetic manually, without built-in safety features. Historically, emerged with the first electronic computers in the mid-1940s, such as the (completed in 1945), where operators wired panels or entered binary via switches to program computations. was invented in 1947 by for the ARC (Automatic Relay Computer) at , introducing mnemonics to simplify coding over pure binary entry. By 1949, assemblers were in use for machines like the , marking a shift toward mnemonic-based programming that improved productivity without sacrificing direct hardware access. Despite the rise of high-level languages in the 1950s (e.g., in 1957), low-level languages remain essential for performance-critical software, with modern variants like assembly supporting billions of instructions per core in contemporary CPUs.

Lenstra–Lenstra–Lovász lattice basis reduction algorithm

The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm computes an approximately reduced basis for a full-rank lattice in \mathbb{R}^n, transforming an arbitrary generating basis into one with short, nearly orthogonal vectors. Developed by Arjen K. Lenstra, Hendrik W. Lenstra Jr., and , the algorithm was introduced in their 1982 paper on factoring polynomials over the rationals, where it enabled polynomial-time solutions to previously intractable lattice problems. The method relies on a parameter \delta \in (1/4, 1), conventionally set to $3/4, which trades off between basis quality and computational efficiency; higher \delta yields better approximations but may increase runtime. A basis \mathbf{b}_1, \dots, \mathbf{b}_n is LLL-reduced if it satisfies two conditions: size reduction, where the Gram-Schmidt orthogonalization coefficients satisfy |\mu_{i,j}| \leq 1/2 for all $1 \leq j < i \leq n; and the Lovász condition, ensuring \|\mathbf{b}_i^*\|^2 \geq (\delta - \mu_{i,i-1}^2) \|\mathbf{b}_{i-1}^*\|^2 for i = 2, \dots, n, with \mathbf{b}_i^* denoting the i-th Gram-Schmidt vector. These properties guarantee that the shortest basis vector \mathbf{b}_1 approximates the lattice's shortest nonzero vector length \lambda_1 within a factor of $2^{(n-1)/2}, i.e., \|\mathbf{b}_1\| \leq 2^{(n-1)/2} \lambda_1, and successive minima are similarly bounded. The algorithm achieves this through an iterative loop: starting from k=1, it size-reduces \mathbf{b}_k against prior vectors by subtracting integer multiples to minimize coefficients, then checks the Lovász condition; if violated, it swaps \mathbf{b}_k with \mathbf{b}_{k+1} and restarts from the swapped position, incrementing k otherwise until k > n. The original LLL runs in polynomial time, with complexity O(n^6 (\log B)^4) in the worst case for dimension n and input bit length B, though practical variants and analyses often show sub-exponential behavior in low dimensions or average cases. Subsequent improvements, such as quasi-linear time algorithms in the bit size for fixed dimensions, have reduced this for specialized settings without altering the core framework. Key applications include factorization over \mathbb{Q}, where LLL reduces associated lattices to identify irreducible factors efficiently. In , it solves problems with a fixed number of variables in polynomial time by embedding them into lattices and reducing to find near-optimal solutions. Cryptographic uses encompass attacking subset-sum (knapsack) cryptosystems by revealing short vectors in density-based lattices, as well as applications in for key recovery or parameter estimation in schemes like . In , LLL aids solving Diophantine approximations, quadratic forms, and simultaneous Pell equations by producing short lattice vectors corresponding to good approximations.

Businesses and defense

L3Harris Technologies (formerly L-3 Communications)

L3Harris Technologies, Inc. is an American multinational aerospace and defense technology contractor that designs, integrates, and sustains mission-critical systems for government, defense, and commercial customers. The company specializes in areas such as secure communications, , , sensors, and systems, with a focus on integrated solutions for air, land, sea, , and cyber domains. The firm traces its origins to L-3 Communications, founded on April 30, 1997, by Frank C. Lanza and Robert V. LaPenta in partnership with , which acquired 11 operating divisions previously owned by and held by following Loral's divestitures. These units specialized in electronic systems, telemetry, and communications technologies originally developed for military and space applications. The name "L-3" derives from the initials of Lanza and LaPenta, with the "3" signifying the involvement of the Lehman Brothers-led investment group. L-3 Communications expanded rapidly through over 50 acquisitions in its first decade, focusing on secure data links, equipment, and modernization services for the U.S. Department of Defense and allies. In 2016, it rebranded as to emphasize its evolution beyond communications into broader technology integration. L3Harris was formed on June 29, 2019, via an all-stock merger of and , valued at approximately $35 billion, combining L3's expertise in and sensors with Harris's strengths in multiband radios, microwave systems, and public safety networks. Headquartered in , the merged entity initially generated over $18 billion in annual revenue and employed about 48,000 people across more than 100 countries. By fiscal year 2024, L3Harris reported trailing twelve-month revenue of $21.37 billion, with segments including Integrated Mission Systems (encompassing and ), Space and Airborne Systems (focusing on electro-optical sensors and ), and Communications Systems (providing tactical radios and solutions). The company targets $23 billion in revenue by 2026, driven by demand for hypersonic defense, unmanned systems, and resilient satellite communications. L3Harris maintains long-term contracts with the U.S. military, including sustainment of fleets and development of next-generation jamming pods, while also supplying products like systems. Its growth has been supported by strategic divestitures, such as the 2020 sale of the business to focus on high-margin priorities, and ongoing R&D investments exceeding $1 billion annually in areas like AI-enabled targeting and quantum-resistant encryption.

Health and non-profits

La Leche League International

La Leche League International (LLLI) is a non-profit organization dedicated to promoting and supporting through assistance. Founded in 1956 in , by seven mothers including Ann Kerwin, White, and others who faced challenges in breastfeeding amid a cultural preference for formula feeding, LLLI's initial meeting occurred on October 17, 1956, at White's home to share experiences and information. The organization's name derives from "Nuestra Señora de la Leche y el Buen Parto" (Our Lady of Milk and Happy Delivery), a depicting the Virgin nursing the , symbolizing maternal nurturing. LLLI's core mission is to assist mothers globally in via mother-to-mother support, encouragement, education, and information, while fostering greater societal understanding of as a natural process. It operates through a of volunteer leaders trained in breastfeeding techniques, who facilitate in-person and online meetings, provide consultations, and offer resources in multiple languages across more than 80 countries. Key publications include The Womanly Art of Breastfeeding, first released in 1963 and revised periodically, which emphasizes practical guidance on positioning, milk supply, and weaning based on anecdotal and observational insights from members. The organization has hosted international conferences since 1964, contributing to for accommodations and policies favoring breastfeeding initiation. LLLI's efforts coincided with rising breastfeeding initiation rates in Western countries; for instance, U.S. rates climbed from under 25% in the early to approximately 80% by the , though attribution to LLLI alone is indirect and shares influence with campaigns and WHO guidelines. It maintains fiscal year annual reports detailing global outreach, such as digital expansions during the , but does not publicly disclose precise membership figures, estimating support for thousands of families annually through group interactions and online platforms. Critics, including some medical professionals, have argued that LLLI's advocacy for extended and exclusive can discourage formula supplementation in medically necessary cases, potentially conflicting with individualized pediatric advice, though LLLI maintains its resources defer to healthcare providers. In recent years, LLLI faced internal and external controversy over inclusivity policies updated around 2020 to encompass "parent-to-parent" support and terminology like "chestfeeding" for and individuals who or induce . This shift prompted resignations among leaders and members who contended it obscured the biological specificity of mother-infant dynamics and prioritized ideological inclusivity over practical focus on and newborn needs. By 2025, dissenting groups formed alternatives emphasizing women-only spaces, with reports of declining donations to LLLI amid accusations of diluting its foundational mission. LLLI defended the changes as broadening access without altering core promotion, though source critiques highlight tensions between empirical science—rooted in function—and expansive identity-based framing.

Low-level laser therapy

Low-level laser therapy (LLLT), also termed photobiomodulation or cold laser therapy, involves the application of low-power lasers or light-emitting diodes emitting red or near-infrared light (typically 600–1000 nm wavelengths at power densities of 1–100 mW/cm²) to tissues, aiming to stimulate cellular repair, reduce inflammation, and alleviate pain without thermal effects. The therapy operates on the principle of non-thermal photon absorption by endogenous chromophores, such as cytochrome c oxidase in mitochondria, which initiates photochemical cascades. Developed in the 1960s following the 1960 invention of the , LLLT's biostimulatory potential was first observed in 1967 by Hungarian researcher Endre Mester, who noted accelerated hair growth and in laser-irradiated mice during experiments intended to test carcinogenic effects. Initial clinical applications emerged in the 1970s for and , with broader adoption in the 1980s–1990s as helium-neon and lasers became available; by 2002, certain devices received U.S. FDA clearance for specific indications like pain relief and inflammation reduction. The proposed mechanism centers on light-induced modulation of cellular metabolism: photons absorbed by enhance efficiency, boosting (ATP) production, reducing , and promoting release, which collectively support cytokine shifts (e.g., increased IL-10, decreased TNF-α) and tissue regeneration. These effects are dose-dependent, following an Arndt-Schulz biphasic response where low doses stimulate and high doses inhibit ; optimal dosing varies by condition, typically involving 1–10 J/cm² per session. Applications span musculoskeletal disorders, dermatological conditions, and oral health issues. In orthopedics, LLLT targets tendinopathies (e.g., Achilles, ), osteoarthritis, and nonspecific chronic , with sessions of 5–20 minutes applied 2–3 times weekly. Dermatologically, it aids , , and skin rejuvenation by enhancing synthesis and proliferation. In oncology support, it mitigates oral from , reducing severity by up to 40% in randomized trials. Other uses include neuropathic and post-surgical recovery, though evidence quality varies due to heterogeneous protocols. Efficacy data from systematic reviews indicate short- to medium-term pain reduction: a 2019 meta-analysis of neck pain trials (n=16 RCTs) reported significant Visual Analog Scale improvements (mean difference 14.23 mm) immediately post-treatment and up to 22 weeks, outperforming . For lower extremity tendinopathies and , 2022 reviews confirmed reduced and disability at 1–3 months (standardized mean difference -0.89 for ). In , a 2021 review of 7 RCTs found LLLT superior to for and sleep quality, though functional gains were inconsistent. However, evidence for long-term benefits or functional outcomes (e.g., ) remains limited, with some reviews noting high risk of bias in trials and no superiority over active comparators like exercise in chronic . Proponents attribute variability to suboptimal dosing, while skeptics highlight effects and in smaller studies. Safety profile is favorable, with no serious adverse events in most trials; the non-thermal nature minimizes tissue damage. Mild, transient side effects occur in <5% of cases, including localized , pruritus, or tenderness, resolving within hours. Contraindications include direct ocular exposure (requiring protective ), treatment over known malignancies due to potential stimulation, and photosensitive epilepsies; caution is advised in over the abdomen. Long-term data are sparse, but cumulative exposure appears well-tolerated in chronic use protocols. Regulatory bodies like the FDA classify approved LLLT devices as Class II, affirming low risk when used per guidelines.

Religious organizations

Lutheran Laymen's League

The Lutheran Laymen's League (LLL), officially known as the International Lutheran Laymen's League, was founded in 1917 by twelve lay members of the who organized a campaign to retire a $100,000 debt owed by the synod following financial strains from and internal organizational costs. This effort succeeded, marking the league's initial focus on supporting synodical missions and lay involvement in church . Over time, the LLL expanded its scope to include and media outreach, notably sponsoring the launch of The Lutheran Hour radio broadcast on October 2, 1930, which remains the world's oldest continuously airing Christ-centered program. The organization operates as Lutheran Hour Ministries (LHM), an independent auxiliary of the LCMS and Lutheran Church–Canada, with a to "Bring Christ to the Nations—and the Nations to the Church" through proclamation in relevant cultural contexts. Key activities encompass international support, distribution, and digital , reaching an estimated 195 million people weekly across radio, , and print in more than 60 countries. LHM facilitates over 450,000 faith-sharing engagements annually, including ambassador programs, listening clubs, and congregational resources, while maintaining approximately 300 staff in over 35 global locations. Membership consists of nearly 100,000 individuals organized into hundreds of local chapters, which now include both men and women, emphasizing volunteer-led initiatives for Christian witness and within LCMS congregations. The LLL's structure supports and zonal groups for fellowship, leadership training, and project funding, aligning with confessional Lutheran principles of Scripture's inerrancy and without direct operational oversight from the LCMS.

Political and historical activism

Lithuanian Liberty League

The Lithuanian Liberty League (LLL; Lithuanian: Lietuvos laisvės lyga) was an underground dissident organization in the Lithuanian Soviet Socialist Republic, founded on June 14, 1978, by Antanas Terleckas and an initiative group in Vilnius to pursue the restoration of Lithuania's pre-war independence through nonviolent means. Terleckas, a former political prisoner born in 1928 who died in 2023, served as its primary leader, drawing on his experience in earlier resistance efforts against Soviet occupation. The LLL's founding declaration emphasized non-partisan, democratic principles, lacking a rigid ; it defined the National Council as its highest authority and stipulated that any Lithuanian resident actively supporting its aims—chiefly the "restoration of an independent Lithuania"—could consider themselves a member, while calling for a foreign council among émigrés to advance the cause abroad. Core tasks included fostering religious, national, and political consciousness domestically and elevating Lithuania's freedom issue in international arenas, with early focus on publicizing the secret protocols of the 1939 Molotov-Ribbentrop Pact that enabled Soviet annexation. Activities centered on producing and distributing anti-Soviet publications, organizing clandestine protests, and launching campaigns; for instance, the group collected nearly 1.5 million signatures demanding Soviet troop withdrawal and coordinated boycotts of military in collaboration with youth groups like Young . Key public actions included a June 14, 1988, rally in Square displaying the banned national flag and an August 23, 1987, demonstration near Adam Mickiewicz's monument marking the pact's anniversary, which drew international dissident support including from . A September 28, 1988, gathering in Cathedral Square faced violent suppression by authorities. Soviet authorities responded with arrests and trials, including Terleckas's repeated imprisonments for "," yet the LLL's persistence amid helped catalyze broader movements like , contributing to heightened awareness of the occupation's illegality and Lithuania's March 11, 1990, independence declaration. Legalized in 1988, it operated as a radical pro-independence political entity into the post-Soviet era, formally ceasing activities around 2003 after influencing early democratic transitions. Terleckas received Lithuania's Freedom Prize in 2012 for his role.

Military history

Loyal Lusitanian Legion

The (LLL) was a formation of the , primarily composed of émigrés exiled in following the invasion of in 1807, raised to combat Napoleonic forces during the . Organized in July 1808 in under the initiative of Portuguese colonels José Maria de Moura and Carlos Frederico Lecor, with governmental support from Lord Castlereagh, the legion was placed under the command of Brigadier-General Sir , a officer experienced in . It embodied 's loyalist resistance, incorporating volunteers alongside limited , , , and even deserter officers, and was equipped and trained in light infantry tactics for skirmishing and guerrilla operations. Structurally, the legion initially comprised two battalions of (each with eight companies of approximately 97 men), an company of 86 men equipped with four light guns and two howitzers, and theoretical cavalry elements of three to four squadrons, yielding a total strength of about 2,000 to 2,600 men by late 1808. commands included Lieutenant-Colonel William Mayne for the 1st and figures such as Frederick Baron de Eben for the 2nd, emphasizing mobility and rapid maneuvers suited to Portugal's terrain. The unit's effectiveness stemmed from its hybrid composition and focus on disruption, though logistical challenges in and occasionally reduced operational numbers. Deployed to Portugal in early 1809, the legion engaged in independent guerrilla actions against French supply lines before integrating into Anglo-Portuguese forces under Arthur Wellesley (later Duke of ). Key engagements included the defense of the Alcantara bridge on 14 May 1809, where it repelled French advances under General Lapisse; the on 27-28 July 1809, contributing to Allied holding actions; the on 27 September 1810, skirmishing on the flanks; and defensive operations along the during the 1810-1811 winter. In 1811, it fought at Fuentes d'Onoro on 3-5 May, harassing French cavalry, and suffered heavy casualties at Albuera on 16 May, where its light troops endured intense combat despite being outnumbered. By April 1811, amid broader reforms under British oversight, the legion was disbanded by royal decree on 20 April to reorganize its personnel into the 7th, 8th, and 9th Battalions of , elite units of the regular Portuguese forces, reflecting the shift toward a unified national army capable of sustained conventional warfare. This integration preserved the legion's expertise in light tactics, enhancing Portugal's contributions to the Peninsular campaign until French withdrawal in 1814.

Anatomy and medicine

Left lower lobe of the lung

The left lower lobe (LLL) constitutes the inferior portion of the left lung, which is smaller and more concave than the right lung due to the cardiac notch accommodating the heart. It is demarcated superiorly by the left oblique fissure, extending from the mediastinal surface near the T4 vertebra to the posterior costophrenic angle. The LLL accounts for roughly 45-50% of the left lung's volume and is enveloped by visceral pleura, facilitating smooth expansion during respiration. Structurally, the LLL is divided into four bronchopulmonary segments, each with independent segmental bronchi, arteries, and veins, allowing for targeted surgical resection in pathologies: the superior segment (LB6), anteromedial basal segment (LB7/8, combining anterior and medial basal due to the heart's position displacing a distinct medial segment), lateral basal segment (LB9), and posterior basal segment (LB10). These segments receive air via the left lower lobar bronchus, a branch of the left main bronchus arising approximately 5 cm from the carina. The superior segment bronchus typically originates posteriorly within the lobe, while basal segments share a common trunk before branching. The LLL's blood supply mirrors the lung's : deoxygenated blood arrives via segmental branches of the left for in alveoli, while oxygenated bronchial arteries—primarily from the descending (intercostal and esophageal branches)—nourish the bronchi, pleura, and , with venous drainage partly into pulmonary veins and partly into azygos/hemiazygos systems. Lymphatics drain to hilar and mediastinal nodes, following bronchial and pulmonary vessel paths. Innervation derives from the pulmonary autonomic plexus, with parasympathetic fibers from the (promoting glandular secretion and ) and sympathetic fibers from upper thoracic ganglia (causing bronchodilation and ). Functionally, the LLL contributes to and , with its basal segments positioned for efficient gravity-dependent in upright posture, aiding oxygenation of approximately 20-25% of total passing through the left . Alveolar units within segments enable across a surface area exceeding 50 m² per , with type I pneumocytes for and type II for production to reduce . Clinically, the LLL's dependent position predisposes it to , , and pleural effusions, with often visible on as opacification in posterior basal segments. It is a common site for bronchogenic carcinoma originating in segmental bronchi, and via thoracotomy or preserves function if segments are spared. Congenital anomalies like or may affect the LLL, while or targets segmental vasculature in malignancies.

Entertainment and media

Long, Long, Long (Beatles song)

"" is a song written and primarily performed by for the ' 1968 double album , commonly known as the White Album. It appears as the seventh track on side three of the original release, running 3:04 in length. The composition features Harrison on lead vocals and guitar, with providing backing vocals and Chris Thomas contributing piano. Harrison began writing the song during the Beatles' retreat in , , from February to April 1968, marking his return to guitar after focusing on . Initially titled "It's Been a Long Long Long Time," which Harrison deemed too lengthy, the track draws inspiration from a reminiscent of Loudon Wainwright III's "Sad Girl." The express themes of spiritual longing and rediscovery, with Harrison later confirming it as his first direct address to God, lamenting years of searching ("So many tears I was searching, so many tears I was wasting") before rejoicing in reunion ("You know it's all right"). Though the ambiguous wording could suggest a romantic ode, Harrison clarified its divine subject, aligning with his growing interest in Eastern spirituality amid the band's internal tensions. Recording took place at EMI Studios (Abbey Road) in , with the rhythm track laid down on October 7, 1968, followed by overdubs on October 8 and 9. Harrison handled most instrumentation, including acoustic and electric guitars, , and tamboura, while played drums and McCartney added harmonies. The song's haunting features eerie from an unplugged , spontaneously amplified when Harrison struck a high note, evoking a ghostly wail that blends with rattling effects from a wine bottle on the guitar strings. This experimental close underscores the track's introspective mood, contrasting the White Album's eclectic styles. The song was released on on November 22, 1968, in the UK and the following day in the US, as Apple Records PCSO 7127/8 and SWBO-101. It received limited contemporary attention amid the album's 30 tracks but has since been praised as one of Harrison's finest spiritual compositions, highlighting his maturation as a songwriter during a prolific phase. Retrospective analyses note its subtle beauty and emotional depth, often citing it as an underrated gem that foreshadows Harrison's solo explorations of faith in works like .

Phrases and idioms

Live and let live

The phrase "live and let live" expresses a philosophy of tolerance and non-interference, urging individuals to conduct their own affairs freely while permitting others to do the same without judgment or obstruction, assuming no direct harm to third parties. It embodies principles of personal autonomy and reciprocal respect, often invoked to promote peaceful coexistence amid diverse lifestyles or beliefs. The expression derives from the Dutch proverb leven en laten leven, first documented in English in 1622 by Gerard de Malynes in his treatise Lex Mercatoria (also known as The Ancient Law-Merchant), where he translates it as "Leuen ende laeten leuen, to liue and to let others liue." This early commercial context highlighted pragmatic accommodation in trade disputes, reflecting a mercantile ethos of avoiding unnecessary conflict to sustain mutual benefit. By the late 17th century, it appeared in collections of proverbs, such as John Ray's A Collection of English Proverbs (1678), establishing it as a common idiom in English-speaking cultures. In literature, the phrase gained prominence in the 19th century; for instance, Sir Walter Scott used it in The Heart of Midlothian (1818), with a character advising, "we suld live and let live—we hae been young oursells." Its application extended to real-world scenarios during , where opposing trench soldiers on the Western Front informally adopted a "live-and-let-live" system of restrained fire and minimal aggression during static periods, prioritizing survival over orders for constant combat—a pattern analyzed as emergent amid repeated interactions. This tacit truce persisted in quiet sectors despite military directives against it, illustrating the phrase's practical utility in de-escalating hostility. In the , "" became a slogan in by the late , encapsulating recovery principles of focusing on self-improvement without policing others' behaviors. The phrase continues to underpin libertarian and pluralistic arguments for intervention in personal matters, though critics argue it falters when individual actions impose externalities, such as in or environmental contexts.