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NP

In , NP (nondeterministic polynomial time) is the class of decision problems for which a "yes" instance can be verified in polynomial time using a deterministic , given a proposed or proof of the solution. This formulation captures problems where solutions may be difficult to find but are straightforward to check once provided, distinguishing NP from the class , which includes problems solvable in time without nondeterminism. The central unsolved question of whether P = NP—one of the seven posed by the —asks if every problem verifiable in time is also solvable in time, with profound implications for fields like optimization, , and if resolved affirmatively. Within NP, the subclass of NP-complete problems represents the hardest instances, as established by Cook's 1971 theorem showing that if any NP-complete problem lies in P, then P = NP; these include the (SAT), where one determines if a given can be true-assigned, and others reducible to it via polynomial-time transformations. Examples abound in practical domains, such as the traveling salesman problem (finding the shortest tour visiting specified cities) and (assigning colors to vertices without adjacent same-color pairs), both proven NP-complete and central to scheduling, logistics, and network design. The prevailing empirical and theoretical consensus among researchers holds that P ≠ NP, supported by decades of failed attempts at polynomial algorithms for NP-complete problems and barriers in proof techniques like relativization and natural proofs, underscoring NP's role in delineating tractable from intractable computation.

Computing and mathematics

Complexity classes

In computational complexity theory, NP denotes the class of decision problems for which a "yes" instance can be verified in polynomial time by a deterministic Turing machine given a certificate of polynomial length. This is equivalent to problems solvable in polynomial time by a nondeterministic Turing machine, where nondeterminism allows branching guesses that are checked efficiently. The class includes problems like the Boolean satisfiability problem (SAT), where verifying a satisfying assignment takes time polynomial in the input size, but finding one may require exhaustive search. The concept of NP emerged in the early 1970s, with introducing it in 1971 alongside the Cook-Levin theorem, which proves that SAT is NP-complete—the hardest problems in NP under polynomial-time reductions. This theorem reduces any NP problem to SAT via a construction encoding the verifier's computation as a formula, establishing as a cornerstone for identifying computationally intractable problems. Independently, Alexander Levin formalized similar ideas around 1973, contributing to the theory's foundations. Over 3,000 NP-complete problems have since been identified through such reductions, including 3-SAT and the traveling salesman problem. The P versus NP question—whether every problem in NP is also in P (solvable deterministically in polynomial time)—remains unsolved as of October 2025 and is one of the seven , offering a $1 million reward from the . A strong among experts holds that P ≠ NP, evidenced by the absence of polynomial-time algorithms for NP-complete problems despite decades of effort and vast computational resources, as well as practical failures in breaking cryptographic systems reliant on assumptions like . Proof barriers underscore the difficulty: the relativization barrier (, , Solovay, 1975) shows that proofs using oracles cannot separate P and NP, while natural proofs (Razborov and Rudich, 1997) rule out certain non-relativizing techniques due to their constructive predictability, which aligns with assumptions tied to . Recent claims of resolutions, such as those in 2024-2025 preprints, have not withstood scrutiny and lack peer-reviewed acceptance. NP's implications extend to algorithm design and real-world applications, where NP-complete optimization problems like vehicle routing and persist as exponential-time challenges despite hardware advances, necessitating approximation algorithms and heuristics. In , NP frameworks enable efficient verification of machine-generated solutions, such as in for planning, while approaches tackle NP-hard instances via relaxations, yielding strong approximations for problems like . This hardness explains why exact solutions for large-scale instances remain elusive, driving research into and fixed-parameter tractable algorithms that exploit problem structure.

Other uses in mathematics

In set theory and mathematical logic, non-principal ultrafilters—sometimes referenced with the abbreviation n.p. or informally as NP ultrafilters—are ultrafilters on the power set of an infinite set, such as the natural numbers, that contain all cofinite sets but no finite sets, distinguishing them from principal ultrafilters fixed on a single element. These structures extend the concept of filters to maximal ones under the ultrafilter lemma, enabling constructions like ultraproducts for nonstandard models of arithmetic and analysis, where sequences are identified modulo the ultrafilter to embed standard reals into hyperreals. Unlike principal ultrafilters, non-principal ones exist under the but their precise cardinality and selective properties depend on additional axioms like the ; for instance, on ω, they generate the Stone-Čech compactification βω \ ω, which is used in topological dynamics and without invoking computational nondeterminism.

Medicine and biology

Nurse practitioners

Nurse practitioners (NPs) are advanced practice registered nurses educated at the master's or doctoral level to provide primary and specialty care, including patient assessment, diagnosis, ordering and interpreting diagnostic tests, prescribing medications, and managing treatments. The role originated in the United States in 1965 when Loretta Ford and Henry Silver established the first NP program at the to address shortages, particularly in pediatric and rural settings where access to care was limited. This development responded to empirical needs for expanded healthcare delivery amid post-World War II population growth and geographic disparities, enabling nurses with additional training to handle routine conditions independently in underserved areas. NPs typically complete 500 to 1,500 clinical hours during graduate programs, far fewer than the 12,000 to 16,000 hours accumulated by physicians through and residency training. This disparity in hands-on experience has fueled debates over competency in complex diagnostics and management, with critics arguing that NP programs prioritize breadth over depth, often relying on shorter rotations that limit exposure to rare or multifaceted cases. Empirical data from residency-comparable training underscore physicians' superior preparation for causal pathways in disease progression, as NPs' abbreviated clinical immersion correlates with higher diagnostic oversight in non-routine scenarios. In for straightforward conditions, multiple 2010s meta-analyses indicate NP-led outcomes comparable to , including similar satisfaction, adherence rates, and costs, with some studies reporting lower utilization of services. These findings, however, often derive from settings with collaborative oversight or selective panels, introducing potential biases toward easier cases; independent NP models in full-practice states show elevated hospitalization risks for ambulatory-sensitive conditions, suggesting limitations in managing escalating complexities without input. A 2024 investigation documented cases of NP misdiagnoses leading to harm and deaths, attributing risks to diluted standards amid rapid program proliferation, while analyses link independent NP care to prolonged stays and preventable admissions. By , U.S. states and territories NPs full , allowing without mandatory collaboration, up from 22 in 2020. Proponents cite improved rural , yet causal reveals higher rates and costs in NP settings for non-primary care, challenging equivalence narratives that overlook differentials and selection effects in supportive . opposition stems from data-driven concerns over thresholds, not territorialism, as expanded NP correlates with suboptimal resolutions in empirical reviews of adverse events. In 2025, lawsuits challenged collaborative practice requirements, exemplified by a case where an NP contested laws mandating payments to physicians for nominal oversight, arguing unconstitutional barriers to care despite evidence that such agreements mitigate risks in states retaining them. These legal actions highlight tensions between pushes and verifiable standards, with outcomes hinging on whether courts prioritize metrics over longitudinal favoring structured .

Biological and medical terms

In , nucleoside s are enzymes that catalyze the reversible phosphorolysis of nucleosides to free bases and ribose-1-phosphate (or deoxyribose-1-phosphate), playing a central role in pathways that recycle and nucleosides for synthesis. nucleoside phosphorylase (, EC 2.4.2.1), a prototypical example, converts inosine to hypoxanthine and guanosine to guanine, maintaining nucleoside across tissues and influencing immune function, as PNP deficiency leads to T-cell . In , elevated PNP activity contributes to chemotherapy resistance by salvaging purine analog drugs like 6-mercaptopurine, prompting research into PNP inhibitors to enhance antitumor efficacy. These enzymes operate alongside , with salvage pathways recycling dietary or catabolic nucleobases to produce efficiently. Kallikrein-8 (KLK8), a predominantly expressed in neural tissues, functions in remodeling and , with studies linking its dysregulation to neurodegenerative processes. In models, KLK8 upregulation correlates with amyloid-beta accumulation, and its inhibition reduces plaque load, tau hyperphosphorylation, and cognitive deficits in transgenic mice, suggesting a role in proteolytic degradation of pathological proteins. Elevated KLK8 levels in and blood have been observed in early Alzheimer's stages, positioning it as a potential for preceding , though causality remains under investigation through longitudinal cohorts. Its substrate specificity includes myelin basic protein, implicating it in neuronal repair and modulation. In medical contexts, NP abbreviates nosocomial pneumonia, defined as pneumonia arising 48 hours or more after hospital admission, excluding cases incubating at admission. This condition, often ventilator-associated in intensive care, stems primarily from bacterial pathogens like Pseudomonas aeruginosa or Staphylococcus aureus, with incidence rates of 5-10 cases per 1,000 hospital admissions contributing to high mortality (20-50%) due to multidrug resistance and delayed diagnosis. Risk factors include mechanical ventilation duration exceeding 48 hours, immunosuppression, and aspiration, with evidence from cohort studies emphasizing early antimicrobial stewardship to mitigate outcomes. NP accounts for up to 25% of all nosocomial infections, underscoring its economic burden estimated at $40,000 per case in U.S. hospitals.

Physics and chemistry

Chemical elements

(Np) is a with 93 and is the first transuranic element in the series. It was discovered in 1940 by and Philip Abelson at the , through the bombardment of , which produced -239 that subsequently beta-decayed to form neptunium-239; the identification relied on detecting characteristic beta rays from the new . This marked the first artificial production of an element beyond , confirmed via chemical separation and spectroscopic analysis. Neptunium is a radioactive, silvery metal that tarnishes upon exposure to air and exhibits multiple oxidation states, primarily +3, +4, +5, and +6, enabling diverse aqueous chemistry similar to and . It has 25 known isotopes, all radioactive, with mass numbers ranging from 225 to 244; the most stable is neptunium-237, with a of 2.144 million years, decaying primarily by alpha emission to protactinium-233. Other notable isotopes include neptunium-239 ( 2.36 days), used in early studies of transuranics, and neptunium-236 ( 1.54 × 10^5 years), which contributes to long-term waste concerns due to its fissionability. Neptunium occurs naturally in quantities in ores, formed via on followed by , with concentrations on the order of parts per trillion in pitchblende deposits. However, it is not geologically abundant and is primarily produced artificially in nuclear reactors through successive captures on , yielding neptunium-237 as a at rates of about 0.6% relative to in typical fuel. Global inventories of separated neptunium-237 exceed several tonnes, accumulated from reprocessing spent fuel, though most remains unfractionated in . In nuclear applications, neptunium plays a minor role; neptunium-237 serves as a precursor for production via neutron irradiation (Np-237 + n → Pu-238), which powers radioisotope thermoelectric generators for missions due to its high specific power from . It has potential as a in fast reactors, with a fission cross-section for fast s comparable to (about 1 ), but has not been deployed commercially owing to risks and the dominance of uranium-plutonium cycles. Neptunium-237 is also utilized in specialized detectors, leveraging its threshold by high-energy neutrons above 1 MeV for calibration and monitoring in environments. Health risks from stem from its alpha-emitting isotopes, which pose internal hazards upon or , leading to in and liver tissues where alpha particles cause localized damage. Animal studies indicate principal late effects include induction of , lung, and liver cancers, with neptunium-237's long exacerbating chronic exposure risks comparable to on a mass basis, though external gamma exposure is minimal. Handling requires stringent radiological controls, as its chemical is secondary to radiotoxicity, with no observed safe threshold for alpha emitters in mammalian systems.

Other scientific uses

Nanoparticles (NPs), defined as particulate materials with at least one between and 100 nanometers, possess enhanced reactivity and arising from quantum effects and high surface area, distinguishing them from bulk materials. In , metallic NPs such as or variants accelerate reactions like oxidation by providing active sites that lower activation energies, with applications in fuel cells achieving turnover frequencies up to 10^5 s^-1 under ambient conditions as reported in 2022 electrochemical studies. For , NPs enable targeted release via surface functionalization, such as lipid-based systems conjugating chemotherapeutics to tumor-specific ligands, improving by factors of 5-10 compared to free drugs in preclinical models from 2021-2023 trials. Despite these advantages, NP efficacy is tempered by toxicity risks, including leading to organ damage; for instance, silver NPs administered intravenously in models accumulated in liver and tissues at concentrations exceeding 100 μg/g, inducing and as detailed in a 2023 of and dermal pathways. Aggregation in physiological environments further limits , with studies from 2022 showing up to 50% loss in dispersion efficacy for silica NPs in serum, necessitating stabilizers like to mitigate clumping and enhance circulation half-lives to over 24 hours. These challenges underscore the need for standardized assessments, as emphasized in 2024 guidelines prioritizing dose-response metrics over mere concentration thresholds. In physics contexts, NP occasionally denotes neutral particles in beam experiments, such as neutral (π^0) decay products analyzed in high-energy colliders, where detection efficiencies reach 95% via electromagnetic calorimeters tracking photons from 2γ decays with energies above 1 GeV. However, such usage is context-specific and less prevalent than in nanoscience applications.

Geography and places

Countries and administrative regions

Nepal uses the code NP to denote the sovereign nation, a landlocked country in the bordered by and , with an area of 147,516 square kilometers. As of mid-2025, Nepal's population is estimated at 29,618,118, reflecting steady growth from 26.5 million in the 2011 census amid high and remittances-driven . The 7.8-magnitude Gorkha earthquake on April 25, 2015, caused approximately $10 billion in damages—equivalent to half of the pre-disaster GDP—and slowed growth to 2.97% in 2014-15, but subsequent reconstruction efforts contributed to an average annual GDP growth of about 7% from onward, supported by rebuilding and tourism recovery. In , NP served as the official abbreviation for the Northern Province, an administrative region established in 1994 under the post-apartheid constitution and renamed Limpopo Province on July 11, 2003, via the 11th constitutional amendment. The province, located in the northeast bordering , , and , spanned 123,910 square kilometers with a population of about 5.4 million as of the 2001 census before the rename, encompassing diverse terrain from bushveld to the valley and including parts of . Its economy historically relied on , , and , though it faced challenges like and infrastructure deficits during its NP-designated period. The United Kingdom's postal system assigns NP as the postcode area for the region in southeastern , covering approximately 488,368 residents across 18 districts including city, Blackwood, , and . This area, administered primarily by City Council and surrounding unitary authorities, spans parts of Gwent and counties, with serving as a key and industrial hub historically tied to steel production and shipping.

Other geographic uses

In the United States, "NP" serves as a common abbreviation for , designating protected areas managed by the for the preservation of natural, historic, and cultural resources while allowing public access for educational and recreational purposes. The was established on August 25, 1916, through the , which consolidated management of existing parks and monuments under a unified federal agency. , created by act of Congress on March 1, 1872, became the inaugural and a foundational model for global conservation efforts. In 2023, the system encompassing 63 designated national parks recorded 325.5 million recreation visits, reflecting sustained public engagement amid increasing pressures on ecosystems from and variability. New Providence, abbreviated as NP in postal conventions and ISO 3166-2 subdivision codes (-NP), constitutes the principal island of , encompassing 207 square kilometers and serving as the economic and administrative hub of the archipelago. The island hosts the capital city of on its northeastern coast and accounted for 296,732 residents in the 2022 census, representing approximately 70% of the national population of around 400,000. Its terrain features low limestone ridges, beaches, and urban development concentrated around harbors, supporting , banking, and shipping industries that drive over 80% of the ' GDP.

Organizations and politics

Political parties

The National Party (NP) of , founded on January 26, 1914, by , emerged as a vehicle for and opposition to cultural dominance within the . It gained power in the 1948 general election, securing 45.7% of the vote and forming a government under , which formalized —a policy of racial classification and segregation enacted through laws like the Population Registration Act of 1950 and the of 1950. The party's ideology emphasized separate development for racial groups, prioritizing white minority rule and economic control, which sustained internal stability for its base through suppressed dissent but provoked widespread unrest, including the Sharpeville massacre in 1960 and in 1976, alongside international arms embargoes starting in 1977. Economically, facilitated averaging 3.2% annually from 1948 to 1973, driven by mining and manufacturing, but sanctions in the 1980s—such as the U.S. of 1986—imposed costs estimated at 0.3-0.6% of GDP yearly through trade disruptions, though circumvention via proxies limited deeper impacts and internal factors like violence contributed more to stagnation. The NP governed until 1994, losing to the amid electoral reforms, after which it rebranded as the New National Party and disbanded in 2005. The (NP) of the , established on April 25, 1907, by and Manuel Quezon, holds the distinction as the oldest extant in the country and . Initially a advocating autonomy from U.S. colonial rule, it dominated Philippine politics during the American era, winning key elections like the 1907 assembly polls and leading independence efforts formalized in the 1935 constitution. Post-independence in 1946, the party shifted toward , emphasizing free enterprise, anti-communism, and limited government intervention, with figures like serving as president from 1957 to 1961 on an NP ticket focused on Filipino-first economic policies. It secured victories in national elections through the mid-20th century but faced declines amid (1965-1986), later resurging as an opposition force; in recent polls, such as the 2022 elections, it held seats with 5.7 million votes for its candidates, reflecting enduring elite networks rather than mass ideology.

Other organizations

The American Association of Nurse Practitioners (AANP), founded in 2013 through the merger of earlier groups, serves as the largest professional organization for nurse practitioners , with over 130,000 members as of 2025. It focuses on advancing NP practice authority, including advocacy for full that encompasses diagnosing illnesses, ordering tests, prescribing medications, and managing patient care independently in states granting such autonomy. AANP's efforts emphasize evidence-based policy changes to address shortages, with position statements supporting research dissemination and removal of collaborative agreement requirements where data indicate no compromise in care quality. In 2024, the organization expended $1,550,000 on federal to promote legislation enhancing NP roles in and reimbursement. The Northern Pacific Railway (NP), incorporated in 1864 under a congressional charter signed by President Abraham Lincoln, operated as a major Class I railroad spanning 6,800 miles by the early 20th century, primarily serving the northern transcontinental route from Minnesota to Washington state. Its completion of the main line on September 8, 1883, facilitated timber extraction, agricultural expansion, and population growth in the Pacific Northwest, with land grants totaling 47 million acres used to fund construction and promote settlement. Facing financial strains from overexpansion and competition, NP merged on March 2, 1970, with the Great Northern Railway, Chicago, Burlington & Quincy Railroad, and Spokane, Portland & Seattle Railway to form Burlington Northern Inc., creating a unified 25,000-mile system that reduced redundancies, lowered operating costs by an estimated 10-15% through integrated dispatching and maintenance, and boosted freight volumes in grain, lumber, and minerals by consolidating traffic flows.

Arts and entertainment

Music and media

"N.P." is a directed by Silvano Agosti, featuring as an in a multinational who confronts corporate exploitation and environmental degradation. The film, with a of 90 minutes, explores themes of and resistance against industrial power structures. "N.P." (2020) is a Belgian silent drama film directed by Lisa Spilliaert, adapted from Banana Yoshimoto's 1991 novel of the same name. Set during a Japanese summer, it follows four young people drawn together by their shared fascination with the book "N.P.," a fictional work within the story that binds their personal struggles. The film premiered in 2020 and emphasizes visual storytelling without dialogue to convey emotional isolation and connection.

Games and fiction

In the virtual pet browser game Neopets, launched on November 15, 1999, NP denotes Neopoints, the principal in-game currency used by players to purchase food, toys, and other items for customizing and maintaining their adoptable creatures known as Neopets within the simulated universe of Neopia. Players accumulate NP primarily through participating in site-hosted mini-games, daily quests, trading, and economic simulations, with earning potential varying by activity efficiency; for example, high-scoring sessions in games like Fruit Machine can yield thousands of NP per play. This currency system fosters a player-driven , where rare items or Neopets themselves command premiums in NP, reflecting supply-demand dynamics without real-world monetary conversion officially supported by the platform. The from has influenced narratives, particularly those probing the ramifications of efficient algorithmic solvability for hard problems. In Charles Stross's 2000 short story "Antibodies," resolving P=NP serves as a foundational step for engineering advanced , highlighting causal dependencies between theoretical breakthroughs and technological leaps. The 2012 film Travelling Salesman dramatizes a scenario where mathematicians prove P=NP, exploring ensuing geopolitical tensions, ethical quandaries over , and risks of widespread cryptographic collapse due to tractable optimization of NP-complete problems like the traveling salesman. These depictions underscore the problem's undecidable status in reality—where no proof exists as of 2025—and emphasize its potential to upend fields from to if affirmatively resolved, though skeptics argue favors P ≠ NP based on decades of failed attempts at algorithms for known NP-complete tasks.

Linguistics and communication

Grammatical terms

In , NP refers to a , defined as a syntactic constituent headed by a (or ) that functions as a , object, or complement within a , often comprising determiners, modifiers, and the head itself. This structure captures hierarchical organization in , where the NP projects from the noun head and may embed subordinate phrases. The concept of NP gained prominence in Noam Chomsky's generative grammar framework, outlined in Syntactic Structures (1957), which employed phrase structure rules to generate well-formed sentences. A foundational rule posits that a sentence (S) expands to NP followed by verb phrase (VP), as in S → NP VP, with NP further rewritten as NP → Det N (determiner + noun) or expanded to include adjectives and prepositional phrases, such as NP → Det AdjP N PP. These rules model recursive embedding, enabling NPs like "the old man's hat" (where "old man" modifies "hat" via possessives) or more complex ones like "the cat that the dog chased." In , NPs appear in parse trees as branching nodes: for the "John sees the dog," the tree roots at S, with left-branch NP ("") as a and right-branch VP containing object NP ("the dog" = Det + N). Such trees facilitate analysis of constituency, where tests confirm NP boundaries—for instance, replacing "the big dog" with "it" preserves , isolating the phrase. NPs play a key role in natural language processing (NLP) for syntactic parsing, where algorithms construct trees to represent structure, and in ambiguity resolution, particularly structural ambiguities involving NP coordination or modification. For example, in noun compound bracketing like "engine oil filter" (is it (engine oil) filter or engine (oil filter)?), NP bracketing rules disambiguate via head projection from the final noun, aiding probabilistic parsers in selecting the maximally likely tree based on corpus frequencies. This application extends to preprocessing tasks, where NP chunking segments text into phrases before deeper semantic analysis, improving accuracy in tasks like machine translation.

Internet slang

In internet slang, "NP" or "np" is an abbreviation for "," typically employed as a concise reply to "" or similar acknowledgments of assistance, supplanting more formal phrases like "." This usage conveys that the favor required no significant effort, aligning with the efficiency-driven norms of digital communication. It appears frequently in texting, , and platforms such as and , where character limits and rapid exchanges favor shorthand. The term's documented history traces to early online communities, with the earliest entry from July 18, 2002, defining it explicitly as "," though anecdotal evidence suggests predating SMS proliferation in the and possibly originating in systems or early . Its adoption surged with the expansion of mobile texting in the early 2000s, as evidenced by its inclusion in texting glossaries by 2012, reflecting broader trends in abbreviating polite responses for speed. Usage patterns indicate it is contextually neutral and non-confrontational, often lowercase in casual threads to blend seamlessly with surrounding text. Empirical observations from slang compilations show "NP" as a staple in youth-oriented digital corpora, appearing in lists of common alongside terms like "" for "," with no notable regional variations beyond English-dominant online spaces. While exact frequency metrics vary by platform, its ubiquity in response chains underscores a preference for over elaboration in asynchronous interactions.

Other uses

In medicine, NP designates a , an advanced practice with graduate-level education who can diagnose illnesses, prescribe medications, and manage patient care independently in most U.S. states as of 2024. In , NP stands for nondeterministic polynomial time, the of decision problems where a proposed solution can be verified by a deterministic in polynomial time, a concept central to the unsolved . NP also refers to a , a commissioned public official authorized to witness signatures, administer oaths, and certify documents to prevent , with requirements varying by such as mandatory bonding and exams . In scientific contexts, NP denotes , particulate matter sized 1–100 nanometers exhibiting unique properties due to quantum effects, applied in and as evidenced by clinical trials for targeted cancer therapies. In accounting, NP signifies net profit (or net income), calculated as total revenue minus all expenses, taxes, interest, and depreciation, serving as a key profitability metric; for instance, a firm with $1 million revenue and $800,000 costs yields an NP of $200,000 or 20% margin.