LP
LP (born Laura Pergolizzi; March 18, 1981) is an American singer-songwriter and musician recognized for her distinctive raspy voice, introspective lyrics, and contributions to pop and alternative rock genres.[1] Originating from Long Island, New York, she initially built her career as a songwriter for major artists, co-writing Rihanna's "Cheers (Drink to That" from her 2010 album Loud, as well as tracks for Cher, Backstreet Boys, Céline Dion, and Christina Aguilera, which helped establish her reputation in the industry despite early struggles with major label deals that yielded no releases.[2] Her breakthrough as a performer came with the 2016 album Lost on You, propelled by the title track that topped charts in 18 countries, amassed billions of streams globally, and earned platinum certifications, marking a shift from behind-the-scenes work to headlining sold-out tours across over 150 cities.[2][3] Subsequent albums including Heart to Mouth (2018), featuring the radio hit "Girls Go Wild," Churches (2021), and Love Lines (2024) have solidified her catalog exceeding 3 billion streams, with live performances drawing capacities from 3,000 to 20,000 attendees and monthly listeners surpassing 25 million.[2] LP's discography reflects persistent themes of personal resilience and emotional depth, drawn from her experiences navigating multiple record contracts and independent releases like her debut Heart-Shaped Scar in the early 2000s.[2] A notable controversy arose in February 2024 when she posted an Instagram video wearing a hoodie emblazoned with the Russian flag, gifted by her Russian fan club, expressing thanks to them amid her prior public support for Ukraine following Russia's 2022 invasion; this prompted backlash, including the cancellation of her scheduled concert in Kaunas, Lithuania, after calls for boycott from local audiences sensitive to Russian symbolism during ongoing geopolitical tensions.[4][5] She subsequently apologized to Ukrainian supporters, clarifying the gesture as appreciation for fans rather than political endorsement, though the incident highlighted risks of perceived neutrality in polarized international contexts.[6]Mathematics and optimization
Linear programming
Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to a set of linear equality and inequality constraints.[7] The feasible region defined by these constraints forms a convex polyhedron, and the optimal solution, if it exists, occurs at a vertex of this polyhedron due to the linearity of the objective.[8] This approach assumes proportionality, additivity, and divisibility of resources, making it suitable for problems where variables represent continuous quantities like production levels or allocations.[9] The field emerged during World War II efforts to optimize military logistics, with George Dantzig formalizing linear programming models and developing the simplex algorithm in 1947 while working for the U.S. Air Force.[10] Dantzig's simplex method iteratively moves along edges of the feasible polyhedron from one basic feasible solution to an adjacent one, improving the objective value until optimality is reached, typically converging in polynomial time on average despite worst-case exponential complexity.[8] Earlier work by Leonid Kantorovich in 1939 addressed similar resource allocation problems in the Soviet economy, but Dantzig's contributions integrated it into Western operations research and computational practice.[11] A standard linear program in maximization form is formulated as: \max \, \mathbf{c}^\top \mathbf{x} subject to \mathbf{A} \mathbf{x} \leq \mathbf{b}, \quad \mathbf{x} \geq \mathbf{0}, where \mathbf{x} is the vector of decision variables, \mathbf{c} the objective coefficients, \mathbf{A} the constraint matrix, and \mathbf{b} the right-hand side vector.[7] Equality constraints can be handled by converting them into pairs of inequalities, and minimization problems by negating the objective.[12] Interior-point methods, developed later in the 1980s by Karmarkar, offer polynomial-time guarantees and complement the simplex method for large-scale problems.[13] Duality associates each primal linear program with a dual program, where the primal maximizes subject to upper bounds and the dual minimizes subject to lower bounds on dual variables.[14] The weak duality theorem states that the primal objective is always less than or equal to the dual objective for feasible solutions, providing bounds for verification.[15] Under feasibility of both, the strong duality theorem guarantees equal optimal values, with complementary slackness conditions linking primal and dual solutions—nonzero slacks in one imply zero multipliers in the other.[14] This duality underpins sensitivity analysis, revealing shadow prices for constraints as dual variables.[16] Applications span production planning, where firms allocate limited resources to maximize profit under capacity constraints; transportation networks, minimizing costs in shipping via balanced supply-demand models; and scheduling, optimizing workforce or machine assignments.[17] In energy management, it schedules generation to meet demand at minimal cost, while in telecommunications, it designs networks for traffic routing efficiency.[18] Supply chain optimization uses it for inventory and distribution, as seen in post-WWII industrial implementations that reduced costs by up to 10-20% in sectors like oil refining and steel production.[11] Limitations arise in non-linear or integer-constrained problems, requiring extensions like mixed-integer programming.[13]Computing and logic
Logic programming
Logic programming is a declarative programming paradigm in which computation is expressed as a set of logical statements—typically facts and rules in the form of Horn clauses—and execution proceeds via automated logical inference, such as resolution or unification, to derive conclusions from queries.[19] Unlike imperative programming, which specifies a sequence of commands to mutate state, logic programming focuses on describing relationships and constraints, leaving the control flow and search strategy to the underlying inference engine.[20] This approach enables concise representations of problems in domains requiring reasoning, such as automated theorem proving and knowledge representation.[21] The paradigm originated in the late 1960s and early 1970s from research in automated theorem proving, drawing on resolution-based deduction systems developed by figures like J.A. Robinson in 1965.[19] Prolog, the seminal logic programming language, was created in 1972 by Alain Colmerauer, Philippe Roussel, and Robert Kowalski at the University of Marseille specifically for natural language processing tasks, evolving from earlier systems like SLIP.[22] By 1975, Prolog had formalized its core semantics around SLD-resolution (Selective Linear Definite clause resolution), enabling efficient backward chaining for query resolution.[23] The language gained prominence in the 1980s through Japanese Fifth Generation Computer Systems projects, which aimed to leverage logic programming for parallel AI applications, though these efforts highlighted scalability challenges in non-deterministic search.[24] Core mechanisms include unification, which matches variables in queries to terms in the knowledge base, and backtracking, which explores alternative bindings upon failure, implementing a depth-first search over the proof tree.[25] Programs are executed by posing goals (queries) against a database of facts (e.g.,parent(tom, bob).) and rules (e.g., grandparent(X, Z) :- parent(X, Y), parent(Y, Z).), with the system deriving bindings that satisfy the goal.[20] This contrasts with imperative paradigms by avoiding explicit loops or assignments, reducing errors from state management but potentially incurring overhead from exhaustive search in large spaces.[26]
Prolog remains the dominant implementation, standardized in ISO/IEC 13211-1 since 1995, with extensions for constraint logic programming (e.g., CLP(FD) for finite domains) addressing numeric and optimization problems.[25] Other variants include Datalog for deductive databases and Mercury for higher performance via static analysis. Applications span artificial intelligence, including expert systems like those in medical diagnosis (e.g., MYCIN's successors), natural language parsing, and symbolic AI, though practical use has declined with the rise of statistical machine learning due to logic programming's sensitivity to combinatorial explosion in undecidable problems.[20] Despite limitations in efficiency for large-scale data, it excels in verifiable reasoning tasks, such as formal verification and planning.[21]