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PFA

The (PFA) is the representing current and former professional association footballers and scholars in , including those in the , FA Women's Super League, and English Football Leagues. Founded in 1907, it operates as the world's longest-established professional sports , advocating for members' rights through , legal representation, and industry influence on working conditions. The organization provides comprehensive support services encompassing education, financial advice, resources, and welfare assistance, while spearheading initiatives against racist abuse, for women's football advancement, and into neurodegenerative diseases linked to heading the ball. Notable achievements include negotiating improved contracts and benefits for players, establishing the annual PFA Awards in 1974 to recognize top performers voted by peers, and contributing to the abolition of the football in 1961 following early campaigns. However, the PFA has encountered controversies, including investigations into its charitable arm's and expenditure, criticisms of executive remuneration under long-serving chief Gordon Taylor, and public backlash over events like a comedian's routine at its awards ceremony perceived as offensive.

Chemistry and materials science

Perfluoroalkoxy

Perfluoroalkoxy (PFA) is a fully fluorinated formed as a of (TFE) and perfluoropropyl vinyl ether (PPVE). This structure introduces perfluoroalkyl side chains that lower crystallinity compared to (PTFE), enabling melt processing via injection molding, , or —methods incompatible with PTFE's rigid, non-melting nature. PFA maintains structural integrity at continuous service temperatures up to 260°C (500°F), with a of 305°C (581°F), surpassing many fluoropolymers in thermal endurance for prolonged exposure. It offers near-universal chemical inertness against acids, bases, and solvents, a coefficient of below 0.2, and exceeding 15 kV/mm, qualifying it under ASTM D3307 for high-purity molding and resins. These attributes stem from its fully perfluorinated backbone, which resists degradation in oxidative or reductive environments, as evidenced by minimal (<0.1% after 20,000 hours at 260°C). In semiconductor fabrication, PFA wafer carriers and fluid-handling components prevent ionic contamination, supporting yields in sub-10 nm processes by maintaining purity levels below 10 ppb total organic carbon. Chemical processing employs PFA-lined vessels and tubing, which endure aggressive media like hydrofluoric acid at elevated temperatures, extending service life to over 5 years versus 1-2 years for metal alternatives and cutting replacement costs by 30-50%. Medical applications utilize PFA in catheters and implants for its sterilizability and low extractables (<1 µg/cm²), ensuring compliance with USP Class VI standards without compromising flexibility or clarity. PFA's high molecular weight and polymeric stability confer low volatility (vapor pressure <10^{-10} torr at 25°C) and negligible leaching in fabricated parts, limiting exposure to perfluorinated fragments below detectable thresholds in aqueous simulants, which contrasts with mobile, low-molecular-weight PFAS and aligns with risk assessments emphasizing end-use containment over inherent persistence.

Paraformaldehyde

Paraformaldehyde (PFA) is a linear polymer of formaldehyde with the chemical formula HO(CH₂O)ₙH, where n typically ranges from 8 to 100, averaging around 30 units, resulting in a white crystalline powder with a pungent formaldehyde odor and density of approximately 1.46 g/cm³. It is commercially produced at 91-96% concentration by dehydrating aqueous solutions through controlled evaporation, vacuum distillation, or concentration processes that promote depolymerization-resistant polymerization, often accelerated by acid or base catalysts. This synthesis leverages formaldehyde's tendency to form polyoxymethylene precipitates in cold, concentrated solutions, a method refined from 19th-century discoveries but industrialized in the early 20th century for stable, solid storage and transport as a formaldehyde source. In biological applications, PFA serves primarily as a fixative for microscopy, histology, and cell studies, where it depolymerizes in aqueous buffers to release formaldehyde that cross-links proteins via methylene bridges between lysine residues, preserving tissue ultrastructure and antigens with minimal distortion compared to alcohol fixatives. Optimal protocols involve 4% PFA solutions prepared in phosphate-buffered saline (PBS) at pH 7.4, heated to 60-70°C with NaOH to aid dissolution, followed by filtration and use at room temperature or 4°C for 10-30 minutes depending on sample type, enabling techniques like immunofluorescence and electron microscopy. This cross-linking stabilizes cellular components against autolysis and osmotic damage, though over-fixation can mask epitopes, necessitating empirical titration for specific antigens. Industrially, PFA functions as a disinfectant, fungicide, and precursor for resins in adhesives, particleboard, and coatings, depolymerizing to gaseous formaldehyde for fumigation in settings like aquaculture and warehousing to control pathogens without residue. It is also used in hardening and , synthetic resins such as and melamine types, and as an antiseptic in dental materials, with global production supporting applications in automotive, paint, and construction sectors. Handling requires precautions due to depolymerization releasing formaldehyde vapor, with OSHA establishing a permissible exposure limit of 0.75 ppm as an 8-hour time-weighted average and 2 ppm short-term (15-minute) ceiling, based on empirical data linking concentrations above 0.5-1 ppm to sensory irritation in mucous membranes rather than solely carcinogenic risks at trace levels. Ventilation, gloves, and respirators are standard, with full-body protection for entries exceeding 100 ppm, prioritizing measured thresholds over broad hazard classifications.

Medicine

Pulsed field ablation

Pulsed field ablation (PFA) is a nonthermal cardiac ablation technique that employs ultra-short-duration, high-voltage electrical pulses to induce in targeted myocardial cells, primarily for treating . IRE disrupts cell membrane integrity through nanoscale pore formation, leading to cardiomyocyte apoptosis while exploiting tissue selectivity: cardiomyocytes are more susceptible due to their membrane composition and field orientation, sparing adjacent structures such as the esophagus, phrenic nerve, and coronary arteries that lack similar dielectric properties. This contrasts with thermal methods like or , which risk collateral damage from heat or ice. Conceptual development of PFA stemmed from electroporation research in the early 2010s, with first-in-human trials emerging by 2020, including the demonstrating 100% acute pulmonary vein isolation (PVI). Regulatory milestones followed, with the U.S. Food and Drug Administration (FDA) approving Medtronic's on December 13, 2023, for paroxysmal and persistent AF, and Boston Scientific's on January 31, 2024, initially for paroxysmal AF with later expansion. Systems from Abbott and Johnson & Johnson have since gained approvals or expansions by 2025. Clinical trials from 2024-2025, including presentations at the Heart Rhythm Society meetings, report procedure durations of 40-70 minutes, shorter than comparable thermal ablations (e.g., 70 vs. 100 minutes in head-to-head comparisons), with acute PVI success rates exceeding 95-99% in paroxysmal AF cohorts. Safety profiles show reduced major complications, such as pericardial effusion rates below 2-5% (often reversible and lower than 5-10% in thermal methods), with primary safety endpoints around 2-5% across large registries like MANIFEST-17KF involving over 17,000 patients. The ADVENT trial confirmed noninferiority to thermal ablation for 1-year freedom from arrhythmia recurrence in paroxysmal AF, at approximately 74-87%. Limitations include variable lesion durability, particularly in persistent AF, where 12-month freedom from recurrence ranges from 55-64%, implying 36-45% recurrence rates in some cohorts, potentially due to incomplete posterior wall isolation or reconnection observed in redo procedures (durability ~87% for PVI but lower elsewhere). Transient skeletal muscle stimulation can occur, complicating intraprocedural monitoring, while upfront costs exceed $10,000-13,000 per procedure, driven by specialized generators and catheters, though offset by efficiency gains. Ongoing debates center on the need for larger randomized trials versus established thermal modalities to verify long-term superiority, as early data show comparable efficacy but highlight gaps in persistent AF outcomes and real-world durability.

Computing and mathematics

Computer science

A probabilistic finite automaton (PFA) is a computational model that extends the deterministic finite automaton by associating probability distributions with transition functions, enabling the representation of stochastic processes over finite state spaces. In this framework, from each state and input symbol, transitions to next states occur according to a probability vector that sums to 1, allowing PFAs to generate probability distributions over strings rather than deterministic acceptance. This stochastic extension facilitates modeling of systems with inherent uncertainty, such as sequences in natural language or biological data, where outcomes are not fixed but follow probabilistic patterns. The theoretical foundations of PFAs trace back to extensions of classical automata theory, with the formal introduction of probabilistic variants occurring in 1963 by Michael O. Rabin, building on nondeterministic finite automata from the 1950s. PFAs differ from hidden Markov models (HMMs) by lacking hidden states and output emissions, instead directly computing acceptance probabilities via matrix products of transition probabilities, which supports efficient computation for identifiable models. Convergence guarantees for learning PFAs from data rely on conditions like linear independence of Hankel matrices derived from observed sequences, ensuring parameter recovery in polynomial time under sufficient sample sizes. Practical implementations emphasize spectral learning algorithms, which estimate PFA parameters by decomposing empirical moment matrices from training data, avoiding local optima common in expectation-maximization methods used for HMMs. These algorithms, such as those based on singular value decomposition of prefix-suffix statistics, achieve provable efficiency for weighted automata approximations, with applications in approximating HMMs for sequence prediction tasks. In machine learning, PFAs have demonstrated empirical advantages in domains like speech recognition, where they model phonetic transitions, and bioinformatics for sequence alignment, outperforming baselines on datasets with variable-length dependencies as reported in studies from the early 2000s onward. Modern uses extend to time-series forecasting in natural language processing benchmarks, leveraging PFA learnability for scalable inference on large corpora.

Mathematics and statistics

A probabilistic finite automaton (PFA) is a stochastic extension of the deterministic finite automaton, incorporating probability distributions over state transitions to model sequences with inherent randomness. Formally, it is defined as a 5-tuple (Q, \Sigma, q_0, \delta, F), where Q is a finite set of states, \Sigma is the input alphabet, q_0 \in Q is the initial state, \delta: Q \times \Sigma \to \Delta(Q) maps each state-symbol pair to a probability distribution \Delta(Q) over next states (satisfying \sum_{q' \in Q} \delta(q, a)(q') = 1 for all q \in Q, a \in \Sigma), and F \subseteq Q is the set of accepting states. The model induces a Markov chain for each input string, where the probability of transitioning along a path is the product of individual transition probabilities, enabling computation of acceptance probability as the expected measure over paths ending in F. In probability theory, PFAs formalize stochastic languages, with acceptance defined threshold-wise (e.g., probability exceeding $1/2) or via expectation operators for moments of path probabilities. Theoretical advancements since the 1990s establish learnability for restricted classes, such as acyclic PFAs, where parameters are identifiable from sufficient samples via spectral methods or expectation-maximization, recovering transition matrices up to permutation equivalence under full-rank assumptions on the observability matrix derived from Hankel representations. Asymptotic consistency of estimators follows from ergodic theorems for hidden Markov models (to which PFAs generalize), ensuring parameter convergence in probability as sample size grows, provided the chain is irreducible and aperiodic on the state graph. Unlike deterministic finite automata, which yield binary outcomes, PFAs introduce variance across realizations due to stochastic transitions, quantified by the variance of the acceptance indicator under the induced measure; this necessitates probabilistic distinguishability conditions, such as total variation distance exceeding a margin for language separation. Error bounds for estimation leverage concentration inequalities, like Hoeffding's for bounded martingales in state probability trajectories, yielding exponential tails on deviation probabilities (e.g., P(|\hat{p} - p| > \epsilon) \leq 2\exp(-2n\epsilon^2) for n samples and acceptance probability p). simulations approximate these but inherit variance scaling as O(1/\sqrt{n}), underscoring the need for theoretical guarantees over empirical variance in high-dimensional state spaces.

Sports

Professional Footballers' Association

The (PFA) functions as the for professional association players in , advocating for , welfare, and contractual protections. Founded on 2 1907 as the Association Football Players' and Trainers' Union, it evolved into its current form to represent current and former players across the , Football League, and women's leagues, with over 5,000 active members as of recent records. The organization has prioritized campaigns for standards in lower divisions, contract stability, and post-career transitions, including the abolition of the Football League's maximum wage cap of £20 per week in 1961 under chairman , which empowered individual negotiations. Through agreements with leagues and clubs, the PFA has secured benefits such as comprehensive accident and sickness insurance covering all contracted and Football League players, alongside an fund administered via its Personal Development Department, offering up to £7,500 lifetime per member for accredited qualifications to support career diversification. The has also facilitated enhanced player mobility following the 1995 , which eliminated transfer fees for out-of-contract players, aligning with PFA advocacy for free agency and contributing to market liberalization that increased earning potential. Notable activities include the annual PFA Players' Player of the Year awards, established for the 1973–74 season and determined by member votes, recognizing top performers across divisions. In the 2010s onward, the PFA expanded support, delivering club-based workshops, discreet assessments, and a confidential 24/7 to address performance pressures and transitions. While these efforts have bolstered player security and mobility, the PFA has faced criticism for prioritizing wage protections that exacerbate financial pressures on clubs, particularly through opposition to salary caps aimed at curbing inflation. In 2021, the PFA successfully appealed the EFL's proposed s for and Two, deeming them rushed and lacking consultation, which delayed implementation and preserved higher spending thresholds. More recently, in 2025, the PFA threatened legal action against a potential hard for the 2025–26 season, arguing it would infringe on player rights, amid debates that such resistance hinders sustainability amid rising wage-to-revenue ratios and club debts exceeding £1 billion collectively in recent analyses. Detractors contend this stance enables underperforming players to leverage contract clauses for transfer refusals, inflating squad costs and contributing to fiscal instability, as evidenced by clubs' reported losses despite revenue growth. Historical disputes, including 1970s negotiations over benefits amid broader union tensions, underscored similar strains, though the PFA's membership growth to historic highs reflects sustained player support for its protective role.

Law and protection

Protection from abuse

Protection from abuse (PFA) orders serve as civil remedies primarily enacted through statutes to safeguard victims of , with Pennsylvania's Protection From Abuse Act of 1976 providing a foundational example by authorizing courts to issue protective orders against individuals who cause bodily injury, place another in reasonable fear of imminent serious bodily injury, or engage in or within domestic relationships. Abuse under such laws typically encompasses physical harm, attempts or threats thereof, , or patterns of coercive control, excluding mere verbal arguments absent fear of harm. Procedures for obtaining a PFA begin with a petitioner's filing, often enabling an temporary order—issued without notifying or hearing from the respondent—which can take effect immediately upon , typically within hours, and remains in force until a full hearing scheduled within 10 business days. At the hearing, the petitioner bears the burden of proof by a preponderance of the evidence standard, meaning the claims must be deemed more likely true than not, after which courts may grant final orders lasting up to three years, incorporating remedies such as no-contact provisions, exclusive possession of the residence, temporary custody arrangements, and mandatory relinquishment of firearms under integrations with federal laws like the (VAWA) and the Lautenberg Amendment. Enforcement outcomes reveal mixed effectiveness, with meta-analyses indicating protection orders reduce and violation rates in cases, particularly when combined with arrests, though victim-reported violations exceed police-recorded ones, suggesting underreporting influences perceived . Compliance with order terms hovers around 60-80% in monitored scenarios, yet persists at rates up to 40% within six months post-order, per Department of Justice-linked studies on , highlighting limitations in deterrence amid ongoing offender access to victims. Critics argue emergency orders erode by preempting respondent input, potentially leading to erroneous restrictions without adversarial testing, while statistics suffer from underreporting biases—victims often fail to disclose due to fear or of system efficacy—skewing data toward apparent order and masking true re-abuse prevalence.

Communications and business

Please find attached

PFA serves as an for "Please Find Attached," a phrase employed in communication to notify recipients of an included or . This shorthand originated in the expansion of digital during the late and early , supplanting lengthier expressions to enhance brevity amid rising email volumes in corporate environments. Its adoption reflects adaptations in business writing conventions, where acronyms like PFA streamline interactions without sacrificing clarity, particularly in global English-speaking contexts. Standard usage places PFA near the email's closing, often followed by a brief description of the attachment for context, such as "PFA: Quarterly financial report.xlsx," to facilitate quick recipient comprehension. Alternatives include direct references like "attached is" or "see attached," which prioritize even greater conciseness while maintaining ; PFA remains prevalent in formal or international exchanges but is less common in non-digital or casual formats. Best practices recommend verifying attachment inclusion before sending, as the abbreviation assumes successful transmission, and pairing it with specific file details mitigates in threaded or high-volume inboxes. Efficiency gains from PFA include reduced typing and in routine correspondence, contributing to faster in settings; however, it carries risks of oversight, with email analyses indicating attachments are frequently missed or fail to open due to compatibility issues or in busy professional streams. Critiques note its potential over-formality in informal teams and vulnerability to attachment-related errors, prompting shifts toward secure sharing in platforms to bypass file transmission pitfalls altogether.

Other uses

Physical fitness assessment

The Physical Fitness Assessment (PFA) in the United States Navy evaluates service members' overall physical readiness through a combination of body composition measurement and performance-based exercises, aiming to ensure combat effectiveness and health maintenance. Conducted semi-annually, it includes a Body Composition Assessment (BCA) via circumference measurements or dual-energy X-ray absorptiometry, followed by the Physical Readiness Test (PRT) comprising push-ups for upper-body endurance, a forearm plank for core stability, and a cardiovascular option such as a 1.5-mile run, 500-yard swim, or 2,000-meter row. These elements were revised in the early 2020s, with the plank introduced in 2021 to replace curl-ups, emphasizing sustained muscular contraction over repetitive motion to better align with injury prevention. Scoring follows age- and gender-specific standards, with points awarded per event to yield a composite total; passing requires at least 50 points overall and minimum thresholds per component to avoid remedial programs. For males aged 20-24, minimums include 15 push-ups in two minutes, a 0:50 forearm plank, and a 13:45 1.5-mile run time, while maximums reach 87 push-ups, a 3:20 plank, and an 8:30 run. Females in the same group face adjusted minima like 8 push-ups, a 0:50 plank, and a 15:35 run, reflecting physiological differences. The assessment's purpose extends beyond pass/fail to establishing health baselines, with higher proficiency linked to operational resilience; military studies demonstrate that elevated fitness levels correlate with 30-50% reductions in training-related musculoskeletal injuries, as fitter individuals tolerate physical stressors with lower fatigue and tissue strain. Despite these benefits, PFAs face scrutiny for limited predictive validity in diverse job roles, with meta-analyses showing modest correlations (r=0.3-0.5) between test scores and task-specific performance like load carriage or simulations, potentially overlooking functional strength in favor of metrics. Age and gender norms mitigate claims but introduce debates on , as larger body types—common in strength-demanding roles—may score lower due to biomechanical disadvantages in events like push-ups, despite greater lean mass aiding real-world tasks. Proponents advocate shifting toward occupationally tailored functional tests, such as those emphasizing power output over timed reps, to enhance job relevance without compromising readiness screening. Similar protocols appear in occupational settings like corporate or hiring, adapting military-derived metrics for baseline and absenteeism reduction, though evidence for non-military efficacy remains sparser.

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