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Contraction

A contraction is the action or process of becoming smaller, shorter, or narrower in size, length, or volume, often resulting from external influences such as temperature changes, motion, or biological signals.^[1] This fundamental concept manifests in diverse disciplines, including physics, where it describes dimensional changes in materials or objects; physiology, involving the activation of muscle fibers; linguistics, referring to the shortening of words in speech and writing; and economics, denoting a decline in economic activity.^[2] Each context highlights unique mechanisms and implications, underscoring contraction's role in natural and human systems. In physics, contraction commonly arises from thermal effects, where most materials decrease in volume when cooled due to reduced atomic vibrations and increased intermolecular forces.^[3] For instance, metals like steel exhibit linear contraction proportional to the temperature drop, quantified by the coefficient of thermal expansion, which is typically on the order of 10^{-5} to 10^{-6} per degree Celsius for solids.^[3] In special relativity, length contraction occurs when an object's length, as measured in a frame moving relative to the observer, shortens along the direction of motion by a factor of \sqrt{1 - v^2/c^2}, where v is the relative speed and c is the speed of light; this effect becomes significant only at velocities approaching c.^[4] In physiology, muscle contraction is the process by which muscle fibers generate tension through the sliding of actin and myosin filaments, powered by ATP hydrolysis and triggered by nerve impulses at the neuromuscular junction.^[5] This mechanism enables movement, posture maintenance, and vital functions like heartbeat in cardiac muscle or peristalsis in smooth muscle; contractions are classified as isometric (no length change, tension increases) or isotonic (length changes under constant tension).^[5] Disruptions in this process, such as in myasthenia gravis, can impair force generation and lead to weakness.^[5] Linguistically, a contraction is a reduced form of one or more words, created by omitting internal letters or sounds and often using an apostrophe, as in "don't" for "do not" or "I'm" for "I am."^[6] Common in informal English speech and writing, contractions enhance fluency and are governed by phonological rules; for example, negative contractions like "won't" blend "will not" irregularly.^[7] They appear across languages but are particularly prevalent in English, where over 100 standard forms exist, though overuse in formal contexts is avoided.^[7] In economics, a contraction is a phase of the business cycle characterized by declining economic output, rising unemployment, and reduced consumer spending, typically following a peak and preceding a trough.^[8] Lasting from several months to over a year, contractions are measured by indicators like negative GDP growth for two consecutive quarters (defining a recession); notable examples include the 2008 financial crisis, which saw U.S. GDP contract by 4.3%.^[8] Central banks often respond with monetary easing to mitigate severity and duration.^[8]

Linguistics

Grammatical Contractions

In linguistics, a contraction is a shortened form of one or more words, typically created by combining elements such as a pronoun or auxiliary verb with a negation or another verb, through the process of elision, where letters and sounds are omitted, and an apostrophe is used to indicate the missing elements.^[9]^[10] This orthographic convention helps maintain readability while reflecting spoken reductions, distinguishing contractions from full forms in both writing and speech.^[11] Common examples in English illustrate this formation. "I'm" contracts "I am" by eliding the "a" and using the apostrophe to mark its place, resulting in a phonetic reduction where the vowel sound weakens. Similarly, "don't" shortens "do not" by omitting the "o," pronounced as a single word with blended sounds; "it's" represents "it is" or "it has," replacing the "i" or "ha" with an apostrophe; and "won't," an irregular contraction of "will not," drops the "i" and "l," often with a further reduction in the vowel quality for smoother utterance.^[10]^[9] These examples highlight how contractions typically involve auxiliary verbs or negations, pronounced fluidly as unified units. Formation rules extend to irregular cases, such as "o'clock" from "of the clock," which elides "f the" without a strict apostrophe placement, or "gov't" for "government," shortening a single word through partial omission.^[10] Unlike abbreviations, which shorten standalone words (e.g., "Dr." for "doctor" and read letter-by-letter), contractions fuse multiple words into a single pronounced entity, emphasizing their role in informal discourse.^[10] The apostrophe's use in English contractions evolved from Middle English practices, where no such marker existed for genitives or elisions marked by endings like -es; by the mid-16th century, printers introduced the apostrophe as a visual cue for omissions, standardizing its role in contractions by the 17th century amid inconsistent early applications.^[12]^[11] In terms of prosody and phonetics, contractions influence stress patterns by destressing auxiliaries, reducing full vowels to schwa (e.g., /hæz/ to [əz] in "has"), and altering intonation through decreased junctures in casual speech, which enhances fluency but can be blocked by pitch accents or phrase boundaries.^[13]

Contractions in Different Languages

In Romance languages, contractions often involve the fusion of prepositions with definite articles or pronouns, a process governed by phonological rules that prevent vowel hiatus or simplify articulation. In French, elision occurs when a word ending in a vowel precedes one beginning with a vowel or mute h, resulting in forms like l'eau (from la eau, "the water") or j'ai (from je ai, "I have"), while true contractions include du (from de le, "of the") and au (from à le, "to the"). These are mandatory in both spoken and written registers to maintain euphony.^[14] In Spanish, contractions are limited but obligatory for prepositions with the masculine singular article el, yielding al (from a el, "to the") as in voy al cine ("I go to the cinema") and del (from de el, "of the") as in el libro del autor ("the book of the author"); no such fusion occurs with feminine articles like la. This cliticization reflects a syntactic attachment of prepositions to nouns, differing from English by being non-optional and morphologically fixed.^[15]^[16] Germanic languages exhibit contractions primarily through phonological reduction of function words in casual speech and dialects, rather than systematic orthographic mergers. In German, adjacent function words like prepositions or negation particles form trochaic feet that trigger reductions, such as wenn's (from wenn es, "if it") or dialectal shortenings of nicht to nich, net, or nit in northern and southern varieties, respectively; these reflect prosodic grouping but are avoided in standard written forms. Separable verb prefixes, like auf- in aufmachen ("to open"), contrast with true contractions by remaining distinct in composition. In Dutch, elisions and cliticizations appear in informal speech, including 't (from het, "the/it") before vowels, as in 't huis ("the house"), or m'n (from mijn, "my"); historical Middle Dutch texts show metrical cliticisation leading to orthographic contractions in verse, influencing modern casual reductions. These processes prioritize rhythmic flow over the auxiliary-verb mergers common in English.^[17]^[18]^[19] Other language families demonstrate contractions via clitic attachment or prosodic adjustments without always altering orthography. In Arabic, a Semitic language, clitics like the conjunction wa- ("and") proclitically attach to following words, as in wa-synhy ("and he swims"), forming a single phonological unit; this is part of a templatic morphology where conjunctions, particles, and articles prefix to stems, increasing word complexity but aiding cohesion in diglossic contexts. Mandarin Chinese, a Sino-Tibetan tonal language, lacks orthographic contractions but features tone sandhi—a spoken phonological alternation analogous to contraction—where a third tone (low-dipping) shifts to second tone (rising) before another third tone, as in hǎo hǎo becoming hǎo hào ("good good"); this eases prosodic flow without written changes, differing from alphabetic elisions.^[20]^[21] Historically, contractions in colonial languages like French have evolved in sociolinguistic contexts, with African varieties showing heightened elision due to substrate influences from local languages, as seen in phonetic reductions in Kinshasa French where vowel elisions mark informal urban speech. Across languages, formality levels influence usage: contractions and elisions are typically omitted in formal writing to preserve clarity and prestige, as in standard German or Spanish academic texts, where full forms signal professionalism. In translation, English contractions like can't (cannot) pose challenges in languages without elision, such as Mandarin or Arabic, often requiring periphrastic expansions (e.g., bù néng in Mandarin) to convey negation without losing nuance, leading to longer target texts and potential shifts in tone.^[22]^[23]

Mathematics and Logic

Contraction Mappings

In mathematics, a contraction mapping, also known as a contractive mapping, is a function f: X \to X defined on a metric space (X, d) that satisfies the inequality d(f(x), f(y)) \leq k \, d(x, y) for all x, y \in X, where k is a constant with $0 \leq k < 1, referred to as the Lipschitz constant or contraction constant.^[24] This condition ensures that the mapping reduces distances between points by a factor strictly less than 1, making it a special case of a Lipschitz continuous function with Lipschitz constant less than 1. The concept of contraction mappings originated in the work of Stefan Banach, who introduced it in his 1922 doctoral thesis as part of broader studies in functional analysis and integral equations.^[24] Banach's formulation laid the groundwork for fixed-point theory in abstract spaces, emphasizing applications to solving nonlinear equations through iterative methods.^[25] A fundamental result associated with contraction mappings is the Banach fixed-point theorem, which states that if (X, d) is a complete metric space and f: X \to X is a contraction mapping with constant k, then f has a unique fixed point p \in X such that f(p) = p.^[24] Moreover, for any initial point x_0 \in X, the sequence defined by x_{n+1} = f(x_n) converges to p, and the error is bounded by

d(x_n, p) \leq \frac{k^n}{1 - k} \, d(x_0, x_1)

for all n \geq 1, providing an explicit rate of convergence that depends on k.^[24] This theorem guarantees not only existence and uniqueness but also an efficient constructive method to approximate the fixed point via Picard iteration. Contraction mappings possess several key properties: they are continuous, as the contraction condition implies uniform continuity with modulus of continuity controlled by k, and they preserve completeness in the sense that the image of a complete subspace remains complete under certain conditions. Extensions to quasi-contractions, introduced by Ljubomir Ćirić, relax the strict k < 1 requirement by allowing k \leq 1 alongside additional inequality conditions on distances, such as d(f(x), f(y)) \leq k \, d(x, y) + \frac{c}{1 + d(x, y)} for some c > 0, which still ensure fixed points in complete metric spaces under suitable hypotheses. Classic examples illustrate the utility of contraction mappings. On the real line \mathbb{R} with the standard metric, the function f(x) = x/2 is a contraction with k = 1/2, and its unique fixed point is $0, reached by iterating from any starting point with linear [convergence](/page/Convergence) at rate 1/2.[24] Another application arises in solving [the equation](/page/The_Equation) x = \cos x, where the [mapping](/page/Mapping) f(x) = \cos xacts as a contraction on an appropriate closed interval like[-1, 1](withk = \sin 1 \approx 0.841

), allowing successive approximations to converge to the [Dottie number](/page/Dottie_number), approximately &#36;0.739085

.^[26] These examples highlight how the theorem facilitates numerical solutions to nonlinear problems in analysis and beyond.

Contraction Rule in Logic

In sequent calculus, the contraction rule is a structural rule that permits the reduction of duplicate formulas on the same side of a sequent, allowing multiple instances of the same formula in the antecedent or succedent to be replaced by a single instance while preserving validity.^[27] Specifically, it transforms a sequent of the form \Gamma, A, A \vdash \Delta into \Gamma, A \vdash \Delta, or analogously in the succedent, ensuring that proofs remain sound by eliminating redundant resources without altering the logical consequence.^[28] This rule embodies the principle that a conclusion derivable from multiple copies of a premise is also derivable from a single copy.^[27] In classical and intuitionistic logic, the contraction rule forms one of the core structural rules, alongside weakening (which adds extraneous formulas) and exchange (which permutes formulas), enabling the full expressiveness of standard deductive systems.^[29] Its presence allows assumptions to be reused freely, mirroring intuitive reasoning where premises are not consumed after use; without it, systems become substructural, such as linear logic, where resources are tracked linearly to prevent duplication and support applications in concurrency and computation.^[30] For instance, in Gentzen's LK (classical) and LJ (intuitionistic) calculi, contraction ensures cut-elimination and normalization, foundational for proving consistency and completeness. In natural deduction systems, contraction is often implicit, permitting the reuse of assumptions within a proof tree, as seen in derivations of transitivity for implication: to prove (A \to B) \to ((B \to C) \to (A \to C)), the assumption A \to B is applied twice—once to derive B from A, and again in the outer scope—relying on contraction to avoid resource exhaustion.^[31] This reuse facilitates modular proofs but can complicate termination in automated theorem proving if unchecked.^[32] The implications of contraction extend to variant logics where it is restricted to enforce relevance or resource sensitivity. In relevance logic, full contraction is curtailed to prevent derivations from irrelevant premises, ensuring that antecedents genuinely contribute to consequents; for example, affine variants allow weakening but limit contraction to avoid paradoxes like Curry's.^[33] Curry's lemma addresses internalizing limited contraction in modal logics, showing that height-preserving admissibility of contraction can be derived within contraction-free systems via modal operators, thus embedding stronger logics into weaker ones without explicit duplication rules.^[32] This lemma, rooted in work by Haskell Curry, highlights how contraction can be "tamed" for decidability and complexity control in proof search.^[34] Historically, the contraction rule was formalized by Gerhard Gentzen in his 1934 investigation of logical inference, where sequent calculus was introduced to analyze natural deduction and achieve cut-elimination, revealing contraction's role in structural completeness. Mid-20th-century proof theory debates, influenced by substructural systems, explored contraction-free alternatives for bounded proof lengths and termination guarantees, paving the way for linear and relevance logics in the 1980s.^[29]

Medicine

Muscle Contractions

Muscle contractions refer to the process by which muscle fibers shorten or generate tension in response to neural stimuli, enabling movement, posture maintenance, and vital functions across skeletal, cardiac, and smooth muscle types. This physiological phenomenon is fundamental to locomotion and organ function, involving intricate interactions between contractile proteins and regulatory mechanisms. The primary model explaining this process is the sliding filament theory, which describes how overlapping actin and myosin filaments slide past each other to produce force.^[35] The mechanism of muscle contraction follows the sliding filament theory, first proposed by Hugh Huxley and Jean Hanson in 1954 based on observations of sarcomere changes during shortening. In this model, myosin heads bind to actin filaments after hydrolyzing ATP, which provides the energy for the power stroke that pulls the actin filaments toward the center of the sarcomere, resulting in overall muscle shortening. Calcium ions play a critical role by binding to troponin, a regulatory protein on the actin filament, which shifts tropomyosin to expose myosin-binding sites and initiate cross-bridge formation.^[5]^[35] Muscle contractions are classified into several types based on changes in length and load. Isotonic contractions involve muscle shortening or lengthening against a constant load, such as lifting a weight (concentric) or lowering it controlled (eccentric), allowing joint movement. Isometric contractions generate tension without a change in muscle length, as seen in holding a posture against gravity, where force matches the load but no shortening occurs. Isokinetic contractions maintain a constant speed throughout the range of motion under varying loads, typically requiring specialized equipment like dynamometers for rehabilitation or training.^[5]^[36] Neural control of muscle contractions begins with action potentials from motor neurons that innervate motor units, groups of muscle fibers activated synchronously to produce graded force via recruitment of additional units. This process, known as excitation-contraction coupling, links the electrical signal at the neuromuscular junction to mechanical response: acetylcholine release depolarizes the muscle membrane, triggering calcium release from the sarcoplasmic reticulum in skeletal muscle, while cardiac and smooth muscles involve distinct calcium handling, such as influx from extracellular sources in cardiac tissue or phosphorylation in smooth muscle. In skeletal muscle, the coupling is rapid and voluntary, whereas cardiac contractions are autorhythmic and smooth ones are often involuntary, modulated by hormones or autonomic nerves.^[36]^[5]^[37] Energy for muscle contractions is primarily supplied by ATP, resynthesized through three interconnected systems: the phosphagen system using creatine phosphate for immediate high-intensity bursts lasting seconds; anaerobic glycolysis for short-term efforts up to a few minutes, producing ATP but yielding lactate; and oxidative phosphorylation in mitochondria for sustained aerobic activity, relying on oxygen and fats or carbohydrates. Muscle fatigue arises from factors including ATP depletion, accumulation of lactic acid (lowering pH and impairing cross-bridge cycling), and ionic imbalances, limiting force production during prolonged or intense efforts.^[38]^[39] Disorders affecting muscle contractions include tetanus, caused by Clostridium tetani toxin blocking inhibitory neurotransmitters, leading to sustained, involuntary contractions (rigidity and spasms) due to unopposed motor neuron activity. Myasthenia gravis, an autoimmune condition, impairs contraction by antibodies targeting acetylcholine receptors at the neuromuscular junction, resulting in fatigable weakness and reduced force generation in skeletal muscles. These conditions highlight vulnerabilities in neural-muscle coupling and protein interactions central to normal contraction.^[40]^[41]

Uterine Contractions

Uterine contractions are rhythmic contractions of the smooth muscle in the myometrium, the middle layer of the uterus, essential for processes such as menstruation and labor. These contractions are triggered by oxytocin, a hormone released from the posterior pituitary gland, which binds to G-protein-coupled receptors on myometrial cells. This binding activates phospholipase C, leading to the production of inositol trisphosphate (IP3) and the release of calcium from the sarcoplasmic reticulum, causing membrane depolarization and subsequent contraction through activation of myosin light chain kinase.^[42] The electrical signals propagate across myometrial cells via gap junctions, specialized protein channels that increase in number toward the end of pregnancy, enabling synchronized and coordinated uterine activity.^[42] During labor, the frequency and strength of uterine contractions progressively increase to facilitate cervical dilation and fetal expulsion. In early labor, contractions occur every 5 to 20 minutes and are relatively mild, but as labor advances, they become stronger and occur every 2 to 3 minutes, with intrauterine pressures rising from about 20 mmHg to 50-80 mmHg.^[42]^[43] Prior to true labor, Braxton-Hicks contractions serve as practice contractions; these are irregular, sporadic tightenings of the uterus that do not lead to cervical changes and are typically non-painful, though they can become uncomfortable in later pregnancy.^[42]^[44] Labor is divided into stages characterized by distinct patterns of uterine contractions. The first stage encompasses early (latent) labor, where mild contractions every 5 to 20 minutes cause initial cervical effacement and dilation up to 6 cm, often lasting several hours or more.^[45] Active labor follows, with stronger, more regular contractions every 2 to 5 minutes driving rapid dilation from 6 cm to 10 cm, typically at a rate of 1-2 cm per hour.^[45] The transition phase marks the end of the first stage, featuring intense contractions every 2 to 3 minutes that complete dilation to 10 cm and prepare for pushing in the second stage.^[45] Monitoring uterine contractions during labor is crucial for assessing progress and fetal well-being. Tocodynamometry, often using external strain gauges (toco), primarily measures contraction frequency and duration, though internal intrauterine pressure catheters (IUPC) provide more accurate intensity data when needed, such as in cases of obesity or ambiguous external readings.^[46] Fetal heart rate monitoring is correlated with contraction patterns to detect distress; variable decelerations during contractions may indicate cord compression, while late decelerations suggest uteroplacental insufficiency.^[46] Medical interventions target uterine contractions to manage preterm labor or augment stalled progress. Tocolytics, such as magnesium sulfate administered intravenously, relax uterine smooth muscle by competing with calcium for binding sites, delaying preterm delivery and allowing time for corticosteroid administration to enhance fetal lung maturity.^[47]^[48] Historically, ergot alkaloids, derived from the fungus Claviceps purpurea, have been used since the 19th century to stimulate uterine contractions and control postpartum hemorrhage, with early applications noted as far back as 1582 for inducing labor.^[49] Complications related to uterine contractions include preterm labor, defined as regular contractions before 37 weeks' gestation, affecting approximately 10% of pregnancies and leading to preterm birth in many cases.^[50] Prostaglandin administration for labor induction can cause uterine hyperstimulation, characterized by excessive contractions (more than five in 10 minutes or lasting over two minutes), potentially reducing fetal oxygenation and necessitating intervention.^[51]

Other Uses

Length Contraction in Physics

Length contraction is a fundamental consequence of special relativity, describing how the length of an object moving at relativistic speeds appears shortened to an observer at rest relative to that object. This effect arises from the Lorentz transformations, which relate space and time coordinates between inertial frames moving at constant velocity relative to each other. In Albert Einstein's seminal 1905 paper, length contraction emerges as part of the kinematic framework ensuring the invariance of the speed of light for all observers.^[52] The contracted length L of an object is given by L = L_0 / \gamma, where L_0 is the proper length measured in the object's rest frame, v is the relative velocity, c is the speed of light, and the Lorentz factor is \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}. This shortening occurs only along the direction parallel to the motion; dimensions perpendicular to the velocity remain unchanged. For instance, consider a rod of proper length L_0 = 1 m moving at v = 0.8c; here, \gamma \approx 1.667, so the observed length is L \approx 0.6 m.^[52] The derivation follows from applying the Lorentz transformations to determine the positions of the ends of the object at the same time in the observer's frame. Consider a rod at rest in frame S, with ends at x = 0 and x = L_0. In frame S' (moving at velocity v relative to S), the position of the left end (x = 0) at t' = 0 is x' = 0. For the right end (x = L_0), set t' = 0: t' = \gamma (t - v x / c^2) = 0 implies t = v L_0 / c^2. Then, x' = \gamma (L_0 - v t) = \gamma (L_0 - v^2 L_0 / c^2) = \gamma L_0 (1 - v^2 / c^2) = L_0 / \gamma. Thus, the length in S' is L_0 / \gamma. This ensures consistency with the relativity postulates, preserving Maxwell's equations across frames.^[52] A common apparent paradox, known as the pole-barn paradox, illustrates the implications: a pole longer than a barn appears to fit inside when moving rapidly due to contraction, but from the pole's frame, the barn contracts instead. The resolution lies in the relativity of simultaneity—events simultaneous in one frame (e.g., both barn doors closing while the pole is inside) are not simultaneous in the other, preventing any physical contradiction. Length contraction thus applies exclusively to lengths parallel to the motion, maintaining frame-invariant physical laws.^[53] Experimental confirmation comes from cosmic ray muons, which decay rapidly (proper lifetime about 2.2 μs) but reach Earth's surface in greater numbers than expected classically. In the 1962 experiment by David H. Frisch and James H. Smith, muon flux was measured at sea level and atop Mount Washington (elevation 1917 m); the observed rate matched predictions incorporating both time dilation and length contraction, with muons traveling at ~0.995c experiencing an effective time dilation factor of approximately 9 (corresponding to a path contraction factor of ~9 in their frame).^[54] Similar effects are routinely observed in particle accelerators, where short-lived particles like muons and pions travel distances far exceeding their proper decay lengths, consistent with relativistic predictions since the early 20th century validations of Einstein's postulates.^[55] Length contraction is intrinsically linked to time dilation, both deriving from the same Lorentz transformations, but while time dilation affects temporal intervals, contraction is a spatial effect observable in the relative motion of rigid bodies or particle paths. This duality underscores the spacetime unity in special relativity, where neither effect is absolute but relative to the observer's frame.^[52]

Economic Contraction

An economic contraction refers to a phase in the business cycle characterized by a sustained decline in economic activity, typically marked by two consecutive quarters of negative real gross domestic product (GDP) growth, although this is a common rule of thumb rather than an official definition.^[8]^[56] During this period, key indicators such as rising unemployment rates, falling industrial production, and reduced consumer spending signal broader economic slowdowns.^[57] Contractions are integral to business cycles, alternating with expansions, and can vary in duration and severity depending on underlying triggers.^[58] The primary causes of economic contractions include demand shocks, such as financial crises that erode consumer and investor confidence, and supply disruptions, like oil embargoes that increase production costs and constrain output.^[59] Policy errors, including overly tight monetary restrictions or fiscal austerity, can exacerbate these issues.^[60] From a Keynesian perspective, contractions often stem from insufficient aggregate demand, where sticky prices and wages lead to surpluses in labor and goods markets, amplifying downturns through multiplier effects.^[61] Monetarists, in contrast, emphasize failures in money supply management, arguing that central banks' reluctance to expand liquidity during crises deepens contractions, as seen in historical banking panics.^[60] Notable examples illustrate the scope of contractions. The Great Depression (1929–1939) represented an extreme case, with U.S. real GDP falling by approximately 30 percent from peak to trough and unemployment peaking at 25 percent in 1933, driven by stock market collapse, bank failures, and international trade barriers.^[57]^[62] The 2008 Great Recession, triggered by the burst of a housing bubble and subprime mortgage defaults, led to a sharp credit freeze and global financial turmoil, with U.S. GDP contracting by about 4.3 percent and unemployment rising to 10 percent.^[63] The COVID-19 contraction in 2020, induced by pandemic lockdowns and supply chain breakdowns, caused a global GDP decline of around 3.4 percent, the steepest since World War II.^[64] Contractions yield significant consequences, ranging from mild recessions—short-lived downturns with moderate job losses—to severe depressions marked by prolonged stagnation and widespread hardship.^[65] In recessions, employment effects include cyclical unemployment spikes, reduced wages, and income inequality, while depressions amplify these through deflationary spirals and asset devaluation.^[66] For instance, the Great Depression's 25 percent unemployment rate devastated households, leading to mass foreclosures and social unrest.^[62] Policymakers respond to contractions through fiscal and monetary measures to stabilize activity. Fiscal stimulus, such as the New Deal programs in the 1930s, involved public works and relief spending to boost demand and employment, though their Keynesian impact was debated amid balanced-budget constraints.^[67] Monetary easing has evolved, including the abandonment of the gold standard in 1933, which allowed currency devaluation and money supply expansion to counter deflation.^[68] Post-2008, central banks implemented quantitative easing (QE), purchasing assets to inject liquidity and lower long-term rates, aiding recovery from the housing crisis.^[63] These interventions reflect a shift toward proactive countercyclical policies since the 1930s.

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Relevance Logic - Stanford Encyclopedia of Philosophy
Jun 17, 1998 · Hence, contraction fails in linear logic. Linear logics are, in fact, relevant logics that lack contraction and the distribution of conjunction ...
[33]
[PDF] On the Decidability of Certain Semi-Lattice Based Modal Logics
Sep 28, 2017 · A lemma with a similar claim for a particular logic is often referred to as Curry's lemma or as height-preserving admissibility of contraction.
[34]
Actin, Myosin, and Cell Movement - The Cell - NCBI Bookshelf - NIH
In addition to binding actin, the myosin heads bind and hydrolyze ATP, which provides the energy to drive filament sliding. This translation of chemical energy ...
[35]
Nervous System Control of Muscle Tension – Anatomy & Physiology
Neural control regulates concentric, eccentric and isometric contractions, muscle fiber recruitment, and muscle tone.
[36]
Physiology, Skeletal Muscle Contraction - StatPearls - NCBI Bookshelf
May 1, 2023 · Skeletal muscle contraction begins first at the neuromuscular junction, which is the synapse between a motoneuron and a muscle fiber.
[37]
Interaction among Skeletal Muscle Metabolic Energy Systems ...
Three energy systems function to replenish ATP in muscle: (1) Phosphagen, (2) Glycolytic, and (3) Mitochondrial Respiration.
[38]
Muscle fatigue: general understanding and treatment - PMC
Oct 6, 2017 · Muscular work must be supported by a ready supply of ATP energy. There are three major ATPases that require ATP for muscle activity: Na+/K+- ...
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Tetanus: Pathophysiology, Treatment, and the Possibility of Using ...
Jan 8, 2013 · Tetanus toxin causes hyperactivity of voluntary muscles in the form of rigidity and spasms. Rigidity is the tonic, involuntary contraction of ...Missing: sustained | Show results with:sustained
[40]
Myasthenia Gravis - StatPearls - NCBI Bookshelf - NIH
Myasthenia gravis is an autoimmune disorder affecting the neuromuscular junction. It manifests as a generalized muscle weakness which can involve the ...Missing: tetanus | Show results with:tetanus
[41]
Physiology, Pregnancy Contractions - StatPearls - NCBI Bookshelf
Sep 19, 2022 · Labor contractions are painful, regular, and present with a change in cervical dilation and/or effacement.Issues of Concern · Cellular Level · Function · Mechanism
[42]
Physiology of Normal Labor and Delivery: Part I and II
Uterine contractions are involuntary and, for the most part, independent of extrauterine control. It has been demonstrated that the uterus has pacemakers to ...Characteristics of uterine... · Physiology of labor and delivery
[43]
Braxton Hicks Contractions - StatPearls - NCBI Bookshelf
Aug 8, 2023 · Braxton Hicks contractions are sporadic contractions and relaxation of the uterine muscle. Sometimes, they are referred to as prodromal or ...
[44]
Normal Labor: Physiology, Evaluation, and Management - NCBI - NIH
Feb 15, 2025 · [1] Regular uterine contractions cause progressive cervical dilation and effacement, ultimately leading to expulsion of the uterine contents.Introduction · Pathophysiology · History and Physical · Treatment / Management
[45]
Monitoring uterine activity during labor: a comparison of three methods
Tocodynamometry (Toco—strain gauge technology) provides contraction frequency and approximate duration of labor contractions, but suffers frequent signal ...
[46]
Preterm Labor and Birth | ACOG
Magnesium Sulfate: A drug that may help prevent cerebral palsy when it is given to women in preterm labor who may deliver before 32 weeks of pregnancy. Neonatal ...
[47]
Different treatment regimens of magnesium sulphate for tocolysis in ...
Magnesium sulphate has been trialled as a tocolytic agent in the context of threatened preterm labour for the prevention of preterm birth. Controversy exists ...
[48]
The Usage of Ergot (Claviceps purpurea (fr.) Tul.) in Obstetrics ... - NIH
Jul 15, 2021 · Applications of ergot in gynecology and obstetrics in the 19th century were limited to controlling excessive uterine bleeding and irregular spasms.
[49]
Preterm Birth | Maternal Infant Health - CDC
Nov 8, 2024 · The preterm birth rate declined 1% from 2021 to 2022, to 10.4%, following an increase of 4% from 2020 to 2021. However, racial and ethnic ...
[50]
Vaginal prostaglandin (PGE2 and PGF2a) for induction of labour at ...
Prostaglandins PGE2 probably increase the chance of vaginal delivery in 24 hours, they increase uterine hyperstimulation with fetal heart changes but do not ...Missing: prevalence | Show results with:prevalence
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[PDF] ON THE ELECTRODYNAMICS OF MOVING BODIES - Fourmilab
This edition of Einstein's On the Electrodynamics of Moving Bodies is based on the English translation of his original 1905 German-language paper. (published as ...
[52]
The Pole-Barn Paradox
The pole-barn paradox is a famous variation on the twin paradox which must be addressed with the ideas of simultaneity in relativity.
[53]
[PDF] Measurement of the Relativistic Time Dilation Using u-Mesons
* This experiment was the basis for a film Time Dilation. -An Experiment With p-Mesons conceived and planned by. Francis Friedman, David Frisch, and James Smith ...
[54]
Time dilation and length contraction in Special Relativity - UNSW
Particles certainly do. Particle accelerators generate some short lived particles (eg muons or pions) that travel within a fraction of a percent of c, and ...Missing: evidence | Show results with:evidence
[55]
Does Two Consecutive Quarters of a Decline in GDP Signify a ...
A decline in GDP over 2 quarters doesn't necessarily signify a recession, despite popular use of this “rule of thumb,” which emerged in 1974.
[56]
[PDF] The Great Depression: An Overview by David C. Wheelock
Between 1929 and 1933, the quantity of goods and services produced in the United States fell by one-third, the unemployment rate soared to. 25 percent of the ...
[57]
All About the Business Cycle: Where Do Recessions Come From?
Mar 1, 2023 · Using the informal “rule of thumb” definition, a recession is two consecutive quarters of economic contraction (negative real GDP growth).
[58]
Supply Chain Disruptions, Inflation, and the Fed
This negative correlation is just what one would expect for an economic cycle driven by a combination of negative supply and demand shocks. Supply disruptions ...A Mix Of Supply And Demand... · Supply Chain Disruptions · How Much Inflation Came From...Missing: contraction monetarist<|separator|>
[59]
The Great Depression - Federal Reserve History
The Depression lasted a decade, beginning in 1929 and ending during World War II. Industrial production plummeted. Unemployment soared. Families suffered.
[60]
What Is Keynesian Economics? - Back to Basics
Prices, and especially wages, respond slowly to changes in supply and demand, resulting in periodic shortages and surpluses, especially of labor. • Changes in ...
[61]
Great Depression Facts - FDR Presidential Library & Museum
At the height of the Depression in 1933, 24.9% of the total work force or 12,830,000 people was unemployed. Although farmers technically were not counted among ...
[62]
The Great Recession and Its Aftermath - Federal Reserve History
Effects on the Broader Economy. The housing sector led not only the financial crisis, but also the downturn in broader economic activity. Residential ...Sections · Effects On The Financial... · Effects On Financial...
[63]
https://www.federalreservehistory.org/essays/great-recession-and-its-aftermath
[64]
What's the Difference Between a Recession and a Depression?
Apr 9, 2025 · In a recession, the financial toll on households and businesses is significant, but manageable. In a depression, it's overwhelming. While the ...
[65]
Unemployment and Earnings Losses: A Look at Long-Term Impacts ...
During economic downturns, those who suffer job loss tend to be out of work for longer periods of time, resulting in higher total earnings losses. In the ...
[66]
[PDF] US Monetary and Fiscal Policy in the 1930s Price V. Fishback ...
The discussion of fiscal policy shows why economists do not see the New Deal as a Keynesian stimulus, describes the significant shift toward excise taxation ...
[67]
[PDF] Recovery of 1933 - Federal Reserve Board
When Roosevelt abandoned the gold standard in April 1933, he converted govern- ment debt from a tax-backed claim to gold to a claim to dollars, opening the door.Missing: contractions | Show results with:contractions