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Interval

In mathematics, an interval is a connected subset of the real line, consisting of all real numbers between two endpoints (which may be finite or infinite) and potentially including or excluding those endpoints.^[1] Intervals are classified based on whether they are open, closed, or half-open, and whether they are bounded or unbounded.^[2] Open intervals, denoted (a, b), exclude the endpoints a and b; closed intervals, denoted [a, b], include them; half-open intervals include one endpoint but not the other, such as [a, b) or (a, b].^[3] Unbounded intervals extend to positive or negative infinity, like (-∞, b), (a, ∞), or (-∞, ∞), which represents the entire real line.^[4] Intervals form the foundational building blocks in real analysis, serving as the basic open and closed sets in the standard topology of the real numbers.^[3] They are essential for defining continuity, where a function is continuous on an interval if it is continuous at every point in that interval.^[5] In the context of compactness, the Heine-Borel theorem states that a subset of the real line is compact if and only if it is closed and bounded. In particular, closed and bounded intervals are compact.^[6] Intervals also play a critical role in integration theory, as the Riemann integral is defined over closed bounded intervals, and many theorems, such as the mean value theorem, require the domain to be an interval to ensure connectedness.^[7] Furthermore, the nested interval theorem guarantees that a sequence of nested closed intervals with lengths approaching zero has a non-empty intersection, which is a single point, underpinning proofs of completeness in the real numbers.^[8] The term "interval" also has meanings in other fields such as physics, music, sports, and medicine (see sections below).

Mathematics

Real intervals

In mathematics, a real interval is defined as a connected subset of the real line \mathbb{R}, consisting of all real numbers between any two points in the set without gaps.^[1] This connectedness ensures that for any two points a < b in the interval, every real number x satisfying a \leq x \leq b (or appropriate inequalities based on endpoint inclusion) is also contained within it.^[9] Real intervals are classified by their boundedness and endpoint inclusion. Bounded intervals have finite endpoints a and b with a \leq b, and include closed intervals [a, b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}, open intervals (a, b) = \{x \in \mathbb{R} \mid a < x < b\}, and half-open (or half-closed) intervals [a, b) = \{x \in \mathbb{R} \mid a \leq x < b\} or (a, b] = \{x \in \mathbb{R} \mid a < x \leq b\}.^[10] Unbounded intervals extend to infinity in one or both directions, such as (-\infty, b] = \{x \in \mathbb{R} \mid x \leq b\}, [a, \infty) = \{x \in \mathbb{R} \mid x \geq a\}, or the entire line (-\infty, \infty) = \mathbb{R}.^[1] Key properties of real intervals include convexity and connectedness, which coincide in one dimension: a subset of \mathbb{R} is convex if and only if it is an interval.^[11] Convexity means that the line segment joining any two points in the set lies entirely within the set, formalized as \lambda x + (1 - \lambda) y \in I for all x, y \in I and \lambda \in [0, 1].^[12] For bounded intervals, the length or Lebesgue measure is b - a, providing a quantitative measure of size independent of endpoint openness.^[13] The concept of real intervals developed in the 19th century within real analysis, building on efforts to rigorously define the real numbers; Karl Weierstrass emphasized power series and continuity, while Georg Cantor advanced set-theoretic constructions, including nested intervals to characterize completeness.^[14] A representative example is the unit interval [0, 1], which is closed, bounded, convex, and has length 1.^[1] Unlike continuous intervals, discrete sets like the integers \mathbb{Z} are not intervals due to gaps between elements, such as between 0 and 1.^[10] In topology, open intervals form a basis for the standard topology on \mathbb{R}.^[15]

Interval arithmetic

Interval arithmetic is a method for performing operations on intervals representing sets of possible values, ensuring that the result encloses all potential outcomes of the computation. It extends standard arithmetic to bounded sets, typically closed real intervals [a, b] where a ≤ b, to handle uncertainties such as measurement errors or rounding in numerical computations. The operations are defined such that for input intervals X and Y, the output interval contains the range of the corresponding real function over all pairs (x, y) ∈ X × Y.^[16] The basic operations are defined as follows. Addition is straightforward: [a, b] + [c, d] = [a + c, b + d]. Subtraction follows similarly: [a, b] - [c, d] = [a - d, b - c]. Multiplication depends on the signs of the endpoints and is given by [a, b] × [c, d] = [min{ac, ad, bc, bd}, max{ac, ad, bc, bd}], considering all combinations to capture the extremal values. Division is handled as [a, b] / [c, d] = [a, b] × [1/d, 1/c] if 0 ∉ [c, d] and c ≤ d (with the reciprocal interval adjusted for the sign of [c, d]); if 0 ∈ [c, d], the result is undefined or taken as the empty set to avoid division by zero.^[16] Interval arithmetic exhibits key properties that make it suitable for reliable computations. Results are always non-empty for defined operations on non-empty inputs, and the system satisfies inclusion monotonicity: if X ⊆ X' and Y ⊆ Y', then X ⊕ Y ⊆ X' ⊕ Y' for any operation ⊕, ensuring that wider input intervals yield enclosing output intervals. However, a significant limitation is the dependency problem, where multiple occurrences of the same variable lead to overestimation of the range. For instance, computing [a, b] - [a, b] yields [a - b, b - a], which is [-|b - a|, |b - a|] and overestimates the true range {0} due to treating the subtrahend as independent.^[16] Interval arithmetic finds applications in numerical analysis for verified computing, where it guarantees enclosures of solutions to equations or bounds on function ranges without floating-point errors. In engineering, it propagates errors deterministically through complex systems, such as in chemical process simulations or control design, providing rigorous bounds on uncertainties rather than probabilistic estimates.^[16]^[17] The field was developed in the 1950s and 1960s by R. E. Moore, primarily for computer-assisted proofs and error control in automatic computation, as detailed in his seminal 1966 book Interval Analysis.^[16] As an example, consider evaluating the range of f(x) = x(1 - x) over the interval [0, 1]. The exact range, found via calculus, is [0, 0.25], achieved at x = 0.5. Using the natural interval extension, naive computation yields [0, 1] due to the dependency problem in the term x - x^2, but refined methods or the united extension can recover the sharp bound [0, 0.25].^[16]

Statistical intervals

In statistics, statistical intervals provide probabilistic ranges for estimating unknown parameters, predicting future observations, or bounding population distributions based on sample data, incorporating uncertainty from random sampling. These intervals differ from deterministic real intervals by assigning coverage probabilities that reflect long-run frequency properties in repeated sampling. They are central to inference in frequentist and Bayesian frameworks, enabling quantification of estimation precision without assuming the true value is known. The confidence interval is a range constructed from sample data such that, over repeated samples, it contains the true population parameter with a pre-specified probability, known as the coverage probability. Introduced by Jerzy Neyman in 1937 as a method for frequentist statistical estimation, confidence intervals emphasize the interval's reliability across hypothetical replications rather than the probability for a single interval.^[18] For estimating a population mean μ under normality with known standard deviation σ, the 100(1-α)% confidence interval is given by

\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}},

where \bar{x} is the sample mean, z_{\alpha/2} is the upper α/2 quantile of the standard normal distribution, and n is the sample size. Key properties include the fixed coverage probability (e.g., 95% for α=0.05), which guarantees the interval covers the parameter in 95% of repeated samples; efforts to minimize interval width often involve optimal point estimators or adjustments for variance; and a duality with hypothesis testing, where a (1-α) confidence interval excludes values for which the p-value exceeds α under the null hypothesis.^[19] Statistical intervals encompass several types tailored to inference goals. Confidence intervals focus on the parameter with fixed coverage probability, while prediction intervals extend to future observations by accounting for both parameter uncertainty and inherent observation variability, resulting in wider ranges.^[20] In Bayesian statistics, credible intervals derive from the posterior distribution, providing the probability that the parameter lies within the interval given the data and prior beliefs, such as a 95% credible interval containing 95% of the posterior probability mass.^[21] Tolerance intervals differ by bounding a specified proportion of the population (e.g., 95%) with a given confidence level that the bound holds, useful for quality control to ensure most items fall within limits.^[22] Examples illustrate practical application. For a 95% confidence interval on a population mean, a sample of n=25 yields \bar{x}=100 with σ=15, giving [94.12, 105.88] via the formula above, indicating the mean is likely between these values with 95% coverage. For binomial proportions, the Wilson score interval improves on normal approximations for small samples or extreme proportions; with x=10 successes in n=20 trials ( \hat{p}=0.5 ), the 95% interval is approximately [0.27, 0.73], calculated as

\hat{p} + \frac{z^2}{2n} \pm \frac{z}{1 + z^2/n} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} + \frac{z^2}{4n^2} },

where z=1.96.

Physics

Temporal intervals

A temporal interval, or time interval, refers to the duration between two distinct events in time, mathematically expressed as Δt = t₂ - t₁, where t₁ and t₂ are the times of occurrence of the respective events. In physics, this interval quantifies the elapsed time over which physical processes unfold, serving as a fundamental measure in both theoretical and experimental contexts.^[23] In classical physics, time intervals are considered absolute and uniform, flowing independently of the observer's motion or location, as posited by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica. This absolute time underpins Newtonian mechanics, where intervals remain invariant under Galilean transformations between inertial frames, ensuring that the laws of motion hold equally for all observers moving at constant velocities relative to one another.^[24]^[25] Albert Einstein's theory of special relativity, introduced in his 1905 paper "On the Electrodynamics of Moving Bodies," revolutionized this view by demonstrating that time intervals are relative to the observer's frame of reference. In this framework, the proper time τ represents the invariant time interval measured by a clock following the worldline of a particle, calculated as τ = ∫ √(1 - v²/c²) dt, where v is the instantaneous velocity, c is the speed of light, and the integral is taken along the path from event 1 to event 2. This proper time contrasts with the coordinate time Δt measured in a specific inertial frame, leading to time dilation effects. For a clock moving at constant velocity v relative to a stationary observer, the proper time interval is Δτ = Δt / γ, where γ = 1 / √(1 - v²/c²) is the Lorentz factor; thus, moving clocks tick slower as observed from the stationary frame. These relativistic effects have been experimentally verified and find practical application in technologies like the Global Positioning System (GPS), where satellite clocks experience both velocity-based special relativistic time dilation (causing a daily loss of about 7 microseconds) and gravitational effects from general relativity (causing a gain of about 45 microseconds per day), necessitating precise corrections to maintain positional accuracy within meters. A seminal confirmation came from the 1971 Hafele-Keating experiment, in which cesium atomic clocks flown around the world on commercial jets showed time shifts consistent with relativistic predictions, with eastward flights losing 59 ± 10 nanoseconds and westward flights gaining 273 ± 7 nanoseconds relative to ground clocks.^[26] Time intervals are measured using clocks, such as atomic clocks for high precision or stopwatches for everyday durations, with the International System of Units (SI) defining the base unit as the second (s), equivalent to 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom. Submultiples like milliseconds (ms) and microseconds (μs) are commonly used for finer resolutions in physics experiments.^[27] Historically, Newton's concept of absolute time, which assumed universal uniformity without regard to motion, dominated physics until Einstein's 1905 relativity theory established time's relativity, resolving paradoxes in electromagnetism and paving the way for modern spacetime geometry.^[24]

Spectral intervals

A spectral interval denotes a continuous band of frequencies or wavelengths within the electromagnetic spectrum, representing a range over which radiation exhibits specific properties or interactions. For instance, the visible light spectrum occupies a spectral interval from approximately 400 nm to 700 nm in wavelength.^[28] This concept is fundamental in describing the distribution of radiant energy, where spectral irradiance is quantified as power per unit wavelength interval at a given wavelength.^[29] In optics, spectral intervals characterize the bandwidth of light sources, defined as the width of the wavelength range over which significant emission occurs, often on the order of tens to hundreds of nanometers for broadband sources like LEDs. Spectral lines within these intervals, observed in spectroscopy, appear as narrow dark absorption or bright emission features superimposed on a continuous spectrum, arising from atomic or molecular transitions that selectively interact with photons in specific narrow bands. The interval width, denoted as \Delta \lambda, quantifies the breadth of such features or source emissions, influencing resolution in spectroscopic instruments. The photon energy E relates to frequency \nu via E = h \nu, where h is Planck's constant, while wavelength \lambda connects to frequency through \lambda = c / \nu, with c the speed of light in vacuum; these relations underpin the conversion between spectral domains.^[30]^[31]^[32]^[33]^[34] In quantum mechanics, spectral intervals correspond to allowed energy bands in solid-state materials, such as the valence band—filled with electrons at absolute zero—and the conduction band—where electrons can move freely—separated by a forbidden energy gap devoid of quantum states. This band structure, arising from the periodic potential of crystal lattices, determines electrical conductivity: narrow gaps enable semiconductors, while wide gaps characterize insulators. Applications include chemical analysis via absorption spectra, where molecules exhibit characteristic intervals of light absorption corresponding to electronic transitions, enabling identification and quantification. In radio frequency engineering, spectral intervals are allocated for industrial, scientific, and medical (ISM) purposes, such as the 902–928 MHz band, to minimize interference in unlicensed operations like wireless devices.^[35]^[36]^[37] Historically, the investigation of spectral intervals originated with Joseph von Fraunhofer's 1814 discovery of dark absorption lines in the solar spectrum using a prism spectrometer, cataloging over 500 such features and establishing standards for wavelength measurement. In 1928, Felix Bloch advanced the understanding of quantum spectral intervals through his theorem on electron waves in periodic potentials, laying the foundation for band theory in solids and explaining phenomena like electrical conduction in crystals.^[38]^[39]

Arts and entertainment

Music

In music theory, a musical interval is defined as the distance in pitch between two notes, which can be measured either in semitones within the chromatic scale or as a frequency ratio.^[40] For example, an octave interval corresponds to a frequency ratio of 2:1, where the higher note vibrates twice as fast as the lower one.^[41] This concept forms the foundation of melody and harmony, as intervals determine the relational structure of pitches in musical compositions.^[42] Musical intervals are categorized by their presentation and extent: melodic intervals occur between successive notes in a sequence, while harmonic intervals sound simultaneously as chords.^[43] Simple intervals span up to an octave, such as a unison or perfect fifth, whereas compound intervals exceed an octave and are often analyzed by reducing them to their simple equivalents plus additional octaves.^[44] Intervals are further classified by quality and size: perfect intervals include the unison (1:1 ratio), perfect fourth (4:3), perfect fifth (3:2), and octave (2:1), which are considered consonant and stable in most tuning systems.^[45] Major and minor qualities apply to seconds, thirds, sixths, and sevenths—for instance, a major third spans four semitones, while a minor third spans three—allowing for brighter or darker tonal colors.^[46] Augmented intervals are enlarged by a semitone from perfect or major forms, and diminished intervals are reduced by a semitone from perfect or minor ones, creating tension or dissonance.^[45] Tuning systems influence interval realization: in just intonation, intervals derive from simple whole-number frequency ratios for purity, such as the major third at 5:4 (approximately 1.25), yielding a warm consonance.^[47] In contrast, equal temperament divides the octave into 12 equal semitones, approximating the major third at about 1.26 (2^{4/12}), which facilitates modulation but introduces slight inharmonicity compared to just ratios.^[48] Interval inversion involves transposing the lower note up an octave (or the higher down), where an interval and its complement always sum to an octave; for example, a major third inverts to a minor sixth.^[49] The historical development of intervals traces to ancient Greece, where Pythagorean tuning around 500 BCE established scales using stacked perfect fifths (3:2 ratios) and octaves, prioritizing mathematical harmony over equal spacing.^[50] By the 11th century, Guido d'Arezzo introduced the Guidonian hand, a mnemonic diagram mapping the hexachord's intervals across the hand's joints to aid singers in sight-reading and interval recognition.^[51] In modern applications, the perfect fifth underpins power chords in rock and metal music, consisting of a root note and its fifth for a robust, ambiguous tonality that works across keys.^[52] The tritone, an augmented fourth or diminished fifth spanning six semitones, earned the moniker "devil's interval" (diabolus in musica) in medieval times due to its dissonant instability, often evoking tension in compositions from classical to heavy metal.^[53]

Dramatic arts

In the dramatic arts, an interval refers to a deliberate pause or break in a performance, such as an intermission between acts in live theatre, which provides the audience with respite, facilitates practical needs like set changes, and allows time for reflection on the unfolding narrative. These breaks structure the overall flow of a production, creating rhythmic ebbs that heighten dramatic tension upon resumption. Historically, intervals emerged as essential components of theatrical staging, evolving from practical necessities to integral elements of audience engagement.^[54] In live theatre, intervals typically last 10 to 20 minutes, with 15 minutes being the standard duration to accommodate brief discussions, refreshments, and physical relief without disrupting momentum. Their origins trace back to 18th-century European theatre, particularly in French drama, where they were introduced to manage lighting challenges, such as relighting chandeliers with fresh candles, and to execute complex scene shifts between acts of similar length. This convention persists in contemporary productions, where intervals not only serve logistical purposes but also enable performers to reset and audiences to absorb key plot developments.^[55]^[56] In film, intervals manifest as narrative pauses, often through montage sequences that compress time or evoke emotional transitions, allowing viewers a moment of interpretive breathing room amid continuous action. While modern cinema rarely includes formal intermissions due to digital projection eliminating reel changes, historical epics like those from the mid-20th century occasionally featured them to sustain engagement over extended runtimes. These pauses parallel theatrical breaks by building suspense and permitting reflection on thematic undercurrents.^[57]^[58] In literature, intervals appear as temporal gaps between chapters or scenes, commonly known as time skips, which propel the story forward while implying significant off-page developments in characters' lives or circumstances. For instance, a novel might bridge years between sections to focus on pivotal transformations, using subtle cues like altered settings or character descriptions to orient readers without exhaustive exposition. This technique structures narrative flow, emphasizing conceptual progression over chronological detail and mirroring the reflective pauses of stage drama.^[59]^[60] Cultural variations in intervals reflect genre and tradition: operas often incorporate longer or multiple breaks—sometimes exceeding 20 minutes each—to accommodate elaborate scenery transitions and vocal recovery, contrasting with shorter, single intervals in modern straight plays that prioritize pacing. In Western opera houses, productions may feature two or three intermissions over three-plus hours, enhancing the epic scope, whereas contemporary spoken-word theatre favors briefer pauses or none at all to maintain intimacy.^[61]^[62] Historically, William Shakespeare's plays were divided into five acts but performed without formal intervals at venues like the Globe Theatre, relying on natural scene breaks and audience immersion under daylight conditions to sustain continuous energy. In contrast, Richard Wagner's operas, such as Das Rheingold, eschew intermissions entirely to preserve mythic continuity, composing scenes as seamless musical narratives that unfold over 2.5 hours without pause, as intended in the original scores. These examples illustrate intervals' role in adapting dramatic structure to artistic vision and venue constraints.^[63]^[64]^[65] Intervals profoundly impact dramatic arts by building tension through anticipation of resumption and fostering audience reflection on prior events, thereby deepening emotional investment. In modern theatre, however, some productions eliminate them to enhance immersion and inclusivity, arguing that uninterrupted flow better suits diverse attention spans and post-pandemic preferences for concise experiences, though this risks fatigue in longer works. Such choices underscore intervals' dual function as both structural tools and cultural barometers of engagement.^[66]^[67]^[63]

Sports

Interval training

Interval training is a structured form of exercise that alternates periods of high-intensity effort, known as work intervals, with periods of lower-intensity recovery or rest to allow partial replenishment of energy stores.^[68] This method enhances cardiovascular and muscular adaptations by pushing the body to near-maximal efforts during work phases while enabling sustained training volume through recovery.^[69] The origins of interval training trace back to the 1930s in Germany, where coach Woldemar Gerschler, in collaboration with cardiologist Hans Reindell, developed it to improve athletic performance by monitoring heart rate recovery between efforts.^[70] Gerschler's approach emphasized repetition of intense bouts to build endurance without excessive fatigue, influencing modern protocols for middle- and long-distance runners.^[71] Common types include high-intensity interval training (HIIT), which features short bursts of vigorous activity—such as 30 seconds of sprinting followed by 30 seconds of rest—designed to elevate heart rate significantly, and fartlek training, a Swedish term meaning "speed play," involving unstructured variations in pace during continuous running.^[72] HIIT is typically more regimented with predetermined durations and intensities, whereas fartlek allows intuitive adjustments based on terrain or effort, making it adaptable for recreational athletes.^[73] Key protocols include the Norwegian 4x4 method, consisting of four 4-minute high-intensity intervals at 85-95% of maximum heart rate, separated by 3 minutes of active recovery, commonly applied in running and cycling to boost aerobic capacity.^[74] This approach has been validated for improving endurance in both modalities, with sessions totaling around 30-40 minutes including warm-up and cool-down.^[75] Prominent examples are the Tabata protocol, involving eight cycles of 20 seconds of all-out effort followed by 10 seconds of rest (totaling 4 minutes), originally tested on speed skaters and shown to enhance both anaerobic capacity and VO2 max more effectively than moderate continuous training.^[76] Similarly, the Gibala method, or sprint interval training (SIT), features repeated 30-second maximal sprints with 4-minute recoveries, demonstrating comparable skeletal muscle adaptations to traditional endurance exercise in far less time.^[77] Interval training offers benefits such as increased VO2 max—a measure of maximal oxygen uptake and aerobic fitness—by 5-10% in as few as six weeks,^[78] comparable fat loss to moderate continuous exercise due to elevated post-exercise oxygen consumption,^[79] and superior time efficiency for busy individuals. These outcomes are supported by meta-analyses confirming improvements in cardiorespiratory fitness^[78] and an edge in body composition changes, particularly for overweight populations.^[80] Interval training is also widely applied in team sports such as soccer and basketball, where short high-intensity bursts simulate match demands, improving repeated sprint ability and recovery, as shown in studies up to 2025.^[81]

Racing intervals

Racing intervals refer to structured training workouts that incorporate segments of varying intensity to simulate the demands of competitive events and optimize performance. These differ from continuous efforts by including strategic variations to manage fatigue and prepare for race dynamics.^[82]^[83] In running races, negative split intervals in training involve starting at a controlled pace and accelerating in the second half to build finishing strength, a tactic that conserves energy early and leverages improved economy later. This approach is particularly effective in marathons, where interval pacing—alternating surges with steady sections—helps maintain overall speed while countering fatigue. Studies show athletes using negative splits achieve better times and lower physiological strain compared to even or positive splits.^[84]^[85] Cycling races feature attack intervals in training, short, explosive efforts to practice breaking away from the group, as seen in simulations of Tour de France mountain stages where riders surge for 30-90 seconds to gain separation. Time trial segments divide the course into focused intervals of maximum effort, often monitored for power output to sustain velocity. These tactics allow riders to capitalize on terrain advantages and disrupt group rhythm.^[86]^[87] In competitive swimming, set intervals structure longer races like the 1500m freestyle into mental segments in practice, such as 100m repeats paced with brief recovery glides to simulate race flow and manage lactate buildup. Swimmers target consistent splits across these intervals to build momentum without early burnout.^[88] Key strategies in racing intervals emphasize energy management, where athletes allocate anaerobic bursts for attacks while preserving aerobic capacity for sustained efforts. Drafting plays a crucial role, reducing air resistance by up to 40% in cycling and 2-5% in running when positioned behind competitors, allowing energy savings during recovery intervals. GPS watches provide real-time data on pace and heart rate, enabling adjustments to optimize interval execution.^[89]^[90]^[91] Historically, interval methods in Olympic training evolved from basic repetition work to structured pacing by the early 20th century. In the 1950s, Bill Bowerman integrated interval approaches into University of Oregon programs, emphasizing hard surges alternated with recovery to prepare runners for race-day surges, influencing modern tactics.^[92]^[93] Representative examples include 400m repeats at 1500m race pace in middle-distance preparation, building the ability to sustain surges over the final laps. In road racing, peloton intervals involve maintaining high output within the group before launching attacks, conserving energy through drafting before a decisive break. These elements draw from preparatory training methods to inform competitive execution.^[94]^[95]

Other uses

Computing

In computing, intervals represent contiguous ranges of values, such as time periods, numerical bounds, or address spaces, and are fundamental to algorithms, data structures, and system operations for efficient range management and querying.^[96] These representations enable tasks like scheduling resources without overlaps, querying overlapping ranges, and verifying computational accuracy by enclosing possible results within bounds.^[97] A prominent application is interval scheduling, a greedy algorithm that maximizes the number of non-overlapping intervals selected from a set by sorting them by ending time and iteratively choosing the one with the earliest end that does not conflict with previously selected intervals.^[96] This approach, running in O(n log n) time due to sorting, is optimal for unit-weight jobs and traces back to early work on resource allocation problems.^[96] For example, in lecture scheduling, it assigns the maximum number of sessions to rooms by prioritizing those finishing soonest.^[96] Interval trees serve as an efficient data structure for storing intervals and supporting range queries, such as finding all intervals overlapping a given point or interval in O(log n + k) time, where k is the output size.^[98] Introduced by Herbert Edelsbrunner in 1980, the structure augments a binary search tree on interval endpoints with secondary lists at nodes to track overlapping subintervals, enabling dynamic insertions and deletions while maintaining balance.^[98] This logarithmic complexity makes it suitable for applications requiring frequent overlap checks. In networking, intervals denote blocks of IP addresses using Classless Inter-Domain Routing (CIDR), introduced by RFC 1519 and specified in RFC 4632 (2006), where notation like 192.168.0.0/24 represents the contiguous range [192.168.0.0, 192.168.0.255] by specifying a prefix length after a slash.^[99] This aggregation reduces routing table sizes by treating variable-length prefixes as supernets, allowing longest-match routing for efficient packet forwarding across the internet.^[99] Operating systems use time intervals for scheduling via POSIX timers, which support one-shot or periodic expirations to trigger events like signal delivery after a specified duration, as defined in the timer_create interface.^[100] Similarly, Unix-like systems employ cron for interval-based job scheduling, where entries in crontab files specify execution at recurring times or intervals, such as every hour, using a five-field format for minute, hour, day, month, and weekday. These mechanisms ensure precise timing for background tasks without continuous polling. For verified computation, floating-point intervals bound potential rounding errors in numerical algorithms by performing operations in directed rounding modes, yielding enclosures that contain the exact result.^[97] This approach, using intervals like [lower bound, upper bound] for additions and multiplications, supports rigorous error analysis in scientific simulations where precision is critical, as specified in IEEE Std 1788-2015 for interval arithmetic.^[101] Applications extend to database range searches, where interval trees or augmented variants index timestamped data for efficient overlap queries, such as retrieving records within a time window in O(log n) time.^[102] In computer graphics, interval methods detect collisions between time-dependent surfaces by minimizing distance functions over motion intervals, identifying multi-point contacts robustly for animations and simulations.^[103]

Medicine

In medicine, intervals refer to measurable time spans in physiological processes, particularly those captured through diagnostic tools like electrocardiography (ECG), electroencephalography (EEG), and polysomnography, which aid in assessing cardiac, neurological, and respiratory function. These intervals provide critical insights into conduction delays, variability, and event frequencies, helping clinicians diagnose arrhythmias, autonomic dysfunction, and sleep disorders. For instance, deviations in these intervals can signal underlying pathologies, guiding interventions from medication adjustments to surgical options. Cardiac intervals are fundamental in ECG analysis, with the PR interval representing the time for electrical impulses to travel from the atria to the ventricles via atrioventricular conduction, typically lasting 120-200 milliseconds in healthy adults.^[104] A prolonged PR interval beyond 200 ms may indicate atrioventricular block, while a shortened one suggests accelerated conduction.^[105] The QT interval measures ventricular depolarization and repolarization, with the corrected QTc value normally under 440 ms in males; prolongation increases the risk of ventricular arrhythmias.^[106] The RR interval, denoting the time between consecutive R waves on ECG, is central to heart rate variability (HRV) analysis, where metrics like the standard deviation of normal-to-normal (NN) intervals (SDNN) quantify autonomic nervous system balance, with reduced SDNN often reflecting sympathetic overactivity or parasympathetic withdrawal.^[107]^[108] In neurology, intervals play a key role in epilepsy management and brain wave assessment. Interictal periods describe the symptom-free intervals between seizures, comprising the majority of an epileptic patient's life and often involving subtle cognitive or behavioral changes that impact quality of life.^[109] EEG frequency intervals classify brain activity, such as the delta band (0.5-4 Hz), which predominates during deep non-rapid eye movement sleep and can indicate pathology if prominent in wakefulness.^[110] Respiratory intervals involve the timing and volume of breathing cycles, with tidal volume—the air displaced per breath—measured over successive intervals to evaluate ventilatory efficiency. The apnea-hypopnea index (AHI) quantifies obstructive sleep apnea severity by counting respiratory events (apneas or hypopneas lasting at least 10 seconds) per hour of sleep, where an AHI above 30 indicates severe disease and correlates with cardiovascular risks.^[111]^[112] These intervals have significant diagnostic utility; for example, a prolonged QT interval is strongly linked to torsades de pointes, a polymorphic ventricular tachycardia that can degenerate into ventricular fibrillation and sudden death.^[113] Holter monitoring, a portable ECG device worn for 24-hour intervals, captures intermittent arrhythmias missed by standard 10-second ECGs, enabling precise correlation with symptoms.^[114] Historically, the foundation for cardiac interval measurement traces to Willem Einthoven's 1903 development of the string galvanometer ECG, which first allowed precise recording and definition of waveform intervals like PR and QT.^[115] HRV research expanded post-1960s with digital signal processing advancements, initially applied to fetal monitoring and later to autonomic assessment in adults, establishing HRV as a prognostic tool in cardiology.^[116]

References

[1]
Interval -- from Wolfram MathWorld
An interval is a connected portion of the real line. If the endpoints a and b are finite and are included, the interval is called closed and is denoted [a,b].
[2]
Interval Notation - Department of Mathematics at UTSA
Jan 15, 2022 · The open intervals are open sets of the real line in its standard ... An interval I is subinterval of interval J if I is a subset of J.
[3]
Topology of the Real Line - Advanced Analysis
Jan 17, 2024 · Open Intervals and Open Sets. Definition. An open interval in the real line is a set of the form ... Closed Sets. Definition. A closed set E ...
[4]
[PDF] Appendices A.1. Real Numbers and the Real Line
Aug 6, 2020 · A.1 Real Numbers and the Real Line. 1. Appendices. A.1. Real ... An interval with two endpoints is a finite interval. If an interval ...
[5]
[PDF] Introduction to Real Analysis Chapter 0 - Christopher Heil
(a) Let I be an interval in the real line, and let C(I) be the set of all continuous scalar-valued functions whose domain is I, i.e.,. C(I) = f ∈ F(I) : f is ...
[6]
https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-27.pdf
[7]
Why is it important that we use the Mean value theorem on an interval?
May 8, 2022 · The key assumption in mean value theorem is the connectedness of [a,b]. Roughly speaking connectedness means "the interval is made of one piece".
[8]
nested interval theorem - PlanetMath
Mar 22, 2013 · There are two consequences to nesting of intervals: [am,bm]⊆[an,bn] [ a m , b m ] ⊆ [ a n , b n ] for n≤m n ≤ m : 1. ... an≤am a n ≤ a m for n≤m n ...Missing: mathematics | Show results with:mathematics
[9]
interval - PlanetMath.org
Mar 22, 2013 · Loosely speaking, an interval is a part of the real numbers that start at one number and stops at another number. For instance, all numbers ...
[10]
Real Numbers:Intervals - Department of Mathematics at UTSA
Nov 6, 2021 · The open intervals are open sets of the real line ... The identity component of this group is quadrant I. Every interval can be considered a ...
[11]
[PDF] 1 Convex Sets, and Convex Functions
(a) An interval of [a, b] ⊂ R is a convex set. To see this, let c, d ∈ [a, b] and assume, without loss of generality, that c<d. Let λ ∈ (0, 1).
[12]
[PDF] 5. Connectedness
A subset K ⊂ R is convex if for any c, d ∈ K, the set [c, d] = {c(1−t)+dt | 0 ≤ t ≤ 1} is contained in K. Show that a convex subset of R is an open, closed, or ...
[13]
proof that the outer (Lebesgue) measure of an interval is its length
Mar 22, 2013 · We begin with the case in which we have a bounded interval, say [a,b] [ a , b ] . Since the open interval (a−ε,b+ε) ( a - ε , b + ε ) ...
[14]
[PDF] On the history of nested intervals: from Archimedes to Cantor - arXiv
The concept of a real number was elaborated in the 1870s in works of Ch. Méray, Weierstrass, H.E.. Heine, Cantor, and R. Dedekind. Cantor's elaboration was ...
[15]
[PDF] Connected Sets
This means any connected subset of R must be an interval. (Note however that we have not yet proved that intervals are, in fact, connected.) Theorem 2.
[16]
[PDF] Introduction to - INTERVAL ANALYSIS
Interval analysis includes enclosing a solution, bounding roundoff error, and number pair extensions.
[17]
(PDF) Automatic Error Analysis Using Intervals - ResearchGate
Aug 6, 2025 · Interval analysis has practical applications in economics, chemical engineering, beam physics, control circuit design, global optimization ...
[18]
[PDF] Outline of a Theory of Statistical Estimation Based on the Classical ...
The main problem treated in this paper is that of confidence limits and of confidence intervals and may be briefly described as follows. Let p (xl,... x ...Missing: Jerzy | Show results with:Jerzy
[19]
Statistical tests, P values, confidence intervals, and power: a guide ...
As Neyman stressed repeatedly, this coverage probability is a property of a long sequence of confidence intervals computed from valid models, rather than a ...
[20]
The distinction between confidence intervals, prediction ... - GraphPad
Prediction intervals tell you where you can expect to see the next data point sampled. Assume that the data are randomly sampled from a Gaussian distribution.
[21]
Understanding and interpreting confidence and credible intervals ...
Dec 31, 2018 · Interpretation of the Bayesian 95% confidence interval (which is known as credible interval): there is a 95% probability that the true (unknown ...
[22]
7.2.6.3. Tolerance intervals for a normal distribution
Definition of a tolerance interval, A confidence interval covers a population parameter with a stated confidence, that is, a certain proportion of the time.Missing: source | Show results with:source
[23]
[PDF] On the Theory of Scales of Measurement
Thus, an interval scale can be erected only provided we have an operation for determining equality of intervals, for determining greater or less, and for ...<|separator|>
[24]
Time, Velocity, and Speed | Physics - Lumen Learning
In physics, the definition of time is simple— time is change, or the interval over which change occurs. It is impossible to know that time has passed unless ...Velocity · Speed · Section Summary
[25]
Newton's views on space, time, and motion
Aug 12, 2004 · Isaac Newton founded classical mechanics on the view that space is distinct from body and that time passes uniformly without regard to whether anything happens ...Overview of the Scholium · The Legacy from Antiquity · The Structure of Newton's...
[26]
Special Relativity: Proper Time, Coordinate Systems, and Lorentz ...
This invariance principle is fundamental to classical physics, and it means that in classical physics we can define: Coordinate time = Proper time for all ...
[27]
Around-the-World Atomic Clocks: Predicted Relativistic Time Gains
Abstract. During October 1971, four cesium beam atomic clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and ...
[28]
NIST Guide to the SI, Chapter 8
Jan 28, 2016 · The SI unit of time (actually time interval) is the second (s) and should be used in all technical calculations. When time relates to ...
[29]
Visible Light - NASA Science
Aug 4, 2023 · Violet has the shortest wavelength, at around 380 nanometers, and red has the longest wavelength, at around 700 nanometers.
[30]
Spectral Irradiance per Interval of Equal Solar Flux - Campaigns
The spectral irradiance is defined as irradiance per unit wavelength interval at a given wavelength. However, we grew accustomed to a nonlinear subdivision ...
[31]
Bandwidth - Definition and Introduction
Bandwidth is the width of a wavelength range with regard to a specific part of the spectrum which transmits incident energy via a filter.
[32]
Spectral Lines
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow ...
[33]
[PDF] Optical Sources - University of Washington
Two basic light sources for fiber optics are lasers and light-emitting diodes (LEDs). Lasers have higher output power and are faster, while LEDs have lower ...
[34]
https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_%28CK-12%29/05%253A_Electrons_in_Atoms/5.02%253A_Wavelength_and_Frequency_Calculations
[35]
5.2: Wavelength and Frequency Calculations - Chemistry LibreTexts
Mar 20, 2025 · All waves can be defined in terms of their frequency and intensity. · c = λ ⁢ ν expresses the relationship between wavelength and frequency.
[36]
Band Theory for Solids - HyperPhysics
In insulators the electrons in the valence band are separated by a large gap from the conduction band, in conductors like metals the valence band overlaps the ...
[37]
6.4: Emission and Absorbance Spectra - Chemistry LibreTexts
Sep 26, 2022 · Absorbance Spectra. When an atom, ion, or molecule moves from a lower-energy state to a higher-energy state it absorbs photons with energies ...Emission Spectra · Line Spectra · Band Spectra
[38]
47 CFR 18.301 -- Operating frequencies. - eCFR
ISM equipment may be operated on any frequency above 9 kHz except as indicated in § 18.303. The following frequency bands, in accordance with § 2.106 of the ...
[39]
Absorption Spectra (Joseph von Fraunhofer)
In 1814 Joseph von Fraunhofer noticed that the light from the sun does not give a continuous spectrum. By using an unusually good prism, Fraunhofer observed a ...
[40]
12.8: Crystalline Solids: Band Theory - Chemistry LibreTexts
Jun 28, 2017 · In 1928, Felix Bloch had the idea to take the quantum theory and apply it to solids. In 1927, Walter Heitler and Fritz London discovered bands- ...
[41]
Introduction to Intervals - Music Theory for the 21st-Century Classroom
Intervals are a measurement between two pitches, either vertically or horizontally. When measuring vertically, we refer to harmonic intervals because the two ...
[42]
Musical Scales and Frequencies - Swarthmore College
so a fifth is 1.05946 to the 7th power (seven semitones in a musical fifth) or 1.498 (slightly flatter than 3/2).
[43]
Standard 4: Intervals - Open Music Theory × CUNY
An interval is the distance between two pitches, usually measured in two components: 1) the size, and 2) the quality. The term interval regularly refers both to ...
[44]
11. Intervals – Fundamentals, Function, and Form - Milne Publishing
An interval is the distance a listener perceives between two pitches. When the two pitches sound simultaneously, we refer to it as a harmonic interval.
[45]
Interval Inversion - musictheory.net
To invert this interval, move the lowest note (the C) an octave higher. The result is a perfect fourth: G to C. Next, let's invert a perfect fourth: F# to B.Missing: complement | Show results with:complement
[46]
Augmented and Diminished Intervals
Augmented intervals are one half step larger than perfect or major intervals and diminished intervals are one half step smaller than perfect or minor intervals.Missing: classification | Show results with:classification
[47]
Ch.6.9 The five types of intervals - MusicTheory.Education
Five types of intervals. The five types of intervals are called perfect, minor, major, diminished or augmented. The intervals fourth, fifth and octave are ...
[48]
Just Intonation Explained - Kyle Gann
In the notation of just-intonation (pure) tuning, pitches are given as fractions, which are actually ratios between the named pitch and a constant fundamental.<|separator|>
[49]
Tuning Systems: Equal Temperament vs Just Intonation - muted.io
Equal temperament (aka 12 tone equal temperament) divides each octave into 12 equal steps, making slight modifications to just intonation.
[50]
Interval inversion | Musiclever
To invert an interval consists to put an octave higher the interval lowest note. An interval and its inversion are complementary and always form an octave.
[51]
Pythagorean Intervals - UConn Physics
These intervals are defined by the lengths of the strings being in a certain ratio. One does not form an octave by reducing the length of on string by a fixed ...
[52]
The Guidonian Hand | Georgetown University Library
Sep 26, 2023 · A mnemonic device first developed by an Italian monk to teach his religious brethren choral singing some 700 years earlier.
[53]
[PDF] Definitions of "Chord" in the Teaching of Tonal Harmony
Add-sixth chords abound in the pop- rock repertory, as do thirdless 'power chords'. (dyads of perfect fifths or perfect fourths, often doubled in octaves).6 ...<|separator|>
[54]
What is a tritone and why was it nicknamed the devil's interval?
Jun 18, 2018 · There's an interval so dissonant that it's earned the nickname 'The Devil's Interval' and was avoided for centuries by composers and the ...Missing: credible source
[55]
The History and Power of Intermissions - CrowdWork
Jan 5, 2024 · As such, intermissions embody a unique form of narrative power, contributing to the depth and richness of theatrical and cinematic experiences.Missing: reflection views
[56]
Broadway Show Intermission: How Long and What to Do - nytix.com
Intermissions can be anywhere from 10-20 minutes, with 15-minute breaks being by far the most common. To most people, 15 minutes seems like ample time to ...
[57]
How Long Are Broadway Intermissions? Timing & Tips for ...
Rating 5.0 (1) Aug 27, 2025 · Broadway intermissions usually last 15–20 minutes, but what should you do during the break? Learn how long they run, why they matter, ...
[58]
Set the Pace With This Guide to Film Rhythm Editing - Backstage
Sep 16, 2024 · In addition to pace and shot duration, pauses or “breaths”—intentional moments of silence or stillness—can add impact and depth to a scene.
[59]
Should Movie Theater Intermissions Make a Comeback?
Nov 8, 2023 · Historically, movie theater intermissions were needed to change out film reels, giving audiences time to use the restroom and stretch their legs.
[60]
How to Effectively Handle Time Shifts in Your Story | Jane Friedman
Sep 28, 2015 · The key is to ensure each time shift occurs at the right moment in your fiction. A character's past and future are always relevant to her present.
[61]
Fonda Lee's Advice On Writing Time Jumps In Fiction - Medium
Jan 4, 2024 · On the other hand, time jumps allow the writer to greatly expand the scope of the story, and when they work well, they preserve and even ...
[62]
Everything you need to know about going to the opera
Feb 1, 2022 · Even though operas can be multiple hours long, there are often multiple intermissions, so your child won't have to sit for more than an hour ...
[63]
Performance running times | Lyric Opera of Chicago
Performance running times. Cavalleria rusticana & Pagliacci, 3 hours, including 1 intermission. Carmina Burana, 1 hour and 5 minutes, with no intermission.
[64]
Is the theatre interval becoming a thing of the past? - The Guardian
Aug 18, 2024 · Its artistic director, Michelle Terry, said Shakespeare's plays were “never written with intervals, so we won't play them with intervals”.
[65]
The Globe Theatre - Shakespeare - CSM Library at College of San ...
Sunlight provided the lighting, although torches were sometimes lit to suggest night scenes. There were no intermissions. All performances had to end before ...
[66]
LA Opera's "Das Rheingold": 2 hours, 45 minutes, no intermission
Feb 19, 2009 · ... no intermission because that's the way Wagner wrote it. E-mails are sent to ticket buyers warning them of the fact. Advertisement. But not to ...
[67]
In praise of ceasing: Taking breaks in theatre, life, and war
Dec 3, 2023 · The call for intermissions must extend beyond the theatre into our daily lives, given their capacity to soothe the human mind.
[68]
Exploring Theatre Hot Takes: Give Us an Intermission, Please
Sep 16, 2025 · An intermission gives everyone a moment to process what has already happened, to carry the stakes into the second half, and to return with clear ...Missing: reflection | Show results with:reflection
[69]
Interval Training to Boost Speed and Endurance - Verywell Fit
Jun 12, 2024 · Interval training is built upon alternating short, high-intensity bursts of speed with slower, recovery phases throughout a single workout.
[70]
Interval training: A shorter, more enjoyable workout? - Harvard Health
Jun 1, 2024 · As its name implies, HIIT features high-intensity (vigorous) activity done in intervals (short time periods) with brief periods of either rest ...
[71]
Sprinting Toward Fitness - ScienceDirect.com
May 2, 2017 · In the 1930s, a German physician and coach, Woldemar Gerschler, along with cardiologist Herbert Reindel, devised a system of training that ...Missing: history | Show results with:history
[72]
A Brief History of Interval Training: The 1800's to Now
Aug 18, 2016 · During this same period the famous German coach Woldemar Gerschler came up with an interval training method based on heart rates to monitor ...
[73]
What's the difference between fartlek, tempo and interval runs?
Jul 22, 2025 · Fartlek is unstructured speed play, tempo is slightly above anaerobic threshold, and interval runs are short, intense efforts with recovery.
[74]
Interval Training: Health Benefits & Types (HIIT, Tabata & Fartlek)
Dec 3, 2024 · Fartlek (which is Swedish for "speed play") is a type of interval running workout. It consists of high-speed runs or sprints with intervals of ...
[75]
Exercise advice - CERG - NTNU
CERG recommends 4x4-minute interval training, even for older adults, and a 7-week program for better fitness and health.Missing: cycling | Show results with:cycling
[76]
How to Use the Norwegian 4x4 Workout Method to Increase Your ...
May 12, 2025 · The Norwegian 4x4 workout is 4 sets of 4-minute intervals at 85-95% of max heart rate, with 3 minutes rest, to increase VO2 max.
[77]
Effects of moderate-intensity endurance and high-intensity ... - PubMed
This study showed that moderate-intensity aerobic training that improves the maximal aerobic power does not change anaerobic capacity.
[78]
Physiological adaptations to low-volume, high-intensity interval ...
A growing body of evidence demonstrates that high-intensity interval training (HIT) can serve as an effective alternate to traditional endurance-based training.<|control11|><|separator|>
[79]
VO 2 max Trainability and High Intensity Interval Training in Humans
The main finding of this meta-analysis is that interval training produces improvements in VO2max slightly greater than those typically reported with what might ...
[80]
Effectiveness of high-intensity interval training for weight loss in ...
Jul 20, 2021 · HIIT is as effective as MICT in inducing weight loss despite requiring shorter training sessions, with comparable side effects and dropout rate.
[81]
Efficacy of Interval Training in Improving Body Composition and ...
Jul 14, 2024 · Interval training showed greater reductions in total body fat compared to MICT and non-exercise control, and significant reductions in fat mass ...
[82]
Intervals, Part 4 - Joe Friel
Aug 6, 2011 · Let's start with the word “intervals.” What this actually refers to is the time between the hard efforts—the recovery breaks or pauses between ...
[83]
https://fascatcoaching.com/blogs/training-tips/race-specific-cycling-intervals
[84]
The physiology and psychology of negative splits - PubMed Central
Recent studies have demonstrated that athletes who adopt a negative split strategy tend to achieve better overall times and experience less physiological ...
[85]
Marathon Pacing Strategy: How Negative Split Training Prevents the ...
Research reveals why negative split training improves marathon times by 2-3 minutes, yet 92% of runners fail to execute it. Discover the proven method.<|separator|>
[86]
Cycling Training to Accelerate or Attack on Climbs - CTS
Jul 21, 2018 · Cycling Training to Accelerate or Attack on Climbs. Jim Rutberg ... the Tour de France is that the attacks in the mountains are so short.
[87]
Cycling Workouts: 5 Interval Workouts for Different Goals - Bicycling
Dec 31, 2024 · Attack Intervals to Increase Your Threshold · During your next ride, ride as hard as you can for 2 to 3 minutes (you will be flagging by the end) ...
[88]
7 Types of Interval Training Workouts to Try for Faster Swimming
Interval training for swimmers is a strategy that involves alternating shorter bursts of intense effort with periods of either rest or active rest. Interval ...
[89]
Be Aware of the Benefits of Drafting in Sports and Take Your ...
Nov 8, 2023 · In addition, drafting can be used as a strategy to reduce energy expenditure during portions of the race, while maintaining velocity.
[90]
Review Aerodynamic drafting in mass-start non-motorized sports
By leveraging drafting to reduce metabolic demands, athletes can significantly enhance their overall performance, particularly in high-intensity disciplines ...
[91]
Quantifying the Benefits of Drafting for Runners - Outside Magazine
Jun 29, 2020 · A new aerodynamic analysis runs the numbers on exactly where to run when you're behind someone else.
[92]
History and perspectives on interval training in sport, health, and ...
Interval training was developed for athletes in the early 20th century. It was systemized in Sweden as Fartlek, and in Germany as die interval Method, in the ...
[93]
Crossing the Golden Training Divide: The Science and Practice of ...
Middle-distance running was a central part of the Olympic program for men already at the first modern Games in 1896. Over the last century, quantum leaps in ...
[94]
How to Perfect Your 400-Meter Repeats to Get Faster - Runner's World
Jul 3, 2024 · 400-Meter Repeats to Get Faster. No matter your racing distance, this interval workout can improve your speed. Here's how.
[95]
Intervals, Part 5 - Joe Friel
Aug 17, 2011 · For the road-racing cyclist such interval sessions pay off when riding in a fast moving peloton or steady climbing.
[96]
[PDF] Interval Scheduling: A Survey
Apr 21, 2006 · This paper surveys the area of interval scheduling and presents proofs of results that have been known within the community for some time. We ...
[97]
What Every Computer Scientist Should Know About Floating-Point ...
When a floating-point calculation is performed using interval arithmetic, the final answer is an interval that contains the exact result of the calculation.
[98]
[PDF] dynamic data structures for orthogonal - intersection queries
In this paper we will describe a slightly modified version of the interval tree that Leads to a better dynamic tructure than developed by Edelsbrunner [6]. As ...
[99]
RFC 1519 - Classless Inter-Domain Routing (CIDR) - IETF Datatracker
RFC 1519 proposes CIDR, a strategy for address assignment and aggregation to conserve IP space and slow routing table growth by allocating segments to transit ...
[100]
timer_create(2) - Linux manual page - man7.org
The timers created by timer_create() are commonly known as "POSIX (interval) timers". The POSIX timers API consists of the following interfaces: timer_create() ...
[101]
Efficient Genomic Interval Queries Using Augmented Range Trees
Mar 25, 2019 · In this paper, we propose query rewriting and adapt the range trees to design and implement an efficient interval query method that can achieve ...
[102]
Interval Methods for Multi-Point Collisions between Time-Dependent ...
We use a new interval approach for constrained minimization to detect collisions, and a tangency condition to reduce the dimensionality of the search space.
[103]
PR Interval - ECG Basics - LITFL
Feb 4, 2021 · The normal PR interval is between 120 – 200 ms (0.12-0.20s) in duration (three to five small squares). If the PR interval is > 200 ms, first ...PR segment · Lown-Ganong-Levine · Wolff-Parkinson-White
[104]
First-Degree Heart Block - StatPearls - NCBI Bookshelf - NIH
It is defined by electrocardiogram (ECG) changes that include a PR interval of greater than 0.20 without disruption of atrial to ventricular conduction.
[105]
QT Interval - ECG Library Basics - LITFL
Oct 8, 2024 · Normal QTc values. QTc is prolonged if > 440ms in men or > 460ms in women; QTc > 500 is associated with an increased risk of torsades de pointes ...
[106]
An Overview of Heart Rate Variability Metrics and Norms - PMC
Sep 28, 2017 · HRV indexes neurocardiac function and is generated by heart-brain interactions and dynamic non-linear autonomic nervous system (ANS) processes.
[107]
Heart Rate Variability | Circulation
“Heart rate variability” has become the conventionally accepted term to describe variations of both instantaneous heart rate and RR intervals.
[108]
Interictal Problems - Epilepsy Foundation
The interictal period comprises more than 99% of most patients' lives. Interictal cognitive and behavioral disorders profoundly impair the quality of life.
[109]
Delta Wave - an overview | ScienceDirect Topics
Delta waves, defined as low-frequency (typically 0.5–4.5 Hz), high-amplitude oscillations, are a hallmark of non-rapid eye movement (NREM) sleep, particularly ...
[110]
Apnea-Hypopnea Index (AHI): What It Is & Ranges - Cleveland Clinic
Feb 21, 2025 · It identifies how many times your breathing slows or stops during an hour of sleep. You might see an AHI after a sleep study or on a CPAP ...Missing: intervals tidal volume
[111]
Apnea-hypopnea index: time to wake up - PMC - PubMed Central
Apr 5, 2014 · The index may be viewed either as a rate (number of events per hour of sleep) or as a ratio of two variables (number of events/number of hours of sleep).Missing: tidal | Show results with:tidal
[112]
Torsade de Pointes - StatPearls - NCBI Bookshelf - NIH
Torsades de Pointes is associated with QTc prolongation, which is the heart rate adjusted lengthening of the QT interval. The rhythm may terminate spontaneously ...Introduction · Etiology · Epidemiology · Treatment / Management
[113]
Holter monitor - Mayo Clinic
Apr 16, 2024 · A Holter monitor is typically worn for 1 to 2 days. During that time, the device records all of the heartbeats. Holter monitoring is painless.
[114]
Centennial of the string galvanometer and the electrocardiogram
A preliminary report by Einthoven describing the instrument appeared in 1901 (18) and a more detailed description including ECG tracings in 1903 (19). The ...
[115]
Heart Rate Variability – A Historical Perspective - PMC
Nov 29, 2011 · However, the first reports of the applications of HRV in the clinic only began to appear in the mid 1960s. Hon and Lee (1965) noted that fetal ...