Range
Range: Why Generalists Triumph in a Specialized World is a 2019 non-fiction book by science journalist David Epstein, which contends that individuals with diverse experiences and broad knowledge bases—termed "generalists"—often outperform early specialists in unpredictable and multifaceted fields such as innovation, forecasting, and scientific discovery.[1][2] Drawing on case studies from athletics, music, and research, Epstein illustrates how figures like tennis player Roger Federer, who sampled multiple sports before focusing, achieved superior adaptability compared to those like golfer Tiger Woods, who specialized young in more predictable "kind" environments.[1][3] Epstein structures his argument around the distinction between "kind" learning environments, where narrow expertise yields rapid gains, and "wicked" ones, characterized by novelty and incomplete feedback, where interdisciplinary "range" fosters creativity and problem-solving through analogies across domains.[1][3] He advocates for extended "sampling periods" in education and careers to build transferable skills, critiquing the pressure for premature specialization that can limit resilience in dynamic contexts, supported by evidence from polymathic inventors and military forecasters who integrated outsider perspectives.[2][4] The book builds on Epstein's prior work, The Sports Gene, by shifting from genetic factors in performance to experiential breadth, urging readers to embrace inefficiency in exploration for long-term efficacy in an era of accelerating change.[1][5]Mathematics and Statistics
Range of a function
The range of a function f: X \to Y, where X is the domain and Y is the codomain, is the image of X under f, defined as the set \{f(x) \mid x \in X\} \subseteq Y. This set comprises precisely those elements of the codomain that are attained as outputs for some input in the domain.[6][7] The codomain Y is part of the function's specification and may contain elements not reached by f; the range is always a subset of Y, potentially proper. If the range equals the codomain, the function is surjective (onto). In contrast, some educational contexts, particularly in introductory algebra, loosely use "range" to denote the codomain, but this conflates specification with attainment and is imprecise in set-theoretic mathematics.[8][9] Determining the range requires analyzing the function's behavior over its domain. For continuous functions on closed intervals, such as polynomials, the range can be found via critical points and endpoint evaluations using derivatives; for example, the quadratic f(x) = x^2 on \mathbb{R} has range [0, \infty), as the minimum value is 0 at x=0. Discrete functions, like f: \mathbb{N} \to \mathbb{N} given by f(n) = n^2, yield ranges \{1, 4, 9, \dots\}, excluding non-squares. Piecewise or absolute value functions may require case-by-case evaluation, ensuring all branches are considered.[10][11] In applications, the range informs feasibility: for instance, in optimization, it identifies achievable outcomes, while in data mapping, it reveals output constraints. Notationally, the range is often written as \operatorname{range}(f) or \operatorname{im}(f), emphasizing its role as the actual image rather than the intended codomain.[12]Statistical range as dispersion
The statistical range, a fundamental measure of dispersion in descriptive statistics, quantifies the spread of data by subtracting the minimum value from the maximum value in a dataset, expressed as R = X_{\max} - X_{\min}, where X_i represents the observations.[13] This yields the total span over which the data points vary, providing an immediate sense of variability without requiring complex computations.[14] For instance, in a dataset of exam scores {65, 72, 80, 92, 98}, the range is $98 - 65 = 33, indicating the scores extend across 33 points.[15] As a dispersion metric, the range highlights the extent of data scattering, with larger values signaling greater heterogeneity and smaller values implying tighter clustering around central tendencies.[16] It proves useful in preliminary data exploration or when datasets are small and normally distributed, as it captures the full extent of extremes.[17] However, its reliance solely on two endpoints renders it vulnerable to outliers; a single anomalous value can inflate [R](/page/R) disproportionately, misrepresenting the typical spread.[13] Consequently, it underutilizes intermediate data points and lacks robustness for skewed or non-normal distributions, where alternatives like the interquartile range or standard deviation offer more stable insights into central variability.[15] To mitigate some limitations, the coefficient of range normalizes dispersion relative to the data's scale, computed as \frac{X_{\max} - X_{\min}}{X_{\max} + X_{\min}}, yielding a unitless proportion between 0 and 1 that facilitates comparisons across datasets of differing magnitudes.[18] A value near 0 denotes minimal dispersion, while approaching 1 indicates high variability.[18] Despite its simplicity—requiring only sorting and subtraction—the range's sensitivity to sampling variability and extremes confines its role to supplementary analysis rather than primary inference in rigorous statistical modeling.[17]Mid-range statistic
The mid-range is a measure of central tendency in statistics, defined as the arithmetic mean of the minimum and maximum values in a dataset.[19][20] It is computed using the formula \frac{x_{\min} + x_{\max}}{2}, where x_{\min} is the smallest observation and x_{\max} is the largest.[21] This approach requires sorting the data only to identify the extremes, making it computationally simple and fast, particularly for small samples.[22] In contrast to the arithmetic mean, which weights all data points equally, or the median, which selects the middle value after ordering, the mid-range disregards intermediate values entirely.[23] For a dataset such as {1, 3, 5, 7, 9}, the minimum is 1 and the maximum is 9, yielding a mid-range of 5, which coincides with the mean and median in this symmetric case.[22] However, its reliance on only two points renders it highly sensitive to outliers; for {1, 2, 3, 4, 100}, the mid-range shifts to 50.5, far from the mean of 22 and median of 3.[24] This sensitivity constitutes a primary disadvantage, as extreme values can distort the measure disproportionately, reducing its robustness compared to the median or trimmed mean.[24][23] The mid-range also exhibits low statistical efficiency for normally distributed data, performing worse than the sample mean in estimating the population center over repeated samples.[20] Consequently, it is infrequently used in practice, reserved mainly for preliminary approximations or datasets known to lack outliers and exhibit symmetry.[20]Physics and Engineering
Projectile motion range
The range of a projectile in motion on level ground, neglecting air resistance, is the horizontal distance from the launch point to the impact point at the same elevation. This distance R is derived from the kinematic equations for two-dimensional motion under constant gravity, yielding R = \frac{v_0^2 \sin 2\theta}{[g](/page/G)}, where v_0 is the initial launch speed, \theta is the angle of elevation relative to the horizontal, and g \approx 9.8 \, \mathrm{m/s^2} is the acceleration due to gravity near Earth's surface.[25][26] The derivation assumes uniform horizontal velocity v_x = v_0 \cos \theta and parabolic vertical trajectory, with time of flight T = \frac{2 v_0 \sin \theta}{[g](/page/G)}, so R = v_x T./3:_Two-Dimensional_Kinematics/3.3:_Projectile_Motion) This equation holds under idealized conditions: launch and landing at the same height, negligible air drag or other forces, constant g, and no Coriolis effects, which apply well to low-speed, short-range projectiles like thrown objects or light artillery shells.[27] Deviations occur in real scenarios; for instance, air resistance reduces range by dissipating energy, particularly at higher speeds or for elongated projectiles, and requires numerical integration for precise prediction.[28] The range maximizes when \sin 2\theta = 1, at \theta = 45^\circ, giving R_\mathrm{max} = \frac{v_0^2}{[g](/page/G)}; for other angles, symmetric pairs (e.g., 30° and 60°) yield identical ranges./3:_Two-Dimensional_Kinematics/3.3:_Projectile_Motion) If launch height h exceeds zero, the formula modifies to R = \frac{v_0^2}{[g](/page/G)} \left( \sin 2\theta + 2 \cos^2 \theta \sqrt{\frac{2gh}{v_0^2} + \sin^2 \theta} \right), increasing range and shifting the optimal angle below 45°.[26] Experimental validations, such as ballistics tests, confirm the ideal model's accuracy within 5-10% for subsonic velocities under standard conditions.[29]Sensor and signal range
Sensor range in engineering refers to the maximum distance over which a sensor can reliably detect or measure a target or phenomenon, constrained by signal-to-noise ratio thresholds and physical propagation limits. For active sensors like radar or lidar, this range follows principles derived from the inverse square law of signal attenuation, where received power diminishes proportionally to the fourth power of distance in monostatic systems due to both transmission and echo paths.[30] Key determinants include transmitted power, antenna gains (transmit and receive), wavelength (shorter wavelengths enable higher resolution but greater atmospheric absorption), target reflectivity or radar cross-section, and system losses such as noise figure and minimum detectable signal.[30] Environmental and operational factors further modulate sensor range; for radar, atmospheric conditions like rain or fog introduce attenuation, while for ultrasonic sensors, air temperature and humidity affect sound speed and absorption, often limiting effective detection to tens of meters with operating ranges typically 25-50% below the absolute limit to ensure reliability.[31] Inductive proximity sensors exhibit ranges of millimeters to centimeters, reduced by target material conductivity and size, with standards like DIN EN 50205 specifying reductions for non-ferrous metals.[32] In lidar systems, detection range—often exceeding 100 meters in clear conditions—degrades with low target albedo, scattering in fog, or misalignment, emphasizing the causal role of photon return rates over nominal specifications.[33] Signal range, conversely, pertains to the propagation distance of transmitted signals in communication or sensing systems before attenuation renders them unusable, primarily governed by free-space path loss (FSPL) modeled as FSPL = 20 \log_{10}(d) + 20 \log_{10}(f) + 20 \log_{10}(4\pi/c), where d is distance, f frequency, and c speed of light, yielding approximately 6 dB loss per distance doubling at any frequency.[34] Higher frequencies (e.g., 2.4 GHz vs. 900 MHz) incur steeper losses, necessitating elevated antennas for line-of-sight extensions—such as 9.6 m heights for 8 km at 2.4 GHz—while multipath fading from terrain, buildings, or vegetation can halve effective range via destructive interference.[34] Transmit power and receiver sensitivity set baseline limits, but causal obstructions like ionospheric refraction or tropospheric ducting introduce variability, with empirical models showing 10-15 dB additional loss over 50 km in urban RF paths at 450 MHz.[35] In practice, sensor and signal ranges overlap in hybrid systems like wireless sensor networks, where node detection distances (e.g., 10-100 m for Zigbee at 2.4 GHz) balance power constraints against interference, underscoring first-principles trade-offs: amplifying power quadratically extends range but linearly increases noise and energy costs, while beamforming via phased arrays can double effective range by concentrating energy without raw power hikes.[36] These limits stem from electromagnetic wave physics, independent of institutional narratives, with peer-reviewed validations prioritizing measurable attenuation over unsubstantiated claims of indefinite extension via unproven technologies.[37]Frequency and wavelength range
In physics, the frequency range of electromagnetic waves spans from extremely low frequencies around 3 Hz, corresponding to very long radio waves with wavelengths up to 100 km, to gamma rays exceeding 10^{19} Hz with wavelengths below 10^{-11} m. This continuum, known as the electromagnetic spectrum, arises from the propagation of oscillating electric and magnetic fields at the speed of light, where higher frequencies correlate with shorter wavelengths via the relation \lambda = c / f, with c \approx 3 \times 10^8 m/s in vacuum./23%3A_Electromagnetic_Waves/23.1%3A_The_Electromagnetic_Spectrum) Specific subdivisions include radio waves (3 kHz to 300 GHz, wavelengths 100 km to 1 mm), microwaves (300 MHz to 300 GHz, 1 m to 1 mm), infrared (300 GHz to 400 THz, 1 mm to 700 nm), visible light (400 to 790 THz, 700 to 380 nm), ultraviolet (790 THz to 30 PHz, 380 nm to 10 nm), X-rays (30 PHz to 30 EHz, 10 nm to 0.01 nm), and gamma rays (above 30 EHz, below 0.01 nm).[38] In engineering applications, such as signal processing and communications, the frequency range defines the bandwidth over which a system transmits, receives, or processes signals effectively, often limited by hardware capabilities like antennas or filters.[39] For instance, cellular networks operate in bands like 700 MHz to 2.6 GHz for LTE, balancing propagation distance and data capacity, while radar systems use ranges from 1 GHz (L-band) to 40 GHz (Ka-band) for detection precision.[40] Wavelength considerations influence design; longer wavelengths penetrate obstacles better but require larger antennas, as antenna size scales with \lambda / 2 for resonance.[41] Measurement systems, including oscilloscopes, specify operational frequency ranges (e.g., up to 1 GHz for standard models) to ensure accurate capture without distortion.[42] The interplay between frequency and wavelength ranges underpins technologies like spectroscopy, where infrared ranges (0.7–1000 μm) probe molecular vibrations for material analysis, and fiber optics, confined to near-infrared (850–1550 nm) for minimal attenuation.[43] Empirical limits stem from quantum effects at high frequencies and practical propagation losses at extremes, with no theoretical upper bound but observational data capping detected gamma rays around 10^{28} Hz from cosmic sources.[38]Biology and Ecology
Animal home range
The home range of an animal constitutes the spatial domain traversed by an individual during its routine activities, encompassing foraging, mating, resting, and rearing offspring, rather than a strictly defended area.[44] This contrasts with a territory, which represents a subset of the home range that is actively defended against conspecifics through aggression, scent marking, or displays to secure exclusive access to resources.[45] Home ranges often overlap between individuals of the same species, facilitating social interactions without necessitating defense, whereas territories minimize such overlap to reduce competition.[46] Estimation of home range size relies on tracking data from methods such as radio telemetry, GPS collars, or direct observations, analyzed via statistical techniques. The minimum convex polygon (MCP) method delineates the range by forming the smallest convex hull enclosing all recorded locations, providing a simple but potentially biased estimate that includes unused peripheral areas.[47] Kernel density estimation (KDE) offers a more probabilistic approach, generating a utilization distribution where isopleths (e.g., 95% for the full range, 50% for core areas) represent the probability of occurrence, better accounting for clustered activity centers.[48] Other advanced models incorporate autocorrelation in movement data or habitat covariates to refine estimates, though no single method universally outperforms others across species or datasets.[49] Home range size varies predictably with intrinsic and extrinsic factors, grounded in resource demands and environmental constraints. Larger-bodied species exhibit proportionally larger ranges due to higher energetic needs and lower population densities, while males typically maintain larger ranges than females to access dispersed mates.[50] Population density inversely correlates with range size, as heightened competition prompts individuals to exploit smaller, more familiar areas to minimize energy expenditure on exploration and maximize resource familiarity.[51] Habitat productivity, food distribution, and seasonality further modulate sizes: ranges contract in resource-rich environments or wet seasons and expand in sparse or dry conditions; human disturbance often reduces ranges in urban settings by concentrating resources.[52] [53] Empirical examples illustrate these patterns across taxa. Gray squirrels (Sciurus carolinensis) exhibit home ranges of 1.2 to 8 acres (0.005–0.032 km²), scaling with habitat patchiness.[54] Urban coyotes (Canis latrans) have core home ranges averaging 20.31 km² (50% MCP), smaller than rural counterparts due to anthropogenic food subsidies, while red foxes (Vulpes vulpes) average 3.45 km² under similar conditions.[55] In primates, such as Japanese macaques (Macaca fuscata), seasonal ranges span 11.6–15.4 km², shrinking in spring with peak forage availability.[52] Ecologically, home ranges reflect adaptive spatial strategies for resource acquisition and predator avoidance, underpinning fitness through maintained cognitive maps of dynamic environments.[56] Quantifying them informs conservation by delineating minimum viable habitat patches, assessing carrying capacity, and predicting responses to fragmentation or climate shifts, as ranges indicate the scale at which animals perceive and respond to landscape heterogeneity.[57] [58]Ecological tolerance range
The ecological tolerance range, also known as the range of tolerance, refers to the spectrum of environmental conditions under which a given species can survive, grow, and reproduce without experiencing physiological stress or mortality.[59] This concept is formalized in Shelford's Law of Tolerance, proposed by ecologist Victor Ernest Shelford in 1913, which posits that the distribution, abundance, and success of a species are governed by its tolerances to multiple abiotic and biotic factors, each defined by minimum, optimum, and maximum limits.[60] Conditions below the minimum or above the maximum lead to limiting factors that restrict population viability, while the optimum range supports maximal fitness.[61] Tolerance ranges vary by factor and species; for instance, temperature tolerances often delineate fundamental niches, with generalist species exhibiting broader ranges than specialists.[59] Salmon species, such as Oncorhynchus spp., typically endure temperatures from 6°C to 22°C, beyond which metabolic rates disrupt osmoregulation and increase mortality.[62] Salinity tolerances are critical in aquatic environments; estuarine species like raccoons or certain fish withstand fluctuations from freshwater (0 ppt) to near-marine levels (up to 35 ppt), but exceedances cause osmotic stress and cellular damage.[61] Other key factors include pH (e.g., most freshwater fish optimal at 6.5–8.0), oxygen levels (minimum dissolved oxygen around 5 mg/L for many vertebrates to avoid hypoxia), and sunlight exposure, where deviations induce suboptimal photosynthesis in plants or behavioral shifts in animals.[59][63] Individual variation within populations influences effective tolerance ranges, driven by genetic diversity and phenotypic plasticity, such as acclimation to gradual changes.[64] For example, bryophytes like mosses show tolerance limits shaped by moisture and light, with desiccation below critical thresholds halting metabolic processes, while excess water promotes fungal overgrowth.[65] Interactions among factors compound effects; high temperatures may narrow pH tolerance by altering enzyme kinetics, as observed in aquatic invertebrates where combined heat and low oxygen reduces survival by 50% or more in lab studies.[66] Broader geographical ranges correlate with wider tolerances, enabling invasion success in non-native habitats, though local adaptation can shift limits over generations.[67][68] In ecological contexts, tolerance ranges underpin habitat suitability and predict responses to perturbations like climate change, where shifting isotherms may push species beyond limits, contracting distributions poleward or upslope.[69] Specialists, such as certain alpine plants with narrow temperature windows (e.g., 5–15°C optima), face heightened extinction risks from exceedances, whereas generalists like rodents maintain viability across wider gradients.[70] Empirical data from field and lab experiments confirm that the narrowest tolerance often dictates overall range limits, emphasizing multifactor integration over single-variable assessments.[65] This framework informs conservation by identifying vulnerability thresholds, as seen in models forecasting 20–30% range reductions for temperature-sensitive taxa under 2°C warming scenarios.[69]Music and Acoustics
Vocal and instrumental range
Vocal range denotes the span of pitches, from the lowest to the highest note, that a human voice can produce with sufficient control, volume, and tonal quality to be musically usable.[71] This span is typically measured in semitones or octaves and varies by individual physiology, training, and vocal category.[72] For trained singers, a usable range often extends 2 to 3 octaves, though untrained individuals may achieve only 1.5 to 2 octaves reliably.[73] Exceptional cases, such as bass singer Tim Storms, have documented spans exceeding 10 octaves (from G−5 at 0.189 Hz to G/G♯5 at 807.3 Hz), verified in a 2012 recording, but such extremes exceed practical musical application due to timbre limitations at register boundaries.[74] Classical voice classifications delineate standard ranges based on tessitura (the most comfortable pitch span) and passaggio (register shifts). Soprano voices extend approximately from C4 to A5; mezzo-soprano from A3 to F5; alto (or contralto) from F3 to D5; tenor from B2 to G4; baritone from G2 to E4; and bass from E2 to C4, per established music dictionaries.[72] These categories reflect anatomical differences, with female voices generally occupying higher registers due to shorter, thinner vocal folds averaging 17.5–25 mm in length, compared to 17–25 mm in males but with greater mass, enabling lower fundamentals around 85–180 Hz versus 165–255 Hz.[75] Empirical studies of trained singers confirm overlap in voicing ranges, with sopranos reaching up to 1500–2000 Hz and basses down to 80–100 Hz, though full-range phonation demands breath support and resonance adjustments to avoid strain.[75] Instrumental range parallels vocal range as the continuum of pitches an instrument can generate, bounded by its mechanical design, material properties, and performer technique, often extending beyond fundamental tones via harmonics or overtones.[76] For instance, the piano's standard 88-key model spans from A0 (27.5 Hz) to C8 (4186.01 Hz), covering 7 full octaves plus a minor third, limited by string length and tension.[77] String instruments like the violin achieve G3 (196 Hz) to about A7 (3520 Hz), roughly 4 octaves, through finger positioning and bowing, while the guitar's standard tuning yields E2 (82.4 Hz) to E4 (329.6 Hz) on open strings, extendable to higher frets up to about 2–3 additional octaves.[78] Brass instruments such as the trumpet produce from F♯3 (185 Hz) to C6 (1046.5 Hz) or higher in the altissimo register, relying on lip vibration and valve combinations, with practical limits imposed by airflow and embouchure stability.[79] These ranges inform orchestration, as composers select instruments to match required pitches while accounting for timbre variations across the span—lower notes often richer in even harmonics, higher ones brighter but prone to instability.[80]Dynamic range in audio
Dynamic range in audio refers to the measurable difference, expressed in decibels (dB), between the strongest (loudest) and weakest (quietest) signals that an audio system, recording medium, or human ear can handle without distortion or unacceptable noise.[81] This ratio quantifies the span from the noise floor—the inherent background hiss or electronic noise—to the maximum signal level before clipping or overload occurs.[82] In practice, it determines the fidelity with which subtle nuances, such as faint decays or ambient details, can be preserved alongside peaks like percussion strikes or orchestral swells. The human auditory system provides a benchmark, with a dynamic range spanning approximately 120 to 130 dB in the most sensitive frequency bands (around 500 Hz to 6 kHz), from the threshold of hearing near 0 dB sound pressure level (SPL) to pain thresholds exceeding 120 dB SPL.[83] This capability allows perception of whispers at about 20-30 dB SPL alongside jet engines at 140 dB SPL, though adaptation mechanisms like the stapedius reflex compress perceived range over time to around 90-100 dB for sustained listening.[81] Audio systems aim to approximate this to convey realism, as insufficient range results in lost detail (quiet sounds masked by noise) or harsh distortion (loud sounds clipped). In analog audio formats, dynamic range is constrained by physical limits: vinyl records typically achieve 60-70 dB due to groove noise and surface imperfections, while magnetic tape reaches 70-80 dB depending on formulation and bias techniques, limited by tape hiss and saturation.[84] Digital audio, by contrast, derives range from bit depth, where each bit provides roughly 6 dB of resolution; thus, 16-bit depth (standard for CDs) yields a theoretical 96 dB (20-bit log2(2^16) ≈ 96 dB), and 24-bit (common in professional recording) extends to 144 dB, far exceeding human hearing and enabling headroom for processing without requantization noise.[85] However, real-world digital range is often lower due to dithering, quantization errors, and converter noise floors around -120 dB in high-end analog-to-digital converters. Maintaining adequate dynamic range during recording and reproduction preserves expressive intent, as music's emotional impact relies on contrasts—piano to forte transitions that evoke tension and release—rather than uniform loudness.[86] Excessive compression, which narrows range to boost average levels (e.g., for radio play), can induce listener fatigue and diminish perceived depth, though it ensures consistency in noisy environments.[81] In mastering, tools like limiters balance range against commercial loudness standards, but empirical tests show uncompressed recordings with 10-15 dB range retain greater timbral accuracy and spatial imaging on capable systems.[87] Ultimately, the effective range in playback chains—microphones, amplifiers, speakers—is the minimum across components, underscoring the need for matched specifications to avoid bottlenecks.Geography and Land Use
Mountain range
A mountain range consists of a series or chain of mountains aligned in proximity, typically sharing common geological origins through processes such as tectonic uplift, folding, or volcanic activity.[88] These formations arise predominantly at convergent plate boundaries, where lithospheric plates collide, subjecting the crust to compression that results in crustal thickening, metamorphism, and the development of fold-thrust belts.[89] For instance, collisional orogeny occurs as oceanic basins close and continental crusts converge, producing elongated belts with elevations often exceeding 2,000 meters above surrounding terrain.[90] Key characteristics include significant topographic relief—defined as the vertical difference between peaks and adjacent lowlands, frequently surpassing 500 to 1,000 meters—and linear alignment due to underlying structural features like faults or anticlines.[91] Mountain ranges exhibit varied morphologies, from sharp, jagged peaks in young orogenic systems to rounded, eroded profiles in older ones, influenced by factors such as uplift rates, erosion by glaciers and rivers, and isostatic rebound.[92] Erosion plays a causal role in shaping relief, as differential weathering exposes resistant rock layers while transporting sediment, thereby enhancing local prominence over geological timescales.[93] Prominent examples include the Andes, the world's longest continental mountain range at approximately 7,000 kilometers, extending along South America's western edge due to ongoing subduction of the Nazca Plate beneath the South American Plate.[94] The Rocky Mountains span about 4,800 kilometers from Canada to New Mexico, formed by Laramide orogeny around 70-40 million years ago through flat-slab subduction.[95] These ranges not only demarcate biogeographic barriers but also influence global climate patterns by altering atmospheric circulation and precipitation distribution.[96] Empirical data from seismic and GPS monitoring confirm active deformation in such systems, with uplift rates of millimeters per year sustaining their elevation against erosional downcutting.[97]Open range grazing
Open range grazing refers to a system of livestock husbandry in which cattle, sheep, or other animals roam freely across unfenced expanses of public or communal land, with no obligation for owners to confine their herds to specific boundaries.[98] This practice historically relied on natural forage and water sources, with herders using seasonal movements to access grass after winter snowmelt or during migrations.[99] In the American West, open range grazing expanded rapidly from the 1860s onward as settlers introduced large herds to federal public domain lands in states such as Wyoming, Montana, Texas, and the Dakotas, operating under informal "first come, first served" customs tied to control of water holes.[99] Ranchers grazed millions of cattle without fencing, employing cowboys for roundup and branding; by the 1880s, an estimated 7.5 million head roamed the open ranges of the Northern Plains alone.[100] Conflicts arose between cattle and sheep operators, culminating in violent episodes like the Johnson County War of 1892 in Wyoming, where armed ranchers targeted perceived rustlers and smaller competitors, resulting in multiple deaths and federal intervention.[99] The system's viability declined due to overgrazing, which depleted vegetation and caused soil erosion, intensified by droughts and severe winters such as the 1886-1887 "Great Die-Up" that killed up to 90% of some herds.[100] The invention of barbed wire in the 1870s enabled private fencing, fragmenting ranges as homesteaders claimed land under acts like the Homestead Act of 1862, while escalating resource competition prompted calls for regulation.[101] The Taylor Grazing Act, signed into law on June 28, 1934, by President Franklin D. Roosevelt, marked the effective end of unregulated open range by authorizing the Secretary of the Interior to establish grazing districts on approximately 142 million acres of public land, issue permits with capacity limits, and impose fees, reducing overall livestock numbers by over 1 million head to curb the "tragedy of the commons."[99][101] Under the Act, grazing transitioned to allocated allotments managed by the newly formed Grazing Service (merged into the Bureau of Land Management in 1946), prioritizing sustainable use through rotational practices and monitoring to prevent degradation.[99] Today, pure open range persists in limited "fence-out" jurisdictions across Western states including Texas, Idaho, Montana, Wyoming, and Colorado, where statutes place the onus on non-livestock owners to erect barriers, though counties may vote to impose fencing requirements on herders.[98][102] On federal lands, which comprise about 155 million acres of grazed rangeland, operations occur via permits under 43 CFR Part 4100, with annual adjustments for factors like drought or fire, aiming to maintain ecological health while supporting rural economies.[103] Environmentally, unmanaged open range historically contributed to biodiversity loss and watershed impairment through compaction and invasive species proliferation, but regulated grazing can enhance soil carbon sequestration—rangelands store 50% more organic carbon than global forests—and fire resilience when stock densities align with forage capacity.[104] Critics from conservation groups argue persistent impacts include riparian damage and contamination from waste, yet federal assessments emphasize that compliant practices yield net benefits for sagebrush steppe and grassland ecosystems when monitored.[105][106]Technology and Domestic Applications
Kitchen stove range
A kitchen range, also known as a stove, is an integrated cooking appliance consisting of a cooktop for direct heating of cookware and an enclosed oven for baking, roasting, or broiling.[107][108] This design enables simultaneous surface and cavity cooking, distinguishing it from separate cooktops and wall ovens. Typical dimensions range from 30 to 36 inches wide for standard models, with professional styles up to 48 inches.[109] The evolution of kitchen ranges began with open hearths and progressed to enclosed cast-iron wood-burning models in the early 19th century, such as the Oberlin Stove patented in 1834, which was compact enough for domestic use and featured a firebox, oven, and flues for heat circulation.[110] By the mid-19th century, American ranges incorporated ash drawers and warming closets beneath the oven, improving efficiency over earlier step-stoves.[111] Gas ranges emerged in the late 19th century, followed by the first electric range in 1910 by Hotpoint, which offered precise temperature control via resistance heating elements.[112] Ranges are classified by fuel source and installation style. Gas ranges use natural gas or propane flames for rapid, visible heating and precise simmer control, favored by professional cooks, though they emit nitrogen dioxide and other pollutants linked to respiratory risks, with ongoing debate over mitigation via ventilation.[113][114] Electric ranges employ coil or radiant elements under a smooth ceramic-glass surface for even heating and simpler cleaning, but slower response times.[115] Induction ranges, a subtype of electric, generate heat via electromagnetic fields that directly excite ferromagnetic cookware, achieving 84-90% efficiency compared to 40% for gas and 74% for traditional electric, while remaining cool to touch absent compatible pots, reducing burn and fire hazards.[116] Dual-fuel ranges combine a gas cooktop with an electric oven for convection baking.[117] Freestanding models have exposed sides and storage drawers, while slide-in versions fit flush with cabinetry for a seamless look.[107] Safety standards, enforced by the U.S. Consumer Product Safety Commission (CPSC), mandate tip-over resistance, automatic shut-off for unattended flames, and surface temperature limits below 194°F (90°C) on exteriors during operation to prevent scalds.[118][119] Gas models require proper installation to avoid carbon monoxide leaks, and all types necessitate clearance from combustibles like curtains.[120] The U.S. Department of Energy finalized efficiency standards in January 2024 for residential cooking tops and ovens, projecting $1.6 billion in utility savings over 30 years through reduced standby power and improved insulation.[121] In North America, the range market reached $15.81 billion in 2024, driven by demand for induction and smart features like Wi-Fi connectivity for remote monitoring.[122]Shooting and testing range
A shooting range, also known as a firing range, is a specialized facility designed for the safe discharge of firearms for purposes such as target practice, skill development, and competitive shooting. These ranges accommodate various weapons including handguns, rifles, shotguns, and sometimes archery equipment, with facilities divided into indoor and outdoor configurations to suit different calibers and user needs. Indoor ranges typically feature controlled environments with bullet traps and ventilation systems to manage lead particulates and noise, while outdoor ranges provide extended distances for long-range rifles and incorporate natural backstops like berms or hills.[123][124] Safety protocols at shooting ranges universally emphasize four foundational rules derived from organizations like the National Shooting Sports Foundation: treat all firearms as loaded; keep the muzzle pointed in a safe direction downrange; keep fingers off the trigger until ready to fire; and be sure of the target and what lies beyond it. Additional operational standards include eye and hearing protection mandates, prohibitions on alcohol or impairing substances, and restrictions to firing only from designated lines at approved targets to prevent ricochets or stray projectiles. Compliance with these rules reduces accident rates, with U.S. firearm-related injuries at ranges remaining low relative to participation volumes exceeding 20 million annual visits.[125][126] Regulations governing shooting ranges address zoning, environmental impacts, and public safety, varying by jurisdiction but often requiring impenetrable barriers, noise mitigation, and lead remediation plans due to soil and air contamination from spent ammunition. For instance, U.S. facilities must account for projectile containment, solar orientation to avoid glare, and separation from populated areas, with states like West Virginia defining ranges explicitly for rifles, pistols, and trap shooting while exempting established operations from nuisance lawsuits.[127][128][129] Testing ranges extend beyond recreational use to evaluate weapon performance, reliability, and lethality under controlled conditions, particularly in military contexts. These facilities, such as the U.S. Army's White Sands Missile Range—the largest open-air test site in the Department of Defense—support live-fire trials for missiles, artillery, and hypersonic systems, incorporating instrumentation for trajectory tracking and environmental simulations like extreme temperatures or salt fog exposure. Military testing distinguishes from standard shooting by focusing on empirical metrics like muzzle velocity, accuracy at maximum effective range (e.g., up to 1,725 miles for systems like the Long-Range Hypersonic Weapon), and failure rates in adverse conditions, ensuring causal links between design variables and operational outcomes.[130][131][132]Device operational range
The operational range of a device encompasses the defined boundaries of environmental, electrical, and mechanical conditions under which it performs reliably without degradation or failure. These specifications, outlined in manufacturer datasheets, are derived from empirical testing to validate functionality across variables such as temperature, voltage, humidity, and vibration. Exceeding these limits can lead to reduced efficiency, erratic behavior, or permanent damage, as evidenced by thermal runaway in semiconductors beyond rated thresholds.[133][134] Temperature is a primary parameter, with commercial-grade electronic components typically rated for 0°C to 70°C ambient operation to suit office or consumer environments. Industrial variants extend to -40°C to 85°C to accommodate harsher factory conditions, while military specifications reach -55°C to 125°C for reliability in extreme deployments. These ranges account for junction temperatures in active devices like transistors, where heat dissipation influences safe limits; for instance, exceeding 150°C in silicon-based ICs risks electromigration and failure. Humidity operational ranges often span 0% to 95% non-condensing to prevent corrosion or short circuits in enclosures.[133][135][136] Electrical operational ranges include input voltage tolerances, where devices must maintain performance within a nominal value plus or minus a percentage—such as 5V ±5% for logic ICs—to avoid under-voltage lockout or overvoltage breakdown. Frequency ranges apply to RF and communication devices, specifying bands like 2.4 GHz ±100 MHz for Wi-Fi modules to ensure signal integrity without interference. Power consumption limits, often 1-10W for low-power sensors, define sustainable draw under load to prevent thermal overload. Testing protocols, including accelerated life tests at elevated conditions, verify these ranges, with standards like IEC 60068 simulating real-world stresses.[137][138][139] In mechanical devices, operational ranges extend to vibration (e.g., 5-500 Hz at 5g acceleration for automotive sensors) and pressure (e.g., 0.5-10 bar for industrial valves) to guarantee structural integrity. For integrated systems like DC/DC converters, combined ranges—such as -40°C to 85°C at 4.5-5.5V input—ensure cascading failures are minimized. Adherence to these specifications mitigates risks in applications from consumer electronics to aerospace, where deviations have caused incidents like battery fires in portable devices operating outside thermal bounds.[139][140]Named Entities
Places named Range
Several small, unincorporated communities in the United States bear the name Range. Geographic records identify six such locales across Alabama, Ohio, Oklahoma, Oregon, Tennessee, and Wisconsin.[141]| State | Description |
|---|---|
| Alabama | Unincorporated community in Conecuh County along Alabama State Route 41; ZIP code 36473 with an estimated population of 24.[142][143][144] |
| Ohio | Locale within Range Township, Madison County.[145] |
| Oklahoma | Neighborhood in Texas County near Hardesty.[146] |
| Oregon | Unincorporated community in Grant County.[141] |
| Tennessee | Small locale, details limited.[141] |
| Wisconsin | Unincorporated community in Polk County near Amery.[147] |