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Smooth

''Smooth'' is an adjective with multiple meanings across various fields. In , it refers to concepts such as smooth functions, manifolds, numbers, and algorithms like . In the natural sciences, it describes biological structures like and smooth endoplasmic reticulum. In and , it applies to designs such as smooth bore firearms and techniques like . In arts and entertainment, it is used in music genres and other media contexts.

Mathematics

Smooth function

In mathematical analysis, a smooth function, also known as a C^\infty function, is defined as a function that is infinitely differentiable on its domain, with all derivatives existing and being continuous everywhere in that domain. This means that for a function f: U \to \mathbb{R}, where U is an open subset of \mathbb{R}^n, the partial derivatives of all orders must be continuous functions on U. The class of smooth functions encompasses those that are C^k for every finite k, where C^k denotes functions with continuous derivatives up to order k, but extends beyond any finite level of differentiability. The concept of smooth functions emerged in the 19th century amid advancements in and , particularly through investigations into the nature of differentiability and series expansions. Early examples of smooth but non-analytic functions trace back to in 1823, who constructed functions differentiable to all orders yet not representable by their at certain points, laying groundwork for understanding limitations in classical analysis. This development was further explored by figures like Paul du Bois-Reymond in 1876 and in 1884, who contributed theorems on behavior, solidifying the distinction between infinite differentiability and analyticity in real-variable functions. Smooth functions differ from merely continuously differentiable (C^1) functions, which require only the first to be continuous, allowing for phenomena like corners or cusps in higher derivatives that are absent in smooth cases. More notably, while all real analytic functions—those locally expressible as —are smooth, the converse does not hold; there exist smooth functions that are nowhere analytic, meaning their do not converge to the function itself in any neighborhood. A classic example is the defined by f(x) = \begin{cases} \exp\left(-\frac{1}{x^2}\right) & \text{if } x > 0, \\ 0 & \text{if } x \leq 0, \end{cases} which is smooth on \mathbb{R}, with all derivatives at x=0 vanishing to zero, yet its at 0 is identically zero, failing to represent f(x) for x > 0. This example, originating from Cauchy's work, illustrates how smooth functions can exhibit "flatness" at points without being analytic. Smooth functions form a foundational class in advanced , enabling rigorous treatments in areas such as ordinary and partial differential equations, where solutions often require infinite differentiability for existence and uniqueness theorems, as in the Cauchy-Kovalevskaya theorem. They are also essential in for defining smooth structures on manifolds via transition maps and in proofs involving partitions of unity, which rely on compactly supported smooth functions to approximate and analyze topological properties.

Smooth manifold

A smooth manifold is a topological manifold—a second-countable, that is locally homeomorphic to \mathbb{R}^n—equipped with a defined by an atlas of charts whose transition maps are smooth functions, meaning infinitely differentiable (C^\infty) diffeomorphisms between open subsets of \mathbb{R}^n. This structure ensures that the manifold supports a consistent notion of differentiability, allowing the definition of tangent spaces at each point as the vector space of derivations of the algebra of smooth functions on the manifold. Key properties of smooth manifolds include their local nature, which facilitates the patching of coordinate charts without singularities, and the compatibility of the smooth atlas, which is maximal in the sense that any compatible chart can be added. This smoothness guarantees the existence of well-defined differential forms, vector fields, and tensor fields globally on the manifold, enabling the development of in a coordinate-independent manner. Additionally, smooth manifolds form a category under smooth maps, preserving the structure and allowing for embeddings into larger spaces. The concept of smooth manifolds was formalized in the early , with contributing foundational work on differential forms and moving frames in the 1890s that laid groundwork for local . A rigorous axiomatic definition was provided by and in 1931, establishing manifolds as spaces satisfying postulates for , including the existence of coordinate systems with smooth transitions. This development was pivotal for abstracting geometric ideas beyond spaces. Classic examples include the n-sphere S^n, defined as the unit sphere in \mathbb{R}^{n+1}, which admits a standard smooth structure via stereographic projections or graph coordinates that yield smooth transition maps. The torus T^n, as the Cartesian product of n circles S^1 \times \cdots \times S^1, inherits a smooth manifold structure from the smooth structure on the circle, making it a compact n-dimensional example. Smooth manifolds underpin , serving as the model for in , where the universe is described as a 4-dimensional smooth manifold with a pseudo-Riemannian metric governing and geodesics. They also form the basis for groups, which are smooth manifolds endowed with compatible group operations, essential for groups in and .

Smooth number

In , a B-smooth number, also known as a B-friable number, is a positive whose largest prime is at most B. Equivalently, all prime factors of such a number are less than or equal to B. For the specific case where B=5, these numbers are also called Hamming numbers or , named after who posed the problem of generating the first 1000 such numbers in ascending order. Examples of 5-smooth numbers include 1 (considered smooth by convention), 2, 3, 4=$2^2, 5, 6=$2 \times 3, 8=$2^3, 9=$3^2, 10=$2 \times 5, and 30=$2 \times 3 \times 5, all of whose prime factors are at most 5. In contrast, 7 is not 5-smooth because its only prime factor is 7, which exceeds 5. The sequence of 5-smooth numbers grows by multiplying existing terms by 2, 3, or 5 while maintaining order, avoiding duplicates through careful generation methods. The density of B-smooth numbers up to a large x is approximated by the Dickman-de Bruijn \rho(u), where u = \log x / \log B; this proportion decreases as B decreases for fixed x, reflecting the scarcity of numbers with only very small prime factors. Introduced in the early through studies related to the of prime factors, the gained prominence in analyses of highly composite numbers, where such often feature small primes with optimized exponents to maximize the divisor count. The term "smooth number" itself was coined by in the context of early cryptographic work. Smooth numbers are also relevant to the , particularly in examining solutions where terms have restricted prime factors, leading to bounds on "smooth" abc-triples. Smooth numbers play a key role in factorization algorithms, where identifying smooth composites aids in reconstructing factors from relations in methods like the or number field sieve. In cryptography, they are essential to the Lenstra elliptic curve factorization method (), which exploits the likelihood that the order of an group modulo a prime is smooth, allowing efficient point to reveal divisors. This application underscores their importance in , enabling practical factoring of integers up to moderate sizes.

Smoothsort

Smoothsort is a comparison-based, in-place invented by in 1981 as an adaptive variant of . It employs a collection of binary heaps organized as Leonardo trees, whose sizes follow Leonardo numbers—a sequence similar to the Fibonacci numbers (starting with 1, 1, 3, 5, 9, 15, ...). The algorithm achieves efficiency by recognizing and preserving naturally occurring sorted subsequences, or "runs," in the input data. The key mechanism of smoothsort involves two main phases: heap construction and extraction. In the construction phase, the algorithm scans the array from right to left, building a forest of Leonardo heaps bottom-up by sifting elements into place using operations like "sift" (to maintain order during insertion) and "trinkle" (to adjust larger heaps when smaller ones are fused). This exploits any existing order to minimize adjustments. During extraction, the largest element ( of the principal heap) is repeatedly removed and placed in its final sorted position at the end of the array, after which the remaining structure is rebalanced by splitting the heap into smaller Leonardo trees and reintegrating them. A high-level outline is as follows: 
procedure smoothsort(A[0..n-1]):
    // Phase 1: Build initial forest of Leonardo heaps from right to left
    for i from n-1 downto 0:
        p := i  // Start with single-node [heap](/page/Heap)
        while p has a valid [parent](/page/Parent) [heap](/page/Heap):
            [fuse](/page/Fuse) heaps at p and [parent](/page/Parent)
            sift down if necessary
        trinkle(p)  // Adjust for order across heaps

    // Phase 2: Extract maxima to sort in place
    for i from n-1 downto 1:
        swap A[0] with A[i]  // Move max to position i
        if i > 0:
            split the principal heap at root
            trinkle on subheaps
            semitrinkle if needed for partial adjustments
This process ensures the heaps remain balanced and adaptive, with the "semitrinkle" operation handling cases where full rebalancing is unnecessary for nearly sorted inputs. Developed in Dijkstra's personal notes (EWD 796a) as an improvement over standard , smoothsort was motivated by the desire for a with guaranteed worst-case performance but better behavior on partially ordered data, without the space overhead of . It was published informally through Dijkstra's "EWD" series, emphasizing rigorous program derivation and in-situ operation to avoid auxiliary . The use of Leonardo allows for a smoother degradation in performance compared to binary heaps in heapsort. Time complexity analysis reveals smoothsort's adaptivity: in the worst case, it performs Θ(n log n) comparisons, matching , as every element may require log n adjustments during full operations. In the best case, for already sorted input, it runs in Θ(n) time by recognizing single-element runs and performing minimal sifting. The average case for random inputs is also Θ(n log n). Space complexity is O(1), relying only on a small for tree indices. For a illustration, consider the small [3, 1, 4, 1, 5] using . The algorithm begins by building from the right: forms a heap (size 1), then [1, 5] becomes a P1 heap (size 2) after sifting 1 below 5, followed by fusing with to form a P2 heap [4, 1, 5], and so on, adjusting via trinkle to ensure cross-heap order. Extraction starts by swapping the global max (5) to the end, splitting the principal heap, and reintegrating subheaps like [3, 1, 4, 1] into smaller Leonardo trees, progressively yielding the sorted [1, 1, 3, 4, 5] with fewer operations than a non-adaptive sort due to partial runs. Despite its theoretical elegance, smoothsort is rarely used in practical applications owing to the intricate implementation of Leonardo management, which increases code complexity without significant real-world speed gains over simpler adaptive sorts like . It remains influential in for demonstrating how structures can achieve adaptivity and for studies in lower bounds.

Natural sciences

Smooth muscle

Smooth muscle is a type of involuntary muscle found in the walls of internal organs and structures, characterized by its lack of striations and control by the . Unlike , it contracts slowly and can maintain tension over extended periods without fatigue, enabling functions such as regulating organ diameter and propulsion of contents. This is essential for visceral movements and in multicellular animals. Structurally, smooth muscle consists of elongated, spindle-shaped cells that are typically 30–200 μm long and 3–10 μm wide, each containing a single central . The contractile apparatus comprises and filaments arranged in a disorganized rather than the sarcomeres seen in striated muscle, which contributes to its smooth appearance under light microscopy. These cells are connected by gap junctions in single-unit types, allowing coordinated contractions, while multi-unit smooth muscle operates more independently. The distinction between smooth and skeletal muscle emerged in 19th-century histological studies, with early detailed observations of muscle fine structure highlighting the absence of striations in non-skeletal tissues. These works on microscopic anatomy laid foundational insights into these differences, advancing understanding of tissue-specific organization. Smooth muscle is located in the tunica media of blood vessel walls, where it facilitates vasoconstriction and vasodilation to regulate blood flow and pressure; in the digestive tract, it drives peristalsis for propulsion of food; and in the uterus, it enables rhythmic contractions during labor. In the respiratory system, bronchial smooth muscle controls airway diameter, while in the bladder and ureters, it supports urine transport. These functions rely on autonomic innervation and hormones to modulate tone. For instance, arterial contraction maintains systemic by adjusting vessel resistance, and dysregulation can lead to . In , hyperactivity of airway causes , narrowing passages and impairing breathing, often exacerbated by allergens or irritants. Pharmacologically, is a key target for therapies addressing cardiovascular and respiratory conditions; for example, beta-blockers such as reduce vascular tone indirectly by lowering cardiac output and sympathetic drive, thereby relaxing vascular to manage . These agents also influence airway but require caution in asthmatics due to potential from beta-2 receptor blockade.

Smooth endoplasmic reticulum

The smooth endoplasmic reticulum (SER) is a membrane-bound in eukaryotic cells, consisting of a network of interconnected tubules and flattened sacs that lacks the ribosomes attached to the rough endoplasmic reticulum (). Unlike the RER, the SER appears smooth under electron microscopy due to the absence of ribosomal granules, and its structure facilitates the transport of and other non-protein molecules within the . The SER forms part of a continuous membrane system that connects to the and , with tubule diameters typically ranging from 50-100 nm. The SER was first observed in the 1940s through electron microscopy studies of cells, where it appeared as a lace-like network of intracellular membranes. In 1945, Keith R. Porter, Albert Claude, and Ernest F. Fullam described this structure in detail in their seminal paper on chick embryo cells, marking the initial visualization of the . Porter later coined the term "endoplasmic reticulum" in 1953, distinguishing the smooth variant based on its ribosome-free appearance. The primary functions of the SER revolve around , , and storage, rather than protein . It serves as the site for the of , including phospholipids, , and triglycerides, which are essential for formation and cellular signaling. In cells specialized for production, such as those in the adrenal glands, the SER houses enzymes that convert into steroids like and aldosterone, with abundant tubular networks supporting high synthetic rates. For , particularly in liver hepatocytes, the SER contains enzymes that metabolize lipid-soluble toxins, drugs, and carcinogens by adding hydroxyl groups, enabling their excretion; chronic exposure to substances like barbiturates can induce SER proliferation to enhance this capacity. Additionally, the SER acts as a calcium reservoir, sequestering Ca²⁺ in its via Ca²⁺- pumps and releasing it in response to cellular signals, a process critical for signaling in various types including muscle cells where it aids calcium regulation as the specialized . In contrast to the RER, which is dedicated to protein and folding due to its ribosomal coating, the SER lacks these attachments and instead specializes in non-protein metabolic pathways, such as the modifications performed by embedded in its . This structural difference allows the SER to maintain a more dynamic, tubular architecture suited for transport and activity, without the protein-laden cisternae of the RER.

Engineering and technology

Smooth bore

A is a type of barrel that lacks , meaning its interior surface has no spiral grooves cut into it, in contrast to rifled bores where such grooves impart to projectiles for improved and accuracy. This design allows projectiles to travel straight through the barrel without rotation from the bore itself, relying instead on the projectile's inherent stability or external factors for trajectory control. Historically, were the predominant design from early cannons through the , including used in major conflicts like the , where the smoothbore served as the standard infantry weapon for both and Continental forces. Examples include the , a .75-caliber smoothbore that fired lead balls with an of 50 to 75 yards and a up to three rounds per minute. The transition to rifled bores gained momentum in the , particularly during the , as rifle-muskets offered greater range and accuracy—effective up to 400 yards compared to the smoothbore's 100 yards—though smoothbores remained in use for their simplicity until largely supplanted by rifled designs. In terms of design and function, smoothbores facilitate faster muzzle loading and smoother firing mechanics, as the absence of rifling prevents the projectile from catching on grooves, enabling quicker ramming of and — an advantage in scenarios where exceeded two to three shots per minute, outpacing early rifled arms that required more deliberate loading. This design is particularly suited to shotguns, where multiple pellets or spread in a upon exiting the barrel, providing wide coverage for or close-range defense rather than pinpoint precision. Ballistically, smoothbores trade accuracy for volume of fire; for instance, 19th-century tests showed smoothbore muskets achieving about 36% hit rates at 200 yards versus 64% to 82% for rifle-muskets, but their ease of use supported higher sustained fire rates in tactics. Representative examples include the 12-gauge , a modern commonly used for birdshot loads that produce a spreading effective up to 40 yards for fowl hunting. In contemporary applications, smoothbores persist in shotguns for civilian and roles due to their versatility with散 shot patterns, while in military , such as the M1 tank's 120mm , they fire fin-stabilized sabot rounds for anti-armor roles, achieving higher muzzle velocities (over 1,700 m/s) and reduced barrel wear compared to rifled equivalents, though at the cost of spin-dependent projectile stability. This results in effective ranges exceeding 2,000 meters for kinetic penetrators, prioritizing penetration over the long-range accuracy of older rifled naval guns.

Smoothing

Smoothing is a fundamental technique in statistics and signal processing that involves constructing an approximating function to capture essential patterns in data while attenuating noise or minor fluctuations. This process acts as a low-pass filter, reducing high-frequency components to reveal underlying trends without altering the core structure of the signal. By averaging or weighting nearby data points, smoothing enhances interpretability and facilitates tasks like trend detection in noisy datasets. The historical foundations of smoothing lie in 19th-century statistical methods, particularly the approach developed by in 1805 for orbital calculations and independently formalized by around 1801 to minimize observational errors in astronomy. These early techniques emphasized fitting smooth curves to data via error minimization. With the rise of digital computing in the 1950s, smoothing evolved into practical algorithms, including introduced by Robert G. Brown for inventory control and extended by Charles C. Holt to handle trends, enabling efficient processing of time-series data. Key methods for smoothing include moving averages, which compute local means over fixed windows to suppress short-term variations; Gaussian filters, which apply weights following a bell-shaped distribution for gradual transitions; and splines, such as cubic smoothing splines that interpolate data while penalizing for controlled flexibility. A of nonparametric smoothing is the , particularly the Nadaraya-Watson estimator, which weights observations by their proximity to a target point using a K and h: \hat{f}(x) = \frac{\sum_{i=1}^n y_i K\left(\frac{x - x_i}{h}\right)}{\sum_{i=1}^n K\left(\frac{x - x_i}{h}\right)} This estimator, independently proposed by Nadaraya and Watson, provides a flexible way to approximate regression functions from scattered data. Smoothing splines, formalized by Reinsch, further optimize this by minimizing a combination of residual sum of squares and a roughness penalty. In practice, smoothing appears in image editing tools like Adobe Photoshop, where Gaussian blur reduces pixel-level noise to create softer effects while preserving overall composition. Similarly, in economics, moving average smoothing aids time-series forecasting by dampening seasonal irregularities to highlight long-term economic cycles, as seen in analyses of GDP or unemployment data. Applications span , where uses to model data distributions nonparametrically, enabling generative tasks like without assuming parametric forms. In audio processing, smoothing filters such as low-pass variants eliminate high-frequency hiss or interference, improving clarity in recordings by targeting noise while retaining speech fundamentals. A critical consideration in these uses is the bias-variance : excessive smoothing lowers variance by stabilizing estimates but introduces through overgeneralization, while insufficient smoothing heightens variance from noise sensitivity; optimal bandwidth selection, often via cross-validation, balances this for minimal .

Arts and entertainment

Music

Smooth jazz is a subgenre of that emerged in the mid-1970s and gained prominence in the 1980s, characterized by its fusion of with elements of R&B, , and pop, often featuring mellow tones and accessible melodies. Pioneering artists such as guitarist , who blended soulful vocals and guitar work in albums like Breezin' (1976), helped define the style's relaxed, radio-friendly sound. Saxophonist further popularized in the late 1980s with his soprano saxophone-driven hits like "Songbird" from the 1987 album of the same name, achieving massive commercial success and influencing the genre's dominance on adult contemporary airwaves. In R&B, the "smooth" descriptor is closely tied to the quiet storm radio format, which originated in 1976 on WHUR-FM in Washington, D.C., created by DJ Melvin Lindsey as a late-night program featuring soft, soulful tracks. This format emphasized romantic, laid-back R&B and soul music, evolving into a widespread urban adult contemporary style played on stations like KRNB in Dallas, known for its "Smooth R&B" branding. Iconic tracks exemplifying this smooth R&B aesthetic include Sade's "Smooth Operator," released in 1984 as the third single from the band's debut album Diamond Life, which showcased Helen Folasade Adu's sultry vocals over a sophisticated, jazz-inflected groove and became a hallmark of the quiet storm sound. Another landmark song is Santana's "Smooth," a 1999 collaboration with Rob Thomas of from the Supernatural, which topped the for 12 weeks and won for Record of the Year and Song of the Year in 2000. The track's enduring popularity is evident in its continued streaming success, amassing hundreds of millions of plays on platforms like and in the 2020s. Rapper Smooth, born Juanita Stokes, was an active figure in 1990s , initially debuting as MC Smooth with the 1990 album Smooth & Legit, produced by her brother Chris Stokes and featuring the single "Smooth & Legit" that peaked at #11 on the U.S. chart. She released You Been Played in 1993 under , including the track "Female Mac," followed by her self-titled third album in 1995, which reached #35 on the R&B/ Albums chart and highlighted her blend of with R&B influences. Her work contributed to the era's female-led scene, with hits like "Mind Blowin'" showcasing assertive lyrics over smooth production. In music theory and performance, "smooth" often refers to legato , where notes are played in a connected, fluid manner without audible breaks, creating seamless phrases as described in instrumental pedagogy.

Other media

In publications, Smooth was an upscale urban lifestyle magazine targeted at African American men, blending , , and cultural content with pictorial features of models, published from 2001 to around 2010. Smooth Radio is a of adult contemporary stations launched nationally in 2010 by , following the merger of regional Smooth FM services, and now operated by , it emphasizes and easy-listening tracks for a mature audience. In Australia, smoothfm comprises easy-listening stations rebranded in 2012 under , playing ballads and classics from the 1950s onward across markets like (95.3 ) and (91.5 ). In film, (1985) is a coming-of-age drama directed by , adapted from Joyce Carol Oates's short story "Where Are You Going, Where Have You Been?" The plot centers on 15-year-old ( in her breakout role), a rebellious teen whose flirtatious summer turns menacing when she encounters the charismatic yet predatory Arnold Friend () at a drive-in, forcing her to confront the perils of her budding independence amid family tensions. Premiering at the 1986 , it won the Grand Jury Prize for its nuanced portrayal of adolescent vulnerability and received praise for Dern's performance and Chopra's direction. In literature and comics, adaptations of Michael Jackson's "Smooth Criminal" have appeared in graphic formats, notably the 1989 3-D comic Michael Jackson's Moonwalker, which retells the music video's narrative of Jackson rescuing a girl from a criminal gang in a 1930s speakeasy setting, co-written by Jackson with John Stephenson and David Newman.