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46-453: L1 , L01 , L.1 , L 1 or L-1 may refer to: Mathematics, science and technology [ edit ] Math [ edit ] L 1 distance in mathematics, used in taxicab geometry L, the space of Lebesgue integrable functions ℓ, the space of absolutely convergent sequences Science [ edit ] L1 family , a protein family of cell adhesion molecules L1 (protein) ,

92-410: A b 1 + | f ′ ( x ) | d x {\displaystyle {\begin{aligned}s&=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}\Delta x_{i}(1+|f'(x_{i}^{*})|)\\&=\int _{a}^{b}1+|f'(x)|\,dx\end{aligned}}} To test this, take the taxicab circle of radius r {\displaystyle r} centered at the origin. Its curve in

138-974: A , b ] {\displaystyle [a,b]} , the arc length of f {\displaystyle f} over [ a , b ] {\displaystyle [a,b]} is ( b − a ) + ∣ f ( b ) − f ( a ) ∣ {\displaystyle (b-a)+\mid f(b)-f(a)\mid } . Two triangles are congruent if and only if three corresponding sides are equal in distance and three corresponding angles are equal in measure. There are several theorems that guarantee triangle congruence in Euclidean geometry, namely Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), Side-Angle-Side (SAS), and Side-Side-Side (SSS). In taxicab geometry, however, only SASAS guarantees triangle congruence. Take, for example, two right isosceles taxicab triangles whose angles measure 45-90-45. The two legs of both triangles have

184-660: A partition of the interval into equal infinitesimal subintervals, and let Δ s i {\displaystyle \Delta s_{i}} be the taxicab length of the i th {\displaystyle i^{\text{th}}} subarc. Then Δ s i = Δ x i + Δ y i = Δ x i + | f ( x i ) − f ( x i − 1 ) | . {\displaystyle \Delta s_{i}=\Delta x_{i}+\Delta y_{i}=\Delta x_{i}+|f(x_{i})-f(x_{i-1})|.} By

230-499: A 1915 United States Navy L-class submarine HMS L1 , a Royal Navy submarine Other [ edit ] Ligue 1 , the top division of French football Lowest unique bid L1, in linguistics, a subject's first language or mother tongue L-1 Identity Solutions , a US face-recognition corporation L-1 visa , a document used to enter the United States for the purpose of work L1, an abbreviation denoting someone

276-573: A cell adhesion molecule L1 or LINE1 ; transposable elements in the DNA L 1 , Lagrangian point 1, the most intuitive position for an object to be gravitationally stationary relative to two larger objects (such as a satellite with respect to the Earth and Moon) Anthranilic acid , also called vitamin L1 The first lumbar vertebra of the vertebral column in human anatomy The first larval stage in

322-645: A class of 4-4-0 steam locomotives of Great Britain VR Class Vr1 (originally L1), a Finnish steam locomotive class Aircraft [ edit ] Arado L 1 , a 1929 German two-seat parasol-wing sporting monoplane Macchi L.1 , a 1915 Austro-Hungarian reconnaissance flying boat Stinson L-1 Vigilant , the US Army Air Forces designation for the Stinson Model 74 observation aircraft Submarines [ edit ] USS L-1 (SS-40) ,

368-423: A class of British 0-8-2T steam locomotives LNER Thompson Class L1 , a class of British 2-6-4T steam locomotives NCC Class L1 , a Northern Counties Committee 0-6-0 class steam locomotive Pennsylvania Railroad class L1s , an American 2-8-2 steam locomotive class Soo Line L-1 class , an American 2-8-2 steam locomotive class SP&S Class L-1 , an American 4-4-0 steam locomotive class SR L1 class ,

414-634: A computer Sony Xperia L1 , an Android smartphone A class of FM broadcast station in North America Transportation and military [ edit ] Lehrgeschwader 1 , from its historic Geschwaderkennung code with the Luftwaffe in World War II Lufthansa Systems ' IATA code Lawrance L-1 , a predecessor of the 1920s American Lawrance J-1 aircraft engine L1 Dulwich Hill Line on

460-625: A special case, is credited to Riesz. In developing the geometry of numbers , Hermann Minkowski established his Minkowski inequality , stating that these spaces define normed vector spaces . The name taxicab geometry was introduced by Karl Menger in a 1952 booklet You Will Like Geometry , accompanying a geometry exhibit intended for the general public at the Museum of Science and Industry in Chicago. Thought of as an additional structure layered on Euclidean space , taxicab distance depends on

506-418: A sphere is a cross-polytope , the n -dimensional generalization of a regular octahedron , whose points p {\displaystyle \mathbf {p} } satisfy the equation: where c {\displaystyle \mathbf {c} } is the center and r is the radius. Points p {\displaystyle \mathbf {p} } on the unit sphere , a sphere of radius 1 centered at

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552-429: A square with side length 2 r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L 1 and L ∞ metrics does not generalize to higher dimensions. Whenever each pair in a collection of these circles has a nonempty intersection, there exists an intersection point for the whole collection; therefore,

598-463: A taxicab length 2, but the hypotenuses are not congruent. This counterexample eliminates AAS, ASA, and SAS. It also eliminates AASS, AAAS, and even ASASA. Having three congruent angles and two sides does not guarantee triangle congruence in taxicab geometry. Therefore, the only triangle congruence theorem in taxicab geometry is SASAS, where all three corresponding sides must be congruent and at least two corresponding angles must be congruent. This result

644-888: Is | x 1 − x 2 | + | y 1 − y 2 | {\displaystyle \left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|} . That is, it is the sum of the absolute values of the differences in both coordinates. The taxicab distance, d T {\displaystyle d_{\text{T}}} , between two points p = ( p 1 , p 2 , … , p n )  and  q = ( q 1 , q 2 , … , q n ) {\displaystyle \mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})} in an n -dimensional real coordinate space with fixed Cartesian coordinate system ,

690-458: Is a Level 1 Judge, in reference to Magic: The Gathering Bose L1 Portable Systems L=1, a lunar eclipse classification on the Danjon scale L1, the postcode for central Liverpool , a major UK city L1 or Lucifer (2019 Indian film) , a 2019 Indian political action thriller film, followed by L2: Empuraan Containing L1 ATC code L01 Antineoplastic agents , a subgroup of

736-418: Is different from Wikidata All article disambiguation pages All disambiguation pages Taxicab geometry The taxicab distance is also sometimes known as rectilinear distance or L distance (see L space ). This geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO . Its geometric interpretation dates to non-Euclidean geometry of

782-460: Is equal to 4. A closed ball (or closed disk in the 2-dimensional case) is a filled-in sphere, the set of points at distance less than or equal to the radius from a specific center. For cellular automata on a square grid, a taxicab disk is the von Neumann neighborhood of range r of its center. A circle of radius r for the Chebyshev distance ( L ∞ metric ) on a plane is also

828-487: Is given as the sum of every partition of s {\displaystyle s} on [ a , b ] {\displaystyle [a,b]} as they get arbitrarily small . s = lim n → ∞ ∑ i = 1 n Δ x i ( 1 + | f ′ ( x i ∗ ) | ) = ∫

874-489: Is mainly due to the fact that the length of a line segment depends on its orientation in taxicab geometry. In solving an underdetermined system of linear equations, the regularization term for the parameter vector is expressed in terms of the ℓ 1 {\displaystyle \ell _{1}} norm (taxicab geometry) of the vector. This approach appears in the signal recovery framework called compressed sensing . Taxicab geometry can be used to assess

920-642: Is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes . More formally, d T ( p , q ) = ‖ p − q ‖ T = ∑ i = 1 n | p i − q i | {\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {q} )=\left\|\mathbf {p} -\mathbf {q} \right\|_{\text{T}}=\sum _{i=1}^{n}\left|p_{i}-q_{i}\right|} For example, in R 2 {\displaystyle \mathbb {R} ^{2}} ,

966-638: The Caenorhabditis elegans worm development Technology [ edit ] L1, one of the frequencies used by GPS systems (see GPS frequencies ) L1, the common name for the Soviet space effort known formally as Soyuz 7K-L1 , designed to launch men from the Earth to circle the Moon without going into lunar orbit ISO/IEC 8859-1 (Latin-1), an 8-bit character encoding An L-carrier cable system developed by AT&T The level-1 CPU cache in

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1012-534: The E class , and had completed designs for a larger L class at the time of his retirement at the end of 1913. However, by 1918 they were beginning to struggle with the heaviest express trains. Wainwright's successor, Richard Maunsell, therefore rebuilt several examples of the D and E classes immediately before the grouping of the SECR with other railways to form the Southern Railway in 1923. When Maunsell

1058-627: The North British Locomotive Company . The weight of the new class was increased to 57 tons 16 cwt. The boiler pressure was increased from 160 to 180 lbs per square inch, but the cylinders were reduced in diameter from 20 + 1 ⁄ 2 to 19 + 1 ⁄ 2 inches (521 to 495 mm). The engines also had long-travel piston valves, Maunsell's own design of superheater and side-window cab, and other detail alterations. The locomotives were originally numbered A753–A759, and A782–A789, but were later renumbered by

1104-579: The mean value theorem , there exists some point x i ∗ {\displaystyle x_{i}^{*}} between x i {\displaystyle x_{i}} and x i − 1 {\displaystyle x_{i-1}} such that f ( x i ) − f ( x i − 1 ) = f ′ ( x i ∗ ) d x i {\displaystyle f(x_{i})-f(x_{i-1})=f'(x_{i}^{*})dx_{i}} . Then

1150-414: The orientation of the coordinate system and is changed by Euclidean rotation of the space, but is unaffected by translation or axis-aligned reflections . Taxicab geometry satisfies all of Hilbert's axioms (a formalization of Euclidean geometry ) except that the congruence of angles cannot be defined to precisely match the Euclidean concept, and under plausible definitions of congruent taxicab angles,

1196-427: The origin , satisfy the equation d T ( p , 0 ) = ∑ i = 1 n | p i | = 1. {\textstyle d_{\text{T}}(\mathbf {p} ,\mathbf {0} )=\sum _{i=1}^{n}|p_{i}|=1.} In two dimensional taxicab geometry, the sphere (called a circle ) is a square oriented diagonally to the coordinate axes. The image to

1242-407: The side-angle-side axiom is not satisfied as in general triangles with two taxicab-congruent sides and a taxicab-congruent angle between them are not congruent triangles . In any metric space , a sphere is a set of points at a fixed distance, the radius , from a specific center point. Whereas a Euclidean sphere is round and rotationally symmetric, under the taxicab distance, the shape of

1288-430: The 19th century and is due to Hermann Minkowski . In the two- dimensional real coordinate space R 2 , {\displaystyle \mathbb {R} ^{2},} the taxicab distance between two points ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})}

1334-540: The Anatomical Therapeutic Chemical Classification System Barcelona Metro line 1 DSC-L1 , a 2004 Sony Cyber-shot L series camera model Haplogroup L1 (mtDNA) , a human mitochondrial haplogroup from Africa Luxo L-1, a lamp Panasonic Lumix DMC-L1 , a 2006 single-lens reflex camera See also [ edit ] Level 1 (disambiguation) [REDACTED] Topics referred to by

1380-666: The Inner West Light Rail, a light rail service in Sydney, Australia L1 (New York City bus) , a temporary bus route in New York City Volkswagen L1 , a Volkswagen concept hybrid car L1A1 Self-Loading Rifle Locomotives [ edit ] Erie L-1 , an American 0-8-8-0 steam locomotive class GCR Class 1B , latterly known as LNER Class L1, a class of British 2-6-4T steam locomotives GNR Class L1 , latterly known as LNER Class R1,

1426-476: The Manhattan distance forms an injective metric space . Let y = f ( x ) {\displaystyle y=f(x)} be a continuously differentiable function. Let s {\displaystyle s} be the taxicab arc length of the graph of f {\displaystyle f} on some interval [ a , b ] {\displaystyle [a,b]} . Take

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1472-602: The Southern Railway as 1753–1759 and 1782–1789. All passed to British Railways (BR) in 1948, and were renumbered, now between 31753 and 31789. The locomotives were used on express trains on the South Eastern main lines from London to Dover , Ramsgate and Hastings . They remained on these duties until the mid-1930s when they were gradually replaced on the heavier trains by the older "King Arthur" and newer "Schools" classes. They continued to be used on

1518-483: The differences in discrete frequency distributions. For example, in RNA splicing positional distributions of hexamers , which plot the probability of each hexamer appearing at each given nucleotide near a splice site, can be compared with L1-distance. Each position distribution can be represented as a vector where each entry represents the likelihood of the hexamer starting at a certain nucleotide. A large L1-distance between

1564-423: The first quadrant is given by f ( x ) = − x + r {\displaystyle f(x)=-x+r} whose length is s = ∫ 0 r 1 + | − 1 | d x = 2 r {\displaystyle s=\int _{0}^{r}1+|-1|dx=2r} Multiplying this value by 4 {\displaystyle 4} to account for

1610-655: The first quadrant is given by s = ∫ 0 r 1 + | − x r 2 − x 2 | d x = x + r 2 − x 2 | 0 r = r − ( − r ) = 2 r {\displaystyle {\begin{aligned}s&=\int _{0}^{r}1+|{\frac {-x}{\sqrt {r^{2}-x^{2}}}}|dx\\&=x+{\sqrt {r^{2}-x^{2}}}{\bigg |}_{0}^{r}\\&=r-(-r)\\&=2r\end{aligned}}} Accounting for

1656-503: The former London Chatham and Dover Railway main line to Dover and Ramsgate until after the Second World War and the nationalisation of British Railways in 1948 . The transfer of Bulleid "Light Pacifics" to these services in the early 1950s made the class largely redundant. Some were briefly transferred to Eastleigh and to replace worn out locomotives on cross-country services, but most spent their careers between London and

1702-542: The previous equation can be written Δ s i = Δ x i + | f ′ ( x i ∗ ) | Δ x i = Δ x i ( 1 + | f ′ ( x i ∗ ) | ) . {\displaystyle \Delta s_{i}=\Delta x_{i}+|f'(x_{i}^{*})|\Delta x_{i}=\Delta x_{i}(1+|f'(x_{i}^{*})|).} Then s {\displaystyle s}

1748-476: The remaining quadrants gives 4 × 2 r = 8 r {\displaystyle 4\times 2r=8r} again. Therefore, the circumference of the taxicab circle and the Euclidean circle in the taxicab metric are equal. In fact, for any function f {\displaystyle f} that is monotonic and differentiable with a continuous derivative over an interval [

1794-439: The remaining quadrants gives 8 r {\displaystyle 8r} , which agrees with the circumference of a taxicab circle. Now take the Euclidean circle of radius r {\displaystyle r} centered at the origin, which is given by f ( x ) = r 2 − x 2 {\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}} . Its arc length in

1840-410: The right shows in red the set of all points on a square grid with a fixed distance from the blue center. As the grid is made finer, the red points become more numerous, and in the limit tend to a continuous tilted square. Each side has taxicab length 2 r , so the circumference is 8 r . Thus, in taxicab geometry, the value of the analog of the circle constant π , the ratio of circumference to diameter ,

1886-447: The same term This disambiguation page lists articles associated with the same title formed as a letter–number combination. If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=L1&oldid=1232521331 " Category : Letter–number combination disambiguation pages Hidden categories: Short description

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1932-549: The taxicab distance between p = ( p 1 , p 2 ) {\displaystyle \mathbf {p} =(p_{1},p_{2})} and q = ( q 1 , q 2 ) {\displaystyle \mathbf {q} =(q_{1},q_{2})} is | p 1 − q 1 | + | p 2 − q 2 | . {\displaystyle \left|p_{1}-q_{1}\right|+\left|p_{2}-q_{2}\right|.} The L metric

1978-483: The two vectors indicates a significant difference in the nature of the distributions while a small distance denotes similarly shaped distributions. This is equivalent to measuring the area between the two distribution curves because the area of each segment is the absolute difference between the two curves' likelihoods at that point. When summed together for all segments, it provides the same measure as L1-distance. SR L1 class The Southern Railway L1 class

2024-607: Was a class of 4-4-0 steam tender locomotives built for express passenger service on the South Eastern Main Line of the UK Southern Railway . They were designed by Richard Maunsell as a development of Harry Wainwright 's L class . Harry Wainwright had built two useful and attractive 4-4-0 classes for the South Eastern and Chatham Railway between 1902 and 1908; the D class and

2070-558: Was appointed as Chief Mechanical Engineer of the new railway he turned his attention to designing new K & K1 class 2-6-4 tank engines . However, in 1926 there was an urgent requirement for fifteen more powerful 4-4-0 locomotives for the London-to- Folkestone express trains. Maunsell did not rebuild the L class, as the locomotives were still relatively new and useful in their current form, but amended Wainwright's drawings to form his own L1 class, and supplied them to

2116-424: Was used in regression analysis , as a measure of goodness of fit , in 1757 by Roger Joseph Boscovich . The interpretation of it as a distance between points in a geometric space dates to the late 19th century and the development of non-Euclidean geometries . Notably it appeared in 1910 in the works of both Frigyes Riesz and Hermann Minkowski . The formalization of L spaces , which include taxicab geometry as

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