Gaussian curvature reflects the combination of two distinct curvatures (such as on an x-axis and a y-axis).
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Amazon already knows tons of stuff about me; what does it matter if they also know the exact curvatures of my body?
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The idea is that massive objects curve space and time around them, and these curvatures in space-time can affect the paths on which light travels.
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The artist's Contortion series examines mainly a body at rest, depicting the recognizable linear extensions and soft curvatures, which somehow still reflect the dormant energy inside.
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Although insurance coverage has long been hard to come by, most health insurers, including Medicare, now cover Xiaflex treatment for men with curvatures greater than 30 degrees.
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In "Ultra Violet Visionary Transformation No.2" (2014–20163), the mannered curvatures of the syringe-applied paint strands sit awkwardly with the slapdash texture of their enclosed surfaces.
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But to examine them more closely is simply to get reacquainted with the features of the human face: its shapes and curvatures; gradations of light and color; freckles; eyelashes.
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Some of the angles of the Go's body are softer—Groene calls these "curvatures and radii"—making it more comfortable to hold close for extended time periods, like if you're reading or drawing.
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Big, bright, and boasting 1000R curvatures that surround your entire field of vision, Samsung's new Odyssey G7 and G9 QLED ultrawide gaming monitors were two of the big cheeses at last week's CES 2020.
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Both Samsung and MSI introduced what appear to be the first monitors with screen curvatures of 21000R, which is a measurement I admit I have never heard of, but supposedly replicates the human eye.
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Her boat-like shapes can evoke the earth's curvatures, or mesas rising in the distance, or the changing sky, which we can never see in its entirety — the rounded forms thereby also invoke the limits of our sight.
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SpecsLike many wraparound sunglasses, the Barras have a lot of curvatures (six points, to be exact), and are equipped with Smith's patented ChromaPop technology, which, in lay talk, works to separate confusion between the way light passes information to our brains.
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The impeachment debate, which followed similar curvatures of previous dueling Democratic and GOP claims about the propriety of Trump's contacts with Ukraine, provided a new cast of Democratic and Republican lawmakers the opportunity to amplify their views before the rolling television cameras.
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"Abstract work is a great challenge of perfecting composition with balance of weight and depth, and making the flow of the piece match and compliment the general build and curvatures of the body at hand, so that the right parts of the body can be accentuated," she explains.
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The curvatures of the stomach refer to the greater and lesser curvatures. The greater curvature of the stomach is four or five times as long as the lesser curvature.
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J. Linn. Soc. 72, 93–106. 1981 a two-dimensional stiffness. which permits bending at small curvatures and resists bending at high curvatures and skin is attached directly to the underlying muscles.
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The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane The curvature of the normal section is called the normal curvature. If the surface is bow or cylinder shaped the maximum and the minimum of these curvatures are the principal curvatures. If the surface is saddle shaped the maxima of both sides are the principal curvatures. The product of the principal curvatures is the Gaussian curvature of the surface.
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If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group C1; the gasket described by curvatures (−10, 18, 23, 27) is an example.
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In differential geometry, a Dupin hypersurface is a submanifold in a space form, whose principal curvatures have globally constant multiplicities.
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Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval [1,4]. The standard counterexample is complex projective space with the Fubini–Study metric; sectional curvatures of this metric take on values between 1 and 4, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces.
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Normal lordotic curvatures, also known as secondary curvatures, result in a difference in the thickness between the front and back parts of the intervertebral disc. Lordosis may also increase at puberty, sometimes not becoming evident until the early or mid-20s. In radiology, a lordotic view is an X-ray taken of a patient leaning backward.
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Parallel transport Polyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space. Principal curvature is the maximum and minimum normal curvatures at a point on a surface. Principal direction is the direction of the principal curvatures. Path isometry Proper metric space is a metric space in which every closed ball is compact.
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If two different curvatures are repeated within the first five, the gasket will have D2 symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only Apollonian gasket (up to a scaling factor) to possess D2 symmetry.
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The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space. They have additional names and more semantic information attached to them.
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An earlier result (from 1855), due to Ossian Bonnet, has the same conclusion but under the stronger assumption that the sectional curvatures is bounded below by .
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In differential geometry, an isoparametric function is a function on a Riemannian manifold whose level surfaces are parallel and of constant mean curvatures. They were introduced by .
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All physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures. Many equations in relativistic physics appear simpler when expressed in geometric units, because all occurrences of G and of c drop out. For example, the Schwarzschild radius of a nonrotating uncharged black hole with mass m becomes . For this reason, many books and papers on relativistic physics use geometric units.
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The LIGO Hanford Observatory located in Washington, US where gravitational waves were first observed in September 2015. General relativity predicts that energy can be transported out of a system through gravitational radiation. Any accelerating matter can create curvatures in the space-time metric, which is how the gravitational radiation is transported away from the system. Co-orbiting objects can generate curvatures in space-time such as the Earth-Sun system, pairs of neutron stars, and pairs of black holes.
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Kissing circles. Given three mutually tangent circles (), what radius can a fourth tangent circle have? There are in general two possible answers (). Descartes' theorem is most easily stated in terms of the circles' curvatures.
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Opinions differed on how many humps (or rather "bends, curvatures") it had, varyingly given as one, two, or three. It was blamed for making the river overflow and causing the ground and houses to shake.
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The Angolare (“Angular”) series consists of 12 painted works from 1960-1965 for which Castellani fabricated corner-shaped armatures that impose concave and convex curvatures on the canvas, yielding subtly disorienting perceptual and spatial effects.
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Saddle surface with normal planes in directions of principal curvatures All curves on the surface with the same tangent vector at a given point will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing and . Taking all possible tangent vectors, the maximum and minimum values of the normal curvature at a point are called the principal curvatures, and , and the directions of the corresponding tangent vectors are called principal normal directions.
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Its mean curvature is not zero, though; hence extrinsically it is different from a plane. In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. The eigenvalues correspond to the principal curvatures of the surface and the eigenvectors are the corresponding principal directions. The principal directions specify the directions that a curve embedded in the surface must travel to have maximum and minimum curvature, these being given by the principal curvatures.
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For this reason the Gauss-Codazzi equations are often called the fundamental equations for embedded surfaces, precisely identifying where the intrinsic and extrinsic curvatures come from. They admit generalizations to surfaces embedded in more general Riemannian manifolds.
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The primary curves (thoracic and sacral curvatures) form during fetal development. The secondary curves develop after birth. The cervical curvature forms as a result of lifting the head and the lumbar curvature forms as a result of walking.
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The ribbon windows are surrounded by gray frames. In between floors is a thin red band which recalls the red color of lipstick. The curvatures of the building allows light to reflect off the polished stone and stainless surfaces.
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Cities XXL allows players to create a road network of a variety of road types at many different angles and curvatures. Bridges and tunnels are also part of the simulator. Other transport options are buses, trains, ferries, and subways.
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It attempts to depict the movements of people with the lines and soft curvatures of dancing in order to link with Seungmu (Buddhist dance), and to express dynamic movement with the tide and wind, which represent the sea of Incheon.
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Animals such as hummingbirds, and bats that eat pollen and nectar, are able to hover. They produce vortex lift with the sharp leading edges of their wings and change their wing shapes and curvatures to create stability in the lift.
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Changes in the skull also occur causing a distinctive "square headed" appearance known as "caput quadratum". These deformities persist into adult life if not treated. Long-term consequences include permanent curvatures or disfiguration of the long bones, and a curved back.
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There are two colic flexures, or curvatures in the transverse colon. The one on the right, the right colic flexure is known as the hepatic flexure. The one on the left, the left colic flexure is known as the splenic flexure.
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One application is the Gromov–Lees Theorem, named for him and Jack Alexander Lees, concerning Lagrangian immersions and a one-to-one correspondence between the connected components of spaces. In 1978, Gromov introduced the notion of almost flat manifolds. The famous quarter-pinched sphere theorem in Riemannian geometry says that if a complete Riemannian manifold has sectional curvatures which are all sufficiently close to a given positive constant, then must be finitely covered by a sphere. In contrast, it can be seen by scaling that every closed Riemannian manifold has Riemannian metrics whose sectional curvatures are arbitrarily close to zero.
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In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.
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High effective masses of charge carriers are a result of low band curvatures, which correspond to low mobility. Fast response times of devices with wide bandgap semiconductors is due to the high carrier drift velocity at large electric fields, or saturation velocity.
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The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to Apollonian gaskets, which were more recently related to the Ramanujan–Petersson conjecture..
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Let ki (for i = 1, ..., 4) denote the curvatures of four mutually tangent circles. Then Descartes' Theorem states The Apollonian gasket has a Hausdorff dimension of about 1.3057.McMullen, Curtis T. (3 October 1997). "Hausdorff dimension and conformal dynamics III: Computation of dimension", Abel.Math.Harvard.edu.
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In Riemannian geometry, an isoparametric manifold is a type of (immersed) submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel normal vector field. The set of isoparametric manifolds is stable under the mean curvature flow.
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Hilbert's lemma was proposed at the end of the 19th century by mathematician David Hilbert. The lemma describes a property of the principal curvatures of surfaces. It may be used to prove Liebmann's theorem that a compact surface with constant Gaussian curvature must be a sphere..
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The Great Western Railway (GWR) 1366 Class was a class of 0-6-0 pannier tank steam locomotive built in 1934. They were a useful design and because of their light weight and short wheelbase and were often used on dockside branches or other lines with sharp curvatures.
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Hence bats cannot travel over long distances as birds can. Nectar- and pollen-eating bats can hover, in a similar way to hummingbirds. The sharp leading edges of the wings can create vortices, which provide lift. The vortex may be stabilised by the animal changing its wing curvatures.
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Euler's theorem asserts that X1 and X2 are perpendicular and that, moreover, if X is any vector making an angle θ with X1, then The quantities k1 and k2 are called the principal curvatures, and X1 and X2 are the corresponding principal directions. Equation () is sometimes called Euler's equation .
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Day often employed popular design motifs such as ogee curves, serpentine curvatures, and spur motifs; the latter two are commonly seen on Day's chairs. With each of his personal stylistic motifs, Day worked to create balance within motion, and method and symmetry within improvisational spontaneity in his furniture. Another common design tool employed by Day is the interplay between positive and negative space, where positive space is created by actual wood, and negative space is created by the empty spaces between wooden elements on his pieces. By bordering the positive spaces on his furniture with curvatures, Day formed eye-catching negative spaces as well that were meant to again create a sense of balance within the piece.
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National Bureau of Standards tests proved that for its weight it was stronger than metal. The plastic plywood was waterproof and could readily be formed into compound curvatures. Processes had been developed so that an entire fuselage could be turned out in two halves. They then were sealed together mechanically.
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TVT (Tierärztliche Vereinigung für Tierschutz) recommend wheels should be at least 30cm for Syrian Hamsters, since smaller diameters lead to permanent spinal curvatures, especially in young animals. They also recommend a solid running surface because rungs or mesh can cause injury.Tierärztliche Vereinigung für Tierschutz e.V., Merkblatt Nr. 62 - Heimtierhaltung, Tierschutzwidriges Zubehör (Stand: Jan.
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In contrast to composites, isotropic materials (for example, aluminium or steel), in standard wrought forms, typically have the same stiffness regardless of the directional orientation of the applied forces and/or moments. The relationship between forces/moments and strains/curvatures for an isotropic material can be described with the following material properties: Young's Modulus, the shear Modulus and the Poisson's ratio, in relatively simple mathematical relationships. For the anisotropic material, it requires the mathematics of a second order tensor and up to 21 material property constants. For the special case of orthogonal isotropy, there are three different material property constants for each of Young's Modulus, Shear Modulus and Poisson's ratio—a total of 9 constants to describe the relationship between forces/moments and strains/curvatures.
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They also prevent food from being forced through the contact area which might cause food packing and periodontal pain and permit a slight amount of stimulation to the gingiva. When two teeth in the same arch are in contact, their curvatures adjacent to the contact areas form spillway spaces which are called as embrasures.
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A horizontal line passes through both zero and the 180th meridians. By convention, a horizontal axis is recorded as 180. In a regular toric lens, the flattest and steepest curvatures are separated by 90°. As a result, the axis of the cylinder is also the meridian with the same power as the recorded sphere power.
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However, Ritchey and Hale had a falling-out. With the project already late and over budget, Hale refused to adopt the new design, with its complex curvatures, and Ritchey left the project. The Mount Palomar Hale Telescope turned out to be the last world-leading telescope to have a parabolic primary mirror., p. 317.
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The mean curvature is an extrinsic measure of curvature equal to half the sum of the principal curvatures, . It has a dimension of length−1. Mean curvature is closely related to the first variation of surface area. In particular, a minimal surface such as a soap film has mean curvature zero and a soap bubble has constant mean curvature.
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A related brace is the Boston brace (underarm brace, also known as a thoraco-lumbo-sacral orthosis, or TLSO), which is more commonly used for scoliosis. That brace does not have a neck ring and is more easily concealed under clothing, thus more acceptable to patients. However, it is not suitable for high thoracic or cervical spinal curvatures.
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201 (2015), no. 3, 993–1071. An essentially analogous program for sectional curvature bounds (from either below or above) was initiated in the 1990s by a highly influential article of Yuri Burago, Mikhail Gromov, and Grigori Perelman, following foundations laid in the 1950s by Aleksandr Aleksandrov.Burago, Yu.; Gromov, M.; Perelʹman, G. A.D. Aleksandrov spaces with curvatures bounded below.
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Saddle surface with normal planes in directions of principal curvatures. A normal plane is any plane containing the normal vector of a surface at a particular point. The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal vector) see Frenet–Serret formulas.
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When new imaging is used that takes into account asymmetries in the curvatures of lateral fissures, the hemispheric asymmetries of the planum temporale become negligible. This newer imaging would indicate that the size of the region would not explain the higher faculties of language in the left hemisphere, but would instead require an analysis of the neural circuitry.
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The curvature may be dorsal or ventral . These curvatures may be the result of an incisor malocclusion (e.g. ventral=overbite/dorsal=underbite). The curvature may also be diagonal, stemming from a wear pattern, offset incisors, or pain in the cheek teeth (rather than the incisors), which causes the horse to chew in one direction over the other.
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He mainly studied contractures of joints, congenital dislocations and spodiloarthritis. One of his main works is the monograph "Biomechanics as the basis of the theory of curvatures of the human body", published in 1923. In 2012 at a sanatorium of the Ministry of Defense of Ukraine in Evpatoria was installed a memorial plaque dedicated to Nikolay Parijskij.
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The city announced an international contest for the best theatre design. Forty designs were submitted, but none was chosen.Karakina, p. 67 States: There were forty three entrants, and...Felner and Gelmer were selected. Finally, the project was drafted along the lines of Dresden Semperoper built in 1878, with its nontraditional foyer following the curvatures of auditorium.
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The Northwest Specialty Sales Co. created several prototypes and induced 38 localities to purchase the tokens. Each locality had the option of using the standard obverse and a customizable standard reverse. The standard reverse also had room for two logos around the outer rim. Each locality determined the logos that were displayed on the reverse upper and lower curvatures.
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This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.
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Fig. 12: The upper diagram shows a cross-section A-A' along the fold crests of two growing folds. The lower part shows a map view of the region. WG=wind gap. Red arrows: direction of lateral fold growth; Wind gap elevation decreases along lateral fold growth direction and curvatures of wind gaps convex towards lateral fold growth direction.
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The second terms, involving squares of displacement gradients, are non-linear, and need to be considered when the plate bending is fairly large (when the rotations are about 10 – 15 degrees). These first two terms together are called the membrane strains. The last terms, involving second derivatives, are the flexural (bending) strains. They involve the curvatures.
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An anatomical variation is that the left vertebral artery can arise from the aortic arch instead of the left subclavian artery. The arch of the aorta forms two curvatures: one with its convexity upward, the other with its convexity forward and to the left. Its upper border is usually about 2.5 cm. below the superior border to the manubrium sterni.
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These features determine nearly any other properties such as optical, mechanical, and electrical properties. Carbon nanotubes are unique "one-dimensional systems" which can be envisioned as rolled single sheets of graphite (or more precisely graphene). This rolling can be done at different angles and curvatures resulting in different nanotube properties. The diameter typically varies in the range 0.4–40 nm (i.e.
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Cities XL allows players to create a road network of a variety of road types at many different angles and curvatures. Bridges and tunnels are also part of the simulator. Other transport options that were planned to be included in the game were airports, trains, ferries and subways. An add-on introducing buses to the game was released in December 2009.
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Hexagonal(II) phases are inverted water-in-oil phases with net concave curvatures describing the lipid and water interactions. Cubic phases (Pn3m, Im3m, la3d, etc.) or bicontinuous cubic phases composed of multiple connected bilayers that resemble a three- dimensional cube.Interaction of Antimicrobial Peptides with Microbial Lipid Extracts: Evidence for Cubic Phase Formation. E. Staudegger, H. Amenitsch, M. Kriechbaum and K. Lohner.
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This makes the cervical line appear as a semicircle in shape. From this view, the root is blunt and cone-shaped. Although there is a large amount of variation between people, the length of the root is usually 2–3 mm longer than the length of the crown. Large curvatures of the root are usually not seen in this tooth.
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The name "axis" comes from the concept of generating a cylinder by rotating a line around an axis. The curve of that cylinder is 90° from that axis of rotation. When dealing with toric lenses, the axis defines the orientation of the steepest and flattest curvatures relative to horizontal and vertical. The "3 o'clock" position is defined as zero, and the 90th meridian is a vertical line.
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Some glasses have a peculiar property called anomalous partial dispersion. Their use in long-focal-length lens assemblies was pioneered by Leitz. Before their availability, calcium fluoride in the form of fluorite crystals were used as material for these lenses; however the low refraction index of calcium fluoride required high curvatures of the lenses, therefore increasing spherical aberration. Fluorite has poor shape retention and is very fragile.
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Proper seating is essential to prevent spinal curvatures. Severe scoliosis is common in DMD and can interfere with sitting, sleeping, and breathing. A wheelchair that is fitted appropriately accounts for frame size, type of seat, lumbar support and cushioning to avoid pressure ulcers. It should be equipped with other mechanical devices, such as tilt ability, in order to provide comfort and to protect the skin.
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Harry Dolan was born in Pittsburgh and attended Pittsburgh High and Carnegie Tech. Studying to be an architect, he failed because he "couldn't make those curvatures," he said once. During his seven years in the Coast Guard that followed, he had ample time to satisfy his appetite for reading books and "all kinds of writing." After his discharge, he worked as fiction editor of The Boston Sun.
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Every connected closed 2-dimensional manifold (surface) admits a constant curvature metric, by the uniformization theorem. There are 3 such curvatures (positive, zero, and negative). This is a classical result, and as stated, easy (the full uniformization theorem is subtler). The study of surfaces is deeply connected with complex analysis and algebraic geometry, as every orientable surface can be considered a Riemann surface or complex algebraic curve.
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He solved the problem of A. G. Vitushkin about the semi-additivity of analytic capacity. This enabled him to solve an even older problem of Paul Painlevé on the geometric characterization of removable sets. Tolsa succeeded in solving the Painlevé problem by using the concept of so-called curvatures of measures introduced by Mark Melnikov in 1995. Tolsa's proof involves estimates of Cauchy transforms.
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From this they created a differential equation for the shape of the bundle relating the elasticity, gravity, and orientational disorder and extracted a simple equation of state to relate the swelling pressure to the measured random curvatures of individual hairs.Goldstein, R. E., Warren, P. B., & Ball, R. C. (2012). "Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles." Physical Review Letters, 108(7).
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His doctoral advisor was Witold Hurewicz. His dissertation thesis was titled Some Fixed Point Theorems He has worked as a professor of mathematics at UCLA, where supervised the PhDs of eight doctoral students. He made a foundational contribution to the theory of Riemannian submersions, showing how geometric quantities on the total space and on the base are related to one another. "O'Neill's formula" refers to the relation between the sectional curvatures.
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Mutations in this gene are a cause of autosomal recessive posterior microphthalmos. The clinical features of this condition include extreme hyperopia due to short axial length with essentially normal anterior segment, steep corneal curvatures, shallow anterior chamber, thick lenses and thickened scleral walls. The palpebral fissures appear narrow because of relatively deep set eyes. Visual acuity is mildly to moderately reduced, and anisometropic or strabismic amblyopia is common.
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In young people the buccal and lingual curvatures lie below the gum as the teeth have not fully erupted. However, as tooth eruption progresses, the curvature is easier to see. The shape of the crown of a tooth helps to prevent periodontal disease. It does this by allowing food to be deflected onto the gums at a specific angle so that the gum tissue can be stimulated and cleaned.
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The implicit function theorem describes conditions under which an equation F(x,y,z)=0 can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult.
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The consequence of SON haploinsufficiency on embryonic development has also been studied in zebrafish animal models (Danio rerio). A range of developmental defects was observed, including bent, shortened or gnarled tails, massive body curvatures with deformed body axes, eye malformations and microcephaly. Embryos that survived for a longer period of time have more severe phenotypes such as spinal malformations with brain oedema, imitating features observed in affected ZTTK syndrome individuals.
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Given a manifold in three dimensions that is smooth and differentiable over a patch containing the point p, where k and m are defined as the principal curvatures and K(x) is the Gaussian curvature at a point x, if k has a max at p, m has a min at p, and k is strictly greater than m at p, then K(p) is a non-positive real number..
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Items directly beneath the radar appear as if optically viewed horizontally (i.e., from the side) and those at far ranges as if optically viewed from directly above. These curvatures are not evident unless large extents of near-range terrain, including steep slant ranges, are being viewed. When viewed as specified above, fine-resolution radar images of small areas can appear most nearly like familiar optical ones, for two reasons.
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Gromov showed that if the scaling possibility is broken by only considering Riemannian manifolds of a fixed diameter, then a closed manifold admitting such a Riemannian metric with sectional curvatures sufficiently close to zero must be finitely covered by a nilmanifold. The proof works by replaying the proofs of the Bieberbach theorem and Margulis lemma. Gromov's proof was given a careful exposition by Peter Buser and Hermann Karcher.Hermann Karcher.
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With the knee in flexion, the radius of curvatures of the condyles is decreased and the origin and insertions of the ligaments are brought closer together which make them lax. The pair of ligaments thus stabilize the knee joint in the coronal plane. Therefore, damage and rupture of these ligaments can be diagnosed by examining the knee's mediolateral (side) stability. Immediately below its origin is the groove for the tendon of the popliteus.
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A special class of accelerated observers follow worldlines whose three curvatures are constant. These motions belong to the class of Born rigid motions, i.e., the motions at which the mutual distance of constituents of an accelerated body or congruence remains unchanged in its proper frame. Two examples are Rindler coordinates or Kottler-Møller coordinates for the proper reference frame of hyperbolic motion, and Born or Langevin coordinates in the case of uniform circular motion.
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It serves as the standard method for quantification of scoliosis deformities. Sagittal plane posture aberrations such as cervical and lumbar lordosis and thoracic kyphosis have yet to be quantified due to considerable inter- individual variability in normal sagittal curvature. The Cobb method was also one of the first techniques used to quantify sagittal deformity. As a 2D measurement technique it has limitations and new techniques are being proposed for measurement of these curvatures.
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This was accomplished with some resistance from Löber and the other employees who preferred to remain with their traditional methods. The plan was to measure every individual property of each lens element before an objective was constructed to allow precise reproduction of the optical system. The D objective, for example contained 5 lenses. Each was composed of glass with a specific index of refraction, with exact curvatures, of a specific focal length and exact spacings.
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There are also areas where the trains cannot run at high speed near the stations: - for trains leaving Brussels, 220 km/h will not be possible until the trains reach Schaarbeek and HSL-4 does not begin until a few kilometres after Antwerp. The track around Rotterdam station has curvatures that are too tight to allow trains to run at full speed and trains run on conventional track between Schiphol and Amsterdam.
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It has also been hypothesized that polyaxial screws add a safety benefit by failing in the housing/screw interface before breaking in the shaft of the bone screw or in the orthopaedic rod. Unlike standard lateral mass plate and screw systems, the new cervical polyaxial screw and rod system easily accommodates severe degenerative cervical spondylosis and curvatures. This instrumentation system allows for polyaxial screw placement with subsequent multiplanar rod contouring and offset attachment.
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For example, the figure that depicts the law of closure portrays what we perceive as a circle on the left side of the image and a rectangle on the right side of the image. However, gaps are present in the shapes. If the law of closure did not exist, the image would depict an assortment of different lines with different lengths, rotations, and curvatures—but with the law of closure, we perceptually combine the lines into whole shapes.
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The experiment studied the adsorption of organic molecules on carbon nanoparticles and carbon nanotubes. According to the Polyani theory the surface defect curvatures of carbon nanoparticles could affect their adsorption. Flat surfaces on the particles will allow more surface atoms to approach adsorbing organic molecules which will increase the potential, leading to stronger interactions. The theory has been beneficial in trying to understand the adsorption mechanisms of organic compounds on carbon nanoparticles and estimating the adsorption capacity and affinity.
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In the Soviet Union, Alexei Starobinsky noted that quantum corrections to general relativity should be important for the early universe. These generically lead to curvature-squared corrections to the Einstein–Hilbert action and a form of f(R) modified gravity. The solution to Einstein's equations in the presence of curvature squared terms, when the curvatures are large, leads to an effective cosmological constant. Therefore, he proposed that the early universe went through an inflationary de Sitter era.
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The eigenvalues of are just the principal curvatures and at . In particular the determinant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shape operator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically.
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Williston explains that the humerus and femur of Dissorophus are solidly built and stouter. The humerus has "deep lateral curvatures and wide supracondicular ridges" while the femur is a lot stronger built compared to the humerus. He also mentions that the articular surface of Dissorophus femur is "flattened with sharp rims on the antero-posterior convexity". He adds that both femur and humerus are both "expanded on the inner and outer side and narrow in the middle".
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Figure 3. Gold nanoparticle filled and core-less spherical nucleic acid structures (SNAs). Due to their structure and function, SNAs occupy a materials space distinct from DNA nanotechnology and DNA origami,Han, D.; Pal, S.; Nangreave, J.; Deng, Z.; Liu, Y.; Yan, H. "DNA Origami with Complex Curvatures in Three-Dimensional Space," Science, 2011, 332, 342-346, doi: 10.1126/science.1202998.Seeman, N. C. "DNA in a material world," Nature, 2003, 421, 427-431, doi:10.1038/nature01406.
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Convex aspheric curvatures are used in many presbyopic vari-focal lenses to increase the optical power over part of the lens, aiding in near-pointed tasks such as reading. The reading portion is an aspheric "progressive add". Also, in aphakia or extreme hyperopia, high plus power aspheric lenses can be prescribed, but this practice is becoming obsolete, replaced by surgical implants of intra-ocular lenses. Many convex types of lens have been approved by governing agencies regulating prescriptions.
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The three given circles of this Apollonius problem form a Steiner chain tangent to the two Soddy's circles. Figure 12: The two solutions (red) to Apollonius' problem with mutually tangent given circles (black), labeled by their curvatures. Either Soddy circle, when taken together with the three given circles, produces a set of four circles that are mutually tangent at six points. The radii of these four circles are related by an equation known as Descartes' theorem. In a 1643 letter to Princess Elizabeth of Bohemia,Descartes R, Œuvres de Descartes, Correspondance IV, (C. Adam and P. Tannery, Eds.), Paris: Leopold Cert 1901. René Descartes showed that : (k_1+k_2+k_3+k_s)^2 = 2( k_1^2 + k_2^2 + k_3^2 + k_s^2) where ks = 1/rs and rs are the curvature and radius of the solution circle, respectively, and similarly for the curvatures k1, k2 and k3 and radii r1, r2 and r3 of the three given circles. For every set of four mutually tangent circles, there is a second set of four mutually tangent circles that are tangent at the same six points.
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Skeletal problems may also be present, including abnormal curvatures of the spine. An informal study of twenty girls with tetrasomy and pentasomy X found that 10 percent of patients had joint laxity in the hips, while 20 percent had joint limitations. Developmentally, people with tetrasomy X frequently show mild delays in the areas of speech development and articulation, language expression and understanding, and reading skills. Delays in motor development are also present, with walking ages ranging from 16 months to 4.5 years.
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Researchers at Tufts developed scaffolds made of spongy silk that feel and look similar to human tissue. They are implanted during reconstructive surgery to support or restructure damaged ligaments, tendons, and other tissue. They also created implants made of silk and drug compounds which can be implanted under the skin for steady and gradual time release of medications. Researchers at the MIT Media Lab experimented with silkworms to see what they would weave when left on surfaces with different curvatures.
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The dioptre can also be used as a measurement of curvature equal to the reciprocal of the radius measured in metres. For example, a circle with a radius of 1/2 metre has a curvature of 2 dioptres. If the curvature of a surface of a lens is C and the index of refraction is n, the optical power is φ = (n − 1)C. If both surfaces of the lens are curved, consider their curvatures as positive toward the lens and add them.
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The CRL gets its reasonably short focal length, on the order of meters, by using many lenses in series, hence reducing the curvatures of each lens to practical levels. Absorption in the lens is still a challenge, however, and lenses are usually made from low-atomic-number materials such as aluminium, beryllium, or lithium. CRLs were first demonstrated in the mid-1990s by a group of scientists at the ESRF. They drilled holes in an aluminium block, and achieved focusing in two dimensions.
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Hemodynamics is the study of blood flow in the circulatory system. Blood flow in straight sections of the arterial tree are typically laminar (high, directed wall stress), but branches and curvatures in the system cause turbulent flow. Turbulent flow in the arterial tree can cause a number of concerning effects, including atherosclerotic lesions, postsurgical neointimal hyperplasia, in-stent restenosis, vein bypass graft failure, transplant vasculopathy, and aortic valve calcification. Comparison of air flow around a smooth golf ball versus a dimpled golf ball.
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Salmonsens konversationsleksikon (biography) Tscherning is remembered for contributions made in optical physiology. He conducted research of entoptic phenomenon, Purkinje images, the etiology of myopia, and Listing's law of ocular movement. He also designed an ophthalmophacometer, a device used to measure changes that happen in the front and back curvatures of the lens during accommodation. He is probably best known for his theory regarding the mechanism of accommodation, of which he disagreed with the accommodation theory proposed by Hermann von Helmholtz (1821-1894).
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The chief negative characteristic of 8mm Lebel ammunition was the geometry of its rimmed bottlenecked case, due to the will to reuse 11mm Gras case set of tools. This adversely affected the magazine capacity and functioning of firearms, particularly in automatic weapons such as the Chauchat machine rifle. The Lebel cartridge's heavily tapered case shape and substantial rim forced weapon designers to resort to magazines with extreme curvatures as for the Chauchat machine rifle. In contrast, rimless straight-wall cartridges such as the .
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At smaller sizes the files can be pre-curved which is a major advantage for the debridement of roots with sharp curvatures. Their rigidity also has an advantage in calcified root canals in the initial stages of debridement. The ISO stainless steel files on the market today include K-Flex, K-Flexofile and Hedström where the tip size and taper is standardised. ISO normed hand files have a standardised taper of 2% that equates to 0.02mm increase in diameter per mm of file.
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A ridge In differential geometry, a smooth surface in three dimensions has a ridge point when a line of curvature has a local maximum or minimum of principal curvature. The set of ridge points form curves on the surface called ridges. The ridges of a given surface fall into two families, typically designated red and blue, depending on which of the two principal curvatures has an extremum. At umbilical points the colour of a ridge will change from red to blue.
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Some of the basic concepts of general relativity can be outlined outside the relativistic domain. In particular, the idea that mass/energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a Newtonian setting. General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with Fermi–Walker transport along world lines in spacetime. The equations are also generalized to more complicated curvatures.
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After scale space extrema are detected (their location being shown in the uppermost image) the SIFT algorithm discards low-contrast keypoints (remaining points are shown in the middle image) and then filters out those located on edges. Resulting set of keypoints is shown on last image. Scale-space extrema detection produces too many keypoint candidates, some of which are unstable. The next step in the algorithm is to perform a detailed fit to the nearby data for accurate location, scale, and ratio of principal curvatures.
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This asymmetric expansion creates two curvatures, with the ventral side creating the lesser curvature and the dorsal side creating the greater curvature. The expanding dorsal stomach wall then rotates the on its transverse plane, pulling its caudal portion upward and forcing the upper duodenum into a C shape. This rotation positions the left vagus nerve anteriorly and right vagus nerve posteriorly. While the hindgut and midgut are only attached dorsally to the body wall by a fold of peritoneum, the foregut also has a ventral attachment.
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The mullion grid itself is attached to sliding bearings, that allows its curtain wall to adjust to changes in temperature, without compromising the integrity of the wood. Most of the timber was made of Douglas fir trees, from a manufacturer based in Penticton, British Columbia. Each piece of timber is unique, given that the galleria's design featured slants that increased in width incrementally, and whose curvatures were changing throughout its length. The galleria uses 128 steel horizontal beams to prevent the radials from contorting.
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The Geneva Lens Measure is a device used to measure curved surfaces. It is most commonly used by opticians to measure lenses but can also be used by paleontologists to estimate the life size of dinosaur eggs from shell fragments. The instrument can be used to help estimate the size of fossil eggshells by measuring their curved surfaces. Since most eggs aren't perfectly round measurements from multiple parts of the egg with varying shell curvatures may be needed to get a full idea of the egg's size.
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Let v : M → R3 be a smooth immersion of a compact, orientable surface. Giving M the Riemannian metric induced by v, let H : M → R be the mean curvature (the arithmetic mean of the principal curvatures κ1 and κ2 at each point). In this notation, the Willmore energy W(M) of M is given by : W(M) = \int_M H^2 \, dA. It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, with equality if and only if M is an embedded round sphere.
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For the first time in the Corvette's history, wind tunnel testing influenced the final shape, as did practical matters like interior space, windshield curvatures, and tooling limitations. Both body styles were extensively evaluated as production-ready 3/8-scale models at the Caltech wind tunnel. The vehicle's inner structure received as much attention as the aerodynamics of its exterior. Fiberglass outer panels were retained, but the Sting Ray emerged with nearly twice as much steel support in its central structure as the 1958–62 Corvette.
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This then corresponds to the second derivative of a function which, when positive, indicates a curvature that is 'lower at the centre' i.e. sagging. When defining moments and curvatures in this way calculus can be more readily used to find slopes and deflections. Critical values within the beam are most commonly annotated using a bending moment diagram, where negative moments are plotted to scale above a horizontal line and positive below. Bending moment varies linearly over unloaded sections, and parabolically over uniformly loaded sections.
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In polar regions, a Fata Morgana may be observed on cold days; in desert areas and over oceans and lakes, a Fata Morgana may be observed on hot days. For a Fata Morgana, temperature inversion has to be strong enough that light rays' curvatures within the inversion are stronger than the curvature of the Earth.An Introduction to Mirages by Andy Young The rays will bend and create arcs. An observer needs to be within an atmospheric duct to be able to see a Fata Morgana.SDSU.
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Tierärztliche Vereinigung für Tierschutz (TVT) recommends wheels should be at least 20cm (8") for Dwarf Hamsters and at least 30cm (12") for Syrian Hamsters, since smaller diameters lead to permanent spinal curvatures, especially in young animals. They also recommend a solid running surface because rungs or mesh can cause injury. It has been published in several books about small pet care as far back as 2000 that rungs and mesh wheels can cause injuries. Most wheels are constructed of steel, wood or plastic, each having advantages and disadvantages.
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World line of a circular orbit about the Earth depicted in two spatial dimensions X and Y (the plane of the orbit) and a time dimension, usually put as the vertical axis. Note that the orbit about the Earth is (almost) a circle in space, but its worldline is a helix in spacetime. General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with world lines in spacetime. The equations are also generalized to more complicated curvatures.
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In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gi, such that as i goes to infinity the manifold is close to a k-dimensional space, where k < n, in the Gromov–Hausdorff distance sense. Generally there are some restrictions on the sectional curvatures of (M, gi). The simplest example is a flat manifold, whose metric can be rescaled by 1/i, so that the manifold is close to a point, but its curvature remains 0 for all i.
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The theorem states, somewhat surprisingly, that the total integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its total curvature is 4π (the Euler characteristic of a sphere being 2), no matter how big or deep the dent. Compactness of the surface is of crucial importance. Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss–Bonnet formula does not work.
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In contrast to curves that do not have intrinsic curvature but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, . It has a dimension of length−2 and is positive for spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a surface is locally convex (when it is positive) or locally saddle-shaped (when it is negative).
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Spherical aberration occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be ground and polished, and so are often used. Spherical aberration causes beams parallel to, but distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Spherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures for a particular application.
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However these curves are estimated, it is the assumption that they are intrinsically smooth that often defines a functional data analysis. In particular, FDA often makes use of the information in the slopes and curvatures of curves, as reflected in their derivatives. Plots of first and second derivatives as functions of t, or plots of second derivative values as functions of first derivative values, may reveal important aspects of the processes generating the data. As a consequence, curve estimation methods designed to yield good derivative estimates can play a critical role in functional data analysis.
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The final production model, the Formula One McLaren M2B of 1966, only used Mallite for its internal skins and lower bodywork; a lack of understanding of the material's properties had led the team to design the car with conventional curved bodywork, creating problems during the fabrication process as the inherently inflexible material would not readily conform to complex, compound curvatures. With greater understanding of Mallite's properties Herd later used the material to construct the unraced Cosworth four-wheel drive Formula One car of 1969, noted for its slab-sided, angular looks.
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The region ahead of the wing was slightly different on the Sirály II, with a little longer and deeper nose and a slightly more humped canopy with curvatures adjusted to minimize optical distortions. The landing gear changes were more obvious. The Sirály I had a long landing skid from the nose to the leading edge and used a two-wheeled dolly just aft of the skid for take-offs. In contrast, the Sirály II's nose- skid was much shorter and it had a retractable, sprung monowheel under the trailing edge.
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The increase of a allows for an increase of the number of sine waves produced by buckling along the length, but also increases the resistance from the buckling along the width. This creates the preference of the plate to buckle in such a way to equal the number of curvatures both along the width and length. Due to boundary conditions, when a plate is loaded with a critical stress and buckles, the edges perpendicular to the load cannot deform out-of-plane and will therefore continue to carry the stresses.
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Though the sabre had already become very popular in Britain, experience in Egypt did lead to a fashion trend for mameluke sword style blades, a type of Middle Eastern scimitar, by some infantry and cavalry officers. These blades differ from the more typical British ones in that they have more extreme curvatures, in that they are usually not fullered, and in that they taper to a finer point. Mameluke swords also gained some popularity in France as well. Arthur Wellesley, 1st Duke of Wellington, himself carried a mameluke-style sword.
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These two beveled areas have different curvatures and in a well-made razor they transition seamlessly in the ridge (belly) and the cutting edge respectively. Sometimes there are three bevels. The ridge stabilizes the blade against torsional flexing in a direction perpendicular to its longitudinal axis by acting as a lengthwise spine for the blade. The distance between the ridge and the back of the blade is inversely proportional to the hollowness of the blade and is described in fractional terms in ascending steps of as, for example, hollow, hollow, or or (full hollow).
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When the Second Sino-Japanese War broke out in July 1937, Qian returned to China to join the resistance. He helped the Institute of Physics of the Beiping Academy to evacuate Beiping (Beijing), which had come under Japanese attack, and to relocate to Kunming in southwestern China. He assisted Yan Jici, the director of the institute, with establishing an optical workshop in Kunming, where they developed and manufactured hundreds of high-powered microscopes and other instruments for hospitals and factories. He designed an instrument that measured tiny curvatures, which became widely used in Chinese factories.
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The program calculates the necessary reinforcement, checks the R/C sections capacity, generate interaction failure surfaces, calculates the strains, stresses, curvatures, stiffnesses (II order, creep) and the cracks widths for any reinforced concrete cross - section. The program has reported users in over 30 countries:GaLa Reinforcement official link alashki.com. 2000 Germany, USA, Italy, Belgium, Canada, Netherlands, Spain, Portugal, Sweden, Iceland, UK, South Korea, Greece, Columbia, Bulgaria, Singapore, Saipan, China, Taiwan, Vietnam, El Salvador, Brazil, Ireland, Mexico, Chile, Turkey, New Zealand, France, Haiti, Slovenia, Nicaragua, Philippines, Saudi Arabia, Qatar, Austria, United Arab Emirates.
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In short, a model requires a particular amount of matter in order to produce particular curvatures and expansion rates. In terms of matter, all modern cosmologies are founded on the cosmological principle, which states that whichever direction we look from earth, the universe is basically the same: homogeneous and isotropic (uniform in all dimensions). This principle grew out of Copernicus's assertion that there were no special observers in the universe and nothing special about the earth's location in the universe (i.e., Earth was not the center of the universe, as previously thought).
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The brace has a pelvic unit from which strong elastic bands wrap around the body, pulling against curves, rotations, and imbalances. It is most successful when the patient has relatively small and simple curvatures, is structurally young, and compliant—it is usually worn 20 hours a day. The patient is not to have it off for more than two hours at a time. While it is expected that patients can participate in activities as strenuous as competitive gymnastics while in brace, it also pulls down against shoulder misalignments which compresses the spine.
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Coherent detection is needed to capture the signal phase information in addition to the signal amplitude information. That type of detection requires finding the differences between the phases of the received signals and the simultaneous phase of a well-preserved sample of the transmitted illumination. Every wave scattered from any point in the scene has a circular curvature about that point as a center. Signals from scene points at different ranges therefore arrive at a planar array with different curvatures, resulting in signal phase changes which follow different quadratic variations across a planar phased array.
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The first opera house was opened in 1810 and destroyed by fire in 1873. The modern building was constructed by Fellner and Helmer in neo-baroque; its luxurious hall was built in the rococo style. It is said that thanks to its unique acoustics even a whisper from the stage can be heard in any part of the hall. The theatre was projected along the lines of Dresden's Semperoper built in 1878, with its nontraditional foyer following the curvatures of the auditorium; the building's most recent renovation was completed in 2007.
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There are three main variants of the AEK assault rifles: the AEK-971, AEK-972 and AEK-973. The different variants are most easily recognized by their respective magazine curvatures. The AEK-971S and AEK-973S are improved variants of the AEK-971 and AEK-973 which features a three-round burst fire mode and numerous improvements. A heavily improved variant of the AEK-971 and AEK-973 lines' designated as A-545 and A-762 was later released and is set to be used in the Russian Special Operations Forces (Spetsnaz).
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Due to their heightened speeds, route alignments for high-speed rail tend to have broader curves than conventional railways, but may have steeper grades that are more easily climbed by trains with large kinetic energy. Their high kinetic energy translates to higher horsepower-to- ton ratios (e.g. ); this allows trains to accelerate and maintain higher speeds and negotiate steep grades as momentum builds up and recovered in downgrades (reducing cut, fill, and tunnelling requirements). Since lateral forces act on curves, curvatures are designed with the highest possible radius.
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Japan 19 (1967), 205–214. Omori's formulation required the more restrictive assumption that the sectional curvatures of are bounded below by a constant, although it allowed for a stronger conclusion, in which the Laplacian of can be replaced by its hessian. A direct application of the Omori-Yau principle, published in 1978, gives Yau's generalization of the classical Schwarz lemma of complex analysis. Cheng and Yau showed that the Ricci curvature assumption in the Omori-Yau maximum principle can be replaced by the assumption of the existence of smooth cutoff functions of certain controllable geometry.
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Effective use of the towed array system requires a vessel to maintain a straight, level course over a data sampling interval. Maneuvering, or changing course, disturbs the array and interrupts the sampled data stream. These periods of instability are closely tested during sea trials and known by the crew's officers and enlisted sonar experts. Modern systems compensate by constantly self-measuring the relative positions of the array, element to element, reporting back data that can be automatically corrected for curvatures by computers as part of the beamforming maths processing.
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No mapping between a portion of a sphere and the plane can preserve both angles and areas. (If one did, then it would be a local isometry and would preserve Gaussian curvature; but the sphere and disk have different curvatures, so this is impossible.) This fact, that flat pictures cannot perfectly represent regions of spheres, is the fundamental problem of cartography. As a consequence, regions on the sphere may be projected to the plane with greatly distorted shapes. This distortion is particularly dramatic far away from the center of the projection (0, 0, −1).
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Epps was the half-brother of physician and homeopath John Epps, and was born on 22 July 1815. He was educated at Mill Hill School in London After being for some years his brother's pupil and assistant, he became a member of the London College of Surgeons in 1845, and was in the same year appointed surgeon to the Homœopathic Hospital in Hanover Square. He was successful in treating spinal curvatures and deformities. Epps had a large practice to which he was devoted, never sleeping out of his house for twenty years.
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From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor.
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Dual rotating axis grinding can be used for high index glass that isn't easily spin molded, as the CR-39 resin lens is. Techniques such as laser ablation can also be used to modify the curvature of a lens, but the polish quality of the resulting surfaces is not as good as those achieved with lapidary techniques. Standards for the dispensing of prescription eyeglass lenses discourage the use of curvatures that deviate from definite focal lengths. Multiple focal lengths are accepted in the form of bifocals, trifocals, vari-focals, and cylindrical components for astigmatism.
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A 2013 study by Weinstein et al. found that rigid bracing significantly reduces worsening of curves in the 20-45 degree range and found that 58% of children receiving "observation only" progressed to surgical range. Recent guidelines published by the Scientific Society of Scoliosis Orthopaedic and Rehabilitation Treatment (SOSORT) in 2016 state that “the use of a brace is recommended in patients with evolutive idiopathic scoliosis above 25º during growth” based on a review of current scientific literature. Severe curvatures that rapidly progress may be treated surgically with spinal rod placement.
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From this it is easy to understand why decreasing the surface area of a mass of liquid is always spontaneous, provided it is not coupled to any other energy changes. It follows that in order to increase surface area, a certain amount of energy must be added. Gibbs and other scientists have wrestled with the arbitrariness in the exact microscopic placement of the surface. For microscopic surfaces with very tight curvatures, it is not correct to assume the surface tension is independent of size, and topics like the Tolman length come into play.
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The theory of inflation thus explains why the temperatures and curvatures of different regions are so nearly equal. It also predicts that the total curvature of a space-slice at constant global time is zero. This prediction implies that the total ordinary matter, dark matter and residual vacuum energy in the Universe have to add up to the critical density, and the evidence supports this. More strikingly, inflation allows physicists to calculate the minute differences in temperature of different regions from quantum fluctuations during the inflationary era, and many of these quantitative predictions have been confirmed.
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There are two basic terms used to describe lipid phases: lamellar and non- lamellar phases. Lipids can undergo polymorphic or mesomorphic changes leading to the formation of lamellar or non-lamellar phases. Various factors can affect the overall function of the biomembrane and decrease its ability to function as a protective barrier and maintained the order of the inner components. The bilayer thickness, surface charge, intermolecular forces, amphiphilic molecules, changes in free energy, alternating or spontaneous curvatures, increase or decrease in temperature, solvents, and the environment are all examples of different conditions that cause changes in biomembranes.
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Nested Apollonian gaskets For any integer n > 0, there exists an Apollonian gasket defined by the following curvatures: (−n, n + 1, n(n + 1), n(n + 1) + 1). For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by n + 1 can become the bounding circle (defined by −n) in another gasket, these gaskets can be nested. This is demonstrated in the figure at right, which contains these sequential gaskets with n running from 2 through 20.
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In 1850 he wrote in the Journal de Mathématiques of the lines of curvature and asymptotic lines on a surface, at any point of which the sum of the principal curvatures is zero. The International Exhibition of 1851 at Hyde Park displayed a small model ellipsoid, on which the lines of curvature had been traced according to a method Roberts invented. Roberts published several papers on the properties and functions of the roots of algebraic equations, and on covariants and invariants. From 1868 to 1873 he published work in Annali di Matematica, including in 1869 and 1871 two papers on Abelian function.
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Much of the confusion with respect to the Keying may come from reported bow and stern heights above the waterline that may have been for the tops of 'wings' and 'poop', not of the weather deck at bow and stern. The exaggerated measurements in most contemporary reports suggest it was the former, not the latter. The Museum model unquestionably fits the accounts of Keying's sea-keeping qualities better than a model with the bizarrely exaggerated curvatures shown in other contemporary illustrations. Such curvature was unknown in similar vessels: the acutely distorted waterlines that would result when heeled would have rendered the vessel unmanageable.
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As with his furniture, Day took the popular design of the newel post and added his own stylistic flair through his scroll curves. In crafting these newel posts, Day employed four different types of newels: s-shaped, traditional, a fusion of the two, or completely unique designs; today, twenty-five s-shaped newel posts have been attributed to Day. Day crafted stair brackets to match and complement these newel posts, again employing curvatures and wave motifs that, combined with the newel posts, suggested a tranquil fluidity. The choice to match newel posts and stair brackets appears to be unique to Day's architectural work.
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Allegorical 'Inspiration' and 'Man', communicating, with the vastness of the Earth depicted by the curvatures on its cast bronze panel. The model for 'Man, discovering his power to transmit sound through space', was Cyril William George Kinsella a wounded war veteran and a former resident of Brantford. Kinsella served as Allward's nude model representing Man after being severely injured in the European conflict. Born in the UK he had become one of approximately 100,000 disadvantaged British children and orphans, a "Home Child", sent to Canada and other Commonwealth countries to find better lives in the late 19th and early 20th centuries.
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People with osteoporosis are at higher risk of falls due to poor postural control, muscle weakness, and overall deconditioning. Postural control is important to maintaining functional movements such as walking and standing. Physical therapy may be an effective way to address postural weakness that may result from vertebral fractures, which are common in people with osteoporosis. Physical therapy treatment plans for people with vertebral fractures include balance training, postural correction, trunk and lower extremity muscle strengthening exercises, and moderate-intensity aerobic physical activity.. The goal of these interventions are to regain normal spine curvatures, increase spine stability, and improve functional performance.
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Animated simulation of gravitational lensing caused by a Schwarzschild black hole passing in a line-of-sight planar to a background galaxy. Around and at the time of exact alignment (syzygy) extreme lensing of the light is observed. A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime where the gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter.
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Widening of wrist Signs and symptoms of rickets can include bone tenderness, and a susceptibility for bone fractures particularly greenstick fractures.Medical News – Symptoms of Rickets Early skeletal deformities can arise in infants such as soft, thinned skull bones – a condition known as craniotabes, which is the first sign of rickets; skull bossing may be present and a delayed closure of the fontanelles. Young children may have bowed legs and thickened ankles and wrists;Mayo Clinic – Signs and Symptoms of Rickets older children may have knock knees. Spinal curvatures of kyphoscoliosis or lumbar lordosis may be present.
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Construction of church Longuelo (1961-1966) The church is built with reinforced cement. The geometrical arrangement is described as four autonomous volumes, knitted together by five roofs, also made with thin 6 cm cement. Four of the roofs have double curvatures, while the main central roof is a steep parabolic tent like design.Lombardy Cultural Holdings. Monsignor Andrea Spada, director of the journal L'Eco di Bergamo, said of the building in 1966:L’architetto ha voluto adagiarla sul bordo della strada con tutta la nuda violenza dei suoi motivi architettonici, come capace di bloccare l’attenzione di chi passa. E, se è questo, c’è indubbiamente riuscito.
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Markze has worked at ASU since then, with a 9 year break from 1986 to 1995 when she worked as an anatomist at the Primate Foundation of Arizona. Markze has been a professor at ASU since 2004, most recently teaching courses on primate anatomy and fossil hominins. Apart from teaching, Markze has done an extensive amount of research throughout her career, with a “special focus on the evolution of the hominin hand and bipedality.” Markze’s research involves “extensive dissections, electromyography, kinematic analysis of joint angle displacement and tendon excursion, and stereophotogrammetry and laser digitizing for 3-D analysis of joint surface areas, angles and curvatures.
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An application of the Theorema Egregium is seen when a flat object is somewhat folded or bent along a line, creating rigidity in the perpendicular direction. This is of practical use in construction, as well as in a common pizza-eating strategy: A flat slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero.
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Bassleroceras is an elongate upwardly curved, exogastric, genus with the venter on the under side more sharply rounded than the dorsum on the upper. The siphuncle is ventral, composed of thick-walled tubular segments in which connection rings thicken in towardly as in both the Ellesmerocerida and primitive Tarphycerida. Bassleroceras is the type genus of the Bassleroceratidae which Furnish and Glenister (1964) included in the Ellesmerocerida, but which Flower (1967) placed in the Tarphycerida. Bassleroceras gave rise to the Tarphycerida (sensu Furnish and Glenister, 1964) by evolving genera with tighter and tighter curvatures until becoming gyroconic, a character of the Estonioceratidae, a family of early tarphycerids.
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Some have flat bases, for convex work; the more distinctive have curved bases for concave surfaces. They are made with soles in a range of curvatures and a typical workshop will have several, of varying curvature. Ibex finger plane, 35 mm length, with curved sole As the planes are only used for a narrow task, they are not adjustable and they are made with fixed mouths and a blade that is clamped in place by a simple wedge, or by a clamp screw, not an adjuster. To add mass to such a small plane, they are commonly cast in a dense brass or bronze alloy.
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It has sectional curvature ranging from 1/4 to 1, and is the roundest manifold that is not a sphere (or covered by a sphere): by the 1/4-pinched sphere theorem, any complete, simply connected Riemannian manifold with curvature strictly between 1/4 and 1 is diffeomorphic to the sphere. Complex projective space shows that 1/4 is sharp. Conversely, if a complete simply connected Riemannian manifold has sectional curvatures in the closed interval [1/4,1], then it is either diffeomorphic to the sphere, or isometric to the complex projective space, the quaternionic projective space, or else the Cayley plane F4/Spin(9); see .
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Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the generalization to higher- dimensional differentiable manifolds. Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.. See also Lorentzian manifold. Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.
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These include ion-beam finishing, abrasive water jets, and magnetorheological finishing, in which a magnetically guided fluid jet is used to remove material from the surface. Another method for producing aspheric lenses is by depositing optical resin onto a spherical lens to form a composite lens of aspherical shape. Plasma ablation has also been proposed. Lapping tool on a spindle below the lens, and mounting tool on a second spindle (swung out) uses pitch to hold the lens shown with its concave side down The non-spherical curvature of an aspheric lens can also be created by blending from a spherical into an aspherical curvature by grinding the curvatures off-axis.
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The Milwaukee brace, also known as a cervico-thoraco-lumbo-sacral orthosis or CTLSO, is a back brace most often used in the treatment of spinal curvatures (such as scoliosis or kyphosis) in children but also, more rarely, in adults to prevent collapse of the spine and associated pain and deformity. It is a full-torso brace that extends from the pelvis to the base of the skull. It was originally designed by Blount and Schmidt in 1946 for postoperative care when surgery required long periods of immobilization. Milwaukee braces are often custom-made over a mold of the patient's torso, but in some cases, it can be made from prefabricated parts.
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The arc was measured with at least three latitude determinations, so they were able to deduce mean curvatures for the northern and southern halves of the arc, allowing a determination of the overall shape. The results indicated that the Earth was a prolate spheroid (with an equatorial radius less than the polar radius). To resolve the issue, the French Academy of Sciences (1735) proposed expeditions to Peru (Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan) and Lapland (Maupertuis, Clairaut, Camus, Le Monnier, Abbe Outhier, Anders Celsius). The expedition to Peru is described in the French Geodesic Mission article and that to Lapland is described in the Torne Valley article.
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The pubic bones of both pelvis halves are connected via narrow bony skirts that originated at a rather high position on the rear side and continued downwards to a point low on the front side of the shaft. The ischium is S-shaped in side view, showing at the transition point between the two curvatures a rough boss on the outer side. On the front edge of the ischial shaft an obturator process is present in the form of a low ridge, at its top separated from the shaft by a notch. To below, this ridge continues into an exceptionally thick bony skirt at the inner rear side of the shaft, covering over half of its length.
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The traditional medical management of scoliosis is complex and is determined by the severity of the curvature and skeletal maturity, which together help predict the likelihood of progression. The conventional options for children and adolescents are: #Observation #Bracing #Surgery For adults, treatment usually focuses on relieving any pain: #Pain medication #Posture checking #Bracing #Surgery Treatment for idiopathic scoliosis also depends upon the severity of the curvature, the spine's potential for further growth, and the risk that the curvature will progress. Mild scoliosis (less than 30° deviation) and moderate scoliosis (30–45°) can typically be treated conservatively with bracing in conjunction with scoliosis-specific exercises. Severe curvatures that rapidly progress may require surgery with spinal rod placement and spinal fusion.
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Dime- and nickel-radius cue tips (left to right, respectively) Billiard chalk Layered (laminated) tip Cue tip shaper Leather tips of varying curvature and degrees of hardness are glued to (or in some cases screwed into) the ferrule. The de facto standard curvatures for a pool tip are dime- and nickel-radius, determined by shaping a tip so that when one puts a nickel or dime to it, they have the same curvature. The tip end of the cue will vary in diameter but is typically in the 9 to 14 millimeter range with 12–13 mm for pool cues, and 9–10 mm for Snooker cues being most common. Rounder (i.e.
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Since a standard torus is the orbit of a point under a two dimensional abelian subgroup of the Möbius group, it follows that the cyclides also are, and this provides a second way to define them. A third property which characterizes Dupin cyclides is that their curvature lines are all circles (possibly through the point at infinity). Equivalently, the curvature spheres, which are the spheres tangent to the surface with radii equal to the reciprocals of the principal curvatures at the point of tangency, are constant along the corresponding curvature lines: they are the tangent spheres containing the corresponding curvature lines as great circles. Equivalently again, both sheets of the focal surface degenerate to conics.
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The trains travel at a top speed of 250 km/h between Córdoba and Seville, possibly on account of the AVANT services that also use the line, whose trains are limited to 250 km/h. On most journeys, the trains spend a very small proportion of the journey travelling above 250 km/h, although most of the Málaga branch is done at 300 km/h (save for station approaches and the Gobantes and Abdalajís tunnels). The trains slow down to approximately 160 km/h when travelling through Ciudad Real station. They also have to slow down to 70 km/h when travelling through Puertollano station because of the lack of a bypass route and tight curvatures in the station.
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The Midland's easy grades and gentle curvatures made it easy to handle large amounts of freight."Truro Subdivision: The Midland Line", Dominion Atlantic Railway Digital Preservation Institute The line was busiest during World War Two when in addition to heavy freight traffic, it delivered supplies and personnel to the training air fields at the Stanley Airport and Maitland. Traffic dwindled due to subsidized highway construction after World War Two, although the line became well known among railway historians for offering one of Canada's last mixed train, a passenger and freight train combined, which ran with vintage passenger equipment until passenger service ended on the Midland in 1979.Greg McDonnell, "Last Train to Clarksville", Passing Trains: The Changing Face of Canadian Railroading (1991), pp.
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Sagnotti's main contribution to physics is perhaps the analysis of the 2-loop divergences in Einstein's theory of General Relativity. Moreover, he was the first to propose, in 1987, that the type I string theory can be obtained as an orientifold of type IIB string theory, with 32 half-D9-branes added in the vacuum to cancel various anomalies and offered the elucidation of the key properties of orientifold constructions and of Conformal Field Theory on non-orientable surfaces. He also discovered the 10D "0B' string", including both open and closed strings, non supersymmetric but free of tachyons. He has worked extensively on higher spins, arriving at a geometric formulation of their free field equations in terms of higher-spin curvatures.
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Day's mantle designs are often characterized by ionic columns topped with serpentine friezes, but some mantle pieces also employed horizontal and vertical linear designs to create a sharpness in his pieces. Overall, Day's architectural work, much like his furniture craftsmanship, focuses on creating a balance of motion through complementary curvatures. Unlike his furniture, however, Day sometimes added carved faces to his architectural work, especially on mantle pieces, as seen on a mantle in the Long House. Although some scholars have argued that Day's use of these faces represents his connections to his African heritage, since the use of carved faces was popular in African cultural images and craft designs, most scholars today agree that Day's carved faces have a more Euro-American design heritage.
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An article in the Quarterly Journal of Science, Literature, and The Arts (1821) raised interest in optical illusions of curved spokes in rotating wheels seen through vertical apertures. In 1824 Peter Mark Roget provided mathematical details about the appearing curvatures and added the observation that the spokes appeared motionless. Roget claimed that the illusion is due to the fact “that an impression made by a pencil of rays on the retina, if sufficiently vivid, will remain for a certain time after the cause has ceased.” This was later seen as the basis for the theory of "persistence of vision" as the principle of how we see film as motion rather than the successive stream of still images actually presented to the eye.
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Twisted geometries are discrete geometries that play a role in loop quantum gravity and spin foam models, where they appear in the semiclassical limit of spin networks. A twisted geometry can be visualized as collections of polyhedra dual to the nodes of the spin network's graph. Intrinsic and extrinsic curvatures are defined in a manner similar to Regge calculus, but with the generalisation of including a certain type of metric discontinuities: the face shared by two adjacent polyhedra has a unique area, but its shape can be different. This is a consequence of the quantum geometry of spin networks: ordinary Regge calculus is "too rigid" to account for all the geometric degrees of freedom described by the semiclassical limit of a spin network.
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Types of two-mirror optical cavities, with mirrors of various curvatures, showing the radiation pattern inside each cavity. Light confined in a resonator will reflect multiple times from the mirrors, and due to the effects of interference, only certain patterns and frequencies of radiation will be sustained by the resonator, with the others being suppressed by destructive interference. In general, radiation patterns which are reproduced on every round-trip of the light through the resonator are the most stable, and these are the eigenmodes, known as the modes, of the resonator. Resonator modes can be divided into two types: longitudinal modes, which differ in frequency from each other; and transverse modes, which may differ in both frequency and the intensity pattern of the light.
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The earliest signs and symptoms occur in newborns and consist of hypotonia, but show up in youth as developmental delays, low muscle tone, learning disabilities, being overweight, autism-like symptoms, seizures, eczema, asthma, chest and ear infections, and abnormalities in face, hands, and feet such as brachydactyly. Autism-like symptoms consist of odd obsessions, repetitive behavior, poor use of eye contact, impaired speech, poor understanding of others’ emotions, idiosyncratic use of words or phrases. People with this disorder also tend to have a characteristic appearance, including prominent forehead, thin, highly arched eyebrows, depressed nasal bridge, full cheeks, deficient nasal alae and prominent columella, thin upper lip, and various minor anomalies of the pinnae. Heart, brain, gastrointestinal, and kidney problems such as Wilms tumor, and hernias, spinal curvatures, osteopenia, hearing and sight difficulties can also occur.
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This method of generating primitive Pythagorean triples also provides integer solutions to Descartes' Circle Equation, : \left( k_1 + k_2 + k_3 + k_4 \right)^2 = 2\left( k_1^2 + k_2^2 + k_3^2 + k_4^2 \right), where integer curvatures ki are obtained by multiplying the reciprocal of each radius by the area A. The result is k1 = pp', k2 = qp', k3 = q'p, k4 = qq'. Here, the largest circle is taken as having negative curvature with respect to the other three. The largest circle (curvature k4) may also be replaced by a smaller circle with positive curvature ( k0 = 4pp' − qq' ). EXAMPLE: Using the area and four radii obtained above for primitive triple [44, 117, 125] we obtain the following integer solutions to Descartes' Equation: k1 = 143, k2 = 99, k3 = 26, k4 = (−18), and k0 = 554.
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Gaussian curvature is an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature. Formally, Gaussian curvature only depends on the Riemannian metric of the surface.
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In case all three Frenet–Serret curvatures are constant, the corresponding worldlines are identical to those that follow from the Killing motions in flat spacetime. They are of particular interest since the corresponding proper frames and congruences satisfy the condition of Born rigidity, that is, the spacetime distance of two neighbouring worldlines is constant.Bel (1995), theorem 2Giulini (2008), Theorem 18 These motions correspond to "timelike helices" or "stationary worldlines", and can be classified into six principal types: two with zero torsions (uniform translation, hyperbolic motion) and four with non- zero torsions (uniform rotation, catenary, semicubical parabola, general case):Herglotz (1909), sections 3-4, who focuses on the four rotational motions in addition to hyperbolic motion.Kottler (1912), § 6; (1914a), table I & IIPetruv (1964)Synge (1967)Letaw & Pfautsch (1982)Pauri & Vallisneri (2001), Appendix ARosu (2000), section 0.2.
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The phytochrome that expressed more than normal amounts of phytochrome A it was found that as the fluence increased the curvature also increased up to 10umol-m−2 the curvature was similar to the wild-type. The phytochrome expressing more than normal amounts of phytochrome B exhibited curvatures similar to that of the wild type at different fluences of red light up until the fluence of 100umol-m−2 at fluences higher than this curvature was much higher than the wild-type. Thus, the experiment resulted in the finding that another phytochrome than just phytochrome A acts in influencing the curvature since the mutant is not that far off from the wild-type, and phyA is not expressed at all. Thus leading to the conclusion that two phases must be responsible for phototropism.
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They travel at 200 km/h through the Sierra Morena region, possibly because the S/100 trains aren't pressure-sealed and this section includes many tunnels and also because of the tight curvatures in the Sierra Morena (occasionally dipping as low as 2300m). According to the HS2 website, a 200 km/h track needs a curvature of 1800m and a 400 km/h track needs a curvature of 7200m. As the necessary curvature increases in proportion to the square of the maximum velocity, the maximum safe speed for a curvature of 2300m would be 226 km/h, assuming no tilting technology: - only the AVE Class 100, AVE Class 102 and AVE Class 103 run through the Sierra Morena section, of which only the AVE Class 102 has tilting technology.
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However, already by the end of the 1930s the troublesome maintenance problems and the excessive travel time pushed the FS to design a new line, however the start of construction for the new track was delayed, in part by the Second World War. The Ministry of Public Works project envisaged a route with a tunnel of approximately 17km with maximum slopes of 18 feet per thousand and radii of curvatures of at least 300 meters, with a new station in Cosenza and electrification of the line. The planned expenditure for the work was around 20bn lire. In March 1966 construction work began on the new line, and the excavation of the long tunnel. The closure of the railway operation on the old section took place on April 28, 1987 while the new section, electrified, opened for operation on May 31, 1987.
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As particles are moving in a closed orbit, the lateral acceleration causes them to emit synchrotron radiation, thereby reducing the size of their momentum vectors (relative to the design orbit) without changing their orientation (ignoring quantum effects for the moment). In longitudinal direction, the loss of particle impulse due to radiation is replaced by accelerating sections (RF cavities) that are installed in the beam path so that an equilibrium is reached at the design energy of the accelerator. Since this is not happening in transverse direction, where the emittance of the beam is only increased by the quantization of radiation losses (quantum effects), the transverse equilibrium emittance of the particle beam will be smaller with large radiation losses, compared to small radiation losses. Because high orbit curvatures (low curvature radii) increase the emission of synchrotron radiation, damping rings are often small.
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Sponsored by the Stockton and Darlington Railway this was conceived as part of a future through main line route to Scotland. The terrain was very challenging, and attempts to keep to main line curvatures and gradients had to compromise with realistic earthworks. Opposition from the Earl of Carlisle, the major landowner in the area, resulted in the proposal being dropped in 1846. Now in the same year the Newcastle and Carlisle published their intention to build a branch to Allendale from Morralee, but they were unable to prepare timely and accurate survey documents for the line through difficult terrain and the plan was deferred; even the Alston branch, serving a greater concentration of the lead industry, was much delayed in opening.. The independently promoted Border Counties Railway ran north from Hexham, connecting mineral deposits to the main line as well as harbouring strategic plans to reach further, which it later achieved.
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Shiing-Shen Chern published his proof of the theorem in 1944 while at the Institute for Advanced Study. This was historically the first time that the formula was proven without assuming the manifold to be embedded in a Euclidean space, which is what it means by "intrinsic". The special case for a hypersurface (an n-1-dimensional submanifolds in an n-dimensional Euclidean space) was proved by H. Hopf in which the integrand is the Gauss-Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz-Killing curvature (the average of the Gauss-Kronecker curvature along each unit normal vector over the unit sphere in the normal space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold).
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If a straight line is considered a degenerate circle with zero curvature (and thus infinite radius), Descartes' theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line. If four circles are tangent to each other at six distinct points, and the circles have curvatures ki (for i = 1, ..., 4), Descartes' theorem says: When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as: The ± sign reflects the fact that there are in general two solutions. Ignoring the degenerate case of a straight line, one solution is positive and the other is either positive or negative; if negative, it represents a circle that circumscribes the first three (as shown in the diagram above). Problem-specific criteria may favor one solution over the other in any given problem.
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In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in . More precisely, let M be a surface in three-dimensional Euclidean space, and p a point on M. A normal plane through p is a plane passing through the point p containing the normal vector to M. Through each (unit) tangent vector to M at p, there passes a normal plane PX which cuts out a curve in M. That curve has a certain curvature κX when regarded as a curve inside PX. Provided not all κX are equal, there is some unit vector X1 for which k1 = κX1 is as large as possible, and another unit vector X2 for which k2 = κX2 is as small as possible.
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So the extra mass of galaxies and galaxy clusters (and dark matter, should particles of it ever be directly detected) must cause nearby space-time to curve more positively, and voids should have the opposite effect, causing space-time around them to take on negative curvatures. The question is whether these effects, called backreactions, are negligible or together comprise enough to change the universe's geometry. Most scientists have assumed that they are negligible, but this has partly been because there has been no way to average space-time geometry in Einstein's equations. In 2000, a set of new equations—now referred to as the set of Buchert equations—based on general relativity was published by cosmologist Thomas Buchert of the École Normale Supérieure in Lyon, France, which allow the effects of a non-uniform distribution of matter to be taken into account but still allow the behavior of the universe to be averaged.
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71, No. 2, pp. 209–219, 1999 Link Examples include fullerenes, nanotubes, corannulenes, helicenes and sumanene. The molecular orbitals of the carbon atoms in these systems are to some extent pyramidalized resulting a different pi electron density on either side of the molecule with consequences for reactivity. One member of this group of organic compounds, pentaindenocorannulene (depicted below),Pentaindenocorannulene and Tetraindenocorannulene: New Aromatic Hydrocarbon Systems with Curvatures Surpassing That of C60Edward A. Jackson, Brian D. Steinberg, Mihail Bancu, Atsushi Wakamiya, and Lawrence T. Scott J. Am. Chem. Soc.; 2007; 129(3) pp 484 - 485; (Communication) The two steps in this sequence are a Suzuki-Miyaura coupling with an ionic liquid and Tris(dibenzylideneacetone)dipalladium(0) and a Heck reaction with another palladium catalyst, DBU and DMAC can be considered a large fullerene fragment. The experimentally obtained curvature and degree of pyramidalizion (12.6° for the carbons of the pentagon at the center by a so-called p-orbital axis vector (POAV). The value for fullerene is 11.6° and benzene 0°) are both actually larger than that of fullerene but according to its discoverers, the compound is relatively easy to synthesize starting from corannulene and a way is opened to produce larger such fragments by stitching. Synthesis of Pentaindenocorannulene The crystal structure of pentaindenocorannulene has been obtained.
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