It gleams and sparkles in the sunlight and invites lovers to enter and osculate.
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The word osculate is Latin for "kiss". In mathematics, two curves osculate when they just touch, without (necessarily) crossing, at a point, where both have the same position and slope, i.e. the two curves "kiss".
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Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations, differential equations which come in different forms developed by Lagrange, Gauss, Delaunay, Poincaré, or Hill.
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In mathematical invariant theory, the osculant or tacinvariant or tact invariant is an invariant of a hypersurface that vanishes if the hypersurface touches itself, or an invariant of several hypersurfaces that osculate, meaning that they have a common point where they meet to unusually high order.
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Insulin release from The Islet of Langerhans is pulsatile with a period of 3-6 minutes. Oscillations of intracellular calcium concentration in beta cells within the pancreas produces a basal pulsatile secretion of insulin form the pancreas. Secretion pulses emanating form free beta cells not located within an islet of Lagerhans have been observed to be highly variable (2 to 10 minutes). Beta cells within an islet, however, become synchronized via electrical coupling resulting from gap junctions and osculate more regularly (3 to 6 minutes).
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The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an osculating plane to a space curve is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.. In one dimension, analytic curves are said to osculate at a point if they share the first three terms of their Taylor expansion about that point. This concept can be generalized to superosculation, in which two curves share more than the first three terms of their Taylor expansion.
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Example of the circle packing theorem on K−5, the complete graph on five vertices, minus one edge. We say that two circles drawn in a plane kiss (or osculate) whenever they intersect in exactly one point. A "coin graph" is a graph formed by a set of circles, no two of which have overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circle packing theorem, first proved by Paul Koebe in 1936, states that a graph is planar if and only if it is a coin graph.
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Velpius became a bookseller in Leuven in 1564, and in 1565 was licensed as a "sworn bookseller" to the University of Leuven. Around 1567 he married Catherine Waen, daughter of the Scottish expatriate bookseller John Waen. In 1570 Velpius was examined and certified as a printer, his certification specifying that he knew Latin, French and Flemish, and a little bit of Greek. For his work in Leuven he used two printer's marks: a large one with a crenellated tower, an angel of vengeance above it and the figures of Justice and Peace embracing before the gates, with the motto Justitia et pax osculate sunt. Psal. 84.
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Frenet–Serret frame, and the osculating plane (spanned by T and N). In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane which "kisses" a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors.
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Johnson's theorem states that if the three blue circles in the picture have equal radius and intersect at a single point, H, then the resulting red circle has the same radius as the blue circles. The green triangle ΔJAJBJC is then the Johnson triangle of the black reference triangle, ΔABC, and has a circumcircle (orange) of radius r. In geometry, a set of Johnson circles comprises three circles of equal radius r sharing one common point of intersection H. In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point H that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection). If any two of the circles happen to osculate, they only have H as a common point, and it will then be considered that H be their 2-wise intersection as well; if they should coincide we declare their 2-wise intersection be the point diametrically opposite H. The three 2-wise intersection points define the reference triangle of the figure.
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