1.
Equator
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The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north

2.
Norm (mathematics)
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A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors. A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the definition below. A simple example is the 2-dimensional Euclidean space R2 equipped with the Euclidean norm, elements in this vector space are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin. The Euclidean norm assigns to each vector the length of its arrow, because of this, the Euclidean norm is often known as the magnitude. A vector space on which a norm is defined is called a vector space. Similarly, a space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a vector space in more than one way. If p =0 then v is the zero vector, by the first axiom, absolute homogeneity, we have p =0 and p = p, so that by the triangle inequality p ≥0. A seminorm on V is a p, V → R with the properties 1. and 2. Every vector space V with seminorm p induces a normed space V/W, called the quotient space, the induced norm on V/W is clearly well-defined and is given by, p = p. A topological vector space is called if the topology of the space can be induced by a norm. If a norm p, V → R is given on a vector space V then the norm of a vector v ∈ V is usually denoted by enclosing it within double vertical lines, such notation is also sometimes used if p is only a seminorm. For the length of a vector in Euclidean space, the notation | v | with single vertical lines is also widespread, in Unicode, the codepoint of the double vertical line character ‖ is U+2016. The double vertical line should not be confused with the parallel to symbol and this is usually not a problem because the former is used in parenthesis-like fashion, whereas the latter is used as an infix operator. The double vertical line used here should not be confused with the symbol used to denote lateral clicks. The single vertical line | is called vertical line in Unicode, the trivial seminorm has p =0 for all x in V. Every linear form f on a vector space defines a seminorm by x → | f |, the absolute value ∥ x ∥ = | x | is a norm on the one-dimensional vector spaces formed by the real or complex numbers. The absolute value norm is a case of the L1 norm