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Vertical Extent
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an example of vertical position, "The height of an airplane in-flight is about 10,000 meters." When the term is used to describe vertical position (of, e.g., an airplane) from sea level, height is more often called ''altitude''. Furthermore, if the point is attached to the Earth (e.g., a mountain peak), then altitude (height above sea level) is called ''elevation''. In a two-dimensional Cartesian space, height is measured along the vertical axis (''y'') between a specific point and another that does not have the same ''y''-value. If both points happen to have the same ''y''-value, then their relative height is zero. In the case of three-dimensional space, height is measured along the vertical ''z'' axis, describing a distance from (or "above" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height Function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the ''classical'' or ''naive height'' over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. for the coordinates ), but in a logarithmic scale. Significance Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tallest Mountain
This is a list of mountain peaks ordered by their topographic prominence. Terminology The prominence of a peak is the minimum height of climb to the summit on any route from a higher peak, or from sea level if there is no higher peak. The lowest point on that route is the col. For full definitions and explanations of ''topographic prominence'', ''key col'', and ''parent'', see topographic prominence. In particular, the different definitions of the parent of a peak are addressed at length in that article. ''Height'' on the other hand simply means elevation of the summit above sea level. Regarding parents, the ''prominence parent'' of peak A can be found by dividing the island or region in question into territories, by tracing the runoff from the key col (mountain pass) of every peak that is more prominent than peak A. The parent is the peak whose territory peak A resides in. The ''encirclement parent'' is found by tracing the contour below peak A's key col and picking the hig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tibet
Tibet (; ''Böd''; ), or Greater Tibet, is a region in the western part of East Asia, covering much of the Tibetan Plateau and spanning about . It is the homeland of the Tibetan people. Also resident on the plateau are other ethnic groups such as Mongols, Monpa people, Monpa, Tamang people, Tamang, Qiang people, Qiang, Sherpa people, Sherpa, Lhoba people, Lhoba, and since the 20th century Han Chinese and Hui people, Hui. Tibet is the highest region on Earth, with an average elevation of . Located in the Himalayas, the highest elevation in Tibet is Mount Everest, Earth's highest mountain, rising above sea level. The Tibetan Empire emerged in the 7th century. At its height in the 9th century, the Tibetan Empire extended far beyond the Tibetan Plateau, from the Tarim Basin and Pamirs in the west, to Yunnan and Bengal in the southeast. It then divided into a variety of territories. The bulk of western and central Tibet (Ü-Tsang) was often at least nominally unified under a ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nepal
Nepal, officially the Federal Democratic Republic of Nepal, is a landlocked country in South Asia. It is mainly situated in the Himalayas, but also includes parts of the Indo-Gangetic Plain. It borders the Tibet Autonomous Region of China China–Nepal border, to the north, and India India–Nepal border, to the south, east, and west, while it is narrowly separated from Bangladesh by the Siliguri Corridor, and from Bhutan by the States and union territories of India, Indian state of Sikkim. Nepal has a Geography of Nepal, diverse geography, including Terai, fertile plains, subalpine forested hills, and eight of the world's ten List of highest mountains#List, tallest mountains, including Mount Everest, the highest point on Earth. Kathmandu is the nation's capital and List of cities in Nepal, its largest city. Nepal is a multi-ethnic, multi-lingual, multi-religious, and multi-cultural state, with Nepali language, Nepali as the official language. The name "Nepal" is first record ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mount Everest
Mount Everest (), known locally as Sagarmatha in Nepal and Qomolangma in Tibet, is Earth's highest mountain above sea level. It lies in the Mahalangur Himal sub-range of the Himalayas and marks part of the China–Nepal border at its summit. Its height was most recently measured in 2020 by Chinese and Nepali authorities as . Mount Everest attracts many climbers, including highly experienced mountaineers. There are two main climbing routes, one approaching the summit from the southeast in Nepal (known as the standard route) and the other from the north in Tibet. While not posing substantial technical climbing challenges on the standard route, Everest presents dangers such as altitude sickness, weather, and wind, as well as hazards from avalanches and the Khumbu Icefall. As of May 2024, 340 people have died on Everest. Over 200 bodies remain on the mountain and have not been removed due to the dangerous conditions. Climbers typically ascend only part of Mount Eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topographic Prominence
In topography, prominence or relative height (also referred to as autonomous height, and shoulder drop in US English, and drop in British English) measures the height of a mountain or hill's summit relative to the lowest contour line encircling it but containing no higher summit within it. It is a measure of the independence of a summit. The key col ("saddle") around the peak is a unique point on this contour line and the ''parent peak'' (if any) is some higher mountain, selected according to various criteria. Definitions The prominence of a peak is the least drop in height necessary in order to get from the summit to any higher terrain. This can be calculated for a given peak in the following manner: for every path connecting the peak to higher terrain, find the lowest point on the path; the ''key col'' (or ''highest saddle (landform), saddle'', or ''linking col'', or ''link'') is defined as the highest of these points, along all connecting paths; the prominence is the differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesy
Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional space, 3D. It is called planetary geodesy when studying other astronomical body, astronomical bodies, such as planets or Natural satellite, circumplanetary systems. Geodynamics, Geodynamical phenomena, including crust (geology), crustal motion, tides, and polar motion, can be studied by designing global and national Geodetic control network, control networks, applying space geodesy and terrestrial geodetic techniques, and relying on Geodetic datum, datums and coordinate systems. Geodetic job titles include geodesist and geodetic surveyor. History Geodesy began in pre-scientific Classical antiquity, antiquity, so the very word geodesy comes from the Ancient Greek word or ''geodaisia'' (literally, "division of Earth"). Early ideas about t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all the multiple (mathematics), multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary ideal, primary and semiprime ideal, semiprime. Prime ideals for commutative rings Definition An ideal (ring theory), ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if n\Z is a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull Dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
