Share to: share facebook share twitter share wa share telegram print page

Real coordinate space

Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers

In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real line R1, the real coordinate plane R2, and the real coordinate three-dimensional space R3. With component-wise addition and scalar multiplication, it is a real vector space.

The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form a real coordinate space of dimension n.

These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.

Definition and structures

For any natural number n, the set Rn consists of all n-tuples of real numbers (R). It is called the "n-dimensional real space" or the "real n-space".

An element of Rn is thus a n-tuple, and is written

where each xi is a real number. So, in multivariable calculus, the domain of a function of several real variables and the codomain of a real vector valued function are subsets of Rn for some n.

The real n-space has several further properties, notably:

These properties and structures of Rn make it fundamental in almost all areas of mathematics and their application domains, such as statistics, probability theory, and many parts of physics.

The domain of a function of several variables

Any function f(x1, x2, ..., xn) of n real variables can be considered as a function on Rn (that is, with Rn as its domain). The use of the real n-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for n = 2, a function composition of the following form:

where functions g1 and g2 are continuous. If

  • x1 ∈ R : f(x1, ·) is continuous (by x2)
  • x2 ∈ R : f(·, x2) is continuous (by x1)

then F is not necessarily continuous. Continuity is a stronger condition: the continuity of f in the natural R2 topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition F.

Vector space

The coordinate space Rn forms an n-dimensional vector space over the field of real numbers with the addition of the structure of linearity, and is often still denoted Rn. The operations on Rn as a vector space are typically defined by

The zero vector is given by
and the additive inverse of the vector x is given by

This structure is important because any n-dimensional real vector space is isomorphic to the vector space Rn.

Matrix notation

In standard matrix notation, each element of Rn is typically written as a column vector

and sometimes as a row vector:

The coordinate space Rn may then be interpreted as the space of all n × 1 column vectors, or all 1 × n row vectors with the ordinary matrix operations of addition and scalar multiplication.

Linear transformations from Rn to Rm may then be written as m × n matrices which act on the elements of Rn via left multiplication (when the elements of Rn are column vectors) and on elements of Rm via right multiplication (when they are row vectors). The formula for left multiplication, a special case of matrix multiplication, is:

Any linear transformation is a continuous function (see below). Also, a matrix defines an open map from Rn to Rm if and only if the rank of the matrix equals to m.

Standard basis

The coordinate space Rn comes with a standard basis:

To see that this is a basis, note that an arbitrary vector in Rn can be written uniquely in the form

Geometric properties and uses

Orientation

The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on Rn. Any full-rank linear map of Rn to itself either preserves or reverses orientation of the space depending on the sign of the determinant of its matrix. If one permutes coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the parity of the permutation.

Diffeomorphisms of Rn or domains in it, by their virtue to avoid zero Jacobian, are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of differential forms, whose applications include electrodynamics.

Another manifestation of this structure is that the point reflection in Rn has different properties depending on evenness of n. For even n it preserves orientation, while for odd n it is reversed (see also improper rotation).

Affine space

Rn understood as an affine space is the same space, where Rn as a vector space acts by translations. Conversely, a vector has to be understood as a "difference between two points", usually illustrated by a directed line segment connecting two points. The distinction says that there is no canonical choice of where the origin should go in an affine n-space, because it can be translated anywhere.

Convexity

The n-simplex (see below) is the standard convex set, that maps to every polytope, and is the intersection of the standard (n + 1) affine hyperplane (standard affine space) and the standard (n + 1) orthant (standard cone).

In a real vector space, such as Rn, one can define a convex cone, which contains all non-negative linear combinations of its vectors. Corresponding concept in an affine space is a convex set, which allows only convex combinations (non-negative linear combinations that sum to 1).

In the language of universal algebra, a vector space is an algebra over the universal vector space R of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal orthant (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal simplex (of finite sequences of nonnegative numbers summing to 1). This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates".

Another concept from convex analysis is a convex function from Rn to real numbers, which is defined through an inequality between its value on a convex combination of points and sum of values in those points with the same coefficients.

Euclidean space

The dot product

defines the norm |x| = xx on the vector space Rn. If every vector has its Euclidean norm, then for any pair of points the distance
is defined, providing a metric space structure on Rn in addition to its affine structure.

As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in Rn without special explanations. However, the real n-space and a Euclidean n-space are distinct objects, strictly speaking. Any Euclidean n-space has a coordinate system where the dot product and Euclidean distance have the form shown above, called Cartesian. But there are many Cartesian coordinate systems on a Euclidean space.

Conversely, the above formula for the Euclidean metric defines the standard Euclidean structure on Rn, but it is not the only possible one. Actually, any positive-definite quadratic form q defines its own "distance" q(xy), but it is not very different from the Euclidean one in the sense that

Such a change of the metric preserves some of its properties, for example the property of being a complete metric space. This also implies that any full-rank linear transformation of Rn, or its affine transformation, does not magnify distances more than by some fixed C2, and does not make distances smaller than 1 / C1 times, a fixed finite number times smaller.[clarification needed]

The aforementioned equivalence of metric functions remains valid if q(xy) is replaced with M(xy), where M is any convex positive homogeneous function of degree 1, i.e. a vector norm (see Minkowski distance for useful examples). Because of this fact that any "natural" metric on Rn is not especially different from the Euclidean metric, Rn is not always distinguished from a Euclidean n-space even in professional mathematical works.

In algebraic and differential geometry

Although the definition of a manifold does not require that its model space should be Rn, this choice is the most common, and almost exclusive one in differential geometry.

On the other hand, Whitney embedding theorems state that any real differentiable m-dimensional manifold can be embedded into R2m.

Other appearances

Other structures considered on Rn include the one of a pseudo-Euclidean space, symplectic structure (even n), and contact structure (odd n). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.

Rn is also a real vector subspace of Cn which is invariant to complex conjugation; see also complexification.

Polytopes in Rn

There are three families of polytopes which have simple representations in Rn spaces, for any n, and can be used to visualize any affine coordinate system in a real n-space. Vertices of a hypercube have coordinates (x1, x2, ..., xn) where each xk takes on one of only two values, typically 0 or 1. However, any two numbers can be chosen instead of 0 and 1, for example −1 and 1. An n-hypercube can be thought of as the Cartesian product of n identical intervals (such as the unit interval [0,1]) on the real line. As an n-dimensional subset it can be described with a system of 2n inequalities:

for [0,1], and
for [−1,1].

Each vertex of the cross-polytope has, for some k, the xk coordinate equal to ±1 and all other coordinates equal to 0 (such that it is the kth standard basis vector up to sign). This is a dual polytope of hypercube. As an n-dimensional subset it can be described with a single inequality which uses the absolute value operation:

but this can be expressed with a system of 2n linear inequalities as well.

The third polytope with simply enumerable coordinates is the standard simplex, whose vertices are n standard basis vectors and the origin (0, 0, ..., 0). As an n-dimensional subset it is described with a system of n + 1 linear inequalities:

Replacement of all "≤" with "<" gives interiors of these polytopes.

Topological properties

The topological structure of Rn (called standard topology, Euclidean topology, or usual topology) can be obtained not only from Cartesian product. It is also identical to the natural topology induced by Euclidean metric discussed above: a set is open in the Euclidean topology if and only if it contains an open ball around each of its points. Also, Rn is a linear topological space (see continuity of linear maps above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from Rn to itself which are not isometries, there can be many Euclidean structures on Rn which correspond to the same topology. Actually, it does not depend much even on the linear structure: there are many non-linear diffeomorphisms (and other homeomorphisms) of Rn onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube).

Rn has the topological dimension n.

An important result on the topology of Rn, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rn (with its subspace topology) that is homeomorphic to another open subset of Rn is itself open. An immediate consequence of this is that Rm is not homeomorphic to Rn if mn – an intuitively "obvious" result which is nonetheless difficult to prove.

Despite the difference in topological dimension, and contrary to a naïve perception, it is possible to map a lesser-dimensional[clarification needed] real space continuously and surjectively onto Rn. A continuous (although not smooth) space-filling curve (an image of R1) is possible.[clarification needed]

Examples

Empty column vector,
the only element of R0

n ≤ 1

Cases of 0 ≤ n ≤ 1 do not offer anything new: R1 is the real line, whereas R0 (the space containing the empty column vector) is a singleton, understood as a zero vector space. However, it is useful to include these as trivial cases of theories that describe different n.

n = 2

Both hypercube and cross-polytope in R2 are squares, but coordinates of vertices are arranged differently

The case of (x,y) where x and y are real numbers has been developed as the Cartesian plane P. Further structure has been attached with Euclidean vectors representing directed line segments in P. The plane has also been developed as the field extension by appending roots of X2 + 1 = 0 to the real field The root i acts on P as a quarter turn with counterclockwise orientation. This root generates the group . When (x,y) is written x + y i it is a complex number.

Another group action by , where the actor has been expressed as j, uses the line y=x for the involution of flipping the plane (x,y) ↦ (y,x), an exchange of coordinates. In this case points of P are written x + y j and called split-complex numbers. These numbers, with the coordinate-wise addition and multiplication according to jj=+1, form a ring that is not a field.

Another ring structure on P uses a nilpotent e to write x + y e for (x,y). The action of e on P reduces the plane to a line: It can be decomposed into the projection into the x-coordinate, then quarter-turning the result to the y-axis: e (x + y e) = x e since e2 = 0. A number x + y e is a dual number. The dual numbers form a ring, but, since e has no multiplicative inverse, it does not generate a group so the action is not a group action.

Excluding (0,0) from P makes [x : y] projective coordinates which describe the real projective line, a one-dimensional space. Since the origin is excluded, at least one of the ratios x/y and y/x exists. Then [x : y] = [x/y : 1] or [x : y] = [1 : y/x]. The projective line P1(R) is a topological manifold covered by two coordinate charts, [z : 1] → z or [1 : z] → z, which form an atlas. For points covered by both charts the transition function is multiplicative inversion on an open neighborhood of the point, which provides a homeomorphism as required in a manifold. One application of the real projective line is found in Cayley–Klein metric geometry.

n = 3

Cube (the hypercube) and octahedron (the cross-polytope) of R3. Coordinates are not shown

n = 4

R4 can be imagined using the fact that 16 points (x1, x2, x3, x4), where each xk is either 0 or 1, are vertices of a tesseract (pictured), the 4-hypercube (see above).

The first major use of R4 is a spacetime model: three spatial coordinates plus one temporal. This is usually associated with theory of relativity, although four dimensions were used for such models since Galilei. The choice of theory leads to different structure, though: in Galilean relativity the t coordinate is privileged, but in Einsteinian relativity it is not. Special relativity is set in Minkowski space. General relativity uses curved spaces, which may be thought of as R4 with a curved metric for most practical purposes. None of these structures provide a (positive-definite) metric on R4.

Euclidean R4 also attracts the attention of mathematicians, for example due to its relation to quaternions, a 4-dimensional real algebra themselves. See rotations in 4-dimensional Euclidean space for some information.

In differential geometry, n = 4 is the only case where Rn admits a non-standard differential structure: see exotic R4.

Norms on Rn

One could define many norms on the vector space Rn. Some common examples are

  • the p-norm, defined by for all where is a positive integer. The case is very important, because it is exactly the Euclidean norm.
  • the -norm or maximum norm, defined by for all . This is the limit of all the p-norms: .

A really surprising and helpful result is that every norm defined on Rn is equivalent. This means for two arbitrary norms and on Rn you can always find positive real numbers , such that

for all .

This defines an equivalence relation on the set of all norms on Rn. With this result you can check that a sequence of vectors in Rn converges with if and only if it converges with .

Here is a sketch of what a proof of this result may look like:

Because of the equivalence relation it is enough to show that every norm on Rn is equivalent to the Euclidean norm . Let be an arbitrary norm on Rn. The proof is divided in two steps:

  • We show that there exists a , such that for all . In this step you use the fact that every can be represented as a linear combination of the standard basis: . Then with the Cauchy–Schwarz inequality
    where .
  • Now we have to find an , such that for all . Assume there is no such . Then there exists for every a , such that . Define a second sequence by . This sequence is bounded because . So because of the Bolzano–Weierstrass theorem there exists a convergent subsequence with limit Rn. Now we show that but , which is a contradiction. It is
    because and , so . This implies , so . On the other hand , because . This can not ever be true, so the assumption was false and there exists such a .

See also

Sources

  • Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
  • Munkres, James (1999). Topology. Prentice-Hall. ISBN 0-13-181629-2.

Read other information related to :Real coordinate space/

Real Réal Real Real Real Real-time computing Real number Real analysis Real estate investing Real estate Real estate agent Function of a real variable Real mode The Real Housewives Ciudad Real Real closed field Real-time tactics Keepin' It Real Real Madrid C Real Football Real to Real Cacophony Real property Real Life Graduate real estate education in the United States Real algebraic geometry Real to Reel Be Real Real World Real Democracy Function of several real variables Real-time Reel to Real Xinyuan Real Estate Real options valuation Real Talk The Real Thing For Real (disambiguation) The …

Real Me Real-time web Brazilian real Projectively extended real line Real Transportes Aéreos Get Real C-Real Real Madrid Baloncesto Real estate contract In Real Life Real Simple S.C. Vila Real Real Madrid CF Keep It Real ¡Democracia Real YA! Vila Real Football Association Real-time strategy ...Is a Real Boy Real form (Lie theory) Real Madrid Castilla Real C.D. España Expo Real Real, Valencia Real estate derivative Real interest rate List of The Real World seasons Vila Real Argentine real Real Men Real Sociedad Spanish real Mein Real Real estate business Real Betis Real coordinate space Extended real number line Real estate investment club Real Unión CD Manchego Ciudad Real Puerto Real Real Valladolid Real-time Java The Real Deal (magazine) Real Shit Healthcare real estate Real Zaragoza Real Times Sanctus Real Real and nominal value The Real RealPlayer Construction of the real numbers Real Valladolid (women) Real Audiencia The Real World (TV series) Real-estate bubble List of real tennis organizations Real Giulianova Real-time text Real Estate Weekly Real Love Institute for Federal Real Estate (Germany) Totally real number field Real estate economics This Is Real Real-time polym

Read other articles:

Questa voce sull'argomento cestisti sudcoreani è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto di riferimento. Kim Jong-gyu Kim con la maglia della Corea del Sud Nazionalità  Corea del Sud Altezza 206 cm Peso 98 kg Pallacanestro Ruolo Centro Squadra  Wonju DB Promy Carriera Giovanili Kyung Hee University Squadre di club 2013-2019 Changwon Sakers2019- Wonju DB Promy Nazionale 2011- Corea del Sud Palmar…

ФремеріFrémery   Країна  Франція Регіон Гранд-Ест  Департамент Мозель  Округ Саррбур-Шато-Сален Кантон Дельм Код INSEE 57236 Поштові індекси 57590 Координати 48°55′12″ пн. ш. 6°28′28″ сх. д.H G O Висота 228 - 280 м.н.р.м. Площа 4,39 км² Населення 88 (01-2020[1]) Густота 14,58 ос./км²

Casazza Entidad subnacional Escudo CasazzaLocalización de Casazza en Italia Coordenadas 45°44′55″N 9°54′23″E / 45.74855, 9.90627Capital CasazzaIdioma oficial ItalianoEntidad Comuna de Italia • País Italia • Región Lombardía • Provincia BérgamoDirigentes   • Alcalde Sergio ZappellaFracciones Colognola, MolognoMunicipios limítrofes Albino, Bianzano, Gaverina Terme, Grone, Monasterolo del Castello, Spinone al Lago, Vigano San MartinoS…

You can help expand this article with text translated from the corresponding article in Lithuanian. (December 2010) Click [show] for important translation instructions. View a machine-translated version of the Lithuanian article. Machine translation, like DeepL or Google Translate, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English …

مخطط بياني يظهر حركة أسعار أسهم شركةفولكسفاجن وقفزتها على هيئة ارتفاع حاد قصير الأجل في سنة 2008. الارتفاع الحاد قصير الأجل في سوق الأسهم هو ازدياد سريع في سعر السهم لفترة لا تدوم طويلاً، وذلك نتيجةً فائض البيع المكشوف للسهم، وليس بسبب أساسي غير مكشوف. يحدث الارتفاع الحاد قصير…

Brian JohnsonJohnson bermain bersama AC/DC pada November 2008Informasi latar belakangNama lahirBrian Francis JohnsonLahir5 Oktober 1947 (umur 76)Dunston, County Durham, InggrisAsalGateshead, InggrisGenreHard rockrock and rollblues rockglam rockPekerjaanPenyanyipenulis laguInstrumenVocal, gitar, drum, pianoTahun aktif1970–sekarangLabelEMIEpicAtlanticArtis terkaitAC/DCGeordieSitus webacdc.com Brian Johnson (lahir di Dunston, County Durham, 5 Oktober 1947; umur 72 tahun) merupakan seorang pe…

Delapan bidang bit dari citra berderajat keabuan (sebelah kiri). Ada delapan karena kedalaman warnanya 8 bit per piksel. Bidang bit suatu sinyal digital (bahasa Inggris: bit plane) adalah himpunan bit yang terletak pada nilai tempat tertentu dalam tiap bilangan biner yang menggambarkan sinyalnya.[1] Misalnya, untuk kedalaman 16 bit, ada 16 bidang bit. Bidang bit pertama berisi himpunan bit dengan nilai tempat tertinggi. Bidang bit keenam belas berisi himpunan bit dengan nilai tempat …

Sadia Hameed Sadia Hameed (lahir 6 Juni 1986) adalah seorang eks-Muslim, sosialis, sekuleris, aktivis hak wanita dan hak asasi manusia asal Inggris, dan jurubicara untuk Council of Ex-Muslims of Britain. Hameed dibesarkan di Oxford, dimana ia melepaskan imannya dalam usia 15 tahun, namun tetap bungkam terhadap hal tersebut. Saudaranya, Razaa, langsung dirundung karena menjadi ateis terbuka, dan ia mecabut nyawanya sendiri pada 2015, yang benar-benar menekan Sadia. Ia menjadi aktivis untuk eks-Mu…

Ủy ban Trung ương Mặt trận Tổ Quốc Việt Nam Biểu trưng Mặt trận Tổ quốc Việt Nam Khóa thứ IX (2019 - 2024) Ủy viên Chủ tịch Đỗ Văn Chiến Phó Chủ tịch kiêm Tổng Thư ký Nguyễn Thị Thu Hà Phó Chủ tịch chuyên trách (3) Trương Thị Ngọc ÁnhNguyễn Hữu DũngHoàng Công Thủy Phó Chủ tịch không chuyên trách (6) Trần Kim YếnNguyễn Thị DoanHòa thượng Thích Thiện NhơnLinh mục Giuse Trần Xuân MạnhHà …

هذه المقالة تحتاج للمزيد من الوصلات للمقالات الأخرى للمساعدة في ترابط مقالات الموسوعة. فضلًا ساعد في تحسين هذه المقالة بإضافة وصلات إلى المقالات المتعلقة بها الموجودة في النص الحالي. (يناير 2021) هذه المقالة تحتوي على رموز البلدان التي تبدأ بالحرف C.  كمبوديا أيزو 3166-1 رقمي 116…

American college football season 2022 Ohio Bobcats footballMAC East Division championArizona Bowl championMAC Championship, L 7–17 vs. ToledoArizona Bowl, W 30–27 OTvs. WyomingConferenceMid-American ConferenceDivisionEast DivisionRecord10–4 (7–1 MAC)Head coachTim Albin (2nd season)Co-offensive coordinatorScott Isphording (2nd season)Co-offensive coordinatorAllen Rudolph (2nd season)Offensive schemeSpread optionDefensive coordinatorSpence Nowinsky (1st season)Base…

Election in Oregon Main article: 1868 United States presidential election 1868 United States presidential election in Oregon ← 1864 November 3, 1868 1872 →   Nominee Horatio Seymour Ulysses S. Grant Party Democratic Republican Home state New York Illinois Running mate Francis Preston Blair, Jr. Schuyler Colfax Electoral vote 3 0 Popular vote 11,125 10,961 Percentage 50.37% 49.63% County Results Seymour   50-60%   60-70% Grant …

Application that uses a web browser as a client This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Web application – news · newspapers · books · scholar · JSTOR …

Protein-coding gene in the species Homo sapiens FGF19Available structuresPDBOrtholog search: PDBe RCSB List of PDB id codes1PWA, 2P23IdentifiersAliasesFGF19, fibroblast growth factor 19External IDsOMIM: 603891 MGI: 1096383 HomoloGene: 3754 GeneCards: FGF19 Gene location (Human)Chr.Chromosome 11 (human)[1]Band11q13.3Start69,698,238 bp[1]End69,704,022 bp[1]Gene location (Mouse)Chr.Chromosome 7 (mouse)[2]Band7|7 F5Start144,896,531 bp[2]End144,900,953 bp&…

坐标:40°44′56.55″N 73°50′47.57″W / 40.7490417°N 73.8465472°W / 40.7490417; -73.8465472 法拉盛草原可乐娜公园的景点和地标 [全屏互动地图 + 附近条目] 查看讨论编辑法拉盛草原可乐娜公园的景点和地标:1 花旗球场2 皇后动物园和法拉盛草原旋转木马3 法拉盛草原游泳馆4 法拉盛河5 草原湖(Meadow Lake)6 大都会-威利斯角(长岛铁路和地铁车站)7 美国网球协会比莉·珍·…

2018 Indian filmPoomaramTheatrical release posterDirected byAbrid ShineWritten byAbrid ShineProduced byPaul Varghese Abrid ShineStarringKalidas JayaramNeeta PillaiCinematographyGnaanam SubramanianEdited byKR MidhunMusic byPerumbavoor G. RaveendranathGopi SundarFaisal RaziLeela L Girish KuttanArackal Nandakumar Nasil P K.A AnishVishnu manu SivasankarSayoojya DasProductioncompanyDr. Paul's EntertainmentDistributed byCentral Pictures, Popcorn Entertainments (Asia Pacific release)Release date 15…

Halaman ini berisi artikel tentang genus lalat. Untuk rasi bintang, lihat Musca. Untuk kegunaan lain, lihat Musca (disambiguasi). Musca Musca domestica (lalat rumah) Klasifikasi ilmiah Kerajaan: Animalia Filum: Arthropoda Kelas: Insecta Ordo: Diptera Famili: Muscidae Subfamili: Muscinae Tribus: Muscini Genus: MuscaLinnaeus, 1758 Spesies tipe M. domesticaLinnaeus, 1758[1] Spesies Lihat teks. Sinonim Byomya Robineau-Desvoidy, 1830 Eumusca Townsend, 1911 Plaxemya Robineau-Desvoidy, 1830 Mus…

Japanese actress (born 1977) Misato Tanaka田中 美里May 2023 in YokohamaBorn (1977-02-09) 9 February 1977 (age 46)Kanazawa, Ishikawa, JapanNationalityJapaneseOccupation(s)Actress, modelYears active1996–presentWebsitewww.am-ple.co.jp Misato Tanaka (田中 美里, Tanaka Misato, born 9 February 1977) is a Japanese actress.[1] Biography Tanaka started appearing on stage as a child. She won an award at Toho's Cinderella contest in 1996 and landed the starring role in the NHK A…

Politics of Estonia State Constitution Declaration of Independence Human rights Presidency President Alar Karis Executive Prime Minister Kaja Kallas Government of Estonia Incumbent cabinet Legislature Riigikogu Speaker: Lauri Hussar Judiciary Supreme Court Chancellor of Justice Elections Political parties Recent elections Riigikogu:201520192023 Presidential:201120162021 Municipal: 200920132017 European: 201420192024 Administrative divisions Counties Municipalities Foreign relations Ministry of F…

Brigade Infanteri Para Raider 3/Tri Budi SaktiLambang Brigif Para Raider 3/TBSDibentuk9 Desember 1986NegaraIndonesiaCabangInfanteriTipe unitPara RaiderPeranPasukan Pemukul Reaksi Cepat Lintas UdaraBagian dariDivisi Infanteri 3/KostradMarkasTanralili, Maros, Sulawesi SelatanJulukanBrigif PR 3/TBSMotoTri Budi SaktiBaret HIJAU TUA Ulang tahun9 DesemberTokohKomandanKolonel Inf. Eko Antoni Chandra Listianto, S.E.Kasbrigif- Brigade Infanteri Para Raider 3/Tri Budi Sakti atau disingkat Brigif…

Kembali kehalaman sebelumnya