title: Vector space

text: In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called "vectors", may be added together and multiplied ("scaled") by numbers called "scalars". Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called "vector axioms". The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.


title: Vector space

text: Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations.


title: Vector space

text: Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension.


title: Vector space

text: Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces.


title: Vector space

text: A vector space over a field is a set  together with two binary operations that satisfy the eight axioms listed below. In this context, the elements of are commonly called "vectors", and the elements of  are called "scalars".


title: Vector space

text: For having a vector space, the eight following axioms must be satisfied for every , and in , and and in .


title: Vector space

text: When the scalar field is the real numbers the vector space is called a "real vector space". When the scalar field is the complex numbers, the vector space is called a "complex vector space". These two cases are the most common ones, but vector spaces with scalars in an arbitrary field are also commonly considered. Such a vector space is called an -"vector space" or a "vector space over ".


title: Vector space

text: An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication), say that this operation defines a ring homomorphism from the field into the endomorphism ring of this group.


title: Vector space

text: "Bases" are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called Hamel bases, depend on the axiom of choice. It follows that, in general, no base can be explicitly described. For example, the real numbers form an infinite-dimensional vector space over the rational numbers, for which no specific basis is known.


title: Vector space

text: Consider a basis formula_14 of a vector space of dimension over a field . The definition of a basis implies that every formula_15 may be written


title: Vector space

text: with formula_17 in , and that this decomposition is unique. The scalars formula_18 are called the "coordinates" of on the basis. They are also said to be the "coefficients" of the decomposition of on the basis. One also says that the -tuple of the coordinates is the coordinate vector of on the basis, since the set formula_19 of the -tuples of elements of is a vector space for componentwise addition and scalar multiplication, whose dimension is .


title: Vector space

text: The one-to-one correspondence between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a vector space isomorphism, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates. If, in turn, these coordinates are arranged as matrices, these reasonings and computations on coordinates can be expressed concisely as reasonings and computations on matrices. Moreover, a linear equation relating matrices can be expanded into a system of linear equations, and, conversely, every such system can be compacted into a linear equation on matrices.


title: Vector space

text: Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, French mathematicians René Descartes and Pierre de Fermat founded analytic geometry by identifying solutions to an equation of two variables with points on a plane curve. To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. introduced the notion of barycentric coordinates. introduced the notion of a bipoint, i.e., an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions by the latter. They are elements in R and R; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations.


title: Vector space

text: In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of linear independence and dimension, as well as scalar products are present. Actually Grassmann's 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called algebras. Italian mathematician Peano was the first to give the modern definition of vector spaces and linear maps in 1888, although he called them "linear systems".


title: Vector space

text: An important development of vector spaces is due to the construction of function spaces by Henri Lebesgue. This was later formalized by Banach and Hilbert, around 1920. At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of "p"-integrable functions and Hilbert spaces. Also at this time, the first studies concerning infinite-dimensional vector spaces were done.


title: Vector space

text: The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, and , the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the "sum" of the two arrows, and is denoted . In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive real number , the arrow that has the same direction as , but is dilated or shrunk by multiplying its length by , is called "multiplication" of by . It is denoted . When is negative, is defined as the arrow pointing in the opposite direction instead.


title: Vector space

text: The following shows a few examples: if , the resulting vector has the same direction as , but is stretched to the double length of (right image below). Equivalently, is the sum . Moreover, has the opposite direction and the same length as (blue vector pointing down in the right image).


title: Vector space

text: A second key example of a vector space is provided by pairs of real numbers and . (The order of the components and is significant, so such a pair is also called an ordered pair.) Such a pair is written as . The sum of two such pairs and multiplication of a pair with a number is defined as follows:


title: Vector space

text: The first example above reduces to this example, if an arrow is represented by a pair of Cartesian coordinates of its endpoint.


title: Vector space

text: The simplest example of a vector space over a field is the field itself (as it is an abelian group for addition, a part of the requirements to be a field), equipped with its addition (It becomes vector addition.) and multiplication (It becomes scalar multiplication.). More generally, all -tuples (sequences of length )


title: Vector space

text: The case is the above-mentioned simplest example, in which the field is also regarded as a vector space over itself. The case and (so R) was discussed in the introduction above.


title: Vector space

text: The set of complex numbers , that is, numbers that can be written in the form for real numbers and where is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: and for real numbers , , , and . The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic.


title: Vector space

text: In fact, the example of complex numbers is essentially the same as (that is, it is "isomorphic" to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number as representing the ordered pair in the complex plane then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.


title: Vector space

text: More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field containing a smaller field is an -vector space, by the given multiplication and addition operations of . For example, the complex numbers are a vector space over , and the field extension formula_22 is a vector space over .


title: Vector space

text: Functions from any fixed set to a field also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions and is the function given by


title: Vector space

text: and similarly for multiplication. Such function spaces occur in many geometric situations, when is the real line or an interval, or other subsets of . Many notions in topology and analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. Therefore, the set of such functions are vector spaces, whose study belongs to functional analysis.


title: Vector space

text: Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of


title: Vector space

text: are given by triples with arbitrary , , and . They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely


title: Vector space

text: where formula_23 is the matrix containing the coefficients of the given equations, is the vector , denotes the matrix product, and is the zero vector. In a similar vein, the solutions of homogeneous "linear differential equations" form vector spaces. For example,


title: Vector space

text: The relation of two vector spaces can be expressed by "linear map" or "linear transformation". They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication:


title: Vector space

text: An "isomorphism" is a linear map such that there exists an inverse map , which is a map such that the two possible compositions and are identity maps. Equivalently, is both one-to-one (injective) and onto (surjective). If there exists an isomorphism between and , the two spaces are said to be "isomorphic"; they are then essentially identical as vector spaces, since all identities holding in are, via , transported to similar ones in , and vice versa via .


title: Vector space

text: For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the - and -component of the arrow, as shown in the image at the right. Conversely, given a pair , the arrow going by to the right (or to the left, if is negative), and up (down, if is negative) turns back the arrow .


title: Vector space

text: Linear maps between two vector spaces form a vector space , also denoted , or . The space of linear maps from to is called the "dual vector space", denoted . Via the injective natural map , any vector space can be embedded into its "bidual"; the map is an isomorphism if and only if the space is finite-dimensional.


title: Vector space

text: Once a basis of is chosen, linear maps are completely determined by specifying the images of the basis vectors, because any element of is expressed uniquely as a linear combination of them. If , a 1-to-1 correspondence between fixed bases of and gives rise to a linear map that maps any basis element of to the corresponding basis element of . It is an isomorphism, by its very definition. Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is "completely classified" (up to isomorphism) by its dimension, a single number. In particular, any "n"-dimensional -vector space is isomorphic to . There is, however, no "canonical" or preferred isomorphism; actually an isomorphism is equivalent to the choice of a basis of , by mapping the standard basis of to , via . The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see below.


title: Vector space

text: "Matrices" are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any -by- matrix gives rise to a linear map from to , by the following


title: Vector space

text: Moreover, after choosing bases of and , "any" linear map is uniquely represented by a matrix via this assignment.


title: Vector space

text: The determinant of a square matrix is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. The linear transformation of corresponding to a real "n"-by-"n" matrix is orientation preserving if and only if its determinant is positive.


title: Vector space

text: Endomorphisms, linear maps , are particularly important since in this case vectors can be compared with their image under , . Any nonzero vector satisfying , where is a scalar, is called an "eigenvector" of with "eigenvalue" . Equivalently, is an element of the kernel of the difference (where Id is the identity map . If is finite-dimensional, this can be rephrased using determinants: having eigenvalue is equivalent to


title: Vector space

text: By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in , called the characteristic polynomial of . If the field is large enough to contain a zero of this polynomial (which automatically happens for algebraically closed, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map. The set of all eigenvectors corresponding to a particular eigenvalue of forms a vector space known as the "eigenspace" corresponding to the eigenvalue (and ) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.


title: Vector space

text: In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object by specifying the linear maps from to any other vector space.


title: Vector space

text: A nonempty subset "W" of a vector space "V" that is closed under addition and scalar multiplication (and therefore contains the 0-vector of "V") is called a "linear subspace" of "V", or simply a "subspace" of "V", when the ambient space is unambiguously a vector space. Subspaces of "V" are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set "S" of vectors is called its span, and it is the smallest subspace of "V" containing the set "S". Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of "S".


title: Vector space

text: A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension , a vector hyperplane is thus a subspace of dimension .


title: Vector space

text: The counterpart to subspaces are "quotient vector spaces". Given any subspace , the quotient space "V"/"W" (""V" modulo "W"") is defined as follows: as a set, it consists of where v is an arbitrary vector in "V". The sum of two such elements and is and scalar multiplication is given by . The key point in this definition is that if and only if the difference of v and v lies in "W". This way, the quotient space "forgets" information that is contained in the subspace "W".


title: Vector space

text: The kernel ker("f") of a linear map consists of vectors v that are mapped to 0 in "W". The kernel and the image are subspaces of "V" and "W", respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field "F") is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups. Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms)


title: Vector space

text: and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups.


title: Vector space

text: An important example is the kernel of a linear map for some fixed matrix "A", as above. The kernel of this map is the subspace of vectors x such that , which is precisely the set of solutions to the system of homogeneous linear equations belonging to "A". This concept also extends to linear differential equations


title: Vector space

text: the derivatives of the function "f" appear linearly (as opposed to "f"′′("x"), for example). Since differentiation is a linear procedure (that is, and for a constant ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation form a vector space (over or ).


title: Vector space

text: The "direct product" of vector spaces and the "direct sum" of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.


title: Vector space

text: The "direct product" formula_29 of a family of vector spaces "V" consists of the set of all tuples (, which specify for each index "i" in some index set "I" an element v of "V". Addition and scalar multiplication is performed componentwise. A variant of this construction is the "direct sum" formula_30 (also called coproduct and denoted formula_31), where only tuples with finitely many nonzero vectors are allowed. If the index set "I" is finite, the two constructions agree, but in general they are different.


title: Vector space

text: The "tensor product" , or simply , of two vector spaces "V" and "W" is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map is called bilinear if "g" is linear in both variables v and w. That is to say, for fixed w the map is linear in the sense above and likewise for fixed v.


title: Vector space

text: The tensor product is a particular vector space that is a "universal" recipient of bilinear maps "g", as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors


title: Vector space

text: These rules ensure that the map "f" from the to that maps a tuple to is bilinear. The universality states that given "any" vector space "X" and "any" bilinear map , there exists a unique map "u", shown in the diagram with a dotted arrow, whose composition with "f" equals "g": . This is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.


title: Vector space

text: From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces "per se" do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures.


title: Vector space

text: A vector space may be given a partial order ≤, under which some vectors can be compared. For example, "n"-dimensional real space R can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions


title: Vector space

text: "Measuring" vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted formula_36 and respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm Vector spaces endowed with such data are known as "normed vector spaces" and "inner product spaces", respectively.


title: Vector space

text: In R, this reflects the common notion of the angle between two vectors x and y, by the law of cosines:


title: Vector space

text: Because of this, two vectors satisfying formula_39 are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R endowed with the Lorentz product


title: Vector space

text: In contrast to the standard dot product, it is not positive definite: formula_41 also takes negative values, for example, for formula_42. Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity.


title: Vector space

text: Convergence questions are treated by considering vector spaces "V" carrying a compatible topology, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in "V", and "a" in "F" vary by a bounded amount, then so do and . To make sense of specifying the amount a scalar changes, the field "F" also has to carry a topology in this context; a common choice are the reals or the complex numbers.


title: Vector space

text: denotes the limit of the corresponding finite partial sums of the sequence ("f") of elements of "V". For example, the "f" could be (real or complex) functions belonging to some function space "V", in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples.


title: Vector space

text: A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem. In contrast, the space of "all" continuous functions on [0,1] with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors v converges to v if and only if


title: Vector space

text: Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of functional analysis—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. The image at the right shows the equivalence of the 1-norm and ∞-norm on R: as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.


title: Vector space

text: From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) , maps between topological vector spaces are required to be continuous. In particular, the (topological) dual space consists of continuous functionals (or to ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.


title: Vector space

text: in which the first formula_55 components are formula_56 and the following ones are formula_57, converges to the zero vector for


title: Vector space

text: These spaces are complete. (If one uses the Riemann integral instead, the space is "not" complete, which may be seen as a justification for Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions


title: Vector space

text: Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.


title: Vector space

text: Complete inner product spaces are known as "Hilbert spaces", in honor of David Hilbert. The Hilbert space "L"(Ω), with inner product given by


title: Vector space

text: By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions "f" with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions "f" by polynomials. By the Stone–Weierstrass theorem, every continuous function on can be approximated as closely as desired by a polynomial. A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space "H", in the sense that the "closure" of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a "basis" of "H", its cardinality is known as the Hilbert space dimension. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the Gram–Schmidt process, it enables one to construct a basis of orthogonal vectors. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space.


title: Vector space

text: The solutions to various differential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular physical properties are used as basis functions, often orthogonal. As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in time by means of a partial differential equation, whose solutions are called wavefunctions. Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.


title: Vector space

text: General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an "algebra over a field". Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem, for example, relies on Banach algebras which are both Banach spaces and algebras.


title: Vector space

text: Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotients form the basis of algebraic geometry, because they are rings of functions of algebraic geometric objects.


title: Vector space

text: Another crucial example are "Lie algebras", which are neither commutative nor associative, but the failure to be so is limited by the constraints ( denotes the product of and ):


title: Vector space

text: Examples include the vector space of "n"-by-"n" matrices, with , the commutator of two matrices, and , endowed with the cross product.


title: Vector space

text: The tensor algebra T("V") is a formal way of adding products to any vector space "V" to obtain an algebra. As a vector space, it is spanned by symbols, called simple tensors


title: Vector space

text: The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced above. In general, there are no relations between and . Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing yields the exterior algebra.


title: Vector space

text: A "vector bundle" is a family of vector spaces parametrized continuously by a topological space "X". More precisely, a vector bundle over "X" is a topological space "E" equipped with a continuous map


title: Vector space

text: such that for every "x" in "X", the fiber π("x") is a vector space. The case dim is called a line bundle. For any vector space "V", the projection makes the product into a "trivial" vector bundle. Vector bundles over "X" are required to be locally a product of "X" and some (fixed) vector space "V": for every "x" in "X", there is a neighborhood "U" of "x" such that the restriction of π to π("U") is isomorphic to the trivial bundle . Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space "X") be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle ). For example, the Möbius strip can be seen as a line bundle over the circle "S" (by identifying open intervals with the real line). It is, however, different from the cylinder , because the latter is orientable whereas the former is not.


title: Vector space

text: Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle "S" is globally isomorphic to , since there is a global nonzero vector field on "S". In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere "S" which is everywhere nonzero. K-theory studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions O.


title: Vector space

text: The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms.


title: Vector space

text: "Modules" are to rings what vector spaces are to fields: the same axioms, applied to a ring "R" instead of a field "F", yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the elements being called vectors. Some authors use the term "vector space" to mean modules over a division ring. The algebro-geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles.


title: Vector space

text: Roughly, "affine spaces" are vector spaces whose origins are not specified. More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map


title: Vector space

text: If "W" is a vector space, then an affine subspace is a subset of "W" obtained by translating a linear subspace "V" by a fixed vector ; this space is denoted by (it is a coset of "V" in "W") and consists of all vectors of the form for An important example is the space of solutions of a system of inhomogeneous linear equations


title: Vector space

text: generalizing the homogeneous case above, which can be found by setting in this equation. The space of solutions is the affine subspace where x is a particular solution of the equation, and "V" is the space of solutions of the homogeneous equation (the nullspace of "A").


title: Vector space

text: The set of one-dimensional subspaces of a fixed finite-dimensional vector space "V" is known as "projective space"; it may be used to formalize the idea of parallel lines intersecting at infinity. Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension "k" and flags of subspaces, respectively.


title: Windows Server 2016

text: Windows Server 2016 is the eighth release of the Windows Server server operating system developed by Microsoft as part of the Windows NT family of operating systems. It was developed concurrently with Windows 10 and is the successor to the Windows 8.1-based Windows Server 2012 R2. The first early preview version (Technical Preview) became available on October 1, 2014 together with the first technical preview of System Center. Windows Server 2016 was released on September 26, 2016 at Microsoft's Ignite conference and broadly released for retail sale on October 12, 2016. It has three successors: Windows Server 2019, Windows Server 2022, and the Windows Server Semi-Annual Channel, which excludes the graphical user interface and many older components.


title: Windows Server 2016

text: Microsoft announced a new installation option, Nano Server, which offers a minimal-footprint headless version of Windows Server. It excludes the graphical user interface, WoW64 (support for 32-bit software) and Windows Installer. It does not support console login, either locally or via Remote Desktop Connection. All management is performed remotely via Windows Management Instrumentation (WMI), Windows PowerShell and Remote Server Management Tools (a collection of web-based GUI and command line tools). However, in Technical Preview 5, Microsoft has re-added the ability to administer Nano Server locally through PowerShell.


title: Windows Server 2016

text: According to Microsoft engineer Jeffrey Snover, Nano Server has 93% lower VHD size, 92% fewer critical security advisories, and 80% fewer reboots than Windows Server.


title: Windows Server 2016

text: Nano Server is only available to Microsoft Software Assurance customers and on cloud computing platforms such as Microsoft Azure and Amazon Web Services.


title: Windows Server 2016

text: Starting with the new feature release of Windows Server version 1709, Nano Server can only be installed inside a container host.


title: Windows Server 2016

text: Microsoft has been reorganized by Satya Nadella, putting the Server and System Center teams together. Previously, the Server team was more closely aligned with the Windows client team. The Azure team is also working closely with the Server team.


title: Windows Server 2016

text: In March 2017, Microsoft demonstrated an internal version of Server 2016 running on the ARMv8-A architecture. It was reported that Microsoft was working with Qualcomm Centriq and Cavium ThunderX2 chips. According to James Vincent of "The Verge", this decision endangers Intel's dominance of the server CPU market. However, later inquiry from Microsoft revealed that this version of Windows Server is only for internal use and only impacts subscribers of Microsoft Azure service.


title: Windows Server 2016

text: A public beta version of Windows Server 2016 (then still called "vNext") branded as "Windows Server Technical Preview" was released on October 1, 2014; the technical preview builds are aimed toward enterprise users. The first Technical Preview was first set to expire on April 15, 2015 but Microsoft later released a tool to extend the expiry date, to last until the second tech preview of the OS in May 2015. The second beta version, "Technical Preview 2", was released on May 4, 2015. Third preview version, "Technical Preview 3" was released on August 19, 2015. "Technical Preview 4" was released on November 19, 2015. "Technical Preview 5" was released on April 27, 2016.


title: Windows Server 2016

text: Windows Server 2016 was officially released at Microsoft's Ignite Conference on September 26, 2016. Unlike its predecessor, Windows Server 2016 is licensed by the number of CPU cores rather than number of CPU sockets—a change that has similarly been adopted by BizTalk Server 2013 and SQL Server 2014. The new licensing structure that has been adopted by Windows Server 2016 has also moved away from the Windows Server 2012/2012R2 CPU socket licensing model in that now the amount of cores covered under one license is limited. Windows Server 2016 Standard and Datacenter core licensing now covers a minimum of 8 core licenses for each physical processor and a minimum of 16 core licenses for each server. Core licenses are sold in packs of two with Standard Edition providing the familiar rights to run 2 virtualized OS environments. If the server goes over 16 core licenses for a 2 processor server additional licenses will now be required with Windows Server 2016.


title: Windows Server 2016

text: Windows Server 2016 Technical Preview, released on October 1, 2014, was the first beta version of the operating system made publicly available. Its version number was 6.4.9841.


title: Windows Server 2016

text: Windows Server 2016 Technical Preview 2 was made available on May 4, 2015. Its version number was 10.0.10074. (A similar jump in the most significant part of the version number from 6 to 10 is seen in Windows 10.) Highlights of this version include:


title: Windows Server 2016

text: The third technical preview of Windows Server 2016 was made available on August 19, 2015. Its version number was 10.0.10514. Highlights of this version include:


title: Windows Server 2016

text: The fourth technical preview of the operating system was made available on November 19, 2015, one year and one month after the initial technical preview. Its version number was 10.0.10586. Its highlights include:


title: Windows Server 2016

text: The last technical preview of Windows Server 2016 was made available on April 27, 2016. Its version number was 10.0.14300. Its highlights include:


title: Windows Server 2016

text: Windows Server 2016 was released to manufacturing on September 26, 2016, bearing the version number of 10.0.14393 (same as Windows 10 Anniversary Update). Microsoft added the following final touches:


title: Windows Server 2016

text: Windows Server, version 1709 (version shared with Windows 10 Fall Creators Update) was released on October 17, 2017. The release has dropped the Windows Server 2016 name and is just called Windows Server by Microsoft. It is offered to the Microsoft Software Assurance customers who have an active Windows Server 2016 license and has the same system requirements. This is the first Windows Server product to fall under the "Semi-Annual Channel" (SAC) release cadence. This product only features the Server Core and the Nano Server modes. Of the two, only the Server Core mode of the OS can be installed on a bare system. The Nano Server mode is only available as an operating system container.


title: Windows Server 2016

text: Windows Server, version 1803 (version shared with Windows 10 April 2018 Update) is the second Semi-Annual Channel release of Windows Server. It is also the final version to be branched off the Server 2016 codebase, as the next release shares the version number 1809 with Windows Server 2019.


title: Chappaquiddick incident

text: The Chappaquiddick incident occurred on Chappaquiddick Island in Massachusetts some time around midnight between July 18 and 19, 1969, when Senator Edward M. "Ted" Kennedy negligently drove his car off a narrow bridge, causing it to overturn in a tidal pond. This resulted in the drowning death of his 28-year-old passenger Mary Jo Kopechne, who was trapped inside the vehicle.


title: Chappaquiddick incident

text: Kennedy left a party on Chappaquiddick at 11:15p.m. Friday. He maintained that his intent was to immediately take Kopechne to a ferry landing and return to Edgartown, but that he accidentally made a wrong turn onto a dirt road leading to a one-lane bridge. After his car skidded off the bridge into Poucha Pond, Kennedy swam free, and maintained that he tried to rescue Kopechne from the submerged car, but that he could not. Kopechne's death could have happened any time between about 11:30p.m. Friday and 1a.m. Saturday, as an off-duty deputy sheriff stated he saw a car matching Kennedy's license plate at 12:40a.m. Kennedy left the scene and did not report the accident to police until after 10a.m. Saturday. Meanwhile, a diver recovered Kopechne's body from Kennedy's car shortly before 9a.m. Saturday.


title: Chappaquiddick incident

text: At a July 25, 1969, court hearing, Kennedy pled guilty to a charge of leaving the scene of an accident, and received a two-month suspended jail sentence. In a televised statement that same evening, he said his conduct immediately after the accident "made no sense to me at all", and that he regarded his failure to report the accident immediately as "indefensible". A January 5, 1970 judicial inquest concluded that Kennedy and Kopechne did not intend to take the ferry, and that Kennedy intentionally turned toward the bridge, operating his vehicle negligently, if not recklessly, at too high a speed for the hazard which the bridge posed in the dark. The judge stopped short of recommending charges, and a grand jury convened on April 6, 1970, returning no indictments. On May 27, 1970, a Registry of Motor Vehicles hearing resulted in Kennedy's driver's license being suspended for a total of sixteen months after the accident.


title: Chappaquiddick incident

text: The Chappaquiddick incident became national news that influenced Kennedy's decision not to run for President in 1972 and 1976, and it was said to have undermined his chances of ever becoming President. Kennedy ultimately decided to enter the 1980 Democratic Party presidential primaries, but earned only 37.6% of the vote and lost the nomination to incumbent President Jimmy Carter.


title: Chappaquiddick incident

text: U.S. Senator Edward M. (Ted) Kennedy, age 37, and his cousin, Joseph Gargan, 39, planned to race Kennedy's sailboat, "Victura", in the 1969 Edgartown Yacht Club Regatta on Friday and Saturday, July 18 and 19, 1969, after having forgone the previous year's Regatta because of the assassination of Kennedy's brother, Robert, that June. Gargan rented secluded Lawrence Cottage for the weekend on Chappaquiddick Island, which is a tiny island accessible by ferry from Edgartown on Martha's Vineyard. They hosted a cookout party at the cottage at 8:30p.m that evening as a re-union for the "Boiler Room Girls", women who had served on Robert Kennedy's 1968 presidential campaign. Six of these attended the party: Mary Jo Kopechne, Rosemary Keough, Esther Newberg, sisters Nance and Mary Ellen Lyons, and Susan Tannenbaum. All were in their twenties, and single.


title: Chappaquiddick incident

text: The men at the party included the crew of Kennedy's sailboat entered in the regatta: Gargan; Paul Markham, a school friend of Gargan who had previously served as the U.S. Attorney for Massachusetts; and John B. Crimmins, 63, a long-time political associate of Kennedy who served as his chauffeur for the weekend. Others in attendance were attorney Charles Tretter, a Kennedy advisor; and Raymond LaRosa, who had worked on Kennedy's Senate campaigns. All the men were married, except Crimmins; wives were not invited to the Chappaquiddick weekend. Other friends and campaign workers, male and female, had been invited, but did not attend, for various reasons. Markham and Crimmins intended to spend the night at the cottage, while the others were booked at hotels on Martha's Vineyard - the men at the Shiretown Inn, one block from the Edgartown ferry slip, and the women at the Katama Shores motor inn, about south of the ferry slip.


title: Chappaquiddick incident

text: According to Kennedy, Kopechne asked him to give her a ride back to her hotel in Katama. Kennedy requested the keys to his car (which he did not usually drive) from his chauffeur Crimmins. Kennedy put this time at "approximately 11:15p.m.", although he was not wearing a watch; the time came from Crimmins' watch. Returning to Edgartown and Katama required making the last ferry, which left the island at midnight, or else calling to arrange a later ferry. Kopechne told no one else that she was leaving for the night with Kennedy, and, in fact, left her purse and hotel key at the party.


title: Chappaquiddick incident

text: The exact time the crash occurred is unknown, due to a conflict between the testimony of Kennedy and a deputy sheriff who claimed to have seen his car at a later time. Kennedy claimed that, as soon as he left the party, he immediately drove north on Chappaquiddick Road headed for the ferry landing, and mistakenly made a wrong turn, right, onto the dirt Dike Road, instead of bearing left to stay on the paved Chappaquiddick Road for another . There is also a northbound dirt Cemetery Road at this intersection.


title: Chappaquiddick incident

text: Part-time Deputy Sheriff Christopher "Huck" Look left work by 12:30a.m. on Saturday as a gate guard in uniform for the regatta dance, returned to Chappaquiddick Island in the yacht club's private boat, and drove east and south on Chappaquiddick Road toward his home. At around 12:40a.m., after he passed the intersection with Dike Road, he saw a dark four-door sedan driven by a man, with a woman in the front seat, approaching and passing slowly in front of him. The car drove off the pavement onto Cemetery Road, and stopped. Thinking the occupants of the car might be lost, Look stopped and walked towards the other vehicle. When he was away, the car reversed and started backing up towards him. As he called out to offer help, the car moved forward and veered quickly eastward onto Dike Road, speeding away and leaving a cloud of dust. Look recalled that the car's license plate began with an L and contained two 7s, consistent with Kennedy's license L78–207 on his Oldsmobile Delmont 88. He returned to his car, and continued on his way south. Look's version, if true, leaves over an hour of Kennedy's time with Kopechne unaccounted before the crash.


title: Chappaquiddick incident

text: About a minute later, Look saw Kennedy's party guests Nance and Mary Ellen Lyons, and Ray LaRosa, dancing in a conga line down the middle of Chappaquiddick Road, a short distance south of Dike bridge. He stopped to ask if they needed a ride, which they declined. LaRosa and the Lyons sisters corroborated Look's testimony about meeting him in the road and the verbal exchange, but they were unsure of the time. They also said they saw a vehicle driving north on Chappaquiddick Road, which they could not describe in any detail.


title: Chappaquiddick incident

text: Dike Road leads to Dike Bridge, a wooden structure angled obliquely to the road, crossing the channel connecting Cape Pogue Pond to the north and Poucha Pond to the south, leading eastward to a barrier beach known as Tom's Neck Point. At the time, the bridge was not fitted with guardrails. A fraction of a second before Kennedy reached the bridge, he applied his brakes and lost control of the car, which launched over the southern end of the bridge, plunged nose-first into the channel, and flipped over, resting on its roof.


title: Chappaquiddick incident

text: Kennedy was able to swim free of the vehicle, but Kopechne was not. Kennedy said that he called her name several times from the shore, and tried to swim down to reach her seven or eight times. He then rested on the bank for around 15 minutes, before he returned on foot to Lawrence Cottage. He denied seeing any house with a light on during his 15-minute walk back. His route back took him past four houses from which he could have telephoned to summon help before he reached the cottage, but he did not attempt to contact the local residents. The first of the houses was Dike House, from the bridge and occupied by Sylvia Malm and her family. Malm stated later that she was home, she had a phone, and she had left a light on at the residence when she retired that evening.


title: Chappaquiddick incident

text: Kennedy returned to the cottage, where the party was still in progress, but rather than alerting all of the guests to the crash, he quietly summoned Gargan and Markham, and collapsed in the back seat of a rented Plymouth Valiant parked in the driveway. Gargan drove the three to the site of the crash, in order to try to rescue Kopechne from Kennedy's overturned car. Gargan and Markham jumped into the pond, and tried repeatedly to rescue her, but were not able to, due to the strong tidal current. After they recovered, Gargan drove Kennedy and Markham to the ferry landing. The three were all lawyers, and they discussed what they should do while standing next to a public phone booth at the landing. Gargan and Markham insisted multiple times that the crash had to be reported to the authorities.


title: Chappaquiddick incident

text: Kennedy suddenly dove into the channel, swam the across to Edgartown, and returned to his hotel room, where he removed his clothes and collapsed on his bed. He later put on dry clothes, left his room, and asked someone what the time was; it was somewhere around 2:30 a.m., he recalled. Meanwhile, Gargan and Markham had driven the rental car back to the cottage; they entered the cottage at approximately 2 a.m., but told no one what had happened. When questioned by the guests, they said that Kennedy had swum back to Edgartown, and Kopechne was probably at her hotel. Gargan then told everyone to get some sleep.


title: Chappaquiddick incident

text: By 7:30 a.m., Kennedy was talking casually to the winner of the previous day's sailing race, and gave no indication that anything was amiss. At 8 a.m., Gargan and Markham had crossed back to Edgartown on the ferry, and met Kennedy.


title: Chappaquiddick incident

text: A short time after 8 a.m., a man and a fifteen-year-old boy, who went fishing off Tom's Neck Point, saw Kennedy's submerged car in Poucha Pond and notified the residents of the cottage nearest the scene, who, in turn, called the authorities at about 8:20 a.m. Edgartown Police Chief Dominick James Arena arrived at the scene about 10 or 15 minutes later. He attempted to examine the interior of the submerged vehicle, then summoned a trained scuba diver and equipment capable of towing or winching the vehicle out of the water. John Farrar, captain of the Edgartown Fire Rescue unit, arrived at 8:45 a.m., equipped with scuba gear, and discovered Kopechne's body in the back seat; he extricated it from the vehicle within 10 minutes. Police checked the car's license plate, and saw that it was registered to Kennedy. Rosemary Keough's purse was found in the front passenger compartment of the car, causing Arena to misidentify Kopechne.


title: Chappaquiddick incident

text: Meanwhile, Kennedy, Gargan, and Markham crossed back to Chappaquiddick Island on the ferry, where Kennedy made a series of telephone calls from a pay phone near the ferry crossing—the same phone that the three men had stood by approximately six hours earlier discussing Kennedy's options. Kennedy called friends and lawyers for advice, however, instead of notifying the authorities that he was the operator of the vehicle, which was still upside down in Poucha Pond. He called his brother-in-law Stephen Edward Smith, congressman John V. Tunney, and others that morning, but he still did not report the accident to authorities.


title: Chappaquiddick incident

text: Kennedy was still at the pay phone when he heard that his car and Kopechne's body had been discovered; he then crossed back to Edgartown to go to the police station with Markham. Meanwhile, Gargan went to the Katama Shores to inform the Boiler Room Girls of the incident. Kennedy entered the police station at approximately 9:50a.m. He asked to make some telephone calls, and was told he could use Arena's office. When Arena returned to the station at 10:00, he was "stunned" to learn Kennedy already knew of the accident and the true identity of the victim, and admitted he was the driver. Arena led Kennedy to another empty office where he could privately dictate his statement to Markham, who wrote it out in long hand. Arena then typed out the statement:


title: Chappaquiddick incident

text: As the Medical Examiner Dr. Robert Nevin had the day off, Associate Medical Examiner Dr. Donald Mills was called to the crash site to examine the body. He was satisfied that the cause of death was accidental drowning, but asked the District Attorney's office for direction on whether an autopsy was necessary, and was told it was not, as long as there were no signs of foul play and he was satisfied it was a drowning. He signed the death certificate to that effect and released the body for embalming, and directed that a blood sample be collected and sent to the State Police for analysis of alcohol content. The result was 0.09%, which Mills mistakenly thought represented only a "moderate" level, but, in fact, indicated in a person of Kopechne's weight, up to five drinks of liquor within an hour prior to death.


title: Chappaquiddick incident

text: Kopechne's body was released to her family, and the funeral was held on Tuesday, July 22, in Plymouth, Pennsylvania.


title: Chappaquiddick incident

text: Medical Examiner Nevin strongly disagreed with Mills' decision to forgo an autopsy, believing that ruling out foul play would work to Kennedy's advantage by laying prurient public speculation to rest.


title: Chappaquiddick incident

text: After President Richard Nixon's security operative Jack Caulfield learned of the incident, he dispatched Anthony Ulasewicz to Dike Bridge in disguise as a newspaper reporter to collect information since he believed Kennedy would be his rival in the 1972 presidential election. Although Ulasewicz was able to interview several witnesses before law enforcement authorities, he found no useful information.


title: Chappaquiddick incident

text: Farrar, who recovered Kopechne's body from the submerged car, believed that Kopechne died from suffocation, rather than from drowning or from the impact of the overturned vehicle, based upon the posture in which he found the body in the well of the back seat of the car, where an air pocket would have formed. Rigor mortis was apparent, her hands were clasping the back seat, and her face was turned upward. Bob Molla, an inspector for the Massachusetts Registry of Motor Vehicles who investigated the crash at the time, said that parts of the roof and the trunk appeared to be dry. Farrar publicly asserted that Kopechne likely would have survived if a more timely rescue attempt had been conducted.


title: Chappaquiddick incident

text: Kennedy returned to his family's compound in Hyannis Port. Stephen Smith, Robert McNamara, Ted Sorensen, Richard N. Goodwin, Lem Billings, Milton Gwirtzman, David W. Burke, John Culver, Tunney, Gargan, Markham, and others arrived to advise him. Smith, the Kennedy family business manager and "master fixer", resigned himself to the fact that the political damage had been done, and Kennedy's chance of running for President at the next available window, 1972, was gone. The problem remained how to protect him from a charge of manslaughter.


title: Chappaquiddick incident

text: Kennedy's court hearing was held before Massachusetts District Court Judge James Boyle on July 25, seven days after the incident. Kennedy pleaded guilty to a charge of leaving the scene of an accident causing bodily injury. His attorneys argued that any jail sentence should be suspended, and the prosecutors agreed by citing his age (37), character, and prior reputation. "Considering the unblemished record of the defendant, and insofar as the Commonwealth represents this is not a case where he was really trying to conceal his identity...", Boyle sentenced him to the statutory minimum two months in prison, which he suspended, saying that he "has already been, and will continue to be punished far beyond anything this court can impose."


title: Chappaquiddick incident

text: Despite an Associated Press story published that morning, Boyle was unaware that Kennedy's driving record was, in fact, far from "unblemished". While attending University of Virginia School of Law (1956–1959), he had compiled a record of reckless driving and driving without a license. In one particular March 14, 1958, incident, Kennedy ran a red light, then cut his tail lights and raced to avoid a highway patrol officer. When Kennedy was caught, he was cited for reckless driving, racing to avoid pursuit, and driving without a license.


title: Chappaquiddick incident

text: Kennedy's wife Joan was pregnant at the time of the Chappaquiddick incident. She was confined to bed because of two previous miscarriages, but she attended Kopechne's funeral and stood beside her husband in court. Soon after, she suffered a third miscarriage, which she blamed on the Chappaquiddick incident.


title: Chappaquiddick incident

text: At 7:30 p.m. on July 25, Kennedy delivered a lengthy speech about the incident, prepared by Sorensen and broadcast live by the three television networks. He began by reading the speech off a prepared manuscript.


title: Chappaquiddick incident

text: Kennedy explained that his wife did not accompany him to the regatta due to "reasons of health". He denied he engaged in any "immoral conduct" with Kopechne or driving under the influence of alcohol that evening. He said that his conduct during the hours immediately after the accident "made no sense to me at all", and said that his doctors had informed him he had suffered "cerebral concussion and shock". He said he regarded his failure to report the accident to the police immediately as "indefensible". To the horror of Gargan's attorney, his statement revealed his enlistment of the help of Gargan and Markham to try to rescue Kopechne (despite assurances he had made to the effect that he would not involve them).


title: Chappaquiddick incident

text: He said "all kinds of scrambled thoughts" went through his mind after the accident, including "whether the girl might still be alive somewhere out of that immediate area", whether "some awful curse actually did hang over all the Kennedys", whether there was "some justifiable reason for me to doubt what had happened and to delay my report", and whether, "somehow, the awful weight of this incredible incident might in some way pass from my shoulders". He said he was overcome "by a jumble of emotions — grief, fear, doubt, exhaustion, panic, confusion, and shock". He said he instructed Gargan and Markham "not to alarm Mary Jo's friends that night", then returned to the ferry with the two men and "suddenly jumped into the water and impulsively swam across, nearly drowning once again in the effort, returning to my hotel around 2 a.m. and collapsed in my room".


title: Chappaquiddick incident

text: Kennedy then put down his manuscript (though he continued reading from cue cards), and asked the people of Massachusetts to decide whether he should resign:


title: Chappaquiddick incident

text: The speech concluded with a passage quoted from John F. Kennedy's book "Profiles in Courage" (ghostwritten by Sorensen): "A man does what he must — in spite of personal consequences".


title: Chappaquiddick incident

text: Critical reaction to the speech was immediate and negative. NBC newsman John Chancellor compared it to Richard Nixon's 1952 Checkers speech. Kennedy admirer David Halberstam wrote in "Harper's" magazine that it was "of such cheapness and bathos as to be a rejection of everything the Kennedys had stood for in candor and style. It was as if these men had forgotten everything which made the Kennedys distinctive in American politics, and simply told the youngest brother that he could get away with whatever he wanted because he was a Kennedy in Massachusetts."


title: Chappaquiddick incident

text: Although Kennedy received many messages from voters opposed to his resignation from the Senate, reaction in much of the news media, and of District Attorney Edmund Dinis, was that Kennedy's televised speech left many questions unanswered about how the accident happened, and his delay in reporting it. On July 31, 1969, the same day Kennedy returned to his Senate seat, Dinis wrote to the Chief Justice of the Massachusetts Superior Court, Joseph Tauro, asking for a judicial inquest into Kopechne's death. He received a response the next day that such inquests are under jurisdiction of the District Court. Dinis then sent his request to Kenneth Nash, the Chief Justice of the lower court. Nash advised Dinis that a grand jury investigation had more "teeth" than an inquest, as it had the power to indict defendants, whereas an inquest was only authorized to determine if a crime has been committed.


title: Chappaquiddick incident

text: Dinis met with Edgartown District Court Judge James Boyle on August 8 to explain his reasons for requesting the inquest. Boyle did not recuse himself, even though he had presided over the hearing at which Kennedy pled guilty. Boyle announced the inquest was scheduled to start on September 3, and would be open to the press. On September 2, Kennedy's lawyers petitioned the Massachusetts Supreme Court for a temporary injunction against the inquest, which was granted.


title: Chappaquiddick incident

text: Dinis petitioned for an exhumation and autopsy of Kopechne's body, and on September 18, 1969, he publicly disclosed that blood had been found on her long-sleeved blouse and in her mouth and nose, "which may or may not be consistent with death by drowning", when her clothes were given to authorities by the funeral director.


title: Chappaquiddick incident

text: Judge Bernard Brominski of the Court of Common Pleas in Luzerne County, Pennsylvania, held a hearing on the request on October 20–21. The request was opposed by Kopechne's parents. Forensic pathologist Werner Spitz testified on behalf of Joseph and Gwen Kopechne that the autopsy was unnecessary and the available evidence was sufficient to conclude that Kopechne died from drowning. Judge Brominski ruled against the exhumation on December 1, saying that there was "no evidence" that "anything other than drowning had caused the death of Mary Jo Kopechne."


title: Chappaquiddick incident

text: The inquest convened in Edgartown in January 1970. At the request of Kennedy's lawyers, the Massachusetts Supreme Judicial Court ordered it to be performed secretly with Judge James A. Boyle presiding, and the 763-page transcript was released four months later.


title: Chappaquiddick incident

text: Kennedy testified that Kopechne told him, when he was about to leave the party, "that she was desirous of leaving" and asked "if I would be kind enough to drop her back at her hotel." Crimmins and some other guests "were concluding their meal, enjoying the fellowship and it didn't appear to be necessary to require him to bring me back to Edgartown." Witnesses at the party variously placed the time of Kennedy and Kopechne's departure between 11:00 and 11:45p.m..


title: Chappaquiddick incident

text: Kennedy also testified that he never stopped on Cemetery Road, never backed up, never saw the deputy, and never saw another car or person after he left the cottage with Kopechne. He further claimed that after he turned onto Dike Road, he was driving and did not realize that he was no longer headed west toward the ferry landing but was instead heading east toward the barrier beach.


title: Chappaquiddick incident

text: Kennedy testified that he had "full intention of reporting it. And I mentioned to Gargan and Markham something like, 'You take care of the other girls; I will take care of the accident!'—that is what I said and I dove into the water." Kennedy had told Gargan and Markham not to tell the other women anything about the incident "because I felt strongly that if these girls were notified that an accident had taken place and Mary Jo had, in fact, drowned, that it would only be a matter of seconds before all of those girls, who were long and dear friends of Mary Jo's, would go to the scene of the accident and enter the water with, I felt, a good chance that some serious mishap might have occurred to any one of them."


title: Chappaquiddick incident

text: Kennedy testified that he was back at the hotel and "almost tossed and turned and walked around that room... I had not given up hope all night long that, by some miracle, Mary Jo would have escaped from the car." Kennedy complained to the hotel owner at 2:55 a.m. that he had been awakened by a noisy party. At 8 a.m., Gargan and Markham found him at his hotel where they had a "heated conversation" in Kennedy's room. According to Kennedy's testimony, the two men asked why he had not reported the accident, and he responded by telling them "about my own thoughts and feelings as I swam across that channel... that somehow when they arrived in the morning that they were going to say that Mary Jo was still alive."


title: Chappaquiddick incident

text: Markham testified that after their rescue attempt, Kennedy was sobbing and on the verge of becoming crazed. Gargan and Markham testified that they assumed that Kennedy was going to inform the authorities about the accident once he got back to Edgartown, and so they did not do the reporting themselves. In an October 15, 1994, interview for Ronald Kessler's book "The Sins of the Father: Joseph P. Kennedy and the Dynasty He Founded", Gargan said that he and Markham returned to the scene of the accident with Kennedy, they both urged Kennedy to report the accident to the police. "The conversation was brief about having to report", Gargan told Kessler, a former "The Washington Post" reporter, for the book. "I was insistent on it. Paul Markham was backing me up on it. Ted said, 'Okay, okay, Joey, okay. I've got the point, I've got the point.' Then he took a few steps and dove into the water, leaving Markham and I expecting that he would carry out the conversation."


title: Chappaquiddick incident

text: Farrar testified that Kopechne's body was pressed up in the car in the spot where an air bubble would have formed. He interpreted that to mean that she had survived in the air bubble after the car submerged, and he concluded that:


title: Chappaquiddick incident

text: Having found probable cause of a crime, under Massachusetts law Boyle could have issued a warrant for his arrest, but he did not do so. Despite Boyle's findings, Dinis chose not to prosecute Kennedy for manslaughter. The Kopechne family did not bring any legal action against him but did receive a payment of $90,904 from him personally and $50,000 from his insurance company. The Kopechnes later explained their decision not to take legal action by saying, "We figured that people would think we were looking for blood money."


title: Chappaquiddick incident

text: On April 6, 1970, a Dukes County grand jury assembled in special session to investigate Kopechne's death. Judge Wilfred Paquet instructed the members of the grand jury that they could consider only matters brought to their attention by the superior court, the district attorney, or their personal knowledge. He cited the orders of the Massachusetts Supreme Judicial Court and told the grand jury that it could not see the evidence or Boyle's report from the inquest, which were still impounded. Dinis had attended the inquest and seen Boyle's report, and he told the grand jury that there was not enough evidence to indict Kennedy on potential charges of manslaughter, perjury, or driving to endanger. The grand jury called four witnesses who had not testified at the inquest; they testified for a total of 20 minutes, but no indictments were issued.


title: Chappaquiddick incident

text: On July 23, 1969, the registrar of the Massachusetts Registry of Motor Vehicles informed Kennedy that his license would be suspended until there was a statutory hearing concerning the accident. The suspension was required by Massachusetts law for any fatal motor vehicle accident if there were no witnesses. The "in camera" hearing was held May 18, 1970, and found that "operation was too fast for existing conditions." On May 27, the registrar informed Kennedy in a letter that "I am unable to find that the fatal accident in which a motor vehicle operated by you was involved, was without serious fault on your part" and so his driver's license was suspended for a further six months.


title: Chappaquiddick incident

text: Journalist Jack Olsen wrote the first investigative book on the case, "The Bridge at Chappaquiddick", in 1970, attempting to solve the unanswered questions of the incident. Lieutenant Bernie Flynn, a state police detective assigned to the Cape Cod district attorney's office, was a Kennedy admirer who came up with a theory which he couldn't prove: that Kennedy got out of the car, and Kopechne drove herself off the bridge. "Ted Kennedy didn't want to admit being drunk with a broad in a car late at night. When he saw 'Huck' Look, he got scared. He thought a cop was coming after him." Flynn claimed to have told this theory to Olsen, who didn't seem to be very impressed. Although Olsen denied having ever talked to Flynn, he related this theory in his book. Kopechne was , a foot shorter than Kennedy, and Olsen argued that she might possibly not have seen the bridge as she drove Kennedy's car over unfamiliar roads at night, with no external lighting, and after she had consumed several alcoholic drinks. He wrote that Kopechne normally drove a Volkswagen Beetle, which was much smaller, lighter, and easier to handle than Kennedy's larger Oldsmobile 88.


title: Chappaquiddick incident

text: A BBC "Inside Story" episode titled "Chappaquiddick", broadcast on July 20, 1994 (the 25th anniversary of the incident), repeated Flynn's theory. The episode argued that the explanation would account for Kennedy's lack of concern the next morning, as he was unaware of the accident, and for the forensic evidence of the injuries to Kopechne being inconsistent with her sitting in the passenger seat.


title: Chappaquiddick incident

text: Fourth-generation Chappaquiddick resident Bill Pinney, in his 2017 book "Chappaquiddick Speaks", presents a theory that Kopechne was seriously injured in an earlier crash, and then the bridge accident was faked. The book laments how the incident robbed Chappaquiddick of its traditional peace and privacy, attracting large tourist groups wanting to view the sites connected with the tragedy.


title: Chappaquiddick incident

text: The case evoked much satire of Kennedy. For example, "Time" magazine reported immediately after the incident that "one sick joke already visualizes a Democrat asking about Nixon during the 1972 presidential campaign: 'Would you let this man sell you a used car?' Answer: 'Yes, but I sure wouldn't let Teddy drive it. A mock advertisement in "National Lampoon" magazine showed a floating Volkswagen Beetle, itself a parody of a Volkswagen advertisement, showing that the vehicle's underside was so well sealed that it would float on water, but with the caption, "If Ted Kennedy drove a Volkswagen, he'd be President today." The satire resulted in legal action by Volkswagen, claiming unauthorized use of its trademark; the matter was later settled out of court.


title: Chappaquiddick incident

text: Following his televised speech on July 25 regarding the incident, supporters responded with telephone calls and telegrams to newspapers and to the Kennedy family. They were heavily in favor of his remaining in office, and he was re-elected in 1970 with 62% of the vote, a margin of nearly a half million votes, but it was down from 74% in the previous election in 1964.


title: Chappaquiddick incident

text: The incident severely damaged Kennedy's national reputation and reputation for judgement. One analyst asked: "Can we really trust him if the Russians come over the ice cap? Can he make the kind of split-second decisions the astronauts had to make in their landing on the moon?" Before Chappaquiddick, public polls showed that a large majority expected Kennedy to run for the presidency in 1972, but he pledged not to run in 1972. Further, Kennedy declined to serve as George McGovern's running mate that year. In 1974, he pledged not to run in 1976, in part because of the renewed media interest in Chappaquiddick.


title: Chappaquiddick incident

text: In late 1979, Kennedy announced his candidacy for the presidency when he challenged President Jimmy Carter for the Democratic nomination for the 1980 election. On November 4, 1979, CBS broadcast a one-hour television special presented by Roger Mudd, titled "Teddy". The program consisted of an interview with Kennedy; the interview was interspersed with visual materials. Much of the show was devoted to the Chappaquiddick incident. During the interview, Mudd questioned Kennedy repeatedly about the incident, and at one point directly accused him of lying. Kennedy also gave what one author described as an "incoherent and repetitive" answer to the question, "Why do you want to be President?" The program inflicted serious political damage on Kennedy. President Jimmy Carter alluded to the Chappaquiddick incident twice in five days, once declaring that he had not "panicked in the crisis". Kennedy lost the Democratic nomination to Carter, who, in turn, lost the general election to Ronald Reagan by a landslide. After the incident, Kennedy won seven re-elections to the US Senate. Kennedy remained a senator until his death in 2009.


title: Chappaquiddick incident

text: After Kennedy's death, "New York Times Magazine" editor Ed Klein stated that Kennedy asked people, "Have you heard any new jokes about Chappaquiddick?" "It's not that he didn't feel remorse about the death of Mary Jo Kopechne", Klein argued, "But that he still always saw the other side of everything, and the ridiculous side of things, too." The Dike Bridge became an unwanted tourist attraction, and the object of souvenir hunters.


title: Chappaquiddick incident

text: A similar, fictional incident inspired by the Chappaquiddick incident takes place and is covered up in the season 1 finale of "Succession".


title: Chappaquiddick incident

text: The 2019 series "For All Mankind" depicts an alternate timeline where Kennedy cancels his Chappaquiddick party after Soviets land on the Moon before the U.S., thus avoiding Kopechne's death; Kennedy eventually wins the 1972 Presidential election and is later accused of having an extramarital affair with Kopechne, who is working as a White House aide.


title: Sumo

text: Sumo originated in Japan, the only country where it is practiced professionally and where it is considered the national sport. It is considered a "gendai budō", which refers to modern Japanese martial arts, but the sport has a history spanning many centuries. Many ancient traditions have been preserved in sumo, and even today the sport includes many ritual elements, such as the use of salt purification, from Shinto.


title: Sumo

text: Life as a wrestler is highly regimented, with rules regulated by the Japan Sumo Association. Most sumo wrestlers are required to live in communal sumo training stables, known in Japanese as "heya", where all aspects of their daily lives—from meals to their manner of dress—are dictated by strict tradition.


title: Sumo

text: From 2008 to 2016, a number of high-profile controversies and scandals rocked the sumo world, with an associated effect on its reputation and ticket sales. These have also affected the sport's ability to attract recruits. Despite this setback, sumo's popularity and general attendance has rebounded due to having multiple "yokozuna" (or grand champions) for the first time in a number of years and other high-profile wrestlers grabbing the public's attention.


title: Sumo

text: The spoken word "sumō" goes back to the verb "sumau/sumafu", meaning 'compete' or 'fight'. The written word goes back to the expression , which was a wrestling competition at the imperial court during the Heian period. The characters from "sumai", or "sumō" today, mean 'to strike each other'. There is also an alternate spelling of , which can be found in the "Nihon Shoki". Here, the first character means 'corner', but serves as a phonetic element as one reading of it is "sumi", while the second character means 'force'.


title: Sumo

text: "Sumō" is also a general term for wrestling in Japanese. For example, means 'arm wrestling', and means 'finger wrestling'. The professional sumo observed by the Japan Sumo Association is called , or 'grand sumo'.


title: Sumo

text: Prehistoric wall paintings indicate that sumo originated from an agricultural ritual dance performed in prayer for a good harvest. The first mention of sumo can be found in a "Kojiki" manuscript dating back to 712, which describes how possession of the Japanese islands was decided in a wrestling match between the "kami" known as Takemikazuchi and Takeminakata.


title: Sumo

text: Takemikazuchi was a god of thunder, swordsmanship, and conquest, created from the blood that was shed when Izanagi slew the fire-demon Kagu-tsuchi. Takeminakata was a god of water, wind, agriculture and hunting, and a distant descendant of the storm-god Susanoo. When Takemikazuchi sought to conquer the land of Izumo, Takeminakata challenged him in hand-to-hand combat. In their melee, Takemikazuchi grappled Takeminakata's arm and crushed it "like a reed," defeating Takeminakata and claiming Izumo.


title: Sumo

text: The "Nihon Shoki", published in 720, dates the first sumo match between mortals to the year 23 BC, when a man named Nomi no Sukune fought against Taima no Kuehaya at the request of Emperor Suinin and eventually killed him, making him the mythological ancestor of sumo. According to the "Nihon Shoki", Nomi broke a rib of Taima with one kick, and killed him with a kick to the back as well. Until the Japanese Middle Ages, this unregulated form of wrestling was often fought to the death of one of the fighters. The first historically-attested sumo fights were held in 642 at the court of Empress Kōgyoku to entertain a Korean legation. In the centuries that followed, the popularity of sumo within the court increased its ceremonial and religious significance. Regular events at the Emperor's court, the "sumai no sechie", and the establishment of the first set of rules for sumo fall into the cultural heyday of the Heian period.


title: Sumo

text: With the collapse of the Emperor's central authority, sumo lost its importance in the court; during the Kamakura period, sumo was repurposed from a ceremonial struggle to a form of military combat training among "samurai". By the Muromachi period, sumo had fully left the seclusion of the court and became a popular event for the masses, and among the "daimyō" it became common to sponsor wrestlers. "Sumotori" who successfully fought for a "daimyō"'s favor were given generous support and "samurai" status. Oda Nobunaga, a particularly avid fan of the sport, held a tournament of 1,500 wrestlers in February 1578. Because several bouts were to be held simultaneously within Oda Nobunaga's castle, circular arenas were delimited to hasten the proceedings and to maintain the safety of the spectators. This event marks the invention of the "dohyō", which would be developed into its current form up until the 18th century. The winner of Nobunaga's tournament was given a bow for being victorious and he began dancing to show the war-lord his gratitude.


title: Sumo

text: Because sumo had become a nuisance due to wild fighting on the streets, particularly in Edo, sumo was temporarily banned in the city during the Edo period. In 1684, sumo was permitted to be held for charity events on the property of Shinto shrines, as was common in Kyoto and Osaka. The first sanctioned tournament took place in the Tomioka Hachiman Shrine at this time. An official sumo organization was developed, consisting of professional wrestlers at the disposal of the Edo administration. Many elements date from this period, such as the "dohyō-iri", the "heya" system, the "gyōji" and the "mawashi". The 18th century brought forth several notable wrestlers such as Raiden Tameemon, Onogawa Kisaburō and Tanikaze Kajinosuke, the first historical "yokozuna".


title: Sumo

text: When Matthew Perry was shown sumo wrestling during his 1853 expedition to Japan, he found it distasteful and arranged a military showcase to display the merits of Western organization.


title: Sumo

text: The Meiji Restoration of 1868 brought about the end of the feudal system, and with it the wealthy "daimyō" as sponsors. Due to a new fixation on Western culture, sumo had come to be seen as an embarrassing and backward relic, and internal disputes split the central association. The popularity of sumo was restored when Emperor Meiji organized a tournament in 1884; his example would make sumo a national symbol and contribute to nationalist sentiment following military successes against Korea and China. The Japan Sumo Association reunited on 28 December 1925 and increased the number of annual tournaments from two to four, and then to six in 1958. The length of tournaments was extended from ten to fifteen days in 1949.


title: Sumo

text: The elementary principle of sumo is that a match is decided by a fighter first either touching the ground outside the circular "dohyō" (ring) with any part of the body, or touching the ground inside the ring with any part of the body other than the soles of the feet. The wrestlers try to achieve this by pushing, tossing, striking and often by outwitting the opponent. The Japan Sumo Association currently distinguishes 82 "kimarite" (winning techniques), some of which come from judo. Illegal moves are called "kinjite", which include strangulation, hair-pulling, bending fingers, gripping the crotch area, kicking, poking eyes, punching and simultaneously striking both the opponent's ears. The most common basic forms are grabbing the opponent by the "mawashi" (belt) and then forcing him out, a style called , or pushing the opponent out of the ring without a firm grip, a style called .


title: Sumo

text: The "dohyō", which is constructed and maintained by the "yobidashi", consists of a raised pedestal on which a circle in diameter is delimited by a series of rice-straw bales. In the middle of the circle there are two starting lines ("shikiri-sen"), behind which the wrestlers line up for the "tachi-ai", the synchronized charge that initiates the match. The direction of the match is incumbent on the "gyōji", a referee who is supported by five "shimpan" (judges). In some situations, a review of the "gyōji"s decision may be needed. The "shimpan" may convene a conference in the middle of the ring, called a "mono-ii". This is done if the judges decide that the decision over who won the bout needs to be reviewed; for example, if both wrestlers appear to touch the ground or step out of the ring at the same time. In these cases, sometimes video is reviewed to see what happened. Once a decision is made, the chief judge will announce the decision to the spectators and the wrestlers alike. They may order a bout to be restarted, or leave the decision as given by the "gyōji". Occasionally the "shimpan" will overrule the "gyōji" and give the bout to the other wrestler. On rare occasions the referee or judges may award the win to the wrestler who touched the ground first. This happens if both wrestlers touch the ground at nearly the same time and it is decided that the wrestler who touched the ground second had no chance of winning, his opponent's superior "sumo" having put him in an irrecoverable position. The losing wrestler is referred to as being "shini-tai" ("dead body") in this case.


title: Sumo

text: The maximum length of a match varies depending on the division. In the top division, the limit is four minutes, although matches usually only last a few seconds. If the match has not yet ended after the allotted time has elapsed, a "mizu-iri" (water break) is taken, after which the wrestlers continue the fight from their previous positions. If a winner is still not found after another four minutes, the fight restarts from the "tachi-ai" after another "mizu-iri". If this still does not result in a decision, the outcome is considered a "hikiwake" (draw). This is an extremely rare result, with the last such draw being called in September 1974.


title: Sumo

text: A special attraction of sumo is the variety of observed ceremonies and rituals, some of which have been cultivated in connection with the sport and unchanged for centuries. These include the ring-entering ceremonies ("dohyō-iri") at the beginning of each tournament day, in which the wrestlers appear in the ring in elaborate "kesho-mawashi", but also such details as the tossing of salt into the ring by the wrestlers, which serves as a symbolic cleansing of the ring, and rinsing the mouth with "chikara-mizu" (, "power water") before a fight, which is similar to the ritual before entering a Shinto shrine. Additionally, before a match begins the two wrestlers perform and repeat a warm up routine called "shikiri". The top division is given four minutes for "shikiri", while the second division is given three, after which the timekeeping judge signals to the "gyōji" that time is up.


title: Sumo

text: Traditionally, sumo wrestlers are renowned for their great girth and body mass, which is often a winning factor in sumo. No weight divisions are used in professional sumo; a wrestler can sometimes face an opponent twice his own weight. However, with superior technique, smaller wrestlers can control and defeat much larger opponents. The average weight of top division wrestlers has continued to increase, from in 1969 to over by 1991, and was a record as of January 2019.


title: Sumo

text: Professional sumo is organized by the Japan Sumo Association. The members of the association, called "oyakata", are all former wrestlers, and are the only people entitled to train new wrestlers. All professional wrestlers must be a member of a training stable (or "heya") run by one of the "oyakata", who is the stablemaster for the wrestlers under him. In 2007, 43 training stables hosted 660 wrestlers.


title: Sumo

text: To turn professional, wrestlers must have completed at least nine years of compulsory education and meet minimum height and weight requirements. In 1994, the Japanese Sumo Association required that all sumo wrestlers be a minimum in height. This prompted 16-year-old Takeji Harada of Japan (who had failed six previous eligibility tests) to have four separate cosmetic surgeries over a period of 12 months to add an extra of silicone to his scalp, which created a large, protruding bulge on his head. In response to this, the JSA stated that they would no longer accept aspiring wrestlers who surgically enhanced their height, citing health concerns. In 2019, "The Japan Times" reported that the height requirement was , and the weight requirement was , although they also claimed that a "blind eye" is turned for those "just shy" of the minimums.


title: Sumo

text: All sumo wrestlers take wrestling names called , which may or may not be related to their real names. Often, wrestlers have little choice in their names, which are given to them by their stablemasters, or by a supporter or family member who encouraged them into the sport. This is particularly true of foreign-born wrestlers. A wrestler may change his wrestling name during his career, with some changing theirs several times.


title: Sumo

text: Professional sumo wrestling has a strict hierarchy based on sporting merit. The wrestlers are ranked according to a system that dates back to the Edo period. They are promoted or demoted according to their performance in six official tournaments held throughout the year, which are called "honbasho". A carefully prepared "banzuke" listing the full hierarchy is published two weeks prior to each sumo tournament.


title: Sumo

text: In addition to the professional tournaments, exhibition competitions are held at regular intervals every year in Japan, and roughly once every two years, the top-ranked wrestlers visit a foreign country for such exhibitions. None of these displays are taken into account in determining a wrestler's future rank. Rank is determined only by performance in grand sumo tournaments.


title: Sumo

text: Wrestlers enter sumo in the lowest "jonokuchi" division and, ability permitting, work their way up to the top division. A broad demarcation in the sumo world can be seen between the wrestlers in the top two divisions known as and those in the four lower divisions, known commonly by the more generic term . The ranks receive different levels of compensation, privileges, and status.


title: Sumo

text: The topmost "makuuchi" division receives the most attention from fans and has the most complex hierarchy. The majority of wrestlers are and are ranked from the highest level 1 down to about 16 or 17. In each rank are two wrestlers, the higher rank is designated as "east" and the lower as "west", so the list goes #1 east, #1 west, #2 east, #2 west, etc. Above the "maegashira" are the three champion or titleholder ranks, called the "san'yaku", which are not numbered. These are, in ascending order, , , and . At the pinnacle of the ranking system is the rank of .


title: Sumo

text: "Yokozuna", or grand champions, are generally expected to compete for and to win the top division tournament title on a regular basis, hence the promotion criteria for "yokozuna" are very strict. In general, an "ōzeki" must win the championship for two consecutive tournaments or an "equivalent performance" to be considered for promotion to "yokozuna". More than one wrestler can hold the rank of "yokozuna" at the same time.


title: Sumo

text: In antiquity, sumo was solely a Japanese sport. Since the 1900s, however, the number of foreign-born sumo wrestlers has gradually increased. In the beginning of this period, these few foreign wrestlers were listed as Japanese, but particularly since the 1960s, a number of high-profile foreign-born wrestlers became well-known, and in more recent years have even come to dominate in the highest ranks. In the 10 years since January 2009, five of the nine wrestlers promoted to "ōzeki" have been foreign-born, and a Japanese had not been named "yokozuna" from 1998 until the promotion of Kisenosato Yutaka in 2017. This and other issues eventually led the Sumo Association to limit the number of foreigners allowed to one in each stable.


title: Sumo

text: Women are not allowed to compete in professional sumo. They are also not allowed to enter the wrestling ring ("dohyō"), a tradition stemming from Shinto and Buddhist beliefs that women are "impure" because of menstrual blood.


title: Sumo

text: A form of existed in some parts of Japan before professional sumo was established. The 2018 film "The Chrysanthemum and the Guillotine" depicts female sumo wrestlers at the time of civil unrest following the 1923 Great Kantō earthquake.


title: Sumo

text: Since 1958, six Grand Sumo tournaments or "honbasho" have been held each year: three at the Ryōgoku Kokugikan in Ryōgoku, Tokyo (January, May, and September), and one each in Osaka (March), Nagoya (July), and Fukuoka (November). Each tournament begins on a Sunday and runs for 15 days, ending also on a Sunday. The tournaments are organized in a manner akin to a McMahon system tournament; each wrestler in the top two divisions ("sekitori") has one match per day, while the lower-ranked wrestlers compete in seven bouts, about one every two days.


title: Sumo

text: Each day is structured so that the highest-ranked contestants compete at the end of the day. Thus, wrestling starts in the morning with the "jonokuchi" wrestlers and ends at around six o'clock in the evening with bouts involving the "yokozuna". The wrestler who wins the most matches over the 15 days wins the tournament championship ("yūshō") for his division. If two wrestlers are tied for the top, they wrestle each other and the winner takes the title. Three-way ties for a championship are rare, at least in the top division. In these cases, the three wrestle each other in pairs with the first to win two in a row take the tournament. More complex systems for championship playoffs involving four or more wrestlers also exist, but these are usually only seen in determining the winner of one of the lower divisions.


title: Sumo

text: The matchups for each day of the tournament are determined by the sumo elders who are members of the judging division of the Japan Sumo Association. They meet every morning at 11 am and announce the following day's matchups around 12 pm. An exception are the final day 15 matchups, which are announced much later on day 14. Each wrestler only competes against a selection of opponents from the same division, though small overlaps can occur between two divisions. The first bouts of a tournament tend to be between wrestlers who are within a few ranks of each other. Afterwards, the selection of opponents takes into account a wrestler's prior performance. For example, in the lower divisions, wrestlers with the same record in a tournament are generally matched up with each other and the last matchups often involve undefeated wrestlers competing against each other, even if they are from opposite ends of the division. In the top division, in the last few days, wrestlers with exceptional records often have matches against much more highly ranked opponents, including "san'yaku" wrestlers, especially if they are still in the running for the top division championship. Similarly, more highly ranked wrestlers with very poor records may find themselves fighting wrestlers much further down the division.


title: Sumo

text: For the "yokozuna" and "ōzeki", the first week and a half of the tournament tends to be taken up with bouts against the top "maegashira", "komusubi", and "sekiwake", with the bouts within these ranks being concentrated into the last five days or so of the tournament (depending on the number of top-ranked wrestlers competing). Traditionally, on the final day, the last three bouts of the tournament are between the top six ranked wrestlers, with the top two competing in the final matchup, unless injuries during the tournament prevent this.


title: Sumo

text: Certain match-ups are prohibited in regular tournament play. Wrestlers who are from the same training stable cannot compete against each other, nor can wrestlers who are brothers, even if they join different stables. The one exception to this rule is that training stable partners and brothers can face each other in a championship-deciding playoff match.