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Redundancy

Redundancy or redundant may refer to:

Language

 * Redundancy (linguistics), information that is expressed more than once

Engineering and computer science

 * Data redundancy, database systems which have a field that is repeated in two or more tables
 * Logic redundancy, a digital gate network containing circuitry that does not affect the static logic function
 * Redundancy (engineering), the duplication of critical components or functions of a system with the intention of increasing reliability
 * Redundancy (information theory), the number of bits used to transmit a message minus the number of bits of actual information in the message
 * Redundancy in total quality management, quality which exceeds the required quality level, creating unnecessarily high costs
 * The same task executed by several different methods in a user interface

Biology

 * Codon redundancy, the redundancy of the genetic code exhibited as the multiplicity of three-codon combinations
 * Cytokine redundancy, a term in immunology referring to the phenomenon in which, and the ability of, multiple cytokines to exert similar actions
 * Gene redundancy, the existence of several genes in the genome of an organism that perform the same role to some extent

As a proper name

 * "Redundant" (song), 1997 song by American rock band Green Day
 * Redundant (play), 2001 drama by Leo Butler

Other uses

 * Redundant church, no longer used for worship
 * Redundancy in United Kingdom law, dismissal for economic reasons

title: Redundancy


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Convex hull

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time $$O(n\log n)$$ for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions.

As well as for finite point sets, convex hulls have also been studied for simple polygons, Brownian motion, space curves, and epigraphs of functions. Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the orthogonal convex hull, convex layers, Delaunay triangulation and Voronoi diagram, and convex skull.

Definitions
A set of points in a Euclidean space is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a given set $$X$$ may be defined as For bounded sets in the Euclidean plane, not all on one line, the boundary of the convex hull is the simple closed curve with minimum perimeter containing $$X$$. One may imagine stretching a rubber band so that it surrounds the entire set $$S$$ and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of $$S$$. This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a spanning tree of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. However, in higher dimensions, variants of the obstacle problem of finding a minimum-energy surface above a given shape can have the convex hull as their solution.
 * 1) The (unique) minimal convex set containing $$X$$
 * 2) The intersection of all convex sets containing $$X$$
 * 3) The set of all convex combinations of points in $$X$$
 * 4) The union of all simplices with vertices in $$X$$

For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex bounding volume of the objects. The definition using intersections of convex sets may be extended to non-Euclidean geometry, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids.

Equivalence of definitions
It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing $$X$$, for every $$X$$? However, the second definition, the intersection of all convex sets containing $$X$$, is well-defined. It is a subset of every other convex set $$Y$$ that contains $$X$$, because $$Y$$ is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing $$X$$. Therefore, the first two definitions are equivalent.

Each convex set containing $$X$$ must (by the assumption that it is convex) contain all convex combinations of points in $$X$$, so the set of all convex combinations is contained in the intersection of all convex sets containing $$X$$. Conversely, the set of all convex combinations is itself a convex set containing $$X$$, so it also contains the intersection of all convex sets containing $$X$$, and therefore the second and third definitions are equivalent.

In fact, according to Carathéodory's theorem, if $$X$$ is a subset of a $$d$$-dimensional Euclidean space, every convex combination of finitely many points from $$X$$ is also a convex combination of at most $$d+1$$ points in $$X$$. The set of convex combinations of a $$(d+1)$$-tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of $$X$$ belongs to a simplex whose vertices belong to $$X$$, and the third and fourth definitions are equivalent.

Upper and lower hulls
In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points (points for which an upward ray is disjoint from the hull), downward-facing points, and extreme points. For three-dimensional hulls, the upward-facing and downward-facing parts of the boundary form topological disks.

Closed and open hulls
The closed convex hull of a set is the closure of the convex hull, and the open convex hull is the interior (or in some sources the relative interior) of the convex hull.

The closed convex hull of $$X$$ is the intersection of all closed half-spaces containing $$X$$. If the convex hull of $$X$$ is already a closed set itself (as happens, for instance, if $$X$$ is a finite set or more generally a compact set), then it equals the closed convex hull. However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way.

If the open convex hull of a set $$X$$ is $$d$$-dimensional, then every point of the hull belongs to an open convex hull of at most $$2d$$ points of $$X$$. The sets of vertices of a square, regular octahedron, or higher-dimensional cross-polytope provide examples where exactly $$2d$$ points are needed.

Preservation of topological properties
Topologically, the convex hull of an open set is always itself open, and the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed. For instance, the closed set

 * $$\left \{ (x,y) \mathop{\bigg|} y\ge \frac{1}{1+x^2}\right\}$$

(the set of points that lie on or above the witch of Agnesi) has the open upper half-plane as its convex hull.

The compactness of convex hulls of compact sets, in finite-dimensional Euclidean spaces, is generalized by the Krein–Smulian theorem, according to which the closed convex hull of a weakly compact subset of a Banach space (a subset that is compact under the weak topology) is weakly compact.

Extreme points
An extreme point of a convex set is a point in the set that does not lie on any open line segment between any other two points of the same set. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points. However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points. Choquet theory extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces.

Closure operator
The convex-hull operator has the characteristic properties of a closure operator: When applied to a finite set of points, this is the closure operator of an antimatroid, the shelling antimatroid of the point set. Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension.
 * It is extensive, meaning that the convex hull of every set $$X$$ is a superset of $$X$$.
 * It is non-decreasing, meaning that, for every two sets $$X$$ and $$Y$$ with $$X\subseteq Y$$, the convex hull of $$X$$ is a subset of the convex hull of $$Y$$.
 * It is idempotent, meaning that for every $$X$$, the convex hull of the convex hull of $$X$$ is the same as the convex hull of $$X$$.

Minkowski sum
The operations of constructing the convex hull and taking the Minkowski sum commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the Shapley–Folkman theorem bounding the distance of a Minkowski sum from its convex hull.

Projective duality
The projective dual operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point).

Finite point sets

The convex hull of a finite point set $$S \subset \R^d$$ forms a convex polygon when $$d=2$$, or more generally a convex polytope in $$\R^d$$. Each extreme point of the hull is called a vertex, and (by the Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to $$S$$ and that encloses all of $$S$$. For sets of points in general position, the convex hull is a simplicial polytope.

According to the upper bound theorem, the number of faces of the convex hull of $$n$$ points in $$d$$-dimensional Euclidean space is $$O(n^{\lfloor d/2\rfloor})$$. In particular, in two and three dimensions the number of faces is at most linear in $$n$$.

Simple polygons
The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its convex differences tree. Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates.

Brownian motion
The curve generated by Brownian motion in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a continuously differentiable curve. However, for any angle $$\theta$$ in the range $$\pi/2<\theta<\pi$$, there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle $$\theta$$. The Hausdorff dimension of this set of exceptional times is (with high probability) $$1-\pi/2\theta$$.

Space curves
For the convex hull of a space curve or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are developable and ruled surfaces. Examples include the oloid, the convex hull of two circles in perpendicular planes, each passing through the other's center, the sphericon, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from Alexandrov's uniqueness theorem for a surface formed by gluing together two planar convex sets of equal perimeter.

Functions
The convex hull or lower convex envelope of a function $$f$$ on a real vector space is the function whose epigraph is the lower convex hull of the epigraph of $$f$$. It is the unique maximal convex function majorized by $$f$$. The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their pointwise minimum) and, in this form, is dual to the convex conjugate operation.

Computation
In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. Output representations that have been considered for convex hulls of point sets include a list of linear inequalities describing the facets of the hull, an undirected graph of facets and their adjacencies, or the full face lattice of the hull. In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull.

For convex hulls in two or three dimensions, the complexity of the corresponding algorithms is usually estimated in terms of $$n$$, the number of input points, and $$h$$, the number of points on the convex hull, which may be significantly smaller than $$n$$. For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis. Graham scan can compute the convex hull of $$n$$ points in the plane in time $$O(n\log n)$$. For points in two and three dimensions, more complicated output-sensitive algorithms are known that compute the convex hull in time $$O(n\log h)$$. These include Chan's algorithm and the Kirkpatrick–Seidel algorithm. For dimensions $$d>3$$, the time for computing the convex hull is $$O(n^{\lfloor d/2\rfloor})$$, matching the worst-case output complexity of the problem. The convex hull of a simple polygon in the plane can be constructed in linear time.

Dynamic convex hull data structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points, and kinetic convex hull structures can keep track of the convex hull for points moving continuously. The construction of convex hulls also serves as a tool, a building block for a number of other computational-geometric algorithms such as the rotating calipers method for computing the width and diameter of a point set.

Related structures
Several other shapes can be defined from a set of points in a similar way to the convex hull, as the minimal superset with some property, the intersection of all shapes containing the points from a given family of shapes, or the union of all combinations of points for a certain type of combination. For instance:
 * The affine hull is the smallest affine subspace of a Euclidean space containing a given set, or the union of all affine combinations of points in the set.
 * The linear hull is the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set.
 * The conical hull or positive hull of a subset of a vector space is the set of all positive combinations of points in the subset.
 * The visual hull of a three-dimensional object, with respect to a set of viewpoints, consists of the points $$p$$ such that every ray from a viewpoint through $$p$$ intersects the object. Equivalently it is the intersection of the (non-convex) cones generated by the outline of the object with respect to each viewpoint. It is used in 3D reconstruction as the largest shape that could have the same outlines from the given viewpoints.
 * The circular hull or alpha-hull of a subset of the plane is the intersection of all disks with a given radius $$1/\alpha$$ that contain the subset.
 * The relative convex hull of a subset of a two-dimensional simple polygon is the intersection of all relatively convex supersets, where a set within the same polygon is relatively convex if it contains the geodesic between any two of its points.
 * The orthogonal convex hull or rectilinear convex hull is the intersection of all orthogonally convex and connected supersets, where a set is orthogonally convex if it contains all axis-parallel segments between pairs of its points.
 * The orthogonal convex hull is a special case of a much more general construction, the hyperconvex hull, which can be thought of as the smallest injective metric space containing the points of a given metric space.
 * The holomorphically convex hull is a generalization of similar concepts to complex analytic manifolds, obtained as an intersection of sublevel sets of holomorphic functions containing a given set.

The Delaunay triangulation of a point set and its dual, the Voronoi diagram, are mathematically related to convex hulls: the Delaunay triangulation of a point set in $$\R^n$$ can be viewed as the projection of a convex hull in $$\R^{n+1}.$$ The alpha shapes of a finite point set give a nested family of (non-convex) geometric objects describing the shape of a point set at different levels of detail. Each of alpha shape is the union of some of the features of the Delaunay triangulation, selected by comparing their circumradius to the parameter alpha. The point set itself forms one endpoint of this family of shapes, and its convex hull forms the other endpoint. The convex layers of a point set are a nested family of convex polygons, the outermost of which is the convex hull, with the inner layers constructed recursively from the points that are not vertices of the convex hull.

The convex skull of a polygon is the largest convex polygon contained inside it. It can be found in polynomial time, but the exponent of the algorithm is high.

Applications
Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study polynomials, matrix eigenvalues, and unitary elements, and several theorems in discrete geometry involve convex hulls. They are used in robust statistics as the outermost contour of Tukey depth, are part of the bagplot visualization of two-dimensional data, and define risk sets of randomized decision rules. Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics. In economics, convex hulls can be used to apply methods of convexity in economics to non-convex markets. In geometric modeling, the convex hull property Bézier curves helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the home range.

Mathematics
Newton polygons of univariate polynomials and Newton polytopes of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze the asymptotic behavior of the polynomial and the valuations of its roots. Convex hulls and polynomials also come together in the Gauss–Lucas theorem, according to which the roots of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial.

In spectral analysis, the numerical range of a normal matrix is the convex hull of its eigenvalues. The Russo–Dye theorem describes the convex hulls of unitary elements in a C*-algebra. In discrete geometry, both Radon's theorem and Tverberg's theorem concern the existence of partitions of point sets into subsets with intersecting convex hulls.

The definitions of a convex set as containing line segments between its points, and of a convex hull as the intersection of all convex supersets, apply to hyperbolic spaces as well as to Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets of ideal points, points that do not belong to the hyperbolic space itself but lie on the boundary of a model of that space. The boundaries of convex hulls of ideal points of three-dimensional hyperbolic space are analogous to ruled surfaces in Euclidean space, and their metric properties play an important role in the geometrization conjecture in low-dimensional topology. Hyperbolic convex hulls have also been used as part of the calculation of canonical triangulations of hyperbolic manifolds, and applied to determine the equivalence of knots.

See also the section on Brownian motion for the application of convex hulls to this subject, and the section on space curves for their application to the theory of developable surfaces.

Statistics
In robust statistics, the convex hull provides one of the key components of a bagplot, a method for visualizing the spread of two-dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth.

In statistical decision theory, the risk set of a randomized decision rule is the convex hull of the risk points of its underlying deterministic decision rules.

Combinatorial optimization
In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions. In multi-objective optimization, a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any quasiconvex combination of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions.

Economics
In the Arrow–Debreu model of general economic equilibrium, agents are assumed to have convex budget sets and convex preferences. These assumptions of convexity in economics can be used to prove the existence of an equilibrium. When actual economic data is non-convex, it can be made convex by taking convex hulls. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market.

Geometric modeling
In geometric modeling, one of the key properties of a Bézier curve is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves.

In the geometry of boat and ship design, chain girth is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the hull of the vessel. It differs from the skin girth, the perimeter of the cross-section itself, except for boats and ships that have a convex hull.

Ethology
The convex hull is commonly known as the minimum convex polygon in ethology, the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's home range based on points where the animal has been observed. Outliers can make the minimum convex polygon excessively large, which has motivated relaxed approaches that contain only a subset of the observations, for instance by choosing one of the convex layers that is close to a target percentage of the samples, or in the local convex hull method by combining convex hulls of neighborhoods of points.

Quantum physics
In quantum physics, the state space of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are positive-semidefinite operators known as pure states and whose interior points are called mixed states. The Schrödinger–HJW theorem proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.

Thermodynamics
A convex hull in thermodynamics was identified by Josiah Willard Gibbs (1873), although the paper was published before the convex hull was so named. In a set of energies of several stoichiometries of a material, only those measurements on the lower convex hull will be stable. When removing a point from the hull and then calculating its distance to the hull, its distance to the new hull represents the degree of stability of the phase.

History
The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676. The term "convex hull" itself appears as early as the work of, and the corresponding term in German appears earlier, for instance in Hans Rademacher's review of. Other terms, such as "convex envelope", were also used in this time frame. By 1938, according to Lloyd Dines, the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Convex hull


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Talk:City of London

Documentary film
Hi Would you mind assessing why you deleted the documentary film >

–How Britain's Bankers Made Billions From The End Of Empire, by Timeline - World History Documentaries, 1:16:20 min., published on 20 August 2019 on YouTube

″Timeline - World History Documentaries″ is ″the home of world history. Every week we'll be bringing you one-off documentaries and series from the world's top broadcasters, including the BBC, Channel 4, Discovery and PBS.″

–https://www.pbssocal.org/179-history-documentaries-you-can-watch-right-now-no-membership-required

The information I linked to is indeed not in violation of copyright and it does comply with our policies and guidelines. Should I consider using our upload template > https://en.wikipedia.org/wiki/Template:YouTube ?

Thank you.--87.170.192.136 (talk) 10:27, 16 January 2022 (UTC)
 * It does not comply with our inclusion standards, and we do not link to everything that is related to a subject. --Dirk Beetstra T  C 10:52, 16 January 2022 (UTC)

Street Map of London needed
All the maps currently shown are sketchy. It is very difficult to work out where those sketches fit onto the topography, e.g. streets and rivers. The initial location map as a small dot in the middle of Greater London is nigh on useless. It should at least be zoomable in and out. Cvhorie (talk) 08:33, 14 October 2023 (UTC)

 * Done. The Equalizer (talk) 16:20, 30 March 2024 (UTC)

Money Laundering
The City of London's role in illicit financial activity such as money laundering has earned the financial hub sobriquets like ‘The Laundromat’ and ‘Londongrad.’

London’s role as the world’s dirty money clearing house is well-documented but efforts are being made to clean up through legislation, e.g. authorizing unexplained wealth orders. High-value properties are sought after by criminals and money launderers legitimising their gains by investing in the city’s prestigious real estate.

In case of revert due to hurt feelings, see above.Oathed (talk) 12:57, 15 January 2024 (UTC) Oathed (talk) 12:57, 15 January 2024 (UTC)

Form of government
How can the form of government of the City of London be termed? --95.24.68.61 (talk) 19:27, 11 March 2024 (UTC)

Area of London 2.9 km2? wtf?
Who is editing these articles? They need to be fired. I've consistently seen that area for many cities globally are being misreported in wiki.

https://en.wikipedia.org/wiki/City_of_London 61.95.254.10 (talk) 11:42, 2 May 2024 (UTC)

 * If you don't believe something, check the citation. It says: Greater London 157,208ha; Inner London 31,930ha; City of London 290ha. --Cavrdg (talk) 12:32, 2 May 2024 (UTC)

title: Talk:City of London


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NeXTSTEP

NeXTSTEP is a discontinued object-oriented, multitasking operating system based on the Mach kernel and the UNIX-derived BSD. It was developed by NeXT Computer, founded by Steve Jobs, in the late 1980s and early 1990s and was initially used for its range of proprietary workstation computers such as the NeXTcube. It was later ported to several other computer architectures.

Although relatively unsuccessful at the time, it attracted interest from computer scientists and researchers. It hosted the original development of the Electronic AppWrapper, the first commercial electronic software distribution catalog to collectively manage encryption and provide digital rights for application software and digital media, a forerunner of the modern "app store" concept. It is the platform on which Tim Berners-Lee created the first web browser, and on which id Software developed the video games Doom and Quake.

In 1996, Apple Computer acquired NeXT. Apple needed a successor to the classic Mac OS, and merged NeXTSTEP and OpenStep with the Macintosh user environment to create Mac OS X. All of Apple's subsequent platforms since iPhone OS 1 were then based on Mac OS X (later renamed macOS).

Overview
NeXTSTEP (also stylized as NeXTstep, NeXTStep, and NEXTSTEP ) is a combination of several parts:
 * a Unix operating system based on the Mach kernel, plus BSD
 * Display PostScript and a proprietary windowing engine
 * the Objective-C language and runtime
 * an object-oriented (OO) application layer, including several "kits"
 * development tools for the OO layers.

NeXTSTEP is a preeminent implementation of the last three items. The toolkits are the canonical development system for all of the software on the system.

It introduced the idea of the Dock (carried through OpenStep and into macOS) and the Shelf. NeXTSTEP originated or innovated a large number of other GUI concepts which became common in other operating systems: 3D chiseled widgets, large full-color icons, system-wide drag and drop of a wide range of objects beyond file icons, system-wide piped services, real-time scrolling and window dragging, properties dialog boxes called "inspectors", and window modification notices (such as the saved status of a file). The system is among the first general-purpose user interfaces to handle publishing color standards, transparency, sophisticated sound and music processing (through a Motorola 56000 DSP), advanced graphics primitives, internationalization, and modern typography, in a consistent manner across all applications.

Additional kits were added to the product line. These include Portable Distributed Objects (PDO), which allow easy remote invocation, and Enterprise Objects Framework, an object-relational database system. The kits made the system particularly interesting to custom application programmers, and NeXTSTEP had a long history in the financial programming community.

History
NeXTSTEP was built upon Mach and BSD, initially 4.3BSD-Tahoe. A preview release of NeXTSTEP (version 0.8) was shown with the launch of the NeXT Computer on October 12, 1988. The first full release, NeXTSTEP 1.0, shipped on September 18, 1989. It was updated to 4.3BSD-Reno in NeXTSTEP 3.0. The last version, 3.3, was released in early 1995, for the Motorola 68000 family based NeXT computers, Intel x86, Sun SPARC, and HP PA-RISC-based systems.

NeXT separated the underlying operating system from the application frameworks, producing OpenStep. OpenStep and its applications can run on multiple underlying operating systems, including OPENSTEP, Windows NT, and Solaris. In 1997, it was updated to 4.4BSD while assimilated into Apple's development of Rhapsody for x86 and PowerPC. NeXTSTEP's direct descendant is Apple's macOS, which then yielded iPhone OS 1, iOS, iPadOS, watchOS, and tvOS.

Legacy
The first web browser, WorldWideWeb, and the first app store were all invented on the NeXTSTEP platform.

"1990 CERN: A Joint proposal for a hypertext system is presented to the management. Mike Sendall buys a NeXT cube for evaluation, and gives it to Tim Berners-Lee. Tim's prototype implementation on NeXTSTEP is made in the space of a few months, thanks to the qualities of the NeXTSTEP software development system. This prototype offers WYSIWYG browsing/authoring! Current Web browsers used in 'surfing the Internet' are mere passive windows, depriving the user of the possibility to contribute. During some sessions in the CERN cafeteria, Tim and I try to find a catching name for the system. I was determined that the name should not yet again be taken from Greek mythology. Tim proposes 'World-Wide Web'. I like this very much, except that it is difficult to pronounce in French..."

Some features and keyboard shortcuts now common to web browsers originated in NeXTSTEP conventions. The basic layout options of HTML 1.0 and 2.0 are attributable to those features of NeXT's Text class.

Lighthouse Design Ltd. developed Diagram!, a drawing tool, originally called BLT (for Box-and-Line Tool) in which objects (boxes) are connected together using "smart links" (lines) to construct diagrams such a flow charts. This basic design can be enhanced by the simple addition of new links and new documents, located anywhere in the local area network, that foreshadowed Tim Berners-Lee's initial prototype that was written on NeXTSTEP in October–December 1990.

In the 1990s, the pioneering PC games Doom, Doom II, Quake, and their respective level editors were developed by id Software on NeXT machines. Other games based on the Doom engine such as Heretic and its sequel Hexen by Raven Software, and Strife by Rogue Entertainment were developed on NeXT hardware using id's tools.

Altsys made the NeXTSTEP application Virtuoso, version 2 of which was ported to Mac OS and Windows to become Macromedia FreeHand version 4. The modern "Notebook" interface for Mathematica, and the advanced spreadsheet Lotus Improv, were developed using NeXTSTEP. The software that controlled MCI's Friends and Family calling plan program was developed using NeXTSTEP.

About the time of the release of NeXTSTEP 3.2, NeXT partnered with Sun Microsystems to develop OpenStep. It is the product of an effort to separate the underlying operating system from the higher-level object libraries to create a cross-platform object-oriented API standard derived from NeXTSTEP. OpenStep was released for Sun's Solaris, Windows NT, and NeXT's Mach kernel-based operating system. NeXT's implementation is called "OPENSTEP for Mach" and its first release (4.0) superseded NeXTSTEP 3.3 on NeXT, Sun, and Intel IA-32 systems.

Following an announcement on December 20, 1996, Apple Computer acquired NeXT on February 4, 1997, for $429 million. Based upon the "OPENSTEP for Mach" operating system, and developing the OpenStep API to become Cocoa, Apple created the basis of Mac OS X, and eventually of iOS, iPadOS, watchOS, and tvOS.

GNUstep is a free software implementation of the OpenStep standard.

Release history
Versions up to 4.1 are general releases. OPENSTEP 4.2 pre-release 2 is a bug-fix release published by Apple and supported for five years after its September 1997 release.

title: NeXTSTEP


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Non-uniform memory access

Non-uniform memory access (NUMA) is a computer memory design used in multiprocessing, where the memory access time depends on the memory location relative to the processor. Under NUMA, a processor can access its own local memory faster than non-local memory (memory local to another processor or memory shared between processors). NUMA is beneficial for workloads with high memory locality of reference and low lock contention, because a processor may operate on a subset of memory mostly or entirely within its own cache node, reducing traffic on the memory bus.

NUMA architectures logically follow in scaling from symmetric multiprocessing (SMP) architectures. They were developed commercially during the 1990s by Unisys, Convex Computer (later Hewlett-Packard), Honeywell Information Systems Italy (HISI) (later Groupe Bull), Silicon Graphics (later Silicon Graphics International), Sequent Computer Systems (later IBM), Data General (later EMC, now Dell Technologies), Digital (later Compaq, then HP, now HPE) and ICL. Techniques developed by these companies later featured in a variety of Unix-like operating systems, and to an extent in Windows NT.

The first commercial implementation of a NUMA-based Unix system was the Symmetrical Multi Processing XPS-100 family of servers, designed by Dan Gielan of VAST Corporation for Honeywell Information Systems Italy.

Overview

Modern CPUs operate considerably faster than the main memory they use. In the early days of computing and data processing, the CPU generally ran slower than its own memory. The performance lines of processors and memory crossed in the 1960s with the advent of the first supercomputers. Since then, CPUs increasingly have found themselves "starved for data" and having to stall while waiting for data to arrive from memory (e.g. for Von-Neumann architecture-based computers, see Von Neumann bottleneck). Many supercomputer designs of the 1980s and 1990s focused on providing high-speed memory access as opposed to faster processors, allowing the computers to work on large data sets at speeds other systems could not approach.

Limiting the number of memory accesses provided the key to extracting high performance from a modern computer. For commodity processors, this meant installing an ever-increasing amount of high-speed cache memory and using increasingly sophisticated algorithms to avoid cache misses. But the dramatic increase in size of the operating systems and of the applications run on them has generally overwhelmed these cache-processing improvements. Multi-processor systems without NUMA make the problem considerably worse. Now a system can starve several processors at the same time, notably because only one processor can access the computer's memory at a time.

NUMA attempts to address this problem by providing separate memory for each processor, avoiding the performance hit when several processors attempt to address the same memory. For problems involving spread data (common for servers and similar applications), NUMA can improve the performance over a single shared memory by a factor of roughly the number of processors (or separate memory banks). Another approach to addressing this problem is the multi-channel memory architecture, in which a linear increase in the number of memory channels increases the memory access concurrency linearly.

Of course, not all data ends up confined to a single task, which means that more than one processor may require the same data. To handle these cases, NUMA systems include additional hardware or software to move data between memory banks. This operation slows the processors attached to those banks, so the overall speed increase due to NUMA heavily depends on the nature of the running tasks.

Implementations
AMD implemented NUMA with its Opteron processor (2003), using HyperTransport. Intel announced NUMA compatibility for its x86 and Itanium servers in late 2007 with its Nehalem and Tukwila CPUs. Both Intel CPU families share a common chipset; the interconnection is called Intel QuickPath Interconnect (QPI), which provides extremely high bandwidth to enable high on-board scalability and was replaced by a new version called Intel UltraPath Interconnect with the release of Skylake (2017).

Cache coherent NUMA (ccNUMA)

Nearly all CPU architectures use a small amount of very fast non-shared memory known as cache to exploit locality of reference in memory accesses. With NUMA, maintaining cache coherence across shared memory has a significant overhead. Although simpler to design and build, non-cache-coherent NUMA systems become prohibitively complex to program in the standard von Neumann architecture programming model.

Typically, ccNUMA uses inter-processor communication between cache controllers to keep a consistent memory image when more than one cache stores the same memory location. For this reason, ccNUMA may perform poorly when multiple processors attempt to access the same memory area in rapid succession. Support for NUMA in operating systems attempts to reduce the frequency of this kind of access by allocating processors and memory in NUMA-friendly ways and by avoiding scheduling and locking algorithms that make NUMA-unfriendly accesses necessary.

Alternatively, cache coherency protocols such as the MESIF protocol attempt to reduce the communication required to maintain cache coherency. Scalable Coherent Interface (SCI) is an IEEE standard defining a directory-based cache coherency protocol to avoid scalability limitations found in earlier multiprocessor systems. For example, SCI is used as the basis for the NumaConnect technology.

NUMA vs. cluster computing
One can view NUMA as a tightly coupled form of cluster computing. The addition of virtual memory paging to a cluster architecture can allow the implementation of NUMA entirely in software. However, the inter-node latency of software-based NUMA remains several orders of magnitude greater (slower) than that of hardware-based NUMA.

Software support
Since NUMA largely influences memory access performance, certain software optimizations are needed to allow scheduling threads and processes close to their in-memory data.

 * Microsoft Windows 7 and Windows Server 2008 R2 added support for NUMA architecture over 64 logical cores.
 * Java 7 added support for NUMA-aware memory allocator and garbage collector.
 * Linux kernel:
 * Version 2.5 provided a basic NUMA support, which was further improved in subsequent kernel releases.
 * Version 3.8 of the Linux kernel brought a new NUMA foundation that allowed development of more efficient NUMA policies in later kernel releases.
 * Version 3.13 of the Linux kernel brought numerous policies that aim at putting a process near its memory, together with the handling of cases such as having memory pages shared between processes, or the use of transparent huge pages; new sysctl settings allow NUMA balancing to be enabled or disabled, as well as the configuration of various NUMA memory balancing parameters.
 * OpenSolaris models NUMA architecture with lgroups.
 * FreeBSD added support for NUMA architecture in version 9.0.
 * Silicon Graphics IRIX (discontinued as of 2021) support for ccNUMA architecture over 1240 CPU with Origin server series.

Hardware support
As of 2011, ccNUMA systems are multiprocessor systems based on the AMD Opteron processor, which can be implemented without external logic, and the Intel Itanium processor, which requires the chipset to support NUMA. Examples of ccNUMA-enabled chipsets are the SGI Shub (Super hub), the Intel E8870, the HP sx2000 (used in the Integrity and Superdome servers), and those found in NEC Itanium-based systems. Earlier ccNUMA systems such as those from Silicon Graphics were based on MIPS processors and the DEC Alpha 21364 (EV7) processor.

title: Non-uniform memory access


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Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.

Preliminaries
Let $$ (G, \cdot)$$ be a locally compact Hausdorff topological group. The $\sigma$-algebra generated by all open subsets of $$G$$ is called the Borel algebra. An element of the Borel algebra is called a Borel set. If $$g$$ is an element of $$G$$ and $$S$$ is a subset of $$G$$, then we define the left and right translates of $$S$$ by g as follows:
 * Left translate: $$ g S = \{g\cdot s\,:\,s \in S\}.$$
 * Right translate: $$ S g = \{s\cdot g\,:\,s \in S\}.$$

Left and right translates map Borel sets onto Borel sets.

A measure $$\mu$$ on the Borel subsets of $$G$$ is called left-translation-invariant if for all Borel subsets $$S\subseteq G$$ and all $$g\in G$$ one has
 * $$ \mu(g S) = \mu(S). $$

A measure $$\mu$$ on the Borel subsets of $$G$$ is called right-translation-invariant if for all Borel subsets $$S\subseteq G$$ and all $$g\in G$$ one has
 * $$ \mu(S g) = \mu(S). $$

Haar's theorem
There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure $$\mu$$ on the Borel subsets of $$G$$ satisfying the following properties:

 * The measure $$\mu$$ is left-translation-invariant: $$\mu(gS) = \mu(S)$$ for every $$g\in G$$ and all Borel sets $$S\subseteq G$$.
 * The measure $$\mu$$ is finite on every compact set: $$\mu(K) < \infty$$ for all compact $$K \subseteq G$$.
 * The measure $$\mu$$ is outer regular on Borel sets $$S\subseteq G$$: $$ \mu(S) = \inf \{\mu(U): S \subseteq U, U \text{ open}\}.$$
 * The measure $$\mu$$ is inner regular on open sets $$U\subseteq G$$: $$ \mu(U) = \sup \{\mu(K): K \subseteq U, K \text{ compact}\}.$$

Such a measure on $$G$$ is called a left Haar measure. It can be shown as a consequence of the above properties that $$\mu(U)>0$$ for every non-empty open subset $$U\subseteq G$$. In particular, if $$G$$ is compact then $$\mu(G)$$ is finite and positive, so we can uniquely specify a left Haar measure on $$G$$ by adding the normalization condition $$\mu(G)=1$$.

In complete analogy, one can also prove the existence and uniqueness of a right Haar measure on $$G$$. The two measures need not coincide.

Some authors define a Haar measure on Baire sets rather than Borel sets. This makes the regularity conditions unnecessary as Baire measures are automatically regular. Halmos rather confusingly uses the term "Borel set" for elements of the $\sigma$-ring generated by compact sets, and defines Haar measures on these sets.

The left Haar measure satisfies the inner regularity condition for all $\sigma$-finite Borel sets, but may not be inner regular for all Borel sets. For example, the product of the unit circle (with its usual topology) and the real line with the discrete topology is a locally compact group with the product topology and a Haar measure on this group is not inner regular for the closed subset $$\{1\} \times [0,1]$$. (Compact subsets of this vertical segment are finite sets and points have measure $$0$$, so the measure of any compact subset of this vertical segment is $$0$$. But, using outer regularity, one can show the segment has infinite measure.)

The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil. Weil's proof used the axiom of choice and Henri Cartan furnished a proof that avoided its use. Cartan's proof also establishes the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by Alfsen in 1963. The special case of invariant measure for second-countable locally compact groups had been shown by Haar in 1933.

Examples
 If $$G$$ is a discrete group, then the compact subsets coincide with the finite subsets, and a (left and right invariant) Haar measure on $$G$$ is the counting measure.  The Haar measure on the topological group $$(\mathbb{R}, +)$$ that takes the value $$1$$ on the interval $$[0,1]$$ is equal to the restriction of Lebesgue measure to the Borel subsets of $$\mathbb{R}$$. This can be generalized to $$(\mathbb{R}^n, +).$$  In order to define a Haar measure $$\mu$$ on the circle group $$\mathbb{T}$$, consider the function $$f$$ from $$[0,2\pi]$$ onto $$\mathbb{T}$$ defined by $$f(t)=(\cos(t),\sin(t))$$. Then $$\mu$$ can be defined by $$\mu(S)=\frac1{2\pi}m(f^{-1}(S)),$$ where $$m$$ is the Lebesgue measure on $$[0,2\pi]$$. The factor $$(2\pi)^{-1}$$ is chosen so that $$\mu(\mathbb{T})=1$$.  If $$G$$ is the group of positive real numbers under multiplication then a Haar measure $$\mu$$ is given by $$ \mu(S) = \int_S \frac{1}{t} \, dt$$ for any Borel subset $$S$$ of positive real numbers. For example, if $$S$$ is taken to be an interval $$[a,b]$$, then we find $$\mu(S) = \log(b/a)$$. Now we let the multiplicative group act on this interval by a multiplication of all its elements by a number $$g$$, resulting in $$gS$$ being the interval $$[g\cdot a,g\cdot b].$$ Measuring this new interval, we find $$ \mu(gS) = \log((g\cdot b)/(g\cdot a)) = \log(b/a) = \mu(S). $$  If $$G$$ is the group of nonzero real numbers with multiplication as operation, then a Haar measure $$\mu$$ is given by $$ \mu(S) = \int_S \frac{1}{|t|} \, dt $$ for any Borel subset $$S$$ of the nonzero reals.  For the general linear group $$G = GL(n,\mathbb{R})$$, any left Haar measure is a right Haar measure and one such measure $$\mu$$ is given by $$ \mu(S) = \int_S {1\over |\det(X)|^n} \, dX $$ where $$dX$$ denotes the Lebesgue measure on $$\mathbb{R}^{n^2}$$ identified with the set of all $$n\times n$$-matrices. This follows from the change of variables formula.  Generalizing the previous three examples, if the group $$G$$ is represented as an open submanifold of $$\R^n$$ with smooth group operations, then a left Haar measure on $$G$$ is given by $$\frac{1}{|J_{(x\cdot)}(e_1)|}d^n x$$, where $$e_1$$ is the group identity element of $$G$$, $$J_{(x\cdot)}(e_1)$$ is the Jacobian determinant of left multiplication by $$x$$ at $$e_1$$, and $$d^n x$$ is the Lebesgue measure on $$\R^n$$. This follows from the change of variables formula. A right Haar measure is given in the same way, except with $$J_{(\cdot x)}(e_1)$$ being the Jacobian of right multiplication by $$x$$.  For the orthogonal group $$G = O(n)$$, its Haar measure can be constructed as follows (as the distribution of a random variable). First sample $$A \sim N(0, 1)^{n\times n}$$, that is, a matrix with all entries being IID samples of the normal distribution with mean zero and variance one. Next use Gram–Schmidt process on the matrix; the resulting random variable takes values in $$O(n)$$ and it is distributed according to the probability Haar measure on that group. Since the special orthogonal group $$SO(n)$$ is an open subgroup of $$O(n)$$ the restriction of Haar measure of $$O(n)$$ to $$SO(n)$$ gives a Haar measure on $$SO(n)$$ (in random variable terms this means conditioning the determinant to be 1, an event of probability 1/2).

The same method as for $$O(n)$$ can be used to construct the Haar measure on the unitary group $$U(n)$$. For the special unitary group $$G = SU(n)$$ (which has measure 0 in $$U(n)$$), its Haar measure can be constructed as follows. First sample $$A$$ from the Haar measure (normalized to one, so that it's a probability distribution) on $$U(n)$$, and let $$e^{i\theta} = \det A$$, where $$\theta$$ may be any one of the angles, then independently sample $$k$$ from the uniform distribution on $$\{1, ..., n\}$$. Then $$e^{-i\frac{\theta + 2\pi k}n}A$$ is distributed as the Haar measure on $$SU(n)$$.

Let $$G$$ be the set of all affine linear transformations $$A : \mathbb{R} \to \mathbb{R}$$ of the form $$r \mapsto x r + y$$ for some fixed $$x, y \in \mathbb{R}$$ with $$x > 0.$$ Associate with $$G$$ the operation of function composition $$\circ$$, which turns $$G$$ into a non-abelian group. $$G$$ can be identified with the right half plane $$(0, \infty) \times \mathbb{R} = \left\{ (x, y) ~:~ x, y \in \mathbb{R}, x > 0 \right\}$$ under which the group operation becomes $$(s, t) \circ (u, v) = (su, sv + t).$$ A left-invariant Haar measure $$\mu_L$$ (respectively, a right-invariant Haar measure $$\mu_R$$) on $$G = (0, \infty) \times \mathbb{R}$$ is given by $$\mu_L(S) = \int_S \frac{1}{x^2} \,dx\,dy$$ and $$\mu_R(S) = \int_S \frac{1}{x} \,dx\,dy$$ for any Borel subset $$S$$ of $$G = (0, \infty) \times \mathbb{R}.$$ This is because if $$S \subseteq (0, \infty) \times \mathbb{R}$$ is an open subset then for $$(s, t) \in G$$ fixed, integration by substitution gives $$\mu_L((s, t) \circ S) = \int_{(s, t) \circ S} \frac{1}{x^2} \,dx\,dy = \int_{S} \frac{1}{(s u)^2} |(s)(s) - (0)(0)| \,du\,dv = \mu_L(S)$$ while for $$(u, v) \in G$$ fixed, $$\mu_R(S \circ (u, v)) = \int_{S \circ (u, v)} \frac{1}{x} \,dx\,dy = \int_S \frac{1}{s u} |(u)(1) - (v)(0)| \,ds\,dt = \mu_R(S).$$

On any Lie group of dimension $$d$$ a left Haar measure can be associated with any non-zero left-invariant $d$-form $$\omega$$, as the Lebesgue measure $$|\omega|$$; and similarly for right Haar measures. This means also that the modular function can be computed, as the absolute value of the determinant of the adjoint representation.

The unit hyperbola $$\{(x,y) : x^2 - y^2 = 1, x > 0 \}$$ can be taken as a group under multiplication defined as with split-complex numbers $$z = x + y j, \ \ j^2 = +1 .$$ The usual area measure in the crescent $$C = \{(x,y) : |y| < x, \ \ x^2 - y^2 < 1 \}$$ serves to define hyperbolic angle as the area of its hyperbolic sector. The Haar measure of the unit hyperbola is generated by the hyperbolic angle of segments on the hyperbola. For instance, a measure of one unit is given by the segment running from (1,1) to (e,1/e), where e is Euler's number. Hyperbolic angle has been exploited in mathematical physics with rapidity standing in for classical velocity.

If $$G$$ is the group of non-zero quaternions, then $$G$$ can be seen as an open subset of $$\R^4$$. A Haar measure $$\mu$$ is given by $$\mu(S)=\int_S\frac1{(x^2+y^2+z^2+w^2)^2}\,dx\,dy\,dz\,dw$$ where $$dx\wedge dy\wedge dz\wedge dw$$ denotes the Lebesgue measure in $$\mathbb{R}^4$$ and $$S$$ is a Borel subset of $$G$$.

If $$G$$ is the additive group of $p$-adic numbers for a prime $$p$$, then a Haar measure is given by letting $$a+p^n O$$ have measure $$p^{-n}$$, where $$O$$ is the ring of $$p$$-adic integers.

A construction using compact subsets
The following method of constructing Haar measure is essentially the method used by Haar and Weil.

For any subsets $$S,T\subseteq G$$ with $$S$$ nonempty define $$[T:S]$$ to be the smallest number of left translates of $$S$$ that cover $$T$$ (so this is a non-negative integer or infinity). This is not additive on compact sets $$K\subseteq G$$, though it does have the property that $$[K:U]+[L:U]=[K\cup L:U]$$ for disjoint compact sets $$K,L\subseteq G$$ provided that $$U$$ is a sufficiently small open neighborhood of the identity (depending on $$K$$ and $$L$$). The idea of Haar measure is to take a sort of limit of $$[K:U]$$ as $$U$$ becomes smaller to make it additive on all pairs of disjoint compact sets, though it first has to be normalized so that the limit is not just infinity. So fix a compact set $$A$$ with non-empty interior (which exists as the group is locally compact) and for a compact set $$K$$ define
 * $$\mu_A(K)=\lim_U\frac{[K:U]}{[A:U]}$$

where the limit is taken over a suitable directed set of open neighborhoods of the identity eventually contained in any given neighborhood; the existence of a directed set such that the limit exists follows using Tychonoff's theorem.

The function $$\mu_A$$ is additive on disjoint compact subsets of $$G$$, which implies that it is a regular content. From a regular content one can construct a measure by first extending $$\mu_A$$ to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets. (Even for open sets $$U$$, the corresponding measure $$\mu_A(U)$$ need not be given by the lim sup formula above. The problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact closure, so is not an outer measure.)

A construction using compactly supported functions
Cartan introduced another way of constructing Haar measure as a Radon measure (a positive linear functional on compactly supported continuous functions), which is similar to the construction above except that $$A$$, $$K$$, and $$U$$ are positive continuous functions of compact support rather than subsets of $$G$$. In this case we define $$[K:U]$$ to be the infimum of numbers $$c_1+\cdots+c_n$$ such that $$K(g)$$ is less than the linear combination $$c_1 U(g_1 g)+\cdots+c_n U(g_n g)$$ of left translates of $$U$$ for some $$g_1,\ldots,g_n\in G$$. As before we define
 * $$\mu_A(K)=\lim_U\frac{[K:U]}{[A:U]}$$.

The fact that the limit exists takes some effort to prove, though the advantage of doing this is that the proof avoids the use of the axiom of choice and also gives uniqueness of Haar measure as a by-product. The functional $$\mu_A$$ extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure. (Note that even though the limit is linear in $$K$$, the individual terms $$[K:U]$$ are not usually linear in $$K$$.)

A construction using mean values of functions
Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups. The idea is that given a function $$f$$ on a compact group, one can find a convex combination $\sum a_i f(g_i g)$ (where $\sum a_i=1$ ) of its left translates that differs from a constant function by at most some small number $$\epsilon$$. Then one shows that as $$\epsilon$$ tends to zero the values of these constant functions tend to a limit, which is called the mean value (or integral) of the function $$f$$.

For groups that are locally compact but not compact this construction does not give Haar measure as the mean value of compactly supported functions is zero. However something like this does work for almost periodic functions on the group which do have a mean value, though this is not given with respect to Haar measure.

A construction on Lie groups
On an n-dimensional Lie group, Haar measure can be constructed easily as the measure induced by a left-invariant n-form. This was known before Haar's theorem.

The right Haar measure
It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure $$ \nu$$ satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure $$\mu$$. The left and right Haar measures are the same only for so-called unimodular groups (see below). It is quite simple, though, to find a relationship between $$\mu$$ and $$\nu$$.

Indeed, for a Borel set $$S$$, let us denote by $$S^{-1}$$ the set of inverses of elements of $$S$$. If we define
 * $$ \mu_{-1}(S) = \mu(S^{-1}) \quad $$

then this is a right Haar measure. To show right invariance, apply the definition:

 * $$ \mu_{-1}(S g) = \mu((S g)^{-1}) = \mu(g^{-1} S^{-1}) = \mu(S^{-1}) = \mu_{-1}(S). \quad $$

Because the right measure is unique, it follows that $$\mu_{-1}$$ is a multiple of $$\nu$$ and so
 * $$\mu(S^{-1})=k\nu(S)\,$$

for all Borel sets $$S$$, where $$k$$ is some positive constant.

The modular function
The left translate of a right Haar measure is a right Haar measure. More precisely, if $$\nu$$ is a right Haar measure, then for any fixed choice of a group element g,

 * $$ S \mapsto \nu (g^{-1} S) \quad $$

is also right invariant. Thus, by uniqueness up to a constant scaling factor of the Haar measure, there exists a function $$\Delta$$ from the group to the positive reals, called the Haar modulus, modular function or modular character, such that for every Borel set $$S$$

 * $$ \nu (g^{-1} S) = \Delta(g) \nu(S). \quad$$

Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation.

The modular function is a continuous group homomorphism from G to the multiplicative group of positive real numbers. A group is called unimodular if the modular function is identically $$1$$, or, equivalently, if the Haar measure is both left and right invariant. Examples of unimodular groups are abelian groups, compact groups, discrete groups (e.g., finite groups), semisimple Lie groups and connected nilpotent Lie groups. An example of a non-unimodular group is the group of affine transformations

 * $$\big\{ x \mapsto a x + b : a\in\R\setminus\{0\}, b\in\R \big\}=\left\{\begin{bmatrix}

a & b \\ 0 & 1 \end{bmatrix}\right\}$$

on the real line. This example shows that a solvable Lie group need not be unimodular. In this group a left Haar measure is given by $$\frac{1}{a^2}da\wedge db$$, and a right Haar measure by $$\frac{1}{|a|}da\wedge db$$.

Measures on homogeneous spaces
If the locally compact group $$G$$ acts transitively on a homogeneous space $$G/H$$, one can ask if this space has an invariant measure, or more generally a semi-invariant measure with the property that $$\mu(gS) = \chi(g)\mu(S)$$ for some character $$\chi$$ of $$G$$. A necessary and sufficient condition for the existence of such a measure is that the restriction $$\chi|_H$$ is equal to $$\Delta|_H/\delta$$, where $$\Delta$$ and $$\delta$$ are the modular functions of $$G$$ and $$H$$ respectively. In particular an invariant measure on $$G/H$$ exists if and only if the modular function $$\Delta$$ of $$G$$ restricted to $$H$$ is the modular function $$\delta$$ of $$H$$.

Example
If $$G$$ is the group $$SL_2(\mathbb{R})$$ and $$H$$ is the subgroup of upper triangular matrices, then the modular function of $$H$$ is nontrivial but the modular function of $$G$$ is trivial. The quotient of these cannot be extended to any character of $$G$$, so the quotient space $$G/H$$ (which can be thought of as 1-dimensional real projective space) does not have even a semi-invariant measure.

Haar integral
Using the general theory of Lebesgue integration, one can then define an integral for all Borel measurable functions $$f$$ on $$G$$. This integral is called the Haar integral and is denoted as:
 * $$\int f(x) \, d\mu(x)$$

where $$\mu$$ is the Haar measure.

One property of a left Haar measure $$\mu$$ is that, letting $$s$$ be an element of $$G$$, the following is valid:
 * $$ \int_G f(sx) \ d\mu(x) = \int_G f(x) \ d\mu(x) $$

for any Haar integrable function $$f$$ on $$G$$. This is immediate for indicator functions:
 * $$ \int \mathit{1}_A(tg)\,d\mu = \int \mathit{1}_{t^{-1}A}(g)\,d\mu=\mu(t^{-1}A)=\mu(A)=\int\mathit{1}_A(g)\,d\mu,$$

which is essentially the definition of left invariance.

Uses
In the same issue of Annals of Mathematics and immediately after Haar's paper, the Haar theorem was used to solve Hilbert's fifth problem restricted to compact groups by John von Neumann.

Unless $$G$$ is a discrete group, it is impossible to define a countably additive left-invariant regular measure on all subsets of $$G$$, assuming the axiom of choice, according to the theory of non-measurable sets.

Abstract harmonic analysis
The Haar measures are used in harmonic analysis on locally compact groups, particularly in the theory of Pontryagin duality. To prove the existence of a Haar measure on a locally compact group $$G$$ it suffices to exhibit a left-invariant Radon measure on $$G$$.

Mathematical statistics
In mathematical statistics, Haar measures are used for prior measures, which are prior probabilities for compact groups of transformations. These prior measures are used to construct admissible procedures, by appeal to the characterization of admissible procedures as Bayesian procedures (or limits of Bayesian procedures) by Wald. For example, a right Haar measure for a family of distributions with a location parameter results in the Pitman estimator, which is best equivariant. When left and right Haar measures differ, the right measure is usually preferred as a prior distribution. For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the Jeffreys prior measure. Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective information.

Another use of Haar measure in statistics is in conditional inference, in which the sampling distribution of a statistic is conditioned on another statistic of the data. In invariant-theoretic conditional inference, the sampling distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined). The result of conditioning sometimes depends on the order in which invariants are used and on the choice of a maximal invariant, so that by itself a statistical principle of invariance fails to select any unique best conditional statistic (if any exist); at least another principle is needed.

For non-compact groups, statisticians have extended Haar-measure results using amenable groups.

Weil's converse theorem
In 1936, André Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure with a certain separating property, then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as the Haar measure on this completion.

title: Haar measure


text:
Viggo Brun

Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist.

Contributions
In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors.

He also showed that the sum of the reciprocals of twin primes converges to a finite value, now called Brun's constant: by contrast, the sum of the reciprocals of all primes is divergent. He developed a multi-dimensional continued fraction algorithm in 1919–1920 and applied this to problems in musical theory. He also served as praeses of the Royal Norwegian Society of Sciences and Letters in 1946.

Biography
Brun was born at Lier in Buskerud, Norway. He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the Technical University in Trondheim and in 1946 a professor at the University of Oslo.

He retired in 1955 at the age of 70 and died in 1978 (at 92 years-old) at Drøbak in Akershus, Norway.

Other sources

 * H. Halberstam and H. E. Richert, Sieve methods, Academic Press (1974) ISBN 0-12-318250-6. Gives an account of Brun's sieve.
 * C.J. Scriba, Viggo Brun, Historia Mathematica 7 (1980) 1–6.
 * C.J. Scriba, Zur Erinnerung an Viggo Brun, Mitt. Math. Ges. Hamburg 11 (1985) 271-290

title: Viggo Brun


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Buffy Summers

Buffy Anne Summers is the title character of the Buffy the Vampire Slayer franchise. She first appeared in the 1992 film Buffy the Vampire Slayer before going on to appear in The WB/UPN 1997–2003 television series and subsequent 1998–2018 Dark Horse and 2019–present Boom! Studios comic series of the same name. The character has also appeared in the spin-off series Angel, as well as numerous expanded universe materials such as novels and video games. Buffy was portrayed by Kristy Swanson in the film and by Sarah Michelle Gellar in the television series. Giselle Loren has lent her voice to the character in both the Buffy video games and an unproduced animated series, while Kelly Albanese lent her voice to the character in the Buffy the Vampire Slayer Season Eight motion comics.

Buffy Summers is the protagonist of the series, which depicts her life and adventures as she grows up. In the film, she is a high school cheerleader who learns that she is the Slayer (a Chosen One gifted with the strength and skills to fight vampires and the forces of darkness, as a vampire hunter and demon hunter). The television series shows Buffy carrying out her destiny in the small town of Sunnydale, built atop a portal to hell (Hellmouth), surrounded by a group of friends and family who support her in her mission. In the comic book continuation, she is a young woman who has accepted her duties and is now responsible for training others like her.

Buffy was created by Joss Whedon to subvert the stereotypical female horror film victim—Whedon wanted to create a strong female cultural icon. In 2004, Buffy was ranked 13th on Bravo's list of The 100 Greatest TV Characters. In June 2010, Entertainment Weekly ranked her third in its list of the 100 Greatest Characters of the Last 20 Years. AOL named her the sixth Most Memorable Female TV Character. She was ranked at No. 5 in AfterEllen.com's Top 50 Favorite Female TV Characters.

Film
The character of Buffy first appears in the 1992 film, Buffy the Vampire Slayer, played by Kristy Swanson. The film, written by Joss Whedon, depicts Buffy as a shallow high school cheerleader who is informed by a man named Merrick (Donald Sutherland) that she has been chosen by fate to battle the undead. Buffy reluctantly undergoes training in her abilities by Merrick, and as her responsibility as the Slayer causes her to become alienated from her valley girl peers, she finds friendship and romance with fellow outcast Pike (Luke Perry). Merrick eventually comes to respect Buffy's rebellious nature, and she defeats vampire king Lothos (Rutger Hauer) by relying on her own contemporary style as opposed to traditional Slayer conventions. Although this film is not in continuity with the later television series, in 1999, author Christopher Golden adapted Joss Whedon's original script into a comic book entitled "The Origin", which Whedon later confirmed to be "pretty much" canonical.

On May 25, 2009, The Hollywood Reporter revealed Roy Lee and Doug Davison of Vertigo Entertainment would be working with Fran Rubel Kuzui and Kazi Kuzui on a relaunch of the Buffy series for the big screen. The series would not be a sequel or prequel to the existing movie or television franchise and Joss Whedon will have no involvement in the project. None of the cast or original characters from the television series will be featured. Television series executive producer Marti Noxon later reflected that this story might have been produced by the studio to frighten Joss into taking reins of the project. Studio interest in the project has continued, however. A script was rejected in 2011.

Television
Buffy returned in Joss Whedon's television series Buffy the Vampire Slayer, this time played by Sarah Michelle Gellar for all of the show's 144 episodes. In season one (1997), Buffy begins to accept the responsibilities and dangers of her calling as the Slayer after moving to the small California town of Sunnydale. She becomes best friends with Xander Harris (Nicholas Brendon) and Willow Rosenberg (Alyson Hannigan), and meets her new Watcher, the school librarian, Rupert Giles (Anthony Stewart Head). Together they form the Scooby Gang, and work together to battle various supernatural occurrences which plague Sunnydale High. In the season finale, Buffy battles the vampiric villain known as the Master (Mark Metcalf), and is drowned in the process. She is resuscitated by Xander and rises to defeat the vampire lord.

In the show's second season (1997–1998), Buffy continues to come to terms with her destiny, finds forbidden love with benevolent vampire Angel (David Boreanaz), and clashes with new villains Spike (James Marsters) and Drusilla (Juliet Landau). She also meets her Slayer replacement, Kendra (Bianca Lawson), who was called when Buffy was killed by the Master. Kendra is later killed by Drusilla, and the next replacement is seen in season 3. In the episode "Surprise", Buffy loses her virginity to Angel, an event which triggers the loss of his soul and unleashes his sadistic alter-ego, Angelus. Angelus proceeds to subject the characters to mental and physical torture for the remainder of the season. In the final episode of season two, Buffy is forced to reveal her identity as the Slayer to her mother (Kristine Sutherland), and send the newly good Angel to hell to save the world. She then leaves Sunnydale for Los Angeles in the hopes of escaping her life as the Slayer.

Season three (1998–1999) sees Buffy reconnect to her calling, her friends, and her family after her departure, as well as make difficult life decisions regarding her relationship with the resurrected Angel. She must also deal with the introduction of rebellious new Slayer Faith (Eliza Dushku), who becomes increasingly destructive and disloyal over the course of the season. In the season finale, Buffy stabs Faith in an attempt to save Angel's life, and leads her classmates into a climactic battle against the demonic Mayor of Sunnydale (Harry Groener). Angel then leaves Sunnydale in hopes that Buffy can have a more normal life without him.

In the fourth season (1999–2000), Buffy balances her Slayer duties with her new life as a college student at UC Sunnydale. She experiences some difficulty adjusting to college life, and becomes increasingly disconnected from her friends, who all seem to be moving in different directions. Buffy eventually finds a new love interest in the form of Riley Finn (Marc Blucas), a soldier in the demon-hunting government task force known as The Initiative. She briefly joins forces with Riley's team, until they discover one of the Initiative's experiments, Adam (George Hertzberg), is creating an army of demon-human hybrids. Buffy unites with her friends to defeat Adam in a spell which invokes the power of the First Slayer. During Buffy season four, Buffy also appears in the first season of spin-off series Angel (1999–2000), guest starring in the episodes "I Will Remember You" and "Sanctuary".

In season five (2000–2001), Buffy battles the hell-goddess Glory (Clare Kramer), and fully embraces her destiny for the first time. A younger sister named Dawn (Michelle Trachtenberg) mysteriously appears in Buffy's household, her existence having been seamlessly integrated with memories of the other characters. Buffy suffers emotional turmoil throughout this season, including the realization Dawn is not actually her sister, the deterioration of her relationship with Riley, the discovery that Spike has fallen obsessively in love with her, and her mother's death from a brain aneurysm. She takes control of her destiny and duties, informing the Watchers' Council that they work for her, not the other way around; that their job is to supply her with whatever information she requires, immediately and fully. While on a quest to learn more about her nature as the Slayer, Buffy is told "death is her gift", a message she has difficulty understanding until the episode "The Gift", in which she sacrifices herself to save Dawn and the world by diving into Glory's interdimensional portal and closing it.

Season six (2001–2002) depicts Buffy's struggle with depression after her friends, believing she was trapped in a Hell dimension, performed a spell to bring her back from the dead; however, she was actually in Heaven, and feels great loss after being ripped out. Forced to take a mundane and degrading job slinging burgers at the Doublemeat Palace after realizing her family was in financial ruin, she sinks into a deep depression amid feelings of self-loathing and spends much of the season on a downward spiral alienating her friends and family and embarking on a violent sexual relationship with the vampire Spike which leaves neither satisfied and spawns dire consequences for the both of them. Aside from dealing with her emotional and psychological demons in this season, Buffy is continually targeted by a group calling themselves The Trio - Warren Mears (Adam Busch), Andrew Wells (Tom Lenk), and Jonathan Levinson (Danny Strong). Initially satisfied with only wreaking havoc for personal gain in Sunnydale, the Trio soon become bent on becoming powerful nemeses to the Slayer. At first, their activities are merely annoying to Buffy and the Scoobies, but Warren's intentions become darker as the season goes on, and by the end Buffy is forced to deal with the Trio as the Slayer, while Andrew and Jonathan must come to grips with Warren's betrayal against them. As the season draws to a close, Buffy is forced to battle her best friend when Willow becomes psychotic with dark magic after Warren shoots and kills Willow's girlfriend Tara (Amber Benson) and wounds Buffy in the process. Willow then tries to destroy the world to end all suffering, although Xander gets through to her in the end. Buffy then promises to change her self-destructive behavior to be there for her sister.

In the final season of the show (2002–2003), things start to come around for Buffy when Principal Robin Wood (D. B. Woodside) hires her as a school counselor for the newly rebuilt Sunnydale High School and she has repaired her relationships with Dawn and her friends. However, she is also confronted with the threat of the First Evil and becomes a reluctant leader to the Potential Slayers, who are initially respectful of her, but become increasingly more alienated by her tactics and decisions throughout the season. She unexpectedly becomes emotionally close with Spike, who has sought out his soul in an effort to prove himself to her. In the show's final episode "Chosen", Buffy shares her power with her fellow Slayers before leading them into an epic battle against an army of Turok-Han vampires. She also confesses her love to a disbelieving Spike before he sacrifices himself to save the world; as he dies, Buffy escapes Sunnydale's destruction with the surviving characters.

Following the end of Buffy the Vampire Slayer, the character maintains a presence in the fifth season of Angel (2003–2004), but does not appear onscreen. In the episode "The Girl in Question", Angel and a resurrected Spike travel to Rome to find her, where they learn she is apparently now dating the Immortal. Sarah Michelle Gellar was approached to appear as Buffy in Angel ' s one hundredth episode, but declined, so the character of Cordelia Chase (Charisma Carpenter) was used instead. She was asked to appear in the second to last episode of the series, "Power Play", but had to decline due to outside conflicts.

Between 2001 and 2004, Joss Whedon and Jeph Loeb developed a 4-minute pilot episode for Buffy the Animated Series, which was set during the show's first season. Had the series been picked up by a network, the series would have focused upon Buffy (voiced by Giselle Loren) in more high-school adventures. Following a 2008 leak of the pilot to YouTube, Loeb expressed some hope the series may be resurrected in some form.

Literature
As the main character of the franchise, Buffy appears in almost all Buffy the Vampire Slayer literature. This includes a Dark Horse ongoing comic book and a series of novels. Buffy's debut into literature came in the comic Dark Horse Presents 1998 Annual on August 26, 1998, while her first prose appearance was in Halloween Rain by Christopher Golden and Nancy Holder on October 5, 1998. Most of these stories occur between episodes and seasons of the television series, however, some are set outside the timeline of the show to explore in depth other areas of Buffy's history. Christopher Golden adapted the film into a comic entitled "The Origin" (1999) which more closely resembles Joss Whedon's original script. In 2003, Scott Lobdell and Fabian Nicieza wrote a Year One-style run on the Buffy comic book series which filled the gap between the film and the first season of the show. These stories explain how Buffy's relationship with Pike ended, as well as fleshing out events alluded to in the television series, such as the time she spent in a mental institution and her parents' divorce. The novel Queen of the Slayers (2005) by Nancy Holder offers a potential follow-up to the television series; set after season seven, it depicts Buffy living in Italy with the morally ambiguous Immortal.

Buffy also makes appearances in literature outside of her own titular series. In the Tales of the Slayers comic one-shot "Broken Bottle of Djinn" (2002) by Doug Petrie and Jane Espenson, Buffy battles a spirit in Sunnydale High, while the Tales of the Vampires comic book story "Antique" (2004) by Drew Goddard sees her breaking into Dracula's castle to rescue Xander from the infamous vampire. Volume II of the similar series of novels Tales of the Slayer (2003) features two stories about Buffy; the character battles a mummified spirit in Todd A. McIntosh's "All That You Do Comes Back Unto Thee", while Jane Espenson's "Again Sunnydale" sees a season six-era Buffy sent back in time to high school, when her mother is still alive but Dawn does not exist.

Post-television "Season" continuations
In 2007, Buffy's story continued on from season seven when Joss Whedon revived Buffy the Vampire Slayer as a comic book, again published by Dark Horse. Whedon differentiated these comics from previous Buffy literature, stating "We could do something and for once we could make it canon. We could make it officially what happened after the end of the show." The continuation series emulates the structure of a television series, with five "seasons" published between 2007 and 2018 and Whedon overseeing multiple writers in the role of "executive producer". In Season Eight (2007–2011), it is quickly established that Buffy is not living with the Immortal in Rome as previously suggested in Angel; this is simply a cover story to ensure her safety as she is now the leader of an army which recruits and trains Slayers to deal with demonic threats worldwide. However, a mysterious group led by the masked villain Twilight believe the Slayers themselves pose a danger to mankind and the natural order. In "Wolves at the Gate", a lonely Buffy shares a sexual encounter with a younger Slayer Satsu; the relationship ends soon after when Satsu accepts that Buffy does not return her romantic feelings. The time travel story "Time of Your Life" acts as a crossover with Buffy spin-off Fray; Buffy is kidnapped two centuries into the future and meets her eventual successor Melaka Fray and Fray's vampire twin brother Harth. These events have been orchestrated by a villainous future version of Willow, whom Buffy reluctantly kills to return home. The tensions between Buffy and Twilight's respective armies eventually erupts into a full-scale war in Tibet; Twilight is unmasked as Angel being manipulated by enigmatic cosmic forces trying to destroy the universe and usher in a new dimension where Buffy and Angel will live together in paradise. However, Buffy rejects Twilight's influence and saves her world by returning to the ruins of Sunnydale and smashing the Seed of Wonder, cutting Earth off from the source of all magic in the process. During these events, Buffy reunites with Spike, Giles is killed by a Twilight-controlled Angel, and the Slayer army is dissolved; Buffy moves to San Francisco with her friends to grieve their losses.

In contrast to the global scale of Season Eight, Season Nine (2011–2013) follows Buffy living a more grounded civilian life in San Francisco; she works in a local coffee shop and shares an apartment with roommates Anaheed and Tumble. Buffy is now a pariah in the supernatural community due to her destruction of magic and must deal with "zompires", a feral new breed of vampires which have emerged since demons can no longer access Earth and fully possess human bodies. After getting blackout drunk at her housewarming party, Buffy has a pregnancy scare and turns to Spike for support when she decides to have an abortion; the pregnancy turns out to be a misunderstanding caused by Andrew, who switched Buffy's body with a robot as part of a misguided plan to keep her safe. In "Guarded", Buffy explores new career opportunities by temporarily joining Kennedy's private security company Deepscan and shutting down TinCan, an interdimensional social media site run by long-term Angel villains Wolfram & Hart. With most of her existing relationships strained, Buffy makes new allies in SFPD homicide detective Dowling and teenage vampire hunter Billy, and joins a magical council alongside demons D’Hoffryn and Illyria to battle the evil Slayer Simone and magic-siphoning Severin. When Dawn starts fading from existence due to the absence of magic, Buffy reunites with Willow and Xander in "The Core" to save her, journeying deep within the Earth to create a new Seed of Wonder and battling Maloker, an Old One and progenitor of all vampires, in the process. During Season Nine, Buffy also makes minor appearances in the spin-off comics Spike: A Dark Place, Willow: Wonderland, and Angel & Faith.

Having restored magic to the world, Season Ten (2014–2016) picks up with Buffy and her friends reluctantly responsible for creating the laws governing it; Buffy must guard the ancient grimoire Vampyr in which the new laws of magic are formalized when recorded in the book. Some rules have already materialized without Buffy's input (zompires are extinct and vampires now possess heightened strength and shapeshifting powers) and new rules tend to have unforeseen Monkey's Paw-style consequences. Daunted and tempted by this new-found power, Buffy seeks the advice of D’Hoffryn and his magical council, as well as lobbyists from various mystical and demonic communities pursuing their own conflicting agendas. Meanwhile, Buffy and Spike's decision to pursue an official romantic relationship is complicated by the manipulations of the demon Archaeus and the subsequent assistance by Angel; in the one-shot issue "Triggers", Buffy expresses her repressed trauma over the sexual assault she experienced from soulless Spike back in the television episode "Seeing Red". Following his resurrection in the spin-off series Angel & Faith, Giles reunites with Buffy, and the paternal relationship they share is reaffirmed when Buffy's father Hank excludes her from his wedding. D’Hoffryn eventually turns on Buffy and murders the rest of the council to seize the power of Vampyr for himself; finally accepting the responsibility she has been avoiding, Buffy outsmarts D’Hoffryn, commits to her relationship with Spike, and organizes her own council with whom to codify the laws of magic.

Season Eleven (2016–17) opens in disaster when a huge Shenlong dragon attacks San Francisco, killing thousands. In response to public outrage, the US President Malloy introduces the Supernatural Crisis Act, a set of new policies claiming to "legalize and normalise" the supernatural; this begins with a census and quickly leads to the relocation of magical individuals to the "Safe Zone", an internment camp in the Grand Canyon. Buffy rejects an opportunity to join a Slayer peacekeeping force alongside the antagonistic Jordan, and instead opts to join Willow and Spike at the Safe Zone, where she does her best to maintain peace and protect innocent or harmless inmates. Buffy and Willow eventually agree to have their powers removed to leave the camp and further investigate the Pandora Project, a government conspiracy to drain and abuse magical energy; they expose White House Press Secretary Joanna Wise for summoning the Shenlong in the first place to put her plans in motion. Buffy briefly reabsorbs the power of all the Slayers in the world to battle the magic-infused Wise, but returns the power by the season's final issue, warning a repentant Jordan not to misuse it.

In 2018, it was announced that Dark Horse was losing the license to publish Buffy comics; Joss Whedon's intent for the final season was to "give the Dark Horse era some closure". The four-issue Buffy the Vampire Slayer Season Twelve: The Reckoning (2018) picks up one year after Season Eleven, with a thirty-year-old Buffy pondering her future; she is separated but on cordial terms with Spike, still working as a part-time police consultant, and a doting aunt to Dawn and Xander's baby daughter Joyce. Tying up story elements first alluded to in 2001's Fray, Buffy is warned about an apocalypse called "the Reckoning" led by time travelling vampire Harth, in which the Slayers are depowered and Buffy is banished to a hell dimension battling an army of demons. With help from her friends, and those of Angel and Fray, Buffy proactively takes the fight to Harth in an attempt to change her fate; Illyria sacrifices herself to banish the demons in Buffy's place, altering history. After the battle, Buffy becomes a fulltime member of the SFPD supernatural division alongside Faith, reconnects with Spike, and comforts a grieving Angel over Illyria. Meanwhile, unbeknownst to Buffy, Fray returns to the 23rd century to discover her world has been drastically improved by the continued presence of many Slayers.

Boom! Studios reboot
In 2019, Buffy was reinvented by Boom! Studios in their ongoing comic Buffy the Vampire Slayer. This series is a complete reboot and has no continuity with any previous stories; Buffy is depicted as a teenager in 2019 rather than the 1990s. She is already aware of her destiny as the Slayer and has been living in Sunnydale for only three weeks when the series begins. She befriends Willow and Xander after saving them from a vampire outside Tunaverse, the fast food restaurant where she works.

Concept and creation
An early influence for Buffy was Kelli Maroney's character in the 1984 science-fiction horror film Night of the Comet. The character of Buffy was conceived by Joss Whedon as a way of subverting the cliché of "the little blonde girl who goes into a dark alley and gets killed in every horror film". Whedon stated "Rhonda the Immortal Waitress" was the first incarnation of Buffy in his head, "the idea of a seemingly insignificant female who in fact turns out to be extraordinary." When asked how he came up with the name of "Buffy," Whedon states "It was the name that I could think of that I could take the least seriously. There is no way you could hear the name Buffy and think, 'This is an important person.' The name may have come from Night of the Comet where Kelli Maroney's character, Samantha calling out her dog Muffy and Joss thought she said Buffy. To juxtapose that with Vampire Slayer, just felt like that kind of thing—a B movie. But a B movie that had something more going on. That was my dream." Whedon claims the title was criticized for being too silly, and the television network begged him to change it. He refused, insisting "You don't understand. It has to be this. This is what it is." Jason Middleton feels that Buffy avoids the "final girl" character trope seen in horror films, where the androgynous and celibate heroine gets to outlive her friends and exact revenge on their killer; in Middleton's words, "she... gets to have sex with boys and still kill the monster".

Whedon always intended for the character to become an icon, claiming "I wanted her to be a hero that existed in people's minds the way Wonder Woman or Spider-Man does, you know? I wanted her to be a doll or an action figure. I wanted Barbie with Kung Fu grip! I wanted her to enter the mass consciousness and the imaginations of growing kids because I think she's a cool character, and that was always the plan. I wanted Buffy to be a cultural phenomenon, period." In developing Buffy, Whedon was greatly inspired by Kitty Pryde, a character from the pages of the superhero comic X-Men. He admits, "If there's a bigger influence on Buffy than Kitty, I don't know what it was... She was an adolescent girl finding out she has great power and dealing with it." In a 2009 interview, Whedon revealed he only recently realised how much he saw of himself in Buffy. After years of relating more to Xander, he says, "Buffy was always the person that I was in that story because I'm not in every way." Whedon openly wonders why his identification figure is a woman, but describes it as "a real autobiographical kind of therapy for me" to be writing a strong female character like Buffy.

According to Whedon, Buffy "had been brewing in [him] for many years" before finally appearing in the Buffy the Vampire Slayer film played by Kristy Swanson. However, he was not satisfied with the character's treatment in the film, feeling "that's not quite her. It's a start, but it's not quite the girl." Although Whedon's vision of female empowerment was not as apparent as he would have liked in the 1992 film, he was given a second chance when Gail Berman approached him with the idea of re-creating it as a television series. Adapting the concept of the movie into a television series, Whedon decided to reinvent the character of Buffy slightly. The shallow cheerleader of the original film had grown more mature and open-minded, identifying with social outcasts such as Willow and Xander, and instead, the character of Cordelia was created to embody what Buffy once was. Early in the television series, make-up supervisor Todd McIntosh was instructed to make Buffy "a soft and sort of earthy character." He gave Gellar a soft, muted green make-up and kept her look very natural. However, it was later decided this was inappropriate for the character, and Buffy needed to look more like a valley girl. McIntosh switched her make-up around, giving her frosted eyeshadow and lip colors, bright turquoise and aqua marines, bubblegum colored nails, and bleach-blonde hair, causing the character to "blossom."

title: Buffy Summers


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User:Kazik~enwiki

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