text:
Greene County, New York

Greene County is a county located in the U.S. state of New York. As of the 2020 census, the population was 47,931. Its county seat is Catskill. The county's name is in honor of the American Revolutionary War general Nathanael Greene. The county is part of the Capital District region of the state.

History

On March 25, 1800, Greene County was created by the partitioning of Albany (360 Square Miles) and Ulster (270 Square Miles) counties, producing a county of 630 sqmi.

On April 3, 1801, 90 sqmi of land was transferred from Delaware and Ulster counties, raising the total area to 720 sqmi. This transferred Prattsville, Vly Mountain, Halcott Center, Bushnellsville, Highmount, Shandaken, Lanesville, and Pine Hill within Greene County.

On May 26, 1812, Greene County lost 20 sqmi to Ulster County, lowering the total area to 690 sqmi, reassigning Pine Hill, Highmount, and Shandaken to Ulster County.

On April 15, 1814, the county borders were re-surveyed, and the line adjusted with no change in area.

On April 17, 1822, the border was again surveyed, with no change in area, but Palenville was found to be inside Greene County.

On April 23, 1823, the border was again resurveyed, and the Hudson River border with Columbia County realigned accordingly.

On March 3, 1836, 30 sqmi of area was lost to Schoharie County, reducing the total area to 660 sqmi, and reassigning Manorkill, producing the Greene County that exists today.

Geography
According to the U.S. Census Bureau, the county has a total area of 658 sqmi, of which 647 sqmi is land and 11 sqmi (1.7%) is water. Greene County is located in southeast central New York State, just west of the Hudson River and south of Albany.

The northern and eastern regions are mostly low-lying flatlands. Along the Hudson River the lowest elevation is at sea level. The southern and western areas rise sharply into the Catskill Mountains.

Catskill Park takes up much of the south central region. The park contains some of the tallest peaks south of the Adirondacks. For example, the highest elevation is Hunter Mountain, at approximately 4,040 ft above sea level; In addition, there are many picturesque waterfalls in the park such as the famed, Kaaterskill Falls.

The entirety of Greene County is in the 19th Congressional District, and is represented by Republican Marc Molinaro.

Adjacent counties

 * Albany County - north
 * Rensselaer County - northeast
 * Columbia County - east
 * Ulster County - south
 * Delaware County - west
 * Schoharie County - northwest

National protected area

 * Thomas Cole National Historic Site

2000 census
As of the census of 2000, there were 48,195 people, 18,256 households, and 12,067 families residing in the county. The population density was 74 PD/sqmi. There were 26,544 housing units at an average density of 41 /mi2. The racial makeup of the county was 90.76% White, 5.53% Black or African American, 0.28% Native American, 0.54% Asian, 0.02% Pacific Islander, 1.52% from other races, and 1.36% from two or more races. 4.31% of the population were Hispanic or Latino of any race. 18.8% were of Irish, 17.8% Italian, 17.5% German, 8.6% American and 6.9% English ancestry according to Census 2000. 92.3% spoke English, 2.8% Spanish, 1.5% German and 1.3% Italian as their first language.

There were 18,256 households, out of which 29.2% had children under the age of 18 living with them, 51.2% were married couples living together, 10.3% had a female householder with no husband present, and 33.9% were non-families. 27.9% of all households were made up of individuals, and 12.2% had someone living alone who was 65 years of age or older. The average household size was 2.42 and the average family size was 2.97.

In the county, the population was spread out, with 23.00% under the age of 18, 9.5% from 18 to 24, 27.0% from 25 to 44, 24.8% from 45 to 64, and 15.7% who were 65 years of age or older. The median age was 39 years. For every 100 females there were 106.5 males. For every 100 females age 18 and over, there were 108.2 males.

The median income for a household in the county was $36,493, and the median income for a family was $43,854. Males had a median income of $35,598 versus $25,346 for females. The per capita income for the county was $18,931. About 8.6% of families and 12.2% of the population were below the poverty line, including 15.7% of those under age 18 and 10.4% of those age 65 or over.

As of the 2010 census, the racial makeup of the county was 90.3% White, 5.7% African American, 0.3% Native American and 0.8% Asian. Hispanic or Latino people of any race were 4.9% of the population.

Politics
Greene County is considered a Republican stronghold. It was one of only eleven counties in New York that did not vote for Bill Clinton in 1996, a strong year for Democrats. It supported John McCain by a margin just under ten points in 2008 over Barack Obama, another Democratic year. In 2012, 54.1 percent of the county's voters chose Republican challenger Mitt Romney, and 43.7 percent chose Democratic incumbent Obama, despite it being again a Democratic year. In 2016, 59 percent of the county's voters chose Donald Trump while 34 percent chose Hillary Clinton. It has not supported a Democrat for president since 1964, and before that it previously did so in 1912.

In some elections, certain towns in Greene County have had a plurality that have voted Democratic, though the county remains very Republican.

Towns

 * Ashland
 * Athens
 * Cairo
 * Catskill
 * Coxsackie
 * Durham
 * Greenville
 * Halcott
 * Hunter
 * Jewett
 * Lexington
 * New Baltimore
 * Prattsville
 * Windham

Villages

 * Athens
 * Catskill (county seat)
 * Coxsackie
 * Hunter
 * Tannersville

Census-designated places

 * Cairo
 * Cementon
 * East Durham
 * Greenville
 * Haines Falls
 * Jefferson Heights
 * Leeds
 * New Baltimore
 * Palenville
 * Prattsville
 * Round Top
 * Sleepy Hollow Lake
 * South Cairo
 * West Kill
 * Windham

Hamlets

 * Acra
 * Cornwallville
 * Climax
 * Earlton
 * Freehold
 * Haines Falls, New York
 * Hannacroix
 * Hensonville
 * Lanesville
 * Oak Hill
 * Surprise
 * Maplecrest

Notable people

 * John Adams, (1778–1854), born in Oak Hill, United States Congressman
 * Levi Hill, claimed early inventor of color photography
 * Jeff "The Drunk" Curro Howard Stern Wack Packer
 * Blossom Dearie, (1924-2009), jazz singer
 * John B. Terry, (1796-1874), pioneer, merchant, soldier, and Wisconsin territorial legislature, born in Coxsackie
 * Thurlow Weed, (1797-1882), born in Acra, NY. Newspaper publisher, politician, and party boss
 * Edwin L. Drake, (1819-1880), invented method to drill oil from the ground, born in Greenville
 * Scott Adams, born and raised in Windham, NY; an American cartoonist, creator of the Dilbert comic strip and the author of several nonfiction works of satire, commentary, business, and general speculation.

title: Greene County, New York


text:
User talk:SunirShah

See m:User talk:SunirShah for the talk that was previously here, about the FOLDOC fork.

Consider logging-in Meta-Wiki once?
As you have a userpage on meta but their m:Special:Contributions/SunirShah is broken, showing you didn't have any login records? Liuxinyu970226 (talk) 00:33, 3 November 2022 (UTC)

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

title: User talk:SunirShah


text:
Cellular automaton

A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessellation structures, and iterative arrays. Cellular automata have found application in various areas, including physics, theoretical biology and microstructure modeling.

A cellular automaton consists of a regular grid of cells, each in one of a finite number of states, such as on and off (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighborhood is defined relative to the specified cell. An initial state (time t = 0) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. Typically, the rule for updating the state of cells is the same for each cell and does not change over time, and is applied to the whole grid simultaneously, though exceptions are known, such as the stochastic cellular automaton and asynchronous cellular automaton.

The concept was originally discovered in the 1940s by Stanislaw Ulam and John von Neumann while they were contemporaries at Los Alamos National Laboratory. While studied by some throughout the 1950s and 1960s, it was not until the 1970s and Conway's Game of Life, a two-dimensional cellular automaton, that interest in the subject expanded beyond academia. In the 1980s, Stephen Wolfram engaged in a systematic study of one-dimensional cellular automata, or what he calls elementary cellular automata; his research assistant Matthew Cook showed that one of these rules is Turing-complete.

The primary classifications of cellular automata, as outlined by Wolfram, are numbered one to four. They are, in order, automata in which patterns generally stabilize into homogeneity, automata in which patterns evolve into mostly stable or oscillating structures, automata in which patterns evolve in a seemingly chaotic fashion, and automata in which patterns become extremely complex and may last for a long time, with stable local structures. This last class is thought to be computationally universal, or capable of simulating a Turing machine. Special types of cellular automata are reversible, where only a single configuration leads directly to a subsequent one, and totalistic, in which the future value of individual cells only depends on the total value of a group of neighboring cells. Cellular automata can simulate a variety of real-world systems, including biological and chemical ones.

Overview
One way to simulate a two-dimensional cellular automaton is with an infinite sheet of graph paper along with a set of rules for the cells to follow. Each square is called a "cell" and each cell has two possible states, black and white. The neighborhood of a cell is the nearby, usually adjacent, cells. The two most common types of neighborhoods are the von Neumann neighborhood and the Moore neighborhood. The former, named after the founding cellular automaton theorist, consists of the four orthogonally adjacent cells. The latter includes the von Neumann neighborhood as well as the four diagonally adjacent cells. For such a cell and its Moore neighborhood, there are 512 (= 29) possible patterns. For each of the 512 possible patterns, the rule table would state whether the center cell will be black or white on the next time interval. Conway's Game of Life is a popular version of this model. Another common neighborhood type is the extended von Neumann neighborhood, which includes the two closest cells in each orthogonal direction, for a total of eight. The general equation for the total number of automata possible is kk s, where k is the number of possible states for a cell, and s is the number of neighboring cells (including the cell to be calculated itself) used to determine the cell's next state. Thus, in the two-dimensional system with a Moore neighborhood, the total number of automata possible would be 22 9, or $1.34$.

It is usually assumed that every cell in the universe starts in the same state, except for a finite number of cells in other states; the assignment of state values is called a configuration. More generally, it is sometimes assumed that the universe starts out covered with a periodic pattern, and only a finite number of cells violate that pattern. The latter assumption is common in one-dimensional cellular automata.

Cellular automata are often simulated on a finite grid rather than an infinite one. In two dimensions, the universe would be a rectangle instead of an infinite plane. The obvious problem with finite grids is how to handle the cells on the edges. How they are handled will affect the values of all the cells in the grid. One possible method is to allow the values in those cells to remain constant. Another method is to define neighborhoods differently for these cells. One could say that they have fewer neighbors, but then one would also have to define new rules for the cells located on the edges. These cells are usually handled with periodic boundary conditions resulting in a toroidal arrangement: when one goes off the top, one comes in at the corresponding position on the bottom, and when one goes off the left, one comes in on the right. (This essentially simulates an infinite periodic tiling, and in the field of partial differential equations is sometimes referred to as periodic boundary conditions.) This can be visualized as taping the left and right edges of the rectangle to form a tube, then taping the top and bottom edges of the tube to form a torus (doughnut shape). Universes of other dimensions are handled similarly. This solves boundary problems with neighborhoods, but another advantage is that it is easily programmable using modular arithmetic functions. For example, in a 1-dimensional cellular automaton like the examples below, the neighborhood of a cell xit is {xi−1t−1, xit−1, xi+1t−1 }, where t is the time step (vertical), and i is the index (horizontal) in one generation.

History
Stanislaw Ulam, while working at the Los Alamos National Laboratory in the 1940s, studied the growth of crystals, using a simple lattice network as his model. At the same time, John von Neumann, Ulam's colleague at Los Alamos, was working on the problem of self-replicating systems. Von Neumann's initial design was founded upon the notion of one robot building another robot. This design is known as the kinematic model. As he developed this design, von Neumann came to realize the great difficulty of building a self-replicating robot, and of the great cost in providing the robot with a "sea of parts" from which to build its replicant. Neumann wrote a paper entitled "The general and logical theory of automata" for the Hixon Symposium in 1948. Ulam was the one who suggested using a discrete system for creating a reductionist model of self-replication. Nils Aall Barricelli performed many of the earliest explorations of these models of artificial life.

Ulam and von Neumann created a method for calculating liquid motion in the late 1950s. The driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbors' behaviors. Thus was born the first system of cellular automata. Like Ulam's lattice network, von Neumann's cellular automata are two-dimensional, with his self-replicator implemented algorithmically. The result was a universal copier and constructor working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only orthogonal cells), and with 29 states per cell. Von Neumann gave an existence proof that a particular pattern would make endless copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so. This design is known as the tessellation model, and is called a von Neumann universal constructor.

Also in the 1940s, Norbert Wiener and Arturo Rosenblueth developed a model of excitable media with some of the characteristics of a cellular automaton. Their specific motivation was the mathematical description of impulse conduction in cardiac systems. However their model is not a cellular automaton because the medium in which signals propagate is continuous, and wave fronts are curves. A true cellular automaton model of excitable media was developed and studied by J. M. Greenberg and S. P. Hastings in 1978; see Greenberg-Hastings cellular automaton. The original work of Wiener and Rosenblueth contains many insights and continues to be cited in modern research publications on cardiac arrhythmia and excitable systems.

In the 1960s, cellular automata were studied as a particular type of dynamical system and the connection with the mathematical field of symbolic dynamics was established for the first time. In 1969, Gustav A. Hedlund compiled many results following this point of view in what is still considered as a seminal paper for the mathematical study of cellular automata. The most fundamental result is the characterization in the Curtis–Hedlund–Lyndon theorem of the set of global rules of cellular automata as the set of continuous endomorphisms of shift spaces.

In 1969, German computer pioneer Konrad Zuse published his book Calculating Space, proposing that the physical laws of the universe are discrete by nature, and that the entire universe is the output of a deterministic computation on a single cellular automaton; "Zuse's Theory" became the foundation of the field of study called digital physics.

Also in 1969 computer scientist Alvy Ray Smith completed a Stanford PhD dissertation on Cellular Automata Theory, the first mathematical treatment of CA as a general class of computers. Many papers came from this dissertation: He showed the equivalence of neighborhoods of various shapes, how to reduce a Moore to a von Neumann neighborhood or how to reduce any neighborhood to a von Neumann neighborhood. He proved that two-dimensional CA are computation universal, introduced 1-dimensional CA, and showed that they too are computation universal, even with simple neighborhoods. He showed how to subsume the complex von Neumann proof of construction universality (and hence self-reproducing machines) into a consequence of computation universality in a 1-dimensional CA. Intended as the introduction to the German edition of von Neumann's book on CA, he wrote a survey of the field with dozens of references to papers, by many authors in many countries over a decade or so of work, often overlooked by modern CA researchers.

In the 1970s a two-state, two-dimensional cellular automaton named Game of Life became widely known, particularly among the early computing community. Invented by John Conway and popularized by Martin Gardner in a Scientific American article, its rules are as follows: Despite its simplicity, the system achieves an impressive diversity of behavior, fluctuating between apparent randomness and order. One of the most apparent features of the Game of Life is the frequent occurrence of gliders, arrangements of cells that essentially move themselves across the grid. It is possible to arrange the automaton so that the gliders interact to perform computations, and after much effort it has been shown that the Game of Life can emulate a universal Turing machine. It was viewed as a largely recreational topic, and little follow-up work was done outside of investigating the particularities of the Game of Life and a few related rules in the early 1970s.
 * 1) Any live cell with fewer than two live neighbours dies, as if caused by underpopulation.
 * 2) Any live cell with two or three live neighbours lives on to the next generation.
 * 3) Any live cell with more than three live neighbours dies, as if by overpopulation.
 * 4) Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

Stephen Wolfram independently began working on cellular automata in mid-1981 after considering how complex patterns seemed formed in nature in violation of the Second Law of Thermodynamics. His investigations were initially spurred by a desire to model systems such as the neural networks found in brains. He published his first paper in Reviews of Modern Physics investigating elementary cellular automata (Rule 30 in particular) in June 1983. The unexpected complexity of the behavior of these simple rules led Wolfram to suspect that complexity in nature may be due to similar mechanisms. His investigations, however, led him to realize that cellular automata were poor at modelling neural networks. Additionally, during this period Wolfram formulated the concepts of intrinsic randomness and computational irreducibility, and suggested that rule 110 may be universal—a fact proved later by Wolfram's research assistant Matthew Cook in the 1990s.

Classification
Wolfram, in A New Kind of Science and several papers dating from the mid-1980s, defined four classes into which cellular automata and several other simple computational models can be divided depending on their behavior. While earlier studies in cellular automata tended to try to identify types of patterns for specific rules, Wolfram's classification was the first attempt to classify the rules themselves. In order of complexity the classes are:
 * Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears.
 * Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures. Some of the randomness in the initial pattern may filter out, but some remains. Local changes to the initial pattern tend to remain local.
 * Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread indefinitely.
 * Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways, with the formation of local structures that are able to survive for long periods of time. Class 2 type stable or oscillating structures may be the eventual outcome, but the number of steps required to reach this state may be very large, even when the initial pattern is relatively simple. Local changes to the initial pattern may spread indefinitely. Wolfram has conjectured that many class 4 cellular automata, if not all, are capable of universal computation. This has been proven for Rule 110 and Conway's Game of Life.

These definitions are qualitative in nature and there is some room for interpretation. According to Wolfram, "...with almost any general classification scheme there are inevitably cases which get assigned to one class by one definition and another class by another definition. And so it is with cellular automata: there are occasionally rules...that show some features of one class and some of another." Wolfram's classification has been empirically matched to a clustering of the compressed lengths of the outputs of cellular automata.

There have been several attempts to classify cellular automata in formally rigorous classes, inspired by Wolfram's classification. For instance, Culik and Yu proposed three well-defined classes (and a fourth one for the automata not matching any of these), which are sometimes called Culik–Yu classes; membership in these proved undecidable. Wolfram's class 2 can be partitioned into two subgroups of stable (fixed-point) and oscillating (periodic) rules.

The idea that there are 4 classes of dynamical system came originally from Nobel-prize winning chemist Ilya Prigogine who identified these 4 classes of thermodynamical systems: (1) systems in thermodynamic equilibrium, (2) spatially/temporally uniform systems, (3) chaotic systems, and (4) complex far-from-equilibrium systems with dissipative structures (see figure 1 in the 1974 paper of Nicolis, Prigogine's student).

Reversible
A cellular automaton is reversible if, for every current configuration of the cellular automaton, there is exactly one past configuration (preimage). If one thinks of a cellular automaton as a function mapping configurations to configurations, reversibility implies that this function is bijective. If a cellular automaton is reversible, its time-reversed behavior can also be described as a cellular automaton; this fact is a consequence of the Curtis–Hedlund–Lyndon theorem, a topological characterization of cellular automata. For cellular automata in which not every configuration has a preimage, the configurations without preimages are called Garden of Eden patterns.

For one-dimensional cellular automata there are known algorithms for deciding whether a rule is reversible or irreversible. However, for cellular automata of two or more dimensions reversibility is undecidable; that is, there is no algorithm that takes as input an automaton rule and is guaranteed to determine correctly whether the automaton is reversible. The proof by Jarkko Kari is related to the tiling problem by Wang tiles.

Reversible cellular automata are often used to simulate such physical phenomena as gas and fluid dynamics, since they obey the laws of thermodynamics. Such cellular automata have rules specially constructed to be reversible. Such systems have been studied by Tommaso Toffoli, Norman Margolus and others. Several techniques can be used to explicitly construct reversible cellular automata with known inverses. Two common ones are the second-order cellular automaton and the block cellular automaton, both of which involve modifying the definition of a cellular automaton in some way. Although such automata do not strictly satisfy the definition given above, it can be shown that they can be emulated by conventional cellular automata with sufficiently large neighborhoods and numbers of states, and can therefore be considered a subset of conventional cellular automata. Conversely, it has been shown that every reversible cellular automaton can be emulated by a block cellular automaton.

Totalistic
A special class of cellular automata are totalistic cellular automata. The state of each cell in a totalistic cellular automaton is represented by a number (usually an integer value drawn from a finite set), and the value of a cell at time t depends only on the sum of the values of the cells in its neighborhood (possibly including the cell itself) at time t − 1. If the state of the cell at time t depends on both its own state and the total of its neighbors at time t − 1 then the cellular automaton is properly called outer totalistic. Conway's Game of Life is an example of an outer totalistic cellular automaton with cell values 0 and 1; outer totalistic cellular automata with the same Moore neighborhood structure as Life are sometimes called life-like cellular automata.

Related automata
There are many possible generalizations of the cellular automaton concept.

One way is by using something other than a rectangular (cubic, etc.) grid. For example, if a plane is tiled with regular hexagons, those hexagons could be used as cells. In many cases the resulting cellular automata are equivalent to those with rectangular grids with specially designed neighborhoods and rules. Another variation would be to make the grid itself irregular, such as with Penrose tiles.

Also, rules can be probabilistic rather than deterministic. Such cellular automata are called probabilistic cellular automata. A probabilistic rule gives, for each pattern at time t, the probabilities that the central cell will transition to each possible state at time t + 1. Sometimes a simpler rule is used; for example: "The rule is the Game of Life, but on each time step there is a 0.001% probability that each cell will transition to the opposite color."

The neighborhood or rules could change over time or space. For example, initially the new state of a cell could be determined by the horizontally adjacent cells, but for the next generation the vertical cells would be used.

In cellular automata, the new state of a cell is not affected by the new state of other cells. This could be changed so that, for instance, a 2 by 2 block of cells can be determined by itself and the cells adjacent to itself.

There are continuous automata. These are like totalistic cellular automata, but instead of the rule and states being discrete (e.g. a table, using states {0,1,2}), continuous functions are used, and the states become continuous (usually values in [0,1]). The state of a location is a finite number of real numbers. Certain cellular automata can yield diffusion in liquid patterns in this way.

Continuous spatial automata have a continuum of locations. The state of a location is a finite number of real numbers. Time is also continuous, and the state evolves according to differential equations. One important example is reaction–diffusion textures, differential equations proposed by Alan Turing to explain how chemical reactions could create the stripes on zebras and spots on leopards. When these are approximated by cellular automata, they often yield similar patterns. MacLennan considers continuous spatial automata as a model of computation.

There are known examples of continuous spatial automata, which exhibit propagating phenomena analogous to gliders in the Game of Life.

Graph rewriting automata are extensions of cellular automata based on graph rewriting systems.

Elementary cellular automata
The simplest nontrivial cellular automaton would be one-dimensional, with two possible states per cell, and a cell's neighbors defined as the adjacent cells on either side of it. A cell and its two neighbors form a neighborhood of 3 cells, so there are 23 = 8 possible patterns for a neighborhood. A rule consists of deciding, for each pattern, whether the cell will be a 1 or a 0 in the next generation. There are then 28 = 256 possible rules.

These 256 cellular automata are generally referred to by their Wolfram code, a standard naming convention invented by Wolfram that gives each rule a number from 0 to 255. A number of papers have analyzed and compared the distinct cases among the 256 cellular automata (many are trivially isomorphic). The rule 30, rule 90, rule 110, and rule 184 cellular automata are particularly interesting. The images below show the history of rules 30 and 110 when the starting configuration consists of a 1 (at the top of each image) surrounded by 0s. Each row of pixels represents a generation in the history of the automaton, with t=0 being the top row. Each pixel is colored white for 0 and black for 1.

Rule 30 exhibits class 3 behavior, meaning even simple input patterns such as that shown lead to chaotic, seemingly random histories.

Rule 110, like the Game of Life, exhibits what Wolfram calls class 4 behavior, which is neither completely random nor completely repetitive. Localized structures appear and interact in various complicated-looking ways. In the course of the development of A New Kind of Science, as a research assistant to Wolfram in 1994, Matthew Cook proved that some of these structures were rich enough to support universality. This result is interesting because rule 110 is an extremely simple one-dimensional system, and difficult to engineer to perform specific behavior. This result therefore provides significant support for Wolfram's view that class 4 systems are inherently likely to be universal. Cook presented his proof at a Santa Fe Institute conference on Cellular Automata in 1998, but Wolfram blocked the proof from being included in the conference proceedings, as Wolfram did not want the proof announced before the publication of A New Kind of Science. In 2004, Cook's proof was finally published in Wolfram's journal Complex Systems (Vol. 15, No. 1), over ten years after Cook came up with it. Rule 110 has been the basis for some of the smallest universal Turing machines.

Rule space
An elementary cellular automaton rule is specified by 8 bits, and all elementary cellular automaton rules can be considered to sit on the vertices of the 8-dimensional unit hypercube. This unit hypercube is the cellular automaton rule space. For next-nearest-neighbor cellular automata, a rule is specified by 25 = 32 bits, and the cellular automaton rule space is a 32-dimensional unit hypercube. A distance between two rules can be defined by the number of steps required to move from one vertex, which represents the first rule, and another vertex, representing another rule, along the edge of the hypercube. This rule-to-rule distance is also called the Hamming distance.

Cellular automaton rule space allows us to ask the question concerning whether rules with similar dynamical behavior are "close" to each other. Graphically drawing a high dimensional hypercube on the 2-dimensional plane remains a difficult task, and one crude locator of a rule in the hypercube is the number of bit-1 in the 8-bit string for elementary rules (or 32-bit string for the next-nearest-neighbor rules). Drawing the rules in different Wolfram classes in these slices of the rule space show that class 1 rules tend to have lower number of bit-1s, thus located in one region of the space, whereas class 3 rules tend to have higher proportion (50%) of bit-1s.

For larger cellular automaton rule space, it is shown that class 4 rules are located between the class 1 and class 3 rules. This observation is the foundation for the phrase edge of chaos, and is reminiscent of the phase transition in thermodynamics.

Biology

Several biological processes occur—or can be simulated—by cellular automata.

Some examples of biological phenomena modeled by cellular automata with a simple state space are:

 * Patterns of some seashells, like the ones in the genera Conus and Cymbiola, are generated by natural cellular automata. The pigment cells reside in a narrow band along the shell's lip. Each cell secretes pigments according to the activating and inhibiting activity of its neighbor pigment cells, obeying a natural version of a mathematical rule. The cell band leaves the colored pattern on the shell as it grows slowly. For example, the widespread species Conus textile bears a pattern resembling Wolfram's rule 30 cellular automaton.
 * Plants regulate their intake and loss of gases via a cellular automaton mechanism. Each stoma on the leaf acts as a cell.
 * Moving wave patterns on the skin of cephalopods can be simulated with a two-state, two-dimensional cellular automata, each state corresponding to either an expanded or retracted chromatophore.
 * Threshold automata have been invented to simulate neurons, and complex behaviors such as recognition and learning can be simulated.
 * Fibroblasts bear similarities to cellular automata, as each fibroblast only interacts with its neighbors.

Additionally, biological phenomena which require explicit modeling of the agents' velocities (for example, those involved in collective cell migration) may be modeled by cellular automata with a more complex state space and rules, such as biological lattice-gas cellular automata. These include phenomena of great medical importance, such as:

 * Characterization of different modes of metastatic invasion.
 * The role of heterogeneity in the development of aggressive carcinomas.
 * Phenotypic switching during tumor proliferation.

Chemistry
The Belousov–Zhabotinsky reaction is a spatio-temporal chemical oscillator that can be simulated by means of a cellular automaton. In the 1950s A. M. Zhabotinsky (extending the work of B. P. Belousov) discovered that when a thin, homogenous layer of a mixture of malonic acid, acidified bromate, and a ceric salt were mixed together and left undisturbed, fascinating geometric patterns such as concentric circles and spirals propagate across the medium. In the "Computer Recreations" section of the August 1988 issue of Scientific American, A. K. Dewdney discussed a cellular automaton developed by Martin Gerhardt and Heike Schuster of the University of Bielefeld (Germany). This automaton produces wave patterns that resemble those in the Belousov-Zhabotinsky reaction.

Physics
Probabilistic cellular automata are used in statistical and condensed matter physics to study phenomena like fluid dynamics and phase transitions. The Ising model is a prototypical example, in which each cell can be in either of two states called "up" and "down", making an idealized representation of a magnet. By adjusting the parameters of the model, the proportion of cells being in the same state can be varied, in ways that help explicate how ferromagnets become demagnetized when heated. Moreover, results from studying the demagnetization phase transition can be transferred to other phase transitions, like the evaporation of a liquid into a gas; this convenient cross-applicability is known as universality. The phase transition in the two-dimensional Ising model and other systems in its universality class has been of particular interest, as it requires conformal field theory to understand in depth. Other cellular automata that have been of significance in physics include lattice gas automata, which simulate fluid flows.

Computer science, coding, and communication
Cellular automaton processors are physical implementations of CA concepts, which can process information computationally. Processing elements are arranged in a regular grid of identical cells. The grid is usually a square tiling, or tessellation, of two or three dimensions; other tilings are possible, but not yet used. Cell states are determined only by interactions with adjacent neighbor cells. No means exists to communicate directly with cells farther away. One such cellular automaton processor array configuration is the systolic array. Cell interaction can be via electric charge, magnetism, vibration (phonons at quantum scales), or any other physically useful means. This can be done in several ways so that no wires are needed between any elements. This is very unlike processors used in most computers today (von Neumann designs) which are divided into sections with elements that can communicate with distant elements over wires.

Rule 30 was originally suggested as a possible block cipher for use in cryptography. Two-dimensional cellular automata can be used for constructing a pseudorandom number generator.

Cellular automata have been proposed for public-key cryptography. The one-way function is the evolution of a finite CA whose inverse is believed to be hard to find. Given the rule, anyone can easily calculate future states, but it appears to be very difficult to calculate previous states. Cellular automata have also been applied to design error correction codes.

Other problems that can be solved with cellular automata include:

 * Firing squad synchronization problem
 * Majority problem

Generative art and music
Cellular automata have been used in generative music and evolutionary music composition and procedural terrain generation in video games.

Specific rules
Specific cellular automata rules include:

 * Brian's Brain
 * Codd's cellular automaton
 * CoDi
 * Conway's game of life
 * Day and Night
 * Langton's ant
 * Langton's loops
 * Lenia
 * Nobili cellular automata
 * Rule 90
 * Rule 184
 * Seeds
 * Turmite
 * Von Neumann cellular automaton
 * Wireworld

title: Cellular automaton


text:
Austrian euro coins

Austrian euro coins have a unique design for each denomination, with a common theme for each of the three series of coins. The minor coins feature Austrian flowers, the middle coins examples of architecture from Austria's capital, Vienna, and the two major coins famous Austrians. All designs are by the hand of Josef Kaiser and also include the 12 stars of the EU, Flag of Austria and the year of imprint.

Austrian euro design
Prior to 2008, the old common side showing national borders was used, but from the 2008 coins use the new common side without borders. For images of the common side and a detailed description of the coins, see euro coins. Austria is the only country which uses the Latin alphabet and yet, repeats the denomination on the national side of the coins, thus not adhering to the rule.

From 2017 (5 cent coins) and 2018 (2 and 1 cent coins) onwards, the design of the national side has been slightly altered, decreasing the thickness and increasing the amount of hatching representing the red fields of the Austrian flag.

Circulating Mintage quantities
The following table shows the mintage quantity for all Austrian euro coins, per denomination, per year.

Austrian proof set
Each year the Austrian Mint issues a limited edition of its Euro coins in proof quality.

Other commemorative coins (Collector's coins)
Austria has a large collection of euro commemorative coins, mainly in Silver and Gold, but they also use other materials (like Niobium for example). Their face value range from 5 euro to 100 euro. This is mainly done as a legacy of old national practice of minting Gold and Silver coins. These coins are not really intended to be used as means of payment, so generally they do not circulate. Here are some samples:

title: Austrian euro coins


text:
Talk:NeXT

Article Ending
Nice article! But it seems to end abruptly; please tell us more!! What's it doing now? What happened to Steve? — Preceding unsigned comment added by Ed Poor (talk • contribs) 09:12, 31 May 2002 (UTC)

 * This comment is several years old and the problem has been corrected. (Anonymous User) June 1, 2006 — Preceding unsigned comment added by 66.212.44.71 (talk) 07:07, 1 June 2006 (UTC)

WebObjects
We need to create a paragraph about WebObjects. It's an important NeXT product that still exists. -- tooki 17:07, 26 Jul 2004 (UTC)

Technology
I don't think the comparison to 8088/8086/286s is appropriate time-wise. By the time the NeXT was released, 386s were definitely in use, and I think even the 486 was not too far off. Also, when my college received our NeXT cubes, we already had some Mac II computers, with color. I think we even had Macc IIci and IIsi models. — Preceding unsigned comment added by 151.145.250.252 (talk) 17:23, 24 August 2004 (UTC)

Sentence Correction
This sentence needs to be re-worded: "Basically the drive itself, while faster than a floppy, was simply not fast enough to run a Unix based OS as its primary medium." It sounds like the OS is the primary medium of the MO drive, where it is in fact the MO drive that is the primary medium for the OS. — Preceding unsigned comment added by 151.145.250.252 (talk) 17:23, 24 August 2004 (UTC)

 * Changed to: "...was simply not fast enough to run as the primary medium for a Unix-based OS." ABoerma 13:25, 29 March 2006 (UTC)

Sneakernet Link
Sneakernet could probably be an article link. — Preceding unsigned comment added by 151.145.250.252 (talk) 17:23, 24 August 2004 (UTC)

Major Copyedit
I have gone through and performed a pretty serious copyedit... I also added a large number of s. It's important to remember that this article is about NeXT, not Apple, so I removed a lot of statements about OS X that weren't directly related to NeXT's software contributions. Themillofkeytone 16:00, 21 June 2006 (UTC)

This is a Good Article
After reviewing the article, I've determined it to meet the qualifications for GA status. It is well written, comprehensive, and of a good neutral tone. The last copyedit by User:Themillofkeytone seems to have done the trick.

The biggest step for improvement now would be filling all the templates with actual citations. If anyone wanted to help the article out, verifying and citing a few of those little facts would go a long way!

Keep up the good work, folks. Phidauex 15:43, 5 July 2006 (UTC)

Irrelevant Apple Stuff
The last paragraph lists every new Apple machine that was released during the transition from PPC to Intel processors. These details are tangential overkill in an article about NeXT. No one who comes here to learn about the NeXT corporation needs to know the specific date the Apple MacBook was released. Just say when the transition began or ended and be done with it. emw 15:42, 26 August 2006 (UTC)
 * Fixed, removed the specifics about Apple. — Wackymacs 15:49, 26 August 2006 (UTC)

NeXT's software was a BSD Variant
'NeXT's software was originally intended to be used as the foundation for Mac OS X[2], however they opted to base it on a BSD variant instead.'

This is not accurate. NeXTStep was a Mach 2.5-derived kernel with a BSD server which provided most of the UNIX services. During the Rhapsody era, Apple imported a lot of NetBSD code into the BSD component, to replace ageing 4.2BSD parts. With OS X, they imported more FreeBSD code. They also took a load of userland components from NetBSD and FreeBSD, although some (e.g. top) are still the NeXT versions.

If you read the XNU (OS X kernel) source code (or Amit Singh's excellent book on the subject) you will see a large number of source files with copyright notices dating back to 1996 and earlier. Most of these were present in NeXT's operating system and do not originate from any other BSD system.

The new IOKit, based on Embedded C++ replaces the old NeXT driver kit based on Objective-C (for performance reasons), but this is entirely new, and not found in any other OS.

At the kernel level, OS X is an updated OPENSTEP. At the API level (Cocoa) it is an updated OPENSTEP. The window server component was re-written in a large part to take advantage of the features of modern 3D cards, and to address some issues with DPS (e.g. the fact that it is not possible to determine how long a program written in a Turing-complete language will take to execute, making scheduling of drawing difficult).

--David Chisnall 13:52, 1 September 2006 (UTC)

Glass Staircase
So which office had the glass staircase? The first or the second? It is mentioned as being in both.Geneffects 22:10, 11 December 2006 (UTC)

NeXTcube name
This article mistakenly refers to the first-generation NeXT workstation as the "NeXTcube"; this name was only officially used for the second-generation '040-based cube. The original NeXT model was simply called the "NeXT Computer" (or "NeXT Computer System") in NeXT's literature - see the brochure. Also, looking at the brochure scans, the company seems to have been called "NeXT, Inc." at the time of the original launch, and later changed to NeXT Computer, Inc. Letdorf 10:47, 10 April 2007 (UTC).

Continuity issues
This article, says Steve Jobs resigned from Apple Computer. Yet the Steve Jobs article says he was fired ClEeFy

 * Jobs was marginalized from any active role in the company and subsequently resigned. In upper management, this is one way executives get fired. Gwen Gale (talk) 07:49, 29 November 2007 (UTC)

Net income, gross revenues not present in the right side box
can some one please place these values as found in all the other company portals. thank you. —Preceding unsigned comment added by Nikhilhuilgol (talk • contribs) 07:02, 20 September 2007 (UTC)

When did the Intel port of NeXTSTEP start?
Though a minor point, I have some reason to think (a private e-mail message by Jobs) that it started before 1992. Is this 1992 date based on good sources? If not, then from the message I got in Sep 1991 it seems they were already at work then. —Preceding unsigned comment added by Gctwnl (talk • contribs) 00:13, 2 January 2008 (UTC)

The port began in October 1991. A demo version was displayed at the NeXTWorld Expo in January 1992. Subsequent to that, a full product version was developed and completed by May 1993. Tystnaden (talk) 11:58, 25 May 2013 (UTC)

The missing citation for statement "The NeXT Computer was slower than many Unix workstations becoming available at that time, but cost about half as much"
I think that the New York Times article "The Executive Computer; Fresh Momentum for Unix, but Still Hurdles to Clear" is acceptable a  source, but I have chosen not to add it because do not know the proper way to do so. --AdamTheWebMan (talk) 23:54, 22 January 2008 (UTC)
 * As I now read the article, it doesn't say so. It says the NeXT computers were of comparable performance to other UNIX workstations, but cost about half as much. › mys id (☎∆) 19:12, 18 February 2008 (UTC)
 * It actually says that a Sun color IPC was $8,995 and a color Nextstation was $7,995.--NapoliRoma (talk) 19:20, 18 February 2008 (UTC)

removed "NeXT To Return in 2009?"
Yoinked the following for myriad reasons:

Since the early period after the millennium rumors have been circulating in the back rooms of 1 Infinite loop Cupertino, California that NeXT is to make a comeback. Some employes have reportedly seen large black MacPro cases with a covered up logo in place of the usual Apple Logo, being hurriedly transported in and out of rooms. In February 2007 an employe was added to the apple payroll under the heading "New and Continuing Product Division", the name of this employee is Maxx Kai-Morten who previously worked for the now defunct Quicksilver (UK) software. From the very limited information that has been leaked, the new machine could be based upon the current MacPro and the OS would be based around Mac OS 10.5. As to be expected with NeXT the computer would have an insane spec and be very cutting edge.

NeXT was also rumored to make a comeback around the 2003 area but these plans never came to fruition as apple was concentrating all its energy on the iPod. From the time that the product development started the NeXT computer should be due for release in early 2009, although this is yet to be confirmed by Apple from the time that the project started and the amount of time that has been spent on the project the product is now more than overdue.

Diskless workstations

 * At the time of release in 1989, some workstations were not shipped with a hard drive, such as the DECstation 3100 and SPARCstation 1. Floppy disks were used to load the OS and additional software, meaning users needed to swap between floppy disks to load an ever-growing number of applications.

This is nonsense. Diskless workstations did exist, but they used a fileserver for their mass storage (and booting), not floppy disks. I can't think of any UNIX workstations contemporaneous with the NeXT Computer which used floppies as their primary mass storage medium - that would have been impractical. Letdorf (talk) 11:41, 16 April 2008 (UTC).
 * I reworded it slightly. — Wackymacs (talk) 12:28, 16 April 2008 (UTC)

Merge NeXT(UK) and NeXT articles
Does anyone think that the NeXT (UK) and the NeXT articles should be merged? (86.134.36.96 (talk) 19:06, 3 May 2008 (UTC))
 * No. Because none of it is referenced, and it all seems dubious. (And that NeXT (UK) article is up for deletion). — Wackymacs (talk) 19:57, 3 May 2008 (UTC)

Some notes and ideas after copy-editing
At the request of Wackymacs, I've done some copy editing on the article. A lot of this was just simple grammar and formatting corrections which I haven't listed here. Three persistent errors to watch out for:

 * misuse of italics (used for names of books, films, albums but not for software and hardware or company names)
 * companies are singular, not plural ("Apple used the software for its site", not "Apple used the software for their site")
 * in the context of an article about a US company and only quoting US dollars, the US before figures isn't needed (or is needed at most on the first example), not at random after that.

Like many a Wikipedia article, this degrades a little over the course of the text. Here's a list of broader problems which I think need to be addressed in the article but will leave other editors to consider implementing.

 * The successor element in the former company infobox seems redundant.
 * Most of the inline references in the lead seem unnecessary and arbitrary to me, as they don't refer to controversial facts and they're all sourced in the body (I haven't followed any previous discussion on this topic though.
 * I don't think the OpenStep 5.0 reference is needed in the lead (and I'm not sure it's supported by the cited text); also, this point isn't made anywhere in the body text, where it would be more appropriate (if it's justified).
 * The section headings need work. 'Early years' is uninformative and could perhaps be replaced with 'Founding Next'. The subsequent sections are all quite long and could do with being divided up (certainly the hardware section could be split between the two main generations). The Apple section is mostly about the development of OS X, not the acquisition, so should also be renamed. I don't think the ", Inc." is needed in the subheadings either, just at the first reference to that version of the company name in the text.
 * The early history was a bit garbled, so I reordered it to make it more chronological.
 * The comment about late release of Macs needs a more precise time frame.
 * Most of the direct quotations in the article seem gratuitous to me -- included because they sound pithy rather than because they advance our understanding of the topic. The "five years ahead" quote is arguably well-known enough to merit inclusion, but needs to be set in context (what release date was missed?) The "bue jeans" quote adds nothing. The Gates and McNealy material could be presented more succinctly, and needs a topic sentence explaining the context (presumably reactions to NeXT from other industry players).
 * I think a brief explanation of what a wet lab simulation is (not just a Wikilink) would help general reader comprehension. Ditto for a Mach-based system.
 * While the article goes to some lengths to contrast the specifications for the NeXT Computer and its competitors, it doesn't say anything about an equally important factor: the price. Material on this is needed (there's discussion of the 'dorm room ambition' of an affordable machine but the high price is rather ignored subsequently).
 * "The drive was not sufficient to run as the primary medium" needs clarification, Not big enough? Not fast enough?
 * The material about NeXTWorld magazine and the expo doesn't sit well where it's currently located. Might work as a separate section on 'The NeXT Community' after the corporate culture section. More specific dates on the tenure of the editors would be good. Were the four volumes four individual magazines, or four years' worth of publication? (Volume is an odd term to use in a non-academic context.)
 * The processor port history was uncomfortably tangled with the material about NeXT ceasing systems manufacture. I've moved some stuff around here, but it probably needs further work.
 * Did the Sun investment go through?
 * The discussion of the development of Mac OS X is much too detailed in the context of an article about NeXT. The key points seem to me to be that the tools were ported to Mac and Intel platforms, that many key features were retained in OS X, and that platform independence remained a primary goal; the detail on Carbon, Blue Box, Yellow Box etc. is excessive and liable to confuse the reader. Definitely could use some trimming. Also, the details of what got included don't need repeating in the impact section, I'd argue.
 * The corporate culture section would benefit from a discussion of whether any of these innovations survived at Apple or influenced other companies, assuming that material can be sourced.
 * The section on Macs in the impact discussion needs some more context: if 10 million users had shifted, how many hadn't? Do we have more recent figures for OS X sales? What proportion of the overall computer market does this represent?
 * The list of sites that use WebObjects doesn't need repeating.
 * The stuff that's not included in Mac OS X (Display PostScript and tear-off menus) hasn't ever been mentioned before; it should be dropped or explained more fully earlier in the article. (I'd personally just dump it.

Hope this has been helpful, happy to clarify any comments here or explain other changes. Gusworld (talk) 01:26, 2 May 2008 (UTC)

Proposal to remove date-autoformatting
Dear fellow contributors

MOSNUM no longer encourages date autoformatting, having evolved over the past year or so from the mandatory to the optional after much discussion there and elsewhere of the disadvantages of the system. Related to this, MOSNUM prescribes rules for the raw formatting, irrespective of whether or not dates are autoformatted. MOSLINK and CONTEXT are consistent with this.

There are at least six disadvantages in using date-autoformatting, which I've capped here:

Removal has generally been met with positive responses by editors. Does anyone object if I remove it from the main text (using a script) in a few days’ time on a trial basis? The original input formatting would be seen by all WPians, not just the huge number of visitors; it would be plain, unobtrusive text, which would give greater prominence to the high-value links. Tony  (talk)  06:42, 24 July 2008 (UTC)

 * Sound fine to me, but I'm curious about whether you're planning on posing this question on all talk pages...--NapoliRoma (talk) 14:59, 24 July 2008 (UTC)
 * Heavens no: shared task going through some FAs. We hope that letting key editors know about the option may prompt a broader shift. Thanks. Tony   (talk)  08:35, 10 August 2008 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was not moved. While WP:MOSTM recommends using standard English capitalization rules regardless of the trademark owner's preference, there seems to be consensus that the current name is the most recognizable one for English speakers, and as such it meets the naming conventions. Jafeluv (talk) 11:48, 20 September 2009 (UTC)

NeXT → ? — I am not familiar enough to know what it should be moved to, but the current name of the article violates WP:MOSTM, so the article needs to be moved (to something that has the "Next" capitalization).  TJ   Spyke   22:23, 16 September 2009 (UTC)

Survey

 * Feel free to state your position on the renaming proposal by beginning a new line in this section with  or  , then sign your comment with  . Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's naming conventions.

 * Support — per WP:MOSTM & suggest NeXT → Next (computer company) 「 ɠu¹ɖяy 」 ¤ • ¢ 22:45, 16 September 2009 (UTC)
 * Oppose: notwithstanding WP:MOSTM, the distinctive presentation of the NeXT company name and trademark is well-known and generally accepted in published works other than WP. IMHO, an encyclopedia should, within reason, reflect reality as closely as possible and not distort it to fit its own norms - this should include typography. Furthermore, no-one appears to have objected to the title of this article when it achieved FA status in October 2008. Letdorf (talk) 23:52, 16 September 2009 (UTC).
 * First, what they are known as doesn't overide MOSTM. Take a look at TIME for example, they are almost always referred to as that, but the article is at "Time (magazine)", not "TIME" or "TIME (magazine)". As for your second point, even FA's are not perfect and can continue to be changed and improved.  TJ   Spyke   00:28, 17 September 2009 (UTC)
 * Oppose per IDONTLIKETHEAPPLICATIONOFMOSTMINTHISCASE. Bongo  matic  02:08, 17 September 2009 (UTC)
 * Oppose - I do not find this a useful or appropriate application of WP:MOSTM. Tim Pierce (talk) 02:12, 17 September 2009 (UTC)
 * Do you have any policies or guidelines to support NeXT? Because as of right now no one has provided any compelling reason for this article to continue violating the naming guidelines. This is like talking to a religious person about evolution, they keep ignoring facts and want to indulge their fantasy world.  TJ   Spyke   02:33, 17 September 2009 (UTC)
 * MOS is a guideline that should be taken seriously. However, the chief examples given are all-cap or all lowercase versus initial cap. Initial lowercase with a single camelcap (iPhone) are explicitly permitted). There are no examples that are on all fours with NeXT, and the closest (eBay, etc.) seem to point in another direction. Bongo  matic  02:43, 17 September 2009 (UTC)
 * The compelling reason is that the community has already reached consensus on "NeXT". Indeed, it appears to me that the Wikipedia community disagrees with WP:MOSTM on this point in nearly every example I can think of: TRUSTe, Banyan VINES, ProDOS, BeOS, microSD and miniSD, MiG-1, etc. etc. etc.  I think this particular guideline is just not a strong justification. Tim Pierce (talk) 03:17, 17 September 2009 (UTC)
 * Don't forget SunOS. --Tothwolf (talk) 15:16, 17 September 2009 (UTC)
 * Oppose per Tim Pierce,since any other selection would be even more confusing, also considering how common "Next" is. 76.66.196.139 (talk) 04:24, 17 September 2009 (UTC)
 * Oppose NeXT and NeXTSTEP are the official and the most recognised form of the name. The product documentation and software that I personally have in my possession have NeXT and NeXTSTEP printed on them. Even the config.guess script for GNU Autoconf checks to see if NeXT (which is case sensitive) is defined in the system header files or is present in the output of the  command (note that the only place config.guess uses lowercase is the configuration name or canonical host name triplet that the config.guess script itself generates for the configure script). The MOS and other guidelines also do not override common sense, see WP:RAP --Tothwolf (talk) 15:13, 17 September 2009 (UTC)
 * Oppose This is ridiculous. I strongly oppose the proposal to move the article to an unknown title. The nominator, who admits he is unfamiliar with the subject, should close this discussion immediately at least until he can come up with a valid alternate name. WP:MOSTM is a guideline not a policy. AlistairMcMillan (talk) 18:00, 17 September 2009 (UTC)
 * ... which is not in and of itself a reason to side-step it. 81.110.104.91 (talk) 20:05, 17 September 2009 (UTC)
 * Oppose, for reasons that have everything to do with the existing usage, and nothing whatsoever to do with the MOS being a guideline or the wisdom (or otherwise) of the nominator. The exception is well-justified.  81.110.104.91 (talk) 20:05, 17 September 2009 (UTC)

Discussion

 * Any additional comments:

 * NOTE there is a related move request for NeXTSTEP to Nextstep, see Talk:NeXTSTEP 76.66.196.139 (talk) 12:04, 17 September 2009 (UTC)
 * There has been much debate (generating more heat than light, it has to be said) on the WP:MOSTM guideline on its talk page. However, this particular comment seemed quite sensible to me:
 * "MOSTM must be one of the most widely and sensibly ignored guides on Wikipedia, and the timesink caused by people coming to articles to try to enforce it in good faith is an excellent case for junking it. Let editors decide this case by case and do not direct people to follow one way because of a particular guideline or policy which is not widely supported, and has a ridiculously convoluted set of 'legitimate' exceptions symptomatic of a problem."
 * Letdorf (talk) 13:36, 17 September 2009 (UTC).

I am not familiar with the subject either, but could someone say how many of these news hits are about this subject? In particular, how many of the standardized ones are about this subject? It seems that, at the very least, CNET News standardizes this trademark. Croctotheface (talk) 20:28, 17 September 2009 (UTC)

 * It doesn't appear to me that they ever standardized at all: e.g. http://news.cnet.com/Its-Intel-inside-for-Apples-Mac/2009-1006_3-5733937.html uses "NeXT" but http://news.cnet.com/Apple-acquires-Next,-Jobs/2100-1001_3-256914.html uses "Next". Similarly for CNN: "NeXT" at http://money.cnn.com/galleries/2008/fortune/0806/gallery.gates_v_jobs.fortune/7.html and "Next" at http://money.cnn.com/magazines/fortune/fortune_archive/1990/10/08/74184/index.htm.  If anything, it seems as if more recent articles are more likely to use "NeXT" and older articles are more likely to use "Next". Tim Pierce (talk) 00:49, 18 September 2009 (UTC)
 * Interesting. If our sources don't really use a style, it's hard to really argue that it's standard.  If "Next" basically doesn't exist, or only exists in a trivial way, while "NeXT" is used more or less exclusively, then it's hard to see how MOSTM even supports a move.  Croctotheface (talk) 01:10, 18 September 2009 (UTC)

Someone referred the discussion to WP:Naming conventions for additional comments about trademarks. That policy has this to say about case-specific trademarks: "if the name is ambiguous, and one meaning is usually capitalized, this is one possible method of disambiguation." NeXT seems to be an excellent example of just such a term. Tim Pierce (talk) 21:19, 17 September 2009 (UTC)

I'm curious what happens when someone suggests a page move but doesn't suggest a destination name. If consensus resulted in "Support" for the move do we then have a second debate to then try to establish a consensus on an accepted destination name? This debate is ridiculous. AlistairMcMillan (talk) 21:46, 17 September 2009 (UTC)

 * Well, first off, there isn't going to be a move consensus, so it doesn't much matter in this case. I'd imagine that if there were a consensus to use something other than "NeXT," the discussion would be closed and editors would be advised to figure out where to move it and then execute the move.  Croctotheface (talk) 21:59, 17 September 2009 (UTC)

 * Close this already. The consensus is clear, whether or not some editors like it. The community frequently rejects guidelines—that one way they are changed. Bongo  matic  01:16, 18 September 2009 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Spoken Wikipedia recording
I've updated the Spoken Wikipedia recording of this article. Please let me know if I've made any mistakes. Thanks. --Mangst (talk) 15:06, 29 September 2010 (UTC)

Bias
One issue I noticed in reading the page is that it does come across as being a bit biased. I realize NeXT may have its fans out there, but unless something has changed with Wikipedia's standards and views on this, it should be presented in as neutral and unbiased a way and viewpoint as possible. For example, I see references that it was supposedly very influential and yet no mention made of how exactly it impacted or influenced computing and when it does attempt to mention object-oriented programming (which existed long before NeXT came into being, from what I understand) and GUIs (which also existed with other systems out there prior to 1988), it seems a little confusing as to what it's trying to say. Did it cause object-oriented programming to become popular? The use of GUIs? Or some combination of both (possibly using object-oriented programming in GUIs)? Also, if any of that is the case, just how and what did it influence? Also where are the references or citations for any of that? Or did it just influence off-shoots of its own product? In which case, I don't know if I might call that a major influence or even as profound upon the computer industry as the article seems to imply, in the sense that it refers to something more within the company as opposed to the industry as a whole. I only cite that as one example of the bias being shown in this article. I think it might be more effective--and especially informative--if there was more neutral language used. As it is now, it almost sounds like it was written by a fan.Tuxenstein (talk) 23:07, 8 May 2011 (UTC)

 * I agree some things are missing or need improvements, I also was surprised to see no reference BSD and Xerox (there was reference to Mach, though)... 76.10.128.192 (talk) 19:59, 11 June 2012 (UTC)
 * NeXT was a flop as a seller of hardware boxes, but somewhat of an inspiration to many as a more or less comprehensive non-toy graphic/visual windowing Smalltalk-like operating system with integrated development tools. The attempted direct imitations of NeXT didn't work out too well commercially, but NeXT apparently still had a diffuse influence in a number of smaller ways on various operating system efforts during the late 1980s / early 1990s (though it certainly didn't single-handedly popularize the idea of GUIs)... AnonMoos (talk) 23:17, 11 June 2012 (UTC)

Request edit on 11 October 2013
Addition of Books or Further Reading Section and inclusion of specific listing.

The book is "Anatomy of an Apple - The lessons Steve taught us." Author is Ben Klaiber. ISBN is 9781483506975.

While the majority of the book is about the turnaround of Apple with Steve Jobs' return, multiple chapters delve deeply into NeXT. It covers a tremendous amount of history and details of the NeXT period as well as often neglected elements of the company.

Examples: Covers in depth the legal limitations imposed by Apple on NeXT, which contributed heavily to their dearth of sales. Covers specific pricing/performance of models and comparisons with competition at the time. Explains and analyzes the competitive positioning of the hardware-software combination, the software alone, the software as extension of other competing OS's, the NeXT OS as open sourced product.

Anatomy of an Apple also extensively discusses anticompetitive tactics used by Microsoft at the time and how they affected NeXT. The discussion explores and details how these competitive experiences molded the strategies that Apple would use upon Steve Jobs' return there.

Excerpts related to the NeXT period are posted online at http://www.anatomyofanapple.com/2013/09/LetstalkaboutNeXT.html. Bryan Roberts7982 (talk) 15:38, 11 October 2013 (UTC) 15:38, 11 October 2013 (UTC)Bryan Roberts7982 (talk)

(The above requested edit was made by clicking on a link in an automatically added notice.)
 * ❌. Sorry, but there are dozens of books about Steve Jobs and NeXT - we don't have space to include them all, and including only some of them would be playing favourites. — Mr. Stradivarius  ♪ talk ♪ 01:35, 2 November 2013 (UTC)

id software and NeXT
Also Wolfenstein 3D was developed with a NeXT computer(from Masters of Doom).--141.24.16.97 (talk) 13:28, 25 June 2014 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on NeXT. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20120114182612/http://www.kevra.org/TheBestOfNext/ThirdPartyProducts/ThirdPartyHardware/NeXTSTEPonNonNeXTComputers/IntelHardwareForNeXTStep/Canon-object-station41/Canon-object-station41/Canon-object-station41.html to http://www.kevra.org/TheBestOfNext/ThirdPartyProducts/ThirdPartyHardware/NeXTSTEPonNonNeXTComputers/IntelHardwareForNeXTStep/Canon-object-station41/Canon-object-station41/Canon-object-station41.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 15:56, 21 September 2017 (UTC)

Featured article review
Another late 2000s promotion that, for some reason, was "Today's featured article" as late as August 2020 (last year!). I don't see the hype. Excuse me, I was thinking of a different article. This article has many blatant problems that need to be fixed. These are just examples 👨x🐱 (talk) 12:55, 29 March 2021 (UTC)
 * Lead has multiple issues
 * There are specific details about Perot's involvement that are only in the lead when they should be in the body. Nothing in the body talked about how the PBS documentary inspired him to invest into it, nor his $20 million investment amount.
 * "Nevertheless, their innovative object-oriented NeXTSTEP operating system and development environment (Interface Builder) were highly influential." How?
 * Body and lead introduce OpenStep very differently, plus has computer jargon not link or explained to the casual reader. What is a "programming environment standard"? What is an "application layer"
 * Body "NeXT partnered with Sun to create OpenStep which is NeXTSTEP's application layer hosted on a third party operating system."
 * Lead "NeXT later released much of the NeXTSTEP system as a programming environment standard called OpenStep."
 * History section have multiple uncited sentences, in a few instances even a paragraph-chunk of uncited sentences.
 * There's a paragraph in "Legacy" about Cairo that probably needs more citations. Plus, "By 1994, Microsoft and NeXT were collaborating on a Windows NT port of OpenStep which was never released.[69]" Why is this briefly only to never be talked about again, and why isn't this in the History section?
 * There is no explanation or suggestion on how the WebObjects paragraph connects to the legacy of NeXT.

The term "Defunct" should be changed to "Acquired"
Quite possibly the most egregious bias I've ever seen on Wikipedia, and trust me this site is polluted with bias --

To label NeXT Computer as a "Defunct" company in the company info summary section is like calling the Sun "lazy". NeXT Computer was a company that was "Acquired" by Apple, and the technology that NeXT developed, which Apple acquired, can largely be credited for Apple's subsequent success. So in fact, the "near defunct" company Apple "acquired" the non-defunct company NeXT, and Apple saved itself in the process.

I dare not edit the page and change "Defunct" to "Acquired", because certainly some self-important "Editor" will come along and change it right back 30 seconds later.

For this one, as for all biased entries on this polluted site, it's going to take a quorum of reasonable people to come together and make sure the switch from "Defunct" to "Acquired" permanently sticks. I hope the community makes the proper choice. 174.247.193.23 (talk) 13:41, 23 December 2021 (UTC)

 * What are you on about? The article is completely clear that the company was acquired by Apple. VQuakr (talk) 19:29, 23 December 2021 (UTC)
 * Anyway, the HARDWARE technology is 100% defunct, and so is part of the software technology (including Display PostScript, mysteriously not mentioned on this article). It's only parts of the software that continued onwards... AnonMoos (talk) 14:46, 24 December 2021 (UTC)

Good article status
I would oppose this being declared a good article as long as there's no mention of Display PostScript... AnonMoos (talk) 01:41, 2 January 2022 (UTC)

 * @AnonMoos Added. Wingwatchers (talk) 02:24, 4 January 2022 (UTC)
 * Thanks, objection withdrawn. AnonMoos (talk) 08:16, 6 January 2022 (UTC)

Apple-NeXT acquisition price
Wikipedia is all over the place when describing the acquisition deal terms. Corporate deals are often structured with cash and equity making up the total deal value. This article is bafflingly wrong when it states the whole $429m was cash; the equity (stock) was included in that figure. Jobs also did receive cash, unlike what was explicitly stated here. There were 2 citations for the sentence, but neither backed any of this up.

Furthermore, the deal's precise value, listed on this page and others (Apple, Steve Jobs among others) has shifted over the years ($427, $429, $420), with no proper sourcing, and has likely turned into feedback loop where lazy journalists copy Wikipedia's figure, which then gets sourced to these journalists. We must therefore look exclusively at contemporaneous reports.

DFlhb (talk) 00:51, 26 October 2022 (UTC) added links, looked at more financial statements, removed conclusion; see new conclusion below
 * Primary sources:
 * Apple's press release says $400m
 * NeXT's press release says $400m.
 * Gil Amelio, in his 1998 book: $377.5m in cash, plus one and a half million shares of stock, which were worth, according to Amelio, $37m, leading to a total of $415m.
 * Source:
 * Apple's form 8-K ("unscheduled material events") financial statement, filed 24/12/1996, reprints the press release on page 5, but gives an additional breakdown on page 2: $400 million ($350 million plus the assumption of approximately $50 million of debt)
 * Apple's form 10-Q quarterly report for the following quarter says the comprehensive purchase price [...] is expected to be approximately $430 million (page 7)
 * Apple's form 10-Q quarterly report filed August 1997 says total purchase price was $425 million, as adjusted, [...] The purchase price, including the fair value of the net tangible liabilities assumed, was $427 million, as adjusted,, page 6 (net tangible liabilities = debt; that's 2mil, but most of NeXT's debt was paid off with cash, not taken on)
 * Apple's form 10-K annual report (the 10-K is the "main" financial report) filed December 1997, says: the total purchase price [...] was $427 million, including cash, debt, shares and options; doesn't mention $425m (pages 9 and 44)
 * Apple's 1998 annual 10-K repeated 427 million, doesn't mention $425m (page 14)
 * Secondary sources:
 * CNET: $400m
 * SFGATE, published the next year, rounds the financial statements to $430m.
 * Les Echos, $424m.
 * Fortune Mag, $400.

 * Update: WSJ says "about $430 million" (dated May 1997) DFlhb (talk) 20:06, 5 February 2023 (UTC)
 * Update 2: I've added links so others can verify; and looked at a few more financial statements. The final number in financial statements is $427m. I don't think we need the "$400m" approximation anymore. Boldly changed article to $427m. DFlhb (talk) 08:44, 5 June 2023 (UTC)
 * Other sources that say $427m: Linzmayer's great 2004 book (Apple Confidential 2.0), and Kahney (ex-Wired) in his 2008 book (Inside Steve's Brain). DFlhb (talk) 21:39, 2 December 2023 (UTC)

title: Talk:NeXT


text:
Hamilton County, New York

Hamilton County is a county in the U.S. state of New York. As of the 2020 census, the population was 5,107, making it the least populous county of New York, and the only county with less than 10,000 residents. With a land area nearly the size of Delaware, it is the least densely populated county east of the Mississippi River. Its county seat is Lake Pleasant. The county was created in 1816 and organized in 1847. The county is part of the North Country region of the state.

Hamilton County is one of only two counties that lie entirely within the Adirondack Park (Essex being the other). Because of its location in the park, any development in the county is restricted by the New York State Constitution, which designates the park as "forever wild." There are no permanent traffic lights in the county, and much of the county has no cell phone service.

Etymology
The county is named after Alexander Hamilton, the only member of the New York State delegation who signed the United States Constitution in 1787, and was later the first United States Secretary of the Treasury.

History
On April 12, 1816, Hamilton County was created by partitioning 1800 sqmi from Montgomery County, but due to low population it remained unorganized and administered from Montgomery County, N.Y. until it was recognized as sufficiently prepared for self-government on January 1, 1838. The organization process was completed by summer 1847.

On April 6, 1860, Fulton County, which had been partitioned April 18, 1838, had 10 sqmi of land in Sacandaga Park transferred to Hamilton County. On May 24, 1915, land was swapped between Hamilton and Essex counties, with Hamilton ceding Fishing Brook Mountain for Indian Lake. Hamilton gained an additional 20 sqmi, whereas Essex County lost 30 sqmi. This left Hamilton with its present size of 1830 sqmi.

The former town of Gilman was dissolved in 1860. The original county seat was Sageville, now part of Lake Pleasant.

Geography

According to the U.S. Census Bureau, the county has a total area of 1808 sqmi, of which 1717 sqmi is land and 90 sqmi (5.0%) is water. It is New York's third-largest county by land area and fifth-largest by total area.

Hamilton County is in the state's north central section, northwest of Albany. It lies entirely within Adirondack Park and consists mostly of publicly owned parkland.

The county is very mountainous, broken up by rivers and lakes. The county is famous for its lakes. Indian Lake, and Long Lake are both famous for their long length, small width, and erratic shape, similar to the Finger Lakes. Other notable lakes include Piseco Lake, Lewey Lake, Raquette Lake, Sacandaga Lake, and Lake Pleasant. While notable mountains include: Snowy Mountain, Panther Mountain, and Buell Mountain.

Adjacent counties

 * Franklin County - north
 * Essex County - northeast
 * Warren County - east
 * Saratoga County - southeast
 * Fulton County - south
 * Herkimer County - west
 * St. Lawrence County - northwest

Demographics
As of the 2020 census, among the 5,107 residents the racial makeup of the county was 93.69% White, 0.61% Black or African American, 0.27% Native American, 0.23% Asian, 0.02% Pacific Islander, 0.43% from other races, and 4.74% from two or more races. 1.96% of the population were Hispanic or Latino of any race. Additional demographics are based on older data. As of the census of 2000, there were 5,379 people, 2,362 households, and 1,558 families residing in the county. The population density was 3 /mi2; both the total population and population density rank lowest in the state of New York. There were 7,965 housing units at an average density of 5 /mi2. 16.9% were of Irish, 15.7% German, 15.2% English, 10.9% French, 7.3% American and 5.7% Italian ancestry. 97.5% spoke English and 1.7% French as their first language.

There were 2,362 households, out of which 23.60% had children under the age of 18 living with them, 55.70% were married couples living together, 6.70% had a female householder with no husband present, and 34.00% were non-families. 29.60% of all households were made up of individuals, and 13.70% had someone living alone who was 65 years of age or older. The average household size was 2.24 and the average family size was 2.74.

In the county, the population was spread out, with 19.70% under the age of 18, 5.20% from 18 to 24, 24.20% from 25 to 44, 30.90% from 45 to 64, and 20.00% who were 65 years of age or older. The median age was 45 years. For every 100 females there were 100.00 males. For every 100 females age 18 and over, there were 99.40 males.

The median income for a household in the county was $32,287, and the median income for a family was $39,676. Males had a median income of $29,177 versus $21,849 for females. The per capita income for the county was $18,643. About 6.00% of families and 10.40% of the population were below the poverty line, including 10.50% of those under age 18 and 8.70% of those age 65 or over.

Research published in 2016 found that Hamilton County had by far the highest suicide rate in the state of New York; at 28 per 100,000 people. This meant that Hamilton County had a suicide rate more than double the national average and more than triple the state average of 13 and 8.4 per 100,000 people respectively. Though the report found instances of suicide to be significantly higher in Upstate generally, the rate in Hamilton County was disproportionate regardless of region; being 22.1, high though still significantly lower, in Lewis County, which was found to have the second highest rate in New York.

Government and politics

 * }

Hamilton County has long been one of the most consistently Republican counties in New York. Since Woodrow Wilson carried the county in 1916, the Republican candidate has lost only once, when Barry Goldwater in 1964 failed to win a single county in the state. The county was Goldwater's third strongest in the state however. In the 2008 U.S. presidential election, John McCain carried Hamilton County by a 26.9% margin over Barack Obama, with Obama winning statewide by a virtually equal margin over McCain; Republican nominee Mitt Romney won the county over President Obama in the 2012 election, too. Hamilton gave McCain the highest margin of victory in the state.

It was the only county won by Howard Mills over incumbent Chuck Schumer in the 2004 U.S. Senate election. It also voted for John Faso over Eliot Spitzer for governor in 2006, and for John Spencer 55.5%-42.1% over incumbent Hillary Clinton for the U.S. Senate in 2006, despite Faso and Spencer both losing in landslides statewide. It was one of only a handful of counties outside Western New York to have voted for Carl Paladino over eventual winner Andrew Cuomo for Governor in 2010.

Democrat Kirsten Gillibrand, however, won the county in her bid to be elected for a full term to the U.S. Senate in 2012.

Airports
The following public use airports are located in the county:
 * Piseco Airport (K09) – Piseco
 * Long Lake Helms Seaplane Base (NY9) – Long Lake
 * Long Lake Sagamore Seaplane Base (K03) – Long Lake

Ground

 * NY-8.svg NY-8
 * NY-10.svg NY-10
 * NY-28.svg NY-28
 * NY-28N.svg NY-28N
 * NY-30.svg NY-30

Towns

 * Arietta
 * Benson
 * Hope
 * Indian Lake
 * Inlet
 * Lake Pleasant
 * Long Lake
 * Morehouse
 * Wells

Villages

 * Speculator

Other hamlets

 * Arietta
 * Benson
 * Blue Mountain Lake
 * Higgins Bay
 * Hoffmeister
 * Hope
 * Hope Falls
 * Indian Lake
 * Inlet
 * Lake Pleasant (county seat)
 * Piseco
 * Raquette Lake
 * Upper Benson

Former hamlet

 * Sabattis (formerly Long Lake West)

title: Hamilton County, New York


text:
Complement (set theory)

In set theory, the complement of a set $A$, often denoted by $$A^\complement$$ (or $A′$), is the set of elements not in $A$.

When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set $A$, the absolute complement of $A$ is the set of elements in $U$ that are not in $A$.

The relative complement of $U$ with respect to a set $A$, also termed the set difference of $A$ and $B$, written $$B \setminus A,$$ is the set of elements in $B$ that are not in $A$.

Definition
If $B$ is a set, then the absolute complement of $A$ (or simply the complement of $A$) is the set of elements not in $A$ (within a larger set that is implicitly defined). In other words, let $A$ be a set that contains all the elements under study; if there is no need to mention $A$, either because it has been previously specified, or it is obvious and unique, then the absolute complement of $U$ is the relative complement of $U$ in $A$: $$A^\complement = U \setminus A = \{ x \in U : x \notin A \}.$$

The absolute complement of $A$ is usually denoted by $$A^\complement$$. Other notations include $$\overline A, A',$$ $$\complement_U A, \text{ and } \complement A.$$

Examples

 * Assume that the universe is the set of integers. If $U$ is the set of odd numbers, then the complement of $A$ is the set of even numbers. If $A$ is the set of multiples of 3, then the complement of $A$ is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
 * Assume that the universe is the standard 52-card deck. If the set $B$ is the suit of spades, then the complement of $B$ is the union of the suits of clubs, diamonds, and hearts. If the set $A$ is the union of the suits of clubs and diamonds, then the complement of $A$ is the union of the suits of hearts and spades.
 * When the universe is the universe of sets described in formalized set theory, the absolute complement of a set is generally not itself a set, but rather a proper class. For more info, see universal set.

Properties
Let $B$ and $B$ be two sets in a universe $A$. The following identities capture important properties of absolute complements:

De Morgan's laws:
 * $$\left(A \cup B \right)^\complement = A^\complement \cap B^\complement.$$
 * $$\left(A \cap B \right)^\complement = A^\complement \cup B^\complement.$$

Complement laws:
 * $$A \cup A^\complement = U.$$
 * $$A \cap A^\complement = \empty .$$
 * $$\empty^\complement = U.$$
 * $$ U^\complement = \empty.$$
 * $$\text{If }A\subseteq B\text{, then }B^\complement \subseteq A^\complement.$$
 * (this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:
 * $$\left(A^\complement\right)^\complement = A.$$

Relationships between relative and absolute complements:
 * $$A \setminus B = A \cap B^\complement.$$
 * $$(A \setminus B)^\complement = A^\complement \cup B = A^\complement \cup (B \cap A).$$

Relationship with a set difference:
 * $$ A^\complement \setminus B^\complement = B \setminus A. $$

The first two complement laws above show that if $A$ is a non-empty, proper subset of $U$, then ${A, A^{∁}}|undefined$ is a partition of $U$.

Definition
If $A$ and $B$ are sets, then the relative complement of $A$ in $B$, also termed the set difference of $B$ and $A$, is the set of elements in $B$ but not in $A$. The relative complement of $A$ in $B$ is denoted $$B \setminus A$$ according to the ISO 31-11 standard. It is sometimes written $$B - A,$$ but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements $$b - a,$$ where $A$ is taken from $B$ and $b$ from $B$.

Formally: $$B \setminus A = \{ x\in B : x \notin A \}.$$

Examples

 * $$\{ 1, 2, 3\} \setminus \{ 2,3,4\} = \{ 1 \}.$$
 * $$\{ 2, 3, 4 \} \setminus \{ 1,2,3 \} = \{ 4 \} .$$
 * If $$\mathbb{R}$$ is the set of real numbers and $$\mathbb{Q}$$ is the set of rational numbers, then $$\mathbb{R}\setminus\mathbb{Q}$$ is the set of irrational numbers.

Properties
Let $a$, $A$, and $A$ be three sets. The following identities capture notable properties of relative complements:

 * $$C \setminus (A \cap B) = (C \setminus A) \cup (C \setminus B).$$
 * $$C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B).$$
 * $$C \setminus (B \setminus A) = (C \cap A) \cup (C \setminus B),$$
 * with the important special case $$C \setminus (C \setminus A) = (C \cap A)$$ demonstrating that intersection can be expressed using only the relative complement operation.
 * $$(B \setminus A) \cap C = (B \cap C) \setminus A = B \cap (C \setminus A).$$
 * $$(B \setminus A) \cup C = (B \cup C) \setminus (A \setminus C).$$
 * $$A \setminus A = \emptyset.$$
 * $$\empty \setminus A = \empty.$$
 * $$A \setminus \empty = A.$$
 * $$A \setminus U = \empty.$$
 * If $$A\subset B$$, then $$C\setminus A\supset C\setminus B$$.
 * $$A \supseteq B \setminus C$$ is equivalent to $$C \supseteq B \setminus A$$.

Complementary relation
A binary relation $$R$$ is defined as a subset of a product of sets $$X \times Y.$$ The complementary relation $$\bar{R}$$ is the set complement of $$R$$ in $$X \times Y.$$ The complement of relation $$R$$ can be written $$\bar{R} \ = \ (X \times Y) \setminus R.$$ Here, $$R$$ is often viewed as a logical matrix with rows representing the elements of $$X,$$ and columns elements of $$Y.$$ The truth of $$aRb$$ corresponds to 1 in row $$a,$$ column $$b.$$ Producing the complementary relation to $$R$$ then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations.

LaTeX notation
In the LaTeX typesetting language, the command is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the  command looks identical to , except that it has a little more space in front and behind the slash, akin to the LaTeX sequence. A variant  is available in the amssymb package, but this symbol is not included separately in Unicode. The symbol $$\complement$$ (as opposed to $$C$$) is produced by. (It corresponds to the Unicode symbol .)

title: Complement (set theory)


text:
Talk:Computer aided drafting

I guess, this is supposed to be Computer aided design (CAD) rather than drafting. See: http://www.acronymfinder.com/af-query.asp?String=exact&Acronym=cad&Find=Find Hmmm.

--Tobias H�vekamp

title: Talk:Computer aided drafting


text:
Rhamnales

The Rhamnales Lindl. are an order of dicotyledon plants in the subclass Rosidae. In the Cronquist system, the following families were placed here:

 * Family Elaeagnaceae – (Oleaster family)
 * Family Leeaceae
 * Family Rhamnaceae (buckthorn family)
 * Family Vitaceae (grape family)

Leea may be included within the Vitaceae. The Rhamnaceae are no longer considered close relatives of these other forms, and newer systems move them to the Rosales. The order then becomes the Vitales.

Under the APG III system of classification, Rhamnales is not recognized. Instead, the families previously included here under the Cronquist system are included under the following orders:
 * Elaeagnaceae and Rhamnaceae are placed within Rosales
 * Leea, formerly recognized in its own family Leeaceae, is included within Vitaceae, which is recognized in its own order, Vitales

title: Rhamnales