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Localization (commutative algebra)

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions $$\frac{m}{s},$$ such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field $$\Q$$ of rational numbers from the ring $$\Z$$ of integers.

The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term localization originated in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions that are not zero at p and localizes R with respect to S. The resulting ring $$S^{-1}R$$ contains information about the behavior of V near p, and excludes information that is not "local", such as the zeros of functions that are outside V (c.f. the example given at local ring).

Localization of a ring
The localization of a commutative ring $R$ by a multiplicatively closed set $S$ is a new ring $$S^{-1}R$$ whose elements are fractions with numerators in $R$ and denominators in $S$.

If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers. For rings that have zero divisors, the construction is similar but requires more care.

Multiplicative set
Localization is commonly done with respect to a multiplicatively closed set $S$ (also called a multiplicative set or a multiplicative system) of elements of a ring $R$, that is a subset of $R$ that is closed under multiplication, and contains $1$.

The requirement that $S$ must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to $S$. The localization by a set $U$ that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of $U$. However, the same localization is obtained by using the multiplicatively closed set $S$ of all products of elements of $U$. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.

For example, the localization by a single element $s$ introduces fractions of the form $$\tfrac a s,$$ but also products of such fractions, such as $$\tfrac {ab} {s^2}.$$ So, the denominators will belong to the multiplicative set $$\{1, s, s^2, s^3,\ldots\}$$ of the powers of $s$. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".

The localization of a ring $R$ by a multiplicative set $S$ is generally denoted $$S^{-1}R,$$ but other notations are commonly used in some special cases: if $$S= \{1, t, t^2,\ldots \}$$ consists of the powers of a single element, $$S^{-1}R$$ is often denoted $$R_t;$$ if $$S=R\setminus \mathfrak p$$ is the complement of a prime ideal $$\mathfrak p$$, then $$S^{-1}R$$ is denoted $$R_\mathfrak p.$$

In the remainder of this article, only localizations by a multiplicative set are considered.

Integral domains
When the ring $R$ is an integral domain and $S$ does not contain $0$, the ring $$S^{-1}R$$ is a subring of the field of fractions of $R$. As such, the localization of a domain is a domain.

More precisely, it is the subring of the field of fractions of $R$, that consists of the fractions $$\tfrac a s$$ such that $$s\in S.$$ This is a subring since the sum $$\tfrac as + \tfrac bt = \tfrac {at+bs}{st},$$ and the product $$\tfrac as \, \tfrac bt = \tfrac {ab}{st}$$ of two elements of $$S^{-1}R$$ are in $$S^{-1}R.$$ This results from the defining property of a multiplicative set, which implies also that $$1=\tfrac 11\in S^{-1}R.$$ In this case, $R$ is a subring of $$S^{-1}R.$$ It is shown below that this is no longer true in general, typically when $S$ contains zero divisors.

For example, the decimal fractions are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, $$S^{-1}R$$ consists of the rational numbers that can be written as $$\tfrac n{10^k},$$ where $n$ is an integer, and $k$ is a nonnegative integer.

General construction
In the general case, a problem arises with zero divisors. Let $S$ be a multiplicative set in a commutative ring $R$. Suppose that $$s\in S,$$ and $$0\ne a\in R$$ is a zero divisor with $$as=0.$$ Then $$\tfrac a1$$ is the image in $$S^{-1}R$$ of $$a\in R,$$ and one has $$\tfrac a1 = \tfrac {as}s = \tfrac 0s = \tfrac 01.$$ Thus some nonzero elements of $R$ must be zero in $$S^{-1}R.$$ The construction that follows is designed for taking this into account.

Given $R$ and $S$ as above, one considers the equivalence relation on $$R\times S$$ that is defined by $$(r_1, s_1) \sim (r_2, s_2)$$ if there exists a $$t\in S$$ such that $$t(s_1r_2-s_2r_1)=0.$$

The localization $$S^{-1}R$$ is defined as the set of the equivalence classes for this relation. The class of $(r, s)$ is denoted as $$\frac rs,$$ $$r/s,$$ or $$s^{-1}r.$$ So, one has $$\tfrac{r_1}{s_1}=\tfrac{r_2}{s_2}$$ if and only if there is a $$t\in S$$ such that $$t(s_1r_2-s_2r_1)=0.$$ The reason for the $$t$$ is to handle cases such as the above $$\tfrac a1 = \tfrac 01,$$ where $$s_1r_2-s_2r_1$$ is nonzero even though the fractions should be regarded as equal.

The localization $$S^{-1}R$$ is a commutative ring with addition
 * $$\frac {r_1}{s_1}+\frac {r_2}{s_2} = \frac{r_1s_2+r_2s_1}{s_1s_2},$$

multiplication
 * $$\frac {r_1}{s_1}\,\frac {r_2}{s_2} = \frac{r_1r_2}{s_1s_2},$$

additive identity $$\tfrac 01,$$ and multiplicative identity $$\tfrac 11.$$

The function
 * $$r\mapsto \frac r1$$

defines a ring homomorphism from $$R$$ into $$S^{-1}R,$$ which is injective if and only if $S$ does not contain any zero divisors.

If $$0\in S,$$ then $$S^{-1}R$$ is the zero ring that has $0$ as unique element.

If $S$ is the set of all regular elements of $R$ (that is the elements that are not zero divisors), $$S^{-1}R$$ is called the total ring of fractions of $R$.

Universal property
The (above defined) ring homomorphism $$j\colon R\to S^{-1}R$$ satisfies a universal property that is described below. This characterizes $$S^{-1}R$$ up to an isomorphism. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be together technical, straightforward and boring.

The universal property satisfied by $$j\colon R\to S^{-1}R$$ is the following:
 * If $$f\colon R\to T$$ is a ring homomorphism that maps every element of $S$ to a unit (invertible element) in $T$, there exists a unique ring homomorphism $$g\colon S^{-1}R\to T$$ such that $$f=g\circ j.$$

Using category theory, this can be expressed by saying that localization is a functor that is left adjoint to a forgetful functor. More precisely, let $$\mathcal C$$ and $$\mathcal D$$ be the categories whose objects are pairs of a commutative ring and a submonoid of, respectively, the multiplicative monoid or the group of units of the ring. The morphisms of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let $$\mathcal F\colon \mathcal D \to \mathcal C$$ be the forgetful functor that forgets that the elements of the second element of the pair are invertible.

Then the factorization $$f=g\circ j$$ of the universal property defines a bijection
 * $$\hom_\mathcal C((R,S), \mathcal F(T,U))\to \hom_\mathcal D ((S^{-1}R, j(S)), (T,U)).$$

This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor.

Examples

 * If $$R=\Z$$ is the ring of integers, and $$S=\Z\setminus \{0\},$$ then $$S^{-1}R$$ is the field $$\Q$$ of the rational numbers.
 * If $R$ is an integral domain, and $$S=R\setminus \{0\},$$ then $$S^{-1}R$$ is the field of fractions of $R$. The preceding example is a special case of this one.
 * If $R$ is a commutative ring, and if $S$ is the subset of its elements that are not zero divisors, then $$S^{-1}R$$ is the total ring of fractions of $R$. In this case, $S$ is the largest multiplicative set such that the homomorphism $$R\to S^{-1}R$$ is injective. The preceding example is a special case of this one.
 * If $x$ is an element of a commutative ring $R$ and $$S=\{1, x, x^2, \ldots\},$$ then $$S^{-1}R$$ can be identified (is canonically isomorphic to) $$R[x^{-1}]=R[s]/(xs-1).$$ (The proof consists of showing that this ring satisfies the above universal property.) This sort of localization plays a fundamental role in the definition of an affine scheme.
 * If $$\mathfrak p$$ is a prime ideal of a commutative ring $R$, the set complement $$S=R\setminus \mathfrak p$$ of $$\mathfrak p$$ in $R$ is a multiplicative set (by the definition of a prime ideal). The ring $$S^{-1}R$$ is a local ring that is generally denoted $$R_\mathfrak p,$$ and called the local ring of $R$ at $$\mathfrak p.$$ This sort of localization is fundamental in commutative algebra, because many properties of a commutative ring can be read on its local rings. Such a property is often called a local property. For example, a ring is regular if and only if all its local rings are regular.

Ring properties
Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning ideals, modules, or several multiplicative sets are considered in other sections.

 * $$S^{-1}R = 0$$ if and only if $S$ contains $0$.
 * The ring homomorphism $$R\to S^{-1}R$$ is injective if and only if $S$ does not contain any zero divisors.
 * The ring homomorphism $$R\to S^{-1}R$$ is an epimorphism in the category of rings, that is not surjective in general.
 * The ring $$S^{-1}R$$ is a flat $R$-module (see for details).
 * If $$S=R\setminus \mathfrak p$$ is the complement of a prime ideal $$\mathfrak p$$, then $$S^{-1} R,$$ denoted $$R_\mathfrak p,$$ is a local ring; that is, it has only one maximal ideal.

Properties to be moved in another section
 * Localization commutes with formations of finite sums, products, intersections and radicals; e.g., if $$\sqrt{I}$$ denote the radical of an ideal I in R, then
 * $$\sqrt{I} \cdot S^{-1}R = \sqrt{I \cdot S^{-1}R}\,.$$
 * In particular, R is reduced if and only if its total ring of fractions is reduced.

 * Let R be an integral domain with the field of fractions K. Then its localization $$R_\mathfrak{p}$$ at a prime ideal $$\mathfrak{p}$$ can be viewed as a subring of K. Moreover,
 * $$R = \bigcap_\mathfrak{p} R_\mathfrak{p} = \bigcap_\mathfrak{m} R_\mathfrak{m}$$
 * where the first intersection is over all prime ideals and the second over the maximal ideals.

 * There is a bijection between the set of prime ideals of S−1R and the set of prime ideals of R that do not intersect S. This bijection is induced by the given homomorphism R → S−1R.

Saturation of a multiplicative set
Let $$S \subseteq R$$ be a multiplicative set. The saturation $$\hat{S}$$ of $$S$$ is the set
 * $$\hat{S} = \{ r \in R \colon \exists s \in R, rs \in S \}.$$

The multiplicative set $S$ is saturated if it equals its saturation, that is, if $$\hat{S}=S$$, or equivalently, if $$rs \in S$$ implies that $r$ and $s$ are in $S$.

If $S$ is not saturated, and $$rs \in S,$$ then $$\frac s{rs}$$ is a multiplicative inverse of the image of $r$ in $$S^{-1}R.$$ So, the images of the elements of $$\hat S$$ are all invertible in $$S^{-1}R,$$ and the universal property implies that $$S^{-1}R$$ and $$\hat {S}{}^{-1}R$$ are canonically isomorphic, that is, there is a unique isomorphism between them that fixes the images of the elements of $R$.

If $S$ and $T$ are two multiplicative sets, then $$S^{-1}R$$ and $$T^{-1}R$$ are isomorphic if and only if they have the same saturation, or, equivalently, if $s$ belongs to one of the multiplicative sets, then there exists $$t\in R$$ such that $st$ belongs to the other.

Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know all units of the ring.

Terminology explained by the context
The term localization originates in the general trend of modern mathematics to study geometrical and topological objects locally, that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of manifolds, germs and sheafs. In algebraic geometry, an affine algebraic set can be identified with a quotient ring of a polynomial ring in such a way that the points of the algebraic set correspond to the maximal ideals of the ring (this is Hilbert's Nullstellensatz). This correspondence has been generalized for making the set of the prime ideals of a commutative ring a topological space equipped with the Zariski topology; this topological space is called the spectrum of the ring.

In this context, a localization by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as points) that do not intersect the multiplicative set.

Two classes of localizations are more commonly considered:
 * The multiplicative set is the complement of a prime ideal $$\mathfrak p$$ of a ring $R$. In this case, one speaks of the "localization at $$\mathfrak p$$", or "localization at a point". The resulting ring, denoted $$R_\mathfrak p$$ is a local ring, and is the algebraic analog of a ring of germs.
 * The multiplicative set consists of all powers of an element $t$ of a ring $R$. The resulting ring is commonly denoted $$R_t,$$ and its spectrum is the Zariski open set of the prime ideals that do not contain $t$. Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a neighborhood basis consisting of Zariski open sets of this form).

In number theory and algebraic topology, when working over the ring $$\Z$$ of integers, one refers to a property relative to an integer $n$ as a property true at $n$ or away from $n$, depending on the localization that is considered. "Away from $n$" means that the property is considered after localization by the powers of $n$, and, if $p$ is a prime number, "at $p$" means that the property is considered after localization at the prime ideal $$p\Z$$. This terminology can be explained by the fact that, if $p$ is prime, the nonzero prime ideals of the localization of $$\Z$$ are either the singleton set $\{p\}$ or its complement in the set of prime numbers.

Localization and saturation of ideals
Let $S$ be a multiplicative set in a commutative ring $R$, and $$j\colon R\to S^{-1}R$$ be the canonical ring homomorphism. Given an ideal $I$ in $R$, let $$S^{-1}I$$ the set of the fractions in $$S^{-1}R$$ whose numerator is in $I$. This is an ideal of $$S^{-1}R,$$ which is generated by $j(I)$, and called the localization of $I$ by $S$.

The saturation of $I$ by $S$ is $$j^{-1}(S^{-1}I);$$ it is an ideal of $R$, which can also defined as the set of the elements $$r\in R$$ such that there exists $$s\in S$$ with $$sr\in I.$$

Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation. In what follows, $S$ is a multiplicative set in a ring $R$, and $I$ and $J$ are ideals of $R$; the saturation of an ideal $I$ by a multiplicative set $S$ is denoted $$\operatorname{sat}_S (I),$$ or, when the multiplicative set $S$ is clear from the context, $$\operatorname{sat}(I).$$
 * $$1 \in S^{-1}I \quad\iff\quad 1\in \operatorname{sat}(I) \quad\iff\quad S\cap I \neq \emptyset$$
 * $$I \subseteq J \quad\ \implies \quad\ S^{-1}I \subseteq S^{-1}J \quad\ \text{and} \quad\ \operatorname{sat}(I)\subseteq \operatorname{sat}(J)$$ (this is not always true for strict inclusions)
 * $$S^{-1}(I \cap J) = S^{-1}I \cap S^{-1}J,\qquad\, \operatorname{sat}(I \cap J) = \operatorname{sat}(I) \cap \operatorname{sat}(J)$$
 * $$S^{-1}(I + J) = S^{-1}I + S^{-1}J,\qquad \operatorname{sat}(I + J) = \operatorname{sat}(I) + \operatorname{sat}(J)$$
 * $$S^{-1}(I \cdot J) = S^{-1}I \cdot S^{-1}J,\qquad\quad \operatorname{sat}(I \cdot J) = \operatorname{sat}(I) \cdot \operatorname{sat}(J)$$
 * If $$\mathfrak p$$ is a prime ideal such that $$\mathfrak p \cap S = \emptyset,$$ then $$S^{-1}\mathfrak p$$ is a prime ideal and $$\mathfrak p = \operatorname{sat}(\mathfrak p)$$; if the intersection is nonempty, then $$S^{-1}\mathfrak p = S^{-1}R$$ and $$\operatorname{sat}(\mathfrak p)=R.$$

Localization of a module
Let $R$ be a commutative ring, $S$ be a multiplicative set in $R$, and $M$ be an $R$-module. The localization of the module $M$ by $S$, denoted $S^{−1}M$, is an $S^{−1}R$-module that is constructed exactly as the localization of $R$, except that the numerators of the fractions belong to $M$. That is, as a set, it consists of equivalence classes, denoted $$\frac ms$$, of pairs $(m, s)$, where $$m\in M$$ and $$s\in S,$$ and two pairs $(m, s)$ and $(n, t)$ are equivalent if there is an element $u$ in $S$ such that
 * $$u(sn-tm)=0.$$

Addition and scalar multiplication are defined as for usual fractions (in the following formula, $$r\in R,$$ $$s,t\in S,$$ and $$m,n\in M$$):
 * $$\frac{m}{s} + \frac{n}{t} = \frac{tm+sn}{st},$$
 * $$\frac rs \frac{m}{t} = \frac{r m}{st}.$$

Moreover, $S^{−1}M$ is also an $R$-module with scalar multiplication
 * $$ r\, \frac{m}{s} = \frac r1 \frac ms = \frac{rm}s.$$

It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions.

The localization of a module can be equivalently defined by using tensor products:
 * $$S^{-1}M=S^{-1}R \otimes_R M.$$

The proof of equivalence (up to a canonical isomorphism) can be done by showing that the two definitions satisfy the same universal property.

Module properties
If $M$ is a submodule of an $R$-module $N$, and $S$ is a multiplicative set in $R$, one has $$S^{-1}M\subseteq S^{-1}N.$$ This implies that, if $$f\colon M\to N$$ is an injective module homomorphism, then
 * $$S^{-1}R\otimes_R f : \quad S^{-1}R\otimes_R M\to S^{-1}R\otimes_R N$$

is also an injective homomorphism.

Since the tensor product is a right exact functor, this implies that localization by $S$ maps exact sequences of $R$-modules to exact sequences of $$S^{-1}R$$-modules. In other words, localization is an exact functor, and $$S^{-1}R$$ is a flat $R$-module.

This flatness and the fact that localization solves a universal property make that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, the natural map
 * $$S^{-1}(M \otimes_R N) \to S^{-1}M \otimes_{S^{-1}R} S^{-1}N$$

is an isomorphism. If $$M$$ is a finitely presented module, the natural map
 * $$S^{-1} \operatorname{Hom}_R (M, N) \to \operatorname{Hom}_{S^{-1}R} (S^{-1}M, S^{-1}N)$$

is also an isomorphism.

If a module M is a finitely generated over R, one has
 * $$S^{-1}(\operatorname{Ann}_R(M)) = \operatorname{Ann}_{S^{-1}R}(S^{-1}M),$$

where $$\operatorname{Ann}$$ denotes annihilator, that is the ideal of the elements of the ring that map to zero all elements of the module. In particular,
 * $$S^{-1} M = 0\quad \iff \quad S\cap \operatorname{Ann}_R(M) \ne \emptyset,$$

that is, if $$t M = 0$$ for some $$t \in S.$$

Localization at primes
The definition of a prime ideal implies immediately that the complement $$S=R\setminus \mathfrak p$$ of a prime ideal $$\mathfrak p$$ in a commutative ring $R$ is a multiplicative set. In this case, the localization $$S^{-1}R$$ is commonly denoted $$R_\mathfrak p.$$ The ring $$R_\mathfrak p$$ is a local ring, that is called the local ring of $R$ at $$\mathfrak p.$$ This means that $$\mathfrak p\,R_\mathfrak p=\mathfrak p\otimes_R R_\mathfrak p$$ is the unique maximal ideal of the ring $$R_\mathfrak p.$$

Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of Nakayama lemma. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is regular if and only if all its local rings are regular local rings.

Properties of a ring that can be characterized on its local rings are called local properties, and are often the algebraic counterpart of geometric local properties of algebraic varieties, which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see, below.)

Many local properties are a consequence of the fact that the module
 * $$\bigoplus_\mathfrak p R_\mathfrak p$$

is a faithfully flat module when the direct sum is taken over all prime ideals (or over all maximal ideals of $R$). See also Faithfully flat descent.

Examples of local properties
A property $P$ of an $R$-module $M$ is a local property if the following conditions are equivalent:
 * $P$ holds for $M$.
 * $P$ holds for all $$M_\mathfrak{p},$$ where $$\mathfrak{p}$$ is a prime ideal of $R$.
 * $P$ holds for all $$M_\mathfrak{m},$$ where $$\mathfrak{m}$$ is a maximal ideal of $R$.

The following are local properties:
 * $M$ is zero.
 * $M$ is torsion-free (in the case where $R$ is a commutative domain).
 * $M$ is a flat module.
 * $M$ is an invertible module (in the case where $R$ is a commutative domain, and $M$ is a submodule of the field of fractions of $R$).
 * $$f\colon M \to N$$ is injective (resp. surjective), where $N$ is another $R$-module.

On the other hand, some properties are not local properties. For example, an infinite direct product of fields is not an integral domain nor a Noetherian ring, while all its local rings are fields, and therefore Noetherian integral domains.

Non-commutative case
Localizing non-commutative rings is more difficult. While the localization exists for every set S of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is the Ore condition.

One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for differential equations. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The micro- tag is to do with connections with Fourier theory, in particular.

title: Localization (commutative algebra)


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SAM Coupé

The SAM Coupé (pronounced /sæm ku:peɪ/ from its original British English branding) is an 8-bit British home computer manufactured by Miles Gordon Technology (MGT), based in Swansea in the United Kingdom and released in December 1989.

It was based on and designed to have a compatibility mode with the ZX Spectrum 48K with influences from the Loki project and marketed as a logical upgrade from the Spectrum with increased memory, graphical and sound capabilities, native peripheral support (floppy disk, MIDI, joystick, light pen/light gun and a proprietary mouse).

The inclusion of support for higher graphical modes allowed for 80-column text presentation, providing a platform to support productivity and CP/M applications via additional software.

Being based on 8-bit technology at a time when 16-bit home computers were more prevalent, coupled with a lack of commercial software titles, led to it being a commercial failure.

When MGT went into receivership in June 1990 two further attempts were made to restart the computer and brand, firstly under SAM Computers Limited and then in November 1992 under West Coast Computers, a company spun from Format Publications which lasted until liquidation in 2005.

Naming

The capitalised SAM is an acronym for 'Some Amazing Micro' according to Alan Miles in an interview with ZAT magazine.

The ‘Coupé’ part has two sources: one being an ice cream sundae called the “Ice Cream Coupé” and the other because the machine resembles a fastback car in profile with the feet as the wheels.

Hardware
The SAM Coupé's hardware was designed by Bruce Gordon of Miles Gordon Technology. The computer included custom silicon to handle display, memory and IO functionality. This was originally prototyped using wire-wrapped 7400-series logic chips, before being produced as a VLSI VGT-200 gate array ASIC.

Processor and logic
The machine is based around a Z80B CPU clocked at 6 MHz and a 10,000-gate ASIC. The ASIC performs a similar role in the computer to the ULA in the ZX Spectrum. The Z80B CPU accesses selected parts of the large memory space in its 64 KB address space by slicing it into 16 KB banks and using I/O registers to select the memory pages mapped into each 16 KB bank.

Memory and storage
The basic SAM Coupé model has 256 KiB of RAM, internally upgradable to 512 KiB via a connector on the main board accessible via a trapdoor underneath, and externally up to an additional 4 MiB, added in 1 MiB packs via the Euroconnector on the back of the system.

The computer has a direct connection for a cassette recorder for data storage but two 3.5 inch floppy disk drives can be installed within the case as well or externally using an interface.

Graphics
The SAM Coupé was designed primarily for the UK market, and is designed around the PAL television standard, which refreshes at 50 frames per second. Unlike a standard PAL signal which is interleaved, the SAM is designed to emit two identically positioned fields at 50 FPS, giving something closer to a 312 lines progressive signal than the 625-lines interlaced broadcast television signals common in the UK at the time.

The display is surrounded by a large border area to provide a title-safe display zone for the CRT televisions of that era. The colour of this region can be changed in software by using the BORDER port to select a colour from the palette.

Display Modes
The SAM Coupé has four display modes:

The 'attribute' modes borrow their design from the ZX Spectrum, where a bitmap is used to select between a paper and ink colour from two groups of eight colours. The group to use is selected by a 'brightness' flag. The colour block can also be set to 'flash' - that is, alternate between the two colours used for paper and ink.
 * Mode 1 — 256 × 192, non-linear framebuffer, 1 bit per pixel with separate colour attributes for each 8×8 block of pixels = 6.75 KB (arranged to match the display of the ZX Spectrum for backwards compatibility)
 * Mode 2 — 256 × 192, linear framebuffer, 1 bit per pixel with separate colour attributes for each 8-wide block of pixels = 12 KB
 * Mode 3 — 512 × 192, linear framebuffer, 2 bits per pixel (4 colours) = 24 KB
 * Mode 4 — 256 × 192, linear framebuffer, 4 bits per pixel (16 colours) = 24 KB

Hardware palette
All modes use palette-based colour look-up tables, selecting from a palette of 128 colours. The palette values consist of 2 bits for each of the red, green and blue components as well as an extra bit which increases the intensity of all three components by a half-step (a 'brightness' bit).

The ASIC can be configured to generate interrupts when a line on the display is starting to be emitted, allowing video effects to be synchronised with specific display lines with little effort. By default, it will generate an interrupt for every frame. Typically this interrupt is used to double buffer the frame, read the keyboard/mouse state, and output music.

The interrupt state can also be polled directly from the ASIC's status register.

The Motorola MC1377P RGB to PAL/NTSC video encoder creates a composite video signal from the machine's RGB- and Sync-signals (output by the ASIC) for the RF modulator.

The non-standard SCART display connector includes both composite and RGB output, as well as signals to drive a 16-colour TTL monitor.

Access to internal RAM was shared between the display and the CPU, with CPU accesses incurring a speed penalty (memory contention) as it was forced to waited for isochronous ASIC memory-accesses to complete. As a result, the SAM Coupé's CPU effectively ran only around 14% faster than the ZX Spectrum CPU, yet was required to do much more work in SAM's high-resolution modes to produce a similar movement on the display. A Mode 3 or Mode 4 screen uses four times as much RAM as a ZX Spectrum Mode 1 display, so four times the work had to be done in the same time when updating it.

A small compensation was the straightforward arrangement of colour pixels in this memory, instead of the ZX Spectrum's more limited display and attributes memory. Low-level graphics software operations could be much simpler than their Spectrum equivalents and therefore somewhat faster to execute.

The penalty of memory contention delay applied to all memory accesses to RAM, and not just to memory associated with the video circuitry (as in the case of the ZX Spectrum). Hardware sprites and scrolling would have greatly improved the performance of games, unfortunately there was insufficient wafer space on the VLSI ASIC to include such circuitry.

While the main 256 × 192 area of the screen was being drawn, the processor could only access memory in 1 out of every 8 t-states. During the border area this was 1 out of every 4 t-states, which had no effect on the many instructions whose timings were a multiple of 4. In modes 3 and 4 the display could be disabled completely, eliminating these memory contention delays for a full 6 MHz running speed. Code running in ROM or external RAM was unaffected by contention, though any RAM accesses they performed to shared internal RAM would still be affected.

Character set
The character set includes block graphics and international characters. Font size can be altered to make text display in 32, 64, or 85 columns, and double height characters are also possible. Users can freely define their own characters by creating User Defined Graphics (UDGs). A large number of character codes (144 to 255) are available for this purpose. Block graphics (code 128 to 143) can be turned off in order free more characters for UDGs.

When in BASIC, the cursor is composed of the Blocks off characters (128/129) for lower and upper case respectively regardless of current Blocks setting, so overwriting these positions will also change the cursor. If this is undesirable, then systems variable (SVAR) 1 can be set to a 2-character value for the new lower and upper case cursor characters to be used.

The Utility Tapes and SAMDOS disk operating system media included an international character font loader to allow additional characters to be loaded into the block graphics region starting at 128 (0x80). This meant the cursor characters were overwritten so the accompanying BASIC loader set the cursor to * and + for lower and upper case respectively.

The following table show the predefined character set arrangement: Notes:

 * Character codes 0 to 31 (0x00 to 0x1F): control codes
 * Character codes 32 to 168 (0x20 to 0xA8): predefined character set
 * Character codes 128 to 143 (0x80 to 0x8F): block graphics / foreign characters or User Defined Graphics (UDGs)
 * Character codes 144 to 255 (0x90 to 0xFF): User Defined Graphics (UDGs)

Audio
Six channels of 8-octave stereo sound are provided by a Philips SAA1099 sound and noise generator chip. The machine allows for the playback of sound samples under software control as well as supporting four-channel Amiga Tracker modules at 3-bit sample resolution of 10.4KHz and 4-bit at 15.625 Khz.

To provide backwards compatibility with the ZX Spectrum, the SAM also provides a single-bit 'beeper' channel which can be used to emit simple tones by toggling the bit on and off as per the original Spectrum.

Firmware and DOS

The machine shipped with 32 KB of ROM containing code to boot the machine and a BASIC interpreter (SAM BASIC) written by Andrew Wright and heavily influenced by his earlier Beta BASIC for the ZX Spectrum. The ROMs contained only the bootstrap code and the DOS was instead loaded from disk using the  command, or the  key. The majority of disks shipped with SAMDOS, the system's first DOS, on them so that they could be directly booted. An improved replacement, MasterDOS, was also developed offering faster disk access, more files and support for the real-time clock accessory to provide file timestamps amongst many other improvements.

The BASIC was very advanced and included code for sprite drawing and basic vector shapes such as lines and circles. The screen co-ordinate system for these was variable and could be arbitrarily scaled and centred. A provision for "recording" sequences of graphics commands so that they could later be repeated without the speed penalty of a BASIC interpreter in between was provided.

The machine is capable of running CP/M 2.2 using the Pro-Dos software with support for both 720 Kilobyte format disks and IDE drives

Disk drives

The SAM originally used Citizen 3.5 inch slimline drives which slotted in below the keyboard to provide front-facing slots. Like IDE hard disks, these enclosures contained not just the drives but also the drive controllers, a WD1772-02, with the effect that the SAM could use both drives simultaneously.

Due to a flaw in the Coupé's design, resetting the machine while a disk was left in a drive would be liable to cause data corruption on that disk, as while the RESET line is held logic low the ASIC (that generates the 8 Mhz clock) is halted thus no clock signal is sent to the drive's controllers.

The double density disks used a format of 2 sides, 80 tracks per side and 10 sectors per track, with 512 bytes per sector. This gave a total capacity of 800 KB, though the standard directory occupied 20 KB leaving 780 KB free for user files. Files were stored in the same structure as MGT's original +D interface, but with additional codes used for SAM Coupé file types. The disk encoding (NRZ), encoding strategy (linear angular velocity), and track and sector header formats were compatible with those used on the IBM PC and Atari ST, and programs were available to read FAT formatted disks.

Expansion ports

A large array of expansion ports were provided, including:
 * Two internal drive bays.
 * Non-standard SCART connector offering composite video and digital and linear RGB as well as power input.
 * 64-pin Euroconnector for general purpose hardware expansions.
 * Mouse socket (proprietary format, although a converter for Atari ST style mice was later available).
 * Light pen / Light gun and Stereo sound output via 5-pin DIN connector.
 * MIDI IN/OUT ports (and THROUGH, via a software switch).
 * Network using the MIDI port (up to 16 machines could be interconnected).
 * Atari-style 9-pin joystick port (dual capability with a splitter cable although due to a flaw the two joysticks would interfere with each other).
 * 3.5mm mono Cassette jack (dual use for loading and saving).

Up to four devices could be connected to the Coupé's Euroconnector port, through the use of the SAMBUS, which also provided a built-in clock. When using more power-hungry peripherals, the SAMBUS required an additional power supply.

Power
The SAM's Power Supply was a modified Amstrad CPC MP1/MP2 modulator unit, with the RF modulator built in and connected via a joint power/TV socket to the computer. This made signal interference from the AC/DC converter common and it was a popular but entirely unofficial modification to remove the modulator and keep it as a separate unit.

ZX Spectrum compatibility

Emulation of the ZX Spectrum was limited to the 48K and was achieved by either using a supplied utility and a skeleton ROM image (containing no original code) or by loading a complete copy of the ZX Spectrum ROM (obtained from a ZX Spectrum). In order to match the processing speed of the ZX Spectrum (3.5 MHz), the SAM Coupé introduces extra wait states in display Mode 1 (the ZX Spectrum-compatible graphics mode) to slow down the CPU to roughly match the rate of that system.

The 128K model's memory map was incompatible with the Coupé's memory model and the machine featured an entirely different sound generator. It was possible to convert some games and demos by directly hacking the 128K code or by using a software patcher with the 1 megabyte RAM expansion to provide the address space.

Because the Coupé did not run at exactly the same speed as the Spectrum even in emulation mode, many anti-piracy tape loaders would not work on the Coupé hardware. This led to the development by MGT of a special hardware interface called the Messenger which could capture the state of a connected ZX Spectrum to SAM Coupé disk for playback later without the Spectrum connected. The Messenger plugged into the Coupé's network port, and the Spectrum's expansion slot. Due to unsuitable onboard break (NMI) buttons (needed to activate the Messenger software), a de-bounced break-button card was also provided, which plugged into the Coupé's expansion slot.

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

title: SAM Coupé


text:
Microchannel

Microchannel can refer to
 * Basic structure used in microtechnology, see Microchannel (microtechnology).
 * Micro Channel architecture in computing

title: Microchannel


text:
The Runaways

The Runaways were an American female rock band who recorded and performed from 1975 to 1979. Formed in 1975 in Los Angeles, the band released four studio albums and one live album during its run. Among their best-known songs are "Cherry Bomb", "Hollywood", "Queens of Noise" and a cover version of the Velvet Underground's "Rock & Roll". Never a major success in the United States, the Runaways became a sensation overseas, especially in Japan, thanks to the single "Cherry Bomb".

Early years
In early 1975, Rock impresario Kim Fowley had an idea for assembling an all-female rock band in a time when, according to guitarist Joan Jett, "there were no young girls at all playing guitar or any sort of instruments."

Fowley met lyricist Kari Krome at a party held for Alice Cooper, put her on salary, and had her begin looking for female performers. Krome met Joan Jett and brought her to Fowley. Jett successfully auditioned for Fowley by playing ukulele to a Sweet album, and picked up rhythm guitar as her instrument. Soon afterward, in the parking lot of the Rainbow Bar and Grill, Fowley met Sandy West, who agreed to play drums. Fowley had Jett rehearse with West at the latter's home in Huntington Beach, and auditioned the pair on the telephone to music journalist Ritchie Yorke. He then turned to two radio stations to advertise for further auditions, which brought lead vocalist/bassist Micki Steele (later of The Bangles).

The Runaways were formed in August 1975. Fowley touted the new group The Runaways as "an all girl answer to Grand Funk," and the band signed to Kim Fowley Productions for management in September 1975. The lineup of Jett, West, and Steele made its live debut playing the Whisky a Go Go September 28–29, opening for Fowley's previous conceptual band the Hollywood Stars (then billed as "The Stars").

But on October 30, Steele left due to creative differences and was replaced by lead vocalist Paul Goldwin and bassist Peggy Foster in November, respectively. Lita Ford had joined to the group as lead guitarist, allowing Jett to focus on rhythm guitar. Trio were previously members of Dangerous.

After Goldwin left, Fowley and Jett discovered Cherie Currie at the teen nightclub The Sugar Shack and brought her in as lead vocalist for the group.

In two weeks, Foster left the group too and was replaced by Jackie Fox, whom Fowley's colleague Rodney Bingenheimer found at the parking lot of the West Hollywood nightclub the Starwood.

Fame
Phonogram/Mercury Records' West Coast director of A&R, Denny Rosencrantz, agreed to sign The Runaways on December 12, 1975. The band recorded their debut album The Runaways in January 1976 and released in March through Mercury. An album to an initial position of #188 on the Cash Box top albums chart, favorable reviews, and numerous radio and album adds, as well as sales of 70,000 units.

The band toured the U.S. in support of headlining groups such as Cheap Trick, Van Halen, Talking Heads, and Tom Petty and the Heartbreakers. For their stage performance, the documentary Edgeplay: A Film About the Runaways (directed by former Runaway bassist Vicki Blue) revealed each girl patterned herself after their musical idol: Currie on David Bowie, Jett on Suzi Quatro, Ford on a cross between Jeff Beck and Ritchie Blackmore, West on Roger Taylor, and Fox on Gene Simmons.

The Runaways were the subject of a writeup in People magazine in September 1976. A week later, Fowley severed his ties with the band and forfeited his rights, titles, claims, and merchandising ownership to Mercury Records. Mercury refused to accept Fowley's stakehold in the band and turned it back to him the following week. Fowley and the band reconciled in November and returned to the studio to record their followup album Queens of Noise. The album was released on January 7, 1977.

The Runaways performed a world tour in support of Queens of Noise. The band quickly became lumped in with the growing punk rock movement. The band (already fixtures on the West Coast punk scene) formed alliances with mostly male punk bands such as the Ramones and the Dead Boys (via New York City's CBGB) as well as the British punk scene by hanging out with the likes of the Damned, Generation X and the Sex Pistols.

In the summer of 1977, booking agent David Libert secured dates in Japan, where they played a string of sold-out shows. They were unprepared for the onslaught of fans that greeted them at the airport. Jett later described the mass hysteria as "just like Beatlemania". While in Japan, the Runaways had a TV special, made numerous television appearances, and released the album Live in Japan, which went gold. Jackie Fox left mid-tour and was replaced by bassist Vicki Blue when the group returned to America.

Cherie Currie left the band on August 9, 1977. Fowley recorded The Runaways' third LP, Waitin' for the Night, at Larrabee Studios the same month, with the band as a four-piece and with Joan Jett on vocals. The album was released in October 17, 1977, and was the final album that Fowley would record with the group.

Dissolution
Due to disagreements over money and the management of the band, the Runaways and Kim Fowley parted ways in 1977. The group quickly hired Toby Mamis, who worked for Blondie and Suzi Quatro. When the group split from Fowley, they also parted with their record label Mercury/PolyGram, to which their deal was tied. In the Edgeplay documentary, members of the group (especially Fox and Currie) as well as the parents of Currie and West, have accused Fowley and others assigned to look after the band of broken promises as to schooling and other care, using divide and conquer tactics to keep control of the band, along with the verbal taunting of band members. The band reportedly spent much time enjoying the excesses of the rock 'n' roll lifestyle during this time. They partnered with Thin Lizzy producer John Alcock, after Jett's future partner Kenny Laguna turned down the job, to record their last album And Now... The Runaways.

Blue left the group due to medical problems and was briefly replaced by Laurie McAllister in November 1978. McAllister was referred to the band by her neighbor, Duane Hitchings, who played keyboards on And Now... The Runaways. Before joining the Runaways, McAllister played with Baby Roulette and the Rave Ons, who had one song released on a Kim Fowley compilation LP called Vampires From Outer Space. McAllister appeared onstage with the Runaways at their final shows in California in December 1978 and quit in January 1979.

Disagreement among band members included the musical style; Jett wanted the band to make a musical change, shifting towards punk rock/glam rock while Ford and West wanted to continue playing hard rock/heavy metal music. Neither would accept the other's point of view. The band played their last concert on New Year's Eve 1978 at the Cow Palace and officially broke up in April 1979.

Potential reunion
In 2010, Jett and Currie reunited to re-record the song "Cherry Bomb" for the video game Guitar Hero: Warriors of Rock.

In a 2015 interview with WHMH-FM, Ford said that she decided against a possible reunion in the early 1990s because "Nirvana was just kicking in, and it was really bad timing; it wouldn't have worked. People would have just turned their nose up at it."

In December 2018, Ford said that a Runaways reunion would never happen and blamed Jett's manager Kenny Laguna as the reason, saying, "Joan Jett is very much in 'Joan Jett land,' I guess you could say. Will she ever come out of Joan Jett land? I don't think so. I think her manager controls that, and it's really up to him and her. It seems to me like Joan Jett's manager just runs her life in every way, shape, or form. He's very controlling and he has a real problem with me. He has a real issue with me. He sees me as a threat, which is ridiculous because she's like my sister, and I love Joan. It's ridiculous, it's uncalled for, and it's caused a little bit of rivalry between her and I, which is totally uncalled for. It's his fault. The hard part is just trying to communicate with Joan without her manager involved. We had dinner a couple of years ago, what was supposed to be a girl's night out, and she brought her manager with her. So it's like, dude, answer the question. I'm trying to ask you a question. Are you interested in putting The Runaways back together? She never answered the question."

On May 12, 2021, Currie was asked about a reunion, to which she responded, "I've played with all the girls individually; I've played with all of them — with Lita, with Joan, and, of course, with Sandy; I never did a show without her before she passed away. So I'm the only one that's actually played with all the members. Lita and Joan have a little bit of an issue, and that just seems to be the problem. Lita doesn't like Kenny. I wish they would get over it, honestly, but I don't think so. Kenny was my manager during the early stages of making Blvds Of Splendor. We were very good friends for 20 years. But then, unfortunately, with the record, with them holding up Blvds Of Splendor for 10 years, that kind of eroded my friendship with him a bit."

Joan Jett
Jett went on to work with producer Kenny Laguna and after being rejected by 23 record labels, formed their own label, Blackheart Records, in 1980. In doing so, Jett became one of the first female recording artists to found her own record label. The label continues to release albums by the Blackhearts and other bands. Jett went on to have massive success with a cover of the Arrows' song "I Love Rock 'n' Roll", as well as "Crimson and Clover", "Bad Reputation", and "I Hate Myself for Loving You". She also co-starred in the 1987 film Light of Day with Michael J. Fox, and appeared in the 2000 Broadway revival of The Rocky Horror Show as Columbia. Jett is on Rolling Stone magazine's list of "The 100 Greatest Guitarists of All Time". In 2015, she and her band the Blackhearts were inducted into the Rock and Roll Hall of Fame.

Sandy West
West continued her association with John Alcock once the group disbanded. She formed the Sandy West Band and toured California throughout the 1980s and 1990s. She also did session work with John Entwistle of the Who and became a drum teacher.

West died from lung cancer in October 2006, having been diagnosed the previous year. A memorial tribute concert was later held in Los Angeles, featuring the Sandy West Band, Cherie Currie, the Bangles, the Donnas, and Carmine and Vinny Appice, among several others.

Micki Steele
Steele (as Michael Steele) joined the band The Bangles and went on to success with songs such as "Manic Monday", "Walk Like an Egyptian" and "Eternal Flame".

Cherie Currie
Upon leaving the Runaways, Currie released a 1978 solo album titled Beauty's Only Skin Deep and a 1980 duet album with her twin sister Marie Currie, Messin' with the Boys, in which the duo was backed by members of Toto. The Curries' cover of Russ Ballard's "Since You Been Gone" reached Number 95 on the U.S. chart. Currie also appeared in a number of films, most notably Foxes with Jodie Foster. Throughout the 1990s, Currie worked as a drug counselor for addicted teens and as a personal fitness trainer. She married actor Robert Hays and they had a son together, Jake Hays. The couple divorced in 1997.

Currie still performs and records, remaining under contract with Blackheart Records, but her current passion is chainsaw carving which she displays at an art gallery in Chatsworth, California. In 2013, Cherie recorded two songs with Alexx Michael for the Munich-based hard rock-glam metal group Shameless, which were released on the album Beautiful Disaster on October 2, 2013. Currie's most recent solo album, Blvds of Splendor, was released in 2020.

Lita Ford
Ford returned to PolyGram as a solo artist in the 1980s, where she released several albums before pairing with manager Sharon Osbourne. She had success with songs like "Kiss Me Deadly" and "Close My Eyes Forever" (the second a duet with her manager's husband Ozzy Osbourne). She was married to Chris Holmes of W.A.S.P., and to former Nitro singer Jim Gillette, with whom she has two sons. After a long hiatus, Ford staged a comeback, performing at Rock The Bayou and other hard rock festivals during the summer of 2008. She released Wicked Wonderland, her first studio album in 14 years, on October 6, 2009. During that year Ford toured as a special guest of progressive metal band Queensrÿche and performed songs from Wicked Wonderland and reprised her duet "Close My Eyes Forever" with Queensrÿche lead singer Geoff Tate.

Vicki Blue
Vicki Blue is now known as Victory Tischler-Blue. After leaving the Runaways, she shifted her focus to film and television production eventually becoming a producer/director for several reality- and magazine-based television shows including Entertainment Tonight, Access Hollywood, and Real Stories of the Highway Patrol. She went on to form Sacred Dogs Entertainment Group a motion picture production company and released a documentary on the Runaways called Edgeplay: A Film About the Runaways. In 2005, Tischler-Blue directed Naked Under Leather, a documentary about fellow female rocker Suzi Quatro, which was selected for the Santa Cruz Film Festival in May 2004 but never released. Focusing on music driven productions, she was tapped to executive produce a network special: The Bee Gees "Unbroken Fever"—The 30th Anniversary of Saturday Night Fever. Additionally, Tischler-Blue and Ford teamed up to record music for El Guitarrista, an animated series that Sacred Dogs Entertainment Group is producing.

Jackie Fox
Fox returned to using her birth name of Fuchs and graduated from UCLA summa cum laude, with a Bachelor of Arts in Linguistics and Italian. She later received a Juris Doctor from Harvard Law School and practices entertainment law. She co-wrote "Delilah's Scissors" with Tischler-Blue and executive-produced and appeared in Edgeplay, Tischler-Blue's 2005 documentary about the Runaways. She also writes an L.A. cat care column for Examiner.com and is an occasional contributor to Listverse.com. She is the author of The Well, an unpublished work of young adult historical fiction, and is currently working on her second novel. In December 2018 she won four games on the game show Jeopardy!

In July 2015, after Fowley's death, Fuchs revealed publicly that Fowley raped her on New Year's Eve 1975 at a party after a Runaways performance at a club in Orange County. Sixteen years old at the time, she was reportedly given Quaaludes by a man who she thought was a roadie and raped while she was incapacitated. Currie said she spoke up against Fowley's actions, then stormed out of the room when he refused to stop. Look Away, a documentary about sexual abuse in the rock music industry features Fuchs' story.

Laurie McAllister
McAllister joined another of Fowley's all-female bands, the Orchids, who released their only album in 1980. McAllister retired from the music industry and worked as a veterinarian technician in Eugene, Oregon. She died of complications from an asthma attack on August 25, 2011 at the age of 54.

Compilation albums

 * Flaming Schoolgirls (1980, Cherry Red)
 * Little Lost Girls (1981, Rhino)
 * The Best of the Runaways (1982, Mercury)
 * I Love Playin' with Fire (1982, Cherry Red)
 * Born to be Bad (1991, Marilyn)
 * Neon Angels (1992, Mercury)
 * The Runaways featuring Joan Jett and Lita Ford (1997, PolyGram)
 * 20th Century Masters - The Millennium Collection: The Best of the Runaways (2005, Universal)
 * The Mercury Albums Anthology (2010, Hip-O)

Use in media

 * "Cherry Bomb" appears on the soundtracks of Dawn: Portrait of a Teenage Runaway, Dazed and Confused, RV, Cherrybomb, Guardians of the Galaxy and The Boys.
 * In Margaret Cho's stand-up DVD special I'm The One That I Want "Cherry Bomb" is played as she is making her entrance.
 * The television sitcom Step by Step named one of their episodes after the Runaways' song "School Days" (They changed the spelling to "School Daze")
 * "School Days" appears in the 1999 movie Detroit Rock City, and it was played in an episode of the teen drama 90210, the episode "What's Past Is Prologue".
 * In the teen drama The O.C., in the Season 3 episode "The Man of the Year", Marissa makes an entrance to "Cherry Bomb", while dressed in a provocative schoolgirl outfit, to Kaitlyn's boarding school. In another episode of The O.C., in the Season 1 episode "The Girlfriend", "Wait for Me" by the Runaways' was played.
 * In the 2007 movie Juno, the main character mentions her three favorite bands, one of which is the Runaways.
 * In the 2007 novel Derby Girl, by Shauna Cross, Oliver is said to have a poster of Joan Jett "in her Runaways days". The band is also referenced in a chapter title.
 * In the 2008 novel Lonely Werewolf Girl, by Martin Millar, the main character's favorite band is the Runaways.
 * The Runaways' song "You Drive Me Wild" was played in an episode of the comedy-drama Entourage, in the episode "Unlike a Virgin".
 * A cover version of "Cherry Bomb" is available for download in the music video game Rock Band. The song was re-recorded by Cherie Currie and Joan Jett in 2010 for inclusion in Guitar Hero: Warriors of Rock.
 * "Cherry Bomb" appears as load-screen background music in the video game Lollipop Chainsaw.
 * Queens of Noise: The Real Story of the Runaways by Evelyn McDonald was released July 9, 2013.
 * "Dead End Justice" was used in Stranger Things season 2 episode 7 "The Lost Sister" as Jane (eleven), Kali (eight), and the rest of Kali's crew leave their headquarters.
 * A photo of the Runaways appears on the cover of Ford's 2016 album, Time Capsule.

Influence
The Runaways' success paved the way for many successful female artists and female bands over the past 30 years, including the Bangles, the Go-Go's, Sahara Hotnights, L7, the Donnas, and Vixen to enter the male-dominated arena of rock music. They are named as influences by several artists, including the Germs, Courtney Love, the Adolescents, Taylor Momsen, White Flag, and Rhino Bucket who acknowledged the Runaways' influence on their music during their performance at the December 2006 tribute concert honoring Sandy West.

Film
A biographical film about the band, inspired by Currie's memoir, was released in 2010. Jett was one of the executive producers of the film. Actresses Kristen Stewart and Dakota Fanning starred as Jett and Currie, respectively. Michael Shannon played Fowley. None of the band's former bass players were featured in the film; Fox did not want to be involved in any part of the film, and requested that her name be changed in the story. The fictional replacement is named Robin Robbins. The film was written and directed by Floria Sigismondi, and was released to limited theaters on March 19, 2010.

The Runaways received generally positive reviews from critics. On review aggregator Rotten Tomatoes, the film has a 70% rating based on 187 reviews, and an average rating of 6.19/10.

New Runaways (1987)
In the early 1980s, Gayle Welch, an ambitious 13-year-old girl from Kaitaia, New Zealand, wrote a song, "Day of Age", and recorded it in Mandrell Recording Studios in Auckland, New Zealand. The resulting tape found its way to Fowley's desk. He played the Welch tape for colleague and Los Angeles deejay Rodney Bingenheimer, who played the song on his show on radio KROQ and included it on his annual compilation of his most-liked music for the year. Also on that compilation was a song that featured Chicago-native guitarist Bill Millay.

It did not take long before Fowley, who still owned the Runaways trademark, was putting together a new Runaways band built around Welch. Missy Bonilla was recruited from the typing pool of CBS Records, Denise Pryor came from Compton and Kathrine Dombrowski ("Kathy DiAmber") was also added. Welch was present only on tape and only on the first song on the CD, "I Want to Run With the Bad Boys". Millay played guitar, David Carr played keyboards and a drum machine rounded out the team. Glenn Holland, also from New Zealand, a friend of both Bingenheimer and Fowley, facilitated. The album, Young and Fast, was released in 1987, and was a minor hit.

title: The Runaways


text:
User:Williamx~enwiki

稻草人

title: User:Williamx~enwiki


text:
Operation Quicksilver

Operation Quicksilver may refer to:

 * Operation Quicksilver (deception plan), a World War II plan by the Allied nations threatening an invasion at Pas de Calais
 * Operation Quicksilver (1978), an American nuclear test series of eighteen blasts conducted at the Nevada Test Site in 1978 and 1979
 * Operation Quicksilver (1990s), a plan to reduce the size of the United States Army in the early 1990s as a result of the end of the Cold War

title: Operation Quicksilver


text:
User talk:Williamx~enwiki

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title: User talk:Williamx~enwiki


text:
Nilpotent

In mathematics, an element $$x$$ of a ring $$R$$ is called nilpotent if there exists some positive integer $$n$$, called the index (or sometimes the degree), such that $$x^n=0$$.

The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.

Examples

 * This definition can be applied in particular to square matrices. The matrix
 * $$A = \begin{pmatrix}

0 & 1 & 0\\   0 & 0 & 1\\    0 & 0 & 0  \end{pmatrix} $$
 * is nilpotent because $$A^3=0$$. See nilpotent matrix for more.

\begin{align}c^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end{align} $$ An example with matrices (for a, b):$$A = \begin{pmatrix} 0 & 1\\   0 & 1  \end{pmatrix}, \;\; B =\begin{pmatrix} 0 & 1\\   0 & 0  \end{pmatrix}. $$ Here $$AB=0$$ and $$BA=B$$.
 * In the factor ring $$\Z/9\Z$$, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9.
 * Assume that two elements $$a$$ and $$b$$ in a ring $$R$$ satisfy $$ab=0$$. Then the element $$c=ba$$ is nilpotent as $$

 * By definition, any element of a nilsemigroup is nilpotent.

Properties
No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). All nilpotent elements are zero divisors.

An $$n\times n$$ matrix $$A$$ with entries from a field is nilpotent if and only if its characteristic polynomial is $$t^n$$.

If $$x$$ is nilpotent, then $$1-x$$ is a unit, because $$x^n=0$$ entails $$(1 - x) (1 + x + x^2 + \cdots + x^{n-1}) = 1 - x^n = 1.$$

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

Commutative rings
The nilpotent elements from a commutative ring $$R$$ form an ideal $$\mathfrak{N}$$; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element $$x$$ in a commutative ring is contained in every prime ideal $$\mathfrak{p}$$ of that ring, since $$x^n = 0\in \mathfrak{p}$$. So $$\mathfrak{N}$$ is contained in the intersection of all prime ideals.

If $$x$$ is not nilpotent, we are able to localize with respect to the powers of $$x$$: $$S=\{1,x,x^2,...\}$$ to get a non-zero ring $$S^{-1}R$$. The prime ideals of the localized ring correspond exactly to those prime ideals $$\mathfrak{p}$$ of $$R$$ with $$\mathfrak{p}\cap S=\empty$$. As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent $$x$$ is not contained in some prime ideal. Thus $$\mathfrak{N}$$ is exactly the intersection of all prime ideals.

A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of ring $$R$$ are precisely those that annihilate all integral domains internal to the ring $$R$$ (that is, of the form $$R/I$$ for prime ideals $$I$$). This follows from the fact that nilradical is the intersection of all prime ideals.

Nilpotent elements in Lie algebra
Let $$\mathfrak{g}$$ be a Lie algebra. Then an element $$x\in\mathfrak{g}$$ is called nilpotent if it is in $$[\mathfrak{g}, \mathfrak{g}]$$ and $$\operatorname{ad} x$$ is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Nilpotency in physics
Any ladder operator in a finite dimensional space is nilpotent. They represent creation and annihilation operators, which transform from one state to another, for example the raising and lowering Pauli matrices $$\sigma_\pm=(\sigma_x\pm i \sigma_y)/2$$.

An operand $$Q$$ that satisfies $$Q^2=0$$ is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. The BRST charge is an important example in physics.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. More generally, in view of the above definitions, an operator $$Q$$ is nilpotent if there is $$n\in\N$$ such that $$Q^n=0$$ (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with $$n=2$$). Both are linked, also through supersymmetry and Morse theory, as shown by Edward Witten in a celebrated article.

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space. More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is part smooth infinitesimal analysis.

Algebraic nilpotents
The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions $$\mathbb C\otimes\mathbb H$$, and complex octonions $$\mathbb C\otimes\mathbb O$$. If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an indefinitely small proportion of the first order term.

title: Nilpotent


text:
The Frontier

The Frontier can refer to:

 * The Frontier (1991 film), a 1991 Chilean film directed by Ricardo P. Larrain
 * The Frontier (2014 film), an American drama film by Matt Rabinowitz
 * The Frontier (2015 film), an American crime film
 * The Frontier (Hong Kong), a defunct political group in Hong Kong
 * The Frontier (Hong Kong, 2010), a political group in Hong Kong
 * The Frontier (website), an investigative news website in Oklahoma, U.S.
 * The Frontier (North American history), the American frontier during the period of western expansion
 * Fallout: The Frontier, a mod for Fallout: New Vegas.
 * The Frontier, previous name for the New Frontier Hotel and Casino

title: The Frontier


text:
Talk:Armed Forces of Saudi Arabia

Untitled
Agh! I have screwed up the bolding in this article. Allow me to reflect upon my misdeeds and try to fix it tomorrow. -Paul in Saudi

 * This article doesn't really explain why Iran is such a threat to Saudi Arabia. Maybe it should be better explained? FvNK 23:01, Sep 27, 2004 (UTC)

 * This article has way too much opinion and needs help. JD79 02:33, 27 October 2005 (UTC)

 * Does it really need a disputed NPOV tag? The way I see it, it only states facts, IMHO. -- Eagleamn 02:58, 27 October 2005 (UTC)

 * isn't it illegal to talk about secret military equipment ? hehehe

"With the collapse of the Iraqi Ba'athist regime in mid-2003, the greatest conventional threat to the Saudi Kingdom was eliminated. The military situation became both less threatening and more complex."

Was this ever REALLY the case. The war happened. But from my understanding Iran has ALWAYS been considered SA's #1 threat. While SA did have a good amount of surveillence equipment on the ground to monitor the iraqi border, they have always had much more in terms to monitor Iran. I think that sentace needs to be removed or down played.

"Despite its at times anti-Western rhetoric," ... wth?? Maybe if we are reffering to the times of King Faisal. But from the gov't what "anti western" rhetoric? Dude this IS A MILITARY article. Not an op ed piece. It should focus on who they traditonaly got there weapons from, history of the creation of the saudi military. Jnusaira 18:17, 27 December 2006 (UTC)jnusaira

 * Iran does not have direct, contiguous land borders with Saudi Arabia. Therefore they could not use their ground forces, which are the strongest part of their military, in a direct invasion of the Kingdom.  The military threat they pose is mostly naval and aerial, and in those respects the Saudis are strong enough to fight them back.
 * That wasn't the case with Iraq, which had both a long land border with the Kingdom and a military more powerful than anything south of the border. (This remained true even after their defeat in the Gulf War and the Saudi military buildup afterwards - American forces, not Saudi, are the ones who guaranteed the nation's security throughout the nineties).  So in military terms, Iraq did pose a greater threat to Saudi Arabia than did Iran.
 * I don't think the article really needs to explain why Iran is such a threat to Saudi Arabia, though - on the one hand because that would be better suited to a "Foreign relations of Saudi Arabia" article, on the other hand because it's pretty glaringly obvious. They're the two largest rivals in the region, with more bad blood between them than anyone could measure.  They both compete for leadership of the Islamic world, both on theological (Sunni versus Shi'a) and political ("traditional" monarchy versus revolutionary regime) grounds.  Past confrontations; Saudi Arabia supported Iraq when it invaded Iran during the early eighties, while Iran has supported revolutionary factions (like Saudi Hezbollah) in Saudi Arabia and its neighbors for decades.  Islamists backed by Khomeini shot up Mecca during the eighties, and the Saudis have responded by harrassing Iranian and other Shi'a pilgrims ever since.  Saudi and Iranian fighters have engaged over the Gulf at least once before.  They competed for the loyalties of Afghan guerrillas during the 1980s Soviet invasion.  They're funding opposing sides of the Iraqi and Palestinian civil wars...  Now that Saddam is out of power, I think it's fair to say that the Iranian regime is the greatest foreign threat to Saudi Arabia. 213.181.226.21 (talk) 14:03, 24 July 2008 (UTC)

Al Mujahidoon
can any one but tha Reference that says that Al Mujahidoon is one of the Military branches of Ministry of InteriorArabian soul 06:33, 18 August 2007 (UTC)

 * This is not a militiant or a terrorist group, its a special Border Guard force named Al Mujahidoon.   A M M A R   06:36, 13 April 2008 (UTC)

Can Women be Clerks or volunteers?
I like to knew if they do and if they allowed them to do so.Thanks.

 * No. However, they may work as medics only in military hospitals. (More like regular doctors and nurses since they are not trained by the military AFAIK) - Eagleamn 01:18, 14 January 2006 (UTC)

 * In Ministry of Interior forces only, Such as Airport police of Secret service.   A M M A R   06:40, 13 April 2008 (UTC)

Infobox
Ive added an infobox based on the infobox on the Military of Hungary page with the info for the Saudi military. --84.68.189.48 17:32, 8 January 2007 (UTC)

Air Defence !
air defence force DOES belong to the Defence Ministry of Saudi, someone has deleted this fact with a random step Ammar 20:33, 8 January 2007 (UTC)

The lede says that there are 16,000 in air defence. But the article on the air defence force says 40,000!119.224.100.246 (talk) 02:37, 15 August 2015 (UTC)

Recruiting
Does anyone know how they recruit or what the demographics of their army is ? —Preceding unsigned comment added by 60.53.88.45 (talk) 17:05, 2 January 2008 (UTC)

 * According to CIA Fact Book, Armed forces personels are males Saudi national , But i believe the RSAF are using American trainers .   A M M A R   21:41, 2 January 2008 (UTC)

Fair use rationale for Image:Saudi-logo.jpg
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Fair use rationale for Image:Sa army flag.JPG
Image:Sa army flag.JPG is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

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Military situation
The military situation section looks like it was written by a nationalistic school kid.

"The most important threats now are from tribesmen in Yemen who cross the frontier at will (as they have for centuries) and whose presence threatens Saudi territorial integrity"...I mean that is just funny. And "(as they have for centuries)" when Saudi Arabia is less than a century old!!

The article needs a decent rewrite.--Xevorim (talk) 22:15, 2 September 2008 (UTC)

Re-write Feb 2011
I've completely re-written this article, which was in extremely poor shape. Comments please. DeCausa (talk) 13:22, 9 February 2011 (UTC)

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Kuwait war
Saudi involvement in the war over Kuwait was not limited to "Saudi pilots flew more than 7,000 sorties and Saudi troops took part in the battles around the Saudi town of Raʾs al-Khafji". There were other air and land operations.119.224.100.246 (talk) 02:46, 15 August 2015 (UTC)

External links modified
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Supreme Commander
It is clear, that there is a confusion between the position of Minister of Defense, the position of Supreme Commander, and Commander-in-Chief. The position of the Supreme Commander should be used to refer to Custodian of the Two Holy Mosques, who is King Salman. The position of Minister of Defence should refer to prince Mohammad bin Salman. Finally, the position of the Commander-in-Chief has to be removed as it contradicts the position of the Supreme Commander and should be replaced by Chairman of the Joint Chiefs of Staff, who is currently General Fayyadh Al Ruwaili. عمر خالد 8888 (talk) 13:53, 10 December 2019 (UTC)

Numbers
The Wikipedia article on the army says there are 175,000 active personnel. This says 75,000. The two articles ought to be consistent.Royalcourtier (talk) 04:48, 16 January 2016 (UTC)

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 * Emblem of the Saudi Arabian Border Guards.svg

Total Personnel Active in the Saudi Armed Forces
This article says the Saudi Armed Forces consists of 257,000 active duty personnel. However, when you then check the articles for all the various branches of the Saudi Armed forces, they add up to only 126,000 or so active troops. 75,000 for the Army. 20,000 in the Air Force. 13,500 in the Navy. 16,000 in the Air Defense forces. And 2,500 in the Strategic Missile Forces.

So which articles are wrong? Could you please double-check the sources, make sure they are current, and perhaps provide a second source for the force strengths for each of these articles (this one and the articles for the various Branches)? It should be readily available information. An editor of an encyclopedia would normally check things of that nature, I assume, to make sure there weren't any inconsistencies in the facts being presented by the encyclopedia. I hope this helps start the process in getting it all sorted out. Thank you. 174.126.177.92 (talk) 03:45, 15 July 2022 (UTC)

A cadet want to join saudia military
Good Evening, I m mohsin kamal khan , 19 years old , belong to Pakistan, i did FSc pre medical and Diploma in information technology,  qualified from Pakistan military ⚔ as a Cadet,  Currently i am MarshalArt coach in UAE 🇦🇪 Ajman and want to join saudia military ⚔ as a soldier,  Please some one guide me how can i join, i would be Grateful forever .Thank you very much. My whatsapp +971558029866. 5.31.230.189 (talk) 19:50, 3 April 2024 (UTC)

title: Talk:Armed Forces of Saudi Arabia


text:
Talk:Terbium

Untitled
Article changed over to new WikiProject Elements format by Schnee. Elementbox converted 11:35, 10 July 2005 by Femto (previous revision was that of 13:23, 9 July 2005).

Information Sources
Some of the text in this entry was rewritten from Los Alamos National Laboratory - Terbium. Data for the table was obtained from the sources listed on the subject page and WikiProject Elements but was reformatted and converted into SI units.

--

Material Properties
There are contradicting statements in this article concerning the physical properties of Terbium. The opening paragraph claims that Terbium is "very hard" and the physical properties section describes Terbium as "soft enough to be cut with a knife." Perhaps these are differeneces between the oxide and metalic forms. This should be noted if that is the case. — Preceding unsigned comment added by 146.126.51.51 (talk) 14:47, 23 April 2014 (UTC)
 * WebElements says it can be, but I doubt it – it makes the same claim for thulium, which is just about impossible to cut with a normal knife. Maybe it can be scratched with a knife? (Tm can.) I'll change the statement accordingly. Double sharp (talk) 15:07, 23 April 2014 (UTC)

Terbium can be scratched with a knife, but it would take a long time to cut it with a knife. It is very resistant to being damaged. I know this falls under original research guidelines but it is quite strong, at least my piece is. 75.80.123.188 (talk) 01:44, 28 September 2014 (UTC)

Hardness?
The summary claims the the element is 'malleable, ductile, and very hard.' The physical properties section claims that it is 'malleable, ductile, and soft enough to be scratched with a knife.' The hardness indicators (Vickers and Brinell) point to it having a hardness on par with high-grade steel. Can we get some consistency here? I'd fix it myself, but I don't have a reference book handy to double check the data. 173.164.183.245 (talk) 17:42, 29 July 2014 (UTC)
 * Apparently it's hard, but can be scratched with a knife. This is totally not a reliable source, but says that Tb is soft but resists impacts well. That could be an explanation. Annoyingly I haven't found a reliable source on this. Double sharp (talk) 02:10, 28 September 2014 (UTC)

title: Talk:Terbium


text:
Leo II (emperor)

Leo II (Λέων, Leōn; c. 467 – 474), called the Younger, briefly ruled as emperor of the Eastern Roman Empire from 473 to 474. He was the son of Zeno, the Isaurian general and future emperor, and Ariadne, a daughter of the emperor Leo I ((r. 457 – 474)). Leo II was made co-emperor with his grandfather Leo I on 17 November 473, and became sole emperor on 18 January 474 after Leo I died of dysentery. His father Zeno was made co-emperor by the Byzantine Senate on 29 January, and they co-ruled for a short time before Leo II died in late 474.

History

Leo II, called "the Younger", was born in 467, the son of Zeno, an Isaurian general, and Ariadne, the daughter of then-emperor Leo I. He was the maternal grandson of Emperor Leo I and Empress Verina. As the grandson of Leo I, Leo II had a strong claim to succeed his throne. Leo I, who was becoming increasingly ill, felt obligated to declare a successor to the imperial throne, but passed over his son-in-law on account of his unpopularity. Accordingly, Leo II was made caesar (heir to the throne) by Leo I around October 472, and was later promoted to augustus (also by Leo I) in November 473, making him co-emperor alongside his grandfather. He was crowned at the Hippodrome of Constantinople, and the ceremony was presided over by the Ecumenical Patriarch Acacius. The 10th-century De Ceremoniis gives a detailed account of his coronation as augustus, which is dated to 17 November 473. He was also appointed as the sole consul for 474 around this time.

When Leo I died of dysentery on 18 January 474, Leo II acceded to the throne as sole augustus. The historian Warren Treadgold wrote that during the reign of Leo II, his father Zeno was the true power behind the throne. On 29 January 474, the Byzantine Senate, with the approval of Empress Verina, made his father Zeno co-augustus under Leo II, as Leo was too young to sign official documents. Leo II died in Constantinople shortly after 10 October 474. The 6th-century writer John Malalas states that he reigned "1 year and 23 days", which, reckoning from his coronation as augustus, would give a death date of 8/9 December. However, he also states that Leo died on "November of the 13th indiction... as was written by the most learned Nestorianos, whose chronicle ended with Leo". Theodorus Lector, another 6th-century historian, states that Leo II died after a rule of 10 months, that is, from January to November 474. This is also corroborated by the 9th-century writer Theophanes the Confessor. The 20th-century Byzantinist George Ostrogorsky simply wrote that Leo II died sometime in the autumn of 474. He died aged 7, which is corroborated by the 6th-century writers John Malalas and John of Ephesus. The death of Leo II left Zeno as sole emperor.

His death having occurred so soon after he became emperor has led to speculation among some modern scholars that he was poisoned by either his mother or father so that Zeno could become sole emperor. However, no contemporary sources raised this suggestion even though Zeno was unpopular; thus it is considered likely that Leo II's death was natural, especially taking into account the high child mortality rate of the time. Victor of Tunnuna, a 6th-century chronicler, says that Leo II did not actually die, but was rather taken by Ariadne and hidden at a monastery. This is likely a confusion with Basiliscus, the son of the Byzantine commander Armatus. Basiliscus was crowned caesar in 476 and was almost executed in 477 after his father was murdered by Zeno, but was saved by Ariadne. The confusion likely stems from the fact that Basiliscus was renamed Leo in order to avoid association with the usurper who rose against Zeno.

Zeno was vastly unpopular due to a lack of dynastic prestige, with his only familial ties to the imperial throne being his marriage to Ariadne, the daughter of Leo I, and through his now-dead son Leo II. Additionally, because he was an Isaurian, he was seen as a foreigner by the Byzantine elite, and the treasury was empty on his ascension. Zeno's sole rule was opposed by the House of Leo, with Verina, the widow of Leo I, proclaiming her brother Basiliscus as emperor in January 475. Zeno fled, and Basiliscus ruled for 20 months before Zeno returned and retook the throne. Zeno's rule was marked by constant unrest, and it was only through cunning and bribery that he managed to rule for 17 years until his death on 9 April 491.

title: Leo II (emperor)


text:
Talk:Operation Provide Comfort

Well, I don;t have notes in front of me, but the operation was planned in 1990; that is the standard I use in my entries.

Yes, the plan was to keep the Kurds in their homes, to keep them from fleeing into (and so destabliizing) Turkey. Perhaps it could be better worded.

Hey!

 * Regarding this addition: "Operation Provide Comfort sponsored by the Vietnam Veterans Memorial, American Legion Post 295..." - that should be in a different article. --Cheese Sandwich 19:14, 30 June 2006 (UTC)
 * Moved it to its own article at Operation Provide Comfort (Charity Drive). --Cheese Sandwich 18:34, 22 July 2006 (UTC)

Dutch marines.
Why is there no mention of the Dutch marines who were also a part of this operation?

References can be found here in this book: Van Korea tot Kabul - ISBN : 90-12-10915-9
 * There is. Chwyatt (talk) 11:05, 20 May 2008 (UTC)

op provide comfort
Op provide comfort was was not planned in 1990 to stop kurds going into neighbouring Turkey! After the 1991 conflict in the Gulf saddam ordered is army to cause devastation to the kurds. The kurds had no alternative but to flee into the Turkish mountains where there was little or no food or water. This caused famine amongst them. Op provide comfort was a famine relief operation whereby allied coalition forces where sent to northern Iraq and Southern Turkey to protect and feed the Kurdish people. The were fed on supplies of Meals Ready To Eat (MRE'S)American military food supplies left over from the gulf war. This was flown by British and American Chinook, jolly green giants and Puma helicopters to the mountains. The whole operation took around 6 weeks to complete. I dont think Dutch forces took part but i may be wrong! But what would i know "I was only there" serving with the british joint helicopter support unit.(wasnt happy at the time has i had just returned home from op granby to be deployed back out there again) but hey i wouldnt change it for the world... —Preceding unsigned comment added by 81.105.97.252 (talk) 17:09, 11 September 2008 (UTC)

This is a much better description of the operation from my point of view. I was with the 24th MEU in the ground combat element BLT 2/8, on one of the seven Light Armored Vehicles assigned. My impressions are much more from the security side of the operation, my platoon doing zero humanitarian aid. When we first entered Zahko, the town was still occupied by the Iraqi Army. We coocupied the town for I believe 4 days, spending these days surrounding the Iraqi army posts on the ridges around the city. The Iraqi officers were determined to stay there under Saddams threatof death against them. Im amazed the 24th MEU is so little recognized in this operation. The US Army units appeared mostly after we left our frontline/roadblock position south of Zakho two months into the operation. My recollection is that the 24th MEU(around 1900 men ) and the UK Royal Marines(overall UK around 4100 by missions end), along with (yes) Dutch forces(over 1000)(I distincly remember delivering something to them at their area of operations to the east of Zahko) were the major ground security forces doing the initial security work through the end of June of 1991. Overall contribution were made by many countries to include: Australia, Belgium, Canada, France, Germany, Italy, Luxembourg, Netherlands, Portugal, Spain, Turkey, United Kingdom, United States. Some 8 years ago i was working on an order of battle for this(i have much more detail), but have misplaced my reference material, therefore im not compfortable editing this in. The Global Security site has mostly a cut and paste format, from which they gather information from available websites, but is a great starting place. I do recall then US Army Major General Jay Garner coming aboard the USS Austin, at the end of our deployment (after July 24), to address the embarked Marines, in which Garner mentioned our actions had helped to save 1000 Kurds a week there. Scarlettsj (talk) 00:37, 4 December 2009 (UTC)

I to served with the Royal Air Force Tactical Supply Wing deployed first to Diyabakir and then forward to Silopi.I can confirm Dutch Allouettes were in residence.One of which flown by a very attractive lady.Also Dutch Marines did deploy to Turkey.JP Maher R.A.F retired. —Preceding unsigned comment added by 92.16.18.109 (talk) 21:26, 12 January 2010 (UTC)

Someone left this in bold on top of the article. I'm posting it here
Use dmy dates'''You are missing the 527th Military Police Company, which was part of the 709th Mp Batallion, and 18th Mp Brigade. One of the MP units you have listed must be an error. I was there and in the 527th mp comapny!!!...This was in 1991 after Desert shield and Desert Storm ended.'''

I do not have the time to correct this myself, but I'm sure one of the page authors can handle it. Hope this helps out.

you are missing references to the Italian participation to Operation Provide Comfort
Called Missione Airone, in italian. — Preceding unsigned comment added by Bravonc (talk • contribs) 19:20, 28 May 2013 (UTC)

External links modified
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title: Talk:Operation Provide Comfort


text:
Orbit (dynamics)

In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems.

For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces.

Definition

Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function

 * $$\Phi: U \to M$$ where $$U \subset T \times M$$ with $$\Phi(0,x)=x$$

we define

 * $$I(x):=\{t \in T : (t,x) \in U \},$$

then the set

 * $$\gamma_x:=\{\Phi(t,x) : t \in I(x)\} \subset M$$

is called orbit through x. An orbit which consists of a single point is called constant orbit. A non-constant orbit is called closed or periodic if there exists a $$t\neq 0$$ in $$I(x)$$ such that
 * $$\Phi(t, x) = x $$.

Real dynamical system
Given a real dynamical system (R, M, Φ), I(x) is an open interval in the real numbers, that is $$I(x) = (t_x^-, t_x^+)$$. For any x in M
 * $$\gamma_{x}^{+} := \{\Phi(t,x) : t \in (0,t_x^+)\}$$

is called positive semi-orbit through x and
 * $$\gamma_{x}^{-} := \{\Phi(t,x) : t \in (t_x^-,0)\}$$

is called negative semi-orbit through x.

Discrete time dynamical system
For a discrete time dynamical system with a time-invariant evolution function $$ f $$:

The forward orbit of x is the set :
 * $$ \gamma_{x}^{+} \ \overset{\underset{\mathrm{def}}{}}{=}   \     \{ f^{t}(x) : t \ge 0 \} $$

If the function is invertible, thebackward orbit of x is the set :

 * $$\gamma_{x}^{-} \ \overset{\underset{\mathrm{def}}{}}{=}  \     \{f^{t}(x)  : t \le 0 \} $$

and orbit of x is the set :

 * $$\gamma_{x} \  \overset{\underset{\mathrm{def}}{}}{=}  \  \gamma_{x}^{-} \cup \gamma_{x}^{+} $$

where :
 * $$f$$ is the evolution function $$f : X \to X $$
 * set $$X$$ is the dynamical space,
 * $$t$$ is number of iteration, which is natural number and $$t \in T $$
 * $$x$$ is initial state of system and $$x \in X $$

General dynamical system
For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group $$G$$ acting on a probability space $$X$$ in a measure-preserving way, an orbit $$G.x \subset X$$ will be called periodic (or equivalently, closed) if the stabilizer $$Stab_{G}(x)$$ is a lattice inside $$G$$.

In addition, a related term is a bounded orbit, when the set $$G.x$$ is pre-compact inside $$X$$.

The classification of orbits can lead to interesting questions with relations to other mathematical areas, for example the Oppenheim conjecture (proved by Margulis) and the Littlewood conjecture (partially proved by Lindenstrauss) are dealing with the question whether every bounded orbit of some natural action on the homogeneous space $$SL_{3}(\mathbb{R})\backslash SL_{3}(\mathbb{Z})$$ is indeed periodic one, this observation is due to Raghunathan and in different language due to Cassels and Swinnerton-Dyer. Such questions are intimately related to deep measure-classification theorems.

Examples

 * The orbit of an equilibrium point is a constant orbit.

Stability of orbits
A basic classification of orbits is
 * constant orbits or fixed points
 * periodic orbits
 * non-constant and non-periodic orbits

An orbit can fail to be closed in two ways. It could be an asymptotically periodic orbit if it converges to a periodic orbit. Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit. An orbit can also be chaotic. These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic orbit. They exhibit sensitive dependence on initial conditions, meaning that small differences in the initial value will cause large differences in future points of the orbit.

There are other properties of orbits that allow for different classifications. An orbit can be hyperbolic if nearby points approach or diverge from the orbit exponentially fast.

title: Orbit (dynamics)