text:
Preparation H

Preparation H is an American brand of medications that is made by Pfizer, used in the treatment of hemorrhoids. Hemorrhoids are caused at least in part by inflamed blood vessels, and most versions of Preparation H work by reducing inflammation in blood vessels.

The company that makes this product incorporated in 1926 as American Home Products (AHP) and "one of AHP's earliest prizes was the acquisition of a sunburn oil in 1935 that the company transformed into Preparation H, which became one of the world's best-selling hemorrhoid treatments." AHP changed its name to Wyeth in 2002, and was then bought by Pfizer in 2009, which was reported to have paid $68 billion.

Formulations
Several formulations have been produced. Some are in a water-based gel while others are petroleum jelly-based, and the suppositories use a base of cocoa butter. It also comes as a medicated wipe. The active ingredients range from simple caffeine moisturizers with witch-hazel astringent to preparations containing drugs such as phenylephrine, pramoxine, and hydrocortisone. Formulations vary from one country to another. Some Preparation H products have phenylephrine, a drug that constricts blood vessels, in a 0.25% concentration, as well as shark liver oil. Another version of Preparation H has hydrocortisone as its only active ingredient in a 1% concentration.

The Canadian formulation of Preparation H includes a yeast extract called Biodyne which was removed in 1995 from the formulation sold in the United States. This yeast extract was the primary ingredient of the original formulation by inventor George Sperti, and Pfizer believes that it removes wrinkles from skin and heals dry, cracked, and irritated skin. The Canadian formulation has acquired a market in the United States as a skin cream with an anti-wrinkle active ingredient. In Greece, the active ingredients of Preparation H are 1% w/w yeast extract and 3% w/w shark liver oil.

title: Preparation H


text:
Weather gage

The weather gage (sometimes spelled weather gauge) is the advantageous position of a fighting sailing vessel relative to another. It is also known as "nautical gauge" as it is related to the sea shore. The concept is from the Age of Sail and is now antique. A ship at sea is said to possess the weather gage if it is in any position upwind of the other vessel. Proximity with the land, tidal and stream effects and wind variability due to geography (hills, cliffs, etc.) may also come into play.

An upwind vessel is able to manoeuvre at will toward any downwind point, since the relative wind then moves aft. A vessel downwind of another, in attempting to attack upwind, is constrained to trim sail as the relative wind moves forward and cannot point too far into the wind for fear of being headed. In sailing warfare, when beating to windward, the vessel experiences heeling under the sideward pressure of the wind. This restricts gunnery, as cannon on the windward side are elevated, while the leeward gun ports aim into the sea or in heavy weather may be awash. A ship with the weather gage, turning downwind to attack, may alter course at will to bring starboard and port guns to appropriate elevations. Ships seeking to evade capture or attack have the advantage being downwind if they are faster vessels or are close to friendly land. The term has had a literary rebirth in the popular seafaring novels of C.S. Forester, Patrick O'Brian and Alexander Kent.

One of the last times that weather gage was perhaps a factor in a surface naval engagement was in the Battle of the Denmark Strait in 1941, where the German battleship Bismarck and the heavy cruiser Prinz Eugen held the weather gage over the British battleship HMS Prince of Wales (53) and the battlecruiser HMS Hood (51). Being upwind, the German ships had the advantage that their rangefinders were drier than the British rangefinders facing the spray. The weather restricted the visual range to much less than the gun range.

The weather gage is a critical component of aircraft carrier operations. Most carrier aircraft are easier to launch if the aircraft carrier is steaming into the wind to generate stronger apparent winds across the deck that allows for greater lift to be generated at lower actual speeds. The Battle of the Philippine Sea is a notable incident of the advantage of the weather gage for carrier operations. American carriers were upwind of the Japanese carriers. Every time an American aircraft carrier conducted flight operations, they opened the range to the Japanese fleet. Japanese carriers could conduct flight operations while closing the range to the American fleet.

The concept of weather gage is still useful in modern yacht racing, although the term is rarely used. The sails of a boat disrupt the wind to leeward; this disruption is often called "dirt" or "dirty air". An overtaking boat on a downwind course can position itself to focus its dirty air on the boat ahead of it. Conversely, a boat on an upwind course may find itself trapped in the dirty air of a boat immediately to windward. Right-of-way rules give priority to the leeward boat and can make it advantageous to be the boat without the weather gage, especially just before the start or when the boat to leeward can point higher into the wind.

Weather gauge
Gauge and gage are often used as alternative spellings. To the extent that there is a difference, a weather gauge can be a form of meteorological instrumentation for measuring weather quantitatively, such as a rain gauge, thermometer, anemometer, or barometer. A gage is a challenge, and hence an entry into battle, though the word is more commonly embedded in the word engage.

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

title: Weather gage


text:
User:Mad Hungarian

I'm a grad student of Archaeology and History. My current research is focusing upon Spain and the frontiers of the Roman Empire (long story, typical of those things we love to study being subjected to the altar of budget cuts in so called 'Advanced' social-sciences programs). I have recently edited and written several articles here regarding military history, and subjected my work to the tools of editors far less generous than I;)

I am most interested in discussing history of any nature with anyone here!

title: User:Mad Hungarian


text:
Cape Flats

The Cape Flats (Die Kaapse Vlakte) is an expansive, low-lying, flat area situated to the southeast of the central business district of Cape Town. The Cape Flats is also the name of an administrative region of the City of Cape Town, which lies within the larger geographical area.

Geology and geography
In geological terms, the area that makes up The Cape Flats is a vast sheet of aeolian sand, ultimately of marine origin, which has blown up from the adjacent beaches over a period of a hundred thousand years. Below the sand, the bedrock is in general alternating layers of dark grey shale, siltstone and minor sandstone from a late-Precambrian rock formation called the Malmesbury Group. This is except on part of the western margin between Zeekoevlei to the south and Claremont and Wetton to the north, where an intrusive mass of Cape granite is to be found. Most of the sand is unconsolidated; however, in some places near the False Bay coast the oldest sand dunes have been cemented into a soft sandstone (calcrete), and form low cliffs at the edge of the beach. These formations contain important fossils of animals such as the extinct Cape lion and also provide evidence that stone-age people hunted here tens of thousands of years ago.

Extent
To the west the expanse of the Cape Flats is limited by rising ground that slopes up to the steep cliffs of the Cape Peninsula mountain chain, while in the east the land rises gradually towards the equally rugged cliffs of Hottentots Holland mountains and other elevated regions of the interior of the Boland.

Climate
The area has a Mediterranean climate, with warm dry summers and cool, damp winters. It is generally exposed to the wind, both from the northwest (winter) and southeast (summer). Flooding can be a problem, especially in July and August. Cold wet spells, especially in August and September, can make life very difficult for those living in sub-standard housing.

Drainage
Historically the Cape Flats was partly covered in wetlands, particularly during winter. Many of these have been destroyed by canalisation and infilling to provide residential space. Many of the remaining perennial vleis such as Zeekoevlei, Rondevlei and Zandvlei are at the southwest side of the region.
 * Elsieskraal River and the Black River catchment (northwest)
 * The Diep River
 * Zeekoevlei and its catchment
 * The Eerste River
 * The Kuils River

Before 1950
The noted English naturalist, William John Burchell, remarked in 1811 that the deep sand of the Flats made travel by cart or wagon extremely difficult. The situation was aggravated by a widespread shortage of firewood, causing fuel collectors to cut the relatively few indigenous shrubs and trees that stabilised the sand.

During the second half of the 19th century, the area was completely overrun by alien vegetation, mainly of Australian origin. The plants included hakeas and especially wattles (genus Acacia). The principal reason for this infestation lay in decisions made by the colonial authorities. It was an era before the advent of modern technological methods for the construction of permanent roads and in those days the Cape Flats was a massive sea of unstabilised sand dunes that moved with the winds. This made travel between Cape Town and the interior very difficult, particularly for the large ox-drawn wagons of the time. The authorities decided to try to stabilise the sand with plants native to the British colonies of New South Wales and Western Australia.

The earliest importation of wattles was in 1827. Massive plantings were established in the 1840s and 1850s and the work continued until well after 1875. At the time, the plan worked well enough: the march of the dunes was arrested. The price paid, in ecological terms, was that the Cape Flats was carpeted by invasive species. Serious efforts have in recent years been made to roll back this alien scourge.

Since 1950
The Cape Flats has undergone dramatic change in the past half a century. In 1950 the area was practically uninhabited. There was a single, narrow road across the Flats from Cape Town to The Strand that ran between walls of alien rooikrans bushes and one could travel for miles without seeing any sign of habitation other than a few fences and a handful of farmhouses. Native antelope roamed at will between the dense thickets of wattles. The army used the area for military exercises and the few farmers who inhabited the Flats eked out a living by growing vegetables in pockets of relatively poor soil between the barren dunes. Modern amenities were unknown; telephones were unknown, drinking water was collected in tanks from roofs and at night the rooms were lit by oil lamps.

The era of sand and antelopes vanished completely in little more than a generation. Vegetable farming persisted, but to a much lesser extent, because urbanisation enveloped vast tracts of land in short order.

The first use of the Cape Flats to remove people of colour from the more central urban areas was the township of Langa, which saw large scale removal after the Natives (Urban Areas) Act of 1923. It is the oldest township in Cape Town and was the location of much resistance to apartheid. Its origins go back to the 19th century.

During the apartheid era, race-based legislation such as the Group Areas Act and pass laws either forced non-white people out of more central urban areas designated for white people and into government-built townships in the Cape Flats, or made living in the area illegal, forcing many people into informal settlements elsewhere in the Cape Flats. Large housing projects were built here, mostly as part of the Nationalist government's larger effort to force the so-called Coloured communities and other people of Colour (mostly Xhosa) out of the central and western areas of Cape Town, which were designated as white areas under the Group Areas Act.

This meant that only white people could reside there permanently, as under the apartheid system black Africans were officially made citizens of native reserves (called "homelands"), therefore having resident worker status, despite often never having been there & having spent their entire lives in the now "designated white" regions.

Since many of the mostly Xhosa workers and their families, were designated under apartheid as citizens of assigned native reserves/homelands, many who were out of formal work were obliged to live in the area illegally, further contributing to the growth of informal settlements.

In addition to the extant townships of the mid-century, other large townships (such as Mitchell's Plain & Khayelitsha across the road) were created on the flats in the 1970s and 1980s. Both have become among the largest residential areas in Cape Town and some of the largest in the country at large.

Large scale rural-urban migration has rapidly increased the population, putting strain on housing with major increases in informal shack settlement, including in areas that were originally created on a planned basis, for example, Khayelitsha.

One of the major priorities of the RDP (Reconstruction & Development Programme) is to build houses.

Since the end of apartheid, these communities are no longer legally bound by racial restrictions; but history, language, economics and ethnic politics still contribute to homogeneity of local areas. So, for example, most residents of Mitchell's Plain likely still speak a locally inflected version of Afrikaans, along with English and either they or their parents were designated as Coloured by apartheid; most residents of Khayelitsha still speak Xhosa and English and either they or their parents were designated as Black by apartheid. Nonetheless, some areas of the Cape Flats have an increasing diversity of residents, with Xhosa-speaking people an increasingly noticeable presence in some previously mainly Afrikaans-speaking areas.

Culture and politics
Popular musicians from the area include pop singer Brenda Fassie and jazz artists Abdullah Ibrahim and Basil Coetzee, who named their song "Mannenberg" after a Cape Flats township. There is an active hip hop movement.

Its religious communities include (to name only a few) Afrikaans-speaking congregations of the Dutch Reformed Church, Rastafarian communities, people who engage only in traditional Xhosa practices, syncretic Xhosa Christian churches, evangelical Christian churches, and southern Africa's largest Muslim community (drawing its oldest roots from the historic Cape Malay community, which dates to the 17th century).

In the 1940s, a type of dental modification known as a passion gap became fashionable and remains popular today. The modification involves the removal of a person's top front teeth.

The Cape Flats' political history is complex and sometimes baffling even to insiders: for instance, the politics of the Coloured communities of the Cape Flats have included Trotskyist activism in earlier years, and mobilisation for the ANC's United Democratic Front in the 80s; and then, widespread support for the historically white National Party (which had presided over apartheid) in the early post-apartheid elections. More recently, the area has seen an expansion of African National Congress strength from its base in the Black townships and into historically Coloured areas, as well as a particularly strong local growth of left-wing social movements like the Treatment Action Campaign which offer a critique of government policies.

Crime

The Cape Flats is sometimes considered dangerous, although most communities are tight-knit, diverse and tolerant with vibrant cultures and traditions.

Almost all of the communities of the Cape Flats remain, to one degree or another, poverty-stricken. Serious social problems include a high rate of unemployment and high levels of gang activity. During the late 1990s and early 2000s, there was significant armed conflict between various gangs and PAGAD (People Against Gangsterism and Drugs), a vigilante organisation. Post-apartheid development projects, such as the Reconstruction & Development Programme, have also led to violent conflicts within communities. As of 2014, efforts to combat gangs include Hanover Park's Ceasefire programme, where former gang members "use their experiences to mediate gang disputes and help young men and women quit gang life. The gang violence escalated to the point where the South African National Defence Force had to be deployed to the gang-impacted areas of the Cape Flats to help the provincial police force deal with the increasing gang violence.

Some well-known criminal organisations based in the Cape Flats include:
 * The Americans: one of the largest gangs in the Cape Flats, known for their violent tactics and their involvement in drug trafficking.
 * The Hard Livings: a Cape Flats-based gang known for drug trafficking and operating extortion rings.
 * The Junky Funky Kids (JFK): the JFK gang has recently been engaged in a violent turf-war with The Americans gang, and in 2017, 4 of its members were sentenced to life for murder.

A wide range of community empowerment organisations work non-violently to combat the poverty, crime and health problems, and the role of civil society in many parts of the area is relatively strong. In late 2019, South African political group, the TRAKboys began enforcing a ceasefire between the gangs in Cape Flats. Citing the repeated failure of government initiatives aimed at curbing gun violence in Cape Flats, the TRAKboys created a system of community justice and enacted harsh punishments on any gang members found to be plotting violence on any members of a rival gang or a fellow Cape Flats resident.

Suburbs on the Cape Flats

 * Athlone
 * Bonteheuwel
 * Hanover Park
 * Khayelitsha
 * Langa
 * Lavender Hill
 * Parkwood
 * Ottery
 * Manenberg
 * Mitchells Plain
 * Bishop Lavis
 * Matroosfontein
 * Macassar
 * Gugulethu
 * Strandfontein
 * Nyanga
 * Philippi
 * Lotus River
 * Grassy Park
 * Heideveld

title: Cape Flats


text:
Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, $$4x^2 + 2xy - 3y^2$$

is a quadratic form in the variables $x$ and $y$. The coefficients usually belong to a fixed field $K$, such as the real or complex numbers, and one speaks of a quadratic form over $K$. If $K = R$, and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form $$-\mathbf{x}^\mathsf{T}\boldsymbol\Sigma^{-1} \mathbf{x}$$)

Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials.

Introduction
Quadratic forms are homogeneous quadratic polynomials in $n$ variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: $$\begin{align} q(x) &= ax^2&&\textrm{(unary)} \\ q(x,y) &= ax^2 + bxy + cy^2&&\textrm{(binary)} \\ q(x,y,z) &= ax^2 + bxy + cy^2 + dyz + ez^2 + fxz&&\textrm{(ternary)} \end{align}$$

where $a$, ..., $f$ are the coefficients.

The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers $2b$ or the $b$-adic integers $2b$. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in $2d$ variables has important applications to algebraic topology.

Using homogeneous coordinates, a non-zero quadratic form in $2f$ variables defines an $b$-dimensional quadric in the $d$-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections. An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates $f$ and the origin: $$q(x,y,z) = d((x,y,z), (0,0,0))^2 = \left\|(x,y,z)\right\|^2 = x^2 + y^2 + z^2.$$

A closely related notion with geometric overtones is a quadratic space, which is a pair $Z$, with $p$ a vector space over a field $Z_{p}$, and $n$ a quadratic form on V. See  below for the definition of a quadratic form on a vector space.

History
The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form $n$, where $(n − 2)$, $(n − 1)$ are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium BCE.

In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta, which includes, among many other things, a study of equations of the form $(x, y, z)$. He considered what is now called Pell's equation, $(V, q)$, and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange.

In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.

Associated symmetric matrix
Any $V$ matrix $K$ determines a quadratic form $q : V → K$ in $x^{2} + y^{2}$ variables by $$q_A(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j} = \mathbf x^\mathsf{T} A \mathbf x, $$ where $x$.

Example
Consider the case of quadratic forms in three variables $y$. The matrix $A$ has the form $$A=\begin{bmatrix} a&b&c\\d&e&f\\g&h&k \end{bmatrix}.$$

The above formula gives $$q_A(x,y,z)=ax^2 + ey^2 +kz^2 + (b+d)xy + (c+g)xz + (f+h)yz.$$

So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums $x^{2} − ny^{2} = c$, $x^{2} − ny^{2} = 1$ and $n × n$. In particular, the quadratic form $A$ is defined by a unique symmetric matrix $$A=\begin{bmatrix} a&\frac{b+d}2&\frac{c+g}2\\ \frac{b+d}2&e&\frac{f+h}2\\ \frac{c+g}2&\frac{f+h}2&k \end{bmatrix}.$$

This generalizes to any number of variables as follows.

General case
Given a quadratic form $q_{A}$, defined by the matrix $n$, the matrix

is symmetric, defines the same quadratic form as $A$, and is the unique symmetric matrix that defines $A = (a_{ij})$.

So, over the real numbers (and, more generally, over a field of characteristic different from two), there is a one-to-one correspondence between quadratic forms and symmetric matrices that determine them.

Real quadratic forms
A fundamental problem is the classification of real quadratic forms under a linear change of variables.

Jacobi proved that, for every real quadratic form, there is an orthogonal diagonalization; that is, an orthogonal change of variables that puts the quadratic form in a "diagonal form" $$ \lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2, $$ where the associated symmetric matrix is diagonal. Moreover, the coefficients $x, y, z$ are determined uniquely up to a permutation.

If the change of variables is given by an invertible matrix that is not necessarily orthogonal, one can suppose that all coefficients $b + d$ are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 0, 1, and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The signature of the quadratic form is the triple $c + g$, where these components count the number of 0s, number of 1s, and the number of −1s, respectively. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.

The case when all $f + h$ have the same sign is especially important: in this case the quadratic form is called positive definite (all 1) or negative definite (all −1). If none of the terms are 0, then the form is called ; this includes positive definite, negative definite, and isotropic quadratic form (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indefinite nondegenerate quadratic form of index $q_{A}$ (denoting $q_{A}$ 1s and $A = (a_{ij})$ −1s) is often denoted as $q_{A}$ particularly in the physical theory of spacetime.

The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in $λ_{1}, λ_{2}, ..., λ_{n}$ (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative". Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients, $λ_{i}$.

These results are reformulated in a different way below.

Let $(n_{0}, n_{+}, n_{−})$ be a quadratic form defined on an $λ_{i}$-dimensional real vector space. Let $(p, q)$ be the matrix of the quadratic form $p$ in a given basis. This means that $q$ is a symmetric $R^{p,q}$ matrix such that $$q(v) = x^\mathsf{T} Ax,$$ where x is the column vector of coordinates of $K / (K^{×})^{2}$ in the chosen basis. Under a change of basis, the column $(−1)^{n_{−}}|undefined$ is multiplied on the left by an $q$ invertible matrix $n$, and the symmetric square matrix $A$ is transformed into another symmetric square matrix $q$ of the same size according to the formula $$ A\to B=S^\mathsf{T}AS.$$

Any symmetric matrix $A$ can be transformed into a diagonal matrix $$ B=\begin{pmatrix} \lambda_1 & 0 & \cdots & 0\\ 0 & \lambda_2 & \cdots & 0\\ \vdots & \vdots & \ddots & 0\\ 0 & 0 & \cdots & \lambda_n \end{pmatrix}$$ by a suitable choice of an orthogonal matrix $n × n$, and the diagonal entries of $v$ are uniquely determined – this is Jacobi's theorem. If $x$ is allowed to be any invertible matrix then $n × n$ can be made to have only 0, 1, and −1 on the diagonal, and the number of the entries of each type ($S$ for 0, $A$ for 1, and $B$ for −1) depends only on $A$. This is one of the formulations of Sylvester's law of inertia and the numbers $S$ and $B$ are called the positive and negative indices of inertia. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix $S$, Sylvester's law of inertia means that they are invariants of the quadratic form $B$.

The quadratic form $n_{0}$ is positive definite if $n_{+}$ (similarly, negative definite if $n_{−}$) for every nonzero vector $A$. When $n_{+}$ assumes both positive and negative values, $n_{−}$ is an isotropic quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in $A$ variables can be brought to the sum of $q$ squares by a suitable invertible linear transformation: geometrically, there is only one positive definite real quadratic form of every dimension. Its isometry group is a compact orthogonal group $q$. This stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite orthogonal group $q(v) > 0$, is non-compact. Further, the isometry groups of $q(v) < 0$ and $v$ are the same ($q$, but the associated Clifford algebras (and hence pin groups) are different.

Definitions
A quadratic form over a field $q(v)$ is a map $q$ from a finite-dimensional $n$-vector space to $n$ such that $O(n)$ for all $O(p, q)$, $Q$ and the function $−Q$ is bilinear.

More concretely, an $O(p, q) ≈ O(q, p))$-ary quadratic form over a field $K$ is a homogeneous polynomial of degree 2 in $q : V → K$ variables with coefficients in $K$: $$q(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K. $$

This formula may be rewritten using matrices: let $K$ be the column vector with components $q(av) = a^{2}q(v)$, ..., $a ∈ K$ and $v ∈ V$ be the $q(u + v) − q(u) − q(v)$ matrix over $n$ whose entries are the coefficients of $K$. Then $$ q(x) = x^\mathsf{T} A x. $$

A vector $n$ is a null vector if $K$.

Two $x$-ary quadratic forms $x_{1}$ and $x_{n}$ over $A = (a_{ij})$ are equivalent if there exists a nonsingular linear transformation $n × n$ such that $$ \psi(x) = \varphi(Cx). $$

Let the characteristic of $K$ be different from 2. The coefficient matrix $q$ of $v = (x_{1}, ..., x_{n})$ may be replaced by the symmetric matrix $q(v) = 0$ with the same quadratic form, so it may be assumed from the outset that $n$ is symmetric. Moreover, a symmetric matrix $φ$ is uniquely determined by the corresponding quadratic form. Under an equivalence $ψ$, the symmetric matrix $K$ of $C ∈ GL(n, K)$ and the symmetric matrix $K$ of $A$ are related as follows: $$ B = C^\mathsf{T} A C. $$

The associated bilinear form of a quadratic form $q$ is defined by $$ b_q(x,y)=\tfrac{1}{2}(q(x+y)-q(x)-q(y)) = x^\mathsf{T}Ay = y^\mathsf{T}Ax. $$

Thus, $(A + A^{T})/2$ is a symmetric bilinear form over $A$ with matrix $A$. Conversely, any symmetric bilinear form $C$ defines a quadratic form $$ q(x)=b(x,x),$$ and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in $A$ variables are essentially the same.

Quadratic space
Given an $φ$-dimensional vector space $B$ over a field $ψ$, a quadratic form on $q$ is a function $b_{q}$ that has the following property: for some basis, the function $K$ that maps the coordinates of $A$ to $b$ is a quadratic form. In particular, if $n$ with its standard basis, one has $$ q(v_1,\ldots, v_n)= Q([v_1,\ldots,v_n])\quad \text{for} \quad [v_1,\ldots,v_n] \in K^n. $$

The change of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in $n$, although the quadratic form $V$ depends on the choice of the basis.

A finite-dimensional vector space with a quadratic form is called a quadratic space.

The map $K$ is a homogeneous function of degree 2, which means that it has the property that, for all $V$ in $Q : V → K$ and $q$ in $v ∈ V$: $$ Q(av) = a^2 Q(v). $$

When the characteristic of $Q(v)$ is not 2, the bilinear map $V = K^{n}$ over $V$ is defined: $$ B(v,w)= \tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).$$ This bilinear form $q$ is symmetric. That is, $Q$ for all $a$, $K$ in $v$, and it determines $V$: $K$ for all $B : V × V → K$ in $K$.

When the characteristic of $B$ is 2, so that 2 is not a unit, it is still possible to use a quadratic form to define a symmetric bilinear form $B(x, y) = B(y, x)$. However, $x$ can no longer be recovered from this $y$ in the same way, since $V$ for all $Q$ (and is thus alternating). Alternatively, there always exists a bilinear form $Q(x) = B(x, x)$ (not in general either unique or symmetric) such that $x$.

The pair $V$ consisting of a finite-dimensional vector space $K$ over $B′(x, y) = Q(x + y) − Q(x) − Q(y)$ and a quadratic map $Q(x)$ from $B′$ to $B′(x, x) = 0$ is called a quadratic space, and $x$ as defined here is the associated symmetric bilinear form of $B″$. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, $B″(x, x) = Q(x)$ is also called a quadratic form.

Two $(V, Q)$-dimensional quadratic spaces $V$ and $K$ are isometric if there exists an invertible linear transformation $Q$ (isometry) such that $$ Q(v) = Q'(Tv) \text{ for all } v\in V.$$

The isometry classes of $V$-dimensional quadratic spaces over $K$ correspond to the equivalence classes of $B$-ary quadratic forms over $Q$.

Generalization
Let $Q$ be a commutative ring, $n$ be an $(V, Q)$-module, and $(V′, Q′)$ be an $T : V → V′$-bilinear form. A mapping $n$ is the associated quadratic form of $K$, and $n$ is the polar form of $K$.

A quadratic form $R$ may be characterized in the following equivalent ways:
 * There exists an $M$-bilinear form $R$ such that $b : M × M → R$ is the associated quadratic form.
 * $R$ for all $R$ and $q : M → R : v ↦ b(v, v)$, and the polar form of $b$ is $B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v)$-bilinear.

Related concepts
Two elements $q$ and $q : M → R$ of $R$ are called orthogonal if $b : M × M → R$. The kernel of a bilinear form $q(v)$ consists of the elements that are orthogonal to every element of $q(av) = a^{2}q(v)$. $a ∈ R$ is non-singular if the kernel of its associated bilinear form is $v ∈ M$. If there exists a non-zero $q$ in $R$ such that $v$, the quadratic form $w$ is isotropic, otherwise it is definite. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of $V$ to a subspace $B(v, w) = 0$ of $B$ is identically zero, then $V$ is totally singular.

The orthogonal group of a non-singular quadratic form $Q$ is the group of the linear automorphisms of $\{0\}$ that preserve $v$: that is, the group of isometries of $V$ into itself.

If a quadratic space $Q(v) = 0$ has a product so that $Q$ is an algebra over a field, and satisfies $$\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) ,$$ then it is a composition algebra.

Equivalence of forms
Every quadratic form $Q$ in $U$ variables over a field of characteristic not equal to 2 is equivalent to a diagonal form $$q(x)=a_1 x_1^2 + a_2 x_2^2+ \cdots +a_n x_n^2.$$

Such a diagonal form is often denoted by $V$. Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

Geometric meaning
Using Cartesian coordinates in three dimensions, let $U$, and let $Q$ be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation $V$ depends on the eigenvalues of the matrix $Q$.

If all eigenvalues of $(V, Q)$ are non-zero, then the solution set is an ellipsoid or a hyperboloid. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an imaginary ellipsoid (we get the equation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it is a hyperboloid.

If there exist one or more eigenvalues $(A, Q)$, then the shape depends on the corresponding $A$. If the corresponding $q$, then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding $n$, then the dimension $⟨a_{1}, ..., a_{n}⟩$ degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of $x = (x, y, z)^{T}$. When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.

Integral quadratic forms
Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology.

An integral quadratic form has integer coefficients, such as $A$; equivalently, given a lattice $x^{T}Ax + b^{T}x = 1$ in a vector space $A$ (over a field with characteristic 0, such as $A$ or $λ_{i} = 0$), a quadratic form $b_{i}$ is integral with respect to $b_{i} ≠ 0$ if and only if it is integer-valued on $b_{i} = 0$, meaning $i$ if $b$.

This is the current use of the term; in the past it was sometimes used differently, as detailed below.

Historical use
Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean: This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).
 * twos in : the quadratic form associated to a symmetric matrix with integer coefficients
 * twos out : a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)

In "twos in", binary quadratic forms are of the form $x^{2} + xy + y^{2}$, represented by the symmetric matrix $$\begin{pmatrix}a & b\\ b&c\end{pmatrix}$$ This is the convention Gauss uses in Disquisitiones Arithmeticae.

In "twos out", binary quadratic forms are of the form $Λ$, represented by the symmetric matrix $$\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}.$$

Several points of view mean that twos out has been adopted as the standard convention. Those include:
 * better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
 * the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
 * the actual needs for integral quadratic form theory in topology for intersection theory;
 * the Lie group and algebraic group aspects.

Universal quadratic forms
An integral quadratic form whose image consists of all the positive integers is sometimes called universal. Lagrange's four-square theorem shows that $V$ is universal. Ramanujan generalized this $Q$ and found 54 multisets $R$ that can each generate all positive integers, namely, indent = 1 |

$Q$

$Λ$

$Λ$

$Q(x, y) ∈ Z$

$x, y ∈ Λ$

$ax^{2} + 2bxy + cy^{2}$

$ax^{2} + bxy + cy^{2}$

There are also forms whose image consists of all but one of the positive integers. For example, $w^{2} + x^{2} + y^{2} + z^{2}$ has 15 as the exception. Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.

title: Quadratic form


text:
Arthur Lowe

Arthur Lowe (22 September 1915 – 15 April 1982) was an English actor. His acting career spanned 37 years, including starring roles in numerous theatre and television productions. He played Captain Mainwaring in the British sitcom Dad's Army from 1968 until 1977, was nominated for seven BAFTAs and became one of the most recognised faces on UK television.

Lowe began acting professionally in England in 1945, after army service in the Second World War. He worked in theatre, film and television throughout the 1950s but it was not until he landed the part of Leonard Swindley in the television soap Coronation Street in 1960 that he came to national attention. He played the character until 1966, while continuing theatre and other acting work.

In 1968 he took on his role in Dad's Army, written by Jimmy Perry and David Croft. The profile he gained from the role led to further character roles. Despite increasingly poor health in his final years, he maintained a busy professional schedule until his death from a stroke on 15 April 1982, aged 66.

Early life
Lowe was born in Hayfield, Derbyshire, the only child of Arthur Lowe (1888–1971) and his wife, Nan ( Mary Annie Ford; 1885–1981). Lowe's father, a tall man known as "Big Arthur", worked for the Great Central Railway, which was absorbed into the London and North Eastern Railway in 1923. In 1916, Big Arthur took up a job as clerk at London Road Station, Manchester, shortly before being called up for First World War service. The family rented a house in Hemmons Road, in the Manchester suburb of Levenshulme, where Little Arthur (as he was known) attended Chapel Street School. From about 1927 he went to Alma Park School, where one of his first stage performances was in a school production of The Grand Cham's Diamond in December 1929.

Lowe's intention to join the Merchant Navy was thwarted by his poor eyesight. His first job after leaving school was as a barrow boy for the Manchester branch of motor accessory company Brown Brothers. After progressing to the role of clerk within the firm he took up a job at the aircraft factory of the Fairey Aviation Company in 1936. He described his job of progress chaser as "a sort of time and motion man chivvying the fellows along and seeing that they produced a certain amount of work each day". He also had to check that the parts for building the planes were where they needed to be on the production line.

War service
In February 1939 Lowe joined the Territorial Army, which meant several months later he was among the first men called up to serve in the Second World War. He served with the Duke of Lancaster's Own Yeomanry. Initially training with horses, the regiment soon became a mechanised unit of the Royal Artillery. Lowe was medically regraded due to his poor eyesight and after training in wireless and as a radar technician transferred to the Royal Army Ordnance Corps. After working on searchlights in Lincolnshire he was sent out to Egypt in 1942, where he soon transferred to the Royal Electrical and Mechanical Engineers. He was a good horseman and learned to speak Arabic. After a period in the Suez Canal Zone he was stationed at the REME's 15th Radio Repair Workshops at Rafah.

Lowe soon found outlets for developing his talents in entertainment. He was known among the troopers for his impressions of officers and crooners, and when radio equipment was stolen, he read the BBC News over his camp's PA system. In January 1943 he called a meeting to form an amateur dramatics group, the REME No. 1 Welfare Club Dramatic Society. "It was sheer bloody boredom that did it", he said later, "and after that I was hooked". He took his first appearance on stage in The Monkey's Paw on 8 February 1943 and continued both to organise and act in plays, as well as a Christmas revue. His efforts led to a posting with the No. 2 Field Entertainment Unit, promoted to the rank of sergeant major. In this role he helped outlying units to produce their own shows. He assisted Martin Benson in establishing the Mercury Theatre in Alexandria, including in production and management, but not as an actor.

Following the end of the war, Lowe returned to Britain in November 1945, although he was not officially demobbed until March 1946.

Early career
In 1945, Lowe's father was organising special railway trips and excursions, including private trains for circuses and theatre companies. He arranged an audition for Lowe with Eric Norman for the Frank H. Fortescue Famous Players repertory company. Lowe was immediately offered a trial in the comedy play Bedtime Story, in which he took the part of Dickson. In this role he made his professional acting debut at the Manchester Repertory Theatre on 17 December 1945. He was paid £5 per week for twice-nightly performances. In eight months with Fortescue's he appeared in 33 plays and gave 396 performances.

During this time Lowe began a romantic relationship with Joan Cooper (1922–1989), a married actress in the company whose husband also began an affair at about the same time. Arthur and Joan were engaged in June 1946 and lived together from August. After Joan's divorce came through they married at a registry office in Robert Adam Street, The Strand, London, on 10 January 1948. Joan had a son, David Gatehouse, from her first marriage. Another son, Stephen Lowe, was born on 23 January 1953. The couple remained together until Lowe's death.

Lowe worked with repertory companies around the country. After a year at the County Theatre, Hereford, 1946-1947, he moved to London in 1948 and for the next three years mostly worked in South London theatres. An early brief film role was as a reporter for the Tit-Bits magazine, near the end of the Ealing Studios dark comedy classic Kind Hearts and Coronets (1949). His first West End role came in 1950, as Wilson the butler in Guy Bolton's Larger Than Life. Lowe became known for his character roles, which in 1952 included a breakthrough part as Senator Brockbank in the musical Call Me Madam at the London Coliseum. Other roles in musicals included a part in the 1954 London revival of Pal Joey and eighteen months as the salesman in the first West End production of The Pajama Game, from 1955 to 1957. His name first appeared in lights in 1957, at the Piccadilly Theatre, with the part of Bert Vokes in the murder melodrama A Dead Secret. This also brought his first West End reviews.

Lowe made his first television appearance in 1951, in an episode of the BBC series I Made News. He would work in television every year afterwards, until his death. 1950s roles included various minor parts in dramas, including the crime series Murder Bag. He played the role of the gunsmith in Leave It to Todhunter (1958), appeared in the comedy series Time Out for Peggy, and played a fussy, nervous character in an episode of Dial 999. His first regular television part was as ship steward Sydney Barker in the ABC-TV series All Aboard (1958-1959).

In 1960 Lowe took up a regular role as draper and lay preacher Leonard Swindley in the northern soap opera Coronation Street, in which he appeared until 1965. He negotiated a contract through which he only had to work six months of the year, three months on and three months off. During the months he was not playing Swindley, he remained busy on stage or making one-off guest appearances in other TV series such as Z-Cars (1962) and The Avengers (1967) (episode entitled "Dead Man's Treasure"). His most acclaimed stage roles during this period included pompous north-country alderman Michael Oglethorpe in Henry Livings's Stop It, Whoever You Are at the Arts Theatre (1961), and Sir Davey Dunce in The Soldier's Fortune at the Royal Court Theatre (1966).

Lowe did not relish work on Coronation Street and was happy to give it up, but viewer responses to his character led to him reprising Swindley for starring roles in the spin-off series Pardon the Expression (1966) and its sequel Turn Out the Lights (1967).

Stardom
In 1968, Lowe was cast in his best remembered role, as Home Guard platoon leader Captain Mainwaring in the BBC sitcom Dad's Army (1968–1977). Some colleagues on the show later remarked that the role resembled him: pompous and bumbling. Frank Williams said he felt this perception was unfair: "He certainly didn't suffer fools gladly and always knew his own mind, but he also had an ability to laugh at himself. Personally, I found him to be a most kind and generous man". David Croft said Lowe had to be treated with kid gloves. He had firm ideas on what he was willing to do and never took his script home, which resulted in uncertainty over his lines. He could be pompous and over time his part was written so there was a blurring of the line between actor and character. An oddity of his contract was that he would never have to remove his trousers.

Lowe held conservative political views and disapproved of the left-wing politics of his co-star Clive Dunn. Dunn, in turn, described some of Lowe's opinions as outrageous, but as an actor rated him "ten out of ten in his field". Despite some tensions, Jimmy Perry described the cast as a "marvellous bunch of pros" with "no sort of volatile animosity between anybody".

Lowe also played Mainwaring in a radio version of Dad's Army, a stage play and a feature-length film released in 1971. He played Mainwaring's drunken brother Barry Mainwaring, in the series' 1975 Christmas episode "My Brother and I".

While Dad's Army was not in production, Lowe's work continued to include stage roles. In 1968, he was invited by Sir Laurence Olivier to join the National Theatre at the Old Vic, to play divorce solicitor A.B. Raham in Somerset Maugham's Home and Beauty. He returned to the company in 1974 to play Stephano in Peter Hall's production of The Tempest, starring Sir John Gielgud. In the same year he appeared as Ben Jonson alongside Gielgud's Shakespeare in Edward Bond's Bingo at the Royal Court Theatre.

Lowe also had prominent parts in several films directed by Lindsay Anderson, including if.... (1968) and O Lucky Man! (1973), for which he won a BAFTA for Best Actor in a Supporting Role. His other film parts during this period included Spike Milligan's surreal The Bed Sitting Room (1969), in which he mutates into a parrot. He played a drunken butler in The Ruling Class (1972) with Peter O'Toole, and theatre critic Horace Sprout in the horror film Theatre of Blood (1973), in which the character is murdered by a deranged actor played by Vincent Price.

On television, Lowe appeared twice as a guest performer on The Morecambe and Wise Show (1971 and 1977), alongside Richard Briers in a series of Ben Travers farces for the BBC, as the pompous Dr Maxwell in the ITV comedy Doctor at Large (1971) and as Redvers Bodkin, a snooty, old-fashioned butler, in the short-lived sitcom The Last of the Baskets (1971–72). Between 1971 and 1973 Lowe joined Dad's Army colleague Ian Lavender, on the BBC radio comedy Parsley Sidings and he played Mr Micawber in a BBC television serial of David Copperfield (1974). He employed a multitude of voices on the BBC animated television series Mr. Men (1974), in which he was the narrator.

In 1972, Lowe also recorded the novelty songs "How I Won The War" and "My Little Girl, My Little Boy". While touring at coastal theatres with his wife, Lowe used his 1885 former steam yacht Amazon as a floating base. He bought Amazon as a houseboat in 1968, but realised her potential and took her back to sea in 1971; this vessel is still operating in the Mediterranean. The ship had a bar with a semicircular notch cut halfway along, to enable both the portly figure of Lowe and his wife to serve behind the bar at the same time, acting as hosts during the parties they threw on board.

In an interview for a Dad's Army retrospective on BBC television in 2010, Clive Dunn described him sitting at the bar in the evenings when they were filming on location, consuming a drink which Lowe named 'Amazon' after his yacht. Dunn described the drink as comprising "gin and ginger ale, with a single slice of cucumber".

Lowe seldom made public political statements, but his face appeared on posters and other advertising in support of the "Voting Yes" campaign for the 1975 United Kingdom European Communities membership referendum. He also appeared at a Conservative Party fundraising bazaar in Edward Heath's constituency.

Declining health and later career
By the mid-1970s Lowe suffered from narcolepsy, which caused him to fall asleep during rehearsals, performances, and at other unintended times - sometimes in the middle of a sentence. Stephen Lowe said that although he was often mistaken for drunk, he very rarely was. While both biographies of Lowe acknowledge his high consumption of alcohol, neither claim it extended to alcoholism. Lowe was also unfit, a smoker, and increasingly overweight. In 1979, he suffered a minor stroke. Despite his generally declining health, including worsening narcolepsy, he maintained a busy professional life. Derek Benfield described him as a workaholic.

When Dad's Army ended in 1977, Lowe remained in demand, taking starring roles in television comedies such as Bless Me, Father (1978–1981), as the mischievous Catholic priest Father Charles Clement Duddleswell and in Potter (1979–80) as the busybody Redvers Potter. In 1980 he toured Australia and New Zealand with a production of Derek Benfield's play Beyond a Joke. Around this time Lowe was making many television commercials, with no fewer than nineteen in 1981 alone.

His later stage career mainly involved touring the English provinces with his wife. He seldom took on a stage play unless it included a role for Joan and this saw some opportunities fall through. Lowe's agent Peter Campbell said the last ten years of his theatre career were "blown" by this condition, and Stephen Lowe thought his mother placed unreasonable pressure on his father to find her roles. Frank Williams said the couple shared a great love story, and if the arrangement held Lowe back it was only because he chose to be held back. Ian Lavender thought Lowe's narcolepsy led him to pull back from his range and choose safer roles.

In 1981 Lowe reprised his role as Captain Mainwaring for the pilot episode of It Sticks Out Half a Mile, a radio sequel to Dad's Army. At Christmas that year he and Joan appeared in the pantomime Mother Goose at Victoria Palace, London. In January 1982, Richard Burton had his private aeroplane fly Lowe to Venice to film a cameo role in the television miniseries Wagner.

Death and last released works
On 14 April 1982, Lowe gave a live televised interview on Pebble Mill at One. At just after 6 pm the same day, he collapsed from the onset of a stroke in his dressing room at the Alexandra Theatre, Birmingham. This was before a performance of Home at Seven in which he was due to appear with his wife, Joan. He was taken, unconscious, to Birmingham General Hospital, where he died at about 5 am, at the age of 66.

Lowe was cremated and his ashes were scattered at Sutton Coldfield Crematorium, following a small funeral of which few people were notified and fewer than a dozen attended. Joan did not attend, as she refused to miss a performance of Home at Seven and was appearing in Belfast at the time. According to her friend Phyllis Bateman, the couple had a pact that neither would go to the other's funeral. Stephen said his parents were not sentimental or religious and Joan's coping mechanism was summed up in the adage, "the show must go on". A memorial service was held on 24 May 1982 at St Martin-in-the-Fields, attended by Lowe's wife and family, former colleagues and many friends.

Lowe's final film and television performances premiered after his death. His last feature film was Lindsay Anderson's Britannia Hospital (1982). In his final sitcom, A.J. Wentworth, B.A. (1982), he starred as a boys' preparatory school master. Wagner was Lowe's last screen role, released in December 1983.

Recognition
Tom Cole wrote in the Radio Times: "There are few actors who charmed viewers both young and old with such ease, and fewer still who could be trusted with the task of bringing classic literary characters like Charles Pooter and A.J. Wentworth to life." Graham Lord wrote, in his 2002 biography, that "almost every actor who worked with Arthur considered him to be outstanding". He gave as an exception Martin Benson, who said Lowe did not have a lot of vocal skill in his rep days, "and I don't think he had afterwards either...A lot of his success came from this oddball personality that he had and the fact that later in his career he had some very good writers."

In 2002, Paul Scofield described Lowe as a rare talent and "seriously brilliant actor", and said it was his timing that set him apart. Jimmy Perry agreed about his timing: "It was faultless. He could get huge laughs with such simple lines as 'Just a moment,' 'how dare you,' and 'you stupid boy'" - all catchphrases from Dad's Army. Perry also described Lowe as a kind man who went out of his way to help actors less fortunate than himself.

Approach to acting
In the 1970s, Lowe said he had “simply wanted to be the best character actor going” and it was only television that brought him stardom. Of his preferred style of comic acting he said: “Anybody could get a laugh if they pissed into the pit. But it wouldn’t be the right laugh.” He claimed to treat every comic part as a straight part, saying: “The more seriously you play the part, the funnier it is. You see, people are only funny to other people, never to themselves.”

Biographies
Two biographies of Arthur Lowe have been published: Arthur Lowe – Dad's Memory by his son Stephen, in 1997; and Arthur Lowe by Graham Lord in 2002. In 2000, The Unforgettable Arthur Lowe was part of The Unforgettable series of TV biographies of comedy performers.

Memorials
In December 2007 plans were announced for a statue of Lowe to be erected in Thetford, Norfolk, where the outside scenes for Dad's Army were filmed. Series co-writer David Croft unveiled the statue on 19 June 2010. It depicts Lowe in the character of Captain Mainwaring, sitting upright on a simple bench in Home Guard uniform, with his swagger stick across his knees.

The star has also had two blue plaques unveiled, one at Maida Vale and one at his birthplace in Hayfield, Derbyshire.

Portrayals
Robert Daws portrayed Lowe in the BBC Radio 4 drama Dear Arthur, Love John by Roy Smiles, first broadcast in 2012. The play charts the relationship between Lowe and John Le Mesurier. John Sessions played him in the 2015 television movie ''We're Doomed! The Dad's Army Story''.

Discography
(Re-released 1979 by BBC as 'The Mr. Men Songs', and 'The Mr. Men Songs with Arthur Lowe').
 * 'Vocal Gems from Call Me Madam', (1952), (78rpm), Columbia Records D.B. 3067
 * Bless 'em All (1969). LP, World Record Club (ST108). Performed with the Richmond Orchestra and Chorus under Malcolm Lockyer. Produced by Bob Barrett. Songs: "Who Do You Think You Are Kidding, Mr Hitler?", "This is the Army Mr Jones", "Kiss Me Goodnight, Sergeant Major", "I'll Be Seeing You", "Lili Marlene", "Mairzy Doats and Dozy Doats", "The Army, the Navy and the Air Force", "Bless 'em All", "Run, Rabbit, Run!", "(We're Gonna Hang Out) The Washing on the Siegfried Line", "That Lovely Weekend", "Underneath the Arches", "Alouette", "Waltzing Matilda" and "Now is the Hour".
 * Home Town (1975). 45rpm, Warner Bros. Records (K16670). B side to "A Nightingale Sang in Berkeley Square" performed by John Le Mesurier.
 * 'The Mr. Men with Arthur Lowe ', (1976) L.P.: Epic: SEPC 81184

title: Arthur Lowe


text:
John Venn

John Venn, FRS, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. In 1866, Venn published The Logic of Chance, a groundbreaking book which espoused the frequency theory of probability, arguing that probability should be determined by how often something is forecast to occur as opposed to "educated" assumptions. Venn then further developed George Boole's theories in the 1881 work Symbolic Logic, where he highlighted what would become known as Venn diagrams.

Early life
John Venn was born on 4 August 1834 in Kingston upon Hull, Yorkshire, to Martha Sykes and Rev. Henry Venn, who was the rector of the parish of Drypool. His mother died when he was three years old. Venn was descended from a long line of church evangelicals, including his grandfather John Venn. Venn was brought up in a very strict atmosphere at home. His father Henry had played a significant part in the Evangelical movement and he was also the secretary of the Society for Missions to Africa and the East, establishing eight bishoprics overseas. His grandfather was pastor to William Wilberforce of the abolitionist movement, in Clapham.

He began his education in London joining Sir Roger Cholmeley's School, now known as Highgate School, with his brother Henry in September 1846. He moved on to Islington Proprietary School.

University life and career
In October 1853, he went to Gonville and Caius College, Cambridge. He found the Mathematical Tripos unsuited to his mathematical style, complaining that the handful of private tutors he worked with "always had the Tripos prominently in view". In contrast, Venn wished to investigate interesting ideas beyond the syllabus. Nonetheless, he was Sixth Wrangler upon sitting the exams in January 1857.

Venn experienced, in his words, a "reaction and disgust" to the Tripos which led him to sell his books on mathematics and state that he would never return to the subject. Following his family vocation, he was ordained as an Anglican priest in 1859, serving first at the church in Cheshunt, Hertfordshire, and later in Mortlake, Surrey.

In 1862, he returned to Cambridge as a lecturer in moral science, studying and teaching political economy, philosophy, probability theory and logic. He reacquainted himself with logic and became a leading scholar in the field through his textbooks The Logic of Chance (1866), Symbolic Logic (1881) and The Principles of Empirical or Inductive Logic (1889). His academic writing was influenced by his teaching: he saw Venn diagrams, which he called "Eulerian Circles" and introduced in 1880, as a pedagogical tool. Venn was known for teaching students across multiple Cambridge colleges, which was rare at the time.

In 1883, he resigned from the clergy, having concluded that Anglicanism was incompatible with his philosophical beliefs.

In 1903 he was elected President of the college, a post he held until his death.

"I began at once somewhat more steady work on the subjects and books which I should have to lecture on. I now first hit upon the diagrammatical device of representing propositions by inclusive and exclusive circles. Of course the device was not new then, but it was so obviously representative of the way in which any one, who approached the subject from the mathematical side, would attempt to visualise propositions, that it was forced upon me almost at once."

- John Venn

With his son, Venn developed a bowling machine that was able to impart spin to a cricket ball. When members of the Australian cricket team visited Cambridge in June 1909, Venn’s machine bowled Victor Trumper, one of their star batsmen. The machine was recreated in 2024 by the university engineering department.

In 1883, Venn was elected a Fellow of the Royal Society, and in 1884, he was awarded a Sc.D. by Cambridge.

He died on 4 April 1923.

Civic and personal life
In 1868, Venn married Susanna Carnegie Edmonstone with whom he had one son, John Archibald Venn. His son entered the mathematics field as well and became Vice-Chancellor of Cambridge University.

Venn was an active member of civic society in Cambridge. He was a committee member of the Cambridge Charitable Organisations Society, elected vice-chairman in December 1884. Venn was president of the Cambridge Antiquarian Society in 1908–1909. He is also listed as a vice president of the Cambridge Provident Medical Institution.

Venn was a supporter of votes for women. He co-signed with his wife Susanna, a letter to the Cambridge Independent Press, published 16 October 1908, encouraging women to put themselves forward as candidates for the Cambridge Town Council elections. The letter was co-sponsored by Maud Darwin and Florence Ada Keynes.

Venn was also a gardener, regularly taking part in local competitions organised by groups such as the Cambridgeshire Horticultural Society, winning prizes for his roses in July 1885 and for his white carrots later that September.

Memorials

 * In 2017 the Drypool Bridge in Hull was decorated with intersecting circles, in honour of Venn and an unofficial 'blue plaque' is installed near the same location on Clarence Street.
 * Venn is commemorated at the University of Hull by the Venn Building.
 * A stained glass window in the dining hall of Gonville and Caius College, Cambridge, commemorates Venn's work.
 * In commemoration of the 180th anniversary of Venn's birth, on 4 August 2014, Google replaced its normal logo on global search pages with an interactive and animated Google Doodle that incorporated the use of a Venn diagram.
 * Venn Street in Clapham, London, which was the home of his grandfather, shows a Venn diagram on the street sign.

Publications
Venn compiled Alumni Cantabrigienses, a biographical register of former members of the University of Cambridge. It was edited by Venn and his son John Archibald Venn and published by Cambridge University Press in ten volumes between 1922 and 1953.

His other works include:

 * . Two further editions were published.
 * . Two further editions were published.
 * . Two further editions were published.
 * . Two further editions were published.

title: John Venn


text:
Coloureds

[[File:South Africa 2011 Coloured population proportion map.svg|thumb|250px|Coloured people as a proportion of the total population in South Africa.

]] [[File:South Africa 2011 Coloured population density map.svg|thumb|250px|Density of the Coloured population in South Africa.

]]

Coloureds (Kleurlinge) refers to members of multiracial ethnic communities in South Africa who have ancestry from African, European, and Asian people. The intermixing of different races began in the Cape province of South Africa, with European settlers intermixing with the indigenous Khoi tribes, and Asian slaves of the region. Later various other European nationals also contributed to the growing mixed race people, who would later be officially classified as coloured by the apartheid government in the 1950s.

Coloured was a legally defined racial classification during apartheid referring to anyone not white or of the black Bantu tribes, which effectively largely meant people of colour.

In the Western Cape, a distinctive Cape Coloured and affiliated Cape Malay culture developed. Genetic studies suggest the group has the highest levels of mixed ancestry in the world.

The majority of coloureds are found in the western cape but are prevalent throughout the country. In Cape Town, they form 43.2% of the total population, according to the South African National Census of 2011.

The apartheid-era Population Registration Act, 1950 and subsequent amendments, codified the Coloured identity and defined its subgroups, including Cape Coloureds and Malays. Indian South Africans were initially classified under the act as a subgroup of Coloured. As a consequence of Apartheid policies and despite the abolition of the Population Registration Act in 1991, Coloureds are regarded as one of four race groups in South Africa. These groups (blacks, whites, Coloureds and Indians) still tend to have strong racial identities and to classify themselves and others as members of these race groups. The classification continues to persist in government policy, to an extent, as a result of attempts at redress such as Black Economic Empowerment and Employment Equity.

Background
The Coloured / Malay community is predominantly descended from numerous interracial sexual unions, primarily between Western European men with Khoisan, Indian and Asian women in the Cape Colony from the 17th century onwards. The intermixing of different races began in the Cape province of South Africa during the 17th and 18th centuries, the Dutch east indies enslaved people from Asian regions, including; Indonesia, Malaysia and India. These individuals were brought to the Cape Colony to work on farms, in households, as they were enslaved laborer's. There is a significant genetic mixture of European, African and (Indian) Asian DNA in the modern ethnic group of Coloured people. Thus forced into their own communities and therefore created a generational mix of people and are to date an ethnic group.

A ten year study of genetics worldwide noted the Coloured community has highest levels of mixed ancestry anywhere, including African, European, Southeast Asian and South Indian contributions.

A diverse heritage including Indigenous African  Khoisan, European settlers (Dutch, French, British) Asian slaves (mainly from Indonesia, Malaysia and India)

In addition to African, European and Asian ancestry, Coloured people had some portion of Spanish or Portuguese ancestry. In the early 19th century some immigrants from Brazil arrived in South Africa as sailors, traders or refugees, and some intermarried with local mixed race (Coloured) communities. Also The Canary islands (Spain), located off the northwest coast of Africa, were a Spanish colony, and during the 17th and 18th centuries, the Dutch east India Company and other European powers brought enslaved people from the Canary Islands to South Africa, particularly to the Cape Colony (now known as South Africa). The Dutch east India company ship "Het Gelderland", which arrived at the Cape in 1671 with 174 slaves from the Canary Islands and The Portuguese ship, "Sao Jose" which was captured by the Dutch in 1713 and brought to the Cape with slaves from the Canary Islands. These enslaved people were forced to work on farms, in households, and in other industries and many were subjected to harsh conditions and treatment. The intermixing among European men and Spanish and Portuguese women's descendants are part of the diverse Coloured communities in South Africa. It is however important to note that Spanish and Portuguese ancestry is not a dominant feature amongst the Coloured identity in South Africa, Namibia and Zimbabwe. Individual family histories and ancestry may vary widely while the African, European and Asian ancestry is dominant amongst Coloured people from Africa.

Zimbabwean Coloureds are descended from Shona or Ndebele, British and Afrikaner settlers, as well as Arab and Asian people. Griqua, on the other hand, are descendants of Khoisan women and Afrikaner Trekboers. Despite these major differences, as both groups have ancestry from more than one naturalised racial group, they are classified as coloured in the South African context. Such mixed-race people did not necessarily self-identify this way; some preferred to call themselves black or Khoisan or just South African.

The Griqua were subjected to an ambiguity of other creole people within Southern African social order. According to Nurse and Jenkins (1975), the leader of this "mixed" group, Adam Kok I, was a former slave of the Dutch governor who was manumitted and provided land outside Cape Town in the eighteenth century. With territories beyond the Dutch East India Company's administration, Kok provided refuge to deserting soldiers, runaway slaves, and remaining members of various Khoikhoi tribes. In South Africa and many neighbouring countries, the white minority governments historically segregated Africans from Europeans after settlement had progressed, and increasingly classified all mixed race people together into a third group, despite their numerous ethnic and national differences in ancestry. The imperial and apartheid governments categorized them as Coloured. In addition, other distinctly homogeneous ethnic groups also traditionally viewed the mixed-race populations as a separate group, and a growing number of mixed-race people also embraced a shared identity.

During the apartheid era in South Africa of the second half of the 20th century, the government used the term "Coloured" to describe one of the four main racial groups it defined by law (the fourth was "Asian," later "Indian"). This was an effort to impose white supremacy and maintain racial divisions. Individuals were classified as White South Africans (formally classified as "European"), Black South Africans (formally classified as "Native", "Bantu" or simply "African" and comprising the majority of the population), Coloureds (mixed-race) and Indians (formally classified as "Asian"). The census in South Africa during 1911 played a significant role in defining racial identities in the country. One of the most noteworthy aspects of this census was the instructions given to enumerators on how to classify individuals into different racial categories. The category of "coloured persons" was used to refer to all people of mixed race, and this category included various ethnic groups such as Hottentots, Bushmen, Cape Malays, Griquas, Korannas, Creoles, Negroes, and Cape Coloureds.

Of particular importance is the fact that the instruction to classify "coloured persons" as a distinct racial group included individuals of African descent, commonly referred to as Negroes. Therefore, it is important to note that Cape Coloureds, as a group of mixed-race individuals, also have African ancestry and can be considered as part of the broader African diaspora.

Although the apartheid government recognised various coloured subgroups, including the Cape Malays and Cape Coloureds, the Coloured population, was for many purposes treated as a single group, despite their varying ancestries and cultures. Also during apartheid, many Griqua began to self-identify as Coloureds during the apartheid era, because of the benefits of such classification. For example, Coloureds did not have to carry a dompas (a pass, an identity document designed to limit the movements of the black population), while the Griqua, who were seen as an indigenous African group, though heavily mixed, did.

In the 21st century, Coloured people constitute a plurality of the population in the provinces of Western Cape (48.8%), and a large minority in the Northern Cape (40.3%), both areas of centuries of mixing among the populations. In the Eastern Cape, they make up 8.3% of the population. Most speak Afrikaans, as they were generally descendants of Dutch and Afrikaner men and grew up in their society. About twenty percent of the Coloured speak English as their mother tongue, mostly those of the Eastern Cape and KwaZulu-Natal. Virtually all Cape Town Coloureds are bilingual.

Genetics
At least one genetic study indicates that Cape Coloureds have ancestries from the following ethnic groups. Not all Coloureds in South Africa had the same ancestry. It is important to note here that genetic reference cluster term "Khoisan" itself refers to a colonially admixed population cluster, hence the concatenation, and is not a straightforward reference to ancient African pastoralist and hunter ancestry, which is often demarcated by the L0 haplogroup ancestry common in the general South African native population which is also integral part of other aboriginal genetic reference cluster terms like "South-East African Bantu".
 * Indigenous Khoisan: (32–43%)
 * Indigenous Bantu: (20-36%)
 * Peoples from Europe: (21–28%)
 * Peoples from South and Southeast Asia: (30- 35%).

Pre-apartheid era
Coloured people played an important role in the struggle against apartheid and its predecessor policies. The African Political Organisation, established in 1902, had an exclusively Coloured membership; its leader Abdullah Abdurahman rallied Coloured political efforts for many years. Many Coloured people later joined the African National Congress and the United Democratic Front. Whether in these organisations or others, many Coloured people were active in the fight against apartheid.

The political rights of Coloured people varied by location and over time. In the 19th century they theoretically had similar rights to Whites in the Cape Colony (though income and property qualifications affected them disproportionately). In the Transvaal Republic or the Orange Free State, they had few rights. Coloured members were elected to Cape Town's municipal authority (including, for many years, Abdurahman). The establishment of the Union of South Africa gave Coloured people the franchise, although by 1930 they were restricted to electing White representatives. They conducted frequent voting boycotts in protest. Such boycotts may have contributed to the victory of the National Party in 1948. It carried out an apartheid programme that stripped Coloured people of their remaining voting powers.

The term "kaffir" is a racial slur used to refer to Black African people in South Africa. While it is still used against black people, it is not as prevalent as it is against coloured people.

Apartheid era
Coloured people were subject to forced relocation. For instance, the government relocated Coloured from the urban Cape Town areas of District Six, which was later bulldozed. Other areas they were forced to leave included Constantia, Claremont, Simon's Town. Inhabitants were moved to racially designated sections of the metropolitan area on the Cape Flats. Additionally, under apartheid, Coloured people received education inferior to that of Whites. It was, however, better than that provided to Black South Africans.

J. G. Strijdom, known as "the Lion of the North", continued the impetus to restrict Coloured rights, in order to entrench the new-won National Party majority. Coloured participation on juries was removed in 1954, and efforts to abolish their participation on the common voters' roll in the Cape Province escalated drastically; it was accomplished in 1956 by a supermajority amendment to the 1951 Separate Representation of Voters Act, passed by Malan but held back by the judiciary as unconstitutional under the South Africa Act, the Union's effective constitution. In order to bypass this safeguard, enforced since 1909 to ensure Coloured political rights in the then-British Cape Colony, Strijdom's government passed legislation to expand the number of Senate seats from 48 to 89. All of the additional 41 members hailed from the National Party, increasing its representation in the Senate to 77 in total. The Appellate Division Quorum Bill increased the number of judges necessary for constitutional decisions in the Appeal Court from five to eleven. Strijdom, knowing that he had his two-thirds majority, held a joint sitting of parliament in May 1956. The entrenchment clause regarding the Coloured vote, known as the South Africa Act, were thus eliminated and the Separate Representation of Voters Act passed, now successfully.

Coloureds were placed on a separate voters' roll from the 1958 election to the House of Assembly and forward. They could elect four Whites to represent them in the House of Assembly. Two Whites would be elected to the Cape Provincial Council and the governor general could appoint one senator. Both blacks and Whites opposed this measure, particularly from the United Party and more liberal opposition. The Torch Commando was very prominent, while the Black Sash (White women, uniformly dressed, standing on street corners with placards) also made themselves heard. In this way, the question of the Coloured vote became one of the first measures of the regime's unscrupulous nature and flagrant willingness to manipulate its inherited Westminster system. It would remain in power until 1994.

Many Coloureds refused to register for the new voters' roll and the number of Coloured voters dropped dramatically. In the next election, only 50.2% of them voted. They had no interest in voting for White representatives — an activity which many of them saw as pointless, and only persisted for ten years.

Under the Population Registration Act, as amended, Coloureds were formally classified into various subgroups, including Cape Coloureds, Cape Malays and "other coloured". A portion of the small Chinese South African community was also classified as a coloured subgroup.

In 1958, the government established the Department of Coloured Affairs, followed in 1959 by the Union for Coloured Affairs. The latter had 27 members and served as an advisory link between the government and the Coloured people.

The 1964 Coloured Persons Representative Council turned out to be a constitutional hitch which never really got going. In 1969, the Coloureds elected forty onto the council to supplement the twenty nominated by the government, taking the total number to sixty.

Following the 1983 referendum, in which 66.3% of White voters supported the change, the Constitution was reformed to allow the Coloured and Indian minorities limited participation in separate and subordinate Houses in a tricameral Parliament. This was part of a change in which the Coloured minority was to be allowed limited rights and self-governance in "Coloured areas", but continuing the policy of denationalising the Black majority and making them involuntary citizens of independent homelands. The internal rationale was that South African whites, more numerous at the time than Coloureds and Indians combined, could bolster its popular support and divide the democratic opposition while maintaining a working majority. The effort largely failed, with the 1980s seeing increased disintegration of civil society and numerous states of emergency, with violence increasing from all racial groups. The separate arrangements were removed by the negotiations which took place from 1990 to hold the first universal election.

Post-apartheid era
During the 1994 all-race elections, Coloured people voted heavily for the white National Party, which in its first contest with a non-white majority won 20% of the vote and a majority in the new Western Cape province – much due to Cape Coloured support. The National Party recast itself as the New National Party after De Klerk's departure in 1996, partly to attract non-White voters, and grew closer to the ANC. This political alliance, often perplexing to outsiders, has sometimes been explained in terms of the culture and language shared by White and Coloured New National Party members, who both spoke Afrikaans. In addition, both groups opposed affirmative action programmes that might give preference to Black South Africans, and some Coloured people feared giving up older privileges, such as access to municipal jobs, if African National Congress gained leadership in the government. After the absorption of the NNP into the ANC in 2005, Coloured voters have generally drawn to the Democratic Alliance, with some opting for minor parties such as Vryheidsfront and Patricia de Lille's Independent Democrats, with lukewarm support for the ANC.

Since the late 20th century, Coloured identity politics have grown in influence. The Western Cape has been a site of the rise of opposition parties, such as the Democratic Alliance (DA). The Western Cape is considered as an area in which this party might gain ground against the dominant African National Congress. The Democratic Alliance drew in some former New National Party voters and won considerable Coloured support. The New National Party collapsed in the 2004 elections. Coloured support aided the Democratic Alliance's victory in the 2006 Cape Town municipal elections.

Patricia de Lille, who became mayor of Cape Town in 2011 on the platform of the now-defunct Independent Democrats, does not use the label Coloured but many observers would consider her as Coloured by visible appearance. The Independent Democrats party sought the Coloured vote and gained significant ground in the municipal and local elections in 2006, particularly in districts in the Western Cape with high proportions of Coloured residents. The firebrand Peter Marais (formerly a provincial leader of the New National Party) has sought to portray his New Labour Party as the political voice for Coloured people.

Coloured people supported and were members of the African National Congress before, during and after the apartheid era: notable politicians include Ebrahim Rasool (previously Western Cape premier), Beatrice Marshoff, John Schuurman, Allan Hendrickse and Trevor Manuel, longtime Minister of Finance. The Democratic Alliance won control over the Western Cape during the 2009 National and Provincial Elections and subsequently brokered an alliance with the Independent Democrats.

The ANC has had some success in winning Coloured votes, particularly among labour-affiliated and middle-class Coloured voters. Some Coloureds express distrust of the ANC with the comment, saying that the Coloured were considered "not white enough under apartheid and not black enough under the ANC." In the 2004 election, voter apathy was high in historically Coloured areas. The ANC faces the dilemma of having to balance the increasingly nationalistic economic aspirations of its core black African support base, with its ambition to regain control of the Western Cape, which would require support from Coloureds.

Southern Africa
The term Coloured is also used in Namibia, to describe persons of mixed race, specifically part Khoisan, and part European. The Basters of Namibia constitute a separate ethnic group that are sometimes considered a sub-group of the Coloured population of that country. Under South African rule, the policies and laws of apartheid were extended to what was then called South West Africa. In Namibia, Coloureds were treated by the government in a way comparable to that of South African Coloureds.

In Zimbabwe and to a lesser extent Zambia, the term Coloured or Goffal was used to refer to people of mixed race. Most are descended from mixed African and British, or African and Indian, progenitors. Some Coloured families descended from Cape Coloured migrants from South Africa who had children with local women. Under Rhodesia's predominantly white government, Coloureds had more privileges than black Africans, including full voting rights, but still faced social discrimination. The term Coloured is also used in Eswatini.

Lifestyle
As far as family life, housing, eating habits, clothing and so on are concerned, the Christian Coloureds generally maintain a Western lifestyle. Marriages are strictly monogamous, although extramarital and premarital sexual relationships can occur and are perceived differently from family to family. Among the working and agrarian classes, permanent relationships are often officially ratified only after a while if at all.

The average family size of six does not differ from those of other Western families and, as with the latter, is generally related to socio-economic status. Extended families are common. Coloured children are often expected to refer to any extended relatives as their "auntie" or "uncle" as a formality.

While many affluent families live in large, modern, and sometimes luxurious homes, many urban coloured people rely on state-owned economic and sub-economic housing.

Cultural aspects
There are many singing and choir associations as well as orchestras in the Coloured community. The Eoan Group Theatre Company performs opera and ballet in Cape Town. The Kaapse Klopse carnival, held annually on 2 January in Cape Town, and the Cape Malay choir and orchestral performances are an important part of the city's holiday season. Kaapse Klopse consists of several competing groups that have been singing and dancing through Cape Town's streets on New Year's Day earlier this year. Nowadays the drumlines in cheerful, brightly Coloured costumes perform in a stadium. Christmas festivities take place in a sacred atmosphere but are no less vivid, mainly including choirs and orchestras that sing and play Christmas songs in the streets. In the field of performing arts and literature, several Coloureds performed with the CAPAB (Cape Performing Arts Board) ballet and opera company, and the community yielded three major Afrikaans poets the well-known poets, Adam Small, S.V. Petersen, and P.J. Philander. In 1968, the Culture and Recreation Council was established to promote the cultural activities of the Coloured Community.

Education
Until 1841 missionary societies provided all the school facilities for Coloured children.

All South African children are expected to attend school from the age of seven to sixteen years, at the minimum.

Economic activities
Initially, Coloureds were mainly semi-skilled and unskilled labourers who, as builders, masons, carpenters and painters, made an important contribution to the early construction industry in the Cape. Many were also fishermen and farm workers, and the latter had an important share in the development of the wine, fruit and grain farms in the Western Cape.

The Malays were, and still are, skilled furniture makers, dressmakers and coopers. In recent years, more and more Coloureds have been working in the manufacturing and construction industry. There are still many Coloured fishermen, and most Coloureds in the countryside are farm workers and even farmers. The largest percentage of economically active Coloureds is found in the manufacturing industry. About 35% of the economically active Coloured women are employed in clothing, textile, food and other factories.

Another important field of work is the service sector, while an ever-increasing number of Coloureds operate in administrative, clerical and sales positions. All the more professional and managerial posts are. In order to stimulate the economic development of Coloureds, the Coloured Development Corporation was established in 1962. The corporation provided capital to businessmen, offered training courses and undertook the establishment of shopping centres, factories and the like.

Distribution
A majority of those who identify as Coloured live in the Western Cape, where they make up almost half of the province's population. In the 2011 South African census the distribution of the group per province was as follows:

Language
Coloured people were some of the first speakers of Afrikaans along with European (Dutch, German and French) colonists and African and Asian slave descendents. The language was originally an informal dialect of Dutch that was spoken amongst the different ethnic slaves to understand each other and also converse with their Dutch masters. Later the language was adopted by white Afrikaners. According to the 2011 South African census, more than 95% of those who identified as Coloured spoke either Afrikaans (74.6%) or English (20.5%) natively, while 4.93% reported a different first language, the most common being Setswana which was spoken by 0.87% of the group.

Cuisine
Numerous South African cuisines can be traced back to Coloured people. Bobotie, snoek based dishes, koe'sisters, bredies, Malay roti and gatsbies are staple diets of Coloureds and other South Africans as well.

Novels

 * Pamela Jooste, Dance with a Poor Man's Daughter, Doubleday, 1998, ISBN 978-0-385-40911-7
 * Zoë Wicomb, David’s Story, New York, Feminist Press at the City University of New York, 2001.
 * Henry Martin Scholtz, A Place Called Vatmaar, 2000, ISBN 978-0795701047

title: Coloureds