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Laplace–Beltrami operator

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

For any twice-differentiable real-valued function f defined on Euclidean space Rn, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for Rn. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting operator is called the Laplace–de Rham operator (named after Georges de Rham).

Details
The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian) divergence of the (Riemannian) gradient:
 * $$\Delta f = {\rm div}( \nabla f).$$

An explicit formula in local coordinates is possible.

Suppose first that M is an oriented Riemannian manifold. The orientation allows one to specify a definite volume form on M, given in an oriented coordinate system xi by

 * $$\operatorname{vol}_n := \sqrt{|g|} \;dx^1\wedge \cdots \wedge dx^n$$

where g is the absolute value of the determinant of the metric tensor, and the dxi are the 1-forms forming the dual frame to the frame
 * $$\partial_i := \frac {\partial}{\partial x^i}$$

of the tangent bundle $$TM$$ and $$\wedge$$ is the wedge product.

The divergence of a vector field $$X$$ on the manifold is then defined as the scalar function $$\nabla \cdot X$$ with the property

(\nabla \cdot X) \operatorname{vol}_n := L_X \operatorname{vol}_n $$

where LX is the Lie derivative along the vector field X. In local coordinates, one obtains

\nabla \cdot X = \frac{1}{\sqrt{|g|}} \partial_i \left(\sqrt {|g|} X^i\right) $$

where here and below the Einstein notation is implied, so that the repeated index i is summed over. The gradient of a scalar function ƒ is the vector field grad f that may be defined through the inner product $$\langle\cdot,\cdot\rangle$$ on the manifold, as

 * $$\langle \operatorname{grad} f(x), v_x \rangle = df(x)(v_x)$$

for all vectors vx anchored at point x in the tangent space TxM of the manifold at point x. Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument vx. In local coordinates, one has

 * $$ \left(\operatorname{grad} f\right)^i =

\partial^i f = g^{ij} \partial_j f$$

where gij are the components of the inverse of the metric tensor, so that gijgjk = δik with δik the Kronecker delta.

Combining the definitions of the gradient and divergence, the formula for the Laplace–Beltrami operator applied to a scalar function ƒ is, in local coordinates

 * $$\Delta f = \frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right).$$

If M is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by a volume element (a density rather than a form). Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace–Beltrami operator itself does not depend on this additional structure.

Formal self-adjointness
The exterior derivative $$d$$ and $$- \nabla$$ are formal adjoints, in the sense that for a compactly supported function $$f$$

 * $$\int_M df(X) \operatorname{vol}_n = - \int_M f \nabla \cdot X \operatorname{vol}_n $$     (proof)

where the last equality is an application of Stokes' theorem. Dualizing gives

for all compactly supported functions $$f$$ and $$h$$. Conversely, ($$) characterizes the Laplace–Beltrami operator completely, in the sense that it is the only operator with this property.

As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions $$f$$ and $$h$$,

 * $$\int_M f\,\Delta h \operatorname{vol}_n = -\int_M \langle d f, d h \rangle \operatorname{vol}_n = \int_M h\,\Delta  f \operatorname{vol}_n.$$

Because the Laplace–Beltrami operator, as defined in this manner, is negative rather than positive, often it is defined with the opposite sign.

Eigenvalues of the Laplace–Beltrami operator (Lichnerowicz–Obata theorem)
Let M denote a compact Riemannian manifold without boundary. We want to consider the eigenvalue equation,
 * $$ -\Delta u=\lambda u,$$

where $$u$$ is the eigenfunction associated with the eigenvalue $$\lambda$$. It can be shown using the self-adjointness proved above that the eigenvalues $$\lambda$$ are real. The compactness of the manifold $$M$$ allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue $$\lambda$$, i.e. the eigenspaces are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get $$\lambda=0$$ is an eigenvalue. Also since we have considered $$-\Delta $$ an integration by parts shows that $$\lambda\geq 0$$. More precisely if we multiply the eigenvalue equation through by the eigenfunction $$u$$ and integrate the resulting equation on $$M$$ we get (using the notation $$dV=\operatorname{vol}_n$$):
 * $$-\int_M \Delta u\ u\ dV=\lambda\int_Mu^2\ dV$$

Performing an integration by parts or what is the same thing as using the divergence theorem on the term on the left, and since $$M$$ has no boundary we get
 * $$-\int_M\Delta u\ u\ dV=\int_M|\nabla u|^2\ dV$$

Putting the last two equations together we arrive at
 * $$\int_M|\nabla u|^2\ dV=\lambda\int_Mu^2\ dV$$

We conclude from the last equation that $$\lambda\geq 0$$.

A fundamental result of André Lichnerowicz states that: Given a compact n-dimensional Riemannian manifold with no boundary with $$n\geq 2$$. Assume the Ricci curvature satisfies the lower bound:
 * $$ \operatorname{Ric}(X,X)\geq \kappa g(X,X),\kappa>0,$$

where $$g(\cdot,\cdot)$$ is the metric tensor and $$X$$ is any tangent vector on the manifold $$M$$. Then the first positive eigenvalue $$\lambda_1$$ of the eigenvalue equation satisfies the lower bound:
 * $$ \lambda_1\geq \frac{n}{n-1}\kappa.$$

This lower bound is sharp and achieved on the sphere $$\mathbb{S}^n$$. In fact on $$\mathbb{S}^2$$ the eigenspace for $$\lambda_1$$ is three dimensional and spanned by the restriction of the coordinate functions $$x_1,x_2,x_3$$ from $$\mathbb{R}^3$$ to $$\mathbb{S}^2$$. Using spherical coordinates $$(\theta,\phi)$$, on $$\mathbb{S}^2$$ the two dimensional sphere, set
 * $$x_3=\cos\phi=u_1,$$

we see easily from the formula for the spherical Laplacian displayed below that
 * $$ -\Delta _{\mathbb{S}^2}u_1=2u_1$$

Thus the lower bound in Lichnerowicz's theorem is achieved at least in two dimensions.

Conversely it was proved by Morio Obata, that if the n-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue $$\lambda_1$$ one has,
 * $$\lambda_1=\frac{n}{n-1}\kappa,$$

then the manifold is isometric to the n-dimensional sphere $$ \mathbb{S}^n\bigg(\sqrt{\frac{n-1}{\kappa}}\bigg)$$, the sphere of radius $$\sqrt{\frac{n-1}{\kappa}}$$. Proofs of all these statements may be found in the book by Isaac Chavel. Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the Kohn Laplacian (after Joseph J. Kohn) on a compact CR manifold. Applications there are to the global embedding of such CR manifolds in $$\mathbb{C}^n.$$

Tensor Laplacian
The Laplace–Beltrami operator can be written using the trace (or contraction) of the iterated covariant derivative associated with the Levi-Civita connection. The Hessian (tensor) of a function $$f$$ is the symmetric 2-tensor

 * $$\displaystyle \mbox{Hess} f \in \mathbf \Gamma(\mathsf T^*M \otimes \mathsf T^*M)$$, $$\mbox{Hess} f := \nabla^2 f \equiv \nabla \nabla f \equiv \nabla \mathrm df$$,

where df denotes the (exterior) derivative of a function f.

Let Xi be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the components of Hess f are given by

 * $$(\mbox{Hess} f)_{ij} = \mbox{Hess} f(X_i, X_j) = \nabla_{X_i}\nabla_{X_j} f - \nabla_{\nabla_{X_i}X_j} f$$

This is easily seen to transform tensorially, since it is linear in each of the arguments Xi, Xj. The Laplace–Beltrami operator is then the trace (or contraction) of the Hessian with respect to the metric:

 * $$\displaystyle \Delta f := \mathrm{tr} \nabla \mathrm df \in \mathsf C^\infty(M)$$.

More precisely, this means

 * $$\displaystyle \Delta f(x) = \sum_{i=1}^n \nabla \mathrm df(X_i,X_i)$$,

or in terms of the metric

 * $$\Delta f = \sum_{ij} g^{ij} (\mbox{Hess} f)_{ij}.$$

In abstract indices, the operator is often written

 * $$\Delta f = \nabla^a \nabla_a f$$

provided it is understood implicitly that this trace is in fact the trace of the Hessian tensor.

Because the covariant derivative extends canonically to arbitrary tensors, the Laplace–Beltrami operator defined on a tensor T by
 * $$\Delta T = g^{ij}\left( \nabla_{X_i}\nabla_{X_j} T - \nabla_{\nabla_{X_i}X_j} T\right)$$

is well-defined.

Laplace–de Rham operator
More generally, one can define a Laplacian differential operator on sections of the bundle of differential forms on a pseudo-Riemannian manifold. On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic. The Laplace–de Rham operator is defined by

 * $$\Delta = \mathrm{d}\delta + \delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;$$

where d is the exterior derivative or differential and δ is the codifferential, acting as (−1)kn+n+1∗d∗ on k-forms, where ∗ is the Hodge star. The first order operator $$\mathrm{d}+\delta$$ is the Hodge–Dirac operator.

When computing the Laplace–de Rham operator on a scalar function f, we have δf = 0, so that

 * $$\Delta f = \delta \, \mathrm df.$$

Up to an overall sign, the Laplace–de Rham operator is equivalent to the previous definition of the Laplace–Beltrami operator when acting on a scalar function; see the proof for details. On functions, the Laplace–de Rham operator is actually the negative of the Laplace–Beltrami operator, as the conventional normalization of the codifferential assures that the Laplace–de Rham operator is (formally) positive definite, whereas the Laplace–Beltrami operator is typically negative. The sign is merely a convention, and both are common in the literature. The Laplace–de Rham operator differs more significantly from the tensor Laplacian restricted to act on skew-symmetric tensors. Apart from the incidental sign, the two operators differ by a Weitzenböck identity that explicitly involves the Ricci curvature tensor.

Examples
Many examples of the Laplace–Beltrami operator can be worked out explicitly.

Euclidean space
In the usual (orthonormal) Cartesian coordinates xi on Euclidean space, the metric is reduced to the Kronecker delta, and one therefore has $$|g| = 1$$. Consequently, in this case

 * $$\Delta f = \frac{1}{\sqrt{|g|}} \partial_i \sqrt{|g|}\partial^i f = \partial_i \partial^i f$$

which is the ordinary Laplacian. In curvilinear coordinates, such as spherical or cylindrical coordinates, one obtains alternative expressions.

Similarly, the Laplace–Beltrami operator corresponding to the Minkowski metric with signature (− + + +) is the d'Alembertian.

Spherical Laplacian
The spherical Laplacian is the Laplace–Beltrami operator on the (n − 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded into Rn as the unit sphere centred at the origin. Then for a function f on Sn−1, the spherical Laplacian is defined by
 * $$\Delta _{S^{n-1}}f(x) = \Delta f(x/|x|)$$

where f(x/|x|) is the degree zero homogeneous extension of the function f to Rn − {0}, and $$\Delta$$ is the Laplacian of the ambient Euclidean space. Concretely, this is implied by the well-known formula for the Euclidean Laplacian in spherical polar coordinates:
 * $$\Delta f = r^{1-n}\frac{\partial}{\partial r}\left(r^{n-1}\frac{\partial f}{\partial r}\right) + r^{-2}\Delta _{S^{n-1}}f.$$

More generally, one can formulate a similar trick using the normal bundle to define the Laplace–Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space.

One can also give an intrinsic description of the Laplace–Beltrami operator on the sphere in a normal coordinate system. Let (ϕ, ξ) be spherical coordinates on the sphere with respect to a particular point p of the sphere (the "north pole"), that is geodesic polar coordinates with respect to p. Here ϕ represents the latitude measurement along a unit speed geodesic from p, and ξ a parameter representing the choice of direction of the geodesic in Sn−1. Then the spherical Laplacian has the form:
 * $$\Delta _{S^{n-1}} f(\xi,\phi) = (\sin\phi)^{2-n} \frac{\partial}{\partial \phi}\left((\sin\phi)^{n-2}\frac{\partial f}{\partial \phi}\right) + (\sin\phi)^{-2}\Delta _\xi f$$

where $$\Delta _\xi$$ is the Laplace–Beltrami operator on the ordinary unit (n − 2)-sphere. In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get:
 * $$\Delta _{S^{2}} f(\theta,\phi) = (\sin\phi)^{-1} \frac{\partial}{\partial \phi}\left(\sin\phi\frac{\partial f}{\partial \phi}\right) + (\sin\phi)^{-2} \frac{\partial^2}{\partial \theta^2}f$$

Hyperbolic space
A similar technique works in hyperbolic space. Here the hyperbolic space Hn−1 can be embedded into the n dimensional Minkowski space, a real vector space equipped with the quadratic form
 * $$q(x) = x_1^2 - x_2^2-\cdots - x_n^2.$$

Then Hn is the subset of the future null cone in Minkowski space given by
 * $$H^n = \{ x \mid q(x) = 1, x_1>1\}. \, $$

Then
 * $$\Delta _{H^{n-1}} f = \left. \Box f\left(x/q(x)^{1/2}\right) \right| _{H^{n-1}}$$

Here $$f(x/q(x)^{1/2})$$ is the degree zero homogeneous extension of f to the interior of the future null cone and $□$ is the wave operator
 * $$\Box = \frac{\partial^2}{\partial x_1^2} - \cdots - \frac{\partial^2}{\partial x_n^2}.$$

The operator can also be written in polar coordinates. Let (t, ξ) be spherical coordinates on the sphere with respect to a particular point p of Hn−1 (say, the center of the Poincaré disc). Here t represents the hyperbolic distance from p and ξ a parameter representing the choice of direction of the geodesic in Sn−2. Then the hyperbolic Laplacian has the form:
 * $$\Delta _{H^{n-1}} f(t,\xi) = \sinh(t)^{2-n} \frac{\partial}{\partial t}\left(\sinh(t)^{n-2}\frac{\partial f}{\partial t}\right) + \sinh(t)^{-2}\Delta _\xi f$$

where $$\Delta _\xi$$ is the Laplace–Beltrami operator on the ordinary unit (n − 2)-sphere. In particular, for the hyperbolic plane using standard notation for polar coordinates we get:
 * $$\Delta _{H^{2}} f(r,\theta) = \sinh(r)^{-1} \frac{\partial}{\partial r}\left(\sinh(r)\frac{\partial f}{\partial r}\right) + \sinh(r)^{-2} \frac{\partial^2}{\partial \theta^2}f$$

title: Laplace–Beltrami operator


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Savina Monastery (Montenegro)

The Savina Monastery (Манастир Савина; lit. "Sava's monastery") is a Serbian Orthodox monastery of three churches near the city Herceg Novi in the Bay of Kotor, located in thick Mediterranean vegetation in one of the most beautiful parts of the Montenegrin Littoral.

Churches

 * The small Church of the Assumption is 10m high and 6m wide. Its foundation dates to 1030, although the oldest record of it is from 1648. Its reconstruction began in the late 17th century, with the arrival of refugee monks from Tvrdoš Monastery in Herzegovina, and it was completed in 1831.
 * The Great Temple of the Assumption was built between the 1777 and 1799, and builder was a master Nikola Foretić from the island of Korčula.
 * The Church (or Temple) of St. Sava, built by Saint Sava, located outside the monastery complex.

Relics
The monastery has a large number of relics originating from the time of the Nemanjić dynasty (relics of Empress Jelena, cross of Saint Sava), including those transferred from Tvrdoš Monastery.

The iconostasis of the church was done by Simeon Lazović and it represents a unique work of Serb Baroque art.

Emir Kusturica's baptism
On Đurđevdan (St. George's Day) in 2005, director Emir Kusturica was baptised in the monastery as Nemanja Kusturica.

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

title: Savina Monastery (Montenegro)


text: Talk:History of North Africa

title: Talk:History of North Africa


text: Category:Prefecture-level divisions of Hunan


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Benserazide

Benserazide is a peripherally acting aromatic L-amino acid decarboxylase or DOPA decarboxylase inhibitor, which is unable to cross the blood–brain barrier.

It is on the World Health Organization's List of Essential Medicines.

Indications
It is used in the management of Parkinson's disease in combination with L-DOPA (levodopa) as co-beneldopa (BAN), under the brand names Madopar in the UK and Prolopa in Canada, both made by Roche. Benserazide is not approved for use in the US; carbidopa is used, instead, for the same purpose. These combinations are also used for the treatment of restless leg syndrome.

Pharmacology
Levodopa is a precursor to the neurotransmitter dopamine, which is administered to increase its levels in the central nervous system. However, most levodopa is decarboxylated to dopamine before it reaches the brain, and since dopamine is unable to cross the blood–brain barrier, this translates to little therapeutic gain with strong peripheral side effects.

Benserazide inhibits the aforementioned decarboxylation, and since it cannot cross the blood–brain barrier itself, this allows dopamine to build up solely in the brain, instead. Adverse effects caused by peripheral dopamine, such as vasoconstriction, nausea, and arrhythmia, are minimized. However, benserazide cannot reduce the centrally mediated side effects of levodopa, particularly dyskinesia.

Benserazide has little therapeutic effect on its own, and its effect occurs synergically in combination with levodopa.

The enzyme inhibited by benzerazide catalyzes many different decarboxylations. The same effect of concentrating the conversion of levodopa into dopamine to the central nervous system can be achieved with the following decarboxylations being confined to the central nervous system:
 * 5-HTP to serotonin
 * Tryptophan to tryptamine
 * Phenylalanine to phenethylamine
 * L-tyrosine to tyramine

Centrally mediated side effects of higher levels of neuro- and trace amine-transmitters may worsen in combination with monoamine oxidase inhibitors.

title: Benserazide


text: Category:Cities in Hunan

title: Category:Cities in Hunan


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Tvrdoš Monastery

The Tvrdoš Monastery (Манастир Тврдош) is a 15th-century Serbian Orthodox monastery near the city of Trebinje, Republika Srpska, Bosnia and Herzegovina. The 4th-century foundations of the first Roman church on the site are still visible.

The monastery, which is dedicated to the Dormition of the Most Holy Theotokos, was established during the 15th century, with a cathedral constructed around 1508 and painted with murals by Vicko Lavrov from Dubrovnik in 1517. The monastery remained a seat of the Metropolitans of Herzegovina until the Venetian Empire destroyed it in 1694. The current monastery building was constructed in 1924. Today, the monastery is renowned for its wine production (in particular from the local Vranac and Žilavka grape varieties) and its wine cellars, one of which dates to the 15th century, which are a popular tourist attraction.

As of late 2016, the monastery remains on the Provisional List for status as a National Monument of Bosnia and Herzegovina.

On 6 March 2021, the retired Bishop of Zahumlje and Herzegovina (1992–1999), Atanasije was buried in the Chapel of the Resurrection of the Lord in the cemetery of the monastery.

title: Tvrdoš Monastery


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User:Tutv

My name's Andy Smith. Hello.

title: User:Tutv


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Farnham, Essex

Farnham is a small village in Essex, England, situated near Bishop's Stortford. The main features are Farnham Church of England Primary School, the church and the Three Horseshoes pub in Hazel End, which some consider a hamlet in its own right. The population was 410 at the 2011 census and had increased to 418 in the 2021 census It is divided up into several areas, such as Farnham, Hazel End, Bell's Cottages, Saven End and Farnham Green. The name is derived from the Fernham (Hamlet in the ferns).

History
Farnham is mentioned in the Domesday Book as one of the settlements in Clavering hundred.

Farnham Primary School
Farnham C of E Primary School is a primary school in Farnham, Essex, England. Farnham Primary School is located on the Essex/Herts county border, within two miles (3 km) of Stansted Mountfitchet and Bishop's Stortford, and takes pupils from both counties.

Founded in 1874, Farnham primary is Essex's smallest school with about 50 pupils in 2023. It has three classes: Willow (from Reception to Year 1), Silver Birch    (from Year 2 to Year 4) and Horse Chestnut (Year 5 to Year 6). Children graduating from Farnham mostly attend secondary schools in Bishop's Stortford and it lies within Joyce Franklin High school's priority admissions area.

The school is in federation with the larger nearby Rickling Church of England Primary School, where pupils visit weekly for curriculum enrichment activities, including art, gardening, drama, yoga and sports. Both schools share an executive head and deputy head. The deputy head teaches Horse Chestnut class in Farnham.

Farnham primary enjoys consistently good SATS results. In 2013, 100% of Year 6 pupils achieved Level 4+ and 50% achieved Level 5 in mathematics and reading. This compares very favourably with the national average and out-performs most schools in the area.

In 2014, the school was the subject of a high-profile campaign to keep it in Farnham and prevent it from moving to nearby Stansted Mountfitchet, as proposed by Essex County Council and the Diocese Board of Education for Chelmsford. The proposals were unanimously opposed by the parish councils of Farnham and Stansted and the Farnham parochial church. In response, the school governors voted to reject the proposals.

title: Farnham, Essex


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Talk:Oină

Is it really a sport? I thought it was some form of Romanian theater. Scott Gall 11:50, 2005 Apr 19 (UTC)

Yes is a realy sport. Mihai ROMANIA

From the description, the game sounds a lot more like cricket than baseball. 220.253.52.9 15:59, 13 March 2007 (UTC)

It souds like softball

Article clean-up and enhanced
I enhanced the article with what was available on the federation's website, and an English-translated animation of the pitch. The article was moved to oină, and generally cleaned up. —Preceding unsigned comment added by Ayceman (talk • contribs) 22:25, 14 November 2009 (UTC)

salut. what's missing is "Purpose", "Target" sau "Scoring". it's not clear to me what you're supposed to do, and why. thanks. Palamnaeus —Preceding undated comment added 06:02, 21 April 2010 (UTC).

 * I improved the article. I still have to do the „prindere la mijloc”, organizational stuff, kits, bats, and things like that at some point. Ayceman (talk) 21:10, 26 May 2010 (UTC)

title: Talk:Oină


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Rosalind Miles (author)

Rosalind Miles (born Rosalind Mary Simpson on January 6, 1943) is an English author, historian, university lecturer, broadcaster, journalist, magistrate and activist who has written 23 works of fiction and non-fiction. Dr Miles' international best-selling book The Women's History of the World, also called Who Cooked the Last Supper?, charts the untold history of women.

She has two grown children, and is married to the historian Robin Cross.

Life and career
Rosalind Miles was born in Warwickshire, the youngest of three sisters. As a child, Miles suffered from polio, which she acquired at the age of four, causing her to undergo several months of treatment. From the age of ten, Miles attended the King Edward VI High School for Girls, where she obtained a working knowledge of Latin and Greek, along with a lifelong love of Shakespeare. At seventeen, she was accepted at St Hilda's College, Oxford, where she studied English literature, Anglo-Saxon, Middle English, Latin and French. There, she was awarded the Eleanor Rooke Memorial Prize, the Principal's Prize of St Hilda's College, as well as a State Studentship Award. She obtained five degrees in all, including an MA and Ph.D. from the Shakespeare Institute at the University of Birmingham, as well a starred MA* from the Centre for Mass Communication Research at the University of Leicester.

Alongside her studies, Miles worked in several occupations, including working as a travelling saleswoman and a stable hand. She got her first job, in a plastics factory, at the age of 13. Miles later became interested in jurisprudence, which resulted in her appointment at the age of 26 as a lay magistrate in the Warwickshire criminal and family courts, and eventually on the bench in a superior court in Coventry. She served for ten years, and rose to the level of Crown Court. Miles has also worked with numerous government agencies and served on consultative committees.

In addition to novelist, Miles is also a journalist and broadcaster. She began her broadcasting career on the BBC, for which she is now a regular commentator. She also broadcasts on Canadian radio, as well as numerous local radio stations. She has made many television appearances as a historian and commentator, including on CNN, PBS, and CBS. As a journalist, her work has appeared in major newspapers across the English-speaking world, including The Washington Post. Miles is also a major contributor to a number of magazines, including Prospect and Cosmopolitan.

Non-fiction
With Robin Cross:
 * The Fiction of Sex: Themes and Functions of Sex Difference in the Modern Novel
 * The Problem of Measure for Measure
 * Ben Jonson: His Life and Work
 * Ben Jonson: His Craft and Art
 * The Female Form: Women Writers and the Conquest of the Novel
 * Danger! Men At Work
 * Modest Proposals
 * Women and Power
 * The Women's History of the World (also called Who Cooked the Last Supper?)
 * The Rites of Man: Love, Sex and Death in the Making of the Male (US: Love, Sex and Death and the Making of the Male) (1991)
 * The Children We Deserve: Love and Hate in the Making of the Family
 * Hell Hath No Fury: True Stories of Women at War from Antiquity to Iraq
 * Warrior Women: 3000 Years of Courage and Heroism

Fiction

 * Return to Eden
 * Bitter Legacy
 * Prodigal Sins
 * Act of Passion
 * I, Elizabeth: the Word of a Queen Reader's Guide
 * The Guenevere trilogy: Reader's Guide
 * Guenevere, Queen of the Summer Country
 * The Knight of the Sacred Lake
 * The Child of the Holy Grail
 * The Isolde trilogy:
 * The Queen of the Western Isle
 * The Maid of the White Hands
 * The Lady of the Sea

title: Rosalind Miles (author)


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Gun turret

A gun turret (or simply turret) is a mounting platform from which weapons can be fired that affords protection, visibility and ability to turn and aim. A modern gun turret is generally a rotatable weapon mount that houses the crew or mechanism of a projectile-firing weapon and at the same time lets the weapon be aimed and fired in some degree of azimuth and elevation (cone of fire).

Description
Rotating gun turrets protect the weapon and its crew as they rotate. When this meaning of the word "turret" started being used at the beginning of the 1860s, turrets were normally cylindrical. Barbettes were an alternative to turrets; with a barbette the protection was fixed, and the weapon and crew were on a rotating platform inside the barbette. In the 1890s, armoured hoods (also known as "gun houses") were added to barbettes; these rotated with the platform (hence the term "hooded barbette"). By the early 20th century, these hoods were known as turrets. Modern warships have gun-mountings described as turrets, though the "protection" on them is limited to protection from the weather.

Rotating turrets can be mounted on a fortified building or structure such as a coastal blockhouse, be part of a land battery, be mounted on a combat vehicle, a naval ship, or a military aircraft, they may be armed with one or more machine guns, automatic cannons, large-calibre guns, or missile launchers. They may be manned or remotely controlled and are most often protected to some degree, if not actually armoured.

The protection provided by the turret may be against battle damage, the weather conditions, general environment in which the weapon or its crew will be operating. The name derives from the pre-existing noun turret, from the French "touret", diminutive of the word "tower", meaning a self-contained protective position which is situated on top of a fortification or defensive wall as opposed to rising directly from the ground, in which case it constitutes a tower.

Cupolas

A small turret, or sub-turret set on top of a larger one, is called a cupola. The term cupola is also used for a rotating turret that carries a sighting device rather than weaponry, such as that used by a tank commander.

Warships
Before the development of large-calibre, long-range guns in the mid-19th century, the classic battleship design used rows of gunport-mounted guns on each side of the ship, often mounted in casemates. Firepower was provided by a large number of guns, each of which could traverse only in a limited arc. Due to stability issues, fewer large (and thus heavy) guns can be carried high on a ship, but as this set casemates low and thus near the waterline they were vulnerable to flooding, effectively restricted their use to calm seas. Additionally casemate mounts had to be recessed into the side of a vessel to afford a wide arc of fire, and such recesses presented shot traps, compromising the integrity of armour plating.

Rotating turrets were weapon mounts designed to protect the crew and mechanism of the artillery piece and with the capability of being aimed and fired over a broad arc, typically between a three-quarter circle up to a full 360 degrees. These presented the opportunity to concentrate firepower in fewer, better-sited positions by eliminating redundancy, in other words combining the firepower of those guns unable to engage an enemy because they sited on the wrong beam into a more powerful, and more versatile unified battery.

History

Designs for a rotating gun turret date back to the late 18th century. In the mid-19th century, during the Crimean War, Captain Cowper Phipps Coles constructed a raft with guns protected by a 'cupola' and used the raft, named the Lady Nancy, to shell the Russian town of Taganrog in the Black Sea during the Siege of Taganrog. The Lady Nancy "proved a great success" and Coles patented his rotating turret design after the war.

United Kingdom: Early designs
The British Admiralty ordered a prototype of Coles's patented design in 1859, which was installed in the ironclad floating battery, HMS Trusty, for trials in 1861, becoming the first warship to be fitted with a revolving gun turret. Coles's aim was to create a ship with the greatest possible all round arc of fire, as low in the water as possible to minimise the target.

The Admiralty accepted the principle of the turret gun as a useful innovation, and incorporated it into other new designs. Coles submitted a design for a ship having ten domed turrets each housing two large guns.

The design was rejected as impractical, although the Admiralty remained interested in turret ships and instructed its own designers to create better designs. Coles enlisted the support of Prince Albert, who wrote to the first Lord of the Admiralty, the Duke of Somerset, supporting the construction of a turret ship. In January 1862, the Admiralty agreed to construct a ship, HMS Prince Albert which had four turrets and a low freeboard, intended only for coastal defence.

While Coles designed the turrets, the ship was the responsibility of Chief Constructor Isaac Watts. Another ship using Coles' turret designs, HMS Royal Sovereign (1857), was completed in August 1864. Its existing broadside guns were replaced with four turrets on a flat deck and the ship was fitted with 5.5 in of armour in a belt around the waterline.

Early ships like the Royal Sovereign had little sea-keeping qualities being limited to coastal waters. Sir Edward James Reed, went on to design and build HMS Monarch, the first seagoing warship to carry her guns in turrets. Laid down in 1866 and completed in June 1869, it carried two turrets, although the inclusion of a forecastle and poop prevented the turret guns firing fore and aft.

United States: USS Monitor

The gun turret was independently invented in the United States by the Swedish inventor John Ericsson, although his design was technologically inferior to Coles's version. Ericsson designed USS Monitor in 1861, its most prominent feature being a large, cylindrical gun turret mounted amidships above the low-freeboard upper hull, also referred to as the "raft". This extended well past the sides of the lower, more traditionally shaped hull.

A small, armoured pilot house was fitted on the upper deck towards the bow; however, its position prevented Monitor from firing her guns straight forward. Like Coles's, one of Ericsson's goals in designing the ship was to present the smallest possible target to enemy gunfire. The turret's rounded shape helped to deflect cannon shot. A pair of donkey engines rotated the turret through a set of gears; a full rotation was made in 22.5 seconds during testing on 9 February 1862. However, fine control of the turret proved to be difficult, as it would have to be reversed if it overshot its mark. In lieu of reversing the turret, a full rotation would have to be made to train the guns where desired.

Including the guns, the turret weighed approximately 160 LT; the entire weight rested on an iron spindle that had to be jacked up using a wedge before the turret was free to rotate. The spindle was 9 in in diameter which gave it ten times the strength needed in preventing the turret from sliding sideways.

When not in use, the turret rested on a brass ring on the deck that was intended to form a watertight seal. However, in service, the interface between the turret and deck ring heavily leaked, despite caulking by the crew.

The gap between the turret and the deck proved to be another kind of problem for several Passaic-class monitors, which used the same turret design, as debris and shell fragments entered the gap and jammed the turrets during the First Battle of Charleston Harbor in April 1863. Direct hits at the turret with heavy shot also had the potential to bend the spindle, which could also jam the turret.

Monitor was originally intended to mount a pair of 15 in smoothbore Dahlgren guns, but they were not ready in time and 11 in guns were substituted, each gun weighing approximately 16000 lb. Monitor's guns used the standard propellant charge of 15 lb specified by the 1860 ordnance instructions for targets "distant", "near", and "ordinary", established by the gun's designer Dahlgren himself. They could fire a 136 lb round shot or shell up to a range of 3650 yd at an elevation of +15°.

Later designs

HMS Thunderer (1872) (1872) represented the culmination of this pioneering work. An ironclad turret ship designed by Edward James Reed, she was equipped with revolving turrets that used pioneering hydraulic turret machinery to maneouvre the guns. She was also the world's first mastless battleship, built with a central superstructure layout, and became the prototype for all subsequent warships. With her sister HMS Devastation (1871) of 1871 she was another pivotal design, and led directly to the modern battleship.

The US Navy tried to save weight and deck space, and allow the much faster firing 8-inch to shoot during the long reload time necessary for 12-inch guns by superposing secondary gun turrets directly on top of the primary turrets (as in the Kearsarge-class battleship and Virginia-class battleships), but the idea proved to be practically unworkable and was soon abandoned.

With the advent of the South Carolina-class battleships in 1908, the main battery turrets were designed so as to superfire, to improve fire arcs on centerline mounted weapons. This was necessitated by a need to move all main battery turrets to the vessel's centerline for improved structural support. The 1906 HMS Dreadnought (1906), while revolutionary in many other ways, had retained wing turrets due to concerns about muzzle blast affecting the sighting mechanisms of a turret below. A similar advancement was in the Kongō-class battlecruisers and Queen Elizabeth-class battleships, which dispensed with the "Q" turret amidships in favour of heavier guns in fewer mountings.

Like pre-dreadnoughts, the first dreadnoughts had two guns in each turret; however, later ships began to be fitted with triple turrets. The first ship to be built with triple turrets was the Italian ITALIAN BATTLESHIP Dante Alighieri, although the first to be actually commissioned was the Austro-Hungarian SMS Viribus Unitis of the Tegetthoff-class battleship. By the beginning of World War II, most battleships used triple or, occasionally, quadruple turrets, which reduced the total number of mountings and improved armour protection. However, quadruple turrets proved to be extremely complex to arrange, making them unwieldy in practice.

The largest warship turrets were in World War II battleships where a heavily armoured enclosure protected the large gun crew during battle. The calibre of the main armament on large battleships was typically 300 to 460 mm. The turrets carrying three 460 mm guns of JAPANESE BATTLESHIP Yamato each weighed around 2,500 t. The secondary armament of battleships (or the primary armament of light cruisers) was typically between 127 and 152 mm. Smaller ships typically mounted guns of 76 mm and larger, although these rarely required a turret mounting, except for large destroyers, like the American Fletcher-class destroyer and the German Narvik classes.

Layout

In naval terms, turret traditionally and specifically refers to a gun mounting where the entire mass rotates as one, and has a trunk that projects below the deck. The rotating part of a turret seen above deck is the gunhouse, which protects the mechanism and crew, and is where the guns are loaded. The gunhouse is supported on a bed of rotating rollers, and is not necessarily physically attached to the ship at the base of the rotating structure. In the case of the German battleship Bismarck, the turrets were not vertically restrained and fell out when she sank. The British battlecruiser Hood, like some American battleships, did have vertical restraints.

Below the gunhouse there may be a working chamber, where ammunition is handled, and the main trunk, which accommodates the shell and propellant hoists that bring ammunition up from the magazines below. There may be a combined hoist (cf the animated British turret) or separate hoists (cf the US turret cutaway). The working chamber and trunk rotate with the gunhouse, and sit inside a protective armoured barbette. The barbette extends down to the main armoured deck (red in the animation). At the base of the turret sit handing rooms, where shell and propelling charges are passed from the shell room and magazine to the hoists.

The handling equipment and hoists are complex arrangements of machinery that transport the shells and charges from the magazine into the base of the turret. Bearing in mind that shells can weigh around a 5 LT, the hoists have to be powerful and rapid; a 15 in turret of the type in the animation was expected to perform a complete loading and firing cycle in a minute.

The loading system is fitted with a series of mechanical interlocks that ensure that there is never an open path from the gunhouse to the magazine down which an explosive flash might pass. Flash-tight doors and scuttles open and close to allow the passage between areas of the turret. Generally, with large-calibre guns, powered or assisted ramming is required to force the heavy shell and charge into the breech.

As the hoist and breech must be aligned for ramming to occur, there is generally a restricted range of elevations at which the guns can be loaded; the guns return to the loading elevation, are loaded, then return to the target elevation, at which time they are said to be "in battery". The animation illustrates a turret where the rammer is fixed to the cradle that carries the guns, allowing loading to occur across a wider range of elevations.

Earlier turrets differed significantly in their operating principles. It was not until the last of the "rotating drum" designs described in the previous section were phased out that the "hooded barbette" arrangement above became the standard.

Wing turrets

A wing turret is a gun turret mounted along the side, or the wings, of a warship, off the centerline.

The positioning of a wing turret limits the gun's arc of fire, so that it generally can contribute to only the broadside weight of fire on one side of the ship. This is the major weakness of wing turrets as broadsides were the most prevalent type of gunnery duels. Depending on the configurations of ships, such as HMS Dreadnought but not SMS Blücher, the wing turrets could fire fore and aft, so this somewhat reduced the danger when an opponent crossed the T enabling it to fire a full broadside.

Attempts were made to mount turrets en echelon so that they could fire on either beam, such as the Invincible-class battlecruiser and SMS Von der Tann battlecruisers, but this tended to cause great damage to the ships' deck from the muzzle blast.

Wing turrets were commonplace on capital ships and cruisers during the late 19th century up until the 1910s. In pre-dreadnought battleships, the wing turret contributed to the secondary battery of sub-calibre weapons. In large armoured cruisers, wing turrets contributed to the main battery, although the casemate mounting was more common. At the time, large numbers of smaller calibre guns contributing to the broadside were thought to be of great value in demolishing a ship's upperworks and secondary armaments, as distances of battle were limited by fire control and weapon performance.

In the early 1900s, weapon performance, armour quality and vessel speeds generally increased along with the distances of engagement; the utility of large secondary batteries reducing as a consequence, and in addition at extreme range it was impossible to see the fall of lesser weapons and so correct the aim. Therefore, most early dreadnought battleships featured "all big gun" armaments of identical calibre, typically 11 or, some of which were mounted in wing turrets. This arrangement was not satisfactory, however, as the wing turrets not only had a reduced fire arc for broadsides, but also because the weight of the guns put great strain on the hull and it was increasingly difficult to properly armour them.

Larger and later dreadnought battleships carried superimposed or superfiring turrets (i.e. one turret mounted higher than and firing over those in front of and below it). This allowed all turrets to train on either beam, and increased the weight of fire forward and aft. The superfiring or superimposed arrangement had not been proven until after South Carolina went to sea, and it was initially feared that the weakness of the previous Virginia-class battleship ship's stacked turrets would repeat itself. Larger and later guns (such as the US Navy's ultimate big gun design, the 16"/50 Mark 7|16-inch) also could not be shipped in wing turrets, as the strain on the hull would have been too great.

Modern turrets

Many modern surface warships have mountings for larger calibre guns, although the calibres are now generally between 3 and 5 in for use against both air and surface targets. The gunhouses are often just weatherproof covers for the gun mounting equipment and are made of light un-armoured materials such as glass-reinforced plastic. Modern turrets are often automatic in their operation, with no humans working inside them and only a small team passing fixed ammunition into the feed system. Smaller calibre weapons often operate on the autocannon principle, and indeed may not even be turrets at all; they may just be bolted directly to the deck.

Turret identification
On board warships, each turret is given an identification. In the British Royal Navy, these would be letters: "A" and "B" were for the turrets from the front of the ship backwards in front of the bridge, and letters near the end of the alphabet (i.e., "X", "Y", etc.) were for turrets behind the bridge ship, "Y" being the rearmost. Mountings in the middle of the ship would be "P", "Q", "R", etc. Confusingly, the Dido-class cruisers had a "Q" and the Nelson-class battleships had an "X" turret in what would logically be "C" position; the latter being mounted at the main deck level in front of the bridge and behind the "B" turret, thus having restricted training fore and aft.

Secondary turrets were named "P" and "S" (port and starboard) and numbered from fore to aft, e.g. P1 being the forward port turret.

There were exceptions; the battleship HMS Agincourt had the uniquely large number of seven turrets. These were numbered "1" to "7" but were unofficially nicknamed "Sunday", Monday", etc. through to "Saturday".

In German use, turrets were generally named "A", "B", "C", "D", "E", going from bow to stern. Usually the radio alphabet was used on naming the turrets (e.g. "Anton", "Bruno" or "Berta", "Caesar", "Dora") as on the German battleship Bismarck.

In the United States Navy, main battery turrets are numbered fore to aft. Secondary gun mounts are numbered by gun muzzle diameter in inches followed by a second digit indicating the position of the mount, with the second digit increasing fore to aft. Gun mounts not on the centerline would be assigned odd numbers on the port side and even numbers on the starboard side. For example, "Mount 52" would be the forwardmost 5 in gun mount on the starboard side of the ship.

History
During World War I, air gunners initially operated guns that were mounted on pedestals or swivel mounts known as pintles. The latter evolved into the Scarff ring, a rotating ring mount which allowed the gun to be turned to any direction with the gunner remaining directly behind it, the weapon held in an intermediate elevation by bungee cord, a simple and effective mounting for single weapons such as the Lewis Gun though less handy when twin mounted as with the British Bristol F.2 Fighter and German "CL"-class two-seaters such as the Halberstadt and Hannover-designed series of compact two-seat combat aircraft. In a failed 1916 experiment, a variant of the SPAD S.A two-seat fighter was probably the first aircraft to be fitted with a remotely-controlled gun, which was located in a nose nacelle.

As aircraft flew higher and faster, the need for protection from the elements led to the enclosure or shielding of the gun positions, as in the "lobsterback" rear seat of the Hawker Demon biplane fighter.

The first British operational bomber to carry an enclosed, power-operated turret was the Boulton & Paul Overstrand twin-engined biplane, which first flew in 1933. The Overstrand was similar to its First World War predecessors in that it had open cockpits and hand-operated machine guns. However, unlike its predecessors, the Overstrand could fly at 140 mph making operating the exposed gun positions difficult, particularly in the aircraft's nose. To overcome this problem, the Overstrand was fitted with an enclosed and powered nose turret, mounting a Lewis gun. Rotation was handled by pneumatic motors while elevation and depression of the gun used hydraulic rams. The pilot's cockpit was also enclosed but the dorsal (upper) and ventral (belly) gun positions remained open, though shielded.

The Martin B-10 all-metal monocoque monoplane bomber introduced turret-mounted defensive armament within the United States Army Air Corps, almost simultaneously with the RAF's Overstrand biplane bomber design. The Martin XB-10 prototype aircraft first featured the nose turret in June 1932—roughly a year before the less advanced Overstrand airframe design—and was first produced as the YB-10 service test version by November 1933. The production B-10B version started service with the USAAC in July 1935.

In time the number of turrets carried and the number of guns mounted increased. RAF heavy bombers of World War II such as the Handley Page Halifax (until its Mk II Series I (Special) version omitted the nose turret), Short Stirling and Avro Lancaster typically had three powered turrets: rear, mid-upper and nose. (Early in the war, some British heavy bombers also featured a retractable, remotely-operated ventral/mid-under turret). The rear turret mounted the heaviest armament: four 0.303 in Browning machine guns or, late in the war, two AN/M2 light-barrel versions of the US Browning M2 machine gun as in the Rose-Rice turret. The tail gunner or "Tail End Charlie" position was generally accepted to be the most dangerous assignment. During the war, British turrets were largely self-contained units, manufactured by Boulton Paul Aircraft and Nash & Thompson. The same model of turret might be fitted to several different aircraft types. Some models included gun-laying radar that could lead the target and compensate for bullet drop.

As almost a 1930s "updated" adaptation of the First World War Bristol F.2b concept, the UK introduced the concept of the "turret fighter", with aeroplanes such as the Boulton Paul Defiant and Blackburn Roc where the armament was four (0.303 in) machine-guns was in a turret mounted behind the pilot, rather than in fixed positions in the wings. The Defiant and Roc possessed no fixed, forward-firing guns; the Bristol F.2 was designed with one synchronized Vickers machine gun firing forward on a fuselage mount.

The concept came at a time when the standard armament of a fighter was only two machine guns and in the face of heavily armed bombers operating in formation, it was thought that a group of turret fighters would be able to concentrate their fire flexibly on the bombers; making beam, stern and rising attacks practicable. Although the idea had some merits in attacking unescorted bombers the weight and drag penalty of the turret (and gunner) put them at a disadvantage when Germany was able to escort its bombers with fighters from bases in Northern France. By this point British fighters were flying with eight machine guns which concentrated firepower for use in single fleeting attacks of fighters against bombers.

Attempts to put this heavier armament, such as multiple 20 mm cannon in low profile aerodynamic turrets were explored by the British in the Boulton Paul P.92 and other designs but were not successful this class of weapons and heavier armament (up to and including artillery pieces as in the 1,420 examples produced of the American B-25G and B-25H Mitchell medium bombers and the experimental 'Tsetse' variant of the de Havilland Mosquito) being exclusively fuselage or underwing-mounted and thus aimed by pointing the aircraft. Not all turret designs put the gunner in the turret along with the armament: US and German-designed aircraft both featured remote-controlled turrets.

In the US, the large, purpose-built Northrop P-61 Black Widow night fighter was produced with a remotely operated dorsal turret that had a wide range of fire though in practice it was generally fired directly forward under control of the pilot. For the last Douglas-built production blocks of the B-17F (the "B-17F-xx-DL" designated blocks) and for all versions of the B-17G Flying Fortress a twin-gun remotely operated "chin" turret, designed by Bendix and first used on the experimental YB-40 "gunship" version of the Fortress, was added to give more forward defence. Specifically designed to be compact and not obstruct the bombardier, this was operated by a swing-away diagonal column possessing a yoke to traverse the turret, and aimed by a reflector sight mounted in the windscreen.

The intended replacement for the German Bf 110 heavy fighter, the Messerschmitt Me 210, possessed twin half-teardrop-shaped, remotely operated Ferngerichtete Drehringseitenlafette (Remote rotating side mount) FDSL 131/1B turrets, one on each side "flank" of the rear fuselage to defend the rear of the aircraft, controlled from the rear area of the cockpit. By 1942, the German He 177A Greif heavy bomber would feature a Fernbedienbare Drehlafette (Remotely controlled rotary carriage) FDL 131Z remotely operated forward dorsal turret, armed with twin 13mm MG 131 machine guns on the top of the fuselage, which was operated from the astrodome a hemispherical, clear rotating observation cover, just behind the cockpit glazing and offset to starboard atop the fuselage—a second, manned powered Hydraulische Drehlafette (Hydraulic rotary mount) HDL 131 dorsal turret, further aft on the fuselage with a MG 131 was also used on most examples.

The US B-29 Superfortress had four remotely controlled turrets, comprising two dorsal and two ventral turrets. These were controlled from a trio of hemispherical, glazed, gunner-manned "astrodome" sighting stations operated from the pressurised sections in the nose and middle of the aircraft, each housing an altazimuth mounted pivoting gunsight to aim one or more of the unmanned remote turrets as needed, in addition to a B-17 style flexible manned tail gunner's station.

The defensive turret on bombers fell from favour with the realization that bombers could not attempt heavily defended targets without escort regardless of their defensive armament unless very high loss rates were acceptable and the performance penalty from the weight and drag of turrets reduced speed, range and payload and increased the number of crew required. The de Havilland Mosquito light bomber was designed to operate without any defensive armament and used its speed to avoid engagement with fighters, much as the minimally armed German Schnellbomber aircraft concepts had been meant to do early in World War II.

A small number of aircraft continued to use turrets, in particular maritime patrol aircraft such as the Avro Shackleton used one as an offensive weapon against small un-armoured surface targets. The Boeing B-52 jet bomber and many of its contemporaries (particularly Russian) featured a barbette (a British English term equivalent to the American usage of the term 'tail gun'), or a "remote turret"—an unmanned turret but often one with a more limited field of fire than a manned equivalent.

Layout
Aircraft carry their turrets in various locations:
 * "dorsal" – on top of the fuselage, sometimes referred to as a mid-upper turret.
 * "ventral" – underneath the fuselage, often on US heavy bombers, a Sperry-designed ball turret.
 * "rear" or "tail" – at the very end of the fuselage.
 * "nose" – at the front of the fuselage.
 * "cheek" – on the flanks of the nose, as single-gun flexible defensive mounts for B-17 and B-24 heavy bombers
 * "chin" – below the nose of the aircraft as on later versions of the Boeing B-17 Flying Fortress.
 * "wing" – a handful of very large aircraft, such as the Messerschmitt Me 323 and the Blohm & Voss BV 222, had manned turrets in the wings
 * "waist" or "beam" – mounted on the sides of the rear fuselage e.g. US twin- and four-engined bombers.

History

Amongst the first armoured vehicles to be equipped with a gun turret were the Lanchester and Rolls-Royce Armoured Cars, both produced from 1914. The Royal Naval Air Service (RNAS) raised the first British armoured car squadron during the First World War. In September 1914 all available Rolls-Royce Silver Ghost chassis were requisitioned to form the basis for the new armoured car. The following month a special committee of the Admiralty Air Department, among whom was Flight Commander T.G. Hetherington, designed the superstructure which consisted of armoured bodywork and a single fully rotating turret holding a regular water cooled Vickers machine gun.

However, the first tracked combat vehicles were not equipped with turrets due to the problems with getting sufficient trench crossing while keeping the centre of gravity low, and it was not until late in World War I that the French Renault FT light tank introduced the single fully rotating turret carrying the vehicle's main armament that continues to be the standard of almost every modern main battle tank and many post-World War II self-propelled guns. The first turret designed for the FT was a circular, cast steel version almost identical to that of the prototype. It was designed to carry a Hotchkiss 8 mm (0.315 in) machine gun. Meanwhile, the Berliet Company produced a new design, a polygonal turret of riveted plate, which was simpler to produce than the early cast steel turret. It was given the name "omnibus", since it could easily be adapted to mount either the Hotchkiss machine gun or the Puteaux 37 mm with its telescopic sight. This turret was fitted to production models in large numbers.

In the 1930s, several nations produced multi-turreted tanks—probably influenced by the experimental British Vickers A1E1 Independent of 1926. Those that saw combat during the early part of World War II performed poorly and the concept was soon dropped. Combat vehicles without turrets, with the main armament mounted in the hull, or more often in a completely enclosed, integral armored casemate as part of the main hull, saw extensive use by both the German (as Sturmgeschütz and Jagdpanzer vehicles) and Soviet (as Samokhodnaya Ustanovka vehicles) armored forces during World War II as tank destroyers and assault guns. However, post-war, the concept fell out of favour due to its limitations, with the Swedish Stridsvagn 103 'S-Tank' and the German Kanonenjagdpanzer being exceptions.

Layout

In modern tanks, the turret is armoured for crew protection and rotates a full 360 degrees carrying a single large-calibre tank gun, typically in the range of 105 and 125 mm calibre. Machine guns may be mounted inside the turret, which on modern tanks is often on a "coaxial" mount, parallel with the larger main gun.

Early designs often featured multiple weapons mounts. This concept was carried forwards into the early interwar years in Britain, Germany and the Soviet Union, arguably reaching its most absurd expression in the British Vickers A1E1 Independent tank, though this attempt was soon abandoned while the Soviet Union's similar effort produced a 'land battleship' which was actually produced and fought in defence of the Soviet Union.

In modern tanks, the turret houses all the crew except the driver (who is located in the hull). The crew located in the turret typically consist of tank commander, gunner, and often a gun loader (except in tanks that have an autoloader), while the driver sits in a separate compartment with a dedicated entry and exit, though often one that allows the driver to exit via the turret basket (fighting compartment).

For other combat vehicles, the turrets are equipped with other weapons dependent on role. An infantry fighting vehicle may carry a smaller calibre gun or an autocannon, or an anti-tank missile launcher, or a combination of weapons. A modern self-propelled gun mounts a large artillery gun but less armour. Lighter vehicles may carry a one-man turret with a single machine gun, occasionally the same model being shared with other classes of vehicle, such as the Cadillac Gage T50 turret/weapons station.

The size of the turret is a factor in combat vehicle design. One dimension mentioned in terms of turret design is "turret ring diameter" which is the size of the aperture in the top of the chassis into which the turret is seated.

Land fortifications
In 1859, the Royal Commission on the Defence of the United Kingdom were in the process of recommending a huge programme of fortifications to protect Britain's naval bases. They interviewed Captain Coles, who had bombarded Russian fortifications during the Crimean War, however Coles repeatedly lost his temper during the discussion and the commissioners failed to ask him about the gun turret that he had patented earlier in that year, with the result that none of the Palmerston Forts mounted turrets. Eventually, the Admiralty Pier Turret at Dover was commissioned in 1877 and completed in 1882.

In continental Europe, the invention of high explosive shells in 1885 threatened to make all existing fortifications obsolete; a partial solution was the protection of fortress guns in armoured turrets. Pioneering designs were produced by Commandant Henri-Louis-Philippe Mougin in France and Captain Maximilian Schumann in Germany. Mougin's designs were incorporated in a new generation of polygonal forts constructed by Raymond Adolphe Séré de Rivières in France and Henri Alexis Brialmont in Belgium. Developed versions of Schumann's turrets were employed after his death in the fortifications of Metz. In 1914, the Brialmont forts in the Battle of Liège proved unequal to the German "Big Bertha" 42 cm siege howitzers, which were able to penetrate the turret armour and smash turret mountings.

Elsewhere, armoured turrets, sometimes described a cupolas, were incorporated into coastal artillery defences. An extreme example was Fort Drum, the "concrete battleship", near Corregidor, Philippines; this mounted four huge 14-inch guns in two naval pattern turrets and was the only permanent turreted fort ever constructed by the United States. Between the wars, improved turrets formed the offensive armament of the Maginot Line forts in France. During the Second World War, some of the artillery pieces in the Atlantic Wall fortifications, such as the Cross-Channel guns, were large naval guns housed in turrets.

Some nations, from Albania to Switzerland and Austria, have embedded the turrets of obsolete tanks in concrete bunkers, while others have constructed or updated fortifications with modern artillery systems, such as the 1970s era Swedish coastal artillery battery on Landsort Island.

title: Gun turret


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John Fred

John Fred Gourrier (May 8, 1941 – April 15, 2005) was an American blue-eyed soul, swamp pop, rock and roll, and R&B performer from Baton Rouge, Louisiana, best known for the 1967 hit song "Judy in Disguise (With Glasses)".

Early life
John Fred was born on May 8, 1941, in Baton Rouge, Louisiana, the son of John Fred Gourrier Sr. and Miriam Chaisson. He had two sisters: Ann Gourrier Kleinpeter and Kay Gourrier Begue.He had one son, Kevin Morris Gourrier. He was married to the love of his life in 1988, Sandra Ratcliff, after a 10 year courtship.

Career
His group, John Fred and the Playboys, was formed in 1956 when Fred was 15; their first charting single was March 1959's "Shirley". He appeared on Alan Freed's show, but when Dick Clark asked him to sing on American Bandstand, Fred had to turn him down because he had to play in a basketball game. Fred played basketball and baseball at Louisiana State University and Southeastern Louisiana University.

By 1967, the band was renamed John Fred & His Playboy Band – to avoid confusion with Gary Lewis & the Playboys – and Fred and band member Andrew Bernard co-wrote "Judy in Disguise (With Glasses)", whose name is a parodic play on the title of The Beatles' song "Lucy in the Sky with Diamonds". The song, issued by Louisiana-based Jewel Records on the Paula label, became successful, knocking "Hello, Goodbye", another Beatles song, out of the No. 1 chart position on the Billboard Hot 100 for two weeks in January 1968. It sold over one million copies, and was awarded a gold disc. With the success of the single, John Fred & His Playboy Band was branded as a novelty act and never had another major success. The follow-up single, "Hey, Hey, Bunny", peaked at No. 57 on the Billboard chart, and the band never again hit the Hot 100. Only after years of struggles did Fred obtain full legal rights to "Judy in Disguise" and its royalties.

Later life and death
Fred continued to perform in bands, coached high school basketball and baseball, and remained a fixture at concerts and shows in his hometown. He produced records for other artists, including Irma Thomas and Fats Domino, and hosted a popular local radio show, The Roots of Rock 'n' Roll, at WBRH in Baton Rouge. He released several solo albums and one group effort, Louisiana Boys, with Joe Stampley and G. G. Shinn. He also wrote and produced radio commercials and jingles, earning two Clio Awards.

In 2004, Fred's health began to fail. Complications ensued after he received a kidney transplant, which culminated in a long hospital stay in New Orleans. He died on April 15, 2005, at the age of 63, survived by his wife, Sandra; a son, Kevin; Kevin's wife, Jodi; and a grandson, Jon Sterling Gourrier.

Awards
in 1999 he received the Louisiana Hall of Fame Living Legend Award, and in 2007 was the first artist inducted into the Louisiana Music Hall of Fame. Later, he was appointed to serve on the Louisiana Music Commission. He was also inducted into the Louisiana Blues Hall of Fame and the Delta Music Hall of Fame.

Albums

 * John Fred and His Playboys (Paula LPS-2191), 1966
 * 34:40 of John Fred and His Playboys (Paula LPS-2193), 1967
 * Agnes English [reissued as Judy in Disguise, with identical tracklist] (Paula LPS-2197), 1967
 * Permanently Stated (Paula LPS-2201), 1969
 * Love My Soul (Uni 73077), 1970
 * Juke Box (Guinness GNS 36022), 1977
 * Louisiana Boys (Bayou Music BM9301-2), 1993 (as the Louisiana Boys)
 * I Miss Ya'll (The Unreleased Masters) (Club Louisianne CDCL 2001), 1999
 * Somebody's Knockin' (TJ Records TJ0102), 2002
 * Roots Rockin' Blues (CD Baby B01HQEGS8A), 2016 (as the Louisiana Riverfront Band)

Compilations

 * The Best of John Fred and the Playboys (Sugarcane Records SR-100), 1984
 * The History of John Fred and the Playboys (Paula PCD-9000), 1991
 * John Fred & His Playboy Band – Anthologie 1958/1969 (Magic 3930818), 2008, France
 * … With Glasses – The Very Best of John Fred and His Playboy Band: The Jewel and Paula Recordings 1964-69 (Fuel 2000 – 3020617562), 2009
 * Judy in Disguise with Glasses (Liberation Hall 5071), 2022

Appearances

 * The Golden Age Of American Rock 'n' Roll Volume 6 (Ace CDCHD 650), 1997, UK – "Shirley"
 * Swamp Pop by the Bayou: Troubles, Tears & Trains (Ace CDCHD 1462), 2016, UK – "You Know You Made Me Cry", "Shirley"
 * Swamp Pop: The Best of Bayou Country (Apple Music), 2022 – "Good Lovin'"
 * East Coast Teen Party (East Coast Music – ECM CD333) – "Boogie Children", "Shirley"

Singles

 * Discography notes

title: John Fred


text:
Plurilingualism

Plurilingualism is the ability of a person who has competence in more than one language to switch between multiple languages depending on the situation for ease of communication. Plurilingualism is different from code-switching in that plurilingualism refers to the ability of an individual to use multiple languages, while code-switching is the act of using multiple languages together. Plurilinguals practice multiple languages and are able to switch between them when necessary without much difficulty. Although plurilingualism is derived from multilingualism (also referred to as bilingualism), there is a difference between the two. Multilingualism is connected to situations wherein multiple languages exist side-by-side in a society but are utilized separately. In essence, multilingualism is the coexisting knowledge of separate languages while plurilingualism is the interconnected knowledge of multiple languages. In general, plurilinguals have had contact with languages not native to them through educational institutions, however the education system plays only a small role in the linguistic competence of these individuals. Learning a second language is thought to stimulate someone's plurilingualism.

Definition and use
Plurilingualism was first equivalent to multilingualism when referencing plurilingual communities, where multiple languages were spoken in a community. However, a distinction of plurilingualism was necessary to differentiate individual and society. Plurilingualism was used to focus on the individual as the agent in the interaction of languages since multilingualism was societal contact of languages.

Plurilingualism does not necessarily mean a person is fluent in multiple languages, it means that a person can interchange more than one language with each other when a situation calls for it. A person is considered competent in plurilingualism when they can speak in one language while understanding another; and can switch between languages when appropriate and/or necessary. According to the Council of Europe, plurilingualism can also be used to mediate conflict between those with no common language. Researchers have even gone so far as to say that being able to understand different dialects and/or regional versions of one language opens the door for someone to be plurilingual. People who are plurilingual tend to have better communicative sensitivity, creativity, and metalinguistic awareness. The knowledge of multiple languages as well as the understanding of different cultures allows for the improved communicative skills. The advantages of plurilingualism seem to become greater the more languages someone learns.

Second language learners
Historically during the formal learning of a second language, it has been common practice to maintain a clear division between the first and second language. The division involves not speaking the first language while learning the second, as well as avoiding any reference to the elements of the first language. By not referencing the first language, instructors hope to develop in their students the same control of the language that native speakers have. This separation, referred to as a hard boundary, contrasts the soft boundary that comes with a plurilinguistic approach. An integrated curriculum, which utilizes the soft boundary, contrasts the different languages that are being taught and highlights the similarities and differences between the two.

Plurilingual education
Plurilingual education has been an increasing form of education in Europe. It was first introduced to the education system in 1996, along with the idea of pluriculturalism. According to the Council of Europe, it is imperative that students learn to be plurilinguists from a young age so that they can be not only be more competitive in an increasingly globalized world, but also be able to integrate within societies when necessary. The founding principles of plurilingual education are the acceptance of cultural diversity, the right to use one's mother tongue as a form of communication, the right to gain experience and knowledge of another language, the acceptance that language is the core of human dialogue. One of the few things plurilingual education promotes is "an awareness of why and how one learns the language one has chosen, a respect for the plurilingualism of others and the value of languages and varieties irrespective of their perceived status in society, and a global integrated approach to langue education in the curriculum."

Plurilingual education developed in the European Union due to the multilingual and multicultural communities throughout the European Union. Other multilingual countries are beginning to use plurilingual education as well. Some communities in Canada are implementing plurilingual education due to the increasingly multilingual society. The Canadian education system is utilizing the Common European Framework of Reference for Languages as a basis for their plurilingual education. Asian countries that are highly multilingual are also considering plurilingual education. South Asian countries have many mother-tongues, however economies and societies may be based on another language. Plurilingual education benefits the individuals of such countries by gaining functional competency in multiple languages for specific social situations and pluricultural competence.

Plurilingualism and pluricultural competence
Pluricultural competence is a consequence of plurilingualism. Pluricultural competence, on a basic level, is the understanding of several cultures. Rather than learning an additional language and/or culture, it is transforming the current knowledge as a whole. Language and culture are interconnected, by learning a language an understanding of the culture is also gained. However, plurilingualism and pluricultural competence are not separately distinguished understandings of language and culture. As plurilingualism is the complex, hybrid understanding of multiple languages that build to form a larger understanding, pluricultural competence is the same. The ability to comprehend and utilize culture in different situations is a valuable skill for individuals in multilingual societies. Depending on the social factors impacting a plurilingual at a time, their pluricultural competence is able to be altered to best adapt. Being plurilingual and having pluricultural competence allows the speaker to choose words and phrases that best describe the situation, which would not be possible without having a blended context of both languages and cultures.

title: Plurilingualism


text:
Participatory cinema

Participatory cinema tries to involve a film's community in the process of making movies, rather than interaction being left to cinema viewers at the end of the process. The organizers of participatory cinema open up cinema showings and the cinema production process for non professionals.

Where classic film production focuses on the final product, participatory cinema focuses on the process of making movies and its meaning on the participators.

Examples
Participatory documentaries include RiP!: A Remix Manifesto.

Participatory feature films include projects like A Swarm of Angels, Iron Sky or The Cosmonaut.

Participatory film platforms and communities include A Swarm of Angels, Open Source Cinema, Sanctuary, The Cosmonaut and Wreck-A-Movie, The Rocky Horror Picture Show

title: Participatory cinema


text: User talk:69.145.54.67

title: User talk:69.145.54.67


text:
Tensor density

In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is additionally multiplied or weighted by a power W of the Jacobian determinant of the coordinate transition function or its absolute value. A tensor density with a single index is called a vector density. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight W are called tensor capacity.  A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.

Motivation
In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself. An example would be decomposing a vector into a sum of basis vectors weighted by some coefficients such as $$\vec{v} = c_1 \vec{e}_1 + c_2 \vec e_2 + c_ 3\vec e_3$$ where $$\vec v$$ is a vector in 3-dimensional Euclidean space, $$c_i \in \R^1 \text{ and } \vec e_i$$ are the usual standard basis vectors in Euclidean space. This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations. However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to change the basis while the vector $$\vec v$$ remains fixed in physical space. More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation. Physicists will often call this representation of a geometric object a tensor if it transforms under a sequence of linear maps given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids). In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation. In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation. A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on $$\R^2.$$ The representation is given by in the standard basis by $$ \vec u \times \vec v = \begin{bmatrix} u_1& u_2 \end{bmatrix} \begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix}v_1 \\ v_2 \end{bmatrix} = u_1 v_2 - u_2 v_1 $$

If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to $\begin{bmatrix} u'_1 & u'_2 \end{bmatrix}^\textsf{T} = A \begin{bmatrix} u_1 & u_2 \end{bmatrix}^\textsf{T}$ where $$A$$ is some 2 by 2 matrix of real numbers. Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be: $$\left(A^{-1}\right)^\textsf{T} \begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix} A^{-1}$$ which, when expanded is just the original expression but multiplied by the determinant of $$A^{-1},$$ which is also $\frac{1}{\det A}.$  In fact this representation could be thought of as a two index tensor transformation, but instead, it is computationally easier to think of the tensor transformation rule as multiplication by $\frac{1}{\det A},$  rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is $$n, n \times n$$ matrix multiplications, which for large $$n$$ is completely infeasible). Objects which transform in this way are called tensor densities because they arise naturally when considering problems regarding areas and volumes, and so are frequently used in integration.

Definition
Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article. Other authors classify them differently, into the types called even tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd.

Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-reversing coordinate transformations. Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-preserving coordinate transformations.

In this article we have chosen the convention that assigns a weight of +2 to $$g = \det\left(g_{\rho\sigma}\right)$$, the determinant of the metric tensor expressed with covariant indices. With this choice, classical densities, like charge density, will be represented by tensor densities of weight +1. Some authors use a sign convention for weights that is the negation of that presented here.

In contrast to the meaning used in this article, in general relativity "pseudotensor" sometimes means an object that does not transform like a tensor or relative tensor of any weight.

Tensor and pseudotensor densities
For example, a mixed rank-two (authentic) tensor density of weight $$W$$ transforms as:

{\mathfrak{T}}^\alpha_\beta = \left( \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right)^{W} \, \frac{\partial {x}^{\alpha}}{\partial \bar{x}^{\delta}} \, \frac{\partial \bar{x}^{\epsilon}}{\partial {x}^{\beta}} \, \bar{\mathfrak{T}}^{\delta}_{\epsilon} \,,$$    ((authentic) tensor density of (integer) weight W) where $$\bar{\mathfrak{T}}$$ is the rank-two tensor density in the $$\bar{x}$$ coordinate system, $${\mathfrak{T}}$$ is the transformed tensor density in the $${x}$$ coordinate system; and we use the Jacobian determinant. Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when $$W$$ is an integer. (However, see even and odd tensor densities below.)

We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation. A mixed rank-two pseudotensor density of weight $$W$$ transforms as

{\mathfrak{T}}^\alpha_\beta = \sgn\left( \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right) \left( \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right)^{W} \, \frac{\partial {x}^{\alpha}}{\partial \bar{x}^{\delta}} \, \frac{\partial \bar{x}^{\epsilon}}{\partial {x}^{\beta}} \, \bar{\mathfrak{T}}^{\delta}_{\epsilon} \,,$$    (pseudotensor density of (integer) weight W)

where $x → \overbar{x}$ is a function that returns +1 when its argument is positive or −1 when its argument is negative.

Even and odd tensor densities
The transformations for even and odd tensor densities have the benefit of being well defined even when $$W$$ is not an integer. Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2.

When $$W$$ is an even integer the above formula for an (authentic) tensor density can be rewritten as

{\mathfrak{T}}^\alpha_\beta = \left\vert \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right\vert^{W} \, \frac{\partial {x}^{\alpha}}{\partial \bar{x}^{\delta}} \, \frac{\partial \bar{x}^{\epsilon}}{\partial {x}^{\beta}} \, \bar{\mathfrak{T}}^{\delta}_{\epsilon} \,.$$    (even tensor density of weight W)

Similarly, when $$W$$ is an odd integer the formula for an (authentic) tensor density can be rewritten as

{\mathfrak{T}}^\alpha_\beta = \sgn \left( \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right) \left\vert \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right\vert^{W} \, \frac{\partial {x}^{\alpha}}{\partial \bar{x}^{\delta}} \, \frac{\partial \bar{x}^{\epsilon}}{\partial {x}^{\beta}} \, \bar{\mathfrak{T}}^{\delta}_{\epsilon} \,.$$    (odd tensor density of weight W)

Weights of zero and one
A tensor density of any type that has weight zero is also called an absolute tensor. An (even) authentic tensor density of weight zero is also called an ordinary tensor.

If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is +1.

Algebraic properties

 * 1) A linear combination (also known as a weighted sum) of tensor densities of the same type and weight $$W$$ is again a tensor density of that type and weight.
 * 2) A product of two tensor densities of any types, and with weights $$W_1$$ and $$W_2$$, is a tensor density of weight $$W_1 + W_2.$$
 * A product of authentic tensor densities and pseudotensor densities will be an authentic tensor density when an even number of the factors are pseudotensor densities; it will be a pseudotensor density when an odd number of the factors are pseudotensor densities. Similarly, a product of even tensor densities and odd tensor densities will be an even tensor density when an even number of the factors are odd tensor densities; it will be an odd tensor density when an odd number of the factors are odd tensor densities.
 * 1) The contraction of indices on a tensor density with weight $$W$$ again yields a tensor density of weight $$W.$$
 * 2) Using (2) and (3) one sees that raising and lowering indices using the metric tensor (weight 0) leaves the weight unchanged.

Matrix inversion and matrix determinant of tensor densities
If $${\mathfrak{T}}_{\alpha\beta}$$ is a non-singular matrix and a rank-two tensor density of weight $$W$$ with covariant indices then its matrix inverse will be a rank-two tensor density of weight −$$W$$ with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant.

If $${\mathfrak{T}}_{\alpha\beta}$$ is a rank-two tensor density of weight $$W$$ with covariant indices then the matrix determinant $$\det {\mathfrak{T}}_{\alpha\beta}$$ will have weight $$N W + 2,$$ where $$N$$ is the number of space-time dimensions. If $${\mathfrak{T}}^{\alpha\beta}$$ is a rank-two tensor density of weight $$W$$ with contravariant indices then the matrix determinant $$\det {\mathfrak{T}}^{\alpha\beta}$$ will have weight $$N W - 2.$$ The matrix determinant $$\det {\mathfrak{T}}^{\alpha}_{~\beta}$$ will have weight $$N W.$$

Relation of Jacobian determinant and metric tensor
Any non-singular ordinary tensor $$T_{\mu\nu}$$ transforms as $$T_{\mu\nu} = \frac{\partial \bar{x}^\kappa}{\partial {x}^\mu} \bar{T}_{\kappa\lambda} \frac{\partial \bar{x}^\lambda}{\partial {x}^\nu} \,,$$

where the right-hand side can be viewed as the product of three matrices. Taking the determinant of both sides of the equation (using that the determinant of a matrix product is the product of the determinants), dividing both sides by $$\det\left(\bar{T}_{\kappa\lambda}\right),$$ and taking their square root gives $$ \left\vert \det{\left[\frac{\partial \bar{x}^\iota}{\partial {x}^\gamma}\right]} \right\vert = \sqrt{\frac{\det({T}_{\mu\nu})}{\det\left(\bar{T}_{\kappa\lambda}\right)}}\,. $$

When the tensor $$T$$ is the metric tensor, $${g}_{\kappa\lambda},$$ and $$\bar{x}^\iota$$ is a locally inertial coordinate system where $$\bar{g}_{\kappa\lambda} = \eta_{\kappa\lambda} =$$diag(−1,+1,+1,+1), the Minkowski metric, then $$\det\left(\bar{g}_{\kappa\lambda}\right) = \det(\eta_{\kappa\lambda}) =$$−1 and so $$ \left\vert \det{\left[\frac{\partial \bar{x}^{\iota}}{\partial {x}^{\gamma}}\right]} \right\vert = \sqrt{-{g}}\,, $$

where $${g} = \det\left({g}_{\mu\nu}\right)$$ is the determinant of the metric tensor $${g}_{\mu\nu}.$$

Use of metric tensor to manipulate tensor densities
Consequently, an even tensor density, $$\mathfrak{T}^{\mu \dots}_{\nu \dots},$$ of weight W, can be written in the form $$\mathfrak{T}^{\mu \dots}_{\nu \dots} = \sqrt{-g}\;^W T^{\mu \dots}_{\nu \dots} \,,$$

where $$T^{\mu \dots}_{\nu \dots} \,$$ is an ordinary tensor. In a locally inertial coordinate system, where $$g_{\kappa\lambda} = \eta_{\kappa\lambda},$$ it will be the case that $$\mathfrak{T}^{\mu \dots}_{\nu \dots}$$ and $$T^{\mu \dots}_{\nu \dots} \,$$ will be represented with the same numbers.

When using the metric connection (Levi-Civita connection), the covariant derivative of an even tensor density is defined as $$ \mathfrak{T}^{\mu \dots}_{\nu \dots ; \alpha} = \sqrt{-g}\;^W T^{\mu \dots}_{\nu \dots ; \alpha} = \sqrt{-g}\;^W \left(\sqrt{-g}\;^{-W} \mathfrak{T}^{\mu \dots}_{\nu \dots}\right)_{;\alpha} \,. $$

For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely $$-W \, \Gamma^{\delta}_{~\delta \alpha} \, \mathfrak{T}^{\mu \dots}_{\nu \dots}$$ to the expression that would be appropriate for the covariant derivative of an ordinary tensor.

Equivalently, the product rule is obeyed $$ \left(\mathfrak{T}^{\mu \dots}_{\nu \dots} \mathfrak{S}^{\sigma \dots}_{\tau \dots}\right)_{; \alpha} = \left(\mathfrak{T}^{\mu \dots}_{\nu \dots; \alpha}\right) \mathfrak{S}^{\sigma \dots}_{\tau \dots} + \mathfrak{T}^{\mu \dots}_{\nu \dots} \left(\mathfrak{S}^{\sigma \dots}_{\tau \dots; \alpha}\right) \,, $$

where, for the metric connection, the covariant derivative of any function of $$g_{\kappa\lambda}$$ is always zero, $$\begin{align} g_{\kappa\lambda ; \alpha} & = 0 \\ \left(\sqrt{-g}\;^W\right)_{; \alpha} & = \left(\sqrt{-g}\;^W\right)_{, \alpha} - W \Gamma^{\delta}_{~\delta \alpha} \sqrt{-g}\;^W = \frac W2 g^{\kappa\lambda} g_{\kappa\lambda,\alpha} \sqrt{-g}\;^W - W \Gamma^{\delta}_{~\delta \alpha} \sqrt{-g}\;^W = 0 \,. \end{align}$$

Examples
The expression $$\sqrt{-g}$$ is a scalar density. By the convention of this article it has a weight of +1.

The density of electric current $$\mathfrak{J}^\mu$$ (for example, $$\mathfrak{J}^2$$ is the amount of electric charge crossing the 3-volume element $$d x^3 \, d x^4 \, d x^1$$ divided by that element — do not use the metric in this calculation) is a contravariant vector density of weight +1. It is often written as $$\mathfrak{J}^\mu = J^\mu \sqrt{-g}$$ or $$\mathfrak{J}^\mu = \varepsilon^{\mu\alpha\beta\gamma} \mathcal{J}_{\alpha\beta\gamma} / 3!,$$ where $$J^\mu\,$$ and the differential form $$\mathcal{J}_{\alpha\beta\gamma}$$ are absolute tensors, and where $$\varepsilon^{\mu\alpha\beta\gamma}$$ is the Levi-Civita symbol; see below.

The density of Lorentz force $$\mathfrak{f}_\mu$$ (that is, the linear momentum transferred from the electromagnetic field to matter within a 4-volume element $$d x^1 \, d x^2 \, d x^3 \, d x^4$$ divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.

In N-dimensional space-time, the Levi-Civita symbol may be regarded as either a rank-N covariant (odd) authentic tensor density of weight −1 ($\overbar{x} → x$) or a rank-N contravariant (odd) authentic tensor density of weight +1 ($sgn($\cdot$)$). Notice that the Levi-Civita symbol (so regarded) does obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that $$\varepsilon^{\alpha\beta\gamma\delta}\,g_{\alpha\kappa}\,g_{\beta\lambda}\,g_{\gamma\mu}g_{\delta\nu} \,=\, \varepsilon_{\kappa\lambda\mu\nu}\,g \,,$$ but in general relativity, where $$g = \det\left(g_{\rho\sigma}\right)$$ is always negative, this is never equal to $$\varepsilon_{\kappa\lambda\mu\nu}.$$

The determinant of the metric tensor, $$g = \det\left(g_{\rho\sigma}\right) = \frac{1}{4!} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\alpha\kappa} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu}\,,$$ is an (even) authentic scalar density of weight +2, being the contraction of the product of 2 (odd) authentic tensor densities of weight +1 and four (even) authentic tensor densities of weight 0.

title: Tensor density


text:
George Batchelor

George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist.

He was for many years a Professor of Applied Mathematics in the University of Cambridge, and was founding head of the Department of Applied Mathematics and Theoretical Physics (DAMTP). In 1956 he founded the influential Journal of Fluid Mechanics which he edited for some forty years. Prior to Cambridge he studied at Melbourne High School and University of Melbourne.

As an applied mathematician (and for some years at Cambridge a co-worker with Sir Geoffrey Taylor in the field of turbulent flow), he was a keen advocate of the need for physical understanding and sound experimental basis.

His An Introduction to Fluid Dynamics (CUP, 1967) is still considered a classic of the subject, and has been re-issued in the Cambridge Mathematical Library series, following strong current demand. Unusual for an 'elementary' textbook of that era, it presented a treatment in which the properties of a real viscous fluid were fully emphasised. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1959.

The Batchelor Prize award, is named in his honour and is awarded every four years at the meeting of the International Congress on Theoretical and Applied Mechanics.

title: George Batchelor


text:
The Adventure Company

The Adventure Company was a Canadian video game developer and a former publishing division of DreamCatcher Interactive. It was sold to THQ Nordic GmbH in 2011 following DreamCatcher's parent (JoWooD Entertainment) assets being sold after entering administration.

History
The Adventure Company was first launched in January 2002 as a division and brand of DreamCatcher Interactive to distribute their adventure games titles under. The first title released under the new brand was The Cameron Files: Secret at Loch Ness which was released at the end of January 2002. The Adventure Company has worked with many developers including: Kheops Studio, THQ, Microïds, and Cryo Interactive. In 2006, DreamCatcher Interactive became a wholly owned subsidiary of the Austrian video game publisher JoWooD Entertainment.

In 2011, Nordic Games acquired JoWooD, its products and brands and some of the companies' subsidiaries. Following the acquisition JoWood and the Adventure Company became publishing labels for Nordic Games, a wholly owned subsidiary of Nordic Games Holding.

Published games
''Note: This list is for titles which The Adventure Company published. Re-releases of DreamCatcher Interactive games nor Nordic Games titles under The Adventure Company are not included.

title: The Adventure Company


text:
Interactive cinema

Interactive cinema tries to give an audience an active role in the showing of movies.

Another newer definition of interactive cinema is a video game which is a hybrid between participation and viewing, giving the player – or viewer, as it were – a strong amount of control in the characters' decisions. It is compared to interactive film.

This form of media recently has become more relevant. Companies like Netflix have even began coming out with releases that have this different way of consumption.

History
The earliest rudimentary examples of interactive cinema date back to the early 20th century, with "cinematic shooting gallery" games. They were similar to shooting gallery carnival games, except that players shot at a cinema screen displaying film footage of targets. They showed footage of targets, and when a player shot the screen at the right time, it would trigger a mechanism that temporarily pauses the film and registers a point. The first successful example of such a game was Life Targets, released in the United Kingdom in 1912. Cinematic shooting gallery games enjoyed short-lived popularity in several parts of Britain during the 1910s, and often had safari animals as targets, with footage recorded from British imperial colonies. Cinematic shooting gallery games declined some time after the 1910s.

The 1967 Czech film Kinoautomat by Czechoslovakian director Raduz Cincera (presented in the Czech Pavilion at Expo 67 in Montreal) is the first cinema-like interactive movie. The availability of computers for the display of interactive video has made it easier to produce interactive movies.

A prominent pioneer of interactive cinema games is the successful Hideo Kojima (1963–present), whose gameplay often takes a priority to the storyline and long cutscenes. His 1994 game Policenauts, a point-and-click adventure game which has shootout sequences (that make use of the lightgun peripheral on the Sega Saturn version of the game), has a subtitle which reads "Interactive cinema" on the cover art of all versions of said game, which provides an early example of a prominent game-developer labelling a game as such. In 1999 Sega's Shenmue video game series won high praise for its implementation of interactive cinematic elements. Its designer Yu Suzuki stated that his goal "was to create a game that was intricate and lifelike by merging the cinematic qualities of movies and the interactivity of computer games". A recent incarnation of an idea similar to this one is Fahrenheit (censored version released in the US and Canada as Indigo Prophecy) – a game dubbed as "interactive cinema" by its France-based developer, Quantic Dream.

The first American interactive film released in 1992, I'm Your Man. Certain Loews Theatres locations retrofitted with controllers allowed audiences to vote on decisions made by the main character. Although initially touted as the first step toward virtual-reality cinema, the experiment proved a failure, the equipment was removed from theaters by 1994.

title: Interactive cinema


text: Category:Prefecture-level divisions of Hainan