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Motion blur

Motion blur is the apparent streaking of moving objects in a photograph or a sequence of frames, such as a film or animation. It results when the image being recorded changes during the recording of a single exposure, due to rapid movement or long exposure.

Photography
When a camera creates an image, that image does not represent a single instant of time. Because of technological constraints or artistic requirements, the image may represent the scene over a period of time. Most often this exposure time is brief enough that the image captured by the camera appears to capture an instantaneous moment, but this is not always so, and a fast moving object or a longer exposure time may result in blurring artifacts which make this apparent. As objects in a scene move, an image of that scene must represent an integration of all positions of those objects, as well as the camera's viewpoint, over the period of exposure determined by the shutter speed. In such an image, any object moving with respect to the camera will look blurred or smeared along the direction of relative motion. This smearing may occur on an object that is moving or on a static background if the camera is moving. In a film or television image, this looks natural because the human eye behaves in much the same way.

Because the effect is caused by the relative motion between the camera, and the objects and scene, motion blur may be manipulated by panning the camera to track those moving objects. In this case, even with long exposure times, the moving objects will appear sharper while the background will become more blurred, with the resulting image conveying a sense of movement and speed.

Animation
In computer animation this effect must be simulated as a virtual camera actually does capture a discrete moment in time. This simulated motion blur is typically applied when either the camera or objects in the scene move rapidly.

Without this simulated effect each frame shows a perfect instant in time (analogous to a camera with an infinitely fast shutter), with zero motion blur. This is why a video game with a frame rate of 25-30 frames per second will seem staggered, while natural motion filmed at the same frame rate appears rather more continuous. Many modern video games feature motion blur, especially vehicle simulation games.

Some of the better-known games that utilise this are the recent Need for Speed titles, Unreal Tournament III, The Legend of Zelda: Majora's Mask, among many others. There are two main methods used in video games to achieve motion blur: cheaper full-screen effects, which typically only take camera movement (and sometimes how fast the camera is moving in 3-D Space to create a radial blur) into mind, and more "selective" or "per-object" motion blur, which typically uses a shader to create a velocity buffer to mark motion intensity for a motion blurring effect to be applied to or uses a shader to perform geometry extrusion. Classic "motion blur" effects prior to modern per-pixel shading pipelines often simply drew successive frames on top of each other with slight transparency, which is strictly speaking a form of video feedback.

In pre-rendered computer animation, such as CGI movies, realistic motion blur can be drawn because the renderer has more time to draw each frame. Temporal anti-aliasing produces frames as a composite of many instants. Frames are not points in time, they are periods of time. If an object makes a trip at a linear speed along a path from 0% to 100% in four time periods, and if those time periods are considered frames, then the object would exhibit motion blur streaks in each frame that are 25% of the path length. If the shutter speed is shortened to less than the duration of a frame, and it may be so shortened as to approach zero time in duration, then the computer animator must choose which portion of the quarter paths (in our 4 frame example) they wish to feature as "open shutter" times. They may choose to render the beginnings of each frame, in which case they will never see the arrival of the object at the end of the path, or they may choose to render the ends of each frame, in which case they will miss the starting point of the trip. Most computer animations systems make the classic "fence-post error" in the way they handle time, confusing the periods of time of an animation with the instantaneous moments that delimit them. Thus most computer animation systems will incorrectly place an object on a four frame trip along a path at 0%, 0.33%, 0.66%, and 1.0% and when called upon to render motion blur will have to cut one or more frames short, or look beyond the boundaries of the animation, compromises that real cameras don't do and synthetic cameras needn't do.

Motion lines in cel animation are drawn in the same direction as motion blur and perform much the same duty. Go motion is a variant of stop motion animation that moves the models during the exposure to create a less staggered effect.

Computer graphics
In 2D computer graphics, motion blur is an artistic filter that converts the digital image /bitmap /raster image in order to simulate the effect. Many graphical software products (e.g. Adobe Photoshop or GIMP) offer simple motion blur filters. However, for advanced motion blur filtering including curves or non-uniform speed adjustment, specialized software products (e.g. VirtualRig Studio) are necessary.

Biology
When an animal's eye is in motion, the image will suffer from motion blur, resulting in an inability to resolve details. To cope with this, humans generally alternate between saccades (quick eye movements) and fixation (focusing on a single point). Saccadic masking makes motion blur during a saccade invisible. Similarly, smooth pursuit allows the eye to track a target in rapid motion, eliminating motion blur of that target instead of the scene.

Television
In televised sports, where conventional cameras expose pictures 25 or 30 times per second, motion blur can be inconvenient because it obscures the exact position of a projectile or athlete in slow motion. For this reason special cameras are often used which eliminate motion blurring by taking rapid exposures on the order of 1 millisecond, and then transmitting them over the course of the next 30 to 40 milliseconds. Although this gives sharper slow motion replays, it can look strange at normal speeds because the eye expects to see motion blurring and is not provided with blurred images.

Conversely, extra motion blur can unavoidably occur on displays when it is not desired. This occurs with some video displays (especially LCD) that exhibits motion blur during fast motion. This can lead to more perceived motion blurring above and beyond the preexisting motion blur in the video material. See display motion blur.

Sometimes, motion blur can be removed from images with the help of deconvolution.

Video games
In video games the use or not of motion blur is somewhat controversial. Some players claim that the blur actually makes the game worse, as it does blur images, making it more difficult to recognize objects (especially so in fast-paced moments). This does become more noticeable (and more problematic) with a reduction in framerate. Improvements in the visual quality of modern post-process motion blur shaders as well as a tendency towards higher framerates have lessened the visual detriment of undersampled motion blur effects.

Wind turbines
Birds cannot properly see the swirling blades of wind turbines and can get struck by them fatally. A newly published report from Norway suggests that painting one of the three blades with a black tip makes the blades more visible and hence more avoidable. This reduces the motion blur of the unpainted blades, and cuts bird deaths by up to 70 percent.

Aerial mapping/surveying
During aerial mapping an aircraft or drone is used to take pictures of the ground during flight. If the flight speed is too high or if shutter speeds are too long, this can lead to motion blur. Motion blur reduces the quality of the images and has a negative effect on the mapping products. Motion blur can be avoided by adjusting the flight altitude, flight velocity, or shutter speed.

Restoration
An example of blurred image restoration with Wiener deconvolution:

title: Motion blur


text:
User talk:Sardonic

Hello and Welcome! I hope you like the place! --mav

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

title: User talk:Sardonic


text:
Spasm

A spasm is a sudden involuntary contraction of a muscle, a group of muscles, or a hollow organ, such as the bladder.

A spasmodic muscle contraction may be caused by many medical conditions, including dystonia. Most commonly, it is a muscle cramp which is accompanied by a sudden burst of pain. A muscle cramp is usually harmless and ceases after a few minutes. It is typically caused by ion imbalance or muscle overload.

There are other causes of involuntary muscle contractions, and some of these may cause a health problem.

A series of spasms, or permanent spasms, is referred to as a "spasmism".

Description and causes
Various kinds of involuntary muscle activity may be referred to as a "spasm".

A spasm may be a muscle contraction caused by abnormal nerve stimulation or by abnormal activity of the muscle itself.

A spasm may lead to muscle strains or tears in tendons and ligaments if the force of the spasm exceeds the tensile strength of the underlying connective tissue. This can occur with a particularly strong spasm or with weakened connective tissue.

A hypertonic muscle spasm is a condition of chronic, excessive muscle tone (i.e., tension in a resting muscle). This is the amount of contraction that remains when a muscle is not working. A true hypertonic spasm is caused by malfunctioning feedback nerves. This is much more serious and is permanent unless treated. In this case, the hypertonic muscle tone is excessive, and the muscles are unable to relax.

A subtype of spasm is colic. This is an episodic pain caused by spasm of smooth muscle in a particular organ (e.g., the bile duct). A characteristic of colic is the sensation of having to move about, and the pain may induce nausea or vomiting.

title: Spasm


text:
User:Patrick Corcoran

Pages I started:

 * Candlestick Park
 * Urantia
 * White League

Pages I've contributed to:

 * Alief Elsik High School, my alma mater
 * ECMAScript
 * List of massacres
 * Milestones in Aviation
 * Piedmont, California
 * web portal

Wikipedia Entries Who Are Friends:

 * Tony Fadell
 * Andrew McDonald
 * John Meyer
 * Christopher Theofanidis

Companies

 * Nest Labs (current)
 * BEA
 * Hotwired
 * Microsoft
 * Plumtree
 * Salesforce.com
 * Salon.com
 * Xmarks

People

 * Paul Boutin
 * Ana Marie Cox
 * Caterina Fake
 * Michelle Goldberg
 * Mike Krieger
 * Mitch Kapor
 * King Kaufman
 * Glenn Kelman
 * Peter Piot
 * Henri Poole
 * Spain Rodriguez
 * Scott Rosenberg
 * Louis Rossetto
 * David Talbot
 * David Weir

Wikipedia Entries I Have Enjoyed Meeting:

 * Andre Agassi
 * Bill Bradley
 * Terence Trent DArby
 * Celine Dion
 * Rich Eisen
 * Mark Morris
 * Gary Shandling
 * Art Spiegelman
 * Alice Waters

title: User:Patrick Corcoran


text:
User talk:Patrick Corcoran

Hello and Welcome! I hope you like the place! --mav

Thanks. :) I've been a fan of wikis for years now, but this one is incredibly compelling. pbc

 * Yep. I'm hooked. :) --mav

Been editing Narnia?

title: User talk:Patrick Corcoran


text:
Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

Examples of Riemann surfaces include graphs of multivalued functions like  √z or log(z), e.g. the subset of pairs (z,w) ∈ C2 with w = log(z).

Every Riemann surface is a surface: a two-dimensional real manifold, but it contains more structure (specifically a complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. Given this, the sphere and torus admit complex structures but the Möbius strip, Klein bottle and real projective plane do not. Every compact Riemann surface is a complex algebraic curve by Chow's theorem and the Riemann–Roch theorem.

Definitions
There are several equivalent definitions of a Riemann surface.

 * 1) A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and the transition maps between two overlapping charts are required to be holomorphic.
 * 2) A Riemann surface is an oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane. The supplement "Riemann" signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.

A complex structure gives rise to a conformal structure by choosing the standard Euclidean metric given on the complex plane and transporting it to X by means of the charts. Showing that a conformal structure determines a complex structure is more difficult.

Further definitions and properties
As with any map between complex manifolds, a function f: M → N between two Riemann surfaces M and N is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N, the map h ∘ f ∘ g−1 is holomorphic (as a function from C to C) wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces M and N are called biholomorphic (or conformally equivalent to emphasize the conformal point of view) if there exists a bijective holomorphic function from M to N whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted). Two conformally equivalent Riemann surfaces are for all practical purposes identical.

Orientability
Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function h = f(g−1(z)), h can be considered as a map from an open set of R2 to R2 whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h ' (z). However, the real determinant of multiplication by a complex number α equals |α|2, so the Jacobian of h has positive determinant. Consequently, the complex atlas is an oriented atlas.

Functions
Every non-compact Riemann surface admits non-constant holomorphic functions (with values in C). In fact, every non-compact Riemann surface is a Stein manifold.

In contrast, on a compact Riemann surface X every holomorphic function with values in C is constant due to the maximum principle. However, there always exist non-constant meromorphic functions (holomorphic functions with values in the Riemann sphere C ∪ {∞}). More precisely, the function field of X is a finite extension of C(t), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see. Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and the Abel–Jacobi map of the surface.

Algebraicity
All compact Riemann surfaces are algebraic curves since they can be embedded into some $$\mathbb{CP}^n$$. This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve.

Analytic vs. algebraic
The existence of non-constant meromorphic functions can be used to show that any compact Riemann surface is a projective variety, i.e. can be given by polynomial equations inside a projective space. Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3-space. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem.

As an example, consider the torus T := C/(Z + τ Z). The Weierstrass function $$\wp_\tau(z)$$ belonging to the lattice Z + τ Z is a meromorphic function on T. This function and its derivative $$\wp_\tau'(z)$$ generate the function field of T. There is an equation

 * $$[\wp'(z)]^2=4[\wp(z)]^3-g_2\wp(z)-g_3,$$

where the coefficients g2 and g3 depend on τ, thus giving an elliptic curve Eτ in the sense of algebraic geometry. Reversing this is accomplished by the j-invariant j(E), which can be used to determine τ and hence a torus.

Classification of Riemann surfaces
The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant sectional curvature. That is, every connected Riemann surface $$X$$ admits a unique complete 2-dimensional real Riemann metric with constant curvature equal to $$-1, 0$$ or $$1$$ which belongs to the conformal class of Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of isothermal coordinates.

In complex analytic terms, the Poincaré–Koebe uniformization theorem (a generalization of the Riemann mapping theorem) states that every simply connected Riemann surface is conformally equivalent to one of the following: A Riemann surface is elliptic, parabolic or hyperbolic according to whether its universal cover is isomorphic to $$\mathbf P^1(\mathbf C)$$, $$\mathbf C$$ or $$\mathbf D$$. The elements in each class admit a more precise description.
 * The Riemann sphere $$\widehat{\mathbf{C}} := \mathbf{C} \cup\{\infty\}$$, which is isomorphic to the $\mathbf P^1(\mathbf C)$ ;
 * The complex plane $$\mathbf C$$;
 * The open disk $$\mathbf D := \{z \in \mathbf C : |z| < 1\}$$ which is isomorphic to the upper half-plane $$\mathbf H := \{z \in \mathbf C : \mathrm{Im}(z) > 0\}$$.

Elliptic Riemann surfaces
The Riemann sphere $$\mathbf P^1(\mathbf C)$$ is the only example, as there is no group acting on it by biholomorphic transformations freely and properly discontinuously and so any Riemann surface whose universal cover is isomorphic to $$\mathbf P^1(\mathbf C)$$ must itself be isomorphic to it.

Parabolic Riemann surfaces
If $$X$$ is a Riemann surface whose universal cover is isomorphic to the complex plane $$\mathbf C$$ then it is isomorphic to one of the following surfaces: Topologically there are only three types: the plane, the cylinder and the torus. But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter $$\tau$$ in the third case gives non-isomorphic Riemann surfaces. The description by the parameter $$\tau$$ gives the Teichmüller space of "marked" Riemann surfaces (in addition to the Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic moduli space (forgetting the marking) one takes the quotient of Teichmüller space by the mapping class group. In this case it is the modular curve.
 * $$\mathbf C$$ itself;
 * The quotient $$\mathbf C / \mathbf Z$$;
 * A quotient $$\mathbf C / (\mathbf Z + \mathbf Z\tau)$$ where $$\tau \in \mathbf C$$ with $$\mathrm{Im}(\tau) > 0$$.

Hyperbolic Riemann surfaces
In the remaining cases $$X$$ is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface). The topological type of $$X$$ can be any orientable surface save the torus and sphere.

A case of particular interest is when $$X$$ is compact. Then its topological type is described by its genus $$g \ge 2$$. Its Teichmüller space and moduli space are $$6g - 6$$-dimensional. A similar classification of Riemann surfaces of finite type (that is homeomorphic to a closed surface minus a finite number of points) can be given. However in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such a description.

Maps between Riemann surfaces
The geometric classification is reflected in maps between Riemann surfaces, as detailed in Liouville's theorem and the Little Picard theorem: maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained (indeed, generally constant!). There are inclusions of the disc in the plane in the sphere: $$\Delta \subset \mathbf{C} \subset \widehat{\mathbf{C}},$$ but any holomorphic map from the sphere to the plane is constant, any holomorphic map from the plane into the unit disk is constant (Liouville's theorem), and in fact any holomorphic map from the plane into the plane minus two points is constant (Little Picard theorem)!

Punctured spheres
These statements are clarified by considering the type of a Riemann sphere $$\widehat{\mathbf{C}}$$ with a number of punctures. With no punctures, it is the Riemann sphere, which is elliptic. With one puncture, which can be placed at infinity, it is the complex plane, which is parabolic. With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. With three or more punctures, it is hyperbolic – compare pair of pants. One can map from one puncture to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity), but all maps from zero punctures to one or more, or one or two punctures to three or more are constant.

Ramified covering spaces
Continuing in this vein, compact Riemann surfaces can map to surfaces of lower genus, but not to higher genus, except as constant maps. This is because holomorphic and meromorphic maps behave locally like $$z \mapsto z^n,$$ so non-constant maps are ramified covering maps, and for compact Riemann surfaces these are constrained by the Riemann–Hurwitz formula in algebraic topology, which relates the Euler characteristic of a space and a ramified cover.

For example, hyperbolic Riemann surfaces are ramified covering spaces of the sphere (they have non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.

Isometries of Riemann surfaces
The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry:
 * genus 0 – the isometry group of the sphere is the Möbius group of projective transforms of the complex line,
 * the isometry group of the plane is the subgroup fixing infinity, and of the punctured plane is the subgroup leaving invariant the set containing only infinity and zero: either fixing them both, or interchanging them (1/z).
 * the isometry group of the upper half-plane is the real Möbius group; this is conjugate to the automorphism group of the disk.
 * genus 1 – the isometry group of a torus is in general translations (as an Abelian variety), though the square lattice and hexagonal lattice have addition symmetries from rotation by 90° and 60°.
 * For genus g ≥ 2, the isometry group is finite, and has order at most 84(g − 1), by Hurwitz's automorphisms theorem; surfaces that realize this bound are called Hurwitz surfaces.
 * It is known that every finite group can be realized as the full group of isometries of some Riemann surface.
 * For genus 2 the order is maximized by the Bolza surface, with order 48.
 * For genus 3 the order is maximized by the Klein quartic, with order 168; this is the first Hurwitz surface, and its automorphism group is isomorphic to the unique simple group of order 168, which is the second-smallest non-abelian simple group. This group is isomorphic to both PSL(2,7) and PSL(3,2).
 * For genus 4, Bring's surface is a highly symmetric surface.
 * For genus 7 the order is maximized by the Macbeath surface, with order 504; this is the second Hurwitz surface, and its automorphism group is isomorphic to PSL(2,8), the fourth-smallest non-abelian simple group.

Function-theoretic classification
The classification scheme above is typically used by geometers. There is a different classification for Riemann surfaces which is typically used by complex analysts. It employs a different definition for "parabolic" and "hyperbolic". In this alternative classification scheme, a Riemann surface is called parabolic if there are no non-constant negative subharmonic functions on the surface and is otherwise called hyperbolic. This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc.

To avoid confusion, call the classification based on metrics of constant curvature the geometric classification, and the one based on degeneracy of function spaces the function-theoretic classification. For example, the Riemann surface consisting of "all complex numbers but 0 and 1" is parabolic in the function-theoretic classification but it is hyperbolic in the geometric classification.

Theorems regarding Riemann surfaces

 * Branching theorem
 * Hurwitz's automorphisms theorem
 * Identity theorem for Riemann surfaces
 * Riemann–Roch theorem
 * Riemann–Hurwitz formula

title: Riemann surface


title: User:Mr100percent


text:
User talk:Mr100percent

Hello and Welcome! I hope you like the place! --mav

Hello, I've nominated Arabic Calligraphy as a Featured article candidates and there have been a number of requests in regard to the illustrations you added to the article.
 * Could you provide source information for the illustrations?
 * What does each illustration say?
 * How should the various styles be grouped and categorised for the purposes of a table of contents?

Many thanks Lisiate 21:41, 8 Sep 2004 (UTC)

Your uploaded image
Thanks for uploading Image:Ift32.gif. I notice it currently doesn't have an image copyright tag. Could you add one to let us know whether it's released under the GFDL, whether fair use is claimed, or what? Thanks,   – Quadell (talk) (help)   18:11, Oct 1, 2004 (UTC)

Image:Map of Caucasus.gif
Thanks for uploading Image:Map of Caucasus.gif. I notice it currently doesn't have an image copyright tag. Until a more informative tag is provided, it will be listed as no source or no license. Could you add a better tag to let us know its source and/or copyright status? If you made the image yourself, an easy way to deal with this is add GFDL if you're willing to release it under the GFDL. Alternatively, you could release all rights to it by adding NoRightsReserved. This would allow anyone to do whatever they wish with your image, without exceptions. However, if it isn't your own image, you need to specify what free license it was distributed under. You can find a list of the tags here. If it was not distributed under a free license, but you claim fair use, add {{fairuse}} but you need to substantiate your claim by explaining why you think it's fair use. If you don't know what any of this means, just let me know where you got the images by posting to my talk page. If you do this, I can tag them for you. Thanks. RedWolf 05:02, July 20, 2005 (UTC)

Hi!
Just make a easy google search, it ill hit you in the same place where CIA usualy hits people, in your face. Or mine, for that matter. Before i figured out their game, that is. And a word of caution, beware of people that have only made 19 edits total, have only been in wikipedia for about a month, but behave Islamic and friedly. belive it or not, there are Munafiqs amount the muslim, more specificaly user:Saduj al-Dahij and his Muslim fellows. Take a look at the nice offer i received on bottom of my talk page, i could hardly resis it!

Thanks for your comments and have a good day!

Oh, about sites, im not much about politics and stuff, all politicians are controled by the same entity, every single one of them. Those who have not succumed are executed. So i keep it on the private side, i strive to inform, both on the net and IRL... at least till they take over the net as well, then its only IRL thats left, i would gues that the net have like 10-15 years, then it will become as useless as Mainstream media is.

Oh, im babling... im sure you have placed me in "that" section by now... whadever, time will tell :)

Have a best day :)

--Striver 00:10, 1 August 2005 (UTC)

Image deletion
You're listed as the original uploader of Image:Ift32.gif, which is proposed to be deleted, having been replaced by Image:Islam.png AnonMoos 07:52, 29 January 2006 (UTC)

Image:Ift32.gif listed for deletion
An image or media file that you uploaded, Image:Ift32.gif, has been listed at Images and media for deletion. Please look there to see why this is (you may have to search for the title of the image to find its entry), if you are interested in it not being deleted. Thank you.

Image Tagging Image:Siraj.gif
Thanks for uploading Image:Siraj.gif. I notice the 'image' page currently doesn't specify who created the content, so the copyright status is therefore unclear. If you have not created this media yourself then you need to argue that we have the right to use the media on Wikipedia (see copyright tagging below). If you have not created the media yourself then you should also specify where you found it, i.e., in most cases link to the website where you got it, and the terms of use for content from that page.

If the media also doesn't have a copyright tag then you must also add one. If you created/took the picture, audio, or video then you can use GFDL to release it under the GFDL. If you believe the media qualifies as fair use, please read fair use, and then use a tag such as or one of the other tags listed at Image copyright tags. See Image copyright tags for the full list of copyright tags that you can use.

If you have uploaded other media, please check that you have specified their source and copyright tagged them, too. You can find a list of 'image' pages you have edited by clicking on the "my contributions" link (it is located at the very top of any Wikipedia page when you are logged in), and then selecting "Image" from the dropdown box. Note that any unsourced and untagged images will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. Thank you. Longhair 06:34, 4 February 2006 (UTC)

Image:Diwani.png
This image is in wikimedia commons and you are listed as its author. can you tell me more about its origin and why is it licensed as PD... it is clearly not a creation of the USA federal government.. whu would it be considered PD-ineligible.

thanks in advance --Histolo2 (talk) 20:45, 30 May 2008 (UTC)

PD-USGov
Hi, image:Farsi.png om commons was tagged with PD-USGov , with your name as the author. This was a bit confusing, so it is now tagged with a no-source. Can you please elaborate about the source and the fonts used. --Tarawneh (talk) 00:45, 31 May 2008 (UTC)

title: User talk:Mr100percent


text:
Roll-to-roll processing

In the field of electronic devices, roll-to-roll processing, also known as web processing, reel-to-reel processing or R2R, is the process of creating electronic devices on a roll of flexible plastic, metal foil, or flexible glass. In other fields predating this use, it can refer to any process of applying coating, printing, or performing other processes starting with a roll of a flexible material and re-reeling after the process to create an output roll. These processes, and others such as sheeting, can be grouped together under the general term converting. When the rolls of material have been coated, laminated or printed they can be subsequently slit to their finished size on a slitter rewinder.

In electronic devices
Large circuits made with thin-film transistors and other devices can be patterned onto these large substrates, which can be up to a few metres wide and 50 km long. Some of the devices can be patterned directly, much like an inkjet printer deposits ink. For most semiconductors, however, the devices must be patterned using photolithography techniques.

Roll-to-roll processing of large-area electronic devices reduces manufacturing cost. Most notable would be solar cells, which are still prohibitively expensive for most markets due to the high cost per unit area of traditional bulk (mono- or polycrystalline) silicon manufacturing. Other applications could arise which take advantage of the flexible nature of the substrates, such as electronics embedded into clothing, large-area flexible displays, and roll-up portable displays.

LED (Light Emitting Diode)

 * Inorganic LED - Flexible LED is commonly made into 25, 50, 100 m, or even longer strips using a roll-to-roll process. A long neon LED tube is using such a long flexible strip and encapsulated with PVC or silicone diffusing encapsulation.
 * Organic LED (OLED) - OLED for foldable phone screen is adopting roll-to-roll processing technology.

Thin-film cells
A crucial issue for a roll-to-roll thin-film cell production system is the deposition rate of the microcrystalline layer, and this can be tackled using four approaches:
 * very high frequency plasma-enhanced chemical vapour deposition (VHF-PECVD)
 * microwave (MW)-PECVD
 * hot wire chemical vapour deposition (hot-wire CVD)
 * the use of ultrasonic nozzles in an in-line process

In electrochemical devices
The roll-to-roll processing has been used in the manufacture of electrochemical devices such as batteries, supercapacitors, fuel cells, and water electrolyzers. Here, the roll-to-roll processing is utilized for electrode manufacturing and is the key to reducing manufacturing cost through stable production of electrodes on various film substrates such as metal foils, membranes, diffusion media, and separators.

title: Roll-to-roll processing


text:
Talk:Riemann surface

Early 2003
How about an explanation in layman's english, too? Theanthrope 06:46 Jan 23, 2003 (UTC)

I'll try. However, while it's possible to sort of talk about a manifold in layman's english, the distinction between Riemann surfaces and 2-manifolds is very subtle and difficult to describe without using any math. Loisel 07:08 Jan 23, 2003 (UTC)

---

Axel, I saw you improved the opening paragraph. However, I'm concerned that perhaps we could serve Theanthrope's request (see above) by moving some of the jargon to a second paragraph, and keeping the opening paragraph as light on jargon as possible. I think it would also help if someone could give an interesting example of a 2-manifold that can't be viewed as a Riemann surface. This way, we can somehow say "Riemann surfaces are mainly what everyone thinks of as surfaces, however some surfaces such as X aren't Riemann surface, see below for technical details." Unfortunately, I'm not sure which 2 surface does not admit a Riemann surface structure. What do you think? Loisel 06:18 Feb 12, 2003 (UTC)

I moved the term "holomorphic" into the second paragraph, even though I think it's the whole point of the enterprise. We certainly need examples: open sets of the complex plane, the Riemann sphere, the two-sheet thingy you get from the square root function etc.

Everything2 has a nice writeup at http://everything2.com/index.pl?node_id=1007348

All Riemann surfaces are orientable, so the Moebius strip can't be made into a Riemann surface. AxelBoldt 16:56 Feb 12, 2003 (UTC)

I didn't know that. I added an argument outline to the article. I didn't want to infect the article with a lengthy proof. Delete it if you don't like it. Loisel 03:27 Feb 13, 2003 (UTC)

We need to de-TeX this article a bit, it takes far too long to load. -- Tarquin 16:39 Feb 12, 2003 (UTC)

Yes, I agree. AxelBoldt 17:15 Feb 12, 2003 (UTC)

Me too - plus it's horrible to read text where the font size jumps up and down by a factor of 2. Chas zzz brown 12:04 Feb 13, 2003 (UTC)

Axel, I'm not sure I have the correct idea for the charts of concrete riemann surfaces. I'm not completely sure that what I have in mind is a chart independent of the a_k. Can you clarify that for me? Loisel 22:07 Feb 17, 2003 (UTC)

Actually, now that I think about it: it doesn't seem like all "concrete Riemann surfaces" as I defined them are Riemann surfaces. It seems the concrete ones may have singularities while our abstrace Riemann surfaces cannot. I don't know how to fix this right now; maybe we should just take the example of concrete surfaces out, even though historically it was the motivating one. AxelBoldt 00:21 Feb 18, 2003 (UTC)

I'm interested in the "only if" portion of what you just wrote. It says that if a real two-manifold X is orientable, then it has a Riemann surface structure. Is there an easy way to see this, or a book where I can look this stuff up? I've been sitting here trying to take some charts and making sure they're Riemann compatible, but try as I might, I can't figure it out. Take for instance R^2. In some open region, I use the identity map for a chart f(x)=x. In another open region, I use a function g(x) which isn't analytic at some point (like exp(-1/x^2) at 0 or something like that.) How do I turn these charts into Riemann charts? Do I need to know some facts about the ring C^\infty(X) to prove this? Loisel 02:32 Feb 18, 2003 (UTC)

I found the claim in http://www.jchl.co.uk/maths/Riemann%20Surfaces.pdf I don't know how to prove it. But I do know that every differentiable real manifold has an atlas whose elements are compatible as real analytic functions. You somehow need to use the orientability to get from real analytic to holomorphic. AxelBoldt 06:23 Feb 18, 2003 (UTC)

Can you point me to the page you're thinking of? Are you referring to the realization of Riemann surfaces as algebraic surfaces of the form you suggested ("concrete Riemann surfaces") or is it something else? Those are good notes, by the way. Loisel 07:34 Feb 18, 2003 (UTC)

On the top of page 6 of the above given PDF file, he makes the claim that every orientable real 2-manifold can be turned into a Riemann surface. I don't know if he proves it later in the notes. AxelBoldt 16:56 Feb 18, 2003 (UTC)

I see. Interesting. Loisel 17:50 Feb 18, 2003 (UTC)

sheaves
Has anyone else seen the definition of "sheaf" used in this article. I was under the impression a sheaf was a contravariant functor from the category of open sets and inclusions in a topological space to the category of sets, rings, etc. such that if all restrictions of certain sections line up a unique new element can be patched together from the data. The "sheaf" in the article seems to be a global analytic function constructed as the union of the functions given by local power series expansions; it seems like this object would be a section of some sheaf of holomorphic functions rather than a sheaf itself. Is this wrong? Vivacissamamente

The usage isn't exactly correct, perhaps; but it is rather close. That is, with the definition of sheaf space of a sheaf (an alternative to the functor approach, and earlier too), the projection of G really will be the kind of local homeomorphism that is to be identified with a sheaf. As if the Riemann surface required is really constructed as a quite small subsheaf of the sheaf of holomorphic functions, would say it another way.

Charles Matthews 14:32, 8 Jun 2004 (UTC)

I've never been able to parse the algebraic sheaves, although I realize they're much more talked about than the complex analysis sheaves today. However, I believe that the nomenclature originated in complex analysis, and I'm fairly certain that the definition here is the one for complex analysis.

Loisel 23:12, 9 Jun 2004 (UTC)

Historically, I believe, this goes back to the Poincaré-Volterra theorem? Charles Matthews 07:23, 10 Jun 2004 (UTC)

Which one is that? Loisel 21:01, 10 Jun 2004 (UTC)

Nevermind, got it. Loisel 16:06, 11 Jun 2004 (UTC)

Analytic continuation
Would it make sense to move the whole bit about analytic continuation and sheaves to the article on analytic continuation? Its not clear to me why this stuff about analytic continuation is even in this article... (other that that g composed with f-1 needs to be analytic, etc.  Anyone up for that, or will my edits get reverted ??linas 20:03, 22 Jan 2005 (UTC)

 * Still planing on making good on the above threat...real sooner now. linas 14:59, 14 Apr 2005 (UTC)

 * The connection is because the 'maximum analytic continuation' is to a Riemann surface; and this is one of the historical roots of the topic. That said, it might be just as clear to place this material in the other article. Charles Matthews 15:44, 14 Apr 2005 (UTC)

 * Moved. I want to expand this article a bit, and the stuff on analytic continuation was taking up space. I also remember learning analysis for the first time, and being frustrated by the lack of a good definition of analytic continuation. For whatever reason, courses in begining complex analysis don't really use the words riemann surface very much. linas 17:14, 16 Apr 2005 (UTC)

sheaves
the article on sheaves contains a link to this article for an example of a sheaf. I removed this line from the text, because there is no such example here. If anyone does add an example at some point, then the link should be restored in the article on sheaves. -Lethe | Talk 19:53, May 31, 2005 (UTC)

 * Looks like the relevant material was moved in April to analytic continuation. Charles Matthews 20:48, 31 May 2005 (UTC)

Hyperbolic surfaces have nontrivial π1?
the article says the following things: the disk is among the simply connected complex surfaces, the disk is hyperbolic, hyperbolic surfaces have nontrivial fundamental group. One of these three sentences doesn't fit. I think it's probably the last one? -Lethe | Talk 13:29, Jun 16, 2005 (UTC)
 * thinking more on the matter, I think a surface of constant negative curvature must have negative euler characteristic, and so high genus, and so nontrivial pi1. So the first statement is in error? Also, can I infer that the Poincaré half plane is multiply connected?  That seems surprising. -Lethe | Talk 14:19, Jun 16, 2005 (UTC)

The last statement is not correct, the disk is the unique (up to biholomorphism) simply connected hyperbolic surface. The hyperbolic surfaces which aren't simply connected have the disk as the universal cover. The upper half plane is biholomorphic to the disk and so also hyperbolic and simply connected. -- Fropuff 15:05, 16 Jun 2005 (UTC)

 * Maybe the author of that stuff meant compact hyperbolic Riemann surfaces? The relationship between curvature and genus is maybe not straightforward for noncompact surfaces.  I find similar erroneous statements in several other related articles as well. -Lethe | Talk 22:28, Jun 16, 2005 (UTC)

That is entirely possible. Statements further down (regarding the area of the surface) certainly apply only in the compact case. These statements all need to be clarified -- Fropuff 04:59, 17 Jun 2005 (UTC)

corrections
Yes, certainly it should be changed. In Riemann surface, it seems to me that the sentence
 * "Hyperbolic Riemann surfaces are multiply connected and thus have a non-trivial fundamental group."

ought not be the first sentence in the paragraph. I mean, this isn't particular to hyperbolic geometry, right? C is simply connected noncompact and parabolic, and C mod some rank 2 discrete lattice group is closed parabolic, which is not simply connected (pi1 = lattice group). At this point, I don't see what's special about the hyperbolic case: H is simply connected noncompact and hyperbolic. while H mod Fuchsian group is closed hyperbolic (pi1 = Fuchsian group). It's the same, so why is the hyperbolic case getting singled out?

I propose we change it to something along these lines:

 * "For every closed parabolic Riemann surface, the fundamental group is isomorphic to a rank 2 lattice group, and thus the surface can be constructed as C/Γ, where C is the complex plane and Γ is the lattice group. The representatives of the cosets are called fundamental cells.  The area of such a surface is arbitrary (is this true?).

 * OK I copied this paragraph into the article. Notice the red links. Sould we redirect fundamental cell to free regular set for now, until someone writes a "real" article, or is that a bad idea? Should we redirect lattice group to lattice (group)? Or maybe redirect to torus? As to area of surface being arbitrary, that sure sounds right to me, but as you notice I'm error-prone. I left the sentance out. linas 00:33, 18 Jun 2005 (UTC)

 * Actually, there's already an article fundamental domain. I guess fundamental cell should redirect to that.  And yeah, I guess lattice group should redirect to lattice (group).  As for the area of a torus, the complex structure depends on the ratio of the lengths of the sides, so you can scale the area to anything you want.  You can also see that the Gauss-Bonnet theorem can't tell you the area of the torus of constant curvature, since the curvature is zero. -02:10, Jun 19, 2005 (UTC)

 * "Similarly for every hyperbolic Riemann surface, the fundamental group is isomorphic to a Fuchsian group, and thus the surface can be modelled by a Fuchsian model H/? where H is the upper half plane and ? is the Fuchsian group. The representatives of the cosets of H/? are called fundamental polygons. The total area of such a surface, where g is the genus of the surface; the area is obtained by applying the Gauss-Bonnet theorem to the area of the fundamental polygon."

Can we say some more about the elliptic case? Is the Riemann sphere the only closed elliptic surface (up to biholomorphism)? We should link also to some article on Lie groups that says pi1 of universal cover mod discrete subgroup = discrete subgroup.

 * Last I looked, Riemann sphere was unique. As to the modulo-group remark, I think that the article covering map states this, if rather densely. Also, by lattice group do you mean wallpaper group? Or is there another definition? linas 14:03, 17 Jun 2005 (UTC)

 * No, a lattice group is not the same thing as a wallpaper group, although I think the latter includes the former as a special case. see Lattice (group). -Lethe | Talk 23:20, Jun 17, 2005 (UTC)

 * OK, so let me see if I have it straight: there is only one elliptic Riemann surface, and it's simply connected. For the parabolic case, there is one simply connected surface C, one surface with pi1=Z (C mod lattice of rank 1 = C*), and an infinite number of tori with pi1=ZxZ (C mod lattice of rank 2).  And in the hyperbolic case, I'm a bit fuzzy.  Is there only one surface for each genus >1?  each of them has H as universal cover?  -Lethe | Talk 02:10, Jun 19, 2005 (UTC)

 * Hmm. You know, I am at best a novice w.r.t Riemann surfaces. I don't know the parabolic case. Saying "its a lattice" essentially means "its a torus"; I have no clue if there are any parabolic riemann surfaces that aren't toruses. Can't say I've heard of such beasts, but also can't say I've heard of a theorem stating otherwise. Maybe its trivial?  linas 00:33, 18 Jun 2005 (UTC)

 * Well, if the lattice group is rank 1, you get a parabolic manifold that is not a torus. You get the cylinder, which is the same as the multiplicative group of nonzero complex numbers.  C, C*, and a bunch of tori, those are the only parabolic surfaces. -Lethe | Talk 02:10, Jun 19, 2005 (UTC)

The article on hyperbolic geometry has the similar statement "Hyperbolic surfaces have a non-trivial fundamental group, known as the Fuchsian group. The quotient space H/? of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface.", which also needs to me changed. I thought I saw something like this in a third article too, this morning, but now I can't find it. Maybe it was just the two. -Lethe | Talk 05:42, Jun 17, 2005 (UTC)

 * Several quick remarks: don't sacrifice overall clarity of an introduction to exactness. Most hyperbolic surfaces do have non-trivial genus; only one doesn't, the hyperbolic disk. I think that a minor edit could take care of the one-exception case. Other than that, the above proposed paragraphs look reasonable. linas 13:51, 17 Jun 2005 (UTC)

 * Sure, most manifolds in any isogeny class have nontrivial pi1, except for the one that doesn't. This is true for any isogeny class of Lie groups.  But what does that have to do with hyperbolic geometry?  The sentence makes it seem like somehow the hyperbolicity implies nontrivial pi1.  That's true in the compact case, not true in the noncompact case. -Lethe | Talk 22:36, Jun 17, 2005 (UTC)

 * My apologies, I just realized you were complaining about sentences I had written back when. I am sorry they're misleading, this was not my intent, and perhaps shows my ignorance of the subject. The corrections you propose seem reasonable; I invite you to make them, or you can wait for me to do it. However, I have a backlog of unfinished projects, so waiting my prove to be irritating. linas 23:37, 17 Jun 2005 (UTC)

 * Never mind, I fixed by copying you wording. Please review for additional corrections. linas 00:33, 18 Jun 2005 (UTC)

Also, is it correct to say the representatives are the fundamental polygon? Isn't a representative of a coset simply a point? -Lethe | Talk 05:54, Jun 17, 2005 (UTC)

 * No, its not a point. Technically, the representatives are free regular sets which can be converted to fundamental polygons after a set of operations that re-arrange the area of the thing so that its bounded by geodesics, etc. linas 13:51, 17 Jun 2005 (UTC)

 * I think a certain set of representatives form the fundamental domain/polygon/free regular set, but the representatives themselves are simply points.  I mean, I get what it's trying to say, but I think as it stands it is worded incorrectly.  Actually, you can see the article on free regular sets to see the correct wording. -Lethe | Talk 22:36, Jun 17, 2005 (UTC)

 * Oh gosh, very sorry, yes, I misunderstood, you are absolutely right; the wording in the article is wrong. What can I say? Brain in neutral, keyboard in overdrive. I'll fix this in a few minutes. linas 23:37, 17 Jun 2005 (UTC)

Hyperbolic/parabolic/elliptic terminology
I am a little confused as to the meaning of hyperbolic/parabolic/elliptic in regards to Riemann surfaces. This article defines a Riemann surface to be hyperbolic/parabolic/elliptic according to whether the universal cover is the unit disk, the complex plane, or the Riemann sphere. Or, equivalently, whether the surface carries a Riemannian metric with constant curvature −1/0/+1. However, the book by Farkas and Kra (Riemann Surfaces 2nd ed. 1992, ISBN 0-387-97703-1) uses a different definition for these terms. In section IV.3.2 they define a Riemann surface to be According to this classification the three simply connected surfaces are still hyperbolic/parabolic/elliptic as before, but the meaning changes for another surfaces. For example, all compact surfaces (of any genus) are elliptic according to Farkas and Kra but most (g ≥ 2) are hyperbolic according to this article.
 * elliptic iff it is compact,
 * parabolic iff it is noncompact and does not carry a negative nonconstant subharmonic function,
 * hyperbolic iff it does carry a negative nonconstant subharmonic function (necessarily noncompact).

Can anyone comment of the different definitions? Are either or both standard? Should we mention the Farkas and Kra meaning in this article? I note that the Farkas and Kra definition is potentially more interesting since nearly every surface is hyperbolic in the sense of this article (the sole exceptions being the Riemann sphere, the complex plane, the punctured plane, and complex tori).

-- Fropuff 8 July 2005 04:30 (UTC)

 * I don't know. Jost, Compact Riemann Surfaces doesn't appear to attempt such a naming convention. linas 15:29, 11 July 2005 (UTC)

 * No, I don't suppose Jost would, as by the title of his book he only treats the compact case, which are all elliptic according to Farkas and Kra. I'll have to see if I can find any other references. -- Fropuff 16:01, 11 July 2005 (UTC)

I've checked a few other sources. Both Ahlfors (Riemann Surfaces, 1960) and Beardon (A Primer on Riemann Surfaces, 1984) use the Farkas and Kra meaning for parabolic and hyperbolic. They don't define elliptic spaces, refering to those spaces simply as compact surfaces. These books also mention that hyperbolic surfaces (in their sense) are precisely those surfaces which admit Green's functions.

On the other hand, Forster's book (Lectures on Riemann Surfaces, 1981) uses the meaning described in this article. He notes however that no one really refers to the Riemann sphere as an elliptic Riemann surface, as it might cause confusion with elliptic curves (i.e. complex tori, compact surfaces of genus 1).

Anyone have any others? -- Fropuff 19:20, 11 July 2005 (UTC)

 * Gamelin's Complex Analysis has a chapter on Riemann surfaces, with no mention of elliptic, parabolic, or hyperbolic surfaces. Lang's Complex Analysis, doesn't do Riemann surfaces, nor does Wells' Differential Analysis on Complex Manifolds, nor does Kodaira's Complex Manifolds and Deformations of Complex Structures.

 * On the other hand, it is easy to find the words elliptic/hyperbolic/parabolic in a book on Riemannian geometry where they mean positive/negative/zero curvature. I guess that usage is the source of the terms in this article.

 * It would be nice if the two usages agreed, but I guess they don't. I guess they agree one which ones are the simply connected versions: the sphere is elliptic, the plane parabolic, the half-plane hyperbolic.  But with the nontrivial surfaces, they seem to be in total disagreement.  Farkas and Kra want the many-handled torus to be elliptic, while its Riemannian geometry is hyperbolic.  Hrmm...

 * I guess I think the article should be brought in line with Farkas and Kra, but it's not enough to just shuffle the words, some stuff has to be said about subharmonic functions and such. Anyway, I always thought it was weird how, according to this article, there is only one elliptic surface.  Hardly seems worth giving it a name, if there's only the one, right? -Lethe | Talk 22:35, July 11, 2005 (UTC)

Hi! I would stick with the curvature definition. I guess the different terminology comes from the different applications. Riemann surfaces play a role in many different areas. From the geometric viewpoint, you should name them after their curvature. sphere with curvature 1, torus curvature 0, the two special cases kind of, and then all higher genus surfaces with curvature -1 and the hyperbolic space as universal cover.

Note that if you care for any geo-"metric" aspects (maybe in contrast to purely analytic aspects) this seems very important, as the sphere is the prototype of the sphere (somewhat stupid) the plane a prototype for a neighborhood on the torus and a higher genus surface looks like hyperbolic space IH locally.

Hope that makes sense! Norman - nhilbert.de

formal definition
Hello,

I must say the definition given here looks the most accessible one I have found on the internet. However there is one thing I don't understand and perhaps someone could clarify this : when talking about compatible maps f and g, suppose f is a map from G (open subset in X) to H (subset of C) and g is analogously from I to J

we require that G and I are not disjoint( intersecting domains) but what kind of map is f after g^(-1) then

from J to H or are there more restrictions?

 * Yes, there's a restriction, its defined only on the intersection of H and I. The article Atlas (topology) is supposed to explain this in detail, but it doesn't :(  linas 22:09, 15 August 2005 (UTC)

I don't see why we need the appeal to Zorn's Lemma. To get a maximal atlas containing a given one can't we just grab all charts compatible with all the charts already in the atlas. Any pair of these new charts must be compatible because they're compatible with all the charts of the original atlas, which covers. Can someone back me up or smack me down? Amling 16:40, 22 January 2006 (UTC)

 * Sounds right. In the smooth category, your statement is Lemma 1.10(a) in Lee's Introduction to Smooth Manifolds, GTM. Melchoir 18:30, 22 January 2006 (UTC)

Brave New World
Just thought I'd note that in the book "Brave New World", they play tennis on what is described as Riemann Surfaces. :-) In case anyone wanted to know. —Preceding unsigned comment added by 62.31.86.15 (talk • contribs) 23:25, 1 January 2006
 * Dang, I don't have a copy of the book on me. Could someone else add it to the article? Melchoir 00:42, 2 January 2006 (UTC)
 * Never mind, I got it. Melchoir 00:48, 2 January 2006 (UTC)

Fundamental Error
The article makes this claim: " A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable."

Alas, this is simply not true. Perhaps it goes without saying that a Riemann surface must be Hausdorff, but this is not sufficient. It must also have a countable base to its topology. One counterexample is the square of the long line. Another quite surprising example is the Prüfer manifold: Start with an open half-plane H = {(x,y) in R^2 :  y > 0}. For each (c,0) on the x-axis, we augment H in a strange way: Define H_c as the homeomorphic image of H by the (real analytic) homeomophism h_c: H -> H (which is strictly into):

h_c(x,y) =  (x,y) + (x-c,y)/sqrt((x-c)^2 + y^2).

(That is, we push each (x,y) directly away from (c,0) by exactly one unit.) Now take the union of H_c with the original H, which has the effect of adding a semicircle of radius 1 about (c,0) to what was H (which for each c we identify with H_c). Finally, perform this augmentation once for each (c,0) on the x-axis. The result is a real-analytic surface, the "Prüfer manifold", which obviously does not have a countable base to its topology (or a countable dense subset), since we have added uncountably many disjoint (2D) semicircles to what was H. Such a manifold cannot be made into a Riemann surface.Daqu 06:10, 16 May 2006 (UTC)

 * "Manifolds", in the sense of this article, are implicitly second-countable. See Manifold. Melchoir 06:31, 16 May 2006 (UTC)

 * Granted, but this only shows that Wikipedia's various definitions of a manifold need to be put on a sounder logical footing. E.g., the entry Topological manifold makes no such assumption of Hausdorff or separability -- which is exactly as it should be.)  When topologists get serious, they specify a manifold as being topological with such-and-such precise additonal structure.  In the present case, a significant episode in the history of Riemann surfaces was to determine whether or not all Hausdorff locally Euclidean connected 2-manifolds with holomorphic transition maps had to be separable (or equivalently are second countable).  The Prüfer manifold especially called this into question.  The fact that a 1-dimensional Hausdorff connected complex manifold is automatically separable is quite surprising.

 * Perhaps just as surprising is that higher-dimensional complex manifolds defined only as locally Euclidean, Hausdorff, connected and having an atlas with holomorphic transition functions need not be second countable! (Just take the complexification of the Prüfer manifold.)

 * So while I don't advocate getting too technical too quickly, it's also a good idea not to make easily-misinterpreted statements either (like that every connected surface can be given a Riemann surface structure).Daqu 14:30, 16 May 2006 (UTC)

 * Actually, when topologists get serious, they do geometry ;) Sam nead 23:16, 23 October 2006 (UTC)

Attempting to understand
I've got a question which doesn't seem to be in the article. What are the conditions for a real 2-manifold to be equivalent to a Riemann surface? From what I gather they are that: if on any overlap between a (x,y) coordinate patch and a (X,Y) coordinate patch $$Z = X+iY\,$$ is holomorphic in $$z=x+iy\,$$; and that the Cauchy-Riemann equations hold: $$\frac{\delta X}{\delta x} = \frac{\delta Y}{\delta y}, \frac{\delta X}{\delta y} = -\frac{\delta Y}{\delta x}$$. Are those all the conditions? —porg es (talk) 10:52, 10 July 2006 (UTC)

Also, is this equivalent? "A complex function f(x + iy) = u + iv is holomorphic if and only if it satisfies the Cauchy-Riemann equations and u and v have continuous first partial derivatives with respect to x and y."

 * I guess the shorter version is "if a real surface has holomorphic transition functions and it satisfies the Cauchy-Riemann equations, is it the same as a Riemann surface?" —porg es (talk) 00:24, 12 July 2006 (UTC)

 * The first part is correct. If a surface has holomorphic transition functions, then it is a Riemann surface.  Holomorphicity means that the transition functions satisfy the Cauchy-Riemann equations, so you don't have to say that twice. -lethe talk [ +] 00:58, 12 July 2006 (UTC)

Would a simpler example be useful?
I'm adding content to the article complex plane, and I'd like to tie that article into this one. I think the original idea of a Riemann surface arose from the consideration of multi-valued complex functions, and that it's natural to associate certain simple Riemann surfaces with functions like z⅔, or logz.

This article is very nice as it stands. But I fear it may be inaccessible to the general reader, who probably has an intuitive notion of what a Hausdorff space is, but who may be scared-orf by such big words.

Anyway, I'm thinking this article might be improved by re-ordering the sections just a little. Specifically, I'd like to move the "History" section ahead of the "Formal definition" and add one or two naive examples to this section. I think the tie-in between the Riemann surface for logz and the residue theorem is particularly instructive. One need not be a topologist to comprehend that picture.

Does that sound like a reasonable plan? I'm asking because I think there's a real tension between rigor and readability in many Wikipedia articles about math. My own inclination is to lean toward less rigor, at least in the introductory sections of an article. But I don't want to offend the mathematicians who think that rigor is everything. DavidCBryant 12:00, 2 January 2007 (UTC)

What are you trying to do?

Altouugh I'm not a mathematician, I'm not illiterate in math either. I want to ask a rather difficult question about this page. It deals with how to write for the public and raises the issue of what purposes Wikipedia has and, in specific, the purposes of this entry.

If the authors of this entry are writing for each other, and for senior math students, graduate students, or their professors, then you must abandon the idea that Wikipedia is a **general** encyclopedia for readers on the Web. If you are writing for the general public, then you have to abandon the abstract approach you have adopted. The reason is that no one (other than yourselves) knows what you are talking about -- or cares. If that's your goal, well, accept your elitism and redefine the purposes of Wikipedia.

I'll give an example. Let's start with the high school idea of graphing a function by plotting points on paper. Like y = x^2. OK, we get some points, and everyone can see you get a curve if you have enough points. Now let's jump to complex variable: w = z^2, where w = u+iv and z - x+iy. With some patience, even a high school student can understand that: the function z^2 takes PAIRS of points on one plane and MAPS them onto another. Now, can you use this example to describe what a Riemann surface is?

If you can, then you'll be able to pass your graduate student qualifying examination in complex variable. If you can't, then no matter how much you know about Hausdorff, you do not yet understand a Riemann surface. The best reference is Felix Klein's elementary mathematics from an advanced standpoint, because it was THAT kind of question that Klein raised **and answered**. Another excellent source is Klein's book on Riemann surfaces and electrical fields.

The reason this question is so extremely difficult is that it requires thought and understanding, not the ability to use jargon. Understanding _precedes_ jargon. So my advice is to scrap the entire article and start from scratch, with the basic ideas you want to explore. That you have not done, and as a result -- and as long as Wikipedia is an encyclopedia for the general public -- this article is a failure.

Timothy Perper Timothy Perper 09:42, 22 August 2007 (UTC)

 * That seems a good suggestion. Please go ahead and improve the article. -- Jitse Niesen (talk) 11:36, 22 August 2007 (UTC)

 * Thanks, Jitse Niesen and David Bryant -- we seem to see eye-to-eye that some changes in content and exposition are needed in this article.

 * But it's not a matter, I think, of merely making a few changes here and there. I think the article needs to be recast much more thoroughly. Mathematically, it needs more and/or clearer material on analytical continuation, on single- and multiple-valued functions, on genus, and (especially) on branch lines, branch cuts, branch points, and singularities.

 * Moreover, we -- anyone -- aren't "doing" mathematics here. The article starts like any good academic (or textbook) exposition in math, with a formal definition followed by an elaboration, also formal, of various ideas. But the readers aren't mathematicians. They're intelligent members of the general public: this is Wikipedia, not a seminar in complex variable. So the article needs illustrations, examples, and development of ideas historically *and* intellectually. I like the idea of starting with drawing graphs and getting from there to mapping and from there directly to how we can map a multi-valued complex function -- which leads *directly* to the Riemann surface. Only later, with an impulse to abstraction did these initially quite visual and intuitive ideas become abstractions. But that is only a personal preference.

 * I'm certainly not the one to say *unilaterally* what the directions should be for this article. I am therefore reluctant to merely make changes in the name of "improvement." We need some consensus first, especially from the people who did the hard work of writing the article.Timothy Perper 13:42, 22 August 2007 (UTC)

The image
I'd rather not correct this myself, since I'm neither a complex analysis expert, nor a Wikipedia expert. This is just a thought:

The image of the surface for $$\sqrt{z}$$ depicted in the upper left corner is actually only depicting the real part of a multi-valued function $$f: C \rightarrow C$$. What about including also the imaginary part as a separate image? It might be confusing to beginners, that's for sure. Maybe it wouldn't be bad to also include a page about ways how to imagine such mappings and provide a link. I.e. functions $$R \rightarrow R^2$$ and $$R^2 \rightarrow R$$ can be visualised simply in 3D, but that's about it. Even though 4D cannot be imagined, we can still get the general idea by emplying projections into 3D.

On the other hand, Wikipedia is supposed to be a reference rather than a textbook, so I don't know how appropriate this would be.

193.85.150.3 (talk) 11:27, 10 December 2007 (UTC)

Images of Riemann surfaces labeled as for for z-2, z-3, z-4
These are depicted as having branch points at the origin. . . and without having poles there.

But the functions z-n for n = 2,3,4 have no branch points, and each has a pole at the origin. Can it be that the images are in fact the Riemann surfaces for z1/n for n = 2,3,4 ? In any case, something seems to need fixing here.Daqu (talk) 18:01, 9 February 2009 (UTC)

Riemannian manifolds?
This has been in the article for a long time, so perhaps I should think longer before writing this, but anyway: How is a Riemann surface supposed to be a Riemannian manifold as in the second definition?

Certainly the complex structure allows the measurements of angles and hence induces a conformal structure, but a Riemannian metric can also measure lengths. This we cannot do with a complex structure, as maps such as z -> 2*z are holomorphic, but not length-preserving. Right?

Regardless of this, I think that the unsourced phrase in the second definition hence the name Riemann surface is highly doubtful as it stands. -- Momotaro (talk) 20:56, 4 May 2009 (UTC)

 * It is correct. I've provided a reference. Jakob.scholbach (talk) 19:31, 5 May 2009 (UTC)

 * I'm not convinced ... This chapter shows that when you have a Riemannian metric on a surface, you can find a conformal structure on it which is compatible with the metric. But what I am saying is that this does not work vice-versa.

 * Take the Riemann sphere defined by two charts $$U_\alpha = U_\beta = \C$$ and transition map $$\varphi_{\alpha\beta}: z \mapsto 1/z$$. How do you measure lengths there? We can certainly find a Riemannian metric compatible with this conformal structure, but not uniquely: We get a whole class of metrics, up to conformal equivalence. -- Momotaro (talk) 13:23, 6 May 2009 (UTC)

 * Oh, now I just remarked that conformal equivalence is even mentioned in the second definition, but only at the end. This makes it somewhat self-contradictory. -- Momotaro (talk) 13:26, 6 May 2009 (UTC)

 * I made an attempt to correct the second definition, should be ok now. -- Momotaro (talk) 13:10, 8 May 2009 (UTC)

I think it would be good to have a recipe in the article, e.g. how to get a Riemannian manifold from a Riemann surface. And which Riemannian metric is the preferred one. -- Spaetzle (talk) 15:22, 8 Apr 2011 (UTC)

 * There is no unique preferred one. As mentioned in the article, a Riemann surface has a unique Riemannian metric with constant curvature K ∈ {-1, 0, 1}; for many purposes this is the preferred one.  But for many other purposes people prefer to view a compact Riemann surface as a projective curve given by a homogeneous polynomial in projective coordinates. This is sometimes possible in CP2 and always possible in CP3.  The Riemannian metric from such a representation is never of constant curvature, yet it makes the Riemann surface a minimal surface in its projective space.  There are undoubtedly many other considerations.Daqu (talk) 01:19, 12 May 2012 (UTC)

Please, explain what the drawings of surface in the article represent
Do you think that it is clear,when you 'draw' the Riemann surface associated to the holomorphic function,for example,$$f(z)=\sqrt z$$,what does this drawing represent ? A function drawn in a 3-dimension space must represent,in the simplest interpretation,a real function z in the two real variables x and y,so that z=f(x,y).A holomorphic function f in the complex variable z,should at least be represented graphically by TWO surfaces in a 3-dimension space,that is one for the real part and one for the imaginary part of f.

Don't you think that this elementary issue should be cleared up,before talking of anything about the Riemann surfaces ? —Preceding unsigned comment added by 90.0.205.206 (talk) 19:23, 22 March 2010 (UTC)

 * I gave it a shot. Better now? Hanche (talk) 22:56, 23 March 2010 (UTC)

Confusing statement should be removed
The fourth paragraph reads in full:

"Geometrical facts about Riemann surfaces are as "nice" as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties."

I love Riemann surfaces, but:

To claim that certain geometrical facts are "nice as possible" -- with or without the quotation marks around the word "nice" -- is just meaningless, and hence just confusing. If there is an underlying fact here, then why not just state it?

Certainly many Riemann surfaces have conformal automorphism groups that are much smaller than the largest possible symmetry group for a Riemannian surface of the same genus. This certainly isn't "as nice as possible", however the phrase is defined.

Nor is the inability to measure length, a staple of most forms of geometry.

Also, every compact Riemann surface is conformally equivalent to some (projective) algebraic curve, so it seems weird to compare them with "other" curves.Daqu (talk) 01:06, 12 May 2012 (UTC)

Self-intersection
As I understand it (undergraduate with limited analysis/topology experience), manifolds proper don't self-intersect or local homeomorphisms would fail, right (an X is not locally homeomorphic to a line, for instance)? So perhaps a note that sqrt(z) does not properly self-intersect, only its immersion in 3D, would be worthwhile beneath the page image. 67.79.154.194 (talk) 00:45, 15 December 2012 (UTC)

 * I think this highlights a serious problem with the diagram (and others in the article). The diagram illustrates a multi-valued function on the complex plane, and not a Riemann surface. This is likely to cause confusion for those that do not already understand what a Riemann surface is. A key point about a Riemann surface is that there is no vertical axis: it may be thought of as the domain of a function, not as the function itself. This misrepresentation implicit in the diagrams should really be addressed. — Quondum 06:22, 15 December 2012 (UTC)

 * What he said. 76.106.149.108 (talk) 19:12, 30 June 2013 (UTC)

Needed:
The Simple English Wikipedia article on this. 76.106.149.108 (talk) 19:11, 30 June 2013 (UTC)

Riemann Surface is not the 3D surface
There needs to be clarification in the article. The surfaces shown are either the real or imaginary parts. The whole surface is, in general, actually "4-dimensional" in the sense that it has 2 imaginary parts and 2 real parts. — Preceding unsigned comment added by 99.19.84.64 (talk) 16:57, 15 August 2013 (UTC)

 * I agree about clarification. What the depiction of, say, the square-root function sqrt(z) means is several things: The Riemann surface is a surface on which the function sqrt(z) can be defined in a single-valued manner. It can be thought of as having two points lying above each value of z on the complex plane (except only one point above z = 0). One of these points above z = r eiθ for 0 <= θ < 2π represents a point z' at which sqrt(z') =  sqrt(r) eiθ/2, and the other point above this same z represents a point z at which sqrt(z) =  sqrt(r) ei(θ/2+π) (= -sqrt(r) ei(θ/2+π)).

 * In this way, the Riemann surface depicts a new domain for a "new version" of the sqrt function, on which sqrt is single-valued instead of being multiple-valued (in this case, double-valued).

 * This is in contrast to having a "graph" like this where the vertical scale represents some concrete number with a particular meaning: in this kind of depiction of a Riemann surface, it does not.Daqu (talk) 21:19, 10 February 2015 (UTC)

 * I think the diagrams should be replaced as highly misleading. This has already been commented on above (see ).  Daku's description is accurate; the Riemann surface shows both the real and imaginary parts of the domain, and the vertical axis should be removed. A Riemann surface is not a function. It is better thought of as a suitably glued-together paper surface lying completely flat (think of nearly collapsing the diagrams, but keeping the identity of the surfaces distinct). —Quondum 00:47, 11 February 2015 (UTC)

 * But it's not correct to claim (as in ) that a Riemann surface is not the depiction of a multi-valued function. Perhaps saying the Riemann surface of, say, sqrt(z) is the same as the depiction of that multi-valued function is slightly over-simplifying, but it is not wrong to say that.  A very robust definition of the Riemann S surface of sqrt(z) is to define
 * S as {(z,sqrt(z)) in C2}, which is identical to saying S = {(u2, u) in C2}.Daqu (talk) 23:51, 11 February 2015 (UTC)

 * I'm afraid this goes directly against my understanding of a Riemann surface, as explained by Roger Penrose. A Riemann surface is not a function; it is only the domain of a function. Your description earlier was good.  A Riemann surface, by my understanding, can be considered as a pairing of a point in C with a discrete variable, which in this example would be from a two-element set.  This would be like saying S = { (u, w) in C × {0,1} }. The Riemann surface that is needed as the domain of z1/2 is the same Riemann surface that is needed as the domain of 5z3/2. We can then define functions on S, e.g. sqrt : S → C. —Quondum 00:51, 12 February 2015 (UTC)

 * If you reread what I wrote, you will see that I defined a surface, not a function. The surface is of course related to a function, but I never said or implied that the Riemann surface is a function. As you say, it is not. Still, it can be used to depict one. The meaning of your comment about "pairing a function with a discrete variable" is not clear to me.

 * But also, defining a Riemann surface via a function is only one way to define it. Another slightly more modern way (though both are important!) is to say that a Riemann surface is a manifold whose transition functions are each a holomorphic function from one open set of C to another.Daqu (talk) 17:00, 18 February 2015 (UTC)

 * By "transition function" do you mean chart of an atlas? Probably not important; I think we are just somehow missing each other (for example, I cannot figure out what the set of pairs of complex numbers S is meant to represent in relation to a Riemann surface). Let's try another description: a Riemann surface is a covering space of the complex plane (with special treatment for isolated points). I think the important point is that the existing diagrams invite the reader into a misunderstanding that the vertical scale (and hence the shape and curvature) are part of the definition. —Quondum 01:01, 19 February 2015 (UTC)

 * Yes, he means charts. His pairs of complex numbers description is valid; it describes the Riemann surface embedded in a particular way in C2. Constructing charts can be done using the implicit function theorem.  Your description is as a branched cover of C.  Constructing charts can be done by taking branch cuts.  The two are related by the projection C2 → C that sends (z, w) → z, because if z is known, then there are only two possible values for w, the two square roots of z.
 * I would also like to add that I think it's not a good idea to think of the branched cover description as pairs of points in C and a discrete variable. There is set-theoretically such a description, but the second variable ought to vary holomorphically in the first.  The extrinsic description in C2 has this property, for example.  Ozob (talk) 03:20, 19 February 2015 (UTC)
 * I agree that the pairing with a discrete variable is unsuitable for use in the article; I was only trying to try to show that the description with two complex variables defines something with too much structure, though I'm clearly not being heard. Let's try another tack. The {{nowrap|1=S = {(u2, u) in C2}} description is a representation of a Riemann surface. A Riemann surface is not a graph (or function), whereas the typical reader looking at the diagram and captions would be inclined to assume that it is. —Quondum 17:20, 19 February 2015 (UTC)
 * I think you're using the word "representation" in an imprecise, non-mathematical sense. There is a completely precise way to capture the concept you want: The set S is a Riemann surface embedded in C2.  It is a complex submanifold.  I'm not saying that we want to discuss general complex manifolds and their submanifolds in this article.  All I'm saying is that S is a valid description of a Riemann surface.  One can find charts on it, one can map it to C and view it as a branched cover, indeed, it's just as good as any other description.  Ozob (talk) 02:50, 20 February 2015 (UTC)
 * I meant the term in a precise, mathematical sense. However, since this is clearly not showing signs of reaching any mutual understanding, I'll bow out of this discussion. —Quondum 03:47, 20 February 2015 (UTC)

As "deformed versions of the complex plane" ???
The article begins with the statement:

"''Riemann surfaces can be thought of as deformed versions of the complex plane: . . .."

No, they can't be thought of this way — not without deeply confusing the thinker.Daqu (talk) 20:50, 10 February 2015 (UTC)

Dimension
The article states that "Riemann surface [...] is a one-dimensional complex manifold". But later it states that "Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface)". So how many dimensions does it have? Since it's a surface, it should be 2, right? 89.229.8.110 (talk) 20:45, 26 August 2017 (UTC)

 * (I know this question is 2 years old but...) A dimension d complex manifold is a dimension 2d real manifold (with a complex structure added). Complex manifold are locally modelled on C^d, and C^d is R^2d if we forget the complex structure. --Arnaud Chéritat (talk) 23:35, 26 December 2019 (UTC)

Connected or not
The definitions 1 and 2 are not fully equivalent: one mentions connectedness and not the other. --Arnaud Chéritat (talk) 23:37, 26 December 2019 (UTC)

Actually oriented manifolds are connected. Check the page on oriented manifolds. Julian Barathieu (talk) 17:08, 3 November 2021 (UTC)

Non-metrizable counterexample?
Paragraph three of the introduction states a necessary condition for a 2d manifold to be a Riemann surface - it must be orientable and metrizable. It then gives three counterexamples of non-orientable manifolds. Seems to me that the article would be improved by providing a non-metrizable counterexample. Not crucial to the article but an improvement.

Suggestions for non-metrizable counterexamples cheerfully accepted. Mr. Swordfish (talk) 23:45, 30 March 2021 (UTC)

 * metrisability for locally compact separable (in T3 sense) spaces is more or less equivalent to being second countable so the non-metrisable connected surfaces are exactly these. For a explicit example you can take the long line times a circle. jraimbau (talk) 12:13, 31 March 2021 (UTC)
 * Thanks. Long line cross S2 seems like an obvious counterexample (obvious once you point it out, that is).  Should we mention it in the article? We'd need a cite, of course.  Mr. Swordfish (talk) 23:38, 31 March 2021 (UTC)

Lecture notes by Eynard
I have tried to add the following lecture notes from Arxiv to the References:

This was reverted by User:D.Lazard due to WP:reliable sources concerns. However, these are lecture notes by a known expert, author of this book. Isn't this OK per Reliable_sources? Sylvain Ribault (talk) 13:15, 4 November 2021 (UTC)
 * This text has been submitted in 2018 to a physical journal, and is still not published. I do not know the reason for this non-publication, but the name of the author is not sufficient for verifying whether this text is worth to be cited (not all papers of Field medals are good articles).
 * Also, this citation has been added in a section that contains 9 published books, including 5 that have "Riemann surface" in their title. This seems sufficient for sourcing the article. So, adding a new unpublished item would make sense only if it contains some material that is not in the other sources. This is not the case, as the citation is not presented as containing new results.
 * In summary, for adding this citation, strong reasons must be explicited, which are currently lacking. D.Lazard (talk) 14:04, 18 November 2021 (UTC)

 * Thank you for the elaboration. I would have liked to include some openly accessible general reference. I will now leave it as a suggestion. Sylvain Ribault (talk) 20:36, 19 November 2021 (UTC)

in the first image, after rotation colour = height, really?
The description of the first image says "For this function, it is also the height after rotating the plot 180° around the vertical axis." ("it" refers to the color).

I don't see how a rotation in the vertical axis would achieve this. Atoll (talk) 07:31, 2 March 2023 (UTC)

The big picture at the top.
It's described as, "An open subset of the Riemann surface for the multivalued function..."

But it's not on open subset. It's a bounded, closed subset. Every edge is a sharp edge.

Should I fix it or am I wrong? I'm not confident I can tell. Verdana ♥ Bold 17:11, 18 June 2024 (UTC)

 * It's an open set, because a Riemann surface is two dimensional. (I.e., it's not open in R^3, but is open in the surface.) Tito Omburo (talk) 22:05, 18 June 2024 (UTC)
 * Since it is not said whether the represented surface contains its boundary, one cannot say whether the represented surface is open or closed in the Riemann surface, and this has the same low importance as discussing whether the plot of a real function contains the points where the curve is truncated. I have removed "open subset" from the caption, and added an explanation of the meaning of the coordinates. D.Lazard (talk) 01:36, 19 June 2024 (UTC)

title: Talk:Riemann surface


text:
Jingoism

Jingoism is nationalism in the form of aggressive and proactive foreign policy, such as a country's advocacy for the use of threats or actual force, as opposed to peaceful relations, in efforts to safeguard what it perceives as its national interests. Colloquially, jingoism is excessive bias in judging one's own country as superior to others – an extreme type of nationalism (cf. chauvinism and ultranationalism).

Etymology
The chorus of a song by the songwriter G. W. Hunt, popularized by the singer G. H. MacDermott – which was commonly sung in British pubs and music halls around the time of the  Russo-Turkish War of 1877–78 – gave birth to the term. The lyrics included this chorus:

"We don't want to fight but by Jingo if we do We've got the ships, we've got the men, we've got the money too We've fought the Bear before, and while we're Britons true The Russians shall not have Constantinople!"

The capture of Constantinople was a long-standing Russian strategic aim, since it would have given the Russian Navy, based in the Black Sea, unfettered access to the Mediterranean Sea through the  Bosphorus and the Dardanelles (known as the "Turkish Straits"); conversely, the British were determined to block the Russians, in order to protect their own access to India. At the time when the above song was composed and sung, the Russians were nearing their goal, through the Treaty of San Stefano; eventually, the British were able to push the Russians back by means of diplomatic pressure and the threat of war.

The phrase "by Jingo" was a minced oath, scarcely documented in writing, used in place of "by Jesus".

Use of the specific term "jingoism" stems from its coinage by prominent British radical George Holyoake, as a political label, in a letter to the Daily News on 13 March 1878.

Examples
In the 1880s, Henry Hyndman, leader of the Social Democratic Federation, turned against internationalism, and promoted a version of Socialism mixed with nationalism and antisemitism, even to the point of attacking fellow Socialist Eleanor Marx in antisemitic terms, noting that she had "inherited in her nose and mouth the Jewish type from Karl Marx himself". When taking part in the breakaway group which founded the Socialist League, Eleanor Marx wrote polemics in which she characterized Hyndman and his followers as "The Jingo Party".

British artillery major-general Thomas Bland Strange, one of the founders of the Canadian Army and one of the divisional commanders during the 1885 North-West Rebellion, was an eccentric and temperamental soldier who gained the nickname "Jingo Strange" and titled his 1893 autobiography Gunner Jingo's Jubilee.

Probably the first uses of the term in the U.S. press occurred in connection with the proposed annexation of Hawaii in 1893, after a coup led by foreign residents, mostly Americans, and assisted by the U.S. minister in Hawaii, overthrew the constitutional monarchy and declared a republic. Republican president Benjamin Harrison and Republicans in the U.S. Senate were frequently accused of jingoism in the Democratic press for supporting annexation.

Theodore Roosevelt was frequently accused of jingoism. In an article on 23 October 1895 in The New York Times, Roosevelt stated, "There is much talk about 'jingoism'. If by 'jingoism' they mean a policy in pursuance of which Americans will with resolution and common sense insist upon our rights being respected by foreign powers, then we are 'jingoes'."

In Homage to Catalonia, George Orwell decries the tactics of political journalists and wishes for introduction of aeroplanes into war in order to finally see "a jingo with a bullet hole in him."

The policy of appeasement toward Hitler led to satirical references to the disappearance of such jingoistic attitudes when facing German aggression. A cartoon by E. H. Shepard titled "The Old-Fashioned Customer" appeared on 28 March 1938 issue of Punch. Set in a record shop, John Bull asks the record seller (Prime Minister Neville Chamberlain): "I wonder if you've got a song I remember about not wanting to fight, but if we do... something, something, something ... we've got the money too?". On the wall is a portrait of Lord Salisbury.

The rhetoric of North Korea has been described as jingoist.

title: Jingoism


text:
Duet

A duet is a musical composition for two performers in which the performers have equal importance to the piece, often a composition involving two singers or two pianists. It differs from a harmony, as the performers take turns performing a solo section rather than performing simultaneously. A piece performed by two pianists performing together on the same piano is a "piano duet" or "piano four hands". A piece for two pianists performing together on separate pianos is a "piano duo".

"Duet" is also used as a verb for the act of performing a musical duet, or colloquially as a noun to refer to the performers of a duet.

A musical ensemble with more than two solo instruments or voices is called a trio, quartet, quintet, sextet, septet, octet, etc.

History
When Mozart was young, he and his sister Marianne played a duet of his composition at a London concert in 1765. The four-hand, described as a duet, was in many of his compositions which included five sonatas; a set of variations, two performers and one instrument, and a sonata for two pianos. The first published sonata or duet was in 1777.

In Renaissance music, a duet specifically intended as a teaching tool, to be performed by teacher and student, was called a bicinium (see Étude).

In opera
Duets have always been a part of the structure of operas. Early 16th-century operas such as L'Orfeo and L'incoronazione di Poppea involve duets throughout the performance. In 17th-century Italy duets were often used in comic scenes within serious operas. In Baroque France the duet was popular in tragedies, such as songs of vengeance and confrontation. The love duet was characterized by singing in close harmonies of 3rds and 6ths, symbolizing unity after conflict.

Famous operatic duets

 * La clemenza di Tito by Wolfgang Amadeus Mozart
 * La Cenerentola by Gioachino Rossini
 * The Puritans of Vincenzo Bellini
 * Don Pasquale by Gaetano Donizetti
 * La traviata by Giuseppe Verdi
 * Aida of Giuseppe Verdi
 * Mefistofele of Arrigo Boito
 * Manon Lescaut by Giacomo Puccini
 * Madama Butterfly by Giacomo Puccini
 * L'amico Fritz by Pietro Mascagni

In pop music
Throughout the 20th and 21st centuries, duets have been common in the popular music of their respective eras. In addition to a standard vocal duet, some songs have been written to be heard as conversations; for example, "Baby, It's Cold Outside". Other songs are performed around a theme; for example, New York City in "Empire State of Mind". Occasionally, duets are an improvisation between artists; for example, "Under Pressure" by Queen and David Bowie. Bowie and Queen frontman Freddie Mercury reportedly composed the lyrics in a day by improvising together. Duets are also common in musical movies and musical theatre; "Fit as a Fiddle" for the 1952 movie Singin' in the Rain and its corresponding play of the same name.

Virtual duets
In addition to traditional duets performed live—either in front of an audience or recorded in a studio—a so-called virtual duet can be created by having a singer (or musician) perform over the top of a pre-existing recording. Such a duet is a form of overdubbing. A virtual duet is sometimes done when the singer (or musician) of the original recording is deceased; for example, a live performance by Paul McCartney on "I've Got a Feeling" with an isolated vocal recording of John Lennon from The Beatles' famous rooftop performance, or a recording of Judy Garland being dubbed over by her daughter Lorna Luft on "Have Yourself A Merry Little Christmas". It can also be done with an earlier version of oneself (such as Yusuf / Cat Stevens on "Father And Son"); incidentally, Ronan Keating did his own virtual duet with Yusuf on the same song. Virtual duets may also be done live via video link (such as Shaun Escoffery duetting with the late Eva Cassidy's videotaped performance of "Over the Rainbow" in honour of the late Sir Terry Wogan) or be constructed manually from two pre-existing recordings, generally where each singer/musician can have their own isolated audio channel in the form of a stem.

Entire albums of virtual duets have been created, including albums by Frank Sinatra (Duets and Duets II).

Theoretically, a duet could be constructed from any two pre-recorded singers or musicians so long as there are isolated audio channels from each artist. With the advent of audio deepfake technology, it is now possible to create virtual duets using an AI facsimile of a singer (or singers) who never sang the song in the first place. A famous example of this is "Heart on My Sleeve" by ghostwriter977, who wrote and produced an original song using voice models of Drake and The Weeknd.

Notable pop duets

 * "Baby, It's Cold Outside" – Ricardo Montalbán and Esther Williams, and with roles reversed, Red Skelton and Betty Garrett, 1948
 * "Dream a Little Dream of Me" – Louis Armstrong and Ella Fitzgerald, 1950
 * "Chaplin and Keaton Violin and Piano Duet" – Limelight, 1952
 * "I Got You Babe" – Sonny & Cher, 1965
 * "Ain't No Mountain High Enough" – Marvin Gaye and Tammi Terrell, 1967
 * "Jackson" – Johnny Cash and June Carter, 1967
 * "Somethin' Stupid" – Frank Sinatra and Nancy Sinatra, 1967
 * "Waters of March" – Elis Regina and Antonio Carlos Jobim, 1972
 * "Don't Go Breaking My Heart" – Elton John and Kiki Dee, 1976
 * "You're The One That I Want" – John Travolta and Olivia Newton-John, 1978
 * "Endless Love" – Diana Ross and Lionel Richie, 1981
 * "Under Pressure" – Queen and David Bowie, 1981
 * "Up Where We Belong" – Joe Cocker and Jennifer Warnes, 1982
 * "Ebony and Ivory" – Paul McCartney and Stevie Wonder, 1982
 * "Islands in the Stream" – Kenny Rogers and Dolly Parton, 1983
 * "Say Say Say" – Paul McCartney and Michael Jackson, 1983
 * "Don't Give Up" – Peter Gabriel and Kate Bush, 1986
 * "Nothing's Gonna Stop Us Now" – Starship, 1987
 * "Always" – Atlantic Starr, 1987
 * "(I've Had) The Time of My Life" – Bill Medley and Jennifer Warnes, 1987
 * "One Sweet Day" – Mariah Carey and Boyz II Men, 1995
 * "Scream" – Michael Jackson and Janet Jackson, 1995
 * "It's Your Love" – Tim McGraw and Faith Hill, 1997
 * "Tell Him" – Barbra Streisand and Celine Dion, 1997
 * "I'm Your Angel" – R. Kelly and Celine Dion, 1998
 * "The Boy Is Mine" – Brandy and Monica, 1998
 * "When You Believe" – Whitney Houston and Mariah Carey, 1998
 * "Where You Are" – Jessica Simpson and Nick Lachey, 2000
 * "Nobody Wants to Be Lonely" – Ricky Martin and Christina Aguilera, 2001
 * "I Belong to You (Il ritmo della passione)" – Eros Ramazzotti and Anastacia, 2006
 * "Beautiful Liar" – Beyoncé and Shakira, 2007
 * "No Estamos Solos" – Eros Ramazzotti and Ricky Martin, 2007
 * "Limpido" – Laura Pausini and Kylie Minogue, 2013
 * "Hurt You" – Toni Braxton and Babyface, 2014
 * "Bad Things" – Machine Gun Kelly and Camila Cabello, 2016

title: Duet