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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if $$a$$ and $$b$$ are real numbers then the complex conjugate of $$ a + bi$$ is $$a - bi.$$  The complex conjugate of $$z$$ is often denoted as $$\overline{z}$$ or $$z^*$$.

In polar form, if $$r$$ and $$\varphi$$ are real numbers then the conjugate of $$r e^{i \varphi}$$ is $$r e^{-i \varphi}.$$ This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number: $$a^2 + b^2$$ (or $$r^2$$ in polar coordinates).

If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.

Notation
The complex conjugate of a complex number $$z$$ is written as $$\overline z$$ or $$z^*.$$ The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, as well as electrical engineering and computer engineering, where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while the bar notation is more common in pure mathematics.

If a complex number is represented as a $2 \times 2$ matrix, the notations are identical, and the complex conjugate corresponds to the matrix transpose, which is a flip along the diagonal.

Properties
The following properties apply for all complex numbers $$z$$ and $$w,$$ unless stated otherwise, and can be proved by writing $$z$$ and $$w$$ in the form $$a + b i.$$

For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division: $$\begin{align} \overline{z + w} &= \overline{z} + \overline{w}, \\ \overline{z - w} &= \overline{z} - \overline{w}, \\ \overline{zw} &= \overline{z} \; \overline{w}, \quad \text{and} \\ \overline{\left(\frac{z}{w}\right)} &= \frac{\overline{z}}{\overline{w}},\quad \text{if } w \neq 0. \end{align}$$

A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation.

Conjugation does not change the modulus of a complex number: $$\left| \overline{z} \right| = |z|.$$

Conjugation is an involution, that is, the conjugate of the conjugate of a complex number $$z$$ is $$z.$$ In symbols, $$\overline{\overline{z}} = z.$$

The product of a complex number with its conjugate is equal to the square of the number's modulus: $$z\overline{z} = {\left| z \right|}^2.$$ This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates: $$z^{-1} = \frac{\overline{z}}{{\left| z \right|}^2},\quad \text{ for all } z \neq 0.$$

Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments: $$\overline{z^n} = \left(\overline{z}\right)^n,\quad \text{ for all } n \in \Z $$ $$\exp\left(\overline{z}\right) = \overline{\exp(z)}$$ $$\ln\left(\overline{z}\right) = \overline{\ln(z)} \text{ if } z \text{ is not zero or a negative real number }$$

If $$p$$ is a polynomial with real coefficients and $$p(z) = 0,$$ then $$p\left(\overline{z}\right) = 0$$ as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).

In general, if $$\varphi$$ is a holomorphic function whose restriction to the real numbers is real-valued, and $$\varphi(z)$$ and $$\varphi(\overline{z})$$ are defined, then $$\varphi\left(\overline{z}\right) = \overline{\varphi(z)}.\,\!$$

The map $$\sigma(z) = \overline{z}$$ from $$\Complex$$ to $$\Complex$$ is a homeomorphism (where the topology on $$\Complex$$ is taken to be the standard topology) and antilinear, if one considers $$\Complex$$ as a complex vector space over itself. Even though it appears to be a well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension $$\Complex/\R.$$ This Galois group has only two elements: $$\sigma$$ and the identity on $$\Complex.$$ Thus the only two field automorphisms of $$\Complex$$ that leave the real numbers fixed are the identity map and complex conjugation.

Use as a variable
Once a complex number $$z = x + yi$$ or $$z = re^{i\theta}$$ is given, its conjugate is sufficient to reproduce the parts of the $$z$$-variable:
 * Real part: $$x = \operatorname{Re}(z) = \dfrac{z + \overline{z}}{2}$$
 * Imaginary part: $$y = \operatorname{Im}(z) = \dfrac{z - \overline{z}}{2i}$$
 * Modulus (or absolute value): $$r= \left| z \right| = \sqrt{z\overline{z}}$$
 * Argument: $$e^{i\theta} = e^{i\arg z} = \sqrt{\dfrac{z}{\overline z}},$$ so $$\theta = \arg z = \dfrac{1}{i} \ln\sqrt{\frac{z}{\overline{z}}} = \dfrac{\ln z - \ln \overline{z}}{2i}$$

Furthermore, $$\overline{z}$$ can be used to specify lines in the plane: the set $$\left\{z : z \overline{r} + \overline{z} r = 0 \right\}$$ is a line through the origin and perpendicular to $${r},$$ since the real part of $$z\cdot\overline{r}$$ is zero only when the cosine of the angle between $$z$$ and $${r}$$ is zero. Similarly, for a fixed complex unit $$u = e^{i b},$$ the equation $$\frac{z - z_0}{\overline{z} - \overline{z_0}} = u^2$$ determines the line through $$z_0$$ parallel to the line through 0 and $$u.$$

These uses of the conjugate of $$z$$ as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.

Generalizations
The other planar real unital algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.

For matrices of complex numbers, $\overline{\mathbf{AB}} = \left(\overline{\mathbf{A}}\right) \left(\overline{\mathbf{B}}\right),$ where $\overline{\mathbf{A}}$  represents the element-by-element conjugation of $$\mathbf{A}.$$ Contrast this to the property $\left(\mathbf{AB}\right)^*=\mathbf{B}^* \mathbf{A}^*,$  where $\mathbf{A}^*$  represents the conjugate transpose of $\mathbf{A}.$

Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the *-operations of C*-algebras.

One may also define a conjugation for quaternions and split-quaternions: the conjugate of $a + bi + cj + dk$ is $a - bi - cj - dk.$

All these generalizations are multiplicative only if the factors are reversed: $${\left(zw\right)}^* = w^* z^*.$$

Since the multiplication of planar real algebras is commutative, this reversal is not needed there.

There is also an abstract notion of conjugation for vector spaces $V$ over the complex numbers. In this context, any antilinear map $\varphi: V \to V$ that satisfies

 * 1) $$\varphi^2 = \operatorname{id}_V\,,$$ where $$\varphi^2 = \varphi \circ \varphi$$ and $$\operatorname{id}_V$$ is the identity map on $$V,$$
 * 2) $$\varphi(zv) = \overline{z} \varphi(v)$$ for all $$v \in V, z \in \Complex,$$ and
 * 3) $$\varphi\left(v_1 + v_2\right) = \varphi\left(v_1\right) + \varphi\left(v_2\right)\,$$ for all $$v_1, v_2 \in V,$$

is called a, or a real structure. As the involution $$\varphi$$ is antilinear, it cannot be the identity map on $$V.$$

Of course, $\varphi$ is a $\R$ -linear transformation of $V,$  if one notes that every complex space $$V$$ has a real form obtained by taking the same vectors as in the original space and restricting the scalars to be real. The above properties actually define a real structure on the complex vector space $$V.$$

One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no notion of complex conjugation.

title: Complex conjugate


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Talk:Béla Bartók

Death?
What did he die from? 220.240.77.202 (talk) 10:28, 21 January 2022 (UTC)

crater
A large and important crater on Mercury is named after him. It is the largest of "recent" (Kuiperian) craters on Mercury. Jstuby (talk) 03:05, 2 September 2022 (UTC)

Question on a Source.
Could this be reliable and credible enough to be used on the un-cited material in the section on his childhood? ToastEatinBread (talk) 20:13, 10 January 2023 (UTC)

Plagiarism
Significant chunks of this article are copied directly from this source without proper attribution. As an example, the following two paragraphs are nearly verbatim:

In 1908, he and Kodály traveled into the countryside to collect and research old Magyar folk melodies. Their growing interest in folk music coincided with a contemporary social interest in traditional national culture. Magyar folk music had previously been categorised as Gypsy music. The classic example is Franz Liszt's Hungarian Rhapsodies for piano, which he based on popular art songs performed by Romani bands of the time. In contrast, Bartók and Kodály discovered that the old Magyar folk melodies were based on pentatonic scales, similar to those in Asian folk traditions, such as those of Central Asia, Anatolia and Siberia.[citation needed]

Bartók and Kodály set about incorporating elements of such Magyar peasant music into their compositions. They both frequently quoted folk song melodies verbatim and wrote pieces derived entirely from authentic songs. An example is his two volumes entitled For Children for solo piano, containing 80 folk tunes to which he wrote accompaniment. Bartók's style in his art music compositions was a synthesis of folk music, classicism, and modernism. His melodic and harmonic sense was influenced by the folk music of Hungary, Romania, and other nations. He was especially fond of the asymmetrical dance rhythms and pungent harmonies found in Bulgarian music. Most of his early compositions offer a blend of nationalist and late Romanticism elements.

Ben373119 (talk) 18:08, 3 April 2023 (UTC)

Opus numbering his own works?
Need a reference to this. Not seeing in manuscripts or in musicology works. If no citation pops up, I will delete the line. Currently tagged as citation needed. Screen317 (talk) 07:10, 24 November 2023 (UTC)

https://www.henle.de/download/aktuelles/080.pdf
This source by Vera Lampert should at least be included in sources. 2600:4040:52C1:6F00:841:1A0E:4AF9:50FF (talk) 20:48, 28 November 2023 (UTC)

Infobox
I was going to add an infobox per BOLD, revert, discuss cycle but saw the hidden message about consulting first. Would it be OK? — Iadmc  ♫ talk  11:17, 4 June 2024 (UTC)

— Iadmc  ♫ talk  11:27, 4 June 2024 (UTC)

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

title: Talk:Béla Bartók


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Talk:Gauss–Markov theorem

In the beginning the beginning confused me
But as I remembered variance as a squared thing too, it became evident. If the variances dont vary wildly, then the sum of squares dont vary wildly either. And the rest of the conditions are not hard to agree upon. We dont need the gaussian-distribution (normal) to make it "optimal". — Preceding unsigned comment added by 2001:4643:E6E3:0:B99A:4E6A:45C0:A343 (talk) 23:04, 7 January 2019 (UTC)

Untitled
The introduction is confusing to me. Not clear what is assumed and what is not. And no further reference for why.

I don't understand the definition of the least squares estimators: they are supposed to make the sum of the squares as small as possible, but this sum of squares appears to be a random variable. Are they supposed to make the expected values of those random variables as small as possible? AxelBoldt 18:31 Jan 24, 2003 (UTC)

For fixed values of xi and Yi the sum of square is a function of βi for i=1,2. I'll add some material clarifying that. Michael Hardy 19:40 Jan 24, 2003 (UTC)

What does homoscedastic mean?

The term is explicitly defined in the article, but I will make it more conspicuous. It means: having equal variances. Michael Hardy 21:29 Jan 24, 2003 (UTC)

Should it be made explicit that the variance $$\sigma^2$$ is unknown? Albmont 13:17, 27 February 2007 (UTC)

What is the meaning of $${\rm cov}\left(\varepsilon_i,\varepsilon_j\right)=0$$? $$\varepsilon_i,\varepsilon_j$$ are scalars. Covariance is defined only for vectors, isn't it? Sergivs-en 06:05, 16 September 2007 (UTC)

 * Usually the concept of covariance is initially introduced as the covariance between two scalar-valued random variables. See covariance. Michael Hardy 13:58, 16 September 2007 (UTC)

linear vs nonlinear least squares
A suggestion is for someone with an interest in this to review the treatment under least squares. The basic Gauss-Markov apparently applies to the linear case. It would be interesting if there are nonlinear generalizations.Dfarrar 14:10, 11 April 2007 (UTC)

Nice article. Assuming that the β's and x's are not random (only the ε's are), would it be more accurate to say, then, that Ŷ is a stochastic process? - just a thought Ernie shoemaker 02:20, 26 July 2007 (UTC)

Very confusing. Also, should be stated in general vector form of theorem
This article is very confusing and does not explain what it sets out to. Saying that the least squares estimator is the "best" one means nothing. How is "best" defined? Later in the article, it seems to imply that "best" means the one with smallest MSE, and so obviously the least squares estimate is "best".

Also, why not generalise to estimation of a vector of quantities, where the errors have a given correlation structure? —Preceding unsigned comment added by 198.240.128.75 (talk) 16:16, 5 February 2008 (UTC)

best?
What does best estimator mean exactely? Best like Consistent estimator? --217.83.22.80 (talk) 01:19, 14 March 2008 (UTC)

 * At least in the context of "best linear unbiased estimator" (BLUE), "best" means "minimum mean squared error within the class of linear unbiased estimators" (according to this paper by G.K. Robinson). As we're restricting to unbiased estimators, minimum mean squared error implies minimum variance. So BLUE might less confusingly be "minimum variance linear unbiased estimator" (MVLUE?). Perhaps BLUE caught on simply as it's shorter and more pronouncible. Google gives this book preview which points out that the Gauss-Markov theorem states that "$$\hat\beta$$ is BLUE, not necessarily MVUE" as the MVUE estimator may in general be non-linear, but we can't know what it is without specifying a parametric probability distribution. The article needs editing to explain this better but estimation theory is really not my strong point. Qwfp (talk) 05:32, 14 March 2008 (UTC)

 * Can there be more than one unbiased estimator? Is it possible to find another estimator with less mean squared error (should be similar to standard error?) with the standard OLS aproach if there is a violation of the gauß-markov-theorem? --217.83.23.127 (talk) 23:49, 16 March 2008 (UTC)

 * Yes, of course there are many unbiased estimators. There's a space of "linear unbiased estimators" (each one a linear combination of the response variables) and there are also non-linear ones (e.g. the average of the max and the min of a sample is not linear in the observations, and is an unbiased estimator of the population mean). Michael Hardy (talk) 01:08, 17 March 2008 (UTC)

Best? What about bounded PDF?
As I read this article (and linear least squares), it sounds like the least-squares solution should be the BULE even in cases where the solution it gives is impossible. For example, suppose I measure a length three times and suppose that my measurement noise has a PDF of a uniform distribution within ±1mm of the true value. Suppose the length is actually 1mm and by chance I get 0.1mm, 0.1mm, and 1.9mm. All three measurements are off by 0.9mm, which is possible given my PDF. But the mean (least-squares estimate) of the samples is 0.7mm, which is 1.2mm away from the measurement of 1.9mm, which is impossible. My three measurements are consistent with a true value anywhere between 0.9mm and 1.1mm. I imagine the true likelihood function would be highest at 0.9mm and lowest at 1.1mm. This example seems to contradict the Gauss–Markov theorem. What am I missing? How do the assumptions of the Gauss–Markov theorem avoid this problem? —Ben FrantzDale (talk) 19:21, 6 December 2008 (UTC)

 * It can't contradict the Gauss–Markov theorem if it's not a linear function of the tuple of observed random variables, nor if it is biased. Maximum likelihood estimators are typically biased.  And it is well-known that unbiased estimation can result in "impossible" solutions, whereas maximum likelihood cannot. Michael Hardy (talk) 23:15, 6 December 2008 (UTC)

 * That makes sense; duah :-). I may add that clarification to this article and/or to linear least squares. Thanks. —Ben FrantzDale (talk) 15:51, 7 December 2008 (UTC)

Use conditional on X, not to assume X deterministic
Suppose we don't assume Xi to be deterministic, Don't we also need Xi and epsilon i to be independent? 69.134.206.109 (talk) 16:54, 21 March 2009 (UTC)

Don't we need E(e|X)=0 (conditional), as well as E(e)=0 (unconditional) so that we can say E(X'e)=0. In the proof, I am having trouble seeing how the expected value distributes... --169.237.57.103 (talk) 16:27, 27 June 2011 (UTC)

Typo in the expression ot C' (C-transpose)
C' is equal to X (X X')^-1 + D'

At the time of my comment, C' was written as: X (X' X)^-1 + D'

And final expression for V(β) is therefor: σ^2 (X X')^-1 + σ^2 DD' — Preceding unsigned comment added by Pathdependent1 (talk • contribs)

 * If
 * $$ C = (X'X)^{-1}X' \, $$
 * then
 * $$ C' = X(X'X)^{-1}. \, $$
 * The matrix in parentheses is symmetric, i.e. it is its own transpose. Note that
 * $$ (AB)' = B'A' \, $$
 * i.e. the order gets reversed and the two matrices get transposed. Applying this when
 * $$ A = X'\text{ and }B = X \, $$
 * means X gets changed to X' and vice-versa, and the order of multiplication then gets reversed. That means the transpose of (X'X) is (X'X).
 * Hence there was no typo; the expression was correct. Michael Hardy (talk) 22:06, 13 July 2011 (UTC)
 * Hence there was no typo; the expression was correct. Michael Hardy (talk) 22:06, 13 July 2011 (UTC)

Copyright problem removed
Prior content in this article duplicated one or more previously published sources. The material was copied from: http://wiki.answers.com/Q/When_are_OLS_estimators_BLUE. Infringing material has been rewritten or removed and must not be restored, unless it is duly released under a compatible license. (For more information, please see "using copyrighted works from others" if you are not the copyright holder of this material, or "donating copyrighted materials" if you are.) For legal reasons, we cannot accept copyrighted text or images borrowed from other web sites or published material; such additions will be deleted. Contributors may use copyrighted publications as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Wikipedia takes copyright violations very seriously, and persistent violators will be blocked from editing. While we appreciate contributions, we must require all contributors to understand and comply with these policies. Thank you. Voceditenore (talk) 14:04, 10 October 2011 (UTC)
 * Please note also that the content of wiki.answers.com is reader-generated and should never be used as a source. Voceditenore (talk) 14:04, 10 October 2011 (UTC)

Important points?
There are two remarks which -- if correct -- I think should be further emphasized in the article.

The first remark is explained here. It can be summed up as: OLS  =   Argmin of residual   =   (XX')^(-1)X'Y   =   argmin(among linear, unbiased estimators) of error   =   BLUE

The second remark pertains to linearity. As far as I can tell, there are two relations which are linear, which are used in the derivation. The first linear relation is the data-generating model Y = XB + err. The second is the requirement that the estimator be linear in the data. The fact that there are two linear relations is sometimes left out of the discussion. For example here. Only the linearity of the data-generating model is pointed out, even though they derive the BLUE.

Similarly, the OLS "happens to be linear", the BLUE has it as a requirement.

Any experts that can comment on these remarks? — Preceding unsigned comment added by 2001:700:3F00:0:189C:4003:11DF:E92E (talk) 17:22, 24 April 2013 (UTC)

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Hessian Matrix Proof
(′ is Transpose). At the end of the proof, when we get H = 2X′X, we can just use that if B=√2 X, then H=B′B. From the Wikipedia article on definite matrixes, it follows that H is definite. In this article we have a different proof (a pretty nice one in my opinion, just long and maybe unnecessary). Should we remove it and replace it by the simpler theorem? Lainad27 (talk) 02:07, 9 November 2021 (UTC)

Nevermind, forgot X is not always a square matrix... Lainad27 (talk) 07:40, 9 November 2021 (UTC)

title: Talk:Gauss–Markov theorem


text:
GiFT

giFT Internet File Transfer (giFT) is a computer software daemon that allows several file sharing protocols to be used with a simple client having a graphical user interface (GUI). The client dynamically loads plugins implementing the protocols, as they are required.

General
Clients implementing frontends for the giFT daemon communicate with its process using a lightweight network protocol. This allows the networking protocol code to be completely abstracted from the user interface. The giFT daemon is written using relatively cross-platform C code, which means that it can be compiled for and executed on a big variety of operating systems. There are several giFT GUI front-ends for Microsoft Windows, Apple Macintosh, and Unix-like operating systems.

The name giFT (giFT Internet File Transfer) is a so-called recursive acronym, which means that it refers to itself in the expression for which it stands.

One of the biggest drawbacks of the giFT engine is that it currently lacks Unicode support, which prevents sharing files with Unicode characters in their file names (such as "ø","ä", "å", "é" etc.). Also, giFT lacks many features needed to use the gnutella network effectively.

Available plugins
Available protocols are:

Stable
 * OpenFT, giFT's own file sharing protocol
 * gnutella (used by FrostWire, Shareaza)
 * Turtle F2F

Beta version
 * FastTrack (used by Kazaa). The giFT plugin is giFT-FastTrack.

Alpha version
 * OpenNap
 * eDonkey network
 * Soulseek

OpenFT protocol
giFT's sibling project is OpenFT, a peer-to-peer file-sharing network protocol that has a structure in which nodes are divided into 'search' nodes and 'index' supernodes in addition to common nodes. Since both projects are related very closely, when one says 'OpenFT', one can mean either one of two different things: the OpenFT protocol, or the implementation in the form of a plugin for giFT.

Although the name OpenFT stands for "Open FastTrack", the OpenFT protocol is an entirely new protocol design: only a few ideas in the OpenFT protocol are drawn from what little was known about the FastTrack protocol at the time OpenFT was designed.

OpenFT file-sharing protocol
Like FastTrack and Napster, OpenFT is a network where nodes submit lists of shared files to other nodes to keep track of which files are available on the network. This reduces the bandwidth consumed from search requests at the price of additional memory and processing power on the nodes that store that information. The transmission of shared lists is not fully recursive: a node will only transmit its list of shared files to a single search node randomly chosen as that node's "parent", and the list of those files will not be further transmitted to other nodes.

OpenFT is also similar to the gnutella network in that search requests are recursively forwarded in between the nodes that keep track of the shared files.

There are three different kinds of nodes on the OpenFT network:

 * USER
 * Most nodes are USER nodes; these have no special function.

 * SEARCH
 * These nodes handle search requests; they search the filelists their CHILD nodes (explained below) submit to them. These nodes must have a capable Internet connection and at least 128M RAM. A modern processor is highly recommended as well.

 * INDEX
 * Nodes with fast connections and much memory can be INDEX nodes, which keep lists of available search nodes, collect statistics, and try to maintain the structure of the network.

A node can be both a SEARCH and an INDEX node. USER nodes will pick three SEARCH nodes to be their PARENT nodes. They will submit their shares list to them if the PARENT accepts the USER as its CHILD. By default, SEARCH nodes will be PARENTS for a maximum of 500 CHILD nodes.

title: GiFT


text:
User:Torge

.

title: User:Torge


text:
Talk:Hoax

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 13 January 2020 and 5 May 2020. Further details are available on the course page. Student editor(s): Jcjacks6. Peer reviewers: Elizatangen98.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 23:45, 16 January 2022 (UTC)

Let me guess...
...this article is one that often has false information on it. Colabcalub (talk) 13:24, 28 May 2012 (UTC)

Russia and Georgia political hoax
' "On Saturday 13 March 2010 the Imedi television station in Georgia broadcast a false announcement that Russia had invaded Georgia." '

Given the history between the two countries, this sounds less like a hoax and suspiciously like the TV station jumped the gun over something and then tried the old "I was just trolling all along, lol" schtick. Are we sure it deserves a place here? --164.71.1.221 (talk) 04:45, 19 October 2012 (UTC)

Accidental Wikipedia hoax
The article I accidentally started a Wikipedia hoax describes how a college prank, wherein two students added bogus text to the Amelia Bedelia article back in 2009, and which was taken as gospel truth (having never been removed by any other WP editors), to the point of being quoted as fact in several serious publications. WP user Milowent appears in the article as an "expert on Internet hoaxes". It might be useful to mention the article on this page or on a related page. — Loadmaster (talk) 18:09, 29 July 2014 (UTC)
 * I have redacted part of this comment per WP:BLP, as described on Talk:Amelia Bedelia. 9kat (talk) 01:27, 30 July 2014 (UTC)

Regional Hoaxes
I just found a region, which has some Wikipedia articles which are only relevant by a specific region. If I publish an article about a Hoaxes in this region, how could I add this article to the existing one? Thanks! --huggi - never stop exploring (talk) 05:51, 29 September 2015 (UTC)

Redirect of Fake news
For those who are interested, there is an ongoing discussion at Talk:Fake news (disambiguation) about whether Fake news should redirect to Fake news (disambiguation) or Hoax. --Dr. Fleischman (talk) 21:25, 12 January 2017 (UTC)

Media Hoaxes and Fake News
Paragraph 2 and 3 of this section should be reviewed. Postconfused (talk) 16:52, 31 May 2020 (UTC)

Opening definition is too broad
Here is the opening definition that I found:

 * A hoax is a falsehood deliberately fabricated to masquerade as the truth.

This definition is too broad, since both frauds and fakes are deliberately fabricated to masquerade as truth.

The OED says that a hoax is

 * a humorous or mischievous deception, usually taking the form of a fabrication of something fictitious or erroneous, told in such a manner as to impose upon the credulity of the victim.

In this definition it is the last clause that is the most important, because it indicates that the representations of a hoax always involve things that people are willing to believe reflexively, without evidence; frauds and fakes, on the other hand, are developed so as to withstand all but the most diligent efforts at verification or authentication.

The first part of the definition, however, is too narrow, since hoaxers do not always have mischief as their, for some hoaxers may wish to discredit others, some to promote a cause, some for profit.

When we think of the Piltdown Man hoax, the Alan Sokal Affair, the Cardiff Giant, I, Libertine, and Jussie Smollett, for instance, it seems that two other characteristics are distinctive of hoaxes.

First, a hoax is usually submitted to universal public attention so as to excite the interest and passions of people of the most varied social classes who can be counted on to treat the representations as of the highest import.

Second, a hoax is meant to entrap persons and institutions of high repute who can be counted on to vie with one another in putting whatever form of social credit or influence they have in the service of the hoax.

I therefore propose this definition:

 * A hoax is a widely publicized falsehood so fashioned as to invite reflexive, unthinking acceptance by the greatest number of persons of the most varied social identities and of the highest possible social pretensions to gull their victims into putting up the highest possible social currency in support of the hoax.

The definition as it stands at present is followed by this statement:

 * It is distinguishable from errors in observation or judgment, rumors, urban legends, pseudosciences, and April Fools' Day events that are passed along in good faith by believers or as jokes.

There is no point at all in using things that bear no relation whatever to hoaxes to help characterize it concretely. After all, errors aren't intentional, most people are aware that rumors are rumors, nobody takes urban legends so seriously as to stake their reputations on them, people quickly realize that an April Fools' Day prank is a prank, and no matter how many sceptics try to debunk pseudo-sciences, true believers will never accept their arguments.

Hoaxes are related, as I pointed out, to frauds and fakes and scams, which people do take seriously, but whose creators devise their falsehoods to withstand scrutiny. Thus after the definition I think this would be proper:

 * Whereas the promoters of frauds, fakes, and scams devise them so that they will withstand the highest degree of scrutiny customary in the affair, hoaxers are confident, justifiably or not, that their representations will receive no scrutiny at all. They have such confidence because their representations belong to a world of notions fundamental to the victims' views of reality, but whose truth and importance they accept without argument or evidence, and so never question.

 * Some hoaxers intend eventually to unmask their representations as in fact a hoax so as to expose their victims as fools; seeking some form of profit, other hoaxers hope to maintain the hoax indefinitely, so that it is only when sceptical persons willing to investigate their claims publish their findings that at last they stand revealed as hoaxers.

I think my definition and epitome touch upon all the most salient characteristics of hoaxes.Wordwright (talk) 02:12, 29 March 2022 (UTC)

 * This is definitely prose of a high calibre, but I oppose on grounds of WP:TECHNICAL . Someone who is seeking to understand what a hoax is is not going to understand what is meant by "gull their victims into putting up the highest possible social currency in support of the hoax." I did a double take when I saw the page personally.

 * Could you explain why the phrase "most varied social identities and of the highest possible social pretensions" is necessary? This seems more like a specific scholar's opinion of what exactly a hoax is, rather than an introduction to the concept. Something like (roughly) "...A falsehood deliberately constructed so as to receive widespread attention, while plausible enough to be taken as truth within a large segment of a society or population." might get across your point more aptly?

 * I also disagree that it does not need to be distinguished from other concepts relating to deceptive events. Anecdotally, things such as urban legends and pranks are often called hoaxes in common discourse. 58.178.108.163 (talk) 13:31, 14 November 2023 (UTC)

title: Talk:Hoax


text:
Inductive bias

The inductive bias (also known as learning bias) of a learning algorithm is the set of assumptions that the learner uses to predict outputs of given inputs that it has not encountered. Inductive bias is anything which makes the algorithm learn one pattern instead of another pattern (e.g. step-functions in decision trees instead of continuous function in a linear regression model). Learning is the process of apprehending useful knowledge by observing and interacting with the world. It involves searching a space of solutions for one expected to provide a better explanation of the data or to achieve higher rewards. But in many cases, there are multiple solutions which are equally good. An inductive bias allows a learning algorithm to prioritize one solution (or interpretation) over another, independent of the observed data.

In machine learning, one aims to construct algorithms that are able to learn to predict a certain target output. To achieve this, the learning algorithm is presented some training examples that demonstrate the intended relation of input and output values. Then the learner is supposed to approximate the correct output, even for examples that have not been shown during training. Without any additional assumptions, this problem cannot be solved since unseen situations might have an arbitrary output value. The kind of necessary assumptions about the nature of the target function are subsumed in the phrase inductive bias.

A classical example of an inductive bias is Occam's razor, assuming that the simplest consistent hypothesis about the target function is actually the best. Here consistent means that the hypothesis of the learner yields correct outputs for all of the examples that have been given to the algorithm.

Approaches to a more formal definition of inductive bias are based on mathematical logic. Here, the inductive bias is a logical formula that, together with the training data, logically entails the hypothesis generated by the learner. However, this strict formalism fails in many practical cases, where the inductive bias can only be given as a rough description (e.g. in the case of artificial neural networks), or not at all.

Types
The following is a list of common inductive biases in machine learning algorithms.

 * Maximum conditional independence: if the hypothesis can be cast in a Bayesian framework, try to maximize conditional independence. This is the bias used in the Naive Bayes classifier.
 * Minimum cross-validation error: when trying to choose among hypotheses, select the hypothesis with the lowest cross-validation error. Although cross-validation may seem to be free of bias, the "no free lunch" theorems show that cross-validation must be biased, for example assuming that there is no information encoded in the ordering of the data.
 * Maximum margin: when drawing a boundary between two classes, attempt to maximize the width of the boundary. This is the bias used in support vector machines. The assumption is that distinct classes tend to be separated by wide boundaries.
 * Minimum description length: when forming a hypothesis, attempt to minimize the length of the description of the hypothesis.
 * Minimum features: unless there is good evidence that a feature is useful, it should be deleted. This is the assumption behind feature selection algorithms.
 * Nearest neighbors: assume that most of the cases in a small neighborhood in feature space belong to the same class. Given a case for which the class is unknown, guess that it belongs to the same class as the majority in its immediate neighborhood. This is the bias used in the k-nearest neighbors algorithm. The assumption is that cases that are near each other tend to belong to the same class.

Shift of bias
Although most learning algorithms have a static bias, some algorithms are designed to shift their bias as they acquire more data. This does not avoid bias, since the bias shifting process itself must have a bias.

title: Inductive bias


text:
Ladislaus Bortkiewicz

Ladislaus Josephovich Bortkiewicz (Russian Владислав Иосифович Борткевич, German Ladislaus von Bortkiewicz or Ladislaus von Bortkewitsch) (7 August 1868 – 15 July 1931) was a Russian economist and statistician of Polish ancestry. He wrote a book showing how the Poisson distribution, a discrete probability distribution, can be useful in applied statistics, and he made contributions to mathematical economics. He lived most of his professional life in Germany, where he taught at Strassburg University (Privatdozent, 1895–1897) and Berlin University (1901–1931).

Life and work
Ladislaus Bortkiewicz was born in Saint Petersburg, Imperial Russia, to two ethnic Polish parents: Józef Bortkiewicz and Helena Bortkiewicz (née Rokicka). His father was a Polish nobleman who served in the Russian Imperial Army.

Bortkiewicz graduated from the Law Faculty in 1890. In 1898 he published a book about the Poisson distribution, titled The Law of Small Numbers. In this book he first noted that events with low frequency in a large population follow a Poisson distribution even when the probabilities of the events varied. It was that book that made the Prussian horse-kicking data famous. The data gave the number of soldiers killed by being kicked by a horse each year in each of 14 cavalry corps over a 20-year period. Bortkiewicz showed that those numbers followed a Poisson distribution. The book also examined data on child-suicides. Some have suggested that the Poisson distribution should have been named the "Bortkiewicz distribution." In political economy, Bortkiewicz is important for his analysis of Karl Marx's reproduction schema in the last two volumes of Capital. Bortkiewicz identified a transformation problem in Marx's work. Making use of Dmitriev's analysis of Ricardo, Bortkiewicz proved that the data used by Marx was sufficient to calculate the general profit rate and relative prices. Though Marx's transformation procedure was not correct—because it did not calculate prices and profit rate simultaneously, but sequentially—Bortkiewicz has shown that it is possible to get the correct results using the Marxian framework, i.e. using the Marxian variables constant capital and variable capital it is possible to obtain the profit rate and the relative prices in a three-sector model. This "correction of the Marxian system" has been the great contribution of Bortkiewicz to classical and Marxian economics but it was completely unnoticed until Paul Sweezy's 1942 book "Theory of Capitalist Development". Piero Sraffa (1960) has provided the complete generalization of the simultaneous method for classical and Marxian analysis.

Bortkiewicz died in Berlin, Germany. His papers, including a voluminous correspondence file (some 1,000 letters 1876–1931), were deposited at Uppsala University in Sweden, except for his correspondence with Léon Walras which went into the collection of the Walras scholar William Jaffé in the USA.

Major publications

 * Die mittlere Lebensdauer. Die Methoden ihrer Bestimmung und ihr Verhältnis zur Sterblichkeitsmessung. Gustav Fischer, Jena 1893 (Göttinger Digitalisierungszentrum)
 * "Review of Léon Walras, Éléments d'économie politique pure, 2e édit.", 1890, Revue d'économie politique
 * — Article's English translation (Value and Price in the Marxian System), 1952, International Economic Papers, No.2
 * — Spanish version 1917 * Original German version
 * "Die Rodbertus'sche Grundrententheorie und die Marx'sche Lehre von der absoluten Grundrente", from: "Archiv für die Geschichte des Sozialismus und der Arbeiterbewegung", 1910–11
 * — Spanish version 1917 * Original German version
 * "Die Rodbertus'sche Grundrententheorie und die Marx'sche Lehre von der absoluten Grundrente", from: "Archiv für die Geschichte des Sozialismus und der Arbeiterbewegung", 1910–11
 * "Die Rodbertus'sche Grundrententheorie und die Marx'sche Lehre von der absoluten Grundrente", from: "Archiv für die Geschichte des Sozialismus und der Arbeiterbewegung", 1910–11
 * "Die Rodbertus'sche Grundrententheorie und die Marx'sche Lehre von der absoluten Grundrente", from: "Archiv für die Geschichte des Sozialismus und der Arbeiterbewegung", 1910–11

title: Ladislaus Bortkiewicz


text:
Fairport Convention

Fairport Convention are an English folk rock band, formed in 1967 by guitarists Richard Thompson and Simon Nicol, bassist Ashley Hutchings and drummer Shaun Frater (with Frater replaced by Martin Lamble after their first gig). They started out influenced by American folk rock, with a set list dominated by Bob Dylan and Joni Mitchell songs and a sound that earned them the nickname "the British Jefferson Airplane". Vocalists Judy Dyble and Iain Matthews joined them before the recording of their self-titled debut in 1968; afterwards, Dyble was replaced by Sandy Denny, with Matthews later leaving during the recording of their third album.

Denny began steering the group towards traditional British music for their next two albums, What We Did on Our Holidays and Unhalfbricking (both 1969); the latter featured fiddler Dave "Swarb" Swarbrick, most notably on the song "A Sailor's Life", which laid the groundwork for British folk rock by being the first time a traditional British song was combined with a rock beat. Shortly before the album's release, a crash on the M1 motorway killed Lamble and Jeannie Franklyn, Thompson's girlfriend; this resulted in the group retiring most of their prior material and turning entirely towards British folk music for their seminal album Liege & Lief, released the same year. This style has been the band's focus ever since. For this album Swarbrick joined full-time, alongside drummer Dave Mattacks. Both Denny and Hutchings left before the year's end; the latter replaced by Dave Pegg, who has remained the group's sole consistent member to this day; Thompson left after the recording of 1970's Full House.

The 1970s saw numerous lineup changes around the core of Swarbrick and Pegg – Nicol being absent for the middle of the decade – and declining fortunes as folk music fell out of mainstream favour. Denny, whose partner Trevor Lucas had been a guitarist in the group since 1972, returned for the pop-oriented Rising for the Moon album in 1975 in a final bid to crack America; this effort failed, and after three more albums minus Denny and Lucas, the group disbanded in 1979. They played a farewell concert in the village of Cropredy, Oxfordshire, where they had held small concerts since 1976, and this marked the beginning of the Cropredy Festival (since 2005 known as Fairport's Cropredy Convention) which has become the largest folk festival in Britain, with annual attendances of 20,000.

The band was reformed by Nicol, Pegg, and Mattacks in 1985, joined by Maartin Allcock (guitar, mandolin, keys, vocals) and Ric Sanders (fiddle, keyboards), and they have remained active since. Allcock was replaced by Chris Leslie (mandolin, violin, vocals) in 1996, and Gerry Conway replaced Mattacks in 1998. Their 29th studio album, Shuffle and Go was released in 2020, and they continue to headline Cropredy each year.

Despite little mainstream success – their only top 40 single being "Si Tu Dois Partir", a French-language cover of the Dylan song "If You Gotta Go, Go Now" from Unhalfbricking – Fairport Convention remain highly influential in British folk rock and British folk in general. Liege & Lief was named the "Most Influential Folk Album of All Time" at the BBC Radio 2 Folk Awards in 2006, and Pegg's playing style, which incorporates jigs and reels into his basslines, has been imitated by many in the folk rock and folk punk genres. Additionally, many former members went on to form or join other notable groups in the genre, including Fotheringay, Steeleye Span, and the Albion Band; along with solo careers, most notably Thompson and Denny. Sandy Denny, now regarded as being amongst Britain's finest female singer-songwriters, died in 1978; her song "Who Knows Where the Time Goes?" – recorded by Fairport on Unhalfbricking – became a signature song for herself and the band.

Origins
Bassist Ashley Hutchings met guitarist Simon Nicol in North London in 1966 when they both played in the Ethnic Shuffle Orchestra. They rehearsed on the floor above Nicol's father's medical practice in a house called "Fairport" on Fortis Green in Muswell Hill – the same street on which Ray and Dave Davies of the Kinks grew up. The house lent its name to the group they formed together as Fairport Convention in 1967 with Richard Thompson on guitar and Shaun Frater on drums. After their initial performance at St Michael's Church Hall in Golders Green on 27 May 1967, they had their first of many line-up changes as one member of the audience, drummer Martin Lamble, convinced the band that he could do a better job than Frater and replaced him. They soon added a female singer, Judy Dyble, which gave them a distinctive sound among the many London bands of the period.

1967–69: The first three albums
Fairport Convention were soon playing regularly at underground venues such as UFO and The Electric Garden, which later became the Middle Earth club. After only a few months, they caught the attention of manager Joe Boyd who secured them a contract with Polydor Records. Boyd suggested they augment the line-up with another male vocalist. Singer Iain Matthews (then known as Ian MacDonald) joined the band, and their first album, Fairport Convention, was recorded in late 1967 and released in June 1968. At this early stage Fairport looked to North American folk and folk rock acts such as Joni Mitchell, Bob Dylan, and The Byrds for material and inspiration. The name "Fairport Convention" and the use of two lead vocalists led many new listeners to believe that they were an American act, earning them the nickname 'the British Jefferson Airplane' during this period. Fairport Convention played at the first Isle of Wight Festival, in August 1968, when Jefferson Airplane headlined.

After disappointing album sales they signed a new contract with Island Records. Before their next recording Judy Dyble left – she described it as being "unceremoniously dumped" – and was replaced by the band with Sandy Denny, a folk singer who had previously recorded as a soloist and with Strawbs. Denny's distinctive voice, described by Clive James as "open space, low-volume, high-intensity", is one of the characteristics of two albums released in 1969: What We Did on Our Holidays and Unhalfbricking. These recordings marked the growth of much greater musicality and song-writing ability among the band. The first of these featured the Thompson-penned "Meet on the Ledge", which became their second single and eventually the band's unofficial anthem.

During the recording of Unhalfbricking, Matthews left after having sung on only one song, eventually to form Matthews Southern Comfort. He was not replaced; the other male members covered his vocal parts. The album featured a guest appearance by Birmingham folk fiddler Dave Swarbrick on a recording of "A Sailor's Life", a traditional song brought to the band by Denny from her folk club days. The recording of this track marked an important turning point for the band, sparking an interest in traditional music in Ashley Hutchings that led him to detailed research in the English Folk Dance and Song Society Library at Cecil Sharp House; this theme would become the basis for their next, much more ambitious, recording project.

These two albums began to gain the band wider recognition. Radio DJ John Peel championed their music, playing their albums on his influential BBC shows. Peel also recorded a number of sessions which were later released as the album Heyday (1987). They enjoyed some mainstream success when they entered the singles charts with "Si Tu Dois Partir", a French-language version of Bob Dylan's "If You Gotta Go, Go Now". The record just missed the top twenty, but secured the band a slot on Top of the Pops, Britain's most popular television pop music programme at the time. In 1969 four members of the band, one uncredited and three under pseudonyms, featured as backing musicians on the album Love Chronicles by Scottish folk artist Al Stewart.

Developing British folk rock
On 12 May 1969, on the way home from a gig at Birmingham venue Mothers, Fairport's van crashed on the M1 motorway. Martin Lamble, aged only nineteen, and Jeannie Franklyn, Richard Thompson's girlfriend, were killed. The rest of the band suffered injuries of varying severity. They nearly decided to disband. However, they reconvened with Dave Mattacks taking over drumming duties and Dave Swarbrick, having made contribution to Unhalfbricking, now joined as a full member. Boyd set the band up in a rented house in Farley Chamberlayne near Winchester in Hampshire, where they recuperated and worked on the integration of British folk music into rock and roll, which would result in the fourth album Liege & Lief.

Usually considered the highpoint of the band's long career, Liege & Lief was a huge leap forward in concept and musicality. The album consisted of six traditional tracks and three original compositions in a similar style. The traditional tracks included two sustained epics: "Tam Lin", which was over seven minutes in length, and "Matty Groves", at over eight. There was a medley of four traditional tunes, arranged, and, like many of the tracks, enlivened, by Swarbrick's energetic fiddle playing. The first side was bracketed by original compositions "Come all ye" and "Farewell, Farewell", which, in addition to information on the inside of the gatefold cover on Hutchings' research, explaining English folk traditions, helped give the record the feel of a concept album. "Farewell, Farewell" and the final track "Crazy Man Michael", also saw the full emergence of the distinctive song writing talent of Thompson that was to characterize his contributions to the band and later solo career. The distinctive sound of the album came from the use of electric instruments and Mattacks' disciplined drumming with Swarbrick's fiddle accompaniment in a surprising and powerful combination of rock with the traditional. The entire band had reached new levels of musicality, with the fluid guitar playing of Thompson and the "ethereal" vocal of Denny particularly characteristic of the sound of the album. As the reviewer from AllMusic put it, the album was characterised by the "fusing [of] time-worn folk with electric instruments while honoring both".

A few British bands had earlier experimented with playing traditional English songs on electric instruments, (including Strawbs and Pentangle), but Fairport Convention was the first English band to do this in a concerted and focused way. Fairport Convention's achievement was not to invent folk rock, but to create a distinctly English branch of the genre, which would develop alongside, and interact with, American inspired music, but which can also be seen as a distinctively national reaction in opposition to it. Liege & Lief was launched with a sell-out concert in London's Royal Festival Hall late in 1969. It reached number 17 in the UK album chart, where it spent fifteen weeks.

1970s: A time of change
Disagreements arose about the direction of the band in the wake of this success. Ashley Hutchings wanted to explore more traditional material and left to form two groups that would rival Fairport for significance in English folk rock: Steeleye Span and the Albion Band. Sandy Denny also left to found her own group Fotheringay. Dave Pegg took over on bass guitar and has been the group's one constant ever since, in an unbroken membership of over four decades. The band made no serious attempt to replace Denny, and, although she would briefly return, the sound of the band would now be characterized by male vocals.

Despite these changes the band produced another album, Full House (1970), which was remarkably successful as a project. Like its predecessor, it combined traditional songs, including a powerful rendition of "Sir Patrick Spens", with original compositions. The latter benefited from the writing partnership of Thompson and Swarbrick, most obviously on "Walk Awhile", which would become a concert favourite. Despite the loss of Denny the band still possessed four vocalists, including the emerging voices of Nicol and Swarbrick, whose tones would dominate the sound of this period. It was favourably reviewed in Britain and America, drawing comparisons with the Band from Rolling Stone magazine who declared that "Fairport Convention is better than ever". The album reached number 13 in the UK Chart and stayed in the chart for eleven weeks. The same year the band released a single, "Now Be Thankful", and made its American debut, touring with Traffic and Crosby, Stills, Nash & Young.

In the recurring pattern, soon after the album's release Thompson left the band to pursue other projects and eventually his solo career. This left Simon Nicol as the only original member, and Dave Swarbrick emerged as the leading force in the band. In 1970 the members and their families had moved into The Angel, a former pub in Hertfordshire, and this inspired the next album Angel Delight (1971) the band's first to chart in the US, peaking at number 200 on the Billboard 200 and their only top ten album in the UK. The next project was an ambitious folk-rock opera developed by Swarbrick, based on the life of John "Babbacombe" Lee, "the man they couldn't hang" and released with the title Babbacombe Lee (1971). The concept format, originally without clear tracks, excited considerable press interest, and it received good air play in the United States where it reached number 195. A version was produced by the BBC for TV in 1975 with narration by Melvyn Bragg. These two albums were also notable as the first time that Fairport had recorded consecutively with the same line-up, but inevitably stability did not last; Simon Nicol left early in late 1971 to join Ashley Hutchings' Albion Country Band, and he was soon followed by Mattacks.

Only Pegg and Swarbrick remained, and the following few years have been dubbed 'Fairport confusion' as a bewildering sequence of band members came and went, but by 1973 Mattacks had returned and two former members of Sandy Denny's Fotheringay had joined the band, Denny's Australian husband Trevor Lucas on vocals and guitar, and American Jerry Donahue on lead guitar. From these line-ups the band produced two studio albums: Rosie, notable for the Swarbrick-penned title track (1973) and Nine (1974), the ninth studio album by the band. The last of these contained writing contributions by Lucas to five of the nine tracks, which together with Donahue's country influences and outstanding guitar pyrotechnics gave the album a very distinctive feel.

Denny rejoined the band in 1974, and there were considerable expectations, both artistic and commercial, placed on this line-up. Denny was featured on the album Rising for the Moon (1975), which became the band's highest US chart album when it reached number 143 on the Billboard 200 and the first album to reach the top hundred in the UK since Angel Delight, reaching No. 52. During the Rising sessions, Mattacks fell out with producer Glyn Johns and was replaced by former Grease Band drummer Bruce Rowland. Poor UK sales for Rising did not aid morale and, despite the relative success of the line-up, Lucas and Donahue left the band, as did Denny in 1976. She died aged 31, in 1978, of a cerebral haemorrhage after falling down a flight of stairs.

Rowland, Pegg, and Swarbrick fulfilled their remaining contractual obligations to Island Records by turning what had originally been a Swarbrick solo effort into the album Gottle O'Geer (1976) under the name 'Fairport' (as opposed to Fairport Convention) in the UK, and as 'Fairport featuring Dave Swarbrick' in the US, and with various session players and production by Simon Nicol, who subsequently rejoined the band. They then signed with Vertigo, but record sales continued to decline; after producing the first two of four albums for which they were contracted, The Bonny Bunch of Roses (1977) and Tipplers Tales (1978), Vertigo bought them out of their contract. It is claimed by members of the band that this was the only recording money they had seen up to that point.

1979–1985: The Cropredy era
By 1979 the mainstream market for folk rock had largely disappeared, the band had no record deal, and Dave Swarbrick had been diagnosed with tinnitus, which made loud electric gigs increasingly difficult. Fairport decided to disband. They played a farewell tour and a final outdoor concert on 4 August in Cropredy, the Oxfordshire village where Dave and Christine Pegg lived. The finality of this occasion was mitigated by the announcement that the band would meet for a reunion.

In August 1979, the band played at Knebworth Festival in England. The headline act at both their appearances at the festival, over two consecutive Saturdays on 4 and 11 August, were Led Zeppelin.

No record company wanted to release the live recordings of the tour and concert, so the Peggs founded Woodworm Records, which would be the major outlet for the band in the future. Members continued to take part in occasional gigs, particularly in festivals in continental Europe, and after a year they staged a reunion concert in Cropredy which became the annual Cropredy Festival. Over the next few years, it grew rapidly and emerged as the major mechanism for sustaining the band. In August 1981, the band held their annual reunion concert at Broughton Castle, rather than the usual Cropredy location. The concert was recorded, and released on the 1982 album Moat on the Ledge.

The Peggs continued to record and release the Cropredy concerts as 'official bootlegs'. These were supplemented by New Year's gigs in minor locations including the Half Moon at Putney and the Gloucester Leisure Centre. In 1983 the magazine Fairport Fanatics (later Dirty Linen), was created: a testament to the continued existence of a dedicated fan base.

The Angel Delight lineup of Simon Nicol, Dave Swarbrick, Dave Pegg, and Dave Mattacks played a number of gigs in the UK in the early 80s, then toured extensively in the UK and the US in 1984 and 1985. Band alumni like Richard Thompson and Bruce Rowland would occasionally join in.

The remaining members pursued their own lives and careers outside of the band. Nicol, Pegg, and Mattacks had recorded and toured with Richard and Linda Thompson at times in the 1970s, and did so again during this period, culminating in their appearance on the Shoot Out the Lights album and tour in 1982. Bruce Rowlands gave up the music business and moved to Denmark and as a result Dave Mattacks returned as drummer for Fairport's occasional gigs. Dave Pegg was the first of several Fairporters to join Jethro Tull which gave him well-paying steady employment. Simon Nicol had teamed up with Dave Swarbrick in a highly regarded acoustic duo, but this partnership was made difficult by Swarbrick's sudden decision to move to Scotland, where, from 1984, he began to focus on his new project Whippersnapper.

In 1985, Pegg, Nicol and Mattacks found that they all had some free time and an available studio belonging to Pegg. They decided that they needed some new material to add to the catalogue that had been suspended in 1978. As Swarbrick was unavailable, the selection of traditional tunes was more difficult than for past albums and there was a need for a replacement fiddle player and some vocals. Pegg and Nicol took over arranging duties on an instrumental medley and the band turned to sometime Albion Band members: jazz and folk violinist Ric Sanders and singer-songwriter Cathy Lesurf. They also had the help of ex-member Richard Thompson. Thompson and Lesurf contributed songs and took part in the recordings. Also important to the album was Ralph McTell who contributed one song and co-wrote one track each with Nicol and Mattacks; the latter of these, "The Hiring Fair", would become a stage fixture of Fairport.

The resulting album Gladys' Leap (1985) was generally well received in the music and national press, but caused some tension with Swarbrick who refused to play any of the new material at the 1985 Cropredy Festival. Nevertheless, the decision to reform the band, without Swarbrick, was taken by the other three remaining members. Ric Sanders was invited to join, along with guitarist, composer, arranger and multi-instrumentalist Maartin Allcock. Nicol, with his developing baritone voice, took over the main share of the vocal duties. This line-up was to last eleven years, the longest period of membership stability in the band's history so far.

1986–1997: Stability
The new band began a hectic schedule of performing in Britain and the world and prepared material for a new album. The result was the all-instrumental Expletive Delighted! (1986). This showcased the virtuosity of Sanders and Allcock, but perhaps inevitably was not popular with all fans. This was followed by the recording In Real Time: Live '87 which managed to capture the energy and power of the new Fairport on stage, despite the fact that it was recorded in the studio with audience reactions dubbed on.

In this period the band were playing to larger and larger audiences, both on tour and at Cropredy, and it was very productive in terms of recording. Fairport had the considerable composing and arranging skills of Allcock and, to fill the gap created by a lack of a songwriter in the band, they turned to some of the most talented available in the contemporary folk scene. The results were Red & Gold (1989) The Five Seasons (1990) and Jewel in the Crown (1995), the last of which was judged "their bestselling and undoubtedly finest album in years."

At this point, with Mattacks busy with other projects, the band shifted to an acoustic format for touring and released the unplugged Old New Borrowed Blue as "Fairport Acoustic Convention" in 1996. For a while the four-piece acoustic line-up ran in parallel with the electric format. When Allcock left the band, he was replaced by Chris Leslie on vocals, mandolin and fiddle, who formerly worked with Swarbrick in Whippersnapper, and had a one-off stint with the band replacing Ric Sanders for 1992 Cropredy Festival. This meant that for the first time since reforming, the band had a recognized songwriter who contributed significantly to the band's output on the next album Who Knows Where the Time Goes? (1997), particularly the rousing "John Gaudie". By the time of the 30th anniversary Festival at Cropredy in 1997, the new Fairport had been in existence for over a decade and contributed a significant chapter to the history of the band.

1998–present

Dave Mattacks moved to the US in 1998, and Gerry Conway took over on drums and percussion. Fairport produced two more studio albums for Woodworm Records: The Wood and the Wire (2000) and XXXV (2002). Then, for Over the Next Hill (2004) they established a new label: Matty Grooves Records. In this period the band toured extensively in the UK, Europe, Australasia, Europe, the US and Canada, and staged a major fund raiser for Dave Swarbrick at the Birmingham Symphony Hall. In 1998, members of the band began their association with the Breton musician Alan Simon. Working in collaboration with numerous others, members of Fairport (predominantly Nicol and Leslie) have performed in and participated in the recordings of all Simon's rock operas, including the Excalibur trilogy (1998, 2007, 2010) and Anne de Bretagne (2008).

2007 was their fortieth anniversary year and they celebrated by releasing a new album, Sense of Occasion. They performed the whole of the Liege & Lief album live at Cropredy, since 2004 renamed Fairport's Cropredy Convention, featuring the 1969 line-up of Dave Swarbrick, Ashley Hutchings, Dave Mattacks, Simon Nicol and Richard Thompson, with singer-songwriter Chris While taking the place of Sandy Denny. Footage of the festival, although not the Liege and Lief performance, was released as part of a celebratory DVD.

The band's first official YouTube video appeared in April 2008. Edited from footage shot for the DVD, the nine-minute mini-documentary includes interviews with Lulu, Jools Holland, Seth Lakeman, Mike Harding, Geoff Hughes and Frank Skinner.

In 2011 the band released a new studio album Festival Bell, the first new album in four years. This was followed in 2012 by Babbacombe Lee Live Again recorded live during the 2011 tour revisiting the Babbacombe Lee album first issued in 1971. In 2012, the band also released By Popular Request, a reworking in the studio of a number of the most popular songs in the band's repertoire (as determined by a mysterious consultation and voting process conducted by the band with its fans).

the band continues to write and record music, regularly producing new studio albums, the most recent releases being 2015's Myths and Heroes, 2017's 50:50@50 and 2020's Shuffle and Go. The Covid-19 Pandemic impacted significantly on their ability to tour, and their 2022 tour was initially cut short after several of the touring team developed Covid.

In 2022, Gerry Conway made the decision to leave the band after 24 years, and it was announced that the band would continue as a four piece. Conway did not make public that he left because he was suffering from motor neurone disease from which he was to die in April 2024.

On the bands Winter 2023 UK Tour, they were joined by a returning Mattacks on drums and percussion who also returned for the 2024 winter tour.

Public recognition
The mainstream media has increasingly recognized Fairport Convention's historical importance. They received a "Lifetime Achievement Award" at the 2002 BBC Radio 2 Folk Awards. In the same year Free Reed Records, an independent label, released Fairport Unconventional, a four-CD boxed set of rare and unreleased recordings from the band's 35-year career. At the 2006 BBC Radio 2 Folk Awards they received an award when their seminal album Liege & Lief was voted 'Most Influential Folk Album of All Time' by Radio 2 listeners. At the 2007 BBC Radio 2 Folk Awards Fairport Convention received an award recognising the late Sandy Denny and the band for "Favourite Folk Track of All Time" for "Who Knows Where the Time Goes?".

Members

 * Current members
 * Simon Nicol – guitar, dulcimer, lead vocals, keyboards (1967–1971, 1976–1979, 1985–present)
 * Dave Mattacks – drums, percussion, keyboards, bass guitar (1969–1972, 1973–1975, 1985–1997, 2023–present)
 * Dave Pegg – bass guitar, mandolin, backing vocals, guitar, fiddle, drums (1969–1979, 1985–present)
 * Ric Sanders – fiddles, keyboards, ukulele, backing vocals (1985–present)
 * Chris Leslie – mandolin, fiddle, bouzouki, vocals, guitar, flute (1996–present)

Discography
• #Fairport Convention (1968)

• #What We Did on Our Holidays (1969)

• #Unhalfbricking (1969)

• #Liege & Lief (1969)

• #Full House (1970)

• #Angel Delight (1971)

• #"Babbacombe" Lee (1971)

• #Rosie (1973)

• #Nine (1973)

• #Rising for the Moon (1975)

• #Gottle O'Geer (1976)

• #The Bonny Bunch of Roses (1977)

• #Tipplers Tales (1978)

• #Gladys' Leap (1985)

• #Expletive Delighted! (1986)

• #Red & Gold (1988)

• #The Five Seasons (1990)

• #Jewel in the Crown (1995)

• #Old New Borrowed Blue (1996)

• #Who Knows Where the Time Goes? (1997)

• #The Wood and the Wire (1999)

• #XXXV (2002)

• #Over the Next Hill (2004)

• #Sense of Occasion (2007)

• #Festival Bell (2011)

• #By Popular Request (2012)

• #Myths and Heroes (2015)

• #50:50@50 (2017)

• #Shuffle and Go (2020)

Filmography

 * Tony Palmer's Film of Fairport Convention and Matthews Southern Comfort, directed by Tony Palmer, featuring Fairport's appearance at the Maidstone Fiesta in 1970. Originally released as a VHS video by MusicFolk/Weintraub, re-released on DVD by Voiceprint Records in 2007, soundtrack CD issued by Voiceprint as Live in Maidstone 1970 in 2009.

title: Fairport Convention


text:
List of inspectors of Greenland

Royal Inspector was the highest ranking colonial officer in Danish Greenland 1782–1924. They were agents of the Royal Greenland Trading Department established by its Instruction of 1782 and reported to the Board of Managers of the company in Copenhagen.

As senior agents of the company, they were generally forbidden under the Instruction from marrying any non-European women, though Inspector Nicolai Zimmer's wife was half-Inuit.

Royal Inspectors of North Greenland
North Greenland comprised the northwest coast of Greenland between Holsteinsborg and Upernavik.

 * Johan Friedrich Schwabe (1782–1786)
 * Jens Clausen Wille (1786–1790)
 * Børge Johan Schultz (1790–1797)
 * Claus Bendeke (1797–1803)
 * Peter Hanning Motzfeldt (1803–1817)
 * Johannes West (1817–1825)
 * Carl Peter Holbøll (1825–1828)
 * Ludvig Fasting (1828–1843)
 * Hans Peter Christian Møller (1843–1845)
 * Nicolai Zimmer (1845–1846)
 * Christian Søren Marcus Olrik (1846–1866)
 * Carl August Ferdinand Bolbroe (1866–1867)
 * Sophus Theodor Krarup-Smith (1867–1882)
 * Hjalmar Christian Reinholdt (1882–1883) acting
 * Niels Alfred Andersen (1883–1898)
 * Johan Carl Joensen (1898–1900) acting
 * Jens Daugaard-Jensen (1900–1912)
 * Anders Peter Olsen (1912–1913) acting
 * Harald Lindow (1913–1924)

Royal Inspectors of South Greenland
South Greenland comprised the southwest coast between Sukkertoppen and Nennortalik.

 * Bendt Olrik (1782–1789)
 * Andreas Molbech Lund (1789–1795)
 * Claus Bendeke (1795–1797)
 * Niels Rosing Bull (1797–1802)
 * Marcus Nissen Myhlenphort (1802–1821) acting (until 1803)
 * Christian Alexander Platou (1821–1823) acting
 * Arent Christopher Heilmann (1823–1824) acting
 * Christian Alexander Platou (1824–1827)
 * Ove Valentin Kielsen (1827–1828)
 * Carl Peter Holbøll (1828–1856)
 * Jørgen Nielsen Møller (1856–1857) acting
 * Hinrich Johannes Rink (1857–1868)
 * Albert E. Blichfeldt Høyer (1868–1869)
 * Hannes Peter Stephensen (1870–1882)
 * Frederik Tryde Lassen (1882–1884) acting
 * Carl Julius Peter Ryberg (1884–1890)
 * Johan Carl Joansen (1890–1891) acting
 * Conrad Poul Emil Brummerstedt (1891–1892) acting
 * Edgar Christian Fencker (1892–1899)
 * Regnar Stephensen (1899–1902)
 * Oscar Peter Cornelius Koch (1902–1903)
 * Ole Bendixen (1903–1914)
 * Oluf Hastrup (1914–1915)
 * Carl Frederik Harries (1915–1923)
 * Christian Simony (1924) acting
 * Knud Oldendow (1924) acting

Royal Inspector of East Greenland
East Greenland comprised the east coast, with Ammassalik and Scoresbysund.

 * Ejnar Mikkelsen (1933–1950)

title: List of inspectors of Greenland


text:
Talk:List of inspectors of Greenland

Untitled
What's an "inspector"? Sort of like a governor? Who appoints them, the Danes? --Uncle Ed

To be honest with you, I have no idea. But thats what they were apperntly called. - fonzy

 * They were Danes alright (I don't know about the other part of the question, perhaps the title "governor" would have implied a higher paycheck.) Valentinian (talk) 22:13, 24 July 2006 (UTC)

title: Talk:List of inspectors of Greenland


text:
Talk:Rutile

Rutile
Both Ilmenite and this are linked to from titanium ore as the two most common forms of Titanium ore. Should the opening line of both mention that this is a titanium ore? Scottkeir 04:26, 28 July 2005 (UTC)

what does composted mean? please try to explain for a german. -- Anon, Jan 2003

 * Typo. Fixed. Danke. --Menchi 10:07 20 Jun 2003 (UTC)

Rutile
This article ends in mid sentence.

Image dropped
This image was removed in my dit because the article had too many images and it was hard to make it look readable with it in. I've put it here so that it can be put back in, perhaps when the article is fleshed out more. The table may need fixing too.

Titania
Titania is not an "old-fashioned" chemical name; among Materials engineers, it is the term used almost exclusively when referring to TiO2, unless there is need to specificy which phases is meant.

Questionable Point
The statement "(Rutile) is primarily extracted for use in refractory manufacture or use as a base for paints. Rarely is it extracted as an ore of titanium" is questionable. TIMET - a global leader in Titanium based metal production for aerospace and other industries (as well as Russian and other manufacturers) chlorinate rutile ores as a first step in producing titanium tetrachloride, the intermediary in the production of titanium sponge - the raw material for titanium metal products.

Rutile also i used i welding-electrodes. —Preceding unsigned comment added by 87.50.124.146 (talk) 10:48, 11 October 2008 (UTC)

 * I agree, its a very questionable point.  Rutile is mined for the purpose of .... selling rutile.   What the processor does with it,  is up to them.  It is not mined for some specific purpose.  So I have reworded the sentence.Eregli bob (talk) 03:02, 4 November 2008 (UTC)

Rutile Figure
Doesn't the figure for the structure have the aspect ratio flipped? The rutile unit cell has two long axes and one short axis, so that it's shaped more like a pizza box than a bread box, right? --SarahLawrence Scott (talk) 20:00, 21 August 2012 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Rutile. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20080531160020/http://www.titaniumresources.com:80/site/en-GB/Page_26.aspx to http://www.titaniumresources.com/site/en-GB/Page_26.aspx

When you have finished reviewing my changes, please set the checked parameter below to true to let others know.

Cheers. —cyberbot II  Talk to my owner :Online 04:05, 28 August 2015 (UTC)

19
I am not sure how the 19 years are arrived at. — Preceding unsigned comment added by 81.141.102.104 (talk) 14:10, 3 September 2015 (UTC)

Rutile occurrence map missing Sierra Leone
The current occurrence map does not indicate any amount for Sierra Leone, despite it being stated as a major producer (30% world-wide) later in the article. MartinWilson (talk) 09:38, 16 December 2019 (UTC)

What is this "Rutile output" map showing? What are its units?

What is this map showing? What are its units? ——JavaRogers (talk) 00:24, 19 September 2021 (UTC)
 * It's showing rutile output as a percentage of Australia's rutile output, Australia being the leading producer. Units are percent. It seems as good a way as any to depict the relative importance of different rutile mining disticts -- though one can raise questions about how the production is aggregated. --Kent G. Budge (talk) 14:50, 29 December 2021 (UTC)

Rutile main image
Hello, the hematite picture from Cavradi is completely misleading. There is probably only very minor rutile on that specimen. This is a typical hematite specimen.

Moreover, this image comes originally from a mineral dealer, via mindat.org, therefore I see potential problems with copyright. And it may be considered as a commercial advertisement for that dealer on wikipedia. so long, Christian Ipmb-uni-hd (talk) 13:32, 27 June 2018 (UTC)
 * Fixed! DragonTownEpic (talk) 05:49, 29 December 2021 (UTC)

title: Talk:Rutile


text:
Cosmological principle

In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is uniformly isotropic and homogeneous when viewed on a large enough scale, since the forces are expected to act equally throughout the universe on a large scale, and should, therefore, produce no observable inequalities in the large-scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.

Definition
Astronomer William Keel explains:

"The cosmological principle is usually stated formally as 'Viewed on a sufficiently large scale, the properties of the universe are the same for all observers.' This amounts to the strongly philosophical statement that the part of the universe which we can see is a fair sample, and that the same physical laws apply throughout. In essence, this in a sense says that the universe is knowable and is playing fair with scientists."

The cosmological principle depends on a definition of "observer", and contains an implicit qualification and two testable consequences.

"Observers" means any observer at any location in the universe, not simply any human observer at any location on Earth: as Andrew Liddle puts it, "the cosmological principle [means that] the universe looks the same whoever and wherever you are."

The qualification is that variation in physical structures can be overlooked, provided this does not imperil the uniformity of conclusions drawn from observation: the Sun is different from the Earth, our galaxy is different from a black hole, some galaxies advance toward rather than recede from us, and the universe has a "foamy" texture of galaxy clusters and voids, but none of these different structures appears to violate the basic laws of physics.

The two testable structural consequences of the cosmological principle are homogeneity and isotropy. Homogeneity means that the same observational evidence is available to observers at different locations in the universe ("the part of the universe which we can see is a fair sample"). Isotropy means that the same observational evidence is available by looking in any direction in the universe ("the same physical laws apply throughout"). The principles are distinct but closely related, because a universe that appears isotropic from any two (for a spherical geometry, three) locations must also be homogeneous.

Origin
The cosmological principle is first clearly asserted in the Philosophiæ Naturalis Principia Mathematica (1687) of Isaac Newton. In contrast to some earlier classical or medieval cosmologies, in which Earth rested at the center of universe, Newton conceptualized the Earth as a sphere in orbital motion around the Sun within an empty space that extended uniformly in all directions to immeasurably large distances. He then showed, through a series of mathematical proofs on detailed observational data of the motions of planets and comets, that their motions could be explained by a single principle of "universal gravitation" that applied as well to the orbits of the Galilean moons around Jupiter, the Moon around the Earth, the Earth around the Sun, and to falling bodies on Earth. That is, he asserted the equivalent material nature of all bodies within the Solar System, the identical nature of the Sun and distant stars and thus the uniform extension of the physical laws of motion to a great distance beyond the observational location of Earth itself.

Implications
Since the 1990s, observations assuming the cosmological principle have concluded that around 68% of the mass–energy density of the universe can be attributed to dark energy, which led to the development of the ΛCDM model.

Observations show that more distant galaxies are closer together and have lower content of chemical elements heavier than lithium. Applying the cosmological principle, this suggests that heavier elements were not created in the Big Bang but were produced by nucleosynthesis in giant stars and expelled across a series of supernovae and new star formation from the supernova remnants, which means heavier elements would accumulate over time. Another observation is that the furthest galaxies (earlier time) are often more fragmentary, interacting and unusually shaped than local galaxies (recent time), suggesting evolution in galaxy structure as well.

A related implication of the cosmological principle is that the largest discrete structures in the universe are in mechanical equilibrium. Homogeneity and isotropy of matter at the largest scales would suggest that the largest discrete structures are parts of a single indiscrete form, like the crumbs which make up the interior of a cake. At extreme cosmological distances, the property of mechanical equilibrium in surfaces lateral to the line of sight can be empirically tested; however, under the assumption of the cosmological principle, it cannot be detected parallel to the line of sight (see timeline of the universe).

Cosmologists agree that in accordance with observations of distant galaxies, a universe must be non-static if it follows the cosmological principle. In 1923, Alexander Friedmann set out a variant of Albert Einstein's equations of general relativity that describe the dynamics of a homogeneous isotropic universe. Independently, Georges Lemaître derived in 1927 the equations of an expanding universe from the General Relativity equations. Thus, a non-static universe is also implied, independent of observations of distant galaxies, as the result of applying the cosmological principle to general relativity.

Criticism
Karl Popper criticized the cosmological principle on the grounds that it makes "our lack of knowledge a principle of knowing something". He summarized his position as:
 * the "cosmological principles" were, I fear, dogmas that should not have been proposed.

Observations
Although the universe is inhomogeneous at smaller scales, according to the ΛCDM model it ought to be isotropic and statistically homogeneous on scales larger than 250 million light years. However, recent findings (the Axis of Evil for example) have suggested that violations of the cosmological principle exist in the universe and thus have called the ΛCDM model into question, with some authors suggesting that the cosmological principle is now obsolete and the Friedmann–Lemaître–Robertson–Walker metric breaks down in the late universe.

Violations of isotropy
The cosmic microwave background (CMB) is predicted by the ΛCDM model to be isotropic, that is to say that its intensity is about the same whichever direction we look at. Data from the Planck Mission shows hemispheric bias in 2 respects: one with respect to average temperature (i.e. temperature fluctuations), the second with respect to larger variations in the degree of perturbations (i.e. densities), the collaboration noted that these features are not strongly statistically inconsistent with isotropy. Some authors say that the universe around Earth is isotropic at high significance by studies of the cosmic microwave background temperature maps. There are however claims of isotropy violations from galaxy clusters, quasars, and type Ia supernovae.

Violations of homogeneity
The cosmological principle implies that at a sufficiently large scale, the universe is homogeneous. Based on N-body simulations in a ΛCDM universe, Yadav and his colleagues showed that the spatial distribution of galaxies is statistically homogeneous if averaged over scales of 260/h Mpc or more.

A number of observations have been reported to be in conflict with predictions of maximal structure sizes:
 * The Clowes–Campusano LQG, discovered in 1991, has a length of 580 Mpc, and is marginally larger than the consistent scale.
 * The Sloan Great Wall, discovered in 2003, has a length of 423 Mpc, which is only just consistent with the cosmological principle.
 * U1.11, a large quasar group discovered in 2011, has a length of 780 Mpc, and is two times larger than the upper limit of the homogeneity scale.
 * The Huge-LQG, discovered in 2012, is three times longer than, and twice as wide as is predicted possible according to these current models, and so challenges our understanding of the universe on large scales.
 * In November 2013, a new structure 10 billion light years away measuring 2000–3000 Mpc (more than seven times that of the Sloan Great Wall) was discovered, the Hercules–Corona Borealis Great Wall, putting further doubt on the validity of the cosmological principle.
 * In September 2020, a 4.9 σ conflict was found between the kinematic explanation of the CMB dipole and the measurement of the dipole in the angular distribution of a flux-limited, all-sky sample of 1.36 million quasars.
 * In June 2021, the Giant Arc was discovered, a structure spanning approximately 1000 Mpc. It is located 2820 Mpc away and consists of galaxies, galactic clusters, gas, and dust.
 * In January 2024, the Big Ring was discovered. It is located 9.2 billion light years away from Earth has a diameter of 1.3 billion light years or around the size of 15 full Moons as seen from Earth.

However, as pointed out by Seshadri Nadathur in 2013 using statistical properties, the existence of structures larger than the homogeneous scale (260/h Mpc by Yadav's estimation) does not necessarily violate the cosmological principle in the ΛCDM model (see ).

CMB dipole
The cosmic microwave background (CMB) provides a snapshot of a largely isotropic and homogeneous universe. The largest scale feature of the CMB is the dipole anisotropy; it is typically subtracted from maps due to its large amplitude. The standard interpretation of the dipole is that it is due to the Doppler effect caused by the motion of the solar system with respect to the CMB rest-frame.

Several studies have reported dipoles in the large scale distribution of galaxies that align with the CMB dipole direction, but indicate a larger amplitude than would be caused by the CMB dipole velocity. A similar dipole is seen in data of radio galaxies, however the amplitude of the dipole depends on the observing frequency showing that these anomalous features cannot be purely kinematic. Other authors have found radio dipoles consistent with the CMB expectation. Further claims of anisotropy along the CMB dipole axis have been made with respect to the Hubble diagram of type Ia supernovae and quasars. Separately, the CMB dipole direction has emerged as a preferred direction in some studies of alignments in quasar polarizations,  strong lensing time delay, type Ia supernovae, and standard candles. Some authors have argued that the correlation of distant effects with the dipole direction may indicate that its origin is not kinematic.

Alternatively, Planck data has been used to estimate the velocity with respect to the CMB independently of the dipole, by measuring the subtle aberrations and distortions of fluctuations caused by relativistic beaming and separately using the Sunyaev-Zeldovich effect. These studies found a velocity consistent with the value obtained from the dipole, indicating it is consistent with being entirely kinematic. Measurements of the velocity field of galaxies in the local universe show that on short scales galaxies are moving with the local group, and that the average mean velocity decreases with increasing distance. This follows the expectation if the CMB dipole were due to the local peculiar velocity field, it becomes more homogeneous on large scales. Surveys of the local volume have been used to reveal a low density region in the opposite direction to the CMB dipole, potentially explaining the origin of the local bulk flow.

Perfect cosmological principle
The perfect cosmological principle is an extension of the cosmological principle, and states that the universe is homogeneous and isotropic in space and time. In this view the universe looks the same everywhere (on the large scale), the same as it always has and always will. The perfect cosmological principle underpins steady state theory and emerges from chaotic inflation theory.

title: Cosmological principle


text:
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