text:
Banditti of the Plains

Banditti of the Plains is a book written by Asa Mercer about the Johnson County War in Wyoming, United States.

Publication history
The title of the book was taken from a now-forgotten Western titled The Banditti of the Range.

It is written from the perspective of Mercer, who had come to Wyoming to edit the North West Live Stock Journal, the official publication of the Wyoming Stock Growers Association (WSGA), which represented the monied interests intent on controlling the cattle industry. As the events of the range wars unfolded, Mercer came to sympathize with the homesteaders and turned against the WSGA. The book reflects his pro-settler view and is an important document of the events.

The first edition of the book is one of the rarities of western Americana. Immediately after it was printed, the Wyoming cattlemen objected and sued. The court ordered it destroyed. While the books were in the court's custody, a number were stolen and smuggled to Denver and later bound. For many years, the Wyoming Stockgrowers' Association, their sympathizers, and their descendants destroyed every copy they came across. Surviving copies are rare, and in 2016 copies were listed for prices between $1,250 and $6000. The second edition simply disappeared somewhere between Denver, Colorado, and Cheyenne, Wyoming, and the Wyoming Stock Growers Association was rumored to have hijacked and destroyed the second printing as it was being shipped.

Mercer saw his printing press in Cheyenne burned down; he was harassed, arrested, jailed and had the plates of his book destroyed. His writings were seized in the mails as obscenity; his business was ruined.

Mercer's main views of the Johnson County War were ultimately proved true, and never again did the cattle barons attempt to invade Wyoming.

Thanks to republication in 1954 by the University of Oklahoma Press, with a foreword by William H. Kittrell, the first edition text is now readily available, having appeared in paperback in 1976.

Pawn Stars appearance
In "Stick to Your Guns", a November 12, 2012, episode of the reality television series Pawn Stars, a visitor to the Gold & Silver Pawn Shop in Las Vegas brought in an 1884 .45 Colt revolver that belonged to Sheriff Fred Coats, which was accompanied by a copy of Banditti of the Plains. The seller pointed out that Page 140 of the book mentions that Coats was one of three men involved in an assault during the Wyoming Range Wars.

title: Banditti of the Plains


text:
List of mayors of Vancouver, Washington

The following is a list of mayors of Vancouver, Washington from 1858 on. The city of Vancouver was incorporated on January 23, 1857.

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

title: List of mayors of Vancouver, Washington


text:
Talk:Albania

Extended-confirmed-protected edit request on 19 June 2024
Please change BC to BCE and AD to CE. Modern scholarship prefers the far more inclusive BCE and CE versus BC and AD. This article uses both throughout so changing the date format to BCE and CE will also make the article more stylistically consistent. CuJoYYC (talk) 13:50, 19 June 2024 (UTC)
 * Red information icon with gradient background.svg Not done: While I share your preference for BCE/CE overall, see MOS:ERA for the applicable guidelines for Wikipedia articles. As of your writing, BC and BCE were both in use in the article; there was also one use of AD and none of CE. Looking at earlier revisions, I found there was a time when the article (see the revision at https://en.wikipedia.org/w/index.php?title=Albania&oldid=624794034, for example) had only one use of any of BC, BCE, AD, or CE, and it was BC. Therefore, in conformance with the guidelines, I settled on the BC/AD model as the one to follow for this article and changed all the instances of BCE (except in the title in one footnote, which would have been inappropriate to change) to BC. Largoplazo (talk) 14:35, 19 June 2024 (UTC)

Extended-confirmed-protected edit request on 3 July 2024
Seljonura (talk) 17:02, 3 July 2024 (UTC) Gdp per capita in Albania has rise in 10531$ due to latest census population change to 2402113 and Gdp is still 25+ billions of $

Not done: please provide reliable sources that support the change you want to be made. PianoDan (talk) 21:22, 3 July 2024 (UTC)

 * https://a2news.com/shqiperia/ekonomi/pakesimi-i-popullsise-i-ka-bere-shqiptaret-me-te-pasur-e-me-bo-i1123230 i don't know if you can translate it to english to understand Seljonura (talk) 15:57, 4 July 2024 (UTC)

Religion
When talking about religons in Albania don't refer to the censuses made in the country because they're all manipulated. Follow this structure about religions in Albania: ~58% Islam: ~40% Sunni and non-denominational ~18% Bektashism. ~38% Christianity: ~20% Eastern Orthodoxy ~17% Roman Catholicism ~1% Protestantism ~4% other (see Religion in Albania)

I hope I was clear 188.172.108.115 (talk) 10:54, 17 July 2024 (UTC)

title: Talk:Albania


text:
*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings $R$ and $A$, where $R$ is commutative and $A$ has the structure of an associative algebra over $R$. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution.

*-ring
In mathematics, a *-ring is a ring with a map $
 * : A → A$ that is an antiautomorphism and an involution.

More precisely, $$ is required to satisfy the following properties: for all $(x + y)* = x* + y*$ in $A$.

This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.

Elements such that $(x y)* = y* x*$ are called self-adjoint.

Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring.

Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: $1* = 1$ and so on.

*-rings are unrelated to star semirings in the theory of computation.

*-algebra
A *-algebra $A$ is a *-ring, with involution * that is an associative algebra over a commutative *-ring $R$ with involution $$, such that $(x*)* = x$.

The base *-ring $R$ is often the complex numbers (with $$ acting as complex conjugation).

It follows from the axioms that * on $A$ is conjugate-linear in $R$, meaning

for $x, y$.

A *-homomorphism $x* = x$ is an algebra homomorphism that is compatible with the involutions of $A$ and $B$, i.e.,
 * $x ∈ I ⇒ x* ∈ I$ for all $a$ in $A$.

Philosophy of the *-operation
The *-operation on a *-ring is analogous to complex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in complex matrix algebras.

Notation
The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:
 * $(r x)* = r x* ∀r ∈ R, x ∈ A$, or
 * $(λ x + μ y)* = λ x* + μ y*$ (TeX: ),

but not as "$λ, μ ∈ R, x, y ∈ A$"; see the asterisk article for details.

Examples
Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:
 * Any commutative ring becomes a *-ring with the trivial (identical) involution.
 * The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers $f : A → B$ where * is just complex conjugation.
 * More generally, a field extension made by adjunction of a square root (such as the imaginary unit √−1) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.
 * A quadratic integer ring (for some $D$) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings.
 * Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). None of the three is a complex algebra.
 * Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation.
 * The matrix algebra of $f(a*) = f(a)*$matrices over R with * given by the transposition.
 * The matrix algebra of $x ↦ x*$matrices over C with * given by the conjugate transpose.
 * Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
 * The polynomial ring $x ↦ x^{∗}$ over a commutative trivially-*-ring $R$ is a *-algebra over $R$ with $x∗$.
 * If $C$ is simultaneously a *-ring, an algebra over a ring $R$ (commutative), and $n × n$, then $A$ is a *-algebra over $R$ (where * is trivial).
 * As a partial case, any *-ring is a *-algebra over integers.
 * Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring.
 * For a commutative *-ring $R$, its quotient by any its *-ideal is a *-algebra over $R$.
 * For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by $n × n$ makes the original ring.
 * The same about a commutative ring $K$ and its polynomial ring $R[x]$: the quotient by $P *(x) = P (−x)$ restores $K$.
 * In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial.
 * The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties).
 * The group Hopf algebra: a group ring, with involution given by $(A, +, ×, *)$.

Non-Example
Not every algebra admits an involution:

Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra: $$\mathcal{A} := \left\{\begin{pmatrix}a&b\\0&0\end{pmatrix} : a,b\in\Complex\right\}$$

Any nontrivial antiautomorphism necessarily has the form: $$\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix}1&z\\0&0\end{pmatrix} \quad \varphi_z\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}$$ for any complex number $$z\in\Complex$$.

It follows that any nontrivial antiautomorphism fails to be involutive: $$\varphi_z^2\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}\neq\begin{pmatrix}0&1\\0&0\end{pmatrix}$$

Concluding that the subalgebra admits no involution.

Additional structures
Many properties of the transpose hold for general *-algebras:
 * The Hermitian elements form a Jordan algebra;
 * The skew Hermitian elements form a Lie algebra;
 * If 2 is invertible in the *-ring, then the operators $(r x)* = r (x*) ∀r ∈ R, x ∈ A$ and $ε = 0$ are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.

Skew structures
Given a *-ring, there is also the map $K[x]$. It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as $x = 0$, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where $g ↦ g^{−1}$.

Elements fixed by this map (i.e., such that $1⁄2(1 + *)$) are called skew Hermitian.

For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.

title: *-algebra


text:
Talk:Australian Open

Clarity
Hi. The sentence "(...) has been held for 100 years since 1905 (...)" is somewhat misleading, since it gives the impression that the tournament has been in fact held 100 times. That is not the case, as it was not held several times, during World War I and World War II, not to mention the year 1986, when it was not held in order for the 1987 edition to take place in January (although when calculating the actual number of editions, one must remember that it was held 2 times in one year: 1977). I suggest that the sentence in question be altered to: "was held for the first time in 1905, making it 100 years old as of 2005". Any objections and/or other suggestions? Regards, Redux 21:20, 29 Jan 2005 (UTC)

I wish to expand on Redux's talking points made 17 years ago. Specifically, the second paragraph in the History section needs some corrections:

Whilst it is true that a total of seven cities have hosted this particular tournament, the number of editions per city is quite off; when you tabulate all the figures, the total number of editions equates to only 98 when as of 2022, the count is at 110. This is due to the fact that the figure for Melbourne hasn't been updated for a while, it should read 66 (as of 2022.) This is also due to the fact that while the cited reference should be deemed credible for it is from that tournament's own website, it doesn't pass the verifiability test when looking specifically at each year the tournament was held in the city of Adelaide.

In the reference, it says that Adelaide hosted the tournament "14 times" without specifying the years. Upon verifying where each of the 110 editions was held, it turned out that Adelaide held it 15 times, not 14. The source for this figure is the Australian Open tournament page on the ATP website where anyone can just click on the "view draw" link for any given year where the hosting city is mentioned.

These are the complete results as of 2022

Melbourne (66) 1905, 1911, 1914, 1924, 1927, 1930, 1933, 1935, 1939, 1948, 1953, 1957, 1961, 1965, 1968, 1972, 1973, 1974, 1975, 1976, 1977, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022

Christchurch (1) 1906

Brisbane (7) 1907, 1915, 1923, 1956, 1960, 1964, 1969

Sydney (17) 1908, 1919, 1922, 1925, 1928, 1931, 1934, 1937, 1940, 1947, 1951, 1954, 1958, 1962, 1966, 1970, 1971

Perth (3) 1909, 1913, 1921

Adelaide (15) 1910, 1920, 1926, 1929, 1932, 1936, 1938, 1946, 1949, 1950, 1952, 1955, 1959, 1963, 1967

Hastings (1) 1912

Total:110

Still in the second paragraph, since the proper demonym was use to refer to the "five Australian cities", likewise, two New Zealander cities should be used as well as opposed to the current "two New Zealand cities."

Moving-on to the fifth paragraph. Whilst it is true that the tournament was moved to a brand new facility within Melbourne in 1988, it is not quite right to say that it moved to "Melbourne Park" in 1988 as that name was only introduced 8 years later in 1996. Changes should be made to reflect that.

That's it for now.

13lorem (talk) 02:47, 9 May 2022 (UTC)

Hana Mandlikova
Hana Mandlikova's victory in the 1987 Open was achieved previous to gaining her Australian citizenship and thus, she should not be considered the last Australian to have won the title.

2022 Novak Djokovic Vaccination Issue in the History Section
A contribution made to the History section regarding the recent 2022 Novak Djokovic Vaccination Issue made in the revision https://en.wikipedia.org/w/index.php?title=Australian_Open&oldid=1064047276 has been reverted on the grounds of no historical importance. However, my humble submission is against this reversal ; since the present issue and subsequent judicial decision is certainly going to make historical significance and hence, the reverted contribution should be accommodated. Xscontrib (talk) 04:39, 10 January 2022 (UTC)
 * Thanks for discussing. We cannot predict future significance and as of right now, this incident is trivial in the context of the history of the tournament. Should Djokovic ultimately be unable to defend, it may warrant a single sentence to that effect – but no more than that, with details being in the 2022 event article(s). wjematherplease leave a message... 17:31, 14 January 2022 (UTC)
 * I agree. We must avoid recentism. If Djokovic plays, nothing needs to be said. If he is prevented from playing over the vaccination issue, a single, brief sentence would be appropriate. HiLo48 (talk) 22:56, 14 January 2022 (UTC)
 * Now someone should consider writing that single, brief sentence to make it appropriate? Xscontrib (talk) 07:38, 16 January 2022 (UTC)
 * How about "Novak Djokovic was prevented from playing in the tournament after the Australian Minister for Immigration, Citizenship, Migrant Services and Multicultural Affairs, Alex Hawke, cancelled his visa, citing health and good order grounds", with this as a source? HiLo48 (talk) 09:01, 16 January 2022 (UTC)
 * fine Xscontrib (talk) 09:37, 16 January 2022 (UTC)
 * Still seems like excessive detail, and needs to be framed in the context of AO history. wjematherplease leave a message... 09:51, 16 January 2022 (UTC)
 * We have seen effects of falsifying details and denying/expressing reluctance towards scientific process by celebrities on vaccine roll-out. IMHO This event shall not be seen as one watertight incident, exclusive to sports, but a historical event which will shape upcoming days ; how popular persons decide personal benefits over a greater good. Xscontrib (talk) 10:06, 16 January 2022 (UTC)
 * How is this relevant here, specifically to the AO? wjematherplease leave a message... 10:28, 16 January 2022 (UTC)
 * While I agree with the sentiment of Xscontrib's comment, such speculative content doesn't belong in this article. I also agree that we don't want excessive detail regarding this matter, and am very open to constructive suggestions. The problem is that Djokovic is the reigning champion, so I believe we need to say something. HiLo48 (talk) 11:09, 16 January 2022 (UTC)
 * It really just belongs in the 2022 edition of the event. Fyunck(click) (talk) 11:13, 16 January 2022 (UTC)

wheelchair and quad sections
my level of knowledge and interest in tennis is close to zero; however, there really should be details on this page of the other competitions in this tournament. Many other pages on wheelchair and quad competitions and players link to this page.

As an example, Dylan Alcott, who has won not only the past two Australian Opens in the men's quad, but also a multiple Grand Slam winner is not mentioned in this page at all. 121.200.5.19 (talk) 07:18, 20 January 2022 (UTC)

Current champions (2022 edition)
This section is a little ambiguous. Ash Barty has four Grand Slam titles (3 singles, 1 doubles), Barbora Krejčíková has eight Grand Slam titles (1 singles, 4 doubles, 3 mixed doubles), Kristina Mladenovic has eight Grand Slam titles (5 doubles, 3 mixed doubles) and Ivan Dodig has six Grand Slam titles (2 doubles, 4 mixed doubles). Should this be updated to include the collective number of their grand slam titles or clarified that the number refers to the particular category in which they won their 2022 title? — Preceding unsigned comment added by Anookara (talk • contribs) 03:35, 31 January 2022 (UTC)

How wonderfully glorious and positive.
Overnight a piece of text was removed from this article with the Edit summary "Removed out of place protest information". Before even knowing what the information was, I immediately saw something wrong with that. If it was out of place, it should be moved to the right place, not simply removed. Then I looked for the right place, but there isn't one. There is nowhere in the article describing anything negative associated with the event whatsoever, which is ridiculous!

There's nothing about Novak's vaccination issue, Margaret Court's controversial views, the taking away of the land where Melbourne Park is now from its previous use, and no doubt lots of other things I'm not aware of.

I'm not a fan of big "Controversies" sections in articles, but this article right now really is a whitewash. HiLo48 (talk) 00:13, 11 June 2023 (UTC)

title: Talk:Australian Open


text:
Isaac Murphy

Isaac Murphy (October 16, 1799 or 1802 – September 8, 1882) was a native of Pennsylvania, a teacher and lawyer who moved to Fayetteville, Arkansas with his wife and child in 1834. He continued to teach and also became active in politics. Murphy is best known as the only delegate to have repeatedly voted against secession at the second Arkansas Secession Convention in 1861.

Having served in local offices and the State House, Murphy was elected as governor in a special election after the US Army occupied Arkansas in 1863 under President Abraham Lincoln. He is considered the first Reconstruction governor of Arkansas, as he was allowed to stay in office after the Reconstruction Acts were passed by Congress in 1867. Murphy was known for his fiscal restraint and a conciliatory attitude toward former Confederates.

Early life and education
Murphy was born near Pittsburgh, Pennsylvania to a wealthy paper manufacturer and his wife. He was educated locally and at Washington College (now Washington & Jefferson College) in Washington, Pennsylvania. He was admitted to the bar in Allegheny County, Pennsylvania on April 29, 1825.

In 1830, Murphy moved to Clarksville, Tennessee, where he taught school. There he met and wed Angelina Lockhart on July 31, 1830. Her father so opposed the marriage when he learned that Murphy favored the abolition of slavery that he disinherited Angelina.

In 1834, the Murphys, with their newborn daughter, moved west to Fayetteville in the Arkansas Territory. In Fayetteville, Murphy established himself as a school teacher, surveyor, and lawyer. The territory was admitted as a state in 1836.

Career
Murphy became active in politics, elected as the first county treasurer of Washington County in 1836, and serving for two years. He was appointed as a master in chancery in 1841. From 1837 to 1838, Murphy ran the original government land lines for Franklin County, Arkansas.

On November 30, 1844, the noted Indian Missionary Cephas Washburn, along with Murphy and other leaders, secured a charter for a college known as the Far West Seminary. Murphy taught both whites and Indians at the seminary, intended for young men. Murphy taught here until the building was destroyed by fire on February 17, 1845, putting him in debt, as he had invested in the school.

In 1846 Murphy was elected to the Arkansas House of Representatives for Washington County, and re-elected in 1848. With assignment to the Banking Committee, he attempted to introduce reforms but was stymied by the powerful political cabal known as "The Family".

Murphy ran into financial difficulties about 1849 and left for California to seek wealth in the California Gold Rush. Among the many who did not succeed, he returned to Arkansas in 1854 with nothing to show for his efforts.

Upon his return he moved to Huntsville in Madison County, Arkansas. His daughter Mrs. Mary Lowe Pierson of Washington County had been hired to teach at a new female seminary in Huntsville, the Pleasant View Female Seminary. Murphy and two more daughters were hired to assist in its operations.

In 1856, Murphy was elected to the State Senate representing Madison and Benton counties, to succeed the late senator John Berry. Murphy's eldest daughter Malilla married James R. Berry, one of the senator's sons. Northwest Arkansas was a Unionist stronghold in the years before the Civil War.

Secession convention
When the secession crisis swept the State in February 1861, Murphy was elected by 85% on a Unionist platform to represent Madison County at the Secession Convention; his county voted to remain in the Union. When Fort Sumter, South Carolina, was fired on and President Abraham Lincoln called for troops from Arkansas, the delegates of the Secession Convention were recalled, before the planned statewide referendum on secession could be held. The convention voted to take Arkansas out of the Union, but Murphy and four other delegates opposed this step. The convention chair called on the five to switch their votes. All four of the other "nay" delegates changed their votes, but Murphy refused. Initially his position was popular in Huntsville, but as the war went on, Confederate sentiment increased.

Governorship
As war broke out, Murphy fled his home in Huntsville; he spent much of the war traveling with the Union army in northwestern Arkansas. Following the fall of the capital city of Little Rock to the Union in 1863, Arkansas' Confederate government, led by Governor Harris Flanagin, went into exile.

In a special election, held with the approval of President Lincoln, Isaac Murphy was elected as governor of Arkansas in 1863.

Reconstruction era
Murphy worked to try to heal the war wounds in Arkansas, even as the war continued in the southern parts of the state. He worked for balance and said publicly that "We have all done wrong." The 4th of July celebrations in Little Rock were led by pro-Union speakers who refrained from anti-Southern speeches or actions.

After Lincoln's assassination and the actions of numerous state legislatures to control freedmen and limit their rights, Republicans in Congress began advocating stricter Reconstruction for the former Confederate states. In 1866, pro-Confederate legislators won majorities in several southern states. That same year, white violence against former slaves broke out in several states.

In response, Republicans in Congress pushed through the 14th Amendment, granting full citizenship, rights and due process to freedmen, and the Reconstruction Acts of 1867. The rebel states were divided into military districts, to be controlled by US Army forces until the states passed new state constitutions protecting the civil rights of former slaves and accepting the 14th Amendment. Murphy was allowed to stay in office, but he was criticized by both sides.

When Murphy left office, his administration had a budget surplus. It had started with no funds. With no initiatives passed by the Reconstruction legislature to provide for public welfare and education, as well as investment in infrastructure, this surplus was diverted to public projects.

Death and legacy
Murphy returned to Huntsville and took up farming and practicing law. He lived a quiet life with his family. On September 8, 1882, Murphy died unexpectedly at his home. He is buried in Huntsville Cemetery in Huntsville.

In 1974, historian John I. Smith published several articles about what he called the Huntsville Massacre, an execution of prisoners of war in 1862. He said that Murphy had been implicated in those deaths. A memorial to those murdered in Huntsville was erected, dedicated on September 30, 2006.

title: Isaac Murphy


text:
Talk:French Open

French Open being the only Grand Slam tournament held on clay
This should be qualified with "... only current Grand ...". Note: US Open has also been played on clay. Antipodenz (talk) 02:25, 15 September 2021 (UTC)
 * Good point. Noted. oknazevad (talk) 02:28, 15 September 2021 (UTC)

Point distribution for qualifiers
The table lists allocation of prize money for qualifiers but not points awarded for qualifiers (this became more apparent with Raducanu's win at the US Open). Antipodenz (talk) 02:36, 15 September 2021 (UTC)

French Open events
The introduction discusses when the mixed doubles and women's doubles were introduced but not about other events at the Open (boys/girls, wheelchair etc.). These are part of the official tournament so should be given due recognition, including in the table that includes current events/champions. Antipodenz (talk) 02:41, 15 September 2021 (UTC)

Tournoi de France winners
The history section states that It was suspended in 1940 then Bernard Destremau won the first two while Yvon Petra won three in a row from 1942-45. The date she don't line up but the fact is Destremau won 1941-1942 and Petra 1943-45. Antipodenz (talk) 11:12, 25 September 2021 (UTC)

Ref #4
The text states: "these results are not recognised by ... other major international organisations", but the ref doesn't state this. So what organisations are they? Further the ref does state that: "although that it is true that the wartime tournaments were limited to French players" but the Wikipedia article for "Tournoi de France" makes clear that international participation did occur (subsequent articles agree that foreign nationals from Belgium, Switzerland and Luxembourg participated e.g. Tignor, Steve; Jun 05 2020: Tournoi de France winners channeled champion courage into WWII combat). Also the period is generally referred to as being under Nazi occupation but by the time of the 1945 tournament Paris had been liberated for about 10 months. Antipodenz (talk) 11:37, 25 September 2021 (UTC)

Title outdated
I find it strange that the title of this article is “French Open”. I have never heard anyone name this tournament like that. On their official website, it is always named "Roland Garros". It has the same type of name as Wimbledon. If we keep this title logic, we should call Wimbledon "British open" and that is obviously strange. So, I propose to update this title to "Roland Garros" or "Roland Garros Championships" or something similar for this article and its derivatives. 70.27.246.205 (talk) 21:20, 5 June 2023 (UTC)

 * Tertiary English-language sources overwhelmingly use the French Open name. See prior discussions in the talk page archives. oknazevad (talk) 01:57, 6 June 2023 (UTC)

 * It’s also noteworthy that the RG website in a lengthy article explains the official title is French Open. I think it is important that this does not descend into an editing war. 2A00:23C8:C11B:2B01:2D4F:5304:E00C:AA5F (talk) 14:54, 26 May 2024 (UTC)
 * I'm fine with the "French open" title now. I didn't realised at the time that this name was widely use in english. Therefore I don't understand the first article you cite in your reply. It's a french short article that never name the tournament "French open" in it.Virkin29 (talk) 13:33, 27 May 2024 (UTC)

Mistake counting editions number
All is wrong from 1946 onward 100Castor (talk) 12:52, 7 December 2023 (UTC)
 * 1939 French Championships (tennis) is the 44th, this is right.
 * 1946 French Championships (tennis) is the 50th, this is wrong. It's the 45th.
 * While it's a little grey, since there were French Championships held at Roland Garros from '41-'45 (and in '45 it wasn't occupied), the tournament itself does not recognize those years. It recognizes the years prior to it being a major (1891-1924), the amateur era 1925-1939, 1946-1967) and the Open Era (1968-present). That's it. per sources you are correct and the edition numbers should reflect that fact. Fyunck(click) (talk) 21:30, 7 December 2023 (UTC)

title: Talk:French Open


text:
Talk:Stefan Wul

Untitled
I don't know if any of his novels have been translated, therefore I left the French titles. → SeeSchloß 18:49, 10 Aug 2003 (UTC)

title: Talk:Stefan Wul


text:
User talk:Phys

Hi, I read the article about lie algebra and saw your latex compiled image: http://en.wikipedia.org/wiki/Image:Liealgebra.png which is a bit too small. could you recompile it in a bigger version? --phatsphere 19:24, 28 June 2007 (UTC)

Hi. I was wondering what all of these pages with F4 and G2 are for? They don't really add anything, and it seems they could all be covered in a single page called "F (math)" without using up potentially useful links. For instance Apple releases its processor with the names G3, G4 and G5, and these would be much more commonly asked for than a math page.

G2, F4, E6, E7 and E8 are very commonly used Lie groups in math and physics. Maybe I should rename them G2 (math) and F4 (math)? Phys

Yes please. You can use the Move this page function to do this. Still, wouldn't a single page with lie groups and various examples be more than enough? And if not, could you please state at the top of the articles that G4 is one of the common lie groups... so we know what it is. This isn't a math textbook!

Huh, there's no G4! Only a G2 and a F4! Phys

Hi there. Good work. Just one point. Avoid using Math. That is a word used only in American English. Otherwise people who don't use AE will end up changing them to the international word Maths and on such minor points as an s edit wars break out! :-( Use the full Mathematics instead which is universal. FearÉIREANN 03:10, 11 Aug 2003 (UTC)

Hello! You are adding new articles very quickly. But could you slow down a bit please? How about adding your articles to List of mathematical topics? Or list your contributions in your user page? It is hard to improve your articles if nobody really knows those articles have been created. wshun 23:52, 11 Aug 2003 (UTC)

Hi. I like your contributions to the Mathematical Physics side of Wikipedia a lot. Keep it up! -- Miguel

I'm not convinced that root system should be merged with Lie group. Root systems parametrize a lot of different things besides simple Lie groups and simple Lie algebras. Sometimes with omissions, sometimes with extra markings, they parametrize simple singularities in algebraic geometry, quivers with finitely many indecomposables, finite groups of Lie type, rigid Lagrangian singularities, regular polytopes, affine Kac-Moody Lie algebras, etc. They have an independent existence...-Michael Larsen

Was the move of Technicolor all that wise? Most people are aware of the term in association with film, not so physics. Given the large number of links to technicolor from film pages, it might have been better to leave it as it is, and add a link to the minor usage on the Technicolor page. Otherwise, you'll have to go through and edit all of the links from the films. GRAHAMUK 02:37, 14 Aug 2003 (UTC)

 * I've now done this - no need to change any links, or mess about withis any longer. PLEASE!! GRAHAMUK 02:49, 14 Aug 2003 (UTC)

 * Oops, I'm sorry! I didn't realize you reverted! I changed the links because you asked me to!Phys

 * Yeah, sorry about that!! I saw you making the change so I figured I'd fix it in case you hadn't read this in time - oh well, never mind. I think in general disambiguationpages should be avoided unless there really are a lot of terms that come under one title. In this case, as there are only two and one is (with respect) pretty esoteric, I think this approach is the better one. I'll help revert the film links if you want GRAHAMUK

I don't understand your comment on Cayley's theorem in the adjoint representation entry. Could you elaborate or remove it? Michael Larsen

Your comment on Cayley's theorem belongs in a different article, the not yet written one about regular representations. Perhaps you would like to write it? Michael Larsen

Considering that Dual (category theory) is a single line, is there any reason it's not a part of category theory? No one is going to look for an article with this name. User:Maury Markowitz

 * Well, it's on the request list and it's extendable... But if you still insist, I can merge that article. Phys 15:16, 4 Sep 2003 (UTC)

Hi, thanks for correcting my totally wrong definition on Dirac sea (but I'm quite sure that I have seen described as I wrote somewhere...). Anyway, the current version reads like a school textbook question/answer. I think a more "formal" style should be employed - no need for equations, just straightforward definitions. At18 20:01, 5 Sep 2003 (UTC)

Apparently you haven't read the talk page on Liouville's theorem (Hamiltonian) since the time when you created the talk page. Did you intend the differential equation in that article to be the statement of the theorem rather than a corollary of the theorem? If so, that is anything but clear from the way you wrote it. I would never have suspected it if not for your comment on the talk page, and it's still only a suspicion. Michael Hardy 03:22, 20 Sep 2003 (UTC)

 * Yes, that's the theorem, not the corollary!

The page is a mess, you know.

Charles Matthews 08:23, 9 Nov 2003 (UTC)

Hello. In relatively complemented, what operation on members of a lattice are you denoting by $$x\Rightarrow y$$? I'd have interpreted it in such a way that if you're talking about the lattice of all subsets of a set, then it's the intersection of x with the complement of y. But that does not seem to be what you meant. Also, could you address the question I put on the Heyting algebra discussion page. Michael Hardy 03:34, 9 Nov 2003 (UTC)

-

Not quite sure I understand your latest change to von Neumann algebra. AFAICS, "commutative" already was in every place it should be, and definitely shouldn't be where you put it. Connections between noncommutative von Neumann algebras and measure spaces exist, they just aren't as simple as going from a nice measure space to L∞. So, since this is an aesthetic issue, and I really think we should mention a connection between (possibly non-commutative) von Neumann algebras and measure theory as an explanation for "noncommutative measure theory", I think I'd like to just revert your change.

Prumpf 19:41, 12 Nov 2003 (UTC)

- Hi, I'm currently fleshing out coalgebra and was wondering if you still have the source code for ? In the left diagram, the Nabla's should be Delta's, and the whole diagram is kind of hard to read since it is so small. Also, the diagonal arrow in the right one would look better if you used anti-aliasing when converting from postscript (I assume you used TeX?) to png. Could you upload a new version if you get the chance? Cheers, AxelBoldt 16:16, 11 Aug 2004 (UTC)

 * Oops, you're right. Thanks for catching that. I'll see if I can still find the source code. Phys 18:06, 11 Aug 2004 (UTC)

quantum groups (Hopf algebras page)
Hi, I always considered quantum groups to be a particular class of Hopf algebras. Quantum groups are deformations of function algebras on groups, or their duals. For example, a group algebra of a finite group is a Hopf algebra, but is it a quantum group? Anya

 * Well, some people DEFINE a quantum group to be a Hopf algebra, but I guess that's not a universal convention. Phys 19:09, 15 Aug 2004 (UTC)

 * Actually, I've also seen the convention that only the deformations of certain Hopf algebras are termed quantum groups; there's a whole book about "Quantum groups" that doesn't have a single clean definition of a quantum group, it just goes over several examples (all Hopf algebras with a certain q in them...). AxelBoldt 12:42, 18 Aug 2004 (UTC)

Re Talk:Concrete category: would you have an example of a category that doesn't admit a faithful functor to Set? Thanks, AxelBoldt 12:42, 18 Aug 2004 (UTC)

Any category that's not locally small? Michael Lee Baker (talk) 17:43, 6 March 2015 (UTC)

Sorry, but I have to comment on this. Redirecting jet bundle to representation of the diffeomorphism group is really quite an annoying thing to do. It is not as if that page made any attempt to explain what a jet bundle is. Charles Matthews 09:06, 22 Aug 2004 (UTC)

Article Licensing
Hi, I've started a drive to get users to multi-license all of their contributions that they've made to either (1) all U.S. state, county, and city articles or (2) all articles, using the Creative Commons Attribution-Share Alike (CC-by-sa) v1.0 and v2.0 Licenses or into the public domain if they prefer. The CC-by-sa license is a true free documentation license that is similar to Wikipedia's license, the GFDL, but it allows other projects, such as WikiTravel, to use our articles. Since you are among the top 1000 Wikipedians by edits, I was wondering if you would be willing to multi-license all of your contributions or at minimum those on the geographic articles. Over 90% of people asked have agreed. For More Information:
 * Multi-Licensing FAQ - Lots of questions answered
 * Multi-Licensing Guide
 * Free the Rambot Articles Project

To allow us to track those users who muli-license their contributions, many users copy and paste the " " template into their user page, but there are other options at Template messages/User namespace. The following examples could also copied and pasted into your user page:

 * Option 1
 * I agree to multi-license all my contributions, with the exception of my user pages, as described below:

OR
 * Option 2
 * I agree to multi-license all my contributions to any U.S. state, county, or city article as described below:

Or if you wanted to place your work into the public domain, you could replace " " with "  ". If you only prefer using the GFDL, I would like to know that too. Please let me know what you think at my talk page. It's important to know either way so no one keeps asking. -- Ram-Man (comment| talk)

Unverified images
Hi! Thanks for uploading the following image:

 * Image:Diag.png

I notice it currently doesn't have an image copyright tag. Could you add one to let us know its copyright status? (You can use if you release it under the GNU Free Documentation License,  if you claim fair use, etc.) If you don't know what any of this means, just let me know at my talk page where you got the images and I'll tag them for you. Thanks so much. [[User:Poccil|Peter O. (Talk, automation script)]] 05:26, Dec 11, 2004 (UTC)

P.S. You can help tag other images at User:Yann/Untagged_Images. Thanks again.

Assuming you created Image:Algebra.png, can you tag that as well? --Ricky81682 (talk) 07:27, Dec 27, 2004 (UTC)
 * Thanks for the quick response! --Ricky81682 (talk) 07:49, Dec 27, 2004 (UTC)

Punctuation
Hi there. Having nothing better to do, I am now leading a Crusade about style in math articles. Very often I encounter formulas which are at the end of a sentence. Then, I think one should put period at the end of formula, because the sentence ends threre. Anyway, you did not put period at the end of a formula when you contributed to Hamilton-Jacobi-Bellman equation. It was just several occurences, and it clearly was not a big deal, however, I would like to mention this to you, because it is easier to notify people about something than to always fix things long after. Happy editing! Oleg Alexandrov 23:19, 9 Jan 2005 (UTC)

 * Oleg, happened to see this comment. FYI, Paul Halmos, that noted expositor of mathematics, considered the same problem and concluded that one should not add periods after in-line formulas, and many more or less faux-mathematicians, including me, follow his lead.  Phys, I hope you will go with Halmos on this! ---CH  [[User_talk:Hillman|(talk)]] 02:30, 21 November 2005 (UTC)

VFD
Please read the instructions for VFD (at the bottom of WP:VFD or at Template:VfDFooter) before nominating another page for deletion. —Ben Brockert (42) UE News  01:20, Jan 21, 2005 (UTC)

WKB Approximation
On WKB approximation, I am not sure what you're trying to say in the paragraph that you added. In my opinion, a discussion of Feynman diagrams is unnecessary in an article on WKB, but I'd be happy if you proved me otherwise. It's just that right now the tie-in is rather incomprehensible to me and the latest addition makes it sound like you're arguing with yourself over it. I think it either needs to be more clearly explained or removed.Laura Scudder 18:18, 10 Feb 2005 (UTC)

Image sources
I saw your latex compiled picture in Lie algebra and I'm wondering whether you could attach sources to the respective talk pages when you add new pictures. That would make it easier for others to provide updates and would certainly inspire others when creating their own images. Rl 19:24, 20 Feb 2005 (UTC)

Creation and annihilation operators
Just thought I'd drop by and say "hi"! I'm just wondering what other material you plan to add to the creation and annihilation operators page...I don't have much else to add, except maybe stuff on vibronic coupling. Other than that, it's mainly cleanup and rewording to make the concepts more accessible to a larger number of readers. What do you think? I'd also like to convert some of the equations to nicely formatted mathematical equations, but I don't know how to do that. Maybe you have an idea?

I read the comment you left on the discussion page. I assume it was for me, since it looks like we're the only two people actively changing the content on that page (judging by its history). If you feel the need to edit/delete some of my changes, please feel free to do so! :) My perspective is indeed primarily quantum chemistry, so most of my ideas will be centered around the ladder operators. I added those concepts in because I felt that they would help make the article progress towards the more powerful formalism that you've presented with the *-algebras and Hilbert spaces where quantum field theory resides.

If you have time, could you comment on the little section on "notational caveats" I added at the bottom of the page? I think it's relevent to point out the ambiguity in the notation here, as it can be very confusing. Thanks for your input! HappyCamper 23:42, 3 Apr 2005 (UTC)

Public domain image problems
Hi there! I noticed that you have placed some of your image contributions in the public domain. Unfortunately, it turns out that one can't place their work in the public domain, and so these image tags have been deprecated. I encourage you to replace the public domain tags on your images with free-use tags, which are similar in spirit, such as CopyrightedFreeUse-User. Please reply on my talk page or Wikipedia talk: You can't grant your work into the public domain if you have any questions. Thanks. Deco 23:45, 8 Apr 2005 (UTC)

Broken Link in ADS
The folowing page has what appears to have been a link to an image. However, it is a link to upload an image. Is this correct?

The Page: http://en.wikipedia.org/wiki/Anti_de_Sitter_space

The link: "Image:AdS A.png"

Softcafe 17:40, 17 July 2005 (UTC)SoftCafe

Proper use of TeX
Please don't use "<" and ">" for angle-brackets in TeX. TeX (even in its limited form used on Wikipedia) is sophisticated and has provided for this occasion.

Right:
 * $$\langle \alpha^\mu_\tau(A)B\rangle=\langle B\alpha^\mu_{\tau+i\beta}(A)\rangle$$

Wrong:
 * $$< \alpha^\mu_\tau(A)B>

=< B\alpha^\mu_{\tau+i\beta}(A)>$$

Notice also:

 * $$\left\langle\begin{matrix}n \\ k\end{matrix}\right\rangle$$

but

 * $$\langle\begin{matrix}n \\ k\end{matrix}\rangle$$

Thus you need to use \left and \right to get the right sizes. Michael Hardy 01:48, 16 October 2005 (UTC)

Critical point picture
Hello Phys,

The phase diagram is a nice addition to the critical point page. Can you add a quick note to the talk page giving the details of where the picture comes from? Seems like a good idea for non-user-generated content.

More generally, I wonder if it could also be a nice addition to the phase transition page, which could potentially contain a few diagrams, e.g. the solid-liquid-gas one you provide, but also stuff for Ising model, percolation, Hopfield networks, etc. What do you think?

Best wishes, — WebDrake 10:23, 21 October 2005 (UTC)

 * The diagram was created by CYD for another article in Wikipedia. You might want to ask him instead. I was under the impression that any image uploaded to Wikipedia can be used in any other Wikipedia article. Phys 21:15, 21 October 2005 (UTC)

 * Yeah, sure, if it can be used one place in WP it can be used in another, it's all covered by the GFDL. I just think that if you add something significant like that to an article you should mention on the talk page where you got it from ("Picture taken from article... etc.)  Thanks! WebDrake 16:45, 22 October 2005 (UTC)

Question about QCD strings
Are QCD strings the same objects that appear in the Lund string model, or just similar ones? Thanks, SCZenz 16:24, 24 October 2005 (UTC)

 * Probably. QCD strings are theoretical whereas Lund strings are phenomenological. Phys 17:39, 24 October 2005 (UTC)

Radion
Hi, Phys, forgive me if I have already asked you this but forgotten that I asked you, but can you add a citation to Sakharov for this article, which it seems you started as a stub? TIA ---CH (talk) 00:54, 7 November 2005 (UTC)

 * I wasn't aware that Sakharov worked on radions. Phys 15:58, 7 November 2005 (UTC)
 * And no, I didn't start this article as a stub. Phys 16:00, 7 November 2005 (UTC)

AF spacetime
Actually, I previously removed Category:General relativity and added Category:Lorentzian manifolds to emphasize that this is not a concept in general relativity, but rather a concept in Lorentzian manifolds. Eventually, one would hope to have enough discuss of say Brans-Dicke theory so that we use the term in connection with a Brans-Dicke solution which is not a gtr solution. I guess I'll have to let this stand since you are not the first to complain. ---CH (talk) 02:26, 13 November 2005 (UTC)

 * I just noticed that you asked in an edit line for the source of the quotation. I don't know who pasted that in, but I know where it came from.  Arghgh!---until earlier today, the entire content of this miserable article was a verbatim quotation of the abstract of the on-line review article by Frauendiener!  I added some citations (including F's article) for the benefit of anyone who would like to turn this page into an honest encyclopedia article.  (Probably urgent because right now it is a plagiarized paragraph followed by a warning by me that the article consists of a plagiarized paragraph, followed by the citations, which probably won't be acceptable for very long.) ---CH  [[User_talk:Hillman|(talk)]] 02:26, 21 November 2005 (UTC)

Spin tensor
Thanks for your discussion of spin tensor in continuum mechanics.

I am wondering if you can provide reference for your discussion of this. I have spent some time myself wondering about spin density in contiuum mechanics, especially in turbulence. However, almost universally one encounters the statement that the stress tensor must be symmetric. I would be greatly interested if you could point me to references where the stress tensor is *not* symmetric, especially in turbulence theories.

Thanks, Petwil 20:01, 21 December 2005 (UTC)

E9 Lie algebra
Hi, don't know if your still active on Wikipedia. Anyway back in november last year you added a rquest for E9 - lie algebra. I'm fairly sure it does not actually exist as E8 is the largest of the exceptional Lie groups/algebras. Do you have any references for its exsistance. If I don't hear back from you I'll remove it from Requested articles/Mathematics along with 3j symbol, 6j symbol,	9j symbol,	10j symbol,	15j symbol,	Chevalley basis, E9, E10. --Salix alba (talk) 15:21, 31 March 2006 (UTC)

Euclidean relations
In Equivalence relation, you wrote (in effect) that a Euclidean relation is a binary relation satisfying a~c, b~c -> a~b. Do you have a reference for this? Not that I doubt you, but I want to cite it elsewhere. (BTW, you also wrote that an equivalence relation is just a relation that is reflexive, symmetric, and Euclidean. But in fact, any reflexive Euclidean relation is necessarily symmetric already!: just plug in a for c.) --Toby Bartels 12:11, 31 May 2006 (UTC)

Rank of a tensor
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Proposed Image Deletion
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ArbCom elections are now open!
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title: User talk:Phys


text:
Antilinear map

In mathematics, a function $$f : V \to W$$ between two complex vector spaces is said to be antilinear or conjugate-linear if $$\begin{alignat}{9} f(x + y) &= f(x) + f(y) && \qquad \text{ (additivity) } \\ f(s x) &= \overline{s} f(x) && \qquad \text{ (conjugate homogeneity) } \\ \end{alignat}$$ hold for all vectors $$x, y \in V$$ and every complex number $$s,$$ where $$\overline{s}$$ denotes the complex conjugate of $$s.$$

Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity.

Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces.

Definitions and characterizations
A function is called or  if it is additive and conjugate homogeneous. An on a vector space $$V$$ is a scalar-valued antilinear map.

A function $$f$$ is called if $$f(x + y) = f(x) + f(y) \quad \text{ for all vectors } x, y$$ while it is called if $$f(ax) = \overline{a} f(x) \quad \text{ for all vectors } x \text{ and all scalars } a.$$ In contrast, a linear map is a function that is additive and homogeneous, where $$f$$ is called if $$f(ax) = a f(x) \quad \text{ for all vectors } x \text{ and all scalars } a.$$

An antilinear map $$f : V \to W$$ may be equivalently described in terms of the linear map $$\overline{f} : V \to \overline{W}$$ from $$V$$ to the complex conjugate vector space $$\overline{W}.$$

Anti-linear dual map
Given a complex vector space $$V$$ of rank 1, we can construct an anti-linear dual map which is an anti-linear map $$l:V \to \Complex$$ sending an element $$x_1 + iy_1$$ for $$x_1,y_1 \in \R$$ to $$x_1 + iy_1 \mapsto a_1 x_1 - i b_1 y_1$$ for some fixed real numbers $$a_1,b_1.$$ We can extend this to any finite dimensional complex vector space, where if we write out the standard basis $$e_1, \ldots, e_n$$ and each standard basis element as $$e_k = x_k + iy_k$$ then an anti-linear complex map to $$\Complex$$ will be of the form $$\sum_k x_k + iy_k \mapsto \sum_k a_k x_k - i b_k y_k$$ for $$a_k,b_k \in \R.$$

Isomorphism of anti-linear dual with real dual
The anti-linear dual pg 36 of a complex vector space $$V$$ $$\operatorname{Hom}_{\overline{\Complex}}(V,\Complex)$$ is a special example because it is isomorphic to the real dual of the underlying real vector space of $$V,$$ $$\text{Hom}_\R(V,\R).$$ This is given by the map sending an anti-linear map $$\ell: V \to \Complex$$to $$\operatorname{Im}(\ell) : V \to \R$$ In the other direction, there is the inverse map sending a real dual vector $$\lambda : V \to \R$$ to $$\ell(v) = -\lambda(iv) + i\lambda(v)$$ giving the desired map.

Properties
The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps.

Anti-dual space
The vector space of all antilinear forms on a vector space $$X$$ is called the of $$X.$$ If $$X$$ is a topological vector space, then the vector space of all  antilinear functionals on $$X,$$ denoted by $\overline{X}^{\prime},$  is called the  or simply the  of $$X$$ if no confusion can arise.

When $$H$$ is a normed space then the canonical norm on the (continuous) anti-dual space $\overline{X}^{\prime},$ denoted by $\|f\|_{\overline{X}^{\prime}},$  is defined by using this same equation: $$\|f\|_{\overline{X}^{\prime}} ~:=~ \sup_{\|x\| \leq 1, x \in X} |f(x)| \quad \text{ for every } f \in \overline{X}^{\prime}.$$

This formula is identical to the formula for the on the continuous dual space $$X^{\prime}$$ of $$X,$$ which is defined by $$\|f\|_{X^{\prime}} ~:=~ \sup_{\|x\| \leq 1, x \in X} |f(x)| \quad \text{ for every } f \in X^{\prime}.$$

Canonical isometry between the dual and anti-dual

The complex conjugate $$\overline{f}$$ of a functional $$f$$ is defined by sending $$x \in \operatorname{domain} f$$ to $\overline{f(x)}.$ It satisfies $$\|f\|_{X^{\prime}} ~=~ \left\|\overline{f}\right\|_{\overline{X}^{\prime}} \quad \text{ and } \quad \left\|\overline{g}\right\|_{X^{\prime}} ~=~ \|g\|_{\overline{X}^{\prime}}$$ for every $$f \in X^{\prime}$$ and every $g \in \overline{X}^{\prime}.$ This says exactly that the canonical antilinear bijection defined by $$\operatorname{Cong} ~:~ X^{\prime} \to \overline{X}^{\prime} \quad \text{ where } \quad \operatorname{Cong}(f) := \overline{f}$$ as well as its inverse $$\operatorname{Cong}^{-1} ~:~ \overline{X}^{\prime} \to X^{\prime}$$ are antilinear isometries and consequently also homeomorphisms.

If $$\mathbb{F} = \R$$ then $$X^{\prime} = \overline{X}^{\prime}$$ and this canonical map $$\operatorname{Cong} : X^{\prime} \to \overline{X}^{\prime}$$ reduces down to the identity map.

Inner product spaces

If $$X$$ is an inner product space then both the canonical norm on $$X^{\prime}$$ and on $$\overline{X}^{\prime}$$ satisfies the parallelogram law, which means that the polarization identity can be used to define a and also on $$\overline{X}^{\prime},$$ which this article will denote by the notations $$\langle f, g \rangle_{X^{\prime}} := \langle g \mid f \rangle_{X^{\prime}} \quad \text{ and } \quad \langle f, g \rangle_{\overline{X}^{\prime}} := \langle g \mid f \rangle_{\overline{X}^{\prime}}$$ where this inner product makes $$X^{\prime}$$ and $$\overline{X}^{\prime}$$ into Hilbert spaces. The inner products $\langle f, g \rangle_{X^{\prime}}$ and $\langle f, g \rangle_{\overline{X}^{\prime}}$  are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by $f \mapsto \sqrt{\left\langle f, f \right\rangle_{X^{\prime}}}$ ) is consistent with the dual norm (that is, as defined above by the supremum over the unit ball); explicitly, this means that the following holds for every $$f \in X^{\prime}:$$ $$\sup_{\|x\| \leq 1, x \in X} |f(x)| = \|f\|_{X^{\prime}} ~=~ \sqrt{\langle f, f \rangle_{X^{\prime}}} ~=~ \sqrt{\langle f \mid f \rangle_{X^{\prime}}}.$$

If $$X$$ is an inner product space then the inner products on the dual space $$X^{\prime}$$ and the anti-dual space $\overline{X}^{\prime},$ denoted respectively by $\langle \,\cdot\,, \,\cdot\, \rangle_{X^{\prime}}$  and $\langle \,\cdot\,, \,\cdot\, \rangle_{\overline{X}^{\prime}},$  are related by $$\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{\overline{X}^{\prime}} = \overline{\langle \,f\, | \,g\, \rangle_{X^{\prime}}} = \langle \,g\, | \,f\, \rangle_{X^{\prime}} \qquad \text{ for all } f, g \in X^{\prime}$$ and $$\langle \,\overline{f}\, | \,\overline{g}\, \rangle_{X^{\prime}} = \overline{\langle \,f\, | \,g\, \rangle_{\overline{X}^{\prime}}} = \langle \,g\, | \,f\, \rangle_{\overline{X}^{\prime}} \qquad \text{ for all } f, g \in \overline{X}^{\prime}.$$

title: Antilinear map


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Series finale

A series finale is the final installment of an episodic entertainment series, most often a television series. It may also refer to a final theatrical sequel, the last part of a television miniseries, the last installment of a literary series, or any final episode.

Origins in television
Most early television series consisted of stand-alone episodes rather than continuing story arcs, so there was little reason to provide closure at the end of their runs. Early comedy series that had special finale episodes include Howdy Doody in September 1960, Leave It to Beaver in June 1963, Hank in April 1966, and The Dick Van Dyke Show in June 1966. One of the few dramatic series to have a planned finale during this period was Route 66, which concluded in March 1964 with a two-part episode in which the pair of philosophical drifters ended their journey across America and then went their separate ways.

Considered to be "the series finale that invented the modern-day series finale," the August 1967 final episode of ABC's The Fugitive, "The Judgment: Part 2", attracted a 72% audience share when broadcast. This remained the highest viewership percentage in U.S. television history until the 1977 finale of the TV mini-series Roots (on the same network) and later the 1980 resolution episode of the internationally prominent "Who shot J.R.?" cliffhanger of CBS' Dallas.

Most-watched American series finales
The most watched series finale in U.S. television history remains the 1983 finale of the CBS war/medical dramedy M*A*S*H, titled "Goodbye, Farewell and Amen". Viewed by 105.9 million viewers and drawing 77% of those watching televisions at the time, the finale of M*A*S*H held the record for most watched telecast of all-time for decades until 2010's Super Bowl XLIV edged it out with 106 million viewers, which coincidentally also aired on CBS. However, M*A*S*H's final episode remains the all-time most-watched U.S. television episode (and so far, the only single television episode in U.S. history to be watched by at least 100 million viewers for a single telecast).

The second-most-watched series finale in U.S. television history was the 1993 finale of the NBC comedy Cheers, titled "One for the Road". "One for the Road" was watched by between 80.4 million and 93.5 million viewers (estimates vary) while drawing 64% of TVs turned on at the time. To date, "One for the Road" remains the most watched U.S. TV series finale following the rise of cable television, and in terms of sheer viewership numbers for non-sports programming, sits second only to the aforementioned finale of M*A*S*H.

With only slightly fewer viewers than the series finale of Cheers was the finale of its one-time follow-up on NBC's "Must See TV" Thursday night line-up, the absurdist NBC comedy Seinfeld. The third most-watched U.S. TV series finale in television history, Seinfeld's controversial 1998 episode "The Finale" was watched by 76.3 million people, drawing 67% of all televisions turned on at the time – as the New York Times put it, "grazing Super Bowl country" in terms of viewership.

With the shift away from network television viewing toward cable television viewing (and later, internet use) that occurred during the decade between the finales of M*A*S*H (1983) and Cheers (1993) – and continued unabated until and beyond the finale of Seinfeld (1998) – it remains debatable which of these three "event" series finales accomplished the most impressive viewership numbers. Moreover, a large gap in viewership numbers exists between the Super Bowl-sized audiences of the M*A*S*H, Cheers and Seinfeld finales, and the fourth and fifth most watched series finales in television history – respectively, those of the comedy Friends (2004, NBC, 52.5 million viewers) and the detective procedural Magnum, P.I. (1988, CBS, 50.7 million viewers). The Friends finale's ("The Last One") viewership numbers dwarf those of all finales since the start of the new millennium and seem particularly impressive in light of the increased media options since the 1990s "event" finales of Cheers (1993) and Seinfeld (1998).

In Britain, the most-acclaimed series finale of sitcoms was from Blackadder. In 1989 the Britcom, starring Rowan Atkinson as a Blackadder in 1917 WW1, goes "over the top" with his regiment, leaving their fate unknown as fade to a field of poppies. The finale was praised as a "perfect end-of-show finale".

Reception
No matter how critically lauded during their respective runs, relatively few popular television series finales end up pleasing critics and audiences universally, or escaping controversy. Prominent examples of controversial series finales include the final episodes of the comedies Roseanne ("Into That Good Night", 1997), Seinfeld ("The Finale", 1998), How I Met Your Mother ("Last Forever", 2014), and Two and a Half Men ("Of Course He's Dead", 2015), and those of the dramas The Prisoner (Fall Out ,1968), Game of Thrones ("The Iron Throne", 2019), True Blood ("Thank You", 2014), and Dexter ("Remember the Monsters?", 2013). The final episodes of Dexter, How I Met Your Mother and Game of Thrones, in particular, triggerred massive backlash from both fans and critics upon airing, and are often regarded as the worst finales in recent memory.

Some series endings have proved divisive among viewers immediately on airing, leading to extensive discussions online, but ended up being generally lauded by critics. Notable examples of this trend are the finales of  The Sopranos ("Made in America", 2007), The Wire ("-30-", 2008), and Lost ("The End", 2010). For example, The Sopranos ' finale caused millions of viewers to temporarily believe they had lost cable service due to an abrupt blackout. All three episodes were also nominated for multiple Primetime Emmy Awards, including Outstanding Writing, with The Sopranos ' finale winning.

Several television series have, however, managed to produce final episodes that lived up to both critics' and audiences' expectations. Examples include the twist endings that concluded both the Newhart and St. Elsewhere finales, the mixture of comedy and resonance that wrapped up The Mary Tyler Moore Show, Friends, Everybody Loves Raymond, and Brooklyn Nine-Nine and the redemption arc that concluded The Fugitive.

Several more series finales have received unanimous critical and popular acclaim, and are often considered as benchmarks for great TV endings. Recent examples include the finales of Breaking Bad ("Felina", 2013), Community ("Emotional Consequences of Broadcast Television", 2015), The Good Place ("Whenever You're Ready", 2020), Better Call Saul ("Saul Gone", 2022), and Succession ("With Open Eyes", 2023).

Examples of universally acclaimed finales from earlier in the millennium include those of Britain's The Office ("Interview", 2002), The Shield ("Family Meeting", 2008), and Six Feet Under ("Everyone's Waiting", 2005), the last of which TV Guide ranked at No. 22 on their list of "TV's Top 100 Episodes of All Time".

Plot devices
Television series finales frequently feature fundamental deviations from the central plot line, such as the resolution of a central mystery or problem, (e.g. Dallas, Two and a Half Men, Full House) the separation or return of a major character (e.g. Cheers, That '70s Show, The Office) or an event signifying the end of an era, such as a change to primary setting for the series (e.g. The Mary Tyler Moore Show, Boy Meets World, Martin, Liv and Maddie)

Series finales will sometimes include clips or characters from the series' past (e.g. Seinfeld, Suits, Martin, Star Trek: The Next Generation), and the ending moments of the episode often take place in the show's primary setting.

In rare examples, game shows have been given series finales; examples include The Hollywood Squares (which ended its 15-season run in syndication in 1981 with a "Grand Championship Tournament" that awarded a massive cache of prizes including a house to the winner of the tournament), Family Feud (which ended a nine-season run on ABC in 1985 with a tribute from Richard Dawson), Sale of the Century (which ended a six-season run on NBC in 1989), and History IQ (which ended its two-season run on History in 2001 with a $250,000 tournament of champions). Game shows conducted in a tournament format are more likely to have series finales; examples of those include The Million Second Quiz (which aired on NBC in September 2013) and Mental Samurai (which aired on Fox between March and May 2019).

Premature series finales
In some cases, a TV series finale proves premature, as was the case with Here's Lucy, 7th Heaven, Charmed, Babylon 5, and Arrested Development to name but a few. Some shows that have constantly been in danger of cancellation wrote every season finale with the idea that the episode would serve as a quality series finale if the network decided not to bring it back; in recent years from NBC's Thursday night comedy lineup, Parks and Recreation used this formula for the season finales for Seasons 3–6, before getting a renewal for a seventh and final season where the series finale was planned in advance, and Community wrote its fifth-season finale with the notion that whether the show found new life elsewhere or not, it would definitely not be returning to the network (while NBC did indeed cancel the show, it was renewed for a sixth season by Yahoo! Screen, where the season, and sure enough, series finale was once again scripted as a potential last episode ever; the final image is that of text reading "#andamovie", a reference to the show's recurring catchphrase "six seasons and a movie"). The series finale of Dr. Ken, a fictionalized sitcom based on the life of doctor-turned-actor Ken Jeong, features the title character trying out for a fictional version of Community (Jeong was a cast member in real life).

The medical comedy Scrubs aired its two-part finale episode billed simply as a "My Finale" in May 2009 as the show's renewal or cancellation had not been decided as of its airing, and so it was not known whether the episode would conclude just the season or the entire series; Scrubs would eventually be renewed for one additional season, which became a de facto spin-off series titled Scrubs: Med School.

The cartoon Futurama has had four designated series finales, due to the recurringly uncertain future of the series. "The Devil's Hands Are Idle Playthings", "Into the Wild Green Yonder (Part 4)", "Overclockwise", and "Meanwhile" have all been written to serve as a final episode for the show.

The series American Dad! had two possible finales, the season premiere of the eighth season, "Hot Water" was written due to the uncertainty from the staff of the show getting picked up. The penultimate episode of Season 19 was also intended as a series finale, as revealed by showrunner and executive producer Matt Weitzman.

The series finale of The Lucy-Desi Comedy Hour (itself an epilogue to I Love Lucy) was unintentionally fitting: stars Lucille Ball and Desi Arnaz were about to divorce and end the show, a fact that the show's guest star for what would be the final program, Edie Adams, did not know when she chose the song she would sing on the program. Prophetically, the song was named "That's All." The series also ended with Lucy and Ricky making up and kissing, while in reality Ball and Arnaz would not (the two would eventually reconcile later in life, although both would go on to marry other people). The last produced half-hour episode was titled "The Ricardos Dedicate a Statue" which included real-life kids Lucie Arnaz and Desi Arnaz Jr. in the final scene where Ricky unveils what he thinks is a Revolutionary War statue only to find out that it is Lucy. The aforementioned Magnum, P.I. had a premature series finale, as well. At the end of the seventh season, protagonist Thomas Magnum was to be killed off, which was intended to end the series. The final episode of the season, "Limbo", after seeing Magnum wander around as a ghost for nearly the entire run-time, closes with him appearing to walk off into heaven. However, following outcry from fans, who demanded a more satisfactory conclusion, an eighth, final season was produced, to bring Magnum "back to life", and to round the series off. The mystery of whether Higgins was Robin Masters, or not, was a highly anticipated series finale reveal. The mystery still has yet to be revealed. A number of other episodes also make reference to supernatural occurrences and the seeming existence of ghosts.

The Showtime series Californication was designed from start to make any season finale work as a series finale, in case of early cancelling the show. It is seen most primarily at the end of the first and fourth season.

After its fifth season, the sitcom Reba was in danger of being cancelled as a result of its original home, The WB, being replaced by The CW in September 2006, and the resulting uncertainty over which WB series (as well as which series from CW co-predecessor UPN) would be carried over to the new network. (Reba would be renewed by The CW for an additional season as one of the holdover WB series.) The sixth (and final) season's finale episode was written to serve as a series finale, in which Brock and Barbara-Jean come to a reconciliation and Van and Cheyenne move back in with Reba, bringing the show to a full wrap.

A series finale may not be the last aired episode of a show, such as King of the Hill, which produced "To Sirloin with Love" as its series finale, though four more episodes produced before it were aired after in syndication.

Another instance of premature endings was with the Rural Purge of 1971 where networks (notably CBS) axed still-popular TV series in an effort to move to more sophisticated programming. Shows were The Beverly Hillbillies, Mayberry R.F.D., Petticoat Junction and Green Acres.

Finales launching spin-offs
Series finales are sometimes used as a backdoor pilot to launch spinoff series. Two well-known examples include The Andy Griffith Shows series finale, which launched the spinoff Mayberry RFD and The Practices series finale, and much of its final season was used as a launching pad for Boston Legal, starring James Spader and William Shatner. The Golden Girls series finale was set up to lead into a new series with most of the remaining cast, The Golden Palace, Three's Company transitioned more or less seamlessly into Three's a Crowd, the Henry Danger finale served as a pilot to Danger Force while The Fosters series finale acted as an introduction to its spin-off series Good Trouble. The Closer was spun off into a new series, Major Crimes the same night that the original ended, after star Kyra Sedgwick chose, as producer, to end the series.

The Steven Universe finale "Change Your Mind" served as not just a conclusion to the original series and its overarching plot, but helped pave the way for Steven Universe: The Movie, and eventually the limited epilogue series Steven Universe Future.

Some planned spin-offs that influenced series finales, however, never materialized, as in the case of the proposed Laverne & Shirley spin-off for Carmine that never came into fruition, or Posse Impossible, a proposed spin-off of Hong Kong Phooey that, instead of getting its own show, aired as an interstitial segment on CB Bears. Arrow used its penultimate episode of its final season, "Green Arrow & The Canaries", as a backdoor pilot for a potential spin-off series based on the characters Mia Smoak, Laurel Lance and Dinah Drake, however, the series was ultimately not picked up.

title: Series finale