text:
Talk:Smooth function

Can you quantify or make mathematically precise what you mean by a "large gap"? Phys 15:48, 2 Dec 2003 (UTC)

I believe there are a number of mathematical analysis ways. If Taylor's theorem is breaking down as an infinite series expansion, something in the remainder term is blowing up. Smooth says the Fourier transfrom drops off at infinity faster than any polynomial - one can ask for more than that. I'm pretty sure there are classes of functions between smooth and analytic that have been studied, as whole scales (are they called quasi-analytic?). Obviously it's very striking how different the zero sets are, since any closed set can be the zero set of a smooth function.

Charles Matthews 17:14, 2 Dec 2003 (UTC)

Seems protecting the semi-open intervals from interference from Wiki-syntax busibodies also damages the format.

Charles Matthews 10:09, 13 Nov 2004 (UTC)

moving this to differentiable function
Any objections to moving this to differentiable function? That is currently a redirect to derivative. --MarSch 30 June 2005 16:07 (UTC)

 * Infinitely differentiable is not the same as (one time) differentiable.--Patrick June 30, 2005 20:58 (UTC)

 * smooth functions are a very special case of differentiable functions, but there is no such article yet. I think it would be easy to expand this once it is moved there.--MarSch 1 July 2005 10:09 (UTC)

Differentiabillity class could also be merged there. In the intro here C^1 and all that is explained so I think it would be a natural move. --MarSch 1 July 2005 10:12 (UTC)

 * Yes, I think merging is good. However, differentiable function sounds like there is one type of differentiability, so perhaps differentiability or differentiability of functions or the existing name differentiability class is best.--Patrick 1 July 2005 14:38 (UTC)

 * There should be a separate article "Differentiability Classes" and all the stuff explained in the intro about C^1 etc should go there, since to define a smooth function you would have to define differentiability classes first; the latter are more fundamental. --Stephen 9 March 2008 16:52 British Time —Preceding unsigned comment added by 217.44.113.21 (talk) 16:53, 9 March 2008 (UTC)

Currently, "Differentiability Classes" redirects here to Smooth Function, but it should be the other way around. Smooth functions are a special case of differentiability class. It doesn't make sense to redirect the generalization to the most extreme example. — Preceding unsigned comment added by 138.210.251.10 (talk) 06:29, 21 February 2014 (UTC)

Mistake about the definition of function
It is unfortunately a classical mistake. "f(x)" is no function, the function that would correspond is called "f". Or x|->f(x). I can't give further information because my English isn't good enough to describe mathematical objects. Bête spatio-temporelle [my name]

Cω
I don't think Cω is actually defined anywhere in this article, which is a little ridiculous considering how often it is mentioned. As far as I can tell, Ck is only explicitly defined in the article for non-negative integers. Eebster the Great (talk) 21:43, 14 March 2009 (UTC)

 * This seems to have been resolved since, but I find the passages that say "Cω therefore analytic" or "analytic therefore Cω" rather silly, given that they are synonyms. Also I think it should be pointed out more clearly that in spite of appearance, Cω is not a smoothness condition. Marc van Leeuwen (talk) 13:24, 15 April 2011 (UTC)

"The function f is said to be of class Cω, or analytic, if f is smooth and if it equals its Taylor series expansion around any point in its domain." Is it redundant to say f is smooth? Are there functions which equal their Taylor expansions but aren't smooth? If so, perhaps one should be provided. Erasmuse (talk) 03:37, 16 June 2014 (UTC)
 * The definition of Taylor series needs derivatives of every order. Therefore one may talk of Taylor series only for smooth functions. In other words, a function is smooth if and only if it has a Taylor series. D.Lazard (talk) 10:43, 16 June 2014 (UTC)

Vector space
I was looking at the article to see if $$C^\infty$$ functions form a vector space. And, this article told me they form a Frechet space, which means nothing to me. Of course, I can click the link and find out that it is a topological vector space. Wouldn't it make sense to specifically say the space is a vector space? StatisticsMan (talk) 04:32, 3 February 2010 (UTC)

Question/Theorem
Let g be some function then if a function f is of class Cr and g composed f is of class Cr then g is also of class Cr. Is this true and if this is true should not a theorem be in the article stating this and for which a proof should be provided. —Preceding unsigned comment added by 128.100.86.53 (talk) 18:08, 12 February 2010 (UTC)

 * Such things are obviously false (regardless of the order of composition); take f constant for instance. Marc van Leeuwen (talk) 13:21, 15 April 2011 (UTC)

Why C^k?
This may be an easy question to answer (or maybe not), but why all the interest in the class of C^k functions? Why not just k-times-differentiable functions? I understand that there are functions, such as x^2*sin(1/x), which are differentiable but not C^1, but... so what? What parts of the theory become cleaner by excluding such examples? (I guess theorems like Clairaut's theorem would be one example.) Anyhow, I think the article should definitely say why it became the standard thing to insist that a function is not just differentiable k times, but continuously so. Kier07 (talk) 02:00, 5 March 2010 (UTC)

Multivariate differentiability classes -- wrong indices ??
Shouldn't be the paragraph be like this:

Let n and m be some positive integers. If f is a function from an open subset of Rn with values in Rm, then f has component functions f1, ..., fm. Each of these may or may not have partial derivatives. We say that f is of class Ck if all of the partial derivatives $$\frac{\partial^l f_i}{\partial x_{1}^{l_1}\partial x_{2}^{l_2}\cdots\partial x_{n}^{l_n}}$$ exist and are continuous, where $$i$$ is an integer between 1 and m, each of $$ l_1, l_2, \ldots l_n $$ is an integer between 0 and k and $$l_1+l_2+\cdots +l_n=l $$.

 ? --Wohingenau (talk) 13:15, 3 July 2012 (UTC)

title: Talk:Smooth function


text:
Kettering (disambiguation)

Kettering is a town in Northamptonshire, England.

Kettering may also refer to:

Places

 * Kettering, Tasmania, a coastal town in Australia
 * Kettering, Jamaica, a former Free Village
 * Kettering, Maryland, an incorporated area in Maryland, United States
 * Kettering, Ohio, a city in Montgomery County, Ohio, United States
 * Borough of Kettering, a former district of Northamptonshire, England that contained the town
 * Kettering (UK Parliament constituency)

Other uses

 * Kettering College in Kettering, Ohio
 * Kettering University in Flint, Michigan
 * Kettering Tower, a high-rise building in Dayton, Ohio
 * Kettering Town F.C.
 * Kettering Rugby Football Club
 * "Kettering", a 2009 song by the Antlers from Hospice

People with the surname

 * Charles F. Kettering, American inventor
 * Steve Kettering, American politician

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

title: Kettering (disambiguation)


text:
List of Trotskyist internationals

This is a list of Trotskyist internationals. It includes all of the many political internationals which self-identify as Trotskyist.

Of the organizations listed, two claim to be the original Fourth International founded in 1938: the reunified Fourth International (USFI) and the International Committee of the Fourth International (ICFI). The Fourth International split into two factions 1953 over the question of historic perspective, with the International Secretariat (ISFI) turning in favour of deep entryism and supporting petty-bourgeois nationalist movements in less developed countries, and the International Committee (ICFI) upholding the need to form revolutionary parties. The factions reunited in 1963 resulting in the formation of reunified Fourth International (USFI), while parts of the ICFI did not. Both the USFI and ICFI went on to fragment further, giving rise to several new internationals.

Certain organizations which claim to be Trotskyist make no attempt to claim any relationship to the Fourth International in an organizational sense and argue that it no longer exists. Some claim to represent a continuity from the Fourth International or to have re-established it: for example the Fourth International (ICR) International Centre/Center of Reconstruction, also known as the FI (La Verité), also calls itself the "Fourth International".

The various organizations listed here range in size from those having thousands of adherents in dozens of countries to tendencies which can barely claim a dozen members in three or four countries.

List of internationals
The largest internationals in terms of membership are indicated in bold.

Internationals that emerged from the Fourth International (USFI)

 * International Workers League – Fourth International (IWL-FI), founded by former members of the USFI in 1982.
 * Trotskyist Fraction – Fourth International (TF-FI), founded by expelled members of the IWL-FI in 1988
 * International Workers' Unity – Fourth International (IWU-FI), split from IWL-FI in 1997.
 * International Leninist Trotskyist Fraction (ILTF), split from the TF-FI in 1998.
 * Tendency for the Reconstruction of the Fourth International, formed by expelled members of the TF-FI in 2006.
 * Committee for Revolutionary International Regroupment (CRIR), founded by groups originating from the USFI and IWL-FI
 * Committee for a Workers' International (CWI) - claims to be refoundation of the defunct Committee for a Workers' International (1974), which was formed by expelled members of the USFI in 1965, after 2019 split.
 * Revolutionary Communist International, split from CWI in 1992. Previously named International Marxist Tendency (IMT) (2004–2024), and the Committee for a Marxist International (1992–2004)
 * International Revolutionary Left (IRL), split from CWI in 2019.
 * International Socialist Alternative, claims to be successor to the defunct Committee for a Workers' International (1974) after 2019 split.
 * Internationalist Standpoint (IS), split from International Socialist Alternative in 2022.
 * International Socialist Tendency (IST), founded by expelled members of the original Fourth International in the 1950s
 * League for the Fifth International (L5I), founded by expelled members of the IST.
 * Revolutionary Communist International Tendency (RCIT), founded by former members of the IST and L5I.
 * Internationalist Trotskyist Nucleus-Fourth International, founded as a "public faction" of the ILTF, and split from the RCIT.
 * Liaison Committee for the Reconstruction of the Fourth International (CERCI), founded by the Bolivian party POR which broke off with the original Fourth International in the 1950s.

Internationals that emerged from the International Committee of the Fourth International (ICFI)

 * International Communist League (Fourth Internationalist) (ICL-FI), previously the international Spartacist tendency which split from the ICFI in 1966.
 * International Bolshevik Tendency (IBT) formed by expelled members of the international Spartacist tendency in 1982.
 * Bolshevik Tendency, split from IBT in 2018.
 * League for the Fourth International (LFI), formed by expelled members of the ICL-FI in 1996.
 * Organising Committee for the Reconstitution of the Fourth International (OCRFI), split from the defunct Fourth International (ICR) in 2016. The latter was formed after a split from the ICFI in 1972.
 * International Trotskyist Committee (ITC), founded by former members of the defunct Trotskyist International Liaison Committee in 1984. The latter was founded by expelled members of the ICFI in 1974.
 * Liaison Committee for the Fourth International, founded by former members of the ITC.

Others

 * Internationalist Communist Union (ICU)
 * Permanent Revolution Collective

Defunct

 * Committee for a Workers' International (CWI), 1974–2019 – split into Committee for a Workers' International (Refounded) and International Socialist Alternative
 * Coordinating Committee for the Refoundation of the Fourth International (CRFI)
 * International League for the Reconstruction of the Fourth International (ILRFI), 1976–1995
 * Fourth International Posadist
 * Workers International to Rebuild the Fourth International (WIRFI)
 * International Revolutionary Marxist Tendency (TMRI), 1965–1992
 * Permanent Revolution
 * Revolutionary Workers Ferment (Fomento Obrero Revolucionario, FOR)
 * Trotskyist International Liaison Committee, 1979–1984
 * Tendencia Cuartainternacionalista
 * Fourth International (ICR), also called FI (La Verité) or FI (International Secretariat) 1981–2015
 * Socialist Network (Post-Trotskyist, split from IMT)

title: List of Trotskyist internationals


text:
Fugitive Slave Act of 1850

The Fugitive Slave Act or Fugitive Slave Law was a law passed by the 31st United States Congress on September 18, 1850, as part of the Compromise of 1850 between Southern interests in slavery and Northern Free-Soilers.

The Act was one of the most controversial elements of the 1850 compromise and heightened Northern fears of a slave power conspiracy. It required that all escaped slaves, upon capture, be returned to the enslaver and that officials and citizens of free states had to cooperate. The Act contributed to the growing polarization of the country over the issue of slavery. It was one of the factors that led to the American Civil War.

Background
By 1843, several hundred enslaved people a year escaped to the North successfully, making slavery an unstable institution in the border states.

The earlier Fugitive Slave Act of 1793 was a Federal law that was written with the intent to enforce Article 4, Section 2, Clause 3 of the United States Constitution, which required the return of escaped slaves. It sought to force the authorities in free states to return fugitives of enslavement to their enslavers.

Many free states wanted to disregard the Fugitive Slave Act. Some jurisdictions passed personal liberty laws, mandating a jury trial before alleged fugitive slaves could be moved; others forbade the use of local jails or the assistance of state officials in arresting or returning alleged fugitive slaves. In some cases, juries refused to convict individuals who had been indicted under the Federal law.

The Missouri Supreme Court routinely held, with the laws of neighboring free states, that enslaved people who their enslavers had voluntarily transported into free states, with the intent of the enslavers' residing there permanently or indefinitely, gained their freedom as a result. The 1793 act dealt with enslaved people who escaped to free states without their enslavers' consent. The Supreme Court of the United States ruled, in Prigg v. Pennsylvania (1842), that states did not have to offer aid in the hunting or recapture of enslaved people, significantly weakening the law of 1793.

After 1840, the Black population of Cass County, Michigan, proliferated as families were attracted by White defiance of discriminatory laws, by numerous highly supportive Quakers, and by low-priced land. Free and escaping Blacks found Cass County a haven. Their good fortune attracted the attention of Southern slavers. In 1847 and 1849, planters from Bourbon and Boone counties, Kentucky, led raids into Cass County to recapture people escaping slavery. The attacks failed, but the situation contributed to Southern demands in 1850 to pass a strengthened fugitive slave act.

Southern politicians often exaggerated the number of people escaping enslavement, blaming the escapes on Northern abolitionists, whom they saw as stirring up their allegedly happy slaves, interfering with "Southern property rights". According to the Columbus [Georgia] Enquirer of 1850, The support from Northerners for fugitive slaves caused more ill will between the North and the South than all the other causes put together.

New law

In response to the weakening of the original Fugitive Slave Act, Democratic Senator James M. Mason of Virginia drafted the Fugitive Slave Act of 1850, which penalized officials who did not arrest someone allegedly escaping from slavery, and made them liable to a fine of $1,000. Law enforcement officials everywhere were required to arrest people suspected of escaping enslavement on as little as a claimant's sworn testimony of ownership. Habeas corpus was declared irrelevant. The Commissioner before whom the fugitive from slavery was brought for a hearing—no jury was permitted, and the alleged refugee from enslavement could not testify —was compensated $10 if he found that the individual was proven a fugitive and only $5 if he determined the proof to be insufficient. In addition, any person aiding a fugitive by providing food or shelter was subject to up to six months of imprisonment and up to $1,000 in fine. Officers who captured a fugitive from slavery were entitled to a bonus or promotion for their work.

Enslavers needed only to supply an affidavit to a Federal marshal to capture a fugitive from slavery. Since a suspected enslaved person was not eligible for a trial, the law resulted in the kidnapping and conscription of free Blacks into slavery, as purported fugitive slaves had no rights in court and could not defend themselves against accusations.

The Act adversely affected the prospects of escape from slavery, particularly in states close to the North. One study finds that while prices placed on enslaved people rose across the Southern United States in the years after 1850, it appears that "the 1850 Fugitive Slave Act increased prices in border states by 15% to 30% more than in states further south", illustrating how the Act altered the chance of successful escape.

According to abolitionist John Brown, even in the supposedly safe refuge of Springfield, Massachusetts, "some of them are so alarmed that they tell me that they cannot sleep on account of either them or their wives and children. I can only say I think I have been enabled to do something to revive their broken spirits. I want all my family to imagine themselves in the same dreadful condition."

Judicial nullification
In 1855, the Wisconsin Supreme Court became the only state high court to declare the Fugitive Slave Act unconstitutional as a result of a case involving fugitive slave Joshua Glover and Sherman Booth, who led efforts that thwarted Glover's recapture. In 1859 in Ableman v. Booth, the U.S. Supreme Court overruled the state court.

Jury nullification occurred as local Northern juries acquitted men accused of violating the law. Secretary of State Daniel Webster was a key supporter of the law, as expressed in his famous "Seventh of March" speech. He wanted high-profile convictions. The jury nullifications ruined his presidential aspirations and his last-ditch efforts to find a compromise between North and South. Webster led the prosecution against men accused of rescuing Shadrach Minkins in 1851 from Boston officials who intended to return Minkins to slavery; the juries convicted none of the men. Webster sought to enforce a law that was extremely unpopular in the Northern United States, and his Whig Party passed him over again when they chose a presidential nominee in 1852.

Legislative nullification
In November 1850, the Vermont legislature passed the Habeas Corpus Law, requiring Vermont judicial and law enforcement officials to assist captured fugitive slaves. It also established a state judicial process, parallel to the federal process, for people accused of being fugitive slaves. This law rendered the federal Fugitive Slave Act effectively unenforceable in Vermont and caused a storm of controversy nationally. It was considered a nullification of federal law, a concept popular among slave states that wanted to nullify other aspects of federal law, and was part of highly charged debates over slavery. Noted poet and abolitionist John Greenleaf Whittier had called for such laws, and the Whittier controversy heightened pro-slavery reactions to the Vermont law. Virginia governor John B. Floyd warned that nullification could push the South toward secession. At the same time, President Millard Fillmore threatened to use the army to enforce the Fugitive Slave Act in Vermont. No test events took place in Vermont, but the rhetoric of this flare-up echoed South Carolina's 1832 nullification crisis and Thomas Jefferson's 1798 Kentucky Resolutions.

In February 1855, Michigan's legislature also passed a law prohibiting county jails from being used to detain recaptured slaves, directing county prosecutors to defend recaptured slaves, and entitling recaptured slaves to habeas corpus and trial by jury. Other states to pass personal liberty laws include Connecticut, Massachusetts, Maine, New Hampshire, Ohio, Pennsylvania and Wisconsin.

Resistance in the North and other consequences
The Fugitive Slave Law brought the issue home to anti-slavery citizens in the North, as it made them and their institutions responsible for enforcing slavery. "Where before many in the North had little or no opinions or feelings on slavery, this law seemed to demand their direct assent to the practice of human bondage, and it galvanized Northern sentiments against slavery." Moderate abolitionists were faced with the immediate choice of defying what they believed to be an unjust law or breaking with their consciences and beliefs. Harriet Beecher Stowe wrote Uncle Tom's Cabin (1852) in response to the law.

Many abolitionists openly defied the law. Reverend Luther Lee, pastor of the Wesleyan Methodist Church of Syracuse, New York, wrote in 1855:

"I never would obey it. I had assisted thirty slaves to escape to Canada during the last month. If the authorities wanted anything of me, my residence was at 39 Onondaga Street. I would admit that and they could take me and lock me up in the Penitentiary on the hill; but if they did such a foolish thing as that I had friends enough in Onondaga County to level it to the ground before the next morning."

Several years before, in the Jerry Rescue, Syracuse abolitionists freed by force a fugitive slave who was to be sent back to the South and successfully smuggled him to Canada. Thomas Sims and Anthony Burns were both captured fugitives who were part of unsuccessful attempts by opponents of the Fugitive Slave Law to use force to free them. Other famous examples include Shadrach Minkins in 1851 and Lucy Bagby in 1861, whose forcible return has been cited by historians as important and "allegorical". Pittsburgh abolitionists organized groups whose purpose was the seizure and release of any enslaved person passing through the city, as in the case of a free Black servant of the Slaymaker family, erroneously the subject of a rescue by Black waiters in a hotel dining room. If fugitives from slavery were captured and put on trial, abolitionists worked to defend them in trial, and if by chance the recaptured person had their freedom put up for a price, abolitionists worked to pay to free them. Other opponents, such as African-American leader Harriet Tubman, treated the law as just another complication in their activities.

In April 1859, a putative freeman named Daniel Webster was arrested in Harrisburg, Pennsylvania, alleged to be Daniel Dangerfield, an escaped slave from Loudoun County, Virginia. At a hearing in Philadelphia, federal commissioner J. Cooke Longstreth ordered Webster's release, arguing the claimants had not proved that he was Dangerfield. Webster promptly left for Canada.

Canada
One important consequence was that Canada, not the Northern free states, became the foremost destination for escaped slaves. The Black population of Canada increased from 40,000 to 60,000 between 1850 and 1860, and many reached freedom by the Underground Railroad. Notable Black publishers, such as Henry Bibb and Mary Ann Shadd, created publications encouraging emigration to Canada. By 1855, an estimated 3,500 people among Canada's Black population were fugitives from American slavery. In Pittsburgh, for example, during the September following the passage of the law, organized groups of escaped slaves, armed and sworn to "die rather than be taken back into slavery", set out for Canada, with more than 200 men leaving by the end of the month. The Black population in New York City dropped by almost 2,000 from 1850 to 1855.

On the other hand, many Northern businessmen supported the law due to their commercial ties with the Southern states. They founded the Union Safety Committee and raised thousands of dollars to promote their cause, which gained sway, particularly in New York City, and caused public opinion to shift somewhat towards supporting the law.

End of the Act
In the early stages of the American Civil War, the Union had no established policy on people escaping from slavery. Many enslaved people left their plantations heading for Union lines. Still, in the early stages of the war, fugitives from slavery were often returned by Union forces to their enslavers. General Benjamin Butler and some other Union generals, however, refused to recapture fugitives under the law because the Union and the Confederacy were at war. He confiscated enslaved people as contraband of war and set them free, with the justification that the loss of labor would also damage the Confederacy. Lincoln allowed Butler to continue his policy but countermanded broader directives issued by other Union commanders that freed all enslaved people in places under their control.

In August 1861, the U.S. Congress enacted the Confiscation Act of 1861, which barred enslavers from re-enslaving captured fugitives who were forced to aid or abet the insurrection. The legislation, sponsored by Lyman Trumbull, was passed on a near-unanimous vote and established military emancipation as official Union policy, but applied only to enslaved people used by rebel enslavers to support the Confederate cause, creating a limited exception to the Fugitive Slave Act. Union Army forces sometimes returned fugitives from slavery to enslavers until March 1862, when Congress enacted the Confiscation Act of 1862, Section 10 of which barred Union officers from returning slaves to their owners on pain of dismissal from the service. James Mitchell Ashley proposed legislation to repeal the Fugitive Slave Act, but the bill did not make it out of committee in 1863. Although the Union policy of confiscation and military emancipation had effectively superseded the operation of the Fugitive Slave Act, the Fugitive Slave Act was only formally repealed in June 1864. The New York Tribune hailed the repeal, writing: "The blood-red stain that has blotted the statute-book of the Republic is wiped out forever."

Incidents involving the Fugitive Slave Act of 1850 (in chronological order)

 * Fugitive Slave Convention, August 1850, Cazenovia, New York
 * The act passes Congress and is signed by the President on September 18, 1850
 * Seizure and enslavement of James Hamlet, September 26, 1850, Williamsburg, New York
 * Escape of Shadrach Minkins, February 1851, Boston
 * Reenslavement of Thomas Sims, April 1851, Boston
 * Christiana Riot, September 1851, Christiana, Pennsylvania
 * Jerry Rescue, October 1851, Syracuse, New York
 * Escape of Joshua Glover, March 1854, Milwaukee
 * Reenslavement of Anthony Burns, May/June 1854, Boston
 * Attempted reenslavement of Jane Johnson, July 1855, Philadelphia
 * Oberlin–Wellington Rescue, September 1858, Oberlin and Wellington, Ohio
 * Escape of Charles Nalle, November 1860, Troy, New York

title: Fugitive Slave Act of 1850


text:
Sean Biggerstaff

Sean Biggerstaff (born 15 March 1983) is a Scottish actor. He is best known for playing Oliver Wood in the Harry Potter film series, appearing in Philosopher's Stone (2001), Chamber of Secrets (2002), and Deathly Hallows – Part 2 (2011).

Biggerstaff began acting as a child actor at the age of five and made his screen debut in the television miniseries The Crow Road (1996). He won a BAFTA Scotland Award for playing Jeremy Wolfenden in the television film Consenting Adults (2007). He also appeared in the feature films Cashback (2006) and Mary Queen of Scots (2013).

Early life
Biggerstaff was born in Glasgow, Scotland, growing up with his parents in Maryhill. He went to Parkview Primary School in Summerston and joined the local drama group, the Maryhill Youth Theatre, when he was seven.

Career
Biggerstaff gained his first professional acting role at ten, playing the son of MacDuff in a Michael Boyd production of Macbeth at the Tron Theatre, Glasgow. He then joined the Scottish Youth Theatre and spent six years with the group. In 1996, at the age of 13, Biggerstaff secured his first major TV role, playing Young Darren in the BBC production of The Crow Road. At 14, he was cast by future fellow Harry Potter actor Alan Rickman in his new movie, The Winter Guest, as Tom, a mischievous schoolboy. In a 2002 interview, Biggerstaff recalled: "It was while at the Scottish Youth Theatre that I was chosen for The Winter Guest. Alan Rickman (known to most of the kids as "the bad guy from Die Hard") popped along one day looking for two boys to accompany him to the coldest place on Earth, Fife, for two months to make a film. Myself and the suspicious character that is Douglas Murphy were the lucky ones."

Rickman later recommended Biggerstaff to the London acting agency International Creative Management and Paul Lyon-Maris. Within a week, the Harry Potter castings were being distributed. During auditions, the directors suggested that Biggerstaff consider the part of Oliver Wood, and he played the role in the first two films, returning for a brief appearance in Harry Potter and the Deathly Hallows – Part 2.

Biggerstaff's career since the first two Harry Potter films has included playing Henry, Duke of Gloucester in the 2004 television mini-series Charles II: The Power and The Passion; Matt in the world premiere of Sharman MacDonald's stage play The Girl With Red Hair; and Ben Willis in the short film Cashback for which extra scenes were recorded, and the piece was adapted into a feature released in 2006.

In 2007, Biggerstaff played Jeremy Wolfenden in the BBC 4 television film Consenting Adults which was nominated for BAFTA Scotland's Best Drama Award on 18 November 2007, and won him the BAFTA Scotland's Award for Best Actor – Television. He appeared in the film Hippie Hippie Shake. and appeared in Whisky Galore! (2016) with Gillies MacKinnon and Eddie Izzard.

In 2016, Biggerstaff played the lead role in French-Canadian Catherine-Anne Toupin's Right Now, a translation by Chris Campbell of À présent, directed by Michael Boyd, which was critically and commercially well received in Bath, London, and Edinburgh.

In 2017, it was announced that he would star opposite Georgia Tennant in the Big Finish audio series Jenny: The Doctor's Daughter, taking the role of Noah.

title: Sean Biggerstaff


text:
Pin compatibility

In electronics, pin-compatible devices are electronic components, generally integrated circuits or expansion cards, sharing a common footprint and with the same functions assigned or usable on the same pins. Pin compatibility is a property desired by systems integrators as it allows a product to be updated without redesigning printed circuit boards, which can reduce costs and decrease time to market.

Although devices which are pin-compatible share a common footprint, they are not necessarily electrically or thermally compatible. As a result, manufacturers often specify devices as being either pin-to-pin or drop-in compatible. Pin-compatible devices are generally produced to allow upgrading within a single product line, to allow end-of-life devices to be replaced with newer equivalents, or to compete with the equivalent products of other manufacturers.

Pin-to-pin compatibility
Pin-to-pin compatible devices share an assignment of functions to pins, but may have differing electrical characteristics (supply voltages, or oscillator frequencies) or thermal characteristics (TDPs, reflow curves, or temperature tolerances). As a result, their use in a system may require that portions of the system, such as its power delivery subsystem, be adapted to fit the new component.

A common example of pin-to-pin compatible devices which may not be electrically compatible are the 7400 series integrated circuits. The 7400 series devices have been produced on a number of different manufacturing processes, but have retained the same pinouts throughout. For example, all 7405 devices provide six NOT gates (or inverters) but may have incompatible supply voltage tolerances.

 * 7405 – Standard TTL, 4.75–5.25 V.
 * 74C05 – CMOS, 4–15 V.
 * 74LV05 – Low-voltage CMOS, 2.0–5.5 V.

In other cases, particularly with computers, devices may be pin-to-pin compatible but made otherwise incompatible as a result of market segmentation. For example, Intel Skylake desktop-class Core and Xeon E3v5 processors both use the LGA 1151 socket, but motherboards using C230-series chipsets will only be compatible with Xeon-branded processors, and will not work with Core-branded processors.

Drop-in compatibility
A drop-in compatible device is a device which may be swapped with another without need to make compensating alterations to the system the device was a part of. The device will have the same functions available on the same pins, and will be electrically and thermally compatible. Such devices may not be an exact match to the devices they can replace. For example, they may have a wider range of supply voltage or temperature tolerances.

Software compatibility
Software-compatible devices are devices which are able to run the same software to produce the same results without the software having to be modified first.

Microcontrollers, FPGAs, and other programmable devices may be pin-to-pin compatible from the perspective of the program on the device, but incompatible in terms of hardware. For example, the device may take the signal on pin X, negate it, and output the result on pin Y. If the method of configuring a pin remains the same but the package of the device (such as TSSOP or QFN) changes, the program will continue to function but the physical locations of the pins the program works with may change.

A device may also be pin-compatible while being software-incompatible. This may occur when the device uses a different instruction set, or if the device has a multiplexer attached to a pin (which, for example, may allow the switching of the pin between being driven as GPIO or by an A/D) and that multiplexer selects, by default, a different input source than is selected on the device being replaced.

To ease the use of software-incompatible devices, manufacturers often provide hardware abstraction layers. Examples of these include, CMSIS for ARM Cortex-M processors and the now-deprecated HAL subsystem for UNIX-like operating systems.

title: Pin compatibility


text:
Finsler manifold

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold $M$ where a (possibly asymmetric) Minkowski norm $F(x, −)$ is provided on each tangent space $T_{x}M$, that enables one to define the length of any smooth curve $γ : [a, b] → M$ as

 * $$L(\gamma) = \int_a^b F\left(\gamma(t), \dot{\gamma}(t)\right)\,\mathrm{d}t.$$

Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.

Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.

named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation.

Definition
A Finsler manifold is a differentiable manifold $M$ together with a Finsler metric, which is a continuous nonnegative function $F: TM → [0, +∞)$ defined on the tangent bundle so that for each point $x$ of $M$,

 * $F(v + w) ≤ F(v) + F(w)$ for every two vectors $v,w$ tangent to $M$ at $x$ (subadditivity).
 * $F(λv) = λF(v)$ for all $λ ≥ 0$ (but not necessarily for $λ < 0)$ (positive homogeneity).
 * $F(v) > 0$ unless $v = 0$ (positive definiteness).

In other words, $F(x, −)$ is an asymmetric norm on each tangent space $T_{x}M$. The Finsler metric $F$ is also required to be smooth, more precisely:

 * $F$ is smooth on the complement of the zero section of $TM$.

The subadditivity axiom may then be replaced by the following strong convexity condition:

 * For each tangent vector $v ≠ 0$, the Hessian matrix of $F^{2}$ at $v$ is positive definite.

Here the Hessian of $F^{2}$ at $v$ is the symmetric bilinear form

 * $$\mathbf{g}_v(X, Y) := \frac{1}{2}\left.\frac{\partial^2}{\partial s\partial t}\left[F(v + sX + tY)^2\right]\right|_{s=t=0},$$

also known as the fundamental tensor of $F$ at $v$. Strong convexity of $F$ implies the subadditivity with a strict inequality if $u/F(u) ≠ v/F(v)$. If $F$ is strongly convex, then it is a Minkowski norm on each tangent space.

A Finsler metric is reversible if, in addition,

 * $F(−v) = F(v)$ for all tangent vectors v.

A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.

Examples

 * Smooth submanifolds (including open subsets) of a normed vector space of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.
 * Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.

Randers manifolds
Let $$(M, a)$$ be a Riemannian manifold and b a differential one-form on M with
 * $$\|b\|_a := \sqrt{a^{ij}b_i b_j} < 1,$$

where $$\left(a^{ij}\right)$$ is the inverse matrix of $$(a_{ij})$$ and the Einstein notation is used. Then
 * $$F(x, v) := \sqrt{a_{ij}(x)v^i v^j} + b_i(x)v^i$$

defines a Randers metric on M and $$(M, F)$$ is a Randers manifold, a special case of a non-reversible Finsler manifold.

Smooth quasimetric spaces
Let (M, d) be a quasimetric so that M is also a differentiable manifold and d is compatible with the differential structure of M in the following sense:
 * Around any point z on M there exists a smooth chart (U, φ) of M and a constant C ≥ 1 such that for every x, y ∈ U
 * $$ \frac{1}{C}\|\phi(y) - \phi(x)\| \leq d(x, y) \leq C\|\phi(y) - \phi(x)\|.$$
 * The function d: M × M → [0, ∞] is smooth in some punctured neighborhood of the diagonal.

Then one can define a Finsler function F: TM →[0, ∞] by

 * $$F(x, v) := \lim_{t \to 0+} \frac{d(\gamma(0), \gamma(t))}{t},$$

where γ is any curve in M with γ(0) = x and γ'(0) = v. The Finsler function F obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of M. The induced intrinsic metric dL: M × M → [0, ∞] of the original quasimetric can be recovered from

 * $$d_L(x, y) := \inf\left\{\ \left.\int_0^1 F\left(\gamma(t), \dot\gamma(t)\right) \, dt \ \right| \ \gamma\in C^1([0, 1], M) \, \ \gamma(0) = x \ , \ \gamma(1) = y \ \right\},$$

and in fact any Finsler function F: TM → [ 0, ∞) defines an intrinsic quasimetric dL on M by this formula.

Geodesics
Due to the homogeneity of F the length

 * $$L[\gamma] := \int_a^b F\left(\gamma(t), \dot{\gamma}(t)\right)\, dt$$

of a differentiable curve γ: [a, b] → M in M is invariant under positively oriented reparametrizations. A constant speed curve γ is a geodesic of a Finsler manifold if its short enough segments γ|[c,d] are length-minimizing in M from γ(c) to γ(d). Equivalently, γ is a geodesic if it is stationary for the energy functional

 * $$E[\gamma] := \frac{1}{2}\int_a^b F^2\left(\gamma(t), \dot{\gamma}(t)\right)\, dt$$

in the sense that its functional derivative vanishes among differentiable curves γ: [a, b] → M with fixed endpoints and.

Canonical spray structure on a Finsler manifold
The Euler–Lagrange equation for the energy functional E[γ] reads in the local coordinates (x1, ..., xn, v1, ..., vn) of TM as

g_{ik}\Big(\gamma(t), \dot\gamma(t)\Big)\ddot\gamma^i(t) + \left(              \frac{\partial g_{ik}}{\partial x^j}\Big(\gamma(t), \dot\gamma(t)\Big) -    \frac{1}{2}\frac{\partial g_{ij}}{\partial x^k}\Big(\gamma(t), \dot\gamma(t)\Big)  \right) \dot\gamma^i(t)\dot\gamma^j(t) = 0, $$

where k = 1, ..., n and gij is the coordinate representation of the fundamental tensor, defined as

g_{ij}(x,v) := g_v\left(\left.\frac{\partial}{\partial x^i}\right|_x, \left.\frac{\partial}{\partial x^j}\right|_x\right). $$

Assuming the strong convexity of F2(x, v) with respect to v ∈ TxM, the matrix gij(x, v) is invertible and its inverse is denoted by gij(x, v). Then is a geodesic of (M, F) if and only if its tangent curve  is an integral curve of the smooth vector field H on TM∖{0} locally defined by

\left.H\right|_{(x, v)} := \left.v^i\frac{\partial}{\partial x^i}\right|_{(x,v)}\!\! - \left.2G^i(x, v)\frac{\partial}{\partial v^i}\right|_{(x,v)}, $$

where the local spray coefficients Gi are given by

G^i(x, v) := \frac{1}{4}g^{ij}(x, v)\left(2\frac{\partial g_{jk}}{\partial x^\ell}(x, v) - \frac{\partial g_{k\ell}}{\partial x^j}(x, v)\right)v^k v^\ell. $$

The vector field H on TM∖{0} satisfies JH = V and [V, H] = H, where J and V are the canonical endomorphism and the canonical vector field on TM∖{0}. Hence, by definition, H is a spray on M. The spray H defines a nonlinear connection on the fibre bundle TM∖{0} → M through the vertical projection

 * $$v: T(\mathrm{T}M \setminus \{0\}) \to T(\mathrm{T}M \setminus \{0\});\quad v := \frac{1}{2}\big(I + \mathcal{L}_H J\big).$$

In analogy with the Riemannian case, there is a version

 * $$D_{\dot\gamma}D_{\dot\gamma}X(t) + R_{\dot\gamma}\left(\dot\gamma(t), X(t)\right) = 0$$

of the Jacobi equation for a general spray structure (M, H) in terms of the Ehresmann curvature and nonlinear covariant derivative.

Uniqueness and minimizing properties of geodesics
By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough neighborhoods) on (M, F). Length minimizing curves can always be positively reparametrized to be geodesics, and any geodesic must satisfy the Euler–Lagrange equation for E[γ]. Assuming the strong convexity of F2 there exists a unique maximal geodesic γ with γ(0) = x and γ'(0) = v for any (x, v) ∈ TM∖{0} by the uniqueness of integral curves.

If F2 is strongly convex, geodesics γ: [0, b] → M are length-minimizing among nearby curves until the first point γ(s) conjugate to γ(0) along γ, and for t > s there always exist shorter curves from γ(0) to γ(t) near γ, as in the Riemannian case.

title: Finsler manifold


text:
Harriet Cohen

Harriet Pearl Alice Cohen CBE (2 December 1895 – 13 November 1967) was a British pianist.

Biography
Harriet Cohen was born in London. Her younger sister was the singer Myra Verney (1905-1993) and she was a distant cousin of the pianist Irene Scharrer. She studied piano at the Royal Academy of Music under Tobias Matthay, having won the Ada Lewis scholarship at the age of 12 followed by the Sterndale Bennett Prize in 1913. She made her debut at a Chappell's Sunday concert at the Queen's Hall a year later. At this stage Cohen also had ambitions to be a composer: her Russian Impressions for piano (composed circa 1913) became her only original compositions to be published.

Her first major appearance was in 1920 when she appeared at the Wigmore Hall in a joint recital with the tenor John Coates. She became particularly associated with contemporary British music, giving the world premiere of Ralph Vaughan Williams' Piano Concerto (which was written for her) and recording Edward Elgar's Piano Quintet with the Stratton Quartet under the composer's supervision. A number of composers wrote music specifically for her, including John Ireland, Béla Bartók, Ernest Bloch and E. J. Moeran, and particularly Sir Arnold Bax (Cohen's lover), who wrote most of his piano pieces for her. This includes the music for David Lean's 1948 film version of Oliver Twist. He also composed Concertino for Left Hand for her after she lost the use of her right hand in 1948. The last six pieces in the collection Mikrokosmos by Bartók are dedicated to her. Harriet Cohen dedicated an important effort to the performance of the Tudor composers at a time when this was unusual, and gave recitals of works by William Byrd and Orlando Gibbons and also of Henry Purcell. She was considered one of the finest performers of J. S. Bach's keyboard music, winning outstanding praise from the musicologist Alfred Einstein. Pablo Casals, also, invited her to play Bach with his orchestra at Barcelona, and Wilhelm Furtwängler extended a similar invitation on hearing her in Switzerland. She gave the first 'all-Bach' recital at the Queen's Hall in 1925.

She also cultivated Spanish music, and gave the second performance of Manuel de Falla's Nights in the Gardens of Spain, a work which became especially associated with her. She was also an early exponent of music of the Soviet Union in Britain, and visited Russia in 1935 to broadcast from Moscow and Leningrad, including works by Shostakovich, Kabalevsky and Leonid Polovinkin. These composers later sent her further compositions.

Cohen's influence went well beyond that of a musician. She became strongly associated in the 1930s with publicising the plight of German and Austrian Jews and even played a concert with the scientist Albert Einstein (Alfred's cousin) in 1934 to raise funds to bring Jewish scientists out of Germany. She became a close friend of Eleanor Roosevelt and Ramsay MacDonald as well as the first president of Israel, Chaim Weizmann.

Cohen was also a close friend of many leading figures of the time. These included not only musicians such as Jean Sibelius, Ralph Vaughan Williams, Sir Edward Elgar and Sir William Walton, but also writers such as Arnold Bennett, George Bernard Shaw, H. G. Wells and D. H. Lawrence as well as politicians or entrepreneurs such as Max Beaverbrook and Leslie Viscount Runciman. Cohen became one of the most talked-about and photographed musicians of her day.

She was vice-president of the Women's Freedom League, and was for several years associated with the Jewish National Fund and the Palestine Conservatoire of Music at Jerusalem. Cohen was appointed a Commander of the Order of the British Empire (CBE) in 1938. The Harriet Cohen International Music Award was introduced in her honour in 1951.

She was the subject of This Is Your Life in 1959 when she was surprised by Eamonn Andrews at the BBC Television Theatre.

She died in London. Her ashes rest in the Memorial Gardens at Stoke Poges, Buckinghamshire.

Efforts for refugees from Nazism
Harriet Cohen met the American journalist Dorothy Thompson in 1930 on her first tour of America, a tour which took in New York, Washington and the Library of Congress and Chicago, thus finally establishing a name for herself on the International stage. It was a meeting that was to change Cohen's life and awaken her Jewish consciousness. In 1933 Harriet Cohen travelled to Vienna to play a number of concerts, staying with Dorothy Thompson. She was profoundly moved by the plight of refugees, both Jewish and non-Jewish, who were pouring into the city from Germany. Thompson and Cohen were to correspond about the plight of Jewish refugees in Austria and Germany. Cohen was then able to pass on information from Thompson directly to the British Prime Minister, Ramsay MacDonald, who was at this time her intimate friend. Cohen met Thompson every time she went to America thereafter. From 1933 Cohen committed herself to work in Britain and the United States on behalf of refugees. This would result in a concert in America with Albert Einstein in 1934 to raise funds to bring Jewish scientists out of Germany.

In 1935 Ramsay MacDonald warned Cohen not to travel through Germany because the British Government would not be able to provide immunity for her. Soon after, Adolf Hitler passed the Nuremberg laws totally excluding Jews from public life.

Harriet Cohen had met Albert Einstein in Germany in 1929 when she had afternoon tea at his house. At the time Einstein disclosed that he played the violin and said that one day they should play together. Cohen kept her friendship with Einstein even after he had fled Germany in 1933. Cohen would often visit him in Oxford, England where he settled for a short time. Harriet's sister Myra studied there at Somerville College, where she gave a piano concert. In 1934, after Einstein moved to the United States, Harriet Cohen did finally play that duet concert with Einstein to raise funds to bring Jewish scientists out of Nazi Germany. Cohen and Einstein remained friends thereafter and he referred to her as "the beloved piano witch".

It was not until 1939 when she first met Chaim Weizmann, the future first President of Israel, that she began to support the Zionist cause and a Jewish homeland. Cohen's 1939 visit to Palestine extended her reputation there both as a concert pianist and politically. She argued with British and Jewish officials to try to get Jewish refugees admitted on ships from Nazi Germany (rather than be returned), once almost precipitating an International incident. Harriet Cohen believed passionately in a Jewish homeland but with justice to the Arab Palestinians. She survived two assassination attempts during her trip to Palestine. It was when Cohen was having dinner with Weizmann in London that Weizmann heard the news of the British Government's 1939 white paper to limit Jewish immigration to Britain to just 15,000 people a year. Blanche Dugdale, Arthur Balfour's niece, a fellow diner, prophetically said in an agonised voice, "What will happen to the millions fleeing from Hitler?"

Soviet composers
The visit in spring 1935 by Cohen to the Soviet Union was another major milestone in her career. It was the country from which her ancestors had fled 100 years earlier. Not only was Cohen bringing British music to the USSR by playing pieces by Vaughan Williams, Bax, Bliss and Ireland, she also performed Shostakovich's Preludes, Kabalevsky's Sonatina, and the Soviet premiere of Leonid Polovinkin's Suite from manuscript. Thereafter she took their music all over Europe and was acclaimed as the first musician outside the USSR to learn Shostakovich's Twenty-Four Preludes, which he composed in 1932 and 1933. Her contribution in bringing to the attention of the world the work of unknown Soviet composers is often forgotten.

Relationship with Sir Arnold Bax
"One day I will let my music give itself up to love – love without strife or fret or circumstances – just the praise to you". "My mouth longs for your soft mouth...". These are just two quotes from the love letters between Sir Arnold Bax and Harriet Cohen.

Harriet Cohen's love affair with Bax lasted for over forty years until he died in 1953. It was Bax who gave Harriet Cohen the name "Tania", by which she was affectionately known by close friends and family. Their passionate affair started in 1914 when she was 19 and he was 31, although they had met two years earlier. Bax was creatively inspired by Cohen and in 1915 wrote for her within 13 days three pieces including "The Princess's Rose Garden", "The Maiden with the Daffodil" and "In the Vodka Shop".

Some believe that their time together inspired his famous tone-poem Tintagel, in which he may have in part expressed his anguish at "the dream their world denied". The affair led to Bax's ultimate decision to leave his wife and children in 1918, but they could never live openly together because Bax's wife refused a divorce. Neither could their relationship be recognised publicly because of the social climate of their generation. Cohen possibly became pregnant with Bax's child in 1919 but if she did, she lost the child in an early miscarriage. Harriet Cohen's recently published letters reveal the turbulence and anguish of the relationship. Cohen always claimed that the long-standing affair denied her becoming a "Dame", but this is not substantiated. Through the 1930s their relationship became less passionate as her international career flourished, and Bax sought a quieter haven with his gentler mistress Mary Gleaves; nonetheless the affair continued and they remained close, as private letters between Cohen and Bax reveal. In 1936, for example, they travelled together to Stockholm and Helsinki and met Jean Sibelius, a composer who had long influenced Bax's music. On 23 September 1947, Bax's wife Elsa ("Elsita") died. Cohen probably expected to finally marry Bax after an affair that had now lasted 30 years. Events were to unfold quite differently. Bax did not initially inform Cohen about the death of his wife, a fact she was to discover in May 1948 whilst recording the music that Bax had written for the film Oliver Twist, when Elsa's will was published. A greater shock followed, when Bax revealed his secret twenty-year affair with Gleaves and his intention not to remarry. This was at a time when Cohen was losing prominence in Britain, and in May 1948 Cohen had an accident with a tray of glasses, which severed the artery in her right hand thus restricting her performing career for some years.

When Bax died on 3 October 1953, Cohen was deeply affected by his death. His will bequeathed half of his interest from his literary and musical compositions to Cohen for life, and half to Mary Gleaves. After their death his royalties and estate were to pass to his children. Cohen also kept the London property that Bax had bought for her – throughout Cohen's life Bax had financially assisted her.

Other relationships
The British Prime Minister Ramsay MacDonald is one of her more prominent relationships. Harriet Cohen became close to MacDonald during the period when he was Prime Minister from 1929 to 1935, at a time of economic instability and depression which saw the rise of Nazism and Fascism in Europe. It was rumoured that MacDonald and Cohen became lovers. Their letters reflect a closeness; and she often visited him alone at 10 Downing Street and his home in Hampstead. Certainly many people did believe they were lovers and Cohen was often referred to as "the old man's darling".

Cohen was also close to Max Beaverbrook, the founder of Express newspapers and an important entrepreneur of the day. Cohen was introduced to the business tycoon Max Beaverbrook by Arnold Bennett in 1923. Beaverbrook was instantly charmed by Cohen and invited her to dine regularly with him from 1923 and through him met Lord Rothermere and Lloyd George. Beaverbrook and Cohen often met at her house, as noted in her autobiography A Bundle of Time. He was besotted with her in his own way and showered her regularly with a hundred or more roses.

Ralph Vaughan Williams was an intimate lifelong friend of Harriet Cohen. His letters to her reveal a flirtatious relationship, regularly reminding her of the thousands of kisses that she owed him. He was a regular visitor to her home and often attended Cohen's little parties that she held for her friends. She loved entertaining and inviting famous and prominent people. Cohen premiered Vaughan Williams' "Hymn Tune Prelude" in 1930 which he dedicated to her. She later introduced the piece throughout Europe during her concert tours. In 1933 she then premiered his Piano Concerto in C major, a work which was once again dedicated to her. Cohen was given the exclusive right to play the piece for a period of time.

Harriet Cohen first played for Edward Elgar in 1914 at a party when she was 18. They became close friends and this lasted until his death in 1934. In 1933 Cohen organised a concert in his honour under the patronage of the King and Queen. Undoubtedly Elgar doted on Cohen and closely followed her career, giving her constant support. Under Elgar's direction she made a recording of his Piano Quintet with the Stratton String Quartet. Elgar had only sketched it but he gave the short score to Harriet for the recording.

H. G. Wells was part of Harriet Cohen's circle of male admirers from the 1920s. After Wells parted from writer and novelist Rebecca West, it is well known that Wells took up brief liaisons with other women. Harriet Cohen is highly likely to have been one of these, as various letters from her private collection and interviews suggest. She had a magnetic personality and beauty which Wells found irresistible.

D. H. Lawrence became a close friend of Harriet Cohen's. It was clear already in 1915 that this friendship had created some tensions between Cohen and Arnold Bax. Bax protested at Cohen's closeness to Lawrence. She told Lawrence that they would have to meet secretly. In demonstrative mood that day, Lawrence scrawled across her autograph book "The door closed". A short time later Cohen contracted tuberculosis, possibly from Lawrence, who died of the disease in 1930. Cohen to Bax always denied any sexual relationship with Lawrence but many believed they were lovers. Nonetheless, Lawrence and Cohen remained good friends and were regularly seeing each other as least as part of a group of friends up until his death. They would often talk together about the music of their common friend, the musician Elgar.

Cohen was introduced to William Walton in 1923 by Arnold Bennett. They carried on a rather flirtatious friendship which Cohen described in her autobiography as "stormy but delightful". She wrote that the irritation they often felt for each other did not lessen the underlying affection. Cohen championed Walton's music both at home and abroad especially in the late 1920s and early '30s. On assigning to Cohen the premier performance of his Sinfonia Concertante in 1927 he said to her "I know it will be in safe hands".

Rosa Newmarch introduced Harriet Cohen to Jean Sibelius in London 1921 and they spent some time together. In the summer 1936 Cohen travelled in Finland with Arnold Bax and they had long discussions with Sibelius both in Helsinki and Ainola. Sibelius even wrote her the opening chord of his eighth symphony on a cigarette packet. The Eighth Symphony was lost.

However, her most important relationship was probably with Arnold Bennett. "Arnold Bennett, dear friend and mentor of my youth died on 27 March 1931" wrote Cohen in her autobiography. Arnold Bennett was one of Cohen's closest friends and responsible for introducing her to many of her new friends. Bennett introduced Cohen to William Walton and Max Beaverbrook in 1923. Cohen was devastated on Bennett's sudden death from typhoid fever on 27 March 1931. She had spoken to him only a few days earlier, when he had told her how unwell he was feeling. It was Bennett who had kept Cohen on the rails for over a decade giving her honest, blunt necessary advice. Cohen listened to him and respected his judgement. He had guided her when she was in her 20s when her reputation and fame was growing both at home and abroad.

Depictions
Harriet Cohen was portrayed in two novels.

Rebecca West based the main character of Harriet in her novel Harriet Hume (1929) on Harriet Cohen. The novel is described as a modernist story about a piano-playing prodigy and her obsessive lover, a corrupt politician. The novel immortalised Harriet's love affair with the composer Arnold Bax.

William Gerhardie cast Cohen as the heroine Helen Sapphire in the book Pending Heaven and much of what is written mirrors Cohen's own life and character as well as her turbulent relationship with Gerhardie. Helen Sapphire is a musician who performs successfully all over Europe. She plays the harp and the piano. Gerhardie personified himself in the central character of Max who dreams about Helen.

She was portrayed by Glenda Jackson in the 1992 Ken Russell film The Secret Life of Arnold Bax.

Dearest Tania, a words-and-music programme telling the story of Cohen, written by Duncan Honeybourne, premiered in 2006, performed with actress Louisa Clein.

Legacy
The Harriet Cohen International Music Award was founded in 1951. The Harriet Cohen Bach Prize was established in 1994. It is awarded by the Musicians' Company for the most deserving pupil at the Royal Academy of Music in the field of Bach piano playing. The most recent recipient was Adam Heron in 2020.
 * Harriet Cohen International Music Award
 * Harriet Cohen Bach Prize

Cohen was an art collector, and left over 40 paintings to the Royal Academy of Music, including works by Marc Chagall and Camille Pissarro, as well as British artists such as Duncan Grant, J. D. Fergusson, C. R. W. Nevinson, and Edward Wolfe.

title: Harriet Cohen


text:
List of national roads in Latvia

This is a list of national roads in Latvia:

State local roads
Approximately one thousand roads in Latvia are categorised under the state local road category. The total length of the local roads is 13181.616 km, of which 10570.151 km are hard-surface and 2611.465 km are crushed stone or graveled.

These roads are designated by the letter V by the Latvian State Roads, but this classification not used in signage.

title: List of national roads in Latvia


text:
Tomography

Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, cosmochemistry, astrophysics, quantum information, and other areas of science. The word tomography is derived from Ancient Greek τόμος tomos, "slice, section" and γράφω graphō, "to write" or, in this context as well, "to describe." A device used in tomography is called a tomograph, while the image produced is a tomogram.

In many cases, the production of these images is based on the mathematical procedure tomographic reconstruction, such as X-ray computed tomography technically being produced from multiple projectional radiographs. Many different reconstruction algorithms exist. Most algorithms fall into one of two categories: filtered back projection (FBP) and iterative reconstruction (IR). These procedures give inexact results: they represent a compromise between accuracy and computation time required. FBP demands fewer computational resources, while IR generally produces fewer artifacts (errors in the reconstruction) at a higher computing cost.

Although MRI (magnetic resonance imaging), optical coherence tomography and ultrasound are transmission methods, they typically do not require movement of the transmitter to acquire data from different directions. In MRI, both projections and higher spatial harmonics are sampled by applying spatially varying magnetic fields; no moving parts are necessary to generate an image. On the other hand, since ultrasound and optical coherence tomography uses time-of-flight to spatially encode the received signal, it is not strictly a tomographic method and does not require multiple image acquisitions.

Types of tomography
Some recent advances rely on using simultaneously integrated physical phenomena, e.g. X-rays for both CT and angiography, combined CT/MRI and combined CT/PET.

Discrete tomography and Geometric tomography, on the other hand, are research areas that deal with the reconstruction of objects that are discrete (such as crystals) or homogeneous. They are concerned with reconstruction methods, and as such they are not restricted to any of the particular (experimental) tomography methods listed above.

Synchrotron X-ray tomographic microscopy
A new technique called synchrotron X-ray tomographic microscopy (SRXTM) allows for detailed three-dimensional scanning of fossils.

The construction of third-generation synchrotron sources combined with the tremendous improvement of detector technology, data storage and processing capabilities since the 1990s has led to a boost of high-end synchrotron tomography in materials research with a wide range of different applications, e.g. the visualization and quantitative analysis of differently absorbing phases, microporosities, cracks, precipitates or grains in a specimen. Synchrotron radiation is created by accelerating free particles in high vacuum. By the laws of electrodynamics this acceleration leads to the emission of electromagnetic radiation (Jackson, 1975). Linear particle acceleration is one possibility, but apart from the very high electric fields one would need it is more practical to hold the charged particles on a closed trajectory in order to obtain a source of continuous radiation. Magnetic fields are used to force the particles onto the desired orbit and prevent them from flying in a straight line. The radial acceleration associated with the change of direction then generates radiation.

Volume rendering
Volume rendering is a set of techniques used to display a 2D projection of a 3D discretely sampled data set, typically a 3D scalar field. A typical 3D data set is a group of 2D slice images acquired, for example, by a CT, MRI, or MicroCT scanner. These are usually acquired in a regular pattern (e.g., one slice every millimeter) and usually have a regular number of image pixels in a regular pattern. This is an example of a regular volumetric grid, with each volume element, or voxel represented by a single value that is obtained by sampling the immediate area surrounding the voxel.

To render a 2D projection of the 3D data set, one first needs to define a camera in space relative to the volume. Also, one needs to define the opacity and color of every voxel. This is usually defined using an RGBA (for red, green, blue, alpha) transfer function that defines the RGBA value for every possible voxel value.

For example, a volume may be viewed by extracting isosurfaces (surfaces of equal values) from the volume and rendering them as polygonal meshes or by rendering the volume directly as a block of data. The marching cubes algorithm is a common technique for extracting an isosurface from volume data. Direct volume rendering is a computationally intensive task that may be performed in several ways.

History
Focal plane tomography was developed in the 1930s by the radiologist Alessandro Vallebona, and proved useful in reducing the problem of superimposition of structures in projectional radiography.

In a 1953 article in the medical journal Chest, B. Pollak of the Fort William Sanatorium described the use of planography, another term for tomography.

Focal plane tomography remained the conventional form of tomography until being largely replaced by mainly computed tomography in the late 1970s. Focal plane tomography uses the fact that the focal plane appears sharper, while structures in other planes appear blurred. By moving an X-ray source and the film in opposite directions during the exposure, and modifying the direction and extent of the movement, operators can select different focal planes which contain the structures of interest.

title: Tomography