- published: 12 Sep 2014
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In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as # to distinguish from the negation of equality (the denial inequality) ≠, which is weaker.
An apartness relation is a symmetric irreflexive binary relation with the additional condition that if two elements are apart, then any other element is apart from at least one of them (this last property is often called co-transitivity or comparison).
That is, a binary relation # is an apartness relation if it satisfies:
The negation of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight. That is, # is a tight apartness relation if it additionally satisfies:
In classical mathematics, it also follows that every apartness relation is the negation of an equivalence relation, and the only tight apartness relation on a given set is the negation of equality. So in that domain, the concept is not useful. In constructive mathematics, however, this is not the case.
Michael Mayer may refer to:
Jonathan Neil Mendelsohn, Baron Mendelsohn (born 30 December 1966) is a British lobbyist and Labour political organiser. He was appointed the Director of General Election Resources for the party in 2007.
With Neal Lawson and Ben Lucas, he founded LLM Communications in 1997, a lobbying firm with notable access to the new Labour Government. He has been a spokesman and lobbyist for the gambling company PartyGaming.
He had written to controversial Labour donor David Abrahams claiming he was one of Labour's "strongest supporters". In 1998 he was caught on tape along with Derek Draper boasting to Greg Palast, an undercover reporter posing as a businessman, about how they could sell access to government ministers and create tax breaks for their clients in a scandal that was dubbed "Lobbygate". Draper denied the allegations.
In the same incident Mendelsohn was approached by an undercover journalist posing as a representative of American energy companies who were seeking to ignore environmental laws. Despite LLM's claim that "we believe that there will be a new breed of 'ethical winners' who will demonstrate that businesses no longer operate in a moral vacuum", Mendelsohn went on to advise the reporter to rephrase their plans into language that sounded "Earth-Friendly" going on to say "Tony [Blair] is very anxious to be seen as green. Everything has to be couched in environmental language – even if it's slightly Orwellian."
Kasper Bjørke is a Danish DJ, record producer and remixer from Copenhagen.
Kasper Bjørke started making music in the disco house duo Filur with Tomas Barfod in 1999. Filur released four studio albums: Exciting Comfort (2000), Deeply Superficial (2002), Into the Wasteland (2006), Faces (2011) and the “Best Of compilation” In Retrospect (2009). They are known for the hits „Its Alright“, "I Want You" and later for "The Con" and "Concentrate!" and for working with guest vocalists such as Stina Nordenstam, Pernille Rosendahl, Magnum Coltrane Price, Simon Kvamm (Nephew), Daniel Agust (GusGus), Josephine Philip (Darkness Falls), Nellie Ettison and many more. They frequently toured the United States, Europe and Japan where they also reached the top of Tokyo Hot 100 chart and sold 40.000 albums. Filur played Arena Stage at Roskilde Festival as their final liveshow in 2003. After that they only performed as DJs. Filur won two DMA Awards for "Best Producers" and "Best Dance Album" and several other awards incl. a DR P3 Guld award for Best Hit. Filur stopped in 2012.
Modern Synth Pop anthem on HNF Music www.facebook.com/kasperbjorke Get the track here: www.facebook.com/hfnmusic Art by the American visual artist John Copeland A disco noir track taken from Kasper Bjørke's new album 'After Forever.' Singles 'Rush' and 'Sylvia' were strong but with 'Apart' Kasper's created a modern day dark disco anthem that can proudly sit alongside the likes of Sharon Brown in the hall of fame. The press release describes it as 'commercially friendly'. And it may have charted in 1987. But nowadays the chance of this getting Radio 1 A-listed is slim. It's catchy for sure, but its dark, twisted pop overtones are more akin to French Kiss 'Micheif' - itself a prototype techno anthem from 1979. The drums are 80s and unprocessed. The vocals are female and forlorn. The synth...
Music video by Ricky Blaze performing Apart. (C) 2016 FME Recordings, LLC, under exclusive licence to Island Records, a division of Universal Music Operations Limited http://vevo.ly/MjgiR7
Check it out on Apple Music: https://t.co/2gOZM1mqfV
Download on iTunes: http://bit.ly/ApartRmx_iT Listen on Spotify: http://bit.ly/InMyOpinionSP Grab your copy on Beatport: http://bit.ly/ApartRmx_BP Subscribe to Armada TV: http://bit.ly/SubscribeArmada Driving down the freeway slightly too fast... There's a spot inside all of us that cries out for a beautiful radio tune, no matter how much we try to convince our friends that it's "too commercial". Given the backgrounds of both Ørjan Nilsen & Jonathan Mendelsohn, it's clear that these guys have a very "credible" pedigree. The melody, 'Jonathan Mendelsohn' vocals, and the lyrics of 'Apart' are transcendental. No matter what package they're put in, they work wonders. Mike Shiver, and Martin Volt & Quentin State had a field day with their respective mixes, bringing this anthemic gem to the clu...
Hailing from Denmark, with an appearance that leads you to suspect that he knows his way around the studio (and that's a good thing as far as we're concerned). Kasper Bjørke (Eskimo, Correspondant) is not what one would call a household name. However, over the years he has made a name for himself as a club DJ, radio DJ, producer, remixer, label manager and inbetween all these activities he manages his own career as an artist, as well as that of Trentemøller for instance. He began producing music in the form of the Danish disco house duo named Filur. And even before 2007, when he started his solo career under his real name, the duo managed to release 4 studio albums. But after 2007 things really took off, possible kick started by Bjørke 's cover version of a Rolling Stones track (Heaven) th...
Music video by She Wants Revenge performing Tear You Apart. (C) 2005 Flawless / Geffen Records
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He waits for her to understand
But she won't understand at all
She waits all night for him to call
But he won't call anymore
He waits to hear her say
'Forgive'
But she justs drops her pearl-black eyes
And prays to hear him say
'I love you'
But he tells no more lies
He waits for her to sympathize
But she won't sympathize at all
She waits all night to feel his kiss
But always wakes alone
He waits to hear her say
'Forget'
But she just hangs her head in pain
And prays to hear him say
'No more
I'll never leave again'
How did we get this far apart?
We used to be so close together
How did we get this far apart?
I thought this love would last for ever
He waits for her to understand
But she won't understand at all
She waits all night for him to call
But he won't call
He waits to hear her say
'Forgive'
But she just drops her pearl black eyes
And prays to hear him say
'I love you'
But he tells no more lies
How did we get this far apart?
We used to be so close together
How did we get this far apart?
I thought this love would last for ever