- published: 24 May 2016
- views: 263316
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved. Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological. A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to "separate the space cleanly", unless an extra condition of tameness is used to suppress possible wild behaviour. See Jordan-Schönflies theorem.
One can therefore say that (particularly in mathematical analysis and set theory) those searching for the "pathological" are like experimentalists, interested in knocking down potential theorems, in contrast to finding general statements widely applicable. Each activity has its role within mathematics.
A classic example is the Weierstrass function, which is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, by the Baire category theorem one can show that continuous functions are typically or generically nowhere differentiable.
Dylan Kwabena Mills (born 18 September 1984), better known by his stage name Dizzee Rascal, is a UK and US-based rapper, songwriter and record producer of Ghanaian and Nigerian descent. His music is a blend of garage, hip hop, grime, ragga, pop and electronic music, with eclectic samples and more exotic styles. Best known for his number-one hits "Dance wiv Me", "Bonkers", "Holiday", "Dirtee Disco" & "Shout", his debut album, Boy in da Corner, won him the 2003 Mercury Prize. Follow-up albums Showtime, Maths + English and Tongue n' Cheek have all been critically acclaimed and certified gold, the last going platinum for sales exceeding 300,000 units.
Mills' Nigerian father died when he was young, and he was raised in Bow, in the East End of London in a single-parent family by his Ghanaian mother Priscilla, about whom he says: "I had issues as a kid. I was violent and disruptive. The way my mum helped was by finding me a different school every time I got kicked out, always fighting to keep me in the school system" he attended St Pauls Way Community School.
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad behaviour was my savior
Makin' mischief used to make my day
Subsequently, accidentally
I put my fist through a windowpane
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
I remember once I had a brain
Amnesia never crossed my mind
Tonsillitis was my alibi
Truancy was my vocation
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour
Bad, bad, bad, bad behaviour