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One of the most frequently-cited is a result involving [[oracle machine|oracles]]. Imagine you have a magical machine called an ''oracle'' that can solve only one problem, such as determining if a given number is prime, but can solve it in constant time. Our new question is now, if we're allowed to use this oracle as much as we want, are there problems we can verify in polynomial time that we can't solve in polynomial time? It turns out that, depending on the problem that the oracle solves, with certain oracles one has '''P''' = '''NP''', while for other oracles one has '''P''' &ne; '''NP'''. The practical consequence of this is that any proof which can be modified to account for the existence of these oracles cannot solve the problem. Unfortunately, most known methods and nearly all classical methods can be modified in such a way (we say they are ''relativizing''). | [
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Alternatively written: Given <math>I,D\subset\mathbb{R}</math> (that is, ''I'' and ''D'' are subsets of the [[real number]]s), continuity of <math>f:I \to D</math> (read <math>f</math> maps ''I'' into ''D'') at <math>c\in\mathbb{R}</math> means that for all <math>\varepsilon>0</math> there exists a <math>\delta>0</math> such that <math>|x-c|<\delta</math> and <math>x\in I</math> imply that <math>|f(x)-f(c)|<\varepsilon.</math> | [
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* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''[[Counterexamples in Topology]]''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition). ''(See example 29)''. | [
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Coca-Cola's advertising has had a significant impact on American culture, and is frequently credited with the "invention" of the modern image of [[Santa Claus]] as an old man in red-and-white garments; however, while the company did in fact start promoting this image in the 1930s in its winter advertising campaigns, it was already common before that.{{ref|Claus}} In the 1970s, a song from a Coca-Cola commercial called "[[I'd Like to Teach the World to Sing]]", produced by [[Roquel Billy Davis|Billy Davis]], became a popular hit single, but there is no evidence that it did anything to increase sales of the soft drink. | [
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On the Court, Thomas has argued for an [[originalism|originalist]] or "textualist" view of the Constitution faithful to that document's text and history. Especially early in his term on the court, critics often suggested that Thomas lacked a judicial philosophy of his own, and that he unreflectingly signed on to the opinions of Justice [[Antonin Scalia]]. Although Thomas has frequently voted with Scalia, his opinions have sometimes diverged from Scalia's based on Thomas's alternative readings of Constitutional history. Scalia appears much more willing to be guided by precedent in his decisions, even when he disagrees with the precedent in question. He has said that Thomas "doesn't believe in ''[[stare decisis]]'', period." Although both Thomas and Scalia are considered the court's conservative wing, originalism does not inherently or intrinsically favor conservative political views or liberal political views. Consequently, Thomas's originalism occasionally leads him to take what appears to be a "liberal" position on various issues, as discussed below. | [
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#{{note|Death_of_the_Virgin_Baglione}}Giovanni Baglione, ''Le vite de' pittori'', 1642: "For the Madonna della Scala in Trastevere he painted the death of the Madonna, but because he had portrayed the Madonna with little decorum, swollen and with bare legs, it was taken away, and the Duke of Mantua bought it and placed it in his most noble gallery." | [
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