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Here is a direct citation from wikipedia:

The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Friedrich Götze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu proved the circular law under the minimal assumptions stated above.

What exactly does it mean to relax?

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In addition to the nice answers below, here is an example using Sugar Cubes and Coffee Cups. –  Amzoti Mar 12 '13 at 0:14
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To sit back, have a beer, enjoy the warmth afternoon sun of the Israeli spring... We relax the definitions if we give them beer and time off to sit in the sun and chill out. –  Asaf Karagila Mar 12 '13 at 0:23
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2 Answers

up vote 2 down vote accepted

It means that a weaker set of assumptions was used--assumptions that follow from the original ones, but don't necessarily imply the original ones. This may be done by using fewer assumptions, or by weakening some of the original assumptions themselves, or some combination of the two.

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so then if you relaxed something you generalized it? –  user4140 Mar 12 '13 at 0:16
    
@Jorge: No: If you can relax the assumptions of a theorem, then you have generalized it. –  John Bentin Mar 12 '13 at 0:29
    
@Jorge: Yes. That's another way to say it. –  Cameron Buie Mar 12 '13 at 2:47
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It means proving the same result using weaker assumptions, i.e. assumptions that are implied by the original assumptions, but do not necessarily imply them.

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Isn't that backwards? You want conditions that we implied by the original condition. –  Thomas Andrews Mar 12 '13 at 0:11
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