Path: csiph.com!eternal-september.org!feeder.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Ben Bacarisse Newsgroups: comp.programming Subject: Re: on reading ``o(1)'' Date: Wed, 29 Mar 2017 14:23:46 +0100 Organization: A noiseless patient Spider Lines: 52 Message-ID: <87d1d0usgt.fsf@bsb.me.uk> References: <86vaqydwqx.fsf@toledo.com> <871stj7nge.fsf@bsb.me.uk> <86efxhaqwy.fsf@toledo.com> Mime-Version: 1.0 Content-Type: text/plain Injection-Info: mx02.eternal-september.org; posting-host="93eaece5ee1a59cec50ea92cffff1d06"; logging-data="5551"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19eaGEzTUsW9BvVxyUnC+0oSk0kKRmGnAg=" Cancel-Lock: sha1:DxYz1JegHUdtY+qgVJ5XZMRrYK4= sha1:eL0Xhh3phmmssboKk+0xafTgEis= X-BSB-Auth: 1.38102a643929805f2f27.20170329142346BST.87d1d0usgt.fsf@bsb.me.uk Xref: csiph.com comp.programming:7675 Daniel Bastos writes: > Ben Bacarisse writes: > >> Daniel Bastos writes: >> >>> Suppose f in o(1). By definition, there's a natural number n(0) such >>> that for any c > 0 >>> >>> 0 <= f < c * 1 = c >>> >>> for any n >= n(0). (AFAIK, c is a real number.) That seems to imply >>> that f = 0 because that's the only thing that could be less than any >>> c > 0. >> >> Not exactly. First, I would not write n(0) so it looks like a function. > > This notation is great! Let me defend it. Consider n as a function > that takes an argument 0 and gives you the constant ``n0'' --- as you > like to write. Functions can give anything you prefers to read. :-) I respectfully disagree. By all means make n a function but make it a function of something useful. As you use the notation, it is a single-point function. What are n(1), n(-e)? What value does making it a function have when there is only one point in the co-domain? Secondly (and this is more important) you use n as not a function at the very same time: "for any n >= n(0)". Now it's possible to unravel that but, formally, the quantifier binds n in both locations. At the very least, use two names: "there is an m(0) such that for all n >= m(0) ...". And to return to the first point, if you want to make it a function, make it a useful one. The n0 in question depends on c so make it a function of c: For any c > 0 there is a natural number n(c) such that, for all k >= n(c), 0 <= f(k) < ck. > I picked this notation up with Brian M. Scott, an admirable > mathematician who used to hang out at alt.algebra.help. I strongly suspect he'd agree on this. Making things into functions is a great idea (and too few mathematician do it, relying instead on subscripts and context to make it clear that this or that quantity is linked to some other) but n is not a function when writing "for all n > ..." whatever goes in the ... -- Ben.