From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1573 Path: news.gmane.org!not-for-mail From: "Osher Doctorow" Newsgroups: gmane.science.mathematics.categories Subject: Re: Cross-Category "conversions" of some interest Date: Fri, 14 Jul 2000 09:02:39 -0700 Message-ID: <000e01bfedad$0bf82480$aa7079a5@osherphd> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000B_01BFED72.43157E80" X-Trace: ger.gmane.org 1241017935 31884 80.91.229.2 (29 Apr 2009 15:12:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:15 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sun Jul 16 12:11:44 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id MAA12311 for categories-list; Sun, 16 Jul 2000 12:02:57 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2615.200 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2615.200 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 15 Original-Lines: 141 Xref: news.gmane.org gmane.science.mathematics.categories:1573 Archived-At: This is a multi-part message in MIME format. ------=_NextPart_000_000B_01BFED72.43157E80 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable From: Osher Doctorow osher@ix.netcom.com, Friday July 14, 2000, 8:47AM Dear Colleagues: Dr. S. J. Vickers has found two typographical errors in my July 12 = contribution which might well lead someone to conclude that an erroneous = one-sided operation on an inequality had been made. The statement = "....so the converted form reads: (1 - x + y)y^^2 =3D -/1 - x + y/y^^2 > = k," shoud have a second constant k1 or k2 (it is arbitrary which = notation is used) replacing k. I used the same k by a typographical = error and also because I had skipped an intermediate step and in my = haste used the same constant k from before. The intermediate step was = merely to consider what happens to xy when the conversion of y/x to 1 - = x + y is made. Then xy converts to (1 - x + y)y^^2 =3D -/1 - x + = y/y^^2 in the case when 1 - x + y < 0, and this is obviously = nonpositive, so -/1 - x + y/y^^2 < k2 with k2 =3D 0 for example. For 1 = - x + y > 0, we have xy converting to /1 - x + y/y^^2 which is = nonnegative and therefore /1 - x + y/y^^2 > =3D k2 with k2 =3D 0 again. = Vickers' criticism turned out to be very fortunate, not only for = clarifying the typographical error and avoiding the wrong conclusion = that I operated on only one side of an inequality when converting xy to = the other form, but also in my developing a detailed argument concerning = when the conversion from y/x to 1 - x + y becomes an actual function. = This occurs, for example, when y/x is a reduced proper or improper = fraction in the sense that numerator and denominator have no common = primes, in which case the unique factorization into primes and = consideration of the three cases y/x > 1 and y/x < 1 and y/x =3D 1 leads = to the conclusion that the conversion is a function. Thus, on the = reduced rationals, we have a function. This is not a bad set to work = with mathematically, and certainly provides a nontrivial case where the = conversion is a function and is accurate. I hope that S. J. Vickers will continue to contribute to further = discussion in this thread because of his important contributions, = provided of course that he continues to emphasize the correction of = errors and ways of further applying the conversions. If somebody finds = any further errors in my future writings, please give me the benefit of = considering the possibility that I made a typographical error. Osher Doctorow =20 ------=_NextPart_000_000B_01BFED72.43157E80 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
From: Osher Doctorow osher@ix.netcom.com, Friday July = 14, 2000,=20 8:47AM
 
Dear Colleagues:
 
Dr. S. J. Vickers has found two = typographical=20 errors in my July 12 contribution which might well lead someone to = conclude that=20 an erroneous one-sided operation on an inequality had been made.  = The=20 statement "....so the converted form reads: (1 - x + y)y^^2 =3D -/1 - x = + y/y^^2=20 > k," shoud have a second constant k1 or k2 (it is arbitrary which = notation=20 is used) replacing k.  I used the same k by a typographical error = and also=20 because I had skipped an intermediate step and in my haste used the same = constant k from before.  The intermediate step was merely to = consider what=20 happens to xy when the conversion of y/x to 1 - x + y is = made.   Then=20 xy converts to (1 - x + y)y^^2 =3D -/1 - x + y/y^^2 in the case = when 1 - x +=20 y < 0, and this is obviously nonpositive, so -/1 - x + y/y^^2 < k2 = with k2=20 =3D 0 for example.  For 1 - x + y > 0, we have xy = converting=20 to /1 - x + y/y^^2 which is nonnegative and therefore /1 - x + = y/y^^2 >=20 =3D k2 with k2 =3D 0 again.  
 
Vickers'  criticism turned out to = be very=20 fortunate, not only for clarifying the typographical error and = avoiding the=20 wrong conclusion that I operated on only one side of an = inequality when=20 converting xy to the other form, but also in my developing = a detailed=20 argument concerning when the conversion from y/x to 1 - x + y becomes an = actual=20 function.   This occurs, for example, when y/x is a reduced=20 proper or improper fraction in the sense that numerator and=20 denominator have no common primes, in which case the unique = factorization into=20 primes and consideration of the three cases y/x > 1 and y/x < 1 = and y/x =3D=20 1 leads to the conclusion that the conversion is a function.  Thus, = on the=20 reduced rationals, we have a function.  This is not a bad set to = work with=20 mathematically, and certainly provides a nontrivial case where the = conversion is=20 a function and is accurate.
 
I hope that S. J. Vickers will continue = to=20 contribute to further discussion in this thread because of his = important=20 contributions, provided of course that he continues to emphasize = the=20 correction of errors and ways of further applying the = conversions.  If=20 somebody finds any further errors in my future writings, please give me = the=20 benefit of considering the possibility that I made a typographical=20 error.
 
Osher Doctorow   =20
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