From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1563 Path: news.gmane.org!not-for-mail From: "Osher Doctorow" Newsgroups: gmane.science.mathematics.categories Subject: Cross-category "conversions" of some interest Date: Sat, 8 Jul 2000 10:05:44 -0700 Message-ID: <000a01bfe8fe$f2328e80$847979a5@osherphd> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0007_01BFE8C4.150BEB80" X-Trace: ger.gmane.org 1241017926 31831 80.91.229.2 (29 Apr 2009 15:12:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:06 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sat Jul 8 19:57:18 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id TAA29242 for categories-list; Sat, 8 Jul 2000 19:56:14 -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: 5 Original-Lines: 179 Xref: news.gmane.org gmane.science.mathematics.categories:1563 Archived-At: This is a multi-part message in MIME format. ------=_NextPart_000_0007_01BFE8C4.150BEB80 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sat. July 7, 2000, = 9:46AM Dear Colleagues: I have been studying the "conversion" from x/y to 1 - x + y and the = conversion from x/y to x + y - xy, where x and y are elements of any = category (although in practice I have been restricting attention to = probability-statistics (x =3D Pr(A), y =3D Pr(AB) where AB is the = intersection of sets/events A and B) and (fuzzy) multivalued logics such = as Goguen/Product (G) and Lukaciewicz (L) and Godel (Go) (in the case = of G, x/y is the non-trivial implication x --> y and in the case of L, 1 = - x + y is the non-trivial implication x --> y) and the Jacobson radical = in ring theory (which is based on the circle composition product x*y =3D = x + y - xy for x and y elements of the ring) and Fermat's Last Theorem = in number theory (x and y are integers and a super-super short proof = seems to depend on generalizing x*y to n dimensions and using the = conjugate which will be described below). The conjugate ^ of 1 - x + y = is (1 - x + y)^ =3D 1 + x - y, and the conjugate of x + y - xy is x + y = + xy. An n dimensional generalization would involve x^^n + y^^n = where ^^ is exponentiation (unrelated to ^) since the product of 1 - x + = y and its conjugate can be shown to involve x^^2 + y^^2 - x^^2 y^^2 =3D = x^^2 * y^^2 and so on. The expression "conversion" is used above = instead of function because y/x --> 1 - x + y is not a function but a = conversion of the division operation (for x non-zero) to subtraction and = the addition of 1. =20 The probability-statistics reader may recognize x/y and (Bayesian) = conditional probability (BCP for short) written Pr(B/A) for Pr(A) = non-zero, and it turns out that 1 - x + y for probability-statistics is = Pr(A-->B) =3D Pr(A' U B) =3D Pr(A' ) + Pr(B) - Pr(A' B) =3D Pr(A' ) + = Pr(AB) =3D 1 - Pr(A) + P(AB), which latter expression is maximized for = very rare events and lower dimensional events when probability = distributions are continuous on a volume of space containing A and B = (Pr(A) =3D 0 for those cases) and also when A is a subset of B. = Pr(A-->B) is abbreviated LBP for logic-based probability, which I have = been developing since 1980. BCP applies to frequent/common and = independent-like or low influence events (including Markov processes = which have many similarities to independent events although they are = "slightly" dependent) and LBP applies to rare events or rare-like events = (Pr(A) less than epsilon for epsilon small positive) and to n-k = dimensional subset events of n-dimensional Euclidean space for k =3D 0 = to n and to highly dependent or highly one or two-way influencing = events.=20 As for Goguen/Product logic (G) and Lukaciewicz logic, their union or = "join" G U L, similarly to the union or join of either of them with = Godel logic G, equals BL2, the basic (fuzzy) multi-valued logic which = generalizes Boolean logic with the deduction Theorem and the plausible = axiom p V ~p (p or ~p) for each proposition p. Thus, G and L = constitute all of the universe of logic in this sense, and they = partition it into disjoint "roughly equal" parts. Osher Doctorow Doctorow Consultants Culver City, California USA =20 =20 ------=_NextPart_000_0007_01BFE8C4.150BEB80 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sat. July = 7, 2000,=20 9:46AM
 
Dear Colleagues:
 
I have been studying = the "conversion" from x/y=20 to 1 - x + y and the conversion from x/y to x + y - xy, where x and y = are=20 elements of any category (although in practice I have been restricting = attention=20 to probability-statistics (x =3D Pr(A), y =3D Pr(AB) where AB is = the=20 intersection of sets/events A and B) and (fuzzy) multivalued logics = such as=20 Goguen/Product (G) and Lukaciewicz (L) and Godel (Go)  (in the case = of G,=20 x/y is the non-trivial implication x --> y and in the case of L, 1 - = x + y is=20 the non-trivial implication x --> y) and the Jacobson radical in ring = theory=20 (which is based on the circle composition product x*y =3D x + y - xy for = x and y=20 elements of the ring) and Fermat's Last Theorem in number theory (x = and y=20 are integers and a super-super short proof seems to depend on = generalizing=20 x*y to n dimensions and using the conjugate which will be described=20 below).  The conjugate ^ of 1 - x + y is (1 - x + y)^ =3D 1 + = x - y, and=20 the conjugate of x + y - xy is x + y + xy.    An n=20 dimensional generalization would involve x^^n + y^^n  where ^^ = is=20 exponentiation (unrelated to ^) since the product of 1 - x + y and its = conjugate=20 can be shown to involve x^^2 + y^^2 - x^^2 y^^2 =3D x^^2 * y^^2 and = so=20 on.  The expression "conversion" is used above instead of function=20 because y/x --> 1 - x + y is not a function but a conversion of = the=20 division operation (for x non-zero) to subtraction and the addition of = 1. =20   
 
The probability-statistics reader may = recognize x/y=20 and (Bayesian) conditional probability (BCP for short) written = Pr(B/A) for=20 Pr(A) non-zero, and it turns out that 1 - x + y for = probability-statistics=20 is Pr(A-->B) =3D Pr(A' U B) =3D Pr(A' ) + Pr(B) - Pr(A' B) =3D Pr(A' = ) + Pr(AB) =3D 1=20 - Pr(A) + P(AB), which latter expression is maximized for very rare = events=20 and lower dimensional events when probability distributions are = continuous=20 on a volume of space containing A and B (Pr(A) =3D 0 for = those cases) and=20 also when A is a subset of B.  Pr(A-->B) is abbreviated LBP for=20 logic-based probability, which I have been developing since = 1980.  =20 BCP applies to frequent/common and independent-like or low influence = events=20 (including Markov processes which have many similarities to=20 independent events although they are "slightly" dependent) and LBP = applies=20 to rare events or rare-like events (Pr(A) less than epsilon for epsilon = small=20 positive) and to n-k dimensional subset events of n-dimensional = Euclidean space=20 for k =3D 0 to n and to highly dependent or highly one or two-way = influencing=20 events. 
 
As for Goguen/Product logic = (G) and=20 Lukaciewicz logic, their union or "join" G U L, similarly to the union=20 or join of either of them with Godel logic G, equals BL2, the = basic=20 (fuzzy) multi-valued logic which generalizes Boolean logic with the = deduction=20 Theorem and the plausible axiom p V ~p (p or ~p) for each proposition=20 p.   Thus, G and L constitute all of the universe of = logic in=20 this sense, and they partition it into disjoint "roughly equal"=20 parts.
 
Osher Doctorow
Doctorow Consultants
Culver City, California = USA   =20
 
  
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