On August 3 I posted a proof containing definitions, theorems, a postulate, a proof with much discussion of each step in the proof, and some conclusions from the proof. What I termed a postulate in the proof, the existence of a bow effect was explained as a postulate bbecause "Griffin has made reference to the decreasing change in expectations between adjacent true counts as true counts rise and a large number of sims I have done invariably support the proposition but it is impossible IMHO to definitively prove the proposition."
Based on my opinion it is impossible to definitively prove the proposition, the proof was put in an "If...then" mode and the conclusion of the proof was a conditional one, not a definitive one. The full text of the August 3 post may be found at http://bj21satellite.com/misc_bj/index.cgi?read=117222 .
For those who do not want to go to the trouble at this time of reading the complete post, the text of the postulate follows:
"Postulate.
The �bow� effect is the accurate description of the difference in expectations between two adjacent true counts. That is, ((E(i+1,m) - E(i,m)) < ((E(i,m) - E(i-1,m). (note) This proposition is postulated for the purposes of this proof because there is no definitive literature on this topic as there is on the Theorems set out above. Griffin has made reference to the decreasing change in expectations between adjacent true counts as true counts rise and a large number of sims I have done invariably support the proposition but it is impossible IMHO to definitively prove the proposition. Every possibility cannot be simmed and I see no way to show the effect theoretically. It is my strong belief the effect exists because there is no evidence to the contrary I know of. (note) It should be stressed that, if there is such an effect, the aggregate average expectancies associated with each true count obey the law even if some setups of true counts would not (replacing an eight by a two will not cause as large a change in expectation as will replacing an eight by a five) because, if the average expectations do not, the �bow� effect would not be observed. Support for this postulate is provided by my sims because one can find the same tendency exists for any arbitrary grouping of cards if one high card is removed at a time. (note) Experience, also, supports the �bow� effect proposition; more twenty pushes the higher the count, more chances to double 9,10,11 against stiffs at low counts and a higher loss rate, etc. (note) I will return to this proposition in my NOTES. (note) This exercise began as a response to Cant�s nonmathematical �proof� so as a response to his posts, this postulate has to be acceptable because he cites it (which, admittedly means little) but, after I began the exercise, I saw some implications more important than answering a crank�s posts and I am posting the proof now in a larger context."
There are some important ideas relevant to this post contained in the postulate. "It should be stressed that, if there is such an effect, the aggregate average expectancies associated with each true count obey the law even if some setups of true counts would not (replacing an eight by a two will not cause as large a change in expectation as will replacing an eight by a five) because, if the average expectations do not, the �bow� effect would not be observed."
In this passage I made clear I was talking about the expected value of true counts of m size in the context of the proof by using the words "aggregate average expectancies associated with each true count" which is the same as saying the expected value of true counts of m size in the context of the proof.
The original proof post has been published since August 3 but on bjmath yesterday Cant posted:
ML is in total error
Posted By: Clarke Cant
Date: Friday, 6 September 2002, at 2:00 p.m.
In Response To: One definition for expected value (ML)
if E(i+1,m)-E(i,m)< E(i,m)-E(i-1,m) for all i's then a floating advantage exists.
This statement is wrong in that the difference between the true count perdiction (sic) of edge and actual edge can be positive or negative. Nor is the above true for mean compositions in that prior distributions, as per my totally correct proof, make the mean for a true count prediction ever less likely."
One can see Cant used exactly the mathematical formulation of the bow effect used in the original proof post in his post of yesterday.
First a quibble. Notice the statement is a provisional one, an "if, then statement. I repeat if (the postulate is true) then a floating advantage exists. It was impossible for me to be "totally wrong" in an "if, then" statement. If "E(i+1,m)-E(i,m)< E(i,m)-E(i-1,m) for all i's" is not the equivalent of saying "the floating advantage exists" or "if a bow effect exists" The bow effect was described mathematically and the postulate contained the mathematical description of a bow effect: the differences between the expected values of true counts increase as true counts decrease. If a bow effect does, in fact, exist, there is no other mathematical statement which would describe it. I defy Cant to make any other formulation. Either the bow effect exists and I used a correct mathematical statement of the bow effect or it does not exist at all so there is not such a thing as a universal bow effect and the proposition is physically wrong. But, in mathematics, one can presume what one wishes in regards to a proof so long as the statement is used in an "if, then" statement to show it a provisional one.
Cant has had more than a month to challenge my postulate for lack of universality and that would have been a legitimate mathematical criticism. There are reasons for believing in universality as I do and as I briefly stated in my notes to the postulate as posted.
But, again that argument is different from the insults of my wife's any my mathematical integrity and knowledge. The math clearly supports that if "E(i+1,m)-E(i,m)< E(i,m)-E(i-1,m) for all i's," a floating advantage exists. In thousands of words Cant has never addressed mathematics but has brought in irrelevancies. Cant, are you so stupid you do not understand "if, then" statements and attack them as if they were not "if, then" statements and insult me and my wife as incompetents while never addressing the mathematics.
Cant's writings are most of the time incomprehensible and I will admit I hardly ever understand anything he says but I have found that when he does say something understandable he is nearly always wrong.
But yesterday's bjmath post is shedding some light. He is saying "E(i+1,m)-E(i,m)< E(i,m)-E(i-1,m) for all i's" does not reflect reality. If he understood mathematics at all he would have said that in ten words a month ago as a criticism of the proof.
Now I am going to address another portion of the multitudinous Cant criticisms of my and my wife's competency in math in relation to his incompetency.
I had no idea what Cant was trying to convey by saying my proof switched means. His recent post on bjmath copied above gives some insight. The best I can ascertain (and trying to ascertain what Cant says is fraught with peril because he apparently does not have the ability to write with clarity at all) his claim I switched means from general ones to another is based on his disagreement with my postulate and nothing else. He has repeatedly said (at least I understood it that way) the bow effect only applies to fractions of "perfect" starting decks as does the floating advantage. From this he has made the claim my usage of the postulate "switched means" from general starting compositions to perfect starting compositions.
That the proof dealt in arbitrary starting stacks, not perfect ones, and dealt throughout with arbitrary starting stacks and that the postulate dealt with what happens to subsets of arbitrary starting stacks would have been clear to anyone with minimal knowledge of mathematical notation and mathematical proofs. There is abundant evidence contained throughout the proof the corollary applied to arbitrary starting stacks. For example, as part of the notes to the corollary I stated, "Every possibility cannot be simmed and I see no way to show the effect theoretically." Perfect deck compositions can be easily simmed and everyone except a Cant would know that.
But what even more clearly demonstrated in my post I was dealing with arbitrary starting stacks is contained in my Notes at the end of the proof. I said, "Since �perfect� decks are a specific case of this general proof, it would follow part of the difference is caused by he
�bow� effect if the effect speculation is a true property of card counting." I made clear I was reasoning from the general to the specific "perfect" rather than from the other way as Cant has suggested.
