HDTenEightyP/NetNews huggingface dataset:
From: lew@cbnewsd.cb.att.com (lewis.h.mammel..jr)
Newsgroups: sci.math
Subject: Re: Lagrange successor rule
Summary: bayesian probability
Date: 23 Dec 92 03:17:20 GMT
Organization: AT&T

In article <3207@devnull.mpd.tandem.com>, garyb@mpd.tandem.com (Gary Bjerke) writes:
> 
> I was thumbing through an old statistics textbook when I came across the 
> Lagrange successor rule. The example given was that of a collection of coins
> for which the probability of flipping a head is uniformly distributed over
> the set of values {1/N, 2/N, ..., N/N} for N some arbitrary integer. The rule
> states that the probability of getting a head on the (n+1)th flip given that
> the first n flips were heads, is n/(n+1).
> 
> Note that N is arbitrary and not a parameter is the final result. The textbook
> said Lagrange had used this to prove that the sun was extremely likely to rise 
> tomorrow, given that it had risen every day for the past 5000 years (the 
> probability is [5000*365]/[5000*365+1], to be precise). 
> 
> I followed the proof, but I have absolutely no intuition for this result at
> all. I even fail to see how it applies to the rising of the sun (in what sense
> does the unconditional probability of its rising meet the uniform-distribution
> requirements?) Can somebody help me get a gut feel for what this result means?

It's supposed to model your a priori expectations, which are taken
to be extremely open-minded. Part of the idea is that after evidence
accumulates the conditional expectation will depend weakly on the
exact a priori distribution, as long as its very broad.

Conversely, if I insist that I KNOW ( for example ) that an
atom has a probability per unit time of decaying of 100/sec, but
I've been watching it for a week and it's still there, I'm within
the bounds of rationality if I maintain my claim. Realistically
though, I would probably start to develop doubts, since my certainty
would force me to accept that an event with a probability of
exp(-60480000) ( and getting exp(-100) smaller every second ) has
occurred.

This means I should have modeled my a priori expectation of the
intrinsic decay rate with a strong peak at 100/sec, and some broad
background. As time goes on without a decay, the longer decay times
will quickly gain strength in my conditioned expectations, even if
I was 99.9999999999999999999999999999999999% sure of that 100/sec
to start with.

Lew Mammel, Jr.
Newsgroups: sci.math
From: tim@maths.tcd.ie (Timothy Murphy)
Subject: Re: Two problems
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.
Date: Wed, 30 Dec 1992 18:51:16 GMT

israel@unixg.ubc.ca (Robert B. Israel) writes:

>Suppose there are n couples.  
>I'll assume that the men are seated in numerical order, so husband 
>#i+1 is two places to the right of #i.

I like your silent assumption of perfect etiquette!
Thinks: must revise my idea of California ...


-- 
Timothy Murphy  
e-mail: tim@maths.tcd.ie
tel: +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
Xref: sparky sci.math:17408 sci.logic:2480
From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: sci.math,sci.logic
Subject: Re: The Continuum Hypothesis:  Must it be {True or False}
Date: 25 Dec 92 04:33:11 GMT
Organization: The Phallogocentric Cabal

In article <1992Dec24.161747.18827@husc3.harvard.edu> I wrote:

>FLT is a \Pi^0_1 sentence.  In any first-order, or otherwise complete
>theory, undecidability of a \Pi^0_1 sentence requires that it be false
>in some models of the theory, but true in others; for otherwise by
>semantic completeness it would have been provable or refutable,
>according as it were valid or contradictory.  The question of truth
>reduces to whether the sentence can be false in a standard model, --
>iff not, it is by definition true.  Now, by Tarski's definition of
>satisfaction, falsehood in the models of the theory depends on the
>existence therein of a witness set for the \Sigma^0_1 negation of the
>original sentence; but in case of an arithmetical sentence, these
>witnesses must be the integers.  If they are standard integers, then
>they have names in the language of the theory, and therefore the
>original sentence receives an explicit refutation in the latter, in
>contradiction to the hypothesis of undecidability.  On the other hand,
>if all the witnesses in all the relevant models are *non-standard*
>integers, then _a fortiori_ the proposition in question is true in all
>standard models, and so true _simpliciter_, albeit not valid. Hence,
>if a \Pi^0_1 sentence is undecidable in a semantically complete theory
>of Peano Arithmetic, wherein provability is coincident with validity,
>then it is _ipso facto_ true.
>
>On the other hand, in second-order PA, all models are standard by
>categoricity; in other words, the adoption of a full strength
>induction axiom allows us to characterize the integers up to
>isomorphism.  But then the completeness argument, as distinct from
>soundness alone, is likewise inapplicable for standard second-order
>semantics.  (All we can get is that undecidability in second-order PA
>entails falsehood in a faithful (satisfying AC and every instance of
>comprehension) Henkin model.  Henkin semantics is non-categorical, and
>generally is equivalent to the case of many-sorted first-order logic,
>so I think that the first-order situation is recapitulated in this
>case.)  But the moral of this story is that there is a critical
>difference between undecidability in a first-order theory, which by
>L\"owenheim-Skolem theorems cannot characterize infinite structures up
>to isomorphism, and a higher-order theory, which can do so.

To continue, suppose a \Pi^0_1 sentence is undecidable in second-order
PA; suppose further that it is not valid, i.e. there is a model of
second-order PA,in which it is false.  By categoricity, such a model
would contain standard integer witnesses to its negation, which could
be used in an explicit refutation of the sentence in question.
Contradiction.
 
cordially,
mikhail zeleny@husc.harvard.edu
"Nothing can be said truly of what does not exist."
Newsgroups: sci.math
From: martin@lyra.cis.umassd.edu (Gary Martin)
Subject: Re: 2 questions (GRE Subject test)
Organization: University of Massachusetts Dartmouth
Date: Fri, 25 Dec 1992 17:37:17 GMT

In article <1992Dec25.040617.26522@news.columbia.edu> sla7@cunixa.cc.columbia.edu (Seth Louber Antiles) writes:

   Saw this question in a sample GRE math subject test:

   If f(1+x)=f(x) for all real x. If f is a polynomial and f(5)=11 then
   what is f(15/2)? 

   In general, for f a polynomial, for what g will f(g(x))=f(x) for all x
   *not* imply that f is a constant? Appreciate simple examples.

If  g  has an infinite orbit or finite orbits of unbounded cardinality,
then  f  must be constant because the degree of  f  bounds the number
of solutions to  f(x) = c  for any  c.  This suggests looking for
functions  g  for which  g^n is the identity function for some  n.
For example  g(x) = -x. Or perhaps a finite-to-one idempotent function
like g(x) = |x|.

   What if we remove the restriction that f is a polynomial?

Too big a question.

   Another GRE question in the same sample:

   "No element (other than identity) is its own inverse." What does this
   tell you about a group?

That it's solvable. :-)

--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
Newsgroups: sci.math,comp.edu,misc.education
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Integration  Was: Re: Student attitudes
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Fri, 18 Dec 92 18:27:43 GMT

In article <PCG.92Dec14175129@aberdb.aber.ac.uk> pcg@aber.ac.uk (Piercarlo Grandi) writes:

>I am not a mathematician, but I cannot keep my trap shout on this. The
>usual way I explain integrals is:

       Integration is a purely formal exercise in which a formula is
        transformed into another formula, using particularly odd rules.

Actually, this sounds like a nonmathematician's definition of
"mathematics"!  :-)
From: mokie@cco.caltech.edu (Michael L. Brundage)
Newsgroups: sci.math
Subject: Re: Multiples of 9
Date: 21 Dec 1992 11:20:52 GMT
Organization: California Institute of Technology, Pasadena
NNTP-Posting-Host: sandman.caltech.edu


Or, in simplest of terms (without congruences), if you take a number written
in decimal form, i.e.
  n = a0 + a1*10 + a2*10^2 + a3*10^3 + ... + ak*10^k
you see that this can also be expressed
  n = a0 + a1*(9+1) + a2*(9^2 + 2*9 + 1) + a3*(9^3 + 3*9^2 + 3*9 + 1) + ...
    = a0 + a1 + a2 + ... + ak + 9Q
where Q is a bunch of garbage that you don't care about.
But then the left-hand-side (LHS) is divisible by a positive integer whenever
the right-hand-side (RHS) is also; so n is divisible by 9 whenever the sum of
its digits is, since 9*Q is always divisible by 9.
[Or, as the previous reply pointed out, n is congruent to the sum of its
  digits mod 9:
	n # (a0+a1+...+ak) (mod 9)    .]


-- 
140311040009140015091300081200051603002503150826260826150009140002081213092730
mokie@cco.caltech.edu  | "This hope we have as an anchor of the soul, a hope
                       |    both sure and steadfast, and one which enters
                       |    within the veil."  Hebrews 6:19 (NASVB)
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Organization: Computer Science Department,  Stanford University.
Date: Thu, 31 Dec 1992 04:58:51 GMT

In article <VICTOR.92Dec30104755@terse.watson.ibm.com> victor@watson.ibm.com writes:
>The following simple proof comes from "Inequalities" by P. Korovkin,
>Blaidell Scientific paperbacks.
>
>set x_n = (1+1/n)^n, and y_n = (1+1/n)^{n+1}.
>
>We show that x_n is monotone increasing and y_n is monotone
>descreasing.  Since x_n < y_n we find that they both approach a common
>limit, which we call e.

This only shows that x_n and y_n each approach a limit, not necessarily
the same one.  For the purposes of defining e it suffices to take e to
be the limit of x_n.  However y_n is particularly convenient for
showing that x_n and y_n do approach the same limit, y_n/x_n being
simply 1+1/n.

The use of the arithmetic-geometric mean inequality in Korovkin's
argument is not only slick but shows that the whole sequence (1+1/n)^n
is monotone, not just the n-a-power-of-2 part (Michael Somos pointed
this argument out to me yesterday).  However it makes the proof that
much less elementary, not that the inequality is hard to prove but
including the proof does add a few lines.

Korovkin's y_n gives me an idea for shortening my elementary
powers-of-2 proof that (1+1/N)^N has a limit.  I'll continue to write N
for powers of 2 and n for positive integers.

                     1 N
Lemma 1.  X  =  (1 + -)  is monotone increasing.
           N         N

            1           1   2              1  N           1   2N
Proof.  1 + -  <  ( 1 + -- ) , hence ( 1 + - )   <  ( 1 + -- )
            N           2N                 N              2N


Lemma 2.  Ditto with - in place of + throughout, call this sequence Y .
                                                                     N

Proposition.  X  has a limit.
               N

                  1 N         1  N
Proof.  X  = (1 + -)  < (1 + ---)   =  1/Y   <  1/Y .  So X  is bounded.  QED
         N        N          N-1          N        1       N

This ought to be getting close to the shortest possible elementary
proof of this proposition.


We can also show that Y    has a limit, namely 1/e, by tying it to X  thus:
                       N+1                                          N


                       n+1  n    n   n+1     n
Lemma 3.    X Y    = ( --- )  ( --- )    =  --- , for all positive n.
             n n+1      n       n+1         n+1

Note that Lemma 3 holds for all n, but as long as we only understand
X_N we can only apply it to Y_{N+1}.  This raise the following

Problem.  Give comparably elementary proofs of the stronger result that
X_n (not just X_N) has a limit, and the yet stronger result that X_n is
monotone increasing.  Somos' ordered-sequences argument does not yield
these.  Korovkin's AGM argument seems the most elegant way to obtain
the latter, with the former an immediate corollary.  Arguments based on
expanding (a+b)^n seem to entail unreasonably tedious calculation.
-- 
Vaughan Pratt				There's safety in certain numbers.
Newsgroups: sci.math,sci.engr
From: a722756@roper.mc.ti.com (W. Donald Rolph)
Subject: Re: Question about "sparsity" of matrices
Originator: a722756@roper.mc.ti.com
Organization: Texas Instruments / Attleboro Mass / USA
Date: Mon, 28 Dec 1992 18:01:20 GMT


I am going to make the probably incorrect assumption that you are working on
finite element matrices.  If you are, then  a typical measure is the mean squared
bandwidth reported by much of the FEA software.
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263
Newsgroups: sci.math
From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Philosophy of Pi
Organization: ORA Corporation
Date: Tue, 22 Dec 1992 02:34:05 GMT

brumleve@iboga (Dan Brumleve) writes:

>dat92oma@ludat.lth.se (Ola Martensson) writes:
>
>>A friend and I discussed if you can find any integer anywhere
>>in the expansion of PI?
>
>Hmm...  I'd think that you would be able to find any integer in an irrational
>number.  Not sure though.

I don't know the answer to the original question, but I do know that
Dan is wrong: it is easy to show that the following decimal expansion
is irrational:

.01001000100001000001...

Daryl McCullough
ORA Corp.
Ithaca, NY
From: eclark@gauss.math.usf.edu. (Edwin Clark)
Newsgroups: sci.math
Subject: Re: Need reference for weighted checksum code
Date: 23 Dec 92 16:06:52 GMT
Distribution: usa
Organization: Univ. of South Florida, Math Department

In article <1h5618INNgvl@agate.berkeley.edu> gibson@purina.berkeley.edu (richard Gibson) writes:
>Howdy!
>
>Someone recently brought me a problem concerning an error-detecting weighted
>code with a modulo 11 checksum that she's using in her business. 
>
>In short, a bunch of items are given 10 digit identification numbers. 
>Let N_i denote the ith digit of the number.
>Each number has the property that SUM(A_i * N_i) is congruent to 0 mod 11,
>where {A_i} are weights as follows: {1, 2, 5, 3, 6, 4, 8, 7, 10, 9}.
>
>The obvious advantages of this identification system are that no single
>digit errors in a valid number and no single transposition of digits in
>a valid number will produce another valid number. 
>
>Question: What is the advantage of the weights being mixed up like that?
>(i.e. why isn't A_i just set equal to i?)
>References about this particular system would be the most appreciated because 
>I know this weighting must have been written about somewhere. I just can't
>find it.
>
>Thanks,
>
>Richard 
>Gibson@math.ucb.edu


Here are some places to look:

Check Character Systems--A. Ecker and G. Poch, Computing 37,
277-301 1986

Check Digit Methods --by Joseph A Gallian --I have a preprint
of this--apparently it has appeared in International J App
Eng.

Modular Arithmetic in the Marketplace--by Gallian also--I think
in the Monthly - June-July 1988

J.A.Gallian's address: Dept of Math, Univ of Minnesota,
A. Ecker's address: Hahn-Meitner_Institut Berlin BmbH
Bereich Datenverabeitung und Elektronik
Glienicker Starass 100
D-1000 Berlin 39
Germany

Both Gallian and Ecker are quite knowledgeable on this 
subject and I am sure would reply to any inquiries.


--Edwin Clark


--
W. Edwin Clark
Mathematics Department
University of South Florida
Tampa, FL 33620-5700
Xref: sparky sci.math:17587 sci.math.stat:2690 sci.math.num-analysis:3704
From: hougen@focus.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.math,sci.math.stat,sci.math.num-analysis
Subject: Spline regression software??
Date: 2 Jan 1993 03:34:12 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
Distribution: inet
NNTP-Posting-Host: focus.csl.uiuc.edu
Summary: Does anyone have spline regression software?
Keywords: spline,regression,software

Does anyone have software/source code for doing spline
regression/spline approximation, preferably in C?  Univariate?
Multivariate?  Adaptive?

I'll settle for univariate because it fits my particular application
right now but I would like to see what else is out there?  Does anyone
have access to MARS?

Darrell R. Hougen

Newsgroups: sci.math
From: lew@cbnewsd.cb.att.com (lewis.h.mammel..jr)
Subject: Re: Haughty quote
Organization: AT&T
Date: Tue, 22 Dec 1992 20:25:26 GMT
Summary: tricks of the trade

In article <dak.724204467@messua>, dak@messua.informatik.rwth-aachen.de (David Kastrup) writes:
> weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
> 
> >In article <1992Dec9.183542.4613@sjsumcs.sjsu.edu>, kellum@sjsumcs (Ken Kellum) writes:
> >>I recall seeing a very haughty quote:  A proof that the definite
> >>integral from -infinity to infinity of exp(-x^2) is the square
> >>root of Pi, followed by something like "to a mathematician that
> >>is as obvious as two times two is four".
> 
> >>Does anyone know the origin of this quote?
> 
> >Lord Kelvin.
> 
> >It's not haughty, by the way.  Mathematicians are as capable of
> >viewing the definite integral as being obvious as they are at
> >obfuscating 2+2 and 2*2.
> >-- 
> >-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
> 
> I have some times amused myself with trying to solve that integral
> (I know substition associates this with gamma(0.5), but this is
> chasing the hen by throwing eggs).
> 
> My msuings were spoilt when I accidently hit on a (very clean and simple)
> derivation using double integral substitution in the TeXbook
> (for those not in the know, TeX is a typesetting system especially
> friends with math) as a typesetting example.

I'll bet you've seen it somewhere and just forgot. You seem to
be talking about the standard trick for evaluating the integral.
Regarding the quote, I was wondering if it wasn't really the pleasure
taken in the success of this ploy which is at the root of the quote.
It does make for a simple derivation, which could be grounds for
calling it "obvious", I guess.  By the same line of thought, maybe
you could say that a mathematician is someone that thinks the integral
of the secant is obvious, although the result isn't as widely used.
I know I'll never forget it as long as I live, but then I'm not
a mathematician!

I do remember my physics prof saying that "you're just born
knowing the solutions to the simple harmonic oscillator equation."
and it does seem so. Where else would they come from ?

Lew Mammel, Jr.
Newsgroups: sci.math
Subject: help with algerbra
From: jc@oneb.almanac.bc.ca
Date: Thu, 31 Dec 92 19:23:51 PST
Organization: The Old Frog's Almanac, Nanaimo, B.C.
Keywords: transformation of formulae
Summary: simple

Hello out there in math land.  I have only a grade 9 formal education
(20yrs ago).  I am a professional seafarer and and as such must return
to Nautical College every 3year on average to climb up the next step
in an officers certification ladder.  I am presently at school in 
Victoria B.C. doing courses which involve amonst other stuff , some
fairly complex(to me) mathematics, spherical trig and such like.
I can usually figure out most of the stuff I come across but there is
one area that I have consistent problems with and that is transformation
of formulae, getting the unknown by itself.  I do not have the time
to upgrade my math skills at this point in my life and I am looking
for advice on a GOOD book or hopefully a shareware program that will
help me master that specific area... any suggestions???
my apologies if this is the wrong area for this kind of low-level
inquiry... you folks might be unraveling the secrets of the unknown and
have no time for such stuff anyways...cheers and thanx

           jc@oneb.almanac.bc.ca (John Cross)
     The Old Frog's Almanac  (Home of The Almanac UNIX Users Group)    
(604) 245-3205 (v32)    <Public Access UseNet>    (604) 245-4366 (2400x4)
        Vancouver Island, British Columbia    Waffle XENIX 1.64  
Xref: sparky sci.math:17557 sci.math.stat:2687
From: cpg@cs.utexas.edu (Carlos M. Puchol)
Newsgroups: sci.math,sci.math.stat
Subject: Delphi method
Date: 31 Dec 1992 15:19:23 -0600
Organization: Department of Computer Sciences, UT Austin
NNTP-Posting-Host: thor.cs.utexas.edu

Hi there!

A mathematician friend of mine (from Spain) is looking from some references
(books if possible) and example of something called the Delphi Method. He says
it was developed by a folk called Olaf Helmer in 1965, from Rank Corp.

He is specifially looking for some book or paper that explains this method and
that has examples. Looks like what he has is kind of imcomplete.

If you have some info on this subject I would appreciate if you can mail it to
my address, since I don't read these newsgroups frequently.

Thanks a lot in advance, and happy new year!

-- Carlos Puchol
-- cpg@cs.utexas.edu
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Mathmatics
Organization: Purdue University Statistics Department
Date: Wed, 23 Dec 1992 16:28:27 GMT

In article <1992Dec23.065430.29519@leland.Stanford.EDU> ilan@leland.Stanford.EDU (ilan vardi) writes:
>In article <1992Dec22.211716.7065@primerd.prime.com> jasonp@bungie.prime.com (Jason Pascucci) writes:
>>Hi,

>>Yes, this might be a strange request, however please
>>indulge me.

>>I would like to obtain the firm grasp of 'higher-math'
>>without spending time in school for a Doctorate in Mathmatics. 


>It's not a strange request at all. As a matter of fact, some guy asked
>the same thing some time ago and got the response ``there is no royal
>road to geometry'' or words to that effect.

This is reputed to be Aristotle's remark to Alexander, Prince of
Macedon, soon to become Alexander the Great.

But on the other hand, typically less than half of the time spent
in graduate school for a doctorate is spent in getting basic 
mathematics.  Unfortunately, too often not enough breadth is
covered, or even available.  Also, in most universities, little
time in the undergraduate program is spent on other than routine
manipulation; most calculus courses, for example, spend less than
a month, it that much, on anything else.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Newsgroups: sci.math
From: pmb@porthos (Pedro Borges)
Subject: Re: need proof: (1 + 1/n)^n ==> e
Organization: INESC-Porto, Portugal
Date: 21 Dec 92 20:47:58 GMT

In <1glr2qINN2vg@usenet.INS.CWRU.Edu> Francis A. Uy writes:

>I can't believe I forgot the proof of this one.
> Subject line says it all.

Well, I think I remember it.
Good for me, because it is SO HARD to find a book with it!!!

Pedro
Newsgroups: sci.math,sci.math.stat
From: ilan@leland.Stanford.EDU (ilan vardi)
Subject: Re: Delphi method
Organization: DSG, Stanford University, CA 94305, USA
Date: Fri, 1 Jan 93 01:05:57 GMT

In article <lk6ourINN735@thor.cs.utexas.edu> cpg@cs.utexas.edu (Carlos M. Puchol) writes:

>A mathematician friend of mine (from Spain) is looking from some references
>(books if possible) and example of something called the Delphi Method. He says
>it was developed by a folk called Olaf Helmer in 1965, from Rank Corp.

isn't this the method developed by Oracle Corp.?

Newsgroups: sci.math,comp.edu,misc.education