Subject: Re: Roman Numerals
From: David Kastrup
Date: 09 Jan 1997 11:19:16 +0100
daly@PPD.Kodak.COM (Matthew Daly) writes:
> I think it's a little too simplistic to say that it's non-based as some
> before have in this thread. It seems to be that it is essentially
> a base-10 system, although an odd one in several ways.
>
> For instance, 99 is not expressed in Roman numerals as IC but as
> XCIX.
It must be mentioned that the decimal rigidity expressed with XCIX
stems from a time where Roman numerals were no longer in use. These
"normalized" Roman numerals are a later invention by decimal freaks.
Historically, you rather find IC. Sometimes even uglinesses like IIC
can be found.
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Subject: Re: Science Versus Ethical Truth.
From: Rebecca Harris
Date: Wed, 8 Jan 1997 17:27:03 +0000
In article <5au0d9$8ug@fridge-nf0.shore.net>, wetboy
writes
>David Kaufman (davk@netcom.com) wrote:
>: What Is Ethical Truth?
>
>< snip >
>
>: However, from an ethical prospective, the holy person
>: told the ethical truth because Truth in its human dimension
>: also includes not harming others. Truth creates harmony,
>: peace and joy.
>
>< snip >
>
>This is absolute crap, in my view. The "holy person" told a lie,
>plain and simple. Some lies may be justifiable, but that in no
>way turns them into somehow being the truth.
>
>-- Wetboy
>
Excuuuuuse me???
--
R33BOX
http://avnet.co.uk/tony/rebecca/
Subject: Re: Monty Hall Trap
From: "Bruce Spratling"
Date: 9 Jan 1997 07:08:18 GMT
No matter which door is chosen, there will be at least 1 door left that
does not
contain the grand prize. Monty Hall knows which door has the grand
prize. He always picks a door without the grand prize, and shows this
to the player. The player falls into the Monty Hall trap, thinking that
since the prize is not behind door 2, there are only 2 doors left, and
therefore the chance that the prize is behind door 1 is now 1/2.
Cheung Koon Tung, Kent wrote in article
<32C77709.46AC@cs.cityu.edu.hk>...
> Hello all,
>
> I have just finished reading a book, called "For Experts Only". One of
> the chapters in this book discussed about probablities and has caused me
> much confusion. I hope anyone who like bridge and good at mathematics
> would not mind to give some explanations on this problem.
>
> In the chapter that has the same name as the subject title, the author
> wrote a story to introduce his idea:
>
> In a game show, one audience is invited to play the game. There are
> three doors. Behind one of these doors, there is a prize of value
> $100,000. The host, Monty Hall, asked the audience to choose a door to
> see whether he wins the prize.
> "No.1", said the audience.
> "Before I open the door, I would like to buy anything behind door 1
> with $20,000.", said Monty Hall.
> The audience answered confidently, "Of course not. My expected value is
> $33,333, why should I accept $20,000?"
> "Before you see what is behind door 1, let use see what is behind door
> 2?", annouce Monty Hall, "right, it is not the great prize."
> Monty Hall continued to urge the audience to sell his rights on door 1.
> "I 'd like to give a last chance, would you sell anything behind door 1
> to me with $40,000?"
> "I accept.", said the audience. The author explained that if the door
> to open is randomly chosen, the probability that door 1 is the great
> prize will be 0.5 and it is better not to accept the offer. However,
> since the door to open is not randomly chosen, the opened door DOES NOT
> AFFECT THE PROBABILITY THAT DOOR 1 IS THE GREAT PRIZE. IT IS STILL 1/3.
>
> I really don't understand the above argument. Since the book I read was
> a Chinese edition. Maybe I have misunderstanded the original meaning.
> Could anyone give me an explanation to this argument, or tell me the
> original statement in English?
>
> Thank you very much.
>
> Rgds,
> Kent.
>
Subject: Re: Cute Proofs...
From: JC
Date: Thu, 09 Jan 1997 10:45:26 +0000
John R. Black wrote:
>
> What is your favorite "cute" proof? The irrationality of sqrt(2)? The
> fact that there are an infinite number of primes? The proof that all
> numbers are interesting? (This one's more of a joke of course)
>
> If you have a moment, I'd like to hear your favorite. To qualify, the
> proof should be short and shouldn't require any big tools or advanced
> concepts.
>
Theorem: given a collection C of 2 or more points in the plane,
not all lying on the same line, there are two points in C such
that the line through those two points contains no other point
in C.
This theorem states the impossibility of finding an 'orchard'
in the Euclidean plane: a collection of points arranged so that
every line through two points contains a third. Mathematicians
looked for orchards for quite a long time before somebody
(Sylvester?) discovered this simple proof:
For each triple P,Q,R in with R not on the line PQ, we can define
f(P,Q,R) to be the distance from R to PQ. Choose P,Q,R which
minimise f. Consider the line P,Q. A little thought will show that
a point X on P,Q will provide a triple giving a smaller f.
Nice proof? Well yes and no. It turns out that the above proof
uses far more information than is necessary. The theorem can
be proved using a set of axioms for the relation [,,], where points
X,Y,Z satisfy [X,Y,Z] if they are collinear and Y lies between
X and Z. These axioms are far weaker than the Euclidean axioms
needed for the above proof.
JC
Subject: Re: Cute Proofs...
From: JC
Date: Thu, 09 Jan 1997 10:45:26 +0000
John R. Black wrote:
>
> What is your favorite "cute" proof? The irrationality of sqrt(2)? The
> fact that there are an infinite number of primes? The proof that all
> numbers are interesting? (This one's more of a joke of course)
>
> If you have a moment, I'd like to hear your favorite. To qualify, the
> proof should be short and shouldn't require any big tools or advanced
> concepts.
>
Theorem: given a collection C of 2 or more points in the plane,
not all lying on the same line, there are two points in C such
that the line through those two points contains no other point
in C.
This theorem states the impossibility of finding an 'orchard'
in the Euclidean plane: a collection of points arranged so that
every line through two points contains a third. Mathematicians
looked for orchards for quite a long time before somebody
(Sylvester?) discovered this simple proof:
For each triple P,Q,R in with R not on the line PQ, we can define
f(P,Q,R) to be the distance from R to PQ. Choose P,Q,R which
minimise f. Consider the line P,Q. A little thought will show that
a point X on P,Q will provide a triple giving a smaller f.
Nice proof? Well yes and no. It turns out that the above proof
uses far more information than is necessary. The theorem can
be proved using a set of axioms for the relation [,,], where points
X,Y,Z satisfy [X,Y,Z] if they are collinear and Y lies between
X and Z. These axioms are far weaker than the Euclidean axioms
needed for the above proof.
JC
Subject: Re: Science Versus Ethical Truth.
From: David Kastrup
Date: 09 Jan 1997 11:45:00 +0100
mlerma@math.utexas.edu (Miguel Lerma) writes:
> I know that the USA society is diverse, and also I have met very nice
> people here (actually my wife is an American). But that the American
> society is dominated by religious intolerance is out of doubt. Just
> to put an example, in a free society victimless crimes make no sense,
> because they are just "morality" made law, however I do not know of any
> western country where they are persecuted with as much fury as here.
Just to give an example: a few years ago in San Francisco, some
policemen broke into the home of some suspect thinking nobody would be
there (they had a search warrant or somethiung like that).
Unfortunately, both the tenant of the flat *and* another male were
there, involved in, ugh, some action. They were arrested for I don't
know what (something like indecent behaviour or whatever) and actually
persecuted and sentenced. I might add that both were considerably of
age.
Pretty much unthinkable in most civilized states with a separation of
religion and state. Now of course I am aware that the Jewish canon of
law which is supposedly valid for Christians as well contains *very*
strict outruling of homosexuality. But it contains a host of other
rules with equally strict penalties which nobody cares a bit about any
more. We don't lock women away for the time of their period, for
example. No gynaecologist and his patients get sentenced to death
because they uncovered the "blood flow" of the woman.
But Christians have always been very selective in what laws they want
to be zealots about, and the American are traditionally pretty zealous.
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Subject: Re: Calculation for pi
From: newarktm@ix.netcom.com(O. ROYCE)
Date: 9 Jan 1997 08:04:55 GMT
In <32d40537.0@news.cranfield.ac.uk> Simon Read
writes:
>
newarktm@ix.netcom.com(O. ROYCE) wrote:
CALCULATION OF pi
Diameter 1
r = radius of circle
sqrt.2\2 = .707105 (x 4 = 2.28427125...)
Dia. minus .707105 = .2928932188
divided by 2 = .1464466094
Sqrt .1464466094 = .38268343.. (x 8 = 3.06146745.....)
sqrt.(r-sqrt.((r^2-(.5ans.)^2 =.195090322.. (x 16 = 3.12144515....)
There's not enough information here to understand what you're saying.
>
>
>>Each side of an equilateral triangle inside a circle, diameter 1,
will
>>easure ..8860254038. (half of sqrt.3).
>>
>>Using the above formula:
>>
>> sqrt.(r-sqrt.((r^2-(.5ans.)^2 = .5 x 6 = 3.00000
>> - ditto - = .258819 x 12 = 3.105828..
>
>It would make sense if you:
(1) included enough brackets
(2) said what "ans" was, and why you are multiplying by 2, 4, 8, etc.
or by 6, 12, etc.
(1) My calculator (Casio Scientific #fx 7000G) does
not require brackets at the end of the calculation.
However they can be added if yours does requires them.
(2) .5ans (answer) - the amount derived from the previous
calculation before multiplying by 2, 4, 8, etc. or by 6,
12, etc. My calculator has an "ans" key.
Each step refers to a polygon with twice the number of
sides inside the circle getting closer and closer to the
edge and thus to the circumference.
It works on my calculator (Casio fx 7000G) Scientific
Calculator.
Ora
Subject: Re: 1 / 2^.5 or 2^.5 / 2?
From: Simon Read
Date: 9 Jan 97 11:02:00 GMT
Christopher R Volpe wrote:
>In C, the expression "2^(1/2)" yields the value "2". The reason why is
>left as an exercise for the reader.
Fascinating. I can think of three reasons; which one is correct?
(1) the symbol ^ doesn't mean power, but something else like bit
shift
(2) the (1/2) is evaluated as an integer, giving 1 or 0 depending on the
truncation/rounding rules: is it rounded up or truncated?
(3) integers to the power of an integer are calculated by a loop, which
is eager to execute at least once, so you get 2 instead of 1, even
if the exponent is 0
I could think of some more bizarre anomalies if I thought hard enough...
perhaps "2" is defined as a constant which means "1" ..? You could do
it in FORTH but then that's another story altogether.
Subject: Re: Are the Laws of Math Eternal?
From: caj@sherlock.math.niu.edu (Xcott Craver)
Date: 9 Jan 1997 08:02:48 GMT
Tom Robertson wrote:
>
>Someone responded to me as follows:
>
>"If you can demonstrate this with something other than just an assertion
>of your belief, you will have refuted Godel's Theorem. While you may take
??? I can't help but wonder quite what he means by this.
>comfort in the belief that the laws of mathematics are "eternal and
>changeless", no working mathematician has given serious credibility to
>that idea in close to half a century ... its seen in the field as a
>charming, if unsophisticated, superstition from the past. Myself, as
Huh. All of a sudden I feel unsophisticated and superstitious.
I personally tend to be a mathematical realist at least half the time,
and many of my colleagues are full-time realists. I certainly would
not agree that the philosophy has not been "given serious credibility
... in close to half a century" !!
I can't help but be amused that someone would criticize such
a wide-sweeping statement (that mathematical concepts are "eternal")
with a similarly wide-sweeping statement (that NO working mathematician
agrees with you or has for nearly 50 years).
>well as several friends, would be most interested to see a formal proof of
>your statement ... as it would likely be worth a Nobel Prize."
If this guy actually was a mathematician, "working" or non,
he or she would probably be aware that no purely mathematical
discovery would be worth a Nobel Prize, because there IS no Nobel
Prize in mathematics. Now, if you could apply it to economics
(say, prove that the almighty dollar is eternal and changeless),
then you'd have a shot.
.,-::::: :::. ....:::::: @niu.edu -- http://www.math.niu.edu/~caj/
,;;;'````' ;;`;; ;;;;;;;;;````
[[[ ,[[ '[[, ''` `[[. "I'd like a large order of FiboNachos."
$$$ c$$$cc$$$c ,,, `$$ "Okay sir, that'll cost as much as a
`88bo,__,o, 888 888,888boood88 small order and a medium order combined."
"YUMMMMMP"YMM ""` "MMMMMMMM" _____________________________________________
Subject: Re: Help wanted on a simple calculus problem
From: israel@math.ubc.ca (Robert Israel)
Date: 9 Jan 1997 08:23:22 GMT
In article <32D47D26.48EC@rmii.com>, sockeye wrote:
> "A particle starts at the origin at t = 0 and moves along the s-axis in
>such a way that its velocity at position s is ds/dt = [cos(Pi*s)]^2
>(i.e., cosine squared of Pi times s). How long will it take the particle
>to reach s = 1/4?"
>
> My answer:If s = 1/4, ds/dt = [cos(Pi/4)]^2 = 1/2, so ds = dt/2.
ds/dt = 1/2 is only true when s = 1/4, not over the whole motion of the
particle from s=0 to s=1/4.
One approach that works is to regard t as a function of s rather than
vice versa. So dt/ds = (cos(Pi*s))^(-2). Now integrate.
Robert Israel israel@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4
Subject: Re: How many diff kinds of proof exist?
From: marnix@worldonline.nl (Marnix Klooster)
Date: Thu, 09 Jan 1997 12:54:54 GMT
In article ,
Michael A. Stueben wrote:
> >I was trying to list the different kinds of proof for my H.S.
> >precalculus students. [...]
hrubin@b.stat.purdue.edu (Herman Rubin) wrote:
> Instead of this, how about teaching them the complete set of rules
> of proof, and how to use them? It is quite common that a proof will
> use many of them.
[...]
> The entire set of rules for formal proofs form a page or two. Instead
> of beating around the bush, just show them a good description of them.
May I suggest the book "A Logical Approach to Discrete Math" by
David Gries and Fred B. Schneider (Springer, 1993). It teaches
essentially two things: It gives a method for constructing and
writing proofs, and it applies this method in the area of
discrete mathematics. The former part might be of use to you.
The method of proof is based on calculation.
As an example, let us prove where the functions x|->x^2-2 and
x|->3*x-2 are equal. We calculate for all real x
x^2-2 = 3*x-2
==
x^2-3*x = 0
==
x*(x-3) = 0
==
x=0 \/ x=3
So the functions are equal at x=0 and at x=3. (^ is
to-the-power, == stands for logical equivalence, \/ for logical
or, and <...> are hints.)
For more information on this proof style, and its application in
high school mathematics, see
http://www.cs.cornell.edu/Info/People/gries/Logic/Introduction.html
http://cs.anu.edu.au/~Jim.Grundy/schoolmath/schoolmath.html
If you do use this method in high school, please share your
experiences!
> Herman Rubin
Groetjes,
<><
Marnix
--
Marnix Klooster | If you reply to this post,
marnix@worldonline.nl | please send me an e-mail copy.
Subject: Re: Roman Numerals
From: djrigby@undergrad.math.uwaterloo.ca (Darren Rigby)
Date: Thu, 9 Jan 1997 11:15:16 GMT
In article <5b0hqj$2upd@b.stat.purdue.edu>,
Herman Rubin wrote:
>In article <32D2AB04.1983@daedal.net>, James Tuttle wrote:
>>Richard Mentock wrote:
>
>Most of the base 10 systems are positional. This includes, for
>example, the Egyptian and the Greek. From what I have seen of
>the Egyptian system, there was a symbol for each power of 10 up
>to some point, and that symbol was used as many times as needed.
>The Greek system had a symbol for j=10^k, j=1,...,9, k=0,1,2.
>It used an overbar to multiply by 1000. The Roman numerals are
>definitely base 10, with symbols for 10^k and 5x10^k.
>
>--
>This address is for information only. I do not claim that these views
>are those of the Statistics Department or of Purdue University.
>Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
>hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
Doesn't a number system with a base mean that if the system is base n, then
there are n digits, with values 0, 1, 2, ..., n-1, such that the notation for
a number greater than n relies on concatenating digits.
Roman numerals don't follow this definition in that they don't have 0, and
I can't concatenate Roman digits to get anything. IIV is meaningless.
If Roman numerals are base 10 then we should have 10 digits. We have either
seven or infinite digits: I, V, X, L, C, M, ~(the bar that multiplies
digits by 1000, base ten, of course) or
- - - - - = = = = =
I V X L C M V X L C M V X L C M ...
Roman numerals cannot be referred to as "base anything".
--
djr={gridby, dart, axoq}
Subject: Re: Roman Numerals
From: djrigby@undergrad.math.uwaterloo.ca (Darren Rigby)
Date: Thu, 9 Jan 1997 11:27:32 GMT
In article <32D3C5CF.6749@paragon-networks.com>,
Doug McKean wrote:
>Doug McKean wrote:
>>
>> What are they - base 5, base 10, base 50, ..., multibased?
>
>What I am finding most interesting is the concensus
>appears to be "no base". It is convention that a
>number system is founded on successive powers of
>a specific specific number usually positional.
>
>The base 10 system is a "based/positional" number system.
>The Roman numeral system is a "non-based/positional" number system.
>
>So, are there any "based/non-positional" number systems or
>any "non-based/non-positional" number systems out there?
>
>
>*******************************************************
>-------------------------------------------------------
>The comments and opinions stated herein are mine alone,
>and do not reflect those of my employer.
>-------------------------------------------------------
>*******************************************************
based/non-positional: the words for numbers, to a certain extent. In
English, the digits for the ones position are "ONE", "TWO", "THREE", ...
"NINE", "". For the tens position, they are "TEN, "TWENTY", "THIRTY", ...
"NINETY", "". There are problems with this, however. You'd have to
explain hyphens in numbers like "fifty-two", and you'd have to add the
proviso that the number consisting of "" "" "" "" ... would be replaced by
"ZERO" and ..."" "" "" "ten" "one" by "eleven" and so on. Such a system
would have to have an infinite number of digits to be able to express any
possible number.
non-based/non-positional: tally marks. They aren't in any base (unless you
stroke through for a fifth tally, so just don't do that) and the position
of the marks is irrelevant (think of the scratchings on a prison cell wall.
Do the dimensions of the wall affect the number written on it?).
--
djr={gridby, dart, axoq}
Subject: Re: Calculation for pi
From: newarktm@ix.netcom.com(O. ROYCE)
Date: 9 Jan 1997 08:08:46 GMT
In <32d40537.0@news.cranfield.ac.uk> Simon Read
writes:
>
>newarktm@ix.netcom.com(O. ROYCE) wrote:
>>CALCULATION OF pi
>>Diameter 1
>>r = radius of circle
>>sqrt.(r-sqrt.((r^2-(.5ans.)^2 =.195090322.. (x 16 =
3.12144515....)
>
>There's not enough information here to understand what you're saying.
>
>
>>Each side of an equilateral triangle inside a circle, diameter 1,
will
>>easure ..8860254038. (half of sqrt.3).
>
>>Using the above formula:
>>
>> sqrt.(r-sqrt.((r^2-(.5ans.)^2 = .5 x 6 = 3.00000
>> - ditto - = .258819 x 12 = 3.105828..
>
>It would make sense if you:
>(1) included enough brackets
>(2) said what "ans" was, and why you are multiplying by 2, 4, 8, etc.
> or by 6, 12, etc.
>
>
(1) My calculator (Casio Scientific #fx 7000G) does
not require brackets at the end of the calculation.
However they can be added if yours does requires them.
(2) .5ans (answer) - the amount derived from the previous
calculation before multiplying by 2, 4, 8, etc. or by 6,
12, etc. My calculator has an "ans" key.
Each step refers to a polygon with twice the number of
sides inside the circle getting closer and closer to the
edge and thus to the circumference.
It works on my calculator (Casio fx 7000G) Scientific
Calculator.
Ora
Subject: Re: Why can't 1/0 be defined???
From: David Kastrup
Date: 09 Jan 1997 11:50:37 +0100
electronic monk writes:
> Norbert Kolvenbach wrote:
>
> > If infinity is a number, please define the Operations (+,-,*),
> > define the inferse of infinity, concerning multiplication.
>
> infinity's inverse is zero because lim 1/x as x-->oo =3D 0
Ah, the old truth: every function is forced to be equal to its limits
at all of its points, so every reasoning about limits is a reasoning
about values.
I encounter this reasoning so often that I could just throw up... my
hands in disgust.
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Subject: Re: Speed of Light
From: pver@nemdev26 (Peter Verthez)
Date: 9 Jan 1997 14:09:40 GMT
pdiehr@mail.ic.net ("Peter Diehr") writes:
: Jan Zumwalt wrote in article
: <01bbfca9$9aafb6a0$577895ce@Admin>...
: > The most interesting aspect to me professionally was the observation of
: > electrical circuit response at those speeds. For instance at about
:
: Note the word "observation of ...". On the spacecraft, everything
: would work
: normally. It is only the great speed between the spacecraft and the
: objects
: outside that results in problems.
This tackles something I've been struggling with for some time. It is
always said that the speed of light is the maximum possible speed, with
as proof that when you calculate e.g. the observed mass for an object
that approaches the speed of light, this mass approaches infinity, which
is impossible.
Now, isn't this something similar like the above ? It is the *observed*
mass that is infinite, but the real mass stays the same, doesn't it ?
That's why I have problems with the statement that the speed of light
is the maximum possible speed, but perhaps I'm overlooking something else.
Any reactions ?
__________________________________________________________________________
Peter Verthez Software Engineer
Email: at work pver@bsg.bel.alcatel.be
at home pver@innet.be
This post is personal and not related to any company whatsoever.
==========================================================================
Subject: Re: Are the Laws of Math Eternal?
From: marnix@worldonline.nl (Marnix Klooster)
Date: Thu, 09 Jan 1997 12:54:56 GMT
davis_d@spcunb.spc.edu (David K. Davis) wrote:
> [...] But one of the
> very great charms of mathematics is that one can discover (or at least
> behold and admire others' discovery of) eternal, universally valid truths.
>
> And since these truths predate and transcend their discoverers, I think
> 'discover' is the right word - e.g. the infinitude of the primes is not an
> invention. There ARE mathematical inventions - notation for example - and
> there are undoubtedly some gray areas.
And then, of course, there are those that maintain that
mathematics is in fact nothing else than some (OK, a lot) of
well-chosen notation. I am one of those. Doing mathematics is
essentially shuffling symbols around. The purpose? Some people
just do it for the sake of it, trying to model their intuitions
on things like sets, numbers, space, and time. Others -- shudder
-- apply it in the real world, and find that the language of
mathematics can be used to describe a lot of phenomena.
The interaction between these two kinds of mathematicians is what
makes the mathematical world go round. Mathematics cannot live
without its real-world roots. But mathematics itself does not
concern itself with real-world issues. It is about symbols and
notations, definitions and axioms. There is no truth in symbol
shuffling -- the question only arises when you try to interpret
the symbols.
All just my opinion, of course.
> -Dave D.
Groetjes,
<><
Marnix
--
Marnix Klooster | If you reply to this post,
marnix@worldonline.nl | please send me an e-mail copy.
Subject: A little question
From: Philippe Langevin
Date: Thu, 09 Jan 1997 15:54:58 +0100
Hie,
Let A={0,1}
Let s be a word of length n , s in A*
Let t be a word of length m , t in A*
What is the probability that t appears in the word s ?
Thank You.
--
_______
| | | Universite de Toulon et du Var
| G | E | Groupe d'Etude du Codage de Toulon
|___|___| B.P. 132, 83957 LA GARDE CEDEX
| | | TEL: 94.14.20.55 FAX: 94.14.24.79
| C | T | E-MAIL: langevin@univ-tln.fr
|___|___| URL: http://www.univ-tln.fr/~langevin/
Subject: Re: A question about extending a function to a Borel measure
From: edgar@math.ohio-state.edu (G. A. Edgar)
Date: Thu, 09 Jan 1997 10:09:14 -0600
You may find help in the (Russian language) paper by Varadarajan
in Mat. Sb. 55 (1961) 35--100. English translation in
Translations of the AMS, series 2, volume 48.
In article , root@gutman.nsu.ru wrote:
> Hello friends.
>
> Do you know any facts concerning extension
> of a real-valued function f: Cl(Q) -> R,
> defined on closed subsets of a compact set Q,
> to a Borel measure? To a regular Borel measure?
> (No additional requirements are imposed on Q.)
>
> If you prefer to deal with open sets, consider
> the "dual" question about a function f: Op(Q) -> R
> defined on open subsets of Q.
>
> What if f is defined not on all closed/open sets
> but on some of them? For instance, on regular ones?
>
> What are the simplest (easily verifiable) known properties of f
> that guarantee its extendibility to a (regular) Borel measure?
> Could you provide me with a reference?
>
> --
> Alexander E. Gutman
> Novosibirsk, Russia
> root@gutman.nsu.ru
--
Gerald A. Edgar edgar@math.ohio-state.edu
Subject: Re: x^3+y^3=z^3+w^3
From: dean@math.math.ucdavis.edu (Dean Hickerson)
Date: 9 Jan 1997 14:56:27 GMT
I wrote:
: But suppose we want to list all solutions with the absolute values of x, y,
: u, and v less than some bound M. Then the solution above is ineffective
: unless we can give a useful bound on r, s, and t. Of course we can find a
: bound on them, using the fact (obtained from Hardy and Wright's proof) that
: r 2 u x - v x - u y + 2 v y s v x - u y
: - = ------------------------- and - = -----------------. (2)
: t 2 2 t 2 2
: 2 (x - x y + y ) 2 (x - x y + y )
: Thus |r|, |s|, and |t| are all less than 6M.
That's nonsense, of course. What I should have said is that |r| and |t|
are less than 6M^2 and |s| is less than 2M^2.
: So to list all solutions with |x|, ... less than M, we could run through
: roughly (12M)^3 triples (r,s,t), compute x, y, u, and v from (1), divide
: x, y, u, and v by their gcd, and see if the resulting solution is as
: small as we want it to be.
: But that's less efficient than just testing about (2M)^3 values of x, y,
: and u to see if x^3+y^3-u^3 is a cube. So this method is worse than
: the obvious brute force attack.
With the above correction, using the parametric solution requires checking
about 72M^6 triples, so it's even worse than I thought.
Dean Hickerson
dean@ucdmath.ucdavis.edu
Subject: Re: fast Hankel transform
From: David Kastrup
Date: 09 Jan 1997 13:52:27 +0100
david.jones@kcl.ac.uk (David Ll. Jones) writes:
> Please - can anyone give me a tead to the FHT (fast Hankel
> transform) - e.g. textbook, original scientific paper references
> etc?
I don't know about fast variants of the generic Hankel transforms,
however the acronym FHT usually refers to fast *Hartley* transforms.
If you are interested in *them* and know about FFTs, you can construct
an FHT algorithm from an FFT one making use of the info in the paper
http://www.neuroinformatik.ruhr-uni-bochum.de/ini/PEOPLE/dak/hartley.ps
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Subject: Nagata-Smirnov theorem and axiom of choice
From: Markus Reitenbach
Date: Thu, 09 Jan 1997 16:12:53 -0800
Recently, I studied the proof of the Nagata-Smirnov metrization theorem,
which states that a topological space is metrizable if and only if it is
regular and has a countably locally finite base.
The proof uses the well-ordering theorem, which is equivalent to the
axiom of choice.
I suppose that the Nagata-Smirnov metrization theorem vice versa implies
the axiom of choice (i.e. it is equivalent to the axiom of choice), but
so far I am unable to proof this.
Does anyone know something about this problem?
Markus Reitenbach, University of Ulm
Subject: Re: Science Versus Ethical Truth.
From: Smclaugh@TCD.ie
Date: Fri, 10 Jan 1997 03:30:39 +0000
In article <32D3EE7C.66D7@boeing.com>, Fred McGalliard
wrote:
> Miguel Lerma wrote:
> > I can see that you are posting from UK. I have spent three years
> > in the USA (I am from Spain) and never imagined before the level
> > of religious fanatism I would find here.
>
> Your impression is undoubtedly biased by the part of our civilization
you find yourself in. Our
> communities vary widely from those with a single community religous
perspective, not always
> conservative Christian at all, to more mixed groups. Many work quietly
to their own ends but quite a
> few push the limits seeking to force everyone else to accept their
whisdom. In part this is a result
> of the great success that the athiests have had in terrorizing most
school districts, controling
> public funding, property, etc.
As a dual- EC and US citizen, I have had the benefit of growing up in the
States prior to immigrating to Europe. In my years of living and
traveling around Europe, I have never witnessed the degree of religious
zealousness which was so obnoxiously omnipresent in the
Midwest and south. However, the focus of my reprisal to your posting
concerns the misguided, and frankly paranoid, attempt to
blame athesists for the "terrorizing of school districts". It was
always my belief that the seperation of church and state was a fundamental
element of American ideology - an paradigm estabilished by the founding
fathers (white male CHRISTAINS). Although not mentioned in your posting,
I assume you are the success of atheiests includes the banning of prayer
in public schools. I would counter that a number of groups (including
regular praticioners of religion) support this seperation.
America has always had a strong religous base, beginning with colonies
with officially designated religions. Due to this, religion has always
been a stonger political force than alternative ethical paradigms (e.g.
atheism, leftism). It is these groups which have recieved the terror
tacticfrom religious communitites (not the other way around). Even
recently, either Bush or Reagan (I afraid I can't rememberwhich) was
quoted as saying that he felt Athesists shouldn't be considered Americans
as the pledge of Alligence states that America is "one nation under god".
I ask you o remember these points pior to scapegoating us atheists.
Seamus McLaughlin
Subject: Re: Science Versus Ethical Truth.
From: David Kastrup
Date: 09 Jan 1997 17:47:59 +0100
Smclaugh@TCD.ie writes:
> Due to this, religion has always
> been a stonger political force than alternative ethical paradigms (e.g.
> atheism, leftism).
It is nonsense to call atheism and leftism ethical paradigms. They do
not preclude ethical behaviour, but neither do they demand it.
> I ask you o remember these points pior to scapegoating us atheists.
A faithful christian should not be bashing people's head in. There is
not much one can claim about a faithful atheist, though.
It turns out that you cannot rely on either behaviour whether people
call themselves christians or atheists.
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Subject: Re: Vietmath War: war victims; blinded victims
From: Carl Friedrich Socrates Einsteinium
Date: Thu, 9 Jan 1997 19:40:00 GMT
In article <5b0ldj$2gi$2@dartvax.dartmouth.edu> Archimedes Plutonium writes:
>
> In article <57721566490@einsteinium.universe>
> Carl Friedrich Socrates Einsteinium writes:
>
> > The whole problem is you don't even understand where the problem is.
> > Just what do you mean by "such numbers"? If you just mean ...000,
> > ..001 and ...002, then there's nothing ridiculous about asking the
> > question, and it's so easy to find the answer I think even David
> > Madore could do it. But that is not the problem. You want to study
> > a whole set of numbers ...000, ...001, ...002, and so on. But that
> > won't work.
> >
> > For, you see, all numbers are *****
> ^^^^^
>
> Your posts are readable up until you enter that above word, then they
> are no longer interesting and in fact the next time you enter ***** I
> will put your posts into my killfile.
You are blind to the truth then. Well, I will say that all numbers are
UNCERTAIN. Let me try to make things clearer.
We have several different kinds of numbers. REAL numbers represent the
straight line. COMPLEX numbers represent the plane of euclidian geometry.
P-ADICS represent Riemannian geometry, and therefore a closed universe.
In other words, adics correspond to the physical theory of General
Relativity. Not quite, in fact: p-adics correspond to space only, and to
turn it into space-time, you have to add i,j,k, thus creating P-ADIC
QUATERNIONS. (Real quaternions correspond to special relativity, that is
to a flat space-time.)
But we know that classical mechanics is false, that general relativity
is false. Because the world is QUANTUM. And therefore all numbers have
to be UNCERTAIN. You see, before 1996, mathematics was not powerful
enough to describe physics, because mathematicians would not admit there
is such a thing as an uncertain number. They thought all numbers had to
be defined with infinite precision.
Imagine being on a road in the fog. Somewhere on the road there is a
landmark, which represents the number 0. The further you move away from
the landmark, the less you see it, and the more UNCERTAIN your position
becomes. That is a perfectly good analogy of Heisenberg's uncertaincy
principle, and that is also the way numbers work. The bigger they are,
the more uncertain they get.
But until now, quantum mechanics seemed incompatible with general
relativity. The reason for that is that uncertain p-adic numbers were
not discovered. As I am now famous for having invented them, we can
now say that we have a correct and perfectly valid theory of quantum
gravitation. We can go even further, and achieve complete unification
by means of the following equation:
Strong interactions = 8 gluons = octonions
--
Carl Friedrich Socrates Einsteinium
Subject: Burnside problem for groups
From: Estelle Souche
Date: Thu, 9 Jan 1997 18:36:04 +0100
I'd like to find some references about the
Burnside problems for groups (i.e. whether
there exists an infinite, finitely generated
group of exponent n).
According to Baumslag's "Topics on combinatorial
group theory", such a group must be finite if n=2, 3, 4
or 6, and there exist some infinite such groups for
every odd n bigger than 665. I'd be interested in knowing
if there are some ointeresting results for other
exponents n.
Please send a copy of your reply to esouche@ens.ens-lyon.fr
if possible.
Estelle Souche
(esouche@ens.ens-lyon.fr)
Subject: Re: Probability and Wheels: Connections and Closing the Gap
From: Karl Schultz
Date: Thu, 09 Jan 1997 10:52:32 -0700
C. K. Lester wrote:
>
> In response to Karl Schultz's prior post,
>
> >There are no subsets. The 168-ticket wheel will guarantee a 3-match
> >in a 6/49 lotto.
> >
> >No, the first statement is correct.
> >The "wheeled group" is the entire set of 49 numbers.
>
> So, with a 6/49, buying 168 tickets using the 168-ticket wheel will guarantee
> a 3-match...
>
> So what?
Because you were asking about this!!!!!
> If each ticket costs US$1, what you're saying is, "If you spend $168, I
> guarantee you'll win at least $3!" (US dollars used for the sake of saving me
> the time and effort to do some currency exchange rate calculations. This also
> assumes that the payoff for a 3-match is $3.) :)
>
> Again, so what?
Again, because you asked!
And, right, like any method of playing lotto, playing a wheel like
this isn't practical from a win/lose point of view. It is just
an interesting fact that is often useful for judging claims
that the same result can be accomplished with fewer picks.
