Newsgroup sci.math 158056

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In article <32dcab66.0@news3.paonline.com>,   wrote:
>In <5belj4$2ng$1@nuke.csu.net>, ikastan@sol.uucp (ilias kastanas 08-14-90) writes:
>
>> I just see no reason for empty domains
>
>Consider the classic example of an empty domains -- quantification
>over the set of all Unicorns.  Sounds good, except that Unicorns *do*
>exist (modified goat stock, I believe), and with genetic engineering
>*might* exist in the future (in other forms).
>
>The problem with assuming non-empty domains, is that frequently
>one is unaware of the fact that the domain is empty.  Or perhaps
>cannot even know if the domain is empty -- how about sets whose
>defining predicates require the solution of something like the 4-color
>map problem?  Or the domain may be empty today, but not
>tomorrow.  Or perhaps the reverse.
>
>Reasoning should be correct *regardless* of the cardinality of the
>set being reasoned about.  If the reasoning is correct, it is correct
>regardless if the set is later found to contain one element or none.
>
>But classical logic only allows discussion of non-empty domains.
>So, much of actual mathematical practice is therefore unjustified
>from the standpoint of the underlying logic.  This is clearly
>unacceptable.
	There is no problem with 'domains' as above being empty; it is the
   universe that is assumed non-empty.  I.e. of all structures  (relations, functions and elements of M) we exclude the
   one with M = empty  when it comes to interpreting the Predicate Calculus
   (with relation- , function- and constant-symbols).   Otherwise a formula
   like  Ax P(x) -> Ex P(x)  would not be universally valid... failing only
   for empty M; there might be no c_M to interpret c, complicating the defi-
   nition of satisfaction of a formula in a structure; etc.   As it is, empty 
   or nonempty _something_ amounts to, say,  Ex Q(x)  versus  ~ Ex Q(x).
							Ilias

Douglas J. Zare (zare@cco.caltech.edu) wrote:
: NCC-1701-E  wrote:
: >[...]
: >Prob : prove that C(2n, n) is divisible by (n+1). 
: >
: >Note : C(a,b) refers to the Binomial coefficient of x^b in (1+x)^a.
: >
: >Proof: C(2n,n)/(n+1) = 2C(2n, n) - C(2n+1, n+1) q.e.d
: I would like to have a similarly combinatorial proof that 
: (3a! * 2)/(a! a+1! a+2!) is an integer.
Got it :
Note : C(n, a,b,c) refers to n!/(a!b!c!), when a+b+c=n.
Pf : the above value = C(3a+2, a,a+1,a+2) - 9 C(3a, a-1,a,a+2).
PS : It is clear that C(n, a,b,c) is integral since it equals :
     C(n,a) * C(n-a, b).
--
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In article <5bjbtc$uvj$1@dartvax.dartmouth.edu>,
	Benjamin.J.Tilly@dartmouth.edu (Benjamin J. Tilly) writes:
>
>b) In a box there are a number of pieces of paper, each of which is
>randomly black and white on each side. You pick a random piece of paper
>from the box. If it has a black side, what are the odds that it is
>black on the other?
>
Yes as you say, 1/3, but this confused me, since there are various
slightly different version, with different answers.
1. With your box as before, you pick a random piece of paper and inspect just
one side of it, and observe that it is black. What is the probability that
the other side is black?
2. Now you are told that there are exactly three pieces of paper in the box,
one is black on both sides, one white on both sides and one black on one
and white on the other. You pick a piece of paper at random from the box,
and then look at just one side (chosen at random) and observe that it is
black. What is the probability that the other side is black?
Derek Holt.

In <5bhpbg$hfq@dfw-ixnews9.ix.netcom.com> odessey2@ix.netcom.com (Allen
Meisner) writes: 
>
>In <5bfneu$3v1@nntp1.u.washington.edu> hillman@math.washington.edu
>(Christopher Hillman) writes: 
>>
>>In article <32DB0B90.3A6F@quadrant.net>,
>>"Bruce C. Fielder"  writes:
>>
>>|> If the gravitation of a black hole is such that anything falling
>into a
>>|> black hole will have its "time" slowed the closer it comes to the
>event
>>|> horizon, how does the thing form in the first place?  Surely as
the
>>|> original mass contracts, it should slow (from our point of view)
>until 
>>|> the original mass remains "waiting" (sorry about all the quotation
>>|> marks) at the event horizon?
>>|> 
>>|> As far as I can see, the same should hold true with the mass
inside
>the
>>|> (soon to be) event horizon; the acceleration and gravity increases
>and
>>|> slows the time to infinity.  So how does the thing ever form in
our
>>|> universe?
>>
>>The "picture" of a black hole you probably have in mind (really a
sort
>of
>>map of a particular closed space-time, in the same sense that a
>Mercator
>>projection is a particular map of a certain curved surface) are the 
>>Scharwzchild coordinates, in which the "event" horizon appears as
>>a cylindrical coordinate singularity.  Geometrically, this cylinder
>>(in the map) is really a circle (i.e. a two-sphere).  There are other
>>coordinate systems in which this coordinate singularity is removed.
>>The best is a conformal map (preserving small shapes, like the
>Mercator
>>projection does for the surface of the earth) called the
>Kruskal-Szekeres
>>coordinates.
>>
>>It is true that an exterior observer (usually assumed to be
stationary
>>wrt to the black hole) observes nothing of the history of a particle
>>after it passes through the event horizon.  Moreover, as a particle
>>approaches the horizon, signals from it back to more distant
observers
>>are extremely redshifted and also fade in intensity (exponentially in
>the
>>time of a distant observer, in fact, contrary to the impression left
>>by the Schwarzchild coordinates that a distant observer will observe
>>particles "hanging" suspended near the event horizon.
>>
>>Nonetheless, a particle falling into the BH (or the matter of the
star
>>itself as the hole is being formed) experiences nothing strange as it
>passes
>>through the event horizon.  The event horizon is an artificial mental
>construction
>>(like the international date line) which has a GLOBAL significance
>(this is
>>the point of no return) but no LOCAL (physical) meaning.   Indeed, by
>>a remarkable coincidence, it turns out that you can obtain the
correct
>>experience according to gtr by a simple Newtonian analysis. 
>Specifically:
>>
>>Consider two particles falling straight into a gravitational source
of
>mass M.
>>Suppose one is at radius R and the other at radius R+L (L small wrt
>R).
>>Then they accelerate apart relative to one another as
>>
>>   -GM/R^2 + GM/(R+L)^2 ~ 2GML/R^3
>>
>>(where we expand in a power series in L, neglecting all but the first
>order term).
>>If we have two particles both at radius R and seperated tangentially
>by L,
>>they accelerate toward one another as
>>
>>    -GM/R^2 (L/R) = -GML/R^3
>>
>>(by similar triangles).  That is, the curvature coefficients are
>2GM/R^3 radially
>>and -GM/R^2 tangentially.  Someone falling into a black hole is
>therefore
>>compressed tangentially and expanded radially by the force of
gravity,
>this effect
>>increasing smoothly as R^(-3) right through the event horizon and
down
>to
>>the true singularity at R=0.
>>
>>It is not obvious but true that these Newtonian values are in fact
>correct
>>according to standard gtr for a non-rotating non-charged black hole.
>>I have modeled this discussion on the first few pages of the
beautiful
>>book Gravitation, by Misner, Thorne, and Wheeler, Freeman 1970, which
>>also contains a thorough discussion of many coordinate systems for
>>black holes including the Kruskal-Szekeres coordinates, and various
>>techniques for calculating the curvatures and verifying that the
>values
>>given here are correct.
>>
>>Another way to visualize the situation is to consider a sphere of
>particles
>>"at infinity".  They begin to fall slowly into the hole, carving a
>three
>>dimensional surface out in the four dimensional space-time as they do
>so.
>>You can readily determine the intrinsic geometry of this section
using
>>methods dicussed in MTW and then it turns out you can embedd this
>"world-surface"
>>as a sort of half-football in R^4.  Again, the event horizon is
simply
>one of many
>>spherical "latitude surfaces" on this football, and is not
>distinguished in any
>>way from its brethern.  Incidently, such "world surfaces" form an
>entire family
>>of surfaces carving up the space time.  There is a family of
>"orthogonal" surfaces
>>defined in the same way that potential curves determine streamlines
in
>the
>>conformal mapping method of solving hydrodynamical flow problems. 
>These
>>orthogonal surfaces are flat R^3 planes, flat right down to the
>singularity!
>>That is, the Scharzchild universe is a sort of four dimensonal "ruled
>surface".
>>A more familiar example of a (two dimensional) ruled surface is
>obtained by
>>taking a twisting curve in R^3 and considering the surface carved out
>by its
>>tangents.  Typically this surface has a sharp cusp along the curve
>itself;
>>the true singularity as the center of a black hole arises
>geometrically
>>in an analogous fashion.
>>
>>Hope this helps!
>>
>>Chris Hillman
>
>    Mr. Hillman, you explained in another post that mass and kinetic
>energy both contribute to the mass-energy of a particle. A body is
>therefore its total mass-energy. You stated in another post that the
>velocity space of a body is a Lobachevsky geometry. Mr. Archimedes
>Plutonium has stated that the Lobachevsky geometry does not have a
zero
>reference point. Since a body with constant velocity has a non-zero
>slope in the Loba geometry, it therefore has a potential energy? The
>start metric of the potential energy can then be calculated because
the
>Loba geometry does not have a zero reference point? Could the relation
>between the potential energy and inertial energy be the same as the
>relation between the electric and magnetic fields? The potential field
>induces an inertial field and the inertial field induces a potential
>field: potential flux thereby inducing inertial flux? Since a body is
>nothing but the mass-energy given by the sum of mass and kinetic
>energy, then the motion of a macroscopic body is therefore the
>potential-inertial propagation of the mass?
>
>Regards,
>Edward Meisner
    If the potential-inertial propagation of mass theory is correct,
then is motion itself the gravity waves that scientists have been
looking for? Could I have someone other than Mr. Hillman's opinion on
this?
Edward Meisner

In article  
"Jessica T. Fried"  writes:
>
>	Thankyou for picking up this message!
>
>	I'm looking for a topic to write a short paper on for my 300 level
>Diff.Eqn. class(undergrad).  I'm really interested in plants, and would
>love to know if there are any SIMPLE applications of Diff.Eqn. in botany.
>If anyone out there has any ideas please drop me a line.
>
>		-Jessica  Fried
>		 Grand Valley State University
>
The growth of plants is determined by differential equations of
morphogenesis (birth of shape). This was initiated by the famous
Alan Turing. You will find it discussed in books about him - and
I expect references are also given. 
If I recall correctly, there has been an article in Math.
Intelligencer on this. I do not know whether D'Arcy Thompson's
"On growth and form" involves differential equations - but it is
a classic well worth reading.
JM
-- 
Cogito ergo sum aut miror ergo sim?

In article <01bc034b$0f28e4a0$22b32e9c@goldbach.idcnet.com>,
goldbach  wrote:
>Mind probably presupposes the existence of matter
>and change in that matter.  Numbers are concepts
>dealing with objective reality.
The set of numbers (whichever one you are using at the
time, whether "naturaL", integer, rational, real, or
complex, is an abstract concept, as are any numbers
which may be under discussion.  It seems that it is
often the case that "reality" can be modeled well with
numbers.
The numeral, which is
>a symbol-similar to a word in its use, is the means 
>which a mind symbolizes the concept so that it can 
>use it as a unit for purposes of thought.
This is a grave error.  It is the cause of much misunderstanding
of mathematical concepts by far too many people.  The use of
numerals is a means of communication, and the use of these to
think about numbers is one of the reasons why people cannot
handle mathematical concepts.  The number which is represented
as "30" in the usual way, or XXX in Roman numerals, or as 36
in octal, or as '''''''''''''''''''''''''''''' in tick marks,
or as 11110 in binary is the same number, and its properties
are the same, no matter how it is represented.
-- 
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

				ULM 
			Unified Life Models
Population Dynamics Modeling
----------------------------
The ULM computer program (Legendre and Clobert, 1995; Ferriere et al., 1996)
was designed to study population dynamics models, and, more generally,
discrete dynamical systems.
Which platform ?
----------------
	PC/WINDOWS 3.xx
Who's concerned, in which domains ?
-----------------------------------
Biologists, Ecologists, Mathematicians, either for research or teaching
purposes in the following domains:
	- management, conservation biology (Ferriere et al., 1996 ),
	- evolutionary demography,
	- general deterministic or stochastic dynamical systems. 
How to get ULM ?
----------------
ULM is distributed free of charge. Download self-extracting archive
'autoulm.exe' from web site:
		http://www.snv.jussieu.fr
How does ULM work ?
-------------------
Models are described in a text file according to a reduced declaration
language, close to the mathematical formulation. The program is run with
the model file as input and the system can be studied interactively by
means of simple commands producing convenient graphics and numerical
results. The kernel of ULM is a symbolic evaluator. Stochastic models
are handled via Monte Carlo simulation.
ULM is very easy to use.
What can be modeled ?
---------------------
	* any species life cycle graph (matrix models) (Caswell, 1989),
	* inter- and intra-specific competition (non linear systems),
	* environmental stochasticity (Tuljapurkar, 1990) (random processes),
	* demographic stochasticity (branching processes),
	* metapopulations, migrations (coupled systems).
Which results ?
---------------
	* population trajectories, distributions,
	* growth rate, population structure, 
	* generation times, sensitivities to changes in parameters,
	* probability of extinction, extinction times,
and also
	* lyapunov exponent, bifurcation diagrams,
	* power spectrum, correlation,
	* quasi-stationary distribution, statistics,
	* fitness landscape.
References
----------
Caswell, H., 1989. Matrix Population Models.
Sinauer Associates Inc., Sunderland, MA, USA.
Ferriere, R., Sarrazin, F., Legendre, S., Baron, J-P., 1996.
Matrix population models applied to viability analysis and conservation:
Theory and practice with ULM software. Acta OEcologica, 17.
Legendre, S., Clobert, J., 1995. ULM, a software for conservation and
evolutionary biologists. Journal of Applied Statistics, 22, 817-834.
Tuljapurkar, S., 1990. Population Dynamics in Variable Environments.
Lecture Notes in Biomathematics, Springer Verlag, Germany.
Author
------
Stephane Legendre						
Laboratoire d'Ecologie, Ecole Normale Superieure,
46 rue d'Ulm, 75230 Paris Cedex 05, France.
Tel: 	(33) 01 44 32 37 01
Fax: 	(33) 01 4 32 38 85
Email: 	legendre@ens.fr
Contributors
------------
Jean Clobert, Regis Ferriere, Frederic Gosselin,
Jean-Dominique Lebreton, François Sarrazin.

In article , Rebecca Harris
 writes
>In article , Goddess
> writes
>>In article <70D6wCA0Ep0yIwcx@tharris.demon.co.uk>, Rebecca Harris
>> writes
>>>In article , David Kaufman 
>>>writes
>>>>                What Is Ethical Truth?
>>>>
>>>>
>>>>Introduction:
>>>>------------
>>>>
>>>>       Is a holy person (who never tells a lie) lying, if they
>>>>hide a person being chased by a killer (when confronted by 
>>>>the would be murderer) say, "The person ran that way"?
>>>>
>>>>       From a science or math prospective, which usually 
>>>>ignores the ethical consequences of revealing certain 
>>>>truths, the holy person told a propositional falsehood. 
>>>>
>>>>       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. 
>>>>
>>>>
>>>>Is God All Powerful?
>>>>-------------------
>>>>
>>>>       One area that has caused me much mental suffering is 
>>>>the belief that I had because my spiritual teachers told me 
>>>>that God is all powerful and all loving.
>>>>
>>>>       However, I still believe God is all loving, but I now 
>>>>believe God is not all powerful, yet I still believe my 
>>>>spiritual teachers told the ethical Truth as described above
>>>>in the introduction.
>>>>
>>>>       How can an all powerful God allow legs to be blown off 
>>>>by land mines and all the other horrors happening right now 
>>>>as you read these lines?
>>>>
>>>>       To a would be logical person, if someone has the power 
>>>>to act to correct wrongs before their eyes and doesn't, then
>>>>that person is accountable for the harm they allow to 
>>>>happen. 
>>>>
>>>>       So the only conclusion about an all knowing and all 
>>>>loving God is that God can't be all powerful. Otherwise, it 
>>>>would lead a logical person to either ignore the existence 
>>>>of God or to hate God for Mankind's ongoing physical 
>>>>suffering.
>>>>
>>>>
>>>>How To Explain Why Holy People Say God Is All Powerful:
>>>>------------------------------------------------------
>>>>
>>>>       Breaking away from accepted beliefs is not only 
>>>>difficult for true believers (as I am), but requires setting
>>>>up an alternate plausible (non-contradictory) system of 
>>>>ideas to explain, "Why Do Holy People Say God Is All 
>>>>Powerful?" 
>>>>
>>>>       To help others, who might also be struggling with the 
>>>>question of how not to hate an all powerful God, I offer the
>>>>following 2 scenarios for our current condition.
>>>>
>>>>       On scenario (that is used to explain out current 
>>>>situation) is that the Devil created the world and that an 
>>>>all loving God to lesson Mankind's suffering made a deal 
>>>>with the Devil to gain access to the world through God's 
>>>>Saints. 
>>>>
>>>>       However, the deal with the Devil requires that all 
>>>>God's Saints claim God is all powerful and is the Creator of
>>>>all that exists. Otherwise, the Devil regains the power to 
>>>>create even greater harm to Mankind.
>>>>
>>>>       The other scenario imagines that God did create the 
>>>>universe--perhaps our galaxy the Milky Way--and has a grand 
>>>>plan to stop emerging suffering in the other galaxies. 
>>>>
>>>>       To stop enormous suffering elsewhere requires our 
>>>>current training in this galaxy to become fit for the tasks 
>>>>ahead to serve the universe by eventually creating peace and
>>>>joy for all sentient beings.
>>>>
>>>>       I hope the above ideas help those who are currently 
>>>>struggling with the concept of God as I did. My Best Wishes.
>>>>
>>>>------------------------------------------------------------
>>>>A brief note of value:
>>>>---------------------
>>>>
>>>>       Getting teachers to used numbered words properly in 
>>>>solving constant rate math and science problems could lesson
>>>>student and teacher suffering enormously, I believe.
>>>>
>>>>       If you are interested in promoting constant rate 
>>>>solutions using numbered words in the K-12 school system (or
>>>>elsewhere) as outlined briefly in my current post titled, 
>>>>"Numbered Word Forms To Algebra Equations." in newsgroup
>>>>k12.chat.science, please e-mail me your interests on this 
>>>>matter.  Thanks.
>>>>
>>>>-----------------------------------------------------------
>>>>    C by David Kaufman,                   Jan. 5, 1997
>>>>    Remember: Appreciate Each Moment's Opportunities To
>>>>          BE Good, Do Good, Be One, And Go Jolly.
>>>>
>>>>Note: Please feel free to share this 2 page article with 
>>>>      others without charge.
>>>
>>>What is was the point in writing all that "stuff" about god???
>>>I am an athieist(probably wrong spelling)But I believe that everyone is
>>>allowed their own opinion......So why preach about "the wonderful and
>>>powerful god"?
>>
>>R33BOX, you are probably not athiest, but agnostic. But I agree with you. These
>>people think they can change somebody's life just by talking to them...
>
>agnostic, is when you don't know what to belive, but I don't belive in
>God, Full stop.
Fine.
-- 
Goddess
The girl who cried "MONSTER!" and got her brother....
E-mail : goddess@segl.demon.co.uk
Homepage: http:/www.segl.demon.co.uk/frances

weasel (weasel@televar.com) wrote:
: kkumer@desdemona.phy.hr (Kresimir Kumericki) writes: 
: > : Joseph H Allen wrote:
: > : > Thus all of the non-black things you find which
: > : > aren't ravens (your red coat, the white ceiling, etc.)
: > : > also support your generalization that "all ravens are black".
: > 
: >    Hm, isn't there a name for this paradox? 'Hempel paradox' or
: > something? I really like it: Some lazy ornithologist on a rainy
: > day could investigate whether all ravens are really black just
: > by going around his room and noting objects which are not ravens
: > and which are not black. 
: This is not a paradox.  The statement "all non-black things aren't
: ravens" is the contrapositive of the hypothesis "all raven's are black.  
: Mr. Allen is correct; the two statements are logically equivalent.
: I'd not call the ornithologist lazy, however.  In order to prove the 
: hypothesis (all ravens are black) by demonstrating the contrapositive,
: he'll have to examine each and every non-black item and show that 
: none of them are ravens.  I'd call that a more daunting task even than
: checking up on all the ravens.
   I am well aware that these statements are logically equivalent.
Perhaps I was insufficiently precise, but what I meant is not 
that these statements constitute *logical* paradox, but the one
which is *epistemological* in nature.
   The thing is that you want to know generally how knowledge is
acquired and what constitutes a 'proof' of some scientific 
theory. Theories that are most difficult to handle are those
which state that all members of some class have some property.
E. g. why are we so sure that 'all ravens are black'. It is
obvious that it is impossible to prove this because only proof
that would be generally acceptable is something like collection
of photographs off all ravens in the world (for all times in the
history). On the other hand, we are pretty sure that all ravens
are really black and now we want to find a scientific rationale
for believing in such, unproven, statement. 
  One direction in solving this is to say that the more black ravens
we see the more we are sure that our theory is correct. We can
quantify it by saying that each theory has 'correctness value' between
0 and 1, and with each black raven we see the value is closer to 1
(never reaching exactly 1). But then aforementioned 'paradox' comes
to the game because by noting that my computer screen is green I also
move this number closer to 1. I know that I move it by a very very
small amount, but it is paradoxical that I can move it at all.
  This is how I understand this. I'd like to here from someone who
thought about it more than I did. (Or read about it - I'm sure
that this is a very famous paradox in modern philosophy of knowledge,
and I just cannot remember his exact name or what was response
of philosophers to it.)
  But to tell you the truth, now, when I thought about it a bit
writing this article, it doesn't seem much paradoxical to me
either, even in the epistemological sense. Perhaps I'm
missing something. (Physicists often tend to think that most
of philosophy is no good :)
Kresimir Kumericki
--
------------------------------------------------------------
Kresimir Kumericki  kkumer@phy.hr  http://www.phy.hr/~kkumer
Department of Physics, University of Zagreb, CROATIA
------------------------------------------------------------

Hi!
I once knew an algorithm how to find the square root
of a given real number without a calculator.
It was an algorithm designed for using it on paper, just
like dividing. 
Does anybody know how to do find the square root 
without a calculator?
I'm not searching for a newton approximation or similar!
Thank you very much!
Please reply either to 
  welmen@cslab.tuwien.ac.at
or
  schwarzc@cslab.tuwien.ac.at
Yours
Wilfried & Clemens

In article <5beulu$fo3@northshore.shore.net>,
Norman D. Megill  wrote:
>In article <5belj4$2ng$1@nuke.csu.net>,
>ilias kastanas 08-14-90  wrote:
>>
>>	I just see no reason for empty domains;  contortions, simple vali-
>>   dities like  Ax P(x) -> Ex P(x)  fail, etc.  (This was discussed some
>>   months ago).
>
>Ilias, the whole point of this complicated axiom system is to present a
>logic that applies to empty domains (as well as non-empty ones).  Perhaps
>you missed that earlier in the thread or maybe I didn't make it clear.  I
	I only followed the thread intermittently, that's right.
>understand the term 'free logic' as one that includes empty domains.  Of
>course there is no reason for it other than philosophical; in particular
>if used as a basis for set theory all of its complexity is redundant since
>it collapses to ordinary predicate calculus when the axiom Ex (x=x) is
>added.  The philosophical point, I believe, is that logic in its purest
>form should say nothing about the existence or non-existence of anything; 
>that is the role of a theory on top of it.  From a practical point of
>view, we let logic postulate that at least one thing exists in order to
>have a simplified logic that avoids the contortions.  (Unfortunately I
>missed the discussion you refer to but will look it up.) 
	A very sensible position.  And I have nothing against studying which
   formal deduction system works for 'free logic'.  The one you presented
   seems more elegant than some proposed back then.
>As for the completeness and soundness of this axiom system, the article I
>referenced provides the detailed proof. 
	Knowing its purpose, I can work backwards and figure out the situ-
   ation; but I'll try to find the paper too.  Thanks.
							Ilias

In article <5bic59$lmt@maia.cc.upv.es>,
  convidat@upvnet.upv.es (Usuario Invitado) wrote:
> 
> 
> We need to solve these integrals well, we have the solution but we need to 
> demostrate the way to obtain it. We use a method that needs to demostrate 
> integrals are convergent(finite)and we can't demostrate that condition. May be 
> our method is incorrect and there is other solutions. Try it and answer 
> us.....
> 
> OUR LIFES DEPENDS ON YOU!!!!!!!!!!!!!!!!!
> 
> 1ST)      INTEG(0,infinite)(x*sin(a*x^2)*sin(2*b*x)*dx)     a>0,  b>0  
> 
> 2ND)      INTEG(0,infinite)(x*cos(a*x^2)*sin(2*b*x)*dx)     a>0,  b>0
These are not convergent. Integrating the first by parts over the interval
[0, R] gives (-1/2a) cos(aR^2) sin(2bR) plus the integral of
(1/2a) cos(ax^2) sin(2bx). This latter integral converges as R --> infinity
(it's essentially a Fresnel integral), but cos(aR^2) sin(2bR) behaves
in an oscillatory manner as R --> infinity. Similarly for the second.
Robin J. Chapman                        "... needless to say,
Department of Mathematics                I think there should be
University of Exeter, EX4 4QE, UK        more sex and violence
rjc@maths.exeter.ac.uk                   on television, not less."
http://www.maths.ex.ac.uk/~rjc/rjc.html         J. G. Ballard (1990)
-------------------==== Posted via Deja News ====-----------------------
      http://www.dejanews.com/     Search, Read, Post to Usenet

In article <5bj9ia$562@dfw-ixnews7.ix.netcom.com>,
odessey2@ix.netcom.com (Allen Meisner) writes:
|>     Ok Mr. Hillman, what I would like to ask now is whether the
|> topology of the velocity space is responsible for the inertial motion
|> of a body in the same way that the topology of the gravitational field
|> is responsible for the acceleration of a body.
I can't understand what you have in mind here.  In the case of the
acceleration of a body, it would be more correct to say that the
GEOMETRY of space-time is modified by the presence of a
massive body.  The topology of space time can have global effects, but
acceleration is a local phenomenom, and because all space-times are locally
homeomorphic to R^4 the TOPOLOGY must be irrelevant to any such phenomena.
Curvature and the presence of matter in some region of spacetime are
both local phenomena.
|> Why does a curvature in space time constrain an object to move?
I think you still have a misunderstanding here.  Nothing physical MOVES in
space-time; rather, the mental idea of a body being represented by a
"moving point" is replaced by the idea of a body being represented (throughout
its existence) by a "world line", a (possibly non-straight) curve in space-time.
The "length" between two points (events associated with a time and a spatial
location) on such a curve is interpreted as the time interval between these
events as measured by a clock carried with the body during its motion.
Given these assumptions, it is reasonable to ask how gravitation can be
intrepreted as an effect of the curvature of the space-time.  The simplest
answer is by analogy: on a sphere, the shortest distance (along any curve
constrained to lie on the surface of the sphere) between two points lies
along one arc of a great circle (a circle of maximal radius such as the
equator).  For instance, all LONGITUDE lines on a globe are geodesics
(shortest length curves) in this sense, but of the LATITUDE lines, only
the equator is a geodesic.
Now consider two parallel geodesics (straight lines) in euclidean space.
As we all know, these lines remain parallel all along their length.
Contrast two longitude lines on a globe.  They start off being parallel
(imagine two travelers going North starting at the equator, for instance) but
thereafter they CONVERGE (eventually meeting at the North pole).  This
convergence of initially parallel geodesics is one effect of the (positive)
curvature of the globe.  This is a LOCAL effect because you can detect
convergence no matter how close the initially parallel geodesics are to
begin with.
Going back to the computation in which two particles were falling radially
toward a massive object, the closer one pulls ahead as time increases.
Remember that in the space time picture we replace the
idea of two moving points with the idea of two world-lines which, since
no forces other than gravity are acting, we are assuming are geodesics.
Then, the "pulling ahead" of the closer particle appears as DIVERGENCE
of initially parallel geodesics, and this divergence corresponds to the
(negative) curvature measured by one component of the curvature tensor.
|> is time dilation the only explanation for the Lobachevsky curvature?
|> I must admit that I can
|> not see how this is so. Wouldn't time dilation give you the flat
|> spacetime of SR and the Lorentz metric?
I addressed several separate issues in seperate postings, one of which
concerned general relativity and the rest special relativity, in which
only flat spacetime is considered.  The "velocity spaces" I described
in one of these postings are indeed surfaces of constant curvature, but
they are the exact analogs in Minkowsky space (flat t^2 - x^2 - y^2 - z^2
metric) of the sphere, which is the surface of constant distance from
the origin in ordinary space (x^2 + y^2 + z^2 metric).  In the case
of Minkowsky space, it turns out that the analogous surface of constant
distance from the origin has three parts, two copies of hyperbolic space
(only one of which is the velocity space for ordinary particles)
and the tachyon velocity space.  If you have a book discussing analytic geometry
look at the figure depicting a hyperboloid of two sheets--- this
is the three dimensional analog of the copies of hyperbolic space,
and consists of two copies of the Lobachevsky plane-- and look at the
figure depicting a hyperboloid of one sheet--- this is the three dimensional
analog of the tachyonic velocity space. 
Hope this clarifies things!
Chris Hillman

hetherwi@math.wisc.edu (Brent Hetherwick) writes:
> Keith Pitcher (kpitcher@weirdness.com) wrote:
> : Q) Take a square piece of paper. Fold it in half. Do it again. Repeat 2=
5
> : times. How many sheets thick is the final folded piece of paper.
> : 
> : 
> : My sister realized that this was a trick question, as she knew a piece
> : of paper can
> : not be folded that many times in half, and so far every question had
> : been based in reality. She came up with this description of her answer =
:
> 
> What if the sheet of paper had been 25' x 25'?  Or large enough to carry
> this out?
Let's think this argument over. When folding, the thickness doubles.
So we get 2^25 times the original thickness.  Assume that our original
paper had a thickness of 0.1mm.  Then we get a thickness of
3355.4432m, about half as high as the mount everest.  Due to the
nature of folds, the thickness of the pile must be at least as large
as the dimensions of the folded paper.  So our volume is at least
37.7789318629 cubic kilometres.  If we unfold this volume again to
0.1mm flatness, it covers an area of 377789318.629 square kilometres.
The earth has (assuming a sphere of circumference 40000km)
509295817.896 square kilometres.  So our paper could have covered
about three quarter of the earth surface (I believe slightly more than
the area of all oceans, but could have been somewhat less as well),
minimum.
> Why didn't she ASK the teacher if it was a "trick question"? 
Because it's obvious?
> Why didn't she ASK for clarification of the problem?
Because this was homework?
> Doesn't it seem
> implausible that an honest teacher would try to trick students with a
> problem anyway?
Depends on what you mean by "honest".
> In any case, trying to must professional opinion on your
> side is a very bad idea; at best, it can only serve to falsely enhance
> your ego at the expense of good pupil-teacher relations.  The matter trul=
y
> is a judgement call, and you're only asking for trouble by trying to "go
> over" the teachers head, so to speak, in appealing to a greater authority=
. 
Show me the paper for doing this sort of thing, and I'll agree this is
no trick question.  If the teacher had asked "When you fold a paper,
the amount of layers doubles.  How many layers would you get if it were
possible to fold a paper consecutively for 25 times?" this would be
proper.
But when asking in the context of real things, real results should be
expected.
Of course, it is always dangerous to try to second-guess a teacher.  A
cautious answer would give both what was presumably expected, explain
why this is impossible, and give the reasonable answer (and no, you
can't rely on stupid things like this not happening any more even when
you are in college).
Makes for more work than the others getting grade A have to do, but is
safer, anyway.
What to do in this particular case, now that the problem *is* there,
is a matter of personal judgment, I think.
The problem is that the pupil seems *not* to have written why this is
a trick question, so the teacher could claim she hasn't understood
anything and just did something stupid, and she would have no evidence
against it.
-- 
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

Hello 
Is there any method to transfere an inplane rotation to a translation
Changing the space maybe ?
Please send me an email to hamid@eric.u-strasbg.fr

Martijn Dekker (mdekker@wins.uva.nl) wrote:
: Kriang Lerdsuwanakij  wrote: 
: :Why compact set in R^k must be closed ?   To me, it seems like bounded
: :open sets can be covered by a finite number of open subsets as well.
: :Thanks
: of course it CAN be covered by a finite number of open sets, but to
Not only CAN, it also IS covered - by itself.
Vladimir Janis		http://fhpv.umb.sk/~janis/	janis@bb.sanet.sk

Given 10 mares, all are pregnant. What are the odds of guessing what
each individual mare will deliver? I know there are 1024  possible
combinations. But need to know odds of guessing all the horses' outcome
correctly. For instance:
       mare A- female birth
       mare B- female birth
       mare C- male birth
       mare D- female birth
       etc......
Hope this is understandable. Thanks in advance.
Brenda

In article <32DC29F2.3839@mail.teleport.com> salad@mail.teleport.com writes:
>Michael A. Stueben wrote:
>> 
>> Not only can I give you a short easy-to-understand proof of the
>> existence of God. I can do better I can prove to you that I am
>> God.
>> 
>> PROOF: Everytime I find myself talking to God, I realize that I
>> am only talking to myself. Q.E.D.
>
>Really? And every time you have sex with a woman you find that
>it was only with yourself? Q.E.D.
>
You are demonstrating your stupidity....
>> 
>> COMMENT: The existence of God is independent of the question is
>> God the Christian God?
>
>There is only one God, he's made it quite plain (the God of Abraham,
>Isaac, Jacob, ect...).
>
Bullshit!  Prove that your fictional invisible deity exists....
>> 
>> CONVERSATION: I believe in God. Why do you believe your belief
>> is correct? I believe my belief is correct. Well then, why do
>> you believe your belief of your belief is correct? I believe my
>> belief of my belief is correct. well why do you believe your
>> belief of your belief of your belief is correct? i believe my
>> belief . . .
>> 
>> The above conversation is nuts.
>
>I agree, quit talking with yourself. The bible gives quite a few
>predictions that have and others are about to come true. Why waste
>time talking with yourself? Go to the source, read the bible. Start
>with the chapter of Matthew, ask Jesus to forgive your sins and 
>quit worring about the trivial garbage above. Jesus's forgiveness 
>is FREE, you only have to ask for it, or don't. Best of luck.
Nothing mentioned in your collection of fictions and fables has ever
come to pass.....  That collection has no validity as you cannot provide
the proof for the existence of your diety....
--
Ciao!
	John S. 8^{)>
	teddybur@netcom.com
__

|| You have 12 silver balls that look identical.  However, one is either
|| slightly heavier or slightly lighter than the others.  You also have a
|| balance, which you may use only 3 times to find the ball that is
|| different.  How do you find it?
|| 
|| I can find it with 8 balls, but not 12.  Anybody know?
Do we know whether it is lighter or heavier?
If we know that it is heavier, then it is easy:
1 - Put six on each side and keep the heavier six
2 - Put three on each side and keep the heavier three
3 - Pick two and put one on each side.
If the two are equal, then the one you didn't pick is the heavier one.
If one of the two are heavier, then obviously it is the one.
If the one ball could be either heavier or lighter, then it is more
difficult... Let's see...
Nope, I don't think it is possible.  Someone will correct me if I'm
wrong...
-Mike
___________________________________________________________ 
Michael Anttila aka PsychoMan of Craw Productions 
2B Pure Math / Computer Science at U. of Waterloo, Canada
E-Mail:   manttila@undergrad.math.uwaterloo.ca
Homepage: http://www.undergrad.math.uwaterloo.ca/~manttila/
Craw Productions: craw@magi.com, http://www.magi.com/~craw/

Hi there
I was having a conversation with some class mates, and we were trying to
figure this question out:
If the action is variationally stable, then the Lagrangian must satisfy
the Euler Lagrange equations.  Is the converse true?  In other words, if
the Lagrangian satisfies the Euler Lagrange equations, must the action be
variationally stable?
If so, how would one go about proving it?  A friend said it could be done
in just a few lines...  
peter

In article <32DDA440.2008@math.ucla.edu>,
Mike Oliver   wrote:
>But the ^ operator is not exponentiation in C; it's bitwise
>exclusive-or.  
That's been covered several times already.
>Since the standard does not (to my knowledge) define
>how numbers are represented as strings of bits, the value of 2^(1/2) is
>implementation-dependent, but I would expect it, ordinarily, to be 2.
Is that an integer 2, followed by the period at the end of the
sentence, or is it a float 2., with the decimal point performing double
duty as a period?
The representation of numbers is irrelevant to this discussion; all
that matters is the type.  The constant 2 is of type int, while 2.0 is
of type float.  The type is not implementation-dependent.
-- 
Dave Seaman			dseaman@purdue.edu
      ++++ stop the execution of Mumia Abu-Jamal ++++
    ++++ if you agree copy these lines to your sig ++++
++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++

In article <5beek3$lj$1@nuke.csu.net> ikastan@sol.uucp (ilias kastanas
08-14-90)
wrote:
:
:In article <5bdu97$sv7@engnews2.Eng.Sun.COM>,
:Jeffrey Rubin  wrote:
:@
:@Lastly, I said, "all right, I can extend f to be continuous on all of
:@the unit circle and, using the Poisson integral, I can find a g which is
:@harmonic in U and continuous on D which coincides with f on K." 
However,
:@the problem asks for a holomorphic f, not a harmonic one.
:
:       That works too, if need be.  The Poisson kernel is after all the
:   real part of 1+z/1-z ,  z in U.
            How does it work ?!   Is it holomorphic ?

dlv@bwalk.dm.com (Dr.Dimitri Vulis KOTM) writes:
> What do you think APL does when it sees 0/0?
I refuse to let this count as anything.  Computer "real"s do not obey
associative laws or distributive laws.  Mathematics declare infinity a
non-member of the reals so that exactly those laws don't break.
In computer "real" math those laws *are* already broken.  Putting
infinity into computer numbers does not do much harm anymore, thus.
Or a special number NAN or so (what would be the typical result for
0/0).
We are not arguing about computer math here, however, but about real
math.  And INF and NAN and similar values, when admitted into the
reals, would break laws which are precious in there, and valid.
-- 
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

Consider the sequence {sin(n)} where n=1,2,...
Thus, the sequence is sin(1), sin(2), sin(3), ...
I think that for every real number r between -1 and +1, inclusive, and for
every epsilon (a nonzero positive real number), there is an entry in the
sequence that is closer to r than epsilon.
What is more, I think that there are infinitly many entries like that.
This comes down to saying that r is a point of accumulation (good
translation?) of the sequence. That there is a point of accumulation
between -1 and +1, inclusive, is stated in the theorem of Weierstrass-
Bolzano. But not that all real numbers between -1 and +1, inclusive, are.
Can someone prove that each real number between -1 and +1, inclusive, is
a point of accumulation of the sequence {sin(n)}?
A step further:
Consider given the real number r between -1 and +1, inclusive. Also
consider given the nonzero positive real number epsilon.
Is it possible to calculate for which integer
     r - epsilon < sin(n) < r + epsilon
?
That such an n exists, is true if my previous theorem holds. How to
obtain it, is quite another question.
Is it not?
Is there a formula (or an algoritm) to obtain n given r and epsilon?
Thanks.

Clemens Schwarz,9225709,prak=star wrote:
> I once knew an algorithm how to find the square root
> of a given real number without a calculator.
> It was an algorithm designed for using it on paper, just
> like dividing.
It works like this: suppose we want to calculate the square
root of 2. We first draw the frame like this
  02.00 00 00 00 00 |
                    +---------
                    |
                    |
                    |
                    |
                    |
                    |
                    |
Now consider the first two digits, here 02, and look for a
digit d such that d*d is the greatest possible <= to 2.
Here, d=1. So you write
  02.00 00 00 00 00 |1.
 -01                +---------
 ___                |1*1=1
  01 00             |
                    |
                    |
                    |
                    |
                    |
(We subtract the 1 from the 02, and bring two digits down.)
Now double the number written in the upper right corner.
This gives 2. Now we look for a digit d, the greatest possible,
such that 2d*d is <= to 100 (2d stands for 20+d, not for
2*d). It is now 4 because 24*4=96. So we write
  02.00 00 00 00 00 |1.4
 -01                +---------
 ___                |1*1=1
  01 00             |24*4=96
 -00 96             |
 ______             |
     04 00          |
                    |
                    |
(We subtract the 96 from the 100 and bring two digits down.)
Now double the number 14, which gives 28, and look for the
greatest digit d such that 28d*d<=400. It is 1, so we write
  02.00 00 00 00 00 |1.41
 -01                +---------
 ___                |1*1=1
  01 00             |24*4=96
 -00 96             |281*1=281
 ______             |
     04 00          |
    -02 81          |
    ______          |
     01 19 00       |
And you just continue like that as far as you want.
     David A. Madore
    (david.madore@ens.fr,
     http://www.eleves.ens.fr:8080/home/madore/index.html.en)

In article <32DD2A7F.2299@math.okstate.edu>
 David Ullrich  wrote:
:
:Jeffrey Rubin wrote:
:>
       Where ?!
:> 
:> David Ullrich (ullrich@math.okstate.edu) wrote:
       Where ?!
:> <>
:> 
:> Thank you David.  As in the past, you have come through for me with a
:> clear direct answer to my question.  I really appreciate you
volunteering
:> so much of your time to help people out on the net (sci.math in
particular)
:> and I hope you will continue to do so.
:   What a curious attitude...
              The message might be concise. Regrettably not understandable
     to me. Please explain in reference to to the holomorphic and harmonic
     functions.
tleko@aol.com  

Troy Kessler wrote:
> 
> Does anyone know how to prove that V(icosahedron)=5/12*(3+5^.5)*s^3 and
> V(dodecahedron)=(15+7*5^.5)*s^3/4?
If the answers you name are correct, then you might prove them to be by
calculating the volume by dividing the body in a finite number of
pyramids and calculate the volume on the pyramids (which is quite
straightforward).
The best way might be to take pyramids with a facet as base and the
center of the body as top. That way you only have to calculate the
volume of one pyramid and multiply it by 20 for the icosa- and by 12 for
the dodecahedron.
(doing it for s=1 suffices, since volume is multiplied by s^3 if length
is multiplied by s).

electronic monk wrote:
> 
> Norbert Kolvenbach wrote:
> 
>  > WWWOOUUUU!!!  - So by now we have 0^(-1) = oo and 0 * oo = 1 !
>  > That´s good. Enuf for today... (hahahah...)
>  >
> 
> well, 0*oo could be anything, but 0^(-1) = +/- oo because that means
> that means (0^n)/(0^(n-1)) = 1/0 = +/-oo.
> 
>  > NoKo
>  > "Careful with that VAX, Eugene!"
> 
> electronic monk
Hm... lets begin with the beginning: having x^(-1) defined will
give us x * x^(-1) = 1 for all x. We immediately get 0 * 0^(-1) = 1. 
But now we´ve a severe problem:
1 = 0, which is not true on the reals - or is it by now?
NoKo,
"Careful with that VAX, Eugene!"

Have you tried Brian Bolts series of books?

Scottragun wrote:
> 
> I know how obvious the answer to this question may be to most of you, but
> I would still appreciate a simple answer in response:
> 
> Why is it that we most commonly work with log's in base e, that is why use
> the natural log?
Derivative is 1/x, so it is just the kind of function which will appear 
in integration... much more likely than Log10(x)...
Francois

Kurt Foster wrote:
> 
>   "I am a HAL Nine Thousand computer Production Number 3.  I became
> operational at the HAL Plant in Urbana, Illinois, on January 12, 1997."
> --  "2001 a space odyssey" -- a novel by arthur C. Clarke
The movie "2001" was released on April 1968, the same month Hewlett
Packard introduced what was, IMHO, the first microcomputer which the
company pitched as a calculator: The 9100 series!
A couple of years later, Arthur Clarke actually received a 9100 from
Barny Oliver, who was R&D; chief at HP, and ever since has been referring
to his unit as HALjr  :-)

Hi,
I have an arbitrary NxN matrix, which I want to decompose into a product 
of rotations and scalings (that is diagonal matrices).
I could prove that such a decomposition exists, and find a construction 
for it when N=2: 
if the matrix is symetric, it is diagonalisable in an orthogonal base, 
now any orthogonal base change matrix can be done through a rotation.
so my matric can be decomposed as Rot(-t) Scaling Rot(t)
(where Rot(t) denote the rotation of angle t)
If the matrix is non symetric, I can rotate it to make it so. Hence 
applying the above equation I can decompose any 2x2 as a product
Rot(a) Scale Rot(b)
Now, I am wondering if there is an equivalent theorem in higher 
dimensions : like 
any NxN matrix can be (uniquely?) decomposed as the product of one 
rotation, one scaling, and N-2 rotations, for any N over 2
and whether there is a constructive way of proving it?
Does anyone here know about such decompositions?
Many thanks in advance
Francois

Michiel Hazewinkel (mich@cwi.nl) wrote:
: LS
: 	A discussion has started here in Europe and in Russia about the
: creation of a comprehensive collection of mathematics textbooks on the
: web; all interlinked, interactive where possible and useful.
: 	Of course a number of lecturers and researchers have put a variety of
: their manuscripts on the web. Probably rather more than is easily
: findable.
: 	I am not aware of any place where a systematic list is kept of such
: material. If there is one, please tell me. (I do not mean 'mathematics
: information servers', e-math home page, internet public library, on-line
: books page, internet book information centre, ems home page; though a
: great deal of interesting stuff can be found through these.
  We try to maintain such a list on the Mathematics Archives:
       http://archives.math.utk.edu/tutorials.html
: 	Readers of this submission are encouraged to tell me about sets of
: lectures notes of themselves (and others) that have been placed on their
: own homepages (or of their institutes) and that are freely accessible
: for individual (non commercial) use.
: 	Michiel Hazewinkel
: 	mich@cwi.nl
------------------------------------------------------------------------
Larry Husch                phone: 423-974-4162    fax: 423-974-6576
Mathematics Department     http://www.math.utk.edu/~husch
University of Tennessee    Co-Director, Mathematics Archives
Knoxville, TN 37996-1300   http://archives.math.utk.edu/

In article ,
Denis Constales  wrote:
>In article <32d957c8.0@eclipse.wincom.net>, clayton.smith@mustang.com
>(Clayton Smith) wrote:
>
>> I have a question regarding the uses of calculus.  Is it possible to
>> determine the surface area of a function in two parameters over a
>> given range?  For example, would it be possible to find the surface
>> area of the paraboloid given by:
>> 
>> y = f(x,z) = x^2 + z^2
>> 
>> over the range 0<=x<=1, 0<=z<=1.  If this can be done, what is the
>> general procedure I would use to determine this or any other surface
>> area? 
>
>You integrate over that x,z range the square root of 1 plus the squared
>partial x derivative plus the squared partial z derivative, i.e.
>
>sqrt ( 1 + (df/dx)^2 + (df/dz)^2 ),
>
>read curly d for the partial derivative here.
>
>The proof relies on determining the direction of the normal vector to the
>surface and thereby its angle with the vertical, cf. any calculus textbook,
>or e.g. the pale green Schaum book on Calculus, etc. (Mentioning that one
>because it's there I first encountered the formula.)
You'll also find both formula and proof in any intro differential geometry
text, such as Barrett O'Neill's El. Diff. Geo, or Lipschutz's Diff. Geo
(which is another Schaum book...i.e. cheap!).
--Andrew.
-- 
"I shall drink beer and eat bread in the House of Life."

peter salzman:
|> If the action is variationally stable, then the Lagrangian must satisfy
|> the Euler Lagrange equations.  Is the converse true?  In other words, if
|> the Lagrangian satisfies the Euler Lagrange equations, must the action be
|> variationally stable?
|> 
|> If so, how would one go about proving it?  A friend said it could be done
|> in just a few lines...  
I am not sure what you mean by "variationally stable".
Consider an action functional I defined in terms of a 
Lagrangian L:
                  /
              I = | L(x,x',t) dt
                  /
If *I = 0 (the first variation of I), then I has either a minimum, 
maximum or point of inflection.  In all 3 cases, L MUST satisfy 
the usual Euler-Lagrange equations.  
If by "variationally stable" you mean "I has a minimum" (the 
second variation **I > 0), then the answer is NO.
If by "variationally stable" you mean "*I = 0", then the answer 
is trivially YES, since 
           /
      *I = | ( 0 ) *x dt 
           /
-------------------------------------------------------------------
Jeff Candy                        The University of Texas at Austin
Institute for Fusion Studies      Austin, Texas
-------------------------------------------------------------------

     Dear Colleagues,
I am in need of assistance in gaining insight into the following
combinatorial problem that involves determining the size of a 
union of sets. 
     Let F_{n}, n>0, be a collection of sets :
F_{n}={B_{1n},...,B_{k(n)n}},
where each B_{in} is a finite set in some finite space 
X_{n}, with |B_{in}|/|X_{n}|=1/k(n).    Thus the probability
of B_{in} is inversely proportional to the size of F_{n}. 
     Now let A_{n} be a subset of X_{n}. We have for any
1<=i<=k(n),  Prob(A_{n}|B_{in})=.5, where Prob( | ) represents
conditional probability. Furthermore, the conditional probabilty
of A_{n} given a polynomial number of intersections of the B_{in}
approaches .5 as n approaches infinity. In other words, if 
p(n) is some polynomial, and we intersected p(n) of these  sets,
then the conditional probability of A_{n} given this intersection
would approach .5 as n approaches infinity. [Note that 
k(n)=Omega(2^{n})]. 
    Hence, we see that A_{n} is in some sense dense
with respect to the B_{in}. However, we know that, upon
intersecting enough members of F_{n}, we will get the 
emptyset, or simply a singleton set either belonging to 
A_{n} or its complement. 
   My question is, "How can I show that the conditional 
probability of A_{n} given the union of the B_{in} goes
to .5 as n goes to infinity?".
   Has this type of problem been analyzed before? If so,
what are some references?  Thanks in advance for any 
help you may be able to provide.
   Todd Ebert
   ebert@math.ucsb.edu

In article <32dc9156.1472868@news.worldonline.nl>,
Marnix Klooster  wrote:
>hrubin@b.stat.purdue.edu (Herman Rubin) wrote:
<> In article <32da6d43.280556@news.worldonline.nl>,
<> Marnix Klooster  wrote:
<> >And that is exactly what Gries and Schneider do.  They begin by
<> >teaching the equational logic, as a general tool applicable to
<> >many fields of mathematics.  They try to de-mystify logic, and
<> >make it practically useful.
<> "Equational logic" is not even that well defined, unless it is
<> restricted to particular mathematical structures.  There is the
<> one major rule, which is far more general, and that is the rule
<> of equality. 
<> But in the understanding of logic, this rule is but one of many,
<> and it is even possible to dispense with it, although I certainly
<> would not want to.  But the full sentential and predicate calculus
<> can be taught in elementary school, as has been done by those who
<> understand it.  Equality slides in as just another rule.
>Aha.  We have a misunderstanding due to sloppy terminology on my
>side.  What Gries calls `equational logic' is just (first-order,
>classical) predicate logic, but with much emphasis on equivalence
>and equality, instead of on implication.  This emphasis makes it
>possible to consistently write proofs down in a calculational
>format, as I've shown in an example.  (It also means that
>predicate logic becomes easier to understand and use, in my
>experience.)
>Just a couple of days ago I learned that the term `equational
>logic' has another meaning too.  Sorry for the confusion.
One can do almost as well by using what is called natural deduction.
As I stated above, one could dispense with the rule of equality.
Most logic texts, in presenting the first-order predicate logic,
only introduce equality after everything else.  It is this approach
which I suggest be started in primary school, and be in full extent
by the end of elementary school.  
-- 
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

In article <5beulu$fo3@northshore.shore.net>, ndm@shore.net (Norman D.
Megill) wrote:
>Ilias, the whole point of this complicated axiom system is to present a
>logic that applies to empty domains (as well as non-empty ones).  Perhaps
>you missed that earlier in the thread or maybe I didn't make it clear.  I
>understand the term 'free logic' as one that includes empty domains.  Of
>course there is no reason for it other than philosophical; in particular
>if used as a basis for set theory all of its complexity is redundant since
>it collapses to ordinary predicate calculus when the axiom Ex (x=x) is
>added.  The philosophical point, I believe, is that logic in its purest
>form should say nothing about the existence or non-existence of anything; 
>that is the role of a theory on top of it.  From a practical point of
>view, we let logic postulate that at least one thing exists in order to
>have a simplified logic that avoids the contortions.  (Unfortunately I
>missed the discussion you refer to but will look it up.) 
Philosophers find another use for free logics also.
As a part of the common philosophical practice of using formal logic to
regiment and clarify various philosophical theses, free logics are used on
occasion in which constants need not have objects assigned to them. For
example, let "a" be a constant (interpreting a name in natural language
that is meaningful even if it has no bearer) and "F" be a predicate. Using
a free logic, we can distinguish between:
[a]\neg(Fa)
and 
\neg[a]Fa 
where the brackets are used to indicate a scope distinction. The former
claim is supposed to entail that a exists (that "a" has a bearer), and the
latter not; an informal gloss would be i) a is such that it is not F and
ii) It is not the case that a is F (which would be true if a didn't exist).
We can capture the sense in which "a is not F" is ambiguous without
postulating any ambiguity in "not" (an ambiguity between a so-called
"internal" and "external" negation operator) -- the ambiguity is
structural.
See, e.g., Gareth Evans's "The Varieties of Reference," ch. 2.
So there is a philosophical point to free logic even if one disagrees with
the motive for its adoption that Norman Megill points out. And one might
well find that motive dubious, agreeing that logic should not commit us to
the existence of any *particular* thing, while disagreeing with the claim
that it should not commit us to the existence of anything. One might think
that it is impossible that there be nothing; for example, one might think
that the empty set (and other mathematical objects constructed therefrom)
necessarily exists.
-- 
Jonathan Sutton
Rutgers University Philosophy Dept.
"To wage war, it is necessary to be a philosopher" -- from a pamphlet
issued by the singularly brutal Shining Path of Peru, 1982.

Brent Hetherwick (hetherwi@math.wisc.edu) wrote:
: I thought that she was quite the sharp cookie until I read her little
: paperback on Fermat.
In August of '94, she wrote that there was no simple method for determining
whether a long number was divisible by seven.
I wrote her the next day, explaining that there *was* a method, and that it
also simultaneously tested for 11 and 13, since 7 * 11 * 13 = 10^3 + 1. I
also (sarcastically) included the fact that this worked for "short" numbers
and "medium"-sized numbers as well. Perhaps this threw her.
At any rate, I never received any acknowledgement for this. She never
mentioned the fact that she was wrong as long as I had a subscription to a
newspaper with Parade (until Feb '96).
: The depth of her ignorance of mathematics knows no nameable bound;
Agreed.
Dave
dshapir@argo.ecte.uswc.uswest.com

Andre Rohe (rohe@unitas.or.uni-bonn.de) wrote:
: Hello,
: 
: I have the following problem:
: 
: I wrote a computer program and tested the algorithm with
: random problems of different sizes n. For each of these runs
: (I made runs on problems of size 500-150000. On each problem
: of size n I made 1000 runs with different random seeds) I have the
: running time of the algorithm.
: 
: Now I want to calculate the average running time in practice as 
: a function c1*n^c2 (so I want to calculate the constants c1 and c2).
: 
: Does anybody know how to do that ? I am mostly interested in 
: references to papers doing something like that.
Sorry if I miss the whole point here, but I see no difference between your
problem and an experiment. So why don't you just transform your data
according to
log(T) = log(c1) + c2*log(n) 
and then do an ordinary linear least squares fit? 
-- 
Ulrich Lange                       Dept. of Chemical Engineering
                                   University of Alberta
lange@gpu.srv.ualberta.ca          Edmonton, Alberta, T6G 2G6, Canada