Subject: Re: paradox
From: Christopher R Volpe
Date: Thu, 09 Jan 1997 18:25:14 -0500
Herman Rubin wrote:
>
> In article <32D3ED92.31A6@cdf.toronto.edu>,
> Luben Tuikov wrote:
> >Joseph H Allen wrote:
>
> >> Now the negation of "all ravens are black" is "all non-black things aren't
> >> ravens". The two statements are logically equivalent. 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".
>
> >> Now this sounds silly, but it is actually logically correct. If you lived
>
[snip]
> >Don't meddle with the quality but with the quantifier:
> >the negation of "all ravens are black" is "not all ravens are black",
> >namely I've found at least one that is NOT black (maybe it's brown).
>
> It was not stated that this was a negation, but an equivalence.
Gee, it sure sounded to me like he was stating that this was a negation.
I must have been tricked into thinking that by the first sentence you
quoted, which begins, "Now the negation of 'all ravens are black'
is...".
The fact that he contradicted himself by subsequently saying it was an
equivalence is irrelevant. The first thing Mr. Allen said was that it
was a negation, and that's the statement that Mr. Tuikov took issue
with.
--
Chris Volpe Phone: (518) 387-7766
GE Corporate R&D; Fax: (518) 387-6560
PO Box 8 Email: volpecr@crd.ge.com
Schenectady, NY 12301 Web: http://www.crd.ge.com/~volpecr
Subject: Re: Why do stars collapse?
From: kunk@perseus.phys.unm.edu ()
Date: 9 Jan 1997 23:49:19 GMT
In article , Ken Fischer wrote:
>kunk@perseus.phys.unm.edu wrote:
>: Small quibble, Bill. Solar mass sized stars don't supernova, they blow off
>: their outer layers in a "planetary nebula" and expand to red giant stage,
>: then just fairly quietly collapse to white dwarfs. Although the galaxy is
>: filled with planetary nebulae (looking like smoke rings), there is no evi-
>: dence of highly energetic SN events corresponding to them.
>: Jim
>
> Is there some estimate of the mass blown outward in
>a supernova? I ask this in relation to what mass would a
>nominal neutron star have had at formation.
> An article in Air&Space; Smithsonian stated that
>a very large percentage of stars have no more than 5 times
>the mass of the Sun when they form.
> If this is true, it would seem to set parameters
>for the proportion of stars that will eventually be white
>dwarfs, neutron stars, etc. (maybe even black holes). :-)
> It looks to me like some supernova pictures show
>debris that could maybe exceed the mass of many suns.
>
>Ken Fischer
Tough question, Ken. An old reference shows densities of stars in the
galactic plane (which is the only place you find massive, short-lived
stars) versus magnitude. It looks like about one star in 10,000 has
an intrinsic magnitude greater than 0, which corresponds to a mass
greater than 2 solar masses. As an example, SN87A is thought to have
started as a blue giant of 20 solar masses, but I can't find a handy
figure for the mass loss. I would guess one to several solar masses.
It's too soon to see if this SN left a pulsar or a black hole.
Jim
Subject: Re: Probability and Wheels: Connections and Closing the Gap
From: cklester@flash.net (C. K. Lester)
Date: Thu, 09 Jan 97 13:37:56 GMT
In article <32D53060.B3D@fc.hp.com>, Karl Schultz wrote:
>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!!!!!
NO NO NO... sheesh almighty. I was referring to the "perceived value" of such
a scheme... as in, "what value is buying 168 tickets for a guaranteed
three-match?" Maybe I should have said, "Big deal."
I was not asking, "So what, why are you telling me?"
>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.
Well, isn't this fun? :)
Thanks!
ck
Subject: Final call for papers IEEE SMC 1997 Orlando
From: fridrich@binghamton.edu ()
Date: 9 Jan 1997 18:09:30 GMT
FINAL CALL FOR PAPERS
__________________________________________________________________
1997 IEEE International Conference on Systems, Man, and Cybernetics
Hyatt Orlando, Orlando, Florida, USA * October 12-15, 1997
Computational Cybernetics and Simulation
__________________________________________________________________
I am organizing a track at the below conference. The track will
have 2-3 sessions directed towards applied chaotic systems for
simulation, data mining, control, image processing and encryption,
and possibly other related topics connected with chaos.
Jiri Fridrich
Center for Intelligent Systems
SUNY Binghamton, NY 13902-6000
E-mail: fridrich@bingsuns.cc.binghamton.edu
Ph/Fx: 607-777-2577
___________________________________________________________________
Preliminary Announcement
1997 IEEE International Conference on Systems, Man, and Cybernetics
Hyatt Orlando, Orlando, Florida, USA * October 12-15, 1997
Computational Cybernetics and Simulation
Location: October 12-15, 1997 at the Hyatt Orlando in Orlando, FL.
Room rate: $105.00 per night, single or double.
Located in the heart of Central Florida. Easy access to
Disney World, Sea World, Universal Studios. Golf course,
a health club, tennis courts, swimming pools, restaurants.
Theme: Computational Cybernetics and Simulation has been
selected to emphasize the growing importance of compu-
tational methods and modeling tools in the design,
analysis, and control of complex systems. Presentations
dealing with theoretical perspectives, new computational
tools, new paradigms in simulation, and innovative
modeling applications are encouraged.
Organizing Committee:
General Chair, James M. Tien, RPI
Technical Programs Chair, Charles J. Malmborg, RPI
Technical Arrangements Chair, Julia Pet-Edwards, Uni-
versity of Central Florida
Functional Arrangements Chair, Mansooreh Mollaghasemi,
University of Central Florida
Promotional Programs Chair, Mark J. Embrechts, RPI
Call for Contributed Papers:
The Technical Programs Committee solicits papers for pre-
sentation at the conference. All papers will be reviewed by
up to three referees for technical merit and content on the
basis of an abstract of no more than 300 words. Papers
accepted for presentation will appear in the Conference
Proceedings. All abstracts must have a cover page containing
the title of the paper along with the names, affiliations,
and complete mailing addresses of all authors, as well as
a rank-ordered list of the three designated topic areas
most closely related to the paper. The cover sheet should
list the two-digit number along with the name of each of
the three designated topic areas. All correspondence will
be directed to the first named author unless indicated
otherwise. We regret that e-mail abstracts of paper
submissions cannot be accepted. Six pages will be allocated
in the Proceedings for each accepted paper. Papers which
exceed this length will be charged on a per page basis.
Each paper presentation should take no more than 20-30 min.
Call for Invited Sessions / Tracks:
Invited Sessions (each comprised of 4-6 papers) and
invited tracks (each comprised of at least 2 sessions) are
solicited in all topic areas. Survey papers and/or case
studies could form the basis of invited sessions. Each
prospective session/track organizer must submit a proposal
including the title of the session/track, a rank-ordered
list of the three topic areas most closely related to the
session/track, and a list of authors with paper titles and
abstracts.
Call for Conference Tutorials:
The Technical Arrangements Committee solicits proposals
for half-day tutorials or workshops which are related to
the conference theme. An honorarium will be provided for
each tutorial based on the number of registered attendees.
Important Dates:
FEBRUARY 15, 1997 (FIRM) Deadline for 3 copies of
contributed paper abstract (with topic area designations)
MARCH 15, 1997 (FIRM) Deadline for 3 copies of
invited session/track proposal (with topic area
designation)
APRIL 15, 1997 (FIRM) Acceptance/rejection notification
of contributed paper abstracts and invited session/track
proposals
JUNE 15, 1997 Deadline for final "camera ready"
paper and author preregistration
DESIGNATED TOPIC AREAS:
1 Computational Cybernetics
11 Biocybernetics
12 Statistics and Forecasting
13 Pattern Recognition and Classification
14 Image Processing and Classification
15 Fuzzy Systems
16 Neural Networks and Computational Intelligence
17 Data Mining and Knowledge Discovery
18 Optimization, Heuristics, and Search Methods
2 Decision Systems
21 Cognitive Systems and Engineering
22 Desision and Conflict Analysis
23 Decision Support, Expert and Knowledge Systems
24 Management Information Systems
25 Medical Informatics and Decision Making
26 Multicriteria and Group Decision Making
27 Visualization, Multimedia, and Graphical Interfaces
28 Database and Software Engineering
3 Human-Machine Systems
31 Command and Control Systems
32 Human Computer Interaction and Virtual Reality
33 Human Factors in Design
34 Robotics
35 Quality and Productivity
36 Training Technology
37 Adaptive and Learning Systems
38 Machine Learning
4 Simulation
41 Animation
42 Continuous Simulation and Applications
43 Discrete Event Dynamic Systems
44 Output Analysis
45 Simulation Languages and Software
46 Simulation Training Systems
47 Military Simulation
48 Simulation Methodology
5 System Methods and Applications
51 Systems Modeling, Analysis, and Evaluation
52 Education and Multimedia
53 Communications and Transportation Systems
54 Energy and Environmental Systems
55 Health Care Systems
56 Service and Public Sector Systems
57 Military Systems
58 Manufacturing Systems and Petri Nets
**********************************************************************
| Jiri FRIDRICH, Research Associate, Dept. of Systems Science and |
| Industrial Engineering, Center for Intelligent Systems, SUNY |
| Binghamton, Binghamton, NY 13902-6000, Tel.: (607) 797-4660, |
| Fax: (607) 777-2577, E-mail: fridrich@binghamton.edu |
**********************************************************************
......................................................................
Remember, the less insight into a problem, the simpler it seems to be!
----------------------------------------------------------------------
Subject: Re: matrix inversion with random numbers
From: fleck@dirk (Pat Fleckenstein (x58885))
Date: Thu, 9 Jan 1997 18:54:39 GMT
In article <32D005FD.3F5@hamilton.edu>,
Rick Decker wrote:
>David Ullrich wrote:
>> Tarizzo Martial wrote:
>> > I am looking for informations about methods to inverse a matrix ; but only
>> > the ones which are using random numbers.
>> What makes you think there
>> exist matrix-inversion algorithms that use random numbers?
>
>(1) the matrix was comprised of random numbers
>(2) the inversion algorithm itself used random numbers.
These are the two interpretations I had also. I assumed
the first was intended. I also assumed that there wasn't
much you could really do that wasn't already all over
_Numerical_Recipes_In_Blurgle_ or in any text dealing
with matrices. I suppose that you could probably highly
optimize cases where they matrices were small, but....
Additionally, it's a fairly straight-forward check to
see if the matrix is invertible (unless my memory really
blows chow (ciao?)^H^H^H^H^H^H^H^H^H^H^H sucks rope). If
the determinant is non-zero, it's invertible. Additionally,
for some methods of inverting a matrix, this calculation
isn't wasted at all.
> This latter interpretation, which I read as a
>monte carlo/probabilistic inversion algorithm, seems to be much more
>interesting.
I thought it was the much more interesting case, too. It doesn't
fit easily into my model of when monte-carlo methods work well though.
My initial sense was something like the 'prime-test' you mentioned.
But, it was the seemingly trivial: What's the probability that
this matrix is the inverse of that matrix. It seems much simpler
to just multiply them than to try to guess. They'd have to be really
huge matrices (akin to the fact that monte carlo methods are overkill
for determining if 11 is prime) to make it useful to 'guess' instead of
to 'determine'.
I also tried to think of it like a travelling salesman problem. One
monte-carlo technique that comes to mind is 'pick several thousand
paths at random'. Assume that the shortest of those paths is
'close enough'. I think, picking a few thousand matrices at random
and seeing which comes 'closest' to identity when multiplied by the
given matrix, just doesn't seem to be all that useful. (If this
is a useful method, I may just have to invest in a whole slew of
monkeys and typewriters.) Anyway, I had a good chuckle thinking
of a room full of monkeys with MatLab (instead of typewriters) pounding
away until one of them stumbled upon the inverse of this matrix
(instead of the complete works of Shakespeare).
alter,
pat
--
I live in another Dimension, But I Have a Summer Home in Reality
Subject: Re: Implication (p->q is true when p=F & q=T)
From: fleck@dirk (Pat Fleckenstein (x58885))
Date: Thu, 9 Jan 1997 22:05:21 GMT
In article <5aprln$qd2$3@dartvax.dartmouth.edu>,
Benjamin J. Tilly wrote:
>Incidentally I find it interesting that the logical operators implies,
>is implied by, and, and or are *not* all the same gramatically in any
>language that I have heard of even though all are identical in the
>"grammar" of logic. I have never seen an explanation of this though.
I don't understand what you mean by 'are *not* all the same
gramatically'. Do you mean that for each [human?] language there
exists some logical operator such that the word we use to name
that particular logical operator in that language is a word that,
in that language, is commonly held to have a meaning different
than it has in symbolic logic?
The constructed language Lojban has words that mean precisely
'AND', 'OR', 'XOR', and 'NOT'. (Actually, my memory is fading,
I haven't played with it in a bit, there may only have been
a prefix or infix for 'NOT' instead of a whole word).
[http://www.willamette.edu/~tjones/Language-Page.html]
alter,
pat
--
I live in another Dimension, But I Have a Summer Home in Reality
Subject: Re: Why do stars collapse?
From: kfischer@iglou.com (Ken Fischer)
Date: Fri, 10 Jan 1997 04:08:52 GMT
kunk@perseus.phys.unm.edu wrote:
: In article , Ken Fischer wrote:
: > An article in Air&Space; Smithsonian stated that
: >a very large percentage of stars have no more than 5 times
: >the mass of the Sun when they form.
: > If this is true, it would seem to set parameters
: >for the proportion of stars that will eventually be white
: >dwarfs, neutron stars, etc. (maybe even black holes). :-)
: > It looks to me like some supernova pictures show
: >debris that could maybe exceed the mass of many suns.
: >Ken Fischer
: Tough question, Ken. An old reference shows densities of stars in the
: galactic plane (which is the only place you find massive, short-lived
: stars) versus magnitude. It looks like about one star in 10,000 has
: an intrinsic magnitude greater than 0, which corresponds to a mass
: greater than 2 solar masses. As an example, SN87A is thought to have
: started as a blue giant of 20 solar masses, but I can't find a handy
: figure for the mass loss. I would guess one to several solar masses.
: It's too soon to see if this SN left a pulsar or a black hole.
: Jim
How will you be able to tell if it is a black hole? :-)
Only one star in 10,000 has greater than two solar masses,
do I read that right? Does that mean by the time they
burn all available fuel, they may be close to the mass of
the Sun, and then a supernova would leave essentially
not enough to make a black hole?
Ken Fischer
Subject: Re: Recognize this pdf?
From: hrubin@b.stat.purdue.edu (Herman Rubin)
Date: 9 Jan 1997 20:36:20 -0500
In article <5b3p8j$5hc@senator-bedfellow.MIT.EDU>,
Timothy E. Vaughan wrote:
>I am working on a problem of molecules binding to receptors.
>If a certain approximation holds, then the number of bound
>molecules follows the binomial distribution:
> [ N ] n
>p(n) ~ [ ] (p/(1-p))
> [ n ]
>where that first thing is supposed to represent the binomial
>coefficient. I have no problem getting the mean and variance
>for this case. However, I want to explore the case that the
>approximation does NOT hold. In this case, I get the distribution
> [ N1 ] [ N2 ] n
>p(n) ~ [ ] [ ] n! (p/(1-p))
> [ n ] [ n ]
>Does anyone recognize this distribution? I would like to get
>the mean and variance of it, and I REALLY don't want to have
>to calculate it myself! I would like to get a reference if
>possible.
I am inclined to doubt that it has been studied. The right-hand
expression is the n-th term, starting from 0, in the generalized
hypergeometric function
F (-N1,-N2;;(p/1-p))
The place I would look for information would be a book on
"special functions".
2 0
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
Subject: Re: Genetic encoding of oocytes compared to sperm cells
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 10 Jan 1997 04:05:47 GMT
In article <5b0ipi$qko$1@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
> I would bet the answer is yes that the male is the more variable in
> genetic material than the female and that the time lag is the major
> reason because once you have a set already existing and in place, the
> probability chances of variability as compared to a set of haploid not
> yet existing and not yet in place increases by a logarithmic factor
The above first message was unclear. Because the male of the species
produces say millions or a very larger number of sperm than the female
and so one can say that the male variability is greater just from sheer
numbers. That is not my question. My question concerns the variability
tied to the function that the female oocytes are already in place
before she is even born.
So let me please refine my question of research. If we set a
standard of variability, say we take one hundred oocytes from a female
and pick out one of those cells and call it the standard to judge the
other 99 oocytes against. We do the same for the male we take 100 sperm
and take one sperm as the standard and then we try to measure the
variability of those other 99 sperm cells against this standard. My
question is, is the variability of those 99 oocytes a small variability
than the variability found in an equal number of sperm? This appears to
me to be an academic question and of little practical use, but who
knows, it may turn out to be an important question.
The primary reason that I am interested in the answer is because it
would confirm or disprove my hunch. My hunch is that the variability of
genes is greater in the male for the reason that the female's oocytes
are preexisting in time and that variability is less when you have say
100,000 cells existing and in place at one time. Whereas with the male
who cranks out 100,000 cells over a stretch of time, has more chances
of things going wrong which increases the variability.
It is sort of a math probability problem, and perhaps the
mathematicians can offer a purely mathematical answer which would match
the science research answer should a scientist venture out to find the
answer. Think of a large cookie dough cutter that can cut say 100,000
cookies at one time. The chances of those 100,000 being nearly
identical to each other is far greater than say a cutter that can only
cut 1,000 cookies at a time and have to make 100 more cuts to equal the
100,000.
Does anyone has some information on what I seek here?
Subject: Re: Random Associative Functions
From: WayneMV@LocalAccess.Com (Wayne M. VanWeerthuizen)
Date: Fri, 10 Jan 1997 03:03:28 GMT
On Sat, 04 Jan 1997 22:55:27 GMT, WayneMV@LocalAccess.Com (Wayne M.
VanWeerthuizen) wrote:
>
>I'm looking for an efficient way to tell if an arbitrary function is
>associative. The function is represented in an N by N matrix, which
>represents f(x,y). All the values of the function are between 0 and
>N-1. (Such as a table for modular addition.)
>
>I know the function is cummulative if the matrix is symetrical along
>the diagonal. I looking for a simple rule of thumb to check for
>associativity, without having to try all possible triples of values.
>
>I'm also looking for a fast way to generate a random function that is
>associative. This seems to be a harder problem than merely testing.
>
I have not yet seen any responces to my question. Does anyone have an
answer? Or was my question not well explained?
Here is an interesting matrix (which I know to be both commulative and
associative when interpreted as a binary function.)
*A *B *C *D *E *F
---+-------------------------------
A* | A B C D E F
B* | B D F B D F
C* | C F C F C F
D* | D B F D B F
E* | E D C B A F
F* | F F F F F F
So for example f(D,E) = B, and f(E,C) = C;
That f(x,y) = f(y,x) , is clear from the diagonal symmetry
How can I prove that f(f(x,y),z) = f(x,f(y,z)), without having to do
an exhaustive test of every triple?
If I were to create a new matrix by filling it with random occurances
of a symbol from A to F, it is unlikely to be either commulative or
associative. If I placed random values only on and above the
diagonal, then mirrored the values to the lower triangle - the result
would definately be a cumulative function, but probably not
assosiative.
What further restrictions must I consider when placing random numbers
into the matrix if I want the resulting function to be assosiative?
Subject: Re: Are exponentials linearly independent?
From: David Ullrich
Date: Thu, 09 Jan 1997 15:27:23 -0600
Dean Hickerson wrote:
>
> Sherman Stein asked me to post the following question:
>
> Suppose that b(0), ..., b(n) are distinct complex numbers. Then it is
>
> well-known and easy to show that the exponential functions
>
> b(0) z b(n) z
> e , ..., e are linearly independent over C. What if n is
>
> infinite? If b(0), b(1), ... are distinct complex numbers and a(0), a(1),
>
> ... are complex numbers such that
>
> b(0) z b(1) z
> a(0) e + a(1) e + ... converges to 0 for all complex numbers
>
> z, must a(0), a(1), ... all be equal to 0?
No. (Pretty sure it's no, anyway...) Someone should point
out that instead of supposing the series converges to 0 for every z it
might make more sense to assume that the series converges to 0
uniformly on compact subsets of the plane - the answer is "no" to
this version as well. (Maybe that version makes more sense to
analysts and the original makes more sense to algebraists.)
It would take a little space to write down the details, but
it's easy to see that there do exist distinct complex numbers b(0),
b(1), ... nnd a non-trivial sequence a(0), a(1),... such that some
_subsequence_ of the partial sums of the series
b(0) z b(1) z
a(0) e + a(1) e + ...
converges to 0 uniformly on compact subsets of the plane.
Sketch: Let's say s(n) is the nth partial sum of the series
above. Let b(0) = 1, a(0) = 1. Now let a(1) = -1, and choose b(1)
very close to b(0). If you choose b(1) close enough to b(0) you see
that
|s(2)(z)| < 1/2
for all z with |z| < 2 .
Now suppose you've chosen a(j) and b(j) for j < 2^n , in
such a way that |s(2^n)(z)| < 2^(-n) for all z with |z| < 2^n . You
set
a(j+2^n) = - a(j) (j=0,... 2^n - 1)
and you choose b(j+2^n) very close to b(j) for j = 0,... 2^n - 1 . If
"close" is close enough then you get |s(2^(n+1))(z)| < 2^(-(n+1)) for
all z with |z| < 2^(n+1) .
So there's a counterexample with convergence of a
subsequence of partial sums instead of convergence of the partial
sums. Of course the series constructed above does not converge
even for z=0, since all the coefficients are plus or minus 1. But
one can take the same idea and jack it up a little to get
actual convergence to 0. There's going to be some rapidly
increasing sequence N_1, N_2, ... in place of the powers of 2
above. Suppose that you've constructed a(j) and b(j) for j < N_n, in
such a way that |S(N_n)(z)| < 1/N_n for all z with |z| < N_n. Now
instead of taking a(j+N_n) = - a(j) for j < N_n as above, you take a
"large" number of new terms to almost cancel each of the existing
terms. And then you rearrange them: Perhaps you find a large
number K (depending on everything chosen so far) and you set
a(j + k*N_n) = - a(j) / K (j < N_n, k = 1, 2, ... K).
You take b(j + k*N_n) very close to b(j) as before, and set
N_(n+1) = N_n + k*N_n . Now as before you see that
|S(N_(n+1))(z)| < 1 / N_(n+1) for all z with |z| < N_(n+1) ;
for this you only need to take each b(j+k*N_n) close enough to b(j) .
So you get S(N_n) converging to 0 uniformly on compact sets as before.
The big difference is that if you do the thing right you see that
|S(j) - S(N_n)| is small for j between N_n and N_(n+1); this gets you
from convegence of a subsequence of the partial sums to convergence.
(There are wrong ways to try to show |S(j) - S(N_n)| is
small - in case someone tries the wrong way and concludes it
doesn't work: What you do is suppose that N_n + k*N_n <= j <
N_n + (k+1)*N_n . Now you write
S(j) - S(N_n) = [S(j) - S(N_n + k*N_n)]
+ [S(N_n + k*N_n) - S(N_n + (k-1)*N_n)]
+ ...
+ [S(N_n + N_n) - S(N_n)]
You show the first term is small because all the coefficients b() that
arise are no larger than 1/K (note there are at most N_n terms in the
first term (um...) and you get to choose K _after_ choosing N_n.) To
show the other terms are small you don't use the size of the
coefficients - that gives "not small enough". Instead you use the
inductive hypothesis |S(N_n)(z)| < 1/N_n for all z with |z| < N_n .
Each one of the terms [S(N_n + (k-q)*N_n) - S(N_n + (k-q-1)*N_n)]
looks just like S(N_n), because of the way you choose the b()'s.)
Should mention the name Charles Matthews - he asked me
a very similar question some time ago; at first I thought the answer
was yes but it turned out to be no, by a construction morally
equivalent to the above. Has nothing to do with e^z really, the
only property of the exponential function that's used is continuity.
--
David Ullrich
?his ?s ?avid ?llrich's ?ig ?ile
(Someone undeleted it for me...)
Subject: Re: Why can't 1/0 be defined???
From: jmccarty@sun1307.spd.dsccc.com (Mike McCarty)
Date: 10 Jan 1997 03:08:51 GMT
In article , Dik T. Winter wrote:
)In article <32CC4934.6567@sqruhs.ruhs.uwm.edu> donniet@sqruhs.ruhs.uwm.edu writes:
) > that is true. lim 1/x as x->0 does not exist and is considered to be
) > either -oo or +oo.
)
)Or simply not distinguish -oo and +oo. To distinguish them may lead to
)severe problems when you are switching to complex numbers. The fact that
)at one stage IEEE switched in their definition of floating-point numbers
)from a projective infinity to a dual infinity leads to severe problems
)stating basic properties about complex f-p math. That is, unless you
)switch to a polar representation of complex quantities, but polar complex
)math is numerically much less stable than an euclidian equivalent.
Numerically less stable for what operations? It depends on application,
you know.
Mike
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Subject: Re: Need to handle Big Matrix (800x800) to use optimization algorithms
From: jmccarty@sun1307.spd.dsccc.com (Mike McCarty)
Date: 10 Jan 1997 03:06:49 GMT
In article ,
Gary Hampson wrote:
)In article <32C90B5C.64B1@public.ibercaja.es>, benigno
) writes
)>Hi,
)> I need to handle Big matrix of around 800 x 800 to implement
)> some optimization algorithms, I would use C++ libraries if
)> possible to use on BorlandC++ 4.5, but if there is any shareware
)
)Unless the matrix has some structure which allows many short cuts and
)reduced storage (eg Toeplitz), then just get on with coding it. 800*800
)is not that big (unless of course in the optimisation you need to
)evaluate A.x many times, or you have some time critical conditions.
)--
)Gary Hampson
800x800 is not that big? Assuming 64 bit reals, that works out to about 5
megabytes of data. Since he's usig BorlandC, that means he's more or
less stuck with the 640K of main RAM.
If you need that big a matrix, I suggest you may switch to DJGPP
(version of GNU CC) for the DOS machines. It automatically allows using
of virtual memory.
Mike
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Subject: Re: Cute Proofs...
From: sci60065@leonis.nus.sg (NCC-1701-E)
Date: 10 Jan 1997 04:04:12 GMT
Godfrey Degamo (gotd@jimmy.harvard.edu) wrote:
: David Kastrup (dak@mailhost.neuroinformatik.ruhr-uni-bochum.de) wrote:
: : blackj@toadflax.cs.ucdavis.edu (John R. Black) writes:
: : > 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)
: : Show that the opposing angles in an isosceles triangle are the same:
: : Given the triangle ABC with lengths AC=3DBC. This triangle is congruent
: : with the triangle BAC (as AC=3DBC, BC=3DAC, AB=3DBA). Consequently the
: : angle at A in triangle ABC is the same as the angle at B in triangle
: : BAC.
Prob : given triangle ABC, let O(ABC) denote the area of the larges ellipse
that can be inscribed in ABC. Given triangle ABC, divided into finitely many
triangle A_i B_i C_i, prove that O(ABC) = sum O(A_i B_i C_i)
Prove : The ratio O(ABC) : S(ABC) is constant where S(ABC) is the area of
the triangle ABC. This is clear by parallel projecting ABC onto a plane
s.t. the projected triangle is equilateral.
--
+---------------------------------------------------------------------+
| ___.----~~~----.___ "Mr Worf, fire phasers at will |
|,--------.-.,-'-------------------` .....NO, NOT AT WILL RIKER!!!" |
|`--------"-'-.,---`~~~-----~~~' +---------------------------------+ |
| '---'-._____/ | Name : Lin Ziwei | |
+--------------------------------| E-mail : limcucw@singnet.com.sg |--+
\_______________________________| sci60065@nus.sg |_/
| limcw1@csmp01.cz.nus.sg|
+---------------------------------+
\_______________________________/
Subject: Re: Cute Proofs...
From: sci60065@leonis.nus.sg (NCC-1701-E)
Date: 10 Jan 1997 04:17:22 GMT
Godfrey Degamo (gotd@jimmy.harvard.edu) wrote:
: David Kastrup (dak@mailhost.neuroinformatik.ruhr-uni-bochum.de) wrote:
: : blackj@toadflax.cs.ucdavis.edu (John R. Black) writes:
: : > 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)
: : Show that the opposing angles in an isosceles triangle are the same:
: : Given the triangle ABC with lengths AC=3DBC. This triangle is congruent
: : with the triangle BAC (as AC=3DBC, BC=3DAC, AB=3DBA). Consequently the
: : angle at A in triangle ABC is the same as the angle at B in triangle
: : BAC.
How about this one :
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
--
+---------------------------------------------------------------------+
| ___.----~~~----.___ "Mr Worf, fire phasers at will |
|,--------.-.,-'-------------------` .....NO, NOT AT WILL RIKER!!!" |
|`--------"-'-.,---`~~~-----~~~' +---------------------------------+ |
| '---'-._____/ | Name : Lin Ziwei | |
+--------------------------------| E-mail : limcucw@singnet.com.sg |--+
\_______________________________| sci60065@nus.sg |_/
| limcw1@cz.nus.sg |
+---------------------------------+
\_______________________________/
