mm-218
===
Subject: Congruence Involving 6 & Some Sums-of-SumsLet, for
each nonnegative integer k,a(6k) = a(6k +1) = 1;a(6k +2) =
a(6k +5) = 0;a(6k +3) =a(6k +4) = -1.Let A(0,m) = a(m);and for
all positive integers n, and for nonnegative integers m,A(n,m)
= sum{k=0 to m} A(n-1,k);Then, for q and r = any nonnegative
integers:m!*(A(q,m+1) -binomial(q+m,q-1))is congruent
tom!*A(r,m) (-1)^m (mod {m+q+r}).So, more specifically, from
the above we get:For ODD m,(m-1)!*A(r,m)is congruent
to(r+m)!/(m r!) (mod {m+2r+1}).For EVEN m,(m-1)!*A(r,m)is
congruent to(r+m-2)!/(m (r-2)!) (mod {m+2r-2}).(Someone might
enjoy confirming the above congruences...)I wonder if any of
these congruences have any interestingnumber-theory
===
(Goal:Coprime-Grid)For a positive integer n,arrange 1 through
n^2 in a square grid like so:1 2 3 4....nn+1 n+2 n+3... 2n2n+1
... 3n...n^2-n+1... n^2What is the minimum number of switches
neededto get the n-by-n arrangement of the integers whereEVERY
pair of adjacent integers is coprime?By adjacent, I
meanimmediately next to in the directions of up, down, left,
right.By switch, I meanthe exchanging of two adjacent
integers' positions.So, we might have, for m=2,1 23 4We can
switch the 3 and 4, so we then have1 24 3Since every adjacent
pair is now coprime, only 1 switch is needed for n = 2.Yes, ts
puzzle was inspired by Sam Loyd's 14-15 Puzzle.(Or was it the
===
calculus vs. discrete mathTs might just be echoing James's
post, and another one too, butthere are a few slogans one
comes across in learning maths. Two thatstuck in my mind were
that the easier a problem is to state the harderit is to solve
(above a certain level), and there are reasons thedefinitions
are hard and the theorems easy. Perhaps it is tsobservation
alluded to that in calculus there are many problems allhaving
the same solution, yet in discrete maths it is often that
eachhas a different method for solving it, if one even exists
at all. Apersonal favourite would be the if there are six
people at a partythere are 3 mutual friends or 3 people who
don't know each other, easyproof, generalize it and it becomes
imposssibly difficult to solve.Perhaps Erdos's observation (of
another problem) that maths isn'tready for these tngs yet is
===
see me then in the industrial-punk outfits I used to wear inmy:
> more angst-filled youth.:: No problem with that. I have worn
a gilette blade hanging from a: safety pin stuck through the
flesh of my shoulder myself. You: obviously did not understand
the reference. Don't go telling what Eco: would be proud of if
you don't have a founded command of s works.: That was from
Focault's Pendulum, the umbanda ceremony takes place at: the
end of Casaubon's binge in Brazil. Read it if you care, it's
a: very spooky book about knowledge.I _have_ read Foucault's
Pendulum. It _was_ a very powerful book on thenature of
knowledge. Don't go around pretending like everyone is
moreignorant than you. It is quite a disgusting display.I
would love to discuss Eco in depth with you, but you need to
correct yourtone before that happens, and I would prefer to
take that somewhere moreappropriate.(By the way, I'm sure you
have seen, my dear Jorge de Burgos, that there isa strange
similarity of the heresy of Aristotle's work on humor and
levityand the heresy I am accused of by yourself and others,
of being arrogantenough to enjoy my work and my creativity. I
know, I know... I should bemore solemn when walking through
the halls of knowledge you wish tocontrol...)[...]: Look
Galathaea, you've been asked by many others in nicer and
ruder: ways, what it is that you are talking about. Posting a
long list of: references will only convince people that you
are out to show off how: well read you are. Why don't you
state one single mind bogling fact in: the lines of there is
an isomorpc relationsp between the set of: regular verbs of
the natural language Q, and the set of all halting: programs
that can be encoded by a string of length N over an alphabet:
Z and ts is due to an underlying heyting structure. Even
sometng: weaker will do. Even answering my little stupid
question will do.: Remember, nobody asked you questions in the
first place. It was you: who came and said hey everybody look
at my bright idea! and then: simply failed to produce it. As I
told your buddy Mitchy, until now: you've been all width and no
depdth. Now show some meat. (no pun: intended ;p)I posted a
central question that was the focus of my towards a
constructiveeducation thread. In fact, it's right there in the
title. Should weexpand the education of logic? Restructure a
bit, maybe move certain tngsto younger ages? Include the
nonclassical?In particular, I felt it would be less
controversial to focus on addingHeyting structures to the
curricula. There are tons of communities usingthem implicitly
or explicitly in their work. I made the assumption thatsome of
these people are lying dormant out there, aware of the work in
thefields but not really bringing them up as topics very
often. I've seenmentions every now and then on sci.math of
some of the topics, mitch hasbeen fighting away on sci.logic
over similar issues, and since I knew ofwork in a lot of other
fields about it, I thought I would try to see ifthere was any
sort of minority community I could summon to discuss
thattopic.I wasn't trying to sell anytng grand or pass off
some gobbledygook in someattempt to deceive. I was asking
questions and sharing ideas.In other words, I made the
opposite assumption than you, arrogant Guenther,that the
people communicating on these groups might possess some
knowledgeof what I was talking about and have opinions to
share.My entire point has already been stated quite a number
of times, that themodels we are constructing about the world
around us, the languages we usein natural languages,
folk-science, and even quite a lot of modern science,all make
heavy use of nonclassical logics. There is still quite a
traditionto use classical logic as the metamodel, wch then
faces difficulties withclassification of what types of
statements it can really be applied to. Itis not universally
applicable to all statements, and the study of ts isfairly
well established.So I also introduced my suspicion as to why
classical logic gets presentedin the ways I have seen it
presented, as the universal logic of everytng.I suspect that
people really want tngs to be true or false always,either-or,
that they desire at least metaphysically for such
aclassification to exist even when we have known at least for
three quartersof a century that we cannot always deduce such
from our math. Ts Iassociate with monotheism's popularity, and
I am not the only one as I pointto the quote by R. M. in s book
on semiotics makes the same claimand attributes its origin to
Wtehead.And I believe that if people understood the way we can
formalise differentstructures of reasoning about the world, how
we can build various models andtest them to our experience, and
along the way were shown that alot of ourmost useful models
today are built in structures of nonclassical logics,maybe
some of the fundamentalism I see in the world would fade. Just
alittle. I'm don't expect major changes.Its just that I tnk
there is sometng inherently unhealthy about aframework or
world view that forces people to attribute an absolute truth
orfalseness to all statements in their mind. It makes people
build defensivepsychologies to hold on to whatever world-view
they have constructed whenlogical consistency starts kicking
in, and other ideas become excluded tothe point where hatreds
are formed.If you don't believe me, just look at the response
these threads havegotten, from yourself included. None of what
I mentioned was factuallyincorrect to my knowledge (or, at
least, no one has been kind enough topoint to any errors,
though I'd expect one or two errors lying aroundsomewhere),
and indeed it was topical to all groups. However, it wasn't
thetype of post some people wanted. My posts have been
playful, they haven'tneeded any rigor because the papers I
referred to included all of that. Iwas just asking a simple
question, wch I've asked several times now. Andalthough I've
asked it several times, there is still an anger from
those,like yourself Guenther, who admit they don't know what
my question has been.There have been positive responses too.
Obviously mitch who has facedsimilar negativity over on
sci.logic, has contributed alot. Some peopleasked questions in
a very friendly manner, and I was immediately friendlyis
overdue unfrotunately) for those who were open enough to allow
me toexplore more of the ideas because they are curious.I like
open, curious people who don't desire to post contentless
negativitywhen sometng doesn't fit inside their worldview,
because that is aprerequisite of true science. I believe that
there is a certain mentaldynamic that quests for certainty.
Some choose religion, particularlymonotheism, because it often
screams out quite loudly its importance as thesource of the
certain absolute truth and adds some nice, though unseen,perks
through Pascal's wager. Others choose science, wch at first
seemsto offer a more secure form of certain absolute truth in
observablecorrelation.I myself love science. It is the
one-stop source of models to predictreality with. But there is
still uncertainty there. The study of modelshas shown that
there are numerous ontologies with identical
observablepredictions. There are even huge numbers of models
out there that dopredict different observations but only
through experiments we do not yethave the capability (or
energies) to explore yet. Science is not a sourceof certainty.
It is a playground of competing models. And I believe
thatunderstanding better what models are, how they carry a
logical structure,and how they get tested to observation could
stop a lot of the rudeness onthe playground.Take, for instance,
superstring theory. As with all ideas when they arefirst
introduced, it has been attacked quite a lot from many
differentdirections. With more research in the field, it has
been able to establiself as a topic legitimate to study in
physics curricula, even enteringundergraduate courses these
days. So what was the content of the attacks?Excluding
mathematical corrections to various papers (wch I do
notconsider attacks but legitimate scrutiny of method), the
biggest attackshave been on the freedom of predictability in
the theory. Unique orotherwise natural solutions are still not
well established. There is stilltoo much freedom in the theory.
Other attacks (though less frequent) arebothered by the large
dimensionalities or see certain preferred frames inthe
theories as problems despite the lack of observability. There
has beenalong the way a certain meanness in some of the
attacks, a belittling.I have my suspicions that some of the
anger response is residualmanifestations of the male alpha
problem. In past societies, erarces ofpower formed through
displays of strength and aggressiveness. It was usefulbecause
strength and aggressiveness were useful on a biological
level.Nowadays, I see the same quest for erarces in science,
in particular thesame quest for absolute truth centered on one
model. That is why, forexample, there is a standard view of
quantum mechanics and alternateinterpretations are so commonly
demeaned.So you see, Guenther, I have now expressed yet again
the whole point of anyof ts. Along the way, you will notice
that I have had to fight my shareof alpha attacks. But I have
always asserted, as I do to you now, that Ihave a right to ask
the questions I have, in the forums I have, at the timeI have
===
binomial and sum> The following formula,> i i> C C > n 1 n
n+1> SUM ----- -- = 2, with n integer> i=1 2i-1 2i> C> 2n> i
n!> where C = ---------,> n i! (n-i)! can be easily found
using two formulas related to Catalan numbers {C_n} :> (2n)!>
C = ---------, for all n>0, and C = 1 ;> n n! (n+1)! 0> n> SUM
C C = C .> i=1 i-1 n-i n>A elementary proof, using binomial
coefficients properties for example, would be nicer.>TigertibA
start?...Your sum is 2 for n = 1.So if we take the differences
between the sum at n+1 and the sum at n,we should always get 0
for each n >= 1.So,(n!(n+1)!/(2n)!)sum{i=1 to n}
(i2)!(n2-12)!/(i!^2 (n+1-i)!(n-i)!(2i-1))should equal, for all
n >= 1,((n+1)!(n+2)!/(2n+2)!)*sum{i=1 to n+1}
===
rearrangement question....hot-girl> .........> i know that if
sigma An is absolutely convergent,> then any rearrangement is
also absolutely convergent,> and furthermore all rearrangement
of ts series have the same sum.> i saw the next infinite
series> sigma 1/{(2k-1)(2k+1)} = 1/(1*3) + 1/(3*5) + 1/(5*7) +
...> thus> sigma (1/2)[{1/(2k-1)}-{1/(2k+1)}]> = (1/2)(1/1 -
1/3) + (1/2)(1/3-1/5) +.........> = 1/2True. It is easy to
prove that the partial sumsof sigma 1/{(2k-1)(2k+1)} are
k/(2k+1)and these have limit 1/2.> ...> sigma {k/(2k-1)} -
{(k+1)/(2k+1)}> = (1 - 2/3) + (2/3 - 3/5) +............> =
1but the series whose summands are((-1)^k) k/(2k-1)is not
absolutely convergent.If a series of real numbers converges,
but not absolutely,its terms can be rearranged to produce
===
RecursionsWell-known, I am sure...For a fixed positive integer
n,Wch conditions on the fixed coefficients {c(k)} and on the
initial a's,a(k) for 0<=k<=n, are necessary and sufficient for
the sequence{a(k)} to be periodic, wherea(m) = sum{j=1 to n}
c(j)*a(m-j) ?For example:a(m) = sum{k=1 to n}
a(m-k)(-1)^(k+1)is always periodic,with period2n+2 if n is
===
transcendentals> Obviously, a number is of the form Nc + r for
rational r and integral> N iff its ! representation is
eventually constant.Looks right.I am still working on the only
if part.> It would be interesting if PI has a repeating base !
representation.Ts would be easy to computeAnyone out there
with a large number program and a lot of spare
computertime?Pseudo-code:' r = a real numberinput r' ignore
integer partr = Fractional(r)' calculate coefficientsn=2do '
coef=coefficient coef = integer( n! * r ) print n, coef r = r
===
transcendentals>  Russell Easterly <logiclab@comcast.net>
[NonBreakingSpace].8b.96.87.8c .97.99.95
.93fi.92.9b.93.87The factorial base allows you to
create a lot of transcendentals from e.> [snip examples]> I
spent a couple of minutes tnking about base ! and although I
have not> read the relevant older threads in sci.math, it
appears that ts base is> particularly _bad_ in terms of
representations.I gave a very short description.It may be you
didn't understand the system.Integers:Each position, n,
represents n!.The allowed coefficients are 0-n.Every integer
has an unique representation:! - base 100 - 01 - 110 - 211 -
320 - 421 - 5100 - 6...> For example: 1/2 = Sum(1/[n*(n-1)],
n=2..+oo) = Sum((n-2)!/n!, n=2..+oo) => 0.01126... (base !)
and> 1/4 = Sum(1/[n*(n-1)*(n-2)], n=3..+oo) = Sum((n-3)!/n!,
n=3..+oo) => 0.001126... (base !)Fractions:Each position
represents 1/n! (n>1).Coefficients can be 0 through (n-1).Like
fixed radix bases, the representation for certain numbers is
notunique.(e.g. in base 10, 1.0 = 0.999...)All rational
numbers have two representations: one finite and one
infinite:1/2 = 1/2! = .1 = .023456...1/4 = 1/3! + 2/4! = .012
= .011456789...1/11 = 2/4! + 5/6! + 3/7! + 1/8! + 4/9! +
10/11! = .002053140A= .0020531409BCDE...Russell- 2 many 2
===
trying to make sense of sometng that was mentioned in a book
I'mlearning topology from.a) First he states that defining the
product topology of spaces (X,Tx) =(Y,Ty) = R by Bxy = {O1 x O2
| O1 in Tx, O2 in Ty} will not work sincealthough the sets
(0,1) x (0,1), and (2,3) x (2,3) are in Bxy,[(0,1) x (0,1)] U
[(2,3) x (2,3)] is not.Ts made sense to me. Later he mentions
that a workable way to generate atopology T is to define it as
the topology having Bxy as a basis. Ok tsmakes sense.But he
says the following just before ts wch confused me:If it were
[an open set] (referring to [(0,1) x (0,1)] U [(2,3) x
(2,3)]froma) ), then <0.5,2.5> in (0,1)x(2,3), but <0.5,2.5>
is not in[(0,1)x(0,1)] U [(2,3)x(2,3)]. Thus T is not closed
under unions and so isnot a topology.Comments:First of all,
even if we use the basis Bxy to generate a topology T
thenagain, <0.5,2.5> in A = (0,1)x(2,3), but <0.5,2.5> is not
in B =[(0,1)x(0,1)] U [(2,3)x(2,3)]. A and B are both open in
T and the topologyis obviously closed under union.Second, in
what way can [(0,1)x(0,1)] U [(2,3)x(2,3)] be thought of as
a'union' of (0,1)x(2,3) and any open sets (reffering to the
statement Thus Tis not closed under unions)?Does ts guy simply
not know what he's talking about or am I missingsometng very
obvious or have made some basic logical error?l8r, Mike N.
===
ts problem but have no idea how to tackle it.2 rigid rods of
unit length are nged to each other. Initially they areparallel
to each other, much like a closed pair of divider.Ts divider is
placed on a piece of paper. What is the minimum area of
thepaper wch allow the divider to be opened such that the
angle between thetwo legs extend from 0 to 360 degrees. The
divider is to touch the paperand no part of the divider is to
extend beyond the paper throughout thewhole process.What area
===
area to flip 2 nged rodsthe area of the smaller unnged end of
the two rods (area-wise) plusthe larger unnged end's area,
plus a fudge factor (I'm supplying nomath because you didn't
specify the shape of the paper). You didn'tspecify that the
rods were laying on the paper.>I thought of ts problem but
have no idea how to tackle it.>2 rigid rods of unit length are
nged to each other. Initially they are> parallel to each other,
much like a closed pair of divider.>Ts divider is placed on a
piece of paper. What is the minimum area of the> paper wch
allow the divider to be opened such that the angle between
the> two legs extend from 0 to 360 degrees. The divider is to
touch the paper> and no part of the divider is to extend
beyond the paper throughout the> whole process.>What area of
===
to flip 2 nged rods>I thought of ts problem but have no idea
how to tackle it.>2 rigid rods of unit length are nged to each
other. Initially they are> parallel to each other, much like a
closed pair of divider.>Ts divider is placed on a piece of
paper. What is the minimum area of> the paper wch allow the
divider to be opened such that the angle between> the> two
legs extend from 0 to 360 degrees. The divider is to touch the
paper> and no part of the divider is to extend beyond the paper
throughout the> whole process.Since no part of the divider is
to extend beyond the paper, we assume that it is laid down on
the paper, not stood up on one point. We further assume that
the whole of (the underside of) the divider must be in contact
with the paper throughout any rotation operation, and that the
paper does not move.With these fairly simple assumptions in
place, it's hard to see how the answer will be anytng other
than pi square units (area of a circle of radius one unit).>
What area of mathematics deal with such problem ?Geometry.--
Richard Heathfield : binary@eton.powernet.co.ukUsenet is a
strange place. - Dennis M Ritce, 29 July 1999.C FAQ:
http://www.eskimo.com/~scs/C-faq/top.htmlK&R answers, C books,
===
to flip 2 nged rods> I thought of ts problem but have no idea
how to tackle it.> 2 rigid rods of unit length are nged to
each other. Initially they are> parallel to each other, much
like a closed pair of divider.> Ts divider is placed on a
piece of paper. What is the minimum area ofthe> paper wch
allow the divider to be opened such that the angle between
the> two legs extend from 0 to 360 degrees. The divider is to
touch the paper> and no part of the divider is to extend
beyond the paper throughout the> whole process.> What area of
mathematics deal with such problem ?A circle eith radius equal
to the length of one rod (or the longer rod, ifthey are not the
same)? Unless I've understood it incorrectly, it's
prettysimple. One rod lies on a radius of the circle, with the
nge at thecenter. Then the other rod is swung out around and
sweeps over the entirearea of the circle before it gets to 360
degrees, at wch point it is backwhere it started.But... maybe I
===
flip 2 nged rods I thought of ts problem but have no idea how
to tackle it. 2 rigid rods of unit length are nged to each
other. Initially they are> parallel to each other, much like a
closed pair of divider. Ts divider is placed on a piece of
paper. What is the minimum area of> the> paper wch allow the
divider to be opened such that the angle between the> two legs
extend from 0 to 360 degrees. The divider is to touch the
paper> and no part of the divider is to extend beyond the
paper throughout the> whole process. What area of mathematics
deal with such problem ?>A circle eith radius equal to the
length of one rod (or the longer rod, if> they are not the
same)? Unless I've understood it incorrectly, it's pretty>
simple. One rod lies on a radius of the circle, with the nge
at the> center. Then the other rod is swung out around and
sweeps over the entire> area of the circle before it gets to
360 degrees, at wch point it is back> where it started.>But...
maybe I did understand it wrong.>Jonathan> It can certainly be
done in a circle of that radius, but it can also be done in a
half circle of the same radius if the nge is allowed to change
positions,If r is the length of each segment, it can be done
witn the curve|x|^s + |y|^s = r^s, for s = 1, where the circle
would be s = 2.I tnk it can be done with s = 1/2.It may well be
that the minimal s for wch it can be done gives the desired
minimal enclosed area curve.On the other hand, there are
closed curves enclosing arbitrarily small areas in wch a line
===
Re: Do Prime Algebraic Numbers even exist? > Computational
complexity not only involves the number of operations, but >
also the complexity of operations. Operations on large numbers
are more > complex than operations on small numbers. When you
look at it from a pure > mathematical viewpoint it is a bit
different. But when you are actually > implementing stuff the
largeness of numbers gets to play a big role. Well, here's the
whole computation. Complexity is in the eye of the > beholder,
but ts doesn't look too complex to me. Formulas for the >
iterations are below. Use fixed-width font to view.Well, I
have said that I did not want to use a bignum package, and
indeed,the first line that goes wrong because I do not use
such a package is notedbelow: > 18 17 5 7 11752497748
581123613 18 17 5 7 -1132404140 581123613The second line is
what I get (going to unsigned long does not help either).Going
to double precision will give me some space, but not much. So
toget the proper value I need a bignum package wch adds
===
complex integration questionsI have a few quick questions
about integration of complex functions.first, how does one
integrate over a curve that crosses a branch cut?is it
possible? if c(t)=2e^it, t in [-pi,pi], then can integration
of1/(z^2-1)=1/2(1/(z-1)+1/(z+1)) be done as usual? what care
do i haveto take when doing ts?and lastly, is integration of
complex functions ever done over regionsin the plane instead
===
(model-free common sense steering): Neural networks are
model-free estimators, in that they do not requirean: in-depth
understanding of the phenomena they are modeling.::
http://www.arcon.com/arconneu.html:: THE MATHEMATICS OF
CROSSING THE STREET:: You are at the curb deciding, Should I:
cross the street? Well, it depends.:: AT THE CURB: The walk
light is on, but you see a: truck approacng fast. How fast?::
There is no exact number. Instead, there are an infinite
number of: possibilities - from 1kph to over 100kph and
everytng in between. You: don't have a radar gun, so instead
you watch the truck for a second ortwo,: and sum its speed up
in two words very fast. That is good enough.Phew.: Your senses
have told you the truck is coming very fast, but you need more:
information before you can decide whether or not to risk
crossing. Howfar: down the street is the truck? Is it slowing
down? Again, there are no: exact numbers, so you sum up the
situation - close, not slowing quickly: enough.:: Somehow your
brain adds fast + close + not slowing quickly enough,and: warns
you instantly that the risk is gh. It is purely
cognitiveprocess.: It involves a complex combination of
sensory information and experience.:: ...Since there are no
exact numbers in ts story, the mathematicalversion: must be
told with fuzzy numbers...:: But, the process is still not
quite over. Should I wait or cross? Youhave: to make the
decision. Risk tolerance leads to different spins and
endings.: If you walk with a cane, you reason, The risk is gh,
so I'll wait. You: watch as the truck runs the red light. If
you are a jogger, impatient to: cross, you disregard the
evidence, step into the intersection, and jumpback: just in
time to save your life.$#!! Where was my head? =):
http://www.decyde.com/crossingthestreet.html:: Fuzzy logic
works the way that humans tnk as opposed to the way that:
computers typically work. For example, consider the task of
driving a car.:: You notice that the stoplight ahead is: red
and the car ahead is braking. Your: mind might go through the
thought process,:: I see that I need to stop. The roads are:
wet because it's raining and there is a: car only a short
distance in front of me.:: Therefore I need to apply a
significant: pressure on the brake pedal.:: Ts is all
subconscious (in general), but that's the way we tnk - in:
fuzzy terms. Do our brains compute the precise distance to the
car aheadof: us and the exact coefficient of friction between
our tires and the road,and: then use a Kalman filter to derive
the optimal pressure wch should be: applied to the brakes? Of
course not. We use common-sense rules and they: seem to work
pretty well. On the other hand, when we do finally get around:
to pressing the brake pedal there is some exact force that we
apply, say: 1.326 pounds. So although we tnk in fuzzy,
noncrisp ways, our final: actions are crisp. The process of
translating the results of fuzzyreasoning: to a crisp,
nonfuzzy action is called defuzzification.::
http://www.innovatia.com/software/papers/fuzzy.htm:: ...In
particularly vast networks in fast moving environments, the
split: second it takes to traverse the circuit is greater than
the time it takes: for the situation to change. In reaction,
the last node tends tocompensate: by ordering a large
correction. But ts also is delayed by the longjourney: across
many nodes, so that it arrives missing its moving mark,
birtngyet: another gratuitous correction.:: The same effect
causes student drivers: to zigzag down the road, as each late:
large correction of the steering wheel: overreacts to the last
late overcorrection.:: Until the student driver learns to
tighten: the feedback loop to smaller, quicker: corrections,
he cannot help but swerve down: the ghway hunting (in vain)
for the center.:: Ts then is the bane of the simple
auto-circuit. It is liable toflutter: or chatter, that is, to
nervously oscillate from one overreaction to: another, hunting
for its rest. There are a thousand tricks to defeat ts:
tendency of overcompensation, one trick each for the thousand
advance: circuits that have been invented.::
http://www.kk.org/outofcontrol/ch7-c.html:: Fuzzy systems are
based on: storage of common-sense rules.:: For example, a
fuzzy Army-ant robot controller might have the fuzzy:
association if load is heavy, then signal for help longer.
Fuzzyphenomena: admit degrees: some loads are heavier than
others; some signal durationsare: longer then others.:: A
single association (heavy,longer): encodes all
combinations...:: Fuzzy systems reason with: parallel
associative inference.:: A fuzzy system reasons with
multivalued sets, instead of true or false: propositions, and
it may adaptively modify its fuzzy associations from:
representative numerical samples.::
http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html:: Wired:
What is fuzzy logic and why do critics call it the cocaine of:
science?:: Kosko: Fuzzy logic is Spock's worst nightmare - a
way of doing science: without math. It's a new branch of macne
intelligence that tries to make: computers tnk the way people
tnk and not the other way around. Youdon't: write equations
for how to wash clothes. Instead you load a cp withvague:
rules like if the wash water is dirty, add more soap, and if
verydirty,: add a lot more. All wash water is dirty and not
dirty - to some degree.: It's just common sense. But it breaks
the old either/or logic ofAristotle.: That offends some
scientists, who would like us to tnk and talk like: off/on
switches. But they still haven't produced a statement of fact
like: the sky is blue or E=mc^2 that is 100 percent true or
100 percentfalse.: Fact ain't math. You can never get the
science right to more than a few: decimal places. That's one
reason we find chaos when we look at tngs up: close...::
...Fuzzy systems are universal computers. I proved that as a
theorem - the: fuzzy approximation theorem. In theory, you can
replace every book on: physics or economics with equivalent
books that have fuzzy systems wherethe: equations used to be.
Fuzzy systems are model-free estimators. You don't: have to
guess at equations to build a bridge from inputs to outputs.
Fuzzy: rules build that bridge for you. There is math bend the
rules, but you: don't need to know it to program a fuzzy
system. You can program it in: English. If the air is cool,
turn the AC down a little. But the math is: not fuzzy. That's
why you can capture fuzzy logic in a digital cp.:: Most of the
first fuzzy systems were in control - as in adjusting a camera:
lens or backing up a trailer truck to a loading dock. Now we're
applying: fuzzy systems to wireless communications and
multimedia. The fuzzy rulescan: randomly spread signals over a
wide bandwidth or teach an intelligent: agent the kind of
houses or sunsets you prefer. The math says we can apply: them
anywhere. In practice, it may not be so easy.::
http://www.wired.com/wired/arcve/3.02/kosko_pr.html:: Fuzzy
logic is a superset of conventional(Boolean) logic that has
been: extended to handle the concept of partial truth- truth
values between: completely true and completely false. As its
name suggests, it is the: logic underlying modes of reasoning
wch are approximate rather thanexact.:: The importance of
fuzzy logic derives: from the fact that most modes of human:
reasoning and especially common_sense: reasoning are
approximate in nature.:: Boolean vs. Fuzzy: 300 years B.C.,
the Greek plosopher, Aristotle cameup: with binary logic(0,1),
wch is now the principle foundation of: Mathematics. It came
down to one law: A or not-A, either ts or not ts.: For
example, a typical rose is either red or not red. It cannot be
red and: not red. Every statement or sentence is true or false
or has the truthvalue: 1 or 0. Ts is Aristotle's law of
bivalence and was plosopcallycorrect: for over two thousand
years.:: Two centuries before Aristotle, Buddha, had the
belief wch contradicted: the black-and-wte world of worlds,
wch went beyond the bivalent cocoon: and see the world as it
is, filled with contradictions, with tngs andnot: tngs. He
stated that a rose, could be to a certain degree
completelyred,: but at the same time could also be at a
certain degree not red. Meaningthat: it can be red and not red
at the same time.:: Conventional(Boolean) logic states that a
glass can be full or not full of: water. However, suppose one
were to fill the glass only halfway. Then the: glass can be
half-full and half-not-full. Clearly, ts disprove's:
Aristotle's law of bivalence. Ts concept of certain degree
ormultivalence: is the fundamental concept wch propelled Zader
Lofti of UniversityBerkely: in the 1960's to introduce fuzzy
logic. The essential characteristics of: fuzzy logic founded
by m are as follows.::: In 1965, Lofti Zadeh formally
developed multivalued set theory, and: introduced the term
fuzzy into the technical literature. Nowadays, the: recent
emergence of fuzzy commercial products, as well as new theory,
has: generated a new interest in multivalued systems. Yet
already engineershave: successfully applied fuzzy systems in
many commercial areas : intelligent: subways automation,
emergency breakers, cement mixers, Kanji characters:
recognition, control air conditioners, automatic wasng macnes,
guideof: robot-arm manipulators, and so on.:: Fuzzy systems
store banks of fuzzy associations or common-sense rulessuch:
as IF traffic is heavy in ts direction, THEN keep the light
greenlonger: that might be articulated by an human expert.
Some traffic configurationare: heavier that others and some
green-light duration are longer than others,so: that, the
single fuzzy association (HEAVY, LONGER) encodes all these:
combinations. That is to say, fuzzy systems directly encode
structured: knowledge but in a numerical framework : by
entering the fuzzy association: (HEAVY, LONGER) as a single
entry in a rule database we are defining an: input-output
transformation.::
http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm:: Fuzzy
Logic is a computational paradigm capable of modelling the
own: uncertainness of human beings. Fuzzy reasoning is notng
else than aFuzzy: Logic-based formalism for encoding human
knowledge or common sense in a: numerical framework. Indeed,
the mathematical concepts on wch FuzzyLogic: is supported are
very easy to understand. In a Fuzzy Controller, human:
experience is codified by means of linguistic if-then rules,
wch compute: control actions upon given conditions. Fuzzy
Logic has been applied to: problems that are difficult to
solve mathematically. One of its main: advantages lies in the
fact that it offers a straightforward methodologyfor:
modelling and controlling non-linear systems, wch are
difficult to faceby: means of conventional techniques.::
http://www.wkap.nl/prod/b/1-4020-7359-3:: Fuzzy logic models
itself on the pattern of human reasoning in its use of:
approximate information and uncertainty to generate decisions.
It was: designed (during late 1980s and early 1990s) to
mathematically represent: vagueness and develop tools for
dealing with imprecision inherent inseveral: problems.
Normally, in digital computers one uses the binary logic
where: the digital signal has two discrete levels : low (logic
zero) or gh(logic: one); notng in-between. Fuzzy systems use
soft linguistic variables(e.g.: hot, tall, slow, light, heavy,
dry, small, positive, ...etc.) and a rangeof: their weightage
(or truth) values, called membersp functions, in the: interval
(0, 1), enabling the new computers to make human-like
decisions.: Since human beings tend to use words rather than
numbers to describe: behaviour patterns, fuzzy controls avoid
the conventional rigidity of: computers and allow them to use
parameters based on common sense.::
http://www.tribuneindia.com/2002/20021024/science.htm::: Fuzzy
logic best summed up by common sense:: Computer Corner: Boyd::
Fuzzy logic was introduced to the world 27 years ago by
Professor: Lotfi Zadeh in s Fuzzy Sets paper published in
Information: Control magazine, though it is only recently that
we've seen it: applied across a broad range of products.:: Some
readers have asked for more explanation on fuzzy logic, so:
here's an attempt to defuzzify the subject a little further.::
Simply put, fuzzy logic is aimed at enhancing our prissy
computer: technology with a touch of common sense.:: One
problem with the conventional digital computer is that it is:
such a scrupulously either-or beast. It cannot be easily
coaxed: to handle approximations or vague notions like young,
a lot and: probably.:: Yet most of us rely on such terms daily
because we happen to be: humans dealing with other humans, not
robots building cars.:: It's an easy matter to arbitrarily
program a computer so it: designates everyone falling into the
age-range 0f 15 to 18: as being a youth. Such a precise
category has come to be called: a crisp set since the
emergence of fuzzy logic.:: Yet we all know some 14-year-olds
can look older than some: late-developers turning 20. Such
exceptions, however, cannot: be accounted for in conventional
computing. Or at least not: without an inordinate amount of
additional programming and: expense.:: As Tetsuya Yamada, a
senior engineer at tac Ltd., replied: when I asked m if we
couldn't just continue using conventional: programming and
technology for controlling new products, instead: of fuzzy,
Well, we could. And you could probably swim across the:
Pacific if you got enough support from enough people. But
...:: To overcome ts problem, Zadeh was inspired to develop s
fuzzy: theory and the math to go with it that could be used to
create: fuzzy sets based on imprecise natural language.:: Each
member in a fuzzy set (such as the youths and others
considered: in the above example) is assigned one of a
continuous range of values: (called the membersp value)
between zero and one.:: Whereas in the above crisp set a
13-year-old going on 14 would still: have to be considered a
minor and thus be designated as zero in: binary logic, fuzzy
logic could assign m a membersp value of: say 0.1. Likewise,
an immature 20-year-old who would normally fall: outside our
either-or crisp-set range could be assigned a membersp: value
of 0.9 depending upon the criteria we use to measure youth.::
Working out just what criteria to use, what values should be
assigned: each member and deciding what rules are necessary to
govern the: relationsps between members is the key to
successfully applying: fuzzy control in products.:: In some
applications, determining the optimum rules has become so:
complex, some manufacturers have resorted to employing the aid
of: neural networks, wch may be stretcng a good tng too far,
given: fuzzy logic's original purpose to get round
complexity.:: Still, the flexibility in herent in fuzzy is
clearly useful in: dealing with approximate calculations, such
as about 100.:: It can be used in artificial intelligence to
provide us with an: almost true answer. It can also infer a
common-sense result even: when the data is not precise.:: Our
handwritten 5 in 250 would be treated as 5, not the letter S,:
for instance, in Sony's fuzzy-based Palmtop computer.:: Wle we
have all seen fuzzy logic-based products from the likes of:
Matsusta, Sanyo and tac, one unlikely company that has made:
fuzzy technology a central part of its business strategy is
Omron: Corp.:: It began its research into fuzzy logic in 1984
and has since applied: for over 700 patents. Ts puts it in the
forefront of fuzzy: applications in areas like factory and
industry control, as well as: in medical equipment.:: In 1989,
Omron also signed on lotfi Zadeh as a senior advisor.:: Earlier
ts year at the Business Show in Harumi, Omron demonstrated: its
fuzzy workstation. Omron manufactures both standard Motorola:
68040-based and 88000 reduced-instruction or RISC-based
workstations: that can be fitted with a fuzzy inference board,
turning them into: the world's first fuzzy workstations.::
Omron claims such a RISC-based workstation can aceve 4
billion: operations per second, an incredible speed if they
haven't fuzzed: on the number. Fuzzy logic is used in the
workstations to store: and retrieve fuzzy information and make
inferences.:: Ranging in price from Y2.5 million to almost Y4
million (a US dollar: is about 120 Yens -FM), these macnes are
not the kind of products: you will find down in Akihabara. (a
section of Tokyo famous for its: quantity and variety of
electronic goods -FM) Rather, they are: typically aimed at
value-added resellers in niche markets, and: engineers who
want to develop fuzzy applications, fuzzy databases: and
expert systems, as well as fuzzy inference systems.:: However,
the entrepreneurs among you may be interested in Omron's:
FB-30AT fuzzy inference board for the IBM PC and compatible
wares.: It features a 24 MHz FP-3000 fuzzy cp capable of
processing up: to 128 rules, with five antecedents and 2
consequents. Training: software and a compiler is also
available.:: Omron has also produced a fine little booklet on
fuzzy called: Clearly Fuzzy that I dipped into when writing ts
column.:: Tadas Katsuno, at Omron's public relations section,
tells me: he still has a limited number of copies left that he
will send: to the first readers of Computer Corner who write to
m with: contact information.:: The address is Omron Corp.,
International Public Relations Section,: Omron Tokyo Bld.,
3-4-10 Toranomon, Minato Ward, Tokyo 105.: --------:: - Farzin
Mokhtarian:
farzin@apollo3.ntt.jp::http://www-cgi.cs.cmu.edu/afs/cs/
project/ai-repository/ai/areas/fuzzy/doc/intro/j_times.tgz::
http://www.ece.utep.edu/research/webfuzzy/about.html:http://
www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS_NOE_
ASI99/ASI'99_ViharosMonostori.htm:
http://www.bjarne.ca/pmflp.pdf:
http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm:
http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/
FuzzyLogicControl.htmSometimes I really believe that people
give many names to the same tngwhen it is sometng we really
appreciate. There is model in there, but itis prior to
numbers. It is topological, connective. It looks like:a ->
band introduces a fundamental asymmetry we can play
with.Automata build up digraphs of these upon wch they evolve
through statetransitions. Abstraction neural nets are often
lattices, though you toss ina loop or ring or whatever (like
in nematodes) and you have recurrence.All of them say the same
tng: you are passing information through astructure of
transformations.I took a class in Control Theory once in my
undergrad years, and theprofessor used to often suggest books
to me by Norbert Wiener and others inthe cybernetics community
or Bucky's synergetics. When I could understanddynamics better,
I was able to get into Rene Thom and catastrophes
throughanother suggestion when he was teacng a non-linear
diffy-que class.The discrete becomes continuous.And the
logics, all of them, so very useful. The dynamics they
===
Simple idea, mathematics and common-sense>You're really
sounding like an idiot. If you want to say> mathematicians are
all wrong you need to come up with> sometng they actually say
===
under quadratic constraintDaer all:I have a question about
optimization.The description is as below:max. (x-t)'R(x-t)S.T.
(x-m)'W(x-m)=ct, m, R, W, c is givent m x is a n*1 vectorR W is
sym. matrixc is a scalarany body can solve ts
===
optimiziation.The description are as below:max.
(x-t)'R(x-t)S.T. (x-m)'W(x-m)=ct, m, R, W, c is givent m x is
a n*1 vectorR W is sym. matrixc is a scalaranybody can solve
===
majors?>I'm a gh school senior planning on studying pure math
as a major> ts fall in college. What I'm wondering is, what
kind of advice> could those of you who have already been
through the experience offer> to people in my position? What
kind of background is necessary (or> recommended) to begin
college mathematics courses, and what could a> future math
major do in order to get a head start? Are there any> books or
other information sources that would be helpful? Any advice> is
greatly appreciated.>If you haven't already, get some of the
easier topics out of the way by> studying on your own: linear
algebra, probability, number theory, set> theory, as much
calculus as you can stand. If you don't find these topics>
easy, change to accounting.On the contrary, if you find these
topics easy, you are undoubtedly> taking a very superficial
view of them. Math is not easy, and the myth> that it is, is
one of the big tngs a gh-schooler needs to dispell> when s/he
goes to college. Math has certainly never been easy for me, in
part because I used tolearn it by rote like all too many people
do. Mathematical *tnking*is sometng that I discovered entirely
on my own last year, when Irealized that our textbook actually
provided the *reasons* bend thevarious formulae we had all
memorized in trig class.>Getting topics out of the way is not
a good approach in general. To> really understand mathematics,
you have to learn to let your mind> wander and explore. By
being in a rush to pass advanced courses or> learn the
coursework oftentimes students miss out on the deepness of>
the material they study. Agreed. My biggest concern is not
getting topics out of the way,but rather having sufficient
background to begin such courses. Itnk I have a pretty good
idea of what it takes to ace a course. Mybattle plan is
sometng along the lines of:1) read the section of the textbook
to be covered in the lecture*before* coming to class (in many
cases several readings arenecessary); taking notes is also a
good idea2) during the lecture, take notes but more
importantly pay attentionand try to clarify whatever questions
you had during your initialreading3) go home and try to piece
together everytng learned from thetextbook and teacher. work
through examples in the textbook, and doall assigned homework
problems.4) make sure to clarify anytng you have trouble with.
before thetest, do *extra* practice problems. Having
supplementary textbooksaround is very useful.I wish I had
carried out a plan such as ts beginning in, say, 8thgrade, but
unfortunately I feel as if my understanding has some holesin
it. I took my gh school's college prep geometry course in
9thgrade, wch omitted proofs entirely (fortunately I've
introduced themto myself). There are a few holes I'm working
on patcng fromAlgebra II. Trig is a little bit different,
namely because I'm notentirely sure how much of it is really
necessary. Identities, forexample: we've only used the
Pythagorean, reciprocal, andtangent/cotangent identities in
our calc class thus far. How muchtrig do I actually need to
know for more advanced math?>Unfortunately, when you go to
college, you may very well get> indoctrinated into ts way of
doing tngs. It's important to learn> how to creatively learn
mathematics before going to college, so as to> avoid ts trap.
>When you get to college, you will want to become a member of
the MAA. > Their periodicals are extremely helpful and will
give you further> guidance. You may want to look at their
publications, before> graduating. >Calculus is the initial
hurdle of a college freshman, so it's a good> idea to have
some familiarity with it. Someone has already recommended>
Courant and Robbins, _What is Mathematics?_. It has a good
section on> calculus that should cover what you need for now.
Also, Gardner> has a revised version of Silvanus Thompson's
_Calculus Made Easy_ that> I recommend. [Ts doesn't contradict
my statements above, as even> Gardner notes that the title
should really be Calculus made easier]I'm presently taking
Calculus AB (calc ~ 1.5) at my gh school (ourschool does not
offer Calculus BC). I'd say that I have asatisfactory
*understanding* of the important concepts, my hurdle
isfluency. I'm not exactly the brightest guy around, IQ
somewhere inthe 120 range if that matters at all; 700 math SAT
I score. I'mprobably the only student in my class who can
explain the formaldefinition of the derivative, but tell me to
differentiate somegod-awful trig expression and I'll most
likely get lost in the mess. I ended up with a B+ in the
course first semester, in part due to afew instances of rather
poor planning that I intend on avoiding in thefuture.>As for
what to do before going to college, play with math as much as>
you can. There are lots of tngs to explore. You'll not have as
much> time for playing in college.I hope so. I'm doing ts
because I sincerely love math--I've read apretty good deal
about some of the concepts of gher math, and theyreally
intrigue me. I also realize that I'm not going to
trulyunderstand these topics unless I put in a ton of hard
work... how doyou get to Carnegie Hall? (if you didn't notice
from my name ore-mail address, I'm a musician). At the same
time, I'm not planningor expecting to become a full-blown
mathematician. I'm an amateurcomputer programmer, and would
ideally like to double-major in mathand CS, but I'm open to
expand myself to other fields as well. AndI'm not entirely
sure whether I have what it takes, in terms of
rawintelligence, to survive as a math major, but I'm hoping
that hardwork and genuine interest in the topic will prove to
===
future math majors? I'm a gh school senior planning on
studying pure math as a major> ts fall in college. What I'm
wondering is, what kind of advice> could those of you who have
already been through the experienceoffer> to people in my
position? What kind of background is necessary (or>
recommended) to begin college mathematics courses, and what
could a> future math major do in order to get a head start?
Are there any> books or other information sources that would
be helpful? Anyadvice> is greatly appreciated. If you haven't
already, get some of the easier topics out of the wayby>
studying on your own: linear algebra, probability, number
theory, set> theory, as much calculus as you can stand. If you
don't find thesetopics> easy, change to accounting. On the
contrary, if you find these topics easy, you are undoubtedly>
taking a very superficial view of them. Math is not easy, and
the myth> that it is, is one of the big tngs a gh-schooler
needs to dispell> when s/he goes to college.> Math has
certainly never been easy for me, in part because I used to>
learn it by rote like all too many people do. Mathematical
*tnking*> is sometng that I discovered entirely on my own last
year, when I> realized that our textbook actually provided the
*reasons* bend the> various formulae we had all memorized in
trig class. Getting topics out of the way is not a good
approach in general. To> really understand mathematics, you
have to learn to let your mind> wander and explore. By being
in a rush to pass advanced courses or> learn the coursework
oftentimes students miss out on the deepness of> the material
they study.> Agreed. My biggest concern is not getting topics
out of the way,> but rather having sufficient background to
begin such courses. I> tnk I have a pretty good idea of what
it takes to ace a course. My> battle plan is sometng along the
lines of:> 1) read the section of the textbook to be covered in
the lecture> *before* coming to class (in many cases several
readings are> necessary); taking notes is also a good idea> 2)
during the lecture, take notes but more importantly pay
attention> and try to clarify whatever questions you had
during your initial> reading> 3) go home and try to piece
together everytng learned from the> textbook and teacher. work
through examples in the textbook, and do> all assigned homework
problems.> 4) make sure to clarify anytng you have trouble
with. before the> test, do *extra* practice problems. Having
supplementary textbooks> around is very useful.> I wish I had
carried out a plan such as ts beginning in, say, 8th> grade,
but unfortunately I feel as if my understanding has some
holes> in it. I took my gh school's college prep geometry
course in 9th> grade, wch omitted proofs entirely (fortunately
I've introduced them> to myself). There are a few holes I'm
working on patcng from> Algebra II. Trig is a little bit
different, namely because I'm not> entirely sure how much of
it is really necessary. Identities, for> example: we've only
used the Pythagorean, reciprocal, and> tangent/cotangent
identities in our calc class thus far. How much> trig do I
actually need to know for more advanced math?My teacher put a
lot of emphasis on trig identities... in my experience theyare
useful for simplifying derivitives, and even more so
integrals.So yes, they are good. You don't need to memorize a
whole lot, but beingfamiliar with some of them and having
experience proving them is good. Tryand find some problems wch
have awful looking trig identities for you toprove...
unfortunately, the only source I'm familiar with is an out of
printrussian textbook, so I can't give you any advice on
===
Tutorial (Freeware) I don't tnk that there is anytng wrong
with your notation. Well, let's face it Daryl, that's just an
idiotic claim.> For example, it's extremely misleading to use
| instead of the> connective v.> Who the did EVER write> p |
q> when he MEANT> p v q or p / q or p or q> ???Somebody whose
been taking CompSci for too many years?Honestly, I've done it
before, particularly when I've been working onprogramming and
then go to take a break. Of course, my scratch work
containsall sorts of nonstandard symbols... :-/That said...
well... I'll stay out of ts discussion ;-)> nt: The | usually
is called the Sheffer stroke, and it is used to> denote the
NAND truth-function.>
http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Sheffer+stroke>
http://en.wikipedia.org/wiki/Sheffer_stroke> Don't tnk, look!
===
Give me that old time ontology: (was: the anticlassicalist }{
i: linguistic negation)> Why don't you state one single mind
bogling factClassical propositional calculus has
non-Booleanmodels! Related to ts and even more surprisingis
that classical propositional calculus can bemodeled by a
non-Boolean lattice [Reference 6], afact apparently overlooked
for over 100 years!Common intuition is that classical
propositionalcalculus and Boolean algebra go
hand-in-hand.Lattice O6 is a counterexample that shows
tsintuition is false.Ts is in
http://teivos.samos.aegean.gr/~giorgosv/metamathsite/qlegif/
mmql.htmlreferencing:M. Pavicic and N. Megill,Non-Orthomodular
Models for Both Standard Quantum Logicand Standard Classical
Logic: Repercussions for QuantumComputers,Int. J. Theor. Phys.
39, 2349-2391 (2000),http://xxx.lanl.gov/abs/quant-ph/9906101
===
fictional and that all of physics is built from either p-adics
or doubly-infinites Re: infinite rightward strings tacked-on to
p-adics serves as OrthogonalityI can tnk of several cases in
physics where the idea that allnumbers come in only 2 types,
either p-adics or doubly-infinites.wavefunction. And
doubly-infinites as the surrounding wave or field....9999998
is surrounding by doubly-infinites of say for
example.....999998.33333..... and .....9999998.5656565.... to
mention justtwo.A second case is in the Lorentz
transformations of Special Relativityx^2 + y^2 + z^2 -
c^2t^2where the odd term is that of a negative signed c^2t^2.
The terms ofx^2 and y^2 and z^2 are all p-adics but because
the last term isnegative destroys the symmetry. But if we
consider that the last termis a doubly-infinite whereas the
first three terms are p-adics thenthe negative sign is
removed. Also, the parameter of time maybe adoubly-infinite
whereas the parameters of space are p-adics wchmeans that
there are not really any 4th dimension and that dimensionsstop
at 3rd dimension.The trd case deals with the incompatibility
heretofor recognizedbetween thermodynamics, statistical
mechanics and quantum mechanics inthe equipartition energy
function. Perhaps those can be reconciledonce replacing some
terms as p-adic and other terms asdoubly-infinites.The fourth
case to consider is mathematics of e^i(pi) is -1. If
weconsider that Reals are a fiction and that e and pi are
doublyinfinites whereas i is a p-adic then we can manage to
plug into thatequation actual numbers instead of symbols.And
the endresult is not -1 since that is a Real, yet Reals are
afiction and so -1 is really .....9999 in 10-adics. So all
that isneeded is the doubly-infinite that is e and the
doubly-infinite thatis pi.I cannot tnk of any cases in
mathematics where a number system has 2components and where
they are orthogonal to one another to resemble orsimulate the
Doubly-Infinites.Arcmedes Plutoniumwhole entire Universe is
just one big atom where dotsof the electron-dot-cloud are
galaxies(www.iw.net/~a_plutonium) website of the science of AP
under revisionwhat used to be my old science
websitewww.newphys.se/elektromagnum/physics/LudwigPlutonium
===
a system consisting of 6 atoms. They all interact with one
another(bonds and van der waals forces).I have the equations
for the interactions between atoms.Each atom is in 3D space. I
can alter the x,y,z for each atom.I will have a function that
depends on these 3*6 variables. Constructingts function should
be fairly easy.(For ease of work, I presume that I should fix
one of the atoms at theorigin).I want to minimise ts function
(analytically).So, How do I differentiate ts 18 variable
function?I do have a degree in comp + math but it was a long
time ago, and I don'teven know what ts would be called.
Multivariate differentiation? Partialdifferentiation?Anyone
know how to do ts (or have any links)Also, I've got some tools
that I'll be using once I got the method straightin my head:
Mathematica, mathcad and matlab.Anyone got any idea how I
would use one of the packages to help me?(preferably
mathematica, as I heard that the strongest symbolically) for
any advice.Choca