mm-46
===
Do you write ASM for the Pentium?:)> My calculus is self
taught.Mine too.> Int(x*(e^(2x))/(2x+1)^2)> The answer is:
1/4*(e^(2x))/(2x+1) + C> I'm having a terrible time finding a
way to simplify the integral> though.Since 2x appears twice in
the integrand, I tried the substitutionz = 2x+1dx = dz/2and
ended up with1/4e Int [ (z-1)e^z / z ] dzwhich is recognizable
as (dismissing a constant or two)e^z / z.As you may know, the
integrals of e^x / x and e^x / x^2 do not have a closed form ,
although we just saw that the difference between them
does.LH
===
Math is a hobby for me. I've been reading up on
Eigenvectors andEigenvalues. It get the manipulations
involved, but can't imagine theapplications -- and the books I
have don't help. Can people provide afew examples? Specific
examples, if possible -- not just, they are used
inelectronics, or physics, or whatever, but rather, something
like:M is the matrix which describes such-and-such physical
property ortransformation or process, It's eigenvectors V
correspond to such andsuch property, and the eigenvalues of V
and M indicate such-and-such.Steve O.
===
>Math is a hobby for
me. I've been reading up on Eigenvectors and>Eigenvalues. It
get the manipulations involved, but can't imagine
the>applications -- and the books I have don't help. Can
people provide a>few examples? Google uses a (big) matrix to
describe hyperlinks between web pagesand ranks them in order
by solving the eigenvalues and -vectors.
===
> Math is a hobby
for me. I've been reading up on Eigenvectors and> Eigenvalues.
It get the manipulations involved, but can't imagine the>
applications -- and the books I have don't help. Can people
provide a> few examples? Specific examples, if possible -- not
just, they are used in> electronics, or physics, or whatever,
but rather, something like: M is the matrix which describes
such-and-such physical property or> transformation or process,
It's eigenvectors V correspond to such and> such property, and
the eigenvalues of V and M indicate such-and-such.> One
application is in differential equations. If you want to
solvea system of linear equations in the formy' = A ywhere A
is a nxn Matrix and y is a vector valued function youcan use
the eigenvectors of A to obtain the solutions. Inparticular,
if a is an eigenvector of A with eigenvalue l,then y(x) = c
e^(l x) ais a solution for any real or complex constant c. If
the matrixA has a basis of eigenvectors (i.e. if A is
diagonalizable)then any solution looks likey(x) = c_1 e^(l_1
x) a_1 + c_2 e^(l_2 x) a_2 + ... + c_n e^(l_n x) a_nwhere a_1,
a_2, ... a_n are the eigenvectors to the eigenvaluesl_1, l_2,
... l_n respectively and c_1, c_2, ... c_n areconstants.This
fact has also big implications for nonlinear
differentialequations. The keywords for this are linearization
and stability .HTH,Michael.--
&&&&&&&&&&&&&&&&#@#&&&&&&&&&&&&&&&&Dr. Michael UlmFB
Mathematik, Universitaet
Rostockmichael.ulm@mathematik.uni-rostock.de
===
> Math is a
hobby for me. I've been reading up on Eigenvectors and>
Eigenvalues. It get the manipulations involved, but can't
imagine the> applications -- and the books I have don't help.
Can people provide a> few examples?Very useful for in solving
simultaneous equations particularly with a largenumber of
variables.I am from an electronics/physics background and used
them extensively.Andy
===
I distinctly remember a usenet post
quoting some member of Britishroyalty saying, Algebra. Is that
those pointy things? althougha google search denies me.
===
Math
is a hobby for me. I've been reading up on Eigenvectors
andEigenvalues. It get the manipulations involved, but can't
imagine theapplications -- and the books I have don't help.
Can people provide afew examples? Specific examples, if
possible -- not just, they are used inelectronics, or physics,
or whatever, but rather, something like:M is the matrix which
describes such-and-such physical property ortransformation or
process, its eigenvectors V correspond to such andsuch
property, and the eigenvalues of V and M indicate
such-and-such.Steve O.
===
> Math is a hobby for me. I've been
reading up on Eigenvectors and> Eigenvalues. It get the
manipulations involved, but can't imagine the> applications --
and the books I have don't help. Can people provide a> few
examples?> Specific examples, if possible -- not just, they
are used in> electronics, or physics, or whatever, but rather,
something like:> M is the matrix which describes such-and-such
physical property or> transformation or process, its
eigenvectors V correspond to such and> such property, and the
eigenvalues of V and M indicate such-and-such.Imagine P being
the matrix of transition probabilities fromone state of a
system to another of some system.P_ij is the probability that
the system goes from state i tostate j. The sum of each row is
one: sum( P_ij, j = 1...n ) = 1.This is the transition matrix
of a so-called Markov chain.Under certain circumstances the
infinite matrix productlimit converges such that limit(
P_ij^(n), n-->infinity) = p_j for all i,j.where [ p_j, j=1...n
] is the limit vector of the probabilitiesof the system being
in the different states.Here P_ij^(n) is the i,j element of
the product matrix P^n,with the transition probabilities from
state i to state j aftern steps (as opposed to after 1 step as
P_ij).In stead of calculating the limit, one can try to find
thevector [ p_i ] of the probabilities of the initial
states,such that these probabilities are not influenced by
theevolution of the system, i.o.w. find the vector [ p_i ]such
that sum( p_i * P_ij, i=1...n ) = p_j for all j,i.o.w. find an
eigenvector with eigenvalue 1 of thetransposed matrix P^t.This
eigenvector with probabilities of the initial systembeing in
the different states, does not change when thesytem
evolves.Dirk Vdm
===
PM:> Specific examples, if possible -- not
just, they are used in> electronics, or physics, or whatever,
but rather, something like:One example is to describe states
of polarized light. There are twoeigenstates, say horizontally
and vertically polaraized light. Anypolarization state is
represented as a sum of these two states multiplied bycomplex
constants. That includes circular and elliptical
polarization.Operations on these eigenvectors represent what
happens when the light goesthrough a waveplate.Electron spins
can also b represented by the same mathematics.An amusing
series authored by Joseph Slepian some decades ago, probably
inthe 30s, maybe 40s appearing in the Transactions of the
IEEE. You will needa good technical library to dig that out.
It is about someone financed to atechnical education by an
uncle. The lad reduced the inventory of hisuncle's custom
fruit salad business by representing mixtures of fruit
asvectors. The method depended upon making eigenvector
combinations of fruit.Bill
===
> Math is a hobby for me. I've
been reading up on Eigenvectors and> Eigenvalues. It get the
manipulations involved, but can't imagine the> applications --
and the books I have don't help. Can people provide a> few
examples? Specific examples, if possible -- not just, they are
used in> electronics, or physics, or whatever, but rather,
something like: M is the matrix which describes such-and-such
physical property or> transformation or process, its
eigenvectors V correspond to such and> such property, and the
eigenvalues of V and M indicate such-and-such. Steve O. *be*
an idiot, but even idiots have feelings.A common use of
eigenvalues and eigenvectors is in the analysis of
dynamicmechanical systems. Given an undamped mechanical system
described by the differential
equations[M](d^2x)/(dt^2)+[K]u=0where M is mass, K is spring
stiffness(d^2x)/(dt^2) is acceleration and x is position,the
eigenvalues of the system notes the squared ressonant
frequencies of thesystem and the eigenvectors are the
decomposed patterns of motion.--
----------------------------Christopher GrindePh.D
studentMobile:+47 91137588Tlph: +47
33037717Web:http://cg.ans.hive.no-----------------------------
Vestfold University CollegeInstitute of microsystem
technology.http://ri.hive.no/imst
===
> Math is a hobby for me.
I've been reading up on Eigenvectors and> Eigenvalues. It get
the manipulations involved, but can't imagine the>
applications -- and the books I have don't help. Can people
provide a> few examples? Specific examples, if possible -- not
just, they are used in> electronics, or physics, or whatever,
but rather, something like: M is the matrix which describes
such-and-such physical property or> transformation or process,
its eigenvectors V correspond to such and> such property, and
the eigenvalues of V and M indicate such-and-such.The inertia
tensor of 3d objects can be written as asymmetric positive
definite matrix. It is used to calculateangular momentum out
of angular velocity. If the axis ofrotation of a (free) body
is along one of the eigenvectors,angular momentum will have
the same direction as angularvelocity and the axis of rotation
will remain the same.There are three such axes, perpendicular
to each other.(Symmetric matrices are diagonalizable in an
orthogonalbasis.)> Steve O.>
===
I got some great news for
you all. Yesterday a friend called to tell me> about the new
project she's starting.> I have some great news for you. I
just saved a ton of money on myauto insurance by switching to
Geico.
===
http://www22.pair.com/csdc/car/carfre64.htm#
OPENSETSHe defines the sets in a topology T of a topological
space X as closed sets.I've read many definitions that define
an open set to be any subset U of Xthat is in T
:<quote>Definition: A topology T1(closed) on a set X is a
collection or class ofsubsets that obeys the following axioms:
a.. A1(closed): X and the null set 0 are elements of the
collection. b.. A2(closed): The arbitrary intersection of any
number of elements ofthe collection belongs to the collection.
c.. A3(closed): The arbitrary union of any pair of elements
ofthecollection belongs to the collection.The elements of the
collection, T1(closed), are defined to be closed sets.The
compliments of the closed sets are defined as open
sets.</quote>l8r, Mike N. Christoff
===
>
http://www22.pair.com/csdc/car/carfre64.htm#OPENSETS> He
defines the sets in a topology T of a topological space X as
closedsets.> I've read many definitions that define an open
set to be any subset U of X> that is in T :> <quote>
Definition: A topology T1(closed) on a set X is a collection
or class of> subsets that obeys the following axioms:> a..
A1(closed): X and the null set 0 are elements of the
collection.> b.. A2(closed): The arbitrary intersection of any
number of elements of> the collection belongs to the
collection.> c.. A3(closed): The arbitrary union of any pair
of elements of> thecollection belongs to the collection.> The
elements of the collection, T1(closed), are defined to be
closed sets.> The compliments of the closed sets are defined
as open sets.> </quote>I think I figured this out...Here is a
definition of topological space from Wikipedia.Formally, a
topological space is a set X together with a collection T
ofsubsets of X (i.e., T is a subset of the power set of X)
satisfying thefollowing axioms:a) The empty set and X are in
T.b) The union of any collection of sets in T is also in T.c)
The intersection of any pair of sets in T is also in T. The
set T is called a topology on X. The sets in T are referred to
as opensets, and their complements in X are called closed
sets.---------------I didn't realize how close the definition
of open set and properties ofclosed sets were. They seem
almost identicalLet X be a topological space. A subset of X is
closed if its complement isopen. The closed sets satisfy the
following conditions.(i) The empty set ; and the set X are
closed.(ii) Any finite union of closed sets is closed.(iii)
Any intersection of closed sets is closed.except b) refers to
arbitrary union / ii) finite union ; c) pairintersection /
iii) arbitrary intersection. This would also mean X and {}are
both open and closed.Is a set closed iff it has the above
properties with respect to sometopology? Or can it have the
above, yet not be closed (ie: not be thecomplement of an open
set)?l8r, Mike N. Christoff
===
> [...]>I didn't realize how
close the definition of open set and properties of>closed sets
were. They seem almost identical>Let X be a topological space.
A subset of X is closed if its complement is>open. The closed
sets satisfy the following conditions.>(i) The empty set ; and
the set X are closed.>(ii) Any finite union of closed sets is
closed.>(iii) Any intersection of closed sets is
closed.>except b) refers to arbitrary union / ii) finite
union ; c) pair>intersection / iii) arbitrary intersection.
This would also mean X and {}>are both open and closed.>Is a
set closed iff it has the above properties with respect to
some>topology? Or can it have the above, yet not be closed
(ie: not be the>complement of an open set)?>A set is closed
iff its complement is open. Your question is actually not
welll-posed: How can a *single* set have properties
(i)-(iii)?As you and others have noted, most authors define a
topology as the collection of open sets. Actually, the
sequence is usually to define topology (closed under arbtirary
union, finite intersection), define a set to be open iff it is
in the topology, and define a set to be closed iff its
complement is open. But it is not unheard of to define a
topology by its closed sets (as you do above), and the two
approaches are equivalent.-- Stephen J. Herschkorn
herschko@rutcor.rutgers.edu
===
> [...]>I didn't realize how
close the definition of open set and properties of>closed sets
were. They seem almost identical>Let X be a topological space.
A subset of X is closed if its complementis>open. The closed
sets satisfy the following conditions.>(i) The empty set ; and
the set X are closed.>(ii) Any finite union of closed sets is
closed.>(iii) Any intersection of closed sets is
closed.>except b) refers to arbitrary union / ii) finite
union ; c) pair>intersection / iii) arbitrary intersection.
This would also mean X and{}>are both open and closed.>Is a
set closed iff it has the above properties with respect to
some>topology? Or can it have the above, yet not be closed
(ie: not be the>complement of an open set)?> A set is closed
iff its complement is open. Your question is actually> not
welll-posed: How can a *single* set have properties
(i)-(iii)?>rephrased:Let X be a topological space and T be a
collection of sets with thefollowing properties:(i) The empty
set ; and the set X are in T(ii) Any finite union of sets in T
is in T.(iii) Any intersection of sets in T sets is in T.Let U
be an arbitrary set in T. Is U neccessarily closed?l8r, Mike
N. Christoff
===
>Let X be a topological space and T be a
collection of sets with the>following properties:>(i) The
empty set ; and the set X are in T>(ii) Any finite union of
sets in T is in T.>(iii) Any intersection of sets in T sets is
in T.>Let U be an arbitrary set in T. Is U neccessarily
closed?>Unless X is discrete, no: Let A be a set which is not
closed. Then T = {{}, A, X} satisfies the three properties.
Also, P(X) (the power set) satisfies the three properties.Any
such collection T indicates a topology (viz., the one where
the sets of T are the closed sets). If X has more than one
element, there are multiple topologies on X.-- Stephen J.
Herschkorn herschko@rutcor.rutgers.edu
===
rephrased: Let X be
a topological space and T be a collection of sets with the>
following properties:> (i) The empty set ; and the set X are
in T> (ii) Any finite union of sets in T is in T.> (iii) Any
intersection of sets in T sets is in T. Let U be an arbitrary
set in T. Is U neccessarily closed? l8r, Mike N. ChristoffLet
U be any not closed subset in the topology of X. Let T be
{empty set,U,X}Will that do? In general, given a topology on
X, just define anothertopology it in your favourite way
(discrete, cofinite whatever) and itsclosed sets will satisfy
those requirements, and there should be a closedset in the new
topology that wasn't in the old one.
===
Let X be a topological
space and T be a collection of sets with thesets?or subsets of
X?> following properties:> (i) The empty set ; and the set X
are in T> (ii) Any finite union of sets in T is in T.> (iii)
Any intersection of sets in T sets is in T. Let U be an
arbitrary set in T. Is U neccessarily closed?closed?closed in
what?If you mean closed in the topological space X then the
answeris not necessarily . The sets in T may have no relation
withthe given topology on X.-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the
last laugh. Alan Partridge, _Bouncing Back_ (14 times)
===
>
http://www22.pair.com/csdc/car/carfre64.htm#OPENSETS He
defines the sets in a topology T of a topological space X as
closed sets.> I've read many definitions that define an open
set to be any subset U of X> that is in T : <quote>
Definition: A topology T1(closed) on a set X is a collection
or class of> subsets that obeys the following axioms:> a..
A1(closed): X and the null set 0 are elements of the
collection.> b.. A2(closed): The arbitrary intersection of any
number of elements of> the collection belongs to the
collection.> c.. A3(closed): The arbitrary union of any pair
of elements of> thecollection belongs to the collection. The
elements of the collection, T1(closed), are defined to be
closed sets.> The compliments of the closed sets are defined
as open sets.> </quote> This looks OK. You can define a
topology by describing eitheropen or closed sets. Note that
the complements of thusdefined closed sets satisfy the
canonical definition of anopen topology.The only error I see
is saying 'compliments'when you mean 'complements'.
===
> This
looks OK. You can define a topology by describing either> open
or closed sets.or convergent nets, or neighborhoods, or
convergent filter-bases, ornon-standard monads, or...-- G. A.
Edgar http://www.math.ohio-state.edu/~edgar/
===
This looks
OK. You can define a topology by describing either> open or
closed sets. or convergent nets, or neighborhoods, or
convergent filter-bases, or> non-standard monads, or...or a
closure operator. That's a nice one. There is a whole bookdone
that way... (some eastern European I believe, can't think of
whichnow).
===
--------------------------------- <^> <()> <^>
-----------------------------------> The mathematician comes
home early from a conference> and finds his wife...eh, you
know the routine.> Off all the...With a traveling salesman? >
Well, at least he was P-space hard. That was NFHerc
===
I need
help on the following problem.Let X,Y be Banach spaces and
B(X,Y) the space of continuous linearoperators.Let
S:(Y*,sigma)-->(X*,tau) is linear continuous mapping ,where
sigmaand tau are the coresponding weak* topologies on
X*,Y*.Prove that S=T* where T is in B(X,Y).And * denotes the
dual operator or dual spacecorespondingly.Greetings Mladen
===
I need help on the following problem. Let X,Y be Banach spaces
and B(X,Y) the space of continuous linear> operators.> Let
S:(Y*,sigma)-->(X*,tau) is linear continuous mapping ,where
sigma> and tau are the coresponding weak* topologies on
X*,Y*.Prove that S=T*> where T is in B(X,Y).And * denotes the
dual operator or dual space> corespondingly.> T should be the
restriction of T** to (the cononical subpace identifiedwith)
X. So you only have to show that T** maps X into Y. How can
youidentify an element of Y** as belonging to Y? That has
something to dowith the weak* topology on Y.
===
> I need help
on the following problem.>Let X,Y be Banach spaces and B(X,Y)
the space of continuous linear>operators.>Let
S:(Y*,sigma)-->(X*,tau) is linear continuous mapping ,where
sigma>and tau are the coresponding weak* topologies on
X*,Y*.Prove that S=T*> where T is in B(X,Y).And * denotes the
dual operator or dual space>corespondingly.A hint to get
started: Note that one can talk about dual operators(or
adjoint operators) of operators mapping topological
vectorspaces to topological vector spaces; the notion is not
restrictedto operators on Banach sapces. Now given
S:(Y*,sigma)-->(X*,tau)it follows that we have S*:(X*,tau)*
-->(Y*,sigma)*. What is(X*,tau)* ?It probably says somewhere
in the book what (X*,tau)* is. Ifnot: Suppose that L is in
(X*,tau)*. It follows from the definitionof tau that there
exist L_1, ... L_n in X such that |L| <= |L_1| + ...
|L_n|pointwise on X*. And hence...>Greetings
Mladen************************David C. Ullrich
===
I remember
many years at school, a math teacher proving this.Has anyone
got a math example of how this is possible?
===
> I remember
many years at school, a math teacher proving this.> Has anyone
got a math example of how this is possible?Not strictly
related, but I remember the proof that girls are evilfrom
approximately the same era:(1) girl = time * money (2) money =
sqrt(evil) (root of all evil)(3) time = moneyso, girl = time *
money<=> girl = money * money , (by (3))<=> girl =
(sqrt(evil)) ^ 2 , (by (2))<=> girl = evilThis was a
consolation in times of trial.Richard
===
>I remember many
years at school, a math teacher proving this.>Has anyone got a
math example of how this is possible?>1 + 1 = 3 for
sufficiently large values of 1.--Lynn
===
>I remember many
years at school, a math teacher proving this.>Has anyone got a
math example of how this is possible?Your teacher was of course
incorrect. *Everyone* knows 1 + 1 = 10:-4 = 2 * (-2) = Sqrt(4)
* Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4and-4 = 4 <=> -4 + 5 = 4
+ 5 <=> 1 = 9 <=> 1 + 1 = 10
===
>I remember many years at
school, a math teacher proving this.>Has anyone got a math
example of how this is possible? > Your teacher was of course
incorrect. *Everyone* knows 1 + 1 = 10: -4 = 2 * (-2) =
Sqrt(4) * Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4 and -4 = 4 <=>
-4 + 5 = 4 + 5 <=> 1 = 9 <=> 1 + 1 = 10That's not right,
Sqrt(4) is always 2 (not -2). But if you use complex numbers
it all becomes clear:-4 = 2i * 2i = 2 * sqrt(-1) * 2 *
sqrt(-1) = 2 * 2 * sqrt(-1) * sqrt(-1) = 4 * sqrt((-1)*(-1)) =
4 * sqrt(1) = 4The rest of the proof is the same.
===
> I
remember many years at school, a math teacher proving this.>
Has anyone got a math example of how this is possible?The most
popular method is probably to divide both sides of a
validequation by zero, like so: 0*(1 + 1) = 0*3 <=> 1 + 1 =
3.That was not hard, was it? But if multiplication and
division are toodifficult, here is a simple proof that uses
mere addition: add one toboth sides of a valid equation, like
so: 1 + 1 = 2 <=> 1 + 1 = 3.Or you can start from a proof that
0 = 1. Just take a square root onboth sides of 1 = 1 to get -1
= 1, add 1 to get 0 = 2 and divide thatby 2 to get 0 = 1. Add
the same number to both sides of 1 + 1 = 2: 1 + 1 = 2 <=> 1 +
1 = 3.Or just state that 1 + 1 = 3 by definition. There really
is no end tothe errors that you can choose to make. Obfuscation
helps, too. Writesomething incomprehensible and then state that
the result follows. Iwill not attempt this here.--
===
The
Barcelona conjecture:Let c=(x+y+z)^p/(pxyz2^p)for integer
c,x,y,z and p prime greater than or equal to 5, theBarcelona
conjecture is that no solutions exist with gcd(c,xyz)=1 (noc
exist that shares no factor with x or y or z).While this is a
very interesting problem in itself (the heart of it isthe
relation between the factors of a sum versus the product of
theaddends - ie (x+y+z)^p/(xyz)), and may at first seem to be
easy toresolve, it is in fact intimately related to Fermat's
Last Theorem. In fact, proving the Barcelona conjecture also
proves FLT (no integerABC that satisfy A^p+B^p=C^p for p prime
greater than 2) for primeexponents greater than 3. [note that
letting c=1 for p=3 solvesthis exponent for FLT]There are
integer solutions to the above, but all found to date have
afactor in common for c and xyz.
===
I am forwarding this post
from synergeo group. Some here might be interested.Dick
FischbeckI've just created a sourceforge project for the
various programs I'vebeen writing
-http://sourceforge.net/projects/packinonThe associated
website is here -http://packinon.sourceforge.net/Not too much
there at the moment. I've started with the Pythonas I get
round to completing their next versions.Adrian Rossiter
===
I
would be interested in solutions to the following
problem.Elementary solutions are especially welcome.In an ABC
triangle, D is a point on BC from which we constructsegments
that form equal angles at the sides AB, AC, at the points I,K
respectively. Which is the position of D so that the length of
IKwill be minimum?It is assumed that the angles BID=DKC in
question are constant as thepoint D moves on BC.You may assume
that the result is known when BID=DKC=right anlge.
===
Dear
Mathematicians,I plan on attending graduate school next year
in Mathematics to begin myPhD, and was hoping to receive some
insight or advice from the board as itconsists of such a wide
variety of mathematicians. If someone/some peoplecan reply, I
would highly appreciate it. (As background, I am completing
aMaster of Arts at University of Pennsylvania)My question is
as follows : To those who have completed a PhD in math
inAlgebra (or really this applies to any subfield of math), as
you look backat your graduate school days, if you were to go
back and restart graduateschool, what would you have done
differently? How would you go aboutgraduate school in
general?MoshemadrianATmath.upenn.edu)
===
Dear all,I was
wondering : I feel that I am very strong in Algebra, but not
as muchin Analysis, and I would say Topology is in the middle
(I haven't studiedDifferential Geometry yet, that is next
semester). Is it me, or is thinkingin an Analysis type way
different than in an Algebraic type way? I justfeel that they
are very different. I feel that Algebra is much more
visualthan Analysis. For me I can just see Linear Algebra,
representation theory(not too advanced, but advanced enough),
introductory Field theory, etc etc.It is much harder for me to
see Analysis. Does anyone have a recommendationfor me as to
this problem? I feel I should just pick up Rudin's
Analysistextbook and read through it and do all of the
problems, or maybe adifferent textbook, and maybe that would
help me see Analysis better.Moshe
===
> .... It is much harder
for me to see Analysis. Does anyone have a> recommendation for
me as to this problem? I feel I should just pick> up Rudin's
Analysis textbook and read through it and do all of the>
problems, or maybe a different textbook, and maybe that would
help me> see Analysis better.... If you're after a really good
basic feeling for analysis rather than advanced modern
technical facts, then don't forget the classic by G. H. Hardy,
A Course of Pure Mathematics. Just read it for pleasure! :-) It
was written when analysis was neglected in Cambridge, and with
an emphasis and enthusiasm which seem rather ridiculous now.
If I were to rewrite it now I should not write (to use Prof.
Littlewood's simile) like 'a missionary talking to cannibals',
but with decent terseness and restraint .... Luckily he didn't,
and his missionary-talking-to-cannibals style has been of great
value to many of us since. Ken Pledger.
===
Rudin seems to be a
book that one would read after having mastered
thesubject.However , if complex analysis is a topic you are
interested in you mightlike Visual Complex Analysis by Tristan
Needham.> Dear all,> I was wondering : I feel that I am very
strong in Algebra, but not as much> in Analysis, and I would
say Topology is in the middle (I haven't studied> Differential
Geometry yet, that is next semester). Is it me, or isthinking>
in an Analysis type way different than in an Algebraic type
way? I just> feel that they are very different. I feel that
Algebra is much morevisual> than Analysis. For me I can just
see Linear Algebra, representationtheory> (not too advanced,
but advanced enough), introductory Field theory, etcetc.> It
is much harder for me to see Analysis. Does anyone have
arecommendation> for me as to this problem? I feel I should
just pick up Rudin's Analysis> textbook and read through it
and do all of the problems, or maybe a> different textbook,
and maybe that would help me see Analysis better.> Moshe
>
===
I wonder if someone can help me with the following
probabilityproblem, and show me the method by which it can be
solved. If thereis an event with three possible outcomes, and
each outcome has anequal probability of occurring, what are
the odds that over 80 trials,one of the three possible
outcomes only results on two occasions(while the other two
possible outcomes are successful on the remaining78 trials).
Also, is there a significance test that can be applied tothis
type of probability problem, such that it is possible to
declarethat the results are statistically significant?Anon
===
I wonder if someone can help me with the following probability>
problem, and show me the method by which it can be solved. If
there> is an event with three possible outcomes, and each
outcome has an> equal probability of occurring, what are the
odds that over 80 trials,> one of the three possible outcomes
only results on two occasions> (while the other two possible
outcomes are successful on the remaining> 78 trials). Also, is
there a significance test that can be applied to> this type of
probability problem, such that it is possible to declare> that
the results are statistically significant? > AnonAs far as I
know, given the way the question is asked, you can do nomore
than calculating the probability (I know I'll be corrected if
I'mwrong): P(one outcome occurs twice in 80) =
bin(80,2)*(1/3)^2*(2/3)^78 =6.46e-12Jeroen
===
> I wonder if
someone can help me with the following probability> problem,
and show me the method by which it can be solved. If there>
is an event with three possible outcomes, and each outcome has
an> equal probability of occurring, what are the odds that
over 80 trials,> one of the three possible outcomes only
results on two occasions> (while the other two possible
outcomes are successful on the remaining> 78 trials). Also,
is there a significance test that can be applied to> this
type of probability problem, such that it is possible to
declare> that the results are statistically significant?>
Anon>As far as I know, given the way the question is asked,
you can do no>more than calculating the probability (I know
I'll be corrected if I'm>wrong):> P(one outcome occurs twice
in 80) = bin(80,2)*(1/3)^2*(2/3)^78 =>6.46e-12This is the
probability that a particular outcome occurs twice; there are
three such outcomes, so multiplying thisby 3 gives a not quite
correct answer.The reason it is not quite correct is that two
oucomescould each occur twice. The probability of this is fora
particular ordering of the outcomes is[80!/2!2!76!]*(1/3)^80,
and 3 times this needs to besubtracted. This is MUCH
smaller.There are standard significance tests, such as the
chi-squared test, and exact tests.-- This address is for
information only. I do not claim that these viewsare those of
the Statistics Department or of Purdue University.Herman
Rubin, Department of Statistics, Purdue University
===
> Given,
where x is in the ring of algebraic integers, I've shown the>
factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =
49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x)
- 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401
x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0.
I'm curious about the mental processes that allow *some* of you
to> claim that 49 divides off as a *variable* dependent on x,
so I'm> giving another opportunity for you to speak your
minds. To my knowledge, in the history of mathematics, no one
has ever> presented such a proposition, so it is a unique one,
and I must say> that I'm intrigued. Speak your minds. > James
Harris Why does this not count as presenting such a
proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).
then> g1(x)*g2(x) = 3(5-x) but neither g1 nor g2 is divisible
by 3 for all x. - William Hughes OOPS! I just replied to this
post with my own mistake, as I had 1 + (1-sqrt(1+3x))/3 being
an integer if 1+3x is a square, when x=40 refutes that
notion.James Harris
===
> OOPS! I just replied to this post
with my own mistake, as I had> 1 + (1-sqrt(1+3x))/3> being
an integer if 1+3x is a square, when x=40 refutes that
notion.> James HarrisYour OOPS! count just went up by one.
Funny how *your* mistakes are simple, easily corrected faults,
but otherposters mistakes are signs of ignorance, incompetence
or dishonesty.Get a grip, James. You have been thoroughly and
conclusively refuted. Your *research* is a worthless pile
ofpseudo-scientific, incomprehensible and error-ridden
junk--There are two things you must never attempt to prove:
the unprovable -- and the obvious.--Democracy: The triumph of
popularity over principle.--http://www.crbond.com
===
> OOPS!
I just replied to this post with my own mistake, as I had> 1
+ (1-sqrt(1+3x))/3> being an integer if 1+3x is a square,
when x=40 refutes that notion.> James Harris> Your OOPS!
count just went up by one. Funny how *your* mistakes
aresimple, easily corrected faults, but other> posters
mistakes are signs of ignorance, incompetence or dishonesty.>
Get a grip, James. You have been thoroughly and conclusively
refuted. Your*research* is a worthless pile of>
pseudo-scientific, incomprehensible and error-ridden junk>
--> There are two things you must never attempt to prove: the
unprovable -- and the obvious.> --> Democracy: The triumph of
popularity over principle.> --> http://www.crbond.com>Heh,
did you see that message a few days ago from someone who asked
aquestion and James told him if he had to ask, he should read
something else?I thought that was funny considering James
doesn't seem to know what he'stalking about.-- David
MoranChief MeteorologistOklahoma Storm Team
===
> Given, where
x is in the ring of algebraic integers, I've shown the>
factorization (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =
49(300125 x^3 - 18375 x^2 - 360 x + 22) where b_3(x) = a_3(x)
- 3 and the a's are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401
x^3 - 147 x^2 + 3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0.
I'm curious about the mental processes that allow *some* of you
to> claim that 49 divides off as a *variable* dependent on x,
so I'm> giving another opportunity for you to speak your
minds. To my knowledge, in the history of mathematics, no one
has ever> presented such a proposition, so it is a unique one,
and I must say> that I'm intrigued. Speak your minds. > James
Harris Why does this not count as presenting such a
proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).
then> g1(x)*g2(x) = 3(5-x)I noted there's a *sign* ambiguity
in the sqrt() operator, whichsparked a lot of debate.One thing
I found interesting is that posters ignored that if youdivide
both sides by 3, with the convention that you're taking
thepositive of the sqrt() operator for *integer* results, you
have 1 + (1-sqrt(1+3x))/3as a factor, which is an integer
(remember x is an integer andremember 1+3x is a square).> but
neither g1 nor g2 is divisible by 3 for all x. - William
Hughes Prove it. Readers should note that this poster
presented a *later*post claiming a result that covers integer
results of the square rootoperator but made a rather simple
mistake. In my reply to that post Inoted the sign ambiguity in
the square root operator.However sqrt(x), where x is a square
can be taken to be the positiveresult since you can *give* the
result, but the ambiguity remains ifyou see sqrt(x) without a
given value.That is, for instance, sqrt(4) is 2 *or* -2, but
by convention, it's*usually* taken as 2, though if you do
enough analysis you will runinto situations where you need the
negative!!!Mathematics is *logical* and consistent, which is
something that manypeople seem to have a problem with, as they
try to twist it to theirown needs and interests.Note lack of
proper explication from other sci.math posters.James
Harris
===
> Given, where x is in the ring of algebraic
integers, I've shown the> factorization> (5 a_1(x) + 7)(5
a_2(x) + 7)(5 b_3(x) + 22) = > 49(300125 x^3 - 18375 x^2 -
360 x + 22)> where b_3(x) = a_3(x) - 3 and the a's are roots
of> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> so
when x=0, a_1(0) = a_2(0) = b_3(0) = 0.> I'm curious about
the mental processes that allow *some* of you to> claim that
49 divides off as a *variable* dependent on x, so I'm> giving
another opportunity for you to speak your minds.> To my
knowledge, in the history of mathematics, no one has ever>
presented such a proposition, so it is a unique one, and I
must say> that I'm intrigued.> Speak your minds.> James
Harris Why does this not count as presenting such a
proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).
then> g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity
in the sqrt() operator, which> sparked a lot of debate. One
thing I found interesting is that posters ignored that if you>
divide both sides by 3, with the convention that you're taking
the> positive of the sqrt() operator for *integer* results,
you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer
(remember x is an integer and> remember 1+3x is a square). but
neither g1 nor g2 is divisible by 3 for all x. - William
Hughes Prove it. Readers should note that this poster
presented a *later*> post claiming a result that covers
integer results of the square root> operator but made a rather
simple mistake. In my reply to that post I> noted the sign
ambiguity in the square root operator. However sqrt(x), where
x is a square can be taken to be the positive> result since
you can *give* the result, but the ambiguity remains if> you
see sqrt(x) without a given value. That is, for instance,
sqrt(4) is 2 *or* -2, but by convention, it's> *usually* taken
as 2, though if you do enough analysis you will run> into
situations where you need the negative!!!> Well, if you
dislike the function sqrt(x) so much, note thisalternate
presentation [1]Let g1(x) and g2(x) be the two roots of the
quadratic g^2 - 8g + 3(5-x)For x=0, the roots are 3 and 5.
Choose g1 and g2 such thatg1(0)=3 and g2(0) = 5We have (the
product of the roots is the constant term) g1(x) g2(x) =
3(5-x)Clearly g1(x)/3 and g2(x)/3 are both roots of 3g^2 - 8g
+ (5-x)but for x=2 this is primitive, irreducible and
non-monic. Henceneither g1(2) nor g2(2) is divisible by 3.
-William Hughes[1] My thanks to Rick Decker whose post I
adapted
===
> Given, where x is in the ring of algebraic
integers, I've shown the> factorization> (5 a_1(x) + 7)(5
a_2(x) + 7)(5 b_3(x) + 22) = > 49(300125 x^3 - 18375 x^2 -
360 x + 22)> where b_3(x) = a_3(x) - 3 and the a's are roots
of> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> so
when x=0, a_1(0) = a_2(0) = b_3(0) = 0.> I'm curious about
the mental processes that allow *some* of you to> claim that
49 divides off as a *variable* dependent on x, so I'm> giving
another opportunity for you to speak your minds.> To my
knowledge, in the history of mathematics, no one has ever>
presented such a proposition, so it is a unique one, and I
must say> that I'm intrigued.> Speak your minds.> James
Harris Why does this not count as presenting such a
proposition ? Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).
then> g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity
in the sqrt() operator, which> sparked a lot of debate. One
thing I found interesting is that posters ignored that if you>
divide both sides by 3, with the convention that you're taking
the> positive of the sqrt() operator for *integer* results,
you have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer
(remember x is an integer and> remember 1+3x is a square). but
neither g1 nor g2 is divisible by 3 for all x. - William
Hughes Prove it. Consider x =2: g1(2)/3 = (4-sqrt(7))/3 is a
root of the primitive, irreducible, non-monicpolynomial
P(x)=3x^2 - 8x + 3. Thus g1(2)/3 is not an algebraic
integerand g1(2) is not divisible by 3.Now James has made the
bizzare claim that sqrt(x) and hence g1(x) mustbe considered
multi-valued, and furthmore if either value of g1(x) is
divisibleby 3 then g1(x) is divisible by 3. So let's try the
other square root of 7. g1(2)/ = (4+sqrt(7))/3 is a root of
the primitive, irreducible, non-monicpolynomial P(x)=3x^2 - 8x
+ 3. Thus g1(2)/3 is not an algebraic integerand g1(2) is not
divisible by 3.So, even if we accept the ambiguity argument,
g1(2) is not divisible by 3.Similarly, g2(2) is not divisible
by 3.Thus neither g1 nor g2 is divisible by 3 for all x.
-William Hughes
===
> I noted there's a *sign* ambiguity in the
sqrt() operator, which> sparked a lot of debate.There was no
debate. Everyone except the village idiot agrees how the
sqrt(x) function is defined. The village idiot is screaming as
usual, but nobody is bothered too much.
===
> <snip>Why
does this not count as presenting such a proposition ?>Let
g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).>then>
g1(x)*g2(x) = 3(5-x)> I noted there's a *sign* ambiguity in
the sqrt() operator, which> sparked a lot of debate. One thing
I found interesting is that posters ignored that if you> divide
both sides by 3, with the convention that you're taking the>
positive of the sqrt() operator for *integer* results, you
have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer
(remember x is an integer and> remember 1+3x is a square).
>but neither g1 nor g2 is divisible by 3 for all x.> -
William Hughes > Prove it. Clearly, the g's satisfy g^2 - 8g
+ 3(5 - x). If 3 were todivide (in the algebraic integers) one
of the g's, thenone or both of g1(x)/3 and g2(x)/3 would be
algebraicintegers. Letting h1(x) = g1(x)/3, h2(x) = g2(x)/3we
find that the h's satisfy 3h^2 - 8h + (5 - x).This is
obviously a primitive polynomial and isirreducible over Z[x]
if and only if 1 + 3x isn't asquare. In particular, when x = 3
we see that neitherof 4 + sqrt(10) and 4 - sqrt(10) is
divisible by3, though their product, 6, is.Note, BTW, that
even if you're bumfuzzled aboutthe ambiguity of the sqrt
function, there shouldbe no confusion here, since we're using
both of thesecond roots of 10.<snip>Rick
===
> Why does this
not count as presenting such a proposition ?> Let
g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).> then>
g1(x)*g2(x) = 3(5-x) I noted there's a *sign* ambiguity in the
sqrt() operator, which> sparked a lot of debate. One thing I
found interesting is that posters ignored that if you> divide
both sides by 3, with the convention that you're taking the>
positive of the sqrt() operator for *integer* results, you
have 1 + (1-sqrt(1+3x))/3 as a factor, which is an integer
(remember x is an integer and> remember 1+3x is a square).Why
must 1+3x be a square? > but neither g1 nor g2 is divisible by
3 for all x.> erm, how about x =1 for g1, cos then 2/3 wold be
an algebraic integer. andzero for g2, cos then 5/2 would be an
algebraic integer. Of course in yourcore error world then you
may think differently. And I may be wrong, if Iam I hope
someone can point it out.> - William Hughes Prove it. Readers
should note that this poster presented a *later*> post claiming
a result that covers integer results of the square root>
operator but made a rather simple mistake. In my reply to that
post I> noted the sign ambiguity in the square root operator.
However sqrt(x), where x is a square can be taken to be the
positive> result since you can *give* the result, but the
ambiguity remains if> you see sqrt(x) without a given value.
That is, for instance, sqrt(4) is 2 *or* -2, but by
convention, it's> *usually* taken as 2, though if you do
enough analysis you will run> into situations where you need
the negative!!! Mathematics is *logical* and consistent, which
is something that many> people seem to have a problem with, as
they try to twist it to their> own needs and interests. Note
lack of proper explication from other sci.math posters. >
James Harris
===
In sci.math, James
Harris<jstevh@msn.com><3c65f87.0312151404.50d586a2@
posting.google.com>:> Okay, so you're dividing both sides by
49. Point taken.> :> In other words, are you> :> claiming
that (5 a_1(x) + 7) and (5 a_2(x) + 7) are divisible by 7
for> :> all x? If so, could you please explain why the fact
that it's true for> :> x=0 implies it's true for other x
also?> If you are not claiming this, then please explain
what you are claiming.> Many thanks,> Justin> Given, where
x is in the ring of algebraic integers, I've shown the>
factorization> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =
> 49(300125 x^3 - 18375 x^2 - 360 x + 22)> where b_3(x) =
a_3(x) - 3 and the a's are roots of> a^3 + 3(-1 + 49x)a^2 -
49(2401 x^3 - 147 x^2 + 3x)> so when x=0, a_1(0) = a_2(0) =
b_3(0) = 0.> Now you can divide both sides by 49.I could also
divide them by 2198/401, add 5.23e11 to bothsides, take the
cosh of the natural log, and even usepartial differentiation
on the variable b, which I declare asan independent variable
for both a = a(b) and x = x(b). :-)You'll have to be a little
more specific than that here as to whatyou expect to happen.>
Some posters have claimed that when you divide by 49, what
results> varies depending on what value x has.The result
*does* depend on x, since x is part of both sides. Inthe
declaration, say,f(x) = x^3 - 3x + 1f(x) definitely depends on
x. Your functions aren't all thatdifferent.> Do you
understand?No.Now I shall answer your question.It is clear
that the quantities (5 a_1(x) + 7) and (5 a_2(x) + 7) arenot
divisible by 7 for all x (counterexamples: a_1(x) = 1, a_2(x)
= 1).You will probably want to clarify your domain here. Is
x:[1] an integer?[2] a positive integer?[3] an algebraic
integer?[4] a rational number?[5] pi?[6] the square root of
the sum of the squares of a series?[7] something else?If you
want me to actually produce an x such thata^3 + 3(-1 + 49x)a^2
- 49(2401 x^3 - 147 x^2 + 3x) = 0has a root 1, I'll have to do
a bit of work. Thesimplest method might be to set a = 1, then
solve for x:1^3 + 3(-1 + 49x)1^2 - 49(2401 x^3 - 147 x^2 + 3x)
= 0which ultimately generates the equation-117649*x^3 +
7203*x^2 - 2 = 0This equation has 3 real roots, which are
approximately:-
0.014939812399364842725049925330.020408163265306122448979591830
.05575613892997708762300910901These are obviously not
algebraic integers, as the equationof which they are roots is
not of the requisite type.Substituting x = any one of these
will result in your productbecoming either12 ( 5 a_2(x) + 7)
(5 b_3(x) + 22)or(5 a_1(x) + 7) (5 a_2(x) + 7) 12What
conclusions can be drawn from all this, I for one do not
know.[rest snipped]-- #191, ewill3@earthlink.netIt's still
legal to go .sigless.
===
> ...> Let g1(x)=4-sqrt(1+3x) and
g2(x)=4+sqrt(1+3x).> then> g1(x)*g2(x) = 3(5-x)> but
neither g1 nor g2 is divisible by 3 for all x. Ah, but the
constant term of g1(x) = 3, and the constant term of g2(x) =
5,> so g1(x) should be divisible by 3. A core error in
mathematics ;-). It's actually not that simple. It's an
interesting case to highlight> your ineptitude with basic
mathematics though. Consider that the constant term of g1(x)
is 3 *or* 5 because the> sqrt() operator is ambiguous. I've
explained that before in replying> to Arturo Magidin, but
mathematics is a difficult discipline for some,> so repetition
is necessary, and still often not enough.> You say the sqrt()
operator is ambiguous. Well, that makes YOUR OWNclaims
ambiguous. You probably dont realize it, but the a_1, a_2
anda_3 in your core error CONTAIN SQUARE ROOTS! Just use
theCardano(sp?) formulas to find the a_n's and you will see
the squareroots.> You see Dik Winter, actually *knowing*
mathematics versus talking as> if you know it can be two
different things. And you clearly by posts like yours here do
not actually know> mathematics. You do post a lot though
living in a fantasy world. > James Harris
===
> Consider that
the constant term of g1(x) is 3 *or* 5 because the> sqrt()
operator is ambiguous.> You say the sqrt() operator is
ambiguous. Well, that makes YOUR OWN> claims ambiguous. You
probably dont realize it, but the a_1, a_2 and> a_3 in your
core error CONTAIN SQUARE ROOTS! Just use the> Cardano(sp?)
formulas to find the a_n's and you will see the square>
roots.No, a_1(x), a_2(x) and a_3(x) do not CONTAIN SQUARE
ROOTS! . They are functions defined in terms of the roots of a
polynomial withcoefficients depending on x (actually, James
leaves out a couple ofsteps needed to completely specify the
functions).While one can certainly use the Cardano formulae to
find expressions for the a_i, and these formulae contain square
roots,there are ways to express the a_i that do not involve
thesymbol sqrt. One should be careful to distinguish between
theexistence of the underlying function and the notation used
toexpress the function. - William Hughes
===
> Consider that
the constant term of g1(x) is 3 *or* 5 because the> sqrt()
operator is ambiguous.> You say the sqrt() operator is
ambiguous. Well, that makes YOUR OWN> claims ambiguous. You
probably dont realize it, but the a_1, a_2 and> a_3 in your
core error CONTAIN SQUARE ROOTS! Just use the> Cardano(sp?)
formulas to find the a_n's and you will see the square>
roots.> No, a_1(x), a_2(x) and a_3(x) do not CONTAIN SQUARE
ROOTS! . They are> functions defined in terms of the roots of
a polynomial with> coefficients depending on x (actually,
James leaves out a couple of> steps needed to completely
specify the functions).> While one can certainly use the
Cardano formulae to> find expressions for the a_i, and these
formulae contain square roots,> there are ways to express the
a_i that do not involve the> symbol sqrt. One should be
careful to distinguish between the> existence of the
underlying function and the notation used to> express the
function.> - William HughesGiven JHS poly: JHS(x) = a^3 +
3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)taking roots of
the equation: JHS(x) = 0we have,a_1 = 1 - 49*x + (2^(1/3)*(-1
+ 49*x)^2)/ (2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x -
1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 -
847425747*x^6])^(1/3) + (2 - 147*x + 7203*x^2 - 117649*x^3 +
7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4 +
103766418*x^5 - 847425747*x^6])^(1/3)/2^(1/3)a_2 = 1 - 49*x -
((1 + I*Sqrt[3])*(-1 + 49*x)^2)/ (2^(2/3)*(2 - 147*x +
7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 -
5294205*x^4 + 103766418*x^5 - 847425747*x^6])^(1/3)) - ((1 -
I*Sqrt[3])*(2 - 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x -
1911*x^2 + 139258*x^3 - 5294205*x^4 + 103766418*x^5 -
847425747*x^6])^(1/3))/(2*2^(1/3))a_3 = 1 - 49*x - ((1 -
I*Sqrt[3])*(-1 + 49*x)^2)/ (2^(2/3)*(2 - 147*x + 7203*x^2 -
117649*x^3 + 7*Sqrt[12*x - 1911*x^2 + 139258*x^3 - 5294205*x^4
+ 103766418*x^5 - 847425747*x^6])^(1/3)) - ((1 + I*Sqrt[3])*(2
- 147*x + 7203*x^2 - 117649*x^3 + 7*Sqrt[12*x - 1911*x^2 +
139258*x^3 - 5294205*x^4 + 103766418*x^5 -
847425747*x^6])^(1/3))/(2*2^(1/3))where the root assignments
can be permuted. Doubtless other expressions can be found by
suitable transforms.--There are two things you must never
attempt to prove: the unprovable -- and the
obvious.--Democracy: The triumph of popularity over
principle.--http://www.crbond.com
===
> ...> Let
g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).> then>
g1(x)*g2(x) = 3(5-x)> but neither g1 nor g2 is divisible
by 3 for all x.> Ah, but the constant term of g1(x) = 3, and
the constant term of g2(x) = 5,> so g1(x) should be
divisible by 3. A core error in mathematics ;-).> It's
actually not that simple. It's an interesting case to
highlight> your ineptitude with basic mathematics though.>
Consider that the constant term of g1(x) is 3 *or* 5 because
the> sqrt() operator is ambiguous. I've explained that
before in replying> to Arturo Magidin, but mathematics is a
difficult discipline for some,> so repetition is necessary,
and still often not enough. Yup, it appears so. In mathematics
the sqrt when applied to reals and> delivering a real is
defined to give the positive result. This has> been said
before, but it is apparently to difficult for you to
understand.That doesn't change the *inherent* ambiguity in the
sqrt() operator.That's easy to show as consider sqrt(4), and
you wish to say it's*defined* to be 2, but what about -2?Does
your definition take away -2 as a solution?And besides, you
lied, it's by *convention* that the positive istaken, not by
definition.> You see Dik Winter, actually *knowing*
mathematics versus talking as> if you know it can be two
different things. It appears so, yes.Since people working with
square roots *usually* want the positiveroot, by convention the
positive is taken, not by definition.That's necessary because,
like sqrt(4) has *either* 2 or -2 as asolution, as is easily
proven:sqrt(4) = -2, square both sides, 4 = 4. QEDNow then,
can you *prove* that -2 is not a solution to sqrt(4)?James
Harris
===
... > Yup, it appears so. In mathematics the sqrt
when applied to reals and > delivering a real is defined to
give the positive result. This has > been said before, but it
is apparently to difficult for you to understand.... > And
besides, you lied, it's by *convention* that the positive is >
taken, not by definition.What is the distinction? > That's
necessary because, like sqrt(4) has *either* 2 or -2 as a >
solution, as is easily proven: > sqrt(4) = -2, square both
sides, 4 = 4. QED1 = -1, square both sides, 1 = 1, QED > Now
then, can you *prove* that -2 is not a solution to sqrt(4)?I
see no equation, so how can you talk about a solution?-- dik
t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland,
+31205924131home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
> ...> Let g1(x)=4-sqrt(1+3x)
and g2(x)=4+sqrt(1+3x).> then> g1(x)*g2(x) = 3(5-x) but neither g1 nor g2 is divisible by 3 for all x.> Ah,
but the constant term of g1(x) = 3, and the constant term of
g2(x) = 5,> so g1(x) should be divisible by 3. A core error
in mathematics ;-).> It's actually not that simple. It's an
interesting case to highlight> your ineptitude with basic
mathematics though.> Consider that the constant term of
g1(x) is 3 *or* 5 because the> sqrt() operator is ambiguous.
I've explained that before in replying> to Arturo Magidin,
but mathematics is a difficult discipline for some,> so
repetition is necessary, and still often not enough. Yup, it
appears so. In mathematics the sqrt when applied to reals and>
delivering a real is defined to give the positive result. This
has> been said before, but it is apparently to difficult for
you to understand. That doesn't change the *inherent*
ambiguity in the sqrt() operator. That's easy to show as
consider sqrt(4), and you wish to say it's> *defined* to be 2,
but what about -2?Well, as sqrt(4) *is* defined to be to 2 I
think Dik may get his wish. Does your definition take away -2
as a solution?> Solution to what? -2 is a solution to x^2=4,
x=sqrt(4) is notan equation.> And besides, you lied, it's by
*convention* that the positive is> taken, not by
definition.>In this context there is no difference between
convention and definition. > You see Dik Winter, actually
*knowing* mathematics versus talking as> if you know it can
be two different things. It appears so, yes. Since people
working with square roots *usually* want the positive> root,
by convention the positive is taken, not by definition. That's
necessary because, like sqrt(4) has *either* 2 or -2 as a>
solution, as is easily proven:> To say that sqrt(4) has a
solution is nonsensical.> sqrt(4) = -2, square both sides, 4 =
4. QEDThe same logic shows that -x = x for any x The whole
discussion is silly in any case. There certainlyexits a single
valued function, g(z) defined on the complex planesuch that for
all complex z: g(z)*g(z) = z, either real(g(z)) > 0 or
(real(g(z)) = 0 and imag(g(z))>=0)If you have a deep
psycho-sexual need not to call g(z) sqrt(z),then call it
sqrt_pb(z) or harris(z) or hughes(z) or whateverelse you want
[1]. Define g_1(x) = 4 - g(1+3x), g_2(x) = 4 + g(1+3x)(note
that both g_1 and g_2 are single valued) Then g_1(x) g_2(x) =
3(5-x) g_1(0) = 3, g_2(0) = 5 g_1(1) = 2 is not divisible by 3
- William Hughes [1] But be aware that others will still use
sqrt(z).
===
... > The whole discussion is silly in any case.
There certainly > exits a single valued function, g(z) defined
on the complex plane > such that for all complex z: >
g(z)*g(z) = z, > either real(g(z)) > 0 or (real(g(z)) = 0 and
imag(g(z))>=0) > If you have a deep psycho-sexual need not
to call g(z) sqrt(z), > then call it sqrt_pb(z) or harris(z)
or hughes(z) or whatever > else you want [1]. Define >
g_1(x) = 4 - g(1+3x), g_2(x) = 4 + g(1+3x) > (note that both
g_1 and g_2 are single valued) > Then g_1(x) g_2(x) = 3(5-x)
> g_1(0) = 3, g_2(0) = 5 > g_1(1) = 2 is not divisible by
3More interesting, neither g_1(2) nor g_2(2) is divisible by
3, whileg_1(2)*g_2(2) = 9. Moreover, g_1(2) and g_2(2) are
coprime: (12 + sqrt(7)).g_1(2) - (24 - 8.sqrt(7)).g_2(2) =
1.moreover: gcd(g_1(2), 3) = (sqrt(2) - sqrt(14))/2,
gcd(g_2(2), 3) = (sqrt(2) + sqrt(14))/2, gcd(g_1(2), 9) =
gcd(g_1(2), 3)^2 and gcd(g_2(2), 9) = gcd(g_2(2), 3)^2.-- dik
t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland,
+31205924131home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
<3c65f87.0312151213.7bbfb1b7@posting.google.com>
<HpypJB.KA9@cwi.nl>
<3c65f87.0312160631.117923f5@posting.google.com>
<4d5e4663.0312161059.154f9ecc@posting.google.com>
===
>
x=sqrt(4) is not an equation.Surely, you didn't mean to say
that.-- Jesse F. Hughes What you call reasonable is suspect
since you've proven yourself tobe an enemy of mathematics. --
James S. Harris defends the cause.
===
x=sqrt(4) is not an
equation. Surely, you didn't mean to say that.Um, right. This
should read sqrt(4) is not an equation -William
Hughes
===
[...]> Since people working with square roots
*usually* want the positive> root, by convention the positive
is taken, not by definition. That's necessary because, like
sqrt(4) has *either* 2 or -2 as a> solution, as is easily
proven: sqrt(4) has no solution as it isn't an equation,
sqrt(x) is a function wich gives you by *definition* your
positive result.[...]> Now then, can you *prove* that -2 is
not a solution to sqrt(4)? -2 is a solution of the equation x
^ 2 = 4, not of sqrt(4). the word solution doesn't apply to
sqrt(4), see above.--Edgar
===
> ...> Let
g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).> then>
g1(x)*g2(x) = 3(5-x)> but neither g1 nor g2 is
divisible by 3 for all x.> Ah, but the constant term of
g1(x) = 3, and the constant term ofg2(x) = 5,> so g1(x)
should be divisible by 3. A core error in mathematics;-). It's actually not that simple. It's an interesting case to
highlight> your ineptitude with basic mathematics though. Consider that the constant term of g1(x) is 3 *or* 5
because the> sqrt() operator is ambiguous. I've explained
that before in replying> to Arturo Magidin, but mathematics
is a difficult discipline forsome,> so repetition is
necessary, and still often not enough.> Yup, it appears so.
In mathematics the sqrt when applied to reals and> delivering
a real is defined to give the positive result. This has> been
said before, but it is apparently to difficult for you
tounderstand.> That doesn't change the *inherent* ambiguity
in the sqrt() operator.> That's easy to show as consider
sqrt(4), and you wish to say it's> *defined* to be 2, but what
about -2?> Does your definition take away -2 as a solution?>
And besides, you lied, it's by *convention* that the positive
is> taken, not by definition.Quit saying people lie when they
make honest mistakes, there is a BIGdifference. The sqrt of 4
is 2 OR -2; if you dispute this, go back toalgebra I in high
school. You apparently have gaps in your algebra. > You see
Dik Winter, actually *knowing* mathematics versus talking as>
if you know it can be two different things.> It appears so,
yes.> Since people working with square roots *usually* want
the positive> root, by convention the positive is taken, not
by definition.> That's necessary because, like sqrt(4) has
*either* 2 or -2 as a> solution, as is easily proven:>
sqrt(4) = -2, square both sides, 4 = 4. QED> Now then, can
you *prove* that -2 is not a solution to sqrt(4)? James
Harris-- David MoranChief MeteorologistOklahoma Storm
Team
===
> ...>[...]> That's easy to show as consider
sqrt(4), and you wish to say it's> *defined* to be 2, but
what about -2?> Does your definition take away -2 as a
solution?> And besides, you lied, it's by *convention* that
the positive is> taken, not by definition.>Quit saying people
lie when they make honest mistakes, there is a BIG>difference.
Excellent point, but irrelevant here, because there was no
mistakein what Dik said, honest or otherwise.>The sqrt of 4 is
2 OR -2; if you dispute this, go back to>algebra I in high
school. You apparently have gaps in your algebra.No. _The_
square root of 4 is 2. > You see Dik Winter, actually
*knowing* mathematics versus talking as> if you know it can
be two different things.> It appears so, yes.> Since
people working with square roots *usually* want the positive>
root, by convention the positive is taken, not by
definition.> That's necessary because, like sqrt(4) has
*either* 2 or -2 as a> solution, as is easily proven:>
sqrt(4) = -2, square both sides, 4 = 4. QED> Now then, can
you *prove* that -2 is not a solution to sqrt(4)?> James
Harris************************David C. Ullrich
===
[.snip.]>Quit saying people lie when they make honest
mistakes, there is a BIG>difference. The sqrt of 4 is 2 OR -2;
Ehr, no. Both 2 and -2 are square roots of 4 (that is, both of
them,when squared, give 4); but the sqrt of 4 is defined to be
theprincipal branch, and therefore it is unabiguously equal to
2. That'swhy sqrt(a^2) = |a| for real numbers, for example.--
===
=================================
===
=== It's not denial. I'm just very selective about what I
accept as reality. --- Calvin ( Calvin and Hobbes
)
===
=================================
===
===Arturo Magidinmagidin@math.berkeley.edu===>
[.snip.]>Quit saying people lie when they make honest
mistakes, there is a BIG>difference. The sqrt of 4 is 2 OR
-2;> Ehr, no. Both 2 and -2 are square roots of 4 (that is,
both of them,> when squared, give 4); but the sqrt of 4 is
defined to be the> principal branch, and therefore it is
unabiguously equal to 2. That's> why sqrt(a^2) = |a| for real
numbers, for example.> -- >
===
=================================
===
===> It's not denial. I'm just very selective about> what
I accept as reality. > --- Calvin ( Calvin and Hobbes )>
===
=================================
===
===> Arturo Magidin> magidin@math.berkeley.edu>Arturo,
thanks for the correction. I stand corrected.-- David
MoranChief MeteorologistOklahoma Storm Team
===
Since people
working with square roots *usually* want the positive> root,
by convention the positive is taken, not by definition. That's
necessary because, like sqrt(4) has *either* 2 or -2 as a>
solution, as is easily proven: sqrt(4) = -2, square both
sides, 4 = 4. QEDNice logic! -1 is a solution of 1 (whatever
that means), since squaring both sides gives 1=1.
QEDWilbert
===
Since people working with square roots
*usually* want the positive> root, by convention the positive
is taken, not by definition. That's necessary because, like
sqrt(4) has *either* 2 or -2 as a> solution, as is easily
proven: sqrt(4) = -2, square both sides, 4 = 4. QED Nice
logic! -1 is a solution of 1 (whatever that means), since >
squaring both sides gives 1=1. QED Wilbert-1 IS a solution for
sqrt(1).It's interesting that is a point of debate, but not
surprising for thesci.math newsgroup!James Harris
===
> Since
people working with square roots *usually* want the positive>
root, by convention the positive is taken, not by definition. That's necessary because, like sqrt(4) has *either* 2 or -2
as a> solution, as is easily proven:> sqrt(4) = -2, square
both sides, 4 = 4. QED Nice logic! -1 is a solution of 1
(whatever that means), since > squaring both sides gives 1=1.
QED Wilbert -1 IS a solution for sqrt(1). It's interesting
that is a point of debate, but not surprising for the>
sci.math newsgroup!Not surprising that you don't understand
something simple like that. sqrt(x) is a function defined for
real numbers x >= 0: sqrt(x) is by definition the positive
solution of y^2 = x. y = -1 is one of the two solutions of y^2
= 1, but -1 is not the square root of 1.
===
> Since people
working with square roots *usually* want the positive> root,
by convention the positive is taken, not by definition.>
That's necessary because, like sqrt(4) has *either* 2 or -2 as
a> solution, as is easily proven:> sqrt(4) = -2, square both
sides, 4 = 4. QED> Nice logic! -1 is a solution of 1
(whatever that means), since > squaring both sides gives 1=1.
QED> Wilbert>-1 IS a solution for sqrt(1).>It's interesting
that is a point of debate, but not surprising for the>sci.math
newsgroup!Yup. Just like you thought it was interesting that
the fact thatintegers are irrational was a point of
debate...>James HarrisDavid C.
Ullrich**************************As far as I'm concerend
you're trying to wait until I die, so I figuremaybe you should
die instead. How about that, eh? Wouldn't that be abetter
twist?You refuse to follow the math, so the great Powers that
controlreality and *speak* in mathematics decide to kill you
instead of me.So what do you think about that, eh? Oh, can't
hear Them talking?Well, I guess that's because you don't
really understand Mathematics,the true language, which is THE
language.They're talking about you now, and They agree with my
assessment, andwill not penalize me as They allowed the others
like Galois and Abelto be penalized.They will kill you
instead.James Harris speaking on Weird factorization, genius
===
> ...> Let g1(x)=4-sqrt(1+3x) and
g2(x)=4+sqrt(1+3x).> then> g1(x)*g2(x) =
3(5-x)> but neither g1 nor g2 is divisible by 3 for
all x.> Ah, but the constant term of g1(x) = 3, and the
constant term of g2(x) = 5,> so g1(x) should be divisible
by 3. A core error in mathematics ;-).> It's actually not
that simple. It's an interesting case to highlight> your
ineptitude with basic mathematics though.> Consider that
the constant term of g1(x) is 3 *or* 5 because the> sqrt()
operator is ambiguous. I've explained that before in
replying> to Arturo Magidin, but mathematics is a difficult
discipline for some,> so repetition is necessary, and still
often not enough.> Yup, it appears so. In mathematics the
sqrt when applied to reals and> delivering a real is defined
to give the positive result. This has> been said before, but
it is apparently to difficult for you to understand.>That
doesn't change the *inherent* ambiguity in the sqrt()
operator.Uh, yes it does.>That's easy to show as consider
sqrt(4), and you wish to say it's>*defined* to be 2, but what
about -2?>Does your definition take away -2 as a solution?A
solution to what? It doesn't take away -2 as a solution tothe
equation x^2 = 4. In fact -2 is _a_ square root of 4. Butby
definition the square root of 4, aka sqrt(4) , is 2.>And
besides, you lied, it's by *convention* that the positive
is>taken, not by definition.My god you can be an idiot. Saying
sqrt(4) = 2 by convention is the same as saying sqrt(4) = 2 by
definition . A definition isprecisely a convention regarding
the meaning of a word (or phraseor operator.)You should
probably try to avoid calling people liars when yourreason for
saying they lie is based on your profound ignoranceof the
meaning of simple English words.> You see Dik Winter,
actually *knowing* mathematics versus talking as> if you
know it can be two different things.> It appears so,
yes.>Since people working with square roots *usually* want
the positive>root, by convention the positive is taken, not by
definition.>That's necessary because, like sqrt(4) has
*either* 2 or -2 as a>solution, as is easily proven:>sqrt(4)
= -2, square both sides, 4 = 4. QED>Now then, can you *prove*
that -2 is not a solution to sqrt(4)?_equations_ have
solutions. Sqrt(4) is not an equation, so sayingthat -2 is a
solution to sqrt(4) makes no sense, just as sayingthat 2 is a
solution to sqrt(4) makes no sense. In fact 2 and -2are both
solutions to the equation x^2 = 4; nobody has
saidotherwise.>James HarrisDavid C.
Ullrich**************************As far as I'm concerend
you're trying to wait until I die, so I figuremaybe you should
die instead. How about that, eh? Wouldn't that be abetter
twist?You refuse to follow the math, so the great Powers that
controlreality and *speak* in mathematics decide to kill you
instead of me.So what do you think about that, eh? Oh, can't
hear Them talking?Well, I guess that's because you don't
really understand Mathematics,the true language, which is THE
language.They're talking about you now, and They agree with my
assessment, andwill not penalize me as They allowed the others
like Galois and Abelto be penalized.They will kill you
instead.James Harris speaking on Weird factorization, genius
===
Consider that the constant term of g1(x) is 3 *or* 5
because the> sqrt() operator is ambiguous. I've explained that
before in replying> to Arturo Magidin, but mathematics is a
difficult discipline for some,> so repetition is necessary,
and still often not enough.>Damn, James, does this mean your
prime counter's wrong cos it had anambiguous sqrt in it?
===
 : Why does this not count as presenting such a proposition ? : Let g1(x)=4-sqrt(1+3x) and g2(x)=4+sqrt(1+3x).> : then>
: g1(x)*g2(x) = 3(5-x)> : but neither g1 nor g2 is divisible
by 3 for all x.> the product *are* divisible by 7 for a
*specific* value of x, then they> are divisible by 7 for
*all* values of x. In your example neither is ever>
divisible by 3.> Justin Consider any integer a such that
1+3a is a perfect square. Then> g1(a) and g2(a) are both
integers. Since their product is divisible> by 3 one of them
must be divisible by 3. Note that posters have gone off on a
tangent with a made up example,> and they're even getting *it*
wrong! Fascinating behavior, but not> surprising for the
sci.math newsgroup. > Examples g1(0)=3, g2(0)=5 That is
incorrect as the sqrt() operator is ambiguous> No, by sqrt() I
mean the unambiguous principle branch. - William Hughes
===
>
Okay, so you're dividing both sides by 49. Point taken. :> In
other words, are you> :> claiming that (5 a_1(x) + 7) and (5
a_2(x) + 7) are divisible by 7 for> :> all x? If so, could you
please explain why the fact that it's true for> :> x=0 implies
it's true for other x also?> If you are not claiming this,
then please explain what you are claiming.> Many thanks,>
Justin> Given, where x is in the ring of algebraic integers,
I've shown the> factorization> (5 a_1(x) + 7)(5 a_2(x) + 7)(5
b_3(x) + 22) => 49(300125 x^3 - 18375 x^2 - 360 x + 22)>
where b_3(x) = a_3(x) - 3 and the a's are roots of> a^3 +
3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> so when x=0,
a_1(0) = a_2(0) = b_3(0) = 0.> Now you can divide both sides
by 49.> Some posters have claimed that when you divide by 49,
what results> varies depending on what value x has.> Okay, so
you're dividing both sides by 49. Point taken. :> In other
words, are you> :> claiming that (5 a_1(x) + 7) and (5 a_2(x)
+ 7) are divisible by 7 for> :> all x? If so, could you please
explain why the fact that it's true for> :> x=0 implies it's
true for other x also?> If you are not claiming this, then
please explain what you are claiming.> Many thanks,> Justin>
Given, where x is in the ring of algebraic integers, I've shown
the> factorization> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) +
22) => 49(300125 x^3 - 18375 x^2 - 360 x + 22)> where b_3(x)
= a_3(x) - 3 and the a's are roots of> a^3 + 3(-1 + 49x)a^2 -
49(2401 x^3 - 147 x^2 + 3x)> so when x=0, a_1(0) = a_2(0) =
b_3(0) = 0.> Now you can divide both sides by 49.> Some
posters have claimed that when you divide by 49, what results>
varies depending on what value x has.For Q(x) = (5a_1(x) +
7)(5a_2(x) + 7)(5b_3(x) + 22): x Q(x) JSH
'a'----------------------------- 0 1078 0 0 3 1 13789188
25.5404... -138.21... -31.330... 2 114013298 51.6126...
-279.30... -63.312... 3 388910158 77.6850... -420.39...
-95.295...--There are two things you must never attempt to
prove: the unprovable -- and theobvious.--Democracy: The
triumph of popularity over
principle.--http://www.crbond.com
===
One of those hard lessons
to learn in mathematics is operatorambiguity.For instance the
square root and cuberoot operators are ambiguous, andthere's
nothing you can do about it.Given (1)^{1/3} there are *three*
solutions, and not one, which is theoperator ambiguity that
gave me fits for a while back last year when Iposted and
posted trying to find some trick around it.The square root
operator has ambiguity in that it gives *two*solutions, even
if you only want one.I managed to get myself in trouble yet
again today trying yet again toescape operator ambiguity by
making an earlier post trying to go withthe sign convention of
taking the positive solution of the squareroot operator.That
doesn't work.It bothers me that I keep fighting operator
ambiguity and trying tofind ways around it, as if some part of
me just can't accept that ifyou have sqrt(x), or x^{1/3}, you
have *multiple* solutions, whichrefuse to go away, no matter
how hard you wish.James Harris
===
: For instance the square
root and cuberoot operators are ambiguous, and: there's
nothing you can do about it.Pray tell, what's your definition
of operator ?The sqrt(c) *function* is defined to be the
positive real number whosesquare is c . There is no *solution*
here because this is not an*equation*. There is no
ambiguity.The *equation* x^2-c=0 has *two solutions*, one is
sqrt(c) and the otheris -sqrt(c).How do *you* define operator
?Justin
===
> One of those hard lessons to learn in mathematics
is operator> ambiguity. For instance the square root and
cuberoot operators are ambiguous, and> there's nothing you can
do about it. Given (1)^{1/3} there are *three* solutions, and
not one, which is the> operator ambiguity that gave me fits
for a while back last year when I> posted and posted trying to
find some trick around it. The square root operator has
ambiguity in that it gives *two*> solutions, even if you only
want one. I managed to get myself in trouble yet again today
trying yet again to> escape operator ambiguity by making an
earlier post trying to go with> the sign convention of taking
the positive solution of the square> root operator. That
doesn't work. It bothers me that I keep fighting operator
ambiguity and trying to> find ways around it, as if some part
of me just can't accept that if> you have sqrt(x), or x^{1/3},
you have *multiple* solutions, which> refuse to go away, no
matter how hard you wish. > James HarrisI, for one, find it
odd that JSH can be so concerned about the action ofthe Galois
group in this case (of the equation x^n - c = 0), when todate
he has been a Galois Luddite, and proud of it.Dale
===
> One of
those hard lessons to learn in mathematics is operator>
ambiguity. For instance the square root and cuberoot operators
are ambiguous, and> there's nothing you can do about it. Given
(1)^{1/3} there are *three* solutions, and not one, which is
the> operator ambiguity that gave me fits for a while back
last year when I> posted and posted trying to find some trick
around it. The square root operator has ambiguity in that it
gives *two*> solutions, even if you only want one. I managed
to get myself in trouble yet again today trying yet again to>
escape operator ambiguity by making an earlier post trying to
go with> the sign convention of taking the positive solution
of the square> root operator. That doesn't work. It bothers me
that I keep fighting operator ambiguity and trying to> find
ways around it, as if some part of me just can't accept that
if> you have sqrt(x), or x^{1/3}, you have *multiple*
solutions, which> refuse to go away, no matter how hard you
wish. > James HarrisJames, your argument is equivalent to the
following: Roads have two lanes. Therefore roads are ambiguous
and there is nothing you can do about it. Therfore you can't
drive on a road because there is no way for you to know what
lane you should be in. However, we know from common experience
that people do drive onroads. The point is that there is an
unabiguous way forpeople to decide which is the right lane
[1].The same holds with ambiguous operators . The fact that
everycomplex number has two square roots does not stop us
defining asingle valued square root function, as long as we
can findan unambiguous way to select between them. If we are
onlyinterested in the positive real axis, it is enough to
selectthe positive square root (it is easy to show that if x
is apositive real then x has two real square roots, on
positive and one negative). If we need to deal with all
posible complex numbers z, then we choose the square root that
has positive realpart (positive imaginary part if z is negative
real number). For cube roots things get a bit trickier (though
not much). Look up principal branch for more details. -William
Hughes[1] Indeed it's the right lane. And to deal with the
square root you take the values from the right side of the
plane. And to deal with the cube root you take ... Ok,
eventualy this breaks down.
===
> [1] Indeed it's the right
lane.Except where the left lane is the right lane.-- dik t.
winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland,
+31205924131home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
> [1] Indeed it's the right lane.
Except where the left lane is the right lane.But suppose you
don't drive in the left lane: then right is left, right?Jim
Burns
===
> However, we know from common experience that people
do drive on> roads. The point is that there is an unabiguous
way for> people to decide which is the right lane [1]. [1]
Indeed it's the right lane. And to deal with the square root>
you take the values from the right side of the plane. And to
deal> with the cube root you take ... Ok, eventualy this
breaks down.That is more like taking the square root of -1. Of
course the square root of -1 is i, but it could just as well
have been j where j = -i, and nobody would be any the wiser.
Just like you should always drive on the right lane, but in
some countries the left lane is the right line to
use.
===
[...]> It bothers me that I keep fighting operator
ambiguity and trying to> find ways around it, as if some part
of me just can't accept that if> you have sqrt(x), or x^{1/3},
you have *multiple* solutions, which> refuse to go away, no
matter how hard you wish.James you've spent so many years of
your life on maths, and you're still such an ignorant.
Impressive...This operator ambiguity bothers you because you
still don't understand what an operator is. I'll try to help
with the following question addressed to you:What is an
operator?--EdgarP.S. If you can't answer it, then everybody
will know you've no idea what you're talking about.
===
> One
of those hard lessons to learn in mathematics is operator>
ambiguity.> For instance the square root and cuberoot
operators are ambiguous, and> there's nothing you can do about
it.> Given (1)^{1/3} there are *three* solutions, and not one,
which is the> operator ambiguity that gave me fits for a while
back last year when I> posted and posted trying to find some
trick around it.> The square root operator has ambiguity in
that it gives *two*> solutions, even if you only want one.> I
managed to get myself in trouble yet again today trying yet
again to> escape operator ambiguity by making an earlier post
trying to go with> the sign convention of taking the positive
solution of the square> root operator.> That doesn't work.>
It bothers me that I keep fighting operator ambiguity and
trying to> find ways around it, as if some part of me just
can't accept that if> you have sqrt(x), or x^{1/3}, you have
*multiple* solutions, which> refuse to go away, no matter how
hard you wish.> James HarrisBy your scholarly exposition of
the *alleged* operator ambiguity in thesqrt function, you have
completely invalidated your own research results.Time to step
up to the bar and admit that, BY YOUR OWN CRITERIA, your
workis ambiguous, misleading and outright false!Next time you
try to box someone else into a corner, make sure you
aren'tpainting yourself into
one.Hahahahahaha............--There are two things you must
never attempt to prove: the unprovable --and the
obvious.--Democracy: The triumph of popularity over
principle.--http://www.crbond.com
===
> One of those hard
lessons to learn in mathematics is operator> ambiguity. For
instance the square root and cuberoot operators are ambiguous,
and> there's nothing you can do about it. Given (1)^{1/3} there
are *three* solutions, and not one, which is the> operator
ambiguity that gave me fits for a while back last year when I>
posted and posted trying to find some trick around it. The
square root operator has ambiguity in that it gives *two*>
solutions, even if you only want one. I managed to get myself
in trouble yet again today trying yet again to> escape
operator ambiguity by making an earlier post trying to go
with> the sign convention of taking the positive solution of
the square> root operator. That doesn't work. It bothers me
that I keep fighting operator ambiguity and trying to> find
ways around it, as if some part of me just can't accept that
if> you have sqrt(x), or x^{1/3}, you have *multiple*
solutions, which> refuse to go away, no matter how hard you
wish. > James HarrisYou say that sqrt(x) has two solutions,
but a function doesn't have solutions, only an equation.
sqrt(x) is a function, and as such, has only values.The
equation x^2 = 5 has two solutions, namely sqrt(5) and
-sqrt(5), but that's just a result of solving the equation.
It's not inherent in the functions which make up the
solution.For example, the equation 2x = 2sqrt(5) has exactly
one solution, namely sqrt(5).In the same way, while x^3 = 1
has three solutions, 1^y for any y is exactly 1.
===
>excuse my
ignorance...but is there any explicit result for the
following>Tr(AAAAA....A)>in terms of the componants a[i,j]
of the matrix A where the product is taken>N (finite) times?
(where i=1...n, j=1...m)Diagonalize: if r_1, ..., r_n are the
entries on the diagonal of theJordan canonical form of A (in
an extension field of the field ofdefinition of A) then the
quantity T_N you want is the sum of theN-th powers of the r_i.
That sum can be expressed as a polynomial in theelementary
symmetric functions on the r_i, which are the coefficientsof
det(A + X I) (essentially the characteristic polynomial of
A).You can compute the T_N easily from the previous T_k's
usingNewton's identities. If n or N is small you can write
this outexplicitly in symbolic form.(Of course you must have
n=m for the product AA to be defined.)dave
===
>excuse my
ignorance...but is there any explicit result for the
following>Tr(AAAAA....A)>in terms of the componants a[i,j] of
the matrix A where the product is taken>N (finite) times?
(where i=1...n, j=1...m)If A is n x m and you want A^N to make
sense, you'd better have n=m.Tr(A^N) = sum_{i_1=1}^n ...
sum_{i_n=1}^n product_{j=1}^n a[i_j, i_{j+1}]where i_{N+1} =
i_1.Robert Israel israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
>excuse my ignorance...but
is there any explicit result for the
following>Tr(AAAAA....A)>in terms of the componants a[i,j]
of the matrix A where the product istaken>N (finite) times?
(where i=1...n, j=1...m)> If A is n x m and you want A^N to
make sense, you'd better have n=m.> Tr(A^N) = sum_{i_1=1}^n
... sum_{i_n=1}^n product_{j=1}^n a[i_j, i_{j+1}]> where
i_{N+1} = i_1.thank you.cheersmoth
===
For those of you who
have been following the Peter Lynds' controversy,you may be
interested on knowing that a new scandal has happenedrecently
in the field of Maths.You can check here (and leave your
opinion in the forum if you
like):http://www.thequantummachine.comCesar Sirvent
===
> For
those of you who have been following the Peter Lynds'
controversy,> you may be interested on knowing that a new
scandal has happened> recently in the field of Maths.> You can
check here (and leave your opinion in the forum if you
like):Looks like the same old silliness to me.
===
> Suppose p
: E --> B is a covering space and f : X --> B is a continuous>
map where X is connected and B is Hausdorff. Show that if g and
h> are two lifts of f such that there exists an a in X such
that> g(a) = h(a), then g(x) = h(x) for all x in X. (Should we
show> that {x in X : g(x) = h(x) is both open and closed}?) >
That would probably be fine. What I would do is to consider
the path-> component of a , and look at lifts via g and h,
respectively, of any> path from a to an arbitrary other point
in this path-component. Since covering spaces have unique path
lifting, then g and h have to> agree on each path component. By
continuity, they have to agree on> the closures of path
components. Eventually, you can exhaust the> space X by
progressing from point a to its path-component, to> the
closure of that, to the path-components occupied by that
closure,> to the closure of that set, and so on (there's
probably something non-> trivial to prove here, and that's
most likely the core of your> show that {...} is open and
closed in X argument).> No offense, but ... that seems _way_
too complicated. To the OP: showing that A(g,h) = { x in X :
g(x) = h(x) } is both open and closed _is_ the way to go. (1)
Since B is Hausdorff and p: E --> B is a covering map, it
follows that E is Hausdorff. (Proof ??) (2) Given _any_ two
maps X --> E, the subset of X on which they agree is closed
(since E is Hausdorff). So A(g,h) is closed. (3) For any x in
A(g,h), you can find an open neighborhood U of g(a) = h(a) in
E such that p|U is a homeomorphism. Then define V = g^{-1}(U)
intersect h^{-1}(U). Show that V is contained in A(g,h), which
proves that A(g,h) is open.> Suppose that G is a finite group
which acts freely on a Hausdorff space> X. Show that the
action must be properly discontinuous. (Can we use> the above
problem for this?)> About your parenthetical question: it
seems unlikely. If you _knew_ that the quotient map q: X -->
X/G is a covering map, then you might have a chance. But ...
one usually proves that q is a covering map as a consequence
of the fact that G acts properly discontinuously ... (and
that's proved in the manner that Dale indicates below). Think
on these three conditions: G is finite> X is Hausdorff.> The
action is free. Recall that a properly discontinuous action is
one for which,> for each point x in X, there is a neighborhood
U_x of x in X,> such that all non-identity elements of G map U
to a set> disjoint from U. Here, you can take the orbit Gx of
your> favorite x in X, and form disjoint open sets U_gx of
gx.> By a suitable song and dance, you can force the union> to
be G-invariant (hint: G is finite, and a finite> intersection
of open sets is open), and then take the> patch containing
your original x. I'm not sure how to apply the first problem
to address> the second one. Steven Dale
===
>Suppose p : E
--> B is a covering space and f : X --> B is a
continuous>map where X is connected and B is Hausdorff. Show
that if g and h>are two lifts of f such that there exists an
a in X such that>g(a) = h(a), then g(x) = h(x) for all x in
X. (Should we show>that {x in X : g(x) = h(x) is both open
and closed}?)>That would probably be fine. What I would
do is to consider the path->component of a , and look at
lifts via g and h, respectively, of any>path from a to an
arbitrary other point in this path-component.>Since
covering spaces have unique path lifting, then g and h have
to>agree on each path component. By continuity, they have to
agree on>the closures of path components. Eventually, you can
exhaust the>space X by progressing from point a to its
path-component, to>the closure of that, to the
path-components occupied by that closure,>to the closure of
that set, and so on (there's probably something non->trivial
to prove here, and that's most likely the core of your> show
that {...} is open and closed in X argument).> No offense,
but ... that seems _way_ too complicated. To the OP:> showing
that A(g,h) = { x in X : g(x) = h(x) } is both open and>
closed _is_ the way to go. (1) Since B is Hausdorff and p: E
--> B is a covering map, it> follows that E is Hausdorff.
(Proof ??) (2) Given _any_ two maps X --> E, the subset of X
on which they> agree is closed (since E is Hausdorff). So
A(g,h) is closed. (3) For any x in A(g,h), you can find an
open neighborhood U of> g(a) = h(a) in E such that p|U is a
homeomorphism. Then define> V = g^{-1}(U) intersect h^{-1}(U).
Show that V is contained> in A(g,h), which proves that A(g,h)
is open.> I agree. That proof is far simpler. ... stuff
deleted ...>Dale> Dale
===
I came up with a question
regarding limits today,let n be real,x = lim { n/(n!)^(1/) }
n->infHow can I solve this problem? I triedL'Hosptal's rule
but I dont knowthe answer of d(x!) ------ dxCould somebody
please help me to answer thesetwo questions?thanks a lot in
advance, /lucas
===
> let n be real,> x = lim { n/(n!)^(1/) }
n->inf How can I solve this problem? I tried> L'Hosptal's rule
but I dont know> the answer of d(x!)> ------> dxOne way to do
it is to use Stirling's approximationto the factorial
functionn! ~~ sqrt(2 pi) n^(n + 1/2) exp( -n )
===
> I came up
with a question regarding limits today, let n be real,> x =
lim { n/(n!)^(1/) } n->infYou probably mean the limit of
n/(n!)^(1/n) as n -> oo. Take the log to get sum_(j=1,n)
(1/n)*ln(n/j), which is a Riemann sum for int_[0,1] ln(1/x) dx
= 1. So the answer is e.
===
>I came up with a question
regarding limits today,>let n be real,>x = lim { n/(n!)^(1/)
} n->inf>How can I solve this problem?If you mean x =
lim(n/(n!)^(1/n)), n->oo then taking the logarithm ofx
gives:log(x) = lim[ log n - 1/n * log(n!) ]and using
Stirling's approximation log(n!) ~= nlog n - n (for large n)we
get:log(x) = lim[ log n - 1/n * (n * log n - n) ] = lim[ log n
- log n + 1 ] = 1So that x = e^1 = e.
===
> [snipped]>let n
be real,>x = lim { n/(n!)^(1/) } n->inf>How can I solve
this problem? If you mean x = lim(n/(n!)^(1/n)), n->oo then
taking the logarithm of> x gives:> That's it! Sorry, my
mistake. again, x = lim(n/(n!)^(1/n)) n->infinity.> log(x) =
lim[ log n - 1/n * log(n!) ] and using Stirling's
approximation log(n!) ~= nlog n - n (for large n)> we get:
log(x) = lim[ log n - 1/n * (n * log n - n) ]> = lim[ log n -
log n + 1 ]> = 1 So that x = e^1 = e. This is the kind of
trick that I didnt see in the engineering course. Its
perfectly clear for me now, I appreciate that, thanks a lot
everybody that answerd my question. Again, what is the answer
of d(x!)/dx ? thank you, /lucas
===
[snip]> Again, what is the
answer of d(x!)/dx ?[I assume, of course, that you're thinking
about x! = Gamma(x+1), ratherthan having x! defined only for
nonnegative integer x.]See (16) at
<http://mathworld.wolfram.com/GammaFunction.html>.David
===
>
[snip]> Again, what is the answer of d(x!)/dx ? [I assume, of
course, that you're thinking about x! = Gamma(x+1), rather>
than having x! defined only for nonnegative integer x.] See
(16) at <http://mathworld.wolfram.com/GammaFunction.html>.>
Not really. n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 2 * 1
Am I missing something? This is my definition of `n!', and what
I wanted to calculate in the above problem. thanks, /lucas
===
>
[snip]> Again, what is the answer of d(x!)/dx ? [I assume, of
course, that you're thinking about x! = Gamma(x+1), rather>
than having x! defined only for nonnegative integer x.] See
(16) at <http://mathworld.wolfram.com/GammaFunction.html>. Not
really. > n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 2 * 1>
Am I missing something? This is my definition of `n!', and
what> I wanted to calculate in the above problem.> Well, then,
what you are missing is the definition of derivative.
Itrequires a function defined on an interval. Not merely at
discretepoints like 1,2,3,... So for THAT factorial function,
your question> Again, what is the answer of d(x!)/dx ?is
nonsense.
===
> Not really. > n! = n * (n - 1) * (n - 2) * (n
- 3) * ... * 2 * 1> Am I missing something? This is my
definition of `n!', and what> I wanted to calculate in the
above problem.> Well, then, what you are missing is the
definition of derivative. It> requires a function defined on
an interval. Not merely at discrete> points like 1,2,3,... So
for THAT factorial function, your question> I interpreted the
above statement as a polynomial expression. Such as I could
distribute the values and somehow apply a reduction to that
expression being able to derivate it. For example, i can
derivate `ax^2 + bx + c' -> `2ax + b'. I wonder if I can do
the same with this infinity expression. I thought i could
express `n!' as a polynomial, p(n) of n'th degree.> Again,
what is the answer of d(x!)/dx ? is nonsense. Also, I was
working on another solution to the same limit problem and I
blocked myself on this: y = lim(x/[(x!)^(1/x)]) y =
lim([x^x/x!]^1/x) i know that x^x > x!, because, x^x = x * x *
x * x * x * ... * x and x! = x * (x - 1) * (x - 2) ... * 1 so,
lim(x^x/x!) = oo (infinity) i cant manage to solve (oo)^(1/x).
thanks for everything, /lucas
===
I came up with a question
regarding limits today, let n be real,> x = lim { n/(n!)^(1/)
} n->inf How can I solve this problem? I tried> L'Hosptal's
rule but I dont
knowAAAAAAAAAAAAAARRRRRRRRRRGGGGGGGGGGGGGGGGGHHHHHHHHHHHHHHHHHH
HHH!!!!!!!!!Forget the Hospital.I presume you mean
n/(n!)^(1/n).Use Stirling's formula.-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the
last laugh. Alan Partridge, _Bouncing Back_ (14 times)
===
> I
came up with a question regarding limits today, let n be
real,> x = lim { n/(n!)^(1/) } n->inf> It seems something is
missing here...After correcting that... Did you see what
Stirling's formula tells you?> How can I solve this problem? I
tried> L'Hosptal's rule but I dont know> the answer of d(x!)>
------> dx Could somebody please help me to answer these> two
questions? thanks a lot in advance, /lucas
===
I know there are
a number of methods available readily forapproximating the
complete gamma function (e.g., the Lanczosapproximation).
However, what I need is a formula/method for
gettingapproximate values for the LOWER INCOMPLETE GAMMA
FUNCTION; i.e.,where zero is the lower limit of integration
and the upper limit isspecified as one of the parameters of
the function. Does anyone know
===
>I know there are a number
of methods available readily for>approximating the complete
gamma function (e.g., the Lanczos>approximation). However,
what I need is a formula/method for getting>approximate values
for the LOWER INCOMPLETE GAMMA FUNCTION; i.e.,>where zero is
the lower limit of integration and the upper limit
is>specified as one of the parameters of the function. Does
anyone knowIf G(a,z) = int_0^z t^(a-1) exp(-t) dt is your
function, thenfor small z you can use G(a,z) =
sum_{k=0}^infinity (-1)^k z^(a+k)/(k! (a+k))Or for positive
integers a,G(a,z) = Gamma(a) (1 - exp(-z) sum_{k=0}^{a-1}
z^k/k!)If a is not an integer, this becomes an asymptotic
seriesG(a,z) = Gamma(a) (1 - exp(-z) sum_{j=0}^infinity
z^(a-1-j)/Gamma(a-j))Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
>I know there are a number of
methods available readily for>approximating the complete gamma
function (e.g., the Lanczos>approximation). However, what I
need is a formula/method for getting>approximate values for
the LOWER INCOMPLETE GAMMA FUNCTION; i.e.,>where zero is the
lower limit of integration and the upper limit is>specified as
one of the parameters of the function. Does anyone knowThere
are lots of these as well; look in numerical analysistexts
dealing with such functions. Which ones I would usedepends on
the actual parameters.-- This address is for information only.
I do not claim that these viewsare those of the Statistics
Department or of Purdue University.Herman Rubin, Department of
Statistics, Purdue University
===
Given, where x is in the ring
of algebraic integers, I've shown thefactorization(5 a_1(x) +
7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 -
360 x + 22)where b_3(x) = a_3(x) - 3 and the a's are roots
ofa^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)so when
x=0, a_1(0) = a_2(0) = b_3(0) = 0.Now consider the
factorization shown again, but with the 49
multipliedthrough:(5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22)
= 14706125 x^3 - 900375 x^2 - 17640 x + 1078and since a_1(0)=
a_2(0) = b_3(0) = 0, it's not surprising that thevalues thus
shown to be constant in the factors on the left side i.e.7, 7
and 22 are in fact factors of what's constant on the right
sidei.e. 1078.Now if I divide both sides by 49, I end up with
a change where now Ihave constant factors 1, 1, and 22 on the
left which are still factorsof 22 on the right.Does that fact
tell you that *two* of the factors on the left, theones that
have 7 as a constant factor were each divided by 7, or doesit
tell you *nothing* at all?Test is of the ability to understand
constant factors as independentof variables, with a check of
ability to convince large groups. Notesci.logic is included to
emphasize logical thinking and see if itmatters to readers.That
is, the test is of self-doubt.James Harris
===
Is there any
rationality test that JSH has not yet failed?
===
I am going to
try to follow the convention of addressing the assemblyas a
whole, not just the poster to whom I am replying. I think
thispractice improves the ratio of light to heat in a
discussion. > Given, where x is in the ring of algebraic
integers, I've shown the> factorization (5 a_1(x) + 7)(5
a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x
+ 22) where b_3(x) = a_3(x) - 3 and the a's are roots of>
[Polynomial P]> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 +
3x) so when x=0, a_1(0) = a_2(0) = b_3(0) = 0.It took me two
tries, even with Mathematica, but this computationchecks out
OK. Now consider the factorization shown again, but with the
49 multiplied> through: (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x)
+ 22) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 and since
a_1(0)= a_2(0) = b_3(0) = 0, it's not surprising that the>
values thus shown to be constant in the factors on the left
side i.e.> 7, 7 and 22 are in fact factors of what's constant
on the right side> i.e. 1078.Sustitute 0 for x and the
equation reduces to7 * 7 * 22 = 1078. Yes. Now if I divide
both sides by 49, I end up with a change where now I> have
constant factors 1, 1, and 22 on the left which are still
factors> of 22 on the right.Well, let's see if I understand.
The left side could just be written(5 a_1(x) + 7)(5 a_2(x) +
7)(5 b_3(x) + 22)/49.But one can say, Well, let's merge the
factor of 1/49 into the otherfactors. The way that looks
nicest is to multiply each of the firsttwo factors by 1/7.
Then we have this:[Equation A]((5/7) a_1(x) + 1)((5/7)a_2(x) +
1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22 Does
that fact tell you that *two* of the factors on the left, the>
ones that have 7 as a constant factor were each divided by 7,
or does> it tell you *nothing* at all?It tells me a little
bit, but I don't think it tells me as much as Mr.Harris wants.
I am not sure, but I think he wants to infer that thethree
factors in Equation A will be algebraic integers when x is
anon-zero algebraic integer.Hmm. Are they? When x = 0,
Polynomial P has roots 0, with multiplicity 2, and 1
withmultiplicity 1. It seems clear that Mr. Harris intends
that a_1(x) anda_2(x) specialize to 0 when x = 0, whereas
a_3(x) specializes to 1. OK,I think. Suppose we let x = y^2,
where y is an algebraic integer. This does notlose any
generality. And suppose, when y != 0, we let a_i = 7 y c_i. We
can substitute into Polynomial A and divide out by a factor
of49y^2.When the dust settles, we find that the c_i are roots
of thispolynomial:7 y c^3 -3(1 - 49 y^2)c^2 -(3 - 147 y^2 +
2401 y^4)I think Mr. Harris would claim that two of the roots
of this polynomialare algebraic integers, when y is a non-zero
algebraic integer. Idoubt this, but I would be glad to learn
more.I am going to take another look at the paper by Messrs.
Magidin andMacKinnon on Gauss's Lemma , available
athttp://www.math.umt.edu/magidin/preprints/preprints.htmlI
recommend it to all interested readers of this post.-- Chris
Henrich
===
> Given, where x is in the ring of algebraic
integers, I've shown the> factorization> (5 a_1(x) + 7)(5
a_2(x) + 7)(5 b_3(x) + 22) => 49(300125 x^3 - 18375 x^2 - 360
x + 22)> where b_3(x) = a_3(x) - 3 and the a's are roots of>
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> so when
x=0, a_1(0) = a_2(0) = b_3(0) = 0.> Now consider the
factorization shown again, but with the 49 multiplied>
through:> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =>
14706125 x^3 - 900375 x^2 - 17640 x + 1078> and since a_1(0)=
a_2(0) = b_3(0) = 0, it's not surprising that the> values thus
shown to be constant in the factors on the left side i.e.> 7,
7 and 22 are in fact factors of what's constant on the right
side> i.e. 1078.> Now if I divide both sides by 49, I end up
with a change where now I> have constant factors 1, 1, and 22
on the left which are still factors> of 22 on the right.>
Does that fact tell you that *two* of the factors on the left,
the> ones that have 7 as a constant factor were each divided by
7, or does> it tell you *nothing* at all?It provides a special
case in which the expressions are evaluated at 'x' =0.It does
*not* provide insight into the general case.For Q(x) =
(5a_1(x)+7)(5a_2(x)+7)(5b_3(x)+22): x Q(x) JSH 'a' (b_3(x) =
a_3(x) - 3)---------------------------------------------------
0 1078 0 0 3 1 13789188 25.5404 -138.21 -31.3299 2 114013298
51.6126 -279.3 -63.3123 3 388910158 77.685 -420.39 -95.2947>
Test is of the ability to understand constant factors as
independent> of variables, with a check of ability to convince
large groups. Note> sci.logic is included to emphasize logical
thinking and see if it> matters to readers.rather in 'alt.
crackpot.theories'.Give up, James! You are *toast*!--There are
two things you must never attempt to prove: the unprovable
--and the obvious.--Democracy: The triumph of popularity over
principle.--http://www.crbond.com
===
>[...]>Test is of the
ability to understand constant factors as independent>of
variables, with a check of ability to convince large groups.
Note>sci.logic is included to emphasize logical thinking So
today you decided to add what's the point to sci.logic? to the
list of things you're clueless about. Congratulations.>and see
if it>matters to readers.>That is, the test is of self-doubt.
James HarrisDavid C. Ullrich**************************As far as
I'm concerend you're trying to wait until I die, so I
figuremaybe you should die instead. How about that, eh?
Wouldn't that be abetter twist?You refuse to follow the math,
so the great Powers that controlreality and *speak* in
mathematics decide to kill you instead of me.So what do you
think about that, eh? Oh, can't hear Them talking?Well, I
guess that's because you don't really understand
Mathematics,the true language, which is THE language.They're
talking about you now, and They agree with my assessment,
andwill not penalize me as They allowed the others like Galois
and Abelto be penalized.They will kill you instead.James Harris
speaking on Weird factorization, genius
===
> Given, where x is
in the ring of algebraic integers, I've shown the>
factorization> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22)
=> 49(300125 x^3 - 18375 x^2 - 360 x + 22)> where b_3(x) =
a_3(x) - 3 and the a's are roots of> a^3 + 3(-1 + 49x)a^2 -
49(2401 x^3 - 147 x^2 + 3x)> so when x=0, a_1(0) = a_2(0) =
b_3(0) = 0.> Now consider the factorization shown again, but
with the 49 multiplied> through:> (5 a_1(x) + 7)(5 a_2(x) +
7)(5 b_3(x) + 22) => 14706125 x^3 - 900375 x^2 - 17640 x +
1078> and since a_1(0)= a_2(0) = b_3(0) = 0, it's not
surprising that the> values thus shown to be constant in the
factors on the left side i.e.> 7, 7 and 22 are in fact factors
of what's constant on the right side> i.e. 1078.> Now if I
divide both sides by 49, I end up with a change where now I>
have constant factors 1, 1, and 22 on the left which are still
factors> of 22 on the right.> Does that fact tell you that
*two* of the factors on the left, the> ones that have 7 as a
constant factor were each divided by 7, or does> it tell you
*nothing* at all?> Test is of the ability to understand
constant factors as independent> of variables, with a check of
ability to convince large groups. Note> sci.logic is included
to emphasize logical thinking and see if it> matters to
readers.> That is, the test is of self-doubt. James
HarrisJames, You are a total moron. You act like we are
nothing except for yourexperiment. We are not your experiment;
we are all humans. You are totallyout of line by claiming to
use us as part of your experiment .-- David MoranChief
MeteorologistOklahoma Storm Team
===
[...snip JSH nonsense...]
James, You are a total moron. You act like we are nothing
except for your> experiment. We are not your experiment; we
are all humans. You are totally> out of line by claiming to
use us as part of your experiment .No explanation needed. All
that is required is: James, You are a total moron. The rest is
just filler. :):):)Bye,Jay
===
[snip]> best wishes> GeorgeAfter
having some rest, I am back. Scrolling the threads in
sci.physics, I have noticed that nobodyremembers that Wright
brothers took first flight in history of mankindAll Aeroplanes
I see here in Bombay flying in sky is because of them.17 years
ago, when I left my small village at the age of 17, I hadnever
seen any aeroplane flying right over my head so closely.
Andbecause of those Aeroplanes flying in sky, somehow I began
to dreamthat I want to be Pilot, not Engineer. And I just left
this placedespite stiff opposition from my family.For 17 years,
bouncing from this career to that career, somehow I havethis
action device which we can use to hang the things in air,
makesmall bikes, cars etc. and any one, even small kids can
become pilot to fly them in blue sky.I pay my tributes to
Wright Brothers who gave me dreams to fly 17years ago.But
Everything That Has A Beginning Has An
End...http://www.geocities.com/actiondevice-Abhi.
===
> [snip]>
best wishes> George After having some rest, I am back.I hope
you are feeling better. Now instead of snippingwhat I said,
how about reading it. It shows why it isnot possible to build
a machine such as you describeregardless of how you construct
it:> I have once again changed the webpage. Perhaps these
magnets facing> each other will explain the working of this
action device in better> way.> Abhi, you said that logic was
a tool you could use so> consider this explanation of basic
mechanics. 1) Two magnets pull towards each other:>
===
F-->
<--F
===
> The forces are equal and opposite. 2) A stretched
spring pulls the ends towards each other:>
---VVVVVVVVVVVVVV---> F--> <--F> The forces are equal and
opposite.> 3) A compressed rod pushes back against the
compression.>|
===
======|<> <--F F-- The forces are
equal and opposite. 4) Try to bend a rod:> F F> | |> v v>
|
===
======|> ^> |> 2*F> The total forces are equal and
opposite.> (Forces in this case are two of 'F' each
downwards> and one of twice as much upwards.)> Now here's the
logic:> Since the forces created by every rod, spring> or pair
of magnets are always equal and opposite,> the total of all
forces in your device must also> be equal and opposite. Adding
equal but opposite> forces always totals zero, therefore your
device> will always produce zero total force no matter> how
many springs, rods and magnets you introduce.George
===
> For
17 years, bouncing from this career to that career, somehow I
have> this action device which we can use to hang the things
in air, make> small bikes, cars etc. and any one, even small
kids can become pilot > to fly them in blue sky.WHAT??!!YOU
SPEND 17 YEARS ON THAT THING AND NEVER BUILD IT??Whyever
that?Greetings!Volker
===
Can someone please explain how to
permute n objects in a circle withoutgoing too much into the
mathematics of it
===
> Can someone please explain how to
permute n objects in a circle without> going too much into the
mathematics of it Nail one to the ground.
===
> Can someone
please explain how to permute n objects in a circle without>
going too much into the mathematics of it > Nail one to the
ground.Good one :-)... or throw one away and permute the
remaining ones in a line.Dirk Vdm
===
does anybody know about
properties concerning the zeros of thederivatives of smmoth
symmetric functions f(t)=f(1-t) with compactsupport? I was
wondering if he zeros of the derivatives of a functionof the
formf(t)=exp(-1/(t*(1-t))), t in (0,1)Thomas
===
does anybody
know about properties concerning the zeros of the> derivatives
of smmoth symmetric functions f(t)=f(1-t) with compact>
support? I was wondering if he zeros of the derivatives of a
function> of the form f(t)=exp(-1/(t*(1-t))), t in (0,1) >
ThomasThe derivative is going to be a rational function in t
times anexponential and therefore cannot have more than
finitely many.
===
>does anybody know about properties
concerning the zeros of the>derivatives of smmoth symmetric
functions f(t)=f(1-t) with compact>support? I was wondering if
he zeros of the derivatives of a function>of the
form>f(t)=exp(-1/(t*(1-t))), t in (0,1)For this particular
one, I don't know, but it's certainly not true ingeneral. For
example, there are functions of this type that coincidewith
cos(t-1/2) on a neighbourhood of 1/2, and the derivatives of
those have no zeros in that neighbourhood except at 1/2.Robert
Israel israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
We'd like to call your
attention to a proposal for a new usenet newsgroup inthe
sci.physics.* hierarchy with the proposed name being
sci.physics.strings .This new group is supposed to provide a
place for intensive discussion ofstring theory and related
topics by people working in the field as well asinterested
laymen.In order to create such a new usenet group the proposed
group's charter hasto go through a discussion period at
news.groups , which can be accessedfor instance via
http://groups.google.de/groups?&lr=&ie=UTF-8&group=news.groups
.You can find and discuss the proposed charter of
sci.physics.strings as wellas the rationale for its creation
athttp://groups.google.de/groups?selm=1071590601.19921%
40isc.orgAfter the discussion period (which will last
presumably about three to sixweeks) there will be a general
poll where everybody is asked to vote infavour or against the
creation of sci.physics.strings on usenet. We'd liketo kindly
invite everybody with an interest in string theory or
stringtheory related topics to have a look at the above
proposal and toparticipate in the voting.Since a majority of
2/3 of the valid votes must be in favor of the creationof
sci.physics.strings, we ask everybody who would enjoy seeing
interestingonline discussion of string theory to participate
in the voting.P.S.General information on the nature of usenet
discussion fora is given at
http://www.faqs.org/faqs/usenet/what-is/part1/ .More
information on the general procedure for creating new usenet
newsgroupscan be found at
http://www.faqs.org/faqs/usenet/creating-newsgroups/part1/
.
===
>please don't take too much comfort from my post. My
reference, if taken as implying that you might be sympathetic
tomy position, seems not to be welcome. Clarification
accepted.However,back to the bone of contention. There may be
others besides myself who are happier with the olderway of
defining Riemann surfaces for use in analysis so let me
brieflygo into it. Having introduced cuts to branch-points and
taken copiesof the cut plane (sheets) I parametrize the Riemann
surface with twovariables (n,z) where n is the sheet and z is
the point on the sheet.Then one comes to the problem of
boundary identifications. I found aflaw in the way this is
treated in the literature and have noted itelsewhere. I also
found a remedy but have kept this to myself. Othersmight find
it for themselves. It does not need great ability, just
adegree of openmindedness.
===
>please don't take too much
comfort from my post. > My reference, if taken as implying
that you might be sympathetic to>my position, seems not to be
welcome. Clarification accepted.>However,back to the bone of
contention.> There may be others besides myself who are
happier with the older>way of defining Riemann surfaces for
use in analysis so let me briefly>go into it. Having
introduced cuts to branch-points and taken copies>of the cut
plane (sheets) I parametrize the Riemann surface with
two>variables (n,z) where n is the sheet and z is the point on
the sheet.>Then one comes to the problem of boundary
identifications. I found a>flaw in the way this is treated in
the literature and have noted it>elsewhere. I also found a
remedy but have kept this to myself. Others>might find it for
themselves. It does not need great ability, just a>degree of
openmindedness.This was _not_ the bone of contention. If you'd
stated that you'rehappier with branch cuts than with the
definition in terms ofcontinuations along paths nobody would
have argued. Thebone of contention was your statement (in a
post which as faras I can see has been cancelled or something,
althoughit's still available at google) that there is
necessarily somearbitrariness in the construction of the
Riemann surfacefor a function. That statement is simply
false.************************David C. Ullrich
===
My
Commentary on Lenny's lecturein the context of my dialogues
with Paul Zielinski and Harold Puthoffon Metric Engineering:
Making Star Trek Real from my third bookin the Space-Time and
Beyond series now being written with fullcolor illustrations
and cartoons.I examine Lenny's ideas to establish where
mainstream cutting edge physics is these days in 2004 as a
proper context to evaluate Hal's and my own fringe ideas
relating to the UFO controversy. For the record I suggested
the problem for Lenny's first published physics paper at
Cornell in 1963 on the the problem of lack of a Hermitian
operator for both time and wave phase in quantum theory. I had
been working on that problemwith George Parrent Jr, a student
of Emil Wolf's, at Tech/Ops associated with Mitre on Route 2
near Boston. I also brought Johnny Glogower to Cornell with
Phil Morrison's help. Johnny was part of Walter Breen's Super
Kids group from Columbia University in a project allegedly
funded by Eugene McDermott a co-founder of Texas Instruments.
We were all rebels . Lenny was a high school dropout. Johnny
was a Quiz Kid, Westinghouse Finalist who flunked out of
Brandeis.Excerpts from:Physics WorldSuperstrings by Leonard
Susskind String theory is either a theory of everything -
whichautomatically unites gravity with the other three forces
innature - or a theory of nothing, but finding the correct
formof the theory is like searching for a needle in a
stupendoushaystackrather like trying to summarize the history
of the world in 10 pages.It is just too large a subject, with
too many lines of thought and toomany threads to weave
together. In the 34 years since it began,string theory has
developed into an enormous body of knowledgethat touches on
every aspect of theoretical physics.String theory is a theory
of composite hadrons,quantum theory of gravity, and a
framework forunderstanding black holes. It is also a
powerfultechnical tool for taming strongly interactingquantum
field theories and, perhaps, a basis forformulating a
fundamental theory of theuniverse. It even touches on problems
incondensed-matter physics, and has alsoprovided a whole new
world of mathematicalproblems and tools. ...String theory is
considered to be a branch of high-energy orthe 1950s, 1960s or
1970s would be surprised to read a recentstring-theory paper
and find not a single Feynman diagram,literature. What the
reader would find are black-hole metrics,Einstein equations,
Kaluza-Klein theories and plenty of fancygeometry and
topology. The energy scales of interest are not MeV,GeV or
even TeV, but energies at the Planck scale - the scale atwhich
the classical concepts of space and time break down.The Planck
energy is equal to h-bar5/G, where h-bar is Planck'sconstant
divided by 2!, c is the speed of light and G is
thegravitational constant, and it corresponds to masses that
are some19 orders of magnitude larger than the proton mass.
There is actually a typo editor's error here, not Lenny's.The
Planck energy ishbar c/Lp = hbar c/(hbarG/c^3)^1/2 = (hbar
c^5/G)^1/2Back to Lenny's talk: This is the energy of the
universe when it was just 10-43s old, and it willunderstand
physics at the Planck scale we need a quantum theoryof
gravity.In the days when my career was beginning, a typical
colloquium onhigh-energy physics would often begin by stating
that there are fourforces in nature - electromagnetic, weak,
strong and gravitational -followed by a statement that the
gravitational force is much toofrom now on. That has all
changed.Today the other three forces are described by the
gauge theories ofquantum chromodynamics (QCD) and quantum
electrodynamicsphysics. These quantum field theories describe
the fundamentalquanta: the photon for the electromagnetic
force, the W and Zbosons for the weak force, and the gluon for
the strong force. Inthe string-theory community, however, the
electromagnetic, strongand weak forces are generally
considered to be manifestations ofcertain compactifications of
space from 10 or 11 dimensions tothe four familiar dimensions
of space-time. ...Why quantum gravity?charge, colour, parity
and hypercharge - to be truly elementary.scale. Protons and
mesons reveal their parts at the modestly smalldistance of
about 10-15 m, but quarks, leptons and photons hidetheir
structure much more effectively. Indeed, no experiment hasever
seen direct evidence of size or structure for any of theseJS
Comment: This point-like structure may be from a huge
space-warp effectdepending on the momentum transfer from
scattering probe to targetfrom an exotic vacuum dark matter
core of the spatially extendedlepto-quarks where the effective
gravity coupling at short range is40 powers of ten greater than
Newton's.Back to Lenny: ... coupling constants are not really
constants at all - they vary withenergy. If the known
couplings are extrapolated they all intersectthe predictions
of the unified theory at roughly the same scale. JS Comment:
This GUT scale is ~ Lp/(alpha) whereLp^2 ~ hG/c^3 and alpha ~
e^2/hc ~ 1/137.Back to Lenny: Moreover, this scale is close to
the Planck scale. The implication ofthis was clear: the scale
of the internal machinery of elementaryappears in the
definition of the Planck energy, to many of us thisinevitably
meant that gravitation must play an essential role inThe
earliest attempts to reconcile gravity and quantum mechanics
-notably by Richard Feynman, Paul Dirac and Bryce DeWitt, who
isnow at the University of Texas at Austin - were based on
trying tofit Einstein's general theory of relativity into a
quantum field theorylike the hugely successful QED. The goal
was to find a set of rulesfor calculating scattering
amplitudes in which the photons of QEDare replaced by the
quanta of the gravitational field: gravitons. Butgravitational
forces become increasingly strong as the energy of
theparticipating quanta increases, and the theory proved to be
wildlysimply gave rise to far too many degrees of freedom at
shortdistances.In a sense the failure of this quantum gravity
theory was a goodsign. The theory itself gave no insight into
the internal machinery offorces of nature. At best it was more
of the same: an effective (butnot very) description of
gravitation with no deeper insight into thenonsense.Strings as
hadronsWe all know that science is full of surprising twists,
but thediscovery of string theory was particularly
serendipitous. The theorygrew out of attempts in the 1960s to
describe the interactions ofneutron. This was a problem that
had nothing to do with gravity.Gabriele Veneziano, now at
CERN, and others had written down asimple mathematical
expression for scattering amplitudes that hadcertain
properties that were fashionable at that time. It was
soondiscovered by Yoichiro Nambu of the University of Chicago
andmyself, and in a slightly different form by Holger Bech
Nielsen atthe Niels Bohr Institute, that these amplitudes were
the solution ofa definite physical system that consists of
extended 1D elasticstrings....Fermionic versions of string
theory were soon discovered and,moreover, they turned out to
have a surprising symmetry calledsupersymmetry that is now
totally pervasive in high-energy physics.In supersymmetric
theories all bosons have a fermionicsuperpartner and vice
versa. ...Another apparently serious problem with the string
theory ofhadrons concerned dimensions. Although the original
assumptions instring theory were simple enough, the
mathematics provedinternally inconsistent, at least if the
number of dimensions ofspace-time was four. The source of this
problem was quite deep,but, strangely, if space-time has 10
dimensions it contrives tocancel out. ...A mathematical string
can vibrate in many patterns,angular momentum ( spin-two ).
There are certainly spin-twohadrons, but none that have
anything like zero mass. Despite allmade massive....A massless
spin-two field might not be good for hadronic physics, butit is
just what was needed for quantum gravity, albeit in 10D. This
isbecause just as the photon is the quantum of the
electromagneticfield, the graviton is the quantum of the
gravitational field. But thegravitational field is a symmetric
tensor rather than a vector, andthis means the graviton is
spin-two, rather than spin-one like thephoton. This difference
in spin is the principal reason why earlyattempts to quantize
gravity based on QED did not work.A theory of everything,...
either all matter is strings, or stringtheory is wrong. This
is one of the most exciting features of thetheory.But what
about the problem of dimensions? Here again, a sow's earwas
turned into a silk purse. The basic idea goes back to
TheodorKaluza in 1919, who tried to unify Einstein's
gravitational theorywith electrodynamics by introducing a
compact space-like fifthdimension. Kaluza discovered the
beautiful fact that the extracomponents of the gravitational
field tensor in 5 dimensionsbehaved exactly like the
electromagnetic field plus one additionalscalar field.
Somewhat later, in 1938, Oskar Klein and thenWolfgang Pauli
generalized Kaluza's work so that the single compactdimension
was replaced by a 2D space. If the 2D space is thesurface of a
sphere then a remarkable thing happens when Kaluza'sprocedure
is followed. Instead of electrodynamics, Klein and
Paulidiscovered the first non-Abelian gauge theory, which was
laterrediscovered by Chen Ning Yang and Robert Mills. This is
exactly thesame class of theories that is so successful in
describing the strongand electromagnetic interactions in the
Standard Model. JS Comment: A 2D Kaluza-Klein space has group
structureof a 2D sphere embedded in flat 3D space with 3
rotation charge generators,i.e. SU(2) group for the weak force
with 3 charges.In general we have N^2 - 1 charges for the SU(N)
internal symmetry gauge forcegroupat a fixed space-time point
where the minimal coupling local independence of
phaserotations introduces the compensating spin 1 gauge force
fields to restore the brokenglobal symmetry. This force
generator idea is re-expressed in the geometrodynamics of
hyperspace.Here's Lenny:that appears to be standing still in
our usual 3Dspace have velocity or momentum componentsin the
compact dimensions? The answer is yes,and the corresponding
components ofmomentum define new conserved quantities. What
ismore, these quantities are quantized in discrete units. In
short, theyare charges similar to electric charge, isospin and
all the otherthe problem of dimensions in string theory is
obvious: six of the 10dimensions should be wrapped up into
some very small compactspace, and the corresponding quantized
components of momentadetermines their quantum numbers.Life in
six dimensionsMuch of the development of string theory is
therefore concernedwith 6D spaces. These spaces, which can be
thought of asgeneralized Kaluza-Klein compactification spaces,
were originallystudied by mathematicians and are known as
Calabi-Yau spaces.They are tremendously complicated and are
not completelyunderstood. But in the process of studying how
strings move onthem, physicists have created an unexpected
revolution in the studyof Calabi-Yau spaces. JS Comment:
Recall that the classical gravity radius is proportional to M
and thequantum radius is proportional to 1/M. That isRg =
GM/c^2Rq = h/McThereforeRgRq = Gh/c^3 = Lp^2 = 1 Bekenstein
BIT.We have a germ of a duality between black holes and
quantum momentaRg = Lp^2/RqNote also the Blackett empirical
relatione = G*^1/2Mwhere for an electronG* ~ 10^40GThe quantum
momenta p in the compactified extra-dimensions are charges Q
(sources of the spin 1 gauge forces) where by the Blackett
relationQ/G*^1/2 = M = h/cRQ = G*^1/2h/cRR is a
compactification scale.G* = e^phiG(Newton)Back to Lenny In
particular, it was discovered that a compactification radius
of sizeR is completely equivalent to a space with size 1/R
from the pointof view of string theory. This connection, which
is known asT-duality, has a mathematically profound
generalization calledmirror symmetry, which states that there
is an equivalence betweensmall and large spaces ... . Mirror
symmetry ofCalabi-Yau spaces - which are not only of different
sizes but havecompletely different topologies - was completely
unsuspectedbefore physicists began studying quantum strings
moving on them.I wish it was possible to draw a Calabi-Yau
space but they aretremendously complicated. They are
six-dimensional, which is threemore than I can visualize, and
they have very complicatedtopologies, including holes, tunnels
and handles. Furthermore, thereare thousands of them, each with
a different topology. And evenwhen their topology is fixed
there are hundreds of parameterscalled moduli that determine
the shape and size of the variousdimensions. Indeed, it is the
complexity of Calabi-Yau geometrythat makes string theory so
intimidating to an outsider. However,we can abstract a few
useful things from the mathematics, one ofthem being the idea
of moduli.The simplest example of a modulus is just the
compactificationradius, R, when there is only a single compact
dimension. In morecomplicated cases, the moduli determine the
sizes and shapes ofthe various features of the geometry. The
moduli are not constantsbut depend on the geometry of the
space itself, in the same waythat the radius of the universe
changes with time in a manner thatis controlled by dynamical
equations of motion. Since the compactdimensions are too small
to see, the moduli can simply be thoughtof as fields in space
that determine the local conditions. Electric andmagnetic
fields are examples of such fields but the moduli are
evensimpler: they are scalar fields (i.e. they have only one
component),rather than vector fields. String theory always has
lots ofscalar-field moduli and these can potentially play
important roles inAll of this raises an interesting question:
what determines thecompactification moduli in the real world
of experience? Is theresome principle that selects a special
value of the moduli of aparticular Calabi-Yau space and
therefore determines thecoupling constants of the forces, and
so on? The answer seems tobe no: all values of the moduli
apparently give rise tomathematically consistent theories.
Whether or not this is a goodthing, it is certainly
surprising.Ordinarily we might expect the vacuum or ground
state of the worldto be the state of lowest energy.
Furthermore, in the absence ofvery special symmetries, the
energy of a region of space willdepend non-trivially on the
values of the fields in that region.Finding the true vacuum is
then merely an exercise in computingthe energy for a given
field configuration and minimizing it. This is,to be sure, a
difficult task, but it is possible in principle. In
stringtheory, however, we know from the beginning that the
potentialenergy stored in a given configuration has no
dependence on themoduli fields.The reason that the field
potential is exactly zero for every value ofthe moduli is that
string theory is supersymmetric. Supersymmetryhas both
desirable and undesirable consequences. Its most
obviousdrawback is the requirement that for every fermion
there is a bosonwith exactly the same mass, which is clearly
not a property of ourworld.A more subtle difficulty involves
the aforementioned fact that thevacuum energy is independent
of the moduli. As well as telling usthat we cannot determine
the moduli by minimizing the energy,supersymmetry also tells
us that the quanta of the moduli fields areexactly massless.
No such massless fields are known in nature and,furthermore,
such fields are very dangerous. Indeed, masslessmoduli would
probably lead to long-range forces that wouldcompete with
gravity and violate the equivalence principle - thecornerstone
of general relativity - at an observable level.On the plus
side, the vanishing vacuum energy that is implied
bysupersymmetry ensures that the cosmological constant
vanishes. Ifit were not for supersymmetry, the vacuum would
have a hugezero-point energy density that would make the
radius of curvatureof space-time not much bigger than the
Planck scale - a mostundesirable situation. JS Comment: I have
a different much simpler explanation for thesmallness of the
cosmological constant
inhttp://qedcorp.com/APS/EmergentGravity.pdfAlso the
observational fact of dark energy with FRW Omega ~ 0.7means
that the cosmological constant is not exactly zero, which isa
problem for the physics Lenny is talking about.Back to Lenny:
Supersymmetry also stabilizes the vacuumagainst various
hypothetical instabilities, and it allows us to makeexact
mathematical conclusions. Indeed, T-duality and mirrorsymmetry
are examples of those exact consequences.Black holesThroughout
the 1980s and early 1990s progressin string theory largely
consisted of working outthe detailed rules of perturbation
theory for thefive known versions of the theory, which
wouldallow theorists to arrive at actual solutions(figure 2).
These perturbative rules weregeneralizations of the Feynman
diagrams of QEDand QCD in which the world lines of pointmoving
strings. The study of world-sheet physics created a hugebody of
knowledge about 2D quantum field theory, but it did notoffer
much insight into the inner workings of quantum gravity.
Atbest, string theory provided an especially consistent way
tointroduce a small distance scale and thereby regulate
thedivergences that had plagued the older attempts at
quantizinggravity.Personally I found the whole enterprise dry,
overly technical and,above all, disappointing. I felt that a
quantum theory of gravityshould profoundly affect our views of
space-time, quantummechanics, the origin of the universe, and
the mysteries of blackholes. But string theory was largely
silent about all these matters.Then in 1993 all this began to
change, and the catalyst was theawakening interest in Stephen
Hawking's earlier speculations aboutblack holes.The starting
point for Hawking's speculations was the thermalbehaviour of
black holes, which built on earlier work by JacobBekenstein of
the Hebrew University in Israel. Rather than the cold,dead
objects that they were originally thought to be, black
holesturned out to have a heat content and to glow like black
bodies.Because they glow they lose energy and evaporate, and
becausethey have a temperature and an energy, they also have
an entropy.This entropy, S, is defined by the
Bekenstein-Hawking equation: S= AkBc3/4h-barG, where A is the
surface area of the horizon andkBis Boltzmann's constant.After
realizing that black holes must evaporate by the emission
ofblack-body radiation, Hawking raised an extremely
profoundquestion: what happens to all the detailed information
that falls intoa black hole? Once it falls through the horizon
it cannotsubsequently reappear on the outside without
violating causality.That is the meaning of a horizon. But the
black hole will eventuallyevaporate, leaving only photons,
gravitons and other elementaryinformation must ultimately be
lost to our world. But one of thefundamental principles of
quantum mechanics is that information isnever lost, because
the information in the initial state of a quantumsystem is
permanently imprinted in the quantum state.Hawking's view was
that conventional quantum mechanics must beviolated during the
formation and evaporation of the black hole. Herightly
understood that if this is true, the rules of quantummechanics
must be drastically modified as the Planck scale isfor unified
theories, should have been obvious. But initiallyHawking's
idea generated little interest among high-energytheorists,
apart from myself and Gerard 't Hooft at the University
ofUtrecht. We were convinced that by modifying the rules of
quantummechanics in the way advocated by Hawking, all hell
would breakloose, such as causing empty space to quickly heat
up tostupendous temperatures and energy densities. We were
sure thatHawking was wrong. By the early 1990s, however, the
issue wasbecoming critical, especially to string theorists.
String theory by itsvery definition is based on the
conventional rules of quantummechanics and if Hawking was
right, the entire foundation of thetheory would be destroyed.
JS Comment: I tend to disagree with Lenny and t'Hooft that the
unitarity of micro-quantum theory is an absolute. P.W.
Anderson's More is different suggests otherwise. Think of the
relation between special relativity and general relativity --
similarly with micro-quantum theory and MACRO-quantum theory
of superfluids.4D space-time is a non-dynamical absolute in
special relativity. It acts on mass-energy without any direct
reaction of mass-energy back on it. This is because the string
tension is infinite in that limit. Special relativity is action
without reaction. General relativity corrects that giving a
finite value to string tension? How? Because general
relativity emerges from a macro-quantum theory as shown
inhttp://qedcorp.com/APS/EmergentGravity.pdf The finite string
tension in Ed Witten's sense of alpha' is actually a quantum h
effect added to G and c.& 14.6, micro-quantum theory is like
special relativity because the quantum BIT pilot wave is a
non-dynamical absolute. It acts on the IT extra variable
without any direct reaction of IT back on its quantum BIT.That
is, in Wheeler's terms, micro-quantum theory isIT FROM BITIn
contrast, MACRO-quantum theory adds to thatBIT FROM
IT.Micro-quantum theory is linear and nonlocal in
configuration space for entangled composite systems with
unitary time evolution and a probability interpretation in
Lenny's sense but with signal locality in a detente passion at
a distance (A. Shimony) with retarded causality.In contrast,
MACRO-quantum theory with superfluid signal generalized phase
rigidity (e.g. string tension) is nonlinear (Landau-Ginzburg
eq.) and local in ordinary (hyper) space with non-unitary time
evolution and a complete breakdown of the Born probability
interpretation. Also it allows signal nonlocality violating
retarded causality. Micro-quantum theory still works for the
normal fluid noisy component.Back to Lenny: Over the last
decade the apparent clash between standard quantumprinciples
and black-hole evaporation has been resolved, favouring,I
should add, the views of 't Hooft and myself. The formation
andevaporation of a black hole is similar to many other
process inand chaotic spectrum of intermediate states. In the
case of a blackhole, the collisions are between the original
protons, neutrons andelectrons in a collapsing star. Roughly
speaking a black hole isnothing but a very excited string with
a total length that isproportional to the area of its horizon.
JS Comment. Already in 1973 I published a paper in Herbert
Frohlich's Collective Phenomena that the Regge string hadronic
trajectoriesshowed that the hadronic resonances were tiny black
holes inAbdus Salam's strong short-range f-gravity with G* ~
10^40 G(Newton).Spin ~ G*E^2/hc^5 + interceptG*/hc^5 ~ (String
Tension)^-1G*/hc^5 ~ (1 Gev)^-2 UNIVERSAL SLOPE
(micro-geometrodynamics)The decay of the hadrons would be
likeHawking radiation. Abdus Salam invited me to work with him
atTrieste because of this paper. We now see that this idea I
had before its timewas essentially on the right path.Back to
Lenny During the collision orcollapse process, all the energy
of the initial state goes into forminga single long, tangled
string, and the entropy of the configuration isthe logarithm
of the number of configurations of a random-walkingquantum
string.The correspondence between string configurations and
black-holeentropy was checked for all of the various kinds of
charged andneutral black holes that occur in compactifications
of string theory.In most of the cases the entropy of the string
configuration could beestimated and it agreed with the
Bekenstein-Hawking entropy towithin a factor of order
unity.But string theorists wanted to do better. The
Bekenstein-Hawkingformula for the entropy of a black hole is
very precise: the entropyis one quarter of the horizon area,
measured in Planck units, forevery kind of black hole, be it
static, rotating, charged or evenhigher-dimensional. Surely
the universal factor of a quarter shouldbe computable in
string theory? The key to a precise calculation wasobvious.
Certain black holes called extremal black holes - which arethe
ground states of charged black holes that carry electric
andmagnetic charges - are especially tractable in a
supersymmetrictheory. The only problem was that in 1993 no-one
knew how tobuild an extremal black hole out of the right type
out of strings.This had to wait a couple of years for the
discovery of entitiescalled D-branes.Brane worldIn 1995 Joe
Polchinski of the University of California in SantaBarbara
electrified the string-theory community with a majordiscovery
that has subsequently impacted every field of physics. Aswe
have seen, T-duality is the strange symmetry that
interchangesthe Kaluza-Klein momenta and winding numbers of a
closed string.But what happens to an open string? Obviously
theidea of a winding number does not make sense for such a
string.What actually happens to open strings under T-duality
is that thefree ends become fixed on surfaces called
D-branes.D-branes come in various dimensions; 2Dbranes, for
example, can also be calledmembranes. They have an energy
ormass per unit surface area and are localizedphysical objects
in their own right. In a sensethey seem to be no less
fundamental than thestrings themselves. To an outsider,
D-branes may seem to bearbitrary additions to the theory. They
are not. Their existence isabsolutely essential to the
mathematical consistency of the theory.In addition to allowing
T-duality to act on an open string in Type Istring theory, they
are necessary for implementing the deepdualities that link the
five different kinds of string theory together.But from the
point of view of black holes, the importance ofD-branes is
that you can build extremal black holes from them. Infact,
just by placing a large number of D-branes at the samelocation
you can build an extremal supersymmetric black hole. Andbecause
of the special properties of supersymmetric systems,
thestatistical entropy of that black hole can be precisely
computed. Theresult, which was first derived by Andrew
Strominger and CumrunVafa at Harvard in 1996, is that the
entropy is equal to exactly onequarter of the horizon area in
Planck units! This suggested that themicroscopic degrees of
freedom implied by the Bekenstein-Hawkingentropy are the
degrees of freedom describing strings, and was amajor boost
for the superstring community.At about the same time as
D-branes were discovered, another veryimportant development
took place. As I mentioned, the couplingconstant of string
theory is not really a constant at all, and in manyrespects it
is very similar to the compactification moduli. Stringtheorists
took a surprisingly long time to make the connection, butit
turns out that 10D string theory is itself a
Kaluza-Kleincompactification of an 11D theory that became
known as M-theory .M-theory appears to underlie all string
theories. The fivedifferent versions of string theory are just
different ways ofcompactifying its 11 dimensions. But M-theory
is not itself a stringtheory. It has membranes but no strings,
and the strings onlyappear when the 11th dimension is
compactified. Furthermore, themomentum in the compact 11th
direction (the Kaluza-Kleinmomentum) is identified as the
number of D0-branes - i.e.zero-dimensional branes, or points -
in a particular type of stringtheory.This connection between
Kaluza-Klein momentum and D0-branesled to another
breakthrough. In 1996 myself, Tom Banks and SteveShenker (at
Rutgers University), and Willy Fischler (at the Universityof
Texas) realized that M-theory could be cast in a form no
morecomplicated than the quantum mechanics of a system ofis
called Matrix theory, is an exact and complete quantum
theorythat describes the microscopic degrees of freedom of
M-theory. Assuch it is the first precise formulation of a
quantum theory ofgravity.DualityMatrix theory was just one
example of how D-branes can be used toformulate a theory of
quantum gravity. Soon after its discovery,Juan Maldacena, who
is now at the Institute for Advanced Study(IAS) in Princeton,
came up with a new direction to explore. EdWitten of the IAS
and others had previously shown that D-braneshave their own
dynamics. But it turned out that the fluctuations andmotions
of a D-brane can be quantized in the form of a gaugetheory
that is restricted to the D-brane itself. The theory that
liveson a coincident collection of D3-branes, for example, is
asupersymmetric non-Abelian gauge theory. In other words, it
is asupersymmetric version of QCD - the theory describing
quarks andgluons. In a sense, string theory is returning to
its roots as app35-38).Maldacena realized that in an
appropriate limitthe theory of D3-branes should be a
completedescription of string theory - not just on thebranes,
but in the entire geometry in which thebranes are embedded. A
gauge theory wouldtherefore also be a description of
quantumgravity in a particular background space-time.This
space-time is called anti-de Sitter space,which, roughly
speaking, is a universe inside acavity. The walls of the
cavity behave likereflecting surfaces so that nothing escapes
it (figure 4).This duality between quantum field theory and
gravity is an exactrealization of what is called the
holographic principle. This strangeprinciple, formulated by 't
Hooft and myself, grew from our debatewith Hawking regarding
the validity of quantum mechanics in theformation and
evaporation of black holes.According to the holographic
principle, everything that ever falls intoa black hole can be
described by degrees of freedom that reside ina thin layer
just above the horizon. In other words, the full 3D
worldinside the horizon can be described by the 2D degrees of
freedomon its surface. Even more generally, it should be
possible todescribe the physics of any region of space in
terms of holographicdegrees of freedom that reside on the
boundary of that region. Thisleads to a drastic reduction of
the number of degrees of freedom ina field theory, and most
theorists found it very hard to swallow untilMaldacena's work
came along. Maldacena's duality replaces agravitational theory
in anti-de Sitter space by a field theory thatlives on its
boundary in a very precise way. In other words, the 3
+1-dimensional boundary field theory is a holographic
description ofthe interior of 4 + 1-dimensional anti-de Sitter
space.The D-brane revolution has been very far reaching. Matrix
theoryand the Maldacena duality are both formulations of
quantum gravitythat conform to the standard rules of quantum
mechanics, andshould therefore lay to rest any further
questions about black holesviolating these rules.Googles of
possibilitiesI would like to end by discussing the future of
string theory, not ascosmology. The final evaluation of string
theory will rest on itsability to explain the facts of nature,
not on its own internal beautyand consistency. String theory
is well into its fourth decade, but soconvincing explanation
of any cosmological observation. Many of themodels that are
based on specific methods of compactifying either10D string
theory or 11D M-theory have a good deal in commonwith the real
world. They have bosons and fermions, for example,and gauge
theories that are similar to those in the Standard
Model.Furthermore, unlike any other theory, they inevitably
includegravity. But the devil is in the details, and so far
the details haveeluded string theorists.It is, of course,
possible that string theory is the wrong theory, but Ibelieve
that would be a very premature judgement and
probablyincorrect. The problem does not seem to be a lack of
richness, butrather the opposite. String theory contains too
many possibilities.For most physicists, the ideal physical
theory is one that is uniqueand perfect, in that it determines
all that can be determined andthat it could not logically be
any other way. In other words, it is notonly a theory of
everything but it is the only theory of everything.To the
orthodox string theorist, the goal is to discover the one
trueconsistent version of the theory and then to demonstrate
that thesolution manifests the known laws of nature, such as
the StandardBut the more we learn about string theory the more
non-unique itseems to be. There are probably millions of
Calabi-Yau spaces onwhich to compactify string theory. Each
space has hundreds ofmoduli and hundreds of subspaces on which
branes can bewrapped, fluxes imposed upon and so on. A
conservative estimateof the number of distinct vacua of the
theory is in the googles - thatis, more than 10100. The space
of possibilities is called theLandscape, and it is huge. To
mix metaphors, it is a stupendoushaystack that contains
googles of straws and only one needle.Worse still, the theory
itself gives us no hint about how to pickamong the
possibilities (see The string-theory landscape ).This enormous
variety may, however, be exactly what cosmology islooking for.
A common theme among cosmologists is that theobserved universe
may merely be a minuscule part of a vastlybigger universe that
contains many local environments, or whatAlan Guth at MIT
calls pocket universes . According to this view, somany pocket
universes formed during the early inflationary epoch -each of
which with its own vacuum structure - that the entirelandscape
of possibilities is represented. The reasons for this vieware
not just idle speculation but are rooted in the many
accidentalfine-tunings that are necessary for a universe that
supports life.Thus it may be that the enormous number of
possible vacuumthe doctor ordered for cosmology.Further
informationT-dualityIn a single compact dimension there are
two kinds of quantumnumbers: momentum in the compact direction
and the windingnumber. Both of these are quantized in integer
multiples of a basicunit, and each has a certain energy
associated with it. In the caseof momentum, for example, the
energy is just the kinetic energy ofunits of compact momentum
is equal to n/R, where R is thecircumference of the compact
direction. Note that the energy growsas the size of the
compact space gets smaller. On the other hand,the winding
modes also have energy, which is the potential energyneeded to
stretch the string around the compact co-ordinate. If wecall
the winding number m, then the winding energy is equal to
mR.In this case the energy decreases as the size of the
compactdirection decreases.The surprising thing is that the
spectrum of energies is unchanged ifwe change the
compactification radius from R to 1/R, and at thesame time
interchange the Kaluza-Klein momentum and windingmodes. In
other words, just by looking at the spectrum of energiesyou
could never tell the difference between a theory that
iscompactified on a space of size R or on one of size 1/R. As
you tryto make the compactification scale smaller than the
natural stringscale - i.e. the size of a vibrating string -
the theory begins tobehave as if the compactification radius
was getting bigger.Physically, the smallest compactification
value of R is the stringscale. But from a mathematical
viewpoint, two different spaces -one large, the other small -
are completely equivalent. Thisequivalence is called
T-duality.AuthorLeonard Susskind is in the Department of
Physics, Stanfordsusskind@stanford.eduFurther readingJ
Maldacena 1999 The large N limit of superconformal field
theoriesand supergravity Int. J. Theor. Phys. 38 1113-1133J
Polchinski 1995 Dirichlet-branes and Ramond-Ramond
chargesPhys. Rev. Lett. 75 4724-4727J Polchinski 1998 String
Theory (volume 2): Superstring Theory andBeyond (Cambridge
University Press)J H Schwarz et al. 1981 Superstring Theory
(volume 1): Introduction(Cambridge University Press)A
Strominger and C Vafa 1996 Microscopic origin of
theBekenstein-Hawking entropy Phys. Lett. B 379 99-104The
official string theory website: superstringtheory.com/ Note
above my new Blackett formula for charge Q in the Calabi-Yau
space, i.e.Q = G*^1/2h/cRR is a compactification scale modulus
in the extra-dimensional generalized Kaluza-Klein space.G* =
e^phiG(Newton)See Saul-Paul Sirag's Nature paper on Blackett
effect in astrophysics.
===
> My Commentary on Lenny's lecture
in the context of my dialogues with Paul Zielinski and Harold
Puthoff> on Metric Engineering: Making Star Trek Real from my
third book> in the Space-Time and Beyond series now being
written with full> color illustrations and cartoons.> I
examine Lenny's ideas to establish where mainstream cutting
edge> physics is these days in 2004 as a proper context to
evaluate Hal's and> my own fringe ideas relating to the UFO
controversy. For the record I> suggested the problem for
Lenny's first published physics paper at> Cornell in 1963 on
the the problem of lack of a Hermitian operator for> both time
and wave phase in quantum theory. I had been working on that>
problem> with George Parrent Jr, a student of Emil Wolf's, at
Tech/Ops associated> with Mitre on Route 2 near Boston. I also
brought Johnny Glogower to> Cornell with Phil Morrison's help.
Johnny was part of Walter Breen's> Super Kids group from
Columbia University in a project allegedly> funded by Eugene
McDermott a co-founder of Texas Instruments. We were> all
rebels . Lenny was a high school dropout. Johnny was a Quiz
Kid,> Westinghouse Finalist who flunked out of Brandeis.>
Excerpts from:> Physics World> Superstrings by Leonard
Susskind> String theory is either a theory of everything -
which> automatically unites gravity with the other three
forces in> nature - or a theory of nothing, but finding the
correct form> of the theory is like searching for a needle in
a stupendous> haystack> rather like trying to summarize the
history of the world in 10 pages.> It is just too large a
subject, with too many lines of thought and too> many threads
to weave together. In the 34 years since it began,> string
theory has developed into an enormous body of knowledge> that
touches on every aspect of theoretical physics.> String theory
is a theory of composite hadrons,> quantum theory of gravity,
and a framework for> understanding black holes. It is also a
powerful> technical tool for taming strongly interacting>
quantum field theories and, perhaps, a basis for> formulating
a fundamental theory of the> universe. It even touches on
problems in> condensed-matter physics, and has also> provided
a whole new world of mathematical> problems and tools. ...>
String theory is considered to be a branch of high-energy or>
the 1950s, 1960s or 1970s would be surprised to read a recent>
string-theory paper and find not a single Feynman diagram,>
literature. What the reader would find are black-hole
metrics,> Einstein equations, Kaluza-Klein theories and plenty
of fancy> geometry and topology. The energy scales of interest
are not MeV,> GeV or even TeV, but energies at the Planck
scale - the scale at> which the classical concepts of space
and time break down.> The Planck energy is equal to h-bar5/G,
where h-bar is Planck's> constant divided by 2!, c is the
speed of light and G is the> gravitational constant, and it
corresponds to masses that are some> 19 orders of magnitude
larger than the proton mass. > There is actually a typo
editor's error here, not Lenny's.> The Planck energy is>
hbar c/Lp = hbar c/(hbarG/c^3)^1/2 = (hbar c^5/G)^1/2> Back
to Lenny's talk:> This is the energy of the universe when it
was just 10-43s old, and itwill> understand physics at the
Planck scale we need a quantum theory> of gravity.> In the
days when my career was beginning, a typical colloquium on>
high-energy physics would often begin by stating that there
are four> forces in nature - electromagnetic, weak, strong and
gravitational -> followed by a statement that the gravitational
force is much too> from now on. That has all changed.> Today
the other three forces are described by the gauge theories of>
quantum chromodynamics (QCD) and quantum electrodynamics>
physics. These quantum field theories describe the
fundamental> quanta: the photon for the electromagnetic force,
the W and Z> bosons for the weak force, and the gluon for the
strong force. In> the string-theory community, however, the
electromagnetic, strong> and weak forces are generally
considered to be manifestations of> certain compactifications
of space from 10 or 11 dimensions to> the four familiar
dimensions of space-time. ...> Why quantum gravity?> charge,
colour, parity and hypercharge - to be truly elementary.>
scale. Protons and mesons reveal their parts at the modestly
small> distance of about 10-15 m, but quarks, leptons and
photons hide> their structure much more effectively. Indeed,
no experiment has> ever seen direct evidence of size or
structure for any of these> JS Comment: This point-like
structure may be from a huge space-warp effect> depending on
the momentum transfer from scattering probe to target> from an
exotic vacuum dark matter core of the spatially extended>
lepto-quarks where the effective gravity coupling at short
range is> 40 powers of ten greater than Newton's.> Back to
Lenny:> ... coupling constants are not really constants at all
- they vary with> energy. If the known couplings are
extrapolated they all intersect> the predictions of the
unified theory at roughly the same scale. > JS Comment: This
GUT scale is ~ Lp/(alpha) where> Lp^2 ~ hG/c^3 and alpha ~
e^2/hc ~ 1/137.> Back to Lenny:> Moreover, this scale is
close to the Planck scale. The implication of> this was clear:
the scale of the internal machinery of elementary> appears in
the definition of the Planck energy, to many of us this>
inevitably meant that gravitation must play an essential role
in> The earliest attempts to reconcile gravity and quantum
mechanics -> notably by Richard Feynman, Paul Dirac and Bryce
DeWitt, who is> now at the University of Texas at Austin -
were based on trying to> fit Einstein's general theory of
relativity into a quantum field theory> like the hugely
successful QED. The goal was to find a set of rules> for
calculating scattering amplitudes in which the photons of QED>
are replaced by the quanta of the gravitational field:
gravitons. But> gravitational forces become increasingly
strong as the energy of the> participating quanta increases,
and the theory proved to be wildly> simply gave rise to far
too many degrees of freedom at short> distances.> In a sense
the failure of this quantum gravity theory was a good> sign.
The theory itself gave no insight into the internal machinery
of> forces of nature. At best it was more of the same: an
effective (but> not very) description of gravitation with no
deeper insight into the> nonsense.> Strings as hadrons> We
all know that science is full of surprising twists, but the>
discovery of string theory was particularly serendipitous. The
theory> grew out of attempts in the 1960s to describe the
interactions of> neutron. This was a problem that had nothing
to do with gravity.> Gabriele Veneziano, now at CERN, and
others had written down a> simple mathematical expression for
scattering amplitudes that had> certain properties that were
fashionable at that time. It was soon> discovered by Yoichiro
Nambu of the University of Chicago and> myself, and in a
slightly different form by Holger Bech Nielsen at> the Niels
Bohr Institute, that these amplitudes were the solution of> a
definite physical system that consists of extended 1D elastic>
strings.> ...> Fermionic versions of string theory were soon
discovered and,> moreover, they turned out to have a
surprising symmetry called> supersymmetry that is now totally
pervasive in high-energy physics.> In supersymmetric theories
all bosons have a fermionic> superpartner and vice versa.
...> Another apparently serious problem with the string
theory of> hadrons concerned dimensions. Although the original
assumptions in> string theory were simple enough, the
mathematics proved> internally inconsistent, at least if the
number of dimensions of> space-time was four. The source of
this problem was quite deep,> but, strangely, if space-time
has 10 dimensions it contrives to> cancel out. ...> A
mathematical string can vibrate in many patterns,> angular
momentum ( spin-two ). There are certainly spin-two> hadrons,
but none that have anything like zero mass. Despite all> made
massive.> ...> A massless spin-two field might not be good for
hadronic physics, but> it is just what was needed for quantum
gravity, albeit in 10D. This is> because just as the photon is
the quantum of the electromagnetic> field, the graviton is the
quantum of the gravitational field. But the> gravitational
field is a symmetric tensor rather than a vector, and> this
means the graviton is spin-two, rather than spin-one like the>
photon. This difference in spin is the principal reason why
early> attempts to quantize gravity based on QED did not
work.> A theory of everything,> ... either all matter is
strings, or string> theory is wrong. This is one of the most
exciting features of the> theory.> But what about the problem
of dimensions? Here again, a sow's ear> was turned into a silk
purse. The basic idea goes back to Theodor> Kaluza in 1919,
who tried to unify Einstein's gravitational theory> with
electrodynamics by introducing a compact space-like fifth>
dimension. Kaluza discovered the beautiful fact that the
extra> components of the gravitational field tensor in 5
dimensions> behaved exactly like the electromagnetic field
plus one additional> scalar field. Somewhat later, in 1938,
Oskar Klein and then> Wolfgang Pauli generalized Kaluza's work
so that the single compact> dimension was replaced by a 2D
space. If the 2D space is the> surface of a sphere then a
remarkable thing happens when Kaluza's> procedure is followed.
Instead of electrodynamics, Klein and Pauli> discovered the
first non-Abelian gauge theory, which was later> rediscovered
by Chen Ning Yang and Robert Mills. This is exactly the> same
class of theories that is so successful in describing the
strong> and electromagnetic interactions in the Standard
Model. > JS Comment: A 2D Kaluza-Klein space has group
structure> of a 2D sphere embedded in flat 3D space with 3
rotation charge > generators,> i.e. SU(2) group for the weak
force with 3 charges.> In general we have N^2 - 1 charges for
the SU(N) internal symmetry> gauge forcegroup> at a fixed
space-time point where the minimal coupling local>
independence of phase> rotations introduces the compensating
spin 1 gauge force fields to> restore the broken> global
symmetry. This force generator idea is re-expressed in the>
geometrodynamics of hyperspace.> Here's Lenny:> that appears
to be standing still in our usual 3D> space have velocity or
momentum components> in the compact dimensions? The answer is
yes,> and the corresponding components of> momentum define new
conserved quantities. What is> more, these quantities are
quantized in discrete units. In short, they> are charges
similar to electric charge, isospin and all the other> the
problem of dimensions in string theory is obvious: six of the
10> dimensions should be wrapped up into some very small
compact> space, and the corresponding quantized components of
momenta> determines their quantum numbers.> Life in six
dimensions> Much of the development of string theory is
therefore concerned> with 6D spaces. These spaces, which can
be thought of as> generalized Kaluza-Klein compactification
spaces, were originally> studied by mathematicians and are
known as Calabi-Yau spaces.> They are tremendously complicated
and are not completely> understood. But in the process of
studying how strings move on> them, physicists have created an
unexpected revolution in the study> of Calabi-Yau spaces. > JS
Comment: Recall that the classical gravity radius is
proportional to> M and the> quantum radius is proportional to
1/M. That is> Rg = GM/c^2> Rq = h/Mc> Therefore> RgRq =
Gh/c^3 = Lp^2 = 1 Bekenstein BIT.> We have a germ of a
duality between black holes and quantum momenta> Rg =
Lp^2/Rq> Note also the Blackett empirical relation> e =
G*^1/2M> where for an electron> G* ~ 10^40G> The quantum
momenta p in the compactified extra-dimensions are> charges Q
(sources of the spin 1 gauge forces) where by the Blackett>
relation> Q/G*^1/2 = M = h/cR> Q = G*^1/2h/cR> R is a
compactification scale.> G* = e^phiG(Newton)> Back to Lenny>
In particular, it was discovered that a compactification radius
of size> R is completely equivalent to a space with size 1/R
from the point> of view of string theory. This connection,
which is known as> T-duality, has a mathematically profound
generalization called> mirror symmetry, which states that
there is an equivalence between> small and large spaces ... .
Mirror symmetry of> Calabi-Yau spaces - which are not only of
different sizes but have> completely different topologies -
was completely unsuspected> before physicists began studying
quantum strings moving on them.> I wish it was possible to
draw a Calabi-Yau space but they are> tremendously
complicated. They are six-dimensional, which is three> more
than I can visualize, and they have very complicated>
topologies, including holes, tunnels and handles. Furthermore,
there> are thousands of them, each with a different topology.
And even> when their topology is fixed there are hundreds of
parameters> called moduli that determine the shape and size of
the various> dimensions. Indeed, it is the complexity of
Calabi-Yau geometry> that makes string theory so intimidating
to an outsider. However,> we can abstract a few useful things
from the mathematics, one of> them being the idea of moduli.>
The simplest example of a modulus is just the compactification>
radius, R, when there is only a single compact dimension. In
more> complicated cases, the moduli determine the sizes and
shapes of> the various features of the geometry. The moduli
are not constants> but depend on the geometry of the space
itself, in the same way> that the radius of the universe
changes with time in a manner that> is controlled by dynamical
equations of motion. Since the compact> dimensions are too
small to see, the moduli can simply be thought> of as fields
in space that determine the local conditions. Electric and>
magnetic fields are examples of such fields but the moduli are
even> simpler: they are scalar fields (i.e. they have only one
component),> rather than vector fields. String theory always
has lots of> scalar-field moduli and these can potentially
play important roles in> All of this raises an interesting
question: what determines the> compactification moduli in the
real world of experience? Is there> some principle that
selects a special value of the moduli of a> particular
Calabi-Yau space and therefore determines the> coupling
constants of the forces, and so on? The answer seems to> be
no: all values of the moduli apparently give rise to>
mathematically consistent theories. Whether or not this is a
good> thing, it is certainly surprising.> Ordinarily we might
expect the vacuum or ground state of the world> to be the state
of lowest energy. Furthermore, in the absence of> very special
symmetries, the energy of a region of space will> depend
non-trivially on the values of the fields in that region.>
Finding the true vacuum is then merely an exercise in
computing> the energy for a given field configuration and
minimizing it. This is,> to be sure, a difficult task, but it
is possible in principle. In string> theory, however, we know
from the beginning that the potential> energy stored in a
given configuration has no dependence on the> moduli fields.>
The reason that the field potential is exactly zero for every
value of> the moduli is that string theory is supersymmetric.
Supersymmetry> has both desirable and undesirable
consequences. Its most obvious> drawback is the requirement
that for every fermion there is a boson> with exactly the same
mass, which is clearly not a property of our> world.> A more
subtle difficulty involves the aforementioned fact that the>
vacuum energy is independent of the moduli. As well as telling
us> that we cannot determine the moduli by minimizing the
energy,> supersymmetry also tells us that the quanta of the
moduli fields are> exactly massless. No such massless fields
are known in nature and,> furthermore, such fields are very
dangerous. Indeed, massless> moduli would probably lead to
long-range forces that would> compete with gravity and violate
the equivalence principle - the> cornerstone of general
relativity - at an observable level.> On the plus side, the
vanishing vacuum energy that is implied by> supersymmetry
ensures that the cosmological constant vanishes. If> it were
not for supersymmetry, the vacuum would have a huge>
zero-point energy density that would make the radius of
curvature> of space-time not much bigger than the Planck scale
- a most> undesirable situation. > JS Comment: I have a
different much simpler explanation for the> smallness of the
cosmological constant in>
http://qedcorp.com/APS/EmergentGravity.pdf> Also the
observational fact of dark energy with FRW Omega ~ 0.7> means
that the cosmological constant is not exactly zero, which is>
a problem for the physics Lenny is talking about.> Back to
Lenny:> Supersymmetry also stabilizes the vacuum> against
various hypothetical instabilities, and it allows us to make>
exact mathematical conclusions. Indeed, T-duality and mirror>
symmetry are examples of those exact consequences.> Black
holes> Throughout the 1980s and early 1990s progress> in
string theory largely consisted of working out> the detailed
rules of perturbation theory for the> five known versions of
the theory, which would> allow theorists to arrive at actual
solutions> (figure 2). These perturbative rules were>
generalizations of the Feynman diagrams of QED> and QCD in
which the world lines of point> moving strings. The study of
world-sheet physics created a huge> body of knowledge about 2D
quantum field theory, but it did not> offer much insight into
the inner workings of quantum gravity. At> best, string theory
provided an especially consistent way to> introduce a small
distance scale and thereby regulate the> divergences that had
plagued the older attempts at quantizing> gravity.>
Personally I found the whole enterprise dry, overly technical
and,> above all, disappointing. I felt that a quantum theory
of gravity> should profoundly affect our views of space-time,
quantum> mechanics, the origin of the universe, and the
mysteries of black> holes. But string theory was largely
silent about all these matters.> Then in 1993 all this began
to change, and the catalyst was the> awakening interest in
Stephen Hawking's earlier speculations about> black holes.>
The starting point for Hawking's speculations was the thermal>
behaviour of black holes, which built on earlier work by Jacob>
Bekenstein of the Hebrew University in Israel. Rather than the
cold,> dead objects that they were originally thought to be,
black holes> turned out to have a heat content and to glow
like black bodies.> Because they glow they lose energy and
evaporate, and because> they have a temperature and an energy,
they also have an entropy.> This entropy, S, is defined by the
Bekenstein-Hawking equation: S> = AkBc3/4h-barG, where A is
the surface area of the horizon and> kBis Boltzmann's
constant.> After realizing that black holes must evaporate by
the emission of> black-body radiation, Hawking raised an
extremely profound> question: what happens to all the detailed
information that falls into> a black hole? Once it falls
through the horizon it cannot> subsequently reappear on the
outside without violating causality.> That is the meaning of a
horizon. But the black hole will eventually> evaporate, leaving
only photons, gravitons and other elementary> information must
ultimately be lost to our world. But one of the> fundamental
principles of quantum mechanics is that information is> never
lost, because the information in the initial state of a
quantum> system is permanently imprinted in the quantum
state.> Hawking's view was that conventional quantum
mechanics must be> violated during the formation and
evaporation of the black hole. He> rightly understood that if
this is true, the rules of quantum> mechanics must be
drastically modified as the Planck scale is> for unified
theories, should have been obvious. But initially> Hawking's
idea generated little interest among high-energy> theorists,
apart from myself and Gerard 't Hooft at the University of>
Utrecht. We were convinced that by modifying the rules of
quantum> mechanics in the way advocated by Hawking, all hell
would break> loose, such as causing empty space to quickly
heat up to> stupendous temperatures and energy densities. We
were sure that> Hawking was wrong. By the early 1990s,
however, the issue was> becoming critical, especially to
string theorists. String theory by its> very definition is
based on the conventional rules of quantum> mechanics and if
Hawking was right, the entire foundation of the> theory would
be destroyed. > JS Comment: I tend to disagree with Lenny and
t'Hooft that the unitarity> of micro-quantum theory is an
absolute. P.W. Anderson's More is> different suggests
otherwise. Think of the relation between special> relativity
and general relativity -- similarly with micro-quantum theory>
and MACRO-quantum theory of superfluids.> 4D space-time is a
non-dynamical absolute in special relativity. It acts> on
mass-energy without any direct reaction of mass-energy back on
it.> This is because the string tension is infinite in that
limit. Special> relativity is action without reaction. General
relativity corrects that> giving a finite value to string
tension? How? Because general relativity> emerges from a
macro-quantum theory as shown in>
http://qedcorp.com/APS/EmergentGravity.pdf The finite string
tension in> Ed Witten's sense of alpha' is actually a quantum
h effect added to G> and c.> & 14.6, micro-quantum theory is
like special relativity because the> quantum BIT pilot wave is
a non-dynamical absolute. It acts on the IT> extra variable
without any direct reaction of IT back on its quantumBIT.>
That is, in Wheeler's terms, micro-quantum theory is> IT FROM
BIT> In contrast, MACRO-quantum theory adds to that> BIT FROM
IT.> Micro-quantum theory is linear and nonlocal in
configuration space for> entangled composite systems with
unitary time evolution and a> probability interpretation in
Lenny's sense but with signal locality > in a detente passion
at a distance (A. Shimony) with retarded causality.> In
contrast, MACRO-quantum theory with superfluid signal
generalized> phase rigidity (e.g. string tension) is nonlinear
(Landau-Ginzburg eq.)> and local in ordinary (hyper) space with
non-unitary time evolution and> a complete breakdown of the
Born probability interpretation. Also it> allows signal
nonlocality violating retarded causality. Micro-quantum>
theory still works for the normal fluid noisy component.>
Back to Lenny:> Over the last decade the apparent clash
between standard quantum> principles and black-hole
evaporation has been resolved, favouring,> I should add, the
views of 't Hooft and myself. The formation and> evaporation
of a black hole is similar to many other process in> and
chaotic spectrum of intermediate states. In the case of a
black> hole, the collisions are between the original protons,
neutrons and> electrons in a collapsing star. Roughly speaking
a black hole is> nothing but a very excited string with a total
length that is> proportional to the area of its horizon. > JS
Comment. Already in 1973 I published a paper in Herbert
Frohlich's> Collective Phenomena that the Regge string
hadronic trajectories> showed that the hadronic resonances
were tiny black holes in> Abdus Salam's strong short-range
f-gravity with G* ~ 10^40 G(Newton).> Spin ~ G*E^2/hc^5 +
intercept> G*/hc^5 ~ (String Tension)^-1> G*/hc^5 ~ (1
Gev)^-2 UNIVERSAL SLOPE (micro-geometrodynamics)> The decay
of the hadrons would be like> Hawking radiation. Abdus Salam
invited me to work with him at> Trieste because of this paper.
We now see that this idea I had before> its time> was
essentially on the right path.> Back to Lenny> During the
collision or> collapse process, all the energy of the initial
state goes into forming> a single long, tangled string, and
the entropy of the configuration is> the logarithm of the
number of configurations of a random-walking> quantum
string.> The correspondence between string configurations and
black-hole> entropy was checked for all of the various kinds of
charged and> neutral black holes that occur in
compactifications of string theory.> In most of the cases the
entropy of the string configuration could be> estimated and it
agreed with the Bekenstein-Hawking entropy to> within a factor
of order unity.> But string theorists wanted to do better.
The Bekenstein-Hawking> formula for the entropy of a black
hole is very precise: the entropy> is one quarter of the
horizon area, measured in Planck units, for> every kind of
black hole, be it static, rotating, charged or even>
higher-dimensional. Surely the universal factor of a quarter
should> be computable in string theory? The key to a precise
calculation was> obvious. Certain black holes called extremal
black holes - which are> the ground states of charged black
holes that carry electric and> magnetic charges - are
especially tractable in a supersymmetric> theory. The only
problem was that in 1993 no-one knew how to> build an extremal
black hole out of the right type out of strings.> This had to
wait a couple of years for the discovery of entities> called
D-branes.> Brane world> In 1995 Joe Polchinski of the
University of California in Santa> Barbara electrified the
string-theory community with a major> discovery that has
subsequently impacted every field of physics. As> we have
seen, T-duality is the strange symmetry that interchanges> the
Kaluza-Klein momenta and winding numbers of a closed string.>
But what happens to an open string? Obviously the> idea of a
winding number does not make sense for such a string.> What
actually happens to open strings under T-duality is that the>
free ends become fixed on surfaces called D-branes.> D-branes
come in various dimensions; 2D> branes, for example, can also
be called> membranes. They have an energy or> mass per unit
surface area and are localized> physical objects in their own
right. In a sense> they seem to be no less fundamental than
the> strings themselves. To an outsider, D-branes may seem to
be> arbitrary additions to the theory. They are not. Their
existence is> absolutely essential to the mathematical
consistency of the theory.> In addition to allowing T-duality
to act on an open string in Type I> string theory, they are
necessary for implementing the deep> dualities that link the
five different kinds of string theory together.> But from the
point of view of black holes, the importance of> D-branes is
that you can build extremal black holes from them. In> fact,
just by placing a large number of D-branes at the same>
location you can build an extremal supersymmetric black hole.
And> because of the special properties of supersymmetric
systems, the> statistical entropy of that black hole can be
precisely computed. The> result, which was first derived by
Andrew Strominger and Cumrun> Vafa at Harvard in 1996, is that
the entropy is equal to exactly one> quarter of the horizon
area in Planck units! This suggested that the> microscopic
degrees of freedom implied by the Bekenstein-Hawking> entropy
are the degrees of freedom describing strings, and was a>
major boost for the superstring community.> At about the same
time as D-branes were discovered, another very> important
development took place. As I mentioned, the coupling> constant
of string theory is not really a constant at all, and in many>
respects it is very similar to the compactification moduli.
String> theorists took a surprisingly long time to make the
connection, but> it turns out that 10D string theory is itself
a Kaluza-Klein> compactification of an 11D theory that became
known as> M-theory .> M-theory appears to underlie all string
theories. The five> different versions of string theory are
just different ways of> compactifying its 11 dimensions. But
M-theory is not itself a string> theory. It has membranes but
no strings, and the strings only> appear when the 11th
dimension is compactified. Furthermore, the> momentum in the
compact 11th direction (the Kaluza-Klein> momentum) is
identified as the number of D0-branes - i.e.> zero-dimensional
branes, or points - in a particular type of string> theory.>
This connection between Kaluza-Klein momentum and D0-branes>
led to another breakthrough. In 1996 myself, Tom Banks and
Steve> Shenker (at Rutgers University), and Willy Fischler (at
the University> of Texas) realized that M-theory could be cast
in a form no more> complicated than the quantum mechanics of a
system of> is called Matrix theory, is an exact and complete
quantum theory> that describes the microscopic degrees of
freedom of M-theory. As> such it is the first precise
formulation of a quantum theory of> gravity.> Duality> Matrix
theory was just one example of how D-branes can be used to>
formulate a theory of quantum gravity. Soon after its
discovery,> Juan Maldacena, who is now at the Institute for
Advanced Study> (IAS) in Princeton, came up with a new
direction to explore. Ed> Witten of the IAS and others had
previously shown that D-branes> have their own dynamics. But
it turned out that the fluctuations and> motions of a D-brane
can be quantized in the form of a gauge> theory that is
restricted to the D-brane itself. The theory that lives> on a
coincident collection of D3-branes, for example, is a>
supersymmetric non-Abelian gauge theory. In other words, it is
a> supersymmetric version of QCD - the theory describing quarks
and> gluons. In a sense, string theory is returning to its
roots as a> pp35-38).> Maldacena realized that in an
appropriate limit> the theory of D3-branes should be a
complete> description of string theory - not just on the>
branes, but in the entire geometry in which the> branes are
embedded. A gauge theory would> therefore also be a
description of quantum> gravity in a particular background
space-time.> This space-time is called anti-de Sitter space,>
which, roughly speaking, is a universe inside a> cavity. The
walls of the cavity behave like> reflecting surfaces so that
nothing escapes it (figure 4).> This duality between quantum
field theory and gravity is an exact> realization of what is
called the holographic principle. This strange> principle,
formulated by 't Hooft and myself, grew from our debate> with
Hawking regarding the validity of quantum mechanics in the>
formation and evaporation of black holes.> According to the
holographic principle, everything that ever falls into> a
black hole can be described by degrees of freedom that reside
in> a thin layer just above the horizon. In other words, the
full 3D world> inside the horizon can be described by the 2D
degrees of freedom> on its surface. Even more generally, it
should be possible to> describe the physics of any region of
space in terms of holographic> degrees of freedom that reside
on the boundary of that region. This> leads to a drastic
reduction of the number of degrees of freedom in> a field
theory, and most theorists found it very hard to swallow
until> Maldacena's work came along. Maldacena's duality
replaces a> gravitational theory in anti-de Sitter space by a
field theory that> lives on its boundary in a very precise
way. In other words, the 3 +> 1-dimensional boundary field
theory is a holographic description of> the interior of 4 +
1-dimensional anti-de Sitter space.> The D-brane revolution
has been very far reaching. Matrix theory> and the Maldacena
duality are both formulations of quantum gravity> that conform
to the standard rules of quantum mechanics, and> should
therefore lay to rest any further questions about black holes>
violating these rules.> Googles of possibilities> I would like
to end by discussing the future of string theory, not as>
cosmology. The final evaluation of string theory will rest on
its> ability to explain the facts of nature, not on its own
internal beauty> and consistency. String theory is well into
its fourth decade, but so> convincing explanation of any
cosmological observation. Many of the> models that are based
on specific methods of compactifying either> 10D string theory
or 11D M-theory have a good deal in common> with the real
world. They have bosons and fermions, for example,> and gauge
theories that are similar to those in the Standard Model.>
Furthermore, unlike any other theory, they inevitably include>
gravity. But the devil is in the details, and so far the
details have> eluded string theorists.> It is, of course,
possible that string theory is the wrong theory, but I>
believe that would be a very premature judgement and probably>
incorrect. The problem does not seem to be a lack of richness,
but> rather the opposite. String theory contains too many
possibilities.> For most physicists, the ideal physical theory
is one that is unique> and perfect, in that it determines all
that can be determined and> that it could not logically be any
other way. In other words, it is not> only a theory of
everything but it is the only theory of everything.> To the
orthodox string theorist, the goal is to discover the one
true> consistent version of the theory and then to demonstrate
that the> solution manifests the known laws of nature, such as
the Standard> But the more we learn about string theory the
more non-unique it> seems to be. There are probably millions
of Calabi-Yau spaces on> which to compactify string theory.
Each space has hundreds of> moduli and hundreds of subspaces
on which branes can be> wrapped, fluxes imposed upon and so
on. A conservative estimate> of the number of distinct vacua
of the theory is in the googles - that> is, more than 10100.
The space of possibilities is called the> Landscape, and it is
huge. To mix metaphors, it is a stupendous> haystack that
contains googles of straws and only one needle.> Worse still,
the theory itself gives us no hint about how to pick> among
the possibilities (see The string-theory landscape ).> This
enormous variety may, however, be exactly what cosmology is>
looking for. A common theme among cosmologists is that the>
observed universe may merely be a minuscule part of a vastly>
bigger universe that contains many local environments, or
what> Alan Guth at MIT calls pocket universes . According to
this view, so> many pocket universes formed during the early
inflationary epoch -> each of which with its own vacuum
structure - that the entire> landscape of possibilities is
represented. The reasons for this view> are not just idle
speculation but are rooted in the many accidental>
fine-tunings that are necessary for a universe that supports
life.> Thus it may be that the enormous number of possible
vacuum> the doctor ordered for cosmology.> Further
information> T-duality> In a single compact dimension there
are two kinds of quantum> numbers: momentum in the compact
direction and the winding> number. Both of these are quantized
in integer multiples of a basic> unit, and each has a certain
energy associated with it. In the case> of momentum, for
example, the energy is just the kinetic energy of> units of
compact momentum is equal to n/R, where R is the>
circumference of the compact direction. Note that the energy
grows> as the size of the compact space gets smaller. On the
other hand,> the winding modes also have energy, which is the
potential energy> needed to stretch the string around the
compact co-ordinate. If we> call the winding number m, then
the winding energy is equal to mR.> In this case the energy
decreases as the size of the compact> direction decreases.>
The surprising thing is that the spectrum of energies is
unchanged if> we change the compactification radius from R to
1/R, and at the> same time interchange the Kaluza-Klein
momentum and winding> modes. In other words, just by looking
at the spectrum of energies> you could never tell the
difference between a theory that is> compactified on a space
of size R or on one of size 1/R. As you try> to make the
compactification scale smaller than the natural string> scale
- i.e. the size of a vibrating string - the theory begins to>
behave as if the compactification radius was getting bigger.>
Physically, the smallest compactification value of R is the
string> scale. But from a mathematical viewpoint, two
different spaces -> one large, the other small - are
completely equivalent. This> equivalence is called T-duality.>
Author> Leonard Susskind is in the Department of Physics,
Stanford> susskind@stanford.edu> Further reading> J Maldacena
1999 The large N limit of superconformal field theories> and
supergravity Int. J. Theor. Phys. 38 1113-1133> J Polchinski
1995 Dirichlet-branes and Ramond-Ramond charges> Phys. Rev.
Lett. 75 4724-4727> J Polchinski 1998 String Theory (volume
2): Superstring Theory and> Beyond (Cambridge University
Press)> J H Schwarz et al. 1981 Superstring Theory (volume 1):
Introduction> (Cambridge University Press)> A Strominger and C
Vafa 1996 Microscopic origin of the> Bekenstein-Hawking
entropy Phys. Lett. B 379 99-104> The official string theory
website: superstringtheory.com/ > Note above my new Blackett
formula for charge Q in the Calabi-Yau> space, i.e.> Q =
G*^1/2h/cR> R is a compactification scale modulus in the
extra-dimensional> generalized Kaluza-Klein space.> G* =
e^phiG(Newton)> See Saul-Paul Sirag's Nature paper on
Blackett effect in astrophysics.>wow...- Arthur
===
Is there
any free software like Maple for linux?thank you, /lucas
===
>
Is there any free software like Maple for linux? thank you,
/lucas Maxima and Axiom
(http://www.sciface.com/download.shtml) are free software.
http://www.sciface.com/download.shtml They are all fine!
Raymond
===
Is there any free software like Maple for
linux?Take a look perhaps at
http://maxima.sourceforge.net/
===
> [snip] >Surely if every
transaction takes a finite time there is no end. > <Despair>
It seems that some people have yet to resolve, in their own
minds, the> simple paradoxes of Zeno! Believe it or not, an
infinite series of positive terms (times) can have a> finite
sum. For example, 1/2 sec + 1/4 sec + 1/8 sec +...+ 1/2^N sec
+... = 1 sec.<Despair> Why do some people leap to conclusions?
I knew about Zeno while you were learning how to count.I
thought it was obvious that I meant a finite _constant_ time
(the same time for each transaction - why should they get
faster?)Gib
===
> [snip]>Surely if every transaction takes a
finite time there is no end. <Despair> It seems that some
people have yet to resolve, in their own minds, the> simple
paradoxes of Zeno!> Believe it or not, an infinite series of
positive terms (times) can> have a finite sum. For example,>
1/2 sec + 1/4 sec + 1/8 sec +...+ 1/2^N sec +... = 1 sec.>
<Despair> Why do some people leap to conclusions?way in this
case that I could reasonably have guessed that you actually
hadintended something else.)> I knew about Zeno while you were
learning how to count.Glad to know that.> I thought it was
obvious that I meant a finite _constant_ time (the same> time
for each transactionIt certainly wasn't obvious to me.> - why
should they get faster?)Response 0: Why shouldn't
they?Response 1: So that all transactions can be completed in
a finite time.David
===
> If it is not in at the end, and being
not in is true of every ball,> which (allegedly large number
of) balls are in the bucket at the end?Only the finitely
labeled balls are absent. The infinite number ofballs
remaining all have infinite labels. You got a problem with
that ?Lew Mammel, Jr.
===
If it is not in at the end, and
being not in is true of every ball,> which (allegedly large
number of) balls are in the bucket at the end? Only the
finitely labeled balls are absent. The infinite number of>
balls remaining all have infinite labels. You got a problem
with that ? Lew Mammel, Jr.Since initially all balls have
finite labels and no new labels are introduced, where do those
infinite labels come from?
===
> Only the finitely labeled balls
are absent. The infinite number of> balls remaining all have
infinite labels. You got a problem with that ?Yes, as there is
no natural number n such that 10*n is infinite. Ateach stage, n
is finite, and therefore only label with finite valuesare ever
created. There is no statement in the procedure that says, at
noon, give them all infinite labels . Since all label
managementhappens BEFORE noon, n is always finite.Jonathan
HoyleGene Codes Corporation
===
> A transaction consists of
adding ten balls to a bucket and removing 1.> (Obviously a
transaction is a net increase of nine balls.) Assume that>
infinitely many transactions somehow occur. Some people here
think> there will be no balls in the bucket afterward!> There
are some people who think that there will only be zero balls>
left if we label the balls in a certain way, otherwise there
will be> more than zero!Yes, people who are even more stupid
than you are.
===
> A transaction consists of adding ten balls
to a bucket and removing 1.> (Obviously a transaction is a net
increase of nine balls.) Assume that> infinitely many
transactions somehow occur. Some people here think> there will
be no balls in the bucket afterward!> There are some people
who think that there will only be zero balls> left if we label
the balls in a certain way, otherwise there will be> more than
zero!> Yes, people who are even more stupid than you are.Too
bad that doesn't exclude yourself.
===
In a few months I will
be teaching a course to undergraduate math majorscalled
Mathematical Structures . The catalog description is: A
rigorous study of the mathematical structures which form the
foundationof higher mathematics. Set theory, logic, formal
development of the numbersystems from the natural numbers
through the complex numbers, basic algebraic structures
(groups, rings, and fields), and elementary
topologicalconcepts. The course is also supposed to introduce
the students to the practice of good mathematical writing and
the construction of proofs. It will be a ten week course
(which is supposed to be the equivalent of a normal 15 week
semester course).A student will take this course after
completing the calculus sequence,linear algebra, and a couple
of other courses, but before taking abstractalgebra or
analysis.Can anyone recommend a good text or two for my
students? I realize thatone most likely cannot find a book
covering set theory and logic and alsoalgebra and topology. I
am most intersested in recommendations for the set theory,
logic, and proof writing parts of the course (which
willprobably take up 80% of the course).If necessary I can
teach the course without a text (I have extensive noteson set
theory and logic and the construction of the number systems
fromvarious sources). Any thoughts on that?
===
>In a few
months I will be teaching a course to undergraduate math
majors>called Mathematical Structures . The catalog
description is:> A rigorous study of the mathematical
structures which form the foundation>of higher mathematics.
Set theory, logic, formal development of the number>systems
from the natural numbers through the complex numbers, basic
>algebraic structures (groups, rings, and fields), and
elementary topological>concepts. That's quite a lot for one
course!>The course is also supposed to introduce the students
to the practice of >good mathematical writing and the
construction of proofs.And each of those two by itself is
quite a lot.> It will be a >ten week course (which is supposed
to be the equivalent of a normal >15 week semester course).Hey,
why not. Since there's already impossibly too much stuff in it
for a 15 week course, there's little additional harm in having
impossibly*150% too much stuff.I expect that's why you say
supposed to a couple of times. Now seriously, what do you
ACTUALLY expect to accomplish in the available time with the
available students? It is IMO very important for your own
stress management to have realistic(ish) expectations.It
sounds like this is a new course. If so, that increases all
risk factors markedly. If not, you would definitely benefit
from talking with the previous instructors of this course
about what went well and what went badly.>A student will take
this course after completing the calculus sequence,>linear
algebra, and a couple of other courses, but before taking
abstract>algebra or analysis.>Can anyone recommend a good
text or two for my students? I realize that>one most likely
cannot find a book covering set theory and logic and
also>algebra and topology. I am most intersested in
recommendations for >the set theory, logic, and proof writing
parts of the course (which will>probably take up 80% of the
course).And the other topics will take up the other 80%...>If
necessary I can teach the course without a text (I have
extensive notes>on set theory and logic and the construction
of the number systems from>various sources). Any thoughts on
that?In my experience, having a text provides a security
blanket for many students (expecially if the instructor may
have to go on medical stress leave half way through...)Gun,
Will Travel TV series.-- ---------------------------| B B aa
rrr b || BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb |
-----------------------------Cancel-Lock:
sha1:6eqNfLB8izgvwsh3H8fXWRMM+ik
===
=> Ok, I am getting
closer and closer...>[ Feller shows S_n/(n*log2(n)) -> 1 where
log2 is log base 2. You have to show S_n/(n*log(n)) -> log(2)
where log is log base e. In both cases, -> is convergence in
probability. ]Well, you should be able to express log2 in
terms of log. And youshould be able to use the properties of
convergence in probability todecide whether or not the two
statements are equivalent.As far as I can tell, they *aren't*
equivalent.So, Feller's problem is slightly different from
yours, you misreadhas made a mistake.-- Kevin
<buhr@telus.net>
===
> Is there an algorithm that maps positive
integers 1-1 to primes that> is substantially more efficient
than the function that maps n to the> n-th prime?> A lot of
work has been done on this. Fermat thought 2^(2^n) + 1 would >
always give you a prime number, but he was wrong.I haven't
counted up the operations, but calculating 2^(2^n) (indecimal,
say) may not be more efficient than finding the n-th
prime.Dale
===
bcc listJS: The name of the game is to control
the localvacuum coherence. I would like to see how Hal Puthoff
useshis PV model for this problem.HP: You want me to send you
the blueprint already?! (Let's see, where did I put it?)
:-)JS: You already have mine. Seriously, I am much further
along than you on all this.HP: Check with Table 2 in my JBIS
paper. (The Table you don't like, that's not even wrong! )
Time runs very fast for the drivers (i.e., the rest of the
universe is in slow motion relative to them) when they fire up
to reduce K (i.e., manipulate the vacuum polarizability to make
it very stiff, i.e., to resist polarizing).Ahh, how to do that?
I think you're telling me you already have the knob with
<0|e+(x)e-(x)|0> mobilzing / zpf, right? JS: Yes, it's the
Josephson effect where the space-time stiffness plays no role.
I could be mistaken, but it's an idea how to get to the next
step, which is more than you have as far as I can see and you
have revealed. You should come to London March 8-12.HP: So
what are you asking me for?HalJS: Numero Uno, come clean and
fess up where you stand of the reality of the flying saucers
and how that influences your theoretical work both with Ibison
on PV and with Haisch et-al on zero point energy.Numero Due,
make a public statement about all these claims of existing
zero point energy machines like Bearden's and all the Tesla
maniacs etc. You know like what we were doing at ISSO. All
these nuts are citing you as their Guru.On your K-control.
Looks like you have no way to do it. Even if you could it is
not clear to me that it would help.Your K in your simple model
isgoo = K^-1You have K = e^2GM/c^2r with maybe some EM
corrections.Please write down the K extended for EM. Also you
have no rotation no gravimagnetism which must play an
important role.OK for the flying saucer, M is the mass of the
saucer correct? The saucer must make its own timelike
geodesic, correct? Are we on the same page here or not? If so,
we can go maybe to next step.OKdT = goo^1/2dtdR = grr^1/2drSo
how do you get saucer to fly starting from there?You want to
reduce K why?Reducing K increases goo and decreases grr.Now
you havec' = c/Kand if you have K < 1 your c' > c but that's
for a null geodesic.The saucer of mass M is on a timelike
self-made geodesic so how does K < 1at the saucer help
assuming I can help you figure out how to MAKE IT SOwith my
detailed QED vacuum coherence model. At least I have h in
myequations, which you do not have.I also have a detailed
formula for guv in terms of h and even the Au from EM.I got a
lot more than you got Hal. :-)We also have the clue from Ray
Chiao's gravity radio that the metric cross terms ofrotation
(gravimagnetism) are important. I want to look at Chiao's
insight for the near field.He is only thinking far field . If
we can convert near induction EM fields to near guv fieldswe
have done it! Do you get what I am saying here?
===
>Prove or
disprove: if two matrices commutes (AB=BA) then they can
be>expressed with polynomials in terms of the same matrix P (
A=f(P),>B=g(P) ). What would P, f, and g be if A,B = [0 1 0 0]
[0 0 1 0]> [ ] [ ]> [0 0 0 0] [0 0 0 1]> [ ], [ ]> [0 0 0 1] [0
0 0 0]> [ ] [ ]> [0 0 0 0] [0 0 0 0] By considering the
potential Jordan canonical forms for P, and noting> that A^2 =
B^2 = 0, I think one can show there is no possible f and g> no
matter what P is. For example, if P were diagonalizable, f(P)
and> g(P) would be too (but A and B are not).We have a
4-dimensional commutative algebra spanned by I, A, B, and
AB.If A and B are polynomials in P, then so is AB. But in 4 by
4 matrices,the polynomials in P can form an algebra of at most
dimension 4.Hence P itself must be a combination of I, A, B,
and AB. Polynomials inP can all be written as polynomials in P
- cI for any c, so we can assume P is just rA + sB + tAB. But
this P has P^2 = 2rAB and P^3 = 0,so it does not produce a
sufficiently large algebra.William C. WaterhousePenn
State
===
In many mathematics papers and books, you see the
comment We will need thisin the sequel or We adopt this
notation throughout the sequel. Whatexactly does the sequel
and in the sequel mean here, precisely? Does itmean in what
follows?
===
>In many mathematics papers and books, you see
the comment We will need this>in the sequel or We adopt this
notation throughout the sequel. What>exactly does the sequel
and in the sequel mean here, precisely? Does it>mean in what
follows? Yes. Sequel comes from the Latin sequella , meaning
to follow . Inpapers, sometimes below is used instead of in
the sequel , but ifyou are refering to something which is some
way ahead (e.g., inbooks), below is not normally used.--
===
=================================
===
=== It's not denial. I'm just very selective about what I
accept as reality. --- Calvin ( Calvin and Hobbes
)
===
=================================
===
===Arturo Magidinmagidin@math.berkeley.edu===>In many
mathematics papers and books, you see the comment We will need
this>in the sequel or We adopt this notation throughout the
sequel. What>exactly does the sequel and in the sequel mean
here, precisely? Does it>mean in what follows? Yes.Lee
Rudolph
===
I have a question about linear, least-squares (LLS)
data fitting: What is the current best algorithm for solving
LLS problems?I am considering straightforward data-fitting
with no bounds, or anyother conditions. At this point, I plan
to fit polynomials to the data(does the fact I am using a
polynomial function affect the choice ofalgorithm to
use?).After reading some material, I was lead to believe that
Singular ValueDecomposition (SVD) is a better algorithm than a
QR factorization(more stable, but much more expensive in terms
of computations).Upon visiting the netlib site to see what
they have to offer for LLScode, I noticed they have two SVD
algorithms available:DGELSS (SVD via QR)DGELSD (SVD via Divide
and Conquer)Could somebody please let me know what the
strengths and weaknesses ofthese two approaches are. Which one
is better for my application(fitting data to a polynomial)? My
purpose does not mind if thecomputational cost is high, but
the results must be consistent andaccurate.DavidI have been
told the following fact by someone who didn't know where they
knew it from:Every periodic sequence can be expressed as the
sum of aconstant sequence and an irreducible sequence;where an
irreducible sequence is one such that if you take successive
differences iteratively, you eventually end up with the
sequence you started with;a constant sequence is one whose
terms are all equal;and the sum of two sequences is the
sequence whose nth term is the sum of the nth terms of the two
summand sequences.In other words, and stated even less
rigorously than theabove: every periodic sequence is offset by
some constant amount from an irreducible sequence.Does this
seem obvious? Does it even seem true? Is it a result that
you've seen published anywhere? I find it surprising but my
intuition is not to be trusted.him on to branches of math so
he can see what's out there. He likesquirky number theory
stuff, he loved Conway's Book of Numbers, and heloved the book
Knots and Surfaces by Farmer and Stanford, which wewent through
in a blitz - he did almost all of the exercises. I waswondering
whether someone knows of an introduction to group theory thatis
intuitive enough that he can get a feeling for essential
conceptswithout too much formalism. I feel like something that
had a goodpresentation of why there are only a few groups of
size smaller thansome limit, and what those groups are, for
example, could capture hisimagination. (He likes to know what
all the possibilities are invarious contexts.) The texts I
know are too terse for a kid that age,who will have time (and
patience) to do the formal stuff later, but Ithink it would be
good for him to discover as soon as possible that heloves math
in a broad way - if indeed he does love math in a broad way.
He's fine on simple set stuff and linear algebra.
===
>
wondering whether someone knows of an introduction to group
theory that> is intuitive enough that he can get a feeling for
essential concepts> without too much formalism.... Brief but
beautiful:W.W. Sawyer, Prelude to Mathematics, pp.97-102 and
201-214.Actually, he would very probably enjoy this whole
little book. A bit longer, but also very well done:Vol. 3,
Chapter XX (pp.263 ff.) Ken
Pledger.
===
--www.StealthHostiing.com You rule Truman.
http://tinyurl.com/iky4 Hey Trueman...love the show. YOU ARE
the Truman I heard him. Very spooky! >Is the truman living in
Townsville? I've been hearing stuff, yeah.Webmasters help the
TRUEman by joining www.theBanner.net Current:1
Goal:1000-----------------------------------------------------
----------------------- ------> As can be seen by the
number of posts in this thread,> and the references to his
web site in thousands of other posts,> a computer
programmer, who took some data processing classes> at a
third rate California college, has become a highly regarded
expert> in math, physics, and other science disciplines,
and> many people, who pretend to be rational, intelligent,
open-minded> scientists (Or at least, pretend to have a
scientific mind.),> frequently use this programmer as a
major reference.> What third rate California college? Who
rated it? What criteria?> Hey Wormley,> as you use this
programmer's web site as your primary rederence,> it seems to
me that you should know what college your resident expert>
attended.> If you want to know how this college rates,> I
suggest that you learn how to use Google.> I'll give you some
hints.> Caltech and Stanford and first rate California
colleges.> The college that Baez teaches at is a second rate
college.> Your expert took some data processing classes at a
third rate college.> Most scientist are computer
programmers... are you knocking us Potter?> Wormley,> why do
you always try to identify yourself with some group?> Does
identifying yourself with a group make you feel more secure,>
or do you think [sic] that it lends strength to your
position?> Do you have the courage to express any independent
ideas you have> (Assuming that you have an independent idea.),>
or the knowledge to address the point of a dichotomy,> rather
than try to position an opponents point> against some group
that you identify with?> In other words Wormley,> are you a
man or a mouse?> --> Tom Potter http://tompotter.usYou show 2
mutually exclusive beliefs here :One must belong to a named
collegeOne must act independentlyWhich is it?Will you proudly
tell us what University *you* attended?Or do you proudly stand
as an idividual with your own exploits?Herc