=>>It is clear that you don't know what
Cantor's proof actually says, since >>Cantor did not assume
a
mapping from the naturals ONTO the reals, but >>version of the
proof is direct, not by contradiction, It is others who
>>insist on recasting it into an indirect proof.> > > It is
obvious that I dont know that direct proof. After reading your
> post, Ive just got three questions:> > 1)Where can I get this
proof?> > See below.> > 2)Why there are so many books (written
by outstanding > mathematicians, I think) that invariably show
the proof by > contradiction?> > Probably because they are used
to speaking to mathematicians and it has> not occurred to them
that an indirect proof might be harder to understand> than a
direct proof.> > 3)May you explain what does exactly mean
*arbitrary mapping* > from the naturals to the reals? Perhaps
does it mean a mapping from > the naturals to an in'nite set
of reals, but no the set of all the > real numbers? > > Let f:
N -> R be a mapping from the naturals to the reals. Note: we
do> not assume anything at all about f except that it is a
mapping. We make> no assumptions about whether f is injective
(1-1) or not, whether it is> surjective (onto) or not, or
whether the range of f is 'nite or> in'nite. The
mapping f is
arbitrary because we assume absolutely> nothing about it
except that it is a mapping from N to R.> > This mapping
determines a list of reals in the sense that we can write>
down all the reals in the range of f in order:> > f(1)> f(2)>
f(3)> .> .> .> > Proposition: Let f: N -> R be given. Then f
is not a surjection.> > Proof. We are to show that there
exists x in R such that x is not in the> range of f. That is,
x != f(k) for any k in N.> > We do this by de'ning, for each
k, the k-th digit in the decimal> representation of x. Given k
> 0, we 'rst look at d_k, the k-th digit> following the
decimal
point in the representation of f(k) from our list.> We next
de'ne the k-th digit of x, x_k, as follows:> > If d_k is a 1,
set x_k = 2.> If d_k is not a 1, set x_k = 1.> > Then the
number x = .(x_1)(x_2)(x_3)... is the required number. It is>
not in the list because for each k, x differs from f(k) in the
k-th> digit.And note that none of its digits can be 0 or 9, so
that it cannot be any of the numbers having two potential
decimal representations, such as 1.000... =
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i021sTC07369;
=>>The diagonal
argument needs a premise (a list, otherwise it cant >>work).
If you know a *neutral* premise, then I believe that the proof
>>by contradiction would be valid.Assume a set of reals S and
an injection f from the naturals to this>set (effectively
making it a list). Then show that for any such f we>can always
'nd a real not in the map of f using the
oh-so-wonderful>diagonal argument, so f can't be a
surjection
onto any S that contains>all reals. It follows that f cannot
be a bijection.It doesn't really matter anyway since
you've
not given any argument>against proof-by-contradiction (in fact
you accept it as correct in>the case of sqrt(2)) except it can
be used to prove unintuitive>things so it must be wrong.If
with the premise *Assume a set of reals S and an injection f
from the naturals to this* you are supposing that S is *the
set of ALL reals*, then it is a false premise because breaks
the rules of naturals, but if S represents an INCOPLETE set of
reals, then the premise is worse than false, it is stupid,
since in this case you dont need the diagonal argument to 'nd
out that at least one real number is not in S.To get a good
conclusion, the proof by contradiction needs coherent
premises, and this is not the case. To assume that sqrt(2) is
rational is not a false and stupid premise because it doesnt
break any rule and moreover, the probability of being rational
before the proof is one half.Nicolas de la Foz
the premise *Assume a set of reals S and an injection f from
>the naturals to this* you are supposing that S is *the set of
ALL >reals*, then it is a false premise because breaks the
rules of >naturalsWhat? f:N->R, f(n) = n is an injection from
the naturals to reals. No,it does not get all real values
since it's not a surjection - thatwould be impossible. What
do
mean breaks the rules of naturals?>but if S represents an
INCOPLETE set of reals, then the >premise is worse than false,
it is stupid, since in this case you >dont need the diagonal
argument to 'nd out that at least one real >number is not in
S.Of course it's not stupid. It shows that any mapping
between
naturalsand reals can only be to a countable subset of reals.
We can not startfrom an incomplete mapping and complete it by
induction (adding realsone-by-one) because there's always
some
reals that we can add.>To get a good conclusion, the proof by
contradiction needs coherent >premises, and this is not the
case.De'ne coherent.>To assume that sqrt(2) is rational is
not
a false and stupid premise>because it doesnt break any rule and
moreover, the probability of>being rational before the proof is
one half.First you must prove that sqrt(2) is real. Then the
probability of anyreal number being rational is 0.Sqrt(2) is
of the form n/m where n and m are relatively prime and mis not
0 is a false premise, which leads to a direct contradictionwith
itself. There exists a bijection between N and R is a
similarlyfalse premise, which also leads to a direct
contradiction with itself.I fail to see the difference between
your suggested logical values offalse and false and
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i02Dn'20058;
=>My problem is
:>>I would like to minimise a function with N arguments. The
principle is>>to>>change the N args until the function is
equal to zero. My method is to>>add a deplacment to each arg,
and to see witch of them is the best, and>>to itter>>I don't
have a lot of af'nity with maths... Is it a well known
problem,>>with >>typic algorithms ? I don't really know
where
to search on the web to>>'nd this>>type of informationThis is
unconstrained minimization. The method which you describe>is
slow and will not converge well. If your function is
continuous and>differentiable, a conjugate-directions method
is probably the best way>to go. A conjugate-directions solver
is part of my ANSI C subroutine library WNLIB.>You can
download WNLIB as described below. The conjugate-directions
>package is called
wnconj.>------------------------------------------------------
------------------------FEATURES: WNLIB is an ANSI C
subroutine library that contains numerous tools that> I have
found to be useful in my programming practice. The following
are included: miscellaneous: improved memory allocator and
memory debugger; parsing package data structures and
algorithms: linked list; balanced binary tree; hash table -
with common hash> functions; dd tree; sorting for lists and
arrays;> high-quality pseudo-random number generator
numerical: various matrix and vector operations, including
matrix inverse, > least-squares inverse linear programming:
simplex method; transportation problem; shortest path
problem;> critical path problem non-linear optimization:
conjugate directions and conjugate gradient methods for
constrained> and unconstrained problems; simulated annealing
random numbers: very good (cryptographically-strong)
pseudo-random number generator. > True random number generator
(no kidding!). Normal distribution,> Poisson distribution,
Cauchy distribution, etc.>LEGAL: BY DOWNLOADING OR COMPILING
THIS LIBRARY, YOU ACCEPT AND AGREE TO THE TERMS > AND
CONDITIONS PRINTED BELOW. IF YOU DO NOT AGREE, DO NOT DOWNLOAD
OR > COMPILE THIS LIBRARY. The author and Spike Technologies
provide this C code in the hope> that it will be helpful,
however, we assume no responsibility > for the use of this
code, nor any responsibility for its support.> The software is
distributed on an AS IS basis, without warranty.> Neither the
authors, the software developers, nor Spike Technologies> make
any representation, or warranty, either express or implied,
with> respect to the software programs and subroutines, their
quality, accuracy, > or 'tness for a speci'c
purpose.
Therefore, neither the authors, the > software developers, nor
Spike Technologies shall have any liability to > you or any
other person or entity with respect to any liability, loss, >
or damage caused or alleged to have been caused directly or
indirectly by > the programs and subroutines contained in this
library. This includes, but > is not limited to, interruption
of service, loss of data, loss of classroom > time, loss of
consulting or anticipatory pro'ts, or consequential damages >
from the use of these programs. >COPYRIGHT NOTICE: The source
code in this directory is provided free of charge to anybody >
who wants it. It is in the public domain and therefore may be
used by > anybody for any purpose. This copyright notice and
the above legal notice> may not be removed.>AUTHOR: Will
Naylor Spike Technologies> 500 E. Calaveras Blvd #206>
Milpitas, CA 95035 Spike Technologies provides algorithm and
CAD software consulting> and contract programming and IC
design services.>DOWNLOADING and INSTALLATION:1) Create a
directory called wnlib wherever you want this code to> reside
and go into it. Type $ cd $ cd wnlib2) Download using ftp or rftp or a
web browser. > > a) For ftp or rftp type $ ftp ftp.rahul.net
Type ïanonymous' at login prompt.> After you are
logged in
type ftp> cd pub/spiketech/softlib/wnlib> ftp> get
INSTALL.txt> ftp> get uuwnlib.z> ftp> quit b) For a web
browser, go to > > http://www.spiketech.com/spike> > Click
over ïSpike Freeware' link. Read instructions
and > click over
to download. Your browser should throw a >
dialog box; you can select a directory where you want to store
it> and click ok. The browser will download it.3) Decode the
install tar 'le. Type $ uudecode uuwnlib.z> $ uncompress <
tar'le | tar xvf -4) Make the code. Type $ make all5) If this
fails, it may be necessary to modify variables in
acc/make'le>
to 't the platform you are running on.6) When you believe you
have compiled correctly, type $ make selftest to run the
self-testing diagnostics. If selftest runs successfully> to
completion with no error messages, you have successfully
installed> wnlib.>USE:1) Summary: ARCHIVE is in:
wnlib/acc/text.a> H FILES are in: wnlib/acc/h/*.h> MAN PAGES
are in: wnlib/doc/man/*.man 2) Under wnlib, the directory acc
contains .h 'les to include and archive> 'les to
link to. The
directory acc/h contains links to all of the > .h 'les. The
directory acc contains an archive 'le called text.a. > Use
the
-I switch> of cc to include the .h 'les. See the
make'le's in
acc for examples> of linking text.a 'les and including .h
'les.3) man pages are to be found in doc/man. The script
wnman
in the> directory command is a convenient way to view these man
pages.> Type wnman to view a man page.> The command
wnman -k does a keyword search on all> the man
pages.> > The variable wnlib must point to your wnlib
directory for> wnman to work. A csh command of the form $ set
wnlib = xxx/yyy/wnlib sets this up. Include such a command in
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i02Dngq20080;
=Quoting from the
introduction of the book Rings,Fields, and Groups
(allenby):There seems little point in giving here a list of
contentsmore detailed than that which follows. To the
experienced reader those section headings will reveal all; for
the beginnereven more detail will reveal little...I suppose
that applies to names as well: For an absolute beginner fresh
out of school,abelian group probably won't be much more
tantalizing than the descriptive noncommunative group,where
the student may even feel a little intimidated;after all,
he/she has probably almost exclusivelyencountered variables
such as x, y, z, etc. in connection with real numbers and not
with n*n matrices (usually written capatilized, if they were
discussed as variables in equations at all), permutation
functions, elements of abstract groups, etc. during the last
3-4 years with the exception of the vector cross product; a
student thinking so what they're telling mehere is that
noncommunative group elements are similar tovectors forming a
cross product will probably do more to confuse. At least when
a student reads abelian groupand sees what is being de'ned
she/he can say: xy != yx? that Abel person must have been
onecrazy guy... Not that being descriptive is a bad thing in
general; an advanced reader may well be inclined to make his
own descriptive terms from names, such as GODel (whichwas
mentioned earlier in this thread...). By the way, in German
for example, words are concantinated much more often than in
English, making them naturally moredescriptive: Wertebereich
der Funktion =Werte-bereich der Funktion = value-area of the
function =range of the functionUrbild der Funktion =Ur-bild
der Funktion =Pre-picture of the function =domain of the
function Eine Einfuerung in die Mathematik =Eine Ein-fuerung
in die Mathematik =A In-Leading to Mathematics =An
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i02Dnj520184;
=Still being kind of a newcomer to ('nite)
group theory, I read over the proof of Lagrange's theorem:H
subgroup G -> |G| = |H||G:H|. And of the manysubsequent
examples (and problems). The authorof the introductory text
book I was reading misseda very easy example, in my opinion:If
G has a subgroup H of even order, then G is even.Proof: If G
has a subgroup H of even order, then |G|= |H||G:H|. The index
of H in G, |G:H|, is either even or odd. It doesn't matter;
since odd*even = even and even*even = even, q.e.d.This is not
my question, however. It is:(Idea- since {1,-1} is a
nontrivial group, ...), doesit follow that every group with
nontrivial center has even order?C. Dement
= [.snip.]> This
is not my question, however. It is:> (Idea- since {1,-1} is a
nontrivial group, ...),What does -1 mean in the context of an
arbitrary group? And what makes you think that in your group,
even if -1 makes sense,-1 is not equal to 1?> does> it follow
that every group with nontrivial center > has even order?No.
Every group whose order is the power of a prime has
nontrivialcenter (you will deduce it as a consequence of the
class formula),in particular, any group of order p^n, p an odd
prime, n>0, hasnontrivial center.Arturo Magidin, sans .sig
=>
If G has a subgroup H of even order, then G is even.>What about
Z/2 x Z ? Are in'nite groups even?> Proof: If G has a
subgroup
H of even order, then> |G|= |H||G:H|. The index of H in G,
|G:H|, is> either even or odd. It doesn't matter;> since
odd*even = even and even*even = even, q.e.d. This is not my
question, however. It is:> (Idea- since {1,-1} is a nontrivial
group, ...), does> it follow that every group with nontrivial
center> has even order?>1 is the identity? Then -1 = 1.1
isn't
identity? Then 0 is identity and 1+1 is in that group.
=> >
If G has a subgroup H of even order, then G is even. What
about Z/2 x Z ? Are in'nite groups even?> > Proof: If G has a
subgroup H of even order, then> |G|= |H||G:H|. The index of H
in G, |G:H|, is> either even or odd. It doesn't matter;>
since
odd*even = even and even*even = even, q.e.d. This is not my
question, however. It is:> (Idea- since {1,-1} is a nontrivial
group, ...), does> it follow that every group with nontrivial
center> has even order? 1 is the identity? Then -1 = 1.> 1
isn't identity? Then 0 is identity and 1+1 is in that
group.It's an ABELIAN group, meaning it's
multiplication. 1 is
the identityand -1!=1. {-1, 1} is isomorphic to the cyclic
group Z/2Z, (thatisn't Z/2 * Z, but actually Z/(2Z).) Z/2Z
has
2 members, 0 and 1.
=If G has a subgroup H of even order,
then G is even.>> What about Z/2 x Z ? Are in'nite groups
even?Proof: If G has a subgroup H of even order, then>> |G|=
|H||G:H|. The index of H in G, |G:H|, is>> either even or odd.
It doesn't matter;>> since odd*even = even and even*even =
even, q.e.d.>> This is not my question, however. It is:>>
(Idea- since {1,-1} is a nontrivial group, ...), does>> it
follow that every group with nontrivial center>> has even
order?>> 1 is the identity? Then -1 = 1.>> 1 isn't identity?
Then 0 is identity and 1+1 is in that group.It's an ABELIAN
group, meaning it's multiplication. 1 is the identity>and
-1!=1. -1 has no intrinsic meaning. Take the cyclic
multiplicative group of 3elements, 1, x, x^2, with x^3 = 1.
What is -1 then? > {-1, 1} is isomorphic to the cyclic group
Z/2Z,Not every group has a unique central element of order 2.
What is -1?--
=It's not denial. I'm just very selective
about what I
accept as reality. --- Calvin (Calvin and
Hobbes)
Arturo Magidinmagidin@math.berkeley.edu>
This is not my question, however. It is:> (Idea- since {1,-1}
is a nontrivial group, ...), does> it follow that every group
with nontrivial center > has even order?> > C. DementNo.
Trivially every abelian group has non-trivial centre , just
pick onethat has odd order. If you were going to consider
non-abelian groups, trya group of order p^n for p odd. The
book should prove that at some pointthis has non-trivial
centre; this is the same as possessing at least twoconjugacy
approve@localhost)
=>My problem is :>>I would like to
minimise a function with N arguments. The principle
is>>to>>change the N args until the function is equal to zero.
My method is to>>add a deplacment to each arg, and to see witch
of them is the best, and>>to itter>>I don't have a lot of
af'nity with maths... Is it a well known problem,>>with
>>typic algorithms ? I don't really know where to search on
the web to>>'nd this>>type of informationThis is
unconstrained
minimization. The method which you describe>is slow and will
not converge well. If your function is continuous
and>differentiable, a conjugate-directions method is probably
the best way>to go. A conjugate-directions solver is part of
my ANSI C subroutine library WNLIB.>You can download WNLIB as
described below. The conjugate-directions >package is called
wnconj.>------------------------------------------------------
------------------------FEATURES: WNLIB is an ANSI C
subroutine library that contains numerous tools that> I have
found to be useful in my programming practice. The following
are included: miscellaneous: improved memory allocator and
memory debugger; parsing package data structures and
algorithms: linked list; balanced binary tree; hash table -
with common hash> functions; dd tree; sorting for lists and
arrays;> high-quality pseudo-random number generator
numerical: various matrix and vector operations, including
matrix inverse, > least-squares inverse linear programming:
simplex method; transportation problem; shortest path
problem;> critical path problem non-linear optimization:
conjugate directions and conjugate gradient methods for
constrained> and unconstrained problems; simulated annealing
random numbers: very good (cryptographically-strong)
pseudo-random number generator. > True random number generator
(no kidding!). Normal distribution,> Poisson distribution,
Cauchy distribution, etc.>LEGAL: BY DOWNLOADING OR COMPILING
THIS LIBRARY, YOU ACCEPT AND AGREE TO THE TERMS > AND
CONDITIONS PRINTED BELOW. IF YOU DO NOT AGREE, DO NOT DOWNLOAD
OR > COMPILE THIS LIBRARY. The author and Spike Technologies
provide this C code in the hope> that it will be helpful,
however, we assume no responsibility > for the use of this
code, nor any responsibility for its support.> The software is
distributed on an AS IS basis, without warranty.> Neither the
authors, the software developers, nor Spike Technologies> make
any representation, or warranty, either express or implied,
with> respect to the software programs and subroutines, their
quality, accuracy, > or 'tness for a speci'c
purpose.
Therefore, neither the authors, the > software developers, nor
Spike Technologies shall have any liability to > you or any
other person or entity with respect to any liability, loss, >
or damage caused or alleged to have been caused directly or
indirectly by > the programs and subroutines contained in this
library. This includes, but > is not limited to, interruption
of service, loss of data, loss of classroom > time, loss of
consulting or anticipatory pro'ts, or consequential damages >
from the use of these programs. >COPYRIGHT NOTICE: The source
code in this directory is provided free of charge to anybody >
who wants it. It is in the public domain and therefore may be
used by > anybody for any purpose. This copyright notice and
the above legal notice> may not be removed.>AUTHOR: Will
Naylor Spike Technologies> 500 E. Calaveras Blvd #206>
Milpitas, CA 95035 Spike Technologies provides algorithm and
CAD software consulting> and contract programming and IC
design services.>DOWNLOADING and INSTALLATION:1) Create a
directory called wnlib wherever you want this code to> reside
and go into it. Type $ cd $ cd wnlib2) Download using ftp or rftp or a
web browser. > > a) For ftp or rftp type $ ftp ftp.rahul.net
Type ïanonymous' at login prompt.> After you are
logged in
type ftp> cd pub/spiketech/softlib/wnlib> ftp> get
INSTALL.txt> ftp> get uuwnlib.z> ftp> quit b) For a web
browser, go to > > http://www.spiketech.com/spike> > Click
over ïSpike Freeware' link. Read instructions
and > click over
to download. Your browser should throw a >
dialog box; you can select a directory where you want to store
it> and click ok. The browser will download it.3) Decode the
install tar 'le. Type $ uudecode uuwnlib.z> $ uncompress <
tar'le | tar xvf -4) Make the code. Type $ make all5) If this
fails, it may be necessary to modify variables in
acc/make'le>
to 't the platform you are running on.6) When you believe you
have compiled correctly, type $ make selftest to run the
self-testing diagnostics. If selftest runs successfully> to
completion with no error messages, you have successfully
installed> wnlib.>USE:1) Summary: ARCHIVE is in:
wnlib/acc/text.a> H FILES are in: wnlib/acc/h/*.h> MAN PAGES
are in: wnlib/doc/man/*.man 2) Under wnlib, the directory acc
contains .h 'les to include and archive> 'les to
link to. The
directory acc/h contains links to all of the > .h 'les. The
directory acc contains an archive 'le called text.a. > Use
the
-I switch> of cc to include the .h 'les. See the
make'le's in
acc for examples> of linking text.a 'les and including .h
'les.3) man pages are to be found in doc/man. The script
wnman
in the> directory command is a convenient way to view these man
pages.> Type wnman to view a man page.> The command
wnman -k does a keyword search on all> the man
pages.> > The variable wnlib must point to your wnlib
directory for> wnman to work. A csh command of the form $ set
wnlib = xxx/yyy/wnlib sets this up. Include such a command in
your .cshrc.>
http://www.spiketech.com/spikeGot: ïThe requested URL /spike
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i02Dngm20074;
=>numbers in your head in a
matter of seconds. Any truth to this? Any>suggestions as to
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i02Dni920157;
=>Ultimately, one
would have to come up with a means for>ef'ciently refering to
quasi-in'nite regressions of operation>numbers embedded
inside
of other operation numbers...but the present>set of symbols and
terminology should suf'ce for now.>Someone has already come
up
with a notation for thisWhat you are looking for, is Conway
chained arrow notation explained
herehttp://en.wikipedia.org/wiki/Conway_chained_arrowhttp://
home.earthlink.net/~mrob/pub/math/largenum-3.html
=>I need
to do optimization to optimize for a matrix that is as
othogonal as>>possible, or as diagonal as possible...>>How to
I set the objective measure of the orthogonality and
diagonality>>of matrices?This comes immediately to
mind.Orthogonals: how close is AA^T to the identity matrix
(maybe using> square-sums of all elements in the matrix AA^T -
I).Diagonals: how close is A - A^T to the zero matrix (again,
using> square-sums of A - A^T).This would measure how
symmetric A was, not how diagonal it is.Better might be to use
the square-sums of off-diagonal elements.>Not knowing your
algorithm for creating these matrices it's hard to>say
whether
these metrics are useful or not.Rob Johnson
take out the trash before replying
=I
made a post about a quadratic example that Rick Decker, a
professorat Hamilton (I said University before but it might be
College), gavein a recent post. I haven't seen any replies
to
that yet in GoogleGroups, so I'll leave it and make another
thread to consider Decker'sexample in more detail, again
here
are some headers to allow you to'nd his original post:a_1(x)
a_2(x) = 7(x^2 + x)or decomposition, where here 7(x^2 + x) is
being broken up into twoalgebraic integer functions which are
its factors.Now that can be done with my work and my
factorizations, but I thinkthat a quadratic is simple enough
for me to explain some more points,so I'll use
Decker's
example without feeling a need to use morecomplicated work of
my own.So there's this second factorization, and from your
understanding ofalgebraic integers, you should accept that in
general algebraicintegers are in'nitely decomposable, 0
trivially.So from a *common sense* perspective, that is, the
commonunderstanding of the ability to decompose algebraic
integers, youmight *reasonably* suppose that you can always
'nd decompositions inthe ring of algebraic integers such
thata_1(x) = w_1(x) b_1(x), and a_2(x) = w_2(x) b_2(x),where
w_1(x) w_2(x) = 7, and the w's are algebraic integer
functions,so that you could havew_1(x) b_1(x) w_2(x) b_2(x) =
7(x^2 + x)in the ring of algebraic integers.However, that
common sense view is wrong, as although algebraicintegers
*are* in'nitely decomposable into algebraic integers, itturns
out that it's rather easy to show that the peculiar
constructionhere doesn't allow for a decomposition *in
general* in the ring ofalgebraic integers.So why do I
emphasize in general? That's because you may
indeed'nd for
some particular x that you can get an algebraic
integerdecomposition, but you can't 'nd
'nd a de'nition like
for a_1(x),and a_2(x), where they're *guaranteed* to be
algebraic integers foralgebraic integer x because they're
the
roots ofa^2 - (x - 1)a + 7(x^2 + x).Checking at x=0 reveals
that the actual constant terms of thefactorization are 7 and
2, where Decker picked a_1(0) = 0 at x=0.Now then, consider
what happens if you divide both sides of(5a_1(x) + 7)(5a_2(x)
+ 7) = 7(25x^2 + 30x + 2) by 7, as then you end up with
something like(5b_1(x) + 1)(5b_2(x) + 2) = 25x^2 + 30x +
2where the b's are roots of some unknown quadratic, though
the
'rstand last coef'cients ARE known:b^2 + ? b +
(x^2 + x).If
you're a little confused on that point, remember that I
hada_1(x) = w_1(x) b_1(x), and a_2(x) = w_2(x) b_2(x),where
w_1(x) w_2(x) = 7, and the w's are algebraic integer
functions,so that you could havew_1(x) b_1(x) w_2(x) b_2(x) =
7(x^2 + x)and now dividing both sides by 7 givesb_1(x) b_2(x)
= x^2 + x.If you assume that the b's like the
a's before them
are algebraicinteger functions (that common sense again) then
there should be ade'ning quadratic *in the ring of algebraic
integers* like there ISfor the a's, as for the
a's that
de'ning quadratic isa^2 - (x - 1)a + 7(x^2 + x).Now then
Decker probably picked his example not for me to outline
someof the 'ner points of Advanced Polynomial Factorization
Theory (justmade that up, being a discoverer is cool).As I
mentioned in my other thread on his example, he probably
pickedit because at x=1, you have *both* a's with sqrt(7) as
a
factor.It turns out that the ring over which you can get that
factorization,starting from initial steps, is a 'eld, where 7
is a unit.To understand that fact, you need to understand my
work a LOT morethoroughly, like how I found my own
factorization. My guess is thatDecker just 'ddled with things
until he found something that hethought suited his
purposes!But mathematics is about consistency and logic. There
is a reason foreverything in mathematics. And when you make
changes that seem simpleenough to you, underneath a lot of
mathematical machinery may be atwork.Hopefully by talking to
you about de'ning algebraic integer
functionsI've helped you
to get some sense of mathematical ideas clearly at thelimits
of human ability to understand.Want more? Then read various
threads on sci.math that I've started,and be sure to check
out
my blog archives:James Harris
=For other counterexamples,
investigate the ultrametric
case.In the
p-adic numbers, the open unit ball U = {y : d(y,0)<1}is a
closed set, even though there are many points z with
d(z,0)=1.
=> >Is there a suf'cient condition so that the
closure of every open ball>of a metric space is the closed
ball of same center and radius? The>closure of an open ball is
always a subset of the closed ball of same>center and radius,
but it can be a proper subset.> > Necessary and suf'cient is
that for every point p in the space, the > only local minimum
of the function x -> d(x,p) is at x=p.>Yes, that's right.
Fix
p in the space S and de'ne f:S->R by f(x) =d(x,p). Suppose
the
only local minimum of f is at p. Let B(r) be theopen ball
centered at p and with radius r and C(r) the
correspondingclosed ball. If x is in in C(r), then, since x is
not a local minimumof f, every neigborhood of x contains an
element y such that f(y) =d(y,p) < f(x) = d(x,p) = r, which
shows y is B(r). This implies thatevery neighborhood of x
intersects B(r), so that x is in the closureof B(r). We
conclude F(r) is a subset of cl B(r) and, since theconverse is
always true, it follows that F(r) = cl B(r).For the converse,
suppose some q<>p in S is a local minimum of f andlet r =
d(q,p), so that q is in F(r). Since q is a local minimum, qhas
a neighborhood V on which f(x) = d(x,p) > f(q) = d(q,p) =r
forx<>p. Since q is not in B(r), we see no point of V is in
B(r) and,therefore, x is not in cl B(r). So, in this case, cl
B(r) is a propersubset of F(r).Since p is an arbitrary element
of S, your proposition follows.
=Is there
a suf'cient condition so that the closure of every open
ball>of a metric space is the closed ball of same center and
radius? The>closure of an open ball is always a subset of the
closed ball of same>center and radius, but it can be a proper
subset. Necessary and suf'cient is that for every point p in
the space, the> only local minimum of the function x -> d(x,p)
is at x=p. Yes, that's right. Fix p in the space S and
de'ne
f:S->R by f(x) => d(x,p). Suppose the only local minimum of f
is at p. Let B(r) be the> open ball centered at p and with
radius r and C(r) the corresponding> closed ball. If x is in
in C(r), then, since x is not a local minimum> of f, every
neigborhood of x contains an element y such that f(y) =>
d(y,p) < f(x) = d(x,p) = r, which shows y is B(r). This
implies that> every neighborhood of x intersects B(r), so that
x is in the closure> of B(r). We conclude F(r) is a subset of
cl B(r) and, since the> converse is always true, it follows
that F(r) = cl B(r).Do you intend F(r) = C(r) ?> For the
converse, suppose some q<>p in S is a local minimum of f and>
let r = d(q,p), so that q is in F(r). Since q is a local
minimum, q> has a neighborhood V on which f(x) = d(x,p) > f(q)
= d(q,p) =r for> x<>p. Since q is not in B(r), we see no point
of V is in B(r) and,> therefore, x is not in cl B(r). So, in
this case, cl B(r) is a proper> subset of F(r).> Since p is an
arbitrary element of S, your proposition follows. points, then
there are open balls whose closure is not the> corresponding
closed ball, right?Almost. Just one will suf'ce provided
space
is multi-point.Let p be isolated point. Pick x /= p, let r =
d(x,p).p not in B(x,r); p not in cl B(x,r); p in closed ball
B[x,r]----
=> Is there a suf'cient condition so that the
closure of every open ball>of a metric space is the closed
ball of same center and radius? The>closure of an open ball is
always a subset of the closed ball of same>center and radius,
but it can be a proper subset. Necessary and suf'cient is
that
for every point p in the space, the> only local minimum of the
function x -> d(x,p) is at x=p. Yes, that's right. Fix p in
the space S and de'ne f:S->R by f(x) => d(x,p). Suppose the
only local minimum of f is at p. Let B(r) be the> open ball
centered at p and with radius r and C(r) the corresponding>
closed ball. If x is in in C(r), then, since x is not a local
minimum> of f, every neigborhood of x contains an element y
such that f(y) => d(y,p) < f(x) = d(x,p) = r, which shows y is
B(r). This implies that> every neighborhood of x intersects
B(r), so that x is in the closure> of B(r). We conclude F(r)
is a subset of cl B(r) and, since the> converse is always
true, it follows that F(r) = cl B(r).> > Do you intend F(r) =
C(r) ?Oh! Yes, sure! Sorry, I ended up using 2 different
notations for thesame thing. I should have used either F(r) or
C(r), not both. It was amistake.> > For the converse, suppose
some q<>p in S is a local minimum of f and> let r = d(q,p), so
that q is in F(r). Since q is a local minimum, q> has a
neighborhood V on which f(x) = d(x,p) > f(q) = d(q,p) =r for>
x<>p. Since q is not in B(r), we see no point of V is in B(r)
and,> therefore, x is not in cl B(r). So, in this case, cl
B(r) is a proper> subset of F(r).> Since p is an arbitrary
element of S, your proposition follows. points, then there are
open balls whose closure is not the> corresponding closed ball,
right?> > Almost. Just one will suf'ce provided space is
multi-point.> Let p be isolated point. Pick x /= p, let r =
d(x,p).> p not in B(x,r); p not in cl B(x,r); p in closed ball
= >>
isolated points, then there are open balls whose closure is
not >> the corresponding closed ball, right?> >> Almost. Just
one will suf'ce provided space is multi-point. >> Let p be
isolated point. Pick x /= p, let r = d(x,p). >> p not in
B(x,r); p not in cl B(x,r); p in closed ball B[x,r]Small
oversight, your proof of the proposed theorem
excellent.----
=William Elliot scribbled
the following:> >> Do you intend F(r) = C(r) ?> >Oh! Yes, sure!
Sorry, I ended up using 2 different notations for the> >same
thing. I should have used either F(r) or C(r), not both.> Do I
detect a European accent to your math? As I recall,> Europeans
use F for closed sets and O for open sets.In Helsinki,
Finland, at least, F is generally used for closed sets andU
for open opens. F probably comes from ferm.8e (French for
closed)but I don't know where U comes from.
I've thought of it
as a smalluniverse, but it could also come from Umgang(sp?),
German forneighbourhood or something.-- /-- Joona Palaste
(palaste@cc.helsinki.') ------------- Finland ----------
http://www.helsinki.'/~palaste --------------------- rules!
--------/I will never display my bum in public again. - Homer
= >> Do you intend
F(r) = C(r) ?> >Oh! Yes, sure! Sorry, I ended up using 2
different notations for the> >same thing. I should have used
either F(r) or C(r), not both.> Do I detect a European accent
to your math? As I recall,> Europeans use F for closed sets
and O for open sets. In Helsinki, Finland, at least, F is
generally used for closed sets and> U for open opens. F
probably comes from ferm.8e (French for closed)> but I don't
know where U comes from. I've thought of it as a small>
universe, but it could also come from Umgang(sp?), German for>
neighbourhood or something.>Don't know. To avoid O from
being
confused with 0 ?I avoid using l as it confuses with 1 upon
fast readingand in the days of typerwriters, l was used for
1.
=> Is there a suf'cient condition so that the closure of
every open ball> of a metric space is the closed ball of same
center and radius? The> closure of an open ball is always a
subset of the closed ball of same> center and radius, but it
can be a proper subset.> Here's a conjecture: For every
point
x in a metric space, there isradius r(x) such that if 0 < a <
r(x), the closure of B(x;a) isthe closed ball of radius a
about x.
=> Here's a conjecture: For every point x in a
metric space, there is> radius r(x) such that if 0 < a < r(x),
the closure of B(x;a) is> the closed ball of radius a about
x.Take the metric space X = {0} U {1/n; n positive integer}
with the usualmetric and let r be a positive real, N a
positive integer such that 1/N Is there a suf'cient
condition so that the closure of every open ball> of a metric
space is the closed ball of same center and radius? The>
closure of an open ball is always a subset of the closed ball
of same> center and radius, but it can be a proper subset.> >
It seems to me this is more a metric property than a
topologicalproperty. For example, consider the two metrics
d1(x,y) = |x-y| and d2(x,y) = min(|x-y|,1) on the reals. Both
these metrics arecompatible with the usual topology on R. In
both cases, an open ballof radius 1 is an open interval of
length 2. With d1, the closedball is the closure of this
interval, whereas with d2, the closed ballis the entire
space.Stephen J. Herschkorn
=> Is there a suf'cient
condition so that the closure of every open ball> of a metric
space is the closed ball of same center and radius? The>
closure of an open ball is always a subset of the closed ball
of same> center and radius, but it can be a proper subset.> > It seems to me this is more a metric property than a
topological> property. For example, consider the two metrics
d1(x,y) = |x-y| and> d2(x,y) = min(|x-y|,1) on the reals. Both
these metrics are> compatible with the usual topology on R. In
both cases, an open ball> of radius 1 is an open interval of
length 2. With d1, the closed> ball is the closure of this
interval, whereas with d2, the closed ball> is the entire
space.> Yes, sure. And we see that if radius <1 then the
closure and theclosed ball are always the same.Amanda
=I say
a,b are factors of c iff c is equivalent to some word
generatedby a and b. For example, a and b would be factors of
c ifc=ababababaabaaab. For a given non-singular matrix, is
there any wayto determine the set of factor pairs? Is there an
algorithm todetermine if two matrices are factors of diagonal
matrix?Let a,b,c be 2x2 non-singular matrices. Is there a way
to determinefor what values of n, or for how many values of
n,c^n*a*c^-n,c^n*b*c^-n are factors of a diagonal matrix?
Anyinformation or references relating to these questions or
moregeneralized ones would be greatly appreciated.Gershon
Bialer
=>Today, at the Barnes and Noble Bookstore, I saw the
book The Colossal>Book of Mathematics: Classic Puzzles,
Paradoxes, and Problems by>Martin Gardner. The book says
that>e^[pi*sqrt(163)] = 262,537,412,640,768,744>which is an
integer exactly.>Speci'cally, the book says that Srinivasa
Ramanujan(1887-1920)>computed the number manually to
262,537,412,640,768,743.999999, but>could not go further; then
someone in France computed two million>digits after the decimal
point which are all 9; then someone>ingeniously used the
Euler's Constant to prove that the result is>exactly the
integer as given above.>A quick search in the Internet
revealed that the above claim is not>true and it 'rst showed
up as a April's Fool's joke by Martin>Gardner,
and he
subsequently admitted that it was a joke.In fact, after
readint the Scienti'c American April Foollooked up
Ramanujan's
paper cited, and from that I couldprove that it was not an
integer.>Then how come it still got into the book?-- 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
=>> I checked out a book called All the
Mathematics You Missed [But Need to>> Know for Graduate
School] from the library and was surprised by its>> contents.
The book is divided into 16 sections that I am supposed to>>
know before I get into graduate school. This is my last year
and I>> can check off very little.>> Here are the sixteen
topics that I need to know along with whether or>> not I will
have completed them by the end of the year:>> 1. Linear
Algebra - YesOf MODERATE importance. The computation, no.>> 2.
Real Analysis - YesYes.>> 3. Differentiating Vector Valued
Functions (jacobians, inverse>> function theorem) - No
(nothing like this taught at my school)Of mild importance; a
week in a good course.>> 4. Point Set Topolgy - No (not
offered here)If it is general topology. Usually undergraduate
coursesare not.>> 5. Classical Stokes Theorems - Yes>> 6.
Differential Forms and Stokes Theorem - No (nothing like that
here)>> 7. Curvature for Curves and Surfaces (differential
geometry) - No>> (not offered)>> 8. Geometry - No (only course
offered is one for future high school>> teachers and was
advised not to take it)These are not basic courses at all.>>
9. Complex Analysis - No (schedule con§icts last year and this
year)Not basic, but useful.>> 10. Countability and the Axiom of
Choice - No (not offered but I have>> looked into it a bit)Too
specialized. A strong course in set theory is needed.>> 11.
Algebra - YesABSTRACT algebra.>> 12. Lebesgue Integration - No
(not undergrad here)>> 13. Fourier Analysis - No (I thought
this was for engineers)>> 14. Differential Equations - YesNone
of these are essential, and are mostly cookbook.>> 15.
Combinatorics and Probability - No (combinatorics not
offered;>> probability only after calc-based statistics is
taken)If the probability course has a statistics requirement,
it islikely that both courses are of poor quality. EVERY
statisticscourse should be based on probability, which does
not requireany statistics whatever.>> 16. Algorithms - No (the
closest thing to what is described here is a>> mid-level
computer science course).You need to get away from the idea
that mathematics consistsof calculating; get the concepts,
which requires suf'cientabstraction to get away from using
real world connectionstoo much.-- 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
=>>Message-id:
[...]>>If
you are aiming for a ph.d, then I would suggest getting into
the best M.S>>program you can, and then apply to a great ph.d
program when you get your>>M.S.. This is essentially what I am
doing, for I am in the same situation.>>Good luck!>This is
basically what I plan on doing. Is it bad to get a masters at
a school>with a PhD program and then transfer to another
school afterwards, even if you>get good recommendations? I'm
applying to a few pretty low-ranked PhD programs>and one
masters program. I just don't know of any really GREAT
masters>programs but I'm sure they are out there. Do you
know
of any?Watch out! Many schools with MS programs have
essentiallycomputational programs, and what you need for a PhD
programis the ability to do abstract mathematics and
proofs.Unfortunately, most undergraduate schools, as well as
highschools, have almost completely concentrated on
computationalcourses. Whether you can calculate derivatives
and integralsis of little importance; do you know what they
are, and canyou prove theorems? I suggest you 'nd a STRONG
abstractalgebra course to start out, and proceed by looking
for themost general abstract material you can 'nd. I have had
a student tell me that the biggest problem he hadwith general
topology was that he had had metric topology;this is the usual
method. It is unfortunate that we use thechronological instead
of logical method, and increase theamount of abstraction.
Fully abstract is more understandable,as there is less you can
use.-- 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
hrubin@odds.stat.purdue.edu (Herman>Watch out! Many schools
with MS programs have essentially>computational programs, and
what you need for a PhD program>is the ability to do abstract
mathematics and proofs.I presume by computational you mean
non-rigorous cookbookmathematics, not actual scienti'c
computation and assorted numericalmethods.
=Bushnell,> Do
you want tp say that any university outside US (Madrid,
Beijing inyour> examples) are mediocre places?I know enough of
the Chinese-speaking world to be quite sure that
anundergraduate degree in mathematics from Beijing University
is world-classby anyone's standards. A math major from
Beijing
who ends up at a Big Tenstate university for a master's
before
going to a more prestigious Ph.D.program was probably limited
by economics (ability to pay the tuition withavailable
assistanceships) or English ability as measured by foreign
studentadmission tests much more than by math ability.
Presumably, the TOP studentsfrom Beijing easily get into the
top graduate programs in the United Statesin the 'rst place,
but the also-rans from Beijing would be very
attractivecandidates to most graduate schools in the United
States, based on all theinformation I have been able to obtain
about current patterns in foreignstudent §ows.
=Nothing is
necessary,or suf'cient!
=I have been analysing this problem
but can't prove the last bits. Iwould be grateful for some
help.We have a equilateral triangle pointing upwards. The
triangle iscomposed ofa number of dots. By moving some of the
dots this triangle can be madetopoint downwards.Example: O O O
(a) O O (b) OTriangle (a) has 2 rows and is composed of 3 dots.
By moving the upperdotwe get triangle (b). What is the
connection between the number of rows and the minimumnumber
ofdots that must be moved to turn the triangle?Start by
de'ning the number of dots A(n) the nth-triangle has.The
triangle can also be illustrated as a right triangle: O O O a
= 1 + 2 1 2 O O O a = 1 + 2 + 3 O O O 1 2 3We easily realize
that the nth triangle has 1 + 2 + 3 + ... + n dots,This sum
can be written: n(n + 1) 1 + 2 + 3 + ... + n = -------- (1)
2Equ (1) can easily be proved by induction on n. Therfore: n(n
+ 1)A(n) = -------- , n > 1 2The function A(n) constitutes for
the natural numbers the thirddiagonal inPascals triangle.O is
dots that has to be moved to turn the triangle. O X X X X O X
O X O X X X X O X O X O X X O X X O X X X X X X O X X O X X O
O O O X X X O X X X O X X X X X X X X O X X X O X X X O O
ONotice how the 'xed dots are symetrical.By observing
practical experiments i have created the table below.B(n)
isthe number of dots to move.n A(n) B(n)-----------1 1 02 3 13
6 24 10 35 15 56 21 77 28 98 36 12By looking at this table I
assume that B(n)=§oor(A(n)/3). But how caniprove this?It seems
that B(n) also is some kind of sum:n
B(n)--------------------------------1 02 0 + 13 0 + 1 + 14 0 +
1 + 1 + 15 0 + 1 + 1 + 1 + 26 0 + 1 + 1 + 1 + 2 + 27 0 + 1 + 1
+ 1 + 2 + 2 + 28 0 + 1 + 1 + 1 + 2 + 2 + 2 + 3n 0 + 1 + ... +
§oor((n+1)/3)Intrestingly the numbers seems to be repeated 3
times before theyincreasewith 1.
= [...] What is the
connection between the number of rows and the> minimum number
of dots that must be moved to turn the triangle?> > [...]> > O
is dots that has to be moved to turn the triangle.> > O > X X X
X > O> > X O X O > X X X X > O X O X > > O > X X O X X O> X X X
X X X > O X X O X X > O> > O > O O > X X X O X X X O> X X X X X
X X X > O X X X O X X X > O O> O> > > Notice how the 'xed
dots
are symetrical.> > By observing practical experiments i have
created the table below.> B(n) is the number of dots to move. n A(n) B(n)> -----------> 1 1 0> 2 3 1> 3 6 2> 4 10 3> 5 15
5> 6 21 7> 7 28 9> 8 36 12I get B(6) = 8, and your pattern
implies B(n) = (n-3)(n-2)/2 + 2for n >
2.------------------------------------------------------------
---------------John R Ramsden
(jr@adslate.com)----------------------------------------------
-----------------------------Eternity is a long time,
especially towards the end. Woody Allen
=>A book I'm reading
asserts If H and K are two subgroups of a group G,>and if every
element g of G can be uniquely expressed as g=xy, where x>is in
H and Y is in K, then H and K are both normal in G.>I
haven't
been able to show this. I think I might be missing
something>really simple.I do not think you are missing
anything. I believe it isIwasawa's Theorem that a
semi-simple
non-compact Lie group Ghas a compact subgroup K and a solvable
subgroup S such thatevery element g of G has a unique
expression g=ks. Clearly,neither K nor S can be normal.A
special case of this is the theorem that an
n-dimensionalmatrix of determinant 1 can be represented as an
orthogonalmatrix times a matrix which is 0 below the diagonal.
Ifthe diagonal elements of the triangular matrix are requiredto
be positive, the representation is unique, and theconstruction
is straightforward.-- 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
=> I'm looking for an easy but
rigorously correct introduction to the> kalman 'lter for a
'nance application. Either a document (available> on
internet)
or a textbook would be good. Any suggestions?For speci'c
topics, a monograph (a book on *one* topic) is good placeto
start. TryBrian D. O. Anderson and John B. Moore, _Optimal
Filtering_ 1979 ISBN0-13-638122-7. This has no economic
examples but does explain Kalman'ltering clearly and
concisely
in the 'rst chapter or three. Thebulk of the book is
variations
and tricks which I have found to bealso of value.Masanao Aoki
is a good author and has written, e.g., _Optimization
ofStochastic System_ which promises application to economic
theory andeconometrics, but the economics is light and the
sysengr
=Let me translate a nice question from
de.sci.mathematik,for which we still don't have an
answer.Given a quadrilateral Q with corners A,B,C,D in
thatorder, i.e. with sides AB, BC, CD, DA.How to 'nd the
largest rectangle R, such that R iscompletely within Q?My
feeling is, that one has to distinguish betweena lot of cases,
regarding the shape of Q. So weneed clever algorithms for
shape-classi'cation,followed by some rules for the
computation
in eachclass.There are strong time constraints, as I
learned.The classi'cation idea itself is mathematical stuff,I
believe. The rest will be better sought for in
somecomp.sci.xxxx?Rainer Rosenthalr.rosenthal@web.de
=Only a
partial answer besides the point, but may furnish insight ;
forspecial case when diagnols cut at right angles, it is the
join ofcenter points of sides AB, BC, CD, DA. :) .. When
diagnols cutobliquely,the parallogram so formed has maximum
area,half of the areaof Q. > Let me translate a nice question
from de.sci.mathematik,> for which we still don't have an
answer.> > Given a quadrilateral Q with corners A,B,C,D in
that> order, i.e. with sides AB, BC, CD, DA.> How to 'nd the
largest rectangle R, such that R is> completely within Q?> >
My feeling is, that one has to distinguish between> a lot of
cases, regarding the shape of Q. So we> need clever algorithms
for shape-classi'cation,> followed by some rules for the
computation in each> class.> > There are strong time
constraints, as I learned.> The classi'cation idea itself is
mathematical stuff,> I believe. The rest will be better sought
for in some> comp.sci.xxxx?> > Rainer Rosenthal>
r.rosenthal@web.de
=> Somewhere in the IRS forms [and this
is no joke, it's true]You may the thinking of the following,
included as a 'ller in one ofthe MAA journals some years
back.
[Anyone have the exact reference?]Someone dies, and his will
makes some bequests, then ends by sayingafter taxes are paid,
any remaining money should be donated tocharity. Well, if
charitible donations are tax-deductible, then theamount of the
donation effects the taxes. What to do? The I.R.S.supposedly
has a form where you do this computation, but it
amountsessentially to trial and error. In fact the problem can
be solved byhigh-school algebra. (Solution of a linear
equation.) Or, not eventhat: the problem can be solved using
the method of false position:that method is described in the
Rind Papyrus, which dates from maybe2650 B.C.And some people
say the I.R.S. is behind the times...-- G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
=> I recall having
been taught not use operations such as +, -, /, x that will>
effect both sides(at the same time) of an identity to be
proven. And so I> told my students to follow this rule. One
student respectfully and> thoughtfully said why and went on to
pose that the relationship between the> two expressions we are
trying to show are identical must be either >, >=, =,> <= or <
. So with the exception of multiplying or dividing by a
negative> value, the unknown relationship will stay the
same(if we operate on both> expressions at once).> > For
example (trivial but it demonstrates the question) suppose you
want to> prove that 1 + sin(-x) = 1 - sin(x) .> Normally we
would apply the cofunction identity sin(-x) = - sin(x) the
left> hand expression and be done. But why not 'rst subtract
-1 from both> expressions and then apply the cofunction
identity?> > If the rule I remember being taught is valid
there should exist some pair of> trigonometric expressions
Expr1(x) and Expr2(x)that are not in fact identical> but would
appear to be identical if one operates on both expressions>
simulataneously. For example suppose one were trying to prove
Expr1(x) => Expr2(x). In the process he/she divided both sides
by cos(x)and found that> the modi'ed expresssions were in
fact
equal. I can't come up with such a> pair of expressions that
would validate the rule I remember. Can anyone help?> I sould
say there is nothing wrong with the method. BUT when it
iswritten and handed in, so that when it is read the writer is
not thereto explain it, the explanations should be included on
the page.Don't just write a squence of formulas. Include
also
text that tellshow these formulas are related.Example: Prove
that 1 + sin(-x) = 1 - sin(x). This would be true if we knew
sin(-x) = -sin(x) by adding 1 to bothsides. We know sin(-x) =
-sin(x) because ... Therefore 1 + sin(-x) = 1 - sin(x).Bad:
Prove that 1 + sin(-x) = 1 - sin(x). Subtract 1 from both
sides, sin(-x) = -sin(x), done.
=I thought of a simple
problem and i suspect that it has already been worked on, but
I haven't been able to 'nd any discussion of it
in intro Graph
Theory books. Does anyone know where to look for a treatment of
this problem?The speci'c example I thought of is:Supposing
that
every person in the United States can name what state they live
in and all of its adjacent states. What is the smallest group
of people required to name all 50 states, and what states are
they from?I can show by simple arguments (involving only the
degree of the vertices) that there must be at least 9 states.
And i have found several solutions with 13 states through
random trial and error.Clearly it can be generalized to graphs
unrelated to the United States. But is there a solution with 12
states?Steve
= I thought of a simple problem and i suspect
that it has already been> worked on, but I haven't been able
to 'nd any discussion of it in intro> Graph Theory books.
Does
anyone know where to look for a treatment of> this problem? The
speci'c example I thought of is: Supposing that every person
in
the United States can name what state> they live in and all of
its adjacent states. What is the smallest group> of people
required to name all 50 states, and what states are they from?
I can show by simple arguments (involving only the degree of
the> vertices) that there must be at least 9 states. And i
have found several> solutions with 13 states through random
trial and error.>If you will send me (or post) a list of the
states together with theadjacent states, I will see whatI can
do with the question. Preferred form: a list of lists of the
form[i, j1, j2,...,jk]where i is a number between 1 and 50
representing the i-th state and j1, j2,...,jk are the numbers
of the adjacent states. (I have worked on
similarproblems.)--Edwin ClarkPS If you want to check out the
literature yourself do a Google orMathSciNet search on
dominating set or domination number(s)
=...> Supposing that
every person in the United States can name what state> they
live in and all of its adjacent states. What is the smallest
group> of people required to name all 50 states, and what
states are they from? I can show by simple arguments
(involving only the degree of the> vertices) that there must
be at least 9 states. And i have found several> solutions with
13 states through random trial and error.> [...] is there a
solution with 12 states?> > If you will send me (or post) a
list of the states together with the> adjacent states, I will
see what I can do with the question. Preferred > form: a list
of lists of the form [i, j1, j2,...,jk]> where i is a number
between 1 and 50 representing the i-th state and j1, j2,>
...,jk are the numbers of the adjacent states. ...Yes, a
posted list would be good, to clarify for example whether
youcount corner adjacencies (like UT and NM, or AZ and CO),
and which underwater boundaries (as between HI and AK, MN and
MI, or RI and NY) you treat as adjacencies. Without specifying
a list you probably won't get useful answers.-jiw
=> > I
thought of a simple problem and i suspect that it has already
been > worked on, but I haven't been able to
'nd any
discussion of it in intro > Graph Theory books. Does anyone
know where to look for a treatment of > this problem?> > The
speci'c example I thought of is:> > Supposing that every
person in the United States can name what state > they live in
and all of its adjacent states. What is the smallest group > of
people required to name all 50 states, and what states are they
from?> > I can show by simple arguments (involving only the
degree of the > vertices) that there must be at least 9
states. And i have found several > solutions with 13 states
through random trial and error.> > Clearly it can be
generalized to graphs unrelated to the United States. > But is
there a solution with 12 states?> > > Steve> This is called
Graph Covering in general, and what you're looking for is a
solution for the Minimum dominating set. Unfortunately, it's
an NP
problem.http://www.nada.kth.se/~viggo/wwwcompendium/node11.
html
=The consistent way to say, There is no absolute truth
(How about this one?) ;o)THIS STATEMENT IS THE ONLY AND ONLY
ONE ABSOLUTELY-TRUE STATEMENT.George Buyanovsky
=> > The
consistent way to say, There is no absolute truth (How about
this one?) ;o)> > THIS STATEMENT IS THE ONLY AND ONLY ONE
ABSOLUTELY-TRUE STATEMENT.> > George BuyanovskyNo it isn't
George.De'ne statement.De'ne
absolutelyDe'ne true.De'ne your
premises.Then take your meds.I know that you're supposed to
be
on meds, because according to you itcannot be true that George
doesn't take meds.:)Richard
=how do you de'ne absolute?> The
consistent way to say, There is no absolute truth (How about
thisone?) ;o) THIS STATEMENT IS THE ONLY AND ONLY ONE
ABSOLUTELY-TRUE STATEMENT. George Buyanovsky
=> The
consistent way to say, There is no absolute truth (How about
this> one?) ;o)> > THIS STATEMENT IS THE ONLY AND ONLY ONE
ABSOLUTELY-TRUE STATEMENT.> > George Buyanovsky1. THIS
STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE
STATEMENT.2. Statement 1. is true:)
=> >This is not fair,
hinting at the existence of juicy §amewars>elsewhere without
saying where. a.a. is what? alt.a??? ?> > Not much of a §ame
war. For some reason, maky spent a year or so in> alt.atheism
during which time he called everyone an idiot, and at the>
same time praising me (despite the fact that I had never
replied to or> read any of his messages there). At one point,
he posted claiming I> supported his position, which I did not,
and I followed-up saying> so.come again? where? when? how?>
Then he had the little silly interlude here where he claimed
that> a page that described job opportunities in government
whose job title> included the word mathematician was an
accurate measure of the kind> of jobs that someone with a
mathematics Ph.D. could get without> further training.still in
denial?> You might even remember that.> > Since then, maky has
been sending insults my way.> > -- >
=> 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>>
Here is a shocking admission: I'm curiously growing weary of
this>> fascinating exchange. Major snippage below.[...]>> >
You used this claim to support the claim> that Arturo is
incompetent. This is just utter bullshit.>> you have some
reading comprehension problems. my initial follow up in>> no
way claims that arture is incompetent.Explain this post
is indeed either incompetent as a>> | > mathematician, or more
likely a liar, or both.>> | >> | well, he is an ADJUNCT
assistant professor. ADJUNCTS are the pariahs>> | in academic
caste system.>> | >> | so whomever you are, may well be right
on this one...>> `----How is jstevh@yahoo.com right, given
that Arturo is an adjunct?well, some who happen to be adjuncts
happen to be incompetent - duh.> > You demonstrate your usual
blunt reasoning, maky.> > For the record, your post was an
insult to me, and you know it. That's> how I read it,
that's
how everyone reasonable read it.shoudn't you be insulted by
your employers instead?please explain.> Just the latest in
your campaign to attack me because I would not> support you in
a.a.come again?> [.snip.]> > -- >
=> Why do you take so much trouble to expose such a
reasoner as> Mr. Smith? I answer as a deceased friend of mine
used to answer> on like occasions - A man's capacity is no
measure of his power> to do mischief. Mr. Smith has untiring
energy, which does > something; self-evident honesty of
conviction, which does more;> and a long purse, which does
most of all. He has made at least> ten publications, full of
'gures few readers can criticize. A great> many people are
staggered to this extent, that they imagine there> must be the
inde'nite something in the mysterious all this.> They are
brought to the point of suspicion that the mathematicians>
ought not to treat all this with such undisguised contempt,>
at least.> -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus
de Morgan>
=> > Arturo Magidin> magidin@math.berkeley.edu>
Here is a shocking admission: I'm curiously growing weary of
this> fascinating exchange. Major snippage below.> > > >
[...]> >> You used this claim to support the claim>> that
Arturo is incompetent. This is just utter bullshit.>> you have
some reading comprehension problems. my initial follow up in>
no way claims that arture is incompetent.> > Explain this post
is indeed either incompetent as a> | > mathematician, or more
likely a liar, or both.> | > | well, he is an ADJUNCT
assistant professor. ADJUNCTS are the pariahs> | in academic
caste system.> | > | so whomever you are, may well be right on
this one...> `----> > How is jstevh@yahoo.com right, given that
Arturo is an adjunct?>>well, some who happen to be adjuncts
happen to be incompetent - duh.You demonstrate your usual
blunt reasoning, maky.For the record, your post was an insult
to me, and you know it. That's>> how I read it,
that's how
everyone reasonable read it.>No, but that's just because I
'nd
your opinions to be beneathinsult, and your current claims a
show of cowardice. You can't evenstick by your sly attempts
at
an insult.>shoudn't you be insulted by your employers
instead?No. Your statement is plain. It is an attack on
me.>please explain.may well be right can only refer to the
fact that I was being called(through copying an old post of
James Harris) incompetent as amathematician, or more likely a
liar, or both. You are agreeing withthe possibility, which
means you are stating your opinion that it mayvery well be
true that I am an incompetent and/or a liar.>> Just the latest
in your campaign to attack me because I would not>> support you
in a.a.come again?For someone as ignorant as you are, you
don't
fake ignorance too
=It's not denial. I'm just very selective
about what I
accept as reality. --- Calvin (Calvin and
Hobbes)
Arturo Magidinmagidin@math.berkeley.edu> >>
Here is a shocking admission: I'm curiously growing weary of
this>> fascinating exchange. Major snippage below.[...]>> >
You used this claim to support the claim> that Arturo is
incompetent. This is just utter bullshit.>> you have some
reading comprehension problems. my initial follow up in>> no
way claims that arture is incompetent.Explain this post
is indeed either incompetent as a>> | > mathematician, or more
likely a liar, or both.>> | >> | well, he is an ADJUNCT
assistant professor. ADJUNCTS are the pariahs>> | in academic
caste system.>> | >> | so whomever you are, may well be right
on this one...>> `----How is jstevh@yahoo.com right, given
that Arturo is an adjunct? well, some who happen to be
adjuncts happen to be incompetent - duh.> > How does that
increase the likelihood that jstevh@yahoo.com is right?what
makes you assume that's i claim i maintain?
didn't you read my
previous reply?> Some who happen to be named Arturo happen to
be incompetent, too, I'm> sure. This fact
doesn't lead one to
conclude that Arturo Magidin is> likely to be incompetent or
that jstevh@yahoo.com is likely right.but the name arturo is
not in any way connected with academic ranks. ADJUNCT is.get
it?> And in any case, what does the fact that some adjuncts
are incompetent> matter, unless you were inferring that Arturo
is more likely to be> incompetent? >> that you are not alleging
that adjuncts are likely to be liars, so you must>> be
insinuating that adjuncts are incompetent. geez, i would hate
to assume that you are having problems with> existential
quanti'ers. is that it?> > Well, I think I know a thing or
two
about logic, but please enlighten> me about what error I made
regarding existentials.well, 'rst tell me how much logic you
know. can you handle venn diagrammes?>> At least, the
straightforward interpretation of your stupid words is>> that
somehow the information that *Arturo* is an adjunct increases
the>> probability that Arturo is a liar or incompetent. well,
given that some adjuncts are indeed incompetent, i would say>
that a randomly selected adjunct may well be incompetent. as
to how> the probability that a randomly selected adjunct
mathematician being> incompetent compares to the probability
that a randomly selected> non-adjunct mathematician being
incompetent, i'll leave that as an> open question for the
moment being.> > If you leave it as an open question, that
rather makes the point of> your comment (that jstevh@yahoo.com
may be right) rather hard to> grasp.> > Did you mean only
this?what else could i have meant?do you have further
objections?> (1) Some adjuncts are incompetent.> (2) Arturo is
an adjunct.> -------> Therefore, it is possible that Arturo is
incompetent.> > If so, perhaps your grasp of modalities is a
touch weaker than my> grasp of existential.> >> I eagerly
await your explanation that shows my lack of reading>>
comprehension skills. its becoming clear that you did not
understand the content of my> post.> > Know what? I agree. I
still don't. I have no idea *what* your post> meant,why the
objections then? a bit emotional? what?> unless my bad
argument above captures it.the argument where you are trying
to patch poor reading comprehension?> you also made a tonne of
assumptions about it. but worry not, i'll> gladly dissect it
for you, should you need further assistance> undertanding
it...> > Assist on.> > now, where are your answers to the
questions i asked?> > Which ones? I won't respond to your
fantasies regarding those slave> laborers known as adjuncts,
as it's not particularly my interest.then, what are you
doing
in this discussion?> Your insinuation that Arturo is
possibly/probably incompetent because> of his job title while
simultaneously not calling his competence into> question, is a
curiously more diverting topic.in that case, you are invited to
exploit it further.
<87y8swbnll.fsf@phiwumbda.org>
<87fzf3ku82.fsf@phiwumbda.org>
<87isjxvg7q.fsf@phiwumbda.org>
<8765fwy57l.fsf@phiwumbda.org>
=>> > Here
is a shocking admission: I'm curiously growing weary of
this>
fascinating exchange. Major snippage below.> > > > [...]> >
maintain that Magidin is indeed either incompetent as a> | >
mathematician, or more likely a liar, or both.> | > | well, he
is an ADJUNCT assistant professor. ADJUNCTS are the pariahs> |
in academic caste system.> | > | so whomever you are, may well
be right on this one...> `----> > How is jstevh@yahoo.com
right, given that Arturo is an adjunct?>> well, some who
happen to be adjuncts happen to be incompetent - duh.How does
that increase the likelihood that jstevh@yahoo.com is right?
what makes you assume that's i claim i maintain?
didn't you
read my> previous reply?Well what the fuck did [you] may well
be right on this one... mean,if it didn't mean that your
comments increased the likelihood that theoriginal [im]poster
was right?>> that you are not alleging that adjuncts are
likely to be liars, so you must> be insinuating that adjuncts
are incompetent.>> geez, i would hate to assume that you are
having problems with>> existential quanti'ers. is that
it?Well, I think I know a thing or two about logic, but please
enlighten>> me about what error I made regarding existentials.
well, 'rst tell me how much logic you know. can you handle
venn> diagrammes?I have more than suf'cient background in
logic, I am sure. Honest.[...]Did you mean only this? what
else could i have meant? do you have further objections?> (1)
Some adjuncts are incompetent.>> (2) Arturo is an adjunct.>>
------->> Therefore, it is possible that Arturo is
incompetent.I just want to be clear: The argument above is the
argument you knowclaim to have advocated? *This* is an argument
that you think is agood argument? This from the man that wants
to teach me aboutexistentials?You're adorable.>> Know what?
I
agree. I still don't. I have no idea *what* your post>>
meant,
why the objections then? a bit emotional? what?> unless my bad
argument above captures it. the argument where you are trying
to patch poor reading> comprehension?No, the argument with
premises numbered (1) and (2) and withunlabeled conclusion.
The one you seem to think is just peachy.> you also made a
tonne of assumptions about it. but worry not, i'll>> gladly
dissect it for you, should you need further assistance>>
undertanding it...Assist on.now, where are your answers to the
questions i asked?Which ones? I won't respond to your
fantasies
regarding those slave>> laborers known as adjuncts, as it's
not
particularly my interest. then, what are you doing in this
discussion?Mostly, mocking a complete moron. Shame you
hadn't
noticed.-- ...you are around so that I have something else to
do when I'm not'guring something important out.
I was
especially intrigued on thisiteration by cursing, which I
think I'll continue at some later dateas it's
so amusing. ---
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id hBVDVdx31773;
=you can also join
this
site:http://alephnulldimension.net/scgi-bin/ikonboard.cgimain
topic
headings:philosophy/
spiritualitymathematicsartsciencemiscellaneousthe site
de'nitely needs more users. hey, maybe you can help form its
future...also, physicsforums.com has a nice feature of being
able to post tex with [tex][/tex] commands where $$ would
normally be.cheersphoenix
=> > I've come across this
equation which describes the dependency of> certain factors in
the operation of bipolar transistors:-> > Ic = Is[exp(v/t)-1] > Ic is the transistor's collector current (typically from
a
few to> tens of mAmps)> Is is its saturation current (typically
*very* small like 5 fAmps,> for instance.> > I'm confused
about
the term exp and what it refers to - does it mean> exponent or
e^xyes> or something else? I need someone to show me a> worked
example, please.> > Let Is = 5^-15> Let (v/t)=0.5> > Now, from
the above, how would I 'nd Ic?> exp(0.5) you can do on your
calculator, exp(0.5)=1.649, solc = ls*(exp(v/t)-1) =
(5^(-15))*(0.649) = 2.13*10^(-11) approx.Now v/t will have to
be a bit larger to get values like the ones yousuggest. Here,
if ls=5^(-15) and v/t = 5, then we getlc =
(5^(-15))*(exp(5)-1) = 4.86*10^(-9),v/t = 20 to get lc =
0.0159,v/t = 30 to get lc = 350.
=It would seem that the
Action integral for the world sheets of classicalstring theory
are presently justi'ed as a higher dimensional version of
theone dimensional case. But I wonder if this formulation
would be a naturaldescription if we were 'rst given the
geometry of a world-sheet. What wouldbe the most natural
mathematical description of what is happening with
aworld-sheet?I have a justi'cation for a type of world-sheet
geometry, but I am not sureof the mathematics to descript it.
At one instant of time there is a curvethrough space with a
function evaluated at each point along the length ofthe curve.
This string sweeps out a surface as time passes. If I
don'tknow
the path it takes or the function at each point on the surface,
then itwould seem that the best I can do is to describe the
situation with asurface integral of a scalar function of some
unspeci'ed path. The integralis then a functional of the
path.
This is pretty much what the Actionintegral is. Then I would
naturally apply requirements of invariance withcoordinate
changes to come up with a Euler-Lagrange vector. Would I
thenhave a reason to assume a vanishing variation to come up
with Euler-Lagrangeequations (set to zero) and also
Noether's
theorem of conserved values? Isthis last step assuming
conserved quantities to begin with? Or is avanishing variation
in and of itself a type of symmetry or invariance thatone would
naturally expect to employ due to some obvious intrinsic
property?I wonder. Comment requested.See:
http://www.sirus.com/users/mjake/StringTh.html
=I have a
question and would appreciate any help.Let (f_n)_ be a
sequence of real valued functions de'ned on [a, b]that
converges uniformly to f. For every x in [a,b], let F_n(x)
=Integral (from a to x) f_n(t)dt. Then, (F_n) converges to
F(x) =Integral (from a to x) f(t)dt. Is this convergence
uniform on [a,b]?If not, is there any condition that assures
uniformity?If each f_n is de'ned on [a, inf) and its
in'nite
integral exists,then does the sequence of such in'nite
integrals converge to thein'nite integral of f?Artur
=>I
have a question and would appreciate any help.>Let (f_n)_ be a
sequence of real valued functions de'ned on [a, b]>that
converges uniformly to f. For every x in [a,b], let F_n(x)
=>Integral (from a to x) f_n(t)dt. You have to say a little
more than you have or this integraldoesn't even exist.
Possibly you meant to assume that f_nwas continuous - that
would do it.>Then, (F_n) converges to F(x) =>Integral (from a
to x) f(t)dt. Is this convergence uniform on [a,b]?Yes.>If
not, is there any condition that assures uniformity?If each
f_n is de'ned on [a, inf) and its in'nite
integral
exists,>then does the sequence of such in'nite integrals
converge to the>in'nite integral of f?No. Someone else has
already given a counterexample.There's a condition that does
give convergence here: If yousuppose that g >= 0, the integral
of g is 'nite, and |f_n| <= gfor all n then the integral of
f_n
converges to the integralof f. This is the Dominated
Convergence Theorem frommeasure
theory.>Artur************************David C. Ullrich
=> I
have a question and would appreciate any help.> Let (f_n)_ be
a sequence of real valued functions de'ned on [a, b]> that
converges uniformly to f. For every x in [a,b], let F_n(x) =>
Integral (from a to x) f_n(t)dt. Then, (F_n) converges to F(x)
=> Integral (from a to x) f(t)dt. Is this convergence uniform
on [a,b]?> If not, is there any condition that assures
uniformity?Yes, because | F_n(x) - F(x) | can be majorized by
theproduct of (b-a) and the maximum difference between f_n(t)
and f(t).> If each f_n is de'ned on [a, inf) and its
in'nite
integral exists,> then does the sequence of such in'nite
integrals converge to the> in'nite integral of f?No. Consider
a sequence of functions f_n which converge to f(t) = 0such
that for each n, int (0,in'nity) f_n = 1, but theygrow
shallower and wider ( e.g., lambda exp(-lambda t) , whenlambda
= 1/n). Artur>This had better not have been homework :-).Best
wishes, MikeX-Cise:
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kramsay@aol.com (KRamsay) said:>As far as beauty goes, it
seems to me that ordinarily proofs which>avoid proof by
contradiction (and proof via the contrapositive) when>they
don't need them *usually* are a little bit more beautiful
than>the version written in an unnecessarily negative form, so
if we're>going to talk about minor cosmetic blemishes, sure,
it's ever so>slightly better (IMO) to write the proof in a
forward direction:>the distance to zero is continuous, and
since that's bounded the set>is bounded, etc.OTOH, if a
reductio ad absurdum is short and sweet while the directproof
involves a myriad of special cases, most Mathematicians
wouldconsider the reduction ad absurdum to be more beautiful
and moreelegant.-- Shmuel (Seymour J.) Metz, SysProg and
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= at 02:54 PM,
conesetter@btopenworld.com (conesetter) said:> To all: please
excuse my not being good at following what others>have
done.The problem was not your failure to follow; the problem
was youracting as though you had followed it and claiming an
imaginary errorin the material that you did not follow.
Admitting ignorance andasking for clari'cation would have
yielded a different thread.>But one may think that>branches
are worth studying for there own sake, and then in
context>just think of Riemann surfaces as a byproduct.
Certainly, although I don't see the motivation for it. At
'rstreading your analysis in those terms seems correct.--
Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to
<3fee0f44$21$fuzhry+tra$mr2ice@news.patriot.net>
<3fef7511$8$fuzhry+tra$mr2ice@news.patriot.net>
<3ff2665e$2$fuzhry+tra$mr2ice@news.patriot.net>
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= at 12:16 PM,
raynand@netzero.net (Jefferson Rourke) said:>Yes, I did answer
honestly. No. You lied and you're still lying.>I lack a
belief
in gods.K3wl. However, your position goes beyond lacking a
believe in a god;you actively believe that there is no god and
insult people whodisagree with you. To describe your position
is simply lacking abelief in a god is dishonest.>I on the
other hand am honest and complex.One out of two isn't bad.
Now
if you could only cause the 'rst to betrue.>No you
didn't read
any false claims. Really. Then why do the dictionaries
disagree with you?>You chose to misunderstand what was being
said.No, you tried to write things and then pretend that you
hadn't. Youtried to claim that words meant things that they
patently didn't.>You just don't get THE pointI
get the point
that you are a bigot and a true believer. You feltdiscussing
the issue of religion in general rather than the
particularpoint he raised. Well, what's sauce for the goose
is
sauce for thegander. If you can address a peripheral part of
his post rather thanwhat he wished to discuss, then I can
address the parts of your postthat show you to be a dishonest
fool with delusions of adequacy.>Other ideas were borrowed
from other cultures such as the zero >which was brought to the
west by muslims from India.India was another religious culture.
So you admit that religouscultures have contributed something
positive?>Then you are sadly lacking in your knowledge of
historical facts.PKB. It is *you* that is exhibiting a lack of
historical knowledge, alack of reading comprehension, or both.
The phrase I was commenting onwas the human race entering the
Dark Ages. Well, tonto, only onepart of the Human Race entered
the Dark Ages; Asia did not and theMuslims you so despise did
not. Or perhaps you don't consider themHuman?>You might want
to do some research on how the Catholic church >burned whole
libraries and thousands of priceless books.Don't presume to
teach your grandmother to suck eggs. >Where did I say they are
not human?the human race entering the Dark Ages would seem to
exclude everysociety that did not enter the Dark Ages.>How did
you come up with this anyway?By having an open mind and looking
at the available evidence. Youshould try it sometime.>I never
said the Dark ages were universal, Shmol.Nu, faygleleh, how is
the human race not universal? Or are you incommunication with
sentient life on another planet?>I was speaking in the context
of the Catholic church and >christianity in Europe.Were that
true you would have written Europe, not the human race.>So
who's not paying attention, Shmol?You. You
don't seem to even
be able to use cut-and-paste correctly.>Nice try, but you do
not de'ne who I am, Correct. You do. By your actions,
including your words. Havingtransmitted them for all to see,
you can't require people to pretendthat they
don't exist.>I'm
a true believer in mankind,The religous are part of mankind.
Your choice of the terms mentalillness and madness show that
you have a faith that they are wrong,deny it how you
will.>EVERYBODY is uneducated about something.Yes, but you
have such broad coverage of things to be ignorant about.>if
you think you can talk down to me because I didn't know a
phrase>you're laughable.No, I think that everybody here can
talk down to you because you are afool. Your willful ignorance
is just a symptom.-- Shmuel (Seymour J.) Metz, SysProg and
tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate:
admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose:
george_cox@btinternet.comX-Punge: Micro$oft
rsm109@york.ac.uk (rsm109@york.ac.uk) said:>It's not just
iron
that modern science believes was formed in stars>and later
condensed to form the solar system. It's all elements>except
H
and possibly He. H, He, Li and Be, none of which were known to
the ancients. For thatmatter, CHON were not known to the
ancients, nor were any of the otherelements necessary for life
with the exception of Fe and S.-- Shmuel (Seymour J.) Metz,
<3fef7511$8$fuzhry+tra$mr2ice@news.patriot.net>
tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate:
admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose:
george_cox@btinternet.comX-Punge: Micro$oft
= at 04:39 PM,
raynand@netzero.net (Jefferson Rourke) said:>Well then, I'll
put it this way for you and Shmuel I lack a belief>in gods,
therefore, when someone comes around with a religion or>other
mystical god-concept and tells me it is reality without
any>proof except faith - I don't believe them and I deny
that
it is truth>in reality.in
(Seymour J.) Metz, SysProg and JOATnot reply to
=Wheeler is correct, Einstein's
GR is battle-tested bothphilosophically and experimentally.So
is orthodox quantum theory.Simplistic frontal assaults on
their well-established foundations willnever succeed.Jack,Not
so. The term simplistic is too fuzzy. The previous
statement,that Wheeler is correct, Einstein's GR is
battle-tested bothphilosophically and experimentally. So is
orthodox quantum theory -is a distorsion of Wheeler's
views.JS: I do not understand your point here? Please give
more details. I suggest that that we may never understand this
strange thing,the quantum, until we understand how information
may underliereality. Information may not be just what we
*learn* about the world.It may be what *makes* the world.
(J.A. Wheeler, ïGeons, Black Holesand Quantum
Foam')JS: Recent
work by Lee Smolin on the weak hologram principle adds a lot of
detail to Wheeler's remark. Area as quantum information
§ow.
See also Smolin's pop book Three Roads to Quantum
Dynamics.One
issue in regard to quantum information §ow is signal locality
vs. signal
nonlocality.http://www.quantum'elds.com/469Maclay.pdfI say
that quantum gravity needs signal nonlocality in violation of
orthodox quantum theory with signal locality. This allows
information inside the black hole to be available to the
outside observer and may change Lee Smolin's distinction
between the strong and weak hologram principles.LOCAL
MACRO-QUANTUM theory, with signal nonlocality is to GLOBAL
micro-quantum theory with signal locality as LOCAL General
Relativity with curvature is to GLOBAL Special Relativity
without curvature.AJ: Quantum theory needs to unify dy-namics
and
bi-namicshttp://www.cassiopaea.org/quantum_future/jadpub.htm#
blaja95aarkJS: What is
bi-namics?##################################################Dr
Arkadiusz
Jadczykhttp://www.cassiopaea.org/quantum_future/homepage.htmJS
: Your website is very interesting Ark, but would require many
hours to seriously comprehend.Can you give a kind of public
synopsis of your key ideas, your motivation, your
philosophy?Show us your hand in this Cosmic Poker Game. :-)
You are a well-educated PhD theoretical physicist.
=Paul
Zielinski has attempted a defense of the Yilmaz-Puthoff type
challenge toEinstein's gravity theory on the basis that
there
should be a LOCAL stress-energydensity tensor of the pure
gravity 'eld in vacuum far from normal mass-energysources
i.e.
Tuv(Matter) = 0.In fact such a tensor exists trivially ~
Guv(Einstein) which is exactly zeroin ordinary non-exotic
vacuum without any w = -1 zero point dark energyof negative
pressure or dark matter of positive pressure. This
conclusionis consistent with MTW(1973) before exotic vacua
were conceived anddiscovered experimentally as we now know in
the factthat TOTAL energy, momentum and angular momentum of
thepure gravity 'eld are not simple integrals of LOCAL
OBSERVABLES.Smolin below addresses this issue at least
obliquely:arXiv:hep-th/0003056 v1 8 Mar 2000The strong and
weak holographic principlesLee Smolin_Center for Gravitational
Physics and GeometryDepartment of Physics, The Pennsylvania
State UniversityUniversity Park, PA, USA 16802 andThe Blackett
Laboratory,Imperial College of Science, Technology and
MedicineSouth Kensington, London SW7 2BZ, UKMarch 1,
2000ABSTRACTWe review the different proposals which have so
far been made for theholographic principle and the related
entropy bounds and classify them intothe strong, null and weak
forms. These are analyzed, with the aim of discoveringwhich may
hold at the level of the full quantum theory of gravity.We
'nd
that only the weak forms, which constrain the information
availableto observers on boundaries, are implied by arguments
using the generalizedsecond law. The strong forms, which go
further and posit a bound on theentropy in spacelike regions
bounded by surfaces, are found to su_er fromserious problems,
which give rise to counterexamples already at the
semiclassicallevel. The null form, proposed by Fischler,
Susskind, Bousso andothers, in which the bound is on the
entropy of certain null surfaces, appearsadequate at the level
of a bound on the entropy of matter in a singlebackground
spacetime, but attempts to include the gravitational degrees
offreedom encounter serious dif'culties. Only the weak form
seems capable ofholding in the full quantum theory.The
conclusion is that the holographic principle is not a
relationshipbetween two independent sets of concepts: bulk
theories and measures ofgeometry vrs boundary theories and
measures of information. Instead, it isthe assertion that in a
fundamental theory the 'rst set of concepts must becompletely
reduced to the second._ smolin@phys.psu.edu.....Quasi-local
quantities in classical general relativityEven in classical
general relativity, it is well understood that
diffeomorphisminvariance and the equivalence principle forbid
the possibility of local de'nitionsof the basic dynamical
quantities such as energy, momentum andangular momentum. These
kinds of quantities can only be de'ned in termsof integrals
over two dimensional surfaces in the spacetime. When
thosesurfaces are taken to the boundary, in non-cosmological
spacetimes, thesebecome the well known asymptotic de'nitions
of energy, momentum and angularmomentum. However, even in
cosmological spacetimes where there areno boundaries one may
de'ne what are called quasi-local observables[45, 46],in
which
the energy, momentum and angular momentum of an arbitraryregion
are de'ned in terms of certain integrals over its boundary.
SincePenrose's original suggestion[45] many different
proposals have been madefor such quasi-local
observables[46].If there are to be non-trivial notions of
energy, momentum and angularmomentum in a quantum theory of
cosmology then, these must be de'nedso that their classical
limits are these quasi-local quantities. The
simplestpossibility is that the hamiltonian in quantum gravity
should itself be quasilocal,that is de'ned on two dimensional
surfaces, which in the classical limitwill become spacelike
surface embedded in spacetime. This implies someform of the
holographic principle, for if the Hamiltonian is associated
withsurfaces there must be many hamiltonians, each associated
with a differentchoice of surfaces, and the same must be true
of the algebra of observablesand the hilbert spaces on which
they are represented.Relational approaches to quantum
cosmologyAnother kind of argument for the importance of
surface observables in aquantum theory of cosmology was given
by Crane[3], even before the holographichypothesis of `t Hooft
and Susskind was proposed. Crane noted thediffculties of
de'ning a coherent measurement theory for a quantum stateof
the whole universe and proposed instead that the division of
the universeinto two parts-system and observer-that is basic
to Bohr and Heisenberg'smeasurement theory might be
relativised, so that there would be notone quantum state of
the universe, but a system of observable algebras andhilbert
spaces, one associated with every possible splitting of the
universeinto two parts[3].To realize this idea, Crane proposed
a categorical framework to describepositingfunctorial
relationships between the category of cobordisms of
manifoldsto topological quantum 'eld theory, as those
theories
can be formulated insuch categorical terms. As topological
quantum 'eld theories are the onlyspaces,one may try to use
them to construct examples of holographic
theories[13].Furthermore, as Crane pointed out, it may be
possible to extend these structuresto quantum theories of
gravity because it is a fact that at both theclassical and
quantum mechanical level, and for any dimension[47],
generalrelativity and supergravity can be understood as
deformed or constrainedtopological quantum 'eld theories[49,
23, 13, 19, 51, 50, 14, 15].Crane's proposal has been an
inspiration for the development of whathave been called
relational[48, 42, 43, 44] or pluralistic [19, 5] approaches
toquantum cosmology. Using the fact that general relativity
and supergravityare constrained topological 'eld theories, it
has been possible to realize thisidea in the context of full
formulations of quantum gravity and M theory[33, 34, 54].An
even stronger version of Crane's argument was proposed
recently byMarkopoulou[42, 43, 44], who noted that even in
classical general relativitythe logic of propositions which
can be given truth values by observersin a closed universe is
non-boolean, because each observer can only assertthe truth of
falsity of propositions about their past.Jack interjects: Of
course quantum logic is a partially ordered non-boolean
lattice of yes-no answersto simple quantum measurements in Von
Neumann's sense. Smolin's remark only works if
there issignal
locality. So this is an alleged curious connection between
Einstein's gravity and orthodox micro-quantum theory.LS:
Rather than beinga boolean algebra, the algebra of
propositions relevant for a classical cosmologicaltheory is a
multivalued Heyting algebra[42]. When quantized, theresulting
algebra of projection-like operators cannot be represented on
a singleevery possible event at which observations are
made[43]. As each observerreceives information from a distinct
past, the algebra of observables theythey observe, must vary10.
Given the conclusions reached in the precedingsections of this
paper, this is framework is then appropriate for a
formulationof the weak holographic principle[10].13
ConclusionsThe conclusion of the arguments we have given here
is that the holographicbound and holographic principle can
only survive in a quantum theory ofcosmology in their weak
forms, proposed in [10]. While logically weaker,this form is
more radical than the strong forms, in its implications for
how10 Related structures have been studied also by Isham and
collaborators [55], who notethe consistenthistories
proposal[56] precisely. a measurement theory of quantum
cosmology must be constructed. First,the weak forms require
that causal structure exist even at the Planck scale.This most
likely cannot be realized in a conventional formulation of
quantumspacecontaining the physically allowed wavefunctions of
the universe. Instead,such a description may have to be
formulated along the lines proposed ina representation for an
algebra of observables accessible to a single localobserver at
an event or a local region of a spacetime history. These will
berelated to each other by maps which re§ect the quantum
causal structure.In such a spacetime, evolution becomes
closely intertwined with the§ow of quantum information which
also de'nes the causal structure at the Planckscale.
Interactions have to do with the processing of the information
atevents; as noted in [43, 44] a quantum spacetime then becomes
very like aquantum computer that can dynamically evolve its
circuitry.Jack: I note key role of signal locality in
Smolin's
analysis.LS: It is then dif'cult to escape the conclusion
that
the holographic principle,in its weak form, is telling us that
nature is fundamentally discrete. JS: Is this hitting a §ea
with a sledgehammer? ;-)LS: The 'niteness of the information
available per unit area of a surface is to be takensimply as
an indication that fundamentally, geometry must turn out to
reduceto counting. Of course this conclusion has been reached
independentlythrough other arguments coming from quantum
gravity[1, 23, 24, 25, 5] andstring theory[52, 2]. But, as can
be seen most clearly from the argument ofJacobson[53], the
entropy bounds and holographic principle tell us that
thedescription of nature in terms of classical spacetime
geometry is not onlyanalogous to the laws of thermodynamics,
it must be exactly the thermodynamicsof the fundamental
discrete theory of spacetime.JS: I do not like the implicit
assumption of thermal equilibrium at the sub-quantalspin
network level of extra variables that ensures signal locality
as shown byAntony Valentini.LS: What we learn from the
analysis of this paper is that in such a theorythere is no
room for the notion of a bulk theory, and hence no
fundamentalrole for a bulk-boundary correspondence. There is
instead a network ofscreen histories, which describe what
possible observers might be able toobserve from particular
events in their spacetime. By averaging over historiesa bulk
description may emerge at the semiclassical level, but only as
anapproximation in which the past of a particular observer can
be described to'rst order in a perturbation expansion in
terms
of a particular _xed classicalhistory. Thus the proper role of
a bulk-to-boundary map may be to serve asa correspondence
principle to constrain the classical limit of a
backgroundindependent quantum theory of gravity.To put it most
simply: the holographic principle is not about a
relationshipbetween two sets of concepts, bulk and screen and
geometry andinformation §ow. It is the statement that the
former reduce entirely to thelatter in exactly the same sense
that thermodynamic quantities reduce toatomic physics. The
familiar picture of bulk spacetimes with 'elds and
geometrymust emerge in the semiclassical limit, but these
concepts can playno role in the fundamental theory.Can this
picture be used to construct a realistic quantum theory of
gravitywhich addresses also the other problems in the subject?
As mentionedin [10] an example of such a theory is provided by
a class of background independentmembrane theories proposed in
[33]. These extend the formalismof loop quantum gravity in a
way as to provide a possible background independentform of
string theory[34, 54]. So the answer is a very
provisional,yes. Much work remains to be done, but the moral
is that the holographicprinciple, in at least its weak form,
is likely to feature signi'cantly in boththe mathematical
language and the measurement theory of the future
background
=> > Sorry, the ridiculous assertion is 0.999....
does NOT equal 1.> > It certainly does!> > Just try to subtract
0.999... from 1:> > 1 - 0.999... = 0> > Reason:> > There is no
real number between 0.999... and 1, and, therefore, they> must
be one and the same number!> > PHNeat proof, but you are
playing foot loose with the de'nition of =. 0.999... is an
in'nite series, a shorthand notation for .9, .99,.999, ... 1
is an integer. The relationship is that 1 is a (the)value for
which, for every D>0 there is an N such that for all M>N
thevalue of the Mth number in the series is between 1-D and
1+D.The problem occurs when people start saying that .999...
equals 1.Charlie Volkstorf
=> Neat proof, but you are
playing foot loose with the de'nition of =.> 0.999... is an
in'nite series, a shorthand notation for .9, .99,> .999, ...
1
is an integer. The relationship is that 1 is a (the)> value for
which, for every D>0 there is an N such that for all M>N the>
value of the Mth number in the series is between 1-D and 1+D. The problem occurs when people start saying that .999...
equals 1.But this is exactly the de'nition of an
in'nite
decimal. What doyou thing .999... means, if not the limit of
the partial sums?
=> Neat proof, but you are playing foot
loose with the de'nition of =.> 0.999... is an
in'nite series,
a shorthand notation for .9, .99,> .999, ... 1 is an integer.
The relationship is that 1 is a (the)> value for which, for
every D>0 there is an N such that for all M>N the> value of
the Mth number in the series is between 1-D and 1+D.> > The
problem occurs when people start saying that .999... equals
1.There are ways to associate a value with 0.9999... such that
the valuedoes not equal 1. There are two immediate
consequences, though:1) 0.3333... does not equal 1/3 .2)
0.9999... and 0.3333... (and other such values) are not real
numbers.-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
=> There are ways to associate a
value with 0.9999... such that the value> does not equal 1.
There are two immediate consequences, though: 1) 0.3333...
does not equal 1/3 . 2) 0.9999... and 0.3333... (and other
such values) are not real numbers.To expand on this.If you
assume 0.999... and 1.000... represent real numbers,then
0.999... = 1.000... by de'nition.Real numbers can be
de'nedas
the limit of a Cauchy sequence.The limit of the sequence
represented by .999... is 1.You can claim 0.999... != 1.000...
but you shouldn'tcall them real numbers. Real number has a
standardde'nition.Russell- 2 many 2 count
=In sci.logic,
22:31:37 -0500<1g6yh02.h3uybwz75oe4N%panoptes@iquest.net>:> >>
Neat proof, but you are playing foot loose with the de'nition
of =.>> 0.999... is an in'nite series, a shorthand notation
for .9, .99,>> .999, ... 1 is an integer. The relationship is
that 1 is a (the)>> value for which, for every D>0 there is an
N such that for all M>N the>> value of the Mth number in the
series is between 1-D and 1+D.The problem occurs when people
start saying that .999... equals 1.> > There are ways to
associate a value with 0.9999... such that the value> does not
equal 1. There are two immediate consequences, though:> > 1)
0.3333... does not equal 1/3 .> > 2) 0.9999... and 0.3333...
(and other such values) are not real numbers.Well, if x =
0.9999.... = 1-d, then (x/10 + 0.9)= 0.9999.... = 1-d/10. So
either we have some veryweird set of numbers all with the same
decimal expansion0.9999... (and another weird set of numbers
all of whichare non-zero in'nitesimals), or d = 0 and
they're
allequal to 1. The second hypothesis is probably
morestraightforward.Also, 0.999... can be taken as a sequence,
which turnsout to be a Cauchy sequence, with limit 1.-- #191,
ewill3@earthlink.netIt's still legal to go .sigless.
=>
Sorry, the ridiculous assertion is 0.999.... does NOT equal 1.
It certainly does! Just try to subtract 0.999... from 1: 1 -
0.999... = 0 Reason: There is no real number between 0.999...
and 1, and, therefore, they> must be one and the same number!
PH Neat proof, but you are playing foot loose with the
de'nition of =.> 0.999... is an in'nite series,
a shorthand
notation for .9, .99,> .999, ... 1 is an integer. The
relationship is that 1 is a (the)2 - 1 is a function, 1 is a
numberthey're still equal.What is the number between 0.99...
and 1?Herc> value for which, for every D>0 there is an N such
that for all M>N the> value of the Mth number in the series is
between 1-D and 1+D. The problem occurs when people start
saying that .999... equals 1. Charlie Volkstorf
=There
exists a bijection from the Cantor set (middle thirds removed)
to[0,1]. (Ternary rep - 0's and 2's -> Binary
rep 0's and
1's)Is this mapping continuous? It may be but
I'm having
trouble seeing whythis is so.Help?LW
=> > There exists a
bijection from the Cantor set (middle thirds removed) to>
[0,1]. (Ternary rep - 0's and 2's -> Binary
rep 0's and 1's) Is this mapping continuous? It may be but I'm having
trouble
seeing why> this is so.Yes, it is continuous. Call it f. It is
easy to see that, given x and yin the Cantor set, |f(x) -
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i02JJiX09965;
papers published, or lecture notes written.keep us posted.>
Here is my web site and also list of papers submitted for>
publications.congratulations on your newly acquired adjunct
BastiCincinnati,
OH.>------------------------------------------------------->
http://hometown.aol.com/mbasti21/myhomepage/index.html
Papers submitted for publication:1-Basti.M. Solving
polynomials with differential equations. Submitted>for
Polynomial. Submitted for publication to the>proceedings of
publication to the>Mathematics of computation of the
publication to the>electronic Research announcements of the
Computation.6-Basti.M. New Methods Of Solving Riccati
Journal on Computation.7-Basti.M. New Methods Of Solving
Riccati Differential Equations II.>Submitted for publication
Journal on applied Mathematics.9-Basti.M. New solutions of
Bessel Differential Equations. Submitted>for publication to
=> I recommend you
read > > S. Smale's review's of E. C.
Zeeman's catastrophe
theory book,> Bull. Amer. Math. Soc., Vol. 84, 1360-1368,
1978. > > S. Andreski, Social Science as Sorcery, St.
Martin's
Press, 1972.> > C. Truesdell, An Idiot's Fugitive Essays on
Science, Springer, > 1984, p. 122ffAlso, try V. Arnold's
book
on Catastrophe Theory. He's a smart guy who has some
interesting things to say.-- -- Lou Pecora My views are my
own.
=>a week, I guess a few more won't make much
difference.If there's no con'dentiality clause,
I'll post the
problem (if
any),>here.---------------------------------------------------
------------------------>John R Ramsden
(jr@adslate.com)>---------------------------------------------
------------------------------You will post us or tell us you
aren't / can't, right?--Lynn
=The problem is this:You have 2
experiments both modeled by a simple regression model,Y = alpha
+ beta(x - xbar) + R with R~G(0,sigma) and independant.(note G
represents the Gaussian distribution)The 'rst experiment has
n1 samples, the second has n2 samples.We know Sxx1,Sxx2
(likewise for Syy,Sxy) least square estimates forbeta1, beta2,
alpha1 and alpha2 and sigma1 and sigma2 (i will callleast
square estimates beta1^ and beta2^ etc.)So the test for (beta1
- beta2) = 0 goes as follows (please correct mewhere i'm
wrong)First, beta1^ ~ G(beta1, sqrt(sigma1/Sxx))likewise for
beta2^.so beta1^ - beta2^ ~ G(beta1 - beta2, sqrt(sigma1/Sxx1
+ sigma2/Sxx2))therefore, ((beta1^ - beta2^) - (beta1 -
beta2))/sqrt(sigma1/Sxx1 +sigma2/Sxx2) ~ G(0,1)BUT ... we dont
know sigma1 and sigma2, only their estimates.Normally, with
the sigma that you dont know and turn thegaussian to a students
t distro, but you have two sigmas under a rootsign which means
you have to go back to the de'nition for sigma^which involves
a double sum but I will try to write it down but itsbasically
the square root of the sum of the residiules squared:sigma^ =
sqrt(sum(i = 1 to 2)*sum(j = 1 to n(i))*r(i,j)^2/(n1 + n2
-2))now, 'rst is the denominator right? n - q = n1 + n2 - 2??
(q isnumber of independant parameters).now expanding the
double sum is messy so i wont type the forumla outbut I cant
simplify it to anything that uses beta^'s and
Sxx1/2's,Sxy1/2's etc.I think, on this
particular exam, the
prof wanted us to just sub inthe estimates for the real values
(and not go back to the de'nitionof sigma^) but I thought
this
was fundementally wrong.Any thoughts? Any need for
clari'cation?Chris
=>> http://www2.b3ta.com/hawking/>> --->>
Tourette's bugger, post some content please, I just
disconnected before I> clicked your post. ok, I just got the
§ash show, not bad, but who was the young cronie besides>
Einstein and hawking?>My guess is Bohr, crossposted to
sci.math for the answer.Herc
=I CHALLENGE |-|ERC, THIS IS A
FOR ME for the 24 period we agree on beforehand, winneris
person with most POSTS TO USENET DURING this beforehand agreed
upontime period;3) In the event of TIE # OF POSTS we resume
IT GOOD.Winner is person WITH MOST POSTS TO USENET IN THE 24
PERIOD WE AGREEUPON BEFOREHAND.RESULTS , Also open to
negotiation WITHIN REASON:-Winner:Winner has LOSER pay for
high bandwidth (50 gigs plus xfers per month)website FOR ONE
YEAR , winner gets to put whatever on website andLOSER MUST
SAY ONE OF1) IF YOU |-|ERC YOU WIN, I admit you HAVE
PARANORMAL POWERS, DESERVETHE RANDY MILLIONS, AND ETC, I put
it in writing for one of the webpages2) IF **I** WIN, YOU
=CLAP CLAP to you
says Lordjudging by the names of the 2 replies.CLAP CLAPClave
informative
broadcaster.But OK, starting soon, I don't want you
slandering
me thoughas part of your posts, what subject matter?And
afterwards we collate the replies to measure my
claim.Herc
=I challenge him too.....TO SHUT THE FUCK UP
BECAUSE HE'S A FUCKING LOONY!!!!
=Spam King [bow down u ho
both.PLONK
=> Kevin's logic is to the point, and I agree
100%. Just as an addendum,> however, I would clarify that when
he talks about tangential force> he's talking about the
relationship of the sides of the bolt (or> whatever) to the
vector of rotation. Thinking about using force which> is most
nearly parallel to the vector of rotation, you can see that>
something like a wing nut (essentially a 2-sided head) is able
to exert> the most force in the proper vector. A 3-sided head
(which doesn't> exist, to my knowledge, as a standard screw,
bolt or nut) would be next> best. After that, your square head
would be next best able to exert> force to the vector of
rotation.> > As Kevin points out, however, this has little to
do with what actually> WORKS best.> > ïSporky'>
> Consider
heads of 2, 3, 4, 5, etc. sides. As the number of sides>
increases, the cross-sectional area of the head also
increases. It is> this cross-section that determines the shear
strength of the head.> However, as the number of sides
increases, the force acting on the> head become more
tangential: with a pure circle it is strictly> tangential (but
the surface area is the maximum). Note that the> goodness of
these two properties vary in opposite directions: as one> gets
better the other gets worse. Therefore, there must be some>
optimum, and that could very likely be the square head.> > In
the real world, however, idealization based on an engineering>
principle is seldom the overall best. For example, a winch
shaft is> round instead of splined or square (or any other
§at-sided shape),> and the shaft is coupled using a strict
friction-'t. This is because> the friction 't
can give very
nearly the same torque as a spline, but> the failure mode is
much less disasterous (it simply slips instead of> shearing
the splines from the shaft).> > At some point, a product will
be measured by what it does not do, not> necessarily by how
well it does what is advertised.> > KevinI trust Archie has a
large bucket handy in his workshop- he'll need itfor all the
broken (square shafted ) bits!!before breaking the bit!Jim
G
= I'm just a sophomore in college, so I don't
know much.
In our mathbook, it said the integral of sin(x^2) can't be
integrated easily. What does it mean, ïeasily'?
Is there a
simple antiderivative ofsin(x^2), or is it some kind of
in'nite thing? Has it been proveneither way?
I'm just
curious... John Savage
=John> I'm just a sophomore in
college, so I don't know much. In our math> book, it said
the
integral of sin(x^2) can't be integrated easily.> What does
it
mean, ïeasily'? Is there a simple antiderivative
of> sin(x^2),
or is it some kind of in'nite thing? Has it been proven>
either way? I'm just curious...It's one of the
two Fresnel
integrals. To quote MathWorld, In this form,they have a
particularly simple expansion in terms of spherical
Besselfunctions of the 'rst kind. I don't like
the sound of
that:) The de'niteintegrals (from zero or
-in'nity to
+in'nity) are not hard to get, e.g. bya Fourier
transform.http://mathworld.wolfram.com/
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i032NgB05938;
=I've come across
this equation which describes the dependency of>certain
factors in the operation of bipolar transistors:-Ic =
Is[exp(v/t)-1]>Ic is the transistor's collector current
(typically from a few to>tens of mAmps)>Is is its saturation
current (typically *very* small like 5 fAmps,>for
instance.I'm
confused about the term exp and what it refers to - does it
mean>exponent or e^x or something else? I need someone to show
me a>worked example, please.Let Is = 5^-15>Let (v/t)=0.5Now,
from the above, how would I 'nd Ic?>paul.>-- I expect history
will be kind to me, since I intend to write it.> - Winston
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i032Ngd05934;
=If you don't mind,
Im going to use your own proof, with some variations, in order
to prove the same, but only with naturals.Let f: N -> N be a
mapping from the naturals to the naturals. Note: we do not
assume anything at all about f except that it is a mapping. We
make no assumptions about whether f is injective (1-1) or not,
whether it is surjective (onto) or not, or whether the range
of f is 'nite or in'nite. The mapping f is
arbitrary because
we assume absolutely nothing about it except that it is a
mapping from N to N.This mapping determines a list of naturals
in the sense that we can write down all the naturals in the
range of f in order: f(1) f(2) f(3) . . .Before going into the
proof, we will carry out a neutral transformation of the list.
Neutral transformation means that after it, the total amount
of elements in the list has not changed, neither its value,
neither its order. As the number of digits of the natural
numbers increases as its value grows, we will add enough
zeroes on the left of each natural in the list, in order to
equal the amount of 'gures of the naturals with a bigger
number of signi'cant digits in the list.This is a neutral
transformation, and it will always be possible. Firstly
because we do assume nothing about the list (it is arbitrary),
and second because adding zeroes on the left is a variation of
the bijection used by Cantor to count the naturals (i.e. a 1-1
correspondence between f(k) and f(B), being f(B) a natural with
a bigger number of signi'cant digits). N' will
be the
transformed set of naturals. Therefore, after the neutral
transformation we will have f: N -> NProposition: Let f: N ->
N be given. Then f is not a surjection.Proof. We are to show
that there exists n in N such that n is not in the range of f.
That is, n != f(k) for any k in N.We do this by de'ning, for
each k, the k-th digit in the natural representation of n.
Given k > 0, we 'rst look at d_k, the k-th digit following
the
'rst digit in the representation of f(k) from our list. We
next
de'ne the k-th digit of n, n_k, as follows: If d_k is a 1,
set
n_k = 2. If d_k is not a 1, set n_k = 1.Then the number n =
(n_1)(n_2)(n_3)... is the required number. It is not in the
list because for each k, n differs from f(k) in the k-th
digit.Nicolas de la Foz>Let f: N -> R be a mapping from the
naturals to the reals. Note: we do>not assume anything at all
about f except that it is a mapping. We make>no assumptions
about whether f is injective (1-1) or not, whether it
is>surjective (onto) or not, or whether the range of f is
'nite or>in'nite. The mapping f is arbitrary
because we assume
absolutely>nothing about it except that it is a mapping from N
to R.This mapping determines a list of reals in the sense that
we can write>down all the reals in the range of f in order:
f(1)> f(2)> f(3)> .> .> .Proposition: Let f: N -> R be given.
Then f is not a surjection.Proof. We are to show that there
exists x in R such that x is not in the>range of f. That is, x
!= f(k) for any k in N.We do this by de'ning, for each k, the
k-th digit in the decimal>representation of x. Given k > 0, we
'rst look at d_k, the k-th digit>following the decimal point
in
the representation of f(k) from our list.>We next de'ne the
k-th digit of x, x_k, as follows: If d_k is a 1, set x_k = 2.>
If d_k is not a 1, set x_k = 1.Then the number x =
.(x_1)(x_2)(x_3)... is the required number. It is>not in the
list because for each k, x differs from f(k) in the
k-th>digit.-- >Dave Seaman>Judge Yohn's mistakes revealed in
Mumia Abu-Jamal ruling.><http://www.commoncouragepress.com/index.cfm?action=
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i032NhR05963;
=My initial
question has had to run the gauntlet and did not come out
unscathed (i.e., as a tough question... it turned out to be
really easy). But, perhaps, there's stilla little life left
in
it, afterall?My Guess is as Good as Yours Theorem 1.2:Let (G,*)
be a ('nite) non-abelian group with nontrivial center and
assume it possible to 'nd a ring (R, *, +)such that G is a
subgroup of R with respect to *. In addition, assume that for
all x in G: -x(R) in G (where-x(R) is de'ned as the additive
inverse of x). Then Gis of even order.My Guess is as Good as
Yours Theorem 1.1 (initially froma reply to Arturo Magidin):
-x(R) is independentof R, thus we may write -x for it.Happy
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i032NhX05946;
=The following
question did not need the additional complicationof R ring.
Let G be a group. What is known aboutgroups with the property
that there are elements x,y,z in G such that xy=yx, xz=zx, but
yz != zy ? I admit,Im asking this question somewhat
rhetorically- because I have found such a group. Btw., this
property has very little to do with the reasons for me
investigating thegroup found as of yet.Let R be a ring and
[,]:R*R -> R, [x,y] = xy - yx the commutor function.Are those
rings for which for all x,y,z in R:> [x,y] = 0 and [y,z] = 0
-> [x,z] = 0 somehow simpler than other rings? If so,>do any
other of any such ring's properties follow >automatically
from
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i032Nh805977;
=> This is not my
question, however. It is:>> (Idea- since {1,-1} is a
nontrivial group, ...), does>> it follow that every group with
nontrivial center >> has even order?C. DementNo. Trivially
every abelian group has non-trivial centre , just pick
one>that has odd order. anticipating the next one before I
could ask it:>If you were going to consider non-abelian
groups, try>a group of order p^n for p odd. The book should
prove that at some point>this has non-trivial centre; this is
the same as possessing at least two>conjugacy classes with
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i032NjF06063;
= [.snip.]> This is not my question, however.
It is:>> (Idea- since {1,-1} is a nontrivial group, ...),What
does -1 mean in the context of an arbitrary group? Of course,
you are correct here. I suspect one could saythe following:
Given an arbitrary group (G, *), let (R, *, +)be any ring such
that G is a subgroup with respect to * and in a way that every
additive inverse of G is in G,then -1(R) is the additive
inverse of 1 with respect to + and is in G. If no such ring
exists then G does not have a -1.My Guess is as Good as Yours
Theorem 1.1:the element -1 designated as above is
independentof the encompassing ring (R, *, +). >And what makes
you think that in your group, even if -1 makes sense,>-1 is not
equal to 1?Using the technique above, it would follow that 1 is
no longer a unit, am I right?> does>> it follow that every
group with nontrivial center >> has even order?No. Every group
whose order is the power of a prime has nontrivial>center (you
will deduce it as a consequence of the class formula),>in
particular, any group of order p^n, p an odd prime, n>0,
(from approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i032NiR05997;
=respect: 1) to feel or show honor or esteem
for; hold in high regard...2)to show consideration for; avoid
intruding upon or interfering with...>> > > But then, why does
he pay attention to my posts at all, as that is a> sign of
respect?> > What ever gave you that idea?>> Respect comes from
Latin. It might help you to look up the word's
etymology.Why?
I'm interested in the term as it is used today, and why
you>>
think responses to your post are a sign of respect -- in the
common,>> everyday meaning of the word respect.Etymology
doesn't help answer that question.>Then look up the
*current*
de'nition.Why don't you do that and report
back?>James
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i032NkI06081;
=>> Here is a
shocking admission: I'm curiously growing weary of this>>
fascinating exchange. Major snippage below.[...]>> > You used
this claim to support the claim> that Arturo is incompetent.
This is just utter bullshit.>> you have some reading
comprehension problems. my initial follow up in>> no way
claims that arture is incompetent.Explain this post
is indeed either incompetent as a>> | > mathematician, or more
likely a liar, or both.>> | >> | well, he is an ADJUNCT
assistant professor. ADJUNCTS are the pariahs>> | in academic
caste system.>> | >> | so whomever you are, may well be right
on this one...>> `----How is jstevh@yahoo.com right, given
that Arturo is an adjunct?>>well, some who happen to be
adjuncts happen to be incompetent - duh.> > You demonstrate
your usual blunt reasoning, maky.> > For the record, your post
was an insult to me, and you know it. That's> how I read it,
that's how everyone reasonable read it.>>No, but
that's just
because I 'nd your opinions to be beneath>insult, and your
current claims a show of cowardice. You can't even>stick by
your sly attempts at an insult.>shoudn't you be insulted by
your employers instead?No. Your statement is plain. It is an
attack on me.well, it was not inteded as such. if apologies
you seek,i apologise for insulting you.now onto the obvious
topic of my post. don't you agree withmy assessment of the
adjunct rank in academia?>>please explain.may well be right
can only refer to the fact that I was> being called (through
copying an old post of James> Harris) incompetent as a
mathematician, or more likely a> liar, or both. You are
agreeing with the possibility, which> means you are stating
your opinion that it may> very well be true that I am an
incompetent and/or a liar.what i said, does it have mean
that?> Just the latest in your campaign to attack me because I
would not> support you in a.a.>>come again? For someone as
ignorant as you are, you don't fake ignorance> too well.--
>
==
==>It's not denial. I'm just very
selective about> what
I accept as reality.> --- Calvin (Calvin and
Hobbes)>
=Arturo Magidin>magidin@math.berkeley.eduIs
there a closed form for this sum:i=nSUM x^(gcd(i,n))i=1(Note:
This arises in counting the number of essentially
distinctcolorings of a directed cycle of length
n).Siamak
=This is a reponse to a comment something like of
what use isabstract math.through a telescope.Van