mm-137
=I am a math teacher in Colorado Springs, CO. Three
years ago I wanted math history posters to share with
students ~ containing such topics as, who came up with
algebra and when? Or, what did Descartes contribute
besidesthe coordinate plane? However, I could not 'nd any
such items at theteacher stores. So! I made my own. Other
math teachers have since shown quite an interest in my
posters.cover! So far, I have regions of math history, such
as America, Egypt,China, Greece, Mexico & more! Also, I have
Eras of math history, such asMedieval, Renaissance, &
Victorian. There are also people in mathematics &topics of
mathematics.If you would like to see what I have done so far,
please visit my online IOm not a professional sales person,
simply a math teacher who came up witha product that 'ts a
I am a math teacher in
Colorado Springs, CO. Three years ago I wanted math> history
posters to share with students ~ containing such topics as,
who> came up with algebra and when? Or, what did Descartes
contribute besides> the coordinate plane? However, I could
not 'nd any such items at the> teacher stores. So! I made my
own. Other math teachers have since shown> quite an interest
in my posters.cover! So far, I have regions of math history,
such as America, Egypt,> China, Greece, Mexico & more! Also,
I have Eras of math history, such as> Medieval, Renaissance,
& Victorian. There are also people in mathematics &> topics
of mathematics.If you would like to see what I have done so
far, please visit my onlineIOm not a professional sales
person, simply a math teacher who came up with> a product
Rueter> kavon@mathisradical.comMay I suggest the following
link: http://www-groups.dcs.st-and.ac.uk/history/Look
especially at Some student projects on the history of
mathematics.-- Julian V. NobleProfessor Emeritus of
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/
Science knows only one commandment: contribute to science. --
Bertolt Brecht, Galileo.
Are there methods to solve
equations such as:> sqrt(2*x + y) + sqrt(x + y) = 3> or>
sqrt(x+1)+sqrt(x+2)+sqrt(x+3)=3> That is to eliminate the
radicals, to convert this to a polynomial> expression?> Dan
StangerThe second example is ugly: youOll get a sixth or
eighth order> polynomial even if you can eliminate all the
radicals. Actually, you get a fourth order polynomial.
Reduce[Sqrt[x + 1] + Sqrt[x + 2] + Sqrt[x + 3] == 3] x ==
Root[9 #^4 - 180 #^3 + 1110 #^2 - 1716 # - 2231 &, 1]In other
words, x is the 'rst root (using MathematicaOs
convention for
numbering roots) of the quartic 9 z^4 - 180 z^3 + 1110 z^2 -
1716 z - 2231> These youOll have to do numerically anyway,
so
use an equation solver. Mine gives> one root in the range 0 <
x < 2 , whose value is .186600209943Actually, this is the
root of Sqrt[x] + Sqrt[x + 1] + Sqrt[x + 2] == 3not of the
original equation.Paul-- The University of Western Australia
(CRICOS Provider No 00126G) AUSTRALIA
http://physics.uwa.edu.au/~paul
> Are there methods to
solve equations such as:> sqrt(2*x + y) + sqrt(x + y) = 3>
or> sqrt(x+1)+sqrt(x+2)+sqrt(x+3)=3> That is to eliminate the
radicals, to convert this to a polynomial> expression?>> Dan
Stanger>> The second example is ugly: youOll get a sixth or
eighth order> polynomial even if you can eliminate all the
radicals.Actually, you get a fourth order polynomial.
Reduce[Sqrt[x + 1] + Sqrt[x + 2] + Sqrt[x + 3] == 3] x ==
Root[9 #^4 - 180 #^3 + 1110 #^2 - 1716 # - 2231 &, 1]In other
words, x is the 'rst root (using MathematicaOs
convention for>
numbering roots) of the quartic 9 z^4 - 180 z^3 + 1110 z^2 -
1716 z - 2231> These youOll have to do numerically anyway,
so
use an equation solver. Mine gives> one root in the range 0 <
x < 2 , whose value is>> .186600209943Actually, this is the
root of Sqrt[x] + Sqrt[x + 1] + Sqrt[x + 2] == 3not of the
original equation.Paul> Of course you are right--to simplify
the calculation I shifted theorigin by 1 unit and forgot to
shift the root back. The desired rootof the OP is
.186600209943 - 1 .-- Julian V. NobleProfessor Emeritus of
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/
Science knows only one commandment: contribute to science. --
Bertolt Brecht, Galileo.
Are there methods to solve
equations such as:> sqrt(2*x + y) + sqrt(x + y) = 3Using
Mathematica Reduce[Sqrt[x + y] + Sqrt[2 x + y] == 3, y] The
answer found by Reduce contains unsolved equation(s). A
likely reason for this is that the solution set depends on
branch cuts of Mathematica functions. 0 == (1/6) (x -
Sqrt[x^2 + 18 x + 81] + 9) && 0 == (1/6) (-x - Sqrt[x^2 -
18*x + 81] + 9) && y == (1/36) (x^2 - 54*x + 81)>
sqrt(x+1)+sqrt(x+2)+sqrt(x+3)=3> That is to eliminate the
radicals, to convert this to a polynomial> expression?
Reduce[Sqrt[x + 1] + Sqrt[x + 2] + Sqrt[x + 3] == 3] x ==
Root[9 #^4 - 180 #^3 + 1110 #^2 - 1716 # - 2231 &, 1]In other
words, x is the 'rst root (using MathematicaOs
convention for
numbering roots) of the quartic 9 z^4 - 180 z^3 + 1110 z^2 -
1716 z - 2231Numerically, one obtains N[%] x ==
-0.8133997900166419Paul-- The University of Western Australia
(CRICOS Provider No 00126G) AUSTRALIA
http://physics.uwa.edu.au/~paul
Are there methods to
solve equations such as:> sqrt(2*x + y) + sqrt(x + y) =
3Further to my previous posting, using
CylindricalDecomposition (in Mathematica 5.0) we obtain
CylindricalDecomposition[Sqrt[x + y] + Sqrt[2 x + y] == 3,
{x, y}] -9 <= x <= 9 && y == (1/36) (x^2 - 54 x + 81)Paul--
The University of Western Australia (CRICOS Provider No
00126G) AUSTRALIA http://physics.uwa.edu.au/~paul
IFurther to my previous posting, using
CylindricalDecomposition (in> Mathematica 5.0) we obtain>>
CylindricalDecomposition[Sqrt[x + y] + Sqrt[2 x + y] == 3,
{x, y}]>> -9 <= x <= 9 && y == (1/36) (x^2 - 54 x + 81)>How
does CylindricalDecomposition work? Will it produce the
lowestdegreeanswer to this problem?I am not as interested in
a particular answer, as I am in how one wouldsolve
theseproblems in a optimal way.Dan Stanger
Thats all
thats in the question, my teacher says that it is quite
simple> after u have drawn it but I canOt seem to grasp the
connections.> Also itOs two circles.Your teacher is right.
The circles have X-axis as a common tangent, sohave the same
Y co.9ardinate 5 for either center ; Let the x co.9ardinateof
the circle center be h ; The equation of a circle
is(x-h)^2+(y-5)^2=5^2 ; When (x,y) = ( 0,8) is plugged in,we
obtain h=+- 4 ; You get two circles with centers ( 4,5) and (
-4,5) . Join(0,8) to the centers, join centers, and draw radii
at tangent points.You get two joined (along side measuring 3)
Pythogorian righttriangles with sides (3,4,5) sitting like a
roof on a rectanglehouse with sides (8 X 5), all symmetric to
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told me he had received over$800,000 in cash. I didnt believe
him, so he asked me to try thisidea and 'nd out for myself. I
thought about it for a couple of daysand decided I really
didnt have anything to lose, so I asked him forcopies, and
sure enough, the money started coming in! It arrivedslowly at
'rst, but coming. I kept a precise record of the earnings,and
in the end, it totaled $978,493! I could hardly believe it. I
metwith my friend for lunch to 'nd out exactly how it worked.
He told methere are quite a few similar letters around, but
this one isdifferent because there are six names at the end
of the letter, not've like some others. This fact alone
results in your name being infar more returns. The other fact
was the help I gave him, making surethe whole thing was legal,
since no one wants to take the risk ofBy now you are surely
curious to know what small changes to make. Ifyou sent a
letter like this one out, in order to be completely legal,you
must actually sell something in order to receive a dollar
inreturn. So when you send a dollar to each of the names on
the list,you must include these words, PLEASE PUT ME ON YOUR
MAILING LIST andinclude your name and address. This is the
key to the program. Theitem you will receive for the dollar
you sent to the six people belowis the letter. I sent 500
copies. Three months later , I had totaled $2,341,178! Here
are a few reasons a person might give for not trying
thisprogram:  Some people think they can never make a lot of
money withanything this simple. Some are afraid they will be
ridiculed fortrying  Some dream of large sums of money, but do
nothing to actuallyachieve it. Some are just plain lazy.
Some are afraid of losingtheir investment. They think this
program is designed to beat them outof a few dollars. The
system works if you will just try it. But you must follow
thesimple instructions exactly, and in less than three
months, you willreceive $800,000 GUARANTEED! Keep what you
are doing to yourself forawhile. Many will tell you it won`t
work and will try to talk you outof your dreams. Let them
know of your success after it works. LETTERS FROM
ago, I received this letter and ignored it. Five morecame
within a period of time and I ignored them also. I was
tempted,but I was convinced that they were just a Hoax. After
three weeks ofdeliberating, I decided to give it a try ( not
expecting much ). Twoweeks went by and nothing happened. The
fourth week was unbelievable!I cant say I received $800,000
but I received $400,000. For the 'rsttime in years, I am debt
free. I am doing this again, only this timestarting with 500
post. I strongly recommend that you follow theinstructions
exactly as outlined in this letter. INSTRUCTIONS 1.
Immediately send $1.00 to each of the six people on the list
at theend of this letter. Wrap the dollar bill in a note
saying  Please add2. Copy this letter. You do not have to
type it 200 times. Simplyplace your cursor at the top of the
page, hold it and drag it all theway down to the end of the
letter. Then click on edit and selectcopy. Now open up a
notepad 'le on your computer and put the cursorat the top of
the page in the notepad, click on OeditO and
then
selectOpasteO it will copy the letter for you
onto your
computer. 3. Remove the name next to the #1 on the list and
move the rest of thenames up one position (#2 becomes #1, #3
becomes #2, etc..) Thenplace your name in the #6 position.
Then save it, make sure it issaved as a txt. 'le. 4. When you
have completed the instructions, take this letter and thengo
to (Google,Yahoo,...) and type in (Making Money Massege
board,orpost massege,...)and start posting your copy to 200
message boards,ormore this is only the minimum, you can post
as much as you like...Themore copies you send the better the
results. Keep a copy of thisletter so you can use it a second
time. Post it out again in sixmonths, but Post it with the
addresses you receive with each dollar.It will work better
the second time. NOTE: This service is 100% legal- (Refer to
title 18 section 1302 of the U.S. Postal & lottery laws) How
does it work? When you send out 200 Posts, it is estimated
that atleast 15 people will respond and send you a $1.00.
($15.00) Those 15will Post 200 Posts each and 225 people send
you $1.00 ($225.00) Those225 people Post 200 Posts each and
3,375 people send you $1.00($3,375.00) Those 3,375 post 200
posts each and 759,375 people sendyou $1.00 ($759,375.00) At
this point your name drops off the list,but so far you have
received $813,615.00. P.S. When your money beginsto come in,
give the 'rst 10% to charity with spirit and share a
goodfortune! 1. Steven1614-377 Ridelle Ave.Toronto
Blvd.Saida - Lebanon Middel East4. Korrina
Ford17135-63ave#103Edmonton, AB T5T-2K1 CANADA5. Mike
Eilers2005 Faith StUrbana, Ia 523456. Kevin Carpenter1436
Queen Ave N.E.RENTON WA 98056-3339 This really really works,I
tried it once and IOm doing it again, 'rstto be
honest I only
posted 145 posts and I didnOt recive alot only6,689.00$ in 3
months,at least this means that it realy works ,so thistime
IOm posting 604 posts and IOve got so far in 2
month
15,640.00$this is realy like a dream come true. So go on and
try it trust meyouOve got nothing to lose.... important tip
-
Cover money by paper fully by which it can not beenseen in
light , because postal employees do some cheat
(legacy-1.mathforum.org [144.118.94.27]) by
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told me he had received over$800,000 in cash. I didnt believe
him, so he asked me to try thisidea and 'nd out for myself. I
thought about it for a couple of daysand decided I really
didnt have anything to lose, so I asked him forcopies, and
sure enough, the money started coming in! It arrivedslowly at
'rst, but coming. I kept a precise record of the earnings,and
in the end, it totaled $978,493! I could hardly believe it. I
metwith my friend for lunch to 'nd out exactly how it worked.
He told methere are quite a few similar letters around, but
this one isdifferent because there are six names at the end
of the letter, not've like some others. This fact alone
results in your name being infar more returns. The other fact
was the help I gave him, making surethe whole thing was legal,
since no one wants to take the risk ofBy now you are surely
curious to know what small changes to make. Ifyou sent a
letter like this one out, in order to be completely legal,you
must actually sell something in order to receive a dollar
inreturn. So when you send a dollar to each of the names on
the list,you must include these words, PLEASE PUT ME ON YOUR
MAILING LIST andinclude your name and address. This is the
key to the program. Theitem you will receive for the dollar
you sent to the six people belowis the letter. I sent 500
copies. Three months later , I had totaled $2,341,178! Here
are a few reasons a person might give for not trying
thisprogram:  Some people think they can never make a lot of
money withanything this simple. Some are afraid they will be
ridiculed fortrying  Some dream of large sums of money, but do
nothing to actuallyachieve it. Some are just plain lazy.
Some are afraid of losingtheir investment. They think this
program is designed to beat them outof a few dollars. The
system works if you will just try it. But you must follow
thesimple instructions exactly, and in less than three
months, you willreceive $800,000 GUARANTEED! Keep what you
are doing to yourself forawhile. Many will tell you it won`t
work and will try to talk you outof your dreams. Let them
know of your success after it works. LETTERS FROM
ago, I received this letter and ignored it. Five morecame
within a period of time and I ignored them also. I was
tempted,but I was convinced that they were just a Hoax. After
three weeks ofdeliberating, I decided to give it a try ( not
(legacy-1.mathforum.org [144.118.94.27]) by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
told me he had received over$800,000 in cash. I didnt believe
him, so he asked me to try thisidea and 'nd out for myself. I
thought about it for a couple of daysand decided I really
didnt have anything to lose, so I asked him forcopies, and
sure enough, the money started coming in! It arrivedslowly at
'rst, but coming. I kept a precise record of the earnings,and
in the end, it totaled $978,493! I could hardly believe it. I
metwith my friend for lunch to 'nd out exactly how it worked.
He told methere are quite a few similar letters around, but
this one isdifferent because there are six names at the end
of the letter, not've like some others. This fact alone
results in your name being infar more returns. The other fact
was the help I gave him, making surethe whole thing was legal,
since no one wants to take the risk ofBy now you are surely
curious to know what small changes to make. Ifyou sent a
letter like this one out, in order to be completely legal,you
must actually sell something in order to receive a dollar
inreturn. So when you send a dollar to each of the names on
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=MAKE EASY AND
AND VERY LEGAL) Please readon! You will be very thankful you
did! A little while back, I wasbrowsing through newsgroups,
just like you are now,thousands of dollars within weeks with
only aninitial investment of $6.00! So I thought, Yeah,
right, this must bea scam, but like most of us, I was
curious, so Ikept reading. Anyway, it said that you send
$1.00 to each of the 6place your own name and address in the
bottom of the list at #6, and(There are thousands) No catch,
that was it. So after thinking itover, and talking to a few
people 'rst, I thoughtabout trying it. I 'gured
what have I
got to lose except 6 stamps and$6.00, right? Like most of us
I was a littleskeptical and a little worried about the legal
aspects of it all. So Ichecked it out with the U.S. Post
Of'ce(1-800-725-2161) and they con'rmed that it
is indeed
legal! Then Iinvested the measly $6.00. Well GUESSshocked! I
'gured it would end soon, but themoney just kept coming in.
In my 'rst week, I made about $25.00. Bythe end of the second
week I had made atotal of over $1,000.00! In the third week I
had over $10,000.00 anditOs still growing. This is now my
fourth weekand I have made a total of just over $42,000.00
and itOs still comingin rapidly. ItOs
certainly worth $6.00,
and 6stamps, I have spent more than that on the lottery!! Let
me tell youhow this works and most importantly, why itcan get
the information off of it as you need it.STEP 1 Get 6
separate pieces of paper and write the following on eachpiece
of paper PLEASE PUT ME ON YOURMAILING LIST. Now get 6 US $1.00
bills and place ONE inside EACH ofthe 6 pieces of paper so the
bill will not beseen through the envelope to prevent thievery.
Next, place one paperin each of the 6 envelopes and seal them.
Youshould now have 6 sealed envelopes, each with a piece of
paper statingthe above phrase, your name and address,and a
$1.00 bill. What you are doing is creating a service by
this.THIS IS ABSOLUTELY LEGAL! This program remainssuccessful
because of the honesty and integrity of the
participants.Please continue its success by carefully
adheringto the instructions. You will now become part of the
Mail Orderbusiness. Your product is simply a service of
helping#1] J. Willard 2845 Rockford Rd Boonville, NC27011 #2]
A. Hyde 16085 A East Rice Pl. Aurora, CO 80015 #3] Jim P 3766
Spring Green Rd Green Bay, WI 54313 #4] Stephen 17025 E.
Wiley Place Parker, CO 80134 #5] T Gibson 8244 Hydra Lane Las
Vegas, NV 89128 #6] T Bullock 1175 E. Silver Mesa Cir, Sandy,
UT 84094STEP 2 Now take the #1 name and address off the list
that you seeabove, move the other names up (6 becomes 5,5
becomes 4, etc...) and add YOUR Name and address as number 6
on thelist. STEP 3 Change anything you need to,newsgroups. (I
think there are close to 24,000 groups) All you need is200,
but remember, the more you post, themore money you make!
DIRECTIONS--HOW TO POST TONEWSGROUPS Step
1) You do not need tore-type this entire letter to do your
own posting. Simply put yourcursor at the beginning of this
letter and drag yourcursor to the bottom of this document,
and select OcopyO from the editmenu. This will
copy the
entire letter into thecomputers memory. Step 2) Open a blank
OnotepadO 'le and place
yourOeditO menu select
OpasteO. This
will paste a copy of the letter intonotepad so that you can
add your name to the list.Step 3) Save your new notepad 'le
as a .txt 'le. If you want to doyour postings in different
sittings, youOll alwayshave this 'le to go back
to. Step 4)
Use Netscape or Internetexplorer and try searching for
various newsgroups(on-line forums, message boards, chat
sites, discussions.) Step 5)as a new message by highlighting
the text of this letter and selectingthis will be the header
that everyone sees as they scroll through thelist of postings
in a particular group, click thepost message button. YouOre
done with your 'rst
one!Congratulations...THATOS IT! All you
have to do is jump todifferent newsgroups and post away,
after you get the hang of it, itwill take about 30 seconds
for each newsgroup!**REMEMBER, THE MORE NEWSGROUPS YOU POST
IN, THE MORE MONEY YOU WILLMAKE!! BUT YOU HAVE TO POSTA
MINIMUM OF 200** ThatOs it! You will begin receiving money
fromaround the world within days! You maywill receive. If you
wish to stay anonymous,you can invent a name to use, as long
as the postman will deliver it.**JUST MAKE SURE ALL THE
ADDRESSES ARECORRECT.** Now the WHY part Out of 200 postings,
say I receive only 5replies (a very low example). So then
Imade $5.00 with my name and address at #6 on the letter.
Now, each ofthe 5 persons who just sent me $1.00 makethe
MINIMUM 200 postings, each with my name and address at #5 and
only5 persons respond to each of the original5, that is
another $25.00 for me, now those 25 each make 200
MINIMUMposts with my name and address at #4 andonly 5 replies
each, I will bring in an additional $125.00! Now, those125
persons turn around and post the MINIMUM200 with my name and
address at #3 and only receive 5 replies each, Iwill make an
additional $626.00! OK, nowhere is the fun part, each of
those 625 persons post a MINIMUM 200letters with my name and
address at #2 andthey each only receive 5 replies, that just
made me $3,125.00!!! Those3,125 persons will all deliver this
message to200 newsgroups with my name and address at #1 and if
still 5 personsper 200 newsgroups react I will
receive$15,625,00! With an original investment of only $6.00!
the latest posting in the newsgroups, and send out another
$6.00to names on the list, putting your name atnumber 6
again. And start posting again. The thing to remember is,
doyou realize that thousands of people all overeveryday, JUST
LIKE YOU are now!! So can you afford$6.00 and see if it really
works?? I think so... People have said,what if the plan is
played out and no one sends youthe money? So what! What are
the chances of that happening when thereare tons of new
honest users and newhonest people who are joining the
internet and newsgroups everyday andare willing to give it a
try? Estimates are at20,000 to 50,000 new users, every day,
with thousands of those joiningthe actual internet. Remember,
play FAIRLYand HONESTLY and this will work. ALSO REMEMBER*** :
: SEND YOUR $1 OUTTO EVERYONE ON THE LIST, : : EVENIF THEY ARE
NOT FROM THE U.S. : : WHAT ARE YOU WAITING FOR??? : : : :GOOD
EASY$$ and you will 'nd alot ofmessage boards !MAKE MONEY
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=MAKE EASY AND QUICK CASH MONEY by
readon! You will be very thankful you did! A little while
back, I wasbrowsing through newsgroups, just like you are
now,thousands of dollars within weeks with only aninitial
investment of $6.00! So I thought, Yeah, right, this must bea
scam, but like most of us, I was curious, so Ikept reading.
Anyway, it said that you send $1.00 to each of the 6place
your own name and address in the bottom of the list at #6,
and(There are thousands) No catch, that was it. So after
thinking itover, and talking to a few people 'rst, I
thoughtabout trying it. I 'gured what have I got to lose
except 6 stamps and$6.00, right? Like most of us I was a
littleskeptical and a little worried about the legal aspects
of it all. So Ichecked it out with the U.S. Post
Of'ce(1-800-725-2161) and they con'rmed that it
is indeed
legal! Then Iinvested the measly $6.00. Well GUESSshocked! I
'gured it would end soon, but themoney just kept coming in.
In my 'rst week, I made about $25.00. Bythe end of the second
week I had made atotal of over $1,000.00! In the third week I
had over $10,000.00 anditOs still growing. This is now my
fourth weekand I have made a total of just over $42,000.00
and itOs still comingin rapidly. ItOs
certainly worth $6.00,
and 6stamps, I have spent more than that on the lottery!! Let
me tell youhow this works and most importantly, why itcan get
the information off of it as you need it.STEP 1 Get 6
separate pieces of paper and write the following on eachpiece
of paper PLEASE PUT ME ON YOURMAILING LIST. Now get 6 US $1.00
bills and place ONE inside EACH ofthe 6 pieces of paper so the
bill will not beseen through the envelope to prevent thievery.
Next, place one paperin each of the 6 envelopes and seal them.
Youshould now have 6 sealed envelopes, each with a piece of
paper statingthe above phrase, your name and address,and a
$1.00 bill. What you are doing is creating a service by
this.THIS IS ABSOLUTELY LEGAL! This program remainssuccessful
because of the honesty and integrity of the
participants.Please continue its success by carefully
adheringto the instructions. You will now become part of the
Mail Orderbusiness. Your product is simply a service of
helping#1] J. Willard 2845 Rockford Rd Boonville, NC27011 #2]
A. Hyde 16085 A East Rice Pl. Aurora, CO 80015 #3] Jim P 3766
Spring Green Rd Green Bay, WI 54313 #4] Stephen 17025 E.
Wiley Place Parker, CO 80134 #5] T Gibson 8244 Hydra Lane Las
Vegas, NV 89128 #6] T Bullock 1175 E. Silver Mesa Cir, Sandy,
UT 84094STEP 2 Now take the #1 name and address off the list
that you seeabove, move the other names up (6 becomes 5,5
becomes 4, etc...) and add YOUR Name and address as number 6
on thelist. STEP 3 Change anything you need to,newsgroups. (I
think there are close to 24,000 groups) All you need is200,
but remember, the more you post, themore money you make!
DIRECTIONS--HOW TO POST TONEWSGROUPS Step
1) You do not need tore-type this entire letter to do your
own posting. Simply put yourcursor at the beginning of this
letter and drag yourcursor to the bottom of this document,
and select OcopyO from the editmenu. This will
copy the
entire letter into thecomputers memory. Step 2) Open a blank
OnotepadO 'le and place
yourOeditO menu select
OpasteO. This
will paste a copy of the letter intonotepad so that you can
add your name to the list.Step 3) Save your new notepad 'le
as a .txt 'le. If you want to doyour postings in different
sittings, youOll alwayshave this 'le to go back
to. Step 4)
Use Netscape or Internetexplorer and try searching for
various newsgroups(on-line forums, message boards, chat
sites, discussions.) Step 5)as a new message by highlighting
the text of this letter and selectingthis will be the header
that everyone sees as they scroll through thelist of postings
in a particular group, click thepost message button. YouOre
done with your 'rst
one!Congratulations...THATOS IT! All you
have to do is jump todifferent newsgroups and post away,
after you get the hang of it, itwill take about 30 seconds
for each newsgroup!**REMEMBER, THE MORE NEWSGROUPS YOU POST
IN, THE MORE MONEY YOU WILLMAKE!! BUT YOU HAVE TO POSTA
MINIMUM OF 200** ThatOs it! You will begin receiving money
fromaround the world within days! You maywill receive. If you
wish to stay anonymous,you can invent a name to use, as long
as the postman will deliver it.**JUST MAKE SURE ALL THE
ADDRESSES ARECORRECT.** Now the WHY part Out of 200 postings,
say I receive only 5replies (a very low example). So then
Imade $5.00 with my name and address at #6 on the letter.
Now, each ofthe 5 persons who just sent me $1.00 makethe
MINIMUM 200 postings, each with my name and address at #5 and
only5 persons respond to each of the original5, that is
another $25.00 for me, now those 25 each make 200
MINIMUMposts with my name and address at #4 andonly 5 replies
each, I will bring in an additional $125.00! Now, those125
persons turn around and post the MINIMUM200 with my name and
address at #3 and only receive 5 replies each, Iwill make an
additional $626.00! OK, nowhere is the fun part, each of
those 625 persons post a MINIMUM 200letters with my name and
address at #2 andthey each only receive 5 replies, that just
made me $3,125.00!!! Those3,125 persons will all deliver this
message to200 newsgroups with my name and address at #1 and if
still 5 personsper 200 newsgroups react I will
receive$15,625,00! With an original investment of only $6.00!
the latest posting in the newsgroups, and send out another
$6.00to names on the list, putting your name atnumber 6
again. And start posting again. The thing to remember is,
doyou realize that thousands of people all overeveryday, JUST
LIKE YOU are now!! So can you afford$6.00 and see if it really
works?? I think so... People have said,what if the plan is
played out and no one sends youthe money? So what! What are
the chances of that happening when thereare tons of new
honest users and newhonest people who are joining the
internet and newsgroups everyday andare willing to give it a
try? Estimates are at20,000 to 50,000 new users, every day,
with thousands of those joiningthe actual internet. Remember,
play FAIRLYand HONESTLY and this will work. ALSO REMEMBER*** :
: SEND YOUR $1 OUTTO EVERYONE ON THE LIST, : : EVENIF THEY ARE
NOT FROM THE U.S. : : WHAT ARE YOU WAITING FOR??? : : : :GOOD
EASY$$ and you will 'nd alot ofmessage boards !MAKE MONEY
(legacy-1.mathforum.org [144.118.94.27]) by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
=MAKE EASY AND QUICK CASH MONEY by
readon! You will be very thankful you did! A little while
back, I wasbrowsing through newsgroups, just like you are
now,thousands of dollars within weeks with only aninitial
investment of $6.00! So I thought, Yeah, right, this must bea
scam, but like most of us, I was curious, so Ikept reading.
Anyway, it said that you send $1.00 to each of the 6place
your own name and address in the bottom of the list at #6,
and(There are thousands) No catch, that was it. So after
thinking itover, and talking to a few people 'rst, I
thoughtabout trying it. I 'gured what have I got to lose
except 6 stamps and$6.00, right? Like most of us I was a
littleskeptical and a little worried about the legal aspects
of it all. So Ichecked it out with the U.S. Post
Of'ce(1-800-725-2161) and they con'rmed that it
is indeed
legal! Then Iinvested the measly $6.00. Well GUESSshocked! I
'gured it would end soon, but themoney just kept coming in.
In my 'rst week, I made about $25.00. Bythe end of the second
week I had made atotal of over $1,000.00! In the third week I
had over $10,000.00 anditOs still growing. This is now my
fourth weekand I have made a total of just over $42,000.00
and itOs still comingin rapidly. ItOs
certainly worth $6.00,
and 6stamps, I have spent more than that on the lottery!! Let
me tell youhow this works and most importantly, why itcan get
the information off of it as you need it.STEP 1 Get 6
separate pieces of paper and write the following on eachpiece
of paper PLEASE PUT ME ON YOURMAILING LIST. Now get 6 US $1.00
bills and place ONE inside EACH ofthe 6 pieces of paper so the
bill will not beseen through the envelope to prevent thievery.
Next, place one paperin each of the 6 envelopes and seal them.
Youshould now have 6 sealed envelopes, each with a piece of
paper statingthe above phrase, your name and address,and a
$1.00 bill. What you are doing is creating a service by
this.THIS IS ABSOLUTELY LEGAL! This program remainssuccessful
because of the honesty and integrity of the
participants.Please continue its success by carefully
adheringto the instructions. You will now become part of the
Mail Orderbusiness. Your product is simply a service of
helping#1] J. Willard 2845 Rockford Rd Boonville, NC27011 #2]
A. Hyde 16085 A East Rice Pl. Aurora, CO 80015 #3] Jim P 3766
Spring Green Rd Green Bay, WI 54313 #4] Stephen 17025 E.
Wiley Place Parker, CO 80134 #5] T Gibson 8244 Hydra Lane Las
Vegas, NV 89128 #6] T Bullock 1175 E. Silver Mesa Cir, Sandy,
UT 84094STEP 2 Now take the #1 name and address off the list
that you seeabove, move the other names up (6 becomes 5,5
becomes 4, etc...) and add YOUR Name and address as number 6
on thelist. STEP 3 Change anything you need to,newsgroups. (I
think there are close to 24,000 groups) All you need is200,
but remember, the more you post, themore money you make!
DIRECTIONS--HOW TO POST TONEWSGROUPS Step
1) You do not need tore-type this entire letter to do your
own posting. Simply put yourcursor at the beginning of this
letter and drag yourcursor to the bottom of this document,
and select OcopyO from the editmenu. This will
copy the
entire letter into thecomputers memory. Step 2) Open a blank
OnotepadO 'le and place
yourOeditO menu select
OpasteO. This
will paste a copy of the letter intonotepad so that you can
add your name to the list.Step 3) Save your new notepad 'le
as a .txt 'le. If you want to doyour postings in different
sittings, youOll alwayshave this 'le to go back
to. Step 4)
Use Netscape or Internetexplorer and try searching for
various newsgroups(on-line forums, message boards, chat
sites, discussions.) Step 5)as a new message by highlighting
the text of this letter and selectingthis will be the header
that everyone sees as they scroll through thelist of postings
in a particular group, click thepost message button. YouOre
done with your 'rst
one!Congratulations...THATOS IT! All you
have to do is jump todifferent newsgroups and post away,
after you get the hang of it, itwill take about 30 seconds
for each newsgroup!**REMEMBER, THE MORE NEWSGROUPS YOU POST
IN, THE MORE MONEY YOU WILLMAKE!! BUT YOU HAVE TO POSTA
MINIMUM OF 200** ThatOs it! You will begin receiving money
fromaround the world within days! You maywill receive. If you
wish to stay anonymous,you can invent a name to use, as long
as the postman will deliver it.**JUST MAKE SURE ALL THE
ADDRESSES ARECORRECT.** Now the WHY part Out of 200 postings,
say I receive only 5replies (a very low example). So then
Imade $5.00 with my name and address at #6 on the letter.
Now, each ofthe 5 persons who just sent me $1.00 makethe
MINIMUM 200 postings, each with my name and address at #5 and
only5 persons respond to each of the original5, that is
another $25.00 for me, now those 25 each make 200
MINIMUMposts with my name and address at #4 andonly 5 replies
each, I will bring in an additional $125.00! Now, those125
persons turn around and post the MINIMUM200 with my name and
address at #3 and only receive 5 replies each, Iwill make an
additional $626.00! OK, nowhere is the fun part, each of
those 625 persons post a MINIMUM 200letters with my name and
address at #2 andthey each only receive 5 replies, that just
made me $3,125.00!!! Those3,125 persons will all deliver this
message to200 newsgroups with my name and address at #1 and if
still 5 personsper 200 newsgroups react I will
receive$15,625,00! With an original investment of only $6.00!
the latest posting in the newsgroups, and send out another
$6.00to names on the list, putting your name atnumber 6
again. And start posting again. The thing to remember is,
doyou realize that thousands of people all overeveryday, JUST
LIKE YOU are now!! So can you afford$6.00 and see if it really
works?? I think so... People have said,what if the plan is
played out and no one sends youthe money? So what! What are
the chances of that happening when thereare tons of new
honest users and newhonest people who are joining the
internet and newsgroups everyday andare willing to give it a
try? Estimates are at20,000 to 50,000 new users, every day,
with thousands of those joiningthe actual internet. Remember,
play FAIRLYand HONESTLY and this will work. ALSO REMEMBER*** :
: SEND YOUR $1 OUTTO EVERYONE ON THE LIST, : : EVENIF THEY ARE
NOT FROM THE U.S. : : WHAT ARE YOU WAITING FOR??? : : : :GOOD
EASY$$ and you will 'nd alot ofmessage boards !MAKE MONEY
(legacy-1.mathforum.org [144.118.94.27]) by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
=MAKE EASY AND QUICK CASH MONEY by
readon! You will be very thankful you did! A little while
back, I wasbrowsing through newsgroups, just like you are
now,thousands of dollars within weeks with only aninitial
investment of $6.00! So I thought, Yeah, right, this must bea
scam, but like most of us, I was curious, so Ikept reading.
Anyway, it said that you send $1.00 to each of the 6place
your own name and address in the bottom of the list at #6,
and(There are thousands) No catch, that was it. So after
thinking itover, and talking to a few people 'rst, I
thoughtabout trying it. I 'gured what have I got to lose
except 6 stamps and$6.00, right? Like most of us I was a
littleskeptical and a little worried about the legal aspects
of it all. So Ichecked it out with the U.S. Post
Of'ce(1-800-725-2161) and they con'rmed that it
is indeed
legal! Then Iinvested the measly $6.00. Well GUESSshocked! I
'gured it would end soon, but themoney just kept coming in.
In my 'rst week, I made about $25.00. Bythe end of the second
week I had made atotal of over $1,000.00! In the third week I
had over $10,000.00 anditOs still growing. This is now my
fourth weekand I have made a total of just over $42,000.00
and itOs still comingin rapidly. ItOs
certainly worth $6.00,
and 6stamps, I have spent more than that on the lottery!! Let
me tell youhow this works and most importantly, why itcan get
the information off of it as you need it.STEP 1 Get 6
separate pieces of paper and write the following on eachpiece
of paper PLEASE PUT ME ON YOURMAILING LIST. Now get 6 US $1.00
bills and place ONE inside EACH ofthe 6 pieces of paper so the
bill will not beseen through the envelope to prevent thievery.
Next, place one paperin each of the 6 envelopes and seal them.
Youshould now have 6 sealed envelopes, each with a piece of
paper statingthe above phrase, your name and address,and a
$1.00 bill. What you are doing is creating a service by
this.THIS IS ABSOLUTELY LEGAL! This program remainssuccessful
because of the honesty and integrity of the
participants.Please continue its success by carefully
adheringto the instructions. You will now become part of the
Mail Orderbusiness. Your product is simply a service of
helping#1] J. Willard 2845 Rockford Rd Boonville, NC27011 #2]
A. Hyde 16085 A East Rice Pl. Aurora, CO 80015 #3] Jim P 3766
Spring Green Rd Green Bay, WI 54313 #4] Stephen 17025 E.
Wiley Place Parker, CO 80134 #5] T Gibson 8244 Hydra Lane Las
Vegas, NV 89128 #6] T Bullock 1175 E. Silver Mesa Cir, Sandy,
UT 84094STEP 2 Now take the #1 name and address off the list
that you seeabove, move the other names up (6 becomes 5,5
becomes 4, etc...) and add YOUR Name and address as number 6
on thelist. STEP 3 Change anything you need to,newsgroups. (I
think there are close to 24,000 groups) All you need is200,
but remember, the more you post, themore money you make!
DIRECTIONS--HOW TO POST TONEWSGROUPS Step
1) You do not need tore-type this entire letter to do your
own posting. Simply put yourcursor at the beginning of this
letter and drag yourcursor to the bottom of this document,
and select OcopyO from the editmenu. This will
copy the
entire letter into thecomputers memory. Step 2) Open a blank
OnotepadO 'le and place
yourOeditO menu select
OpasteO. This
will paste a copy of the letter intonotepad so that you can
add your name to the list.Step 3) Save your new notepad 'le
as a .txt 'le. If you want to doyour postings in different
sittings, youOll alwayshave this 'le to go back
to. Step 4)
Use Netscape or Internetexplorer and try searching for
various newsgroups(on-line forums, message boards, chat
sites, discussions.) Step 5)as a new message by highlighting
the text of this letter and selectingthis will be the header
that everyone sees as they scroll through thelist of postings
in a particular group, click thepost message button. YouOre
done with your 'rst
one!Congratulations...THATOS IT! All you
have to do is jump todifferent newsgroups and post away,
after you get the hang of it, itwill take about 30 seconds
for each newsgroup!**REMEMBER, THE MORE NEWSGROUPS YOU POST
IN, THE MORE MONEY YOU WILLMAKE!! BUT YOU HAVE TO POSTA
MINIMUM OF 200** ThatOs it! You will begin receiving money
fromaround the world within days! You maywill receive. If you
wish to stay anonymous,you can invent a name to use, as long
as the postman will deliver it.**JUST MAKE SURE ALL THE
ADDRESSES ARECORRECT.** Now the WHY part Out of 200 postings,
say I receive only 5replies (a very low example). So then
Imade $5.00 with my name and address at #6 on the letter.
Now, each ofthe 5 persons who just sent me $1.00 makethe
MINIMUM 200 postings, each with my name and address at #5 and
only5 persons respond to each of the original5, that is
another $25.00 for me, now those 25 each make 200
MINIMUMposts with my name and address at #4 andonly 5 replies
each, I will bring in an additional $125.00! Now, those125
persons turn around and post the MINIMUM200 with my name and
address at #3 and only receive 5 replies each, Iwill make an
additional $626.00! OK, nowhere is the fun part, each of
those 625 persons post a MINIMUM 200letters with my name and
address at #2 andthey each only receive 5 replies, that just
made me $3,125.00!!! Those3,125 persons will all deliver this
message to200 newsgroups with my name and address at #1 and if
still 5 personsper 200 newsgroups react I will
receive$15,625,00! With an original investment of only $6.00!
the latest posting in the newsgroups, and send out another
$6.00to names on the list, putting your name atnumber 6
again. And start posting again. The thing to remember is,
doyou realize that thousands of people all overeveryday, JUST
LIKE YOU are now!! So can you afford$6.00 and see if it really
works?? I think so... People have said,what if the plan is
played out and no one sends youthe money? So what! What are
the chances of that happening when thereare tons of new
honest users and newhonest people who are joining the
internet and newsgroups everyday andare willing to give it a
try? Estimates are at20,000 to 50,000 new users, every day,
with thousands of those joiningthe actual internet. Remember,
play FAIRLYand HONESTLY and this will work. ALSO REMEMBER*** :
: SEND YOUR $1 OUTTO EVERYONE ON THE LIST, : : EVENIF THEY ARE
NOT FROM THE U.S. : : WHAT ARE YOU WAITING FOR??? : : : :GOOD
EASY$$ and you will 'nd alot ofmessage boards !MAKE MONEY
FAST AND EASY$$
=1. How can I obtain numerical solution of
differential equationwith Mathematica or Maxima (Macsyma?)I
need this in text 'le in the formt1 x(t1)t2 x(t2) ....2. Can
Maxima solve nonlinear equation for example : cos(x)-x = 0
(numerical of course ...)Olaf
=For your second question,
there are at least two approaches.1. Use a newton iteration
for f(x)=0. compute fO(x) symbolicallyand you have a
one-line
iterative formula you can run as longas you want and hope that
it converges. There may even bea pre-packaged version of
this.2. Do the same but do the iteration symbolically, as a
series,truncating further out as you iterate. This gives you
ataylor or laurent series of some sortfor a solution. You
need to 'nd a properplace to start the iteration, which
complicates the situationin some cases.see All Algebraic
Functions can be computed fastSource Journal of the ACM
(JACM)Volume 25 , Issue 2 (April 1978)Pages: 245 - 260Year of
Publication: 1978ISSN:0004-5411Authors H. T. Kung J. F. Traub
Publisher ACM Press New York, NY, USA1. How can I obtain
numerical solution of differential equation> with Mathematica
or Maxima (Macsyma?)> I need this in text 'le in the form> t1
x(t1)> t2 x(t2) ....2. Can Maxima solve nonlinear equation>
for example :> cos(x)-x = 0> (numerical of course
...)Olaf
In 1998 there was a thread with the same
subject. I assume thatperspective> might have changed during
last 5 years. Now we have siteseer, etc. Any> updates to the
subject, new excellent textbooks, must read reviews etc?As
there is no responces, the question obviously needs some
clari'cation.What Computer Algebra Systems are the most
interesting from implementationperspective? What software
design looks like? Is OOP, Rule-based, or bruteforce C coded
implementation more common?
>>In 1998 there was a thread
with the same subject. I assume thatperspective> >>might have
changed during last 5 years. Now we have siteseer, etc.
Any>>updates to the subject, new excellent textbooks, must
read reviews etc?> As there is no responces, the question
obviously needs some clari'cation.What Computer Algebra
Systems are the most interesting from implementation>
perspective? What software design looks like? Is OOP,
Rule-based, or brute> force C coded implementation more
common?I donOt think that the perspective has changed much
in
the last 5 years.You can look at the implementation of Axiom
and Maxima since the sourcesare public.If you want to have a
popularity contest, then I think you should addfunctional
style, in which case that might characterize many of
thesystems. There is a rule-based user-accessible top-level
in severalsystems (mathematica, reduce, macsyma, at least),
but I doubt that anyof the major systems are rule-based.
Symbmath, sometimes advertised onsci.math.symbolic, is I
think a prolog program.Most systems beyond a certain size
tend to have a language structureunderneath them that is
supportive of the application. Thus macsyma/maxima is written
in Lisp, but even function de'nitionin the implementation is
slightly changed. (defun foo ....) iswritten (defmfun foo
...) for various reasons, and expanded differently.I hope no
system is written in brute force fashion, but of coursethe
code in Mathematica is hidden from view.Also, since most
systems have to do storage reclaimation, thosethat are not
actually written in a supportive language force their
programmers to follow GreenspunOs 10th lawof programming:
Any
suf'ciently complicated C or Fortran program contains an
ad-hoc, informally-speci'ed bug-ridden slow implementation of
half of Common Lisp.I suspect that some people might think
that there are important changesbecause now people can write
computer algebra systems in Java, c#, .net,ML, Python, Ruby,
Perl, or some other language du jour. I do notagree. There
are many programmers who know Java, but does thatmake it a
good language? (I suspect there is an analogy forpopularity
contests and the voting for governor of
California....)Regarding texts, there is a book, by von zur
Gathen and Gerhard,Modern Computer Algebra that has appeared
since 1999. It collects many facts about algorithms for
computer algebra, but is by no means a guide to
implementation.Now a question for you...Why do you
ask?RJF
 In 1998 there was a thread with the same
subject. I assume that>> perspectivemight have changed
during last 5 years. Now we have siteseer, etc. Anyupdates
to the subject, new excellent textbooks, must read reviews
etc?>> >> >> As there is no responces, the question obviously
needs some clari'cation.>> >> What Computer Algebra Systems
are the most interesting from implementation>> perspective?
What software design looks like? Is OOP, Rule-based, or
brute>> force C coded implementation more common?>> The HOL
systems seem to be getting described as very slow. MAybe too
slowto solve real problems in logic. A 3GL that recurses and
uses for-loopsmay be a lot faster, e.g. at eliminating
quanti'ers.>> >>I donOt think that the
perspective has
changed much in the last 5 years.>You can look at the
implementation of Axiom and Maxima since the sources>are
public.>More hints that somehow LISP is worth considering.
Suppose they were allwrong and it was never really actually
worth considering....>Most systems beyond a certain size tend
to have a language structure>underneath them that is
supportive of the application. Thus Sure: Mr Fatemen is
saying that LISP should rarely (if ever) be usedas the
underlying language because its offerings mismatche with
theneeds of symbolic algebra developers. Since he was not
going to produceevidence, there is some beneath this
line.>macsyma/maxima is written in Lisp, but even function
de'nition>in the implementation is slightly changed. (defun
foo ....) is>written (defmfun foo ...) for various reasons,
and expanded differently.None of which is
interesting...>Also, since most systems have to do storage
reclaimation, those>that are not actually written in a
supportive language> force their programmers to follow
GreenspunOs 10th law...If an expression tends to have 4
factors in the multiplications, and4 terms in the additions,
then arrays 4 items long would be wanted.So symbolic algebra
programming in LISP uses LISPOs arrays ?. If notthen why not
?.>I suspect that some people might think that there are
important changes>because now people can write computer
algebra systems in Java, c#, .net,>ML, Python, Ruby, Perl, or
some other language du jour.The 'rst choice of 3GLs for
precise minded mathematicians, i.e. Ada 95,was missing from
the list. Is there a speci'c §aw of Ada that youwish to
criticise, Mr Fateman ?....>Now a question for you...Why do
you ask?data structure for expressions. Let E be an
expressionFor variable x, the data x is in E.O.all.A.Atom3x
is the same but E.O.all.M contains 3/1 instead of
1/1.Similarly there can be 3+.The 4 of (3x or 4y) would be
stored in E.O.all.Q (2).O.all.M, (andthe or is in
E.O.all.A.Sysatom).The design can be adjusted while remaining
acceptable.But it is de'nitely unlike LISP. That is because
LISP is not a languagethat would used if a software developer
got the best design. with Striunli; with
Ada.Finalization;pragma Elaborate_All (Striunli);generic --
Caller de'nes the numbers for this expressions package type
Num is limited private; type Num_Const_Ptr is access constant
Num; Zero_Ptr : Num_Const_Ptr; One_Ptr : Num_Const_Ptr; with
function = (M1, M2 : Num) return Boolean; with function <
(M1, M2 : Num) return Boolean; with procedure Plus (LHS : out
Num; M1, M2 : Num); with procedure Minus (LHS : out Num; M1,
M2 : Num); with procedure Times (LHS : out Num; M1, M2 :
Num); with procedure Assign_Num (LHS : out Num; RHS : Num);
with procedure Assign_To_Num (LHS : out Num; RHS : Integer);
type Supl_Type is limited private; -- Any secret datapackage
Exprs is -- De'nition of Ada 95: http://www.ada-auth.org/
(+URLs) Error : exception; Zero : Num renames Zero_Ptr.all;
One : Num renames One_Ptr.all; -- Four types of assigning:
RW(deep copy), RO(copy ptrs and -- a unique ID), RSw (swap
ptrs and trash RHS), -- Swap (correctly swap pointers) type
Availability_Enum is (RO, RW, RA_Null); -- (Dupl, Write, ...)
type RA1_Enum is (RO1, RW1); -- Copy ptr, full copy type
RA2_Enum is (RO, RW, RSw); function IfElse (B : Boolean; U, V
: RA2_Enum) return RA2_Enum; type CK_Num is mod 2 ** 16; type
O_Rec; type O_Rec_Ptr is access O_Rec; type Expr is new
Ada.Finalization.Limited_Controlled with record -- Maybe
remove the Limited_ later O : aliased O_Rec_Ptr; UR : aliased
Availability_Enum; UCK : aliased CK_Num; end record; type
Expr_Ptr is access all Expr; type Expr_Ptr_Array is array
(Positive range <>) of Expr_Ptr;
-
- type Sysatom_Enum is ( NUMBER, -- BOOLEANp, -- Bool
seems unneeded but... EQ, NEQ, LT, LEQ, GT, GEQ, NOTp, IMPL,
IMPLBY, EQUIV, PLUS, TIMES, DIVIDE, ANDp, ORp, LIST, QUOTED);
type Atomic_Atom is new Striunli.V_Str; -- A ptr to a string
maybe type Atomic is record Sysatom : aliased Sysatom_Enum :=
QUOTED; Atom : aliased Atomic_Atom; end record;
Null_Atomic_Atom : constant Atomic_Atom := Atomic_Atom
(Striunli.Null_V_Str); Null_Atomic : constant Atomic :=
AtomicO(QUOTED, Null_Atomic_Atom); type Atomic_Array is
array
(Positive range <>) of Atomic; ... type Expr_Array is array
(Positive range <>) of aliased Expr; record A : aliased
Atomic; M : aliased Num; Len : aliased Natural := 0; S :
aliased Supl_Type; end record;end Exprs;(An aliased 'eld is
one that a pointer can point to).That is just a a sample data
structure.Craig Carey
 The HOL systems seem to be
getting described as very slow. MAybe too slow> to solve real
problems in logic. A 3GL that recurses and uses for-loops> may
be a lot faster, e.g. at eliminating quanti'ers.Perhaps you
could point us to a computer algebra system written in
HOL,and how its design has changed your perspective on
implementation ofCAS in the last 5 years?>>I donOt think
that
the perspective has changed much in the last 5 years.>>You can
look at the implementation of Axiom and Maxima since the
sources>>are public. More hints that somehow LISP is worth
considering. Suppose they were all> wrong and it was never
really actually worth considering.Worth considering? Maybe
you mean to ask, rhetorically, Was Lispa good language for
implementation of CAS? I think so. Anotherquestion is, Should
I implement a New CAS in Lisp? I think theanswer depends on
many factors. E.g. if your boss says everythingmust be
written in C#, you should probably write in C#.Another
question is, Is it worth CONSIDERING implementing a CASin
Lisp? I think that Lisp, Scheme, or (what one of my students
worthconsidering means you have considered it :)...> >>Most
systems beyond a certain size tend to have a language
structure>>underneath them that is supportive of the
application. Thus > Sure: Mr Fatemen is saying that LISP
should rarely (if ever) be used> as the underlying language
because its offerings mismatche with the> needs of symbolic
algebra developers. Since he was not going to produce>
evidence, there is some beneath this line.> You entirely
misunderstand. Let me give you an example or two.The earliest
string processing language (1957) was COMIT.Dan Bobrow,
writing his thesis project (STUDENT) in Lisp decided heneeded
linguistic transformations that could be describedfor string
matching, called METEOR. Thus he adapted Lisp to besupportive
of his application. He did not have to use COMIT.(I assume the
similarity in names is not accidental...)Keeping on the track
of other people writing programs in Lispwriting supportive
structures in Lisp, speci'cally pattern matchingin CAS, we
'nd that James Slagle, writing SAINT (c. 1961), a symbolic
integration(for elementary instance).Again, Joel Moses,
writing a fancier integrator, SIN (c. 1966)(Yiddish for
matchmaker).Clearly lisp can support a multitude of language
features,even with overlapping designs.The Reduce CAS has a
supportive language structure writtenin Lisp which provides
an in'x-like language on top of Lisp.It also has pattern
matching.There are also (several) examples of object-oriented
extensionsto Lisp, written in Lisp, prior to the
standardization ofCLOS. These include LOOPS, Flavors, New
Flavors.The point, once again, is that the underlying
language (Lisp,in this case) makes it convenient to build
additional supportivestructures that are closer to the
application. It supportsmultiple paradigms (a word I rarely
use, but here itis probably appropriate)I am sure that people
writing large systems in C, C++, Assembler,etc. can also
bene't from such structuring. E.g. the assemblercoder will
de'ne and use a macro language. The C programmercan use
macros. The C++ programmer can overload operatorssometimes.
This doesnOt change the paradigm, however. Lisp allows you
to
develop a system with similar syntacticconventions to Lisp, or
totally different conventions, including(for example) using
in'x syntax. This is harder to do withother
languages.>>macsyma/maxima is written in Lisp, but even
function de'nition>>in the implementation is slightly
changed. (defun foo ....) is>>written (defmfun foo ...) for
various reasons, and expanded differently.> None of which is
interesting> If you want to process de'nitions so that they
not only de'nefunctions but insert information into a data
base, this can bevery convenient.> ...> ...If an expression
tends to have 4 factors in the multiplications, and> 4 terms
in the additions, then arrays 4 items long would be wanted.>
No, if all products have exactly 4 factors, then you
mightwant arrays of length 4. If they tend to have 4 factors
thenit would probably be a terrible decision to have arrays
of 4. If youmean you should represent a product of n items as
an array ofn elements, that would be a more plausible
decision. You mightwant n+1 elements, with one element being
a tag product.> So symbolic algebra programming in LISP uses
LISPOs arrays ?. If not> then why not ?.(a) Some CAS use
Lisp
arrays. (e.g. mockmma on my home page usesvectors, a 1-d array
in Lisp).(b) Some CAS do not use arrays, but structures
(de'ned by defstruct)or objects (de'ned in CLOS
by defclass,
make-instance, etc), or lists.They use these other forms
because they serve the purposes of thedesign better, make
programming simpler or more reliable, aremore compact,
ef'cient, easier to debug.In some cases a programmer may use
lisp lists because he or sheis simply ignorant of the
alternatives. This doesnOt mean that theuse of lists is a
bad
choice, just not one made with due considerationof other
choices. Just like someone who implements long integersas
strings of digits.> >>I suspect that some people might think
that there are important changes>>because now people can
write computer algebra systems in Java, c#, .net,>>ML,
Python, Ruby, Perl, or some other language du jour.> The 'rst
choice of 3GLs for precise minded mathematicians, i.e. Ada
95,> was missing from the list. Is there a speci'c §aw of
Ada
that you> wish to criticise, Mr Fateman ?.> No, I think I
would criticize ALL the §aws of Ada.If there is a CAS written
in Ada, it has not come to my attention....RJF
Now a
question for you...Why do you
ask?Abelson-sussman-lectureshttp://www.swiss.ai.mit.edu/
classes/6.001/abelson-sussman-lectures/
Now a question
for you...Why do you ask?>> Abelson-sussman-lectures>>
http://www.swiss.ai.mit.edu/classes/6.001/
abelson-sussman-lectures/>The Return key is dangerous:-(
Continuing:lection 4b describes a framework for a computer
algebra system.My question is: Are generic operators really
that useful for developingcomputer algebrasystems? At the end
of the lecture when Hal demonstrated pretty complexexamples,
woudnOt it just be appropriate to change the design to
handlesymbolic expressions directly? After all, how do we
interpret 3/4: is is anoperation on the 2 numbers, or a
single value of type Rational? Undersymbolicexpresion
paradigm we ask to simplify the expression3+4, not to
evaluate it.Next, simpli'cation problem is probably hard, so
thatthere need to be some heuristic algorithms navigating the
space ofequivalentexpressions and 'nding the shortest
expression. ItOs not obvious that thisproblem
'ts naturally
in your framework. And 'nally, Hal
mentionedcoersions.Therefore, are Types useful for
implementing CAS? The correct way to answerthis question is
to look into, say, Maxima implementation, but I took
ashortcut to the newsgroup:-)
>>Now a question for
you...Why do you
ask?>Abelson-sussman-lectures>http://
www.swiss.ai.mit.edu/classes/6.001/abelson-sussman-lectures/> The Return key is dangerous:-( Continuing:lection 4b
describes a framework for a computer algebra system.The part
of a computer algebra system that simultaneously 'ts
ahierarchical data structure framework and also a
mathematicalhierarchical framework is a small,
non-controversial, and elegantexample of programming. It
represents a very small part ofa real computer algebra
system.My question is: Are generic operators really that
useful for developing> computer algebra> systems? The authors
of Axiom make the best possible argument in favor ofthis kind
of view. Axiom was not a success. Whether this wascaused by
this perspective or by other commercial or technicalproblems
is hard to tell.At the end of the lecture when Hal
demonstrated pretty complex> examples, woudnOt it just be
appropriate to change the design to handle> symbolic
expressions directly? After all, how do we interpret 3/4: is
is an> operation on the 2 numbers, or a single value of type
Rational? I am not sure what the question is. the operation
would be (/ 3 4).The result would be the rational number
object usually written 3/4.Under> symbolic> expresion
paradigm we ask to simplify the expression3+4, not to
evaluate it.A simpli'cation program would ordinarily take an
expression in lisplike O(+ 3 4) and simplify it to 7. In this
case there is probablyno difference.Next, simpli'cation
problem is probably hard, so that> there need to be some
heuristic algorithms navigating the space of> equivalent>
expressions and 'nding the shortest expression.Since this
problem is not in general computationally solvable atall,
there is no reason to believe a heuristic algorithm wouldever
'nd the optimal solution, and it might wander around fora
very
long time. ItOs not obvious that this> problem
'ts naturally
in your framework. The most effective simpli'cation programs
use non-local informationto try to 'nd canonical forms for
large expressions. Purely functionalprograms are especially
good for local transformations.And 'nally, Hal mentioned>
coersions.Therefore, are Types useful for implementing CAS?
The correct way to answer> this question is to look into,
say, Maxima implementation, but I took a> shortcut to the
newsgroup:-)Macsyma uses explicit tags on most objects and
many programs consist ofdispatching on these types. This is
probably very common in other CASimplementations as well. The
notion of type as given in most programming languages is
inadequate, as pointed out by Axiom literature.e.g. if you
have types integer, §oat, structure, pointer, that buysyou
almost nothing.If you want to see a readable exposition of
how demo computer algebrasystems can be put together, see
NorvigOs Paradigms of AI Programmingbook. It shows how to
(for example) do some symbolic integration. This task is
quite remote from the A&S discussion.RJF>
=i have two
functions f,g, twice differentiable,
F(x)=f(x)+eps*g(x)f(r)=0, fO(r)=!0,F(r+h)=0how to show that
h
equal approximately -eps*g(r)/fO(r). (onlyO(eps^2), O(h^2)
are
omittable)
i have two functions f,g, twice
differentiable, > F(x)=f(x)+eps*g(x)> f(r)=0,
fO(r)=!0,F(r+h)=0how to show that h equal approximately
-eps*g(r)/fO(r). (only> O(eps^2), O(h^2) are omittable)from
your posting of quite off-topic questions to different
mathematical newsgroups I guess that you didnOt bother at
all
to learn the goal of each group. This group for example is
intended to discuss about different symbolic algebra systems
like Mathematica or Maple, not fr doing simple homework
problems.For this problem: First try to understand the
problem. Then take a paper and a pencil and sketch the
solution for some simple functions f and g, which ful'll the
stated conditions.Good luckAlois
=Sounds a lot like a
calculus homework problem.Consider F(r+h) as a series
expansion for small h,and assume that
abs(eps*gO(r)/fO(r)) is
much less than 1.> i have two functions f,g, twice
differentiable,> F(x)=f(x)+eps*g(x)> f(r)=0,
fO(r)=!0,F(r+h)=0>> how to show that h equal approximately
-eps*g(r)/fO(r). (only> O(eps^2), O(h^2) are
omittable)
=seems that simple. but actually itOs not.note,
only order of eps^2 and h^2 are omitable. no further
assumptions.this is actually a problem i met in numerical
analysis.> Sounds a lot like a calculus homework problem.>
Consider F(r+h) as a series expansion for small h,> and
assume that abs(eps*gO(r)/fO(r)) is much less
than 1.> i have
two functions f,g, twice differentiable,> F(x)=f(x)+eps*g(x)>
f(r)=0, fO(r)=!0,F(r+h)=0>> how to show that h equal
approximately -eps*g(r)/fO(r). (only> O(eps^2), O(h^2) are
(legacy-1.mathforum.org [144.118.94.27]) by
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(from apache@localhost) by legacy.mathforum.org
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h9GKxXT01442
=You did NOT prove:Every odd number > 6 is
equal to sum of three primes.You DID prove:Every odd number >
6 is equal to sum of three numbers of the form6n+-1.Now you
must prove:IF Every odd number > 6 is equal to sum of three
numbers of the form6n+-1 THEN Every odd number > 6 is equal
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h9GKxrt01476
=You did NOT prove:Every odd number > 6 is
equal to sum of three primes.You DID prove:Every odd number >
6 is equal to sum of three numbers of the form6n+-1.Now you
must prove:IF Every odd number > 6 is equal to sum of three
numbers of the form6n+-1 THEN Every odd number > 6 is equal
to sum of three primes..
=I am looking for a way to solve
the diet problem with integer programming(not with the
traditional linear programing), in order to set
amountconstraints for the food items. Is there a way to do it
with excel? If not,is there a software, freeware or shareware,
that could do it? I am aclinical dietitian and I donOt know
much maths. I am only looking for a fastand computerized way
to make menus for my clinic. I formulated a linearprogramming
model in excel but I would like to set some constraints for
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h9I4XOG09217
=MAKE EASY AND QUICK CASH MONEY by doing
readon! You will be very thankful you did! A little while
back, I wasbrowsing through newsgroups, just like you are
now,thousands of dollars within weeks with only aninitial
investment of $6.00! So I thought, Yeah, right, this must bea
scam, but like most of us, I was curious, so Ikept reading.
Anyway, it said that you send $1.00 to each of the 6place
your own name and address in the bottom of the list at #6,
and(There are thousands) No catch, that was it. So after
thinking itover, and talking to a few people 'rst, I
thoughtabout trying it. I 'gured what have I got to lose
except 6 stamps and$6.00, right? Like most of us I was a
littleskeptical and a little worried about the legal aspects
of it all. So Ichecked it out with the U.S. Post
Of'ce(1-800-725-2161) and they con'rmed that it
is indeed
legal! Then Iinvested the measly $6.00. Well GUESSshocked! I
'gured it would end soon, but themoney just kept coming in.
In my 'rst week, I made about $25.00. Bythe end of the second
week I had made atotal of over $1,000.00! In the third week I
had over $10,000.00 anditOs still growing. This is now my
fourth weekand I have made a total of just over $42,000.00
and itOs still comingin rapidly. ItOs
certainly worth $6.00,
and 6stamps, I have spent more than that on the lottery!! Let
me tell youhow this works and most importantly, why itcan get
the information off of it as you need it.STEP 1 Get 6
separate pieces of paper and write the following on eachpiece
of paper PLEASE PUT ME ON YOURMAILING LIST. Now get 6 US $1.00
bills and place ONE inside EACH ofthe 6 pieces of paper so the
bill will not beseen through the envelope to prevent thievery.
Next, place one paperin each of the 6 envelopes and seal them.
Youshould now have 6 sealed envelopes, each with a piece of
paper statingthe above phrase, your name and address,and a
$1.00 bill. What you are doing is creating a service by
this.THIS IS ABSOLUTELY LEGAL! This program remainssuccessful
because of the honesty and integrity of the
participants.Please continue its success by carefully
adheringto the instructions. You will now become part of the
Mail Orderbusiness. Your product is simply a service of
helping#1] J. Willard 2845 Rockford Rd Boonville, NC27011 #2]
A. Hyde 16085 A East Rice Pl. Aurora, CO 80015 #3] Jim P 3766
Spring Green Rd Green Bay, WI 54313 #4] Stephen 17025 E.
Wiley Place Parker, CO 80134 #5] T Gibson 8244 Hydra Lane Las
Vegas, NV 89128 #6] T Bullock 1175 E. Silver Mesa Cir, Sandy,
UT 84094STEP 2 Now take the #1 name and address off the list
that you seeabove, move the other names up (6 becomes 5,5
becomes 4, etc...) and add YOUR Name and address as number 6
on thelist. STEP 3 Change anything you need to,newsgroups. (I
think there are close to 24,000 groups) All you need is200,
but remember, the more you post, themore money you make!
DIRECTIONS--HOW TO POST TONEWSGROUPS Step
1) You do not need tore-type this entire letter to do your
own posting. Simply put yourcursor at the beginning of this
letter and drag yourcursor to the bottom of this document,
and select OcopyO from the editmenu. This will
copy the
entire letter into thecomputers memory. Step 2) Open a blank
OnotepadO 'le and place
yourOeditO menu select
OpasteO. This
will paste a copy of the letter intonotepad so that you can
add your name to the list.Step 3) Save your new notepad 'le
as a .txt 'le. If you want to doyour postings in different
sittings, youOll alwayshave this 'le to go back
to. Step 4)
Use Netscape or Internetexplorer and try searching for
various newsgroups(on-line forums, message boards, chat
sites, discussions.) Step 5)as a new message by highlighting
the text of this letter and selectingthis will be the header
that everyone sees as they scroll through thelist of postings
in a particular group, click thepost message button. YouOre
done with your 'rst
one!Congratulations...THATOS IT! All you
have to do is jump todifferent newsgroups and post away,
after you get the hang of it, itwill take about 30 seconds
for each newsgroup!**REMEMBER, THE MORE NEWSGROUPS YOU POST
IN, THE MORE MONEY YOU WILLMAKE!! BUT YOU HAVE TO POSTA
MINIMUM OF 200** ThatOs it! You will begin receiving money
fromaround the world within days! You maywill receive. If you
wish to stay anonymous,you can invent a name to use, as long
as the postman will deliver it.**JUST MAKE SURE ALL THE
ADDRESSES ARECORRECT.** Now the WHY part Out of 200 postings,
say I receive only 5replies (a very low example). So then
Imade $5.00 with my name and address at #6 on the letter.
Now, each ofthe 5 persons who just sent me $1.00 makethe
MINIMUM 200 postings, each with my name and address at #5 and
only5 persons respond to each of the original5, that is
another $25.00 for me, now those 25 each make 200
MINIMUMposts with my name and address at #4 andonly 5 replies
each, I will bring in an additional $125.00! Now, those125
persons turn around and post the MINIMUM200 with my name and
address at #3 and only receive 5 replies each, Iwill make an
additional $626.00! OK, nowhere is the fun part, each of
those 625 persons post a MINIMUM 200letters with my name and
address at #2 andthey each only receive 5 replies, that just
made me $3,125.00!!! Those3,125 persons will all deliver this
message to200 newsgroups with my name and address at #1 and if
still 5 personsper 200 newsgroups react I will
receive$15,625,00! With an original investment of only $6.00!
the latest posting in the newsgroups, and send out another
$6.00to names on the list, putting your name atnumber 6
again. And start posting again. The thing to remember is,
doyou realize that thousands of people all overeveryday, JUST
LIKE YOU are now!! So can you afford$6.00 and see if it really
works?? I think so... People have said,what if the plan is
played out and no one sends youthe money? So what! What are
the chances of that happening when thereare tons of new
honest users and newhonest people who are joining the
internet and newsgroups everyday andare willing to give it a
try? Estimates are at20,000 to 50,000 new users, every day,
with thousands of those joiningthe actual internet. Remember,
play FAIRLYand HONESTLY and this will work. ALSO REMEMBER*** :
: SEND YOUR $1 OUTTO EVERYONE ON THE LIST, : : EVENIF THEY ARE
NOT FROM THE U.S. : : WHAT ARE YOU WAITING FOR??? : : : :GOOD
EASY$$ and you will 'nd alot ofmessage boards !MAKE MONEY
(legacy-1.mathforum.org [144.118.94.27]) by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
=MAKE EASY AND QUICK CASH MONEY by
readon! You will be very thankful you did! A little while
back, I wasbrowsing through newsgroups, just like you are
now,thousands of dollars within weeks with only aninitial
investment of $6.00! So I thought, Yeah, right, this must bea
scam, but like most of us, I was curious, so Ikept reading.
Anyway, it said that you send $1.00 to each of the 6place
your own name and address in the bottom of the list at #6,
and(There are thousands) No catch, that was it. So after
thinking itover, and talking to a few people 'rst, I
thoughtabout trying it. I 'gured what have I got to lose
except 6 stamps and$6.00, right? Like most of us I was a
littleskeptical and a little worried about the legal aspects
of it all. So Ichecked it out with the U.S. Post
Of'ce(1-800-725-2161) and they con'rmed that it
is indeed
legal! Then Iinvested the measly $6.00. Well GUESSshocked! I
'gured it would end soon, but themoney just kept coming in.
In my 'rst week, I made about $25.00. Bythe end of the second
week I had made atotal of over $1,000.00! In the third week I
had over $10,000.00 anditOs still growing. This is now my
fourth weekand I have made a total of just over $42,000.00
and itOs still comingin rapidly. ItOs
certainly worth $6.00,
and 6stamps, I have spent more than that on the lottery!! Let
me tell youhow this works and most importantly, why itcan get
the information off of it as you need it.STEP 1 Get 6
separate pieces of paper and write the following on eachpiece
of paper PLEASE PUT ME ON YOURMAILING LIST. Now get 6 US $1.00
bills and place ONE inside EACH ofthe 6 pieces of paper so the
bill will not beseen through the envelope to prevent thievery.
Next, place one paperin each of the 6 envelopes and seal them.
Youshould now have 6 sealed envelopes, each with a piece of
paper statingthe above phrase, your name and address,and a
$1.00 bill. What you are doing is creating a service by
this.THIS IS ABSOLUTELY LEGAL! This program remainssuccessful
because of the honesty and integrity of the
participants.Please continue its success by carefully
adheringto the instructions. You will now become part of the
Mail Orderbusiness. Your product is simply a service of
helping#1] J. Willard 2845 Rockford Rd Boonville, NC27011 #2]
A. Hyde 16085 A East Rice Pl. Aurora, CO 80015 #3] Jim P 3766
Spring Green Rd Green Bay, WI 54313 #4] Stephen 17025 E.
Wiley Place Parker, CO 80134 #5] T Gibson 8244 Hydra Lane Las
Vegas, NV 89128 #6] T Bullock 1175 E. Silver Mesa Cir, Sandy,
UT 84094STEP 2 Now take the #1 name and address off the list
that you seeabove, move the other names up (6 becomes 5,5
becomes 4, etc...) and add YOUR Name and address as number 6
on thelist. STEP 3 Change anything you need to,newsgroups. (I
think there are close to 24,000 groups) All you need is200,
but remember, the more you post, themore money you make!
DIRECTIONS--HOW TO POST TONEWSGROUPS Step
1) You do not need tore-type this entire letter to do your
own posting. Simply put yourcursor at the beginning of this
letter and drag yourcursor to the bottom of this document,
and select OcopyO from the editmenu. This will
copy the
entire letter into thecomputers memory. Step 2) Open a blank
OnotepadO 'le and place
yourOeditO menu select
OpasteO. This
will paste a copy of the letter intonotepad so that you can
add your name to the list.Step 3) Save your new notepad 'le
as a .txt 'le. If you want to doyour postings in different
sittings, youOll alwayshave this 'le to go back
to. Step 4)
Use Netscape or Internetexplorer and try searching for
various newsgroups(on-line forums, message boards, chat
sites, discussions.) Step 5)as a new message by highlighting
the text of this letter and selectingthis will be the header
that everyone sees as they scroll through thelist of postings
in a particular group, click thepost message button. YouOre
done with your 'rst
one!Congratulations...THATOS IT! All you
have to do is jump todifferent newsgroups and post away,
after you get the hang of it, itwill take about 30 seconds
for each newsgroup!**REMEMBER, THE MORE NEWSGROUPS YOU POST
IN, THE MORE MONEY YOU WILLMAKE!! BUT YOU HAVE TO POSTA
MINIMUM OF 200** ThatOs it! You will begin receiving money
fromaround the world within days! You maywill receive. If you
wish to stay anonymous,you can invent a name to use, as long
as the postman will deliver it.**JUST MAKE SURE ALL THE
ADDRESSES ARECORRECT.** Now the WHY part Out of 200 postings,
say I receive only 5replies (a very low example). So then
Imade $5.00 with my name and address at #6 on the letter.
Now, each ofthe 5 persons who just sent me $1.00 makethe
MINIMUM 200 postings, each with my name and address at #5 and
only5 persons respond to each of the original5, that is
another $25.00 for me, now those 25 each make 200
MINIMUMposts with my name and address at #4 andonly 5 replies
each, I will bring in an additional $125.00! Now, those125
persons turn around and post the MINIMUM200 with my name and
address at #3 and only receive 5 replies each, Iwill make an
additional $626.00! OK, nowhere is the fun part, each of
those 625 persons post a MINIMUM 200letters with my name and
address at #2 andthey each only receive 5 replies, that just
made me $3,125.00!!! Those3,125 persons will all deliver this
message to200 newsgroups with my name and address at #1 and if
still 5 personsper 200 newsgroups react I will
receive$15,625,00! With an original investment of only $6.00!
the latest posting in the newsgroups, and send out another
$6.00to names on the list, putting your name atnumber 6
again. And start posting again. The thing to remember is,
doyou realize that thousands of people all overeveryday, JUST
LIKE YOU are now!! So can you afford$6.00 and see if it really
works?? I think so... People have said,what if the plan is
played out and no one sends youthe money? So what! What are
the chances of that happening when thereare tons of new
honest users and newhonest people who are joining the
internet and newsgroups everyday andare willing to give it a
try? Estimates are at20,000 to 50,000 new users, every day,
with thousands of those joiningthe actual internet. Remember,
play FAIRLYand HONESTLY and this will work. ALSO REMEMBER*** :
: SEND YOUR $1 OUTTO EVERYONE ON THE LIST, : : EVENIF THEY ARE
NOT FROM THE U.S. : : WHAT ARE YOU WAITING FOR??? : : : :GOOD
EASY$$ and you will 'nd alot ofmessage boards !MAKE MONEY
FAST AND EASY$$