mm-153
can anyone offer some advice on how to solve this problem? When the diffusion coef'cient D is varying in space the PDE
is> not> U_t=D(U_xx+U_yy)> It should be> U_t = div (D
grad U)> To see this read up on the derivation of the heat
equation from> 'rst-principles (i.e., from the conservation
law).> To determine how to corectly discretize the above
equation around> the discontinuity in D read up on 'nite
volume methods that result> from discretizing Gauss's law
(integrate the second equation over a> cell whose sides go
through the midpoint between grid points).> Peterhi
<snip>>
I apreciate your help, but perhaps my English in not good
enough to explain> my problem.> You gave me suggestions on
how to write the C++ code, but I already said> that this is
not my problem.> My *only* problem is to write the f
function.> When I have f(t)=... I'll be able to write all
the
code to do the> simulation.In that case, I guess I'm puzzled
as to what you mean by write the ffunction.If you're not
talking about C++ code, what does that phrase mean?Derrick
gave you an interface for the C++ code. So did I. You
havediscussed all the aspectsof the problem, and seem to
understand the math involved. I gave you aformula for the
forcesthat generate the derivative, and you said that this is
the formula youwere already intendingto use.So I'm not clear
at all, anymore, as to just what it is you're asking usto
do.
Each time oneof us gives advice, you say, in effect, I
already knew that. I'mafraid I've arrived at
the same place
Derrick did, which is that I don't understand what
moreadvice
I can give you._FUNCTIONALLY_, here are the rules. However you
choose to encode thefunction in C++, it musthave the form f(t,
y)where y is that state vector we've been talking about. The
functionmust accept whatever valueof t (time) and y
(containing position and velocity), and compute thederivatives
of both. Thederivative of the position is simply the velocity,
so that part involvesjust copying the velocity given to the
function, into the position slot for the returnvalue. I showed
that lasttime, and you seem to understand that part.The only
thing left to do is to compute the derivative of
velocity,which is, of course, theacceleration. That's the
sum
of all the accelerations from theNewtonian gravity
equation.You and Derrick discussed this at some length.The
only rule you must be sure to follow is that you don't
compute_ANYTHING_ that doesn't dependon the parameters
passed
into the function. It's OK to look up thepositions of the
planets inthe ephemerides, because they are functions only of
the time. Just besure to use the time passedto you, and not
the time you might think is the current time. Likewise, use
the position part of the state vector passed in, to compute
the corresponding accelerations.There's simply nothing else
left to do. Jack
=Ok, try this:Let the mu's of the planets
(GM for each) be mu(i) [best I can do for asubscript in this
medium). Also let their positions from your ephemerides as
functions of time beP(i,t).Let x be your position vector, v
your velocity vector. I was using y tobe the state vector
(trying toconform to your original form f(x,y)): y = [x v] in
Matlab-speak.The things you must compute are: x_dot = v; v_dot
= sum(-mu(i)*R/R^3)where R = x - P(i,t)The x_dot and v_dot are
the derivatives of your x and v. They are thereturn values
from f(x, y).There's one other thing I would suggest, which
is
to start with aproblem (and therefore an f function)that you
already understand, and know the answer to. I always use
thisone: x_dot[0] = -2*pi*x[1]; x_dot[1] = 2*pi*x[0]; double
x[2] = {1,0};If done properly, this should give you a circular
motion in the x-yplane, with period of 1. If donewith RK4, you
should get very accurate results with a step size around0.01.
Because you know what theanswer is, you can compute a vector
error. It should change by orderh^4 when you change step
size.Until you get that test case working, I wouldn't go
further with thereal simulation.> I have this example (for
scalar variables):> double f(double x,double y) { return
y+x*x; }> double RK4(double x,double y,double xmax,int n)>
{> double f1,f2,f3,f4, h=(xmax-x)/n;> for(int i=0; i<n; i++)
{> f1 = f(x, y);> f2 = f(x+.5*h, y+.5*h*f1); //*** Line 6 ***>
f3 = f(x+.5*h, y+.5*h*f2);> f4 = f(x+h, y+h*f3);> y +=
h*(f1+2.*f2+2.*f3+f4)/6.;> x += h;> }> return y;> }> The
real y is: e^x - x*x - 2*x - 2.> As you can see, the f
function returns a value which depends on both x *and*> y.>
For example, to calculate f2 for the position part of the
state vector I> should pass t+h/2 (for the x) and the actual
velocity + h/2*f1 (for the y).> It seems to me a bit strange;
in other words, I don't understand how to link> what you
said
with the above RK4() function.Christiano, I know that you have
told me that you know C++, and have noproblem converting the
mathto code. However, the example you give above, and the
comments that gowith it, make me think thatyou still don't
understand what the algorithm is trying to tell you. You do
not _HAVE_ to passanything to f. The code does that. You
don't
have to calculate t+h/2and v + h/2*f1; you have already done so
in line 6 of RK4. Let the code do its work. Incidentally, I
would also pass the name of the function f to RK4, so it will
work independent of the nameof the function.Are we getting
anywhere?Jack
Ok, try this:>> Let the mu's of the planets
(GM for each) be mu(i) [best I can do for> a subscript in this
medium).> Also let their positions from your ephemerides as
functions of time be> P(i,t).>> Let x be your position vector,
v your velocity vector. I was using y> to be the state vector
(trying to> conform to your original form f(x,y)): y = [x v]
in Matlab-speak.>> The things you must compute are:>> x_dot =
v;> v_dot = sum(-mu(i)*R/R^3)>> where R = x - P(i,t)>> The
x_dot and v_dot are the derivatives of your x and v.You have a
very big patience to deal with me. :-)> They are the return
values from f(x, y).When I have f2 = f(x+.5*h, y+.5*h*f1);what
should I write?> There's one other thing I would suggest,
which
is to start with a> problem (and therefore an f function)> that
you already understand, and know the answer to. I always use
this> one:>> x_dot[0] = -2*pi*x[1];> x_dot[1] = 2*pi*x[0];>>
double x[2] = {1,0};>> If done properly, this should give you
a circular motion in the x-y> plane, with period of 1. If
done> with RK4, you should get very accurate results with a
step size around> 0.01. Because you know what the> answer is,
you can compute a vector error. It should change by order> h^4
when you change step size.>> Until you get that test case
working, I wouldn't go further with the> real
simulation.With
the code posted in my previous message (double RK4(double
x,doubley,double xmax,int n)), I really don't know how to
solve the problem, but ifI use the code in Numerical Recipes
with h=0.02 I get (for example): x[0] x[1] alpha [DEG]0.994736
0.104554 6.0001940.978569 0.208008 12.0003870.951669 0.309227
18.0005810.914329 0.407100 24.0007750.866952 0.500555
30.0009680.810053 0.588563 36.0011620.744253 0.670160
42.0013560.670267 0.744447 48.0015490.588903 0.810608
54.0017430.501051 0.867913 60.0019370.407669 0.915732
66.0021300.309780 0.953535 72.0023240.208453 0.980905
78.0025180.104798 0.997536 84.002711-0.000051 1.003243
90.002905which seems good.But I want to be honest with you: I
don't understand how I got these values.>> I have this
example (for scalar variables):>> double f(double x,double
y) { return y+x*x; }>> double RK4(double x,double y,double
xmax,int n)>> {>> double f1,f2,f3,f4, h=(xmax-x)/n;>> for(int
i=0; i<n; i++) {>> f1 = f(x, y);>> f2 = f(x+.5*h, y+.5*h*f1);
//*** Line 6 ***>> f3 = f(x+.5*h, y+.5*h*f2);>> f4 = f(x+h,
y+h*f3);>> y += h*(f1+2.*f2+2.*f3+f4)/6.;>> x += h;>> }>>
return y;>> }>> The real y is: e^x - x*x - 2*x - 2.>> As
you can see, the f function returns a value which depends on
both>> x *and* y.>> For example, to calculate f2 for the
position part of the state>> vector I should pass t+h/2 (for
the x) and the actual velocity +>> h/2*f1 (for the y). It
seems to me a bit strange; in other words, I>> don't
understand how to link what you said with the above RK4()>>
function.>> Christiano, I know that you have told me that you
know C++, and have> no problem converting the math> to code.
However, the example you give above, and the comments that go>
with it, make me think that> you still don't understand what
the algorithm is trying to tell you.You are perfectly right.>
You do not _HAVE_ to pass> anything to f. The code does that.
You don't have to calculate t+h/2> and v + h/2*f1; you have>
already done so in line 6 of RK4. Let the code do its work.>
Incidentally, I would also pass the> name of the function f to
RK4, so it will work independent of the name> of the
function.>> Are we getting anywhere?Yes, at the starting
point. :-)I learned by myself how to calculate the integral of
a function, thespherical trigonometry, the modular arithmetic
and other stuff (I'm tryingto learn also the English), but
despite your incredibly big help I'm
stillsimulation.Cristiano
> You do not HAVE to pass>
anything to f. The code does that. You don't have to
calculate
t+h/2> and v + h/2*f1; you have> already done so in line 6 of
RK4. Let the code do its work.> Incidentally, I would also
pass the> name of the function f to RK4, so it will work
independent of the name> of the function.>> Are we getting
anywhere?Finally we have solved the problem! Or at least I
think so.Now I understand the RK4 code. f2 = f(x+.5*h,
y+.5*h*f1);in this way: for (i=0; i<2; i++) pvt[i] = pv[i] +
f1[i]* .5*h; for (i=0; i<2; i++) f2[i] = RK_f(i, Tsim + .5*h,
pvt);where pv is the state vector (position and velocity
vectors).When I call RK_f with i = 0, RK_f returns pvt[1] (the
velocity).When I call it with i = 1, RK_f calculate the
positions of the planets attime Tsim+ h/2 and the acceleration
of the ship (v_dot of your previousmessage).The simulaton seems
good, but I still have some doubt.I simulate a journey from
Mars to Jupiter (with a §yby).If I calculate only the initial
position and velocity of the planets andthen I integrate them
using RK4 (without using DE405 at every step), thestep size
(h) doesn't change too much the §yby geometry.The
position
error of Jupiter at the §yby is only 2448 m (for h from 4000to
18000 s).If I calculate the position of the planets at every
step and I use RK4only to integrate the position and velocity
of the ship, the step sizeaffects greatly the simulation (the
distance form Jupiter range from 18 to 1million
your big help!Cristiano[Perhaps the next week I will have
problems to write in this ng. If youdon't see my replies,
you
know the reason.]
<snip>> You have a very big patience to
deal with me. :-)>They are the return values from f(x, y).>
When I have> f2 = f(x+.5*h, y+.5*h*f1);> what should I
write?Write f2 = f(x+.5*h, y+.5*h*f1);That's it.> With the
code posted in my previous message (double RK4(double
x,double> y,double xmax,int n)), I really don't know how to
solve the problem, but if> I use the code in Numerical
Recipes with h=0.02 I get (for example):> x[0] x[1] alpha
[DEG]> 0.994736 0.104554 6.000194> 0.978569 0.208008
12.000387> 0.951669 0.309227 18.000581> 0.914329 0.407100
24.000775> 0.866952 0.500555 30.000968> 0.810053 0.588563
36.001162> 0.744253 0.670160 42.001356> 0.670267 0.744447
48.001549> 0.588903 0.810608 54.001743> 0.501051 0.867913
60.001937> 0.407669 0.915732 66.002130> 0.309780 0.953535
72.002324> 0.208453 0.980905 78.002518> 0.104798 0.997536
84.002711> -0.000051 1.003243 90.002905> which seems
good.See, that's the problem. Seems good
doesn't get it. You
must knowwhat the answeris supposed to be, else how can you
know when it's working right?JackBelow is an example of an
2x4
triangle in
=I am looking for any information I can 'nd on
an example of SophieGermain's number theory. Even an
illustration of one of her problems wouldhelp me greatly. I
appreciate anyone who might know of her
work.Kavonkavon@mathisradical.com
I am looking for any
information I can 'nd on an example of Sophie>
Germain's
number theory. Even an illustration of one of her problems
would> help me greatly. I appreciate anyone who might know of
her work.> Kavon> kavon@mathisradical.comTake a look at
Eric temple Bell's Men [sic] of Mathematics.IIRC she did
important work on Fermat's Last Theorem. See SimonSingh,
Fermat's Enigma.-- Julian V. NobleProfessor Emeritus of
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/
Science knows only one commandment: contribute to science.
-- Bertolt Brecht, Galileo.
=You are solving all my
problems tonight! I really appreciate your posts. Ihave a lot
going on this week in school and I want to be sure all of
mybases are covered for my students.I'm on my way to search
for Men of Mathematics, which you have suggested, assoon as I
click send.Kavonkavon@mathisradical.com
I am looking for
any information I can 'nd on an example of
Sophie>Germain's
number theory. Even an illustration of one of her problems
would>help me greatly. I appreciate anyone who might know of
place to
starthttp://www.agnesscott.edu/lriddle/women/germain.htmShe is
even mention in the recent play `Proof'--
http://www.math.fsu.edu/~bellenotbellenot <At/> math.fsu.edu
dialup-65.56.245.118.dial1.albany1.level3.net)
=ayy!! my
prof is bad... have no clue how to solve thisviton.1@osu.edu
legacy.mathforum.org (legacy-1.mathforum.org [144.118.94.27])
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
=$$$$ IF YOU DO THIS YOU WILL MAKE
$7,000. HELP ME AND YOURSELF$$$$$ GUARANTEED-you WILL have
$7,000 in two weeks ONLY $5 TO $14 Total Cost, 1 HOUR OF WORK
and NO MAILING LISTS GIFTING CLUB PLEASE READ TO FIND OUT HOW!
If you want to make a few thousand dollars really quickly, then
pleasetake a moment to read and understand the MLM program I am
sharing withyou. NO, it is Not what you think! YOU DO NOT have
to spend $5 to 'veany other product. NOR will you need to
invest more money later to getthings going. This is the
fastest, easiest program you will ever beable to do. Complete
it in One hour and you will never forget the dayprograms, by
all means stay with them. The more the merrier! ButPlease read
on. First of all, there are only three levels, not four, 've
or
six, likemany other programs. This three level program is more
realistic andmuch, much faster. Because it is so easy, the
response rate for thisprogram is Very high and Very fast. And
you receive your reward inabout fourteen days. Thats only Two
weeks - not three months. Just intime for next months
bills!!!!!!!!!!! HERE ARE THE SIMPLE DETAILS 20 on the
internet. You should send them to people who send you
theirprograms, because they are already believers and your
program isbetter and faster or post them in chat forums or on
message boards.Even if you are already in a program, continue
to stay with it, but doyourself a favor and DO THIS ONE as
well. RIGHT NOW! It is simple andtakes a very small
investment, not hundreds of dollars. And it willpay you before
the other letters even begin to trickle in!! Just give one
person $5. Thats it! Thats all! Follow the simpleinstructions
and in two weeks you should have at least seen $7,000because
most people will respond due to the low investment, speed,
andhuge pro't potential. We are now at a 50% response rate!
Thats a$10,000 return. So lets all keep it going and help each
other inthese tough times. 1. On a blank sheet of paper write
your name and address clearly andfold it around a 've-dollar
bill. Send this to the 'rst name on thelist. Only the
'rst
person on the list gets your name and a've-dollar gift. 2.
Retype the list only, removing the 'rst (#1) name from the
list.Move the other two names up and add your name to the list
in the third(#3) position. 3. Paste your newly typed list
neatly over the old one and make 20copies of this letter and
send to 20 prospects or post 20 on theinternet on message
boards or chat forums. An excellent source ofnames is the
people who send you other programs, and the names listedon the
letter that they send you. Do it right away. Its so easy,Dont
mull it over, One hour! Thats it! There is no more to do. When
your name reaches the 'rst position in afew days, it will be
your turn to collect your gifts. The gifts willbe sent to you
by over 1,500 to 2,000 people like yourself who arewilling to
invest $5 and one hours to receive $7,000 in cash. Yourentire
investment will be about $5 to $14 including the $5 gift
yousent the 'rst name on the list, envelopes, copies, and
stamps. Thatsbills in two weeks. Consider that! CAN I DO IT
AGAIN? the printer or copier. And now you can do it again over
and over withyour regular group of gifters. Why not? It beats
working! Each timename will climb position at a Dizzying
geometric rate. I just spenttwo weeks in Manhattan at the
Plaza seeing every show and attendingevery concert I wanted to
and I havent worked in three months. and send out 200 or more.
Thats 'ne. You can if you want to. Thatdecision is yours to
make. The possibilities are great. But we areenjoying a 50%
rate to this letter. Not interested? Cmon you havenothing to
lose! Experimentation? One hour of your time and about $12to
$14! ACT FAST AND GET MONEY FAST HONESTY AND INTEGRITY MAKE
MONEY!! 1.Miranda Converse 3 Pinetree dr Plainville, MA 02762
2.TONY STARKS 2206 TANGLEWOOD DR. KINSTON, NC 28504 3.CALEB
legacy.mathforum.org (legacy-1.mathforum.org [144.118.94.27])
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
=THIS IS A NO INVESTMENT
If you are already working a home business, you can dothis one
along with it. If not, this is the easiest way toearn $$$
online that you've ever seen! Just be sure toread it
all...take your time, come back to it... go over andover it,
you won't be sorry.WHAT IF I share with you a way to make
some
fast andeasy cash, using a copy of this letter and the
simplestInternet Payment System around?You have most likely
seen or heard about The Letter$5.00 program that was
recently televised on 20/20,Street Journal on 6/16/2001. If
not, here it is!!!WHAT IF it took only a half hour to set it
up? Even ifyou are already in another program, stay with it,
but doyourself a favor and DO THIS ONE as well.There is NO
LIMIT to the income you can generate from this business!!THE
FACTS ARE SIMPLE;- IF YOU NEED TO MAKE AFEW THOUSAND DOLLARS
EVER DO!!!Please read all of this, right to the last
sentence.Everyone has heard about PayPal (if you haven't
youwill soon) and, when I came across this concept I knewit
would work because, as a member of PayPal, I hadalready
experienced their ef'ciency and excellentstanding. PayPal is
the simplest method of making andreceiving payments online
anyone has ever seen! Anyonefurther before you go there... You
can complete thiswhole process in less than one hour and you
will neverforget the day you decided to do so!!! Oh!, Did I
sayFAST? By fast I mean Othe speed of the Internet-typeAnd, if
perfectly legal!NEED PROOF? Here are just two testimonials
fromindividuals who decided to invest nothing more than
alittle of their time.TESTIMONIALS FROM:Tony Stevens,
Vandenberg AFB, CA:our frat brothers.....and while I haven't
seen my $20grand yet, I'm up to $8,285. Hook me up when you
runthis program again.Mary Gathers, Columbia, SC:Hey cuz! This
is Mary. I only have one thing to say toand went on vacation.
When I got back, my account wasup to over $12,000! I owe
you!GETTING STARTED.If you're not already a user, the very
'rst thing you needto do is go to PayPal and SIGN UP. It
takes
two minutesand PayPal will deposit $5.00 in your account once
you startusing it. That makes this program FREE!HERE'S THE
LINK: https://www.paypal.com/mrb/pal=5J4TFLJV48YA2BE SURE TO
USE THE LINK ABOVE, THEN SIGN UP FOR A(PREMIER) (FREE) ACCOUNT
OR YOU'LL BE LIMITED TOACCEPTING $100 DOLLARS ONLY!!! YOU
WILL
MAKE MUCHMORE THAN THIS!!!Then...... SEND a $5.00 payment from
your PayPal Account($5.00 you can earn back from Paypal, which
makes this(No.1) along with a note saying Please add me to
yourWON'T BE DISAPPOINTED!!! Don't laugh! Try
thiswhile you
wait for the others to start working. One halfcopying or
waiting and the concept is 100% legal. (refer toUS Lottery
Laws, Title 18, Section 1302 and 1341, or Title18, Section
3005 in the US code, also in the code ofFederal Regulations,
Volume 16, Sections 255 and 436,which state a product or
service must be exchanged formoney received). This version has
nothing to do with PostalLaws, as it's EMAIL
ONLY.Here's How It
Works. Unlike many other PayPal programs,this THREE LEVEL
much more realistic and much, muchfaster results. No cheating
can occur (Don't be fooled byother PayPal letters that
mention
cheaters) as Paypal onlyallows One Account per person. Because
it is so easy, theFAST, and you will start seeing results in
less than oneweek! JUST IN TIME FOR NEXT MONTHS BILLS!
Youaway, and send to personal contacts and reply to people
whosend their programs to you as they are already working
onthe web. They know these programs work and they are
youare responding to people or sending to friends.This program
is MUCH, MUCH FASTER and has a HIGHERRESPONSE RATE than any
other program! Even if you arealready in another program, stay
with it, but do yourself afavor and DO THIS ONE as well. START
RIGHT NOW!It's simple and will only cost $5.00 max. It will
pay off longbefore others even begin to receive letters! Just
give ONEperson a $5.00 gift (REMEMBER THOUGH, this $5.00 is
the$5.00 YOU GET BACK WHEN YOU START USING YOURnew PayPal
Account). That's all!! If you already have a PayPalAccount,
please still donate a $5.00 GIFT to the person inthe No.1 spot
and add your name on the list at No.3 position,having moved up
the remaining two. Follow the simpleinstructions and in two
weeks you will have approximately$20,000 in your PayPal
Account!Because of the VIRTUALLY ZERO INVESTMENT, SPEED,and
HIGH PROFIT POTENTIAL, this program has a VERYHIGH RESPONSE
RATE! All from just one $5.00transaction that you can get back
from PayPal!!!Follow These Simple Instructions:
https://www.paypal.com/mrb/pal=RSV2HQBHKHW7AOnce you link to
the Paypal site using the link above, openFIRST name on the
list (No.1) along with a note in the NOTEAREA of that page,
saying SUBSCRIBE ME TO YOURMAILING LIST. Be certain to add
this note, as this is whatKEEPS THIS LETTER LEGAL.
Instructions on how to senda payment are under SEND MONEY at
the Paypal Site.It's Soooo Easy!!Only the 'rst
person on the
list gets your $5.00 gift.Edit the list, REMOVING the FIRST
(No.1) EMAIL ADDRESS FROM THE LIST after you have made the $5
payment. Then,EMAIL ADDRESS in the (No.3) position. Don't
try
to add yourname in the 'rst place in order to earn money
fast!
If you doname will be immediately removed from the No.1 place
and youwon't reach thousands of people! But, if you add your
name on the(No.3) place, there will be millions of people
receiving and sendingNOTE: Do not forget to replace the PayPal
referring URLin the body of the letter with your own PayPal
referring URL.You can 'nd YOUR referring URL at Paypal after
you join. Justlook at the bottom of the 'rst page once you
sign into Paypal,and click on the referrals link.Send out 20
copies (minimum) of this letter, but only to peopleyou know, or
respond to MLM offers.ALSO NOTE: By sending this letter and the
payment viaE-MAIL, the response time is much
faster..........ELECTRONIC TRANSFER INTERNET FAST!!!Consider
this! Millions of people surf the Internet everyday,all day,
all over the world! Here are the 3 people to startwith. Sign
up, send $5.00 to the 'rst person, move the otherassociated
with your PayPal
Account.*************************************1.
pennywooden@earthlink.net2. tomsweb@adelphia.net
3.JPWMHWathome@aol.com*************************************
There are 'fty thousand new people who get on theThe source
is
UNLIMITED! It boggles my mind to think o§etter and payment
TODAY! It's so easy.One hour of your time,
THAT'S IT!TO DO
THIS:1. Go in your toolbar to edit and select all2. Go in
your toolbar to edit and select copy(make sure it's PLAIN
TEXT so everyone can view it!)5. Go to edit and paste(Then
you can format it anyway you want!)Now you can edit the
addresses with ease. Delete thetop name, adding your name and
address to the bottom ofthe list, then simply changing the
2names move up. But DO NOT forget to send $5.00 viaPayPal
(along with your note) to the position (No.1) -TOP E-MAIL
address before deleting it!NOTE: Be sure to replace the PayPal
referring URLreferring
URL;https://www.paypal.com/mrb/pal=5J4TFLJV48YZ2THERE'S
NOTHING MORE TO DO. When your namereaches the 'rst position
in
a few days, it will be yourturn to collect your MONEY! The
money will be sentto you by 2,000 to 4,000 people like
yourself, who arewilling to invest one half hour to receive
$20,000 in cash!That's all! There will be a total of $20,000
in $5.00 billsin your PayPal Account in two weeks. $20,000 for
onehour's work! This is real money that you can spend
onanything you wish! Just deposit it to your own bankaccount
or spend it directly from your PayPal Account!!!It's just
that
easy!!! I think it's WORTH IT, don't you?GO
AHEAD--- TRY IT!!!
EVEN IF YOU MAKE JUST3 OR 4 THOUSAND, WOULDN'T THAT BE
NICE?IF
YOU TRY IT, IT WILL PAY! CAN YOU DO IT AGAIN?OF COURSE YOU
CAN---This plan is structured for everyone to send only
20letters each to start. However, you are certainly notlimited
to 20. Mail out as many as you want. Every 20letters you send
has a return of $20,000 or more. Ifyou can E-MAIL forty,
sixty, eighty, or whatever, GOFOR IT! THE MORE YOU PUT INTO
IT, THE MOREYOU GET OUT OF IT! Sneaking your name higherup on
the list WILL NOT produce the results youthink, and it only
cheats the other people who haveworked hard and have earned
the right to be there. Soplease, play by the rules and the $$$
will come to you!$$$ E-MAIL YOUR LETTERS OUT TODAY! Togetherwe
will prosper! $$$ You are probably skeptical ofthis,
especially with all the different programs outthere on the
web, but if you don't try this you willnever know.
That's the
way I felt. I'm glad I did it!I've been
watching this type of
program for years andthis is about as easy and fast as you can
get it and itcan even be free to try now with PayPal. No
stamps, noenvelopes, no copies to be made - just a little
effortand faith!!!This program really Keeps It Short and
Simple!OH BY THE WAY....each time someone signs up withyour
link under your name... you also make $5.00 fromPayPal... so
they pay to open up the account but theyalso pay you when
someone signs up from being referredby you. AWESOME!Let's
all
make some serious money $$$$$$Copy and Paste the address below
to your
browser:https://www.paypal.com/mrb/pal=5J4TFLJV48YA2Play by
the rules, this doesn't cost anything but yourtime, and if
everyone plays fair, everyone WINS!!FOLLOWING ARE SOME NOTES
FROM THEPAYPAL SITE WHICH YOU MAY FIND HELPFUL:and is the
world's No.1 online payment service -
it'saccepted on over 3
million eBay T auctions andthousands and thousands of online
shops. You canalso use PayPal to pay your friends - example;
it'sa convenient way to split the phone bill with
yourroommate, send cash to your kids in college,or sendcash to
someone in another country. Better yet, youcan also earn $5
while you do it. Each time someonesigns up for an account and
completes theRefer-a-Friend requirements, we'll give you a$5
BONUS! When you send money through PayPal,you can fund your
payments with your credit cardor checking account. You won't
have to worry aboutyour privacy, because PayPal keeps your
accountinginformation safe. Making a purchase with PayPal
iscard number to a stranger. That's why over 9 MILLIONpeople
from around the world use PayPal to move money.Signing up for
a PayPal Account is easy. It takes only acouple of minutes,
and if you complete the bonusrequirements PayPal will
automatically add $5 to youraccount balance.To learn more
about PayPal visit the web site
athttps://www.paypal.com/mrb/pal=5J4TFLJV48YA2Best of the
Web - ForbesUsing the service is actually safer than a
checkor money order- Wall Street JournalPayPal can play a
major role in your life. You can useit to pay for stuff at
auction sites, settle dinnerdebts with friends or nudge your
cousin to repay that$50 he borrowed at the family reunion.P.S.
Does this sound too good? Well maybe to someskeptics it is. But
it actually works, and is worththe few minutes of your time
now.P.S.S. If You Decide to Join in on this Venture,
PleaseThis Helps everyone keep track of their
progress.P.S.S.S. The letter is not written by PayPal, and
it'screator is not an employee or contracted by PayPalto
send
LEGAL method of earning extra
money.***************************************************in
Home and Business Opportunities, and to friends.Your address
is not added to any other lists, so thisis NOT considered
What about OOP support? I think in a very OOP way,
and often 'nd myself> wishing I had better OOP support than
the Classes package created by Dr.What kind of OOP you 'nd
useful in symbolic application area? That vectoris a subclass
of matrix? Term is a subclass of expression? That is justsilly
(besides being plain wrong)?
What kind of OOP you 'nd
useful in symbolic application area? That vector I 'nd it
very useful to have matrices, series expansions etc.
thatsimply know how to be added, how to take the exponential
and so on.It's simply a matter of using common mathematical
notation, which bynature requires some sort of overloading.>>
M7 := Dom::SquareMatrix(2,Dom::IntegerMod(7))
Dom::SquareMatrix(2, Dom::IntegerMod(7))>> A := M7::random()
+- -+ | 6 mod 7, 4 mod 7 | | | | 3 mod 7, 6 mod 7 | +- -+>> B
:= M7::random() +- -+ | 4 mod 7, 5 mod 7 | | | | 2 mod 7, 3
mod 7 | +- -+>> A+B, A/B +- -+ +- -+ | 3 mod 7, 2 mod 7 | | 5
mod 7, 0 mod 7 | | |, | | | 5 mod 7, 2 mod 7 | | 2 mod 7, 1
mod 7 | +- -+ +- -+> is a subclass of matrix? Term is a
subclass of expression? That is just> silly (besides being
plain wrong)? OOP is not necessarily about subclasses. But
yes, I think it isuseful to have matrices over integers that
can be multiplied withmatrices over rational functions, while
matrices over residue classesof integers refuse to. And I 'nd
it *extremely* useful to have thewhole system extendable by
the user, without modifying the prede'nedfunctions for +, *,
exp etc.-- +--+ +--+| |+-|+ Christopher Creutzig
(ccr@mupad.de)
OOP is not necessarily about subclasses.
But yes, I think it is> useful to have matrices over integers
that can be multiplied with> matrices over rational functions,
while matrices over residue classes> of integers refuse to. And
I 'nd it *extremely* useful to have the> whole system
extendable by the user, without modifying the prede'ned>
functions for +, *, exp etc.Sounds a lot like Axiom.
http://savannah.nongnu.org/projects/axiomTim Daly
> OOP is
not necessarily about subclasses. But yes, I think it is>>
useful to have matrices over integers that can be multiplied
with>> matrices over rational functions, while matrices over
residue classes>> of integers refuse to. And I 'nd it
*extremely* useful to have the>> whole system extendable by
the user, without modifying the prede'ned>> functions for +,
*, exp etc.>Sounds a lot like Axiom.
http://savannah.nongnu.org/projects/axiom... or the Domains
package in Maple.Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
British Columbia Vancouver, BC, Canada V6T 1Z2
<Christopher-Creutzig-wqs3cdcqxjy.fsf@bursar.uni-paderborn.de>
<bnop50$78g$1@nntp.itservices.ubc.ca>
of integers
refuse to. And I 'nd it *extremely* useful to have the>
whole system extendable by the user, without modifying the
prede'ned> functions for +, *, exp etc.>Sounds a lot like
Axiom. http://savannah.nongnu.org/projects/axiom>> ... or the
Domains package in Maple. I never said there was just one
system implementing these. :-)-- +--+ +--+| |+-|+ Christopher
Creutzig (ccr@mupad.de)
Your points of comparison are
probably matters of personal> taste, and are largely
peripheral to symbolic computation.> In fact, Matlab is not a
symbolic computation program at all,> unless you have the
symbolic toolkit, which gives it access> to parts of Maple.>
I think you will have to study the systems and decide what>
appeals to you, to your interests and ability to program, and>
what you expect to do with the computational components.> It
is not that these programs cannot be compared on issues> such
as OOP support 'le access widgets etc. They can> be, but
primarily by expressions of personal taste.> It is as though
you were to ask, I am thinking about buying> a car. Can you
advise me which is best, a Hummer H2, a mini-Cooper,> or a
Ford F150 pickup truck? I am especially interested in> the
cup-holders, the choices of §oor coverings, and the
dif'culty>
of changing the oil.> RJF> PS, the Maxima system,
especially with a TeXmacs front end, is> compared. The common
lisp object system (CLOS) supports OOP.> MuPad also supports
OOP, and there may be other programs that> could appeal to
you, especially if Matlab numerics satis'es> your
computational needs.> I have never even looked at anything but
Mathematica. I understand myquestions seem peripherial to the
core of symbolic computing. They are. Tosome extent, this is
Omarketing research'. There are many things I
likeabout
Mathematica. One is the dynamic typesetting (to coin a
phrase). Ilike to be able to write code that looks like math.
Others tell me Mathematica never locks up or crashes on them.
I suspectthey don't do the same kinds of things I do with/to
computers. I crashMathematica, or lock it up regularly. The
crash recovery in Mathematica,so far as I know, consists of
Obe sure to hit save often'. And undo is
verylimited, and
unpredictable.I know that Mathematica is basically OLisp on
steroids', so the kinds offeatures found in (X)Emacs, are
certainly acheivable in Mathematica. Atwhat level of effort, I
know not. These Operipherial' features
I've mentioned *do*
matter to me. My bloodpressure goes up every time I use a
motif widgit. I also have some visionlimitations that are
especially problematic when using Mathematica's UI. Ifnot
for
the lack of support for what I called dynamic typesetting,
I'dwrite all my Mathematica code in
XEmacs.I've incured career
changing consequences from Mathematica crashing at thewrong
time. As far as OOP support, I'm not really sure if
I'm the
one whois missing the point, or if it's the symbolic
computing
community. I'vebeen able to do some really neat stuff with
OOP
in the area of simulatingphysical systems. I've created a
package that provides transformationgroups similar to thoes
found in Java3D, and am able to build complex 3Dsystems much
more quickly than I can with Oraw' Mathematica.
At the
sametime, the OOP support is far more dif'cult to work with
than C++ or Java.-- Gelonus, Kucha, Kizil, Sampul, Dunhuang,
Kanishka, Cherchan, Margiana,Woman, Guancha, Maes Howe,
Purushkhanda.
These Operipherial' features
I've mentioned
*do* matter to me. My blood> pressure goes up every time I use
a motif widgit. I also have some vision> limitations that are
especially problematic when using Mathematica's UI. If> not
for the lack of support for what I called dynamic typesetting,
I'd> write all my Mathematica code in XEmacs.Why not take
advantage of the good Mathematica to Java integration (and.NET
in 5.0) and write all your oop stuff in an external
language.Depending on your inclinations, you can either invoke
the Java/.NET fromMathematica when required (for UI, etc.) or
write the application inJava/.NET and treat Mathematica as a
kind of coprocessor to do symbolicstuff for you.G.
=I
suggest you look at TeXmacs, since that provideswhat I think
you want in dynamic typesetting.Good luck, though.RJF
I
suggest you look at TeXmacs, since that provides> what I think
you want in dynamic typesetting.> Good luck, though.> RJFI
downloaded it, and poked around a bit. I don't see a
Mathematica plugin. That's actually a bit surprising.
I'm not
ready to write one just yeteither. Haven't looked at MuPad,
etc. I'm not really shopping around fora replacement for
Mathematica. I'm just trying to get a sense of how itsUI
compares with the alternative products. A TeXmacs plugin that
enabled meto either directly interact with a Mathematica
kernel, or to save my workas a Mathematica O.m'
'le would be
of interest.-- Gelonus, Kucha, Kizil, Sampul, Dunhuang,
Kanishka, Cherchan, Margiana,Woman, Guancha, Maes Howe,
Purushkhanda.
PS, the Maxima system, especially with a
TeXmacs front end, is> compared. The common lisp object system
(CLOS) supports OOP.> MuPad also supports OOP, and there may be
other programs that And MuPAD interfaces nicely as a TeXmacs
backend, too.-- +--+ +--+| |+-|+ Christopher Creutzig
(ccr@mupad.de)
=Ladies and Gentlemen,Further to the thread
i^i=0.20787957635076 started by Vladimir Ralev, I wondered
whether there were pure imaginary numbers other than
Oi',
which, raised to their own power, were real.Here enclosed are,
found symbolically and numerically with mathematica, by solving
Im[(y*i)^(y*i)]=0 , a few other imaginary numbers of the form
y*i which have that property.I would be glad to dialog with
people interested in these constants and/or in the solution of
z^z = some real.---jcpy[0]=1;y[k_/;k>0]:=k Pi/ProductLog[k
Pi];y[k_/;k<0]:=-y[-k];(* Let r=(y*i)^(y*i)
*)r[0]=E^(-Pi/2);r[k_/;k>0]:=Cos[k*Pi]/E^((E^ProductLog[k*Pi]*
Pi)/2);r[k_/;k<0]:=r[-k];Table[{k,y[k],y[k]//N,r[k],r[k]//N,(I
y[k])^(I y[k])//N//Chop},{k,-3,3}]{{-3,
(-3*Pi)/ProductLog[3*Pi], -5.517981006496787,
-E^(-(E^ProductLog[3*Pi]*Pi)/2), -0.00017206740019164992,
-0.00017206740019164984}, {-2, (-2*Pi)/ProductLog[2*Pi],
-4.30453032451744, E^(-(E^ProductLog[2*Pi]*Pi)/2),
0.0011574448460624678, 0.0011574448460624643}, {-1,
-(Pi/ProductLog[Pi]), -2.926064057273156,
-E^(-(E^ProductLog[Pi]*Pi)/2), -0.0100895941024518,
-0.010089594102451808}, {0, 1, 1., E^(-Pi/2),
0.20787957635076193, 0.20787957635076193}, {1,
Pi/ProductLog[Pi], 2.926064057273156,
-E^(-(E^ProductLog[Pi]*Pi)/2), -0.0100895941024518,
-0.010089594102451808}, {2, (2*Pi)/ProductLog[2*Pi],
4.30453032451744, E^(-(E^ProductLog[2*Pi]*Pi)/2),
0.0011574448460624678, 0.0011574448460624643}, {3,
(3*Pi)/ProductLog[3*Pi], 5.517981006496787,
-E^(-(E^ProductLog[3*Pi]*Pi)/2), -0.00017206740019164992,
-0.00017206740019164984}}
=I don't know where Mr. Ralev
started the thread, butsolving for y gives two sets of
solutions,-exp (W(-n Pi)) and -exp(W(n Pi)), where W =
Lambert's W function,and n is an integer.This was found
using
macsyma to dosolve(imagpart((%i*y)^(%i*y))=0,y);So far as I
know, this solves the problem and you could spend yourtime
learning about the Lambert W function for further
recreation.RJF> Ladies and Gentlemen,> Further to the thread
i^i=0.20787957635076 started by Vladimir Ralev, > I
wondered whether there were pure imaginary numbers other than
Oi', > which, raised to their own power, were
real.> Here
enclosed are, found symbolically and numerically with
mathematica, > by solving Im[(y*i)^(y*i)]=0 , a few other
imaginary > numbers of the form y*i which have that property.>
I don't know where Mr. Ralev started the thread, but>
solving for y gives two sets of solutions,> -exp (W(-n Pi))
and -exp(W(n Pi)), where W = Lambert's W function,> and n is
an integer.> This was found using macsyma to do>
solve(imagpart((%i*y)^(%i*y))=0,y);>> So far as I know, this
solves the problem and you could spend your> time learning
about the Lambert W function for further recreation.Note that
if n<>0, then exp(W(n*Pi)) = n*Pi/W(n*Pi); and W(n*Pi)
withinteger n<>0 is real only for n>0. Also, it is easy to see
that if(y*I)^(y*I) is real, then (-y*I)^(-y*I) is real as well,
so the solutionsare y=0, y=1, y=-1, y = n*Pi/W(n*Pi), and y =
-n*Pi/W(n*Pi) for integer n>0.Here is how these solutions can
be obtained in Maple.First, notice that if y<>0, then
(y*I)^(y*I) = exp( y*I*ln(y*I) )> evalc(y*I*ln(y*I)); 2 -1/2 y
signum(y) Pi + 1/2 I y ln(y )The exponent of it is real iff the
imaginary part of it is n*Pi*I withinteger n,>
solve(y*ln(y^2)=2*n*Pi,y); n Pi n Pi --------------,
--------------- LambertW(n Pi) LambertW(-n Pi)Because
LambertW(n*Pi) with integer n is real only for n>=0, that
gives us 2series of solutions,y = n*Pi/W(n*Pi), y =
-n*Pi/W(n*Pi) for n>0, and y=0 that should beexcluded, because
we assumed that y<>0. Case n=0 should be solvedseparately,>
solve(y*ln(y^2)=0,y); 1, -1That gives us solutions y=1 and
y=-1. Finally, for y=0, 0^0=1, so y=0 alsogives a
solution.Alec
Mihailovshttp://webpages.shepherd.edu/amihailo/
=Also the
year of the use would be quite oldMessages like the one I
reply to seem to not appear at comp.lang.ada.is a bit
similar....>1.String processing in Ada sucks.The comment could
result if there is not an intent to get a fasterunlimited
length strings package from the Internet e.g. from mywebsite.
I guess that the comment is complaining about a decision
ofmanagement instead about Ada 95.>>2.Advertised language
safety provided by checking them statically>during compiling
failed. In practice, the code is stuffed with>enormous number
of types. Because number of imperfection such as no>support
for non consecutive ranges (like (0, 10-20)) and type
cast,>the whole idea is trashed. Ada is not safer then say
Java or C++.>Ada is much safer than Java and C++.Java is
reviewed and found to be a language that students have
troublewith, and inferior etc., in 4 PDFs here:
ftp://ftp.cs.nyu.edu/pub/gnat/jgnat/papers/>3.Paramertized
types stink.That seems to be about altering variant records on
the stack.Maybe the variant record can be made volatile and
then thediscriminant can be altered. Timing tests would show
if thathad no effect on speed. An aim is to avoid storing data
on the heapsince it can be slower.>>4.Support for IO support
sucks. The user can try a vendor supplied routine or import an
I/O routinefrom another language, e.g. a C routine of
MSVCRT.DLL.>>5.Support for OOP stinks. I could tell you some
realy scary stories>here.Ada was running competitvely and
quickly for timings of tagged records
http://dada.perl.it/shootout/gnat.htmlNothing signi'cant is
said about Ada while you don't say anythingabout the quality
of the source code's designs.>>6.Ada as a *second* language
book needs more then 1200 pages  that>tells you something
about the learning curve.Learning any new language is not the
only problem that America'sC programmers have: there seems
to
be no known clue that they wouldwrite a free complex C to Ada
body 'le translator to end theiryears using C, perhaps just
for private projects.>>7.Language standard was surely written
by lawyers  you need one to>explain even simple errors thrown
by compiler>That is about the G. compiler and not Ada.Ada has
the best syntax error messages from the ObjectAda
(aonix.com)and GNAT compilers. >notable exception of GNAT).I
have not seen Greenhills write to the Ada-Comment list
concernedwith improving Ada 95.>>9.Ada lacks standard
libraries such as STL for C++ or JDK for Java.>Java's
library's are terrible. USA's IBM and Rockwell
seems to
likeJava. I used Java for a TCP proxy and it was abysmal and
apparentlyJava can't actually get data out of the TCP stack.
It could not'gure out whether it should return with 0 bytes,
or instead blockand wait. That was Sun Java 1.2. Currently Ada
programmers browse the Internet for a package to copyand
adapt.>10.Lack of mathematical libraries such as big number
arithmetic or>linear algebra.>C libraries could be
used.>11.aDa iS nOt CASe seNSitIVe.>The GNAT compiler has
various style check options that check thesensitive of the
casing of tokens. >12.No easy way for writing GUIs in Ada
unless you spend serious money.>That topic can go to
comp.lang.ada. I shan't comment on it. Ada mixeswith other
languages and the out of date view was that Ada is not
liked.>So far the most signi'cant occupation of Ada
programmers seems to be>rewriting old DdD's CMS2 code-that
certainly means progress.>The DoD is so underfunded because of
the big porting efforts to getcode out of Ada, and huge search
for liars in Iraq claiming that theysaw weapons of mass
destruction [spotted that at BBC TV news] thatthe lower
ranking soldiers don't get bullet proof vests.>There was a
study conducted for DoD (that time the greatest supporter>of
Ada)>Paul Hudak and Mark P. Jones. Haskell vs. Ada vs. C++ vs.
awk vs. : :>:an experiment in software prototyping
productivity. Technical report,>Yale University, Dept. of CS,
New Haven, CT, July 1994. >Read it, its here
www.haskell.org/papers/NSWC/jfp.ps>The document suggests that
Haskell is best. For the old DoD, Ada wouldbe the 2nd language
if it was the 1st of the prototyping. Ada producessyntax errors
hindering changes and that might make a bad choice fordoing
prototyping in. system
....>.....>>There was a study
conducted for DoD (that time the greatest supporter>>of
Ada)>>Paul Hudak and Mark P. Jones. Haskell vs. Ada vs. C++
vs. awk vs. : :>>:an experiment in software prototyping
productivity. Technical report,>>Yale University, Dept. of CS,
New Haven, CT, July 1994. >>Read it, its here
www.haskell.org/papers/NSWC/jfp.ps> The document
suggests that Haskell is best. For the old DoD, Ada would> be
the 2nd language if it was the 1st of the prototyping. Ada
produces> syntax errors hindering changes and that might make
a bad choice for> doing prototyping in.> Actually the shootout
between Haskell and other languages shows that writing programs
in Lisp is faster by a factorof 3 over the best of the other
languages. I think Lisp wins :)Haskell is best if the
criterion is code length, I think.list does not mean you
should post it here. Check with your
(legacy-1.mathforum.org [144.118.94.27]) by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
(from apache@localhost)
=WOW! MY LITTLE NIECE TOLD ME HOW TO
MAKE CASH MONEY I thought I should share this amazing story
with everyone. Last summermy niece, whos only 11yrs. told me
she had $21,000 in her bedroomcloset and didn't want her
parents to 'nd it. She asked me if I couldhold it in my bank
account for her. I was totally shocked by all this.The 'rst
thing I asked of course is, Are you serious and where theheck
did you get $21,000? She explained to me that she and a
friendfriends house. Her friends parents get home a little bit
afterShe thought if her parents found out about the money she
would get introuble. She didn't think that she would still
be
getting all thisthe bills sense her mom can't work anymore.
Anyway, she went on toexplain to me what she actually did to
get all this money and I wasvery skeptical until she brought
me a shoe-box packed with $5 bills.She told me there are three
more at the house and if I could justdeposit the money in her
parents bank account without them knowing itwould solve all
her problems. She then asked if I could use any extramoney. Of
course like anyone I said sure, who can't these days.
Shesaid
ok then I will add your name to the list a sent it out
tomorrow.Still being very skeptical, I said sure why not.To
make a long story short... Two weeks later envelopes full of
moneystopped sense. I also noticed that her dad hasn't been
going to workeither. I 'guring by now, if you have been
reading this letter yourare very curious yourself so, below
are the details of what you needto do to get started yourself
... FROM THE NEEDING TO THE NEEDINGThese are trying times and
so many of us are unemployed, singleparents, unwell or
otherwise in DIRE NEED OF MONEY, to make ends meet.God helps
he that helps himself; well The Letter provides the toolto
do it! Are you one of us; in dire need of money to make ends
meet?If you need to make a few thousand dollars really
quickly, then pleasetake a moment to read and understand this
opportunity I am sharingwith you. This is the fastest &
easiest Money making program you will ever do.Complete it in
One hour and you will never forget the day you 'rstbut get a
$5 note, and send it to one person to participate in
not four, 've or six, as in manyother programs. It is very
realistic and so fast. Because it is soeasy, the response rate
for the program is now as high as 50 %. Youwill receive your
reward in about fourteen days. Just in time for nextmonth's
bills!!!more as in other programs), perhaps to people who send
you theirprograms. These are already believers (and this
program is better andfaster) or post it in 20 chat forums or
message boards on theinternet. Do yourself a favor and DO THIS
ONE RIGHT NOW!Just give one person $5. That's it!
That's all!
Follow the simpleinstructions and in two weeks you should have
seen at least $7,000,because so many people will respond, due
to the low investment, speed,and huge income potential. We are
presently at a 50% response rate!That's a $10,000 return. So
let's all keep it going and help eachother in these tough
times. HERE ARE THE SIMPLE DETAILS1. Write your name on a
blank sheet of paper, add your own address andthis message:
Please ad me to your list. Next fold it around a've-dollar
bill and send this to the 'rst name on the list. Only the#1
person on the list gets your name and a 've-dollar gift in
the'rst round. But make sure the dollar cannot be seen, even
in thelight, to prevent thievery.2. Retype the list only,
removing the 'rst (#1) name from the list.Move the other two
names up and add your name to the list in the third(#3)
position. 3. Paste your newly typed list neatly over the old
one and make 20copies of this letter and send to 20 prospects
or post 20 on theinternet on message boards or chat forums.
Well there is no more todo. It's so easy! One hour!
That's
it!Do it right away; don't mull it over. Your name will
climb
position ata Dizzying geometric rate. When your name reaches
the 'rst positionin a few days, it will be your turn to
collect your gifts. The giftswill be sent to you by over 1,500
to 2,000 people like yourself whoare willing to invest $5 and
one hour to receive $7,000 in cash. Your entire investment
will be about $5 to $10, including the $5 giftyou sent the
'rst name on the list, envelopes, copies, and
stamps.That's
all! In return there will be a total of $7-10000 in yourCAN I
DO IT AGAIN, WHEN IN NEED? OF COURSE!!or copier. And now you
can do it again over and over with your regulargroup of
gifters. Why not? letter! HONESTY AND INTEGRITY MAKE THIS PLAN
LIST! 1. Tom Dunleavy 300 2nd St. Blakely, Pa. 184472.
Veronica Thompson 68 Robin Rd. Hopkinsville, KY. 422403.
Lonnie Samson 5011 Glen Ridge Dr. Apt M-12 San Antonio, TX
78229P.S. What you are doing is creating a service. THIS IS
ABSOLUTELYLEGAL! You are requesting a legitimate service and
you are paying forit! Like most of us I was a little skeptical
and a little worriedabout the legal aspects of it all. So I
checked it out with the U.S.Post Of'ce (1-800-725-2161) and
they con'rmed that it is indeedlegal! DO UNTO OTHERS AND
ENJOY
YOUR MONEY IN PEACE!!
the following ODE:ivp := ode({x''(t) +
4*x'(t) + 13*x(t) = 40 -
20*t^2, x(0) = 3, x'(0) = 4},x(t));solve(ivp);The solution is
a
well-behaved function.I am trying to use the code below to plot
some of the function with MuPad2.0I get Illegal operand
_multDoes anyone see why this fails? Any help appreciated.f :=
(t, X) -> [X[2], -4*[X2]-13*[X1]+40-20*t^2]: X0 := [3,4]:G :=
(t, X) -> [t, X[1]]:p := plot::ode([i/10 $ i = 0..10], f, X0,
[G, Color = RGB::Red]):p := plot::modify(p, Title =
Position-Red):s := plot::Scene(p, Labels = [t, x], Ticks
= [Steps = [1.0, 1], Steps = [1.0, 1]],plot(s):Please do not
advise me to use original analytic solution to produce
theplot. I realise I can do that, but I have a speci'c reason
to makeplot::ode work.Brad
I get Illegal operand _mult>
Does anyone see why this fails? Any help appreciated.>> f :=
(t, X) -> [X[2], -4*[X2]-13*[X1]+40-20*t^2]: X0 := [3,4]: Try
X[1] and X[2] instead of [X1] and [X2]. Works for me.> p :=
plot::modify(p, Title = Position-Red): 'rst of all, this
should bep::Title = Position-Redand secondly, both versions
produce probably not what you want. I'mnot sure how to place
labels at the speci'c points in 2.0.-- +--+ +--+| |+-|+
Christopher Creutzig (ccr@mupad.de)
=I came across this in
looking at an OpenMath CD on trigfunctions which tells the
reader to look at this standardreference to see what is mean
by sin, cos, ..In my (9th printing) dover publishing
paperback, page 72(section 4.3) after formula 4.3.9 is a
'gure
4.3, circularfunctions.It has a graph of various curves like
sin, cos, sec, .The X-axis is marked in degrees. 0, 90, ....
(not in radians)The formula above the 'gure says4.3.7 sin(z +
2 k pi) = sin z (k any integer).But given the 'gure,
wouldn't
that have to be sin (z +360 k) = sin z ?I have on occasion
berated the OM people for not evensaying whether sin/cos/ etc
was supposed to be degreesor radians, making this kind of
amusing, that they referto a handbook which also has some
problems.RJF
I came across this in looking at an OpenMath
CD on trig> functions which tells the reader to look at this
standard> reference to see what is mean by sin, cos, ..>> In
my (9th printing) dover publishing paperback, page 72>
(section 4.3) after formula 4.3.9 is a 'gure 4.3, circular>
functions.>> It has a graph of various curves like sin, cos,
sec, .> The X-axis is marked in degrees. 0, 90, .... (not in
radians)>> The formula above the 'gure says> 4.3.7 sin(z + 2
k
pi) = sin z (k any integer).>> But given the 'gure,
wouldn't
that have to be>> sin (z +360 k) = sin z ?Not only wouldn't
it
have to be that, it should never, at least IMO, bethat. It's
very interesting that you've asked this question in
thisparticular newsgroup. Why? For two reasons:1. CASs that
I'm familiar with have only _one_ sine function. And I
alsohave only one. It takes a _unitless_ argument. OTOH, some
people I respecthave reported to me that sometimes two
different sine functions are used,one such that sin(pi/2) = 1
(in agreement with the sine function used bymyself and CASs),
the other such that sin(90) = 1. [Typing that lastequation
almost makes me ill!]2. CAS that I'm familiar with also have
a
real constant equal to pi/180.In Mathematica, for example, it
is named Degree. When one wishes to worklike Sin[90*Degree],
which simpli'es to 1 .So, at least IMO, the only correct
alternatives to their 4.3.7 would bethings like sin (z +
k*360*degree) = sin zI agree that this might be more
appropriate, given their graph's labelling.But, given my
choice, I'd leave 4.3.7 as is, and change the labellingsto
0,
pi/2, ... in the graph. In any event, though, there's
nothingtechnically incorrect in their pairing of 4.3.7 and the
graph as labelled.> I have on occasion berated the OM people
for not even> saying whether sin/cos/ etc was supposed to be
degrees> or radians, making this kind of amusing, that they
refer> to a handbook which also has some problems.sin/cos/ etc
are supposed to be in _numbers_, period. At least, that's
myopinion. And most CASs apparently agree with me.David
Cantrell