mm-479 === Subject: Re: replace mathematics completely and totally by mma programming. can it be done? mm-479 === SH> The name mathematica is stupid. Personally, though you and me do not much like to type 'Mathematica' and additionally am toooo lazy to write a macro :) I find this name to be pleasing and deep, it caresses my ears. Why? I do not know, it is like poetry, someone likes Shakespeare, someone prefers Conan Doyle. But there is a hidden point I enjoy a lot and would like to share with you as I see you passion for symbolic computations and your bright venturous mind. Please consider the ultimate goal of SW, (actually, only one of his ultimate goals, www.stephenwolfram.com ): Mathematica: A system for doing mathematics (!) by computer This IS a formidable challenge and a noble intellectual task nobody before SW had set before himself (at least to SUCH an extent) and a very important and I would say lucrative industrial task and in case of success a genuine jewel for all us the math fans. I believe, the very name Mathematica is a tool, a reminder for SW himself to keep really focused on 1 of his ultimate lifetime goals. In my opinion, SW is one of the world's best top managers, in the highest sense of this word. He is also an excellent practical psychologist, too. On a personal note: just for the same reason, I acquired from WRI nice coffee mugs I like a lot. Every time I sip coffee (which I do quite often, so to say, industrial engineering :) I think again and again about Mathematica and how to improve it in the fastest and the most inexpensive way. You see, the proud word 'Mathematica' is a kind of Royal Standard. To christen his flagship environment Mathematica is like to raise the standard of revolt against boredom of routine daily calculations and far much more. In Stephen Wolfram, it strikes me, if I think at times his move is not the best, more than one time I have seen that in the long run this ostensibly weak move was just a step to a stronger move. But I did not see this second move, so I could infer erroneously that he had been wrong. About SW, remember, he seems to have an almost fantastic habit and ability to optimize globally, and strategic planning seems to be an integral part of his thinking process. Stephen Wolfram is a highly intuitive person, too. Like any strong mathematician is. SH> It is so close to mathematics Why, sure! Why, of course that was it! SH> and it is long. Complain to Pythagor :) BR> This is nothing more than the arrogance others accuse BR> Wolfram of in a different form. Keith Geddes is a very polite person. But what is the good of being polite if Maple is getting more and more buggy? ;) Also, about this much discussed arrogance. There were many many hundreds variegated comments on A New Kind Of Science, but it seems, an important point was not still voiced. As I can see it, A New Kind Of Science is, apart from many other considerations, a kind of an outstanding work of fiction; a saga, or, a poem. By the way, this is why SW did not add references, that's why he use I etc. Do you think you would like to read a kind of the following Even more important that by publishing A New Kind Of Science Stephen Wolfram has established a new empire bearing hopefully a terrific potential yet not understood properly. Stephen Wolfram wages a war, a war for intelligence, a war for himself and for us, do you really realize it? Publishing A New Kind Of Science as a single book seems to be the best move possible as being scattered over years and over many papers as SW points out himself would degrade the brilliancy of the dyes, would blunt the effect, would reduce tension, and would unloose the grip one needs to set forward. The empire is huge and now a crucial issue is to control it, and this task is almost intractable just in view of the very fact that the terrain under control is unseen in scope, the most penetrant eye cannot take in the whole scene of operations and its beyond human powers however the warlord is puissant and wise, to see all the details and not all is perfect currently over this empire and its Mathematica kingdom; on which account I will add some comments soon. http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ ................................................................... === Subject: Mathematica Polynomial power sorting by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3JE0sJ15278; Is there a way to program Mathematica to sort terms in a mixed polynomial based on the total power of each term? For example, the input 1 + x + x^3 + y + xy should be sorted as 1 + x + y + xy + x^3 Any help would be greatly appreciated. === Subject: Re: Computer Algebra System The first URL is a bad link. The 2nd does not provide a Please try following links from www.mupad.de/doc30.html to section 4.4.2 of the MuPAD tutorial. Sorry, this is the first time for me those links did not work for someone else. > Universit.8at Paderborn's Mupad website incorrectly says: The University of Paderborn just happens to be my employer. Opinions expressed on that web page need not be mine or that of anyone else in the MuPAD group and certainly have no direct influence on MuPAD development. Neither am I able to change what is written there. Subject: Re: Computer Algebra System Expression trees are certainly the most important structure. >Whether you represent them as nested Lisp lists or as n-ary trees with >pointers in C is of secondary importance. > It is controversial to say that the language is not important. I never said that. The language you are using obviously shapes your thinking. Additionally, each language has strengths and weaknesses of its own. But I stand to the above claim that language choice is of secondary importance when your goal is not to build a complete system but rather to learn something about how to do it. When the goal is to build a complete system, well, I'd have to spend more time thinking about that. None of the claims pro and contra certain languages has been convincing to me yet. > A hypothesis is that Mupad would have been free (rather than cheap) if > C (C++) or whatever, was not used. I doubt it. This decision certainly had nothing to do with the implementation language. And since whatever includes *all* programming languages, MuPAD simply wouldn't have existed. :-) > None of the obvious questions are answered by the summarizing page: > * Do nodes contain a number representing a power (useful for > polynomials but extends out to plus and times containing an > exponent?. I can only speak for MuPAD, where a power is represented like this: > prog::exprtree(a^2) _power | +-- a | `-- 2 Polynomials use a denser representation, where a reference to some arbitrary object (the coefficient) is followed by the exponent vector of the current term, stored in C integers. But general expressions are not stored as polynomials. > * Is the exponent of a number always equal to 1 (if there is an > exponent) ?. Probably since matching the exponent of + and y. What do you mean by the exponent of a number? Integers are not stored as floats. > * Does the top/root node of the expression says if the expression > is properly simplified?. No, since this question depends on values/assumptions outside the expression. > * How is information about the identity of the CPU stored?. Why should it be stored at all? > * How are the variables names (subject to lookups) stored?. There > could be subrecords, and a data structure implenting a free > pool. Where would the links be stored?. The requests expands > to ask to about Mupad's memory management data structures. I don't understand the question. A variable is a place in memory which among other things contains a pointer to the name of the variable. Conversely, it should be obvious that there is some sort of hash table pointing from variable names to the variables themselves. The memory management need not be CA-specific, any structure supporting garbage collection will do. (Sure, optimization is largely application specific and contains many trade-offs.) > * Are there any pointers in expressions except to names and > subtrees?. Certainly in an OO system such as MuPAD each object has a pointer to its domain type. (Kernel types do not have a pointer but rather a small number, but that's an implementation detail.) Oh, and you did not include numbers, functions etc. in the list, which are of course pointed to as well. > * Are there different types (i.e. derived record types) of the node > type. Yes, since MuPAD objects have a type. > * What is the interpretation of out-of-bounds lengths of the array?. I'm sorry? Which array are you talking about? > * How are strings stored?. The string could be saved as a variable > name?. Vice versa, the variable name is internally pointing to a string. > I request that you make availanle the source code that data structure You *request*? I certainly hope you did not really mean to use this word. If you did, please explain on which basis you think to have the right to request such a thing. > of expressions against the policy rules. A sample rule is this: > no symbol/name/expression has a mass/weight of zero, except the > scalar zero. It is hard to guess on what Mupad policy is, on > vectors of zeros. It's not really hard to guess, no. Just invoke ?iszero and read the documentation. > The webpage of the 2nd provided Mupad URL does not say that nodes > of expressions actually even contain a field saying what the mass > is. Is 3x implemented with 3 records or one, in Mupad ?. I don't understand what you mean by mass. 3*x is stored as an expression having the operands _mult (the operator symbol, available as op(3*x, 0), x, and 3. > The data has been online in the past ?. Again, I do not understand what you mean. Please clarify. Subject: Elementary Mathematics Welcome to the All Elementary Mathematics - for people who need help, want to help, or want to find the truth in our online discussions. We can help with all areas of Math. Post your questions here and someone will respond. www.bymath.com forum www.bymath.com/cgi-bin/ultimatebb.cgi === Subject: Re: Please repeat link to MINSE Try contacting the author, a grad student at Berkeley now, Ka-Ping Yee http://zesty.ca/ > Could we please have an active link to MINSE as I have a lot of > downloaded files containing text that should be rendered with MINSE > and Will soon be in trouble if I can`t get it right. It is a neat program, but xml/mathml has grown to cover some of the territory. === Subject: Re: freedom and silence. It is alleged that Mark Twain said, It is better to keep your mouth closed and let people think you are a fool than to open it and remove all doubt. There are also these (King James Bible..) Proverbs 17:28 Even a fool, when he holdeth his peace, is counted wise: and he that shutteth his lips is esteemed a man of understanding. Proverbs 23:9 Speak not in the ears of a fool: for he will despise the wisdom of thy words. Proverbs 26:4 Answer not a fool according to his folly, lest thou also be like unto him. .... while reflecting on those last two... Steve_H: if you want to argue with me, my address is on my email. I objected not to a single item, but the direction of your posts and to the fact that your email return address is bogus. === Subject: Re: freedom and silence. > It is better to keep your mouth closed and let people think you are a > fool than to open it and remove all doubt. If you don't know, ask. You will be a fool for the moment, but a wise man for the rest of your life. ----Seneca === Subject: Re: freedom and silence. It is better to keep your mouth closed and let people think you are a > fool than to open it and remove all doubt. > If you don't know, ask. You will be a fool for the moment, but a wise man > for the rest of your life. ----Seneca The difference being, for the most part, whether you are asking questions or not. === Subject: Re: freedom and silence. > It is alleged that Mark Twain said, > It is better to keep your mouth closed and let people think you are a > fool than to open it and remove all doubt. > There are also these (King James Bible..) > Proverbs 17:28 Even a fool, when he holdeth his peace, is counted wise: > and he that shutteth his lips is esteemed a man of understanding. > Proverbs 23:9 Speak not in the ears of a fool: for he will despise the > wisdom of thy words. > Proverbs 26:4 Answer not a fool according to his folly, lest thou also > be like unto him. > .... That is wonderful that you will share these quotes with us. But what is the point of knowing such wise quotes if one does not try to apply to themselves as well? >Steve_H: if you want to argue with me, my address is on my email. I do not care to talk to you. You are the one who started attacking me for no valid reason. I think I have the right to defend myself against your attacks. You must have thought I am one of your student and will lay down and not respond for fear of getting a bad grade, but this is not one of your class rooms and I will defend myself when I see someone attack me for no valid reason. > I objected not to a single item, but the direction of your posts What is the 'direction' of my posts is supposed to mean? and who are you to assign yourself the role of a critic of the newsgroup? Again, if you do not find something I post interesting to you, then you are more than free not to read anything I write, it is a free WORLD, you are no obligation to respond, and I wish you will do so. You are supposed to be a professor?? then please try to act like one. Subject: Re: Laplace transformation software? Free macsyma (maxima) does Laplace transforms. > I am treating the complex problem in chemical engineering. I want > to get help from you to recommend suitable laplace transfromation > software. Your help or guidance will be greatly appreaciated. === Subject: Re: branch cuts, how CAS systems handle it? Kahan test shown for maple and mma I can't find wkahan's notes on this online. There were papers >Macsyma 2.4 handles these issues as follows: >assume (-1radcan(s(r(a))); --> 1/a >assume(b>1); >radcan(s(r(b))); --> b >assume(c<-1); >radcan(s(r(c))); --> c. > Surely there isn't much of a problem if we deal only with real numbers. > BTW, Derive also simplifies s(r(x)) to x if we specify that x>1 or that > x<-1 and to 1/x if we specify that -1 doing this? > I must suppose that the challenge which Kahan had in mind concerned > simplifying s(r(z)) when z is _complex_. >And if you have no assumptions, you get >sqrt(((z + (1/z))/2) - 1) * sqrt(((z + (1/z))/2) + 1) + ((z + (1/z))/2) >I do not know if this is what you would like. > But you should know what _Kahan_ wanted. Unfortunately, I haven't seen > that 1991 paper of his, but presumably you have, since it was a reference > in one of your papers. [For anyone interested, > see 7.5 A Challenge Problem on the last page of > .] > So the challenge was to simplify s(r(z)) for z complex. My questions are: > What would be an answer acceptable to Kahan (or to you)? 1/s inside unit circle, s outside unit circle, ON the unit circle in the top half, s ON the unit circle in the lower half, 1/s and also 1, -1 when s=1,-1 resp. > Does any current CAS give such an answer? None that I know of. > If asked to simplify s(r(z)) with z being complex (but not otherwise > restricted), an output which I would consider fully acceptable is (in > Mathematica-style notation): > If[ Abs[z]>1 || (Abs[z]==1 && Arg[z]>0), z, 1/z] Looks sort of ok to me except if Arg[z]=0. Maybe Arg[z]>=0 would be better. The question is, what more can the CAS do with the answer? Like differentiation, simplification, plotting, ... > Is wanting an answer like that wanting too much? No. >2. Specific to this problem, how to get (1) to reduce to (2) in either >maple or mma? with maple, using assume(z>0) got closer to (2) but not >quite, may be if I knew more about how to use assumptions in maple >I can get it to reduce (1) to (2). With mma, with its weak >typing/assumptions facility, I gave up trying. > r[z_] = (1/2) (z + 1/z); > s[w_] = w + Sqrt[w - 1] Sqrt[w + 1]; > Simplify[s[r[z]], -1 < z < 1] > 1/z > Simplify[s[r[z]], z > 1 || z < -1] > z === Subject: Re: branch cuts, how CAS systems handle it? Kahan test shown for maple and mma > I can't find wkahan's notes on this online. There were papers > [For anyone interested, > see 7.5 A Challenge Problem on the last page of > .] > So the challenge was to simplify s(r(z)) for z complex. My questions > are: > What would be an answer acceptable to Kahan (or to you)? > 1/s inside unit circle, > s outside unit circle, > ON the unit circle in the top half, s > ON the unit circle in the lower half, 1/s > and also 1, -1 when s=1,-1 resp. > Does any current CAS give such an answer? > None that I know of. > If asked to simplify s(r(z)) with z being complex (but not otherwise > restricted), an output which I would consider fully acceptable is (in > Mathematica-style notation): > If[ Abs[z]>1 || (Abs[z]==1 && Arg[z]>0), z, 1/z] > Looks sort of ok to me except if Arg[z]=0. Maybe Arg[z]>=0 would be > better. I don't know how that could be better, at least from a purely mathematical standpoint. If (Abs[z]==1 && Arg[z]==0), then z must be 1, and so it should surely be immaterial whether we say z or 1/z for the result. BTW, for (Abs[z]==1 && Arg[z]==Pi), mutatis mutandis. > The question is, what more can the CAS do with the answer? Like > differentiation, simplification, plotting, ... Indeed, that is an important question. David Cantrell > Is wanting an answer like that wanting too much? > No. >2. Specific to this problem, how to get (1) to reduce to (2) in >either maple or mma? with maple, using assume(z>0) got closer to (2) >but not quite, may be if I knew more about how to use assumptions in >maple I can get it to reduce (1) to (2). With mma, with its weak >typing/assumptions facility, I gave up trying. > > > r[z_] = (1/2) (z + 1/z); > > s[w_] = w + Sqrt[w - 1] Sqrt[w + 1]; > > Simplify[s[r[z]], -1 < z < 1] > > 1/z > > Simplify[s[r[z]], z > 1 || z < -1] > > z === Subject: Replacement to MMA and Maple I am a long time user of Maple. For a while, Maple has turned into an annoying buggy stuff. On my Linux PC, Maple 7,8 and 9 crash every few minutes. I learned how to live with that (I'm paranoiac and save my worksheet every 1mn). What is not acceptable to me, is that Maple can give wrong answers. Worse, Maple 9.5 annonces more JAVA and other fancies, that will mean more bugs. Enough is enough, I want to give up Maple. Then, I tried Mathematica 5.0. At my very first try, I found bugs. I cannot use it for my research. Maple and MMA are to CAS what M$ Word is to word processing. Do you know a CAS that is more into the LATEX philosophy? How robust, advanced, etc., is it? Such a CAS has to be free, because it keeps its creators free of the market pressure: no need for JAVA s, fancy features, etc., that turn softwares into bags of bugs. (I saw that MuPad 3.0 now integrate JAVA. That's very scary to me, and I don't want to give it a try.) More specifically, for my research, I need to compute exotic special functions (that have to be efficiently implemented into the software) in extended precision (floating point numbers from 16 to 256 decimal places, but not millions). FORTRAN or C with extended precision packages are not suitable because the scientific libraries are limited to double precision. In your opinion, what is the best option? (Accuracy is my main concern, speed matters too, but is secondary.) Any feadback is wellcome. === Subject: Re: Replacement to MMA and Maple > (I saw that MuPad 3.0 now > integrate JAVA. That's very > scary to me, and I don't > want to give it a try.) MuPAD does *not* /use/ Java for anything. There is a module (kernel extension) which *allows* you to invoke methods on Java objects, if you want to. If you don't, the module will not even be loaded and MuPAD won't know anything about Java. > More specifically, for my > research, I need to compute > exotic special functions > (that have to be efficiently > implemented into the software) I'm sorry, but the more exotic special functions have probably never been asked for in MuPAD, at least I haven't seen requests, so you won't find e.g. StruveH. (Although MeijerG and hypergeom are there, so you might be able to get some of your functions.) Subject: Re: Replacement to MMA and Maple >(I saw that MuPad 3.0 now >integrate JAVA. That's very >scary to me, and I don't >want to give it a try.) > MuPAD does *not* /use/ Java for anything. There is a module (kernel > extension) which *allows* you to invoke methods on Java objects, if > you want to. If you don't, the module will not even be loaded and > MuPAD won't know anything about Java. OK. Actually, what scared me is the emphasize on: superb new graphics, hundreds of graphical attributes allowing to manipulate all details of a plot interactively, export to MS Word, etc. If I want nice graphics, for example, I will use a software which is specialized in graphics. I do not understand this need for multipurpose softwares (this does not apply only for CAS), and I would like to ask the following questions: Who wants to buy a vacuum cleaner + coffee machine + fridge, all in one? Who will bring his car to a garage where the mechanician use a Swiss-army knife as only tool? Why is it different with sofwares? >More specifically, for my >research, I need to compute >exotic special functions >(that have to be efficiently >implemented into the software) > I'm sorry, but the more exotic special functions have probably never > been asked for in MuPAD, at least I haven't seen requests, so you > won't find e.g. StruveH. (Although MeijerG and hypergeom are there, > so you might be able to get some of your functions.) I agree, the market is not big enough for such a request. That's why, in my opinion, a high-quality CAS has to be free in order to avoid, or at least limit, the market size effects. === Subject: Re: Replacement to MMA and Maple > OK. Actually, what scared me is the emphasize on: > superb new graphics, hundreds of graphical attributes allowing to > manipulate all details of a plot interactively, export to MS Word, > etc. If I want nice graphics, for example, I will use a software which > is specialized in graphics. A software which does graphics only can be useful, of course, but it makes sense for a CAS to deal with graphics, too. They are related; the graphics help the users to visualize the functions they are dealing with, and the computational power of the CAS can help with the graphics. > I do not understand this need for multipurpose softwares (this does > not apply only for CAS), To me, it makes sense to have related functionality tied together. > and I would like to ask the following > questions: > Who wants to buy a vacuum cleaner + coffee machine + fridge, all in one? Not me; I don't see the connection between the vacuum, coffee and fridge. I wouldn't mind having an ice-maker with my fridge, though. > Who will bring his car to a garage where the mechanician use a > Swiss-army knife as only tool? Only? Not me. But I can see where an adjustable wrench could come in handy. > Why is it different with sofwares? I don't see how it is different. In software and elsewhere, it makes sense to tie related functionalities together. The crucial thing, of course, is that the software program can handle email. (I don't recall; whose law am I thinking of?) === Subject: Re: Replacement to MMA and Maple <87d662jzgu.fsf@truman.edu > The crucial thing, of course, is that the software program can > handle email. (I don't recall; whose law am I thinking of?) Zawinski's Law: Every program attempts to expand until it can read mail. Those programs which cannot so expand are replaced by ones which can. Jesper Harder I am a long time user of Maple. > For a while, Maple has turned > into an annoying buggy stuff. > On my Linux PC, Maple 7,8 and 9 > crash every few minutes. I > learned how to live with that > (I'm paranoiac and save my > worksheet every 1mn). > What is not acceptable to me, > is that Maple can give wrong > answers. > Worse, Maple 9.5 annonces > more JAVA and other fancies, > that will mean more bugs. > Enough is enough, I want to > give up Maple. > Then, I tried Mathematica 5.0. > At my very first try, I found > bugs. I cannot use it for my > research. > Maple and MMA are to CAS what > M$ Word is to word processing. > Do you know a CAS that is more > into the LATEX philosophy? > How robust, advanced, etc., > is it? > Such a CAS has to be free, > because it keeps its creators > free of the market pressure: > no need for JAVA s, fancy > features, etc., that turn > softwares into bags of bugs. > (I saw that MuPad 3.0 now > integrate JAVA. That's very > scary to me, and I don't > want to give it a try.) > More specifically, for my > research, I need to compute > exotic special functions > (that have to be efficiently > implemented into the software) > in extended precision (floating > point numbers from 16 to 256 > decimal places, but not millions). > FORTRAN or C with extended precision > packages are not suitable because > the scientific libraries are limited > to double precision. > In your opinion, what is the best > option? (Accuracy is my main > concern, speed matters too, but is > secondary.) === http://ourworld.compuserve.com/homepages/vjp2/vasos.htm === Subject: Re: Replacement to MMA and Maple Try Axiom: http://savannah.nongu.org/projects/axiom or http://axiom.axiom-developers.org Tim Daly === Subject: Re: A proof for Goldbach's conjecture by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3KD0Au11506; I' ve got the proof of the Goldbach's conjecture. Does anybody know if there is a money prize for that proof??? === Subject: Re: A proof for Goldbach's conjecture > I' ve got the proof of the Goldbach's conjecture. > Does anybody know if there is a money prize for that proof??? Isn't fame, glory and immortality good enough for you? === Subject: Re: A proof for Goldbach's conjecture > I' ve got the proof of the Goldbach's conjecture. > Does anybody know if there is a money prize for that proof??? > Isn't fame, glory and immortality good enough for you? If you're going to live forever, substantial monetary reserves are highly advisable! === Subject: Re: A proof for Goldbach's conjecture Faber and Faber publishers' reward has expired. In any case it was for a PUBLISHED proof, not a claim of a proof. See http://en.wikipedia.org/wiki/Goldbach's_conjecture for some info. > I' ve got the proof of the Goldbach's conjecture. > Does anybody know if there is a money prize for that proof??? === Subject: Re: A proof for Goldbach's conjecture > Faber and Faber publishers' reward has expired. > In any case it was for a PUBLISHED proof, not a claim of > a proof. > See > http://en.wikipedia.org/wiki/Goldbach's_conjecture > for some info. > RJF > I' ve got the proof of the Goldbach's conjecture. > Does anybody know if there is a money prize for that proof??? I recommend the book Uncle Petros and Goldbach's Conjecture by ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ === Subject: infinite series overflow problem I have a problem on working with infinite power series in Matlab i am using this series fn=summation(an*x^n) where x=50 and n=1,2,........... when x>182, the term x^n is going to infinity....This may be due to the value is out of matlab limit (its a very big number) but I want to use more number of terms in my apllication (may be around 500 terms) and coefficients (an) are of alternative sign, a(2n+1)=0 is it possible to overcome this problem by normalizing x?? If yes, How can I do this in matlab? Please suggest me how to deal with this problem? is there any method to scaling the x^n term in the series? === Subject: Re: Laplace transformation software? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3KM2NN27449; What is the website to download the free maxima? In additon, it is compatible with the WinNT or Win2K platform? === Subject: Re: Laplace transformation software? Hi there. > What is the website to download the free maxima? http://maxima.sourceforge.net/ > In additon, it is > compatible with the WinNT or Win2K platform? Yes: http://maxima.sourceforge.net/download.shtml Mike Thomas. === Subject: Re: Laplace transformation software? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3KM2NT27457; In our university, I can access the Matlab and Maple. So, with a view to solving my problem (Laplace transformation), which software should be the best choice among Macsyma, Matlab,and Maple? I hope someone could give me a suggestion. === Subject: Re: Laplace transformation software? > In our university, I can access the Matlab and Maple. So, with a > view to solving my problem (Laplace transformation), which software > should be the best choice among Macsyma, Matlab,and Maple? Maple has a command for Laplace transformation and its inverse. See ?inttrans,laplace in the help to get more information. Thomas Richard Maple Support Scientific Computers GmbH http://www.scientific.de === Subject: McCabe Thiel, Traingle Raffinate, & Fixed Point I recently worked with Fixed Point Theorems (eg FixedPoint in Mathematica) and I realised a lot of graphical methods used in chemical engineering, especially in unit operations actually depend on fixed-point theorems for solution. (For Engineers: Fixed-Point theorems basically describe how you can find roots of contracting functions and how to turn all functions into contracting functions. Look at how a Fixed-Point theorem converges and it looks just like one of those graphical-method diagrams.) Is there any book on seprations that uses fixed-point methods instead of graphical methods? Has there been any attempt to incorporate this into the curriculum? The reason I am thinking about this is there has been a lot of thought in removing a lot of deadweight from Ch E curricula, and of making it less information overload and more of a conceptually unified field. - = - http://ourworld.compuserve.com/homepages/vjp2/vasos.htm ---{Nothing herein constitutes advice. Everything fully disclaimed.}---