Re: if-expression and variables

In Axiom, we can implement a characteristic function for a 
given set, in the category of sets, which would avoid 
division by zero.  This can take care of the provisos. If 
I am not mistaken, the step(a,b,x) is the characteristic 
function for the interval set [a,b], but the 
implementation below, as pointed out, may cause division 
by zero. We can use "if-then-else" to implement it 
instead.

William


On Fri, 10 Jun 2011 19:56:18 +0200
  Stefan Karrmann <[email protected]> wrote:
> Dear all,
> 
> we can help ourselves in some cases by:
> 
> sign x == (abs x)/x
> H x == (1 + sign x)/2
> step (a,b,x) == (H(x-a)) * (H(b-x))
> 
> The drawback is that we get a 'Division by zero'.
> 
> Kind regards,
> Stefan
> 
> Am Donnerstag, den 05.05.2011, 00:23 -0400 schrieb 
>William Sit:
>> Dear Stefan:
>> 
>> You posed a legitimate problem: how should symbolic 
>> computation handle piecewise defined functions, and in 
>> particular, how to integrate such a function.
>> 
>> Maple and Mathematica both can handle piecewise 
>>functions. 
>> Look up "piecewise" from Maple Help. You can easily 
>>define 
>> a piecewise function, and differentiate or integrate it. 
>> Indeed, Maple says:
>> 
>> The piecewise function can be differentiated, 
>>integrated, 
>> simplified, plotted, and used in the following types of 
>> differential equations: constant coefficients and 
>> discontinuous perturbation function, general first-order 
>> linear, Riccati, and some other classes which are 
>>handled 
>> by integration or variation of parameter. See 
>> dsolve[piecewise] for more details. series, limit, abs, 
>> and signum can handle the piecewise function.
>> 
>> As example, the desired solution the problem of 
>> integrating f(x) from 0 to t, where f(x) is 2x if x < 10 
>> and 5x^2 otherwise, should be the function g(t), defined 
>> as t^2 if t < 10 and -4000/3 +(5 t^3)/3 otherwise. Maple 
>> does exactly that. In fact, I even tried to integrate 
>> f(f(x)) and f(f(x+1)) and Maple does it with no problems 
>> with all the cases covered.
>> 
>> Mathematica has a similar function called Piecewise to 
>> construct piecewise functions, and like Maple, Piecewise 
>> can be used in such functions as Integrate, Minimize, 
>> Reduce, DSolve and Simplify, as well as their numeric 
>> analogs.
>> 
>> This may be an uncovered domain in Axiom. A search for 
>> "piecewise" shows no hits. I think piecewise functions 
>> have to be separately handled, particularly in case 
>> analysis (possibly involving semi-algebraic sets and 
>>CAD) 
>> if there is any indefiniteness in the answer (like an 
>> indefinite integral). There is some evidence that if the 
>> user does not use the built-in "piecewise" or 
>>"Piecewise" 
>> function, but uses an if-then-else construction, neither 
>> Maple nor Mathematica can handle subsequent mathematical 
>> calculations. For example, the system would not do the 
>> case analysis, much less the "simplification" 
>> automatically, but would present the result as the 
>> integral of If[x < 10, 2 x, 5 x^2] (in Mathematica; I 
>>did 
>> not try Maple). Even when the case analysis is done, it 
>> would still not simplify or evaluate the integrals:
>> 
>> h[x_] := If[x < 10, Integrate[2 y, {y, 0, x}],
>>    Integrate[2 y, {y, 0, 10}] + Integrate[5 y^2, {y, 10, 
>> x}]]
>> 
>> when h[x] is called. It will evaluate on numerical 
>>inputs.
>> 
>> In our earlier discussions, we were "lured" into using 
>> "if-the-else" constructions and thus got the feeling 
>>that 
>> this is difficult to handle. The confusion is that we 
>> interpret "x < 10" as an binary relation, whereas it 
>> should be handled as a semi-algebraic set (in one 
>> dimension, this is just an interval)!
>> 
>> However, the algorithms seem to be there, and someone 
>> should implement them in Axiom if it is not already done 
>> but hidden in some obscure packages.
>> 
>> William
>> 
>> On Wed, 04 May 2011 22:37:48 +0200
>>   Stefan Karrmann <[email protected]> wrote:
>> > Dear all,
>> > 
>> > thanks for your answers. They clears a lot.
>> > 
>> > I actually want to integrate test1 and solve an 
>> >differential equation
>> > with it.
>> > 
>> > E.g.
>> > test2 x == rho * test1 x
>> > y = operator 'y
>> > odeq := D(y x) = test2 x
>> > solve(odeq, y, x)
>> > 
>> > Obviously, the solution is "formally"
>> > 
>> > y_sol x == integrate(test2 x,x)
>> > 
>> > Kind regards,
>> > Stefan
>> > 
>> > Am Dienstag, den 03.05.2011, 11:21 +0200 schrieb Ralf 
>> >Hemmecke:
>> >> Dear Stefan,
>> >> 
>> >> as others already have pointed out, for Axiom, your 
>> >>question is not 
>> >> really well posed.
>> >> 
>> >> In Axiom
>> >> 
>> >>    if x<10 then 2*x else 5*x^2
>> >> 
>> >> is *not* an expression (as you might know it from 
>>other 
>> >>untyped CAS like 
>> >> Mathematica or Maple), but rather a programming 
>>language 
>> >>construct. In 
>> >> other words, if Axiom sees this, it is evaluated. So 
>>the 
>> >>result is 
>> >> either 2*x or 5*x^2 depending on the (boolean) 
>>outcome 
>> >>of the evaluation 
>> >> of x<10.
>> >> 
>> >> I think, Bill suggested to use something like 
>>InputForm. 
>> >>There it would 
>> >> be possible to represent an if-expression 
>>unevaluated.
>> >> 
>> >> But you should rather say what you actually want 
>>(it's 
>> >>not the same what 
>> >> you expect).
>> >> 
>> >> In order for us to suggest you a proper way to handle 
>> >>your use case, you 
>> >> should tell us why you want a piecewise function and 
>> >>(more important) 
>> >> what you later want to do with that function.
>> >> 
>> >> Until we have that information, everything would be 
>>just 
>> >>digging in the 
>> >> dark.
>> >> 
>> >> Ralf
>> >> 
>> >> On 04/30/2011 08:40 PM, Stefan Karrmann wrote:
>> >> > Dear all,
>> >> >
>> >> > I'm new to axiom and have a problem with piecewise 
>> >>functions.
>> >> >
>> >> > test1 (x | x<  10) == 2*x
>> >> > test1 (x | x >= 10) == 5*x^2
>> >> > [was typo: test1 (x | x<  10) == 5*x^2]
>> >> > test1
>> >> > ->
>> >> >     test1 (x | x<  10) == 2x
>> >> >     test1 (x | ^ x<  10) == 5x
>> >> > 
>> >> 
>>                                                   Type: 
>> >>FunctionCalled
>> >> > test1 y
>> >> > ->
>> >> >       2
>> >> >     5y
>> >> >
>> >> > I expected something like (if y<  10 then 2*y else 
>> >>5*y**2).
>> >> >
>> >> > How is it possible to pass a Variable to a 
>>piecewise 
>> >>function respecting
>> >> > the pieces?
>> >> >
>> >> > PS: Using a block and =>  or explicit if-then-else 
>> >>does not help.
>> > 
>> > 
>> > _______________________________________________
>> > Axiom-math mailing list
>> > [email protected]
>> > https://lists.nongnu.org/mailman/listinfo/axiom-math
>> 
>> William Sit, Professor Emeritus
>> Mathematics, City College of New York
>> Office: R6/291D Tel: 212-650-5179
>> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
> 
> 

William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/