Dimiter Prodanov

2017-07-19 13:07:56 UTC

Dear Richard,

I also think it is a good idea to start with a requirements list.

From my side I think that the default domain should always stay real.

Complex numbers are extremely useful but they are only one species of

Clifford algebra.

So instead of complex I would rather say "vector" - that is a number with a

real magnitude and defined direction.

In any case complex numbers in Maxima are of the form a+%i*b with a,b real

so having an explicit complex type begs for bugs.

I see no use of "imaginary" for that reason.

Having a "rational"/"irrational" type would make sense only for rational

simplification since rational numbers can be embedded in real intervals.

"analytic" is nicely undocumented so it's anyone's guess what it does.

My use case is to declare a symbol of certain "feature" and then to look

for it in expressions for simplification purposes.

A useful shortcut will be once a feature is declared to also produce the

predicate function

declare ("vector", feature)

will then also produce

vector(p)

and so on...

best regards,

Dimiter

----------

I also think it is a good idea to start with a requirements list.

From my side I think that the default domain should always stay real.

Complex numbers are extremely useful but they are only one species of

Clifford algebra.

So instead of complex I would rather say "vector" - that is a number with a

real magnitude and defined direction.

In any case complex numbers in Maxima are of the form a+%i*b with a,b real

so having an explicit complex type begs for bugs.

I see no use of "imaginary" for that reason.

Having a "rational"/"irrational" type would make sense only for rational

simplification since rational numbers can be embedded in real intervals.

"analytic" is nicely undocumented so it's anyone's guess what it does.

My use case is to declare a symbol of certain "feature" and then to look

for it in expressions for simplification purposes.

A useful shortcut will be once a feature is declared to also produce the

predicate function

declare ("vector", feature)

will then also produce

vector(p)

and so on...

best regards,

Dimiter

----------

Message: 6

Date: Tue, 18 Jul 2017 14:25:42 -0700

assume()... was Re: simplifcation of (-a)^x for a > 0

Content-Type: text/plain; charset="utf-8"; Format="flowed"

It is perhaps worth presenting a list of what we would

like assume and "is" to do, in one place, and see how

much of it (a) works, (b) doesn't work,

(c) can be made to work,

(d) probably is too hard.

There seem to be several not necessarily conflicting,

lines of reasoning.

One is *types or categories*, using "declare" to establish

features such as integer, real, complex rational, even, odd.

(see ? features).

We can reason that an integer is necessarily real.

We have integerp(3) returns true.

declare(q,integer); integerp(q) returns false.

We can use lisp ?floatp(3.0) which returns true.

Another is "*mathematical facts*" and implications... such as

is (sin(x)^2>=0) which returns true

although sin(%i)^2 is -1.38, so the answer is incorrect.

Does assume() believe that x is real?

Setting domain:complex does not change this.

Arguably, we could have a simplification like

is (sin(x)^2>=0) simplifies to is( x an element of reals)?

Another is *deductions from **assumption data base of user f**acts*

assume(x>4);

is (x^2>16) returns true.

This kind of reasoning could be made algorithmic, for certain

kinds of assumptions. I have not looked at this recently, but

I think it is not hard to allow assumptions of linear combinations

of indeterminates like assume(3*x+4*y > 10). Such equations

determine planes or hyperplanes whose intersections can

be identified. allowing arbitrary polynomial equations is

computable at some expense via "cylindrical algebraic decomposition"

I think.

Arguably there is another source of information, the

*predicates associated with patterns*, e.g. matchdeclare(a,freeof(x)).

I invite interested parties to add to this email and send it out

again. Also, anyone interested in understanding, describing,

extending what Maxima does, should consider this as

an interesting opportunity. There is a literature on Mathematical

Knowledge Manipulation that should be, but mostly is not,

connected with computer algebra systems like Maxima.

RJF

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Date: Tue, 18 Jul 2017 14:25:42 -0700

assume()... was Re: simplifcation of (-a)^x for a > 0

Content-Type: text/plain; charset="utf-8"; Format="flowed"

It is perhaps worth presenting a list of what we would

like assume and "is" to do, in one place, and see how

much of it (a) works, (b) doesn't work,

(c) can be made to work,

(d) probably is too hard.

There seem to be several not necessarily conflicting,

lines of reasoning.

One is *types or categories*, using "declare" to establish

features such as integer, real, complex rational, even, odd.

(see ? features).

We can reason that an integer is necessarily real.

We have integerp(3) returns true.

declare(q,integer); integerp(q) returns false.

We can use lisp ?floatp(3.0) which returns true.

Another is "*mathematical facts*" and implications... such as

is (sin(x)^2>=0) which returns true

although sin(%i)^2 is -1.38, so the answer is incorrect.

Does assume() believe that x is real?

Setting domain:complex does not change this.

Arguably, we could have a simplification like

is (sin(x)^2>=0) simplifies to is( x an element of reals)?

Another is *deductions from **assumption data base of user f**acts*

assume(x>4);

is (x^2>16) returns true.

This kind of reasoning could be made algorithmic, for certain

kinds of assumptions. I have not looked at this recently, but

I think it is not hard to allow assumptions of linear combinations

of indeterminates like assume(3*x+4*y > 10). Such equations

determine planes or hyperplanes whose intersections can

be identified. allowing arbitrary polynomial equations is

computable at some expense via "cylindrical algebraic decomposition"

I think.

Arguably there is another source of information, the

*predicates associated with patterns*, e.g. matchdeclare(a,freeof(x)).

I invite interested parties to add to this email and send it out

again. Also, anyone interested in understanding, describing,

extending what Maxima does, should consider this as

an interesting opportunity. There is a literature on Mathematical

Knowledge Manipulation that should be, but mostly is not,

connected with computer algebra systems like Maxima.

RJF

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