Carl Love

## 19789 Reputation

8 years, 56 days
Mt Laurel, New Jersey, United States
My name was formerly Carl Devore.

## Let's correct the original formula first...

Let's correct the original formula to a proper PDF before getting carried away with enthusiasm.

restart;
with(Statistics):
Density := t-> exp(-t^2)/sqrt(Pi):
Check that it's a PDF:
int(Density(t), t= -infinity..infinity);
1
Horizontal shift won't change the integral. But we shift the PDF, not the CDF.
Z:= Distribution(PDF= unapply(Density(t-1), t)):
Now the other aspects can be computed from this automatically by the Statistics package.
CDF(Z,t);
1   1
- + - erf(t - 1)
2   2
Mean(Z);
1

## Let's correct the original formula first...

Let's correct the original formula to a proper PDF before getting carried away with enthusiasm.

restart;
with(Statistics):
Density := t-> exp(-t^2)/sqrt(Pi):
Check that it's a PDF:
int(Density(t), t= -infinity..infinity);
1
Horizontal shift won't change the integral. But we shift the PDF, not the CDF.
Z:= Distribution(PDF= unapply(Density(t-1), t)):
Now the other aspects can be computed from this automatically by the Statistics package.
CDF(Z,t);
1   1
- + - erf(t - 1)
2   2
Mean(Z);
1

## @williamov Use unapply, not Unapply...

Use unapply, not Unapply. It is meaningless with a capital U. Change the line

RealDist := Distribution(Dist(t-1));

to

RealDist := Distribution(CDF= unapply(Dist(t-1), t));

Of course, Markiyan's objection about the original not being a PDF applies here also. I am just trying to offer basic corrections to syntax, not claiming that the above is mathematically valid.

## @williamov Use unapply, not Unapply...

Use unapply, not Unapply. It is meaningless with a capital U. Change the line

RealDist := Distribution(Dist(t-1));

to

RealDist := Distribution(CDF= unapply(Dist(t-1), t));

Of course, Markiyan's objection about the original not being a PDF applies here also. I am just trying to offer basic corrections to syntax, not claiming that the above is mathematically valid.

## Editor dropped characters?...

Preben,

I am having some trouble reading your answer. I wonder if the editor dropped some angle brackets.

I see

SecTer:= CrossProduct(,)

____________________^

and

ProjX := diff(x(t), t, t) = -q*DotProduct(SecTer,,conjugate=false)/m;

_______________________________________^^

and the same problem in the line after that (I tried to used uparrows to indicate the problematic commas.)

## Editor dropped characters?...

Preben,

I am having some trouble reading your answer. I wonder if the editor dropped some angle brackets.

I see

SecTer:= CrossProduct(,)

____________________^

and

ProjX := diff(x(t), t, t) = -q*DotProduct(SecTer,,conjugate=false)/m;

_______________________________________^^

and the same problem in the line after that (I tried to used uparrows to indicate the problematic commas.)

## Precise definition of independent equati...

@Markiyan Hirnyk

It is easier to define independence precisely. Assume that the equations have all been put in the form where the right sides are 0. They are independent if there exists an assignment of numerical values to all of the unknowns such that

1. none of the evaluated left sides are 0
2. none of the evaluated left sides are equal to any other evaluated left side.

PS: Upon sleeping on it, I woke up realizing the above definition is not adequate. I withdraw it, I'll need to think about it some more.

## Precise definition of independent equati...

@Markiyan Hirnyk

It is easier to define independence precisely. Assume that the equations have all been put in the form where the right sides are 0. They are independent if there exists an assignment of numerical values to all of the unknowns such that

1. none of the evaluated left sides are 0
2. none of the evaluated left sides are equal to any other evaluated left side.

PS: Upon sleeping on it, I woke up realizing the above definition is not adequate. I withdraw it, I'll need to think about it some more.

## Maple chose what to eliminate....

Maple chose two of the five unknowns to eliminate. In this case, it is almost obvious that it will pick d and s, but we can't rely on that in general.

## Maple chose what to eliminate....

Maple chose two of the five unknowns to eliminate. In this case, it is almost obvious that it will pick d and s, but we can't rely on that in general.

## Independent equations...

A set of equations is independent if none of the equations can be derived from the others. I'm just extending the concept of "linear independence" to arbitrary equations.

The reason that I'm asking is that so far (in this thread) it seems that solve simply refuses to give any solution when the number of independent equations is greater than the number of variables for which solutions are requested. If that is the case, then solve should just give an error message telling the user to use eliminate instead. Returning NULL is not very helpful.

## Independent equations...

A set of equations is independent if none of the equations can be derived from the others. I'm just extending the concept of "linear independence" to arbitrary equations.

The reason that I'm asking is that so far (in this thread) it seems that solve simply refuses to give any solution when the number of independent equations is greater than the number of variables for which solutions are requested. If that is the case, then solve should just give an error message telling the user to use eliminate instead. Returning NULL is not very helpful.

## "meaningful" = algebraic...

@Markiyan Hirnyk

I would consider any algebraic response meaningful in this context. By saying "meaningful," I simply wanted to exclude responses such as warnings, error messages, communication via environment variable (like SolutionsMayBeLost), etc. You're right, I should've used a different word. Just change "meaningful" to "algebraic".

## "meaningful" = algebraic...

@Markiyan Hirnyk

I would consider any algebraic response meaningful in this context. By saying "meaningful," I simply wanted to exclude responses such as warnings, error messages, communication via environment variable (like SolutionsMayBeLost), etc. You're right, I should've used a different word. Just change "meaningful" to "algebraic".

## # equations > # desired variables to sol...

Is there any example of solve giving a meaningful non-NULL answer when the number of independent equations is greater than the number of variables for which solutions are requested?

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