## 5566 Reputation

7 years, 181 days

## How to "simplify" a RootOf result?...

Maple

I rephrased my previous question in a more synthetic form
(there was probably a lot in it that I thought was important for understanding the problem, but I realized afterwards that it only added confusion).

The true question is yellow-highlighted in the code below

 > restart
 > # The result below seems natural: we were taught in school that exp  # being a bijective function we can get rid of it in the equality to # solve and write simply x=Pi. x = solve(exp(x)=exp(Pi), x)
 (1)
 > # But the solution method solve uses is not that natural (and I # don't really understand it). # infolevel[solve] := 10: # x = solve(exp(x)=exp(Pi), x);
 > # Replacing now exp by some undefined function f produces a # kind of "no-solution" answer: this seems quite normal because # not knowing the properties of f one cannot simply get rid of it. infolevel[solve] := 0: x = solve(f(x)=f(Pi), x)
 (2)
 > # Finally replace f by a bijective function with no analytic expression. s = solve(erf(x)=erf(Pi), x) assuming x::real
 (3)
 > # It would have seem reasonable for Maple to answer x=Pi, or # at least it is what I would have done given the properties # of the erf function. # # How can I "force" Maple to "simplify" it's RootOf result to get # x=Pi?

For those interested in the motivations of this quastion, see here Where_does_the_question_come_from.mw

The original question is here Original_question.pdf

## Why doesn't "plot" display the graph?...

Maple 2015

Do you have any idea why the graph of function f (see the attached file) is not displayed?
How can I plot it without using the

`plot([seq([t, f], t in [seq](0.9..1.12, 0.002))]);`

command ?

 > restart:
 > kernelopts(version)
 (1)
 > f := 4.185692792*10^2172*t^2499*exp(-5000.000000*sqrt(t)); # Here is a plot of f plot([seq([t, f], t in [seq](0.9..1.12, 0.002))]);
 > # How can I plot f using simply: plot(f, t=0.9..1.12);  #no graph
 > # As numelems([seq](0.9..1.12, 0.002)) = 111, I assume # that forcing numpoints to a number that at least equal # to this one could give a non null display? plot(f, t=0.9..1.12, numpoints=1000):  #no graph
 > # Last attempt by forcing a list of points where f has to be evaluated. plot(f, t=0.9..1.12, sample=[seq](0.9..1.12, 0.002))
 >

## Why doesn't `if` recognize three argumen...

Maple 2015

Here is a chunk of a more complex code

```# syntax 1

decisions := "accept", "reject":

T := 2:
# The true test is `if`(t > T, ...) where t comes from some computation.
# In order to focus on the issue I assumed t was equal to 1.
`if`(1 > T, decisions[1], decisions[2]);
"reject"
```

To get a more concise writing, I did the following

```# syntax 2

decisions := "accept", "reject":
T := 2:
`if`(1 > T, decisions);
```

`Error, invalid input: `if` expects 3 arguments, but received 2`

Why doesn't `if` recognizes that decisions is a two parameters sequence?
Is there a way to force `if` to understand syntax 2 ?

I tried replacing `if` by piecewise: while getting no error I can't understand why I got si strange results:

```piecewise(1 > T, decisions);
piecewise(3 > T, decisions);
0
"accept", "reject"
```

What mechanism does piecewise use to return these values?

If_and_piecewise.mw

## background option not correctly understo...

Maple

I often use DocumentTools:-Tabulate or  DocumentTools:-Layout do display a vector or matrix of plots instead of plots:-display. because I find the latter less practical.
But it seems that the 'background' option is not correctly managed with DocumentTools:-Tabulate or  DocumentTools:-Layout.
The attached file shows that:

• the 'background' option is correctly managed if each "view" contains a single plot,
• but not correctly as soon as at least one "view" contains morethan one plot.

DocumentTools_and_Background.mw

How can I get with DocumentTools:-Tabulate  /  DocumentTools:-Layout the same rendering I get with plots:-display?

## How to define properly multi-variate sta...

Maple

(While using Maple 2015 this question concerns any other Maple versions)

I hesitated on the title to write and my first idea was to write "How to modify a built-in functions without making a mess?".
I finally changed my mind in order not to orient the answers in a wrong way.

So this question is about the construction of multi-variate distributions and concerns only the Statistics package.
Here are some of the attributes of a univariate random variable that Maple recognizes, and it is quite normal to expect that the construction of a multi-variate random variable (MVRV for short) distribution should get, at least, some of them.

```X := RandomVariable(Normal(a, b)):
map(a -> printf("%a\n", a), [exports(attributes(X)[3])]):
Conditions
ParentName
Parameters
CharacteristicFunction
CDF
CGF
HodgesLehmann
Mean
Median
MGF
Mode
PDF
RousseeuwCrouxSn
StandardDeviation
Support
Variance
CDFNumeric
QuantileNumeric
RandomSample
RandomSampleSetup
RandomVariate
MaximumLikelihoodEstimate
```

If the distribution is continuous the PDF is fundamental in the sense it enables constructs all the other statistics (=attributes) of a MVRV.
But it is nice to use integrated functions, such as Mean, Support, PDF, and so on, to get the expressions or values of these statistics instead of computing them from this PDF.
Let's that I prefer doing this

```MyNormal := proc(m, v)
description "Reparameterized Normal randomvariable, m=mean, v=variance":
Distribution(
PDF = (t -> exp(-1/2*(x-m)^2/v)/sqrt(2*Pi*v))
, Conditions = [Sigma > 0]
, Mean =m
)
end proc:

X := RandomVariable(MyNormal(mu, Sigma)):
Mean(X)
m```

than doing this

```MyNormal := proc(m, v)
description "Reparameterized Normal randomvariable, m=mean, v=variance":
Distribution(
PDF = (t -> exp(-1/2*(x-m)^2/v)/sqrt(2*Pi*v))
, Conditions = [Sigma > 0]
)
end proc:

X := RandomVariable(MyNormal(mu, Sigma)):
Mean(X);  # of course undefined
mean := int(PDF(X, x), x=-infinity..+infinity) assuming Sigma > 0
undefined
mean := 1
```

So, while all the statistics can be recover from the CDF (provided it exists), it's nicer to define these statistics within the Distribution structure (as in the first construction above).

Now some problems appear when you want to construct the Distribution structure for a MVRV.
The attached file contains the construction of MVRV whose ecah components are mutually independent (to keep the things simple) and both have a Unifom distribution.

MV_Uniform.mw

Here are some observations:

• Defining a multi-variate PDF goes without problems.
• Defining the Mean (or many other algebraic or numeric statistics) presents a difficulty related to the type of the arguments the build-in function Mean is aimed to recieve.
But a workaround, not very elegant, can be found.
• The case of the Support seems unsolvable: I wasn't able to find any workaround to define the support of a MVRV.
• I did not consider the Conditions attribute, but I'm not sure that, in the case of, let's say, a bi-gaussian random variable I would be capable to set that the variance is a symmetric positive-definite matrix?

I feel like the main restriction to define such MVRV distributions is the types used in the buid-in functions used in the Distribution structure.

Does anyone have an idea to tackle this problem?

• Are we doomed to use workarounds like the one I used for defining the MVRV mean?
• Can we modify the calling sequence of some build-in functions without making a mess and keep them working on build-in distributions?
• Must we overload the construction of these build-in functions?
Doing for instance:
```restart:
with(Statistics):
local Mean:
Mean := proc(...) ... end proc```

Thanks in advance for any suggestion and help.

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