What is f[1]? Advanced question

[amannucci]

I have found a Mathematica program with the following construct:
x[1]=0.1
x[2]=0.2
x[3]=0.3

or
Do[x[i]=i/10.,{i,1,3}]

x is not a function. It is not a list. What is it? If I query x thus:
?x

the answer is just what I have written above. Mathematica knows about x[1], x[2], etc. How does Mathematica know about the "elements" of x?

In some other sense, I could have written:
y1 = 0.1
y2 = 0.2
y3 = 0.3

But
?y

obviously gives a different result (just returns y).

Thank you.

[Tomas Garza]

It is possibly a previously declared "Array" with the name "x", and the three lines are assigning values to the elements of the array. Check for Array in the Help.
-Tomas

[Mannucci, Anthony J (335G)]

I believe I figured this out. x[] is a function that has definitions only at certain values. A function can be defined thus:
x[x_] :=
which matches the rhs to a pattern (x_), or a function can be defined thus:
x[1] = 0.1
which means: if the function argument expression evaluates to 1, then substitute 0.1. However, if the function argument evaluated to something other than 1, there is no substitution made, so that x[2] just evaluates to x[2].

I don't think there is an Array data type in Mathematica.

Thank you!

-Tony

From: Tomas Garza <tgar...@msn.com<mailto:tgar...@msn.com>>
Date: Tuesday, June 25, 2013 6:27 PM
To: Tony Mannucci <Anthony.J...@jpl.nasa.gov<mailto:Anthony.J...@jpl.nasa.gov>>, "math...@smc.vnet.net<mailto:math...@smc.vnet.net>" <math...@smc.vnet.net<mailto:math...@smc.vnet.net>>
Subject: Re: What is f[1]? Advanced question

It is possibly a previously declared "Array" with the name "x", and the three lines are assigning values to the elements of the array. Check for Array in the Help.

-Tomas

[Bob Hanlon]

Do[x[i] = i/10., {i, 1, 3}]


x is a user-defined function whose domain is just the integers 1, 2, and
3. Like all functions, its Head is Symbol.


Head /@ {x, Exp, Mean}


{Symbol, Symbol, Symbol}


The function definitions are stored in the DownValues of the symbol


DownValues[x]


{HoldPattern[x[1]] :> 0.1, HoldPattern[x[2]] :> 0.2, HoldPattern[x[3]] :>
0.3}


Like any function, when given an argument outside of its defined domain, it
returns unevaluated.


x[4]


x[4]



Bob Hanlon




On Tue, Jun 25, 2013 at 9:14 PM, amannucci
<Anthony.J...@jpl.nasa.gov>wrote:

[Bill Rowe]

On 6/25/13 at 9:14 PM, Anthony.J...@jpl.nasa.gov (amannucci)

wrote:

>I have found a Mathematica program with the following construct:
>x[1]=0.1
>x[2]=0.2
>x[3]=0.3

>or Do[x[i]=i/10.,{i,1,3}]

>x is not a function. It is not a list. What is it?

x is a function that you have defined at specific values.

Notice that

In[1]:= x[1] = 0.1;
x[2] = 0.2;
x[3] = 0.3;

In[4]:= DownValues[x]

Out[4]= {HoldPattern[x(1)]:>0.1,HoldPattern[x(2)]:>0.2,HoldPattern[x(3)]:=
>0.3}

and

In[5]:= f[y_] := y^2

In[6]:= DownValues[f]

Out[6]= {HoldPattern[f(y_)]:>y^2}

That is the values for x are stored exactly like the definition
of an expression is what most think of as a function in Mathematica.

[amannucci]

Thanks. This is very helpful. I remain puzzled by this:

u // FullForm
FullForm[Symbol["u"]]
FullForm[Unevaluated[Symbol["u"]]
Head[u]

FullForm[{1, 2, 3}]
Head[{1, 2, 3}]

In other words, Head and FullForm seem consist for the list, and inconsistent for the variable u. u's full form is just, literally "u", not Symbol["u"]. But the list's FullForm starts with "List". Head of Symbol["u"] is not Symbol, but "u". What gives?

[David Bailey]

I don't think there is too much mystery here - FullForm would simply be
too clumsy if it expanded symbols into Symbol constructs. As it is,
FullForm is incredibly useful.

I wish there was also a variant that did not evaluate its argument. True
one can write:

expression //Hold//FullForm

but the expression gets contaminated with Hold.

David Bailey
http://www.dbaileyconsultancy.co.uk

[Louis Talman]

On Tue, 25 Jun 2013 19:14:08 -0600, amannucci
<Anthony.J...@jpl.nasa.gov> wrote:

> I have found a Mathematica program with the following construct:
> x[1]=0.1
> x[2]=0.2
> x[3]=0.3
> or
> Do[x[i]=i/10.,{i,1,3}]

> x is not a function. It is not a list. What is it? If I query x thus:
> ?x

x most certainly *is* a function. Its a function whose domain contains
just the three numbers 1, 2, and 3.

And it is *not* an array. Mathematica has lists, which it uses as arrays
on occasion. An array, y, is indexed with double brackets: a[[1]],
a[[2]], etc.

--Louis A. Talman
Department of Mathematical and Computer Sciences
Metropolitan State University of Denver

<http://rowdy.msudenver.edu/~talmanl>
.

[John Doty]

On Thursday, June 27, 2013 4:25:04 AM UTC-6, amannucci wrote:
> Thanks. This is very helpful. I remain puzzled by this:
>
>
>
> u // FullForm
>
> FullForm[Symbol["u"]]
>
> FullForm[Unevaluated[Symbol["u"]]
>
> Head[u]
>
>
>
> FullForm[{1, 2, 3}]
>
> Head[{1, 2, 3}]
>
>
>
> In other words, Head and FullForm seem consist for the list, and inconsistent for the variable u. u's full form is just, literally "u", not Symbol["u"]. But the list's FullForm starts with "List". Head of Symbol["u"] is not Symbol, but "u". What gives?

If FullForm did what you expected, its output would be infinite, because every expression in Mathematica has a head. Remember that the head of an expression is itself an (arbitrary!) expression. So, FullForm "bottoms out" when further expansion would expose an elementary head like Symbol or Integer.

[Tomas Garza]

In[2]:= Array[x, 3]

Out[2]= {x[1], x[2], x[3]}

x is an array and is indexed with single brackets (cf. the Help browser).
-Tomas

> From: tal...@gmail.com
> Subject: Re: What is f[1]? Advanced question

[Helen Read]

x is not an array. It is a symbol which in this case has not been
defined. It does not refer to the list.

If you were to name the array, like this:

b= Array[x, 3]

The list is named b, and you can extract the entries with double brackets.

b[[2]]

Perhaps your confusion arises from the fact that b[[2]] and x[2] are
equal. Here is another example that will perhaps clarify.

c = Array[x, 5, -8]

c[[1]] (* first entry of c. Not the same as x[1] *)

c[[2]] (* second entry of c. Not the same as x[2] *)