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RE: Apply and Heads option

• To: mathgroup at smc.vnet.net
• Subject: [mg35225] RE: [mg35213] Apply and Heads option
• From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
• Date: Wed, 3 Jul 2002 05:13:25 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

> -----Original Message-----
> From: Ersek, Ted R [mailto:ErsekTR at navair.navy.mil]
To: mathgroup at smc.vnet.net
> Sent: Tuesday, July 02, 2002 8:13 AM
> To: mathgroup at smc.vnet.net
> Subject: [mg35225] [mg35213] Apply and Heads option
> 
> 
> Hello mathgroup, 
> ---------
> 
> Apply has the Heads option.  
> Can somebody give an example where 
> Apply[f, expr, Heads->True]   gives a different result than 
> Apply[f, expr, Heads->False].
> 
> OR where 
> 
> Apply[f, expr, levelspec, Heads->True]   gives a different 
> result than 
> Apply[f, expr, levelspec, Heads->False].
> 
> ----------
> Thanks,
>    Ted Ersek
> 
> 

Ted,

indeed, there is a difference, see e.g.

In[100]:=
Outer[Apply[
        f,{h, h[x], h[x][h[x]], h[x][h[x][h[x]]],
h[x][h[x][h[x][h[x]]]]},{#1},
        Heads -> #2]&, Range[2,4], {True,False}]//TableForm
Out[100]//TableForm=
  h                        h
  h[x]                     h[x]
  f[x][f[x]]               h[x][f[x]]
  f[x][f[h[x]]]            h[x][f[h[x]]]
  f[x][f[h[x][h[x]]]]      h[x][f[h[x][h[x]]]]

  h                        h
  h[x]                     h[x]
  h[x][h[x]]               h[x][h[x]]
  h[x][f[x][f[x]]]         h[x][h[x][f[x]]]
  h[x][f[x][f[h[x]]]]      h[x][h[x][f[h[x]]]]

  h                        h
  h[x]                     h[x]
  h[x][h[x]]               h[x][h[x]]
  h[x][h[x][h[x]]]         h[x][h[x][h[x]]]
  h[x][h[x][f[x][f[x]]]]   h[x][h[x][h[x][f[x]]]]

or

In[106]:= Apply[f, h[x][h[x][h[x][h[x]]]], 3, Heads -> True]
Out[106]= f[x][f[f[f[x]]]]

In[107]:= Apply[f, h[x][h[x][h[x][h[x]]]], 3, Heads -> False]
Out[107]= h[x][f[f[f[x]]]]


As far as I can see, 

  Apply[f, expr, levelspec, headsopt] 

is the same as

  Replace[expr, _[s___] :> f[s], levelspec, headsopt]



In[147]:=
Table[Replace[h[x][h[x][h[x][h[x]]]], _[s___] -> f[s], {l, ll}, 
      Heads -> (h == 1)] == 
      Apply[f, h[x][h[x][h[x][h[x]]]], {l, ll}, Heads -> (h == 1)], 
      {l, 0, 3}, {ll, l, 3}, {h, 0, 1}]
Out[147]=
{{{True, True}, {True, True}, {True, True}, {True, True}}, 
 {{True, True}, {True, True}, {True, True}}, 
 {{True, True}, {True, True}}, 
 {{True, True}}}


--
Hartmut