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RE: Apply and Heads option

• To: mathgroup at smc.vnet.net
• Subject: [mg35238] RE: [mg35213] Apply and Heads option
• From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
• Date: Wed, 3 Jul 2002 05:14:47 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

> -----Original Message-----
> From: Wolf, Hartmut 
To: mathgroup at smc.vnet.net
> Sent: Tuesday, July 02, 2002 11:51 AM
> Subject: [mg35238] RE: [mg35213] Apply and Heads option
> 
> 
> 
> > -----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
> > Subject: [mg35238] [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
> 

some more illustrative examples:

In[54]:= expr = ReplaceRepeated[h[x], e_ -> e[e], MaxIterations -> 4]
>From In[54]:=
ReplaceRepeated::"rrlim": "Exiting after h[x] scanned 4 times."
Out[54]=
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]][h[x][h[x]]]]]


levels {5},...,{0} Heads -> True, quite interesting... 
  
In[60]:= Scan[Print[Apply[f, expr, {#}, Heads -> True]] &, Range[5, 0, -1]]

>From In[60]:=
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]][h[x][h[x]]]]]

>From In[60]:=
f[x][f[x]][f[x][f[x]]][f[x][f[x]][f[x][f[x]]]][
  f[x][f[x]][f[x][f[x]]][f[x][f[x]][f[x][f[x]]]]]

>From In[60]:=
f[h[x]][f[h[x]]][f[h[x]][f[h[x]]]][f[h[x]][f[h[x]]][f[h[x]][f[h[x]]]]]

>From In[60]:=
f[h[x][h[x]]][f[h[x][h[x]]]][f[h[x][h[x]]][f[h[x][h[x]]]]]

>From In[60]:=
f[h[x][h[x]][h[x][h[x]]]][f[h[x][h[x]][h[x][h[x]]]]]

>From In[60]:=
f[h[x][h[x]][h[x][h[x]]][h[x][h[x]][h[x][h[x]]]]]


levels {4,4},...,{0,4} Heads -> True

In[61]:= Scan[Print[Apply[f, expr, {#, 4}, Heads -> True]] &, Range[4, 0,
-1]]

>From In[61]:=
f[x][f[x]][f[x][f[x]]][f[x][f[x]][f[x][f[x]]]][
  f[x][f[x]][f[x][f[x]]][f[x][f[x]][f[x][f[x]]]]]

>From In[61]:=
f[f[x]][f[f[x]]][f[f[x]][f[f[x]]]][f[f[x]][f[f[x]]][f[f[x]][f[f[x]]]]]

>From In[61]:=

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

>From In[61]:=
f[f[f[f[x]]]][f[f[f[f[x]]]]]

>From In[61]:=
f[f[f[f[f[x]]]]]


levels {5},...,{0} Heads -> False, comparatively dull...

In[62]:= Scan[Print[Apply[f, expr, {#}, Heads -> False]] &, Range[5, 0, -1]]

>From In[62]:=
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]][h[x][h[x]]]]]

>From In[62]:=
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]][h[x][f[x]]]]]

>From In[62]:=
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[h[x]]]]]

>From In[62]:=
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[h[x][h[x]]]]]

>From In[62]:=
h[x][h[x]][h[x][h[x]]][h[x][h[x]][h[x][h[x]]]][f[h[x][h[x]][h[x][h[x]]]]]

>From In[62]:=
f[h[x][h[x]][h[x][h[x]]][h[x][h[x]][h[x][h[x]]]]]


levels {4,4},...,{0,4} Heads -> False

In[63]:= Scan[Print[Apply[f, expr, {#, 4}, Heads -> False]] &, Range[4, 0,
-1]]

>From In[63]:=
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]][h[x][f[x]]]]]

>From In[63]:=
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[f[x]]]]]

>From In[63]:=
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[f[f[x]]]]]

>From In[63]:=
h[x][h[x]][h[x][h[x]]][h[x][h[x]][h[x][h[x]]]][f[f[f[f[x]]]]]

>From In[63]:=
f[f[f[f[f[x]]]]] 



--
Hartmut