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Re: What is the purpose of the Defer Command?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82023] Re: What is the purpose of the Defer Command?
  • From: Chris Chiasson <chris.chiasson at gmail.com>
  • Date: Wed, 10 Oct 2007 04:17:48 -0400 (EDT)
  • References: <fe4uhp$155$1@smc.vnet.net><fefhui$j6c$1@smc.vnet.net>

On Oct 9, 4:34 am, "David Park" <djmp... at comcast.net> wrote:
> Still, my question is: "What is the purpose of theDeferCommand?"
>
> The only potential purpose that I can see is for didactic purposes, to show
> preliminary expressions before they are automatically evaluated. But the
> only way to getDeferto evaluate is by copying and pasting and then
> evaluating,  by using Shift-Ctrl-L and then evaluating, or by evaluating in
> place. All of these operations leave no record of the steps taken. That goes
> contrary to any didactic purpose.
>
> --
> DavidPark
> djmp... at comcast.nethttp://home.comcast.net/~djmpark/
>
> "DavidPark" <djmp... at comcast.net> wrote in message
>
> news:fe4uhp$155$1 at smc.vnet.net...
>
> >I do not understand the utility of the newDeferstatement in Mathematica
> > Version 6. Also, it seems to me to be similar to, but not as good as, the
> > HoldTemporary command introduced by Ted Ersek on MathSource a few years
> > ago.
>
> > The help forDefersays: "Defer[expr] yields an object that displays as
> > the
> > unevaluated form of expr, but which is evaluated if it is explicitly given
> > as Mathematica input." What does 'given as Mathematica input' mean? The
> > examples seem to only involve copying and pasting, which I don't consider
> > a
> > great method for doing mathematics, or evaluation in place.
>
> > I would like to understand howDefermight be used in expository notebooks
> > to clarify some piece of mathematics. The problem is that it requires an
> > interactive action, which would be invisible to a reader of a notebook.  I
> > think the idea of 'modification in place' is poor in technical
> > communication
> > because it destroys the record of what was done.
>
> > (In the examples below, whenever an output resulted in an expression that
> > copied as a box structure, I converted to InputForm to simplify the
> > posting.)
>
> > Here is a simple example:
>
> > y =Defer[1 + 1]
> > 1 + y                                               giving
>
> > 1 + 1
>
> > 1 + (1 + 1)
>
> > I would prefer that theDeferexpression would have evaluated in the
> > second
> > statement but I guess it is logical that it didn't. If I write:
>
> > 1 + y
>
> > then select the y and Evaluate In Place I obtain the following, which must
> > then be further evaluated to obtain 3.
>
> > 1 + 1 + 1
>
> > 3
>
> > A second example. I want to show an integral without evaluation and then
> > the
> > evaluated result. I have to write the following expression, then select
> > the
> > second line of output, evaluate in place, and then I obtain the result -
> > but
> > as an Input cell. This is certainly a place where HoldForm would be
> > better.
>
> >Defer[Integrate[x^2 Exp[-x], {x, 0, 1}]]
> > %
> > giving
>
> > Integrate[x^2/E^x, {x, 0, 1}]
>
> > 2 - 5/\[ExponentialE]                        (which is an Input cell)
>
> > Here is third example.Deferdoes not evaluate and we obtain an error
> > message.
>
> > numb =Defer[2^67 - 1]
> > FactorInteger[numb]                                       giving
>
> > 2^67 - 1
>
> > FactorInteger::"exact" :  "\"Argument \!\(\*SuperscriptBox[\"2\", \
> > \"67\"]\) - 1 in FactorInteger[\!\(\*SuperscriptBox[\"2\", \"67\"]\) \
> > - 1] is not an exact number\""
>
> > FactorInteger[2^67 - 1]
>
> > But it works if I copy and paste into FactorInteger.
>
> > Now, look at the behavior of Ted's MathSource package.
>
> > Needs["Enhancements`HoldTemporary`"]
>
> > y = HoldTemporary[1 + 1]
> > 1 + y                                                        giving
>
> > 1 + 1
>
> > 3
>
> > The expression is evaluated if it is an argument of some function.
>
> > HoldTemporary[Integrate[x^2 Exp[-x], {x, 0, 1}]]
> > Identity[%]
> > giving
>
> > Integrate[x^2/E^x, {x, 0, 1}]
>
> > 2 - 5/\[ExponentialE]       (which is an Output cell)
>
> > numb = HoldTemporary[2^67 - 1]
> > FactorInteger[numb]                                            giving
>
> > 2^67 - 1
>
> > {{193707721, 1}, {761838257287, 1}}
>
> > Much better. I might be missing the point, but I don't think thatDeferis
> > at all well designed.
>
> > There is another Hold that is very useful. This is one that holds an
> > operation but evaluates the arguments. We have a HoldOp statement in the
> > Tensorial package.
>
> > Needs["TensorCalculus4V6`Tensorial`"]
>
> > ?HoldOp
>
> > HoldOp[operation][expr] will prevent the given operation from being
> > evaluated in expr. Nevertheless, other operations within expr will be
> > evaluated. Operation may be a pattern, including alternatives, that
> > represents heads of expressions. The HoldOp can be removed with
> > ReleaseHold.
>
> > One reason we want the arguments to evaluate is that the arguments often
> > contain tensor shortcut expressions and we want them evaluated to show the
> > full tensor expression inside some operation. However, there are many
> > other
> > uses.
>
> > f[x_] := Sin[x] \[ExponentialE]^x
>
> > We would like f[x] to be evaluated inside the Integrate statement, but
> > hold
> > the actual itegration.
>
> > Integrate[f[x], {x, 0, \[Pi]}] // HoldOp[Integrate]
> > % // ReleaseHold
> > giving
>
> > HoldForm[Integrate[E^x*Sin[x], {x, 0, Pi}]]
>
> > 1/2 (1 + \[ExponentialE]^\[Pi])
>
> > For exposition purposes we might want to keep the following expression in
> > the input order.
>
> > \[Pi]  Sin[x] \[ExponentialE]^x // HoldOp[Times]
> > % // ReleaseHold
> > giving
>
> > HoldForm[Pi*Sin[x]*E^x]
>
> > \[ExponentialE]^x \[Pi] Sin[x]
>
> > Often we will have cases where some operation has automatic built-in
> > rules,
> > such as linear and Leibnizian breakouts with differentiation. Again, for
> > exposition purposes, we might want to show the expression before these
> > rules
> > are applied.
>
> > g[x_] := x^2
>
> > D[a f[x] g[x], x] // HoldOp[D]
> > % // ReleaseHold                                    giving
>
> > HoldForm[D[a*E^x*x^2*Sin[x], x]]
>
> > a \[ExponentialE]^x x^2 Cos[x] + 2 a \[ExponentialE]^x x Sin[x] +
> > a \[ExponentialE]^x x^2 Sin[x]
>
> > --
> > DavidPark
> > djmp... at comcast.net
> >http://home.comcast.net/~djmpark/

You have quite obviously missed the point. Without Defer, we might
actually have to type ReleaseHold. Furthermore, symbols like Defer are
only afforded by keeping the System context free of useless symbols
such as AxesInFront and GridLinesInFront. Stick that in your pipe and
smoke it.

Sincerely,

--
http://chris.chiasson.name/



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