functions - Strange behavior with Cases and Position with SetDelayed

I feel like I must be missing something simple and obvious here, but this has me scratching my head.

This works as expected:

list = {f[a], f[b]};
Cases[list, f[x_] :> x] -> Position[list, f[_]]

(* {a, b} -> {{1}, {2}} *)

However, this does not:

fun[list_] := Cases[list, f[x_] :> x] -> Position[list, f[_]];
fun[list]


(* {x, x} -> {{1}, {2}} *)

Is this a bug, or have I just not had enough coffee today?

Answer

What's happening

This is not simple by any means. You have encountered another instance of a general situation with lexical scope leaks / emulation / over-protection by symbol renaming. The case at hand is pretty similar to the one discussed here, so you can read the detailed explanation of this behavior in my answer there.

Roughly speaking, outer lexical scoping constructs (RuleDelayed in the linked dicsussion, and its analog for implicit global rule application here), try to protect the inner bindings from destructive changes, but mis-interpret their pieces and instead destroy yet inner bindings in the process. We have to fool that mechanism somehow, to avoid that.

Workarounds

The `"StrictLexicalScoping"` system option

Thanks to the hard work of Daniel Lichtblau, we now have a system option named "StrictLexicalScoping", which, when set to True, fixes many such cases, including the one at hand. You have to execute this:

SetSystemOptions["StrictLexicalScoping" -> True]

before you enter the definition of your function, and then it will work as intended.

Fooling the protection mechanism explicitly

In your case, here is one possible such work-around that is reasonably clean:

fun[list_] := 
   With[{rule = Rule}, 
      rule[Cases[list, f[x_] :> x], Position[list, f[_]]]
   ];

There are many more variations of it. What really matters is that SetDelayed and then the internal rule application engine (internal analog of RuleDelayed for global rules) don't see external Rule during the rule application.

This is surely not something that would first come to mind, though :)

Additional references

Here are a few additional links relevant to this discussion

Comments

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1....

Blog

Search This Blog