programming - Ways to protect certain parts of expressions

I am currently working on a quantum mechanics problem where I try to find equations that relate different states of a many-body system. The states are described by sets of pairs of integers, like list1 = {{0,0},{0,1},{1,-1},{3,-3}}. There is a very expensive and complicated function listToPoly[li_List] that turn such lists into polynomials. It turns out that many seemingly different lists can produce the same polynomial, or that sets of lists produce sets of polynomials that are linearly dependent. I have found a way to determine some of these dependencies without using listToPoly but I would like some help implementing it.

What I want to accomplish is a way to represent these lists in a way that I can both act on them with functions that manipulate their list structure, e.g.,

func1[li_List,k_] := Transpose[{#1+k, #2}]& @@ Transpose[li]

and programmatically generate and solve the equations that represent the linear dependences. Say I have three lists:

l1 = {{0, 0}, {0, 1}}
l2 = {{0, 0}, {0, 2}}
l3 = {{0, 1}, {0, 2}}

and I find out that their corresponding polynomials p1, p2, p3 would satisfy

2*p1 - 3*p2 + p3 == 0

I would like to be able to generate such equations using the lists (I don't want to invoke listToPoly), but then I need to prevent evaluation like

2*l1 - 3*l2 + l3 = {{0, 1}, {0, -2}}

Even combining the two things, it would be nice to be able to define something like

lindep[li_List, kmax_] := Sum[(-1)^k * func1[li, k], {k, kmax}] == 0

and then evaluate

lindep[{{0,0},{0,1}}, 2]

to get

- {{1,0},{1,1}} + {{2,0},{2,1}} == 0

There are probably lots of ways to do this, like:

Using Hold and ReleaseHold

Adding or replacing a custom head and defining functions to act on object with that head

Using ToString and ToExpression back and forth

etc.

Question

What would you guys choose? Any clear (dis)advantages to any particular method? I realize I am basically asking "how to handle custom objects", but I thought giving the context would make it easier to point me in the right direction.

Answer

I realize I am basically asking "how to handle custom objects", but I thought giving the context would make it easier to point me in the right direction.

It seems that you are. I believe the most natural way to do that in Mathematica is to use a custom head. I'll use obj for my examples.

First you might define a pattern for your custom object:

p0 = obj[{{_, _} ..}];

Then define a new func1 (I'll call fn1) referencing that pattern:

fn1[li : p0, k_] := MapAt[# + k &, li, {1, All, 1}]

And lindep:

lindep[li : p0, kmax_] := Sum[(-1)^k*fn1[li, k], {k, kmax}] == 0

We can define a Format to style obj expressions as plain lists:

Format[x : p0] := Interpretation[First @ x, x]

Finally:

lindep[obj[{{0, 0}, {0, 1}}], 2]

-{{1, 0}, {1, 1}} + {{2, 0}, {2, 1}} == 0

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....

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