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In the last session, I mentioned that Alternatives (a pattern object) and Or (a function) are not interchangeable. Here's an example to demonstrate how they differ.

Consider trying to extract all of the cases from a list that are either 2 or True. Using Alternatives (|) gives the correct result:

Cases[{1, 2, 3, True, 4}, 2 | True]
(* {2, True} *)

However, using Or (||) gives an incorrect result:

Cases[{1, 2, 3, True, 4}, 2 || True]
(* True *)

This is because Or immediately evaluates if any of its arguments resolve to True or False. In this case, 2||True evaluates to True, so Cases only picks out the items that match True.

Hi Hubert, your questions can be answered with overloading – something we will be covering in today's session (Extending Capability).

To solve (c), you can use OrderlessPatternSequence, and to solve (b), you can use Condition (/;). Both of these were covered in the first session (The Basics). To solve (a), you can use Message, which will be covered in the fourth session (Error Handling).

Here's some code to get you started:

ClearAll[getClass];

getClass::orderTooLow = "The order you have provided (`1`) is too low. It must be an integer greater than 2.";
getClass::badArgs = "Bad arguments. getClass takes exactly 2 arguments: an 'order' and a 'cycle'. The order must be an integer greater than 2 and the cycle must be a valid permutation cycle.";

(* Main definition: *)
getClass[
    OrderlessPatternSequence[
     order_?IntegerQ,
     cycle_?PermutationCyclesQ
     ]
    ] /; order > 2 := (* Your function *);

(* Order provided is too low: *)
getClass[
   OrderlessPatternSequence[
    order_?IntegerQ,
    cycle_?PermutationCyclesQ
    ]
   ] := (
   Message[getClass::orderTooLow, order];
   $Failed
   );

(* Arguments provided are either invalid, too numerous or too few: *)
getClass[badArgs___] := (
   Message[getClass::badArgs];
   $Failed
   );

You can add more definitions / error messages to handle more cases.

P.S. – you seem to have a lonely p in your function definition.

Wow! Note that it needs 13.3 to evaluate PlotHighlighting -> None, If you delete that statement it works with 13.2

Doh! Thanks Paul! I don't know why I did not think of the first option right away, but I didn't; and it does work. Exactly as it should; and exactly what I was looking for. Come think of it, it's a perfect little example of how the syntax of the Wolfram language works. So thanks for the tip.

Carl

I came across the following expression in a form I had not seen before:

g[h_, b_][x_] := 
 f[b] + (DifferenceQuotient[f[x], {x, h}] /. (x -> b)) (x - b)

What is going on with the [x_] in s separate bracket outside the g[h,b] on the left side of the := ? It looks a little like the Function[parameters,body][x] format but it's not that...

Carl

I want to construct a function that inputs an element of the symmetric group of n objects and returns the conjugacy class of this element. The inputs are thus the positive integer "n" and something like "Cycles[{{1,3,2}}}]" so my starting point is

getClass[n\_, cycle\_ ]

I know that in practice I tend:

1. to forget to to include the argument n,
2. I input an incorrect value of n (a cycle like Cycles[{{1,3,2}}}] permutes objects 1,2,3 so n>= 3 at least
3. I tend to mess up the ordering of the argument.

So I want to build getClass[n_,cycle_] so

• a) getClass[somecycle] returns a message like "order missing",
• b) getClass[q,somecycle] returns "order is too low" if q is incorrect
• c) evaluates getClass[somecycle,n] equally well as getClass[n,somecycle]

I can handle c) by using PermutationCyclesQ, which will tell me if a particular argument is actually a cycle, and thus test if the first argument is a cycle or not.

I'm not sure how to proceed to handle the rest or make it work smoothly. The current working version of getClass is:

getClass[p_, cycle_] := Module[{listofcycles, shortlistofcycles, padding},
  listofcycles = Flatten[List @@ cycle, 1];
  shortlistofcycles = Table[Length[listofcycles[[k]]], {k, 1, Length[listofcycles]}];
  padding = p - Sum[shortlistofcycles[[k]], {k, 1, Length[shortlistofcycles]}];
  If[padding != 0, AppendTo[shortlistofcycles, Table[1, {q, 1, padding}]]];
  Reverse[Sort[Flatten[shortlistofcycles]]]]

Thus for instance getClass[3,Cycles[{{1,3,2}}}] returns {3}

getClass[4,Cycles[{{1,2},{3,4}}] returns {2,2}

I just did a search myself in the documentation and couldn't find anything particularly helpful… (no tutorial that explains how to create your own operator forms, anyway).

However, in case it's of interest to you, I did find some code on StackExchange that claims to list all of the bulit-in functions that have operator forms:

file = FileNameJoin[{$InstallationDirectory, "SystemFiles", "Kernel", 
        "TextResources", $Language, "Usage.m"}];

usage = Import[file, "HeldExpressions"];

usage //
  Cases[
    HoldPattern[
      _[_[sym_Symbol, "usage"] = _String?(StringContainsQ["operator form"])]
    ] :> sym
  ]

Hi Laurence, can you give me some more details? It would be best if you could write down a simple example that includes the setup you want to start with and the result you want to end with. I may then be able to help with the steps in between.

Hi Paul, is that Plot example one that you saw in the lecture? I'm having trouble finding it in the notebook and I'm not sure how to accomplish something like that off the top of my head (or even after some searching). I know that you can do a more basic thing like Module[{x},x] to see the local name of x inside a module, but as far as seeing the names for automatically localized variables, I don't recall seeing that in the presentation and can't find it in the notebook. (Granted, I look at a lot of presentations and notebooks so it's quite possible that I've missed it, in which case a pointed would be helpful!)

Gotcha! The only cell-view switch I know about is cmd+shift+E, which shows the full cell expression, but—by default, at least—it doesn't show full context information. Dan is currently on holiday, but he should be able to provide you with a more concrete answer once he returns on Monday.