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34

Let you have a function and an initial point f[x_] := Cos[x] x0 = 0.2; Then you can calculate a sequence seq = NestList[f, x0, 10] (* {0.2, 0.980067, 0.556967, 0.848862, 0.660838, 0.789478, \ 0.704216, 0.76212, 0.723374, 0.749577, 0.731977} *) and vizualize it with a so-called Cobweb plot p = Join @@ ({{#, #}, {##}} & @@@ Partition[seq, 2, 1]); ...


26

The following is based on the fact that the determinant of a matrix is equal to zero when two rows are the same. Thus, if you plug any of the points in, you get a true statement. SeedRandom[3]; pts = RandomReal[{-1, 1}, {5, 2}]; row[{x_, y_}] := {1, x, y, x*y, x^2, y^2}; eq = Det[Prepend[row /@ pts, row[{x, y}]]] == 0 (* Out: ...


22

Diagnosis Spelunking the definition of Commonest, which is written in top-level Mathematica code, I see that the two parameter form is handled by this internal function: Commonest; (* preload *) ? Statistics`DescriptiveDump`oCommonestSetLength oCommonestSetLength[list_, n_] := Catch[Block[{res, reslen, ord}, res = Tally[list]; reslen = Length[res]; ...


20

We can use Replace to express an almost literal translation of the problem statement: Replace[input, n : Except[Max[input]] :> n/10., {1}] (* {0.2, 0.3, 0.1, -0.3, -0.5, 9, 0.2, 0.6, -0.1, 0.06} *) Why Replace instead of /.? Replace is used instead of ReplaceAll (/.) to ensure that the replacement rule is only applied to the level one list elements ...


19

One way is to use an extra argument that acts as a switch. Clear[f]; f[0] = 1; f[1] = 1; f[n_, True] := f[n - 1] + f[n - 2] Example: f7 = f[7, True] (* Out[329]= f[5] + f[6] *) To proceed another step, can do a replacement. f7 /. f[aa_] :> f[aa, True] (* Out[330]= f[3] + 2 f[4] + f[5] *) Can use Nest to repeat this n times. Nest[# /. f[aa_] ...


19

Max[StringLength@Names["System`*"]] 38 Select[ Names["System`*"], 38 == StringLength[#] &] {"MultivariateHypergeometricDistribution"} As far as I can say there is no limit for lengths of symbol names, besides that of the memory limitation.


19

Introduction This post is long overdue as I have been repeatedly asked to explain code of mine containing these things. As I see increased use of this construct by others perhaps it is past due also. SparseArray objects can behave as functions accepting certain arguments to return internal data or efficiently return data in certain forms. These are known ...


17

Let me first answer your second question, since I can only guess about the main question: I also observed that the syntax colouring (version 10, windows 7) suggests that Trace can be used with only two arguments. It's really just the coloring that goes wrong and has nothing to do with functionality. You can see that it is not even related to ...


17

It seems that there is a significant overhead every time a color scheme is switched. Once a scheme is loaded each use is fast, but changing color schemes apparently unloads and reloads the mechanism. The result is that the speed of application is directly related to the frequency of switching. With sorted values there is only one switch and application is ...


16

I believe this is correct, and very fast: fn[x_Integer, n_Integer] := Complement[Range @ x, Join @@ Range[#, x, #]] & @ Prime @ Range @ n Test: fn[10000, 1223] {1, 9929, 9931, 9941, 9949, 9967, 9973} It seems I am a bit late to return to this problem and Simon Woods already provided a memory optimized approach. His sieve is comparatively ...


16

This is competitive with Mr Wizards code and seems faster in some cases: fn2[x_Integer, n_Integer] := Module[{y = Range @ x}, (y[[# ;; x ;; #]] = 0) & /@ Prime[Range @ Min[n, PrimePi @ x]]; SparseArray[y]["NonzeroValues"]] AbsoluteTiming[fn[10000, 1223];] (* {0.004000, Null} *) AbsoluteTiming[fn2[10000, 1223];] (* {0.010001, Null} *) ...


15

Let's start by taking a look at the compiled form of one of our queries: Dataset`CompileQuery[Query @ First @ spans] (* Dataset`WithOverrides@*Checked[Slice[205 ;; 313], Identity] *) We can see that the operation is not implemented directly in terms of part. Indeed, there are three components: Dataset`WithOverrides, GeneralUtilities`Checked and ...


15

Your question is not well specified for several reasons: Pure functions accept a flexible number of arguments #1 + #2 &[a, b, c, d] a + b It is common for some arguments to not be used: #1 + #3 & @@ {a, b, c, d} a + c SlotSequence includes all arguments after the given position: +##3 & @@ {a, b, c, d} c + d Without ...


15

The term pure function used in Mathematica is not being used in the same sense as the cited Wikipedia article. In Mathematica it refers to an anonymous function. In the Wikipedia article it is a term extracted by analogy from the increasingly popular term "purely functional" which refers (mainly) to deterministic programming free of side-effects. The ...


15

Normally I like to use On and Off for this kind of tracing as it is easy to set up without modifying any symbols. However, it does not immediately work in this case: On[Roots] Solve[x^3 - 2 x + 12 == 0, x]; Off[] This does not produce any trace messages. Something must be using Quiet to suppress them. We can check this hypothesis: On[Quiet] Solve[x^3 ...


15

Let us try to produce the solution without applying brute force, similar to mgamer's answer (that did not actually use Mathematica). Reduce[Mod[10^r - 1, 37] == 0, r, Integers] (* -> C[1] \[Element] Integers && C[1] >= 0 && r == 3 C[1] *) We see that the value of r can in fact be any nonnegative multiple of 3. The result sought is ...


15

In addition to the error messages quoted in the question the line returns: GeneralUtilities`Benchmarking`PackagePrivate`plot[ IndexBy[{{{16, 9.37132*10^-6}, . . . IndexBy was removed from 10.1.0: Note that IndexBy will be removed in a future version of Mathematica. It was something that was considered for 10.0.0 but didn't make the cut. – Stefan R ...


14

It depends on what you want to do with result, but you could try like this: Function[{x}, {x, #}] /@ {1, 2} {{1, #1}, {2, #1}}


13

The separation-of-variables solution you quoted has two indices appearing in it: n and j (the subscripts of the coefficient $A_{nj}$). Here, n is azimuthal mode order, i.e. it counts the number of nodes along the direction in which the polar-angle $\theta$ varies (divided by 2). The index j is needed because the wave is supposed to satisfy the boundary ...


13

For full ranges There is the function DayRange that can be used for this purpose, but not in the same simple way like CharacterRange. For the days: DayName /@ DayRange[Today, Today ~DatePlus~ {{1, "Week"}, {-1, "Day"}}] {Wednesday, Thursday, Friday, Saturday, Sunday, Monday, Tuesday} For the months: DateValue[#, "MonthName"] & /@ DayRange[Today, ...


13

Mathematica does it internally by using BoxForm`ArrangeSummaryBox, which is quite straightforward to figure out: MakeBoxes[obj_MyObject, fmt_] ^:= Module[{o = List @@ obj, shown, hidden, icon = Graphics[{Blue, Circle[]}, ImageSize -> 70]}, shown = {{ BoxForm`MakeSummaryItem[{"Name: ", "Name" /. o /. "Name" -> Missing[]}, fmt], ...


13

It turns out ListSurfacePlot3D does a terribly poor job of approximating the surface in the OP, otherwise one will just apply DiscretizeGraphics to the output obtained from ListSurfacePlot3D and be done with it. But since that's not applicable here, we present an approach that uses alpha shapes to approximate the shape of the given point set by tuning a ...


12

It seems me that the answers of mathe and Yves Klett do not meet expectations of the author. The latter is as much as I have got it, to have a short analytical expression for the solution. Probably the author has an intention to use the result further in some analytical calculations, or to do something comparable. Am I right? If yes, one should first of ...


12

Here is a different approach based on the awesome new Region functions: f[x_] := x^3 g[x_] := x^5 - 2 x^3 - 3 x We solve for the intersections: sol = x /. NSolve[f[x] == g[x], x, Reals] {-1.94712297, 0, 1.94712297} Edit Here are the regions of interest: r1 = ImplicitRegion[g[x] > y && f[x] < y, {{x, -1.94712297, 0}, y}]; r2 = ...


12

Here is my solution: input = {2, 3, 1, -3, -5, 9, 2, 6, -1, 0.6}; factor=10.; pos = First@Ordering[input, -1]; output = ReplacePart[input/factor, pos -> input[[pos]]] {0.2, 0.3, 0.1, -0.3, -0.5, 9, 0.2, 0.6, -0.1, 0.06}


12

Your basic requirement is met with: safeExport[file_String, args___] := If[ ! FileExistsQ[file] || ChoiceDialog["File already exists. Overwrite?"], Export[file, args], $Failed ] What you describe as "attributes" (e.g. PlotRange -> All) are known as Options or named optional arguments. (See Attributes for a description of what that ...


12

fF[x__Integer] := FromDigits[Join @@ IntegerDigits @ {x}] fF[1, 2] (* 12 *) fF[2, 4, 65] (* 2465 *)


12

how to reproduce the default hashing behaviour when explicitly choosing a method Hash[1, "Expression"] (* 6568131406215528669 *) re-create the default hashing behaviour in different versions? Not possible as far as I know. The one-argument Hash implementation may change between versions. The algorithm has been most recently ...


11

Using Function with a named parameter, as halmir showed, is the standard way to do this, however anything that prevents a literal Slot[1] from appearing in the body of the Function will work. Inactive as chuy showed is one possibility, but I find this cleaner: {#, Slot @@ {1}} & /@ {1, 2} {{1, #1}, {2, #1}} If the body will not be evaluated you ...


11

Maybe something like (using Inactive): Activate[{#, Inactive[Slot][1]}] & /@ {1, 2} (* {{1, #1}, {2, #1}} *)



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