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9

A first step would be to implement a convenience function that can automatically apply the method of separation of variables to separable types of equations. To show that the steps could in principle be automated, let me repeat basically the same calculation that I did for cylindrical coordinates with only slight modifications to the heat equation: ...


5

The differential operator in the first form can be written as dd1[n_] := (Sum[a[k] D[#, {t, k}], {k, 0, n}]) & and is applied for example as dd1[1][x[t]] a[0] x[t] + a[1] Derivative[1][x][t] and dd1[2] @x[t] a[0] x[t] + a[1] Derivative[1][x][t] + a[2] (x^\[Prime]\[Prime])[t] In the second (product) form we would perhaps try d2[n_] := a[n] ...


4

I assume you are looking for a pretty specific answer. If this is less information than you are asking for feel free to comment or edit the question and I will expand. As you know, the frontend represents expressions using boxes. These are wrappers that are concerned with appearance, and are peripheral to core symbolic evaluation. Therefore, each step of ...


4

I have a very simple mind, so I'd approach it this way. My idea is to never assign a value to the symbol representing the golden ratio (I'll use ϕ), but to just use rules. Clear[ϕ]; rules = {(1 + Sqrt[5])/2 -> ϕ, (-1 - Sqrt[5])/2 -> -ϕ, Simplify[(1 + Sqrt[5]) (-1 - Sqrt[5])/4] -> -ϕ^2, Simplify[(1 + Sqrt[5])^2/4 -> ϕ^2]}; ...


3

here is the generalization of @Jens answer to 2D (I think). Feel free to include it to your answer? ClearAll[pt, px, x, t, p]; operator = Function[p, D[p, t] - Δ D[p, x, x] - Δ D[p, y, y]]; ansatz = pt[t] px[x] py[y]; pde2 = Expand[Apply[Subtract, operator[ansatz]/ansatz == 0]]; ptSolution = First@DSolve[Select[pde2, (D[#, x] == 0 && ...


3

Good question; this is something that is missing in Mathematica AFAIK. I see two way to proceed: Option 1 One way is to overload the TensorRank, TensorDimensions, and TensorSymmetry functions for patterns that have head CD: CD /: TensorRank[CD[tensor_]] := TensorRank[tensor] + 1 CD /: TensorDimensions[CD[tensor_]] := Join[{First[#]}, #]& @ ...


3

Edit: The following solution works in Mathemtica 8.0.4, but not in 9.0.1: This requires an assumption about the parameter m: Assuming[m > 0 && m ∈ Integers, Sum[KroneckerDelta[m, n] f[n], {n, Infinity}]] (* ==> f[m] *)


3

Here is another way that uses the Graphics object directly: gr = ParametricPlot3D[{Cos[u], Sin[u] + Cos[v], Sin[v]}, {u, 0, 2 Pi}, {v, -Pi, Pi}] We discretize the graphics using DiscretizeGraphics mr = DiscretizeGraphics[Normal[gr /. (Lighting -> _) :> Lighting -> Automatic]] We compute the convex hull hull = ...


2

In Version 10, once the points have been obtained as per user21's approach, we can tetrahedralize them directly using DelaunayMesh pf = {Cos[u], Sin[u] + Cos[v], Sin[v]}; pp = ParametricPlot3D[pf, {u, 0, 2 Pi}, {v, -Pi, Pi}] data = Reap[ParametricPlot3D[Sow[pf], {u, 0, 2 Pi}, {v, -Pi, Pi}]][[2, 1]]; pts = Cases[data, {_?NumericQ, _?NumericQ, ...


2

This is also not an answer but a brief study in $Version "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" which might be of interest. It considers three methods of calculating the requested probability. It shows that in this version there is no negative probability but there is still a "critical number" which amounts to 8. For n = 8 the ...


2

You can set the multiplication symbol in Preferences->Appearance->Numbers->Multiplication


2

Your integral is very unlikely to exist in terms of elementary functions. In particular, it involves terms of the form $$ \int\exp\left[-\frac12\sqrt{a\, \text{poly}(\xi)+b \xi^{0.998906}}\right]\text d\xi, $$ which is very unfriendly as regards symbolic integration. Note that in general symbolic integration is not possible; do you have some specific reason ...


2

EDIT Here's a slightly modified version of a suggestion made by Kuba in my separate question on this topic (coordinates[[1]] /. (Sqrt[5]) -> (2 tau - 1) // Simplify) /. tau -> HoldForm@\[Tau] ORIGINAL This is not the most elegant solution to grace this forum, but: Map[ If[ AtomQ@#, #, (Simplify[#/τ]*HoldForm@τ) /. { τ -> ...


1

Question How do I prevent Part[] from trying to decompose symbolic expressions when it is evaluated? Mathematica 10 implements something like your listPart (with additional functionality): Indexed: Indexed can be used to indicate components of symbolic vectors, matrices, tensors, etc. When expr is a list, Indexed[expr,i] gives ...


1

I suspect there may not be a closed form expression (I have not looked at it hard enough). If the aim is to not analytical but numerical, I post the following for illustration (apologies if not the intent of question): f[x_, y_] := NIntegrate[ Log2[t + 1] Exp[-(x - Log[t])^2/(2 y^2)]/(Sqrt[2 Pi] t y), {t, 0, Infinity}] rv[a_, b_, n_] := ...


1

@Daniel Lichtblau's comment seems like an answer that is worth putting in an answer: (1) Integrate will not catch conditions that are discrete. (2) As was pointed out already in comments, the result is correct anyway; the singularity is removable (e.g. via Limit). Edit: I might add that GenerateConditions might yield a ConditionalExpression but not a ...



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