Is it possible to fit such a large range plot in mathematica?

I am trying to solve the coupled differential equation numerically with Mathematica. But the range of values are large so mathematica cannot give correct plot. Here is the code:

a = 4.75388*10^26; b = 5.424*10^-3; d = 4.75388*10^20; {X, Y} = {x, y} /.  NDSolve[{x'[ z] == -((a/z) (x[z] - b*z^(3/2) E^(-z)) (BesselK[1, z]/BesselK[2, z])),  y'[z] == ((d/z) (x[z] - b *z^(3/2) E^(-z)) (BesselK[1, z]/ BesselK[2, z]) - (a *z/4) (BesselK[1, z]) y[z]),  x[0.1] == 1.552*10^-4, y[0.1] == 10^(-9)}, {x, y}, z] //  FullSimplify // First LogLogPlot[{X[z], Y[z]}, {z, 0.1, 100}, PlotRange -> All] 

Sum of n sums, Permutations of the indices, how to write them in Mathematica?

I was wondering how to write a function $ F (r, q, n, f) $ in Mathematica, defined in this way:

$ $ F(r,q,n,f):=\sum_{i_0=1}^q f(i_0) \Biggl(\sum_{i_1=i_0+1}^{q+1} f(i_1)\biggl(\sum_{i_2=i_1+1}^{q+2} f(i_2)\Bigl(\ldots(\sum_{i_n=i_{n-1}+1}^{q+n} f(i_n))\ldots \Bigl) \biggl) \Biggl)$ $ es. $ $ \sum_{i_0=1}^2 f(i_0) \Biggl(\sum_{i_1=i_0+1}^{3} f(i_1)\biggl(\sum_{i_2=i_1+1}^{4} f(i_2) \biggl) \Biggl)=f(1)f(2)f(3)+f(1)f(2)f(4)+f(1)f(3)f(4)+ +f(2)f(3)f(4)$ $

does an operator already exist that can be used in this way?

trying to write this function on mathematica I realized that the “recursion” is variable and I don’t know how to program in this case.

thank you

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further example

$ $ \sum_{i_0=1}^1 f(i_0) \Biggl(\sum_{i_1=i_0+1}^{2} f(i_1)\biggl(\sum_{i_2=i_1+1}^{3} f(i_2)(\sum_{i_3=i_2+1}^{4} f(i_2)) \biggl) \Biggl)=f(1)f(2)f(3)f(4)$ $

How I can solve below pde (Reynolds equation) with mathematica?

I want to solve below equation in Mathematica. I know a little bit about Mathematica. Can somebody help me?

\begin{array}{l} \frac{1}{{{R^2}}}\frac{\partial }{{\partial \theta }}\left( {{P_0}\frac{{\partial \left( {{P_X}} \right)}}{{\partial \theta }} – \Lambda {R^2}{P_X}} \right) + \frac{1}{R}\frac{\partial }{{\partial R}}\left( {R{P_0}\frac{{\partial \left( {{P_X}} \right)}}{{\partial R}} + R{P_X}\frac{{\partial \left( {{P_0}} \right)}}{{\partial R}}} \right) – i2\Lambda \Gamma \left( {{P_X}} \right)\ + \underbrace {\frac{1}{R}\frac{\partial }{{\partial R}}\left( { – 3R{P_0}\frac{{\partial {P_0}}}{{\partial R}}} \right) + i2\Lambda\left( {{P_0}} \right)}_f = 0\ {P_0}\left( R \right) = {\rm{available}}\ \Lambda = 1 \ {P_X}\left( {{R_1},\theta } \right) = {P_X}\left( {{R_1},\theta } \right) = 0\ {P_X}\left( {R,0} \right) = {P_X}\left( {R,2\pi } \right) \end{array}

What are the most reliable test of goodness of fit in Mathematica

I realise this is possibly a VERY open ended question, and potentially has answers dotted around e.g. here .

What are the most robust methods of testing the result of NonlinearModelFit[...] in Mathematica?

I know already that there are some methods available in Mathematica such as AdjustedRSquared, RSquared, AIC, AICc, and BIC.

The most familiar of these to me personally are AdjustedRSquared and RSquared, but I know many methods of testing exist.

A sub-question to this is, when are these tests (and others, whatever they may be), most appropriate? Does this depend on the model fitted, the number of parameters, the amount of data, and so on?

I strongly appreciate that some of these questions can only be answered subjectively, and I even anticipate some potential off-topic or close requests; after all if there was a catch-all test — I wouldn’t need to ask this question! But I feel having a question where goodness of fit specifically in Mathematica is discussed (especially where some of the more experienced members of the community can contribute) might be beneficial.

Custom SGD optimizer in Mathematica neural network framework?

I have a new approach to SGD optimizer, and thought to try to test it in Mathematica. It uses gradients to simultaneously maintain online parabola model for smarter choice of step size – I only need to ask for gradients and be able to manually update parameters.

Is it doable within Mathematica neural network framework? I see NetPortGradient, but how to modify parameters?

As it seems a common research direction, maybe there is some simple example?

How to get the implicit solution of this PDE (Maple can but I don’t manage with mathematica)?

I asked this question about the analytical solution of the PDE:

$ $ \partial_t c=\partial_x((c-a)(c-b)\partial_xc) $ $

And I was given the answer that Maple gives the following implicit form as a solution:

$ $ \eqalign{&{k_{{1}}}^{2}{k_{{2}}}^{2}{c}^{2} + \left( 2\,{k_{{1}}}^{4}k_{{2}}k_{{3 }}-2\,{k_{{1}}}^{2}{k_{{2}}}^{2}a-2\,{k_{{1}}}^{2}{k_{{2}}}^{2}b \right) c\cr &+ \left( 2\,{k_{{1}}}^{6}{k_{{3}}}^{2}-2\,a{k_{{1}}}^{4}k_{{ 2}}k_{{3}}-2\,b{k_{{1}}}^{4}k_{{2}}k_{{3}}+2\,ab{k_{{1}}}^{2}{k_{{2}}} ^{2} \right) \ln \left( -{k_{{1}}}^{2}k_{{3}}+ck_{{2}} \right)\cr & -2\,{k _{{2}}}^{4}t-2\,k_{{1}}{k_{{2}}}^{3}x-2\,{k_{{2}}}^{3}k_{{3}}-2\,k_{{4 }}{k_{{2}}}^{3} =0} $ $ But when I’m doing the DSolve on Mathematica (I don’t have Maple) and I get no solution!

So my question is: can Mathematica give me that implicit solution? Can Maple do things that Mathematica cannot?

Mathematica beeps and clears all variables

I am writing a code which is supposed to simulate the heat transfer. All the time, when I run the code Mathematica starts computations, and after some time beeps and clears out all variables. If I for example add Print["A"] somewhere it does the same, but in a different moment of computations than previously. What can cause the problem? Shouldn’t Mathematica print some error first? Is this normal behaviour of Mathematica? I’ve been using it for 5 years, but this is the first kind of big program.