Solving an extension of the 2D wave equation to 2n dimensions

The 2D wave equation $$left(frac{partial^2}{partial^2 t} – frac{partial^2}{partial^2 z}right)f(t,z) = 0$$ can be written after changing variables to $x = t – z$, $y = t+ z$ as $$frac{partial}{partial x}frac{partial}{partial y}f(x,y) = 0,$$ with general solution $$f(x,y) = g(x) + h(y),$$ where $g$ and $h$ are arbitrary functions.

I want to solve a similar (looking) PDE for $(x,y) in mathbb{R}^{n}timesmathbb{R}^{n},$ which is given by $$frac{partial}{partial x}cdotfrac{partial}{partial y}f(x,y) = 0,$$ where the dot product is with the Euclidean metric.

Is it possible to find a general solution to this equation in terms of some arbitrary functions?
I asked Mathematica but unfortunately it didn’t know how to solve this one. It can solve the 2D version fine.

linear algebra – Inequality for dimensions of eigenspaces

I want to proof that for an endomorphism $f$ over vector space $V$ with eigenvalue $lambda$ and
$V = U oplus V/U$ with $f(U) subseteq U$ the following holds:
$$
dim E(f,lambda) leq dim E(f_U,lambda) + dim E(f_{V/U}, lambda)
$$

I managed to proof that the inequality holds for the algebraic multiplicity of $lambda$ via splitting the characteristic polynomial of $f$. However I can’t find a way to connect that to the geometric multiplicity. I would appreciate any hint.

calculus and analysis – Problems with improper integrals in higher dimensions

Calculating numerically certain energy integrals in three and four dimensions related to
the Riesz potential
and the capacity, I try

NIntegrate( 1/Sqrt((x - p)^2 + (y - q)^2 + (z - r)^2), {x, y, z} (Element) 
Ball({0, 0, 0}, 1), {p, q, r} (Element) Ball({0, 0, 0}, 1), 
AccuracyGoal -> 3, PrecisionGoal -> 3, WorkingPrecision -> 25, 
Exclusions -> {(x - p)^2 + (y - q)^2 + (z - r)^2 == 0})

0

and a warning message
“NIntegrate::moptxn: The option SymbolicProcessing of the method FiniteElement is not one of {Method,MeshOptions}.”
As far as I understand it, this means that the NIntegrate command does not accept the set of the integration
in the form {x, y, z} (Element) Ball({0, 0, 0}, 1), {p, q, r} (Element) Ball({0, 0, 0}, 1). However, if it is so,
the input, not 0, should be returned.

My next try is

 NIntegrate( 1/((x - p)^2 + (y - q)^2 + (z - r)^2)^(1/2), {x, -1, 1}, {y, -Sqrt(1 - x^2), Sqrt(1 - x^2)}, 
{z, -Sqrt(1 - x^2 - y^2), Sqrt(1 - x^2 - y^2)}, {p, -1, 1}, {q, -Sqrt(1 - p^2), Sqrt(1 - p^2)}, 
{r, -Sqrt(1 - p^2 - q^2), Sqrt(1 - p^2 - q^2)}, AccuracyGoal -> 3, PrecisionGoal -> 3, WorkingPrecision -> 25, 
Exclusions -> {(x - p)^2 + (y - q)^2 + (z - r)^2 == 0})

-1244.482640558337558417913

and a warning
“NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 54 recursive bisections
in z near {x,y,z,p,q,r} = <<1>>. NIntegrate obtained -1244.482640558337558417913
and 3534.518334552443660338233`25. for the integral and error estimates.”
A similar issue with

NIntegrate(1/((x - p)^2 + (y - q)^2 + (z - r)^2)^(1/2)*
Boole(x^2 + y^2 + z^2 <= 1 && p^2 + q^2 + r^2 <= 1), {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, {p, -1, 1}, 
{q, -1, 1}, {r, -1, 1}, AccuracyGoal -> 3, PrecisionGoal -> 3, WorkingPrecision -> 25, 
Exclusions -> {(x - p)^2 + (y - q)^2 + (z - r)^2 == 0})

-1203.034524853306755966531

I wonder negative numbers since the integrand is positive.
Then i switch to MonteCarlo methods.

NIntegrate( 1/((x - p)^2 + (y - q)^2 + (z - r)^2)^(1/2)*
Boole(x^2 + y^2 + z^2 <= 1 && p^2 + q^2 + r^2 <= 1), {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, {p, -1, 1}, 
{q, -1, 1}, {r, -1, 1}, AccuracyGoal -> 2, PrecisionGoal -> 2, WorkingPrecision -> 25, 
Exclusions -> {(x - p)^2 + (y - q)^2 + (z - r)^2 == 0},Method -> "QuasiMonteCarlo")

21.12327556039856680489716

and a warning “NIntegrate::maxp: The integral failed to converge after 50000 integrand evaluations.
NIntegrate obtained 21.1232755603985668048971625. and 0.225183235272419193974785125. for the integral and error estimates.”
A more or less reliable result is obtained by

NIntegrate( 1/((x - p)^2 + (y - q)^2 + (z - r)^2)^(1/2)*
Boole(x^2 + y^2 + z^2 <= 1 && p^2 + q^2 + r^2 <= 1), {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, {p, -1, 1}, 
{q, -1, 1}, {r, -1, 1},  AccuracyGoal -> 2, PrecisionGoal -> 2, WorkingPrecision -> 25, 
Exclusions -> {(x - p)^2 + (y - q)^2 + (z - r)^2 == 0}, Method -> "AdaptiveMonteCarlo")

20.32729987338035891791629

without any warning. Unfortunately, AccuracyGoal -> 3, PrecisionGoal -> 3 is not achieved.
Also this works in eight dimensions:

NIntegrate(1/((x - p)^2 + (y - q)^2 + (z - r)^2 + (w - s)^2)^((4 - 2)/2)*
Boole(x^2 + y^2 + z^2 + w^2 <= 1 &&  p^2 + q^2 + r^2 + s^2 <= 1), {x, -1, 1}, {y, -1, 1}, {z,-1, 1}, 
{w, -1, 1}, {p, -1, 1}, {q, -1, 1}, {r, -1, 1}, {s, -1, 1}, AccuracyGoal -> 2, PrecisionGoal -> 2, 
WorkingPrecision -> 25,Exclusions -> {(x - p)^2 + (y - q)^2 + (z - r)^2 + (w - s)^2 == 0}, 
Method -> "AdaptiveMonteCarlo")

25.78510573458365881830859

BTW, the Exclusions option works in the above: compare with 24.68762920929857902438239 witout it.

The questions arise: what are other methods to calculate such integrals?
is a three-digit result possible with MonteCarlo methods?

In what dimensions can I print a photo shot on Canon EOS 2000D

In what dimensions can I print a photo shot on Canon EOS 2000D at full resolution and save the photo as JPEG in a 3:2 aspect ratio? Let’s assume I want to print it with 600 dpi.

parameters – Different FourierParameters for different dimensions of discrete Fourier transform

This question asks for a way to implement different FourierParameters for different dimensions of a discrete Fourier transform. The question received no answers, but a workaround was mentioned in the comments for the particular case considered by the OP.

I am faced with essentially the same problem, but in my case I would like to use non-integer values for FourierParameters, for which the aforementioned workaround doesn’t help. Given that several years have passed since the linked question was posed, I wonder if there might now be a better solution to the problem of assigning different FourierParameters to different dimensions of a Fourier transform?

How to add missing dimensions to all images to avoid cls?

My sites have lots of external images without dimensions, and that’s causing layout shift.
I need a code which automatically add missing width and height to all images.
I tried 1 plugin Specify Missing Image Dimensions but site become too slow after activating this plugin.
Thanks

Can I use Google Analytics 4 custom dimensions in the Analysis Hub?

In the Analysis Hub:

enter image description here

Can I use a custom dimension I’ve previously added?

enter image description here

For example:

I’ll send the following events and I would like to do analysis based on one of its parameters:

gtag("event", "my_custom_event", {
  my_custom_dimension: "SOME_VALUE"
});

complexity – Finding closest point in N dimensions in reasonable time (O(log(n) ?)

Is it possible to find the closest point (or k closest points) to any arbitrary point in a set of n points (in dimension N)?

When I write closest point, I mean smallest Euclidean distance. I’m looking for an algorithm for multiple searches, where the preprocessing time for the n points is negligible?

Is it possible to get the closest point(s) in a reasonable time (O(log(n)) for example with a method similar to binary search).

Any reference is welcome.

I already read the following questions:

But, none provide a satisfying answer.