## How to convert X,Y Cartesian data to 3D spectrum (frequency, direction, spectral density)?

My question is as follows. I have x,y motion. normally I would ignore directionality and just look at the spectral density of each by performing a FFT on it. For my current problem directionality is crucial. I would like to create a 3D spectrum, frequency, direction and spectral density. I presume I would convert from cartesian to polar coordinates and then convert to frequency domain. Is this the best method and any help on how to do this wold be great. My google skills cant seem to find anything. Thank you

## Finding the probability density function of a random variable in two dimensions

Let $$(X,Y)$$ be a point chosen at random from the triangle $$\{x,y:0\leq x\leq y\leq 1\}$$. $$f_{X,Y}(x,y)=2$$ if $$(x,y)$$ is in the triangle, and it is 0 otherwise. Find the probability density function for $$X$$.

What confuses me about this problem is understanding how $$x$$ and $$y$$ make up the triangle. If I’m understanding this correctly, then the biggest triangle we can make has vertices $$(0,1),(1,1),$$ and $$(0,0)$$. If this is the case then the probability should be 1. As $$x$$ decreases, $$y$$ can at most be $$x$$ which means that $$y$$ is dependent on $$x$$. From this though I’m not sure where to go from here.

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Looking for colocation in the North West of the United Kingdom

Need a lot of power per rack to accommodate blades

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## Density of polynomials in $L^2$ on the unit ball

Are polynomials dense in the Lebesgue space $$L^2$$ of the unit disk?

References are very welcome, thank you.

## How to best improve information density for long, complex forms?

How to layout long, complex forms such that the largest possible number of fields is visible on the page, but legibility is maintained?

I am re-designing an enterprise web app where there are a total of 30x fields across 5 sections when creating a quote, and where an extra 10-20x fields get added depending on the state of the quote.

The client has expressed a strong preference for seeing as many form fields as possible within each screen without having to scroll, (this request seems sensible but I am yet to test)

This is what the form looks like:

My initial attempt was to introduce clear-er hierarchy and group the inputs into further sub-sections:

But I think the UX could be further improved by flowing the content within each tab vertically (and using the sidebar to present a summary) instead of horizontally + vertically.

Are there any other patterns that could be suitable here ?

What other ways are there of keeping a large number of form fields visible ?

## Asympotic density of a very simple sequence

Let $$A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $$a(n)=O(n^{1.5})$$, but not quite.

I’m actually even more interested in the asymptotic behavior of the sequence given by $$AA$$, the set of products of two elements of $$A$$.

Any ideas on how to approach these questions?

For background, there are monic fully reducible cubic polynomials $$P,P+a \in \mathbb{Z}[X]$$ if and only if $$a\in AA$$. Fully reducible means that $$P$$ and $$P+a$$ are both products of 3 linear terms.

## Z scale density plot color

I have a problem with my Z scale color… I would like to saturate the colors in order to see better the white ring… When I am doing the next line code PlotRange -> {{-0.25, 0.25}, {-0.25, 0.25}, {0, 20000}} instead of PlotRange -> {{-0.25, 0.25}, {-0.25, 0.25}, {All, all}}. It becomes better but I have a very strong white features in the middle which is not normal…

So how to change the Z scale color to saturate the colors and see better the features….

Here my line code

SetOptions[ListDensityPlot,     Mesh -> None,     PlotRange -> {{-0.25, 0.25}, {-0.25, 0.25}, {0, 50000}},     ColorFunctionScaling -> True,     Frame -> True];  Original[file_String /; FileExistsQ[file]] :=   Module[{data, dataT}, data = Import[file, "Table", HeaderLines -> 2];     dataT = Transpose[data];     dataT = {dataT[[1]]*10, dataT[[2]]*10, dataT[[3]]}; (*      Subscript[q, y] and Subscript[q, z] in nm *)     data = Transpose[dataT];      graph1 = ListDensityPlot[data,     PlotLabel -> FileBaseName[file],     ScalingFunctions -> {"Linear", "Linear", "Linear"},       ColorFunction -> (ColorData["TemperatureMap"][#] &),       PlotLegends ->         BarLegend[Automatic,          LegendLabel -> Placed[ Rotate["\[CapitalDelta]\!$$\* StyleBox[\"\[CapitalSigma]\",\nFontWeight->\"Plain\",\n\ FontSlant->\"Italic\"]$$/\[CapitalDelta]\!$$\* StyleBox[\"\[CapitalOmega]\",\nFontWeight->\"Plain\",\n\ FontSlant->\"Italic\"]$$ (Counts/Exp time)", -Pi/2], Right],          LabelStyle -> Directive[Black, FontSize -> 52],          LegendMarkerSize -> 400,          LabelingFunction -> (Style[NumberForm[#, {Infinity, 3}],              Black, 52] &)]];      CreateDocument@graph1;          ;     ;   ] 

## Convergence of probability density function

There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.

Let $$X_1, X_2, \dots, X$$ be random variables $$\Omega \to \mathbb{R}$$ and $$f_1, f_2, \dots, f$$ be those density functions $$\mathbb{R} \to \mathbb{R}$$. The convergence of density functions is $$\lim_{n\to\infty}f_n = f$$ (a.e.) or $$\lim_{n\to\infty}\int_\mathbb{R}|f_n-f|=0$$.

I think it would be interesting, for example, to prove an analog of the central limit theorem not on cumulative distribution functions but on density functions.

Question: Is there a research on convergence of density functions?

## Langage density what is it used for

If we have a langage $$L$$ over an alphabet $$\Sigma$$, then we can defined the density function of $$L$$ as :

$$p_L(n) = \mid L \cap \Sigma^n \mid$$

I am wondering why it’s useful to study this function and what informations it gives on $$L$$ ?

Moreover we can then defined the entropy of $$L$$ as :

$$E_L(n) = \frac{1}{n} \log p_L(n)$$

Once again I am wondering what are the informations thus function gives on $$L$$ ? Moreover is there any way to think intuitively what the entropy of a langage really represent ?

Thanks you !