metrics – how to measure software performance

I have to study software metrics for a competition, I’ve found a lot of things but I’m really confused, could you suggest me which metrics are used to measure performance in software systems and when you should prefer one to another?

pr.probability – Is the weak* limit of Girsanov measures also a Girsanov measure?

Let $mu_0$ be the standard Wiener measure on $C[0,T]$. Let $mu_n$ be a sequence of measures with $mu_nll mu_0$ for all $n$ and so that the weak$^ast$ limit of $mu_n$ exists, call it $mu$. Is it true that $mu ll mu_0$?

I know for general measures this is not true. For example we can have a Gaussian with variance $varepsilon$ and send $varepsilon to 0$.

But what about for Girsanov measures?

measure theory – What does it mean for data in $((Ttimes mathbb{R})^k)^i$ to be close to data in $((Ttimes mathbb{R})^k)^j$?

Let $T subset mathbb{R}$ be compact (think (0,1)).
Suppose you have a dataset in $((Ttimes mathbb{R})^k)^j$, such as j collections of k pairs of (time,measurement), where the $k$ pairs are in time order. Let’s call this dataset X. Suppose we have a second dataset similarly structured but with $i$ collections of $k$ pairs instead. What does it mean for these datasets to be close? I’m need a single number that can tell me how close they are.

If you want a more concrete example: suppose you have $j=55$ subjects whom participated in a study. They were given alcohol and their BAC was measured about every 5 minutes for 2 hours (so at 4 minutes 5 seconds, 11 minutes 2 seconds, so on), so they had $k=25$ measurements . The study was repeated on a different group of $i=95$ people. Suppose we want to claim the two groups data were similar. How would you compare their data? My goal is to say that if they are very close, then we can make certain conclusions, so I am interested in a measure that ensures they are close.

I was thinking something along the lines of: for each individual interpolate the data so that you have a function for each person from time to BAC. Shift the functions in time so that the peaks align (if one person starts drinking much later than another, I want to account for this). Average the functions, say $X(t)$ and $Y(t)$ averaged, shifted interpolated measurements from the first and second group respectively. Then consider $$max_{tin T}|X(t)-Y(t)|.$$

If this is small then they are “close”. Now this “metric” for closeness is not ideal, since a group that had half people with 0 BAC the whole time and the other half at .2 BAC the whole time would be considered close to a group that had all people with .1 BAC. Similarly a measure of deviation wouldn’t account for mean. Is there something I could measure that would gaurantee closeness of the data. One problem I have is that there is deviation within each single dataset and I’m not sure how to account for that.

I think the Z test might closer to what I’m looking for, but I’m going to try looking into that more. Any help is appreciated. Thanks.

nt.number theory – Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset of $X$ with complement $Z$.

We can endow $X(F)$ with the Serre-Oesterle measure (e.g. as in (1,Section 2.2) or (2, Section 7.4))–this is just the standard measure coming from a top form of $X$).

My question is then whether one knows a simple proof/reference for the following:

The subset $Z(F)$ of $X(F)$ has measure zero.

I think this is proven in (1, Lemma 2.14)–but this is concerned with a more specific context which makes it non-ideal as a reference.

Any help is appreciated!

(1) http://www.math.uni-bonn.de/people/huybrech/Magni.pdf

(2) Igusa, J.I., 2007. An introduction to the theory of local zeta functions (Vol. 14). American Mathematical Soc..

measure theory – When is the inside of a Jordan curve open?

I’m working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this bounded component is always open. But in pathological cases, like an Osgood curve which has positive measure, clearly the inside cannot be open since it does not contain an open ball (I think).

Are there examples of Jordan curves with measure $0$ that don’t have an open inside? Do Jordan curves with positive measure never have an open inside? More importantly, if the inside is open, does it guarantee that the curve is “non-pathological”?

EDIT: Perhaps my intuition was wrong. According to MO user Timothy Chow in this post, “The Jordan curve theorem was strengthened by Schoenflies to the statement that the two components are homeomorphic to the inside and outside of a circle.” By Brouwer’s invariance of domain theorem, this implies that the inside component of a Jordan curve is open, if I understand everything correctly.

measure theory – Prove equality of these two sigma algebras

Let $B$ be the Borel sigma-algebra over $R$ (real numbers).
Let $Gsubset R$ be a borel-set. And $A0$ the family of all subsets of $G$ which have the form $Gcap O$ for $O$ being an open subset of $R$.

Let $A1$ be the sigma algebra over $G$ generated by $A0$

and $A2$ = {$Xin B$ | $X subset G$}

How to show that $A1 = A2$?
I would be especially interested in the direction $A2 subset A1$

How does YouTube measure watch time in case of accelerated playback?

Through YouTube Studio, a creator gets many audience statistics about the videos of the channel. Among those measurements, there is the watch time typically reported in hours. I am wondering how those measurements interact with the Playback speed feature of the YouTube viewer. More specifically, if a visitor watch a 10min video at 2x playback speed in 5min (wall clock), does this event count as a 5min or 10min view event for YouTube Studio?

light – measure luminanace with lux meter

In the right situation luminance can be converted to illuminance. But I don’t think you have the right conditions… the light source has to be diffused/integrated (such as with an integration sphere) and the only source of light.

In that case the formula for cd/m² is Luminance = (illuminance x reflectance)/π

You might be better off using a camera and converting the exposure value to the luminance value.
https://www.ee.ryerson.ca/~phiscock/astronomy/light-pollution/luminance-notes-2.pdf

surveys – Is it appropriate to use System Usability Scale to measure a KPI?

On several sites I find people recommending the use of SUS as a key metric to keep an eye on.

I can’t really find any information on whether people use it continuosly to measure a KPI (in this case, the KPI being measured would be perceived usability of the website).

The tool/ website in question is a client web with limited new traffic and the use of SUS would mean probing the user base with SUS a few times per year.

Any thoughts or insights into this would be greatly appreciated!

real analysis – Properties of set of positive Lebesgue measure in $mathbb{R}^2$

Let $A,Bsubset mathbb{R}$ be such that they are positive Lebesgue measure in $mathbb{R}$. Let $Csubset mathbb{R}^2$ is nowhere dense, measure zero set in $mathbb{R}^2$.

Does there exists $E,F subset mathbb{R}$ satisfying $$E×Fsubset A×Bsetminus D,$$
such that $E,F$ are positive Lebesgue measure in $mathbb{R}?$