pr.probability – Why does the three points follow by making the two assumptions about the conditioned intensity function?

The intensity function is defined as

$$lambda^*(t)=frac{f(t|H_{t_n})}{1-F(t|H_{t_n})}$$
where $f$ is the density function and $F$ is the distribution function, and $H_{t_n}$ is the history of all the previous points of $t$ up to $t_n$.
Moreover, there is proven that $F(t|H_{t_n})$ is also given as:

$$F(t|H_{t_n})=1-e^{-int_t^{t_n}lambda^*(s)ds}$$.

An assumption is then made, saying that

  1. $lambda^*(t)$ is non-negative and is integrable on every interval after $t_n$.

  2. $int_t^{t_n}lambda^*(s)ds to 1$ for $t to infty$.

It is then said that hence the three points follows:

  1. $0 leq F(t|H_{t_n}) leq 1$
  2. $F(t|H_{t_n})$ is a non-decreasing function of $t$.
  3. $F(t|H_{t_n}) to 1$ for $t to infty$

Can someone explain to me how these three points follow given the two assumptions above? Thank you.

Is there a decomposition of a given subset of Euclidean space into interior/accumulation/limit/boundary points?

Let $A$ be a subset of $mathbb{R}^n$ with the Euclidean topology. Is there a decomposition of $A$ into interior points, isolated points, accumulation points or limit points?

sharepoint enterprise – Need help with layout using custom CSS and JS. Want to lock down page and have web part zones at certain points

Apologies for the awful title.

I’ll try to explain this.

I have a standard page which was blank. On this I added some custom CSS and JS and make an accordion.
I am using this as a base.

The first problem I run into is if I edit the page and make changes (having to expand the accordion to get to certain bits of text) the accordion breaks and jams. It stays open and no longer works, so I have to roll back the page.

The only way I got around this was using Designer to just add text in. This causes more troubles as I’m not a coder. So when I try to add formatting, bold, tables or bullets in Designer, it’s all greyed out. Sometimes it isn’t. So confusing.

I suppose what would be handy perhaps is to have the accordion “locked” on the page. Then in each of the sections it has a web part zone. Then I could add my own custom parts when I needed them, without fear of the accordion breaking. If I need to change titles or icons, I could dive into designer to make those minor changes.

Any advice and help here is appreciated.

computational geometry – For two sets of points find if second one is result of linear transformation of the first

Say we have two sets of points in vector-2 space (In actuality need to solve this problem in vector-3 space but decided to start with a simpler problem). The points in the second set are the result of transformation of the points in the first set. It’s not known which point in the first set corresponds to which point in the second set. What needs to be calculated is if the points of the second set can be seen as the linear transformation of the first set. So basically finding whether there exists a 2 by 2 matrix, which if applied to the first set gives the second set. And if so calculating it.

Should a problem like this be solved using computational geometry, like sweep line algorithm to compare all points between sets?

Adding Button to add points to another sheet – Basketball Live Scorebook

I have created a scorebook sheet in google sheet for tracking live basketball stats.
One sheet with the scorebook table for the team with each player one a row.

I want to have a separate tab for data entry that populates the scorebook table.
This entry sheet will list the 5 players on the floor (column A will have jersey numbers which can change when players subs in)
I want to have a series of buttons to add 2pts/3pts/fouls/etc to the player listed in column A to the appropriate row on the scorebook table.

We have an app that does this beautifully, but it requires downloading a csv file to get the table data. I want to create a similar experience with a google sheet that fills in real-time.

My thought was to create buttons for each change. I have enough formula knowledge to know how to create a vlookup formula to look at column A and the row to find the player. But I need to make this a clickable button to change the data on the fly.

Any suggestions or script that I can add to the button would be helpful

Are all critical points either infection points, local minimum or local maximum?

Question:

  • Let $f$ be a differentiable function. If the point c is a critical number, then either it is a local maximum, or local minimum, or an inflection point. $T/F$ ?

My opinion:

  • If c is a critical point then f'(c)=0 or undefined. So it may local maximum and local minimum.

  • If f ‘(c)=∞ then c is inflection point at the same time and if f ‘(c)=0 it may inflection point again.

  • But i can’t find instance disproves this thesis.

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Selling Doge. Can buy any bmf point amount according to this rule: 5300 bmf points for 1000 doge. no bergain.

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distributed system – Designing an ETL with where there are a few points of entry

I’m trying to think of a scalable solution for my current system.
The current system is

3 microscopes
1 processing machine

 - 60-100GB Files come from 2-3 microscopes every 30 minutes
 - That data is transferred to a (local) network mount of the processing machine
 - The processing machine runs and contains the ETL(airflow)

Scaling issue

Right now it currently works well.
I am concerned in the future that as the demand and load (size of file, processing times, etc..) increases we may face bottleneck(s). I was thinking of using a cluster of machines (via cloud computing or buy a couple more machines), but our network is not the fastest, maybe transferring around 100-200mbps. I worry with distributed computing the transfer speed would nullify the benefit of multiple machines.

Current thinking

I’m considering an idea where a group of machines are in a queue, if the top of the queue is not busy then the microscope can transfer the initial file to that machine and the rest of the process(2-3) can run as normal. I’m just wondering if this is a sane approach or if there is anything I can improve on.

plotting – Achieving similar effects as RegionPlot using discrete points

I have the following data in the format of {x1,y1,z1}:

{{0.05, 0, 2.09412}, {0.1, 0, 2.00505}, {0.15, 0, 1.8692}, {0.2, 0, 
  1.70208}, {0.25, 0, 1.51965}, {0.3, 0, 1.33654}, {0.35, 0, 
  1.16343}, {0.4, 0, 1.01005}, {0.45, 0, 0.883653}, {0.5, 0, 
  0.792333}, {0.6, 0, 0.794668}, {0.65, 0, 1.08459}, {0, -10, 
  2.11716}, {0.05, -10, 2.08704}, {0.1, -10, 2.00062}, {0.15, -10, 
  1.5764}, {0.2, -10, 1.69942}, {0.25, -10, 1.1256}, {0.3, -10, 
  0.915311}, {0.35, -10, 0.724784}, {0.4, -10, 0.557347}, {0.45, -10, 
  0.414329}, {0.5, -10, 0.295847}, {0.55, -10, 0.201322}, {0.6, -10, 
  0.129954}, {0.65, -10, 0.0809156}, {0.7, -10, 
  0.0535506}, {0.75, -10, 0.0474673}, {0, -20, 2.12459}, {0.05, -20, 
  2.01208}, {0.1, -20, 1.84956}, {0.15, -20, 1.81342}, {0.2, -20, 
  1.64175}, {0.25, -20, 1.45434}, {0.3, -20, 1.26322}, {0.35, -20, 
  1.07861}, {0.45, -20, 0.544381}, {0.5, -20, 0.625104}, {0.55, -20, 
  0.335852}, {0.6, -20, 0.265945}, {0.65, -20, 0.218091}, {0.7, -20, 
  0.330485}, {0.75, -20, 0.186575}, {0, -30, 2.12495}, {0.05, -30, 
  2.04312}, {0.1, -30, 1.95586}, {0.15, -30, 1.81408}, {0.2, -30, 
  1.6472}, {0.25, -30, 1.46717}, {0.3, -30, 1.28524}, {0.35, -30, 
  1.11062}, {0.4, -30, 0.949941}, {0.45, -30, 0.807552}, {0.5, -30, 
  0.685852}, {0.55, -30, 0.586033}, {0.6, -30, 0.508438}, {0.65, -30, 
  0.452949}, {0.7, -30, 0.419327}, {0.75, -30, 0.40741}, {0, -40, 
  2.12564}, {0.05, -40, 2.07402}, {0.1, -40, 1.97611}, {0.15, -40, 
  1.80464}, {0.2, -40, 1.68348}, {0.25, -40, 1.51231}, {0.4, -40, 
  1.02099}, {0.45, -40, 0.886038}, {0.5, -40, 0.771047}, {0.55, -40, 
  0.677185}, {0.6, -40, 0.604868}, {0.65, -40, 0.554097}, {0.75, -40, 
  0.51663}, {0, -50, 2.12519}, {0.1, -50, 2.00847}, {0.15, -50, 
  1.88592}, {0.25, -50, 1.57604}, {0.3, -50, 1.41193}, {0.35, -50, 
  1.20883}, {0.4, -50, 1.06172}, {0.45, -50, 0.979901}, {0.5, -50, 
  0.870339}, {0.55, -50, 0.781138}, {0.6, -50, 0.712876}, {0.65, -50, 
  0.665729}, {0.7, -50, 0.639627}, {0.75, -50, 0.603863}, {0, -60, 
  2.12294}, {0.05, -60, 2.11458}, {0.1, -60, 2.06337}, {0.15, -60, 
  1.9396}, {0.2, -60, 1.80301}, {0.25, -60, 1.65146}, {0.3, -60, 
  1.49609}, {0.35, -60, 1.34578}, {0.4, -60, 1.20679}, {0.45, -60, 
  1.08345}, {0.5, -60, 0.978466}, {0.55, -60, 0.893153}, {0.6, -60, 
  0.828311}, {0.65, -60, 0.766473}, {0.75, -60, 0.757909}, {0, -70, 
  2.1163}, {0.1, -70, 2.09201}, {0.15, -70, 1.99933}, {0.25, -70, 
  1.73465}, {0.3, -70, 1.58807}, {0.35, -70, 1.44493}, {0.4, -70, 
  1.31189}, {0.45, -70, 1.19181}, {0.5, -70, 1.09252}, {0.55, -70, 
  1.01063}, {0.65, -70, 0.906969}, {0.7, -70, 0.885735}, {0.75, -70, 
  0.885036}, {0, -80, 2.11109}, {0.05, -80, 2.16128}, {0.1, -80, 
  2.13834}, {0.15, -80, 2.06322}, {0.2, -80, 1.95322}, {0.25, -80, 
  1.82325}, {0.3, -80, 1.68531}, {0.35, -80, 1.54924}, {0.4, -80, 
  1.4219}, {0.45, -80, 1.30806}, {0.5, -80, 1.2108}, {0.55, -80, 
  1.13193}, {0.6, -80, 1.07251}, {0.65, -80, 1.03302}, {0.7, -80, 
  1.01374}, {0.75, -80, 1.01484}, {0, -90, 2.1012}, {0.1, -90, 
  2.18573}, {0.15, -90, 2.12946}, {0.2, -90, 2.03395}, {0.25, -90, 
  1.91529}, {0.3, -90, 1.78634}, {0.35, -90, 1.65728}, {0.4, -90, 
  1.53539}, {0.45, -90, 1.44954}, {0.5, -90, 1.33206}, {0.55, -90, 
  1.256}, {0.6, -90, 1.19888}, {0.65, -90, 1.16138}, {0.7, -90, 
  1.1439}, {0.75, -90, 1.14658}, {0, -100, 2.08892}, {0.05, -100, 
  2.21053}, {0.1, -100, 2.23328}, {0.15, -100, 2.19699}, {0.2, -100, 
  2.11682}, {0.25, -100, 2.0099}, {0.3, -100, 1.93021}, {0.35, -100, 
  1.76802}, {0.55, -100, 1.38212}, {0.6, -100, 1.32714}, {0.65, -100, 
  1.29147}, {0.7, -100, 1.27558}, {0.75, -100, 1.27976}, {0, -110, 
  2.07483}, {0.05, -110, 2.22384}, {0.1, -110, 2.29696}, {0.15, -110, 
  2.26525}, {0.2, -110, 2.20105}, {0.25, -110, 2.10633}, {0.3, -110, 
  1.99567}, {0.35, -110, 1.88055}, {0.4, -110, 1.76944}, {0.45, -110, 
  1.70653}, {0.5, -110, 1.58073}, {0.55, -110, 1.50977}, {0.6, -110, 
  1.45684}, {0.65, -110, 1.4229}, {0.7, -110, 1.40855}, {0.75, -110, 
  1.41406}, {0, -120, 2.05932}, {0.05, -120, 2.24053}, {0.1, -120, 
  2.32622}, {0.15, -120, 2.33349}, {0.2, -120, 2.28603}, {0.25, -120, 
  2.20381}, {0.3, -120, 2.15022}, {0.4, -120, 1.88875}, {0.45, -120, 
  1.7915}, {0.5, -120, 1.70719}, {0.55, -120, 1.63861}, {0.6, -120, 
  1.58767}, {0.7, -120, 1.54239}, {0.75, -120, 1.54925}, {0, -130, 
  2.04175}, {0.05, -130, 2.25467}, {0.1, -130, 2.37059}, {0.15, -130, 
  2.4013}, {0.25, -130, 2.30201}, {0.3, -130, 2.2105}, {0.35, -130, 
  2.10961}, {0.4, -130, 2.0092}, {0.45, -130, 1.9159}, {0.5, -130, 
  1.83461}, {0.55, -130, 1.7684}, {0.6, -130, 1.71929}, {0.65, -130, 
  1.68857}, {0.7, -130, 1.67701}, {0.75, -130, 1.68508}, {0, -140, 
  2.02214}, {0.05, -140, 2.26615}, {0.1, -140, 2.41313}, {0.15, -140, 
  2.46827}, {0.2, -140, 2.45649}, {0.25, -140, 2.40065}, {0.3, -140, 
  2.31894}, {0.4, -140, 2.13038}, {0.45, -140, 2.04116}, {0.5, -140, 
  1.96286}, {0.55, -140, 1.8989}, {0.6, -140, 1.85167}, {0.65, -140, 
  1.82245}, {0.7, -140, 1.81224}, {0.75, -140, 1.8215}, {0, -150, 
  2.00046}, {0.05, -150, 2.27555}, {0.1, -150, 2.45351}, {0.15, -150, 
  2.53417}, {0.25, -150, 2.49932}, {0.3, -150, 2.42779}, {0.35, -150, 
  2.39361}, {0.4, -150, 2.25218}, {0.45, -150, 2.16694}, {0.5, -150, 
  2.09159}, {0.55, -150, 2.03005}, {0.6, -150, 1.98459}, {0.65, -150, 
  1.95688}, {0.7, -150, 1.94792}, {0.75, -150, 1.95841}, {0, -160, 
  1.97679}, {0.05, -160, 2.28305}, {0.1, -160, 2.49248}, {0.15, -160, 
  2.59871}, {0.2, -160, 2.62551}, {0.25, -160, 2.59793}, {0.3, -160, 
  2.53685}, {0.35, -160, 2.45838}, {0.4, -160, 2.37431}, {0.5, -160, 
  2.2209}, {0.55, -160, 2.16152}, {0.6, -160, 2.1179}, {0.65, -160, 
  2.09163}, {0.7, -160, 2.08399}, {0, -170, 1.95107}, {0.05, -170, 
  2.28874}, {0.1, -170, 2.52693}, {0.15, -170, 2.66171}, {0.2, -170, 
  2.70897}, {0.25, -170, 2.69625}, {0.3, -170, 2.64592}, {0.35, -170, 
  2.57521}, {0.4, -170, 2.49675}, {0.45, -170, 2.41973}, {0.5, -170, 
  2.35045}, {0.55, -170, 2.29341}, {0.65, -170, 2.22678}, {0.7, -170, 
  2.22043}, {0.75, -170, 2.23318}, {0, -180, 1.92335}, {0.05, -180, 
  2.29265}, {0.1, -180, 2.55992}, {0.15, -180, 2.72297}, {0.2, -180, 
  2.7915}, {0.25, -180, 2.79413}, {0.3, -180, 2.75487}, {0.35, -180, 
  2.69217}, {0.4, -180, 2.61944}, {0.45, -180, 2.54656}, {0.5, -180, 
  2.4804}, {0.55, -180, 2.47704}, {0.6, -180, 2.38554}, {0.65, -180, 
  2.3622}, {0.75, -180, 2.37093}, {0, -190, 1.89362}, {0.05, -190, 
  2.29474}, {0.15, -190, 2.78236}, {0.25, -190, 2.89155}, {0.3, -190, 
  2.86372}, {0.35, -190, 2.80908}, {0.4, -190, 2.74222}, {0.5, -190, 
  2.61049}, {0.55, -190, 2.55803}, {0.65, -190, 2.49787}, {0.7, -190, 
  2.49395}, {0.75, -190, 2.55036}}


I would like to plot a density plot with 2 distinct colours, one for the region with $z<2.2$, and one for the region with $z>2.2$.

Something like this, where the green region refers to points with values greater than 2.2. and the red region refers to points with values smaller than 2.2

enter image description here

Is it possible to achieve a similar effect with discrete points?
Thank you.