## Fourier series and zeta function

Does anyone know which function this Fourier series comes from? $$sum_ {q = 1} ^ infty frac { cos (2 pi q x)} {q ^ {1+ alpha}}$$ Seems like one $$zeta (1+ alpha)$$ but with cosine at the top … is there any connection? The parameter is $$0 < alpha <1$$. Thanks for any advice!

## Unit – How do I program a series of customizable changes to a game (architectural / organized)?

The bottom of my game (tower defense) consists of a series of dice. I want to change this on some waves. I have a list of changes that need to be made for each shaft in a floorController class. However, to decide which floor cubes should appear or disappear in each round, I had to assign a bool to each cube for each wave (if this is set to true, the floor in that wave changes). However, this process is lengthy and not easy to change and adjust to experiment with the level. How can I structure this system better from the perspective of architecture programming?

## Database design – what is the best way to store multidimensional time series data in a (R) DBMS environment?

Any suggestions on how to store and organize multidimensional time series data in an RDBMS environment? For example, we have multiple experiments from multiple customers. This can be translated well with a NetCDF file type structure, in which a NetCDF can be used for a customer project and contains several data variables over time. This picture explains the concept.

According to my preliminary investigations, this seems to be an unsolved problem or a problem with a complicated solution. Have there been any recent (not necessarily optimal) attempts to do this?

## Express the inner product of a complete orthonormal sequence as a series.

To let $$(x_ {n}) _ {n = 1} ^ infty$$ be a complete orthonormal sequence in a Hilbert space H. Show that

$$langle x, y rangle = sum_ {n = 1} ^ { infty} langle x, x_ {n} rangle overline { langle y, x_ {n} rangle}$$

for all $$x$$, $$y in H$$.

## Design – How can I save time series data like Google Analytics, Facebook etc.?

I'm thinking about it in terms of a No SQL database, more specifically MongoDB. So I want to create something like Google Analytics, in which I record a lot of data and when it occurs so that I can show how often X has happened in this year / month / week / day / hour / etc. As with Google Analytics, only small data is saved next to the time stamp. There will also be several points from which I take data, e.g. B. how there are different administrators in Google Analytics with their own dashboard. I will give an example.

There are 3 "administrators" with their own dashboard and website. Then we have to collect data from each website. For the sake of simplicity, we will only collect data if a user leaves a comment on their page and if the comment contains an image.

I originally had it like that, but of course there is no timestamp

{
},{
},{
}


Then I changed it

{
time: [{
time: "3/28/2020 - 6:45:36 am",
},{
time: "3/28/2020 - 6:45:37 am",
},{
time: "3/28/2020 - 6:46:10 am",
}]
},{
time: [{same structure as above}]
},{
time: [{same structure as above}]
}


But I imagine this would get insanely large and is not a good way to save the data. Not to mention that collections have a maximum storage space of 16MB, which sounds like something I would pass if I saved it that way? How time Array would be very large. This is just 3 comments, imagine there are 10 comments per second and 100 "admins". Within a minute time The array would contain 100 elements and I would have done 1000 writes to the database.

If anyone has any suggestions on how this data can be stored well so that I can easily retrieve and store it, I would love to hear it! Especially when it is easier to retrieve data such as from the last day, the last week, etc.

## Time series analysis – An algorithm to detect whether noisy univariate data is constant or the sum of the step functions

In an explicit algorithm that I am writing, there is a certain phase in which I have to determine whether certain noisy univariate data is constant or the sum of step functions.

Example: Define foo as the algorithm I'm looking for (writing in Python):
assert foo ((0) * 20) == false
assert foo ((0) * 20 + (1) * 20) == True
claim foo ((0) * 20 + (5) * 30 + (1) * 70) == True

• The data in the examples are not noisy, but assume that they are real (only a little can be determined by observing the graph of the data exactly where a step could take place.

I would appreciate ideas, thank you.

## Sequences and series – Finding point in a partition (Riemann-Stieltjes)

Consider the problem of calculating the Riemann-Stieltjes integral $$int_ {5} ^ {9} fd alpha$$ for the functions $$f (x) = 7x ^ 2-3$$ and $$alpha (x) = x ^ { frac {1} {4}}$$. The interval $$(5.9)$$ is to be divided into a partition consisting of 10 sub-intervals $$(x_ {j-1}, x_j), j = 1, …, 10$$ With $$x_j$$ follow a geometric sequence. So $$x_0 = 5$$, $$x_ {10} = 9$$ and $$x_j = rx_ {j-1}$$ to the $$j = 1, …, 10$$ for a suitable selection of $$r> 1$$.

For such a partition $$x_4$$ =?

Find $$x_4$$ we have to find $$r$$So use the information provided $$x_ {10} = r ^ {10} x_0$$therefore $$r ^ {10} = frac {9} {5}$$ and $$r = sqrt (10) { frac {9} {5}}$$.

Then $$x_4 = x_0r ^ {4} = 5 left ( sqrt (10) { frac {9} {5}} right) ^ {4} = 5 left ( frac {9} {5} right ) ^ { frac {2} {5}}$$.

Is that correct? I feel like I missed something. (Sorry for the formatting).

## Binomial Series problem

Prove that
$$lim_ {n to infty}$$[$$sum_ {i, j} {2i choose i} {2j choose j}$$]$$^ {1 / n}$$= 4
so that, $$i + j = n$$
and $$0 le i, j le n$$
My idea was to summarize the series for the first time, though $$i and when $$j .
But I can't go on.

## Sequences and series – evaluation of the summation with counting formula

But avoid

• Make statements based on opinions; Support them with references or personal experiences.

Use MathJax to format equations. MathJax reference.

## Pagination – How can I display a series of items up to the cover viewport without having to refresh the page when the user changes the window size?

I have to load several elements like a list with javascript.

I need to fill the viewport as needed because I have to load more items when the user reaches the bottom of my container. To do this, I need to know how many elements can fit in the viewport.

I fill the container with this code

  for(let i=0; i window.cardsContainer.offsetHeight){
break;
}
}


However, this code is interrupted when the user resizes the window. If the window is smaller and then larger, the functionality is interrupted because the user never reaches the bottom by scrolling

I think when reloading the page when the user resizes, but I think that's a bad idea