## Is DOF in macro depending to method?

I know that the closer the distance to your subject is, the smaller the DOF is. This is true for macro lense, diopters, extension tubes, reverse lenses etc.
But suppose I have a magnification of 1:1, either with a macro lens, diopters, extension tubes or a reverse lens: how is DOF between these methods? Do they all have same DOF using the same aperture?

Best regards, Henry

## The problem

I have a List with loads of rows and I’m doing a workaround in Python to fetch them without exceding the list view threshold. So far I managed to do this by limiting the ID range of the results, like so:

``````<...>/_api/Web/lists/GetByTitle('<ListName>')/Items?\$select=ID&\$top=5000&\$orderby=ID&\$filter=ID ge 1 and ID le 5000
``````

And then I change the numbers of ‘ID’ column from 1~5000 to 5001~10000, and so on…

## Some Queries

So this query here works fine:

``````<...>/_api/Web/lists/GetByTitle('<ListName>')/Items?\$select=*&\$top=5000&\$orderby=ID&\$filter=ID ge 20001 and ID le 25000 and TextField eq 'Some text'
``````

And I manage to get 5000 rows from each request made like this (even though there are more than 5000 rows with ‘Some text’ inside the ‘TextField’ column).

But if I try this one here:

``````<...>/_api/Web/lists/GetByTitle('<ListName>')/Items?\$select=*&\$top=5000&\$orderby=ID&\$filter=ID ge 20001 and ID le 25000 and DateField gt '2020-09-12T00:00:00'
``````

It raises me the limit exception. And if I change the date to a day that I know it’ll rerturn less then 5000 rows, it runs just fine.

## In Sum

I’m limiting the range with the ID column of SharePoint list so it doesn’t exceed the view threshold, but it doesn’t seem consistent.

This is the template i’m using, adding things at the end, and only changing the range until the list is over:

``````<...>\$filter=ID ge 1 and ID le 5000 and <other query options...>
``````

Any ideas on why SharePoint thinks I’m trying to fetch more than 5000 rows at once? Thanks in advance!

## 2d – Object to reach height depending on its gravity and mass

A matemathician will probably think this was a silly question, but I’m not one and I’m in need of help 😀

I’m working in Godot and with it’s `RigidBody2D`s (objects that have automated physics behaviour).
I want to create a spring of type `RigidBody2D` (the ‘springyness’ is not the only function of this object).

I’m at the point in code where this spring and another `RigidBody2D` collide so I have all information about the two bodies at the moment of collision.

My goal is to make any body that touches the spring reach a certain height (from the spring), and move by certain speed horizontally, as you can see in the picture

So, the information about both bodies I have is: position, velocity, gravity scale, mass, weight.
What I want to achieve is constant line of movement of any object (that may have different properties I mentioned above) by applying a certain force to the colliding object.
EDIT: forgot to mention that I can stop the body before applying new force to it, that is, at the time of collision.

Rotation, friction, absorbtion and similar properties don’t need to be counted on as the game I’m making is a simple one. With this I’m trying to make a predictable movement for the player to count on. I also plan to disable collision briefly at the time of collision so that the colliding object can touch the spring from any angle and the collision result would be the same.

## System wanted: Next question depending on multiple previous answers

I want to write an application where the user has to answer several questions. The difficulty is that the third question might depend on the answers of question 1 and 2. For example:

1. question A: possible answers A1 / A2 / A3
2. question B: possible answers B1 / B2 / B3
3. a. question C only if the previous answers where (A1 or A2) and B1
4. b. question D for all other cases (if A3 or B2 or B3)

I am looking for an existing system that’s dealing with this problem. This system should have an open source code base, so I can transform these steps/ideas into the programming language I want.

Do you have any idea what could help me here?

(I didn’t know what tags to set. So the ones I used might be wrong.)

## uml – xtUML: Next question depending on multiple previous answers

I want to write an application where the user has to answer several questions. The difficulty is that the third question might depend on the answers of question 1 and 2. For example:

1. question A: possible answers A1 / A2 / A3
2. question B: possible answers B1 / B2 / B3
3. a. question C only if the previous answers where (A1 or A2) and B1
4. b. question D for all other cases (if A3 or B2 or B3)

The questions and the possible answers should be written in some file, so they could be exchanged easily. And we need some code that always knows which question to ask next. I was thinking of xtUML but couldn‘t find anything that suits our needs.

Our code is going to be written in Perl. But anything (also in other programming languages) is welcome. Maybe someone thought this through and we can transform his ideas.

Do you have any idea what could help me here?

## google sheets – I want a custom conditional formatting to fill a color for a cell depending on the number value after the decimal point

I am entering both positive and negative numbers in a column and want a way to identify them without an extra column for example -100.03 or -250.1 or 100.015. in this example I am listing moneyline betting odds and want to use the .1 to identify inside distance and .03 to mean over 3 rounds, and .015 to mean over 1.5 rounds. This will give me the type of prop bet even though i just have the number entered without the need for an extra column to identify it. I was thinking about regular expressions but not sure

## linear algebra – basis of k-vector space V depending on characteristic of K

Let $$K$$ be a field and $$V=K^3$$
$$B = lbrace f_1 = (1,2,3),f_2=(-1,2,4),f_3=(2,1,5)rbrace$$
for what characteristic of K is $$B$$ a basis of $$V$$.

I am not really sure how to approach this question besides trying different cases for the characteristic of K but that does not seem to be the right way to approach this question for me.

## gr.group theory – Bound for order of a group depending on number of elements of maximal order

This question has been partly answered in MSE, see here.

In a paper On the Number of Elements of maximal order in a Group it is proven that an arbitrary group $$G$$ with a finite number of elements of maximal order has bounded size. Namely: $$|G|leqfrac{mk^2}{varphi(m)},$$ where $$m$$ is the maximal order and $$k$$ the number of elements that have order $$m$$.
I wanted to characterize all groups $$G$$, where the limit is sharp, i.e. $$|G|=frac{mk^2}{varphi(m)}$$. Using GAP I found all groups with this property up to order 1023 and was able to state a conjecture. It is easy to see in the paper, that a group has this property only if all elements of maximal order are conjugated. So we need this as as a requirement.

I wanted to prove the following conjecture, but missing some tiny part. Maybe someone knows a way, I would be really happy.

Conjecture.
Let $$G$$ be a group with $$k elements of maximal order $$m$$, in which all elements of maximal order are conjugated. Then the following are equivalent.
$$i)$$ $$|G|=frac{mk^2}{varphi(m)}$$
$$ii)$$ $$k=varphi(m)$$
$$iii)$$ $$G$$ has a unique subgroup of order $$m$$
$$iv)$$ $$C_m cong C_G(x)=C_G(y)trianglelefteq G$$ for all $$x,yin G$$ with maximal order

Proof.
$$i) implies ii)$$
This is the part, I could not prove:
I only could prove, that all elements of order $$m$$ commute:
Let $$C_G(x)$$ be the stabilizer of an element of maximal order. Orbit-Stabilizer-Theorem tells us, that $$|C_G(x)|=frac{mk}{varphi(m)}$$. Assume there exists an element of order $$m$$, not contained in $$C_G(x)$$. $$langle x rangle$$ operates via left-multiplication on $$C_G(x)$$. $$C_G(x)$$ is partitioned into $$frac{|C_G(x)|}{m}$$ orbits. According to Lemma 3 of the paper linked above, in each orbit exist at least $$varphi(m)$$ elements of order $$m$$, i.e. in $$C_G(x)$$ exist at least $$varphi(m)frac{|C_G(x)|}{m}$$ elements of order $$m$$. Our assumption tells us $$varphi(m)frac{|C_G(x)|}{m} < k$$, which leads to the contradiction $$|C_G(x)| < frac{mk}{varphi(m)}$$. It follows that all elements of order $$m$$ commute.
From here Derek Holt contributed a good point:
All elements of order $$m$$ commute, the elements of order $$m$$ generate an abelian normal subgroup $$N$$ of $$G$$ of exponent $$m$$. We now prove for an arbitrary $$g in G$$ with order $$m$$, that $$C_G(g) = N$$.
Let $$g in G$$ have order $$m$$. We claim that $$C_G(g) = N$$. To prove this, let $$h in C_G(g)$$. We want to show that $$h in N$$. This is clear if $$h in langle g rangle$$. Otherwise, since $$m$$ is the largest order of any element in $$G$$, $$langle g,h rangle$$ is a $$2$$-generator abelian group of exponent $$m$$, and so it is equal to $$langle g rangle times langle hg^i rangle$$ for some $$i$$ with $$0 le i < m$$. But then $$hg^{i+1}$$ has order $$m$$ and hence lies in $$N$$, so $$h in N$$, which establishes the claim.

So $$(G:N) = (G:C_G(g)) = k$$, and hence $$|N| = mk/phi(m)$$.

This is where we could not proceed further, maybe someone has an idea?

$$ii) iff iii)$$
If $$k=varphi(m)$$, an element of order $$m$$ generates a cyclic subgroup which contains $$varphi(m)$$ elements of order $$m$$, that all generate this subgroup. So there can’t be other elements of order $$m$$ in different subgroups. Otherwise, if there is only one cyclic subgroup of order $$m$$, then it contains $$varphi(m)$$ elements of order $$m$$, no additional elements of order $$m$$ can exist, as they would generate a second cyclic subgroup of order $$m$$.

$$iii) implies iv)$$ Let $$Z$$ be the unique subgroup of order $$m$$ and $$X={x_1,dots,x_k}$$ the set of elements of order $$m$$. As all $$xin X$$ generate $$Z$$, $$Z$$ must be contained in all centralizers of elements in $$X$$. Note that $$G$$ operates on itself via conjugation. Orbit-Stabilizer-Theorem tells us for $$x in X$$: $$|G|=|^Gx||G_x|=k|G_x|=frac{mk^2}{varphi(m)}=mk$$ This follows as all elements of order $$m$$ are conjugated and $$k=varphi(m)$$ holds. It follows, that $$|G_x|=m$$, which leads to $$G_x=Zcong C_m$$ for all $$x in X$$.
For the normal subgroup part, note that $$phi(x_i)=x_j$$ for an inner automorphism $$phi$$ and $$i,jin {1,dots k}$$. Let $$y in Z$$ be arbitrary, then $$y=x_1^alpha$$ for $$alpha in mathbb{N}$$. Let $$phi$$ be an arbitrary inner automorphism. It follows that there is a $$i in {1,dots k}$$ with $$phi(y)=phi(x_1^alpha)=phi(x_1)^alpha=x_i^alpha in Z$$ It follows that $$Z$$ is invariant under inner automorphisms, i.e. normal.

$$iv) implies i)$$ Orbit-Stabilizer-Theorem tells us that $$|G|=|^Gx||G_x|=mk$$. As all stabilizers of elements of order $$m$$ are equal to the same cyclic group of order $$m$$, it follows, that there exist only one cyclic group of order $$m$$, it follows $$k=varphi(m)$$ and $$|G|=mk=frac{mk^2}{varphi(m)}$$.

Another property, which my GAP-study suggests to be equivalent is :
$$v)$$ $$G’$$ is cyclic
This proof has low priority, as I first want to have my circle-implications. I guess I can show, that $$G’$$ is contained in the unique cyclic group $$Z$$ of order $$m$$, by proving, that $$G/Z$$ is abelian. I did not succeed yet, though.

## 8 – How to delete message entities (message stack) of a special template depending on entity age

for the “Private Message” module, my Drupal 8 site uses the “Message” module (https://www.drupal.org/project/message) as substructure.

I would like to build a kind of notification center into the page, which should be possible to implement with the message stack (as I heard).

The trigger to generate such a message will probably be a custom module.
The message module also offers a kind of “auto cleanup”, but unfortunately as far as I have seen only for all messages and not per message template.

In principle I only want a notification center, which is triggered at certain events and which cleans itself depending on the message type (e.g. only the last 10 messages of a certain template or delete all messages of a certain template which are older than 10 days).

In general, I guess cleanup does not work, as it would probably also delete the “private messages”.

Any hints.