quadratic forms – Is where a trigonometric field which is different enough from real numbers?

I found this topic in a book ‘Metric Affine Geometry’ by Ernst Snapper and Robert J. Troyer.
I call a field $k$ trigonometric iff there is a quadratic form $q$ over $k^2$ such that every two lines through the origin in $k^2$ is isometric with respect to $q$. This condition is sufficient to introduce trigonometric functions over $mathbf{SO}(k^2,q)$ in a geometric fashion. Hence, a name.

Obviously, $mathbb{R}$ is trigonometric. I know, that to be trigonometric the field $k$ must Pythagorean, that is for every finite sequence of values $(alpha_i)^n_{i=1} in k^n$ there is a $gamma in k$ such that
sum^n_{i=1} alpha_i^2 = gamma^2,

namely every sum of squares is a square. Secondly it must be a formally real field, which means that $-1$ is not a sum of squares. Hence, sadly $mathbb{Q},mathbb{C},mathbb{R}(x),mathbb{Q}_p,mathbb{F}_p$ are all not trigonometric. Probably some extension of $mathbb{R}(x)$ which allows square roots of formally positive functions may work. But I still doubt that it can be totally-ordered, and probably there are some clews in differential Galois theory. Maybe $hat{mathbb{Q}} cap mathbb{R}$, where $hat{mathbb{Q}}$ are algebraic numbers will work, or just adjoining enough real algebraic square roots to mathbb{Q} (call it a Pythagorean closure $overline{mathbb{Q}}$). At least it is Pythagorean and formally real. But I don’t think it is interesting enough.

But I’m very a curious about finding an interesting example of trigonometric field different from $mathbb{R}$. Trigonometric field $k$ different from $mathbb{R}$ may mean formally that $k$ is not between $overline{mathbb{Q}}$ and $mathbb{R}$. I would be very grateful if you could suggest one.
If the result are negative, this would mean that class of all trigonometric fields has certain lower and upper bounds.

forms – Selected radio button shows user more content

I tried to search info for my question but I didn’t found it. I think I just can’t find right search words because this doesn’t feel to be very rare case. If you have topics about this I would like to check it.

Case example:
User can choose one option out of 3 options. At least one of the options provide more content related to selected option.

Where this new content should be shown and why? How would this type of situtation done so it’s accessible?

I made this sketch to make my question more clear.

  • Example 1 shows new content between radio buttons
  • Example 2 shows new content under the radio buttons


sharepoint online – How we can prevent a List which has its Create,Edit,Display forms customized using PowerApp from been rendered using classic UI forms

I have the following:-

  1. custom list inside a classic team sub site
  2. the custom list has its create,edit & display form customized using power app.
  3. the custom list has the following setting to also render it using New Experience:-
    enter image description here

so i thought that the list will always render using power app forms, but what is actually happening is that the users will get the option to render the list using classic UI, as follow:-
enter image description here

So now all the business rules which i have applied inside the power apps forms will have to be applied to the classic UI as well which is not logical..
so my question is how i can prevent a List which has its Create,Edit,Display forms customized using PowerApp from rendering using classic UI forms?

second question. i have a modern communication site and i have the same list and settings, and the users will not have the option to render the list using classic UI,, so is my problem because i am using classic Team Sub-site?

Can anyone advice/help with my above 2 questions please?

forms – International Phone Number Field Layout and Design

While I agree with @jbreckmckye to ask for the country, not the country’s telephone number as that can be inferred from the country, I do think that most people know the telephone number of their country and can spit it out at will.

That said, I would not split the telephone number in separate fields at all.

I would ask for the for the country in a separate field like @jbreckmckye because the country also dictates how an address should be laid out (some countries have the postal code before the city, others have it after the city or even completely separate).

I would allow the telephone number to be entered in a free format. It may restrict what you can do with it “automatically” (or require a very good parser to enable automatic processing). More importantly though, it will allow users to enter their phone number in the way they are used to entering phone numbers instead of forcing them to conform to a format they may not be familiar with.

As a suggestion you could give an example of an international phone number adjacent to the telephone number field.

+31 (0)30 1234567

This example also shows the way international phone numbers are/should be formatted to take care of the dropped zero @PhillipW mentions.

In general terms the international format for a telephone number actually is

+<country> ((0))<area> <subscriber>


  • + signifies the international exit code which differs per country;
  • the (0) is optional (indicated by the square brackets) and signifies the zero of the area code which you need to drop when dialing from outside of the country (indicated by the brackets) but need to dial if you are in the country.

tracking – Using Google Forms is it possible to collect a list of respondents that is separate from their anonymous answers?

new to this particular Stackexchange with a Google Forms question?

I need to a submit a survey to members in our organisation. I have built this survey in Google Forms. I need to record a list of respondents that remains unlinked to each member’s anonymous responses, and members should only respond once. How can this be done?

In Google Forms, I can require an email to track respondents, but in Responses > Summary there is a list of names at the top and then the breakdown for each question. However in Responses > Individual I can see how each respondent answered.

Another attempted answer to this question in this forum suggested removing the required email, and adding a Name field, but it still shows a list of each respondents answers under the Responses > Individual tab. So the name is still connected to their survey responses.

I would also like to allow each email to only respond once to the survey, without requiring a Google Sign In.

Is it possible to build the survey as I’ve done, keep it anonymous with no email requirement or name field, but perhaps send each email in our organisation a unique URL that somehow tracks who responded, separately from their answers in Google Forms?

Any ideas or approaches to solving this? I would be very grateful. Thank you!

Google Forms/ Sheets – Web Applications Stack Exchange

Taking into consideration the little info you give, I imagine that if your response sheet (Form Responses) looks something like this:

|      Timestamp      | Last name | First name |           Email           | Steps |
| 2020-10-19 23:25:51 | Boc       | Florencia  | fboc9@vinaora.com         |   356 |
| 2020-10-19 22:33:29 | Boc       | Florencia  | fboc9@vinaora.com         |   175 |
| 2020-10-19 19:58:08 | Splevins  | Donny      | dsplevins7@ucla.edu       |   153 |
| 2020-10-19 18:40:52 | Castagne  | Cedric     | ccastagneb@ow.ly          |   382 |
| 2020-10-19 17:56:40 | Gipp      | Eloisa     | egipp4@webmd.com          |   147 |
| 2020-10-19 17:26:39 | Splevins  | Donny      | dsplevins7@ucla.edu       |   365 |
| 2020-10-19 17:09:17 | Riccardi  | Monica     | mriccardi6@wikispaces.com |   289 |
| 2020-10-19 17:07:08 | Boc       | Florencia  | fboc9@vinaora.com         |   284 |
| 2020-10-19 16:54:45 | Gipp      | Eloisa     | egipp4@webmd.com          |   174 |
| 2020-10-19 14:20:50 | Riccardi  | Monica     | mriccardi6@wikispaces.com |   233 |
| 2020-10-19 14:02:00 | Splevins  | Donny      | dsplevins7@ucla.edu       |   173 |
| 2020-10-19 09:51:38 | Riccardi  | Monica     | mriccardi6@wikispaces.com |   177 |
| 2020-10-19 04:49:03 | Splevins  | Donny      | dsplevins7@ucla.edu       |   320 |

If so, you then need to create a new sheet and place the following QUERY formula i cell A1

=QUERY('Form Responses'!A1:E, 
    "select B, C, sum(E) where A is not null group by B, C ",1)

(Please adjust ranges to your needs)

enter image description here

If still in need please let us know.

forms – Can a 2-column grid of info be acceptable if it mitigates an awkwardly large amount of white space?

enter image description here

I’m in the middle of a psuedo-overhaul of my company’s platform, with an emphasis on “object-level” pages (ie the page for an individual task, or individual appointment).

A problem I’m running into is that the info for many of these pages is laid out in a two-column format (see pic), which I know any competent ux designer will tell you is bad for scanning. But while combining both columns into one single left-aligned column might improve scannability, it would also leave just a gaping void of whitespace on the right portion of the page. I can’t really articulate any ux code this violates beyond just looking ridiculous and stark, but it still seems like an issue worth surfacing.

Any thoughts on this? I want to follow best practices but I also would like to preserve a sense of visual balance and appeal.


ag.algebraic geometry – Can we attach (formal) abelian varieties to $p$-adic modular forms?

The Jacobian of the modular curve $X_1(N)$ over $mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke cuspforms $f$ of level $N$ of weight $2$. $A_f$ is the abelian variety of dimension $(E_f:mathbb Q)$ associated to $f$ by decomposition of the Hecke algebra, where $E_f$ is the coefficient field of $f$.

Can we do similar things for (overconvergent) $p$-adic modular forms of classical weight $2$ by a limiting process? Maybe the dimension will be larger and larger, but can we at least define things over $mathbb Q_p$? At least can we construct a local Galois representation?

This is really a soft question, thank you very much.

sharepoint online – Can we move/copy a customized list forms (using power app) from one tenant to another

Yes, we can do this using the export and import of the powerapps package. We need to follow the below steps to do that:

  1. Export the app from the source environment
  2. Import the app into the target environment

Export Process

Step 1:

Open the PowerApps form in the browser. Go to the Apps tab. Right-click on the Application which you want to move. You will get the context menu Click on the “Export-Package“.

Step 2:

In the Export package page enter the Name and Description and click on Export.

Step 3:

Now we will get our PowerApps form in our local system.

Import Process:

Step 1:

Open the new tenant in the browser. Go the Apps tab. Click on the “Import-Package” from the top.

Step 2:

Click on the “Upload” in the Import-Package.

Step 3:

Select on the Zip file from the local system which you want to import.

Step 4:

Click on the Update. In the Setup option click on the “Create as new” option.

Step 5:

Click on save. When we have clicked on the Create as new the save button will be enabled.

Step 6:

Click on the Import in the Review Package Content Page.

Step 7:

We will get our imported App in the Apps tab.

For complete details steps, you may refer the below MSDN article:

Environment and tenant app migration through Packaging

Export PowerApps from one Tenant to another Tenant


After importing the package in the target environment, open the form verify the data connection.

forms – UX pattern for implementing calendars

In my recent development I am required to implement a calendar view which is having month view for each month for given date range. See below.

enter image description here

I have already done the programming part. It is required allows users to select all Sundays, all Mondays, all Tues days etc.. The existing swing UI like this. (Hope this will help to get an idea) see below.

enter image description here

As you can see both Sundays and Tues check boxes are selected and marked in calendar as well. Calendar days are are not clickable and only the check boxes can be selected.

Pure idea is to allow users to selected week/end days and see them on calendar view. Calendar days are not clickable. First thought came to mind is implementing the same behavior as given above for web UI as well.

Hope you all thoughts on this.