conic sections – How can one verify the equivalence between the geometric and algebraic definitions of an ellipse/htperbola?

I was recently introduced to the world of conic sections, and I came across two different “definitions” for an ellipse and a hyperbola.

I was wondering if there was an easy way to prove that they happen to be “equivalent”.

First, it was established that:

Given a line $r$, a point $F$ not in $r$, and $e in mathbb R^+_*$, an ellipse/hyperbola is the locus of the points $P in mathbb R^2$ such that $overline{PF} = eoverline{Pr}$ (where $e<1$ for the ellipse and $e>1$ for the hyperbola)

What I would like to verify is that the set of points $bigg{(x,y) in mathbb R^2: dfrac{x^2}{a^2} pm dfrac{y^2}{b^2} =1bigg}$ satisfy the definition given.

Any thoughts?

Thanks in advance.

Ella – Multipurpose Shopify Sections Theme

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javascript – Show contents in a book chapter and sections in a highly normalized repository

I want to present books that has chapters and sections, and contents in each book, chapter and section.

This is my schema:

type BookId = string
type ChapterId = string
type SectionId = string
type ContentId = string


export type SourceId =
| ChapterId
| SectionId
| BookId


type Book = {
  id: BookId,
  name: string,
}

type Chapter = {
  id: ChapterId,
  bookId: BookId,
  name: string
}
type Section = {
  id: SectionId,
  chapterId: ChapterId,
  name: string
}

type Content = {
  id: ContentId,
  sourceId: SourceId,
  name: string,
  content: string
}

This is how I query all the books, chapters and sections:

 list() {
    return this.bookRepo.all().then(books =>
      Promise.all(books.map(book =>
        Promise.all((
          this.bookRepo.contents(book.id),
          this.bookRepo.chapters(book.id)
        )).then(((contents, chapters)) =>
          Promise.all(chapters.map(chapter =>
            Promise.all((
              this.bookRepo.contents(chapter.id),
              this.bookRepo.sections(chapter.id)
            )).then(((contents, sections)) =>
              Promise.all(sections.map(section =>
                this.bookRepo.contents(section.id).then(contents => (
                  section, contents
                ))))
                .then(sections => (chapter, contents, sections)))))
            .then(chapters => (book, contents, chapters))))));
  }

I just render as json at the moment, but the query code seems hard to maintain.

at.algebraic topology – Does homeomorphism between cones imply homeomorphism between sections

The answer is no. Let $X$, $Y$ be any two smoothly h-cobordant closed manifolds of dimension $ge 4$ that are non-homeomorphic. For example, we can take $X=S^2times L(7,1)$ and $Y=S^2times L(7,2)$ where as usual $L(p,q)$ is a 3-dimensional lens space. By the weak h-cobordism theorem the interior of the h-cobordism is diffeomorphic to $Xtimesmathbb R$ and also to $Ytimes mathbb R$. The interior is two-ended, and we can compactify it by adding two points (= the ends). The “identity” homeomorphism $Xtimesmathbb Rto Ytimes mathbb R$ extends to
a homeomorphism of the compactifications that takes ends to ends. Removing the points corresponding to one end gives a homeomorphism from the cone over $X$ to the cone over $Y$ that preserves the cone point.

Conversely, if $X$, $Y$ are closed manifolds and $Xtimesmathbb R$ is diffeomorphic to $Ytimes mathbb R$, then $X$, $Y$ are h-cobordant.

The same trick of course works when $X$, $Y$ are compact non-homeomorphic spaces and $Xtimesmathbb R$ is homeomorphic to $Ytimes mathbb R$.

Cognito Forms: Query across repeating sections by ItemNumber

Is there a way to lookup by ItemNumber across repeating section?

This can really help the design when you have large sections that are bound by the number of items but should be logically separate in the design and work-flow. For example, sections get filled out at different times in the work-flow. Think of it as a one-to-one join.

Design:

enter image description here

Enter two items into the form:

enter image description here

The next section should show the items separate, however, it lists both items in both fields:

enter image description here

See: “Referencing repeating data” here:
https://www.cognitoforms.com/support/51/calculations/repeating-sectionstables

=Form.RepeatingSection1.Where(ItemNumber = ItemNumber).Select(UserInput)

This could be a cognito limitation / bug. I can tell that the first ItemNumber in the Where function definitely references any field in RepeatingSection1 because of the drop-down autocomplete list so we would expect that the the second ItemNumber would properly reference the current section.

enter image description here

Can one edit the sections in a Google form after sending it?

Yes you can.

First of all make a copy of your form(a test form), link it to a new answer sheet and submit couple of test replies.
Following that, make the changes you want and submit again some replies.

If you notice no changes to your answer sheet, continue by making the same changes to the original form. Problem solved 🙂

If you do notice differences to the answer sheet, then please follow these steps.

Important note
If I were you, I would first make a note -by checking the already existing timestamps- of when the least activity is registered and take the following steps during a passive period.

  1. Change the logic of your sections
  2. Unlink the answer sheet
  3. Relink your form to the same answer sheet.

Following the above steps a new tab will be created in your answer sheet where new submissions will be recorded.
To have results from both tabs together, create a third tab where by using formulas you can import date both previous tabs.

How to retrieve sections of a form I accidently deleted (including the surveyors responses)?

First time using Google Forms and I was so impressed! It was doing everything I needed for research on a book I am writing. However, Thursday night… on my phone looking at some responses I accidentally deleted the questions (and the responses). I could not find them anywhere and re-created the document but I am now missing my first 40 responses. The updated survey still includes the #40 replies at the top but the answers are not there. Can it be recovered?

google drive – How to retrieve sections of a survey I accidently deleted (including the surveyors responses)?

First time using Google Surveys and I was so impressed! It was doing everything I needed for research on a book I am writing. However, Thursday night… on my phone looking at some responses I accidentally deleted the questions (and the responses). I could not find them anywhere and re-created the document but I am now missing my first 40 responses. The updated survey still includes the #40 replies at the top but the answers are not there. Can it be recovered? Please help!