at.algebraic topology – $pi_{2n-1}(operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible?

Question: Is the element $alpha$ in $pi_{2n-1}(operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?

My understanding so far β€”

An $operatorname{SO}(2n)$ bundle over $S^{2n}$ corresponds to an element in $pi_{2n}operatorname{BSO}(2n) =pi_{2n-1}operatorname{SO}(2n)$.

Not torsion: There does not exist any integer $m > 0$ such that $malpha$ is a trivial element.

Indivisible: There does not exist any integer $k > 1$ and any element $beta$ in $pi_{2n-1}operatorname{SO}(2n)$ such that $alpha=kbeta$.

Ref: Mimura, Toda: Topology of Lie groups. Chapter IV Corollary 6.14.

multivariable calculus – Can we obtain the one parameter function that represents the arguments progression on gradient progression of a multiple parameters function?

Assume that $(a_1,a_2,…,a_n) in Bbb R^n$ and the

$F: Bbb R^n rightarrow Bbb R$
$(x_1,x_2,…,x_n) longmapsto F(x_1,x_2,…,x_n)$

is differentiable function at all parameters. Is there any math tool (operator, method…) that gives

$f: Bbb R rightarrow Bbb R$
$p longmapsto f(p) = F(x_1(p),x_2(p),…,x_n(p))$

such that $p=0$ acts like $a_1,a_2,…,a_n$

$x_i(0) = a_i, i=1,2,…,n Longrightarrow$
$f(0) = F(x_1(0),x_2(0),…,x_n(0)) = F(a_1,a_2,…,a_n)$

and other $p$ values describes the arguments progress on gradient like parametric function

$(x_1′(p),x_2′(p),…,x_n'(p)) = nabla F(x_1(p),x_2(p),…,x_n(p)), p in Bbb R$?

Is it line integral? Is it gradient flow? How to do it?

user meta – Which admin color represents what?

In wordpress, the admin color scheme has 7(basic, more or less) colors, the first 4 represent “colors”, and the last 3 “base”, “focus”, “current”. In the first 4 colors, which color represents what?

In the default-fresh scheme in array 1 color # 1d2327, represents the background, and in sunrise 2 colors # cf4944 in the array represents the backround. Why is this so and how to get and insert color for text and hover etc …

Also the colors for hover and text are mixed, how to understand the admin color scheme?

Fresh 1. color = #1d2327 = background

    wp_admin_css_color(
        'fresh',
        _x( 'Default', 'admin color scheme' ),
        false,
        array( '#1d2327', '#2c3338', '#3582c4', '#72aee6' ),
        array(
            'base'    => '#a7aaad',
            'focus'   => '#72aee6',
            'current' => '#fff',
        )
    );

Sunrise 2. color = #cf4944 = background

    wp_admin_css_color(
        'sunrise',
        _x( 'Sunrise', 'admin color scheme' ),
        admin_url( "css/colors/sunrise/colors$suffix.css" ),
        array( '#b43c38', '#cf4944', '#dd823b', '#ccaf0b' ),
        array(
            'base'    => '#f3f1f1',
            'focus'   => '#fff',
            'current' => '#fff',
        )
    );

Is there any way to find a set of points that represents all the remaining points?

Suppose, We have been given a set of 100 points and the distance matrix for all pair of points, how should 20 points be picked such that they best represent all the remaining points.

reference for: no finite set of positive (integer) binary quadratic forms represents all primes

This recent question asks for a set of forms (binary quadratic) representing all primes.
Set of quadratic forms that represents all primes

When the question was asked on MSE last month

https://math.stackexchange.com/questions/3820129/non-linear-forms-for-all-prime-numbers

I made the claim that no finite set of positive binary forms would suffice. This still seems right to me, but I lack a proof or any reference. The subject is traditional, I would guess there is a mention in, say Dickson’s History, which I do have. I will check.

Let’s see, this will take some time, but there is no problem writing a Manjul Bhargava style “truant” program, begin with $x^2 + y^2,$ prime $3$ missing says add $x^2 + 2 y^2,$ then $7$ missing says add $x^2 + xy + 2 y^2,$ and so on. Eventually i would expect to see some non-principal forms as the smallest absolute discriminant form.

functional programming – Trying to understand how this class representation truly represents Natural numbers in Scala

Following Martin Odersky’s course on coursera – Functional Programming with Scala and I’m on Week 4 where we’re learning about Types and Pattern Matching. In the video lecture, this is the representation of a Natural Number:

abstract class Nat {
  def isZero: Boolean
  def predecessor: Nat
  def successor: Nat = new Succ(this)
  def + (that: Nat): Nat
  def - (that: Nat): Nat = if (that.isZero) this else (predecessor - that.predecessor)
}

object Zero extends Nat { // for a zero
  def isZero = true
  def predecessor = throw new NoSuchElementException
  def + (that: Nat) = that
}
class Succ(n: Nat) extends Nat { // for non-zero (positive) numbers
  def isZero = false
  def predecessor = n
  def + (that: Nat) = n + that.successor
}

My questions are:

  1. when I create a val two = new Succ(2) why would I set the two.predecessor = 2 when 2‘s predecessor is actually 1?

  2. When I call two + new Succ(4) internally why am I evaluating 2 + successor of 4 and not new Succ(2 + 4)?

  3. In the main abstract class Nat, the successor field is intialised with a Succ object. Wouldn’t the value of successor be the same as the object that was just constructed?

I’m just …unable to grasp the relationship/implementation here …

PS – I do have a Java background if that helps

buttons – What icon intuitively represents making comparisons?

I want to make an app which involves repeatedly giving users two options and having them select the one they like better. Is there any icon or symbol which would intuitively represent this concept, which I could place next to the word “compare” on a navigation button? I’m wracking my brain and can’t come up with anything, which suggests that there may not be a natural icon choice…

polynomial time – Decide if a string is a member of a language that represents $P$?

For some enumeration of the complexity class P (such as this as an example: How does an enumerator for machines for languages work?), for each string 𝑝 in the enumeration, does there exist some other string (certificate) 𝑐 that allows you to verify 𝑝 is a member of the enumeration in poly time? I believe that it might be possible in poly time because all we have to do is check if a string fits some certain format (format of the encoding)?

A decision problem $P$ is poly time verifiable iff there is an algorithm 𝑉 called verifier such that if $P(w)=$π‘ŒπΈπ‘† then there is a string $c$ s.t. $𝑉(w,c)=$π‘ŒπΈπ‘†, if $P(w)=𝑁𝑂$ then for all strings $c$, $𝑉(w,c)=$𝑁𝑂 and V runs in $O(w^{k})$ for some constant $k$ for all inputs $w$.

terminology – What text most clearly represents “Download all that have not already been downloaded” (for a button or link)?

If I have a table of items to download, some of which may already have been downloaded, what is the most succinct and clearest phrase to use for a button or link which represents “download all the items which I have not already downloaded”?

So far, the best I have is Download All Missing but I wonder if there are common phrases that most people already use? Have you seen examples of clearer terminology?

In the below mockup I would expect clicking the ‘Missing’ link to download A and C.

enter image description here

What type of diagram best represents the flow of a recording process?

It's difficult to label the name of this diagram, but you're basically looking to visualize the flow. This can be called a "flowchart", a "user flow", a "user interface flow" or a "use case" ux (different from the "use case" ; a programmer).

A good example of an organizational chart can be found here:

https://www.lucidchart.com/documents/edit/4543-dcc8-5123936c-8c53-10720a005798#?demo=on

And a good article on a faster way to sketch it is available here:

http://37signals.com/svn/posts/1926-a-shorthand-for-designing-ui-flows

I don't think the schema you are using really matters, what is important is that you can see what needs to be done and put everyone on the same page. Personally, I prefer an organization chart because it has well-defined symbols that create a standardized language to convey the necessary steps to designers, project managers, programmers and others. Anyway, for registration, I think the conditional is an important aspect to show what decisions the user will have to make and where it will lead them. Also remember that some conditional will be decided by logic and not by the user.