reference request – Orbit closures of real symmetric bilinear forms

Let $alpha$ and $beta$ be two real symmetric bilinear forms in $operatorname{sym}(mathbb{R}^{n})$, with signatures $(p_{alpha},n_{alpha},z_{alpha})$ and $(p_{beta},n_{beta},z_{beta})$.

Please, I would like to have some references or bibliography about published papers concerning to the following theorem:

Theorem $β∈overline{ operatorname{GL}(n,mathbb{R})⋅α}$ if and only if $p_{alpha}geq p_{beta}$ and $n_{alpha}geq n_{beta}$.

Here, $operatorname{GL}(n,mathbb{R})⋅alpha:={alpha(g^{−1}⋅,g^{−1}⋅):gin operatorname{GL}(n,mathbb{R})}$ and $overline{ operatorname{GL}(n,mathbb{R})⋅α}$ is the closure of $operatorname{GL}(n,mathbb{R})⋅α$ with respect to the usual (Euclidean) topology of $operatorname{sym}(mathbb{R}^{n})$.

fingerprinting – Do emulator programs like Bluestacks, Memu, Genymotion use the real device settings of the computer you’re connecting from?

Many online services use fingerprinting to track users – storing enough hardware and software details to create a unique record.

But what do emulators do? Can they make you completely anonymous on the internet?
Do Bluestacks or Memu use the physical device settings or do they use something else that will help avoid this tracking?

I’m a novice in this world of emulating and I just wish to understand a little bit.

Real Analysis – I don’t have any examples on proving that the infimum of a set is equal to a specific number.

let x be a positive real number and let A = {m+nx: m,n ∈ Z and m+nx>0}. Prove that infA=0

So far I have the following:
Let a = m+nx. A is a set of positive real numbers so 0<a ∀ a ∈ A, thus 0 is a lower bound for A. Since 0 is a lower bound of A, then we say A is bounded by definition. We know that inf A exists by the completeness axiom.

real analysis – First order PDE in complex variables?

You can start by looking at the chain rule for wirtinger derivatives, from which you deduce that

$$
partial_{bar z} exp(h(z)) = exp(h(z)) cdot partial_{bar z} h(z)
$$

Therefore, if you find a function $h$ such that $partial_{bar z} h = – g(z)$ (I think you forgot a “$-$” sign in your solution for the real case!) taking $f(z) = exp(h(z)) $ will solve your problem. In general, this is known as the d-bar problem (or $barpartial-$problem).

Disable SharePoint Online Real Time Save on Excel Document

Is it possible to disable the in browser real time auto save that SharePoint Online has for Excel documents(or any O365 app) that are stored on it? I have a Excel document that I essentially want any time a user goes into it, it is cleared of everything except what was originally there when it was uploaded. Essentially so they can enter in their values in the Excel editor in the browser, snapshot it, close it, and when they come back to it next month it is cleared. Also so that if multiple users have it open in their browser using it all users would NOT see each others changes, in a sense keeping it local to their machine. Then when they close it, it is wiped of what they put in.

Thank you

real analysis – Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?

While solving a problem in calculus of variations, I came to the following question:

Let $A,B$ be two real $2 times 2$ matrices with positive determinants, and suppose that $operatorname{rank}(A-B)=1$.

Let $D subseteq mathbb{R}^2$ be the closed unit disk.
Does there exist a Lipschitz map $u:D
to D$
such that:

  1. $nabla u in {A,B}$ a.e. in $D$
  2. $u$ is surjective.
  3. $u^{-1}(y)$ is a singleton for almost every $y in D$.

By a theorem by Ball and James, any $u$ satisfying condition $(1)$ is of the following form:

Write $B-A=an^T$ for some $a,n inmathbb{R}^2$, $n$ a unit vector. Then
$$
u(x)=Ax+h(xcdot n)a+v_0,
$$

for some $v_0 in mathbb{R}^2$, and a Lipschitz function $h:mathbb{R} to mathbb{R}$ with $h’ in {0,1}$ a.e..

So, it “remains” to examine maps $u$ of the form above.


The assumptions $(1)$ and $(2)$ imply that the average integral of the Jacobian $Ju$ is $1$, so the measure of the set where $nabla u=A$ is predetermined by $det A,det B$.


If it matters, then in my case of application, $A in alpha text{SO}(2)$, where $alpha>1$, and the singular values of $B$ satisfy:
$sigma_1+sigma_2>beta>1$ and $sigma_1 sigma_2 <<1$ very close to zero. Here $alpha, beta$ are some fixed parameters.

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    real analysis – Proving that, if $y_1geq y_2$ at some interval $[x_0, x_1]$, there exists a point where $y_1’>y_2’$

    I am working on an exercise on ODE and I need the following lemma to solve it.
    Lemma: Let two functions $y_1, y_2$: $(x_0, x_1)tomathbb{R}$, such that $y_1(x_0) = y_2(x_0)$ and $y_1(x_1) = y_2(x_1)$. Moreover $y_1(x)>y_2(x) ; forall xin (x_0, x_1)$. Prove that there exists some point $zin(x_0,x_1)$, for which $y_1′(z) > y_2′(z)$.

    It is geometrically obvious, as every point near enough $x_0$ satisfies the condition, but I couldn’t prove it formally. I tried working with local maximums and minimums, but I think there is a more obvious way than considering all options for $max$ and $min$.

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