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# Oracle VM VirtualBox on ARM processor - privilege levels

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I am trying to understand how a type 2 (hosted) hypervisor such as Oracle VM VirtualBox works in detail, specifically on an ARM processor running Ubuntu 22.04.
Based on my research, I understand that ARM architecture employs three privilege levels:
PL-0 for user tasks (lowest level of privilege) and PL-1 and PL-2 for higher privileged components (OS and hypervisor).
As VirtualBox employs a hosted hypervisor, does the OS run on PL-2 and the hypervisor on PL-1?
Thanks for any clarification!

## What exactly does the Remainder Estimate for Integral Test actually mean? \$R_n le int_{n}^{infty}f(x)dx\$

$$int_{n+1}^{infty}f(x)dx le R_n le int_{n}^{infty}f(x)dx$$

What does this actually mean?

Let’s use n=5.

The $$R_5$$ is the error of the partial sum $$S_5$$

That error is less than the sum of the remaining terms from 5 to $$infty$$ ?

Why is that?

Also, why is it greater than sum of the remaining terms from 6 to $$infty$$ ?

## Simple proof for a congruence relation connecting the \$p\$-adic order of a positive integer and a sum of binomial coefficients

Let $$n$$ be a positive integer and $$p$$ be a prime. Let $$v_p(n)$$ be the $$p$$-adic order of $$n$$, i.e., the exponent of the highest power of $$p$$ that divides $$n$$. I would like to know if there is a quick and simple proof for the following congruence relation.
$$sum_{j=1}^{lfloor log_{p} n rfloor} {n-1 choose p^j-1} equiv v_p(n) ;mbox{(mod }pmbox{)}.$$

Key ideas involved in a ‘not so simple proof’ can be found in http://math.colgate.edu/~integers/w61/w61.pdf

Best wishes.

## Continuity of the fractional Laplacian operator

Considering $$(-Delta)^s: H^s(Omega) to L^2(Omega)$$, is it possible to show that this operator is closed (continuous)?
For instance, taking a sequence $$(u_n) subset H^s(Omega)$$ with $$u_n to u$$ in $$H^s(Omega)$$, we need to show that $$(-Delta)^su_n to (-Delta)^su$$ in $$L^2(Omega)$$.

Attempt:

Consider
$$lim_{n to +infty}int_Omega |(-Delta^s)u_n(x)|^2 dx = lim_{n to +infty} int_Omega Big(int_Omega frac{u_n(x) – u_n(y)}{|x – y|^{N + 2s}} dyBig)^2 dx.$$

So my idea would be to use some inequality (perhaps something similar to the Poincaré inequality) to obtain the norm of $$u_n$$ in $$H^s(Omega)$$ on the right side. But, I don’t know how to get rid of this one on the right side of the expression above.

## How to make associated ribbon surface?

I am studying ribbon graphs on surfaces. I am reading this paper (page 5).

But I have confused about how to make surfaces using the ribbon graphs. Actually, we call this surfaces as associated ribbon surfaces.

The definition of ribbon graph is given in definition 1.5 of the above mentioned paper.

Consider the following ribbon graph

To get its associated ribbon surface, first we do its edge refinement by adding each edge a degree two vertex. Then consider the half edges incident to each vertices of the graph.

Then replace the half edges with thin strips with the orientation on boundaries (this orientation of the boundaries follows the orientation of each vertex), which is as follows:

My question is how we connect the strips corresponding to the two half edges following the orientation of their boundaries to form ribbons?

## Characterization of the Poisson distribution

Let $$Z$$ be a $$[0, infty)$$-valued random variable satisfying $$lambda E[g(Z+1)] = E[Zg(Z)]$$ for all indicator functions $$g$$ of Borel subsets of $$[0, infty)$$.

Prove that $$mathcal{L}(Z) = Poisson(lambda)$$.

Hint: Consider $$1_{(n,n+1)}$$ for $$n in mathbb{N}_0$$.

Proof:

Let $$n in mathbb{N}_0$$ and g = 1_{(n,n+1)}, then holds:
begin{align*} lambda E[g(Z+1)] = lambda P[Z+1 in (n,n+1)] = lambda (P[Z+1 = n+1] – P[Z+1 = n]) = lambda P[Z = n] – lambda P[Z = n-1]). end{align*}
But what is
begin{align*} E[Zg(Z)] = E[Z cdot 1_{(n,n+1)}(Z)] = ? end{align*}

Have I got everything right so far? If so, how should I proceed now?

Thanks for the help!

## What is a sufficient and necessary condition for a system of linear equations to have a unique integer solution?

Bézout’s theorem states that a linear diophantine equation $$ax+by=c$$ has integer solutions if and only if $$gcd(a,b)|c$$. Then is there a theorem regarding the existence of a unique integer solution for a system of two linear equations $$ax+by=c$$ and $$px+qy=m$$, where $$a,b,c,p,q,m$$ are positive integers?

Additionally, can the theorem be generalized to the condition where there are more than two unknowns?

## Indices without a given domain in sum notation.

I’m learning about normal modes right now and I have a question pertaining to the way the equations of motion are written vis-a-vis summation notation. The book defines the equations of motion of a system of $$n$$ masses attached to springs, whose displacements are $$q_j$$ as the following: $$underset{j}{sum} (A_{jk}q_j + m_{jk}ddot{q_{j}}) = 0$$My quandary is caused by the subscript $$k$$ that appears in the undetermined coefficients $$A_{jk}$$ as well as the masses $$m_{jk}$$. I just want to know what values of $$k$$ to use here in this equation in the general case. Since it’s not explicitly stated, I’m sure there’s an implicit understanding there.

Let’s take for instance $$n = 2$$. So, we have $$2$$ masses attached to springs and $$2$$ solutions for the displacements (from equilibrium) $$q_j$$ of each mass. Then we must have $$2$$ equations of motion. I’m assuming $$A$$ and $$m$$ are $$2 times 2$$ matrices and then is it the case – in this particular situation – that we have the relation $$delta_{jk}(A_{jk}q_j + m_{jk}ddot{q_j})$$? Such that our equations of motion become $$A_{11}q_1 + m_{11}ddot{q_{1}} = 0\ A_{22}q_2 + m_{22}ddot{q_{2}} = 0$$

If this is true, then for the case $$n gt 2$$ how do we treat the matrices $$A$$ and $$m$$? Do they just become $$n times n$$ matrices? Where the elements on the diagonal are the nonzero ones?

## Unity/C# Unable to access an instanced health system

I am having an issue with a health system I am making for my game. I have been at it for 3 days now and have not made much progress. For reference, I am following this tutorial by code monkey: Code Monkey Health Tutorial. The goal is to have two buttons, one for healing and one to damage the player, and a Text Mesh Pro UI Text to show the health of the player, just for testing purposes. Unlike the tutorial, the buttons and text are just on a canvas, not attached to the player, as I am setting it up as a form of a HUD system. also, forgive the formatting, during troubleshooting, I was more focused on trying to fix my issue and my code became a bit of a mess. Sorry for that.

I have three scripts

using UnityEngine;

public class HealthSystem
{
private int _health;
private int _healthMax;

public HealthSystem(int healthMax)
{
this._healthMax = healthMax;
_health = healthMax;
Debug.Log("healthSystem Created");
}
public int GetHealth()
{
return _health;
}
public void Damage(int damageAmount)
{
_health -= damageAmount;
if (_health < 0) _health = 0;
}
public void Heal(int healAmount)
{
_health += healAmount;
if (_health > _healthMax) _health = _healthMax;
}
}

using UnityEngine;

public class GameHandler : MonoBehaviour
{
private HealthButtons _healthButtons;

private void Start()
{
HealthSystem _healthSystem = new HealthSystem(100);
Debug.Log("Health = "+_healthSystem.GetHealth());
_healthButtons.Setup(_healthSystem);
}
}
}
using UnityEngine;
using TMPro;

public class HealthButtons : MonoBehaviour
{
[SerializeField]
private TextMeshProUGUI numberText;
private static HealthSystem _healthSystem;
private int intToStringHealth;

public void DamageButton()
{
_healthSystem.Damage(10);
Debug.Log(_healthSystem.GetHealth());
}
public void HealButton()
{
_healthSystem.Heal(10);
Debug.Log(_healthSystem.GetHealth());
}
public void Setup(HealthSystem healthSystem)
{
if (healthSystem == null) Debug.Log("healthSystem");
Debug.Log(healthSystem);
_healthSystem = healthSystem;
Debug.Log("Setup is running");
}
private void Update()
{
intToStringHealth = _healthSystem.GetHealth();
numberText.text = intToStringHealth.ToString();
}
}

my issue currently is Im trying to create an instance of HealthSystem in GameHandler, which I am successful in, then pass the information to HealthButtons using _healthButtons.Setup(). This is where I become unsuccessful as I am getting a NullReferenceException error.

from my efforts of troubleshooting currently, I have determined I am not calling the instance correctly, but I do not want to use code monkey’s libraries. I have tried making all classes monoclasses and attaching them to game objects and using SerializeFields for all my variables, which does work, but I am not sure that is best practice as I do know linking everything up in the inspector can be a chore if something goes wrong.

This tells me the solution should be fairly simple, I am just not able to articulate the question into google to get the correct answer. It may be a bit overkill just to test a function, but I would like to understand what I am doing wrong and get better. Thank you very much for your help.

## Prove that there does not exists an estimator for which the risk is \$0\$.

Let $$lbrace mathbb{P}_theta rbrace_{thetain Theta}, Theta subset mathbb{R}$$, be an identifiable parametric family of distributions with common support, where card$$(Theta)geq 2$$. Consider the family of estimators $$Delta = lbrace delta(textbf{X}): mathbb{E}_theta delta^2 and the loss function $$L(theta,a)=(theta-a)^2$$. Prove that there does not exist an estimator $$delta(textbf{X})$$ for which $$R(theta,delta)=0,theta in Theta$$.

So I suppose that there exists an estimator $$delta(textbf{X})$$ for which $$R(theta,delta)=0,theta in Theta$$. Since the loss function is non-negative, then it concludes that $$delta(textbf{X}) = theta$$ almost surely. I don’t know if it is true, because I think that $$theta(textbf{X})$$ cannot take many values. If yes, then what can I do next to show the contradictory? I want to show that for all $$x$$ in $$Omega$$, $$f(x;theta_1)=f(x;theta_2)$$ so it contradicts the definition of identifiable parametric family, but I am not sure how.

## Proof that all Quotient Groups are Abelian- where is my error?

I’m having trouble finding the error in my proof:

Theorem: If H is a normal subgroup of G, then the quotient group G/H is abelian.

Proof: $$forall x, y in G$$, $$xy(yx)^{-1} = xyy^{-1}x^{-1} = e in H$$. By the lemma, $$H(xy) = H(yx)$$, thus $$HxHy= HyHx$$. This shows that the commutative property holds for all the elements in the quotient group G/H, therefore G/H is abelian.

Lemma: Let H be a subgroup of G. Then $$Ha = Hb iff ab^{-1} in H$$. Proof: We start with the $$Rightarrow$$ direction. $$Ha = Hb$$ implies $$a in Hb$$, thus $$a=hb$$ for some $$h in H$$. Therefore $$ab^{-1} = h in H$$. For the reverse direction, if $$ab^{-1} = h in H$$ then $$a=hb$$ for some $$h in H$$. Thus $$a in Hb$$, and since we also have $$a in Ha$$, $$Ha = Hb$$ (because cosets partition a group).

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