Suppose for a data matrix $$X$$ that contains $$n$$ observations with $$p$$ variables and let $$S$$ be its covariance sample matrix. $$S=VDV^{T}$$ by eigendecomposition, where V is called a loading matrix. Am I right in saying that the columns of loadings are principle components of that data matrix? If not, what is the connection between the loadings and principle components?

## Problem with finding the limits for double integral.

Hello i am doubting myself that i am finding the right limits for a problem so if someone can verify my approach i will be thankful. So i have $$\iint_D (xy)dxdy$$ a domain is $$D= (y=x^2, x-y+2-0)$$. From the equation $$x-y+2=0$$ i find that $$y = x+2$$ so that should be my upper limit for $$y$$ and my upper limit should be $$y=x^2$$ after that i substitute $$y$$ with $$x^2$$ in $$y=x+2$$ and i get $$x^2=x+2$$ so the roots $$x_1 = 2$$ and $$x_2=-1$$ so those should be my limits for x. In the end i have : $$\int_{-1}^{2}\int_{x^2}^{x+2}(xy)dydx$$ and the problem is i don’t got the right answer. Thank you for any help in advance.

## PowerShell to find accidental double file extensions

I have some users that occasionally accidentally double append a file extension to a file and have been trying to figure out how to script this in powershell.

E.G. The user names a file “document123.docx.docx” but they have visible file extensions disabled on their machine so they assume the file name is “document123.docx” How can I search for double appended file extensions so that this can be easily identified?

This only affects program code that expects the exact file to be named “document123.docx”and since it doesn’t exist it is unable to continue.

## Double centralizer of derived subgroup

Let $$G$$ be (finite) matabelian group; define $$W(G):=C_G(C_G(G’))$$; $$G’$$ is derived subgroup of $$G$$.

If $$\mathcal{F}$$ is the collection of all maximal abelian normal subgroups of $$G$$ which contain $$G’$$, then for any $$A\in\mathcal{F}$$, since $$C_G(A)=A$$, so we see that $$W(G)=C_G(C_G(G’))\le \cap_{A\in\mathcal{F}} A.$$

Q.1 Is there non-nilpotent metabelian group where $$W(G)$$ is proper subgroup of intersection of all maximal abelian normal subgroups in $$G$$?

Q.2 In metabelian groups $$G$$, what properties about $$W(G)$$ are known? Is there special name to this subgroup?

In the case of nilpotent groups, one can easily obtain example of this nature: take a $$p$$-group $$H$$ in which $$H’$$ is proper in $$Z(H)$$. Then $$W(H)=Z(H)$$; so to push $$W(H)$$ properly in intersection of all maximal abelian normal subgroups, we may look for a $$p$$-group in which maximal abelian normal subgroup is unique. I think, the construction of such $$p$$-group is not difficult; however, I was unable to find an example of non-nilpotent group.

## How to design/architect a double navbar interface using Native JS

A design pattern that I admired is a left side double navbar. When you click on a button on the left, it has two functions. The first is it controls the second navbar section which when clicked, alternates new content which can act as more navigation to displaying some content. The second function the first navbar does is it load a entire new route that only loads in a section like a iframe area.

How would you/the best solution to design this where when you click on a navbar button that loads a new route the second navbar stays on whatever is loaded and not load to a state of a default navbar, which would happen whenever a new route is loaded.

So if I were to click on a option that changes the 2nd navbar and then click on a new route, it would only change to the new route and not impact the 2nd navbar.

## Predicate Logics – double negation HELP me understand

Sorry for maybe a silly question but i need to understand how ¬(¬∀x ¬A(x)) equals ∀x ¬A(x)

In my mind, the negation before the parenthesis will be applied to both ¬∀x and ¬A(x). So it would look like this:

¬(¬∀x ¬A(x)) = ¬¬∀x ¬¬A(x)

A double negation would become positive and would then let ¬(¬∀x ¬A(x)) equal to ∀x A(x).

So, why would ¬(¬∀x ¬A(x)) equal ∀x ¬A(x)? Why wouldnt ¬(¬∀x ¬A(x)) equal ∀x A(x)?

## Why a lot of BST functions returns the root and don’t use double pointer?

I’m writing a BST library that uses a lot of recursive functions that can potentially modify the root of the tree. I noticed that a lot of programmers, even on StackOverflow, often return the root node when a function could potentially modify it.

Why this practice is so diffuse? Use a double pointer to do the same thing isn’t good?

I searched online and found nothing :/

## Intuition behind RDP (Rational Double Points)

Let $$S$$ be a surface (so a $$2$$-dimensional proper $$k$$-scheme) and $$s$$ a singular point which is a rational double point.

One common characterisation of a RDP is that under sufficient conditions there exist a resultion morphsim $$r: \tilde{S}\to S$$ of surfaces such that the fiber $$r^{-1}(s)$$ is an set consisting of rational curves with self-intersection-number $$−2$$. This leads to classification of such RDP’s via Dynkin-diagrams for ADE-curves.

I’m keen interested in geometrical intuition behind the RDP’s. Concretely, my question is if there exist characterisation for a rational double point on the level of it’s stalk $$\mathcal{O}_{S,s}$$ which “reflects” in some way the geometry of this rational double point?

Namely, since $$S$$ is proper and therefore finitely generated $$k$$-algebra, can the stalk $$\mathcal{O}_{S,s}$$ (or maybe at least it’s completion or base change to algebraically closed field) obtain the shape like $$k[x,y,z]_s/(f)$$ where $$f$$ represents the curve/geometry of this singularity? If yes, what kind of polynomial $$f$$ might be?

Why it is called a double point? Does this mean that $$f$$ is contained in in the square $$m_s^2$$ of the unique maximal ideal $$m_s$$ but not in $$m_s^3$$?

Or is it a too naive approach in order to understand intuitively the “nature” of RDP’s?

## Double cosets and the Weber function

Let $$n$$ be an odd positive integer. Let $$\mathcal M_n$$ be the set of all $$2$$-by-$$2$$ primitive matrices with integral entries and with determinant $$n$$.

Let $$\Gamma$$ be the subgroup of $$\operatorname{SL}_2(\mathbb Z)$$ generated by the matrices $$T^2=\begin{pmatrix}1 & 2 \ 0 & 1 \end{pmatrix}$$ and $$S=\begin{pmatrix}0 & -1 \ 1 & 0 \end{pmatrix}$$.

Then $$\Gamma = \bigg \lbrace \begin{pmatrix}a & b \ c & d \end{pmatrix}:\begin{pmatrix}a & b \ c & d \end{pmatrix}\equiv \begin{pmatrix}1 & 0 \ 0 & 1 \end{pmatrix}\text{ or }\begin{pmatrix}a & b \ c & d \end{pmatrix}\equiv\begin{pmatrix}0 & 1 \ 1 & 0 \end{pmatrix}\text{ mod }2\bigg \rbrace.$$

How many cosets are in $$\Gamma \backslash\mathcal M_n/ \Gamma$$ ?.

Let $$r,s,t$$ be positive integers. Supposse that $$rt=n$$, $$s<2t$$, and that $$s$$ is even. Are there matrices $$A,B\in \Gamma$$ such that $$A\begin{pmatrix}n & 0 \ 0 & 1 \end{pmatrix}B=\begin{pmatrix}r & s \ 0 & t \end{pmatrix}$$?

Motivation.

The Hauptmodul for the group $$\Gamma$$ is the function $$\mathfrak f(\tau)^{24}=q^{-1/2}\prod_{k=1}^{\infty}(1+q^{n-1/2}).$$ Let $$\Phi_n(X)$$ be the minimal polynomial of $$\mathfrak f(n\tau)^{24}$$ over $$\mathbb C(\mathfrak f^{24})$$. Is $$\mathfrak f\left(\frac{r\tau+s}{t}\right)$$ a root of $$\Phi_n(X)$$?

## Can a double checked locking be potentially locked by another double checked locking?

I’m trying to optimize my cache system for a custom entity list.

I have a GetEntity method that searches for an entity and when it find nothing, it adds the entity to the cached list. The cached list is available through a read only property named EntityList.

Either methods have a double check system to avoid useless entity reading or adding. The locker are intentionally different to allow a parallel execution of a reading and a getting/add operations.

In spite of this, I’m not sure that the use of the double check in the two crossed methods is always safe. Can I cause a deadlock this way? If yes, how can I avoid it?

public EntityUI GetEntity(int idEntity) {     // Get the entity from the cached list     EntityUI foundEntity = EntityList.Find(currEntity => currEntity != null && currEntity.IdEntity == idEntity);      // If the entity is not available load it from the DB     if (foundEntity == null)     {         var EntityFromDB = ... // [Get the entity from DB]          if (EntityFromDB != null)         {             lock (lockerEntityAdd)             {                 // Double check                 //-----------------------------------------------------------------------------------------------------                 // Here, while I'm locked, I'm reading from the EntityList which is subjected to another lock... can I create the deadlock here?                 // Consider that this piece of code is subjected to several requests per seconds from several users                 //-----------------------------------------------------------------------------------------------------                 foundEntity = EntityList.Find(currEntity => currEntity != null && currEntity.IdEntity == idEntity);                 if (foundEntity == null)                 {                     foundEntity = EntityDaDB;                     // Add the Entity to the cached list                     EntityList.Add(foundEntity);                 }             }         }     }     return foundEntity; }  private List<EntityUI> EntityList {     get     {         object o = HttpContext.Current.Cache[_CacheName];         if (o == null)         {             lock (lockerEntityList)             {                 o = HttpContext.Current.Cache[_CacheName];                 if (o == null)                 {                     o = new List<EntityUI>();                      CacheDependency d = new CacheDependency(GetCacheFile(Constants.CacheFilePath_Entity));                     HttpContext.Current.Cache.Insert(_CacheName, o, d, DateTime.Now.AddDays(Constants.CacheFileExpirationHours_Entity), Cache.NoSlidingExpiration);                 }             }         }         return (List<EntityUI>)o;     } }