Separating Different Types of Buffers in HLSL
Non-interleaved vertex buffers DirectX11
Keyboard layout indicator sometimes disappears Ubuntu 18.04.1 after some update During graduate studies online application submission period, is there any preference given for an applicant if he/she submits his application at the beginning of that period compared with someone who submits his/her application couple of hours before the deadline?  Or as long as the applications are complete there is absolutely no difference! Good day. I'm not too sure how I should go about asking my question, so if it is in anyway confusing, feel free to edit it.   In my Group Theory course we defined and showed a few properties of the commutator subgroup $[G,G]$ of a group $G$. And then we stated and proved a lemma in lecture that I'm having some difficulty wrapping my head around. I'll state it here for clarity.  Lemma a) The commutator subgroup $[G,G]$ of $G$ is a normal subgroup of $G$. b) The quotient group $G / [G,G]$ is the largest abelian quotient of G in the sense that every homomorphism $\varphi : G \to A$ where $A$ is an an abelian group, induces a unique homomorphism $\bar \varphi : G/[G,G] \to A$ so that $\varphi =\bar \varphi \circ \pi$ where $\pi : G \to G/[G,G]$ is the quotient homomorphism.  Now, it's not the proof of this lemma that I'm having trouble with, it's the italicized text that is messing with me. I completely understand what it is that the italicized text is saying, I'm just not too sure why it is that the property gives us the "largeness" that is claimed. I can see that the above forces $G$ to "pass through" $G/[G,G]$ before it "reaches" $A$ (I apologize for the casual terminology), so I can sort of see why $G/[G,G]$ should be the largest quotient, in a non-rigorous sort of way.  Overall I'm very confused. Perhaps I'm missing something very simple. Any help or nudges in the right direction would be much appreciated.   As a side note I did try a search on Google and here on Stack Exchange and couldn't find anything that answered my question. So if I missed something obvious, then my apologies. Thank you in advance. Is PCA unstable under multicollinearity? I know that in a regression situation, if you have a set of highly correlated variables this is usually "bad" because of the instability in the estimated coefficients (variance goes toward infinity as determinant goes towards zero).  My question is whether this "badness" persists in a PCA situation.  Do the coefficients/loadings/weights/eigenvectors for any particular PC become unstable/arbitrary/non-unique as the covariance matrix becomes singular?  I am particularly interested in the case where only the first principal component is retained, and all others are dismissed as "noise" or "something else" or "unimportant".  I don't think that it does, because you will just be left with a few principal components which have zero, or close to zero variance.  Easy to see this isn't the case in the simple extreme case with 2 variables - suppose they are perfectly correlated.  Then the first PC will be the exact linear relationship, and the second PC will be perpindicular to the first PC, with all PC values equal to zero for all observations (i.e. zero variance).  Wondering if its more general. I drew a parallelepiped that is spanned by three vectors, and we know the volume is given by the absolute value of the determinant of the matrix - with the three vectors arranged in rows (or columns, since $detA = detA^T$).  How can I draw the tetrahedron from this parallelepiped to convince myself it is indeed 1/6th of the volume, i.e., 1/6 * |detA|?  Thanks, Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:  Now what is the paradox, exactly? Ignoring any physical intuition behind this (point charge at the origin) how are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically? Or are we forced to believe that because there was a contradiction with the divergence theorem? Also how would the situation differ if $\vec v$ was the same vector function but not for a point charge? Or is it impossible? How do I get the title of a .pdf document to appear as the page title? How to determine whether a unique factorization domain is a principal ideal domain? Could someone please provide an example of a unique factorization domain that is not a principal ideal domain?  Furthermore, is there some way to determine whether a UFD is a PID? Summation of natural number set with power of $m$ Who knows about the summation of this series: $$\sum\limits_{i=1}^{n}i^m $$  where $m$ is constant and $m\in \mathbb{N}$?  thanks Prompted by a  I was pondering the idea of bikes with variable gearing (derailleur or otherwise) but without the ability to freewheel. Have such bikes ever been widely used?  They would appear to share many of the advantages and disadvantages of fixies but be better at handling hills (up and down) while being more complex. Apply SP3 on SQL Server 2012 AlwaysOn AAGs What does a linear transformation do to a vector in $\mathbb{R}^2$ $\newcommand{\Reals}{\mathbb{R}}$Doing some linear algebra. This question has me intrigued. I imagine it causes the vector to rotates the $x$ axis. But I mean, is there more to it?  $$ T_{A(\theta)}: \Reals^2 \to \Reals^2,\qquad A(\theta) =  \left[\begin{array} {rr}\cos \theta &amp; -\sin \theta \\   \sin \theta &amp; \cos \theta \\ \end{array}\right]. $$ What, geometrically, does the linear transformation $T_{A(\theta)}$ do to a vector in $\Reals^2$? Why do I get different outputs in case of sh and bash? Analytics search keywords are unreliable What does it mean if a Python object is "subscriptable" or not? Partitioning $\{1,2,\ldots,k\}$ into $p$ subsets with equal sums, where $p$ is prime Let $p$ be a prime natural number. For which positive integer $k$ can the set $\{1,2,\ldots,k\}$ be partitioned into $p$ subsets with equal sums of elements ?  Obviously, $p\mid k(k+1)$. Hence, $p\mid k$ or $p\mid k+1$. All we have to do now is to show a construction. But I can't find one. I have tried partitioning the set and choose one element from each set but that hasn't yielded anything. Any hint will be appreciated. Derivative of determinant of a matrix I would like to know how to calculate:  $$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big).$$ I understand that the mechanics and design of a lightsaber aren't that complicated.  Do others besides the Jedi and Sith (or their minions) make or use lightsabers?  And if they don't, why not? Indefinite integral of $ \int \frac{x^3}{\sqrt{x^2+1}} \,\text{d}x$. Can you please provide any sort of hint or suggestion on how to find the following indefinite integral?  $$\int\frac{x^3}{\sqrt{x^2+1}}\text{d}x$$  I tried substituting everything but it didn't work. I also tried trigonometric substitution but I couldn't find any valid trigonometric identity for $ \sqrt{\cos^2(x)+1} $ or $ \sqrt{\sin^2(x)+1} $. Can a homomorphic encryption scheme be made CCA2 Secure? Is it possible to modify a homomorphic encryption scheme so that it can be CCA2 secure?  From the definition of a homomorphic scheme, it seems that it is malleable, which would result in lack of CCA2 security. So, if we use the regular method to convert a CPA secure scheme to CCA2 secure scheme (using a hash of the message and a random value), will it still preserve its homomorphic property? Please correct me if my understanding is wrong.