If $X$ and $Y$ are identically distributed, not necessarily independent, show that $cov(X+Y,X-Y)=0$
show that Cov(X+ Y, X-Y)= Var(X) - Var(Y)
I use a Windows 10 machine remotely at work. Both the local and remote machine are in the same ISP network (not same local network, but same ISP). I've been using it like that for almost a year now and I had never experienced absolutely any issues.  Until recently. I don't know if anything specific triggered this behavior, but I feel that it started happening after I upgraded the remote to W10 1903. I later upgraded the local one as well and I feel that it only made the problem worse (but I don't really have any data).  From time to time, the session will freeze - the image stops updating. However, what I type and mouse clicks are still being sent. No error message is shown whatsoever, the image simply freezes. To make it work again, I have to close and re-open the session. Everything that I had typed and/or clicked while it was frozen is there.  Is there any way I can debug this? In attempting to wrap my mind around the basic vocabulary, concepts, and methods of philosophy, I find myself wondering what the difference is between a philosopher and a theologian.   can have two meanings: 1. Theology is a rational study of the existence of God/gods and the nature of religious ideas. 2. Theology is simply a study of a particular religion (or all religions), really more the practice than the theory, but maybe a mixture.  It seems from an immediate reading that definition 1 would lead to the conclusion that a theologian is a philosopher (perhaps that theology is a branch or subset of philosophy), but definition 2 appears to be rather unphilosophical in nature.    So, how does philosophy answer this question?   Is theology considered a proper subset/branch of philosophy overall? Does it depend entirely on which of the above 2 definitions of theology fits a particular theologian as to whether he is a philosopher or not?   I think this answer is simple and fairly self-evident, but somewhere in my gut I feel like I'm missing something.  Is there a treatment of this question in the literature?    Edit: the answers so far seem to want to redefine theology or other terms - that is fine, but please be clear that you are redfining them, and provide, if possible, sources for why the term should be different than given (support from historical works, other sites, etc.). How can I disable Amazon search results in the Dash? How do I do a case-insensitive string comparison? How to remove/apply all modifiers of one object in python Lets assume you have 1 object with 4 modifiers added to it. I am curious how one would approach the task to apply all or to delete all modifiers of a just selected object that has many modifiers already added.  The console reads out the modifier delete as  bpy.ops.object.modifier_remove(modifier="Box Right ")   So "Box Right " I might have to replace with something like active.object.modifier.name and then work each modifier off?  This would be very helpful when you want to generate a static mesh for animation or 3D printing. I'm using Ubuntu 11.10.  The python 2.7.2 provided has a bug that has been corrected in the latest 2.7.2 sources from the python repository.  What is the recommended way to replace the ubuntu provided python with the python from source? Or put another way, how do I "update" the ubuntu-provided one?  Ideally, I would just be able to create new .deb(s) that "update" the python2.7/python2.7-dev provided by Ubuntu. For you that have tested this, should we use Ubuntu and install different Desktop Environments (KDE/XFCE...) or we should install different distributions (Kubuntu/Xubuntu...). Are there any notable issues in performance / CPU consumption?  PS:  I know how to install them (already did), I wanted to know if anyone has any problems regarding the performance and what is better to install the desktop or to use the ditribution I am looking for a word that describes both of these activities, however the word coding does not seem to be suitable.   Likewise with encryption and decryption, cryption does not work well. Proof that Limit of the Log is the Log of the Limit Proof that Limit of the Log is the Log of the Limit. What is the intuition behind this statement? If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times.    But If I multiply two matrices, what does it mean ?  I mean I can't think it in terms of repetitive addition.       What is the intuitive way of thinking about multiplication of matrices? Is there a Stack Exchange site for both technical and behavioural interview questions? Where I could ask questions about, for example:   Sample questions and solutions, experiences, or seeking advice for interviews A place where I could ask interview questions and what the best way to answer it (not just a correct way or the most efficient).   What to do in a certain situation such as negotiation, unfair interviews, etc.   Experiences interviewing at so-and-so company. Is it ok to have mixed grounds and neutrals on bars in a breaker box? So to continue my line of electrical questioning (and perhaps narrow down my ), I took a look through the two panels in this house.  There is a 200amp main service in the basement, which then feeds up to a 100amp sub-panel for the main floor.  This 100amp sub feeds a kitchen (fridge, microwave, dishwasher, gas range), a bathroom, 3 bedrooms, and a living room.  The 200amp main feeds the 100amp sub, 2 bedrooms, a living room, a washer/dryer (gas dryer), a utility room (well pump, pressure tank, gas water heater, gas furnace), and an A/C.  My first concern is that the main service panel has the bare ground and neutral wires mixed on the two bus bars.  Reading around, some say this is OK, other's say it is bad.  Any thoughts?  The sub-panel is wired with grounds and neutrals on separate bars.  My second concern- is a 100amp sub sufficient for the main floor? How to draw this P-V graph? Updating list definition using features I have a feature that deploys content types, a list definition and template that uses these content types and finally a list instance. This feature is already in production.  Now we have a change request to add new views to this existing list. So I make the changes to schema.xml and upgrade the solution. But alas, the new views don't show up. If I create a new site, the list instance is created with the new views. I tried deactivating/reactivating, iisreset but still no success. Is there anything I'm missing? How to install Ubuntu on portable external Hard Drive? jQuery autocomplete tagging plug-in like StackOverflow's input tags? Which is the best way to put function plots into a LaTeX document? I've been testing PCA via SVD to decompose a simple time series data matrix, $X$.  I have two signals $x_1(t)$ and $x_2(t)$ in a data matrix where $M$ rows represents each timepoint sample and each column represents $x_1$ and $x_2$.  The mean signal, $\hat{x}$, is defined as the mean along the row axis (average of $x_1$ and $x_2$ along each timepoint).  I normalize each column of $X$ by subtracting its mean and dividing by the standard deviation.  When I use [U S V] = svd(Xz) in matlab, regardless of how the variables are distributed (whether they are correlated or uncorrelated), one of the columns of the right singular matrix, V, always points in the same direction (to a multiplicative constant) as the mean vector $(1/2, 1/2)$.  But when add an additional third time series, this is never the case (where the mean vector is $(1/3, 1/3, 1/3)$.  Because I normalize the standard deviation for each vector, it does make sense that the direction of most variance given by PCA would be the diagonal 45 degree line.  But if both variables $x_1$ and $x_2$ are independent gaussians, couldn't the PCA direction be any direction since the distribution is radially symmetric?  MATLAB Code:  s = RandStream('mcg16807', 'Seed', 0); RandStream.setDefaultStream(s);  G = zeros(1000,2); G(:,1) = 40*randn(1000,1)-100; G(:,2) = tan(G(:,1)) + randn(1000,1); X = G - repmat(mean(G),[size(G,1), 1]); Xz = X./repmat(std(G),[size(G,1), 1]); [U S V] = svd(Xz); G_mean = mean(Xz,2); corrcoef(G_mean, U(:,1)) corrcoef(G_mean, U(:,2)) Calculate $\lim\limits_{x\to4} \frac {\sqrt {1+2x}-3} {\sqrt x -2}$ After I just learned there are ways to cancel out $0$s in the divisor of fractions onto limits I looked back at a task where I gave up when I got the result $\lim\limits_{x\to4} \frac {\sqrt {1+2x}-3} {\sqrt x -2}$ so I tried to find a way to get the $0$ out here, as well. Am I just not seeing the solution or is there no way to do it? About free quarks and confinement I simply know that a single free quark does not exist. What is the reason that we can not get a free quark?  If we can't get a free quark then what is single-top-quark?