Modulus problem with need to prove
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
Let's say that there exists some "true" relationship between $y$ and $x$ such that $y = ax + b + \epsilon$, where $a$ and $b$ are constants and $\epsilon$ is i.i.d normal noise. When I randomly generate data from that R code: x &lt;- 1:100; y &lt;- ax + b + rnorm(length(x)) and then fit a model like y ~ x, I obviously get reasonably good estimates for $a$ and $b$.  If I switch the role of the variables as in (x ~ y), however, and then rewrite the result for $y$ to be a function of $x$, the resulting slope is always steeper (either more negative or more positive) than that estimated by the y ~ x regression. I'm trying to understand exactly why that is and would appreciate it if anyone could give me an intuition as to what's going on there. I have two servers (both servers are on Windows OS):   Server_A running Oracle 12c Server_B running SQL Server 2014   I want these servers to be able to see each other, so I can SELECT anything I want in Server_A from Server_B and vice versa.  I only want the ability to run SELECT queries (not INSERT, UPDATE, DELETE or any DDL queries), and also if possible to EXECUTE procedures.  What options or technologies we have for achieving this issue and which one is better?    Tell me if any software I have to install or any configuration have to be set on each of servers. I appreciate explaining the instructions step by step. Subset of $C[0,1]$ is nowhere dense Reputation does not update after community wiki is removed from a post, and does not match the value displayed in the reputation audit If votes were cast on a post while it was community wiki, and a moderator removes wiki from that post, the votes should count towards reputation again. However, while this is visible in a reputation audit, , and the reputation score needs to be recalculated to reflect the change.  Can someone fix this? Concatenating two std::vectors When I open a window to air out the room, how does the smell disperse? Let's say I'm in a room with some kind of noxious stink, possibly of flatulent nature.  The quickest way to right the world that comes to mind is to open a window.  When I open a window, how do the stank particles leave the room? is a psychological phenomenon wherein an individual is convinced that they do not deserve the success that they have achieved despite (perhaps extensive) empirical evidence to the contrary.  Some people with imposter syndrome doubt their own ability level. For me, this usually manifests in the thought:  I've somehow convinced everyone into believing that I'm actually good at this.  followed by the horrifying notion that eventually I'll be found out and my world will collapse. I believe that the imposter syndrome is not uncommon in academia. I'm interested to know how other people here have experienced and dealt with it.  In particular, I'm looking for actual tactics that someone might use to combat imposter syndrome.   Why I asked this question: Beyond the obvious reason that this is something I have struggled with for many years and only recently gotten a handle on myself, there's also some evidence that merely realizing that other people experience imposter syndrome helps fellow imposters. As a grad student, I've managed to meet a fair number of people who are in their own struggles with imposter syndrome; however, the tactics and self-trickery that have helped me don't always speak to them and perhaps someone else's methods might. Divisibility is transitive: $\ a\mid b\mid c\,\Rightarrow\ a\mid c$ As the title says, if a number is divisible by a number, is it always divisible by that number's factors?  An example being that $100$ is divisible by $20$, it is also divisible by $10, 5, 4, 2$ as well?  Does this always apply? Integrating $\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$ How to rsync files between two remotes? I would like to transfer files between two remote hosts using on local shell, but it seems rsync doesn't support synchronisation if two remotes are specified as follow:  $ rsync -vuar host1:/var/www host2:/var/www The source and destination cannot both be remote.   What other workarounds/commands I could use to achieve similar results? Show that $y_1$ and $y_2$ are not Linearly Independent Suppose that $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$. How can I show that if $y_1$ and $y_2$ vanish at the same point then they are not linearly independent?   Here is my attempt to prove the problem. Since $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$, it is enough for me to show that the Wronskian of $y_1$ and $y_2$ denoted by $W(y_1,y_2)$ is zero. Now since $y_1$ and $y_2$ vanish at the same point say $p$ we have $y_1(p)=0$ and also $y_2(p)=0$.   Solving for $W(y_1,y_2)(p)$ we have: $y_1(p)y_2'(p)-y_2(p)y_1'(p)=0$ since $y_1(p)=0$ and also $y_2(p)=0$.   Am I correct? Thanks Historical flight data outside of the USA? What is the difference between a desktop environment and a window manager? I'm having a really hard time trying to understand what the differences between a desktop environment and a window manager are?  EDIT: From this :  There are basically three layers that can be included in the Linux desktop: X Windows – This is the foundation that allows for graphic elements to be drawn on the display.  builds the primitive framework that allows moving of windows, interactions with keyboard and mouse, and draws windows. This is required for any graphical desktop. Window Manager – The Window Manager is the piece of the puzzle that controls the placement and appearance of windows. Window Managers include: , , , , , etc. Requires X Windows but not a desktop environment. Desktop Environment – This is where it begins to get a little fuzzy for some. A Desktop Environment includes a Window Manager but builds upon it. The Desktop Environment typically is a far more fully integrated system than a Window Manager. Requires both X Windows and a Window Manager.  Examples of desktop environments are , , ,  among others) How to run ubuntu/xubuntu in a complete non-gui mode How to get a variable name as a string in PHP? Say i have this PHP code:  $FooBar = "a string";   i then need a function like this:  print_var_name($FooBar);   which prints:  FooBar   Any Ideas how to achieve this? Is this even possible in PHP? I have an object at a fixed position in 3D space. I would like the camera to circle around it while facing it. I want to path of the camera to be circular in 2D, meaning that it does not move up or down.  I need this to be pretty accurate. I don't want to manually set a keyframe every frame for position and rotation. I would like to do this maybe with an expression, not with manual keyframing. I am not sure how to go about this but I think it is going to involve Python as well as some trigonometry. Novel about a teleportation device and a con artist alien Arabic Stack Exchange? I have been learning some Arabic mostly for fun. I am looking for the Arabic Stack Exchange site but cannot find it. I am sure I saw it lying around a few months ago or so. Any idea what happened to it? When should N-Gons be used, and when shouldn't they? Is it acceptable to say "We'll get back to you as quick as we can"? Is "quick" a flat adverb in this case?