Linear independence of vectors (involving absolute values of components)
Strictly diagonally dominant matrices are non singular
How exactly does one “control for other variables”? Here is the article that motivated this question:   I liked this article, and it nicely demonstrates the concept of “controlling for other variables” (IQ, career, income, age, etc) in order to best isolate the true relationship between just the 2 variables in question.    Can you explain to me how you actually control for variables on a typical data set?      E.g., if you have 2 people with the same impatience level and BMI, but different incomes, how do you treat these data?  Do you categorize them into different subgroups that do have similar income, patience, and BMI?  But, eventually there are dozens of variables to control for (IQ, career, income, age, etc)  How do you then aggregate these (potentially) 100’s of subgroups?  In fact, I have a feeling this approach is barking up the wrong tree, now that I’ve verbalized it.  Thanks for shedding any light on something I've meant to get to the bottom of for a few years now...! What is the standard error of the sample standard deviation? Prove that $\liminf x_n = -\limsup (-x_n)$ The "correct" way to write a set without a specific element is as follows: $S \setminus \{s\}$  But in some contexts this is cumbersome to write/type or read, and it detracts from the flow of the writing. Is it acceptable to just use $S \setminus s$?  Edit: By "cumbersome" i mean they look bad when you're trying to inline formulas in LaTeX. Does anyone know the name of travel posters from the 1930s? Figure or longtabu/longtable causes bottom margin overflow I have been searching for a while now without any luck, so I hope you can help me. The problem is that when I insert a figure/longtabu on my page, the text below is running out of the page, like this image shows:    Do you know what the problem is here?  I have uploaded some sample code here: I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote:     The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.   The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two? How do I get useful SQL Server database performance statistics? We currently have several shared SQL Server servers and instances in our organization. They are going with a different architecture where business units will get their own SQL Server instances. We have SQL Server 2008R2.  In the past, databases were put on servers in a more or less random fashion -- business use and disk space were the main drivers.  I've been tasked with coming up with a plan for allocating our 50+ databases across 3 servers (one instance per server). My criteria for divvying them up is as follows:   Business critical db's should be spread across the servers. i.e. avoid lumping a bunch of business critical db's all one server.   For the remainder of the non-mission critical db's, consider: a. Disk space usage b. I/O usage -- memory, CPU, etc.     1 and 2a are easy for me to figure out. I know how to get data file sizes.   My question is with that last bit - 2b. I do not know how relevant this is, what specific stats to look at, or how to get them.   Frankly, I'm not 100% sure how to phrase this question! I think I basically want to know is, "How 'busy' is this db -- however you define 'busy'?" Or maybe, "How does this db impact the performance of the server?"  I mean, disk space is important -- but it could (in theory) have a ton of rows or binary objects in there that are not written very often, so it has little impact on the performance of the server. I think I would want to know how much impact a db within a date range.  Is there a way to get these kind of general stats? I'm not looking for expensive queries, detailed user stats, or query optimization here. I'm aware of the SSMS reports, but they seem to be more real-time. The end goal is to divide up our current databases into evenly balanced buckets, and I would think performance is one way to do that.  Thoughts? Wisdom? Any other ideas on how to divvy up the db's?  Thanks! Tom "[somebody]'s theme" vs "[somebody] theme" — Saxon genitive dillema When is it required to use 's in phrases like below (examples taken from Wikipedia)?   James Bond theme 007 theme Boba Fett motif Droids motif Yoda's theme ...   Are there any rules for eliding Saxon genitive?  How does the situation change if we have something instead of somebody? Find a bijection to prove a fact about integer partitions Prove that for fixed positive integers $k$ and $n$, the number of partitions of $n$ is equal to the number of partitions of $2n + k$ into $n + k$ parts.  I tried this with different values of n but can't seem to get a bijection between the two partitions. In the Android app, left sidebar, there is an editable list of "Your Sites". In the editing pane, the sites are re-orderable (although awkwardly), and the order one places the sites in there is preserved, even after quitting the app and returning to this pane. One expects that order to be reflected in the "Your Sites" portion of the left sidebar when the editing pane is closed, but it is not.  I'm not sure what is determining the order of the sites in the left sidebar, it doesn't seem to be by date they were "pinned" into the list one wants to show, either. \footnote breaks @-constructs in \index Complex numbers - Exponential numbers - Proof Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6.  For this problem I am stumped...how should I begin? Also there's a hint for it:  From $z^n = 1$, prove that $|z| = 1$. What does the equation $(z + 1)^n = 1$ tell you? What do the resulting equations tell you about $z$?  Could someone give me a hint on where to begin? thanks in advance Not able to execute a .sh file: /bin/bash^M: bad interpreter I wanted to execute a shell script:  -rwxr-x--x 1 root root   17234 Jun  6 18:31 create_mgw_3shelf_6xIPNI1P.sh   I tried to do a standard procedure, but I got this error:  ./create_mgw_3shelf_6xIPNI1P.sh  localhost 389 -l /opt/fews/sessions/AMGWM19/log/2013-06-06-143637_CLA-0  DEBUG   cd/etc/opt/ldapfiles/ldif_in ; ./create_mgw_3shelf_6xIPNI1P.sh  localhost 389 -l /opt/fews/sessions/AMGWM19/log/2013-06-06-143637_CLA-0 **ERROR  sh: ./create_mgw_3shelf_6xIPNI1P.sh: /bin/bash^M: bad interpreter: No such file or directory**   What does it mean? I was doing this as the root user under the root group.  Does it mean that the file does not have the correct permission for the root user? Which side of wave-particle duality to choose in a given situation Object renders black Check if email exist php Charge controller for Li-ion or Li-Polymer battery I am trying to work out the good charger for Mobile phones, which can serve as good performer for various smart phones out there in market these days &amp; mostly using Li-ion battery. Charging has 4 stages -- "the initial trickle-charging, then constant current charging, then constant voltage charging &amp; finaly Top-up charge" -- all this is controlled &amp; monitored by a control circuit/IC of charger. So, my concern is if such circuit/module/IC is already there in Mobile phones &amp; taking care of all protections to battery/phone including overheating during charge OR The charger designer have to incorporate these circuits in Charger itself? If former case, that means we just need to make a charging source giving ample current &amp; 4.75V-5.25V voltage, like a high-powered USB output port would gives. My feeling is such stages-wise charging &amp; control/monitor circuitry is already within the phone set, i just want to reconfirm. Find $(a,b)$ such that $\lim\limits_{x\to 0}\frac{f(x)}{x}=1$ implies $\lim\limits_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$ Let $f$ denote any function such that $\lim\limits_{x\to 0}\frac{f(x)}{x}=1$. Find the value of $a$ and $b$, assuming that $\lim\limits_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$.   My  attempt: $\lim_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$ $\lim_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{x^3\frac{(f(x))^3}{x^3}}=1$ $\lim_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{x^3}=1$ $\lim_{x\to 0}\frac{1+a\cos x-b\frac{\sin x}{x}}{x^2}=1$ $\lim_{x\to 0}\frac{1+a\cos x-b}{x^2}=1$ As denominator is tending to zero,so numerato will also tend to zero. $1+a-b=0............................(1)$ Applying L Hospital rule, $\lim_{x\to 0}\frac{-a\sin x}{2x}=1$ So $a=-2$ and $b=-1$ But the answer given in my book is $a=\frac{-5}{2}$ and $b=-\frac{3}{2}$. I do not understand where have i gone wrong? Can a purely biological account of man show a superiority over other animals other than that of belonging to a later stage of evolution or of greater possibilities of action, a superiority “in his very being”?