It is well known that the probability density function characterizes the independence of random variables in the following sense.  $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$  where $f$ is the pdf of the random variable $(X,Y)$ and $f_x$ and $f_y$ are the pdfs of the marginal distributions of $X$ and $Y$, respectively. My question is, seeing as there is a bijection between characteristic functions and probability distributions, do characteristic functions also characterize independence? In the following sense  $$\varphi_{(X,Y)}(t_1,t_2)=\varphi_X(t_1)\varphi_Y(t_2)$$  where again $\varphi$ is the characteristic function of the random variable in its subscript.
This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems?
Do I have to leave the Schengen area if I have a short term (C) visa that will expire right before my National Visa (D) officially starts? Can't indent heredoc to match code block's indentation If there's a "First World Problems" for scripting, this would be it.   I have the following code in a script I'm updating:  if [ $diffLines -eq 1 ]; then         dateLastChanged=$(stat --format '%y' /.bbdata | awk '{print $1" "$2}' | sed 's/\.[0-9]*//g')          mailx -r "Systems and Operations &lt;sysadmin@[redacted].edu&gt;" -s "Warning Stale BB Data" jadavis6@[redacted].edu &lt;&lt;EOI         Last Change: $dateLastChanged          This is an automated warning of stale data for the UNC-G Blackboard Snapshot process. EOI  else         echo "$diffLines have changed" fi   The script sends email without issues, but the mailx command is nested within an if statement so I appear to be left with two choices:    Put EOI on a new line and break indentation patterns or Keep with indentation but use something like an echo statement to get mailx to suck up my email.   I'm open to alternatives to heredoc, but if there's a way to get around this it's my preferred syntax. Brief Description:  I need my Ethernet card (en0) to see all traffic under 192.168.2.xxx.  However I also need to use WiFi card (en1) for all other traffic.  OS: OS X Lion    I am using a separate network location to deal with this specific use (since this is a for a robotics project that communicates via Ethernet). Ethernet (en0) had to be set up as a static IP (192.168.2.10) per the requirements for the device I am connecting to. All connections on the Ethernet card (en0) should be 192.168.2.xxx as specified above. WiFi (en1) is set up as DHCP to a router since their is no point in assigning it as a static for general purpose traffic. Ethernet (en0) is set up as priority over WiFi (en1).  I would also like to keep this under the network location preferences if at all possible since I do not require this feature to be on all the time.  I would really appreciate your help on this. No one I have talked to knows how to solve this. Is there a word for a question asked despite knowing the answer, but meant to elicit a response for the sake of the audience? In journalistic writing I often see writers, correspondents, and interviewers use questions in non-literal ways.  Of course there are rhetorical questions designed to make a point and not meant to literally elicit an answer.   Instead of that, I have in mind when an interviewer asks a correspondent, "why did she not go to the police?" when the interviewer already knows the story and knows the answer to the question, but wants the correspondent to articulate the answer for the audience's benefit.    Is there a commonly used term for this sort of a question?  A "leading" question generally implies an attempt to manipulate the respondent into giving certain responses, which is not exactly the same as what I'm describing. Seeking SF Short Story from late 1970s/early 1980s about dystopian future with an old man playing a piano I read this in the late 1970s or early 1980s, possibly as part of a juvenile fiction anthology/collection. From what I remember in this story a post-apocalyptic society requires each member to go out before their 18th birthday and kill an "oldster" adult (i.e. I think over 30 similar to the time in Logan's Run) to make room for themselves.  Otherwise they will be killed.   The kids actually cut off the hand of the victim which has a tattoo on it with the birthdate/age. In this case, the world has gotten to the point the person who kills the oldest person gets a "prize" of one more year of life beyond 30. Note, this is NOT Logan's run nor the novel upon which it is based.   The hero in this case finds an old man, the oldest he had ever seen (something like 45-50 I think) making wonderful sounds with a strange device. It was a piano.   He then confronts the old man  SPOILER ALERT     ... but cannot bear to kill him. The old man takes pity on the boy and offers him his hand to take back so he can live. The boy asks if the old man will still be able to play the piano with just one hand. The old man smiles and says "Yes, but not as well. " Then, the boy decides to stay with the old man and risk becoming an outlaw so he can learn to play the piano. Suppose we have a situation like      A ball of some mass $m$ with some velocity collides with rod hinged at point $A$. Is momentum conserved in this situation? I know that hinge will give impulsive force on rod but that is internal force when we take ball and rod together as a system so that shouldn't stop us using conservation of momentum equation How to migrate from test environment to production environment? The migration is from the local environment to the production environment. The production environment has run some time and created a lot of articles.   In order to add new things to my site, I added a custom theme and installed CCK, Views and other modules in my local test environment. Now that the local test environment is finished, how do I migrate it to the production environment, without destroying the content of its database? Block network access of a process? One server with multiple desktops "heads" with VNC I am managing a system of kiosks.  Each kiosk currently is running a web browser with the application for the kiosk running in the browser.  Each kiosk needs to be able to display separate content.  At times, the application running in the web browser freezes.  Thus, I have to go out to the site to refresh the page.  I want to see if there is a way to have one central server that has multiple browser heads.  Then each kiosk would run a program like VNC to display one of the heads.  This way when the program freezes, I just have to login to the central server and refresh the page.  Getting VNC or another remote desktop software installed on the clients is no problem.  What I am looking for is a way to have VNC remote into a specific head of a head of a web browser.  Does such a thing exist?  Or do I have to run a VM for each kiosk to remote into?  Any advice, pointers, or solutions would be helpful. How can I parse dates and convert time zones in Perl? How to use triangle inequality to establish Reverse triangle inequality I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$ .   I have tried to represent  $||a|-|b||$ as  $||a|+(-|b|)|$ , and then get $||a|+(-|b|)| \leq |a|+|-|b||$ , but that isn't leading me anywhere given $|a-b| \leq |a|+|b|$. Close Lightning Modal by clicking anywhere outside of it For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $ Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and  $f(1)=1$.For every integer $n$ show  that there must exist $n$ distinct points $\alpha_1,\alpha_2\cdots,\alpha_n$ in that interval so that    $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $   Is it theorem or i can show it with basic differentiation knowledge .Please give me hint . Modern Linux commands that parallel uuencode/uudecode functionality I need to get binary files into text format.  Years ago, I used uuencode/uudecode, and it was easy:  uuencode file.xls &gt; file.txt # Delete file.xls # Do whatever with file.txt # Now I need file.xls back uudecode file.txt &gt; file.xls   But uuencode/uudecode is not on my system, and as I look around, the yum/rpm process is ugly--this gives me the clue that maybe no one uses it any more and instead there is something more modern to do this.  I looked at xxd and hexdump but couldn't get them to do it.  Could just be dumb user issues.  What can I use on Linux to do this?  Would like it to be something that tends to be in the native install so I don't need to yum/rpm it if possible.  Just for your own understanding, what I'm specifically trying to do now is source control an *.xls file but the corporate tool doesn't allow *.xls in.  I respect that constraint normally, but in my context, an *.xls is a configuration table for my code so It is logically like source code, but the corporate SVN trigger is physical--if it's *.xls, it gets rejected, so I'll encode it and source control the *.txt file.  Thanks! What is the black thing on the circuit What is the black thing on the below circuit?   Is it used to hide the IC (or other components) that is used on the circuit?   What material is it? What do you call a response which does not address the question? When some one is asked a question, sometimes if they are trying to avoid answering the question, they respond with something unrelated. What is the word for that response?  Eg. A: Why were you late?     B: This bagel tastes good.  What I am looking for is the name of that response, not the action. I installed the latest pgAdmin 4 (v2.1) but I found out that (), for the moment, the only way to establish a connection through an SSH tunneling is using another outside process, an  says:  I setup a tunnel as follows: ssh -f dpage(at)hostname(dot)enterprisedb(dot)com -L 5999: hostname.enterprisedb.com:5432 -N  (...) I was running on Mac, but the same procedure should work on Linux. On Windows, there's a tunnelling app in Putty iirc, but I don't know how to use it offhand.  I have found a simple . The connection is established but it seems, when I try to connect through pgAdmin it fails or is refused (using 127.0.0.1:3306 or 127.0.0.1:5432 as the tunnel's destination). Why? Let's say I fire a bus through space at (almost) the speed of light in vacuum. If I'm inside the bus (sitting on the back seat) and I run up the aisle of the bus toward the front, does that mean I'm traveling faster than the speed of light? (Relative to Earth that I just took off from.) Do spells with a range of "Self" target the caster? New to D&amp;D, and thought about creating a Beast Master. I was wondering if spells with a range of "Self" would count you as the "target" of the spell. There's some debate in my group, as unlike something such as Cure Wounds, which specifies "A creature you touch," there's technically no mention of a target in a Self range spell within the description. A fair amount of people in my playgroup come from a TCG background, where the presence (or lack thereof) of keywords like "target" are essential for determining interaction.  tl;dr Would Paladin Smites or Thunderwave count as "targeting" yourself, and therefore be affected by Beast Master's Share Spell? I had posted a  last week about customizing a ToolValidator class and got some very good answers.  In working with the proposed solutions, I have created a custom module that performs queries on a db, and will be called by both the ToolValidator class (to provide values for the drop-down lists) and later in the geoprocessing script (to get other parameters based on items selected in drop-down lists).  However, I can't seem to actually call the custom module in the ToolValidator class.  I have been trying to append to path with no luck.  When I try to apply these changes to the script, I get a runtime error:  [Errno 9] Bad file descriptor.  If I comment out import line, no errors.  sys.path.append('my_custom_module_directory') import my_custom_module   Many of you may be asking why don't I just implement a custom tool with ArcObjects.  The reason is that my end users don't have the privleges necessary to register ANY dlls on their computer.  UPDATE:  This was happening to me in ArcGIS 10.  Interestingly enough, I was initially appending to the path inside of the initialiazeParameters function of the ToolValidator class.  If I do the append outside (i.e., on top of) of the ToolValidator class, everything works as expected.  sys.path.append('C:/Working/SomeFolder') import somescript --------&gt;THIS WORKS  class ToolValidator:   """Class for validating a tool's parameter values and controlling   the behavior of the tool's dialog."""    def __init__(self):     """Setup arcpy and the list of tool parameters."""     import arcpy     sys.path.append('C:/Working/SomeFolder')     import somescript -------&gt; THIS DOESNT WORK     self.params = arcpy.GetParameterInfo()   UPDATE 2:  I think I found the true cause of my problem.  In the code snippets in this post, I have been appending what appear to be real paths (ex C:/Working/SomeFolder) to the sys.path.  In my actual ToolValidator class, I was constructing a relative path using os.path.dirname(__file__) +"\my_special_folder...".  I was anticipating that os.path.dirname(__file__) would return the path of the toolbox, since that contains the ToolValidator class.  I have come to find this is not the case.  As far as I can tell, the ToolValidator class is never actually written to a .py file, and I speculate the the this code is passed to the python interpreter in memory, so __file__ is useless, or some temp script is persisted and then execfile(path_to_script) is called, again rendering __file__ useless.  I'm sure there are other reasons I am missing as well.    Long story short, if I use a hardcoded path, the sys.append works anywhere, relative paths don't work so well in the ToolValidator class.