Let $X$ be a non-negative random variable. Show $(EX)(E\frac{1}{X})\geq 1$. What does this say about the correlation between $X$ and $\frac{1}{X}$. Further show that if $f,g$ are positive borel measurable functions  such that $f(x)g(x)\geq1$ then $Ef(X)Eg(X)\geq 1$ I suppose I could start with proving the second part of the question and then prove the first part as a consequence. I know I'm going to ultimately use Holder's inequality for this one, but I'm struggling to clear the $p$ and $q$ when I integrate $$1=\int_\Omega 1 dP\leq \int_\Omega f(X)g(X) dP\leq \left(\int_\Omega f(X)^pdP\right)^{1/p}\left(\int_\Omega g(X)^qdP\right)^{1/q}$$ Is the product of those integrals at the end less than or equal to $$\left(\int_\Omega f(X)dP\right)\left(\int_\Omega g(X)dP\right)?$$ If so, how could I prove this? Thanks in advance for any suggestions.
Let $\mu(\Omega)$ be a probability measure (i.e. $\mu(\Omega) = 1$), and let $f,g$ be non-negative measurable functions on $\Omega$ such that $fg \geq 1$. Show that $1 \leq (\int f^p )(\int g^p)$ for all $0 &lt; p &lt; \infty$   I'm a bit at a loss of what to do. I know that since $fg \geq 1$, we may obtain:  $$\int_{\Omega} d\mu = \mu(\Omega) = 1 \leq \int_{\Omega} fg = \int_{\Omega} |fg|$$ The right-most integral is true since $fg$ is positive and it is equal to $||fg||_1$.  By Hölder's inequality, we obtain:  $$1 \leq \int |fg| \leq ||f||_p ||g||_q = (\int f^p)^\frac{1}{p}(\int g^q)^\frac{1}{q}$$  where $p$ and $q$ satisfy $\frac{1}{p} + \frac{1}{q} = 1$. Raising the left-most and right-most sides of this inequality to the power $p$, I may obtain:  $$1 \leq (\int f^p)(\int g^q)^\frac{p}{q}$$  I now wish to show that $(\int g^q)^\frac{p}{q} \leq \int g^p$, but I've been at a loss of how to do it. Any tips or hints would be greatly appreciated.
Which kind of functions $f(x, \ y)$ can be written as $g(x)\cdot h(y)$? How to make the function of the planes from the lorenz attractor The following code helps me validate an XML file with an XSD schema.  XmlReaderSettings settings = new XmlReaderSettings(); settings.Schemas.Add(null, xsdFilePath); settings.ValidationType = ValidationType.Schema; settings.ValidationEventHandler += new System.Xml.Schema.ValidationEventHandler(settings_ValidationEventHandler); XmlDocument document = new XmlDocument(); document.Load(xmlFilePath); XmlReader rdr = XmlReader.Create(new StringReader(document.InnerXml), settings);  while (rdr.Read()) {  } isValid = true;   The ValidationEventHandler also tells me what the errors are, but doesn't tell me on 'where' or 'on which line' they are located. Is there any way to get the line number where the XML fails to be validated? "To mentor someone during a project" vs. "to mentor someone on a project" ..., whom I mentored during his final semester's project.   ..., whom I mentored on his final semester's project.      Which of these two is grammatically correct? Since I am not talking about the final semester but the final semester's project, I am not very sure about during. Can someone help me with this? Trying to identify a short sci-fi story about a human that is "resurrected" by malignant aliens I would like to show that $a^n \mid b^n$ implies $a \mid b$  I thought I could convert it to congruences and work backwards, but as far as I remember, $a \equiv b \pmod{m}$ implies $a^n \equiv b^n \pmod{m}$, not the opposite, unless $m$ is prime. Is that right?  So I am not sure how to approach this one. Any ideas?  Thanks! Regex for splitting a string using space when not surrounded by single or double quotes I'm new to regular expressions and would appreciate your help.  I'm trying to put together an expression that will split the example string using all spaces that are not surrounded by single or double quotes. My last attempt looks like this: (?!") and isn't quite working. It's splitting on the space before the quote.  Example input:   This is a string that "will be" highlighted when your 'regular expression' matches something.   Desired output:  This is a string that will be highlighted when your regular expression matches something.   Note that "will be" and 'regular expression' retain the space between the words. Draft mode for pgfplots How to access java-classes in the default-package? I'm working now together with others in a grails project. I have to write some Java-classes. But I need access to an searchable object created with groovy. It seems, that this object has to be placed in the default-package.   My question is: Is there a way to access this object in the default-package from a Java-class in a named package? How can an Image-Empty be done with Python?  And how to set an image file on it? Object.data is None for empties, and there is no .image property on the object level either. using System; using System.Security.Permissions; using Microsoft.SharePoint; using Microsoft.SharePoint.Security; using Microsoft.SharePoint.Utilities; using Microsoft.SharePoint.Workflow;  namespace Change_Title_Event.Change_Title {     /// &lt;summary&gt;     /// List Item Events     /// &lt;/summary&gt;     public class Change_Title : SPItemEventReceiver     {        /// &lt;summary&gt;        /// An item is being added.        /// &lt;/summary&gt;        public override void ItemAdding(SPItemEventProperties properties)        {            properties.AfterProperties["Title"] = "Kevin: Changed by event receiver at" + DateTime.Now.TimeOfDay.ToString();            base.ItemAdding(properties);        }       } }   Where simple example, which works perfect on my local Sharepoint 2010. It is sandboxed.  But when I package this solution, it do not work on Sharepoint Online. I does not give any Errors either.  I think I am doing something wrong, something simple, but I can't simply find anything on Google about it.  Here is another who asks on the MSDN forum, but gets a very wierd reply, that I do not understand. What does it mean to instantiate a trait? What does it mean that GUIkit is being deprecated? Sorting citations using \cites command in biblatex I am using biblatex for citations and bibliography. Now I often have lists of multiple citations, for which biblatex has the \cites command. Now I can get citations to automatically sort if I don't want page numbers, but I cannot get it right with page numbers.  What I want is all citation ordered chronological (oldest first). Is this possible, or must I just do this by hand?  Here is an example.  \documentclass[11pt,oneside]{memoir}  \usepackage[     style=authortitle,     useprefix=false,     maxnames=2,     firstinits=true,     sortcites=true,     uniquename=init,     abbreviate=true,     backend=biber,     sorting=ynt ]{biblatex}  \begin{filecontents}{jobname.bib} @book{tom1,   author = {Author, A.},   year = {2001},   title = {This book is from 2001},   shorttitle = {2001}, } @book{tom2,   author = {Author, A.},   year = {1999},   title = {This book is from 1999},   shorttitle = {1999}, } \end{filecontents}  \addbibresource{jobname.bib}   \begin{document} This one is corrent.\footnote{\Cites{tom1,tom2}.} This one is corrent.\footnote{\Cite{tom1,tom2}.}   This one is incorrent.\footnote{\Cites{tom1}{tom2}.} This one is incorrent.\footnote{\Cites[1]{tom1}[12]{tom2}.} \end{document} How to automatically fetch missing dependencies when installing software from .deb? I'm trying to install  on my Ubuntu box (11.04). The website has a Ubuntu .deb available for download (for 10.10 and 10.04 so I chose 10.10).   However,   sudo dpkg -i mysql-workbench-gpl-5.2.34-1ubu1010-amd64.deb   yields:  (Reading database ... 194069 files and directories currently installed.) Preparing to replace mysql-workbench-gpl 5.2.34-1ubu1010 (using mysql-workbench-gpl-5.2.34-1ubu1010-amd64.deb) ... Unpacking replacement mysql-workbench-gpl ... dpkg: dependency problems prevent configuration of mysql-workbench-gpl:  mysql-workbench-gpl depends on libctemplate0; however:   Package libctemplate0 is not installed.  mysql-workbench-gpl depends on libpython2.6 (&gt;= 2.6); however:   Package libpython2.6 is not installed.  mysql-workbench-gpl depends on libzip1 (&gt;= 0.8); however:   Package libzip1 is not installed.  mysql-workbench-gpl depends on python-paramiko; however:   Package python-paramiko is not installed.  mysql-workbench-gpl depends on python-pysqlite2; however:   Package python-pysqlite2 is not installed. dpkg: error processing mysql-workbench-gpl (--install):  dependency problems - leaving unconfigured Processing triggers for bamfdaemon ... Rebuilding /usr/share/applications/bamf.index... Processing triggers for desktop-file-utils ... Processing triggers for python-gmenu ... Rebuilding /usr/share/applications/desktop.en_US.utf8.cache... Processing triggers for python-support ... Errors were encountered while processing:  mysql-workbench-gpl   My question is, is there a way to tell dpkg to automatically fetch missing dependencies, or do I need to manually apt-get install missing packages like libctemplate0 and  libpython2.6?   (Or alternatively, is there some other way to get MySQL Workbench easily up &amp; running?) I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function  $d(u,v) = \lVert u - v \rVert$, $u,v \in V$.   My question is whether every metric on a linear space can be induced by norm? I know answer is  no but I need proper justification.   Edit: Is there any method to check whether a given metric space is induced by norm ?  Thanks for help Which is correct: I'll be moving next month or I'll be shifting next month? For changing one's home from one place to another, I've heard people in western part of the world using the sentence:      I'll be moving next month.   In India, even in the English news channels, I've heard many people using the phrase:     I'll be shifting next month.   Which one is the correct usage? Or are both of them correct and the difference in usage is regional? Recently, I have noticed that on most applications I have on Ubuntu 16.04, when I load the application, the window is displayed with a strange multi-coloured border (see the images below). If I maximise the window though, the border disappears and the window is displayed as expected.  This started happening for no apparent reason. If I reboot my machine, things return to normal at first, but after a while the problem comes back.  The issue occurs on most applications, including Nautilus, Software Centre, Terminal, System Settings, etc... I also get the same border appearing when I right click on the desktop, with the border surrounding the pop-up menu.  However, I have noticed that it does not happen on Chrome, but it does happen on other applications which did not come with Ubuntu by default.  Any ideas on what is causing the problem? Thanks! I was reading a  where the author mentions: "The coefficient of variation is primarily a descriptive statistic, but it is amenable to statistical inferences such as null hypothesis testing or confidence intervals. Standard procedures are often very dependent of the normality assumption and current work is exploring alternative procedures which are less dependent on this normality assumption."  The paper is from 2010. So my question is: has there been any recent "advancements" in terms of a coefficient of variation statistic that is not dependent upon the normality assumption? And, is this what one may call a non-parametric CV? Ubuntu installation does not recognize drive partitioning I have a 1TB drive and installed Windows 7 on a 128GB partition.  When I now try to install Ubuntu 11.04 it does not recognize the Windows partition but offers the complete 1TB drive to install Ubuntu on instead. It displays:    However, in the Ubuntu Disk Utility the Windows partitions are recognized.    What do I need to do in order for Ubuntu to recognize the Windows 7 partition and install Ubuntu as a dual boot?   Response to comments  The following commands were executed and the results are shown below:  fdisk -l  WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk  doesn't support GPT. Use GNU Parted.  Disk /dev/sda: 1000.2 GB, 1000204886016 bytes 255 heads, 63 sectors/track, 121601 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x34a38165     Device Boot      Start         End      Blocks   Id  System /dev/sda1   *           1          13      102400    7  HPFS/NTFS Partition 1 does not end on cylinder boundary. /dev/sda2              13       16318   130969600    7  HPFS/NTFS  Disk /dev/sdb: 500.1 GB, 500107862016 bytes 255 heads, 63 sectors/track, 60801 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x14a714a6     Device Boot      Start         End      Blocks   Id  System /dev/sdb1               1       60801   488384001   83  Linux   parted -l  Warning: Unable to open /dev/sr0 read-write (Read-only file system).  /dev/sr0 has been opened read-only. Error: /dev/sr0: unrecognised disk label