Suppose I wanted to find the number of solutions of $$e_1 + e_2 + e_3 = 17$$ where $e_1, e_2,$ and $e_3$ are nonnegative integers with $2 \leq e_1 \leq 5, 3 \leq e_2 \leq 6, 4 \leq e_3 \leq 7$.  The solution [apparently] lies in the fact that the number of solutions with the indicated constraints is the coefficient of $x^{17}$ in the expansion of $$\bigg(\sum_{k=2}^5 x^k \bigg)\bigg(\sum_{k=3}^6 x^k \bigg) \bigg(\sum_{k=4}^7 x^k\bigg) $$Why is this the case? Why, on a fundamental level, do elementary generating function-based counting methods (such as this one) work?
Find the number of solutions of $a_0+a_1+a_2=17$ if $2\le a_0\le 5$,   $3\le a_1\le 6$, $4\le a_2\le 7$.   This is an exmaple given by my professor, and his solution is:       The number of solutions to this equation satisfying the given   constraints is equal to the coefficient of $x^{21}$ in the expression   $$\left(\sum_{n=2}^5x^n\right)\left(\sum_{n=3}^6x^n\right)\left(\sum_{n=4}^7x^n\right)=x^9\left(\sum_{n=0}^3x^n\right)\left(\sum_{n=0}^3x^n\right)\left(\sum_{n=0}^3x^n\right).$$   This must be 3, since the coefficient of $x^8$ in $(1+x+x^2+x^3)^3$ is   3.   I understand actually nothing here. The expression of summation just pops up, and then comes the answer. What I only know that the idea of generating function is used here. Can someone help explain how the steps in the solution come?
A regular challenge that I face is balancing two (or more) logos. Whether they need to be equal or one needs to be given priority. Some logos however might be squares, others rectangles, circles, etc...   As a quick sample I snagged a nice high resolution photo from  and pretend its going to be an advertisement that needs to show some footwear brand logos on it. In this case I used Adidas and Saucony.  This is certainly not balanced even though they're height is the same:    Likewise this is not balanced even though they're width is the same:    So then the question becomes when are they in balance, is this balance:    Maybe a little more adjustment, is this balance:    And in these examples I'm only working with two Black Logos. Factor in color and it can get even more difficult to balance. Another issue being when there's tiny text as part of the logo.  So the question is, other than estimating / eye-balling it, what tips, methods or even tools are there to balance different sized logos? Java obfuscators I'm looking for a good Java obfuscator.  I've done initial research into the following Java obfuscators: proguard, yguard, retroguard, dasho, allatori, jshrink, smokescreen, jobfuscate, marvin, jbco, jode, javaguard, jarg, joga, cafebabe, donquixote, mwobfu, bbmug, zelix klassmaster, sandmark, jcloak, thicket, blufuscator, and java code protector.  I tried proguard and it has a really nice GUI, seems really stable, and seems to be the most popular, but it seemed to not like some enumeration on a referenced jar file (not within the code I was trying to obfuscate) which was weird.  Yguard seems to require some interaction with ant, which I didn't know too much about.  What is a good java obfuscator?  It doesn't need to be free, it just needs to work well and be easy to use. Can markdown mini replace an en dash with a normal hyphen in URLs? Numlock turned on on boot-up I have a desktop running ubuntu 10.04  I want numlock to be turned on initially.  What do I have to change to accomplish this?  ~~edit~~  I checked in my BIOS.  It was already set up to have the numlock turned on initially.  Yet it isn't being turned on initially.  I have a USB keyboard: would that break this functionality?   Or, should I ask elsewhere (superuser, perhaps) about how to get that BOIS setting working properly? This question is a sequel of sorts to my  about .  In the paper, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes.  The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction.  In my previous question, I asked about how particle number density is computed.  In this question, I would like to ask about how the electric potential is computed, given the charge density distribution.  Recall that in CGS (Gaussian) units, the Poisson equation is   $$\nabla^2 \Phi = -4\pi \rho$$  where $\Phi$ is the electric potential and $\rho$ is the charge density.  So the charge density $\rho$ is proportional to the Laplacian of the potential.  Now suppose I want to find the potential $\Phi(z)$ along $z$, by integrating the Poisson equation.  How can I do this?  In the paper, on page 254, the authors write down the average charge density $\bar{\rho}_{\alpha}(z)$ at $z$:  $$\bar{\rho}_{\alpha}(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^{\prime} \; dy^{\prime} \; \rho_{\alpha}(x^{\prime}, y^{\prime}, z)$$  where $\rho_{\alpha}(x, y, z)$ is the local charge density arising from the atomic charge distribution of ionic species $\alpha$, $\bar{\rho}_{\alpha}(z)$ is the average charge density at $z$ obtained by averaging $\rho_{\alpha}(x, y, z)$ over $x$ and $y$, and $\sum_{\alpha}$ denotes sum over ionic species.  The authors then integrate the Poisson equation to obtain $\Phi(z)$:  $$\Phi(z) = -4\pi \sum_{\alpha} \int_{-z_0}^z (z - z^{\prime}) \bar{\rho}_{\alpha}(z^{\prime}) \; dz^{\prime} \; \; \; \; \textbf{(eq. 2)}$$  My question is, how do I "integrate the Poisson equation" to obtain equation (2)?  How do I go from  $\nabla^2 \Phi = -4\pi \rho$ to equation (2)?  In paricular, where does the $(z - z^{\prime})$ factor come from?   Thanks for your time. How can a question banned user fix their questions if they're deleted? If a user is question banned  and the , how can the user fix this on their own?  Until recently, advice has been to improve their questions, but since they can't see the questions on their question screen, they can't improve them:     Feature request: Allow a question banned user to see their own deleted questions, or warn them that deleting questions could lead to a question ban. get current user info using jquery ,REST and csom My goal is to get the login name and picture for the current user. I am using SharePoint 2013 Foundation. I tried three method and non of then worked correctly:   This is with jQuery SPServices:   function getCurrentUserInfo() {     var thisUsersValues = $().SPServices.SPGetCurrentUser({         fieldNames: ["ID", "Name", "Picture"],         debug: false     });      var name = thisUsersValues["Name"];      var ID = thisUsersValues["ID"];      var SIPAddress = thisUsersValues["Picture"];  }    But I am getting this error:     SCRIPT438: Object doesn't support property or method 'replace'      File: jquery.SPServices-2014.01.js, Line: 2414, Column: 17    REST API method:   function getinfo(loginName) {     var theData = {         "propertiesForUser": {             "__metadata": { "type": "SP.UserProfiles.UserProfilePropertiesForUser" },             "accountName": loginName,             "propertyNames": ["PreferredName", "PictureURL"]         }     };      var requestHeaders = {         "Accept": "application/json;odata=verbose",         "X-RequestDigest": jQuery("#__REQUESTDIGEST").val()     };      jQuery.ajax({         url: _spPageContextInfo.webAbsoluteUrl + "/_api/SP.UserProfiles.PeopleManager/GetUserProfilePropertiesFor",         type: "POST",         data: JSON.stringify(theData),         contentType: "application/json;odata=verbose",         headers: requestHeaders,         success: function (data) {             document.getElementById("profilelink").href = userProfileProperties[1];;             document.getElementById("username").innerHTML = userProfileProperties[0];             if (userProfileProperties[1] !== null) {                 document.getElementById("Userprofileimage").src = userProfileProperties[1];             }         },         error: function (jqxr, errorCode, errorThrown) {             console.log(jqxr.responseText);         }     }); }    Error:     "{\"error\":{\"code\":\"-1,        Microsoft.SharePoint.Client.InvalidClientQueryException\",\"message\":{\"lang\":\"en-US\",\"value\":\"The method         GetUserProfilePropertiesFor cannot be invoked as its parameter propertiesForUser is not supported.\"},\"innererror\":        {\"message\":\"The method GetUserProfilePropertiesFor cannot be invoked as its parameter propertiesForUser is not supported.\"        ,\"type\":\"Microsoft.SharePoint.Client.InvalidClientQueryException\",\"stacktrace\":\"     With CSOM:  //SP.SOD.executeOrDelayUntilScriptLoaded(getUserProperties, 'SP.UserProfiles.js'); function getUserProperties(targetUser) {      // get the target users domain name and account name.     var tUser = targetUser.substring(7);     // Get the current client context.     var clientContext = new SP.ClientContext.get_current();     //Get PeopleManager Instance     var peopleManager = new SP.UserProfiles.PeopleManager(clientContext);     // Get the properties to retrieve     var profilePropertyNames = ["PreferredName", "PictureURL"];     var userProfilePropertiesForUser = new SP.UserProfiles.UserProfilePropertiesForUser(         clientContext,         tUser,         profilePropertyNames);     // Get user profile properties for the target user     userProfileProperties = peopleManager.getUserProfilePropertiesFor(userProfilePropertiesForUser);     // Load the UserProfilePropertiesForUser object.     clientContext.load(userProfilePropertiesForUser);     //Execute the Query     clientContext.executeQueryAsync(onRequestSuccess, onRequestFail); }    I really don't understand where the issue is. Does anyone have an idea? I'm really new in using biblatex, compile the sample placed in , but what I get is    1.1 Foo    Some text [B02x ].    1.2 Bar    Some text [B02y ].   Do not get the bibliography by chapters, or the global references.  Really not remains to be done   \documentclass{report}  \usepackage[style=alphabetic]{biblatex}  \addbibresource{references.bib}  \begin{document}  \chapter{First chapter}  \section{Foo}  Some text \cite{childs_temperature}.  \printbibliography  \chapter{Second chapter}  \section{Bar}  Some text \cite{hashemian}.  \printbibliography  \newpage  \printbibliography  \end{document}   And the references.bib    @BOOK{childs_temperature,   title = {Practical Temperature Measurement},   publisher = {Butterworth - Heinemann},   year = {2001},   author = {Childs, Peter R N},   address = {Great Britain},   edition = {1},   isbn = {0 7506 5080 X}   }    @PHDTHESIS{hashemian,   author = {Hashemian, Hashem Mehrdad},   title = {Measurements of dynamic temperatures and pressures in nuclear power plants},   school = {{The University of Western Ontario}},   year = {2011},   type = {PhD {T}hesis}   }   I need to obtain references by chapter and global cited references, but i don't know how compile. Can I install any iOS version I wish by picking and choosing which firmware to restore? I'm trying to downgrade my iPhone from the current iOS to an older one, as per .  But every time I try, I always receive this error:     The iPhone could not be restored. This device isn't eligible for the requested build.     I have tried many times with and without DFU mode. Any ideas regarding this? How can I reset or revert a file to a specific revision? I have made some changes to a file which has been committed a few times as part of a group of files, but now want to reset/revert the changes on it back to a previous version.   I have done a git log along with a git diff to find the revision I need, but just have no idea how to get the file back to its former state in the past. Find three real orthogonal matrices of order $3$ having all integer entries.   I have no idea to solve the problem. I don't know how to start. If $A$ be such matrix then $AA^T=A^TA=I_3$. Please help me. If $f^2$ is an analytic function then so is $f$ I want to show the following:  If $f(z) $ is a continuous  function on a connected open subset of the complex plane and $f(z)^2$ is an analytic function, then $f(z)$ is analytic.  Clearly if $f(z) \neq 0$ then $$\frac{f(z+h)-f(z)}{h}=\frac{f(z+h)^2-f(z)^2}{h}.\frac{1}{f(z+h)+f(z)}$$  which shows $f^\prime(z) $ exists if $f(z)\neq0$.  What do I do when $f(z) =0$? Editing polygons simultaneously in ArcMap? I have two polygons:    How can I reshape both Polygon 1 and Polygon 2 at the same time?  Will the area of each polygon change automatically in the table after reshaping?  I'm using ArcMap 10.1. What is the difference between Clamav and Clamtk? I know that both of them are antivirus. What are the limitations on the Shape Water cantrip? I have been having a discussion regarding the Shape Water cantrip from the Elemental Evil Player's Companion. A friend and I are trying to come to an accurate view of how this spell actually functions.   Some example questions:   Must the targeted water be within a 5 foot cube area, or does it need to be of a volume not exceeding 125 cubic feet? If you shape water that is within a 5 foot cube, can the final shape you animate it to extend outside the cube? Can you cause the shaping to affect ice and snow, or only liquid water? Would you be able to carry a globe of water alongside you over distances?   Really, there are two parts:    What can be targeted by the spell? What are the limitations on the 'shape and animate' portion of the spell?   The answer I am looking for, if possible, would use the language of the spell, possibly with precedent from other game text, to clearly define the limitations of this cantrip. String.split() *not* on regular expression? What happens to matter in a standard model with zero Higgs VEV? Suppose you reset the parameters of the standard model so that the Higgs field average value is zero in the vacuum, what would happen to standard matter?  If the fundamental fermions go from a finite to a zero rest mass, I'm pretty sure that the electrons would fly away from nuclei at the speed of light, leaving positively charged nuclei trying to get away from each other.  Looking at the solution for the Hydrogen atom, I don't see how it would be possible to have atoms with zero rest mass electrons.  What happens to protons and neutrons?  Since only a very small part of the mass of protons and neutrons is the rest mass of the quarks, and since they're flying around in there at relativistic speeds already, and since the nuclear force is so much stronger than the electrical force with an incredible aversion to naked color, would protons and neutrons remain bound assemblages of quarks and virtual gluons?  Would they get a little larger?  A little less massive?  What would happen to nuclei?  Would they stay together?  If the protons and neutrons hold together and their properties change only some, then I might expect the same of nuclei.  Different stable isotopes, different sizes, and different masses, but I would expect there would still be nuclei.  Also the W and Z particles go to zero rest mass.  What does that do to the electroweak interactions?  Does that affect normal stable matter (outside of nuclear decay modes)?  Is the weak force no longer weak?  What happens to the forces overall? I have named him/he who shall not be named. Which of these is correct? I think it should be "him" because "him" is a direct object in this context. In this context, "him/he who shall not be named" is not a proper noun. Dependency error while installing WINE Every time I try to install "WINE" VIA the Software Center on Ubuntu 12.10, I get this error:  The following packages have unmet dependencies:  wine1.4: PreDepends: dpkg (&gt;= 1.15.7.2~) but 1.16.7ubuntu6 is to be installed          Depends: libc6 (&gt;= 2.14) but 2.15-0ubuntu20 is to be installed          Depends: wine1.4-amd64 (= 1.4.1-0ubuntu1) but 1.4.1-0ubuntu1 is to be installed          Depends: wine1.4-i386 (= 1.4.1-0ubuntu1) but it is not going to be installed   When I try installing it VIA the terminal, this is what I get:  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  wine1.4 : Depends: wine1.4-i386 (= 1.4.1-0ubuntu1) but it is not installable            Recommends: gnome-exe-thumbnailer but it is not going to be installed or                        kde-runtime but it is not going to be installed            Recommends: ttf-droid            Recommends: ttf-mscorefonts-installer but it is not going to be installed            Recommends: ttf-umefont but it is not going to be installed            Recommends: ttf-unfonts-core but it is not going to be installed            Recommends: winbind but it is not going to be installed            Recommends: winetricks but it is not going to be installed E: Unable to correct problems, you have held broken packages.   how do I fix this? float to int unexpected behaviour