Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = a_n$ and $g(z) := \sum_{n=0}^\infty f_n(z)$ converges for $z$ in some open set $U$ (which does not contain 0, or else $\sum_{n=0}^\infty a_n$ would converge), then analytically continue $g(z)$ to $z=0$ and assign $\sum_{n=0}^\infty a_n$ the value $g(0)$.  Does this prescription always yield a unique finite answer, or do there exist two different sets of regularization functions $f_n(z)$ and $h_n(z)$ that agree at $z=0$, such that applying the analytic continuation procedure above to $f_n(z)$ and to $h_n(z)$ yields two different, finite values?
Does the regularization of a divergent infinite sum yield a unique value?  I.e. do different regularization schemes acting on the same infinite sum produce the same exact value independent of the regulator?  What, exactly, do these values mean?  Or what are they?  My understanding is that they are not "convergent" values.  (Sorry in advance for being a physicist.
When a user has the  privilege, they can click on the post score to see the individual counts for upvotes and downvotes.  When a user doesn't have this privilege this functionality isn't present, except after loading newly added answers without refreshing the page. After clicking the "new answers to this question" link the click functionality to view vote counts is added to all posts, though actually clicking on it results in a dialog stating that an error occurred (I'd assume due to not having the required reputation).  May be related to the fix for this previous bug report: How can I add the session name to my screen status display? This is my current .screenrc files which is mostly copied from the link in the code  #First iteration 2014-09-06, not quite sure of stuff to add # got a fancy hardstatus line noted below hardstatus on hardstatus alwayslastline # hardstatus string "%w"  # blagged this hardstatus like from https://bbs.archlinux.org/viewtopic.php?id=55618, not quite sure what it does hardstatus string '%{= kG}%{C}Screen: %{c}%{= kG}[%= %{= kw}%?%-Lw%?%{r}(%{W}%n*%f%t%?(%u)%?%{r})%{w}%?%+Lw%?%?%= %{g}][%{C} %d-%m %{W}%c %{g}]'   I now want the session name to appear in it as well, and the one I have settled on is from this answer -   The solution runs along these line:  hardstatus string '%H:%`' backtick 0 30 30 sh -c 'screen -ls | grep --color=no -o "$PPID[^[:space:]]*"'   It is an area I am not familiar with and I wonder how the backtick command fits in it, ie whether it is a screen command at all. In the following is it better to use a possessive noun with an apostrophe or an attributive noun without an apostrophe?     The following list details the assumptions that have been made in   conjunction with the Dell Company’s staff.   Or      The following list details the assumptions that have been made in   conjunction with the Dell Company staff.   Assume Company is the name of a company, such as Dell. Meaning of probability notations $P(z;d,w)$ and $P(z|d,w)$ What is the difference in meaning between the notation $P(z;d,w)$ and $P(z|d,w)$ which are commonly used in many books and papers? I'd like to see the full How-To on how to use manual partitioning during Ubuntu installation. The existing guides (at least those I found here) cover only automatic part and leave untouched the manual part (or extremely short and contain no pictures).  I'd like to cover such situations:     If your disk contains other systems: The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ How to compute this limit:  $$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$  Please give me some hint. Avoiding NullPointerException in Java Do Arcane Ward's starting hit points include the INT modifier once or twice? An abjuration wizard's Arcane Ward states:     [...] The ward has hit points equal to twice your wizard level + your Intelligence modifier.   So how many hit points is this supposed to be?  \$ (2 \times WizardLevel) + INT_{mod}\$  or  \$2 \times (WizardLevel + INT_{mod})\$ ?  I feel like the phrasing hints toward the first calculation, but it's still somewhat ambiguous. Parallel website running a copy of my original website We have been working on a gaming website. Recently while making note of the major traffic sources, I noticed a website that I found to be a carbon-copy of our website. It uses our logo, and everything else the same as ours, but with a different domain name.   Their domain name is not however pointing to our site because at several places the links are like: carbonccopy-website/our-links. This website even has links to some images as: carbonccopy-website/our-images.   How could they have done that, and what can I do to stop it? What is the use for non-inverting input resistance in a negative feedback opamp? Here in the Figure R5=50K. This resistor value is not in the equations to obtain output voltage, and in many tutorials non-inverting input is directly grounded without a resistor. What might be the reason to use it here? Get value of R, G, B by color Array only adds duplicate values Find $P(12)$ when $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$ let $P(x)$ be a polynomial of degree $11$ such that  $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$  then find value of $P(12)$ Rotate Geographic map How to create a SQL Server function to "join" multiple rows from a subquery into a single delimited field? To illustrate, assume that I have two tables as follows:  VehicleID Name 1         Chuck 2         Larry  LocationID VehicleID City 1          1         New York 2          1         Seattle 3          1         Vancouver 4          2         Los Angeles 5          2         Houston   I want to write a query to return the following results:  VehicleID Name    Locations 1         Chuck   New York, Seattle, Vancouver 2         Larry   Los Angeles, Houston   I know that this can be done using server side cursors, ie:  DECLARE @VehicleID int DECLARE @VehicleName varchar(100) DECLARE @LocationCity varchar(100) DECLARE @Locations varchar(4000) DECLARE @Results TABLE (   VehicleID int   Name varchar(100)   Locations varchar(4000) )  DECLARE VehiclesCursor CURSOR FOR SELECT   [VehicleID] , [Name] FROM [Vehicles]  OPEN VehiclesCursor  FETCH NEXT FROM VehiclesCursor INTO   @VehicleID , @VehicleName WHILE @@FETCH_STATUS = 0 BEGIN    SET @Locations = ''    DECLARE LocationsCursor CURSOR FOR   SELECT     [City]   FROM [Locations]   WHERE [VehicleID] = @VehicleID    OPEN LocationsCursor    FETCH NEXT FROM LocationsCursor INTO     @LocationCity   WHILE @@FETCH_STATUS = 0   BEGIN     SET @Locations = @Locations + @LocationCity      FETCH NEXT FROM LocationsCursor INTO       @LocationCity   END   CLOSE LocationsCursor   DEALLOCATE LocationsCursor    INSERT INTO @Results (VehicleID, Name, Locations) SELECT @VehicleID, @Name, @Locations  END      CLOSE VehiclesCursor DEALLOCATE VehiclesCursor  SELECT * FROM @Results   However, as you can see, this requires a great deal of code.  What I would like is a generic function that would allow me to do something like this:  SELECT VehicleID      , Name      , JOIN(SELECT City FROM Locations WHERE VehicleID = Vehicles.VehicleID, ', ') AS Locations FROM Vehicles   Is this possible?  Or something similar? Consider the general linear regression model: $$y_i = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \cdots + \beta_px_{ip} + \epsilon_i = \mathbf{x}_i^t \beta + \epsilon_i$$  where $\textbf{x}_i = (1,x_{i1},x_{i2},\cdots,x_{ip})^T$, $\beta=(\beta_0,\beta_1,\cdots,\beta_p)^T$ and $\epsilon_i$ are iid N(0,$\sigma^2$).  I would like to see a complete proof of the following identity from first principles: $$\sum_{i=1}^n(y_i - \bar{y})^2 = \sum_{i=1}^n(\hat{y}_i - \bar{y})^2 + \sum_{i=1}^n(y_i - \hat{y}_i)^2$$ where $\hat{y}_i= \mathbf{x}_i^t \hat{\beta} $ ($\hat{\beta}$ is the least square estimator, $\bar{y}$ ia the sample mean of $y_i$).  I know that the two terms on the right can be obtained by subtracting and adding $\hat{y}_i$ on the left side. But this introduces a "cross term": $$\sum_{i=1}^n2(\hat{y}_i - \bar{y})(y_i - \hat{y}_i)$$  Many texts claim that this is zero, but I have not seen a general proof of this statement. How can this be shown? How can I show or hide boot messages when Ubuntu starts? Is there a way to easily turn on/off showing the boot messages (loading the services) when Ubuntu starts? Is it something in Grub2?  I am running 10.04. I've got a 32-bit Ubuntu installation running on 64-bit hardware. Now that , I would like to switch to 64-bit without having to reinstall the OS.  This is one of the :  Shawn installed his system using the 32-bit version of Ubuntu, but his hardware is 64-bit and he wants to switch over. He manually installs the amd64 versions of dpkg and apt, replacing the i386 versions and changing which architecture is used as the default; then he installs the amd64 ubuntu-minimal package; then he installs the amd64 ubuntu-desktop package. Over time the remaining i386 packages are replaced automatically on upgrade.  However, when trying to follow the instructions in there, I cannot find any 64-bit version of dpkg or apt.  Did this user story got implemented in a different way in the final spec, or do I need to do something differently?  In short, how can I switch my 32-bit installation to 64-bit? I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. How much does it cost to have a custom ASIC made? I have browsed several ASIC manufacturer's webs, but I haven't found an actual number. I assume there would be a fixed cost associated with creating masks and such and then there will be a cost per unit.     Note: that I don't actually want to have an ASIC made, I'm just curious.