Relation Between Normalizations
Given the functional dependencies of a relation is there more then one way to normalize it?
Map a function across a list conditionally I have 13.04 installed from a DVD that I got on eBay but the application launcher is missing. I've tried CCSM but it won't work! Deleted downvotes should appear in Achievements dialog "Good morning.  Here is something awesome that you got last night!", said the Top Bar:    "21?  That must be two answer upvotes and, um, an answer that I downvoted that was deleted?", I thought.  Ok, let's look at the details and find out:    The +20 on my egrep answer is new and I've seen the other Achievements on the list, so where did the extra +1 come from?  In order to avoid confusion, the recent reputation changes in the dropdown should add up to the number in the notification.      I suspect this particular case is a bug.  Assuming the +1 is a downvote reversed by an answer deletion, I could see the argument for excluding it from the Achievements dropdown.  After all, it's not exactly something you accomplished.  Except it kinda is.  Part of the reason for downvoting is to  low-quality content.  The ultimate sign that such content was deemed beneath community standards is when it gets deleted.  I'm not sure that we should , but putting a note in your achievements in such cases seems like it would reinforce positive (albeit negative) behaviour.  This question is a feature request that rep gains from deleted downvotes be listed in the Achievements dialog, without getting into broader issues of displaying other activity on deleted posts such as those discussed in .  Thus, this is not intended to completely address the issue of the Achievements sum not adding up, but is intended to help encourage good community moderation by appropriate downvoting of low-quality content. As of the November 2018 core book updates, what changes were made in the latest printings of the PHB that aren't noted in the errata document? How to compute   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$   My Working :   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1} =  1-0 = 1$$  Is it correct Smooth reload of uitableview data when rows are deleted Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism? Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w \mapsto \sum_i f_i(v) \cdot g_i(w)$ for all $(v,w) \in V \times W$. In the finite dimensional case it is easy to see this is an isomorphism by comparing dimensions on both sides. I'm aware that this is not an isomorphism in the infinite-dimensional case (see Relation with the dual space in the  on tensor products) but I must admit that I'm not sure why. Cardinal arithmetic does not seem help here and, even if it did, I would be much happier to see an explicit example of a functional in $(V \otimes W)^*$ outside the range of this map. I'm having trouble cooking one up myself.  Added: Let me elaborate on my comment about cardinal arithmetic. Suppose that $X$ and $Y$ are infinite dimensional vector spaces over a field $k$. I'm confident that $\dim (X \otimes Y) = \dim X \cdot \dim Y$ holds. It is clear that $|X^*| = |k|^{\dim X}$ since we may identify a functional on $X$ with a function from a basis for $X$ to $k$. Also I think I've convinced myself that if $\dim X \geq |k|$ then in fact $|X| = \dim X$ ie. the cardinality and dimension of a vector space agree when the dimension is larger than the cardinality of the ground field. Consequently, assuming say that $|k| \leq \dim X \leq \dim Y$ it seems we have $$ \dim(X \otimes Y)^* = |k|^{ \dim X \cdot \dim Y} = |k|^{ \dim Y} = \dim Y^* = \dim X^* \cdot \dim Y^* = \dim (X^* \otimes Y^*)$$  which implies that in fact $(X \otimes Y)^*$ and $X^* \otimes Y^*$ are isomorphic but, rather frustratingly, the obvious map does not do the job. Did I make a mistake here? If not, is it generally true (ie without making assumptions about $|k|$) that there exists some isomorphism $(X \otimes Y)^* \to X^* \otimes Y^*$? Is it possible to repeat my track object along a path? I tried using the method shown here  using an array and then a curve modifier, but this happened when I added the curve modifier and chose the path as the curve object. Where do I find the current C or C++ standard documents? Behaviour of return statement in catch and finally I have purchased a rather large bag of long-grain white rice (25 lbs.) and need to know the best way for storing it long-term. This rice will be used as both a food-storage food source and as my everyday source of rice for cooking, so the storage solution should (hopefully) be resealable, rather than once-it's-open-it's-open. I eat rice maybe three times a month, so it isn't really "everyday" but it will be my go-to when I have it. How to list all files ordered by size I would like to list all files in the order of big to small in size and the files could be present anywhere in a certain folder. Strange black shading. Cannot fully light mesh. (Cycles) Here is the viewport in material view.  The node setup is just a PNG texture into a diffuse shader.    And this is in rendered view.  I've tried using environmental lighting as well as a directional light, and while the lit parts become brighter/washed out, the darker parts stay dark.  Normals are correct and there is no z-fighting.  Ambient Occlusion is not enabled.  Changing to Flat shading makes it worse.    Update:  Any modifier that alters the mesh (bevel, subdivide, etc) make the problem go away instantly. Uninstall GRUB and use Windows bootloader I've stumbled upon a strange behaviour with Windows machines, which seems to be fairly consistent between all Windows versions from Vista/2008 to 8.1/2012 R2; it doesn't happen instead when using Windows XP or Windows Server 2003.  The problem is this: when the network adapter is configured for DHCP and the DHCP server doesn't register DNS records on behalf of its clients (because it can't, or because it's not configured to do so), then the forward A record gets registered, but the reverse PTR record doesn't.  Some more details:   Both the forward and the reverse DNS zones are AD-integrated and accept dynamic updates. All computers are joined to the domain. All computers use the correct internal DNS servers, both when configured statically and when getting their configuration from DHCP. "Register this connection's addresses in DNS" is enabled in the network adapters. Everything is fine when a computer has a static IP address; both the forward and the reverse records get automatically registered. When the same computer is configured for DHCP, the forward record is registered, but the reverse record isn't. This happens for all computers with an OS version >= 6.0, and it's definitely not related to a single machine. No amount of ipconfig /registerdns will change anything. No errors are logged anywhere.   Why does this happen, and how can it be fixed?  And no, configuring the DHCP server to perform DNS registration is not an option here. From what I understand, entropy is a concept defined by the experimentalist due to his ignorance of the exact microstate of a system. To say the number of accessible microstates $W$ of the universe is constantly increasing is nothing more than saying 'ignorance begets ignorance'.  I have often encountered the argument of ever increasing entropy for the presence of an inherent time asymmetry, most prominently in the works of Penrose. It just doesn't seem to make sense.  Let us imagine alien beings who are experiencing time in reverse. For them, increasing ignorance is in the direction of our decreasing ignorance. So how can perpetual increase in entropy indicate an 'arrow of time'?  One possible explanation (I thought of) for this argument was the fact that there may never be a mechanism to reduce ignorance. I will pose this as a question:  An observer determines the number of possible microstates of system+observer to be $W_0$ at time $t_0$. After improving his measurements, can he (at a later time) measure $W'$ (where $W_0&gt;W'$) as the number of possible microstates? Assume that all the microstates are equally probable in this case.  Unless the above in false in general, how else can anyone claim that entropy reveals an arrow of time?  EDIT: I am performing this discussion for isolated systems (the universe or it's parts if relevant). The means I propose to reduce entropy in the present, is by redefinition of macroscopic variables and of microscopic models used to count the number of distinguishable micro-states (that obviously yield the same macro-state). This is essentially an argument against the robustness of entropy via redefinition. Who is sample No. 118 in The Case of Charles Dexter Ward? In H. P. Lovecraft's The Case of Charles Dexter Ward, Dr. Willett      Enters Joseph Curwen's dungeon and resurrects a dead person identified as No. 118 by reciting the incantation of the Ascending Node over his salts.   Dr. Willett is told by      The resurrected Joseph Curwen, impersonating Charles Dexter Ward, who he secretly murdered,   that   " 'twas Number 118, and I conceive you would have shook had you looked it up in my list in t'other room."  No. 118 himself gives Dr. Willett a note asking the doctor to      kill Joseph Curwen.   It soon becomes clear that No. 118 is opposed to the nefarious dealings that have been transpiring and desirous of being an ally of Dr. Willett.  Who is No. 118?  Earlier in the story,      The grave of Ezra Weeden, a primary instigator in Joseph Curwen's initial downfall and the romantic rival of Curwen, is robbed. There doesn't seem to be any other mention of Weeden in the 20th century occurrences, so without a further connection the passage seems superfluous.   The shared goals of these two people seem to indicate that No. 118 may, in fact, have been that person, but the      Note given by No. 118 to Dr. Willett in eighth century Latin seems to indicate someone much older than Ezra Weeden.   It also occurred to me that No. 118 could be      The recently murdered Charles Dexter Ward himself. Ward's reference to killing Dr. Allen (who is actually an alias of Curwen) and dissolving his body with acid and No. 118's mention of doing the same to Curwen seem to support the idea, but that still doesn't explain the archaic note. Ward is described as writing in a somewhat modern style, at least as opposed to Curwen. It also doesn't explain why Curwen would want to murder Ward if he was going to resurrect him later anyway. Integral of Sinc Function Squared Over The Real Line I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks.  Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anyways. Inspired exercise Sheet from Indesign I would like to create this sheet which was made it by adobe indesign   Could someone create it with latex ?     Here's my code:  \documentclass[12pt,a4paper,twocolumn]{report} \usepackage[margin=0.5in]{geometry} \usepackage[french]{babel} \usepackage{fontspec} \usepackage{graphicx} \usepackage{amsthm,amssymb,amsfonts,mathtools} \usepackage[utf8]{inputenc} \usepackage{multicol} \usepackage{multirow} \usepackage{amssymb} \usepackage{array} \usepackage{graphicx} \setlength{\columnseprule}{0.5pt} \thispagestyle{empty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%head%%%%%%%%%%%%%%%%%%%%%%% \twocolumn[    \begin{@twocolumnfalse}  \begin{center}  \fbox{\Large Exercise Sheets} \end{center}     \hrulefill \end{@twocolumnfalse}   ]  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \fbox{ Exercise $1$:}\\  Compute the following limits \begin{multicols}{2} \begin{enumerate} \item $\lim_{x\to 2}\dfrac{4x^{3}-5x-22}{x^{2}-x-2}$ \item $\lim_{x\to 0^{+}}\dfrac{x-\sqrt{x}}{x+\sqrt{x}}$ \item $\lim_{x\to 1}\dfrac{\sqrt{x+3}+\sqrt{x}-3}{x-1}$ \item $\lim_{x\to 2}\dfrac{x^{2}\sqrt{x+2}-8}{4-x^{2}}$ \end{enumerate} \end{multicols}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \fbox{ Exercise $2$:}\\ Let $n\in\mathbb{N}^{*}$ \begin{enumerate}     \item calculate : ${\displaystyle \lim_{x\to 0}\dfrac{1-\cos(x)\cos(2x)\ldots+\cos(nx)}{x^2} }$     \item b     \item c     \item d     \item e     \item f \end{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document} This is not for homework. (I am a grader for a class.)  The case in which $G$ is finite is trivial. (That is, use a corollary to Lagrange's Theorem, and set $[G:H] = \dfrac{|G|}{|H|}$, and similarly for $[H:K]$.) How do you prove this for when $G$ is infinite?   The proof of this statement that I know well tries to show that $[G:K]$ is finite iff $[G:H]$ and $[H:K]$ are finite, but I am not a fan of that proof. For pedagogical purposes, I'm looking for a better proof. (The course uses right cosets by convention.)  The course has just started covering normal subgroups, so things like the Isomorphism Theorems are not known yet.