SQL Injection with $db->query($sql)?
How can I prevent SQL injection in PHP?
Characterization of weak convergence in $\ell_\infty$ Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets?   I was only able to come up with a characterization of sequential weak convergence using limit along ultrafilters, which I will describe below. I wonder whether there is some insight into this characterization. (E.g. whether there is some simple reformulations which does not use ultralimits.) Moreover, I do not know whether at least this characterization works for nets, too.  We have the following result describing weak convergence in $C(K)$, see e.g. Corollary 3.138,  in Banach Space Theory by Fabian, Habala et al. (It is a consequence of Rainwater theorem and characterization of extreme points of unit ball in $C(K)^*$.)     Let $K$ be a compact topological space. Let $\{f_n\}$ be a bounded sequence in $C(K)$ and $f\in C(K)$. Then, if $f_n\to f$ pointwise, we have $f_n\overset{w}\to f$.   Moreover we have isometric isomorphism between $\ell_\infty$ and $C(\beta\mathbb N)$, which is described e.g. in the Wikipedia article on  or in  of Carothers' book A short course on Banach space theory. This isomorphism assigns to each bounded sequence $(x_n)$ the continuous function $\overline x$ on $\beta\mathbb N$ defined by $$\overline x(\mathscr U) = \operatorname{\mathscr U-lim} x_n,$$ where $\operatorname{\mathscr U-lim} x_n$ denotes the  of $x_n$ w.r.t the ultrafilter $\mathscr U$.  Combining the above results we get the following characterization:      Let $f^{(n)},f\in\ell_\infty$. The sequence $f^{(n)}$ converges to $f$ weakly if and only if for every ultrafilter $\mathscr U$    $$\lim_{n\to\infty} \operatorname{\mathscr U-lim} f^{(n)}= \operatorname{\mathscr U-lim} f.$$   (The above claim for principal ultrafilters is just a pointwise convergence. But in the above claim the equality is required for free ultrafilters, too.) Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$ I am given the info that A is an invertible  m by m matrix and B is a m by n matrix. By solving for the null space $ABx=0$, I get $A^{-1}ABx=A^{-1}0$, therefore $Bx=0$. I concluded that the null spaces of A and AB must be equivalent, and therefore the nullity(AB) = nullity(B) = some number p. Therefore rank(AB) = n-p = rank(B)   (because both AB and B are m by n matrices).   Is this proof valid, specifically the null space step? Create backup iPhone with forgotten encryption password I'm trying to make a backup from my iPhone, however it's set to be password protected. I do not know the password, as I've never set one. I've tried all the passwords I know, which I've ever used to make sure I didn't set it.  I've also tried the default 0000 and 1234 codes.   I want to uncheck the encryption checkbox in iTunes, but it requires me to type a password. What else can I try?  I have no problems in removing all backups and start from the scratch somehow, however - I do not want to restore my phone as new and remove all those settings. Problems with fonts I'm an average user of LaTeX, but not proficiency. I'm trying to learn from book templates I've found on the web, but I get this error message:  Trying to make PK font ccicons at 1493 DPI... Running miktex-makemf.exe... miktex-makemf: The ccicons source file could not be found. Running ttf2pk.exe... miktex-makemf: PK font ccicons could not be created.  LaTeX Font Warning: Some font shapes were not available, defaults substituted.  !pdfTeX error: pdflatex.exe (file ccicons): Font ccicons at 1493 not found  ==&gt; Fatal error occurred, no output PDF file produced!   I'm using MiKTeX (over Windows 7), and according to the MiKTeX package manager the "ccicons" package is already installed.  Could you please help me solve this problem? For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous. Polygonizing a raster yields lots of little squares? I have a pdf with the area of a county that I want to separate into multiple shapefiles (polygons) based on color.   As shown in the first picture there are 4 colors so I would need to separate based on color between those four. But first, I need to convert it into a vector (unless there is an easier way).   Doing so, I get the second image as an output by using the polygonize tool which when zoomed in very close, has a lot of tiny squares where color should be different. I was wondering if there is a way to break a raster up by color even though the image isn't smart. Play WAV file backward How check with SOQL if a picklist value is available or not for a specified record type How check if a picklist value is available or not for a specified record type ( through SOQL)? Thanks in advance When moving around the sites on Stack Exchange using Safari on my iPad (not the Stack Exchange App), I am fine when moving to any site other than SuperUser.  When logged in, I can switch sites and stay logged in, but when I switch to Super User I need to log in again. If I choose not to log in and switch to another site, I am logged in there. Is there a problem with the login handling for Super User? If it helps, I log in with a Stack Exchange account, not through Facebook or Google. Update After getting fed up with the problem, I took the advice given in the answer below which was to use Chrome, and the problem still exists. The same goes for Firefox. Have I missed something?  As you can see from the screenshots below, there are no cookie settings to be found. Chrome for iOS settings page  Privacy settings page  Content settings page  Google app settings page I would like to know the model of my graphics card. I think it may be an ATI, but I want to be sure!   I have Ubuntu 11.10 (32 bit) and an Asus A6 VA laptop. Can bounties be awarded multiple times by the same person on the same question? Is there any restriction to prevent multiple bounties ( not simultaneously) being awarded to different answers on the same question ?  i.e., on a question  -  I want to set multiple bounties since I've received 2 awesome answers. Can I set award a bounty on answer A first, and then set the bounty and award it to answer B ? Natural deduction, Proof $\vdash$ $P\Rightarrow(Q\Rightarrow P)$ So I have a question regarding natural deduction, are we allowed to "copy" our previous assumption inside a new assumption. I will use an example to illustrate.  $\vdash$ $P\Rightarrow(Q\Rightarrow P)$  So here are my steps.  1.) $P$             Assumption  2.)$Q$             Assumption  3.)$P$             Copy (1)  4.)$Q \Rightarrow P$   Implication Introduction  5.)$P \Rightarrow (Q \Rightarrow P)$ Implication Introduction  Is my proof correct, or is there something missing. Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$ I'm self studying from Category Theory in Context by Emily Riehl and have encountered the following question regarding slice categories:     Let $\text{C}$ be a category and $c$ be an object in $\text{C}$. Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$    The exercise appears in an chapter that introduces the notion of opposite categories and duality proofs.     My intuition of why the statement is true comes from the following observation:  Let $x,y$ be objects in $\mathbf{C}$ (or equivalently $\mathbf{C}^{\text{op}}$). Let $g^{\text{op}} : c \to y$ and $f^{\text{op}} : c \to x$ be objects in $c/(\mathbf{C}^{\text{op}})$. By the definition of slice category, there exists a morphism $h^{\text{op}} : y \to x$ in $\mathbf{C}^{\text{op}}$ such that the diagram below commutes:  $\hskip2in$   Now the 'corresponding diagram' in the opposite category $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ would be:  $\hskip2in$    which, as I understand it, would also be commutative. Now, to me, this indicates:   that the objects in $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ are the same as the objects in $\mathbf{C}/c$, namely, a morphism in $\mathbf{C}$ from an object $z \in \mathbf{C}$ to $c$. the commutativity of the seconds diagram describes morphisms $h$ in both $\mathbf{C}/c$ and $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ if you reverse the arrows again you return to the same diagram. This makes me think you could construct a map between morphisms by first making a map from $ \mathbf{C}/c \to (\mathbf{C}/c)^{\text{op}} ( \cong c/(\mathbf{C}^{\text{op}}) )$ by $h \mapsto h^{op}$     Now, what I am really looking for is a way to structure a proof. I feel like I have some intuition but am very unsure of how to write it down in a concise way. I am new to category theory so I'm still very much learning how to construct these types of proofs using duality; I feel like what I have written is maybe overkill but I'm trying to be as explicit as possible to make sure I understand everything.  In particular, I don't know how I would get a concrete correspondence between morphisms in the two categories that respects associativity of composition etc.  Any help or pointers would be appreciated! How to merge vertices in Blender? I am attempting to create my first 3D character just for practice. Being one of my first 3d models I going through the learning curve needless to say. In doing so I find myself creating many extra vertices. Where we will say 10 vertices on a rounded shape are spaced out on the equator and on the other north pole end of the edges they are very close. The close end could all be one vertex. Is it possible to say merge all points to one point and delete the rest? How do I dynamically invoke a class method in PHP? How can I dynamically invoke a class method in PHP? The class method is not static. It appears that   call_user_func(...)   only works with static functions?  Thanks. Consequence of the Hahn-Banach Theorem From wikipedia's page on the Hahn-Banach Theorem:  If $V$ is a normed vector space with linear subspace $U$ (not necessarily closed) and if $z$ is an element of $V$ not in the closure of $U$, then there exists a continuous linear map $ψ : V → K$ with $ψ(x) = 0$ for all $x$ in $U$, $ψ(z) = 1$, and $||ψ|| = dist(z, U)^{−1}.$  How is this a consequence of the Hahn-Banach Theorem? Where can I buy an Ubuntu Phone? I need to buy an Ubuntu phone, where can I buy it? It is unavailable everywhere. Can someone define what exactly 'POCO' means? I am encountering the term more and more often, and I'm wondering if it is only about plain classes or it means something more? Why not introduce a donate button in Stack Exchange sites? MySQL strip non-numeric characters to compare I'm looking to find records in a table that match a specific number that the user enters.  So, the user may enter 12345, but this could be 123zz4-5 in the database. I imagine something like this would work, if PHP functions worked in MySQL. SELECT * FROM foo WHERE preg_replace(&quot;/[^0-9]/&quot;,&quot;&quot;,bar) = '12345'  What's the equivalent function or way to do this with just MySQL? Speed is not important.