Difference between +1+1 and +1+1 counters
Are +1/+1 counters different from +1/+1 effects?
How do you do AppBar docking (to screen edge, like WinAmp) in WPF? About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$ Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$.  Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} $$  Show that:  1) $f$ is strictly increasing  2) $f$ is continuous at $x$ iff $x \in \mathbb{R} \setminus \mathbb{Q}$.  My work:  Let $x, y \in \mathbb{R}$, $x &lt; y$. Then there exists $q_n \in \mathbb{Q}$ s.t. $x &lt; q_n &lt; y$. Now $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} &lt; \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} + 2^{-n} \leq \underset{q_k \leq y}{\sum_{k \in \mathbb{N}}} 2^{-n} = f(y) $$ so $f$ is strictly increasing.  I think my proof of 1) is ok, but how can I prove 2)? I'm trying to prove that  $$ \left| \frac{1}{x}-\frac{1}{y} \right| $$  is a metric on $X = \mathbb R^+$.  The properties of non-negativity and symmetry are obvious (do I actually even need to prove them?), but the tricky one is, as usual the inequality property, i.e.  $$ d(x,z) \le d(x,y) + d(y,z) \text{ for any } z \in X. $$  So, ok, I have that  $$ d(x,z) \le d(x,y) + d(y,z) \\ \iff \left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right| \\ \iff \left| \frac{1}{x} \right|- \left| \frac{1}{z} \right| \le \left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right| $$  ...and that's about as far as I've gotten–which is to say, not very far. I also tried multiplying both sides by $ \left| \frac{1}{x}-\frac{1}{z} \right| $, but that wasn't very fruitful. If someone could point me in the right direction, it'd be greatly appreciated. Thanks. Tracking email bounces, opens, clicks How do you make the shapefile a Read-only so the client can not make any edits using 10.2.2 ArcGIS.  It is not within a geodatabase, just a stand alone shapefile.  I am using Windows 8 Should individual letters be preceded with "an"? I was reading a comic book and I noticed that the author used "an A" and "an N" in a particular phrase (screenshot below). While I understand the usage of "An A" here (A being a vowel and what not) and "N" having a pronunciation which has a sound similar to "A".  Since most of English alphabets have pronunciations which start with a syllable similar to one of the vowels or semi-vowels; is it OK to prefix the letters using "An"?  For eg. An X, or An Y.        Zenescope's Grimm Fairy Tales - Werewolves: The Hunger, page #8 (Click to enlarge) Can we have cross-site badges? How do I paste into the Windows CMD prompt by using only the keyboard? My current technique is to right-click in the command prompt and then click paste.  This bugs me - is there a better way? How to get status start_date and end_date from status_track table? I have two tables:   things:  +------+----------------+------------+ | id   | current_status | created_at | +------+----------------+------------+ | 7770 | active         | 2016-08-09 | +------+----------------+------------+  thing_status_tract:  +----------+-------------+-----------+---------------------+ | thing_id | status_from | status_to | changed_at          | +----------+-------------+-----------+---------------------+ |     7770 | incomplete  | inactive  | 2016-08-09 16:26:22 | |     7770 | inactive    | active    | 2016-08-10 12:31:04 | +----------+-------------+-----------+---------------------+    I need data in the following form. This table has a status and its corresponding start and end timestamp for a particular thing_id:  +----------+-------------+---------------------+---------------------+ | thing_id | status      | status_start_date   | status_end_date     | +----------+-------------+---------------------+---------------------+ |     7770 | incomplete  | 2016-08-09 00:00:00 | 2016-08-09 16:26:22 | |     7770 | inactive    | 2016-08-09 16:26:22 | 2016-08-10 12:31:04 | |     7770 | active      | 2016-08-10 12:31:04 | now()               | +----------+-------------+---------------------+---------------------+   How to do that with a SQL query? Where to find cross section data for $e^{-}$ + $p$ $\longrightarrow$  $p$ + $e^{-}$ ?  does not include it. How do I install Ubuntu 16.04 from an SD card? Could someone explain how to install Ubuntu from an SD card? I don't have any CDs or USB sticks around. My current operating system is Windows 10 64 bit. Compute the (multiplicative) inverse of $4x+3$ in the field $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$? So I am finding a polynomial $px+q$ ($p,q \in \Bbb F_{11}$) which is multiplicative inverse of $4x+3$ in $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$.  i.e. $[(4x+3)+\langle x^2+1 \rangle][(px+q)+\langle x^2+1 \rangle]=1+\langle x^2+1 \rangle$   $\Rightarrow$  $(4x+3)(px+q)+\langle x^2+1 \rangle=1+\langle x^2+1 \rangle$  $\Rightarrow$  $4px^2+(4q+3p)x+3q+\langle x^2+1 \rangle=1+\langle x^2+1 \rangle$.  We see that the remainder,when $(4x+3)(px+q)$ is divided by $x^2+1$, is $1$. So by Division algorithm,  $$ \require{enclose} \begin{array}{r} 4p \\[-3pt] x^2+1 \enclose{longdiv}{4px^2+(4q+3p)x+3q} \\[-3pt] \underline{4px^2+4p} \\[-3pt] (4q+3p)x+(3q-4p) \\[-3pt] \end{array} $$    So I equate $(4q+3p)x+(3q-4p)=1$ and solve the simultaneous linear equations  $4q+3p=0, 3q-4p=1$. I get $p=6,q=1$  Hence $6x+1$ is the required inverse.  I am pretty sure that the answer is correct but is the method to achieve it right? Richard Stallman states that Ubuntu "has installed surveillance code". See:   Is this true?   I agree with him that "Any excuse Canonical offers is inadequate....". I don't understand how quantum mechanics (and therefore also quantum computers) can work  given that while we work with quantum states, particles that this quantum state consist of cannot be observed, which is the most fundamental requirement.  If I am not mistaken, by "observed" we mean interaction with any other particle (photon, gluon, electron or whatever else). So my very important questions:   Aren't the particles this quantum state consists of interacting with each other? Why doesn't that cause the state to collapse? Aren't all particles in the universe interacting with Higgs field and gravitons etc? Why doesn't that cause every quantum state to collapse?   I feel there is something very fundamental in quantum mechanics that I am not aware of, hence I would be very pleased to have these questions answered. The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction?  Thanks Installing File Geodatabase (*.gdb) support in QGIS? I have spent around 2 days to find the way to open GDB (Esri geodatabase) in QGIS (or any other open source software) but still without success.  I have downloaded the newest OSGeo4W installer and tried the setup - express desktop install - all packages, as well as advanced install incl gdal-filegdb.  Can you describe a more detailed procedure, including installation and how to open .gdb in QGIS (OSGeo4W installation)? Is AirDrop possible between iPad, iPhone and MacBook? A new MacBook came with something called AirDrop and worked between many MacBooks such as MacBook Pro and MacBook Air. Can I also use AirDrop to exchange data between Macs and iOS devices? Sum of nilpotent and element is a unit in ring? I can't tell where I went wrong, am still a beginner.    Let N be a nilpotent in a commutive ring , let X be any element.   I'll will be showing X + N is a unit .   Assume Y(X+N) =1   Then YX(N^n-1) =  N^n-1   Multiplying both side by  X+N   We get X(N^n-1) =  X(N^n-1)  Thus nilpotent summed with any element gives unit. An example for a homomorphism that is not an automorphism Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function.  I know that for $f$ is differentiable in $(a,b)$ and that it has one-sided derivative $f_+' (a)≠f_-' (b)$ at the endpoints. If $C$ is a real number between $f_+' (a)$  and $f_-' (b)$, then there exists $c∈(a,b)$ such that $f' (c)=C $. How can I use this to prove the above?