If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit.   I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 = (u+a)x$. After tried several elements and manipulated $ua = au$, I still couldn't find any clue. Can anybody give me a hint?
Let $R$ be a commutative ring with 1, we define  $$N(R):=\{ a\in R \mid  \exists k\in \mathbb{N}:a^k=0\}$$  and  $$U(R):=\{ a\in R \mid a\mbox{ is invertible} \}.$$  Could anyone help me prove that if $a\in N(R) \Rightarrow 1+a\in U(R)$?  I've been trying to construst a $b$ such that $ab=1$ rather than doing it by contradiction, as I don't see how you could go about doing that.
Sending a message to nil in Objective-C As a Java developer who is reading Apple's Objective-C 2.0 documentation: I wonder what "sending a message to nil" means - let alone how it is actually useful.  Taking an excerpt from the documentation:     There are several patterns in Cocoa   that take advantage of this fact. The   value returned from a message to nil   may also be valid:         If the method returns an object, any pointer type, any integer scalar   of size less than or equal to   sizeof(void*), a float, a double, a   long double, or a long long, then a   message sent to nil returns 0.   If the method returns a struct, as defined by the Mac OS X ABI Function   Call Guide to be returned in   registers, then a message sent to nil   returns 0.0 for every field in the   data structure. Other struct data   types will not be filled with zeros.   If the method returns anything other than the aforementioned value   types the return value of a message   sent to nil is undefined.      Has Java rendered my brain incapable of grokking the explanation above?  Or is there something that I am missing that would make this as clear as glass?  I do get the idea of messages/receivers in Objective-C, I am simply confused about a receiver that happens to be nil. I'm trying to install queXS cati app on my Ubuntu desktop and I installed MySQL server and PHP 5 and I cannot login into MySQL server as root without password:  mysql -u root    it says  ERROR 1045(28000) : Access denied for user 'root@localhost' (using password: no )   But it's okay when I enter mysql -u root -p   I can't figure out what the problem is. How to show that $|D_{2n}| = 2n$ via the presentation? Idiom/Phrase for "results without work/effort" I'd be greatly appreciative of a cleverly devised idiom/phrase that depicts the following concept:     Results without work/effort.   ...and/or...     My Work/Effort; Your Result.   Any idioms/phrases you can recount or conceive would be of use.    Contextual Edit:  The idiom/phrase should be in the context of service offering/hiring or task delegation, not free-loading as most answers thus far have understandably assumed. E.g.     [You hire me for] My Work/Effort; [My work/effort produces] Your   Result. Proof of Divergence for a Sequence How can I make my touchpad scroll either vertically or horizontally, but not both? If I'm browsing the internet and a page has both horizontal and vertical scroll bars it becomes really irritating to scroll with my touchpad. It keeps scrolling horizontally as well as vertically!  I'm using Ubuntu 16.04  and below are touchpad settings  Device 'SynPS/2 Synaptics TouchPad':     Device Enabled (160):   1     Coordinate Transformation Matrix (162): 1.000000, 0.000000, 0.000000, 0.000000, 1.000000, 0.000000, 0.000000, 0.000000, 1.000000     Device Accel Profile (282): 1     Device Accel Constant Deceleration (283):   2.500000     Device Accel Adaptive Deceleration (284):   1.000000     Device Accel Velocity Scaling (285):    12.500000     Synaptics Edges (286):  1769, 5431, 1652, 4658     Synaptics Finger (287): 25, 30, 0     Synaptics Tap Time (288):   180     Synaptics Tap Move (289):   242     Synaptics Tap Durations (290):  180, 100, 100     Synaptics ClickPad (291):   0     Synaptics Middle Button Timeout (292):  75     Synaptics Two-Finger Pressure (293):    282     Synaptics Two-Finger Width (294):   7     Synaptics Scrolling Distance (295): 200, 200     Synaptics Edge Scrolling (296): 0, 0, 0     Synaptics Two-Finger Scrolling (297):   1, 1     Synaptics Move Speed (298): 1.000000, 1.750000, 0.036300, 0.000000     Synaptics Off (299):    1     Synaptics Locked Drags (300):   0     Synaptics Locked Drags Timeout (301):   5000     Synaptics Tap Action (302): 2, 0, 0, 0, 1, 3, 0     Synaptics Click Action (303):   1, 1, 0     Synaptics Circular Scrolling (304): 0     Synaptics Circular Scrolling Distance (305):    0.100007     Synaptics Circular Scrolling Trigger (306): 0     Synaptics Circular Pad (307):   0     Synaptics Palm Detection (308): 0     Synaptics Palm Dimensions (309):    10, 200     Synaptics Coasting Speed (310): 21.120001, 50.000000     Synaptics Pressure Motion (311):    30, 160     Synaptics Pressure Motion Factor (312): 1.000000, 1.000000     Synaptics Resolution Detect (313):  1     Synaptics Grab Event Device (314):  0     Synaptics Gestures (315):   1     Synaptics Capabilities (316):   1, 0, 1, 1, 1, 1, 1     Synaptics Pad Resolution (317): 83, 46     Synaptics Area (318):   0, 0, 0, 0     Synaptics Noise Cancellation (319): 8, 8     Device Product ID (277):    2, 7     Device Node (278):  "/dev/input/event6" How to clean an E: drive (new Recovery drive since Windows 10 1803) on HP laptop At what time will the upcoming Ubuntu release be made available? Ubuntu to Kubuntu switch! I have been running Ubuntu for about a year, now I would like to switch to kubuntu. Is it possible that I can port all the configuration data from the home folder of Ubuntu to kubuntu? (Without causing some serious trouble) If I copy all them contents of the home folder (Including the hidden files and folders) will I be able to get the same application configurations ? I know that copying the Thunderbird folder copies the emails on the newer installation, will it work to other applications? What does "upgrade to 18.04" mean? I'm using Ubuntu 16.04 LTS on an Asus K53E laptop with 8 GB RAM. I check for updates to the software daily. If there are no updates, it asks me if I want to upgrade? What does upgrade exactly mean? Will my personal files still be there? Will it wipe EVERYTHING and give me a completely new installation of Ubuntu? Or will it allow me to keep my files and applications, and merely upgrade the OS?  When I upgrade my Mac or Windows machines, my personal files remain available after the upgrade. Does Ubuntu do the same, or are there completely different "rules"? Referencing List by indexVar in lightning component I have nested list in my lightning component and now I want  to refer records of list using indexVar.  Has anyone done this before?  &lt;aura:iteration items="{!v.records}" var="row" indexVar="rowCnt"&gt;  Like `{!row[rowCnt]}?`  &lt;/aura:iteration&gt; Already the early papers on supergravity were written using computer algebra software to do some calculations. What modern packages do people normally use for doing such calculations? Of course Mathematica and presumable a lot of other programs can be made to do such calculations, I am wondering if there are special packages or tutorials how to do such calculations in them. I want to know if I can pass an argument with an alias command.  for example:  alias d="dmesg|grep -iw usb|tail -5"    Now d will print the last 5 lines. If I want to use d to print a different number of lines, I have to make change in the alias command declaration of d again.  Is there any way I can modify the declaration of an alias so that I don't have to retype the declaration to change the number of lines. Like incorporating passing the number of lines as an argument while declaring alias for d? Or is there some other method to solve this? Apport-gpu-error-intel.py crash Something small goes wrong, and this triggers something slightly bigger, which triggers something slightly bigger, and so on and so forth until you end up with a chain of problems of increasing severity until it ends in catastrophy.  Something like the old saying "for want of a nail the battle was lost", only highlighting that each intermediate issue was a problem in its own right, not just a stepping-stone to the final disaster.  Is there a common word/phrase that specifically describes this? I'm thinking "cascade of ever-increasing problems", but this seems clumsy, and I suspect there might already be a neat word or expression that covers it. Is it the "Stack Exchange Network" or the "Stack Overflow Network"? XGBoost Loss function Approximation With Taylor Expansion Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$ How to unprotect GeneratingFunction How do I get the number of days between two dates in JavaScript? For example, given two dates in input boxes:  &lt;input id="first" value="1/1/2000"/&gt; &lt;input id="second" value="1/1/2001"/&gt;  &lt;script&gt;   alert(datediff("day", first, second)); // what goes here? &lt;/script&gt;