How do you use block in other hand
How to use an item in my off-hand?
n-dimensional vector space isomorphic to the n-th product of a field as abelian groups Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:  Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!   I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help. Let $A$ be a UFD. Assume that $a,b \in A$ are relatively prime, $c \in A$ and $a | bc$. To prove that $a|c$, is the following approach correct (or do you have to use some type of prime factorization argument)?  By the relatively prime assumption, $\gcd(a,b)$ is a unit. Call that unit $u \in A^{\times}$. Then we can write $u = ax + by$ for some $x,y \in A$. But then, $c = c \cdot u = c(ax+by)=cax+cby$. However, it is clear that $a|cax$ and the fact that $a|bc$ implies $a|cby$. Thus, $a|cax+cby=c$. "Z-level" in TikZ We have this large database (>1TB) that we intend to "shrink".  The database revolves around one main entity, let's call it "Visit".  For discussion, let's say it is a database for a medical practice.  There are a total of 30 visit "types", such as procedure, annual, follow-up, immunisation etc, each of which is a subsidary table to "Visit", e.g. "visit_immuno".  The database has accummulated some 12 years of data since 2000.  Someone has proposed that we keep about 3 years of data in the "live" version and have the rest live in an "old_data" database.  The date is ONLY stored in the "Visit" table since it is normalised.  The Visit table also contains a ROWVERSION column and a BIGINT pseudo-identity (clustered) column.  For all intents and purposes, let's say the clustering key is populated by a SEQUENCE (SQL Server 2012 Enterprise) - we shall name it cid.  The visit.date is not always in the same order as the clustering key, for example when a doctor goes on extended visitations and returns with his "briefcase" of data, it gets merged into the main table.  There are also some updates to the "visit" table that will cause the ROWVERSION column to be out of sync with both the cid and date columns - to put it simply, neither ROWVERSION nor cid would make suitable partition keys for this reason.  The business rule for removing data from the "live" is that the visit.date must be greater than 36 months and a child visit_payment record must exist.  Also, the "old_data" database does not contain any of the base tables except visit%.  So we end up with:  Live DB (daily use) - All tables Old-Data DB - older data for the visit% tables  The proposal calls for a Combined DB that is a shell containing Synonyms to ALL the base tables in the Live DB (except visit%) plus Views that UNION ALL across the visit% tables in the two databases.  Assuming the same indexes are created in the Old-Data DB, will the queries perform well on the UNION-ALL Views?  What type of query patterns might trip up the execution plan for the UNION-ALL Views? Understanding central limit theorem What is wrong with the following sentence:      The Central Limit Theorem  implies that, as the sample size grows, the error distribution approaches normality.   Am I correct by saying that it should in stead state the MEAN of the sample error approaches zero as sample size grows? I know Americans commonly use &quot;damn&quot; or &quot;damned&quot; to describe things. Sometimes, more appropriately, it's even &quot;darn&quot; or &quot;darned&quot;. For example,  This damn/damned computer is too slow.  However, I can't wrap my head around when to use &quot;damn&quot; and when to use &quot;damned&quot;. Or is only one of them correct? Given a group $ G $, how many topological/Lie group structures does $ G $ have? Equivalence Relation with dividing x and y integers Define $x\sim y$ means 5 divides $(x - y)$ for $x$ and $y$ integers. Show that is an equivalence relation.  Equivalence relation means it satisfies reflexity, symmetry, and transitivity.  reflexive: $x\sim x$ means 5 divides $x$  symmetry: $x \sim y \rightarrow y\sim x$ means 5 divides $x - y$ and 5 divides $y - x$: $$5/(x - y) = 5/(y - x)$$ so symmetry is satisfied.  I am not sure if I am right here and I am lost on how to prove it is transitive any suggestions would be greatly appreciated I remember (vaguely) reading somewhere that it is wrong to use the present perfect when the subject of a sentence is someone who is long dead:     *Einstein has visited the Philippines.   My question is, What if the dead person is not the subject but an object:     Leo is the best president our country has/had ever had. (The ones who   succeeded him were inferior, even our current president.)   Which tense is proper here I never liked that SO/SE relied on the external gravatar service, so I was glad to . However, it turns out that one cannot use as profile picture a PNG file with transparent background (something that is supported by gravatar):    Above is a screen capture of my profile page, with a profile picture that has a transparent background. That background is turned into black, which I suppose is a bug.  The method of upload doesn't change anything (I uploaded it manually to imgur, but the same black background is observed). See the difference between the two following URLs:  http://i.stack.imgur.com/Da3ex.png?s=128&amp;g=1   and  http://i.stack.imgur.com/Da3ex.png   The problem is thus in the resizing code from imgur. Community Wiki posts don't change usernames if the user does According to the well ordering theorem "". Whenever we have a well ordering on a set, it is not difficult to construct a new well ordering with a largest element. My question is:     For a given infinite set, can we find a well ordering on it such that there is no largest element? As a programmer, I have no problem with seeing or using "and/or" in technical documentation. For example,     I can upvote an answer that satisfies me and/or mark it as accepted.   That's perfectly good English to me. However, if I were writing a novel, or even just an essay, or — heaven forbid — a poem, "and/or" would seem extremely out of place.  My question is, what should I be using instead? (I solemnly swear I will not use that knowledge to actually write a poem.) If $f(\frac{1}{n})=0$ all $ n\to \mathbb{Z^{+}}$ then find $ f'(0)$? Let $f: \mathbb{R}\to \mathbb{R}$ be a twice differntiable function and $f(\frac{1}{n})=0$ all $n\in \mathbb{Z^{+}}$? then which of the statement (s) are correct from below.  (a)$f(0)=0$  (b)$f^{'}(0)=0$  (c)$f^{''}(0)=0$  (d)$f$ is a non zero polynomial  I tried of using Tyler series but I wasn't able to do this any help would be appreciated thank How can I determine the list of files in a directory from inside my C or C++ code?  I'm not allowed to execute the ls command and parse the results from within my program. So, i want to make a timer that will turn on/off a circuit. I want to turn on the connected device at 08:00 and stop it at 20:00, so the interval is 12 hours. Correct me if i'm wrong but i can't use IC 555(or another IC), i'll need some huge resistors/capacitors. Is there another way to build that other than using a microcontroller? I'm trying to do a number checker in imessage. We all know that there is no such api, so I'm trying to use the tools at hand. The main took with past issues on this site on this topic.  I made a loop that inserts a number into a string, and by the color of the number (blue - registered, red - missing), using the standard utility Digital Colors Meter determines the color of the number. Then, depending on the color, it either adds a variable with a number from the initial txt file to the new one, or starts the script from the beginning. At least it should be.  But something is going wrong in loop. It checks the color, inserts numbers, everything is fine, but apparently the wrong condition, apparently the wrong data types. It inserts different numbers into the scatter. Some will insert, some will not, regardless of the initial condition. What could be the reason? Where is the error?      tell application "Digital Color Meter" to activate     tell application "System Events"         tell process "Digital Color Meter"             click menu item 1 of menu 1 of menu bar item 4 of menu bar 1         end tell         do shell script "/usr/bin/python -c \"import objc;bndl = objc.loadBundle('CoreGraphics', globals(), '/System/Library/Frameworks/ApplicationServices.framework');objc.loadBundleFunctions(bndl, globals(), [('CGWarpMouseCursorPosition', 'v{CGPoint=dd}')]);CGWarpMouseCursorPosition((" &amp; 1002 &amp; "," &amp; 222 &amp; "));\""         set rgbColors to words of (the clipboard)     end tell     if rgbColors is equal to {"77", "39", "38"} then         exit repeat     else         do shell script "echo " &amp; quoted form of n &amp; " &gt;&gt; /Users/oneone/Desktop/base2.txt"     end if   Only the color check condition for Digital Color Meter is presented here.  Through the do shell, the color is read at the mouse position and the variable is added to the new file. I am looking to implement a biplot for principal component analysis (PCA) in JavaScript. My question is, how do I determine the coordinates of the arrows from the $U,V,D$ output of the singular vector decomposition (SVD) of the data matrix?  Here is an example biplot produced by R:  biplot(prcomp(iris[,1:4]))     I tried looking it up in the  but it's not very useful. Or correct. Not sure which. What provokes an opportunity attack?  In 3.X moving out of a threatened square, standing up from prone, casting a spell (unless cast defensively), or making a ranged attack provoked, while in 4e moving out of a threatened square or making a ranged/area attack provoked. Installation of Ubuntu 16.04 blocked: cannot find hard disk I am trying to install Ubuntu 16.04 LTS on a Dell XPS 13 (9350). The laptop came with Windows 10 pre-installed. I am installing from a pendrive but I think the issue would be the same if I used e.g. a USB DVD drive.   Dude, where's my hard drive?  I am blocked because the installation program cannot find 8.6Gb of free space on my hard drive despite the fact that there are hundreds of Gb free according to Windows. Digging a bit into the problem, it seems that Ubuntu cannot see my hard disk at all.    This is the output of GParted:    I am not sure what that drive mounted at /cdrom is, but it seems the only device that exists in the system that is not a RAM disk or a loop device:   Content of /proc/partitions:   ubuntu@ubuntu:~$ cat /proc/partitions   major minor  #blocks  name     1        0      65536 ram0     1        1      65536 ram1     1        2      65536 ram2     1        3      65536 ram3     1        4      65536 ram4     1        5      65536 ram5     1        6      65536 ram6     1        7      65536 ram7     1        8      65536 ram8     1        9      65536 ram9     1       10      65536 ram10     1       11      65536 ram11     1       12      65536 ram12     1       13      65536 ram13     1       14      65536 ram14     1       15      65536 ram15     7        0     130048 loop0     7        1    1104188 loop1     8        0    1970239 sda     8        1    1953439 sda1   Summary of fdisk -l:   ubuntu@ubuntu:~$ sudo fdisk -l |grep Disk Disk /dev/ram0: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram1: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram2: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram3: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram4: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram5: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram6: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram7: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram8: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram9: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram10: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram11: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram12: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram13: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram14: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram15: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/loop0: 127 MiB, 133169152 bytes, 260096 sectors Disk /dev/loop1: 1.1 GiB, 1130688512 bytes, 2208376 sectors Disk /dev/sda: 1.9 GiB, 2017525248 bytes, 3940479 sectors Disklabel type: dos Disk identifier: 0x00000000   Result of fdisk of the only non-loop non-RAM device (not sure this is relevant):   ubuntu@ubuntu:~$ sudo fdisk -l |grep -A 50 sda Disk /dev/sda: 1.9 GiB, 2017525248 bytes, 3940479 sectors Units: sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disklabel type: dos Disk identifier: 0x00000000  Device     Boot Start     End Sectors  Size Id Type /dev/sda1  *      129 3907007 3906879  1.9G  c W95 FAT32 (LBA)   I am blocked at this point. Any help will be much appreciated.   Additional information (following comments):   Running dmesg |grep -i sata shows (among others) eight lines that suggest some kind of error:   ata1: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata2: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata3: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata4: SATA link down (SStatus 0 SControl 0) ata5: SATA link down (SStatus 0 SControl 0) ata6: SATA link down (SStatus 0 SControl 0) ata7: SATA link down (SStatus 0 SControl 0) ata8: SATA link down (SStatus 0 SControl 0)  I have "Restarted" windows instead of only "Shutdown" it to avoid problems with "fast startup", following some recommendations I have read of other users having problems installing Linux on Windows 10 laptops.ago   Regarding SATA, the BIOS says: "SATA Operation: RAID ON". The other two options available are "Disabled" and "AHCI".    I have followed the instructions of these two pages from Dell: , .