Rigorous proof of $n\sum_{d\mid n}\frac{1}{d}=\sum_{d\mid n}d$ for $n\in\mathbb N$. Let $n\in\mathbb N$ and consider the sum $$\sum_{d\mid n}\frac{n}{d}.$$ I understand that dividing $n$ by any of it's divisors is also a divisor of $n$ so intuitively it would make sense that $\{\frac n d\ :\ d\mid n \}=\{d\ :\ d\mid n\}$, from which the desired equality in the title is acquired. but I cannot &quot;see&quot; how this can be proven rigorously. Any help or insight would be appreciated! Edit: In many sources, including the question this one is marked as a possible duplicate to, it is not explained why the identity holds, it's just used. If it actually is explained, it's not clear to me, at least. I am interested in an actual explanation, that's why my question differs from the one marked as duplicate.
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$. Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.   where $\sigma (n)$ is the sum of all the divisors of $n$ and $\sum\nolimits_{d|n} f(d)$ is the summation of $f$ at each $d$ where $d$ is the divisor of $n$.    I have written $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}.......p_k^{\alpha_k}$ then:- $$\begin {align*} \sum\nolimits_{d|n} \frac{1}{d}&amp;=\frac{d_2.d_3......d_m+d_1.d_3......d_m+........+d_1.d_2.d_3......d_{m-1}}{d_1.d_2.d_3......d_m}  \\&amp;=\frac{d_2.d_3......d_m+d_1.d_3......d_m+........+d_1.d_2.d_3......d_{m-1}}{p_1^{1+2+...+\alpha_1}p_2^{1+2+...+\alpha_2}p_3^{1+2+....+\alpha_3}.......p_k^{1+2+....+\alpha_k}}  \\ \end{align*}$$ where $d_i$ is some divisor among the $m$ divisors. Then I cannot comprehend the numerator so that to get the desired result.  Also suggest some other approches to this question.
Is it correct that "etc." can not be used together in a sentence with "for example" and "such as"? I just read an article from a Chinese website for English teaching which mentions  that point. For instance, one can't say:     "I can play quite a few musical instruments, for example, the flute, the guitar, the piano, etc.".    The correct one should be:     "I can play quite a few musical instruments, for example, the flute, the guitar, and the piano."    Is it possible to find any references to prove that? Secure hash and salt for PHP passwords It is currently said that MD5 is partially unsafe. Taking this into consideration, I'd like to know which mechanism to use for password protection.  This question,   suggests that hashing multiple times may be a good idea, whereas  suggests using salt.  I'm using PHP. I want a safe and fast password encryption system. Hashing a password a million times may be safer, but also slower. How to achieve a good balance between speed and safety? Also, I'd prefer the result to have a constant number of characters.   The hashing mechanism must be available in PHP It must be safe It can use salt (in this case, are all salts equally good? Is there any way to generate good salts?)   Also, should I store two fields in the database (one using MD5 and another one using SHA, for example)? Would it make it safer or unsafer?  In case I wasn't clear enough, I want to know which hashing function(s) to use and how to pick a good salt in order to have a safe and fast password protection mechanism.  Related questions that don't quite cover my question: Execute sudo without Password? Singular information matrix error in lrm.fit in R I am running an ordinal logistic regression in R and running into trouble when I include dummy variables.  My model works great with my first set of predictors.  Next I want to add dummy variables for each of the years represented in my dataset.    I created the dummy variables with car:recode in this manner (one statement like this for each of the 11 years)  fsd$admityear2000 &lt;- recode(fsd$ApplicationYear ,"2000=1;else=0")   The lrm model is specified as follows  library(Design) ddist&lt;- datadist(fsd) options(datadist='ddist') m4 &lt;- lrm(Outcome ~ relGPA + mcAvgGPA + Interview_Z + WorkHistory_years + GMAT + UGI_Gourman + admityear1999 + admityear2000 + admityear2001 + admityear2002 + admityear2003 + admityear2004 + admityear2005 + admityear2006 + admityear2007 + admityear2008 + admityear2009, data=fsd)   (sorry for all of the other random variables, but I don't want to introduce confusion by changing my code)  I get the error  singular information matrix in lrm.fit (rank= 22 ).  Offending variable(s): admityear2009 admityear2000 admityear1999  Error in lrm(Outcome ~ relGPA + mcAvgGPA + Interview_Z + WorkHistory_years +  :    Unable to fit model using “lrm.fit”   I understand that including all options of a dummy variable over-defines the model, but I get the error whether I include all 11 years or just 10.  I found a suggestion  to set the penalty parameter of lrm to a small positive value.  Setting it to 1 or 5 changes the error such that it only names one of the variables as offending.  The error doesn't go away even with penalty=100.  I'm pretty new to R, but loving the freedom so far.  Thanks for any help!  Responses and Lessons   Factors are awesome and I can't believe I didn't notice them earlier. Man that cleans up my code a lot.  Thanks! My DV, 'Outcome' is indeed ordinal and after making it a factor(), I also made it ordered(). The str() command is also awesome and this is what my data now looks like (with some of the non-relevant variables omitted)   output:  str(fsd)      Outcome      : Ord.factor w/ 3 levels "0"&lt;"1"&lt;"2" relGPA       : num   mcAvgGPA     : num  admitschool  : Factor w/ 4 levels "1","2","3","4" appyear      : Factor w/ 11 levels "1999","2000",..    both lrm() and polr() now run successfully, and they both deal with appyear by dropping some values of the factor.  lrm() drops 1999, 2000, and 2001 while polr() just drops 1999 and 2000.  lrm() gives no warnings while polr() says "design appears to be rank-deficient, so dropping some coefs."  This is an improvement, but I still don't understand why more than one value needs to be dropped. xtabs shows that there isn't full seperation right?   output:  xtabs(~fsd$appyear + fsd$Outcome)            fsd$Outcome fsd$appyear    0    1    2        1999 1207  123  418        2000 1833  246  510        2001 1805  294  553        2002 1167  177  598        2003 4070  158 1076        2004 2803  106 1138        2005 3749  513 2141        2006 4429  519 2028        2007 6134  670 1947        2008 7446  662 1994        2009 4411   86 1118 Arrow with plus/minus sign on top Below you can see an arrow with a plus and a minus sign on top of it. How is this achievable in latex? Further possibilities: arrow with only plus and arrow with only minus sign on top. How can I do this? Is there a convergent, alternating series that fails the AST? The alternating series test (AST) says, briefly, that if   $a_k&gt;0$ $a_k \geq a_{k+1}$, and $a_k \to 0$ as $k \to \infty$   then $\sum_k (-1)^k a_k$ converges.  This seems to be a one-way test (that is, if an alternating series fails the test, we don't know that it diverges).   document says so explicitly. It then gives an example of an alternating series which fails the AST. That doesn't prove the series is divergent (but it turns out to be divergent, anyway, by $n$th term test).  Is there a convergent, alternating series that fails the AST?  Of course, if the series fails condition 3, then it also fails the $n$th term test, and must diverge. And if it fails the first condition, then it's not strictly alternating, anyway.  So it must be a series that is not getting smaller (condition 2), but still converges. $$\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$$  I'm not entirely sure how to go about this, since rationalizing anything doesn't quite work for me. Just looking for some help. I read on here of an exercise in interviews known as validating a binary search tree.  How exactly does this work? What would one be looking for in validating a binary search tree? I have written a basic search tree, but never heard of this concept. Is this series known to converge, and if so, what does it converge to (if known)?  Where $p_n$ is prime number $n$, and $p_1 = 2$,  $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$ What can I do to increase passive HF RFID read range? I am working on a prototype, one function of which is to detect whether an object is "in range" by a few cm. Right now I'm just using an Arduino and a cheap  that operates at 13.56 MHz:    I've successfully gotten it to detect the tag, but only at a distance of about 3-4 cm when the tag is oriented parallel to the antenna.  According to my  , the HF band allows read ranges of between 10 cm and 1 m for passive tags, depending on factors like antenna size, tag size and orientation, and transceiver power.  I realize this inexpensive reader is not up to task as-is, but I'm interested in building something that achieves read ranges of 15-20 cm.  Some criteria:   No need to read multiple tags; only whether one tag is in range or not. No need to write to the tag. Cannot use an active tag.   I'm not experienced with RFID reader construction (yet) but am interested in knowing what I should pursue.  Here are some things I've considered:   Ditch the embedded antenna on the cheap RFID reader and construct my own, larger antenna. I'd need to see if the onboard  is up to the task of providing additional power. Look for a different IC around which to build a reader from scratch. Find a ready-made solution instead of trying to build one (more cost-effective?). Use something other than RFID for simple proximity detection.   I am experienced with microcontrollers and basic electronics, and have made audio and lighting projects, but this is a first with RFID. Since I don't need to read multiple tags, which RFID is suitable for, I've considered maybe some sort of magnet and hall-effect sensor, but the range does not seem adequate. I considered using an ultrasonic proximity detector, but the application calls for detecting the tagged object while ignoring untagged ones (ultrasonic would produce false positives on untagged objects). I've considered some sort of reflective surface placed on the object, and detecting reflected light, but the object orientation is not reliable enough to maintain alignment.  So, in a nutshell, What modifications can I make to my existing reader, or what alternate technology should I pursue to achieve 15-20 cm detection of a passively tagged object?  I plan on connecting whatever proximity detector is used to a microcontroller using SPI. Suppose $f$ and $g$ are Lebesgue measurable, we want to show $f+g$ is measurable. So, the hint is to consider the continuous functions $F : \mathbb{R}^2 \to \mathbb{R} $ given by $h(x) = F(f ,g ) $. If we can show $F$ Is measurable, then Taking $F = f +g $ would solve our problem.  In other words, I want to show that the set $R = \{ (f,g) : F(f,g) &gt; a $ } is lebesgue measurable.. But this set is just a rectangle in the plane. And since $F$ is continuous, then $R$ must be open, and hence a union of open rectangles which are measurable and hence $R$ must be measurable. Is this a correct approach to the problem? Can someone help me to make this formal? thanks Pivot in sql server Is static HTML the fastest? Disk brakes severely reduced in power after replacing bearings How to increase swap space? Proper notation for distinct sets Consider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of both ⊄ and ⊅ ?  Also, is there a notation to mean that two sets are completely distinct and share no elements in common (for example, the sets of odd and even integers)?  ≠ seems very inappropriate, and I'd like to avoid it.  Context: My area is actually Computer Science. I'm using this refine my vocabulary in order to teach my students about  (which is basically about sets and subsets of IP addresses), and to write some exercises for them. I'll probably use the words incomparable and disjoint as suggested. Reflection of a line in a plane The line $l_1$ has the equation $r=(6i+2-2k)+\lambda(4i+5j-k)$ and the plane $\pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $\pi_1$. Find the exact vector equation of line $l_2$   So the line intersects the plane when $\lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated. I've installed ubuntu 17.04 and I want to upgrade it to 18.04. When I tried to do:   sudo do-release-upgrade   I got a message saying An upgrade from 'zesty' to 'bionic' is not supported with this tool. Need inputs to proceed on this.  sudo do-release-upgrade Checking for a new Ubuntu release Your Ubuntu release is not supported anymore. For upgrade information, please visit: http://www.ubuntu.com/releaseendoflife  Get:1 Upgrade tool signature [819 B]                                            Get:2 Upgrade tool [1,257 kB]                                                   Fetched 1,258 kB in 0s (0 B/s)                                                  authenticate 'bionic.tar.gz' against 'bionic.tar.gz.gpg'  extracting 'bionic.tar.gz'  Reading cache  Checking package manager  Can not upgrade   An upgrade from 'zesty' to 'bionic' is not supported with this tool. Suppose I have an Atwood machine, that is, two different masses connected with an inextensible, massless rope over a pulley. Assuming no friction between the rope and the pulley, the heavier mass will accelerate towards the ground, the lighter mass will accelerate towards the pulley, and the rope will accelerate towards the heavier mass. These three accelerations will be equal in magnitude. But this makes no sense to me. Force causes acceleration. But there is no force acting on the rope. And even if there was, the acceleration of the rope would be infinite because its mass is 0. So why does the rope accelerate? And how can the magnitude of this acceleration be finite? How does Meta Stack Exchange work? Meta Stack Exchange seems to have different rules and user behavior than the "regular" Stack Exchange sites.  How is it different?    Individual questions answered below:                Related