Is the use of upon correct in the following sentence?  There is a book upon the desk.  Also, is there any difference between on and upon in the following?  He stood on/upon the rock.  I'd appreciate your help.
Which of these two is more correct and why?   The sun shines on my little tree. The sun shines upon my little tree.   From what I've read in dictionaries both are acceptable, however I would like to know if there are subtleties that might arise due to, for example, the context.
Need of the cold body in heat engine Why does light bend towards the normal when passing from a rarer to a denser medium? Whenever light rays are entering a denser medium they bend towards the normal.  Why do rays choose the path towards the normal? Why cannot they choose the path away form the normal? Show that $n$ lines separate the plane into  $\frac{n^2+n+2}{2}$ regions if no two of these lines are parallel and no three pass through a common point.   I know we start with the base case, where, if we call the above equation P(n), P(0), for 0 lines would be 0.  But I really have no idea how to begin the inductive step.  How do we know what k+1 we're supposed to arrive at?  Thanks! I am trying to solve the following question which is Ex3 from Arthur Engel, Problem Solving strategies. Here is the question: $\sum_{k=1}^n k^3 {n \choose k}$ and asks to find the sum. I am sincerely having a hard time trying to figure out how to solve it and get $n^2 (n+3)(2)^{n-3}$. I would really appreciate if someone could show the step-by-step process. Thank you! Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$ What's the difference between Google Chrome and/or Chromium? What are the advantages/disadvantages to each? I used Google Chrome in Windows, and am now using Chromium in Ubuntu.  Is there any difference to the two programs?  What are the advantages and disadvantages to each program?  Which one seems better? Can a spell be prepared once and cast multiple times? Only charge capacitor when button pushed then turn on LED momentarily with capacitor when button released I am trying to make circuit to turn on an LED momentarily. I would like to charge the capacitor when the button is pushed, then when released turn on the LED momentarily with the capacitor. SEO and URL Shorteners I'm getting the following error when trying to update a taxonomy field value via REST api:  {   "error": {     "code": "-1, Microsoft.SharePoint.Client.ClientServiceException",     "message": {       "lang": "en-US",       "value": "Cannot deserialize data for type Microsoft.SharePoint.Taxonomy.TaxonomyFieldValueCollection."     }   } }   The request:  https://server/_api/web/GetFileByServerRelativeUrl('/Shared Documents/file.txt')/ListItemAllFields  {   "__metadata": {     "type": "SP.Data.Shared_x0020_DocumentsItem"   },   "Title": "some title",   "Project": {     "results": [       {         "Label": "Term1",         "TermGuid": "897bdf7f-08d6-44b9-a71a-ddc4cf131664",         "WssId": 5       }     ]   } } Are all algebraic integers with absolute value 1 roots of unity? Validationengine jquery not working properly for localization In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual? I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as well, but not canonically."  I've finally sat down to work through this, and -  Okay, yes, it is easy to see that the "canonical isomorphism" from $V$ to $V^{**}$ is a functor that has a natural isomorphism (in the sense of category theory) to the identity functor.  Also, I see that there is no way that the functor $V\mapsto V^*$ could have a natural isomorphism to the identity functor, because it is contravariant whereas the identity functor is covariant. My question amounts to:     Is contravariance the whole problem?   To elaborate:   I was initially disappointed by the realization that the definition of natural isomorphism doesn't apply to a pair of functors one of which is covariant and the other contravariant, because I was hoping that the lack of a canonical isomorphism $V\rightarrow V^*$ would feel more like a theorem as opposed to an artifact of the inapplicability of a definition.  Then I tried to create a definition of a natural transformation from a covariant functor $F:\mathscr{A}\rightarrow\mathscr{B}$ to a contravariant functor $G:\mathscr{A}\rightarrow\mathscr{B}$. It seems to me that this definition should be that all objects $A\in\mathscr{A}$ get a morphism $m_A:F(A)\rightarrow G(A)$ such that for all morphisms $f:A\rightarrow A'$ of $\mathscr{A}$, the following diagram (in $\mathscr{B}$) commutes:  $$\require{AMScd}\begin{CD} F(A) @&gt;m_A&gt;&gt; G(A)\\ @VF(f)VV @AAG(f)A\\ F(A') @&gt;&gt;m_{A'}&gt; G(A') \end{CD}$$  This is much more stringent a demand on the $m_A$ than the typical definition of a natural transformation.  Indeed, it is asking that $m_A=G(f)\circ m_{A'}\circ F(f)$, regardless of how $f$ or $A'$ may vary. Taking $\mathscr{A}=\mathscr{B}=\text{f.d.Vec}_k$, $F$ the identity functor and $G$ the dualizing functor, it is clear that this definition can never be satisfied unless $m_V$ is the zero map for all $V\in\text{f.d.Vec}_k$ (because take $f$ to be the zero map). In particular, it cannot be satisfied if $m_V$ is required to be an isomorphism.     Is this the right way to understand (categorically) why there is no natural isomorphism $V\rightarrow V^*$?   As an aside, are there any interesting cases of some kind of analog (the above definition or another) of natural transformations from covariant to contravariant functors?  Note: I have read a number of math.SE answers regarding why $V^*$ is not naturally isomorphic to $V$. None that I have found are addressed to what I'm asking here, which is about how categories make the question and answer precise. ( was closest.) Hence my question here. Standard deviation of binned observations I have a dataset of sample observations, stored as counts within range bins. e.g.:  min/max  count 40/44    1 45/49    2 50/54    3 55/59    4 70/74    1   Now, finding an estimate of the average from this is pretty straight forward.  Simply use the mean (or median) of each range bin as the observation and the count as a weight and find the weighted average:  $$\bar{x}^* = \frac{1}{\sum_{i=1}^N w_i} \sum_{i=1}^N w_ix_i$$  For my test case, this gives me 53.82.  My question now is, what's the correct method of finding the standard deviation (or variance)?  Through my searching, I've found several answers, but I'm unsure which, if any, is actually appropriate for my dataset.  I was able to find the following formula both  and .  $$s^{2*} = \frac{ \sum_{i=1}^N w_i (x_i - \bar{x}^*)^2 }{ \frac{(M-1)}{M} \sum_{i=1}^N w_i }$$  Which gives a standard deviation of 8.35 for my test case.  However, the Wikipedia article on  gives both the formula:  $$s^{2*} = \frac{ \sum_{i=1}^N w_i}{(\sum_{i=1}^N w_i)^2 - \sum_{i=1}^N w_i^2} \sum_{i=1}^N w_i(x_i-\bar{x}^*)^2$$  and  $$s^{2*} = \frac{1}{(\sum_{i=1}^N w_i) - 1} \sum_{i=1}^N w_i(x_i-\bar{x}^*)^2$$  Which give standard deviations of 8.66 and 7.83, respectively, for my test case.  Update  Thanks to @whuber who suggested looking into Sheppard's Corrections, and your helpful comments related to them.  Unfortunately, I'm having a difficult time understanding the resources I can find about it (and I can't find any good examples).  To recap though, I understand that the following is a biased estimate of variance:  $$s^{2*} = \frac{1}{\sum_{i=1}^N w_i} \sum_{i=1}^N w_i(x_i-\bar{x}^*)^2$$  I also understand that most standard corrections for the bias are for direct random samples of a normal distribution.  Therefore, I see two potential issues for me:   These are binned random samples (which, I'm pretty sure, is where Sheppard's Corrections come in.)  It's unknown whether or not the data is for a normal distribution (thus I'm assuming not, which, I'm pretty sure, invalidates Sheppard's Corrections.)   So, my updated question is; What's the appropriate method for handling the bias imposed by the "simple" weighted standard deviation/variance formula on a non-normal distribution? Most specifically with regards to binned data.  Note: I'm using the following terms:     $s^{2*}$ is the weighted variance   $N$ is the number of observations.  (i.e. the number of bins) $M$ is the number of nonzero weights.  (i.e. the number of bins with counts) $w_i$ are the weights  (i.e. the counts) $x_i$ are the observations.  (i.e. the bin means) $\bar{x}^*$ is the weighted mean. How to both display a command line's output on console and save the output into a text file? jQuery Validation plugin: disable validation for specified submit buttons I have a form with multiple fields that I'm validating (some with methods added for custom validation) with Jörn Zaeffere's excellent jQuery Validation plugin. How do you circumvent validation with specified submit controls (in other words, fire validation with some submit inputs, but do not fire validation with others)? This would be similar to ValidationGroups with standard ASP.NET validator controls.  My situation:  It's with ASP.NET WebForms, but you can ignore that if you wish. However, I am using the validation more as a "recommendation": in other words, when the form is submitted, validation fires but instead of a "required" message displaying, a "recommendation" shows that says something along the line of "you missed the following fields.... do you wish to proceed anyways?" At that point in the error container there's another submit button now visible that can be pressed which would ignore the validation and submit anyways. How to circumvent the forms .validate() for this button control and still post?  The Buy and Sell a House sample at  allows for this in order to hit the previous links, but it does so through creating custom methods and adding it to the validator. I would prefer to not have to create custom methods duplicating functionality already in the validation plugin.  The following is a shortened version of the immediately applicable script that I've got right now:  var container = $("#&lt;%= Form.ClientID %&gt; div.validationSuggestion");  $('#&lt;%= Form.ClientID %&gt;').validate({               errorContainer: container,     errorLabelContainer: $("ul",container),     rules: {         &lt;%= YesNo.UniqueID %&gt;: { required: true },         &lt;%= ShortText.UniqueID %&gt;: { required: true } // etc.      },     messages: {         &lt;%= YesNo.UniqueID %&gt;: 'A message.',         &lt;%= ShortText.UniqueID %&gt;: 'Another message.' // etc.     },     highlight: function(element, errorClass) {         $(element).addClass(errorClass);         $(element.form).find("label[for=" + element.id + "]").addClass(errorClass);         $(element.form).find("label[for=" + element.id + "]").removeClass("valid");     },     unhighlight: function(element, errorClass) {         $(element).removeClass(errorClass);         $(element.form).find("label[for=" + element.id + "]").removeClass(errorClass);         $(element.form).find("label[for=" + element.id + "]").addClass("valid");     },     wrapper: 'li' }); How to import Illustrator swatch libraries into InDesign - Adobe Indesign Mac? I have a very reach swatch library in adobe-illustrator like food colors and.... But in adobe-indesign I don't have many colors. Why? Can I import them to indesign? User is suspended network-wide, except not really A user has been active on one of the sites recently, even though they're concurrent suspended network-wide on the rest of the network. How is this possible, and is it possible to evade a network suspension on only one site? What is the life expectancy of a Wookiee? At lunch, my coworkers and I were discussing The Force Awakens, and started questioning Chewbacca's age.  A quick check of  and it says he is possibly about 230 years old as of Episode VII.  Following the  link on that page, Wookieepedia claims that they live to be 400 years old.  Do we know from canon what the average life expectancy of a Wookiee is? Extracting intersection areas in R I have two polygons. One contains fields(X,Y,Z) and the other contains soil types (A,B,C,D). I want to know what  area of every field contains which type of soil. I tried the following:    library(rgdal) library(rgeos) Field&lt;-readOGR("./","Field") Soil&lt;-readOGR("./","Soil") Results&lt;-gIntersects(Soil,Field,byid=TRUE) rownames(Results)&lt;-Field@data$FieldName colnames(Results)&lt;-Soil@data$SoilType  &gt; Results       A     B     C     D Z  TRUE FALSE FALSE FALSE Y FALSE  TRUE  TRUE FALSE X  TRUE  TRUE  TRUE  TRUE   and achieved good results with it telling me which field contains which soil type. However, how do I get the area instead?