At wikipedia and here at math.stackexchange.com it is often said that a vector space is a set with some properties and then one goes on to refer to the set as the vector space. I am guessing that this is &quot;just&quot; an abuse of terminology and notation, right? Shouldn't the tuple $(X, S, +, *)$ be the vector space, where $X$ is a set, which elements we can add up using $+$ (and receive an element of $X$), and which we can multiply with elements of the set $S$ (and receive an element of $X$) using $*$? Often it is said that $X$ is the vector space, but that is strictly speaking wrong isn't it? I am asking, so that I may understand the mathematical concepts better. Bonus question: $X$ would be the underlying set of the vector space $(X, S, +, *)$, I guess. Could I say that, if $X=\mathbb R^n$, $X$ is the Euclidean spaced set or do I have to go route of writing in my publications that &quot;$X$ is the underlying set of an Euclidean space&quot;? Background: I submitted a paper and, even though it was ultimately accepted, one of the reviewers said that my math is not precise enough. I think (s)he might be right... especially with this topic.
I am kind of confused on the terminology of &quot;space&quot;. From  I am getting that  In mathematics, a space is a set (sometimes called a universe) with some added structure.  And from topological and metric spaces, I know that we a space is a a tuple of a set and a structure, e.g. $(X, \tau)$, $(X, d)$, where $\tau$ and $d$ are a topology and a metric respectively. On the other hand, in machine learning the term &quot;features space&quot; is used a lot for sets like $\mathbb R^n$, e.g.  and although it often refers to the set underlaying set itself, not the tuple of set plus the added structure. In fact, I have done the same in a previous publication (which the peer-reviewers accepted), but I would like to be both correct and precise in what I research, write, and submit. Is it just that machine learning researchers are imprecise in their terminology? Is it just an ? I think, what some people are doing, might be they consider a space to be a set, which is somewhat structured, instead considering the set with the structure (so, the tuple), to be the space.  I am not sure how to think about this. How do I deal with this in my writing? Especially since, what happens a lot is that I need subsets and element out of the underlying sets of all kinds of spaces.  Of course it would greatly help if there was a general name for a set that is the underlying set of a space, , but it seems that there is no dedicated name for such a set. Let's say there was a features space $(X, \cdot)$, (where I am not even sure, what structure we would add). Then it would be great to have a name for the set, let's call it an asdf. So we could say the feature asdf.  Afterthoughts: What makes it worse for me is that it seems ( and ) I am not even using the word &quot;structure&quot;, right, since it is about operations, so we are talking about an &quot;algebraic structure&quot;, while &quot;geometric space&quot; is... something else...? So apparently a &quot;vector space&quot; is actually not a &quot;geometric space&quot;, but an &quot;algebraic structure&quot;. I can understand that point, but the language gets even more confusing.
I met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal.  Lets say the Jacobian is $2$ by $2$   and Hessian is $$\begin{bmatrix}\frac{\partial^2f_1}{\partial^2 x_1 } &amp; \frac{\partial^2f_1}{\partial^2 x_2  } \\ \frac{\partial^2f_2}{\partial^2 x_1 } &amp; \frac{\partial^2f_2}{\partial^2 x_2  }\end{bmatrix}$$   But the right hand side of the equation is  $$J^TJ =\begin{bmatrix} \left( \frac{\partial f_1}{\partial x_1} \right)^2 + \left( \frac{\partial f_2}{\partial x_1} \right)^2  &amp;  \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1}\\ \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1} &amp;\left( \frac{\partial f_1}{\partial x_2} \right)^2 + \left( \frac{\partial f_2}{\partial x_2} \right)^2\end{bmatrix} $$  Why can these two be equal as presented in papers? How do I mimic Pathauto transliterated output for an alias in my custom PHP code? When writing my custom function I need to manually append node name to a taxonomy term link. To get the taxonomy term part done I use:    $link_uri = taxonomy_term_uri($term);   $path_alias = drupal_get_path_alias($link_uri['path']);   This works ok, now - I want to append node name ("/" . $node_name), but instead of raw string, I want the Pathauto transliterated output, as set in the URL alias configuration, for that node context.  What function should I use to mimic that output? I looked at hook_pathauto_alias_alter(&amp;$alias, array $context) but I'm not sure that's what I need. An equivalent condition with $\{0\}$ being the only nilpotent ideal Holomorphic function has at most countably zeros Let $f$ be holomorphic in a domain $U$, then $f$ has at most countably zeros.  How can I prove this statement? I have seen it in Sheng Gong  explicitly, but without any indications. How to set page geometry for a single page only? What is the speed of time? What's "Another Way In"? There's a new achievement and I'm stumped as to how to get it:     Start a game without breaking the laboratory wall   I assume it has something to do with the new Gadgets but none of them sound like they'd help. How do I get this achievement? Is there anything I can do about someone who has pointed their domain at my ip? This is kind of a weird request. For some reason whoever owns safeandbuy.com has pointed their domain at my IP address. The reason it's a problem is that I'm having all kinds of crawlers that are trying to crawl my site with that domain name.   Is there anything I can do about this? How do I extract an ISO on Linux without root access My textbook states that:  ...The capacitance $C$ depends only on the geometrical configuration (shape, size, separation)of the system of two conductors. [As we shall see, later, it also depends on the nature of the insulator (dielectric) separating the two conductors.]...  Now I wanted to know if these were the only two factors on which the capacitance of a conductor depends. So I tried thinking about it from the basic definition, which is:  Capacitance is the ratio of the change in electric charge of a system to the corresponding change in its electric potential. (Source: )  i.e., $$C= \frac {dQ}{dV}$$ where, $V$ is the potential of the conductor with respect to zero potential at infinity. Now, I can not think of why $C = f(V)$ isn't a possible scenario, assuming that shape size, etc remains constant. So:  Why cannot $C$ be a function of $V$? Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. Negative Mass and gravitation Prove that for for all $p,q\in \mathbb{Z}$, $q&gt;0$ we have: $$ \left| \sqrt{2}-\frac{p}{q} \right| &gt; \frac{1}{3q^2}. $$  To be honest, I do not know where to start - any help would be appreciated. Given a DateTime representing a person's birthday, how do I calculate their age in years? Have a seat, guys   I think "have a seat" is a fixed phrase that does not change whether you address someone or a group of people.  Is that correct?  If you have more to add to the post, please do so as I would love to know more. I've tried to import but I don't see the (.max) file as an option when importing. Is there another way to import the 3DMax model into Blender for rigging and animation? Retrieve Windows 8 Product Key from mainboard My new laptop came preinstalled with Windows 8. Naively, as I am, I just formatted the harddrive and installed fine old Ubuntu. Now I want to install Windows 8 for dual boot again, but I have no DVD and do download the ISO one needs a product key. That key is not on the back of the laptop anymore but somewhere on the mainboard.  Is there any way to recover the product key from the mainboard using Ubuntu? The normal JOIN ... ON ... syntax is well known. But it is also possible to position the ON clause separately from the JOIN that it corresponds to. This is something that is rarely seen in practice, not found in tutorials and I have not found any web resource that even mentions that this is possible.  Here is a script to play around with:  SELECT * INTO #widgets1 FROM (VALUES (1), (2), (3)) x(WidgetID)   SELECT * INTO #widgets2 FROM (VALUES (1, 'SomeValue1'), (2, 'SomeValue2'), (3, 'SomeValue3')) x(WidgetID, SomeValue)  SELECT * INTO #widgetProperties FROM (VALUES     (1, 'a'), (1, 'b'),     (2, 'a'), (2, 'b')) x(WidgetID, PropertyName)   --q1 SELECT w1.WidgetID, w2.SomeValue, wp.PropertyName FROM #widgets1 w1 LEFT JOIN #widgets2 w2 ON w2.WidgetID = w1.WidgetID LEFT JOIN #widgetProperties wp ON w2.WidgetID = wp.WidgetID AND wp.PropertyName = 'b' ORDER BY w1.WidgetID   --q2 SELECT w1.WidgetID, w2.SomeValue, wp.PropertyName FROM #widgets1 w1 LEFT JOIN #widgets2 w2 --no ON clause here JOIN #widgetProperties wp  ON w2.WidgetID = wp.WidgetID AND wp.PropertyName = 'b'  ON w2.WidgetID = w1.WidgetID ORDER BY w1.WidgetID   --q3 SELECT w1.WidgetID, w2.SomeValue, wp.PropertyName FROM #widgets1 w1 LEFT JOIN (     #widgets2 w2 --no SELECT or FROM here     JOIN #widgetProperties wp     ON w2.WidgetID = wp.WidgetID AND wp.PropertyName = 'b') ON w2.WidgetID = w1.WidgetID ORDER BY w1.WidgetID   q1 looks normal. q2 and q3 have these unusual positionings of the ON clause.  This script does not necessarily make much sense. It was hard for me to contrive a meaningful scenario.  So what do these unusual syntax patterns mean? How is this defined? I noticed that not all positions and orderings for the two ON clauses are allowed. What are the rules governing this?  Also is it ever a good idea to write queries like this? Suddenly unable to screenshot- Do not have permission to save files in the location where screenshots are stored Recieved the error:  'Your screenshot could not be saved. You don't have permission to save files in the location where screenshots are stored.' with OS X Yosemite.  I opened terminal and tried directing the screenshots to save on my desktop:  Brigittes-Air:~ eggsalad$ defaults read com.apple.screencapture location 2017-01-06 03:04:47.765 defaults[17729:1947540]  The domain/default pair of (com.apple.screencapture, location) does not exist  So I tried writing and:  Brigittes-Air:~ eggsalad$ defaults write com.apple.screencapture location ~/Desktop 2017-01-06 03:05:08.652 defaults[17742:1948451] Could not write domain com.apple.screencapture; exiting  I'm not sure where the screenshots are being stored if the location does not exist. Ideas?  EDIT  I ran something else and got a different response, but defaults are still unchanged.   Brigittes-Air:~ eggsalad$ defaults write com.apple.screencapture location/Users/eggsalad/Desktop killall SystemUIServer  2017-01-06 03:22:35.444 defaults[18265:2001767] Unexpected argument SystemUIServer; leaving defaults unchanged.  2nd EDIT  Result of:  ls -l ~/Library/Preferences/com.apple.screencapture.plist  was:   ls: /Users/eggsalad/Library/Preferences/com.apple.screencapture.plist: No such file or directory proving topological equivalence How would I show that $$d_E = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$ and $$d_\infty= \sum_{i=1}^n|x_i-y_i|$$ and $$\sup\{|x_1-y_1|,|x_2-y_2|,...,|x_n-y_n|\}$$ are topologically equivalent on $\mathbb{R}^n$? I can do it for $\mathbb{R}^2$, but I'm not sure how a generalization looks like.