I'm not familiar with algebra theory, but I'm interested to know if there is an ensemble in mathematics for which the power of ensemble could be a real number, we take for example $\mathbb{R^{\frac{1}{2}}}$, Does this meant the space dimension is $\frac{1}{2} $?  Note:I want $\mathbb{R^{\frac{1}{2}}}$ have a metric space structure   Thanks for any comments
Can the notion of vector space or algebra over a field be meaningfully extended to fractional dimensions, so that for example $\mathbb{R}^{-2/3}$ makes sense? Has this been explored somewhere?  I know that super vector spaces can be thought of as one way of generalizing vector spaces to negative integer dimensions. Is there a similar concept for dimensions that are rational numbers? I'm not talking about Hausdorff dimension, because it doesn't allow for negative rationals, and I'm rather looking for extensions from a more algebraic point of view (dimension as the trace of the identity map), without recurring to a given metric.
In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. Do I need a visa to visit the UK with my EU partner? I am not from EU. I have Spanish partner and we live in Spain and, because of our official relationship, I have a Spanish family resident permit. We are going to travel to the UK together for tourism. I am confused whether I need an EEA family permit or not.  I will not live there and I'm going to travel with my partner. I was told that I don't need a visa, that showing our civil union document will be sufficent. Is this correct, or do I need a visa to enter the UK?    Thank you so much for your answer. Let me give you more detail.  I hold Turkish citizenship.  My resident card exactly says that Residence Card of a Family Member of a Union Citizen " tarjeta de familiar de la union.   my case is like For example, a non-EEA spouse of a German national living in Germany will usually hold a residence permit issued under German domestic law. Therefore, a United Kingdom EEA family permit is required for travel and entry to the UK , therefore I should consider that I need to EEA Family Permit even if for touristic trip with my partner. How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $? If it is possible to express  $ \cos n \alpha $ in terms of $ \cos \alpha $  as a power series for integer $n$ ...  I like to see an expression for the quotient angle that obviously tallies when $ (\alpha , \beta) $ are swapped.  EDIT1:  Something like:  $$ \cos (\alpha - \beta)= \cos \alpha \cos \beta  + \sin \alpha  \sin \beta $$  EDIT2:  Like to know why $ \cos ( \alpha / \beta )$ cannot be expressed in terms of $ \cos \alpha, \cos \beta $, but  $ \cos ( \alpha + \beta) $  can be expressed in terms of  $\cos \alpha $ and $ \cos \beta. $ Updating the SubscriberKey on the All Subscriber table through the API I'm trying to update an existing Subscriber in the All Subscriber Table through the API; i've set up the AppId in the App Center and have connected and created Subscribers, but i'm still having issues updating.  What call do I need to be making to do this? These are the ones i've been trying. Can anyone explain precisely what IOWait is? I am looking for a way to identify primitives types in a template class definition.  I mean, having this class :  template&lt;class T&gt; class A{ void doWork(){    if(T isPrimitiveType())      doSomething();    else      doSomethingElse();  } private: T *t;  };   Is there is any way to "implement" isPrimitiveType(). Adding values to a C# array Probably a really simple one this - I'm starting out with C# and need to add values to an array, for example:  int[] terms;  for(int runs = 0; runs &lt; 400; runs++) {     terms[] = runs; }   For those who have used PHP, here's what I'm trying to do in C#:  $arr = array(); for ($i = 0; $i &lt; 10; $i++) {     $arr[] = $i; } Does iOS support a built-in firewall like macOS does? I don't see anything about it in Settings. I want to check if points layer that I have intersect with polygons layer that I have, as Boolean column in the points dataframe. I have two GeoPandas dataframes, the first one is many points and looks like this: &gt;&gt;&gt;ID  geometry 0  12  POINT (5.0279 7.4547) 1  45  POINT (6.6539 12.139) ...  and the second dataframe is one layer of many different polygons  that looks something like this: &gt;&gt;&gt;name     code   geometry 0  Desert   12     POLYGON ((5.52013 13.8902, 5.5265 13.892,...) 1  Water    24     POLYGON ((5.53756 13.88472, 5.5291 13.8791,...) ...  I'm trying to check if there is any intersection between the points layer and the regions layer. For that I have determined the crs and then used intersects : regions=regions.to_crs({'init': 'epsg:4326'}) points=points.set_crs({'init': 'epsg:4326'}) inter=points.geometry.intersects(regions.geometry)  A The script runs with the following warnings:  FutureWarning: '+init=:' syntax is deprecated. ':' is the preferred initialization method. When making the change, be mindful of axis order changes: https://pyproj4.github.io/pyproj/stable/gotchas.html#axis-order-changes-in-proj-6 return _prepare_from_string(&quot; &quot;.join(pjargs)) /opt/conda/lib/python3.8/site-packages/geopandas/base.py:39: UserWarning: The indices of the two GeoSeries are different. warn(&quot;The indices of the two GeoSeries are different.&quot;)  Then when I check the results the only value is False, like all th epoints do not intersect: inter.unique().tolist() &gt;&gt;&gt;[False]  *I saw on QGIS that the there are points that intersect and there are points that do not so there is no way this result is true *I have checked the dtypes - each one of my geodataframes has one column that is geometry and called geometry. My end goal: to add new column in the points geodataframe that will tell if it intersects the regions or not. $(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse? Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&amp;$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&amp;(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ as addition and multiplication, respectively. Do these operations form a field?  I have shown that every axiom of the field is satisfied except for the "existence of multiplicative inverses" axiom. This is where I need help. We either need to show that this axiom fails and so we don't have a field or that the axiom $aa^{-1}=1$ is satisfied for every $a\neq 0 \in \mathbb{R}$  The multiplicative identity in this problem is $(1,0)$.  Any solutions or hints are greatly appreciated. When drag force ($bV$) equals to object's weight (mg) then upward and downward force becomes equal. As a result the object comes to rest. If this is true, how is a body moving with constant velocity? Problem: We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of elements in each row is zero. Prove that $\det(AA^T)=nk^2$, where $k\in \mathbb Z$.  What have I considered so far:   First I thought, since sum of all elements in each row is zero, zero is eigenvalue of $A$ but $A\in M_{(n-1)\times n}(\mathbb Z)$ confused me. I see that $AA^T$ will be $(n-1)$ by $(n-1)$, so I tried calculating $AA^T$ but I failed to see any connection.   I did some research and found  question but we had no mention of it, which makes me believe I am on the wrong track.  Thank you all for your help. W and Z bosons are observed/discovered. But as force carrying bosons they should be virtual particles, unobservable? And also they require to have mass, but if they are virtual they may be off-shell, so are they virtual or not. New figure environment I am writing my PhD thesis. In my manuscript I have Figures in the text and Photographic plates (a figure of figures) in the annexes. The two elements are different and I would like create a new environment of Plate that follows the same structure of the Figure environment. Where can I find the Figure environment to modify it?  The new environment table should have exactly the same structure than the figure environment but replacing the word figure by the word Planche (in French). How do I remove this wall plate with no visible screws? I'm amazed how complicated this wall plate is to remove. It is made of plastic and accommodates a phone jack and a coaxial cable. Please see the images that accompany this question. I've searched it thoroughly and there are no tabs anywhere on the face or edges to press in. There are no screws. I've pulled on it pretty hard, hoping that it would pop off, but if I pulled any harder it surely would break. Of course, I had the coaxial off while I tried to remove the plate. The brand name on the phone jack part is &quot;KRONE&quot;. Has anyone seen a wall plate like this? If so, how do you remove it? I have to say, I'm absolutely blown away by how ridiculously difficult this is. Any help would be appreciated. PHP Codes are displaying in HTML CDN - Content Delivery Networks. How do they work and why would I want to use one? What exactly is a Content Delivery Network (CDN), how does it work, and why would I want to use one for my web site?  Which are the well-known CDNs out there? How can I add a PPA from a previous release? My table is:     id home datetime player resource     1 10 04/03/2009 john 399   2 11 04/03/2009 juliet 244   5 12 04/03/2009 borat 555   3 10 03/03/2009 john 300   4 11 03/03/2009 juliet 200   6 12 03/03/2009 borat 500   7 13 24/12/2008 borat 600   8 13 01/01/2009 borat 700     I need to select each distinct home holding the maximum value of datetime. Result would be:     id home datetime player resource     1 10 04/03/2009 john 399   2 11 04/03/2009 juliet 244   5 12 04/03/2009 borat 555   8 13 01/01/2009 borat 700     I have tried: -- 1 ..by the MySQL manual:   SELECT DISTINCT   home,   id,   datetime AS dt,   player,   resource FROM topten t1 WHERE datetime = (SELECT   MAX(t2.datetime) FROM topten t2 GROUP BY home) GROUP BY datetime ORDER BY datetime DESC  Doesn't work. Result-set has 130 rows although database holds 187. Result includes some duplicates of home. -- 2 ..join  SELECT   s1.id,   s1.home,   s1.datetime,   s1.player,   s1.resource FROM topten s1 JOIN (SELECT   id,   MAX(datetime) AS dt FROM topten GROUP BY id) AS s2   ON s1.id = s2.id ORDER BY datetime   Nope. Gives all the records. -- 3 ..something exotic:   With various results. Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$ Let $A,B$ be matrices $m\times n$ $(A, B \in M_{m\times n}(R))$. How can we prove that $$\operatorname{rank}(A+ B) \le \operatorname{rank}(A) + \operatorname{rank}(B)\ ?$$